A quadruple-porosity model for transient production analysis of multiple-fractured horizontal wells in shale gas reservoirs

Abstract

Multiple types of pores are present in shale gas reservoirs, including organic pores of nano scale, non-organic pores, natural and hydraulic fractures. Gas flow in different types of pores is controlled by different mechanisms, and Darcy’s law is not able to describe all these processes adequately. This paper presents a “quadruple-porosity” model and corresponding analytical solutions to describe the different permeable media and simulate transient production behavior of multiple-fractured horizontal wells in shale gas reservoirs. Dimensionless transient production decline curves are plotted, and characteristic bilinear flow and linear flow periods are identified based on the analysis of type curves. Sensitivity analysis of transient production dynamics suggests that desorption of absorbed gas, Knudsen diffusive flow, gas slippage and parameters related to hydraulic fractures have significant influence on the production dynamics of a multiple-fractured horizontal well in shale gas reservoirs. The model provides insights into multiple shale gas flow mechanisms and production prediction of shale gas reservoirs.

Appendices

Appendix A: non-organic system solution

The Laplace transformation of the dimensionless governing equation of non-organic system, i.e., Eq. (27), yields the following equation in the Laplace domain:

$$\frac{{\partial^{2} \overline{m}_{\text{mD}} }}{{\partial z_{\text{D}}^{ 2} }} = \frac{{3s\omega_{\text{m}} }}{{\lambda_{\text{mf}} \beta_{\text{m}} }}\overline{m}_{\text{mD}}$$

(49)

Corresponding dimensionless boundary conditions, i.e., Eqs. (29) and (30), can also be transformed into the Laplace domain:

$$\left. {\frac{{\partial \overline{m}_{\text{mD}} }}{{\partial z_{\text{D}} }}} \right|_{{z_{\text{D}} = 0}} = 0$$

(50)

$$\left. {\overline{m}_{\text{mD}} } \right|_{{z_{\text{D}} = 1}} = \overline{m}_{\text{fD}}$$

(51)

Equation (49) together with Eqs. (50) and (51) compose a boundary value problem, and the general solution of Eq.(49) can be obtained as follows:

$$\bar{m}_{\text{mD}} = A_{1} \cosh \left( {z_{\text{D}} \sqrt {\frac{{3s\omega_{\text{m}} }}{{\lambda_{\text{mf}} \beta_{\text{m}} }}} } \right) + B_{1} \sinh \left( {z_{\text{D}} \sqrt {\frac{{3s\omega_{\text{m}} }}{{\lambda_{\text{mf}} \beta_{\text{m}} }}} } \right)$$

(52)

where A ₁ and B ₁ are coefficients which can be determined with corresponding boundary conditions of non-organic system.

Taking derivative of Eq. (52) with respect to z _D and substituting it into Eq. (50), the coefficient B ₁ can be obtained as follows:

$$B_{1} = 0$$

(53)

Using the boundary condition given by Eq. (51), the coefficient A ₁ can be obtained as follows:

$$A_{1} = \frac{{\overline{m}_{\text{fD}} }}{{\cosh \left( {\sqrt {\frac{{3s\omega_{\text{m}} }}{{\lambda_{\text{mf}} \beta_{\text{m}} }}} } \right)}}$$

(54)

The final solution of the dimensionless mathematical model for non-organic system thus can be obtained by substituting Eqs. (53) and (54) into Eq. (52):

$$\bar{m}_{\text{mD}} = \frac{{\cosh \left( {z_{\text{D}} \sqrt {\frac{{3s\omega_{\text{m}} }}{{\lambda_{\text{mf}} \beta_{\text{m}} }}} } \right)}}{{\cosh \left( {\sqrt {\frac{{3s\omega_{\text{m}} }}{{\lambda_{\text{mf}} \beta_{\text{m}} }}} } \right)}}\bar{m}_{\text{fD}}$$

(55)

