Page 124 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

106 Thomas Russell et al.

Zone 1 (X w , X 1 T , X m ) contains the zone where the entirety of
the initial attached concentration is detached. As such, the suspension
concentration in this zone is initially constant with X.
The initial condition for the suspended concentration in zone 1 is
given by the first line of Eq. (3.63).
Integrating Eq. (3.76) by separation of variables yields:
p ffiffiffi p ffiffiffiffiffiffiffiffi
CX; TÞ 5 e Λð X2 X1TÞ : (3.77)
ð
Integrating Eq. (3.69) yields the strained concentration distribution in
this zone:
1 h p ffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffi p ffiffiffiffiffiffiffiffi i
S s X; Tð Þ 5 p ffiffiffiffi Λ X 1 1 2 Λ X 1 T 1 1 e Λð X1 X1TÞ : (3.78)
Λ X
Zones 2 (X w , X , X m , X m , X 1 T , X i ) and 3 (X m , X , X i ,
X 1 T , X i ) both contain suspended particles detached on the X-axis
between points X m and X i such that the initial suspended concentration is
governed by the critical retention function. The initial value of the sus-
pended concentration will be a function of X for these zones.
For both of these zones, Eq. (3.76) is integrated by separation of vari-
ables using the middle line of Eq. (3.63) as the initial condition. This
yields:
q p ffiffiffi p ffiffiffiffiffiffiffiffi

ð
CX; TÞ 5 S ai 2 S cr p ffiffiffiffiffiffiffiffiffiffiffiffiffi e Λð X2 X1TÞ : (3.79)
2πr i X 1 T
The strained concentration is derived by integrating Eq. (3.69) as an
Ordinary Differential Equation (ODE). Zone 2 will inherit the particles
strained in zone 1. The solution (Eq. (3.78)) thus acts as an initial condi-
tion for solving the straining kinetics Eq. (3.69). The solution is:

1 h p ffiffiffiffi p ffiffiffiffiffiffi p ffiffiffi p ffiffiffiffiffi i
S s X; Tð Þ 5 p ffiffiffiffi Λ X 1 1 2 Λ X m 1 1 e Λð X2 X mÞ
Λ X
Λ Λ X ð T q 2Λ X1T
p
p
ffiffiffi
ffiffiffiffiffiffiffiffi
1 p e S ai 2 S a p ffiffiffiffiffiffiffiffiffiffiffiffiffi e dT:
ffiffiffiffi
2 X X m 2X 2πr i X 1 T
(3.80)
The first term here corresponds to the strained particles inherited
from zone 1. The second term is the straining of particles detached in