Locally Conservative Continuous Galerkin FEM for Pressure Equation in Two-Phase Flow Model in Subsurfaces

Abstract

A typical two-phase model for subsurface flow couples the Darcy equation for pressure and a transport equation for saturation in a nonlinear manner. In this paper, we study a combined method consisting of continuous Galerkin finite element methods (CGFEMs) followed by a post-processing technique for Darcy equation and finite volume method (FVM) with upwind schemes for the saturation transport equation, in which the coupled nonlinear problem is solved in the framework of operator decomposition. The postprocessing technique is applied to CGFEM solutions to obtain locally conservative fluxes which ensures accuracy and robustness of the FVM solver for the saturation transport equation. We applied both upwind scheme and upwind scheme with slope limiter for FVM on triangular meshes in order to eliminate the non-physical oscillations. Various numerical examples are presented to demonstrate the performance of the overall methodology.

Continuous data assimilation for displacement in a porous medium

“…The numerical approximation that is utilized for simulation of the data assimilation adopts a strategy described in [17], which develops an algorithm for simulating a standard two-phase flow and transport model. The domain Ω is discretized into a collection of nonoverlapping rectangles τ ∈ T h such that Ω = τ ∈T h τ (see fig.…”

“…Thus, (4.3c) is a step to produce a locally conservative normal flux −κ( θh )∇ Ψd,h (t) • n in the sense of (4.5). We refer to [17] for an extensive discussion on how this property is established.…”

“…The nonlinear algebraic system must be solved by some iterative techniques at every time level. Formal discussions of the construction and techniques for efficiently solving such an algebraic system can be found in many texts (see for example [14,17]). However, there exists a fast-slow relation between the pressure and concentration profiles [3].…”