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Transport in Porous Media

, Volume 126, Issue 3, pp 779–806 | Cite as

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

  • Mohamed F. El-AminEmail author
  • Jisheng Kou
  • Shuyu Sun
Article
  • 64 Downloads

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.

Keywords

Shale-gas Porous media Stress sensitivity Stability analysis Mixed finite element 

1 Introduction

Numerical simulation of gas transport in underground formations (such as shale formations) is a computationally challenging because it requires a high accuracy and a local mass conservation constraint. For example, the transport in geological formations involves long durations; therefore, any small error in each step may cause a huge error in the long run. Also, the heterogeneity, anisotropy, and discontinuity of the medium properties require special treatments for efficient approximations of advection, diffusion, dispersion, and chemical reactions. Some numerical techniques failed to preserve the physical and mathematical principles which lead to erroneous results (Riviere and Wheeler 2000; Arbogast and Huang 2006; Arbogast et al. 1997). Therefore, it is very important to examine the efficiency of the numerical schemes by studying solution properties such as numerical stability.

It is well known that the permeability of shale reservoirs is very low compared to that of the conventional reservoirs. Existence of fracture networks including natural fractures and their connection to the well may enhance the efficiency of the gas production. Presently, the advancement in drilling technology allows the use of horizontal wells that can provide a larger area of contact with gas-bearing formations. In such technology, hydraulic fracking has been used to artificially create extensive fractures in the vicinity of the wellbore. Mathematical models of flow and transport in fractured media may be extended to model the transport of gas between the shale matrix blocks and the fractures. One of the common models to simulate the transport in fracture media is the dual-porosity, dual-porosity dual-permeability (DPDP) model which built mainly based on Darcy’s law. However, the flow in shale formations deviates from Darcy’s law because of gas slippage and inertial flow in nano-size pores in shale matrix (Salama et al. 2017). The framework of the gas transport in fractured shale formation is described by the early established models of transport in fractured porous media. Warren and Root (1963) presented an idealized dual-continuum model of transport in fractured porous media. Bustin et al. (2008) have studied the permeability behavior using the standard dual-continuum approach. Ozkan et al. (2010) have incorporated Darcy and diffusive flows in the matrix and stress-dependent permeability in the fractures and have ignored sorption processes. Also, Moridis et al. (2010) have employed the standard dual-continuum model to describe some mechanisms of transport in Kerogen. Wu and Fakcharoenphol (2011) have developed general reservoir simulators using a generalized dual-continuum methodology with ignoring the adsorption and desorption processes. Guo et al. (2014) presented a DPDP model including the adsorption mechanism to simulate gas transport in shale formations. Javadpour et al. (2007) and Javadpour (2009) presented a model to describe gas transport in shale matrix by combining several physical phenomena such as Knudsen diffusion, slip flow, and desorption. Screening improved recovery methods in tight-oil formations by injecting and producing through fractures have been presented by Singh and Cai (2018). Nanoporous structure and gas occurrence of organic-rich shales have been studied by Qi et al. (2017). Hu et al. (2019) presented a multiscale model for methane transport mechanisms in shale-gas reservoirs. Shen et al. (2018) presented a numerical study of methane diffusion and adsorption in shale rocks using the dusty gas model in TOUGH2/EOS7C-ECBM. Pressure transient behaviors of hydraulically fractured horizontal shale-gas wells were studied by using DPDP model by Li et al. (2018). Liu et al. (2019) developed a fully coupled fracture equivalent continuum-dual-porosity model for hydro-mechanical process in fractured shale-gas reservoirs. El-Amin et al. (2017) presented a comparative study of the gas transport in shale by a DPDP model and a single-domain model. Also, El-Amin (2017) and El-Amin et al. (2017) introduced analytical solutions for the fractional derivative model of gas transport in porous media. Moreover, an intensive review on the recent achievements of transport in tight and shale formations has been presented by Salama et al. (2017).

Gas production causes reduction in the pressure which in turn may affect the stress state of the solid. The change in the stress state stimulates the rock and may affect the fluid flow. This sort of reservoirs is called stress-sensitive reservoirs. The tight naturally fractured reservoirs are the most sensitive to changing stresses (Holcomb et al. 1994). The solid deformation in the oil/gas reservoirs has been described by the poroelasticity theory which has been established, developed, and used by several authors such as Terzaghi (1936), Biot (1941, 1956a, b, 1962), Biot and Willis (1957), Safai and Pinder (1979), Safai et al. (1980), and Narasimhan and Witherspoon (1977).

The numerical simulation is an important tool which allows engineers and scientists to predict shale-gas production, test, and optimize appropriate intervention strategies for gas recovery. The MFEM (Brezzi et al. 1985; Brezzi and Fortin 1991; Nakshatrala et al. 2006; Raviart and Thomas 1975) is one of the effective numerical methods in reservoir simulation because it is locally conservative and can be extended to a higher-order approximation. In the MFEM, the approximation to velocity can be more accurate than the approximation to pressure. For example, the enhanced linear Brezzi–Douglas–Marini (BDM1) space is used for the velocity approximation, while the piecewise constant space is used for the pressure approximation. El-Amin et al. (2018) have used the MFEM with stability analysis to simulate the problem of natural gas transport in a low-permeability reservoir without considering fractures. The mathematical model contains slippage effect, adsorption and diffusion mechanisms.

