Page 115 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

Formation Damage by Fines Migration: Mathematical and Laboratory Modeling, Field Cases 97

Table 3.1 Solutions for suspended and strained concentrations, and impedance
ahead and behind the particle front
Line Term Zone Solution
1 X . αT e 2αΛT
C(X,T)
2 X , αT 0
3 X . αT 1 2 e 2αΛT
S s (X,T)
4 X , αT 1 2 e 2ΛX

5 T , α 1 1 1 βφΔσ cr 1 2 Λ 1 2 1 2 Λ 1 2 αT e 2αΛT
J(T) 1 h i
6 T . 1 e 2Λ
α 1 1 βφΔσ cr 1 2 Λ 2 Λ

The dimensionless pressure drop J(T), referred to as the impedance,
can be solved by integrating Eq. (3.45) directly using separation of
variables:

1 @P ð 1
ð
JTðÞ 5 2 dX 5 1 1 βφΔσ S s X; TÞdX: (3.54)
ð
@X
0 0
Substituting the expression for the strained concentration and integrat-
ing make it possible to express the impedance explicitly as:
" ! #
8
1 1 1
> 2αΛT
> 11βφΔσ cr 12 2 12 2αT e ; T ,
Λ Λ α
>
>
<
JT ðÞ5 ! : (3.55)
1 e 2Λ 1
>
11βφΔσ cr 12 1 T $
>
> ;
Λ Λ α
>
:
The two solutions correspond to times before and after the passing of
the suspended particle front. The latter of the two solutions is indepen-
dent of time, as after time 1/α, the core no longer contains any sus-
pended particles. Table 3.1 summarizes the solutions for suspended and
strained particle concentrations, and impedance.
3.3.3 Qualitative analysis of the solution
Fig. 3.11A shows the X-T plane with the two regions in which the solu-
tion is presented. Figs. 3.11B and C show the profiles of the suspended
and strained particle concentrations across the core. The profiles are taken
at the following moments: initially (T 5 0), and before and after perme-
ability stabilization (T a and T b , respectively).