Page 110 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

92 Thomas Russell et al.

Assuming that the rock permeability is a function of both the strained and
attached concentrations, a Taylor’s series expansion can be used to derive:
k 0
2
5 1 1 β σ a 1 β σ s 1 O σ ; (3.29)
s
a
k σ a ; σ s Þ
ð
where the small magnitude of both concentrations typically allows
2
neglecting all terms of order σ and above.
Substituting Eq. (3.29) into Darcy’s law for flow in porous media pro-
duces the modified Darcy’s law accounting for permeability decline due
to both strained and attached particles (Bedrikovetsky et al., 2011a; Pang
and Sharma, 1997):
k @p
U 52 : (3.30)
μ c ðÞ 1 1 β σ s 1 β σ a Þ @x
ð
a
s
Note that the fluid viscosity is a function of the suspended particle
concentration. In almost all applications of fines migration, this depen-
dency is small and thus is neglected. The use of Darcy’s law requires the
assumption of steady state, laminar, and incompressible flow.
Usually, it is also assumed that the coating of grains by attached parti-
cles causes insignificant permeability damage compared to that caused by
the straining of particles in the pore throats. Therefore, β a ,, β s , and
the modified Darcy’s law can be expressed simply as:

k @p
U 52 : (3.31)
μ 1 1 β σ s Þ @x
ð
s
The mass balance equation, combined with expressions for the strain-
ing and detachment rates and the modified Darcy’s law, presents a closed
mathematical system capable of modeling the process of fines migration
in porous media and its effect on rock permeability.

3.3 FINES MIGRATION RESULTING FROM HIGH
FLUID VELOCITIES

In Section 3.2 it was shown that increasing fluid velocity can result
in particle detachment through an increase in the hydrodynamic drag
force. In the following section, the mathematical and laboratory modeling
of fines migration induced by high fluid velocities will be presented. In
Section 3.3.1 the concept of slow particle migration and the drift delay