On the Conservation Properties in Multiple Scale Coupling and Simulation for Darcy Flow with Hyperbolic-Transport in Complex Flows

Abstract

We present and discuss a novel approach to deal with conservation properties for the simulation of nonlinear complex porous media flows in the presence of: 1) multiscale heterogeneity structures appearing in the elliptic-pressure-velocity and in the rock geology model, and 2) multiscale wave structures resulting from shock waves and rarefaction interactions from the nonlinear hyperbolictransport model. For the pressure-velocity Darcy flow problem, we revisit a recent high-order and volumetric residual-based Lagrange multipliers saddle point problem to impose local mass conservation on convex polygons. We clarify and improve conservation properties on applications. For the hyperbolic-transport problem we introduce a new locally conservative Lagrangian-Eulerian finite volume method. For the purpose of this work, we recast our method within the Crandall and Majda treatment of the stability and convergence properties of conservation-form, monotone difference, in which the scheme converges to the physical weak solution satisfying the entropy condition. This multiscale coupling approach was applied to several nontrivial examples to show that we are computing qualitatively correct reference solutions. We combine these procedures for the simulation of the fundamental two-phase flow problem with high-contrast multiscale porous medium, but recalling state-of-the-art paradigms on the of notion of solution in related multiscale applications. This is a first step to deal with out-of-reach multiscale systems with traditional techniques. We provide robust numerical examples for verifying the theory and illustrating the capabilities of the approach being presented.

An exponential integration generalized multiscale finite element method for parabolic problems

“…Approximations of solutions of problem (1) and many other interesting questions have been considered in the literature. In particular we mention [14,26,2,33] and references therein. We focus our discussion on the numerical computation of solutions of this problem.…”

“…We consider the case of time marching schemes where, in each iteration, a large ill-conditioned linear problem has to be solved. We mention the recent papers [12,20,13,9,2,30,29] where different time dependent problems have been considered within the GMsFEM framework. Unfortunately, the loss of stability (due to the high-contrast in the coefficients) requires the reduction of the time step size (inversely proportional to the contrast in the coefficient) which ends up reducing the gain obtained by using the GMsFEM method to improve the overall computational time to obtain the final time solution.…”