An Improved Sucker Rod Pumping System Model and Swabbing...

16
Research Article An Improved Sucker Rod Pumping System Model and Swabbing Parameters Optimized Design Weicheng Li , 1,2 Shimin Dong , 1 and Xiurong Sun 1 1 School of Mechanical Engineering, Yanshan University, Qinhuangdao, 066004, China 2 School of Engineering, King’s College, University of Aberdeen, Aberdeen, AB24 3UE Scotland, UK Correspondence should be addressed to Shimin Dong; [email protected] Received 28 December 2017; Accepted 15 October 2018; Published 30 October 2018 Academic Editor: Gen Q. Xu Copyright © 2018 Weicheng Li 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. Considering the impact of fluid flowing into pump on sucker rod pumping system (SRPS) dynamic behaviors, an improved SRPS model with new boundary model is presented, which is a fluid-solid coupled model with the interactions among surface transmission, rod string longitudinal vibration, plunger motion, and fluid flow. A uniform algorithm is adopted instead of the mixed iteration algorithm for the surface transmission and downhole rod string vibration submodels, to reduce the difficulties of solving the entire SRPS model. e dynamic response comparison is executed between the improved model and the current model, and the results show that it will bring a calculation error on pump load and pump fullness if the progress of fluid flowing into the pump (PFFP) is ignored. Based on this improved model, a multitarget optimization model is proposed and the dynamic behavior of SRPS is improved with the optimized swabbing parameters. 1. Introduction e SRPS is widely used in oil fields. It comprises three parts: surface transmission unit converting rotational motion into linear motion, sucker rod string as a joint between surface and downhole, and reciprocating pump exploiting the oil (see Figure 1). e importance of predicting the dynamic responses of SRPS is to determine the operating situation and oil production [1]. For this equipment is usually set up in an open-air environment, and the test data device mounted on it is usually broken, especially the main working subsystem which is located nearly one kilometer or more downhole, making it difficult to be tested. erefore, a more accurate SRPS simulation model should be established although the research in this field of study has been carried out widely. Due to the slender rod string moving upwards and down- wards all time, an intense longitudinal vibration is produced. According to Figure 1, the SRPS model can be divided into rod string longitudinal vibration model, surface transmission model, and downhole pumping model. Commonly, this model is solved using the rod string longitudinal vibration equation as the foundation, surface and downhole model as boundary conditions. e most successful model of rod string longitudinal vibration is the Gibbs’s wave equation, and based on that, the models are studied specifically on the enhancement of the surface and downhole boundaries with different operating conditions [2–4]. e surface boundary condition that includes the motor speed variations has been extensively used and is more applicable in practice [5, 6] whereas the downhole boundary condition has been contin- uously improved. However, further study is still required due to the inconsistent alteration and complication encountered in downhole operation. e downhole boundary condition actually is a model describing the pump operation. e pump operating mode can be divided into upstroke and downstroke. During upstroke, as the plunger moves upwards, the pump pressure will decrease and the fluid will not be sucked into the pump until this pressure drop down to pump inlet pressure. For the downstroke, the pressure will increase with the plunger goes down and the fluid will begin to be drained out when the pressure equals the pump outlet pressure. Based on the SRPS’s operating state, the first universal downhole boundary model was divided into four phases with a vague formulation Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 4746210, 15 pages https://doi.org/10.1155/2018/4746210

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Research ArticleAn Improved Sucker Rod Pumping System Model and SwabbingParameters Optimized Design

Weicheng Li 12 Shimin Dong 1 and Xiurong Sun 1

1School of Mechanical Engineering Yanshan University Qinhuangdao 066004 China2School of Engineering Kingrsquos College University of Aberdeen Aberdeen AB24 3UE Scotland UK

Correspondence should be addressed to Shimin Dong ysudshm163com

Received 28 December 2017 Accepted 15 October 2018 Published 30 October 2018

Academic Editor Gen Q Xu

Copyright copy 2018 Weicheng Li 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

Considering the impact of fluid flowing into pump on sucker rod pumping system (SRPS) dynamic behaviors an improvedSRPS model with new boundary model is presented which is a fluid-solid coupled model with the interactions among surfacetransmission rod string longitudinal vibration plunger motion and fluid flow A uniform algorithm is adopted instead of themixed iteration algorithm for the surface transmission and downhole rod string vibration submodels to reduce the difficulties ofsolving the entire SRPS modelThe dynamic response comparison is executed between the improvedmodel and the currentmodeland the results show that it will bring a calculation error on pump load and pump fullness if the progress of fluid flowing into thepump (PFFP) is ignored Based on this improved model a multitarget optimization model is proposed and the dynamic behaviorof SRPS is improved with the optimized swabbing parameters

1 Introduction

The SRPS is widely used in oil fields It comprises three partssurface transmission unit converting rotational motion intolinear motion sucker rod string as a joint between surfaceand downhole and reciprocating pump exploiting the oil(see Figure 1) The importance of predicting the dynamicresponses of SRPS is to determine the operating situation andoil production [1] For this equipment is usually set up in anopen-air environment and the test data device mounted onit is usually broken especially the main working subsystemwhich is located nearly one kilometer or more downholemaking it difficult to be tested Therefore a more accurateSRPS simulation model should be established although theresearch in this field of study has been carried out widelyDue to the slender rod string moving upwards and down-wards all time an intense longitudinal vibration is producedAccording to Figure 1 the SRPS model can be divided intorod string longitudinal vibration model surface transmissionmodel and downhole pumping model Commonly thismodel is solved using the rod string longitudinal vibrationequation as the foundation surface and downhole model

as boundary conditions The most successful model of rodstring longitudinal vibration is the Gibbsrsquos wave equationand based on that the models are studied specifically on theenhancement of the surface and downhole boundaries withdifferent operating conditions [2ndash4] The surface boundarycondition that includes the motor speed variations has beenextensively used and is more applicable in practice [5 6]whereas the downhole boundary condition has been contin-uously improved However further study is still required dueto the inconsistent alteration and complication encounteredin downhole operation

The downhole boundary condition actually is a modeldescribing the pump operation The pump operating modecan be divided into upstroke and downstroke Duringupstroke as the plunger moves upwards the pump pressurewill decrease and the fluid will not be sucked into the pumpuntil this pressure drop down to pump inlet pressure Forthe downstroke the pressure will increase with the plungergoes down and the fluid will begin to be drained out whenthe pressure equals the pump outlet pressure Based on theSRPSrsquos operating state the first universal downhole boundarymodel was divided into four phases with a vague formulation

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 4746210 15 pageshttpsdoiorg10115520184746210

2 Mathematical Problems in Engineering

Dynamic liquid level

Pump inlet

Pump outlet

Plunger

Pump depth

Rod string

Suspension point

ldquoHorse headrdquoBeam

Motor

Belt

Reduction gear box

Crank

Link rod

Figure 1 Sucker rod pumping system

[2] In an ideal condition during the fully loaded upstrokemovement the pump load was set equivalent to the fluidload pump load was set to zero for the unloaded downstrokemovement [7 8] As the pump operation phases are highlyrelated to pump pressure it was revised with an explicitformation deduced by the interaction between pump outletpressure and the pressure in the pump barrel [9ndash14] asfollows

119865119875119871 (119905) = 119860119901 (119901119889 minus 119901) minus 119860119903119901119889 (1)

where FPL is the pump load NAp andAr are the cross sectionarea of plunger and rod string respectively m2 pd is the pumpoutlet pressure pa p is the pump pressure pa

For the pd is always considered as a constant pressure thekey of this research is to establish an accurate pump pressuremodel which consists of four phases as shown in Figure 2With considering of the gas this downhole boundary modelis improved as shown in (2)When the pump is at phases 1 and3 the pressure variation obeys the rules of gas state equationAs for in phases 2 and 4 the pump pressure is taken as the psand pd separately

119901 = ( 119871119900119892119871119900119892 + 119906119901 minus int119905119904

0(119902119860119901))

119876 119901119889 phase 1119901 = 119901119904 phase 2119901 = ( 119871119892119871119892 minus 119871 119904 + 119906119901 minus int119905119905

119905119906(119902119860119901))

119876

119901119904 phase 3119901 = 119901119889 phase 4

(2)

phase 1 phase 2 phase 3 phase 4

gas e

xpan

sion

pum

ping

flui

d

gas c

ompr

essio

n

disc

harg

e flui

d

Figure 2 Four phases of pump operation

where 119871119900119892 and 119871119892 are the gas column length when plungeris arriving at bottom dead center and top dead centerrespectively m up is the plunger displacement m Ls is thepump stroke displacement m Lp is the plunger length m 120583is the fluid dynamic viscosity pasdots 120575 is the clearance betweenplunger and pump barrel m Dd is the pump diameter mts and tt are the open time of standing valve and travellingvalve respectively s tu is the upstroke time q is the liquidinstantaneous leakage volume

In this formula the principle that the gasoil ratio ofclearance volume (the space volume when plunger arrives atbottom dead center) equals the gasoil ratio of pump inletis applied However at this time the pump pressure shouldbe the same as the pump outlet pressure Thus it is revisedwith the pump outlet pressure and this improved model hasbeen widely used until now [15ndash18] However the currentdownhole boundary model still exists some shortcomingsneed to be improved Due to ignoring the progress of PFFPit is established with the hypothesis of regarding the pumpis filled with the fluid whose gasliquid ratio always equalsthat at pump intake as well as the one in the clearancevolume This assumption is not applicable for the oil wellwith insufficient oil well deliverability (OWD) which will

Mathematical Problems in Engineering 3

Motorrotation

Surfacetransmission

Rod string longitudinal

vibration

Plunger motionPump pressure variationFluid motion

Figure 3 SRPS coupled model sketch

result in the pumping fluid being unable to keep pace withthe plunger and cause an incomplete fullness Meanwhile apump load calculation error will be produced with keepingthe pump pressure as constant when fluid flows into thepump

The SRPS model lays the foundation for optimizingthe swabbing parameters to improve the system operationstatus except for predicting and evaluating its dynamicresponse Miska et al 1997 [19] propose a computer-aidedoptimization method relying on a simple linear algebraicsystem model so as to minimize the energy consumptionFiru et al 2003 [20] present an improve optimization cri-terion including eight operational parameters to achieve themaximum system efficiency Liu and Qi 2011 [21] apply fluidflow characteristics in coalbed methane reservoirs to estimatethe production capacity and build the system efficiencyoptimization model combined with SRPS performance Theabove optimization models are built with a simplified pumpload then an improved system efficiency optimization modelis built jointing with formula (2) [22 23] However thecurrent optimization models ignore the effect of swabbingparameters on pump fullness in that the SRPS modelrsquosrestriction Besides that single target optimization cannotmake an accurate and comprehensive presentation for SRPSwhose pumping progress is complex multicomponent andinteractive

In this paper firstly an improved SRPSmodel is presentedwith the new downhole boundary model considering themovement of fluid flowing into pump with gas instantaneousdissolution and evolution Secondly for the new downholeboundary model a nonlinear fluid-solid coupled model willincrease the complexity of this improved SRPS model Aunified numerical algorithm is applied on the whole modelto decrease the calculation time instead of traditional mixediteration algorithm Thirdly a multiobjective optimizationmodel is proposed based on the improved SRPS modelFourthly the surface dynamometer card is collected to verifythe improvedmodelrsquos accuracy Fifthly the dynamic responsecomparison on the current SRPS model and improved SRPSmodel is executed Finally the optimization program isapplied on a test well and the results are given before andafter optimization

2 Integrated Simulation Model

The improved SRPS model is a nonlinear fluid-solid coupledmodel consisted of drive transmission and loadThemotor isused to generate power and its rotationmotion is transformedinto linear motion of suspension point This linear motionis passed on to the plunger through rod string longitudinalvibration meanwhile plunger motion has an influence onpumping fluid by regulating pump pressure On the contraryfluid motion affects pump pressure which is also the mainfactor in determining pump load This pump load andother system loads are transferred to motor output shaft bysurface and downhole transmission causing impact on themotor motion The above coupled process is depicted inFigure 3 In this paper the SRPSmodel is subdivided into rodstring longitudinal vibration model corresponding surfaceboundary and downhole boundary model so as to have aclear description

21 Sucker Rod Longitudinal Vibration Model While theSRPS is operating the motion of arbitrary micro elementof the slender rod string produced intense longitudinalvibration can be composed of the following two parts (1)following the movement of suspension point and (2) relativemovement to the suspension point Referring to Figure 4 thecoordinate system is built based on the top dead center ofhorse head as the origin Then the stress balance terms of rodstring element and top and bottom boundary conditions areused to establish the sucker rod longitudinal vibration modelwith the assumption of ignoring the deformation of tube andthe vibration of liquid column in a vertical well

12059721199061205971199052 minus 119864119903120588119903 12059721199061205971199092+ 120583120588119903119860119903

120597119906120597119905= minus1198892119906119886 (119905)1198891199052 minus 120583120588119903119860119903

119889119906119886 (119905)119889119905 + 119892119906 (119909 119905)|119909=0 = 119906119886 (119905)119864119903119860119903

120597119906 (119909 119905)12059711990910038161003816100381610038161003816100381610038161003816119909=119871119903

= 119865119875119871 (119905)

(3)

4 Mathematical Problems in Engineering

Top dead center

x

dx

Lru(xt)

FPL(t)

ua(t)

rAr(d2ua

dt2+

2u

t2)dx

(dua

dt+

u

t)dx

ErAru

x+

x(ErAr

u

x)dx

rArgdx

ErAru

x

Figure 4 Mechanical model of rod string longitudinal vibration

Med

mMef

Figure 5 Surface apparatus equivalent motion model

where u is the displacement of the rod string at arbitrarydepth and time m Er and 120588r are the elasticity modulus anddensity of rod string respectively pa and kgm3 ua is thesuspension point displacement m

22 Surface Boundary Model The surface boundary condi-tion refers to the motion of suspension point as in (3) Withthe assumption that the surface transmission mechanismconsists of belt-reducing gear and four-bar linkage as a singledegree of freedom system it can be represented as a functionof motor angle Later the surface apparatus motion model isset up using Lagrange equation taking motor output shaft asequivalent component its motion as generalized coordinatesand set 12 orsquoclock as a reference direction (Figure 5)

119868119890119898 + 12 119898

2 119889119868119890119889120579119898 = 119872119890119889 minus119872119890119891

1205791198981003816100381610038161003816119905=0 = 0119898

10038161003816100381610038161003816120579=0= 1205960

(4)

LC

LP

LR

LL

LK

LI

LH

LAFRL

LRmdash crankLPmdash link rodLCmdash beam backLAmdash beam front

Figure 6 Four-bar linkage

With the equivalent rotating inertia

119868119890 = sum119868119895 (120596119895119898

)2 +sum119898119895 ( V119895119898

)2

(5)

The semiempirical formulation of motor driving torqueMed deduced from the speedtorque characteristic is adoptedfromWu et al [24]

119872119890119889 = 2120582119896119872119867120576119888120596119899 [120596119899 minus 119898]12057621198881205962

119899 + [120596119899 minus 119898]2 (6)

where 120582k is the motor overload coefficient 120576c is the motorcritical-slip ratio 120596n is the motor synchronous angularvelocity rads

119872119867 = 9550119875119867119899119867

120576119888 = 120576119903 (120582119896 + radic1205822119896minus 1)

120596119899 = 2120587119891119911(7)

where z is the motor pole pairs f is the power frequency HzPH is the motor rated power kW nH is the motor rated speedminminus1 120576r is the motor rated-slip ratio

The equivalent resistance torque is derived in accordancewith the force balance of four-bar linkage (Figure 6) writtenas

119872119890119891 = 1119894119887119892120576119887 [119878119879119891 (119865119877119871 minus 119865119887) 1205761198961119901 minus119872119888 sin (120579 minus 120579120591)] (8)

with the angle of crank

120579 = 120579119898119894119887119892(9)

Mathematical Problems in Engineering 5

where ibg is the transmission ratio of the belt-reducinggear system 120576p is the transmission efficiency from crank tosuspension point STf is torque factorm FRL is the suspensionpoint load N Fb is the structural unbalance weight NMc isthe maximum balancing torque of the crank Nsdotm

The displacement of suspension point can be expressedas motor angle based on the geometry relation shown inFigure 6 [6] The surface boundary motion model is

119906119886 (119905) = arccos [1198712119862 + 1198712

119870 minus (119871119877 + 119871119875)22119871119862119871119870

]minus arccos(1198712

119862 + 1198712119871 minus 1198712

1198752119871119862119871119871

) minus arcsin(119871119877119871119871

sdot sin(2120587 minus 1205790 minus 120579119898119894119887119892

+ arcsin( 119871119868119871119870

)))(10)

23 Downhole Boundary Model This new model is animproved method from the current boundary models con-sidering the interaction among pump pressure fluid flow intopump gas dissolution and evolution pressure drop due tofluid gravitational potential energy inertia head loss frictionhead loss and local head loss The new boundary modelconsists of pressure variation model and fluid flow into pumpmodel Some assumptions are made for building this model

(1) Bottom dead point of plunger as the origin position(2) The pump inlet pressure ps and outlet pressure pd are

kept constant for one cycle of SRPS operation

Pressure Variation Model The gas state equations are built asfollows ( (11) and (12)) when the displacement of plunger is upor up+dup

119901 [(119906119901 + 119871119900119892 minus 119871119891 minus 119871119896)119860119901] = 119885119873119876119879 (11)

(119901 + 119889119901)sdot [(119906119901 + 119889119906119901 + 119871119900119892 minus 119871119891 minus 119871119896 minus 119889119871119891 minus 119889119871119896)119860119901]= (119885 + 120597119885120597119901 119889119901) (119873 + 119889119873)119876119879

(12)

where N is the gas molar number mol Q is the natural gasconstant J(molsdotK) T is the temperature in pump barrel ∘CZ is the natural gas compressibility factor Lk is the equivalentlength of pump leakage m Lf is the liquid level in the pumpm

Dividing (11) by (12) and ignoring the second order smallquantities then the variation of pump pressure with plungerdisplacement is obtained

119889119901119889119906119901

= (1119873) (119889119873119889119906119901) + (1 (119906119901 + 119871119900119892 minus 119871119891 minus 119871119896)) (119889 (119871119891 + 119871119896) 119889119906119901 minus 1)119901minus1 minus (1119885) (119889119885119889119901) (13)

The above equation is converted into a time varyingfunction with the purpose of facilitating it with the rod stringlongitudinal vibration equation

119889119901119889119905 = (1119873) (119889119873119889119905) (1V119901) + (1 (119906119901 + 119871119900119892 minus 119871119891 minus 119871119896)) ((119889 (119871119891 + 119871119896) 119889119905) (1V119901) minus 1)119901minus1 minus (1119885) (119889119885119889119901) V119901(14)

where vp is the plunger velocity msFrom (14) dN is composed of the following sections(1)Themolar number of gas is released from or dissolved

into oil as pressure varies

119889119873119903 = minus 119885119879119901119904119905119885119904119905119879119904119905119901 (120572 sdot 119889119901)sdot 103119860119901 (119871119891 + 119871119900 minus 119871119900119892) (1 minus 120576119908)119861119900119881119898119900119897

(15)

where pst is the standard pressure pa Zst is the standard natu-ral gas compressibility factorTst is the standard temperature∘C 120572 is the solubility coefficient of natural gas m3 (m3sdotpa)

120576w is the water content Vmol is the molar volume m3 Bo isthe crude oil volume factor

(2)Themolar number of gas with fluid being sucked intothe pump corresponding to the pump pressure is

119889119873119904 = 103119860119901119877119869119889119871119891119881119898119900119897

(16)

where RJ is the transient gas liquid ratio in the pump m3m3(3)Themolar number of gaswith leaked fluid at the pump

pressure is

119889119873V = 103119860119901119877119869119889119871V119881119898119900119897

(17)

6 Mathematical Problems in Engineering

f-f

L

Lf

La

a-a

Figure 7 Process of fluid flows into the pump

where

119877119869 = (1 minus 120576119908) (119877119901 minus 119877119904) 119901119904119905119885119879119879119904119905119885119904119905119901119877119904 = 120572 (119901 minus 119901119904119905)

(18)

where Rp is the production gas oil ratio m3m3 Rs is the gasoil ratio at pump intake m3m3

Fluid Flowing into Pump Model Figure 7 shows the pumpoperation when the gas-liquid flow is drawn into the pumpSection a-a indicates the cross section of standing valve holeSupposing the section f -f is the liquid level at arbitrarytime neglect the process of gas bubbling from fluid byconsidering only the pump gas to occupy the upper spaceof f -f The one-dimensional unsteady flow equation based onBernoulli equation is established to describe the flow state ofpumping liquid In the Bernoulli equation the inertia headloss friction head loss and local head loss are all considered

119901119904120588g + V21198862119892 = 119901120588g + V2

1198912g + (119871119900 minus 119871119900119892 + 119871119891 + 119871V) + ℎ119886

+ ℎV + ℎ119891

(19)

where

ℎ119886 = 1119892 (119871119900 minus 119871119900119892 + 119871119891 + 119871V119860119891119860119901

) 119889V119891119889119905ℎ119891 = 64120583V119891 (119871119900 minus 119871119900119892 + 119871119891)21198632

119889120588119892

ℎV = 11198921205762 (119860119901119860119891

)2 V21198862

V119886 = 119860119901119860119891

V119891

(20)

where 120576 is the flow coefficient of standing valveThen (19) is converted into differential forms

119891 = V119891

V119891 = (1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V119860119891119860119901

)minus1119901119904120588 minus 119901120588+ V2

1198912 [11986021199011198602119891

minus 119860211990112057621198602

119891

minus 64120583 (1198710 minus 119881119900119892119860119901 + 119871119891)1198632119889V119891120588

minus 1] minus 119892(1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V)

(21)

Normally the fluid in the barrel nomatter in which formwill all be discharged except the one in dead spaceThereforethe fluid flowing out of the model with opening travelingvalve is not proposed The pump outlet pressure generatedby several kilometers liquid column is large compared to thehydraulic loss causing the ignorance of hydraulic loss whenfluid flows out Based on the statement mentioned above thenew downhole boundary model is expressed as follows andalso is divided into four phases corresponding to the phasesshown in Figure 2

1198781199051198861199051199061199041 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199042 af ter 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = ((V119891 sdot 119860119901 + 119902) (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)]sdot V119901

Mathematical Problems in Engineering 7

1198781199051198861199051199061199043 before 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199044 af ter 119900119901119890119899119892119894119899 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = 0

(22)

where

119881119898119900119897 = 224119885119879119901119904119905119879119904119905119901 (23)

3 Integrated Numerical Algorithm

For the current SRPS model the surface transmission modelis solved by numerical integration method [25] While thefinite difference method is used to describe the rod stringlongitudinal vibration [2 26 27] Since there is no fixedsolution between surface and downholemodel it needs cycliciteration to handle thewholemodel which is time consuming[25] Considering the interaction within plunger motionfluid flow into pump and pump pressure the improved SRPSmodel has a higher nonlinear degree and thus increasesthe difficulty of solving it In order to reduce the solvingtime the wave equation of rod string longitudinal vibrationis converted into ordinary differential equations by modalsuperposition method Then the surface and downholeboundary model are now in the form of ordinary differentialequations at which the whole model can be solved by Runge-Kutta method directly Besides that solving the rod stringlongitudinal vibration equation is also solving the forcevibration response of rod string where (3) is converted intothe following form

120588119903119860119903

12059721199061205971199052 minus 120588119903119864119903

12059721199061205971199092+ 120583120597119906120597119905 = 120588119903119860119903119891 (119909 119905) (24)

where

119891 (119909 119905) = minus1198892119906119886 (119905)1198891199052 minus 120583120588119903119860119903

119889119906119886 (119905)119889119905 + 119892 (25)

And its solution can be expressed in the function of timeand space by separating the variables

119906 (119909 119905) = infinsum119894=1

Φ119894 (119909) 119902119894 (119905) (26)

Therefore (24) can be expressed asinfinsum119894=1

119860119903120588119903Φ119894 (119909) 119894 (119905)minus infinsum

119894=1

119864119903119860119903

1198892Φ119894 (119909)1198891199092119902119894 (119905) + infinsum

119894=1

120583Φ119894 (119909) 119894 (119905)= 119860119903120588119903119891 (119909 119905)

(27)

Multiply the above equation by Φi(x) and it is integratedalong the rod length Then the following is derived

119860119903120588119903119894 (119905) int119871119903

0Φ119894

2 (119909) 119889119909minus 119864119903119860119903119902119894 (119905) int119871119903

0Φ119894 (119909)Φ119894

10158401015840 (119909) 119889119909+ 120583119894 (119905) int119871119903

0Φ119894

2 (119909) 119889119909= 119860119903120588119903 int119871119903

0Φ119894 (119909) 119891 (119909 119905) 119889119909

(28)

Its mode shapes and natural frequencies are

Φ119894 (119909) = radic 2120588119903119860119903

sin((2119894 minus 1) 1205872119871119903

119909)119901119899119894 = (2119894 minus 1) 120587radic1198641199031205881199032119871119903

(29)

Then (28) can be simplified as follows using mode shapeorthogonality

119894 (119905) + 120583119860119903120588119903 119894 (119905) + ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 119902119894 (119905) = 119865119894 (30)

where

119865119894 = (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903(2119894 minus 1) 120587+ 119865119875119871 (119905) radic 2119860119903120588119903119871119903

sin ((2119894 minus 1) 1205872 )(31)

8 Mathematical Problems in Engineering

Let ith-order forced vibration displacement response andvelocity response be xi1 and xi2 respectivelyThen (30) can beexpressed as the following form

1198941 (119905) = 1199091198942 (119905)1198942 (119905) = minus 1205831198601199031205881199031199091198942 (119905) minus ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 1199091198941 (119905) + 119865119894

(32)

Then suspension point load is derived

119865119877119871 (119905) = 119864119903119860119903

120597119906 (0 119905)120597119909 + 119865119903

= 119864119903119860119903radic 2120588119903119860119903119871119903

119873sum119894=1

(2119894 minus 1) 1205872119871119903

1199091198941 (119905) + 119865119903

(33)

The displacement and velocity of plunger are

119906119901 (119905) = 119906119886 (119905) minus 119906 (119871 119905)= 119906119886 (119905)minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198941 (119905) minus 119894=2lowast119895sum119894=1

1199091198941 (119905))V119901 (119905) = 119886 (119905) minus 120597119906 (119871 119905)120597119905

= 119886 (119905)minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198942 (119905) minus 119894=2lowast119895sum119894=1

1199091198942 (119905))

(34)

Then the integrated numerical model is given as follows

119898 = 120596119898

119898 = 1119868119890 [[2120582119896119872119867120576119888120596119899 [120596119899 minus 119898]12057621198881205962

119899 + [120596119899 minus 119898]2 minus 1119894119887119892120578119887119892

[119878119879119891 (119865119877119871 minus 119865119887) 1205781198961119862119871 minus119872119888 sin (120579 minus 120579120591)] minus 121205962

119898

119889119868119890119889120579119898]]119906119886 (119905) = arccos[1198712

119862 + 1198712119870 minus (119871119877 + 119871119875)22119871119862119871119870

] minus arccos(1198712119862 + 1198712

119871 minus 11987121198752119871119862119871119871

)minus arcsin(119871119877119871119871

sin(2120587 minus 1205790 minus 120579119898119894119887119892

+ arcsin ( 119871119868119871119870

)))11990911 (119905) = 11990912 (119905)12 (119905) = minus 12058311986011990312058811990311990912 (119905) minus ( 1205872119871119903

)2 119864119903120588119903 11990911 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903120587+ (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

21 (119905) = 11990922 (119905)22 (119905) = minus 12058311986011990312058811990311990922 (119905) minus ( 31205872119871119903

)2 119864119903120588119903 11990921 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic21198601199031205881199031198711199033120587minus (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

1198941 (119905) = 1199091198942 (119905)1198942 (119905) = minus 1205831198601199031205881199031199091198942 (119905) minus ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 1199091198941 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903(2119894 minus 1) 120587+ (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

sin ((2119894 minus 1) 1205872 )

Mathematical Problems in Engineering 9

119865119877119871 (119905) = 119864119903119860119903radic 2120588119903119860119903119871119903

119873sum119894=1

(2119894 minus 1) 1205872119871119903

1199091198941 (119905) + 119865119903

119906119901 (119905) = 119906119886 (119905) minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198941 (119905) minus 119894=2lowast119895sum119894=1

1199091198941 (119905))

V119901 (119905) = 119886 (119905) minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198942 (119905) minus 119894=2lowast119895sum119894=1

1199091198942 (119905))1198781199051198861199051199061199041 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199042 119886119891119905119890119903 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = ((V119891 sdot 119860119901 + 119902) (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)]sdot V119901

1198781199051198861199051199061199043 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199044 119886119891119905119890119903 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = 0119889119871119891119889119905 = V119891

119889V119891119889119905 = (1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V119860119891119860119901

)minus1119901119904120588 minus 119901120588 + V21198912 [119860

21199011198602119891

minus 119860211990112057621198602

119891

minus 64120583 (1198710 minus 119881119900119892119860119901 + 119871119891)1198632V119891120588 minus 1]minus 119892(1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V)(35)

4 Optimization Model

41 Optimization Goal As mostly developed oil-field movesinto the mid and late stage and the OWD begins todecline gradually energy-saving production-increasing andreducing load variation as much as possible are particularlyimportant Then we take pump fullness epf suspension pointload amplitude FRLA crank torque standard deviation Mcsdand motor input power average 119875119898 as the optimization goalto build a multitarget model Suspension point load andcrank torque can be solved directly by (35) The motor input

power and pump fullness calculation formula are deduced asfollows

119875119898 = 119872119890119889119898 + 1198750

+ [( 1120578H minus 1)119875119867 minus 1198750](119872119890119889119898119875119867

)2 (36)

119890119901119891 = 119871119891 minus int119905119906

0119902119889119905 minus (119871119900 minus 119871119900119892)119906119901119906

(37)

10 Mathematical Problems in Engineering

where P0 is the no-load power of motor kW PH is the motorrated power kW 120578H is the motor rated efficiency kW upuis the plunger displacement when it is arriving at dead topcenter

So the objective function is

Ω(119890119901119891 (X) 119865119877119871119860 (X) 119872119888119904119889 (X) 119875119898 (X))= 1198701119890119901119891 (X) + 1198702119865119877119871119860 (X) + 1198703119872119888119904119889 (X)+ 1198704119875119898 (X)

(38)

where K1 K2 K3 and K4 are the weight coefficients

42 Design Variables The objective function can beexpressed as the function of the swabbing parameters whenthe SRPS type and oil well basic parameters are confirmedIn this paper the swabbing parameters denote stroke Sstroke frequency ns pump diameter Dd pump depth Lpdcrank balance radius rc and rod string combination (the jthrod string diameter and length are dj and Lj respectively119896 = 1 2 119898) Then the design variables are shown asfollows

X = 119878 119899119904 119863119889 119871119901119889 (119889119895 119871119895 119895 = 1 2 sdot sdot sdot 119898) (39)

43 Constraints

(a) Crank Balance Degree Crank balance degree indicates theload fluctuation to a certain degree and it needs to be kept ata high value

095 le 119872119888119896119906119872119888119896119889

le 1 (40)

where Mcku and Mckd are the maximum crank torque whenplunger is at upstroke and downstroke respectively Nsdotm

(b) Ground Device Carrying Capacity The suspensionpoint load crank torque and motor torque at anytime [0 119879] do not go beyond the allowable rangemax(F119877119871) min(F119877119871) max(119872119888) min(119872119888) max(119872119890119889)min(119872119890119889) for the given type

min (119865119877119871) le 119865119877119871 le max (119865119877119871)min (119872119888119896) le 119872119888119896 le max (119872119888119896)min (119872119890119889) le 119872119890119889 le max (119872119890119889)

(41)

(c) Rod String StrengthThemaximumandminimum stress ofany point x along the rod string does not exceed permissiblestress range in one cycle

[120590min] le 120590119903119904 le [120590max] (42)

(d) Swabbing Parameters Each oil well has different limit onthe swabbing parameters in accordance with the device typeand actual operation hence their allowable variation rangesare adjustable

Figure 8 Dynamometer sensor

44 Optimization Algorithm In summary this multivariableoptimization model is established with nonlinear restrictionand nonlinear objective function So as to seek the bestresults the genetic algorithm is applied to solve it

5 Test and Verification

Surface dynamometer card is a closed graph recordingpolished rod loads versus rod displacement over a SRPScycle which is generally collected and taken as an indexto estimate the operation of SRPS In this paper based onthe dynamometer sensor shown in Figure 8 four test wellsare used to validate the improved SRPS model The oil wellparameters are listed in Table 1 and the simulation and fieldtest results are given in Figure 9 According to the plottedcurves the simulation results are basically consistent withthat in measured and the improved SRPS model is accurateenough to be proposed for engineering practices

6 Dynamic Response Comparison

Pump dynamometer card is a closed graph recording plungerloads versus plunger displacement over a SRPS cycle It isdetermined by multifactors such as stroke stroke frequencypump diameter pump depth dynamic liquid level and theother oil well parameters The difference between the currentSRPS model and the improved SRPS model proposed in thispaper is whether to consider the PFFP In view of this theoil operating status is divided into two forms as follows(1) the fluid always keeps pace with the plunger when itis being sucked into the pump (2) the fluid is unable tofollowwith the plungerThen two oil wells are selected well1with sufficient OWD and well2 with insufficient OWDFigure 10 describes the plunger and fluid velocity duringupstroke based on the improved SRPS model It can beconcluded that when the well has sufficient OWD and thefluid possesses good ability of keeping pace with the plungerwhen the well has insufficient OWD the fluid velocity lagsbehind the one of plunger at the beginning whereas it is

Mathematical Problems in Engineering 11

Table 1 Oil well basic parameters

Well 1 2 3 4Motor type YD280S-8 Y250M-6 Y250M-6 YD280S-6Pumping unit type CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HBStroke length (m) 3 3 3 3Stroke frequency (minminus1) 35 3 4 6Pump diameter (mm) 57 44 38 44Pump clearance level 1 2 2 1Pump depth (m) 890 1377 1138 1430Sucker-rod string (mm timesm) 25lowast890 19lowast673+22lowast704 19lowast672+22lowast466 19lowast720+22lowast710middle depth of reservoir (m) 1000 1492 1569 1600Crude oil density (kgm3) 795 857 857 850Water content () 95 97 92 98Dynamic liquid level (m) 880 1347 757 1340Casing pressure (Pa) 02 03 03 02Oil pressure (Pa) 03 03 03 02Fluid dynamic viscosity (Pasdots) 0006 0007 0007 0006Gas oil ratio (m3 m3) 20 19 40 80Plunger length (m) 12 12 12 12Clearance length (m) 05 05 05 05

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(a) Well 1

1 2 30Polished rod displacement (m)

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

simulationmeasured

(b) Well 2

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(c) Well 3

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(d) Well 4Figure 9 Suspension point dynamometer card comparison between the simulated and measured

12 Mathematical Problems in Engineering

FluidPlunger

minus05

00

05

10

15Ve

loci

ty(

ms

)

2 4 60Time (s)

(a) Well 1

minus02

00

02

04

06

Velo

city

(m)

4 8 120Time (s)

FluidPlunger

(b) Well 2

Figure 10 Plunger and fluid velocity

Current SRPS model 970Improved SRPS model 981

minus10

0

10

20

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(a) Well 1

Current SRPS model 724Improved SRPS model 775

minus10

0

10

20

40

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(b) Well 2

Figure 11 Comparison of pump dynamometer card

faster after a certain time period Then the comparisons ofpump dynamometer card are executed by the two modelsmeanwhile their individual pump fullness result is given atthe end of the legend (Figure 11)

Pump dynamometer card is very important method andnormally used to diagnose the pump operations particularlywhose shape is more similar to a rectangle the pump iscloser to the fullness [28] Hence from Figure 11 it canbe known that the pump fullness of current SRPS modelis higher than that of the improved one depending on thequalitative judgment and this conclusion is consistent withthe quantitative calculation result meanwhile this gap forthe well 2 is bigger than well 1 According the above

description we can know that the pump fullness calculationresult is on the high side if the PFFP is not considered andthis phenomenon will becomemore obvious for the well withinsufficient OWD

The load presented by the left upper right and lowerborderline of pump dynamometer card is the results of pumpmoving from phase 1 to 4 in turn Therefore its upperborderline describes the pump load when fluid is pumpedinto the barrel From Figure 11(a) it can be found thatthe upper borderline load simulated by the improved SRPSmodel is significantly larger than the one of current ForFigure 11(b) this difference can be neglected Based on (1)and (2) the pump pressure keeps constant as ps when fluid is

Mathematical Problems in Engineering 13

Before optimizationAfter optimization

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

1 2 30Polished rod displacement (m)

(a) Suspension point dynamometer card

minus10

0

10

20

30

40

Cran

k to

rque

(kNmiddotm

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(b) Crank torque

0

5

10

15

20

Mot

or in

put p

ower

(kW

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(c) Motor input power

Figure 12 Dynamic response comparison before and after optimization

sucked into the pump and then the corresponding pump loadis a fixed value Due to the new model takes into account ofPFFP a pressure drop will be brought which will vary withthe fluid velocity Combined with (1) this pressure drop willlead to the pump load increases Figure 10 shows that thefluid velocity of well 1 is larger than that of well 2 andthis is the reason why the difference of upper borderline loadbetween the two models is obvious This illustrates the erroron the upper borderline load of pump dynamometer carddepending on the velocity at which the fluid flows into thepump

7 Optimization Test

Based on the above models a simulation and optimizationsoftware is developed byMATLABOnewell with insufficient

OWD is tested Its original swabbing parameters are asfollows stroke S is 3 m stroke frequency ns is 3 minminus1pump diameter Dd is 57mm pump depth Lpd is 900m crankbalance radius rc is 12 m and rod string combination djtimesLj is 25 mm times 524 m+22 mm times 376 m And its swabbingparameters after optimizing are as follows stroke S is 3 mstroke frequency ns is 25 minminus1 pump diameterDd is 57mmpump depth Lpd is 990 m crank balance radius rc is 09 mand rod string combination djtimes Lj is 22 mm times 426 m+19 mmtimes 564 m The comparisons before and after optimization areshown in Figure 12 and Table 2

From the above comparison results some conclusions areobtained

(1) After optimization the maximum and minimumof suspension point load are all decreased and the load

14 Mathematical Problems in Engineering

Table 2 Specific parameters comparison before and after optimization

Suspension point load Crank torque Motor input power PumpfullnessMaximum Minimum Amplitude Standard

deviationBalancedegree Maximum Minimum mean

Beforeoptimization 555 kN 190 kN 365 kN 107 kN 854 159 kW 09 kW 62 kW 725

Afteroptimization 441 kN 132 kN 309 kN 94 kN 978 106 kW 10 kW 54 kW 908

darr 205 darr 305 darr 153 darr 121 uarr 145 darr 333 uarr 111 darr 129 uarr 252 amplitude is lowered by 153 It can contribute to enhancethe rod string life and prolong the maintenance period

(2) After optimization the standard deviation of cranktorque is reduced by 121 and its balance degree is raisedby 145 It illustrates that the load torque fluctuation is cutdown which canminimize damage to transmission parts andimprove the motor efficiency

(3) After optimization the pump fullness is improvedby 252 It is conducive to improve pump efficiency andproduction

(4) After optimization it plays a role of peak shavingand valley filling for motor input power and extends theoperational life meanwhile power saving rate is 129

8 Conclusions

(1) In this article an improved SRPS model is presentedconsidering the couple effect of pumping fluid and plungermotion on the dynamic response of SRPS instead of theexisting models that assume the pumping fluid volumeis always equal to plunger travelling volume The Runge-Kutta method is applied to solve the whole system modeltrough transforming the rod string longitudinal vibrationequation into ordinary differential equations And the SRPSmodelrsquos precision has been validated by adopting surfacedynamometer card

(2) Two oil wells are served to compare the differencebetween the current SRPS model and the improved oneThe results indicate that the current SRPS model is relativelylow in calculating pump fullness and this gap will increasewith the reduction of OWD the influence of fluid flowinginto the pump on pump load cannot be ignored when thefluid velocity is high Therefore the PFFP is necessary to beconsidered in order to improve the simulation accuracy

(3) On the basis of the improved SRPS model a multitar-get optimization model is proposed in purpose of improvingproduction decreasing load fluctuation and saving energy Bycomparison the optimal scheme can achieve the decreasingof maximum and minimum suspension point load cranktorque fluctuation and energy consumption as well asimproving the system balance degree and pump fullness Insummary it improves the dynamic behavior of SRPS

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

National Natural Science Foundation of China (Grantno 51174175) China Scholarship Council (Grant no201708130108) Hebei Natural Science Foundation (Grant noE201703101) are acknowledged

References

[1] T A Aliev A H Rzayev G A Guluyev T A Alizada and NE Rzayeva ldquoRobust technology and system for management ofsucker rod pumping units in oil wellsrdquoMechanical Systems andSignal Processing vol 99 pp 47ndash56 2018

[2] S Gibbs ldquoPredicting the Behavior of Sucker-Rod PumpingSystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 1963

[3] D R Doty and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo SPE Journal vol 23 no 1 pp 33ndash41 1983

[4] I N Shardakov and I NWasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[5] S G Gibbs ldquoComputing gearbox torque and motor loading forbeam pumping units with consideration of inertia effectsrdquo SPEJ vol 27 pp 1153ndash1159 1975

