Simulation of Coupled Flow and Mechanical Deformation Using IMplicit Pressure-Displacement Explicit Saturation (IMPDES) Scheme

El-amin, Mohamed Fathy (KAUST) | Negara, Ardiansyah Kusuma (King Abdullah University of Science and Technology) | Salama, Amgad (King Abdullah U of Science & Tech) | Sun, Shuyu

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The problem of coupled structural deformation with two-phase flow in porous media is solved numerically using cellcentered finite difference (CCFD) method. In order to solve the system of governed partial differential equations, the implicit pressure explicit saturation (IMPES) scheme that governs flow equations is combined with the the implicit displacement scheme. The combined scheme may be called IMplicit Pressure-Displacement Explicit Saturation (IMPDES). The pressure distribution for each cell along the entire domain is given by the implicit difference equation. Also, the deformation equations are discretized implicitly. Using the obtained pressure, velocity is evaluated explicitly, while, using the upwind scheme, the saturation is obtained explicitly. Moreover, the stability analysis of the present scheme has been introduced and the stability condition is determined.

Introduction
Coupling of solid deformation with fluid flow in porous media is one of the most challinging numerical issues in reservoir engineering applications such as CO2 sequestration and enhanced oil recovery. Several schemes; such as, fully implicit, IMplicit-EXplicit (IMEX), operator splitting, and IMplicit Pressure Explicit Saturation (IMPES); are widely used to solve the time-dependent partial differential equations that govern fluid flow in porous media. However, only few works address the issues related to mathematical and numerical modeling of the coupled flow-deformation problem. Furthermore, IMPES scheme is widely used for simulating the problem of two-phase flow in porous media. In spite of the fact that fully implicit method [1-5] is computationally expensive, it is unconditionally stable. The IMEX method [6-8] is more stable compares to the fully implicit method because it considers the linear terms implicitly and solves the other terms explicitly. The IMEX scheme is used to solve the ordinary differential equations resulting from the spatial discretization of the time-dependent partial differential equations. The operator splitting method [9-11] is used to simplify the original problem into a simpler form by the time-lag dimension. Furthermore, the iterative operator splitting [11, 12] has been considered as a part of the step of the iteration in the fully implicit method. The sequential method [13], on the other hand, is a modified version of the
IMPES method since the saturation is also evaluated implicitly. Comparing to the fully implicit method, the computational cost of the sequential method is less; therefore it is suitable for the large size models where stability becomes an important consideration.