Page 153 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

Formation Damage by Fines Migration: Mathematical and Laboratory Modeling, Field Cases 131

(A) T
X =X w +αT X = X w + T

T crI
T mJ D
T mI B C
A
X w X mI X mJ X crI X

(B) T
X = X w +αT X =X mI +αT X =X mJ +αT X =X crI +αT 9
T I 8 X =X w + T
2
7
T crI
6
T mJ 0
T 3 4 5 1
T mI
3
X w X mI X 3 X mJ X crI X I X
Figure 3.21 (A) Particle and (B) front trajectories in different flow zones in (X,T) plane
(X: dimensionless spatial coordinate, T: dimensionless time).

Suspended concentration ahead of the salinity front is obtained by
substitution of the trajectory coordinate X 5 X w 1 T into formulae for c
(X,T) in zones 3, 1, and 0:
1
ð
c X w 1T;TÞ
8 0 1
> Λ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
> ffiffiffiffi p
>
> p X2 ;
@ ð A
S aI =φexp 2 ffiffiffiffiffiffi 12αÞX1αX w T,T 1
>
>
> X w
>
>
<
5 0 0 11 0 1 :
1 q Λ p ffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
> ffiffiffiffi;γ
> @ @ p AA @ p X2 A
φ
> S aI 2S cr I exp 2 ffiffiffiffiffiffi ð 12αÞX1αX w ;T 1 ,T,T 3
>
> 2πr e X X w
>
>
>
>
0; T.T 3
:
(3.133)
Fig. 3.21A shows the moments T 1 and T 3 of arrival of the boundaries
between zones 3, 1, and 0 to the salinity front:
X mJ 2 X w X crI 2 X w
T 3 5 ; T 1 5 : (3.134)
1 2 α 1 2 α