The current paper is devoted to employing the MFEM to simulate the problem of shale-gas transport using the DPDP model including the rock stress-sensitivity. We provide a stability analysis of the MFEM and establish a mathematical foundation, in addition to selective numerical tests. The sections of this paper are arranged as follows: Sect. 2 is devoted to developing the governing mathematical model. In Sect. 3, we introduce the model validation. The MFEM formulation is provided in Sect. 4. The stability analysis has been considered in Sect. 5. Physical and numerical discussions are presented in Sect. 6, and we present the conclusions in Sect. 7.

2 Modeling and Formulation

The modeling of transport in fractured porous media can be described by two main methods; (1) dual-continua (e.g., DPDP) and (2) discrete-fracture models (DFM). In this study, we consider the DPDP model. Warren and Root (1963) have developed an idealized model to study the behavior flow in fractured porous media based on DPDP models. The dual-continua models (such DPDP model) treat the matrix blocks and the fractures as two separate continua, that are theoretically interconnected by fluid mass transfer across their interfaces. They considered the matrix parts as idealized uniform cubical blocks distributed uniformly and separated by fractures, as shown in Fig. 1. Each part, either fracture or matrix has different porosity and permeability values.
Fig. 1

Warren-Root’s idealization fractured porous medium

The DPDP model consists of two groups of equations, one group is for gas transport in matrix blocks and the second is for transport in fractures. Each group of equations includes the related physics occurring in matrix or fracture. For instance, the adsorbed gas and the free gas coexist in the matrix blocks of shale. So, the model of mass accumulation in matrix has the free gas per volume of bulk rock, \(\phi _m \rho _m\), the adsorbed gas accumulation which is represented by the Langmuir isotherm model (Guo et al. 2014; Cui et al. 2009; Shabro et al. 2011; Freeman et al. 2012; Civan et al. 2011),
$$\begin{aligned} \frac{(1-\phi _m)\rho _s M_w V_L p_m}{V_\mathrm{std}(P_L+p_m)} \end{aligned}$$
(1)
where \(\rho _m\) is the gas density in the matrix; \(\phi _m\) is the matrix porosity; \(\rho _s\) is the solid density; \(M_w\) is the pseudo molecular mass of gas; \(p_m\) is the pressure in matrix; \(V_L\) is the Langmuir volume; \(P_L\) is the Langmuir pressure; and \(V_\mathrm{std}\) is the volume under standard conditions. Also, one can define the gas mass density as,
$$\begin{aligned} \rho _{g,d} = \frac{p_d M_w}{ZRT} , \quad \quad d=m,f. \end{aligned}$$
(2)
where m and f stand for matrix and fracture, respectively. T is the temperature; R is the universal gas constant; and Z is the gas compressibility factor. The factor Z can be estimated by the Peng–Robinson equation of state (Firoozabadi 2015),
$$\begin{aligned} Z^3 - (1-B)Z^2 + (A-3B^3-2B)Z-(AB-B^2-B^3) = 0 \end{aligned}$$
(3)
where
$$\begin{aligned} A= & {} \frac{a_Tp}{R^2T^2}, \quad B=\frac{b_Tp}{RT}. \end{aligned}$$
(4)
$$\begin{aligned} a_T= & {} 0.45724 \frac{R^2T_c^2}{p_c}, \quad b_T = 0.0778 \frac{RT_c}{p_c}. \end{aligned}$$
(5)
where \(T_c\) is the critical temperature and \(p_c\) is the critical pressure. In low-permeability formations, Klinkenberg effect, which is described by the apparent permeability, takes place (Ertekin et al. 1986) and is given by,
$$\begin{aligned} k_{\mathrm{app},d} = k_{0,d} \left( 1+\frac{b_d}{p_d}\right) , \quad d=m,f, \end{aligned}$$
(6)
where \(b_d\) is the factor of slippage and \(k_{0,d}\) is the intrinsic permeability. The slippage factor in the mass balance equation in the fracture system is not significant; however, it is considered for consistency of modeling and analysis to be comparable with the slippage factor in the mass balance equation for flow in the matrix system. The effect of the slippage factor in the fracture system is negligibly small (i.e., \(\frac{b_f}{p_f}<<1\)), so, it has no effect on the flow. The MFEM with stability analysis are eventually depending on the similarity between the two equations of matrix domain and fracture domain, as shown below. The DPDP model consists of matrix mass conservation equation, and fracture mass conservation equation. Assuming that the flow is isothermal and single-phase, with neglecting gravity. The DPDP model of gas transport in fracture shale strata is represented as,
$$\begin{aligned} f_1(p_m) \frac{\partial p_m}{\partial t} - \nabla \cdot \left[ \frac{\rho _{g,m}(p_m)}{\mu } k_{0,m}\left( 1 + \frac{b_m}{p_m} \right) \nabla p_m\right] = -S(p_m,p_f), \end{aligned}$$
