Page 104 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

86 Thomas Russell et al.

3.2.2.1 Internal filter cake of multilayer with
mono-sized fine particles
This model assumes that the drag, lift, gravitational, and electrostatic forces
describe the mechanical equilibrium for attached particles as outlined in
the previous section. Here, all attached particles are assumed to have the
same size and electrostatic properties. The pore space is modeled as a bun-
dle of square, parallel capillaries with the same size. An attached particle
with acting forces and corresponding lever arms is shown in Fig. 3.7.
Particles attach to the pore walls in discrete layers. As more particle
layers appear, the cross-sectional area within the pore that is available for
flow decreases. Thus, the flow velocity increases, increasing the drag and
lifting forces. This is captured in the equations for these two forces with
an inverse dependency on the pore size (Eqs. (3.7 and 3.8)). Therefore, as
more layers of particles appear, the condition for mechanical equilibrium
shifts further toward detachment. At some critical internal cake thickness,
h c , any additional layers of attached particles would be unstable and
detach. This cake thickness, when upscaled, defines the maximum con-
centration of particles that can be attached for given values of total fluid
velocity, fluid salinity, pH, etc.
The assumption of spherical particles, accompanied by the assumption
that the particles will arrange evenly on the pore space, makes it possible
to approximate the lever arm ratio l d /l n as O3. Given this assumption, the
torque balance can be evaluated as:
p ffiffiffi
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
4πr 3 ρμU 3 3ωπμr U
F e 1 s Δρg 2 χr 3 5 s : (3.16)
3 s ð H22h c Þ 3 H 2 2h c

Figure 3.7 Mechanical equilibrium of fines on the rock surface: drag, lift, electro-
static, and gravitational forces exerting on the particle at the cake surface
(Bedrikovetsky et al., 2011a).