A Control-Volume, Finite-Element Method for Local Mesh Refinement in Thermal Reservoir Simulation

Abstract

Summary This paper describes a control-volume, finite-element technique for coupling coarse grids with local fine meshes. The pressure is treated in a finite-element manner, while the mobility terms are upstream weighted in the usual way. This requires identification of the cell volumes and edges that are consistent with the linear finite-element discretization of the pressure. To ensure that the pressure equation yields an M matrix, various conditions are required for the type of triangulation allowed. Because the form of the equations is similar to the usual finite-difference discretization, standard techniques can be used to solve the Jacobian. The local mesh-refinement method is demonstrated on some thermal reservoir simulation problems, and computational results are presented. Significant savings in execution times are obtained while predictions similar to global fine-mesh runs are given. Introduction Local and dynamic mesh refinement are useful methods for obtaining high resolution of shock fronts and near-wellbore flow in reservoir simulation. This is particularly important in modeling cyclic steam injection, which is used to recover high-viscosity crude oil. Typically, control-volume-based, finite-difference methods are used to discretize the system of equations representing multiphase heat and mass transfer in a porous medium. These equations are highly nonlinear and of mixed hyperbolic/parabolic type. Phase upstream weighting is used for mobility terms. In certain cases, phase upstream weighting can be shown to converge to the phase upstream weighting can be shown to converge to the physically correct solution that satisfies the entropy condition. physically correct solution that satisfies the entropy condition. If the usual finite-element method (FEM) could be used, it would be a simple matter to generate a locally refined discretization. It is difficult, however, to combine upstream weighting and the FEM for multidimensional, multiphase flows. Use of an asymmetric weighting function is possible, but this does not ensure that saturations will remain positive in the absence of interphase mass transfer. This positive condition is especially important in thermal problems, where interphase mass transfer occurs. In this situation, problems, where interphase mass transfer occurs. In this situation, the appearance of negative saturation values indicates violation of a thermodynamic constraint. Consequently, because multiphase flows with heat transfer are very complex, it is desirable to retain the easy physical interpretation of control-volume methods, while at the same time obtaining some of the intrinsic flexibility of the FEM. Although the techniques used in this paper are quite general, we focus on the problem of local mesh refinement in reservoir simulation. In particular, coupling the local fine mesh to the global coarse mesh will be considered. Two methods were proposed for achieving this coupling. One technique uses a very-low-order coupling for the pressure. In fact, the truncation error for this scheme is formally of 0 (1/), where is a measure of the grid size. A more subtle analysis, however, shows that though this scheme is convergent, the convergence can be slow in any region within O[ log(1/)] of the interface. This technique has the advantage that in the single-phase limit, the pressure is given by the solution of an M matrix, which ensures that no nonphysical local maxima and minima can occur. Although this method is often effective, some flow situations give inaccurate results. An alternative is to use interpolation to obtain a high-order coupling. This reduces the diagonal dominance of the Jacobian, however, which can lead to difficulties for an iterative solver unless some of the terms are evaluated explicitly. As well, in the single-phase limit, the pressure matrix is not an M matrix, which can have undesirable pressure matrix is not an M matrix, which can have undesirable consequences. The control-volume FEM was proposed as a technique for allowing flexible grids in Navier-Stokes fluid-flow simulations. In reservoir simulation, Dalen suggested a similar idea. This method is also similar to the box method in semiconductor device modeling. Varga proposed similar ideas. In the following, we apply the control-volume FEM to thermal reservoir simulation problems, subject to the constraint that in the single-phase limit, problems, subject to the constraint that in the single-phase limit, the pressure equation must be an M matrix. As previously mentioned, this ensures that the pressure solution retains physically reasonable behavior. An added bonus is that, when viewed appropriately, the discrete equations have the same form as the usual finite-difference, control-volume formulation. Provided that an existing simulation code has a facility for arbitrary cell connectivity, it is a simple matter to convert a finite-difference code to a control-volume, finite-element formulation.