# Transient Pressure Behavior of a Horizontal Well in a Naturally Fractured Gas Reservoir with Dual-Permeability Flow and Stress Sensitivity Effect

Authors: Guiqin Wang; Rui Zhang; Liangliang Cui
Journal: Geofluids
 (2022)

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## Abstract

Dual-permeability flow and stress sensitivity effect are two fundamental issues that have been widely investigated in transient pressure analysis for horizontal wells. However, few attempts have been made to simulate the combined effects of dual-permeability flow and stress-dependent fracture permeability on the pressure transient dynamics of a horizontal well in a naturally fractured gas reservoir. In this approach, an analytical model is proposed to integrate the complexities of pressure-dependent PVT properties, dual-permeability flow behavior, and stress-dependent fracture permeability characteristics. The nonlinearity of the mathematical model is weakened by using Pedrosa’s transform formulation. Then, the Laplace integral transformation and separation of variables are applied to solve the model. Based on the solution of the mathematical model, a series of new-type curves are drawn to make a precise observation of different flow regimes. The main differences between the proposed model and the traditional models are discussed, and the effects of the permeability modulus of fractures, storability ratio, interporosity flow factor, and skin factor on transient pressure response are also examined. The results show that there are obvious differences in transient pressure dynamic curves between the proposed model and traditional models. The stress sensitivity effect plays a significant role in the intermediate flow period and the late-time pseudoradial flow period. The dual-permeability flow behavior mainly affects the early transient and interporosity flow stages. The proposed model can accurately simulate the transient pressure behaviors of a horizontal well in a naturally fractured gas reservoir with a dual-permeability flow and stress sensitivity effect. The novel model can be used to interpret pressure signals with accurate matching results and more reasonable interpreted parameters.


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## Body

## 1. Introduction

Transient pressure analysis of a horizontal well in a naturally fractured gas reservoir is greatly affected by fracture seepage parameters and stress-dependent fracture permeability. For naturally fractured reservoirs, the fractures are always with heterogeneities [1, 2]. An experimental study on fracture stress sensitivities proves that stress-dependent fracture permeability significantly affects the transient pressure response [3]. So, it is pretty essential to propose a comprehensive model to capture the transient behavior of horizontal wells in naturally fractured gas reservoirs.Because it is often impossible to describe the complex fractures precisely, continuum models are proposed to capture the flow behavior of this kind of reservoir. Much research has been done on theoretical models of vertical wells in naturally fractured reservoirs. Barenblatt et al. [4] proposed a classic double-porosity and single-permeability model to study vertical well production in porous media reservoirs. This model assumes that a naturally fractured reservoir is composed of two completely overlapping continua, porous matrix, and fractures. Warren and Root [5] expanded Barenblatt et al.’s approach to cover the independent physical properties of fracture and matrix. In their model, naturally fractured formation is formed by matrix blocks, which is separated by uniform and orthogonal fractures. Besides, the pseudosteady-state interporosity flow is firstly adopted to simulate mass transfer between fracture and matrix systems. After that, Kazemi et al. [6], de Swaan [7], Raghavan and Ohaeri [8], Serra et al. [9], Jalali and Ershaghi [10], Wu and Pruess [11], Bui et al. [12], Wu et al. [13], Kuchuk et al. [14], Jia et al. [2], and Wang et al. [15] proposed their own dual-media model for naturally fractured reservoirs with the consideration of transient interporosity flow behavior. These models assume that the fracture system is the only flow pathway directly connected with wellbore by ignoring the flow from the matrix system to the wellbore. Because the dual-porosity and single-permeability model is no longer applicable in naturally fractured reservoirs, a great deal of work have instead been directed at using dual-porosity and dual-permeability models [14, 16–18] to describe the flow behavior, which assumes that both the fracture and matrix systems are the flow pathway directly connected with the wellbore, also considered the pseudosteady-state and transient-state interporosity flow between matrix and fracture systems. However, most of the models ignored the effect of the stress-dependent fracture permeability on transient pressure response. Although a great deal of work has been done on theoretical models of naturally fractured gas wells considering stress sensitivity of fracture permeability, most of them are restricted to the dual-porosity and single-permeability flow problem [3, 15, 19–21].In recent years, horizontal wells have been increasingly applied to some naturally fractured gas reservoirs. Research on the transient pressure behavior of this kind of well has become increasingly popular among engineers [22–30]. However, transient pressure analysis for horizontal wells is commonly performed assuming that permeability for natural fractures remains constant, which might not be physically applicable for stress-sensitive reservoirs. Besides, dual-permeability flow is seldom considered in their models. In general, the big challenge of analyzing the transient pressure response of a horizontal well is that the dual-permeability flow behavior and stress-dependent fracture permeability should be all incorporated in the mathematical models.This paper presented a novel semianalytical model to examine the combined effects of dual-permeability flow behavior and stress-dependent fracture permeability on the transient pressure response of a horizontal well in a naturally fractured gas reservoir. The nonlinearity of the governing equations caused by the stress sensitivity of fracture permeability is eliminated using Pedrosa’s [31] transform formulation. With Laplace transform and separation of variables, we got the analytical solution of the mathematical model. A series of new transient pressure dynamic curves are drawn to observe different flow regimes based on the solution. Then, differences between the proposed model and traditional models are discussed and the effects of some critical parameters on transient pressure response are also analyzed with the proposed model.


## 2. Methodology

### 2.1. Model Assumption

As shown in Figure1, the naturally fractured gas reservoir is composed of fracture and matrix systems and the physical properties of the two systems are independent. A radial cylindrical dual-porosity and dual-permeability medium reservoir is considered in which a single horizontal well is located at the center, completely penetrating the formation. The matrix/fracture flow is schematically described in Figure 2. In this study, the proposed model assumes that both the fracture and matrix systems are the flow pathway directly connected with the wellbore and fluids in the fracture and matrix systems first flow into the horizontal wellbore, followed by the matrix-fracture interporosity flow. Some simplifying physical model assumptions for the derivation of the governing equation are listed as the horizontal well produced with the constant production rate in a naturally fractured gas reservoir. The external boundaries of the top and bottom are assumed to be closed, and the lateral boundary is assumed to be infinite. The matrix-fracture interporosity flow in the reservoir is described by the pseudosteady-state model [5, 25, 26]. Fluid flow follows the law of Darcy seepage, and stress-dependent fracture permeability is considered. Also, capillary and gravity forces are neglected to simplify the model.Figure 1
Schematic of naturally fractured reservoir.Figure 2
Dual-porosity and dual-permeability flow scheme.


