Sign-preserving second-order IMPEC time discretization and its application in compressible miscible displacement with Darcy-Forchheimer models

Abstract

In this paper, we construct sign-preserving second-order IMplicit Pressure Explicit Concentration (IMPEC) time methods for generalized coupled non-Darcy flow and transport problems in petroleum engineering, extending the algorithm given in [10] which is only applicable to Darcy flows. We use interior penalty discontinuous Galerkin (IPDG) methods for spatial discretization, and develop bound-preserving technique to obtain physically relevant numerical approximations. The sign-preserving second-order IMPEC method is an important innovation. The method is based on the framework of the second-order strong-stability-preserving Runge-Kutta (SSP-RK2) method. The basic idea is to treat the pressure equation implicitly and the concentration equation explicitly so as to obtain a first-order time integration. Then we introduce a correction stage to compensate the accuracy, maintaining the physical bounds of the numerical cell averages. Unfortunately, the above algorithm is not applicable to non-Darcy problems. There are two main difficulties. Firstly, since the velocity equation is nonlinear and time-independent, all variables in the equation must be calculated at the same time level. However, the treatment in [10] will yield an extremely complicated algorithm and significantly large computational cost, and some iterations whose convergence may not be available if the solutions are not smooth. In our scheme, we linearize the velocity equation and the numerical solutions are not at the same time level, leading to first-order accurate solutions. Therefore, we adopt a completely different approach from [10] to derive the correction stage, keeping the physical bounds of the numerical solutions. Secondly, though with the new correction stage, it is still not easy to solve velocity equations in some stages. In this paper, we construct a direct solver for the velocity equation to save computational cost. Numerical experiments will be given to demonstrate that the improved sign-preserving second-order IMPEC scheme can reduce the computational cost significantly compared with explicit schemes if the diffusion coefficient D is small in the concentration equation. The proposed method also yields much larger CFL number compared with first-order IMPEC schemes. Moreover, the effectiveness of the bound-preserving technique will also be verified.