Validation of the multiscale mixed finite‐element method

Abstract

SUMMARYSubsurface reservoirs generally have a complex description in terms of both geometry and geology. This poses a continuing challenge in modeling and simulation of petroleum reservoirs owing to variations of static and dynamic properties at different length scales. Multiscale methods constitute a promising approach that enables efficient simulation of geological models while retaining a level of detail in heterogeneity that would not be possible via conventional upscaling methods. Multiscale methods developed to solve coupled flow equations for reservoir simulation are based on a hierarchical strategy in which the pressure equation is solved on a coarsened grid and the transport equation is solved on the fine grid, and the two equations are treated as a decoupled system. In particular, the multiscale mixed finite-element (MsMFE) method attempts to capture subgrid geological heterogeneity directly into the coarse-scale equations via a set of numerically computed basis functions. These basis functions are able to capture the predominant multiscale information and are coupled through a global formulation to provide good approximation of the subsurface flow solution. In the literature, the general formulation of the MsMFE method for incompressible two-phase and compressible three-phase flow has mainly addressed problems with idealized flow physics. In this paper, we first outline a recent formulation that accounts for compressibility, gravity, and spatially dependent rockfluid parameters. Then, we validate the method by evaluating its computational efficiency and accuracy on a series of representative benchmark tests that have a high degree of realism with respect to flow physics, heterogeneity in the petrophysical models, and geometry/topology of the corner-point grids. In particular, the MsMFE method is validated and compared against an industry-standard fine-scale solver. The fine-scale flux, pressure, and saturation fields computed by the multiscale simulation show a noteworthy improvement in resolution and accuracy compared with coarse-scale models.

Generalized multiscale finite element method for highly heterogeneous compressible flow

“…[16,42,43]. Using other types of multiscale methods for the compressible flow can be found in [31,27,5,34,37,6], but there is few evidence that these methods (most of which are the variants of MsFEM) can deal with arbitrarily complicated porous media. Therefore it is necessary to systemically study the GMsFEM for the nonlinear compressible flow in high-contrast media.…”

A Comparison of Mixed Multiscale Finite Element Methods for Multiphase Transport in Highly Heterogeneous Media

“…The mixed and mortar multiscale method are based on first‐order discretization for the elliptic pressure equation, a conservative velocity field can be obtained for the transport equation of the two‐phase system. It has shown a great success in simulating very practical nonlinear two phase and three phase flows (Aarnes, 2004; Pal et al., 2015; Singh et al., 2019). The multiscale finite volume method (Jenny et al., 2003) is based on the second‐order discretization of the pressure equation and is motivated by the multiscale finite element method (MsFEM), but it can also yield conservative velocity field and thus can be used for two‐phase simulation.…”

“…mixed multiscale finite element methods (MMsFEM) (Aarnes, 2004;Aarnes & Efendiev, 2008;Chen & Hou, 2003; E. T. Chung, Efendiev, & Leung, 2015b;Pal et al, 2015;Singh et al, 2019), mortar multiscale methods (Peszyńska, 2005;Peszyńska et al, 2002;Todd et al, 2007), discontinuous Galerkin (DG) methods (Cockburn et al, 2002;Du & Chung, 2018;Kim et al, 2013), and postprocessing methods (Bush & Ginting, 2013;Odsaeter et al, 2017). Among these methods, the mixed and mortar mixed multiscale method and the multiscale finite volume method are the most popular methods for highly heterogeneous two-phase simulations.…”

A Multiscale Restriction-Smoothed Basis Method for Compressible Black-Oil Models

“…On the other hand, the multiscale mixed finite-element (MsMFE) method [10] is the method that has come the longest way in incorporating realistic descriptions of reservoir geology, see [1-3, 42, 46]. However, except for the somewhat idealized black-oil and compressible flow cases studied in [26,27,31,46], the MsMFE method has so far only been demonstrated to work well for incompressible flow.…”