[6] J Svinos ldquoExact Kinematic Analysis of Pumping Unitsrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionSan Francisco California 1983

[7] D Schafer and J Jennings ldquoAn Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical Modelsrdquoin Proceedings of the SPE Annual Technical Conference andExhibition pp 27ndash30 Dallas Texas 1987

[8] G Takacs Sucker-Rod PumpingManual Pennwell Books Tulsa2003

[9] G W Wang S S Rahman and G Y Yang ldquoAn improvedmodel for the sucker rod pumping systemrdquo in Proceedings of the11thAustralasian FluidMechanics Conference pp 14ndash18HobartAustralia December 1992

[10] S D L Lekia andR D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystemsrdquo SPE Production amp Facilities vol 10 no 1 pp 26ndash331995

[11] J Lea and P Pattillo ldquoInterpretation of Calculated Forces onSucker Rodsrdquo SPE Production amp Facilities vol 10 no 01 pp 41ndash45 1995

Mathematical Problems in Engineering 15

[12] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[13] L Guo-hua H Shun-li Y Zhi et al ldquoA prediction model fora new deep-rod pumping systemrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 75ndash80 2011

[14] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[15] O Becerra J Gamboa and F Kenyery ldquoModelling a DoublePiston Pumprdquo in Proceedings of the SPE International ThermalOperations and Heavy Oil Symposium and International Hor-izontal Well Technology Conference Calgary Alberta Canada2002

[16] A L Podio J Gomez A J Mansure et al ldquoLaboratoryinstrumented sucker-rod pumprdquo J Pet Technol vol 53 no 05pp 104ndash113 2003

[17] Z H Gu H Q Peng and H Y Geng ldquoAnalysis and mea-surement of gas effect on pumping efficiencyrdquoChina PetroleumMachinery vol 34 no 02 pp 64ndash69 2006

[18] M Xing ldquoResponse analysis of longitudinal vibration of suckerrod string considering rod bucklingrdquo Advances in EngineeringSoftware vol 99 pp 49ndash58 2016

[19] S Miska A Sharaki and J M Rajtar ldquoA simple model forcomputer-aided optimization and design of sucker-rod pump-ing systemsrdquo Journal of Petroleum Science and Engineering vol17 no 3-4 pp 303ndash312 1997

[20] L S Firu T Chelu and C Militaru-Petre ldquoAmodern approachto the optimum design of sucker-rod pumping systemrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash9 DenverColorado 2003

[21] X F Liu and Y G Qi ldquoA modern approach to the selectionof sucker rod pumping systems in CBM wellsrdquo Journal ofPetroleum Science and Engineering vol 76 no 3-4 pp 100ndash1082011

[22] S M Dong N N Feng and Z J Ma ldquoSimulating maximumof system efficiency of rod pumping wellsrdquo Journal of SystemSimulation vol 20 no 13 pp 3533ndash3537 2008

[23] M Xing and S Dong ldquoA New Simulation Model for a Beam-Pumping System Applied in Energy Saving and Resource-Consumption Reductionrdquo SPE Production amp Operations vol30 no 02 pp 130ndash140 2015

[24] Y XWu Y Li andH LiuElectric Motor andDriving ChemicalIndustry Press Beijing 2008

[25] S M Dong Computer Simulation of Dynamic Parameters ofRod Pumping System Optimization Petroleum Industry PressBeijing Chinese 2003

[26] F Yavuz J F Lea J C Cox et al ldquoWave equation simulationof fluid pound and gas interferencerdquo in Proceedings of the SPEProduction Operations Symposium pp 16ndash19 Oklahoma CityOklahoma April 2005

[27] D Y Wang and H Z Liu ldquoDynamic modeling and analysis ofsucker rod pumping system in a directional well Mechanismand Machine Sciencerdquo in Proceedings of the ASIAN MMS 2016amp CCMMS pp 1115ndash1127 2017

[28] F M A Barreto M Tygel A F Rocha and C K MorookaldquoAutomatic downhole card generation and classificationrdquo inProceedings of the SPE Annual Technical Conference pp 6ndash9Denver Colorado 1996

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Page 2: An Improved Sucker Rod Pumping System Model and Swabbing ...downloads.hindawi.com/journals/mpe/2018/4746210.pdf · An Improved Sucker Rod Pumping System Model and Swabbing Parameters

2 Mathematical Problems in Engineering

Dynamic liquid level

Pump inlet

Pump outlet

Plunger

Pump depth

Rod string

Suspension point

ldquoHorse headrdquoBeam

Motor

Belt

Reduction gear box

Crank

Link rod

Figure 1 Sucker rod pumping system

[2] In an ideal condition during the fully loaded upstrokemovement the pump load was set equivalent to the fluidload pump load was set to zero for the unloaded downstrokemovement [7 8] As the pump operation phases are highlyrelated to pump pressure it was revised with an explicitformation deduced by the interaction between pump outletpressure and the pressure in the pump barrel [9ndash14] asfollows

119865119875119871 (119905) = 119860119901 (119901119889 minus 119901) minus 119860119903119901119889 (1)

where FPL is the pump load NAp andAr are the cross sectionarea of plunger and rod string respectively m2 pd is the pumpoutlet pressure pa p is the pump pressure pa

For the pd is always considered as a constant pressure thekey of this research is to establish an accurate pump pressuremodel which consists of four phases as shown in Figure 2With considering of the gas this downhole boundary modelis improved as shown in (2)When the pump is at phases 1 and3 the pressure variation obeys the rules of gas state equationAs for in phases 2 and 4 the pump pressure is taken as the psand pd separately

119901 = ( 119871119900119892119871119900119892 + 119906119901 minus int119905119904

0(119902119860119901))

119876 119901119889 phase 1119901 = 119901119904 phase 2119901 = ( 119871119892119871119892 minus 119871 119904 + 119906119901 minus int119905119905

119905119906(119902119860119901))

119876

119901119904 phase 3119901 = 119901119889 phase 4

(2)

phase 1 phase 2 phase 3 phase 4

gas e

xpan

sion

pum

ping

flui

d

gas c

ompr

essio

n

disc

harg

e flui

d

Figure 2 Four phases of pump operation

where 119871119900119892 and 119871119892 are the gas column length when plungeris arriving at bottom dead center and top dead centerrespectively m up is the plunger displacement m Ls is thepump stroke displacement m Lp is the plunger length m 120583is the fluid dynamic viscosity pasdots 120575 is the clearance betweenplunger and pump barrel m Dd is the pump diameter mts and tt are the open time of standing valve and travellingvalve respectively s tu is the upstroke time q is the liquidinstantaneous leakage volume

In this formula the principle that the gasoil ratio ofclearance volume (the space volume when plunger arrives atbottom dead center) equals the gasoil ratio of pump inletis applied However at this time the pump pressure shouldbe the same as the pump outlet pressure Thus it is revisedwith the pump outlet pressure and this improved model hasbeen widely used until now [15ndash18] However the currentdownhole boundary model still exists some shortcomingsneed to be improved Due to ignoring the progress of PFFPit is established with the hypothesis of regarding the pumpis filled with the fluid whose gasliquid ratio always equalsthat at pump intake as well as the one in the clearancevolume This assumption is not applicable for the oil wellwith insufficient oil well deliverability (OWD) which will

Mathematical Problems in Engineering 3

Motorrotation

Surfacetransmission

Rod string longitudinal

vibration

Plunger motionPump pressure variationFluid motion

Figure 3 SRPS coupled model sketch

result in the pumping fluid being unable to keep pace withthe plunger and cause an incomplete fullness Meanwhile apump load calculation error will be produced with keepingthe pump pressure as constant when fluid flows into thepump

The SRPS model lays the foundation for optimizingthe swabbing parameters to improve the system operationstatus except for predicting and evaluating its dynamicresponse Miska et al 1997 [19] propose a computer-aidedoptimization method relying on a simple linear algebraicsystem model so as to minimize the energy consumptionFiru et al 2003 [20] present an improve optimization cri-terion including eight operational parameters to achieve themaximum system efficiency Liu and Qi 2011 [21] apply fluidflow characteristics in coalbed methane reservoirs to estimatethe production capacity and build the system efficiencyoptimization model combined with SRPS performance Theabove optimization models are built with a simplified pumpload then an improved system efficiency optimization modelis built jointing with formula (2) [22 23] However thecurrent optimization models ignore the effect of swabbingparameters on pump fullness in that the SRPS modelrsquosrestriction Besides that single target optimization cannotmake an accurate and comprehensive presentation for SRPSwhose pumping progress is complex multicomponent andinteractive

In this paper firstly an improved SRPSmodel is presentedwith the new downhole boundary model considering themovement of fluid flowing into pump with gas instantaneousdissolution and evolution Secondly for the new downholeboundary model a nonlinear fluid-solid coupled model willincrease the complexity of this improved SRPS model Aunified numerical algorithm is applied on the whole modelto decrease the calculation time instead of traditional mixediteration algorithm Thirdly a multiobjective optimizationmodel is proposed based on the improved SRPS modelFourthly the surface dynamometer card is collected to verifythe improvedmodelrsquos accuracy Fifthly the dynamic responsecomparison on the current SRPS model and improved SRPSmodel is executed Finally the optimization program isapplied on a test well and the results are given before andafter optimization

2 Integrated Simulation Model

The improved SRPS model is a nonlinear fluid-solid coupledmodel consisted of drive transmission and loadThemotor isused to generate power and its rotationmotion is transformedinto linear motion of suspension point This linear motionis passed on to the plunger through rod string longitudinalvibration meanwhile plunger motion has an influence onpumping fluid by regulating pump pressure On the contraryfluid motion affects pump pressure which is also the mainfactor in determining pump load This pump load andother system loads are transferred to motor output shaft bysurface and downhole transmission causing impact on themotor motion The above coupled process is depicted inFigure 3 In this paper the SRPSmodel is subdivided into rodstring longitudinal vibration model corresponding surfaceboundary and downhole boundary model so as to have aclear description

21 Sucker Rod Longitudinal Vibration Model While theSRPS is operating the motion of arbitrary micro elementof the slender rod string produced intense longitudinalvibration can be composed of the following two parts (1)following the movement of suspension point and (2) relativemovement to the suspension point Referring to Figure 4 thecoordinate system is built based on the top dead center ofhorse head as the origin Then the stress balance terms of rodstring element and top and bottom boundary conditions areused to establish the sucker rod longitudinal vibration modelwith the assumption of ignoring the deformation of tube andthe vibration of liquid column in a vertical well

12059721199061205971199052 minus 119864119903120588119903 12059721199061205971199092+ 120583120588119903119860119903

120597119906120597119905= minus1198892119906119886 (119905)1198891199052 minus 120583120588119903119860119903

119889119906119886 (119905)119889119905 + 119892119906 (119909 119905)|119909=0 = 119906119886 (119905)119864119903119860119903

120597119906 (119909 119905)12059711990910038161003816100381610038161003816100381610038161003816119909=119871119903

= 119865119875119871 (119905)

(3)

4 Mathematical Problems in Engineering

Top dead center

x

dx

Lru(xt)

FPL(t)

ua(t)

rAr(d2ua

dt2+

2u

t2)dx

(dua

dt+

u

t)dx

ErAru

x+

x(ErAr

u

x)dx

rArgdx

ErAru

x

Figure 4 Mechanical model of rod string longitudinal vibration

Med

mMef

Figure 5 Surface apparatus equivalent motion model

where u is the displacement of the rod string at arbitrarydepth and time m Er and 120588r are the elasticity modulus anddensity of rod string respectively pa and kgm3 ua is thesuspension point displacement m

22 Surface Boundary Model The surface boundary condi-tion refers to the motion of suspension point as in (3) Withthe assumption that the surface transmission mechanismconsists of belt-reducing gear and four-bar linkage as a singledegree of freedom system it can be represented as a functionof motor angle Later the surface apparatus motion model isset up using Lagrange equation taking motor output shaft asequivalent component its motion as generalized coordinatesand set 12 orsquoclock as a reference direction (Figure 5)

119868119890119898 + 12 119898

2 119889119868119890119889120579119898 = 119872119890119889 minus119872119890119891

1205791198981003816100381610038161003816119905=0 = 0119898

10038161003816100381610038161003816120579=0= 1205960

(4)

LC

LP

LR

LL

LK

LI

LH

LAFRL

LRmdash crankLPmdash link rodLCmdash beam backLAmdash beam front

Figure 6 Four-bar linkage

With the equivalent rotating inertia

119868119890 = sum119868119895 (120596119895119898

)2 +sum119898119895 ( V119895119898

)2

(5)

The semiempirical formulation of motor driving torqueMed deduced from the speedtorque characteristic is adoptedfromWu et al [24]

119872119890119889 = 2120582119896119872119867120576119888120596119899 [120596119899 minus 119898]12057621198881205962

119899 + [120596119899 minus 119898]2 (6)

where 120582k is the motor overload coefficient 120576c is the motorcritical-slip ratio 120596n is the motor synchronous angularvelocity rads

119872119867 = 9550119875119867119899119867

120576119888 = 120576119903 (120582119896 + radic1205822119896minus 1)

120596119899 = 2120587119891119911(7)

where z is the motor pole pairs f is the power frequency HzPH is the motor rated power kW nH is the motor rated speedminminus1 120576r is the motor rated-slip ratio

The equivalent resistance torque is derived in accordancewith the force balance of four-bar linkage (Figure 6) writtenas

119872119890119891 = 1119894119887119892120576119887 [119878119879119891 (119865119877119871 minus 119865119887) 1205761198961119901 minus119872119888 sin (120579 minus 120579120591)] (8)

with the angle of crank

120579 = 120579119898119894119887119892(9)

Mathematical Problems in Engineering 5

where ibg is the transmission ratio of the belt-reducinggear system 120576p is the transmission efficiency from crank tosuspension point STf is torque factorm FRL is the suspensionpoint load N Fb is the structural unbalance weight NMc isthe maximum balancing torque of the crank Nsdotm

The displacement of suspension point can be expressedas motor angle based on the geometry relation shown inFigure 6 [6] The surface boundary motion model is

119906119886 (119905) = arccos [1198712119862 + 1198712

119870 minus (119871119877 + 119871119875)22119871119862119871119870

]minus arccos(1198712

119862 + 1198712119871 minus 1198712

1198752119871119862119871119871

) minus arcsin(119871119877119871119871

sdot sin(2120587 minus 1205790 minus 120579119898119894119887119892

+ arcsin( 119871119868119871119870

)))(10)

23 Downhole Boundary Model This new model is animproved method from the current boundary models con-sidering the interaction among pump pressure fluid flow intopump gas dissolution and evolution pressure drop due tofluid gravitational potential energy inertia head loss frictionhead loss and local head loss The new boundary modelconsists of pressure variation model and fluid flow into pumpmodel Some assumptions are made for building this model

(1) Bottom dead point of plunger as the origin position(2) The pump inlet pressure ps and outlet pressure pd are

kept constant for one cycle of SRPS operation

Pressure Variation Model The gas state equations are built asfollows ( (11) and (12)) when the displacement of plunger is upor up+dup

119901 [(119906119901 + 119871119900119892 minus 119871119891 minus 119871119896)119860119901] = 119885119873119876119879 (11)

(119901 + 119889119901)sdot [(119906119901 + 119889119906119901 + 119871119900119892 minus 119871119891 minus 119871119896 minus 119889119871119891 minus 119889119871119896)119860119901]= (119885 + 120597119885120597119901 119889119901) (119873 + 119889119873)119876119879

(12)

where N is the gas molar number mol Q is the natural gasconstant J(molsdotK) T is the temperature in pump barrel ∘CZ is the natural gas compressibility factor Lk is the equivalentlength of pump leakage m Lf is the liquid level in the pumpm

Dividing (11) by (12) and ignoring the second order smallquantities then the variation of pump pressure with plungerdisplacement is obtained

119889119901119889119906119901

= (1119873) (119889119873119889119906119901) + (1 (119906119901 + 119871119900119892 minus 119871119891 minus 119871119896)) (119889 (119871119891 + 119871119896) 119889119906119901 minus 1)119901minus1 minus (1119885) (119889119885119889119901) (13)

The above equation is converted into a time varyingfunction with the purpose of facilitating it with the rod stringlongitudinal vibration equation

119889119901119889119905 = (1119873) (119889119873119889119905) (1V119901) + (1 (119906119901 + 119871119900119892 minus 119871119891 minus 119871119896)) ((119889 (119871119891 + 119871119896) 119889119905) (1V119901) minus 1)119901minus1 minus (1119885) (119889119885119889119901) V119901(14)

where vp is the plunger velocity msFrom (14) dN is composed of the following sections(1)Themolar number of gas is released from or dissolved

into oil as pressure varies

119889119873119903 = minus 119885119879119901119904119905119885119904119905119879119904119905119901 (120572 sdot 119889119901)sdot 103119860119901 (119871119891 + 119871119900 minus 119871119900119892) (1 minus 120576119908)119861119900119881119898119900119897

(15)

where pst is the standard pressure pa Zst is the standard natu-ral gas compressibility factorTst is the standard temperature∘C 120572 is the solubility coefficient of natural gas m3 (m3sdotpa)

120576w is the water content Vmol is the molar volume m3 Bo isthe crude oil volume factor

(2)Themolar number of gas with fluid being sucked intothe pump corresponding to the pump pressure is

119889119873119904 = 103119860119901119877119869119889119871119891119881119898119900119897

(16)

where RJ is the transient gas liquid ratio in the pump m3m3(3)Themolar number of gaswith leaked fluid at the pump

pressure is

119889119873V = 103119860119901119877119869119889119871V119881119898119900119897

(17)

6 Mathematical Problems in Engineering

f-f

L

Lf

La

a-a

Figure 7 Process of fluid flows into the pump

where

119877119869 = (1 minus 120576119908) (119877119901 minus 119877119904) 119901119904119905119885119879119879119904119905119885119904119905119901119877119904 = 120572 (119901 minus 119901119904119905)

(18)

where Rp is the production gas oil ratio m3m3 Rs is the gasoil ratio at pump intake m3m3

Fluid Flowing into Pump Model Figure 7 shows the pumpoperation when the gas-liquid flow is drawn into the pumpSection a-a indicates the cross section of standing valve holeSupposing the section f -f is the liquid level at arbitrarytime neglect the process of gas bubbling from fluid byconsidering only the pump gas to occupy the upper spaceof f -f The one-dimensional unsteady flow equation based onBernoulli equation is established to describe the flow state ofpumping liquid In the Bernoulli equation the inertia headloss friction head loss and local head loss are all considered

119901119904120588g + V21198862119892 = 119901120588g + V2

1198912g + (119871119900 minus 119871119900119892 + 119871119891 + 119871V) + ℎ119886

+ ℎV + ℎ119891

(19)

where

ℎ119886 = 1119892 (119871119900 minus 119871119900119892 + 119871119891 + 119871V119860119891119860119901

) 119889V119891119889119905ℎ119891 = 64120583V119891 (119871119900 minus 119871119900119892 + 119871119891)21198632

119889120588119892

ℎV = 11198921205762 (119860119901119860119891

)2 V21198862

V119886 = 119860119901119860119891

V119891

(20)

where 120576 is the flow coefficient of standing valveThen (19) is converted into differential forms

119891 = V119891

V119891 = (1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V119860119891119860119901

)minus1119901119904120588 minus 119901120588+ V2

1198912 [11986021199011198602119891

minus 119860211990112057621198602

119891

minus 64120583 (1198710 minus 119881119900119892119860119901 + 119871119891)1198632119889V119891120588

minus 1] minus 119892(1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V)

(21)

Normally the fluid in the barrel nomatter in which formwill all be discharged except the one in dead spaceThereforethe fluid flowing out of the model with opening travelingvalve is not proposed The pump outlet pressure generatedby several kilometers liquid column is large compared to thehydraulic loss causing the ignorance of hydraulic loss whenfluid flows out Based on the statement mentioned above thenew downhole boundary model is expressed as follows andalso is divided into four phases corresponding to the phasesshown in Figure 2

1198781199051198861199051199061199041 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199042 af ter 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = ((V119891 sdot 119860119901 + 119902) (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)]sdot V119901

Mathematical Problems in Engineering 7

1198781199051198861199051199061199043 before 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199044 af ter 119900119901119890119899119892119894119899 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = 0

(22)

where

119881119898119900119897 = 224119885119879119901119904119905119879119904119905119901 (23)

3 Integrated Numerical Algorithm

For the current SRPS model the surface transmission modelis solved by numerical integration method [25] While thefinite difference method is used to describe the rod stringlongitudinal vibration [2 26 27] Since there is no fixedsolution between surface and downholemodel it needs cycliciteration to handle thewholemodel which is time consuming[25] Considering the interaction within plunger motionfluid flow into pump and pump pressure the improved SRPSmodel has a higher nonlinear degree and thus increasesthe difficulty of solving it In order to reduce the solvingtime the wave equation of rod string longitudinal vibrationis converted into ordinary differential equations by modalsuperposition method Then the surface and downholeboundary model are now in the form of ordinary differentialequations at which the whole model can be solved by Runge-Kutta method directly Besides that solving the rod stringlongitudinal vibration equation is also solving the forcevibration response of rod string where (3) is converted intothe following form

120588119903119860119903

12059721199061205971199052 minus 120588119903119864119903

12059721199061205971199092+ 120583120597119906120597119905 = 120588119903119860119903119891 (119909 119905) (24)

where

119891 (119909 119905) = minus1198892119906119886 (119905)1198891199052 minus 120583120588119903119860119903

119889119906119886 (119905)119889119905 + 119892 (25)

And its solution can be expressed in the function of timeand space by separating the variables

119906 (119909 119905) = infinsum119894=1

Φ119894 (119909) 119902119894 (119905) (26)

Therefore (24) can be expressed asinfinsum119894=1

119860119903120588119903Φ119894 (119909) 119894 (119905)minus infinsum

119894=1

119864119903119860119903

1198892Φ119894 (119909)1198891199092119902119894 (119905) + infinsum

119894=1

120583Φ119894 (119909) 119894 (119905)= 119860119903120588119903119891 (119909 119905)

(27)

Multiply the above equation by Φi(x) and it is integratedalong the rod length Then the following is derived

119860119903120588119903119894 (119905) int119871119903

0Φ119894

2 (119909) 119889119909minus 119864119903119860119903119902119894 (119905) int119871119903

0Φ119894 (119909)Φ119894

10158401015840 (119909) 119889119909+ 120583119894 (119905) int119871119903

0Φ119894

2 (119909) 119889119909= 119860119903120588119903 int119871119903

0Φ119894 (119909) 119891 (119909 119905) 119889119909

(28)

Its mode shapes and natural frequencies are

Φ119894 (119909) = radic 2120588119903119860119903

sin((2119894 minus 1) 1205872119871119903

119909)119901119899119894 = (2119894 minus 1) 120587radic1198641199031205881199032119871119903

(29)

Then (28) can be simplified as follows using mode shapeorthogonality

119894 (119905) + 120583119860119903120588119903 119894 (119905) + ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 119902119894 (119905) = 119865119894 (30)

where

119865119894 = (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903(2119894 minus 1) 120587+ 119865119875119871 (119905) radic 2119860119903120588119903119871119903

sin ((2119894 minus 1) 1205872 )(31)

8 Mathematical Problems in Engineering

Let ith-order forced vibration displacement response andvelocity response be xi1 and xi2 respectivelyThen (30) can beexpressed as the following form

1198941 (119905) = 1199091198942 (119905)1198942 (119905) = minus 1205831198601199031205881199031199091198942 (119905) minus ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 1199091198941 (119905) + 119865119894

(32)

Then suspension point load is derived

119865119877119871 (119905) = 119864119903119860119903

120597119906 (0 119905)120597119909 + 119865119903

= 119864119903119860119903radic 2120588119903119860119903119871119903

119873sum119894=1

(2119894 minus 1) 1205872119871119903

1199091198941 (119905) + 119865119903

(33)

The displacement and velocity of plunger are

119906119901 (119905) = 119906119886 (119905) minus 119906 (119871 119905)= 119906119886 (119905)minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198941 (119905) minus 119894=2lowast119895sum119894=1

1199091198941 (119905))V119901 (119905) = 119886 (119905) minus 120597119906 (119871 119905)120597119905

= 119886 (119905)minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198942 (119905) minus 119894=2lowast119895sum119894=1

1199091198942 (119905))

(34)

Then the integrated numerical model is given as follows

119898 = 120596119898

119898 = 1119868119890 [[2120582119896119872119867120576119888120596119899 [120596119899 minus 119898]12057621198881205962

119899 + [120596119899 minus 119898]2 minus 1119894119887119892120578119887119892

[119878119879119891 (119865119877119871 minus 119865119887) 1205781198961119862119871 minus119872119888 sin (120579 minus 120579120591)] minus 121205962

119898

119889119868119890119889120579119898]]119906119886 (119905) = arccos[1198712

119862 + 1198712119870 minus (119871119877 + 119871119875)22119871119862119871119870

] minus arccos(1198712119862 + 1198712

119871 minus 11987121198752119871119862119871119871

)minus arcsin(119871119877119871119871

sin(2120587 minus 1205790 minus 120579119898119894119887119892

+ arcsin ( 119871119868119871119870

)))11990911 (119905) = 11990912 (119905)12 (119905) = minus 12058311986011990312058811990311990912 (119905) minus ( 1205872119871119903

)2 119864119903120588119903 11990911 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903120587+ (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

21 (119905) = 11990922 (119905)22 (119905) = minus 12058311986011990312058811990311990922 (119905) minus ( 31205872119871119903

)2 119864119903120588119903 11990921 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic21198601199031205881199031198711199033120587minus (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

1198941 (119905) = 1199091198942 (119905)1198942 (119905) = minus 1205831198601199031205881199031199091198942 (119905) minus ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 1199091198941 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903(2119894 minus 1) 120587+ (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

sin ((2119894 minus 1) 1205872 )

Mathematical Problems in Engineering 9

119865119877119871 (119905) = 119864119903119860119903radic 2120588119903119860119903119871119903

119873sum119894=1

(2119894 minus 1) 1205872119871119903

1199091198941 (119905) + 119865119903

119906119901 (119905) = 119906119886 (119905) minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198941 (119905) minus 119894=2lowast119895sum119894=1

1199091198941 (119905))

V119901 (119905) = 119886 (119905) minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198942 (119905) minus 119894=2lowast119895sum119894=1

1199091198942 (119905))1198781199051198861199051199061199041 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199042 119886119891119905119890119903 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = ((V119891 sdot 119860119901 + 119902) (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)]sdot V119901

1198781199051198861199051199061199043 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199044 119886119891119905119890119903 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = 0119889119871119891119889119905 = V119891

119889V119891119889119905 = (1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V119860119891119860119901

)minus1119901119904120588 minus 119901120588 + V21198912 [119860

21199011198602119891

minus 119860211990112057621198602

119891

minus 64120583 (1198710 minus 119881119900119892119860119901 + 119871119891)1198632V119891120588 minus 1]minus 119892(1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V)(35)

4 Optimization Model

41 Optimization Goal As mostly developed oil-field movesinto the mid and late stage and the OWD begins todecline gradually energy-saving production-increasing andreducing load variation as much as possible are particularlyimportant Then we take pump fullness epf suspension pointload amplitude FRLA crank torque standard deviation Mcsdand motor input power average 119875119898 as the optimization goalto build a multitarget model Suspension point load andcrank torque can be solved directly by (35) The motor input

power and pump fullness calculation formula are deduced asfollows

119875119898 = 119872119890119889119898 + 1198750

+ [( 1120578H minus 1)119875119867 minus 1198750](119872119890119889119898119875119867

)2 (36)

119890119901119891 = 119871119891 minus int119905119906

0119902119889119905 minus (119871119900 minus 119871119900119892)119906119901119906

(37)

10 Mathematical Problems in Engineering

where P0 is the no-load power of motor kW PH is the motorrated power kW 120578H is the motor rated efficiency kW upuis the plunger displacement when it is arriving at dead topcenter

So the objective function is

Ω(119890119901119891 (X) 119865119877119871119860 (X) 119872119888119904119889 (X) 119875119898 (X))= 1198701119890119901119891 (X) + 1198702119865119877119871119860 (X) + 1198703119872119888119904119889 (X)+ 1198704119875119898 (X)

(38)

where K1 K2 K3 and K4 are the weight coefficients

42 Design Variables The objective function can beexpressed as the function of the swabbing parameters whenthe SRPS type and oil well basic parameters are confirmedIn this paper the swabbing parameters denote stroke Sstroke frequency ns pump diameter Dd pump depth Lpdcrank balance radius rc and rod string combination (the jthrod string diameter and length are dj and Lj respectively119896 = 1 2 119898) Then the design variables are shown asfollows

X = 119878 119899119904 119863119889 119871119901119889 (119889119895 119871119895 119895 = 1 2 sdot sdot sdot 119898) (39)

43 Constraints

(a) Crank Balance Degree Crank balance degree indicates theload fluctuation to a certain degree and it needs to be kept ata high value

095 le 119872119888119896119906119872119888119896119889

le 1 (40)

where Mcku and Mckd are the maximum crank torque whenplunger is at upstroke and downstroke respectively Nsdotm

(b) Ground Device Carrying Capacity The suspensionpoint load crank torque and motor torque at anytime [0 119879] do not go beyond the allowable rangemax(F119877119871) min(F119877119871) max(119872119888) min(119872119888) max(119872119890119889)min(119872119890119889) for the given type

min (119865119877119871) le 119865119877119871 le max (119865119877119871)min (119872119888119896) le 119872119888119896 le max (119872119888119896)min (119872119890119889) le 119872119890119889 le max (119872119890119889)

(41)

(c) Rod String StrengthThemaximumandminimum stress ofany point x along the rod string does not exceed permissiblestress range in one cycle

[120590min] le 120590119903119904 le [120590max] (42)

(d) Swabbing Parameters Each oil well has different limit onthe swabbing parameters in accordance with the device typeand actual operation hence their allowable variation rangesare adjustable

Figure 8 Dynamometer sensor

44 Optimization Algorithm In summary this multivariableoptimization model is established with nonlinear restrictionand nonlinear objective function So as to seek the bestresults the genetic algorithm is applied to solve it

5 Test and Verification

Surface dynamometer card is a closed graph recordingpolished rod loads versus rod displacement over a SRPScycle which is generally collected and taken as an indexto estimate the operation of SRPS In this paper based onthe dynamometer sensor shown in Figure 8 four test wellsare used to validate the improved SRPS model The oil wellparameters are listed in Table 1 and the simulation and fieldtest results are given in Figure 9 According to the plottedcurves the simulation results are basically consistent withthat in measured and the improved SRPS model is accurateenough to be proposed for engineering practices

6 Dynamic Response Comparison

Pump dynamometer card is a closed graph recording plungerloads versus plunger displacement over a SRPS cycle It isdetermined by multifactors such as stroke stroke frequencypump diameter pump depth dynamic liquid level and theother oil well parameters The difference between the currentSRPS model and the improved SRPS model proposed in thispaper is whether to consider the PFFP In view of this theoil operating status is divided into two forms as follows(1) the fluid always keeps pace with the plunger when itis being sucked into the pump (2) the fluid is unable tofollowwith the plungerThen two oil wells are selected well1with sufficient OWD and well2 with insufficient OWDFigure 10 describes the plunger and fluid velocity duringupstroke based on the improved SRPS model It can beconcluded that when the well has sufficient OWD and thefluid possesses good ability of keeping pace with the plungerwhen the well has insufficient OWD the fluid velocity lagsbehind the one of plunger at the beginning whereas it is

Mathematical Problems in Engineering 11

Table 1 Oil well basic parameters

Well 1 2 3 4Motor type YD280S-8 Y250M-6 Y250M-6 YD280S-6Pumping unit type CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HBStroke length (m) 3 3 3 3Stroke frequency (minminus1) 35 3 4 6Pump diameter (mm) 57 44 38 44Pump clearance level 1 2 2 1Pump depth (m) 890 1377 1138 1430Sucker-rod string (mm timesm) 25lowast890 19lowast673+22lowast704 19lowast672+22lowast466 19lowast720+22lowast710middle depth of reservoir (m) 1000 1492 1569 1600Crude oil density (kgm3) 795 857 857 850Water content () 95 97 92 98Dynamic liquid level (m) 880 1347 757 1340Casing pressure (Pa) 02 03 03 02Oil pressure (Pa) 03 03 03 02Fluid dynamic viscosity (Pasdots) 0006 0007 0007 0006Gas oil ratio (m3 m3) 20 19 40 80Plunger length (m) 12 12 12 12Clearance length (m) 05 05 05 05

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(a) Well 1

1 2 30Polished rod displacement (m)

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

simulationmeasured

(b) Well 2

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(c) Well 3

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(d) Well 4Figure 9 Suspension point dynamometer card comparison between the simulated and measured

12 Mathematical Problems in Engineering

FluidPlunger

minus05

00

05

10

15Ve

loci

ty(

ms

)

2 4 60Time (s)

(a) Well 1

minus02

00

02

04

06

Velo

city

(m)

4 8 120Time (s)

FluidPlunger

(b) Well 2

Figure 10 Plunger and fluid velocity

Current SRPS model 970Improved SRPS model 981

minus10

0

10

20

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(a) Well 1

Current SRPS model 724Improved SRPS model 775

minus10

0

10

20

40

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(b) Well 2

Figure 11 Comparison of pump dynamometer card

faster after a certain time period Then the comparisons ofpump dynamometer card are executed by the two modelsmeanwhile their individual pump fullness result is given atthe end of the legend (Figure 11)

Pump dynamometer card is very important method andnormally used to diagnose the pump operations particularlywhose shape is more similar to a rectangle the pump iscloser to the fullness [28] Hence from Figure 11 it canbe known that the pump fullness of current SRPS modelis higher than that of the improved one depending on thequalitative judgment and this conclusion is consistent withthe quantitative calculation result meanwhile this gap forthe well 2 is bigger than well 1 According the above

description we can know that the pump fullness calculationresult is on the high side if the PFFP is not considered andthis phenomenon will becomemore obvious for the well withinsufficient OWD

The load presented by the left upper right and lowerborderline of pump dynamometer card is the results of pumpmoving from phase 1 to 4 in turn Therefore its upperborderline describes the pump load when fluid is pumpedinto the barrel From Figure 11(a) it can be found thatthe upper borderline load simulated by the improved SRPSmodel is significantly larger than the one of current ForFigure 11(b) this difference can be neglected Based on (1)and (2) the pump pressure keeps constant as ps when fluid is

Mathematical Problems in Engineering 13

Before optimizationAfter optimization

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

1 2 30Polished rod displacement (m)

(a) Suspension point dynamometer card

minus10

0

10

20

30

40

Cran

k to

rque

(kNmiddotm

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(b) Crank torque

0

5

10

15

20

Mot

or in

put p

ower

(kW

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(c) Motor input power

Figure 12 Dynamic response comparison before and after optimization

sucked into the pump and then the corresponding pump loadis a fixed value Due to the new model takes into account ofPFFP a pressure drop will be brought which will vary withthe fluid velocity Combined with (1) this pressure drop willlead to the pump load increases Figure 10 shows that thefluid velocity of well 1 is larger than that of well 2 andthis is the reason why the difference of upper borderline loadbetween the two models is obvious This illustrates the erroron the upper borderline load of pump dynamometer carddepending on the velocity at which the fluid flows into thepump

7 Optimization Test

Based on the above models a simulation and optimizationsoftware is developed byMATLABOnewell with insufficient

OWD is tested Its original swabbing parameters are asfollows stroke S is 3 m stroke frequency ns is 3 minminus1pump diameter Dd is 57mm pump depth Lpd is 900m crankbalance radius rc is 12 m and rod string combination djtimesLj is 25 mm times 524 m+22 mm times 376 m And its swabbingparameters after optimizing are as follows stroke S is 3 mstroke frequency ns is 25 minminus1 pump diameterDd is 57mmpump depth Lpd is 990 m crank balance radius rc is 09 mand rod string combination djtimes Lj is 22 mm times 426 m+19 mmtimes 564 m The comparisons before and after optimization areshown in Figure 12 and Table 2

From the above comparison results some conclusions areobtained

(1) After optimization the maximum and minimumof suspension point load are all decreased and the load

14 Mathematical Problems in Engineering

Table 2 Specific parameters comparison before and after optimization

Suspension point load Crank torque Motor input power PumpfullnessMaximum Minimum Amplitude Standard

deviationBalancedegree Maximum Minimum mean

Beforeoptimization 555 kN 190 kN 365 kN 107 kN 854 159 kW 09 kW 62 kW 725

Afteroptimization 441 kN 132 kN 309 kN 94 kN 978 106 kW 10 kW 54 kW 908

darr 205 darr 305 darr 153 darr 121 uarr 145 darr 333 uarr 111 darr 129 uarr 252 amplitude is lowered by 153 It can contribute to enhancethe rod string life and prolong the maintenance period

(2) After optimization the standard deviation of cranktorque is reduced by 121 and its balance degree is raisedby 145 It illustrates that the load torque fluctuation is cutdown which canminimize damage to transmission parts andimprove the motor efficiency

(3) After optimization the pump fullness is improvedby 252 It is conducive to improve pump efficiency andproduction

(4) After optimization it plays a role of peak shavingand valley filling for motor input power and extends theoperational life meanwhile power saving rate is 129

8 Conclusions

(1) In this article an improved SRPS model is presentedconsidering the couple effect of pumping fluid and plungermotion on the dynamic response of SRPS instead of theexisting models that assume the pumping fluid volumeis always equal to plunger travelling volume The Runge-Kutta method is applied to solve the whole system modeltrough transforming the rod string longitudinal vibrationequation into ordinary differential equations And the SRPSmodelrsquos precision has been validated by adopting surfacedynamometer card

(2) Two oil wells are served to compare the differencebetween the current SRPS model and the improved oneThe results indicate that the current SRPS model is relativelylow in calculating pump fullness and this gap will increasewith the reduction of OWD the influence of fluid flowinginto the pump on pump load cannot be ignored when thefluid velocity is high Therefore the PFFP is necessary to beconsidered in order to improve the simulation accuracy

(3) On the basis of the improved SRPS model a multitar-get optimization model is proposed in purpose of improvingproduction decreasing load fluctuation and saving energy Bycomparison the optimal scheme can achieve the decreasingof maximum and minimum suspension point load cranktorque fluctuation and energy consumption as well asimproving the system balance degree and pump fullness Insummary it improves the dynamic behavior of SRPS

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

National Natural Science Foundation of China (Grantno 51174175) China Scholarship Council (Grant no201708130108) Hebei Natural Science Foundation (Grant noE201703101) are acknowledged

References

[1] T A Aliev A H Rzayev G A Guluyev T A Alizada and NE Rzayeva ldquoRobust technology and system for management ofsucker rod pumping units in oil wellsrdquoMechanical Systems andSignal Processing vol 99 pp 47ndash56 2018

[2] S Gibbs ldquoPredicting the Behavior of Sucker-Rod PumpingSystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 1963

[3] D R Doty and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo SPE Journal vol 23 no 1 pp 33ndash41 1983

[4] I N Shardakov and I NWasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[5] S G Gibbs ldquoComputing gearbox torque and motor loading forbeam pumping units with consideration of inertia effectsrdquo SPEJ vol 27 pp 1153ndash1159 1975

[6] J Svinos ldquoExact Kinematic Analysis of Pumping Unitsrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionSan Francisco California 1983

[7] D Schafer and J Jennings ldquoAn Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical Modelsrdquoin Proceedings of the SPE Annual Technical Conference andExhibition pp 27ndash30 Dallas Texas 1987

[8] G Takacs Sucker-Rod PumpingManual Pennwell Books Tulsa2003

[9] G W Wang S S Rahman and G Y Yang ldquoAn improvedmodel for the sucker rod pumping systemrdquo in Proceedings of the11thAustralasian FluidMechanics Conference pp 14ndash18HobartAustralia December 1992

[10] S D L Lekia andR D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystemsrdquo SPE Production amp Facilities vol 10 no 1 pp 26ndash331995

[11] J Lea and P Pattillo ldquoInterpretation of Calculated Forces onSucker Rodsrdquo SPE Production amp Facilities vol 10 no 01 pp 41ndash45 1995

Mathematical Problems in Engineering 15

[12] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[13] L Guo-hua H Shun-li Y Zhi et al ldquoA prediction model fora new deep-rod pumping systemrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 75ndash80 2011

[14] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[15] O Becerra J Gamboa and F Kenyery ldquoModelling a DoublePiston Pumprdquo in Proceedings of the SPE International ThermalOperations and Heavy Oil Symposium and International Hor-izontal Well Technology Conference Calgary Alberta Canada2002

[16] A L Podio J Gomez A J Mansure et al ldquoLaboratoryinstrumented sucker-rod pumprdquo J Pet Technol vol 53 no 05pp 104ndash113 2003

[17] Z H Gu H Q Peng and H Y Geng ldquoAnalysis and mea-surement of gas effect on pumping efficiencyrdquoChina PetroleumMachinery vol 34 no 02 pp 64ndash69 2006

[18] M Xing ldquoResponse analysis of longitudinal vibration of suckerrod string considering rod bucklingrdquo Advances in EngineeringSoftware vol 99 pp 49ndash58 2016

[19] S Miska A Sharaki and J M Rajtar ldquoA simple model forcomputer-aided optimization and design of sucker-rod pump-ing systemsrdquo Journal of Petroleum Science and Engineering vol17 no 3-4 pp 303ndash312 1997