(7)
and
$$\begin{aligned} f_2(p_f) \frac{\partial p_f}{\partial t} - \nabla \cdot \left[ \frac{\rho _{g,f}(p_f)}{\mu } k_{0,f}\left( 1 + \frac{b_f}{p_f} \right) \nabla p_f\right] = S(p_m,p_f) - Q(p_f), \end{aligned}$$
(8)
where
$$\begin{aligned} f_1(p_m) = \frac{M_w}{ZRT} \left[ \phi _m(p_m) + \phi '_m(p_m) p_m \right] + \frac{M_wV_L\rho _s}{V_\mathrm{std} } \left( \frac{P_L(1-\phi _m(p_m))}{(P_L + p_m)^2} - \frac{\phi '_m(p_m)}{P_L + p_m} \right) ,\nonumber \\ \end{aligned}$$
(9)
and
$$\begin{aligned} f_2(p_f) = \frac{M_w}{ZRT} \left[ \phi _f(p_f) + \phi '_f(p_f) p_f \right] , \end{aligned}$$
(10)
such that \(\phi _m\) and \(\phi _f\) are the matrix and fractures porosities, respectively.
$$\begin{aligned} \phi '_d(p_d) = \frac{\hbox {d}\phi _d}{\hbox {d}p_d} , \quad d=m,f, \end{aligned}$$
(11)
\(S(p_m,p_f)\) is a theoretical transfer term to connect the matrix domain to the fracture domain, and defined by,
$$\begin{aligned} S(p_m,p_f) = \frac{\sigma \rho _{g,m} k_m}{\mu } \left( p_m - p_f\right) , \end{aligned}$$
(12)
The source of the production well is given by,
$$\begin{aligned} Q(p_f) = \frac{\theta \rho _{g,f} k_f}{\mu \ln \frac{re}{r_w}} \left( p_{fe} - p_{w}\right) , \end{aligned}$$
(13)
such that \(r_w\), \(r_e\) and \(p_{fe}\) are, respectively, the well-radius, the drainage radius, and the average fracture pressure around the well. In the case of the production well is located at the center of the field, \(\theta =2\pi \), while, if it is located at the corner of the field, \(\theta =\pi /2\). Given the constant, \(r_c\), the drainage radius is represented by,
$$\begin{aligned} r_e = 0.14 \sqrt{(\Delta x)^2 + (\Delta y)^2}, \end{aligned}$$
(14)
The coefficient of crossflow between the matrix and fracture domains is defined by Warren and Root (1963),
$$\begin{aligned} \sigma = \frac{2n(n+1)}{l^2} \end{aligned}$$
(15)
such that l is given by,
$$\begin{aligned} l = \frac{3L_xL_yL_z}{L_xL_y+L_yL_z+L_xL_z}, \end{aligned}$$
(16)
where \(L_x, L_y\) and \(L_z\) are the fracture spacing of xy and z, respectively, and n is the set of normal fractures.
In this study, we consider the effect of rock stress-sensitivity using the concept of effective-stress (e.g., Terzaghi 1936; Biot and Willis 1957). The porosity in terms of the mean effective-stress is given by.
$$\begin{aligned} \phi _d (\sigma '_d(p_d)) = \phi _{r,d} + (\phi _{0,d} - \phi _{r,d}) \exp (- a \sigma '_d), \quad d = m,f, \end{aligned}$$
(17)
where \(\sigma '_m\) and \(\sigma '_f\) are the effective-stress in matrix and fracture, respectively, that are defined by:
$$\begin{aligned} \sigma '_d = \sigma _d + \alpha _d p_d, \quad d = m,f, \end{aligned}$$
(18)
such that \(\sigma _m\) and \(\sigma _f\) are the mean total stress in matrix and fracture, respectively. \(\alpha _d\) is the Biot’s effective parameter. Combining (17) and (18), one can write,
$$\begin{aligned} \phi _d (p_d) = \phi _{r,d} + C_{1,d} \exp (-a p_d), \quad C_{1,d}=(\phi _{0,d} - \phi _{r,d}) \exp (- a \sigma _d), \quad d = m,f. \end{aligned}$$
(19)
The difference \((\phi _{0,d} - \phi _{r,d} )\) is positive if the porosity increases and negative if the porosity decreases. The coefficient \(C_{1,d}\) is relatively small depending on the variation of the porosity. Therefore, \(C_{1,d}\) is positive when porosity increasing and negative if porosity decreasing.
Moreover, based on the following definitions, \(b_m\) and \(b_f\) are treated as constants (Javadpour et al. 2007; Javadpour 2009),
$$\begin{aligned} b_m = \sqrt{\frac{8\pi RT}{M_w} } \frac{1}{r} \left( \frac{2 }{\alpha } - 0.995 \right) \mu _g, \end{aligned}$$
(20)
and
$$\begin{aligned} b_f = \sqrt{\frac{\pi RT \phi _{0,f}}{M_w k_{0,f} }} \ \mu _g \end{aligned}$$
(21)
The Knudsen diffusion coefficient is defined as,
$$\begin{aligned} D_{kf} = \sqrt{\frac{\pi RT k_{0,f} \phi _{0,f}}{M_w}}. \end{aligned}$$
(22)
Initially, both matrix pressure and fracture pressure are equals. Therefore, the initial conditions can be written as,
$$\begin{aligned} p_m(\cdot ,0) = p_f(\cdot ,0) = p_0 \quad \mathrm{in} \quad \Omega _m \cup \Omega _f. \end{aligned}$$
(23)
The boundary condition which represents the pressure on the production well is given as,
$$\begin{aligned} p_f(\cdot ,t) = p_w \quad \mathrm{on} \quad \Gamma ^f_D \times (0,T), \end{aligned}$$
(24)
The boundary conditions for matrix domain are,
$$\begin{aligned} \mathbf{u }_m \cdot \mathbf{n } = 0 \quad \mathrm{on} \quad \Gamma _N^m \cup \Gamma _N^f \times (0,T), \end{aligned}$$
(25)
while the boundary conditions for fracture domain are,
$$\begin{aligned} \mathbf{u }_f \cdot \mathbf{n } = 0 \quad \mathrm{on} \quad \Gamma ^f_N \times (0,T), \end{aligned}$$
(26)