Appendix B: organic system solution

The Laplace transformation of the dimensionless governing equation of organic system, i.e., Eq. (31), yields the following equation in the Laplace domain:

$$\frac{{\partial^{2} \bar{m}_{\text{kD}} }}{{\partial z_{\text{D}}^{ 2} }} = \frac{{3s\omega_{\text{k}} }}{{\lambda_{\text{kf}} \beta_{\text{k}} }}\left( {1 + \sigma_{\text{k}} } \right)\overline{m}_{\text{kD}}$$

(56)

Corresponding dimensionless boundary conditions, i.e., Eqs. (33) and (34), can also be transformed into the Laplace domain:

$$\left. {\frac{{\partial \bar{m}_{\text{kD}} }}{{\partial z_{\text{D}} }}} \right|_{{z_{\text{D}} = 0}} = 0$$

(57)

$$\left. {\bar{m}_{\text{kD}} } \right|_{{z_{\text{D}} = 1}} = \bar{m}_{\text{fD}}$$

(58)

Equation (56) together with Eqs. (57) and (58) also compose a boundary value problem, and the general solution of Eq. (56) can be obtained as follows:

$$\bar{m}_{\text{kD}} = A_{2} \cosh \left( {z_{\text{D}} \sqrt {\frac{{3s\omega_{\text{k}} \left( {1 + \sigma_{\text{k}} } \right)}}{{\lambda_{\text{kf}} \beta_{\text{k}} }}} } \right) + B_{2} \sinh \left( {z_{\text{D}} \sqrt {\frac{{3s\omega_{\text{k}} \left( {1 + \sigma_{\text{k}} } \right)}}{{\lambda_{\text{kf}} \beta_{\text{k}} }}} } \right)$$

(59)

where A ₂ and B ₂ are coefficients which can be determined with corresponding boundary conditions of organic system.

Taking derivative of Eq. (59) with respect to z _D and substituting it into Eq. (57), the coefficient B ₂ can be obtained as follows:

$$B_{2} = 0$$

(60)

With the boundary condition given by Eq. (58), the coefficient A ₂ can be obtained as follows:

$$A_{2} = \frac{{\bar{m}_{\text{fD}} }}{{\cosh \left( {\sqrt {\frac{{3s\omega_{\text{k}} \left( {1 + \sigma_{\text{k}} } \right)}}{{\lambda_{\text{kf}} \beta_{\text{k}} }}} } \right)}}$$

(61)

The final solution of the dimensionless mathematical model for organic system thus can be obtained by substituting Eqs. (60) and (61) into Eq. (59):

$$\bar{m}_{\text{kD}} = \frac{{\cosh \left( {z_{\text{D}} \sqrt {\frac{{3s\omega_{\text{k}} \left( {1 + \sigma_{\text{k}} } \right)}}{{\lambda_{\text{kf}} \beta_{\text{k}} }}} } \right)}}{{\cosh \left( {\sqrt {\frac{{3s\omega_{\text{k}} \left( {1 + \sigma_{\text{k}} } \right)}}{{\lambda_{\text{kf}} \beta_{\text{k}} }}} } \right)}}\bar{m}_{\text{fD}}$$

(62)

Appendix C: natural fracture system solution

The Laplace transformation of the dimensionless governing equation of natural fracture system, i.e., Eq. (35), yields the following equation in the Laplace domain:

$$\frac{{\partial^{2} \bar{m}_{\text{fD}} }}{{\partial y_{\text{D}}^{ 2} }} = s\omega_{\text{f}} \bar{m}_{\text{fD}} + \frac{{\lambda_{\text{mf}} \beta_{\text{m}} }}{3}\left. {\frac{{\partial \bar{m}_{\text{mD}} }}{{\partial z_{\text{D}} }}} \right|_{{z_{\text{D}} = 1}} + \frac{{\lambda_{\text{kf}} \beta_{\text{k}} }}{3}\left. {\frac{{\partial \bar{m}_{\text{kD}} }}{{\partial z_{\text{D}} }}} \right|_{{z_{\text{D}} = 1}}$$

(63)

