Page 155 - gas transport in porous media

Sahimi et al.
148
The predictions of Koval’s model are in good agreement with the experimental
data of Blackwell et al. (1959). Despite its success, Koval’s model does suffer from
two fundamental shortcomings. One is the empirical nature of Eq. (8.9). The model’s
second shortcoming is the inadequacy of H for describing the heterogeneity of a
field-scale porous medium which is characterized by large-scale variations in the
permeabilities and long-range correlations in their distribution.
The second model is due to Todd and Longstaff (1972). In their model the average
concentration C s of the solvent is described by
¯
∂C s ∂f s
+ = 0 (8.10)
∂t D ∂x D
where f s represents the average of f s and is a function of C s , and x D and t D are
¯
dimensionless distance and time, respectively. This equation is, of course, a limiting
case of a convective-diffusion equation in which dispersion has been neglected. Todd
and Longstaff assumed that
1−ς ς
µ eo = µ µ (8.11)
o m
with a similar expression for µ es ,where µ m is given by Eq. (8.1) in which C s , instead
¯
∼
of, C s is used, and ς = 2/3. f s is given by
C s
¯
f s = (8.12)
−1
C s + M e 1 − C s
¯
One finds that the average concentration C s moves with a speed df s /dC s . There-
¯
fore, the leading edge of the finger, where C s = 0, moves at a speed df s /dC s =
µ eo /µ es , whereas the trailing edge, where C s = 0, moves at a speed
df s /dC s = µ es /µ eo . Despite some success, the model of Todd and Longstaff (1972)
¯
suffers from the same shortcomings as those of Koval’s model. However, because
of their simplicity, and despite their shortcomings, these two models have been used
heavily in the petroleum industry.
More recent models, which are more or less based on the same type of averaged
continuum equations, are those of Vossoughi et al. (1984), Newley (1987), Fayers
(1988), Odeh and Cohen (1989), and Fayers et al. (1990). Although these mod-
els may give an adequate fit to the production/effluent data in an (either one- or
two-dimensional) experiment, they often lead to quite different predictions of the
pressure field during the unstable displacement process. In the evaluation of the aver-
aged models of viscous fingering, their performance in the two-dimensional model,
where viscous instability occurs, has been considered (Newley, 1987; Fayers et al.,
1990). To distinguish between such models experimentally, it is necessary to mea-
sure the pressure field directly during unstable miscible displacements, an extremely
difficult task to achieve for either two-dimensional linear or other geometries (see,
however, Sorbie et al., 1995). Alternatively, the computed pressure distribution may