### 2.2. Mathematical Model

The PVT properties, such as fluid viscosity and volume factor of the gas phase, are quite sensitive to formation pressure. In this section, the pseudopressure transformation is used to capture pressure-dependent PVT properties and reduce the nonlinearity of governing differential equations. The definitions of pseudopressure and pseudotime are given by(1)ψj=2∫0ppμgZdp,j=f,m,ta=∫0tμgictiμgpctpdt,where p is the pressure, MPa; ψf is the pseudopressure of the fracture, MPa2/(mPa·s); ψm is the pseudo pressure of the matrix, MPa2/(mPa·s); t is the time, h; ta is the pseudotime; μg is the gas viscosity, mPa·s; ct is the total compressibility coefficient, MPa−1; and Z is the gas compressibility factor.To describe the degree of stress sensitivity and its influence to fracture permeability, the concept of pseudopermeability modulusγf is defined as [19, 26]
(2)γf=1kf∂kf∂ψf.Equation (2) can be further written as
(3)kf=kfie−γfψi−ψf,where kf is the permeability of fracture, mD; kfi is the initial permeability of fracture, mD; ψi is the initial pseudopressure, MPa2/(mPa·s); and γf is the pseudopermeability modulus of the fracture, (mPa·s)/MPa2.To establish and solve the model, a radial cylindrical system (r) is used to describe the flow of the fracture and matrix system. With consideration of the dual-porosity and dual-permeability flow behavior and stress-dependent fracture permeability, the governing differential equation of the complex system can be described as follows:For the fracture system,(4)∂2ψf∂r2+1r∂ψf∂r+γf∂ψf∂r2+kfvikfhi∂2ψf∂z2+γf∂ψf∂z2+αkmkfhiψm−ψf=eγfψi−ψfϕfμgctf3.6kfhi∂ψf∂t.For the matrix system,(5)∂2ψm∂r2+1r∂ψm∂r+kmvkmh∂2ψm∂z2−αkmkmhψm−ψf=ϕmμgctm3.6kmh∂ψm∂t.For the initial condition,(6)ψft=0=ψmt=0=0.In the inner boundary condition,(7)limε⟶0limr⟶04πhμgε∫zw−ε/2zw+ε/2kmhr∂ψm∂r+kfhie−γfψi−ψfr∂ψf∂rdz=pscTZpTscqsc,z−zw≤ε2.The top and bottom boundaries are assumed to be closed and given by(8)∂ψm∂zz=0=∂ψf∂zz=0=0,∂ψm∂zz=h=∂ψf∂zz=h=0The lateral boundary condition is assumed to be infinite and expressed as(9)limr⟶∞ψm=limr⟶∞ψf=ψi,where ctf is the total compressibility of the fracture, MPa−1; ctm is the total compressibility of the fracture, MPa−1; h is the reservoir thickness, m; kfhi is the initial horizontal permeability of the fracture, mD; kfvi is the initial vertical permeability of the fracture, mD; kmh is the horizontal permeability of the matrix, mD; kmv is the vertical permeability of the matrix, mD; p is pressure, MPa; psc is the pressure at standard condition, MPa; qsc is the surface gas production rate, 104 m3/d; r is the radial distance, m; T is temperature, K; Tsc is the temperature at standard condition, K; z is the vertical distance from the bottom, m; zw is the vertical distance of the horizontal well from the bottom, m; ε is a variable in the z direction, m; ϕf is the porosity of fracture; ϕm is the porosity of the matrix; and α is the geometric shape factor of matrix block, m−2.To make the equations homogeneous, some dimensionless variables are defined and tabulated in Table1. Taking the dimensionless variables into equations (4)–(9), one can obtain the dimensionless differential equations.Table 1
Definitions of the parameters used in this work.
ParametersSymbolDefinitionDimensionless pseudopressure of the fractureψfDψfD=78.489kmh+kfhih/Tqscψi−ψfDimensionless pseudopressure of the matrixψmDψmD=78.489kmh+kfhih/Tqscψi−ψmDimensionless vertical distancezDzD=z/hDimensionless pseudotimetDtD=3.6kmh+kfhi/μrw2ϕmCmt+ϕfCfttaDimensionless reservoir thicknesshDhD=h/rwe−Skfhi/kfpiDimensionless radial distancerDrD=r/rwe−SDimensionless wellbore storage coefficientCDCD=Cs/6.2832ϕfCft+ϕmCmthrw2Dimensionless pseudopermeability modulusγfDγfD=Tqsc/78.489kfhihγfThe permeability ratio of the fracture system to the sum of the fracture and matrix systemκκ=kfhi/kfhi+kmhCapacitance coefficient of the fractureωfωf=ϕfCft/ϕfCft+ϕmCmtInterporosity flow factor of matrix system into the fracture systemλmfλmf=αmkmhrw2/kfhi+kmhFor the fracture system,(10)κ∂2ψfD∂rD2+1rD∂ψfD∂rD−γfD∂ψfD∂rD2+1hD2∂2ψfD∂zD2−γfD∂ψfD∂zD2=eγfDψfDωfe−2S∂ψfD∂tD+λmfe−2SψfD−ψmD.For the matrix system,(11)1−κ∂2ψmD∂rD2+1rD∂ψmD∂rD+1hD2∂2ψmD∂zD2−λmfe−2SψmD−ψfD=ωme−2S∂ψmD∂tD.In the initial condition,(12)ψfDtD=0=ψmDtD=0=0.In the inner boundary condition,(13)limεD⟶0limrD⟶0∫zwD−εD/2zwD+εD/2κrD∂ψmD∂rD+1−κrDe−γfDψfD∂ψfD∂rDdzD=−12,zD−zwD≤εD2.In the outer boundary conditions,(14)∂ψmD∂zDzD=0=∂ψfD∂zDzD=0=0,(15)∂ψmD∂zDzD=1=∂ψfD∂zDzD=1=0,(16)limrD⟶∞ψmD=limrD⟶∞ψfD=0.


### 2.3. Solution to the Mathematical Model

It should be noted that equations (10) and (13) are strongly nonlinear with the consideration of the stress sensitivity of fracture permeability. This is because the stress-dependent fracture permeability is a function of the pseudopressure of the fracture. However, the pressure of fracture is an unknown parameter. Therefore, the mathematical model cannot be solved analytically. In this work, the Pedrosa [31] variable substitution and regular perturbation method are firstly deployed to alleviate the nonlinearity. Then, the Laplace transformation and separation of variables are adopted to address the linearized model. Thus, the model can be solved in the Laplace space and the Stehfest and Harald [32] numerical inversion is used to calculate the pressure in real space.


#### 2.3.1. Linearization of the Flow Equation

To linearize the mathematical model, the Pedrosa transformation is employed in this section and given by(17)ψfDrD,tDtD=0=−1γfDln1−γfDηfDrD,tD,where ηfD is an intermediate variable called the perturbation deformation function.After the Pedrosa transformation, equations (10)–(16) can be rewritten as
(18)κ∂2ηfD∂rD2+1rD∂ηfD∂rD+1hD2∂2ηfD∂zD2=e−2S11−γfDηfDωf∂ηfD∂tD+1−ωf∂ψmD∂tD,(19)1−κ∂2ψmD∂rD2+1rD∂ψmD∂rD+1hD2∂2ψmD∂zD2=e−2S1−ωf∂ψmD∂tD+λmf1γfDln1−γfDηfD−ψmD,(20)ηfDrD,tDtD=0=ψmDrD,tDtD=0=0,(21)limεD⟶0limrD⟶0∫zwD−εD/2zwD+εD/21−κrD∂ψmD∂rD+κrD∂ηfD∂rDdzD=−12,zD−zwD≤εD2,(22)∂ηfD∂zDzD=1=∂ψmD∂zDzD=1=0,(23)∂ηfD∂zDzD=0=∂ψmD∂zDzD=0=0,(24)limrD⟶∞ψmD=limrD⟶∞ηfD=0.According to the regular perturbation theory, equations (18)–(24) can be simplified as
(25)κ∂2ηfD0∂rD2+1rD∂ηfD0∂rD+1hD2∂2ηfD0∂zD2=ωfe−2S∂ηfD0∂tD+λmfe−2SηfD0−ψmD,(26)1−κ∂2ψmD∂rD2+1rD∂ψmD∂rD+1hD2∂2ψmD∂zD2=1−ωfe−2S∂ψmD∂tD+λmfe−2SηfD0−ψmD,(27)ηfD0rD,tDtD=0=ψmDrD,tDtD=0=0,(28)limεD⟶0limrD⟶0∫zwD−εD/2zwD+εD/21−κrD∂ψmD∂rD+κrD∂ηfD0∂rDdzD=−12,zD−zwD≤εD2,(29)∂ηfD0∂zDzD=1=∂ψmD∂zDzD=1=0,(30)∂ηfD0∂zDzD=0=∂ψmD∂zDzD=0=0,(31)limrD⟶∞ψmD=limrD⟶∞ηfD0=0.


