Research ArticleA Two-Phase Flow Model for Pressure TransientAnalysis of Water Injection Well considering WaterImbibition in Natural Fractured Reservoirs

Mengmeng Li 1 Qi Li12 Gang Bi2 and Jiaen Lin2

1College of Petroleum Engineering China University of Petroleum Beijing 102249 Beijing China2College of Petroleum Engineering Xirsquoan Shiyou University 710065 Xirsquoan China

Correspondence should be addressed to Mengmeng Li 2015312041studentcupeducn

Received 14 May 2018 Revised 30 July 2018 Accepted 19 August 2018 Published 4 September 2018

Academic Editor Gilberto Espinosa-Paredes

Copyright copy 2018 Mengmeng 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

The pressure injection falloff test for water injection well has the advantages of briefness and convenience with no effect on theoil production It has been widely used in the oil field Tremendous attention has been focused on oil-water two-phase flow modelbased on the Perrine-Martin theory However the saturation gradient is not considered in the Perrine-Martin method whichmay result in errors in computation Moreover water imbibition is important for water flooding in natural fractured reservoirswhile the pressure transient analysis model has rarely considered water imbibition In this paper we proposed a semianalytical oil-water two-phase flow imbibition model for pressure transient analysis of a water injection well in natural fractured reservoirs Theparameters in this model including total compressibility coefficient interporosity flow coefficient and total mobility change withwater saturation The model was solved by Laplace transform finite-difference (LTFD) method coupled with the quasi-stationarymethod Based on the solution the model was verified by the analytical method and a field water injection test The features oftypical curves and the influences of the parameters on the typical curves were analyzed Results show that the shape of pressurecurves for single phase flow resembles two-phase flow but the position of the two-phase flow curves is on the upper right of thesingle phase flow curvesThe skin factor and wellbore storage coefficient mainly influence the peak value of the pressure derivativesand the straight line of the early period The shape factor has a major effect on the position of the ldquodiprdquo of pressure derivativesThe imbibition rate coefficient mainly influences the whole system radial flow period of the curves This work provides valuableinformation in the design and evaluation of stimulation treatments in natural fractured reservoirs

1 Introduction

Themethods for pressure transient analysis of water injectionwell mostly still use the conventional well test method ofsingle phase flow Perrine [1] proposed to apply the singlephase theory into the two-phase or multiphase well testtheory by substitution of the single phase compressibilityand mobility into the sum of total mobility and total com-pressibility of the multiphase systemMartin [2] provided thetheoretical validation to Perrinersquos theory and indicated thatthe saturation gradient was neglected for Perrinersquos methodThe totalmobility skin factor and average formation pressurecan be obtained based on this method If it is a two-phaseflow well test model of a water injection well a multizonecomposite reservoir model [3ndash8] is used due to the different

properties of the injected fluid and the reservoir fluidWithineach region the properties of the reservoir and fluid areconstant but itmay be different for the two or three regions ina composite reservoirThe two or three zones for a compositereservoir were divided according to the formation and fluidproperties Some scholars [9ndash13] developed a two- or three-zone composite reservoir well test model by finite-differenceor Laplace transformation method to get the pressure curvesfor water injection and pressure falloff period Howeverthese models assumed no saturation gradients within oil andwater transition zone and the zone boundaries were virtuallystationary before the well shut-in

Actually there were saturation discontinuities betweenthe two or three fluid banks in a composite reservoir Asaturation gradient was formed after the water injection due

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 2896251 12 pageshttpsdoiorg10115520182896251

2 Mathematical Problems in Engineering

to the differences in oil and water properties Weinstein [14]investigated the pressure-falloff data with a numerical modeland suggested that the location of the fluid banks couldbe determined based on the Buckley-Leverett [15] frontal-advance equations Sosa et al examined the influence ofsaturation gradients on water injection pressure-falloff testsaccording to a two-phase radial numerical simulator [16] Itwas further demonstrated that the radius of the fluid banksthat was flooded completely by water should be estimatedfrom the Buckley-Leverett theory rather than the techniquementioned in the previous literatures [17ndash19]

According to the Buckley-Leverett theory the locationof the front was estimated at any time during the injectionperiod The fluid bank can be discretized into a series ofbanks so the water saturation distribution and the gradualchange in fluid properties due to saturation gradients inthe reservoir were obtained [20ndash24] Chen [25] derived anapproximate analytical solution for the pressure responseduring water injectionfalloff test based on the front trackingmethod and Buckley-Leverett theory Boughrara [26] addeda two-phase term which represents the existence of the two-phase zone and the movement of the water front to theanalytical single phase well test solution based on Buckley-Leverett equations for vertical and horizontal water injectionwells Zheng [27] developed a modified P-M approach fornumerical well testing analysis of oil and water two-phaseflowing reservoir on account of Buckley-Leverett equationsThe water saturation distribution for a water injection well atany time can be obtained with the Buckley-Leverett frontal-advance equations [28ndash30]

Based on the quasi-stationary method [31ndash33] the sat-uration was decoupled from the pressure with the Buckley-Leverett theory Therefore the water saturation can be con-sidered as constants when the diffusivity equation was solvedsimultaneously for water injection well test analysis On thebasis of the Buckley-Leverett equations De Swaan calculatedthe rate of water imbibition in a fracture surrounded bymatrix in a convolution form [34 35] Kazemi [36] proposedan analytical solution for Buckley-Leverett equations in afracture surrounded by matrix undergoing imbibition Byextending the linear Buckley-Leverett formula consideringimbibition to a radial flow system for the naturally fracturedreservoir [37] the water saturation can be estimated accord-ing to the radial flow model

In this work an oil-water two-phase flow imbibitionwell test model of a water injection well in natural frac-tured reservoirs was developed and solved by the LTFDmethod coupledwith the quasi-stationarymethodTheLTFDmethod transformed the equations into Laplace domainand eliminated the need for time discretization thus themethod was semianalytical in time and the stability andconvergence problems in temporal domain were avoided[38 39] According to analysis of the pressure falloff datathe features of the typical curves of oil-water two-phase flowconsidering water imbibition were analyzed The effect ofthe imbibition rate coefficient skin factor wellbore storagecoefficient shape factor and boundary conditions on thetypical curves was also investigated New ideas had been putforward in accordance with the analysis above

2 Model Description

In order to calculate the saturation and pressure of the dif-fusivity equation in the two-phase pressure transient modelsimultaneously the quasi-stationary method was adopted bydecoupling the saturation from the pressure on account of theBuckley-Leverett water displacement theory

21 Water Saturation Model The water saturation distribu-tion in the fracture and matrix considering water imbibi-tion was estimated by integrating empirical matrix fracturetransfer functions into the Buckley-Leverett equation Basedon the assumption that the cumulative oil recovery from apiece of rock surrounded by water is a continuousmonotonicfunction of time and converges to a finite limit Arnofsky et al[34] proposed an exponential equation for oil recovery esti-mation of water displacement considering water imbibitionin a fractured reservoir as shown

119877 = 119877infin (1 minus 119890minus119877119888119905) (1)

where

119877infin = 0119898 (1 minus 119904119900119903119898 minus 119904119908119888119898) (2)

119877 = 0119898 (119904119898 minus 119904119908119888119898) (3)

Considering the gradual variation of the water saturationin surrounding fractures De Swaan [35] calculated the rateof water imbibition in a fracture surrounded by matrix ina convolution form Kazemi [36] proposed an analyticalsolution for a linear flow system that accounts for saturationchanges in fracture andmatrix Shimamoto [37] extended theformula to a radial flow system for the Warrant-Root modelwith the volume conservation in a fracture

The water saturation change in the fracture can beexpressed as

minus q2120587119903ℎ120597119904119908119891120597119903 minus 119877119888119877infin int119905

0119890minus119877119888(119905minus120591) 120597119904119908119891120597120591 119889120591 = 0119891 120597119904119908119891120597119905 (4)

And the water saturation in the matrix can be written as

119877119888119877infin int1199050119890minus119877119888(119905minus120591) 120597119904119908119891120597120591 119889120591 = 0119898 120597119904119908119898120597119905 (5)

The initial and boundary condition

119904119908119891 (119903 = 0 119905) = 1 (6)

119904119908119891 (119903 119905 = 0) = 0 (7)

Introducing Laplace transform the water saturation inLaplace domain can be expressed as

119904119908119891 = 11199111 119890minus(1199111120573(119877119888+1199111)+1199111120572) (8)

119904119908119898 = 119904119908119898 (119903 119905 = 0)1199111 + 119877119888119877infin0119898119890minus(1199111120573(119877119888+1199111)+1199111120572)1199111 (119877119888 + 1199111) (9)

Mathematical Problems in Engineering 3

where z1 is Laplace variable

120572 = 120587ℎ1199032119902 0119891 (10)

120573 = 120587ℎ1199032119902 119877119888119877infin (11)

The water saturation in fracture and matrix in the realdomain can be obtained by inversion of Laplace transform

119904119908119891 (119903 119905) = 0 (119905 lt 120572)119890minus120573 [119890minus119877119888(119905minus120572)1198680 (2radic120573119877119888 (119905 minus 120572)) + 119877119888 int119905

120572119890minus119877119888(120591minus120572)1198680 (2radic120573119877119888 (120591 minus 120572)) 119889120591] (119905 ge 120572) (12)

119904119908119898 (119903 119905) = 119904119908119888119898 (119905 lt 120572)119904119908119888119898 + (1 minus 119904119900119903119898 minus 119904119908119888119898) 119877119888119890minus120573 int119905

120572119890minus119877119888(120591minus120572)1198680 (2radic120573119877119888 (120591 minus 120572)) 119889120591 (119905 ge 120572) (13)

22 Pressure Transient Model

221 Physical Model The naturally fractured reservoir isdescribed by theWarren-Root model with constant tempera-ture and uniform initial pressure The reservoir is assumedto be a homogeneous horizontal radial reservoir with anupper and lower sealed boundary The slightly compressiblefluids (water and oil) flowing in the matrix and fracturesystem obey Darcyrsquos law and the mass exchange betweenmatrix and fracture system is assumed to be pseudosteady[40] The gravity effect of the 2D system is neglected Afully penetrating water injection well with constant injectionrate is located in the center of the reservoir As the waterinjection into the formation the water saturation changeswith time and space A moving interface divided the radialformation into two regions as shown in Figure 1 From thefigure rf indicates the position of water drive front Region1 is the water invaded region and region 2 is the uninvadedregion

222 Mathematical Model After the water saturation distri-bution at any time during the injection period was obtainedthe pressure equations can be solved by the quasi-stationaryand LTFD method According to the double porosity modelproposed by Warren and Root an oil-water two-phaseflow well test model considering the saturation gradientswithin each region with a moving boundary between theregions for a water injection well was proposed as fol-lows

Fracture system

1119903 120597120597119903 (119903119896119891119896119903119900119891120583119900119861119900

120597119901119891120597119903 ) minus 120591119900119898119891 = 120597120597119905 (0119891119878119900119891119861119900 ) (14)

1119903 120597120597119903 (119903119896119891119896119903119908119891120583119908119861119908

120597119901119891120597119903 ) minus 120591119908119898119891 = 120597120597119905 (0119891119878119908119891119861119908 ) (15)

Matrix system

120591119900119898119891 = 120597120597119905 (0119898119878119900119898119861119900 ) (16)

120591119908119898119891 = 120597120597119905 (0119898119878119908119898119861119908 ) (17)

Matrix and fracture transfer equations

120591119900119898119891 = 119865119904119896119898119896119903119900119898120583119900119861119900 (119901119891 minus 119901119898) (18)

120591119908119898119891 = 119865119904119896119898119896119903119908119898120583119908119861119908 (119901119891 minus 119901119898) (19)

Substituting (16) and (17) into (14) and (15) and (18) and(19) yields

1119903 120597120597119903 (119903119872119905119891120597119901119891120597119903 ) = 0119891119862119905119891 120597119901119891120597119905 + 0119898119862119905119898 120597119901119898120597119905 (20)

0119898119862119905119898 120597119901119898120597119905 = 119865119904119872119905119906 (119901119891 minus 119901119898) (21)

where 119872119905119891 denotes the total mobility of fracture systemand 119872119905119906 is the mobility between fracture and matrix Thesubscript ldquourdquo represents ldquoupstream saturationrdquo For waterinjection period the water flows from fracture to matrixso water saturation in the fracture is used for the relativepermeability evaluation (see (23)) Inversely water saturation

4 Mathematical Problems in Engineering

in the matrix is applied to calculate the relative permeabilityduring the falloff period (see (24))

119872119905119891 = 119896119891(119896119903119900119891 (119878119908119891)120583119900 + 119896119903119908119891 (119878119908119891)120583119908 ) (22)

119872119905119906 = 119896119898(119896119903119900119898 (119878119908119891)120583119900 + 119896119903119908119898 (119878119908119891)120583119908 ) (23)

119872119905119906 = 119896119898 (119896119903119900119898 (119878119908119898)120583119900 + 119896119903119908119898 (119878119908119898)120583119908 ) (24)

In order to simplify the calculation a set of dimensionlessvariables were introduced as listed in Table 1 Equations (20)and (21) and the initial and boundary conditions can bewritten in dimensionless form

1119903119863120597120597119903119863 (119903119863119872119905119891119863

120597119901119891119863120597119903119863 ) = 1205961 120597119901119891119863120597119905119863 + 1205962 120597119901119898119863120597119905119863 (25)

1205962 120597119901119898119863120597119905119863 = 120582 (119901119891119863 minus 119901119898119863) (26)

The initial condition

119901119891119863 (119903119863 119905119863 = 0) = 0 (27)

119901119898119863 (119903119863 119905119863 = 0) = 0 (28)

The inner boundary condition considering the skin effectis

119901119908119863 = (119901119891119863 minus 119878119903119863120597119901119891119863120597119903119863 )119903119863=1

(29)

The inner boundary condition considering wellbore stor-age effect is

119862119863119889119901119908119863119889119905119863 minus (119903119863120597119901119891119863120597119903119863 )119903119863=1

= 1 (30)

The constant pressure outer boundary condition is

119901119891119863 (119903119863 = 119903119890119863 119905119863) = 0 (31)

119901119898119863 (119903119863 = 119903119890119863 119905119863) = 0 (32)

The no flow outer boundary condition is

(120597119901119891119863120597119903119863 )119903119863=119903119890119863

= 0 (33)

(120597119901119898119863120597119903119863 )119903119863=119903119890119863

= 0 (34)

Figure 1 Schematic of two-zone composite reservoir

To simplify the computation a logarithmic transform z =ln(rD) and the Laplace transform with the quasi-stationaryassumption were used Thus (25)ndash(34) can be rewritten as

11198902119911 119889119889119911 (119872119905119891119863119889119901119891119863119889119911 )

= 1205961 [119906119901119891119863 minus 119901119891119863 (119911 0)]+ 1205962 [119906119901119898119863 minus 119901119898119863 (119911 0)]

(35)

1205962 [119906119901119898119863 minus 119901119898119863 (119911 0)] = 120582 (119901119891119863 minus 119901119898119863) (36)

The initial condition is

119901119891119863 (119905119863 = 0) = 0 (37)

119901119898119863 (119905119863 = 0) = 0 (38)

The inner boundary condition considering the skin effectis

119901119908119863 = (119901119891119863 minus 119878119889119901119891119863119889119911 )119911=0

(39)

The inner boundary condition considering wellbore stor-age effect is

(119889119901119891119863119889119911 )119911=0

minus 119862119863 [119906119901119908119863 minus 119901119908119863 (119905119863 = 0)] = minus1119906 (40)

The constant pressure outer boundary condition is

119901119891119863 (119911 = 119911119890) = 0 (41)

119901119898119863 (119911 = 119911119890) = 0 (42)

Mathematical Problems in Engineering 5

Table 1 Dimensionless variable for pressure calculation

119872119905119891 = 119896119891 (119896119903119900119891 (119878119908119891 = 1)120583119900 + 119896119903119908119891 (119878119908119891 = 1)

120583119908 ) 119872119905119891119863 = 119872119905119891119872119905119891

119901119891119863 = 2120587119872119905119891ℎ119902119861119908 (119901119891 minus 119901119894) 120582 = 1198651198781199032119908119872119905119906119872119905119891119901119898119863 = 2120587119872119905119891ℎ119902119861119908 (119901119898 minus 119901119894) 119903119863 = 119903119903119908119862119905119891 = 119862119903 + 119862119908119878119908119894 + 119862119900 (1 minus 119878119908119894) 1205961 = 0119891119862119905119891

(0119891119862119905119891 + 0119898119862119905119898)119862119905119898 = 119862119903 + 119862119908119878119908119888119898 + 119862119900 (1 minus 119878119908119888119898) 1205962 = 0119898119862119905119898(0119891119862119905119891 + 0119898119862119905119898)119905119863 = 119872119905119891

(0119891119862119905119891 + 0119898119862119905119898) 1199032119908 119862119863 = 1198622120587 (0119891119862119905119891 + 0119898119862119905119898) ℎ1199032119908

The no flow outer boundary condition is

(119889119901119891119863119889119911 )119911=119911119890

= 0 (43)

(119889119901119898119863119889119911 )119911=119911119890

= 0 (44)

The pressure change near the wellbore was relativelylarger than the area away from the bottom of the well Sothe formation was divided into n grid blocks in radial direc-tion according to the point-centered logarithmic griddingmethod which was shown as the dotted blue line in Figure 1By using finite-difference approximation (35)ndash(44) can bewritten as

119886119894119901119891119863119894 + 119887119894119901119891119863119894+1 + 119888119894119901119891119863119894+2 = 119889119894 (45)

119901119898119863119894 = 120582119901119891119863119894 + 1205962119901119898119863 (119905119863 = 0)1205962119906 + 120582 (46)

where

119886119894 = 120583119908(119896119903119900119891 (119878119908119891 (119903119894minus1 119905))120583119900 + 119896119903119908119891 (119878119908119891 (119903119894minus1 119905))120583119908 ) (47)

119887119894 = minus[119886119894 + 119888119894

+ (120596112059621199062 + 1205961120582119906 + 12059621205821199061205962119906 + 120582 )119894

1198902119894Δ119911Δ1199112](48)

119888119894 = 120583119908(119896119903119900119891 (119878119908119891 (119903119894 119905))120583119900 + 119896119903119908119891 (119878119908119891 (119903119894 119905))120583119908 ) (49)

119889119894 = minus[1198902119894Δ119911Δ11991121205961119894119901119891119863119894 (119905119863 = 0)

+ 1198902119894Δ119911Δ1199112 ( 12059621205821205962119906 + 120582)119894 119901119898119863119894 (119905119863 = 0)](50)

The inner boundary condition is

1198901119901119908119863 + 11989021199011198911198630 + 11989031199011198911198631 = 1198908 (51)

11989041199011198911198630 + 11989051199011198911198631 = 1198909 (52)

The outer boundary condition is

1198906119901119891119863119899minus1 minus 1198907119901119891119863119899 = 0 (53)

The coefficients for different boundary conditions areshown in Table 2

The pressure difference equations ((45) and (46)) can bewritten in a matrix form as

M997888rarr119909 = 997888rarr119887 (54)

where M is a large sparse matrix 997888x is the unknown pressurevector and

997888b is the known vector related to the initial and

6 Mathematical Problems in Engineering

Table 2 Coefficients with different boundary conditions

Boundary conditions Coefficients

Constant pressure outer boundary1198901 = 1 1198902 = minus(1 + 119878Δ119911 ) 1198903 = 119878Δ119911 1198904 = 1 + Δ119911119862119863119906 + 119878119862119863119906 1198905 = minus119878119862119863119906 minus 1 1198906 = 0

1198907 = 1 1198908 = 0 1198909 = Δz119906 minus Δz119862119863119901119908119863(119905119863 = 0)No flow outer boundary

1198901 = 1 1198902 = minus(1 + 119878Δ119911 ) 1198903 = 119878Δ119911 1198904 = 1 + Δ119911119862119863119906 + 1198781198621198631199061198905 = minus119878119862119863119906 minus 1 1198906 = 119886119899minus1 + 119888119899minus11198907 = 119887119899minus1 1198908 = 0 1198909 = Δz119906 minus Δz119862119863119901119908119863(119905119863 = 0)

boundary conditionsThematrixM and vectors997888x and997888b can

be explicitly given as

M

=

[[[[[[[[[[[[[[[[[[[[[[[[[[[

1198901 1198902 1198903 0 0 00 1198904 1198905 0 0 00 0 1198861 1198871 1198881 00 0 0 1198862 1198872 1198882

d

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

sdot sdot sdot

d

d

sdot sdot sdot

0 0 00 0 00 0 00 0 0

d

d d

119886119899minus3 119887119899minus3 119888119899minus3119886119899minus2 119887119899minus2 119888119899minus20 1198906 1198907

]]]]]]]]]]]]]]]]]]]]]]]]]]]

(55)

997888rarr119909 =

[[[[[[[[[[[[[[[[[

119901119908119863119901119891119863011990111989111986311199011198911198632

119901119891119863119899minus1

]]]]]]]]]]]]]]]]]

(56)

997888rarr119887 =

[[[[[[[[[[[[[[[[[

1198908119890911988911198892

119889119899minus1

]]]]]]]]]]]]]]]]]

(57)

Krwmkrom

KrwfKrof

0

02

04

06

08

1

Kro

K

rw

02 04 06 08 10Sw

Figure 2 Curves of relative permeability of oil and water in fractureand matrix

The unknown vector 997888x was calculated by using theStehfest algorithm Therefore the dimensionless pressureand pressure derivatives [41] at the bottom of the wellborecan be plotted according to the solutions of the formulatedmathematical model for the water injection system Thebasic parameters and the relative permeability data for thecalculation of pressure curves are illustrated in Table 3 andFigure 2

3 Verification of the Mathematical Model

Because little was available in the literatures for pressurebehavior of a water injection well considering saturationgradient and water imbibition in dual-porosity reservoirby LTFD method the model solutions were validated bysimplifying the model into a single phase water injectionwell in a dual-porosity system which was compared to theconventional analytical solution [42] by commercial softwareas shown in Figure 3 From the figure the LTFD results showgood agreement with the analytical solution

Moreover a field water injection test was used to furtherverify the proposedmodelThe basic information of thewaterinjection well and the relative permeability data was listed inTable 4 and Figure 4 And Figure 5 shows the pressure curves

Mathematical Problems in Engineering 7

tDPwDminusLTFD methoddPwDminusLTFD method

PwDminusanalytical methoddPwDminusanalytical method

10minus2

10minus1

100

101

101

102

102

103

104

105

106

107

108

PwD

dPw

D

Figure 3 Comparison of the results by LTFDmethod and analyticalmethod

Table 3 Basic parameters for calculation of pressure curves

Parameter Value UnitsFormation height 10 mFracture porosity 003 fractionMatrix porosity 027 fractionWellbore radius 01 mFracture permeability 06 120583m2Matrix permeability 6 times 10minus5 120583m2Irreducible water saturation 20 Residual oil saturation 14 Oil volume factor 103 fractionWater volume factor 1 fractionShape factor 6 times 10minus4 1cm2

Oil viscosity 1 mPasWater viscosity 05 mPasrock compressibility 6 times 10minus4 1MPaoil compressibility 1 times 10minus3 1MPawater compressibility 4 times 10minus4 1MPaInjection rate 30 m3dayInitial pressure 14 MPaSkin factor 01 fractionImbibition rate coefficient 01 1dayWellbore storage coefficient 001 1m3MPa

of the well test data comparing with the results in our modelFrom the figure it is found that our results match well withthe field test data

4 Model Features Analysis

The pressure curves of oil-water two-phase flow consider-ing water imbibition and the saturation gradients duringinjection period by the LTFD method were plotted in thissection The behaviors of the typical curves were analyzed by

Table 4 Basic information of the water injection well

Parameter Value UnitsFormation height 126 mFracture porosity 003 fractionMatrix porosity 0268 fractionWellbore radius 012 mOil volume factor 1163 fractionWater volume factor 1 fractionOil viscosity 96 mPasWater viscosity 05 mPasrock compressibility 5 times 10minus4 1MPaoil compressibility 48 times 10minus3 1MPawater compressibility 39 times 10minus4 1MPaInjection rate 256 m3dayInitial pressure 15 MPa

KrwKro

0

02

04

06

08

1

Kro

K

rw

02 04 06 08 10Sw

Figure 4 The relative permeability of oil and water for the waterinjection well

comparing the typical curves of single phase flow with thetwo-phase flow curves in the fractured reservoirs as follows

41 Features of Typical Curves Figure 6 illustrates the pres-sure curves of single phase flow and oil-water two-phaseflow in fractured reservoirs by LTFD method without waterimbibition According to the feature of the curves theformation flow can be divided into three stages The firststage is the flow in the fracture system the second stage isthe flow between the fracture and the matrix system andthe third stage is the flow in the whole system including thefracture andmatrixThe figure shows that the shape of curvesfor single phase flow and two-phase flow is similar but theposition of the two-phase flow curves is on the upper rightof the single phase flow curves The reason is that when oiland water exist in the whole system the total mobility of oiland water is lower than the single phase and the bottom-hole pressure increases more quickly as the fluid is injected

8 Mathematical Problems in Engineering

tDPwDminusField testdPwDminusField test

PwDminusLTFD methoddPwDminusLTFD method

101

102

103

104

105

106

107

100

101

102

PwD

dPw

D

Figure 5 Comparison of the results in this work and the fieldinjection test

tD

PwDminussingle phasedPwDminussingle phase

PwDminustwo-phasedPwDminustwo-phase

101

102

103

104

105

106

107

108

10minus1

100

101

102

PwD

dPw

D

Figure 6 Typical curves of single phase and two-phase flow duringinjection period

into the well Thus the pressure derivative curves of two-phase flow move upward compared to the single phase flowcurves Moreover due to the decrease of the total mobilitythe interporosity flow capacity between matrix and fracturesystem decreases when the single phase flow becomes two-phase flow Thus the ldquodiprdquo of the two-phase flow curvesoccurs later than that of the single phase flow curves

42 Effects of Parameters The key factors influencing thepressure behavior of water injection period including skinfactor wellbore storage coefficient shape factor water imbibi-tion rate coefficient and boundary conditions were discussedin detail in this part

421 Skin Factor Figure 7 shows the effect of skin factoron the pressure curves of the water injection well The solid

tD

S=minus1S=minus1S=1

S=1S=10S=10

S=minus1S=minus1S=1

S=1S=10S=10

102

104

106

108

10minus1

100

101

102

PwD

dPw

D

Figure 7 Influence of different skin factors on pressure transientbehavior

lines are the curves of single phase flow and the circlesare oil-water two-phase flow curves The red green andblue colors indicate that the shape factor values are -1 1and 10 respectively From the figure the skin factor mainlyinfluences the peak value of the pressure derivatives As thevalue of skin factor increases the peak value gets higherand the time for peak value occurrence becomes later whichindicates that the reservoir is more severely contaminatedThe pressure derivative curves of two-phase flow movetoward the upper right of the single phase flow curves whichshows the same change rule with Figure 6 When the skinfactor becomes larger the peak value difference between thesingle phase and two-phase decreases and the inclination ofthe curves after the peak value increases

422 Wellbore Storage Coefficient Figure 8 is the influencesof wellbore storage coefficient on the pressure and pressurederivative curves of the water injection well The red greenand blue colors indicate that wellbore storage coefficientvalues are 001 01 and 1 respectively As seen the maindifference of the curves occurs on the straight line of earlyperiod As the wellbore storage coefficient increases thestraight line section becomes longer and the time of peakvalue occurs later Besides the wellbore storage effect willcover up the radial flow stage in the fracture system directlygoing to the transitional flow stage between the fracture andmatrix The pressure derivative curves of two-phase flowmove upper right to the single phase flow curves after thestraight line of wellbore storage stage which agrees with thechange rule of Figure 6

423 Shape Factor The influences of shape factor on thepressure curves of the water injection well are shown inFigure 9 The red green and blue colors indicate thatshape factor values are 10minus5 10minus4 and 10minus3 respectivelyBy comparison of the curves it is found that the shape

Mathematical Problems in Engineering 9

tD10

210

410

610

810

minus3

10minus2

10minus1

100

101

102

PwD

dPw

D

C=001C=001C=01

C=01C=1C=1

C=001C=001C=01

C=01C=1C=1

Figure 8 Influence of different wellbore storage coefficients onpressure transient behavior

tD

Fs=1lowast10minus5Fs=1lowast10minus5Fs=1lowast10minus4Fs=1lowast10minus4Fs=1lowast10minus3Fs=1lowast10minus3

Fs=1lowast10minus5Fs=1lowast10minus5Fs=1lowast10minus4Fs=1lowast10minus4Fs=1lowast10minus3Fs=1lowast10minus3

102

104

106

108

10minus1

100

101

102

PwD

dPw

D

Figure 9 Influence of different shape factors on pressure transientbehavior

factor mainly has an effect on the position of the ldquodiprdquo ofthe pressure derivatives When the value of shape factorgets greater the ldquodiprdquo appears earlier and the ldquodiprdquo movescloser to the left The main reason is that the shape factoris a component of the equation defining the interporosityparameter The higher the shape factor is the easier the fluidexchange between the fracture and matrix is and the earlierthe transitional flow stage occurs It also demonstrates thatthe pressure derivative curves of two-phase flow move upperright to the single phase flow curves which is in line with thechange rule of Figure 6

tDRc=0Rc=0Rc=001

Rc=001Rc=005Rc=005

Rc=01Rc=01Rc=05

Rc=05Rc=1Rc=1

102

104

106

108

10minus2

10minus1

100

101

102

PwD

dPw

DFigure 10 Pressure behavior of different imbibition rate coefficientswith no flow outer boundary

