Page 162 - gas transport in porous media

155
Chapter 8: Gas Injection and Fingering in Porous Media
radial Hele-Shaw Cell (Kopf-Sill and Homsy, 1988; Meiburg and Homsy, 1988). At
the same time, explanation of fingering in Hele-Shaw cells is typically based on the
assumption of the presence of a perturbation in the system. Moreover, some studies
(Christie and Bond, 1987; Fanchi and Christianson, 1989; Fanchi, 1990) suggest that
non-linear dynamics of a miscible displacement may provide an alternative source
for the disturbances in porous media.
Kelkar and Gupta (1988) reported that they were unable to initiate a finger with-
out introducing some type of perturbation, such as a permeability variation or a
concentration perturbation. As described above, Araktingi and Orr (1990) intro-
duced randomness in their model. These works illustrate the approach taken by
many researchers in the field: Identify a parameter in the system with values that
exhibit some degree of spatial randomness. The randomness then becomes the source
of the perturbation. A problem with this approach is its dependence on the ad hoc
incorporation of a probabilistically-distributed variable.
We now provide a more quantitative discussion of stability analysis of miscible
displacements. The main goals of this discussion are to illustrate, (i) how a stability
analysis is actually carried out, and (ii) how far such an analysis can take one.
For simplicity, we assume that the dispersion coefficients D L and D T are related
to the flow velocities υ x and υ y through the following relations (Sahimi, 1995):

(α L − α T )υ x 2
D L = D m + α L |υ| + (8.21)
|υ|
(α L − α T )υ y 2
D T = D m + α T |υ| + (8.22)
|υ|
For convenience, we follow the notation in Yortsos and Zeybek (1988). The gov-
erning equations are the CD and the continuity equations, coupled with Darcy’s law:

∂C ∂C ∂C ∂ ∂C ∂ ∂C
φ + υ x + υ y = D L + D T (8.23)
∂t ∂x ∂y ∂x ∂x ∂y ∂y

∂υ x ∂υ y
+ = 0 (8.24)
∂x ∂y
k ∂P k ∂P
υ x =− , υ y =− (8.25)
µ ∂x µ ∂y
Implicit in the above continuum description is the assumption that the local Peclet
number υl/D m is small.
The base-state solution (i.e., the solution to unperturbed or mean quantities), cor-
responding to a constant injection rate in a rectilinear flow geometry, is given by the
well-known diffusive profile

1 ζ
C = erfc √ (8.26)
2 2 t