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International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.452 SNIP: 0.68 CiteScore™: 1.18

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.v9.i2.80
pages 243-256

A MULTILEVEL MULTISCALE MIMETIC (M3) METHOD FOR AN ANISOTROPIC INFILTRATION PROBLEM

Konstantin Lipnikov
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
David Moulton
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
Daniil Svyatskiy
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

ABSTRACT

Modeling of multiphase flow in highly heterogeneous porous media must capture a broad range of spatial and temporal scales that are strongly influenced by the complex structure of the subsurface environment. However, the most popular discretization methods are based on a two-point flux approximation that is only accurate for simple geometries, such as orthogonal meshes with mesh-aligned diagonal tensor permeabilities. In more realistic situations, such as sloping layers with anisotropic permeabilities, these methods may provide qualitatively plausible results, but with O(1) errors. The family of mimetic finite-difference discretizations uses a mimetic approximation of the flux to provide a robust and accurate discretization in these more complex geometries. In addition, the increasing use of large-scale models with high-resolution realizations of parameter fields is driving the need for efficient multiscale methods. Typically, these methods are based on two-level concepts that do not effectively capture the coupling of local and global information critical to anisotropic features. To address these problems we have recently developed a new hierarchical approach, dubbed the multilevel multiscale mimetic (M 3) method, which builds on the mimetic methodology. The M 3 method is locally mass conserving at all levels in its hierarchy, it supports unstructured polygonal grids and full tensor permeabilities, and it can achieve large coarsening factors. To highlight the advantages of the mimetic flux approximation, as well as the flexibility and efficiency of the M 3 method, we consider water infiltration into a two-dimensional layered anisotropic medium. The mesh is aligned with the sloping layers, not the coordinate axes. First, we present a comparison of the water infiltration resulting from the two-point and mimetic flux approximations. Then, we demonstrate that with an efficient temporal updating strategy for the coarsening parameters, fine-scale accuracy of prominent features in the flow is maintained by the M 3 method.

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