Page 176 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

152 Thomas Russell et al.

Next, the pseudo-salinity is solved by integrating Eq. (3.178) by sepa-
ration of variables:
1; X . X w 1 T
(
Γ 5 X2T : (3.185)
0
ε
e ; X , X w 1 T
The suspended concentration is then solved using the mass balance
Eq. (3.174) by substituting the expressions for the straining rate and the
attached concentration:

@C @C αΛC @S cr X; ΓÞ
ð
1 α 52 p p : (3.186)
ffiffiffiffi ffiffiffiffiffiffi 2
@T @X 2 X X w @T
As before, this equation is solved separately for the three solution
regions. For all points ahead of the particle concentration front, along
parametric curves given by:

dX
5 α: (3.187)
dT
Eq. (3.186) reduces to the ODE:
dC αΛC @S cr X; ΓÞ
ð
ffiffiffiffi ffiffiffiffiffiffi 2
52 p p : (3.188)
dT 2 X X w @T
For the region ahead of the salinity front, salinity is constant, so the
critical retention function does not change with time. Using the initial
condition (3.182), this equation can be integrated to yield:
p ffiffiffi p ffiffiffiffiffiffiffiffiffiffi
CX; TÞ 5 e 2Λð X2 X2αTÞ ð S aI 2 S a ðT 5 0ÞÞ: (3.189)
ð
Following the reasoning presented in Section 3.6.2, it can be shown
that the suspended concentration is continuous across the salinity shock.
As such, the solution (Eq. (3.189)) is used as an initial condition for the
second region, between the salinity and concentration shock.
Integrating Eq. (3.188) using separation of variables with the initial
condition:
p ffiffiffi p ffiffiffiffiffiffiffiffiffiffi
CX 5 T; TÞ 5 e 2Λð X2 X2αTÞ ð S aI 2 S a ðT 5 0ÞÞ; (3.190)
ð