# A Two-Phase Flow Model for Pressure Transient Analysis of Water Injection Well considering Water Imbibition in Natural Fractured Reservoirs

Authors: Mengmeng Li; Qi Li; Gang Bi; Jiaen Lin
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

The pressure injection falloff test for water injection well has the advantages of briefness and convenience, with no effect on the oil production. It has been widely used in the oil field. Tremendous attention has been focused on oil-water two-phase flow model based on the Perrine-Martin theory. However, the saturation gradient is not considered in the Perrine-Martin method, which may result in errors in computation. Moreover, water imbibition is important for water flooding in natural fractured reservoirs, while 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. The parameters in this model, including total compressibility coefficient, interporosity flow coefficient, and total mobility, change with water saturation. The model was solved by Laplace transform finite-difference (LTFD) method coupled with the quasi-stationary method. Based on the solution, the model was verified by the analytical method and a field water injection test. The features of typical curves and the influences of the parameters on the typical curves were analyzed. Results show that the shape of pressure curves for single phase flow resembles two-phase flow, but the position of the two-phase flow curves is on the upper right of the single phase flow curves. The skin factor and wellbore storage coefficient mainly influence the peak value of the pressure derivatives and the straight line of the early period. The shape factor has a major effect on the position of the “dip” of pressure derivatives. The imbibition rate coefficient mainly influences the whole system radial flow period of the curves. This work provides valuable information in the design and evaluation of stimulation treatments in natural fractured reservoirs.


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## Body

## 1. Introduction

The methods for pressure transient analysis of water injection well mostly still use the conventional well test method of single phase flow. Perrine [1] proposed to apply the single phase theory into the two-phase or multiphase well test theory by substitution of the single phase compressibility and mobility into the sum of total mobility and total compressibility of the multiphase system. Martin [2] provided the theoretical validation to Perrine’s theory and indicated that the saturation gradient was neglected for Perrine’s method. The total mobility, skin factor, and average formation pressure can be obtained based on this method. If it is a two-phase flow well test model of a water injection well, a multizone composite reservoir model [3–8] is used due to the different properties of the injected fluid and the reservoir fluid. Within each region, the properties of the reservoir and fluid are constant, but it may be different for the two or three regions in a composite reservoir. The two or three zones for a composite reservoir were divided according to the formation and fluid properties. Some scholars [9–13] developed a two- or three-zone composite reservoir well test model by finite-difference or Laplace transformation method to get the pressure curves for water injection and pressure falloff period. However, these models assumed no saturation gradients within oil and water transition zone and the zone boundaries were virtually stationary before the well shut-in.Actually, there were saturation discontinuities between the two or three fluid banks in a composite reservoir. A saturation gradient was formed after the water injection due to the differences in oil and water properties. Weinstein [14] investigated the pressure-falloff data with a numerical model and suggested that the location of the fluid banks could be determined based on the Buckley-Leverett [15] frontal-advance equations. Sosa et al. examined the influence of saturation gradients on water injection pressure-falloff tests according to a two-phase radial numerical simulator [16]. It was further demonstrated that the radius of the fluid banks that was flooded completely by water should be estimated from the Buckley-Leverett theory rather than the technique mentioned in the previous literatures [17–19].According to the Buckley-Leverett theory, the location of the front was estimated at any time during the injection period. The fluid bank can be discretized into a series of banks, so the water saturation distribution and the gradual change in fluid properties due to saturation gradients in the reservoir were obtained [20–24]. Chen [25] derived an approximate analytical solution for the pressure response during water injection/falloff test based on the front tracking method and Buckley-Leverett theory. Boughrara [26] added a two-phase term, which represents the existence of the two-phase zone and the movement of the water front, to the analytical single phase well test solution based on Buckley-Leverett equations for vertical and horizontal water injection wells. Zheng [27] developed a modified P-M approach for numerical well testing analysis of oil and water two-phase flowing reservoir on account of Buckley-Leverett equations. The water saturation distribution for a water injection well at any time can be obtained with the Buckley-Leverett frontal-advance equations [28–30].Based on the quasi-stationary method [31–33], the saturation was decoupled from the pressure with the Buckley-Leverett theory. Therefore, the water saturation can be considered as constants when the diffusivity equation was solved simultaneously for water injection well test analysis. On the basis of the Buckley-Leverett equations, De Swaan calculated the rate of water imbibition in a fracture surrounded by matrix in a convolution form [34, 35]. Kazemi [36] proposed an analytical solution for Buckley-Leverett equations in a fracture surrounded by matrix undergoing imbibition. By extending the linear Buckley-Leverett formula considering imbibition to a radial flow system for the naturally fractured reservoir [37], the water saturation can be estimated according to the radial flow model.In this work, an oil-water two-phase flow imbibition well test model of a water injection well in natural fractured reservoirs was developed and solved by the LTFD method coupled with the quasi-stationary method. The LTFD method transformed the equations into Laplace domain and eliminated the need for time discretization; thus, the method was semianalytical in time and the stability and convergence problems in temporal domain were avoided [38, 39]. According to analysis of the pressure falloff data, the features of the typical curves of oil-water two-phase flow considering water imbibition were analyzed. The effect of the imbibition rate coefficient, skin factor, wellbore storage coefficient, shape factor, and boundary conditions on the typical curves was also investigated. New ideas had been put forward in accordance with the analysis above.


## 2. Model Description

In order to calculate the saturation and pressure of the diffusivity equation in the two-phase pressure transient model simultaneously, the quasi-stationary method was adopted by decoupling the saturation from the pressure on account of the Buckley-Leverett water displacement theory.


### 2.1. Water Saturation Model

The water saturation distribution in the fracture and matrix considering water imbibition was estimated by integrating empirical matrix fracture transfer functions into the Buckley-Leverett equation. Based on the assumption that the cumulative oil recovery from a piece of rock surrounded by water is a continuous monotonic function of time and converges to a finite limit, Arnofsky et al. [34] proposed an exponential equation for oil recovery estimation of water displacement considering water imbibition in a fractured reservoir as shown:(1)R=R∞1-e-Rctwhere(2)R∞=∅m1-sorm-swcm(3)R=∅msm-swcmConsidering the gradual variation of the water saturation in surrounding fractures, De Swaan [35] calculated the rate of water imbibition in a fracture surrounded by matrix in a convolution form. Kazemi [36] proposed an analytical solution for a linear flow system that accounts for saturation changes in fracture and matrix. Shimamoto [37] extended the formula to a radial flow system for the Warrant-Root model with the volume conservation in a fracture.The water saturation change in the fracture can be expressed as(4)-q2πrh∂swf∂r-RcR∞∫0te-Rct-τ∂swf∂τdτ=∅f∂swf∂tAnd the water saturation in the matrix can be written as(5)RcR∞∫0te-Rct-τ∂swf∂τdτ=∅m∂swm∂tThe initial and boundary condition(6)swfr=0,t=1(7)swfr,t=0=0Introducing Laplace transform, the water saturation in Laplace domain can be expressed as(8)s-wf=1z1e-z1β/Rc+z1+z1α(9)s-wm=swmr,t=0z1+RcR∞∅me-z1β/Rc+z1+z1αz1Rc+z1where z1 is Laplace variable,(10)α=πhr2q∅f(11)β=πhr2qRcR∞The water saturation in fracture and matrix in the real domain can be obtained by inversion of Laplace transform(12)swfr,t=0t<αe-βe-Rct-αI02βRct-α+Rc∫αte-Rcτ-αI02βRcτ-αdτt≥α(13)swmr,t=swcmt<αswcm+1-sorm-swcmRce-β∫αte-Rcτ-αI02βRcτ-αdτt≥α


