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We develop an Eulerian-Lagrangian localized adjoint method (ELLAM)-mixed finite element method (MFEM) solution technique for accurate numerical simulation of coupled systems of partial differential equations (PDEs), which describe complex fluid flow processes in porous media. An ELLAM, which was shown previously to outperform many widely used methods in the context of linear convection-diffusion PDEs, is presented to solve the transport equation for concentration. Since accurate fluid velocities are crucial in numerical simulations, an MFEM is used to solve the pressure equation for the pressure and Darcy velocity. This minimizes the numerical difficulties occurring in standard methods for approximating velocities caused by differentiation of the pressure and then multiplication by rough coefficients. The ELLAM-MFEM solution technique significantly reduces temporal errors, symmetrizes the governing transport equation, eliminates nonphysical oscillation and/or excessive numerical dispersion in many simulators, conserves mass, and treats boundary conditions accurately. Numerical experiments show that the ELLAM-MFEM solution technique simulates miscible displacements of incompressible fluid flows in porous media accurately with fairly coarse spatial grids and very large time steps, which are one or two orders of magnitude larger than the time steps used in many methods. Moreover, the ELLAM-MFEM solution technique can treat large mobility ratios, discontinuous permeabilities and porosities, anisotropic dispersion in tensor form, and point sources and sinks.

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Modeling and numerical simulations of Carbonate karst reservoirs is a challenging problem because of the presence of vugs and caves which are connected through fracture networks at multiple scales. In this paper, we propose a unified approach to this problem by using the Stokes-Brinkman equations which combine both Stokes and Darcy flows. These equations are capable of representing porous media (porous rock) as well as free-flow regions (fractures, vugs, and caves) in a single system of equations. The Stokes-Brinkman equations also generalize the traditional Darcy-Stokes coupling without sacrificing the modeling rigor. Thus, it allows us to use a single set of equations to represent multiphysics phenomena on multiple scales. The local Stokes-Brinkman equations are used to perform accurate scale-up. We present numerical results for permeable rock matrix populated with elliptical vugs and we consider upscaling to two different coarse-scale grids-5Â5 and 10Â10. Both constant and variable background permeability matrices are considered and the effect the vugs have on the overall permeability is evaluated. The Stokes-Brinkman equations are also used to study several vug/cave configurations which are typical of Tahe oilfield in China.