The Computation of Nonclassical Shock Waves in Porous Media with a Heterogeneous Multiscale Method: The Multidimensional Case

Abstract

We consider two-phase flow problems in porous media with overshooting waves. If higher order effects are neglected, such problems are governed by a hyperbolic conservation law with non-convex flux function as macroscale model. It is well known, that there can be multiple weak solutions. To ensure uniqueness, we use on the microscale either a kinetic relation or in a second approach an extended system, taking rate-dependent capillary pressure effects into account. We present a new multidimensional mass-conserving numerical method to solve the macroscale model. The method belongs to the class of Heterogeneous Multiscale Methods in the sense of E&Engquist (2003) and is L ∞-stable. A key part of the approximation is a novel numerical flux function for the multidimensional setting, which captures undercompressive waves and generalizes the one-dimensional approach of Boutin et al. (2008). Furthermore, we improve the overall computational complexity, using a data-based approach. Finally, we validate the numerical method and test it on several infiltration problems.

Constraint-aware neural networks for Riemann problems

“…In this section we test the constraint-aware methods described in Section 4 for the cubic ux model, i.e. we seek to learn the Riemann solver (21), with κ = 0.75 in the kinetic relation (20). The resulting networks are compared with their standard/constraint-unaware counterparts.…”

“…with u − > 0, while selecting solutions that satisfy the kinetic relation (20). The kinetic relation comes with a corresponding, so-called companion function (see [23]), which is de ned in this instance by…”

On Darcy- and Brinkman-type models for two-phase flow in asymptotically flat domains

“…Moreover, the spreading speed of the invading fluid U ≈ 1.27 is smaller than that using the VE-model u ≈ 1.33. Mathematically, the phenomenon of saturation overshoots is identified by undercompressive waves, which are known to be slower than classical compressive waves [21]. Yortsos and Salin [30] expect that sharp bounds on the spreading speed might be found using these waves.…”

Surrogate modeling of multiscale models using kernel methods

“…Then, is the missing microscale information, which is fed back into the macroscale model using a specific flux function (cf. , § 6.1] for details).…”

“…Other quantities like the permeability matrix P 2 R 2 2 are suitably given. This model allows to produce the observed overshoot effects as the solutions may contain nonclassical shock waves [69,72]. Unfortunately, besides lack of a unique solution, Equation (15) alone does not produce the observed overshoots as microscale information is missing.…”