Richardson's Iteration With Dynamic Parameters and the SIP Incomplete Factorization for the Solution of Linear Systems of Equations

Abstract

Abstract Reservoir simulation yields a system of linear algebraic equations, Ap = q, that may be solved by Richardson's iterative method, p(k+1)= p(k)+ tkr(k), where r(k)= q − Ap(k) is the residual and t0, … tk are acceleration parameters. The incomplete factorization, Kα, of the strongly implicit procedure (SIP) yields an improvement of Richardson's method, p(k+1)=p(k)+tkKα−1r(k). Parameter α originates from SIP. The product of the L and U factors produced by SIP gives Kα = LU. The best values of the tk acceleration parameters may be computed dynamically by an efficient algorithm; the best value of α must be found by trial and error, which is not hard for only one value. The advantages of the method are (1) it always converges, (2) with the exception of the α parameter, parameters are computed dynamically, and (3) convergence is efficient for test problems characterized by heterogeneities and transmissibilities varying over 10 orders of magnitude. The test problems originate from field data and were suggested by industry personnel as particularly difficult. Dynamic computation of parameters is also a feature of the conjugate gradient method, but the iteration described here does not require A to be symmetric. Matrix Kα−1 must be such that the real part of each eigenvalue is nonnegative, or the real part of each is nonpositive, but not both positive and negative. It is in this sense that the method always converges. This condition is satisfied by many simulator-generated matrices. The method also may be applied to matrices arising from the simulation of other processes, such as chemical flooding.