Taking derivatives of Eqs. (55) and (62) with respect to z _D and setting z _D equal to 1, we can obtain the following equations:

$$\left. {\frac{{\partial \bar{m}_{\text{mD}} }}{{\partial z_{\text{D}} }}} \right|_{{z_{\text{D}} = 1}} = \frac{{\sqrt {\frac{{3s\omega_{\text{m}} }}{{\lambda_{\text{mf}} \beta_{\text{m}} }}} \sinh \left( {\sqrt {\frac{{3s\omega_{\text{m}} }}{{\lambda_{\text{mf}} \beta_{\text{m}} }}} } \right)}}{{\cosh \left( {\sqrt {\frac{{3s\omega_{\text{m}} }}{{\lambda_{\text{mf}} \beta_{\text{m}} }}} } \right)}}\bar{m}_{\text{fD}}$$

(64)

$$\left. {\frac{{\partial \bar{m}_{\text{kD}} }}{{\partial z_{\text{D}} }}} \right|_{{z_{\text{D}} = 1}} = \frac{{\sqrt {\frac{{3s\omega_{\text{k}} \left( {1 + \sigma_{\text{k}} } \right)}}{{\lambda_{\text{kf}} \beta_{\text{k}} }}} \sinh \left( {\sqrt {\frac{{3s\omega_{\text{k}} \left( {1 + \sigma_{\text{k}} } \right)}}{{\lambda_{\text{kf}} \beta_{\text{k}} }}} } \right)}}{{\cosh \left( {\sqrt {\frac{{3s\omega_{\text{k}} \left( {1 + \sigma_{\text{k}} } \right)}}{{\lambda_{\text{kf}} \beta_{\text{k}} }}} } \right)}}\bar{m}_{\text{fD}}$$

(65)

Substitution of Eqs. (64) and (65) into Eq. (63) yields:

$$\frac{{\partial^{2} \bar{m}_{\text{fD}} }}{{\partial y_{\text{D}}^{ 2} }} - sf_{1} \left( s \right)\bar{m}_{\text{fD}} = 0$$

(66)

where $f_{1} \left( s \right) = \omega_{\text{f}} + \frac{{\lambda_{\text{mf}} \beta_{\text{m}} }}{3s}\sqrt {\frac{{3s\omega_{\text{m}} }}{{\lambda_{\text{mf}} \beta_{\text{m}} }}} \tanh \left( {\sqrt {\frac{{3s\omega_{\text{m}} }}{{\lambda_{\text{mf}} \beta_{\text{m}} }}} } \right) + \frac{{\lambda_{\text{kf}} \beta_{\text{k}} }}{3s}\sqrt {\frac{{3s\omega_{\text{k}} \left( {1 + \sigma_{\text{k}} } \right)}}{{\lambda_{\text{kf}} \beta_{\text{k}} }}} \tanh \left( {\sqrt {\frac{{3s\omega_{\text{k}} \left( {1 + \sigma_{\text{k}} } \right)}}{{\lambda_{\text{kf}} \beta_{\text{k}} }}} } \right)$.

Corresponding dimensionless boundary conditions, i.e., Eqs. (37) and (38), can also be transformed into the Laplace domain:

$$\left. {\frac{{\partial \bar{m}_{\text{fD}} }}{{\partial y_{\text{D}} }}} \right|_{{y_{\text{D}} = L_{\text{FD}} /2}} = 0$$

(67)

$$\left. {\bar{m}_{\text{fD}} } \right|_{{y_{\text{D}} = w_{\text{FD}} /2}} = \bar{m}_{\text{FD}}$$

(68)

The general solution of Eq. (66) is:

$$\bar{m}_{\text{fD}} = A_{3} \cosh \left( {y_{\text{D}} \sqrt {sf_{1} \left( s \right)} } \right) + B_{3} \sinh \left( {y_{\text{D}} \sqrt {sf_{1} \left( s \right)} } \right)$$

(69)

where A ₃ and B ₃ are coefficients which can be later determined with corresponding boundary conditions of natural fracture system.

