In this paper, an arbitrary Lagrangian and Eulerian method (ALE) has been applied to the coupled model for oil/water flow in deforming porous media. The ALE frame referred to porous solid and fluid flow was derived and a consistent coordinate frame description for the governing formation was established. When the mesh motion is equal to the velocity of the deformed porous media, a simpler formulation will be presented which reduces the difficulty of numerical solution. In the numerical testing, it was applied to the simulation of water injection in reservoir. Some simulation results, which demonstrate the validity of the ALE formulation, have been obtained.
Such independent variables 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 solid phase and fluid phase are difficult to distinguish and they should be regarded as an overlap continuums. So a consistent coordinate frame description for this kind of coupled models is necessary, and many results have been presented. The mass equation, introduced by J. Bear[1], employed Eulerian description through the continuum theory for multi-phase flow. Copper [2] derived the one-dimension governing equations of groundwater low in Eulerian and deforming coordinates. In the deforming coordinates the form of governing equations doesn't involve the convective term Equation (Available in full paper) which is introduced by the motion of porous media. The two-phase flow in deformed porous media was simulated under deforming coordinate by Shifeng Xue [3]. When the assumption with a steady and little displacement is given, some simply results have been obtained with Eulerian mesh. The governing equations for oil/water flow in deformed porous media are derived on the assumption that Equations (Available in full paper)
Although the deforming ordinates concept is proposed, the consistent coordinate frame descriptions for three-dimension flow in deformed porous and finite deformation have 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.
The assumptions for coupling of fluid flow and porous media model are as follows:
The reservoir rocks are isotropic and deformable and they are composed of solid skeleton and pore space;
The solid matrix is perfectly Druck-Prager constitutive law;
The fluid flow in porous media is based on the Darcy law;
The seepage is under isothermal conditions. The physical adsorption and chemistry effect are ignored between the porous media and fluid flow.
The papers[10, 11] have proposed the the ALE concept.
