Page 105 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

Formation Damage by Fines Migration: Mathematical and Laboratory Modeling, Field Cases 87

For rectangular pores, the critical retention function can be related to
the critical cake thickness as per:
" #
2
h c
σ cr 5 1 2 12 1 2 φ φ: (3.17)
c
H
where φ c is the porosity of the internal filter cake formed by attached
particles. This parameter is often unknown and must be determined
empirically.
Removing the gravitational and lift forces from the torque balance
and substituting the resulting expression into Eq. (3.17) yield the expres-
sion for the critical retention function:
" p #
ffiffiffi 2 2
1 3ωπμr U
σ cr U; γ; pH; TÞ 5 1 2 1 s 1 2 φ φ: (3.18)
ð
c
2 2F e γ; pH; Tð ÞH
The above equation provides the qualitative descriptions of particle
detachment as outlined above. This equation predicts a decrease in σ cr
both with an increase in fluid velocity and a decrease in fluid salinity,
which will decrease the electrostatic force. Both of these dependencies
are monotonic. Typical forms of the critical retention function plotted
against the fluid velocity and fluid salinity are presented in Fig. 3.8.
Figs. 3.8A and B demonstrate the monotonic decline of σ cr both with
increases to velocity and decreases in salinity. Both forms also show the
highly nonlinear dependency of the critical retention function on the
injection conditions.

σ cr (U,γ) σ cr (U,γ)
Δσ cr,1 Δσ cr,1
Δσ Δσ cr,2
cr,2

(A) (B)
0 U 1 U 2 U 3 U max U 0 γ 3 γ 2 γ 1 γ
Figure 3.8 The critical retention function as a function of (A) the fluid velocity and
(B) the fluid salinity (σ cr : critical retention function, U: fluid velocity, γ: fluid salinity,
Δσ cr : detached particle concentration).