Schlumberger Technology Corporation

Chevron U.s.a. Inc.

Schlumberger Canada Limited

John R Wallis

Hamdi A Tchelepi

Gary K Li

Wen Hsiung Chen

Pallav Sarma

Gareth Shaw

Hui Cao

Choongyong Han

Mark Schrader

A computer-implemented system, method, and software are provided for solving linear systems of equations arising in reservoir simulation. A linear system of fluid flow equations, including a plurality of unknown variables, is provided that represents fluid flow properties in a geological formation of a subterranean reservoir. Block diagonal scaling is applied to the linear system of fluid flow equations. Threshold incomplete lower-upper approximate factorization is applied to obtain a preconditioning matrix, which can be used to iteratively solve unknown variables associated with the system of fluid flow equations. Preconditioning can be used directly as part of a preconditioned Krylov subspace method or alternatively as a 2stage preconditioner in a CPR algorithm.

System and method for performing reservoir simulation using preconditioning

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 (&#xc3;), which comprises a transpose of a Jacobian matrix associated with the linear system of fluid flow equations, is constructed. A constrained pressure residual preconditioner Mis constructed responsive to the matrix (&#xc3;). Matrix equation (&#xc3;)y=d is then solved using the constrained pressure residual preconditioner M.

Constrained pressure residual preconditioner for efficient solution of the adjoint equation

A method, system, a program storage device and apparatus are disclosed for conducting a reservoir simulation, using a reservoir model of a region of interest, wherein the region of interest has been gridded into cells. Each cell has one or more unknown variable. Each cell has a node. A graph of the nodes is represented by a sparse matrix. The graph is an initially decomposed into a pre-specified number of domains, such that each cell exists in at least one domain. The cells and domains are numbered. Each cell has a key, the key of each cell is the set of domain numbers to which the cell belongs. Each cell has a class, the class of each cell being the number of elements in the cell. The cells are grouped into connectors, each connector being the set of cells that share the same key. Each connector having a connector class, the connector class being the number of elements in the key of the connector. Each connector having only one higher-order neighbor connector is merged with such higher-order neighbor connector. The class of all locally maximum class connectors is reset to the maximum class of held by any connector. The maximum class connector is forced to contain only one cell. The connectors are ordered in increasing order of class. Interpolation operator and restriction operator are constructed from the ordered connectors. The interpolation operator and restriction operator are used to construct a coarse grid. The coarse grid may be used to determine the unknown variables of the cells.

Apparatus, method and system for improved reservoir simulation using an algebraic cascading class linear solver

A method, system and apparatus are disclosed for conducting a reservoir simulation, using a reservoir model of a gridded region of interest. The grid of the region of interest includes one or more types of cells, the type of cell being distinguished by the number of unknown variables representing properties of the cells. The cells share a common variable as an unknown variable. The method includes the steps of identifying different cell types for the grid; constructing an overall matrix for the reservoir model based on the different cell types; at least partially decoupling the common variable from the other unknown variables in the matrix by using a reduction process to yield a reduced matrix; mathematically breaking up the variables in the reduced matrix into k subsets by cell types; applying an overlapping multiplicative Schwartz procedure to the reduced matrix to obtain a preconditioner and using the preconditioner to solve for the unknown variables.

Apparatus, method and system for improved reservoir simulation using a multiplicative overlapping Schwarz preconditioning for adaptive implicit linear systems

A Linear Solver method and apparatus, embodied in a Simulator and adapted for solving systems of non-linear partial differential equations and systems of linear equations representing physical characteristics of an oil and/or gas reservoir, includes receiving a first signal representing physical characteristics of a reservoir, obtaining a residual vector {right arrow over (r)},from the first signal (representing errors associated with a system of nonlinear equations describing the reservoir) and a first matrix A,(representing the sensitivity of the residual vector to changes in a system of nonlinear equations), recursively decomposing matrix A,into a lower block triangular matrix, an upper block triangular matrix, and a diagonal matrix, and generating a second matrix M,that is an approximation to matrix A,. A solution to the systems of non-linear partial differential equations may then be found by using certain values that were used to produce the matrix M,, and that solution does not require the direct computation of A,{right arrow over (x)}&equals;{right arrow over (b)} (representing the system of linear equations) as required by conventional methods.

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Sugar Land, TX, United States of America

John R Wallis