3 Model Validation

The model of Knudsen diffusion in nanopores in terms of the pressure gradient has been developed by Roy and Raju (2003). In the following, we present some useful formulae. The diffusion mass-flux nanopores with neglecting viscous effect is represented as (Javadpour et al. 2007; Roy and Raju 2003),
$$\begin{aligned} J_D =\frac{M D_K}{RT} \nabla P \end{aligned}$$
The mass flux through a circular tube with neglecting the effect of entrance length is derived from Hagen–Poiseuille’s equation as,
$$\begin{aligned} J_a= -\frac{\rho _r^2}{8\mu } \nabla P \end{aligned}$$
The ideal gas law was used to link density to pressure in order to integrate over the length to obtain (Javadpour et al. 2007),
$$\begin{aligned} J_a = - \frac{r^2}{8\mu } \frac{\rho _{avg}}{L} \nabla P \end{aligned}$$
where \(\nabla P \) is the pressure drop between inlet and outlet. \(\rho _{avg}\) is the algebraic average of the density at inlet and outlet.
Invoking the compressibility factor to represent the non-ideality, the flux becomes,
$$\begin{aligned} J_a = - \frac{r^2}{8\mu } \frac{\gamma }{Z} P\nabla P. \end{aligned}$$
The integration of flux over the length-scale becomes unsolvable analytically; therefore, numerical calculations takes place.
The slip effect on the boundary is required for the case of nanopores (Javadpour et al. 2007). The slip velocity in a tube is corrected by the coefficient (Brown et al. 1946),
$$\begin{aligned} F=1+\sqrt{8\mu \gamma } \frac{\mu }{rP{avg}} (\frac{2}{\alpha }-1) \end{aligned}$$
where \(\alpha \) is the tangential momentum accommodation coefficient. In order to determine \(\alpha \) for specific systems, experimental measurements are needed. Experiment for a homogeneous porous medium was conducted by Roy and Raju (2003) for gas flow in a relatively cylindrical straight nanopores in a thick membrane. Argon gas was injected into the nanopores at different pressure gradients, and the total mass flux was measured. Our integral model results give a good match with the experimental results for \(\alpha = 0.8\) (see Fig. 2). Also, Javadpour et al. (2007) has estimated the same value of \(\alpha = 0.8\) to fit to the model.
Fig. 2

Model verification against experiment data of Roy and Raju (2003)

4 Mixed Finite Element Approximation

The mixed method being discussed here can be directly adapted to those developed in the previous section for nonzero pressure p. The mixed finite element formulations are stated as follows:

Find \(p^h_m,p^h_f\in W_h\) and \(u^h_m,u^h_f\in V_h\) such that,
$$\begin{aligned}&\displaystyle (f_1(p^h_m) \frac{\partial p^h_m}{\partial t}, \varphi ) +(\nabla \cdot \mathbf{u }^h_m,\varphi ) +({\mathcal {S}}(p^h_m,p^h_f),\varphi )=0 \end{aligned}$$
(27)
$$\begin{aligned}&\displaystyle (\mathbf{D }_m(p^h_m)^{-1}\mathbf{u }^h_m,\omega )= (p^h_m, \nabla \cdot \omega ), \end{aligned}$$
(28)
$$\begin{aligned}&\displaystyle (f_2(p^h_f) \frac{\partial p^h_f}{\partial t},\varphi ) +( \nabla \cdot \mathbf{u }^h_f,\varphi ) -({\mathcal {S}}(p^h_m,p^h_f),\varphi )= - (Q(p^h_f),\varphi ), \end{aligned}$$
(29)
$$\begin{aligned}&\displaystyle (\mathbf{D }_f(p^h_f)^{-1}\mathbf{u }^h_f,\omega )= (p^h_f, \nabla \cdot \omega ) - \left\langle p_w, \omega \right\rangle _{\Gamma ^D_f}, \end{aligned}$$
(30)
for any \(\varphi \in W_h\) and \(\omega \in V_h\).

More details and the stability analysis are presented in Appendices A and B.

In order to obtain an explicit formulation for the flux, we employ a quadrature rule along with the MFE method to Arbogast et al. (1997). The total time interval [0, T] is divided into \(N_T\) time steps with length \(\Delta t^n = t^{n+1} - t^n\). The superscript \(n+1\) denotes for the current time step, while n denotes for the previous one. We use a backward Euler semi-implicit discretization for the time derivatives terms. The following scheme has been developed,
$$\begin{aligned}&\left( f_1(p^{h,n}_m) \frac{p^{h,n+1}_m - p^{h,n}_m}{\Delta t}, \varphi \right) \nonumber \\&\qquad +\,(\nabla \cdot \mathbf{u }^{h,n+1}_m,\varphi ) +\left( {\mathcal {S}}(p^{h,n+1}_m,p^{h,n}_f),\varphi \right) =0 \end{aligned}$$
(31)
$$\begin{aligned}&\left( \mathbf{D }_m(p^{h,n}_m)^{-1}\mathbf{u }^{h,n+1}_m,\omega \right) = \left( p^{h,n+1}_m, \nabla \cdot \omega \right) , \end{aligned}$$
(32)
$$\begin{aligned}&\left( f_2(p^{h,n}_f) \frac{p^{h,n+1}_f - p^{h,n}_f}{\Delta t},\varphi \right) +\left( \nabla \cdot \mathbf{u }^{h,n+1}_f,\varphi \right) \nonumber \\&\qquad -\left( {\mathcal {S}}(p^{h,n+1}_m,p^{h,n+1}_f),\varphi \right) = - (Q(p^{h,n+1}_f),\varphi ), \end{aligned}$$
(33)
$$\begin{aligned}&\left( \mathbf{D }_f(p^{h,n}_f)^{-1}\mathbf{u }^{h,n+1}_f,\omega \right) = (p^{h,n+1}_f, \nabla \cdot \omega ) - \left\langle p_w, \omega \right\rangle _{\Gamma ^D_f}, \end{aligned}$$
(34)
Given \(p^{h,n}_m\) and \(p^{h,n}_f\), the numerical procedure to calculate pressure and velocity is presented here,
  1. 1.

