Fully Implicit Versus Dynamic Implicit Reservoir Simulation
Authors: Vinsome P.K.W. 
Published: March 2010

Abstract
The Dynamic Implicit method is described and compared with the Fully Implicit method. The Dynamic Implicit method is a procedure to be used in numerical reservoir simulation for automatically switching between fully implicit and IMPES depending on the amount of /low through a grid block face. Some additional related methods are described:A general degree LDU factorization/or iterative inversion of the pressure equations arising in numerical reservoir simulation.Some modifications of the COMBINATIVE iterative method to lower the storage requirements and increase the efficiency of the inversion of large Jacobian matrices.An automatic damping procedure to increase the stability of Newton-Raphson iteration.
Illustrative examples are given.

Introduction
The Dynamic Implicit method is similar to the Adaptive Implicit method(1) inasmuch as it attempts to achieve the stability of the Fully Implicit method with reduced computing cost. It works by solving fully implicitly for flows across grid block faces only where required, and by using the IMPES procedure elsewhere. However, the implementation of the Dynamic Implicit method is completely different from the Adaptive Implicit method. The Jacobian matrix solution technique is an integral part of the method, so we begin by describing the solution technique for the linearized Jacobian equations.

Solution of the Jacobian Equations
The solution technique is a modification of the COMBINATIVE method(2). The steps involved for a black-oil Jacobian where solution is for pressure and two saturations (or pressure, one saturation, and a solution gas ratio) are as follows.Preprocess each block line of the Jacobian matrix with a Gaussian Elimination so that the diagonal block is reduced to diagonal form, as shown in Figure 1. This step requires very little work and involves no approximations. In the original COMBINATIVE method the diagonal block was reduced only to lower triangular form. Reduction to diagonal form generally results in one less iteration for convergence.Temporarily assume that the elements marked ‘E’ in Figure 1 can be neglected. The pressure equations are then totally decoupled from the rest of the Jacobian and can be removed and solved for separately as in IMPES.Solve the pressure equations, which involves inversion of a matrix containing only one equation per grid mode. In the original COMBINATIVE method this was done by Gaussian Elimination. However, in this modified COMBINATIVE method an iterative technique containing a general degree LDU factorization accelerated with ORTHOMIN(3) is used. The factorization employed is described in the next section. Solution of these pressure equations therefore requires iterations, which we term "inner iterations". The solution is only a first approximation to the pressures, because of the neglected ‘E’ elements.Form the new Jacobian residual using the first approximation to the pressures found in Step 3.Form a first degree outer LDU factorization of the whole of the Jacobian and invert it to obtain the saturations and an additive correction to the pressures. In the original COMBINATIVE method a second-degree factorization was used in this step, and could also be employed here, but first degree is sufficient for black-oil problems.