#### 2.3.2. Solution of the Proposed Model

To derive the analytical solution of the model, the mathematical model is translated into the Laplace domain with respect totD:
(32)LηfDrD,tD=η¯fDrD,u=∫0+∞ηfDrD,tDe−utDdtD,(33)LψmDrD,tD=ψ¯mDrD,u=∫0+∞ψmDrD,tDe−utDdtD,where u is the Laplace transform variable.With equations (32) and (33) and taking the Laplace transform of equations (25)–(31), one can obtain the dimensionless mathematical model in the Laplace space:For the fracture system,(34)∂2η¯fD∂rD2+1rD∂η¯fD∂rD+1hD2∂2η¯fD∂zD2+A1κη¯fD+A2κψ¯mD=0.For the matrix system,(35)∂2ψ¯mD∂rD2+1rD∂ψ¯mD∂rD+1hD2∂2ψ¯mD∂zD2+A21−κη¯fD+A31−κψ¯mD=0,where A1=−λmf+uωfe−2S, A2=λmfe−2S, and A3=−λmf+u1−ωfe−2S.In the inner boundary condition,(36)limεD⟶0limrD⟶0∫zwD−εD/2zwD+εD/21−κrD∂ψ¯mD∂rD+κrD∂η¯fD0∂rDdzD=−12u,zD−zwD≤εD2.In the outer boundary conditions,(37)∂ψ¯mD∂zDzD=0=∂η¯fD0∂zDzD=0=0,(38)∂ψ¯mD∂zDzD=1=∂η¯fD0∂zDzD=1=0,(39)limrD⟶∞ψ¯mD=limrD⟶∞η¯fD0=0.This section uses the separation of variables to solve the dual-porosity and dual-permeability modeling of a horizontal well in a naturally fractured reservoir. With the separation of variables, the dimensionless pseudopressure in the Laplace space can be separated by [25]
(40)η¯jD=R¯jrDZ¯jzD,j=m,f.Substituting equation (40) into equations (34) and (35), one can obtain
(41)hD2R¯j′′+1/rDR¯j′−σR¯jR¯j=−Z¯j′′Z¯j=λ.According to equation (41), the fluid flow in the horizontal direction can be written as
(42)R¯j′′+1rDR¯j′−ξR¯j=0,(43)ξ=σ+λhD2.And the fluid flow in the vertical direction is(44)Z¯j′′+λZ¯j=0.Without consideration of the flow in thez-direction and taking equation (42) into equations (34) and (35), we have
(45)∂2R¯fD∂rD2+1rD∂R¯fD∂rD+A1κR¯fD+A2κR¯mD=0,(46)∂2R¯mD∂rD2+1rD∂R¯mD∂rD+A21−κR¯fD+A31−κR¯mD=0.Under the infinite external boundary of side, the solutions of equations (45) and (46) can be expressed by
(47)R¯f=AfK0σrD,(48)R¯m=AmK0σrD.Substituting equations (47) and (48) into (45) and (46), we have
(49)σK0σrDAf+A1κK0σrDAf+A2κK0σrDAm=0,(50)σK0σrDAm+A21−κK0σrDAf+A31−κK0σrDAm=0.Because the modeling must have solutions, the coefficientsAm and Af cannot be zero, so the term σ in equation (50) can be given by
(51)σ=−κA3+1−κA1±κA3+1−κA12−4κ1−κA1A3−A222κ1−κ.With equation (49)–(51), the general solutions of equations (45) and (46) can be expressed by
(52)R¯f=Af,1K0σ1rD+Af,2K0σ2rD,(53)R¯m=Am,1K0σ1rD+Am,2K0σ2rD,(54)Am,1=a1Af,1,Am,2=a2Af,2,(55)a1=−κσ1+A1A2,a2=−κσ2+A1A2.Considering the fluid flow in thez-direction, the general solutions of equations (52) and (53) can be given as
(56)R¯f=Af,1K0ξ1rD+Af,2K0ξ2rD,(57)Rm=a1Af,1K0ξ1rD+a2Af,2K0ξ2rD,(58)ξ1=σ1+λhD2,ξ2=σ2+λhD2.Combined with the boundary conditions, the termsAf,1 and Af,2 in equation (56) are
(59)Af,2=1u1−a1K0ξ1κ+1−κa21−a1K0ξ1−κ+1−κa11−a2K0ξ2,Af,1=1κ+1−κa1u−κ+1−κa2κ+1−κa1Af,2.The general solution of equation (44) can be expressed by
(60)Z¯j=CcosλzD+DsinλzD.Substituting equation (60) into equations (37) and (38), we have
(61)D=0,λ=λn=nπ2,n=0,1,2,⋯,C=12cosλnzwD,λn=nπ2,n=0,1,2,⋯,so the solution in the vertical direction is
(62)Z¯j=12cosλzwDcosλzD.Combined with equations (56), (57), and (62), the pressure solution for a three-dimensional volumetric source can be obtained and expressed by
(63)η¯fD=∑n=0∞R¯fn⋅Z¯fn.According to the superposition principle, the dimensionless pressure in the Laplace space for constant-rate production can be obtained by integrating equation (63) along with the direction of horizontal wellbore.
(64)η¯sD=∑n=0∞∫−L/2/rwL/2/rwR¯fndxD⋅Z¯w.Equation (64) is the dimensionless pseudopressure solution expression considering the skin effect. The wellbore storage effect can be incorporated in the abovementioned solution using Duhamel’s principle [25], and the bottom-hole pressure solution is
(65)η¯wD=η¯sD1+CDu2η¯sD.η¯wD in equation (65) is the dimensionless pressure solution in the Laplace space, and in real space, the dimensionless pressure ηwD can be obtained using Stehfest numerical inversion [32]. After that, utilizing the inverse transformation of equation (66), the bottom-hole pressure response for a horizontal well incorporating the stress-dependent permeability of fracture system can be obtained:
(66)ψwDrD,tD=−1γfDln1−γfDηwDrD,tD.


## 2.1. Model Assumption

As shown in Figure1, the naturally fractured gas reservoir is composed of fracture and matrix systems and the physical properties of the two systems are independent. A radial cylindrical dual-porosity and dual-permeability medium reservoir is considered in which a single horizontal well is located at the center, completely penetrating the formation. The matrix/fracture flow is schematically described in Figure 2. In this study, the proposed model assumes that both the fracture and matrix systems are the flow pathway directly connected with the wellbore and fluids in the fracture and matrix systems first flow into the horizontal wellbore, followed by the matrix-fracture interporosity flow. Some simplifying physical model assumptions for the derivation of the governing equation are listed as the horizontal well produced with the constant production rate in a naturally fractured gas reservoir. The external boundaries of the top and bottom are assumed to be closed, and the lateral boundary is assumed to be infinite. The matrix-fracture interporosity flow in the reservoir is described by the pseudosteady-state model [5, 25, 26]. Fluid flow follows the law of Darcy seepage, and stress-dependent fracture permeability is considered. Also, capillary and gravity forces are neglected to simplify the model.Figure 1
Schematic of naturally fractured reservoir.Figure 2
Dual-porosity and dual-permeability flow scheme.


