Well-Testing Model for Dual-Porosity Reservoir considering Stress-Sensitivity and Elastic Outer Boundary Condition

Stress sensitivity and the elastic outer boundary (EOB) condition have a great e ﬀ ect on the analysis of the characteristics of the ﬂ uid ﬂ ow in a reservoir. When researchers analyzed the characteristics of the ﬂ uid ﬂ ow, they have considered the stress sensitivity and the EOB condition separately but have not considered them simultaneously. Therefore, errors are inevitable during the analysis of well testing. The main object of this work is to present a well-testing model for stress-sensitivity dual-porosity reservoir (DPR) with EOB to improve the accuracy of the analysis of well-testing data. To this end, in this paper, we established a well-testing model for the DPR, considering the stress sensitivity and the EOB simultaneously


Introduction
During the development of oil and gas reservoirs, pressure drops and effective stress increases, so their permeability changes.Stress sensitivity has an effect on accurately evaluating the characteristics of wells and reservoirs.
Many researchers have researched the stress sensitivity of reservoirs and established models considering it.Experimental researches on oil reservoirs and coalbed methane reservoirs were also carried out [1][2][3][4][5][6][7][8][9][10].Their experiments determined the effect of the formation stress on the permeability and its relationship.In order to study the relationship between the compressibility of the pore-fracture system and its effective stress, Zhang et al. [11] conducted an experiment with the nuclear magnetic resonance technique, calculated the stress sensitivity of the pore and fracture, and discussed the variation of its heterogeneity.Based on many compaction researches of rocks, Chilingar et al. [12] argued that well testing could be wrong due to the plastic deformation that increases the effective stress in undercompacted overpressured reservoirs, and thus, many erroneously condemned overpressured reservoirs should be reexamined and reevaluated, and techniques should be developed to recover the oil and gas from these stress-sensitivity reservoirs.Guo et al. [13] and Meng et al. [14] presented semianalytical models to evaluate production features in the stress-sensitivity carbonate gas reservoirs, which were considered triple-porosity media composed of matrix, fractures, and vugs.Aguilera [15] proposed a material-balance equation considering the effective compressibility of fractures and matrix, and Wang et al. [16,17], Luo et al. [18], and Mo et al. [19] established the fluid flow model for the stress-sensitivity fracture reservoirs.Zhang et al. [20] suggested the well-testing model for the stress-sensitivity DPR, where its thickness and fluid property vary in the radial direction, and Tian et al. [21] presented the well-testing model for a multiregion radially composite reservoir considering the stress sensitivity.Ren and Guo [22] researched the general method for analyzing nonsteady flow, and Jelmert and Toverud [23] obtained the approximate analytical solution for stress-sensitivity deformable reservoirs and drew the corresponding type curve.Xia et al. [24] established the numerical model, considering nonlinear filter features, wellbore storage, and skin, and proposed the general method for evaluating the effect of fracture on the volume strain of a vertical well based on the well-testing and production data.Zhu and Liang [25] derived the production equation considering stress sensitivity and analyzed the effect of stress sensitivity on production.Moradi et al. [26] studied three-dimensional transient pressure features and pressure drop in the stress-sensitivity reservoirs during the production of hydrocarbons, and Zhu et al. [27] proposed the analytical model for stress-sensitivity coalbed methane reservoirs to study the permeability evolution during production.Based on a combination of geomechanics and flow characteristics, Li et al. [28] presented the production model for triple-porosity media, taking the adsorption and desorption of gas, slip flow, Knudsen, interfacial diffusions, and stress sensitivity into account.Chen et al. [29] established the mathematical model for tight gas reservoirs considering threshold pressure gradient, gas slipping, and stress sensitivity, and Xue et al. [30] conducted research on tight sandstone gas reservoirs with water produced and showed that strong stress sensitivity appeared as a result and increased with the water production.Zhao et al. [31], Jabbari et al. [32], Samaniego and Villalobos [33], and Zhang et al. [34] carried out research on the stress-sensitivity of fracture reservoirs, and Wang and Wang [35] proposed the mathematical model considering the effect of slipping and stress sensitivity in fractured gas reservoirs.Huang et al. [36] presented the transient flow model for horizontal wells in stress-sensitivity composite reservoirs, and Li et al. [37] presented the dual-porosity media model for horizontal wells in fractured tight gas reservoirs with stress sensitivity.Yuan et al. [38] suggested a general solution for oil flow, taking multiphase flow, the properties of stress-sensitivity reservoirs, and the changes in operation conditions into consideration.Li et al. [39] and Wang et al. [40] researched the characteristics of fluid flow in a stress-sensitivity tight reservoir, where they considered the reservoir as an outer unstimulated reservoir volume