424 Imbibition Rate Coefficient with No Flow Outer Bound-ary Condition Figure 10 describes the pressure behaviorof the water injection well under no flow outer boundarycondition when the water imbibition rate coefficient valuesare 0 001 005 01 05 and 1 respectively From the figurethe imbibition rate coefficient mainly has an effect on thewhole system (matrix and fracture system) radial flow periodof the curves As the imbibition rate coefficient becomeslarger there are more fluid exchange between the fractureand matrix and more fluid supply from the matrix to thefracture The water imbibition process is slow so there isa second interporosity period after the whole system radialflow period And with the increase of water imbibition ratecoefficient the pressure derivative curves dip downwardearlier and the degree for the curves dropping down becomeslarger After that the pressure derivative curves rise up to auniform line with the same slope due to the influence of theno flow outer boundary

425 Imbibition Rate Coefficient with Constant PressureBoundary Condition Figure 11 shows the pressure and pres-sure derivative curves of the water injection well underconstant pressure boundary condition when the water imbi-bition rate coefficient values are 0 001 005 01 05 and1 respectively By comparison and analysis it is concludedthat the pressure derivative curve is a straight line at thewhole system radial flow period when no water imbibition isconsideredWhen considering water imbibition the pressurederivatives decline at the whole system radial flow periodThe pressure derivatives drop downward more and earlierwith the increase of imbibition rate coefficient After that thecurves also drop down due to the constant pressure outerboundary effect

10 Mathematical Problems in Engineering

tD

Rc=0Rc=0Rc=001

Rc=001Rc=005Rc=005

Rc=01Rc=01Rc=05

Rc=05Rc=1Rc=1

102

104

106

108

10minus2

10minus1

100

101

102

PwD

dPw

D

Figure 11 Pressure behavior of different imbibition rate coefficientswith constant pressure boundary

5 Conclusions

In this work a semianalytical two-phase flow model consid-ering water imbibition and the saturation gradients withineach region during injection period was proposed to analyzethe pressure transient behavior of the water injection well infractured reservoirs Validation of the presented model wasperformed by analytical method and a field water injectiontest The model features were investigated by sensitivityanalysis Some conclusions were drawn as follows

(1) The shape of the pressure curves for the proposedmodel is similar to the single phase flowmodel whilethe position of the curves for the two-phase flowmodel is on the upper right of the single phase flowmodel

(2) Themain influence of skin factor on the curves occurson the peak value of the pressure derivatives As theskin factor becomes larger the peak value differencebetween single phase and two-phase pressure curvesdecreases and the inclination of the curves after thepeak value increases

(3) The wellbore storage coefficient mainly has an effecton the straight line of the early period for the pressurecurves The shape factor mainly has an effect on theposition of the ldquodiprdquo of the pressure derivatives whichis the same as the interporosity parameter

(4) The imbibition rate coefficient mainly influences thewhole system radial flow period of the pressurecurves The pressure derivative curves dip downwardearlier and drop down to a greater degree with theincrease of imbibition rate coefficient

Nomenclature

R Recovery fractionR119888 Imbibition rate coefficient 1day119877infin Ultimate cumulative oil recovery fraction0 Porosity fraction119904119900119903 Residual oil saturation fraction119904119908119888 Irreducible water saturation fraction119904119908 Water saturation fractionz Logarithmic transform variable dimensionlessu Laplace variable dimensionlessq Displacement rate m3dt Time hk Permeability 1205831198982119896119903 Relative permeability fractionh Formation height mr Radial distance m119903119908 Wellbore radius mp Pressure MPa120583 Viscosity mPasdotsB Formation volume factor dimensionless120591 Fracture matrix transfer term m3dF119904 Shape factor 1m2119862119905 Total compressibility 1MPa119872119905 Total mobility 1MPa120582 Interporosity flow coefficient fraction120596 Storativity ratio fractionS Skin factor dimensionlessC Wellbore storage coefficient m3MPa

Subscripts

f Fracturem Matrixo Oilw Watert Totale ExternalD Dimensionless variable

Superscripts

- Laplace-transformed variable Endpoint property997888rarr Vector symbol

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

The authors gratefully acknowledge the support of theNational Basic Research 973 Program of China (Grant No

Mathematical Problems in Engineering 11

2015CB250900) the National Natural Science Foundationof China (Grant No 51704237) and the Research Projectsof Shaanxi Provincial Education Department (Grant No13JS090)

References

[1] R L Perrine Analysis of Pressure-buildup Curves vol 56American Petroleum Institute 1956

[2] J C Martin ldquoSimplified Equations of Flow in Gas Drive Reser-voirs and the Theoretical Foundation of Multiphase PressureBuildup Analysesrdquo Society of Petroleum Engineers vol 216 pp321ndash323 1959

[3] W Hurst ldquoInterference between Oil Fieldrdquo Transactions of theMetallurgical Society of AIME vol 219 no 8 pp 175ndash192 1960

[4] MMortada ldquoOilfield Interference in Aquifers of Non-UniformPropertiesrdquo Journal of Petroleum Technology vol 12 no 12 pp55ndash57 2013

[5] T Loucks and E Guerrero ldquoPressure Drop in a CompositeReservoirrdquo SPE Journal vol 1 no 03 pp 170ndash176 2013

[6] H Bixel and H Van Poollen ldquoPressure Drawdown and Buildupin the Presence of Radial Discontinuitiesrdquo SPE Journal vol 7no 03 pp 301ndash309 2013

[7] H Ramey ldquoApproximate Solutions For Unsteady LiquidFlow InComposite Reservoirsrdquo Journal of Canadian Petroleum Technol-ogy vol 9 no 01 2013

[8] Q Deng R Nie Y Jia et al ldquoPressure transient behavior of afractured well in multi-region composite reservoirsrdquo Journal ofPetroleum Science and Engineering vol 158 pp 535ndash553 2017

[9] H Kazemi LMerrill and J Jargon ldquoProblems in Interpretationof Pressure Fall-OffTests in ReservoirsWithAndWithout FluidBanksrdquo Journal of PetroleumTechnology vol 24 no 09 pp 1147ndash1156 2013

[10] L Merrill H Kazemi and W B Gogarty ldquoPressure FalloffAnalysis in Reservoirs With Fluid Banksrdquo Journal of PetroleumTechnology vol 26 no 07 pp 809ndash818 2013

[11] A Satman M Eggenschwiler R Tang and R HJ ldquoAnanalytical study of transient flow in systemswith radial disconti-nuitiesrdquo in Proceedings of the SPE9399 presented at SPE AnnualTechnical Conference and Exhibition Dallas Texas TX USASeptember 1980

[12] A Al-Bemani and I Ershaghi ldquoTwo-Phase Flow InterporosityEffects on Pressure Transient Test Response in Naturally Frac-tured Reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Dallas Texas

[13] Z Chen W Yu X Liao X Zhao Y Chen and K SepehrnoorildquoA Two-Phase Flow Model for Fractured Horizontal Well withComplex Fracture Networks Transient Analysis in FlowbackPeriodrdquo in Proceedings of the SPE Liquids-Rich Basins Confer-ence - North America Midland Texas USA

[14] H G Weinstein ldquoCold Waterflooding a Warm Reservoirrdquo inProceedings of the Fall Meeting of the Society of PetroleumEngineers of AIME Houston Texas

[15] S E Buckley andM C Leverett ldquoMechanism of fluid displace-ment in sandsrdquoTransactions of the AIME vol 146 no 1 pp 107ndash116 2013

[16] A Sosa R Raghavan and T Limon ldquoEffect of Relative Perme-ability andMobility Ratio on Pressure Falloff Behaviorrdquo Journalof Petroleum Technology vol 33 no 06 pp 1125ndash1135 2013

[17] P Hazebroek H Rainbow and CS Matthews ldquoPressure Fall-Off in Water Injection Wellsrdquo Transactions of the MetallurgicalSociety of AIME vol 213 pp 250ndash260 1958

[18] R Carter ldquoPressure Behavior of a Limited Circular CompositeReservoirrdquo SPE Journal vol 6 no 04 pp 328ndash334 2013

[19] A Odeh ldquoFlowTest Analysis for aWell with Radial Discontinu-ityrdquo Journal of Petroleum Technology vol 21 no 02 pp 207ndash2102013

[20] M Abbaszadeh and M Kamal ldquoPressure-Transient Testing ofWater-InjectionWellsrdquo SPE Reservoir Engineering vol 4 no 01pp 115ndash124 2013

[21] N Yeh and R Agarwal ldquoPressure Transient Analysis of Injec-tionWells in ReservoirsWithMultiple Fluid Banksrdquo in Proceed-ings of the SPE Annual Technical Conference and Exhibition SanAntonio Texas

[22] M M Levitan ldquoApplication of water injectionfalloff tests forreservoir appraisal New analytical solution method for two-phase variable rate problemsrdquo SPE Journal vol 8 no 4 pp 341ndash349 2003

[23] T Nanba and R Horne ldquoEstimation of Water and Oil RelativePermeabilities From Pressure Transient Analysis of WaterInjectionWell Datardquo in Proceedings of the SPE Annual TechnicalConference and Exhibition San Antonio Texas

[24] R B Bratvold andRNHorne ldquoAnalysis of pressure-falloff testsfollowing cold-water injectionrdquo SPE Formation Evaluation vol5 no 3 pp 293ndash302 1990

[25] S Chen G Li and A C Reynolds ldquoAnalytical Solution forInjection-Falloff-Production Testrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition San AntonioTexas USA

[26] A A Boughrara A M M Peres S Chen A A V Machadoand A C Reynolds ldquoApproximate analytical solutions for thepressure response at a water-injection wellrdquo SPE Journal vol12 no 1 pp 19ndash34 2007

[27] S Zheng and w xu ldquoNew approaches for analyzing transientpressure from oil and water two-phase flowing reservoirrdquo inProceedings of the Kuwait International Petroleum Conferenceand Exhibition Kuwait City Kuwait

[28] N A El-Khatib ldquoTransient Pressure Behavior of CompositeReservoirs with Moving Boundariesrdquo in Proceedings of theMiddle East Oil Show and Conference Bahrain

[29] A Dastan M M Kamal Y Hwang F Suleen and S MorsyldquoFalloff Testing Under Multiphase Flow Conditions in Natu-rally Fractured Reservoirsrdquo in Proceedings of the SPE WesternRegional Meeting Garden Grove California USA

[30] C n Machado and A C Reynolds ldquoApproximate semi-analytical solution for injection-falloff-production well test ananalytical tool for the in situ estimation of relative permeabilitycurvesrdquo Transport in Porous Media vol 121 no 1 pp 207ndash2312018

[31] T Barkve ldquoAnalytical study of reservoir pressure during water-injection well testsrdquo Society of Petroleum Engineers of AIME(Paper) SPE 1986

[32] T S Ramakrishnan and F J Kuchuk ldquoTesting injection wellswith rate and pressure datardquo SPE Formation Evaluation vol 9no 3 pp 228ndash236 1994

[33] L G Thompson and A C Reynolds ldquoWell testing for radiallyheterogeneous reservoirs under single and multiphase flowconditionsrdquo SPE Formation Evaluation vol 12 no 1 pp 57ndash641997

12 Mathematical Problems in Engineering

[34] JS Aronofsky L Masse and SG Natanson ldquoA Model for theMechanism of Oil Recovery from the Porous Matrix Due toWater Invasion in Fractured Reservoirsrdquo Trans AIME vol 213pp 17ndash19 1958

[35] A de Swaan ldquoTheory of Waterflooding in Fractured Reser-voirsrdquo SPE Journal vol 18 no 02 pp 117ndash122 2013

[36] H Kazemi J Gilman and A Elsharkawy ldquoAnalytical andNumerical Solution of Oil Recovery From Fractured ReservoirsWith Empirical Transfer Functions (includes associated papers25528 and 25818)rdquo SPE Reservoir Engineering vol 7 no 02 pp219ndash227 2013

[37] T Shimamoto D Kuramoto N Arihara and T Onishi ldquoA NewPressure Transient Analysis Model for Water Injection Wellin Dual-Porosity Reservoirsrdquo in Proceedings of the SPE100881presented at SPE Asia Pacific Oil ampamp Gas Conference andExhibition Adelaide Australia 2006

[38] G J Moridis D A McVay D L Reddell and T A BlasingameldquoLaplace transform finite difference (LTFD) numerical methodfor the simulation of compressible liquid flow in reservoirsrdquo SPEAdvanced Technology Series vol 2 no 2 pp 122ndash131 1994

[39] A D Habte and M Onur ldquoLaplace-transform finite-differenceand quasistationary solution method for water-injectionfallofftestsrdquo SPE Journal vol 19 no 3 pp 398ndash409 2014

[40] J Warren and P Root ldquoThe behavior of naturally fracturedreservoirsrdquo SPE Journal vol 3 no 3 pp 245ndash255 2013

[41] D Bourdet J Ayoub and Y Pirard ldquoUse of Pressure Derivativein Well Test Interpretationrdquo SPE Formation Evaluation vol 4no 02 pp 293ndash302 2013

[42] M Mavor and H Cinco-Ley ldquoTransient Pressure BehaviorOf Naturally Fractured Reservoirsrdquo in Proceedings of the SPECalifornia Regional Meeting Ventura California

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Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 3

where z1 is Laplace variable

120572 = 120587ℎ1199032119902 0119891 (10)

120573 = 120587ℎ1199032119902 119877119888119877infin (11)

The water saturation in fracture and matrix in the realdomain can be obtained by inversion of Laplace transform

119904119908119891 (119903 119905) = 0 (119905 lt 120572)119890minus120573 [119890minus119877119888(119905minus120572)1198680 (2radic120573119877119888 (119905 minus 120572)) + 119877119888 int119905

120572119890minus119877119888(120591minus120572)1198680 (2radic120573119877119888 (120591 minus 120572)) 119889120591] (119905 ge 120572) (12)

119904119908119898 (119903 119905) = 119904119908119888119898 (119905 lt 120572)119904119908119888119898 + (1 minus 119904119900119903119898 minus 119904119908119888119898) 119877119888119890minus120573 int119905

120572119890minus119877119888(120591minus120572)1198680 (2radic120573119877119888 (120591 minus 120572)) 119889120591 (119905 ge 120572) (13)

22 Pressure Transient Model

221 Physical Model The naturally fractured reservoir isdescribed by theWarren-Root model with constant tempera-ture and uniform initial pressure The reservoir is assumedto be a homogeneous horizontal radial reservoir with anupper and lower sealed boundary The slightly compressiblefluids (water and oil) flowing in the matrix and fracturesystem obey Darcyrsquos law and the mass exchange betweenmatrix and fracture system is assumed to be pseudosteady[40] The gravity effect of the 2D system is neglected Afully penetrating water injection well with constant injectionrate is located in the center of the reservoir As the waterinjection into the formation the water saturation changeswith time and space A moving interface divided the radialformation into two regions as shown in Figure 1 From thefigure rf indicates the position of water drive front Region1 is the water invaded region and region 2 is the uninvadedregion

222 Mathematical Model After the water saturation distri-bution at any time during the injection period was obtainedthe pressure equations can be solved by the quasi-stationaryand LTFD method According to the double porosity modelproposed by Warren and Root an oil-water two-phaseflow well test model considering the saturation gradientswithin each region with a moving boundary between theregions for a water injection well was proposed as fol-lows

Fracture system

1119903 120597120597119903 (119903119896119891119896119903119900119891120583119900119861119900

120597119901119891120597119903 ) minus 120591119900119898119891 = 120597120597119905 (0119891119878119900119891119861119900 ) (14)

1119903 120597120597119903 (119903119896119891119896119903119908119891120583119908119861119908

120597119901119891120597119903 ) minus 120591119908119898119891 = 120597120597119905 (0119891119878119908119891119861119908 ) (15)

Matrix system

120591119900119898119891 = 120597120597119905 (0119898119878119900119898119861119900 ) (16)

120591119908119898119891 = 120597120597119905 (0119898119878119908119898119861119908 ) (17)

Matrix and fracture transfer equations

120591119900119898119891 = 119865119904119896119898119896119903119900119898120583119900119861119900 (119901119891 minus 119901119898) (18)

120591119908119898119891 = 119865119904119896119898119896119903119908119898120583119908119861119908 (119901119891 minus 119901119898) (19)

Substituting (16) and (17) into (14) and (15) and (18) and(19) yields

1119903 120597120597119903 (119903119872119905119891120597119901119891120597119903 ) = 0119891119862119905119891 120597119901119891120597119905 + 0119898119862119905119898 120597119901119898120597119905 (20)

0119898119862119905119898 120597119901119898120597119905 = 119865119904119872119905119906 (119901119891 minus 119901119898) (21)

where 119872119905119891 denotes the total mobility of fracture systemand 119872119905119906 is the mobility between fracture and matrix Thesubscript ldquourdquo represents ldquoupstream saturationrdquo For waterinjection period the water flows from fracture to matrixso water saturation in the fracture is used for the relativepermeability evaluation (see (23)) Inversely water saturation

4 Mathematical Problems in Engineering

in the matrix is applied to calculate the relative permeabilityduring the falloff period (see (24))

119872119905119891 = 119896119891(119896119903119900119891 (119878119908119891)120583119900 + 119896119903119908119891 (119878119908119891)120583119908 ) (22)

119872119905119906 = 119896119898(119896119903119900119898 (119878119908119891)120583119900 + 119896119903119908119898 (119878119908119891)120583119908 ) (23)

119872119905119906 = 119896119898 (119896119903119900119898 (119878119908119898)120583119900 + 119896119903119908119898 (119878119908119898)120583119908 ) (24)

In order to simplify the calculation a set of dimensionlessvariables were introduced as listed in Table 1 Equations (20)and (21) and the initial and boundary conditions can bewritten in dimensionless form

1119903119863120597120597119903119863 (119903119863119872119905119891119863

120597119901119891119863120597119903119863 ) = 1205961 120597119901119891119863120597119905119863 + 1205962 120597119901119898119863120597119905119863 (25)

1205962 120597119901119898119863120597119905119863 = 120582 (119901119891119863 minus 119901119898119863) (26)

The initial condition

119901119891119863 (119903119863 119905119863 = 0) = 0 (27)

119901119898119863 (119903119863 119905119863 = 0) = 0 (28)

The inner boundary condition considering the skin effectis

119901119908119863 = (119901119891119863 minus 119878119903119863120597119901119891119863120597119903119863 )119903119863=1

(29)

The inner boundary condition considering wellbore stor-age effect is

119862119863119889119901119908119863119889119905119863 minus (119903119863120597119901119891119863120597119903119863 )119903119863=1

= 1 (30)

The constant pressure outer boundary condition is

119901119891119863 (119903119863 = 119903119890119863 119905119863) = 0 (31)

119901119898119863 (119903119863 = 119903119890119863 119905119863) = 0 (32)

The no flow outer boundary condition is

(120597119901119891119863120597119903119863 )119903119863=119903119890119863

= 0 (33)

(120597119901119898119863120597119903119863 )119903119863=119903119890119863

= 0 (34)

Figure 1 Schematic of two-zone composite reservoir

To simplify the computation a logarithmic transform z =ln(rD) and the Laplace transform with the quasi-stationaryassumption were used Thus (25)ndash(34) can be rewritten as

11198902119911 119889119889119911 (119872119905119891119863119889119901119891119863119889119911 )

= 1205961 [119906119901119891119863 minus 119901119891119863 (119911 0)]+ 1205962 [119906119901119898119863 minus 119901119898119863 (119911 0)]

(35)

1205962 [119906119901119898119863 minus 119901119898119863 (119911 0)] = 120582 (119901119891119863 minus 119901119898119863) (36)

The initial condition is

119901119891119863 (119905119863 = 0) = 0 (37)

119901119898119863 (119905119863 = 0) = 0 (38)

The inner boundary condition considering the skin effectis

119901119908119863 = (119901119891119863 minus 119878119889119901119891119863119889119911 )119911=0

(39)

The inner boundary condition considering wellbore stor-age effect is

(119889119901119891119863119889119911 )119911=0

minus 119862119863 [119906119901119908119863 minus 119901119908119863 (119905119863 = 0)] = minus1119906 (40)

The constant pressure outer boundary condition is

119901119891119863 (119911 = 119911119890) = 0 (41)

119901119898119863 (119911 = 119911119890) = 0 (42)

Mathematical Problems in Engineering 5

Table 1 Dimensionless variable for pressure calculation

119872119905119891 = 119896119891 (119896119903119900119891 (119878119908119891 = 1)120583119900 + 119896119903119908119891 (119878119908119891 = 1)

120583119908 ) 119872119905119891119863 = 119872119905119891119872119905119891

119901119891119863 = 2120587119872119905119891ℎ119902119861119908 (119901119891 minus 119901119894) 120582 = 1198651198781199032119908119872119905119906119872119905119891119901119898119863 = 2120587119872119905119891ℎ119902119861119908 (119901119898 minus 119901119894) 119903119863 = 119903119903119908119862119905119891 = 119862119903 + 119862119908119878119908119894 + 119862119900 (1 minus 119878119908119894) 1205961 = 0119891119862119905119891

(0119891119862119905119891 + 0119898119862119905119898)119862119905119898 = 119862119903 + 119862119908119878119908119888119898 + 119862119900 (1 minus 119878119908119888119898) 1205962 = 0119898119862119905119898(0119891119862119905119891 + 0119898119862119905119898)119905119863 = 119872119905119891

(0119891119862119905119891 + 0119898119862119905119898) 1199032119908 119862119863 = 1198622120587 (0119891119862119905119891 + 0119898119862119905119898) ℎ1199032119908

The no flow outer boundary condition is

(119889119901119891119863119889119911 )119911=119911119890

= 0 (43)

(119889119901119898119863119889119911 )119911=119911119890

= 0 (44)

The pressure change near the wellbore was relativelylarger than the area away from the bottom of the well Sothe formation was divided into n grid blocks in radial direc-tion according to the point-centered logarithmic griddingmethod which was shown as the dotted blue line in Figure 1By using finite-difference approximation (35)ndash(44) can bewritten as

119886119894119901119891119863119894 + 119887119894119901119891119863119894+1 + 119888119894119901119891119863119894+2 = 119889119894 (45)

119901119898119863119894 = 120582119901119891119863119894 + 1205962119901119898119863 (119905119863 = 0)1205962119906 + 120582 (46)

where

119886119894 = 120583119908(119896119903119900119891 (119878119908119891 (119903119894minus1 119905))120583119900 + 119896119903119908119891 (119878119908119891 (119903119894minus1 119905))120583119908 ) (47)

119887119894 = minus[119886119894 + 119888119894

+ (120596112059621199062 + 1205961120582119906 + 12059621205821199061205962119906 + 120582 )119894

1198902119894Δ119911Δ1199112](48)

119888119894 = 120583119908(119896119903119900119891 (119878119908119891 (119903119894 119905))120583119900 + 119896119903119908119891 (119878119908119891 (119903119894 119905))120583119908 ) (49)

119889119894 = minus[1198902119894Δ119911Δ11991121205961119894119901119891119863119894 (119905119863 = 0)

+ 1198902119894Δ119911Δ1199112 ( 12059621205821205962119906 + 120582)119894 119901119898119863119894 (119905119863 = 0)](50)

The inner boundary condition is

1198901119901119908119863 + 11989021199011198911198630 + 11989031199011198911198631 = 1198908 (51)

11989041199011198911198630 + 11989051199011198911198631 = 1198909 (52)

The outer boundary condition is

1198906119901119891119863119899minus1 minus 1198907119901119891119863119899 = 0 (53)

The coefficients for different boundary conditions areshown in Table 2

The pressure difference equations ((45) and (46)) can bewritten in a matrix form as

M997888rarr119909 = 997888rarr119887 (54)

where M is a large sparse matrix 997888x is the unknown pressurevector and

997888b is the known vector related to the initial and

6 Mathematical Problems in Engineering

Table 2 Coefficients with different boundary conditions

Boundary conditions Coefficients

Constant pressure outer boundary1198901 = 1 1198902 = minus(1 + 119878Δ119911 ) 1198903 = 119878Δ119911 1198904 = 1 + Δ119911119862119863119906 + 119878119862119863119906 1198905 = minus119878119862119863119906 minus 1 1198906 = 0

1198907 = 1 1198908 = 0 1198909 = Δz119906 minus Δz119862119863119901119908119863(119905119863 = 0)No flow outer boundary

1198901 = 1 1198902 = minus(1 + 119878Δ119911 ) 1198903 = 119878Δ119911 1198904 = 1 + Δ119911119862119863119906 + 1198781198621198631199061198905 = minus119878119862119863119906 minus 1 1198906 = 119886119899minus1 + 119888119899minus11198907 = 119887119899minus1 1198908 = 0 1198909 = Δz119906 minus Δz119862119863119901119908119863(119905119863 = 0)

boundary conditionsThematrixM and vectors997888x and997888b can

be explicitly given as

M

=

[[[[[[[[[[[[[[[[[[[[[[[[[[[

1198901 1198902 1198903 0 0 00 1198904 1198905 0 0 00 0 1198861 1198871 1198881 00 0 0 1198862 1198872 1198882

d

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

sdot sdot sdot

d

d

sdot sdot sdot

0 0 00 0 00 0 00 0 0

d

d d

119886119899minus3 119887119899minus3 119888119899minus3119886119899minus2 119887119899minus2 119888119899minus20 1198906 1198907

]]]]]]]]]]]]]]]]]]]]]]]]]]]

(55)

997888rarr119909 =

[[[[[[[[[[[[[[[[[

119901119908119863119901119891119863011990111989111986311199011198911198632

119901119891119863119899minus1

]]]]]]]]]]]]]]]]]

(56)

997888rarr119887 =

[[[[[[[[[[[[[[[[[

1198908119890911988911198892

119889119899minus1

]]]]]]]]]]]]]]]]]

(57)

Krwmkrom

KrwfKrof

0

02

04

06

08

1

Kro

K

rw

02 04 06 08 10Sw

Figure 2 Curves of relative permeability of oil and water in fractureand matrix

The unknown vector 997888x was calculated by using theStehfest algorithm Therefore the dimensionless pressureand pressure derivatives [41] at the bottom of the wellborecan be plotted according to the solutions of the formulatedmathematical model for the water injection system Thebasic parameters and the relative permeability data for thecalculation of pressure curves are illustrated in Table 3 andFigure 2

3 Verification of the Mathematical Model

Because little was available in the literatures for pressurebehavior of a water injection well considering saturationgradient and water imbibition in dual-porosity reservoirby LTFD method the model solutions were validated bysimplifying the model into a single phase water injectionwell in a dual-porosity system which was compared to theconventional analytical solution [42] by commercial softwareas shown in Figure 3 From the figure the LTFD results showgood agreement with the analytical solution

Moreover a field water injection test was used to furtherverify the proposedmodelThe basic information of thewaterinjection well and the relative permeability data was listed inTable 4 and Figure 4 And Figure 5 shows the pressure curves

Mathematical Problems in Engineering 7

tDPwDminusLTFD methoddPwDminusLTFD method

PwDminusanalytical methoddPwDminusanalytical method

10minus2

10minus1

100

101

101

102

102

103

104

105

106

107

108

PwD

dPw

D

Figure 3 Comparison of the results by LTFDmethod and analyticalmethod

Table 3 Basic parameters for calculation of pressure curves

Parameter Value UnitsFormation height 10 mFracture porosity 003 fractionMatrix porosity 027 fractionWellbore radius 01 mFracture permeability 06 120583m2Matrix permeability 6 times 10minus5 120583m2Irreducible water saturation 20 Residual oil saturation 14 Oil volume factor 103 fractionWater volume factor 1 fractionShape factor 6 times 10minus4 1cm2

Oil viscosity 1 mPasWater viscosity 05 mPasrock compressibility 6 times 10minus4 1MPaoil compressibility 1 times 10minus3 1MPawater compressibility 4 times 10minus4 1MPaInjection rate 30 m3dayInitial pressure 14 MPaSkin factor 01 fractionImbibition rate coefficient 01 1dayWellbore storage coefficient 001 1m3MPa

of the well test data comparing with the results in our modelFrom the figure it is found that our results match well withthe field test data

4 Model Features Analysis

The pressure curves of oil-water two-phase flow consider-ing water imbibition and the saturation gradients duringinjection period by the LTFD method were plotted in thissection The behaviors of the typical curves were analyzed by

Table 4 Basic information of the water injection well

Parameter Value UnitsFormation height 126 mFracture porosity 003 fractionMatrix porosity 0268 fractionWellbore radius 012 mOil volume factor 1163 fractionWater volume factor 1 fractionOil viscosity 96 mPasWater viscosity 05 mPasrock compressibility 5 times 10minus4 1MPaoil compressibility 48 times 10minus3 1MPawater compressibility 39 times 10minus4 1MPaInjection rate 256 m3dayInitial pressure 15 MPa

