Simulation of Two-Dimensional Waterflooding By Using Mixed Finite Elements

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Abstract
A new method to simulate incompressible diphasic flow in two dimensions (2D) is presented. Its distinctive features include (1) a reformulation of the basic equation using the premise of a global pressure and (2) approximation of convective terms by an upwind scheme for discontinuous finite elements. A mixed finite-element method approximates both the scalar functions (pressure and saturation) and the vector functions (tool velocity field and capillary diffusion vector). The pressure (resp. the saturation) is approximated by a discontinuous function piecewise constant (resp. linear) on the elements of the mesh. A basis of divergence-free vectors is used in the pressure equation, which accelerates computation. Several test examples, which include gravity and capillary effects, are presented. presented. Introduction Several approaches have been proposed recently to improve on the classical finite-differences treatment of 2D diphasic simulation. Typical examples of these are the finite-element interior-penalty method, the sampling method, the moving point method, and a continuous finite-element method. New characteristics are presented in our approach to incompressible two-phase flow. First, the basic equations of diphasic flow are reformulated in terms of only two dependent variables, the (reduced) water saturation. Sr, and a global pressure, p, governing the global oil/water flow vector, q0. The introduction of p leads to noticeable simplifications in the pressure equation. Simultaneously, several auxiliary functions or notations are introduced to simplify the equation. Those functions are computed from the usual relative-permeability and capillary-pressure curves. At this point it becomes apparent that the possible capillary heterogeneity effects can be dealt with in the same way as gravity. Either capillary or gravity effects lead, in the saturation equation, to a transport term that can be a non-monotone function of saturation, and therefore, require a special treatment. The pressure equation is discretized by a mixed finite-element methods: piecewise-constant pressure. ph, on each element (triangular or rectangular) of the mesh and finite dimensional global flow vector, q0h, with continuous nominal component across the edges of the mesh. At this stage, we take advantage of the incompressibility of the fluids by using a null-divergence vector basis for the computation of q0h. This leads to a significant reduction in the size of the corresponding equation, and therefore of the computing time. Moreover, the pressure need not be computed systematically and appears as a byproduct. The saturation is approximated by a discontinuous piecewise linear function, and the capillary flow vector piecewise linear function, and the capillary flow vector is approximated in the same manner as q0h. The convective term is treated by an upwind scheme for discontinuous finite elements that is based on the works of Lesaint-Raviart and Godunov. Note that the scheme is conservative, yields an exact mass balance for each fluid, and works with or without capillary pressure. The method's first demonstration involves use of a test case without capillary diffusion to investigate grid-orientation effects and to compare results with a finite-difference solution. A second series of runs includes capillary and gravity effects. The method provides a good description of sharp fronts. with a computational-cost increase factor of 3 to 3.5 for an equal number of degrees of freedom for saturation. Because of the discontinuous character of the basis functions used, adequate representation methods are required for display of the final saturation map. Pressure Equation Pressure Equation The pressure equation is written in terms of the global pressure, p, for the case of incompressible fluids. pressure, p, for the case of incompressible fluids. This pressure has the following properties. 1. For a given water saturation level, the knowledge of the global pressure is equivalent to the knowledge of either the water pressure or the oil pressure. 2. The value of the global pressure is intermediate between the water and oil pressures, and hence differs from either of them at most by the capillary pressure. SPEJ P. 382