## 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.

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## References

Arbogast, T., Huang, C.: A fully mass and volume conserving implementation of a characteristic method for transport problems. SIAM J. Sci. Comput.

**28**(6), 2001–2022 (2006)Arbogast, T., Wheeler, M.F., Yotov, I.: Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J Num. Anal.

**34**(2), 828–852 (1997)Biot, M.A.: General theory of three dimensional consolidation. J. Appl. Phys.

**12**, 155–164 (1941)Biot, M.A.: Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range. J. Acoust. Soc. Am

**28**, 168–178 (1956)Biot, M.A.: Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range. J. Acoust. Soc. Am

**28**, 179–191 (1956a)Biot, D.: Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys.

**33**, 1482–1498 (1962b)Biot, M.A., Willis, D.: The elastic coefficients of the theory of consolidation. J. Appl. Mech.

**24**, 594–601 (1957)Brezzi, F., Fortin, V.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)

Brezzi, F., Douglas, J., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math.

**47**, 217–235 (1985)Brown, G., Dinardo, A., Cheng, G., Sherwood, T.: The flow of gases in pipes at low pressures. J. Appl. Phys.

**17**, 802–813 (1946)Bustin, A., Bustin, R., Cui, X.: Importance of fabric on the production of gas shales. In: Unconventional reservoirs conference in Colorado, USA, 10–12 February. SPE-114167-MS (2008)

Civan, F., Rai, C.S., Sondergeld, C.H.: Shale-gas permeability and diffusivity inferred by improved formulation of relevant retention and transport mechanisms. Transp. Porous Media

**86**(3), 925–944 (2011)Cui, X., Bustin, A.M.M., Bustin, R.M.: Measurements of gas permeability and diffusivity of tight reservoir rocks: different approaches and their applications. Geofluids.

**9**(3), 208–223 (2009)El-Amin, M.F.: Analytical solution of the apparent-permeability gas-transport equation in porous media. Eur. Phys. J. Plus

**132**, 129–135 (2017)El-Amin, M.F., Amir, S., Salama, A., Urozayev, D., Sun, S.: Comparative study of shale-gas production using single- and dual-continuum approaches. J. Pet. Sci. Eng.

**157**, 894–905 (2017)El-Amin, M.F., Radwan, A., Sun, S.: Analytical solution for fractional derivative gas-flow equation in porous media. Results Phys.

**7**, 2432–2438 (2017)El-Amin, M.F., Kou, J., Sun, S.: Mixed finite element simulation with stability analysis for gas transport in low-permeability reservoirs. Energies

**11**(1), 208–226 (2018)Ertekin, T., King, G.R., Schwerer, F.C.: Dynamic gas slippage: a unique dual-mechanism approach to the flow of gas in tight formations. SPE Formation Evaluation (Feb.) 43–52 (1986)

Esmaili, S., Mohaghegh, S.D.: Full field reservoir modeling of shale assets using advanced data-driven analytics. Geosci. Front.

**7**, 11–20 (2016)Firoozabadi, A.: Thermodynamics and Applications in Hydrocarbon Reservoirs and Production. McGraw-Hill Education - Europe, USA (2015)

Freeman, C., Moridis, G., Michael, G.: Measurement, modeling, and diagnostics of flowing gas composition changes in shale gas well. In: Latin America and Caribbean Petroleum Engineering Conference in Mexico City, Mexico. 16–18 April. SPE-153391-MS (2012)

Guo, C., Wei, M., Chen, H., He, X., Bai, B.: Improved numerical simulation for shale gas reservoirs. In: Offshore Conference-Asia, 25–28 March. Kuala Lumpur, Malaysia (2014)

Holcomb, D.J., Brown, S.R., Lorenz, J.C., Olsson, W.A., Teufel, L.W., Warpinski, N.R.: Geomechanics of horizontally-drilled, stress-sensitive, naturally-fractured reservoirs. Technical Report, SAND-94-1743, Sandia National Labs, United States (1994)

Hu, X., Yu, W., Liu, M., Wang, M., Wang, W.: A multiscale model for methane transport mechanisms in shale gas reservoirs. J. Pet. Sci. Eng.

**172**, 40–49 (2019)Javadpour, F., Fisher, D., Unsworth, M.: Nanoscale gas flow in shale gas sediments. J. Can. Pet. Technol. 46(10) (2007)

Javadpour, F.: Nanopores and apparent permeability of gas flow in mudrocks (shales and siltstone). J. Can. Pet. Technol.

**48**(8), 16–21 (2009)Li, D., Wang, J.Y., Zha, W., Lu, D.: Pressure transient behaviors of hydraulically fractured horizontal shale-gas wells by using dual-porosity and dual-permeability model. J. Pet. Sci. Eng.