    Update the thermodynamical variables explicitly;

     
  2. 2.

    Solve Eqs. (31)–(32) to obtain \(p^{h,n+1}_m\) and \(\mathbf{u }^{h,n+1}_m\);

     
  3. 3.

    Solve Eq. (33)—(B.14) to obtain \(p^{h,n+1}_f\) and \(\mathbf{u }^{h,n+1}_f\).

     

5 Stability Analysis

In this section, we carry out the stability analysis of the proposed MFE method, which ensures that the discrete solutions are bounded in a physically reasonable range. The key issue encountered in the stability analysis is the nonlinearity of the matrix and fracture pressures. In order to resolve this issue, we need to define some auxiliary functions and analyze their boundedness. The following auxiliary functions are obtained by integrating \(f_1(p^h)\) and \(f_2(p^h)\) over \(p^h\),
$$\begin{aligned} \begin{array}{l} F_1(p^h_m) =\left( \phi _{r,m} + C_{1,m} \right) \frac{M_w}{ZRT} p^h_m e^{-ap^h_m} + \frac{M_wV_L\rho _s}{V_\mathrm{std} } \left[ C_{1,m} \left( a e^{aP_L} (1+P_L) \right. \right. \\ \left. \left. Ei(-x)|^{a(P_L + p^h_m)}_{aP_L} +\frac{P_L e^{-ap^h_m}}{P_L + p^h_m} -1 \right) + (1- \phi _{r,m})\frac{p^h_m}{P_L + p^h_m} \right] , \end{array} \end{aligned}$$
(35)
and
$$\begin{aligned} F_2(p^h_f) =\left( \phi _{r,f} + C_{1,f} \right) \frac{M_w}{ZRT} p^h_f e^{-ap^h_f}, \end{aligned}$$
(36)
where Ei(x) is a special function called the exponent integral function. As stated above in the model formulation that the coefficient \(C_{1,d}, d=m,f\) is positive in the case of increasing porosity and negative in the case of decreasing porosity.

5.1 Increasing Matrix Porosity (\(C_{1,m}>0\))

Note that the value of the exponent integral function \(Ei(-x)\) is negative and (small for big values of x); therefore, the quantity \(Ei(-x) |^{a(P_L + p^h_m)}_{aP_L}\) has always a positive value which has lower and upper bounds, namely,
$$\begin{aligned} C^u_{2,m} \ge Ei(-x) |^{a(P_L + p^h_m)}_{a P_L} \ge C^l_{2,m}. \end{aligned}$$
(37)
where \(C^l_{2,m},C^u_{2,m}>0\). On the other hand, in shale reservoir, it is well known that the initial pressure has the maximum value, i.e., \(max(p^h_m) = P_0\), while, the pressure of the well has the minimum value, i.e., \(min(p^h_m) = P_w\). Therefore, we have,
$$\begin{aligned} C^u_3= & {} \frac{1}{P_L+p_w} \ge \frac{1}{P_L+p^h_m} \ge \frac{1}{P_L+p_0} = C^l_3, \end{aligned}$$
(38)
$$\begin{aligned} C^u_4= & {} e^{-ap_w} \ge e^{-ap^h_m} \ge e^{-ap_0} = C^l_4. \end{aligned}$$
(39)
and,
$$\begin{aligned} C^u_5 = \frac{e^{-ap_w}}{P_L+p_w} \ge \frac{e^{-ap^h_m}}{P_L+p^h_m} \ge \frac{e^{-ap_0}}{P_L+p_0} = C^l_5. \end{aligned}$$
(40)
where \(C^l_3, C^u_3, C^l_4, C^u_4, C^l_5,C^u_5\ge 0\). Therefore, for the case of increasing matrix porosity, \(C_{1,m}>0\), and sufficiently large \(p^h_m\), and holding (37)–(40), the following coefficient is positive, i.e.,
$$\begin{aligned}&a e^{aP_L} \left( 1+P_L \right) Ei(-x) |^{a(P_L + p^h_m)}_{aP_L} +\frac{P_L e^{-ap^h_m}}{P_L + p^h_m} -1 > \end{aligned}$$
(41)
$$\begin{aligned}&a e^{aP_L} \left( 1+P_L \right) C^l_{2,m} + P_LC^l_5 -1 =C_{3,m}>0. \end{aligned}$$
(42)
Therefore,
$$\begin{aligned}&(F_1(p^h_m), p^h_m)\ge (\phi _{r,m} + C_{1,m}) C^l_4 \frac{M_w}{ZRT} (p^h_m, p^h_m)\nonumber \\&\qquad +\frac{M_wV_L\rho _s}{V_\mathrm{std} } \left[ C_{1,m} C_{3,m} + (1- \phi _{r,m}) C^l_3 (p^h_m, p^h_m) \right] \nonumber \\&\quad \ge \left( \phi _{r,m} + C_{1,m} \right) C^l_4 \frac{M_w}{ZRT} (p^h_m, p^h_m) +\frac{M_wV_L\rho _s}{V_\mathrm{std} } (1- \phi _{r,m}) C^l_3 (p^h_m, p^h_m) \nonumber \\&\quad = \left[ \left( \phi _{r,m} + C_{1,m} \right) C^l_4 \frac{M_w}{ZRT} +\frac{M_wV_L\rho _s}{V_\mathrm{std} } (1- \phi _{r,m}) C^l_3 \right] \Vert p^h_m\Vert ^2 = \gamma _1 \Vert p^h_m\Vert ^2, \end{aligned}$$
(43)
such that,
$$\begin{aligned} \gamma _1 = \left( \phi _{r,m} + C_{1,m} \right) C^l_4 \frac{M_w}{ZRT} +\frac{M_wV_L\rho _s}{V_\mathrm{std} } (1- \phi _{r,m}) C^l_3, \end{aligned}$$
(44)

