Page 121 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

Formation Damage by Fines Migration: Mathematical and Laboratory Modeling, Field Cases 103

The rate of the particle capture by straining is given by the kinetics
equation:
@σ s
jj:
5 λcU (3.61)
@t
Darcy’s law accounts for permeability damage due to straining:

k @p
U 52 : (3.62)
μ 1 1 βσ s Þ @r
ð
As particle detachment is instantaneous, the initial condition for the
suspended concentration is the difference between the attached concen-
tration and the critical retention function:
8
σ ai
φ
> ; r w , r , r m
>
>
>
>
>
!
>
q
>
<
t 5 0 : cr; 0Þ 5 σ ai 2 σ a , (3.63)
ð
2πr
>
>
> ;
> r m , r , r i
> φ
>
>
>
:
0 r . r i
where
q
r i 5 : (3.64)
jj
2π U i
As before, the strained concentration begins at zero:
t 5 0:σ s 5 0: (3.65)
The boundary condition for this system is given at the boundary
between the damaged and undamaged zones:
r 5 r i :cr i ; tÞ 5 0; p 5 p i : (3.66)
ð
Introducing the following dimensionless variables:
2

r qt σ a σ cr σ s
X 5 ; T 5 ; Λ 5 λr e ; S a 5 ; S cr 5 S s 5 ;
r e πφr 2 φσ ai φσ ai φσ ai
e
c 2πkp
C 5 ; P 5 ; (3.67)
σ ai μqtðÞ