Page 164 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

142 Thomas Russell et al.

long stabilization times could not be justified by typical ranges of diffusion
coefficients found in the literature. To account for the atypically slow rate
of diffusion, these authors proposed the use of the Nernst-Planck (NP)
equation for diffusion of charged ionic species among charged surfaces
(Joekar-Niasar and Mahani, 2016; Zheng and Wei, 2011). The electro-
static influence of the charged surfaces coupled with the electrostatic gra-
dient induced by the ionic concentration gradient will oppose the ion
transfer and hence decrease the net ionic transfer rate with the bulk
solution.
The NP equation has so far not been applied to incorporate delayed
detachment explicitly into models for fines migration. Instead, authors
have opted for simplified relationships between the attached concentra-
tion and the critical retention function.
Suppose that the attached concentration is governed by the critical
retention function, but their equilibrium will occur only after some time
τ. This can be expressed as:

ð
ð
σ a x; t 1 τð Þ 5 σ cr U; γ x; tÞÞ: (3.144)
By taking a Taylor’s series expansion of the attached concentration at
time t, around time t 1 τ and discarding all but the first two terms in the
expansion, the following expression can be derived:
ð
τ @σ a x; tÞ 5 σ cr U; γ x; tðð ÞÞ 2 σ a x; tÞ: (3.145)
ð
@t
The parameter τ is referred to as the delay time. This equation will
replace Eq. (3.35) in the system of Eqs. (3.33 3.36, and 3.92) to incor-
porate the effect of nonequilibrium or delayed detachment into the sys-
tem for fines migration.

3.6.2 Exact solution for 1D problem accounting for delay
with detachment
The solution for the system of Eqs. (3.33, 3.34, 3.36, and 3.92) coupled
with the new nonequilibrium Eq. (3.145) for particle detachment will
mostly follow the solution procedure outlined in Section 3.5.1.
First, let us introduce the dimensionless delay factor:
Uτ
ε 5 : (3.146)
φL