region and an inner stimulated reservoir volume region.Jiang and Yang [41] presented the fully coupled fluid flow and geomechanics model in fractured shale gas reservoirs, which characterized stress-sensitivity productivity, and Guo et al. [42] proposed transient pressure and decline rate analysis in shale gas reservoir, considering multiflow mechanics including desorption, diffusion, Darcy's flow, and stress sensitivity.Wu et al. [43], Xu et al. [44], Zhang and Yang [45,46], and Zongxiao et al. [47] presented well-testing models for multifractured horizontal stresssensitivity reservoirs.Wu et al. [48], Ji et al. [49], Huang et al. [50], and Wu et al. [51] established the well-testing models for the multifractured horizontal well in the stresssensitivity tight reservoirs.Wang et al. [52], Liu et al. [53], and Du et al. [54] presented the well-testing models for the multifractured horizontal wells in the stress-sensitivity shale gas, and Chen et al. [55] and Wang et al. [56] presented the mathematical model of the transient pressure features in the stress-sensitivity coalbed methane reservoirs.Liu [57] discussed the well-testing model for multifractured horizontal wells, considering the effect of stress sensitivity and the threshold pressure gradient in a low-permeable reservoir.Zhang et al. [58] built the analytical well-testing model for a low-permeable reservoir with consideration of anisotropy, heterogeneity, stress sensitivity, wellbore storage, and skin, and Wang et al. [59] presented the analytical model for multilateral horizontal wells considering the combination of multiple branches and stress sensitivity in low-permeability natural fracture reservoirs.Cao et al. [60] and Yan et al. [61] researched the transient pressure behavior of wells, considering the effects of sand production and stress sensitivity simultaneously.Zhang and Tong [62] studied the transient pressure response of fractal media in stress-sensitive lowpermeability reservoirs.Gao et al. [63] established a welltesting model of pressure buildup considering stress sensitivity and the hysteresis effect in deep-water composite reservoirs with high temperature and pressure.Shovkun and Espinoza [64] conducted the coupled fluid flow and geomechanics simulation for the coal and shale reservoirs with consideration of the impact of desorption-induced stress, shear failure, and fines migration.The well-testing models relevant to the stress sensitivity mentioned above have been considered only for reservoirs with ideal outer boundaries.However, there are few ideal outer boundaries in reality.Therefore, when the outer boundary is considered the ideal boundary, the reservoirs cannot be described objectively, and errors are inevitable.
SSP makes its heterogeneity stronger in heterogeneous reservoirs.Although such strong heterogeneity gives for detailed characterization of different aspects of the reservoir, it may be tedious and time-consuming in real application.In order to simply describe the stress-sensitivity reservoir, many researchers have introduced a conception of the permeability modulus to ensure the robustness of the analytical solution [20,36,40,47,48,52,58,59].
Several researchers studied the fluid flow characteristics of the reservoirs with the EOB.Li et al. [65] studied the seepage model for the homogeneous reservoir by introducing the EOB, and Li et al. [66] discussed an elastic boundary value problem of the extended modified Bessel equation introducing the elastic boundary value condition, established the seepage model for fractal homogeneous reservoirs with the EOB, and obtained its solution.Kim et al. [67] built the well-testing model for the DPRs with the EOB 2 Geofluids considering skin and wellbore storage and introducing effective well radius, and Zheng et al. [68] established the dualporosity media seepage model for shale reservoirs with the EOB taking the adsorption and desorption processes into account.Zhao and Min [69] presented the non-Newtonian power-law fluid percolation model with the EOB.The stress sensitivity of the reservoir is not considered in the models with the EOB mentioned above.Both stress sensitivity and the EOB condition significantly affect the characteristics of fluid flow in reservoirs, but stress sensitivity and the elastic boundary condition were not considered simultaneously in previous researches.The precedent researchers have considered either fluid flow in the stress-sensitivity reservoirs with the ideal boundary conditions or the elasticity of the boundary in the reservoirs without considering stress sensitivity.This affects the analysis of well-testing data and results in considerable errors.
In this paper, the well-testing model for DPR is established considering the stress sensitivity and EOB condition simultaneously, and its solution is obtained to improve the accuracy of the analysis of well-testing data.On the basis of the consideration of the EOB condition and SSP, the seepage model for the DPR with the EOB is built using the continuity equation, motion equation, state equation, and interporosity flow equation between matrix and fracture.By applying effective well radius, Pedrosa's transformation, perturbation transformation, and the Laplace transformation, an analytical solution in the Laplace space is obtained, and curves of PPD are drawn by numerically inverting them.This work may be significant for evaluating more accurately the parameters of wells and reservoirs using well testing.

