Research Article Calculation Analysis of Pressure Wave ...
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Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 318912, 17 pages http://dx.doi.org/10.1155/2013/318912 Research Article Calculation Analysis of Pressure Wave Velocity in Gas and Drilling Mud Two-Phase Fluid in Annulus during Drilling Operations Yuanhua Lin, 1 Xiangwei Kong, 2 Yijie Qiu, 2 and Qiji Yuan 1 1 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500, China 2 School of Petroleum Engineering, Southwest Petroleum University, Chengdu, Sichuan 610500, China Correspondence should be addressed to Xiangwei Kong; [email protected] Received 4 June 2013; Revised 6 July 2013; Accepted 7 July 2013 Academic Editor: Waqar Khan Copyright © 2013 Yuanhua Lin et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Investigation of propagation characteristics of a pressure wave is of great significance to the solution of the transient pressure problem caused by unsteady operations during management pressure drilling operations. With consideration of the important factors such as virtual mass force, drag force, angular frequency, gas influx rate, pressure, temperature, and well depth, a united wave velocity model has been proposed based on pressure gradient equations in drilling operations, gas-liquid two-fluid model, the gas-drilling mud equations of state, and small perturbation theory. Solved by adopting the Runge-Kutta method, calculation results indicate that the wave velocity and void fraction have different values with respect to well depth. In the annulus, the drop of pressure causes an increase in void fraction along the flow direction. e void fraction increases first slightly and then sharply; correspondingly the wave velocity first gradually decreases and then slightly increases. In general, the wave velocity tends to increase with the increase in back pressure and the decrease of gas influx rate and angular frequency, significantly in low range. Taking the virtual mass force into account, the dispersion characteristic of the pressure wave weakens obviously, especially at the position close to the wellhead. 1. Introduction One of the future trends of the petroleum industry is the exploration and development of high pressure, low perme- ability reservoirs [1]. Drilling-related issues such as excessive mud cost, wellbore ballooning/breathing, kick-detection lim- itations, difficulty in avoiding gross overbalance conditions, differentially stuck pipe, and resulting well-control issues together contribute to the application of managed pressure drilling (MPD) technology [2]. MPD technology has the ability to quickly react to the expected drilling problems and formations pressures uncertainties, reduce nonproductive time and mitigating drilling hazards, and offer a considerable amount of tangible benefits while drilling in extremely narrow fracture/pore pressure windows [3]. It allows drilling operations to proceed where conventional drilling is easy to cause formation damage or considered uneconomical, of high risk, or even impossible [4]. Although drilling oper- ators try to avoid the risk of influxes, occasionally there are influxes for various reasons. Gas influx occurs whenever the pressure of a gas-bearing formation exceeds the pressure at the bottom of a wellbore. Since the subsequent intrusion of gas displaces drilling mud, it decreases the pressure in the wellbore and makes gas enter even faster [5, 6]. If it is not counteracted in time, the unstable effect can escalate into a blowout creating severe financial losses, environmental contamination, and potential loss of human lives. e basic principle of MPD well control is to keep the bottomhole pressure (BHP) as constant as possible at a value that is at least equal to the formation pressure [7]. MPD is a class of techniques that allow precise management of BHP under both static and dynamic conditions through a combination of controlling the flow rate, mud density, and back pressure (or wellhead pressure) on the fluid returns (choke manifold) of
Transcript of Research Article Calculation Analysis of Pressure Wave ...
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 318912 17 pageshttpdxdoiorg1011552013318912
Research ArticleCalculation Analysis of Pressure Wave Velocity inGas and Drilling Mud Two-Phase Fluid in Annulus duringDrilling Operations
Yuanhua Lin1 Xiangwei Kong2 Yijie Qiu2 and Qiji Yuan1
1 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation Southwest Petroleum UniversityChengdu Sichuan 610500 China
2 School of Petroleum Engineering Southwest Petroleum University Chengdu Sichuan 610500 China
Correspondence should be addressed to Xiangwei Kong m13880214723163com
Received 4 June 2013 Revised 6 July 2013 Accepted 7 July 2013
Academic Editor Waqar Khan
Copyright copy 2013 Yuanhua Lin et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Investigation of propagation characteristics of a pressure wave is of great significance to the solution of the transient pressureproblem caused by unsteady operations during management pressure drilling operations With consideration of the importantfactors such as virtual mass force drag force angular frequency gas influx rate pressure temperature and well depth a unitedwave velocity model has been proposed based on pressure gradient equations in drilling operations gas-liquid two-fluid modelthe gas-drilling mud equations of state and small perturbation theory Solved by adopting the Runge-Kutta method calculationresults indicate that the wave velocity and void fraction have different values with respect to well depth In the annulus the dropof pressure causes an increase in void fraction along the flow direction The void fraction increases first slightly and then sharplycorrespondingly thewave velocity first gradually decreases and then slightly increases In general thewave velocity tends to increasewith the increase in back pressure and the decrease of gas influx rate and angular frequency significantly in low range Taking thevirtual mass force into account the dispersion characteristic of the pressure wave weakens obviously especially at the position closeto the wellhead
1 Introduction
One of the future trends of the petroleum industry is theexploration and development of high pressure low perme-ability reservoirs [1] Drilling-related issues such as excessivemud cost wellbore ballooningbreathing kick-detection lim-itations difficulty in avoiding gross overbalance conditionsdifferentially stuck pipe and resulting well-control issuestogether contribute to the application of managed pressuredrilling (MPD) technology [2] MPD technology has theability to quickly react to the expected drilling problems andformations pressures uncertainties reduce nonproductivetime andmitigating drilling hazards and offer a considerableamount of tangible benefits while drilling in extremelynarrow fracturepore pressure windows [3] It allows drillingoperations to proceed where conventional drilling is easyto cause formation damage or considered uneconomical
of high risk or even impossible [4] Although drilling oper-ators try to avoid the risk of influxes occasionally there areinfluxes for various reasons Gas influx occurs whenever thepressure of a gas-bearing formation exceeds the pressure atthe bottom of a wellbore Since the subsequent intrusionof gas displaces drilling mud it decreases the pressure inthe wellbore and makes gas enter even faster [5 6] If itis not counteracted in time the unstable effect can escalateinto a blowout creating severe financial losses environmentalcontamination and potential loss of human lives The basicprinciple of MPD well control is to keep the bottomholepressure (BHP) as constant as possible at a value that is atleast equal to the formation pressure [7] MPD is a classof techniques that allow precise management of BHP underboth static and dynamic conditions through a combination ofcontrolling the flow rate mud density and back pressure (orwellhead pressure) on the fluid returns (choke manifold) of
2 Mathematical Problems in Engineering
the enclosed and pressurized fluid system [8 9] As a resultof the ability to manipulate the back pressure MPD offersthe capability to improve safety and well control throughearly detection of influxes and losses in a microflux levelreduces the risk of influx and thus the chance of blowoutsand controls an influx dynamically without conventionallyshutting-in [10]
During the managed pressure drilling process all theunsteady operations such as adjustment of choke wellheadback pressure controls [11] tripping inout [12] shutting-in[13] and mud pump rate changes [14] will cause generationand propagation of pressure waves which would threatenthe whole drilling system from the wellhead facilities tothe bottomhole drilling equipment and the formation [15]In the analysis of an influx well and formulation of wellcontrol scheme the dynamic effects of these operationsappeared as pressure fluctuations should be accounted for[16] As a basic parameter of pressure fluctuation the pressurewave velocity has close relevance with the determination oftransient pressure and safe operating parameters for wellcontrol However the gas influx is more troublesome for thehigher compressibility and lower density of influx gas thanthe single phase drilling mud [17] The works of Bacon etal [18] had demonstrated that compressibility effects of agas influx can be significant during an applied-back-pressuredynamic MPD well control response and can impact thewell control process Hence this paper considered the prop-agation behavior of pressure wave in gas-drilling mud two-phase flow in the annulus to provide reference for the MPDoperations
Well control includes not only the handling but also earlydetection of a gas influx Besides the transient pressure prob-lem mentioned previously investigation of the propagationcharacteristics of pressure wave is of great significance tothe early detection of gas influx [19] Many scholars believethat the pressure fluctuations procedure contains a wealth ofinformation about the flow Thus characteristics of pressurewave can be easily used to measure some important parame-ters in the two-phase flow [20] In the late 1970s the formerSoviet All-Union Drilling Technology Research Institutebegan to study characteristics of pressure wave propagationvelocity in gas-liquid two-phase flow to detect early gasinflux and achieved some important results [21] Accordingto the functional relationship between the pressure wavepropagation velocity in gas-liquid two-phase flow and the gasvoid fraction Li et al [22] presented a method of detectingthe gas influxes rate and the height of gas migration earlyafter gas influxes into the wellbore Furthermore mud pulsetelemetry is the most common method of data transmissionused by measurement-while-drilling and the transmissionvelocity of the pulse is a basic parameter for this kind of datatransmission mode [23]
The pressure wave discussed in this paper can be trans-mitted as a pressure perturbation along the direction offlow in wellbore which propagates with the speed of soundin the mud and gas two-phase drilling fluid Due to thecompressibility of the gas phase the changes in interfacebetween the gas and drilling mud and the momentum andenergy transfer between two phases it is complicated to
predict the pressure wave velocity in gas and drilling mudtwo-phase flow Since the 1940s many experimental andtheoretical studies have been performed Experimental testswere conducted to inspect the contributions of fluctuationand flow characteristics on pressure wave Ruggles et al[24] firstly performed the experimental investigation on thedispersion property of pressure wave propagation in air-water bubbly flow It was demonstrated that the propagationspeed of pressure wave varies over a range of values forthe given state depending on the angular frequency of thepressure wave Legius et al [25] tested the propagation ofpressure waves in bubbly and slug flow The experimentalresult is similar to the calculation result of the Nguyen modeland simulation result of the Sophy-2 package Concludedform Miyazaki and Nakajima experiments [26] in Nitrogen-Mercury two-phase system the slip between the phasesplays a very important role in the mechanism of pressurewave propagation From experimental investigation Bai [27]found that the fractal dimension correlation dimension andthe Kolmogorov entropy have close relationship with flowregime and the fractal dimension will be greater than 15when the flow is annular with high gas velocity The char-acteristics of pressure wave propagation in bubbly and slugflow in a vertical pipe were investigated experimentally byHuang et al in detail [28] It confirmed that the propagationvelocity is greatly affected by the gas void fraction and angularfrequency of the pressure disturbance and the superficialvelocity of flowing medium has almost no effect on thepropagation velocity Also there are some widely acceptedmodels including elasticity model homogeneous mixturemodel and continuummodel for pressure wave propagationin gas-liquid two-phase flow Tangren et al [29] took thetwo-phase media as a homogeneous fluid and presented thesolution model concerning the problem of pressure wavevelocity in two-phase flow at low gas void fraction Wallis[30] studied the propagation mechanism of pressure wavesand derived the propagation velocity in a bubbly flow andseparated flow using the homogeneous model in which thetwo-phase mixture is treated as a compressible fluid withsuitably averaged properties Nguyen et al [31] applied theelastic theory to predict the propagation velocity of pressurewaves in several different flow regimes The comparisonbetween the calculation results and available experimentaldata suggests its success at low void fraction Mecredy andHamilton [32] derived a detailed continuummodel for soundwave propagation in gas-liquid flow by using six separateconservation equations to describe the flow of the vaporand liquid phases This so-called two-fluid representationallows for nonequilibriummass heat and momentum trans-fer between the phases Results indicated that in a bubblyflow high angular frequency waves travelled an order ofmagnitude faster than low angular frequency waves Withthe development of hydrodynamics the two-fluid model iswidely used to determine the propagation velocity of thepressure wave in two-phase flow as it can provide a generaldispersion relation valid for arbitrary flow regimes includingeffects of the interphase mass heat and momentum transferArdron and Duffey [33] developed a model for sound-wavepropagation in nonequilibrium vapor-liquid flows which
Mathematical Problems in Engineering 3
predicts sound speeds and wave attenuations dependent onlyon measurable flow properties on the basis of two-fluidconservation equations Ruggles et al [24] Xu and Chen[34] Chung et al [35] Huang et al [28] and Bai et al [27]investigated the propagation velocity behavior of pressurewave via two-fluid model and small perturbation theoryand the predicted results show good agreement with theexperimental data In recent years some new researchesand models that are especially important in this area weredeveloped Xu and Gong [36] proposed a united model topredict wave velocity for different flow patterns In this unitedmodel the effect of a virtual mass coefficient was taken intoconsideration The propagation of pressure wave during thecondensation of R404A and R134a refrigerants in pipe mini-channels that undergo periodic hydrodynamic disturbanceswas given by Kuczynski [37 38] Li et al [39] simulated thecondensation of gas oxygen in subcooled liquid oxygen andthe corresponding mixing process in pump pipeline with theapplication of thermal phase changemodel in ComputationalFluidDynamics codeCFX and investigated the pressure wavepropagation characteristics in two-phase flow pipeline forliquid-propellant rocket based on a proposed pressure wavepropagation model and the predicted flow parameters Basedon the unified theory of Kanagawa et al [40] the nonlinearwave equation for pressure wave propagation in liquidscontaining gas bubbles is rederivedOn the basis of numericalsimulation of the gaseous oxygen and liquid oxygen conden-sation process with the thermal phase model in ANSYS CFXChen et al [41] solved the pressure wave propagation velocityin pump pipeline via the dispersion equation derived fromensemble average two-fluid model Li and He [42] developedan improved slug flow tracking model and analyzed thevariation rule of the pressure wave along the pipeline andinfluence of the variation of initial inlet liquid flow rate andgas flow rate in horizontal air-water slug flow with transientair flow rate Meanwhile the compressibility effects of gashad been noticed in the research field of propagation ofpressure waves in the drilling industry and some effortshave been made Li et al [22] established the relationshipbetween wave velocity and gas void fraction according tothe empirical formula of the homogeneous mixture modelpresented by Martin and Padmanatbhan [43] and frequencyresponse model presented by Henry [44] By applying theunsteady flow dynamic theory Liu et al [45] derived thepressure wave velocity calculation formula for gas-drillingmud-solid three-phase flowbased on the continuity equationWang and Zhang [46] studied the pressure pulsation inmud and set up a model for calculating the amplitude ofpressure pulsation when pressure wave is transmitted indrilling-fluid channel especially drilling hose with differentinside diameters However some efforts have been made thepressure wave velocity is usually determined by empiricalformula In the past researches the influencing factors forpressure wave propagation were simulated and analyzed withthe mathematical model however the variation of wavevelocity and gas void at different depth of wellbore was notconsidered In addition the current researches are limited intheir assumption and neglect the flow pattern translation andinterphase forces along the annulus Up to now no complete
mathematical model of pressure wave in an annulus withvariations of gas void flow pattern temperature and backpressure during MPD operations has been derived
The object of the present work is to study the velocityfor the transmission of pressure disturbance in the two-phase drilling fluid in the form of a pressure wave inannulus during MPD operations In this paper in additionto the pressure temperature and the void fraction in theannulus the compressibility of the gas phase the virtualmass force and the changes of interface in two phases arealso taken into consideration By introducing the pressuregradient equations in MPD operations gas-liquid two-fluidmodel the gas-drilling mud equations of state (EOS) andsmall perturbation theory a united model for predictingpressure wave velocity in gas and drilling mud in an annulusis developed The model can be used to predict the wavevelocity of various annulus positions at different influx ratesapplied back pressures and angular frequencies with a fullconsideration of drilling mud compressibility and interphaseforces
2 The Mathematical Model
21 The Basic Equation In this paper the two-fluid modeland the pressure gradient equation along the flow direction inthe annulus are combined to study the pressure wave velocityin MPD operations Drilling fluid contains clay cuttingsbarite other solids and so forth The solid particles aresmall and uniformly distributed therefore drilling fluid isconsidered to be a pseudohomogeneous liquid and the influxnatural gas is considered to be the gas phase The followingassumptions are made
(i) the two-phase flow is treated as one dimension(ii) no mass transfers between the gas and drilling mud(iii) the flow pattern in an annulus is either bubbly or slug
flowAs shown in Figure 1 the gas and drilling mud two-
phase fluid travels along the annulus in the drilling processThe fluid flows along the annulus in ldquominus119911rdquo direction and theannulus is formed by the casing and drill string
Themomentum conservation equation for gas and liquidtwo-phase flow is
sum119865
119911= 120588
119898119860119889119911
119889V119898
119889119905
(1)
The mass conservation equations are120597
120597119905
(120601
119866120588
119866) +
120597
120597119911
(120601
119866120588
119866V119866) = 0
120597
120597119905
(120601
119871120588
119871) +
120597
120597119911
(120601
119871120588
119871V119871) = 0
(2)
The momentum conservation equations for gas are120597
120597119905
(120601
119866120588
119866V119866) +
120597
120597119911
(120601
119866120588
119866V2119866) +
120597
120597119911
(120601
119866120588
119866)
minus
120597
120597119911
[120601
119866(120591
119891119903
119866+ 120591
Re119866) +119872Gi minus 4
120591
119866
119863
] = 0
(3)
4 Mathematical Problems in Engineering
Casing
Drill string
Flow direction
Gas
Drilling mud
z
Figure 1 The schematic of gas and drilling mud two-phase flow inannulus
The momentum conservation equations for liquid are
120597
120597119905
(120601
119871120588
119871V119871) +
120597
120597119911
(120601
119871120588
119871V2119871) +
120597
120597119911
(120601
119871120588
119871)
minus
120597
120597119911
[120601
119871(120591
119891119903
119871+ 120591
Re119871) +119872Li minus 4
120591
119871
119863
] = 0
(4)
The interphase forces include virtual mass force dragforce and the wall shear stress The transfer of momentumbetween the gas and drilling mud phases119872Gi and119872Li can bewritten as follows
119872Gi = minus119872ndLi minus119872
119889
Li + (120591119891119903
Li + 120591ReLi )
120597120601
119871
120597119911
+
120597 (120601
119866120590
119904)
120597119911
+
120597 (120601
119866119901Gi)
120597119911
minus 120601
119866
120597 (119901Li)
120597119911
119872Li = 119872ndLi +119872
119889
Li + 119901Li120597 (120601
119871)
120597119911
minus (120591
119891119903
Li + 120591ReLi )
120597120601
119871
120597119911
(5)
The momentum transfer term caused by the gas-liquidinterfacial relative acceleration motion (ie the virtual massforce) can be expressed as
119872
ndLi = 119888VM120601119866120588119871120572VM minus 01120601
119866120588
119871V119904
120597V119904
120597119911
minus 119888
1198981120588
119871V2119904
120597120601
119866
120597119911
(6)
where 1198881198981= 01 and v
119904is the slip velocity V
119904= V119866minus V119871
The momentum transfer term caused by the drag forceprovided by Park et al [47] can be described as
119872
119889
Li =3
8
119862
119863
119903
120588
119871119877
119866V2119904
(7)
The pressure difference between the liquid interface andliquid is defined by the following formula presented by Parket al
119901Li minus 119901119871 = minus119888119901120588119871V2
119904 (8)
where 119888119901= 025
The proportion of gas phase in the interface between thegas and drilling mud is rather small in that the pressuredifference between the gas interface and gas is not very highOmitting the pressure difference the gas interface pressure119875Gi can be written as
119901Gi minus 119901119866 asymp 0 (9)
According to the equations of Arnold [48] the pressureof the drilling mud is described as follows
119901
119871= 119901 minus 025120588
119871120601
119866V2119904
(10)
In fact the shear stress and the interphase shear stress arevery small relative to the Reynolds stress Also the Reynoldsstress in gas phase can be omitted relative to the interphaseforce Hence it can be described as
120591
119891119903
119866asymp 120591
119891119903
Li asymp 120591119891119903
119871asymp 120591
119866asymp 120591
Re119866asymp 0
(11)
The Reynolds stress and interfacial average Reynoldsstress can be obtained by
120591
Re119871= minus119888
119903120588
119871V2119904
120601
119866
120601
119871
120591
ReLi = minus119888119903120588119871V
2
119904
(12)
where 119888119903= 02
The wall shear stress of liquid phase is given by Wallis inthe following form [30]
120591
119871= 05119891
119871120588
119871V2119871
(13)
22 Physical Equations
221 Equations of State for Gas The equation of state (EOS)for gas can be expressed as follows
120588
119866=
119901
(119885
119866sdot 119877 sdot 119879)
(14)
where Z119866is the compression factor of gas
The formula presented by Dranchuk and Abou-Kassemhas been used to solve the gas deviation factor under thecondition of low and medium pressure (119901 lt 35MPa) [49]
119885
119866= 1 + (03051 minus
10467
119879
119903
minus
05783
119879
3
119903
)120588
119903
+ (05353 minus
026123
119879
119903
minus
06816
119879
3
119903
)120588
2
119903
(15)
where 119879119903= 119879119879
119888 119901119903= 119901119901
119888 and 120588
119903= 027119901
119903119885
119866119879
119903
The formula presented by Yarborough and Hall has beenadopted to solve the gas deviation factor under the conditionof high pressure (119901 ge 35MPa) [50]
119885
119866=
006125119875
119903119879
minus1
119903exp (minus12 (1 minus 119879minus1
119903) 2)
119884
(16)
Mathematical Problems in Engineering 5
where Y is given by
minus 006125119901
119903119879
minus1
119903exp (minus12(1 minus 119879minus1
119903)
2
) +
119884 + 119884
2
+ 119884
3
+ 119884
4
(1 minus 119884)
3
= (1476119879
minus1
119903minus 976119879
minus2
119903+ 458119879
minus3
119903) 119884
2
minus (907119879
minus1
119903minus 2422119879
minus2
119903+ 424119879
minus3
119903) 119884
(218+282119879minus1
119903
)
(17)
222 Equations of State for Liquid Under different tem-peratures and pressures the density of drilling mud can beobtained by the empirical formulas
If 119879 lt 130 ∘C the density of drillingmud can be obtainedby the following equation
120588
119871= 120588
0(1 + 4 times 10
minus10
119901
119871minus 4 times 10
minus5
119879 minus 3 times 10
minus6
119879
2
)
(18)
If 119879 ge 130 ∘C the density of drilling mud is
120588
119871= 120588
0(1 + 4 times 10
minus10
119875
119871minus 4 times 10
minus5
119879
minus3 times 10
minus6
119879
2
+ 04(
119879 minus 130
119879
)
2
)
(19)
223 Correlation of Temperature Distribution The tempera-ture of the drilling mud at different depths of the annulus canbe determined by the relationship presented by Hasan andKabir [51]
119879 = 119879
119890119894+ 119865 [1 minus 119890
(119911119887ℎ
minus119911)119860
] (minus
119892 sin 120579119892
119888119869119888
119901119898
+ 119873 + 119892
119879sin 120579)
+ 119890
(119911119887ℎ
minus119911)119860
(119879
119891119887ℎminus 119879
119890119887ℎ)
(20)
23 Flow Pattern Analysis Based on the analysis of flowcharacteristics in the enclosed drilling system it can be safelyassumed that the flowpattern in an annulus is either bubbly orslug flow [52] The pattern transition criteria for bubbly flowand slug flow given by Orkiszewski are used to judge the flowpattern in the gas-drilling mud two-phase flow [53]
For bubbly flow119902
119866
119902
119898
lt 119871
119861 (21)
For slug flow119902
119866
119902
119898
gt 119871
119861 119873
119866119881lt 119871
119878 (22)
where q119898is the volumetric flow rate of two-phase flow 119902
119898=
119902
119866+ 119902
119871
The dimensionless numbers L119904and L
119861are defined as
119871
119904= 50 + 36119873
119866119881
119902
119871
119902
119866
119871
119861= 1071 minus
07277V2119898
119863
(23)
where
119873
119866119881= V119904(
120588
119871
119892120590
119904
)
025
(24)
Flow parameters such as void fraction mixture densityand virtual mass force coefficient are discussed for a specificflow pattern
The correlation between void fraction and liquid holdupis expressed as
120601
119866+ 120601
119871= 1 (25)
231 Bubbly Flow As for bubbly flow the density of gas anddrilling mud mixture is described as the following formula
120588
119898= 120601
119871120588
119871+ 120601
119866120588
119866 (26)
The void fraction is determined by the following formula
120601
119866=
1
2
[
[
1 +
119902
119898
V119904119860
minus
radic
(1 +
119902
119898
V119904119860
)
2
minus
4119902
119866
V119904119860
]
]
(27)
The coefficient of virtual mass force 119862vm for bubbly flowcan be expressed as follows [54]
119862vm = 05
1 + 2120601
119866
1 minus 120601
119866
(28)
The coefficient of resistance coefficientC119863for bubbly flow
can be expressed as
119862
119863=
4119877
119887
3
radic
119892 (120588
119871minus 120588
119866)
120590
119904
[
1 + 1767(1 minus 120601
119866)
97
1867(1 minus 120601
119866)
15]
2
(29)
The friction pressure gradient for bubbly flow can beobtained from the following equation
120591
119891= 119891
120588
119871V2119871
2119863
(30)
232 Slug Flow As for slug flow distribution coefficient ofgas in the liquid phase is
119862
0=
000252lg (103120583119871)
119860
138minus 0782 + 0232lgV
119898minus 0428lg119860
(31)
The average density of the mixture for slug flow isdetermined by
120588
119898=
119882
119898+ 120588
119871V119904119860
119902
119898+ V119904119860
+ 119862
0120588
119871 (32)
The void fraction for slug flow is determined by thefollowing formula
120601
119866=
119902
119866
119902
119866+ 119902
119871
(33)
6 Mathematical Problems in Engineering
The coefficient of virtual mass force 119862vm for slug flow canbe expressed as follows
119862vm = 33 + 17
3119871
119902minus 3119877
119902
3119871
119902minus 119877
119902
(34)
The coefficient of resistance coefficient C119863for slug flow
can be expressed as
119862
119863= 110120601
3
119871119877
119887
(35)
The friction pressure gradient of bubbly flow can beobtained from the following equation
120591
119891= 119891
120588
119871V2119898
2119863
(
119902
119871+ V119904119860
119902
119898+ V119904119860
+ 119862
0)
(36)
24 Annulus Characteristic Analysis The annulus effectivediameter proposed by Sanchez [55] is used in the hydrauliccalculation of annulus
119863 =
120587 (119863
2
119900minus 119863
2
119894) 4
120587 (119863
119900+ 119863
119894) 4
= 119863
119900minus 119863
119894
(37)
The effective roughness of the annulus can be expressedas
119896
119890= 119896
119900
119863
119900
119863
119900+ 119863
119894
+ 119896
119894
119863
119894
119863
119900+ 119863
119894
(38)
where theD119894andD
119900are the diameters of inner pipe and outer
pipe respectively the k119900and k119894are the roughnesses of outer
pipe and the inner pipe respectively
3 The United Model Developed
The total pressure drop gradient is the sum of pressure dropgradients due to potential energy change and kinetic energyand frictional loss From (1) the equation used to calculatethe pressure gradient of gas and drilling mud flow within theannulus can be written as
119889119901
119889119911
= 120588
119898119892 sin 120579 minus
120591
119908120587119863
119860
minus 120588
119898V119898
119889V119898
119889119911
(39)
Assuming the compressibility of the gas is only related tothe pressure in the annulus the kinetic energy or accelerationterm in the previous equation can be simplified to
120588
119898V119898
119889V119898
119889119911
= minus
120588
119898V119898V119904119892
119901
119889119901
119889119911
= minus
119882
119898119902
119892
119860
2119901
119889119901
119889119911
(40)
Substituting (39) into (40) the total pressure drop gra-dient along the flow direction within the annulus can beexpressed as
119889119901
119889119911
=
120588
119898119892 + 120591
119891
1 minus119882
119898119902
119866 (119860
2119901)
(41)
It is assumed that the gas obeys to the EOS (14) and thecompressibility of drilling mud can be obtained by adopting
the simplified EOS ((18) and (19)) which neglects the thermalexpansion of liquid The sonic speed of gas phase c
119866and that
of liquid phase c119871can be presented in the following form
119889119901
119871
119889120588
119871
= 119888
2
119871
119889119901
119866
119889120588
119866
= 119888
2
119866
(42)
By introducing (42) the hydrodynamic equations of two-fluid model ((2)ndash(4)) can be written in the matrix form
119860
120597120585
120597119905
+ 119861
120597120585
120597119911
= 119862120585
(43)
where A is the matrix of parameters considered in relation totime B is the matrix of parameters considered in relation tothe spatial coordinate and C is the vector of extractions
By introducing the small disturbance theory the distur-bance of the state variable 120585(120601
119866 119901 V119866 V119871)
119879 can be written as
120585 = 120585
0+ 120575120585 exp [119894 (119908119905 minus 119896119905)] (44)
In (44) k is the wave number and 119908 is the angularfrequency of the disturbances Substituting (44) into (43)gives homogenous linear equations concerning the expres-sion (120575120601
119866 120575119901 120575V
119866 120575V119871)
119879 According to the solvable conditionof the homogenous linear equations that the determinant ofthe equations is zero dispersion equation of pressure wavecan be expressed in the following form
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119872
1119872
2119872
3119872
4
minus120588
119871119908
1 minus 120601
119866
119888
2
119871
119908 0 minus119896 (1 minus 120601
119866) 120588
119871
119872
5119872
6119872
7119872
8
119872
9119872
10119872
8119872
11
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
= 0 (45)
whereM1ndashM11can be illustrated by
119872
1= (120588
119866+ 119888
119901120601
119866120588
119871
V2119904
119888
2
119866
)119908
119872
2=
120601
119866
119888
2
119866
[1 minus 119888
119901120601
119871]
V2119904
119888
2
119871
119908
119872
3= minus[120601
119866120588
119866119896 + 2119888
119901120601
119866120601
119871120588
119871
V119904
119888
2
119871
119908]
119872
4= 22119888
119901120601
119866120601
119871120588
119871
V119904
119888
2
119871
119908
119872
5= 120588
119871V2119903119896 (minus120601
119866119888
119901+ 119888
119903minus 119888
119894+ 119888
1198982)
119872
6= minus120601
119866119896[1 minus 120601
119871
119888
119901V2119904
119888
2
119871
+ 119888
119894
V2119904
119888
2
119871
]
Mathematical Problems in Engineering 7
119872
7= 120601
119866(120588
119866+ 119888vm120588119871) 119908 minus 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904+
4
119863
119891
119866119908120588
119866V119866)
119872
8= minus119888vm120601119866120588119871119908 + 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904)
119872
9= 120588
119871V2119904119896 (120601
119871119888
119901minus 2119888
119903minus 119888
1198982)
119872
10= minus119896(120601
119871+ 119888
119903120601
119866
V2119904
119888
2
119871
)
119872
11= 120588
119871[120601
119871+ 120601
119866119888vm] 119908 minus 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904+
4
119863
119891
119871120588
119871V119871)
(46)
where 1198881198982= 01 119888
119901= 025 119888
119894= 03 and 119888
119903= 02
By solving the dispersion equation indicated previouslyfour roots can be obtained Because the wavelengths asso-ciated with two of the roots are too short to allow the two-fluid medium to be treated as a continuum the two rootsshould be omitted As for the two remaining roots one ofthem represents a pressure wave that transmited along theaxis z and the other represents a pressure wave transmits inthe opposite direction in accordance with the direction of theflow along the annulusThe real value of wave number can beused to determine thewave velocity cThewave velocity in thetwo-phase fluid can be determined by the following model
119888 =
1003816
1003816
1003816
1003816
(119908119877
+
(119896)) minus (119908119877
minus
(119896))
1003816
1003816
1003816
1003816
2
(47)
4 Solution of the United Model
Obtaining the analytical solution of themathematical modelsconcerned with flow pattern void fraction characteristicparameters and pressure drop gradient is generally impossi-ble for two-phase flow In this paper the Runge-Kuttamethod(R-K4) is used to discretize the theoretical model
We can obtain pressure temperature gas velocity drillingmud velocity and void fraction at different annulus depthsby R-K4 The solution of pressure drop gradient equation(41) can be seen as an initial-value problem of the ordinarydifferential equation
119889119901
119889119911
= 119865 (119911 119901)
119901 (119911
0) = 119875
0
(48)
With the initial value (z0p0) and the function F(z p)
(50)ndash(53) can be obtained by
119896
1= 119865 (119911
0 119901
0) (49)
119896
2= 119865(119911
0+
ℎ
2
119901
0+
ℎ
2
119896
1) (50)
119896
3= 119865(119911
0+
ℎ
2
119901
0+
ℎ
2
119896
2) (51)
119896
4= 119865 (119911
0+ ℎ 119901
0+ ℎ119896
3) (52)
where h is the step of depthThe pressure on the nod 119894 = 119894 +1can be obtained by
119901
1= 119901
0+ Δ119901 = 119901
0+
ℎ
6
(119896
1+ 2119896
2+ 2119896
3+ 119896
4)
(53)
In the present work the mathematical model and pres-sure wave velocity calculation model are solved by a person-ally compiled code on VBNET (Version 2010) The solutionprocedure for the wave velocity in the annulus is shown inFigure 1 At initial time the wellhead back pressure wellheadtemperature wellbore structure well depth gas and drillingmud properties and so forth are known On the node ithe pressure gradient temperature and the void fraction canbe obtained by adopting R-K4 Then the determinant (45)is calculated based on the calculated parameters Omittingthe two unreasonable roots the pressure wave velocity atdifferent depths of the annulus in MPD operations can besolved by (47)The process is repeated until the pressure wavevelocity of every position in the wellbore is obtained as shownin Figure 2
The developed model takes full consideration of theinterfacial interaction and the virtual mass force Owing tothe complex conditions of the annulus in MPD operationsmeasurement of wave velocity in the actual drilling processis very difficult In order to verify the united model thepredicted pressure waves are compared with the results ofprevious simulated experimental investigations presented forgas and drillingmud by Liu et al [45] in Figure 3(a) and by Liet al [22] in Figure 3(b) The lines represent the calculationresults and the points represent the experimental data
The comparisons reveal that the developed united modelfits well with the experimental data Thus the united modelcan be used to accurately predict wave velocity at differentwellhead back pressures and gas influx rates (the influx rateof gas in the bottomhole) in MPD operations
5 Analysis and Discussion
The drilling system described is an enclosed system Theschematic diagram of the gas influx process is illustrated inFigure 4 The drilling mud is pumped from surface storagedown the drill pipe Returns from the wellbore annulus travelback through surface processing where drilling solids areremoved to surface storage The key equipments include thefollowing
(i) The Rotating Control Device (RCD) The rotatingcontrol device provides the seal between atmosphereand wellbore while allowing pipe movement anddiverting returns flow In conjunction with the flowcontrol choke the RCD provides the ability to applyback pressure on the annulus during an MPD opera-tion
(ii) Choke The MPD choke manifold provides anadjustable choke system which is used to dynamicallycontrol the required BHP by means of applyingsurface BP
(iii) Coriolis Meter A Coriolis meter is used to accu-rately measure the mass flow rate of fluid exiting
8 Mathematical Problems in Engineering
Grid division
Start
Delete two roots
End
Initial conditions
Meet demand
i = i + 1
Solve eq (20) for T
Solve eq (15)ndash(17) for Zg
Judging flow pattern by eq (21) and eq (22)
Bubbly flow pattern Slug flow pattern
Solve determinant eq (45) for four roots
Calculate F(z p) by eq (48)
Calculate k1 minus k4Pi+1 = Pi + ΔPzi+1 = zi + h
zi+1 gt HNo
No
Yes
Yes
Solve eq (18) (19) for120588L
Solve eq ( 47) for c
Solve eq (14)120588G
Calculate 120588m 120601G f and CDCalculate 120588m 120601G f and CD CvmCvm
Figure 2 Solution procedure for wave velocity in MPD operations
the annulus The ability to measure return flowaccurately is essential for the applied back pressure
(iv) Pressure Sensor A pressure sensor is used to measuresurface back pressure on the wellhead
The gas and drilling mud flow rate measured by the Cori-olis meter and the back pressure measured by the pressuresensor are the initial data for annulus pressure calculationThe well used for calculation is a gas well in XinjiangUygur Autonomous Region Northwest China The wellborestructure well design parameters (depths and diameters)gas and drilling mud properties (density and viscosity) andoperational conditions of calculation well are displayed inTable 1
The drilling mud mixed with gas is taken as a two-phaseflow medium The propagation velocity of pressure wave inthe gas-drillingmud flow is calculated and discussed by usingthe established model and well parameters
51 Effect of Back Pressure on Wave Velocity Increase inapplied back pressure is the most common approach fordynamicwell controlThewellbore can be seen as an enclosed
Table 1 Parameters of calculation well
Type Property Value
Mud Dynamic viscosity (Pa sdot s) 0056Density (kgcm3) 1460
Gas Relative density 065Viscosity (Pa sdot s) 114 times 10minus5
StringElastic modulus of string (Pa) 207 times 1011
Poisson ratio of string 03Roughness (m) 154 times 10minus7
Surface condition Surface temperature (K) 298Atmosphere pressure (MPa) 0101
pressurized system The pressure at different depths of theannulus varied with the change of back pressure Accordingto the EOS of gas and drilling mud the influence of pressureon gas volume is much greater than that of drilling mudfor the greater compressibility of gas So the gas voidfraction changes with the variation of gas volume at thenearly constant flow rate of drilling mud at different annularpressures Meanwhile the pressure wave velocity is sensitive
Mathematical Problems in Engineering 9
5000 2 4 6 8 10
750
1000
1250
1500c
(ms
)
This paper p = 30MPa w = 50Hz = Re
120601G ()
CvmLiu et al [45] p = 30MPa
(a) Comparison with the experimental data of Liu
0 20 40 60 80 1000
40
80
120
160
200
c(m
s)
This paper p = 01MPa w = 50Hz = Re
120601G ()
CvmLi et al [22] p = 01MPa
(b) Comparison with the experimental data of Li
Figure 3 Experimental verification by comparison with previous experimental data
Drill pipe
Annulus
RCD
Formation
Mud pump
Drill collar
Coriolis meterChoke
Casing
Pressure sensors
120601 2159 mm times G105
1206011778mm (120601 78 mm) times G105
120601127 mm (120601 78 mm) times G105
z0 m
3600 m
4000 m
QL
Figure 4 Schematic diagram of the gas influx process
to the gas void fraction As a result when the back pressureat the wellhead is changed by adjusting the choke the wavevelocity in the two-phase drilling fluid and the distributionof void fraction at different depth of the annulus will divergeThe calculation results affected by the back pressure arepresented
Figures 5 and 6 show the distributions of void fractionand variations of wave velocity along the flow directionin the annulus when the back pressure at the wellhead is01MPa 08MPa 15MPa 25MPa 50MPa and 65MPa
respectively It can be seen that the void fraction significantlyincreases along the flow direction in the annulus Converselythe wave velocity shows a remarkable decrease tendency Forinstance BP = 15MPa at the wellhead the wave velocityis 9012ms and the void fraction is 0611 while in thebottomhole the wave velocity reaches 73142ms and voidfraction is reduced to 0059 However a sudden increase inwave velocity is observed in Figure 5 when the back pressureis 01MPa at the position close towellhead as the void fractionis further increased Moreover the influence of back pressure
10 Mathematical Problems in Engineering
10075500 254000
3000
2000
1000
0
BP = 01 MPaBP = 08 MPaBP = 15 MPa
BP = 25 MPa
BP = 65 MPaBP = 5 MPa
H(m
)
Q = 9m3h w = 50Hz = Re
120601G ()
Cvm
Figure 5 Void fraction distribution at different back pressures
on the void fraction is larger at the position close to thewellhead than that in the bottomhole
This can be explained from the viewpoints of mixturedensity and compressibility of two-phase fluid and the pres-sure drop along the flow direction in the wellbore In thelow void fraction range the gas phase is dispersed in theliquid as bubble so the wave velocity is influenced greatly bythe added gas phase According to the EOS if gas invadesinto the wellbore with a small amount in the bottomholethe density of the drilling mud has little variation while thecompressibility increases obviously which makes the wavevelocity decreased Then the two-phase drilling mud flowfrom the bottomhole to the wellhead along the annulus witha drop of pressure caused by potential energy change kineticenergy and frictional loss which leads to an increase in voidfractionWhen the void fraction is increased both the densityand the compressibility of the two-phase fluid change slightlyresulting in a flat decrease in wave velocity along the flowdirection With the further increasing in void fraction thebubbly flow turns into slug flow according to the flow patterntransient criteria of Orkiszewski which shows that the flowpattern is dependent on the void fraction Generally speak-ing the gas-drilling mud slug flow is composed of liquidand gas slugs In the preliminary slug flow the liquid slugsare much longer than the gas slugs so the wave velocity isdetermined primarily by the wave velocities in the liquidslugs which are hardly affected by the void fraction At theposition close to the wellhead the pressure of two-phase flowfluid can be reduced to a low value which approaches theback pressure The compressibility of gas will be improvedat the low pressure It results in significant increase in void
0 250 500 750 10004000
3000
2000
1000
0
H(m
)BP = 01 MPaBP = 08 MPaBP = 15 MPa
BP = 25 MPa
BP = 65 MPaBP = 5 MPa
c (ms)
Q = 9m3h w = 50Hz = ReCvm
Figure 6 Wave velocity variations at different back pressures
fraction As the void fraction increases greatly at the positionclose to the wellhead gas slugs becomemuch longer than theliquid slugs It is assumed that the actual wave velocity in slugflow happens to play a leading role when the void fractionincreases to some extent For the decrease in liquid holdupin the gas slug with the increasing in void fraction the wavevelocity in the gas slug is infinitely close to that in pure gaswhich is equivalent to an increase in wave velocity in the gasslug Meanwhile the gas content in liquid slug also increasesand results in a lower wave velocity in the liquid slug As aresult a slight increase in wave velocity appears
Figures 7 and 8 present the influence of back pressureon the void fraction and wave velocity with respect to theparameters of well depth As the back pressure increases thevoid fraction at different depth of the annulus is reducedgradually while the wave velocity in the two-phase flow tendsto increase Analytical results show that the increased backpressure is equivalent to be applied to the entire encloseddrilling fluid cyclical systemThe pressure transmits from thewellhead to the bottomhole therefore the annular pressurein the entire wellbore is increased According to the EOSthe density of gas increases and the compressibility of gasdecreases with the increasing of gas pressure So the lossof interphase momentum and energy exchange is reducedand the interphase momentum exchange is promoted Itcontributes to the increase in wave velocity with the increasein gas pressure In addition due to lower compressibility oftwo-phase flow medium under high pressure the increasetendency of pressure wave velocity and the decrease tendencyof void fraction are slowed down in the high back pressurerange
Mathematical Problems in Engineering 11
0
25
50
75
100
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
0 25 75 105
Q = 9m3h w = 50Hz = Re
120601G
()
Cvm
Figure 7 Effect of back pressure on the void fraction
52 Effect of Gas Influx Rate on Wave Velocity Figures 9and 10 graphically interpret the distributions of void fractionand variations of wave velocity along the flow direction inthe annulus When gas influx occurs in the bottomhole gasinvades into the wellbore and migrates from the bottomholeto the wellhead along the flow direction At a low gas influxrate it is extremely obvious that the void fraction and wavevelocity first slightly change in a comparatively smooth valuethen change sharply It is because of the rapid expansion ofgas volume with the decreasing in pressure near the wellheadthat the void fraction increases sharply and the wave velocitydecreases obviously at the same time But under the highbottomhole pressure (up to 50MPa) the compressibilityof the gas is low This results in a slight change in voidfraction and wave velocity at the position far away from thewellhead Since the compressible component increases withthe increase in the gas influx rate the compressibility of thegas and drilling mud two-phase fluid is improved So thevariations of void fraction and wave velocity become moreprominent Also the void fraction still shows an increasetendency that is steady first and then sharp It acts inaccordance with the variation of void fraction at a low gasinflux rate At a low gas influx rate if the void fraction atthe position close to the wellhead can not increase to a highextent the wave velocity always shows a decrease tendencyAt a high gas influx rate such as the 119876 = 8312m3h thewave velocity tends to increase because the void fraction inthe wellhead is increased to a high extent In conclusionthe wave pressure is sensitive to the void fraction and thevoid fraction is dominated by influx rate and pressure in theannulus especially the influx rate
0 25 75 1050
300
600
900
1200
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
c(m
s)
Q = 9m3h w = 50Hz = ReCvm
Figure 8 Effect of back pressure on the pressure wave velocity
With the influx of gas the mixing of gas and drillingmud occurs in the annulus and the corresponding interfacialtransfer of momentum and mass causes an increase in gasphase void fraction and a decrease in pressure wave velocityas shown in Figures 11 and 12 Within the range of low gasinflux rate the wave velocity decreases significantly It isbecause of this that the compressibility of the gas increasesremarkably and the medium appears to be of high elasticitythough the density of gas-drilling mud two-phase flowchanges slightly With the increase in the gas influx rate andcorresponding increase in the void fraction in the annulusthe compressibility of the two-phase unceasingly increaseswhich promotes the momentum and energy exchange inthe interface So the pressure wave continuously decreasesWhen the void fraction increases to some extent following theincrease in the gas influx rate the decrease in wave velocityin the liquid slug is gradually less than the increase in wavevelocity in the gas slug thus the decrease of wave velocityis slowed down for the growth of gas slug Especially in thewellhead a slight increase in wave velocity is observed
53 Effect of Virtual Mass Force on Wave Velocity Figure 13illustrates the effect of virtual mass force on the wave velocitywith the parameter of gas influx rate As shown in Figure 15under the same gas influx rate (0594m3h 1098m3h and1768m3h resp) the wave velocity curve of 119862vm = Rediverges from the curve of 119862vm = 0 at the position close tothewellhead whereas thewave velocity curve of the two typesis almost coincided below the position of 119867 = 500m Thisdivergence between the two types of curves is connected withthe fact that significance of influence of interphase virtual
12 Mathematical Problems in Engineering
0 25 50 75 1004000
3000
2000
1000
0
Q = 0058m3h Q = 1228m3hQ = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
H(m
)
BP = 01 MPaw = 50Hz = Re
120601G ()
Cvm
Figure 9 Void fraction distribution at different gas influx rates
mass forces increases together with the increase of the relativemotion in the interphase It is evident that the bottomholepressure is hundreds of times higher than the wellheadpressure As a result the void fraction at the position closeto the wellhead increases sharply meanwhile the interphasemomentum and energy exchanges are promoted The virtualmass force can be described as the transfer of momentumbetween the gas phase and drilling mud phase caused bythe relative motion in the interphase If the relative motionin the interphase is quite weak the value of virtual massforce will intend to approach 0 and the influence on thewave velocity can be ignored However if the relative motionis rather intense the effect of virtual mass force on thewave velocity should not be ignored Furthermore taking thevirtual mass force into account the dispersion characteristicof the pressure wave weakens obviously Compared with thepressure wave velocity calculated by ignoring the effect ofvirtual mass force the calculated pressure wave velocity witha consideration of the virtual mass force is lower Thereforeit is necessary to analyze the effect of the virtual mass forceon the wave velocity at the position close to the wellhead inMPD operations
54 Effect of Disturbance Angular Frequency onWave VelocityFigure 14 presents the pressure wave velocity in the annulusat different angular frequencies According to Figures 5 and9 the void fraction first slightly increases in the bottomholeand then sharply increases along the flow direction at theposition close to the wellhead At a fixed angular frequencythis phenomenon results in an overall decrease in the pressurewave velocity except for the section near the wellhead At the
0 500 1000 1500 20004000
3000
2000
1000
0
c (ms)
H(m
)Q = 0058m3h Q = 1228m3h
Q = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
BP = 01 MPaw = 50Hz = ReCvm
Figure 10 Wave velocity variations at different gas influx rates
section close to the wellhead an opposite change trend ofpressure wave velocity is observed for the transition of flowpattern from bubbly flow to slug flow due to the continuousincrease in void fraction along the flow direction In additionit can be clearly seen from the curves that the wave velocityincreases accompanying with the increase in the angularfrequency above the position of 119867 = 500m This propertyis not very distinct at the position below 119867 = 500m for thelow void fraction
Figure 15 shows the effect of frequency on the wavevelocity in the gas-drilling mud flow (119862vm = 0)The curves ofthe wave velocity of different positions reveal that the prop-agation velocity of pressure disturbances increases togetherwith the growth of the angular frequency (0 lt 119908 lt 500Hz)It proves that the pressure wave has an obvious dispersioncharacteristic in the two-phase flow As the angular frequencyincreases in the range of less than 500Hz there is sufficienttime to carry out the exchange of energy and momentumbetween two phases It achieves a state of mechanical andthermodynamic equilibrium between the two phases Sothe wave velocity increases gradually with the increase inthe angular frequency at different depths of the annulus Itis considered that the wave velocity is mainly affected bythe interphase mechanical and thermodynamic equilibriumat low anglular frequencies When the angular frequencyreaches the value of 119908 = 500Hz the pressure wave velocityachieves a constant value and remains on this level regardlessof the further growth in angular frequency 119908 With theincrease in angular frequency there is not enough time forenergy and momentum exchange between the gas-drillingmud two phases to reach themechanical and thermodynamic
Mathematical Problems in Engineering 13
0
25
50
75
100
125
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
Q (m3h)
120601G
()
0 25 5 75 10
BP = 01 MPaw = 50Hz = ReCvm
Figure 11 Effect of gas influx rate on the void fraction
equilibrium state and thus the wave dispersion does notexist At a high angular frequency the wave velocity is mainlydominated by the mechanical and thermal nonequilibriumin the flow and keeps almost unchanged when the angularfrequency is further increased This is consistent with theinfluences of the angular frequency in the horizontal pipe[28] Moreover the effect of virtual mass force is also shownin this figure (119862vm = Re) It is observed that the wave velocityis significantly reduced when the virtual mass force is takeninto consideration in the calculation of the wave velocity atthe position close to the wellhead such as 119867 = 0m It canbe explained by the reduction of dispersion characteristic ofa pressure wave Along the wellbore from the wellhead to thebottomhole the distinction gradually decreases
6 Conclusions
With full consideration of the important factors such asvirtual mass force drag force gas void fraction pressuretemperature and angular frequency a united wave velocitymodel has been proposed based on pressure drop gradientequations in MPD operations gas-liquid two-fluid modelthe gas-drilling mud equations of state and small pertur-bation theory Solved by the fourth-order explicit Runge-Kutta method the model is used to predict wave velocityfor different back pressures and gas influx rates in MPDoperations The main conclusions can be summarized asfollows
(1) With the introduction of virtual mass force and dragforce the united model agrees well with the previousexperimental data The united model can be used to
0
280
560
840
1120
1400
Q (m3h)0 25 5 75 10
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
c(m
s)
BP = 01 MPaw = 50Hz = ReCvm
Figure 12 Effect of gas influx rate on the pressure wave velocity
accurately calculate the wave velocity in the annulusThe application of the model will be beneficial tofurther study the wave velocity at different gas influxrates and back pressures in MPD operations reducenonproductive times and provide a reference for thedrilling operations in extremely narrow porefracturewindows existing conditions
(2) The wave velocity and void fraction have differentvalues with respect to well depth In the annulusthe drop of pressure causes an increase in the voidfraction along the flow direction The void fractionincreases first slightly and then sharply Correspond-ingly the wave velocity first gradually decreases in thbubbly flow and preliminary slug flow and then thewave velocity slightly increases accompanying withthe increase in the relative length ratio of gas slugto the liquid slug for the continuous increase in voidfraction at the position close to the wellhead Theminimum wave velocity appears in the long gas slugflow
(3) When the back pressure in the MPD operationsincreases the void fraction at different depths of theannulus is reduced gradually while the wave velocityin the two-phase flow tends to increase Moreoverthe influence of back pressure on the void fraction isgreater at a position close to the wellhead than that inthe bottomhole because of the great decrease in thepressure from the bottomhole to the wellhead Alsothe effect of back pressure on void fraction and wavevelocity is decreased in the high back pressure range
14 Mathematical Problems in Engineering
0 300 600 900 12004000
3000
2000
1000
225
180
135
90
45
0
BP = 01 MPaw = 50Hz
Q = 0594m3h = ReQ = 0594m3h = 0
Q = 1098m3h = ReQ = 1768m3h = ReQ = 1098m3h = 0
Q = 1768m3h = 0
c (ms)
H(m
)
CvmCvm
Cvm
Cvm
Cvm
Cvm
Figure 13 Effect of virtual mass force on the pressure wave velocity
(4) The wave velocity is sensitive to the void fraction butthe void fraction is dominated by gas influx rate andpressure in the annulus especially the gas influx rateSince the compressibility of the gas and drilling mudtwo-phase fluid is improved with the increase in theinflux rate the void fraction increases greatly andthe wave velocity decreases significantly within thelow gas influx rate range When the void fraction isincreased to some extent following the increase in thegas influx rate the decrease of wave velocity is slowedfor the growth of gas slug Especially at the wellheada slight increase in wave velocity is observed at a highgas influx rate for the sharp increase in void fraction
(5) It is necessary to analyze the effect of the virtualmass force on the wave velocity in MPD operationsEspecially at the position close to the wellhead theeffect of virtual mass force is more prominent for theintense phase interactions Taking the virtual massforce into account the dispersion characteristic of thepressure wave weakens obviously Compared with theresults calculated by ignoring the effect of virtualmassforce the calculated pressure wave velocity with aconsideration of the virtual mass force is lower
(6) The effect of angular frequency on the propagationvelocity of pressure wave appears at a low angularfrequency The propagation velocity of pressure dis-turbances increases together with the growth of theangular frequency (0 lt 119908 lt 500Hz) When theangular frequency reaches the value of 119908 = 500Hzthe pressure wave velocity achieves a constant value
0 240 480 720 960 12004000
3000
2000
1000
560
420
280
140
0
Q = 0727m3h BP = 01 MPa
w = 50Hz = Rew = 50Hz = 0w = 300Hz = 0
w = 800Hz = 0w = 1600Hz = 0w = 5000Hz = 0
c (ms)
H(m
)CvmCvmCvm
CvmCvmCvm
Figure 14 Wave velocity variations at different angular frequencies
and remains on this level regardless of the furthergrowth in angular frequency 119908
Nomenclature
A Annulus effective cross area (m2)c119866 Wave velocity in gas (ms)
c119871 Wave velocity in Drilling mud (ms)
C119863 The coefficient of drag force
D Annulus effective diameter (m)D119894 Diameter of the inner pipe (m)
D119900 Diameter of the outer pipe (m)
f119866 Shear stresses coefficient of gas interface
f119871 Shear stresses coefficient of drilling mud
interfaceF119911 Pressure applied to the two-phase flow (N)
g Acceleration due to gravity (m2s)g119888 Conversion factor
g119879 Geothermal temperature gradient (Km)
h One discrete length of annulus (m)k119890 Annulus effective roughness (m)
k119900 Roughness of outer pipe (m)
k119894 Roughness of inner pipe (m)
H The depth of annulus (m)L119902 Air bubbles length of slug (m)
119872Gi Momentum transfer in gas interface (Nm3)119872Li Momentum transfer in liquid interface (Nm3)119872
ndLi The none-drag-force (virtual mass force)
(Nm3)119872
119889
Li The momentum transfer term caused by thedrag force (Nm3)
Mathematical Problems in Engineering 15
100 1000 100000
450
900
1350
1800
0w (Hz)
0 m = 0 0 m = Re1000 m = 0
2000 m = 0
4000 m = 0
1000 m = Re2000 m = Re4000 m = Re
c(m
s)
Q = 0727m3h BP = 01 MPaCvmCvm
CvmCvmCvm
CvmCvmCvm
Figure 15 Effect of angular frequency on wave velocity
N Parameter combiningThompsonenergy effects
119901Gi Gas interface pressure (MPa)119901Li Liquid interface pressure (MPa)p Pressure (MPa)p0 Initial pressure of annulus (MPa)
p119888 Critical pressure (kPa)
p119871 Pressure of drilling mud (MPa)
p119903 Reduced pressure
q119898 Volumetric flow rate of gas and drilling
mud (m3s)q119866 Velocity of the gas (m3s)
q119871 Velocity of the drilling mud (m3s)
Q119871 Drilling mud rate within the annulus
(m3h)r Average diameter of the bubble (m)R Constant of EOS (JKgK)R119902 Air bubbles radius of slug (m)
R119887 The bubble diameter (m)
T Temperature (K)T119888 Critical temperature (K)
T119903 Reduced temperature
T119890119894 Undisturbed formation temperature at a
depth (K)T119890119887ℎ
Undisturbed formation temperature ofwellhead (K)
T119891119887ℎ
Undisturbed formation temperature atthe bottomhole (K)
v119898 Gas and drilling mud flow velocity
(ms)
v119866 Gas flow velocity (ms)
v119904 Slip velocity (ms)
v119904119892 Superficial gas velocity (ms)
v119871 Drilling mud flow velocity (ms)
W119898 Mass flow velocity (kgm3)
Z119866 Gas deviation factor
z0 Initial calculation height of annulus (m)
z119887ℎ Total well depth from surface (m)
z Variable of distance from surface (m)
Greek Letters
120588
119866 Gas density (kgm3)
120591
119891119903
Li Shear stresses of drilling mud noninterfacial(Nm2)
120591
119891119903
119871 Shear stresses of drilling mud interface (Nm2)
120591
Re119871 Reynolds stress of drilling mud (Nm2)
120591
Re119871 Reynolds stress of drilling mud interface
(Nm2)120591
Re119866 Reynolds stress of gas interface (Nm2)
120591
119891119903
119866 Shear stresses of gas interface (Nm2)
120572VM Acceleration due to virtual mass forcecoefficient (ms2)
120591
119871 Shear stresses of drilling mud along wall
(Nm2)120591
119866 Shear stresses of gas along wall (Nm2)
120591
Re119866 Reynolds stress of gas (Nm2)
120591
119891 Frictional pressure gradient (Pam)
120591
119908 Frictional pressure coefficient
120588
119898 Gas and drilling density (kgm3)
120588
0 Density under standard atmospheric pressure
(kgm3)120588
119871 Drilling mud density (kgm3)
120590
119904 Surface tension (Nm2)
Φ
119866 Gas void fraction
Φ
119871 Drilling mud holdup
120588
119903 Reduced density
120583
119871 Viscosity of drilling mud (Pa sdot s)
Subscripts
BHP Bottomhole pressureBP Back pressureEOS Equations of stateMPD Managed pressure drillingRCD The rotating control deviceR-K4 The fourth-order explicit Runge-Kutta
Subscripts of Graph
BP Back pressure (MPa)c Wave velocity in gas and drilling mud
two-phase flow (ms)119862vm The coefficient of virtual mass force
16 Mathematical Problems in Engineering
H Depth of annulus (m)Q Gas influx rate at the bottomhole (m3h)Re The value of virtual mass force
coefficient according to (28) or (34)119908 Angle frequency (Hz)Φ
119866 Void fraction ()
Acknowledgment
Research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 and 51274170)Without their support this work would not have beenpossible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of airinjection in annuli to manage pressure during cementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling conferenceand Exhibition San Diego Calif USA March 2012
[4] W A Bacon Consideration of compressibility effects for applied-back-pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Tex USA 2011
[5] J Choe C Hans and J Wold ldquoA modified two-phase well-control model and its computer applications as a training andeducational toolrdquo SPE Computer Applications vol 2 pp 14ndash201997
[6] J A Tarvin IWalton and PWand ldquoAnalysis of a gas kick takenin a deep well drilled with oil-based mudrdquo in Proceedings ofthe Annual Technical Conference and Exhibition of the Societyof Petroleum Engineers pp 255ndash264 Dallas Tex USA 1991
[7] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[8] J Zhou and G Nygaard ldquoAutomatic model-based controlscheme for stabilizing pressure during dual-gradient drillingrdquoJournal of Process Control vol 21 no 8 pp 1138ndash1147 2011
[9] A Ochoa O Acevedo L Nieto J E Lambarria and H PerezldquoSuccessful application of MPD (Managed Pressure Drilling)for prevention controland detection of borehole ballooning intight gas reservoir in Cuervito Field Mexicordquo in Proceedings ofthe Canadian Unconventional Resources Conference (CURC rsquo11)pp 172ndash185 November 2011
[10] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[11] R Ashena and J Moghadasi ldquoBottom hole pressure estimationusing evolved neural networks by real coded ant colony opti-mization and genetic algorithmrdquo Journal of Petroleum Scienceand Engineering vol 77 no 3-4 pp 375ndash385 2011
[12] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvanced Materials Research vol 468-471 pp 1736ndash1742 2012
[13] I Zubizarreta ldquoPore pressure evolution core damage and trip-ping out schedulesf a computational fluid dynamics approachrdquoin Proceedings of the SPEIADC Drilling Conference and Exhibi-tion Amsterdam The Netherlands March 2013
[14] C H Whitson ldquoCyclic shut-in eliminates liquid-loading in gaswellsrdquo in Proceedings of the SPEEAGE European Unconven-tional Resources Conference and Exhibition Vienna AustriaMarch 2012
[15] G M de Oliveira A T Franco C O R Negrao A LMartins and R A Silva ldquoModeling and validation of pressurepropagation in drilling fluids pumped into a closed wellrdquoJournal of Petroleum Science and Engineering vol 103 pp 61ndash71 2012
[16] Y Sato and H Kanki ldquoFormulas for compression wave andoscillating flow in circular piperdquo Applied Acoustics vol 69 no1 pp 1ndash11 2008
[17] J Xie B Yu X Zhang Q Shao and X Song ldquoNumericalsimulation of gas-liquid-solid three-phase flow in deep wellsrdquoAdvances in Mechanical Engineering vol 2013 Article ID951298 10 pages 2013
[18] W Bacon A Tong O Gabaldon C Sugden and P V Surya-narayana ldquoAn improved dynamic well control response to a gasinflux in managed pressure drilling operationsrdquo in Proceedingsof the IADCSPE Drilling Conference and Exhibition San DiegoCalifornia USA March 2012
[19] J I Hage and D ter Avest ldquoBorehole acoustics applied to kickdetectionrdquo Journal of Petroleum Science and Engineering vol 12no 2 pp 157ndash166 1994
[20] A M R Hopkins Fluid dynamics and surface pressure fluctua-tions of two-dimensional turbulent boundary layers over denselydistributed surface roughness [PhD thesis] Virginia PolytechnicInstitute and State University Blacksburg Va USA 2010
[21] Z S Fu ldquoThe gas cut detect technology during drilling in SovietUnionrdquo Petroleum Drilling Techniques vol 18 no 1 pp 19ndash211990
[22] X F Li C X Guan and X X Sui ldquoThe theory of gas influxdetection of pressure wave and its applicationrdquo Acta PetroleiSinica vol 18 no 3 pp 128ndash133 1997
[23] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[24] A E Ruggles R T Lahey Jr D A Drew and H A ScartonldquoInvestigation of the propagation of pressure perturbations inbubbly airwater flowsrdquo Journal of Heat Transfer vol 110 no 2pp 494ndash499 1988
[25] H J W M Legius H E A Van Den Akker and T NarumoldquoMeasurements on wave propagation and bubble and slugvelocities in cocurrent upward two-phase flowrdquo ExperimentalThermal and Fluid Science vol 15 no 3 pp 267ndash278 1997
[26] K Miyazaki and I Nakajima ldquoPropagation of pressure wavein NItrogen-Mercury two-phase systemrdquo Journal of NuclearScience and Technology vol 9 no 5 pp 315ndash318 1972
[27] B Bai L Guo and X Chen ldquoPressure fluctuation for air-watertwo-phase flowrdquo Journal of Hydrodynamics vol 18 no 4 pp476ndash482 2003
[28] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
Mathematical Problems in Engineering 17
[29] R F Tangren C H Dodge and H S Seifert ldquoCompressibilityeffects in two-phase flowrdquo Journal of Applied Physics vol 20 no7 pp 637ndash645 1949
[30] G B Wallis One Dimensional Two Phase Flow McGraw HillNew York NY USA 1969
[31] D L Nguyen E R F Winter and M Greiner ldquoSonic velocityin two-phase systemsrdquo International Journal ofMultiphase Flowvol 7 no 3 pp 311ndash320 1981
[32] R C Mecredy and L J Hamilton ldquoThe effects of nonequilib-rium heat mass and momentum transfer on two-phase soundspeedrdquo International Journal of Heat and Mass Transfer vol 15no 1 pp 61ndash72 1972
[33] K H Ardron and R B Duffey ldquoAcoustic wave propagationin a flowing liquid-vapour mixturerdquo International Journal ofMultiphase Flow vol 4 no 3 pp 303ndash322 1978
[34] J Xu and T Chen ldquoAcoustic wave prediction in flowing steam-water two-phase mixturerdquo International Journal of Heat andMass Transfer vol 43 no 7 pp 1079ndash1088 2000
[35] N ChungW Lin B Pei and Y Hsu ldquoModel for sound velocityin a two-phase air-water bubbly flowrdquo Nuclear Technology vol99 no 1 pp 80ndash89 1992
[36] X Xu and J Gong ldquoA unitedmodel for predicting pressurewavespeeds in oil and gas two-phase pipeflowsrdquo Journal of PetroleumScience and Engineering vol 60 no 3-4 pp 150ndash160 2008
[37] W Kuczynski ldquoCharacterization of pressure-wave propagationduring the condensation of R404A and R134a refrigerantsin pipe mini-channels that undergo periodic hydrodynamicdisturbancerdquo International Journal of Heat and Fluid Flow vol40 pp 135ndash150 2013
[38] W Kuczynski ldquoModeling of the propagation of a pressure waveduring the condensation process of R134a refrigerant in a pipeminichannel under the periodic conditions of hydrodynamicdisturbancesrdquo International Journal of Heat and Mass Transfervol 56 pp 715ndash723 2013
[39] Y Li C Li E Chen and Y Ying ldquoPressure wave propaga-tion characteristics in a two-phase flow pipeline for liquid-propellant rocketrdquoAerospace Science and Technology vol 15 no6 pp 453ndash464 2011
[40] T Kanagawa M Watanabe T Yano and S Fujikawa ldquoNon-linear wave equations for pressure wave propagation in liquidscontaining gas bubbles (comparison between two-fluid modeland mixture model)rdquo Journal of Fluid Science and Technologyvol 6 no 6 pp 836ndash849 2011
[41] EChen Y Li andYYing ldquoNumerical investigation onpressurewave propagation speed of gas-liquid two-phase flow in pumppipelinerdquo Journal of Aerospace Power vol 25 no 4 pp 754ndash7602010
[42] F Li and L He ldquoInvestigation on pressure wave propagationcharacteristics in slug flow field with increasing gas flow raterdquoPetroleum Geology and Recovery Efficiency vol 11 no 5 pp 78ndash80 2004
[43] C SMartin andM Padmanabhan ldquoPressure pulse propagationin two-component slug flowrdquo Journal of Fluids Engineering vol101 no 1 pp 44ndash52 1979
[44] R E Henry ldquoPressure wave propagation in two-phase mix-turesrdquo Chemical Engineering Progress vol 66 no 1 pp 1ndash101969
[45] X Liu B Li and Y Yue ldquoTransmission behavior of mud-pressure pulse alongwell borerdquo Journal ofHydrodynamics B vol19 no 2 pp 236ndash240 2007
[46] XWang and J Zhang ldquoThe research of pressure wave Pulsationin mud pulse transmittingrdquo Journal of Chongqing University ofScience and Technology vol 14 no 2 pp 55ndash58 2012
[47] J-W Park D A Drew and R T Lahey Jr ldquoThe analysisof void wave propagation in adiabatic monodispersed bubblytwo-phase flows using an ensemble-averaged two-fluid modelrdquoInternational Journal ofMultiphase Flow vol 24 no 7 pp 1205ndash1244 1998
[48] G Arnold Entropy and objectivity as constraints upons constitu-tive equations for two-fluid modeling of multiphase flow [PhDthesis] Rensselaer Polytechnic Institute New York NY USA1998
[49] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[50] L Yarborough and K R Hall ldquoHow to solve equation of statefor z-factorsrdquoOil and Gas Journal vol 72 no 7 pp 86ndash88 1974
[51] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86 pp 127ndash136 2012
[52] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[53] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[54] M Ishii and K Mishima ldquoTwo-fluid model and hydrodynamicconstitutive relationsrdquo Nuclear Engineering and Design vol 82no 2-3 pp 107ndash126 1984
[55] M J Sanchez Comparison of correlations for predicting pressurelosses in vertical multiphase annular flow [MS thesis] Univer-sity of Tulsa Tulsa Okla USA 1972
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
the enclosed and pressurized fluid system [8 9] As a resultof the ability to manipulate the back pressure MPD offersthe capability to improve safety and well control throughearly detection of influxes and losses in a microflux levelreduces the risk of influx and thus the chance of blowoutsand controls an influx dynamically without conventionallyshutting-in [10]
