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5.3. Finite difference/finite element method
For the purposes of computational efficiency, a coupled, two-step finite
difference/finite element numerical method was used to compute k * . In order to
adequately resolve centimeter-thick DBs within meter-scale patterns, finely discretized
grids a few hundred cells on a side were generally necessary, with each cell being
assigned one of two isotropic permeability values: kmatrix or kband (e.g. Figure 6.6). Again
for the sake of computational efficiency, two coarsening steps were used to compute the
final, upscaled permeability tensor. For example, to compute k * for a pattern meshed at
400 × 400 cells, the first coarsening step might be to a 20 x 20 cell system (yielding 400
intermediate permeability tensors) and then to the final k * in the second step. An efficient
finite difference method was used to make the first coarsening step. The second step to k *
required a finite element procedure that is less efficient, but capable of handling the
matrix of full permeability tensors produced during the first step. Repeat testing revealed
that the final k * computed is relatively insensitive to the size of the coarsening steps used,
varying by less than 10% for the patterns studied.
6. Effective permeability results
The governing equations, boundary conditions and numerical methods detailed above
were used to calculate 2-D effective principal permeabilities for each of the three
characteristic DB patterns described for the Aztec. For the simpler, more regular parallel
and cross-hatch examples, we used idealized patterns to assess how effective
permeability varies as a function of the relevant physical variables—band thickness,
spacing and intersection angle—and present analytical solutions for comparison where
possible. For the more generally irregular anastomosing pattern, which exhibits
substantial spatial variability that defies analytical treatment, we assess effective
permeability for a representative outcrop pattern. In order to emphasize reductions in
permeability attributable to DBs, all permeability values reported below and in the
figures are normalized by that of the undeformed sandstone matrix (~ 0.1 to 1.0 darcys)
to yield dimensionless ratios of relative permeability. Thus in the analyses that follow,
DBs are assigned isotropic internal permeability values of 10 -2 and 10 -3 , representing the
mid range of values for DBs as reported by other workers and summarized above.
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