Operator Splitting for Three-Phase Flow in Heterogeneous Porous Media

Abstract

We describe an operator splitting technique based on physics rather than on dimension for the numerical solution of a nonlinear system of partial differential equations which models three-phase flow through heterogeneous porous media. The model for three-phase flow considered in this work takes into account capillary forces, general relations for the relative permeability functions and variable porosity and permeability fields. In our numerical procedure a high resolution, nonoscillatory, second order, conservative central difference scheme is used for the approximation of the nonlinear system of hyperbolic conservation laws modeling the convective transport of the fluid phases. This scheme is combined with locally conservative mixed finite elements for the numerical solution of the parabolic and elliptic problems associated with the diffusive transport of fluid phases and the pressure-velocity problem. This numerical procedure has been used to investigate the existence and stability of nonclassical shock waves (called transitional or undercompressive shock waves) in two-dimensional heterogeneous flows, thereby extending previous results for one-dimensional flow problems. Numerical experiments indicate that the operator splitting technique discussed here leads to computational efficiency and accurate numerical results.

An $h$-Adaptive Operator Splitting Method for Two-Phase Flow in 3D Heterogeneous Porous Media

“…During saturation time steps in which we do not solve for the velocity, we find a velocity field by extrapolation from the previous two available velocity solutions. A similar method is described by Abreu et al in [1] where the pressure system is only solved once per fixed number of saturation time steps. A better approach would let the length of the macro time steps depend adaptively on the changes incurred in the saturation since the last update.…”

“…Such methods have previously been developed (see, for example, [1,18,19]) but with a fixed number of saturation time steps between each solution of the flow field. We will here make the timing of solving the flow equations adaptive using a new a posteriori criterion (see Theorem 3.1 below) relating the change in the velocity to the change in the saturation since the flow equations were solved last.…”

Modeling 3D Magma Dynamics Using a Discontinuous Galerkin Method

“…Therefore, the computational wall time can be significantly reduced if the pressure is solved only when necessary. Such ideas have been pursued with solving the pressure equation once per fixed number of time steps [1] or utilizing the particular form of elliptic equations [8]. In this work we explore an adaptive indicator of the pressure equation, driven by a user defined tolerance δ * , and use this indicator along with a projective pressure estimation to estimate the pressure for cases where an exact solver is not required.…”

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

“…Numerous methods for determining the constitutive relations of capillary pressure curves can be found within the petroleum engineering literature, see e.g., Bentsen and Anli (1976), Hilfer (2006), Thomeer (1960). In flow modeling communities, these relations have been incorporated into numerical models which allow for capillary effects (Abreu et al 2008(Abreu et al , 2009Douglas et al 1997). Because of the difficulty in obtaining numerical solutions of the governing equations in a fully implicit manner, operator splitting methods have been shown to be suitable techniques for solving the partial differential equations (Abreu et al 2008(Abreu et al , 2009Douglas et al 1997).…”

“…In flow modeling communities, these relations have been incorporated into numerical models which allow for capillary effects (Abreu et al 2008(Abreu et al , 2009Douglas et al 1997). Because of the difficulty in obtaining numerical solutions of the governing equations in a fully implicit manner, operator splitting methods have been shown to be suitable techniques for solving the partial differential equations (Abreu et al 2008(Abreu et al , 2009Douglas et al 1997). Such methods typically involve splitting the governing equations into three separate parts.…”