During the managed pressure drilling process all theunsteady operations such as adjustment of choke wellheadback pressure controls [11] tripping inout [12] shutting-in[13] and mud pump rate changes [14] will cause generationand propagation of pressure waves which would threatenthe whole drilling system from the wellhead facilities tothe bottomhole drilling equipment and the formation [15]In the analysis of an influx well and formulation of wellcontrol scheme the dynamic effects of these operationsappeared as pressure fluctuations should be accounted for[16] As a basic parameter of pressure fluctuation the pressurewave velocity has close relevance with the determination oftransient pressure and safe operating parameters for wellcontrol However the gas influx is more troublesome for thehigher compressibility and lower density of influx gas thanthe single phase drilling mud [17] The works of Bacon etal [18] had demonstrated that compressibility effects of agas influx can be significant during an applied-back-pressuredynamic MPD well control response and can impact thewell control process Hence this paper considered the prop-agation behavior of pressure wave in gas-drilling mud two-phase flow in the annulus to provide reference for the MPDoperations
Well control includes not only the handling but also earlydetection of a gas influx Besides the transient pressure prob-lem mentioned previously investigation of the propagationcharacteristics of pressure wave is of great significance tothe early detection of gas influx [19] Many scholars believethat the pressure fluctuations procedure contains a wealth ofinformation about the flow Thus characteristics of pressurewave can be easily used to measure some important parame-ters in the two-phase flow [20] In the late 1970s the formerSoviet All-Union Drilling Technology Research Institutebegan to study characteristics of pressure wave propagationvelocity in gas-liquid two-phase flow to detect early gasinflux and achieved some important results [21] Accordingto the functional relationship between the pressure wavepropagation velocity in gas-liquid two-phase flow and the gasvoid fraction Li et al [22] presented a method of detectingthe gas influxes rate and the height of gas migration earlyafter gas influxes into the wellbore Furthermore mud pulsetelemetry is the most common method of data transmissionused by measurement-while-drilling and the transmissionvelocity of the pulse is a basic parameter for this kind of datatransmission mode [23]
The pressure wave discussed in this paper can be trans-mitted as a pressure perturbation along the direction offlow in wellbore which propagates with the speed of soundin the mud and gas two-phase drilling fluid Due to thecompressibility of the gas phase the changes in interfacebetween the gas and drilling mud and the momentum andenergy transfer between two phases it is complicated to
predict the pressure wave velocity in gas and drilling mudtwo-phase flow Since the 1940s many experimental andtheoretical studies have been performed Experimental testswere conducted to inspect the contributions of fluctuationand flow characteristics on pressure wave Ruggles et al[24] firstly performed the experimental investigation on thedispersion property of pressure wave propagation in air-water bubbly flow It was demonstrated that the propagationspeed of pressure wave varies over a range of values forthe given state depending on the angular frequency of thepressure wave Legius et al [25] tested the propagation ofpressure waves in bubbly and slug flow The experimentalresult is similar to the calculation result of the Nguyen modeland simulation result of the Sophy-2 package Concludedform Miyazaki and Nakajima experiments [26] in Nitrogen-Mercury two-phase system the slip between the phasesplays a very important role in the mechanism of pressurewave propagation From experimental investigation Bai [27]found that the fractal dimension correlation dimension andthe Kolmogorov entropy have close relationship with flowregime and the fractal dimension will be greater than 15when the flow is annular with high gas velocity The char-acteristics of pressure wave propagation in bubbly and slugflow in a vertical pipe were investigated experimentally byHuang et al in detail [28] It confirmed that the propagationvelocity is greatly affected by the gas void fraction and angularfrequency of the pressure disturbance and the superficialvelocity of flowing medium has almost no effect on thepropagation velocity Also there are some widely acceptedmodels including elasticity model homogeneous mixturemodel and continuummodel for pressure wave propagationin gas-liquid two-phase flow Tangren et al [29] took thetwo-phase media as a homogeneous fluid and presented thesolution model concerning the problem of pressure wavevelocity in two-phase flow at low gas void fraction Wallis[30] studied the propagation mechanism of pressure wavesand derived the propagation velocity in a bubbly flow andseparated flow using the homogeneous model in which thetwo-phase mixture is treated as a compressible fluid withsuitably averaged properties Nguyen et al [31] applied theelastic theory to predict the propagation velocity of pressurewaves in several different flow regimes The comparisonbetween the calculation results and available experimentaldata suggests its success at low void fraction Mecredy andHamilton [32] derived a detailed continuummodel for soundwave propagation in gas-liquid flow by using six separateconservation equations to describe the flow of the vaporand liquid phases This so-called two-fluid representationallows for nonequilibriummass heat and momentum trans-fer between the phases Results indicated that in a bubblyflow high angular frequency waves travelled an order ofmagnitude faster than low angular frequency waves Withthe development of hydrodynamics the two-fluid model iswidely used to determine the propagation velocity of thepressure wave in two-phase flow as it can provide a generaldispersion relation valid for arbitrary flow regimes includingeffects of the interphase mass heat and momentum transferArdron and Duffey [33] developed a model for sound-wavepropagation in nonequilibrium vapor-liquid flows which
Mathematical Problems in Engineering 3
predicts sound speeds and wave attenuations dependent onlyon measurable flow properties on the basis of two-fluidconservation equations Ruggles et al [24] Xu and Chen[34] Chung et al [35] Huang et al [28] and Bai et al [27]investigated the propagation velocity behavior of pressurewave via two-fluid model and small perturbation theoryand the predicted results show good agreement with theexperimental data In recent years some new researchesand models that are especially important in this area weredeveloped Xu and Gong [36] proposed a united model topredict wave velocity for different flow patterns In this unitedmodel the effect of a virtual mass coefficient was taken intoconsideration The propagation of pressure wave during thecondensation of R404A and R134a refrigerants in pipe mini-channels that undergo periodic hydrodynamic disturbanceswas given by Kuczynski [37 38] Li et al [39] simulated thecondensation of gas oxygen in subcooled liquid oxygen andthe corresponding mixing process in pump pipeline with theapplication of thermal phase changemodel in ComputationalFluidDynamics codeCFX and investigated the pressure wavepropagation characteristics in two-phase flow pipeline forliquid-propellant rocket based on a proposed pressure wavepropagation model and the predicted flow parameters Basedon the unified theory of Kanagawa et al [40] the nonlinearwave equation for pressure wave propagation in liquidscontaining gas bubbles is rederivedOn the basis of numericalsimulation of the gaseous oxygen and liquid oxygen conden-sation process with the thermal phase model in ANSYS CFXChen et al [41] solved the pressure wave propagation velocityin pump pipeline via the dispersion equation derived fromensemble average two-fluid model Li and He [42] developedan improved slug flow tracking model and analyzed thevariation rule of the pressure wave along the pipeline andinfluence of the variation of initial inlet liquid flow rate andgas flow rate in horizontal air-water slug flow with transientair flow rate Meanwhile the compressibility effects of gashad been noticed in the research field of propagation ofpressure waves in the drilling industry and some effortshave been made Li et al [22] established the relationshipbetween wave velocity and gas void fraction according tothe empirical formula of the homogeneous mixture modelpresented by Martin and Padmanatbhan [43] and frequencyresponse model presented by Henry [44] By applying theunsteady flow dynamic theory Liu et al [45] derived thepressure wave velocity calculation formula for gas-drillingmud-solid three-phase flowbased on the continuity equationWang and Zhang [46] studied the pressure pulsation inmud and set up a model for calculating the amplitude ofpressure pulsation when pressure wave is transmitted indrilling-fluid channel especially drilling hose with differentinside diameters However some efforts have been made thepressure wave velocity is usually determined by empiricalformula In the past researches the influencing factors forpressure wave propagation were simulated and analyzed withthe mathematical model however the variation of wavevelocity and gas void at different depth of wellbore was notconsidered In addition the current researches are limited intheir assumption and neglect the flow pattern translation andinterphase forces along the annulus Up to now no complete
mathematical model of pressure wave in an annulus withvariations of gas void flow pattern temperature and backpressure during MPD operations has been derived
The object of the present work is to study the velocityfor the transmission of pressure disturbance in the two-phase drilling fluid in the form of a pressure wave inannulus during MPD operations In this paper in additionto the pressure temperature and the void fraction in theannulus the compressibility of the gas phase the virtualmass force and the changes of interface in two phases arealso taken into consideration By introducing the pressuregradient equations in MPD operations gas-liquid two-fluidmodel the gas-drilling mud equations of state (EOS) andsmall perturbation theory a united model for predictingpressure wave velocity in gas and drilling mud in an annulusis developed The model can be used to predict the wavevelocity of various annulus positions at different influx ratesapplied back pressures and angular frequencies with a fullconsideration of drilling mud compressibility and interphaseforces
2 The Mathematical Model
21 The Basic Equation In this paper the two-fluid modeland the pressure gradient equation along the flow direction inthe annulus are combined to study the pressure wave velocityin MPD operations Drilling fluid contains clay cuttingsbarite other solids and so forth The solid particles aresmall and uniformly distributed therefore drilling fluid isconsidered to be a pseudohomogeneous liquid and the influxnatural gas is considered to be the gas phase The followingassumptions are made
(i) the two-phase flow is treated as one dimension(ii) no mass transfers between the gas and drilling mud(iii) the flow pattern in an annulus is either bubbly or slug
flowAs shown in Figure 1 the gas and drilling mud two-
phase fluid travels along the annulus in the drilling processThe fluid flows along the annulus in ldquominus119911rdquo direction and theannulus is formed by the casing and drill string
Themomentum conservation equation for gas and liquidtwo-phase flow is
sum119865
119911= 120588
119898119860119889119911
119889V119898
119889119905
(1)
The mass conservation equations are120597
120597119905
(120601
119866120588
119866) +
120597
120597119911
(120601
119866120588
119866V119866) = 0
120597
120597119905
(120601
119871120588
119871) +
120597
120597119911
(120601
119871120588
119871V119871) = 0
(2)
The momentum conservation equations for gas are120597
120597119905
(120601
119866120588
119866V119866) +
120597
120597119911
(120601
119866120588
119866V2119866) +
120597
120597119911
(120601
119866120588
119866)
minus
120597
120597119911
[120601
119866(120591
119891119903
119866+ 120591
Re119866) +119872Gi minus 4
120591
119866
119863
] = 0
(3)
4 Mathematical Problems in Engineering
Casing
Drill string
Flow direction
Gas
Drilling mud
z
Figure 1 The schematic of gas and drilling mud two-phase flow inannulus
The momentum conservation equations for liquid are
120597
120597119905
(120601
119871120588
119871V119871) +
120597
120597119911
(120601
119871120588
119871V2119871) +
120597
120597119911
(120601
119871120588
119871)
minus
120597
120597119911
[120601
119871(120591
119891119903
119871+ 120591
Re119871) +119872Li minus 4
120591
119871
119863
] = 0
(4)
The interphase forces include virtual mass force dragforce and the wall shear stress The transfer of momentumbetween the gas and drilling mud phases119872Gi and119872Li can bewritten as follows
119872Gi = minus119872ndLi minus119872
119889
Li + (120591119891119903
Li + 120591ReLi )
120597120601
119871
120597119911
+
120597 (120601
119866120590
119904)
120597119911
+
120597 (120601
119866119901Gi)
120597119911
minus 120601
119866
120597 (119901Li)
120597119911
119872Li = 119872ndLi +119872
119889
Li + 119901Li120597 (120601
119871)
120597119911
minus (120591
119891119903
Li + 120591ReLi )
120597120601
119871
120597119911
(5)
The momentum transfer term caused by the gas-liquidinterfacial relative acceleration motion (ie the virtual massforce) can be expressed as
119872
ndLi = 119888VM120601119866120588119871120572VM minus 01120601
119866120588
119871V119904
120597V119904
120597119911
minus 119888
1198981120588
119871V2119904
120597120601
119866
120597119911
(6)
where 1198881198981= 01 and v
119904is the slip velocity V
119904= V119866minus V119871
The momentum transfer term caused by the drag forceprovided by Park et al [47] can be described as
119872
119889
Li =3
8
119862
119863
119903
120588
119871119877
119866V2119904
(7)
The pressure difference between the liquid interface andliquid is defined by the following formula presented by Parket al
119901Li minus 119901119871 = minus119888119901120588119871V2
119904 (8)
where 119888119901= 025
The proportion of gas phase in the interface between thegas and drilling mud is rather small in that the pressuredifference between the gas interface and gas is not very highOmitting the pressure difference the gas interface pressure119875Gi can be written as
119901Gi minus 119901119866 asymp 0 (9)
According to the equations of Arnold [48] the pressureof the drilling mud is described as follows
119901
119871= 119901 minus 025120588
119871120601
119866V2119904
(10)
In fact the shear stress and the interphase shear stress arevery small relative to the Reynolds stress Also the Reynoldsstress in gas phase can be omitted relative to the interphaseforce Hence it can be described as
120591
119891119903
119866asymp 120591
119891119903
Li asymp 120591119891119903
119871asymp 120591
119866asymp 120591
Re119866asymp 0
(11)
The Reynolds stress and interfacial average Reynoldsstress can be obtained by
120591
Re119871= minus119888
119903120588
119871V2119904
120601
119866
120601
119871
120591
ReLi = minus119888119903120588119871V
2
119904
(12)
where 119888119903= 02
The wall shear stress of liquid phase is given by Wallis inthe following form [30]
120591
119871= 05119891
119871120588
119871V2119871
(13)
22 Physical Equations
221 Equations of State for Gas The equation of state (EOS)for gas can be expressed as follows
120588
119866=
119901
(119885
119866sdot 119877 sdot 119879)
(14)
where Z119866is the compression factor of gas
The formula presented by Dranchuk and Abou-Kassemhas been used to solve the gas deviation factor under thecondition of low and medium pressure (119901 lt 35MPa) [49]
119885
119866= 1 + (03051 minus
10467
119879
119903
minus
05783
119879
3
119903
)120588
119903
+ (05353 minus
026123
119879
119903
minus
06816
119879
3
119903
)120588
2
119903
(15)
where 119879119903= 119879119879
119888 119901119903= 119901119901
119888 and 120588
119903= 027119901
119903119885
119866119879
119903
The formula presented by Yarborough and Hall has beenadopted to solve the gas deviation factor under the conditionof high pressure (119901 ge 35MPa) [50]
119885
119866=
006125119875
119903119879
minus1
119903exp (minus12 (1 minus 119879minus1
119903) 2)
119884
(16)
Mathematical Problems in Engineering 5
where Y is given by
minus 006125119901
119903119879
minus1
119903exp (minus12(1 minus 119879minus1
119903)
2
) +
119884 + 119884
2
+ 119884
3
+ 119884
4
(1 minus 119884)
3
= (1476119879
minus1
119903minus 976119879
minus2
119903+ 458119879
minus3
119903) 119884
2
minus (907119879
minus1
119903minus 2422119879
minus2
119903+ 424119879
minus3
119903) 119884
(218+282119879minus1
119903
)
(17)
222 Equations of State for Liquid Under different tem-peratures and pressures the density of drilling mud can beobtained by the empirical formulas
If 119879 lt 130 ∘C the density of drillingmud can be obtainedby the following equation
120588
119871= 120588
0(1 + 4 times 10
minus10
119901
119871minus 4 times 10
minus5
119879 minus 3 times 10
minus6
119879
2
)
(18)
If 119879 ge 130 ∘C the density of drilling mud is
120588
119871= 120588
0(1 + 4 times 10
minus10
119875
119871minus 4 times 10
minus5
119879
minus3 times 10
minus6
119879
2
+ 04(
119879 minus 130
119879
)
2
)
(19)
223 Correlation of Temperature Distribution The tempera-ture of the drilling mud at different depths of the annulus canbe determined by the relationship presented by Hasan andKabir [51]
119879 = 119879
119890119894+ 119865 [1 minus 119890
(119911119887ℎ
minus119911)119860
] (minus
119892 sin 120579119892
119888119869119888
119901119898
+ 119873 + 119892
119879sin 120579)
+ 119890
(119911119887ℎ
minus119911)119860
(119879
119891119887ℎminus 119879
119890119887ℎ)
(20)
23 Flow Pattern Analysis Based on the analysis of flowcharacteristics in the enclosed drilling system it can be safelyassumed that the flowpattern in an annulus is either bubbly orslug flow [52] The pattern transition criteria for bubbly flowand slug flow given by Orkiszewski are used to judge the flowpattern in the gas-drilling mud two-phase flow [53]
For bubbly flow119902
119866
119902
119898
lt 119871
119861 (21)
For slug flow119902
119866
119902
119898
gt 119871
119861 119873
119866119881lt 119871
119878 (22)
where q119898is the volumetric flow rate of two-phase flow 119902
119898=
119902
119866+ 119902
119871
The dimensionless numbers L119904and L
119861are defined as
119871
119904= 50 + 36119873
119866119881
119902
119871
119902
119866
119871
119861= 1071 minus
07277V2119898
119863
(23)
where
119873
119866119881= V119904(
120588
119871
119892120590
119904
)
025
(24)
Flow parameters such as void fraction mixture densityand virtual mass force coefficient are discussed for a specificflow pattern
The correlation between void fraction and liquid holdupis expressed as
120601
119866+ 120601
119871= 1 (25)
231 Bubbly Flow As for bubbly flow the density of gas anddrilling mud mixture is described as the following formula
120588
119898= 120601
119871120588
119871+ 120601
119866120588
119866 (26)
The void fraction is determined by the following formula
120601
119866=
1
2
[
[
1 +
119902
119898
V119904119860
minus
radic
(1 +
119902
119898
V119904119860
)
2
minus
4119902
119866
V119904119860
]
]
(27)
The coefficient of virtual mass force 119862vm for bubbly flowcan be expressed as follows [54]
119862vm = 05
1 + 2120601
119866
1 minus 120601
119866
(28)
The coefficient of resistance coefficientC119863for bubbly flow
can be expressed as
119862
119863=
4119877
119887
3
radic
119892 (120588
119871minus 120588
119866)
120590
119904
[
1 + 1767(1 minus 120601
119866)
97
1867(1 minus 120601
119866)
15]
2
(29)
The friction pressure gradient for bubbly flow can beobtained from the following equation
120591
119891= 119891
120588
119871V2119871
2119863
(30)
232 Slug Flow As for slug flow distribution coefficient ofgas in the liquid phase is
119862
0=
000252lg (103120583119871)
119860
138minus 0782 + 0232lgV
119898minus 0428lg119860
(31)
The average density of the mixture for slug flow isdetermined by
120588
119898=
119882
119898+ 120588
119871V119904119860
119902
119898+ V119904119860
+ 119862
0120588
119871 (32)
The void fraction for slug flow is determined by thefollowing formula
120601
119866=
119902
119866
119902
119866+ 119902
119871
(33)
6 Mathematical Problems in Engineering
The coefficient of virtual mass force 119862vm for slug flow canbe expressed as follows
119862vm = 33 + 17
3119871
119902minus 3119877
119902
3119871
119902minus 119877
119902
(34)
The coefficient of resistance coefficient C119863for slug flow
can be expressed as
119862
119863= 110120601
3
119871119877
119887
(35)
The friction pressure gradient of bubbly flow can beobtained from the following equation
120591
119891= 119891
120588
119871V2119898
2119863
(
119902
119871+ V119904119860
119902
119898+ V119904119860
+ 119862
0)
(36)
24 Annulus Characteristic Analysis The annulus effectivediameter proposed by Sanchez [55] is used in the hydrauliccalculation of annulus
119863 =
120587 (119863
2
119900minus 119863
2
119894) 4
120587 (119863
119900+ 119863
119894) 4
= 119863
119900minus 119863
119894
(37)
The effective roughness of the annulus can be expressedas
119896
119890= 119896
119900
119863
119900
119863
119900+ 119863
119894
+ 119896
119894
119863
119894
119863
119900+ 119863
119894
(38)
where theD119894andD
119900are the diameters of inner pipe and outer
pipe respectively the k119900and k119894are the roughnesses of outer
pipe and the inner pipe respectively
3 The United Model Developed
The total pressure drop gradient is the sum of pressure dropgradients due to potential energy change and kinetic energyand frictional loss From (1) the equation used to calculatethe pressure gradient of gas and drilling mud flow within theannulus can be written as
119889119901
119889119911
= 120588
119898119892 sin 120579 minus
120591
119908120587119863
119860
minus 120588
119898V119898
119889V119898
119889119911
(39)
Assuming the compressibility of the gas is only related tothe pressure in the annulus the kinetic energy or accelerationterm in the previous equation can be simplified to
120588
119898V119898
119889V119898
119889119911
= minus
120588
119898V119898V119904119892
119901
119889119901
119889119911
= minus
119882
119898119902
119892
119860
2119901
119889119901
119889119911
(40)
Substituting (39) into (40) the total pressure drop gra-dient along the flow direction within the annulus can beexpressed as
119889119901
119889119911
=
120588
119898119892 + 120591
119891
1 minus119882
119898119902
119866 (119860
2119901)
(41)
It is assumed that the gas obeys to the EOS (14) and thecompressibility of drilling mud can be obtained by adopting
the simplified EOS ((18) and (19)) which neglects the thermalexpansion of liquid The sonic speed of gas phase c
119866and that
of liquid phase c119871can be presented in the following form
119889119901
119871
119889120588
119871
= 119888
2
119871
119889119901
119866
119889120588
119866
= 119888
2
119866
(42)
By introducing (42) the hydrodynamic equations of two-fluid model ((2)ndash(4)) can be written in the matrix form
119860
120597120585
120597119905
+ 119861
120597120585
120597119911
= 119862120585
(43)
where A is the matrix of parameters considered in relation totime B is the matrix of parameters considered in relation tothe spatial coordinate and C is the vector of extractions
By introducing the small disturbance theory the distur-bance of the state variable 120585(120601
119866 119901 V119866 V119871)
119879 can be written as
120585 = 120585
0+ 120575120585 exp [119894 (119908119905 minus 119896119905)] (44)
In (44) k is the wave number and 119908 is the angularfrequency of the disturbances Substituting (44) into (43)gives homogenous linear equations concerning the expres-sion (120575120601
119866 120575119901 120575V
119866 120575V119871)
119879 According to the solvable conditionof the homogenous linear equations that the determinant ofthe equations is zero dispersion equation of pressure wavecan be expressed in the following form
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119872
1119872
2119872
3119872
4
minus120588
119871119908
1 minus 120601
119866
119888
2
119871
119908 0 minus119896 (1 minus 120601
119866) 120588
119871
119872
5119872
6119872
7119872
8
119872
9119872
10119872
8119872
11
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
= 0 (45)
whereM1ndashM11can be illustrated by
119872
1= (120588
119866+ 119888
119901120601
119866120588
119871
V2119904
119888
2
119866
)119908
119872
2=
120601
119866
119888
2
119866
[1 minus 119888
119901120601
119871]
V2119904
119888
2
119871
119908
119872
3= minus[120601
119866120588
119866119896 + 2119888
119901120601
119866120601
119871120588
119871
V119904
119888
2
119871
119908]
119872
4= 22119888
119901120601
119866120601
119871120588
119871
V119904
119888
2
119871
119908
119872
5= 120588
119871V2119903119896 (minus120601
119866119888
119901+ 119888
119903minus 119888
119894+ 119888
1198982)
119872
6= minus120601
119866119896[1 minus 120601
119871
119888
119901V2119904
119888
2
119871
+ 119888
119894
V2119904
119888
2
119871
]
Mathematical Problems in Engineering 7
119872
7= 120601
119866(120588
119866+ 119888vm120588119871) 119908 minus 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904+
4
119863
119891
119866119908120588
119866V119866)
119872
8= minus119888vm120601119866120588119871119908 + 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904)
119872
9= 120588
119871V2119904119896 (120601
119871119888
119901minus 2119888
119903minus 119888
1198982)
119872
10= minus119896(120601
119871+ 119888
119903120601
119866
V2119904
119888
2
119871
)
119872
11= 120588
119871[120601
119871+ 120601
119866119888vm] 119908 minus 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904+
4
119863
119891
119871120588
119871V119871)
(46)
where 1198881198982= 01 119888
119901= 025 119888
119894= 03 and 119888
119903= 02
By solving the dispersion equation indicated previouslyfour roots can be obtained Because the wavelengths asso-ciated with two of the roots are too short to allow the two-fluid medium to be treated as a continuum the two rootsshould be omitted As for the two remaining roots one ofthem represents a pressure wave that transmited along theaxis z and the other represents a pressure wave transmits inthe opposite direction in accordance with the direction of theflow along the annulusThe real value of wave number can beused to determine thewave velocity cThewave velocity in thetwo-phase fluid can be determined by the following model
119888 =
1003816
1003816
1003816
1003816
(119908119877
+
(119896)) minus (119908119877
minus
(119896))
1003816
1003816
1003816
1003816
2
(47)
4 Solution of the United Model
Obtaining the analytical solution of themathematical modelsconcerned with flow pattern void fraction characteristicparameters and pressure drop gradient is generally impossi-ble for two-phase flow In this paper the Runge-Kuttamethod(R-K4) is used to discretize the theoretical model
We can obtain pressure temperature gas velocity drillingmud velocity and void fraction at different annulus depthsby R-K4 The solution of pressure drop gradient equation(41) can be seen as an initial-value problem of the ordinarydifferential equation
119889119901
119889119911
= 119865 (119911 119901)
119901 (119911
0) = 119875
0
(48)
With the initial value (z0p0) and the function F(z p)
(50)ndash(53) can be obtained by
119896
1= 119865 (119911
0 119901
0) (49)
119896
2= 119865(119911
0+
ℎ
2
119901
0+
ℎ
2
119896
1) (50)
119896
3= 119865(119911
0+
ℎ
2
119901
0+
ℎ
2
119896
2) (51)
119896
4= 119865 (119911
0+ ℎ 119901
0+ ℎ119896
3) (52)
where h is the step of depthThe pressure on the nod 119894 = 119894 +1can be obtained by
119901
1= 119901
0+ Δ119901 = 119901
0+
ℎ
6
(119896
1+ 2119896
2+ 2119896
3+ 119896
4)
(53)
In the present work the mathematical model and pres-sure wave velocity calculation model are solved by a person-ally compiled code on VBNET (Version 2010) The solutionprocedure for the wave velocity in the annulus is shown inFigure 1 At initial time the wellhead back pressure wellheadtemperature wellbore structure well depth gas and drillingmud properties and so forth are known On the node ithe pressure gradient temperature and the void fraction canbe obtained by adopting R-K4 Then the determinant (45)is calculated based on the calculated parameters Omittingthe two unreasonable roots the pressure wave velocity atdifferent depths of the annulus in MPD operations can besolved by (47)The process is repeated until the pressure wavevelocity of every position in the wellbore is obtained as shownin Figure 2
The developed model takes full consideration of theinterfacial interaction and the virtual mass force Owing tothe complex conditions of the annulus in MPD operationsmeasurement of wave velocity in the actual drilling processis very difficult In order to verify the united model thepredicted pressure waves are compared with the results ofprevious simulated experimental investigations presented forgas and drillingmud by Liu et al [45] in Figure 3(a) and by Liet al [22] in Figure 3(b) The lines represent the calculationresults and the points represent the experimental data
The comparisons reveal that the developed united modelfits well with the experimental data Thus the united modelcan be used to accurately predict wave velocity at differentwellhead back pressures and gas influx rates (the influx rateof gas in the bottomhole) in MPD operations
5 Analysis and Discussion
The drilling system described is an enclosed system Theschematic diagram of the gas influx process is illustrated inFigure 4 The drilling mud is pumped from surface storagedown the drill pipe Returns from the wellbore annulus travelback through surface processing where drilling solids areremoved to surface storage The key equipments include thefollowing
(i) The Rotating Control Device (RCD) The rotatingcontrol device provides the seal between atmosphereand wellbore while allowing pipe movement anddiverting returns flow In conjunction with the flowcontrol choke the RCD provides the ability to applyback pressure on the annulus during an MPD opera-tion
(ii) Choke The MPD choke manifold provides anadjustable choke system which is used to dynamicallycontrol the required BHP by means of applyingsurface BP
(iii) Coriolis Meter A Coriolis meter is used to accu-rately measure the mass flow rate of fluid exiting
8 Mathematical Problems in Engineering
Grid division
Start
Delete two roots
End
Initial conditions
Meet demand
i = i + 1
Solve eq (20) for T
Solve eq (15)ndash(17) for Zg
Judging flow pattern by eq (21) and eq (22)
Bubbly flow pattern Slug flow pattern
Solve determinant eq (45) for four roots
Calculate F(z p) by eq (48)
Calculate k1 minus k4Pi+1 = Pi + ΔPzi+1 = zi + h
zi+1 gt HNo
No
Yes
Yes
Solve eq (18) (19) for120588L
Solve eq ( 47) for c
Solve eq (14)120588G
Calculate 120588m 120601G f and CDCalculate 120588m 120601G f and CD CvmCvm
Figure 2 Solution procedure for wave velocity in MPD operations
the annulus The ability to measure return flowaccurately is essential for the applied back pressure
(iv) Pressure Sensor A pressure sensor is used to measuresurface back pressure on the wellhead
The gas and drilling mud flow rate measured by the Cori-olis meter and the back pressure measured by the pressuresensor are the initial data for annulus pressure calculationThe well used for calculation is a gas well in XinjiangUygur Autonomous Region Northwest China The wellborestructure well design parameters (depths and diameters)gas and drilling mud properties (density and viscosity) andoperational conditions of calculation well are displayed inTable 1
The drilling mud mixed with gas is taken as a two-phaseflow medium The propagation velocity of pressure wave inthe gas-drillingmud flow is calculated and discussed by usingthe established model and well parameters
51 Effect of Back Pressure on Wave Velocity Increase inapplied back pressure is the most common approach fordynamicwell controlThewellbore can be seen as an enclosed
Table 1 Parameters of calculation well
Type Property Value
Mud Dynamic viscosity (Pa sdot s) 0056Density (kgcm3) 1460
Gas Relative density 065Viscosity (Pa sdot s) 114 times 10minus5
StringElastic modulus of string (Pa) 207 times 1011
Poisson ratio of string 03Roughness (m) 154 times 10minus7
Surface condition Surface temperature (K) 298Atmosphere pressure (MPa) 0101
pressurized system The pressure at different depths of theannulus varied with the change of back pressure Accordingto the EOS of gas and drilling mud the influence of pressureon gas volume is much greater than that of drilling mudfor the greater compressibility of gas So the gas voidfraction changes with the variation of gas volume at thenearly constant flow rate of drilling mud at different annularpressures Meanwhile the pressure wave velocity is sensitive
Mathematical Problems in Engineering 9
5000 2 4 6 8 10
750
1000
1250
1500c
(ms
)
This paper p = 30MPa w = 50Hz = Re
120601G ()
CvmLiu et al [45] p = 30MPa
(a) Comparison with the experimental data of Liu
0 20 40 60 80 1000
40
80
120
160
200
c(m
s)
This paper p = 01MPa w = 50Hz = Re
120601G ()
CvmLi et al [22] p = 01MPa
(b) Comparison with the experimental data of Li
Figure 3 Experimental verification by comparison with previous experimental data
Drill pipe
Annulus
RCD
Formation
Mud pump
Drill collar
Coriolis meterChoke
Casing
Pressure sensors
120601 2159 mm times G105
1206011778mm (120601 78 mm) times G105
120601127 mm (120601 78 mm) times G105
z0 m
3600 m
4000 m
QL
Figure 4 Schematic diagram of the gas influx process
to the gas void fraction As a result when the back pressureat the wellhead is changed by adjusting the choke the wavevelocity in the two-phase drilling fluid and the distributionof void fraction at different depth of the annulus will divergeThe calculation results affected by the back pressure arepresented
Figures 5 and 6 show the distributions of void fractionand variations of wave velocity along the flow directionin the annulus when the back pressure at the wellhead is01MPa 08MPa 15MPa 25MPa 50MPa and 65MPa
respectively It can be seen that the void fraction significantlyincreases along the flow direction in the annulus Converselythe wave velocity shows a remarkable decrease tendency Forinstance BP = 15MPa at the wellhead the wave velocityis 9012ms and the void fraction is 0611 while in thebottomhole the wave velocity reaches 73142ms and voidfraction is reduced to 0059 However a sudden increase inwave velocity is observed in Figure 5 when the back pressureis 01MPa at the position close towellhead as the void fractionis further increased Moreover the influence of back pressure
10 Mathematical Problems in Engineering
10075500 254000
3000
2000
1000
0
BP = 01 MPaBP = 08 MPaBP = 15 MPa
BP = 25 MPa
BP = 65 MPaBP = 5 MPa
H(m
)
Q = 9m3h w = 50Hz = Re
120601G ()
Cvm
Figure 5 Void fraction distribution at different back pressures
on the void fraction is larger at the position close to thewellhead than that in the bottomhole
This can be explained from the viewpoints of mixturedensity and compressibility of two-phase fluid and the pres-sure drop along the flow direction in the wellbore In thelow void fraction range the gas phase is dispersed in theliquid as bubble so the wave velocity is influenced greatly bythe added gas phase According to the EOS if gas invadesinto the wellbore with a small amount in the bottomholethe density of the drilling mud has little variation while thecompressibility increases obviously which makes the wavevelocity decreased Then the two-phase drilling mud flowfrom the bottomhole to the wellhead along the annulus witha drop of pressure caused by potential energy change kineticenergy and frictional loss which leads to an increase in voidfractionWhen the void fraction is increased both the densityand the compressibility of the two-phase fluid change slightlyresulting in a flat decrease in wave velocity along the flowdirection With the further increasing in void fraction thebubbly flow turns into slug flow according to the flow patterntransient criteria of Orkiszewski which shows that the flowpattern is dependent on the void fraction Generally speak-ing the gas-drilling mud slug flow is composed of liquidand gas slugs In the preliminary slug flow the liquid slugsare much longer than the gas slugs so the wave velocity isdetermined primarily by the wave velocities in the liquidslugs which are hardly affected by the void fraction At theposition close to the wellhead the pressure of two-phase flowfluid can be reduced to a low value which approaches theback pressure The compressibility of gas will be improvedat the low pressure It results in significant increase in void
0 250 500 750 10004000
3000
2000
1000
0
H(m
)BP = 01 MPaBP = 08 MPaBP = 15 MPa
BP = 25 MPa
BP = 65 MPaBP = 5 MPa
c (ms)
Q = 9m3h w = 50Hz = ReCvm
Figure 6 Wave velocity variations at different back pressures
fraction As the void fraction increases greatly at the positionclose to the wellhead gas slugs becomemuch longer than theliquid slugs It is assumed that the actual wave velocity in slugflow happens to play a leading role when the void fractionincreases to some extent For the decrease in liquid holdupin the gas slug with the increasing in void fraction the wavevelocity in the gas slug is infinitely close to that in pure gaswhich is equivalent to an increase in wave velocity in the gasslug Meanwhile the gas content in liquid slug also increasesand results in a lower wave velocity in the liquid slug As aresult a slight increase in wave velocity appears
Figures 7 and 8 present the influence of back pressureon the void fraction and wave velocity with respect to theparameters of well depth As the back pressure increases thevoid fraction at different depth of the annulus is reducedgradually while the wave velocity in the two-phase flow tendsto increase Analytical results show that the increased backpressure is equivalent to be applied to the entire encloseddrilling fluid cyclical systemThe pressure transmits from thewellhead to the bottomhole therefore the annular pressurein the entire wellbore is increased According to the EOSthe density of gas increases and the compressibility of gasdecreases with the increasing of gas pressure So the lossof interphase momentum and energy exchange is reducedand the interphase momentum exchange is promoted Itcontributes to the increase in wave velocity with the increasein gas pressure In addition due to lower compressibility oftwo-phase flow medium under high pressure the increasetendency of pressure wave velocity and the decrease tendencyof void fraction are slowed down in the high back pressurerange
Mathematical Problems in Engineering 11
0
25
50
75
100
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
0 25 75 105
Q = 9m3h w = 50Hz = Re
120601G
()
Cvm
Figure 7 Effect of back pressure on the void fraction
52 Effect of Gas Influx Rate on Wave Velocity Figures 9and 10 graphically interpret the distributions of void fractionand variations of wave velocity along the flow direction inthe annulus When gas influx occurs in the bottomhole gasinvades into the wellbore and migrates from the bottomholeto the wellhead along the flow direction At a low gas influxrate it is extremely obvious that the void fraction and wavevelocity first slightly change in a comparatively smooth valuethen change sharply It is because of the rapid expansion ofgas volume with the decreasing in pressure near the wellheadthat the void fraction increases sharply and the wave velocitydecreases obviously at the same time But under the highbottomhole pressure (up to 50MPa) the compressibilityof the gas is low This results in a slight change in voidfraction and wave velocity at the position far away from thewellhead Since the compressible component increases withthe increase in the gas influx rate the compressibility of thegas and drilling mud two-phase fluid is improved So thevariations of void fraction and wave velocity become moreprominent Also the void fraction still shows an increasetendency that is steady first and then sharp It acts inaccordance with the variation of void fraction at a low gasinflux rate At a low gas influx rate if the void fraction atthe position close to the wellhead can not increase to a highextent the wave velocity always shows a decrease tendencyAt a high gas influx rate such as the 119876 = 8312m3h thewave velocity tends to increase because the void fraction inthe wellhead is increased to a high extent In conclusionthe wave pressure is sensitive to the void fraction and thevoid fraction is dominated by influx rate and pressure in theannulus especially the influx rate
0 25 75 1050
300
600
900
1200
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
c(m
s)
Q = 9m3h w = 50Hz = ReCvm
Figure 8 Effect of back pressure on the pressure wave velocity
With the influx of gas the mixing of gas and drillingmud occurs in the annulus and the corresponding interfacialtransfer of momentum and mass causes an increase in gasphase void fraction and a decrease in pressure wave velocityas shown in Figures 11 and 12 Within the range of low gasinflux rate the wave velocity decreases significantly It isbecause of this that the compressibility of the gas increasesremarkably and the medium appears to be of high elasticitythough the density of gas-drilling mud two-phase flowchanges slightly With the increase in the gas influx rate andcorresponding increase in the void fraction in the annulusthe compressibility of the two-phase unceasingly increaseswhich promotes the momentum and energy exchange inthe interface So the pressure wave continuously decreasesWhen the void fraction increases to some extent following theincrease in the gas influx rate the decrease in wave velocityin the liquid slug is gradually less than the increase in wavevelocity in the gas slug thus the decrease of wave velocityis slowed down for the growth of gas slug Especially in thewellhead a slight increase in wave velocity is observed
53 Effect of Virtual Mass Force on Wave Velocity Figure 13illustrates the effect of virtual mass force on the wave velocitywith the parameter of gas influx rate As shown in Figure 15under the same gas influx rate (0594m3h 1098m3h and1768m3h resp) the wave velocity curve of 119862vm = Rediverges from the curve of 119862vm = 0 at the position close tothewellhead whereas thewave velocity curve of the two typesis almost coincided below the position of 119867 = 500m Thisdivergence between the two types of curves is connected withthe fact that significance of influence of interphase virtual
12 Mathematical Problems in Engineering
0 25 50 75 1004000
3000
2000
1000
0
Q = 0058m3h Q = 1228m3hQ = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
H(m
)
BP = 01 MPaw = 50Hz = Re
120601G ()
Cvm
Figure 9 Void fraction distribution at different gas influx rates
mass forces increases together with the increase of the relativemotion in the interphase It is evident that the bottomholepressure is hundreds of times higher than the wellheadpressure As a result the void fraction at the position closeto the wellhead increases sharply meanwhile the interphasemomentum and energy exchanges are promoted The virtualmass force can be described as the transfer of momentumbetween the gas phase and drilling mud phase caused bythe relative motion in the interphase If the relative motionin the interphase is quite weak the value of virtual massforce will intend to approach 0 and the influence on thewave velocity can be ignored However if the relative motionis rather intense the effect of virtual mass force on thewave velocity should not be ignored Furthermore taking thevirtual mass force into account the dispersion characteristicof the pressure wave weakens obviously Compared with thepressure wave velocity calculated by ignoring the effect ofvirtual mass force the calculated pressure wave velocity witha consideration of the virtual mass force is lower Thereforeit is necessary to analyze the effect of the virtual mass forceon the wave velocity at the position close to the wellhead inMPD operations
54 Effect of Disturbance Angular Frequency onWave VelocityFigure 14 presents the pressure wave velocity in the annulusat different angular frequencies According to Figures 5 and9 the void fraction first slightly increases in the bottomholeand then sharply increases along the flow direction at theposition close to the wellhead At a fixed angular frequencythis phenomenon results in an overall decrease in the pressurewave velocity except for the section near the wellhead At the
0 500 1000 1500 20004000
3000
2000
1000
0
c (ms)
H(m
)Q = 0058m3h Q = 1228m3h
Q = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
BP = 01 MPaw = 50Hz = ReCvm
Figure 10 Wave velocity variations at different gas influx rates
section close to the wellhead an opposite change trend ofpressure wave velocity is observed for the transition of flowpattern from bubbly flow to slug flow due to the continuousincrease in void fraction along the flow direction In additionit can be clearly seen from the curves that the wave velocityincreases accompanying with the increase in the angularfrequency above the position of 119867 = 500m This propertyis not very distinct at the position below 119867 = 500m for thelow void fraction
Figure 15 shows the effect of frequency on the wavevelocity in the gas-drilling mud flow (119862vm = 0)The curves ofthe wave velocity of different positions reveal that the prop-agation velocity of pressure disturbances increases togetherwith the growth of the angular frequency (0 lt 119908 lt 500Hz)It proves that the pressure wave has an obvious dispersioncharacteristic in the two-phase flow As the angular frequencyincreases in the range of less than 500Hz there is sufficienttime to carry out the exchange of energy and momentumbetween two phases It achieves a state of mechanical andthermodynamic equilibrium between the two phases Sothe wave velocity increases gradually with the increase inthe angular frequency at different depths of the annulus Itis considered that the wave velocity is mainly affected bythe interphase mechanical and thermodynamic equilibriumat low anglular frequencies When the angular frequencyreaches the value of 119908 = 500Hz the pressure wave velocityachieves a constant value and remains on this level regardlessof the further growth in angular frequency 119908 With theincrease in angular frequency there is not enough time forenergy and momentum exchange between the gas-drillingmud two phases to reach themechanical and thermodynamic
Mathematical Problems in Engineering 13
0
25
50
75
100
125
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
Q (m3h)
120601G
()
0 25 5 75 10
BP = 01 MPaw = 50Hz = ReCvm
Figure 11 Effect of gas influx rate on the void fraction
equilibrium state and thus the wave dispersion does notexist At a high angular frequency the wave velocity is mainlydominated by the mechanical and thermal nonequilibriumin the flow and keeps almost unchanged when the angularfrequency is further increased This is consistent with theinfluences of the angular frequency in the horizontal pipe[28] Moreover the effect of virtual mass force is also shownin this figure (119862vm = Re) It is observed that the wave velocityis significantly reduced when the virtual mass force is takeninto consideration in the calculation of the wave velocity atthe position close to the wellhead such as 119867 = 0m It canbe explained by the reduction of dispersion characteristic ofa pressure wave Along the wellbore from the wellhead to thebottomhole the distinction gradually decreases
6 Conclusions
With full consideration of the important factors such asvirtual mass force drag force gas void fraction pressuretemperature and angular frequency a united wave velocitymodel has been proposed based on pressure drop gradientequations in MPD operations gas-liquid two-fluid modelthe gas-drilling mud equations of state and small pertur-bation theory Solved by the fourth-order explicit Runge-Kutta method the model is used to predict wave velocityfor different back pressures and gas influx rates in MPDoperations The main conclusions can be summarized asfollows
(1) With the introduction of virtual mass force and dragforce the united model agrees well with the previousexperimental data The united model can be used to
0
280
560
840
1120
1400
Q (m3h)0 25 5 75 10
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
c(m
s)
BP = 01 MPaw = 50Hz = ReCvm
Figure 12 Effect of gas influx rate on the pressure wave velocity
accurately calculate the wave velocity in the annulusThe application of the model will be beneficial tofurther study the wave velocity at different gas influxrates and back pressures in MPD operations reducenonproductive times and provide a reference for thedrilling operations in extremely narrow porefracturewindows existing conditions
(2) The wave velocity and void fraction have differentvalues with respect to well depth In the annulusthe drop of pressure causes an increase in the voidfraction along the flow direction The void fractionincreases first slightly and then sharply Correspond-ingly the wave velocity first gradually decreases in thbubbly flow and preliminary slug flow and then thewave velocity slightly increases accompanying withthe increase in the relative length ratio of gas slugto the liquid slug for the continuous increase in voidfraction at the position close to the wellhead Theminimum wave velocity appears in the long gas slugflow
(3) When the back pressure in the MPD operationsincreases the void fraction at different depths of theannulus is reduced gradually while the wave velocityin the two-phase flow tends to increase Moreoverthe influence of back pressure on the void fraction isgreater at a position close to the wellhead than that inthe bottomhole because of the great decrease in thepressure from the bottomhole to the wellhead Alsothe effect of back pressure on void fraction and wavevelocity is decreased in the high back pressure range
14 Mathematical Problems in Engineering
0 300 600 900 12004000
3000
2000
1000
225
180
135
90
45
0
BP = 01 MPaw = 50Hz
Q = 0594m3h = ReQ = 0594m3h = 0
Q = 1098m3h = ReQ = 1768m3h = ReQ = 1098m3h = 0
Q = 1768m3h = 0
c (ms)
H(m
)
CvmCvm
Cvm
Cvm
Cvm
Cvm
Figure 13 Effect of virtual mass force on the pressure wave velocity
(4) The wave velocity is sensitive to the void fraction butthe void fraction is dominated by gas influx rate andpressure in the annulus especially the gas influx rateSince the compressibility of the gas and drilling mudtwo-phase fluid is improved with the increase in theinflux rate the void fraction increases greatly andthe wave velocity decreases significantly within thelow gas influx rate range When the void fraction isincreased to some extent following the increase in thegas influx rate the decrease of wave velocity is slowedfor the growth of gas slug Especially at the wellheada slight increase in wave velocity is observed at a highgas influx rate for the sharp increase in void fraction
(5) It is necessary to analyze the effect of the virtualmass force on the wave velocity in MPD operationsEspecially at the position close to the wellhead theeffect of virtual mass force is more prominent for theintense phase interactions Taking the virtual massforce into account the dispersion characteristic of thepressure wave weakens obviously Compared with theresults calculated by ignoring the effect of virtualmassforce the calculated pressure wave velocity with aconsideration of the virtual mass force is lower
(6) The effect of angular frequency on the propagationvelocity of pressure wave appears at a low angularfrequency The propagation velocity of pressure dis-turbances increases together with the growth of theangular frequency (0 lt 119908 lt 500Hz) When theangular frequency reaches the value of 119908 = 500Hzthe pressure wave velocity achieves a constant value
0 240 480 720 960 12004000
3000
2000
1000
560
420
280
140
0
Q = 0727m3h BP = 01 MPa
w = 50Hz = Rew = 50Hz = 0w = 300Hz = 0
w = 800Hz = 0w = 1600Hz = 0w = 5000Hz = 0
c (ms)
H(m
)CvmCvmCvm
CvmCvmCvm
Figure 14 Wave velocity variations at different angular frequencies
and remains on this level regardless of the furthergrowth in angular frequency 119908
Nomenclature
A Annulus effective cross area (m2)c119866 Wave velocity in gas (ms)
c119871 Wave velocity in Drilling mud (ms)
C119863 The coefficient of drag force
D Annulus effective diameter (m)D119894 Diameter of the inner pipe (m)
D119900 Diameter of the outer pipe (m)
f119866 Shear stresses coefficient of gas interface
f119871 Shear stresses coefficient of drilling mud
interfaceF119911 Pressure applied to the two-phase flow (N)
g Acceleration due to gravity (m2s)g119888 Conversion factor
g119879 Geothermal temperature gradient (Km)
h One discrete length of annulus (m)k119890 Annulus effective roughness (m)
k119900 Roughness of outer pipe (m)
k119894 Roughness of inner pipe (m)
H The depth of annulus (m)L119902 Air bubbles length of slug (m)
119872Gi Momentum transfer in gas interface (Nm3)119872Li Momentum transfer in liquid interface (Nm3)119872
ndLi The none-drag-force (virtual mass force)
(Nm3)119872
119889
Li The momentum transfer term caused by thedrag force (Nm3)
Mathematical Problems in Engineering 15
100 1000 100000
450
900
1350
1800
0w (Hz)
0 m = 0 0 m = Re1000 m = 0
2000 m = 0
4000 m = 0
1000 m = Re2000 m = Re4000 m = Re
c(m
s)
Q = 0727m3h BP = 01 MPaCvmCvm
CvmCvmCvm
CvmCvmCvm
Figure 15 Effect of angular frequency on wave velocity
N Parameter combiningThompsonenergy effects
119901Gi Gas interface pressure (MPa)119901Li Liquid interface pressure (MPa)p Pressure (MPa)p0 Initial pressure of annulus (MPa)
p119888 Critical pressure (kPa)
p119871 Pressure of drilling mud (MPa)
p119903 Reduced pressure
q119898 Volumetric flow rate of gas and drilling
mud (m3s)q119866 Velocity of the gas (m3s)
q119871 Velocity of the drilling mud (m3s)
Q119871 Drilling mud rate within the annulus
(m3h)r Average diameter of the bubble (m)R Constant of EOS (JKgK)R119902 Air bubbles radius of slug (m)
R119887 The bubble diameter (m)
T Temperature (K)T119888 Critical temperature (K)
T119903 Reduced temperature
T119890119894 Undisturbed formation temperature at a
depth (K)T119890119887ℎ
Undisturbed formation temperature ofwellhead (K)
T119891119887ℎ
Undisturbed formation temperature atthe bottomhole (K)
v119898 Gas and drilling mud flow velocity
(ms)
v119866 Gas flow velocity (ms)
v119904 Slip velocity (ms)
v119904119892 Superficial gas velocity (ms)
v119871 Drilling mud flow velocity (ms)
W119898 Mass flow velocity (kgm3)
Z119866 Gas deviation factor
z0 Initial calculation height of annulus (m)
z119887ℎ Total well depth from surface (m)
z Variable of distance from surface (m)
Greek Letters
120588
119866 Gas density (kgm3)
120591
119891119903
Li Shear stresses of drilling mud noninterfacial(Nm2)
120591
119891119903
119871 Shear stresses of drilling mud interface (Nm2)
120591
Re119871 Reynolds stress of drilling mud (Nm2)
120591
Re119871 Reynolds stress of drilling mud interface
(Nm2)120591
Re119866 Reynolds stress of gas interface (Nm2)
120591
119891119903
119866 Shear stresses of gas interface (Nm2)
120572VM Acceleration due to virtual mass forcecoefficient (ms2)
120591
119871 Shear stresses of drilling mud along wall
(Nm2)120591
119866 Shear stresses of gas along wall (Nm2)
120591
Re119866 Reynolds stress of gas (Nm2)
120591
119891 Frictional pressure gradient (Pam)
120591
119908 Frictional pressure coefficient
120588
119898 Gas and drilling density (kgm3)
120588
0 Density under standard atmospheric pressure
(kgm3)120588
119871 Drilling mud density (kgm3)
120590
119904 Surface tension (Nm2)
Φ
119866 Gas void fraction
Φ
119871 Drilling mud holdup
120588
119903 Reduced density
120583
119871 Viscosity of drilling mud (Pa sdot s)
Subscripts
BHP Bottomhole pressureBP Back pressureEOS Equations of stateMPD Managed pressure drillingRCD The rotating control deviceR-K4 The fourth-order explicit Runge-Kutta
Subscripts of Graph
BP Back pressure (MPa)c Wave velocity in gas and drilling mud
two-phase flow (ms)119862vm The coefficient of virtual mass force
16 Mathematical Problems in Engineering
H Depth of annulus (m)Q Gas influx rate at the bottomhole (m3h)Re The value of virtual mass force
coefficient according to (28) or (34)119908 Angle frequency (Hz)Φ
119866 Void fraction ()
Acknowledgment
Research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 and 51274170)Without their support this work would not have beenpossible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of airinjection in annuli to manage pressure during cementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling conferenceand Exhibition San Diego Calif USA March 2012
[4] W A Bacon Consideration of compressibility effects for applied-back-pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Tex USA 2011
[5] J Choe C Hans and J Wold ldquoA modified two-phase well-control model and its computer applications as a training andeducational toolrdquo SPE Computer Applications vol 2 pp 14ndash201997
[6] J A Tarvin IWalton and PWand ldquoAnalysis of a gas kick takenin a deep well drilled with oil-based mudrdquo in Proceedings ofthe Annual Technical Conference and Exhibition of the Societyof Petroleum Engineers pp 255ndash264 Dallas Tex USA 1991
[7] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[8] J Zhou and G Nygaard ldquoAutomatic model-based controlscheme for stabilizing pressure during dual-gradient drillingrdquoJournal of Process Control vol 21 no 8 pp 1138ndash1147 2011
[9] A Ochoa O Acevedo L Nieto J E Lambarria and H PerezldquoSuccessful application of MPD (Managed Pressure Drilling)for prevention controland detection of borehole ballooning intight gas reservoir in Cuervito Field Mexicordquo in Proceedings ofthe Canadian Unconventional Resources Conference (CURC rsquo11)pp 172ndash185 November 2011
[10] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[11] R Ashena and J Moghadasi ldquoBottom hole pressure estimationusing evolved neural networks by real coded ant colony opti-mization and genetic algorithmrdquo Journal of Petroleum Scienceand Engineering vol 77 no 3-4 pp 375ndash385 2011
[12] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvanced Materials Research vol 468-471 pp 1736ndash1742 2012
[13] I Zubizarreta ldquoPore pressure evolution core damage and trip-ping out schedulesf a computational fluid dynamics approachrdquoin Proceedings of the SPEIADC Drilling Conference and Exhibi-tion Amsterdam The Netherlands March 2013
[14] C H Whitson ldquoCyclic shut-in eliminates liquid-loading in gaswellsrdquo in Proceedings of the SPEEAGE European Unconven-tional Resources Conference and Exhibition Vienna AustriaMarch 2012
[15] G M de Oliveira A T Franco C O R Negrao A LMartins and R A Silva ldquoModeling and validation of pressurepropagation in drilling fluids pumped into a closed wellrdquoJournal of Petroleum Science and Engineering vol 103 pp 61ndash71 2012
[16] Y Sato and H Kanki ldquoFormulas for compression wave andoscillating flow in circular piperdquo Applied Acoustics vol 69 no1 pp 1ndash11 2008
[17] J Xie B Yu X Zhang Q Shao and X Song ldquoNumericalsimulation of gas-liquid-solid three-phase flow in deep wellsrdquoAdvances in Mechanical Engineering vol 2013 Article ID951298 10 pages 2013
[18] W Bacon A Tong O Gabaldon C Sugden and P V Surya-narayana ldquoAn improved dynamic well control response to a gasinflux in managed pressure drilling operationsrdquo in Proceedingsof the IADCSPE Drilling Conference and Exhibition San DiegoCalifornia USA March 2012
[19] J I Hage and D ter Avest ldquoBorehole acoustics applied to kickdetectionrdquo Journal of Petroleum Science and Engineering vol 12no 2 pp 157ndash166 1994
[20] A M R Hopkins Fluid dynamics and surface pressure fluctua-tions of two-dimensional turbulent boundary layers over denselydistributed surface roughness [PhD thesis] Virginia PolytechnicInstitute and State University Blacksburg Va USA 2010
[21] Z S Fu ldquoThe gas cut detect technology during drilling in SovietUnionrdquo Petroleum Drilling Techniques vol 18 no 1 pp 19ndash211990
[22] X F Li C X Guan and X X Sui ldquoThe theory of gas influxdetection of pressure wave and its applicationrdquo Acta PetroleiSinica vol 18 no 3 pp 128ndash133 1997
[23] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[24] A E Ruggles R T Lahey Jr D A Drew and H A ScartonldquoInvestigation of the propagation of pressure perturbations inbubbly airwater flowsrdquo Journal of Heat Transfer vol 110 no 2pp 494ndash499 1988
[25] H J W M Legius H E A Van Den Akker and T NarumoldquoMeasurements on wave propagation and bubble and slugvelocities in cocurrent upward two-phase flowrdquo ExperimentalThermal and Fluid Science vol 15 no 3 pp 267ndash278 1997
[26] K Miyazaki and I Nakajima ldquoPropagation of pressure wavein NItrogen-Mercury two-phase systemrdquo Journal of NuclearScience and Technology vol 9 no 5 pp 315ndash318 1972
[27] B Bai L Guo and X Chen ldquoPressure fluctuation for air-watertwo-phase flowrdquo Journal of Hydrodynamics vol 18 no 4 pp476ndash482 2003
[28] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
Mathematical Problems in Engineering 17
[29] R F Tangren C H Dodge and H S Seifert ldquoCompressibilityeffects in two-phase flowrdquo Journal of Applied Physics vol 20 no7 pp 637ndash645 1949
[30] G B Wallis One Dimensional Two Phase Flow McGraw HillNew York NY USA 1969
[31] D L Nguyen E R F Winter and M Greiner ldquoSonic velocityin two-phase systemsrdquo International Journal ofMultiphase Flowvol 7 no 3 pp 311ndash320 1981
[32] R C Mecredy and L J Hamilton ldquoThe effects of nonequilib-rium heat mass and momentum transfer on two-phase soundspeedrdquo International Journal of Heat and Mass Transfer vol 15no 1 pp 61ndash72 1972
[33] K H Ardron and R B Duffey ldquoAcoustic wave propagationin a flowing liquid-vapour mixturerdquo International Journal ofMultiphase Flow vol 4 no 3 pp 303ndash322 1978
[34] J Xu and T Chen ldquoAcoustic wave prediction in flowing steam-water two-phase mixturerdquo International Journal of Heat andMass Transfer vol 43 no 7 pp 1079ndash1088 2000
[35] N ChungW Lin B Pei and Y Hsu ldquoModel for sound velocityin a two-phase air-water bubbly flowrdquo Nuclear Technology vol99 no 1 pp 80ndash89 1992
[36] X Xu and J Gong ldquoA unitedmodel for predicting pressurewavespeeds in oil and gas two-phase pipeflowsrdquo Journal of PetroleumScience and Engineering vol 60 no 3-4 pp 150ndash160 2008
[37] W Kuczynski ldquoCharacterization of pressure-wave propagationduring the condensation of R404A and R134a refrigerantsin pipe mini-channels that undergo periodic hydrodynamicdisturbancerdquo International Journal of Heat and Fluid Flow vol40 pp 135ndash150 2013
[38] W Kuczynski ldquoModeling of the propagation of a pressure waveduring the condensation process of R134a refrigerant in a pipeminichannel under the periodic conditions of hydrodynamicdisturbancesrdquo International Journal of Heat and Mass Transfervol 56 pp 715ndash723 2013
[39] Y Li C Li E Chen and Y Ying ldquoPressure wave propaga-tion characteristics in a two-phase flow pipeline for liquid-propellant rocketrdquoAerospace Science and Technology vol 15 no6 pp 453ndash464 2011
[40] T Kanagawa M Watanabe T Yano and S Fujikawa ldquoNon-linear wave equations for pressure wave propagation in liquidscontaining gas bubbles (comparison between two-fluid modeland mixture model)rdquo Journal of Fluid Science and Technologyvol 6 no 6 pp 836ndash849 2011
[41] EChen Y Li andYYing ldquoNumerical investigation onpressurewave propagation speed of gas-liquid two-phase flow in pumppipelinerdquo Journal of Aerospace Power vol 25 no 4 pp 754ndash7602010
[42] F Li and L He ldquoInvestigation on pressure wave propagationcharacteristics in slug flow field with increasing gas flow raterdquoPetroleum Geology and Recovery Efficiency vol 11 no 5 pp 78ndash80 2004
[43] C SMartin andM Padmanabhan ldquoPressure pulse propagationin two-component slug flowrdquo Journal of Fluids Engineering vol101 no 1 pp 44ndash52 1979
[44] R E Henry ldquoPressure wave propagation in two-phase mix-turesrdquo Chemical Engineering Progress vol 66 no 1 pp 1ndash101969
[45] X Liu B Li and Y Yue ldquoTransmission behavior of mud-pressure pulse alongwell borerdquo Journal ofHydrodynamics B vol19 no 2 pp 236ndash240 2007
[46] XWang and J Zhang ldquoThe research of pressure wave Pulsationin mud pulse transmittingrdquo Journal of Chongqing University ofScience and Technology vol 14 no 2 pp 55ndash58 2012
[47] J-W Park D A Drew and R T Lahey Jr ldquoThe analysisof void wave propagation in adiabatic monodispersed bubblytwo-phase flows using an ensemble-averaged two-fluid modelrdquoInternational Journal ofMultiphase Flow vol 24 no 7 pp 1205ndash1244 1998
[48] G Arnold Entropy and objectivity as constraints upons constitu-tive equations for two-fluid modeling of multiphase flow [PhDthesis] Rensselaer Polytechnic Institute New York NY USA1998
[49] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[50] L Yarborough and K R Hall ldquoHow to solve equation of statefor z-factorsrdquoOil and Gas Journal vol 72 no 7 pp 86ndash88 1974
[51] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86 pp 127ndash136 2012
[52] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[53] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[54] M Ishii and K Mishima ldquoTwo-fluid model and hydrodynamicconstitutive relationsrdquo Nuclear Engineering and Design vol 82no 2-3 pp 107ndash126 1984
[55] M J Sanchez Comparison of correlations for predicting pressurelosses in vertical multiphase annular flow [MS thesis] Univer-sity of Tulsa Tulsa Okla USA 1972
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Mathematical Problems in Engineering 3
predicts sound speeds and wave attenuations dependent onlyon measurable flow properties on the basis of two-fluidconservation equations Ruggles et al [24] Xu and Chen[34] Chung et al [35] Huang et al [28] and Bai et al [27]investigated the propagation velocity behavior of pressurewave via two-fluid model and small perturbation theoryand the predicted results show good agreement with theexperimental data In recent years some new researchesand models that are especially important in this area weredeveloped Xu and Gong [36] proposed a united model topredict wave velocity for different flow patterns In this unitedmodel the effect of a virtual mass coefficient was taken intoconsideration The propagation of pressure wave during thecondensation of R404A and R134a refrigerants in pipe mini-channels that undergo periodic hydrodynamic disturbanceswas given by Kuczynski [37 38] Li et al [39] simulated thecondensation of gas oxygen in subcooled liquid oxygen andthe corresponding mixing process in pump pipeline with theapplication of thermal phase changemodel in ComputationalFluidDynamics codeCFX and investigated the pressure wavepropagation characteristics in two-phase flow pipeline forliquid-propellant rocket based on a proposed pressure wavepropagation model and the predicted flow parameters Basedon the unified theory of Kanagawa et al [40] the nonlinearwave equation for pressure wave propagation in liquidscontaining gas bubbles is rederivedOn the basis of numericalsimulation of the gaseous oxygen and liquid oxygen conden-sation process with the thermal phase model in ANSYS CFXChen et al [41] solved the pressure wave propagation velocityin pump pipeline via the dispersion equation derived fromensemble average two-fluid model Li and He [42] developedan improved slug flow tracking model and analyzed thevariation rule of the pressure wave along the pipeline andinfluence of the variation of initial inlet liquid flow rate andgas flow rate in horizontal air-water slug flow with transientair flow rate Meanwhile the compressibility effects of gashad been noticed in the research field of propagation ofpressure waves in the drilling industry and some effortshave been made Li et al [22] established the relationshipbetween wave velocity and gas void fraction according tothe empirical formula of the homogeneous mixture modelpresented by Martin and Padmanatbhan [43] and frequencyresponse model presented by Henry [44] By applying theunsteady flow dynamic theory Liu et al [45] derived thepressure wave velocity calculation formula for gas-drillingmud-solid three-phase flowbased on the continuity equationWang and Zhang [46] studied the pressure pulsation inmud and set up a model for calculating the amplitude ofpressure pulsation when pressure wave is transmitted indrilling-fluid channel especially drilling hose with differentinside diameters However some efforts have been made thepressure wave velocity is usually determined by empiricalformula In the past researches the influencing factors forpressure wave propagation were simulated and analyzed withthe mathematical model however the variation of wavevelocity and gas void at different depth of wellbore was notconsidered In addition the current researches are limited intheir assumption and neglect the flow pattern translation andinterphase forces along the annulus Up to now no complete
mathematical model of pressure wave in an annulus withvariations of gas void flow pattern temperature and backpressure during MPD operations has been derived
The object of the present work is to study the velocityfor the transmission of pressure disturbance in the two-phase drilling fluid in the form of a pressure wave inannulus during MPD operations In this paper in additionto the pressure temperature and the void fraction in theannulus the compressibility of the gas phase the virtualmass force and the changes of interface in two phases arealso taken into consideration By introducing the pressuregradient equations in MPD operations gas-liquid two-fluidmodel the gas-drilling mud equations of state (EOS) andsmall perturbation theory a united model for predictingpressure wave velocity in gas and drilling mud in an annulusis developed The model can be used to predict the wavevelocity of various annulus positions at different influx ratesapplied back pressures and angular frequencies with a fullconsideration of drilling mud compressibility and interphaseforces
2 The Mathematical Model
21 The Basic Equation In this paper the two-fluid modeland the pressure gradient equation along the flow direction inthe annulus are combined to study the pressure wave velocityin MPD operations Drilling fluid contains clay cuttingsbarite other solids and so forth The solid particles aresmall and uniformly distributed therefore drilling fluid isconsidered to be a pseudohomogeneous liquid and the influxnatural gas is considered to be the gas phase The followingassumptions are made
(i) the two-phase flow is treated as one dimension(ii) no mass transfers between the gas and drilling mud(iii) the flow pattern in an annulus is either bubbly or slug
flowAs shown in Figure 1 the gas and drilling mud two-
phase fluid travels along the annulus in the drilling processThe fluid flows along the annulus in ldquominus119911rdquo direction and theannulus is formed by the casing and drill string
Themomentum conservation equation for gas and liquidtwo-phase flow is
sum119865
119911= 120588
119898119860119889119911
119889V119898
119889119905
(1)
The mass conservation equations are120597
120597119905
(120601
119866120588
119866) +
120597
120597119911
(120601
119866120588
119866V119866) = 0
120597
120597119905
(120601
119871120588
119871) +
120597
120597119911
(120601
119871120588
119871V119871) = 0
(2)
The momentum conservation equations for gas are120597
120597119905
(120601
119866120588
119866V119866) +
120597
120597119911
(120601
119866120588
119866V2119866) +
120597
120597119911
(120601
119866120588
119866)
minus
120597
120597119911
[120601
119866(120591
119891119903
119866+ 120591
Re119866) +119872Gi minus 4
120591
119866
119863
] = 0
(3)
4 Mathematical Problems in Engineering
Casing
Drill string
Flow direction
Gas
Drilling mud
z
Figure 1 The schematic of gas and drilling mud two-phase flow inannulus
The momentum conservation equations for liquid are
120597
120597119905
(120601
119871120588
119871V119871) +
120597
120597119911
(120601
119871120588
119871V2119871) +
120597
120597119911
(120601
119871120588
119871)
minus
120597
120597119911
[120601
119871(120591
119891119903
119871+ 120591
Re119871) +119872Li minus 4
120591
119871
119863
] = 0
(4)
The interphase forces include virtual mass force dragforce and the wall shear stress The transfer of momentumbetween the gas and drilling mud phases119872Gi and119872Li can bewritten as follows
119872Gi = minus119872ndLi minus119872
119889
Li + (120591119891119903
Li + 120591ReLi )
120597120601
119871
120597119911
+
120597 (120601
119866120590
119904)
120597119911
+
120597 (120601
119866119901Gi)
120597119911
minus 120601
119866
120597 (119901Li)
120597119911
119872Li = 119872ndLi +119872
119889
Li + 119901Li120597 (120601
119871)
120597119911
minus (120591
119891119903
Li + 120591ReLi )
120597120601
119871
120597119911
(5)
The momentum transfer term caused by the gas-liquidinterfacial relative acceleration motion (ie the virtual massforce) can be expressed as
119872
ndLi = 119888VM120601119866120588119871120572VM minus 01120601
119866120588
119871V119904
120597V119904
120597119911
minus 119888
1198981120588
119871V2119904
120597120601
119866
120597119911
(6)
where 1198881198981= 01 and v
119904is the slip velocity V
119904= V119866minus V119871
The momentum transfer term caused by the drag forceprovided by Park et al [47] can be described as
119872
119889
Li =3
8
119862
119863
119903
120588
119871119877
119866V2119904
(7)
The pressure difference between the liquid interface andliquid is defined by the following formula presented by Parket al
119901Li minus 119901119871 = minus119888119901120588119871V2
119904 (8)
where 119888119901= 025
The proportion of gas phase in the interface between thegas and drilling mud is rather small in that the pressuredifference between the gas interface and gas is not very highOmitting the pressure difference the gas interface pressure119875Gi can be written as
119901Gi minus 119901119866 asymp 0 (9)
According to the equations of Arnold [48] the pressureof the drilling mud is described as follows
119901
119871= 119901 minus 025120588
119871120601
119866V2119904
(10)
In fact the shear stress and the interphase shear stress arevery small relative to the Reynolds stress Also the Reynoldsstress in gas phase can be omitted relative to the interphaseforce Hence it can be described as
120591
119891119903
119866asymp 120591
119891119903
Li asymp 120591119891119903
119871asymp 120591
119866asymp 120591
Re119866asymp 0
(11)
The Reynolds stress and interfacial average Reynoldsstress can be obtained by
120591
Re119871= minus119888
119903120588
119871V2119904
120601
119866
120601
119871
120591
ReLi = minus119888119903120588119871V
2
119904
(12)
where 119888119903= 02
The wall shear stress of liquid phase is given by Wallis inthe following form [30]
