Page 146 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

Formation Damage by Fines Migration: Mathematical and Laboratory Modeling, Field Cases 125

Inserting Eq. (3.113) in the basic governing system mentioned above
yields the following system of equations in radial coordinates:
@ αq @c
ð φc 1 σ s 1 σ a Þ 1 5 0; (3.114)
@t 2πr @r
αλcq
@σ s
5 ; (3.115)
@t 2πr
q

σ a 5 σ cr ; γ ; (3.116)
2πr
@γ q @γ
φ 1 5 0; (3.117)
@t 2πr @r
q k @p
52 : (3.118)
2πr μ 1 1 βσÞ @r
ð
Introducing the following parameters:
2 γ2γ
qt r σ s σ a σ cr πkp J
T- ;X- ;S s - ;S a - ;S cr - ;Λ-λr w ;P- ; Γ 5 ;
πr e φ φ φ φ qμ γ 2γ
2
r e
I J
(3.119)
where r e is the drainage radius, X is the dimensionless radial coordinate,
and r w is the wellbore radius. This transforms the system
(Eqs. (3.114 3.118)) into the following dimensionless form:

ð
@ C 1 S s 1 S a Þ @C
1 α 5 0; (3.120)
@T @X
αΛC
@S s
ffiffiffiffi ffiffiffiffiffiffi ;
5 p p (3.121)
@T 2 X X w

q
S a 5 S cr p ffiffiffiffi ; Γ ; (3.122)
2πr e X
@Γ @Γ
1 5 0; (3.123)
@T @X
1 2 @P
52 ; (3.124)
X ð 1 1 βφS s Þ @X

Mass balance Eq. (3.120) for axisymmetric flow is the same as the
p ffiffiffiffi
one for linear flow, given by Eq. (3.42). Appearance of X in the