A Block Preconditioned Conjugate Gradienttype Iterative Solver for Linear Systems in Thermal Reservoir Simulation
Abstract
A preconditioned residualnormreducing iterative solver is described. Based on a truncated form of the generalizedconjugategradient method for nonsymmetric systems of linear equations, the iterative scheme is very effective for linear systems generated in reservoir simulation of thermal oil recovery processes. As a consequence of employing an adaptive implicit finitedifference scheme to solve the model equations, the number of variables per cellblock varies dynamically over the grid. The data structure allows for 5 and 9point operators in the areal model, 5point in the crosssectional model, and 7 and 11point operators in the threedimensional model. Blockdiagonalscaling of the linear system, done prior to iteration, is found to have a significant effect on the rate of convergence. BlockincompleteLUdecomposition (BILU) and blocksymmetricGaussSeidel (BSGS) methods, which result in no fillin, are used as preconditioning procedures. A full factorization is done on the well terms, and the cells are ordered in a manner which minimizes the fillin in the wellcolumn due to this factorization. The convergence criterion for the linear (inner) iteration is linked to that of the nonlinear (Newton) iteration, thereby enhancing the efficiency of the computation. The algorithm, with both BILU and BSGS preconditioners, is evaluated in the context of a variety of thermal simulation problems. The solver is robust and can be used with little or no user intervention.
 Publication:

Journal of Computational Physics
 Pub Date:
 November 1986
 DOI:
 10.1016/00219991(86)901142
 Bibcode:
 1986JCoPh..67...37B