5.2 Decreasing Matrix Porosity (\(C_{1,m}<0\))

It is clear that in the case of decreasing matrix porosity, \(C_{1,m}<0\), \(C_{1,m}<\phi _{r,m}\), so, we always have, \(\phi _{r,m} - C_{1,m}>0\). Holding the assumption,
$$\begin{aligned} a e^{aP_L} \left( 1+P_L \right) Ei(-x) |^{a(P_L + p^h_m)}_{aP_L} +\frac{P_L e^{-ap^h_m}}{P_L + p^h_m} -1 <0 \end{aligned}$$
(45)
there exist a positive constant \(\gamma _1\), such that,
$$\begin{aligned} (F_1(p^h_m), p^h_m)\ge \gamma _1\Vert p^h_m\Vert ^2. \end{aligned}$$
(46)

5.3 Increasing/Decreasing Fracture Porosity (\(C_{1,f}>0\)/\(C_{1,f}<0\))

We have,
$$\begin{aligned} C^u_4 = e^{-ap_w} \ge e^{-ap^h_f} \ge e^{-ap_0} = C^l_4. \end{aligned}$$
(47)
where \(C^l_4,C^u_4>0\). Therefore, for the case of increasing fracture porosity, \(C_{1,f}>0\), and sufficiently large \(p^h_f\), Therefore,
$$\begin{aligned} (F_2(p^h_f), p^h_f)\ge \left( \phi _{r,f} + C_{1,f} \right) \frac{M_w}{ZRT} C^l_4 (p^h_f, p^h_f) = \gamma _3 \Vert p^h_f\Vert ^2, \end{aligned}$$
(48)
As stated above, for the case of the decreasing porosity, \(C_{1,f}<0\), the coefficient \(( \phi _{r,f} + C_{1,f} )\) remains positive, then the above inequality holds. Therefore, for both cases of increasing or decreasing fracture porosity, namely, \(C_{1,f}>0\) or \(C_{1,f}<0\), and sufficiently large \(p^h_f\), there exist a positive constant,
$$\begin{aligned} \gamma _3 = \left( \phi _{r,f} + C_{1,f} \right) \frac{M_w}{ZRT} C^l_4, \end{aligned}$$
(49)
such that,
$$\begin{aligned} (F_2(p^h_f), p^h_f)\ge \gamma _3\Vert p^h_f\Vert ^2. \end{aligned}$$
(50)

5.4 Boundedness of the Production Term

Since \(p_{fe}\) is bounded by \(p_w\) and \(p_0\), i.e., \(p_w \le p_{fe} \le p_0\), then, \(0<p_{fe}- p_w \le p_0 - p_w =C_p\), such that \(C_p\) is positive number. Holding the conditions,
$$\begin{aligned} 0< \rho _{g*}\le \rho _g\le \rho _g^*, ~~0<k_{f*}\le k_f\le k_f^*,~~0<\mu _{g*}\le \mu _g\le \mu _g^* \end{aligned}$$
(51)
One may find,
$$\begin{aligned} 0 < \frac{k_f(p^h_f) \rho _g(p^h_f) [p_{fe} - p_w]}{\mu _g} \le \frac{k_f^* \rho _g^* C_p}{\mu _g^*} = C_Q >0. \end{aligned}$$
(52)
Thus, \(Q(p^h_f)\) is bounded.
Table 1

Stability parameters for various values of the stress-sensitivity parameter a

a

1E3

1

1E−3

1E−5

1E−7

\(\gamma _1\)

3.9311e−09

3.9311E−9

3.9311E−9

3.9311e−09

1.1798e−07

\(\gamma _3\)

0

0

0

4.0342e−54

2.1043e−09

Table 2

Physical parameters of the model

Parameter

Value

Unit

Description

\(k_{0,m}\)

1E−04

md

Matrix initial permeability

\(k_{0,f}\)

10

md

Fracture initial permeability

\(\phi _{m,0}\)

0.05

Dimensionless

Matrix initial porosity

\(\phi _{m,r}\)

0.049

Dimensionless

Matrix reference porosity

\(\phi _{f,0}\)

0.001

Dimensionless

Fractures initial porosity

\(\phi _{f,r}\)

0.0009

Dimensionless

Fractures reference porosity

R

8.314

\(m^3\)Pa/mol K

Gas constant

\(p_0\)

10.4

MPa

Initial reservoir pressure

\(p_w\)

3.45

MPa

Bottom hole pressure

\(M_w\)

0.016

kg/mol

Molecular weight of methane

\(V_\mathrm{std}\)

0.0224

m\(^3\)/mol

Standard gas volume

\(P_L\)

2.07

MPa

Langmuir pressure

\(V_L\)

2.83E−03

m\(^3\)/kg

Langmuir volume

\(\rho _s\)

2550

kg/m\(^3\)

Shale rock density

\(\mu \)

1.02E−05

Pa s

Initial gas viscosity

\(r_w\)

0.1

m

Wellbore radius

\(L_x\)

0.2

m

Fracture spacing

\(\alpha \)

0.8

Dimensionless

Constant

\(\alpha _m(\alpha _f)\)

