Numerical Modeling and Simulation of Shale-Gas Transport with Geomechanical Effect

Abstract

Throughout this study, we present a dual-continuum model of transport of the natural gas in shale formations. The model includes several physical mechanisms such as diffusion, adsorption and rock stress sensitivity. The slippage has a clear effect in the low-permeability formations which can be described by the apparent permeability. The adsorption mechanism has been modeled by the Langmuir isotherm. The porosity-stress model has been used to describe stress state of the rocks. The thermodynamics deviation factor is calculated using the equation of state of Peng–Robinson. The governing differential system has been solved numerically using the mixed finite element method (MFEM). The stability of the MFEM has been investigated theoretically and numerically. A semi-implicit scheme is employed to solve the two coupled pressure equations, while the thermodynamic calculations are conducted explicitly. Moreover, numerical experiments are performed under the corresponding physical parameters of the model. Some represented results are shown in graphs including the rates of production as well as the pressures and the apparent permeability profiles.

Appendices

Appendix A: Preliminaries

In the case of the parabolic system, there are two natural variables, a scalar variable and it’s flux. Each of these two variables belongs to a given finite element space. The mixed method is developed to approximate both variables simultaneously and to give a higher-order approximation for them. In general, mixed finite element method has two different finite element spaces. An important compatibility condition must hold for the two finite element spaces. This compatibility guarantees the stability, the consistency and the convergence of the mixed method and furthermore by adding additional constraints to the numerical discretization. These requirements are based on properties of the corresponding continuum problem, i.e., the partial differential equations (PDEs). The inner product in $\Omega$ is defined as,

$$\begin{aligned} (f,g)_\Omega = \int _\Omega f(\mathbf{x }) g(\mathbf{x }) \hbox {d}\mathbf{V } \quad \forall f,g: \Omega \rightarrow {\mathbb {R}}, \end{aligned}$$

(A.1)

and the inner product on $\partial \Omega$ is defined as,

$$\begin{aligned} \langle f,g \rangle _{\partial \Omega } = \int _{\partial \Omega } f g dS \quad \forall f,g: \partial \Omega \rightarrow {\mathbb {R}} \end{aligned}$$

(A.2)

Given,

$$\begin{aligned} {\mathbb {L}}^2(\Omega ) = \{f:\Omega \rightarrow {\mathbb {R}}: \int _\Omega f^2 \hbox {d}x < +\infty \}, \end{aligned}$$

(A.3)

we are looking for the largest possible space to include a solution for the weak formulations even the strong form has no solution. ${\mathbb {L}}^2(\Omega )$ is the largest Hilbert space such that, $(\mathbf{D }^{-1} \mathbf{u },w)$ is well defined. Also, if $(p,\nabla \cdot w)$ is well defined, then, $p,\nabla \cdot w \in {\mathbb {L}}^2(\Omega )$. It is required, $p,\varphi \in {\mathbb {L}}^2(\Omega )$ and $\mathbf{u },w \in {\mathbb {H}}(\hbox {div},\Omega )$. The space ${\mathbb {H}}(\hbox {div},\Omega )$ is composed of those functions $\mathbf{u }$ for which it holds $\nabla \cdot \mathbf{u } \in {\mathbb {L}}^2(\Omega )$, i.e., $\mathbf{u } \in {\mathbb {H}}(\hbox {div},\Omega )$ where

$$\begin{aligned} {\mathbb {H}}(\hbox {div},\Omega ) = \{\mathbf{u }: \nabla \cdot \mathbf{u } \in {\mathbb {L}}^2(\Omega ) \}, \end{aligned}$$

(A.4)

which is a Hilbert space with norm given by

$$\begin{aligned} \Vert \mathbf{u }\Vert ^2_{\mathbf{H }(\mathrm{div},\Omega )} = \Vert \mathbf{u }\Vert ^2\, +\, \Vert \nabla \cdot \mathbf{u }\Vert ^2\, , \end{aligned}$$

(A.5)

which means that not only the function w needs to be square-integrable but also it’s divergence $\nabla \cdot \mathbf{u }$ must hold square-integrability over $\Omega$. In case of the scalar function, we choose to be in the space $L^2(\Omega )$. Similarly, if $w \in {\mathbb {L}}^2(\Omega )$ and $\nabla \cdot \mathbf{u }\in {\mathbb {L}}^2(\Omega )$ hold, we have, $\mathbf{u } \in {\mathbb {H}}(\hbox {div},\Omega )$. In the context of mixed finite element, ${\mathbb {H}}(\hbox {div},\Omega )$ is the typical finite element space.