### 2.2. Pressure Transient Model

#### 2.2.1. Physical Model

The naturally fractured reservoir is described by the Warren-Root model with constant temperature and uniform initial pressure. The reservoir is assumed to be a homogeneous horizontal radial reservoir with an upper and lower sealed boundary. The slightly compressible fluids (water and oil) flowing in the matrix and fracture system obey Darcy’s law, and the mass exchange between matrix and fracture system is assumed to be pseudosteady [40]. The gravity effect of the 2D system is neglected. A fully penetrating water injection well with constant injection rate is located in the center of the reservoir. As the water injection into the formation, the water saturation changes with time and space. A moving interface divided the radial formation into two regions, as shown in Figure 1. From the figure, rf indicates the position of water drive front. Region 1 is the water invaded region and region 2 is the uninvaded region.Figure 1
Schematic of two-zone composite reservoir.


#### 2.2.2. Mathematical Model

After the water saturation distribution at any time during the injection period was obtained, the pressure equations can be solved by the quasi-stationary and LTFD method. According to the double porosity model proposed by Warren and Root, an oil-water two-phase flow well test model considering the saturation gradients within each region with a moving boundary between the regions for a water injection well was proposed as follows:Fracture system(14)1r∂∂rrkfkrofμoBo∂pf∂r-τomf=∂∂t∅fSofBo(15)1r∂∂rrkfkrwfμwBw∂pf∂r-τwmf=∂∂t∅fSwfBwMatrix system(16)τomf=∂∂t∅mSomBo(17)τwmf=∂∂t∅mSwmBwMatrix and fracture transfer equations(18)τomf=FskmkromμoBopf-pm(19)τwmf=FskmkrwmμwBwpf-pmSubstituting (16) and (17) into (14) and (15) and (18) and (19) yields(20)1r∂∂rrMtf∂pf∂r=∅fCtf∂pf∂t+∅mCtm∂pm∂t(21)∅mCtm∂pm∂t=FsMtupf-pmwhere Mtf denotes the total mobility of fracture system and Mtu is the mobility between fracture and matrix. The subscript “u” represents “upstream saturation”. For water injection period, the water flows from fracture to matrix, so water saturation in the fracture is used for the relative permeability evaluation (see (23)). Inversely, water saturation in the matrix is applied to calculate the relative permeability during the falloff period (see (24)).(22)Mtf=kfkrofSwfμo+krwfSwfμw(23)Mtu=kmkromSwfμo+krwmSwfμw(24)Mtu=kmkromSwmμo+krwmSwmμwIn order to simplify the calculation, a set of dimensionless variables were introduced as listed in Table1. Equations (20) and (21) and the initial and boundary conditions can be written in dimensionless form(25)1rD∂∂rDrDMtfD∂pfD∂rD=ω1∂pfD∂tD+ω2∂pmD∂tD(26)ω2∂pmD∂tD=λpfD-pmDTable 1
Dimensionless variable for pressure calculation.
M t f ^ = k f k r o f S w f = 1 μ o + k r w f S w f = 1 μ w M t f D = M t f M t f ^ p f D = 2 π M t f ^ h q B w p f - p i λ = F S r w 2 M t u M t f ^ p m D = 2 π M t f ^ h q B w p m - p i r D = r r w C t f ^ = C r + C w S w i + C o 1 - S w i ω 1 = ∅ f C t f ∅ f C t f ^ + ∅ m C t m ^ C t m ^ = C r + C w S w c m + C o 1 - S w c m ω 2 = ∅ m C t m ∅ f C t f ^ + ∅ m C t m ^ t D = M t f ^ ∅ f C t f ^ + ∅ m C t m ^ r w 2 C D = C 2 π ∅ f C t f ^ + ∅ m C t m ^ h r w 2The initial condition(27)pfDrD,tD=0=0(28)pmDrD,tD=0=0The inner boundary condition considering the skin effect is(29)pwD=pfD-SrD∂pfD∂rDrD=1The inner boundary condition considering wellbore storage effect is(30)CDdpwDdtD-rD∂pfD∂rDrD=1=1The constant pressure outer boundary condition is(31)pfDrD=reD,tD=0(32)pmDrD=reD,tD=0The no flow outer boundary condition is(33)∂pfD∂rDrD=reD=0(34)∂pmD∂rDrD=reD=0To simplify the computation, a logarithmic transform,z=ln⁡(rD), and the Laplace transform with the quasi-stationary assumption were used. Thus, (25)–(34) can be rewritten as(35)1e2zddzMtfDdp-fDdz=ω1up-fD-pfDz,0+ω2up-mD-pmDz,0(36)ω2up-mD-pmDz,0=λp-fD-p-mDThe initial condition is(37)pfDtD=0=0(38)pmDtD=0=0The inner boundary condition considering the skin effect is(39)p-wD=p-fD-Sdp-fDdzz=0The inner boundary condition considering wellbore storage effect is(40)dp-fDdzz=0-CDup-wD-pwDtD=0=-1uThe constant pressure outer boundary condition is(41)p-fDz=ze=0(42)p-mDz=ze=0The no flow outer boundary condition is(43)dp-fDdzz=ze=0(44)dp-mDdzz=ze=0The pressure change near the wellbore was relatively larger than the area away from the bottom of the well. So the formation was divided into n grid blocks in radial direction according to the point-centered logarithmic gridding method, which was shown as the dotted blue line in Figure1. By using finite-difference approximation, (35)–(44) can be written as(45)aip-fDi+bip-fDi+1+cip-fDi+2=di(46)p-mDi=λp-fDi+ω2pmDtD=0ω2u+λwhere(47)ai=μwkrofSwfri-1,tμo+krwfSwfri-1,tμw(48)bi=-ai+ci+ω1ω2u2+ω1λu+ω2λuω2u+λie2iΔzΔz2(49)ci=μwkrofSwfri,tμo+krwfSwfri,tμw(50)di=-e2iΔzΔz2ω1ipfDitD=0+e2iΔzΔz2ω2λω2u+λipmDitD=0The inner boundary condition is(51)e1p-wD+e2p-fD0+e3p-fD1=e8(52)e4p-fD0+e5p-fD1=e9The outer boundary condition is(53)e6p-fDn-1-e7p-fDn=0The coefficients for different boundary conditions are shown in Table2.Table 2
Coefficients with different boundary conditions.
Boundary conditions Coefficients Constant pressure outer boundary e 1 = 1, e2=-1+SΔz, e3=SΔz, e 4 = 1 + Δ z C D u + S C D u, e5=-SCDu-1, e6=0 e 7 = 1, e8=0, e9=Δzu-ΔzCDpwDtD=0  No flow outer boundary e 1 = 1, e2=-1+SΔz, e3=SΔz, e4=1+ΔzCDu+SCDu e 5 = - S C D u - 1, e6=an-1+cn-1 e 7 = b n - 1, e8=0, e9=Δzu-ΔzCDpwDtD=0The pressure difference equations ((45) and (46)) can be written in a matrix form as(54)Mx→=b→where M is a large sparse matrix, x⃑ is the unknown pressure vector, and b⃑ is the known vector related to the initial and boundary conditions. The matrix M and vectors x⃑ and b⃑ can be explicitly given as(55)M=e1e2e30000e4e500000a1b1c10000a2b2c2⋮⋱000000000000000000⋯⋱⋱⋯000000000000⋱⋮⋱⋱an-3bn-3cn-3an-2bn-2cn-20e6e7(56)x→=p-wDp-fD0p-fD1p-fD2⋮⋮p-fDn-1(57)b→=e8e9d1d2⋮⋮dn-1The unknown vectorx⃑ was calculated by using the Stehfest algorithm. Therefore, the dimensionless pressure and pressure derivatives [41] at the bottom of the wellbore can be plotted according to the solutions of the formulated mathematical model for the water injection system. The basic parameters and the relative permeability data for the calculation of pressure curves are illustrated in Table 3 and Figure 2.Table 3
Basic parameters for calculation of pressure curves.
Parameter Value Units Formation height 10 m Fracture porosity 0.03 fraction Matrix porosity 0.27 fraction Wellbore radius 0.1 m Fracture permeability 0.6 μm2 Matrix permeability 6×10-5 μm2 Irreducible water saturation 20 % Residual oil saturation 14 % Oil volume factor 1.03 fraction Water volume factor 1 fraction Shape factor 6×10-4 1/cm2 Oil viscosity 1 mPa.s Water viscosity 0.5 mPa.s rock compressibility 6×10-4 1/MPa oil compressibility 1×10-3 1/MPa water compressibility 4×10-4 1/MPa Injection rate 30 m3/day Initial pressure 14 MPa Skin factor 0.1 fraction Imbibition rate coefficient 0.1 1/day Wellbore storage coefficient 0.01 1m3/MPaFigure 2
Curves of relative permeability of oil and water in fracture and matrix.


