Page 186 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

Formation Damage by Fines Migration: Mathematical and Laboratory Modeling, Field Cases 161

The corresponding differential form of Eq. (3.221) results in:
dϕ 5 fdT 2 sdX; (3.223)

and the lines of constant stream-function are stream-lines of the flow:
dX f
5 : (3.224)
dT s

Time derivative, dT can be found from Eq. (3.223) as:
dϕ sdX
dT 5 1 : (3.225)
f f
The equality of second mixed derivatives of function T 5 T(X,ϕ)in
Eq. (3.225) yields:

@FU; γÞ 1 @U 5 0:
ð
@ϕ @X (3.226)
where,
1 s
U 5 ; FU; γÞ 52 : (3.227)
ð
ð
fs; γÞ fs; γÞ
ð
Eq. (3.226) is the transformed form of Eq. (3.211) in (X,ϕ) plane.
Using Green’s theorem over any arbitrary domain ϖ with a continu-
ous boundary to Eq. (3.212) and applying to Eq. (3.223):
@γ

I I I ZZ
0 5 ðÞdT 2 γsðÞdX 5 γ fdT 2 sdXÞ 5 γdϕ 5 dXdϕ:
γf
ð
@ϖ @ϖ @ϖ ϖ @X
(3.228)
transforms the salt transport equation into (X,ϕ)-coordinates.
@γ
5 0: (3.229)
@X
In the same way, applying Green’s theorem to Eqs. (3.214) and
(3.215) transforms the fines transport and straining rate equations into the
new coordinate system:
@c @S s
1 5 0; (3.230)
@X @ϕ
Λc
@S s 5 p : (3.231)
@ϕ 2 X
ffiffiffiffi