1

Dimensionless

Biot’s effective parameter

\(\sigma _m (\sigma _f)\)

30

MPa

Mean total stress

5.5 Sensitivity of Stability Parameters

Based on the primary parameters that are given in Table 2, we can evaluate the coefficients of the stability analysis, namely, \(\gamma _1\) and \(\gamma _3\) that appear in the above analysis. We need first to estimate \(C_{1,m}, C^u_{2,m}, C^u_{4}, C^l_3, C^l_4, C^u_3\). Both the coefficients, \( C_{1,m}=(\phi _{0,m} - \phi _{r,m}) e^{- a \sigma _m}\) and \(C_{1,f}=(\phi _{0,f} - \phi _{r,f}) e^{- a \sigma _f} \) are very small positive numbers that depend on the value of the parameter a. For example, in the case of \(a=1\), \(C_{1,m}\) and \(C_{1,f}\) approach zero, so, in this case the effect of the stress-sensitivity vanishes. We stated the stability coefficients in Table 1, for various values of the constant a. It is clear from this table that as the stress-sensitivity parameter a decreases the stability parameters, \(\gamma _1\) and \(\gamma _3\) increase. In general, one may note that all stability parameters have bounded values.

6 Numerical Experiments

In order to examine the performance of the proposed scheme, we present some numerical investigations. In the following subsections, we present and discuss the physical and computational parameters in the first subsection. In the second subsection, we introduce the stress-sensitivity analysis. Finally, we provide some physical results.

6.1 Physical and Computational Parameters

All physical parameters used in the computations are stated in Table 2 (Bustin et al. 2008; Guo et al. 2014). The DPDP model has a specific permeability and porosity used for each continuum. The porosity of the matrix and fractures are 0.05 and 0.001, respectively. The initial permeability of the matrix is 1\(\times 10^{-4}\) md and initial permeability of the fractures is 10 md. The reservoir depth is of 1665 m with a pressure gradient of 11990 (Pa/m (0.53 psi/ft) and temperature gradient of 0.0198 K\(^\circ \)/m (0.065 K\(^\circ \)/ft). The initial pressure of the reservoir is of 20 mPa and temperature of the reservoir is of 353 K\(^\circ \) (Bustin et al. 2008). We use 20 \(\times \) 20 m, 10 \(\times \) 10 m and 1 \(\times \) 1 m domains and discretize them uniformly. Figure 3 shows a 2D reservoir model. For seeking simplification and due to the domain symmetry, we only consider one-quarter of the whole reservoir.
Fig. 3

2D reservoir model

Fig. 4

Pressure/Velocity distribution of matrix (upper left) and fractures (upper right) and apparent permeability of the matrix (lower left) and fractures (lower right) for a domain of size 20 \(\times \) 20 m

Fig. 5

Matrix apparent permeability profile against time for a domain of size 1 \(\times \) 1 m

Fig. 6

Fractures apparent permeability profile against time for a domain of size 1 \(\times \) 1 m

Fig. 7

Pressure distribution in the matrix for a domain of size 40 \(\times \) 40 m

Fig. 8

Average velocity distribution in the matrix for a domain of size 10 \(\times \) 10 m

6.2 Physical Results

Firstly, let us consider a domain of size 20 \(\times \) 20 m. In the upper left of Fig. 4, the distribution of the matrix pressure along with the matrix average velocity is shown. The fractures pressure with the average fracture velocity distribution is shown in the upper right corner of the same figure. These two graphs illustrate how matrix and fractures pressures decrease gradually around the production well. The corresponding apparent permeabilities of the matrix (lower left) and fractures (lower right) are shown in Fig. 4. It can be seen from these figures that far away from the well, the apparent permeability decreases gradually. One may notice that the high values of the apparent permeability are close to the well because it has a reverse relationship with the pressure, Eq. (6). The matrix and fractures apparent permeability profiles are plotted against time for a domain of size of 1 \(\times \) 1 m in Figs. 5 and 6, respectively. It is clear from these two figures that as the production time increases the apparent permeability increases. Another example for a domain of size 10 \(\times \) 10 m is also provided. One may notice that the production well is located at the upper right corner of the domain. The matrix pressure distribution of this case is shown in Fig. 7, while, its average velocity distribution of the matrix is shown in Fig. 8. Streamlines in fractures for the domain of size 1 \(\times \) 1 m is shown in Fig. 9, while the production well is located at the lower right corner. Its average velocity distribution of fractures is shown in Fig. 10. Figure 11 illustrates the production rate profile against time of production. In Fig. 12, the cumulative production rate profiles are plotted against time of production with and without stress. This figure shows that the stress effect reduces the cumulative production rate.
Fig. 9

Streamlines in fractures for a domain of size 1 \(\times \) 1 m

Fig. 10

Average velocity distribution in fractures for a domain of size 1 \(\times \) 1 m

Fig. 11

Production rate profile against time of production

Fig. 12

Cumulative production rate profile against time of production with and without stresses

The current calculations (in particular, gas production rate and cumulative production, Figs. 11 and 12) are qualitatively comparable to actual field data represented in Figs. 14, 15 and 16 of Esmaili and Mohaghegh (2016), and in Fig. 4a of Yu and Sepehrnoori (2013). However, the quantitative comparison is not adequate because we consider 2D small-scale reservoir in our simulation. Full comparison to actual production data remains for a future work.
Fig. 13

Pressure variation with time at specific points (top) fractures and (bottom) matrix blocks

Fig. 14

Porosity variation with time at specific points (top) fractures and (bottom) matrix blocks

