Nested Factorization

Abstract

Members SPE-AIME 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 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 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 published methods in terms both of computing speed storage requirements. Introduction A requirement common to methods for the iterative solution of large sparse sets of linear equations (1) is a procedure for computing an approximate solution to the residual equation (2) where rm = b - A.xm, is the residual after m iterations, and B is an approximation to A chosen so that the solution of (2) may be obtained efficlently. The search direction, qm, is used to update the solution, for example using (3) The updating procedure of equation (3) is used by older methods such as LSORC (1) and SIP (2). More recently, the Conjugate Gradient method (3), which in effect uses (4) where the scalar constants ai are chosen to minimize r.A-1.r, has been used to good effect on problems where the coefficient matrix, A, is symmetric. For asymmetric matrices, the Orthomin procedure (4), which minimizes r.r, provides an effective alternative. The effectiveness of all the above methods depends critically on how closely the preconditioning matrix, B, approximates to A. In this paper, preconditioning matrix, B, approximates to A. In this paper, we describe a new technique, called Nested Factorization, for constructing the preconditioning matrix in finite difference applications. The method differs from incomplete Cholesky factorizations, which are normally used to precondition the Conjugate Gradient and Orthomin procedures, in that B is not the product of strictly lower and upper triangular factors. Instead block lower and upper triangular factors are constructed by a procedure which adds one dimension at a time to the preconditioning matrix. In the remainder of this paper, the abbreviation ICCG is used to denote all methods which use an incomplete Cholesky factorization to precondition the Conjugate Gradient method. Nested Factorization was introduced and fully described by Appleyard et al. . It has subsequently been described by Ponting et al. in a revision of the original manuscript on mass conserving methods by Cheshire et al.