120591
119871= 05119891
119871120588
119871V2119871
(13)
22 Physical Equations
221 Equations of State for Gas The equation of state (EOS)for gas can be expressed as follows
120588
119866=
119901
(119885
119866sdot 119877 sdot 119879)
(14)
where Z119866is the compression factor of gas
The formula presented by Dranchuk and Abou-Kassemhas been used to solve the gas deviation factor under thecondition of low and medium pressure (119901 lt 35MPa) [49]
119885
119866= 1 + (03051 minus
10467
119879
119903
minus
05783
119879
3
119903
)120588
119903
+ (05353 minus
026123
119879
119903
minus
06816
119879
3
119903
)120588
2
119903
(15)
where 119879119903= 119879119879
119888 119901119903= 119901119901
119888 and 120588
119903= 027119901
119903119885
119866119879
119903
The formula presented by Yarborough and Hall has beenadopted to solve the gas deviation factor under the conditionof high pressure (119901 ge 35MPa) [50]
119885
119866=
006125119875
119903119879
minus1
119903exp (minus12 (1 minus 119879minus1
119903) 2)
119884
(16)
Mathematical Problems in Engineering 5
where Y is given by
minus 006125119901
119903119879
minus1
119903exp (minus12(1 minus 119879minus1
119903)
2
) +
119884 + 119884
2
+ 119884
3
+ 119884
4
(1 minus 119884)
3
= (1476119879
minus1
119903minus 976119879
minus2
119903+ 458119879
minus3
119903) 119884
2
minus (907119879
minus1
119903minus 2422119879
minus2
119903+ 424119879
minus3
119903) 119884
(218+282119879minus1
119903
)
(17)
222 Equations of State for Liquid Under different tem-peratures and pressures the density of drilling mud can beobtained by the empirical formulas
If 119879 lt 130 ∘C the density of drillingmud can be obtainedby the following equation
120588
119871= 120588
0(1 + 4 times 10
minus10
119901
119871minus 4 times 10
minus5
119879 minus 3 times 10
minus6
119879
2
)
(18)
If 119879 ge 130 ∘C the density of drilling mud is
120588
119871= 120588
0(1 + 4 times 10
minus10
119875
119871minus 4 times 10
minus5
119879
minus3 times 10
minus6
119879
2
+ 04(
119879 minus 130
119879
)
2
)
(19)
223 Correlation of Temperature Distribution The tempera-ture of the drilling mud at different depths of the annulus canbe determined by the relationship presented by Hasan andKabir [51]
119879 = 119879
119890119894+ 119865 [1 minus 119890
(119911119887ℎ
minus119911)119860
] (minus
119892 sin 120579119892
119888119869119888
119901119898
+ 119873 + 119892
119879sin 120579)
+ 119890
(119911119887ℎ
minus119911)119860
(119879
119891119887ℎminus 119879
119890119887ℎ)
(20)
23 Flow Pattern Analysis Based on the analysis of flowcharacteristics in the enclosed drilling system it can be safelyassumed that the flowpattern in an annulus is either bubbly orslug flow [52] The pattern transition criteria for bubbly flowand slug flow given by Orkiszewski are used to judge the flowpattern in the gas-drilling mud two-phase flow [53]
For bubbly flow119902
119866
119902
119898
lt 119871
119861 (21)
For slug flow119902
119866
119902
119898
gt 119871
119861 119873
119866119881lt 119871
119878 (22)
where q119898is the volumetric flow rate of two-phase flow 119902
119898=
119902
119866+ 119902
119871
The dimensionless numbers L119904and L
119861are defined as
119871
119904= 50 + 36119873
119866119881
119902
119871
119902
119866
119871
119861= 1071 minus
07277V2119898
119863
(23)
where
119873
119866119881= V119904(
120588
119871
119892120590
119904
)
025
(24)
Flow parameters such as void fraction mixture densityand virtual mass force coefficient are discussed for a specificflow pattern
The correlation between void fraction and liquid holdupis expressed as
120601
119866+ 120601
119871= 1 (25)
231 Bubbly Flow As for bubbly flow the density of gas anddrilling mud mixture is described as the following formula
120588
119898= 120601
119871120588
119871+ 120601
119866120588
119866 (26)
The void fraction is determined by the following formula
120601
119866=
1
2
[
[
1 +
119902
119898
V119904119860
minus
radic
(1 +
119902
119898
V119904119860
)
2
minus
4119902
119866
V119904119860
]
]
(27)
The coefficient of virtual mass force 119862vm for bubbly flowcan be expressed as follows [54]
119862vm = 05
1 + 2120601
119866
1 minus 120601
119866
(28)
The coefficient of resistance coefficientC119863for bubbly flow
can be expressed as
119862
119863=
4119877
119887
3
radic
119892 (120588
119871minus 120588
119866)
120590
119904
[
1 + 1767(1 minus 120601
119866)
97
1867(1 minus 120601
119866)
15]
2
(29)
The friction pressure gradient for bubbly flow can beobtained from the following equation
120591
119891= 119891
120588
119871V2119871
2119863
(30)
232 Slug Flow As for slug flow distribution coefficient ofgas in the liquid phase is
119862
0=
000252lg (103120583119871)
119860
138minus 0782 + 0232lgV
119898minus 0428lg119860
(31)
The average density of the mixture for slug flow isdetermined by
120588
119898=
119882
119898+ 120588
119871V119904119860
119902
119898+ V119904119860
+ 119862
0120588
119871 (32)
The void fraction for slug flow is determined by thefollowing formula
120601
119866=
119902
119866
119902
119866+ 119902
119871
(33)
6 Mathematical Problems in Engineering
The coefficient of virtual mass force 119862vm for slug flow canbe expressed as follows
119862vm = 33 + 17
3119871
119902minus 3119877
119902
3119871
119902minus 119877
119902
(34)
The coefficient of resistance coefficient C119863for slug flow
can be expressed as
119862
119863= 110120601
3
119871119877
119887
(35)
The friction pressure gradient of bubbly flow can beobtained from the following equation
120591
119891= 119891
120588
119871V2119898
2119863
(
119902
119871+ V119904119860
119902
119898+ V119904119860
+ 119862
0)
(36)
24 Annulus Characteristic Analysis The annulus effectivediameter proposed by Sanchez [55] is used in the hydrauliccalculation of annulus
119863 =
120587 (119863
2
119900minus 119863
2
119894) 4
120587 (119863
119900+ 119863
119894) 4
= 119863
119900minus 119863
119894
(37)
The effective roughness of the annulus can be expressedas
119896
119890= 119896
119900
119863
119900
119863
119900+ 119863
119894
+ 119896
119894
119863
119894
119863
119900+ 119863
119894
(38)
where theD119894andD
119900are the diameters of inner pipe and outer
pipe respectively the k119900and k119894are the roughnesses of outer
pipe and the inner pipe respectively
3 The United Model Developed
The total pressure drop gradient is the sum of pressure dropgradients due to potential energy change and kinetic energyand frictional loss From (1) the equation used to calculatethe pressure gradient of gas and drilling mud flow within theannulus can be written as
119889119901
119889119911
= 120588
119898119892 sin 120579 minus
120591
119908120587119863
119860
minus 120588
119898V119898
119889V119898
119889119911
(39)
Assuming the compressibility of the gas is only related tothe pressure in the annulus the kinetic energy or accelerationterm in the previous equation can be simplified to
120588
119898V119898
119889V119898
119889119911
= minus
120588
119898V119898V119904119892
119901
119889119901
119889119911
= minus
119882
119898119902
119892
119860
2119901
119889119901
119889119911
(40)
Substituting (39) into (40) the total pressure drop gra-dient along the flow direction within the annulus can beexpressed as
119889119901
119889119911
=
120588
119898119892 + 120591
119891
1 minus119882
119898119902
119866 (119860
2119901)
(41)
It is assumed that the gas obeys to the EOS (14) and thecompressibility of drilling mud can be obtained by adopting
the simplified EOS ((18) and (19)) which neglects the thermalexpansion of liquid The sonic speed of gas phase c
119866and that
of liquid phase c119871can be presented in the following form
119889119901
119871
119889120588
119871
= 119888
2
119871
119889119901
119866
119889120588
119866
= 119888
2
119866
(42)
By introducing (42) the hydrodynamic equations of two-fluid model ((2)ndash(4)) can be written in the matrix form
119860
120597120585
120597119905
+ 119861
120597120585
120597119911
= 119862120585
(43)
where A is the matrix of parameters considered in relation totime B is the matrix of parameters considered in relation tothe spatial coordinate and C is the vector of extractions
By introducing the small disturbance theory the distur-bance of the state variable 120585(120601
119866 119901 V119866 V119871)
119879 can be written as
120585 = 120585
0+ 120575120585 exp [119894 (119908119905 minus 119896119905)] (44)
In (44) k is the wave number and 119908 is the angularfrequency of the disturbances Substituting (44) into (43)gives homogenous linear equations concerning the expres-sion (120575120601
119866 120575119901 120575V
119866 120575V119871)
119879 According to the solvable conditionof the homogenous linear equations that the determinant ofthe equations is zero dispersion equation of pressure wavecan be expressed in the following form
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119872
1119872
2119872
3119872
4
minus120588
119871119908
1 minus 120601
119866
119888
2
119871
119908 0 minus119896 (1 minus 120601
119866) 120588
119871
119872
5119872
6119872
7119872
8
119872
9119872
10119872
8119872
11
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
= 0 (45)
whereM1ndashM11can be illustrated by
119872
1= (120588
119866+ 119888
119901120601
119866120588
119871
V2119904
119888
2
119866
)119908
119872
2=
120601
119866
119888
2
119866
[1 minus 119888
119901120601
119871]
V2119904
119888
2
119871
119908
119872
3= minus[120601
119866120588
119866119896 + 2119888
119901120601
119866120601
119871120588
119871
V119904
119888
2
119871
119908]
119872
4= 22119888
119901120601
119866120601
119871120588
119871
V119904
119888
2
119871
119908
119872
5= 120588
119871V2119903119896 (minus120601
119866119888
119901+ 119888
119903minus 119888
119894+ 119888
1198982)
119872
6= minus120601
119866119896[1 minus 120601
119871
119888
119901V2119904
119888
2
119871
+ 119888
119894
V2119904
119888
2
119871
]
Mathematical Problems in Engineering 7
119872
7= 120601
119866(120588
119866+ 119888vm120588119871) 119908 minus 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904+
4
119863
119891
119866119908120588
119866V119866)
119872
8= minus119888vm120601119866120588119871119908 + 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904)
119872
9= 120588
119871V2119904119896 (120601
119871119888
119901minus 2119888
119903minus 119888
1198982)
119872
10= minus119896(120601
119871+ 119888
119903120601
119866
V2119904
119888
2
119871
)
119872
11= 120588
119871[120601
119871+ 120601
119866119888vm] 119908 minus 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904+
4
119863
119891
119871120588
119871V119871)
(46)
where 1198881198982= 01 119888
119901= 025 119888
119894= 03 and 119888
119903= 02
By solving the dispersion equation indicated previouslyfour roots can be obtained Because the wavelengths asso-ciated with two of the roots are too short to allow the two-fluid medium to be treated as a continuum the two rootsshould be omitted As for the two remaining roots one ofthem represents a pressure wave that transmited along theaxis z and the other represents a pressure wave transmits inthe opposite direction in accordance with the direction of theflow along the annulusThe real value of wave number can beused to determine thewave velocity cThewave velocity in thetwo-phase fluid can be determined by the following model
119888 =
1003816
1003816
1003816
1003816
(119908119877
+
(119896)) minus (119908119877
minus
(119896))
1003816
1003816
1003816
1003816
2
(47)
4 Solution of the United Model
Obtaining the analytical solution of themathematical modelsconcerned with flow pattern void fraction characteristicparameters and pressure drop gradient is generally impossi-ble for two-phase flow In this paper the Runge-Kuttamethod(R-K4) is used to discretize the theoretical model
We can obtain pressure temperature gas velocity drillingmud velocity and void fraction at different annulus depthsby R-K4 The solution of pressure drop gradient equation(41) can be seen as an initial-value problem of the ordinarydifferential equation
119889119901
119889119911
= 119865 (119911 119901)
119901 (119911
0) = 119875
0
(48)
With the initial value (z0p0) and the function F(z p)
(50)ndash(53) can be obtained by
119896
1= 119865 (119911
0 119901
0) (49)
119896
2= 119865(119911
0+
ℎ
2
119901
0+
ℎ
2
119896
1) (50)
119896
3= 119865(119911
0+
ℎ
2
119901
0+
ℎ
2
119896
2) (51)
119896
4= 119865 (119911
0+ ℎ 119901
0+ ℎ119896
3) (52)
where h is the step of depthThe pressure on the nod 119894 = 119894 +1can be obtained by
119901
1= 119901
0+ Δ119901 = 119901
0+
ℎ
6
(119896
1+ 2119896
2+ 2119896
3+ 119896
4)
(53)
In the present work the mathematical model and pres-sure wave velocity calculation model are solved by a person-ally compiled code on VBNET (Version 2010) The solutionprocedure for the wave velocity in the annulus is shown inFigure 1 At initial time the wellhead back pressure wellheadtemperature wellbore structure well depth gas and drillingmud properties and so forth are known On the node ithe pressure gradient temperature and the void fraction canbe obtained by adopting R-K4 Then the determinant (45)is calculated based on the calculated parameters Omittingthe two unreasonable roots the pressure wave velocity atdifferent depths of the annulus in MPD operations can besolved by (47)The process is repeated until the pressure wavevelocity of every position in the wellbore is obtained as shownin Figure 2
The developed model takes full consideration of theinterfacial interaction and the virtual mass force Owing tothe complex conditions of the annulus in MPD operationsmeasurement of wave velocity in the actual drilling processis very difficult In order to verify the united model thepredicted pressure waves are compared with the results ofprevious simulated experimental investigations presented forgas and drillingmud by Liu et al [45] in Figure 3(a) and by Liet al [22] in Figure 3(b) The lines represent the calculationresults and the points represent the experimental data
The comparisons reveal that the developed united modelfits well with the experimental data Thus the united modelcan be used to accurately predict wave velocity at differentwellhead back pressures and gas influx rates (the influx rateof gas in the bottomhole) in MPD operations
5 Analysis and Discussion
The drilling system described is an enclosed system Theschematic diagram of the gas influx process is illustrated inFigure 4 The drilling mud is pumped from surface storagedown the drill pipe Returns from the wellbore annulus travelback through surface processing where drilling solids areremoved to surface storage The key equipments include thefollowing
(i) The Rotating Control Device (RCD) The rotatingcontrol device provides the seal between atmosphereand wellbore while allowing pipe movement anddiverting returns flow In conjunction with the flowcontrol choke the RCD provides the ability to applyback pressure on the annulus during an MPD opera-tion
(ii) Choke The MPD choke manifold provides anadjustable choke system which is used to dynamicallycontrol the required BHP by means of applyingsurface BP
(iii) Coriolis Meter A Coriolis meter is used to accu-rately measure the mass flow rate of fluid exiting
8 Mathematical Problems in Engineering
Grid division
Start
Delete two roots
End
Initial conditions
Meet demand
i = i + 1
Solve eq (20) for T
Solve eq (15)ndash(17) for Zg
Judging flow pattern by eq (21) and eq (22)
Bubbly flow pattern Slug flow pattern
Solve determinant eq (45) for four roots
Calculate F(z p) by eq (48)
Calculate k1 minus k4Pi+1 = Pi + ΔPzi+1 = zi + h
zi+1 gt HNo
No
Yes
Yes
Solve eq (18) (19) for120588L
Solve eq ( 47) for c
Solve eq (14)120588G
Calculate 120588m 120601G f and CDCalculate 120588m 120601G f and CD CvmCvm
Figure 2 Solution procedure for wave velocity in MPD operations
the annulus The ability to measure return flowaccurately is essential for the applied back pressure
(iv) Pressure Sensor A pressure sensor is used to measuresurface back pressure on the wellhead
The gas and drilling mud flow rate measured by the Cori-olis meter and the back pressure measured by the pressuresensor are the initial data for annulus pressure calculationThe well used for calculation is a gas well in XinjiangUygur Autonomous Region Northwest China The wellborestructure well design parameters (depths and diameters)gas and drilling mud properties (density and viscosity) andoperational conditions of calculation well are displayed inTable 1
The drilling mud mixed with gas is taken as a two-phaseflow medium The propagation velocity of pressure wave inthe gas-drillingmud flow is calculated and discussed by usingthe established model and well parameters
51 Effect of Back Pressure on Wave Velocity Increase inapplied back pressure is the most common approach fordynamicwell controlThewellbore can be seen as an enclosed
Table 1 Parameters of calculation well
Type Property Value
Mud Dynamic viscosity (Pa sdot s) 0056Density (kgcm3) 1460
Gas Relative density 065Viscosity (Pa sdot s) 114 times 10minus5
StringElastic modulus of string (Pa) 207 times 1011
Poisson ratio of string 03Roughness (m) 154 times 10minus7
Surface condition Surface temperature (K) 298Atmosphere pressure (MPa) 0101
pressurized system The pressure at different depths of theannulus varied with the change of back pressure Accordingto the EOS of gas and drilling mud the influence of pressureon gas volume is much greater than that of drilling mudfor the greater compressibility of gas So the gas voidfraction changes with the variation of gas volume at thenearly constant flow rate of drilling mud at different annularpressures Meanwhile the pressure wave velocity is sensitive
Mathematical Problems in Engineering 9
5000 2 4 6 8 10
750
1000
1250
1500c
(ms
)
This paper p = 30MPa w = 50Hz = Re
120601G ()
CvmLiu et al [45] p = 30MPa
(a) Comparison with the experimental data of Liu
0 20 40 60 80 1000
40
80
120
160
200
c(m
s)
This paper p = 01MPa w = 50Hz = Re
120601G ()
CvmLi et al [22] p = 01MPa
(b) Comparison with the experimental data of Li
Figure 3 Experimental verification by comparison with previous experimental data
Drill pipe
Annulus
RCD
Formation
Mud pump
Drill collar
Coriolis meterChoke
Casing
Pressure sensors
120601 2159 mm times G105
1206011778mm (120601 78 mm) times G105
120601127 mm (120601 78 mm) times G105
z0 m
3600 m
4000 m
QL
Figure 4 Schematic diagram of the gas influx process
to the gas void fraction As a result when the back pressureat the wellhead is changed by adjusting the choke the wavevelocity in the two-phase drilling fluid and the distributionof void fraction at different depth of the annulus will divergeThe calculation results affected by the back pressure arepresented
Figures 5 and 6 show the distributions of void fractionand variations of wave velocity along the flow directionin the annulus when the back pressure at the wellhead is01MPa 08MPa 15MPa 25MPa 50MPa and 65MPa
respectively It can be seen that the void fraction significantlyincreases along the flow direction in the annulus Converselythe wave velocity shows a remarkable decrease tendency Forinstance BP = 15MPa at the wellhead the wave velocityis 9012ms and the void fraction is 0611 while in thebottomhole the wave velocity reaches 73142ms and voidfraction is reduced to 0059 However a sudden increase inwave velocity is observed in Figure 5 when the back pressureis 01MPa at the position close towellhead as the void fractionis further increased Moreover the influence of back pressure
10 Mathematical Problems in Engineering
10075500 254000
3000
2000
1000
0
BP = 01 MPaBP = 08 MPaBP = 15 MPa
BP = 25 MPa
BP = 65 MPaBP = 5 MPa
H(m
)
Q = 9m3h w = 50Hz = Re
120601G ()
Cvm
Figure 5 Void fraction distribution at different back pressures
on the void fraction is larger at the position close to thewellhead than that in the bottomhole
This can be explained from the viewpoints of mixturedensity and compressibility of two-phase fluid and the pres-sure drop along the flow direction in the wellbore In thelow void fraction range the gas phase is dispersed in theliquid as bubble so the wave velocity is influenced greatly bythe added gas phase According to the EOS if gas invadesinto the wellbore with a small amount in the bottomholethe density of the drilling mud has little variation while thecompressibility increases obviously which makes the wavevelocity decreased Then the two-phase drilling mud flowfrom the bottomhole to the wellhead along the annulus witha drop of pressure caused by potential energy change kineticenergy and frictional loss which leads to an increase in voidfractionWhen the void fraction is increased both the densityand the compressibility of the two-phase fluid change slightlyresulting in a flat decrease in wave velocity along the flowdirection With the further increasing in void fraction thebubbly flow turns into slug flow according to the flow patterntransient criteria of Orkiszewski which shows that the flowpattern is dependent on the void fraction Generally speak-ing the gas-drilling mud slug flow is composed of liquidand gas slugs In the preliminary slug flow the liquid slugsare much longer than the gas slugs so the wave velocity isdetermined primarily by the wave velocities in the liquidslugs which are hardly affected by the void fraction At theposition close to the wellhead the pressure of two-phase flowfluid can be reduced to a low value which approaches theback pressure The compressibility of gas will be improvedat the low pressure It results in significant increase in void
0 250 500 750 10004000
3000
2000
1000
0
H(m
)BP = 01 MPaBP = 08 MPaBP = 15 MPa
BP = 25 MPa
BP = 65 MPaBP = 5 MPa
c (ms)
Q = 9m3h w = 50Hz = ReCvm
Figure 6 Wave velocity variations at different back pressures
fraction As the void fraction increases greatly at the positionclose to the wellhead gas slugs becomemuch longer than theliquid slugs It is assumed that the actual wave velocity in slugflow happens to play a leading role when the void fractionincreases to some extent For the decrease in liquid holdupin the gas slug with the increasing in void fraction the wavevelocity in the gas slug is infinitely close to that in pure gaswhich is equivalent to an increase in wave velocity in the gasslug Meanwhile the gas content in liquid slug also increasesand results in a lower wave velocity in the liquid slug As aresult a slight increase in wave velocity appears
Figures 7 and 8 present the influence of back pressureon the void fraction and wave velocity with respect to theparameters of well depth As the back pressure increases thevoid fraction at different depth of the annulus is reducedgradually while the wave velocity in the two-phase flow tendsto increase Analytical results show that the increased backpressure is equivalent to be applied to the entire encloseddrilling fluid cyclical systemThe pressure transmits from thewellhead to the bottomhole therefore the annular pressurein the entire wellbore is increased According to the EOSthe density of gas increases and the compressibility of gasdecreases with the increasing of gas pressure So the lossof interphase momentum and energy exchange is reducedand the interphase momentum exchange is promoted Itcontributes to the increase in wave velocity with the increasein gas pressure In addition due to lower compressibility oftwo-phase flow medium under high pressure the increasetendency of pressure wave velocity and the decrease tendencyof void fraction are slowed down in the high back pressurerange
Mathematical Problems in Engineering 11
0
25
50
75
100
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
0 25 75 105
Q = 9m3h w = 50Hz = Re
120601G
()
Cvm
Figure 7 Effect of back pressure on the void fraction
52 Effect of Gas Influx Rate on Wave Velocity Figures 9and 10 graphically interpret the distributions of void fractionand variations of wave velocity along the flow direction inthe annulus When gas influx occurs in the bottomhole gasinvades into the wellbore and migrates from the bottomholeto the wellhead along the flow direction At a low gas influxrate it is extremely obvious that the void fraction and wavevelocity first slightly change in a comparatively smooth valuethen change sharply It is because of the rapid expansion ofgas volume with the decreasing in pressure near the wellheadthat the void fraction increases sharply and the wave velocitydecreases obviously at the same time But under the highbottomhole pressure (up to 50MPa) the compressibilityof the gas is low This results in a slight change in voidfraction and wave velocity at the position far away from thewellhead Since the compressible component increases withthe increase in the gas influx rate the compressibility of thegas and drilling mud two-phase fluid is improved So thevariations of void fraction and wave velocity become moreprominent Also the void fraction still shows an increasetendency that is steady first and then sharp It acts inaccordance with the variation of void fraction at a low gasinflux rate At a low gas influx rate if the void fraction atthe position close to the wellhead can not increase to a highextent the wave velocity always shows a decrease tendencyAt a high gas influx rate such as the 119876 = 8312m3h thewave velocity tends to increase because the void fraction inthe wellhead is increased to a high extent In conclusionthe wave pressure is sensitive to the void fraction and thevoid fraction is dominated by influx rate and pressure in theannulus especially the influx rate
0 25 75 1050
300
600
900
1200
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
c(m
s)
Q = 9m3h w = 50Hz = ReCvm
Figure 8 Effect of back pressure on the pressure wave velocity
With the influx of gas the mixing of gas and drillingmud occurs in the annulus and the corresponding interfacialtransfer of momentum and mass causes an increase in gasphase void fraction and a decrease in pressure wave velocityas shown in Figures 11 and 12 Within the range of low gasinflux rate the wave velocity decreases significantly It isbecause of this that the compressibility of the gas increasesremarkably and the medium appears to be of high elasticitythough the density of gas-drilling mud two-phase flowchanges slightly With the increase in the gas influx rate andcorresponding increase in the void fraction in the annulusthe compressibility of the two-phase unceasingly increaseswhich promotes the momentum and energy exchange inthe interface So the pressure wave continuously decreasesWhen the void fraction increases to some extent following theincrease in the gas influx rate the decrease in wave velocityin the liquid slug is gradually less than the increase in wavevelocity in the gas slug thus the decrease of wave velocityis slowed down for the growth of gas slug Especially in thewellhead a slight increase in wave velocity is observed
53 Effect of Virtual Mass Force on Wave Velocity Figure 13illustrates the effect of virtual mass force on the wave velocitywith the parameter of gas influx rate As shown in Figure 15under the same gas influx rate (0594m3h 1098m3h and1768m3h resp) the wave velocity curve of 119862vm = Rediverges from the curve of 119862vm = 0 at the position close tothewellhead whereas thewave velocity curve of the two typesis almost coincided below the position of 119867 = 500m Thisdivergence between the two types of curves is connected withthe fact that significance of influence of interphase virtual
12 Mathematical Problems in Engineering
0 25 50 75 1004000
3000
2000
1000
0
Q = 0058m3h Q = 1228m3hQ = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
H(m
)
BP = 01 MPaw = 50Hz = Re
120601G ()
Cvm
Figure 9 Void fraction distribution at different gas influx rates
mass forces increases together with the increase of the relativemotion in the interphase It is evident that the bottomholepressure is hundreds of times higher than the wellheadpressure As a result the void fraction at the position closeto the wellhead increases sharply meanwhile the interphasemomentum and energy exchanges are promoted The virtualmass force can be described as the transfer of momentumbetween the gas phase and drilling mud phase caused bythe relative motion in the interphase If the relative motionin the interphase is quite weak the value of virtual massforce will intend to approach 0 and the influence on thewave velocity can be ignored However if the relative motionis rather intense the effect of virtual mass force on thewave velocity should not be ignored Furthermore taking thevirtual mass force into account the dispersion characteristicof the pressure wave weakens obviously Compared with thepressure wave velocity calculated by ignoring the effect ofvirtual mass force the calculated pressure wave velocity witha consideration of the virtual mass force is lower Thereforeit is necessary to analyze the effect of the virtual mass forceon the wave velocity at the position close to the wellhead inMPD operations
54 Effect of Disturbance Angular Frequency onWave VelocityFigure 14 presents the pressure wave velocity in the annulusat different angular frequencies According to Figures 5 and9 the void fraction first slightly increases in the bottomholeand then sharply increases along the flow direction at theposition close to the wellhead At a fixed angular frequencythis phenomenon results in an overall decrease in the pressurewave velocity except for the section near the wellhead At the
0 500 1000 1500 20004000
3000
2000
1000
0
c (ms)
H(m
)Q = 0058m3h Q = 1228m3h
Q = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
BP = 01 MPaw = 50Hz = ReCvm
Figure 10 Wave velocity variations at different gas influx rates
section close to the wellhead an opposite change trend ofpressure wave velocity is observed for the transition of flowpattern from bubbly flow to slug flow due to the continuousincrease in void fraction along the flow direction In additionit can be clearly seen from the curves that the wave velocityincreases accompanying with the increase in the angularfrequency above the position of 119867 = 500m This propertyis not very distinct at the position below 119867 = 500m for thelow void fraction
Figure 15 shows the effect of frequency on the wavevelocity in the gas-drilling mud flow (119862vm = 0)The curves ofthe wave velocity of different positions reveal that the prop-agation velocity of pressure disturbances increases togetherwith the growth of the angular frequency (0 lt 119908 lt 500Hz)It proves that the pressure wave has an obvious dispersioncharacteristic in the two-phase flow As the angular frequencyincreases in the range of less than 500Hz there is sufficienttime to carry out the exchange of energy and momentumbetween two phases It achieves a state of mechanical andthermodynamic equilibrium between the two phases Sothe wave velocity increases gradually with the increase inthe angular frequency at different depths of the annulus Itis considered that the wave velocity is mainly affected bythe interphase mechanical and thermodynamic equilibriumat low anglular frequencies When the angular frequencyreaches the value of 119908 = 500Hz the pressure wave velocityachieves a constant value and remains on this level regardlessof the further growth in angular frequency 119908 With theincrease in angular frequency there is not enough time forenergy and momentum exchange between the gas-drillingmud two phases to reach themechanical and thermodynamic
Mathematical Problems in Engineering 13
0
25
50
75
100
125
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
Q (m3h)
120601G
()
0 25 5 75 10
BP = 01 MPaw = 50Hz = ReCvm
Figure 11 Effect of gas influx rate on the void fraction
equilibrium state and thus the wave dispersion does notexist At a high angular frequency the wave velocity is mainlydominated by the mechanical and thermal nonequilibriumin the flow and keeps almost unchanged when the angularfrequency is further increased This is consistent with theinfluences of the angular frequency in the horizontal pipe[28] Moreover the effect of virtual mass force is also shownin this figure (119862vm = Re) It is observed that the wave velocityis significantly reduced when the virtual mass force is takeninto consideration in the calculation of the wave velocity atthe position close to the wellhead such as 119867 = 0m It canbe explained by the reduction of dispersion characteristic ofa pressure wave Along the wellbore from the wellhead to thebottomhole the distinction gradually decreases
6 Conclusions
With full consideration of the important factors such asvirtual mass force drag force gas void fraction pressuretemperature and angular frequency a united wave velocitymodel has been proposed based on pressure drop gradientequations in MPD operations gas-liquid two-fluid modelthe gas-drilling mud equations of state and small pertur-bation theory Solved by the fourth-order explicit Runge-Kutta method the model is used to predict wave velocityfor different back pressures and gas influx rates in MPDoperations The main conclusions can be summarized asfollows
(1) With the introduction of virtual mass force and dragforce the united model agrees well with the previousexperimental data The united model can be used to
0
280
560
840
1120
1400
Q (m3h)0 25 5 75 10
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
c(m
s)
BP = 01 MPaw = 50Hz = ReCvm
Figure 12 Effect of gas influx rate on the pressure wave velocity
accurately calculate the wave velocity in the annulusThe application of the model will be beneficial tofurther study the wave velocity at different gas influxrates and back pressures in MPD operations reducenonproductive times and provide a reference for thedrilling operations in extremely narrow porefracturewindows existing conditions
(2) The wave velocity and void fraction have differentvalues with respect to well depth In the annulusthe drop of pressure causes an increase in the voidfraction along the flow direction The void fractionincreases first slightly and then sharply Correspond-ingly the wave velocity first gradually decreases in thbubbly flow and preliminary slug flow and then thewave velocity slightly increases accompanying withthe increase in the relative length ratio of gas slugto the liquid slug for the continuous increase in voidfraction at the position close to the wellhead Theminimum wave velocity appears in the long gas slugflow
(3) When the back pressure in the MPD operationsincreases the void fraction at different depths of theannulus is reduced gradually while the wave velocityin the two-phase flow tends to increase Moreoverthe influence of back pressure on the void fraction isgreater at a position close to the wellhead than that inthe bottomhole because of the great decrease in thepressure from the bottomhole to the wellhead Alsothe effect of back pressure on void fraction and wavevelocity is decreased in the high back pressure range
14 Mathematical Problems in Engineering
0 300 600 900 12004000
3000
2000
1000
225
180
135
90
45
0
BP = 01 MPaw = 50Hz
Q = 0594m3h = ReQ = 0594m3h = 0
Q = 1098m3h = ReQ = 1768m3h = ReQ = 1098m3h = 0
Q = 1768m3h = 0
c (ms)
H(m
)
CvmCvm
Cvm
Cvm
Cvm
Cvm
Figure 13 Effect of virtual mass force on the pressure wave velocity
(4) The wave velocity is sensitive to the void fraction butthe void fraction is dominated by gas influx rate andpressure in the annulus especially the gas influx rateSince the compressibility of the gas and drilling mudtwo-phase fluid is improved with the increase in theinflux rate the void fraction increases greatly andthe wave velocity decreases significantly within thelow gas influx rate range When the void fraction isincreased to some extent following the increase in thegas influx rate the decrease of wave velocity is slowedfor the growth of gas slug Especially at the wellheada slight increase in wave velocity is observed at a highgas influx rate for the sharp increase in void fraction
(5) It is necessary to analyze the effect of the virtualmass force on the wave velocity in MPD operationsEspecially at the position close to the wellhead theeffect of virtual mass force is more prominent for theintense phase interactions Taking the virtual massforce into account the dispersion characteristic of thepressure wave weakens obviously Compared with theresults calculated by ignoring the effect of virtualmassforce the calculated pressure wave velocity with aconsideration of the virtual mass force is lower
(6) The effect of angular frequency on the propagationvelocity of pressure wave appears at a low angularfrequency The propagation velocity of pressure dis-turbances increases together with the growth of theangular frequency (0 lt 119908 lt 500Hz) When theangular frequency reaches the value of 119908 = 500Hzthe pressure wave velocity achieves a constant value
0 240 480 720 960 12004000
3000
2000
1000
560
420
280
140
0
Q = 0727m3h BP = 01 MPa
w = 50Hz = Rew = 50Hz = 0w = 300Hz = 0
w = 800Hz = 0w = 1600Hz = 0w = 5000Hz = 0
c (ms)
H(m
)CvmCvmCvm
CvmCvmCvm
Figure 14 Wave velocity variations at different angular frequencies
and remains on this level regardless of the furthergrowth in angular frequency 119908
Nomenclature
A Annulus effective cross area (m2)c119866 Wave velocity in gas (ms)
c119871 Wave velocity in Drilling mud (ms)
C119863 The coefficient of drag force
D Annulus effective diameter (m)D119894 Diameter of the inner pipe (m)
D119900 Diameter of the outer pipe (m)
f119866 Shear stresses coefficient of gas interface
f119871 Shear stresses coefficient of drilling mud
interfaceF119911 Pressure applied to the two-phase flow (N)
g Acceleration due to gravity (m2s)g119888 Conversion factor
g119879 Geothermal temperature gradient (Km)
h One discrete length of annulus (m)k119890 Annulus effective roughness (m)
k119900 Roughness of outer pipe (m)
k119894 Roughness of inner pipe (m)
H The depth of annulus (m)L119902 Air bubbles length of slug (m)
119872Gi Momentum transfer in gas interface (Nm3)119872Li Momentum transfer in liquid interface (Nm3)119872
ndLi The none-drag-force (virtual mass force)
(Nm3)119872
119889
Li The momentum transfer term caused by thedrag force (Nm3)
Mathematical Problems in Engineering 15
100 1000 100000
450
900
1350
1800
0w (Hz)
0 m = 0 0 m = Re1000 m = 0
2000 m = 0
4000 m = 0
1000 m = Re2000 m = Re4000 m = Re
c(m
s)
Q = 0727m3h BP = 01 MPaCvmCvm
CvmCvmCvm
CvmCvmCvm
Figure 15 Effect of angular frequency on wave velocity
N Parameter combiningThompsonenergy effects
119901Gi Gas interface pressure (MPa)119901Li Liquid interface pressure (MPa)p Pressure (MPa)p0 Initial pressure of annulus (MPa)
p119888 Critical pressure (kPa)
p119871 Pressure of drilling mud (MPa)
p119903 Reduced pressure
q119898 Volumetric flow rate of gas and drilling
mud (m3s)q119866 Velocity of the gas (m3s)
q119871 Velocity of the drilling mud (m3s)
Q119871 Drilling mud rate within the annulus
(m3h)r Average diameter of the bubble (m)R Constant of EOS (JKgK)R119902 Air bubbles radius of slug (m)
R119887 The bubble diameter (m)
T Temperature (K)T119888 Critical temperature (K)
T119903 Reduced temperature
T119890119894 Undisturbed formation temperature at a
depth (K)T119890119887ℎ
Undisturbed formation temperature ofwellhead (K)
T119891119887ℎ
Undisturbed formation temperature atthe bottomhole (K)
v119898 Gas and drilling mud flow velocity
(ms)
v119866 Gas flow velocity (ms)
v119904 Slip velocity (ms)
v119904119892 Superficial gas velocity (ms)
v119871 Drilling mud flow velocity (ms)
W119898 Mass flow velocity (kgm3)
Z119866 Gas deviation factor
z0 Initial calculation height of annulus (m)
z119887ℎ Total well depth from surface (m)
z Variable of distance from surface (m)
Greek Letters
120588
119866 Gas density (kgm3)
120591
119891119903
Li Shear stresses of drilling mud noninterfacial(Nm2)
120591
119891119903
119871 Shear stresses of drilling mud interface (Nm2)
120591
Re119871 Reynolds stress of drilling mud (Nm2)
120591
Re119871 Reynolds stress of drilling mud interface
(Nm2)120591
Re119866 Reynolds stress of gas interface (Nm2)
120591
119891119903
119866 Shear stresses of gas interface (Nm2)
120572VM Acceleration due to virtual mass forcecoefficient (ms2)
120591
119871 Shear stresses of drilling mud along wall
(Nm2)120591
119866 Shear stresses of gas along wall (Nm2)
120591
Re119866 Reynolds stress of gas (Nm2)
120591
119891 Frictional pressure gradient (Pam)
120591
119908 Frictional pressure coefficient
120588
119898 Gas and drilling density (kgm3)
120588
0 Density under standard atmospheric pressure
(kgm3)120588
119871 Drilling mud density (kgm3)
120590
119904 Surface tension (Nm2)
Φ
119866 Gas void fraction
Φ
119871 Drilling mud holdup
120588
119903 Reduced density
120583
119871 Viscosity of drilling mud (Pa sdot s)
Subscripts
BHP Bottomhole pressureBP Back pressureEOS Equations of stateMPD Managed pressure drillingRCD The rotating control deviceR-K4 The fourth-order explicit Runge-Kutta
Subscripts of Graph
BP Back pressure (MPa)c Wave velocity in gas and drilling mud
two-phase flow (ms)119862vm The coefficient of virtual mass force
16 Mathematical Problems in Engineering
H Depth of annulus (m)Q Gas influx rate at the bottomhole (m3h)Re The value of virtual mass force
coefficient according to (28) or (34)119908 Angle frequency (Hz)Φ
119866 Void fraction ()
Acknowledgment
Research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 and 51274170)Without their support this work would not have beenpossible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of airinjection in annuli to manage pressure during cementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling conferenceand Exhibition San Diego Calif USA March 2012
[4] W A Bacon Consideration of compressibility effects for applied-back-pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Tex USA 2011
[5] J Choe C Hans and J Wold ldquoA modified two-phase well-control model and its computer applications as a training andeducational toolrdquo SPE Computer Applications vol 2 pp 14ndash201997
[6] J A Tarvin IWalton and PWand ldquoAnalysis of a gas kick takenin a deep well drilled with oil-based mudrdquo in Proceedings ofthe Annual Technical Conference and Exhibition of the Societyof Petroleum Engineers pp 255ndash264 Dallas Tex USA 1991
[7] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[8] J Zhou and G Nygaard ldquoAutomatic model-based controlscheme for stabilizing pressure during dual-gradient drillingrdquoJournal of Process Control vol 21 no 8 pp 1138ndash1147 2011
[9] A Ochoa O Acevedo L Nieto J E Lambarria and H PerezldquoSuccessful application of MPD (Managed Pressure Drilling)for prevention controland detection of borehole ballooning intight gas reservoir in Cuervito Field Mexicordquo in Proceedings ofthe Canadian Unconventional Resources Conference (CURC rsquo11)pp 172ndash185 November 2011
[10] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[11] R Ashena and J Moghadasi ldquoBottom hole pressure estimationusing evolved neural networks by real coded ant colony opti-mization and genetic algorithmrdquo Journal of Petroleum Scienceand Engineering vol 77 no 3-4 pp 375ndash385 2011
[12] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvanced Materials Research vol 468-471 pp 1736ndash1742 2012
[13] I Zubizarreta ldquoPore pressure evolution core damage and trip-ping out schedulesf a computational fluid dynamics approachrdquoin Proceedings of the SPEIADC Drilling Conference and Exhibi-tion Amsterdam The Netherlands March 2013
[14] C H Whitson ldquoCyclic shut-in eliminates liquid-loading in gaswellsrdquo in Proceedings of the SPEEAGE European Unconven-tional Resources Conference and Exhibition Vienna AustriaMarch 2012
[15] G M de Oliveira A T Franco C O R Negrao A LMartins and R A Silva ldquoModeling and validation of pressurepropagation in drilling fluids pumped into a closed wellrdquoJournal of Petroleum Science and Engineering vol 103 pp 61ndash71 2012
[16] Y Sato and H Kanki ldquoFormulas for compression wave andoscillating flow in circular piperdquo Applied Acoustics vol 69 no1 pp 1ndash11 2008
[17] J Xie B Yu X Zhang Q Shao and X Song ldquoNumericalsimulation of gas-liquid-solid three-phase flow in deep wellsrdquoAdvances in Mechanical Engineering vol 2013 Article ID951298 10 pages 2013
[18] W Bacon A Tong O Gabaldon C Sugden and P V Surya-narayana ldquoAn improved dynamic well control response to a gasinflux in managed pressure drilling operationsrdquo in Proceedingsof the IADCSPE Drilling Conference and Exhibition San DiegoCalifornia USA March 2012
[19] J I Hage and D ter Avest ldquoBorehole acoustics applied to kickdetectionrdquo Journal of Petroleum Science and Engineering vol 12no 2 pp 157ndash166 1994
[20] A M R Hopkins Fluid dynamics and surface pressure fluctua-tions of two-dimensional turbulent boundary layers over denselydistributed surface roughness [PhD thesis] Virginia PolytechnicInstitute and State University Blacksburg Va USA 2010
[21] Z S Fu ldquoThe gas cut detect technology during drilling in SovietUnionrdquo Petroleum Drilling Techniques vol 18 no 1 pp 19ndash211990
[22] X F Li C X Guan and X X Sui ldquoThe theory of gas influxdetection of pressure wave and its applicationrdquo Acta PetroleiSinica vol 18 no 3 pp 128ndash133 1997
[23] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[24] A E Ruggles R T Lahey Jr D A Drew and H A ScartonldquoInvestigation of the propagation of pressure perturbations inbubbly airwater flowsrdquo Journal of Heat Transfer vol 110 no 2pp 494ndash499 1988
[25] H J W M Legius H E A Van Den Akker and T NarumoldquoMeasurements on wave propagation and bubble and slugvelocities in cocurrent upward two-phase flowrdquo ExperimentalThermal and Fluid Science vol 15 no 3 pp 267ndash278 1997
[26] K Miyazaki and I Nakajima ldquoPropagation of pressure wavein NItrogen-Mercury two-phase systemrdquo Journal of NuclearScience and Technology vol 9 no 5 pp 315ndash318 1972
[27] B Bai L Guo and X Chen ldquoPressure fluctuation for air-watertwo-phase flowrdquo Journal of Hydrodynamics vol 18 no 4 pp476ndash482 2003
[28] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
Mathematical Problems in Engineering 17
[29] R F Tangren C H Dodge and H S Seifert ldquoCompressibilityeffects in two-phase flowrdquo Journal of Applied Physics vol 20 no7 pp 637ndash645 1949
[30] G B Wallis One Dimensional Two Phase Flow McGraw HillNew York NY USA 1969
[31] D L Nguyen E R F Winter and M Greiner ldquoSonic velocityin two-phase systemsrdquo International Journal ofMultiphase Flowvol 7 no 3 pp 311ndash320 1981
[32] R C Mecredy and L J Hamilton ldquoThe effects of nonequilib-rium heat mass and momentum transfer on two-phase soundspeedrdquo International Journal of Heat and Mass Transfer vol 15no 1 pp 61ndash72 1972
[33] K H Ardron and R B Duffey ldquoAcoustic wave propagationin a flowing liquid-vapour mixturerdquo International Journal ofMultiphase Flow vol 4 no 3 pp 303ndash322 1978
[34] J Xu and T Chen ldquoAcoustic wave prediction in flowing steam-water two-phase mixturerdquo International Journal of Heat andMass Transfer vol 43 no 7 pp 1079ndash1088 2000
[35] N ChungW Lin B Pei and Y Hsu ldquoModel for sound velocityin a two-phase air-water bubbly flowrdquo Nuclear Technology vol99 no 1 pp 80ndash89 1992
[36] X Xu and J Gong ldquoA unitedmodel for predicting pressurewavespeeds in oil and gas two-phase pipeflowsrdquo Journal of PetroleumScience and Engineering vol 60 no 3-4 pp 150ndash160 2008
[37] W Kuczynski ldquoCharacterization of pressure-wave propagationduring the condensation of R404A and R134a refrigerantsin pipe mini-channels that undergo periodic hydrodynamicdisturbancerdquo International Journal of Heat and Fluid Flow vol40 pp 135ndash150 2013
[38] W Kuczynski ldquoModeling of the propagation of a pressure waveduring the condensation process of R134a refrigerant in a pipeminichannel under the periodic conditions of hydrodynamicdisturbancesrdquo International Journal of Heat and Mass Transfervol 56 pp 715ndash723 2013
[39] Y Li C Li E Chen and Y Ying ldquoPressure wave propaga-tion characteristics in a two-phase flow pipeline for liquid-propellant rocketrdquoAerospace Science and Technology vol 15 no6 pp 453ndash464 2011
[40] T Kanagawa M Watanabe T Yano and S Fujikawa ldquoNon-linear wave equations for pressure wave propagation in liquidscontaining gas bubbles (comparison between two-fluid modeland mixture model)rdquo Journal of Fluid Science and Technologyvol 6 no 6 pp 836ndash849 2011
[41] EChen Y Li andYYing ldquoNumerical investigation onpressurewave propagation speed of gas-liquid two-phase flow in pumppipelinerdquo Journal of Aerospace Power vol 25 no 4 pp 754ndash7602010
[42] F Li and L He ldquoInvestigation on pressure wave propagationcharacteristics in slug flow field with increasing gas flow raterdquoPetroleum Geology and Recovery Efficiency vol 11 no 5 pp 78ndash80 2004
[43] C SMartin andM Padmanabhan ldquoPressure pulse propagationin two-component slug flowrdquo Journal of Fluids Engineering vol101 no 1 pp 44ndash52 1979
[44] R E Henry ldquoPressure wave propagation in two-phase mix-turesrdquo Chemical Engineering Progress vol 66 no 1 pp 1ndash101969
[45] X Liu B Li and Y Yue ldquoTransmission behavior of mud-pressure pulse alongwell borerdquo Journal ofHydrodynamics B vol19 no 2 pp 236ndash240 2007
[46] XWang and J Zhang ldquoThe research of pressure wave Pulsationin mud pulse transmittingrdquo Journal of Chongqing University ofScience and Technology vol 14 no 2 pp 55ndash58 2012
[47] J-W Park D A Drew and R T Lahey Jr ldquoThe analysisof void wave propagation in adiabatic monodispersed bubblytwo-phase flows using an ensemble-averaged two-fluid modelrdquoInternational Journal ofMultiphase Flow vol 24 no 7 pp 1205ndash1244 1998
[48] G Arnold Entropy and objectivity as constraints upons constitu-tive equations for two-fluid modeling of multiphase flow [PhDthesis] Rensselaer Polytechnic Institute New York NY USA1998
[49] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[50] L Yarborough and K R Hall ldquoHow to solve equation of statefor z-factorsrdquoOil and Gas Journal vol 72 no 7 pp 86ndash88 1974
[51] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86 pp 127ndash136 2012
[52] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[53] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[54] M Ishii and K Mishima ldquoTwo-fluid model and hydrodynamicconstitutive relationsrdquo Nuclear Engineering and Design vol 82no 2-3 pp 107ndash126 1984
[55] M J Sanchez Comparison of correlations for predicting pressurelosses in vertical multiphase annular flow [MS thesis] Univer-sity of Tulsa Tulsa Okla USA 1972
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Casing
Drill string
Flow direction
Gas
Drilling mud
z
Figure 1 The schematic of gas and drilling mud two-phase flow inannulus
The momentum conservation equations for liquid are
120597
120597119905
(120601
119871120588
119871V119871) +
120597
120597119911
(120601
119871120588
119871V2119871) +
120597
120597119911
(120601
119871120588
119871)
minus
120597
120597119911
[120601
119871(120591
119891119903
119871+ 120591
Re119871) +119872Li minus 4
120591
119871
119863
] = 0
(4)
The interphase forces include virtual mass force dragforce and the wall shear stress The transfer of momentumbetween the gas and drilling mud phases119872Gi and119872Li can bewritten as follows
119872Gi = minus119872ndLi minus119872
119889
Li + (120591119891119903
Li + 120591ReLi )
120597120601
119871
120597119911
+
120597 (120601
119866120590
119904)
120597119911
+
120597 (120601
119866119901Gi)
120597119911
minus 120601
119866
120597 (119901Li)
120597119911
119872Li = 119872ndLi +119872
119889
Li + 119901Li120597 (120601
119871)
120597119911
minus (120591
119891119903
Li + 120591ReLi )
120597120601
119871
120597119911
(5)
The momentum transfer term caused by the gas-liquidinterfacial relative acceleration motion (ie the virtual massforce) can be expressed as
119872
ndLi = 119888VM120601119866120588119871120572VM minus 01120601
119866120588
119871V119904
120597V119904
120597119911
minus 119888
1198981120588
119871V2119904
120597120601
119866
120597119911
(6)
where 1198881198981= 01 and v
119904is the slip velocity V
119904= V119866minus V119871
The momentum transfer term caused by the drag forceprovided by Park et al [47] can be described as
119872
119889
Li =3
8
119862
119863
119903
120588
119871119877
119866V2119904
(7)
The pressure difference between the liquid interface andliquid is defined by the following formula presented by Parket al
119901Li minus 119901119871 = minus119888119901120588119871V2
119904 (8)
where 119888119901= 025
The proportion of gas phase in the interface between thegas and drilling mud is rather small in that the pressuredifference between the gas interface and gas is not very highOmitting the pressure difference the gas interface pressure119875Gi can be written as
119901Gi minus 119901119866 asymp 0 (9)
According to the equations of Arnold [48] the pressureof the drilling mud is described as follows
119901
119871= 119901 minus 025120588
119871120601
119866V2119904
(10)
In fact the shear stress and the interphase shear stress arevery small relative to the Reynolds stress Also the Reynoldsstress in gas phase can be omitted relative to the interphaseforce Hence it can be described as
120591
119891119903
119866asymp 120591
119891119903
Li asymp 120591119891119903
119871asymp 120591
119866asymp 120591
Re119866asymp 0
(11)
The Reynolds stress and interfacial average Reynoldsstress can be obtained by
120591
Re119871= minus119888
119903120588
119871V2119904
120601
119866
120601
119871
120591
ReLi = minus119888119903120588119871V
2
119904
(12)
where 119888119903= 02
The wall shear stress of liquid phase is given by Wallis inthe following form [30]
120591
119871= 05119891
119871120588
119871V2119871
(13)
22 Physical Equations
221 Equations of State for Gas The equation of state (EOS)for gas can be expressed as follows
120588
119866=
119901
(119885
119866sdot 119877 sdot 119879)
(14)
where Z119866is the compression factor of gas
The formula presented by Dranchuk and Abou-Kassemhas been used to solve the gas deviation factor under thecondition of low and medium pressure (119901 lt 35MPa) [49]
119885
119866= 1 + (03051 minus
10467
119879
119903
minus
05783
119879
3
119903
)120588
119903
+ (05353 minus
026123
119879
119903
minus
06816
119879
3
119903
)120588
2
119903
(15)
where 119879119903= 119879119879
119888 119901119903= 119901119901
119888 and 120588
119903= 027119901
119903119885
119866119879
119903
The formula presented by Yarborough and Hall has beenadopted to solve the gas deviation factor under the conditionof high pressure (119901 ge 35MPa) [50]
119885
119866=
006125119875
119903119879
minus1
119903exp (minus12 (1 minus 119879minus1
119903) 2)
119884
(16)
Mathematical Problems in Engineering 5
where Y is given by
minus 006125119901
119903119879
minus1
119903exp (minus12(1 minus 119879minus1
119903)
2
) +
119884 + 119884
2
+ 119884
3
+ 119884
4
(1 minus 119884)
3
= (1476119879
minus1
119903minus 976119879
minus2
119903+ 458119879
minus3
119903) 119884
2
minus (907119879
minus1
119903minus 2422119879
minus2
119903+ 424119879
minus3
119903) 119884
(218+282119879minus1
119903
)
(17)
222 Equations of State for Liquid Under different tem-peratures and pressures the density of drilling mud can beobtained by the empirical formulas
If 119879 lt 130 ∘C the density of drillingmud can be obtainedby the following equation
120588
119871= 120588
0(1 + 4 times 10
minus10
119901
119871minus 4 times 10
minus5
119879 minus 3 times 10
minus6
119879
2
)
(18)
If 119879 ge 130 ∘C the density of drilling mud is
120588
119871= 120588
0(1 + 4 times 10
minus10
119875
119871minus 4 times 10
minus5
119879
minus3 times 10
minus6
119879
2
+ 04(
119879 minus 130
119879
)
2
)
(19)
223 Correlation of Temperature Distribution The tempera-ture of the drilling mud at different depths of the annulus canbe determined by the relationship presented by Hasan andKabir [51]
119879 = 119879
119890119894+ 119865 [1 minus 119890
(119911119887ℎ
minus119911)119860
] (minus
119892 sin 120579119892
119888119869119888
119901119898
+ 119873 + 119892
119879sin 120579)
+ 119890
(119911119887ℎ
minus119911)119860
(119879
119891119887ℎminus 119879
119890119887ℎ)
(20)
23 Flow Pattern Analysis Based on the analysis of flowcharacteristics in the enclosed drilling system it can be safelyassumed that the flowpattern in an annulus is either bubbly orslug flow [52] The pattern transition criteria for bubbly flowand slug flow given by Orkiszewski are used to judge the flowpattern in the gas-drilling mud two-phase flow [53]
For bubbly flow119902
119866
119902
119898
lt 119871
119861 (21)
For slug flow119902
119866
119902
119898
gt 119871
119861 119873
119866119881lt 119871
119878 (22)
where q119898is the volumetric flow rate of two-phase flow 119902
119898=
119902
119866+ 119902
119871
The dimensionless numbers L119904and L
119861are defined as
119871
119904= 50 + 36119873
119866119881
119902
119871
119902
119866
119871
119861= 1071 minus
07277V2119898
119863
(23)
where
119873
119866119881= V119904(
120588
119871
119892120590
119904
)
025
(24)
Flow parameters such as void fraction mixture densityand virtual mass force coefficient are discussed for a specificflow pattern
The correlation between void fraction and liquid holdupis expressed as
120601
119866+ 120601
119871= 1 (25)
231 Bubbly Flow As for bubbly flow the density of gas anddrilling mud mixture is described as the following formula
120588
119898= 120601
119871120588
119871+ 120601
119866120588
119866 (26)
The void fraction is determined by the following formula
120601
119866=
1
2
[
[
1 +
119902
119898
V119904119860
minus
radic
(1 +
119902
119898
V119904119860
)
2
minus
4119902
119866
V119904119860
]
]
(27)
The coefficient of virtual mass force 119862vm for bubbly flowcan be expressed as follows [54]
119862vm = 05
1 + 2120601
119866
1 minus 120601
119866
(28)
The coefficient of resistance coefficientC119863for bubbly flow
can be expressed as
119862
119863=
4119877
119887
3
radic
119892 (120588
119871minus 120588
119866)
120590
119904
[
1 + 1767(1 minus 120601
119866)
97
1867(1 minus 120601
119866)
15]
2
(29)
The friction pressure gradient for bubbly flow can beobtained from the following equation
120591
119891= 119891
120588
119871V2119871
2119863
(30)
232 Slug Flow As for slug flow distribution coefficient ofgas in the liquid phase is
119862
0=
000252lg (103120583119871)
119860
138minus 0782 + 0232lgV
119898minus 0428lg119860
(31)
The average density of the mixture for slug flow isdetermined by
120588
119898=
119882
119898+ 120588
119871V119904119860
119902
119898+ V119904119860
+ 119862
0120588
119871 (32)
The void fraction for slug flow is determined by thefollowing formula
120601
119866=
119902
119866
119902
119866+ 119902
119871
(33)
6 Mathematical Problems in Engineering
The coefficient of virtual mass force 119862vm for slug flow canbe expressed as follows
119862vm = 33 + 17
3119871
119902minus 3119877
119902
3119871
119902minus 119877
119902
(34)
The coefficient of resistance coefficient C119863for slug flow
can be expressed as
119862
119863= 110120601
3
119871119877
119887
(35)
The friction pressure gradient of bubbly flow can beobtained from the following equation
120591
119891= 119891
120588
119871V2119898
2119863
(
119902
119871+ V119904119860
119902
119898+ V119904119860
+ 119862
0)
(36)
24 Annulus Characteristic Analysis The annulus effectivediameter proposed by Sanchez [55] is used in the hydrauliccalculation of annulus
119863 =
120587 (119863
2
119900minus 119863
2
119894) 4
120587 (119863
119900+ 119863
119894) 4
= 119863
119900minus 119863
119894
(37)
The effective roughness of the annulus can be expressedas
119896
119890= 119896
119900
119863
119900
119863
119900+ 119863
119894
+ 119896
119894
119863
119894
119863
119900+ 119863
119894
(38)
where theD119894andD
119900are the diameters of inner pipe and outer
pipe respectively the k119900and k119894are the roughnesses of outer
pipe and the inner pipe respectively
3 The United Model Developed
The total pressure drop gradient is the sum of pressure dropgradients due to potential energy change and kinetic energyand frictional loss From (1) the equation used to calculatethe pressure gradient of gas and drilling mud flow within theannulus can be written as
119889119901
119889119911
= 120588
119898119892 sin 120579 minus
120591
119908120587119863
119860
minus 120588
119898V119898
119889V119898
119889119911
(39)
Assuming the compressibility of the gas is only related tothe pressure in the annulus the kinetic energy or accelerationterm in the previous equation can be simplified to
120588
119898V119898
119889V119898
119889119911
= minus
120588
119898V119898V119904119892
119901
119889119901
119889119911
= minus
119882
119898119902
119892
119860
2119901
119889119901
119889119911
(40)
Substituting (39) into (40) the total pressure drop gra-dient along the flow direction within the annulus can beexpressed as
119889119901
119889119911
=
120588
119898119892 + 120591
119891
1 minus119882
119898119902
119866 (119860
2119901)
(41)
It is assumed that the gas obeys to the EOS (14) and thecompressibility of drilling mud can be obtained by adopting
the simplified EOS ((18) and (19)) which neglects the thermalexpansion of liquid The sonic speed of gas phase c
119866and that
of liquid phase c119871can be presented in the following form
119889119901
119871
119889120588
119871
= 119888
2
119871
119889119901
119866
119889120588
119866
= 119888
2
119866
(42)
By introducing (42) the hydrodynamic equations of two-fluid model ((2)ndash(4)) can be written in the matrix form
119860
120597120585
120597119905
+ 119861
120597120585
120597119911
= 119862120585
(43)
where A is the matrix of parameters considered in relation totime B is the matrix of parameters considered in relation tothe spatial coordinate and C is the vector of extractions
By introducing the small disturbance theory the distur-bance of the state variable 120585(120601
119866 119901 V119866 V119871)
119879 can be written as
120585 = 120585
0+ 120575120585 exp [119894 (119908119905 minus 119896119905)] (44)
In (44) k is the wave number and 119908 is the angularfrequency of the disturbances Substituting (44) into (43)gives homogenous linear equations concerning the expres-sion (120575120601
119866 120575119901 120575V
119866 120575V119871)
119879 According to the solvable conditionof the homogenous linear equations that the determinant ofthe equations is zero dispersion equation of pressure wavecan be expressed in the following form
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119872
1119872
2119872
3119872
4
minus120588
119871119908
1 minus 120601
119866
119888
2
119871
119908 0 minus119896 (1 minus 120601
119866) 120588
119871
119872
5119872
6119872
7119872
8
119872
9119872
10119872
8119872
11
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
= 0 (45)
whereM1ndashM11can be illustrated by
119872
1= (120588
119866+ 119888
119901120601
119866120588
119871
V2119904
119888
2
119866
)119908
119872
2=
120601
119866
119888
2
119866
[1 minus 119888
119901120601
119871]
V2119904
119888
2
119871
119908
119872
3= minus[120601
119866120588
119866119896 + 2119888
119901120601
119866120601
119871120588
119871
V119904
119888
2
119871
119908]
119872
4= 22119888
119901120601
119866120601
119871120588
119871
V119904
119888
2
119871
119908
119872
5= 120588
119871V2119903119896 (minus120601
119866119888
119901+ 119888
119903minus 119888
119894+ 119888
1198982)
119872
6= minus120601
119866119896[1 minus 120601
119871
119888
119901V2119904
119888
2
119871
+ 119888
119894
V2119904
119888
2
119871
]
Mathematical Problems in Engineering 7
119872
7= 120601
119866(120588
119866+ 119888vm120588119871) 119908 minus 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904+
4
119863
119891
119866119908120588
119866V119866)
119872
8= minus119888vm120601119866120588119871119908 + 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904)
119872
9= 120588
119871V2119904119896 (120601
119871119888
119901minus 2119888
119903minus 119888
1198982)
119872
10= minus119896(120601
119871+ 119888
119903120601
119866
V2119904
119888
2
119871
)
119872
11= 120588
119871[120601
119871+ 120601
119866119888vm] 119908 minus 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904+
4
119863
119891
119871120588
119871V119871)
(46)
where 1198881198982= 01 119888
119901= 025 119888
119894= 03 and 119888
119903= 02
By solving the dispersion equation indicated previouslyfour roots can be obtained Because the wavelengths asso-ciated with two of the roots are too short to allow the two-fluid medium to be treated as a continuum the two rootsshould be omitted As for the two remaining roots one ofthem represents a pressure wave that transmited along theaxis z and the other represents a pressure wave transmits inthe opposite direction in accordance with the direction of theflow along the annulusThe real value of wave number can beused to determine thewave velocity cThewave velocity in thetwo-phase fluid can be determined by the following model
119888 =
1003816
1003816
1003816
1003816
(119908119877
+
(119896)) minus (119908119877
minus
(119896))
1003816
1003816
1003816
1003816
2
(47)
4 Solution of the United Model
Obtaining the analytical solution of themathematical modelsconcerned with flow pattern void fraction characteristicparameters and pressure drop gradient is generally impossi-ble for two-phase flow In this paper the Runge-Kuttamethod(R-K4) is used to discretize the theoretical model
We can obtain pressure temperature gas velocity drillingmud velocity and void fraction at different annulus depthsby R-K4 The solution of pressure drop gradient equation(41) can be seen as an initial-value problem of the ordinarydifferential equation
119889119901
119889119911
= 119865 (119911 119901)
119901 (119911
0) = 119875
0
(48)
With the initial value (z0p0) and the function F(z p)
(50)ndash(53) can be obtained by
119896
1= 119865 (119911
0 119901
0) (49)
119896
2= 119865(119911
0+
ℎ
2
119901
0+
ℎ
2
119896
1) (50)
119896
3= 119865(119911
0+
ℎ
2
119901
0+
ℎ
2
119896
2) (51)
119896
4= 119865 (119911
0+ ℎ 119901
0+ ℎ119896
3) (52)
where h is the step of depthThe pressure on the nod 119894 = 119894 +1can be obtained by
119901
1= 119901
0+ Δ119901 = 119901
0+
ℎ
6
(119896
1+ 2119896
2+ 2119896
3+ 119896
4)
(53)
In the present work the mathematical model and pres-sure wave velocity calculation model are solved by a person-ally compiled code on VBNET (Version 2010) The solutionprocedure for the wave velocity in the annulus is shown inFigure 1 At initial time the wellhead back pressure wellheadtemperature wellbore structure well depth gas and drillingmud properties and so forth are known On the node ithe pressure gradient temperature and the void fraction canbe obtained by adopting R-K4 Then the determinant (45)is calculated based on the calculated parameters Omittingthe two unreasonable roots the pressure wave velocity atdifferent depths of the annulus in MPD operations can besolved by (47)The process is repeated until the pressure wavevelocity of every position in the wellbore is obtained as shownin Figure 2
The developed model takes full consideration of theinterfacial interaction and the virtual mass force Owing tothe complex conditions of the annulus in MPD operationsmeasurement of wave velocity in the actual drilling processis very difficult In order to verify the united model thepredicted pressure waves are compared with the results ofprevious simulated experimental investigations presented forgas and drillingmud by Liu et al [45] in Figure 3(a) and by Liet al [22] in Figure 3(b) The lines represent the calculationresults and the points represent the experimental data
The comparisons reveal that the developed united modelfits well with the experimental data Thus the united modelcan be used to accurately predict wave velocity at differentwellhead back pressures and gas influx rates (the influx rateof gas in the bottomhole) in MPD operations
5 Analysis and Discussion
The drilling system described is an enclosed system Theschematic diagram of the gas influx process is illustrated inFigure 4 The drilling mud is pumped from surface storagedown the drill pipe Returns from the wellbore annulus travelback through surface processing where drilling solids areremoved to surface storage The key equipments include thefollowing
(i) The Rotating Control Device (RCD) The rotatingcontrol device provides the seal between atmosphereand wellbore while allowing pipe movement anddiverting returns flow In conjunction with the flowcontrol choke the RCD provides the ability to applyback pressure on the annulus during an MPD opera-tion
(ii) Choke The MPD choke manifold provides anadjustable choke system which is used to dynamicallycontrol the required BHP by means of applyingsurface BP
(iii) Coriolis Meter A Coriolis meter is used to accu-rately measure the mass flow rate of fluid exiting
8 Mathematical Problems in Engineering
Grid division
Start
Delete two roots
End
Initial conditions
Meet demand
i = i + 1
Solve eq (20) for T
Solve eq (15)ndash(17) for Zg
Judging flow pattern by eq (21) and eq (22)
Bubbly flow pattern Slug flow pattern
Solve determinant eq (45) for four roots
Calculate F(z p) by eq (48)
Calculate k1 minus k4Pi+1 = Pi + ΔPzi+1 = zi + h
zi+1 gt HNo
No
Yes
Yes
Solve eq (18) (19) for120588L
Solve eq ( 47) for c
Solve eq (14)120588G
Calculate 120588m 120601G f and CDCalculate 120588m 120601G f and CD CvmCvm
Figure 2 Solution procedure for wave velocity in MPD operations
the annulus The ability to measure return flowaccurately is essential for the applied back pressure
(iv) Pressure Sensor A pressure sensor is used to measuresurface back pressure on the wellhead
The gas and drilling mud flow rate measured by the Cori-olis meter and the back pressure measured by the pressuresensor are the initial data for annulus pressure calculationThe well used for calculation is a gas well in XinjiangUygur Autonomous Region Northwest China The wellborestructure well design parameters (depths and diameters)gas and drilling mud properties (density and viscosity) andoperational conditions of calculation well are displayed inTable 1
The drilling mud mixed with gas is taken as a two-phaseflow medium The propagation velocity of pressure wave inthe gas-drillingmud flow is calculated and discussed by usingthe established model and well parameters
51 Effect of Back Pressure on Wave Velocity Increase inapplied back pressure is the most common approach fordynamicwell controlThewellbore can be seen as an enclosed
Table 1 Parameters of calculation well
Type Property Value
Mud Dynamic viscosity (Pa sdot s) 0056Density (kgcm3) 1460
Gas Relative density 065Viscosity (Pa sdot s) 114 times 10minus5
StringElastic modulus of string (Pa) 207 times 1011
Poisson ratio of string 03Roughness (m) 154 times 10minus7
Surface condition Surface temperature (K) 298Atmosphere pressure (MPa) 0101
pressurized system The pressure at different depths of theannulus varied with the change of back pressure Accordingto the EOS of gas and drilling mud the influence of pressureon gas volume is much greater than that of drilling mudfor the greater compressibility of gas So the gas voidfraction changes with the variation of gas volume at thenearly constant flow rate of drilling mud at different annularpressures Meanwhile the pressure wave velocity is sensitive
Mathematical Problems in Engineering 9
5000 2 4 6 8 10
750
1000
1250
1500c
(ms
)
This paper p = 30MPa w = 50Hz = Re
120601G ()
CvmLiu et al [45] p = 30MPa
(a) Comparison with the experimental data of Liu
0 20 40 60 80 1000
40
80
120
160
200
c(m
s)
This paper p = 01MPa w = 50Hz = Re
120601G ()
CvmLi et al [22] p = 01MPa
(b) Comparison with the experimental data of Li
Figure 3 Experimental verification by comparison with previous experimental data
Drill pipe
Annulus
RCD
Formation
Mud pump
Drill collar
Coriolis meterChoke
Casing
Pressure sensors
120601 2159 mm times G105
1206011778mm (120601 78 mm) times G105
120601127 mm (120601 78 mm) times G105
z0 m
3600 m
4000 m
QL
Figure 4 Schematic diagram of the gas influx process
to the gas void fraction As a result when the back pressureat the wellhead is changed by adjusting the choke the wavevelocity in the two-phase drilling fluid and the distributionof void fraction at different depth of the annulus will divergeThe calculation results affected by the back pressure arepresented
Figures 5 and 6 show the distributions of void fractionand variations of wave velocity along the flow directionin the annulus when the back pressure at the wellhead is01MPa 08MPa 15MPa 25MPa 50MPa and 65MPa
respectively It can be seen that the void fraction significantlyincreases along the flow direction in the annulus Converselythe wave velocity shows a remarkable decrease tendency Forinstance BP = 15MPa at the wellhead the wave velocityis 9012ms and the void fraction is 0611 while in thebottomhole the wave velocity reaches 73142ms and voidfraction is reduced to 0059 However a sudden increase inwave velocity is observed in Figure 5 when the back pressureis 01MPa at the position close towellhead as the void fractionis further increased Moreover the influence of back pressure
10 Mathematical Problems in Engineering
10075500 254000
3000
2000
1000
0
BP = 01 MPaBP = 08 MPaBP = 15 MPa
BP = 25 MPa
BP = 65 MPaBP = 5 MPa
H(m
)
Q = 9m3h w = 50Hz = Re
120601G ()
Cvm
Figure 5 Void fraction distribution at different back pressures
on the void fraction is larger at the position close to thewellhead than that in the bottomhole
This can be explained from the viewpoints of mixturedensity and compressibility of two-phase fluid and the pres-sure drop along the flow direction in the wellbore In thelow void fraction range the gas phase is dispersed in theliquid as bubble so the wave velocity is influenced greatly bythe added gas phase According to the EOS if gas invadesinto the wellbore with a small amount in the bottomholethe density of the drilling mud has little variation while thecompressibility increases obviously which makes the wavevelocity decreased Then the two-phase drilling mud flowfrom the bottomhole to the wellhead along the annulus witha drop of pressure caused by potential energy change kineticenergy and frictional loss which leads to an increase in voidfractionWhen the void fraction is increased both the densityand the compressibility of the two-phase fluid change slightlyresulting in a flat decrease in wave velocity along the flowdirection With the further increasing in void fraction thebubbly flow turns into slug flow according to the flow patterntransient criteria of Orkiszewski which shows that the flowpattern is dependent on the void fraction Generally speak-ing the gas-drilling mud slug flow is composed of liquidand gas slugs In the preliminary slug flow the liquid slugsare much longer than the gas slugs so the wave velocity isdetermined primarily by the wave velocities in the liquidslugs which are hardly affected by the void fraction At theposition close to the wellhead the pressure of two-phase flowfluid can be reduced to a low value which approaches theback pressure The compressibility of gas will be improvedat the low pressure It results in significant increase in void
