Operator Splitting Multiscale Finite Volume Element Method for Two-Phase Flow with Capillary Pressure

Abstract

A numerical method used for solving a two-phase flow problem as found in typical oil recovery is investigated in the setting of physics-based two-level operator splitting. The governing equations involve an elliptic differential equation coupled with a parabolic convection-dominated equation which poses a severe restriction for obtaining fully implicit numerical solutions. Furthermore, strong heterogeneity of the porous medium over many length scales adds to the complications for effectively solving the system. One viable approach is to split the system into three sub-systems: the elliptic, the hyperbolic, and the parabolic equation, respectively. In doing so, we allow for the use of appropriate numerical discretization for each type of equation and the careful exchange of information between them. We propose to use the multiscale finite volume element method (MsFVEM) for the elliptic and parabolic equations, and a nonoscillatory difference scheme for the hyperbolic equation. Performance of this procedure is confirmed through several numerical experiments.

Enhanced multiscale mixed methods for two-phase flows in high-contrast porous media

“…Many applications to two-phase flows can be found, for example, in (EFENDIEV et al, 2006;KROGSTAD;LIE, 2006;GANIS et al, 2014c;LIE, 2016;GALVIS, 2016;DURÁN et al, 2020) and references therein. The coupling of the multiscale flow and transport problem has already been treated by operator splitting techniques PEREIRA, 1997;FURTADO et al, 2011;HILL, 2020) and implicit formulations TCHELEPI, 2006;TCHELEPI, 2019;GANIS et al, 2014c).…”

The Multiscale Perturbation Method for Two-Phase Reservoir Flow Problems

“…In order to explain our approach, consider the operator splitting scheme for two-phase flows [20,21,22]. If the above mentioned multiscale mixed method is applied to solve the pressure equation, then a set of multiscale basis functions has to be, in principle, recomputed every time the solution algorithm calls for an updated velocity field.…”

A multiscale Robin-coupled implicit method for two-phase flows in high-contrast formations

“…Our focus in this work is the solution of nonlinear two-phase flow models. In the literature the coupling of multiscale flow and transport problems has been treated by explicit operator splitting techniques [18,19] and implicit formulations [20,21]. We have considered operator splitting techniques with explicit approximations for the transport problem in previous works [15,17].…”