Page 155 - Formation Damage during Improved Oil Recovery Fundamentals and Applications
P. 155
Formation Damage by Fines Migration: Mathematical and Laboratory Modeling, Field Cases 133
2
c X w 1T;TÞ
ð
0 1
8
Λ p
> ffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
> X2
@ p ð A
> 12αÞX1αX w ; T,X mI 2X w
S aI =φ exp 2 ffiffiffiffiffiffi
>
X w
>
>
>
>
>
> 0 1
>
1 q
>
>
>
> ffiffiffiffi;γ
> S cr @ p A
> 12αÞφ I
ð 2πr e X
>
>
>
>
>
>
0 1
>
Λ
>
>
> p ffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
> X2
@ p ð A
> 1S aI =φ exp 2 ffiffiffiffiffiffi 12αÞX1αX w ; X mI 2X w ,T,T mI
>
> X w
>
>
>
>
> 0 1 0 0 11
>
1 q 1 q
>
>
>
> ffiffiffiffi;γ 1 ffiffiffiffi;γ
> S cr @ p I A @ S aI 2S cr @ p I AA
> 12αÞφ φ
ð 2πr e X 2πr e X
>
>
>
>
>
>
0 1
>
>
> Λ p
> ffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
> X2
@ p A
> ð 12αÞX1αX w ; T mI ,T,X mJ 2X w
exp 2 ffiffiffiffiffiffi
>
X w
>
<
5 2 0 1 0 13 0 0 11;
>
> 1 q q 1 q
>
> p ffiffiffiffi;γ p ffiffiffiffi;γ 1 p ffiffiffiffi;γ
> 4 S cr @ A 2S cr @ A5 @ S aI 2S cr @ AA
> 12αÞφ I J φ I
ð 2πr e X 2πr e X 2πr e X
>
>
>
>
>
> 0 1
>
Λ
>
> p
> ffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
> X2
@ p ð A
> exp 2 ffiffiffiffiffiffi 12αÞX1αX w ; X mJ 2X w ,T,X crI 2X w
>
> X w
>
>
>
>
> 2 0 13 0 0 11
>
1 q 1 q
>
>
>
> ffiffiffiffi;γ 1 ffiffiffiffi;γ
> 4 S aI 2S cr @ p J A5 @ S aI 2S cr @ p I AA
> 12αÞφ φ
> ð 2πr e X 2πr e X
>
>
>
>
>
0 1
>
>
> Λ p
> ffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
> @ p X2 ð 12αÞX1αX w A ; X crI 2X w ,T,T crI
exp 2 ffiffiffiffiffiffi
>
>
> X w
>
>
>
>
2 0 13
>
>
>
> 1 q
> ffiffiffiffi;γ
> ;
> 4 S aI 2S cr @ p J A5 T.T crI
> 12αÞφ
: ð 2πr e X
(3.139)
where,
X mJ 2 X w X mI 2 X w X crI 2 X w
T mJ 5 ; T mI 5 ; T crI 5 : (3.140)
1 2 α 1 2 α 1 2 α
Eq. (3.139) is the boundary condition for the solution of Eq. (3.132)
in zones 4 9in Fig. 3.21B. Applying the method of characteristics yields
explicit formulae for the suspended concentration in zones 4 to 9 (rows
11 to 16 in Table 3.8).
