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The use of parallel iterative techniques for the linear equations associated with reservoir simulation was investigated. A technique is described in which a lower dimensional problem is used as a correction to the preconditioned generalized conjugate residual method with the domain decomposition technique as the overall preconditioner. For the cases investigated the method proved to be both robust and efficient.
Three example matrices, including a highly heterogeneous example, an example from the SPE/SIAM comparative solution project, and a nonsymmetric IMPES reservoir simulation example, are presented to validate the method. Several comparisons are made with other well-known preconditioners including incomplete LU and reduced system/incomplete LU factorizations. For the examples considered the domain decomposition technique was the most efficient in a nonparallel environment; in a parallel computational environment the algorithm was several times more efficient than the other techniques. An analysis was made concerning both vector and parallel computational aspects of the domain decomposition method. Numerical experiments showed that the effect of the number of domains on the convergence rate is small. Corrections using a lower dimensional problem, known as line corrections, were found to be necessary for the rapid convergence of the method. Finally, the three dimensional domain decomposition algorithm was efficiently implemented in parallel computational environments using multitasking and microtasking on both the CRAY K/MP 48 and IBM 3090/400 four-way parallel supercomputers.
Solution of the following system of partial differential equations from petroleum reservoir simulation with finite differences can yield a nonsymmetric, generally positive definite, matrix:
(1)
where,
P = oil phase pressure,
C, C, C = phase transmissibilities,
q, q, q = phase source and sinkterms,
phi = reservoir porosity,
S, S, S = phase saturations,
B, B, B = phase density terms,
P, P = capillary pressure terms.
R = gas-oil mass transfer term
Omega = boundary of reservoir.
This system of equations is often reduced to a single finite difference equation involving the unknown Po, the oil phase pressure, using the IMPES reduction scheme. The single equation varies from either elliptic to parabolic in nature depending on the compressibility of the fluids in the reservoir.
The resultant matrix for the finite difference numerical solution is large, banded, and sparse.
P. 293^
