Numerical Simulations For Complex Flow System In Porous Media

Complex flow system in porous media describe the miscible displacement process, such as the oil and gas migration in complex strata and groundwater contamination transmission. Therefore, the establishment of accurate, highly efficient numerical methods and complete theoretical system of numerical analysis has important theoretical value and application prospects for deeply revealing the mechanism of percolation and guiding science engineering practice. However, many standard numerical methods do not work or work poorly for complex flow system due to the fact the shape of the flow area is irregular and the global regularity of the solution is usually low.Base on the Lagrange and Crouzeix-Raviart type finite element space’s ca-pability of recognizing and approximating the interface, we develop numerical methods and establish rigorous numerical analysis theoretical system for a class of complex flow system (second order elliptic interface flow system) with homoge-neous and nonhomogeneous interface jump condition in different complex strata. We use the conservation characteristics of finite volume element method and the flexible mesh of the discontinuous Galerkin finite element method in these nu-merical methods. Accordingly, we achieve an efficient numerical simulation of complex flow system, provide a solid scientific foundation for engineering prac-tice, and make up for the deficiency of current theoretical research, improve the system of theoretical analysis of such problem. The main results are as follows:(1) We construct piecewise linear finite element space of Lagrange type on interface element and apply conforming or nonconforming linear finite element space on non-interface element. We proposed a discontinuous Galerkin immersed finite volume element method for isotropic and anisotropic second order elliptic interface problem with homogeneous interface condition respectively, combing with the discontinuous volume element method. The existence and uniqueness of the solutions of the discrete scheme are proved. The optimal error estimates in energy norm and suboptimal error estimates in L2 norm are derived.(2) We transform the isotropic second order elliptic interface flow models with nonhomogeneous interface conditions to homogeneous interface conditions using the level set technology. Enforcing the jump condition into the finite ele-ment space involved the interface element, we construct the piecewise linear finite element space of Lagrange and Crouzeix-Raviart type on each element. Based on the space’s capability of recognizing and approximating the interface, we develop the corresponding partially penalty immersed interface finite element methods and their optimal-order error analysis.(3)Numerical examples are presented to illustrate the theoretical results.