Constrained pressure residual preconditioner for efficient solution of the adjoint equation

摘要

A method, system and computer program product is disclosed for using a constrained pressure residual (CPR) preconditioner to solve adjoint models. A linear system of fluid flow equations comprising a plurality of variables that represent fluid flow properties in a geological formation of a subterranean reservoir is provided. Matrix (Ã)T, which comprises a transpose of a Jacobian matrix associated with the linear system of fluid flow equations, is constructed. A constrained pressure residual preconditioner MCPRA−1 is constructed responsive to the matrix (Ã)T. Matrix equation (Ã)Ty=d is then solved using the constrained pressure residual preconditioner MCPRA−1.

主权项

1. A computer-implemented method for solving adjoint models, the method comprising:
(a) Providing, to a computer, a linear system of fluid flow equations comprising a plurality of variables that represent fluid flow properties in a geological formation of a subterranean reservoir; (b) constructing a matrix (Ã)T, through the computer, comprising a transpose of a Jacobian matrix, the transpose of the Jacobian matrix being associated with the linear system of fluid flow equations; (c) constructing a constrained pressure residual preconditioner MCPRA−1, through the computer, responsive to the matrix (Ã)T wherein MCPRA−1 is a two-stage preconditioner constructed as:
MCPRA−1=C(ApT)−1CT[I−(Ã)T({circumflex over (M)})−1]+({circumflex over (M)})−1; and (d) solving, through the computer, matrix equation (Ã)Ty=d using the constrained pressure residual preconditioner MCPRA−1.