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where kmax and kmin are the maximum and minimum principal effective permeability
values, kp is the permeability parallel to the bands when θ=0°, kn is the permeability
normal to the bands when θ =0°, and k+ is the isotropic permeability when half of the
bands are set orthogonal to the other half (θ =90°). Since the analytical solution for k+
only holds for low percentages of DBs, we use k+ as calculated by our numerical method
in the equations above to find the maximum and minimum effective permeability values
as functions of wb/W, θ and kb/km.
Figure 6.11 shows the results for kb/km=10 -2 over the entire range of wb/W for
intersection angles of θ =15°, 30°, 60°and 90°. Both principal effective permeability
values decrease as wb/W increases, with the reduction in kmin being most pronounced. The
degree of this permeability anisotropy also decreases as θ increases, with an isotropic
permeability drop attained at θ =90°. For kb/km=10 -3 , the anisotropy introduced is even
more pronounced, approaching 2 orders of magnitude as θ approaches 0° and the pattern
becomes essentially parallel. For the generally high intersection angles (~80) and low
volume percents (~10%) exhibited for cross-hatch DB patterns in the Aztec, relatively
isotropic permeability reductions ranging from about 0.7 to1.5 orders of magnitude
would be expected.
6.3. Anastomosing
No analytical solutions exist for the systematic, but irregular anastomosing pattern, so
the purely numerical method was used to calculate effective permeability tensors for a
finely discretized version of a real, 12 x 15 m DB outcrop pattern mapped in the Aztec on
a low altitude balloon photograph (Figure 6.12). Not surprisingly, the maximum effective
permeability is oriented roughly parallel to the mean trend of the bands, while minimum
permeability is directed roughly perpendicular. We also compared the effective
permeability of this real anastomosing pattern to an idealized parallel pattern using the
same average spacing and wb/W ratio of 15%. As expected, effective band-parallel
*
p
permeability (k ) was somewhat lower for the anastomosing pattern—0.79 versus 0.85 at
kb/km=10
-2 —due presumably to band connectivity and associated compartmentalization
*
along trend. On the other hand, effective band-normal permeability ( k ) was significantly
higher for the anastomosing pattern—0.26 versus 0.06 at kb/km=10
-2 —due presumably to
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n