Nested Factorization

Abstract

ABSTRACT A new iterative technique for the solution of the linear equations arising in finite difference reservoir simulations is described. The method comprises a novel preconditioning for the Conjugate Gradient and Orthomin iteration procedures. Nested Factorization differs from the more commonly used incomplete Cholesky factorizations in that it does not form the preconditioning matrix from strictly upper and lower factors. Instead, it constructs block lower and upper factors using a procedure which adds one dimension at a time to the preconditioning matrix. The factorization procedure conserves material exactly for each phase at each linear iteration, and accomodates non-neighbour connections (arising from the treatment of faults, completing the circle in 3D coning studies, numerical aquifers, dual porosity/permeability systems etc.) in a natural way. A number of versions of the Nested Factorization algorithm are compared with other published methods for a series of 2D and 3D problems. It is found that the Nested Factorization algorithms are all roughly comparable and that all compare very favourably with other published methods in terms both of computing speed and storage requirements.