Fig. 15

Porosity change against horizontal lines (top) and vertical lines (bottom)

Fig. 16

The production rate for various values of n

Fig. 17

The production rate for various values of \(\alpha \)

Fig. 18

The production rate for various values of \(\theta \)

Fig. 19

The production rate for various values of \(b_f\)

Fig. 20

The production rate for various values of \(b_m\)

Fig. 21

The production rate of 60 \(\times \) 60 m domain after 1000 days

Fig. 22

The cumulative production of 60 \(\times \) 60 m domain after 1000 days

Now, let us carry out some numerical experiments to see the effect of changing properties such as pressures and porosities. Figure 13 gives an indication that the fracture continuum slop is larger than the matrix continuum slop for the same points at the beginning of the simulation for approximately the first 5 days. After that, the matrix continuum slop becomes bigger than the fracture continuum slop. Figure 14 gives a similar indication to Fig. 13. The fracture continuum slop is larger than the matrix continuum slop for the same points at the beginning of the simulation for approximately the first 5 days. After that, the matrix continuum slop becomes larger than the fracture continuum slop here. This may be interpreted as; the gas is produced from the production well, the reservoir sediments move closer to each other causing porosity reduction. Figure 15 shows that as we proceed closer to the production well, the porosity decreases more, which is consistent with the physical behavior of such system.

Figure 16 illustrates the influence of the factor n on the production rate. This figure indicates that the production rate increases as the coefficient n increases approximately in the first 100 day of the production, and after that the opposite is true. The reason of that is the dependance of the transfer term on the factor n in terms of the shape factor \(\sigma \) (Eq. (15)).

Figure 17 shows the production rate for various values of the coefficient, \(\alpha \). This figure indicates that the production rate increases as \(\alpha \) increases. Brown et al. (1946) proposed the coefficient, \(F = 1 + \sqrt{8\pi \gamma } \frac{\mu }{rp} (\frac{2}{\alpha }-1)\), to correct the slip velocity in a tube. It is clear from this formula that \(\alpha \) has an opposite effect on the slip effect which, in turn, improves the flow on the boundary. This may interpret the previous observation. Javadpour et al. (2007) has estimated a value for \(\alpha = 0.8\). A constructed model gives a good match with experimental results of Roy and Raju (2003), with the same value for \(\alpha = 0.8\) (El-Amin et al. 2017).

The production rate profiles are plotted in Fig. 18 against production time for various values of \(\theta \). This figure indicates that in the case of the production well is located in the center (\(\theta =2\pi \)), the production rate is higher.

The production rate profiles are plotted in Fig. 19 against production time for various values of \(b_f\), while, they are plotted in Fig. 20 for various values of \(b_m\). Figure 19 indicates that the slippage factor \(b_f\) has no effect on the flow; however, the slippage factor \(b_m\) has a significant effect on the flow as seen in Fig. 20.

The production rate and cumulative production of 60 \(\times \) 60 m domain after 1000 days are plotted, respectively, in Figs. 21 and 22. It is clear from these two figures that the domain size has a positive effect on the production rate and the cumulative production. It is noteworthy that the cumulative production depends on the domain size, the fracture and matrix permeability, and the other model parameters.

7 Conclusions

This paper is devoted to construct a DPDP model under geomechanical effect and develop MFE technique with a new stability analysis for shale-gas simulation. The dual-porosity dual-permeability model with the slippage effect and the apparent permeability has been considered in the model. The gas adsorption on the pore surface of matrix is described by the Langmuir isotherm model. The thermodynamics deviation factor and the Peng–Robinson equation of state are also used for the thermodynamics calculations. The MFEM is developed to simulate the problem under consideration; and theoretical and numerically stability analysis of the MFEM is introduced. Stability conditions of the MFEM are stated and estimated. The boundedness of the stability parameters that are related to the physical parameters are proved. The governing coupled two pressure equations are solved using a semi-implicit scheme and the thermodynamic calculations are carried out explicitly. Numerical experiments are conducted for various values of model parameters, the reservoir size, and the production time. Results such as cumulative rate and variations in pressures and apparent permeability are represented in graphs. We found that the apparent permeability decreases gradually as it goes away from the production well, while the opposite is true for the pressure. Variation in the apparent permeability occurs based on Klinkenberg effect, in which the apparent permeability depends on 1 / p. We also found that the effect of the stress reduces the cumulative production rate. Variation in the porosity is related to the stress-sensitivity effect based on Eq. (21). However, it is not necessarily that the stress reduces the cumulative production rate. These variations depend on the values of corresponding physical parameters. For example, the coefficient \(C_{1,d}\) is relatively small depending on the variation of the porosity. It has a positive value in the case of porosity increasing while it has a negative value in the case of porosity decreasing. We provided a numerical discussion about selecting these parameters and other stress-sensitivity parameters. It has been found that the factor n is a major parameter of the gas transfer between matrix and fractures. We also observed that as \(\alpha \) increases the production rate and the cumulative production increase. Moreover, from the results, we concluded that when the production well is located in the center of the field the production rate will be higher. Finally, we found out that the slippage factor in the fracture system \(b_f\) has no effect on the flow, however, the slippage factor in the matrix system \(b_m\) has a significant effect on the flow.

Notes

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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.College of EngineeringEffat UniversityJeddahKingdom of Saudi Arabia
  2. 2.School of Mathematics and StatisticsHubei Engineering UniversityXiaoganChina
  3. 3.Division of Physical Sciences and Engineering (PSE)King Abdullah University of Science and Technology (KAUST)Thuwal, JeddahKingdom of Saudi Arabia

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