[20] L S Firu T Chelu and C Militaru-Petre ldquoAmodern approachto the optimum design of sucker-rod pumping systemrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash9 DenverColorado 2003

[21] X F Liu and Y G Qi ldquoA modern approach to the selectionof sucker rod pumping systems in CBM wellsrdquo Journal ofPetroleum Science and Engineering vol 76 no 3-4 pp 100ndash1082011

[22] S M Dong N N Feng and Z J Ma ldquoSimulating maximumof system efficiency of rod pumping wellsrdquo Journal of SystemSimulation vol 20 no 13 pp 3533ndash3537 2008

[23] M Xing and S Dong ldquoA New Simulation Model for a Beam-Pumping System Applied in Energy Saving and Resource-Consumption Reductionrdquo SPE Production amp Operations vol30 no 02 pp 130ndash140 2015

[24] Y XWu Y Li andH LiuElectric Motor andDriving ChemicalIndustry Press Beijing 2008

[25] S M Dong Computer Simulation of Dynamic Parameters ofRod Pumping System Optimization Petroleum Industry PressBeijing Chinese 2003

[26] F Yavuz J F Lea J C Cox et al ldquoWave equation simulationof fluid pound and gas interferencerdquo in Proceedings of the SPEProduction Operations Symposium pp 16ndash19 Oklahoma CityOklahoma April 2005

[27] D Y Wang and H Z Liu ldquoDynamic modeling and analysis ofsucker rod pumping system in a directional well Mechanismand Machine Sciencerdquo in Proceedings of the ASIAN MMS 2016amp CCMMS pp 1115ndash1127 2017

[28] F M A Barreto M Tygel A F Rocha and C K MorookaldquoAutomatic downhole card generation and classificationrdquo inProceedings of the SPE Annual Technical Conference pp 6ndash9Denver Colorado 1996

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Page 3: An Improved Sucker Rod Pumping System Model and Swabbing ...downloads.hindawi.com/journals/mpe/2018/4746210.pdf · An Improved Sucker Rod Pumping System Model and Swabbing Parameters

Mathematical Problems in Engineering 3

Motorrotation

Surfacetransmission

Rod string longitudinal

vibration

Plunger motionPump pressure variationFluid motion

Figure 3 SRPS coupled model sketch

result in the pumping fluid being unable to keep pace withthe plunger and cause an incomplete fullness Meanwhile apump load calculation error will be produced with keepingthe pump pressure as constant when fluid flows into thepump

The SRPS model lays the foundation for optimizingthe swabbing parameters to improve the system operationstatus except for predicting and evaluating its dynamicresponse Miska et al 1997 [19] propose a computer-aidedoptimization method relying on a simple linear algebraicsystem model so as to minimize the energy consumptionFiru et al 2003 [20] present an improve optimization cri-terion including eight operational parameters to achieve themaximum system efficiency Liu and Qi 2011 [21] apply fluidflow characteristics in coalbed methane reservoirs to estimatethe production capacity and build the system efficiencyoptimization model combined with SRPS performance Theabove optimization models are built with a simplified pumpload then an improved system efficiency optimization modelis built jointing with formula (2) [22 23] However thecurrent optimization models ignore the effect of swabbingparameters on pump fullness in that the SRPS modelrsquosrestriction Besides that single target optimization cannotmake an accurate and comprehensive presentation for SRPSwhose pumping progress is complex multicomponent andinteractive

In this paper firstly an improved SRPSmodel is presentedwith the new downhole boundary model considering themovement of fluid flowing into pump with gas instantaneousdissolution and evolution Secondly for the new downholeboundary model a nonlinear fluid-solid coupled model willincrease the complexity of this improved SRPS model Aunified numerical algorithm is applied on the whole modelto decrease the calculation time instead of traditional mixediteration algorithm Thirdly a multiobjective optimizationmodel is proposed based on the improved SRPS modelFourthly the surface dynamometer card is collected to verifythe improvedmodelrsquos accuracy Fifthly the dynamic responsecomparison on the current SRPS model and improved SRPSmodel is executed Finally the optimization program isapplied on a test well and the results are given before andafter optimization

2 Integrated Simulation Model

The improved SRPS model is a nonlinear fluid-solid coupledmodel consisted of drive transmission and loadThemotor isused to generate power and its rotationmotion is transformedinto linear motion of suspension point This linear motionis passed on to the plunger through rod string longitudinalvibration meanwhile plunger motion has an influence onpumping fluid by regulating pump pressure On the contraryfluid motion affects pump pressure which is also the mainfactor in determining pump load This pump load andother system loads are transferred to motor output shaft bysurface and downhole transmission causing impact on themotor motion The above coupled process is depicted inFigure 3 In this paper the SRPSmodel is subdivided into rodstring longitudinal vibration model corresponding surfaceboundary and downhole boundary model so as to have aclear description

21 Sucker Rod Longitudinal Vibration Model While theSRPS is operating the motion of arbitrary micro elementof the slender rod string produced intense longitudinalvibration can be composed of the following two parts (1)following the movement of suspension point and (2) relativemovement to the suspension point Referring to Figure 4 thecoordinate system is built based on the top dead center ofhorse head as the origin Then the stress balance terms of rodstring element and top and bottom boundary conditions areused to establish the sucker rod longitudinal vibration modelwith the assumption of ignoring the deformation of tube andthe vibration of liquid column in a vertical well

12059721199061205971199052 minus 119864119903120588119903 12059721199061205971199092+ 120583120588119903119860119903

120597119906120597119905= minus1198892119906119886 (119905)1198891199052 minus 120583120588119903119860119903

119889119906119886 (119905)119889119905 + 119892119906 (119909 119905)|119909=0 = 119906119886 (119905)119864119903119860119903

120597119906 (119909 119905)12059711990910038161003816100381610038161003816100381610038161003816119909=119871119903

= 119865119875119871 (119905)

(3)

4 Mathematical Problems in Engineering

Top dead center

x

dx

Lru(xt)

FPL(t)

ua(t)

rAr(d2ua

dt2+

2u

t2)dx

(dua

dt+

u

t)dx

ErAru

x+

x(ErAr

u

x)dx

rArgdx

ErAru

x

Figure 4 Mechanical model of rod string longitudinal vibration

Med

mMef

Figure 5 Surface apparatus equivalent motion model

where u is the displacement of the rod string at arbitrarydepth and time m Er and 120588r are the elasticity modulus anddensity of rod string respectively pa and kgm3 ua is thesuspension point displacement m

22 Surface Boundary Model The surface boundary condi-tion refers to the motion of suspension point as in (3) Withthe assumption that the surface transmission mechanismconsists of belt-reducing gear and four-bar linkage as a singledegree of freedom system it can be represented as a functionof motor angle Later the surface apparatus motion model isset up using Lagrange equation taking motor output shaft asequivalent component its motion as generalized coordinatesand set 12 orsquoclock as a reference direction (Figure 5)

119868119890119898 + 12 119898

2 119889119868119890119889120579119898 = 119872119890119889 minus119872119890119891

1205791198981003816100381610038161003816119905=0 = 0119898

10038161003816100381610038161003816120579=0= 1205960

(4)

LC

LP

LR

LL

LK

LI

LH

LAFRL

LRmdash crankLPmdash link rodLCmdash beam backLAmdash beam front

Figure 6 Four-bar linkage

With the equivalent rotating inertia

119868119890 = sum119868119895 (120596119895119898

)2 +sum119898119895 ( V119895119898

)2

(5)

The semiempirical formulation of motor driving torqueMed deduced from the speedtorque characteristic is adoptedfromWu et al [24]

119872119890119889 = 2120582119896119872119867120576119888120596119899 [120596119899 minus 119898]12057621198881205962

119899 + [120596119899 minus 119898]2 (6)

where 120582k is the motor overload coefficient 120576c is the motorcritical-slip ratio 120596n is the motor synchronous angularvelocity rads

119872119867 = 9550119875119867119899119867

120576119888 = 120576119903 (120582119896 + radic1205822119896minus 1)

120596119899 = 2120587119891119911(7)

where z is the motor pole pairs f is the power frequency HzPH is the motor rated power kW nH is the motor rated speedminminus1 120576r is the motor rated-slip ratio

The equivalent resistance torque is derived in accordancewith the force balance of four-bar linkage (Figure 6) writtenas

119872119890119891 = 1119894119887119892120576119887 [119878119879119891 (119865119877119871 minus 119865119887) 1205761198961119901 minus119872119888 sin (120579 minus 120579120591)] (8)

with the angle of crank

120579 = 120579119898119894119887119892(9)

Mathematical Problems in Engineering 5

where ibg is the transmission ratio of the belt-reducinggear system 120576p is the transmission efficiency from crank tosuspension point STf is torque factorm FRL is the suspensionpoint load N Fb is the structural unbalance weight NMc isthe maximum balancing torque of the crank Nsdotm

The displacement of suspension point can be expressedas motor angle based on the geometry relation shown inFigure 6 [6] The surface boundary motion model is

119906119886 (119905) = arccos [1198712119862 + 1198712

119870 minus (119871119877 + 119871119875)22119871119862119871119870

]minus arccos(1198712

119862 + 1198712119871 minus 1198712

1198752119871119862119871119871

) minus arcsin(119871119877119871119871

sdot sin(2120587 minus 1205790 minus 120579119898119894119887119892

+ arcsin( 119871119868119871119870

)))(10)

23 Downhole Boundary Model This new model is animproved method from the current boundary models con-sidering the interaction among pump pressure fluid flow intopump gas dissolution and evolution pressure drop due tofluid gravitational potential energy inertia head loss frictionhead loss and local head loss The new boundary modelconsists of pressure variation model and fluid flow into pumpmodel Some assumptions are made for building this model

(1) Bottom dead point of plunger as the origin position(2) The pump inlet pressure ps and outlet pressure pd are

kept constant for one cycle of SRPS operation

Pressure Variation Model The gas state equations are built asfollows ( (11) and (12)) when the displacement of plunger is upor up+dup

119901 [(119906119901 + 119871119900119892 minus 119871119891 minus 119871119896)119860119901] = 119885119873119876119879 (11)

(119901 + 119889119901)sdot [(119906119901 + 119889119906119901 + 119871119900119892 minus 119871119891 minus 119871119896 minus 119889119871119891 minus 119889119871119896)119860119901]= (119885 + 120597119885120597119901 119889119901) (119873 + 119889119873)119876119879

(12)

where N is the gas molar number mol Q is the natural gasconstant J(molsdotK) T is the temperature in pump barrel ∘CZ is the natural gas compressibility factor Lk is the equivalentlength of pump leakage m Lf is the liquid level in the pumpm

Dividing (11) by (12) and ignoring the second order smallquantities then the variation of pump pressure with plungerdisplacement is obtained

119889119901119889119906119901

= (1119873) (119889119873119889119906119901) + (1 (119906119901 + 119871119900119892 minus 119871119891 minus 119871119896)) (119889 (119871119891 + 119871119896) 119889119906119901 minus 1)119901minus1 minus (1119885) (119889119885119889119901) (13)

The above equation is converted into a time varyingfunction with the purpose of facilitating it with the rod stringlongitudinal vibration equation

119889119901119889119905 = (1119873) (119889119873119889119905) (1V119901) + (1 (119906119901 + 119871119900119892 minus 119871119891 minus 119871119896)) ((119889 (119871119891 + 119871119896) 119889119905) (1V119901) minus 1)119901minus1 minus (1119885) (119889119885119889119901) V119901(14)

where vp is the plunger velocity msFrom (14) dN is composed of the following sections(1)Themolar number of gas is released from or dissolved

into oil as pressure varies

119889119873119903 = minus 119885119879119901119904119905119885119904119905119879119904119905119901 (120572 sdot 119889119901)sdot 103119860119901 (119871119891 + 119871119900 minus 119871119900119892) (1 minus 120576119908)119861119900119881119898119900119897

(15)

where pst is the standard pressure pa Zst is the standard natu-ral gas compressibility factorTst is the standard temperature∘C 120572 is the solubility coefficient of natural gas m3 (m3sdotpa)

120576w is the water content Vmol is the molar volume m3 Bo isthe crude oil volume factor

(2)Themolar number of gas with fluid being sucked intothe pump corresponding to the pump pressure is

119889119873119904 = 103119860119901119877119869119889119871119891119881119898119900119897

(16)

where RJ is the transient gas liquid ratio in the pump m3m3(3)Themolar number of gaswith leaked fluid at the pump

pressure is

119889119873V = 103119860119901119877119869119889119871V119881119898119900119897

(17)

6 Mathematical Problems in Engineering

f-f

L

Lf

La

a-a

Figure 7 Process of fluid flows into the pump

where

119877119869 = (1 minus 120576119908) (119877119901 minus 119877119904) 119901119904119905119885119879119879119904119905119885119904119905119901119877119904 = 120572 (119901 minus 119901119904119905)

(18)

where Rp is the production gas oil ratio m3m3 Rs is the gasoil ratio at pump intake m3m3

Fluid Flowing into Pump Model Figure 7 shows the pumpoperation when the gas-liquid flow is drawn into the pumpSection a-a indicates the cross section of standing valve holeSupposing the section f -f is the liquid level at arbitrarytime neglect the process of gas bubbling from fluid byconsidering only the pump gas to occupy the upper spaceof f -f The one-dimensional unsteady flow equation based onBernoulli equation is established to describe the flow state ofpumping liquid In the Bernoulli equation the inertia headloss friction head loss and local head loss are all considered

119901119904120588g + V21198862119892 = 119901120588g + V2

1198912g + (119871119900 minus 119871119900119892 + 119871119891 + 119871V) + ℎ119886

+ ℎV + ℎ119891

(19)

where

ℎ119886 = 1119892 (119871119900 minus 119871119900119892 + 119871119891 + 119871V119860119891119860119901

) 119889V119891119889119905ℎ119891 = 64120583V119891 (119871119900 minus 119871119900119892 + 119871119891)21198632

119889120588119892

ℎV = 11198921205762 (119860119901119860119891

)2 V21198862

V119886 = 119860119901119860119891

V119891

(20)

where 120576 is the flow coefficient of standing valveThen (19) is converted into differential forms

119891 = V119891

V119891 = (1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V119860119891119860119901

)minus1119901119904120588 minus 119901120588+ V2

1198912 [11986021199011198602119891

minus 119860211990112057621198602

119891

minus 64120583 (1198710 minus 119881119900119892119860119901 + 119871119891)1198632119889V119891120588

minus 1] minus 119892(1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V)

(21)

Normally the fluid in the barrel nomatter in which formwill all be discharged except the one in dead spaceThereforethe fluid flowing out of the model with opening travelingvalve is not proposed The pump outlet pressure generatedby several kilometers liquid column is large compared to thehydraulic loss causing the ignorance of hydraulic loss whenfluid flows out Based on the statement mentioned above thenew downhole boundary model is expressed as follows andalso is divided into four phases corresponding to the phasesshown in Figure 2

1198781199051198861199051199061199041 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199042 af ter 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = ((V119891 sdot 119860119901 + 119902) (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)]sdot V119901

Mathematical Problems in Engineering 7

1198781199051198861199051199061199043 before 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199044 af ter 119900119901119890119899119892119894119899 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = 0

(22)

where

119881119898119900119897 = 224119885119879119901119904119905119879119904119905119901 (23)

3 Integrated Numerical Algorithm

For the current SRPS model the surface transmission modelis solved by numerical integration method [25] While thefinite difference method is used to describe the rod stringlongitudinal vibration [2 26 27] Since there is no fixedsolution between surface and downholemodel it needs cycliciteration to handle thewholemodel which is time consuming[25] Considering the interaction within plunger motionfluid flow into pump and pump pressure the improved SRPSmodel has a higher nonlinear degree and thus increasesthe difficulty of solving it In order to reduce the solvingtime the wave equation of rod string longitudinal vibrationis converted into ordinary differential equations by modalsuperposition method Then the surface and downholeboundary model are now in the form of ordinary differentialequations at which the whole model can be solved by Runge-Kutta method directly Besides that solving the rod stringlongitudinal vibration equation is also solving the forcevibration response of rod string where (3) is converted intothe following form

120588119903119860119903

12059721199061205971199052 minus 120588119903119864119903

12059721199061205971199092+ 120583120597119906120597119905 = 120588119903119860119903119891 (119909 119905) (24)

where

119891 (119909 119905) = minus1198892119906119886 (119905)1198891199052 minus 120583120588119903119860119903

119889119906119886 (119905)119889119905 + 119892 (25)

And its solution can be expressed in the function of timeand space by separating the variables

119906 (119909 119905) = infinsum119894=1

Φ119894 (119909) 119902119894 (119905) (26)

Therefore (24) can be expressed asinfinsum119894=1

119860119903120588119903Φ119894 (119909) 119894 (119905)minus infinsum

119894=1

119864119903119860119903

1198892Φ119894 (119909)1198891199092119902119894 (119905) + infinsum

119894=1

120583Φ119894 (119909) 119894 (119905)= 119860119903120588119903119891 (119909 119905)

(27)

Multiply the above equation by Φi(x) and it is integratedalong the rod length Then the following is derived

119860119903120588119903119894 (119905) int119871119903

0Φ119894

2 (119909) 119889119909minus 119864119903119860119903119902119894 (119905) int119871119903

0Φ119894 (119909)Φ119894

10158401015840 (119909) 119889119909+ 120583119894 (119905) int119871119903

0Φ119894

2 (119909) 119889119909= 119860119903120588119903 int119871119903

0Φ119894 (119909) 119891 (119909 119905) 119889119909

(28)

Its mode shapes and natural frequencies are

Φ119894 (119909) = radic 2120588119903119860119903

sin((2119894 minus 1) 1205872119871119903

119909)119901119899119894 = (2119894 minus 1) 120587radic1198641199031205881199032119871119903

(29)

Then (28) can be simplified as follows using mode shapeorthogonality

119894 (119905) + 120583119860119903120588119903 119894 (119905) + ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 119902119894 (119905) = 119865119894 (30)

where

119865119894 = (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903(2119894 minus 1) 120587+ 119865119875119871 (119905) radic 2119860119903120588119903119871119903

sin ((2119894 minus 1) 1205872 )(31)

8 Mathematical Problems in Engineering

Let ith-order forced vibration displacement response andvelocity response be xi1 and xi2 respectivelyThen (30) can beexpressed as the following form

1198941 (119905) = 1199091198942 (119905)1198942 (119905) = minus 1205831198601199031205881199031199091198942 (119905) minus ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 1199091198941 (119905) + 119865119894

(32)

Then suspension point load is derived

119865119877119871 (119905) = 119864119903119860119903

120597119906 (0 119905)120597119909 + 119865119903

= 119864119903119860119903radic 2120588119903119860119903119871119903

119873sum119894=1

(2119894 minus 1) 1205872119871119903

1199091198941 (119905) + 119865119903

(33)

The displacement and velocity of plunger are

119906119901 (119905) = 119906119886 (119905) minus 119906 (119871 119905)= 119906119886 (119905)minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198941 (119905) minus 119894=2lowast119895sum119894=1

1199091198941 (119905))V119901 (119905) = 119886 (119905) minus 120597119906 (119871 119905)120597119905

= 119886 (119905)minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198942 (119905) minus 119894=2lowast119895sum119894=1

1199091198942 (119905))

(34)

Then the integrated numerical model is given as follows

119898 = 120596119898

119898 = 1119868119890 [[2120582119896119872119867120576119888120596119899 [120596119899 minus 119898]12057621198881205962

119899 + [120596119899 minus 119898]2 minus 1119894119887119892120578119887119892

[119878119879119891 (119865119877119871 minus 119865119887) 1205781198961119862119871 minus119872119888 sin (120579 minus 120579120591)] minus 121205962

119898

119889119868119890119889120579119898]]119906119886 (119905) = arccos[1198712

119862 + 1198712119870 minus (119871119877 + 119871119875)22119871119862119871119870

] minus arccos(1198712119862 + 1198712

119871 minus 11987121198752119871119862119871119871

)minus arcsin(119871119877119871119871

sin(2120587 minus 1205790 minus 120579119898119894119887119892

+ arcsin ( 119871119868119871119870

)))11990911 (119905) = 11990912 (119905)12 (119905) = minus 12058311986011990312058811990311990912 (119905) minus ( 1205872119871119903

)2 119864119903120588119903 11990911 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903120587+ (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

21 (119905) = 11990922 (119905)22 (119905) = minus 12058311986011990312058811990311990922 (119905) minus ( 31205872119871119903

)2 119864119903120588119903 11990921 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic21198601199031205881199031198711199033120587minus (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

1198941 (119905) = 1199091198942 (119905)1198942 (119905) = minus 1205831198601199031205881199031199091198942 (119905) minus ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 1199091198941 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903(2119894 minus 1) 120587+ (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

sin ((2119894 minus 1) 1205872 )

Mathematical Problems in Engineering 9

119865119877119871 (119905) = 119864119903119860119903radic 2120588119903119860119903119871119903

119873sum119894=1

(2119894 minus 1) 1205872119871119903

1199091198941 (119905) + 119865119903

119906119901 (119905) = 119906119886 (119905) minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198941 (119905) minus 119894=2lowast119895sum119894=1

1199091198941 (119905))

V119901 (119905) = 119886 (119905) minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198942 (119905) minus 119894=2lowast119895sum119894=1

1199091198942 (119905))1198781199051198861199051199061199041 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199042 119886119891119905119890119903 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = ((V119891 sdot 119860119901 + 119902) (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)]sdot V119901

1198781199051198861199051199061199043 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199044 119886119891119905119890119903 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = 0119889119871119891119889119905 = V119891

119889V119891119889119905 = (1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V119860119891119860119901

)minus1119901119904120588 minus 119901120588 + V21198912 [119860

21199011198602119891

minus 119860211990112057621198602

119891

minus 64120583 (1198710 minus 119881119900119892119860119901 + 119871119891)1198632V119891120588 minus 1]minus 119892(1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V)(35)

4 Optimization Model

41 Optimization Goal As mostly developed oil-field movesinto the mid and late stage and the OWD begins todecline gradually energy-saving production-increasing andreducing load variation as much as possible are particularlyimportant Then we take pump fullness epf suspension pointload amplitude FRLA crank torque standard deviation Mcsdand motor input power average 119875119898 as the optimization goalto build a multitarget model Suspension point load andcrank torque can be solved directly by (35) The motor input

power and pump fullness calculation formula are deduced asfollows

119875119898 = 119872119890119889119898 + 1198750

+ [( 1120578H minus 1)119875119867 minus 1198750](119872119890119889119898119875119867

)2 (36)

119890119901119891 = 119871119891 minus int119905119906

0119902119889119905 minus (119871119900 minus 119871119900119892)119906119901119906

(37)

10 Mathematical Problems in Engineering

where P0 is the no-load power of motor kW PH is the motorrated power kW 120578H is the motor rated efficiency kW upuis the plunger displacement when it is arriving at dead topcenter

So the objective function is

Ω(119890119901119891 (X) 119865119877119871119860 (X) 119872119888119904119889 (X) 119875119898 (X))= 1198701119890119901119891 (X) + 1198702119865119877119871119860 (X) + 1198703119872119888119904119889 (X)+ 1198704119875119898 (X)

(38)

where K1 K2 K3 and K4 are the weight coefficients

42 Design Variables The objective function can beexpressed as the function of the swabbing parameters whenthe SRPS type and oil well basic parameters are confirmedIn this paper the swabbing parameters denote stroke Sstroke frequency ns pump diameter Dd pump depth Lpdcrank balance radius rc and rod string combination (the jthrod string diameter and length are dj and Lj respectively119896 = 1 2 119898) Then the design variables are shown asfollows

X = 119878 119899119904 119863119889 119871119901119889 (119889119895 119871119895 119895 = 1 2 sdot sdot sdot 119898) (39)

43 Constraints

(a) Crank Balance Degree Crank balance degree indicates theload fluctuation to a certain degree and it needs to be kept ata high value

095 le 119872119888119896119906119872119888119896119889

le 1 (40)

where Mcku and Mckd are the maximum crank torque whenplunger is at upstroke and downstroke respectively Nsdotm

(b) Ground Device Carrying Capacity The suspensionpoint load crank torque and motor torque at anytime [0 119879] do not go beyond the allowable rangemax(F119877119871) min(F119877119871) max(119872119888) min(119872119888) max(119872119890119889)min(119872119890119889) for the given type

min (119865119877119871) le 119865119877119871 le max (119865119877119871)min (119872119888119896) le 119872119888119896 le max (119872119888119896)min (119872119890119889) le 119872119890119889 le max (119872119890119889)

(41)

(c) Rod String StrengthThemaximumandminimum stress ofany point x along the rod string does not exceed permissiblestress range in one cycle

[120590min] le 120590119903119904 le [120590max] (42)

(d) Swabbing Parameters Each oil well has different limit onthe swabbing parameters in accordance with the device typeand actual operation hence their allowable variation rangesare adjustable

Figure 8 Dynamometer sensor

44 Optimization Algorithm In summary this multivariableoptimization model is established with nonlinear restrictionand nonlinear objective function So as to seek the bestresults the genetic algorithm is applied to solve it

5 Test and Verification

Surface dynamometer card is a closed graph recordingpolished rod loads versus rod displacement over a SRPScycle which is generally collected and taken as an indexto estimate the operation of SRPS In this paper based onthe dynamometer sensor shown in Figure 8 four test wellsare used to validate the improved SRPS model The oil wellparameters are listed in Table 1 and the simulation and fieldtest results are given in Figure 9 According to the plottedcurves the simulation results are basically consistent withthat in measured and the improved SRPS model is accurateenough to be proposed for engineering practices

6 Dynamic Response Comparison

Pump dynamometer card is a closed graph recording plungerloads versus plunger displacement over a SRPS cycle It isdetermined by multifactors such as stroke stroke frequencypump diameter pump depth dynamic liquid level and theother oil well parameters The difference between the currentSRPS model and the improved SRPS model proposed in thispaper is whether to consider the PFFP In view of this theoil operating status is divided into two forms as follows(1) the fluid always keeps pace with the plunger when itis being sucked into the pump (2) the fluid is unable tofollowwith the plungerThen two oil wells are selected well1with sufficient OWD and well2 with insufficient OWDFigure 10 describes the plunger and fluid velocity duringupstroke based on the improved SRPS model It can beconcluded that when the well has sufficient OWD and thefluid possesses good ability of keeping pace with the plungerwhen the well has insufficient OWD the fluid velocity lagsbehind the one of plunger at the beginning whereas it is

Mathematical Problems in Engineering 11

Table 1 Oil well basic parameters

Well 1 2 3 4Motor type YD280S-8 Y250M-6 Y250M-6 YD280S-6Pumping unit type CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HBStroke length (m) 3 3 3 3Stroke frequency (minminus1) 35 3 4 6Pump diameter (mm) 57 44 38 44Pump clearance level 1 2 2 1Pump depth (m) 890 1377 1138 1430Sucker-rod string (mm timesm) 25lowast890 19lowast673+22lowast704 19lowast672+22lowast466 19lowast720+22lowast710middle depth of reservoir (m) 1000 1492 1569 1600Crude oil density (kgm3) 795 857 857 850Water content () 95 97 92 98Dynamic liquid level (m) 880 1347 757 1340Casing pressure (Pa) 02 03 03 02Oil pressure (Pa) 03 03 03 02Fluid dynamic viscosity (Pasdots) 0006 0007 0007 0006Gas oil ratio (m3 m3) 20 19 40 80Plunger length (m) 12 12 12 12Clearance length (m) 05 05 05 05

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(a) Well 1

1 2 30Polished rod displacement (m)

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

simulationmeasured

(b) Well 2

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(c) Well 3

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(d) Well 4Figure 9 Suspension point dynamometer card comparison between the simulated and measured

12 Mathematical Problems in Engineering

FluidPlunger

minus05

00

05

10

15Ve

loci

ty(

ms

)

2 4 60Time (s)

(a) Well 1

minus02

00

02

04

06

Velo

city

(m)

4 8 120Time (s)

FluidPlunger

(b) Well 2

Figure 10 Plunger and fluid velocity

Current SRPS model 970Improved SRPS model 981

minus10

0

10

20

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(a) Well 1

Current SRPS model 724Improved SRPS model 775

minus10

0

10

20

40

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(b) Well 2

Figure 11 Comparison of pump dynamometer card

faster after a certain time period Then the comparisons ofpump dynamometer card are executed by the two modelsmeanwhile their individual pump fullness result is given atthe end of the legend (Figure 11)

Pump dynamometer card is very important method andnormally used to diagnose the pump operations particularlywhose shape is more similar to a rectangle the pump iscloser to the fullness [28] Hence from Figure 11 it canbe known that the pump fullness of current SRPS modelis higher than that of the improved one depending on thequalitative judgment and this conclusion is consistent withthe quantitative calculation result meanwhile this gap forthe well 2 is bigger than well 1 According the above

description we can know that the pump fullness calculationresult is on the high side if the PFFP is not considered andthis phenomenon will becomemore obvious for the well withinsufficient OWD

The load presented by the left upper right and lowerborderline of pump dynamometer card is the results of pumpmoving from phase 1 to 4 in turn Therefore its upperborderline describes the pump load when fluid is pumpedinto the barrel From Figure 11(a) it can be found thatthe upper borderline load simulated by the improved SRPSmodel is significantly larger than the one of current ForFigure 11(b) this difference can be neglected Based on (1)and (2) the pump pressure keeps constant as ps when fluid is

Mathematical Problems in Engineering 13

Before optimizationAfter optimization

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

1 2 30Polished rod displacement (m)

(a) Suspension point dynamometer card

minus10

0

10

20

30

40

Cran

k to

rque

(kNmiddotm

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(b) Crank torque

0

5

10

15

20

Mot

or in

put p

ower

(kW

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(c) Motor input power

Figure 12 Dynamic response comparison before and after optimization

sucked into the pump and then the corresponding pump loadis a fixed value Due to the new model takes into account ofPFFP a pressure drop will be brought which will vary withthe fluid velocity Combined with (1) this pressure drop willlead to the pump load increases Figure 10 shows that thefluid velocity of well 1 is larger than that of well 2 andthis is the reason why the difference of upper borderline loadbetween the two models is obvious This illustrates the erroron the upper borderline load of pump dynamometer carddepending on the velocity at which the fluid flows into thepump

7 Optimization Test

Based on the above models a simulation and optimizationsoftware is developed byMATLABOnewell with insufficient

OWD is tested Its original swabbing parameters are asfollows stroke S is 3 m stroke frequency ns is 3 minminus1pump diameter Dd is 57mm pump depth Lpd is 900m crankbalance radius rc is 12 m and rod string combination djtimesLj is 25 mm times 524 m+22 mm times 376 m And its swabbingparameters after optimizing are as follows stroke S is 3 mstroke frequency ns is 25 minminus1 pump diameterDd is 57mmpump depth Lpd is 990 m crank balance radius rc is 09 mand rod string combination djtimes Lj is 22 mm times 426 m+19 mmtimes 564 m The comparisons before and after optimization areshown in Figure 12 and Table 2

From the above comparison results some conclusions areobtained

(1) After optimization the maximum and minimumof suspension point load are all decreased and the load

14 Mathematical Problems in Engineering

Table 2 Specific parameters comparison before and after optimization

Suspension point load Crank torque Motor input power PumpfullnessMaximum Minimum Amplitude Standard

deviationBalancedegree Maximum Minimum mean

Beforeoptimization 555 kN 190 kN 365 kN 107 kN 854 159 kW 09 kW 62 kW 725

Afteroptimization 441 kN 132 kN 309 kN 94 kN 978 106 kW 10 kW 54 kW 908

darr 205 darr 305 darr 153 darr 121 uarr 145 darr 333 uarr 111 darr 129 uarr 252 amplitude is lowered by 153 It can contribute to enhancethe rod string life and prolong the maintenance period

(2) After optimization the standard deviation of cranktorque is reduced by 121 and its balance degree is raisedby 145 It illustrates that the load torque fluctuation is cutdown which canminimize damage to transmission parts andimprove the motor efficiency

(3) After optimization the pump fullness is improvedby 252 It is conducive to improve pump efficiency andproduction

(4) After optimization it plays a role of peak shavingand valley filling for motor input power and extends theoperational life meanwhile power saving rate is 129

8 Conclusions

(1) In this article an improved SRPS model is presentedconsidering the couple effect of pumping fluid and plungermotion on the dynamic response of SRPS instead of theexisting models that assume the pumping fluid volumeis always equal to plunger travelling volume The Runge-Kutta method is applied to solve the whole system modeltrough transforming the rod string longitudinal vibrationequation into ordinary differential equations And the SRPSmodelrsquos precision has been validated by adopting surfacedynamometer card

(2) Two oil wells are served to compare the differencebetween the current SRPS model and the improved oneThe results indicate that the current SRPS model is relativelylow in calculating pump fullness and this gap will increasewith the reduction of OWD the influence of fluid flowinginto the pump on pump load cannot be ignored when thefluid velocity is high Therefore the PFFP is necessary to beconsidered in order to improve the simulation accuracy

(3) On the basis of the improved SRPS model a multitar-get optimization model is proposed in purpose of improvingproduction decreasing load fluctuation and saving energy Bycomparison the optimal scheme can achieve the decreasingof maximum and minimum suspension point load cranktorque fluctuation and energy consumption as well asimproving the system balance degree and pump fullness Insummary it improves the dynamic behavior of SRPS

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

National Natural Science Foundation of China (Grantno 51174175) China Scholarship Council (Grant no201708130108) Hebei Natural Science Foundation (Grant noE201703101) are acknowledged

References

[1] T A Aliev A H Rzayev G A Guluyev T A Alizada and NE Rzayeva ldquoRobust technology and system for management ofsucker rod pumping units in oil wellsrdquoMechanical Systems andSignal Processing vol 99 pp 47ndash56 2018

[2] S Gibbs ldquoPredicting the Behavior of Sucker-Rod PumpingSystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 1963

[3] D R Doty and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo SPE Journal vol 23 no 1 pp 33ndash41 1983

[4] I N Shardakov and I NWasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[5] S G Gibbs ldquoComputing gearbox torque and motor loading forbeam pumping units with consideration of inertia effectsrdquo SPEJ vol 27 pp 1153ndash1159 1975

[6] J Svinos ldquoExact Kinematic Analysis of Pumping Unitsrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionSan Francisco California 1983

[7] D Schafer and J Jennings ldquoAn Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical Modelsrdquoin Proceedings of the SPE Annual Technical Conference andExhibition pp 27ndash30 Dallas Texas 1987

[8] G Takacs Sucker-Rod PumpingManual Pennwell Books Tulsa2003

[9] G W Wang S S Rahman and G Y Yang ldquoAn improvedmodel for the sucker rod pumping systemrdquo in Proceedings of the11thAustralasian FluidMechanics Conference pp 14ndash18HobartAustralia December 1992

[10] S D L Lekia andR D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystemsrdquo SPE Production amp Facilities vol 10 no 1 pp 26ndash331995

[11] J Lea and P Pattillo ldquoInterpretation of Calculated Forces onSucker Rodsrdquo SPE Production amp Facilities vol 10 no 01 pp 41ndash45 1995

Mathematical Problems in Engineering 15

[12] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[13] L Guo-hua H Shun-li Y Zhi et al ldquoA prediction model fora new deep-rod pumping systemrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 75ndash80 2011

[14] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[15] O Becerra J Gamboa and F Kenyery ldquoModelling a DoublePiston Pumprdquo in Proceedings of the SPE International ThermalOperations and Heavy Oil Symposium and International Hor-izontal Well Technology Conference Calgary Alberta Canada2002

[16] A L Podio J Gomez A J Mansure et al ldquoLaboratoryinstrumented sucker-rod pumprdquo J Pet Technol vol 53 no 05pp 104ndash113 2003

[17] Z H Gu H Q Peng and H Y Geng ldquoAnalysis and mea-surement of gas effect on pumping efficiencyrdquoChina PetroleumMachinery vol 34 no 02 pp 64ndash69 2006

[18] M Xing ldquoResponse analysis of longitudinal vibration of suckerrod string considering rod bucklingrdquo Advances in EngineeringSoftware vol 99 pp 49ndash58 2016

[19] S Miska A Sharaki and J M Rajtar ldquoA simple model forcomputer-aided optimization and design of sucker-rod pump-ing systemsrdquo Journal of Petroleum Science and Engineering vol17 no 3-4 pp 303ndash312 1997

[20] L S Firu T Chelu and C Militaru-Petre ldquoAmodern approachto the optimum design of sucker-rod pumping systemrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash9 DenverColorado 2003

[21] X F Liu and Y G Qi ldquoA modern approach to the selectionof sucker rod pumping systems in CBM wellsrdquo Journal ofPetroleum Science and Engineering vol 76 no 3-4 pp 100ndash1082011

[22] S M Dong N N Feng and Z J Ma ldquoSimulating maximumof system efficiency of rod pumping wellsrdquo Journal of SystemSimulation vol 20 no 13 pp 3533ndash3537 2008

[23] M Xing and S Dong ldquoA New Simulation Model for a Beam-Pumping System Applied in Energy Saving and Resource-Consumption Reductionrdquo SPE Production amp Operations vol30 no 02 pp 130ndash140 2015

[24] Y XWu Y Li andH LiuElectric Motor andDriving ChemicalIndustry Press Beijing 2008

[25] S M Dong Computer Simulation of Dynamic Parameters ofRod Pumping System Optimization Petroleum Industry PressBeijing Chinese 2003

[26] F Yavuz J F Lea J C Cox et al ldquoWave equation simulationof fluid pound and gas interferencerdquo in Proceedings of the SPEProduction Operations Symposium pp 16ndash19 Oklahoma CityOklahoma April 2005

[27] D Y Wang and H Z Liu ldquoDynamic modeling and analysis ofsucker rod pumping system in a directional well Mechanismand Machine Sciencerdquo in Proceedings of the ASIAN MMS 2016amp CCMMS pp 1115ndash1127 2017

[28] F M A Barreto M Tygel A F Rocha and C K MorookaldquoAutomatic downhole card generation and classificationrdquo inProceedings of the SPE Annual Technical Conference pp 6ndash9Denver Colorado 1996

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Page 4: An Improved Sucker Rod Pumping System Model and Swabbing ...downloads.hindawi.com/journals/mpe/2018/4746210.pdf · An Improved Sucker Rod Pumping System Model and Swabbing Parameters

4 Mathematical Problems in Engineering

Top dead center

x

dx

Lru(xt)

FPL(t)

ua(t)

rAr(d2ua

dt2+

2u

t2)dx

(dua

dt+

u

t)dx

ErAru

x+

x(ErAr

u

x)dx

rArgdx

ErAru

x

Figure 4 Mechanical model of rod string longitudinal vibration

Med

mMef

Figure 5 Surface apparatus equivalent motion model

where u is the displacement of the rod string at arbitrarydepth and time m Er and 120588r are the elasticity modulus anddensity of rod string respectively pa and kgm3 ua is thesuspension point displacement m

22 Surface Boundary Model The surface boundary condi-tion refers to the motion of suspension point as in (3) Withthe assumption that the surface transmission mechanismconsists of belt-reducing gear and four-bar linkage as a singledegree of freedom system it can be represented as a functionof motor angle Later the surface apparatus motion model isset up using Lagrange equation taking motor output shaft asequivalent component its motion as generalized coordinatesand set 12 orsquoclock as a reference direction (Figure 5)

119868119890119898 + 12 119898

2 119889119868119890119889120579119898 = 119872119890119889 minus119872119890119891

1205791198981003816100381610038161003816119905=0 = 0119898

10038161003816100381610038161003816120579=0= 1205960

(4)

LC

LP

LR

LL

LK

LI

LH

LAFRL

LRmdash crankLPmdash link rodLCmdash beam backLAmdash beam front

Figure 6 Four-bar linkage

With the equivalent rotating inertia

119868119890 = sum119868119895 (120596119895119898

)2 +sum119898119895 ( V119895119898

)2

(5)

The semiempirical formulation of motor driving torqueMed deduced from the speedtorque characteristic is adoptedfromWu et al [24]

119872119890119889 = 2120582119896119872119867120576119888120596119899 [120596119899 minus 119898]12057621198881205962

119899 + [120596119899 minus 119898]2 (6)

where 120582k is the motor overload coefficient 120576c is the motorcritical-slip ratio 120596n is the motor synchronous angularvelocity rads

119872119867 = 9550119875119867119899119867

120576119888 = 120576119903 (120582119896 + radic1205822119896minus 1)

120596119899 = 2120587119891119911(7)

where z is the motor pole pairs f is the power frequency HzPH is the motor rated power kW nH is the motor rated speedminminus1 120576r is the motor rated-slip ratio

The equivalent resistance torque is derived in accordancewith the force balance of four-bar linkage (Figure 6) writtenas

119872119890119891 = 1119894119887119892120576119887 [119878119879119891 (119865119877119871 minus 119865119887) 1205761198961119901 minus119872119888 sin (120579 minus 120579120591)] (8)

with the angle of crank

120579 = 120579119898119894119887119892(9)

Mathematical Problems in Engineering 5

where ibg is the transmission ratio of the belt-reducinggear system 120576p is the transmission efficiency from crank tosuspension point STf is torque factorm FRL is the suspensionpoint load N Fb is the structural unbalance weight NMc isthe maximum balancing torque of the crank Nsdotm

The displacement of suspension point can be expressedas motor angle based on the geometry relation shown inFigure 6 [6] The surface boundary motion model is

