Page 68 - Principles of Applied Reservoir Simulation 2E

Part I: Reservoir Engineering Primer 53

6.3 Linear Stability Analysis

The stability of frontal advance is determined by considering the rate of
growth of a perturbation at the front. We first express the frontal advance
velocity Eqs. (6.17) and (6.31) in the general form

dx f a + $x f
. / ^L (632)
di j + 8jt f
where the coefficients are independent of time and frontal location. Equation
(6.32) is a nonlinear, first-order differential equation. Imposing a slight
perturbation on the front location gives

+ e) a + e)
d(x f + $(x f
' = _i__ (6J3)
dt y +b(x f + K)
The velocity of propagation of the perturbation is given by the difference
between Eqs. (6.33) and (6.32):

de a + $x f f + pe a + $x f
= • -*- (6.34)
dt j + ox f + 6s y + bx f
Combining fractions and simplifying yields

j£ (6.35)
dt

Further simplification is achieved by recognizing that the perturbation is slight
so that we have the approximation
1 8s c
5 w 1 — for 6s « y + ox f
i + . S£ T + 5^/ f (6.36)
y +8^
Substituting Eq. (6.36) into Eq. (6.35) gives

8s
1- (637)