A General Formulation of Linearization Methods Used In Reservoir Simulation with Applications to Adaptive Implicit Methods

Abstract

Abstract A general method to solve the set of nonlinear difference equations for multiphase flow in a reservoir is presented in this paper. Newton's method, applied to the fully implicit, FI, flow difference equations, is considered the general method from which classical formulations, IMPES, SS, SEQ, LSI, SI, are obtained in a systematic approach. It is demonstrated, by decomposing the Jacobian matrix of the Fully Implicit method into a summation of matrices, [J]FI = [T] + [T′]p,S + [Pc′]+ [γ]p,S + [q′]p,S + [(ϕbS′)], that the Jacobian matrix of any formulation is a particular form of [J]FI. The level of implicitness of each method becomes evident from its Jacobian matrix composition, IMPES method possess the lowest implicitness, [J]IMPES = [T] + [(ϕbS′)], and Jacobians for any other formulation can be obtained by adding to [J]IMPES proper matrices taken from [J]FI. The general method here presented is used to formulate an Adaptive Implicit Method, AIM. Corrections for implicitness are applied locally, at a gridblock level. Details of this procedure are discussed, along with the main points to consider when developing an AIM simulator.