Patent ID: 6230101
Filing Date: 2001-05-08
Classification: G06T

Abstract:
In a simulator, a method for solving a system of linear equations representing reservoir conditions, comprising the steps of:(1) receiving a first signal representing physical characteristics of a reservoir;(2) obtaining, from the received signal, a residual vector r.sub.0 representing errors associated with a system of nonlinear equations describing the reservoir;(3) obtaining, from the received signal, a first matrix A.sub.0 representing sensitivity of the residual vector r.sub.0 to changes in the system of nonlinear equations describing the reservoir;(4) decomposing matrix A.sub.0 in accordance with the following steps:(a) letting i=0,(b) if matrix A.sub.i can be efficiently factored exactly, letting n=i, else,(i) letting A.sub.i =L.sub.i +A.sub.i+1 +U.sub.i where L.sub.i is a lower block diagonal matrix and U.sub.i is an upper block diagonal matrix,(ii) incrementing i, and(iii) repeating step (4)(b) for matrix A.sub.i+1 ;(5) generating a second matrix M.sub.0 that is an approximation to matrix A.sub.0 in accordance with the following steps:(a) letting M.sub.n =A.sub.n -D where D is a diagonal matrix defined by a specified operation on the matrix given by ##EQU5##(b) generating, for i=(n-1) to 1, M.sub.i =(L.sub.i +M.sub.i+1)(I+M.sub.i+1.sup.-1 U.sub.i), and(c) letting M.sub.0 =(L.sub.0 M.sub.1.sup.-1 +I)(M.sub.1 +U.sub.0);(6) generating, for k=1 until convergence, the following values:(a) y=(L.sub.0 M.sub.1.sup.-1 +I).sup.-1 r.sub.k-1,(b) q=(M.sub.1 +U.sub.0).sup.-1 y,(c) z.sub.i =M.sub.i+1.sup.-1 U.sub.i q(for i=1 , . . . , n-1), ##EQU6##(e) r.sub.k+1 by accelerating the value A.sub.0 M.sub.0.sup.-1 r.sub.k-1 ;(7) generating an approximate solution to the system of linear equations representing reservoir conditions using a value of said r.sub.k+1 that converged; and(8) generating a second signal representing physical characteristics of the reservoir.