Page 112 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

94 Thomas Russell et al.

attached particles where the suspended particles are now transported by
the reduced velocity U s 5 αU:
@ @c
ð φc 1 σ s 1 σ a Þ 1 αU 5 0: (3.33)
@t @x
Similarly, as the rate of straining is proportional to particle flux, the
relationship for the straining rate becomes:
@σ s
5 λcαU: (3.34)
@t
The attached particle concentration is expressed by the maximum
retention function, which will vary with the bulk fluid velocity:
ðÞ:
σ a 5 σ cr U (3.35)
Finally, the effect of strained particles is incorporated into the modi-
fied Darcy’s law:

k 0 @p
U 52 : (3.36)
μ 1 1 βσ s Þ @x
ð
The four Eqs. (3.33 3.36) constitute a closed system of equations in
the four unknowns c, σ a , σ s , and p which describe the detachment,
migration, and straining of fine particles in porous media.
During a coreflood with velocity altered from U 0 to U 1 with
U 1 . U 0 , the critical retention function will decrease by an amount
Δσ 5 σ cr (U 0 ) σ cr (U 1 ). The detached concentration will immediately
enter the colloidal suspension. These conditions give the initial state of
the suspended and attached particle concentrations:
Δσ
t 5 0:c 5 ; σ a 5 σ cr U 1 Þ: (3.37)
ð
φ
As the process of straining is considered to be irreversible, any particles
that were strained prior to the injection period under consideration (i.e.,
prior to time t 5 0) can simply be considered as a part of the initial per-
meability k 0 . Doing so allows defining the initial condition for the
strained particle concentration as follows:
t 5 0:σ s 5 0: (3.38)
The boundary conditions correspond to the absence of suspended par-
ticles in the injected fluid:

x 5 0:c 5 0: (3.39)