Therefore, the seeking solution is,

$$\begin{aligned} (p, \mathbf{u }) \in {\mathbb {L}}^2(\Omega ) \times {\mathbb {H}}(\hbox {div},\Omega ), \end{aligned}$$

(A.6)

such that p and $\mathbf{u }$ are smooth. For numerical discretization, we define the two discretized spaces,

$$\begin{aligned} W_h \subset {\mathbb {L}}^2(\Omega ), \quad \mathbf{dim} ( W_h)<+\infty \end{aligned}$$

(A.7)

and,

$$\begin{aligned} V_h \subset {\mathbb {H}}(\hbox {div},\Omega ), \quad \mathbf{dim} ( V_h)<+\infty \end{aligned}$$

(A.8)

The condition $V_h \subset {\mathbb {H}}(\hbox {div},\Omega )$ translates into a continuity condition over the inter-element boundaries $E \in {\mathcal {E}}_h$ of the mesh ${\mathcal {K}}_h$. On other words, one requires that the normal component $\mathbf{u } \cdot \mathbf{n }$ is continuous across the inter-element boundaries. We seek the velocity $\mathbf{u }_h \in V_h$ and the pressure $p_h \in W_h$.

On the other hand, the $\mathbf{RT}_r$ elements are designed to approximate ${\mathbb {H}}(\hbox {div},\Omega )$ (Raviart and Thomas 1975). We consider the $\mathbf{RT}_r$ space for which,

$$\begin{aligned} W_h = \{ w \in {\mathbb {L}}^2(\Omega ): w|_{E} \in {\mathbb {P}}_0(E), E \in {\mathcal {E}}_h\} \end{aligned}$$

(A.9)

and,

$$\begin{aligned} V_h = \{ \mathbf{u } \in {\mathbb {H}}(\hbox {div},\Omega ): \mathbf{u }|_{E} \in {\mathbb {P}}_1(E), E \in {\mathcal {E}}_h\} \end{aligned}$$

(A.10)

where w is discontinuous piecewise constant and $\mathbf{u }$ is piecewise linear.

Appendix B: MFEM Approximation

Let $\Omega _m$ and $\Omega _f$ are, respectively, the matrix and fracture in a polygonal/polyhedral Lipschitz domain $\Omega \subset {\mathbb {R}}^d, d \in \{ 1,2,3 \}$ (on which we define the standard $L^2(\Omega )$ space such that ${\mathbb {L}}^2(\Omega ) \equiv \left( L^2(\Omega ) \right) ^d$), with the boundaries, $\partial \Omega _m = \Gamma ^m_D \cup \Gamma ^m_N$ and $\partial \Omega _f = \Gamma ^f_D \cup \Gamma ^f_N$. One may write the above dual-porosity model in the following general form,

$$\begin{aligned}&\displaystyle f_1(p_m) \frac{\partial p_m}{\partial t} + \nabla \cdot \mathbf{u }_m = -S(p_m,p_f) \quad \mathrm{in} \quad \Omega _m \times (0,T), \end{aligned}$$

(B.1)

$$\begin{aligned}&\displaystyle \mathbf{D }_m(p_m)^{-1} \mathbf{u }_m = - \nabla p_m \quad \mathrm{in} \quad \Omega _m \times (0,T), \end{aligned}$$

(B.2)

$$\begin{aligned}&\displaystyle f_2(p_f) \frac{\partial p_f}{\partial t} + \nabla \cdot \mathbf{u }_f = S(p_m,p_f) - Q(p_f) \quad \mathrm{in} \quad \Omega _f \times (0,T), \end{aligned}$$