EOB Condition
Li et al. [65] defined an elastic coefficient of the reservoir as follows: where ε P r is the rate of the relative change of the pressure difference P with respect to r. p is the function of time (t) and position (x, y, and z).p = p x, y, z, t , 3 The farther the distance from a well is, the lower the pressure difference is.Thus, the change direction of pressure difference is opposite to the change direction of the position.
Therefore, the elastic coefficient of the outer boundary in a cylinder reservoir can be defined as follows [67]: From Eq. ( 4), the EOB condition may be obtained as follows: = 0, 5 Equations ( 5) or ( 6) are elastic outer conditions.

Physical Model
(1) Reservoir consists of the natural fracture and matrix, and fluid flows through both the natural fracture and matrix system.That is, the reservoir has dual porosity and dual permeability characteristics (2) The permeability of the natural fracture system is considered stress sensitivity This physical model is suitable for matrix-fracture dualporosity reservoirs, including naturally fractured carbonate reservoirs.This requires that the reservoir with a uniform thickness be on a horizontal plane.

Mathematical Model.
Using the state equation, motion equation, interporosity flow equation, and continuity equation, a seepage differential equation for dual porosity reservoir considering the stress sensitivity is obtained.For convenience, the seepage model is nondimensionalized using dimensionless variables.The detailed derivation of the mathematical model is provided in Appendix A.
The seepage differential equation for the matrix system in the natural fracture reservoir is as follows.
The seepage differential equation for the natural fracture system in the natural fracture reservoir is as follows.
The initial condition is as follows: The inner boundary condition is as follows: The outer boundary condition is as follows: Introducing the dimensionless variables, Eqs. ( 7)~( 12) are nondimensionalized, and these equations are arranged using r De = r D e S and T D = t D /C D .
The dimensionless seepage differential equation for the natural fracture system in the natural fracture reservoir is as follows.
The dimensionless seepage differential equation for the matrix system in the natural fracture reservoir is as follows.
The dimensionless initial condition is as follows: The dimensionless inner boundary condition is as follows: The dimensionless outer boundary condition is as follows: 3.3.Solution of Model.The seepage differential equation (Eq.( 13)) for fracture system and inner boundary condition (Eq.( 15)) have strong nonlinearity.Therefore, Pedrosa's transformation and perturbation transformation are applied so as to linearize these equations.Laplace transformation is applied to obtain its solution in the Laplace domain.The detailed derivation process is provided in Appendix B. Applying Pedrosa's transformation, perturbation transformation, and the Laplace transformation, the following equations are obtained.
The seepage differential equation for natural fracture system is as follows: The seepage differential equation for matrix system is as follows: The inner boundary condition is as follows: The outer boundary condition is as follows: The special solutions to Eqs. ( 19) and ( 20) are obtained, and based on the special solutions, a general solution to Eqs. (19) and ( 20) is obtained.From the special solutions, the general solution to Eqs. ( 19) and ( 20) may be expressed as follows. where Substituting the general solution Eq. ( 24) into the inner boundary condition (21) and outer boundary condition (23), the following equations are obtained. Geofluids where From Eqs. ( 26) to (27), A f and B f are obtained as follows.
From Eq. ( 22), ξ wD0 is obtained by numerically inverting the Laplace transformation (Eq.( 34)).Many papers related to the welltesting model have used the Stehfest algorithm because of its simplicity and high accuracy.The algorithm proposed by Kim et al. [70] has lower numerical oscillation and higher accuracy than the Stehfest algorithm, and thus, this paper uses the algorithm proposed by Kim et al. [70], where n = 25, a = 6 5, k = 2, and σ = 0.
Bottom-hole PPD is calculated with the following equations.Γ = 0, 0 1, 1 000.Figure 1 shows the result of comparison of the two models.The solid lines represent our model, and the circle marks the dual porosity without consideration of stress sensitivity [67].From Figure 1, it can be seen that the two models agree well, which shows that our model is valid.

