Efficient and Thermodynamically Consistent Numerical Schemes for Porous Media Flow and Multicomponent Transport

Abstract

In the first part, efficient and fully mass-conservative numerical schemes for gas flow and multicomponent transport in fractured porous media are studied. The gas flow and multicomponent transport in fracture is rigorously derived and described by reduced fracture modeling. A new IMplicit Pressure Explicit Concentration (IMPEC) scheme is derived for the gas flow and multicomponent transport in fractured porous media. Compared with IMPEC schemes in the literature by which mass-conservation of all species may not be guaranteed, the proposed scheme overcomes the drawbacks of the conventional IMPEC methods and is thermodynamically consistent and computationally efficient. The second part presents an efficient convex splitting scheme for numerical simulation of multicomponent two-phase fluids mixture in a closed domain at constant temperature, which is modeled by diffuse interface theory coupling with the Van der Waals and the Peng-Robinson equations of state (EOS). The proposed numerical algorithm avoids the indefiniteness of the Hessian matrix arising from the second-order derivative of homogeneous contribution of total Helmholtz free energy; it is extremely efficient and easy-to-implement as well. This scheme is unconditionally componentwise energy stable and naturally results in unconditional stability for the Van der Waals model. For the Peng-Robinson EOS, it is unconditionally stable by introducing a physics-preserving correction term, which is analogous to the attractive term in the Van der Waals EOS. The third part studies the dynamical modeling of composition variation under gravity in the framework of the modified Helmholtz free energy coupling with the realistic equations of state. An efficient, easy-to-implement, thermodynamically consistent, and robust numerical scheme is derived for the models. This numerical scheme for multicomponent systems is rigorously proved to be unconditionally stable. The implementation is straightforward based on the single-component system and is not required to choose a reference species for multicomponent fluids. The present scheme is computationally efficient and saves computer memory.