Page 139 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

118 Thomas Russell et al.

Dimensionless pressure drop across the core J, called impedance, is cal-
culated from Darcy’s law (Eq. (3.98)):
ð 1
JTðÞ 5 P 0; TÞ 2 P 1; TÞ 5 1 1 βφΔσ S s X; TÞdX: (3.110)
ð
ð
ð
0
Substituting the expression for strained concentration into Eq. (3.110)
and integrating in X yield:
" #
8
1 ð 12αÞ
>
> exp 2αΛTÞ21Þ1
11βΔσ T1 ð ð ð exp 2αΛTÞ21Þ ;T,1
ð
>
>
> Λ αΛ
>
>
>
>
>
>
>
"
>
>
> 1 ð 12αÞ
>
11βΔσ 11 ð ð
> exp 2αΛTÞ21Þ1
< Λ αΛ
JTðÞ5 :
! ! !#
>
>
exp 2αΛTÞ2exp αΛ
> 12T 1
>
> ð ; 1,T,
>
> 12α α
>
>
>
> " #
1 1
>
>
>
>
ð
> 11βΔσ 11 ð exp 2ΛÞ21Þ ; T.
Λ α
>
:
(3.111)
3.5.1.1 Qualitative analysis of the model
Figs. 3.16B, C, and D present profiles of suspended, strained, and attached
concentrations, respectively, in four different moments. Moment T 1 is
before the salinity front arrival at the outlet (X 5 1), T 2 5 1 PVI is the
exact moment when the injected salinity reaches the outlet, T 3 is before
the particle front arrival at the outlet, and T 4 is at some point afterward.
First, consider the behavior of profiles in Figs. 3.16B and C. Consider
two points X 1 , X 2 at the moment T 1 in zone 1. Suspended particles at
point X 1 are submitted to deep bed filtration longer than those at point
X 2 ; the released concentration C 2 behind the salinity front is constant.
Therefore, suspended concentration C at point X 1 is lower than that at
point X 2 . Suspended concentration monotonically increases along the
core in zone 1.
Straining at point X 1 has occurred for longer than at point X 2 , so the
strained concentration at point X 1 is higher than that at point X 2 . It fol-
lows that the strained concentration monotonically decreases along the
core in zone 1.