Results and Discussion
4.1.2.Verification Using Case Data.For verification, case data presented by Wang et al. [71] is used, which is obtained from an oil well in the Tahe oilfield.Figure 2 shows the result of matching with case data.As seen in Figure 2 3, the larger the value of γ D , the higher the PPD curves.As time increases, the effect of the permeability modulus on PPD curves increases.When γ D ≠ 0, horizontal line of value 0.5 does not appear in the middle stage of the pressure derivative curve.This means that the SSP affects the analysis of the well-testing data.Γ increases, the higher the late stage of pressure curves gets from the horizontal line to the curve of the closed boundary and the higher the "hump" of the pressure derivative curve is.However, the outer boundary condition between the closed boundary and constant pressure boundary has not been considered in the model of the reservoir with the ideal outer boundary (infinite boundary, closed boundary, and constant pressure boundary).Thus, our model has generality in comparison to the model with an ideal outer boundary.Therefore, this means that the accuracy of well-testing analysis can be enhanced.

Effect of C D e 2S
on the PPD Curves.C D e 2S is a dimensionless quantity, which considers wellbore storage and skin simultaneously and shows degrees of improvement or damage to the well.
Figure 5 shows the effect of C D e 2S (C D e 2S = 1 000, 2 000, 5 000, and 10 000) on the PPD curves.Other parameters are as follows: S = 1, γ D = 0 01, ε P D Γ = 100, λ = 10 −5 , ω = 10 −5 , R De = 2 000, and K = 0 999.C D e 2S affects the middle stage of the pressure curve.The larger the value of C D e 2S increases, the higher the pressure derivative curves are in the middle stage and the earlier the effect of boundary appears on the pressure derivative curves.4.2.4.Effect of Skin S on the PPD Curves.Figure 6 shows the effect of skin S (S = −3, -1, 0, 1, 3) on the PPD curves.Other parameters are as follows: γ D = 0 01, ε 10, R D = 5 000, and K = 0 999.Skin S affects the middle and late stages of the PPD.The smaller value of S decreases, the upper the PPD curves are, the higher "hump" of the pressure derivative curve gets, and the later it appears.Other parameters are as follows: γ D = 0 01, ε 10, S = 1, R D = 5 000, and K = 0 999.The interporosity flow coefficient λ affects the middle stage of the PPD curves.The larger the value of λ increases, the upper the PPD curves are, the higher the "hump" of the pressure derivative curve gets, the deeper the concave of the pressure derivative curve is, and the later the concave appears.

