The ALE Method for Oil/Water Two-Phase Flow in Deforming Porous Media
Authors: Liang S. , Yin J.Y. , Xue S.F. 
Published: March 2010

Abstract
The main objective of this paper is to establish a consistent geometrical frame focusing on the coupling between hydro-mechanical aspects. An arbitrary Lagrangian and Eulerian method (ALE) has been used to deal with the numerical simulation for oil/water flow in deforming porous media. When the mesh motion is equal to the velocity of the deformed porous media, the convection term referred to as the relative velocity between fluid flow and the skeleton deformation is removed from the Finite Element Method (FEM) formula. Therefore, some difficulty in FEM numerical simulation has been solved. A classical oil/water flow coupled with solid deformation is simulated by ALE code with an iterative scheme. An ALE with a fully implicit and sequential iterative algorithm [Implicit Pressure Explicit Saturation (IMPES) method], which calculates the oil phase pressure and water phase saturation with least squares FEM, is used to simulate the oil/water two-phase fluid flow. Compared with general reservoir simulation, the ALE method could reach a consistent frame description or a more rigorous definition of the parameters for the solid phase and the flow phase. This may be very important for larger deformations in stratum induced by petroleum production and tracking the moving interface in multi-phase fluid flow.

Introduction
Such independent variables such as stress, strain and deformation in a continuum are described by Lagrangian or Eulerian coordinates. The Lagrangian description, generally, is employed extensively for solid mechanics, and the Eulerian description is preferred for situations which involve large flows and large distortions such as fluid mechanics. The coupling models or mathematical equations for two-phase flow in deforming porous media belong to the inner coupling. It is recognized that the independent variables for a solid phase and a fluid phase are difficult to distinguish and they should be regarded as overlapping contiuums. So a consistent coordinate frame description for this kind of coupled model is necessary, and many results have been presented. The mass equation, introduced by Bear(1), employed an Eulerian description through continuum theory for multi-phase flow. Cooper(2) derived the one-dimensional governing equations of groundwater flow in Eulerian and deforming coordinates. In the deforming coordinates, the form of governing equations does not involve the convective term which is introduced by the motion of porous media. The two-phase flow in deformed porous media was simulated under deforming coordinates by Xue(3).
When the assumption with a steady and a small displacement is given, some results have been obtained with Eulerian mesh. The governing equations for oil/water flow in deformed porous media are derived using the assumption that presented by a number of authors (4–9).
Although the deforming ordinates concept is proposed, a consistent coordinate frame description for three-dimensional flow in deformed porous and finite deformations has not been satisfactorily resolved. The main object of this paper is to build a consistent coordinate frame description through the ALE theory for the coupling of two-phase flow and deforming porous media. Furthermore, an injection water example is given with the finite element method.