## 2.2. Mathematical Model

The PVT properties, such as fluid viscosity and volume factor of the gas phase, are quite sensitive to formation pressure. In this section, the pseudopressure transformation is used to capture pressure-dependent PVT properties and reduce the nonlinearity of governing differential equations. The definitions of pseudopressure and pseudotime are given by(1)ψj=2∫0ppμgZdp,j=f,m,ta=∫0tμgictiμgpctpdt,where p is the pressure, MPa; ψf is the pseudopressure of the fracture, MPa2/(mPa·s); ψm is the pseudo pressure of the matrix, MPa2/(mPa·s); t is the time, h; ta is the pseudotime; μg is the gas viscosity, mPa·s; ct is the total compressibility coefficient, MPa−1; and Z is the gas compressibility factor.To describe the degree of stress sensitivity and its influence to fracture permeability, the concept of pseudopermeability modulusγf is defined as [19, 26]
(2)γf=1kf∂kf∂ψf.Equation (2) can be further written as
(3)kf=kfie−γfψi−ψf,where kf is the permeability of fracture, mD; kfi is the initial permeability of fracture, mD; ψi is the initial pseudopressure, MPa2/(mPa·s); and γf is the pseudopermeability modulus of the fracture, (mPa·s)/MPa2.To establish and solve the model, a radial cylindrical system (r) is used to describe the flow of the fracture and matrix system. With consideration of the dual-porosity and dual-permeability flow behavior and stress-dependent fracture permeability, the governing differential equation of the complex system can be described as follows:For the fracture system,(4)∂2ψf∂r2+1r∂ψf∂r+γf∂ψf∂r2+kfvikfhi∂2ψf∂z2+γf∂ψf∂z2+αkmkfhiψm−ψf=eγfψi−ψfϕfμgctf3.6kfhi∂ψf∂t.For the matrix system,(5)∂2ψm∂r2+1r∂ψm∂r+kmvkmh∂2ψm∂z2−αkmkmhψm−ψf=ϕmμgctm3.6kmh∂ψm∂t.For the initial condition,(6)ψft=0=ψmt=0=0.In the inner boundary condition,(7)limε⟶0limr⟶04πhμgε∫zw−ε/2zw+ε/2kmhr∂ψm∂r+kfhie−γfψi−ψfr∂ψf∂rdz=pscTZpTscqsc,z−zw≤ε2.The top and bottom boundaries are assumed to be closed and given by(8)∂ψm∂zz=0=∂ψf∂zz=0=0,∂ψm∂zz=h=∂ψf∂zz=h=0The lateral boundary condition is assumed to be infinite and expressed as(9)limr⟶∞ψm=limr⟶∞ψf=ψi,where ctf is the total compressibility of the fracture, MPa−1; ctm is the total compressibility of the fracture, MPa−1; h is the reservoir thickness, m; kfhi is the initial horizontal permeability of the fracture, mD; kfvi is the initial vertical permeability of the fracture, mD; kmh is the horizontal permeability of the matrix, mD; kmv is the vertical permeability of the matrix, mD; p is pressure, MPa; psc is the pressure at standard condition, MPa; qsc is the surface gas production rate, 104 m3/d; r is the radial distance, m; T is temperature, K; Tsc is the temperature at standard condition, K; z is the vertical distance from the bottom, m; zw is the vertical distance of the horizontal well from the bottom, m; ε is a variable in the z direction, m; ϕf is the porosity of fracture; ϕm is the porosity of the matrix; and α is the geometric shape factor of matrix block, m−2.To make the equations homogeneous, some dimensionless variables are defined and tabulated in Table1. Taking the dimensionless variables into equations (4)–(9), one can obtain the dimensionless differential equations.Table 1
Definitions of the parameters used in this work.
ParametersSymbolDefinitionDimensionless pseudopressure of the fractureψfDψfD=78.489kmh+kfhih/Tqscψi−ψfDimensionless pseudopressure of the matrixψmDψmD=78.489kmh+kfhih/Tqscψi−ψmDimensionless vertical distancezDzD=z/hDimensionless pseudotimetDtD=3.6kmh+kfhi/μrw2ϕmCmt+ϕfCfttaDimensionless reservoir thicknesshDhD=h/rwe−Skfhi/kfpiDimensionless radial distancerDrD=r/rwe−SDimensionless wellbore storage coefficientCDCD=Cs/6.2832ϕfCft+ϕmCmthrw2Dimensionless pseudopermeability modulusγfDγfD=Tqsc/78.489kfhihγfThe permeability ratio of the fracture system to the sum of the fracture and matrix systemκκ=kfhi/kfhi+kmhCapacitance coefficient of the fractureωfωf=ϕfCft/ϕfCft+ϕmCmtInterporosity flow factor of matrix system into the fracture systemλmfλmf=αmkmhrw2/kfhi+kmhFor the fracture system,(10)κ∂2ψfD∂rD2+1rD∂ψfD∂rD−γfD∂ψfD∂rD2+1hD2∂2ψfD∂zD2−γfD∂ψfD∂zD2=eγfDψfDωfe−2S∂ψfD∂tD+λmfe−2SψfD−ψmD.For the matrix system,(11)1−κ∂2ψmD∂rD2+1rD∂ψmD∂rD+1hD2∂2ψmD∂zD2−λmfe−2SψmD−ψfD=ωme−2S∂ψmD∂tD.In the initial condition,(12)ψfDtD=0=ψmDtD=0=0.In the inner boundary condition,(13)limεD⟶0limrD⟶0∫zwD−εD/2zwD+εD/2κrD∂ψmD∂rD+1−κrDe−γfDψfD∂ψfD∂rDdzD=−12,zD−zwD≤εD2.In the outer boundary conditions,(14)∂ψmD∂zDzD=0=∂ψfD∂zDzD=0=0,(15)∂ψmD∂zDzD=1=∂ψfD∂zDzD=1=0,(16)limrD⟶∞ψmD=limrD⟶∞ψfD=0.


## 2.3. Solution to the Mathematical Model

It should be noted that equations (10) and (13) are strongly nonlinear with the consideration of the stress sensitivity of fracture permeability. This is because the stress-dependent fracture permeability is a function of the pseudopressure of the fracture. However, the pressure of fracture is an unknown parameter. Therefore, the mathematical model cannot be solved analytically. In this work, the Pedrosa [31] variable substitution and regular perturbation method are firstly deployed to alleviate the nonlinearity. Then, the Laplace transformation and separation of variables are adopted to address the linearized model. Thus, the model can be solved in the Laplace space and the Stehfest and Harald [32] numerical inversion is used to calculate the pressure in real space.


### 2.3.1. Linearization of the Flow Equation

To linearize the mathematical model, the Pedrosa transformation is employed in this section and given by(17)ψfDrD,tDtD=0=−1γfDln1−γfDηfDrD,tD,where ηfD is an intermediate variable called the perturbation deformation function.After the Pedrosa transformation, equations (10)–(16) can be rewritten as
(18)κ∂2ηfD∂rD2+1rD∂ηfD∂rD+1hD2∂2ηfD∂zD2=e−2S11−γfDηfDωf∂ηfD∂tD+1−ωf∂ψmD∂tD,(19)1−κ∂2ψmD∂rD2+1rD∂ψmD∂rD+1hD2∂2ψmD∂zD2=e−2S1−ωf∂ψmD∂tD+λmf1γfDln1−γfDηfD−ψmD,(20)ηfDrD,tDtD=0=ψmDrD,tDtD=0=0,(21)limεD⟶0limrD⟶0∫zwD−εD/2zwD+εD/21−κrD∂ψmD∂rD+κrD∂ηfD∂rDdzD=−12,zD−zwD≤εD2,(22)∂ηfD∂zDzD=1=∂ψmD∂zDzD=1=0,(23)∂ηfD∂zDzD=0=∂ψmD∂zDzD=0=0,(24)limrD⟶∞ψmD=limrD⟶∞ηfD=0.According to the regular perturbation theory, equations (18)–(24) can be simplified as
(25)κ∂2ηfD0∂rD2+1rD∂ηfD0∂rD+1hD2∂2ηfD0∂zD2=ωfe−2S∂ηfD0∂tD+λmfe−2SηfD0−ψmD,(26)1−κ∂2ψmD∂rD2+1rD∂ψmD∂rD+1hD2∂2ψmD∂zD2=1−ωfe−2S∂ψmD∂tD+λmfe−2SηfD0−ψmD,(27)ηfD0rD,tDtD=0=ψmDrD,tDtD=0=0,(28)limεD⟶0limrD⟶0∫zwD−εD/2zwD+εD/21−κrD∂ψmD∂rD+κrD∂ηfD0∂rDdzD=−12,zD−zwD≤εD2,(29)∂ηfD0∂zDzD=1=∂ψmD∂zDzD=1=0,(30)∂ηfD0∂zDzD=0=∂ψmD∂zDzD=0=0,(31)limrD⟶∞ψmD=limrD⟶∞ηfD0=0.