Effect of Storage
Ratio ω on the PPD Curves.Figure 9 shows the effect of the storage ratio ω (ω = 10 −1 , 10 -2 , and 10 -3 ) on the PPD curves.Other parameters are as follows: γ D = 0 01, ε P D Γ = 1, λ = 10 −4 , C D e 2S = 10, S = 1, R D = 5 000, and K = 0 999.ω affects both the early and middle periods of flow.The smaller the value of ω gets, the upper the pressure curves are, the higher the "hump" of the pressure derivative curve gets, the deeper the concave is, and the earlier the concave appears.

Effect of Permeability Ratio K on the PPD Curves.
Figure 10 shows the effect of the permeability ratio K (K = 0 999, 0.9, 0.85, and 0.8) on the PPD curves.Other parameters are as follows: γ D = 0 001, ε 1 000, S = 0, and R D = 500.K affects both the early and middle periods of flow.The smaller the value of K, the upper the pressure curves are, and the higher the "hump" of the pressure derivative curve gets, the deeper the concave are.
Both the SSP and the condition of elasticity of the outer boundary greatly affect the well-testing analysis, but the previous papers have considered these effects individually and have not considered them simultaneously, which may result in considerable errors in the well-testing analysis.The well-testing model proposed in our paper considers the effect of the SSP and the condition of elasticity of the outer boundary simultaneously.Thus, this model may improve the accuracy of the well-testing analysis for the DPR.In reality, there is not only the dual-porosity reservoir for the vertical well but also the triple-porosity reservoir for the vertical well and the dual-porosity reservoir for the    7 Geofluids horizontal well.However, this model is limited to the dualporosity model for vertical wells.The model proposed in this paper may be improved into dual-porosity and triple-porosity models for horizontal and inclined wells.We are going to study dual-porosity and the triple-porosity dualpermeability model for horizontal and inclined wells, considering stress sensitivity and EOB.

Conclusions
In this paper, the well-testing model for the DPR is presented, considering SSP and EOB simultaneously.
(1) The seepage differential equation for the DPR considering stress sensitivity and EOB simultaneously is established using the continuity equation, motion equation, state equation, and interporosity flow equation between matrix and fracture (2) The dimensionless seepage differential equation is obtained using dimensionless variables (3) Applying Pedrosa's transformation and perturbation transformation, the nonlinearity of the nonlinear seepage differential equation is decreased (4) The analytical solution in the Laplace space is obtained by using the Laplace transformation, and the type curves of PPD are drawn by applying the Laplace numerical inversion [70] (5) This model is verified by comparing it to the model in [67] with the EOB without consideration of SSP and using case data (6) Through analysis of the sensitivity of parameters, it can be seen that both the SSP and EOB conditions affect the analysis of well-testing data This model may improve the accuracy of the analysis of the well-testing data as compared with the previous welltesting model for DPR.

Appendix A. Mathematical Model
In order to describe the degree of the SSP of the reservoir and its influence, Pedrosa [72] introduced the concept of a permeability modulus for the homogenous reservoir.As the reservoir pressure decreases, the opening of the natural fracture decreases.Therefore, the permeability of the natural fracture also decreases.
The permeability modulus is defined as follows: Integrating Eq. (A.1),

Geofluids
It is assumed that the fluid flows in the matrix and natural fracture system obey Darcy's law.The following equation may be obtained, which shows the fluid flow in a natural fracture.
Substituting Eq. (A.2) into Eq.(A.3), The motion equation of fluid flow in a matrix system is as follows.
To describe the changes in oil density in the natural fracture and matrix system during the production of oil, the following equations are applied, respectively.
To describe the changes in porosity of the natural fracture and matrix system during the production of oil, the following equations are applied, respectively.
It is assumed that the interporosity flow is pseudosteady state.Then, the following equation should be satisfied [73]: The continuity equation of fluid flow in the natural fracture system may be obtained as follows: The continuity equation of fluid flow in the matrix system may be obtained as follows: In order to obtain a seepage differential equation for the matrix system, Eqs.(A.5), (A.7), (A.9), and (A.10) are substituted into Eq.(A.12).
Equation (A.13) is a seepage differential equation for the matrix system in the natural fracture reservoir.
) is a seepage differential equation for the natural fracture system considering the SSP in the natural fracture reservoir.
The reservoir has uniform pressure at the initial stage of oil production.Therefore, the initial condition is as follows.
Inner boundary condition is as follows.
Considering the outer boundary as the elastic boundary, the outer boundary condition can be obtained as follows.
For convenience, the following dimensionless variables are defined.
The dimensionless pressure is The storage ratio is ω = w .The dimensionless permeability modulus is γ D = qBμ/ 2πk f i h γ.