## 2.1. Water Saturation Model

The water saturation distribution in the fracture and matrix considering water imbibition was estimated by integrating empirical matrix fracture transfer functions into the Buckley-Leverett equation. Based on the assumption that the cumulative oil recovery from a piece of rock surrounded by water is a continuous monotonic function of time and converges to a finite limit, Arnofsky et al. [34] proposed an exponential equation for oil recovery estimation of water displacement considering water imbibition in a fractured reservoir as shown:(1)R=R∞1-e-Rctwhere(2)R∞=∅m1-sorm-swcm(3)R=∅msm-swcmConsidering the gradual variation of the water saturation in surrounding fractures, De Swaan [35] calculated the rate of water imbibition in a fracture surrounded by matrix in a convolution form. Kazemi [36] proposed an analytical solution for a linear flow system that accounts for saturation changes in fracture and matrix. Shimamoto [37] extended the formula to a radial flow system for the Warrant-Root model with the volume conservation in a fracture.The water saturation change in the fracture can be expressed as(4)-q2πrh∂swf∂r-RcR∞∫0te-Rct-τ∂swf∂τdτ=∅f∂swf∂tAnd the water saturation in the matrix can be written as(5)RcR∞∫0te-Rct-τ∂swf∂τdτ=∅m∂swm∂tThe initial and boundary condition(6)swfr=0,t=1(7)swfr,t=0=0Introducing Laplace transform, the water saturation in Laplace domain can be expressed as(8)s-wf=1z1e-z1β/Rc+z1+z1α(9)s-wm=swmr,t=0z1+RcR∞∅me-z1β/Rc+z1+z1αz1Rc+z1where z1 is Laplace variable,(10)α=πhr2q∅f(11)β=πhr2qRcR∞The water saturation in fracture and matrix in the real domain can be obtained by inversion of Laplace transform(12)swfr,t=0t<αe-βe-Rct-αI02βRct-α+Rc∫αte-Rcτ-αI02βRcτ-αdτt≥α(13)swmr,t=swcmt<αswcm+1-sorm-swcmRce-β∫αte-Rcτ-αI02βRcτ-αdτt≥α


## 2.2. Pressure Transient Model

### 2.2.1. Physical Model

The naturally fractured reservoir is described by the Warren-Root model with constant temperature and uniform initial pressure. The reservoir is assumed to be a homogeneous horizontal radial reservoir with an upper and lower sealed boundary. The slightly compressible fluids (water and oil) flowing in the matrix and fracture system obey Darcy’s law, and the mass exchange between matrix and fracture system is assumed to be pseudosteady [40]. The gravity effect of the 2D system is neglected. A fully penetrating water injection well with constant injection rate is located in the center of the reservoir. As the water injection into the formation, the water saturation changes with time and space. A moving interface divided the radial formation into two regions, as shown in Figure 1. From the figure, rf indicates the position of water drive front. Region 1 is the water invaded region and region 2 is the uninvaded region.Figure 1
Schematic of two-zone composite reservoir.