KrwKro

0

02

04

06

08

1

Kro

K

rw

02 04 06 08 10Sw

Figure 4 The relative permeability of oil and water for the waterinjection well

comparing the typical curves of single phase flow with thetwo-phase flow curves in the fractured reservoirs as follows

41 Features of Typical Curves Figure 6 illustrates the pres-sure curves of single phase flow and oil-water two-phaseflow in fractured reservoirs by LTFD method without waterimbibition According to the feature of the curves theformation flow can be divided into three stages The firststage is the flow in the fracture system the second stage isthe flow between the fracture and the matrix system andthe third stage is the flow in the whole system including thefracture andmatrixThe figure shows that the shape of curvesfor single phase flow and two-phase flow is similar but theposition of the two-phase flow curves is on the upper rightof the single phase flow curves The reason is that when oiland water exist in the whole system the total mobility of oiland water is lower than the single phase and the bottom-hole pressure increases more quickly as the fluid is injected

8 Mathematical Problems in Engineering

tDPwDminusField testdPwDminusField test

PwDminusLTFD methoddPwDminusLTFD method

101

102

103

104

105

106

107

100

101

102

PwD

dPw

D

Figure 5 Comparison of the results in this work and the fieldinjection test

tD

PwDminussingle phasedPwDminussingle phase

PwDminustwo-phasedPwDminustwo-phase

101

102

103

104

105

106

107

108

10minus1

100

101

102

PwD

dPw

D

Figure 6 Typical curves of single phase and two-phase flow duringinjection period

into the well Thus the pressure derivative curves of two-phase flow move upward compared to the single phase flowcurves Moreover due to the decrease of the total mobilitythe interporosity flow capacity between matrix and fracturesystem decreases when the single phase flow becomes two-phase flow Thus the ldquodiprdquo of the two-phase flow curvesoccurs later than that of the single phase flow curves

42 Effects of Parameters The key factors influencing thepressure behavior of water injection period including skinfactor wellbore storage coefficient shape factor water imbibi-tion rate coefficient and boundary conditions were discussedin detail in this part

421 Skin Factor Figure 7 shows the effect of skin factoron the pressure curves of the water injection well The solid

tD

S=minus1S=minus1S=1

S=1S=10S=10

S=minus1S=minus1S=1

S=1S=10S=10

102

104

106

108

10minus1

100

101

102

PwD

dPw

D

Figure 7 Influence of different skin factors on pressure transientbehavior

lines are the curves of single phase flow and the circlesare oil-water two-phase flow curves The red green andblue colors indicate that the shape factor values are -1 1and 10 respectively From the figure the skin factor mainlyinfluences the peak value of the pressure derivatives As thevalue of skin factor increases the peak value gets higherand the time for peak value occurrence becomes later whichindicates that the reservoir is more severely contaminatedThe pressure derivative curves of two-phase flow movetoward the upper right of the single phase flow curves whichshows the same change rule with Figure 6 When the skinfactor becomes larger the peak value difference between thesingle phase and two-phase decreases and the inclination ofthe curves after the peak value increases

422 Wellbore Storage Coefficient Figure 8 is the influencesof wellbore storage coefficient on the pressure and pressurederivative curves of the water injection well The red greenand blue colors indicate that wellbore storage coefficientvalues are 001 01 and 1 respectively As seen the maindifference of the curves occurs on the straight line of earlyperiod As the wellbore storage coefficient increases thestraight line section becomes longer and the time of peakvalue occurs later Besides the wellbore storage effect willcover up the radial flow stage in the fracture system directlygoing to the transitional flow stage between the fracture andmatrix The pressure derivative curves of two-phase flowmove upper right to the single phase flow curves after thestraight line of wellbore storage stage which agrees with thechange rule of Figure 6

423 Shape Factor The influences of shape factor on thepressure curves of the water injection well are shown inFigure 9 The red green and blue colors indicate thatshape factor values are 10minus5 10minus4 and 10minus3 respectivelyBy comparison of the curves it is found that the shape

Mathematical Problems in Engineering 9

tD10

210

410

610

810

minus3

10minus2

10minus1

100

101

102

PwD

dPw

D

C=001C=001C=01

C=01C=1C=1

C=001C=001C=01

C=01C=1C=1

Figure 8 Influence of different wellbore storage coefficients onpressure transient behavior

tD

Fs=1lowast10minus5Fs=1lowast10minus5Fs=1lowast10minus4Fs=1lowast10minus4Fs=1lowast10minus3Fs=1lowast10minus3

Fs=1lowast10minus5Fs=1lowast10minus5Fs=1lowast10minus4Fs=1lowast10minus4Fs=1lowast10minus3Fs=1lowast10minus3

102

104

106

108

10minus1

100

101

102

PwD

dPw

D

Figure 9 Influence of different shape factors on pressure transientbehavior

factor mainly has an effect on the position of the ldquodiprdquo ofthe pressure derivatives When the value of shape factorgets greater the ldquodiprdquo appears earlier and the ldquodiprdquo movescloser to the left The main reason is that the shape factoris a component of the equation defining the interporosityparameter The higher the shape factor is the easier the fluidexchange between the fracture and matrix is and the earlierthe transitional flow stage occurs It also demonstrates thatthe pressure derivative curves of two-phase flow move upperright to the single phase flow curves which is in line with thechange rule of Figure 6

tDRc=0Rc=0Rc=001

Rc=001Rc=005Rc=005

Rc=01Rc=01Rc=05

Rc=05Rc=1Rc=1

102

104

106

108

10minus2

10minus1

100

101

102

PwD

dPw

DFigure 10 Pressure behavior of different imbibition rate coefficientswith no flow outer boundary

424 Imbibition Rate Coefficient with No Flow Outer Bound-ary Condition Figure 10 describes the pressure behaviorof the water injection well under no flow outer boundarycondition when the water imbibition rate coefficient valuesare 0 001 005 01 05 and 1 respectively From the figurethe imbibition rate coefficient mainly has an effect on thewhole system (matrix and fracture system) radial flow periodof the curves As the imbibition rate coefficient becomeslarger there are more fluid exchange between the fractureand matrix and more fluid supply from the matrix to thefracture The water imbibition process is slow so there isa second interporosity period after the whole system radialflow period And with the increase of water imbibition ratecoefficient the pressure derivative curves dip downwardearlier and the degree for the curves dropping down becomeslarger After that the pressure derivative curves rise up to auniform line with the same slope due to the influence of theno flow outer boundary

425 Imbibition Rate Coefficient with Constant PressureBoundary Condition Figure 11 shows the pressure and pres-sure derivative curves of the water injection well underconstant pressure boundary condition when the water imbi-bition rate coefficient values are 0 001 005 01 05 and1 respectively By comparison and analysis it is concludedthat the pressure derivative curve is a straight line at thewhole system radial flow period when no water imbibition isconsideredWhen considering water imbibition the pressurederivatives decline at the whole system radial flow periodThe pressure derivatives drop downward more and earlierwith the increase of imbibition rate coefficient After that thecurves also drop down due to the constant pressure outerboundary effect

10 Mathematical Problems in Engineering

tD

Rc=0Rc=0Rc=001

Rc=001Rc=005Rc=005

Rc=01Rc=01Rc=05

Rc=05Rc=1Rc=1

102

104

106

108

10minus2

10minus1

100

101

102

PwD

dPw

D

Figure 11 Pressure behavior of different imbibition rate coefficientswith constant pressure boundary

5 Conclusions

In this work a semianalytical two-phase flow model consid-ering water imbibition and the saturation gradients withineach region during injection period was proposed to analyzethe pressure transient behavior of the water injection well infractured reservoirs Validation of the presented model wasperformed by analytical method and a field water injectiontest The model features were investigated by sensitivityanalysis Some conclusions were drawn as follows

(1) The shape of the pressure curves for the proposedmodel is similar to the single phase flowmodel whilethe position of the curves for the two-phase flowmodel is on the upper right of the single phase flowmodel

(2) Themain influence of skin factor on the curves occurson the peak value of the pressure derivatives As theskin factor becomes larger the peak value differencebetween single phase and two-phase pressure curvesdecreases and the inclination of the curves after thepeak value increases

(3) The wellbore storage coefficient mainly has an effecton the straight line of the early period for the pressurecurves The shape factor mainly has an effect on theposition of the ldquodiprdquo of the pressure derivatives whichis the same as the interporosity parameter

(4) The imbibition rate coefficient mainly influences thewhole system radial flow period of the pressurecurves The pressure derivative curves dip downwardearlier and drop down to a greater degree with theincrease of imbibition rate coefficient

Nomenclature

R Recovery fractionR119888 Imbibition rate coefficient 1day119877infin Ultimate cumulative oil recovery fraction0 Porosity fraction119904119900119903 Residual oil saturation fraction119904119908119888 Irreducible water saturation fraction119904119908 Water saturation fractionz Logarithmic transform variable dimensionlessu Laplace variable dimensionlessq Displacement rate m3dt Time hk Permeability 1205831198982119896119903 Relative permeability fractionh Formation height mr Radial distance m119903119908 Wellbore radius mp Pressure MPa120583 Viscosity mPasdotsB Formation volume factor dimensionless120591 Fracture matrix transfer term m3dF119904 Shape factor 1m2119862119905 Total compressibility 1MPa119872119905 Total mobility 1MPa120582 Interporosity flow coefficient fraction120596 Storativity ratio fractionS Skin factor dimensionlessC Wellbore storage coefficient m3MPa

Subscripts

f Fracturem Matrixo Oilw Watert Totale ExternalD Dimensionless variable

Superscripts

- Laplace-transformed variable Endpoint property997888rarr Vector symbol

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

The authors gratefully acknowledge the support of theNational Basic Research 973 Program of China (Grant No

Mathematical Problems in Engineering 11

2015CB250900) the National Natural Science Foundationof China (Grant No 51704237) and the Research Projectsof Shaanxi Provincial Education Department (Grant No13JS090)

References

[1] R L Perrine Analysis of Pressure-buildup Curves vol 56American Petroleum Institute 1956

[2] J C Martin ldquoSimplified Equations of Flow in Gas Drive Reser-voirs and the Theoretical Foundation of Multiphase PressureBuildup Analysesrdquo Society of Petroleum Engineers vol 216 pp321ndash323 1959

[3] W Hurst ldquoInterference between Oil Fieldrdquo Transactions of theMetallurgical Society of AIME vol 219 no 8 pp 175ndash192 1960

[4] MMortada ldquoOilfield Interference in Aquifers of Non-UniformPropertiesrdquo Journal of Petroleum Technology vol 12 no 12 pp55ndash57 2013

[5] T Loucks and E Guerrero ldquoPressure Drop in a CompositeReservoirrdquo SPE Journal vol 1 no 03 pp 170ndash176 2013

[6] H Bixel and H Van Poollen ldquoPressure Drawdown and Buildupin the Presence of Radial Discontinuitiesrdquo SPE Journal vol 7no 03 pp 301ndash309 2013

[7] H Ramey ldquoApproximate Solutions For Unsteady LiquidFlow InComposite Reservoirsrdquo Journal of Canadian Petroleum Technol-ogy vol 9 no 01 2013

[8] Q Deng R Nie Y Jia et al ldquoPressure transient behavior of afractured well in multi-region composite reservoirsrdquo Journal ofPetroleum Science and Engineering vol 158 pp 535ndash553 2017

[9] H Kazemi LMerrill and J Jargon ldquoProblems in Interpretationof Pressure Fall-OffTests in ReservoirsWithAndWithout FluidBanksrdquo Journal of PetroleumTechnology vol 24 no 09 pp 1147ndash1156 2013

[10] L Merrill H Kazemi and W B Gogarty ldquoPressure FalloffAnalysis in Reservoirs With Fluid Banksrdquo Journal of PetroleumTechnology vol 26 no 07 pp 809ndash818 2013

[11] A Satman M Eggenschwiler R Tang and R HJ ldquoAnanalytical study of transient flow in systemswith radial disconti-nuitiesrdquo in Proceedings of the SPE9399 presented at SPE AnnualTechnical Conference and Exhibition Dallas Texas TX USASeptember 1980

[12] A Al-Bemani and I Ershaghi ldquoTwo-Phase Flow InterporosityEffects on Pressure Transient Test Response in Naturally Frac-tured Reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Dallas Texas

[13] Z Chen W Yu X Liao X Zhao Y Chen and K SepehrnoorildquoA Two-Phase Flow Model for Fractured Horizontal Well withComplex Fracture Networks Transient Analysis in FlowbackPeriodrdquo in Proceedings of the SPE Liquids-Rich Basins Confer-ence - North America Midland Texas USA

[14] H G Weinstein ldquoCold Waterflooding a Warm Reservoirrdquo inProceedings of the Fall Meeting of the Society of PetroleumEngineers of AIME Houston Texas

[15] S E Buckley andM C Leverett ldquoMechanism of fluid displace-ment in sandsrdquoTransactions of the AIME vol 146 no 1 pp 107ndash116 2013

[16] A Sosa R Raghavan and T Limon ldquoEffect of Relative Perme-ability andMobility Ratio on Pressure Falloff Behaviorrdquo Journalof Petroleum Technology vol 33 no 06 pp 1125ndash1135 2013

[17] P Hazebroek H Rainbow and CS Matthews ldquoPressure Fall-Off in Water Injection Wellsrdquo Transactions of the MetallurgicalSociety of AIME vol 213 pp 250ndash260 1958

[18] R Carter ldquoPressure Behavior of a Limited Circular CompositeReservoirrdquo SPE Journal vol 6 no 04 pp 328ndash334 2013

[19] A Odeh ldquoFlowTest Analysis for aWell with Radial Discontinu-ityrdquo Journal of Petroleum Technology vol 21 no 02 pp 207ndash2102013

[20] M Abbaszadeh and M Kamal ldquoPressure-Transient Testing ofWater-InjectionWellsrdquo SPE Reservoir Engineering vol 4 no 01pp 115ndash124 2013

[21] N Yeh and R Agarwal ldquoPressure Transient Analysis of Injec-tionWells in ReservoirsWithMultiple Fluid Banksrdquo in Proceed-ings of the SPE Annual Technical Conference and Exhibition SanAntonio Texas

[22] M M Levitan ldquoApplication of water injectionfalloff tests forreservoir appraisal New analytical solution method for two-phase variable rate problemsrdquo SPE Journal vol 8 no 4 pp 341ndash349 2003

[23] T Nanba and R Horne ldquoEstimation of Water and Oil RelativePermeabilities From Pressure Transient Analysis of WaterInjectionWell Datardquo in Proceedings of the SPE Annual TechnicalConference and Exhibition San Antonio Texas

[24] R B Bratvold andRNHorne ldquoAnalysis of pressure-falloff testsfollowing cold-water injectionrdquo SPE Formation Evaluation vol5 no 3 pp 293ndash302 1990

[25] S Chen G Li and A C Reynolds ldquoAnalytical Solution forInjection-Falloff-Production Testrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition San AntonioTexas USA

[26] A A Boughrara A M M Peres S Chen A A V Machadoand A C Reynolds ldquoApproximate analytical solutions for thepressure response at a water-injection wellrdquo SPE Journal vol12 no 1 pp 19ndash34 2007

[27] S Zheng and w xu ldquoNew approaches for analyzing transientpressure from oil and water two-phase flowing reservoirrdquo inProceedings of the Kuwait International Petroleum Conferenceand Exhibition Kuwait City Kuwait

[28] N A El-Khatib ldquoTransient Pressure Behavior of CompositeReservoirs with Moving Boundariesrdquo in Proceedings of theMiddle East Oil Show and Conference Bahrain

[29] A Dastan M M Kamal Y Hwang F Suleen and S MorsyldquoFalloff Testing Under Multiphase Flow Conditions in Natu-rally Fractured Reservoirsrdquo in Proceedings of the SPE WesternRegional Meeting Garden Grove California USA

[30] C n Machado and A C Reynolds ldquoApproximate semi-analytical solution for injection-falloff-production well test ananalytical tool for the in situ estimation of relative permeabilitycurvesrdquo Transport in Porous Media vol 121 no 1 pp 207ndash2312018

[31] T Barkve ldquoAnalytical study of reservoir pressure during water-injection well testsrdquo Society of Petroleum Engineers of AIME(Paper) SPE 1986

[32] T S Ramakrishnan and F J Kuchuk ldquoTesting injection wellswith rate and pressure datardquo SPE Formation Evaluation vol 9no 3 pp 228ndash236 1994

[33] L G Thompson and A C Reynolds ldquoWell testing for radiallyheterogeneous reservoirs under single and multiphase flowconditionsrdquo SPE Formation Evaluation vol 12 no 1 pp 57ndash641997

12 Mathematical Problems in Engineering

[34] JS Aronofsky L Masse and SG Natanson ldquoA Model for theMechanism of Oil Recovery from the Porous Matrix Due toWater Invasion in Fractured Reservoirsrdquo Trans AIME vol 213pp 17ndash19 1958

[35] A de Swaan ldquoTheory of Waterflooding in Fractured Reser-voirsrdquo SPE Journal vol 18 no 02 pp 117ndash122 2013

[36] H Kazemi J Gilman and A Elsharkawy ldquoAnalytical andNumerical Solution of Oil Recovery From Fractured ReservoirsWith Empirical Transfer Functions (includes associated papers25528 and 25818)rdquo SPE Reservoir Engineering vol 7 no 02 pp219ndash227 2013

[37] T Shimamoto D Kuramoto N Arihara and T Onishi ldquoA NewPressure Transient Analysis Model for Water Injection Wellin Dual-Porosity Reservoirsrdquo in Proceedings of the SPE100881presented at SPE Asia Pacific Oil ampamp Gas Conference andExhibition Adelaide Australia 2006

[38] G J Moridis D A McVay D L Reddell and T A BlasingameldquoLaplace transform finite difference (LTFD) numerical methodfor the simulation of compressible liquid flow in reservoirsrdquo SPEAdvanced Technology Series vol 2 no 2 pp 122ndash131 1994

[39] A D Habte and M Onur ldquoLaplace-transform finite-differenceand quasistationary solution method for water-injectionfallofftestsrdquo SPE Journal vol 19 no 3 pp 398ndash409 2014

[40] J Warren and P Root ldquoThe behavior of naturally fracturedreservoirsrdquo SPE Journal vol 3 no 3 pp 245ndash255 2013

[41] D Bourdet J Ayoub and Y Pirard ldquoUse of Pressure Derivativein Well Test Interpretationrdquo SPE Formation Evaluation vol 4no 02 pp 293ndash302 2013

[42] M Mavor and H Cinco-Ley ldquoTransient Pressure BehaviorOf Naturally Fractured Reservoirsrdquo in Proceedings of the SPECalifornia Regional Meeting Ventura California

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4 Mathematical Problems in Engineering

in the matrix is applied to calculate the relative permeabilityduring the falloff period (see (24))

119872119905119891 = 119896119891(119896119903119900119891 (119878119908119891)120583119900 + 119896119903119908119891 (119878119908119891)120583119908 ) (22)

119872119905119906 = 119896119898(119896119903119900119898 (119878119908119891)120583119900 + 119896119903119908119898 (119878119908119891)120583119908 ) (23)

119872119905119906 = 119896119898 (119896119903119900119898 (119878119908119898)120583119900 + 119896119903119908119898 (119878119908119898)120583119908 ) (24)

In order to simplify the calculation a set of dimensionlessvariables were introduced as listed in Table 1 Equations (20)and (21) and the initial and boundary conditions can bewritten in dimensionless form

1119903119863120597120597119903119863 (119903119863119872119905119891119863

120597119901119891119863120597119903119863 ) = 1205961 120597119901119891119863120597119905119863 + 1205962 120597119901119898119863120597119905119863 (25)

1205962 120597119901119898119863120597119905119863 = 120582 (119901119891119863 minus 119901119898119863) (26)

The initial condition

119901119891119863 (119903119863 119905119863 = 0) = 0 (27)

119901119898119863 (119903119863 119905119863 = 0) = 0 (28)

The inner boundary condition considering the skin effectis

119901119908119863 = (119901119891119863 minus 119878119903119863120597119901119891119863120597119903119863 )119903119863=1

(29)

The inner boundary condition considering wellbore stor-age effect is

119862119863119889119901119908119863119889119905119863 minus (119903119863120597119901119891119863120597119903119863 )119903119863=1

= 1 (30)

The constant pressure outer boundary condition is

119901119891119863 (119903119863 = 119903119890119863 119905119863) = 0 (31)

119901119898119863 (119903119863 = 119903119890119863 119905119863) = 0 (32)

The no flow outer boundary condition is

(120597119901119891119863120597119903119863 )119903119863=119903119890119863

= 0 (33)

(120597119901119898119863120597119903119863 )119903119863=119903119890119863

= 0 (34)

Figure 1 Schematic of two-zone composite reservoir

To simplify the computation a logarithmic transform z =ln(rD) and the Laplace transform with the quasi-stationaryassumption were used Thus (25)ndash(34) can be rewritten as

11198902119911 119889119889119911 (119872119905119891119863119889119901119891119863119889119911 )

= 1205961 [119906119901119891119863 minus 119901119891119863 (119911 0)]+ 1205962 [119906119901119898119863 minus 119901119898119863 (119911 0)]

(35)

1205962 [119906119901119898119863 minus 119901119898119863 (119911 0)] = 120582 (119901119891119863 minus 119901119898119863) (36)

The initial condition is

119901119891119863 (119905119863 = 0) = 0 (37)

119901119898119863 (119905119863 = 0) = 0 (38)

The inner boundary condition considering the skin effectis

119901119908119863 = (119901119891119863 minus 119878119889119901119891119863119889119911 )119911=0

(39)

The inner boundary condition considering wellbore stor-age effect is

(119889119901119891119863119889119911 )119911=0

minus 119862119863 [119906119901119908119863 minus 119901119908119863 (119905119863 = 0)] = minus1119906 (40)

The constant pressure outer boundary condition is

119901119891119863 (119911 = 119911119890) = 0 (41)

119901119898119863 (119911 = 119911119890) = 0 (42)

Mathematical Problems in Engineering 5

Table 1 Dimensionless variable for pressure calculation

119872119905119891 = 119896119891 (119896119903119900119891 (119878119908119891 = 1)120583119900 + 119896119903119908119891 (119878119908119891 = 1)

120583119908 ) 119872119905119891119863 = 119872119905119891119872119905119891

119901119891119863 = 2120587119872119905119891ℎ119902119861119908 (119901119891 minus 119901119894) 120582 = 1198651198781199032119908119872119905119906119872119905119891119901119898119863 = 2120587119872119905119891ℎ119902119861119908 (119901119898 minus 119901119894) 119903119863 = 119903119903119908119862119905119891 = 119862119903 + 119862119908119878119908119894 + 119862119900 (1 minus 119878119908119894) 1205961 = 0119891119862119905119891

(0119891119862119905119891 + 0119898119862119905119898)119862119905119898 = 119862119903 + 119862119908119878119908119888119898 + 119862119900 (1 minus 119878119908119888119898) 1205962 = 0119898119862119905119898(0119891119862119905119891 + 0119898119862119905119898)119905119863 = 119872119905119891

(0119891119862119905119891 + 0119898119862119905119898) 1199032119908 119862119863 = 1198622120587 (0119891119862119905119891 + 0119898119862119905119898) ℎ1199032119908

The no flow outer boundary condition is

(119889119901119891119863119889119911 )119911=119911119890

= 0 (43)

(119889119901119898119863119889119911 )119911=119911119890

= 0 (44)

The pressure change near the wellbore was relativelylarger than the area away from the bottom of the well Sothe formation was divided into n grid blocks in radial direc-tion according to the point-centered logarithmic griddingmethod which was shown as the dotted blue line in Figure 1By using finite-difference approximation (35)ndash(44) can bewritten as

119886119894119901119891119863119894 + 119887119894119901119891119863119894+1 + 119888119894119901119891119863119894+2 = 119889119894 (45)

119901119898119863119894 = 120582119901119891119863119894 + 1205962119901119898119863 (119905119863 = 0)1205962119906 + 120582 (46)

where

119886119894 = 120583119908(119896119903119900119891 (119878119908119891 (119903119894minus1 119905))120583119900 + 119896119903119908119891 (119878119908119891 (119903119894minus1 119905))120583119908 ) (47)

119887119894 = minus[119886119894 + 119888119894

+ (120596112059621199062 + 1205961120582119906 + 12059621205821199061205962119906 + 120582 )119894

1198902119894Δ119911Δ1199112](48)

119888119894 = 120583119908(119896119903119900119891 (119878119908119891 (119903119894 119905))120583119900 + 119896119903119908119891 (119878119908119891 (119903119894 119905))120583119908 ) (49)

119889119894 = minus[1198902119894Δ119911Δ11991121205961119894119901119891119863119894 (119905119863 = 0)

+ 1198902119894Δ119911Δ1199112 ( 12059621205821205962119906 + 120582)119894 119901119898119863119894 (119905119863 = 0)](50)

The inner boundary condition is

1198901119901119908119863 + 11989021199011198911198630 + 11989031199011198911198631 = 1198908 (51)

11989041199011198911198630 + 11989051199011198911198631 = 1198909 (52)

The outer boundary condition is

1198906119901119891119863119899minus1 minus 1198907119901119891119863119899 = 0 (53)

The coefficients for different boundary conditions areshown in Table 2

The pressure difference equations ((45) and (46)) can bewritten in a matrix form as

M997888rarr119909 = 997888rarr119887 (54)

where M is a large sparse matrix 997888x is the unknown pressurevector and

997888b is the known vector related to the initial and

6 Mathematical Problems in Engineering

Table 2 Coefficients with different boundary conditions

Boundary conditions Coefficients

Constant pressure outer boundary1198901 = 1 1198902 = minus(1 + 119878Δ119911 ) 1198903 = 119878Δ119911 1198904 = 1 + Δ119911119862119863119906 + 119878119862119863119906 1198905 = minus119878119862119863119906 minus 1 1198906 = 0

1198907 = 1 1198908 = 0 1198909 = Δz119906 minus Δz119862119863119901119908119863(119905119863 = 0)No flow outer boundary

1198901 = 1 1198902 = minus(1 + 119878Δ119911 ) 1198903 = 119878Δ119911 1198904 = 1 + Δ119911119862119863119906 + 1198781198621198631199061198905 = minus119878119862119863119906 minus 1 1198906 = 119886119899minus1 + 119888119899minus11198907 = 119887119899minus1 1198908 = 0 1198909 = Δz119906 minus Δz119862119863119901119908119863(119905119863 = 0)

boundary conditionsThematrixM and vectors997888x and997888b can

be explicitly given as

M

=

[[[[[[[[[[[[[[[[[[[[[[[[[[[

1198901 1198902 1198903 0 0 00 1198904 1198905 0 0 00 0 1198861 1198871 1198881 00 0 0 1198862 1198872 1198882

d

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

sdot sdot sdot

d

d

sdot sdot sdot

0 0 00 0 00 0 00 0 0

d

d d

119886119899minus3 119887119899minus3 119888119899minus3119886119899minus2 119887119899minus2 119888119899minus20 1198906 1198907

]]]]]]]]]]]]]]]]]]]]]]]]]]]

(55)

997888rarr119909 =

[[[[[[[[[[[[[[[[[

119901119908119863119901119891119863011990111989111986311199011198911198632

119901119891119863119899minus1

]]]]]]]]]]]]]]]]]

(56)

997888rarr119887 =

[[[[[[[[[[[[[[[[[

1198908119890911988911198892

119889119899minus1

]]]]]]]]]]]]]]]]]

(57)

Krwmkrom

KrwfKrof

0

02

04

06

08

1

Kro

K

rw

02 04 06 08 10Sw

Figure 2 Curves of relative permeability of oil and water in fractureand matrix

The unknown vector 997888x was calculated by using theStehfest algorithm Therefore the dimensionless pressureand pressure derivatives [41] at the bottom of the wellborecan be plotted according to the solutions of the formulatedmathematical model for the water injection system Thebasic parameters and the relative permeability data for thecalculation of pressure curves are illustrated in Table 3 andFigure 2

3 Verification of the Mathematical Model

Because little was available in the literatures for pressurebehavior of a water injection well considering saturationgradient and water imbibition in dual-porosity reservoirby LTFD method the model solutions were validated bysimplifying the model into a single phase water injectionwell in a dual-porosity system which was compared to theconventional analytical solution [42] by commercial softwareas shown in Figure 3 From the figure the LTFD results showgood agreement with the analytical solution

Moreover a field water injection test was used to furtherverify the proposedmodelThe basic information of thewaterinjection well and the relative permeability data was listed inTable 4 and Figure 4 And Figure 5 shows the pressure curves

Mathematical Problems in Engineering 7

tDPwDminusLTFD methoddPwDminusLTFD method

PwDminusanalytical methoddPwDminusanalytical method

10minus2

10minus1

100

101

101

102

102

103

104

105

106

107

108

PwD

dPw

D

Figure 3 Comparison of the results by LTFDmethod and analyticalmethod

Table 3 Basic parameters for calculation of pressure curves

Parameter Value UnitsFormation height 10 mFracture porosity 003 fractionMatrix porosity 027 fractionWellbore radius 01 mFracture permeability 06 120583m2Matrix permeability 6 times 10minus5 120583m2Irreducible water saturation 20 Residual oil saturation 14 Oil volume factor 103 fractionWater volume factor 1 fractionShape factor 6 times 10minus4 1cm2