119906119886 (119905) = arccos [1198712119862 + 1198712

119870 minus (119871119877 + 119871119875)22119871119862119871119870

]minus arccos(1198712

119862 + 1198712119871 minus 1198712

1198752119871119862119871119871

) minus arcsin(119871119877119871119871

sdot sin(2120587 minus 1205790 minus 120579119898119894119887119892

+ arcsin( 119871119868119871119870

)))(10)

23 Downhole Boundary Model This new model is animproved method from the current boundary models con-sidering the interaction among pump pressure fluid flow intopump gas dissolution and evolution pressure drop due tofluid gravitational potential energy inertia head loss frictionhead loss and local head loss The new boundary modelconsists of pressure variation model and fluid flow into pumpmodel Some assumptions are made for building this model

(1) Bottom dead point of plunger as the origin position(2) The pump inlet pressure ps and outlet pressure pd are

kept constant for one cycle of SRPS operation

Pressure Variation Model The gas state equations are built asfollows ( (11) and (12)) when the displacement of plunger is upor up+dup

119901 [(119906119901 + 119871119900119892 minus 119871119891 minus 119871119896)119860119901] = 119885119873119876119879 (11)

(119901 + 119889119901)sdot [(119906119901 + 119889119906119901 + 119871119900119892 minus 119871119891 minus 119871119896 minus 119889119871119891 minus 119889119871119896)119860119901]= (119885 + 120597119885120597119901 119889119901) (119873 + 119889119873)119876119879

(12)

where N is the gas molar number mol Q is the natural gasconstant J(molsdotK) T is the temperature in pump barrel ∘CZ is the natural gas compressibility factor Lk is the equivalentlength of pump leakage m Lf is the liquid level in the pumpm

Dividing (11) by (12) and ignoring the second order smallquantities then the variation of pump pressure with plungerdisplacement is obtained

119889119901119889119906119901

= (1119873) (119889119873119889119906119901) + (1 (119906119901 + 119871119900119892 minus 119871119891 minus 119871119896)) (119889 (119871119891 + 119871119896) 119889119906119901 minus 1)119901minus1 minus (1119885) (119889119885119889119901) (13)

The above equation is converted into a time varyingfunction with the purpose of facilitating it with the rod stringlongitudinal vibration equation

119889119901119889119905 = (1119873) (119889119873119889119905) (1V119901) + (1 (119906119901 + 119871119900119892 minus 119871119891 minus 119871119896)) ((119889 (119871119891 + 119871119896) 119889119905) (1V119901) minus 1)119901minus1 minus (1119885) (119889119885119889119901) V119901(14)

where vp is the plunger velocity msFrom (14) dN is composed of the following sections(1)Themolar number of gas is released from or dissolved

into oil as pressure varies

119889119873119903 = minus 119885119879119901119904119905119885119904119905119879119904119905119901 (120572 sdot 119889119901)sdot 103119860119901 (119871119891 + 119871119900 minus 119871119900119892) (1 minus 120576119908)119861119900119881119898119900119897

(15)

where pst is the standard pressure pa Zst is the standard natu-ral gas compressibility factorTst is the standard temperature∘C 120572 is the solubility coefficient of natural gas m3 (m3sdotpa)

120576w is the water content Vmol is the molar volume m3 Bo isthe crude oil volume factor

(2)Themolar number of gas with fluid being sucked intothe pump corresponding to the pump pressure is

119889119873119904 = 103119860119901119877119869119889119871119891119881119898119900119897

(16)

where RJ is the transient gas liquid ratio in the pump m3m3(3)Themolar number of gaswith leaked fluid at the pump

pressure is

119889119873V = 103119860119901119877119869119889119871V119881119898119900119897

(17)

6 Mathematical Problems in Engineering

f-f

L

Lf

La

a-a

Figure 7 Process of fluid flows into the pump

where

119877119869 = (1 minus 120576119908) (119877119901 minus 119877119904) 119901119904119905119885119879119879119904119905119885119904119905119901119877119904 = 120572 (119901 minus 119901119904119905)

(18)

where Rp is the production gas oil ratio m3m3 Rs is the gasoil ratio at pump intake m3m3

Fluid Flowing into Pump Model Figure 7 shows the pumpoperation when the gas-liquid flow is drawn into the pumpSection a-a indicates the cross section of standing valve holeSupposing the section f -f is the liquid level at arbitrarytime neglect the process of gas bubbling from fluid byconsidering only the pump gas to occupy the upper spaceof f -f The one-dimensional unsteady flow equation based onBernoulli equation is established to describe the flow state ofpumping liquid In the Bernoulli equation the inertia headloss friction head loss and local head loss are all considered

119901119904120588g + V21198862119892 = 119901120588g + V2

1198912g + (119871119900 minus 119871119900119892 + 119871119891 + 119871V) + ℎ119886

+ ℎV + ℎ119891

(19)

where

ℎ119886 = 1119892 (119871119900 minus 119871119900119892 + 119871119891 + 119871V119860119891119860119901

) 119889V119891119889119905ℎ119891 = 64120583V119891 (119871119900 minus 119871119900119892 + 119871119891)21198632

119889120588119892

ℎV = 11198921205762 (119860119901119860119891

)2 V21198862

V119886 = 119860119901119860119891

V119891

(20)

where 120576 is the flow coefficient of standing valveThen (19) is converted into differential forms

119891 = V119891

V119891 = (1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V119860119891119860119901

)minus1119901119904120588 minus 119901120588+ V2

1198912 [11986021199011198602119891

minus 119860211990112057621198602

119891

minus 64120583 (1198710 minus 119881119900119892119860119901 + 119871119891)1198632119889V119891120588

minus 1] minus 119892(1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V)

(21)

Normally the fluid in the barrel nomatter in which formwill all be discharged except the one in dead spaceThereforethe fluid flowing out of the model with opening travelingvalve is not proposed The pump outlet pressure generatedby several kilometers liquid column is large compared to thehydraulic loss causing the ignorance of hydraulic loss whenfluid flows out Based on the statement mentioned above thenew downhole boundary model is expressed as follows andalso is divided into four phases corresponding to the phasesshown in Figure 2

1198781199051198861199051199061199041 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199042 af ter 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = ((V119891 sdot 119860119901 + 119902) (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)]sdot V119901

Mathematical Problems in Engineering 7

1198781199051198861199051199061199043 before 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199044 af ter 119900119901119890119899119892119894119899 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = 0

(22)

where

119881119898119900119897 = 224119885119879119901119904119905119879119904119905119901 (23)

3 Integrated Numerical Algorithm

For the current SRPS model the surface transmission modelis solved by numerical integration method [25] While thefinite difference method is used to describe the rod stringlongitudinal vibration [2 26 27] Since there is no fixedsolution between surface and downholemodel it needs cycliciteration to handle thewholemodel which is time consuming[25] Considering the interaction within plunger motionfluid flow into pump and pump pressure the improved SRPSmodel has a higher nonlinear degree and thus increasesthe difficulty of solving it In order to reduce the solvingtime the wave equation of rod string longitudinal vibrationis converted into ordinary differential equations by modalsuperposition method Then the surface and downholeboundary model are now in the form of ordinary differentialequations at which the whole model can be solved by Runge-Kutta method directly Besides that solving the rod stringlongitudinal vibration equation is also solving the forcevibration response of rod string where (3) is converted intothe following form

120588119903119860119903

12059721199061205971199052 minus 120588119903119864119903

12059721199061205971199092+ 120583120597119906120597119905 = 120588119903119860119903119891 (119909 119905) (24)

where

119891 (119909 119905) = minus1198892119906119886 (119905)1198891199052 minus 120583120588119903119860119903

119889119906119886 (119905)119889119905 + 119892 (25)

And its solution can be expressed in the function of timeand space by separating the variables

119906 (119909 119905) = infinsum119894=1

Φ119894 (119909) 119902119894 (119905) (26)

Therefore (24) can be expressed asinfinsum119894=1

119860119903120588119903Φ119894 (119909) 119894 (119905)minus infinsum

119894=1

119864119903119860119903

1198892Φ119894 (119909)1198891199092119902119894 (119905) + infinsum

119894=1

120583Φ119894 (119909) 119894 (119905)= 119860119903120588119903119891 (119909 119905)

(27)

Multiply the above equation by Φi(x) and it is integratedalong the rod length Then the following is derived

119860119903120588119903119894 (119905) int119871119903

0Φ119894

2 (119909) 119889119909minus 119864119903119860119903119902119894 (119905) int119871119903

0Φ119894 (119909)Φ119894

10158401015840 (119909) 119889119909+ 120583119894 (119905) int119871119903

0Φ119894

2 (119909) 119889119909= 119860119903120588119903 int119871119903

0Φ119894 (119909) 119891 (119909 119905) 119889119909

(28)

Its mode shapes and natural frequencies are

Φ119894 (119909) = radic 2120588119903119860119903

sin((2119894 minus 1) 1205872119871119903

119909)119901119899119894 = (2119894 minus 1) 120587radic1198641199031205881199032119871119903

(29)

Then (28) can be simplified as follows using mode shapeorthogonality

119894 (119905) + 120583119860119903120588119903 119894 (119905) + ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 119902119894 (119905) = 119865119894 (30)

where

119865119894 = (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903(2119894 minus 1) 120587+ 119865119875119871 (119905) radic 2119860119903120588119903119871119903

sin ((2119894 minus 1) 1205872 )(31)

8 Mathematical Problems in Engineering

Let ith-order forced vibration displacement response andvelocity response be xi1 and xi2 respectivelyThen (30) can beexpressed as the following form

1198941 (119905) = 1199091198942 (119905)1198942 (119905) = minus 1205831198601199031205881199031199091198942 (119905) minus ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 1199091198941 (119905) + 119865119894

(32)

Then suspension point load is derived

119865119877119871 (119905) = 119864119903119860119903

120597119906 (0 119905)120597119909 + 119865119903

= 119864119903119860119903radic 2120588119903119860119903119871119903

119873sum119894=1

(2119894 minus 1) 1205872119871119903

1199091198941 (119905) + 119865119903

(33)

The displacement and velocity of plunger are

119906119901 (119905) = 119906119886 (119905) minus 119906 (119871 119905)= 119906119886 (119905)minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198941 (119905) minus 119894=2lowast119895sum119894=1

1199091198941 (119905))V119901 (119905) = 119886 (119905) minus 120597119906 (119871 119905)120597119905

= 119886 (119905)minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198942 (119905) minus 119894=2lowast119895sum119894=1

1199091198942 (119905))

(34)

Then the integrated numerical model is given as follows

119898 = 120596119898

119898 = 1119868119890 [[2120582119896119872119867120576119888120596119899 [120596119899 minus 119898]12057621198881205962

119899 + [120596119899 minus 119898]2 minus 1119894119887119892120578119887119892

[119878119879119891 (119865119877119871 minus 119865119887) 1205781198961119862119871 minus119872119888 sin (120579 minus 120579120591)] minus 121205962

119898

119889119868119890119889120579119898]]119906119886 (119905) = arccos[1198712

119862 + 1198712119870 minus (119871119877 + 119871119875)22119871119862119871119870

] minus arccos(1198712119862 + 1198712

119871 minus 11987121198752119871119862119871119871

)minus arcsin(119871119877119871119871

sin(2120587 minus 1205790 minus 120579119898119894119887119892

+ arcsin ( 119871119868119871119870

)))11990911 (119905) = 11990912 (119905)12 (119905) = minus 12058311986011990312058811990311990912 (119905) minus ( 1205872119871119903

)2 119864119903120588119903 11990911 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903120587+ (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

21 (119905) = 11990922 (119905)22 (119905) = minus 12058311986011990312058811990311990922 (119905) minus ( 31205872119871119903

)2 119864119903120588119903 11990921 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic21198601199031205881199031198711199033120587minus (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

1198941 (119905) = 1199091198942 (119905)1198942 (119905) = minus 1205831198601199031205881199031199091198942 (119905) minus ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 1199091198941 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903(2119894 minus 1) 120587+ (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

sin ((2119894 minus 1) 1205872 )

Mathematical Problems in Engineering 9

119865119877119871 (119905) = 119864119903119860119903radic 2120588119903119860119903119871119903

119873sum119894=1

(2119894 minus 1) 1205872119871119903

1199091198941 (119905) + 119865119903

119906119901 (119905) = 119906119886 (119905) minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198941 (119905) minus 119894=2lowast119895sum119894=1

1199091198941 (119905))

V119901 (119905) = 119886 (119905) minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198942 (119905) minus 119894=2lowast119895sum119894=1

1199091198942 (119905))1198781199051198861199051199061199041 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199042 119886119891119905119890119903 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = ((V119891 sdot 119860119901 + 119902) (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)]sdot V119901

1198781199051198861199051199061199043 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199044 119886119891119905119890119903 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = 0119889119871119891119889119905 = V119891

119889V119891119889119905 = (1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V119860119891119860119901

)minus1119901119904120588 minus 119901120588 + V21198912 [119860

21199011198602119891

minus 119860211990112057621198602

119891

minus 64120583 (1198710 minus 119881119900119892119860119901 + 119871119891)1198632V119891120588 minus 1]minus 119892(1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V)(35)

4 Optimization Model

41 Optimization Goal As mostly developed oil-field movesinto the mid and late stage and the OWD begins todecline gradually energy-saving production-increasing andreducing load variation as much as possible are particularlyimportant Then we take pump fullness epf suspension pointload amplitude FRLA crank torque standard deviation Mcsdand motor input power average 119875119898 as the optimization goalto build a multitarget model Suspension point load andcrank torque can be solved directly by (35) The motor input

power and pump fullness calculation formula are deduced asfollows

119875119898 = 119872119890119889119898 + 1198750

+ [( 1120578H minus 1)119875119867 minus 1198750](119872119890119889119898119875119867

)2 (36)

119890119901119891 = 119871119891 minus int119905119906

0119902119889119905 minus (119871119900 minus 119871119900119892)119906119901119906

(37)

10 Mathematical Problems in Engineering

where P0 is the no-load power of motor kW PH is the motorrated power kW 120578H is the motor rated efficiency kW upuis the plunger displacement when it is arriving at dead topcenter

So the objective function is

Ω(119890119901119891 (X) 119865119877119871119860 (X) 119872119888119904119889 (X) 119875119898 (X))= 1198701119890119901119891 (X) + 1198702119865119877119871119860 (X) + 1198703119872119888119904119889 (X)+ 1198704119875119898 (X)

(38)

where K1 K2 K3 and K4 are the weight coefficients

42 Design Variables The objective function can beexpressed as the function of the swabbing parameters whenthe SRPS type and oil well basic parameters are confirmedIn this paper the swabbing parameters denote stroke Sstroke frequency ns pump diameter Dd pump depth Lpdcrank balance radius rc and rod string combination (the jthrod string diameter and length are dj and Lj respectively119896 = 1 2 119898) Then the design variables are shown asfollows

X = 119878 119899119904 119863119889 119871119901119889 (119889119895 119871119895 119895 = 1 2 sdot sdot sdot 119898) (39)

43 Constraints

(a) Crank Balance Degree Crank balance degree indicates theload fluctuation to a certain degree and it needs to be kept ata high value

095 le 119872119888119896119906119872119888119896119889

le 1 (40)

where Mcku and Mckd are the maximum crank torque whenplunger is at upstroke and downstroke respectively Nsdotm

(b) Ground Device Carrying Capacity The suspensionpoint load crank torque and motor torque at anytime [0 119879] do not go beyond the allowable rangemax(F119877119871) min(F119877119871) max(119872119888) min(119872119888) max(119872119890119889)min(119872119890119889) for the given type

min (119865119877119871) le 119865119877119871 le max (119865119877119871)min (119872119888119896) le 119872119888119896 le max (119872119888119896)min (119872119890119889) le 119872119890119889 le max (119872119890119889)

(41)

(c) Rod String StrengthThemaximumandminimum stress ofany point x along the rod string does not exceed permissiblestress range in one cycle

[120590min] le 120590119903119904 le [120590max] (42)

(d) Swabbing Parameters Each oil well has different limit onthe swabbing parameters in accordance with the device typeand actual operation hence their allowable variation rangesare adjustable

Figure 8 Dynamometer sensor

44 Optimization Algorithm In summary this multivariableoptimization model is established with nonlinear restrictionand nonlinear objective function So as to seek the bestresults the genetic algorithm is applied to solve it

5 Test and Verification

Surface dynamometer card is a closed graph recordingpolished rod loads versus rod displacement over a SRPScycle which is generally collected and taken as an indexto estimate the operation of SRPS In this paper based onthe dynamometer sensor shown in Figure 8 four test wellsare used to validate the improved SRPS model The oil wellparameters are listed in Table 1 and the simulation and fieldtest results are given in Figure 9 According to the plottedcurves the simulation results are basically consistent withthat in measured and the improved SRPS model is accurateenough to be proposed for engineering practices

6 Dynamic Response Comparison

Pump dynamometer card is a closed graph recording plungerloads versus plunger displacement over a SRPS cycle It isdetermined by multifactors such as stroke stroke frequencypump diameter pump depth dynamic liquid level and theother oil well parameters The difference between the currentSRPS model and the improved SRPS model proposed in thispaper is whether to consider the PFFP In view of this theoil operating status is divided into two forms as follows(1) the fluid always keeps pace with the plunger when itis being sucked into the pump (2) the fluid is unable tofollowwith the plungerThen two oil wells are selected well1with sufficient OWD and well2 with insufficient OWDFigure 10 describes the plunger and fluid velocity duringupstroke based on the improved SRPS model It can beconcluded that when the well has sufficient OWD and thefluid possesses good ability of keeping pace with the plungerwhen the well has insufficient OWD the fluid velocity lagsbehind the one of plunger at the beginning whereas it is

Mathematical Problems in Engineering 11

Table 1 Oil well basic parameters

Well 1 2 3 4Motor type YD280S-8 Y250M-6 Y250M-6 YD280S-6Pumping unit type CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HBStroke length (m) 3 3 3 3Stroke frequency (minminus1) 35 3 4 6Pump diameter (mm) 57 44 38 44Pump clearance level 1 2 2 1Pump depth (m) 890 1377 1138 1430Sucker-rod string (mm timesm) 25lowast890 19lowast673+22lowast704 19lowast672+22lowast466 19lowast720+22lowast710middle depth of reservoir (m) 1000 1492 1569 1600Crude oil density (kgm3) 795 857 857 850Water content () 95 97 92 98Dynamic liquid level (m) 880 1347 757 1340Casing pressure (Pa) 02 03 03 02Oil pressure (Pa) 03 03 03 02Fluid dynamic viscosity (Pasdots) 0006 0007 0007 0006Gas oil ratio (m3 m3) 20 19 40 80Plunger length (m) 12 12 12 12Clearance length (m) 05 05 05 05

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(a) Well 1

1 2 30Polished rod displacement (m)

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

simulationmeasured

(b) Well 2

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(c) Well 3

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(d) Well 4Figure 9 Suspension point dynamometer card comparison between the simulated and measured

12 Mathematical Problems in Engineering

FluidPlunger

minus05

00

05

10

15Ve

loci

ty(

ms

)

2 4 60Time (s)

(a) Well 1

minus02

00

02

04

06

Velo

city

(m)

4 8 120Time (s)

FluidPlunger

(b) Well 2

Figure 10 Plunger and fluid velocity

Current SRPS model 970Improved SRPS model 981

minus10

0

10

20

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(a) Well 1

Current SRPS model 724Improved SRPS model 775

minus10

0

10

20

40

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(b) Well 2

Figure 11 Comparison of pump dynamometer card

faster after a certain time period Then the comparisons ofpump dynamometer card are executed by the two modelsmeanwhile their individual pump fullness result is given atthe end of the legend (Figure 11)

Pump dynamometer card is very important method andnormally used to diagnose the pump operations particularlywhose shape is more similar to a rectangle the pump iscloser to the fullness [28] Hence from Figure 11 it canbe known that the pump fullness of current SRPS modelis higher than that of the improved one depending on thequalitative judgment and this conclusion is consistent withthe quantitative calculation result meanwhile this gap forthe well 2 is bigger than well 1 According the above

description we can know that the pump fullness calculationresult is on the high side if the PFFP is not considered andthis phenomenon will becomemore obvious for the well withinsufficient OWD

The load presented by the left upper right and lowerborderline of pump dynamometer card is the results of pumpmoving from phase 1 to 4 in turn Therefore its upperborderline describes the pump load when fluid is pumpedinto the barrel From Figure 11(a) it can be found thatthe upper borderline load simulated by the improved SRPSmodel is significantly larger than the one of current ForFigure 11(b) this difference can be neglected Based on (1)and (2) the pump pressure keeps constant as ps when fluid is

Mathematical Problems in Engineering 13

Before optimizationAfter optimization

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

1 2 30Polished rod displacement (m)

(a) Suspension point dynamometer card

minus10

0

10

20

30

40

Cran

k to

rque

(kNmiddotm

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(b) Crank torque

0

5

10

15

20

Mot

or in

put p

ower

(kW

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(c) Motor input power

Figure 12 Dynamic response comparison before and after optimization

sucked into the pump and then the corresponding pump loadis a fixed value Due to the new model takes into account ofPFFP a pressure drop will be brought which will vary withthe fluid velocity Combined with (1) this pressure drop willlead to the pump load increases Figure 10 shows that thefluid velocity of well 1 is larger than that of well 2 andthis is the reason why the difference of upper borderline loadbetween the two models is obvious This illustrates the erroron the upper borderline load of pump dynamometer carddepending on the velocity at which the fluid flows into thepump

7 Optimization Test

Based on the above models a simulation and optimizationsoftware is developed byMATLABOnewell with insufficient

OWD is tested Its original swabbing parameters are asfollows stroke S is 3 m stroke frequency ns is 3 minminus1pump diameter Dd is 57mm pump depth Lpd is 900m crankbalance radius rc is 12 m and rod string combination djtimesLj is 25 mm times 524 m+22 mm times 376 m And its swabbingparameters after optimizing are as follows stroke S is 3 mstroke frequency ns is 25 minminus1 pump diameterDd is 57mmpump depth Lpd is 990 m crank balance radius rc is 09 mand rod string combination djtimes Lj is 22 mm times 426 m+19 mmtimes 564 m The comparisons before and after optimization areshown in Figure 12 and Table 2

From the above comparison results some conclusions areobtained

(1) After optimization the maximum and minimumof suspension point load are all decreased and the load

14 Mathematical Problems in Engineering

Table 2 Specific parameters comparison before and after optimization

Suspension point load Crank torque Motor input power PumpfullnessMaximum Minimum Amplitude Standard

deviationBalancedegree Maximum Minimum mean

Beforeoptimization 555 kN 190 kN 365 kN 107 kN 854 159 kW 09 kW 62 kW 725

Afteroptimization 441 kN 132 kN 309 kN 94 kN 978 106 kW 10 kW 54 kW 908

darr 205 darr 305 darr 153 darr 121 uarr 145 darr 333 uarr 111 darr 129 uarr 252 amplitude is lowered by 153 It can contribute to enhancethe rod string life and prolong the maintenance period

(2) After optimization the standard deviation of cranktorque is reduced by 121 and its balance degree is raisedby 145 It illustrates that the load torque fluctuation is cutdown which canminimize damage to transmission parts andimprove the motor efficiency

(3) After optimization the pump fullness is improvedby 252 It is conducive to improve pump efficiency andproduction

(4) After optimization it plays a role of peak shavingand valley filling for motor input power and extends theoperational life meanwhile power saving rate is 129

8 Conclusions

(1) In this article an improved SRPS model is presentedconsidering the couple effect of pumping fluid and plungermotion on the dynamic response of SRPS instead of theexisting models that assume the pumping fluid volumeis always equal to plunger travelling volume The Runge-Kutta method is applied to solve the whole system modeltrough transforming the rod string longitudinal vibrationequation into ordinary differential equations And the SRPSmodelrsquos precision has been validated by adopting surfacedynamometer card

(2) Two oil wells are served to compare the differencebetween the current SRPS model and the improved oneThe results indicate that the current SRPS model is relativelylow in calculating pump fullness and this gap will increasewith the reduction of OWD the influence of fluid flowinginto the pump on pump load cannot be ignored when thefluid velocity is high Therefore the PFFP is necessary to beconsidered in order to improve the simulation accuracy

(3) On the basis of the improved SRPS model a multitar-get optimization model is proposed in purpose of improvingproduction decreasing load fluctuation and saving energy Bycomparison the optimal scheme can achieve the decreasingof maximum and minimum suspension point load cranktorque fluctuation and energy consumption as well asimproving the system balance degree and pump fullness Insummary it improves the dynamic behavior of SRPS

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

National Natural Science Foundation of China (Grantno 51174175) China Scholarship Council (Grant no201708130108) Hebei Natural Science Foundation (Grant noE201703101) are acknowledged

References

[1] T A Aliev A H Rzayev G A Guluyev T A Alizada and NE Rzayeva ldquoRobust technology and system for management ofsucker rod pumping units in oil wellsrdquoMechanical Systems andSignal Processing vol 99 pp 47ndash56 2018

[2] S Gibbs ldquoPredicting the Behavior of Sucker-Rod PumpingSystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 1963

[3] D R Doty and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo SPE Journal vol 23 no 1 pp 33ndash41 1983

[4] I N Shardakov and I NWasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[5] S G Gibbs ldquoComputing gearbox torque and motor loading forbeam pumping units with consideration of inertia effectsrdquo SPEJ vol 27 pp 1153ndash1159 1975

[6] J Svinos ldquoExact Kinematic Analysis of Pumping Unitsrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionSan Francisco California 1983

[7] D Schafer and J Jennings ldquoAn Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical Modelsrdquoin Proceedings of the SPE Annual Technical Conference andExhibition pp 27ndash30 Dallas Texas 1987

[8] G Takacs Sucker-Rod PumpingManual Pennwell Books Tulsa2003

[9] G W Wang S S Rahman and G Y Yang ldquoAn improvedmodel for the sucker rod pumping systemrdquo in Proceedings of the11thAustralasian FluidMechanics Conference pp 14ndash18HobartAustralia December 1992

[10] S D L Lekia andR D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystemsrdquo SPE Production amp Facilities vol 10 no 1 pp 26ndash331995

[11] J Lea and P Pattillo ldquoInterpretation of Calculated Forces onSucker Rodsrdquo SPE Production amp Facilities vol 10 no 01 pp 41ndash45 1995

Mathematical Problems in Engineering 15

[12] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[13] L Guo-hua H Shun-li Y Zhi et al ldquoA prediction model fora new deep-rod pumping systemrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 75ndash80 2011

[14] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[15] O Becerra J Gamboa and F Kenyery ldquoModelling a DoublePiston Pumprdquo in Proceedings of the SPE International ThermalOperations and Heavy Oil Symposium and International Hor-izontal Well Technology Conference Calgary Alberta Canada2002

[16] A L Podio J Gomez A J Mansure et al ldquoLaboratoryinstrumented sucker-rod pumprdquo J Pet Technol vol 53 no 05pp 104ndash113 2003

[17] Z H Gu H Q Peng and H Y Geng ldquoAnalysis and mea-surement of gas effect on pumping efficiencyrdquoChina PetroleumMachinery vol 34 no 02 pp 64ndash69 2006

[18] M Xing ldquoResponse analysis of longitudinal vibration of suckerrod string considering rod bucklingrdquo Advances in EngineeringSoftware vol 99 pp 49ndash58 2016

[19] S Miska A Sharaki and J M Rajtar ldquoA simple model forcomputer-aided optimization and design of sucker-rod pump-ing systemsrdquo Journal of Petroleum Science and Engineering vol17 no 3-4 pp 303ndash312 1997

[20] L S Firu T Chelu and C Militaru-Petre ldquoAmodern approachto the optimum design of sucker-rod pumping systemrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash9 DenverColorado 2003

[21] X F Liu and Y G Qi ldquoA modern approach to the selectionof sucker rod pumping systems in CBM wellsrdquo Journal ofPetroleum Science and Engineering vol 76 no 3-4 pp 100ndash1082011

[22] S M Dong N N Feng and Z J Ma ldquoSimulating maximumof system efficiency of rod pumping wellsrdquo Journal of SystemSimulation vol 20 no 13 pp 3533ndash3537 2008

[23] M Xing and S Dong ldquoA New Simulation Model for a Beam-Pumping System Applied in Energy Saving and Resource-Consumption Reductionrdquo SPE Production amp Operations vol30 no 02 pp 130ndash140 2015

[24] Y XWu Y Li andH LiuElectric Motor andDriving ChemicalIndustry Press Beijing 2008

[25] S M Dong Computer Simulation of Dynamic Parameters ofRod Pumping System Optimization Petroleum Industry PressBeijing Chinese 2003

[26] F Yavuz J F Lea J C Cox et al ldquoWave equation simulationof fluid pound and gas interferencerdquo in Proceedings of the SPEProduction Operations Symposium pp 16ndash19 Oklahoma CityOklahoma April 2005

[27] D Y Wang and H Z Liu ldquoDynamic modeling and analysis ofsucker rod pumping system in a directional well Mechanismand Machine Sciencerdquo in Proceedings of the ASIAN MMS 2016amp CCMMS pp 1115ndash1127 2017

[28] F M A Barreto M Tygel A F Rocha and C K MorookaldquoAutomatic downhole card generation and classificationrdquo inProceedings of the SPE Annual Technical Conference pp 6ndash9Denver Colorado 1996

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Page 5: An Improved Sucker Rod Pumping System Model and Swabbing ...downloads.hindawi.com/journals/mpe/2018/4746210.pdf · An Improved Sucker Rod Pumping System Model and Swabbing Parameters

Mathematical Problems in Engineering 5

where ibg is the transmission ratio of the belt-reducinggear system 120576p is the transmission efficiency from crank tosuspension point STf is torque factorm FRL is the suspensionpoint load N Fb is the structural unbalance weight NMc isthe maximum balancing torque of the crank Nsdotm

The displacement of suspension point can be expressedas motor angle based on the geometry relation shown inFigure 6 [6] The surface boundary motion model is

119906119886 (119905) = arccos [1198712119862 + 1198712

119870 minus (119871119877 + 119871119875)22119871119862119871119870

]minus arccos(1198712

119862 + 1198712119871 minus 1198712

1198752119871119862119871119871

) minus arcsin(119871119877119871119871

sdot sin(2120587 minus 1205790 minus 120579119898119894119887119892

+ arcsin( 119871119868119871119870

)))(10)

23 Downhole Boundary Model This new model is animproved method from the current boundary models con-sidering the interaction among pump pressure fluid flow intopump gas dissolution and evolution pressure drop due tofluid gravitational potential energy inertia head loss frictionhead loss and local head loss The new boundary modelconsists of pressure variation model and fluid flow into pumpmodel Some assumptions are made for building this model

(1) Bottom dead point of plunger as the origin position(2) The pump inlet pressure ps and outlet pressure pd are

kept constant for one cycle of SRPS operation

Pressure Variation Model The gas state equations are built asfollows ( (11) and (12)) when the displacement of plunger is upor up+dup

119901 [(119906119901 + 119871119900119892 minus 119871119891 minus 119871119896)119860119901] = 119885119873119876119879 (11)

(119901 + 119889119901)sdot [(119906119901 + 119889119906119901 + 119871119900119892 minus 119871119891 minus 119871119896 minus 119889119871119891 minus 119889119871119896)119860119901]= (119885 + 120597119885120597119901 119889119901) (119873 + 119889119873)119876119879

(12)

where N is the gas molar number mol Q is the natural gasconstant J(molsdotK) T is the temperature in pump barrel ∘CZ is the natural gas compressibility factor Lk is the equivalentlength of pump leakage m Lf is the liquid level in the pumpm

Dividing (11) by (12) and ignoring the second order smallquantities then the variation of pump pressure with plungerdisplacement is obtained

119889119901119889119906119901

= (1119873) (119889119873119889119906119901) + (1 (119906119901 + 119871119900119892 minus 119871119891 minus 119871119896)) (119889 (119871119891 + 119871119896) 119889119906119901 minus 1)119901minus1 minus (1119885) (119889119885119889119901) (13)

The above equation is converted into a time varyingfunction with the purpose of facilitating it with the rod stringlongitudinal vibration equation

119889119901119889119905 = (1119873) (119889119873119889119905) (1V119901) + (1 (119906119901 + 119871119900119892 minus 119871119891 minus 119871119896)) ((119889 (119871119891 + 119871119896) 119889119905) (1V119901) minus 1)119901minus1 minus (1119885) (119889119885119889119901) V119901(14)

where vp is the plunger velocity msFrom (14) dN is composed of the following sections(1)Themolar number of gas is released from or dissolved

into oil as pressure varies

119889119873119903 = minus 119885119879119901119904119905119885119904119905119879119904119905119901 (120572 sdot 119889119901)sdot 103119860119901 (119871119891 + 119871119900 minus 119871119900119892) (1 minus 120576119908)119861119900119881119898119900119897

(15)

where pst is the standard pressure pa Zst is the standard natu-ral gas compressibility factorTst is the standard temperature∘C 120572 is the solubility coefficient of natural gas m3 (m3sdotpa)

120576w is the water content Vmol is the molar volume m3 Bo isthe crude oil volume factor

(2)Themolar number of gas with fluid being sucked intothe pump corresponding to the pump pressure is

119889119873119904 = 103119860119901119877119869119889119871119891119881119898119900119897

(16)

where RJ is the transient gas liquid ratio in the pump m3m3(3)Themolar number of gaswith leaked fluid at the pump

pressure is

119889119873V = 103119860119901119877119869119889119871V119881119898119900119897

(17)

6 Mathematical Problems in Engineering

f-f

L

Lf

La

a-a

Figure 7 Process of fluid flows into the pump

where

119877119869 = (1 minus 120576119908) (119877119901 minus 119877119904) 119901119904119905119885119879119879119904119905119885119904119905119901119877119904 = 120572 (119901 minus 119901119904119905)

(18)

where Rp is the production gas oil ratio m3m3 Rs is the gasoil ratio at pump intake m3m3

Fluid Flowing into Pump Model Figure 7 shows the pumpoperation when the gas-liquid flow is drawn into the pumpSection a-a indicates the cross section of standing valve holeSupposing the section f -f is the liquid level at arbitrarytime neglect the process of gas bubbling from fluid byconsidering only the pump gas to occupy the upper spaceof f -f The one-dimensional unsteady flow equation based onBernoulli equation is established to describe the flow state ofpumping liquid In the Bernoulli equation the inertia headloss friction head loss and local head loss are all considered

119901119904120588g + V21198862119892 = 119901120588g + V2

1198912g + (119871119900 minus 119871119900119892 + 119871119891 + 119871V) + ℎ119886

+ ℎV + ℎ119891

(19)

where

ℎ119886 = 1119892 (119871119900 minus 119871119900119892 + 119871119891 + 119871V119860119891119860119901

) 119889V119891119889119905ℎ119891 = 64120583V119891 (119871119900 minus 119871119900119892 + 119871119891)21198632

119889120588119892

ℎV = 11198921205762 (119860119901119860119891

)2 V21198862

V119886 = 119860119901119860119891

V119891

(20)

where 120576 is the flow coefficient of standing valveThen (19) is converted into differential forms

119891 = V119891

V119891 = (1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V119860119891119860119901

)minus1119901119904120588 minus 119901120588+ V2

1198912 [11986021199011198602119891

minus 119860211990112057621198602

119891

minus 64120583 (1198710 minus 119881119900119892119860119901 + 119871119891)1198632119889V119891120588

minus 1] minus 119892(1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V)

(21)

Normally the fluid in the barrel nomatter in which formwill all be discharged except the one in dead spaceThereforethe fluid flowing out of the model with opening travelingvalve is not proposed The pump outlet pressure generatedby several kilometers liquid column is large compared to thehydraulic loss causing the ignorance of hydraulic loss whenfluid flows out Based on the statement mentioned above thenew downhole boundary model is expressed as follows andalso is divided into four phases corresponding to the phasesshown in Figure 2

1198781199051198861199051199061199041 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199042 af ter 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = ((V119891 sdot 119860119901 + 119902) (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)]sdot V119901

Mathematical Problems in Engineering 7

1198781199051198861199051199061199043 before 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199044 af ter 119900119901119890119899119892119894119899 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = 0

(22)

where

119881119898119900119897 = 224119885119879119901119904119905119879119904119905119901 (23)

3 Integrated Numerical Algorithm

For the current SRPS model the surface transmission modelis solved by numerical integration method [25] While thefinite difference method is used to describe the rod stringlongitudinal vibration [2 26 27] Since there is no fixedsolution between surface and downholemodel it needs cycliciteration to handle thewholemodel which is time consuming[25] Considering the interaction within plunger motionfluid flow into pump and pump pressure the improved SRPSmodel has a higher nonlinear degree and thus increasesthe difficulty of solving it In order to reduce the solvingtime the wave equation of rod string longitudinal vibrationis converted into ordinary differential equations by modalsuperposition method Then the surface and downholeboundary model are now in the form of ordinary differentialequations at which the whole model can be solved by Runge-Kutta method directly Besides that solving the rod stringlongitudinal vibration equation is also solving the forcevibration response of rod string where (3) is converted intothe following form

120588119903119860119903

12059721199061205971199052 minus 120588119903119864119903

12059721199061205971199092+ 120583120597119906120597119905 = 120588119903119860119903119891 (119909 119905) (24)

where

119891 (119909 119905) = minus1198892119906119886 (119905)1198891199052 minus 120583120588119903119860119903

119889119906119886 (119905)119889119905 + 119892 (25)

And its solution can be expressed in the function of timeand space by separating the variables

119906 (119909 119905) = infinsum119894=1

Φ119894 (119909) 119902119894 (119905) (26)

Therefore (24) can be expressed asinfinsum119894=1

119860119903120588119903Φ119894 (119909) 119894 (119905)minus infinsum

119894=1

119864119903119860119903

1198892Φ119894 (119909)1198891199092119902119894 (119905) + infinsum

119894=1

120583Φ119894 (119909) 119894 (119905)= 119860119903120588119903119891 (119909 119905)

(27)

Multiply the above equation by Φi(x) and it is integratedalong the rod length Then the following is derived

119860119903120588119903119894 (119905) int119871119903

0Φ119894

2 (119909) 119889119909minus 119864119903119860119903119902119894 (119905) int119871119903

0Φ119894 (119909)Φ119894

10158401015840 (119909) 119889119909+ 120583119894 (119905) int119871119903

0Φ119894

2 (119909) 119889119909= 119860119903120588119903 int119871119903

0Φ119894 (119909) 119891 (119909 119905) 119889119909

(28)

Its mode shapes and natural frequencies are

Φ119894 (119909) = radic 2120588119903119860119903

sin((2119894 minus 1) 1205872119871119903

119909)119901119899119894 = (2119894 minus 1) 120587radic1198641199031205881199032119871119903

(29)

Then (28) can be simplified as follows using mode shapeorthogonality

119894 (119905) + 120583119860119903120588119903 119894 (119905) + ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 119902119894 (119905) = 119865119894 (30)

where

119865119894 = (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903(2119894 minus 1) 120587+ 119865119875119871 (119905) radic 2119860119903120588119903119871119903

sin ((2119894 minus 1) 1205872 )(31)

8 Mathematical Problems in Engineering

Let ith-order forced vibration displacement response andvelocity response be xi1 and xi2 respectivelyThen (30) can beexpressed as the following form

1198941 (119905) = 1199091198942 (119905)1198942 (119905) = minus 1205831198601199031205881199031199091198942 (119905) minus ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 1199091198941 (119905) + 119865119894

(32)

Then suspension point load is derived

119865119877119871 (119905) = 119864119903119860119903

120597119906 (0 119905)120597119909 + 119865119903

= 119864119903119860119903radic 2120588119903119860119903119871119903

119873sum119894=1

(2119894 minus 1) 1205872119871119903

1199091198941 (119905) + 119865119903

(33)

The displacement and velocity of plunger are

119906119901 (119905) = 119906119886 (119905) minus 119906 (119871 119905)= 119906119886 (119905)minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198941 (119905) minus 119894=2lowast119895sum119894=1

1199091198941 (119905))V119901 (119905) = 119886 (119905) minus 120597119906 (119871 119905)120597119905

= 119886 (119905)minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198942 (119905) minus 119894=2lowast119895sum119894=1

1199091198942 (119905))

(34)

Then the integrated numerical model is given as follows

119898 = 120596119898

119898 = 1119868119890 [[2120582119896119872119867120576119888120596119899 [120596119899 minus 119898]12057621198881205962

119899 + [120596119899 minus 119898]2 minus 1119894119887119892120578119887119892

[119878119879119891 (119865119877119871 minus 119865119887) 1205781198961119862119871 minus119872119888 sin (120579 minus 120579120591)] minus 121205962

119898

119889119868119890119889120579119898]]119906119886 (119905) = arccos[1198712

119862 + 1198712119870 minus (119871119877 + 119871119875)22119871119862119871119870

] minus arccos(1198712119862 + 1198712

119871 minus 11987121198752119871119862119871119871

)minus arcsin(119871119877119871119871

sin(2120587 minus 1205790 minus 120579119898119894119887119892

+ arcsin ( 119871119868119871119870

)))11990911 (119905) = 11990912 (119905)12 (119905) = minus 12058311986011990312058811990311990912 (119905) minus ( 1205872119871119903

)2 119864119903120588119903 11990911 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903120587+ (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

21 (119905) = 11990922 (119905)22 (119905) = minus 12058311986011990312058811990311990922 (119905) minus ( 31205872119871119903

)2 119864119903120588119903 11990921 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic21198601199031205881199031198711199033120587minus (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

