Page 175 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

Formation Damage by Fines Migration: Mathematical and Laboratory Modeling, Field Cases 151

The system of equations becomes:

ð
@ C 1 S s 1 S a Þ @C
1 α 5 0; (3.174)
@T @X
αΛC
@S s
ffiffiffiffi ffiffiffiffiffiffi ;
5 p p (3.175)
@T 2 X X w
0
S a 5 S cr X; Γð Þ; (3.176)
@Γ @Γ
1 5 0; (3.177)
@T @X
@Γ 0
ε 5 Γ 2 Γ ; (3.178)
0
@T
1 2 @P
52 : (3.179)
X ð 1 1 βφS s Þ @X
The initial conditions for this system correspond to high-salinity fluid,
and an absence of strained particles:

T 5 0:Γ 5 1; S s 5 0: (3.180)
The attached concentration begins at some initial value S aI for all X.
However, as the fluid is assumed to be incompressible, any particle
detachment due to velocity at high salinity will occur instantaneously.
Hence, the initial attached concentration can be given as:

S aI ; S cr X; Γ 5 1Þ . S aI

0
ð
S a X; T 5 0ð Þ 5 ; (3.181)
0
0
S cr X; Γ 5 1Þ; S cr X; Γ 5 1Þ , S aI
ð
ð
The initial value of suspended concentration will be given by:
CX; T 5 0Þ 5 S aI 2 S a X; T 5 0ð Þ: (3.182)
ð
The boundary condition is given by the injection of low-salinity water
with an absence of suspended particles:
X 5 0:Γ 5 0; C 5 0: (3.183)
The solution procedure is presented for an arbitrary form of the maxi-
mum retention function. First, the salt mass balance Eq. (3.177) is solved
using the method of characteristics to yield:

1; X . X w 1 T
Γ 5 : (3.184)
0; X , X w 1 T