Oil viscosity 1 mPasWater viscosity 05 mPasrock compressibility 6 times 10minus4 1MPaoil compressibility 1 times 10minus3 1MPawater compressibility 4 times 10minus4 1MPaInjection rate 30 m3dayInitial pressure 14 MPaSkin factor 01 fractionImbibition rate coefficient 01 1dayWellbore storage coefficient 001 1m3MPa

of the well test data comparing with the results in our modelFrom the figure it is found that our results match well withthe field test data

4 Model Features Analysis

The pressure curves of oil-water two-phase flow consider-ing water imbibition and the saturation gradients duringinjection period by the LTFD method were plotted in thissection The behaviors of the typical curves were analyzed by

Table 4 Basic information of the water injection well

Parameter Value UnitsFormation height 126 mFracture porosity 003 fractionMatrix porosity 0268 fractionWellbore radius 012 mOil volume factor 1163 fractionWater volume factor 1 fractionOil viscosity 96 mPasWater viscosity 05 mPasrock compressibility 5 times 10minus4 1MPaoil compressibility 48 times 10minus3 1MPawater compressibility 39 times 10minus4 1MPaInjection rate 256 m3dayInitial pressure 15 MPa

KrwKro

0

02

04

06

08

1

Kro

K

rw

02 04 06 08 10Sw

Figure 4 The relative permeability of oil and water for the waterinjection well

comparing the typical curves of single phase flow with thetwo-phase flow curves in the fractured reservoirs as follows

41 Features of Typical Curves Figure 6 illustrates the pres-sure curves of single phase flow and oil-water two-phaseflow in fractured reservoirs by LTFD method without waterimbibition According to the feature of the curves theformation flow can be divided into three stages The firststage is the flow in the fracture system the second stage isthe flow between the fracture and the matrix system andthe third stage is the flow in the whole system including thefracture andmatrixThe figure shows that the shape of curvesfor single phase flow and two-phase flow is similar but theposition of the two-phase flow curves is on the upper rightof the single phase flow curves The reason is that when oiland water exist in the whole system the total mobility of oiland water is lower than the single phase and the bottom-hole pressure increases more quickly as the fluid is injected

8 Mathematical Problems in Engineering

tDPwDminusField testdPwDminusField test

PwDminusLTFD methoddPwDminusLTFD method

101

102

103

104

105

106

107

100

101

102

PwD

dPw

D

Figure 5 Comparison of the results in this work and the fieldinjection test

tD

PwDminussingle phasedPwDminussingle phase

PwDminustwo-phasedPwDminustwo-phase

101

102

103

104

105

106

107

108

10minus1

100

101

102

PwD

dPw

D

Figure 6 Typical curves of single phase and two-phase flow duringinjection period

into the well Thus the pressure derivative curves of two-phase flow move upward compared to the single phase flowcurves Moreover due to the decrease of the total mobilitythe interporosity flow capacity between matrix and fracturesystem decreases when the single phase flow becomes two-phase flow Thus the ldquodiprdquo of the two-phase flow curvesoccurs later than that of the single phase flow curves

42 Effects of Parameters The key factors influencing thepressure behavior of water injection period including skinfactor wellbore storage coefficient shape factor water imbibi-tion rate coefficient and boundary conditions were discussedin detail in this part

421 Skin Factor Figure 7 shows the effect of skin factoron the pressure curves of the water injection well The solid

tD

S=minus1S=minus1S=1

S=1S=10S=10

S=minus1S=minus1S=1

S=1S=10S=10

102

104

106

108

10minus1

100

101

102

PwD

dPw

D

Figure 7 Influence of different skin factors on pressure transientbehavior

lines are the curves of single phase flow and the circlesare oil-water two-phase flow curves The red green andblue colors indicate that the shape factor values are -1 1and 10 respectively From the figure the skin factor mainlyinfluences the peak value of the pressure derivatives As thevalue of skin factor increases the peak value gets higherand the time for peak value occurrence becomes later whichindicates that the reservoir is more severely contaminatedThe pressure derivative curves of two-phase flow movetoward the upper right of the single phase flow curves whichshows the same change rule with Figure 6 When the skinfactor becomes larger the peak value difference between thesingle phase and two-phase decreases and the inclination ofthe curves after the peak value increases

422 Wellbore Storage Coefficient Figure 8 is the influencesof wellbore storage coefficient on the pressure and pressurederivative curves of the water injection well The red greenand blue colors indicate that wellbore storage coefficientvalues are 001 01 and 1 respectively As seen the maindifference of the curves occurs on the straight line of earlyperiod As the wellbore storage coefficient increases thestraight line section becomes longer and the time of peakvalue occurs later Besides the wellbore storage effect willcover up the radial flow stage in the fracture system directlygoing to the transitional flow stage between the fracture andmatrix The pressure derivative curves of two-phase flowmove upper right to the single phase flow curves after thestraight line of wellbore storage stage which agrees with thechange rule of Figure 6

423 Shape Factor The influences of shape factor on thepressure curves of the water injection well are shown inFigure 9 The red green and blue colors indicate thatshape factor values are 10minus5 10minus4 and 10minus3 respectivelyBy comparison of the curves it is found that the shape

Mathematical Problems in Engineering 9

tD10

210

410

610

810

minus3

10minus2

10minus1

100

101

102

PwD

dPw

D

C=001C=001C=01

C=01C=1C=1

C=001C=001C=01

C=01C=1C=1

Figure 8 Influence of different wellbore storage coefficients onpressure transient behavior

tD

Fs=1lowast10minus5Fs=1lowast10minus5Fs=1lowast10minus4Fs=1lowast10minus4Fs=1lowast10minus3Fs=1lowast10minus3

Fs=1lowast10minus5Fs=1lowast10minus5Fs=1lowast10minus4Fs=1lowast10minus4Fs=1lowast10minus3Fs=1lowast10minus3

102

104

106

108

10minus1

100

101

102

PwD

dPw

D

Figure 9 Influence of different shape factors on pressure transientbehavior

factor mainly has an effect on the position of the ldquodiprdquo ofthe pressure derivatives When the value of shape factorgets greater the ldquodiprdquo appears earlier and the ldquodiprdquo movescloser to the left The main reason is that the shape factoris a component of the equation defining the interporosityparameter The higher the shape factor is the easier the fluidexchange between the fracture and matrix is and the earlierthe transitional flow stage occurs It also demonstrates thatthe pressure derivative curves of two-phase flow move upperright to the single phase flow curves which is in line with thechange rule of Figure 6

tDRc=0Rc=0Rc=001

Rc=001Rc=005Rc=005

Rc=01Rc=01Rc=05

Rc=05Rc=1Rc=1

102

104

106

108

10minus2

10minus1

100

101

102

PwD

dPw

DFigure 10 Pressure behavior of different imbibition rate coefficientswith no flow outer boundary

424 Imbibition Rate Coefficient with No Flow Outer Bound-ary Condition Figure 10 describes the pressure behaviorof the water injection well under no flow outer boundarycondition when the water imbibition rate coefficient valuesare 0 001 005 01 05 and 1 respectively From the figurethe imbibition rate coefficient mainly has an effect on thewhole system (matrix and fracture system) radial flow periodof the curves As the imbibition rate coefficient becomeslarger there are more fluid exchange between the fractureand matrix and more fluid supply from the matrix to thefracture The water imbibition process is slow so there isa second interporosity period after the whole system radialflow period And with the increase of water imbibition ratecoefficient the pressure derivative curves dip downwardearlier and the degree for the curves dropping down becomeslarger After that the pressure derivative curves rise up to auniform line with the same slope due to the influence of theno flow outer boundary

425 Imbibition Rate Coefficient with Constant PressureBoundary Condition Figure 11 shows the pressure and pres-sure derivative curves of the water injection well underconstant pressure boundary condition when the water imbi-bition rate coefficient values are 0 001 005 01 05 and1 respectively By comparison and analysis it is concludedthat the pressure derivative curve is a straight line at thewhole system radial flow period when no water imbibition isconsideredWhen considering water imbibition the pressurederivatives decline at the whole system radial flow periodThe pressure derivatives drop downward more and earlierwith the increase of imbibition rate coefficient After that thecurves also drop down due to the constant pressure outerboundary effect

10 Mathematical Problems in Engineering

tD

Rc=0Rc=0Rc=001

Rc=001Rc=005Rc=005

Rc=01Rc=01Rc=05

Rc=05Rc=1Rc=1

102

104

106

108

10minus2

10minus1

100

101

102

PwD

dPw

D

Figure 11 Pressure behavior of different imbibition rate coefficientswith constant pressure boundary

5 Conclusions

In this work a semianalytical two-phase flow model consid-ering water imbibition and the saturation gradients withineach region during injection period was proposed to analyzethe pressure transient behavior of the water injection well infractured reservoirs Validation of the presented model wasperformed by analytical method and a field water injectiontest The model features were investigated by sensitivityanalysis Some conclusions were drawn as follows

(1) The shape of the pressure curves for the proposedmodel is similar to the single phase flowmodel whilethe position of the curves for the two-phase flowmodel is on the upper right of the single phase flowmodel

(2) Themain influence of skin factor on the curves occurson the peak value of the pressure derivatives As theskin factor becomes larger the peak value differencebetween single phase and two-phase pressure curvesdecreases and the inclination of the curves after thepeak value increases

(3) The wellbore storage coefficient mainly has an effecton the straight line of the early period for the pressurecurves The shape factor mainly has an effect on theposition of the ldquodiprdquo of the pressure derivatives whichis the same as the interporosity parameter

(4) The imbibition rate coefficient mainly influences thewhole system radial flow period of the pressurecurves The pressure derivative curves dip downwardearlier and drop down to a greater degree with theincrease of imbibition rate coefficient

Nomenclature

R Recovery fractionR119888 Imbibition rate coefficient 1day119877infin Ultimate cumulative oil recovery fraction0 Porosity fraction119904119900119903 Residual oil saturation fraction119904119908119888 Irreducible water saturation fraction119904119908 Water saturation fractionz Logarithmic transform variable dimensionlessu Laplace variable dimensionlessq Displacement rate m3dt Time hk Permeability 1205831198982119896119903 Relative permeability fractionh Formation height mr Radial distance m119903119908 Wellbore radius mp Pressure MPa120583 Viscosity mPasdotsB Formation volume factor dimensionless120591 Fracture matrix transfer term m3dF119904 Shape factor 1m2119862119905 Total compressibility 1MPa119872119905 Total mobility 1MPa120582 Interporosity flow coefficient fraction120596 Storativity ratio fractionS Skin factor dimensionlessC Wellbore storage coefficient m3MPa

Subscripts

f Fracturem Matrixo Oilw Watert Totale ExternalD Dimensionless variable

Superscripts

- Laplace-transformed variable Endpoint property997888rarr Vector symbol

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

The authors gratefully acknowledge the support of theNational Basic Research 973 Program of China (Grant No

Mathematical Problems in Engineering 11

2015CB250900) the National Natural Science Foundationof China (Grant No 51704237) and the Research Projectsof Shaanxi Provincial Education Department (Grant No13JS090)

References

[1] R L Perrine Analysis of Pressure-buildup Curves vol 56American Petroleum Institute 1956

[2] J C Martin ldquoSimplified Equations of Flow in Gas Drive Reser-voirs and the Theoretical Foundation of Multiphase PressureBuildup Analysesrdquo Society of Petroleum Engineers vol 216 pp321ndash323 1959

[3] W Hurst ldquoInterference between Oil Fieldrdquo Transactions of theMetallurgical Society of AIME vol 219 no 8 pp 175ndash192 1960

[4] MMortada ldquoOilfield Interference in Aquifers of Non-UniformPropertiesrdquo Journal of Petroleum Technology vol 12 no 12 pp55ndash57 2013

[5] T Loucks and E Guerrero ldquoPressure Drop in a CompositeReservoirrdquo SPE Journal vol 1 no 03 pp 170ndash176 2013

[6] H Bixel and H Van Poollen ldquoPressure Drawdown and Buildupin the Presence of Radial Discontinuitiesrdquo SPE Journal vol 7no 03 pp 301ndash309 2013

[7] H Ramey ldquoApproximate Solutions For Unsteady LiquidFlow InComposite Reservoirsrdquo Journal of Canadian Petroleum Technol-ogy vol 9 no 01 2013

[8] Q Deng R Nie Y Jia et al ldquoPressure transient behavior of afractured well in multi-region composite reservoirsrdquo Journal ofPetroleum Science and Engineering vol 158 pp 535ndash553 2017

[9] H Kazemi LMerrill and J Jargon ldquoProblems in Interpretationof Pressure Fall-OffTests in ReservoirsWithAndWithout FluidBanksrdquo Journal of PetroleumTechnology vol 24 no 09 pp 1147ndash1156 2013

[10] L Merrill H Kazemi and W B Gogarty ldquoPressure FalloffAnalysis in Reservoirs With Fluid Banksrdquo Journal of PetroleumTechnology vol 26 no 07 pp 809ndash818 2013

[11] A Satman M Eggenschwiler R Tang and R HJ ldquoAnanalytical study of transient flow in systemswith radial disconti-nuitiesrdquo in Proceedings of the SPE9399 presented at SPE AnnualTechnical Conference and Exhibition Dallas Texas TX USASeptember 1980

[12] A Al-Bemani and I Ershaghi ldquoTwo-Phase Flow InterporosityEffects on Pressure Transient Test Response in Naturally Frac-tured Reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Dallas Texas

[13] Z Chen W Yu X Liao X Zhao Y Chen and K SepehrnoorildquoA Two-Phase Flow Model for Fractured Horizontal Well withComplex Fracture Networks Transient Analysis in FlowbackPeriodrdquo in Proceedings of the SPE Liquids-Rich Basins Confer-ence - North America Midland Texas USA

[14] H G Weinstein ldquoCold Waterflooding a Warm Reservoirrdquo inProceedings of the Fall Meeting of the Society of PetroleumEngineers of AIME Houston Texas

[15] S E Buckley andM C Leverett ldquoMechanism of fluid displace-ment in sandsrdquoTransactions of the AIME vol 146 no 1 pp 107ndash116 2013

[16] A Sosa R Raghavan and T Limon ldquoEffect of Relative Perme-ability andMobility Ratio on Pressure Falloff Behaviorrdquo Journalof Petroleum Technology vol 33 no 06 pp 1125ndash1135 2013

[17] P Hazebroek H Rainbow and CS Matthews ldquoPressure Fall-Off in Water Injection Wellsrdquo Transactions of the MetallurgicalSociety of AIME vol 213 pp 250ndash260 1958

[18] R Carter ldquoPressure Behavior of a Limited Circular CompositeReservoirrdquo SPE Journal vol 6 no 04 pp 328ndash334 2013

[19] A Odeh ldquoFlowTest Analysis for aWell with Radial Discontinu-ityrdquo Journal of Petroleum Technology vol 21 no 02 pp 207ndash2102013

[20] M Abbaszadeh and M Kamal ldquoPressure-Transient Testing ofWater-InjectionWellsrdquo SPE Reservoir Engineering vol 4 no 01pp 115ndash124 2013

[21] N Yeh and R Agarwal ldquoPressure Transient Analysis of Injec-tionWells in ReservoirsWithMultiple Fluid Banksrdquo in Proceed-ings of the SPE Annual Technical Conference and Exhibition SanAntonio Texas

[22] M M Levitan ldquoApplication of water injectionfalloff tests forreservoir appraisal New analytical solution method for two-phase variable rate problemsrdquo SPE Journal vol 8 no 4 pp 341ndash349 2003

[23] T Nanba and R Horne ldquoEstimation of Water and Oil RelativePermeabilities From Pressure Transient Analysis of WaterInjectionWell Datardquo in Proceedings of the SPE Annual TechnicalConference and Exhibition San Antonio Texas

[24] R B Bratvold andRNHorne ldquoAnalysis of pressure-falloff testsfollowing cold-water injectionrdquo SPE Formation Evaluation vol5 no 3 pp 293ndash302 1990

[25] S Chen G Li and A C Reynolds ldquoAnalytical Solution forInjection-Falloff-Production Testrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition San AntonioTexas USA

[26] A A Boughrara A M M Peres S Chen A A V Machadoand A C Reynolds ldquoApproximate analytical solutions for thepressure response at a water-injection wellrdquo SPE Journal vol12 no 1 pp 19ndash34 2007

[27] S Zheng and w xu ldquoNew approaches for analyzing transientpressure from oil and water two-phase flowing reservoirrdquo inProceedings of the Kuwait International Petroleum Conferenceand Exhibition Kuwait City Kuwait

[28] N A El-Khatib ldquoTransient Pressure Behavior of CompositeReservoirs with Moving Boundariesrdquo in Proceedings of theMiddle East Oil Show and Conference Bahrain

[29] A Dastan M M Kamal Y Hwang F Suleen and S MorsyldquoFalloff Testing Under Multiphase Flow Conditions in Natu-rally Fractured Reservoirsrdquo in Proceedings of the SPE WesternRegional Meeting Garden Grove California USA

[30] C n Machado and A C Reynolds ldquoApproximate semi-analytical solution for injection-falloff-production well test ananalytical tool for the in situ estimation of relative permeabilitycurvesrdquo Transport in Porous Media vol 121 no 1 pp 207ndash2312018

[31] T Barkve ldquoAnalytical study of reservoir pressure during water-injection well testsrdquo Society of Petroleum Engineers of AIME(Paper) SPE 1986

[32] T S Ramakrishnan and F J Kuchuk ldquoTesting injection wellswith rate and pressure datardquo SPE Formation Evaluation vol 9no 3 pp 228ndash236 1994

[33] L G Thompson and A C Reynolds ldquoWell testing for radiallyheterogeneous reservoirs under single and multiphase flowconditionsrdquo SPE Formation Evaluation vol 12 no 1 pp 57ndash641997

12 Mathematical Problems in Engineering

[34] JS Aronofsky L Masse and SG Natanson ldquoA Model for theMechanism of Oil Recovery from the Porous Matrix Due toWater Invasion in Fractured Reservoirsrdquo Trans AIME vol 213pp 17ndash19 1958

[35] A de Swaan ldquoTheory of Waterflooding in Fractured Reser-voirsrdquo SPE Journal vol 18 no 02 pp 117ndash122 2013

[36] H Kazemi J Gilman and A Elsharkawy ldquoAnalytical andNumerical Solution of Oil Recovery From Fractured ReservoirsWith Empirical Transfer Functions (includes associated papers25528 and 25818)rdquo SPE Reservoir Engineering vol 7 no 02 pp219ndash227 2013

[37] T Shimamoto D Kuramoto N Arihara and T Onishi ldquoA NewPressure Transient Analysis Model for Water Injection Wellin Dual-Porosity Reservoirsrdquo in Proceedings of the SPE100881presented at SPE Asia Pacific Oil ampamp Gas Conference andExhibition Adelaide Australia 2006

[38] G J Moridis D A McVay D L Reddell and T A BlasingameldquoLaplace transform finite difference (LTFD) numerical methodfor the simulation of compressible liquid flow in reservoirsrdquo SPEAdvanced Technology Series vol 2 no 2 pp 122ndash131 1994

[39] A D Habte and M Onur ldquoLaplace-transform finite-differenceand quasistationary solution method for water-injectionfallofftestsrdquo SPE Journal vol 19 no 3 pp 398ndash409 2014

[40] J Warren and P Root ldquoThe behavior of naturally fracturedreservoirsrdquo SPE Journal vol 3 no 3 pp 245ndash255 2013

[41] D Bourdet J Ayoub and Y Pirard ldquoUse of Pressure Derivativein Well Test Interpretationrdquo SPE Formation Evaluation vol 4no 02 pp 293ndash302 2013

[42] M Mavor and H Cinco-Ley ldquoTransient Pressure BehaviorOf Naturally Fractured Reservoirsrdquo in Proceedings of the SPECalifornia Regional Meeting Ventura California

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

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Mathematical Problems in Engineering 5

Table 1 Dimensionless variable for pressure calculation

119872119905119891 = 119896119891 (119896119903119900119891 (119878119908119891 = 1)120583119900 + 119896119903119908119891 (119878119908119891 = 1)

120583119908 ) 119872119905119891119863 = 119872119905119891119872119905119891

119901119891119863 = 2120587119872119905119891ℎ119902119861119908 (119901119891 minus 119901119894) 120582 = 1198651198781199032119908119872119905119906119872119905119891119901119898119863 = 2120587119872119905119891ℎ119902119861119908 (119901119898 minus 119901119894) 119903119863 = 119903119903119908119862119905119891 = 119862119903 + 119862119908119878119908119894 + 119862119900 (1 minus 119878119908119894) 1205961 = 0119891119862119905119891

(0119891119862119905119891 + 0119898119862119905119898)119862119905119898 = 119862119903 + 119862119908119878119908119888119898 + 119862119900 (1 minus 119878119908119888119898) 1205962 = 0119898119862119905119898(0119891119862119905119891 + 0119898119862119905119898)119905119863 = 119872119905119891

(0119891119862119905119891 + 0119898119862119905119898) 1199032119908 119862119863 = 1198622120587 (0119891119862119905119891 + 0119898119862119905119898) ℎ1199032119908

The no flow outer boundary condition is

(119889119901119891119863119889119911 )119911=119911119890

= 0 (43)

(119889119901119898119863119889119911 )119911=119911119890

= 0 (44)

The pressure change near the wellbore was relativelylarger than the area away from the bottom of the well Sothe formation was divided into n grid blocks in radial direc-tion according to the point-centered logarithmic griddingmethod which was shown as the dotted blue line in Figure 1By using finite-difference approximation (35)ndash(44) can bewritten as

119886119894119901119891119863119894 + 119887119894119901119891119863119894+1 + 119888119894119901119891119863119894+2 = 119889119894 (45)

119901119898119863119894 = 120582119901119891119863119894 + 1205962119901119898119863 (119905119863 = 0)1205962119906 + 120582 (46)

where

119886119894 = 120583119908(119896119903119900119891 (119878119908119891 (119903119894minus1 119905))120583119900 + 119896119903119908119891 (119878119908119891 (119903119894minus1 119905))120583119908 ) (47)

119887119894 = minus[119886119894 + 119888119894

+ (120596112059621199062 + 1205961120582119906 + 12059621205821199061205962119906 + 120582 )119894

1198902119894Δ119911Δ1199112](48)

119888119894 = 120583119908(119896119903119900119891 (119878119908119891 (119903119894 119905))120583119900 + 119896119903119908119891 (119878119908119891 (119903119894 119905))120583119908 ) (49)

119889119894 = minus[1198902119894Δ119911Δ11991121205961119894119901119891119863119894 (119905119863 = 0)

+ 1198902119894Δ119911Δ1199112 ( 12059621205821205962119906 + 120582)119894 119901119898119863119894 (119905119863 = 0)](50)

The inner boundary condition is

1198901119901119908119863 + 11989021199011198911198630 + 11989031199011198911198631 = 1198908 (51)

11989041199011198911198630 + 11989051199011198911198631 = 1198909 (52)

The outer boundary condition is

1198906119901119891119863119899minus1 minus 1198907119901119891119863119899 = 0 (53)

The coefficients for different boundary conditions areshown in Table 2

The pressure difference equations ((45) and (46)) can bewritten in a matrix form as

M997888rarr119909 = 997888rarr119887 (54)

where M is a large sparse matrix 997888x is the unknown pressurevector and

997888b is the known vector related to the initial and

6 Mathematical Problems in Engineering

Table 2 Coefficients with different boundary conditions

Boundary conditions Coefficients

Constant pressure outer boundary1198901 = 1 1198902 = minus(1 + 119878Δ119911 ) 1198903 = 119878Δ119911 1198904 = 1 + Δ119911119862119863119906 + 119878119862119863119906 1198905 = minus119878119862119863119906 minus 1 1198906 = 0

1198907 = 1 1198908 = 0 1198909 = Δz119906 minus Δz119862119863119901119908119863(119905119863 = 0)No flow outer boundary

1198901 = 1 1198902 = minus(1 + 119878Δ119911 ) 1198903 = 119878Δ119911 1198904 = 1 + Δ119911119862119863119906 + 1198781198621198631199061198905 = minus119878119862119863119906 minus 1 1198906 = 119886119899minus1 + 119888119899minus11198907 = 119887119899minus1 1198908 = 0 1198909 = Δz119906 minus Δz119862119863119901119908119863(119905119863 = 0)

boundary conditionsThematrixM and vectors997888x and997888b can

be explicitly given as

M

=

[[[[[[[[[[[[[[[[[[[[[[[[[[[

1198901 1198902 1198903 0 0 00 1198904 1198905 0 0 00 0 1198861 1198871 1198881 00 0 0 1198862 1198872 1198882

d

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

sdot sdot sdot

d

d

sdot sdot sdot

0 0 00 0 00 0 00 0 0

d

d d

119886119899minus3 119887119899minus3 119888119899minus3119886119899minus2 119887119899minus2 119888119899minus20 1198906 1198907

]]]]]]]]]]]]]]]]]]]]]]]]]]]

(55)

997888rarr119909 =

[[[[[[[[[[[[[[[[[

119901119908119863119901119891119863011990111989111986311199011198911198632

119901119891119863119899minus1

]]]]]]]]]]]]]]]]]

(56)

997888rarr119887 =

[[[[[[[[[[[[[[[[[

1198908119890911988911198892

119889119899minus1

]]]]]]]]]]]]]]]]]

(57)

Krwmkrom

KrwfKrof

0

02

04

06

08

1

Kro

K

rw

02 04 06 08 10Sw

Figure 2 Curves of relative permeability of oil and water in fractureand matrix

The unknown vector 997888x was calculated by using theStehfest algorithm Therefore the dimensionless pressureand pressure derivatives [41] at the bottom of the wellborecan be plotted according to the solutions of the formulatedmathematical model for the water injection system Thebasic parameters and the relative permeability data for thecalculation of pressure curves are illustrated in Table 3 andFigure 2

3 Verification of the Mathematical Model

Because little was available in the literatures for pressurebehavior of a water injection well considering saturationgradient and water imbibition in dual-porosity reservoirby LTFD method the model solutions were validated bysimplifying the model into a single phase water injectionwell in a dual-porosity system which was compared to theconventional analytical solution [42] by commercial softwareas shown in Figure 3 From the figure the LTFD results showgood agreement with the analytical solution

Moreover a field water injection test was used to furtherverify the proposedmodelThe basic information of thewaterinjection well and the relative permeability data was listed inTable 4 and Figure 4 And Figure 5 shows the pressure curves

Mathematical Problems in Engineering 7

tDPwDminusLTFD methoddPwDminusLTFD method

PwDminusanalytical methoddPwDminusanalytical method

10minus2

10minus1

100

101

101

102

102

103

104

105

106

107

108

PwD

dPw

D

Figure 3 Comparison of the results by LTFDmethod and analyticalmethod

Table 3 Basic parameters for calculation of pressure curves

Parameter Value UnitsFormation height 10 mFracture porosity 003 fractionMatrix porosity 027 fractionWellbore radius 01 mFracture permeability 06 120583m2Matrix permeability 6 times 10minus5 120583m2Irreducible water saturation 20 Residual oil saturation 14 Oil volume factor 103 fractionWater volume factor 1 fractionShape factor 6 times 10minus4 1cm2

Oil viscosity 1 mPasWater viscosity 05 mPasrock compressibility 6 times 10minus4 1MPaoil compressibility 1 times 10minus3 1MPawater compressibility 4 times 10minus4 1MPaInjection rate 30 m3dayInitial pressure 14 MPaSkin factor 01 fractionImbibition rate coefficient 01 1dayWellbore storage coefficient 001 1m3MPa

of the well test data comparing with the results in our modelFrom the figure it is found that our results match well withthe field test data

4 Model Features Analysis

The pressure curves of oil-water two-phase flow consider-ing water imbibition and the saturation gradients duringinjection period by the LTFD method were plotted in thissection The behaviors of the typical curves were analyzed by

Table 4 Basic information of the water injection well

Parameter Value UnitsFormation height 126 mFracture porosity 003 fractionMatrix porosity 0268 fractionWellbore radius 012 mOil volume factor 1163 fractionWater volume factor 1 fractionOil viscosity 96 mPasWater viscosity 05 mPasrock compressibility 5 times 10minus4 1MPaoil compressibility 48 times 10minus3 1MPawater compressibility 39 times 10minus4 1MPaInjection rate 256 m3dayInitial pressure 15 MPa

KrwKro

0

02

04

06

08

1

Kro

K

rw

02 04 06 08 10Sw

Figure 4 The relative permeability of oil and water for the waterinjection well

comparing the typical curves of single phase flow with thetwo-phase flow curves in the fractured reservoirs as follows

41 Features of Typical Curves Figure 6 illustrates the pres-sure curves of single phase flow and oil-water two-phaseflow in fractured reservoirs by LTFD method without waterimbibition According to the feature of the curves theformation flow can be divided into three stages The firststage is the flow in the fracture system the second stage isthe flow between the fracture and the matrix system andthe third stage is the flow in the whole system including thefracture andmatrixThe figure shows that the shape of curvesfor single phase flow and two-phase flow is similar but theposition of the two-phase flow curves is on the upper rightof the single phase flow curves The reason is that when oiland water exist in the whole system the total mobility of oiland water is lower than the single phase and the bottom-hole pressure increases more quickly as the fluid is injected

8 Mathematical Problems in Engineering

tDPwDminusField testdPwDminusField test

PwDminusLTFD methoddPwDminusLTFD method

101

102

103

104

105

106

107

100

101

102

PwD

dPw

D

Figure 5 Comparison of the results in this work and the fieldinjection test

tD

PwDminussingle phasedPwDminussingle phase

PwDminustwo-phasedPwDminustwo-phase

101

102

103

104

105

106

107

108

10minus1

100

101

102

PwD

dPw

D

Figure 6 Typical curves of single phase and two-phase flow duringinjection period

into the well Thus the pressure derivative curves of two-phase flow move upward compared to the single phase flowcurves Moreover due to the decrease of the total mobilitythe interporosity flow capacity between matrix and fracturesystem decreases when the single phase flow becomes two-phase flow Thus the ldquodiprdquo of the two-phase flow curvesoccurs later than that of the single phase flow curves

42 Effects of Parameters The key factors influencing thepressure behavior of water injection period including skinfactor wellbore storage coefficient shape factor water imbibi-tion rate coefficient and boundary conditions were discussedin detail in this part

421 Skin Factor Figure 7 shows the effect of skin factoron the pressure curves of the water injection well The solid

tD

S=minus1S=minus1S=1

S=1S=10S=10

S=minus1S=minus1S=1

S=1S=10S=10

102

104

106

108

10minus1

100

101

102

PwD

dPw

D

Figure 7 Influence of different skin factors on pressure transientbehavior

lines are the curves of single phase flow and the circlesare oil-water two-phase flow curves The red green andblue colors indicate that the shape factor values are -1 1and 10 respectively From the figure the skin factor mainlyinfluences the peak value of the pressure derivatives As thevalue of skin factor increases the peak value gets higherand the time for peak value occurrence becomes later whichindicates that the reservoir is more severely contaminatedThe pressure derivative curves of two-phase flow movetoward the upper right of the single phase flow curves whichshows the same change rule with Figure 6 When the skinfactor becomes larger the peak value difference between thesingle phase and two-phase decreases and the inclination ofthe curves after the peak value increases

422 Wellbore Storage Coefficient Figure 8 is the influencesof wellbore storage coefficient on the pressure and pressurederivative curves of the water injection well The red greenand blue colors indicate that wellbore storage coefficientvalues are 001 01 and 1 respectively As seen the maindifference of the curves occurs on the straight line of earlyperiod As the wellbore storage coefficient increases thestraight line section becomes longer and the time of peakvalue occurs later Besides the wellbore storage effect willcover up the radial flow stage in the fracture system directlygoing to the transitional flow stage between the fracture andmatrix The pressure derivative curves of two-phase flowmove upper right to the single phase flow curves after thestraight line of wellbore storage stage which agrees with thechange rule of Figure 6

423 Shape Factor The influences of shape factor on thepressure curves of the water injection well are shown inFigure 9 The red green and blue colors indicate thatshape factor values are 10minus5 10minus4 and 10minus3 respectivelyBy comparison of the curves it is found that the shape

Mathematical Problems in Engineering 9

tD10

210

410

610

810

minus3

10minus2

10minus1

100

101

102

PwD

dPw

D

C=001C=001C=01

C=01C=1C=1

C=001C=001C=01

C=01C=1C=1

Figure 8 Influence of different wellbore storage coefficients onpressure transient behavior

tD

Fs=1lowast10minus5Fs=1lowast10minus5Fs=1lowast10minus4Fs=1lowast10minus4Fs=1lowast10minus3Fs=1lowast10minus3

Fs=1lowast10minus5Fs=1lowast10minus5Fs=1lowast10minus4Fs=1lowast10minus4Fs=1lowast10minus3Fs=1lowast10minus3

102

104

106

108

10minus1

100

101

102

PwD

dPw

D

Figure 9 Influence of different shape factors on pressure transientbehavior

factor mainly has an effect on the position of the ldquodiprdquo ofthe pressure derivatives When the value of shape factorgets greater the ldquodiprdquo appears earlier and the ldquodiprdquo movescloser to the left The main reason is that the shape factoris a component of the equation defining the interporosityparameter The higher the shape factor is the easier the fluidexchange between the fracture and matrix is and the earlierthe transitional flow stage occurs It also demonstrates thatthe pressure derivative curves of two-phase flow move upperright to the single phase flow curves which is in line with thechange rule of Figure 6

tDRc=0Rc=0Rc=001

Rc=001Rc=005Rc=005

Rc=01Rc=01Rc=05

Rc=05Rc=1Rc=1

102

104

106

108

10minus2

10minus1

100

101

102

PwD

dPw

DFigure 10 Pressure behavior of different imbibition rate coefficientswith no flow outer boundary

424 Imbibition Rate Coefficient with No Flow Outer Bound-ary Condition Figure 10 describes the pressure behaviorof the water injection well under no flow outer boundarycondition when the water imbibition rate coefficient valuesare 0 001 005 01 05 and 1 respectively From the figurethe imbibition rate coefficient mainly has an effect on thewhole system (matrix and fracture system) radial flow periodof the curves As the imbibition rate coefficient becomeslarger there are more fluid exchange between the fractureand matrix and more fluid supply from the matrix to thefracture The water imbibition process is slow so there isa second interporosity period after the whole system radialflow period And with the increase of water imbibition ratecoefficient the pressure derivative curves dip downwardearlier and the degree for the curves dropping down becomeslarger After that the pressure derivative curves rise up to auniform line with the same slope due to the influence of theno flow outer boundary

425 Imbibition Rate Coefficient with Constant PressureBoundary Condition Figure 11 shows the pressure and pres-sure derivative curves of the water injection well underconstant pressure boundary condition when the water imbi-bition rate coefficient values are 0 001 005 01 05 and1 respectively By comparison and analysis it is concludedthat the pressure derivative curve is a straight line at thewhole system radial flow period when no water imbibition isconsideredWhen considering water imbibition the pressurederivatives decline at the whole system radial flow periodThe pressure derivatives drop downward more and earlierwith the increase of imbibition rate coefficient After that thecurves also drop down due to the constant pressure outerboundary effect

10 Mathematical Problems in Engineering

tD

Rc=0Rc=0Rc=001

Rc=001Rc=005Rc=005

Rc=01Rc=01Rc=05

Rc=05Rc=1Rc=1

102

104

106

108

10minus2

10minus1

100

101

102

PwD

dPw

D

Figure 11 Pressure behavior of different imbibition rate coefficientswith constant pressure boundary

5 Conclusions

In this work a semianalytical two-phase flow model consid-ering water imbibition and the saturation gradients withineach region during injection period was proposed to analyzethe pressure transient behavior of the water injection well infractured reservoirs Validation of the presented model wasperformed by analytical method and a field water injectiontest The model features were investigated by sensitivityanalysis Some conclusions were drawn as follows

(1) The shape of the pressure curves for the proposedmodel is similar to the single phase flowmodel whilethe position of the curves for the two-phase flowmodel is on the upper right of the single phase flowmodel

(2) Themain influence of skin factor on the curves occurson the peak value of the pressure derivatives As theskin factor becomes larger the peak value differencebetween single phase and two-phase pressure curvesdecreases and the inclination of the curves after thepeak value increases

(3) The wellbore storage coefficient mainly has an effecton the straight line of the early period for the pressurecurves The shape factor mainly has an effect on theposition of the ldquodiprdquo of the pressure derivatives whichis the same as the interporosity parameter

(4) The imbibition rate coefficient mainly influences thewhole system radial flow period of the pressurecurves The pressure derivative curves dip downwardearlier and drop down to a greater degree with theincrease of imbibition rate coefficient

Nomenclature

R Recovery fractionR119888 Imbibition rate coefficient 1day119877infin Ultimate cumulative oil recovery fraction0 Porosity fraction119904119900119903 Residual oil saturation fraction119904119908119888 Irreducible water saturation fraction119904119908 Water saturation fractionz Logarithmic transform variable dimensionlessu Laplace variable dimensionlessq Displacement rate m3dt Time hk Permeability 1205831198982119896119903 Relative permeability fractionh Formation height mr Radial distance m119903119908 Wellbore radius mp Pressure MPa120583 Viscosity mPasdotsB Formation volume factor dimensionless120591 Fracture matrix transfer term m3dF119904 Shape factor 1m2119862119905 Total compressibility 1MPa119872119905 Total mobility 1MPa120582 Interporosity flow coefficient fraction120596 Storativity ratio fractionS Skin factor dimensionlessC Wellbore storage coefficient m3MPa

Subscripts

f Fracturem Matrixo Oilw Watert Totale ExternalD Dimensionless variable

Superscripts

- Laplace-transformed variable Endpoint property997888rarr Vector symbol

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

The authors gratefully acknowledge the support of theNational Basic Research 973 Program of China (Grant No

Mathematical Problems in Engineering 11

2015CB250900) the National Natural Science Foundationof China (Grant No 51704237) and the Research Projectsof Shaanxi Provincial Education Department (Grant No13JS090)

References

[1] R L Perrine Analysis of Pressure-buildup Curves vol 56American Petroleum Institute 1956

[2] J C Martin ldquoSimplified Equations of Flow in Gas Drive Reser-voirs and the Theoretical Foundation of Multiphase PressureBuildup Analysesrdquo Society of Petroleum Engineers vol 216 pp321ndash323 1959

[3] W Hurst ldquoInterference between Oil Fieldrdquo Transactions of theMetallurgical Society of AIME vol 219 no 8 pp 175ndash192 1960

[4] MMortada ldquoOilfield Interference in Aquifers of Non-UniformPropertiesrdquo Journal of Petroleum Technology vol 12 no 12 pp55ndash57 2013

[5] T Loucks and E Guerrero ldquoPressure Drop in a CompositeReservoirrdquo SPE Journal vol 1 no 03 pp 170ndash176 2013

[6] H Bixel and H Van Poollen ldquoPressure Drawdown and Buildupin the Presence of Radial Discontinuitiesrdquo SPE Journal vol 7no 03 pp 301ndash309 2013

[7] H Ramey ldquoApproximate Solutions For Unsteady LiquidFlow InComposite Reservoirsrdquo Journal of Canadian Petroleum Technol-ogy vol 9 no 01 2013

[8] Q Deng R Nie Y Jia et al ldquoPressure transient behavior of afractured well in multi-region composite reservoirsrdquo Journal ofPetroleum Science and Engineering vol 158 pp 535ndash553 2017

[9] H Kazemi LMerrill and J Jargon ldquoProblems in Interpretationof Pressure Fall-OffTests in ReservoirsWithAndWithout FluidBanksrdquo Journal of PetroleumTechnology vol 24 no 09 pp 1147ndash1156 2013

[10] L Merrill H Kazemi and W B Gogarty ldquoPressure FalloffAnalysis in Reservoirs With Fluid Banksrdquo Journal of PetroleumTechnology vol 26 no 07 pp 809ndash818 2013

[11] A Satman M Eggenschwiler R Tang and R HJ ldquoAnanalytical study of transient flow in systemswith radial disconti-nuitiesrdquo in Proceedings of the SPE9399 presented at SPE AnnualTechnical Conference and Exhibition Dallas Texas TX USASeptember 1980

[12] A Al-Bemani and I Ershaghi ldquoTwo-Phase Flow InterporosityEffects on Pressure Transient Test Response in Naturally Frac-tured Reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Dallas Texas

[13] Z Chen W Yu X Liao X Zhao Y Chen and K SepehrnoorildquoA Two-Phase Flow Model for Fractured Horizontal Well withComplex Fracture Networks Transient Analysis in FlowbackPeriodrdquo in Proceedings of the SPE Liquids-Rich Basins Confer-ence - North America Midland Texas USA

[14] H G Weinstein ldquoCold Waterflooding a Warm Reservoirrdquo inProceedings of the Fall Meeting of the Society of PetroleumEngineers of AIME Houston Texas

[15] S E Buckley andM C Leverett ldquoMechanism of fluid displace-ment in sandsrdquoTransactions of the AIME vol 146 no 1 pp 107ndash116 2013

[16] A Sosa R Raghavan and T Limon ldquoEffect of Relative Perme-ability andMobility Ratio on Pressure Falloff Behaviorrdquo Journalof Petroleum Technology vol 33 no 06 pp 1125ndash1135 2013

[17] P Hazebroek H Rainbow and CS Matthews ldquoPressure Fall-Off in Water Injection Wellsrdquo Transactions of the MetallurgicalSociety of AIME vol 213 pp 250ndash260 1958

[18] R Carter ldquoPressure Behavior of a Limited Circular CompositeReservoirrdquo SPE Journal vol 6 no 04 pp 328ndash334 2013

[19] A Odeh ldquoFlowTest Analysis for aWell with Radial Discontinu-ityrdquo Journal of Petroleum Technology vol 21 no 02 pp 207ndash2102013

[20] M Abbaszadeh and M Kamal ldquoPressure-Transient Testing ofWater-InjectionWellsrdquo SPE Reservoir Engineering vol 4 no 01pp 115ndash124 2013

[21] N Yeh and R Agarwal ldquoPressure Transient Analysis of Injec-tionWells in ReservoirsWithMultiple Fluid Banksrdquo in Proceed-ings of the SPE Annual Technical Conference and Exhibition SanAntonio Texas

[22] M M Levitan ldquoApplication of water injectionfalloff tests forreservoir appraisal New analytical solution method for two-phase variable rate problemsrdquo SPE Journal vol 8 no 4 pp 341ndash349 2003

[23] T Nanba and R Horne ldquoEstimation of Water and Oil RelativePermeabilities From Pressure Transient Analysis of WaterInjectionWell Datardquo in Proceedings of the SPE Annual TechnicalConference and Exhibition San Antonio Texas

[24] R B Bratvold andRNHorne ldquoAnalysis of pressure-falloff testsfollowing cold-water injectionrdquo SPE Formation Evaluation vol5 no 3 pp 293ndash302 1990

[25] S Chen G Li and A C Reynolds ldquoAnalytical Solution forInjection-Falloff-Production Testrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition San AntonioTexas USA

[26] A A Boughrara A M M Peres S Chen A A V Machadoand A C Reynolds ldquoApproximate analytical solutions for thepressure response at a water-injection wellrdquo SPE Journal vol12 no 1 pp 19ndash34 2007

[27] S Zheng and w xu ldquoNew approaches for analyzing transientpressure from oil and water two-phase flowing reservoirrdquo inProceedings of the Kuwait International Petroleum Conferenceand Exhibition Kuwait City Kuwait

[28] N A El-Khatib ldquoTransient Pressure Behavior of CompositeReservoirs with Moving Boundariesrdquo in Proceedings of theMiddle East Oil Show and Conference Bahrain

[29] A Dastan M M Kamal Y Hwang F Suleen and S MorsyldquoFalloff Testing Under Multiphase Flow Conditions in Natu-rally Fractured Reservoirsrdquo in Proceedings of the SPE WesternRegional Meeting Garden Grove California USA

[30] C n Machado and A C Reynolds ldquoApproximate semi-analytical solution for injection-falloff-production well test ananalytical tool for the in situ estimation of relative permeabilitycurvesrdquo Transport in Porous Media vol 121 no 1 pp 207ndash2312018

[31] T Barkve ldquoAnalytical study of reservoir pressure during water-injection well testsrdquo Society of Petroleum Engineers of AIME(Paper) SPE 1986

[32] T S Ramakrishnan and F J Kuchuk ldquoTesting injection wellswith rate and pressure datardquo SPE Formation Evaluation vol 9no 3 pp 228ndash236 1994

[33] L G Thompson and A C Reynolds ldquoWell testing for radiallyheterogeneous reservoirs under single and multiphase flowconditionsrdquo SPE Formation Evaluation vol 12 no 1 pp 57ndash641997

12 Mathematical Problems in Engineering

[34] JS Aronofsky L Masse and SG Natanson ldquoA Model for theMechanism of Oil Recovery from the Porous Matrix Due toWater Invasion in Fractured Reservoirsrdquo Trans AIME vol 213pp 17ndash19 1958

[35] A de Swaan ldquoTheory of Waterflooding in Fractured Reser-voirsrdquo SPE Journal vol 18 no 02 pp 117ndash122 2013

[36] H Kazemi J Gilman and A Elsharkawy ldquoAnalytical andNumerical Solution of Oil Recovery From Fractured ReservoirsWith Empirical Transfer Functions (includes associated papers25528 and 25818)rdquo SPE Reservoir Engineering vol 7 no 02 pp219ndash227 2013

[37] T Shimamoto D Kuramoto N Arihara and T Onishi ldquoA NewPressure Transient Analysis Model for Water Injection Wellin Dual-Porosity Reservoirsrdquo in Proceedings of the SPE100881presented at SPE Asia Pacific Oil ampamp Gas Conference andExhibition Adelaide Australia 2006

[38] G J Moridis D A McVay D L Reddell and T A BlasingameldquoLaplace transform finite difference (LTFD) numerical methodfor the simulation of compressible liquid flow in reservoirsrdquo SPEAdvanced Technology Series vol 2 no 2 pp 122ndash131 1994

[39] A D Habte and M Onur ldquoLaplace-transform finite-differenceand quasistationary solution method for water-injectionfallofftestsrdquo SPE Journal vol 19 no 3 pp 398ndash409 2014

[40] J Warren and P Root ldquoThe behavior of naturally fracturedreservoirsrdquo SPE Journal vol 3 no 3 pp 245ndash255 2013

[41] D Bourdet J Ayoub and Y Pirard ldquoUse of Pressure Derivativein Well Test Interpretationrdquo SPE Formation Evaluation vol 4no 02 pp 293ndash302 2013

[42] M Mavor and H Cinco-Ley ldquoTransient Pressure BehaviorOf Naturally Fractured Reservoirsrdquo in Proceedings of the SPECalifornia Regional Meeting Ventura California

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

6 Mathematical Problems in Engineering

Table 2 Coefficients with different boundary conditions

Boundary conditions Coefficients

Constant pressure outer boundary1198901 = 1 1198902 = minus(1 + 119878Δ119911 ) 1198903 = 119878Δ119911 1198904 = 1 + Δ119911119862119863119906 + 119878119862119863119906 1198905 = minus119878119862119863119906 minus 1 1198906 = 0

1198907 = 1 1198908 = 0 1198909 = Δz119906 minus Δz119862119863119901119908119863(119905119863 = 0)No flow outer boundary

1198901 = 1 1198902 = minus(1 + 119878Δ119911 ) 1198903 = 119878Δ119911 1198904 = 1 + Δ119911119862119863119906 + 1198781198621198631199061198905 = minus119878119862119863119906 minus 1 1198906 = 119886119899minus1 + 119888119899minus11198907 = 119887119899minus1 1198908 = 0 1198909 = Δz119906 minus Δz119862119863119901119908119863(119905119863 = 0)

boundary conditionsThematrixM and vectors997888x and997888b can

be explicitly given as

M

=

[[[[[[[[[[[[[[[[[[[[[[[[[[[

1198901 1198902 1198903 0 0 00 1198904 1198905 0 0 00 0 1198861 1198871 1198881 00 0 0 1198862 1198872 1198882

d

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

sdot sdot sdot

d

d

sdot sdot sdot

0 0 00 0 00 0 00 0 0

d

d d

119886119899minus3 119887119899minus3 119888119899minus3119886119899minus2 119887119899minus2 119888119899minus20 1198906 1198907

]]]]]]]]]]]]]]]]]]]]]]]]]]]

(55)

997888rarr119909 =

[[[[[[[[[[[[[[[[[

119901119908119863119901119891119863011990111989111986311199011198911198632

119901119891119863119899minus1

]]]]]]]]]]]]]]]]]

(56)

997888rarr119887 =

[[[[[[[[[[[[[[[[[

1198908119890911988911198892

119889119899minus1

]]]]]]]]]]]]]]]]]

(57)

Krwmkrom

KrwfKrof

0

02

04

06

08

1

Kro

K

rw

02 04 06 08 10Sw

Figure 2 Curves of relative permeability of oil and water in fractureand matrix

The unknown vector 997888x was calculated by using theStehfest algorithm Therefore the dimensionless pressureand pressure derivatives [41] at the bottom of the wellborecan be plotted according to the solutions of the formulatedmathematical model for the water injection system Thebasic parameters and the relative permeability data for thecalculation of pressure curves are illustrated in Table 3 andFigure 2

3 Verification of the Mathematical Model

Because little was available in the literatures for pressurebehavior of a water injection well considering saturationgradient and water imbibition in dual-porosity reservoirby LTFD method the model solutions were validated bysimplifying the model into a single phase water injectionwell in a dual-porosity system which was compared to theconventional analytical solution [42] by commercial softwareas shown in Figure 3 From the figure the LTFD results showgood agreement with the analytical solution

Moreover a field water injection test was used to furtherverify the proposedmodelThe basic information of thewaterinjection well and the relative permeability data was listed inTable 4 and Figure 4 And Figure 5 shows the pressure curves

Mathematical Problems in Engineering 7

tDPwDminusLTFD methoddPwDminusLTFD method

PwDminusanalytical methoddPwDminusanalytical method

10minus2

10minus1

100

101

101

102

102

103

104

105

106

107

108

PwD

dPw

D

Figure 3 Comparison of the results by LTFDmethod and analyticalmethod

Table 3 Basic parameters for calculation of pressure curves

Parameter Value UnitsFormation height 10 mFracture porosity 003 fractionMatrix porosity 027 fractionWellbore radius 01 mFracture permeability 06 120583m2Matrix permeability 6 times 10minus5 120583m2Irreducible water saturation 20 Residual oil saturation 14 Oil volume factor 103 fractionWater volume factor 1 fractionShape factor 6 times 10minus4 1cm2

Oil viscosity 1 mPasWater viscosity 05 mPasrock compressibility 6 times 10minus4 1MPaoil compressibility 1 times 10minus3 1MPawater compressibility 4 times 10minus4 1MPaInjection rate 30 m3dayInitial pressure 14 MPaSkin factor 01 fractionImbibition rate coefficient 01 1dayWellbore storage coefficient 001 1m3MPa

of the well test data comparing with the results in our modelFrom the figure it is found that our results match well withthe field test data

4 Model Features Analysis

The pressure curves of oil-water two-phase flow consider-ing water imbibition and the saturation gradients duringinjection period by the LTFD method were plotted in thissection The behaviors of the typical curves were analyzed by

Table 4 Basic information of the water injection well

Parameter Value UnitsFormation height 126 mFracture porosity 003 fractionMatrix porosity 0268 fractionWellbore radius 012 mOil volume factor 1163 fractionWater volume factor 1 fractionOil viscosity 96 mPasWater viscosity 05 mPasrock compressibility 5 times 10minus4 1MPaoil compressibility 48 times 10minus3 1MPawater compressibility 39 times 10minus4 1MPaInjection rate 256 m3dayInitial pressure 15 MPa

KrwKro

0

02

04

06

08

1

Kro

K

rw

02 04 06 08 10Sw

Figure 4 The relative permeability of oil and water for the waterinjection well

comparing the typical curves of single phase flow with thetwo-phase flow curves in the fractured reservoirs as follows

41 Features of Typical Curves Figure 6 illustrates the pres-sure curves of single phase flow and oil-water two-phaseflow in fractured reservoirs by LTFD method without waterimbibition According to the feature of the curves theformation flow can be divided into three stages The firststage is the flow in the fracture system the second stage isthe flow between the fracture and the matrix system andthe third stage is the flow in the whole system including thefracture andmatrixThe figure shows that the shape of curvesfor single phase flow and two-phase flow is similar but theposition of the two-phase flow curves is on the upper rightof the single phase flow curves The reason is that when oiland water exist in the whole system the total mobility of oiland water is lower than the single phase and the bottom-hole pressure increases more quickly as the fluid is injected

8 Mathematical Problems in Engineering

tDPwDminusField testdPwDminusField test

PwDminusLTFD methoddPwDminusLTFD method

101

102

103

104

105

106

107

100

101

102

PwD

dPw

D

Figure 5 Comparison of the results in this work and the fieldinjection test

tD

PwDminussingle phasedPwDminussingle phase

PwDminustwo-phasedPwDminustwo-phase

101

102

103

104

105

106

107

108

10minus1

100

101

102

PwD

dPw

D

Figure 6 Typical curves of single phase and two-phase flow duringinjection period

into the well Thus the pressure derivative curves of two-phase flow move upward compared to the single phase flowcurves Moreover due to the decrease of the total mobilitythe interporosity flow capacity between matrix and fracturesystem decreases when the single phase flow becomes two-phase flow Thus the ldquodiprdquo of the two-phase flow curvesoccurs later than that of the single phase flow curves

42 Effects of Parameters The key factors influencing thepressure behavior of water injection period including skinfactor wellbore storage coefficient shape factor water imbibi-tion rate coefficient and boundary conditions were discussedin detail in this part

421 Skin Factor Figure 7 shows the effect of skin factoron the pressure curves of the water injection well The solid

tD

S=minus1S=minus1S=1

S=1S=10S=10

S=minus1S=minus1S=1

S=1S=10S=10

102

104

106

108

10minus1

100

101

102

PwD

dPw

D

Figure 7 Influence of different skin factors on pressure transientbehavior

lines are the curves of single phase flow and the circlesare oil-water two-phase flow curves The red green andblue colors indicate that the shape factor values are -1 1and 10 respectively From the figure the skin factor mainlyinfluences the peak value of the pressure derivatives As thevalue of skin factor increases the peak value gets higherand the time for peak value occurrence becomes later whichindicates that the reservoir is more severely contaminatedThe pressure derivative curves of two-phase flow movetoward the upper right of the single phase flow curves whichshows the same change rule with Figure 6 When the skinfactor becomes larger the peak value difference between thesingle phase and two-phase decreases and the inclination ofthe curves after the peak value increases

422 Wellbore Storage Coefficient Figure 8 is the influencesof wellbore storage coefficient on the pressure and pressurederivative curves of the water injection well The red greenand blue colors indicate that wellbore storage coefficientvalues are 001 01 and 1 respectively As seen the maindifference of the curves occurs on the straight line of earlyperiod As the wellbore storage coefficient increases thestraight line section becomes longer and the time of peakvalue occurs later Besides the wellbore storage effect willcover up the radial flow stage in the fracture system directlygoing to the transitional flow stage between the fracture andmatrix The pressure derivative curves of two-phase flowmove upper right to the single phase flow curves after thestraight line of wellbore storage stage which agrees with thechange rule of Figure 6

423 Shape Factor The influences of shape factor on thepressure curves of the water injection well are shown inFigure 9 The red green and blue colors indicate thatshape factor values are 10minus5 10minus4 and 10minus3 respectivelyBy comparison of the curves it is found that the shape

Mathematical Problems in Engineering 9

tD10

210

410

610

810

minus3

10minus2

10minus1

100

101

102

PwD

dPw

D

C=001C=001C=01

C=01C=1C=1

C=001C=001C=01

C=01C=1C=1

Figure 8 Influence of different wellbore storage coefficients onpressure transient behavior

tD

Fs=1lowast10minus5Fs=1lowast10minus5Fs=1lowast10minus4Fs=1lowast10minus4Fs=1lowast10minus3Fs=1lowast10minus3

Fs=1lowast10minus5Fs=1lowast10minus5Fs=1lowast10minus4Fs=1lowast10minus4Fs=1lowast10minus3Fs=1lowast10minus3

102

104

106

108

10minus1

100

101

102

PwD

dPw

D

Figure 9 Influence of different shape factors on pressure transientbehavior

factor mainly has an effect on the position of the ldquodiprdquo ofthe pressure derivatives When the value of shape factorgets greater the ldquodiprdquo appears earlier and the ldquodiprdquo movescloser to the left The main reason is that the shape factoris a component of the equation defining the interporosityparameter The higher the shape factor is the easier the fluidexchange between the fracture and matrix is and the earlierthe transitional flow stage occurs It also demonstrates thatthe pressure derivative curves of two-phase flow move upperright to the single phase flow curves which is in line with thechange rule of Figure 6

tDRc=0Rc=0Rc=001

Rc=001Rc=005Rc=005

Rc=01Rc=01Rc=05

Rc=05Rc=1Rc=1

102

104

106

108

10minus2

10minus1

100

101

102

PwD

dPw

DFigure 10 Pressure behavior of different imbibition rate coefficientswith no flow outer boundary

424 Imbibition Rate Coefficient with No Flow Outer Bound-ary Condition Figure 10 describes the pressure behaviorof the water injection well under no flow outer boundarycondition when the water imbibition rate coefficient valuesare 0 001 005 01 05 and 1 respectively From the figurethe imbibition rate coefficient mainly has an effect on thewhole system (matrix and fracture system) radial flow periodof the curves As the imbibition rate coefficient becomeslarger there are more fluid exchange between the fractureand matrix and more fluid supply from the matrix to thefracture The water imbibition process is slow so there isa second interporosity period after the whole system radialflow period And with the increase of water imbibition ratecoefficient the pressure derivative curves dip downwardearlier and the degree for the curves dropping down becomeslarger After that the pressure derivative curves rise up to auniform line with the same slope due to the influence of theno flow outer boundary

