Ponting, D.K., Ponting, D.K., Atomic Energy Research Establishment Foster, B.A., Atomic Energy Research Establishment Naccache, P.F., Atomic Energy Research Establishment Nicholas, M.O., Atomic Energy Research Establishment Pollard, R.K., Pollard, R.K., Atomic Energy Research Establishment Rae, J., SPE, Atomic Energy Research Establishment Banks, D., British Natl. Oil Corp. Walsh, S.K., Atomic Energy Establishment
Abstract
This paper describes Program for Oil Reservoir Simulation (PORES), an efficient general-purpose black-oil simulator now in production use for modeling North Sea fields. The frilly implicit finite-difference equations are solved for each time step with a Newton-Raphson procedure. The resulting large sets of linear equations are procedure. The resulting large sets of linear equations are usually solved simultaneously by a new and powerful iterative method that uses a preconditioned conjugate gradient algorithm with an enforced column-sum condition to accelerate convergence. A sequential solution option is available, and direct matrix inversion methods also are provided. Gas condensate problems are handled by a variable switching technique. Four examples are presented to illustrate the power and efficiency of the presented to illustrate the power and efficiency of the program. program. Introduction
PORES is a general-purpose three-phase fully implicit PORES is a general-purpose three-phase fully implicit finite-difference oil-reservoir simulator. It was developed to exploit the stability of the fully implicit equations to provide a package capable of efficiently solving widely differing types of problems without the danger of numerical instability. The program has been in production use for the U.K. Dept. of Energy (DEn), the British Natl. Oil Corp., and the British Gas Corp. for the past 2 years. This paper describes the main features of the PORES program.
The finite-difference equations describing the flow of oil, water, and gas are formulated in terms of residuals associated with each phase for every cell of the reservoir. The resulting fully implicit equations are linearized by a Newton-Raphson iteration scheme. Convergence of the nonlinear iterations is monitored by substituting successive iterates into the finite-difference equations and evaluating the root mean square (RMS) residual.
The linear equations that arise at each nonlinear iteration can be solved by iterative or direct methods. The iterative method is a preconditioned conjugate gradient algorithm with an enforced column-sum condition to ensure material conservation and to accelerate the iteration process. The form of preconditioning used is particularly process. The form of preconditioning used is particularly suited to reservoir simulation problems that contain an inherent directionality. The direct solver is based on Gaussian elimination with D4 ordering. Both types of solution include the effects of off-band terms arising from the inclusion of multilayer wells. The effect of flow between non-neighboring pairs of reservoir gridblocks can also be included. The linear equations may be solved fully simultaneously or a number of phases may be treated sequentially. The sequential method performs an approximate factorization of the Jacobian matrix while employing a column-sum condition to maintain material balance. Although the sequential option is efficient on easy problems, we have found that many North Sea applications require the extra stringency of a simultaneous solution.
Computer core requirements can be reduced by allocating storage for only the active cells in the reservoir. For large problems this can reduce memory requirements by 50% and in exceptional cases by as much as 75%.
The program contains a number of additional features to improve user convenience.A micro- or minicomputer-based color graphics facility enables interrogation of an interface file produced from a simulation run. produced from a simulation run.One option generates directional pseudo functions on a prescribed coarse mesh from the results of a fine-grid simulation.An aquifer model based on fault connections between large aquifer blocks and any number of reservoir gridblocks is available, in addition to analytical models.
JPT
P. 544
On the Theory of Relaxation