0 250 500 750 10004000
3000
2000
1000
0
H(m
)BP = 01 MPaBP = 08 MPaBP = 15 MPa
BP = 25 MPa
BP = 65 MPaBP = 5 MPa
c (ms)
Q = 9m3h w = 50Hz = ReCvm
Figure 6 Wave velocity variations at different back pressures
fraction As the void fraction increases greatly at the positionclose to the wellhead gas slugs becomemuch longer than theliquid slugs It is assumed that the actual wave velocity in slugflow happens to play a leading role when the void fractionincreases to some extent For the decrease in liquid holdupin the gas slug with the increasing in void fraction the wavevelocity in the gas slug is infinitely close to that in pure gaswhich is equivalent to an increase in wave velocity in the gasslug Meanwhile the gas content in liquid slug also increasesand results in a lower wave velocity in the liquid slug As aresult a slight increase in wave velocity appears
Figures 7 and 8 present the influence of back pressureon the void fraction and wave velocity with respect to theparameters of well depth As the back pressure increases thevoid fraction at different depth of the annulus is reducedgradually while the wave velocity in the two-phase flow tendsto increase Analytical results show that the increased backpressure is equivalent to be applied to the entire encloseddrilling fluid cyclical systemThe pressure transmits from thewellhead to the bottomhole therefore the annular pressurein the entire wellbore is increased According to the EOSthe density of gas increases and the compressibility of gasdecreases with the increasing of gas pressure So the lossof interphase momentum and energy exchange is reducedand the interphase momentum exchange is promoted Itcontributes to the increase in wave velocity with the increasein gas pressure In addition due to lower compressibility oftwo-phase flow medium under high pressure the increasetendency of pressure wave velocity and the decrease tendencyof void fraction are slowed down in the high back pressurerange
Mathematical Problems in Engineering 11
0
25
50
75
100
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
0 25 75 105
Q = 9m3h w = 50Hz = Re
120601G
()
Cvm
Figure 7 Effect of back pressure on the void fraction
52 Effect of Gas Influx Rate on Wave Velocity Figures 9and 10 graphically interpret the distributions of void fractionand variations of wave velocity along the flow direction inthe annulus When gas influx occurs in the bottomhole gasinvades into the wellbore and migrates from the bottomholeto the wellhead along the flow direction At a low gas influxrate it is extremely obvious that the void fraction and wavevelocity first slightly change in a comparatively smooth valuethen change sharply It is because of the rapid expansion ofgas volume with the decreasing in pressure near the wellheadthat the void fraction increases sharply and the wave velocitydecreases obviously at the same time But under the highbottomhole pressure (up to 50MPa) the compressibilityof the gas is low This results in a slight change in voidfraction and wave velocity at the position far away from thewellhead Since the compressible component increases withthe increase in the gas influx rate the compressibility of thegas and drilling mud two-phase fluid is improved So thevariations of void fraction and wave velocity become moreprominent Also the void fraction still shows an increasetendency that is steady first and then sharp It acts inaccordance with the variation of void fraction at a low gasinflux rate At a low gas influx rate if the void fraction atthe position close to the wellhead can not increase to a highextent the wave velocity always shows a decrease tendencyAt a high gas influx rate such as the 119876 = 8312m3h thewave velocity tends to increase because the void fraction inthe wellhead is increased to a high extent In conclusionthe wave pressure is sensitive to the void fraction and thevoid fraction is dominated by influx rate and pressure in theannulus especially the influx rate
0 25 75 1050
300
600
900
1200
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
c(m
s)
Q = 9m3h w = 50Hz = ReCvm
Figure 8 Effect of back pressure on the pressure wave velocity
With the influx of gas the mixing of gas and drillingmud occurs in the annulus and the corresponding interfacialtransfer of momentum and mass causes an increase in gasphase void fraction and a decrease in pressure wave velocityas shown in Figures 11 and 12 Within the range of low gasinflux rate the wave velocity decreases significantly It isbecause of this that the compressibility of the gas increasesremarkably and the medium appears to be of high elasticitythough the density of gas-drilling mud two-phase flowchanges slightly With the increase in the gas influx rate andcorresponding increase in the void fraction in the annulusthe compressibility of the two-phase unceasingly increaseswhich promotes the momentum and energy exchange inthe interface So the pressure wave continuously decreasesWhen the void fraction increases to some extent following theincrease in the gas influx rate the decrease in wave velocityin the liquid slug is gradually less than the increase in wavevelocity in the gas slug thus the decrease of wave velocityis slowed down for the growth of gas slug Especially in thewellhead a slight increase in wave velocity is observed
53 Effect of Virtual Mass Force on Wave Velocity Figure 13illustrates the effect of virtual mass force on the wave velocitywith the parameter of gas influx rate As shown in Figure 15under the same gas influx rate (0594m3h 1098m3h and1768m3h resp) the wave velocity curve of 119862vm = Rediverges from the curve of 119862vm = 0 at the position close tothewellhead whereas thewave velocity curve of the two typesis almost coincided below the position of 119867 = 500m Thisdivergence between the two types of curves is connected withthe fact that significance of influence of interphase virtual
12 Mathematical Problems in Engineering
0 25 50 75 1004000
3000
2000
1000
0
Q = 0058m3h Q = 1228m3hQ = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
H(m
)
BP = 01 MPaw = 50Hz = Re
120601G ()
Cvm
Figure 9 Void fraction distribution at different gas influx rates
mass forces increases together with the increase of the relativemotion in the interphase It is evident that the bottomholepressure is hundreds of times higher than the wellheadpressure As a result the void fraction at the position closeto the wellhead increases sharply meanwhile the interphasemomentum and energy exchanges are promoted The virtualmass force can be described as the transfer of momentumbetween the gas phase and drilling mud phase caused bythe relative motion in the interphase If the relative motionin the interphase is quite weak the value of virtual massforce will intend to approach 0 and the influence on thewave velocity can be ignored However if the relative motionis rather intense the effect of virtual mass force on thewave velocity should not be ignored Furthermore taking thevirtual mass force into account the dispersion characteristicof the pressure wave weakens obviously Compared with thepressure wave velocity calculated by ignoring the effect ofvirtual mass force the calculated pressure wave velocity witha consideration of the virtual mass force is lower Thereforeit is necessary to analyze the effect of the virtual mass forceon the wave velocity at the position close to the wellhead inMPD operations
54 Effect of Disturbance Angular Frequency onWave VelocityFigure 14 presents the pressure wave velocity in the annulusat different angular frequencies According to Figures 5 and9 the void fraction first slightly increases in the bottomholeand then sharply increases along the flow direction at theposition close to the wellhead At a fixed angular frequencythis phenomenon results in an overall decrease in the pressurewave velocity except for the section near the wellhead At the
0 500 1000 1500 20004000
3000
2000
1000
0
c (ms)
H(m
)Q = 0058m3h Q = 1228m3h
Q = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
BP = 01 MPaw = 50Hz = ReCvm
Figure 10 Wave velocity variations at different gas influx rates
section close to the wellhead an opposite change trend ofpressure wave velocity is observed for the transition of flowpattern from bubbly flow to slug flow due to the continuousincrease in void fraction along the flow direction In additionit can be clearly seen from the curves that the wave velocityincreases accompanying with the increase in the angularfrequency above the position of 119867 = 500m This propertyis not very distinct at the position below 119867 = 500m for thelow void fraction
Figure 15 shows the effect of frequency on the wavevelocity in the gas-drilling mud flow (119862vm = 0)The curves ofthe wave velocity of different positions reveal that the prop-agation velocity of pressure disturbances increases togetherwith the growth of the angular frequency (0 lt 119908 lt 500Hz)It proves that the pressure wave has an obvious dispersioncharacteristic in the two-phase flow As the angular frequencyincreases in the range of less than 500Hz there is sufficienttime to carry out the exchange of energy and momentumbetween two phases It achieves a state of mechanical andthermodynamic equilibrium between the two phases Sothe wave velocity increases gradually with the increase inthe angular frequency at different depths of the annulus Itis considered that the wave velocity is mainly affected bythe interphase mechanical and thermodynamic equilibriumat low anglular frequencies When the angular frequencyreaches the value of 119908 = 500Hz the pressure wave velocityachieves a constant value and remains on this level regardlessof the further growth in angular frequency 119908 With theincrease in angular frequency there is not enough time forenergy and momentum exchange between the gas-drillingmud two phases to reach themechanical and thermodynamic
Mathematical Problems in Engineering 13
0
25
50
75
100
125
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
Q (m3h)
120601G
()
0 25 5 75 10
BP = 01 MPaw = 50Hz = ReCvm
Figure 11 Effect of gas influx rate on the void fraction
equilibrium state and thus the wave dispersion does notexist At a high angular frequency the wave velocity is mainlydominated by the mechanical and thermal nonequilibriumin the flow and keeps almost unchanged when the angularfrequency is further increased This is consistent with theinfluences of the angular frequency in the horizontal pipe[28] Moreover the effect of virtual mass force is also shownin this figure (119862vm = Re) It is observed that the wave velocityis significantly reduced when the virtual mass force is takeninto consideration in the calculation of the wave velocity atthe position close to the wellhead such as 119867 = 0m It canbe explained by the reduction of dispersion characteristic ofa pressure wave Along the wellbore from the wellhead to thebottomhole the distinction gradually decreases
6 Conclusions
With full consideration of the important factors such asvirtual mass force drag force gas void fraction pressuretemperature and angular frequency a united wave velocitymodel has been proposed based on pressure drop gradientequations in MPD operations gas-liquid two-fluid modelthe gas-drilling mud equations of state and small pertur-bation theory Solved by the fourth-order explicit Runge-Kutta method the model is used to predict wave velocityfor different back pressures and gas influx rates in MPDoperations The main conclusions can be summarized asfollows
(1) With the introduction of virtual mass force and dragforce the united model agrees well with the previousexperimental data The united model can be used to
0
280
560
840
1120
1400
Q (m3h)0 25 5 75 10
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
c(m
s)
BP = 01 MPaw = 50Hz = ReCvm
Figure 12 Effect of gas influx rate on the pressure wave velocity
accurately calculate the wave velocity in the annulusThe application of the model will be beneficial tofurther study the wave velocity at different gas influxrates and back pressures in MPD operations reducenonproductive times and provide a reference for thedrilling operations in extremely narrow porefracturewindows existing conditions
(2) The wave velocity and void fraction have differentvalues with respect to well depth In the annulusthe drop of pressure causes an increase in the voidfraction along the flow direction The void fractionincreases first slightly and then sharply Correspond-ingly the wave velocity first gradually decreases in thbubbly flow and preliminary slug flow and then thewave velocity slightly increases accompanying withthe increase in the relative length ratio of gas slugto the liquid slug for the continuous increase in voidfraction at the position close to the wellhead Theminimum wave velocity appears in the long gas slugflow
(3) When the back pressure in the MPD operationsincreases the void fraction at different depths of theannulus is reduced gradually while the wave velocityin the two-phase flow tends to increase Moreoverthe influence of back pressure on the void fraction isgreater at a position close to the wellhead than that inthe bottomhole because of the great decrease in thepressure from the bottomhole to the wellhead Alsothe effect of back pressure on void fraction and wavevelocity is decreased in the high back pressure range
14 Mathematical Problems in Engineering
0 300 600 900 12004000
3000
2000
1000
225
180
135
90
45
0
BP = 01 MPaw = 50Hz
Q = 0594m3h = ReQ = 0594m3h = 0
Q = 1098m3h = ReQ = 1768m3h = ReQ = 1098m3h = 0
Q = 1768m3h = 0
c (ms)
H(m
)
CvmCvm
Cvm
Cvm
Cvm
Cvm
Figure 13 Effect of virtual mass force on the pressure wave velocity
(4) The wave velocity is sensitive to the void fraction butthe void fraction is dominated by gas influx rate andpressure in the annulus especially the gas influx rateSince the compressibility of the gas and drilling mudtwo-phase fluid is improved with the increase in theinflux rate the void fraction increases greatly andthe wave velocity decreases significantly within thelow gas influx rate range When the void fraction isincreased to some extent following the increase in thegas influx rate the decrease of wave velocity is slowedfor the growth of gas slug Especially at the wellheada slight increase in wave velocity is observed at a highgas influx rate for the sharp increase in void fraction
(5) It is necessary to analyze the effect of the virtualmass force on the wave velocity in MPD operationsEspecially at the position close to the wellhead theeffect of virtual mass force is more prominent for theintense phase interactions Taking the virtual massforce into account the dispersion characteristic of thepressure wave weakens obviously Compared with theresults calculated by ignoring the effect of virtualmassforce the calculated pressure wave velocity with aconsideration of the virtual mass force is lower
(6) The effect of angular frequency on the propagationvelocity of pressure wave appears at a low angularfrequency The propagation velocity of pressure dis-turbances increases together with the growth of theangular frequency (0 lt 119908 lt 500Hz) When theangular frequency reaches the value of 119908 = 500Hzthe pressure wave velocity achieves a constant value
0 240 480 720 960 12004000
3000
2000
1000
560
420
280
140
0
Q = 0727m3h BP = 01 MPa
w = 50Hz = Rew = 50Hz = 0w = 300Hz = 0
w = 800Hz = 0w = 1600Hz = 0w = 5000Hz = 0
c (ms)
H(m
)CvmCvmCvm
CvmCvmCvm
Figure 14 Wave velocity variations at different angular frequencies
and remains on this level regardless of the furthergrowth in angular frequency 119908
Nomenclature
A Annulus effective cross area (m2)c119866 Wave velocity in gas (ms)
c119871 Wave velocity in Drilling mud (ms)
C119863 The coefficient of drag force
D Annulus effective diameter (m)D119894 Diameter of the inner pipe (m)
D119900 Diameter of the outer pipe (m)
f119866 Shear stresses coefficient of gas interface
f119871 Shear stresses coefficient of drilling mud
interfaceF119911 Pressure applied to the two-phase flow (N)
g Acceleration due to gravity (m2s)g119888 Conversion factor
g119879 Geothermal temperature gradient (Km)
h One discrete length of annulus (m)k119890 Annulus effective roughness (m)
k119900 Roughness of outer pipe (m)
k119894 Roughness of inner pipe (m)
H The depth of annulus (m)L119902 Air bubbles length of slug (m)
119872Gi Momentum transfer in gas interface (Nm3)119872Li Momentum transfer in liquid interface (Nm3)119872
ndLi The none-drag-force (virtual mass force)
(Nm3)119872
119889
Li The momentum transfer term caused by thedrag force (Nm3)
Mathematical Problems in Engineering 15
100 1000 100000
450
900
1350
1800
0w (Hz)
0 m = 0 0 m = Re1000 m = 0
2000 m = 0
4000 m = 0
1000 m = Re2000 m = Re4000 m = Re
c(m
s)
Q = 0727m3h BP = 01 MPaCvmCvm
CvmCvmCvm
CvmCvmCvm
Figure 15 Effect of angular frequency on wave velocity
N Parameter combiningThompsonenergy effects
119901Gi Gas interface pressure (MPa)119901Li Liquid interface pressure (MPa)p Pressure (MPa)p0 Initial pressure of annulus (MPa)
p119888 Critical pressure (kPa)
p119871 Pressure of drilling mud (MPa)
p119903 Reduced pressure
q119898 Volumetric flow rate of gas and drilling
mud (m3s)q119866 Velocity of the gas (m3s)
q119871 Velocity of the drilling mud (m3s)
Q119871 Drilling mud rate within the annulus
(m3h)r Average diameter of the bubble (m)R Constant of EOS (JKgK)R119902 Air bubbles radius of slug (m)
R119887 The bubble diameter (m)
T Temperature (K)T119888 Critical temperature (K)
T119903 Reduced temperature
T119890119894 Undisturbed formation temperature at a
depth (K)T119890119887ℎ
Undisturbed formation temperature ofwellhead (K)
T119891119887ℎ
Undisturbed formation temperature atthe bottomhole (K)
v119898 Gas and drilling mud flow velocity
(ms)
v119866 Gas flow velocity (ms)
v119904 Slip velocity (ms)
v119904119892 Superficial gas velocity (ms)
v119871 Drilling mud flow velocity (ms)
W119898 Mass flow velocity (kgm3)
Z119866 Gas deviation factor
z0 Initial calculation height of annulus (m)
z119887ℎ Total well depth from surface (m)
z Variable of distance from surface (m)
Greek Letters
120588
119866 Gas density (kgm3)
120591
119891119903
Li Shear stresses of drilling mud noninterfacial(Nm2)
120591
119891119903
119871 Shear stresses of drilling mud interface (Nm2)
120591
Re119871 Reynolds stress of drilling mud (Nm2)
120591
Re119871 Reynolds stress of drilling mud interface
(Nm2)120591
Re119866 Reynolds stress of gas interface (Nm2)
120591
119891119903
119866 Shear stresses of gas interface (Nm2)
120572VM Acceleration due to virtual mass forcecoefficient (ms2)
120591
119871 Shear stresses of drilling mud along wall
(Nm2)120591
119866 Shear stresses of gas along wall (Nm2)
120591
Re119866 Reynolds stress of gas (Nm2)
120591
119891 Frictional pressure gradient (Pam)
120591
119908 Frictional pressure coefficient
120588
119898 Gas and drilling density (kgm3)
120588
0 Density under standard atmospheric pressure
(kgm3)120588
119871 Drilling mud density (kgm3)
120590
119904 Surface tension (Nm2)
Φ
119866 Gas void fraction
Φ
119871 Drilling mud holdup
120588
119903 Reduced density
120583
119871 Viscosity of drilling mud (Pa sdot s)
Subscripts
BHP Bottomhole pressureBP Back pressureEOS Equations of stateMPD Managed pressure drillingRCD The rotating control deviceR-K4 The fourth-order explicit Runge-Kutta
Subscripts of Graph
BP Back pressure (MPa)c Wave velocity in gas and drilling mud
two-phase flow (ms)119862vm The coefficient of virtual mass force
16 Mathematical Problems in Engineering
H Depth of annulus (m)Q Gas influx rate at the bottomhole (m3h)Re The value of virtual mass force
coefficient according to (28) or (34)119908 Angle frequency (Hz)Φ
119866 Void fraction ()
Acknowledgment
Research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 and 51274170)Without their support this work would not have beenpossible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of airinjection in annuli to manage pressure during cementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling conferenceand Exhibition San Diego Calif USA March 2012
[4] W A Bacon Consideration of compressibility effects for applied-back-pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Tex USA 2011
[5] J Choe C Hans and J Wold ldquoA modified two-phase well-control model and its computer applications as a training andeducational toolrdquo SPE Computer Applications vol 2 pp 14ndash201997
[6] J A Tarvin IWalton and PWand ldquoAnalysis of a gas kick takenin a deep well drilled with oil-based mudrdquo in Proceedings ofthe Annual Technical Conference and Exhibition of the Societyof Petroleum Engineers pp 255ndash264 Dallas Tex USA 1991
[7] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[8] J Zhou and G Nygaard ldquoAutomatic model-based controlscheme for stabilizing pressure during dual-gradient drillingrdquoJournal of Process Control vol 21 no 8 pp 1138ndash1147 2011
[9] A Ochoa O Acevedo L Nieto J E Lambarria and H PerezldquoSuccessful application of MPD (Managed Pressure Drilling)for prevention controland detection of borehole ballooning intight gas reservoir in Cuervito Field Mexicordquo in Proceedings ofthe Canadian Unconventional Resources Conference (CURC rsquo11)pp 172ndash185 November 2011
[10] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[11] R Ashena and J Moghadasi ldquoBottom hole pressure estimationusing evolved neural networks by real coded ant colony opti-mization and genetic algorithmrdquo Journal of Petroleum Scienceand Engineering vol 77 no 3-4 pp 375ndash385 2011
[12] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvanced Materials Research vol 468-471 pp 1736ndash1742 2012
[13] I Zubizarreta ldquoPore pressure evolution core damage and trip-ping out schedulesf a computational fluid dynamics approachrdquoin Proceedings of the SPEIADC Drilling Conference and Exhibi-tion Amsterdam The Netherlands March 2013
[14] C H Whitson ldquoCyclic shut-in eliminates liquid-loading in gaswellsrdquo in Proceedings of the SPEEAGE European Unconven-tional Resources Conference and Exhibition Vienna AustriaMarch 2012
[15] G M de Oliveira A T Franco C O R Negrao A LMartins and R A Silva ldquoModeling and validation of pressurepropagation in drilling fluids pumped into a closed wellrdquoJournal of Petroleum Science and Engineering vol 103 pp 61ndash71 2012
[16] Y Sato and H Kanki ldquoFormulas for compression wave andoscillating flow in circular piperdquo Applied Acoustics vol 69 no1 pp 1ndash11 2008
[17] J Xie B Yu X Zhang Q Shao and X Song ldquoNumericalsimulation of gas-liquid-solid three-phase flow in deep wellsrdquoAdvances in Mechanical Engineering vol 2013 Article ID951298 10 pages 2013
[18] W Bacon A Tong O Gabaldon C Sugden and P V Surya-narayana ldquoAn improved dynamic well control response to a gasinflux in managed pressure drilling operationsrdquo in Proceedingsof the IADCSPE Drilling Conference and Exhibition San DiegoCalifornia USA March 2012
[19] J I Hage and D ter Avest ldquoBorehole acoustics applied to kickdetectionrdquo Journal of Petroleum Science and Engineering vol 12no 2 pp 157ndash166 1994
[20] A M R Hopkins Fluid dynamics and surface pressure fluctua-tions of two-dimensional turbulent boundary layers over denselydistributed surface roughness [PhD thesis] Virginia PolytechnicInstitute and State University Blacksburg Va USA 2010
[21] Z S Fu ldquoThe gas cut detect technology during drilling in SovietUnionrdquo Petroleum Drilling Techniques vol 18 no 1 pp 19ndash211990
[22] X F Li C X Guan and X X Sui ldquoThe theory of gas influxdetection of pressure wave and its applicationrdquo Acta PetroleiSinica vol 18 no 3 pp 128ndash133 1997
[23] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[24] A E Ruggles R T Lahey Jr D A Drew and H A ScartonldquoInvestigation of the propagation of pressure perturbations inbubbly airwater flowsrdquo Journal of Heat Transfer vol 110 no 2pp 494ndash499 1988
[25] H J W M Legius H E A Van Den Akker and T NarumoldquoMeasurements on wave propagation and bubble and slugvelocities in cocurrent upward two-phase flowrdquo ExperimentalThermal and Fluid Science vol 15 no 3 pp 267ndash278 1997
[26] K Miyazaki and I Nakajima ldquoPropagation of pressure wavein NItrogen-Mercury two-phase systemrdquo Journal of NuclearScience and Technology vol 9 no 5 pp 315ndash318 1972
[27] B Bai L Guo and X Chen ldquoPressure fluctuation for air-watertwo-phase flowrdquo Journal of Hydrodynamics vol 18 no 4 pp476ndash482 2003
[28] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
Mathematical Problems in Engineering 17
[29] R F Tangren C H Dodge and H S Seifert ldquoCompressibilityeffects in two-phase flowrdquo Journal of Applied Physics vol 20 no7 pp 637ndash645 1949
[30] G B Wallis One Dimensional Two Phase Flow McGraw HillNew York NY USA 1969
[31] D L Nguyen E R F Winter and M Greiner ldquoSonic velocityin two-phase systemsrdquo International Journal ofMultiphase Flowvol 7 no 3 pp 311ndash320 1981
[32] R C Mecredy and L J Hamilton ldquoThe effects of nonequilib-rium heat mass and momentum transfer on two-phase soundspeedrdquo International Journal of Heat and Mass Transfer vol 15no 1 pp 61ndash72 1972
[33] K H Ardron and R B Duffey ldquoAcoustic wave propagationin a flowing liquid-vapour mixturerdquo International Journal ofMultiphase Flow vol 4 no 3 pp 303ndash322 1978
[34] J Xu and T Chen ldquoAcoustic wave prediction in flowing steam-water two-phase mixturerdquo International Journal of Heat andMass Transfer vol 43 no 7 pp 1079ndash1088 2000
[35] N ChungW Lin B Pei and Y Hsu ldquoModel for sound velocityin a two-phase air-water bubbly flowrdquo Nuclear Technology vol99 no 1 pp 80ndash89 1992
[36] X Xu and J Gong ldquoA unitedmodel for predicting pressurewavespeeds in oil and gas two-phase pipeflowsrdquo Journal of PetroleumScience and Engineering vol 60 no 3-4 pp 150ndash160 2008
[37] W Kuczynski ldquoCharacterization of pressure-wave propagationduring the condensation of R404A and R134a refrigerantsin pipe mini-channels that undergo periodic hydrodynamicdisturbancerdquo International Journal of Heat and Fluid Flow vol40 pp 135ndash150 2013
[38] W Kuczynski ldquoModeling of the propagation of a pressure waveduring the condensation process of R134a refrigerant in a pipeminichannel under the periodic conditions of hydrodynamicdisturbancesrdquo International Journal of Heat and Mass Transfervol 56 pp 715ndash723 2013
[39] Y Li C Li E Chen and Y Ying ldquoPressure wave propaga-tion characteristics in a two-phase flow pipeline for liquid-propellant rocketrdquoAerospace Science and Technology vol 15 no6 pp 453ndash464 2011
[40] T Kanagawa M Watanabe T Yano and S Fujikawa ldquoNon-linear wave equations for pressure wave propagation in liquidscontaining gas bubbles (comparison between two-fluid modeland mixture model)rdquo Journal of Fluid Science and Technologyvol 6 no 6 pp 836ndash849 2011
[41] EChen Y Li andYYing ldquoNumerical investigation onpressurewave propagation speed of gas-liquid two-phase flow in pumppipelinerdquo Journal of Aerospace Power vol 25 no 4 pp 754ndash7602010
[42] F Li and L He ldquoInvestigation on pressure wave propagationcharacteristics in slug flow field with increasing gas flow raterdquoPetroleum Geology and Recovery Efficiency vol 11 no 5 pp 78ndash80 2004
[43] C SMartin andM Padmanabhan ldquoPressure pulse propagationin two-component slug flowrdquo Journal of Fluids Engineering vol101 no 1 pp 44ndash52 1979
[44] R E Henry ldquoPressure wave propagation in two-phase mix-turesrdquo Chemical Engineering Progress vol 66 no 1 pp 1ndash101969
[45] X Liu B Li and Y Yue ldquoTransmission behavior of mud-pressure pulse alongwell borerdquo Journal ofHydrodynamics B vol19 no 2 pp 236ndash240 2007
[46] XWang and J Zhang ldquoThe research of pressure wave Pulsationin mud pulse transmittingrdquo Journal of Chongqing University ofScience and Technology vol 14 no 2 pp 55ndash58 2012
[47] J-W Park D A Drew and R T Lahey Jr ldquoThe analysisof void wave propagation in adiabatic monodispersed bubblytwo-phase flows using an ensemble-averaged two-fluid modelrdquoInternational Journal ofMultiphase Flow vol 24 no 7 pp 1205ndash1244 1998
[48] G Arnold Entropy and objectivity as constraints upons constitu-tive equations for two-fluid modeling of multiphase flow [PhDthesis] Rensselaer Polytechnic Institute New York NY USA1998
[49] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[50] L Yarborough and K R Hall ldquoHow to solve equation of statefor z-factorsrdquoOil and Gas Journal vol 72 no 7 pp 86ndash88 1974
[51] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86 pp 127ndash136 2012
[52] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[53] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[54] M Ishii and K Mishima ldquoTwo-fluid model and hydrodynamicconstitutive relationsrdquo Nuclear Engineering and Design vol 82no 2-3 pp 107ndash126 1984
[55] M J Sanchez Comparison of correlations for predicting pressurelosses in vertical multiphase annular flow [MS thesis] Univer-sity of Tulsa Tulsa Okla USA 1972
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
where Y is given by
minus 006125119901
119903119879
minus1
119903exp (minus12(1 minus 119879minus1
119903)
2
) +
119884 + 119884
2
+ 119884
3
+ 119884
4
(1 minus 119884)
3
= (1476119879
minus1
119903minus 976119879
minus2
119903+ 458119879
minus3
119903) 119884
2
minus (907119879
minus1
119903minus 2422119879
minus2
119903+ 424119879
minus3
119903) 119884
(218+282119879minus1
119903
)
(17)
222 Equations of State for Liquid Under different tem-peratures and pressures the density of drilling mud can beobtained by the empirical formulas
If 119879 lt 130 ∘C the density of drillingmud can be obtainedby the following equation
120588
119871= 120588
0(1 + 4 times 10
minus10
119901
119871minus 4 times 10
minus5
119879 minus 3 times 10
minus6
119879
2
)
(18)
If 119879 ge 130 ∘C the density of drilling mud is
120588
119871= 120588
0(1 + 4 times 10
minus10
119875
119871minus 4 times 10
minus5
119879
minus3 times 10
minus6
119879
2
+ 04(
119879 minus 130
119879
)
2
)
(19)
223 Correlation of Temperature Distribution The tempera-ture of the drilling mud at different depths of the annulus canbe determined by the relationship presented by Hasan andKabir [51]
119879 = 119879
119890119894+ 119865 [1 minus 119890
(119911119887ℎ
minus119911)119860
] (minus
119892 sin 120579119892
119888119869119888
119901119898
+ 119873 + 119892
119879sin 120579)
+ 119890
(119911119887ℎ
minus119911)119860
(119879
119891119887ℎminus 119879
119890119887ℎ)
(20)
23 Flow Pattern Analysis Based on the analysis of flowcharacteristics in the enclosed drilling system it can be safelyassumed that the flowpattern in an annulus is either bubbly orslug flow [52] The pattern transition criteria for bubbly flowand slug flow given by Orkiszewski are used to judge the flowpattern in the gas-drilling mud two-phase flow [53]
For bubbly flow119902
119866
119902
119898
lt 119871
119861 (21)
For slug flow119902
119866
119902
119898
gt 119871
119861 119873
119866119881lt 119871
119878 (22)
where q119898is the volumetric flow rate of two-phase flow 119902
119898=
119902
119866+ 119902
119871
The dimensionless numbers L119904and L
119861are defined as
119871
119904= 50 + 36119873
119866119881
119902
119871
119902
119866
119871
119861= 1071 minus
07277V2119898
119863
(23)
where
119873
119866119881= V119904(
120588
119871
119892120590
119904
)
025
(24)
Flow parameters such as void fraction mixture densityand virtual mass force coefficient are discussed for a specificflow pattern
The correlation between void fraction and liquid holdupis expressed as
120601
119866+ 120601
119871= 1 (25)
231 Bubbly Flow As for bubbly flow the density of gas anddrilling mud mixture is described as the following formula
120588
119898= 120601
119871120588
119871+ 120601
119866120588
119866 (26)
The void fraction is determined by the following formula
120601
119866=
1
2
[
[
1 +
119902
119898
V119904119860
minus
radic
(1 +
119902
119898
V119904119860
)
2
minus
4119902
119866
V119904119860
]
]
(27)
The coefficient of virtual mass force 119862vm for bubbly flowcan be expressed as follows [54]
119862vm = 05
1 + 2120601
119866
1 minus 120601
119866
(28)
The coefficient of resistance coefficientC119863for bubbly flow
can be expressed as
119862
119863=
4119877
119887
3
radic
119892 (120588
119871minus 120588
119866)
120590
119904
[
1 + 1767(1 minus 120601
119866)
97
1867(1 minus 120601
119866)
15]
2
(29)
The friction pressure gradient for bubbly flow can beobtained from the following equation
120591
119891= 119891
120588
119871V2119871
2119863
(30)
232 Slug Flow As for slug flow distribution coefficient ofgas in the liquid phase is
119862
0=
000252lg (103120583119871)
119860
138minus 0782 + 0232lgV
119898minus 0428lg119860
(31)
The average density of the mixture for slug flow isdetermined by
120588
119898=
119882
119898+ 120588
119871V119904119860
119902
119898+ V119904119860
+ 119862
0120588
119871 (32)
The void fraction for slug flow is determined by thefollowing formula
120601
119866=
119902
119866
119902
119866+ 119902
119871
(33)
6 Mathematical Problems in Engineering
The coefficient of virtual mass force 119862vm for slug flow canbe expressed as follows
119862vm = 33 + 17
3119871
119902minus 3119877
119902
3119871
119902minus 119877
119902
(34)
The coefficient of resistance coefficient C119863for slug flow
can be expressed as
119862
119863= 110120601
3
119871119877
119887
(35)
The friction pressure gradient of bubbly flow can beobtained from the following equation
120591
119891= 119891
120588
119871V2119898
2119863
(
119902
119871+ V119904119860
119902
119898+ V119904119860
+ 119862
0)
(36)
24 Annulus Characteristic Analysis The annulus effectivediameter proposed by Sanchez [55] is used in the hydrauliccalculation of annulus
119863 =
120587 (119863
2
119900minus 119863
2
119894) 4
120587 (119863
119900+ 119863
119894) 4
= 119863
119900minus 119863
119894
(37)
The effective roughness of the annulus can be expressedas
119896
119890= 119896
119900
119863
119900
119863
119900+ 119863
119894
+ 119896
119894
119863
119894
119863
119900+ 119863
119894
(38)
where theD119894andD
119900are the diameters of inner pipe and outer
pipe respectively the k119900and k119894are the roughnesses of outer
pipe and the inner pipe respectively
3 The United Model Developed
The total pressure drop gradient is the sum of pressure dropgradients due to potential energy change and kinetic energyand frictional loss From (1) the equation used to calculatethe pressure gradient of gas and drilling mud flow within theannulus can be written as
119889119901
119889119911
= 120588
119898119892 sin 120579 minus
120591
119908120587119863
119860
minus 120588
119898V119898
119889V119898
119889119911
(39)
Assuming the compressibility of the gas is only related tothe pressure in the annulus the kinetic energy or accelerationterm in the previous equation can be simplified to
120588
119898V119898
119889V119898
119889119911
= minus
120588
119898V119898V119904119892
119901
119889119901
119889119911
= minus
119882
119898119902
119892
119860
2119901
119889119901
119889119911
(40)
Substituting (39) into (40) the total pressure drop gra-dient along the flow direction within the annulus can beexpressed as
119889119901
119889119911
=
120588
119898119892 + 120591
119891
1 minus119882
119898119902
119866 (119860
2119901)
(41)
It is assumed that the gas obeys to the EOS (14) and thecompressibility of drilling mud can be obtained by adopting
the simplified EOS ((18) and (19)) which neglects the thermalexpansion of liquid The sonic speed of gas phase c
119866and that
of liquid phase c119871can be presented in the following form
119889119901
119871
119889120588
119871
= 119888
2
119871
119889119901
119866
119889120588
119866
= 119888
2
119866
(42)
By introducing (42) the hydrodynamic equations of two-fluid model ((2)ndash(4)) can be written in the matrix form
119860
120597120585
120597119905
+ 119861
120597120585
120597119911
= 119862120585
(43)
where A is the matrix of parameters considered in relation totime B is the matrix of parameters considered in relation tothe spatial coordinate and C is the vector of extractions
By introducing the small disturbance theory the distur-bance of the state variable 120585(120601
119866 119901 V119866 V119871)
119879 can be written as
120585 = 120585
0+ 120575120585 exp [119894 (119908119905 minus 119896119905)] (44)
In (44) k is the wave number and 119908 is the angularfrequency of the disturbances Substituting (44) into (43)gives homogenous linear equations concerning the expres-sion (120575120601
119866 120575119901 120575V
119866 120575V119871)
119879 According to the solvable conditionof the homogenous linear equations that the determinant ofthe equations is zero dispersion equation of pressure wavecan be expressed in the following form
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119872
1119872
2119872
3119872
4
minus120588
119871119908
1 minus 120601
119866
119888
2
119871
119908 0 minus119896 (1 minus 120601
119866) 120588
119871
119872
5119872
6119872
7119872
8
119872
9119872
10119872
8119872
11
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
= 0 (45)
whereM1ndashM11can be illustrated by
119872
1= (120588
119866+ 119888
119901120601
119866120588
119871
V2119904
119888
2
119866
)119908
119872
2=
120601
119866
119888
2
119866
[1 minus 119888
119901120601
119871]
V2119904
119888
2
119871
119908
119872
3= minus[120601
119866120588
119866119896 + 2119888
119901120601
119866120601
119871120588
119871
V119904
119888
2
119871
119908]
119872
4= 22119888
119901120601
119866120601
119871120588
119871
V119904
119888
2
119871
119908
119872
5= 120588
119871V2119903119896 (minus120601
119866119888
119901+ 119888
119903minus 119888
119894+ 119888
1198982)
119872
6= minus120601
119866119896[1 minus 120601
119871
119888
119901V2119904
119888
2
119871
+ 119888
119894
V2119904
119888
2
119871
]
Mathematical Problems in Engineering 7
119872
7= 120601
119866(120588
119866+ 119888vm120588119871) 119908 minus 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904+
4
119863
119891
119866119908120588
119866V119866)
119872
8= minus119888vm120601119866120588119871119908 + 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904)
119872
9= 120588
119871V2119904119896 (120601
119871119888
119901minus 2119888
119903minus 119888
1198982)
119872
10= minus119896(120601
119871+ 119888
119903120601
119866
V2119904
119888
2
119871
)
119872
11= 120588
119871[120601
119871+ 120601
119866119888vm] 119908 minus 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904+
4
119863
119891
119871120588
119871V119871)
(46)
where 1198881198982= 01 119888
119901= 025 119888
119894= 03 and 119888
119903= 02
By solving the dispersion equation indicated previouslyfour roots can be obtained Because the wavelengths asso-ciated with two of the roots are too short to allow the two-fluid medium to be treated as a continuum the two rootsshould be omitted As for the two remaining roots one ofthem represents a pressure wave that transmited along theaxis z and the other represents a pressure wave transmits inthe opposite direction in accordance with the direction of theflow along the annulusThe real value of wave number can beused to determine thewave velocity cThewave velocity in thetwo-phase fluid can be determined by the following model
119888 =
1003816
1003816
1003816
1003816
(119908119877
+
(119896)) minus (119908119877
minus
(119896))
1003816
1003816
1003816
1003816
2
(47)
4 Solution of the United Model
Obtaining the analytical solution of themathematical modelsconcerned with flow pattern void fraction characteristicparameters and pressure drop gradient is generally impossi-ble for two-phase flow In this paper the Runge-Kuttamethod(R-K4) is used to discretize the theoretical model
We can obtain pressure temperature gas velocity drillingmud velocity and void fraction at different annulus depthsby R-K4 The solution of pressure drop gradient equation(41) can be seen as an initial-value problem of the ordinarydifferential equation
119889119901
119889119911
= 119865 (119911 119901)
119901 (119911
0) = 119875
0
(48)
With the initial value (z0p0) and the function F(z p)
(50)ndash(53) can be obtained by
119896
1= 119865 (119911
0 119901
0) (49)
119896
2= 119865(119911
0+
ℎ
2
119901
0+
ℎ
2
119896
1) (50)
119896
3= 119865(119911
0+
ℎ
2
119901
0+
ℎ
2
119896
2) (51)
119896
4= 119865 (119911
0+ ℎ 119901
0+ ℎ119896
3) (52)
where h is the step of depthThe pressure on the nod 119894 = 119894 +1can be obtained by
119901
1= 119901
0+ Δ119901 = 119901
0+
ℎ
6
(119896
1+ 2119896
2+ 2119896
3+ 119896
4)
(53)
In the present work the mathematical model and pres-sure wave velocity calculation model are solved by a person-ally compiled code on VBNET (Version 2010) The solutionprocedure for the wave velocity in the annulus is shown inFigure 1 At initial time the wellhead back pressure wellheadtemperature wellbore structure well depth gas and drillingmud properties and so forth are known On the node ithe pressure gradient temperature and the void fraction canbe obtained by adopting R-K4 Then the determinant (45)is calculated based on the calculated parameters Omittingthe two unreasonable roots the pressure wave velocity atdifferent depths of the annulus in MPD operations can besolved by (47)The process is repeated until the pressure wavevelocity of every position in the wellbore is obtained as shownin Figure 2
The developed model takes full consideration of theinterfacial interaction and the virtual mass force Owing tothe complex conditions of the annulus in MPD operationsmeasurement of wave velocity in the actual drilling processis very difficult In order to verify the united model thepredicted pressure waves are compared with the results ofprevious simulated experimental investigations presented forgas and drillingmud by Liu et al [45] in Figure 3(a) and by Liet al [22] in Figure 3(b) The lines represent the calculationresults and the points represent the experimental data
The comparisons reveal that the developed united modelfits well with the experimental data Thus the united modelcan be used to accurately predict wave velocity at differentwellhead back pressures and gas influx rates (the influx rateof gas in the bottomhole) in MPD operations
5 Analysis and Discussion
The drilling system described is an enclosed system Theschematic diagram of the gas influx process is illustrated inFigure 4 The drilling mud is pumped from surface storagedown the drill pipe Returns from the wellbore annulus travelback through surface processing where drilling solids areremoved to surface storage The key equipments include thefollowing
(i) The Rotating Control Device (RCD) The rotatingcontrol device provides the seal between atmosphereand wellbore while allowing pipe movement anddiverting returns flow In conjunction with the flowcontrol choke the RCD provides the ability to applyback pressure on the annulus during an MPD opera-tion
(ii) Choke The MPD choke manifold provides anadjustable choke system which is used to dynamicallycontrol the required BHP by means of applyingsurface BP
(iii) Coriolis Meter A Coriolis meter is used to accu-rately measure the mass flow rate of fluid exiting
8 Mathematical Problems in Engineering
Grid division
Start
Delete two roots
End
Initial conditions
Meet demand
i = i + 1
Solve eq (20) for T
Solve eq (15)ndash(17) for Zg
Judging flow pattern by eq (21) and eq (22)
Bubbly flow pattern Slug flow pattern
Solve determinant eq (45) for four roots
Calculate F(z p) by eq (48)
Calculate k1 minus k4Pi+1 = Pi + ΔPzi+1 = zi + h
zi+1 gt HNo
No
Yes
Yes
Solve eq (18) (19) for120588L
Solve eq ( 47) for c
Solve eq (14)120588G
Calculate 120588m 120601G f and CDCalculate 120588m 120601G f and CD CvmCvm
Figure 2 Solution procedure for wave velocity in MPD operations
the annulus The ability to measure return flowaccurately is essential for the applied back pressure
(iv) Pressure Sensor A pressure sensor is used to measuresurface back pressure on the wellhead
The gas and drilling mud flow rate measured by the Cori-olis meter and the back pressure measured by the pressuresensor are the initial data for annulus pressure calculationThe well used for calculation is a gas well in XinjiangUygur Autonomous Region Northwest China The wellborestructure well design parameters (depths and diameters)gas and drilling mud properties (density and viscosity) andoperational conditions of calculation well are displayed inTable 1
The drilling mud mixed with gas is taken as a two-phaseflow medium The propagation velocity of pressure wave inthe gas-drillingmud flow is calculated and discussed by usingthe established model and well parameters
51 Effect of Back Pressure on Wave Velocity Increase inapplied back pressure is the most common approach fordynamicwell controlThewellbore can be seen as an enclosed
Table 1 Parameters of calculation well
Type Property Value
Mud Dynamic viscosity (Pa sdot s) 0056Density (kgcm3) 1460
Gas Relative density 065Viscosity (Pa sdot s) 114 times 10minus5
StringElastic modulus of string (Pa) 207 times 1011
Poisson ratio of string 03Roughness (m) 154 times 10minus7
Surface condition Surface temperature (K) 298Atmosphere pressure (MPa) 0101
pressurized system The pressure at different depths of theannulus varied with the change of back pressure Accordingto the EOS of gas and drilling mud the influence of pressureon gas volume is much greater than that of drilling mudfor the greater compressibility of gas So the gas voidfraction changes with the variation of gas volume at thenearly constant flow rate of drilling mud at different annularpressures Meanwhile the pressure wave velocity is sensitive
Mathematical Problems in Engineering 9
5000 2 4 6 8 10
750
1000
1250
1500c
(ms
)
This paper p = 30MPa w = 50Hz = Re
120601G ()
CvmLiu et al [45] p = 30MPa
(a) Comparison with the experimental data of Liu
0 20 40 60 80 1000
40
80
120
160
200
c(m
s)
This paper p = 01MPa w = 50Hz = Re
120601G ()
CvmLi et al [22] p = 01MPa
(b) Comparison with the experimental data of Li
Figure 3 Experimental verification by comparison with previous experimental data
Drill pipe
Annulus
RCD
Formation
Mud pump
Drill collar
Coriolis meterChoke
Casing
Pressure sensors
120601 2159 mm times G105
1206011778mm (120601 78 mm) times G105
120601127 mm (120601 78 mm) times G105
z0 m
3600 m
4000 m
QL
Figure 4 Schematic diagram of the gas influx process
to the gas void fraction As a result when the back pressureat the wellhead is changed by adjusting the choke the wavevelocity in the two-phase drilling fluid and the distributionof void fraction at different depth of the annulus will divergeThe calculation results affected by the back pressure arepresented
Figures 5 and 6 show the distributions of void fractionand variations of wave velocity along the flow directionin the annulus when the back pressure at the wellhead is01MPa 08MPa 15MPa 25MPa 50MPa and 65MPa
respectively It can be seen that the void fraction significantlyincreases along the flow direction in the annulus Converselythe wave velocity shows a remarkable decrease tendency Forinstance BP = 15MPa at the wellhead the wave velocityis 9012ms and the void fraction is 0611 while in thebottomhole the wave velocity reaches 73142ms and voidfraction is reduced to 0059 However a sudden increase inwave velocity is observed in Figure 5 when the back pressureis 01MPa at the position close towellhead as the void fractionis further increased Moreover the influence of back pressure
10 Mathematical Problems in Engineering
10075500 254000
3000
2000
1000
0
BP = 01 MPaBP = 08 MPaBP = 15 MPa
BP = 25 MPa
BP = 65 MPaBP = 5 MPa
H(m
)
Q = 9m3h w = 50Hz = Re
120601G ()
Cvm
Figure 5 Void fraction distribution at different back pressures
on the void fraction is larger at the position close to thewellhead than that in the bottomhole
This can be explained from the viewpoints of mixturedensity and compressibility of two-phase fluid and the pres-sure drop along the flow direction in the wellbore In thelow void fraction range the gas phase is dispersed in theliquid as bubble so the wave velocity is influenced greatly bythe added gas phase According to the EOS if gas invadesinto the wellbore with a small amount in the bottomholethe density of the drilling mud has little variation while thecompressibility increases obviously which makes the wavevelocity decreased Then the two-phase drilling mud flowfrom the bottomhole to the wellhead along the annulus witha drop of pressure caused by potential energy change kineticenergy and frictional loss which leads to an increase in voidfractionWhen the void fraction is increased both the densityand the compressibility of the two-phase fluid change slightlyresulting in a flat decrease in wave velocity along the flowdirection With the further increasing in void fraction thebubbly flow turns into slug flow according to the flow patterntransient criteria of Orkiszewski which shows that the flowpattern is dependent on the void fraction Generally speak-ing the gas-drilling mud slug flow is composed of liquidand gas slugs In the preliminary slug flow the liquid slugsare much longer than the gas slugs so the wave velocity isdetermined primarily by the wave velocities in the liquidslugs which are hardly affected by the void fraction At theposition close to the wellhead the pressure of two-phase flowfluid can be reduced to a low value which approaches theback pressure The compressibility of gas will be improvedat the low pressure It results in significant increase in void
0 250 500 750 10004000
3000
2000
1000
0
H(m
)BP = 01 MPaBP = 08 MPaBP = 15 MPa
BP = 25 MPa
BP = 65 MPaBP = 5 MPa
c (ms)
Q = 9m3h w = 50Hz = ReCvm
Figure 6 Wave velocity variations at different back pressures
fraction As the void fraction increases greatly at the positionclose to the wellhead gas slugs becomemuch longer than theliquid slugs It is assumed that the actual wave velocity in slugflow happens to play a leading role when the void fractionincreases to some extent For the decrease in liquid holdupin the gas slug with the increasing in void fraction the wavevelocity in the gas slug is infinitely close to that in pure gaswhich is equivalent to an increase in wave velocity in the gasslug Meanwhile the gas content in liquid slug also increasesand results in a lower wave velocity in the liquid slug As aresult a slight increase in wave velocity appears
Figures 7 and 8 present the influence of back pressureon the void fraction and wave velocity with respect to theparameters of well depth As the back pressure increases thevoid fraction at different depth of the annulus is reducedgradually while the wave velocity in the two-phase flow tendsto increase Analytical results show that the increased backpressure is equivalent to be applied to the entire encloseddrilling fluid cyclical systemThe pressure transmits from thewellhead to the bottomhole therefore the annular pressurein the entire wellbore is increased According to the EOSthe density of gas increases and the compressibility of gasdecreases with the increasing of gas pressure So the lossof interphase momentum and energy exchange is reducedand the interphase momentum exchange is promoted Itcontributes to the increase in wave velocity with the increasein gas pressure In addition due to lower compressibility oftwo-phase flow medium under high pressure the increasetendency of pressure wave velocity and the decrease tendencyof void fraction are slowed down in the high back pressurerange
Mathematical Problems in Engineering 11
0
25
50
75
100
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
0 25 75 105
Q = 9m3h w = 50Hz = Re
120601G
()
Cvm
Figure 7 Effect of back pressure on the void fraction
52 Effect of Gas Influx Rate on Wave Velocity Figures 9and 10 graphically interpret the distributions of void fractionand variations of wave velocity along the flow direction inthe annulus When gas influx occurs in the bottomhole gasinvades into the wellbore and migrates from the bottomholeto the wellhead along the flow direction At a low gas influxrate it is extremely obvious that the void fraction and wavevelocity first slightly change in a comparatively smooth valuethen change sharply It is because of the rapid expansion ofgas volume with the decreasing in pressure near the wellheadthat the void fraction increases sharply and the wave velocitydecreases obviously at the same time But under the highbottomhole pressure (up to 50MPa) the compressibilityof the gas is low This results in a slight change in voidfraction and wave velocity at the position far away from thewellhead Since the compressible component increases withthe increase in the gas influx rate the compressibility of thegas and drilling mud two-phase fluid is improved So thevariations of void fraction and wave velocity become moreprominent Also the void fraction still shows an increasetendency that is steady first and then sharp It acts inaccordance with the variation of void fraction at a low gasinflux rate At a low gas influx rate if the void fraction atthe position close to the wellhead can not increase to a highextent the wave velocity always shows a decrease tendencyAt a high gas influx rate such as the 119876 = 8312m3h thewave velocity tends to increase because the void fraction inthe wellhead is increased to a high extent In conclusionthe wave pressure is sensitive to the void fraction and thevoid fraction is dominated by influx rate and pressure in theannulus especially the influx rate
0 25 75 1050
300
600
900
1200
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
c(m
s)
Q = 9m3h w = 50Hz = ReCvm
Figure 8 Effect of back pressure on the pressure wave velocity
With the influx of gas the mixing of gas and drillingmud occurs in the annulus and the corresponding interfacialtransfer of momentum and mass causes an increase in gasphase void fraction and a decrease in pressure wave velocityas shown in Figures 11 and 12 Within the range of low gasinflux rate the wave velocity decreases significantly It isbecause of this that the compressibility of the gas increasesremarkably and the medium appears to be of high elasticitythough the density of gas-drilling mud two-phase flowchanges slightly With the increase in the gas influx rate andcorresponding increase in the void fraction in the annulusthe compressibility of the two-phase unceasingly increaseswhich promotes the momentum and energy exchange inthe interface So the pressure wave continuously decreasesWhen the void fraction increases to some extent following theincrease in the gas influx rate the decrease in wave velocityin the liquid slug is gradually less than the increase in wavevelocity in the gas slug thus the decrease of wave velocityis slowed down for the growth of gas slug Especially in thewellhead a slight increase in wave velocity is observed
53 Effect of Virtual Mass Force on Wave Velocity Figure 13illustrates the effect of virtual mass force on the wave velocitywith the parameter of gas influx rate As shown in Figure 15under the same gas influx rate (0594m3h 1098m3h and1768m3h resp) the wave velocity curve of 119862vm = Rediverges from the curve of 119862vm = 0 at the position close tothewellhead whereas thewave velocity curve of the two typesis almost coincided below the position of 119867 = 500m Thisdivergence between the two types of curves is connected withthe fact that significance of influence of interphase virtual
12 Mathematical Problems in Engineering
0 25 50 75 1004000
3000
2000
1000
0
Q = 0058m3h Q = 1228m3hQ = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
H(m
)
BP = 01 MPaw = 50Hz = Re
120601G ()
Cvm
Figure 9 Void fraction distribution at different gas influx rates
mass forces increases together with the increase of the relativemotion in the interphase It is evident that the bottomholepressure is hundreds of times higher than the wellheadpressure As a result the void fraction at the position closeto the wellhead increases sharply meanwhile the interphasemomentum and energy exchanges are promoted The virtualmass force can be described as the transfer of momentumbetween the gas phase and drilling mud phase caused bythe relative motion in the interphase If the relative motionin the interphase is quite weak the value of virtual massforce will intend to approach 0 and the influence on thewave velocity can be ignored However if the relative motionis rather intense the effect of virtual mass force on thewave velocity should not be ignored Furthermore taking thevirtual mass force into account the dispersion characteristicof the pressure wave weakens obviously Compared with thepressure wave velocity calculated by ignoring the effect ofvirtual mass force the calculated pressure wave velocity witha consideration of the virtual mass force is lower Thereforeit is necessary to analyze the effect of the virtual mass forceon the wave velocity at the position close to the wellhead inMPD operations
54 Effect of Disturbance Angular Frequency onWave VelocityFigure 14 presents the pressure wave velocity in the annulusat different angular frequencies According to Figures 5 and9 the void fraction first slightly increases in the bottomholeand then sharply increases along the flow direction at theposition close to the wellhead At a fixed angular frequencythis phenomenon results in an overall decrease in the pressurewave velocity except for the section near the wellhead At the
0 500 1000 1500 20004000
3000
2000
1000
0
c (ms)
H(m
)Q = 0058m3h Q = 1228m3h
Q = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
BP = 01 MPaw = 50Hz = ReCvm
Figure 10 Wave velocity variations at different gas influx rates
section close to the wellhead an opposite change trend ofpressure wave velocity is observed for the transition of flowpattern from bubbly flow to slug flow due to the continuousincrease in void fraction along the flow direction In additionit can be clearly seen from the curves that the wave velocityincreases accompanying with the increase in the angularfrequency above the position of 119867 = 500m This propertyis not very distinct at the position below 119867 = 500m for thelow void fraction
Figure 15 shows the effect of frequency on the wavevelocity in the gas-drilling mud flow (119862vm = 0)The curves ofthe wave velocity of different positions reveal that the prop-agation velocity of pressure disturbances increases togetherwith the growth of the angular frequency (0 lt 119908 lt 500Hz)It proves that the pressure wave has an obvious dispersioncharacteristic in the two-phase flow As the angular frequencyincreases in the range of less than 500Hz there is sufficienttime to carry out the exchange of energy and momentumbetween two phases It achieves a state of mechanical andthermodynamic equilibrium between the two phases Sothe wave velocity increases gradually with the increase inthe angular frequency at different depths of the annulus Itis considered that the wave velocity is mainly affected bythe interphase mechanical and thermodynamic equilibriumat low anglular frequencies When the angular frequencyreaches the value of 119908 = 500Hz the pressure wave velocityachieves a constant value and remains on this level regardlessof the further growth in angular frequency 119908 With theincrease in angular frequency there is not enough time forenergy and momentum exchange between the gas-drillingmud two phases to reach themechanical and thermodynamic
Mathematical Problems in Engineering 13
0
25
50
75
100
125
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
Q (m3h)
120601G
()
0 25 5 75 10
BP = 01 MPaw = 50Hz = ReCvm
Figure 11 Effect of gas influx rate on the void fraction
equilibrium state and thus the wave dispersion does notexist At a high angular frequency the wave velocity is mainlydominated by the mechanical and thermal nonequilibriumin the flow and keeps almost unchanged when the angularfrequency is further increased This is consistent with theinfluences of the angular frequency in the horizontal pipe[28] Moreover the effect of virtual mass force is also shownin this figure (119862vm = Re) It is observed that the wave velocityis significantly reduced when the virtual mass force is takeninto consideration in the calculation of the wave velocity atthe position close to the wellhead such as 119867 = 0m It canbe explained by the reduction of dispersion characteristic ofa pressure wave Along the wellbore from the wellhead to thebottomhole the distinction gradually decreases
6 Conclusions
With full consideration of the important factors such asvirtual mass force drag force gas void fraction pressuretemperature and angular frequency a united wave velocitymodel has been proposed based on pressure drop gradientequations in MPD operations gas-liquid two-fluid modelthe gas-drilling mud equations of state and small pertur-bation theory Solved by the fourth-order explicit Runge-Kutta method the model is used to predict wave velocityfor different back pressures and gas influx rates in MPDoperations The main conclusions can be summarized asfollows
(1) With the introduction of virtual mass force and dragforce the united model agrees well with the previousexperimental data The united model can be used to
0
280
560
840
1120
1400
Q (m3h)0 25 5 75 10
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
c(m
s)
BP = 01 MPaw = 50Hz = ReCvm
Figure 12 Effect of gas influx rate on the pressure wave velocity
accurately calculate the wave velocity in the annulusThe application of the model will be beneficial tofurther study the wave velocity at different gas influxrates and back pressures in MPD operations reducenonproductive times and provide a reference for thedrilling operations in extremely narrow porefracturewindows existing conditions
(2) The wave velocity and void fraction have differentvalues with respect to well depth In the annulusthe drop of pressure causes an increase in the voidfraction along the flow direction The void fractionincreases first slightly and then sharply Correspond-ingly the wave velocity first gradually decreases in thbubbly flow and preliminary slug flow and then thewave velocity slightly increases accompanying withthe increase in the relative length ratio of gas slugto the liquid slug for the continuous increase in voidfraction at the position close to the wellhead Theminimum wave velocity appears in the long gas slugflow
(3) When the back pressure in the MPD operationsincreases the void fraction at different depths of theannulus is reduced gradually while the wave velocityin the two-phase flow tends to increase Moreoverthe influence of back pressure on the void fraction isgreater at a position close to the wellhead than that inthe bottomhole because of the great decrease in thepressure from the bottomhole to the wellhead Alsothe effect of back pressure on void fraction and wavevelocity is decreased in the high back pressure range
14 Mathematical Problems in Engineering
0 300 600 900 12004000
3000
2000
1000
225
180
135
90
45
0
BP = 01 MPaw = 50Hz
Q = 0594m3h = ReQ = 0594m3h = 0
Q = 1098m3h = ReQ = 1768m3h = ReQ = 1098m3h = 0
Q = 1768m3h = 0
c (ms)
H(m
)
CvmCvm
Cvm
Cvm
Cvm
Cvm
Figure 13 Effect of virtual mass force on the pressure wave velocity
(4) The wave velocity is sensitive to the void fraction butthe void fraction is dominated by gas influx rate andpressure in the annulus especially the gas influx rateSince the compressibility of the gas and drilling mudtwo-phase fluid is improved with the increase in theinflux rate the void fraction increases greatly andthe wave velocity decreases significantly within thelow gas influx rate range When the void fraction isincreased to some extent following the increase in thegas influx rate the decrease of wave velocity is slowedfor the growth of gas slug Especially at the wellheada slight increase in wave velocity is observed at a highgas influx rate for the sharp increase in void fraction
(5) It is necessary to analyze the effect of the virtualmass force on the wave velocity in MPD operationsEspecially at the position close to the wellhead theeffect of virtual mass force is more prominent for theintense phase interactions Taking the virtual massforce into account the dispersion characteristic of thepressure wave weakens obviously Compared with theresults calculated by ignoring the effect of virtualmassforce the calculated pressure wave velocity with aconsideration of the virtual mass force is lower
(6) The effect of angular frequency on the propagationvelocity of pressure wave appears at a low angularfrequency The propagation velocity of pressure dis-turbances increases together with the growth of theangular frequency (0 lt 119908 lt 500Hz) When theangular frequency reaches the value of 119908 = 500Hzthe pressure wave velocity achieves a constant value
0 240 480 720 960 12004000
3000
2000
1000
560
420
280
140
0
Q = 0727m3h BP = 01 MPa
w = 50Hz = Rew = 50Hz = 0w = 300Hz = 0
w = 800Hz = 0w = 1600Hz = 0w = 5000Hz = 0
c (ms)
H(m
)CvmCvmCvm
CvmCvmCvm
Figure 14 Wave velocity variations at different angular frequencies
and remains on this level regardless of the furthergrowth in angular frequency 119908
Nomenclature
A Annulus effective cross area (m2)c119866 Wave velocity in gas (ms)
c119871 Wave velocity in Drilling mud (ms)
C119863 The coefficient of drag force
D Annulus effective diameter (m)D119894 Diameter of the inner pipe (m)
D119900 Diameter of the outer pipe (m)
f119866 Shear stresses coefficient of gas interface
f119871 Shear stresses coefficient of drilling mud
interfaceF119911 Pressure applied to the two-phase flow (N)
g Acceleration due to gravity (m2s)g119888 Conversion factor
g119879 Geothermal temperature gradient (Km)
h One discrete length of annulus (m)k119890 Annulus effective roughness (m)
k119900 Roughness of outer pipe (m)
k119894 Roughness of inner pipe (m)
H The depth of annulus (m)L119902 Air bubbles length of slug (m)
119872Gi Momentum transfer in gas interface (Nm3)119872Li Momentum transfer in liquid interface (Nm3)119872
ndLi The none-drag-force (virtual mass force)
(Nm3)119872
119889
Li The momentum transfer term caused by thedrag force (Nm3)
Mathematical Problems in Engineering 15
100 1000 100000
450
900
1350
1800
0w (Hz)
0 m = 0 0 m = Re1000 m = 0
2000 m = 0
4000 m = 0
1000 m = Re2000 m = Re4000 m = Re
c(m
s)
Q = 0727m3h BP = 01 MPaCvmCvm
CvmCvmCvm
CvmCvmCvm
Figure 15 Effect of angular frequency on wave velocity
N Parameter combiningThompsonenergy effects
119901Gi Gas interface pressure (MPa)119901Li Liquid interface pressure (MPa)p Pressure (MPa)p0 Initial pressure of annulus (MPa)
p119888 Critical pressure (kPa)
p119871 Pressure of drilling mud (MPa)
p119903 Reduced pressure
q119898 Volumetric flow rate of gas and drilling
mud (m3s)q119866 Velocity of the gas (m3s)
q119871 Velocity of the drilling mud (m3s)
Q119871 Drilling mud rate within the annulus
(m3h)r Average diameter of the bubble (m)R Constant of EOS (JKgK)R119902 Air bubbles radius of slug (m)
R119887 The bubble diameter (m)
T Temperature (K)T119888 Critical temperature (K)
T119903 Reduced temperature
T119890119894 Undisturbed formation temperature at a
depth (K)T119890119887ℎ
Undisturbed formation temperature ofwellhead (K)
T119891119887ℎ
Undisturbed formation temperature atthe bottomhole (K)
v119898 Gas and drilling mud flow velocity
(ms)
v119866 Gas flow velocity (ms)
v119904 Slip velocity (ms)
v119904119892 Superficial gas velocity (ms)
v119871 Drilling mud flow velocity (ms)
W119898 Mass flow velocity (kgm3)
Z119866 Gas deviation factor
z0 Initial calculation height of annulus (m)
z119887ℎ Total well depth from surface (m)
z Variable of distance from surface (m)
Greek Letters
120588
119866 Gas density (kgm3)
120591
119891119903
Li Shear stresses of drilling mud noninterfacial(Nm2)
120591
119891119903
119871 Shear stresses of drilling mud interface (Nm2)
120591
Re119871 Reynolds stress of drilling mud (Nm2)
120591
Re119871 Reynolds stress of drilling mud interface
(Nm2)120591
Re119866 Reynolds stress of gas interface (Nm2)
120591
119891119903
119866 Shear stresses of gas interface (Nm2)
120572VM Acceleration due to virtual mass forcecoefficient (ms2)
120591
119871 Shear stresses of drilling mud along wall
(Nm2)120591
119866 Shear stresses of gas along wall (Nm2)
120591
Re119866 Reynolds stress of gas (Nm2)
120591
119891 Frictional pressure gradient (Pam)
120591
119908 Frictional pressure coefficient
120588
119898 Gas and drilling density (kgm3)
120588
0 Density under standard atmospheric pressure
(kgm3)120588
119871 Drilling mud density (kgm3)
120590
119904 Surface tension (Nm2)
Φ
119866 Gas void fraction
Φ
119871 Drilling mud holdup
120588
119903 Reduced density
120583
119871 Viscosity of drilling mud (Pa sdot s)
Subscripts
BHP Bottomhole pressureBP Back pressureEOS Equations of stateMPD Managed pressure drillingRCD The rotating control deviceR-K4 The fourth-order explicit Runge-Kutta
Subscripts of Graph
BP Back pressure (MPa)c Wave velocity in gas and drilling mud
two-phase flow (ms)119862vm The coefficient of virtual mass force
16 Mathematical Problems in Engineering
H Depth of annulus (m)Q Gas influx rate at the bottomhole (m3h)Re The value of virtual mass force
coefficient according to (28) or (34)119908 Angle frequency (Hz)Φ
119866 Void fraction ()
Acknowledgment
Research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 and 51274170)Without their support this work would not have beenpossible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of airinjection in annuli to manage pressure during cementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling conferenceand Exhibition San Diego Calif USA March 2012
[4] W A Bacon Consideration of compressibility effects for applied-back-pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Tex USA 2011
[5] J Choe C Hans and J Wold ldquoA modified two-phase well-control model and its computer applications as a training andeducational toolrdquo SPE Computer Applications vol 2 pp 14ndash201997
[6] J A Tarvin IWalton and PWand ldquoAnalysis of a gas kick takenin a deep well drilled with oil-based mudrdquo in Proceedings ofthe Annual Technical Conference and Exhibition of the Societyof Petroleum Engineers pp 255ndash264 Dallas Tex USA 1991
[7] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[8] J Zhou and G Nygaard ldquoAutomatic model-based controlscheme for stabilizing pressure during dual-gradient drillingrdquoJournal of Process Control vol 21 no 8 pp 1138ndash1147 2011
[9] A Ochoa O Acevedo L Nieto J E Lambarria and H PerezldquoSuccessful application of MPD (Managed Pressure Drilling)for prevention controland detection of borehole ballooning intight gas reservoir in Cuervito Field Mexicordquo in Proceedings ofthe Canadian Unconventional Resources Conference (CURC rsquo11)pp 172ndash185 November 2011
[10] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[11] R Ashena and J Moghadasi ldquoBottom hole pressure estimationusing evolved neural networks by real coded ant colony opti-mization and genetic algorithmrdquo Journal of Petroleum Scienceand Engineering vol 77 no 3-4 pp 375ndash385 2011
[12] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvanced Materials Research vol 468-471 pp 1736ndash1742 2012
[13] I Zubizarreta ldquoPore pressure evolution core damage and trip-ping out schedulesf a computational fluid dynamics approachrdquoin Proceedings of the SPEIADC Drilling Conference and Exhibi-tion Amsterdam The Netherlands March 2013
[14] C H Whitson ldquoCyclic shut-in eliminates liquid-loading in gaswellsrdquo in Proceedings of the SPEEAGE European Unconven-tional Resources Conference and Exhibition Vienna AustriaMarch 2012
[15] G M de Oliveira A T Franco C O R Negrao A LMartins and R A Silva ldquoModeling and validation of pressurepropagation in drilling fluids pumped into a closed wellrdquoJournal of Petroleum Science and Engineering vol 103 pp 61ndash71 2012
[16] Y Sato and H Kanki ldquoFormulas for compression wave andoscillating flow in circular piperdquo Applied Acoustics vol 69 no1 pp 1ndash11 2008
[17] J Xie B Yu X Zhang Q Shao and X Song ldquoNumericalsimulation of gas-liquid-solid three-phase flow in deep wellsrdquoAdvances in Mechanical Engineering vol 2013 Article ID951298 10 pages 2013
[18] W Bacon A Tong O Gabaldon C Sugden and P V Surya-narayana ldquoAn improved dynamic well control response to a gasinflux in managed pressure drilling operationsrdquo in Proceedingsof the IADCSPE Drilling Conference and Exhibition San DiegoCalifornia USA March 2012
[19] J I Hage and D ter Avest ldquoBorehole acoustics applied to kickdetectionrdquo Journal of Petroleum Science and Engineering vol 12no 2 pp 157ndash166 1994
[20] A M R Hopkins Fluid dynamics and surface pressure fluctua-tions of two-dimensional turbulent boundary layers over denselydistributed surface roughness [PhD thesis] Virginia PolytechnicInstitute and State University Blacksburg Va USA 2010
[21] Z S Fu ldquoThe gas cut detect technology during drilling in SovietUnionrdquo Petroleum Drilling Techniques vol 18 no 1 pp 19ndash211990
[22] X F Li C X Guan and X X Sui ldquoThe theory of gas influxdetection of pressure wave and its applicationrdquo Acta PetroleiSinica vol 18 no 3 pp 128ndash133 1997
[23] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[24] A E Ruggles R T Lahey Jr D A Drew and H A ScartonldquoInvestigation of the propagation of pressure perturbations inbubbly airwater flowsrdquo Journal of Heat Transfer vol 110 no 2pp 494ndash499 1988
[25] H J W M Legius H E A Van Den Akker and T NarumoldquoMeasurements on wave propagation and bubble and slugvelocities in cocurrent upward two-phase flowrdquo ExperimentalThermal and Fluid Science vol 15 no 3 pp 267ndash278 1997
[26] K Miyazaki and I Nakajima ldquoPropagation of pressure wavein NItrogen-Mercury two-phase systemrdquo Journal of NuclearScience and Technology vol 9 no 5 pp 315ndash318 1972
[27] B Bai L Guo and X Chen ldquoPressure fluctuation for air-watertwo-phase flowrdquo Journal of Hydrodynamics vol 18 no 4 pp476ndash482 2003
[28] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
Mathematical Problems in Engineering 17
[29] R F Tangren C H Dodge and H S Seifert ldquoCompressibilityeffects in two-phase flowrdquo Journal of Applied Physics vol 20 no7 pp 637ndash645 1949
[30] G B Wallis One Dimensional Two Phase Flow McGraw HillNew York NY USA 1969
[31] D L Nguyen E R F Winter and M Greiner ldquoSonic velocityin two-phase systemsrdquo International Journal ofMultiphase Flowvol 7 no 3 pp 311ndash320 1981
[32] R C Mecredy and L J Hamilton ldquoThe effects of nonequilib-rium heat mass and momentum transfer on two-phase soundspeedrdquo International Journal of Heat and Mass Transfer vol 15no 1 pp 61ndash72 1972
[33] K H Ardron and R B Duffey ldquoAcoustic wave propagationin a flowing liquid-vapour mixturerdquo International Journal ofMultiphase Flow vol 4 no 3 pp 303ndash322 1978
[34] J Xu and T Chen ldquoAcoustic wave prediction in flowing steam-water two-phase mixturerdquo International Journal of Heat andMass Transfer vol 43 no 7 pp 1079ndash1088 2000
[35] N ChungW Lin B Pei and Y Hsu ldquoModel for sound velocityin a two-phase air-water bubbly flowrdquo Nuclear Technology vol99 no 1 pp 80ndash89 1992
