Page 174 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

150 Thomas Russell et al.

To incorporate this, we propose that the attached concentration is
governed by:
0
ð
σ a x; tð Þ 5 σ cr Ux; tÞ; γ x; tÞÞ: (3.165)
ð
ð
0
where γ is a new variable, referred to as the pseudo-salinity, which is
related to the bulk fluid salinity by:
0
ð
γ x; t 1 τð Þ 5 γ x; tÞ: (3.166)
This parallels the initial introduction of delayed detachment given in
Section 3.6.1. It follows that using a Taylor’s series expansion, a relation-
ship can be derived for the pseudo-salinity:
@γ 0
τ 5 γ 2 γ : (3.167)
0
@t
The Eqs. (3.165) and (3.167) satisfy the condition that changes to the
fluid velocity result in an immediate change in the attached concentra-
tion, while changes to the fluid salinity have a delayed effect according to
a nonequilibrium relation.
The full system of equations can then be given:
@ αq @c
ð φc 1 σ s 1 σ a Þ 1 5 0; (3.168)
@t 2πr @r
αλcq
@σ s
5 ; (3.169)
@t 2πr

q
0
σ a 5 σ cr ; γ ; (3.170)
2πr
@γ q @γ
φ 1 5 0; (3.171)
@t 2πr @r
q k @p
52 : (3.172)
2πr μ 1 1 βσ s Þ @r
ð
Using the dimensionless coordinates outlined in Section 3.5,

2
qt r σ s σ a σ cr πkp
T- ; X- ; S s - ; S a - ; S cr - ; Λ-λr w ; P- ;
πr φ r e φ φ φ qμ
2
e
γ 2 γ qτ
Γ 5 J ; ε- :
γ 2 γ πr φ
2
I J e
(3.173)