### 2.3.2. Solution of the Proposed Model

To derive the analytical solution of the model, the mathematical model is translated into the Laplace domain with respect totD:
(32)LηfDrD,tD=η¯fDrD,u=∫0+∞ηfDrD,tDe−utDdtD,(33)LψmDrD,tD=ψ¯mDrD,u=∫0+∞ψmDrD,tDe−utDdtD,where u is the Laplace transform variable.With equations (32) and (33) and taking the Laplace transform of equations (25)–(31), one can obtain the dimensionless mathematical model in the Laplace space:For the fracture system,(34)∂2η¯fD∂rD2+1rD∂η¯fD∂rD+1hD2∂2η¯fD∂zD2+A1κη¯fD+A2κψ¯mD=0.For the matrix system,(35)∂2ψ¯mD∂rD2+1rD∂ψ¯mD∂rD+1hD2∂2ψ¯mD∂zD2+A21−κη¯fD+A31−κψ¯mD=0,where A1=−λmf+uωfe−2S, A2=λmfe−2S, and A3=−λmf+u1−ωfe−2S.In the inner boundary condition,(36)limεD⟶0limrD⟶0∫zwD−εD/2zwD+εD/21−κrD∂ψ¯mD∂rD+κrD∂η¯fD0∂rDdzD=−12u,zD−zwD≤εD2.In the outer boundary conditions,(37)∂ψ¯mD∂zDzD=0=∂η¯fD0∂zDzD=0=0,(38)∂ψ¯mD∂zDzD=1=∂η¯fD0∂zDzD=1=0,(39)limrD⟶∞ψ¯mD=limrD⟶∞η¯fD0=0.This section uses the separation of variables to solve the dual-porosity and dual-permeability modeling of a horizontal well in a naturally fractured reservoir. With the separation of variables, the dimensionless pseudopressure in the Laplace space can be separated by [25]
(40)η¯jD=R¯jrDZ¯jzD,j=m,f.Substituting equation (40) into equations (34) and (35), one can obtain
(41)hD2R¯j′′+1/rDR¯j′−σR¯jR¯j=−Z¯j′′Z¯j=λ.According to equation (41), the fluid flow in the horizontal direction can be written as
(42)R¯j′′+1rDR¯j′−ξR¯j=0,(43)ξ=σ+λhD2.And the fluid flow in the vertical direction is(44)Z¯j′′+λZ¯j=0.Without consideration of the flow in thez-direction and taking equation (42) into equations (34) and (35), we have
(45)∂2R¯fD∂rD2+1rD∂R¯fD∂rD+A1κR¯fD+A2κR¯mD=0,(46)∂2R¯mD∂rD2+1rD∂R¯mD∂rD+A21−κR¯fD+A31−κR¯mD=0.Under the infinite external boundary of side, the solutions of equations (45) and (46) can be expressed by
(47)R¯f=AfK0σrD,(48)R¯m=AmK0σrD.Substituting equations (47) and (48) into (45) and (46), we have
(49)σK0σrDAf+A1κK0σrDAf+A2κK0σrDAm=0,(50)σK0σrDAm+A21−κK0σrDAf+A31−κK0σrDAm=0.Because the modeling must have solutions, the coefficientsAm and Af cannot be zero, so the term σ in equation (50) can be given by
(51)σ=−κA3+1−κA1±κA3+1−κA12−4κ1−κA1A3−A222κ1−κ.With equation (49)–(51), the general solutions of equations (45) and (46) can be expressed by
(52)R¯f=Af,1K0σ1rD+Af,2K0σ2rD,(53)R¯m=Am,1K0σ1rD+Am,2K0σ2rD,(54)Am,1=a1Af,1,Am,2=a2Af,2,(55)a1=−κσ1+A1A2,a2=−κσ2+A1A2.Considering the fluid flow in thez-direction, the general solutions of equations (52) and (53) can be given as
(56)R¯f=Af,1K0ξ1rD+Af,2K0ξ2rD,(57)Rm=a1Af,1K0ξ1rD+a2Af,2K0ξ2rD,(58)ξ1=σ1+λhD2,ξ2=σ2+λhD2.Combined with the boundary conditions, the termsAf,1 and Af,2 in equation (56) are
(59)Af,2=1u1−a1K0ξ1κ+1−κa21−a1K0ξ1−κ+1−κa11−a2K0ξ2,Af,1=1κ+1−κa1u−κ+1−κa2κ+1−κa1Af,2.The general solution of equation (44) can be expressed by
(60)Z¯j=CcosλzD+DsinλzD.Substituting equation (60) into equations (37) and (38), we have
(61)D=0,λ=λn=nπ2,n=0,1,2,⋯,C=12cosλnzwD,λn=nπ2,n=0,1,2,⋯,so the solution in the vertical direction is
(62)Z¯j=12cosλzwDcosλzD.Combined with equations (56), (57), and (62), the pressure solution for a three-dimensional volumetric source can be obtained and expressed by
(63)η¯fD=∑n=0∞R¯fn⋅Z¯fn.According to the superposition principle, the dimensionless pressure in the Laplace space for constant-rate production can be obtained by integrating equation (63) along with the direction of horizontal wellbore.
(64)η¯sD=∑n=0∞∫−L/2/rwL/2/rwR¯fndxD⋅Z¯w.Equation (64) is the dimensionless pseudopressure solution expression considering the skin effect. The wellbore storage effect can be incorporated in the abovementioned solution using Duhamel’s principle [25], and the bottom-hole pressure solution is
(65)η¯wD=η¯sD1+CDu2η¯sD.η¯wD in equation (65) is the dimensionless pressure solution in the Laplace space, and in real space, the dimensionless pressure ηwD can be obtained using Stehfest numerical inversion [32]. After that, utilizing the inverse transformation of equation (66), the bottom-hole pressure response for a horizontal well incorporating the stress-dependent permeability of fracture system can be obtained:
(66)ψwDrD,tD=−1γfDln1−γfDηwDrD,tD.


## 2.3.1. Linearization of the Flow Equation

To linearize the mathematical model, the Pedrosa transformation is employed in this section and given by(17)ψfDrD,tDtD=0=−1γfDln1−γfDηfDrD,tD,where ηfD is an intermediate variable called the perturbation deformation function.After the Pedrosa transformation, equations (10)–(16) can be rewritten as
(18)κ∂2ηfD∂rD2+1rD∂ηfD∂rD+1hD2∂2ηfD∂zD2=e−2S11−γfDηfDωf∂ηfD∂tD+1−ωf∂ψmD∂tD,(19)1−κ∂2ψmD∂rD2+1rD∂ψmD∂rD+1hD2∂2ψmD∂zD2=e−2S1−ωf∂ψmD∂tD+λmf1γfDln1−γfDηfD−ψmD,(20)ηfDrD,tDtD=0=ψmDrD,tDtD=0=0,(21)limεD⟶0limrD⟶0∫zwD−εD/2zwD+εD/21−κrD∂ψmD∂rD+κrD∂ηfD∂rDdzD=−12,zD−zwD≤εD2,(22)∂ηfD∂zDzD=1=∂ψmD∂zDzD=1=0,(23)∂ηfD∂zDzD=0=∂ψmD∂zDzD=0=0,(24)limrD⟶∞ψmD=limrD⟶∞ηfD=0.According to the regular perturbation theory, equations (18)–(24) can be simplified as
(25)κ∂2ηfD0∂rD2+1rD∂ηfD0∂rD+1hD2∂2ηfD0∂zD2=ωfe−2S∂ηfD0∂tD+λmfe−2SηfD0−ψmD,(26)1−κ∂2ψmD∂rD2+1rD∂ψmD∂rD+1hD2∂2ψmD∂zD2=1−ωfe−2S∂ψmD∂tD+λmfe−2SηfD0−ψmD,(27)ηfD0rD,tDtD=0=ψmDrD,tDtD=0=0,(28)limεD⟶0limrD⟶0∫zwD−εD/2zwD+εD/21−κrD∂ψmD∂rD+κrD∂ηfD0∂rDdzD=−12,zD−zwD≤εD2,(29)∂ηfD0∂zDzD=1=∂ψmD∂zDzD=1=0,(30)∂ηfD0∂zDzD=0=∂ψmD∂zDzD=0=0,(31)limrD⟶∞ψmD=limrD⟶∞ηfD0=0.


