Application of the Alternating Direction Explicit Procedure To Two-Dimensional Natural Gas Reservoirs

Abstract

Abstract The alternating direction explicit procedure (ADEP) makes use of the boundary conditions to reduce multi-dimensional problems to a series of one-dimensional problems. The method, previously applied to reservoirs containing only an undersaturated oil, bas now been extended to cover the case of natural gas reservoirs. Although this involves solving a non-linear partial differential equation, application of the procedure is straight- forward and no calculational problems were encountered. Introduction There has been a growing interest in formulating mathematical models of petroleum and natural gas reservoirs - models which permit the engineer to examine and evaluate the physical and economic consequences of various alternative production policies. The tremendous reduction in the cost of solving such models in recent years has made possible their use as an almost routine management tool. This reduction is the result not only of improved computer hardware but also of the development of more efficient mathematical techniques. The present paper is concerned with the application of a recently proposed numerical method (ADEP) to two-dimensional gas reservoirs. STATEMENT OF THE PROBLEM Given a two-dimensional Region R, bounded by a closed Curve C (Fig. 1) such that the behavior on Curve is known, the differential mass balance for each fluid phase in R, neglecting gravitational effects and assuming Darcy's law for fluid transport, can be written as: A typical and common set of boundary conditions is given by (1) / = 0 on Curve C where r is the direction normal to Curve C; (2) p is known throughout Region R at some time t; and (3) wf is known for all x, y and t. Physically, this represents the case where the reservoir is bounded by impermeable media, where the initial pressure throughout the reservoir at the beginning of gas production is known and where the production rate at each well is specified at all times. For single-phase reservoirs, the problem is to determine the pressure throughout Region R at all times; for multi-phase reservoirs, in addition to the pressure, the value of the fluid saturation is also required. Since analytical solutions to this equation for the general case are not available, we must resort to numerical integrating techniques, using finite-difference approximations. SPEJ P. 137ˆ