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Using Nanofluids to Control Fines Migration in Porous Systems 181

nanoparticles detachment behaviors during the postflush of the brine
(Yuan et al., 2017a,c).
As nanoparticles pass through porous medium, they are adsorbed and
strained at the stagnant points on the pore-throat surfaces, which can be
confirmed by the reduction of effluent-nanoparticle concentrations
from the injection source (Zhang et al., 2013; Li et al., 2015). The mass-
balance equation of nanoparticles flowing through permeable media, con-
sidering their deposition onto rock grains and straining into pore-throats,
can be written as,
@C NP @C NP 1 @σ NP 1 @S NP
1 1 1 5 0 (4.1)
@x D @t D φ @t D φ @t D
x
where, x D 5 ; t D 5 q inj t .
L φAL
Nanoparticles straining rate can be expressed by classical filtration
kinetics (Gruedes et al., 2006; Massoudieh and Ginn, 2010). Until nano-
particles adsorption reaches the maximum retention concentration (Yuan,
2017a), the classic particle-capture kinetics can be applied to quantify the
transient attachment rates of nanoparticles (Vafai, 2005).

@S NP @σ NP
5 λ s C NP φL; 5 λ ad C NP φL (4.2)
@t D @t D
h 2 i
μr U 2
when, σ NP , σ NP;max ; σ NP;max1 5 1 2 ð NP Þ φð1 2 S or Þ.
2φr P F e;max y
During the stage of brine postflush, there are no changes of fluid
salinity. Hence, it can be concluded that any changes of nanoparticles
retention concentration are only attributed to the decrease of average
fluid density, because of the changes of nanoparticle concentrations. The
average fluid density is expressed as a weighted average of nanoparticles
and carrier water density:

ρ 5 ρ 1 2 C NP Þ 1 ρ NP C NP 5 ρ 1 C NP ρ NP 2 ρ w (4.3)

ð
w
w
The maximum retention concentration of nanoparticles becomes a
function of injected nanoparticles concentration. The detachment of
nanoparticles occurs instantly along with the abrupt changes of flowing
nanoparticles concentration. Thus, the mass-balance equation of nanopar-
ticles during the postflush of brine could be expressed as follows:

@C NP 1 σ NP;max1 2 σ NP;max2 @C NP
1 1 1 5 0 (4.4)
@x D φ C NP @t D