Mixed Finite Element Simulation with Stability Analysis for Gas Transport in Low-Permeability Reservoirs

Abstract

Natural gas exists in considerable quantities in tight reservoirs. Tight formations are rocks with very tiny or poorly connected pors that make flow through them very difficult, i.e., the permeability is very low. The mixed finite element method (MFEM), which is locally conservative, is suitable to simulate the flow in porous media. This paper is devoted to developing a mixed finite element (MFE) technique to simulate the gas transport in low permeability reservoirs. The mathematical model, which describes gas transport in low permeability formations, contains slippage effect, as well as adsorption and diffusion mechanisms. The apparent permeability is employed to represent the slippage effect in low-permeability formations. The gas adsorption on the pore surface has been described by Langmuir isotherm model, while the Peng-Robinson equation of state is used in the thermodynamic calculations. Important compatibility conditions must hold to guarantee the stability of the mixed method by adding additional constraints to the numerical discretization. The stability conditions of the MFE scheme has been provided. A theorem and three lemmas on the stability analysis of the mixed finite element method (MFEM) have been established and proven. A semi-implicit scheme is developed to solve the governing equations. Numerical experiments are carried out under various values of the physical parameters.

Chemical Potential-Based Modeling of Shale Gas Transport

“…Modeling of gas flow in shale media plays a crucial role in predicting shale gas production [10][11][12][13][14][15][16][17][18][19]. For gas flow in tight porous media, the most remarkable phenomenon is the so-called Klinkenberg effect [20], which results from slip flow of gas molecules through very small pores.…”

“…This effect leads to the apparent permeability that is generally greater than the absolute permeability of a porous medium [14,15]. By using the apparent permeability, the shale gas flow equation can be simply formulated as the form of Darcy's law, which states that gas velocity is proportional to the pressure gradient [12][13][14][15][16][17][18][19]21].…”

Theoretical stability analysis of mixed finite element model of shale-gas flow with geomechanical effect

“…Girault et al [21] have introduced a priori error estimates for a discretized poro-elastic-elastic system, in which the flow pressure equation is discretized by either a continuous Galerkin scheme or a mixed scheme, while, elastic displacement equations are discretized by a continuous Galerkin scheme. El-Amin et al [22] have used the MFEM with stability analysis to simulate the problem of natural gas transport in a low-permeability reservoir without considering fractures. The authors [23] extended their work to cover fractured porous media and the rock stress-sensitivity with considered stability analysis of the MFEM.…”