### 2.2.2. Mathematical Model

After the water saturation distribution at any time during the injection period was obtained, the pressure equations can be solved by the quasi-stationary and LTFD method. According to the double porosity model proposed by Warren and Root, an oil-water two-phase flow well test model considering the saturation gradients within each region with a moving boundary between the regions for a water injection well was proposed as follows:Fracture system(14)1r∂∂rrkfkrofμoBo∂pf∂r-τomf=∂∂t∅fSofBo(15)1r∂∂rrkfkrwfμwBw∂pf∂r-τwmf=∂∂t∅fSwfBwMatrix system(16)τomf=∂∂t∅mSomBo(17)τwmf=∂∂t∅mSwmBwMatrix and fracture transfer equations(18)τomf=FskmkromμoBopf-pm(19)τwmf=FskmkrwmμwBwpf-pmSubstituting (16) and (17) into (14) and (15) and (18) and (19) yields(20)1r∂∂rrMtf∂pf∂r=∅fCtf∂pf∂t+∅mCtm∂pm∂t(21)∅mCtm∂pm∂t=FsMtupf-pmwhere Mtf denotes the total mobility of fracture system and Mtu is the mobility between fracture and matrix. The subscript “u” represents “upstream saturation”. For water injection period, the water flows from fracture to matrix, so water saturation in the fracture is used for the relative permeability evaluation (see (23)). Inversely, water saturation in the matrix is applied to calculate the relative permeability during the falloff period (see (24)).(22)Mtf=kfkrofSwfμo+krwfSwfμw(23)Mtu=kmkromSwfμo+krwmSwfμw(24)Mtu=kmkromSwmμo+krwmSwmμwIn order to simplify the calculation, a set of dimensionless variables were introduced as listed in Table1. Equations (20) and (21) and the initial and boundary conditions can be written in dimensionless form(25)1rD∂∂rDrDMtfD∂pfD∂rD=ω1∂pfD∂tD+ω2∂pmD∂tD(26)ω2∂pmD∂tD=λpfD-pmDTable 1
Dimensionless variable for pressure calculation.
M t f ^ = k f k r o f S w f = 1 μ o + k r w f S w f = 1 μ w M t f D = M t f M t f ^ p f D = 2 π M t f ^ h q B w p f - p i λ = F S r w 2 M t u M t f ^ p m D = 2 π M t f ^ h q B w p m - p i r D = r r w C t f ^ = C r + C w S w i + C o 1 - S w i ω 1 = ∅ f C t f ∅ f C t f ^ + ∅ m C t m ^ C t m ^ = C r + C w S w c m + C o 1 - S w c m ω 2 = ∅ m C t m ∅ f C t f ^ + ∅ m C t m ^ t D = M t f ^ ∅ f C t f ^ + ∅ m C t m ^ r w 2 C D = C 2 π ∅ f C t f ^ + ∅ m C t m ^ h r w 2The initial condition(27)pfDrD,tD=0=0(28)pmDrD,tD=0=0The inner boundary condition considering the skin effect is(29)pwD=pfD-SrD∂pfD∂rDrD=1The inner boundary condition considering wellbore storage effect is(30)CDdpwDdtD-rD∂pfD∂rDrD=1=1The constant pressure outer boundary condition is(31)pfDrD=reD,tD=0(32)pmDrD=reD,tD=0The no flow outer boundary condition is(33)∂pfD∂rDrD=reD=0(34)∂pmD∂rDrD=reD=0To simplify the computation, a logarithmic transform,z=ln⁡(rD), and the Laplace transform with the quasi-stationary assumption were used. Thus, (25)–(34) can be rewritten as(35)1e2zddzMtfDdp-fDdz=ω1up-fD-pfDz,0+ω2up-mD-pmDz,0(36)ω2up-mD-pmDz,0=λp-fD-p-mDThe initial condition is(37)pfDtD=0=0(38)pmDtD=0=0The inner boundary condition considering the skin effect is(39)p-wD=p-fD-Sdp-fDdzz=0The inner boundary condition considering wellbore storage effect is(40)dp-fDdzz=0-CDup-wD-pwDtD=0=-1uThe constant pressure outer boundary condition is(41)p-fDz=ze=0(42)p-mDz=ze=0The no flow outer boundary condition is(43)dp-fDdzz=ze=0(44)dp-mDdzz=ze=0The pressure change near the wellbore was relatively larger than the area away from the bottom of the well. So the formation was divided into n grid blocks in radial direction according to the point-centered logarithmic gridding method, which was shown as the dotted blue line in Figure1. By using finite-difference approximation, (35)–(44) can be written as(45)aip-fDi+bip-fDi+1+cip-fDi+2=di(46)p-mDi=λp-fDi+ω2pmDtD=0ω2u+λwhere(47)ai=μwkrofSwfri-1,tμo+krwfSwfri-1,tμw(48)bi=-ai+ci+ω1ω2u2+ω1λu+ω2λuω2u+λie2iΔzΔz2(49)ci=μwkrofSwfri,tμo+krwfSwfri,tμw(50)di=-e2iΔzΔz2ω1ipfDitD=0+e2iΔzΔz2ω2λω2u+λipmDitD=0The inner boundary condition is(51)e1p-wD+e2p-fD0+e3p-fD1=e8(52)e4p-fD0+e5p-fD1=e9The outer boundary condition is(53)e6p-fDn-1-e7p-fDn=0The coefficients for different boundary conditions are shown in Table2.Table 2
Coefficients with different boundary conditions.
Boundary conditions Coefficients Constant pressure outer boundary e 1 = 1, e2=-1+SΔz, e3=SΔz, e 4 = 1 + Δ z C D u + S C D u, e5=-SCDu-1, e6=0 e 7 = 1, e8=0, e9=Δzu-ΔzCDpwDtD=0  No flow outer boundary e 1 = 1, e2=-1+SΔz, e3=SΔz, e4=1+ΔzCDu+SCDu e 5 = - S C D u - 1, e6=an-1+cn-1 e 7 = b n - 1, e8=0, e9=Δzu-ΔzCDpwDtD=0The pressure difference equations ((45) and (46)) can be written in a matrix form as(54)Mx→=b→where M is a large sparse matrix, x⃑ is the unknown pressure vector, and b⃑ is the known vector related to the initial and boundary conditions. The matrix M and vectors x⃑ and b⃑ can be explicitly given as(55)M=e1e2e30000e4e500000a1b1c10000a2b2c2⋮⋱000000000000000000⋯⋱⋱⋯000000000000⋱⋮⋱⋱an-3bn-3cn-3an-2bn-2cn-20e6e7(56)x→=p-wDp-fD0p-fD1p-fD2⋮⋮p-fDn-1(57)b→=e8e9d1d2⋮⋮dn-1The unknown vectorx⃑ was calculated by using the Stehfest algorithm. Therefore, the dimensionless pressure and pressure derivatives [41] at the bottom of the wellbore can be plotted according to the solutions of the formulated mathematical model for the water injection system. The basic parameters and the relative permeability data for the calculation of pressure curves are illustrated in Table 3 and Figure 2.Table 3
Basic parameters for calculation of pressure curves.
Parameter Value Units Formation height 10 m Fracture porosity 0.03 fraction Matrix porosity 0.27 fraction Wellbore radius 0.1 m Fracture permeability 0.6 μm2 Matrix permeability 6×10-5 μm2 Irreducible water saturation 20 % Residual oil saturation 14 % Oil volume factor 1.03 fraction Water volume factor 1 fraction Shape factor 6×10-4 1/cm2 Oil viscosity 1 mPa.s Water viscosity 0.5 mPa.s rock compressibility 6×10-4 1/MPa oil compressibility 1×10-3 1/MPa water compressibility 4×10-4 1/MPa Injection rate 30 m3/day Initial pressure 14 MPa Skin factor 0.1 fraction Imbibition rate coefficient 0.1 1/day Wellbore storage coefficient 0.01 1m3/MPaFigure 2
Curves of relative permeability of oil and water in fracture and matrix.


