Tests of the Stability and Time-Step Sensitivity of Semi-Implicit Reservoir Stimulation Techniques

Abstract

Abstract A reservoir simulation technique that employs semi-implicit approximations to relative permeabilities exhibits excellent stability and permeabilities exhibits excellent stability and convergence characteristics when applied to water- or gas-coning problems. Recent workers in this area have made a simplifying assumption in order to linearize the flow terms of the semi-implicit finite-difference equations. This paper describes a method of solving efficiently paper describes a method of solving efficiently the nonlinear form of the equations and demonstrates that time-step sensitivity is reduced by iterating on the nonlinear terms. In addition, it addresses the problem of allocating a well's production among multiple grid blocks. Example problems include both water-coning and gas-percolation applications. Introduction Multiphase reservoir simulators traditionally have employed finite-difference approximations in which relative permeabilities are evaluated explicitly at the beginning of each time step. Simulators of this type are capable of handling many reservoir studies in a perfectly satisfactory fashion, but they are incapable of solving economically problems characterized by high flow velocities. Included in this category are the studies of such phenomena as well coning and gas percolation. The difficulty in such problems is a stability limitation imposed by the use of explicit relative permeabilities. In an attempt to overcome this permeabilities. In an attempt to overcome this limitation, Blair and Weinaug developed a simulator that employed implicitly evaluated relative permeabilities. The increased stability of their permeabilities. The increased stability of their equations allowed the use of time steps much larger than previously possible, but this was counteracted by an increase in the computational work per time step and an increased difficulty in the iterative solution of the difference equations. While the net result was a significant advance in the solution of coning problems, improvements still were needed to increase the dependability and decrease the cost of obtaining solutions for such problems. More recently, two papers were published describing a method that employs semi-implicit relative permeabilities. This method is greatly superior to the fully implicit method, both in computational effort and maximum time-step size. In developing this method, the previous workers made a simplifying assumption to obtain linear finite-difference equations. We have developed a reservoir simulator based on the nonlinear form of the semi-implicit finite-difference equations. This paper describes the techniques used in the simulator and presents the results of some tests conducted with it. These include time-step sensitivity studies and tests of alternate production allocation methods. Some of these tests compare the nonlinear form of the semi-implicit method with the linear form. SPEJ P. 253