Page 154 - Formation Damage during Improved Oil Recovery Fundamentals and Applications
P. 154
132 Thomas Russell et al.
Next, the suspension concentration released over the salinity front
2 1
X 5 X w 1 T is calculated. i.e., c (X,X X w ) c (X,X X w ) 5 [c].
Kinetics Eq. (3.121) yields the continuous strained concentration. The
salinity front speed is equal to unity. Particle mass balance on the salinity
front follows from condition (Eq. (3.106)):
σ a 5 α 2 1Þ c ½:
½ ð (3.135)
According to Eq. (3.122), the attached concentrations ahead and
behind the salinity front are:
8 !
q
>
< p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi;γ q
1
2
S a 5 S cr 2πr e X w 1T I ; U .U crI ; S a 5S cr p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi;γ J :
2πr e X w 1T
>
:
S aI; U ,U crI
(3.136)
The jump in attached concentration across the salinity front
X 5 X w 1 T is:
1 2
S a X; X w 1 Tð Þ 2 S a X; X w 1 TÞ
ð
8
0;
> X w , X , X mI
>
0 1
>
>
>
1 q
>
>
> ffiffiffiffi ; γ ;
φ
> S cr @ p A X mI , X , X mJ
> I
> 2πr e X
>
>
>
>
>
2 0 1 0 13
<
5 1 q q :
> 4 S cr @ p ffiffiffiffi ; γ I A 2 S cr @ p ffiffiffiffi ; γ J A5 ; X mJ , X , X crI
> φ
> 2πr e X 2πr e X
>
>
>
>
2 0 13
>
>
>
> 1 q
>
> ffiffiffiffi ; γ ; X crI , X , N
φ
> 4 S aI 2 S cr @ p J A5
>
> 2πr e X
:
(3.137)
Substituting Eq. (3.137) into the mass balance condition on the salin-
ity front (Eq. (3.135)) yields the expression for the suspended concentra-
tion c released by the salinity front
2 1
c X w 1Tð Þ2c X w 1TÞ
ð
q q
1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi;γ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi;γ 21
5 S a p I 2S a p J ð 12αÞ ;
2πr e X w 1T 2πr e X w 1T
(3.138)
