Page 184 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

Formation Damage by Fines Migration: Mathematical and Laboratory Modeling, Field Cases 159

The kinetic rate of fines straining is proportional to the advective flux
of suspended fines:
@σ s
5 λðγ; σ s ÞcfU: (3.207)
@t
A modified Darcy’s law accounts for permeability damage due to
particle straining is given as:

q k rw ðs;γ;σ s Þ k ro ðs; γÞ @ p w kk ro ðs;γÞ @ σ WO γðÞ cos θJÞ
ð
52 k 1 2 p ffiffiffiffiffiffiffiffi :
2πr μ μ @ r k=φμ @ r
w o o
(3.208)

q k rw ðs; γ;σ s Þ k ro ðs;γÞ @ p w
52 k 1 : (3.209)
2πr μ μ @ r
w o
Introducing the following dimensionless variables:
2
qt r σ a σ s kπp w
T 5 ; X 5 ; Λ 5 λr e ; S a 5 ; S s 5 ; P w 5 ; (3.210)
πr φ φ φ qμ
2
e r e o
transforms the system of Eqs. (3.201, 3.204 3.207, and 3.209) in the
following dimensionless form:
@s @f
1 5 0; (3.211)
@T @X
@ðγsÞ @ γfðÞ
1 5 0; (3.212)
@T @X

q
S a 5 S cr p ffiffiffiffi ; γ ; (3.213)
2πr e X
@ @ cfðÞ
ð sc 1 S a 1 S s Þ 1 5 0; (3.214)
@T @X
@S s 1
5 p Λðγ; σ s Þcf ; (3.215)
ffiffiffiffi
@T 2 X
@P w 2 1
; (3.216)
5
@X 2 μ k rw ðs;γ;σ s Þ
ð
o μ w 1 k ro s; γÞ 2X
The result is a system of six equations which determines six
unknowns: saturation s, salinity γ, attached fines concentration S a , strained
fines concentration fines S s , suspended fines concentration c, and water