A cost-limiting factor of numerical reservoir simulation is the execution timespent on solving large systems of equations.The Symbolic D4 method is a newdirect solver which uses less memory and is significantly faster than otherdirect methods. This method decomposes the coefficient matrix symbolically, based on the structural patterns generated by D4 ordering. The reduced systemoccupies a single Quadrant of the original matrix, requiring fewer floatingpoint operations to solve.
Execution time and memory requirements are compared between the Symbolic D4method and band algorithm using PCSIM Version 1.0, a three-phase, three-dimensional black oil simulator for the IBM PC. One, two andthree-dimensional applications are investigated_ It is shown that the Symbolic D4 method is many times faster than the band algorithm, while using a fractionof the memory. Performance also improves with increasing grid Size, making the Symbolic D4 method ideally suited for multidimensional simulation on smallercomputers with memory and/or speed limitations. Larger grids can be addressedand results obtained more rapidly.
Differential flow equations, typically used for mass and energy balances inreservoir Simulation, are spatially discretized on a reservoir grid. Spatialvariables, such as pressure, temperature and saturation, Will generally have adifferent numerical value at each element (i, j, k) in the three-dimensionalreservoir grid (NX, NY, NZ). Finite difference equations (FDE's) relate spatialvariables at each grid element (i, j, k) to the values in adjacentelements.
Finite difference equations with the block standard form
Equations 1(available in full paper)
Are commonly used in reservoir simulation. The dependent variables, X, arealgebraicly related to their neighbouring values using the block differencecertificients, Z, B. D, E, F. H, Sand Q. An FDE set at each element (i, j, k)yields a system of linear equations, one set for each of the N unknowns,
Equations 2 & 3(available in full paper)
Conventional solution algorithms are based on a certifficient matrix A1 which has a banded structure based an a natural ordering of thegrid elements (Figure 1a). Numbering the grid elements consecutively alongalternate diagonasis(1), will partition the coefficient matrix Intowell defined quadrants (Figure 1b) The (diagonal quadrants are strictly blockdiagonal. The alternate diagonal, or D4 ordering gives rise to a precise matrixstructure which allows partial symbolic decomposition(2) into upperand lower block triangular matrices. Forward and backward substitutions canthen be used to obtain the solution.
A. D4 mapping is defined for the natural index of each grid element, ijk,
Equations 4(available in full paper) to its corresponding D4 Index. The inverse D4 mapping follows directly.
Equations 5 & 6(available in full paper)
The equivalent D4 ordered block difference system (2). has the vectorquantities,
Equations 7(available in full paper) and a corresponding coefficient matrix with nonzero block elements,
Equations 8(available in full paper)
LU decomposition of A Into upper and lower block triangular matrices,
Equations 9(available in full paper)
Will have the general forms illustrated In Figure 2.