## 2.3.2. Solution of the Proposed Model

To derive the analytical solution of the model, the mathematical model is translated into the Laplace domain with respect totD:
(32)LηfDrD,tD=η¯fDrD,u=∫0+∞ηfDrD,tDe−utDdtD,(33)LψmDrD,tD=ψ¯mDrD,u=∫0+∞ψmDrD,tDe−utDdtD,where u is the Laplace transform variable.With equations (32) and (33) and taking the Laplace transform of equations (25)–(31), one can obtain the dimensionless mathematical model in the Laplace space:For the fracture system,(34)∂2η¯fD∂rD2+1rD∂η¯fD∂rD+1hD2∂2η¯fD∂zD2+A1κη¯fD+A2κψ¯mD=0.For the matrix system,(35)∂2ψ¯mD∂rD2+1rD∂ψ¯mD∂rD+1hD2∂2ψ¯mD∂zD2+A21−κη¯fD+A31−κψ¯mD=0,where A1=−λmf+uωfe−2S, A2=λmfe−2S, and A3=−λmf+u1−ωfe−2S.In the inner boundary condition,(36)limεD⟶0limrD⟶0∫zwD−εD/2zwD+εD/21−κrD∂ψ¯mD∂rD+κrD∂η¯fD0∂rDdzD=−12u,zD−zwD≤εD2.In the outer boundary conditions,(37)∂ψ¯mD∂zDzD=0=∂η¯fD0∂zDzD=0=0,(38)∂ψ¯mD∂zDzD=1=∂η¯fD0∂zDzD=1=0,(39)limrD⟶∞ψ¯mD=limrD⟶∞η¯fD0=0.This section uses the separation of variables to solve the dual-porosity and dual-permeability modeling of a horizontal well in a naturally fractured reservoir. With the separation of variables, the dimensionless pseudopressure in the Laplace space can be separated by [25]
(40)η¯jD=R¯jrDZ¯jzD,j=m,f.Substituting equation (40) into equations (34) and (35), one can obtain
(41)hD2R¯j′′+1/rDR¯j′−σR¯jR¯j=−Z¯j′′Z¯j=λ.According to equation (41), the fluid flow in the horizontal direction can be written as
(42)R¯j′′+1rDR¯j′−ξR¯j=0,(43)ξ=σ+λhD2.And the fluid flow in the vertical direction is(44)Z¯j′′+λZ¯j=0.Without consideration of the flow in thez-direction and taking equation (42) into equations (34) and (35), we have
(45)∂2R¯fD∂rD2+1rD∂R¯fD∂rD+A1κR¯fD+A2κR¯mD=0,(46)∂2R¯mD∂rD2+1rD∂R¯mD∂rD+A21−κR¯fD+A31−κR¯mD=0.Under the infinite external boundary of side, the solutions of equations (45) and (46) can be expressed by
(47)R¯f=AfK0σrD,(48)R¯m=AmK0σrD.Substituting equations (47) and (48) into (45) and (46), we have
(49)σK0σrDAf+A1κK0σrDAf+A2κK0σrDAm=0,(50)σK0σrDAm+A21−κK0σrDAf+A31−κK0σrDAm=0.Because the modeling must have solutions, the coefficientsAm and Af cannot be zero, so the term σ in equation (50) can be given by
(51)σ=−κA3+1−κA1±κA3+1−κA12−4κ1−κA1A3−A222κ1−κ.With equation (49)–(51), the general solutions of equations (45) and (46) can be expressed by
(52)R¯f=Af,1K0σ1rD+Af,2K0σ2rD,(53)R¯m=Am,1K0σ1rD+Am,2K0σ2rD,(54)Am,1=a1Af,1,Am,2=a2Af,2,(55)a1=−κσ1+A1A2,a2=−κσ2+A1A2.Considering the fluid flow in thez-direction, the general solutions of equations (52) and (53) can be given as
(56)R¯f=Af,1K0ξ1rD+Af,2K0ξ2rD,(57)Rm=a1Af,1K0ξ1rD+a2Af,2K0ξ2rD,(58)ξ1=σ1+λhD2,ξ2=σ2+λhD2.Combined with the boundary conditions, the termsAf,1 and Af,2 in equation (56) are
(59)Af,2=1u1−a1K0ξ1κ+1−κa21−a1K0ξ1−κ+1−κa11−a2K0ξ2,Af,1=1κ+1−κa1u−κ+1−κa2κ+1−κa1Af,2.The general solution of equation (44) can be expressed by
(60)Z¯j=CcosλzD+DsinλzD.Substituting equation (60) into equations (37) and (38), we have
(61)D=0,λ=λn=nπ2,n=0,1,2,⋯,C=12cosλnzwD,λn=nπ2,n=0,1,2,⋯,so the solution in the vertical direction is
(62)Z¯j=12cosλzwDcosλzD.Combined with equations (56), (57), and (62), the pressure solution for a three-dimensional volumetric source can be obtained and expressed by
(63)η¯fD=∑n=0∞R¯fn⋅Z¯fn.According to the superposition principle, the dimensionless pressure in the Laplace space for constant-rate production can be obtained by integrating equation (63) along with the direction of horizontal wellbore.
(64)η¯sD=∑n=0∞∫−L/2/rwL/2/rwR¯fndxD⋅Z¯w.Equation (64) is the dimensionless pseudopressure solution expression considering the skin effect. The wellbore storage effect can be incorporated in the abovementioned solution using Duhamel’s principle [25], and the bottom-hole pressure solution is
(65)η¯wD=η¯sD1+CDu2η¯sD.η¯wD in equation (65) is the dimensionless pressure solution in the Laplace space, and in real space, the dimensionless pressure ηwD can be obtained using Stehfest numerical inversion [32]. After that, utilizing the inverse transformation of equation (66), the bottom-hole pressure response for a horizontal well incorporating the stress-dependent permeability of fracture system can be obtained:
(66)ψwDrD,tD=−1γfDln1−γfDηwDrD,tD.


## 3. New-Type Curves

In this work, a novel semi-analytical model is presented to examine the combined effects of dual-permeability flow behavior and stress-dependent fracture permeability on the transient pressure response of a horizontal well in a naturally fractured gas reservoir. A series of new transient pressure dynamic curves are drawn to observe different flow regimes based on the solution of the proposed model. The differences between the proposed model and traditional models are discussed, and the effects of some critical parameters on transient pressure response are also analyzed with the proposed model. Furthermore, it provides an efficient method for field engineers and related research and further to interpret pressure signals with accurate matching results and more reasonable interpreted parameters. After that, the effects of some critical parameters, including the dimensionless permeability modulus, the storability ratio of fracture, interporosity flow factor of matrix system into fracture system, and skin factor, on the characteristics of the type curves are examined and analyzed.


### 3.1. Flow Regime Identification

Figure3 depicts the standard-type curves of wellbore pressure responses for a horizontal well located at a naturally fractured gas reservoir considering dual-permeability flow and stress-dependent fracture permeability. Basic data used to generate the type curves are listed in Table 2. An entire transient flow process is clearly shown, and the following six main flow stages can be recognized:
(1)
Pure wellbore storage stage: this period is characterized by a slope of 1 on the pressure and pressure derivative curves and governed by the wellbore storage coefficient,C(2)
Skin effect transition stage: the pressure derivative curve exhibits like a “hump.” The peak of the “hump” is dominated by the skin factor,S(3)
Early radial flow stage: this period is present whenever the wellbore storage coefficient,C, and the horizontal wellbore length, L, are suitable. During this period, the pressure derivative curve is a horizontal line with a value of “1/4LD”(4)
Early linear flow stage: it can be identified by a half-slope trend on the pressure derivative curve. During this period, gas flows linearly from the formation to natural fractures(5)
Interporosity flow stage from the matrix system to the fracture system: this process is characterized by a concave in the pressure derivative curve. The concave shape is controlled by the storability ratio of fracture,ωf, and the interporosity flow coefficient from the matrix system to the fracture system, λmf(6)
External boundary response stage: during this period, the pressures in the matrix and fracture systems reach a dynamic balance state. This period is marked by a slope of 0.5 on the pressure derivative curve without considering stress-dependent fracture permeability. However, the pressure derivative curve is no longer a horizontal line with a value of “0.5” but exhibits an upward tendency due to the effect of stress-dependent fracture permeabilityFigure 3
The transient responses with the log-log plot.Table 2
Parameters used for model validation.
ParametersSymbolValueDimensionless wellbore storage coefficientCD1×10−4Skin factorS0.1Dimensionless reservoir thicknesshD400Dimensionless horizontal section positionzwD0.5Dimensionless pseudopermeability modulusγfD0.08The permeability ratio of the fracture system to the sum of the fracture and matrix systemκ0.9Capacitance coefficient of the fractureωf0.05Interporosity flow factor of the matrix system into the fracture systemλmf0.01