## 2.2.1. Physical Model

The naturally fractured reservoir is described by the Warren-Root model with constant temperature and uniform initial pressure. The reservoir is assumed to be a homogeneous horizontal radial reservoir with an upper and lower sealed boundary. The slightly compressible fluids (water and oil) flowing in the matrix and fracture system obey Darcy’s law, and the mass exchange between matrix and fracture system is assumed to be pseudosteady [40]. The gravity effect of the 2D system is neglected. A fully penetrating water injection well with constant injection rate is located in the center of the reservoir. As the water injection into the formation, the water saturation changes with time and space. A moving interface divided the radial formation into two regions, as shown in Figure 1. From the figure, rf indicates the position of water drive front. Region 1 is the water invaded region and region 2 is the uninvaded region.Figure 1
Schematic of two-zone composite reservoir.


## 2.2.2. Mathematical Model

After the water saturation distribution at any time during the injection period was obtained, the pressure equations can be solved by the quasi-stationary and LTFD method. According to the double porosity model proposed by Warren and Root, an oil-water two-phase flow well test model considering the saturation gradients within each region with a moving boundary between the regions for a water injection well was proposed as follows:Fracture system(14)1r∂∂rrkfkrofμoBo∂pf∂r-τomf=∂∂t∅fSofBo(15)1r∂∂rrkfkrwfμwBw∂pf∂r-τwmf=∂∂t∅fSwfBwMatrix system(16)τomf=∂∂t∅mSomBo(17)τwmf=∂∂t∅mSwmBwMatrix and fracture transfer equations(18)τomf=FskmkromμoBopf-pm(19)τwmf=FskmkrwmμwBwpf-pmSubstituting (16) and (17) into (14) and (15) and (18) and (19) yields(20)1r∂∂rrMtf∂pf∂r=∅fCtf∂pf∂t+∅mCtm∂pm∂t(21)∅mCtm∂pm∂t=FsMtupf-pmwhere Mtf denotes the total mobility of fracture system and Mtu is the mobility between fracture and matrix. The subscript “u” represents “upstream saturation”. For water injection period, the water flows from fracture to matrix, so water saturation in the fracture is used for the relative permeability evaluation (see (23)). Inversely, water saturation in the matrix is applied to calculate the relative permeability during the falloff period (see (24)).(22)Mtf=kfkrofSwfμo+krwfSwfμw(23)Mtu=kmkromSwfμo+krwmSwfμw(24)Mtu=kmkromSwmμo+krwmSwmμwIn order to simplify the calculation, a set of dimensionless variables were introduced as listed in Table1. Equations (20) and (21) and the initial and boundary conditions can be written in dimensionless form(25)1rD∂∂rDrDMtfD∂pfD∂rD=ω1∂pfD∂tD+ω2∂pmD∂tD(26)ω2∂pmD∂tD=λpfD-pmDTable 1
Dimensionless variable for pressure calculation.
M t f ^ = k f k r o f S w f = 1 μ o + k r w f S w f = 1 μ w M t f D = M t f M t f ^ p f D = 2 π M t f ^ h q B w p f - p i λ = F S r w 2 M t u M t f ^ p m D = 2 π M t f ^ h q B w p m - p i r D = r r w C t f ^ = C r + C w S w i + C o 1 - S w i ω 1 = ∅ f C t f ∅ f C t f ^ + ∅ m C t m ^ C t m ^ = C r + C w S w c m + C o 1 - S w c m ω 2 = ∅ m C t m ∅ f C t f ^ + ∅ m C t m ^ t D = M t f ^ ∅ f C t f ^ + ∅ m C t m ^ r w 2 C D = C 2 π ∅ f C t f ^ + ∅ m C t m ^ h r w 2The initial condition(27)pfDrD,tD=0=0(28)pmDrD,tD=0=0The inner boundary condition considering the skin effect is(29)pwD=pfD-SrD∂pfD∂rDrD=1The inner boundary condition considering wellbore storage effect is(30)CDdpwDdtD-rD∂pfD∂rDrD=1=1The constant pressure outer boundary condition is(31)pfDrD=reD,tD=0(32)pmDrD=reD,tD=0The no flow outer boundary condition is(33)∂pfD∂rDrD=reD=0(34)∂pmD∂rDrD=reD=0To simplify the computation, a logarithmic transform,z=ln⁡(rD), and the Laplace transform with the quasi-stationary assumption were used. Thus, (25)–(34) can be rewritten as(35)1e2zddzMtfDdp-fDdz=ω1up-fD-pfDz,0+ω2up-mD-pmDz,0(36)ω2up-mD-pmDz,0=λp-fD-p-mDThe initial condition is(37)pfDtD=0=0(38)pmDtD=0=0The inner boundary condition considering the skin effect is(39)p-wD=p-fD-Sdp-fDdzz=0The inner boundary condition considering wellbore storage effect is(40)dp-fDdzz=0-CDup-wD-pwDtD=0=-1uThe constant pressure outer boundary condition is(41)p-fDz=ze=0(42)p-mDz=ze=0The no flow outer boundary condition is(43)dp-fDdzz=ze=0(44)dp-mDdzz=ze=0The pressure change near the wellbore was relatively larger than the area away from the bottom of the well. So the formation was divided into n grid blocks in radial direction according to the point-centered logarithmic gridding method, which was shown as the dotted blue line in Figure1. By using finite-difference approximation, (35)–(44) can be written as(45)aip-fDi+bip-fDi+1+cip-fDi+2=di(46)p-mDi=λp-fDi+ω2pmDtD=0ω2u+λwhere(47)ai=μwkrofSwfri-1,tμo+krwfSwfri-1,tμw(48)bi=-ai+ci+ω1ω2u2+ω1λu+ω2λuω2u+λie2iΔzΔz2(49)ci=μwkrofSwfri,tμo+krwfSwfri,tμw(50)di=-e2iΔzΔz2ω1ipfDitD=0+e2iΔzΔz2ω2λω2u+λipmDitD=0The inner boundary condition is(51)e1p-wD+e2p-fD0+e3p-fD1=e8(52)e4p-fD0+e5p-fD1=e9The outer boundary condition is(53)e6p-fDn-1-e7p-fDn=0The coefficients for different boundary conditions are shown in Table2.Table 2
Coefficients with different boundary conditions.
Boundary conditions Coefficients Constant pressure outer boundary e 1 = 1, e2=-1+SΔz, e3=SΔz, e 4 = 1 + Δ z C D u + S C D u, e5=-SCDu-1, e6=0 e 7 = 1, e8=0, e9=Δzu-ΔzCDpwDtD=0  No flow outer boundary e 1 = 1, e2=-1+SΔz, e3=SΔz, e4=1+ΔzCDu+SCDu e 5 = - S C D u - 1, e6=an-1+cn-1 e 7 = b n - 1, e8=0, e9=Δzu-ΔzCDpwDtD=0The pressure difference equations ((45) and (46)) can be written in a matrix form as(54)Mx→=b→where M is a large sparse matrix, x⃑ is the unknown pressure vector, and b⃑ is the known vector related to the initial and boundary conditions. The matrix M and vectors x⃑ and b⃑ can be explicitly given as(55)M=e1e2e30000e4e500000a1b1c10000a2b2c2⋮⋱000000000000000000⋯⋱⋱⋯000000000000⋱⋮⋱⋱an-3bn-3cn-3an-2bn-2cn-20e6e7(56)x→=p-wDp-fD0p-fD1p-fD2⋮⋮p-fDn-1(57)b→=e8e9d1d2⋮⋮dn-1The unknown vectorx⃑ was calculated by using the Stehfest algorithm. Therefore, the dimensionless pressure and pressure derivatives [41] at the bottom of the wellbore can be plotted according to the solutions of the formulated mathematical model for the water injection system. The basic parameters and the relative permeability data for the calculation of pressure curves are illustrated in Table 3 and Figure 2.Table 3
Basic parameters for calculation of pressure curves.
Parameter Value Units Formation height 10 m Fracture porosity 0.03 fraction Matrix porosity 0.27 fraction Wellbore radius 0.1 m Fracture permeability 0.6 μm2 Matrix permeability 6×10-5 μm2 Irreducible water saturation 20 % Residual oil saturation 14 % Oil volume factor 1.03 fraction Water volume factor 1 fraction Shape factor 6×10-4 1/cm2 Oil viscosity 1 mPa.s Water viscosity 0.5 mPa.s rock compressibility 6×10-4 1/MPa oil compressibility 1×10-3 1/MPa water compressibility 4×10-4 1/MPa Injection rate 30 m3/day Initial pressure 14 MPa Skin factor 0.1 fraction Imbibition rate coefficient 0.1 1/day Wellbore storage coefficient 0.01 1m3/MPaFigure 2
Curves of relative permeability of oil and water in fracture and matrix.


