A Thermodynamically Consistent Modeling and Numerical Framework for Non‐Isothermal Incompressible Two‐Phase Flow in Porous Media: Entropy Stability and Energy Conservation

ABSTRACT In this paper, we focus on numerical modeling of coupled processes of heat transfer and two‐phase flow in porous media, which play a crucial role in many fields, particularly in thermally enhanced oil recovery and geothermal production. We first introduce a thermodynamically consistent numerical modeling framework for non‐isothermal incompressible immiscible two‐phase flow in porous media, which integrates the energy conservation equation with the newly developed two‐phase flow equations. Applying the Gibbs fundamental relation, we rigorously derive an entropy equation, which demonstrates that the model obeys the second law of thermodynamics. To resolve numerical challenging aspects resulting from the inherent nonlinearity and strong coupling of the model, we apply subtle implicit and explicit mixed treatments and the energy factorization approach, in order to design a linearized and decoupled time marching scheme. The spatial discretization is constructed using the cell‐centered finite volume method with carefully designed treatments. In particular, the averaging and upwind strategies are applied for discretizing the energy conservation equation to enforce the local energy conservation and the entropy stability (i.e., the adherence to the second law of thermodynamics). Taking advantage of the discrete versions of the Gibbs relation and the specific mean and difference splitting rules, we derive a discrete counterpart of the second law of thermodynamics, which yields the entropy stability without any restriction on time step sizes. Numerical experiments are performed to demonstrate the features and capabilities of the proposed scheme.