Page 136 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

116 Thomas Russell et al.

Solving Eq. (3.105) by separation of variables, accounting for initial
condition (Eq. (3.99)), yields C(X,T) 5 0 ahead of the salinity front.
To find a solution between the suspended particle and the salinity front,
zone (αT , X , T), it is necessary to calculate the boundary condition for
suspended particle concentration behind the front X 5 T. Consider the mass
balance condition on the salinity front (Bedrikovetsky, 1993; Lake, 2010):

½ C 1 S s 1 S a ŠD 5 α C ½Š; (3.106)

where D is the shock front velocity, and jump of the quantity A is the
1
difference between A-values ahead and behind the front: [A] 5 A -A 2
(Polyanin and Zaitsev, 2011; Polyanin and Manzhirov, 2007). As it follows
from the kinetics Eq. (3.95), the strained concentration is continuous
across the shock. The values of the attached concentration ahead and
behind the shock follow from the solution (Eq. (3.102)), and the value of
the suspended concentration ahead of the shock follows from the solution
to Eq. (3.103) for zone 0:

1
2
1

C 5 0; S a 5 S cr γ ; S a 5 S cr γ ; (3.107)
I J
and as the shock is being evaluated across the line X 5 T, the shock veloc-
ity equals one.
Substituting Eq. (3.107) into Eq. (3.106) yields the suspended concen-
tration behind the salinity front:
1
C 5 : (3.108)
ð 1 2 αÞφ
Solving Eq. (3.105) by separation of variables, accounting for the
boundary condition for suspended particle concentration behind the front
X 5 T, yields:

1 T 2 X
CX; TÞ 5 exp 2αΛ : (3.109)
ð
ð 1 2 αÞφ 1 2 α
Using the method of characteristics for Eq. (3.103) along with the
boundary condition (Eq. (3.100)) yields a zero suspended concentration
C(X,T) 5 0 behind the suspended particle front X 5 αT, zone 2.
The strained concentration is obtained from Eq. (3.95) by integration
of the suspended concentration in time. Table 3.5 presents a summary of
the solutions for salt transport, and attached, suspended, and strained con-
centrations in the different zones.