## 3. Verification of the Mathematical Model

Because little was available in the literatures for pressure behavior of a water injection well considering saturation gradient and water imbibition in dual-porosity reservoir by LTFD method, the model solutions were validated by simplifying the model into a single phase water injection well in a dual-porosity system, which was compared to the conventional analytical solution [42] by commercial software as shown in Figure 3. From the figure, the LTFD results show good agreement with the analytical solution.Figure 3
Comparison of the results by LTFD method and analytical method.Moreover, a field water injection test was used to further verify the proposed model. The basic information of the water injection well and the relative permeability data was listed in Table4 and Figure 4. And Figure 5 shows the pressure curves of the well test data comparing with the results in our model. From the figure, it is found that our results match well with the field test data.Table 4
Basic information of the water injection well.
Parameter Value Units Formation height 12.6 m Fracture porosity 0.03 fraction Matrix porosity 0.268 fraction Wellbore radius 0.12 m Oil volume factor 1.163 fraction Water volume factor 1 fraction Oil viscosity 9.6 mPa.s Water viscosity 0.5 mPa.s rock compressibility 5×10-4 1/MPa oil compressibility 4.8×10-3 1/MPa water compressibility 3.9×10-4 1/MPa Injection rate 256 m3/day Initial pressure 15 MPaFigure 4
The relative permeability of oil and water for the water injection well.Figure 5
Comparison of the results in this work and the field injection test.


## 4. Model Features Analysis

The pressure curves of oil-water two-phase flow considering water imbibition and the saturation gradients during injection period by the LTFD method were plotted in this section. The behaviors of the typical curves were analyzed by comparing the typical curves of single phase flow with the two-phase flow curves in the fractured reservoirs as follows.


### 4.1. Features of Typical Curves

Figure6 illustrates the pressure curves of single phase flow and oil-water two-phase flow in fractured reservoirs by LTFD method without water imbibition. According to the feature of the curves, the formation flow can be divided into three stages. The first stage is the flow in the fracture system, the second stage is the flow between the fracture and the matrix system, and the third stage is the flow in the whole system, including the fracture and matrix. The figure shows that the shape of curves for single phase flow and two-phase flow is similar, but the position of the two-phase flow curves is on the upper right of the single phase flow curves. The reason is that when oil and water exist in the whole system, the total mobility of oil and water is lower than the single phase, and the bottom-hole pressure increases more quickly as the fluid is injected into the well. Thus, the pressure derivative curves of two-phase flow move upward compared to the single phase flow curves. Moreover, due to the decrease of the total mobility, the interporosity flow capacity between matrix and fracture system decreases when the single phase flow becomes two-phase flow. Thus, the “dip” of the two-phase flow curves occurs later than that of the single phase flow curves.Figure 6
Typical curves of single phase and two-phase flow during injection period.


### 4.2. Effects of Parameters

The key factors influencing the pressure behavior of water injection period including skin factor, wellbore storage coefficient, shape factor, water imbibition rate coefficient, and boundary conditions were discussed in detail in this part.


#### 4.2.1. Skin Factor

Figure7 shows the effect of skin factor on the pressure curves of the water injection well. The solid lines are the curves of single phase flow and the circles are oil-water two-phase flow curves. The red, green, and blue colors indicate that the shape factor values are -1, 1, and 10, respectively. From the figure, the skin factor mainly influences the peak value of the pressure derivatives. As the value of skin factor increases, the peak value gets higher and the time for peak value occurrence becomes later which indicates that the reservoir is more severely contaminated. The pressure derivative curves of two-phase flow move toward the upper right of the single phase flow curves, which shows the same change rule with Figure 6. When the skin factor becomes larger, the peak value difference between the single phase and two-phase decreases, and the inclination of the curves after the peak value increases.Figure 7
Influence of different skin factors on pressure transient behavior.