Similarly, taking derivative of Eq. (69) with respect to y _D and substituting it into Eq. (67), we can obtain the following equation:

$$A_{3} \sqrt {sf_{1} \left( s \right)} \sinh \left( {\frac{{L_{\text{FD}} }}{2}\sqrt {sf_{1} \left( s \right)} } \right) + B_{3} \sqrt {sf_{1} \left( s \right)} \cosh \left( {\frac{{L_{\text{FD}} }}{2}\sqrt {sf_{1} \left( s \right)} } \right) = 0$$

(70)

Substitution of Eq. (69) into the Eq. (68) yields:

$$A_{3} \cosh \left( {\frac{{w_{\text{FD}} }}{2}\sqrt {sf_{1} \left( s \right)} } \right) + B_{3} \sinh \left( {\frac{{w_{\text{FD}} }}{2}\sqrt {sf_{1} \left( s \right)} } \right) = \bar{m}_{\text{FD}}$$

(71)

The coefficients A ₃ and B ₃ can be determined by solving Eqs. (70) and (71) simultaneously:

$$A_{3} = \bar{m}_{\text{FD}} \frac{{\cosh \left( {\frac{{L_{\text{FD}} }}{2}\sqrt {sf_{1} \left( s \right)} } \right)}}{{\cosh \left[ {\left( {\frac{{L_{\text{FD}} - w_{\text{FD}} }}{2}} \right)\sqrt {sf_{1} \left( s \right)} } \right]}}$$

(72)

$$B_{3} = - \bar{m}_{\text{FD}} \frac{{\sinh \left( {\frac{{L_{\text{FD}} }}{2}\sqrt {sf_{1} \left( s \right)} } \right)}}{{\cosh \left[ {\left( {\frac{{L_{\text{FD}} - w_{\text{FD}} }}{2}} \right)\sqrt {sf_{1} \left( s \right)} } \right]}}$$

(73)

The final solution of the dimensionless mathematical model for natural fracture system thus can be obtained by substituting Eqs. (72) and (73) into Eq. (69):

$$\bar{m}_{\text{fD}} = \bar{m}_{\text{FD}} \frac{{\cosh \left[ {\left( {\frac{{L_{\text{FD}} }}{2} - y_{\text{D}} } \right)\sqrt {sf_{1} \left( s \right)} } \right]}}{{\cosh \left[ {\left( {\frac{{L_{\text{FD}} - w_{\text{FD}} }}{2}} \right)\sqrt {sf_{1} \left( s \right)} } \right]}}$$

(74)

Appendix D: hydraulic fracture system solution

The Laplace transformation of the dimensionless governing equation of hydraulic fracture system, i.e., Eq. (39), yields the following equation in the Laplace domain:

$$\frac{{\partial^{2} \bar{m}_{\text{FD}} }}{{\partial x_{\text{D}}^{ 2} }} + \frac{2}{{C_{\text{FD}} }}\left. {\frac{{\partial \bar{m}_{\text{fD}} }}{{\partial y_{\text{D}} }}} \right|_{{y_{\text{D}} = w_{\text{FD}} /2}} = \frac{{s\omega_{\text{f}} }}{{\eta_{\text{D}} }}\bar{m}_{\text{FD}}$$

(75)

Taking derivative of Eq. (74) with respect to y _D and setting y _D equal to w _FD/2, we can obtain the following equations:

$$\left. {\frac{{\partial \bar{m}_{fD} }}{{\partial y_{\text{D}} }}} \right|_{{y_{\text{D}} = w_{{{\text{FD}}/2}} }} = - \bar{m}_{\text{FD}} \sqrt {sf_{1} \left( s \right)} \tanh \left[ {\left( {\frac{{L_{\text{FD}} - w_{\text{FD}} }}{2}} \right)\sqrt {sf_{1} \left( s \right)} } \right]$$

(76)

Substitution of Eq. (76) into Eq. (75) yields:

$$\frac{{\partial^{2} \bar{m}_{\text{FD}} }}{{\partial x_{\text{D}}^{ 2} }} = sf_{2} \left( s \right)\bar{m}_{\text{FD}}$$

(77)

where $f_{2} \left( s \right) = \left\{ {\frac{{\omega_{f} }}{{\eta_{D} }} + \frac{{2\sqrt {sf_{1} \left( s \right)} }}{{sC_{FD} }}\tanh \left[ {\left( {\frac{{L_{FD} - w_{FD} }}{2}} \right)\sqrt {sf_{1} \left( s \right)} } \right]} \right\}$.

