Local Mesh Refinement and Modeling of Faults and Pinchouts

Abstract

Summary Completely flexible gridblock connections are necessary for local mesh refinement and modeling of faults and pinchouts. For practical purposes, it is necessary to combine such arbitrary connectivity with an purposes, it is necessary to combine such arbitrary connectivity with an adaptive implicit technique. In this paper, the methods used to discretize the flow equations for unusual geometries is discussed, and a rigorous analysis is given. A completely general, sparse, incomplete LU iterative solver is described that can solve Jacobian matrices with arbitrary connectivity. Various test results are presented. Introduction Recently, there has been some interest in the use of local mesh refinement in reservoir simulation. This technique can provide greater resolution in areas of interest provided that modifications to the usual Cartesian grid provided that modifications to the usual Cartesian grid structure are permitted. Modeling of near-wellbore effects and fractures in very large fields can be carried out effectively with this method. Fine-scale effects of many geological layers near wells can also be studied with this technique. In addition, simulation of nonsealing faults and pinchouts requires similar modification of the usual grid pinchouts requires similar modification of the usual grid structure. Completely flexible gridblock connections are necessary for local mesh refinement and modeling of faults and pinchouts. This means that the computational molecule pinchouts. This means that the computational molecule associated with any given cell is not fixed and can have any kind of pattern. For practical purposes, it is necessary to combine such arbitrary connectivity with an adaptive implicit technique. Adaptive implicitness is useful because such widely varying grid sizes will require fully implicit treatment in some regions while an implicit-pressure/ explicit-saturation method will be adequate in regions with large gridblocks. To solve the Jacobian matrix that arises from such simulations with arbitrary connectivity, a completely general, sparse, incomplete-factorization iterative solver is required. This solver allows a variable number of unknowns per cell (arising from the adaptive implicit technique) and uses minimal storage (no zeros stored). The incomplete factorization is carried out in a block sense, with all unknowns in a given cell treated as a single unit. This means that the symbolic factorization is independent of the degree of implicitness. It is necessary to discuss the methods used to discretize the flow equations for local mesh refinement, and a complete error analysis is given in the Appendix. The results of various test simulations are given that show the applicability of local mesh refinement. In particular, we show that coning effects can be modeled on coarse Cartesian grids. Fault and pinchout modeling can also be carried out by means of the non-neighbor connection facility. Discretization for Unusual Grid Configurations The method of discretizing the conservation equations for unusual geometries is similar to that proposed by Heinemann et al. Essentially, an integral- or control-volume approach is used that is inherently mass conservative. The volume of each cell is simply the geometric volume, while the flux into a cell is given by face area multiplied by velocity perpendicular to the face. The distance used in the flux calculation is measured in the coordinate distance perpendicular to the face. This method will be demonstrated for the case of local mesh refinement. Other cases (faults) use generalizations of this technique. Consider the configuration shown in Fig. 1 that could result from a locally refined mesh. For notational convenience, we will assume in the following that kkr/ is equal to unity. This implies that the fluid velocity, v, is simply proportional to the pressure gradient. Thus the total flux into Cell 1 (Fig. 1) is given by: P2 - P1 P2 - P1Ut = ------------ + ---------- yx zx x P4 -P1+----------- y4 z.........................(1)x Note that we have not attempted to use interpolation to account for any y-direction variation in the pressure across Cell 1. Interpolation tends to reduce diagonal dominance of the pressure equation and can cause difficulties for iterative solvers. Although formal truncation-error analysis applied to Eq. 1 shows an O(1/h) error (h is of the size of x, v), it is possible to show that this method generates 0(h) approximate pressures. p. 275