Incomplete Gaussian Elimination as a Preconditioning for Generalized Conjugate Gradient Acceleration

Abstract

Wallis, J.R., Scientific Software- lntercomp Abstract This paper discusses the application of preconditioned generalized conjugate gradient preconditioned generalized conjugate gradient acceleration to fully implicit thermal simulation. The preconditioning step utilizes incomplete Gaussian elimination (IGE) to form an approximate factorization of the Jacobian matrix. The implementation allowsany finite difference approximationany grid block orderingassociated well constraint equationsany level of incomplete factorizationreduced system preconditioningconstrained residual preconditioning Acceleration procedures includeORTHOMINORTHORES IGE preconditioning and its implementation is discussed with respect to five, seven, nine or eleven-point finite difference approximations and optional well constraint equations. Numerical results were obtained using a thermal model allowing any number of components. The model's implicit formulation requires the solution of a linear system of equations in which each grid block has Nc + 1 unknowns. Test problems involved combustion, steam drive and cyclic steam stimulation processes, some of which exhibited ill-conditioning, processes, some of which exhibited ill-conditioning, negative transmissabilities and high transmissability ratios. Numerical results indicate how different grid block orderings, different levels of incomplete factorization and different acceleration procedures affect convergence, storage and work requirements. Introduction Simulation of thermally enhanced oil recovery processes using a fully implicit treatment of processes using a fully implicit treatment of component concentrations, phase saturations, pressure and temperature requires solution of pressure and temperature requires solution of large systems of linear equations. These systems result from the application of Newton's method to the solution of finite difference approximations to a set of mass and energy balance equations and constraint equations for each reservoir grid block. Generally, the unknowns are ordered on a grid block basis and the Jacobian matrix is partitioned into (Nc + 1) × (Nc + 1) submatrix partitioned into (Nc + 1) × (Nc + 1) submatrix elements. Solution is accomplished by direct or iterative techniques. The work and storage involved is strongly dependent on the finite difference approximation and the matrix solution scheme. The most frequently used finite-difference operators are the five-point (2D) and seven-point (3D) schemes which have been shown to exhibit grid orientation effects in steamflooding and in in-situ combustion. Direct solution is enhanced by the two-cyclic property of the associated Jacobian. In this case a reduced-bandwidth approach which utilizes a grid block ordering is generally used. In order to reduce grid orientation effects the nine-point (2D) and eleven-point (3D) operators have also been employed. The corresponding Jacobians are not two-cyclic and direct solution may be accomplished by a dissection method (nested, one-way). However, for large three dimensional multi-component problems (seven-point or eleven-point), the work and storage requirements for direct solution make this approach impractical due to the lack of economic and/or computer resources. Iterative techniques such as LSOR5 or conjugate gradients have proven highly successful in solving symmetric positive definite systems but are not generally appropriate for the highly nonsymmetric, non-diagonally dominant matrices which arise in thermal simulation. Recent developments have yielded generalizations of the conjugate residual and conjugate gradient methods which are applicable to non-symmetric systems. p. 325