Geofluids
Permeability ratio of fracture system to the sum of fracture and matrix system Introducing these dimensionless variables, the dimensionless equations can be obtained from Eqs. (A.13)~(A.18).
The dimensionless seepage differential equations for the natural fracture system and matrix system in the natural fracture reservoir are as follows.
Dimensionless initial condition: Dimensionless inner boundary condition A 24 Dimensionless outer boundary condition: The dimensionless well radius is introduced as follows: Taking T D = t D /C D , the dimensionless seepage differential equations for the natural fracture system and matrix system in the natural fracture reservoir can be written as follows. De Dimensionless initial condition:

B. Solution of Model
The seepage differential equation (Eq.(A.27)) for the fracture system and inner boundary condition (Eq.(A.30)) has strong nonlinearity.Therefore, to linearize these equations, Pedrosa's transformation and perturbation transformation are applied.Pedrosa's transformation is as follows [72]: Applying Pedrosa's transformation to Eqs. (A.27) and (A.29) and rearranging, the following equations are obtained.
The seepage differential equations for the natural fracture and matrix system in the natural fracture reservoir are as follows.
Initial condition: Inner boundary condition: Outer boundary condition: The perturbation transformations are as follows [72]: As the value of γ D is very small (γ D < <1), the solution of zero-order perturbation can satisfy the accuracy requirement.Thus, seepage differential equations, initial conditions, and inner and outer boundary conditions are written as follows.
Seepage differential equation for natural fracture system: Seepage differential equation for matrix system: Initial condition: Outer boundary condition: The Laplace transformations are defined as follows: The Laplace transformations of Eqs.(B.11)~(B.16)are taken with respect to T D .
The equations for the natural fracture and matrix system in the Laplace space can be written as follows.
Inner boundary condition in the Laplace space is as follows.
Outer boundary condition in the Laplace space is as follows. where From Eqs. (B.24) to (B.25), the following equations are obtained.
Substituting ξ f D0 = B f I 0 σr and p mD = B m I 0 σr into Eqs.(B.19) and (B.20), the above results are also obtained.
From the special solutions, the general solution to Eqs. (B.19) and (B.20) may be expressed as follows.
Taking A m = − σ + A 1 /A 2 A f , B m = − σ + A 1 /A 2 B f , and Taking where Oil compressibility, Pa -1 K: Permeability ratio of fracture the sum of fracture and matrix system, fraction k m , k f : Permeability of matrix and natural fracture system, respectively, m 2 k f i : Permeability of natural fracture under initial condition, m 2 h: Reservoir thickness, m p: Reservoir pressure, Pa R: Outer boundary radius, m R D : Dimensionless outer boundary radius, dimensionless p m , p f : Pressure in matrix system and natural fracture system, respectively, Pa p i : Initial reservoir pressure, Pa p w : Wellbore pressure, Pa q: Well rate, m 3 /s q * : Mass flow velocity between matrix and fracture system in the reservoir of unit volume, kg/(m 3 s) r: Radius, m r w : Well radius, m S: Skin, dimensionless t: Time, s t D : Dimensionless time, dimensionless z: Laplace variable, dimensionless α: Shape factor between matrix and fracture, m -2 ρ m , ρ f : Oil density at pressure p m in the matrix system and at pressure p f in natural fracture system, respectively, kg/m 3 ρ o : Oil density under initial condition, kg/m 3 γ: Permeability modulus, Pa -1 γ D : Dimensionless permeability modulus, dimensionless λ: Dimensionless interporosity flow coefficient, dimensionless ω: Storage ratio, dimensionless μ: Viscosity of oil, Pa•s φ m , φ f : Porosity of matrix and natural respectively, fraction φ mi , φ f i : Porosity of matrix and natural fracture system under initial condition, respectively, fraction v m , v f : Flow velocity of fluid in matrix and natural fracture system, respectively, m/s.

