An Approximation to Miscible Fluid Flows in Porous Media with Point Sources and Sinks by an Eulerian--Lagrangian Localized Adjoint Method and Mixed Finite Element Methods

Abstract

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.

An Eulerian-Lagrangian Formulation for Compositional Flow in Porous Media

“…We then multiply equation (3.1) by these weighting functions z(x, t) and integrate the resulting equation to obtain an Eulerian-Lagrangian method [17,8,10,19]…”

“…They generate accurate numerical solutions and significantly reduce the numerical diffusion and grid-orientation effect present in upwind methods, even if large time steps are used. Eulerian-Lagrangian methods have been developed for miscible displacement in enhanced oil recovery [6,7,8,9,10] and for immiscible displacement in water flooding [11,12]. Because of increased efficiency, more components can be modeled using Eulerian-Lagrangian methods.…”

“…(4.1) can be reformulated into a similar form to single-phase flow [8,10]. We utilize momentum balance to define a barycentric component velocity …”

Modified Method of Characteristics Combined with Finite Volume Element Methods for Incompressible Miscible Displacement Problems in Porous Media

“…For the test problems, we have taken data from [28]. The spatial domain is Ω = (0, 1000) × (0, 1000) ft 2 , the time period is [0, 3600] days, and viscosity of oil is (0) = 1.0 cp.…”

Numerical Homogenization of a Nonlinearly Coupled Elliptic–parabolic System, Reduced Basis Method, and Application to Nuclear Waste Storage

“…In particular, for the numerical tests of the coupled system we use a naive P 1 -finite element method in space combined with the implicit Euler method in time. For more efficient and modern methods, we refer the reader to 24,14,15,26,27,8,9 . The main contribution of this section (a numerical method for the computation of D * ) can indeed be combined with any strategy to solve the advection-diffusion equation (2.12).…”