[36] X Xu and J Gong ldquoA unitedmodel for predicting pressurewavespeeds in oil and gas two-phase pipeflowsrdquo Journal of PetroleumScience and Engineering vol 60 no 3-4 pp 150ndash160 2008
[37] W Kuczynski ldquoCharacterization of pressure-wave propagationduring the condensation of R404A and R134a refrigerantsin pipe mini-channels that undergo periodic hydrodynamicdisturbancerdquo International Journal of Heat and Fluid Flow vol40 pp 135ndash150 2013
[38] W Kuczynski ldquoModeling of the propagation of a pressure waveduring the condensation process of R134a refrigerant in a pipeminichannel under the periodic conditions of hydrodynamicdisturbancesrdquo International Journal of Heat and Mass Transfervol 56 pp 715ndash723 2013
[39] Y Li C Li E Chen and Y Ying ldquoPressure wave propaga-tion characteristics in a two-phase flow pipeline for liquid-propellant rocketrdquoAerospace Science and Technology vol 15 no6 pp 453ndash464 2011
[40] T Kanagawa M Watanabe T Yano and S Fujikawa ldquoNon-linear wave equations for pressure wave propagation in liquidscontaining gas bubbles (comparison between two-fluid modeland mixture model)rdquo Journal of Fluid Science and Technologyvol 6 no 6 pp 836ndash849 2011
[41] EChen Y Li andYYing ldquoNumerical investigation onpressurewave propagation speed of gas-liquid two-phase flow in pumppipelinerdquo Journal of Aerospace Power vol 25 no 4 pp 754ndash7602010
[42] F Li and L He ldquoInvestigation on pressure wave propagationcharacteristics in slug flow field with increasing gas flow raterdquoPetroleum Geology and Recovery Efficiency vol 11 no 5 pp 78ndash80 2004
[43] C SMartin andM Padmanabhan ldquoPressure pulse propagationin two-component slug flowrdquo Journal of Fluids Engineering vol101 no 1 pp 44ndash52 1979
[44] R E Henry ldquoPressure wave propagation in two-phase mix-turesrdquo Chemical Engineering Progress vol 66 no 1 pp 1ndash101969
[45] X Liu B Li and Y Yue ldquoTransmission behavior of mud-pressure pulse alongwell borerdquo Journal ofHydrodynamics B vol19 no 2 pp 236ndash240 2007
[46] XWang and J Zhang ldquoThe research of pressure wave Pulsationin mud pulse transmittingrdquo Journal of Chongqing University ofScience and Technology vol 14 no 2 pp 55ndash58 2012
[47] J-W Park D A Drew and R T Lahey Jr ldquoThe analysisof void wave propagation in adiabatic monodispersed bubblytwo-phase flows using an ensemble-averaged two-fluid modelrdquoInternational Journal ofMultiphase Flow vol 24 no 7 pp 1205ndash1244 1998
[48] G Arnold Entropy and objectivity as constraints upons constitu-tive equations for two-fluid modeling of multiphase flow [PhDthesis] Rensselaer Polytechnic Institute New York NY USA1998
[49] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[50] L Yarborough and K R Hall ldquoHow to solve equation of statefor z-factorsrdquoOil and Gas Journal vol 72 no 7 pp 86ndash88 1974
[51] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86 pp 127ndash136 2012
[52] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[53] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[54] M Ishii and K Mishima ldquoTwo-fluid model and hydrodynamicconstitutive relationsrdquo Nuclear Engineering and Design vol 82no 2-3 pp 107ndash126 1984
[55] M J Sanchez Comparison of correlations for predicting pressurelosses in vertical multiphase annular flow [MS thesis] Univer-sity of Tulsa Tulsa Okla USA 1972
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
The coefficient of virtual mass force 119862vm for slug flow canbe expressed as follows
119862vm = 33 + 17
3119871
119902minus 3119877
119902
3119871
119902minus 119877
119902
(34)
The coefficient of resistance coefficient C119863for slug flow
can be expressed as
119862
119863= 110120601
3
119871119877
119887
(35)
The friction pressure gradient of bubbly flow can beobtained from the following equation
120591
119891= 119891
120588
119871V2119898
2119863
(
119902
119871+ V119904119860
119902
119898+ V119904119860
+ 119862
0)
(36)
24 Annulus Characteristic Analysis The annulus effectivediameter proposed by Sanchez [55] is used in the hydrauliccalculation of annulus
119863 =
120587 (119863
2
119900minus 119863
2
119894) 4
120587 (119863
119900+ 119863
119894) 4
= 119863
119900minus 119863
119894
(37)
The effective roughness of the annulus can be expressedas
119896
119890= 119896
119900
119863
119900
119863
119900+ 119863
119894
+ 119896
119894
119863
119894
119863
119900+ 119863
119894
(38)
where theD119894andD
119900are the diameters of inner pipe and outer
pipe respectively the k119900and k119894are the roughnesses of outer
pipe and the inner pipe respectively
3 The United Model Developed
The total pressure drop gradient is the sum of pressure dropgradients due to potential energy change and kinetic energyand frictional loss From (1) the equation used to calculatethe pressure gradient of gas and drilling mud flow within theannulus can be written as
119889119901
119889119911
= 120588
119898119892 sin 120579 minus
120591
119908120587119863
119860
minus 120588
119898V119898
119889V119898
119889119911
(39)
Assuming the compressibility of the gas is only related tothe pressure in the annulus the kinetic energy or accelerationterm in the previous equation can be simplified to
120588
119898V119898
119889V119898
119889119911
= minus
120588
119898V119898V119904119892
119901
119889119901
119889119911
= minus
119882
119898119902
119892
119860
2119901
119889119901
119889119911
(40)
Substituting (39) into (40) the total pressure drop gra-dient along the flow direction within the annulus can beexpressed as
119889119901
119889119911
=
120588
119898119892 + 120591
119891
1 minus119882
119898119902
119866 (119860
2119901)
(41)
It is assumed that the gas obeys to the EOS (14) and thecompressibility of drilling mud can be obtained by adopting
the simplified EOS ((18) and (19)) which neglects the thermalexpansion of liquid The sonic speed of gas phase c
119866and that
of liquid phase c119871can be presented in the following form
119889119901
119871
119889120588
119871
= 119888
2
119871
119889119901
119866
119889120588
119866
= 119888
2
119866
(42)
By introducing (42) the hydrodynamic equations of two-fluid model ((2)ndash(4)) can be written in the matrix form
119860
120597120585
120597119905
+ 119861
120597120585
120597119911
= 119862120585
(43)
where A is the matrix of parameters considered in relation totime B is the matrix of parameters considered in relation tothe spatial coordinate and C is the vector of extractions
By introducing the small disturbance theory the distur-bance of the state variable 120585(120601
119866 119901 V119866 V119871)
119879 can be written as
120585 = 120585
0+ 120575120585 exp [119894 (119908119905 minus 119896119905)] (44)
In (44) k is the wave number and 119908 is the angularfrequency of the disturbances Substituting (44) into (43)gives homogenous linear equations concerning the expres-sion (120575120601
119866 120575119901 120575V
119866 120575V119871)
119879 According to the solvable conditionof the homogenous linear equations that the determinant ofthe equations is zero dispersion equation of pressure wavecan be expressed in the following form
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119872
1119872
2119872
3119872
4
minus120588
119871119908
1 minus 120601
119866
119888
2
119871
119908 0 minus119896 (1 minus 120601
119866) 120588
119871
119872
5119872
6119872
7119872
8
119872
9119872
10119872
8119872
11
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
= 0 (45)
whereM1ndashM11can be illustrated by
119872
1= (120588
119866+ 119888
119901120601
119866120588
119871
V2119904
119888
2
119866
)119908
119872
2=
120601
119866
119888
2
119866
[1 minus 119888
119901120601
119871]
V2119904
119888
2
119871
119908
119872
3= minus[120601
119866120588
119866119896 + 2119888
119901120601
119866120601
119871120588
119871
V119904
119888
2
119871
119908]
119872
4= 22119888
119901120601
119866120601
119871120588
119871
V119904
119888
2
119871
119908
119872
5= 120588
119871V2119903119896 (minus120601
119866119888
119901+ 119888
119903minus 119888
119894+ 119888
1198982)
119872
6= minus120601
119866119896[1 minus 120601
119871
119888
119901V2119904
119888
2
119871
+ 119888
119894
V2119904
119888
2
119871
]
Mathematical Problems in Engineering 7
119872
7= 120601
119866(120588
119866+ 119888vm120588119871) 119908 minus 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904+
4
119863
119891
119866119908120588
119866V119866)
119872
8= minus119888vm120601119866120588119871119908 + 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904)
119872
9= 120588
119871V2119904119896 (120601
119871119888
119901minus 2119888
119903minus 119888
1198982)
119872
10= minus119896(120601
119871+ 119888
119903120601
119866
V2119904
119888
2
119871
)
119872
11= 120588
119871[120601
119871+ 120601
119866119888vm] 119908 minus 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904+
4
119863
119891
119871120588
119871V119871)
(46)
where 1198881198982= 01 119888
119901= 025 119888
119894= 03 and 119888
119903= 02
By solving the dispersion equation indicated previouslyfour roots can be obtained Because the wavelengths asso-ciated with two of the roots are too short to allow the two-fluid medium to be treated as a continuum the two rootsshould be omitted As for the two remaining roots one ofthem represents a pressure wave that transmited along theaxis z and the other represents a pressure wave transmits inthe opposite direction in accordance with the direction of theflow along the annulusThe real value of wave number can beused to determine thewave velocity cThewave velocity in thetwo-phase fluid can be determined by the following model
119888 =
1003816
1003816
1003816
1003816
(119908119877
+
(119896)) minus (119908119877
minus
(119896))
1003816
1003816
1003816
1003816
2
(47)
4 Solution of the United Model
Obtaining the analytical solution of themathematical modelsconcerned with flow pattern void fraction characteristicparameters and pressure drop gradient is generally impossi-ble for two-phase flow In this paper the Runge-Kuttamethod(R-K4) is used to discretize the theoretical model
We can obtain pressure temperature gas velocity drillingmud velocity and void fraction at different annulus depthsby R-K4 The solution of pressure drop gradient equation(41) can be seen as an initial-value problem of the ordinarydifferential equation
119889119901
119889119911
= 119865 (119911 119901)
119901 (119911
0) = 119875
0
(48)
With the initial value (z0p0) and the function F(z p)
(50)ndash(53) can be obtained by
119896
1= 119865 (119911
0 119901
0) (49)
119896
2= 119865(119911
0+
ℎ
2
119901
0+
ℎ
2
119896
1) (50)
119896
3= 119865(119911
0+
ℎ
2
119901
0+
ℎ
2
119896
2) (51)
119896
4= 119865 (119911
0+ ℎ 119901
0+ ℎ119896
3) (52)
where h is the step of depthThe pressure on the nod 119894 = 119894 +1can be obtained by
119901
1= 119901
0+ Δ119901 = 119901
0+
ℎ
6
(119896
1+ 2119896
2+ 2119896
3+ 119896
4)
(53)
In the present work the mathematical model and pres-sure wave velocity calculation model are solved by a person-ally compiled code on VBNET (Version 2010) The solutionprocedure for the wave velocity in the annulus is shown inFigure 1 At initial time the wellhead back pressure wellheadtemperature wellbore structure well depth gas and drillingmud properties and so forth are known On the node ithe pressure gradient temperature and the void fraction canbe obtained by adopting R-K4 Then the determinant (45)is calculated based on the calculated parameters Omittingthe two unreasonable roots the pressure wave velocity atdifferent depths of the annulus in MPD operations can besolved by (47)The process is repeated until the pressure wavevelocity of every position in the wellbore is obtained as shownin Figure 2
The developed model takes full consideration of theinterfacial interaction and the virtual mass force Owing tothe complex conditions of the annulus in MPD operationsmeasurement of wave velocity in the actual drilling processis very difficult In order to verify the united model thepredicted pressure waves are compared with the results ofprevious simulated experimental investigations presented forgas and drillingmud by Liu et al [45] in Figure 3(a) and by Liet al [22] in Figure 3(b) The lines represent the calculationresults and the points represent the experimental data
The comparisons reveal that the developed united modelfits well with the experimental data Thus the united modelcan be used to accurately predict wave velocity at differentwellhead back pressures and gas influx rates (the influx rateof gas in the bottomhole) in MPD operations
5 Analysis and Discussion
The drilling system described is an enclosed system Theschematic diagram of the gas influx process is illustrated inFigure 4 The drilling mud is pumped from surface storagedown the drill pipe Returns from the wellbore annulus travelback through surface processing where drilling solids areremoved to surface storage The key equipments include thefollowing
(i) The Rotating Control Device (RCD) The rotatingcontrol device provides the seal between atmosphereand wellbore while allowing pipe movement anddiverting returns flow In conjunction with the flowcontrol choke the RCD provides the ability to applyback pressure on the annulus during an MPD opera-tion
(ii) Choke The MPD choke manifold provides anadjustable choke system which is used to dynamicallycontrol the required BHP by means of applyingsurface BP
(iii) Coriolis Meter A Coriolis meter is used to accu-rately measure the mass flow rate of fluid exiting
8 Mathematical Problems in Engineering
Grid division
Start
Delete two roots
End
Initial conditions
Meet demand
i = i + 1
Solve eq (20) for T
Solve eq (15)ndash(17) for Zg
Judging flow pattern by eq (21) and eq (22)
Bubbly flow pattern Slug flow pattern
Solve determinant eq (45) for four roots
Calculate F(z p) by eq (48)
Calculate k1 minus k4Pi+1 = Pi + ΔPzi+1 = zi + h
zi+1 gt HNo
No
Yes
Yes
Solve eq (18) (19) for120588L
Solve eq ( 47) for c
Solve eq (14)120588G
Calculate 120588m 120601G f and CDCalculate 120588m 120601G f and CD CvmCvm
Figure 2 Solution procedure for wave velocity in MPD operations
the annulus The ability to measure return flowaccurately is essential for the applied back pressure
(iv) Pressure Sensor A pressure sensor is used to measuresurface back pressure on the wellhead
The gas and drilling mud flow rate measured by the Cori-olis meter and the back pressure measured by the pressuresensor are the initial data for annulus pressure calculationThe well used for calculation is a gas well in XinjiangUygur Autonomous Region Northwest China The wellborestructure well design parameters (depths and diameters)gas and drilling mud properties (density and viscosity) andoperational conditions of calculation well are displayed inTable 1
The drilling mud mixed with gas is taken as a two-phaseflow medium The propagation velocity of pressure wave inthe gas-drillingmud flow is calculated and discussed by usingthe established model and well parameters
51 Effect of Back Pressure on Wave Velocity Increase inapplied back pressure is the most common approach fordynamicwell controlThewellbore can be seen as an enclosed
Table 1 Parameters of calculation well
Type Property Value
Mud Dynamic viscosity (Pa sdot s) 0056Density (kgcm3) 1460
Gas Relative density 065Viscosity (Pa sdot s) 114 times 10minus5
StringElastic modulus of string (Pa) 207 times 1011
Poisson ratio of string 03Roughness (m) 154 times 10minus7
Surface condition Surface temperature (K) 298Atmosphere pressure (MPa) 0101
pressurized system The pressure at different depths of theannulus varied with the change of back pressure Accordingto the EOS of gas and drilling mud the influence of pressureon gas volume is much greater than that of drilling mudfor the greater compressibility of gas So the gas voidfraction changes with the variation of gas volume at thenearly constant flow rate of drilling mud at different annularpressures Meanwhile the pressure wave velocity is sensitive
Mathematical Problems in Engineering 9
5000 2 4 6 8 10
750
1000
1250
1500c
(ms
)
This paper p = 30MPa w = 50Hz = Re
120601G ()
CvmLiu et al [45] p = 30MPa
(a) Comparison with the experimental data of Liu
0 20 40 60 80 1000
40
80
120
160
200
c(m
s)
This paper p = 01MPa w = 50Hz = Re
120601G ()
CvmLi et al [22] p = 01MPa
(b) Comparison with the experimental data of Li
Figure 3 Experimental verification by comparison with previous experimental data
Drill pipe
Annulus
RCD
Formation
Mud pump
Drill collar
Coriolis meterChoke
Casing
Pressure sensors
120601 2159 mm times G105
1206011778mm (120601 78 mm) times G105
120601127 mm (120601 78 mm) times G105
z0 m
3600 m
4000 m
QL
Figure 4 Schematic diagram of the gas influx process
to the gas void fraction As a result when the back pressureat the wellhead is changed by adjusting the choke the wavevelocity in the two-phase drilling fluid and the distributionof void fraction at different depth of the annulus will divergeThe calculation results affected by the back pressure arepresented
Figures 5 and 6 show the distributions of void fractionand variations of wave velocity along the flow directionin the annulus when the back pressure at the wellhead is01MPa 08MPa 15MPa 25MPa 50MPa and 65MPa
respectively It can be seen that the void fraction significantlyincreases along the flow direction in the annulus Converselythe wave velocity shows a remarkable decrease tendency Forinstance BP = 15MPa at the wellhead the wave velocityis 9012ms and the void fraction is 0611 while in thebottomhole the wave velocity reaches 73142ms and voidfraction is reduced to 0059 However a sudden increase inwave velocity is observed in Figure 5 when the back pressureis 01MPa at the position close towellhead as the void fractionis further increased Moreover the influence of back pressure
10 Mathematical Problems in Engineering
10075500 254000
3000
2000
1000
0
BP = 01 MPaBP = 08 MPaBP = 15 MPa
BP = 25 MPa
BP = 65 MPaBP = 5 MPa
H(m
)
Q = 9m3h w = 50Hz = Re
120601G ()
Cvm
Figure 5 Void fraction distribution at different back pressures
on the void fraction is larger at the position close to thewellhead than that in the bottomhole
This can be explained from the viewpoints of mixturedensity and compressibility of two-phase fluid and the pres-sure drop along the flow direction in the wellbore In thelow void fraction range the gas phase is dispersed in theliquid as bubble so the wave velocity is influenced greatly bythe added gas phase According to the EOS if gas invadesinto the wellbore with a small amount in the bottomholethe density of the drilling mud has little variation while thecompressibility increases obviously which makes the wavevelocity decreased Then the two-phase drilling mud flowfrom the bottomhole to the wellhead along the annulus witha drop of pressure caused by potential energy change kineticenergy and frictional loss which leads to an increase in voidfractionWhen the void fraction is increased both the densityand the compressibility of the two-phase fluid change slightlyresulting in a flat decrease in wave velocity along the flowdirection With the further increasing in void fraction thebubbly flow turns into slug flow according to the flow patterntransient criteria of Orkiszewski which shows that the flowpattern is dependent on the void fraction Generally speak-ing the gas-drilling mud slug flow is composed of liquidand gas slugs In the preliminary slug flow the liquid slugsare much longer than the gas slugs so the wave velocity isdetermined primarily by the wave velocities in the liquidslugs which are hardly affected by the void fraction At theposition close to the wellhead the pressure of two-phase flowfluid can be reduced to a low value which approaches theback pressure The compressibility of gas will be improvedat the low pressure It results in significant increase in void
0 250 500 750 10004000
3000
2000
1000
0
H(m
)BP = 01 MPaBP = 08 MPaBP = 15 MPa
BP = 25 MPa
BP = 65 MPaBP = 5 MPa
c (ms)
Q = 9m3h w = 50Hz = ReCvm
Figure 6 Wave velocity variations at different back pressures
fraction As the void fraction increases greatly at the positionclose to the wellhead gas slugs becomemuch longer than theliquid slugs It is assumed that the actual wave velocity in slugflow happens to play a leading role when the void fractionincreases to some extent For the decrease in liquid holdupin the gas slug with the increasing in void fraction the wavevelocity in the gas slug is infinitely close to that in pure gaswhich is equivalent to an increase in wave velocity in the gasslug Meanwhile the gas content in liquid slug also increasesand results in a lower wave velocity in the liquid slug As aresult a slight increase in wave velocity appears
Figures 7 and 8 present the influence of back pressureon the void fraction and wave velocity with respect to theparameters of well depth As the back pressure increases thevoid fraction at different depth of the annulus is reducedgradually while the wave velocity in the two-phase flow tendsto increase Analytical results show that the increased backpressure is equivalent to be applied to the entire encloseddrilling fluid cyclical systemThe pressure transmits from thewellhead to the bottomhole therefore the annular pressurein the entire wellbore is increased According to the EOSthe density of gas increases and the compressibility of gasdecreases with the increasing of gas pressure So the lossof interphase momentum and energy exchange is reducedand the interphase momentum exchange is promoted Itcontributes to the increase in wave velocity with the increasein gas pressure In addition due to lower compressibility oftwo-phase flow medium under high pressure the increasetendency of pressure wave velocity and the decrease tendencyof void fraction are slowed down in the high back pressurerange
Mathematical Problems in Engineering 11
0
25
50
75
100
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
0 25 75 105
Q = 9m3h w = 50Hz = Re
120601G
()
Cvm
Figure 7 Effect of back pressure on the void fraction
52 Effect of Gas Influx Rate on Wave Velocity Figures 9and 10 graphically interpret the distributions of void fractionand variations of wave velocity along the flow direction inthe annulus When gas influx occurs in the bottomhole gasinvades into the wellbore and migrates from the bottomholeto the wellhead along the flow direction At a low gas influxrate it is extremely obvious that the void fraction and wavevelocity first slightly change in a comparatively smooth valuethen change sharply It is because of the rapid expansion ofgas volume with the decreasing in pressure near the wellheadthat the void fraction increases sharply and the wave velocitydecreases obviously at the same time But under the highbottomhole pressure (up to 50MPa) the compressibilityof the gas is low This results in a slight change in voidfraction and wave velocity at the position far away from thewellhead Since the compressible component increases withthe increase in the gas influx rate the compressibility of thegas and drilling mud two-phase fluid is improved So thevariations of void fraction and wave velocity become moreprominent Also the void fraction still shows an increasetendency that is steady first and then sharp It acts inaccordance with the variation of void fraction at a low gasinflux rate At a low gas influx rate if the void fraction atthe position close to the wellhead can not increase to a highextent the wave velocity always shows a decrease tendencyAt a high gas influx rate such as the 119876 = 8312m3h thewave velocity tends to increase because the void fraction inthe wellhead is increased to a high extent In conclusionthe wave pressure is sensitive to the void fraction and thevoid fraction is dominated by influx rate and pressure in theannulus especially the influx rate
0 25 75 1050
300
600
900
1200
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
c(m
s)
Q = 9m3h w = 50Hz = ReCvm
Figure 8 Effect of back pressure on the pressure wave velocity
With the influx of gas the mixing of gas and drillingmud occurs in the annulus and the corresponding interfacialtransfer of momentum and mass causes an increase in gasphase void fraction and a decrease in pressure wave velocityas shown in Figures 11 and 12 Within the range of low gasinflux rate the wave velocity decreases significantly It isbecause of this that the compressibility of the gas increasesremarkably and the medium appears to be of high elasticitythough the density of gas-drilling mud two-phase flowchanges slightly With the increase in the gas influx rate andcorresponding increase in the void fraction in the annulusthe compressibility of the two-phase unceasingly increaseswhich promotes the momentum and energy exchange inthe interface So the pressure wave continuously decreasesWhen the void fraction increases to some extent following theincrease in the gas influx rate the decrease in wave velocityin the liquid slug is gradually less than the increase in wavevelocity in the gas slug thus the decrease of wave velocityis slowed down for the growth of gas slug Especially in thewellhead a slight increase in wave velocity is observed
53 Effect of Virtual Mass Force on Wave Velocity Figure 13illustrates the effect of virtual mass force on the wave velocitywith the parameter of gas influx rate As shown in Figure 15under the same gas influx rate (0594m3h 1098m3h and1768m3h resp) the wave velocity curve of 119862vm = Rediverges from the curve of 119862vm = 0 at the position close tothewellhead whereas thewave velocity curve of the two typesis almost coincided below the position of 119867 = 500m Thisdivergence between the two types of curves is connected withthe fact that significance of influence of interphase virtual
12 Mathematical Problems in Engineering
0 25 50 75 1004000
3000
2000
1000
0
Q = 0058m3h Q = 1228m3hQ = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
H(m
)
BP = 01 MPaw = 50Hz = Re
120601G ()
Cvm
Figure 9 Void fraction distribution at different gas influx rates
mass forces increases together with the increase of the relativemotion in the interphase It is evident that the bottomholepressure is hundreds of times higher than the wellheadpressure As a result the void fraction at the position closeto the wellhead increases sharply meanwhile the interphasemomentum and energy exchanges are promoted The virtualmass force can be described as the transfer of momentumbetween the gas phase and drilling mud phase caused bythe relative motion in the interphase If the relative motionin the interphase is quite weak the value of virtual massforce will intend to approach 0 and the influence on thewave velocity can be ignored However if the relative motionis rather intense the effect of virtual mass force on thewave velocity should not be ignored Furthermore taking thevirtual mass force into account the dispersion characteristicof the pressure wave weakens obviously Compared with thepressure wave velocity calculated by ignoring the effect ofvirtual mass force the calculated pressure wave velocity witha consideration of the virtual mass force is lower Thereforeit is necessary to analyze the effect of the virtual mass forceon the wave velocity at the position close to the wellhead inMPD operations
54 Effect of Disturbance Angular Frequency onWave VelocityFigure 14 presents the pressure wave velocity in the annulusat different angular frequencies According to Figures 5 and9 the void fraction first slightly increases in the bottomholeand then sharply increases along the flow direction at theposition close to the wellhead At a fixed angular frequencythis phenomenon results in an overall decrease in the pressurewave velocity except for the section near the wellhead At the
0 500 1000 1500 20004000
3000
2000
1000
0
c (ms)
H(m
)Q = 0058m3h Q = 1228m3h
Q = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
BP = 01 MPaw = 50Hz = ReCvm
Figure 10 Wave velocity variations at different gas influx rates
section close to the wellhead an opposite change trend ofpressure wave velocity is observed for the transition of flowpattern from bubbly flow to slug flow due to the continuousincrease in void fraction along the flow direction In additionit can be clearly seen from the curves that the wave velocityincreases accompanying with the increase in the angularfrequency above the position of 119867 = 500m This propertyis not very distinct at the position below 119867 = 500m for thelow void fraction
Figure 15 shows the effect of frequency on the wavevelocity in the gas-drilling mud flow (119862vm = 0)The curves ofthe wave velocity of different positions reveal that the prop-agation velocity of pressure disturbances increases togetherwith the growth of the angular frequency (0 lt 119908 lt 500Hz)It proves that the pressure wave has an obvious dispersioncharacteristic in the two-phase flow As the angular frequencyincreases in the range of less than 500Hz there is sufficienttime to carry out the exchange of energy and momentumbetween two phases It achieves a state of mechanical andthermodynamic equilibrium between the two phases Sothe wave velocity increases gradually with the increase inthe angular frequency at different depths of the annulus Itis considered that the wave velocity is mainly affected bythe interphase mechanical and thermodynamic equilibriumat low anglular frequencies When the angular frequencyreaches the value of 119908 = 500Hz the pressure wave velocityachieves a constant value and remains on this level regardlessof the further growth in angular frequency 119908 With theincrease in angular frequency there is not enough time forenergy and momentum exchange between the gas-drillingmud two phases to reach themechanical and thermodynamic
Mathematical Problems in Engineering 13
0
25
50
75
100
125
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
Q (m3h)
120601G
()
0 25 5 75 10
BP = 01 MPaw = 50Hz = ReCvm
Figure 11 Effect of gas influx rate on the void fraction
equilibrium state and thus the wave dispersion does notexist At a high angular frequency the wave velocity is mainlydominated by the mechanical and thermal nonequilibriumin the flow and keeps almost unchanged when the angularfrequency is further increased This is consistent with theinfluences of the angular frequency in the horizontal pipe[28] Moreover the effect of virtual mass force is also shownin this figure (119862vm = Re) It is observed that the wave velocityis significantly reduced when the virtual mass force is takeninto consideration in the calculation of the wave velocity atthe position close to the wellhead such as 119867 = 0m It canbe explained by the reduction of dispersion characteristic ofa pressure wave Along the wellbore from the wellhead to thebottomhole the distinction gradually decreases
6 Conclusions
With full consideration of the important factors such asvirtual mass force drag force gas void fraction pressuretemperature and angular frequency a united wave velocitymodel has been proposed based on pressure drop gradientequations in MPD operations gas-liquid two-fluid modelthe gas-drilling mud equations of state and small pertur-bation theory Solved by the fourth-order explicit Runge-Kutta method the model is used to predict wave velocityfor different back pressures and gas influx rates in MPDoperations The main conclusions can be summarized asfollows
(1) With the introduction of virtual mass force and dragforce the united model agrees well with the previousexperimental data The united model can be used to
0
280
560
840
1120
1400
Q (m3h)0 25 5 75 10
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
c(m
s)
BP = 01 MPaw = 50Hz = ReCvm
Figure 12 Effect of gas influx rate on the pressure wave velocity
accurately calculate the wave velocity in the annulusThe application of the model will be beneficial tofurther study the wave velocity at different gas influxrates and back pressures in MPD operations reducenonproductive times and provide a reference for thedrilling operations in extremely narrow porefracturewindows existing conditions
(2) The wave velocity and void fraction have differentvalues with respect to well depth In the annulusthe drop of pressure causes an increase in the voidfraction along the flow direction The void fractionincreases first slightly and then sharply Correspond-ingly the wave velocity first gradually decreases in thbubbly flow and preliminary slug flow and then thewave velocity slightly increases accompanying withthe increase in the relative length ratio of gas slugto the liquid slug for the continuous increase in voidfraction at the position close to the wellhead Theminimum wave velocity appears in the long gas slugflow
(3) When the back pressure in the MPD operationsincreases the void fraction at different depths of theannulus is reduced gradually while the wave velocityin the two-phase flow tends to increase Moreoverthe influence of back pressure on the void fraction isgreater at a position close to the wellhead than that inthe bottomhole because of the great decrease in thepressure from the bottomhole to the wellhead Alsothe effect of back pressure on void fraction and wavevelocity is decreased in the high back pressure range
14 Mathematical Problems in Engineering
0 300 600 900 12004000
3000
2000
1000
225
180
135
90
45
0
BP = 01 MPaw = 50Hz
Q = 0594m3h = ReQ = 0594m3h = 0
Q = 1098m3h = ReQ = 1768m3h = ReQ = 1098m3h = 0
Q = 1768m3h = 0
c (ms)
H(m
)
CvmCvm
Cvm
Cvm
Cvm
Cvm
Figure 13 Effect of virtual mass force on the pressure wave velocity
(4) The wave velocity is sensitive to the void fraction butthe void fraction is dominated by gas influx rate andpressure in the annulus especially the gas influx rateSince the compressibility of the gas and drilling mudtwo-phase fluid is improved with the increase in theinflux rate the void fraction increases greatly andthe wave velocity decreases significantly within thelow gas influx rate range When the void fraction isincreased to some extent following the increase in thegas influx rate the decrease of wave velocity is slowedfor the growth of gas slug Especially at the wellheada slight increase in wave velocity is observed at a highgas influx rate for the sharp increase in void fraction
(5) It is necessary to analyze the effect of the virtualmass force on the wave velocity in MPD operationsEspecially at the position close to the wellhead theeffect of virtual mass force is more prominent for theintense phase interactions Taking the virtual massforce into account the dispersion characteristic of thepressure wave weakens obviously Compared with theresults calculated by ignoring the effect of virtualmassforce the calculated pressure wave velocity with aconsideration of the virtual mass force is lower
(6) The effect of angular frequency on the propagationvelocity of pressure wave appears at a low angularfrequency The propagation velocity of pressure dis-turbances increases together with the growth of theangular frequency (0 lt 119908 lt 500Hz) When theangular frequency reaches the value of 119908 = 500Hzthe pressure wave velocity achieves a constant value
0 240 480 720 960 12004000
3000
2000
1000
560
420
280
140
0
Q = 0727m3h BP = 01 MPa
w = 50Hz = Rew = 50Hz = 0w = 300Hz = 0
w = 800Hz = 0w = 1600Hz = 0w = 5000Hz = 0
c (ms)
H(m
)CvmCvmCvm
CvmCvmCvm
Figure 14 Wave velocity variations at different angular frequencies
and remains on this level regardless of the furthergrowth in angular frequency 119908
Nomenclature
A Annulus effective cross area (m2)c119866 Wave velocity in gas (ms)
c119871 Wave velocity in Drilling mud (ms)
C119863 The coefficient of drag force
D Annulus effective diameter (m)D119894 Diameter of the inner pipe (m)
D119900 Diameter of the outer pipe (m)
f119866 Shear stresses coefficient of gas interface
f119871 Shear stresses coefficient of drilling mud
interfaceF119911 Pressure applied to the two-phase flow (N)
g Acceleration due to gravity (m2s)g119888 Conversion factor
g119879 Geothermal temperature gradient (Km)
h One discrete length of annulus (m)k119890 Annulus effective roughness (m)
k119900 Roughness of outer pipe (m)
k119894 Roughness of inner pipe (m)
H The depth of annulus (m)L119902 Air bubbles length of slug (m)
119872Gi Momentum transfer in gas interface (Nm3)119872Li Momentum transfer in liquid interface (Nm3)119872
ndLi The none-drag-force (virtual mass force)
(Nm3)119872
119889
Li The momentum transfer term caused by thedrag force (Nm3)
Mathematical Problems in Engineering 15
100 1000 100000
450
900
1350
1800
0w (Hz)
0 m = 0 0 m = Re1000 m = 0
2000 m = 0
4000 m = 0
1000 m = Re2000 m = Re4000 m = Re
c(m
s)
Q = 0727m3h BP = 01 MPaCvmCvm
CvmCvmCvm
CvmCvmCvm
Figure 15 Effect of angular frequency on wave velocity
N Parameter combiningThompsonenergy effects
119901Gi Gas interface pressure (MPa)119901Li Liquid interface pressure (MPa)p Pressure (MPa)p0 Initial pressure of annulus (MPa)
p119888 Critical pressure (kPa)
p119871 Pressure of drilling mud (MPa)
p119903 Reduced pressure
q119898 Volumetric flow rate of gas and drilling
mud (m3s)q119866 Velocity of the gas (m3s)
q119871 Velocity of the drilling mud (m3s)
Q119871 Drilling mud rate within the annulus
(m3h)r Average diameter of the bubble (m)R Constant of EOS (JKgK)R119902 Air bubbles radius of slug (m)
R119887 The bubble diameter (m)
T Temperature (K)T119888 Critical temperature (K)
T119903 Reduced temperature
T119890119894 Undisturbed formation temperature at a
depth (K)T119890119887ℎ
Undisturbed formation temperature ofwellhead (K)
T119891119887ℎ
Undisturbed formation temperature atthe bottomhole (K)
v119898 Gas and drilling mud flow velocity
(ms)
v119866 Gas flow velocity (ms)
v119904 Slip velocity (ms)
v119904119892 Superficial gas velocity (ms)
v119871 Drilling mud flow velocity (ms)
W119898 Mass flow velocity (kgm3)
Z119866 Gas deviation factor
z0 Initial calculation height of annulus (m)
z119887ℎ Total well depth from surface (m)
z Variable of distance from surface (m)
Greek Letters
120588
119866 Gas density (kgm3)
120591
119891119903
Li Shear stresses of drilling mud noninterfacial(Nm2)
120591
119891119903
119871 Shear stresses of drilling mud interface (Nm2)
120591
Re119871 Reynolds stress of drilling mud (Nm2)
120591
Re119871 Reynolds stress of drilling mud interface
(Nm2)120591
Re119866 Reynolds stress of gas interface (Nm2)
120591
119891119903
119866 Shear stresses of gas interface (Nm2)
120572VM Acceleration due to virtual mass forcecoefficient (ms2)
120591
119871 Shear stresses of drilling mud along wall
(Nm2)120591
119866 Shear stresses of gas along wall (Nm2)
120591
Re119866 Reynolds stress of gas (Nm2)
120591
119891 Frictional pressure gradient (Pam)
120591
119908 Frictional pressure coefficient
120588
119898 Gas and drilling density (kgm3)
120588
0 Density under standard atmospheric pressure
(kgm3)120588
119871 Drilling mud density (kgm3)
120590
119904 Surface tension (Nm2)
Φ
119866 Gas void fraction
Φ
119871 Drilling mud holdup
120588
119903 Reduced density
120583
119871 Viscosity of drilling mud (Pa sdot s)
Subscripts
BHP Bottomhole pressureBP Back pressureEOS Equations of stateMPD Managed pressure drillingRCD The rotating control deviceR-K4 The fourth-order explicit Runge-Kutta
Subscripts of Graph
BP Back pressure (MPa)c Wave velocity in gas and drilling mud
two-phase flow (ms)119862vm The coefficient of virtual mass force
16 Mathematical Problems in Engineering
H Depth of annulus (m)Q Gas influx rate at the bottomhole (m3h)Re The value of virtual mass force
coefficient according to (28) or (34)119908 Angle frequency (Hz)Φ
119866 Void fraction ()
Acknowledgment
Research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 and 51274170)Without their support this work would not have beenpossible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of airinjection in annuli to manage pressure during cementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling conferenceand Exhibition San Diego Calif USA March 2012
[4] W A Bacon Consideration of compressibility effects for applied-back-pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Tex USA 2011
[5] J Choe C Hans and J Wold ldquoA modified two-phase well-control model and its computer applications as a training andeducational toolrdquo SPE Computer Applications vol 2 pp 14ndash201997
[6] J A Tarvin IWalton and PWand ldquoAnalysis of a gas kick takenin a deep well drilled with oil-based mudrdquo in Proceedings ofthe Annual Technical Conference and Exhibition of the Societyof Petroleum Engineers pp 255ndash264 Dallas Tex USA 1991
[7] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[8] J Zhou and G Nygaard ldquoAutomatic model-based controlscheme for stabilizing pressure during dual-gradient drillingrdquoJournal of Process Control vol 21 no 8 pp 1138ndash1147 2011
[9] A Ochoa O Acevedo L Nieto J E Lambarria and H PerezldquoSuccessful application of MPD (Managed Pressure Drilling)for prevention controland detection of borehole ballooning intight gas reservoir in Cuervito Field Mexicordquo in Proceedings ofthe Canadian Unconventional Resources Conference (CURC rsquo11)pp 172ndash185 November 2011
[10] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[11] R Ashena and J Moghadasi ldquoBottom hole pressure estimationusing evolved neural networks by real coded ant colony opti-mization and genetic algorithmrdquo Journal of Petroleum Scienceand Engineering vol 77 no 3-4 pp 375ndash385 2011
[12] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvanced Materials Research vol 468-471 pp 1736ndash1742 2012
[13] I Zubizarreta ldquoPore pressure evolution core damage and trip-ping out schedulesf a computational fluid dynamics approachrdquoin Proceedings of the SPEIADC Drilling Conference and Exhibi-tion Amsterdam The Netherlands March 2013
[14] C H Whitson ldquoCyclic shut-in eliminates liquid-loading in gaswellsrdquo in Proceedings of the SPEEAGE European Unconven-tional Resources Conference and Exhibition Vienna AustriaMarch 2012
[15] G M de Oliveira A T Franco C O R Negrao A LMartins and R A Silva ldquoModeling and validation of pressurepropagation in drilling fluids pumped into a closed wellrdquoJournal of Petroleum Science and Engineering vol 103 pp 61ndash71 2012
[16] Y Sato and H Kanki ldquoFormulas for compression wave andoscillating flow in circular piperdquo Applied Acoustics vol 69 no1 pp 1ndash11 2008
[17] J Xie B Yu X Zhang Q Shao and X Song ldquoNumericalsimulation of gas-liquid-solid three-phase flow in deep wellsrdquoAdvances in Mechanical Engineering vol 2013 Article ID951298 10 pages 2013
[18] W Bacon A Tong O Gabaldon C Sugden and P V Surya-narayana ldquoAn improved dynamic well control response to a gasinflux in managed pressure drilling operationsrdquo in Proceedingsof the IADCSPE Drilling Conference and Exhibition San DiegoCalifornia USA March 2012
[19] J I Hage and D ter Avest ldquoBorehole acoustics applied to kickdetectionrdquo Journal of Petroleum Science and Engineering vol 12no 2 pp 157ndash166 1994
[20] A M R Hopkins Fluid dynamics and surface pressure fluctua-tions of two-dimensional turbulent boundary layers over denselydistributed surface roughness [PhD thesis] Virginia PolytechnicInstitute and State University Blacksburg Va USA 2010
[21] Z S Fu ldquoThe gas cut detect technology during drilling in SovietUnionrdquo Petroleum Drilling Techniques vol 18 no 1 pp 19ndash211990
[22] X F Li C X Guan and X X Sui ldquoThe theory of gas influxdetection of pressure wave and its applicationrdquo Acta PetroleiSinica vol 18 no 3 pp 128ndash133 1997
[23] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[24] A E Ruggles R T Lahey Jr D A Drew and H A ScartonldquoInvestigation of the propagation of pressure perturbations inbubbly airwater flowsrdquo Journal of Heat Transfer vol 110 no 2pp 494ndash499 1988
[25] H J W M Legius H E A Van Den Akker and T NarumoldquoMeasurements on wave propagation and bubble and slugvelocities in cocurrent upward two-phase flowrdquo ExperimentalThermal and Fluid Science vol 15 no 3 pp 267ndash278 1997
[26] K Miyazaki and I Nakajima ldquoPropagation of pressure wavein NItrogen-Mercury two-phase systemrdquo Journal of NuclearScience and Technology vol 9 no 5 pp 315ndash318 1972
[27] B Bai L Guo and X Chen ldquoPressure fluctuation for air-watertwo-phase flowrdquo Journal of Hydrodynamics vol 18 no 4 pp476ndash482 2003
[28] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
Mathematical Problems in Engineering 17
[29] R F Tangren C H Dodge and H S Seifert ldquoCompressibilityeffects in two-phase flowrdquo Journal of Applied Physics vol 20 no7 pp 637ndash645 1949
[30] G B Wallis One Dimensional Two Phase Flow McGraw HillNew York NY USA 1969
[31] D L Nguyen E R F Winter and M Greiner ldquoSonic velocityin two-phase systemsrdquo International Journal ofMultiphase Flowvol 7 no 3 pp 311ndash320 1981
[32] R C Mecredy and L J Hamilton ldquoThe effects of nonequilib-rium heat mass and momentum transfer on two-phase soundspeedrdquo International Journal of Heat and Mass Transfer vol 15no 1 pp 61ndash72 1972
[33] K H Ardron and R B Duffey ldquoAcoustic wave propagationin a flowing liquid-vapour mixturerdquo International Journal ofMultiphase Flow vol 4 no 3 pp 303ndash322 1978
[34] J Xu and T Chen ldquoAcoustic wave prediction in flowing steam-water two-phase mixturerdquo International Journal of Heat andMass Transfer vol 43 no 7 pp 1079ndash1088 2000
[35] N ChungW Lin B Pei and Y Hsu ldquoModel for sound velocityin a two-phase air-water bubbly flowrdquo Nuclear Technology vol99 no 1 pp 80ndash89 1992
[36] X Xu and J Gong ldquoA unitedmodel for predicting pressurewavespeeds in oil and gas two-phase pipeflowsrdquo Journal of PetroleumScience and Engineering vol 60 no 3-4 pp 150ndash160 2008
[37] W Kuczynski ldquoCharacterization of pressure-wave propagationduring the condensation of R404A and R134a refrigerantsin pipe mini-channels that undergo periodic hydrodynamicdisturbancerdquo International Journal of Heat and Fluid Flow vol40 pp 135ndash150 2013
[38] W Kuczynski ldquoModeling of the propagation of a pressure waveduring the condensation process of R134a refrigerant in a pipeminichannel under the periodic conditions of hydrodynamicdisturbancesrdquo International Journal of Heat and Mass Transfervol 56 pp 715ndash723 2013
[39] Y Li C Li E Chen and Y Ying ldquoPressure wave propaga-tion characteristics in a two-phase flow pipeline for liquid-propellant rocketrdquoAerospace Science and Technology vol 15 no6 pp 453ndash464 2011
[40] T Kanagawa M Watanabe T Yano and S Fujikawa ldquoNon-linear wave equations for pressure wave propagation in liquidscontaining gas bubbles (comparison between two-fluid modeland mixture model)rdquo Journal of Fluid Science and Technologyvol 6 no 6 pp 836ndash849 2011
[41] EChen Y Li andYYing ldquoNumerical investigation onpressurewave propagation speed of gas-liquid two-phase flow in pumppipelinerdquo Journal of Aerospace Power vol 25 no 4 pp 754ndash7602010
[42] F Li and L He ldquoInvestigation on pressure wave propagationcharacteristics in slug flow field with increasing gas flow raterdquoPetroleum Geology and Recovery Efficiency vol 11 no 5 pp 78ndash80 2004
[43] C SMartin andM Padmanabhan ldquoPressure pulse propagationin two-component slug flowrdquo Journal of Fluids Engineering vol101 no 1 pp 44ndash52 1979
[44] R E Henry ldquoPressure wave propagation in two-phase mix-turesrdquo Chemical Engineering Progress vol 66 no 1 pp 1ndash101969
[45] X Liu B Li and Y Yue ldquoTransmission behavior of mud-pressure pulse alongwell borerdquo Journal ofHydrodynamics B vol19 no 2 pp 236ndash240 2007
[46] XWang and J Zhang ldquoThe research of pressure wave Pulsationin mud pulse transmittingrdquo Journal of Chongqing University ofScience and Technology vol 14 no 2 pp 55ndash58 2012
[47] J-W Park D A Drew and R T Lahey Jr ldquoThe analysisof void wave propagation in adiabatic monodispersed bubblytwo-phase flows using an ensemble-averaged two-fluid modelrdquoInternational Journal ofMultiphase Flow vol 24 no 7 pp 1205ndash1244 1998
[48] G Arnold Entropy and objectivity as constraints upons constitu-tive equations for two-fluid modeling of multiphase flow [PhDthesis] Rensselaer Polytechnic Institute New York NY USA1998
[49] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[50] L Yarborough and K R Hall ldquoHow to solve equation of statefor z-factorsrdquoOil and Gas Journal vol 72 no 7 pp 86ndash88 1974
[51] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86 pp 127ndash136 2012
[52] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[53] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[54] M Ishii and K Mishima ldquoTwo-fluid model and hydrodynamicconstitutive relationsrdquo Nuclear Engineering and Design vol 82no 2-3 pp 107ndash126 1984
[55] M J Sanchez Comparison of correlations for predicting pressurelosses in vertical multiphase annular flow [MS thesis] Univer-sity of Tulsa Tulsa Okla USA 1972
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
119872
7= 120601
119866(120588
119866+ 119888vm120588119871) 119908 minus 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904+
4
119863
119891
119866119908120588
119866V119866)
119872
8= minus119888vm120601119866120588119871119908 + 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904)
119872
9= 120588
119871V2119904119896 (120601
119871119888
119901minus 2119888
119903minus 119888
1198982)
119872
10= minus119896(120601
119871+ 119888
119903120601
119866
V2119904
119888
2
119871
)
119872
11= 120588
119871[120601
119871+ 120601
119866119888vm] 119908 minus 119894 (
3
4
119888
119863
119903
120588
119871120601
119866V119904+
4
119863
119891
119871120588
119871V119871)
(46)
where 1198881198982= 01 119888
119901= 025 119888
119894= 03 and 119888
119903= 02
By solving the dispersion equation indicated previouslyfour roots can be obtained Because the wavelengths asso-ciated with two of the roots are too short to allow the two-fluid medium to be treated as a continuum the two rootsshould be omitted As for the two remaining roots one ofthem represents a pressure wave that transmited along theaxis z and the other represents a pressure wave transmits inthe opposite direction in accordance with the direction of theflow along the annulusThe real value of wave number can beused to determine thewave velocity cThewave velocity in thetwo-phase fluid can be determined by the following model
119888 =
1003816
1003816
1003816
1003816
(119908119877
+
(119896)) minus (119908119877
minus
(119896))
1003816
1003816
1003816
1003816
2
(47)
4 Solution of the United Model
Obtaining the analytical solution of themathematical modelsconcerned with flow pattern void fraction characteristicparameters and pressure drop gradient is generally impossi-ble for two-phase flow In this paper the Runge-Kuttamethod(R-K4) is used to discretize the theoretical model
We can obtain pressure temperature gas velocity drillingmud velocity and void fraction at different annulus depthsby R-K4 The solution of pressure drop gradient equation(41) can be seen as an initial-value problem of the ordinarydifferential equation
119889119901
119889119911
= 119865 (119911 119901)
119901 (119911
0) = 119875
0
(48)
With the initial value (z0p0) and the function F(z p)
(50)ndash(53) can be obtained by
119896
1= 119865 (119911
0 119901
0) (49)
119896
2= 119865(119911
0+
ℎ
2
119901
0+
ℎ
2
119896
1) (50)
119896
3= 119865(119911
0+
ℎ
2
119901
0+
ℎ
2
119896
2) (51)
119896
4= 119865 (119911
0+ ℎ 119901
0+ ℎ119896
3) (52)
where h is the step of depthThe pressure on the nod 119894 = 119894 +1can be obtained by
119901
1= 119901
0+ Δ119901 = 119901
0+
ℎ
6
(119896
1+ 2119896
2+ 2119896
3+ 119896
4)
(53)
In the present work the mathematical model and pres-sure wave velocity calculation model are solved by a person-ally compiled code on VBNET (Version 2010) The solutionprocedure for the wave velocity in the annulus is shown inFigure 1 At initial time the wellhead back pressure wellheadtemperature wellbore structure well depth gas and drillingmud properties and so forth are known On the node ithe pressure gradient temperature and the void fraction canbe obtained by adopting R-K4 Then the determinant (45)is calculated based on the calculated parameters Omittingthe two unreasonable roots the pressure wave velocity atdifferent depths of the annulus in MPD operations can besolved by (47)The process is repeated until the pressure wavevelocity of every position in the wellbore is obtained as shownin Figure 2
The developed model takes full consideration of theinterfacial interaction and the virtual mass force Owing tothe complex conditions of the annulus in MPD operationsmeasurement of wave velocity in the actual drilling processis very difficult In order to verify the united model thepredicted pressure waves are compared with the results ofprevious simulated experimental investigations presented forgas and drillingmud by Liu et al [45] in Figure 3(a) and by Liet al [22] in Figure 3(b) The lines represent the calculationresults and the points represent the experimental data
The comparisons reveal that the developed united modelfits well with the experimental data Thus the united modelcan be used to accurately predict wave velocity at differentwellhead back pressures and gas influx rates (the influx rateof gas in the bottomhole) in MPD operations
5 Analysis and Discussion
The drilling system described is an enclosed system Theschematic diagram of the gas influx process is illustrated inFigure 4 The drilling mud is pumped from surface storagedown the drill pipe Returns from the wellbore annulus travelback through surface processing where drilling solids areremoved to surface storage The key equipments include thefollowing
(i) The Rotating Control Device (RCD) The rotatingcontrol device provides the seal between atmosphereand wellbore while allowing pipe movement anddiverting returns flow In conjunction with the flowcontrol choke the RCD provides the ability to applyback pressure on the annulus during an MPD opera-tion
(ii) Choke The MPD choke manifold provides anadjustable choke system which is used to dynamicallycontrol the required BHP by means of applyingsurface BP
(iii) Coriolis Meter A Coriolis meter is used to accu-rately measure the mass flow rate of fluid exiting
8 Mathematical Problems in Engineering
Grid division
Start
Delete two roots
End
Initial conditions
Meet demand
i = i + 1
Solve eq (20) for T
Solve eq (15)ndash(17) for Zg
Judging flow pattern by eq (21) and eq (22)
Bubbly flow pattern Slug flow pattern
Solve determinant eq (45) for four roots
Calculate F(z p) by eq (48)
Calculate k1 minus k4Pi+1 = Pi + ΔPzi+1 = zi + h
zi+1 gt HNo
No
Yes
Yes
Solve eq (18) (19) for120588L
Solve eq ( 47) for c
Solve eq (14)120588G
Calculate 120588m 120601G f and CDCalculate 120588m 120601G f and CD CvmCvm
Figure 2 Solution procedure for wave velocity in MPD operations
the annulus The ability to measure return flowaccurately is essential for the applied back pressure
(iv) Pressure Sensor A pressure sensor is used to measuresurface back pressure on the wellhead
The gas and drilling mud flow rate measured by the Cori-olis meter and the back pressure measured by the pressuresensor are the initial data for annulus pressure calculationThe well used for calculation is a gas well in XinjiangUygur Autonomous Region Northwest China The wellborestructure well design parameters (depths and diameters)gas and drilling mud properties (density and viscosity) andoperational conditions of calculation well are displayed inTable 1
The drilling mud mixed with gas is taken as a two-phaseflow medium The propagation velocity of pressure wave inthe gas-drillingmud flow is calculated and discussed by usingthe established model and well parameters
51 Effect of Back Pressure on Wave Velocity Increase inapplied back pressure is the most common approach fordynamicwell controlThewellbore can be seen as an enclosed
Table 1 Parameters of calculation well
Type Property Value
Mud Dynamic viscosity (Pa sdot s) 0056Density (kgcm3) 1460
Gas Relative density 065Viscosity (Pa sdot s) 114 times 10minus5
StringElastic modulus of string (Pa) 207 times 1011
Poisson ratio of string 03Roughness (m) 154 times 10minus7
Surface condition Surface temperature (K) 298Atmosphere pressure (MPa) 0101
pressurized system The pressure at different depths of theannulus varied with the change of back pressure Accordingto the EOS of gas and drilling mud the influence of pressureon gas volume is much greater than that of drilling mudfor the greater compressibility of gas So the gas voidfraction changes with the variation of gas volume at thenearly constant flow rate of drilling mud at different annularpressures Meanwhile the pressure wave velocity is sensitive
Mathematical Problems in Engineering 9
5000 2 4 6 8 10
750
1000
1250
1500c
(ms
)
This paper p = 30MPa w = 50Hz = Re
120601G ()
CvmLiu et al [45] p = 30MPa
(a) Comparison with the experimental data of Liu
0 20 40 60 80 1000
40
80
120
160
200
c(m
s)
This paper p = 01MPa w = 50Hz = Re
120601G ()
CvmLi et al [22] p = 01MPa
(b) Comparison with the experimental data of Li
Figure 3 Experimental verification by comparison with previous experimental data
Drill pipe
Annulus
RCD
Formation
Mud pump
Drill collar
Coriolis meterChoke
Casing
Pressure sensors
120601 2159 mm times G105
1206011778mm (120601 78 mm) times G105
120601127 mm (120601 78 mm) times G105
z0 m
3600 m
4000 m
QL
Figure 4 Schematic diagram of the gas influx process
to the gas void fraction As a result when the back pressureat the wellhead is changed by adjusting the choke the wavevelocity in the two-phase drilling fluid and the distributionof void fraction at different depth of the annulus will divergeThe calculation results affected by the back pressure arepresented
Figures 5 and 6 show the distributions of void fractionand variations of wave velocity along the flow directionin the annulus when the back pressure at the wellhead is01MPa 08MPa 15MPa 25MPa 50MPa and 65MPa
respectively It can be seen that the void fraction significantlyincreases along the flow direction in the annulus Converselythe wave velocity shows a remarkable decrease tendency Forinstance BP = 15MPa at the wellhead the wave velocityis 9012ms and the void fraction is 0611 while in thebottomhole the wave velocity reaches 73142ms and voidfraction is reduced to 0059 However a sudden increase inwave velocity is observed in Figure 5 when the back pressureis 01MPa at the position close towellhead as the void fractionis further increased Moreover the influence of back pressure
10 Mathematical Problems in Engineering
10075500 254000
3000
2000
1000
0
BP = 01 MPaBP = 08 MPaBP = 15 MPa
BP = 25 MPa
BP = 65 MPaBP = 5 MPa
H(m
)
Q = 9m3h w = 50Hz = Re
120601G ()
Cvm
Figure 5 Void fraction distribution at different back pressures
on the void fraction is larger at the position close to thewellhead than that in the bottomhole
This can be explained from the viewpoints of mixturedensity and compressibility of two-phase fluid and the pres-sure drop along the flow direction in the wellbore In thelow void fraction range the gas phase is dispersed in theliquid as bubble so the wave velocity is influenced greatly bythe added gas phase According to the EOS if gas invadesinto the wellbore with a small amount in the bottomholethe density of the drilling mud has little variation while thecompressibility increases obviously which makes the wavevelocity decreased Then the two-phase drilling mud flowfrom the bottomhole to the wellhead along the annulus witha drop of pressure caused by potential energy change kineticenergy and frictional loss which leads to an increase in voidfractionWhen the void fraction is increased both the densityand the compressibility of the two-phase fluid change slightlyresulting in a flat decrease in wave velocity along the flowdirection With the further increasing in void fraction thebubbly flow turns into slug flow according to the flow patterntransient criteria of Orkiszewski which shows that the flowpattern is dependent on the void fraction Generally speak-ing the gas-drilling mud slug flow is composed of liquidand gas slugs In the preliminary slug flow the liquid slugsare much longer than the gas slugs so the wave velocity isdetermined primarily by the wave velocities in the liquidslugs which are hardly affected by the void fraction At theposition close to the wellhead the pressure of two-phase flowfluid can be reduced to a low value which approaches theback pressure The compressibility of gas will be improvedat the low pressure It results in significant increase in void
0 250 500 750 10004000
3000
2000
1000
0
H(m
)BP = 01 MPaBP = 08 MPaBP = 15 MPa
BP = 25 MPa
BP = 65 MPaBP = 5 MPa
c (ms)
Q = 9m3h w = 50Hz = ReCvm
Figure 6 Wave velocity variations at different back pressures
fraction As the void fraction increases greatly at the positionclose to the wellhead gas slugs becomemuch longer than theliquid slugs It is assumed that the actual wave velocity in slugflow happens to play a leading role when the void fractionincreases to some extent For the decrease in liquid holdupin the gas slug with the increasing in void fraction the wavevelocity in the gas slug is infinitely close to that in pure gaswhich is equivalent to an increase in wave velocity in the gasslug Meanwhile the gas content in liquid slug also increasesand results in a lower wave velocity in the liquid slug As aresult a slight increase in wave velocity appears
Figures 7 and 8 present the influence of back pressureon the void fraction and wave velocity with respect to theparameters of well depth As the back pressure increases thevoid fraction at different depth of the annulus is reducedgradually while the wave velocity in the two-phase flow tendsto increase Analytical results show that the increased backpressure is equivalent to be applied to the entire encloseddrilling fluid cyclical systemThe pressure transmits from thewellhead to the bottomhole therefore the annular pressurein the entire wellbore is increased According to the EOSthe density of gas increases and the compressibility of gasdecreases with the increasing of gas pressure So the lossof interphase momentum and energy exchange is reducedand the interphase momentum exchange is promoted Itcontributes to the increase in wave velocity with the increasein gas pressure In addition due to lower compressibility oftwo-phase flow medium under high pressure the increasetendency of pressure wave velocity and the decrease tendencyof void fraction are slowed down in the high back pressurerange
Mathematical Problems in Engineering 11
0
25
50
75
100
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
0 25 75 105
Q = 9m3h w = 50Hz = Re
120601G
()
Cvm
Figure 7 Effect of back pressure on the void fraction
52 Effect of Gas Influx Rate on Wave Velocity Figures 9and 10 graphically interpret the distributions of void fractionand variations of wave velocity along the flow direction inthe annulus When gas influx occurs in the bottomhole gasinvades into the wellbore and migrates from the bottomholeto the wellhead along the flow direction At a low gas influxrate it is extremely obvious that the void fraction and wavevelocity first slightly change in a comparatively smooth valuethen change sharply It is because of the rapid expansion ofgas volume with the decreasing in pressure near the wellheadthat the void fraction increases sharply and the wave velocitydecreases obviously at the same time But under the highbottomhole pressure (up to 50MPa) the compressibilityof the gas is low This results in a slight change in voidfraction and wave velocity at the position far away from thewellhead Since the compressible component increases withthe increase in the gas influx rate the compressibility of thegas and drilling mud two-phase fluid is improved So thevariations of void fraction and wave velocity become moreprominent Also the void fraction still shows an increasetendency that is steady first and then sharp It acts inaccordance with the variation of void fraction at a low gasinflux rate At a low gas influx rate if the void fraction atthe position close to the wellhead can not increase to a highextent the wave velocity always shows a decrease tendencyAt a high gas influx rate such as the 119876 = 8312m3h thewave velocity tends to increase because the void fraction inthe wellhead is increased to a high extent In conclusionthe wave pressure is sensitive to the void fraction and thevoid fraction is dominated by influx rate and pressure in theannulus especially the influx rate
0 25 75 1050
300
600
900
1200
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
c(m
s)
Q = 9m3h w = 50Hz = ReCvm
Figure 8 Effect of back pressure on the pressure wave velocity
With the influx of gas the mixing of gas and drillingmud occurs in the annulus and the corresponding interfacialtransfer of momentum and mass causes an increase in gasphase void fraction and a decrease in pressure wave velocityas shown in Figures 11 and 12 Within the range of low gasinflux rate the wave velocity decreases significantly It isbecause of this that the compressibility of the gas increasesremarkably and the medium appears to be of high elasticitythough the density of gas-drilling mud two-phase flowchanges slightly With the increase in the gas influx rate andcorresponding increase in the void fraction in the annulusthe compressibility of the two-phase unceasingly increaseswhich promotes the momentum and energy exchange inthe interface So the pressure wave continuously decreasesWhen the void fraction increases to some extent following theincrease in the gas influx rate the decrease in wave velocityin the liquid slug is gradually less than the increase in wavevelocity in the gas slug thus the decrease of wave velocityis slowed down for the growth of gas slug Especially in thewellhead a slight increase in wave velocity is observed
53 Effect of Virtual Mass Force on Wave Velocity Figure 13illustrates the effect of virtual mass force on the wave velocitywith the parameter of gas influx rate As shown in Figure 15under the same gas influx rate (0594m3h 1098m3h and1768m3h resp) the wave velocity curve of 119862vm = Rediverges from the curve of 119862vm = 0 at the position close tothewellhead whereas thewave velocity curve of the two typesis almost coincided below the position of 119867 = 500m Thisdivergence between the two types of curves is connected withthe fact that significance of influence of interphase virtual
12 Mathematical Problems in Engineering
0 25 50 75 1004000
3000
2000
1000
0
Q = 0058m3h Q = 1228m3hQ = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
H(m
)
BP = 01 MPaw = 50Hz = Re
120601G ()
Cvm
Figure 9 Void fraction distribution at different gas influx rates
mass forces increases together with the increase of the relativemotion in the interphase It is evident that the bottomholepressure is hundreds of times higher than the wellheadpressure As a result the void fraction at the position closeto the wellhead increases sharply meanwhile the interphasemomentum and energy exchanges are promoted The virtualmass force can be described as the transfer of momentumbetween the gas phase and drilling mud phase caused bythe relative motion in the interphase If the relative motionin the interphase is quite weak the value of virtual massforce will intend to approach 0 and the influence on thewave velocity can be ignored However if the relative motionis rather intense the effect of virtual mass force on thewave velocity should not be ignored Furthermore taking thevirtual mass force into account the dispersion characteristicof the pressure wave weakens obviously Compared with thepressure wave velocity calculated by ignoring the effect ofvirtual mass force the calculated pressure wave velocity witha consideration of the virtual mass force is lower Thereforeit is necessary to analyze the effect of the virtual mass forceon the wave velocity at the position close to the wellhead inMPD operations
54 Effect of Disturbance Angular Frequency onWave VelocityFigure 14 presents the pressure wave velocity in the annulusat different angular frequencies According to Figures 5 and9 the void fraction first slightly increases in the bottomholeand then sharply increases along the flow direction at theposition close to the wellhead At a fixed angular frequencythis phenomenon results in an overall decrease in the pressurewave velocity except for the section near the wellhead At the
0 500 1000 1500 20004000
3000
2000
1000
0
c (ms)
H(m
)Q = 0058m3h Q = 1228m3h
Q = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
BP = 01 MPaw = 50Hz = ReCvm
Figure 10 Wave velocity variations at different gas influx rates
section close to the wellhead an opposite change trend ofpressure wave velocity is observed for the transition of flowpattern from bubbly flow to slug flow due to the continuousincrease in void fraction along the flow direction In additionit can be clearly seen from the curves that the wave velocityincreases accompanying with the increase in the angularfrequency above the position of 119867 = 500m This propertyis not very distinct at the position below 119867 = 500m for thelow void fraction
Figure 15 shows the effect of frequency on the wavevelocity in the gas-drilling mud flow (119862vm = 0)The curves ofthe wave velocity of different positions reveal that the prop-agation velocity of pressure disturbances increases togetherwith the growth of the angular frequency (0 lt 119908 lt 500Hz)It proves that the pressure wave has an obvious dispersioncharacteristic in the two-phase flow As the angular frequencyincreases in the range of less than 500Hz there is sufficienttime to carry out the exchange of energy and momentumbetween two phases It achieves a state of mechanical andthermodynamic equilibrium between the two phases Sothe wave velocity increases gradually with the increase inthe angular frequency at different depths of the annulus Itis considered that the wave velocity is mainly affected bythe interphase mechanical and thermodynamic equilibriumat low anglular frequencies When the angular frequencyreaches the value of 119908 = 500Hz the pressure wave velocityachieves a constant value and remains on this level regardlessof the further growth in angular frequency 119908 With theincrease in angular frequency there is not enough time forenergy and momentum exchange between the gas-drillingmud two phases to reach themechanical and thermodynamic
Mathematical Problems in Engineering 13
0
25
50
75
100
125
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
Q (m3h)
120601G
()
0 25 5 75 10
BP = 01 MPaw = 50Hz = ReCvm
Figure 11 Effect of gas influx rate on the void fraction
equilibrium state and thus the wave dispersion does notexist At a high angular frequency the wave velocity is mainlydominated by the mechanical and thermal nonequilibriumin the flow and keeps almost unchanged when the angularfrequency is further increased This is consistent with theinfluences of the angular frequency in the horizontal pipe[28] Moreover the effect of virtual mass force is also shownin this figure (119862vm = Re) It is observed that the wave velocityis significantly reduced when the virtual mass force is takeninto consideration in the calculation of the wave velocity atthe position close to the wellhead such as 119867 = 0m It canbe explained by the reduction of dispersion characteristic ofa pressure wave Along the wellbore from the wellhead to thebottomhole the distinction gradually decreases
6 Conclusions
With full consideration of the important factors such asvirtual mass force drag force gas void fraction pressuretemperature and angular frequency a united wave velocitymodel has been proposed based on pressure drop gradientequations in MPD operations gas-liquid two-fluid modelthe gas-drilling mud equations of state and small pertur-bation theory Solved by the fourth-order explicit Runge-Kutta method the model is used to predict wave velocityfor different back pressures and gas influx rates in MPDoperations The main conclusions can be summarized asfollows
(1) With the introduction of virtual mass force and dragforce the united model agrees well with the previousexperimental data The united model can be used to
0
280
560
840
1120
1400
Q (m3h)0 25 5 75 10
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
c(m
s)
BP = 01 MPaw = 50Hz = ReCvm
Figure 12 Effect of gas influx rate on the pressure wave velocity
accurately calculate the wave velocity in the annulusThe application of the model will be beneficial tofurther study the wave velocity at different gas influxrates and back pressures in MPD operations reducenonproductive times and provide a reference for thedrilling operations in extremely narrow porefracturewindows existing conditions
(2) The wave velocity and void fraction have differentvalues with respect to well depth In the annulusthe drop of pressure causes an increase in the voidfraction along the flow direction The void fractionincreases first slightly and then sharply Correspond-ingly the wave velocity first gradually decreases in thbubbly flow and preliminary slug flow and then thewave velocity slightly increases accompanying withthe increase in the relative length ratio of gas slugto the liquid slug for the continuous increase in voidfraction at the position close to the wellhead Theminimum wave velocity appears in the long gas slugflow
(3) When the back pressure in the MPD operationsincreases the void fraction at different depths of theannulus is reduced gradually while the wave velocityin the two-phase flow tends to increase Moreoverthe influence of back pressure on the void fraction isgreater at a position close to the wellhead than that inthe bottomhole because of the great decrease in thepressure from the bottomhole to the wellhead Alsothe effect of back pressure on void fraction and wavevelocity is decreased in the high back pressure range
14 Mathematical Problems in Engineering
0 300 600 900 12004000
3000
2000
1000
225
180
135
90
45
0
BP = 01 MPaw = 50Hz
Q = 0594m3h = ReQ = 0594m3h = 0
Q = 1098m3h = ReQ = 1768m3h = ReQ = 1098m3h = 0
Q = 1768m3h = 0
c (ms)
H(m
)
CvmCvm
Cvm
Cvm
Cvm
Cvm
Figure 13 Effect of virtual mass force on the pressure wave velocity
(4) The wave velocity is sensitive to the void fraction butthe void fraction is dominated by gas influx rate andpressure in the annulus especially the gas influx rateSince the compressibility of the gas and drilling mudtwo-phase fluid is improved with the increase in theinflux rate the void fraction increases greatly andthe wave velocity decreases significantly within thelow gas influx rate range When the void fraction isincreased to some extent following the increase in thegas influx rate the decrease of wave velocity is slowedfor the growth of gas slug Especially at the wellheada slight increase in wave velocity is observed at a highgas influx rate for the sharp increase in void fraction
