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On the conservation properties in multiple scale coupling and simulation for Darcy flow with hyperbolic-transport in complex flows. (English) Zbl 1454.35271

Summary: 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 the following: (1) multiscale heterogeneity structures appearing in the elliptic-pressure-velocity and in the rock geology model and (2) multiscale wave structures resulting from interactions of shock waves and rarefaction from the nonlinear hyperbolic-transport 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 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.

MSC:

35Q35 PDEs in connection with fluid mechanics
35J20 Variational methods for second-order elliptic equations
35L67 Shocks and singularities for hyperbolic equations
76M12 Finite volume methods applied to problems in fluid mechanics
76M10 Finite element methods applied to problems in fluid mechanics
76L05 Shock waves and blast waves in fluid mechanics
76S05 Flows in porous media; filtration; seepage
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