Page 148 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

Formation Damage by Fines Migration: Mathematical and Laboratory Modeling, Field Cases 127

the maximum retention concentration are removed. In some region near
to the wellbore X w , X , X mI , the velocity is sufficient to detach all parti-
cles. At some distance far enough away from the wellbore, X . X crI ,no
particles are mobilized at all. These distances are defined by:
2 2 2

r w q q
X w 5 ; X mI 5 ; X mJ 5 ;
r e 2πU mI r e 2πU mJ r e
(3.125)
2

q
X crI 5 ; S cr U crI ; γ 5 S aI ;
I
2πU crI r e
where U mI is the maximum velocity for which no particles can be held
on the grain surface for initial water salinity, U mJ is the maximum velocity
for which no particles can be held on the grain surface for injected water
salinity, U crI is the critical particle detachment velocity for initial water
salinity, X mI is dimensionless radius where water velocity is equal to U mI ,
X mJ is dimensionless radius where the fluid velocity is equal to U mJ , X crI is
the dimensionless radius corresponding to velocity U crI , where initial
attached concentration is equal to the critical value; X w is the dimension-
less radial coordinate for r 5 r w . The relationship U 5 U(X) translates typi-
cal velocities U mI and U crI in Eq. (3.125) into positions where those
velocities are reached at X mI and X crI .
Fine particles detached due to high velocity near to the wellbore are
removed from the attached state into suspension at the beginning of
injection (t 510). Therefore, initial suspension is constant near to the
wellbore at X w , X , X mI , and zero outside the fines lifting zone for
X . X crI . The remaining attached concentration is equal to maximum
retention function in the intermediate zone X mI , X , X crI . The initial
conditions for the system (Eq. (3.120 3.124)) are:
(
0; X w , X , X mI
0 1
1 q
T 5 0: S a X; 0ð Þ 5 S cr @ p ffiffiffiffi ; γ I A ; X mI , X , X crI ;
φ 2πr e X
S aI =φ; X . X crI
; (3.126)
(
S aI =φ; X w , X , X mI
0 0 11
1 q
cX; 0Þ 5 @ S aI 2 S cr @ p ffiffiffiffi ; γ AA ; X mI , X , X crI
ð
φ 2πr e X I
0; X . X crI
Γ 5 1