A structure-preserving numerical scheme for non-isothermal compressible flow in porous media: Energy conservation and entropy stability

Numerical simulation of non-isothermal flow in porous media has great applications in unconventional gas recovery. Reliable and promising numerical methods should be compatible with the first and second laws of thermodynamics. Toward this goal, by taking molar density and temperature as the primary unknowns, we propose an alternative formulation of the realistic non-isothermal gas flow model, which is proved to obey the first and second laws of thermodynamics. On the basis of this formulation, we propose an energy conservative and entropy stable numerical scheme; that is, the scheme can preserve the first law of thermodynamics (the energy conservation law) as well as the second law of thermodynamics (entropy stability). An important ingredient of the scheme is to derive a new discrete chemical potential utilizing the energy factorization approach, which can preserve the dissipation of original Helmholtz free energy. Some subtle temporal and spatial treatments have also been introduced to deal with the inherent coupling relationships between multiple variables. Using the local analysis method, we prove that the scheme can be compatible with both local and global forms of the first and second laws of thermodynamics. Numerical results validate our theoretical analysis and demonstrate conservative and stable features of the proposed scheme.