#### 4.2.2. Wellbore Storage Coefficient

Figure8 is the influences of wellbore storage coefficient on the pressure and pressure derivative curves of the water injection well. The red, green, and blue colors indicate that wellbore storage coefficient values are 0.01, 0.1, and 1, respectively. As seen, the main difference of the curves occurs on the straight line of early period. As the wellbore storage coefficient increases, the straight line section becomes longer and the time of peak value occurs later. Besides, the wellbore storage effect will cover up the radial flow stage in the fracture system, directly going to the transitional flow stage between the fracture and matrix. The pressure derivative curves of two-phase flow move upper right to the single phase flow curves after the straight line of wellbore storage stage, which agrees with the change rule of Figure 6.Figure 8
Influence of different wellbore storage coefficients on pressure transient behavior.


#### 4.2.3. Shape Factor

The influences of shape factor on the pressure curves of the water injection well are shown in Figure9. The red, green, and blue colors indicate that shape factor values are 10−5, 10−4, and 10−3, respectively. By comparison of the curves, it is found that the shape factor mainly has an effect on the position of the “dip” of the pressure derivatives. When the value of shape factor gets greater, the “dip” appears earlier and the “dip” moves closer to the left. The main reason is that the shape factor is a component of the equation defining the interporosity parameter. The higher the shape factor is, the easier the fluid exchange between the fracture and matrix is and the earlier the transitional flow stage occurs. It also demonstrates that the pressure derivative curves of two-phase flow move upper right to the single phase flow curves, which is in line with the change rule of Figure 6.Figure 9
Influence of different shape factors on pressure transient behavior.


#### 4.2.4. Imbibition Rate Coefficient with No Flow Outer Boundary Condition

Figure10 describes the pressure behavior of the water injection well under no flow outer boundary condition when the water imbibition rate coefficient values are 0, 0.01, 0.05, 0.1, 0.5, and 1, respectively. From the figure, the imbibition rate coefficient mainly has an effect on the whole system (matrix and fracture system) radial flow period of the curves. As the imbibition rate coefficient becomes larger, there are more fluid exchange between the fracture and matrix, and more fluid supply from the matrix to the fracture. The water imbibition process is slow, so there is a second interporosity period after the whole system radial flow period. And with the increase of water imbibition rate coefficient, the pressure derivative curves dip downward earlier and the degree for the curves dropping down becomes larger. After that, the pressure derivative curves rise up to a uniform line with the same slope due to the influence of the no flow outer boundary.Figure 10
Pressure behavior of different imbibition rate coefficients with no flow outer boundary.


#### 4.2.5. Imbibition Rate Coefficient with Constant Pressure Boundary Condition

Figure11 shows the pressure and pressure derivative curves of the water injection well under constant pressure boundary condition when the water imbibition rate coefficient values are 0, 0.01, 0.05, 0.1, 0.5, and 1, respectively. By comparison and analysis, it is concluded that the pressure derivative curve is a straight line at the whole system radial flow period when no water imbibition is considered. When considering water imbibition, the pressure derivatives decline at the whole system radial flow period. The pressure derivatives drop downward more and earlier with the increase of imbibition rate coefficient. After that, the curves also drop down due to the constant pressure outer boundary effect.Figure 11
Pressure behavior of different imbibition rate coefficients with constant pressure boundary.


## 4.1. Features of Typical Curves

Figure6 illustrates the pressure curves of single phase flow and oil-water two-phase flow in fractured reservoirs by LTFD method without water imbibition. According to the feature of the curves, the formation flow can be divided into three stages. The first stage is the flow in the fracture system, the second stage is the flow between the fracture and the matrix system, and the third stage is the flow in the whole system, including the fracture and matrix. The figure shows that the shape of curves for single phase flow and two-phase flow is similar, but the position of the two-phase flow curves is on the upper right of the single phase flow curves. The reason is that when oil and water exist in the whole system, the total mobility of oil and water is lower than the single phase, and the bottom-hole pressure increases more quickly as the fluid is injected into the well. Thus, the pressure derivative curves of two-phase flow move upward compared to the single phase flow curves. Moreover, due to the decrease of the total mobility, the interporosity flow capacity between matrix and fracture system decreases when the single phase flow becomes two-phase flow. Thus, the “dip” of the two-phase flow curves occurs later than that of the single phase flow curves.Figure 6
Typical curves of single phase and two-phase flow during injection period.


## 4.2. Effects of Parameters

The key factors influencing the pressure behavior of water injection period including skin factor, wellbore storage coefficient, shape factor, water imbibition rate coefficient, and boundary conditions were discussed in detail in this part.


### 4.2.1. Skin Factor

Figure7 shows the effect of skin factor on the pressure curves of the water injection well. The solid lines are the curves of single phase flow and the circles are oil-water two-phase flow curves. The red, green, and blue colors indicate that the shape factor values are -1, 1, and 10, respectively. From the figure, the skin factor mainly influences the peak value of the pressure derivatives. As the value of skin factor increases, the peak value gets higher and the time for peak value occurrence becomes later which indicates that the reservoir is more severely contaminated. The pressure derivative curves of two-phase flow move toward the upper right of the single phase flow curves, which shows the same change rule with Figure 6. When the skin factor becomes larger, the peak value difference between the single phase and two-phase decreases, and the inclination of the curves after the peak value increases.Figure 7
Influence of different skin factors on pressure transient behavior.


### 4.2.2. Wellbore Storage Coefficient

Figure8 is the influences of wellbore storage coefficient on the pressure and pressure derivative curves of the water injection well. The red, green, and blue colors indicate that wellbore storage coefficient values are 0.01, 0.1, and 1, respectively. As seen, the main difference of the curves occurs on the straight line of early period. As the wellbore storage coefficient increases, the straight line section becomes longer and the time of peak value occurs later. Besides, the wellbore storage effect will cover up the radial flow stage in the fracture system, directly going to the transitional flow stage between the fracture and matrix. The pressure derivative curves of two-phase flow move upper right to the single phase flow curves after the straight line of wellbore storage stage, which agrees with the change rule of Figure 6.Figure 8
Influence of different wellbore storage coefficients on pressure transient behavior.


### 4.2.3. Shape Factor

The influences of shape factor on the pressure curves of the water injection well are shown in Figure9. The red, green, and blue colors indicate that shape factor values are 10−5, 10−4, and 10−3, respectively. By comparison of the curves, it is found that the shape factor mainly has an effect on the position of the “dip” of the pressure derivatives. When the value of shape factor gets greater, the “dip” appears earlier and the “dip” moves closer to the left. The main reason is that the shape factor is a component of the equation defining the interporosity parameter. The higher the shape factor is, the easier the fluid exchange between the fracture and matrix is and the earlier the transitional flow stage occurs. It also demonstrates that the pressure derivative curves of two-phase flow move upper right to the single phase flow curves, which is in line with the change rule of Figure 6.Figure 9
Influence of different shape factors on pressure transient behavior.


