Page 240 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

214 Bin Yuan and Rouzbeh G. Moghanloo

in Eq. (A.2) by the
Replacing the coefficients of C 1;t D and C 2;t D
numerators in Eq. (A.3) and substitution of total derivatives from
Eq. (A.6):

ð
ð λ 1 A 1 1 λ 2 A 2 ÞC 1;ξ 1 λ 1 C 1 1 λ 2 C 2 ÞC 2;ξ 1 λ 1 E 1 1 λ 2 E 2 Þx D;ξ 5 0
ð
(A.7)
in Eq. (A.2)
Alternatively, replace the coefficients of C 1;x D and C 1;x D
by the numerators in Eq. (A.3) and substitution of total derivatives from
Eq. (A.6):
ð
ð λ 1 B 1 1 λ 2 B 2 ÞC 1;ξ 1 λ 1 D 1 1 λ 2 D 2 ÞC 2;ξ 1 λ 1 E 1 1 λ 2 E 2 Þt D;ξ 5 0 (A.8)
ð
For system of hyperbolic partial differential equations, two roots to the
system of Eqs. (A.7-A.8) and two roots obtained from the system of Eq. (A.4)
form the four-coupled ordinary differential equations which may be
integrated simultaneously for (C 1 , C 2 )and ðx D ; t D Þ from an initial curve. The
system of four above equations (Eq. A.4, A.7 and A.8) are summarized as,
8

A 1 C 1;ξ 1 C 1 C 2;ξ 1 E 1 x D;ξ λ 1 1 A 2 C 1;ξ 1 C 2 C 2;ξ 1 E 2 x D;ξ λ 2 5 0
>
>

B 1 C 1;ξ 1 D 1 C 2;ξ 1 E 1 t D;ξ λ 1 1 B 2 C 1;ξ 1 D 2 C 2;ξ 1 E 2 t D;ξ λ 2 5 0
<
ð
ð A 1 σ 6 2 B 1 Þλ 1 1 A 2 σ 6 2 B 2 Þλ 2 5 0
>
>
:
ð C 1 σ 6 2 D 1 Þλ 1 1 C 2 σ 6 2 D 2 Þλ 2 5 0
ð
(A.9)
For nontrial value of two unknowns λ 1 and λ 2 , we can choose any
two equations from the system Eq. (A.9), and set the determinant of the
coefficient matrix be zero. The below system Eq. (A.10) is the determi-
nant of coefficient matrix of Eq. (A.9):
8
ð A 2 B 1 2 A 1 B 2 ÞC 1;ξ 1 A 2 C 1 2 A 1 C 2 Þσ 6 1 B 1 C 2 2 B 2 C 1 ފC 2;ξ
ð
ð ½
>
>
1 A 2 E 1 2 A 1 E 2 Þσ 6 1 B 1 E 2 2 B 2 E 1 ފx D;ξ 5 0
ð ½
>
ð
>
>
>
ð ½ A 1 C 2 2 A 2 C 1 Þσ 6 1 A 2 D 1 2 A 1 D 2 ފC 1;ξ 1 C 2 D 1 2 C 1 D 2 ÞC 2;ξ
>
> ð ð
>
>
1 C 2 E 1 2 C 1 E 2 Þσ 6 1 D 1 E 2 2 D 2 E 1 ފx D;ξ 5 0
<
ð ½
ð
ð ½
ð ½ A 2 B 1 2 A 1 B 2 Þσ 6 ŠC 1;ξ 1 A 2 D 1 2 A 1 D 2 Þσ 6 1 B 1 D 2 2 B 2 D 1 ފC 2;ξ
ð
>
>
1 A 2 E 1 2 A 1 E 2 Þσ 6 1 B 1 E 2 2 B 2 E 1 ފt D;ξ 5 0
>
> ð ½ ð
>
>
>
ð
ð
> ð ½ B 1 C 2 2 B 2 C 1 Þσ 6 1 B 2 D 1 2 B 1 D 2 ފC 1;ξ 1 C 2 D 1 2 C 1 D 2 ÞC 2;ξ
>
>
1 C 2 E 1 2 C 1 E 2 Þσ 6 1 D 1 E 2 2 D 2 E 1 ފt D;ξ 5 0
:
ð ½
ð
(A.10)
As for the system of Eq. (A.11), it is a coupled four-equation system
with four unknows, C 1;ξ C 2;ξ t D;ξ x D;ξ . To obtain nonzero solutions, the