1198941 (119905) = 1199091198942 (119905)1198942 (119905) = minus 1205831198601199031205881199031199091198942 (119905) minus ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 1199091198941 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903(2119894 minus 1) 120587+ (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

sin ((2119894 minus 1) 1205872 )

Mathematical Problems in Engineering 9

119865119877119871 (119905) = 119864119903119860119903radic 2120588119903119860119903119871119903

119873sum119894=1

(2119894 minus 1) 1205872119871119903

1199091198941 (119905) + 119865119903

119906119901 (119905) = 119906119886 (119905) minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198941 (119905) minus 119894=2lowast119895sum119894=1

1199091198941 (119905))

V119901 (119905) = 119886 (119905) minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198942 (119905) minus 119894=2lowast119895sum119894=1

1199091198942 (119905))1198781199051198861199051199061199041 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199042 119886119891119905119890119903 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = ((V119891 sdot 119860119901 + 119902) (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)]sdot V119901

1198781199051198861199051199061199043 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199044 119886119891119905119890119903 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = 0119889119871119891119889119905 = V119891

119889V119891119889119905 = (1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V119860119891119860119901

)minus1119901119904120588 minus 119901120588 + V21198912 [119860

21199011198602119891

minus 119860211990112057621198602

119891

minus 64120583 (1198710 minus 119881119900119892119860119901 + 119871119891)1198632V119891120588 minus 1]minus 119892(1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V)(35)

4 Optimization Model

41 Optimization Goal As mostly developed oil-field movesinto the mid and late stage and the OWD begins todecline gradually energy-saving production-increasing andreducing load variation as much as possible are particularlyimportant Then we take pump fullness epf suspension pointload amplitude FRLA crank torque standard deviation Mcsdand motor input power average 119875119898 as the optimization goalto build a multitarget model Suspension point load andcrank torque can be solved directly by (35) The motor input

power and pump fullness calculation formula are deduced asfollows

119875119898 = 119872119890119889119898 + 1198750

+ [( 1120578H minus 1)119875119867 minus 1198750](119872119890119889119898119875119867

)2 (36)

119890119901119891 = 119871119891 minus int119905119906

0119902119889119905 minus (119871119900 minus 119871119900119892)119906119901119906

(37)

10 Mathematical Problems in Engineering

where P0 is the no-load power of motor kW PH is the motorrated power kW 120578H is the motor rated efficiency kW upuis the plunger displacement when it is arriving at dead topcenter

So the objective function is

Ω(119890119901119891 (X) 119865119877119871119860 (X) 119872119888119904119889 (X) 119875119898 (X))= 1198701119890119901119891 (X) + 1198702119865119877119871119860 (X) + 1198703119872119888119904119889 (X)+ 1198704119875119898 (X)

(38)

where K1 K2 K3 and K4 are the weight coefficients

42 Design Variables The objective function can beexpressed as the function of the swabbing parameters whenthe SRPS type and oil well basic parameters are confirmedIn this paper the swabbing parameters denote stroke Sstroke frequency ns pump diameter Dd pump depth Lpdcrank balance radius rc and rod string combination (the jthrod string diameter and length are dj and Lj respectively119896 = 1 2 119898) Then the design variables are shown asfollows

X = 119878 119899119904 119863119889 119871119901119889 (119889119895 119871119895 119895 = 1 2 sdot sdot sdot 119898) (39)

43 Constraints

(a) Crank Balance Degree Crank balance degree indicates theload fluctuation to a certain degree and it needs to be kept ata high value

095 le 119872119888119896119906119872119888119896119889

le 1 (40)

where Mcku and Mckd are the maximum crank torque whenplunger is at upstroke and downstroke respectively Nsdotm

(b) Ground Device Carrying Capacity The suspensionpoint load crank torque and motor torque at anytime [0 119879] do not go beyond the allowable rangemax(F119877119871) min(F119877119871) max(119872119888) min(119872119888) max(119872119890119889)min(119872119890119889) for the given type

min (119865119877119871) le 119865119877119871 le max (119865119877119871)min (119872119888119896) le 119872119888119896 le max (119872119888119896)min (119872119890119889) le 119872119890119889 le max (119872119890119889)

(41)

(c) Rod String StrengthThemaximumandminimum stress ofany point x along the rod string does not exceed permissiblestress range in one cycle

[120590min] le 120590119903119904 le [120590max] (42)

(d) Swabbing Parameters Each oil well has different limit onthe swabbing parameters in accordance with the device typeand actual operation hence their allowable variation rangesare adjustable

Figure 8 Dynamometer sensor

44 Optimization Algorithm In summary this multivariableoptimization model is established with nonlinear restrictionand nonlinear objective function So as to seek the bestresults the genetic algorithm is applied to solve it

5 Test and Verification

Surface dynamometer card is a closed graph recordingpolished rod loads versus rod displacement over a SRPScycle which is generally collected and taken as an indexto estimate the operation of SRPS In this paper based onthe dynamometer sensor shown in Figure 8 four test wellsare used to validate the improved SRPS model The oil wellparameters are listed in Table 1 and the simulation and fieldtest results are given in Figure 9 According to the plottedcurves the simulation results are basically consistent withthat in measured and the improved SRPS model is accurateenough to be proposed for engineering practices

6 Dynamic Response Comparison

Pump dynamometer card is a closed graph recording plungerloads versus plunger displacement over a SRPS cycle It isdetermined by multifactors such as stroke stroke frequencypump diameter pump depth dynamic liquid level and theother oil well parameters The difference between the currentSRPS model and the improved SRPS model proposed in thispaper is whether to consider the PFFP In view of this theoil operating status is divided into two forms as follows(1) the fluid always keeps pace with the plunger when itis being sucked into the pump (2) the fluid is unable tofollowwith the plungerThen two oil wells are selected well1with sufficient OWD and well2 with insufficient OWDFigure 10 describes the plunger and fluid velocity duringupstroke based on the improved SRPS model It can beconcluded that when the well has sufficient OWD and thefluid possesses good ability of keeping pace with the plungerwhen the well has insufficient OWD the fluid velocity lagsbehind the one of plunger at the beginning whereas it is

Mathematical Problems in Engineering 11

Table 1 Oil well basic parameters

Well 1 2 3 4Motor type YD280S-8 Y250M-6 Y250M-6 YD280S-6Pumping unit type CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HBStroke length (m) 3 3 3 3Stroke frequency (minminus1) 35 3 4 6Pump diameter (mm) 57 44 38 44Pump clearance level 1 2 2 1Pump depth (m) 890 1377 1138 1430Sucker-rod string (mm timesm) 25lowast890 19lowast673+22lowast704 19lowast672+22lowast466 19lowast720+22lowast710middle depth of reservoir (m) 1000 1492 1569 1600Crude oil density (kgm3) 795 857 857 850Water content () 95 97 92 98Dynamic liquid level (m) 880 1347 757 1340Casing pressure (Pa) 02 03 03 02Oil pressure (Pa) 03 03 03 02Fluid dynamic viscosity (Pasdots) 0006 0007 0007 0006Gas oil ratio (m3 m3) 20 19 40 80Plunger length (m) 12 12 12 12Clearance length (m) 05 05 05 05

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(a) Well 1

1 2 30Polished rod displacement (m)

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

simulationmeasured

(b) Well 2

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(c) Well 3

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(d) Well 4Figure 9 Suspension point dynamometer card comparison between the simulated and measured

12 Mathematical Problems in Engineering

FluidPlunger

minus05

00

05

10

15Ve

loci

ty(

ms

)

2 4 60Time (s)

(a) Well 1

minus02

00

02

04

06

Velo

city

(m)

4 8 120Time (s)

FluidPlunger

(b) Well 2

Figure 10 Plunger and fluid velocity

Current SRPS model 970Improved SRPS model 981

minus10

0

10

20

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(a) Well 1

Current SRPS model 724Improved SRPS model 775

minus10

0

10

20

40

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(b) Well 2

Figure 11 Comparison of pump dynamometer card

faster after a certain time period Then the comparisons ofpump dynamometer card are executed by the two modelsmeanwhile their individual pump fullness result is given atthe end of the legend (Figure 11)

Pump dynamometer card is very important method andnormally used to diagnose the pump operations particularlywhose shape is more similar to a rectangle the pump iscloser to the fullness [28] Hence from Figure 11 it canbe known that the pump fullness of current SRPS modelis higher than that of the improved one depending on thequalitative judgment and this conclusion is consistent withthe quantitative calculation result meanwhile this gap forthe well 2 is bigger than well 1 According the above

description we can know that the pump fullness calculationresult is on the high side if the PFFP is not considered andthis phenomenon will becomemore obvious for the well withinsufficient OWD

The load presented by the left upper right and lowerborderline of pump dynamometer card is the results of pumpmoving from phase 1 to 4 in turn Therefore its upperborderline describes the pump load when fluid is pumpedinto the barrel From Figure 11(a) it can be found thatthe upper borderline load simulated by the improved SRPSmodel is significantly larger than the one of current ForFigure 11(b) this difference can be neglected Based on (1)and (2) the pump pressure keeps constant as ps when fluid is

Mathematical Problems in Engineering 13

Before optimizationAfter optimization

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

1 2 30Polished rod displacement (m)

(a) Suspension point dynamometer card

minus10

0

10

20

30

40

Cran

k to

rque

(kNmiddotm

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(b) Crank torque

0

5

10

15

20

Mot

or in

put p

ower

(kW

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(c) Motor input power

Figure 12 Dynamic response comparison before and after optimization

sucked into the pump and then the corresponding pump loadis a fixed value Due to the new model takes into account ofPFFP a pressure drop will be brought which will vary withthe fluid velocity Combined with (1) this pressure drop willlead to the pump load increases Figure 10 shows that thefluid velocity of well 1 is larger than that of well 2 andthis is the reason why the difference of upper borderline loadbetween the two models is obvious This illustrates the erroron the upper borderline load of pump dynamometer carddepending on the velocity at which the fluid flows into thepump

7 Optimization Test

Based on the above models a simulation and optimizationsoftware is developed byMATLABOnewell with insufficient

OWD is tested Its original swabbing parameters are asfollows stroke S is 3 m stroke frequency ns is 3 minminus1pump diameter Dd is 57mm pump depth Lpd is 900m crankbalance radius rc is 12 m and rod string combination djtimesLj is 25 mm times 524 m+22 mm times 376 m And its swabbingparameters after optimizing are as follows stroke S is 3 mstroke frequency ns is 25 minminus1 pump diameterDd is 57mmpump depth Lpd is 990 m crank balance radius rc is 09 mand rod string combination djtimes Lj is 22 mm times 426 m+19 mmtimes 564 m The comparisons before and after optimization areshown in Figure 12 and Table 2

From the above comparison results some conclusions areobtained

(1) After optimization the maximum and minimumof suspension point load are all decreased and the load

14 Mathematical Problems in Engineering

Table 2 Specific parameters comparison before and after optimization

Suspension point load Crank torque Motor input power PumpfullnessMaximum Minimum Amplitude Standard

deviationBalancedegree Maximum Minimum mean

Beforeoptimization 555 kN 190 kN 365 kN 107 kN 854 159 kW 09 kW 62 kW 725

Afteroptimization 441 kN 132 kN 309 kN 94 kN 978 106 kW 10 kW 54 kW 908

darr 205 darr 305 darr 153 darr 121 uarr 145 darr 333 uarr 111 darr 129 uarr 252 amplitude is lowered by 153 It can contribute to enhancethe rod string life and prolong the maintenance period

(2) After optimization the standard deviation of cranktorque is reduced by 121 and its balance degree is raisedby 145 It illustrates that the load torque fluctuation is cutdown which canminimize damage to transmission parts andimprove the motor efficiency

(3) After optimization the pump fullness is improvedby 252 It is conducive to improve pump efficiency andproduction

(4) After optimization it plays a role of peak shavingand valley filling for motor input power and extends theoperational life meanwhile power saving rate is 129

8 Conclusions

(1) In this article an improved SRPS model is presentedconsidering the couple effect of pumping fluid and plungermotion on the dynamic response of SRPS instead of theexisting models that assume the pumping fluid volumeis always equal to plunger travelling volume The Runge-Kutta method is applied to solve the whole system modeltrough transforming the rod string longitudinal vibrationequation into ordinary differential equations And the SRPSmodelrsquos precision has been validated by adopting surfacedynamometer card

(2) Two oil wells are served to compare the differencebetween the current SRPS model and the improved oneThe results indicate that the current SRPS model is relativelylow in calculating pump fullness and this gap will increasewith the reduction of OWD the influence of fluid flowinginto the pump on pump load cannot be ignored when thefluid velocity is high Therefore the PFFP is necessary to beconsidered in order to improve the simulation accuracy

(3) On the basis of the improved SRPS model a multitar-get optimization model is proposed in purpose of improvingproduction decreasing load fluctuation and saving energy Bycomparison the optimal scheme can achieve the decreasingof maximum and minimum suspension point load cranktorque fluctuation and energy consumption as well asimproving the system balance degree and pump fullness Insummary it improves the dynamic behavior of SRPS

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

National Natural Science Foundation of China (Grantno 51174175) China Scholarship Council (Grant no201708130108) Hebei Natural Science Foundation (Grant noE201703101) are acknowledged

References

[1] T A Aliev A H Rzayev G A Guluyev T A Alizada and NE Rzayeva ldquoRobust technology and system for management ofsucker rod pumping units in oil wellsrdquoMechanical Systems andSignal Processing vol 99 pp 47ndash56 2018

[2] S Gibbs ldquoPredicting the Behavior of Sucker-Rod PumpingSystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 1963

[3] D R Doty and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo SPE Journal vol 23 no 1 pp 33ndash41 1983

[4] I N Shardakov and I NWasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[5] S G Gibbs ldquoComputing gearbox torque and motor loading forbeam pumping units with consideration of inertia effectsrdquo SPEJ vol 27 pp 1153ndash1159 1975

[6] J Svinos ldquoExact Kinematic Analysis of Pumping Unitsrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionSan Francisco California 1983

[7] D Schafer and J Jennings ldquoAn Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical Modelsrdquoin Proceedings of the SPE Annual Technical Conference andExhibition pp 27ndash30 Dallas Texas 1987

[8] G Takacs Sucker-Rod PumpingManual Pennwell Books Tulsa2003

[9] G W Wang S S Rahman and G Y Yang ldquoAn improvedmodel for the sucker rod pumping systemrdquo in Proceedings of the11thAustralasian FluidMechanics Conference pp 14ndash18HobartAustralia December 1992

[10] S D L Lekia andR D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystemsrdquo SPE Production amp Facilities vol 10 no 1 pp 26ndash331995

[11] J Lea and P Pattillo ldquoInterpretation of Calculated Forces onSucker Rodsrdquo SPE Production amp Facilities vol 10 no 01 pp 41ndash45 1995

Mathematical Problems in Engineering 15

[12] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[13] L Guo-hua H Shun-li Y Zhi et al ldquoA prediction model fora new deep-rod pumping systemrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 75ndash80 2011

[14] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[15] O Becerra J Gamboa and F Kenyery ldquoModelling a DoublePiston Pumprdquo in Proceedings of the SPE International ThermalOperations and Heavy Oil Symposium and International Hor-izontal Well Technology Conference Calgary Alberta Canada2002

[16] A L Podio J Gomez A J Mansure et al ldquoLaboratoryinstrumented sucker-rod pumprdquo J Pet Technol vol 53 no 05pp 104ndash113 2003

[17] Z H Gu H Q Peng and H Y Geng ldquoAnalysis and mea-surement of gas effect on pumping efficiencyrdquoChina PetroleumMachinery vol 34 no 02 pp 64ndash69 2006

[18] M Xing ldquoResponse analysis of longitudinal vibration of suckerrod string considering rod bucklingrdquo Advances in EngineeringSoftware vol 99 pp 49ndash58 2016

[19] S Miska A Sharaki and J M Rajtar ldquoA simple model forcomputer-aided optimization and design of sucker-rod pump-ing systemsrdquo Journal of Petroleum Science and Engineering vol17 no 3-4 pp 303ndash312 1997

[20] L S Firu T Chelu and C Militaru-Petre ldquoAmodern approachto the optimum design of sucker-rod pumping systemrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash9 DenverColorado 2003

[21] X F Liu and Y G Qi ldquoA modern approach to the selectionof sucker rod pumping systems in CBM wellsrdquo Journal ofPetroleum Science and Engineering vol 76 no 3-4 pp 100ndash1082011

[22] S M Dong N N Feng and Z J Ma ldquoSimulating maximumof system efficiency of rod pumping wellsrdquo Journal of SystemSimulation vol 20 no 13 pp 3533ndash3537 2008

[23] M Xing and S Dong ldquoA New Simulation Model for a Beam-Pumping System Applied in Energy Saving and Resource-Consumption Reductionrdquo SPE Production amp Operations vol30 no 02 pp 130ndash140 2015

[24] Y XWu Y Li andH LiuElectric Motor andDriving ChemicalIndustry Press Beijing 2008

[25] S M Dong Computer Simulation of Dynamic Parameters ofRod Pumping System Optimization Petroleum Industry PressBeijing Chinese 2003

[26] F Yavuz J F Lea J C Cox et al ldquoWave equation simulationof fluid pound and gas interferencerdquo in Proceedings of the SPEProduction Operations Symposium pp 16ndash19 Oklahoma CityOklahoma April 2005

[27] D Y Wang and H Z Liu ldquoDynamic modeling and analysis ofsucker rod pumping system in a directional well Mechanismand Machine Sciencerdquo in Proceedings of the ASIAN MMS 2016amp CCMMS pp 1115ndash1127 2017

[28] F M A Barreto M Tygel A F Rocha and C K MorookaldquoAutomatic downhole card generation and classificationrdquo inProceedings of the SPE Annual Technical Conference pp 6ndash9Denver Colorado 1996

Hindawiwwwhindawicom Volume 2018

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Page 6: An Improved Sucker Rod Pumping System Model and Swabbing ...downloads.hindawi.com/journals/mpe/2018/4746210.pdf · An Improved Sucker Rod Pumping System Model and Swabbing Parameters

6 Mathematical Problems in Engineering

f-f

L

Lf

La

a-a

Figure 7 Process of fluid flows into the pump

where

119877119869 = (1 minus 120576119908) (119877119901 minus 119877119904) 119901119904119905119885119879119879119904119905119885119904119905119901119877119904 = 120572 (119901 minus 119901119904119905)

(18)

where Rp is the production gas oil ratio m3m3 Rs is the gasoil ratio at pump intake m3m3

Fluid Flowing into Pump Model Figure 7 shows the pumpoperation when the gas-liquid flow is drawn into the pumpSection a-a indicates the cross section of standing valve holeSupposing the section f -f is the liquid level at arbitrarytime neglect the process of gas bubbling from fluid byconsidering only the pump gas to occupy the upper spaceof f -f The one-dimensional unsteady flow equation based onBernoulli equation is established to describe the flow state ofpumping liquid In the Bernoulli equation the inertia headloss friction head loss and local head loss are all considered

119901119904120588g + V21198862119892 = 119901120588g + V2

1198912g + (119871119900 minus 119871119900119892 + 119871119891 + 119871V) + ℎ119886

+ ℎV + ℎ119891

(19)

where

ℎ119886 = 1119892 (119871119900 minus 119871119900119892 + 119871119891 + 119871V119860119891119860119901

) 119889V119891119889119905ℎ119891 = 64120583V119891 (119871119900 minus 119871119900119892 + 119871119891)21198632

119889120588119892

ℎV = 11198921205762 (119860119901119860119891

)2 V21198862

V119886 = 119860119901119860119891

V119891

(20)

where 120576 is the flow coefficient of standing valveThen (19) is converted into differential forms

119891 = V119891

V119891 = (1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V119860119891119860119901

)minus1119901119904120588 minus 119901120588+ V2

1198912 [11986021199011198602119891

minus 119860211990112057621198602

119891

minus 64120583 (1198710 minus 119881119900119892119860119901 + 119871119891)1198632119889V119891120588

minus 1] minus 119892(1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V)

(21)

Normally the fluid in the barrel nomatter in which formwill all be discharged except the one in dead spaceThereforethe fluid flowing out of the model with opening travelingvalve is not proposed The pump outlet pressure generatedby several kilometers liquid column is large compared to thehydraulic loss causing the ignorance of hydraulic loss whenfluid flows out Based on the statement mentioned above thenew downhole boundary model is expressed as follows andalso is divided into four phases corresponding to the phasesshown in Figure 2

1198781199051198861199051199061199041 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199042 af ter 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = ((V119891 sdot 119860119901 + 119902) (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)]sdot V119901

Mathematical Problems in Engineering 7

1198781199051198861199051199061199043 before 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199044 af ter 119900119901119890119899119892119894119899 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = 0

(22)

where

119881119898119900119897 = 224119885119879119901119904119905119879119904119905119901 (23)

3 Integrated Numerical Algorithm

For the current SRPS model the surface transmission modelis solved by numerical integration method [25] While thefinite difference method is used to describe the rod stringlongitudinal vibration [2 26 27] Since there is no fixedsolution between surface and downholemodel it needs cycliciteration to handle thewholemodel which is time consuming[25] Considering the interaction within plunger motionfluid flow into pump and pump pressure the improved SRPSmodel has a higher nonlinear degree and thus increasesthe difficulty of solving it In order to reduce the solvingtime the wave equation of rod string longitudinal vibrationis converted into ordinary differential equations by modalsuperposition method Then the surface and downholeboundary model are now in the form of ordinary differentialequations at which the whole model can be solved by Runge-Kutta method directly Besides that solving the rod stringlongitudinal vibration equation is also solving the forcevibration response of rod string where (3) is converted intothe following form

120588119903119860119903

12059721199061205971199052 minus 120588119903119864119903

12059721199061205971199092+ 120583120597119906120597119905 = 120588119903119860119903119891 (119909 119905) (24)

where

119891 (119909 119905) = minus1198892119906119886 (119905)1198891199052 minus 120583120588119903119860119903

119889119906119886 (119905)119889119905 + 119892 (25)

And its solution can be expressed in the function of timeand space by separating the variables

119906 (119909 119905) = infinsum119894=1

Φ119894 (119909) 119902119894 (119905) (26)

Therefore (24) can be expressed asinfinsum119894=1

119860119903120588119903Φ119894 (119909) 119894 (119905)minus infinsum

119894=1

119864119903119860119903

1198892Φ119894 (119909)1198891199092119902119894 (119905) + infinsum

119894=1

120583Φ119894 (119909) 119894 (119905)= 119860119903120588119903119891 (119909 119905)

(27)

Multiply the above equation by Φi(x) and it is integratedalong the rod length Then the following is derived

119860119903120588119903119894 (119905) int119871119903

0Φ119894

2 (119909) 119889119909minus 119864119903119860119903119902119894 (119905) int119871119903

0Φ119894 (119909)Φ119894

10158401015840 (119909) 119889119909+ 120583119894 (119905) int119871119903

0Φ119894

2 (119909) 119889119909= 119860119903120588119903 int119871119903

0Φ119894 (119909) 119891 (119909 119905) 119889119909

(28)

Its mode shapes and natural frequencies are

Φ119894 (119909) = radic 2120588119903119860119903

sin((2119894 minus 1) 1205872119871119903

119909)119901119899119894 = (2119894 minus 1) 120587radic1198641199031205881199032119871119903

(29)

Then (28) can be simplified as follows using mode shapeorthogonality

119894 (119905) + 120583119860119903120588119903 119894 (119905) + ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 119902119894 (119905) = 119865119894 (30)

where

119865119894 = (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903(2119894 minus 1) 120587+ 119865119875119871 (119905) radic 2119860119903120588119903119871119903

sin ((2119894 minus 1) 1205872 )(31)

8 Mathematical Problems in Engineering

Let ith-order forced vibration displacement response andvelocity response be xi1 and xi2 respectivelyThen (30) can beexpressed as the following form

1198941 (119905) = 1199091198942 (119905)1198942 (119905) = minus 1205831198601199031205881199031199091198942 (119905) minus ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 1199091198941 (119905) + 119865119894

(32)

Then suspension point load is derived

119865119877119871 (119905) = 119864119903119860119903

120597119906 (0 119905)120597119909 + 119865119903

= 119864119903119860119903radic 2120588119903119860119903119871119903

119873sum119894=1

(2119894 minus 1) 1205872119871119903

1199091198941 (119905) + 119865119903

(33)

The displacement and velocity of plunger are

119906119901 (119905) = 119906119886 (119905) minus 119906 (119871 119905)= 119906119886 (119905)minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198941 (119905) minus 119894=2lowast119895sum119894=1

1199091198941 (119905))V119901 (119905) = 119886 (119905) minus 120597119906 (119871 119905)120597119905

= 119886 (119905)minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198942 (119905) minus 119894=2lowast119895sum119894=1

1199091198942 (119905))

(34)

Then the integrated numerical model is given as follows

119898 = 120596119898

119898 = 1119868119890 [[2120582119896119872119867120576119888120596119899 [120596119899 minus 119898]12057621198881205962

119899 + [120596119899 minus 119898]2 minus 1119894119887119892120578119887119892

[119878119879119891 (119865119877119871 minus 119865119887) 1205781198961119862119871 minus119872119888 sin (120579 minus 120579120591)] minus 121205962

119898

119889119868119890119889120579119898]]119906119886 (119905) = arccos[1198712

119862 + 1198712119870 minus (119871119877 + 119871119875)22119871119862119871119870

] minus arccos(1198712119862 + 1198712

119871 minus 11987121198752119871119862119871119871

)minus arcsin(119871119877119871119871

sin(2120587 minus 1205790 minus 120579119898119894119887119892

+ arcsin ( 119871119868119871119870

)))11990911 (119905) = 11990912 (119905)12 (119905) = minus 12058311986011990312058811990311990912 (119905) minus ( 1205872119871119903

)2 119864119903120588119903 11990911 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903120587+ (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

21 (119905) = 11990922 (119905)22 (119905) = minus 12058311986011990312058811990311990922 (119905) minus ( 31205872119871119903

)2 119864119903120588119903 11990921 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic21198601199031205881199031198711199033120587minus (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

1198941 (119905) = 1199091198942 (119905)1198942 (119905) = minus 1205831198601199031205881199031199091198942 (119905) minus ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 1199091198941 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903(2119894 minus 1) 120587+ (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

sin ((2119894 minus 1) 1205872 )

Mathematical Problems in Engineering 9

119865119877119871 (119905) = 119864119903119860119903radic 2120588119903119860119903119871119903

119873sum119894=1

(2119894 minus 1) 1205872119871119903

1199091198941 (119905) + 119865119903

119906119901 (119905) = 119906119886 (119905) minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198941 (119905) minus 119894=2lowast119895sum119894=1

1199091198941 (119905))

V119901 (119905) = 119886 (119905) minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198942 (119905) minus 119894=2lowast119895sum119894=1

1199091198942 (119905))1198781199051198861199051199061199041 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199042 119886119891119905119890119903 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = ((V119891 sdot 119860119901 + 119902) (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)]sdot V119901

1198781199051198861199051199061199043 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199044 119886119891119905119890119903 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = 0119889119871119891119889119905 = V119891

119889V119891119889119905 = (1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V119860119891119860119901

)minus1119901119904120588 minus 119901120588 + V21198912 [119860

21199011198602119891

minus 119860211990112057621198602

119891

minus 64120583 (1198710 minus 119881119900119892119860119901 + 119871119891)1198632V119891120588 minus 1]minus 119892(1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V)(35)

4 Optimization Model

41 Optimization Goal As mostly developed oil-field movesinto the mid and late stage and the OWD begins todecline gradually energy-saving production-increasing andreducing load variation as much as possible are particularlyimportant Then we take pump fullness epf suspension pointload amplitude FRLA crank torque standard deviation Mcsdand motor input power average 119875119898 as the optimization goalto build a multitarget model Suspension point load andcrank torque can be solved directly by (35) The motor input

power and pump fullness calculation formula are deduced asfollows

119875119898 = 119872119890119889119898 + 1198750

+ [( 1120578H minus 1)119875119867 minus 1198750](119872119890119889119898119875119867

)2 (36)

119890119901119891 = 119871119891 minus int119905119906

0119902119889119905 minus (119871119900 minus 119871119900119892)119906119901119906

(37)

10 Mathematical Problems in Engineering

where P0 is the no-load power of motor kW PH is the motorrated power kW 120578H is the motor rated efficiency kW upuis the plunger displacement when it is arriving at dead topcenter

So the objective function is

Ω(119890119901119891 (X) 119865119877119871119860 (X) 119872119888119904119889 (X) 119875119898 (X))= 1198701119890119901119891 (X) + 1198702119865119877119871119860 (X) + 1198703119872119888119904119889 (X)+ 1198704119875119898 (X)

(38)

where K1 K2 K3 and K4 are the weight coefficients

42 Design Variables The objective function can beexpressed as the function of the swabbing parameters whenthe SRPS type and oil well basic parameters are confirmedIn this paper the swabbing parameters denote stroke Sstroke frequency ns pump diameter Dd pump depth Lpdcrank balance radius rc and rod string combination (the jthrod string diameter and length are dj and Lj respectively119896 = 1 2 119898) Then the design variables are shown asfollows

X = 119878 119899119904 119863119889 119871119901119889 (119889119895 119871119895 119895 = 1 2 sdot sdot sdot 119898) (39)

43 Constraints

(a) Crank Balance Degree Crank balance degree indicates theload fluctuation to a certain degree and it needs to be kept ata high value

095 le 119872119888119896119906119872119888119896119889

le 1 (40)

where Mcku and Mckd are the maximum crank torque whenplunger is at upstroke and downstroke respectively Nsdotm

(b) Ground Device Carrying Capacity The suspensionpoint load crank torque and motor torque at anytime [0 119879] do not go beyond the allowable rangemax(F119877119871) min(F119877119871) max(119872119888) min(119872119888) max(119872119890119889)min(119872119890119889) for the given type

min (119865119877119871) le 119865119877119871 le max (119865119877119871)min (119872119888119896) le 119872119888119896 le max (119872119888119896)min (119872119890119889) le 119872119890119889 le max (119872119890119889)

(41)

(c) Rod String StrengthThemaximumandminimum stress ofany point x along the rod string does not exceed permissiblestress range in one cycle

[120590min] le 120590119903119904 le [120590max] (42)

(d) Swabbing Parameters Each oil well has different limit onthe swabbing parameters in accordance with the device typeand actual operation hence their allowable variation rangesare adjustable

Figure 8 Dynamometer sensor

44 Optimization Algorithm In summary this multivariableoptimization model is established with nonlinear restrictionand nonlinear objective function So as to seek the bestresults the genetic algorithm is applied to solve it

5 Test and Verification

Surface dynamometer card is a closed graph recordingpolished rod loads versus rod displacement over a SRPScycle which is generally collected and taken as an indexto estimate the operation of SRPS In this paper based onthe dynamometer sensor shown in Figure 8 four test wellsare used to validate the improved SRPS model The oil wellparameters are listed in Table 1 and the simulation and fieldtest results are given in Figure 9 According to the plottedcurves the simulation results are basically consistent withthat in measured and the improved SRPS model is accurateenough to be proposed for engineering practices

6 Dynamic Response Comparison

Pump dynamometer card is a closed graph recording plungerloads versus plunger displacement over a SRPS cycle It isdetermined by multifactors such as stroke stroke frequencypump diameter pump depth dynamic liquid level and theother oil well parameters The difference between the currentSRPS model and the improved SRPS model proposed in thispaper is whether to consider the PFFP In view of this theoil operating status is divided into two forms as follows(1) the fluid always keeps pace with the plunger when itis being sucked into the pump (2) the fluid is unable tofollowwith the plungerThen two oil wells are selected well1with sufficient OWD and well2 with insufficient OWDFigure 10 describes the plunger and fluid velocity duringupstroke based on the improved SRPS model It can beconcluded that when the well has sufficient OWD and thefluid possesses good ability of keeping pace with the plungerwhen the well has insufficient OWD the fluid velocity lagsbehind the one of plunger at the beginning whereas it is

Mathematical Problems in Engineering 11

Table 1 Oil well basic parameters

Well 1 2 3 4Motor type YD280S-8 Y250M-6 Y250M-6 YD280S-6Pumping unit type CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HBStroke length (m) 3 3 3 3Stroke frequency (minminus1) 35 3 4 6Pump diameter (mm) 57 44 38 44Pump clearance level 1 2 2 1Pump depth (m) 890 1377 1138 1430Sucker-rod string (mm timesm) 25lowast890 19lowast673+22lowast704 19lowast672+22lowast466 19lowast720+22lowast710middle depth of reservoir (m) 1000 1492 1569 1600Crude oil density (kgm3) 795 857 857 850Water content () 95 97 92 98Dynamic liquid level (m) 880 1347 757 1340Casing pressure (Pa) 02 03 03 02Oil pressure (Pa) 03 03 03 02Fluid dynamic viscosity (Pasdots) 0006 0007 0007 0006Gas oil ratio (m3 m3) 20 19 40 80Plunger length (m) 12 12 12 12Clearance length (m) 05 05 05 05

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(a) Well 1

1 2 30Polished rod displacement (m)

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

simulationmeasured

(b) Well 2

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(c) Well 3

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(d) Well 4Figure 9 Suspension point dynamometer card comparison between the simulated and measured

12 Mathematical Problems in Engineering

FluidPlunger

minus05

00

05

10

15Ve

loci

ty(

ms

)

2 4 60Time (s)

(a) Well 1

minus02

00

02

04

06

Velo

city

(m)

4 8 120Time (s)

FluidPlunger

(b) Well 2

Figure 10 Plunger and fluid velocity

Current SRPS model 970Improved SRPS model 981

minus10

0

10

20

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(a) Well 1

Current SRPS model 724Improved SRPS model 775

minus10

0

10

20

40

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(b) Well 2

Figure 11 Comparison of pump dynamometer card

faster after a certain time period Then the comparisons ofpump dynamometer card are executed by the two modelsmeanwhile their individual pump fullness result is given atthe end of the legend (Figure 11)

Pump dynamometer card is very important method andnormally used to diagnose the pump operations particularlywhose shape is more similar to a rectangle the pump iscloser to the fullness [28] Hence from Figure 11 it canbe known that the pump fullness of current SRPS modelis higher than that of the improved one depending on thequalitative judgment and this conclusion is consistent withthe quantitative calculation result meanwhile this gap forthe well 2 is bigger than well 1 According the above

description we can know that the pump fullness calculationresult is on the high side if the PFFP is not considered andthis phenomenon will becomemore obvious for the well withinsufficient OWD

The load presented by the left upper right and lowerborderline of pump dynamometer card is the results of pumpmoving from phase 1 to 4 in turn Therefore its upperborderline describes the pump load when fluid is pumpedinto the barrel From Figure 11(a) it can be found thatthe upper borderline load simulated by the improved SRPSmodel is significantly larger than the one of current ForFigure 11(b) this difference can be neglected Based on (1)and (2) the pump pressure keeps constant as ps when fluid is

Mathematical Problems in Engineering 13

Before optimizationAfter optimization

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

1 2 30Polished rod displacement (m)

(a) Suspension point dynamometer card

minus10

0

10

20

30

40

Cran

k to

rque

(kNmiddotm

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(b) Crank torque

0

5

10

15

20

Mot

or in

put p

ower

(kW

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(c) Motor input power

Figure 12 Dynamic response comparison before and after optimization

sucked into the pump and then the corresponding pump loadis a fixed value Due to the new model takes into account ofPFFP a pressure drop will be brought which will vary withthe fluid velocity Combined with (1) this pressure drop willlead to the pump load increases Figure 10 shows that thefluid velocity of well 1 is larger than that of well 2 andthis is the reason why the difference of upper borderline loadbetween the two models is obvious This illustrates the erroron the upper borderline load of pump dynamometer carddepending on the velocity at which the fluid flows into thepump

7 Optimization Test

Based on the above models a simulation and optimizationsoftware is developed byMATLABOnewell with insufficient

OWD is tested Its original swabbing parameters are asfollows stroke S is 3 m stroke frequency ns is 3 minminus1pump diameter Dd is 57mm pump depth Lpd is 900m crankbalance radius rc is 12 m and rod string combination djtimesLj is 25 mm times 524 m+22 mm times 376 m And its swabbingparameters after optimizing are as follows stroke S is 3 mstroke frequency ns is 25 minminus1 pump diameterDd is 57mmpump depth Lpd is 990 m crank balance radius rc is 09 mand rod string combination djtimes Lj is 22 mm times 426 m+19 mmtimes 564 m The comparisons before and after optimization areshown in Figure 12 and Table 2

From the above comparison results some conclusions areobtained

(1) After optimization the maximum and minimumof suspension point load are all decreased and the load

14 Mathematical Problems in Engineering

Table 2 Specific parameters comparison before and after optimization

Suspension point load Crank torque Motor input power PumpfullnessMaximum Minimum Amplitude Standard

deviationBalancedegree Maximum Minimum mean

Beforeoptimization 555 kN 190 kN 365 kN 107 kN 854 159 kW 09 kW 62 kW 725

Afteroptimization 441 kN 132 kN 309 kN 94 kN 978 106 kW 10 kW 54 kW 908

darr 205 darr 305 darr 153 darr 121 uarr 145 darr 333 uarr 111 darr 129 uarr 252 amplitude is lowered by 153 It can contribute to enhancethe rod string life and prolong the maintenance period

(2) After optimization the standard deviation of cranktorque is reduced by 121 and its balance degree is raisedby 145 It illustrates that the load torque fluctuation is cutdown which canminimize damage to transmission parts andimprove the motor efficiency

(3) After optimization the pump fullness is improvedby 252 It is conducive to improve pump efficiency andproduction

(4) After optimization it plays a role of peak shavingand valley filling for motor input power and extends theoperational life meanwhile power saving rate is 129

8 Conclusions

(1) In this article an improved SRPS model is presentedconsidering the couple effect of pumping fluid and plungermotion on the dynamic response of SRPS instead of theexisting models that assume the pumping fluid volumeis always equal to plunger travelling volume The Runge-Kutta method is applied to solve the whole system modeltrough transforming the rod string longitudinal vibrationequation into ordinary differential equations And the SRPSmodelrsquos precision has been validated by adopting surfacedynamometer card

(2) Two oil wells are served to compare the differencebetween the current SRPS model and the improved oneThe results indicate that the current SRPS model is relativelylow in calculating pump fullness and this gap will increasewith the reduction of OWD the influence of fluid flowinginto the pump on pump load cannot be ignored when thefluid velocity is high Therefore the PFFP is necessary to beconsidered in order to improve the simulation accuracy

(3) On the basis of the improved SRPS model a multitar-get optimization model is proposed in purpose of improvingproduction decreasing load fluctuation and saving energy Bycomparison the optimal scheme can achieve the decreasingof maximum and minimum suspension point load cranktorque fluctuation and energy consumption as well asimproving the system balance degree and pump fullness Insummary it improves the dynamic behavior of SRPS

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

National Natural Science Foundation of China (Grantno 51174175) China Scholarship Council (Grant no201708130108) Hebei Natural Science Foundation (Grant noE201703101) are acknowledged

References

[1] T A Aliev A H Rzayev G A Guluyev T A Alizada and NE Rzayeva ldquoRobust technology and system for management ofsucker rod pumping units in oil wellsrdquoMechanical Systems andSignal Processing vol 99 pp 47ndash56 2018

[2] S Gibbs ldquoPredicting the Behavior of Sucker-Rod PumpingSystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 1963

[3] D R Doty and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo SPE Journal vol 23 no 1 pp 33ndash41 1983

[4] I N Shardakov and I NWasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[5] S G Gibbs ldquoComputing gearbox torque and motor loading forbeam pumping units with consideration of inertia effectsrdquo SPEJ vol 27 pp 1153ndash1159 1975

[6] J Svinos ldquoExact Kinematic Analysis of Pumping Unitsrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionSan Francisco California 1983

[7] D Schafer and J Jennings ldquoAn Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical Modelsrdquoin Proceedings of the SPE Annual Technical Conference andExhibition pp 27ndash30 Dallas Texas 1987

[8] G Takacs Sucker-Rod PumpingManual Pennwell Books Tulsa2003

[9] G W Wang S S Rahman and G Y Yang ldquoAn improvedmodel for the sucker rod pumping systemrdquo in Proceedings of the11thAustralasian FluidMechanics Conference pp 14ndash18HobartAustralia December 1992

[10] S D L Lekia andR D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystemsrdquo SPE Production amp Facilities vol 10 no 1 pp 26ndash331995

[11] J Lea and P Pattillo ldquoInterpretation of Calculated Forces onSucker Rodsrdquo SPE Production amp Facilities vol 10 no 01 pp 41ndash45 1995

Mathematical Problems in Engineering 15

[12] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[13] L Guo-hua H Shun-li Y Zhi et al ldquoA prediction model fora new deep-rod pumping systemrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 75ndash80 2011

[14] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[15] O Becerra J Gamboa and F Kenyery ldquoModelling a DoublePiston Pumprdquo in Proceedings of the SPE International ThermalOperations and Heavy Oil Symposium and International Hor-izontal Well Technology Conference Calgary Alberta Canada2002

[16] A L Podio J Gomez A J Mansure et al ldquoLaboratoryinstrumented sucker-rod pumprdquo J Pet Technol vol 53 no 05pp 104ndash113 2003

[17] Z H Gu H Q Peng and H Y Geng ldquoAnalysis and mea-surement of gas effect on pumping efficiencyrdquoChina PetroleumMachinery vol 34 no 02 pp 64ndash69 2006

