Page 69 - Principles of Applied Reservoir Simulation 2E

54 Principles of Applied Reservoir Simulation

Keeping only terms to first order in e and simplifying gives

(6.38)
dt ~ (y

Equation (6.38) has the solution

(6.39)

where e 0 is an integration constant, and
By — 8ot
2
(y+5* ^ 6 40^
— 7 )
T =
If T is negative, the perturbation decays exponentially. If T is greater than zero,
the perturbation grows exponentially. Finally, if T equals zero, the perturbation
does not propagate because de/dt - 0 in Eq. (6.38).
We can now examine the stability of a displacement front. Comparing
Eq. (6.32) with (6.31) lets us make the identifications

a =

X w (p 0 -p w ) g sine (6 42)
-

T = ML (6.43)

8 = (1-M) (6.44)

The resulting expression for the growth of a perturbation is
& M (l-MX$,-<D 2 )+M.(p 0 -pJgsin0
(6 45)
'

Equation (6.45) agrees with Eq. (7-104) in Collins [1961].
Zero growth rate of a perturbation is determined by setting the derivative
de/dt = 0 in Eq. (6.45). The resulting condition for zero growth rate is
(1- AfX*i-* 2 )+ M,(p e -p w )gsine = 0 (6.46)