Corresponding dimensionless boundary conditions, i.e., Eqs. (41) and (42), can also be transformed into the Laplace domain:

$$\left. {\frac{{\partial \bar{m}_{{F{\text{D}}}} }}{{\partial x_{\text{D}} }}} \right|_{{x_{\text{D}} = 0}} = - \frac{\pi }{{sC_{\text{FD}} }}$$

(78)

$$\left. {\frac{{\partial \bar{m}_{\text{FD}} }}{{\partial x_{\text{D}} }}} \right|_{{x_{\text{D}} = 1}} = 0$$

(79)

The general solution of Eq. (77) can be obtained as follows:

$$\bar{m}_{\text{FD}} = A_{4} \cosh \left( {x_{\text{D}} \sqrt {sf_{2} \left( s \right)} } \right) + B_{4} \sinh \left( {x_{\text{D}} \sqrt {sf_{2} \left( s \right)} } \right)$$

(80)

where A ₄ and B ₄ are coefficients which can be later determined with corresponding boundary conditions of hydraulic fracture system.

Taking derivative of Eq. (80) with respect to x _D, and we can obtain:

$$\frac{{\partial \bar{m}_{\text{FD}} }}{{\partial x_{\text{D}} }} = A_{4} \sqrt {sf_{2} \left( s \right)} \sinh \left( {x_{\text{D}} \sqrt {sf_{2} \left( s \right)} } \right) + B_{4} \sqrt {sf_{2} \left( s \right)} \cosh \left( {x_{\text{D}} \sqrt {sf_{2} \left( s \right)} } \right)$$

(81)

Substitution of Eq. (81) into Eq. (78) yields:

$$B_{4} \sqrt {sf_{2} \left( s \right)} = - \frac{\pi }{{sC_{\text{FD}} }}$$

(82)

Thus we can get:

$$B_{4} = - \frac{\pi }{{s\sqrt {sf_{2} \left( s \right)} C_{\text{FD}} }}$$

(83)

Substitution of Eqs. (81) and (83) into Eq. (79) yields:

$$A_{4} \sinh \left( {\sqrt {sf_{2} \left( s \right)} } \right) - \frac{\pi }{{s\sqrt {sf_{2} \left( s \right)} C_{\text{FD}} }}\cosh \left( {\sqrt {sf_{2} \left( s \right)} } \right) = 0$$

(84)

Thus we can get:

$$A_{4} = \frac{\pi }{{C_{\text{FD}} }}\frac{1}{{s\sqrt {sf_{2} \left( s \right)} }}\coth \left( {\sqrt {sf_{2} \left( s \right)} } \right)$$

(85)

Then the dimensionless pressure response for hydraulic fracture system is obtained to be:

$$\bar{m}_{\text{FD}} = \frac{\pi }{{C_{\text{FD}} }}\frac{1}{{s\sqrt {sf_{2} \left( s \right)} }}\left[ {\coth \left( {\sqrt {sf_{2} \left( s \right)} } \right)\cosh \left( {x_{\text{D}} \sqrt {sf_{2} \left( s \right)} } \right) - \sinh \left( {x_{\text{D}} \sqrt {sf_{2} \left( s \right)} } \right)} \right]$$

(86)

After some mathematical manipulations, Eq. (86) can be rewritten as:

$$\bar{m}_{\text{FD}} = \frac{\pi }{{C_{\text{FD}} }}\frac{1}{{s\sqrt {sf_{2} \left( s \right)} }}\frac{{\cosh \left[ {\left( {x_{\text{D}} - 1} \right)\sqrt {sf_{2} \left( s \right)} } \right]}}{{\sinh \left( {\sqrt {sf_{2} \left( s \right)} } \right)}}$$

(87)