Page 150 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

Formation Damage by Fines Migration: Mathematical and Laboratory Modeling, Field Cases 129

zero for X w , X , X mJ in areas A and B, and equal to maximum retention
function for X . X mJ in areas C and D:

8
0; X w , X , X mJ ; T . X 2 X w
>
>
0 1
<
S a X; Tð Þ 5 1 q ffiffiffiffi ; γ ; :
φ
> S cr @ p J A X mJ , X , N; T . X 2 X w
> 2πr e X
:
(3.130)
Therefore, the attached concentration corresponds to the value of the
maximum retention function of γ I ahead of salinity front X-X w 5 T, and
of γ J behind this front. The attached concentrations are steady state,
which cancels S a in the accumulative term of the mass balance
Eq. (3.120). Substituting the straining rate (Eq. (3.121)) into Eq. (3.120)
yields:
@c @c αΛc
1 α 52 p p (3.131)
ffiffiffiffi ffiffiffiffiffiffi :
@T @X 2 X X w
In zones 0, 1, and 3 (Fig. 3.20B), Eq. (3.131) subject to initial condi-
tions (Eq. (3.126)) is solved by the method of characteristics. Here, time
T is set as the parameter along the characteristic lines:

dX dc αΛc
ffiffiffiffi ffiffiffiffiffiffi ;
5 α; X 2 X w 2 X 0 5 α T 2 T 0 Þ; 52 p p (3.132)
ð
dT dT 2 X X w
where X 0 corresponds to the intersection of the characteristic line with
the X-axis. Formula (Eq. (3.126)) for the suspended concentration for
three X-intervals provides initial conditions for the ordinary differential
Eqs. (3.132) in zones 3, 1, and 0, respectively.
Separation of variables in Eq. (3.132) yields explicit formulae for sus-
pended concentration. In zone 0, the initial suspended concentration is
zero. The solution along characteristic lines yields zero suspended con-
centration in the overall zone 0.
In zone 3, the initial suspended concentration c is a constant given by
Eq. (3.126). The solution c(X,T) along characteristics is given by the for-
mula listed in the tenth row of Table 3.8. The solution is independent of
X in zone 3. The solution in zone 1 with X-distributed initial suspended
concentration given by formula (Eq. (3.126)) is presented in the formula
given in the ninth row of Table 3.8.