Linearization Techniques of Reservoir Simulation Equations: Fully Implicit Cases
Authors: Farkas Eva 
Published: December 2007

Abstract
The complexity and nonlinearity of reservoir simulation equations make it possible to apply a great number of linearization techniques. The SPE comparison projects and some other papers compare models for various problem classes. These papers generally compare the results and the speed of runs. The main object of this paper is to examine and compare the models based on the numerical background of the linearization techniques. A general comparison method is proposed that can be applied to analyze these techniques.
Recently the proposed comparison method has been applied to examine IMPES and IMPEM techniques. Three primary sets of the two basic types (the "saturation/phase concentration"-type and the "overall density"-type) have been examined. It has been revealed that the direct IMPES and IMPEM methods result in exactly the same pressure distribution, while the saturations and the masses are different. By applying the primary set transformation techniques, the saturation and mass distributions of the direct IMPES and IMPEM techniques can be transformed from one into the other.
The fully implicit methods are compared for a 3P black oil system, using 1D radial grid. The relationships, and convergence behavior of the implicit techniques are discussed. An analytical approximation of the implicit part of the Jacobian matrix is suggested for "overall density" type primary sets. An object-oriented C++ program has been used to compare the linearization techniques.

Introduction
The governing equations of reservoir simulators consist of a set of nonlinear partial differential equations coupled with algebraic, i.e. constraint equations. The number of unknown functions of the models can be quite large, depending on the complexity of the fluid—rock system described. This possibility has yielded a number of equivalent modeling techniques, and these apply different groups of unknown variables as primary sets.
The widely used IMPES technique was developed in the early sixties. This technique separates the parabolic and hyperbolic parts of the model equations to obtain an equation that has only one unknown function: the pressure. After the solution of this equation implicitly for the pressure, the saturations can be calculated explicitly from the mass conservation equations. This direct sequential technique was originally developed to simulate black oil models. All the coefficients of a black oil model depend on pressure and saturations; thus, it was a natural choice to select pressure and saturations as primary variables of IMPES black oil models. Since the seventies, there has been a growing demand to use compositional models. In compositional cases the number of model equations (unknown functions) can be numerous, and the nonlinearity of the governing equations is more severe than in black oil models. The coefficients of the model equations, besides the pressure, and the saturations, depend on the phase composition, too. In the seventies a number of IMPES-type techniques were developed to simulate compositional models, but all these models used iterative techniques. It was in the early eighties, when a method was published, which was suitable for separating the pressure equation correctly from the set of governing differential equations. Thus, a direct sequential method could be also derived for compositional cases, too.
Due to the explicit calculation of the saturations (concentration), the IMPES type models have time step limits. Implicit models, besides the pressure, also calculate other variables implicitly. These models allow greater time steps, but the drawback is the greater computing time requirement (and numerical dispersion). There have been many linearization technique for handling nonlinearities, and creating implicit schemes. The fully implicit methods are often applied besides or together with the IMPES type methods. A fully implicit technique was proposed for two-phase system at the end of sixties. The first fully implicit compositional model was published at the end of the seventies. A number of fully implicit models have been published since then. Because of the considerable computing requirement of an iteration step, to achieve a fast convergence is especially important in compositional cases. P. 87^