Page 177 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

Formation Damage by Fines Migration: Mathematical and Laboratory Modeling, Field Cases 153

yields:
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffi p ffiffiffiffiffiffiffiffiffiffi
Λ X1α X2TÞ22 X1 X2αT
ð
CX;TÞ5e ð S aI 2S a ðT 50ÞÞ
ð
ð p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p
p ffiffiffi p ffiffiffiffiffiffiffiffiffiffi T Λ αT1T 0 12αÞ2 ffiffiffiffiffiffiffiffiffiffiffiffiffi @S cr TðÞ
ð
2e ½ 2Λð X2 X2αTފ e ð T 0 12αÞ dT:
0 @T
(3.191)
Note that the integral term contains the constant T 0 defined by the
solution of Eq. (3.187):
X 2 X w 2 X 0 5 α T 2 T 0 Þ; (3.192)
ð
where on the salinity shock X w 1 X 0 5 T 0 and so X 5 αT 1 T 0 (1-α).
This constant is used to eliminate X from the integral such that the
integrand is a function of T only. For cases where the integral can be eval-
uated explicitly, the constant can be substituted following integration.
For the region behind the concentration shock, the original PDE
(Eq. (3.186)) is reduced to the ODE:
dC ΛC 1 @S a
ffiffiffiffi ffiffiffiffiffiffi 2
52 p p : (3.193)
dX 2 X X w α @T
Along the parametric curves given by:
dT 1
5 : (3.194)
dX α
Integrating Eq. (3.193) subject to the boundary condition
(Eq. (3.183)) yields:
p ffiffiffi p
ffiffiffiffiffi ð X
e 2Λð X2 X wÞ p ffiffiffi p ffiffiffiffiffi @S cr XðÞ
CX; TÞ 52 e Λð X2 X wÞ dX: (3.195)
ð
α @T
X w
The solution here is clearly presented in an implicit form due to the
requirement of a continuous function S cr (U,γ ) of which there is yet no
accepted general form. This requirement complicates generating results,
and for particular cases of the critical retention function will lead to a
need for numerically solving the integrals in Eqs. (3.191 and 3.195).
The strained concentration is solved by integrating Eq. (3.175) as an
ordinary differential equation. The dimensionless pressure drop is then
calculated by integrating Eq. (3.179). The results are summarized in
Table 3.12.