### 3.2. Comparisons with the Traditional Models

So far, few attempts have been made to quantify the combined effects of dual-permeability flow behavior and stress-dependent fracture permeability on the transient pressure behavior of a horizontal well in fractured gas reservoirs. The main difference between the proposed and traditional models [25, 26] is that the dual-permeability flow behavior and stress-dependent fracture permeability are all incorporated in the new model. In this section, we simultaneously simulated the pressure response of both the proposed and conventional models using the same group of formation and well parameters in Table 2.Figure4 shows the comparison results of transient responses for the new model with the solution presented by Nie et al. [25]. The main difference between the two models is that stress-dependent fracture permeability is not considered in the model of Nie et al. [25]. As shown in Figure 4, fracture permeability stress sensitivity is found to significantly affect the middle stream flow period and the late-time pseudoradial flow period. There exist obvious differences during late-time pseudoradial flow period. The dimensionless pressure derivative curve exhibits a horizontal line with a value of 0.5 in the model of Nie et al. [25]; however, the derivative curve is no longer horizontal but bends upward in the new model. In addition, the location of the dimensionless pressure and pressure derivative curves during the middle stream flow period is higher in our model. This is because an additional pressure drop will be required to maintain a constant flow rate when the stress-sensitivity effect is taken into account.Figure 4
Comparison of the results of the proposed model with that of Nie et al. [25].We also compared the proposed model with the solution of a single-permeability model [26] based on the same reservoir properties and fracture parameters. The stress-dependent fracture permeability is considered both in two models. As can be seen in Figure 5, the combined effects of dual-permeability flow behavior and stress-dependent fracture permeability play a significant role in the early radial flow stage, the early linear flow stage, and the interporosity flow stage of the matrix system to the fracture system. Dimensionless pressure of the proposed model is lower than that of the single-permeability model [26] during the three flow periods. Besides, the concave in the pressure derivative curve of the dual-permeability model is shallower than that of single-permeability model. This is because the single-permeability model assumes that the fracture system is the only flow pathway directly connected with wellbore; however, the fluid supply from the matrix system to the wellbore is not considered. The dual-permeability flow behavior will accelerate energy supplement in the matrix during production compared with the single-permeability model.Figure 5
Comparison of the results of the proposed model with that of Li et al. [26].


### 3.3. Sensitivity Analysis

Based on the proposed model, the influences of stress sensitivity of fracture permeability, storativity ratio of the fracture, interporosity flow factor, and skin factor on pressure response are discussed. Except for the parameters analyzed, other parameters are the same and are shown in Table2.


#### 3.3.1. Effect of the Permeability Modulus of the Fracture

The stress-sensitivity effect can be determined with the dimensionless permeability modulus,γfD. Figure 6 shows the pressure and pressure derivative curves for γfD=0.04, 0.06, and 0.08. As stated, with increasing the value of γfD, the slope of the derivative curves increases during the intermediate and late time period. This is because the pressure drop increases as the dimensionless permeability modulus increases and fluid flow will be difficult and more gas is left in the reservoir. The stress sensitivity reflects the damage of permeability, and a larger dimensionless permeability modulus will increase the damage of permeability. Consequently, the permeability stress sensitivity of the fracture decreases the cumulative production. So, it is believed that reasonable producing pressure differential is excellent for reducing the negative effect of stress sensitivity on gas productivity in the development of fractured gas reservoirs.Figure 6
Effect of dimensionless permeability modulus,γfD, on type curves.


#### 3.3.2. Effect of the Storability Ratio of the Fracture

The effect of the storability ratio of the fracture,ωf, on transient behavior is shown in Figure 7. As shown in Figure 7, the storability ratio of the fracture not only determines the duration and the depth of the concave but also has a significant effect on the early flow regimes (early radial and early linear flow stage). It can be clearly observed that the larger the ωf is, the deeper and wider the concave in dimensionless pressure derivative curve. In addition, the dimensionless pressure curve becomes higher with the increase of the storability ratio of the fracture. This is because the storativity ratio of fracture reflects the relative capacity of fluid stored in the fracture system; a smaller storability ratio of the fracture is the response of relative abundant reserves in the matrix system. The pressure drop should increase to maintain the constant production rate when increasing the storability ratio of the fracture.Figure 7
Effect of the storability ratio of the fracture,ωf, on type curves.


#### 3.3.3. Effect of the Interporosity Flow Factor

The effect of the interporosity flow factor of the matrix system to the fracture system,λmf, on pressure response is shown in Figure 8. According to the definition of the interporosity flow factor, the λmf represents the starting time of the flow exchange from the matrix system to the fracture system. The larger the λmf is, the earlier the time of the interporosity flow period is. Besides, the late-time pseudoradial flow period would be masked if the λmf is large enough.Figure 8
Effect of the interporosity flow factor,λmf, on type curves.


#### 3.3.4. Effect of the Skin Factor

The effect of the skin factor,S, on pressure response is shown in Figure 9. As shown in Figure 9, skin factor plays a significant role in the early transient flow period. A larger skin factor leads to a higher location of dimensionless pressure and pressure derivative curves. This is because the incremental value of the skin factor results in the increasing additional filtration resistance and the skin effect transition period will last longer. The larger the skin factor is, the slower the pressure wave propagates to the external boundary and the larger the pressure drop is.Figure 9
Effect of skin factor,S, on type curves.


## 3.1. Flow Regime Identification

Figure3 depicts the standard-type curves of wellbore pressure responses for a horizontal well located at a naturally fractured gas reservoir considering dual-permeability flow and stress-dependent fracture permeability. Basic data used to generate the type curves are listed in Table 2. An entire transient flow process is clearly shown, and the following six main flow stages can be recognized:
(1)
Pure wellbore storage stage: this period is characterized by a slope of 1 on the pressure and pressure derivative curves and governed by the wellbore storage coefficient,C(2)
Skin effect transition stage: the pressure derivative curve exhibits like a “hump.” The peak of the “hump” is dominated by the skin factor,S(3)
Early radial flow stage: this period is present whenever the wellbore storage coefficient,C, and the horizontal wellbore length, L, are suitable. During this period, the pressure derivative curve is a horizontal line with a value of “1/4LD”(4)
Early linear flow stage: it can be identified by a half-slope trend on the pressure derivative curve. During this period, gas flows linearly from the formation to natural fractures(5)
Interporosity flow stage from the matrix system to the fracture system: this process is characterized by a concave in the pressure derivative curve. The concave shape is controlled by the storability ratio of fracture,ωf, and the interporosity flow coefficient from the matrix system to the fracture system, λmf(6)
External boundary response stage: during this period, the pressures in the matrix and fracture systems reach a dynamic balance state. This period is marked by a slope of 0.5 on the pressure derivative curve without considering stress-dependent fracture permeability. However, the pressure derivative curve is no longer a horizontal line with a value of “0.5” but exhibits an upward tendency due to the effect of stress-dependent fracture permeabilityFigure 3
The transient responses with the log-log plot.Table 2
Parameters used for model validation.
ParametersSymbolValueDimensionless wellbore storage coefficientCD1×10−4Skin factorS0.1Dimensionless reservoir thicknesshD400Dimensionless horizontal section positionzwD0.5Dimensionless pseudopermeability modulusγfD0.08The permeability ratio of the fracture system to the sum of the fracture and matrix systemκ0.9Capacitance coefficient of the fractureωf0.05Interporosity flow factor of the matrix system into the fracture systemλmf0.01


## 3.2. Comparisons with the Traditional Models

So far, few attempts have been made to quantify the combined effects of dual-permeability flow behavior and stress-dependent fracture permeability on the transient pressure behavior of a horizontal well in fractured gas reservoirs. The main difference between the proposed and traditional models [25, 26] is that the dual-permeability flow behavior and stress-dependent fracture permeability are all incorporated in the new model. In this section, we simultaneously simulated the pressure response of both the proposed and conventional models using the same group of formation and well parameters in Table 2.Figure4 shows the comparison results of transient responses for the new model with the solution presented by Nie et al. [25]. The main difference between the two models is that stress-dependent fracture permeability is not considered in the model of Nie et al. [25]. As shown in Figure 4, fracture permeability stress sensitivity is found to significantly affect the middle stream flow period and the late-time pseudoradial flow period. There exist obvious differences during late-time pseudoradial flow period. The dimensionless pressure derivative curve exhibits a horizontal line with a value of 0.5 in the model of Nie et al. [25]; however, the derivative curve is no longer horizontal but bends upward in the new model. In addition, the location of the dimensionless pressure and pressure derivative curves during the middle stream flow period is higher in our model. This is because an additional pressure drop will be required to maintain a constant flow rate when the stress-sensitivity effect is taken into account.Figure 4
Comparison of the results of the proposed model with that of Nie et al. [25].We also compared the proposed model with the solution of a single-permeability model [26] based on the same reservoir properties and fracture parameters. The stress-dependent fracture permeability is considered both in two models. As can be seen in Figure 5, the combined effects of dual-permeability flow behavior and stress-dependent fracture permeability play a significant role in the early radial flow stage, the early linear flow stage, and the interporosity flow stage of the matrix system to the fracture system. Dimensionless pressure of the proposed model is lower than that of the single-permeability model [26] during the three flow periods. Besides, the concave in the pressure derivative curve of the dual-permeability model is shallower than that of single-permeability model. This is because the single-permeability model assumes that the fracture system is the only flow pathway directly connected with wellbore; however, the fluid supply from the matrix system to the wellbore is not considered. The dual-permeability flow behavior will accelerate energy supplement in the matrix during production compared with the single-permeability model.Figure 5
Comparison of the results of the proposed model with that of Li et al. [26].