**164**, 531–545 (2018)Liu, J., Wang, J.G., Gao, F., Leung, C.F., Ma, Z.: A fully coupled fracture equivalent continuum-dual porosity model for hydro-mechanical process in fractured shale gas reservoirs. Comput. Geotech.

**106**, 143–160 (2019)Moridis, G., Blasingame, T., Freeman, C.: Analysis of mechanisms of flow in fractured tight-gas and shale-gas reservoirs. In: Latin American and Caribbean Petroleum Engineering Conference1-3 December. SPE-139250-MS (2010)

Nakshatrala, K.B., Turner, D.Z., Hjelmstad, K.D., Masud, A.: A mixed stabilized finite element formulation for Darcy flow based on a multiscale decomposition of the solution. Comput. Meth. Appl. Mech. Eng.

**195**(2006), 4036–4049 (2006)Narasimhan, T.N., Witherspoon, P.: Numerical model for saturated-unsaturated flow in deformable porous media 1. Theory. Water Resour. Res.

**13**, 657–664 (1977)Ozkan, E., Raghavan, R., Apaydin, O.: Modeling of fluid transfer from shale matrix to fracture network. In: Annual Technical Conference and Exhibition, Lima, Peru, 1–3 December. SPE-134830-MS (2010)

Qi, Y., Ju, Y., Jia, T., Zhu, H., Cai, J.: Nanoporous structure and gas occurrence of organic-rich shales. J. Nanosci. Nanotechnol.

**17**(9), 6942–6950 (2017)Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods (Proceedings of Conference, Consiglio Naz. delle Ricerche (C.N.R.), Rome, : Lecture Notes in Mathematics, vol. 606. Springer, Berlin,

**1977**, 292–315 (1975)Riviere, B., Wheeler, M.F.: A discontinuous Galerkin method applied to nonlinear parabolic equations. In: Discontinuous Galerkin methods, pp. 231–244. Springer, Berlin (2000)

Roy, S., Raju, R.: Modeling gas flow through microchannels and nanopores. J. Appl. Phys.

**93**, 4870–4879 (2003)Safai, N.M., Pinder, G.F.: Vertical and horizontal land deformation in a desaturating porous medium. Adv. Water Resour.

**2**, 19–25 (1979)Safai, N.M., Pinder, G.F.: Vertical and horizontal land deformation due to fluid withdrawal. Int. J. Numer. Anal. Methods Geomech.

**4**, 131–142 (1980)Salama, A., El-Amin, M.F., Kumar, K., Sun, S.: Flow and transport in tight and shale formations, Geofluids, Article ID 4251209. (2017)

Shabro, V., Torres-Verdin, C., Javadpour, F.: Numerical simulation of shale-gas production: From pore-scale modeling of slip-flow, Knudsen diffusion, and Langmuir desorption to reservoir modeling of compressible fluid, North American Unconventional Gas Conference and Exhibition in Texas, USA. 14–16 June. SPE-144355-MS (2011)

Shen, W., Zheng, L., Oldenburg, C.M., Cihan, A., Wan, J., Tokunaga, T.K.: Methane diffusion and adsorption in shale rocks: a numerical study using the dusty gas model in TOUGH2/EOS7C-ECBM. Transp Porous Med

**123**, 521–531 (2018)Singh, H., Cai, J.: Screening improved recovery methods in tight-oil formations by injecting and producing through fractures. Int. J. Heat Mass Transf.

**116**, 977–993 (2018)Terzaghi, K.: The shearing resistance of saturated soils and the angle between the planes of shear. In: Proceedings of International Conference on Soil Mechanics and Foundation Engineering, 1, 54–56, Harvard University Press, Cambridge (1936)

Warren, J.E., Root, P.J.: The behavior of naturally fractured reservoirs. Soc. Petrol. Eng. J.

**3**, 245–255 (1963)Wu Y.S., Fakcharoenphol P.: A unified mathematical model for unconventional reservoir simulation. In: EUROPEC/EAGE Annual Conference and Exhibition in Vienna, Austria, 23–26 May. SPE-142884-MS (2011)

Yu, W., Sepehrnoori, K.: Optimization of multiple hydraulically fractured horizontal wells in unconventional gas reservoirs, J. Petrol. Eng. 2013, Article ID 151898 (2013)

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## 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,

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

Given,

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

which is a Hilbert space with norm given by

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,

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

and,

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,

and,

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,

where

and

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,

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,

for any \(\varphi \in W_h\) and \(\omega \in V_h\).

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El-Amin, M.F., Kou, J. & Sun, S. Numerical Modeling and Simulation of Shale-Gas Transport with Geomechanical Effect.
*Transp Porous Med* **126, **779–806 (2019). https://doi.org/10.1007/s11242-018-1206-z

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### Keywords

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