[18] M Xing ldquoResponse analysis of longitudinal vibration of suckerrod string considering rod bucklingrdquo Advances in EngineeringSoftware vol 99 pp 49ndash58 2016

[19] S Miska A Sharaki and J M Rajtar ldquoA simple model forcomputer-aided optimization and design of sucker-rod pump-ing systemsrdquo Journal of Petroleum Science and Engineering vol17 no 3-4 pp 303ndash312 1997

[20] L S Firu T Chelu and C Militaru-Petre ldquoAmodern approachto the optimum design of sucker-rod pumping systemrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash9 DenverColorado 2003

[21] X F Liu and Y G Qi ldquoA modern approach to the selectionof sucker rod pumping systems in CBM wellsrdquo Journal ofPetroleum Science and Engineering vol 76 no 3-4 pp 100ndash1082011

[22] S M Dong N N Feng and Z J Ma ldquoSimulating maximumof system efficiency of rod pumping wellsrdquo Journal of SystemSimulation vol 20 no 13 pp 3533ndash3537 2008

[23] M Xing and S Dong ldquoA New Simulation Model for a Beam-Pumping System Applied in Energy Saving and Resource-Consumption Reductionrdquo SPE Production amp Operations vol30 no 02 pp 130ndash140 2015

[24] Y XWu Y Li andH LiuElectric Motor andDriving ChemicalIndustry Press Beijing 2008

[25] S M Dong Computer Simulation of Dynamic Parameters ofRod Pumping System Optimization Petroleum Industry PressBeijing Chinese 2003

[26] F Yavuz J F Lea J C Cox et al ldquoWave equation simulationof fluid pound and gas interferencerdquo in Proceedings of the SPEProduction Operations Symposium pp 16ndash19 Oklahoma CityOklahoma April 2005

[27] D Y Wang and H Z Liu ldquoDynamic modeling and analysis ofsucker rod pumping system in a directional well Mechanismand Machine Sciencerdquo in Proceedings of the ASIAN MMS 2016amp CCMMS pp 1115ndash1127 2017

[28] F M A Barreto M Tygel A F Rocha and C K MorookaldquoAutomatic downhole card generation and classificationrdquo inProceedings of the SPE Annual Technical Conference pp 6ndash9Denver Colorado 1996

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Page 7: An Improved Sucker Rod Pumping System Model and Swabbing ...downloads.hindawi.com/journals/mpe/2018/4746210.pdf · An Improved Sucker Rod Pumping System Model and Swabbing Parameters

Mathematical Problems in Engineering 7

1198781199051198861199051199061199043 before 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199044 af ter 119900119901119890119899119892119894119899 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = 0

(22)

where

119881119898119900119897 = 224119885119879119901119904119905119879119904119905119901 (23)

3 Integrated Numerical Algorithm

For the current SRPS model the surface transmission modelis solved by numerical integration method [25] While thefinite difference method is used to describe the rod stringlongitudinal vibration [2 26 27] Since there is no fixedsolution between surface and downholemodel it needs cycliciteration to handle thewholemodel which is time consuming[25] Considering the interaction within plunger motionfluid flow into pump and pump pressure the improved SRPSmodel has a higher nonlinear degree and thus increasesthe difficulty of solving it In order to reduce the solvingtime the wave equation of rod string longitudinal vibrationis converted into ordinary differential equations by modalsuperposition method Then the surface and downholeboundary model are now in the form of ordinary differentialequations at which the whole model can be solved by Runge-Kutta method directly Besides that solving the rod stringlongitudinal vibration equation is also solving the forcevibration response of rod string where (3) is converted intothe following form

120588119903119860119903

12059721199061205971199052 minus 120588119903119864119903

12059721199061205971199092+ 120583120597119906120597119905 = 120588119903119860119903119891 (119909 119905) (24)

where

119891 (119909 119905) = minus1198892119906119886 (119905)1198891199052 minus 120583120588119903119860119903

119889119906119886 (119905)119889119905 + 119892 (25)

And its solution can be expressed in the function of timeand space by separating the variables

119906 (119909 119905) = infinsum119894=1

Φ119894 (119909) 119902119894 (119905) (26)

Therefore (24) can be expressed asinfinsum119894=1

119860119903120588119903Φ119894 (119909) 119894 (119905)minus infinsum

119894=1

119864119903119860119903

1198892Φ119894 (119909)1198891199092119902119894 (119905) + infinsum

119894=1

120583Φ119894 (119909) 119894 (119905)= 119860119903120588119903119891 (119909 119905)

(27)

Multiply the above equation by Φi(x) and it is integratedalong the rod length Then the following is derived

119860119903120588119903119894 (119905) int119871119903

0Φ119894

2 (119909) 119889119909minus 119864119903119860119903119902119894 (119905) int119871119903

0Φ119894 (119909)Φ119894

10158401015840 (119909) 119889119909+ 120583119894 (119905) int119871119903

0Φ119894

2 (119909) 119889119909= 119860119903120588119903 int119871119903

0Φ119894 (119909) 119891 (119909 119905) 119889119909

(28)

Its mode shapes and natural frequencies are

Φ119894 (119909) = radic 2120588119903119860119903

sin((2119894 minus 1) 1205872119871119903

119909)119901119899119894 = (2119894 minus 1) 120587radic1198641199031205881199032119871119903

(29)

Then (28) can be simplified as follows using mode shapeorthogonality

119894 (119905) + 120583119860119903120588119903 119894 (119905) + ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 119902119894 (119905) = 119865119894 (30)

where

119865119894 = (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903(2119894 minus 1) 120587+ 119865119875119871 (119905) radic 2119860119903120588119903119871119903

sin ((2119894 minus 1) 1205872 )(31)

8 Mathematical Problems in Engineering

Let ith-order forced vibration displacement response andvelocity response be xi1 and xi2 respectivelyThen (30) can beexpressed as the following form

1198941 (119905) = 1199091198942 (119905)1198942 (119905) = minus 1205831198601199031205881199031199091198942 (119905) minus ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 1199091198941 (119905) + 119865119894

(32)

Then suspension point load is derived

119865119877119871 (119905) = 119864119903119860119903

120597119906 (0 119905)120597119909 + 119865119903

= 119864119903119860119903radic 2120588119903119860119903119871119903

119873sum119894=1

(2119894 minus 1) 1205872119871119903

1199091198941 (119905) + 119865119903

(33)

The displacement and velocity of plunger are

119906119901 (119905) = 119906119886 (119905) minus 119906 (119871 119905)= 119906119886 (119905)minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198941 (119905) minus 119894=2lowast119895sum119894=1

1199091198941 (119905))V119901 (119905) = 119886 (119905) minus 120597119906 (119871 119905)120597119905

= 119886 (119905)minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198942 (119905) minus 119894=2lowast119895sum119894=1

1199091198942 (119905))

(34)

Then the integrated numerical model is given as follows

119898 = 120596119898

119898 = 1119868119890 [[2120582119896119872119867120576119888120596119899 [120596119899 minus 119898]12057621198881205962

119899 + [120596119899 minus 119898]2 minus 1119894119887119892120578119887119892

[119878119879119891 (119865119877119871 minus 119865119887) 1205781198961119862119871 minus119872119888 sin (120579 minus 120579120591)] minus 121205962

119898

119889119868119890119889120579119898]]119906119886 (119905) = arccos[1198712

119862 + 1198712119870 minus (119871119877 + 119871119875)22119871119862119871119870

] minus arccos(1198712119862 + 1198712

119871 minus 11987121198752119871119862119871119871

)minus arcsin(119871119877119871119871

sin(2120587 minus 1205790 minus 120579119898119894119887119892

+ arcsin ( 119871119868119871119870

)))11990911 (119905) = 11990912 (119905)12 (119905) = minus 12058311986011990312058811990311990912 (119905) minus ( 1205872119871119903

)2 119864119903120588119903 11990911 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903120587+ (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

21 (119905) = 11990922 (119905)22 (119905) = minus 12058311986011990312058811990311990922 (119905) minus ( 31205872119871119903

)2 119864119903120588119903 11990921 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic21198601199031205881199031198711199033120587minus (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

1198941 (119905) = 1199091198942 (119905)1198942 (119905) = minus 1205831198601199031205881199031199091198942 (119905) minus ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 1199091198941 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903(2119894 minus 1) 120587+ (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

sin ((2119894 minus 1) 1205872 )

Mathematical Problems in Engineering 9

119865119877119871 (119905) = 119864119903119860119903radic 2120588119903119860119903119871119903

119873sum119894=1

(2119894 minus 1) 1205872119871119903

1199091198941 (119905) + 119865119903

119906119901 (119905) = 119906119886 (119905) minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198941 (119905) minus 119894=2lowast119895sum119894=1

1199091198941 (119905))

V119901 (119905) = 119886 (119905) minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198942 (119905) minus 119894=2lowast119895sum119894=1

1199091198942 (119905))1198781199051198861199051199061199041 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199042 119886119891119905119890119903 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = ((V119891 sdot 119860119901 + 119902) (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)]sdot V119901

1198781199051198861199051199061199043 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199044 119886119891119905119890119903 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = 0119889119871119891119889119905 = V119891

119889V119891119889119905 = (1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V119860119891119860119901

)minus1119901119904120588 minus 119901120588 + V21198912 [119860

21199011198602119891

minus 119860211990112057621198602

119891

minus 64120583 (1198710 minus 119881119900119892119860119901 + 119871119891)1198632V119891120588 minus 1]minus 119892(1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V)(35)

4 Optimization Model

41 Optimization Goal As mostly developed oil-field movesinto the mid and late stage and the OWD begins todecline gradually energy-saving production-increasing andreducing load variation as much as possible are particularlyimportant Then we take pump fullness epf suspension pointload amplitude FRLA crank torque standard deviation Mcsdand motor input power average 119875119898 as the optimization goalto build a multitarget model Suspension point load andcrank torque can be solved directly by (35) The motor input

power and pump fullness calculation formula are deduced asfollows

119875119898 = 119872119890119889119898 + 1198750

+ [( 1120578H minus 1)119875119867 minus 1198750](119872119890119889119898119875119867

)2 (36)

119890119901119891 = 119871119891 minus int119905119906

0119902119889119905 minus (119871119900 minus 119871119900119892)119906119901119906

(37)

10 Mathematical Problems in Engineering

where P0 is the no-load power of motor kW PH is the motorrated power kW 120578H is the motor rated efficiency kW upuis the plunger displacement when it is arriving at dead topcenter

So the objective function is

Ω(119890119901119891 (X) 119865119877119871119860 (X) 119872119888119904119889 (X) 119875119898 (X))= 1198701119890119901119891 (X) + 1198702119865119877119871119860 (X) + 1198703119872119888119904119889 (X)+ 1198704119875119898 (X)

(38)

where K1 K2 K3 and K4 are the weight coefficients

42 Design Variables The objective function can beexpressed as the function of the swabbing parameters whenthe SRPS type and oil well basic parameters are confirmedIn this paper the swabbing parameters denote stroke Sstroke frequency ns pump diameter Dd pump depth Lpdcrank balance radius rc and rod string combination (the jthrod string diameter and length are dj and Lj respectively119896 = 1 2 119898) Then the design variables are shown asfollows

X = 119878 119899119904 119863119889 119871119901119889 (119889119895 119871119895 119895 = 1 2 sdot sdot sdot 119898) (39)

43 Constraints

(a) Crank Balance Degree Crank balance degree indicates theload fluctuation to a certain degree and it needs to be kept ata high value

095 le 119872119888119896119906119872119888119896119889

le 1 (40)

where Mcku and Mckd are the maximum crank torque whenplunger is at upstroke and downstroke respectively Nsdotm

(b) Ground Device Carrying Capacity The suspensionpoint load crank torque and motor torque at anytime [0 119879] do not go beyond the allowable rangemax(F119877119871) min(F119877119871) max(119872119888) min(119872119888) max(119872119890119889)min(119872119890119889) for the given type

min (119865119877119871) le 119865119877119871 le max (119865119877119871)min (119872119888119896) le 119872119888119896 le max (119872119888119896)min (119872119890119889) le 119872119890119889 le max (119872119890119889)

(41)

(c) Rod String StrengthThemaximumandminimum stress ofany point x along the rod string does not exceed permissiblestress range in one cycle

[120590min] le 120590119903119904 le [120590max] (42)

(d) Swabbing Parameters Each oil well has different limit onthe swabbing parameters in accordance with the device typeand actual operation hence their allowable variation rangesare adjustable

Figure 8 Dynamometer sensor

44 Optimization Algorithm In summary this multivariableoptimization model is established with nonlinear restrictionand nonlinear objective function So as to seek the bestresults the genetic algorithm is applied to solve it

5 Test and Verification

Surface dynamometer card is a closed graph recordingpolished rod loads versus rod displacement over a SRPScycle which is generally collected and taken as an indexto estimate the operation of SRPS In this paper based onthe dynamometer sensor shown in Figure 8 four test wellsare used to validate the improved SRPS model The oil wellparameters are listed in Table 1 and the simulation and fieldtest results are given in Figure 9 According to the plottedcurves the simulation results are basically consistent withthat in measured and the improved SRPS model is accurateenough to be proposed for engineering practices

6 Dynamic Response Comparison

Pump dynamometer card is a closed graph recording plungerloads versus plunger displacement over a SRPS cycle It isdetermined by multifactors such as stroke stroke frequencypump diameter pump depth dynamic liquid level and theother oil well parameters The difference between the currentSRPS model and the improved SRPS model proposed in thispaper is whether to consider the PFFP In view of this theoil operating status is divided into two forms as follows(1) the fluid always keeps pace with the plunger when itis being sucked into the pump (2) the fluid is unable tofollowwith the plungerThen two oil wells are selected well1with sufficient OWD and well2 with insufficient OWDFigure 10 describes the plunger and fluid velocity duringupstroke based on the improved SRPS model It can beconcluded that when the well has sufficient OWD and thefluid possesses good ability of keeping pace with the plungerwhen the well has insufficient OWD the fluid velocity lagsbehind the one of plunger at the beginning whereas it is

Mathematical Problems in Engineering 11

Table 1 Oil well basic parameters

Well 1 2 3 4Motor type YD280S-8 Y250M-6 Y250M-6 YD280S-6Pumping unit type CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HBStroke length (m) 3 3 3 3Stroke frequency (minminus1) 35 3 4 6Pump diameter (mm) 57 44 38 44Pump clearance level 1 2 2 1Pump depth (m) 890 1377 1138 1430Sucker-rod string (mm timesm) 25lowast890 19lowast673+22lowast704 19lowast672+22lowast466 19lowast720+22lowast710middle depth of reservoir (m) 1000 1492 1569 1600Crude oil density (kgm3) 795 857 857 850Water content () 95 97 92 98Dynamic liquid level (m) 880 1347 757 1340Casing pressure (Pa) 02 03 03 02Oil pressure (Pa) 03 03 03 02Fluid dynamic viscosity (Pasdots) 0006 0007 0007 0006Gas oil ratio (m3 m3) 20 19 40 80Plunger length (m) 12 12 12 12Clearance length (m) 05 05 05 05

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(a) Well 1

1 2 30Polished rod displacement (m)

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

simulationmeasured

(b) Well 2

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(c) Well 3

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(d) Well 4Figure 9 Suspension point dynamometer card comparison between the simulated and measured

12 Mathematical Problems in Engineering

FluidPlunger

minus05

00

05

10

15Ve

loci

ty(

ms

)

2 4 60Time (s)

(a) Well 1

minus02

00

02

04

06

Velo

city

(m)

4 8 120Time (s)

FluidPlunger

(b) Well 2

Figure 10 Plunger and fluid velocity

Current SRPS model 970Improved SRPS model 981

minus10

0

10

20

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(a) Well 1

Current SRPS model 724Improved SRPS model 775

minus10

0

10

20

40

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(b) Well 2

Figure 11 Comparison of pump dynamometer card

faster after a certain time period Then the comparisons ofpump dynamometer card are executed by the two modelsmeanwhile their individual pump fullness result is given atthe end of the legend (Figure 11)

Pump dynamometer card is very important method andnormally used to diagnose the pump operations particularlywhose shape is more similar to a rectangle the pump iscloser to the fullness [28] Hence from Figure 11 it canbe known that the pump fullness of current SRPS modelis higher than that of the improved one depending on thequalitative judgment and this conclusion is consistent withthe quantitative calculation result meanwhile this gap forthe well 2 is bigger than well 1 According the above

description we can know that the pump fullness calculationresult is on the high side if the PFFP is not considered andthis phenomenon will becomemore obvious for the well withinsufficient OWD

The load presented by the left upper right and lowerborderline of pump dynamometer card is the results of pumpmoving from phase 1 to 4 in turn Therefore its upperborderline describes the pump load when fluid is pumpedinto the barrel From Figure 11(a) it can be found thatthe upper borderline load simulated by the improved SRPSmodel is significantly larger than the one of current ForFigure 11(b) this difference can be neglected Based on (1)and (2) the pump pressure keeps constant as ps when fluid is

Mathematical Problems in Engineering 13

Before optimizationAfter optimization

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

1 2 30Polished rod displacement (m)

(a) Suspension point dynamometer card

minus10

0

10

20

30

40

Cran

k to

rque

(kNmiddotm

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(b) Crank torque

0

5

10

15

20

Mot

or in

put p

ower

(kW

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(c) Motor input power

Figure 12 Dynamic response comparison before and after optimization

sucked into the pump and then the corresponding pump loadis a fixed value Due to the new model takes into account ofPFFP a pressure drop will be brought which will vary withthe fluid velocity Combined with (1) this pressure drop willlead to the pump load increases Figure 10 shows that thefluid velocity of well 1 is larger than that of well 2 andthis is the reason why the difference of upper borderline loadbetween the two models is obvious This illustrates the erroron the upper borderline load of pump dynamometer carddepending on the velocity at which the fluid flows into thepump

7 Optimization Test

Based on the above models a simulation and optimizationsoftware is developed byMATLABOnewell with insufficient

OWD is tested Its original swabbing parameters are asfollows stroke S is 3 m stroke frequency ns is 3 minminus1pump diameter Dd is 57mm pump depth Lpd is 900m crankbalance radius rc is 12 m and rod string combination djtimesLj is 25 mm times 524 m+22 mm times 376 m And its swabbingparameters after optimizing are as follows stroke S is 3 mstroke frequency ns is 25 minminus1 pump diameterDd is 57mmpump depth Lpd is 990 m crank balance radius rc is 09 mand rod string combination djtimes Lj is 22 mm times 426 m+19 mmtimes 564 m The comparisons before and after optimization areshown in Figure 12 and Table 2

From the above comparison results some conclusions areobtained

(1) After optimization the maximum and minimumof suspension point load are all decreased and the load

14 Mathematical Problems in Engineering

Table 2 Specific parameters comparison before and after optimization

Suspension point load Crank torque Motor input power PumpfullnessMaximum Minimum Amplitude Standard

deviationBalancedegree Maximum Minimum mean

Beforeoptimization 555 kN 190 kN 365 kN 107 kN 854 159 kW 09 kW 62 kW 725

Afteroptimization 441 kN 132 kN 309 kN 94 kN 978 106 kW 10 kW 54 kW 908

darr 205 darr 305 darr 153 darr 121 uarr 145 darr 333 uarr 111 darr 129 uarr 252 amplitude is lowered by 153 It can contribute to enhancethe rod string life and prolong the maintenance period

(2) After optimization the standard deviation of cranktorque is reduced by 121 and its balance degree is raisedby 145 It illustrates that the load torque fluctuation is cutdown which canminimize damage to transmission parts andimprove the motor efficiency

(3) After optimization the pump fullness is improvedby 252 It is conducive to improve pump efficiency andproduction

(4) After optimization it plays a role of peak shavingand valley filling for motor input power and extends theoperational life meanwhile power saving rate is 129

8 Conclusions

(1) In this article an improved SRPS model is presentedconsidering the couple effect of pumping fluid and plungermotion on the dynamic response of SRPS instead of theexisting models that assume the pumping fluid volumeis always equal to plunger travelling volume The Runge-Kutta method is applied to solve the whole system modeltrough transforming the rod string longitudinal vibrationequation into ordinary differential equations And the SRPSmodelrsquos precision has been validated by adopting surfacedynamometer card

(2) Two oil wells are served to compare the differencebetween the current SRPS model and the improved oneThe results indicate that the current SRPS model is relativelylow in calculating pump fullness and this gap will increasewith the reduction of OWD the influence of fluid flowinginto the pump on pump load cannot be ignored when thefluid velocity is high Therefore the PFFP is necessary to beconsidered in order to improve the simulation accuracy

(3) On the basis of the improved SRPS model a multitar-get optimization model is proposed in purpose of improvingproduction decreasing load fluctuation and saving energy Bycomparison the optimal scheme can achieve the decreasingof maximum and minimum suspension point load cranktorque fluctuation and energy consumption as well asimproving the system balance degree and pump fullness Insummary it improves the dynamic behavior of SRPS

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

National Natural Science Foundation of China (Grantno 51174175) China Scholarship Council (Grant no201708130108) Hebei Natural Science Foundation (Grant noE201703101) are acknowledged

References

[1] T A Aliev A H Rzayev G A Guluyev T A Alizada and NE Rzayeva ldquoRobust technology and system for management ofsucker rod pumping units in oil wellsrdquoMechanical Systems andSignal Processing vol 99 pp 47ndash56 2018

[2] S Gibbs ldquoPredicting the Behavior of Sucker-Rod PumpingSystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 1963

[3] D R Doty and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo SPE Journal vol 23 no 1 pp 33ndash41 1983

[4] I N Shardakov and I NWasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[5] S G Gibbs ldquoComputing gearbox torque and motor loading forbeam pumping units with consideration of inertia effectsrdquo SPEJ vol 27 pp 1153ndash1159 1975

[6] J Svinos ldquoExact Kinematic Analysis of Pumping Unitsrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionSan Francisco California 1983

[7] D Schafer and J Jennings ldquoAn Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical Modelsrdquoin Proceedings of the SPE Annual Technical Conference andExhibition pp 27ndash30 Dallas Texas 1987

[8] G Takacs Sucker-Rod PumpingManual Pennwell Books Tulsa2003

[9] G W Wang S S Rahman and G Y Yang ldquoAn improvedmodel for the sucker rod pumping systemrdquo in Proceedings of the11thAustralasian FluidMechanics Conference pp 14ndash18HobartAustralia December 1992

[10] S D L Lekia andR D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystemsrdquo SPE Production amp Facilities vol 10 no 1 pp 26ndash331995

[11] J Lea and P Pattillo ldquoInterpretation of Calculated Forces onSucker Rodsrdquo SPE Production amp Facilities vol 10 no 01 pp 41ndash45 1995

Mathematical Problems in Engineering 15

[12] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[13] L Guo-hua H Shun-li Y Zhi et al ldquoA prediction model fora new deep-rod pumping systemrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 75ndash80 2011

[14] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[15] O Becerra J Gamboa and F Kenyery ldquoModelling a DoublePiston Pumprdquo in Proceedings of the SPE International ThermalOperations and Heavy Oil Symposium and International Hor-izontal Well Technology Conference Calgary Alberta Canada2002

[16] A L Podio J Gomez A J Mansure et al ldquoLaboratoryinstrumented sucker-rod pumprdquo J Pet Technol vol 53 no 05pp 104ndash113 2003

[17] Z H Gu H Q Peng and H Y Geng ldquoAnalysis and mea-surement of gas effect on pumping efficiencyrdquoChina PetroleumMachinery vol 34 no 02 pp 64ndash69 2006

[18] M Xing ldquoResponse analysis of longitudinal vibration of suckerrod string considering rod bucklingrdquo Advances in EngineeringSoftware vol 99 pp 49ndash58 2016

[19] S Miska A Sharaki and J M Rajtar ldquoA simple model forcomputer-aided optimization and design of sucker-rod pump-ing systemsrdquo Journal of Petroleum Science and Engineering vol17 no 3-4 pp 303ndash312 1997

[20] L S Firu T Chelu and C Militaru-Petre ldquoAmodern approachto the optimum design of sucker-rod pumping systemrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash9 DenverColorado 2003

[21] X F Liu and Y G Qi ldquoA modern approach to the selectionof sucker rod pumping systems in CBM wellsrdquo Journal ofPetroleum Science and Engineering vol 76 no 3-4 pp 100ndash1082011

[22] S M Dong N N Feng and Z J Ma ldquoSimulating maximumof system efficiency of rod pumping wellsrdquo Journal of SystemSimulation vol 20 no 13 pp 3533ndash3537 2008

[23] M Xing and S Dong ldquoA New Simulation Model for a Beam-Pumping System Applied in Energy Saving and Resource-Consumption Reductionrdquo SPE Production amp Operations vol30 no 02 pp 130ndash140 2015

[24] Y XWu Y Li andH LiuElectric Motor andDriving ChemicalIndustry Press Beijing 2008

[25] S M Dong Computer Simulation of Dynamic Parameters ofRod Pumping System Optimization Petroleum Industry PressBeijing Chinese 2003

[26] F Yavuz J F Lea J C Cox et al ldquoWave equation simulationof fluid pound and gas interferencerdquo in Proceedings of the SPEProduction Operations Symposium pp 16ndash19 Oklahoma CityOklahoma April 2005

[27] D Y Wang and H Z Liu ldquoDynamic modeling and analysis ofsucker rod pumping system in a directional well Mechanismand Machine Sciencerdquo in Proceedings of the ASIAN MMS 2016amp CCMMS pp 1115ndash1127 2017

[28] F M A Barreto M Tygel A F Rocha and C K MorookaldquoAutomatic downhole card generation and classificationrdquo inProceedings of the SPE Annual Technical Conference pp 6ndash9Denver Colorado 1996

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Page 8: An Improved Sucker Rod Pumping System Model and Swabbing ...downloads.hindawi.com/journals/mpe/2018/4746210.pdf · An Improved Sucker Rod Pumping System Model and Swabbing Parameters

8 Mathematical Problems in Engineering

Let ith-order forced vibration displacement response andvelocity response be xi1 and xi2 respectivelyThen (30) can beexpressed as the following form

1198941 (119905) = 1199091198942 (119905)1198942 (119905) = minus 1205831198601199031205881199031199091198942 (119905) minus ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 1199091198941 (119905) + 119865119894

(32)

Then suspension point load is derived

119865119877119871 (119905) = 119864119903119860119903

120597119906 (0 119905)120597119909 + 119865119903

= 119864119903119860119903radic 2120588119903119860119903119871119903

119873sum119894=1

(2119894 minus 1) 1205872119871119903

1199091198941 (119905) + 119865119903

(33)

The displacement and velocity of plunger are

119906119901 (119905) = 119906119886 (119905) minus 119906 (119871 119905)= 119906119886 (119905)minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198941 (119905) minus 119894=2lowast119895sum119894=1

1199091198941 (119905))V119901 (119905) = 119886 (119905) minus 120597119906 (119871 119905)120597119905

= 119886 (119905)minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198942 (119905) minus 119894=2lowast119895sum119894=1

1199091198942 (119905))

(34)

Then the integrated numerical model is given as follows

119898 = 120596119898

119898 = 1119868119890 [[2120582119896119872119867120576119888120596119899 [120596119899 minus 119898]12057621198881205962

119899 + [120596119899 minus 119898]2 minus 1119894119887119892120578119887119892

[119878119879119891 (119865119877119871 minus 119865119887) 1205781198961119862119871 minus119872119888 sin (120579 minus 120579120591)] minus 121205962

119898

119889119868119890119889120579119898]]119906119886 (119905) = arccos[1198712

119862 + 1198712119870 minus (119871119877 + 119871119875)22119871119862119871119870

] minus arccos(1198712119862 + 1198712

119871 minus 11987121198752119871119862119871119871

)minus arcsin(119871119877119871119871

sin(2120587 minus 1205790 minus 120579119898119894119887119892

+ arcsin ( 119871119868119871119870

)))11990911 (119905) = 11990912 (119905)12 (119905) = minus 12058311986011990312058811990311990912 (119905) minus ( 1205872119871119903

)2 119864119903120588119903 11990911 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903120587+ (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

21 (119905) = 11990922 (119905)22 (119905) = minus 12058311986011990312058811990311990922 (119905) minus ( 31205872119871119903

)2 119864119903120588119903 11990921 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic21198601199031205881199031198711199033120587minus (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

1198941 (119905) = 1199091198942 (119905)1198942 (119905) = minus 1205831198601199031205881199031199091198942 (119905) minus ((2119894 minus 1) 1205872119871119903

)2 119864119903120588119903 1199091198941 (119905) + (d2119906119886 (119905)d1199052 + 120583d119906119886 (119905)

d119905 ) 2radic2119860119903120588119903119871119903(2119894 minus 1) 120587+ (119860119901 (119901119889 minus 119901) minus 119860119903119901119889 + 119865119891)radic 2119860119903120588119903119871119903

sin ((2119894 minus 1) 1205872 )

Mathematical Problems in Engineering 9

119865119877119871 (119905) = 119864119903119860119903radic 2120588119903119860119903119871119903

119873sum119894=1

(2119894 minus 1) 1205872119871119903

1199091198941 (119905) + 119865119903

119906119901 (119905) = 119906119886 (119905) minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198941 (119905) minus 119894=2lowast119895sum119894=1

1199091198941 (119905))

V119901 (119905) = 119886 (119905) minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198942 (119905) minus 119894=2lowast119895sum119894=1

1199091198942 (119905))1198781199051198861199051199061199041 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199042 119886119891119905119890119903 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = ((V119891 sdot 119860119901 + 119902) (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)]sdot V119901

1198781199051198861199051199061199043 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199044 119886119891119905119890119903 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = 0119889119871119891119889119905 = V119891

119889V119891119889119905 = (1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V119860119891119860119901

)minus1119901119904120588 minus 119901120588 + V21198912 [119860

21199011198602119891

minus 119860211990112057621198602

119891

minus 64120583 (1198710 minus 119881119900119892119860119901 + 119871119891)1198632V119891120588 minus 1]minus 119892(1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V)(35)

4 Optimization Model

41 Optimization Goal As mostly developed oil-field movesinto the mid and late stage and the OWD begins todecline gradually energy-saving production-increasing andreducing load variation as much as possible are particularlyimportant Then we take pump fullness epf suspension pointload amplitude FRLA crank torque standard deviation Mcsdand motor input power average 119875119898 as the optimization goalto build a multitarget model Suspension point load andcrank torque can be solved directly by (35) The motor input

power and pump fullness calculation formula are deduced asfollows

119875119898 = 119872119890119889119898 + 1198750

+ [( 1120578H minus 1)119875119867 minus 1198750](119872119890119889119898119875119867

)2 (36)

119890119901119891 = 119871119891 minus int119905119906

0119902119889119905 minus (119871119900 minus 119871119900119892)119906119901119906

(37)

10 Mathematical Problems in Engineering

where P0 is the no-load power of motor kW PH is the motorrated power kW 120578H is the motor rated efficiency kW upuis the plunger displacement when it is arriving at dead topcenter

So the objective function is

Ω(119890119901119891 (X) 119865119877119871119860 (X) 119872119888119904119889 (X) 119875119898 (X))= 1198701119890119901119891 (X) + 1198702119865119877119871119860 (X) + 1198703119872119888119904119889 (X)+ 1198704119875119898 (X)

(38)

where K1 K2 K3 and K4 are the weight coefficients

42 Design Variables The objective function can beexpressed as the function of the swabbing parameters whenthe SRPS type and oil well basic parameters are confirmedIn this paper the swabbing parameters denote stroke Sstroke frequency ns pump diameter Dd pump depth Lpdcrank balance radius rc and rod string combination (the jthrod string diameter and length are dj and Lj respectively119896 = 1 2 119898) Then the design variables are shown asfollows

X = 119878 119899119904 119863119889 119871119901119889 (119889119895 119871119895 119895 = 1 2 sdot sdot sdot 119898) (39)

43 Constraints

(a) Crank Balance Degree Crank balance degree indicates theload fluctuation to a certain degree and it needs to be kept ata high value

095 le 119872119888119896119906119872119888119896119889

le 1 (40)

where Mcku and Mckd are the maximum crank torque whenplunger is at upstroke and downstroke respectively Nsdotm

(b) Ground Device Carrying Capacity The suspensionpoint load crank torque and motor torque at anytime [0 119879] do not go beyond the allowable rangemax(F119877119871) min(F119877119871) max(119872119888) min(119872119888) max(119872119890119889)min(119872119890119889) for the given type

min (119865119877119871) le 119865119877119871 le max (119865119877119871)min (119872119888119896) le 119872119888119896 le max (119872119888119896)min (119872119890119889) le 119872119890119889 le max (119872119890119889)

(41)

(c) Rod String StrengthThemaximumandminimum stress ofany point x along the rod string does not exceed permissiblestress range in one cycle

[120590min] le 120590119903119904 le [120590max] (42)

(d) Swabbing Parameters Each oil well has different limit onthe swabbing parameters in accordance with the device typeand actual operation hence their allowable variation rangesare adjustable

Figure 8 Dynamometer sensor

44 Optimization Algorithm In summary this multivariableoptimization model is established with nonlinear restrictionand nonlinear objective function So as to seek the bestresults the genetic algorithm is applied to solve it

5 Test and Verification

Surface dynamometer card is a closed graph recordingpolished rod loads versus rod displacement over a SRPScycle which is generally collected and taken as an indexto estimate the operation of SRPS In this paper based onthe dynamometer sensor shown in Figure 8 four test wellsare used to validate the improved SRPS model The oil wellparameters are listed in Table 1 and the simulation and fieldtest results are given in Figure 9 According to the plottedcurves the simulation results are basically consistent withthat in measured and the improved SRPS model is accurateenough to be proposed for engineering practices

6 Dynamic Response Comparison

Pump dynamometer card is a closed graph recording plungerloads versus plunger displacement over a SRPS cycle It isdetermined by multifactors such as stroke stroke frequencypump diameter pump depth dynamic liquid level and theother oil well parameters The difference between the currentSRPS model and the improved SRPS model proposed in thispaper is whether to consider the PFFP In view of this theoil operating status is divided into two forms as follows(1) the fluid always keeps pace with the plunger when itis being sucked into the pump (2) the fluid is unable tofollowwith the plungerThen two oil wells are selected well1with sufficient OWD and well2 with insufficient OWDFigure 10 describes the plunger and fluid velocity duringupstroke based on the improved SRPS model It can beconcluded that when the well has sufficient OWD and thefluid possesses good ability of keeping pace with the plungerwhen the well has insufficient OWD the fluid velocity lagsbehind the one of plunger at the beginning whereas it is

Mathematical Problems in Engineering 11

Table 1 Oil well basic parameters

Well 1 2 3 4Motor type YD280S-8 Y250M-6 Y250M-6 YD280S-6Pumping unit type CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HBStroke length (m) 3 3 3 3Stroke frequency (minminus1) 35 3 4 6Pump diameter (mm) 57 44 38 44Pump clearance level 1 2 2 1Pump depth (m) 890 1377 1138 1430Sucker-rod string (mm timesm) 25lowast890 19lowast673+22lowast704 19lowast672+22lowast466 19lowast720+22lowast710middle depth of reservoir (m) 1000 1492 1569 1600Crude oil density (kgm3) 795 857 857 850Water content () 95 97 92 98Dynamic liquid level (m) 880 1347 757 1340Casing pressure (Pa) 02 03 03 02Oil pressure (Pa) 03 03 03 02Fluid dynamic viscosity (Pasdots) 0006 0007 0007 0006Gas oil ratio (m3 m3) 20 19 40 80Plunger length (m) 12 12 12 12Clearance length (m) 05 05 05 05

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(a) Well 1

1 2 30Polished rod displacement (m)

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

simulationmeasured

(b) Well 2

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(c) Well 3

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(d) Well 4Figure 9 Suspension point dynamometer card comparison between the simulated and measured

12 Mathematical Problems in Engineering

FluidPlunger

minus05

00

05

10

15Ve

loci

ty(

ms

)

2 4 60Time (s)

(a) Well 1

minus02

00

02

04

06

Velo

city

(m)

4 8 120Time (s)

FluidPlunger

(b) Well 2

Figure 10 Plunger and fluid velocity

Current SRPS model 970Improved SRPS model 981

minus10

0

10

20

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(a) Well 1

Current SRPS model 724Improved SRPS model 775

minus10

0

10

20

40

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(b) Well 2

Figure 11 Comparison of pump dynamometer card

faster after a certain time period Then the comparisons ofpump dynamometer card are executed by the two modelsmeanwhile their individual pump fullness result is given atthe end of the legend (Figure 11)

Pump dynamometer card is very important method andnormally used to diagnose the pump operations particularlywhose shape is more similar to a rectangle the pump iscloser to the fullness [28] Hence from Figure 11 it canbe known that the pump fullness of current SRPS modelis higher than that of the improved one depending on thequalitative judgment and this conclusion is consistent withthe quantitative calculation result meanwhile this gap forthe well 2 is bigger than well 1 According the above

description we can know that the pump fullness calculationresult is on the high side if the PFFP is not considered andthis phenomenon will becomemore obvious for the well withinsufficient OWD

The load presented by the left upper right and lowerborderline of pump dynamometer card is the results of pumpmoving from phase 1 to 4 in turn Therefore its upperborderline describes the pump load when fluid is pumpedinto the barrel From Figure 11(a) it can be found thatthe upper borderline load simulated by the improved SRPSmodel is significantly larger than the one of current ForFigure 11(b) this difference can be neglected Based on (1)and (2) the pump pressure keeps constant as ps when fluid is

Mathematical Problems in Engineering 13

Before optimizationAfter optimization

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

1 2 30Polished rod displacement (m)

(a) Suspension point dynamometer card

minus10

0

10

20

30

40

Cran

k to

rque

(kNmiddotm

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(b) Crank torque

0

5

10

15

20

Mot

or in

put p

ower

(kW

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(c) Motor input power

Figure 12 Dynamic response comparison before and after optimization

sucked into the pump and then the corresponding pump loadis a fixed value Due to the new model takes into account ofPFFP a pressure drop will be brought which will vary withthe fluid velocity Combined with (1) this pressure drop willlead to the pump load increases Figure 10 shows that thefluid velocity of well 1 is larger than that of well 2 andthis is the reason why the difference of upper borderline loadbetween the two models is obvious This illustrates the erroron the upper borderline load of pump dynamometer carddepending on the velocity at which the fluid flows into thepump

7 Optimization Test

Based on the above models a simulation and optimizationsoftware is developed byMATLABOnewell with insufficient

OWD is tested Its original swabbing parameters are asfollows stroke S is 3 m stroke frequency ns is 3 minminus1pump diameter Dd is 57mm pump depth Lpd is 900m crankbalance radius rc is 12 m and rod string combination djtimesLj is 25 mm times 524 m+22 mm times 376 m And its swabbingparameters after optimizing are as follows stroke S is 3 mstroke frequency ns is 25 minminus1 pump diameterDd is 57mmpump depth Lpd is 990 m crank balance radius rc is 09 mand rod string combination djtimes Lj is 22 mm times 426 m+19 mmtimes 564 m The comparisons before and after optimization areshown in Figure 12 and Table 2

From the above comparison results some conclusions areobtained

(1) After optimization the maximum and minimumof suspension point load are all decreased and the load

14 Mathematical Problems in Engineering

Table 2 Specific parameters comparison before and after optimization

Suspension point load Crank torque Motor input power PumpfullnessMaximum Minimum Amplitude Standard

deviationBalancedegree Maximum Minimum mean

Beforeoptimization 555 kN 190 kN 365 kN 107 kN 854 159 kW 09 kW 62 kW 725

Afteroptimization 441 kN 132 kN 309 kN 94 kN 978 106 kW 10 kW 54 kW 908

darr 205 darr 305 darr 153 darr 121 uarr 145 darr 333 uarr 111 darr 129 uarr 252 amplitude is lowered by 153 It can contribute to enhancethe rod string life and prolong the maintenance period

(2) After optimization the standard deviation of cranktorque is reduced by 121 and its balance degree is raisedby 145 It illustrates that the load torque fluctuation is cutdown which canminimize damage to transmission parts andimprove the motor efficiency

(3) After optimization the pump fullness is improvedby 252 It is conducive to improve pump efficiency andproduction

(4) After optimization it plays a role of peak shavingand valley filling for motor input power and extends theoperational life meanwhile power saving rate is 129

8 Conclusions

(1) In this article an improved SRPS model is presentedconsidering the couple effect of pumping fluid and plungermotion on the dynamic response of SRPS instead of theexisting models that assume the pumping fluid volumeis always equal to plunger travelling volume The Runge-Kutta method is applied to solve the whole system modeltrough transforming the rod string longitudinal vibrationequation into ordinary differential equations And the SRPSmodelrsquos precision has been validated by adopting surfacedynamometer card

(2) Two oil wells are served to compare the differencebetween the current SRPS model and the improved oneThe results indicate that the current SRPS model is relativelylow in calculating pump fullness and this gap will increasewith the reduction of OWD the influence of fluid flowinginto the pump on pump load cannot be ignored when thefluid velocity is high Therefore the PFFP is necessary to beconsidered in order to improve the simulation accuracy

(3) On the basis of the improved SRPS model a multitar-get optimization model is proposed in purpose of improvingproduction decreasing load fluctuation and saving energy Bycomparison the optimal scheme can achieve the decreasingof maximum and minimum suspension point load cranktorque fluctuation and energy consumption as well asimproving the system balance degree and pump fullness Insummary it improves the dynamic behavior of SRPS