### 4.2.4. Imbibition Rate Coefficient with No Flow Outer Boundary Condition

Figure10 describes the pressure behavior of the water injection well under no flow outer boundary condition when the water imbibition rate coefficient values are 0, 0.01, 0.05, 0.1, 0.5, and 1, respectively. From the figure, the imbibition rate coefficient mainly has an effect on the whole system (matrix and fracture system) radial flow period of the curves. As the imbibition rate coefficient becomes larger, there are more fluid exchange between the fracture and matrix, and more fluid supply from the matrix to the fracture. The water imbibition process is slow, so there is a second interporosity period after the whole system radial flow period. And with the increase of water imbibition rate coefficient, the pressure derivative curves dip downward earlier and the degree for the curves dropping down becomes larger. After that, the pressure derivative curves rise up to a uniform line with the same slope due to the influence of the no flow outer boundary.Figure 10
Pressure behavior of different imbibition rate coefficients with no flow outer boundary.


### 4.2.5. Imbibition Rate Coefficient with Constant Pressure Boundary Condition

Figure11 shows the pressure and pressure derivative curves of the water injection well under constant pressure boundary condition when the water imbibition rate coefficient values are 0, 0.01, 0.05, 0.1, 0.5, and 1, respectively. By comparison and analysis, it is concluded that the pressure derivative curve is a straight line at the whole system radial flow period when no water imbibition is considered. When considering water imbibition, the pressure derivatives decline at the whole system radial flow period. The pressure derivatives drop downward more and earlier with the increase of imbibition rate coefficient. After that, the curves also drop down due to the constant pressure outer boundary effect.Figure 11
Pressure behavior of different imbibition rate coefficients with constant pressure boundary.


## 4.2.1. Skin Factor

Figure7 shows the effect of skin factor on the pressure curves of the water injection well. The solid lines are the curves of single phase flow and the circles are oil-water two-phase flow curves. The red, green, and blue colors indicate that the shape factor values are -1, 1, and 10, respectively. From the figure, the skin factor mainly influences the peak value of the pressure derivatives. As the value of skin factor increases, the peak value gets higher and the time for peak value occurrence becomes later which indicates that the reservoir is more severely contaminated. The pressure derivative curves of two-phase flow move toward the upper right of the single phase flow curves, which shows the same change rule with Figure 6. When the skin factor becomes larger, the peak value difference between the single phase and two-phase decreases, and the inclination of the curves after the peak value increases.Figure 7
Influence of different skin factors on pressure transient behavior.


## 4.2.2. Wellbore Storage Coefficient

Figure8 is the influences of wellbore storage coefficient on the pressure and pressure derivative curves of the water injection well. The red, green, and blue colors indicate that wellbore storage coefficient values are 0.01, 0.1, and 1, respectively. As seen, the main difference of the curves occurs on the straight line of early period. As the wellbore storage coefficient increases, the straight line section becomes longer and the time of peak value occurs later. Besides, the wellbore storage effect will cover up the radial flow stage in the fracture system, directly going to the transitional flow stage between the fracture and matrix. The pressure derivative curves of two-phase flow move upper right to the single phase flow curves after the straight line of wellbore storage stage, which agrees with the change rule of Figure 6.Figure 8
Influence of different wellbore storage coefficients on pressure transient behavior.


## 4.2.3. Shape Factor

The influences of shape factor on the pressure curves of the water injection well are shown in Figure9. The red, green, and blue colors indicate that shape factor values are 10−5, 10−4, and 10−3, respectively. By comparison of the curves, it is found that the shape factor mainly has an effect on the position of the “dip” of the pressure derivatives. When the value of shape factor gets greater, the “dip” appears earlier and the “dip” moves closer to the left. The main reason is that the shape factor is a component of the equation defining the interporosity parameter. The higher the shape factor is, the easier the fluid exchange between the fracture and matrix is and the earlier the transitional flow stage occurs. It also demonstrates that the pressure derivative curves of two-phase flow move upper right to the single phase flow curves, which is in line with the change rule of Figure 6.Figure 9
Influence of different shape factors on pressure transient behavior.


## 4.2.4. Imbibition Rate Coefficient with No Flow Outer Boundary Condition

Figure10 describes the pressure behavior of the water injection well under no flow outer boundary condition when the water imbibition rate coefficient values are 0, 0.01, 0.05, 0.1, 0.5, and 1, respectively. From the figure, the imbibition rate coefficient mainly has an effect on the whole system (matrix and fracture system) radial flow period of the curves. As the imbibition rate coefficient becomes larger, there are more fluid exchange between the fracture and matrix, and more fluid supply from the matrix to the fracture. The water imbibition process is slow, so there is a second interporosity period after the whole system radial flow period. And with the increase of water imbibition rate coefficient, the pressure derivative curves dip downward earlier and the degree for the curves dropping down becomes larger. After that, the pressure derivative curves rise up to a uniform line with the same slope due to the influence of the no flow outer boundary.Figure 10
Pressure behavior of different imbibition rate coefficients with no flow outer boundary.


## 4.2.5. Imbibition Rate Coefficient with Constant Pressure Boundary Condition

Figure11 shows the pressure and pressure derivative curves of the water injection well under constant pressure boundary condition when the water imbibition rate coefficient values are 0, 0.01, 0.05, 0.1, 0.5, and 1, respectively. By comparison and analysis, it is concluded that the pressure derivative curve is a straight line at the whole system radial flow period when no water imbibition is considered. When considering water imbibition, the pressure derivatives decline at the whole system radial flow period. The pressure derivatives drop downward more and earlier with the increase of imbibition rate coefficient. After that, the curves also drop down due to the constant pressure outer boundary effect.Figure 11
Pressure behavior of different imbibition rate coefficients with constant pressure boundary.


## 5. Conclusions

In this work, a semianalytical two-phase flow model considering water imbibition and the saturation gradients within each region during injection period was proposed to analyze the pressure transient behavior of the water injection well in fractured reservoirs. Validation of the presented model was performed by analytical method and a field water injection test. The model features were investigated by sensitivity analysis. Some conclusions were drawn as follows:(1)
The shape of the pressure curves for the proposed model is similar to the single phase flow model, while the position of the curves for the two-phase flow model is on the upper right of the single phase flow model.(2)
The main influence of skin factor on the curves occurs on the peak value of the pressure derivatives. As the skin factor becomes larger, the peak value difference between single phase and two-phase pressure curves decreases, and the inclination of the curves after the peak value increases.(3)
The wellbore storage coefficient mainly has an effect on the straight line of the early period for the pressure curves. The shape factor mainly has an effect on the position of the “dip” of the pressure derivatives which is the same as the interporosity parameter.(4)
The imbibition rate coefficient mainly influences the whole system radial flow period of the pressure curves. The pressure derivative curves dip downward earlier and drop down to a greater degree with the increase of imbibition rate coefficient.



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*Source: 2896251-2018-09-04.xml*