( 3 )( 4 ) 5 )
Reservoir has a uniform thickness of h Isotropic fluid flow in a single phase is assumed (The effects of gravity and capillary are negligible

Γ
of Outer Boundary on the PPD Curves.According to Li et al. [65], ε P D Γ ⟶ 0 reflects the closed boundary, ε P D Γ ⟶ +∞ reflects the constant pressure boundary, R D ⟶ ∞ reflects the infinite boundary, and 0 < ε P D Γ < +∞ reflects between the closed boundary and constant pressure boundary.

Figure 1 : 5 GeofluidsFigure 4 Γ
Figure 1: Result of comparison of our model to the model without consideration of stress sensitivity.

4. 2 . 5 .
Effect of Dimensionless Radius R D on the PPD Curves.

0
Case data (pressure) Case data (pressure derivative) Paper model (pressure) Paper model (pressure derivative)

Figure 2 :Figure 3 :
Figure 2: Result of matching with case data.

Figure 4 :
Figure 4: Effect of elastic coefficient ε P D Γ of outer boundary on the PPD curves.

Figure 5 :
Figure 5: Effect of C D e 2S on the PPD curves.

Figure 6 :
Figure 6: Effect of skin S on the PPD curves.

Figure 7 :
Figure 7: Effect of dimensionless radius R D on the PPD curves.

8 Figure 10 :
Figure 10: Effect of permeability ratio K on the PPD curves.

−
Kr De e γ D p f D ∂p f D ∂r De + 1 − K r De ∂p mD ∂r De r De =1 = 1, A 30p wD = p f D 1, T D = p mD 1, T D A 31Dimensionless outer boundary conditionε P f D Γ p f D + r De ∂p f D ∂r De r De =R De = ε P f D Γ p mD + r De ∂p mD ∂r De r De =R De = 0, A32where R De = R D e S .

39 B f = d 1 z −c 1 d 2 + c 2 d 1 B 40 From
1 = ε P f D Γ K 0 σR De − R De σK 1 σR De , B37d 2 = ε P f D Γ I 0 σR De + R De σI 1 σR De B38From (B.33) to (B.34),A f and B f are obtained as follows.A f = −d 2 z −c 1 d 2 + c 2 d 1 , B Eq. (B.22), ξ wD0 = ξ f D0 1, T D = A f K 0 σ + B f I 0 σ B 41NomenclatureB:Volume factor, dimensionless C:Wellbore storage coefficient, m 3 /Pa C D :Dimensionless wellbore storage, dimensionlessε P f D Γ A f K 0 σR De + B f I 0 σR De + A f R De − σ K 1 σR De + B f R De σI 1 σR De = 0, B31A f ε P f D Γ K 0 σR De − R De σK 1 σR De + B f ε P f D Γ I 0 σR De + R De σI 1 σR De = 0 B 3212 Geofluids C m , C f : Compressibility of matrix and natural fracture system, respectively, Pa -1 C mt :Total compressibility of matrix system and oil,C mt = C m + φ m C o , Pa -1 C ft :Total compressibility of natural fracture system and oil, C ft = C f + φ f C o , Pa -1 C o :
r De , T D T D=0 = p mD r De , T D T D=0 AK 0 σr and y 2 = BI 0 σr .Using 11 Geofluids these special solutions, σ and coefficients A and B may be obtained.Substituting ξ f D0 = A f K 0 σr and p mD = A m K 0 σr into Eqs.(B.19) and (B.20) and arranging , and A 4 = λe −2S /1 − K .The special solutions of Eqs.(B.19) and (B.20) may be expressed by y 1 = 4A 2 A 3 /2 and substituting the general solution (B.28) into the inner boundary condition (B.21), the following equation is obtained.