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

National Natural Science Foundation of China (Grantno 51174175) China Scholarship Council (Grant no201708130108) Hebei Natural Science Foundation (Grant noE201703101) are acknowledged

References

[1] T A Aliev A H Rzayev G A Guluyev T A Alizada and NE Rzayeva ldquoRobust technology and system for management ofsucker rod pumping units in oil wellsrdquoMechanical Systems andSignal Processing vol 99 pp 47ndash56 2018

[2] S Gibbs ldquoPredicting the Behavior of Sucker-Rod PumpingSystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 1963

[3] D R Doty and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo SPE Journal vol 23 no 1 pp 33ndash41 1983

[4] I N Shardakov and I NWasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[5] S G Gibbs ldquoComputing gearbox torque and motor loading forbeam pumping units with consideration of inertia effectsrdquo SPEJ vol 27 pp 1153ndash1159 1975

[6] J Svinos ldquoExact Kinematic Analysis of Pumping Unitsrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionSan Francisco California 1983

[7] D Schafer and J Jennings ldquoAn Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical Modelsrdquoin Proceedings of the SPE Annual Technical Conference andExhibition pp 27ndash30 Dallas Texas 1987

[8] G Takacs Sucker-Rod PumpingManual Pennwell Books Tulsa2003

[9] G W Wang S S Rahman and G Y Yang ldquoAn improvedmodel for the sucker rod pumping systemrdquo in Proceedings of the11thAustralasian FluidMechanics Conference pp 14ndash18HobartAustralia December 1992

[10] S D L Lekia andR D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystemsrdquo SPE Production amp Facilities vol 10 no 1 pp 26ndash331995

[11] J Lea and P Pattillo ldquoInterpretation of Calculated Forces onSucker Rodsrdquo SPE Production amp Facilities vol 10 no 01 pp 41ndash45 1995

Mathematical Problems in Engineering 15

[12] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[13] L Guo-hua H Shun-li Y Zhi et al ldquoA prediction model fora new deep-rod pumping systemrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 75ndash80 2011

[14] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[15] O Becerra J Gamboa and F Kenyery ldquoModelling a DoublePiston Pumprdquo in Proceedings of the SPE International ThermalOperations and Heavy Oil Symposium and International Hor-izontal Well Technology Conference Calgary Alberta Canada2002

[16] A L Podio J Gomez A J Mansure et al ldquoLaboratoryinstrumented sucker-rod pumprdquo J Pet Technol vol 53 no 05pp 104ndash113 2003

[17] Z H Gu H Q Peng and H Y Geng ldquoAnalysis and mea-surement of gas effect on pumping efficiencyrdquoChina PetroleumMachinery vol 34 no 02 pp 64ndash69 2006

[18] M Xing ldquoResponse analysis of longitudinal vibration of suckerrod string considering rod bucklingrdquo Advances in EngineeringSoftware vol 99 pp 49ndash58 2016

[19] S Miska A Sharaki and J M Rajtar ldquoA simple model forcomputer-aided optimization and design of sucker-rod pump-ing systemsrdquo Journal of Petroleum Science and Engineering vol17 no 3-4 pp 303ndash312 1997

[20] L S Firu T Chelu and C Militaru-Petre ldquoAmodern approachto the optimum design of sucker-rod pumping systemrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash9 DenverColorado 2003

[21] X F Liu and Y G Qi ldquoA modern approach to the selectionof sucker rod pumping systems in CBM wellsrdquo Journal ofPetroleum Science and Engineering vol 76 no 3-4 pp 100ndash1082011

[22] S M Dong N N Feng and Z J Ma ldquoSimulating maximumof system efficiency of rod pumping wellsrdquo Journal of SystemSimulation vol 20 no 13 pp 3533ndash3537 2008

[23] M Xing and S Dong ldquoA New Simulation Model for a Beam-Pumping System Applied in Energy Saving and Resource-Consumption Reductionrdquo SPE Production amp Operations vol30 no 02 pp 130ndash140 2015

[24] Y XWu Y Li andH LiuElectric Motor andDriving ChemicalIndustry Press Beijing 2008

[25] S M Dong Computer Simulation of Dynamic Parameters ofRod Pumping System Optimization Petroleum Industry PressBeijing Chinese 2003

[26] F Yavuz J F Lea J C Cox et al ldquoWave equation simulationof fluid pound and gas interferencerdquo in Proceedings of the SPEProduction Operations Symposium pp 16ndash19 Oklahoma CityOklahoma April 2005

[27] D Y Wang and H Z Liu ldquoDynamic modeling and analysis ofsucker rod pumping system in a directional well Mechanismand Machine Sciencerdquo in Proceedings of the ASIAN MMS 2016amp CCMMS pp 1115ndash1127 2017

[28] F M A Barreto M Tygel A F Rocha and C K MorookaldquoAutomatic downhole card generation and classificationrdquo inProceedings of the SPE Annual Technical Conference pp 6ndash9Denver Colorado 1996

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Page 9: An Improved Sucker Rod Pumping System Model and Swabbing ...downloads.hindawi.com/journals/mpe/2018/4746210.pdf · An Improved Sucker Rod Pumping System Model and Swabbing Parameters

Mathematical Problems in Engineering 9

119865119877119871 (119905) = 119864119903119860119903radic 2120588119903119860119903119871119903

119873sum119894=1

(2119894 minus 1) 1205872119871119903

1199091198941 (119905) + 119865119903

119906119901 (119905) = 119906119886 (119905) minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198941 (119905) minus 119894=2lowast119895sum119894=1

1199091198941 (119905))

V119901 (119905) = 119886 (119905) minus radic 2119860119903120588119903119871119903

(119894=2lowast119895+1sum119894=1

1199091198942 (119905) minus 119894=2lowast119895sum119894=1

1199091198942 (119905))1198781199051198861199051199061199041 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199042 119886119891119905119890119903 119900119901119890119899119894119899119892 119905ℎ119890 119904119905119886119899119889119894119899119892 V119886119897V119890119889119901119889119905 = ((V119891 sdot 119860119901 + 119902) (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)]sdot V119901

1198781199051198861199051199061199043 119887119890119891119900119903119890 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = (119902 (119860119901 sdot V119901)) (1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892) + (103119860119901 sdot 119877119869) (119873 sdot 119881119898119900119897)) minus 1 (119906119901 minus 119871119891 minus 119871V + 119871119900119892)[119901minus1 minus 1119885 sdot 119889119885119889119901 + 103120572119873 sdot 119885119879119901119904119905119879119904119905119901 sdot (119871V + 119871119891 + 119871119900 minus 119871119900119892) 119904119901119881119898119900119897 sdot 119860119901 sdot (1 minus 120576119908)] sdot V119901

1198781199051198861199051199061199044 119886119891119905119890119903 119900119901119890119899119894119899119892 119905ℎ119890 119905119903119886V119890119897119894119899119892 V119886119897V119890119889119901119889119905 = 0119889119871119891119889119905 = V119891

119889V119891119889119905 = (1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V119860119891119860119901

)minus1119901119904120588 minus 119901120588 + V21198912 [119860

21199011198602119891

minus 119860211990112057621198602

119891

minus 64120583 (1198710 minus 119881119900119892119860119901 + 119871119891)1198632V119891120588 minus 1]minus 119892(1198710 minus 119881119900119892119860119901

+ 119871119891 + 119871V)(35)

4 Optimization Model

41 Optimization Goal As mostly developed oil-field movesinto the mid and late stage and the OWD begins todecline gradually energy-saving production-increasing andreducing load variation as much as possible are particularlyimportant Then we take pump fullness epf suspension pointload amplitude FRLA crank torque standard deviation Mcsdand motor input power average 119875119898 as the optimization goalto build a multitarget model Suspension point load andcrank torque can be solved directly by (35) The motor input

power and pump fullness calculation formula are deduced asfollows

119875119898 = 119872119890119889119898 + 1198750

+ [( 1120578H minus 1)119875119867 minus 1198750](119872119890119889119898119875119867

)2 (36)

119890119901119891 = 119871119891 minus int119905119906

0119902119889119905 minus (119871119900 minus 119871119900119892)119906119901119906

(37)

10 Mathematical Problems in Engineering

where P0 is the no-load power of motor kW PH is the motorrated power kW 120578H is the motor rated efficiency kW upuis the plunger displacement when it is arriving at dead topcenter

So the objective function is

Ω(119890119901119891 (X) 119865119877119871119860 (X) 119872119888119904119889 (X) 119875119898 (X))= 1198701119890119901119891 (X) + 1198702119865119877119871119860 (X) + 1198703119872119888119904119889 (X)+ 1198704119875119898 (X)

(38)

where K1 K2 K3 and K4 are the weight coefficients

42 Design Variables The objective function can beexpressed as the function of the swabbing parameters whenthe SRPS type and oil well basic parameters are confirmedIn this paper the swabbing parameters denote stroke Sstroke frequency ns pump diameter Dd pump depth Lpdcrank balance radius rc and rod string combination (the jthrod string diameter and length are dj and Lj respectively119896 = 1 2 119898) Then the design variables are shown asfollows

X = 119878 119899119904 119863119889 119871119901119889 (119889119895 119871119895 119895 = 1 2 sdot sdot sdot 119898) (39)

43 Constraints

(a) Crank Balance Degree Crank balance degree indicates theload fluctuation to a certain degree and it needs to be kept ata high value

095 le 119872119888119896119906119872119888119896119889

le 1 (40)

where Mcku and Mckd are the maximum crank torque whenplunger is at upstroke and downstroke respectively Nsdotm

(b) Ground Device Carrying Capacity The suspensionpoint load crank torque and motor torque at anytime [0 119879] do not go beyond the allowable rangemax(F119877119871) min(F119877119871) max(119872119888) min(119872119888) max(119872119890119889)min(119872119890119889) for the given type

min (119865119877119871) le 119865119877119871 le max (119865119877119871)min (119872119888119896) le 119872119888119896 le max (119872119888119896)min (119872119890119889) le 119872119890119889 le max (119872119890119889)

(41)

(c) Rod String StrengthThemaximumandminimum stress ofany point x along the rod string does not exceed permissiblestress range in one cycle

[120590min] le 120590119903119904 le [120590max] (42)

(d) Swabbing Parameters Each oil well has different limit onthe swabbing parameters in accordance with the device typeand actual operation hence their allowable variation rangesare adjustable

Figure 8 Dynamometer sensor

44 Optimization Algorithm In summary this multivariableoptimization model is established with nonlinear restrictionand nonlinear objective function So as to seek the bestresults the genetic algorithm is applied to solve it

5 Test and Verification

Surface dynamometer card is a closed graph recordingpolished rod loads versus rod displacement over a SRPScycle which is generally collected and taken as an indexto estimate the operation of SRPS In this paper based onthe dynamometer sensor shown in Figure 8 four test wellsare used to validate the improved SRPS model The oil wellparameters are listed in Table 1 and the simulation and fieldtest results are given in Figure 9 According to the plottedcurves the simulation results are basically consistent withthat in measured and the improved SRPS model is accurateenough to be proposed for engineering practices

6 Dynamic Response Comparison

Pump dynamometer card is a closed graph recording plungerloads versus plunger displacement over a SRPS cycle It isdetermined by multifactors such as stroke stroke frequencypump diameter pump depth dynamic liquid level and theother oil well parameters The difference between the currentSRPS model and the improved SRPS model proposed in thispaper is whether to consider the PFFP In view of this theoil operating status is divided into two forms as follows(1) the fluid always keeps pace with the plunger when itis being sucked into the pump (2) the fluid is unable tofollowwith the plungerThen two oil wells are selected well1with sufficient OWD and well2 with insufficient OWDFigure 10 describes the plunger and fluid velocity duringupstroke based on the improved SRPS model It can beconcluded that when the well has sufficient OWD and thefluid possesses good ability of keeping pace with the plungerwhen the well has insufficient OWD the fluid velocity lagsbehind the one of plunger at the beginning whereas it is

Mathematical Problems in Engineering 11

Table 1 Oil well basic parameters

Well 1 2 3 4Motor type YD280S-8 Y250M-6 Y250M-6 YD280S-6Pumping unit type CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HBStroke length (m) 3 3 3 3Stroke frequency (minminus1) 35 3 4 6Pump diameter (mm) 57 44 38 44Pump clearance level 1 2 2 1Pump depth (m) 890 1377 1138 1430Sucker-rod string (mm timesm) 25lowast890 19lowast673+22lowast704 19lowast672+22lowast466 19lowast720+22lowast710middle depth of reservoir (m) 1000 1492 1569 1600Crude oil density (kgm3) 795 857 857 850Water content () 95 97 92 98Dynamic liquid level (m) 880 1347 757 1340Casing pressure (Pa) 02 03 03 02Oil pressure (Pa) 03 03 03 02Fluid dynamic viscosity (Pasdots) 0006 0007 0007 0006Gas oil ratio (m3 m3) 20 19 40 80Plunger length (m) 12 12 12 12Clearance length (m) 05 05 05 05

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(a) Well 1

1 2 30Polished rod displacement (m)

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

simulationmeasured

(b) Well 2

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(c) Well 3

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(d) Well 4Figure 9 Suspension point dynamometer card comparison between the simulated and measured

12 Mathematical Problems in Engineering

FluidPlunger

minus05

00

05

10

15Ve

loci

ty(

ms

)

2 4 60Time (s)

(a) Well 1

minus02

00

02

04

06

Velo

city

(m)

4 8 120Time (s)

FluidPlunger

(b) Well 2

Figure 10 Plunger and fluid velocity

Current SRPS model 970Improved SRPS model 981

minus10

0

10

20

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(a) Well 1

Current SRPS model 724Improved SRPS model 775

minus10

0

10

20

40

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(b) Well 2

Figure 11 Comparison of pump dynamometer card

faster after a certain time period Then the comparisons ofpump dynamometer card are executed by the two modelsmeanwhile their individual pump fullness result is given atthe end of the legend (Figure 11)

Pump dynamometer card is very important method andnormally used to diagnose the pump operations particularlywhose shape is more similar to a rectangle the pump iscloser to the fullness [28] Hence from Figure 11 it canbe known that the pump fullness of current SRPS modelis higher than that of the improved one depending on thequalitative judgment and this conclusion is consistent withthe quantitative calculation result meanwhile this gap forthe well 2 is bigger than well 1 According the above

description we can know that the pump fullness calculationresult is on the high side if the PFFP is not considered andthis phenomenon will becomemore obvious for the well withinsufficient OWD

The load presented by the left upper right and lowerborderline of pump dynamometer card is the results of pumpmoving from phase 1 to 4 in turn Therefore its upperborderline describes the pump load when fluid is pumpedinto the barrel From Figure 11(a) it can be found thatthe upper borderline load simulated by the improved SRPSmodel is significantly larger than the one of current ForFigure 11(b) this difference can be neglected Based on (1)and (2) the pump pressure keeps constant as ps when fluid is

Mathematical Problems in Engineering 13

Before optimizationAfter optimization

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

1 2 30Polished rod displacement (m)

(a) Suspension point dynamometer card

minus10

0

10

20

30

40

Cran

k to

rque

(kNmiddotm

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(b) Crank torque

0

5

10

15

20

Mot

or in

put p

ower

(kW

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(c) Motor input power

Figure 12 Dynamic response comparison before and after optimization

sucked into the pump and then the corresponding pump loadis a fixed value Due to the new model takes into account ofPFFP a pressure drop will be brought which will vary withthe fluid velocity Combined with (1) this pressure drop willlead to the pump load increases Figure 10 shows that thefluid velocity of well 1 is larger than that of well 2 andthis is the reason why the difference of upper borderline loadbetween the two models is obvious This illustrates the erroron the upper borderline load of pump dynamometer carddepending on the velocity at which the fluid flows into thepump

7 Optimization Test

Based on the above models a simulation and optimizationsoftware is developed byMATLABOnewell with insufficient

OWD is tested Its original swabbing parameters are asfollows stroke S is 3 m stroke frequency ns is 3 minminus1pump diameter Dd is 57mm pump depth Lpd is 900m crankbalance radius rc is 12 m and rod string combination djtimesLj is 25 mm times 524 m+22 mm times 376 m And its swabbingparameters after optimizing are as follows stroke S is 3 mstroke frequency ns is 25 minminus1 pump diameterDd is 57mmpump depth Lpd is 990 m crank balance radius rc is 09 mand rod string combination djtimes Lj is 22 mm times 426 m+19 mmtimes 564 m The comparisons before and after optimization areshown in Figure 12 and Table 2

From the above comparison results some conclusions areobtained

(1) After optimization the maximum and minimumof suspension point load are all decreased and the load

14 Mathematical Problems in Engineering

Table 2 Specific parameters comparison before and after optimization

Suspension point load Crank torque Motor input power PumpfullnessMaximum Minimum Amplitude Standard

deviationBalancedegree Maximum Minimum mean

Beforeoptimization 555 kN 190 kN 365 kN 107 kN 854 159 kW 09 kW 62 kW 725

Afteroptimization 441 kN 132 kN 309 kN 94 kN 978 106 kW 10 kW 54 kW 908

darr 205 darr 305 darr 153 darr 121 uarr 145 darr 333 uarr 111 darr 129 uarr 252 amplitude is lowered by 153 It can contribute to enhancethe rod string life and prolong the maintenance period

(2) After optimization the standard deviation of cranktorque is reduced by 121 and its balance degree is raisedby 145 It illustrates that the load torque fluctuation is cutdown which canminimize damage to transmission parts andimprove the motor efficiency

(3) After optimization the pump fullness is improvedby 252 It is conducive to improve pump efficiency andproduction

(4) After optimization it plays a role of peak shavingand valley filling for motor input power and extends theoperational life meanwhile power saving rate is 129

8 Conclusions

(1) In this article an improved SRPS model is presentedconsidering the couple effect of pumping fluid and plungermotion on the dynamic response of SRPS instead of theexisting models that assume the pumping fluid volumeis always equal to plunger travelling volume The Runge-Kutta method is applied to solve the whole system modeltrough transforming the rod string longitudinal vibrationequation into ordinary differential equations And the SRPSmodelrsquos precision has been validated by adopting surfacedynamometer card

(2) Two oil wells are served to compare the differencebetween the current SRPS model and the improved oneThe results indicate that the current SRPS model is relativelylow in calculating pump fullness and this gap will increasewith the reduction of OWD the influence of fluid flowinginto the pump on pump load cannot be ignored when thefluid velocity is high Therefore the PFFP is necessary to beconsidered in order to improve the simulation accuracy

(3) On the basis of the improved SRPS model a multitar-get optimization model is proposed in purpose of improvingproduction decreasing load fluctuation and saving energy Bycomparison the optimal scheme can achieve the decreasingof maximum and minimum suspension point load cranktorque fluctuation and energy consumption as well asimproving the system balance degree and pump fullness Insummary it improves the dynamic behavior of SRPS

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

National Natural Science Foundation of China (Grantno 51174175) China Scholarship Council (Grant no201708130108) Hebei Natural Science Foundation (Grant noE201703101) are acknowledged

References

[1] T A Aliev A H Rzayev G A Guluyev T A Alizada and NE Rzayeva ldquoRobust technology and system for management ofsucker rod pumping units in oil wellsrdquoMechanical Systems andSignal Processing vol 99 pp 47ndash56 2018

[2] S Gibbs ldquoPredicting the Behavior of Sucker-Rod PumpingSystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 1963

[3] D R Doty and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo SPE Journal vol 23 no 1 pp 33ndash41 1983

[4] I N Shardakov and I NWasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[5] S G Gibbs ldquoComputing gearbox torque and motor loading forbeam pumping units with consideration of inertia effectsrdquo SPEJ vol 27 pp 1153ndash1159 1975

[6] J Svinos ldquoExact Kinematic Analysis of Pumping Unitsrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionSan Francisco California 1983

[7] D Schafer and J Jennings ldquoAn Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical Modelsrdquoin Proceedings of the SPE Annual Technical Conference andExhibition pp 27ndash30 Dallas Texas 1987

[8] G Takacs Sucker-Rod PumpingManual Pennwell Books Tulsa2003

[9] G W Wang S S Rahman and G Y Yang ldquoAn improvedmodel for the sucker rod pumping systemrdquo in Proceedings of the11thAustralasian FluidMechanics Conference pp 14ndash18HobartAustralia December 1992

[10] S D L Lekia andR D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystemsrdquo SPE Production amp Facilities vol 10 no 1 pp 26ndash331995

[11] J Lea and P Pattillo ldquoInterpretation of Calculated Forces onSucker Rodsrdquo SPE Production amp Facilities vol 10 no 01 pp 41ndash45 1995

Mathematical Problems in Engineering 15

[12] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[13] L Guo-hua H Shun-li Y Zhi et al ldquoA prediction model fora new deep-rod pumping systemrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 75ndash80 2011

[14] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[15] O Becerra J Gamboa and F Kenyery ldquoModelling a DoublePiston Pumprdquo in Proceedings of the SPE International ThermalOperations and Heavy Oil Symposium and International Hor-izontal Well Technology Conference Calgary Alberta Canada2002

[16] A L Podio J Gomez A J Mansure et al ldquoLaboratoryinstrumented sucker-rod pumprdquo J Pet Technol vol 53 no 05pp 104ndash113 2003

[17] Z H Gu H Q Peng and H Y Geng ldquoAnalysis and mea-surement of gas effect on pumping efficiencyrdquoChina PetroleumMachinery vol 34 no 02 pp 64ndash69 2006

[18] M Xing ldquoResponse analysis of longitudinal vibration of suckerrod string considering rod bucklingrdquo Advances in EngineeringSoftware vol 99 pp 49ndash58 2016

[19] S Miska A Sharaki and J M Rajtar ldquoA simple model forcomputer-aided optimization and design of sucker-rod pump-ing systemsrdquo Journal of Petroleum Science and Engineering vol17 no 3-4 pp 303ndash312 1997

[20] L S Firu T Chelu and C Militaru-Petre ldquoAmodern approachto the optimum design of sucker-rod pumping systemrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash9 DenverColorado 2003

[21] X F Liu and Y G Qi ldquoA modern approach to the selectionof sucker rod pumping systems in CBM wellsrdquo Journal ofPetroleum Science and Engineering vol 76 no 3-4 pp 100ndash1082011

[22] S M Dong N N Feng and Z J Ma ldquoSimulating maximumof system efficiency of rod pumping wellsrdquo Journal of SystemSimulation vol 20 no 13 pp 3533ndash3537 2008

[23] M Xing and S Dong ldquoA New Simulation Model for a Beam-Pumping System Applied in Energy Saving and Resource-Consumption Reductionrdquo SPE Production amp Operations vol30 no 02 pp 130ndash140 2015

[24] Y XWu Y Li andH LiuElectric Motor andDriving ChemicalIndustry Press Beijing 2008

[25] S M Dong Computer Simulation of Dynamic Parameters ofRod Pumping System Optimization Petroleum Industry PressBeijing Chinese 2003

[26] F Yavuz J F Lea J C Cox et al ldquoWave equation simulationof fluid pound and gas interferencerdquo in Proceedings of the SPEProduction Operations Symposium pp 16ndash19 Oklahoma CityOklahoma April 2005

[27] D Y Wang and H Z Liu ldquoDynamic modeling and analysis ofsucker rod pumping system in a directional well Mechanismand Machine Sciencerdquo in Proceedings of the ASIAN MMS 2016amp CCMMS pp 1115ndash1127 2017

[28] F M A Barreto M Tygel A F Rocha and C K MorookaldquoAutomatic downhole card generation and classificationrdquo inProceedings of the SPE Annual Technical Conference pp 6ndash9Denver Colorado 1996

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Mathematical Problems in Engineering

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Page 10: An Improved Sucker Rod Pumping System Model and Swabbing ...downloads.hindawi.com/journals/mpe/2018/4746210.pdf · An Improved Sucker Rod Pumping System Model and Swabbing Parameters

10 Mathematical Problems in Engineering

where P0 is the no-load power of motor kW PH is the motorrated power kW 120578H is the motor rated efficiency kW upuis the plunger displacement when it is arriving at dead topcenter

So the objective function is

Ω(119890119901119891 (X) 119865119877119871119860 (X) 119872119888119904119889 (X) 119875119898 (X))= 1198701119890119901119891 (X) + 1198702119865119877119871119860 (X) + 1198703119872119888119904119889 (X)+ 1198704119875119898 (X)

(38)

where K1 K2 K3 and K4 are the weight coefficients

42 Design Variables The objective function can beexpressed as the function of the swabbing parameters whenthe SRPS type and oil well basic parameters are confirmedIn this paper the swabbing parameters denote stroke Sstroke frequency ns pump diameter Dd pump depth Lpdcrank balance radius rc and rod string combination (the jthrod string diameter and length are dj and Lj respectively119896 = 1 2 119898) Then the design variables are shown asfollows

X = 119878 119899119904 119863119889 119871119901119889 (119889119895 119871119895 119895 = 1 2 sdot sdot sdot 119898) (39)

43 Constraints

(a) Crank Balance Degree Crank balance degree indicates theload fluctuation to a certain degree and it needs to be kept ata high value

095 le 119872119888119896119906119872119888119896119889

le 1 (40)

where Mcku and Mckd are the maximum crank torque whenplunger is at upstroke and downstroke respectively Nsdotm

(b) Ground Device Carrying Capacity The suspensionpoint load crank torque and motor torque at anytime [0 119879] do not go beyond the allowable rangemax(F119877119871) min(F119877119871) max(119872119888) min(119872119888) max(119872119890119889)min(119872119890119889) for the given type

min (119865119877119871) le 119865119877119871 le max (119865119877119871)min (119872119888119896) le 119872119888119896 le max (119872119888119896)min (119872119890119889) le 119872119890119889 le max (119872119890119889)

(41)

(c) Rod String StrengthThemaximumandminimum stress ofany point x along the rod string does not exceed permissiblestress range in one cycle

[120590min] le 120590119903119904 le [120590max] (42)

(d) Swabbing Parameters Each oil well has different limit onthe swabbing parameters in accordance with the device typeand actual operation hence their allowable variation rangesare adjustable

Figure 8 Dynamometer sensor

44 Optimization Algorithm In summary this multivariableoptimization model is established with nonlinear restrictionand nonlinear objective function So as to seek the bestresults the genetic algorithm is applied to solve it

5 Test and Verification

Surface dynamometer card is a closed graph recordingpolished rod loads versus rod displacement over a SRPScycle which is generally collected and taken as an indexto estimate the operation of SRPS In this paper based onthe dynamometer sensor shown in Figure 8 four test wellsare used to validate the improved SRPS model The oil wellparameters are listed in Table 1 and the simulation and fieldtest results are given in Figure 9 According to the plottedcurves the simulation results are basically consistent withthat in measured and the improved SRPS model is accurateenough to be proposed for engineering practices

6 Dynamic Response Comparison

Pump dynamometer card is a closed graph recording plungerloads versus plunger displacement over a SRPS cycle It isdetermined by multifactors such as stroke stroke frequencypump diameter pump depth dynamic liquid level and theother oil well parameters The difference between the currentSRPS model and the improved SRPS model proposed in thispaper is whether to consider the PFFP In view of this theoil operating status is divided into two forms as follows(1) the fluid always keeps pace with the plunger when itis being sucked into the pump (2) the fluid is unable tofollowwith the plungerThen two oil wells are selected well1with sufficient OWD and well2 with insufficient OWDFigure 10 describes the plunger and fluid velocity duringupstroke based on the improved SRPS model It can beconcluded that when the well has sufficient OWD and thefluid possesses good ability of keeping pace with the plungerwhen the well has insufficient OWD the fluid velocity lagsbehind the one of plunger at the beginning whereas it is

Mathematical Problems in Engineering 11

Table 1 Oil well basic parameters

Well 1 2 3 4Motor type YD280S-8 Y250M-6 Y250M-6 YD280S-6Pumping unit type CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HBStroke length (m) 3 3 3 3Stroke frequency (minminus1) 35 3 4 6Pump diameter (mm) 57 44 38 44Pump clearance level 1 2 2 1Pump depth (m) 890 1377 1138 1430Sucker-rod string (mm timesm) 25lowast890 19lowast673+22lowast704 19lowast672+22lowast466 19lowast720+22lowast710middle depth of reservoir (m) 1000 1492 1569 1600Crude oil density (kgm3) 795 857 857 850Water content () 95 97 92 98Dynamic liquid level (m) 880 1347 757 1340Casing pressure (Pa) 02 03 03 02Oil pressure (Pa) 03 03 03 02Fluid dynamic viscosity (Pasdots) 0006 0007 0007 0006Gas oil ratio (m3 m3) 20 19 40 80Plunger length (m) 12 12 12 12Clearance length (m) 05 05 05 05

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(a) Well 1

1 2 30Polished rod displacement (m)

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

simulationmeasured

(b) Well 2

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(c) Well 3

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(d) Well 4Figure 9 Suspension point dynamometer card comparison between the simulated and measured

12 Mathematical Problems in Engineering

FluidPlunger

minus05

00

05

10

15Ve

loci

ty(

ms

)

2 4 60Time (s)

(a) Well 1

minus02

00

02

04

06

Velo

city

(m)

4 8 120Time (s)

FluidPlunger

(b) Well 2

Figure 10 Plunger and fluid velocity

Current SRPS model 970Improved SRPS model 981

minus10

0

10

20

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(a) Well 1

Current SRPS model 724Improved SRPS model 775

minus10

0

10

20

40

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(b) Well 2

Figure 11 Comparison of pump dynamometer card

faster after a certain time period Then the comparisons ofpump dynamometer card are executed by the two modelsmeanwhile their individual pump fullness result is given atthe end of the legend (Figure 11)

Pump dynamometer card is very important method andnormally used to diagnose the pump operations particularlywhose shape is more similar to a rectangle the pump iscloser to the fullness [28] Hence from Figure 11 it canbe known that the pump fullness of current SRPS modelis higher than that of the improved one depending on thequalitative judgment and this conclusion is consistent withthe quantitative calculation result meanwhile this gap forthe well 2 is bigger than well 1 According the above

description we can know that the pump fullness calculationresult is on the high side if the PFFP is not considered andthis phenomenon will becomemore obvious for the well withinsufficient OWD

The load presented by the left upper right and lowerborderline of pump dynamometer card is the results of pumpmoving from phase 1 to 4 in turn Therefore its upperborderline describes the pump load when fluid is pumpedinto the barrel From Figure 11(a) it can be found thatthe upper borderline load simulated by the improved SRPSmodel is significantly larger than the one of current ForFigure 11(b) this difference can be neglected Based on (1)and (2) the pump pressure keeps constant as ps when fluid is

Mathematical Problems in Engineering 13

Before optimizationAfter optimization

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

1 2 30Polished rod displacement (m)

(a) Suspension point dynamometer card

minus10

0

10

20

30

40

Cran

k to

rque

(kNmiddotm

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(b) Crank torque

0

5

10

15

20

Mot

or in

put p

ower

(kW

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(c) Motor input power

Figure 12 Dynamic response comparison before and after optimization

sucked into the pump and then the corresponding pump loadis a fixed value Due to the new model takes into account ofPFFP a pressure drop will be brought which will vary withthe fluid velocity Combined with (1) this pressure drop willlead to the pump load increases Figure 10 shows that thefluid velocity of well 1 is larger than that of well 2 andthis is the reason why the difference of upper borderline loadbetween the two models is obvious This illustrates the erroron the upper borderline load of pump dynamometer carddepending on the velocity at which the fluid flows into thepump

7 Optimization Test

Based on the above models a simulation and optimizationsoftware is developed byMATLABOnewell with insufficient

OWD is tested Its original swabbing parameters are asfollows stroke S is 3 m stroke frequency ns is 3 minminus1pump diameter Dd is 57mm pump depth Lpd is 900m crankbalance radius rc is 12 m and rod string combination djtimesLj is 25 mm times 524 m+22 mm times 376 m And its swabbingparameters after optimizing are as follows stroke S is 3 mstroke frequency ns is 25 minminus1 pump diameterDd is 57mmpump depth Lpd is 990 m crank balance radius rc is 09 mand rod string combination djtimes Lj is 22 mm times 426 m+19 mmtimes 564 m The comparisons before and after optimization areshown in Figure 12 and Table 2

From the above comparison results some conclusions areobtained

(1) After optimization the maximum and minimumof suspension point load are all decreased and the load

14 Mathematical Problems in Engineering

Table 2 Specific parameters comparison before and after optimization

Suspension point load Crank torque Motor input power PumpfullnessMaximum Minimum Amplitude Standard

deviationBalancedegree Maximum Minimum mean

Beforeoptimization 555 kN 190 kN 365 kN 107 kN 854 159 kW 09 kW 62 kW 725

Afteroptimization 441 kN 132 kN 309 kN 94 kN 978 106 kW 10 kW 54 kW 908

darr 205 darr 305 darr 153 darr 121 uarr 145 darr 333 uarr 111 darr 129 uarr 252 amplitude is lowered by 153 It can contribute to enhancethe rod string life and prolong the maintenance period

(2) After optimization the standard deviation of cranktorque is reduced by 121 and its balance degree is raisedby 145 It illustrates that the load torque fluctuation is cutdown which canminimize damage to transmission parts andimprove the motor efficiency

(3) After optimization the pump fullness is improvedby 252 It is conducive to improve pump efficiency andproduction

(4) After optimization it plays a role of peak shavingand valley filling for motor input power and extends theoperational life meanwhile power saving rate is 129

8 Conclusions

(1) In this article an improved SRPS model is presentedconsidering the couple effect of pumping fluid and plungermotion on the dynamic response of SRPS instead of theexisting models that assume the pumping fluid volumeis always equal to plunger travelling volume The Runge-Kutta method is applied to solve the whole system modeltrough transforming the rod string longitudinal vibrationequation into ordinary differential equations And the SRPSmodelrsquos precision has been validated by adopting surfacedynamometer card

(2) Two oil wells are served to compare the differencebetween the current SRPS model and the improved oneThe results indicate that the current SRPS model is relativelylow in calculating pump fullness and this gap will increasewith the reduction of OWD the influence of fluid flowinginto the pump on pump load cannot be ignored when thefluid velocity is high Therefore the PFFP is necessary to beconsidered in order to improve the simulation accuracy

(3) On the basis of the improved SRPS model a multitar-get optimization model is proposed in purpose of improvingproduction decreasing load fluctuation and saving energy Bycomparison the optimal scheme can achieve the decreasingof maximum and minimum suspension point load cranktorque fluctuation and energy consumption as well asimproving the system balance degree and pump fullness Insummary it improves the dynamic behavior of SRPS

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

National Natural Science Foundation of China (Grantno 51174175) China Scholarship Council (Grant no201708130108) Hebei Natural Science Foundation (Grant noE201703101) are acknowledged

References

[1] T A Aliev A H Rzayev G A Guluyev T A Alizada and NE Rzayeva ldquoRobust technology and system for management ofsucker rod pumping units in oil wellsrdquoMechanical Systems andSignal Processing vol 99 pp 47ndash56 2018

[2] S Gibbs ldquoPredicting the Behavior of Sucker-Rod PumpingSystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 1963

[3] D R Doty and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo SPE Journal vol 23 no 1 pp 33ndash41 1983

[4] I N Shardakov and I NWasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[5] S G Gibbs ldquoComputing gearbox torque and motor loading forbeam pumping units with consideration of inertia effectsrdquo SPEJ vol 27 pp 1153ndash1159 1975

[6] J Svinos ldquoExact Kinematic Analysis of Pumping Unitsrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionSan Francisco California 1983

[7] D Schafer and J Jennings ldquoAn Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical Modelsrdquoin Proceedings of the SPE Annual Technical Conference andExhibition pp 27ndash30 Dallas Texas 1987

[8] G Takacs Sucker-Rod PumpingManual Pennwell Books Tulsa2003

[9] G W Wang S S Rahman and G Y Yang ldquoAn improvedmodel for the sucker rod pumping systemrdquo in Proceedings of the11thAustralasian FluidMechanics Conference pp 14ndash18HobartAustralia December 1992

[10] S D L Lekia andR D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystemsrdquo SPE Production amp Facilities vol 10 no 1 pp 26ndash331995

[11] J Lea and P Pattillo ldquoInterpretation of Calculated Forces onSucker Rodsrdquo SPE Production amp Facilities vol 10 no 01 pp 41ndash45 1995

Mathematical Problems in Engineering 15

[12] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[13] L Guo-hua H Shun-li Y Zhi et al ldquoA prediction model fora new deep-rod pumping systemrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 75ndash80 2011

[14] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[15] O Becerra J Gamboa and F Kenyery ldquoModelling a DoublePiston Pumprdquo in Proceedings of the SPE International ThermalOperations and Heavy Oil Symposium and International Hor-izontal Well Technology Conference Calgary Alberta Canada2002

[16] A L Podio J Gomez A J Mansure et al ldquoLaboratoryinstrumented sucker-rod pumprdquo J Pet Technol vol 53 no 05pp 104ndash113 2003

[17] Z H Gu H Q Peng and H Y Geng ldquoAnalysis and mea-surement of gas effect on pumping efficiencyrdquoChina PetroleumMachinery vol 34 no 02 pp 64ndash69 2006

[18] M Xing ldquoResponse analysis of longitudinal vibration of suckerrod string considering rod bucklingrdquo Advances in EngineeringSoftware vol 99 pp 49ndash58 2016

[19] S Miska A Sharaki and J M Rajtar ldquoA simple model forcomputer-aided optimization and design of sucker-rod pump-ing systemsrdquo Journal of Petroleum Science and Engineering vol17 no 3-4 pp 303ndash312 1997

[20] L S Firu T Chelu and C Militaru-Petre ldquoAmodern approachto the optimum design of sucker-rod pumping systemrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash9 DenverColorado 2003

[21] X F Liu and Y G Qi ldquoA modern approach to the selectionof sucker rod pumping systems in CBM wellsrdquo Journal ofPetroleum Science and Engineering vol 76 no 3-4 pp 100ndash1082011

[22] S M Dong N N Feng and Z J Ma ldquoSimulating maximumof system efficiency of rod pumping wellsrdquo Journal of SystemSimulation vol 20 no 13 pp 3533ndash3537 2008

[23] M Xing and S Dong ldquoA New Simulation Model for a Beam-Pumping System Applied in Energy Saving and Resource-Consumption Reductionrdquo SPE Production amp Operations vol30 no 02 pp 130ndash140 2015

[24] Y XWu Y Li andH LiuElectric Motor andDriving ChemicalIndustry Press Beijing 2008

[25] S M Dong Computer Simulation of Dynamic Parameters ofRod Pumping System Optimization Petroleum Industry PressBeijing Chinese 2003

[26] F Yavuz J F Lea J C Cox et al ldquoWave equation simulationof fluid pound and gas interferencerdquo in Proceedings of the SPEProduction Operations Symposium pp 16ndash19 Oklahoma CityOklahoma April 2005

[27] D Y Wang and H Z Liu ldquoDynamic modeling and analysis ofsucker rod pumping system in a directional well Mechanismand Machine Sciencerdquo in Proceedings of the ASIAN MMS 2016amp CCMMS pp 1115ndash1127 2017

[28] F M A Barreto M Tygel A F Rocha and C K MorookaldquoAutomatic downhole card generation and classificationrdquo inProceedings of the SPE Annual Technical Conference pp 6ndash9Denver Colorado 1996

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 11: An Improved Sucker Rod Pumping System Model and Swabbing ...downloads.hindawi.com/journals/mpe/2018/4746210.pdf · An Improved Sucker Rod Pumping System Model and Swabbing Parameters

Mathematical Problems in Engineering 11

Table 1 Oil well basic parameters

Well 1 2 3 4Motor type YD280S-8 Y250M-6 Y250M-6 YD280S-6Pumping unit type CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HB CYJ10-3-53HBStroke length (m) 3 3 3 3Stroke frequency (minminus1) 35 3 4 6Pump diameter (mm) 57 44 38 44Pump clearance level 1 2 2 1Pump depth (m) 890 1377 1138 1430Sucker-rod string (mm timesm) 25lowast890 19lowast673+22lowast704 19lowast672+22lowast466 19lowast720+22lowast710middle depth of reservoir (m) 1000 1492 1569 1600Crude oil density (kgm3) 795 857 857 850Water content () 95 97 92 98Dynamic liquid level (m) 880 1347 757 1340Casing pressure (Pa) 02 03 03 02Oil pressure (Pa) 03 03 03 02Fluid dynamic viscosity (Pasdots) 0006 0007 0007 0006Gas oil ratio (m3 m3) 20 19 40 80Plunger length (m) 12 12 12 12Clearance length (m) 05 05 05 05

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(a) Well 1

1 2 30Polished rod displacement (m)

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

simulationmeasured

(b) Well 2

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(c) Well 3

1 2 30Polished rod displacement (m)

0

20

40

60

Polis

hed

rod

load

(kN

)

simulationmeasured

(d) Well 4Figure 9 Suspension point dynamometer card comparison between the simulated and measured

12 Mathematical Problems in Engineering

FluidPlunger

minus05

00

05

10

15Ve

loci

ty(

ms

)

2 4 60Time (s)

(a) Well 1

minus02

00

02

04

06

Velo

city

(m)

4 8 120Time (s)

FluidPlunger

(b) Well 2

Figure 10 Plunger and fluid velocity

Current SRPS model 970Improved SRPS model 981

minus10

0

10

20

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(a) Well 1

Current SRPS model 724Improved SRPS model 775

minus10

0

10

20

40

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(b) Well 2

Figure 11 Comparison of pump dynamometer card

faster after a certain time period Then the comparisons ofpump dynamometer card are executed by the two modelsmeanwhile their individual pump fullness result is given atthe end of the legend (Figure 11)