425 Imbibition Rate Coefficient with Constant PressureBoundary Condition Figure 11 shows the pressure and pres-sure derivative curves of the water injection well underconstant pressure boundary condition when the water imbi-bition rate coefficient values are 0 001 005 01 05 and1 respectively By comparison and analysis it is concludedthat the pressure derivative curve is a straight line at thewhole system radial flow period when no water imbibition isconsideredWhen considering water imbibition the pressurederivatives decline at the whole system radial flow periodThe pressure derivatives drop downward more and earlierwith the increase of imbibition rate coefficient After that thecurves also drop down due to the constant pressure outerboundary effect

10 Mathematical Problems in Engineering

tD

Rc=0Rc=0Rc=001

Rc=001Rc=005Rc=005

Rc=01Rc=01Rc=05

Rc=05Rc=1Rc=1

102

104

106

108

10minus2

10minus1

100

101

102

PwD

dPw

D

Figure 11 Pressure behavior of different imbibition rate coefficientswith constant pressure boundary

5 Conclusions

In this work a semianalytical two-phase flow model consid-ering water imbibition and the saturation gradients withineach region during injection period was proposed to analyzethe pressure transient behavior of the water injection well infractured reservoirs Validation of the presented model wasperformed by analytical method and a field water injectiontest The model features were investigated by sensitivityanalysis Some conclusions were drawn as follows

(1) The shape of the pressure curves for the proposedmodel is similar to the single phase flowmodel whilethe position of the curves for the two-phase flowmodel is on the upper right of the single phase flowmodel

(2) Themain influence of skin factor on the curves occurson the peak value of the pressure derivatives As theskin factor becomes larger the peak value differencebetween single phase and two-phase pressure curvesdecreases and the inclination of the curves after thepeak value increases

(3) The wellbore storage coefficient mainly has an effecton the straight line of the early period for the pressurecurves The shape factor mainly has an effect on theposition of the ldquodiprdquo of the pressure derivatives whichis the same as the interporosity parameter

(4) The imbibition rate coefficient mainly influences thewhole system radial flow period of the pressurecurves The pressure derivative curves dip downwardearlier and drop down to a greater degree with theincrease of imbibition rate coefficient

Nomenclature

R Recovery fractionR119888 Imbibition rate coefficient 1day119877infin Ultimate cumulative oil recovery fraction0 Porosity fraction119904119900119903 Residual oil saturation fraction119904119908119888 Irreducible water saturation fraction119904119908 Water saturation fractionz Logarithmic transform variable dimensionlessu Laplace variable dimensionlessq Displacement rate m3dt Time hk Permeability 1205831198982119896119903 Relative permeability fractionh Formation height mr Radial distance m119903119908 Wellbore radius mp Pressure MPa120583 Viscosity mPasdotsB Formation volume factor dimensionless120591 Fracture matrix transfer term m3dF119904 Shape factor 1m2119862119905 Total compressibility 1MPa119872119905 Total mobility 1MPa120582 Interporosity flow coefficient fraction120596 Storativity ratio fractionS Skin factor dimensionlessC Wellbore storage coefficient m3MPa

Subscripts

f Fracturem Matrixo Oilw Watert Totale ExternalD Dimensionless variable

Superscripts

- Laplace-transformed variable Endpoint property997888rarr Vector symbol

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

The authors gratefully acknowledge the support of theNational Basic Research 973 Program of China (Grant No

Mathematical Problems in Engineering 11

2015CB250900) the National Natural Science Foundationof China (Grant No 51704237) and the Research Projectsof Shaanxi Provincial Education Department (Grant No13JS090)

References

[1] R L Perrine Analysis of Pressure-buildup Curves vol 56American Petroleum Institute 1956

[2] J C Martin ldquoSimplified Equations of Flow in Gas Drive Reser-voirs and the Theoretical Foundation of Multiphase PressureBuildup Analysesrdquo Society of Petroleum Engineers vol 216 pp321ndash323 1959

[3] W Hurst ldquoInterference between Oil Fieldrdquo Transactions of theMetallurgical Society of AIME vol 219 no 8 pp 175ndash192 1960

[4] MMortada ldquoOilfield Interference in Aquifers of Non-UniformPropertiesrdquo Journal of Petroleum Technology vol 12 no 12 pp55ndash57 2013

[5] T Loucks and E Guerrero ldquoPressure Drop in a CompositeReservoirrdquo SPE Journal vol 1 no 03 pp 170ndash176 2013

[6] H Bixel and H Van Poollen ldquoPressure Drawdown and Buildupin the Presence of Radial Discontinuitiesrdquo SPE Journal vol 7no 03 pp 301ndash309 2013

[7] H Ramey ldquoApproximate Solutions For Unsteady LiquidFlow InComposite Reservoirsrdquo Journal of Canadian Petroleum Technol-ogy vol 9 no 01 2013

[8] Q Deng R Nie Y Jia et al ldquoPressure transient behavior of afractured well in multi-region composite reservoirsrdquo Journal ofPetroleum Science and Engineering vol 158 pp 535ndash553 2017

[9] H Kazemi LMerrill and J Jargon ldquoProblems in Interpretationof Pressure Fall-OffTests in ReservoirsWithAndWithout FluidBanksrdquo Journal of PetroleumTechnology vol 24 no 09 pp 1147ndash1156 2013

[10] L Merrill H Kazemi and W B Gogarty ldquoPressure FalloffAnalysis in Reservoirs With Fluid Banksrdquo Journal of PetroleumTechnology vol 26 no 07 pp 809ndash818 2013

[11] A Satman M Eggenschwiler R Tang and R HJ ldquoAnanalytical study of transient flow in systemswith radial disconti-nuitiesrdquo in Proceedings of the SPE9399 presented at SPE AnnualTechnical Conference and Exhibition Dallas Texas TX USASeptember 1980

[12] A Al-Bemani and I Ershaghi ldquoTwo-Phase Flow InterporosityEffects on Pressure Transient Test Response in Naturally Frac-tured Reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Dallas Texas

[13] Z Chen W Yu X Liao X Zhao Y Chen and K SepehrnoorildquoA Two-Phase Flow Model for Fractured Horizontal Well withComplex Fracture Networks Transient Analysis in FlowbackPeriodrdquo in Proceedings of the SPE Liquids-Rich Basins Confer-ence - North America Midland Texas USA

[14] H G Weinstein ldquoCold Waterflooding a Warm Reservoirrdquo inProceedings of the Fall Meeting of the Society of PetroleumEngineers of AIME Houston Texas

[15] S E Buckley andM C Leverett ldquoMechanism of fluid displace-ment in sandsrdquoTransactions of the AIME vol 146 no 1 pp 107ndash116 2013

[16] A Sosa R Raghavan and T Limon ldquoEffect of Relative Perme-ability andMobility Ratio on Pressure Falloff Behaviorrdquo Journalof Petroleum Technology vol 33 no 06 pp 1125ndash1135 2013

[17] P Hazebroek H Rainbow and CS Matthews ldquoPressure Fall-Off in Water Injection Wellsrdquo Transactions of the MetallurgicalSociety of AIME vol 213 pp 250ndash260 1958

[18] R Carter ldquoPressure Behavior of a Limited Circular CompositeReservoirrdquo SPE Journal vol 6 no 04 pp 328ndash334 2013

[19] A Odeh ldquoFlowTest Analysis for aWell with Radial Discontinu-ityrdquo Journal of Petroleum Technology vol 21 no 02 pp 207ndash2102013

[20] M Abbaszadeh and M Kamal ldquoPressure-Transient Testing ofWater-InjectionWellsrdquo SPE Reservoir Engineering vol 4 no 01pp 115ndash124 2013

[21] N Yeh and R Agarwal ldquoPressure Transient Analysis of Injec-tionWells in ReservoirsWithMultiple Fluid Banksrdquo in Proceed-ings of the SPE Annual Technical Conference and Exhibition SanAntonio Texas

[22] M M Levitan ldquoApplication of water injectionfalloff tests forreservoir appraisal New analytical solution method for two-phase variable rate problemsrdquo SPE Journal vol 8 no 4 pp 341ndash349 2003

[23] T Nanba and R Horne ldquoEstimation of Water and Oil RelativePermeabilities From Pressure Transient Analysis of WaterInjectionWell Datardquo in Proceedings of the SPE Annual TechnicalConference and Exhibition San Antonio Texas

[24] R B Bratvold andRNHorne ldquoAnalysis of pressure-falloff testsfollowing cold-water injectionrdquo SPE Formation Evaluation vol5 no 3 pp 293ndash302 1990

[25] S Chen G Li and A C Reynolds ldquoAnalytical Solution forInjection-Falloff-Production Testrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition San AntonioTexas USA

[26] A A Boughrara A M M Peres S Chen A A V Machadoand A C Reynolds ldquoApproximate analytical solutions for thepressure response at a water-injection wellrdquo SPE Journal vol12 no 1 pp 19ndash34 2007

[27] S Zheng and w xu ldquoNew approaches for analyzing transientpressure from oil and water two-phase flowing reservoirrdquo inProceedings of the Kuwait International Petroleum Conferenceand Exhibition Kuwait City Kuwait

[28] N A El-Khatib ldquoTransient Pressure Behavior of CompositeReservoirs with Moving Boundariesrdquo in Proceedings of theMiddle East Oil Show and Conference Bahrain

[29] A Dastan M M Kamal Y Hwang F Suleen and S MorsyldquoFalloff Testing Under Multiphase Flow Conditions in Natu-rally Fractured Reservoirsrdquo in Proceedings of the SPE WesternRegional Meeting Garden Grove California USA

[30] C n Machado and A C Reynolds ldquoApproximate semi-analytical solution for injection-falloff-production well test ananalytical tool for the in situ estimation of relative permeabilitycurvesrdquo Transport in Porous Media vol 121 no 1 pp 207ndash2312018

[31] T Barkve ldquoAnalytical study of reservoir pressure during water-injection well testsrdquo Society of Petroleum Engineers of AIME(Paper) SPE 1986

[32] T S Ramakrishnan and F J Kuchuk ldquoTesting injection wellswith rate and pressure datardquo SPE Formation Evaluation vol 9no 3 pp 228ndash236 1994

[33] L G Thompson and A C Reynolds ldquoWell testing for radiallyheterogeneous reservoirs under single and multiphase flowconditionsrdquo SPE Formation Evaluation vol 12 no 1 pp 57ndash641997

12 Mathematical Problems in Engineering

[34] JS Aronofsky L Masse and SG Natanson ldquoA Model for theMechanism of Oil Recovery from the Porous Matrix Due toWater Invasion in Fractured Reservoirsrdquo Trans AIME vol 213pp 17ndash19 1958

[35] A de Swaan ldquoTheory of Waterflooding in Fractured Reser-voirsrdquo SPE Journal vol 18 no 02 pp 117ndash122 2013

[36] H Kazemi J Gilman and A Elsharkawy ldquoAnalytical andNumerical Solution of Oil Recovery From Fractured ReservoirsWith Empirical Transfer Functions (includes associated papers25528 and 25818)rdquo SPE Reservoir Engineering vol 7 no 02 pp219ndash227 2013

[37] T Shimamoto D Kuramoto N Arihara and T Onishi ldquoA NewPressure Transient Analysis Model for Water Injection Wellin Dual-Porosity Reservoirsrdquo in Proceedings of the SPE100881presented at SPE Asia Pacific Oil ampamp Gas Conference andExhibition Adelaide Australia 2006

[38] G J Moridis D A McVay D L Reddell and T A BlasingameldquoLaplace transform finite difference (LTFD) numerical methodfor the simulation of compressible liquid flow in reservoirsrdquo SPEAdvanced Technology Series vol 2 no 2 pp 122ndash131 1994

[39] A D Habte and M Onur ldquoLaplace-transform finite-differenceand quasistationary solution method for water-injectionfallofftestsrdquo SPE Journal vol 19 no 3 pp 398ndash409 2014

[40] J Warren and P Root ldquoThe behavior of naturally fracturedreservoirsrdquo SPE Journal vol 3 no 3 pp 245ndash255 2013

[41] D Bourdet J Ayoub and Y Pirard ldquoUse of Pressure Derivativein Well Test Interpretationrdquo SPE Formation Evaluation vol 4no 02 pp 293ndash302 2013

[42] M Mavor and H Cinco-Ley ldquoTransient Pressure BehaviorOf Naturally Fractured Reservoirsrdquo in Proceedings of the SPECalifornia Regional Meeting Ventura California

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

Mathematical Problems in Engineering 7

tDPwDminusLTFD methoddPwDminusLTFD method

PwDminusanalytical methoddPwDminusanalytical method

10minus2

10minus1

100

101

101

102

102

103

104

105

106

107

108

PwD

dPw

D

Figure 3 Comparison of the results by LTFDmethod and analyticalmethod

Table 3 Basic parameters for calculation of pressure curves

Parameter Value UnitsFormation height 10 mFracture porosity 003 fractionMatrix porosity 027 fractionWellbore radius 01 mFracture permeability 06 120583m2Matrix permeability 6 times 10minus5 120583m2Irreducible water saturation 20 Residual oil saturation 14 Oil volume factor 103 fractionWater volume factor 1 fractionShape factor 6 times 10minus4 1cm2

Oil viscosity 1 mPasWater viscosity 05 mPasrock compressibility 6 times 10minus4 1MPaoil compressibility 1 times 10minus3 1MPawater compressibility 4 times 10minus4 1MPaInjection rate 30 m3dayInitial pressure 14 MPaSkin factor 01 fractionImbibition rate coefficient 01 1dayWellbore storage coefficient 001 1m3MPa

of the well test data comparing with the results in our modelFrom the figure it is found that our results match well withthe field test data

4 Model Features Analysis

The pressure curves of oil-water two-phase flow consider-ing water imbibition and the saturation gradients duringinjection period by the LTFD method were plotted in thissection The behaviors of the typical curves were analyzed by

Table 4 Basic information of the water injection well

Parameter Value UnitsFormation height 126 mFracture porosity 003 fractionMatrix porosity 0268 fractionWellbore radius 012 mOil volume factor 1163 fractionWater volume factor 1 fractionOil viscosity 96 mPasWater viscosity 05 mPasrock compressibility 5 times 10minus4 1MPaoil compressibility 48 times 10minus3 1MPawater compressibility 39 times 10minus4 1MPaInjection rate 256 m3dayInitial pressure 15 MPa

KrwKro

0

02

04

06

08

1

Kro

K

rw

02 04 06 08 10Sw

Figure 4 The relative permeability of oil and water for the waterinjection well

comparing the typical curves of single phase flow with thetwo-phase flow curves in the fractured reservoirs as follows

41 Features of Typical Curves Figure 6 illustrates the pres-sure curves of single phase flow and oil-water two-phaseflow in fractured reservoirs by LTFD method without waterimbibition According to the feature of the curves theformation flow can be divided into three stages The firststage is the flow in the fracture system the second stage isthe flow between the fracture and the matrix system andthe third stage is the flow in the whole system including thefracture andmatrixThe figure shows that the shape of curvesfor single phase flow and two-phase flow is similar but theposition of the two-phase flow curves is on the upper rightof the single phase flow curves The reason is that when oiland water exist in the whole system the total mobility of oiland water is lower than the single phase and the bottom-hole pressure increases more quickly as the fluid is injected

8 Mathematical Problems in Engineering

tDPwDminusField testdPwDminusField test

PwDminusLTFD methoddPwDminusLTFD method

101

102

103

104

105

106

107

100

101

102

PwD

dPw

D

Figure 5 Comparison of the results in this work and the fieldinjection test

tD

PwDminussingle phasedPwDminussingle phase

PwDminustwo-phasedPwDminustwo-phase

101

102

103

104

105

106

107

108

10minus1

100

101

102

PwD

dPw

D

Figure 6 Typical curves of single phase and two-phase flow duringinjection period

into the well Thus the pressure derivative curves of two-phase flow move upward compared to the single phase flowcurves Moreover due to the decrease of the total mobilitythe interporosity flow capacity between matrix and fracturesystem decreases when the single phase flow becomes two-phase flow Thus the ldquodiprdquo of the two-phase flow curvesoccurs later than that of the single phase flow curves

42 Effects of Parameters The key factors influencing thepressure behavior of water injection period including skinfactor wellbore storage coefficient shape factor water imbibi-tion rate coefficient and boundary conditions were discussedin detail in this part

421 Skin Factor Figure 7 shows the effect of skin factoron the pressure curves of the water injection well The solid

tD

S=minus1S=minus1S=1

S=1S=10S=10

S=minus1S=minus1S=1

S=1S=10S=10

102

104

106

108

10minus1

100

101

102

PwD

dPw

D

Figure 7 Influence of different skin factors on pressure transientbehavior

lines are the curves of single phase flow and the circlesare oil-water two-phase flow curves The red green andblue colors indicate that the shape factor values are -1 1and 10 respectively From the figure the skin factor mainlyinfluences the peak value of the pressure derivatives As thevalue of skin factor increases the peak value gets higherand the time for peak value occurrence becomes later whichindicates that the reservoir is more severely contaminatedThe pressure derivative curves of two-phase flow movetoward the upper right of the single phase flow curves whichshows the same change rule with Figure 6 When the skinfactor becomes larger the peak value difference between thesingle phase and two-phase decreases and the inclination ofthe curves after the peak value increases

422 Wellbore Storage Coefficient Figure 8 is the influencesof wellbore storage coefficient on the pressure and pressurederivative curves of the water injection well The red greenand blue colors indicate that wellbore storage coefficientvalues are 001 01 and 1 respectively As seen the maindifference of the curves occurs on the straight line of earlyperiod As the wellbore storage coefficient increases thestraight line section becomes longer and the time of peakvalue occurs later Besides the wellbore storage effect willcover up the radial flow stage in the fracture system directlygoing to the transitional flow stage between the fracture andmatrix The pressure derivative curves of two-phase flowmove upper right to the single phase flow curves after thestraight line of wellbore storage stage which agrees with thechange rule of Figure 6

423 Shape Factor The influences of shape factor on thepressure curves of the water injection well are shown inFigure 9 The red green and blue colors indicate thatshape factor values are 10minus5 10minus4 and 10minus3 respectivelyBy comparison of the curves it is found that the shape

Mathematical Problems in Engineering 9

tD10

210

410

610

810

minus3

10minus2

10minus1

100

101

102

PwD

dPw

D

C=001C=001C=01

C=01C=1C=1

C=001C=001C=01

C=01C=1C=1

Figure 8 Influence of different wellbore storage coefficients onpressure transient behavior

tD

Fs=1lowast10minus5Fs=1lowast10minus5Fs=1lowast10minus4Fs=1lowast10minus4Fs=1lowast10minus3Fs=1lowast10minus3

Fs=1lowast10minus5Fs=1lowast10minus5Fs=1lowast10minus4Fs=1lowast10minus4Fs=1lowast10minus3Fs=1lowast10minus3

102

104

106

108

10minus1

100

101

102

PwD

dPw

D

Figure 9 Influence of different shape factors on pressure transientbehavior

factor mainly has an effect on the position of the ldquodiprdquo ofthe pressure derivatives When the value of shape factorgets greater the ldquodiprdquo appears earlier and the ldquodiprdquo movescloser to the left The main reason is that the shape factoris a component of the equation defining the interporosityparameter The higher the shape factor is the easier the fluidexchange between the fracture and matrix is and the earlierthe transitional flow stage occurs It also demonstrates thatthe pressure derivative curves of two-phase flow move upperright to the single phase flow curves which is in line with thechange rule of Figure 6

tDRc=0Rc=0Rc=001

Rc=001Rc=005Rc=005

Rc=01Rc=01Rc=05

Rc=05Rc=1Rc=1

102

104

106

108

10minus2

10minus1

100

101

102

PwD

dPw

DFigure 10 Pressure behavior of different imbibition rate coefficientswith no flow outer boundary

424 Imbibition Rate Coefficient with No Flow Outer Bound-ary Condition Figure 10 describes the pressure behaviorof the water injection well under no flow outer boundarycondition when the water imbibition rate coefficient valuesare 0 001 005 01 05 and 1 respectively From the figurethe imbibition rate coefficient mainly has an effect on thewhole system (matrix and fracture system) radial flow periodof the curves As the imbibition rate coefficient becomeslarger there are more fluid exchange between the fractureand matrix and more fluid supply from the matrix to thefracture The water imbibition process is slow so there isa second interporosity period after the whole system radialflow period And with the increase of water imbibition ratecoefficient the pressure derivative curves dip downwardearlier and the degree for the curves dropping down becomeslarger After that the pressure derivative curves rise up to auniform line with the same slope due to the influence of theno flow outer boundary

425 Imbibition Rate Coefficient with Constant PressureBoundary Condition Figure 11 shows the pressure and pres-sure derivative curves of the water injection well underconstant pressure boundary condition when the water imbi-bition rate coefficient values are 0 001 005 01 05 and1 respectively By comparison and analysis it is concludedthat the pressure derivative curve is a straight line at thewhole system radial flow period when no water imbibition isconsideredWhen considering water imbibition the pressurederivatives decline at the whole system radial flow periodThe pressure derivatives drop downward more and earlierwith the increase of imbibition rate coefficient After that thecurves also drop down due to the constant pressure outerboundary effect

10 Mathematical Problems in Engineering

tD

Rc=0Rc=0Rc=001

Rc=001Rc=005Rc=005

Rc=01Rc=01Rc=05

Rc=05Rc=1Rc=1

102

104

106

108

10minus2

10minus1

100

101

102

PwD

dPw

D

Figure 11 Pressure behavior of different imbibition rate coefficientswith constant pressure boundary

5 Conclusions

In this work a semianalytical two-phase flow model consid-ering water imbibition and the saturation gradients withineach region during injection period was proposed to analyzethe pressure transient behavior of the water injection well infractured reservoirs Validation of the presented model wasperformed by analytical method and a field water injectiontest The model features were investigated by sensitivityanalysis Some conclusions were drawn as follows

(1) The shape of the pressure curves for the proposedmodel is similar to the single phase flowmodel whilethe position of the curves for the two-phase flowmodel is on the upper right of the single phase flowmodel

(2) Themain influence of skin factor on the curves occurson the peak value of the pressure derivatives As theskin factor becomes larger the peak value differencebetween single phase and two-phase pressure curvesdecreases and the inclination of the curves after thepeak value increases

(3) The wellbore storage coefficient mainly has an effecton the straight line of the early period for the pressurecurves The shape factor mainly has an effect on theposition of the ldquodiprdquo of the pressure derivatives whichis the same as the interporosity parameter

(4) The imbibition rate coefficient mainly influences thewhole system radial flow period of the pressurecurves The pressure derivative curves dip downwardearlier and drop down to a greater degree with theincrease of imbibition rate coefficient

Nomenclature

R Recovery fractionR119888 Imbibition rate coefficient 1day119877infin Ultimate cumulative oil recovery fraction0 Porosity fraction119904119900119903 Residual oil saturation fraction119904119908119888 Irreducible water saturation fraction119904119908 Water saturation fractionz Logarithmic transform variable dimensionlessu Laplace variable dimensionlessq Displacement rate m3dt Time hk Permeability 1205831198982119896119903 Relative permeability fractionh Formation height mr Radial distance m119903119908 Wellbore radius mp Pressure MPa120583 Viscosity mPasdotsB Formation volume factor dimensionless120591 Fracture matrix transfer term m3dF119904 Shape factor 1m2119862119905 Total compressibility 1MPa119872119905 Total mobility 1MPa120582 Interporosity flow coefficient fraction120596 Storativity ratio fractionS Skin factor dimensionlessC Wellbore storage coefficient m3MPa

Subscripts

f Fracturem Matrixo Oilw Watert Totale ExternalD Dimensionless variable

Superscripts

- Laplace-transformed variable Endpoint property997888rarr Vector symbol

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

The authors gratefully acknowledge the support of theNational Basic Research 973 Program of China (Grant No

Mathematical Problems in Engineering 11

2015CB250900) the National Natural Science Foundationof China (Grant No 51704237) and the Research Projectsof Shaanxi Provincial Education Department (Grant No13JS090)

References

[1] R L Perrine Analysis of Pressure-buildup Curves vol 56American Petroleum Institute 1956

[2] J C Martin ldquoSimplified Equations of Flow in Gas Drive Reser-voirs and the Theoretical Foundation of Multiphase PressureBuildup Analysesrdquo Society of Petroleum Engineers vol 216 pp321ndash323 1959

[3] W Hurst ldquoInterference between Oil Fieldrdquo Transactions of theMetallurgical Society of AIME vol 219 no 8 pp 175ndash192 1960

[4] MMortada ldquoOilfield Interference in Aquifers of Non-UniformPropertiesrdquo Journal of Petroleum Technology vol 12 no 12 pp55ndash57 2013

[5] T Loucks and E Guerrero ldquoPressure Drop in a CompositeReservoirrdquo SPE Journal vol 1 no 03 pp 170ndash176 2013

[6] H Bixel and H Van Poollen ldquoPressure Drawdown and Buildupin the Presence of Radial Discontinuitiesrdquo SPE Journal vol 7no 03 pp 301ndash309 2013

[7] H Ramey ldquoApproximate Solutions For Unsteady LiquidFlow InComposite Reservoirsrdquo Journal of Canadian Petroleum Technol-ogy vol 9 no 01 2013

[8] Q Deng R Nie Y Jia et al ldquoPressure transient behavior of afractured well in multi-region composite reservoirsrdquo Journal ofPetroleum Science and Engineering vol 158 pp 535ndash553 2017

[9] H Kazemi LMerrill and J Jargon ldquoProblems in Interpretationof Pressure Fall-OffTests in ReservoirsWithAndWithout FluidBanksrdquo Journal of PetroleumTechnology vol 24 no 09 pp 1147ndash1156 2013

[10] L Merrill H Kazemi and W B Gogarty ldquoPressure FalloffAnalysis in Reservoirs With Fluid Banksrdquo Journal of PetroleumTechnology vol 26 no 07 pp 809ndash818 2013

[11] A Satman M Eggenschwiler R Tang and R HJ ldquoAnanalytical study of transient flow in systemswith radial disconti-nuitiesrdquo in Proceedings of the SPE9399 presented at SPE AnnualTechnical Conference and Exhibition Dallas Texas TX USASeptember 1980

[12] A Al-Bemani and I Ershaghi ldquoTwo-Phase Flow InterporosityEffects on Pressure Transient Test Response in Naturally Frac-tured Reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Dallas Texas

[13] Z Chen W Yu X Liao X Zhao Y Chen and K SepehrnoorildquoA Two-Phase Flow Model for Fractured Horizontal Well withComplex Fracture Networks Transient Analysis in FlowbackPeriodrdquo in Proceedings of the SPE Liquids-Rich Basins Confer-ence - North America Midland Texas USA

[14] H G Weinstein ldquoCold Waterflooding a Warm Reservoirrdquo inProceedings of the Fall Meeting of the Society of PetroleumEngineers of AIME Houston Texas

[15] S E Buckley andM C Leverett ldquoMechanism of fluid displace-ment in sandsrdquoTransactions of the AIME vol 146 no 1 pp 107ndash116 2013

[16] A Sosa R Raghavan and T Limon ldquoEffect of Relative Perme-ability andMobility Ratio on Pressure Falloff Behaviorrdquo Journalof Petroleum Technology vol 33 no 06 pp 1125ndash1135 2013

[17] P Hazebroek H Rainbow and CS Matthews ldquoPressure Fall-Off in Water Injection Wellsrdquo Transactions of the MetallurgicalSociety of AIME vol 213 pp 250ndash260 1958

[18] R Carter ldquoPressure Behavior of a Limited Circular CompositeReservoirrdquo SPE Journal vol 6 no 04 pp 328ndash334 2013

[19] A Odeh ldquoFlowTest Analysis for aWell with Radial Discontinu-ityrdquo Journal of Petroleum Technology vol 21 no 02 pp 207ndash2102013

[20] M Abbaszadeh and M Kamal ldquoPressure-Transient Testing ofWater-InjectionWellsrdquo SPE Reservoir Engineering vol 4 no 01pp 115ndash124 2013

[21] N Yeh and R Agarwal ldquoPressure Transient Analysis of Injec-tionWells in ReservoirsWithMultiple Fluid Banksrdquo in Proceed-ings of the SPE Annual Technical Conference and Exhibition SanAntonio Texas

[22] M M Levitan ldquoApplication of water injectionfalloff tests forreservoir appraisal New analytical solution method for two-phase variable rate problemsrdquo SPE Journal vol 8 no 4 pp 341ndash349 2003

[23] T Nanba and R Horne ldquoEstimation of Water and Oil RelativePermeabilities From Pressure Transient Analysis of WaterInjectionWell Datardquo in Proceedings of the SPE Annual TechnicalConference and Exhibition San Antonio Texas

[24] R B Bratvold andRNHorne ldquoAnalysis of pressure-falloff testsfollowing cold-water injectionrdquo SPE Formation Evaluation vol5 no 3 pp 293ndash302 1990

[25] S Chen G Li and A C Reynolds ldquoAnalytical Solution forInjection-Falloff-Production Testrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition San AntonioTexas USA

[26] A A Boughrara A M M Peres S Chen A A V Machadoand A C Reynolds ldquoApproximate analytical solutions for thepressure response at a water-injection wellrdquo SPE Journal vol12 no 1 pp 19ndash34 2007

[27] S Zheng and w xu ldquoNew approaches for analyzing transientpressure from oil and water two-phase flowing reservoirrdquo inProceedings of the Kuwait International Petroleum Conferenceand Exhibition Kuwait City Kuwait

[28] N A El-Khatib ldquoTransient Pressure Behavior of CompositeReservoirs with Moving Boundariesrdquo in Proceedings of theMiddle East Oil Show and Conference Bahrain

[29] A Dastan M M Kamal Y Hwang F Suleen and S MorsyldquoFalloff Testing Under Multiphase Flow Conditions in Natu-rally Fractured Reservoirsrdquo in Proceedings of the SPE WesternRegional Meeting Garden Grove California USA

[30] C n Machado and A C Reynolds ldquoApproximate semi-analytical solution for injection-falloff-production well test ananalytical tool for the in situ estimation of relative permeabilitycurvesrdquo Transport in Porous Media vol 121 no 1 pp 207ndash2312018

[31] T Barkve ldquoAnalytical study of reservoir pressure during water-injection well testsrdquo Society of Petroleum Engineers of AIME(Paper) SPE 1986

[32] T S Ramakrishnan and F J Kuchuk ldquoTesting injection wellswith rate and pressure datardquo SPE Formation Evaluation vol 9no 3 pp 228ndash236 1994

[33] L G Thompson and A C Reynolds ldquoWell testing for radiallyheterogeneous reservoirs under single and multiphase flowconditionsrdquo SPE Formation Evaluation vol 12 no 1 pp 57ndash641997

12 Mathematical Problems in Engineering

[34] JS Aronofsky L Masse and SG Natanson ldquoA Model for theMechanism of Oil Recovery from the Porous Matrix Due toWater Invasion in Fractured Reservoirsrdquo Trans AIME vol 213pp 17ndash19 1958

[35] A de Swaan ldquoTheory of Waterflooding in Fractured Reser-voirsrdquo SPE Journal vol 18 no 02 pp 117ndash122 2013

[36] H Kazemi J Gilman and A Elsharkawy ldquoAnalytical andNumerical Solution of Oil Recovery From Fractured ReservoirsWith Empirical Transfer Functions (includes associated papers25528 and 25818)rdquo SPE Reservoir Engineering vol 7 no 02 pp219ndash227 2013

[37] T Shimamoto D Kuramoto N Arihara and T Onishi ldquoA NewPressure Transient Analysis Model for Water Injection Wellin Dual-Porosity Reservoirsrdquo in Proceedings of the SPE100881presented at SPE Asia Pacific Oil ampamp Gas Conference andExhibition Adelaide Australia 2006

[38] G J Moridis D A McVay D L Reddell and T A BlasingameldquoLaplace transform finite difference (LTFD) numerical methodfor the simulation of compressible liquid flow in reservoirsrdquo SPEAdvanced Technology Series vol 2 no 2 pp 122ndash131 1994

[39] A D Habte and M Onur ldquoLaplace-transform finite-differenceand quasistationary solution method for water-injectionfallofftestsrdquo SPE Journal vol 19 no 3 pp 398ndash409 2014

[40] J Warren and P Root ldquoThe behavior of naturally fracturedreservoirsrdquo SPE Journal vol 3 no 3 pp 245ndash255 2013

[41] D Bourdet J Ayoub and Y Pirard ldquoUse of Pressure Derivativein Well Test Interpretationrdquo SPE Formation Evaluation vol 4no 02 pp 293ndash302 2013

[42] M Mavor and H Cinco-Ley ldquoTransient Pressure BehaviorOf Naturally Fractured Reservoirsrdquo in Proceedings of the SPECalifornia Regional Meeting Ventura California

Hindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Mathematical Problems in Engineering

tDPwDminusField testdPwDminusField test

PwDminusLTFD methoddPwDminusLTFD method

101

102

103

104

105

106

107

100

101

102

PwD

dPw

D

Figure 5 Comparison of the results in this work and the fieldinjection test

tD

PwDminussingle phasedPwDminussingle phase

PwDminustwo-phasedPwDminustwo-phase

101

102

103

104

105

106

107

108

10minus1

100

101

102

PwD

dPw

D

Figure 6 Typical curves of single phase and two-phase flow duringinjection period

into the well Thus the pressure derivative curves of two-phase flow move upward compared to the single phase flowcurves Moreover due to the decrease of the total mobilitythe interporosity flow capacity between matrix and fracturesystem decreases when the single phase flow becomes two-phase flow Thus the ldquodiprdquo of the two-phase flow curvesoccurs later than that of the single phase flow curves

42 Effects of Parameters The key factors influencing thepressure behavior of water injection period including skinfactor wellbore storage coefficient shape factor water imbibi-tion rate coefficient and boundary conditions were discussedin detail in this part

421 Skin Factor Figure 7 shows the effect of skin factoron the pressure curves of the water injection well The solid

tD

S=minus1S=minus1S=1

S=1S=10S=10

S=minus1S=minus1S=1

S=1S=10S=10

102

104

106

108

10minus1

100

101

102

PwD

dPw

D

Figure 7 Influence of different skin factors on pressure transientbehavior

lines are the curves of single phase flow and the circlesare oil-water two-phase flow curves The red green andblue colors indicate that the shape factor values are -1 1and 10 respectively From the figure the skin factor mainlyinfluences the peak value of the pressure derivatives As thevalue of skin factor increases the peak value gets higherand the time for peak value occurrence becomes later whichindicates that the reservoir is more severely contaminatedThe pressure derivative curves of two-phase flow movetoward the upper right of the single phase flow curves whichshows the same change rule with Figure 6 When the skinfactor becomes larger the peak value difference between thesingle phase and two-phase decreases and the inclination ofthe curves after the peak value increases

422 Wellbore Storage Coefficient Figure 8 is the influencesof wellbore storage coefficient on the pressure and pressurederivative curves of the water injection well The red greenand blue colors indicate that wellbore storage coefficientvalues are 001 01 and 1 respectively As seen the maindifference of the curves occurs on the straight line of earlyperiod As the wellbore storage coefficient increases thestraight line section becomes longer and the time of peakvalue occurs later Besides the wellbore storage effect willcover up the radial flow stage in the fracture system directlygoing to the transitional flow stage between the fracture andmatrix The pressure derivative curves of two-phase flowmove upper right to the single phase flow curves after thestraight line of wellbore storage stage which agrees with thechange rule of Figure 6

423 Shape Factor The influences of shape factor on thepressure curves of the water injection well are shown inFigure 9 The red green and blue colors indicate thatshape factor values are 10minus5 10minus4 and 10minus3 respectivelyBy comparison of the curves it is found that the shape

Mathematical Problems in Engineering 9

tD10

210

410

610

810

minus3

10minus2

10minus1

100

101

102

PwD

dPw

D

C=001C=001C=01

C=01C=1C=1

C=001C=001C=01

C=01C=1C=1

Figure 8 Influence of different wellbore storage coefficients onpressure transient behavior

tD

Fs=1lowast10minus5Fs=1lowast10minus5Fs=1lowast10minus4Fs=1lowast10minus4Fs=1lowast10minus3Fs=1lowast10minus3

Fs=1lowast10minus5Fs=1lowast10minus5Fs=1lowast10minus4Fs=1lowast10minus4Fs=1lowast10minus3Fs=1lowast10minus3

102

104

106

108

10minus1

100

101

102

PwD

dPw

D

Figure 9 Influence of different shape factors on pressure transientbehavior

factor mainly has an effect on the position of the ldquodiprdquo ofthe pressure derivatives When the value of shape factorgets greater the ldquodiprdquo appears earlier and the ldquodiprdquo movescloser to the left The main reason is that the shape factoris a component of the equation defining the interporosityparameter The higher the shape factor is the easier the fluidexchange between the fracture and matrix is and the earlierthe transitional flow stage occurs It also demonstrates thatthe pressure derivative curves of two-phase flow move upperright to the single phase flow curves which is in line with thechange rule of Figure 6

tDRc=0Rc=0Rc=001

Rc=001Rc=005Rc=005

Rc=01Rc=01Rc=05

Rc=05Rc=1Rc=1

102

104

106

108

10minus2

10minus1

100

101

102

PwD

dPw

DFigure 10 Pressure behavior of different imbibition rate coefficientswith no flow outer boundary

424 Imbibition Rate Coefficient with No Flow Outer Bound-ary Condition Figure 10 describes the pressure behaviorof the water injection well under no flow outer boundarycondition when the water imbibition rate coefficient valuesare 0 001 005 01 05 and 1 respectively From the figurethe imbibition rate coefficient mainly has an effect on thewhole system (matrix and fracture system) radial flow periodof the curves As the imbibition rate coefficient becomeslarger there are more fluid exchange between the fractureand matrix and more fluid supply from the matrix to thefracture The water imbibition process is slow so there isa second interporosity period after the whole system radialflow period And with the increase of water imbibition ratecoefficient the pressure derivative curves dip downwardearlier and the degree for the curves dropping down becomeslarger After that the pressure derivative curves rise up to auniform line with the same slope due to the influence of theno flow outer boundary

425 Imbibition Rate Coefficient with Constant PressureBoundary Condition Figure 11 shows the pressure and pres-sure derivative curves of the water injection well underconstant pressure boundary condition when the water imbi-bition rate coefficient values are 0 001 005 01 05 and1 respectively By comparison and analysis it is concludedthat the pressure derivative curve is a straight line at thewhole system radial flow period when no water imbibition isconsideredWhen considering water imbibition the pressurederivatives decline at the whole system radial flow periodThe pressure derivatives drop downward more and earlierwith the increase of imbibition rate coefficient After that thecurves also drop down due to the constant pressure outerboundary effect

10 Mathematical Problems in Engineering

tD

Rc=0Rc=0Rc=001

Rc=001Rc=005Rc=005

Rc=01Rc=01Rc=05

Rc=05Rc=1Rc=1

102

104

106

108

10minus2

10minus1

100

101

102

PwD

dPw

D

Figure 11 Pressure behavior of different imbibition rate coefficientswith constant pressure boundary

5 Conclusions

In this work a semianalytical two-phase flow model consid-ering water imbibition and the saturation gradients withineach region during injection period was proposed to analyzethe pressure transient behavior of the water injection well infractured reservoirs Validation of the presented model wasperformed by analytical method and a field water injectiontest The model features were investigated by sensitivityanalysis Some conclusions were drawn as follows

(1) The shape of the pressure curves for the proposedmodel is similar to the single phase flowmodel whilethe position of the curves for the two-phase flowmodel is on the upper right of the single phase flowmodel

(2) Themain influence of skin factor on the curves occurson the peak value of the pressure derivatives As theskin factor becomes larger the peak value differencebetween single phase and two-phase pressure curvesdecreases and the inclination of the curves after thepeak value increases

(3) The wellbore storage coefficient mainly has an effecton the straight line of the early period for the pressurecurves The shape factor mainly has an effect on theposition of the ldquodiprdquo of the pressure derivatives whichis the same as the interporosity parameter

(4) The imbibition rate coefficient mainly influences thewhole system radial flow period of the pressurecurves The pressure derivative curves dip downwardearlier and drop down to a greater degree with theincrease of imbibition rate coefficient

Nomenclature

R Recovery fractionR119888 Imbibition rate coefficient 1day119877infin Ultimate cumulative oil recovery fraction0 Porosity fraction119904119900119903 Residual oil saturation fraction119904119908119888 Irreducible water saturation fraction119904119908 Water saturation fractionz Logarithmic transform variable dimensionlessu Laplace variable dimensionlessq Displacement rate m3dt Time hk Permeability 1205831198982119896119903 Relative permeability fractionh Formation height mr Radial distance m119903119908 Wellbore radius mp Pressure MPa120583 Viscosity mPasdotsB Formation volume factor dimensionless120591 Fracture matrix transfer term m3dF119904 Shape factor 1m2119862119905 Total compressibility 1MPa119872119905 Total mobility 1MPa120582 Interporosity flow coefficient fraction120596 Storativity ratio fractionS Skin factor dimensionlessC Wellbore storage coefficient m3MPa

Subscripts

f Fracturem Matrixo Oilw Watert Totale ExternalD Dimensionless variable

Superscripts

- Laplace-transformed variable Endpoint property997888rarr Vector symbol

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

The authors gratefully acknowledge the support of theNational Basic Research 973 Program of China (Grant No

Mathematical Problems in Engineering 11

2015CB250900) the National Natural Science Foundationof China (Grant No 51704237) and the Research Projectsof Shaanxi Provincial Education Department (Grant No13JS090)

References

[1] R L Perrine Analysis of Pressure-buildup Curves vol 56American Petroleum Institute 1956

[2] J C Martin ldquoSimplified Equations of Flow in Gas Drive Reser-voirs and the Theoretical Foundation of Multiphase PressureBuildup Analysesrdquo Society of Petroleum Engineers vol 216 pp321ndash323 1959

[3] W Hurst ldquoInterference between Oil Fieldrdquo Transactions of theMetallurgical Society of AIME vol 219 no 8 pp 175ndash192 1960

[4] MMortada ldquoOilfield Interference in Aquifers of Non-UniformPropertiesrdquo Journal of Petroleum Technology vol 12 no 12 pp55ndash57 2013

[5] T Loucks and E Guerrero ldquoPressure Drop in a CompositeReservoirrdquo SPE Journal vol 1 no 03 pp 170ndash176 2013

[6] H Bixel and H Van Poollen ldquoPressure Drawdown and Buildupin the Presence of Radial Discontinuitiesrdquo SPE Journal vol 7no 03 pp 301ndash309 2013

[7] H Ramey ldquoApproximate Solutions For Unsteady LiquidFlow InComposite Reservoirsrdquo Journal of Canadian Petroleum Technol-ogy vol 9 no 01 2013

[8] Q Deng R Nie Y Jia et al ldquoPressure transient behavior of afractured well in multi-region composite reservoirsrdquo Journal ofPetroleum Science and Engineering vol 158 pp 535ndash553 2017

[9] H Kazemi LMerrill and J Jargon ldquoProblems in Interpretationof Pressure Fall-OffTests in ReservoirsWithAndWithout FluidBanksrdquo Journal of PetroleumTechnology vol 24 no 09 pp 1147ndash1156 2013

[10] L Merrill H Kazemi and W B Gogarty ldquoPressure FalloffAnalysis in Reservoirs With Fluid Banksrdquo Journal of PetroleumTechnology vol 26 no 07 pp 809ndash818 2013

[11] A Satman M Eggenschwiler R Tang and R HJ ldquoAnanalytical study of transient flow in systemswith radial disconti-nuitiesrdquo in Proceedings of the SPE9399 presented at SPE AnnualTechnical Conference and Exhibition Dallas Texas TX USASeptember 1980

[12] A Al-Bemani and I Ershaghi ldquoTwo-Phase Flow InterporosityEffects on Pressure Transient Test Response in Naturally Frac-tured Reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Dallas Texas

[13] Z Chen W Yu X Liao X Zhao Y Chen and K SepehrnoorildquoA Two-Phase Flow Model for Fractured Horizontal Well withComplex Fracture Networks Transient Analysis in FlowbackPeriodrdquo in Proceedings of the SPE Liquids-Rich Basins Confer-ence - North America Midland Texas USA

[14] H G Weinstein ldquoCold Waterflooding a Warm Reservoirrdquo inProceedings of the Fall Meeting of the Society of PetroleumEngineers of AIME Houston Texas

[15] S E Buckley andM C Leverett ldquoMechanism of fluid displace-ment in sandsrdquoTransactions of the AIME vol 146 no 1 pp 107ndash116 2013

[16] A Sosa R Raghavan and T Limon ldquoEffect of Relative Perme-ability andMobility Ratio on Pressure Falloff Behaviorrdquo Journalof Petroleum Technology vol 33 no 06 pp 1125ndash1135 2013

[17] P Hazebroek H Rainbow and CS Matthews ldquoPressure Fall-Off in Water Injection Wellsrdquo Transactions of the MetallurgicalSociety of AIME vol 213 pp 250ndash260 1958

[18] R Carter ldquoPressure Behavior of a Limited Circular CompositeReservoirrdquo SPE Journal vol 6 no 04 pp 328ndash334 2013

[19] A Odeh ldquoFlowTest Analysis for aWell with Radial Discontinu-ityrdquo Journal of Petroleum Technology vol 21 no 02 pp 207ndash2102013

[20] M Abbaszadeh and M Kamal ldquoPressure-Transient Testing ofWater-InjectionWellsrdquo SPE Reservoir Engineering vol 4 no 01pp 115ndash124 2013

[21] N Yeh and R Agarwal ldquoPressure Transient Analysis of Injec-tionWells in ReservoirsWithMultiple Fluid Banksrdquo in Proceed-ings of the SPE Annual Technical Conference and Exhibition SanAntonio Texas

[22] M M Levitan ldquoApplication of water injectionfalloff tests forreservoir appraisal New analytical solution method for two-phase variable rate problemsrdquo SPE Journal vol 8 no 4 pp 341ndash349 2003

[23] T Nanba and R Horne ldquoEstimation of Water and Oil RelativePermeabilities From Pressure Transient Analysis of WaterInjectionWell Datardquo in Proceedings of the SPE Annual TechnicalConference and Exhibition San Antonio Texas

[24] R B Bratvold andRNHorne ldquoAnalysis of pressure-falloff testsfollowing cold-water injectionrdquo SPE Formation Evaluation vol5 no 3 pp 293ndash302 1990

[25] S Chen G Li and A C Reynolds ldquoAnalytical Solution forInjection-Falloff-Production Testrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition San AntonioTexas USA

[26] A A Boughrara A M M Peres S Chen A A V Machadoand A C Reynolds ldquoApproximate analytical solutions for thepressure response at a water-injection wellrdquo SPE Journal vol12 no 1 pp 19ndash34 2007

[27] S Zheng and w xu ldquoNew approaches for analyzing transientpressure from oil and water two-phase flowing reservoirrdquo inProceedings of the Kuwait International Petroleum Conferenceand Exhibition Kuwait City Kuwait

[28] N A El-Khatib ldquoTransient Pressure Behavior of CompositeReservoirs with Moving Boundariesrdquo in Proceedings of theMiddle East Oil Show and Conference Bahrain

[29] A Dastan M M Kamal Y Hwang F Suleen and S MorsyldquoFalloff Testing Under Multiphase Flow Conditions in Natu-rally Fractured Reservoirsrdquo in Proceedings of the SPE WesternRegional Meeting Garden Grove California USA

[30] C n Machado and A C Reynolds ldquoApproximate semi-analytical solution for injection-falloff-production well test ananalytical tool for the in situ estimation of relative permeabilitycurvesrdquo Transport in Porous Media vol 121 no 1 pp 207ndash2312018

[31] T Barkve ldquoAnalytical study of reservoir pressure during water-injection well testsrdquo Society of Petroleum Engineers of AIME(Paper) SPE 1986

[32] T S Ramakrishnan and F J Kuchuk ldquoTesting injection wellswith rate and pressure datardquo SPE Formation Evaluation vol 9no 3 pp 228ndash236 1994

[33] L G Thompson and A C Reynolds ldquoWell testing for radiallyheterogeneous reservoirs under single and multiphase flowconditionsrdquo SPE Formation Evaluation vol 12 no 1 pp 57ndash641997

12 Mathematical Problems in Engineering

[34] JS Aronofsky L Masse and SG Natanson ldquoA Model for theMechanism of Oil Recovery from the Porous Matrix Due toWater Invasion in Fractured Reservoirsrdquo Trans AIME vol 213pp 17ndash19 1958

[35] A de Swaan ldquoTheory of Waterflooding in Fractured Reser-voirsrdquo SPE Journal vol 18 no 02 pp 117ndash122 2013

[36] H Kazemi J Gilman and A Elsharkawy ldquoAnalytical andNumerical Solution of Oil Recovery From Fractured ReservoirsWith Empirical Transfer Functions (includes associated papers25528 and 25818)rdquo SPE Reservoir Engineering vol 7 no 02 pp219ndash227 2013

[37] T Shimamoto D Kuramoto N Arihara and T Onishi ldquoA NewPressure Transient Analysis Model for Water Injection Wellin Dual-Porosity Reservoirsrdquo in Proceedings of the SPE100881presented at SPE Asia Pacific Oil ampamp Gas Conference andExhibition Adelaide Australia 2006

[38] G J Moridis D A McVay D L Reddell and T A BlasingameldquoLaplace transform finite difference (LTFD) numerical methodfor the simulation of compressible liquid flow in reservoirsrdquo SPEAdvanced Technology Series vol 2 no 2 pp 122ndash131 1994

[39] A D Habte and M Onur ldquoLaplace-transform finite-differenceand quasistationary solution method for water-injectionfallofftestsrdquo SPE Journal vol 19 no 3 pp 398ndash409 2014

[40] J Warren and P Root ldquoThe behavior of naturally fracturedreservoirsrdquo SPE Journal vol 3 no 3 pp 245ndash255 2013

[41] D Bourdet J Ayoub and Y Pirard ldquoUse of Pressure Derivativein Well Test Interpretationrdquo SPE Formation Evaluation vol 4no 02 pp 293ndash302 2013

[42] M Mavor and H Cinco-Ley ldquoTransient Pressure BehaviorOf Naturally Fractured Reservoirsrdquo in Proceedings of the SPECalifornia Regional Meeting Ventura California

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

Mathematical Problems in Engineering 9

tD10

210

410

610

810

minus3

10minus2

10minus1

100

101

102

PwD

dPw

D

C=001C=001C=01

C=01C=1C=1

C=001C=001C=01

C=01C=1C=1

Figure 8 Influence of different wellbore storage coefficients onpressure transient behavior

tD

Fs=1lowast10minus5Fs=1lowast10minus5Fs=1lowast10minus4Fs=1lowast10minus4Fs=1lowast10minus3Fs=1lowast10minus3

Fs=1lowast10minus5Fs=1lowast10minus5Fs=1lowast10minus4Fs=1lowast10minus4Fs=1lowast10minus3Fs=1lowast10minus3

102

104

106

108

10minus1

100

101

102

PwD

dPw

D

Figure 9 Influence of different shape factors on pressure transientbehavior

factor mainly has an effect on the position of the ldquodiprdquo ofthe pressure derivatives When the value of shape factorgets greater the ldquodiprdquo appears earlier and the ldquodiprdquo movescloser to the left The main reason is that the shape factoris a component of the equation defining the interporosityparameter The higher the shape factor is the easier the fluidexchange between the fracture and matrix is and the earlierthe transitional flow stage occurs It also demonstrates thatthe pressure derivative curves of two-phase flow move upperright to the single phase flow curves which is in line with thechange rule of Figure 6

tDRc=0Rc=0Rc=001

Rc=001Rc=005Rc=005

Rc=01Rc=01Rc=05

Rc=05Rc=1Rc=1

102

104

106

108

10minus2

10minus1

100

101

102

PwD

dPw

DFigure 10 Pressure behavior of different imbibition rate coefficientswith no flow outer boundary

424 Imbibition Rate Coefficient with No Flow Outer Bound-ary Condition Figure 10 describes the pressure behaviorof the water injection well under no flow outer boundarycondition when the water imbibition rate coefficient valuesare 0 001 005 01 05 and 1 respectively From the figurethe imbibition rate coefficient mainly has an effect on thewhole system (matrix and fracture system) radial flow periodof the curves As the imbibition rate coefficient becomeslarger there are more fluid exchange between the fractureand matrix and more fluid supply from the matrix to thefracture The water imbibition process is slow so there isa second interporosity period after the whole system radialflow period And with the increase of water imbibition ratecoefficient the pressure derivative curves dip downwardearlier and the degree for the curves dropping down becomeslarger After that the pressure derivative curves rise up to auniform line with the same slope due to the influence of theno flow outer boundary

425 Imbibition Rate Coefficient with Constant PressureBoundary Condition Figure 11 shows the pressure and pres-sure derivative curves of the water injection well underconstant pressure boundary condition when the water imbi-bition rate coefficient values are 0 001 005 01 05 and1 respectively By comparison and analysis it is concludedthat the pressure derivative curve is a straight line at thewhole system radial flow period when no water imbibition isconsideredWhen considering water imbibition the pressurederivatives decline at the whole system radial flow periodThe pressure derivatives drop downward more and earlierwith the increase of imbibition rate coefficient After that thecurves also drop down due to the constant pressure outerboundary effect

10 Mathematical Problems in Engineering

tD

Rc=0Rc=0Rc=001

Rc=001Rc=005Rc=005

Rc=01Rc=01Rc=05

Rc=05Rc=1Rc=1

102

104

106

108

10minus2

10minus1

100

101

102

PwD

dPw

D

Figure 11 Pressure behavior of different imbibition rate coefficientswith constant pressure boundary

5 Conclusions

In this work a semianalytical two-phase flow model consid-ering water imbibition and the saturation gradients withineach region during injection period was proposed to analyzethe pressure transient behavior of the water injection well infractured reservoirs Validation of the presented model wasperformed by analytical method and a field water injectiontest The model features were investigated by sensitivityanalysis Some conclusions were drawn as follows

(1) The shape of the pressure curves for the proposedmodel is similar to the single phase flowmodel whilethe position of the curves for the two-phase flowmodel is on the upper right of the single phase flowmodel

(2) Themain influence of skin factor on the curves occurson the peak value of the pressure derivatives As theskin factor becomes larger the peak value differencebetween single phase and two-phase pressure curvesdecreases and the inclination of the curves after thepeak value increases

(3) The wellbore storage coefficient mainly has an effecton the straight line of the early period for the pressurecurves The shape factor mainly has an effect on theposition of the ldquodiprdquo of the pressure derivatives whichis the same as the interporosity parameter

(4) The imbibition rate coefficient mainly influences thewhole system radial flow period of the pressurecurves The pressure derivative curves dip downwardearlier and drop down to a greater degree with theincrease of imbibition rate coefficient

Nomenclature

R Recovery fractionR119888 Imbibition rate coefficient 1day119877infin Ultimate cumulative oil recovery fraction0 Porosity fraction119904119900119903 Residual oil saturation fraction119904119908119888 Irreducible water saturation fraction119904119908 Water saturation fractionz Logarithmic transform variable dimensionlessu Laplace variable dimensionlessq Displacement rate m3dt Time hk Permeability 1205831198982119896119903 Relative permeability fractionh Formation height mr Radial distance m119903119908 Wellbore radius mp Pressure MPa120583 Viscosity mPasdotsB Formation volume factor dimensionless120591 Fracture matrix transfer term m3dF119904 Shape factor 1m2119862119905 Total compressibility 1MPa119872119905 Total mobility 1MPa120582 Interporosity flow coefficient fraction120596 Storativity ratio fractionS Skin factor dimensionlessC Wellbore storage coefficient m3MPa

Subscripts

f Fracturem Matrixo Oilw Watert Totale ExternalD Dimensionless variable

Superscripts

- Laplace-transformed variable Endpoint property997888rarr Vector symbol

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

The authors gratefully acknowledge the support of theNational Basic Research 973 Program of China (Grant No

Mathematical Problems in Engineering 11

2015CB250900) the National Natural Science Foundationof China (Grant No 51704237) and the Research Projectsof Shaanxi Provincial Education Department (Grant No13JS090)

References

[1] R L Perrine Analysis of Pressure-buildup Curves vol 56American Petroleum Institute 1956

[2] J C Martin ldquoSimplified Equations of Flow in Gas Drive Reser-voirs and the Theoretical Foundation of Multiphase PressureBuildup Analysesrdquo Society of Petroleum Engineers vol 216 pp321ndash323 1959

[3] W Hurst ldquoInterference between Oil Fieldrdquo Transactions of theMetallurgical Society of AIME vol 219 no 8 pp 175ndash192 1960

[4] MMortada ldquoOilfield Interference in Aquifers of Non-UniformPropertiesrdquo Journal of Petroleum Technology vol 12 no 12 pp55ndash57 2013

[5] T Loucks and E Guerrero ldquoPressure Drop in a CompositeReservoirrdquo SPE Journal vol 1 no 03 pp 170ndash176 2013

[6] H Bixel and H Van Poollen ldquoPressure Drawdown and Buildupin the Presence of Radial Discontinuitiesrdquo SPE Journal vol 7no 03 pp 301ndash309 2013

[7] H Ramey ldquoApproximate Solutions For Unsteady LiquidFlow InComposite Reservoirsrdquo Journal of Canadian Petroleum Technol-ogy vol 9 no 01 2013

[8] Q Deng R Nie Y Jia et al ldquoPressure transient behavior of afractured well in multi-region composite reservoirsrdquo Journal ofPetroleum Science and Engineering vol 158 pp 535ndash553 2017

[9] H Kazemi LMerrill and J Jargon ldquoProblems in Interpretationof Pressure Fall-OffTests in ReservoirsWithAndWithout FluidBanksrdquo Journal of PetroleumTechnology vol 24 no 09 pp 1147ndash1156 2013

[10] L Merrill H Kazemi and W B Gogarty ldquoPressure FalloffAnalysis in Reservoirs With Fluid Banksrdquo Journal of PetroleumTechnology vol 26 no 07 pp 809ndash818 2013

[11] A Satman M Eggenschwiler R Tang and R HJ ldquoAnanalytical study of transient flow in systemswith radial disconti-nuitiesrdquo in Proceedings of the SPE9399 presented at SPE AnnualTechnical Conference and Exhibition Dallas Texas TX USASeptember 1980

[12] A Al-Bemani and I Ershaghi ldquoTwo-Phase Flow InterporosityEffects on Pressure Transient Test Response in Naturally Frac-tured Reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Dallas Texas

