Page 98 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

80 Thomas Russell et al.

lever arm can be calculated from the geometrical relationship between
p ffiffiffiffiffiffiffiffiffiffiffiffiffi
2
the particle radius and the normal lever arm, l d 5 r 2 l . For Young’s
2
s n
modulus typical for silica and kaolinite, corresponding to more frequent
cases of attached fines in sandstone rocks, the contact deformation radius
is very small (Kalantariasl and Bedrikovetsky, 2013), so:
r s .. l n ; l d Dr s ; (3.5)
Eq. (3.5) shows that in this case, the electrostatic force significantly
exceeds drag under the mechanical equilibrium.
Assuming the particle to be spherical, the gravitational force can be
expressed as:
4 3
F g 5 πr Δρg; (3.6)
s
3
where Δρ is the difference of densities between the particle and the
water, and g is the gravitational acceleration.
Drag and lift forces are the hydrodynamic forces exerting on the parti-
cle and mainly dependent on the carrier fluid velocity. Expressions for lift
and drag forces are:
s ffiffiffiffiffiffiffiffiffiffi
ρμu 3
F L 5 χr 3 ; (3.7)
s 3
r
p
2
ϖπμr u
F d 5 s ; (3.8)
r p
respectively, where χ is the lift coefficient, which Kang et al. (2004)
reported to be equal to 89.5 while Altmann and Ripperger (1997)
reported a value of 1190, ρ is the fluid density, μ is the fluid viscosity,
ω is the drag coefficient, and r p is the pore size. The geometrical shape
coefficients χ and ω can be calculated using computational fluid dynamics
modeling (CFD).
Eqs. (3.7) and (3.8) show that both of the hydrodynamic forces
increase with increasing fluid velocity, which will favor the conditions for
particle detachment from the surface.
The electrostatic force is calculated from the extended DLVO
(Derjaguin-Landau-Verwey-Overbeek) theory that takes into account the
interacting energies between the particles and the grain surface. The total
energy is the sum of the London-Van der Waals, Electrical Double Layer,