Pump dynamometer card is very important method andnormally used to diagnose the pump operations particularlywhose shape is more similar to a rectangle the pump iscloser to the fullness [28] Hence from Figure 11 it canbe known that the pump fullness of current SRPS modelis higher than that of the improved one depending on thequalitative judgment and this conclusion is consistent withthe quantitative calculation result meanwhile this gap forthe well 2 is bigger than well 1 According the above

description we can know that the pump fullness calculationresult is on the high side if the PFFP is not considered andthis phenomenon will becomemore obvious for the well withinsufficient OWD

The load presented by the left upper right and lowerborderline of pump dynamometer card is the results of pumpmoving from phase 1 to 4 in turn Therefore its upperborderline describes the pump load when fluid is pumpedinto the barrel From Figure 11(a) it can be found thatthe upper borderline load simulated by the improved SRPSmodel is significantly larger than the one of current ForFigure 11(b) this difference can be neglected Based on (1)and (2) the pump pressure keeps constant as ps when fluid is

Mathematical Problems in Engineering 13

Before optimizationAfter optimization

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

1 2 30Polished rod displacement (m)

(a) Suspension point dynamometer card

minus10

0

10

20

30

40

Cran

k to

rque

(kNmiddotm

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(b) Crank torque

0

5

10

15

20

Mot

or in

put p

ower

(kW

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(c) Motor input power

Figure 12 Dynamic response comparison before and after optimization

sucked into the pump and then the corresponding pump loadis a fixed value Due to the new model takes into account ofPFFP a pressure drop will be brought which will vary withthe fluid velocity Combined with (1) this pressure drop willlead to the pump load increases Figure 10 shows that thefluid velocity of well 1 is larger than that of well 2 andthis is the reason why the difference of upper borderline loadbetween the two models is obvious This illustrates the erroron the upper borderline load of pump dynamometer carddepending on the velocity at which the fluid flows into thepump

7 Optimization Test

Based on the above models a simulation and optimizationsoftware is developed byMATLABOnewell with insufficient

OWD is tested Its original swabbing parameters are asfollows stroke S is 3 m stroke frequency ns is 3 minminus1pump diameter Dd is 57mm pump depth Lpd is 900m crankbalance radius rc is 12 m and rod string combination djtimesLj is 25 mm times 524 m+22 mm times 376 m And its swabbingparameters after optimizing are as follows stroke S is 3 mstroke frequency ns is 25 minminus1 pump diameterDd is 57mmpump depth Lpd is 990 m crank balance radius rc is 09 mand rod string combination djtimes Lj is 22 mm times 426 m+19 mmtimes 564 m The comparisons before and after optimization areshown in Figure 12 and Table 2

From the above comparison results some conclusions areobtained

(1) After optimization the maximum and minimumof suspension point load are all decreased and the load

14 Mathematical Problems in Engineering

Table 2 Specific parameters comparison before and after optimization

Suspension point load Crank torque Motor input power PumpfullnessMaximum Minimum Amplitude Standard

deviationBalancedegree Maximum Minimum mean

Beforeoptimization 555 kN 190 kN 365 kN 107 kN 854 159 kW 09 kW 62 kW 725

Afteroptimization 441 kN 132 kN 309 kN 94 kN 978 106 kW 10 kW 54 kW 908

darr 205 darr 305 darr 153 darr 121 uarr 145 darr 333 uarr 111 darr 129 uarr 252 amplitude is lowered by 153 It can contribute to enhancethe rod string life and prolong the maintenance period

(2) After optimization the standard deviation of cranktorque is reduced by 121 and its balance degree is raisedby 145 It illustrates that the load torque fluctuation is cutdown which canminimize damage to transmission parts andimprove the motor efficiency

(3) After optimization the pump fullness is improvedby 252 It is conducive to improve pump efficiency andproduction

(4) After optimization it plays a role of peak shavingand valley filling for motor input power and extends theoperational life meanwhile power saving rate is 129

8 Conclusions

(1) In this article an improved SRPS model is presentedconsidering the couple effect of pumping fluid and plungermotion on the dynamic response of SRPS instead of theexisting models that assume the pumping fluid volumeis always equal to plunger travelling volume The Runge-Kutta method is applied to solve the whole system modeltrough transforming the rod string longitudinal vibrationequation into ordinary differential equations And the SRPSmodelrsquos precision has been validated by adopting surfacedynamometer card

(2) Two oil wells are served to compare the differencebetween the current SRPS model and the improved oneThe results indicate that the current SRPS model is relativelylow in calculating pump fullness and this gap will increasewith the reduction of OWD the influence of fluid flowinginto the pump on pump load cannot be ignored when thefluid velocity is high Therefore the PFFP is necessary to beconsidered in order to improve the simulation accuracy

(3) On the basis of the improved SRPS model a multitar-get optimization model is proposed in purpose of improvingproduction decreasing load fluctuation and saving energy Bycomparison the optimal scheme can achieve the decreasingof maximum and minimum suspension point load cranktorque fluctuation and energy consumption as well asimproving the system balance degree and pump fullness Insummary it improves the dynamic behavior of SRPS

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

National Natural Science Foundation of China (Grantno 51174175) China Scholarship Council (Grant no201708130108) Hebei Natural Science Foundation (Grant noE201703101) are acknowledged

References

[1] T A Aliev A H Rzayev G A Guluyev T A Alizada and NE Rzayeva ldquoRobust technology and system for management ofsucker rod pumping units in oil wellsrdquoMechanical Systems andSignal Processing vol 99 pp 47ndash56 2018

[2] S Gibbs ldquoPredicting the Behavior of Sucker-Rod PumpingSystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 1963

[3] D R Doty and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo SPE Journal vol 23 no 1 pp 33ndash41 1983

[4] I N Shardakov and I NWasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[5] S G Gibbs ldquoComputing gearbox torque and motor loading forbeam pumping units with consideration of inertia effectsrdquo SPEJ vol 27 pp 1153ndash1159 1975

[6] J Svinos ldquoExact Kinematic Analysis of Pumping Unitsrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionSan Francisco California 1983

[7] D Schafer and J Jennings ldquoAn Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical Modelsrdquoin Proceedings of the SPE Annual Technical Conference andExhibition pp 27ndash30 Dallas Texas 1987

[8] G Takacs Sucker-Rod PumpingManual Pennwell Books Tulsa2003

[9] G W Wang S S Rahman and G Y Yang ldquoAn improvedmodel for the sucker rod pumping systemrdquo in Proceedings of the11thAustralasian FluidMechanics Conference pp 14ndash18HobartAustralia December 1992

[10] S D L Lekia andR D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystemsrdquo SPE Production amp Facilities vol 10 no 1 pp 26ndash331995

[11] J Lea and P Pattillo ldquoInterpretation of Calculated Forces onSucker Rodsrdquo SPE Production amp Facilities vol 10 no 01 pp 41ndash45 1995

Mathematical Problems in Engineering 15

[12] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[13] L Guo-hua H Shun-li Y Zhi et al ldquoA prediction model fora new deep-rod pumping systemrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 75ndash80 2011

[14] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[15] O Becerra J Gamboa and F Kenyery ldquoModelling a DoublePiston Pumprdquo in Proceedings of the SPE International ThermalOperations and Heavy Oil Symposium and International Hor-izontal Well Technology Conference Calgary Alberta Canada2002

[16] A L Podio J Gomez A J Mansure et al ldquoLaboratoryinstrumented sucker-rod pumprdquo J Pet Technol vol 53 no 05pp 104ndash113 2003

[17] Z H Gu H Q Peng and H Y Geng ldquoAnalysis and mea-surement of gas effect on pumping efficiencyrdquoChina PetroleumMachinery vol 34 no 02 pp 64ndash69 2006

[18] M Xing ldquoResponse analysis of longitudinal vibration of suckerrod string considering rod bucklingrdquo Advances in EngineeringSoftware vol 99 pp 49ndash58 2016

[19] S Miska A Sharaki and J M Rajtar ldquoA simple model forcomputer-aided optimization and design of sucker-rod pump-ing systemsrdquo Journal of Petroleum Science and Engineering vol17 no 3-4 pp 303ndash312 1997

[20] L S Firu T Chelu and C Militaru-Petre ldquoAmodern approachto the optimum design of sucker-rod pumping systemrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash9 DenverColorado 2003

[21] X F Liu and Y G Qi ldquoA modern approach to the selectionof sucker rod pumping systems in CBM wellsrdquo Journal ofPetroleum Science and Engineering vol 76 no 3-4 pp 100ndash1082011

[22] S M Dong N N Feng and Z J Ma ldquoSimulating maximumof system efficiency of rod pumping wellsrdquo Journal of SystemSimulation vol 20 no 13 pp 3533ndash3537 2008

[23] M Xing and S Dong ldquoA New Simulation Model for a Beam-Pumping System Applied in Energy Saving and Resource-Consumption Reductionrdquo SPE Production amp Operations vol30 no 02 pp 130ndash140 2015

[24] Y XWu Y Li andH LiuElectric Motor andDriving ChemicalIndustry Press Beijing 2008

[25] S M Dong Computer Simulation of Dynamic Parameters ofRod Pumping System Optimization Petroleum Industry PressBeijing Chinese 2003

[26] F Yavuz J F Lea J C Cox et al ldquoWave equation simulationof fluid pound and gas interferencerdquo in Proceedings of the SPEProduction Operations Symposium pp 16ndash19 Oklahoma CityOklahoma April 2005

[27] D Y Wang and H Z Liu ldquoDynamic modeling and analysis ofsucker rod pumping system in a directional well Mechanismand Machine Sciencerdquo in Proceedings of the ASIAN MMS 2016amp CCMMS pp 1115ndash1127 2017

[28] F M A Barreto M Tygel A F Rocha and C K MorookaldquoAutomatic downhole card generation and classificationrdquo inProceedings of the SPE Annual Technical Conference pp 6ndash9Denver Colorado 1996

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 12: An Improved Sucker Rod Pumping System Model and Swabbing ...downloads.hindawi.com/journals/mpe/2018/4746210.pdf · An Improved Sucker Rod Pumping System Model and Swabbing Parameters

12 Mathematical Problems in Engineering

FluidPlunger

minus05

00

05

10

15Ve

loci

ty(

ms

)

2 4 60Time (s)

(a) Well 1

minus02

00

02

04

06

Velo

city

(m)

4 8 120Time (s)

FluidPlunger

(b) Well 2

Figure 10 Plunger and fluid velocity

Current SRPS model 970Improved SRPS model 981

minus10

0

10

20

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(a) Well 1

Current SRPS model 724Improved SRPS model 775

minus10

0

10

20

40

30

Pum

p lo

ad (k

N)

1 2 30Pump displacement (m)

(b) Well 2

Figure 11 Comparison of pump dynamometer card

faster after a certain time period Then the comparisons ofpump dynamometer card are executed by the two modelsmeanwhile their individual pump fullness result is given atthe end of the legend (Figure 11)

Pump dynamometer card is very important method andnormally used to diagnose the pump operations particularlywhose shape is more similar to a rectangle the pump iscloser to the fullness [28] Hence from Figure 11 it canbe known that the pump fullness of current SRPS modelis higher than that of the improved one depending on thequalitative judgment and this conclusion is consistent withthe quantitative calculation result meanwhile this gap forthe well 2 is bigger than well 1 According the above

description we can know that the pump fullness calculationresult is on the high side if the PFFP is not considered andthis phenomenon will becomemore obvious for the well withinsufficient OWD

The load presented by the left upper right and lowerborderline of pump dynamometer card is the results of pumpmoving from phase 1 to 4 in turn Therefore its upperborderline describes the pump load when fluid is pumpedinto the barrel From Figure 11(a) it can be found thatthe upper borderline load simulated by the improved SRPSmodel is significantly larger than the one of current ForFigure 11(b) this difference can be neglected Based on (1)and (2) the pump pressure keeps constant as ps when fluid is

Mathematical Problems in Engineering 13

Before optimizationAfter optimization

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

1 2 30Polished rod displacement (m)

(a) Suspension point dynamometer card

minus10

0

10

20

30

40

Cran

k to

rque

(kNmiddotm

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(b) Crank torque

0

5

10

15

20

Mot

or in

put p

ower

(kW

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(c) Motor input power

Figure 12 Dynamic response comparison before and after optimization

sucked into the pump and then the corresponding pump loadis a fixed value Due to the new model takes into account ofPFFP a pressure drop will be brought which will vary withthe fluid velocity Combined with (1) this pressure drop willlead to the pump load increases Figure 10 shows that thefluid velocity of well 1 is larger than that of well 2 andthis is the reason why the difference of upper borderline loadbetween the two models is obvious This illustrates the erroron the upper borderline load of pump dynamometer carddepending on the velocity at which the fluid flows into thepump

7 Optimization Test

Based on the above models a simulation and optimizationsoftware is developed byMATLABOnewell with insufficient

OWD is tested Its original swabbing parameters are asfollows stroke S is 3 m stroke frequency ns is 3 minminus1pump diameter Dd is 57mm pump depth Lpd is 900m crankbalance radius rc is 12 m and rod string combination djtimesLj is 25 mm times 524 m+22 mm times 376 m And its swabbingparameters after optimizing are as follows stroke S is 3 mstroke frequency ns is 25 minminus1 pump diameterDd is 57mmpump depth Lpd is 990 m crank balance radius rc is 09 mand rod string combination djtimes Lj is 22 mm times 426 m+19 mmtimes 564 m The comparisons before and after optimization areshown in Figure 12 and Table 2

From the above comparison results some conclusions areobtained

(1) After optimization the maximum and minimumof suspension point load are all decreased and the load

14 Mathematical Problems in Engineering

Table 2 Specific parameters comparison before and after optimization

Suspension point load Crank torque Motor input power PumpfullnessMaximum Minimum Amplitude Standard

deviationBalancedegree Maximum Minimum mean

Beforeoptimization 555 kN 190 kN 365 kN 107 kN 854 159 kW 09 kW 62 kW 725

Afteroptimization 441 kN 132 kN 309 kN 94 kN 978 106 kW 10 kW 54 kW 908

darr 205 darr 305 darr 153 darr 121 uarr 145 darr 333 uarr 111 darr 129 uarr 252 amplitude is lowered by 153 It can contribute to enhancethe rod string life and prolong the maintenance period

(2) After optimization the standard deviation of cranktorque is reduced by 121 and its balance degree is raisedby 145 It illustrates that the load torque fluctuation is cutdown which canminimize damage to transmission parts andimprove the motor efficiency

(3) After optimization the pump fullness is improvedby 252 It is conducive to improve pump efficiency andproduction

(4) After optimization it plays a role of peak shavingand valley filling for motor input power and extends theoperational life meanwhile power saving rate is 129

8 Conclusions

(1) In this article an improved SRPS model is presentedconsidering the couple effect of pumping fluid and plungermotion on the dynamic response of SRPS instead of theexisting models that assume the pumping fluid volumeis always equal to plunger travelling volume The Runge-Kutta method is applied to solve the whole system modeltrough transforming the rod string longitudinal vibrationequation into ordinary differential equations And the SRPSmodelrsquos precision has been validated by adopting surfacedynamometer card

(2) Two oil wells are served to compare the differencebetween the current SRPS model and the improved oneThe results indicate that the current SRPS model is relativelylow in calculating pump fullness and this gap will increasewith the reduction of OWD the influence of fluid flowinginto the pump on pump load cannot be ignored when thefluid velocity is high Therefore the PFFP is necessary to beconsidered in order to improve the simulation accuracy

(3) On the basis of the improved SRPS model a multitar-get optimization model is proposed in purpose of improvingproduction decreasing load fluctuation and saving energy Bycomparison the optimal scheme can achieve the decreasingof maximum and minimum suspension point load cranktorque fluctuation and energy consumption as well asimproving the system balance degree and pump fullness Insummary it improves the dynamic behavior of SRPS

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

National Natural Science Foundation of China (Grantno 51174175) China Scholarship Council (Grant no201708130108) Hebei Natural Science Foundation (Grant noE201703101) are acknowledged

References

[1] T A Aliev A H Rzayev G A Guluyev T A Alizada and NE Rzayeva ldquoRobust technology and system for management ofsucker rod pumping units in oil wellsrdquoMechanical Systems andSignal Processing vol 99 pp 47ndash56 2018

[2] S Gibbs ldquoPredicting the Behavior of Sucker-Rod PumpingSystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 1963

[3] D R Doty and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo SPE Journal vol 23 no 1 pp 33ndash41 1983

[4] I N Shardakov and I NWasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[5] S G Gibbs ldquoComputing gearbox torque and motor loading forbeam pumping units with consideration of inertia effectsrdquo SPEJ vol 27 pp 1153ndash1159 1975

[6] J Svinos ldquoExact Kinematic Analysis of Pumping Unitsrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionSan Francisco California 1983

[7] D Schafer and J Jennings ldquoAn Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical Modelsrdquoin Proceedings of the SPE Annual Technical Conference andExhibition pp 27ndash30 Dallas Texas 1987

[8] G Takacs Sucker-Rod PumpingManual Pennwell Books Tulsa2003

[9] G W Wang S S Rahman and G Y Yang ldquoAn improvedmodel for the sucker rod pumping systemrdquo in Proceedings of the11thAustralasian FluidMechanics Conference pp 14ndash18HobartAustralia December 1992

[10] S D L Lekia andR D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystemsrdquo SPE Production amp Facilities vol 10 no 1 pp 26ndash331995

[11] J Lea and P Pattillo ldquoInterpretation of Calculated Forces onSucker Rodsrdquo SPE Production amp Facilities vol 10 no 01 pp 41ndash45 1995

Mathematical Problems in Engineering 15

[12] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[13] L Guo-hua H Shun-li Y Zhi et al ldquoA prediction model fora new deep-rod pumping systemrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 75ndash80 2011

[14] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[15] O Becerra J Gamboa and F Kenyery ldquoModelling a DoublePiston Pumprdquo in Proceedings of the SPE International ThermalOperations and Heavy Oil Symposium and International Hor-izontal Well Technology Conference Calgary Alberta Canada2002

[16] A L Podio J Gomez A J Mansure et al ldquoLaboratoryinstrumented sucker-rod pumprdquo J Pet Technol vol 53 no 05pp 104ndash113 2003

[17] Z H Gu H Q Peng and H Y Geng ldquoAnalysis and mea-surement of gas effect on pumping efficiencyrdquoChina PetroleumMachinery vol 34 no 02 pp 64ndash69 2006

[18] M Xing ldquoResponse analysis of longitudinal vibration of suckerrod string considering rod bucklingrdquo Advances in EngineeringSoftware vol 99 pp 49ndash58 2016

[19] S Miska A Sharaki and J M Rajtar ldquoA simple model forcomputer-aided optimization and design of sucker-rod pump-ing systemsrdquo Journal of Petroleum Science and Engineering vol17 no 3-4 pp 303ndash312 1997

[20] L S Firu T Chelu and C Militaru-Petre ldquoAmodern approachto the optimum design of sucker-rod pumping systemrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash9 DenverColorado 2003

[21] X F Liu and Y G Qi ldquoA modern approach to the selectionof sucker rod pumping systems in CBM wellsrdquo Journal ofPetroleum Science and Engineering vol 76 no 3-4 pp 100ndash1082011

[22] S M Dong N N Feng and Z J Ma ldquoSimulating maximumof system efficiency of rod pumping wellsrdquo Journal of SystemSimulation vol 20 no 13 pp 3533ndash3537 2008

[23] M Xing and S Dong ldquoA New Simulation Model for a Beam-Pumping System Applied in Energy Saving and Resource-Consumption Reductionrdquo SPE Production amp Operations vol30 no 02 pp 130ndash140 2015

[24] Y XWu Y Li andH LiuElectric Motor andDriving ChemicalIndustry Press Beijing 2008

[25] S M Dong Computer Simulation of Dynamic Parameters ofRod Pumping System Optimization Petroleum Industry PressBeijing Chinese 2003

[26] F Yavuz J F Lea J C Cox et al ldquoWave equation simulationof fluid pound and gas interferencerdquo in Proceedings of the SPEProduction Operations Symposium pp 16ndash19 Oklahoma CityOklahoma April 2005

[27] D Y Wang and H Z Liu ldquoDynamic modeling and analysis ofsucker rod pumping system in a directional well Mechanismand Machine Sciencerdquo in Proceedings of the ASIAN MMS 2016amp CCMMS pp 1115ndash1127 2017

[28] F M A Barreto M Tygel A F Rocha and C K MorookaldquoAutomatic downhole card generation and classificationrdquo inProceedings of the SPE Annual Technical Conference pp 6ndash9Denver Colorado 1996

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 13: An Improved Sucker Rod Pumping System Model and Swabbing ...downloads.hindawi.com/journals/mpe/2018/4746210.pdf · An Improved Sucker Rod Pumping System Model and Swabbing Parameters

Mathematical Problems in Engineering 13

Before optimizationAfter optimization

0

20

40

60Po

lishe

d ro

d lo

ad (k

N)

1 2 30Polished rod displacement (m)

(a) Suspension point dynamometer card

minus10

0

10

20

30

40

Cran

k to

rque

(kNmiddotm

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(b) Crank torque

0

5

10

15

20

Mot

or in

put p

ower

(kW

)

0 2Crank angle (rad)

Before optimizationAfter optimization

(c) Motor input power

Figure 12 Dynamic response comparison before and after optimization

sucked into the pump and then the corresponding pump loadis a fixed value Due to the new model takes into account ofPFFP a pressure drop will be brought which will vary withthe fluid velocity Combined with (1) this pressure drop willlead to the pump load increases Figure 10 shows that thefluid velocity of well 1 is larger than that of well 2 andthis is the reason why the difference of upper borderline loadbetween the two models is obvious This illustrates the erroron the upper borderline load of pump dynamometer carddepending on the velocity at which the fluid flows into thepump

7 Optimization Test

Based on the above models a simulation and optimizationsoftware is developed byMATLABOnewell with insufficient

OWD is tested Its original swabbing parameters are asfollows stroke S is 3 m stroke frequency ns is 3 minminus1pump diameter Dd is 57mm pump depth Lpd is 900m crankbalance radius rc is 12 m and rod string combination djtimesLj is 25 mm times 524 m+22 mm times 376 m And its swabbingparameters after optimizing are as follows stroke S is 3 mstroke frequency ns is 25 minminus1 pump diameterDd is 57mmpump depth Lpd is 990 m crank balance radius rc is 09 mand rod string combination djtimes Lj is 22 mm times 426 m+19 mmtimes 564 m The comparisons before and after optimization areshown in Figure 12 and Table 2

From the above comparison results some conclusions areobtained

(1) After optimization the maximum and minimumof suspension point load are all decreased and the load

14 Mathematical Problems in Engineering

Table 2 Specific parameters comparison before and after optimization

Suspension point load Crank torque Motor input power PumpfullnessMaximum Minimum Amplitude Standard

deviationBalancedegree Maximum Minimum mean

Beforeoptimization 555 kN 190 kN 365 kN 107 kN 854 159 kW 09 kW 62 kW 725

Afteroptimization 441 kN 132 kN 309 kN 94 kN 978 106 kW 10 kW 54 kW 908

darr 205 darr 305 darr 153 darr 121 uarr 145 darr 333 uarr 111 darr 129 uarr 252 amplitude is lowered by 153 It can contribute to enhancethe rod string life and prolong the maintenance period

(2) After optimization the standard deviation of cranktorque is reduced by 121 and its balance degree is raisedby 145 It illustrates that the load torque fluctuation is cutdown which canminimize damage to transmission parts andimprove the motor efficiency

(3) After optimization the pump fullness is improvedby 252 It is conducive to improve pump efficiency andproduction

(4) After optimization it plays a role of peak shavingand valley filling for motor input power and extends theoperational life meanwhile power saving rate is 129

8 Conclusions

(1) In this article an improved SRPS model is presentedconsidering the couple effect of pumping fluid and plungermotion on the dynamic response of SRPS instead of theexisting models that assume the pumping fluid volumeis always equal to plunger travelling volume The Runge-Kutta method is applied to solve the whole system modeltrough transforming the rod string longitudinal vibrationequation into ordinary differential equations And the SRPSmodelrsquos precision has been validated by adopting surfacedynamometer card

(2) Two oil wells are served to compare the differencebetween the current SRPS model and the improved oneThe results indicate that the current SRPS model is relativelylow in calculating pump fullness and this gap will increasewith the reduction of OWD the influence of fluid flowinginto the pump on pump load cannot be ignored when thefluid velocity is high Therefore the PFFP is necessary to beconsidered in order to improve the simulation accuracy

(3) On the basis of the improved SRPS model a multitar-get optimization model is proposed in purpose of improvingproduction decreasing load fluctuation and saving energy Bycomparison the optimal scheme can achieve the decreasingof maximum and minimum suspension point load cranktorque fluctuation and energy consumption as well asimproving the system balance degree and pump fullness Insummary it improves the dynamic behavior of SRPS

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

National Natural Science Foundation of China (Grantno 51174175) China Scholarship Council (Grant no201708130108) Hebei Natural Science Foundation (Grant noE201703101) are acknowledged

References

[1] T A Aliev A H Rzayev G A Guluyev T A Alizada and NE Rzayeva ldquoRobust technology and system for management ofsucker rod pumping units in oil wellsrdquoMechanical Systems andSignal Processing vol 99 pp 47ndash56 2018

[2] S Gibbs ldquoPredicting the Behavior of Sucker-Rod PumpingSystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 1963

[3] D R Doty and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo SPE Journal vol 23 no 1 pp 33ndash41 1983

[4] I N Shardakov and I NWasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[5] S G Gibbs ldquoComputing gearbox torque and motor loading forbeam pumping units with consideration of inertia effectsrdquo SPEJ vol 27 pp 1153ndash1159 1975

[6] J Svinos ldquoExact Kinematic Analysis of Pumping Unitsrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionSan Francisco California 1983

[7] D Schafer and J Jennings ldquoAn Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical Modelsrdquoin Proceedings of the SPE Annual Technical Conference andExhibition pp 27ndash30 Dallas Texas 1987

[8] G Takacs Sucker-Rod PumpingManual Pennwell Books Tulsa2003

[9] G W Wang S S Rahman and G Y Yang ldquoAn improvedmodel for the sucker rod pumping systemrdquo in Proceedings of the11thAustralasian FluidMechanics Conference pp 14ndash18HobartAustralia December 1992

[10] S D L Lekia andR D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystemsrdquo SPE Production amp Facilities vol 10 no 1 pp 26ndash331995

[11] J Lea and P Pattillo ldquoInterpretation of Calculated Forces onSucker Rodsrdquo SPE Production amp Facilities vol 10 no 01 pp 41ndash45 1995

Mathematical Problems in Engineering 15

[12] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[13] L Guo-hua H Shun-li Y Zhi et al ldquoA prediction model fora new deep-rod pumping systemrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 75ndash80 2011

[14] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[15] O Becerra J Gamboa and F Kenyery ldquoModelling a DoublePiston Pumprdquo in Proceedings of the SPE International ThermalOperations and Heavy Oil Symposium and International Hor-izontal Well Technology Conference Calgary Alberta Canada2002

[16] A L Podio J Gomez A J Mansure et al ldquoLaboratoryinstrumented sucker-rod pumprdquo J Pet Technol vol 53 no 05pp 104ndash113 2003

[17] Z H Gu H Q Peng and H Y Geng ldquoAnalysis and mea-surement of gas effect on pumping efficiencyrdquoChina PetroleumMachinery vol 34 no 02 pp 64ndash69 2006

[18] M Xing ldquoResponse analysis of longitudinal vibration of suckerrod string considering rod bucklingrdquo Advances in EngineeringSoftware vol 99 pp 49ndash58 2016

[19] S Miska A Sharaki and J M Rajtar ldquoA simple model forcomputer-aided optimization and design of sucker-rod pump-ing systemsrdquo Journal of Petroleum Science and Engineering vol17 no 3-4 pp 303ndash312 1997

[20] L S Firu T Chelu and C Militaru-Petre ldquoAmodern approachto the optimum design of sucker-rod pumping systemrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash9 DenverColorado 2003

[21] X F Liu and Y G Qi ldquoA modern approach to the selectionof sucker rod pumping systems in CBM wellsrdquo Journal ofPetroleum Science and Engineering vol 76 no 3-4 pp 100ndash1082011

[22] S M Dong N N Feng and Z J Ma ldquoSimulating maximumof system efficiency of rod pumping wellsrdquo Journal of SystemSimulation vol 20 no 13 pp 3533ndash3537 2008

[23] M Xing and S Dong ldquoA New Simulation Model for a Beam-Pumping System Applied in Energy Saving and Resource-Consumption Reductionrdquo SPE Production amp Operations vol30 no 02 pp 130ndash140 2015

[24] Y XWu Y Li andH LiuElectric Motor andDriving ChemicalIndustry Press Beijing 2008

[25] S M Dong Computer Simulation of Dynamic Parameters ofRod Pumping System Optimization Petroleum Industry PressBeijing Chinese 2003

[26] F Yavuz J F Lea J C Cox et al ldquoWave equation simulationof fluid pound and gas interferencerdquo in Proceedings of the SPEProduction Operations Symposium pp 16ndash19 Oklahoma CityOklahoma April 2005

[27] D Y Wang and H Z Liu ldquoDynamic modeling and analysis ofsucker rod pumping system in a directional well Mechanismand Machine Sciencerdquo in Proceedings of the ASIAN MMS 2016amp CCMMS pp 1115ndash1127 2017

[28] F M A Barreto M Tygel A F Rocha and C K MorookaldquoAutomatic downhole card generation and classificationrdquo inProceedings of the SPE Annual Technical Conference pp 6ndash9Denver Colorado 1996

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 14: An Improved Sucker Rod Pumping System Model and Swabbing ...downloads.hindawi.com/journals/mpe/2018/4746210.pdf · An Improved Sucker Rod Pumping System Model and Swabbing Parameters

14 Mathematical Problems in Engineering

Table 2 Specific parameters comparison before and after optimization

Suspension point load Crank torque Motor input power PumpfullnessMaximum Minimum Amplitude Standard

deviationBalancedegree Maximum Minimum mean

Beforeoptimization 555 kN 190 kN 365 kN 107 kN 854 159 kW 09 kW 62 kW 725

Afteroptimization 441 kN 132 kN 309 kN 94 kN 978 106 kW 10 kW 54 kW 908

darr 205 darr 305 darr 153 darr 121 uarr 145 darr 333 uarr 111 darr 129 uarr 252 amplitude is lowered by 153 It can contribute to enhancethe rod string life and prolong the maintenance period

(2) After optimization the standard deviation of cranktorque is reduced by 121 and its balance degree is raisedby 145 It illustrates that the load torque fluctuation is cutdown which canminimize damage to transmission parts andimprove the motor efficiency

(3) After optimization the pump fullness is improvedby 252 It is conducive to improve pump efficiency andproduction

(4) After optimization it plays a role of peak shavingand valley filling for motor input power and extends theoperational life meanwhile power saving rate is 129

8 Conclusions

(1) In this article an improved SRPS model is presentedconsidering the couple effect of pumping fluid and plungermotion on the dynamic response of SRPS instead of theexisting models that assume the pumping fluid volumeis always equal to plunger travelling volume The Runge-Kutta method is applied to solve the whole system modeltrough transforming the rod string longitudinal vibrationequation into ordinary differential equations And the SRPSmodelrsquos precision has been validated by adopting surfacedynamometer card

(2) Two oil wells are served to compare the differencebetween the current SRPS model and the improved oneThe results indicate that the current SRPS model is relativelylow in calculating pump fullness and this gap will increasewith the reduction of OWD the influence of fluid flowinginto the pump on pump load cannot be ignored when thefluid velocity is high Therefore the PFFP is necessary to beconsidered in order to improve the simulation accuracy

(3) On the basis of the improved SRPS model a multitar-get optimization model is proposed in purpose of improvingproduction decreasing load fluctuation and saving energy Bycomparison the optimal scheme can achieve the decreasingof maximum and minimum suspension point load cranktorque fluctuation and energy consumption as well asimproving the system balance degree and pump fullness Insummary it improves the dynamic behavior of SRPS

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

National Natural Science Foundation of China (Grantno 51174175) China Scholarship Council (Grant no201708130108) Hebei Natural Science Foundation (Grant noE201703101) are acknowledged

References

[1] T A Aliev A H Rzayev G A Guluyev T A Alizada and NE Rzayeva ldquoRobust technology and system for management ofsucker rod pumping units in oil wellsrdquoMechanical Systems andSignal Processing vol 99 pp 47ndash56 2018

[2] S Gibbs ldquoPredicting the Behavior of Sucker-Rod PumpingSystemsrdquo Journal of Petroleum Technology vol 15 no 07 pp769ndash778 1963

[3] D R Doty and Z Schmidt ldquoImproved model for sucker rodpumpingrdquo SPE Journal vol 23 no 1 pp 33ndash41 1983

[4] I N Shardakov and I NWasserman ldquoNumerical modelling oflongitudinal vibrations of a sucker rod stringrdquo Journal of Soundand Vibration vol 329 no 3 pp 317ndash327 2010

[5] S G Gibbs ldquoComputing gearbox torque and motor loading forbeam pumping units with consideration of inertia effectsrdquo SPEJ vol 27 pp 1153ndash1159 1975

[6] J Svinos ldquoExact Kinematic Analysis of Pumping Unitsrdquo in Pro-ceedings of the SPE Annual Technical Conference and ExhibitionSan Francisco California 1983

[7] D Schafer and J Jennings ldquoAn Investigation of Analyticaland Numerical Sucker Rod Pumping Mathematical Modelsrdquoin Proceedings of the SPE Annual Technical Conference andExhibition pp 27ndash30 Dallas Texas 1987

[8] G Takacs Sucker-Rod PumpingManual Pennwell Books Tulsa2003

[9] G W Wang S S Rahman and G Y Yang ldquoAn improvedmodel for the sucker rod pumping systemrdquo in Proceedings of the11thAustralasian FluidMechanics Conference pp 14ndash18HobartAustralia December 1992

[10] S D L Lekia andR D Evans ldquoA coupled rod and fluid dynamicmodel for predicting the behavior of sucker-rod pumpingsystemsrdquo SPE Production amp Facilities vol 10 no 1 pp 26ndash331995

[11] J Lea and P Pattillo ldquoInterpretation of Calculated Forces onSucker Rodsrdquo SPE Production amp Facilities vol 10 no 01 pp 41ndash45 1995

Mathematical Problems in Engineering 15

[12] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[13] L Guo-hua H Shun-li Y Zhi et al ldquoA prediction model fora new deep-rod pumping systemrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 75ndash80 2011

[14] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[15] O Becerra J Gamboa and F Kenyery ldquoModelling a DoublePiston Pumprdquo in Proceedings of the SPE International ThermalOperations and Heavy Oil Symposium and International Hor-izontal Well Technology Conference Calgary Alberta Canada2002

[16] A L Podio J Gomez A J Mansure et al ldquoLaboratoryinstrumented sucker-rod pumprdquo J Pet Technol vol 53 no 05pp 104ndash113 2003

[17] Z H Gu H Q Peng and H Y Geng ldquoAnalysis and mea-surement of gas effect on pumping efficiencyrdquoChina PetroleumMachinery vol 34 no 02 pp 64ndash69 2006

[18] M Xing ldquoResponse analysis of longitudinal vibration of suckerrod string considering rod bucklingrdquo Advances in EngineeringSoftware vol 99 pp 49ndash58 2016

[19] S Miska A Sharaki and J M Rajtar ldquoA simple model forcomputer-aided optimization and design of sucker-rod pump-ing systemsrdquo Journal of Petroleum Science and Engineering vol17 no 3-4 pp 303ndash312 1997

[20] L S Firu T Chelu and C Militaru-Petre ldquoAmodern approachto the optimum design of sucker-rod pumping systemrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash9 DenverColorado 2003

[21] X F Liu and Y G Qi ldquoA modern approach to the selectionof sucker rod pumping systems in CBM wellsrdquo Journal ofPetroleum Science and Engineering vol 76 no 3-4 pp 100ndash1082011

[22] S M Dong N N Feng and Z J Ma ldquoSimulating maximumof system efficiency of rod pumping wellsrdquo Journal of SystemSimulation vol 20 no 13 pp 3533ndash3537 2008

[23] M Xing and S Dong ldquoA New Simulation Model for a Beam-Pumping System Applied in Energy Saving and Resource-Consumption Reductionrdquo SPE Production amp Operations vol30 no 02 pp 130ndash140 2015

[24] Y XWu Y Li andH LiuElectric Motor andDriving ChemicalIndustry Press Beijing 2008

[25] S M Dong Computer Simulation of Dynamic Parameters ofRod Pumping System Optimization Petroleum Industry PressBeijing Chinese 2003

[26] F Yavuz J F Lea J C Cox et al ldquoWave equation simulationof fluid pound and gas interferencerdquo in Proceedings of the SPEProduction Operations Symposium pp 16ndash19 Oklahoma CityOklahoma April 2005

[27] D Y Wang and H Z Liu ldquoDynamic modeling and analysis ofsucker rod pumping system in a directional well Mechanismand Machine Sciencerdquo in Proceedings of the ASIAN MMS 2016amp CCMMS pp 1115ndash1127 2017

[28] F M A Barreto M Tygel A F Rocha and C K MorookaldquoAutomatic downhole card generation and classificationrdquo inProceedings of the SPE Annual Technical Conference pp 6ndash9Denver Colorado 1996

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 15: An Improved Sucker Rod Pumping System Model and Swabbing ...downloads.hindawi.com/journals/mpe/2018/4746210.pdf · An Improved Sucker Rod Pumping System Model and Swabbing Parameters

Mathematical Problems in Engineering 15

[12] P A Lollback G Y Wang and S S Rahman ldquoAn alternativeapproach to the analysis of sucker-rod dynamics in vertical anddeviated wellsrdquo Journal of Petroleum Science and Engineeringvol 17 no 3-4 pp 313ndash320 1997

[13] L Guo-hua H Shun-li Y Zhi et al ldquoA prediction model fora new deep-rod pumping systemrdquo Journal of Petroleum Scienceand Engineering vol 80 no 1 pp 75ndash80 2011

[14] L-M Lao and H Zhou ldquoApplication and effect of buoyancy onsucker rod string dynamicsrdquo Journal of Petroleum Science andEngineering vol 146 pp 264ndash271 2016

[15] O Becerra J Gamboa and F Kenyery ldquoModelling a DoublePiston Pumprdquo in Proceedings of the SPE International ThermalOperations and Heavy Oil Symposium and International Hor-izontal Well Technology Conference Calgary Alberta Canada2002

[16] A L Podio J Gomez A J Mansure et al ldquoLaboratoryinstrumented sucker-rod pumprdquo J Pet Technol vol 53 no 05pp 104ndash113 2003

[17] Z H Gu H Q Peng and H Y Geng ldquoAnalysis and mea-surement of gas effect on pumping efficiencyrdquoChina PetroleumMachinery vol 34 no 02 pp 64ndash69 2006

[18] M Xing ldquoResponse analysis of longitudinal vibration of suckerrod string considering rod bucklingrdquo Advances in EngineeringSoftware vol 99 pp 49ndash58 2016

[19] S Miska A Sharaki and J M Rajtar ldquoA simple model forcomputer-aided optimization and design of sucker-rod pump-ing systemsrdquo Journal of Petroleum Science and Engineering vol17 no 3-4 pp 303ndash312 1997

[20] L S Firu T Chelu and C Militaru-Petre ldquoAmodern approachto the optimum design of sucker-rod pumping systemrdquo in SPEAnnual Technical Conference and Exhibition pp 1ndash9 DenverColorado 2003

[21] X F Liu and Y G Qi ldquoA modern approach to the selectionof sucker rod pumping systems in CBM wellsrdquo Journal ofPetroleum Science and Engineering vol 76 no 3-4 pp 100ndash1082011

[22] S M Dong N N Feng and Z J Ma ldquoSimulating maximumof system efficiency of rod pumping wellsrdquo Journal of SystemSimulation vol 20 no 13 pp 3533ndash3537 2008

[23] M Xing and S Dong ldquoA New Simulation Model for a Beam-Pumping System Applied in Energy Saving and Resource-Consumption Reductionrdquo SPE Production amp Operations vol30 no 02 pp 130ndash140 2015

[24] Y XWu Y Li andH LiuElectric Motor andDriving ChemicalIndustry Press Beijing 2008

[25] S M Dong Computer Simulation of Dynamic Parameters ofRod Pumping System Optimization Petroleum Industry PressBeijing Chinese 2003

[26] F Yavuz J F Lea J C Cox et al ldquoWave equation simulationof fluid pound and gas interferencerdquo in Proceedings of the SPEProduction Operations Symposium pp 16ndash19 Oklahoma CityOklahoma April 2005

[27] D Y Wang and H Z Liu ldquoDynamic modeling and analysis ofsucker rod pumping system in a directional well Mechanismand Machine Sciencerdquo in Proceedings of the ASIAN MMS 2016amp CCMMS pp 1115ndash1127 2017

[28] F M A Barreto M Tygel A F Rocha and C K MorookaldquoAutomatic downhole card generation and classificationrdquo inProceedings of the SPE Annual Technical Conference pp 6ndash9Denver Colorado 1996

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Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom