Page 109 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

Formation Damage by Fines Migration: Mathematical and Laboratory Modeling, Field Cases 91

The typical form assumed for the filtration coefficient follows from
the principles of Langmuir adsorption curves (Wang et al., 2017; Yang
and Balhoff, 2017). Assuming that there are a fixed and finite number of
vacancies available for particle straining, the concentration of these vacan-
cies is denoted σ m . When there are no strained particles, the filtration
coefficient has some initial value λ 0 . As particles begin to strain, the like-
lihood of suspended particles encountering a vacancy decreases; therefore,
the filtration coefficient decreases. The filtration coefficient then takes the
following form:

σ s
λσ ðÞ 5 λ 0 1 2 : (3.25)
σ m
When the strained concentration is significantly smaller than the num-
ber of vacancies, the filtration coefficient will only be weakly dependent
on the strained concentration. In this case, it is often assumed that the
filtration coefficient is constant altogether, and so Eq. (3.24) simplifies to:
@σ s
5 λcU: (3.26)
@t
The result of the analysis of particle detachment presented in the pre-
vious section is the maximum retention function:
σ a 5 σ cr U; γð Þ; (3.27)

which characterizes the changes to the attached concentration based on
shifts in the torque balance. When equilibrium between the attached con-
centration and the maximum retention function is reached, no particle
detachment occurs. As discussed earlier, the finiteness of the detachment
rate is still a matter of debate. What is known is that once the attached
concentration has reached the value of the maximum retention function,
the detachment rate will be zero.
The mass balance equation for salt neglecting the diffusion/dispersion is:
@γ @γ
φ 1 U 5 0: (3.28)
@t @x
With given expressions for the straining and detachment rates, a
system of equations can be produced that allows solving for the three par-
ticle concentrations and salinity. The remaining component of modeling
fines migration is to determine the effect of the in-situ particles on the
permeability of the rock.