(B.3)

$$\begin{aligned}&\displaystyle \mathbf{D }_f(p_f)^{-1} \mathbf{u }_f = - \nabla p_f \quad \mathrm{in} \quad \Omega _f \times (0,T), \end{aligned}$$

(B.4)

where

$$\begin{aligned} \mathbf{D }_m(p_m)=\frac{\rho (p_m)}{\mu } k_{0,m} \left( 1+\frac{b_m}{p_m}\right) , \end{aligned}$$

(B.5)

and

$$\begin{aligned} \mathbf{D }_f(p_f)=\frac{\rho (p_f)}{\mu } k_{0,f} \left( 1+\frac{b_f}{p_f}\right) . \end{aligned}$$

(B.6)

The functions $\mathbf{D }_m(p_m)^{-1}$ and $\mathbf{D }_f(p_f)^{-1}$ are moved to the left hand side to avoid discontinuity when we integrate $\nabla p_m$ and $\nabla p_f$ by parts. Selecting any $\varphi \in W_h$ and $\omega \in V_h$, the mixed finite element weak formulation can be written in the following form,

$$\begin{aligned}&\displaystyle (f_1(p_m) \frac{\partial p_m}{\partial t}, \varphi ) +(\nabla \cdot \mathbf{u }_m,\varphi ) +({\mathcal {S}}(p_m,p_f),\varphi )=0 \end{aligned}$$

(B.7)

$$\begin{aligned}&\displaystyle (\mathbf{D }_m(p_m)^{-1}\mathbf{u }_m,\omega )= (p_m, \nabla \cdot \omega ), \end{aligned}$$

(B.8)

$$\begin{aligned}&\displaystyle (f_2(p_f) \frac{\partial p_f}{\partial t},\varphi ) +( \nabla \cdot \mathbf{u }_f,\varphi ) -({\mathcal {S}}(p_m,p_f),\varphi )= - (Q(p_f),\varphi ), \end{aligned}$$

(B.9)

$$\begin{aligned}&\displaystyle (\mathbf{D }_f(p_f)^{-1}\mathbf{u }_f,\omega )= (p_f, \nabla \cdot \omega ) - \left\langle p_w, \omega \right\rangle _{\Gamma ^D_f}. \end{aligned}$$

(B.10)

Now, let the approximating subspace duality $V_{h}\subset H(\Omega ;\hbox {div})$ and $W_{h} \subset L^{2}(\Omega )$ be the r-th order ($r\ge 0$) Raviart–Thomas space (RT$_r$) on the partition ${\mathcal {T}}_h$. The mixed finite element formulations are stated as below: find $p^h_m,p^h_f\in W_h$ and $u^h_m,u^h_f\in V_h$ such that,

$$\begin{aligned}&\displaystyle (f_1(p^h_m) \frac{\partial p^h_m}{\partial t}, \varphi ) +(\nabla \cdot \mathbf{u }^h_m,\varphi ) +({\mathcal {S}}(p^h_m,p^h_f),\varphi )=0 \end{aligned}$$

(B.11)

$$\begin{aligned}&\displaystyle (\mathbf{D }_m(p^h_m)^{-1}\mathbf{u }^h_m,\omega )= (p^h_m, \nabla \cdot \omega ), \end{aligned}$$

(B.12)

$$\begin{aligned}&\displaystyle (f_2(p^h_f) \frac{\partial p^h_f}{\partial t},\varphi ) +( \nabla \cdot \mathbf{u }^h_f,\varphi ) -({\mathcal {S}}(p^h_m,p^h_f),\varphi )= - (Q(p^h_f),\varphi ), \end{aligned}$$

(B.13)

$$\begin{aligned}&\displaystyle (\mathbf{D }_f(p^h_f)^{-1}\mathbf{u }^h_f,\omega )= (p^h_f, \nabla \cdot \omega ) - \left\langle p_w, \omega \right\rangle _{\Gamma ^D_f}, \end{aligned}$$

(B.14)

for any $\varphi \in W_h$ and $\omega \in V_h$.