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

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## 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

*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),

*l*is given by,

*x*,

*y*and

*z*, respectively, and

*n*is the set of normal fractures.

## 3 Model Validation

## 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:

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

*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,

## 5 Stability Analysis

*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 (\(C_{1,m}>0\))

*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,

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

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

### 5.4 Boundedness of the Production Term

Stability parameters for various values of the stress-sensitivity parameter *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 |

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 |

| 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

### 6.2 Physical Results

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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