(5) It is necessary to analyze the effect of the virtualmass force on the wave velocity in MPD operationsEspecially at the position close to the wellhead theeffect of virtual mass force is more prominent for theintense phase interactions Taking the virtual massforce into account the dispersion characteristic of thepressure wave weakens obviously Compared with theresults calculated by ignoring the effect of virtualmassforce the calculated pressure wave velocity with aconsideration of the virtual mass force is lower
(6) The effect of angular frequency on the propagationvelocity of pressure wave appears at a low angularfrequency The propagation velocity of pressure dis-turbances increases together with the growth of theangular frequency (0 lt 119908 lt 500Hz) When theangular frequency reaches the value of 119908 = 500Hzthe pressure wave velocity achieves a constant value
0 240 480 720 960 12004000
3000
2000
1000
560
420
280
140
0
Q = 0727m3h BP = 01 MPa
w = 50Hz = Rew = 50Hz = 0w = 300Hz = 0
w = 800Hz = 0w = 1600Hz = 0w = 5000Hz = 0
c (ms)
H(m
)CvmCvmCvm
CvmCvmCvm
Figure 14 Wave velocity variations at different angular frequencies
and remains on this level regardless of the furthergrowth in angular frequency 119908
Nomenclature
A Annulus effective cross area (m2)c119866 Wave velocity in gas (ms)
c119871 Wave velocity in Drilling mud (ms)
C119863 The coefficient of drag force
D Annulus effective diameter (m)D119894 Diameter of the inner pipe (m)
D119900 Diameter of the outer pipe (m)
f119866 Shear stresses coefficient of gas interface
f119871 Shear stresses coefficient of drilling mud
interfaceF119911 Pressure applied to the two-phase flow (N)
g Acceleration due to gravity (m2s)g119888 Conversion factor
g119879 Geothermal temperature gradient (Km)
h One discrete length of annulus (m)k119890 Annulus effective roughness (m)
k119900 Roughness of outer pipe (m)
k119894 Roughness of inner pipe (m)
H The depth of annulus (m)L119902 Air bubbles length of slug (m)
119872Gi Momentum transfer in gas interface (Nm3)119872Li Momentum transfer in liquid interface (Nm3)119872
ndLi The none-drag-force (virtual mass force)
(Nm3)119872
119889
Li The momentum transfer term caused by thedrag force (Nm3)
Mathematical Problems in Engineering 15
100 1000 100000
450
900
1350
1800
0w (Hz)
0 m = 0 0 m = Re1000 m = 0
2000 m = 0
4000 m = 0
1000 m = Re2000 m = Re4000 m = Re
c(m
s)
Q = 0727m3h BP = 01 MPaCvmCvm
CvmCvmCvm
CvmCvmCvm
Figure 15 Effect of angular frequency on wave velocity
N Parameter combiningThompsonenergy effects
119901Gi Gas interface pressure (MPa)119901Li Liquid interface pressure (MPa)p Pressure (MPa)p0 Initial pressure of annulus (MPa)
p119888 Critical pressure (kPa)
p119871 Pressure of drilling mud (MPa)
p119903 Reduced pressure
q119898 Volumetric flow rate of gas and drilling
mud (m3s)q119866 Velocity of the gas (m3s)
q119871 Velocity of the drilling mud (m3s)
Q119871 Drilling mud rate within the annulus
(m3h)r Average diameter of the bubble (m)R Constant of EOS (JKgK)R119902 Air bubbles radius of slug (m)
R119887 The bubble diameter (m)
T Temperature (K)T119888 Critical temperature (K)
T119903 Reduced temperature
T119890119894 Undisturbed formation temperature at a
depth (K)T119890119887ℎ
Undisturbed formation temperature ofwellhead (K)
T119891119887ℎ
Undisturbed formation temperature atthe bottomhole (K)
v119898 Gas and drilling mud flow velocity
(ms)
v119866 Gas flow velocity (ms)
v119904 Slip velocity (ms)
v119904119892 Superficial gas velocity (ms)
v119871 Drilling mud flow velocity (ms)
W119898 Mass flow velocity (kgm3)
Z119866 Gas deviation factor
z0 Initial calculation height of annulus (m)
z119887ℎ Total well depth from surface (m)
z Variable of distance from surface (m)
Greek Letters
120588
119866 Gas density (kgm3)
120591
119891119903
Li Shear stresses of drilling mud noninterfacial(Nm2)
120591
119891119903
119871 Shear stresses of drilling mud interface (Nm2)
120591
Re119871 Reynolds stress of drilling mud (Nm2)
120591
Re119871 Reynolds stress of drilling mud interface
(Nm2)120591
Re119866 Reynolds stress of gas interface (Nm2)
120591
119891119903
119866 Shear stresses of gas interface (Nm2)
120572VM Acceleration due to virtual mass forcecoefficient (ms2)
120591
119871 Shear stresses of drilling mud along wall
(Nm2)120591
119866 Shear stresses of gas along wall (Nm2)
120591
Re119866 Reynolds stress of gas (Nm2)
120591
119891 Frictional pressure gradient (Pam)
120591
119908 Frictional pressure coefficient
120588
119898 Gas and drilling density (kgm3)
120588
0 Density under standard atmospheric pressure
(kgm3)120588
119871 Drilling mud density (kgm3)
120590
119904 Surface tension (Nm2)
Φ
119866 Gas void fraction
Φ
119871 Drilling mud holdup
120588
119903 Reduced density
120583
119871 Viscosity of drilling mud (Pa sdot s)
Subscripts
BHP Bottomhole pressureBP Back pressureEOS Equations of stateMPD Managed pressure drillingRCD The rotating control deviceR-K4 The fourth-order explicit Runge-Kutta
Subscripts of Graph
BP Back pressure (MPa)c Wave velocity in gas and drilling mud
two-phase flow (ms)119862vm The coefficient of virtual mass force
16 Mathematical Problems in Engineering
H Depth of annulus (m)Q Gas influx rate at the bottomhole (m3h)Re The value of virtual mass force
coefficient according to (28) or (34)119908 Angle frequency (Hz)Φ
119866 Void fraction ()
Acknowledgment
Research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 and 51274170)Without their support this work would not have beenpossible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of airinjection in annuli to manage pressure during cementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling conferenceand Exhibition San Diego Calif USA March 2012
[4] W A Bacon Consideration of compressibility effects for applied-back-pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Tex USA 2011
[5] J Choe C Hans and J Wold ldquoA modified two-phase well-control model and its computer applications as a training andeducational toolrdquo SPE Computer Applications vol 2 pp 14ndash201997
[6] J A Tarvin IWalton and PWand ldquoAnalysis of a gas kick takenin a deep well drilled with oil-based mudrdquo in Proceedings ofthe Annual Technical Conference and Exhibition of the Societyof Petroleum Engineers pp 255ndash264 Dallas Tex USA 1991
[7] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[8] J Zhou and G Nygaard ldquoAutomatic model-based controlscheme for stabilizing pressure during dual-gradient drillingrdquoJournal of Process Control vol 21 no 8 pp 1138ndash1147 2011
[9] A Ochoa O Acevedo L Nieto J E Lambarria and H PerezldquoSuccessful application of MPD (Managed Pressure Drilling)for prevention controland detection of borehole ballooning intight gas reservoir in Cuervito Field Mexicordquo in Proceedings ofthe Canadian Unconventional Resources Conference (CURC rsquo11)pp 172ndash185 November 2011
[10] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[11] R Ashena and J Moghadasi ldquoBottom hole pressure estimationusing evolved neural networks by real coded ant colony opti-mization and genetic algorithmrdquo Journal of Petroleum Scienceand Engineering vol 77 no 3-4 pp 375ndash385 2011
[12] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvanced Materials Research vol 468-471 pp 1736ndash1742 2012
[13] I Zubizarreta ldquoPore pressure evolution core damage and trip-ping out schedulesf a computational fluid dynamics approachrdquoin Proceedings of the SPEIADC Drilling Conference and Exhibi-tion Amsterdam The Netherlands March 2013
[14] C H Whitson ldquoCyclic shut-in eliminates liquid-loading in gaswellsrdquo in Proceedings of the SPEEAGE European Unconven-tional Resources Conference and Exhibition Vienna AustriaMarch 2012
[15] G M de Oliveira A T Franco C O R Negrao A LMartins and R A Silva ldquoModeling and validation of pressurepropagation in drilling fluids pumped into a closed wellrdquoJournal of Petroleum Science and Engineering vol 103 pp 61ndash71 2012
[16] Y Sato and H Kanki ldquoFormulas for compression wave andoscillating flow in circular piperdquo Applied Acoustics vol 69 no1 pp 1ndash11 2008
[17] J Xie B Yu X Zhang Q Shao and X Song ldquoNumericalsimulation of gas-liquid-solid three-phase flow in deep wellsrdquoAdvances in Mechanical Engineering vol 2013 Article ID951298 10 pages 2013
[18] W Bacon A Tong O Gabaldon C Sugden and P V Surya-narayana ldquoAn improved dynamic well control response to a gasinflux in managed pressure drilling operationsrdquo in Proceedingsof the IADCSPE Drilling Conference and Exhibition San DiegoCalifornia USA March 2012
[19] J I Hage and D ter Avest ldquoBorehole acoustics applied to kickdetectionrdquo Journal of Petroleum Science and Engineering vol 12no 2 pp 157ndash166 1994
[20] A M R Hopkins Fluid dynamics and surface pressure fluctua-tions of two-dimensional turbulent boundary layers over denselydistributed surface roughness [PhD thesis] Virginia PolytechnicInstitute and State University Blacksburg Va USA 2010
[21] Z S Fu ldquoThe gas cut detect technology during drilling in SovietUnionrdquo Petroleum Drilling Techniques vol 18 no 1 pp 19ndash211990
[22] X F Li C X Guan and X X Sui ldquoThe theory of gas influxdetection of pressure wave and its applicationrdquo Acta PetroleiSinica vol 18 no 3 pp 128ndash133 1997
[23] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[24] A E Ruggles R T Lahey Jr D A Drew and H A ScartonldquoInvestigation of the propagation of pressure perturbations inbubbly airwater flowsrdquo Journal of Heat Transfer vol 110 no 2pp 494ndash499 1988
[25] H J W M Legius H E A Van Den Akker and T NarumoldquoMeasurements on wave propagation and bubble and slugvelocities in cocurrent upward two-phase flowrdquo ExperimentalThermal and Fluid Science vol 15 no 3 pp 267ndash278 1997
[26] K Miyazaki and I Nakajima ldquoPropagation of pressure wavein NItrogen-Mercury two-phase systemrdquo Journal of NuclearScience and Technology vol 9 no 5 pp 315ndash318 1972
[27] B Bai L Guo and X Chen ldquoPressure fluctuation for air-watertwo-phase flowrdquo Journal of Hydrodynamics vol 18 no 4 pp476ndash482 2003
[28] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
Mathematical Problems in Engineering 17
[29] R F Tangren C H Dodge and H S Seifert ldquoCompressibilityeffects in two-phase flowrdquo Journal of Applied Physics vol 20 no7 pp 637ndash645 1949
[30] G B Wallis One Dimensional Two Phase Flow McGraw HillNew York NY USA 1969
[31] D L Nguyen E R F Winter and M Greiner ldquoSonic velocityin two-phase systemsrdquo International Journal ofMultiphase Flowvol 7 no 3 pp 311ndash320 1981
[32] R C Mecredy and L J Hamilton ldquoThe effects of nonequilib-rium heat mass and momentum transfer on two-phase soundspeedrdquo International Journal of Heat and Mass Transfer vol 15no 1 pp 61ndash72 1972
[33] K H Ardron and R B Duffey ldquoAcoustic wave propagationin a flowing liquid-vapour mixturerdquo International Journal ofMultiphase Flow vol 4 no 3 pp 303ndash322 1978
[34] J Xu and T Chen ldquoAcoustic wave prediction in flowing steam-water two-phase mixturerdquo International Journal of Heat andMass Transfer vol 43 no 7 pp 1079ndash1088 2000
[35] N ChungW Lin B Pei and Y Hsu ldquoModel for sound velocityin a two-phase air-water bubbly flowrdquo Nuclear Technology vol99 no 1 pp 80ndash89 1992
[36] X Xu and J Gong ldquoA unitedmodel for predicting pressurewavespeeds in oil and gas two-phase pipeflowsrdquo Journal of PetroleumScience and Engineering vol 60 no 3-4 pp 150ndash160 2008
[37] W Kuczynski ldquoCharacterization of pressure-wave propagationduring the condensation of R404A and R134a refrigerantsin pipe mini-channels that undergo periodic hydrodynamicdisturbancerdquo International Journal of Heat and Fluid Flow vol40 pp 135ndash150 2013
[38] W Kuczynski ldquoModeling of the propagation of a pressure waveduring the condensation process of R134a refrigerant in a pipeminichannel under the periodic conditions of hydrodynamicdisturbancesrdquo International Journal of Heat and Mass Transfervol 56 pp 715ndash723 2013
[39] Y Li C Li E Chen and Y Ying ldquoPressure wave propaga-tion characteristics in a two-phase flow pipeline for liquid-propellant rocketrdquoAerospace Science and Technology vol 15 no6 pp 453ndash464 2011
[40] T Kanagawa M Watanabe T Yano and S Fujikawa ldquoNon-linear wave equations for pressure wave propagation in liquidscontaining gas bubbles (comparison between two-fluid modeland mixture model)rdquo Journal of Fluid Science and Technologyvol 6 no 6 pp 836ndash849 2011
[41] EChen Y Li andYYing ldquoNumerical investigation onpressurewave propagation speed of gas-liquid two-phase flow in pumppipelinerdquo Journal of Aerospace Power vol 25 no 4 pp 754ndash7602010
[42] F Li and L He ldquoInvestigation on pressure wave propagationcharacteristics in slug flow field with increasing gas flow raterdquoPetroleum Geology and Recovery Efficiency vol 11 no 5 pp 78ndash80 2004
[43] C SMartin andM Padmanabhan ldquoPressure pulse propagationin two-component slug flowrdquo Journal of Fluids Engineering vol101 no 1 pp 44ndash52 1979
[44] R E Henry ldquoPressure wave propagation in two-phase mix-turesrdquo Chemical Engineering Progress vol 66 no 1 pp 1ndash101969
[45] X Liu B Li and Y Yue ldquoTransmission behavior of mud-pressure pulse alongwell borerdquo Journal ofHydrodynamics B vol19 no 2 pp 236ndash240 2007
[46] XWang and J Zhang ldquoThe research of pressure wave Pulsationin mud pulse transmittingrdquo Journal of Chongqing University ofScience and Technology vol 14 no 2 pp 55ndash58 2012
[47] J-W Park D A Drew and R T Lahey Jr ldquoThe analysisof void wave propagation in adiabatic monodispersed bubblytwo-phase flows using an ensemble-averaged two-fluid modelrdquoInternational Journal ofMultiphase Flow vol 24 no 7 pp 1205ndash1244 1998
[48] G Arnold Entropy and objectivity as constraints upons constitu-tive equations for two-fluid modeling of multiphase flow [PhDthesis] Rensselaer Polytechnic Institute New York NY USA1998
[49] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[50] L Yarborough and K R Hall ldquoHow to solve equation of statefor z-factorsrdquoOil and Gas Journal vol 72 no 7 pp 86ndash88 1974
[51] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86 pp 127ndash136 2012
[52] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[53] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[54] M Ishii and K Mishima ldquoTwo-fluid model and hydrodynamicconstitutive relationsrdquo Nuclear Engineering and Design vol 82no 2-3 pp 107ndash126 1984
[55] M J Sanchez Comparison of correlations for predicting pressurelosses in vertical multiphase annular flow [MS thesis] Univer-sity of Tulsa Tulsa Okla USA 1972
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
Grid division
Start
Delete two roots
End
Initial conditions
Meet demand
i = i + 1
Solve eq (20) for T
Solve eq (15)ndash(17) for Zg
Judging flow pattern by eq (21) and eq (22)
Bubbly flow pattern Slug flow pattern
Solve determinant eq (45) for four roots
Calculate F(z p) by eq (48)
Calculate k1 minus k4Pi+1 = Pi + ΔPzi+1 = zi + h
zi+1 gt HNo
No
Yes
Yes
Solve eq (18) (19) for120588L
Solve eq ( 47) for c
Solve eq (14)120588G
Calculate 120588m 120601G f and CDCalculate 120588m 120601G f and CD CvmCvm
Figure 2 Solution procedure for wave velocity in MPD operations
the annulus The ability to measure return flowaccurately is essential for the applied back pressure
(iv) Pressure Sensor A pressure sensor is used to measuresurface back pressure on the wellhead
The gas and drilling mud flow rate measured by the Cori-olis meter and the back pressure measured by the pressuresensor are the initial data for annulus pressure calculationThe well used for calculation is a gas well in XinjiangUygur Autonomous Region Northwest China The wellborestructure well design parameters (depths and diameters)gas and drilling mud properties (density and viscosity) andoperational conditions of calculation well are displayed inTable 1
The drilling mud mixed with gas is taken as a two-phaseflow medium The propagation velocity of pressure wave inthe gas-drillingmud flow is calculated and discussed by usingthe established model and well parameters
51 Effect of Back Pressure on Wave Velocity Increase inapplied back pressure is the most common approach fordynamicwell controlThewellbore can be seen as an enclosed
Table 1 Parameters of calculation well
Type Property Value
Mud Dynamic viscosity (Pa sdot s) 0056Density (kgcm3) 1460
Gas Relative density 065Viscosity (Pa sdot s) 114 times 10minus5
StringElastic modulus of string (Pa) 207 times 1011
Poisson ratio of string 03Roughness (m) 154 times 10minus7
Surface condition Surface temperature (K) 298Atmosphere pressure (MPa) 0101
pressurized system The pressure at different depths of theannulus varied with the change of back pressure Accordingto the EOS of gas and drilling mud the influence of pressureon gas volume is much greater than that of drilling mudfor the greater compressibility of gas So the gas voidfraction changes with the variation of gas volume at thenearly constant flow rate of drilling mud at different annularpressures Meanwhile the pressure wave velocity is sensitive
Mathematical Problems in Engineering 9
5000 2 4 6 8 10
750
1000
1250
1500c
(ms
)
This paper p = 30MPa w = 50Hz = Re
120601G ()
CvmLiu et al [45] p = 30MPa
(a) Comparison with the experimental data of Liu
0 20 40 60 80 1000
40
80
120
160
200
c(m
s)
This paper p = 01MPa w = 50Hz = Re
120601G ()
CvmLi et al [22] p = 01MPa
(b) Comparison with the experimental data of Li
Figure 3 Experimental verification by comparison with previous experimental data
Drill pipe
Annulus
RCD
Formation
Mud pump
Drill collar
Coriolis meterChoke
Casing
Pressure sensors
120601 2159 mm times G105
1206011778mm (120601 78 mm) times G105
120601127 mm (120601 78 mm) times G105
z0 m
3600 m
4000 m
QL
Figure 4 Schematic diagram of the gas influx process
to the gas void fraction As a result when the back pressureat the wellhead is changed by adjusting the choke the wavevelocity in the two-phase drilling fluid and the distributionof void fraction at different depth of the annulus will divergeThe calculation results affected by the back pressure arepresented
Figures 5 and 6 show the distributions of void fractionand variations of wave velocity along the flow directionin the annulus when the back pressure at the wellhead is01MPa 08MPa 15MPa 25MPa 50MPa and 65MPa
respectively It can be seen that the void fraction significantlyincreases along the flow direction in the annulus Converselythe wave velocity shows a remarkable decrease tendency Forinstance BP = 15MPa at the wellhead the wave velocityis 9012ms and the void fraction is 0611 while in thebottomhole the wave velocity reaches 73142ms and voidfraction is reduced to 0059 However a sudden increase inwave velocity is observed in Figure 5 when the back pressureis 01MPa at the position close towellhead as the void fractionis further increased Moreover the influence of back pressure
10 Mathematical Problems in Engineering
10075500 254000
3000
2000
1000
0
BP = 01 MPaBP = 08 MPaBP = 15 MPa
BP = 25 MPa
BP = 65 MPaBP = 5 MPa
H(m
)
Q = 9m3h w = 50Hz = Re
120601G ()
Cvm
Figure 5 Void fraction distribution at different back pressures
on the void fraction is larger at the position close to thewellhead than that in the bottomhole
This can be explained from the viewpoints of mixturedensity and compressibility of two-phase fluid and the pres-sure drop along the flow direction in the wellbore In thelow void fraction range the gas phase is dispersed in theliquid as bubble so the wave velocity is influenced greatly bythe added gas phase According to the EOS if gas invadesinto the wellbore with a small amount in the bottomholethe density of the drilling mud has little variation while thecompressibility increases obviously which makes the wavevelocity decreased Then the two-phase drilling mud flowfrom the bottomhole to the wellhead along the annulus witha drop of pressure caused by potential energy change kineticenergy and frictional loss which leads to an increase in voidfractionWhen the void fraction is increased both the densityand the compressibility of the two-phase fluid change slightlyresulting in a flat decrease in wave velocity along the flowdirection With the further increasing in void fraction thebubbly flow turns into slug flow according to the flow patterntransient criteria of Orkiszewski which shows that the flowpattern is dependent on the void fraction Generally speak-ing the gas-drilling mud slug flow is composed of liquidand gas slugs In the preliminary slug flow the liquid slugsare much longer than the gas slugs so the wave velocity isdetermined primarily by the wave velocities in the liquidslugs which are hardly affected by the void fraction At theposition close to the wellhead the pressure of two-phase flowfluid can be reduced to a low value which approaches theback pressure The compressibility of gas will be improvedat the low pressure It results in significant increase in void
0 250 500 750 10004000
3000
2000
1000
0
H(m
)BP = 01 MPaBP = 08 MPaBP = 15 MPa
BP = 25 MPa
BP = 65 MPaBP = 5 MPa
c (ms)
Q = 9m3h w = 50Hz = ReCvm
Figure 6 Wave velocity variations at different back pressures
fraction As the void fraction increases greatly at the positionclose to the wellhead gas slugs becomemuch longer than theliquid slugs It is assumed that the actual wave velocity in slugflow happens to play a leading role when the void fractionincreases to some extent For the decrease in liquid holdupin the gas slug with the increasing in void fraction the wavevelocity in the gas slug is infinitely close to that in pure gaswhich is equivalent to an increase in wave velocity in the gasslug Meanwhile the gas content in liquid slug also increasesand results in a lower wave velocity in the liquid slug As aresult a slight increase in wave velocity appears
Figures 7 and 8 present the influence of back pressureon the void fraction and wave velocity with respect to theparameters of well depth As the back pressure increases thevoid fraction at different depth of the annulus is reducedgradually while the wave velocity in the two-phase flow tendsto increase Analytical results show that the increased backpressure is equivalent to be applied to the entire encloseddrilling fluid cyclical systemThe pressure transmits from thewellhead to the bottomhole therefore the annular pressurein the entire wellbore is increased According to the EOSthe density of gas increases and the compressibility of gasdecreases with the increasing of gas pressure So the lossof interphase momentum and energy exchange is reducedand the interphase momentum exchange is promoted Itcontributes to the increase in wave velocity with the increasein gas pressure In addition due to lower compressibility oftwo-phase flow medium under high pressure the increasetendency of pressure wave velocity and the decrease tendencyof void fraction are slowed down in the high back pressurerange
Mathematical Problems in Engineering 11
0
25
50
75
100
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
0 25 75 105
Q = 9m3h w = 50Hz = Re
120601G
()
Cvm
Figure 7 Effect of back pressure on the void fraction
52 Effect of Gas Influx Rate on Wave Velocity Figures 9and 10 graphically interpret the distributions of void fractionand variations of wave velocity along the flow direction inthe annulus When gas influx occurs in the bottomhole gasinvades into the wellbore and migrates from the bottomholeto the wellhead along the flow direction At a low gas influxrate it is extremely obvious that the void fraction and wavevelocity first slightly change in a comparatively smooth valuethen change sharply It is because of the rapid expansion ofgas volume with the decreasing in pressure near the wellheadthat the void fraction increases sharply and the wave velocitydecreases obviously at the same time But under the highbottomhole pressure (up to 50MPa) the compressibilityof the gas is low This results in a slight change in voidfraction and wave velocity at the position far away from thewellhead Since the compressible component increases withthe increase in the gas influx rate the compressibility of thegas and drilling mud two-phase fluid is improved So thevariations of void fraction and wave velocity become moreprominent Also the void fraction still shows an increasetendency that is steady first and then sharp It acts inaccordance with the variation of void fraction at a low gasinflux rate At a low gas influx rate if the void fraction atthe position close to the wellhead can not increase to a highextent the wave velocity always shows a decrease tendencyAt a high gas influx rate such as the 119876 = 8312m3h thewave velocity tends to increase because the void fraction inthe wellhead is increased to a high extent In conclusionthe wave pressure is sensitive to the void fraction and thevoid fraction is dominated by influx rate and pressure in theannulus especially the influx rate
0 25 75 1050
300
600
900
1200
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
c(m
s)
Q = 9m3h w = 50Hz = ReCvm
Figure 8 Effect of back pressure on the pressure wave velocity
With the influx of gas the mixing of gas and drillingmud occurs in the annulus and the corresponding interfacialtransfer of momentum and mass causes an increase in gasphase void fraction and a decrease in pressure wave velocityas shown in Figures 11 and 12 Within the range of low gasinflux rate the wave velocity decreases significantly It isbecause of this that the compressibility of the gas increasesremarkably and the medium appears to be of high elasticitythough the density of gas-drilling mud two-phase flowchanges slightly With the increase in the gas influx rate andcorresponding increase in the void fraction in the annulusthe compressibility of the two-phase unceasingly increaseswhich promotes the momentum and energy exchange inthe interface So the pressure wave continuously decreasesWhen the void fraction increases to some extent following theincrease in the gas influx rate the decrease in wave velocityin the liquid slug is gradually less than the increase in wavevelocity in the gas slug thus the decrease of wave velocityis slowed down for the growth of gas slug Especially in thewellhead a slight increase in wave velocity is observed
53 Effect of Virtual Mass Force on Wave Velocity Figure 13illustrates the effect of virtual mass force on the wave velocitywith the parameter of gas influx rate As shown in Figure 15under the same gas influx rate (0594m3h 1098m3h and1768m3h resp) the wave velocity curve of 119862vm = Rediverges from the curve of 119862vm = 0 at the position close tothewellhead whereas thewave velocity curve of the two typesis almost coincided below the position of 119867 = 500m Thisdivergence between the two types of curves is connected withthe fact that significance of influence of interphase virtual
12 Mathematical Problems in Engineering
0 25 50 75 1004000
3000
2000
1000
0
Q = 0058m3h Q = 1228m3hQ = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
H(m
)
BP = 01 MPaw = 50Hz = Re
120601G ()
Cvm
Figure 9 Void fraction distribution at different gas influx rates
mass forces increases together with the increase of the relativemotion in the interphase It is evident that the bottomholepressure is hundreds of times higher than the wellheadpressure As a result the void fraction at the position closeto the wellhead increases sharply meanwhile the interphasemomentum and energy exchanges are promoted The virtualmass force can be described as the transfer of momentumbetween the gas phase and drilling mud phase caused bythe relative motion in the interphase If the relative motionin the interphase is quite weak the value of virtual massforce will intend to approach 0 and the influence on thewave velocity can be ignored However if the relative motionis rather intense the effect of virtual mass force on thewave velocity should not be ignored Furthermore taking thevirtual mass force into account the dispersion characteristicof the pressure wave weakens obviously Compared with thepressure wave velocity calculated by ignoring the effect ofvirtual mass force the calculated pressure wave velocity witha consideration of the virtual mass force is lower Thereforeit is necessary to analyze the effect of the virtual mass forceon the wave velocity at the position close to the wellhead inMPD operations
54 Effect of Disturbance Angular Frequency onWave VelocityFigure 14 presents the pressure wave velocity in the annulusat different angular frequencies According to Figures 5 and9 the void fraction first slightly increases in the bottomholeand then sharply increases along the flow direction at theposition close to the wellhead At a fixed angular frequencythis phenomenon results in an overall decrease in the pressurewave velocity except for the section near the wellhead At the
0 500 1000 1500 20004000
3000
2000
1000
0
c (ms)
H(m
)Q = 0058m3h Q = 1228m3h
Q = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
BP = 01 MPaw = 50Hz = ReCvm
Figure 10 Wave velocity variations at different gas influx rates
section close to the wellhead an opposite change trend ofpressure wave velocity is observed for the transition of flowpattern from bubbly flow to slug flow due to the continuousincrease in void fraction along the flow direction In additionit can be clearly seen from the curves that the wave velocityincreases accompanying with the increase in the angularfrequency above the position of 119867 = 500m This propertyis not very distinct at the position below 119867 = 500m for thelow void fraction
Figure 15 shows the effect of frequency on the wavevelocity in the gas-drilling mud flow (119862vm = 0)The curves ofthe wave velocity of different positions reveal that the prop-agation velocity of pressure disturbances increases togetherwith the growth of the angular frequency (0 lt 119908 lt 500Hz)It proves that the pressure wave has an obvious dispersioncharacteristic in the two-phase flow As the angular frequencyincreases in the range of less than 500Hz there is sufficienttime to carry out the exchange of energy and momentumbetween two phases It achieves a state of mechanical andthermodynamic equilibrium between the two phases Sothe wave velocity increases gradually with the increase inthe angular frequency at different depths of the annulus Itis considered that the wave velocity is mainly affected bythe interphase mechanical and thermodynamic equilibriumat low anglular frequencies When the angular frequencyreaches the value of 119908 = 500Hz the pressure wave velocityachieves a constant value and remains on this level regardlessof the further growth in angular frequency 119908 With theincrease in angular frequency there is not enough time forenergy and momentum exchange between the gas-drillingmud two phases to reach themechanical and thermodynamic
Mathematical Problems in Engineering 13
0
25
50
75
100
125
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
Q (m3h)
120601G
()
0 25 5 75 10
BP = 01 MPaw = 50Hz = ReCvm
Figure 11 Effect of gas influx rate on the void fraction
equilibrium state and thus the wave dispersion does notexist At a high angular frequency the wave velocity is mainlydominated by the mechanical and thermal nonequilibriumin the flow and keeps almost unchanged when the angularfrequency is further increased This is consistent with theinfluences of the angular frequency in the horizontal pipe[28] Moreover the effect of virtual mass force is also shownin this figure (119862vm = Re) It is observed that the wave velocityis significantly reduced when the virtual mass force is takeninto consideration in the calculation of the wave velocity atthe position close to the wellhead such as 119867 = 0m It canbe explained by the reduction of dispersion characteristic ofa pressure wave Along the wellbore from the wellhead to thebottomhole the distinction gradually decreases
6 Conclusions
With full consideration of the important factors such asvirtual mass force drag force gas void fraction pressuretemperature and angular frequency a united wave velocitymodel has been proposed based on pressure drop gradientequations in MPD operations gas-liquid two-fluid modelthe gas-drilling mud equations of state and small pertur-bation theory Solved by the fourth-order explicit Runge-Kutta method the model is used to predict wave velocityfor different back pressures and gas influx rates in MPDoperations The main conclusions can be summarized asfollows
(1) With the introduction of virtual mass force and dragforce the united model agrees well with the previousexperimental data The united model can be used to
0
280
560
840
1120
1400
Q (m3h)0 25 5 75 10
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
c(m
s)
BP = 01 MPaw = 50Hz = ReCvm
Figure 12 Effect of gas influx rate on the pressure wave velocity
accurately calculate the wave velocity in the annulusThe application of the model will be beneficial tofurther study the wave velocity at different gas influxrates and back pressures in MPD operations reducenonproductive times and provide a reference for thedrilling operations in extremely narrow porefracturewindows existing conditions
(2) The wave velocity and void fraction have differentvalues with respect to well depth In the annulusthe drop of pressure causes an increase in the voidfraction along the flow direction The void fractionincreases first slightly and then sharply Correspond-ingly the wave velocity first gradually decreases in thbubbly flow and preliminary slug flow and then thewave velocity slightly increases accompanying withthe increase in the relative length ratio of gas slugto the liquid slug for the continuous increase in voidfraction at the position close to the wellhead Theminimum wave velocity appears in the long gas slugflow
(3) When the back pressure in the MPD operationsincreases the void fraction at different depths of theannulus is reduced gradually while the wave velocityin the two-phase flow tends to increase Moreoverthe influence of back pressure on the void fraction isgreater at a position close to the wellhead than that inthe bottomhole because of the great decrease in thepressure from the bottomhole to the wellhead Alsothe effect of back pressure on void fraction and wavevelocity is decreased in the high back pressure range
14 Mathematical Problems in Engineering
0 300 600 900 12004000
3000
2000
1000
225
180
135
90
45
0
BP = 01 MPaw = 50Hz
Q = 0594m3h = ReQ = 0594m3h = 0
Q = 1098m3h = ReQ = 1768m3h = ReQ = 1098m3h = 0
Q = 1768m3h = 0
c (ms)
H(m
)
CvmCvm
Cvm
Cvm
Cvm
Cvm
Figure 13 Effect of virtual mass force on the pressure wave velocity
(4) The wave velocity is sensitive to the void fraction butthe void fraction is dominated by gas influx rate andpressure in the annulus especially the gas influx rateSince the compressibility of the gas and drilling mudtwo-phase fluid is improved with the increase in theinflux rate the void fraction increases greatly andthe wave velocity decreases significantly within thelow gas influx rate range When the void fraction isincreased to some extent following the increase in thegas influx rate the decrease of wave velocity is slowedfor the growth of gas slug Especially at the wellheada slight increase in wave velocity is observed at a highgas influx rate for the sharp increase in void fraction
(5) It is necessary to analyze the effect of the virtualmass force on the wave velocity in MPD operationsEspecially at the position close to the wellhead theeffect of virtual mass force is more prominent for theintense phase interactions Taking the virtual massforce into account the dispersion characteristic of thepressure wave weakens obviously Compared with theresults calculated by ignoring the effect of virtualmassforce the calculated pressure wave velocity with aconsideration of the virtual mass force is lower
(6) The effect of angular frequency on the propagationvelocity of pressure wave appears at a low angularfrequency The propagation velocity of pressure dis-turbances increases together with the growth of theangular frequency (0 lt 119908 lt 500Hz) When theangular frequency reaches the value of 119908 = 500Hzthe pressure wave velocity achieves a constant value
0 240 480 720 960 12004000
3000
2000
1000
560
420
280
140
0
Q = 0727m3h BP = 01 MPa
w = 50Hz = Rew = 50Hz = 0w = 300Hz = 0
w = 800Hz = 0w = 1600Hz = 0w = 5000Hz = 0
c (ms)
H(m
)CvmCvmCvm
CvmCvmCvm
Figure 14 Wave velocity variations at different angular frequencies
and remains on this level regardless of the furthergrowth in angular frequency 119908
Nomenclature
A Annulus effective cross area (m2)c119866 Wave velocity in gas (ms)
c119871 Wave velocity in Drilling mud (ms)
C119863 The coefficient of drag force
D Annulus effective diameter (m)D119894 Diameter of the inner pipe (m)
D119900 Diameter of the outer pipe (m)
f119866 Shear stresses coefficient of gas interface
f119871 Shear stresses coefficient of drilling mud
interfaceF119911 Pressure applied to the two-phase flow (N)
g Acceleration due to gravity (m2s)g119888 Conversion factor
g119879 Geothermal temperature gradient (Km)
h One discrete length of annulus (m)k119890 Annulus effective roughness (m)
k119900 Roughness of outer pipe (m)
k119894 Roughness of inner pipe (m)
H The depth of annulus (m)L119902 Air bubbles length of slug (m)
119872Gi Momentum transfer in gas interface (Nm3)119872Li Momentum transfer in liquid interface (Nm3)119872
ndLi The none-drag-force (virtual mass force)
(Nm3)119872
119889
Li The momentum transfer term caused by thedrag force (Nm3)
Mathematical Problems in Engineering 15
100 1000 100000
450
900
1350
1800
0w (Hz)
0 m = 0 0 m = Re1000 m = 0
2000 m = 0
4000 m = 0
1000 m = Re2000 m = Re4000 m = Re
c(m
s)
Q = 0727m3h BP = 01 MPaCvmCvm
CvmCvmCvm
CvmCvmCvm
Figure 15 Effect of angular frequency on wave velocity
N Parameter combiningThompsonenergy effects
119901Gi Gas interface pressure (MPa)119901Li Liquid interface pressure (MPa)p Pressure (MPa)p0 Initial pressure of annulus (MPa)
p119888 Critical pressure (kPa)
p119871 Pressure of drilling mud (MPa)
p119903 Reduced pressure
q119898 Volumetric flow rate of gas and drilling
mud (m3s)q119866 Velocity of the gas (m3s)
q119871 Velocity of the drilling mud (m3s)
Q119871 Drilling mud rate within the annulus
(m3h)r Average diameter of the bubble (m)R Constant of EOS (JKgK)R119902 Air bubbles radius of slug (m)
R119887 The bubble diameter (m)
T Temperature (K)T119888 Critical temperature (K)
T119903 Reduced temperature
T119890119894 Undisturbed formation temperature at a
depth (K)T119890119887ℎ
Undisturbed formation temperature ofwellhead (K)
T119891119887ℎ
Undisturbed formation temperature atthe bottomhole (K)
v119898 Gas and drilling mud flow velocity
(ms)
v119866 Gas flow velocity (ms)
v119904 Slip velocity (ms)
v119904119892 Superficial gas velocity (ms)
v119871 Drilling mud flow velocity (ms)
W119898 Mass flow velocity (kgm3)
Z119866 Gas deviation factor
z0 Initial calculation height of annulus (m)
z119887ℎ Total well depth from surface (m)
z Variable of distance from surface (m)
Greek Letters
120588
119866 Gas density (kgm3)
120591
119891119903
Li Shear stresses of drilling mud noninterfacial(Nm2)
120591
119891119903
119871 Shear stresses of drilling mud interface (Nm2)
120591
Re119871 Reynolds stress of drilling mud (Nm2)
120591
Re119871 Reynolds stress of drilling mud interface
(Nm2)120591
Re119866 Reynolds stress of gas interface (Nm2)
120591
119891119903
119866 Shear stresses of gas interface (Nm2)
120572VM Acceleration due to virtual mass forcecoefficient (ms2)
120591
119871 Shear stresses of drilling mud along wall
(Nm2)120591
119866 Shear stresses of gas along wall (Nm2)
120591
Re119866 Reynolds stress of gas (Nm2)
120591
119891 Frictional pressure gradient (Pam)
120591
119908 Frictional pressure coefficient
120588
119898 Gas and drilling density (kgm3)
120588
0 Density under standard atmospheric pressure
(kgm3)120588
119871 Drilling mud density (kgm3)
120590
119904 Surface tension (Nm2)
Φ
119866 Gas void fraction
Φ
119871 Drilling mud holdup
120588
119903 Reduced density
120583
119871 Viscosity of drilling mud (Pa sdot s)
Subscripts
BHP Bottomhole pressureBP Back pressureEOS Equations of stateMPD Managed pressure drillingRCD The rotating control deviceR-K4 The fourth-order explicit Runge-Kutta
Subscripts of Graph
BP Back pressure (MPa)c Wave velocity in gas and drilling mud
two-phase flow (ms)119862vm The coefficient of virtual mass force
16 Mathematical Problems in Engineering
H Depth of annulus (m)Q Gas influx rate at the bottomhole (m3h)Re The value of virtual mass force
coefficient according to (28) or (34)119908 Angle frequency (Hz)Φ
119866 Void fraction ()
Acknowledgment
Research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 and 51274170)Without their support this work would not have beenpossible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of airinjection in annuli to manage pressure during cementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling conferenceand Exhibition San Diego Calif USA March 2012
[4] W A Bacon Consideration of compressibility effects for applied-back-pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Tex USA 2011
[5] J Choe C Hans and J Wold ldquoA modified two-phase well-control model and its computer applications as a training andeducational toolrdquo SPE Computer Applications vol 2 pp 14ndash201997
[6] J A Tarvin IWalton and PWand ldquoAnalysis of a gas kick takenin a deep well drilled with oil-based mudrdquo in Proceedings ofthe Annual Technical Conference and Exhibition of the Societyof Petroleum Engineers pp 255ndash264 Dallas Tex USA 1991
[7] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[8] J Zhou and G Nygaard ldquoAutomatic model-based controlscheme for stabilizing pressure during dual-gradient drillingrdquoJournal of Process Control vol 21 no 8 pp 1138ndash1147 2011
[9] A Ochoa O Acevedo L Nieto J E Lambarria and H PerezldquoSuccessful application of MPD (Managed Pressure Drilling)for prevention controland detection of borehole ballooning intight gas reservoir in Cuervito Field Mexicordquo in Proceedings ofthe Canadian Unconventional Resources Conference (CURC rsquo11)pp 172ndash185 November 2011
[10] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[11] R Ashena and J Moghadasi ldquoBottom hole pressure estimationusing evolved neural networks by real coded ant colony opti-mization and genetic algorithmrdquo Journal of Petroleum Scienceand Engineering vol 77 no 3-4 pp 375ndash385 2011
[12] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvanced Materials Research vol 468-471 pp 1736ndash1742 2012
[13] I Zubizarreta ldquoPore pressure evolution core damage and trip-ping out schedulesf a computational fluid dynamics approachrdquoin Proceedings of the SPEIADC Drilling Conference and Exhibi-tion Amsterdam The Netherlands March 2013
[14] C H Whitson ldquoCyclic shut-in eliminates liquid-loading in gaswellsrdquo in Proceedings of the SPEEAGE European Unconven-tional Resources Conference and Exhibition Vienna AustriaMarch 2012
[15] G M de Oliveira A T Franco C O R Negrao A LMartins and R A Silva ldquoModeling and validation of pressurepropagation in drilling fluids pumped into a closed wellrdquoJournal of Petroleum Science and Engineering vol 103 pp 61ndash71 2012
[16] Y Sato and H Kanki ldquoFormulas for compression wave andoscillating flow in circular piperdquo Applied Acoustics vol 69 no1 pp 1ndash11 2008
[17] J Xie B Yu X Zhang Q Shao and X Song ldquoNumericalsimulation of gas-liquid-solid three-phase flow in deep wellsrdquoAdvances in Mechanical Engineering vol 2013 Article ID951298 10 pages 2013
[18] W Bacon A Tong O Gabaldon C Sugden and P V Surya-narayana ldquoAn improved dynamic well control response to a gasinflux in managed pressure drilling operationsrdquo in Proceedingsof the IADCSPE Drilling Conference and Exhibition San DiegoCalifornia USA March 2012
[19] J I Hage and D ter Avest ldquoBorehole acoustics applied to kickdetectionrdquo Journal of Petroleum Science and Engineering vol 12no 2 pp 157ndash166 1994
[20] A M R Hopkins Fluid dynamics and surface pressure fluctua-tions of two-dimensional turbulent boundary layers over denselydistributed surface roughness [PhD thesis] Virginia PolytechnicInstitute and State University Blacksburg Va USA 2010
[21] Z S Fu ldquoThe gas cut detect technology during drilling in SovietUnionrdquo Petroleum Drilling Techniques vol 18 no 1 pp 19ndash211990
[22] X F Li C X Guan and X X Sui ldquoThe theory of gas influxdetection of pressure wave and its applicationrdquo Acta PetroleiSinica vol 18 no 3 pp 128ndash133 1997
[23] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[24] A E Ruggles R T Lahey Jr D A Drew and H A ScartonldquoInvestigation of the propagation of pressure perturbations inbubbly airwater flowsrdquo Journal of Heat Transfer vol 110 no 2pp 494ndash499 1988
[25] H J W M Legius H E A Van Den Akker and T NarumoldquoMeasurements on wave propagation and bubble and slugvelocities in cocurrent upward two-phase flowrdquo ExperimentalThermal and Fluid Science vol 15 no 3 pp 267ndash278 1997
[26] K Miyazaki and I Nakajima ldquoPropagation of pressure wavein NItrogen-Mercury two-phase systemrdquo Journal of NuclearScience and Technology vol 9 no 5 pp 315ndash318 1972
[27] B Bai L Guo and X Chen ldquoPressure fluctuation for air-watertwo-phase flowrdquo Journal of Hydrodynamics vol 18 no 4 pp476ndash482 2003
[28] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
Mathematical Problems in Engineering 17
[29] R F Tangren C H Dodge and H S Seifert ldquoCompressibilityeffects in two-phase flowrdquo Journal of Applied Physics vol 20 no7 pp 637ndash645 1949
[30] G B Wallis One Dimensional Two Phase Flow McGraw HillNew York NY USA 1969
[31] D L Nguyen E R F Winter and M Greiner ldquoSonic velocityin two-phase systemsrdquo International Journal ofMultiphase Flowvol 7 no 3 pp 311ndash320 1981
[32] R C Mecredy and L J Hamilton ldquoThe effects of nonequilib-rium heat mass and momentum transfer on two-phase soundspeedrdquo International Journal of Heat and Mass Transfer vol 15no 1 pp 61ndash72 1972
[33] K H Ardron and R B Duffey ldquoAcoustic wave propagationin a flowing liquid-vapour mixturerdquo International Journal ofMultiphase Flow vol 4 no 3 pp 303ndash322 1978
[34] J Xu and T Chen ldquoAcoustic wave prediction in flowing steam-water two-phase mixturerdquo International Journal of Heat andMass Transfer vol 43 no 7 pp 1079ndash1088 2000
[35] N ChungW Lin B Pei and Y Hsu ldquoModel for sound velocityin a two-phase air-water bubbly flowrdquo Nuclear Technology vol99 no 1 pp 80ndash89 1992
[36] X Xu and J Gong ldquoA unitedmodel for predicting pressurewavespeeds in oil and gas two-phase pipeflowsrdquo Journal of PetroleumScience and Engineering vol 60 no 3-4 pp 150ndash160 2008
[37] W Kuczynski ldquoCharacterization of pressure-wave propagationduring the condensation of R404A and R134a refrigerantsin pipe mini-channels that undergo periodic hydrodynamicdisturbancerdquo International Journal of Heat and Fluid Flow vol40 pp 135ndash150 2013
[38] W Kuczynski ldquoModeling of the propagation of a pressure waveduring the condensation process of R134a refrigerant in a pipeminichannel under the periodic conditions of hydrodynamicdisturbancesrdquo International Journal of Heat and Mass Transfervol 56 pp 715ndash723 2013
[39] Y Li C Li E Chen and Y Ying ldquoPressure wave propaga-tion characteristics in a two-phase flow pipeline for liquid-propellant rocketrdquoAerospace Science and Technology vol 15 no6 pp 453ndash464 2011
[40] T Kanagawa M Watanabe T Yano and S Fujikawa ldquoNon-linear wave equations for pressure wave propagation in liquidscontaining gas bubbles (comparison between two-fluid modeland mixture model)rdquo Journal of Fluid Science and Technologyvol 6 no 6 pp 836ndash849 2011
[41] EChen Y Li andYYing ldquoNumerical investigation onpressurewave propagation speed of gas-liquid two-phase flow in pumppipelinerdquo Journal of Aerospace Power vol 25 no 4 pp 754ndash7602010
[42] F Li and L He ldquoInvestigation on pressure wave propagationcharacteristics in slug flow field with increasing gas flow raterdquoPetroleum Geology and Recovery Efficiency vol 11 no 5 pp 78ndash80 2004
[43] C SMartin andM Padmanabhan ldquoPressure pulse propagationin two-component slug flowrdquo Journal of Fluids Engineering vol101 no 1 pp 44ndash52 1979
[44] R E Henry ldquoPressure wave propagation in two-phase mix-turesrdquo Chemical Engineering Progress vol 66 no 1 pp 1ndash101969
[45] X Liu B Li and Y Yue ldquoTransmission behavior of mud-pressure pulse alongwell borerdquo Journal ofHydrodynamics B vol19 no 2 pp 236ndash240 2007
[46] XWang and J Zhang ldquoThe research of pressure wave Pulsationin mud pulse transmittingrdquo Journal of Chongqing University ofScience and Technology vol 14 no 2 pp 55ndash58 2012
[47] J-W Park D A Drew and R T Lahey Jr ldquoThe analysisof void wave propagation in adiabatic monodispersed bubblytwo-phase flows using an ensemble-averaged two-fluid modelrdquoInternational Journal ofMultiphase Flow vol 24 no 7 pp 1205ndash1244 1998
[48] G Arnold Entropy and objectivity as constraints upons constitu-tive equations for two-fluid modeling of multiphase flow [PhDthesis] Rensselaer Polytechnic Institute New York NY USA1998
[49] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[50] L Yarborough and K R Hall ldquoHow to solve equation of statefor z-factorsrdquoOil and Gas Journal vol 72 no 7 pp 86ndash88 1974
[51] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86 pp 127ndash136 2012
[52] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[53] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[54] M Ishii and K Mishima ldquoTwo-fluid model and hydrodynamicconstitutive relationsrdquo Nuclear Engineering and Design vol 82no 2-3 pp 107ndash126 1984
[55] M J Sanchez Comparison of correlations for predicting pressurelosses in vertical multiphase annular flow [MS thesis] Univer-sity of Tulsa Tulsa Okla USA 1972
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
5000 2 4 6 8 10
750
1000
1250
1500c
(ms
)
This paper p = 30MPa w = 50Hz = Re
120601G ()
CvmLiu et al [45] p = 30MPa
(a) Comparison with the experimental data of Liu
0 20 40 60 80 1000
40
80
120
160
200
c(m
s)
This paper p = 01MPa w = 50Hz = Re
120601G ()
CvmLi et al [22] p = 01MPa
(b) Comparison with the experimental data of Li
Figure 3 Experimental verification by comparison with previous experimental data
Drill pipe
Annulus
RCD
Formation
Mud pump
Drill collar
Coriolis meterChoke
Casing
Pressure sensors
120601 2159 mm times G105
1206011778mm (120601 78 mm) times G105
120601127 mm (120601 78 mm) times G105
z0 m
3600 m
4000 m
QL
Figure 4 Schematic diagram of the gas influx process
to the gas void fraction As a result when the back pressureat the wellhead is changed by adjusting the choke the wavevelocity in the two-phase drilling fluid and the distributionof void fraction at different depth of the annulus will divergeThe calculation results affected by the back pressure arepresented
Figures 5 and 6 show the distributions of void fractionand variations of wave velocity along the flow directionin the annulus when the back pressure at the wellhead is01MPa 08MPa 15MPa 25MPa 50MPa and 65MPa
respectively It can be seen that the void fraction significantlyincreases along the flow direction in the annulus Converselythe wave velocity shows a remarkable decrease tendency Forinstance BP = 15MPa at the wellhead the wave velocityis 9012ms and the void fraction is 0611 while in thebottomhole the wave velocity reaches 73142ms and voidfraction is reduced to 0059 However a sudden increase inwave velocity is observed in Figure 5 when the back pressureis 01MPa at the position close towellhead as the void fractionis further increased Moreover the influence of back pressure
10 Mathematical Problems in Engineering
10075500 254000
3000
2000
1000
0
BP = 01 MPaBP = 08 MPaBP = 15 MPa
BP = 25 MPa
BP = 65 MPaBP = 5 MPa
H(m
)
Q = 9m3h w = 50Hz = Re
120601G ()
Cvm
Figure 5 Void fraction distribution at different back pressures
on the void fraction is larger at the position close to thewellhead than that in the bottomhole
This can be explained from the viewpoints of mixturedensity and compressibility of two-phase fluid and the pres-sure drop along the flow direction in the wellbore In thelow void fraction range the gas phase is dispersed in theliquid as bubble so the wave velocity is influenced greatly bythe added gas phase According to the EOS if gas invadesinto the wellbore with a small amount in the bottomholethe density of the drilling mud has little variation while thecompressibility increases obviously which makes the wavevelocity decreased Then the two-phase drilling mud flowfrom the bottomhole to the wellhead along the annulus witha drop of pressure caused by potential energy change kineticenergy and frictional loss which leads to an increase in voidfractionWhen the void fraction is increased both the densityand the compressibility of the two-phase fluid change slightlyresulting in a flat decrease in wave velocity along the flowdirection With the further increasing in void fraction thebubbly flow turns into slug flow according to the flow patterntransient criteria of Orkiszewski which shows that the flowpattern is dependent on the void fraction Generally speak-ing the gas-drilling mud slug flow is composed of liquidand gas slugs In the preliminary slug flow the liquid slugsare much longer than the gas slugs so the wave velocity isdetermined primarily by the wave velocities in the liquidslugs which are hardly affected by the void fraction At theposition close to the wellhead the pressure of two-phase flowfluid can be reduced to a low value which approaches theback pressure The compressibility of gas will be improvedat the low pressure It results in significant increase in void
0 250 500 750 10004000
3000
2000
1000
0
H(m
)BP = 01 MPaBP = 08 MPaBP = 15 MPa
BP = 25 MPa
BP = 65 MPaBP = 5 MPa
c (ms)
Q = 9m3h w = 50Hz = ReCvm
Figure 6 Wave velocity variations at different back pressures
fraction As the void fraction increases greatly at the positionclose to the wellhead gas slugs becomemuch longer than theliquid slugs It is assumed that the actual wave velocity in slugflow happens to play a leading role when the void fractionincreases to some extent For the decrease in liquid holdupin the gas slug with the increasing in void fraction the wavevelocity in the gas slug is infinitely close to that in pure gaswhich is equivalent to an increase in wave velocity in the gasslug Meanwhile the gas content in liquid slug also increasesand results in a lower wave velocity in the liquid slug As aresult a slight increase in wave velocity appears
Figures 7 and 8 present the influence of back pressureon the void fraction and wave velocity with respect to theparameters of well depth As the back pressure increases thevoid fraction at different depth of the annulus is reducedgradually while the wave velocity in the two-phase flow tendsto increase Analytical results show that the increased backpressure is equivalent to be applied to the entire encloseddrilling fluid cyclical systemThe pressure transmits from thewellhead to the bottomhole therefore the annular pressurein the entire wellbore is increased According to the EOSthe density of gas increases and the compressibility of gasdecreases with the increasing of gas pressure So the lossof interphase momentum and energy exchange is reducedand the interphase momentum exchange is promoted Itcontributes to the increase in wave velocity with the increasein gas pressure In addition due to lower compressibility oftwo-phase flow medium under high pressure the increasetendency of pressure wave velocity and the decrease tendencyof void fraction are slowed down in the high back pressurerange
Mathematical Problems in Engineering 11
0
25
50
75
100
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
0 25 75 105
Q = 9m3h w = 50Hz = Re
120601G
()
Cvm
Figure 7 Effect of back pressure on the void fraction
52 Effect of Gas Influx Rate on Wave Velocity Figures 9and 10 graphically interpret the distributions of void fractionand variations of wave velocity along the flow direction inthe annulus When gas influx occurs in the bottomhole gasinvades into the wellbore and migrates from the bottomholeto the wellhead along the flow direction At a low gas influxrate it is extremely obvious that the void fraction and wavevelocity first slightly change in a comparatively smooth valuethen change sharply It is because of the rapid expansion ofgas volume with the decreasing in pressure near the wellheadthat the void fraction increases sharply and the wave velocitydecreases obviously at the same time But under the highbottomhole pressure (up to 50MPa) the compressibilityof the gas is low This results in a slight change in voidfraction and wave velocity at the position far away from thewellhead Since the compressible component increases withthe increase in the gas influx rate the compressibility of thegas and drilling mud two-phase fluid is improved So thevariations of void fraction and wave velocity become moreprominent Also the void fraction still shows an increasetendency that is steady first and then sharp It acts inaccordance with the variation of void fraction at a low gasinflux rate At a low gas influx rate if the void fraction atthe position close to the wellhead can not increase to a highextent the wave velocity always shows a decrease tendencyAt a high gas influx rate such as the 119876 = 8312m3h thewave velocity tends to increase because the void fraction inthe wellhead is increased to a high extent In conclusionthe wave pressure is sensitive to the void fraction and thevoid fraction is dominated by influx rate and pressure in theannulus especially the influx rate
0 25 75 1050
300
600
900
1200
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
c(m
s)
Q = 9m3h w = 50Hz = ReCvm
Figure 8 Effect of back pressure on the pressure wave velocity
With the influx of gas the mixing of gas and drillingmud occurs in the annulus and the corresponding interfacialtransfer of momentum and mass causes an increase in gasphase void fraction and a decrease in pressure wave velocityas shown in Figures 11 and 12 Within the range of low gasinflux rate the wave velocity decreases significantly It isbecause of this that the compressibility of the gas increasesremarkably and the medium appears to be of high elasticitythough the density of gas-drilling mud two-phase flowchanges slightly With the increase in the gas influx rate andcorresponding increase in the void fraction in the annulusthe compressibility of the two-phase unceasingly increaseswhich promotes the momentum and energy exchange inthe interface So the pressure wave continuously decreasesWhen the void fraction increases to some extent following theincrease in the gas influx rate the decrease in wave velocityin the liquid slug is gradually less than the increase in wavevelocity in the gas slug thus the decrease of wave velocityis slowed down for the growth of gas slug Especially in thewellhead a slight increase in wave velocity is observed
53 Effect of Virtual Mass Force on Wave Velocity Figure 13illustrates the effect of virtual mass force on the wave velocitywith the parameter of gas influx rate As shown in Figure 15under the same gas influx rate (0594m3h 1098m3h and1768m3h resp) the wave velocity curve of 119862vm = Rediverges from the curve of 119862vm = 0 at the position close tothewellhead whereas thewave velocity curve of the two typesis almost coincided below the position of 119867 = 500m Thisdivergence between the two types of curves is connected withthe fact that significance of influence of interphase virtual
12 Mathematical Problems in Engineering
0 25 50 75 1004000
3000
2000
1000
0
Q = 0058m3h Q = 1228m3hQ = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
H(m
)
BP = 01 MPaw = 50Hz = Re
120601G ()
Cvm
Figure 9 Void fraction distribution at different gas influx rates
mass forces increases together with the increase of the relativemotion in the interphase It is evident that the bottomholepressure is hundreds of times higher than the wellheadpressure As a result the void fraction at the position closeto the wellhead increases sharply meanwhile the interphasemomentum and energy exchanges are promoted The virtualmass force can be described as the transfer of momentumbetween the gas phase and drilling mud phase caused bythe relative motion in the interphase If the relative motionin the interphase is quite weak the value of virtual massforce will intend to approach 0 and the influence on thewave velocity can be ignored However if the relative motionis rather intense the effect of virtual mass force on thewave velocity should not be ignored Furthermore taking thevirtual mass force into account the dispersion characteristicof the pressure wave weakens obviously Compared with thepressure wave velocity calculated by ignoring the effect ofvirtual mass force the calculated pressure wave velocity witha consideration of the virtual mass force is lower Thereforeit is necessary to analyze the effect of the virtual mass forceon the wave velocity at the position close to the wellhead inMPD operations
54 Effect of Disturbance Angular Frequency onWave VelocityFigure 14 presents the pressure wave velocity in the annulusat different angular frequencies According to Figures 5 and9 the void fraction first slightly increases in the bottomholeand then sharply increases along the flow direction at theposition close to the wellhead At a fixed angular frequencythis phenomenon results in an overall decrease in the pressurewave velocity except for the section near the wellhead At the
0 500 1000 1500 20004000
3000
2000
1000
0
c (ms)
H(m
)Q = 0058m3h Q = 1228m3h
Q = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
BP = 01 MPaw = 50Hz = ReCvm
Figure 10 Wave velocity variations at different gas influx rates
section close to the wellhead an opposite change trend ofpressure wave velocity is observed for the transition of flowpattern from bubbly flow to slug flow due to the continuousincrease in void fraction along the flow direction In additionit can be clearly seen from the curves that the wave velocityincreases accompanying with the increase in the angularfrequency above the position of 119867 = 500m This propertyis not very distinct at the position below 119867 = 500m for thelow void fraction
Figure 15 shows the effect of frequency on the wavevelocity in the gas-drilling mud flow (119862vm = 0)The curves ofthe wave velocity of different positions reveal that the prop-agation velocity of pressure disturbances increases togetherwith the growth of the angular frequency (0 lt 119908 lt 500Hz)It proves that the pressure wave has an obvious dispersioncharacteristic in the two-phase flow As the angular frequencyincreases in the range of less than 500Hz there is sufficienttime to carry out the exchange of energy and momentumbetween two phases It achieves a state of mechanical andthermodynamic equilibrium between the two phases Sothe wave velocity increases gradually with the increase inthe angular frequency at different depths of the annulus Itis considered that the wave velocity is mainly affected bythe interphase mechanical and thermodynamic equilibriumat low anglular frequencies When the angular frequencyreaches the value of 119908 = 500Hz the pressure wave velocityachieves a constant value and remains on this level regardlessof the further growth in angular frequency 119908 With theincrease in angular frequency there is not enough time forenergy and momentum exchange between the gas-drillingmud two phases to reach themechanical and thermodynamic
Mathematical Problems in Engineering 13
0
25
50
75
100
125
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
Q (m3h)
120601G
()
0 25 5 75 10
BP = 01 MPaw = 50Hz = ReCvm
Figure 11 Effect of gas influx rate on the void fraction
equilibrium state and thus the wave dispersion does notexist At a high angular frequency the wave velocity is mainlydominated by the mechanical and thermal nonequilibriumin the flow and keeps almost unchanged when the angularfrequency is further increased This is consistent with theinfluences of the angular frequency in the horizontal pipe[28] Moreover the effect of virtual mass force is also shownin this figure (119862vm = Re) It is observed that the wave velocityis significantly reduced when the virtual mass force is takeninto consideration in the calculation of the wave velocity atthe position close to the wellhead such as 119867 = 0m It canbe explained by the reduction of dispersion characteristic ofa pressure wave Along the wellbore from the wellhead to thebottomhole the distinction gradually decreases
6 Conclusions
With full consideration of the important factors such asvirtual mass force drag force gas void fraction pressuretemperature and angular frequency a united wave velocitymodel has been proposed based on pressure drop gradientequations in MPD operations gas-liquid two-fluid modelthe gas-drilling mud equations of state and small pertur-bation theory Solved by the fourth-order explicit Runge-Kutta method the model is used to predict wave velocityfor different back pressures and gas influx rates in MPDoperations The main conclusions can be summarized asfollows
(1) With the introduction of virtual mass force and dragforce the united model agrees well with the previousexperimental data The united model can be used to
0
280
560
840
1120
1400
Q (m3h)0 25 5 75 10
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
c(m
s)
BP = 01 MPaw = 50Hz = ReCvm
Figure 12 Effect of gas influx rate on the pressure wave velocity
accurately calculate the wave velocity in the annulusThe application of the model will be beneficial tofurther study the wave velocity at different gas influxrates and back pressures in MPD operations reducenonproductive times and provide a reference for thedrilling operations in extremely narrow porefracturewindows existing conditions
(2) The wave velocity and void fraction have differentvalues with respect to well depth In the annulusthe drop of pressure causes an increase in the voidfraction along the flow direction The void fractionincreases first slightly and then sharply Correspond-ingly the wave velocity first gradually decreases in thbubbly flow and preliminary slug flow and then thewave velocity slightly increases accompanying withthe increase in the relative length ratio of gas slugto the liquid slug for the continuous increase in voidfraction at the position close to the wellhead Theminimum wave velocity appears in the long gas slugflow
(3) When the back pressure in the MPD operationsincreases the void fraction at different depths of theannulus is reduced gradually while the wave velocityin the two-phase flow tends to increase Moreoverthe influence of back pressure on the void fraction isgreater at a position close to the wellhead than that inthe bottomhole because of the great decrease in thepressure from the bottomhole to the wellhead Alsothe effect of back pressure on void fraction and wavevelocity is decreased in the high back pressure range
14 Mathematical Problems in Engineering
0 300 600 900 12004000
3000
2000
1000
225
180
135
90
45
0
BP = 01 MPaw = 50Hz
Q = 0594m3h = ReQ = 0594m3h = 0
Q = 1098m3h = ReQ = 1768m3h = ReQ = 1098m3h = 0
Q = 1768m3h = 0
c (ms)
H(m
)
CvmCvm
Cvm
Cvm
Cvm
Cvm
Figure 13 Effect of virtual mass force on the pressure wave velocity
(4) The wave velocity is sensitive to the void fraction butthe void fraction is dominated by gas influx rate andpressure in the annulus especially the gas influx rateSince the compressibility of the gas and drilling mudtwo-phase fluid is improved with the increase in theinflux rate the void fraction increases greatly andthe wave velocity decreases significantly within thelow gas influx rate range When the void fraction isincreased to some extent following the increase in thegas influx rate the decrease of wave velocity is slowedfor the growth of gas slug Especially at the wellheada slight increase in wave velocity is observed at a highgas influx rate for the sharp increase in void fraction
(5) It is necessary to analyze the effect of the virtualmass force on the wave velocity in MPD operationsEspecially at the position close to the wellhead theeffect of virtual mass force is more prominent for theintense phase interactions Taking the virtual massforce into account the dispersion characteristic of thepressure wave weakens obviously Compared with theresults calculated by ignoring the effect of virtualmassforce the calculated pressure wave velocity with aconsideration of the virtual mass force is lower
(6) The effect of angular frequency on the propagationvelocity of pressure wave appears at a low angularfrequency The propagation velocity of pressure dis-turbances increases together with the growth of theangular frequency (0 lt 119908 lt 500Hz) When theangular frequency reaches the value of 119908 = 500Hzthe pressure wave velocity achieves a constant value
0 240 480 720 960 12004000
3000
2000
1000
560
420
280
140
0
Q = 0727m3h BP = 01 MPa
w = 50Hz = Rew = 50Hz = 0w = 300Hz = 0
w = 800Hz = 0w = 1600Hz = 0w = 5000Hz = 0
c (ms)
H(m
)CvmCvmCvm
CvmCvmCvm
Figure 14 Wave velocity variations at different angular frequencies
and remains on this level regardless of the furthergrowth in angular frequency 119908
Nomenclature
A Annulus effective cross area (m2)c119866 Wave velocity in gas (ms)
c119871 Wave velocity in Drilling mud (ms)
C119863 The coefficient of drag force
D Annulus effective diameter (m)D119894 Diameter of the inner pipe (m)
D119900 Diameter of the outer pipe (m)
f119866 Shear stresses coefficient of gas interface
f119871 Shear stresses coefficient of drilling mud
interfaceF119911 Pressure applied to the two-phase flow (N)
g Acceleration due to gravity (m2s)g119888 Conversion factor
g119879 Geothermal temperature gradient (Km)
h One discrete length of annulus (m)k119890 Annulus effective roughness (m)
k119900 Roughness of outer pipe (m)
k119894 Roughness of inner pipe (m)
H The depth of annulus (m)L119902 Air bubbles length of slug (m)
119872Gi Momentum transfer in gas interface (Nm3)119872Li Momentum transfer in liquid interface (Nm3)119872
ndLi The none-drag-force (virtual mass force)
(Nm3)119872
119889
Li The momentum transfer term caused by thedrag force (Nm3)
Mathematical Problems in Engineering 15
100 1000 100000
450
900
1350
1800
0w (Hz)
0 m = 0 0 m = Re1000 m = 0
2000 m = 0
4000 m = 0
1000 m = Re2000 m = Re4000 m = Re
c(m
s)
Q = 0727m3h BP = 01 MPaCvmCvm
CvmCvmCvm
CvmCvmCvm
Figure 15 Effect of angular frequency on wave velocity
N Parameter combiningThompsonenergy effects
119901Gi Gas interface pressure (MPa)119901Li Liquid interface pressure (MPa)p Pressure (MPa)p0 Initial pressure of annulus (MPa)
p119888 Critical pressure (kPa)
p119871 Pressure of drilling mud (MPa)
p119903 Reduced pressure
q119898 Volumetric flow rate of gas and drilling
mud (m3s)q119866 Velocity of the gas (m3s)
q119871 Velocity of the drilling mud (m3s)
Q119871 Drilling mud rate within the annulus
(m3h)r Average diameter of the bubble (m)R Constant of EOS (JKgK)R119902 Air bubbles radius of slug (m)
R119887 The bubble diameter (m)
T Temperature (K)T119888 Critical temperature (K)
T119903 Reduced temperature
T119890119894 Undisturbed formation temperature at a
depth (K)T119890119887ℎ
Undisturbed formation temperature ofwellhead (K)
T119891119887ℎ
Undisturbed formation temperature atthe bottomhole (K)
v119898 Gas and drilling mud flow velocity
(ms)
v119866 Gas flow velocity (ms)
v119904 Slip velocity (ms)
v119904119892 Superficial gas velocity (ms)
v119871 Drilling mud flow velocity (ms)
W119898 Mass flow velocity (kgm3)
Z119866 Gas deviation factor
z0 Initial calculation height of annulus (m)
z119887ℎ Total well depth from surface (m)
z Variable of distance from surface (m)
Greek Letters
120588
119866 Gas density (kgm3)
120591
119891119903
Li Shear stresses of drilling mud noninterfacial(Nm2)
120591
119891119903
119871 Shear stresses of drilling mud interface (Nm2)
120591
Re119871 Reynolds stress of drilling mud (Nm2)
120591
Re119871 Reynolds stress of drilling mud interface
(Nm2)120591
Re119866 Reynolds stress of gas interface (Nm2)
120591
119891119903
119866 Shear stresses of gas interface (Nm2)
120572VM Acceleration due to virtual mass forcecoefficient (ms2)
120591
119871 Shear stresses of drilling mud along wall
(Nm2)120591
119866 Shear stresses of gas along wall (Nm2)
120591
Re119866 Reynolds stress of gas (Nm2)
120591
119891 Frictional pressure gradient (Pam)
120591
119908 Frictional pressure coefficient
120588
119898 Gas and drilling density (kgm3)
120588
0 Density under standard atmospheric pressure
(kgm3)120588
119871 Drilling mud density (kgm3)
120590
119904 Surface tension (Nm2)
Φ
119866 Gas void fraction
Φ
119871 Drilling mud holdup
120588
119903 Reduced density
120583
119871 Viscosity of drilling mud (Pa sdot s)
Subscripts
BHP Bottomhole pressureBP Back pressureEOS Equations of stateMPD Managed pressure drillingRCD The rotating control deviceR-K4 The fourth-order explicit Runge-Kutta
Subscripts of Graph
BP Back pressure (MPa)c Wave velocity in gas and drilling mud
two-phase flow (ms)119862vm The coefficient of virtual mass force
16 Mathematical Problems in Engineering
H Depth of annulus (m)Q Gas influx rate at the bottomhole (m3h)Re The value of virtual mass force
coefficient according to (28) or (34)119908 Angle frequency (Hz)Φ
119866 Void fraction ()
Acknowledgment
Research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 and 51274170)Without their support this work would not have beenpossible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of airinjection in annuli to manage pressure during cementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling conferenceand Exhibition San Diego Calif USA March 2012
[4] W A Bacon Consideration of compressibility effects for applied-back-pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Tex USA 2011
[5] J Choe C Hans and J Wold ldquoA modified two-phase well-control model and its computer applications as a training andeducational toolrdquo SPE Computer Applications vol 2 pp 14ndash201997
[6] J A Tarvin IWalton and PWand ldquoAnalysis of a gas kick takenin a deep well drilled with oil-based mudrdquo in Proceedings ofthe Annual Technical Conference and Exhibition of the Societyof Petroleum Engineers pp 255ndash264 Dallas Tex USA 1991
[7] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[8] J Zhou and G Nygaard ldquoAutomatic model-based controlscheme for stabilizing pressure during dual-gradient drillingrdquoJournal of Process Control vol 21 no 8 pp 1138ndash1147 2011
[9] A Ochoa O Acevedo L Nieto J E Lambarria and H PerezldquoSuccessful application of MPD (Managed Pressure Drilling)for prevention controland detection of borehole ballooning intight gas reservoir in Cuervito Field Mexicordquo in Proceedings ofthe Canadian Unconventional Resources Conference (CURC rsquo11)pp 172ndash185 November 2011
[10] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[11] R Ashena and J Moghadasi ldquoBottom hole pressure estimationusing evolved neural networks by real coded ant colony opti-mization and genetic algorithmrdquo Journal of Petroleum Scienceand Engineering vol 77 no 3-4 pp 375ndash385 2011
[12] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvanced Materials Research vol 468-471 pp 1736ndash1742 2012
[13] I Zubizarreta ldquoPore pressure evolution core damage and trip-ping out schedulesf a computational fluid dynamics approachrdquoin Proceedings of the SPEIADC Drilling Conference and Exhibi-tion Amsterdam The Netherlands March 2013
[14] C H Whitson ldquoCyclic shut-in eliminates liquid-loading in gaswellsrdquo in Proceedings of the SPEEAGE European Unconven-tional Resources Conference and Exhibition Vienna AustriaMarch 2012
[15] G M de Oliveira A T Franco C O R Negrao A LMartins and R A Silva ldquoModeling and validation of pressurepropagation in drilling fluids pumped into a closed wellrdquoJournal of Petroleum Science and Engineering vol 103 pp 61ndash71 2012
[16] Y Sato and H Kanki ldquoFormulas for compression wave andoscillating flow in circular piperdquo Applied Acoustics vol 69 no1 pp 1ndash11 2008
[17] J Xie B Yu X Zhang Q Shao and X Song ldquoNumericalsimulation of gas-liquid-solid three-phase flow in deep wellsrdquoAdvances in Mechanical Engineering vol 2013 Article ID951298 10 pages 2013
[18] W Bacon A Tong O Gabaldon C Sugden and P V Surya-narayana ldquoAn improved dynamic well control response to a gasinflux in managed pressure drilling operationsrdquo in Proceedingsof the IADCSPE Drilling Conference and Exhibition San DiegoCalifornia USA March 2012
[19] J I Hage and D ter Avest ldquoBorehole acoustics applied to kickdetectionrdquo Journal of Petroleum Science and Engineering vol 12no 2 pp 157ndash166 1994
[20] A M R Hopkins Fluid dynamics and surface pressure fluctua-tions of two-dimensional turbulent boundary layers over denselydistributed surface roughness [PhD thesis] Virginia PolytechnicInstitute and State University Blacksburg Va USA 2010
[21] Z S Fu ldquoThe gas cut detect technology during drilling in SovietUnionrdquo Petroleum Drilling Techniques vol 18 no 1 pp 19ndash211990
[22] X F Li C X Guan and X X Sui ldquoThe theory of gas influxdetection of pressure wave and its applicationrdquo Acta PetroleiSinica vol 18 no 3 pp 128ndash133 1997
[23] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[24] A E Ruggles R T Lahey Jr D A Drew and H A ScartonldquoInvestigation of the propagation of pressure perturbations inbubbly airwater flowsrdquo Journal of Heat Transfer vol 110 no 2pp 494ndash499 1988
[25] H J W M Legius H E A Van Den Akker and T NarumoldquoMeasurements on wave propagation and bubble and slugvelocities in cocurrent upward two-phase flowrdquo ExperimentalThermal and Fluid Science vol 15 no 3 pp 267ndash278 1997
[26] K Miyazaki and I Nakajima ldquoPropagation of pressure wavein NItrogen-Mercury two-phase systemrdquo Journal of NuclearScience and Technology vol 9 no 5 pp 315ndash318 1972
[27] B Bai L Guo and X Chen ldquoPressure fluctuation for air-watertwo-phase flowrdquo Journal of Hydrodynamics vol 18 no 4 pp476ndash482 2003
[28] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
Mathematical Problems in Engineering 17
[29] R F Tangren C H Dodge and H S Seifert ldquoCompressibilityeffects in two-phase flowrdquo Journal of Applied Physics vol 20 no7 pp 637ndash645 1949
[30] G B Wallis One Dimensional Two Phase Flow McGraw HillNew York NY USA 1969
[31] D L Nguyen E R F Winter and M Greiner ldquoSonic velocityin two-phase systemsrdquo International Journal ofMultiphase Flowvol 7 no 3 pp 311ndash320 1981
[32] R C Mecredy and L J Hamilton ldquoThe effects of nonequilib-rium heat mass and momentum transfer on two-phase soundspeedrdquo International Journal of Heat and Mass Transfer vol 15no 1 pp 61ndash72 1972
[33] K H Ardron and R B Duffey ldquoAcoustic wave propagationin a flowing liquid-vapour mixturerdquo International Journal ofMultiphase Flow vol 4 no 3 pp 303ndash322 1978
[34] J Xu and T Chen ldquoAcoustic wave prediction in flowing steam-water two-phase mixturerdquo International Journal of Heat andMass Transfer vol 43 no 7 pp 1079ndash1088 2000
[35] N ChungW Lin B Pei and Y Hsu ldquoModel for sound velocityin a two-phase air-water bubbly flowrdquo Nuclear Technology vol99 no 1 pp 80ndash89 1992
[36] X Xu and J Gong ldquoA unitedmodel for predicting pressurewavespeeds in oil and gas two-phase pipeflowsrdquo Journal of PetroleumScience and Engineering vol 60 no 3-4 pp 150ndash160 2008
[37] W Kuczynski ldquoCharacterization of pressure-wave propagationduring the condensation of R404A and R134a refrigerantsin pipe mini-channels that undergo periodic hydrodynamicdisturbancerdquo International Journal of Heat and Fluid Flow vol40 pp 135ndash150 2013
[38] W Kuczynski ldquoModeling of the propagation of a pressure waveduring the condensation process of R134a refrigerant in a pipeminichannel under the periodic conditions of hydrodynamicdisturbancesrdquo International Journal of Heat and Mass Transfervol 56 pp 715ndash723 2013
[39] Y Li C Li E Chen and Y Ying ldquoPressure wave propaga-tion characteristics in a two-phase flow pipeline for liquid-propellant rocketrdquoAerospace Science and Technology vol 15 no6 pp 453ndash464 2011
[40] T Kanagawa M Watanabe T Yano and S Fujikawa ldquoNon-linear wave equations for pressure wave propagation in liquidscontaining gas bubbles (comparison between two-fluid modeland mixture model)rdquo Journal of Fluid Science and Technologyvol 6 no 6 pp 836ndash849 2011
[41] EChen Y Li andYYing ldquoNumerical investigation onpressurewave propagation speed of gas-liquid two-phase flow in pumppipelinerdquo Journal of Aerospace Power vol 25 no 4 pp 754ndash7602010
[42] F Li and L He ldquoInvestigation on pressure wave propagationcharacteristics in slug flow field with increasing gas flow raterdquoPetroleum Geology and Recovery Efficiency vol 11 no 5 pp 78ndash80 2004
[43] C SMartin andM Padmanabhan ldquoPressure pulse propagationin two-component slug flowrdquo Journal of Fluids Engineering vol101 no 1 pp 44ndash52 1979
[44] R E Henry ldquoPressure wave propagation in two-phase mix-turesrdquo Chemical Engineering Progress vol 66 no 1 pp 1ndash101969
[45] X Liu B Li and Y Yue ldquoTransmission behavior of mud-pressure pulse alongwell borerdquo Journal ofHydrodynamics B vol19 no 2 pp 236ndash240 2007
[46] XWang and J Zhang ldquoThe research of pressure wave Pulsationin mud pulse transmittingrdquo Journal of Chongqing University ofScience and Technology vol 14 no 2 pp 55ndash58 2012
[47] J-W Park D A Drew and R T Lahey Jr ldquoThe analysisof void wave propagation in adiabatic monodispersed bubblytwo-phase flows using an ensemble-averaged two-fluid modelrdquoInternational Journal ofMultiphase Flow vol 24 no 7 pp 1205ndash1244 1998
[48] G Arnold Entropy and objectivity as constraints upons constitu-tive equations for two-fluid modeling of multiphase flow [PhDthesis] Rensselaer Polytechnic Institute New York NY USA1998
[49] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[50] L Yarborough and K R Hall ldquoHow to solve equation of statefor z-factorsrdquoOil and Gas Journal vol 72 no 7 pp 86ndash88 1974
[51] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86 pp 127ndash136 2012
[52] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[53] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[54] M Ishii and K Mishima ldquoTwo-fluid model and hydrodynamicconstitutive relationsrdquo Nuclear Engineering and Design vol 82no 2-3 pp 107ndash126 1984
[55] M J Sanchez Comparison of correlations for predicting pressurelosses in vertical multiphase annular flow [MS thesis] Univer-sity of Tulsa Tulsa Okla USA 1972
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
10075500 254000
3000
2000
1000
0
BP = 01 MPaBP = 08 MPaBP = 15 MPa
BP = 25 MPa
BP = 65 MPaBP = 5 MPa
H(m
)
Q = 9m3h w = 50Hz = Re
120601G ()
Cvm
Figure 5 Void fraction distribution at different back pressures
on the void fraction is larger at the position close to thewellhead than that in the bottomhole
This can be explained from the viewpoints of mixturedensity and compressibility of two-phase fluid and the pres-sure drop along the flow direction in the wellbore In thelow void fraction range the gas phase is dispersed in theliquid as bubble so the wave velocity is influenced greatly bythe added gas phase According to the EOS if gas invadesinto the wellbore with a small amount in the bottomholethe density of the drilling mud has little variation while thecompressibility increases obviously which makes the wavevelocity decreased Then the two-phase drilling mud flowfrom the bottomhole to the wellhead along the annulus witha drop of pressure caused by potential energy change kineticenergy and frictional loss which leads to an increase in voidfractionWhen the void fraction is increased both the densityand the compressibility of the two-phase fluid change slightlyresulting in a flat decrease in wave velocity along the flowdirection With the further increasing in void fraction thebubbly flow turns into slug flow according to the flow patterntransient criteria of Orkiszewski which shows that the flowpattern is dependent on the void fraction Generally speak-ing the gas-drilling mud slug flow is composed of liquidand gas slugs In the preliminary slug flow the liquid slugsare much longer than the gas slugs so the wave velocity isdetermined primarily by the wave velocities in the liquidslugs which are hardly affected by the void fraction At theposition close to the wellhead the pressure of two-phase flowfluid can be reduced to a low value which approaches theback pressure The compressibility of gas will be improvedat the low pressure It results in significant increase in void
0 250 500 750 10004000
3000
2000
1000
0
H(m
)BP = 01 MPaBP = 08 MPaBP = 15 MPa
BP = 25 MPa
BP = 65 MPaBP = 5 MPa
c (ms)
Q = 9m3h w = 50Hz = ReCvm
Figure 6 Wave velocity variations at different back pressures
fraction As the void fraction increases greatly at the positionclose to the wellhead gas slugs becomemuch longer than theliquid slugs It is assumed that the actual wave velocity in slugflow happens to play a leading role when the void fractionincreases to some extent For the decrease in liquid holdupin the gas slug with the increasing in void fraction the wavevelocity in the gas slug is infinitely close to that in pure gaswhich is equivalent to an increase in wave velocity in the gasslug Meanwhile the gas content in liquid slug also increasesand results in a lower wave velocity in the liquid slug As aresult a slight increase in wave velocity appears
Figures 7 and 8 present the influence of back pressureon the void fraction and wave velocity with respect to theparameters of well depth As the back pressure increases thevoid fraction at different depth of the annulus is reducedgradually while the wave velocity in the two-phase flow tendsto increase Analytical results show that the increased backpressure is equivalent to be applied to the entire encloseddrilling fluid cyclical systemThe pressure transmits from thewellhead to the bottomhole therefore the annular pressurein the entire wellbore is increased According to the EOSthe density of gas increases and the compressibility of gasdecreases with the increasing of gas pressure So the lossof interphase momentum and energy exchange is reducedand the interphase momentum exchange is promoted Itcontributes to the increase in wave velocity with the increasein gas pressure In addition due to lower compressibility oftwo-phase flow medium under high pressure the increasetendency of pressure wave velocity and the decrease tendencyof void fraction are slowed down in the high back pressurerange
Mathematical Problems in Engineering 11
0
25
50
75
100
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
0 25 75 105
Q = 9m3h w = 50Hz = Re
120601G
()
Cvm
Figure 7 Effect of back pressure on the void fraction
52 Effect of Gas Influx Rate on Wave Velocity Figures 9and 10 graphically interpret the distributions of void fractionand variations of wave velocity along the flow direction inthe annulus When gas influx occurs in the bottomhole gasinvades into the wellbore and migrates from the bottomholeto the wellhead along the flow direction At a low gas influxrate it is extremely obvious that the void fraction and wavevelocity first slightly change in a comparatively smooth valuethen change sharply It is because of the rapid expansion ofgas volume with the decreasing in pressure near the wellheadthat the void fraction increases sharply and the wave velocitydecreases obviously at the same time But under the highbottomhole pressure (up to 50MPa) the compressibilityof the gas is low This results in a slight change in voidfraction and wave velocity at the position far away from thewellhead Since the compressible component increases withthe increase in the gas influx rate the compressibility of thegas and drilling mud two-phase fluid is improved So thevariations of void fraction and wave velocity become moreprominent Also the void fraction still shows an increasetendency that is steady first and then sharp It acts inaccordance with the variation of void fraction at a low gasinflux rate At a low gas influx rate if the void fraction atthe position close to the wellhead can not increase to a highextent the wave velocity always shows a decrease tendencyAt a high gas influx rate such as the 119876 = 8312m3h thewave velocity tends to increase because the void fraction inthe wellhead is increased to a high extent In conclusionthe wave pressure is sensitive to the void fraction and thevoid fraction is dominated by influx rate and pressure in theannulus especially the influx rate
0 25 75 1050
300
600
900
1200
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
c(m
s)
Q = 9m3h w = 50Hz = ReCvm
Figure 8 Effect of back pressure on the pressure wave velocity
With the influx of gas the mixing of gas and drillingmud occurs in the annulus and the corresponding interfacialtransfer of momentum and mass causes an increase in gasphase void fraction and a decrease in pressure wave velocityas shown in Figures 11 and 12 Within the range of low gasinflux rate the wave velocity decreases significantly It isbecause of this that the compressibility of the gas increasesremarkably and the medium appears to be of high elasticitythough the density of gas-drilling mud two-phase flowchanges slightly With the increase in the gas influx rate andcorresponding increase in the void fraction in the annulusthe compressibility of the two-phase unceasingly increaseswhich promotes the momentum and energy exchange inthe interface So the pressure wave continuously decreasesWhen the void fraction increases to some extent following theincrease in the gas influx rate the decrease in wave velocityin the liquid slug is gradually less than the increase in wavevelocity in the gas slug thus the decrease of wave velocityis slowed down for the growth of gas slug Especially in thewellhead a slight increase in wave velocity is observed
53 Effect of Virtual Mass Force on Wave Velocity Figure 13illustrates the effect of virtual mass force on the wave velocitywith the parameter of gas influx rate As shown in Figure 15under the same gas influx rate (0594m3h 1098m3h and1768m3h resp) the wave velocity curve of 119862vm = Rediverges from the curve of 119862vm = 0 at the position close tothewellhead whereas thewave velocity curve of the two typesis almost coincided below the position of 119867 = 500m Thisdivergence between the two types of curves is connected withthe fact that significance of influence of interphase virtual
12 Mathematical Problems in Engineering
0 25 50 75 1004000
3000
2000
1000
0
Q = 0058m3h Q = 1228m3hQ = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
H(m
)
BP = 01 MPaw = 50Hz = Re
120601G ()
Cvm
Figure 9 Void fraction distribution at different gas influx rates
mass forces increases together with the increase of the relativemotion in the interphase It is evident that the bottomholepressure is hundreds of times higher than the wellheadpressure As a result the void fraction at the position closeto the wellhead increases sharply meanwhile the interphasemomentum and energy exchanges are promoted The virtualmass force can be described as the transfer of momentumbetween the gas phase and drilling mud phase caused bythe relative motion in the interphase If the relative motionin the interphase is quite weak the value of virtual massforce will intend to approach 0 and the influence on thewave velocity can be ignored However if the relative motionis rather intense the effect of virtual mass force on thewave velocity should not be ignored Furthermore taking thevirtual mass force into account the dispersion characteristicof the pressure wave weakens obviously Compared with thepressure wave velocity calculated by ignoring the effect ofvirtual mass force the calculated pressure wave velocity witha consideration of the virtual mass force is lower Thereforeit is necessary to analyze the effect of the virtual mass forceon the wave velocity at the position close to the wellhead inMPD operations
54 Effect of Disturbance Angular Frequency onWave VelocityFigure 14 presents the pressure wave velocity in the annulusat different angular frequencies According to Figures 5 and9 the void fraction first slightly increases in the bottomholeand then sharply increases along the flow direction at theposition close to the wellhead At a fixed angular frequencythis phenomenon results in an overall decrease in the pressurewave velocity except for the section near the wellhead At the
0 500 1000 1500 20004000
3000
2000
1000
0
c (ms)
H(m
)Q = 0058m3h Q = 1228m3h
Q = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
BP = 01 MPaw = 50Hz = ReCvm
Figure 10 Wave velocity variations at different gas influx rates
section close to the wellhead an opposite change trend ofpressure wave velocity is observed for the transition of flowpattern from bubbly flow to slug flow due to the continuousincrease in void fraction along the flow direction In additionit can be clearly seen from the curves that the wave velocityincreases accompanying with the increase in the angularfrequency above the position of 119867 = 500m This propertyis not very distinct at the position below 119867 = 500m for thelow void fraction
Figure 15 shows the effect of frequency on the wavevelocity in the gas-drilling mud flow (119862vm = 0)The curves ofthe wave velocity of different positions reveal that the prop-agation velocity of pressure disturbances increases togetherwith the growth of the angular frequency (0 lt 119908 lt 500Hz)It proves that the pressure wave has an obvious dispersioncharacteristic in the two-phase flow As the angular frequencyincreases in the range of less than 500Hz there is sufficienttime to carry out the exchange of energy and momentumbetween two phases It achieves a state of mechanical andthermodynamic equilibrium between the two phases Sothe wave velocity increases gradually with the increase inthe angular frequency at different depths of the annulus Itis considered that the wave velocity is mainly affected bythe interphase mechanical and thermodynamic equilibriumat low anglular frequencies When the angular frequencyreaches the value of 119908 = 500Hz the pressure wave velocityachieves a constant value and remains on this level regardlessof the further growth in angular frequency 119908 With theincrease in angular frequency there is not enough time forenergy and momentum exchange between the gas-drillingmud two phases to reach themechanical and thermodynamic
Mathematical Problems in Engineering 13
0
25
50
75
100
125
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
Q (m3h)
120601G
()
0 25 5 75 10
BP = 01 MPaw = 50Hz = ReCvm
Figure 11 Effect of gas influx rate on the void fraction
equilibrium state and thus the wave dispersion does notexist At a high angular frequency the wave velocity is mainlydominated by the mechanical and thermal nonequilibriumin the flow and keeps almost unchanged when the angularfrequency is further increased This is consistent with theinfluences of the angular frequency in the horizontal pipe[28] Moreover the effect of virtual mass force is also shownin this figure (119862vm = Re) It is observed that the wave velocityis significantly reduced when the virtual mass force is takeninto consideration in the calculation of the wave velocity atthe position close to the wellhead such as 119867 = 0m It canbe explained by the reduction of dispersion characteristic ofa pressure wave Along the wellbore from the wellhead to thebottomhole the distinction gradually decreases
6 Conclusions
With full consideration of the important factors such asvirtual mass force drag force gas void fraction pressuretemperature and angular frequency a united wave velocitymodel has been proposed based on pressure drop gradientequations in MPD operations gas-liquid two-fluid modelthe gas-drilling mud equations of state and small pertur-bation theory Solved by the fourth-order explicit Runge-Kutta method the model is used to predict wave velocityfor different back pressures and gas influx rates in MPDoperations The main conclusions can be summarized asfollows
(1) With the introduction of virtual mass force and dragforce the united model agrees well with the previousexperimental data The united model can be used to
0
280
560
840
1120
1400
Q (m3h)0 25 5 75 10
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
c(m
s)
BP = 01 MPaw = 50Hz = ReCvm
Figure 12 Effect of gas influx rate on the pressure wave velocity
accurately calculate the wave velocity in the annulusThe application of the model will be beneficial tofurther study the wave velocity at different gas influxrates and back pressures in MPD operations reducenonproductive times and provide a reference for thedrilling operations in extremely narrow porefracturewindows existing conditions
(2) The wave velocity and void fraction have differentvalues with respect to well depth In the annulusthe drop of pressure causes an increase in the voidfraction along the flow direction The void fractionincreases first slightly and then sharply Correspond-ingly the wave velocity first gradually decreases in thbubbly flow and preliminary slug flow and then thewave velocity slightly increases accompanying withthe increase in the relative length ratio of gas slugto the liquid slug for the continuous increase in voidfraction at the position close to the wellhead Theminimum wave velocity appears in the long gas slugflow
(3) When the back pressure in the MPD operationsincreases the void fraction at different depths of theannulus is reduced gradually while the wave velocityin the two-phase flow tends to increase Moreoverthe influence of back pressure on the void fraction isgreater at a position close to the wellhead than that inthe bottomhole because of the great decrease in thepressure from the bottomhole to the wellhead Alsothe effect of back pressure on void fraction and wavevelocity is decreased in the high back pressure range
14 Mathematical Problems in Engineering
0 300 600 900 12004000
3000
2000
1000
225
180
135
90
45
0
BP = 01 MPaw = 50Hz
Q = 0594m3h = ReQ = 0594m3h = 0
Q = 1098m3h = ReQ = 1768m3h = ReQ = 1098m3h = 0
Q = 1768m3h = 0
c (ms)
H(m
)
CvmCvm
Cvm
Cvm
Cvm
Cvm
Figure 13 Effect of virtual mass force on the pressure wave velocity
(4) The wave velocity is sensitive to the void fraction butthe void fraction is dominated by gas influx rate andpressure in the annulus especially the gas influx rateSince the compressibility of the gas and drilling mudtwo-phase fluid is improved with the increase in theinflux rate the void fraction increases greatly andthe wave velocity decreases significantly within thelow gas influx rate range When the void fraction isincreased to some extent following the increase in thegas influx rate the decrease of wave velocity is slowedfor the growth of gas slug Especially at the wellheada slight increase in wave velocity is observed at a highgas influx rate for the sharp increase in void fraction
(5) It is necessary to analyze the effect of the virtualmass force on the wave velocity in MPD operationsEspecially at the position close to the wellhead theeffect of virtual mass force is more prominent for theintense phase interactions Taking the virtual massforce into account the dispersion characteristic of thepressure wave weakens obviously Compared with theresults calculated by ignoring the effect of virtualmassforce the calculated pressure wave velocity with aconsideration of the virtual mass force is lower
(6) The effect of angular frequency on the propagationvelocity of pressure wave appears at a low angularfrequency The propagation velocity of pressure dis-turbances increases together with the growth of theangular frequency (0 lt 119908 lt 500Hz) When theangular frequency reaches the value of 119908 = 500Hzthe pressure wave velocity achieves a constant value
0 240 480 720 960 12004000
3000
2000
1000
560
420
280
140
0
Q = 0727m3h BP = 01 MPa
w = 50Hz = Rew = 50Hz = 0w = 300Hz = 0
w = 800Hz = 0w = 1600Hz = 0w = 5000Hz = 0
c (ms)
H(m
)CvmCvmCvm
CvmCvmCvm
Figure 14 Wave velocity variations at different angular frequencies
and remains on this level regardless of the furthergrowth in angular frequency 119908
Nomenclature
A Annulus effective cross area (m2)c119866 Wave velocity in gas (ms)
c119871 Wave velocity in Drilling mud (ms)
C119863 The coefficient of drag force
D Annulus effective diameter (m)D119894 Diameter of the inner pipe (m)
D119900 Diameter of the outer pipe (m)
f119866 Shear stresses coefficient of gas interface
f119871 Shear stresses coefficient of drilling mud
interfaceF119911 Pressure applied to the two-phase flow (N)
g Acceleration due to gravity (m2s)g119888 Conversion factor
g119879 Geothermal temperature gradient (Km)
h One discrete length of annulus (m)k119890 Annulus effective roughness (m)
k119900 Roughness of outer pipe (m)
k119894 Roughness of inner pipe (m)
H The depth of annulus (m)L119902 Air bubbles length of slug (m)
119872Gi Momentum transfer in gas interface (Nm3)119872Li Momentum transfer in liquid interface (Nm3)119872
ndLi The none-drag-force (virtual mass force)
(Nm3)119872
119889
Li The momentum transfer term caused by thedrag force (Nm3)
Mathematical Problems in Engineering 15
100 1000 100000
450
900
1350
1800
0w (Hz)
0 m = 0 0 m = Re1000 m = 0
2000 m = 0
4000 m = 0
1000 m = Re2000 m = Re4000 m = Re
c(m
s)
Q = 0727m3h BP = 01 MPaCvmCvm
CvmCvmCvm
CvmCvmCvm
Figure 15 Effect of angular frequency on wave velocity
N Parameter combiningThompsonenergy effects
119901Gi Gas interface pressure (MPa)119901Li Liquid interface pressure (MPa)p Pressure (MPa)p0 Initial pressure of annulus (MPa)
p119888 Critical pressure (kPa)
p119871 Pressure of drilling mud (MPa)
p119903 Reduced pressure
q119898 Volumetric flow rate of gas and drilling
mud (m3s)q119866 Velocity of the gas (m3s)
q119871 Velocity of the drilling mud (m3s)
Q119871 Drilling mud rate within the annulus
(m3h)r Average diameter of the bubble (m)R Constant of EOS (JKgK)R119902 Air bubbles radius of slug (m)
R119887 The bubble diameter (m)
T Temperature (K)T119888 Critical temperature (K)
T119903 Reduced temperature
T119890119894 Undisturbed formation temperature at a
depth (K)T119890119887ℎ
Undisturbed formation temperature ofwellhead (K)
T119891119887ℎ
Undisturbed formation temperature atthe bottomhole (K)
v119898 Gas and drilling mud flow velocity
(ms)
v119866 Gas flow velocity (ms)
v119904 Slip velocity (ms)
v119904119892 Superficial gas velocity (ms)
v119871 Drilling mud flow velocity (ms)
W119898 Mass flow velocity (kgm3)
Z119866 Gas deviation factor
z0 Initial calculation height of annulus (m)
z119887ℎ Total well depth from surface (m)
z Variable of distance from surface (m)
Greek Letters
120588
119866 Gas density (kgm3)
120591
119891119903
Li Shear stresses of drilling mud noninterfacial(Nm2)
120591
119891119903
119871 Shear stresses of drilling mud interface (Nm2)
120591
Re119871 Reynolds stress of drilling mud (Nm2)
120591
Re119871 Reynolds stress of drilling mud interface
(Nm2)120591
Re119866 Reynolds stress of gas interface (Nm2)
120591
119891119903
119866 Shear stresses of gas interface (Nm2)
120572VM Acceleration due to virtual mass forcecoefficient (ms2)
120591
119871 Shear stresses of drilling mud along wall
(Nm2)120591
119866 Shear stresses of gas along wall (Nm2)
120591
Re119866 Reynolds stress of gas (Nm2)
120591
119891 Frictional pressure gradient (Pam)
120591
119908 Frictional pressure coefficient
120588
119898 Gas and drilling density (kgm3)
120588
0 Density under standard atmospheric pressure
(kgm3)120588
119871 Drilling mud density (kgm3)
120590
119904 Surface tension (Nm2)
Φ
119866 Gas void fraction
Φ
119871 Drilling mud holdup
120588
119903 Reduced density
120583
119871 Viscosity of drilling mud (Pa sdot s)
Subscripts
BHP Bottomhole pressureBP Back pressureEOS Equations of stateMPD Managed pressure drillingRCD The rotating control deviceR-K4 The fourth-order explicit Runge-Kutta
Subscripts of Graph
BP Back pressure (MPa)c Wave velocity in gas and drilling mud
two-phase flow (ms)119862vm The coefficient of virtual mass force
16 Mathematical Problems in Engineering
H Depth of annulus (m)Q Gas influx rate at the bottomhole (m3h)Re The value of virtual mass force
coefficient according to (28) or (34)119908 Angle frequency (Hz)Φ
119866 Void fraction ()
Acknowledgment
Research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 and 51274170)Without their support this work would not have beenpossible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of airinjection in annuli to manage pressure during cementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling conferenceand Exhibition San Diego Calif USA March 2012
[4] W A Bacon Consideration of compressibility effects for applied-back-pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Tex USA 2011
[5] J Choe C Hans and J Wold ldquoA modified two-phase well-control model and its computer applications as a training andeducational toolrdquo SPE Computer Applications vol 2 pp 14ndash201997
[6] J A Tarvin IWalton and PWand ldquoAnalysis of a gas kick takenin a deep well drilled with oil-based mudrdquo in Proceedings ofthe Annual Technical Conference and Exhibition of the Societyof Petroleum Engineers pp 255ndash264 Dallas Tex USA 1991
[7] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[8] J Zhou and G Nygaard ldquoAutomatic model-based controlscheme for stabilizing pressure during dual-gradient drillingrdquoJournal of Process Control vol 21 no 8 pp 1138ndash1147 2011
[9] A Ochoa O Acevedo L Nieto J E Lambarria and H PerezldquoSuccessful application of MPD (Managed Pressure Drilling)for prevention controland detection of borehole ballooning intight gas reservoir in Cuervito Field Mexicordquo in Proceedings ofthe Canadian Unconventional Resources Conference (CURC rsquo11)pp 172ndash185 November 2011
[10] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[11] R Ashena and J Moghadasi ldquoBottom hole pressure estimationusing evolved neural networks by real coded ant colony opti-mization and genetic algorithmrdquo Journal of Petroleum Scienceand Engineering vol 77 no 3-4 pp 375ndash385 2011
[12] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvanced Materials Research vol 468-471 pp 1736ndash1742 2012
[13] I Zubizarreta ldquoPore pressure evolution core damage and trip-ping out schedulesf a computational fluid dynamics approachrdquoin Proceedings of the SPEIADC Drilling Conference and Exhibi-tion Amsterdam The Netherlands March 2013
[14] C H Whitson ldquoCyclic shut-in eliminates liquid-loading in gaswellsrdquo in Proceedings of the SPEEAGE European Unconven-tional Resources Conference and Exhibition Vienna AustriaMarch 2012
[15] G M de Oliveira A T Franco C O R Negrao A LMartins and R A Silva ldquoModeling and validation of pressurepropagation in drilling fluids pumped into a closed wellrdquoJournal of Petroleum Science and Engineering vol 103 pp 61ndash71 2012
[16] Y Sato and H Kanki ldquoFormulas for compression wave andoscillating flow in circular piperdquo Applied Acoustics vol 69 no1 pp 1ndash11 2008
[17] J Xie B Yu X Zhang Q Shao and X Song ldquoNumericalsimulation of gas-liquid-solid three-phase flow in deep wellsrdquoAdvances in Mechanical Engineering vol 2013 Article ID951298 10 pages 2013
[18] W Bacon A Tong O Gabaldon C Sugden and P V Surya-narayana ldquoAn improved dynamic well control response to a gasinflux in managed pressure drilling operationsrdquo in Proceedingsof the IADCSPE Drilling Conference and Exhibition San DiegoCalifornia USA March 2012
[19] J I Hage and D ter Avest ldquoBorehole acoustics applied to kickdetectionrdquo Journal of Petroleum Science and Engineering vol 12no 2 pp 157ndash166 1994
[20] A M R Hopkins Fluid dynamics and surface pressure fluctua-tions of two-dimensional turbulent boundary layers over denselydistributed surface roughness [PhD thesis] Virginia PolytechnicInstitute and State University Blacksburg Va USA 2010
[21] Z S Fu ldquoThe gas cut detect technology during drilling in SovietUnionrdquo Petroleum Drilling Techniques vol 18 no 1 pp 19ndash211990
[22] X F Li C X Guan and X X Sui ldquoThe theory of gas influxdetection of pressure wave and its applicationrdquo Acta PetroleiSinica vol 18 no 3 pp 128ndash133 1997
[23] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[24] A E Ruggles R T Lahey Jr D A Drew and H A ScartonldquoInvestigation of the propagation of pressure perturbations inbubbly airwater flowsrdquo Journal of Heat Transfer vol 110 no 2pp 494ndash499 1988
[25] H J W M Legius H E A Van Den Akker and T NarumoldquoMeasurements on wave propagation and bubble and slugvelocities in cocurrent upward two-phase flowrdquo ExperimentalThermal and Fluid Science vol 15 no 3 pp 267ndash278 1997
[26] K Miyazaki and I Nakajima ldquoPropagation of pressure wavein NItrogen-Mercury two-phase systemrdquo Journal of NuclearScience and Technology vol 9 no 5 pp 315ndash318 1972
[27] B Bai L Guo and X Chen ldquoPressure fluctuation for air-watertwo-phase flowrdquo Journal of Hydrodynamics vol 18 no 4 pp476ndash482 2003
[28] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
Mathematical Problems in Engineering 17
[29] R F Tangren C H Dodge and H S Seifert ldquoCompressibilityeffects in two-phase flowrdquo Journal of Applied Physics vol 20 no7 pp 637ndash645 1949
[30] G B Wallis One Dimensional Two Phase Flow McGraw HillNew York NY USA 1969
[31] D L Nguyen E R F Winter and M Greiner ldquoSonic velocityin two-phase systemsrdquo International Journal ofMultiphase Flowvol 7 no 3 pp 311ndash320 1981
[32] R C Mecredy and L J Hamilton ldquoThe effects of nonequilib-rium heat mass and momentum transfer on two-phase soundspeedrdquo International Journal of Heat and Mass Transfer vol 15no 1 pp 61ndash72 1972
[33] K H Ardron and R B Duffey ldquoAcoustic wave propagationin a flowing liquid-vapour mixturerdquo International Journal ofMultiphase Flow vol 4 no 3 pp 303ndash322 1978
[34] J Xu and T Chen ldquoAcoustic wave prediction in flowing steam-water two-phase mixturerdquo International Journal of Heat andMass Transfer vol 43 no 7 pp 1079ndash1088 2000
[35] N ChungW Lin B Pei and Y Hsu ldquoModel for sound velocityin a two-phase air-water bubbly flowrdquo Nuclear Technology vol99 no 1 pp 80ndash89 1992
[36] X Xu and J Gong ldquoA unitedmodel for predicting pressurewavespeeds in oil and gas two-phase pipeflowsrdquo Journal of PetroleumScience and Engineering vol 60 no 3-4 pp 150ndash160 2008
[37] W Kuczynski ldquoCharacterization of pressure-wave propagationduring the condensation of R404A and R134a refrigerantsin pipe mini-channels that undergo periodic hydrodynamicdisturbancerdquo International Journal of Heat and Fluid Flow vol40 pp 135ndash150 2013
[38] W Kuczynski ldquoModeling of the propagation of a pressure waveduring the condensation process of R134a refrigerant in a pipeminichannel under the periodic conditions of hydrodynamicdisturbancesrdquo International Journal of Heat and Mass Transfervol 56 pp 715ndash723 2013
[39] Y Li C Li E Chen and Y Ying ldquoPressure wave propaga-tion characteristics in a two-phase flow pipeline for liquid-propellant rocketrdquoAerospace Science and Technology vol 15 no6 pp 453ndash464 2011
[40] T Kanagawa M Watanabe T Yano and S Fujikawa ldquoNon-linear wave equations for pressure wave propagation in liquidscontaining gas bubbles (comparison between two-fluid modeland mixture model)rdquo Journal of Fluid Science and Technologyvol 6 no 6 pp 836ndash849 2011
[41] EChen Y Li andYYing ldquoNumerical investigation onpressurewave propagation speed of gas-liquid two-phase flow in pumppipelinerdquo Journal of Aerospace Power vol 25 no 4 pp 754ndash7602010
[42] F Li and L He ldquoInvestigation on pressure wave propagationcharacteristics in slug flow field with increasing gas flow raterdquoPetroleum Geology and Recovery Efficiency vol 11 no 5 pp 78ndash80 2004
[43] C SMartin andM Padmanabhan ldquoPressure pulse propagationin two-component slug flowrdquo Journal of Fluids Engineering vol101 no 1 pp 44ndash52 1979
[44] R E Henry ldquoPressure wave propagation in two-phase mix-turesrdquo Chemical Engineering Progress vol 66 no 1 pp 1ndash101969
[45] X Liu B Li and Y Yue ldquoTransmission behavior of mud-pressure pulse alongwell borerdquo Journal ofHydrodynamics B vol19 no 2 pp 236ndash240 2007
[46] XWang and J Zhang ldquoThe research of pressure wave Pulsationin mud pulse transmittingrdquo Journal of Chongqing University ofScience and Technology vol 14 no 2 pp 55ndash58 2012
[47] J-W Park D A Drew and R T Lahey Jr ldquoThe analysisof void wave propagation in adiabatic monodispersed bubblytwo-phase flows using an ensemble-averaged two-fluid modelrdquoInternational Journal ofMultiphase Flow vol 24 no 7 pp 1205ndash1244 1998
[48] G Arnold Entropy and objectivity as constraints upons constitu-tive equations for two-fluid modeling of multiphase flow [PhDthesis] Rensselaer Polytechnic Institute New York NY USA1998
[49] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[50] L Yarborough and K R Hall ldquoHow to solve equation of statefor z-factorsrdquoOil and Gas Journal vol 72 no 7 pp 86ndash88 1974
[51] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86 pp 127ndash136 2012
[52] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[53] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[54] M Ishii and K Mishima ldquoTwo-fluid model and hydrodynamicconstitutive relationsrdquo Nuclear Engineering and Design vol 82no 2-3 pp 107ndash126 1984
[55] M J Sanchez Comparison of correlations for predicting pressurelosses in vertical multiphase annular flow [MS thesis] Univer-sity of Tulsa Tulsa Okla USA 1972
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
0
25
50
75
100
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
0 25 75 105
Q = 9m3h w = 50Hz = Re
120601G
()
Cvm
Figure 7 Effect of back pressure on the void fraction
52 Effect of Gas Influx Rate on Wave Velocity Figures 9and 10 graphically interpret the distributions of void fractionand variations of wave velocity along the flow direction inthe annulus When gas influx occurs in the bottomhole gasinvades into the wellbore and migrates from the bottomholeto the wellhead along the flow direction At a low gas influxrate it is extremely obvious that the void fraction and wavevelocity first slightly change in a comparatively smooth valuethen change sharply It is because of the rapid expansion ofgas volume with the decreasing in pressure near the wellheadthat the void fraction increases sharply and the wave velocitydecreases obviously at the same time But under the highbottomhole pressure (up to 50MPa) the compressibilityof the gas is low This results in a slight change in voidfraction and wave velocity at the position far away from thewellhead Since the compressible component increases withthe increase in the gas influx rate the compressibility of thegas and drilling mud two-phase fluid is improved So thevariations of void fraction and wave velocity become moreprominent Also the void fraction still shows an increasetendency that is steady first and then sharp It acts inaccordance with the variation of void fraction at a low gasinflux rate At a low gas influx rate if the void fraction atthe position close to the wellhead can not increase to a highextent the wave velocity always shows a decrease tendencyAt a high gas influx rate such as the 119876 = 8312m3h thewave velocity tends to increase because the void fraction inthe wellhead is increased to a high extent In conclusionthe wave pressure is sensitive to the void fraction and thevoid fraction is dominated by influx rate and pressure in theannulus especially the influx rate
0 25 75 1050
300
600
900
1200
BP (MPa)
H = 0mH = 500m
H = 250m
H = 2000mH = 1000mH = 4000m
c(m
s)
Q = 9m3h w = 50Hz = ReCvm
Figure 8 Effect of back pressure on the pressure wave velocity
With the influx of gas the mixing of gas and drillingmud occurs in the annulus and the corresponding interfacialtransfer of momentum and mass causes an increase in gasphase void fraction and a decrease in pressure wave velocityas shown in Figures 11 and 12 Within the range of low gasinflux rate the wave velocity decreases significantly It isbecause of this that the compressibility of the gas increasesremarkably and the medium appears to be of high elasticitythough the density of gas-drilling mud two-phase flowchanges slightly With the increase in the gas influx rate andcorresponding increase in the void fraction in the annulusthe compressibility of the two-phase unceasingly increaseswhich promotes the momentum and energy exchange inthe interface So the pressure wave continuously decreasesWhen the void fraction increases to some extent following theincrease in the gas influx rate the decrease in wave velocityin the liquid slug is gradually less than the increase in wavevelocity in the gas slug thus the decrease of wave velocityis slowed down for the growth of gas slug Especially in thewellhead a slight increase in wave velocity is observed
53 Effect of Virtual Mass Force on Wave Velocity Figure 13illustrates the effect of virtual mass force on the wave velocitywith the parameter of gas influx rate As shown in Figure 15under the same gas influx rate (0594m3h 1098m3h and1768m3h resp) the wave velocity curve of 119862vm = Rediverges from the curve of 119862vm = 0 at the position close tothewellhead whereas thewave velocity curve of the two typesis almost coincided below the position of 119867 = 500m Thisdivergence between the two types of curves is connected withthe fact that significance of influence of interphase virtual
12 Mathematical Problems in Engineering
0 25 50 75 1004000
3000
2000
1000
0
Q = 0058m3h Q = 1228m3hQ = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
H(m
)
BP = 01 MPaw = 50Hz = Re
120601G ()
Cvm
Figure 9 Void fraction distribution at different gas influx rates
mass forces increases together with the increase of the relativemotion in the interphase It is evident that the bottomholepressure is hundreds of times higher than the wellheadpressure As a result the void fraction at the position closeto the wellhead increases sharply meanwhile the interphasemomentum and energy exchanges are promoted The virtualmass force can be described as the transfer of momentumbetween the gas phase and drilling mud phase caused bythe relative motion in the interphase If the relative motionin the interphase is quite weak the value of virtual massforce will intend to approach 0 and the influence on thewave velocity can be ignored However if the relative motionis rather intense the effect of virtual mass force on thewave velocity should not be ignored Furthermore taking thevirtual mass force into account the dispersion characteristicof the pressure wave weakens obviously Compared with thepressure wave velocity calculated by ignoring the effect ofvirtual mass force the calculated pressure wave velocity witha consideration of the virtual mass force is lower Thereforeit is necessary to analyze the effect of the virtual mass forceon the wave velocity at the position close to the wellhead inMPD operations
54 Effect of Disturbance Angular Frequency onWave VelocityFigure 14 presents the pressure wave velocity in the annulusat different angular frequencies According to Figures 5 and9 the void fraction first slightly increases in the bottomholeand then sharply increases along the flow direction at theposition close to the wellhead At a fixed angular frequencythis phenomenon results in an overall decrease in the pressurewave velocity except for the section near the wellhead At the
0 500 1000 1500 20004000
3000
2000
1000
0
c (ms)
H(m
)Q = 0058m3h Q = 1228m3h
Q = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
BP = 01 MPaw = 50Hz = ReCvm
Figure 10 Wave velocity variations at different gas influx rates
section close to the wellhead an opposite change trend ofpressure wave velocity is observed for the transition of flowpattern from bubbly flow to slug flow due to the continuousincrease in void fraction along the flow direction In additionit can be clearly seen from the curves that the wave velocityincreases accompanying with the increase in the angularfrequency above the position of 119867 = 500m This propertyis not very distinct at the position below 119867 = 500m for thelow void fraction
Figure 15 shows the effect of frequency on the wavevelocity in the gas-drilling mud flow (119862vm = 0)The curves ofthe wave velocity of different positions reveal that the prop-agation velocity of pressure disturbances increases togetherwith the growth of the angular frequency (0 lt 119908 lt 500Hz)It proves that the pressure wave has an obvious dispersioncharacteristic in the two-phase flow As the angular frequencyincreases in the range of less than 500Hz there is sufficienttime to carry out the exchange of energy and momentumbetween two phases It achieves a state of mechanical andthermodynamic equilibrium between the two phases Sothe wave velocity increases gradually with the increase inthe angular frequency at different depths of the annulus Itis considered that the wave velocity is mainly affected bythe interphase mechanical and thermodynamic equilibriumat low anglular frequencies When the angular frequencyreaches the value of 119908 = 500Hz the pressure wave velocityachieves a constant value and remains on this level regardlessof the further growth in angular frequency 119908 With theincrease in angular frequency there is not enough time forenergy and momentum exchange between the gas-drillingmud two phases to reach themechanical and thermodynamic
Mathematical Problems in Engineering 13
0
25
50
75
100
125
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
Q (m3h)
120601G
()
0 25 5 75 10
BP = 01 MPaw = 50Hz = ReCvm
Figure 11 Effect of gas influx rate on the void fraction
equilibrium state and thus the wave dispersion does notexist At a high angular frequency the wave velocity is mainlydominated by the mechanical and thermal nonequilibriumin the flow and keeps almost unchanged when the angularfrequency is further increased This is consistent with theinfluences of the angular frequency in the horizontal pipe[28] Moreover the effect of virtual mass force is also shownin this figure (119862vm = Re) It is observed that the wave velocityis significantly reduced when the virtual mass force is takeninto consideration in the calculation of the wave velocity atthe position close to the wellhead such as 119867 = 0m It canbe explained by the reduction of dispersion characteristic ofa pressure wave Along the wellbore from the wellhead to thebottomhole the distinction gradually decreases
6 Conclusions
With full consideration of the important factors such asvirtual mass force drag force gas void fraction pressuretemperature and angular frequency a united wave velocitymodel has been proposed based on pressure drop gradientequations in MPD operations gas-liquid two-fluid modelthe gas-drilling mud equations of state and small pertur-bation theory Solved by the fourth-order explicit Runge-Kutta method the model is used to predict wave velocityfor different back pressures and gas influx rates in MPDoperations The main conclusions can be summarized asfollows
(1) With the introduction of virtual mass force and dragforce the united model agrees well with the previousexperimental data The united model can be used to
0
280
560
840
1120
1400
Q (m3h)0 25 5 75 10
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
c(m
s)
BP = 01 MPaw = 50Hz = ReCvm
Figure 12 Effect of gas influx rate on the pressure wave velocity
accurately calculate the wave velocity in the annulusThe application of the model will be beneficial tofurther study the wave velocity at different gas influxrates and back pressures in MPD operations reducenonproductive times and provide a reference for thedrilling operations in extremely narrow porefracturewindows existing conditions
(2) The wave velocity and void fraction have differentvalues with respect to well depth In the annulusthe drop of pressure causes an increase in the voidfraction along the flow direction The void fractionincreases first slightly and then sharply Correspond-ingly the wave velocity first gradually decreases in thbubbly flow and preliminary slug flow and then thewave velocity slightly increases accompanying withthe increase in the relative length ratio of gas slugto the liquid slug for the continuous increase in voidfraction at the position close to the wellhead Theminimum wave velocity appears in the long gas slugflow
(3) When the back pressure in the MPD operationsincreases the void fraction at different depths of theannulus is reduced gradually while the wave velocityin the two-phase flow tends to increase Moreoverthe influence of back pressure on the void fraction isgreater at a position close to the wellhead than that inthe bottomhole because of the great decrease in thepressure from the bottomhole to the wellhead Alsothe effect of back pressure on void fraction and wavevelocity is decreased in the high back pressure range
14 Mathematical Problems in Engineering
0 300 600 900 12004000
3000
2000
1000
225
180
135
90
45
0
BP = 01 MPaw = 50Hz
Q = 0594m3h = ReQ = 0594m3h = 0
Q = 1098m3h = ReQ = 1768m3h = ReQ = 1098m3h = 0
Q = 1768m3h = 0
c (ms)
H(m
)
CvmCvm
Cvm
Cvm
Cvm
Cvm
Figure 13 Effect of virtual mass force on the pressure wave velocity
(4) The wave velocity is sensitive to the void fraction butthe void fraction is dominated by gas influx rate andpressure in the annulus especially the gas influx rateSince the compressibility of the gas and drilling mudtwo-phase fluid is improved with the increase in theinflux rate the void fraction increases greatly andthe wave velocity decreases significantly within thelow gas influx rate range When the void fraction isincreased to some extent following the increase in thegas influx rate the decrease of wave velocity is slowedfor the growth of gas slug Especially at the wellheada slight increase in wave velocity is observed at a highgas influx rate for the sharp increase in void fraction
(5) It is necessary to analyze the effect of the virtualmass force on the wave velocity in MPD operationsEspecially at the position close to the wellhead theeffect of virtual mass force is more prominent for theintense phase interactions Taking the virtual massforce into account the dispersion characteristic of thepressure wave weakens obviously Compared with theresults calculated by ignoring the effect of virtualmassforce the calculated pressure wave velocity with aconsideration of the virtual mass force is lower
(6) The effect of angular frequency on the propagationvelocity of pressure wave appears at a low angularfrequency The propagation velocity of pressure dis-turbances increases together with the growth of theangular frequency (0 lt 119908 lt 500Hz) When theangular frequency reaches the value of 119908 = 500Hzthe pressure wave velocity achieves a constant value
0 240 480 720 960 12004000
3000
2000
1000
560
420
280
140
0
Q = 0727m3h BP = 01 MPa
w = 50Hz = Rew = 50Hz = 0w = 300Hz = 0
w = 800Hz = 0w = 1600Hz = 0w = 5000Hz = 0
c (ms)
H(m
)CvmCvmCvm
CvmCvmCvm
Figure 14 Wave velocity variations at different angular frequencies
and remains on this level regardless of the furthergrowth in angular frequency 119908
Nomenclature
A Annulus effective cross area (m2)c119866 Wave velocity in gas (ms)
c119871 Wave velocity in Drilling mud (ms)
C119863 The coefficient of drag force
D Annulus effective diameter (m)D119894 Diameter of the inner pipe (m)
D119900 Diameter of the outer pipe (m)
f119866 Shear stresses coefficient of gas interface
f119871 Shear stresses coefficient of drilling mud
interfaceF119911 Pressure applied to the two-phase flow (N)
g Acceleration due to gravity (m2s)g119888 Conversion factor
g119879 Geothermal temperature gradient (Km)
h One discrete length of annulus (m)k119890 Annulus effective roughness (m)
k119900 Roughness of outer pipe (m)
k119894 Roughness of inner pipe (m)
H The depth of annulus (m)L119902 Air bubbles length of slug (m)
119872Gi Momentum transfer in gas interface (Nm3)119872Li Momentum transfer in liquid interface (Nm3)119872
ndLi The none-drag-force (virtual mass force)
(Nm3)119872
119889
Li The momentum transfer term caused by thedrag force (Nm3)
Mathematical Problems in Engineering 15
100 1000 100000
450
900
1350
1800
0w (Hz)
0 m = 0 0 m = Re1000 m = 0
2000 m = 0
4000 m = 0
1000 m = Re2000 m = Re4000 m = Re
c(m
s)
Q = 0727m3h BP = 01 MPaCvmCvm
CvmCvmCvm
CvmCvmCvm
Figure 15 Effect of angular frequency on wave velocity
N Parameter combiningThompsonenergy effects
119901Gi Gas interface pressure (MPa)119901Li Liquid interface pressure (MPa)p Pressure (MPa)p0 Initial pressure of annulus (MPa)
p119888 Critical pressure (kPa)
p119871 Pressure of drilling mud (MPa)
p119903 Reduced pressure
q119898 Volumetric flow rate of gas and drilling
mud (m3s)q119866 Velocity of the gas (m3s)
q119871 Velocity of the drilling mud (m3s)
Q119871 Drilling mud rate within the annulus
(m3h)r Average diameter of the bubble (m)R Constant of EOS (JKgK)R119902 Air bubbles radius of slug (m)
R119887 The bubble diameter (m)
T Temperature (K)T119888 Critical temperature (K)
T119903 Reduced temperature
T119890119894 Undisturbed formation temperature at a
depth (K)T119890119887ℎ
Undisturbed formation temperature ofwellhead (K)
T119891119887ℎ
Undisturbed formation temperature atthe bottomhole (K)
v119898 Gas and drilling mud flow velocity
(ms)
v119866 Gas flow velocity (ms)
v119904 Slip velocity (ms)
v119904119892 Superficial gas velocity (ms)
v119871 Drilling mud flow velocity (ms)
W119898 Mass flow velocity (kgm3)
Z119866 Gas deviation factor
z0 Initial calculation height of annulus (m)
z119887ℎ Total well depth from surface (m)
z Variable of distance from surface (m)
Greek Letters
120588
119866 Gas density (kgm3)
120591
119891119903
Li Shear stresses of drilling mud noninterfacial(Nm2)
120591
119891119903
119871 Shear stresses of drilling mud interface (Nm2)
120591
Re119871 Reynolds stress of drilling mud (Nm2)
120591
Re119871 Reynolds stress of drilling mud interface
(Nm2)120591
Re119866 Reynolds stress of gas interface (Nm2)
120591
119891119903
119866 Shear stresses of gas interface (Nm2)
120572VM Acceleration due to virtual mass forcecoefficient (ms2)
120591
119871 Shear stresses of drilling mud along wall
(Nm2)120591
119866 Shear stresses of gas along wall (Nm2)
120591
Re119866 Reynolds stress of gas (Nm2)
120591
119891 Frictional pressure gradient (Pam)
120591
119908 Frictional pressure coefficient
120588
119898 Gas and drilling density (kgm3)
120588
0 Density under standard atmospheric pressure
(kgm3)120588
119871 Drilling mud density (kgm3)
120590
119904 Surface tension (Nm2)
Φ
119866 Gas void fraction
Φ
119871 Drilling mud holdup
120588
119903 Reduced density
120583
119871 Viscosity of drilling mud (Pa sdot s)
Subscripts
BHP Bottomhole pressureBP Back pressureEOS Equations of stateMPD Managed pressure drillingRCD The rotating control deviceR-K4 The fourth-order explicit Runge-Kutta
Subscripts of Graph
BP Back pressure (MPa)c Wave velocity in gas and drilling mud
two-phase flow (ms)119862vm The coefficient of virtual mass force
16 Mathematical Problems in Engineering
H Depth of annulus (m)Q Gas influx rate at the bottomhole (m3h)Re The value of virtual mass force
coefficient according to (28) or (34)119908 Angle frequency (Hz)Φ
119866 Void fraction ()
Acknowledgment
Research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 and 51274170)Without their support this work would not have beenpossible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of airinjection in annuli to manage pressure during cementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling conferenceand Exhibition San Diego Calif USA March 2012
[4] W A Bacon Consideration of compressibility effects for applied-back-pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Tex USA 2011
[5] J Choe C Hans and J Wold ldquoA modified two-phase well-control model and its computer applications as a training andeducational toolrdquo SPE Computer Applications vol 2 pp 14ndash201997
[6] J A Tarvin IWalton and PWand ldquoAnalysis of a gas kick takenin a deep well drilled with oil-based mudrdquo in Proceedings ofthe Annual Technical Conference and Exhibition of the Societyof Petroleum Engineers pp 255ndash264 Dallas Tex USA 1991
[7] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[8] J Zhou and G Nygaard ldquoAutomatic model-based controlscheme for stabilizing pressure during dual-gradient drillingrdquoJournal of Process Control vol 21 no 8 pp 1138ndash1147 2011
[9] A Ochoa O Acevedo L Nieto J E Lambarria and H PerezldquoSuccessful application of MPD (Managed Pressure Drilling)for prevention controland detection of borehole ballooning intight gas reservoir in Cuervito Field Mexicordquo in Proceedings ofthe Canadian Unconventional Resources Conference (CURC rsquo11)pp 172ndash185 November 2011
[10] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[11] R Ashena and J Moghadasi ldquoBottom hole pressure estimationusing evolved neural networks by real coded ant colony opti-mization and genetic algorithmrdquo Journal of Petroleum Scienceand Engineering vol 77 no 3-4 pp 375ndash385 2011
[12] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvanced Materials Research vol 468-471 pp 1736ndash1742 2012
[13] I Zubizarreta ldquoPore pressure evolution core damage and trip-ping out schedulesf a computational fluid dynamics approachrdquoin Proceedings of the SPEIADC Drilling Conference and Exhibi-tion Amsterdam The Netherlands March 2013
[14] C H Whitson ldquoCyclic shut-in eliminates liquid-loading in gaswellsrdquo in Proceedings of the SPEEAGE European Unconven-tional Resources Conference and Exhibition Vienna AustriaMarch 2012
[15] G M de Oliveira A T Franco C O R Negrao A LMartins and R A Silva ldquoModeling and validation of pressurepropagation in drilling fluids pumped into a closed wellrdquoJournal of Petroleum Science and Engineering vol 103 pp 61ndash71 2012
[16] Y Sato and H Kanki ldquoFormulas for compression wave andoscillating flow in circular piperdquo Applied Acoustics vol 69 no1 pp 1ndash11 2008
[17] J Xie B Yu X Zhang Q Shao and X Song ldquoNumericalsimulation of gas-liquid-solid three-phase flow in deep wellsrdquoAdvances in Mechanical Engineering vol 2013 Article ID951298 10 pages 2013
[18] W Bacon A Tong O Gabaldon C Sugden and P V Surya-narayana ldquoAn improved dynamic well control response to a gasinflux in managed pressure drilling operationsrdquo in Proceedingsof the IADCSPE Drilling Conference and Exhibition San DiegoCalifornia USA March 2012
[19] J I Hage and D ter Avest ldquoBorehole acoustics applied to kickdetectionrdquo Journal of Petroleum Science and Engineering vol 12no 2 pp 157ndash166 1994
[20] A M R Hopkins Fluid dynamics and surface pressure fluctua-tions of two-dimensional turbulent boundary layers over denselydistributed surface roughness [PhD thesis] Virginia PolytechnicInstitute and State University Blacksburg Va USA 2010
[21] Z S Fu ldquoThe gas cut detect technology during drilling in SovietUnionrdquo Petroleum Drilling Techniques vol 18 no 1 pp 19ndash211990
[22] X F Li C X Guan and X X Sui ldquoThe theory of gas influxdetection of pressure wave and its applicationrdquo Acta PetroleiSinica vol 18 no 3 pp 128ndash133 1997
[23] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[24] A E Ruggles R T Lahey Jr D A Drew and H A ScartonldquoInvestigation of the propagation of pressure perturbations inbubbly airwater flowsrdquo Journal of Heat Transfer vol 110 no 2pp 494ndash499 1988
[25] H J W M Legius H E A Van Den Akker and T NarumoldquoMeasurements on wave propagation and bubble and slugvelocities in cocurrent upward two-phase flowrdquo ExperimentalThermal and Fluid Science vol 15 no 3 pp 267ndash278 1997
[26] K Miyazaki and I Nakajima ldquoPropagation of pressure wavein NItrogen-Mercury two-phase systemrdquo Journal of NuclearScience and Technology vol 9 no 5 pp 315ndash318 1972
[27] B Bai L Guo and X Chen ldquoPressure fluctuation for air-watertwo-phase flowrdquo Journal of Hydrodynamics vol 18 no 4 pp476ndash482 2003
[28] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
Mathematical Problems in Engineering 17
[29] R F Tangren C H Dodge and H S Seifert ldquoCompressibilityeffects in two-phase flowrdquo Journal of Applied Physics vol 20 no7 pp 637ndash645 1949
[30] G B Wallis One Dimensional Two Phase Flow McGraw HillNew York NY USA 1969
[31] D L Nguyen E R F Winter and M Greiner ldquoSonic velocityin two-phase systemsrdquo International Journal ofMultiphase Flowvol 7 no 3 pp 311ndash320 1981
[32] R C Mecredy and L J Hamilton ldquoThe effects of nonequilib-rium heat mass and momentum transfer on two-phase soundspeedrdquo International Journal of Heat and Mass Transfer vol 15no 1 pp 61ndash72 1972
[33] K H Ardron and R B Duffey ldquoAcoustic wave propagationin a flowing liquid-vapour mixturerdquo International Journal ofMultiphase Flow vol 4 no 3 pp 303ndash322 1978
[34] J Xu and T Chen ldquoAcoustic wave prediction in flowing steam-water two-phase mixturerdquo International Journal of Heat andMass Transfer vol 43 no 7 pp 1079ndash1088 2000
[35] N ChungW Lin B Pei and Y Hsu ldquoModel for sound velocityin a two-phase air-water bubbly flowrdquo Nuclear Technology vol99 no 1 pp 80ndash89 1992
[36] X Xu and J Gong ldquoA unitedmodel for predicting pressurewavespeeds in oil and gas two-phase pipeflowsrdquo Journal of PetroleumScience and Engineering vol 60 no 3-4 pp 150ndash160 2008
[37] W Kuczynski ldquoCharacterization of pressure-wave propagationduring the condensation of R404A and R134a refrigerantsin pipe mini-channels that undergo periodic hydrodynamicdisturbancerdquo International Journal of Heat and Fluid Flow vol40 pp 135ndash150 2013
[38] W Kuczynski ldquoModeling of the propagation of a pressure waveduring the condensation process of R134a refrigerant in a pipeminichannel under the periodic conditions of hydrodynamicdisturbancesrdquo International Journal of Heat and Mass Transfervol 56 pp 715ndash723 2013
[39] Y Li C Li E Chen and Y Ying ldquoPressure wave propaga-tion characteristics in a two-phase flow pipeline for liquid-propellant rocketrdquoAerospace Science and Technology vol 15 no6 pp 453ndash464 2011
[40] T Kanagawa M Watanabe T Yano and S Fujikawa ldquoNon-linear wave equations for pressure wave propagation in liquidscontaining gas bubbles (comparison between two-fluid modeland mixture model)rdquo Journal of Fluid Science and Technologyvol 6 no 6 pp 836ndash849 2011
[41] EChen Y Li andYYing ldquoNumerical investigation onpressurewave propagation speed of gas-liquid two-phase flow in pumppipelinerdquo Journal of Aerospace Power vol 25 no 4 pp 754ndash7602010
[42] F Li and L He ldquoInvestigation on pressure wave propagationcharacteristics in slug flow field with increasing gas flow raterdquoPetroleum Geology and Recovery Efficiency vol 11 no 5 pp 78ndash80 2004
[43] C SMartin andM Padmanabhan ldquoPressure pulse propagationin two-component slug flowrdquo Journal of Fluids Engineering vol101 no 1 pp 44ndash52 1979
[44] R E Henry ldquoPressure wave propagation in two-phase mix-turesrdquo Chemical Engineering Progress vol 66 no 1 pp 1ndash101969
[45] X Liu B Li and Y Yue ldquoTransmission behavior of mud-pressure pulse alongwell borerdquo Journal ofHydrodynamics B vol19 no 2 pp 236ndash240 2007
[46] XWang and J Zhang ldquoThe research of pressure wave Pulsationin mud pulse transmittingrdquo Journal of Chongqing University ofScience and Technology vol 14 no 2 pp 55ndash58 2012
[47] J-W Park D A Drew and R T Lahey Jr ldquoThe analysisof void wave propagation in adiabatic monodispersed bubblytwo-phase flows using an ensemble-averaged two-fluid modelrdquoInternational Journal ofMultiphase Flow vol 24 no 7 pp 1205ndash1244 1998
[48] G Arnold Entropy and objectivity as constraints upons constitu-tive equations for two-fluid modeling of multiphase flow [PhDthesis] Rensselaer Polytechnic Institute New York NY USA1998
[49] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[50] L Yarborough and K R Hall ldquoHow to solve equation of statefor z-factorsrdquoOil and Gas Journal vol 72 no 7 pp 86ndash88 1974
[51] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86 pp 127ndash136 2012
[52] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[53] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[54] M Ishii and K Mishima ldquoTwo-fluid model and hydrodynamicconstitutive relationsrdquo Nuclear Engineering and Design vol 82no 2-3 pp 107ndash126 1984
[55] M J Sanchez Comparison of correlations for predicting pressurelosses in vertical multiphase annular flow [MS thesis] Univer-sity of Tulsa Tulsa Okla USA 1972
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
0 25 50 75 1004000
3000
2000
1000
0
Q = 0058m3h Q = 1228m3hQ = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
H(m
)
BP = 01 MPaw = 50Hz = Re
120601G ()
Cvm
Figure 9 Void fraction distribution at different gas influx rates
mass forces increases together with the increase of the relativemotion in the interphase It is evident that the bottomholepressure is hundreds of times higher than the wellheadpressure As a result the void fraction at the position closeto the wellhead increases sharply meanwhile the interphasemomentum and energy exchanges are promoted The virtualmass force can be described as the transfer of momentumbetween the gas phase and drilling mud phase caused bythe relative motion in the interphase If the relative motionin the interphase is quite weak the value of virtual massforce will intend to approach 0 and the influence on thewave velocity can be ignored However if the relative motionis rather intense the effect of virtual mass force on thewave velocity should not be ignored Furthermore taking thevirtual mass force into account the dispersion characteristicof the pressure wave weakens obviously Compared with thepressure wave velocity calculated by ignoring the effect ofvirtual mass force the calculated pressure wave velocity witha consideration of the virtual mass force is lower Thereforeit is necessary to analyze the effect of the virtual mass forceon the wave velocity at the position close to the wellhead inMPD operations
54 Effect of Disturbance Angular Frequency onWave VelocityFigure 14 presents the pressure wave velocity in the annulusat different angular frequencies According to Figures 5 and9 the void fraction first slightly increases in the bottomholeand then sharply increases along the flow direction at theposition close to the wellhead At a fixed angular frequencythis phenomenon results in an overall decrease in the pressurewave velocity except for the section near the wellhead At the
0 500 1000 1500 20004000
3000
2000
1000
0
c (ms)
H(m
)Q = 0058m3h Q = 1228m3h
Q = 3427m3hQ = 8312m3h
Q = 0288m3hQ = 0594m3h
BP = 01 MPaw = 50Hz = ReCvm
Figure 10 Wave velocity variations at different gas influx rates
section close to the wellhead an opposite change trend ofpressure wave velocity is observed for the transition of flowpattern from bubbly flow to slug flow due to the continuousincrease in void fraction along the flow direction In additionit can be clearly seen from the curves that the wave velocityincreases accompanying with the increase in the angularfrequency above the position of 119867 = 500m This propertyis not very distinct at the position below 119867 = 500m for thelow void fraction
Figure 15 shows the effect of frequency on the wavevelocity in the gas-drilling mud flow (119862vm = 0)The curves ofthe wave velocity of different positions reveal that the prop-agation velocity of pressure disturbances increases togetherwith the growth of the angular frequency (0 lt 119908 lt 500Hz)It proves that the pressure wave has an obvious dispersioncharacteristic in the two-phase flow As the angular frequencyincreases in the range of less than 500Hz there is sufficienttime to carry out the exchange of energy and momentumbetween two phases It achieves a state of mechanical andthermodynamic equilibrium between the two phases Sothe wave velocity increases gradually with the increase inthe angular frequency at different depths of the annulus Itis considered that the wave velocity is mainly affected bythe interphase mechanical and thermodynamic equilibriumat low anglular frequencies When the angular frequencyreaches the value of 119908 = 500Hz the pressure wave velocityachieves a constant value and remains on this level regardlessof the further growth in angular frequency 119908 With theincrease in angular frequency there is not enough time forenergy and momentum exchange between the gas-drillingmud two phases to reach themechanical and thermodynamic
Mathematical Problems in Engineering 13
0
25
50
75
100
125
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
Q (m3h)
120601G
()
0 25 5 75 10
BP = 01 MPaw = 50Hz = ReCvm
Figure 11 Effect of gas influx rate on the void fraction
equilibrium state and thus the wave dispersion does notexist At a high angular frequency the wave velocity is mainlydominated by the mechanical and thermal nonequilibriumin the flow and keeps almost unchanged when the angularfrequency is further increased This is consistent with theinfluences of the angular frequency in the horizontal pipe[28] Moreover the effect of virtual mass force is also shownin this figure (119862vm = Re) It is observed that the wave velocityis significantly reduced when the virtual mass force is takeninto consideration in the calculation of the wave velocity atthe position close to the wellhead such as 119867 = 0m It canbe explained by the reduction of dispersion characteristic ofa pressure wave Along the wellbore from the wellhead to thebottomhole the distinction gradually decreases
6 Conclusions
With full consideration of the important factors such asvirtual mass force drag force gas void fraction pressuretemperature and angular frequency a united wave velocitymodel has been proposed based on pressure drop gradientequations in MPD operations gas-liquid two-fluid modelthe gas-drilling mud equations of state and small pertur-bation theory Solved by the fourth-order explicit Runge-Kutta method the model is used to predict wave velocityfor different back pressures and gas influx rates in MPDoperations The main conclusions can be summarized asfollows
(1) With the introduction of virtual mass force and dragforce the united model agrees well with the previousexperimental data The united model can be used to
0
280
560
840
1120
1400
Q (m3h)0 25 5 75 10
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
c(m
s)
BP = 01 MPaw = 50Hz = ReCvm
Figure 12 Effect of gas influx rate on the pressure wave velocity
accurately calculate the wave velocity in the annulusThe application of the model will be beneficial tofurther study the wave velocity at different gas influxrates and back pressures in MPD operations reducenonproductive times and provide a reference for thedrilling operations in extremely narrow porefracturewindows existing conditions
(2) The wave velocity and void fraction have differentvalues with respect to well depth In the annulusthe drop of pressure causes an increase in the voidfraction along the flow direction The void fractionincreases first slightly and then sharply Correspond-ingly the wave velocity first gradually decreases in thbubbly flow and preliminary slug flow and then thewave velocity slightly increases accompanying withthe increase in the relative length ratio of gas slugto the liquid slug for the continuous increase in voidfraction at the position close to the wellhead Theminimum wave velocity appears in the long gas slugflow
(3) When the back pressure in the MPD operationsincreases the void fraction at different depths of theannulus is reduced gradually while the wave velocityin the two-phase flow tends to increase Moreoverthe influence of back pressure on the void fraction isgreater at a position close to the wellhead than that inthe bottomhole because of the great decrease in thepressure from the bottomhole to the wellhead Alsothe effect of back pressure on void fraction and wavevelocity is decreased in the high back pressure range
14 Mathematical Problems in Engineering
0 300 600 900 12004000
3000
2000
1000
225
180
135
90
45
0
BP = 01 MPaw = 50Hz
Q = 0594m3h = ReQ = 0594m3h = 0
Q = 1098m3h = ReQ = 1768m3h = ReQ = 1098m3h = 0
Q = 1768m3h = 0
c (ms)
H(m
)
CvmCvm
Cvm
Cvm
Cvm
Cvm
Figure 13 Effect of virtual mass force on the pressure wave velocity
(4) The wave velocity is sensitive to the void fraction butthe void fraction is dominated by gas influx rate andpressure in the annulus especially the gas influx rateSince the compressibility of the gas and drilling mudtwo-phase fluid is improved with the increase in theinflux rate the void fraction increases greatly andthe wave velocity decreases significantly within thelow gas influx rate range When the void fraction isincreased to some extent following the increase in thegas influx rate the decrease of wave velocity is slowedfor the growth of gas slug Especially at the wellheada slight increase in wave velocity is observed at a highgas influx rate for the sharp increase in void fraction
(5) It is necessary to analyze the effect of the virtualmass force on the wave velocity in MPD operationsEspecially at the position close to the wellhead theeffect of virtual mass force is more prominent for theintense phase interactions Taking the virtual massforce into account the dispersion characteristic of thepressure wave weakens obviously Compared with theresults calculated by ignoring the effect of virtualmassforce the calculated pressure wave velocity with aconsideration of the virtual mass force is lower
(6) The effect of angular frequency on the propagationvelocity of pressure wave appears at a low angularfrequency The propagation velocity of pressure dis-turbances increases together with the growth of theangular frequency (0 lt 119908 lt 500Hz) When theangular frequency reaches the value of 119908 = 500Hzthe pressure wave velocity achieves a constant value
0 240 480 720 960 12004000
3000
2000
1000
560
420
280
140
0
Q = 0727m3h BP = 01 MPa
w = 50Hz = Rew = 50Hz = 0w = 300Hz = 0
w = 800Hz = 0w = 1600Hz = 0w = 5000Hz = 0
c (ms)
H(m
)CvmCvmCvm
CvmCvmCvm
Figure 14 Wave velocity variations at different angular frequencies
and remains on this level regardless of the furthergrowth in angular frequency 119908
Nomenclature
A Annulus effective cross area (m2)c119866 Wave velocity in gas (ms)
c119871 Wave velocity in Drilling mud (ms)
C119863 The coefficient of drag force
D Annulus effective diameter (m)D119894 Diameter of the inner pipe (m)
D119900 Diameter of the outer pipe (m)
f119866 Shear stresses coefficient of gas interface
f119871 Shear stresses coefficient of drilling mud
interfaceF119911 Pressure applied to the two-phase flow (N)
g Acceleration due to gravity (m2s)g119888 Conversion factor
g119879 Geothermal temperature gradient (Km)
h One discrete length of annulus (m)k119890 Annulus effective roughness (m)
k119900 Roughness of outer pipe (m)
k119894 Roughness of inner pipe (m)
H The depth of annulus (m)L119902 Air bubbles length of slug (m)
119872Gi Momentum transfer in gas interface (Nm3)119872Li Momentum transfer in liquid interface (Nm3)119872
ndLi The none-drag-force (virtual mass force)
(Nm3)119872
119889
Li The momentum transfer term caused by thedrag force (Nm3)
Mathematical Problems in Engineering 15
100 1000 100000
450
900
1350
1800
0w (Hz)
0 m = 0 0 m = Re1000 m = 0
2000 m = 0
4000 m = 0
1000 m = Re2000 m = Re4000 m = Re
c(m
s)
Q = 0727m3h BP = 01 MPaCvmCvm
CvmCvmCvm
CvmCvmCvm
Figure 15 Effect of angular frequency on wave velocity
N Parameter combiningThompsonenergy effects
119901Gi Gas interface pressure (MPa)119901Li Liquid interface pressure (MPa)p Pressure (MPa)p0 Initial pressure of annulus (MPa)
p119888 Critical pressure (kPa)
p119871 Pressure of drilling mud (MPa)
p119903 Reduced pressure
q119898 Volumetric flow rate of gas and drilling
mud (m3s)q119866 Velocity of the gas (m3s)
q119871 Velocity of the drilling mud (m3s)
Q119871 Drilling mud rate within the annulus
(m3h)r Average diameter of the bubble (m)R Constant of EOS (JKgK)R119902 Air bubbles radius of slug (m)
R119887 The bubble diameter (m)
T Temperature (K)T119888 Critical temperature (K)
T119903 Reduced temperature
T119890119894 Undisturbed formation temperature at a
depth (K)T119890119887ℎ
Undisturbed formation temperature ofwellhead (K)
T119891119887ℎ
Undisturbed formation temperature atthe bottomhole (K)
v119898 Gas and drilling mud flow velocity
(ms)
v119866 Gas flow velocity (ms)
v119904 Slip velocity (ms)
v119904119892 Superficial gas velocity (ms)
v119871 Drilling mud flow velocity (ms)
W119898 Mass flow velocity (kgm3)
Z119866 Gas deviation factor
z0 Initial calculation height of annulus (m)
z119887ℎ Total well depth from surface (m)
z Variable of distance from surface (m)
Greek Letters
120588
119866 Gas density (kgm3)
120591
119891119903
Li Shear stresses of drilling mud noninterfacial(Nm2)
120591
119891119903
119871 Shear stresses of drilling mud interface (Nm2)
120591
Re119871 Reynolds stress of drilling mud (Nm2)
120591
Re119871 Reynolds stress of drilling mud interface
(Nm2)120591
Re119866 Reynolds stress of gas interface (Nm2)
120591
119891119903
119866 Shear stresses of gas interface (Nm2)
120572VM Acceleration due to virtual mass forcecoefficient (ms2)
120591
119871 Shear stresses of drilling mud along wall
(Nm2)120591
119866 Shear stresses of gas along wall (Nm2)
120591
Re119866 Reynolds stress of gas (Nm2)
120591
119891 Frictional pressure gradient (Pam)
120591
119908 Frictional pressure coefficient
120588
119898 Gas and drilling density (kgm3)
120588
0 Density under standard atmospheric pressure
(kgm3)120588
119871 Drilling mud density (kgm3)
120590
119904 Surface tension (Nm2)
Φ
119866 Gas void fraction
Φ
119871 Drilling mud holdup
120588
119903 Reduced density
120583
119871 Viscosity of drilling mud (Pa sdot s)
Subscripts
BHP Bottomhole pressureBP Back pressureEOS Equations of stateMPD Managed pressure drillingRCD The rotating control deviceR-K4 The fourth-order explicit Runge-Kutta
Subscripts of Graph
BP Back pressure (MPa)c Wave velocity in gas and drilling mud
two-phase flow (ms)119862vm The coefficient of virtual mass force
16 Mathematical Problems in Engineering
H Depth of annulus (m)Q Gas influx rate at the bottomhole (m3h)Re The value of virtual mass force
coefficient according to (28) or (34)119908 Angle frequency (Hz)Φ
119866 Void fraction ()
Acknowledgment
Research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 and 51274170)Without their support this work would not have beenpossible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of airinjection in annuli to manage pressure during cementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling conferenceand Exhibition San Diego Calif USA March 2012
[4] W A Bacon Consideration of compressibility effects for applied-back-pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Tex USA 2011
[5] J Choe C Hans and J Wold ldquoA modified two-phase well-control model and its computer applications as a training andeducational toolrdquo SPE Computer Applications vol 2 pp 14ndash201997
[6] J A Tarvin IWalton and PWand ldquoAnalysis of a gas kick takenin a deep well drilled with oil-based mudrdquo in Proceedings ofthe Annual Technical Conference and Exhibition of the Societyof Petroleum Engineers pp 255ndash264 Dallas Tex USA 1991
[7] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[8] J Zhou and G Nygaard ldquoAutomatic model-based controlscheme for stabilizing pressure during dual-gradient drillingrdquoJournal of Process Control vol 21 no 8 pp 1138ndash1147 2011
[9] A Ochoa O Acevedo L Nieto J E Lambarria and H PerezldquoSuccessful application of MPD (Managed Pressure Drilling)for prevention controland detection of borehole ballooning intight gas reservoir in Cuervito Field Mexicordquo in Proceedings ofthe Canadian Unconventional Resources Conference (CURC rsquo11)pp 172ndash185 November 2011
[10] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[11] R Ashena and J Moghadasi ldquoBottom hole pressure estimationusing evolved neural networks by real coded ant colony opti-mization and genetic algorithmrdquo Journal of Petroleum Scienceand Engineering vol 77 no 3-4 pp 375ndash385 2011
[12] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvanced Materials Research vol 468-471 pp 1736ndash1742 2012
[13] I Zubizarreta ldquoPore pressure evolution core damage and trip-ping out schedulesf a computational fluid dynamics approachrdquoin Proceedings of the SPEIADC Drilling Conference and Exhibi-tion Amsterdam The Netherlands March 2013
[14] C H Whitson ldquoCyclic shut-in eliminates liquid-loading in gaswellsrdquo in Proceedings of the SPEEAGE European Unconven-tional Resources Conference and Exhibition Vienna AustriaMarch 2012
[15] G M de Oliveira A T Franco C O R Negrao A LMartins and R A Silva ldquoModeling and validation of pressurepropagation in drilling fluids pumped into a closed wellrdquoJournal of Petroleum Science and Engineering vol 103 pp 61ndash71 2012
[16] Y Sato and H Kanki ldquoFormulas for compression wave andoscillating flow in circular piperdquo Applied Acoustics vol 69 no1 pp 1ndash11 2008
[17] J Xie B Yu X Zhang Q Shao and X Song ldquoNumericalsimulation of gas-liquid-solid three-phase flow in deep wellsrdquoAdvances in Mechanical Engineering vol 2013 Article ID951298 10 pages 2013
[18] W Bacon A Tong O Gabaldon C Sugden and P V Surya-narayana ldquoAn improved dynamic well control response to a gasinflux in managed pressure drilling operationsrdquo in Proceedingsof the IADCSPE Drilling Conference and Exhibition San DiegoCalifornia USA March 2012
[19] J I Hage and D ter Avest ldquoBorehole acoustics applied to kickdetectionrdquo Journal of Petroleum Science and Engineering vol 12no 2 pp 157ndash166 1994
[20] A M R Hopkins Fluid dynamics and surface pressure fluctua-tions of two-dimensional turbulent boundary layers over denselydistributed surface roughness [PhD thesis] Virginia PolytechnicInstitute and State University Blacksburg Va USA 2010
[21] Z S Fu ldquoThe gas cut detect technology during drilling in SovietUnionrdquo Petroleum Drilling Techniques vol 18 no 1 pp 19ndash211990
[22] X F Li C X Guan and X X Sui ldquoThe theory of gas influxdetection of pressure wave and its applicationrdquo Acta PetroleiSinica vol 18 no 3 pp 128ndash133 1997
[23] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[24] A E Ruggles R T Lahey Jr D A Drew and H A ScartonldquoInvestigation of the propagation of pressure perturbations inbubbly airwater flowsrdquo Journal of Heat Transfer vol 110 no 2pp 494ndash499 1988
[25] H J W M Legius H E A Van Den Akker and T NarumoldquoMeasurements on wave propagation and bubble and slugvelocities in cocurrent upward two-phase flowrdquo ExperimentalThermal and Fluid Science vol 15 no 3 pp 267ndash278 1997
[26] K Miyazaki and I Nakajima ldquoPropagation of pressure wavein NItrogen-Mercury two-phase systemrdquo Journal of NuclearScience and Technology vol 9 no 5 pp 315ndash318 1972
[27] B Bai L Guo and X Chen ldquoPressure fluctuation for air-watertwo-phase flowrdquo Journal of Hydrodynamics vol 18 no 4 pp476ndash482 2003
[28] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
Mathematical Problems in Engineering 17
[29] R F Tangren C H Dodge and H S Seifert ldquoCompressibilityeffects in two-phase flowrdquo Journal of Applied Physics vol 20 no7 pp 637ndash645 1949
[30] G B Wallis One Dimensional Two Phase Flow McGraw HillNew York NY USA 1969
[31] D L Nguyen E R F Winter and M Greiner ldquoSonic velocityin two-phase systemsrdquo International Journal ofMultiphase Flowvol 7 no 3 pp 311ndash320 1981
[32] R C Mecredy and L J Hamilton ldquoThe effects of nonequilib-rium heat mass and momentum transfer on two-phase soundspeedrdquo International Journal of Heat and Mass Transfer vol 15no 1 pp 61ndash72 1972
[33] K H Ardron and R B Duffey ldquoAcoustic wave propagationin a flowing liquid-vapour mixturerdquo International Journal ofMultiphase Flow vol 4 no 3 pp 303ndash322 1978
[34] J Xu and T Chen ldquoAcoustic wave prediction in flowing steam-water two-phase mixturerdquo International Journal of Heat andMass Transfer vol 43 no 7 pp 1079ndash1088 2000
[35] N ChungW Lin B Pei and Y Hsu ldquoModel for sound velocityin a two-phase air-water bubbly flowrdquo Nuclear Technology vol99 no 1 pp 80ndash89 1992
[36] X Xu and J Gong ldquoA unitedmodel for predicting pressurewavespeeds in oil and gas two-phase pipeflowsrdquo Journal of PetroleumScience and Engineering vol 60 no 3-4 pp 150ndash160 2008
[37] W Kuczynski ldquoCharacterization of pressure-wave propagationduring the condensation of R404A and R134a refrigerantsin pipe mini-channels that undergo periodic hydrodynamicdisturbancerdquo International Journal of Heat and Fluid Flow vol40 pp 135ndash150 2013
[38] W Kuczynski ldquoModeling of the propagation of a pressure waveduring the condensation process of R134a refrigerant in a pipeminichannel under the periodic conditions of hydrodynamicdisturbancesrdquo International Journal of Heat and Mass Transfervol 56 pp 715ndash723 2013
[39] Y Li C Li E Chen and Y Ying ldquoPressure wave propaga-tion characteristics in a two-phase flow pipeline for liquid-propellant rocketrdquoAerospace Science and Technology vol 15 no6 pp 453ndash464 2011
[40] T Kanagawa M Watanabe T Yano and S Fujikawa ldquoNon-linear wave equations for pressure wave propagation in liquidscontaining gas bubbles (comparison between two-fluid modeland mixture model)rdquo Journal of Fluid Science and Technologyvol 6 no 6 pp 836ndash849 2011
[41] EChen Y Li andYYing ldquoNumerical investigation onpressurewave propagation speed of gas-liquid two-phase flow in pumppipelinerdquo Journal of Aerospace Power vol 25 no 4 pp 754ndash7602010
[42] F Li and L He ldquoInvestigation on pressure wave propagationcharacteristics in slug flow field with increasing gas flow raterdquoPetroleum Geology and Recovery Efficiency vol 11 no 5 pp 78ndash80 2004
[43] C SMartin andM Padmanabhan ldquoPressure pulse propagationin two-component slug flowrdquo Journal of Fluids Engineering vol101 no 1 pp 44ndash52 1979
[44] R E Henry ldquoPressure wave propagation in two-phase mix-turesrdquo Chemical Engineering Progress vol 66 no 1 pp 1ndash101969
[45] X Liu B Li and Y Yue ldquoTransmission behavior of mud-pressure pulse alongwell borerdquo Journal ofHydrodynamics B vol19 no 2 pp 236ndash240 2007
[46] XWang and J Zhang ldquoThe research of pressure wave Pulsationin mud pulse transmittingrdquo Journal of Chongqing University ofScience and Technology vol 14 no 2 pp 55ndash58 2012
[47] J-W Park D A Drew and R T Lahey Jr ldquoThe analysisof void wave propagation in adiabatic monodispersed bubblytwo-phase flows using an ensemble-averaged two-fluid modelrdquoInternational Journal ofMultiphase Flow vol 24 no 7 pp 1205ndash1244 1998
[48] G Arnold Entropy and objectivity as constraints upons constitu-tive equations for two-fluid modeling of multiphase flow [PhDthesis] Rensselaer Polytechnic Institute New York NY USA1998
[49] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[50] L Yarborough and K R Hall ldquoHow to solve equation of statefor z-factorsrdquoOil and Gas Journal vol 72 no 7 pp 86ndash88 1974
[51] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86 pp 127ndash136 2012
[52] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[53] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[54] M Ishii and K Mishima ldquoTwo-fluid model and hydrodynamicconstitutive relationsrdquo Nuclear Engineering and Design vol 82no 2-3 pp 107ndash126 1984
[55] M J Sanchez Comparison of correlations for predicting pressurelosses in vertical multiphase annular flow [MS thesis] Univer-sity of Tulsa Tulsa Okla USA 1972
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
0
25
50
75
100
125
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
Q (m3h)
120601G
()
0 25 5 75 10
BP = 01 MPaw = 50Hz = ReCvm
Figure 11 Effect of gas influx rate on the void fraction
equilibrium state and thus the wave dispersion does notexist At a high angular frequency the wave velocity is mainlydominated by the mechanical and thermal nonequilibriumin the flow and keeps almost unchanged when the angularfrequency is further increased This is consistent with theinfluences of the angular frequency in the horizontal pipe[28] Moreover the effect of virtual mass force is also shownin this figure (119862vm = Re) It is observed that the wave velocityis significantly reduced when the virtual mass force is takeninto consideration in the calculation of the wave velocity atthe position close to the wellhead such as 119867 = 0m It canbe explained by the reduction of dispersion characteristic ofa pressure wave Along the wellbore from the wellhead to thebottomhole the distinction gradually decreases
6 Conclusions
With full consideration of the important factors such asvirtual mass force drag force gas void fraction pressuretemperature and angular frequency a united wave velocitymodel has been proposed based on pressure drop gradientequations in MPD operations gas-liquid two-fluid modelthe gas-drilling mud equations of state and small pertur-bation theory Solved by the fourth-order explicit Runge-Kutta method the model is used to predict wave velocityfor different back pressures and gas influx rates in MPDoperations The main conclusions can be summarized asfollows
(1) With the introduction of virtual mass force and dragforce the united model agrees well with the previousexperimental data The united model can be used to
0
280
560
840
1120
1400
Q (m3h)0 25 5 75 10
H = 0mH = 500mH = 2000m
H = 250mH = 1000mH = 4000m
c(m
s)
BP = 01 MPaw = 50Hz = ReCvm
Figure 12 Effect of gas influx rate on the pressure wave velocity
accurately calculate the wave velocity in the annulusThe application of the model will be beneficial tofurther study the wave velocity at different gas influxrates and back pressures in MPD operations reducenonproductive times and provide a reference for thedrilling operations in extremely narrow porefracturewindows existing conditions
(2) The wave velocity and void fraction have differentvalues with respect to well depth In the annulusthe drop of pressure causes an increase in the voidfraction along the flow direction The void fractionincreases first slightly and then sharply Correspond-ingly the wave velocity first gradually decreases in thbubbly flow and preliminary slug flow and then thewave velocity slightly increases accompanying withthe increase in the relative length ratio of gas slugto the liquid slug for the continuous increase in voidfraction at the position close to the wellhead Theminimum wave velocity appears in the long gas slugflow
(3) When the back pressure in the MPD operationsincreases the void fraction at different depths of theannulus is reduced gradually while the wave velocityin the two-phase flow tends to increase Moreoverthe influence of back pressure on the void fraction isgreater at a position close to the wellhead than that inthe bottomhole because of the great decrease in thepressure from the bottomhole to the wellhead Alsothe effect of back pressure on void fraction and wavevelocity is decreased in the high back pressure range
14 Mathematical Problems in Engineering
0 300 600 900 12004000
3000
2000
1000
225
180
135
90
45
0
BP = 01 MPaw = 50Hz
Q = 0594m3h = ReQ = 0594m3h = 0
Q = 1098m3h = ReQ = 1768m3h = ReQ = 1098m3h = 0
Q = 1768m3h = 0
c (ms)
H(m
)
CvmCvm
Cvm
Cvm
Cvm
Cvm
Figure 13 Effect of virtual mass force on the pressure wave velocity
(4) The wave velocity is sensitive to the void fraction butthe void fraction is dominated by gas influx rate andpressure in the annulus especially the gas influx rateSince the compressibility of the gas and drilling mudtwo-phase fluid is improved with the increase in theinflux rate the void fraction increases greatly andthe wave velocity decreases significantly within thelow gas influx rate range When the void fraction isincreased to some extent following the increase in thegas influx rate the decrease of wave velocity is slowedfor the growth of gas slug Especially at the wellheada slight increase in wave velocity is observed at a highgas influx rate for the sharp increase in void fraction
(5) It is necessary to analyze the effect of the virtualmass force on the wave velocity in MPD operationsEspecially at the position close to the wellhead theeffect of virtual mass force is more prominent for theintense phase interactions Taking the virtual massforce into account the dispersion characteristic of thepressure wave weakens obviously Compared with theresults calculated by ignoring the effect of virtualmassforce the calculated pressure wave velocity with aconsideration of the virtual mass force is lower
(6) The effect of angular frequency on the propagationvelocity of pressure wave appears at a low angularfrequency The propagation velocity of pressure dis-turbances increases together with the growth of theangular frequency (0 lt 119908 lt 500Hz) When theangular frequency reaches the value of 119908 = 500Hzthe pressure wave velocity achieves a constant value
0 240 480 720 960 12004000
3000
2000
1000
560
420
280
140
0
Q = 0727m3h BP = 01 MPa
w = 50Hz = Rew = 50Hz = 0w = 300Hz = 0
w = 800Hz = 0w = 1600Hz = 0w = 5000Hz = 0
c (ms)
H(m
)CvmCvmCvm
CvmCvmCvm
Figure 14 Wave velocity variations at different angular frequencies
and remains on this level regardless of the furthergrowth in angular frequency 119908
Nomenclature
A Annulus effective cross area (m2)c119866 Wave velocity in gas (ms)
c119871 Wave velocity in Drilling mud (ms)
C119863 The coefficient of drag force
D Annulus effective diameter (m)D119894 Diameter of the inner pipe (m)
D119900 Diameter of the outer pipe (m)
f119866 Shear stresses coefficient of gas interface
f119871 Shear stresses coefficient of drilling mud
interfaceF119911 Pressure applied to the two-phase flow (N)
g Acceleration due to gravity (m2s)g119888 Conversion factor
g119879 Geothermal temperature gradient (Km)
h One discrete length of annulus (m)k119890 Annulus effective roughness (m)
k119900 Roughness of outer pipe (m)
k119894 Roughness of inner pipe (m)
H The depth of annulus (m)L119902 Air bubbles length of slug (m)
119872Gi Momentum transfer in gas interface (Nm3)119872Li Momentum transfer in liquid interface (Nm3)119872
ndLi The none-drag-force (virtual mass force)
(Nm3)119872
119889
Li The momentum transfer term caused by thedrag force (Nm3)
Mathematical Problems in Engineering 15
100 1000 100000
450
900
1350
1800
0w (Hz)
0 m = 0 0 m = Re1000 m = 0
2000 m = 0
4000 m = 0
1000 m = Re2000 m = Re4000 m = Re
c(m
s)
Q = 0727m3h BP = 01 MPaCvmCvm
CvmCvmCvm
CvmCvmCvm
Figure 15 Effect of angular frequency on wave velocity
N Parameter combiningThompsonenergy effects
119901Gi Gas interface pressure (MPa)119901Li Liquid interface pressure (MPa)p Pressure (MPa)p0 Initial pressure of annulus (MPa)
p119888 Critical pressure (kPa)
p119871 Pressure of drilling mud (MPa)
p119903 Reduced pressure
q119898 Volumetric flow rate of gas and drilling
mud (m3s)q119866 Velocity of the gas (m3s)
q119871 Velocity of the drilling mud (m3s)
Q119871 Drilling mud rate within the annulus
(m3h)r Average diameter of the bubble (m)R Constant of EOS (JKgK)R119902 Air bubbles radius of slug (m)
R119887 The bubble diameter (m)
T Temperature (K)T119888 Critical temperature (K)
T119903 Reduced temperature
T119890119894 Undisturbed formation temperature at a
depth (K)T119890119887ℎ
Undisturbed formation temperature ofwellhead (K)
T119891119887ℎ
Undisturbed formation temperature atthe bottomhole (K)
v119898 Gas and drilling mud flow velocity
(ms)
v119866 Gas flow velocity (ms)
v119904 Slip velocity (ms)
v119904119892 Superficial gas velocity (ms)
v119871 Drilling mud flow velocity (ms)
W119898 Mass flow velocity (kgm3)
Z119866 Gas deviation factor
z0 Initial calculation height of annulus (m)
z119887ℎ Total well depth from surface (m)
z Variable of distance from surface (m)
Greek Letters
120588
119866 Gas density (kgm3)
120591
119891119903
Li Shear stresses of drilling mud noninterfacial(Nm2)
120591
119891119903
119871 Shear stresses of drilling mud interface (Nm2)
120591
Re119871 Reynolds stress of drilling mud (Nm2)
120591
Re119871 Reynolds stress of drilling mud interface
(Nm2)120591
Re119866 Reynolds stress of gas interface (Nm2)
120591
119891119903
119866 Shear stresses of gas interface (Nm2)
120572VM Acceleration due to virtual mass forcecoefficient (ms2)
120591
119871 Shear stresses of drilling mud along wall
(Nm2)120591
119866 Shear stresses of gas along wall (Nm2)
120591
Re119866 Reynolds stress of gas (Nm2)
120591
119891 Frictional pressure gradient (Pam)
120591
119908 Frictional pressure coefficient
120588
119898 Gas and drilling density (kgm3)
120588
0 Density under standard atmospheric pressure
(kgm3)120588
119871 Drilling mud density (kgm3)
120590
119904 Surface tension (Nm2)
Φ
119866 Gas void fraction
Φ
119871 Drilling mud holdup
120588
119903 Reduced density
120583
119871 Viscosity of drilling mud (Pa sdot s)
Subscripts
BHP Bottomhole pressureBP Back pressureEOS Equations of stateMPD Managed pressure drillingRCD The rotating control deviceR-K4 The fourth-order explicit Runge-Kutta
Subscripts of Graph
BP Back pressure (MPa)c Wave velocity in gas and drilling mud
two-phase flow (ms)119862vm The coefficient of virtual mass force
16 Mathematical Problems in Engineering
H Depth of annulus (m)Q Gas influx rate at the bottomhole (m3h)Re The value of virtual mass force
coefficient according to (28) or (34)119908 Angle frequency (Hz)Φ
119866 Void fraction ()
Acknowledgment
Research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 and 51274170)Without their support this work would not have beenpossible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of airinjection in annuli to manage pressure during cementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling conferenceand Exhibition San Diego Calif USA March 2012
[4] W A Bacon Consideration of compressibility effects for applied-back-pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Tex USA 2011
[5] J Choe C Hans and J Wold ldquoA modified two-phase well-control model and its computer applications as a training andeducational toolrdquo SPE Computer Applications vol 2 pp 14ndash201997
[6] J A Tarvin IWalton and PWand ldquoAnalysis of a gas kick takenin a deep well drilled with oil-based mudrdquo in Proceedings ofthe Annual Technical Conference and Exhibition of the Societyof Petroleum Engineers pp 255ndash264 Dallas Tex USA 1991
[7] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[8] J Zhou and G Nygaard ldquoAutomatic model-based controlscheme for stabilizing pressure during dual-gradient drillingrdquoJournal of Process Control vol 21 no 8 pp 1138ndash1147 2011
[9] A Ochoa O Acevedo L Nieto J E Lambarria and H PerezldquoSuccessful application of MPD (Managed Pressure Drilling)for prevention controland detection of borehole ballooning intight gas reservoir in Cuervito Field Mexicordquo in Proceedings ofthe Canadian Unconventional Resources Conference (CURC rsquo11)pp 172ndash185 November 2011
[10] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[11] R Ashena and J Moghadasi ldquoBottom hole pressure estimationusing evolved neural networks by real coded ant colony opti-mization and genetic algorithmrdquo Journal of Petroleum Scienceand Engineering vol 77 no 3-4 pp 375ndash385 2011
[12] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvanced Materials Research vol 468-471 pp 1736ndash1742 2012
[13] I Zubizarreta ldquoPore pressure evolution core damage and trip-ping out schedulesf a computational fluid dynamics approachrdquoin Proceedings of the SPEIADC Drilling Conference and Exhibi-tion Amsterdam The Netherlands March 2013
[14] C H Whitson ldquoCyclic shut-in eliminates liquid-loading in gaswellsrdquo in Proceedings of the SPEEAGE European Unconven-tional Resources Conference and Exhibition Vienna AustriaMarch 2012
[15] G M de Oliveira A T Franco C O R Negrao A LMartins and R A Silva ldquoModeling and validation of pressurepropagation in drilling fluids pumped into a closed wellrdquoJournal of Petroleum Science and Engineering vol 103 pp 61ndash71 2012
[16] Y Sato and H Kanki ldquoFormulas for compression wave andoscillating flow in circular piperdquo Applied Acoustics vol 69 no1 pp 1ndash11 2008
[17] J Xie B Yu X Zhang Q Shao and X Song ldquoNumericalsimulation of gas-liquid-solid three-phase flow in deep wellsrdquoAdvances in Mechanical Engineering vol 2013 Article ID951298 10 pages 2013
[18] W Bacon A Tong O Gabaldon C Sugden and P V Surya-narayana ldquoAn improved dynamic well control response to a gasinflux in managed pressure drilling operationsrdquo in Proceedingsof the IADCSPE Drilling Conference and Exhibition San DiegoCalifornia USA March 2012
[19] J I Hage and D ter Avest ldquoBorehole acoustics applied to kickdetectionrdquo Journal of Petroleum Science and Engineering vol 12no 2 pp 157ndash166 1994
[20] A M R Hopkins Fluid dynamics and surface pressure fluctua-tions of two-dimensional turbulent boundary layers over denselydistributed surface roughness [PhD thesis] Virginia PolytechnicInstitute and State University Blacksburg Va USA 2010
[21] Z S Fu ldquoThe gas cut detect technology during drilling in SovietUnionrdquo Petroleum Drilling Techniques vol 18 no 1 pp 19ndash211990
[22] X F Li C X Guan and X X Sui ldquoThe theory of gas influxdetection of pressure wave and its applicationrdquo Acta PetroleiSinica vol 18 no 3 pp 128ndash133 1997
[23] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[24] A E Ruggles R T Lahey Jr D A Drew and H A ScartonldquoInvestigation of the propagation of pressure perturbations inbubbly airwater flowsrdquo Journal of Heat Transfer vol 110 no 2pp 494ndash499 1988
[25] H J W M Legius H E A Van Den Akker and T NarumoldquoMeasurements on wave propagation and bubble and slugvelocities in cocurrent upward two-phase flowrdquo ExperimentalThermal and Fluid Science vol 15 no 3 pp 267ndash278 1997
[26] K Miyazaki and I Nakajima ldquoPropagation of pressure wavein NItrogen-Mercury two-phase systemrdquo Journal of NuclearScience and Technology vol 9 no 5 pp 315ndash318 1972
[27] B Bai L Guo and X Chen ldquoPressure fluctuation for air-watertwo-phase flowrdquo Journal of Hydrodynamics vol 18 no 4 pp476ndash482 2003
[28] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
Mathematical Problems in Engineering 17
[29] R F Tangren C H Dodge and H S Seifert ldquoCompressibilityeffects in two-phase flowrdquo Journal of Applied Physics vol 20 no7 pp 637ndash645 1949
[30] G B Wallis One Dimensional Two Phase Flow McGraw HillNew York NY USA 1969
[31] D L Nguyen E R F Winter and M Greiner ldquoSonic velocityin two-phase systemsrdquo International Journal ofMultiphase Flowvol 7 no 3 pp 311ndash320 1981
[32] R C Mecredy and L J Hamilton ldquoThe effects of nonequilib-rium heat mass and momentum transfer on two-phase soundspeedrdquo International Journal of Heat and Mass Transfer vol 15no 1 pp 61ndash72 1972
[33] K H Ardron and R B Duffey ldquoAcoustic wave propagationin a flowing liquid-vapour mixturerdquo International Journal ofMultiphase Flow vol 4 no 3 pp 303ndash322 1978
[34] J Xu and T Chen ldquoAcoustic wave prediction in flowing steam-water two-phase mixturerdquo International Journal of Heat andMass Transfer vol 43 no 7 pp 1079ndash1088 2000
[35] N ChungW Lin B Pei and Y Hsu ldquoModel for sound velocityin a two-phase air-water bubbly flowrdquo Nuclear Technology vol99 no 1 pp 80ndash89 1992
[36] X Xu and J Gong ldquoA unitedmodel for predicting pressurewavespeeds in oil and gas two-phase pipeflowsrdquo Journal of PetroleumScience and Engineering vol 60 no 3-4 pp 150ndash160 2008
[37] W Kuczynski ldquoCharacterization of pressure-wave propagationduring the condensation of R404A and R134a refrigerantsin pipe mini-channels that undergo periodic hydrodynamicdisturbancerdquo International Journal of Heat and Fluid Flow vol40 pp 135ndash150 2013
[38] W Kuczynski ldquoModeling of the propagation of a pressure waveduring the condensation process of R134a refrigerant in a pipeminichannel under the periodic conditions of hydrodynamicdisturbancesrdquo International Journal of Heat and Mass Transfervol 56 pp 715ndash723 2013
[39] Y Li C Li E Chen and Y Ying ldquoPressure wave propaga-tion characteristics in a two-phase flow pipeline for liquid-propellant rocketrdquoAerospace Science and Technology vol 15 no6 pp 453ndash464 2011
[40] T Kanagawa M Watanabe T Yano and S Fujikawa ldquoNon-linear wave equations for pressure wave propagation in liquidscontaining gas bubbles (comparison between two-fluid modeland mixture model)rdquo Journal of Fluid Science and Technologyvol 6 no 6 pp 836ndash849 2011
[41] EChen Y Li andYYing ldquoNumerical investigation onpressurewave propagation speed of gas-liquid two-phase flow in pumppipelinerdquo Journal of Aerospace Power vol 25 no 4 pp 754ndash7602010
[42] F Li and L He ldquoInvestigation on pressure wave propagationcharacteristics in slug flow field with increasing gas flow raterdquoPetroleum Geology and Recovery Efficiency vol 11 no 5 pp 78ndash80 2004
[43] C SMartin andM Padmanabhan ldquoPressure pulse propagationin two-component slug flowrdquo Journal of Fluids Engineering vol101 no 1 pp 44ndash52 1979
[44] R E Henry ldquoPressure wave propagation in two-phase mix-turesrdquo Chemical Engineering Progress vol 66 no 1 pp 1ndash101969
[45] X Liu B Li and Y Yue ldquoTransmission behavior of mud-pressure pulse alongwell borerdquo Journal ofHydrodynamics B vol19 no 2 pp 236ndash240 2007
[46] XWang and J Zhang ldquoThe research of pressure wave Pulsationin mud pulse transmittingrdquo Journal of Chongqing University ofScience and Technology vol 14 no 2 pp 55ndash58 2012
[47] J-W Park D A Drew and R T Lahey Jr ldquoThe analysisof void wave propagation in adiabatic monodispersed bubblytwo-phase flows using an ensemble-averaged two-fluid modelrdquoInternational Journal ofMultiphase Flow vol 24 no 7 pp 1205ndash1244 1998
[48] G Arnold Entropy and objectivity as constraints upons constitu-tive equations for two-fluid modeling of multiphase flow [PhDthesis] Rensselaer Polytechnic Institute New York NY USA1998
[49] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[50] L Yarborough and K R Hall ldquoHow to solve equation of statefor z-factorsrdquoOil and Gas Journal vol 72 no 7 pp 86ndash88 1974
[51] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86 pp 127ndash136 2012
[52] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[53] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[54] M Ishii and K Mishima ldquoTwo-fluid model and hydrodynamicconstitutive relationsrdquo Nuclear Engineering and Design vol 82no 2-3 pp 107ndash126 1984
[55] M J Sanchez Comparison of correlations for predicting pressurelosses in vertical multiphase annular flow [MS thesis] Univer-sity of Tulsa Tulsa Okla USA 1972
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
0 300 600 900 12004000
3000
2000
1000
225
180
135
90
45
0
BP = 01 MPaw = 50Hz
Q = 0594m3h = ReQ = 0594m3h = 0
Q = 1098m3h = ReQ = 1768m3h = ReQ = 1098m3h = 0
Q = 1768m3h = 0
c (ms)
H(m
)
CvmCvm
Cvm
Cvm
Cvm
Cvm
Figure 13 Effect of virtual mass force on the pressure wave velocity
(4) The wave velocity is sensitive to the void fraction butthe void fraction is dominated by gas influx rate andpressure in the annulus especially the gas influx rateSince the compressibility of the gas and drilling mudtwo-phase fluid is improved with the increase in theinflux rate the void fraction increases greatly andthe wave velocity decreases significantly within thelow gas influx rate range When the void fraction isincreased to some extent following the increase in thegas influx rate the decrease of wave velocity is slowedfor the growth of gas slug Especially at the wellheada slight increase in wave velocity is observed at a highgas influx rate for the sharp increase in void fraction
(5) It is necessary to analyze the effect of the virtualmass force on the wave velocity in MPD operationsEspecially at the position close to the wellhead theeffect of virtual mass force is more prominent for theintense phase interactions Taking the virtual massforce into account the dispersion characteristic of thepressure wave weakens obviously Compared with theresults calculated by ignoring the effect of virtualmassforce the calculated pressure wave velocity with aconsideration of the virtual mass force is lower
(6) The effect of angular frequency on the propagationvelocity of pressure wave appears at a low angularfrequency The propagation velocity of pressure dis-turbances increases together with the growth of theangular frequency (0 lt 119908 lt 500Hz) When theangular frequency reaches the value of 119908 = 500Hzthe pressure wave velocity achieves a constant value
0 240 480 720 960 12004000
3000
2000
1000
560
420
280
140
0
Q = 0727m3h BP = 01 MPa
w = 50Hz = Rew = 50Hz = 0w = 300Hz = 0
w = 800Hz = 0w = 1600Hz = 0w = 5000Hz = 0
c (ms)
H(m
)CvmCvmCvm
CvmCvmCvm
Figure 14 Wave velocity variations at different angular frequencies
and remains on this level regardless of the furthergrowth in angular frequency 119908
Nomenclature
A Annulus effective cross area (m2)c119866 Wave velocity in gas (ms)
c119871 Wave velocity in Drilling mud (ms)
C119863 The coefficient of drag force
D Annulus effective diameter (m)D119894 Diameter of the inner pipe (m)
D119900 Diameter of the outer pipe (m)
f119866 Shear stresses coefficient of gas interface
f119871 Shear stresses coefficient of drilling mud
interfaceF119911 Pressure applied to the two-phase flow (N)
g Acceleration due to gravity (m2s)g119888 Conversion factor
g119879 Geothermal temperature gradient (Km)
h One discrete length of annulus (m)k119890 Annulus effective roughness (m)
k119900 Roughness of outer pipe (m)
k119894 Roughness of inner pipe (m)
H The depth of annulus (m)L119902 Air bubbles length of slug (m)
119872Gi Momentum transfer in gas interface (Nm3)119872Li Momentum transfer in liquid interface (Nm3)119872
ndLi The none-drag-force (virtual mass force)
(Nm3)119872
119889
Li The momentum transfer term caused by thedrag force (Nm3)
Mathematical Problems in Engineering 15
100 1000 100000
450
900
1350
1800
0w (Hz)
0 m = 0 0 m = Re1000 m = 0
2000 m = 0
4000 m = 0
1000 m = Re2000 m = Re4000 m = Re
c(m
s)
Q = 0727m3h BP = 01 MPaCvmCvm
CvmCvmCvm
CvmCvmCvm
Figure 15 Effect of angular frequency on wave velocity
N Parameter combiningThompsonenergy effects
119901Gi Gas interface pressure (MPa)119901Li Liquid interface pressure (MPa)p Pressure (MPa)p0 Initial pressure of annulus (MPa)
p119888 Critical pressure (kPa)
p119871 Pressure of drilling mud (MPa)
p119903 Reduced pressure
q119898 Volumetric flow rate of gas and drilling
mud (m3s)q119866 Velocity of the gas (m3s)
q119871 Velocity of the drilling mud (m3s)
Q119871 Drilling mud rate within the annulus
(m3h)r Average diameter of the bubble (m)R Constant of EOS (JKgK)R119902 Air bubbles radius of slug (m)
R119887 The bubble diameter (m)
T Temperature (K)T119888 Critical temperature (K)
T119903 Reduced temperature
T119890119894 Undisturbed formation temperature at a
depth (K)T119890119887ℎ
Undisturbed formation temperature ofwellhead (K)
T119891119887ℎ
Undisturbed formation temperature atthe bottomhole (K)
v119898 Gas and drilling mud flow velocity
(ms)
v119866 Gas flow velocity (ms)
v119904 Slip velocity (ms)
v119904119892 Superficial gas velocity (ms)
v119871 Drilling mud flow velocity (ms)
W119898 Mass flow velocity (kgm3)
Z119866 Gas deviation factor
z0 Initial calculation height of annulus (m)
z119887ℎ Total well depth from surface (m)
z Variable of distance from surface (m)
Greek Letters
120588
119866 Gas density (kgm3)
120591
119891119903
Li Shear stresses of drilling mud noninterfacial(Nm2)
120591
119891119903
119871 Shear stresses of drilling mud interface (Nm2)
120591
Re119871 Reynolds stress of drilling mud (Nm2)
120591
Re119871 Reynolds stress of drilling mud interface
(Nm2)120591
Re119866 Reynolds stress of gas interface (Nm2)
120591
119891119903
119866 Shear stresses of gas interface (Nm2)
120572VM Acceleration due to virtual mass forcecoefficient (ms2)
120591
119871 Shear stresses of drilling mud along wall
(Nm2)120591
119866 Shear stresses of gas along wall (Nm2)
120591
Re119866 Reynolds stress of gas (Nm2)
120591
119891 Frictional pressure gradient (Pam)
120591
119908 Frictional pressure coefficient
120588
119898 Gas and drilling density (kgm3)
120588
0 Density under standard atmospheric pressure
(kgm3)120588
119871 Drilling mud density (kgm3)
120590
119904 Surface tension (Nm2)
Φ
119866 Gas void fraction
Φ
119871 Drilling mud holdup
120588
119903 Reduced density
120583
119871 Viscosity of drilling mud (Pa sdot s)
Subscripts
BHP Bottomhole pressureBP Back pressureEOS Equations of stateMPD Managed pressure drillingRCD The rotating control deviceR-K4 The fourth-order explicit Runge-Kutta
Subscripts of Graph
BP Back pressure (MPa)c Wave velocity in gas and drilling mud
two-phase flow (ms)119862vm The coefficient of virtual mass force
16 Mathematical Problems in Engineering
H Depth of annulus (m)Q Gas influx rate at the bottomhole (m3h)Re The value of virtual mass force
coefficient according to (28) or (34)119908 Angle frequency (Hz)Φ
119866 Void fraction ()
Acknowledgment
Research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 and 51274170)Without their support this work would not have beenpossible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of airinjection in annuli to manage pressure during cementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling conferenceand Exhibition San Diego Calif USA March 2012
[4] W A Bacon Consideration of compressibility effects for applied-back-pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Tex USA 2011
[5] J Choe C Hans and J Wold ldquoA modified two-phase well-control model and its computer applications as a training andeducational toolrdquo SPE Computer Applications vol 2 pp 14ndash201997
[6] J A Tarvin IWalton and PWand ldquoAnalysis of a gas kick takenin a deep well drilled with oil-based mudrdquo in Proceedings ofthe Annual Technical Conference and Exhibition of the Societyof Petroleum Engineers pp 255ndash264 Dallas Tex USA 1991
[7] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[8] J Zhou and G Nygaard ldquoAutomatic model-based controlscheme for stabilizing pressure during dual-gradient drillingrdquoJournal of Process Control vol 21 no 8 pp 1138ndash1147 2011
[9] A Ochoa O Acevedo L Nieto J E Lambarria and H PerezldquoSuccessful application of MPD (Managed Pressure Drilling)for prevention controland detection of borehole ballooning intight gas reservoir in Cuervito Field Mexicordquo in Proceedings ofthe Canadian Unconventional Resources Conference (CURC rsquo11)pp 172ndash185 November 2011
[10] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[11] R Ashena and J Moghadasi ldquoBottom hole pressure estimationusing evolved neural networks by real coded ant colony opti-mization and genetic algorithmrdquo Journal of Petroleum Scienceand Engineering vol 77 no 3-4 pp 375ndash385 2011
[12] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvanced Materials Research vol 468-471 pp 1736ndash1742 2012
[13] I Zubizarreta ldquoPore pressure evolution core damage and trip-ping out schedulesf a computational fluid dynamics approachrdquoin Proceedings of the SPEIADC Drilling Conference and Exhibi-tion Amsterdam The Netherlands March 2013
[14] C H Whitson ldquoCyclic shut-in eliminates liquid-loading in gaswellsrdquo in Proceedings of the SPEEAGE European Unconven-tional Resources Conference and Exhibition Vienna AustriaMarch 2012
[15] G M de Oliveira A T Franco C O R Negrao A LMartins and R A Silva ldquoModeling and validation of pressurepropagation in drilling fluids pumped into a closed wellrdquoJournal of Petroleum Science and Engineering vol 103 pp 61ndash71 2012
[16] Y Sato and H Kanki ldquoFormulas for compression wave andoscillating flow in circular piperdquo Applied Acoustics vol 69 no1 pp 1ndash11 2008
[17] J Xie B Yu X Zhang Q Shao and X Song ldquoNumericalsimulation of gas-liquid-solid three-phase flow in deep wellsrdquoAdvances in Mechanical Engineering vol 2013 Article ID951298 10 pages 2013
[18] W Bacon A Tong O Gabaldon C Sugden and P V Surya-narayana ldquoAn improved dynamic well control response to a gasinflux in managed pressure drilling operationsrdquo in Proceedingsof the IADCSPE Drilling Conference and Exhibition San DiegoCalifornia USA March 2012
[19] J I Hage and D ter Avest ldquoBorehole acoustics applied to kickdetectionrdquo Journal of Petroleum Science and Engineering vol 12no 2 pp 157ndash166 1994
[20] A M R Hopkins Fluid dynamics and surface pressure fluctua-tions of two-dimensional turbulent boundary layers over denselydistributed surface roughness [PhD thesis] Virginia PolytechnicInstitute and State University Blacksburg Va USA 2010
[21] Z S Fu ldquoThe gas cut detect technology during drilling in SovietUnionrdquo Petroleum Drilling Techniques vol 18 no 1 pp 19ndash211990
[22] X F Li C X Guan and X X Sui ldquoThe theory of gas influxdetection of pressure wave and its applicationrdquo Acta PetroleiSinica vol 18 no 3 pp 128ndash133 1997
[23] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[24] A E Ruggles R T Lahey Jr D A Drew and H A ScartonldquoInvestigation of the propagation of pressure perturbations inbubbly airwater flowsrdquo Journal of Heat Transfer vol 110 no 2pp 494ndash499 1988
[25] H J W M Legius H E A Van Den Akker and T NarumoldquoMeasurements on wave propagation and bubble and slugvelocities in cocurrent upward two-phase flowrdquo ExperimentalThermal and Fluid Science vol 15 no 3 pp 267ndash278 1997
[26] K Miyazaki and I Nakajima ldquoPropagation of pressure wavein NItrogen-Mercury two-phase systemrdquo Journal of NuclearScience and Technology vol 9 no 5 pp 315ndash318 1972
[27] B Bai L Guo and X Chen ldquoPressure fluctuation for air-watertwo-phase flowrdquo Journal of Hydrodynamics vol 18 no 4 pp476ndash482 2003
[28] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
Mathematical Problems in Engineering 17
[29] R F Tangren C H Dodge and H S Seifert ldquoCompressibilityeffects in two-phase flowrdquo Journal of Applied Physics vol 20 no7 pp 637ndash645 1949
[30] G B Wallis One Dimensional Two Phase Flow McGraw HillNew York NY USA 1969
[31] D L Nguyen E R F Winter and M Greiner ldquoSonic velocityin two-phase systemsrdquo International Journal ofMultiphase Flowvol 7 no 3 pp 311ndash320 1981
[32] R C Mecredy and L J Hamilton ldquoThe effects of nonequilib-rium heat mass and momentum transfer on two-phase soundspeedrdquo International Journal of Heat and Mass Transfer vol 15no 1 pp 61ndash72 1972
[33] K H Ardron and R B Duffey ldquoAcoustic wave propagationin a flowing liquid-vapour mixturerdquo International Journal ofMultiphase Flow vol 4 no 3 pp 303ndash322 1978
[34] J Xu and T Chen ldquoAcoustic wave prediction in flowing steam-water two-phase mixturerdquo International Journal of Heat andMass Transfer vol 43 no 7 pp 1079ndash1088 2000
[35] N ChungW Lin B Pei and Y Hsu ldquoModel for sound velocityin a two-phase air-water bubbly flowrdquo Nuclear Technology vol99 no 1 pp 80ndash89 1992
[36] X Xu and J Gong ldquoA unitedmodel for predicting pressurewavespeeds in oil and gas two-phase pipeflowsrdquo Journal of PetroleumScience and Engineering vol 60 no 3-4 pp 150ndash160 2008
[37] W Kuczynski ldquoCharacterization of pressure-wave propagationduring the condensation of R404A and R134a refrigerantsin pipe mini-channels that undergo periodic hydrodynamicdisturbancerdquo International Journal of Heat and Fluid Flow vol40 pp 135ndash150 2013
[38] W Kuczynski ldquoModeling of the propagation of a pressure waveduring the condensation process of R134a refrigerant in a pipeminichannel under the periodic conditions of hydrodynamicdisturbancesrdquo International Journal of Heat and Mass Transfervol 56 pp 715ndash723 2013
[39] Y Li C Li E Chen and Y Ying ldquoPressure wave propaga-tion characteristics in a two-phase flow pipeline for liquid-propellant rocketrdquoAerospace Science and Technology vol 15 no6 pp 453ndash464 2011
[40] T Kanagawa M Watanabe T Yano and S Fujikawa ldquoNon-linear wave equations for pressure wave propagation in liquidscontaining gas bubbles (comparison between two-fluid modeland mixture model)rdquo Journal of Fluid Science and Technologyvol 6 no 6 pp 836ndash849 2011
[41] EChen Y Li andYYing ldquoNumerical investigation onpressurewave propagation speed of gas-liquid two-phase flow in pumppipelinerdquo Journal of Aerospace Power vol 25 no 4 pp 754ndash7602010
[42] F Li and L He ldquoInvestigation on pressure wave propagationcharacteristics in slug flow field with increasing gas flow raterdquoPetroleum Geology and Recovery Efficiency vol 11 no 5 pp 78ndash80 2004
[43] C SMartin andM Padmanabhan ldquoPressure pulse propagationin two-component slug flowrdquo Journal of Fluids Engineering vol101 no 1 pp 44ndash52 1979
[44] R E Henry ldquoPressure wave propagation in two-phase mix-turesrdquo Chemical Engineering Progress vol 66 no 1 pp 1ndash101969
[45] X Liu B Li and Y Yue ldquoTransmission behavior of mud-pressure pulse alongwell borerdquo Journal ofHydrodynamics B vol19 no 2 pp 236ndash240 2007
[46] XWang and J Zhang ldquoThe research of pressure wave Pulsationin mud pulse transmittingrdquo Journal of Chongqing University ofScience and Technology vol 14 no 2 pp 55ndash58 2012
[47] J-W Park D A Drew and R T Lahey Jr ldquoThe analysisof void wave propagation in adiabatic monodispersed bubblytwo-phase flows using an ensemble-averaged two-fluid modelrdquoInternational Journal ofMultiphase Flow vol 24 no 7 pp 1205ndash1244 1998
[48] G Arnold Entropy and objectivity as constraints upons constitu-tive equations for two-fluid modeling of multiphase flow [PhDthesis] Rensselaer Polytechnic Institute New York NY USA1998
[49] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[50] L Yarborough and K R Hall ldquoHow to solve equation of statefor z-factorsrdquoOil and Gas Journal vol 72 no 7 pp 86ndash88 1974
[51] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86 pp 127ndash136 2012
[52] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[53] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[54] M Ishii and K Mishima ldquoTwo-fluid model and hydrodynamicconstitutive relationsrdquo Nuclear Engineering and Design vol 82no 2-3 pp 107ndash126 1984
[55] M J Sanchez Comparison of correlations for predicting pressurelosses in vertical multiphase annular flow [MS thesis] Univer-sity of Tulsa Tulsa Okla USA 1972
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
100 1000 100000
450
900
1350
1800
0w (Hz)
0 m = 0 0 m = Re1000 m = 0
2000 m = 0
4000 m = 0
1000 m = Re2000 m = Re4000 m = Re
c(m
s)
Q = 0727m3h BP = 01 MPaCvmCvm
CvmCvmCvm
CvmCvmCvm
Figure 15 Effect of angular frequency on wave velocity
N Parameter combiningThompsonenergy effects
119901Gi Gas interface pressure (MPa)119901Li Liquid interface pressure (MPa)p Pressure (MPa)p0 Initial pressure of annulus (MPa)
p119888 Critical pressure (kPa)
p119871 Pressure of drilling mud (MPa)
p119903 Reduced pressure
q119898 Volumetric flow rate of gas and drilling
mud (m3s)q119866 Velocity of the gas (m3s)
q119871 Velocity of the drilling mud (m3s)
Q119871 Drilling mud rate within the annulus
(m3h)r Average diameter of the bubble (m)R Constant of EOS (JKgK)R119902 Air bubbles radius of slug (m)
R119887 The bubble diameter (m)
T Temperature (K)T119888 Critical temperature (K)
T119903 Reduced temperature
T119890119894 Undisturbed formation temperature at a
depth (K)T119890119887ℎ
Undisturbed formation temperature ofwellhead (K)
T119891119887ℎ
Undisturbed formation temperature atthe bottomhole (K)
v119898 Gas and drilling mud flow velocity
(ms)
v119866 Gas flow velocity (ms)
v119904 Slip velocity (ms)
v119904119892 Superficial gas velocity (ms)
v119871 Drilling mud flow velocity (ms)
W119898 Mass flow velocity (kgm3)
Z119866 Gas deviation factor
z0 Initial calculation height of annulus (m)
z119887ℎ Total well depth from surface (m)
z Variable of distance from surface (m)
Greek Letters
120588
119866 Gas density (kgm3)
120591
119891119903
Li Shear stresses of drilling mud noninterfacial(Nm2)
120591
119891119903
119871 Shear stresses of drilling mud interface (Nm2)
120591
Re119871 Reynolds stress of drilling mud (Nm2)
120591
Re119871 Reynolds stress of drilling mud interface
(Nm2)120591
Re119866 Reynolds stress of gas interface (Nm2)
120591
119891119903
119866 Shear stresses of gas interface (Nm2)
120572VM Acceleration due to virtual mass forcecoefficient (ms2)
120591
119871 Shear stresses of drilling mud along wall
(Nm2)120591
119866 Shear stresses of gas along wall (Nm2)
120591
Re119866 Reynolds stress of gas (Nm2)
120591
119891 Frictional pressure gradient (Pam)
120591
119908 Frictional pressure coefficient
120588
119898 Gas and drilling density (kgm3)
120588
0 Density under standard atmospheric pressure
(kgm3)120588
119871 Drilling mud density (kgm3)
120590
119904 Surface tension (Nm2)
Φ
119866 Gas void fraction
Φ
119871 Drilling mud holdup
120588
119903 Reduced density
120583
119871 Viscosity of drilling mud (Pa sdot s)
Subscripts
BHP Bottomhole pressureBP Back pressureEOS Equations of stateMPD Managed pressure drillingRCD The rotating control deviceR-K4 The fourth-order explicit Runge-Kutta
Subscripts of Graph
BP Back pressure (MPa)c Wave velocity in gas and drilling mud
two-phase flow (ms)119862vm The coefficient of virtual mass force
16 Mathematical Problems in Engineering
H Depth of annulus (m)Q Gas influx rate at the bottomhole (m3h)Re The value of virtual mass force
coefficient according to (28) or (34)119908 Angle frequency (Hz)Φ
119866 Void fraction ()
Acknowledgment
Research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 and 51274170)Without their support this work would not have beenpossible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of airinjection in annuli to manage pressure during cementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling conferenceand Exhibition San Diego Calif USA March 2012
[4] W A Bacon Consideration of compressibility effects for applied-back-pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Tex USA 2011
[5] J Choe C Hans and J Wold ldquoA modified two-phase well-control model and its computer applications as a training andeducational toolrdquo SPE Computer Applications vol 2 pp 14ndash201997
[6] J A Tarvin IWalton and PWand ldquoAnalysis of a gas kick takenin a deep well drilled with oil-based mudrdquo in Proceedings ofthe Annual Technical Conference and Exhibition of the Societyof Petroleum Engineers pp 255ndash264 Dallas Tex USA 1991
[7] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[8] J Zhou and G Nygaard ldquoAutomatic model-based controlscheme for stabilizing pressure during dual-gradient drillingrdquoJournal of Process Control vol 21 no 8 pp 1138ndash1147 2011
[9] A Ochoa O Acevedo L Nieto J E Lambarria and H PerezldquoSuccessful application of MPD (Managed Pressure Drilling)for prevention controland detection of borehole ballooning intight gas reservoir in Cuervito Field Mexicordquo in Proceedings ofthe Canadian Unconventional Resources Conference (CURC rsquo11)pp 172ndash185 November 2011
[10] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[11] R Ashena and J Moghadasi ldquoBottom hole pressure estimationusing evolved neural networks by real coded ant colony opti-mization and genetic algorithmrdquo Journal of Petroleum Scienceand Engineering vol 77 no 3-4 pp 375ndash385 2011
[12] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvanced Materials Research vol 468-471 pp 1736ndash1742 2012
[13] I Zubizarreta ldquoPore pressure evolution core damage and trip-ping out schedulesf a computational fluid dynamics approachrdquoin Proceedings of the SPEIADC Drilling Conference and Exhibi-tion Amsterdam The Netherlands March 2013
[14] C H Whitson ldquoCyclic shut-in eliminates liquid-loading in gaswellsrdquo in Proceedings of the SPEEAGE European Unconven-tional Resources Conference and Exhibition Vienna AustriaMarch 2012
[15] G M de Oliveira A T Franco C O R Negrao A LMartins and R A Silva ldquoModeling and validation of pressurepropagation in drilling fluids pumped into a closed wellrdquoJournal of Petroleum Science and Engineering vol 103 pp 61ndash71 2012
[16] Y Sato and H Kanki ldquoFormulas for compression wave andoscillating flow in circular piperdquo Applied Acoustics vol 69 no1 pp 1ndash11 2008
[17] J Xie B Yu X Zhang Q Shao and X Song ldquoNumericalsimulation of gas-liquid-solid three-phase flow in deep wellsrdquoAdvances in Mechanical Engineering vol 2013 Article ID951298 10 pages 2013
[18] W Bacon A Tong O Gabaldon C Sugden and P V Surya-narayana ldquoAn improved dynamic well control response to a gasinflux in managed pressure drilling operationsrdquo in Proceedingsof the IADCSPE Drilling Conference and Exhibition San DiegoCalifornia USA March 2012
[19] J I Hage and D ter Avest ldquoBorehole acoustics applied to kickdetectionrdquo Journal of Petroleum Science and Engineering vol 12no 2 pp 157ndash166 1994
[20] A M R Hopkins Fluid dynamics and surface pressure fluctua-tions of two-dimensional turbulent boundary layers over denselydistributed surface roughness [PhD thesis] Virginia PolytechnicInstitute and State University Blacksburg Va USA 2010
[21] Z S Fu ldquoThe gas cut detect technology during drilling in SovietUnionrdquo Petroleum Drilling Techniques vol 18 no 1 pp 19ndash211990
[22] X F Li C X Guan and X X Sui ldquoThe theory of gas influxdetection of pressure wave and its applicationrdquo Acta PetroleiSinica vol 18 no 3 pp 128ndash133 1997
[23] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[24] A E Ruggles R T Lahey Jr D A Drew and H A ScartonldquoInvestigation of the propagation of pressure perturbations inbubbly airwater flowsrdquo Journal of Heat Transfer vol 110 no 2pp 494ndash499 1988
[25] H J W M Legius H E A Van Den Akker and T NarumoldquoMeasurements on wave propagation and bubble and slugvelocities in cocurrent upward two-phase flowrdquo ExperimentalThermal and Fluid Science vol 15 no 3 pp 267ndash278 1997
[26] K Miyazaki and I Nakajima ldquoPropagation of pressure wavein NItrogen-Mercury two-phase systemrdquo Journal of NuclearScience and Technology vol 9 no 5 pp 315ndash318 1972
[27] B Bai L Guo and X Chen ldquoPressure fluctuation for air-watertwo-phase flowrdquo Journal of Hydrodynamics vol 18 no 4 pp476ndash482 2003
[28] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
Mathematical Problems in Engineering 17
[29] R F Tangren C H Dodge and H S Seifert ldquoCompressibilityeffects in two-phase flowrdquo Journal of Applied Physics vol 20 no7 pp 637ndash645 1949
[30] G B Wallis One Dimensional Two Phase Flow McGraw HillNew York NY USA 1969
[31] D L Nguyen E R F Winter and M Greiner ldquoSonic velocityin two-phase systemsrdquo International Journal ofMultiphase Flowvol 7 no 3 pp 311ndash320 1981
[32] R C Mecredy and L J Hamilton ldquoThe effects of nonequilib-rium heat mass and momentum transfer on two-phase soundspeedrdquo International Journal of Heat and Mass Transfer vol 15no 1 pp 61ndash72 1972
[33] K H Ardron and R B Duffey ldquoAcoustic wave propagationin a flowing liquid-vapour mixturerdquo International Journal ofMultiphase Flow vol 4 no 3 pp 303ndash322 1978
[34] J Xu and T Chen ldquoAcoustic wave prediction in flowing steam-water two-phase mixturerdquo International Journal of Heat andMass Transfer vol 43 no 7 pp 1079ndash1088 2000
[35] N ChungW Lin B Pei and Y Hsu ldquoModel for sound velocityin a two-phase air-water bubbly flowrdquo Nuclear Technology vol99 no 1 pp 80ndash89 1992
[36] X Xu and J Gong ldquoA unitedmodel for predicting pressurewavespeeds in oil and gas two-phase pipeflowsrdquo Journal of PetroleumScience and Engineering vol 60 no 3-4 pp 150ndash160 2008
[37] W Kuczynski ldquoCharacterization of pressure-wave propagationduring the condensation of R404A and R134a refrigerantsin pipe mini-channels that undergo periodic hydrodynamicdisturbancerdquo International Journal of Heat and Fluid Flow vol40 pp 135ndash150 2013
[38] W Kuczynski ldquoModeling of the propagation of a pressure waveduring the condensation process of R134a refrigerant in a pipeminichannel under the periodic conditions of hydrodynamicdisturbancesrdquo International Journal of Heat and Mass Transfervol 56 pp 715ndash723 2013
[39] Y Li C Li E Chen and Y Ying ldquoPressure wave propaga-tion characteristics in a two-phase flow pipeline for liquid-propellant rocketrdquoAerospace Science and Technology vol 15 no6 pp 453ndash464 2011
[40] T Kanagawa M Watanabe T Yano and S Fujikawa ldquoNon-linear wave equations for pressure wave propagation in liquidscontaining gas bubbles (comparison between two-fluid modeland mixture model)rdquo Journal of Fluid Science and Technologyvol 6 no 6 pp 836ndash849 2011
[41] EChen Y Li andYYing ldquoNumerical investigation onpressurewave propagation speed of gas-liquid two-phase flow in pumppipelinerdquo Journal of Aerospace Power vol 25 no 4 pp 754ndash7602010
[42] F Li and L He ldquoInvestigation on pressure wave propagationcharacteristics in slug flow field with increasing gas flow raterdquoPetroleum Geology and Recovery Efficiency vol 11 no 5 pp 78ndash80 2004
[43] C SMartin andM Padmanabhan ldquoPressure pulse propagationin two-component slug flowrdquo Journal of Fluids Engineering vol101 no 1 pp 44ndash52 1979
[44] R E Henry ldquoPressure wave propagation in two-phase mix-turesrdquo Chemical Engineering Progress vol 66 no 1 pp 1ndash101969
[45] X Liu B Li and Y Yue ldquoTransmission behavior of mud-pressure pulse alongwell borerdquo Journal ofHydrodynamics B vol19 no 2 pp 236ndash240 2007
[46] XWang and J Zhang ldquoThe research of pressure wave Pulsationin mud pulse transmittingrdquo Journal of Chongqing University ofScience and Technology vol 14 no 2 pp 55ndash58 2012
[47] J-W Park D A Drew and R T Lahey Jr ldquoThe analysisof void wave propagation in adiabatic monodispersed bubblytwo-phase flows using an ensemble-averaged two-fluid modelrdquoInternational Journal ofMultiphase Flow vol 24 no 7 pp 1205ndash1244 1998
[48] G Arnold Entropy and objectivity as constraints upons constitu-tive equations for two-fluid modeling of multiphase flow [PhDthesis] Rensselaer Polytechnic Institute New York NY USA1998
[49] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[50] L Yarborough and K R Hall ldquoHow to solve equation of statefor z-factorsrdquoOil and Gas Journal vol 72 no 7 pp 86ndash88 1974
[51] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86 pp 127ndash136 2012
[52] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[53] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[54] M Ishii and K Mishima ldquoTwo-fluid model and hydrodynamicconstitutive relationsrdquo Nuclear Engineering and Design vol 82no 2-3 pp 107ndash126 1984
[55] M J Sanchez Comparison of correlations for predicting pressurelosses in vertical multiphase annular flow [MS thesis] Univer-sity of Tulsa Tulsa Okla USA 1972
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Mathematical Problems in Engineering
H Depth of annulus (m)Q Gas influx rate at the bottomhole (m3h)Re The value of virtual mass force
coefficient according to (28) or (34)119908 Angle frequency (Hz)Φ
119866 Void fraction ()
Acknowledgment
Research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 and 51274170)Without their support this work would not have beenpossible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of airinjection in annuli to manage pressure during cementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling conferenceand Exhibition San Diego Calif USA March 2012
[4] W A Bacon Consideration of compressibility effects for applied-back-pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Tex USA 2011
[5] J Choe C Hans and J Wold ldquoA modified two-phase well-control model and its computer applications as a training andeducational toolrdquo SPE Computer Applications vol 2 pp 14ndash201997
[6] J A Tarvin IWalton and PWand ldquoAnalysis of a gas kick takenin a deep well drilled with oil-based mudrdquo in Proceedings ofthe Annual Technical Conference and Exhibition of the Societyof Petroleum Engineers pp 255ndash264 Dallas Tex USA 1991
[7] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[8] J Zhou and G Nygaard ldquoAutomatic model-based controlscheme for stabilizing pressure during dual-gradient drillingrdquoJournal of Process Control vol 21 no 8 pp 1138ndash1147 2011
[9] A Ochoa O Acevedo L Nieto J E Lambarria and H PerezldquoSuccessful application of MPD (Managed Pressure Drilling)for prevention controland detection of borehole ballooning intight gas reservoir in Cuervito Field Mexicordquo in Proceedings ofthe Canadian Unconventional Resources Conference (CURC rsquo11)pp 172ndash185 November 2011
[10] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the North Africa Technical Conference andExhibition Cairo Egypt February 2012
[11] R Ashena and J Moghadasi ldquoBottom hole pressure estimationusing evolved neural networks by real coded ant colony opti-mization and genetic algorithmrdquo Journal of Petroleum Scienceand Engineering vol 77 no 3-4 pp 375ndash385 2011
[12] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvanced Materials Research vol 468-471 pp 1736ndash1742 2012
[13] I Zubizarreta ldquoPore pressure evolution core damage and trip-ping out schedulesf a computational fluid dynamics approachrdquoin Proceedings of the SPEIADC Drilling Conference and Exhibi-tion Amsterdam The Netherlands March 2013
[14] C H Whitson ldquoCyclic shut-in eliminates liquid-loading in gaswellsrdquo in Proceedings of the SPEEAGE European Unconven-tional Resources Conference and Exhibition Vienna AustriaMarch 2012
[15] G M de Oliveira A T Franco C O R Negrao A LMartins and R A Silva ldquoModeling and validation of pressurepropagation in drilling fluids pumped into a closed wellrdquoJournal of Petroleum Science and Engineering vol 103 pp 61ndash71 2012
[16] Y Sato and H Kanki ldquoFormulas for compression wave andoscillating flow in circular piperdquo Applied Acoustics vol 69 no1 pp 1ndash11 2008
[17] J Xie B Yu X Zhang Q Shao and X Song ldquoNumericalsimulation of gas-liquid-solid three-phase flow in deep wellsrdquoAdvances in Mechanical Engineering vol 2013 Article ID951298 10 pages 2013
[18] W Bacon A Tong O Gabaldon C Sugden and P V Surya-narayana ldquoAn improved dynamic well control response to a gasinflux in managed pressure drilling operationsrdquo in Proceedingsof the IADCSPE Drilling Conference and Exhibition San DiegoCalifornia USA March 2012
[19] J I Hage and D ter Avest ldquoBorehole acoustics applied to kickdetectionrdquo Journal of Petroleum Science and Engineering vol 12no 2 pp 157ndash166 1994
[20] A M R Hopkins Fluid dynamics and surface pressure fluctua-tions of two-dimensional turbulent boundary layers over denselydistributed surface roughness [PhD thesis] Virginia PolytechnicInstitute and State University Blacksburg Va USA 2010
[21] Z S Fu ldquoThe gas cut detect technology during drilling in SovietUnionrdquo Petroleum Drilling Techniques vol 18 no 1 pp 19ndash211990
[22] X F Li C X Guan and X X Sui ldquoThe theory of gas influxdetection of pressure wave and its applicationrdquo Acta PetroleiSinica vol 18 no 3 pp 128ndash133 1997
[23] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[24] A E Ruggles R T Lahey Jr D A Drew and H A ScartonldquoInvestigation of the propagation of pressure perturbations inbubbly airwater flowsrdquo Journal of Heat Transfer vol 110 no 2pp 494ndash499 1988
[25] H J W M Legius H E A Van Den Akker and T NarumoldquoMeasurements on wave propagation and bubble and slugvelocities in cocurrent upward two-phase flowrdquo ExperimentalThermal and Fluid Science vol 15 no 3 pp 267ndash278 1997
[26] K Miyazaki and I Nakajima ldquoPropagation of pressure wavein NItrogen-Mercury two-phase systemrdquo Journal of NuclearScience and Technology vol 9 no 5 pp 315ndash318 1972
[27] B Bai L Guo and X Chen ldquoPressure fluctuation for air-watertwo-phase flowrdquo Journal of Hydrodynamics vol 18 no 4 pp476ndash482 2003
[28] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
Mathematical Problems in Engineering 17
[29] R F Tangren C H Dodge and H S Seifert ldquoCompressibilityeffects in two-phase flowrdquo Journal of Applied Physics vol 20 no7 pp 637ndash645 1949
[30] G B Wallis One Dimensional Two Phase Flow McGraw HillNew York NY USA 1969
[31] D L Nguyen E R F Winter and M Greiner ldquoSonic velocityin two-phase systemsrdquo International Journal ofMultiphase Flowvol 7 no 3 pp 311ndash320 1981
[32] R C Mecredy and L J Hamilton ldquoThe effects of nonequilib-rium heat mass and momentum transfer on two-phase soundspeedrdquo International Journal of Heat and Mass Transfer vol 15no 1 pp 61ndash72 1972
[33] K H Ardron and R B Duffey ldquoAcoustic wave propagationin a flowing liquid-vapour mixturerdquo International Journal ofMultiphase Flow vol 4 no 3 pp 303ndash322 1978
[34] J Xu and T Chen ldquoAcoustic wave prediction in flowing steam-water two-phase mixturerdquo International Journal of Heat andMass Transfer vol 43 no 7 pp 1079ndash1088 2000
[35] N ChungW Lin B Pei and Y Hsu ldquoModel for sound velocityin a two-phase air-water bubbly flowrdquo Nuclear Technology vol99 no 1 pp 80ndash89 1992
[36] X Xu and J Gong ldquoA unitedmodel for predicting pressurewavespeeds in oil and gas two-phase pipeflowsrdquo Journal of PetroleumScience and Engineering vol 60 no 3-4 pp 150ndash160 2008
[37] W Kuczynski ldquoCharacterization of pressure-wave propagationduring the condensation of R404A and R134a refrigerantsin pipe mini-channels that undergo periodic hydrodynamicdisturbancerdquo International Journal of Heat and Fluid Flow vol40 pp 135ndash150 2013
[38] W Kuczynski ldquoModeling of the propagation of a pressure waveduring the condensation process of R134a refrigerant in a pipeminichannel under the periodic conditions of hydrodynamicdisturbancesrdquo International Journal of Heat and Mass Transfervol 56 pp 715ndash723 2013
[39] Y Li C Li E Chen and Y Ying ldquoPressure wave propaga-tion characteristics in a two-phase flow pipeline for liquid-propellant rocketrdquoAerospace Science and Technology vol 15 no6 pp 453ndash464 2011
[40] T Kanagawa M Watanabe T Yano and S Fujikawa ldquoNon-linear wave equations for pressure wave propagation in liquidscontaining gas bubbles (comparison between two-fluid modeland mixture model)rdquo Journal of Fluid Science and Technologyvol 6 no 6 pp 836ndash849 2011
[41] EChen Y Li andYYing ldquoNumerical investigation onpressurewave propagation speed of gas-liquid two-phase flow in pumppipelinerdquo Journal of Aerospace Power vol 25 no 4 pp 754ndash7602010
[42] F Li and L He ldquoInvestigation on pressure wave propagationcharacteristics in slug flow field with increasing gas flow raterdquoPetroleum Geology and Recovery Efficiency vol 11 no 5 pp 78ndash80 2004
[43] C SMartin andM Padmanabhan ldquoPressure pulse propagationin two-component slug flowrdquo Journal of Fluids Engineering vol101 no 1 pp 44ndash52 1979
[44] R E Henry ldquoPressure wave propagation in two-phase mix-turesrdquo Chemical Engineering Progress vol 66 no 1 pp 1ndash101969
[45] X Liu B Li and Y Yue ldquoTransmission behavior of mud-pressure pulse alongwell borerdquo Journal ofHydrodynamics B vol19 no 2 pp 236ndash240 2007
[46] XWang and J Zhang ldquoThe research of pressure wave Pulsationin mud pulse transmittingrdquo Journal of Chongqing University ofScience and Technology vol 14 no 2 pp 55ndash58 2012
[47] J-W Park D A Drew and R T Lahey Jr ldquoThe analysisof void wave propagation in adiabatic monodispersed bubblytwo-phase flows using an ensemble-averaged two-fluid modelrdquoInternational Journal ofMultiphase Flow vol 24 no 7 pp 1205ndash1244 1998
[48] G Arnold Entropy and objectivity as constraints upons constitu-tive equations for two-fluid modeling of multiphase flow [PhDthesis] Rensselaer Polytechnic Institute New York NY USA1998
[49] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[50] L Yarborough and K R Hall ldquoHow to solve equation of statefor z-factorsrdquoOil and Gas Journal vol 72 no 7 pp 86ndash88 1974
[51] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86 pp 127ndash136 2012
[52] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[53] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[54] M Ishii and K Mishima ldquoTwo-fluid model and hydrodynamicconstitutive relationsrdquo Nuclear Engineering and Design vol 82no 2-3 pp 107ndash126 1984
[55] M J Sanchez Comparison of correlations for predicting pressurelosses in vertical multiphase annular flow [MS thesis] Univer-sity of Tulsa Tulsa Okla USA 1972
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 17
[29] R F Tangren C H Dodge and H S Seifert ldquoCompressibilityeffects in two-phase flowrdquo Journal of Applied Physics vol 20 no7 pp 637ndash645 1949
[30] G B Wallis One Dimensional Two Phase Flow McGraw HillNew York NY USA 1969
[31] D L Nguyen E R F Winter and M Greiner ldquoSonic velocityin two-phase systemsrdquo International Journal ofMultiphase Flowvol 7 no 3 pp 311ndash320 1981
[32] R C Mecredy and L J Hamilton ldquoThe effects of nonequilib-rium heat mass and momentum transfer on two-phase soundspeedrdquo International Journal of Heat and Mass Transfer vol 15no 1 pp 61ndash72 1972
[33] K H Ardron and R B Duffey ldquoAcoustic wave propagationin a flowing liquid-vapour mixturerdquo International Journal ofMultiphase Flow vol 4 no 3 pp 303ndash322 1978
[34] J Xu and T Chen ldquoAcoustic wave prediction in flowing steam-water two-phase mixturerdquo International Journal of Heat andMass Transfer vol 43 no 7 pp 1079ndash1088 2000
[35] N ChungW Lin B Pei and Y Hsu ldquoModel for sound velocityin a two-phase air-water bubbly flowrdquo Nuclear Technology vol99 no 1 pp 80ndash89 1992
[36] X Xu and J Gong ldquoA unitedmodel for predicting pressurewavespeeds in oil and gas two-phase pipeflowsrdquo Journal of PetroleumScience and Engineering vol 60 no 3-4 pp 150ndash160 2008
[37] W Kuczynski ldquoCharacterization of pressure-wave propagationduring the condensation of R404A and R134a refrigerantsin pipe mini-channels that undergo periodic hydrodynamicdisturbancerdquo International Journal of Heat and Fluid Flow vol40 pp 135ndash150 2013
[38] W Kuczynski ldquoModeling of the propagation of a pressure waveduring the condensation process of R134a refrigerant in a pipeminichannel under the periodic conditions of hydrodynamicdisturbancesrdquo International Journal of Heat and Mass Transfervol 56 pp 715ndash723 2013
[39] Y Li C Li E Chen and Y Ying ldquoPressure wave propaga-tion characteristics in a two-phase flow pipeline for liquid-propellant rocketrdquoAerospace Science and Technology vol 15 no6 pp 453ndash464 2011
[40] T Kanagawa M Watanabe T Yano and S Fujikawa ldquoNon-linear wave equations for pressure wave propagation in liquidscontaining gas bubbles (comparison between two-fluid modeland mixture model)rdquo Journal of Fluid Science and Technologyvol 6 no 6 pp 836ndash849 2011
[41] EChen Y Li andYYing ldquoNumerical investigation onpressurewave propagation speed of gas-liquid two-phase flow in pumppipelinerdquo Journal of Aerospace Power vol 25 no 4 pp 754ndash7602010
[42] F Li and L He ldquoInvestigation on pressure wave propagationcharacteristics in slug flow field with increasing gas flow raterdquoPetroleum Geology and Recovery Efficiency vol 11 no 5 pp 78ndash80 2004
[43] C SMartin andM Padmanabhan ldquoPressure pulse propagationin two-component slug flowrdquo Journal of Fluids Engineering vol101 no 1 pp 44ndash52 1979
[44] R E Henry ldquoPressure wave propagation in two-phase mix-turesrdquo Chemical Engineering Progress vol 66 no 1 pp 1ndash101969
[45] X Liu B Li and Y Yue ldquoTransmission behavior of mud-pressure pulse alongwell borerdquo Journal ofHydrodynamics B vol19 no 2 pp 236ndash240 2007
[46] XWang and J Zhang ldquoThe research of pressure wave Pulsationin mud pulse transmittingrdquo Journal of Chongqing University ofScience and Technology vol 14 no 2 pp 55ndash58 2012
[47] J-W Park D A Drew and R T Lahey Jr ldquoThe analysisof void wave propagation in adiabatic monodispersed bubblytwo-phase flows using an ensemble-averaged two-fluid modelrdquoInternational Journal ofMultiphase Flow vol 24 no 7 pp 1205ndash1244 1998
[48] G Arnold Entropy and objectivity as constraints upons constitu-tive equations for two-fluid modeling of multiphase flow [PhDthesis] Rensselaer Polytechnic Institute New York NY USA1998
[49] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[50] L Yarborough and K R Hall ldquoHow to solve equation of statefor z-factorsrdquoOil and Gas Journal vol 72 no 7 pp 86ndash88 1974
[51] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86 pp 127ndash136 2012
[52] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[53] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[54] M Ishii and K Mishima ldquoTwo-fluid model and hydrodynamicconstitutive relationsrdquo Nuclear Engineering and Design vol 82no 2-3 pp 107ndash126 1984
[55] M J Sanchez Comparison of correlations for predicting pressurelosses in vertical multiphase annular flow [MS thesis] Univer-sity of Tulsa Tulsa Okla USA 1972
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
