Page 244 - gas transport in porous media

Stockman
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were not distributed efficiently, and the 100 × 100 pixel image had to be stretched
into a 400×400×20 LB automaton to achieve the correct aspect ratio. The geometry
was then reflected in the flow direction, to assure continuity of flow fields. Finally,
after the automaton reached a steady-state flow, the geometry was cloned up to 16
times in the flow direction (to provide a long path for method-of-moments dispersion,
as described in Section 13.2), and a slug of solute introduced. Second, the irregu-
lar surfaces induced the instability described in Section 13.4.3, which limited the
maximum Pe to ∼700. (This study was completed before the technique described in
Section 13.4.3 was available.)
Figure 13.13 compares the LB estimated dispersion coefficient, with the experi-
mental dye dispersion measurements. Despite all the limitations, the LB technique
gave reasonable agreement. The two subsamples give significantly different results,
but the differences are comparable to the differences created by other assumptions –
e.g., the assumption of asymmetric vs. symmetric disposition of the apertures, which
is an inherent uncertainty in the Reynolds equation modeling used by Detwiler et al.
(2000).

13.6 CONCLUSIONS

Lattice Boltzmann modeling provides a powerful technique for investigating gas
permeability and dispersion in complex geometries. Results averaged over many
pore scales can be used to estimate macroscopic transport properties. However, the
method is not a panacea, and for many geologic problems, pore-scale modeling
is far too calculation-intensive. Significant problems include the need to define a
representative volume; the fixed CWL number, often implying that only hours of
physical time can be modeled with days of computer computation; and the restriction
to “modest” Pe and Re (<1000 in a typical problem). Nonetheless, LB can model
many coupled phenomena that are difficult to approach with traditional methods, such
as finite difference and finite element techniques.

Answers to Self-Study Examples:
2
(1a) Yes. From Section 13.2.2, viscous equilibrium requires ∼ L /ν =
2
(26 − 2) /0.01 = 57600. (The right and left, and top and bottom sites of the
channel are solid, so the channel is 24 lu wide.)
6
(1b) First part: (sites = 26×26×24)×(60000 ts)/(9·10 ts·sites/s) = 108 s.
6
Second part: (sites = 26 × 26 × 24 × 48) × (60000 ts)/(35 · 10 ts · sites/s) =
1335 s.
(1c) First part: memory to store vectors: (sites = 26 × 26 × 24) · (19 vectors) ×
(4 bytes/vector) = 1233024 bytes = 1.18 MB. There is a smaller amount of
memory required to store pointers, 26 · 26 · 24 bytes required to store solid
site information, and (sites = 26 × 26 × 24)× (3 speed components) × (4
bytes/component) = 0.185 MB required to store the x, y, z components of the flow
velocity at each site.