A New Iterative Solution Technique for Reservoir Simulation Equations on Locally Refined Grids
- Clemens W. Brand (Mining U. Leoben) | Zoltan E. Heinemann (Mining U. Leoben)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- November 1990
- Document Type
- Journal Paper
- 555 - 560
- 1990. Society of Petroleum Engineers
- 4.1.5 Processing Equipment, 4.3.4 Scale, 4.1.2 Separation and Treating, 5.5 Reservoir Simulation, 5.1.5 Geologic Modeling
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This paper presents an incomplete LU (ILU) factorization technique coupled with generalized conjugate-gradient acceleration especially designed for linear equations resulting from locally refined grids. The factorization is based on a special ordering scheme. This repeated red/black (RRB) ordering can cope naturally with grid irregularities introduced by local refinement. The scheme is also very effective on regular grids, where the condition number (a measure of the convergence rate) of the preconditioned linear system increases asymptotically with the number of gridblocks more slowly than it does for conventional ILU preconditionings. For large linear systems, fewer iterations are needed to reach convergence. This makes the method highly competitive compared with other techniques even on regular grids, although the new ordering scheme requires more data handling than standard orderings. Results for several idealized test cases [implicit-pressure, explicit-saturation (IMPES) equations] show the new method to be faster than standard iterative methods.
Proper ordering schemes can reduce the computational work Proper ordering schemes can reduce the computational work required to solve the large systems of linear equations generated by reservoir simulation. For direct solution methods, besides the standard natural or row-wise ordering, diagonal and red/black-type schemes are used (D2 and D4 ordering 1). These ordering schemes reduce the number of arithmetic operations by a factor of about two or four in 2D grid systems and even more for 3D systems.
More complicated ordering schemes (nested dissection) reduce the amount of computation for nxn grids to an asymptotically proportional n operations. In comparison, the number of proportional n operations. In comparison, the number of operations for D2 or D4 ordering is an asymptotically proportional n. For large grid systems and especially for 3D examples, however, the computational work for direct elimination becomes too timeconsuming. Iterative methods are more appropriate for large systems of equations. It is well known, however, that the ordering also influences the convergence of iterative methods.
A serious drawback of any complicated ordering is the overhead costs in data storage (additional pointer arrays required) and data access (indirect addressing necessary). The sparse-matrix structure is more complicated than the well-known fixed-bandwidth, fixedblock-size form obtained by natural ordering on a regular grid. Es-pecially when vector and parallel computing techniques are considered, keeping the matrix structure as regular as possible may be desirable. Today, however, simulators generate Jacobian matrices with a much more complex structure than block-banded matrices with constant bandwidth. More sophisticated EOR models (compositional and thermal), grid-refinement techniques, and adaptive implicitness are some features that contribute to the complexity of the sparse-matrix pattern.
If the linear solver of the simulation program must cope with all these matrix irregularities, use of a sophisticated ordering scheme could reduce the number of arithmetic operations or speed the convergence rate.
In this paper, the performance for reservoir simulation purposes of an ordering based on the repeated application of a red/black method is examined. This RRB ordering was described in Ref. 7 for discretizations obtained on regular n x n grids. It generates an ILU factorization and leads to an iterative method with attractive convergence properties. This ordering is generalized to locally refined grids and 3D block systems.
The main idea of this ordering scheme is to order the unknowns or gridpoints in a nested sequence of red/black orderings. The algorithm partitions the unknowns into several groups. Each group covers the whole grid with an increasingly coarser pattern. This allows locally refined grids to be handled and offers possibilities of vectorization.
This paper describes the ordering scheme and the incomplete factorization method. Numerical tests demonstrate the faster convergence rate of the new method, especially for large matrices.
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