Page 164 - gas transport in porous media

Chapter 8: Gas Injection and Fingering in Porous Media
1.2

0.8 L =1. 157
c
Growth rate

0.4

0.5
0.
0 2 4 6
Wave number
Figure 8.4. Step-profile results for the growth rate of fingers (after Yortsos and Zeybek, 1988)

(R = lnM > 0), large wavelengths are unstable, while a strong stabilization due to
transverse dispersion is exerted at smaller wavelengths. A cutoff wave number can
be identified
R
α c = √ (8.36)
2(ε + ε)
As expected, α c increases with increasingly unfavorable mobility, and with an
increase in the ratio of longitudinal and transverse dispersion (1/ε). However, the
limits of the continuum description should be kept in mind. The size of the most
unstable disturbance scales with the characteristic length, which for large enough
flow rates becomes equal to the dispersivity α L , which is normally a multiple of
the typical pore size (or the length scale of the heterogeneities). It is apparent that a
possibleconflictmaydevelopbetweentheaboveresultandthecontinuumdescription,
precluding meaningful predictions over scales of the order of the microscale.
While the limit L c = 0 leads to expected results, a distinct sensitivity develops for
L c = 0 (see Figure 8.4). This effect is present only due to the velocity dependence
of the dispersion coefficients, which can be best quantified in terms of the following
combination

RL c R √
B c tanh − ε − 1 (8.37)
2 2
The following results may then be shown (Yortsos and Zeybek, 1988):
(i) When B c < 0 (which is always the case if L c = 0), at small enough viscosity
ratio and for L c = 0, the cutoff wave number is finite:
R
α c = √ (−B c ) (8.38)
2 ε
although it increases as L c or B c does.