A multiscale mixed finite element method applied to the simulation of two-phase flows

Abstract

The multiscale hybrid mixed finite element method (MHM-H(div)), previously developed for Darcy's problems, is extended for coupled flow/pressure and transport system of two-phase flow equations on heterogeneous media under the effect of gravitational segregation. It is combined with an implicit transport solver in a sequential fully implicit (SFI) manner. The MHM-H(div) method is designed to cope with the complex geometry and inherent multiscale nature of the phenomena. The discretizations are based on a general domain partition formed by polyhedral subregions, where a hierarchy of meshes and approximation spaces are considered. The multiscale approach is applied to the flux/pressure kernel making use of coarse scale normal fluxes between subregions (trace variable). The fine-scale features inside each subregion are determined by resolving completely independent local Neumann problems, the boundary conditions being set by the trace variable, by the mixed finite element method using fine flux and pressure representations. These properties imply that the MHM-H(div) can be interpreted as a classical mixed formulation of the model problem in the whole domain, based on a H(div)conforming space with normal components over the macro-partition interfaces constrained by the trace space, and showing divergence compatibility with the pressure space. Consequently, local mass conservation is observed at the micro-scale elements inside the subregions, an essential property for flows in heterogeneous media, and divergence-free constraint strongly enforced for incompressible flows. The efficient use of static condensation leads to a global system to be solved only in terms of primary degrees of freedom associated with the trace variable and of a piecewise constant pressure for each subregion. This procedure allows a substantial reduction of the dominant computational costs associated with the flux/pressure kernel embedded in the numerical model. An iterative coupling technique is adopted to solve the two-phase flow equations using a shared integration point memory implementation model. At each SFI time step, the efficiency iterative method for the transport equations is improved using a Quasi-Newton method with a simple but effective nonlinear acceleration. The numerical examples show that the proposed scheme is able to solve challenging coupled flow and transport problems.

A posteriori error estimator for a multiscale hybrid mixed method for Darcy's flows

“…Hierarchical shape functions for local FE spaces in elements are useful for simplicity of implementing and assembling the constrained FE spaces . The description of theses algorithms for a variety of mixed formulations of Darcy's flows in different situations can be found in the following works: for accuracy enhancement in the presence of deformed elements, 20,21 for ‐adaptive strategies, 22,23 for the transition between quadrilateral and triangular facets when combining different element geometry in the same computational mesh, 24 and for multiscale flow in porous media and elasticity simulations 1,25,26 …”

“…for the MHM-H(div)- 𝛾 method deformed elements, 20,21 for hp-adaptive strategies, 22,23 for the transition between quadrilateral and triangular facets when combining different element geometry in the same computational mesh, 24 and for multiscale flow in porous media and elasticity simulations. 1,25,26…”

Flux-Mortar Mixed Finite Element Methods on NonMatching Grids

“…The flux-mortar mixed finite element method is related to the subgrid upscaling method proposed in [7]. Moreover, the method has similarities with the multiscale hybrid-mixed (MHM) method with local mixed solves [20,21] and we refer the interested reader to [15,Sec. 1] for an exposition of these relations.…”

A block preconditioner for two‐phase flow in porous media by mixed hybrid finite elements

“…Despite a few drawbacks, such as the enlargement of the stencil or the breach of the discrete maximum principle, 11 MHFE and the pioneering nonhybridized version, the mixed finite element (MFE) method, have witnessed an increasing popularity in the last decade. Recent significant applications include coupled flow-poromechanics, [12][13][14][15][16] deformation models, 17,18 multiphase [19][20][21][22][23] and coupled Stokes-Darcy 24 flow in porous media. An MHFE discretization of the two-phase flow model in porous media, which is similar to the one introduced in this work, was proposed by Fučík and Mikyška, 25 but it relied on a sequential splitting technique to handle the inherent coupling of the governing equations.…”