Some Practical Considerations in the Numerical Solution of Two-Dimensional Reservoir Problems

Abstract

Abstract A study was made of numerical techniques for solving the large sets of simultaneous equations that arise in the mathematical modeling of oil reservoir behavior. It was found that noniterative techniques, such as the Alternating Direction Implicit (ADI) method, as well as some other finite difference approximations, produce oscillatory or unsmooth results for large time steps. Estimates of time step sizes sufficient to avoid such behavior are given. A comparison was made of the Point Successive Over-Relaxation (PSOR), Two-Line Cyclic Chebyshev Semi-Iterative SOR (2LCC), and iterative ADI methods, with respect to speed of solution of a test problem. It was found that, when applicable, iterative ADI is fastest for problems involving many points, while 2LCC is preferable for smaller problems. Introduction With the advent of high speed, large memory, digital computers, there has been an increasing emphasis on the development of improved methods for simulating and predicting reservoir performance. Two-dimensional, three-phase reservoir models with various combinations of PVT effects, as well as gravity and capillary forces, are common throughout the industry. Such models are also available through consulting firms, to anyone desiring to use them. Three-dimensional models will probably be practical in only a few years. We conducted a study of some of the numerical methods used for solving the large sets of simultaneous equations that arise in such models. A typical set of equations for a reservoir model is shown below: ..............(1) .............(2) .............(3) ..............(4) where a = 5.615 cu ft/bbl. In addition to Eqs. 1 through 4, one also would have to specify the conditions at the boundaries of the reservoir or aquifer being studied. Equations such as these are normally approximated by finite difference techniques and solved numerically because of their complexity. In deciding how to solve such equations, a number of decisions must be made. It is not our intention to cover all facets of the problem, but rather to concentrate on one of the important aspects, such as solving Eq. 1. SPEJ P. 185ˆ