structural geology, propagation mechanics and - Stanford School of ...

contiguous areas. This type of characteristic pattern replication lends itself naturally to
numerical solution using spatially periodic boundary conditions, as detailed below.
It is important to note that spatial periodicity does not imply fractal scaling. Based on
our observations of DBs in the Aztec, we can state that their characteristic patterns
definitely are not fractal. Also, it is not necessary for actual pattern periodicity in nature
to be nearly as distinct and regular as idealized in Figure 6.7 for the numerical methods
described here to work effectively.
Using the two-scale representation of k and combining Darcy’s law with continuity
yields the pressure equation for steady state, incompressible flow:
[ ( c,f
) ⋅ ∇ = 0
∇ ⋅ k p]
(4)
where k varies on both the fine and coarse scales. Our intent is to calculate an
effective permeability tensor, k * , that varies only on the c-scale but that implicitly
accounts for permeability variations on the f-scale. This allows us to model the system
using an equation of the form of (4) but with k * (c) replacing k (c,f). The solution to this
“upscaled” equation is much less computationally demanding than the solution of
equation (4), because variations on the fine scale f need not be resolved. The essential
step is to transform k (c,f) to k * (c) by calculating the effective permeability tensor of a c-
scale block containing f-scale heterogeneities (DBs). It has been shown (e.g. Bourgeat,
1984)) that homogenized upscaling is exact when two distinct spatial scales of
permeability variation exist. Subsequent work (e.g. Durlofsky, 1991) has demonstrated
that upscaling can also be applied with reasonable accuracy to more realistic situations
such as represented by DBs in the Aztec, where distinct spatial scales of permeability
variation cannot be defined.
5.2. Boundary conditions and solution
In order to drive large-scale flow, pressure variation on the coarse scale c is imposed.
This can be expressed generally as:
p = p 0 + G ⋅ (c − c 0 ) (5)
where c0 is the center of the region under study, p0 is the fluid pressure at c0, and G is
the local fluid pressure gradient ( ∇p ) over the c-scale at c0.
The average flow through a
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