  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 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 newlocally 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.
