Page 133 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

Formation Damage by Fines Migration: Mathematical and Laboratory Modeling, Field Cases 113

Including the mass balance for solute concentration in the system of
Eqs. (3.33 3.36) presented in Section 3.3 allows for a full system of
equations with five equations and five unknowns, c, σ a , σ s , γ, and p.
Let us introduce the following dimensionless parameters and the full
system of equations for single-phase flow with varying salinity accounting
for fines mobilization due to salinity change in the dimensionless form.

x Ut c σ a σ s
X- ; T- ; C- ; S a - ; S s - ;
L φL Δσ φΔσ φΔσ

γ 2 γ

k 0 J
Δσ 5 S cr γ 2 S cr γ ; Λ-λL; P- p; Γ 5 ; (3.93)

I
J
UμL γ 2 γ
I J
@ @C
ð C 1 S s 1 S a Þ 1 α 5 0; (3.94)
@T @X
@S s
5 αΛC; (3.95)
@T
S a 5 S cr ðΓÞ; (3.96)
@Γ @Γ
1 5 0; (3.97)
@T @X
1 @P
1 52 ; (3.98)
1 1 βΔσφS s @X

where Γ is the dimensionless parameter for salinity.
Initial conditions correspond to an absence of suspended and strained
particles, salinity of the formation water, and an initial attached concen-
tration given by the value of the maximum retention function for the
reservoir conditions

T 5 0:C 5 0; Γ 5 1; S a 5 S cr γ ; S s 5 0: (3.99)
I
Boundary conditions correspond to injection of particle-free water
with given salinity:
X 5 0:C 5 0; Γ 5 0: (3.100)

The five Eqs. (3.94 3.98) subject to initial and boundary conditions
(Eqs. (3.99 and 3.100)) determine unknowns C, S a , S s , Γ, and P. The salt
transport Eq. (3.97) separates from the rest of the system, and as such can
be solved separately. Pressure P(X,T) is determined by integration of