[13] Z Chen W Yu X Liao X Zhao Y Chen and K SepehrnoorildquoA Two-Phase Flow Model for Fractured Horizontal Well withComplex Fracture Networks Transient Analysis in FlowbackPeriodrdquo in Proceedings of the SPE Liquids-Rich Basins Confer-ence - North America Midland Texas USA

[14] H G Weinstein ldquoCold Waterflooding a Warm Reservoirrdquo inProceedings of the Fall Meeting of the Society of PetroleumEngineers of AIME Houston Texas

[15] S E Buckley andM C Leverett ldquoMechanism of fluid displace-ment in sandsrdquoTransactions of the AIME vol 146 no 1 pp 107ndash116 2013

[16] A Sosa R Raghavan and T Limon ldquoEffect of Relative Perme-ability andMobility Ratio on Pressure Falloff Behaviorrdquo Journalof Petroleum Technology vol 33 no 06 pp 1125ndash1135 2013

[17] P Hazebroek H Rainbow and CS Matthews ldquoPressure Fall-Off in Water Injection Wellsrdquo Transactions of the MetallurgicalSociety of AIME vol 213 pp 250ndash260 1958

[18] R Carter ldquoPressure Behavior of a Limited Circular CompositeReservoirrdquo SPE Journal vol 6 no 04 pp 328ndash334 2013

[19] A Odeh ldquoFlowTest Analysis for aWell with Radial Discontinu-ityrdquo Journal of Petroleum Technology vol 21 no 02 pp 207ndash2102013

[20] M Abbaszadeh and M Kamal ldquoPressure-Transient Testing ofWater-InjectionWellsrdquo SPE Reservoir Engineering vol 4 no 01pp 115ndash124 2013

[21] N Yeh and R Agarwal ldquoPressure Transient Analysis of Injec-tionWells in ReservoirsWithMultiple Fluid Banksrdquo in Proceed-ings of the SPE Annual Technical Conference and Exhibition SanAntonio Texas

[22] M M Levitan ldquoApplication of water injectionfalloff tests forreservoir appraisal New analytical solution method for two-phase variable rate problemsrdquo SPE Journal vol 8 no 4 pp 341ndash349 2003

[23] T Nanba and R Horne ldquoEstimation of Water and Oil RelativePermeabilities From Pressure Transient Analysis of WaterInjectionWell Datardquo in Proceedings of the SPE Annual TechnicalConference and Exhibition San Antonio Texas

[24] R B Bratvold andRNHorne ldquoAnalysis of pressure-falloff testsfollowing cold-water injectionrdquo SPE Formation Evaluation vol5 no 3 pp 293ndash302 1990

[25] S Chen G Li and A C Reynolds ldquoAnalytical Solution forInjection-Falloff-Production Testrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition San AntonioTexas USA

[26] A A Boughrara A M M Peres S Chen A A V Machadoand A C Reynolds ldquoApproximate analytical solutions for thepressure response at a water-injection wellrdquo SPE Journal vol12 no 1 pp 19ndash34 2007

[27] S Zheng and w xu ldquoNew approaches for analyzing transientpressure from oil and water two-phase flowing reservoirrdquo inProceedings of the Kuwait International Petroleum Conferenceand Exhibition Kuwait City Kuwait

[28] N A El-Khatib ldquoTransient Pressure Behavior of CompositeReservoirs with Moving Boundariesrdquo in Proceedings of theMiddle East Oil Show and Conference Bahrain

[29] A Dastan M M Kamal Y Hwang F Suleen and S MorsyldquoFalloff Testing Under Multiphase Flow Conditions in Natu-rally Fractured Reservoirsrdquo in Proceedings of the SPE WesternRegional Meeting Garden Grove California USA

[30] C n Machado and A C Reynolds ldquoApproximate semi-analytical solution for injection-falloff-production well test ananalytical tool for the in situ estimation of relative permeabilitycurvesrdquo Transport in Porous Media vol 121 no 1 pp 207ndash2312018

[31] T Barkve ldquoAnalytical study of reservoir pressure during water-injection well testsrdquo Society of Petroleum Engineers of AIME(Paper) SPE 1986

[32] T S Ramakrishnan and F J Kuchuk ldquoTesting injection wellswith rate and pressure datardquo SPE Formation Evaluation vol 9no 3 pp 228ndash236 1994

[33] L G Thompson and A C Reynolds ldquoWell testing for radiallyheterogeneous reservoirs under single and multiphase flowconditionsrdquo SPE Formation Evaluation vol 12 no 1 pp 57ndash641997

12 Mathematical Problems in Engineering

[34] JS Aronofsky L Masse and SG Natanson ldquoA Model for theMechanism of Oil Recovery from the Porous Matrix Due toWater Invasion in Fractured Reservoirsrdquo Trans AIME vol 213pp 17ndash19 1958

[35] A de Swaan ldquoTheory of Waterflooding in Fractured Reser-voirsrdquo SPE Journal vol 18 no 02 pp 117ndash122 2013

[36] H Kazemi J Gilman and A Elsharkawy ldquoAnalytical andNumerical Solution of Oil Recovery From Fractured ReservoirsWith Empirical Transfer Functions (includes associated papers25528 and 25818)rdquo SPE Reservoir Engineering vol 7 no 02 pp219ndash227 2013

[37] T Shimamoto D Kuramoto N Arihara and T Onishi ldquoA NewPressure Transient Analysis Model for Water Injection Wellin Dual-Porosity Reservoirsrdquo in Proceedings of the SPE100881presented at SPE Asia Pacific Oil ampamp Gas Conference andExhibition Adelaide Australia 2006

[38] G J Moridis D A McVay D L Reddell and T A BlasingameldquoLaplace transform finite difference (LTFD) numerical methodfor the simulation of compressible liquid flow in reservoirsrdquo SPEAdvanced Technology Series vol 2 no 2 pp 122ndash131 1994

[39] A D Habte and M Onur ldquoLaplace-transform finite-differenceand quasistationary solution method for water-injectionfallofftestsrdquo SPE Journal vol 19 no 3 pp 398ndash409 2014

[40] J Warren and P Root ldquoThe behavior of naturally fracturedreservoirsrdquo SPE Journal vol 3 no 3 pp 245ndash255 2013

[41] D Bourdet J Ayoub and Y Pirard ldquoUse of Pressure Derivativein Well Test Interpretationrdquo SPE Formation Evaluation vol 4no 02 pp 293ndash302 2013

[42] M Mavor and H Cinco-Ley ldquoTransient Pressure BehaviorOf Naturally Fractured Reservoirsrdquo in Proceedings of the SPECalifornia Regional Meeting Ventura California

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

10 Mathematical Problems in Engineering

tD

Rc=0Rc=0Rc=001

Rc=001Rc=005Rc=005

Rc=01Rc=01Rc=05

Rc=05Rc=1Rc=1

102

104

106

108

10minus2

10minus1

100

101

102

PwD

dPw

D

Figure 11 Pressure behavior of different imbibition rate coefficientswith constant pressure boundary

5 Conclusions

In this work a semianalytical two-phase flow model consid-ering water imbibition and the saturation gradients withineach region during injection period was proposed to analyzethe pressure transient behavior of the water injection well infractured reservoirs Validation of the presented model wasperformed by analytical method and a field water injectiontest The model features were investigated by sensitivityanalysis Some conclusions were drawn as follows

(1) The shape of the pressure curves for the proposedmodel is similar to the single phase flowmodel whilethe position of the curves for the two-phase flowmodel is on the upper right of the single phase flowmodel

(2) Themain influence of skin factor on the curves occurson the peak value of the pressure derivatives As theskin factor becomes larger the peak value differencebetween single phase and two-phase pressure curvesdecreases and the inclination of the curves after thepeak value increases

(3) The wellbore storage coefficient mainly has an effecton the straight line of the early period for the pressurecurves The shape factor mainly has an effect on theposition of the ldquodiprdquo of the pressure derivatives whichis the same as the interporosity parameter

(4) The imbibition rate coefficient mainly influences thewhole system radial flow period of the pressurecurves The pressure derivative curves dip downwardearlier and drop down to a greater degree with theincrease of imbibition rate coefficient

Nomenclature

R Recovery fractionR119888 Imbibition rate coefficient 1day119877infin Ultimate cumulative oil recovery fraction0 Porosity fraction119904119900119903 Residual oil saturation fraction119904119908119888 Irreducible water saturation fraction119904119908 Water saturation fractionz Logarithmic transform variable dimensionlessu Laplace variable dimensionlessq Displacement rate m3dt Time hk Permeability 1205831198982119896119903 Relative permeability fractionh Formation height mr Radial distance m119903119908 Wellbore radius mp Pressure MPa120583 Viscosity mPasdotsB Formation volume factor dimensionless120591 Fracture matrix transfer term m3dF119904 Shape factor 1m2119862119905 Total compressibility 1MPa119872119905 Total mobility 1MPa120582 Interporosity flow coefficient fraction120596 Storativity ratio fractionS Skin factor dimensionlessC Wellbore storage coefficient m3MPa

Subscripts

f Fracturem Matrixo Oilw Watert Totale ExternalD Dimensionless variable

Superscripts

- Laplace-transformed variable Endpoint property997888rarr Vector symbol

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

The authors gratefully acknowledge the support of theNational Basic Research 973 Program of China (Grant No

Mathematical Problems in Engineering 11

2015CB250900) the National Natural Science Foundationof China (Grant No 51704237) and the Research Projectsof Shaanxi Provincial Education Department (Grant No13JS090)

References

[1] R L Perrine Analysis of Pressure-buildup Curves vol 56American Petroleum Institute 1956

[2] J C Martin ldquoSimplified Equations of Flow in Gas Drive Reser-voirs and the Theoretical Foundation of Multiphase PressureBuildup Analysesrdquo Society of Petroleum Engineers vol 216 pp321ndash323 1959

[3] W Hurst ldquoInterference between Oil Fieldrdquo Transactions of theMetallurgical Society of AIME vol 219 no 8 pp 175ndash192 1960

[4] MMortada ldquoOilfield Interference in Aquifers of Non-UniformPropertiesrdquo Journal of Petroleum Technology vol 12 no 12 pp55ndash57 2013

[5] T Loucks and E Guerrero ldquoPressure Drop in a CompositeReservoirrdquo SPE Journal vol 1 no 03 pp 170ndash176 2013

[6] H Bixel and H Van Poollen ldquoPressure Drawdown and Buildupin the Presence of Radial Discontinuitiesrdquo SPE Journal vol 7no 03 pp 301ndash309 2013

[7] H Ramey ldquoApproximate Solutions For Unsteady LiquidFlow InComposite Reservoirsrdquo Journal of Canadian Petroleum Technol-ogy vol 9 no 01 2013

[8] Q Deng R Nie Y Jia et al ldquoPressure transient behavior of afractured well in multi-region composite reservoirsrdquo Journal ofPetroleum Science and Engineering vol 158 pp 535ndash553 2017

[9] H Kazemi LMerrill and J Jargon ldquoProblems in Interpretationof Pressure Fall-OffTests in ReservoirsWithAndWithout FluidBanksrdquo Journal of PetroleumTechnology vol 24 no 09 pp 1147ndash1156 2013

[10] L Merrill H Kazemi and W B Gogarty ldquoPressure FalloffAnalysis in Reservoirs With Fluid Banksrdquo Journal of PetroleumTechnology vol 26 no 07 pp 809ndash818 2013

[11] A Satman M Eggenschwiler R Tang and R HJ ldquoAnanalytical study of transient flow in systemswith radial disconti-nuitiesrdquo in Proceedings of the SPE9399 presented at SPE AnnualTechnical Conference and Exhibition Dallas Texas TX USASeptember 1980

[12] A Al-Bemani and I Ershaghi ldquoTwo-Phase Flow InterporosityEffects on Pressure Transient Test Response in Naturally Frac-tured Reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Dallas Texas

[13] Z Chen W Yu X Liao X Zhao Y Chen and K SepehrnoorildquoA Two-Phase Flow Model for Fractured Horizontal Well withComplex Fracture Networks Transient Analysis in FlowbackPeriodrdquo in Proceedings of the SPE Liquids-Rich Basins Confer-ence - North America Midland Texas USA

[14] H G Weinstein ldquoCold Waterflooding a Warm Reservoirrdquo inProceedings of the Fall Meeting of the Society of PetroleumEngineers of AIME Houston Texas

[15] S E Buckley andM C Leverett ldquoMechanism of fluid displace-ment in sandsrdquoTransactions of the AIME vol 146 no 1 pp 107ndash116 2013

[16] A Sosa R Raghavan and T Limon ldquoEffect of Relative Perme-ability andMobility Ratio on Pressure Falloff Behaviorrdquo Journalof Petroleum Technology vol 33 no 06 pp 1125ndash1135 2013

[17] P Hazebroek H Rainbow and CS Matthews ldquoPressure Fall-Off in Water Injection Wellsrdquo Transactions of the MetallurgicalSociety of AIME vol 213 pp 250ndash260 1958

[18] R Carter ldquoPressure Behavior of a Limited Circular CompositeReservoirrdquo SPE Journal vol 6 no 04 pp 328ndash334 2013

[19] A Odeh ldquoFlowTest Analysis for aWell with Radial Discontinu-ityrdquo Journal of Petroleum Technology vol 21 no 02 pp 207ndash2102013

[20] M Abbaszadeh and M Kamal ldquoPressure-Transient Testing ofWater-InjectionWellsrdquo SPE Reservoir Engineering vol 4 no 01pp 115ndash124 2013

[21] N Yeh and R Agarwal ldquoPressure Transient Analysis of Injec-tionWells in ReservoirsWithMultiple Fluid Banksrdquo in Proceed-ings of the SPE Annual Technical Conference and Exhibition SanAntonio Texas

[22] M M Levitan ldquoApplication of water injectionfalloff tests forreservoir appraisal New analytical solution method for two-phase variable rate problemsrdquo SPE Journal vol 8 no 4 pp 341ndash349 2003

[23] T Nanba and R Horne ldquoEstimation of Water and Oil RelativePermeabilities From Pressure Transient Analysis of WaterInjectionWell Datardquo in Proceedings of the SPE Annual TechnicalConference and Exhibition San Antonio Texas

[24] R B Bratvold andRNHorne ldquoAnalysis of pressure-falloff testsfollowing cold-water injectionrdquo SPE Formation Evaluation vol5 no 3 pp 293ndash302 1990

[25] S Chen G Li and A C Reynolds ldquoAnalytical Solution forInjection-Falloff-Production Testrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition San AntonioTexas USA

[26] A A Boughrara A M M Peres S Chen A A V Machadoand A C Reynolds ldquoApproximate analytical solutions for thepressure response at a water-injection wellrdquo SPE Journal vol12 no 1 pp 19ndash34 2007

[27] S Zheng and w xu ldquoNew approaches for analyzing transientpressure from oil and water two-phase flowing reservoirrdquo inProceedings of the Kuwait International Petroleum Conferenceand Exhibition Kuwait City Kuwait

[28] N A El-Khatib ldquoTransient Pressure Behavior of CompositeReservoirs with Moving Boundariesrdquo in Proceedings of theMiddle East Oil Show and Conference Bahrain

[29] A Dastan M M Kamal Y Hwang F Suleen and S MorsyldquoFalloff Testing Under Multiphase Flow Conditions in Natu-rally Fractured Reservoirsrdquo in Proceedings of the SPE WesternRegional Meeting Garden Grove California USA

[30] C n Machado and A C Reynolds ldquoApproximate semi-analytical solution for injection-falloff-production well test ananalytical tool for the in situ estimation of relative permeabilitycurvesrdquo Transport in Porous Media vol 121 no 1 pp 207ndash2312018

[31] T Barkve ldquoAnalytical study of reservoir pressure during water-injection well testsrdquo Society of Petroleum Engineers of AIME(Paper) SPE 1986

[32] T S Ramakrishnan and F J Kuchuk ldquoTesting injection wellswith rate and pressure datardquo SPE Formation Evaluation vol 9no 3 pp 228ndash236 1994

[33] L G Thompson and A C Reynolds ldquoWell testing for radiallyheterogeneous reservoirs under single and multiphase flowconditionsrdquo SPE Formation Evaluation vol 12 no 1 pp 57ndash641997

12 Mathematical Problems in Engineering

[34] JS Aronofsky L Masse and SG Natanson ldquoA Model for theMechanism of Oil Recovery from the Porous Matrix Due toWater Invasion in Fractured Reservoirsrdquo Trans AIME vol 213pp 17ndash19 1958

[35] A de Swaan ldquoTheory of Waterflooding in Fractured Reser-voirsrdquo SPE Journal vol 18 no 02 pp 117ndash122 2013

[36] H Kazemi J Gilman and A Elsharkawy ldquoAnalytical andNumerical Solution of Oil Recovery From Fractured ReservoirsWith Empirical Transfer Functions (includes associated papers25528 and 25818)rdquo SPE Reservoir Engineering vol 7 no 02 pp219ndash227 2013

[37] T Shimamoto D Kuramoto N Arihara and T Onishi ldquoA NewPressure Transient Analysis Model for Water Injection Wellin Dual-Porosity Reservoirsrdquo in Proceedings of the SPE100881presented at SPE Asia Pacific Oil ampamp Gas Conference andExhibition Adelaide Australia 2006

[38] G J Moridis D A McVay D L Reddell and T A BlasingameldquoLaplace transform finite difference (LTFD) numerical methodfor the simulation of compressible liquid flow in reservoirsrdquo SPEAdvanced Technology Series vol 2 no 2 pp 122ndash131 1994

[39] A D Habte and M Onur ldquoLaplace-transform finite-differenceand quasistationary solution method for water-injectionfallofftestsrdquo SPE Journal vol 19 no 3 pp 398ndash409 2014

[40] J Warren and P Root ldquoThe behavior of naturally fracturedreservoirsrdquo SPE Journal vol 3 no 3 pp 245ndash255 2013

[41] D Bourdet J Ayoub and Y Pirard ldquoUse of Pressure Derivativein Well Test Interpretationrdquo SPE Formation Evaluation vol 4no 02 pp 293ndash302 2013

[42] M Mavor and H Cinco-Ley ldquoTransient Pressure BehaviorOf Naturally Fractured Reservoirsrdquo in Proceedings of the SPECalifornia Regional Meeting Ventura California

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

Mathematical Problems in Engineering 11

2015CB250900) the National Natural Science Foundationof China (Grant No 51704237) and the Research Projectsof Shaanxi Provincial Education Department (Grant No13JS090)

References

[1] R L Perrine Analysis of Pressure-buildup Curves vol 56American Petroleum Institute 1956

[2] J C Martin ldquoSimplified Equations of Flow in Gas Drive Reser-voirs and the Theoretical Foundation of Multiphase PressureBuildup Analysesrdquo Society of Petroleum Engineers vol 216 pp321ndash323 1959

[3] W Hurst ldquoInterference between Oil Fieldrdquo Transactions of theMetallurgical Society of AIME vol 219 no 8 pp 175ndash192 1960

[4] MMortada ldquoOilfield Interference in Aquifers of Non-UniformPropertiesrdquo Journal of Petroleum Technology vol 12 no 12 pp55ndash57 2013

[5] T Loucks and E Guerrero ldquoPressure Drop in a CompositeReservoirrdquo SPE Journal vol 1 no 03 pp 170ndash176 2013

[6] H Bixel and H Van Poollen ldquoPressure Drawdown and Buildupin the Presence of Radial Discontinuitiesrdquo SPE Journal vol 7no 03 pp 301ndash309 2013

[7] H Ramey ldquoApproximate Solutions For Unsteady LiquidFlow InComposite Reservoirsrdquo Journal of Canadian Petroleum Technol-ogy vol 9 no 01 2013

[8] Q Deng R Nie Y Jia et al ldquoPressure transient behavior of afractured well in multi-region composite reservoirsrdquo Journal ofPetroleum Science and Engineering vol 158 pp 535ndash553 2017

[9] H Kazemi LMerrill and J Jargon ldquoProblems in Interpretationof Pressure Fall-OffTests in ReservoirsWithAndWithout FluidBanksrdquo Journal of PetroleumTechnology vol 24 no 09 pp 1147ndash1156 2013

[10] L Merrill H Kazemi and W B Gogarty ldquoPressure FalloffAnalysis in Reservoirs With Fluid Banksrdquo Journal of PetroleumTechnology vol 26 no 07 pp 809ndash818 2013

[11] A Satman M Eggenschwiler R Tang and R HJ ldquoAnanalytical study of transient flow in systemswith radial disconti-nuitiesrdquo in Proceedings of the SPE9399 presented at SPE AnnualTechnical Conference and Exhibition Dallas Texas TX USASeptember 1980

[12] A Al-Bemani and I Ershaghi ldquoTwo-Phase Flow InterporosityEffects on Pressure Transient Test Response in Naturally Frac-tured Reservoirsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Dallas Texas

[13] Z Chen W Yu X Liao X Zhao Y Chen and K SepehrnoorildquoA Two-Phase Flow Model for Fractured Horizontal Well withComplex Fracture Networks Transient Analysis in FlowbackPeriodrdquo in Proceedings of the SPE Liquids-Rich Basins Confer-ence - North America Midland Texas USA

[14] H G Weinstein ldquoCold Waterflooding a Warm Reservoirrdquo inProceedings of the Fall Meeting of the Society of PetroleumEngineers of AIME Houston Texas

[15] S E Buckley andM C Leverett ldquoMechanism of fluid displace-ment in sandsrdquoTransactions of the AIME vol 146 no 1 pp 107ndash116 2013

[16] A Sosa R Raghavan and T Limon ldquoEffect of Relative Perme-ability andMobility Ratio on Pressure Falloff Behaviorrdquo Journalof Petroleum Technology vol 33 no 06 pp 1125ndash1135 2013

[17] P Hazebroek H Rainbow and CS Matthews ldquoPressure Fall-Off in Water Injection Wellsrdquo Transactions of the MetallurgicalSociety of AIME vol 213 pp 250ndash260 1958

[18] R Carter ldquoPressure Behavior of a Limited Circular CompositeReservoirrdquo SPE Journal vol 6 no 04 pp 328ndash334 2013

[19] A Odeh ldquoFlowTest Analysis for aWell with Radial Discontinu-ityrdquo Journal of Petroleum Technology vol 21 no 02 pp 207ndash2102013

[20] M Abbaszadeh and M Kamal ldquoPressure-Transient Testing ofWater-InjectionWellsrdquo SPE Reservoir Engineering vol 4 no 01pp 115ndash124 2013

[21] N Yeh and R Agarwal ldquoPressure Transient Analysis of Injec-tionWells in ReservoirsWithMultiple Fluid Banksrdquo in Proceed-ings of the SPE Annual Technical Conference and Exhibition SanAntonio Texas

[22] M M Levitan ldquoApplication of water injectionfalloff tests forreservoir appraisal New analytical solution method for two-phase variable rate problemsrdquo SPE Journal vol 8 no 4 pp 341ndash349 2003

[23] T Nanba and R Horne ldquoEstimation of Water and Oil RelativePermeabilities From Pressure Transient Analysis of WaterInjectionWell Datardquo in Proceedings of the SPE Annual TechnicalConference and Exhibition San Antonio Texas

[24] R B Bratvold andRNHorne ldquoAnalysis of pressure-falloff testsfollowing cold-water injectionrdquo SPE Formation Evaluation vol5 no 3 pp 293ndash302 1990

[25] S Chen G Li and A C Reynolds ldquoAnalytical Solution forInjection-Falloff-Production Testrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition San AntonioTexas USA

[26] A A Boughrara A M M Peres S Chen A A V Machadoand A C Reynolds ldquoApproximate analytical solutions for thepressure response at a water-injection wellrdquo SPE Journal vol12 no 1 pp 19ndash34 2007

[27] S Zheng and w xu ldquoNew approaches for analyzing transientpressure from oil and water two-phase flowing reservoirrdquo inProceedings of the Kuwait International Petroleum Conferenceand Exhibition Kuwait City Kuwait

[28] N A El-Khatib ldquoTransient Pressure Behavior of CompositeReservoirs with Moving Boundariesrdquo in Proceedings of theMiddle East Oil Show and Conference Bahrain

[29] A Dastan M M Kamal Y Hwang F Suleen and S MorsyldquoFalloff Testing Under Multiphase Flow Conditions in Natu-rally Fractured Reservoirsrdquo in Proceedings of the SPE WesternRegional Meeting Garden Grove California USA

[30] C n Machado and A C Reynolds ldquoApproximate semi-analytical solution for injection-falloff-production well test ananalytical tool for the in situ estimation of relative permeabilitycurvesrdquo Transport in Porous Media vol 121 no 1 pp 207ndash2312018

[31] T Barkve ldquoAnalytical study of reservoir pressure during water-injection well testsrdquo Society of Petroleum Engineers of AIME(Paper) SPE 1986

[32] T S Ramakrishnan and F J Kuchuk ldquoTesting injection wellswith rate and pressure datardquo SPE Formation Evaluation vol 9no 3 pp 228ndash236 1994

[33] L G Thompson and A C Reynolds ldquoWell testing for radiallyheterogeneous reservoirs under single and multiphase flowconditionsrdquo SPE Formation Evaluation vol 12 no 1 pp 57ndash641997

12 Mathematical Problems in Engineering

[34] JS Aronofsky L Masse and SG Natanson ldquoA Model for theMechanism of Oil Recovery from the Porous Matrix Due toWater Invasion in Fractured Reservoirsrdquo Trans AIME vol 213pp 17ndash19 1958

[35] A de Swaan ldquoTheory of Waterflooding in Fractured Reser-voirsrdquo SPE Journal vol 18 no 02 pp 117ndash122 2013

[36] H Kazemi J Gilman and A Elsharkawy ldquoAnalytical andNumerical Solution of Oil Recovery From Fractured ReservoirsWith Empirical Transfer Functions (includes associated papers25528 and 25818)rdquo SPE Reservoir Engineering vol 7 no 02 pp219ndash227 2013

[37] T Shimamoto D Kuramoto N Arihara and T Onishi ldquoA NewPressure Transient Analysis Model for Water Injection Wellin Dual-Porosity Reservoirsrdquo in Proceedings of the SPE100881presented at SPE Asia Pacific Oil ampamp Gas Conference andExhibition Adelaide Australia 2006

[38] G J Moridis D A McVay D L Reddell and T A BlasingameldquoLaplace transform finite difference (LTFD) numerical methodfor the simulation of compressible liquid flow in reservoirsrdquo SPEAdvanced Technology Series vol 2 no 2 pp 122ndash131 1994

[39] A D Habte and M Onur ldquoLaplace-transform finite-differenceand quasistationary solution method for water-injectionfallofftestsrdquo SPE Journal vol 19 no 3 pp 398ndash409 2014

[40] J Warren and P Root ldquoThe behavior of naturally fracturedreservoirsrdquo SPE Journal vol 3 no 3 pp 245ndash255 2013

[41] D Bourdet J Ayoub and Y Pirard ldquoUse of Pressure Derivativein Well Test Interpretationrdquo SPE Formation Evaluation vol 4no 02 pp 293ndash302 2013

[42] M Mavor and H Cinco-Ley ldquoTransient Pressure BehaviorOf Naturally Fractured Reservoirsrdquo in Proceedings of the SPECalifornia Regional Meeting Ventura California

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

12 Mathematical Problems in Engineering

[34] JS Aronofsky L Masse and SG Natanson ldquoA Model for theMechanism of Oil Recovery from the Porous Matrix Due toWater Invasion in Fractured Reservoirsrdquo Trans AIME vol 213pp 17ndash19 1958

[35] A de Swaan ldquoTheory of Waterflooding in Fractured Reser-voirsrdquo SPE Journal vol 18 no 02 pp 117ndash122 2013

[36] H Kazemi J Gilman and A Elsharkawy ldquoAnalytical andNumerical Solution of Oil Recovery From Fractured ReservoirsWith Empirical Transfer Functions (includes associated papers25528 and 25818)rdquo SPE Reservoir Engineering vol 7 no 02 pp219ndash227 2013

[37] T Shimamoto D Kuramoto N Arihara and T Onishi ldquoA NewPressure Transient Analysis Model for Water Injection Wellin Dual-Porosity Reservoirsrdquo in Proceedings of the SPE100881presented at SPE Asia Pacific Oil ampamp Gas Conference andExhibition Adelaide Australia 2006

[38] G J Moridis D A McVay D L Reddell and T A BlasingameldquoLaplace transform finite difference (LTFD) numerical methodfor the simulation of compressible liquid flow in reservoirsrdquo SPEAdvanced Technology Series vol 2 no 2 pp 122ndash131 1994

[39] A D Habte and M Onur ldquoLaplace-transform finite-differenceand quasistationary solution method for water-injectionfallofftestsrdquo SPE Journal vol 19 no 3 pp 398ndash409 2014

[40] J Warren and P Root ldquoThe behavior of naturally fracturedreservoirsrdquo SPE Journal vol 3 no 3 pp 245ndash255 2013

[41] D Bourdet J Ayoub and Y Pirard ldquoUse of Pressure Derivativein Well Test Interpretationrdquo SPE Formation Evaluation vol 4no 02 pp 293ndash302 2013

[42] M Mavor and H Cinco-Ley ldquoTransient Pressure BehaviorOf Naturally Fractured Reservoirsrdquo in Proceedings of the SPECalifornia Regional Meeting Ventura California

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

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