## 3.3. Sensitivity Analysis

Based on the proposed model, the influences of stress sensitivity of fracture permeability, storativity ratio of the fracture, interporosity flow factor, and skin factor on pressure response are discussed. Except for the parameters analyzed, other parameters are the same and are shown in Table2.


### 3.3.1. Effect of the Permeability Modulus of the Fracture

The stress-sensitivity effect can be determined with the dimensionless permeability modulus,γfD. Figure 6 shows the pressure and pressure derivative curves for γfD=0.04, 0.06, and 0.08. As stated, with increasing the value of γfD, the slope of the derivative curves increases during the intermediate and late time period. This is because the pressure drop increases as the dimensionless permeability modulus increases and fluid flow will be difficult and more gas is left in the reservoir. The stress sensitivity reflects the damage of permeability, and a larger dimensionless permeability modulus will increase the damage of permeability. Consequently, the permeability stress sensitivity of the fracture decreases the cumulative production. So, it is believed that reasonable producing pressure differential is excellent for reducing the negative effect of stress sensitivity on gas productivity in the development of fractured gas reservoirs.Figure 6
Effect of dimensionless permeability modulus,γfD, on type curves.


### 3.3.2. Effect of the Storability Ratio of the Fracture

The effect of the storability ratio of the fracture,ωf, on transient behavior is shown in Figure 7. As shown in Figure 7, the storability ratio of the fracture not only determines the duration and the depth of the concave but also has a significant effect on the early flow regimes (early radial and early linear flow stage). It can be clearly observed that the larger the ωf is, the deeper and wider the concave in dimensionless pressure derivative curve. In addition, the dimensionless pressure curve becomes higher with the increase of the storability ratio of the fracture. This is because the storativity ratio of fracture reflects the relative capacity of fluid stored in the fracture system; a smaller storability ratio of the fracture is the response of relative abundant reserves in the matrix system. The pressure drop should increase to maintain the constant production rate when increasing the storability ratio of the fracture.Figure 7
Effect of the storability ratio of the fracture,ωf, on type curves.


### 3.3.3. Effect of the Interporosity Flow Factor

The effect of the interporosity flow factor of the matrix system to the fracture system,λmf, on pressure response is shown in Figure 8. According to the definition of the interporosity flow factor, the λmf represents the starting time of the flow exchange from the matrix system to the fracture system. The larger the λmf is, the earlier the time of the interporosity flow period is. Besides, the late-time pseudoradial flow period would be masked if the λmf is large enough.Figure 8
Effect of the interporosity flow factor,λmf, on type curves.


### 3.3.4. Effect of the Skin Factor

The effect of the skin factor,S, on pressure response is shown in Figure 9. As shown in Figure 9, skin factor plays a significant role in the early transient flow period. A larger skin factor leads to a higher location of dimensionless pressure and pressure derivative curves. This is because the incremental value of the skin factor results in the increasing additional filtration resistance and the skin effect transition period will last longer. The larger the skin factor is, the slower the pressure wave propagates to the external boundary and the larger the pressure drop is.Figure 9
Effect of skin factor,S, on type curves.


## 3.3.1. Effect of the Permeability Modulus of the Fracture

The stress-sensitivity effect can be determined with the dimensionless permeability modulus,γfD. Figure 6 shows the pressure and pressure derivative curves for γfD=0.04, 0.06, and 0.08. As stated, with increasing the value of γfD, the slope of the derivative curves increases during the intermediate and late time period. This is because the pressure drop increases as the dimensionless permeability modulus increases and fluid flow will be difficult and more gas is left in the reservoir. The stress sensitivity reflects the damage of permeability, and a larger dimensionless permeability modulus will increase the damage of permeability. Consequently, the permeability stress sensitivity of the fracture decreases the cumulative production. So, it is believed that reasonable producing pressure differential is excellent for reducing the negative effect of stress sensitivity on gas productivity in the development of fractured gas reservoirs.Figure 6
Effect of dimensionless permeability modulus,γfD, on type curves.


## 3.3.2. Effect of the Storability Ratio of the Fracture

The effect of the storability ratio of the fracture,ωf, on transient behavior is shown in Figure 7. As shown in Figure 7, the storability ratio of the fracture not only determines the duration and the depth of the concave but also has a significant effect on the early flow regimes (early radial and early linear flow stage). It can be clearly observed that the larger the ωf is, the deeper and wider the concave in dimensionless pressure derivative curve. In addition, the dimensionless pressure curve becomes higher with the increase of the storability ratio of the fracture. This is because the storativity ratio of fracture reflects the relative capacity of fluid stored in the fracture system; a smaller storability ratio of the fracture is the response of relative abundant reserves in the matrix system. The pressure drop should increase to maintain the constant production rate when increasing the storability ratio of the fracture.Figure 7
Effect of the storability ratio of the fracture,ωf, on type curves.


## 3.3.3. Effect of the Interporosity Flow Factor

The effect of the interporosity flow factor of the matrix system to the fracture system,λmf, on pressure response is shown in Figure 8. According to the definition of the interporosity flow factor, the λmf represents the starting time of the flow exchange from the matrix system to the fracture system. The larger the λmf is, the earlier the time of the interporosity flow period is. Besides, the late-time pseudoradial flow period would be masked if the λmf is large enough.Figure 8
Effect of the interporosity flow factor,λmf, on type curves.


## 3.3.4. Effect of the Skin Factor

The effect of the skin factor,S, on pressure response is shown in Figure 9. As shown in Figure 9, skin factor plays a significant role in the early transient flow period. A larger skin factor leads to a higher location of dimensionless pressure and pressure derivative curves. This is because the incremental value of the skin factor results in the increasing additional filtration resistance and the skin effect transition period will last longer. The larger the skin factor is, the slower the pressure wave propagates to the external boundary and the larger the pressure drop is.Figure 9
Effect of skin factor,S, on type curves.


## 4. Conclusions

This paper provided a semianalytical model to investigate the combined effects of dual-permeability flow behavior and stress-dependent fracture permeability on the transient pressure response for a horizontal well in a naturally fractured gas reservoir. The main conclusions of this work are as follows:(i)
A horizontal production well in a naturally fractured gas reservoir with consideration of stress sensitivity effect may exhibit six flow stages: pure wellbore storage stage, skin effect transition stage, early radial flow stage, early linear flow stage, interporosity flow stage, and external boundary response stage(ii)
The stress-dependent fracture permeability imposes effects on the intermediate flow period and the late-time pseudoradial flow period; the existence of dual-permeability flow behavior can make the stress sensitivity effect more significant(iii)
The storability ratio of the fracture,ωf, mainly affects the duration and the depth of the concave, and a larger ωf leads to a deeper and wider concave in dimensionless pressure derivative curve. In addition, the storability ratio plays a significant role in the early radial flow stage and early linear flow stage. The interporosity flow factor, λmf, mainly affects the starting time of the flow exchange from the matrix to the fracture. The larger the value of λmf, the earlier the occurrence of the interporosity flow period(iv)
The proposed model is suitable for various naturally fractured gas reservoirs and can interpret pressure signals with accurate matching results and more reasonable interpreted parameters



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*Source: 5993701-2022-04-30.xml*