Page 238 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

212 Bin Yuan and Rouzbeh G. Moghanloo

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APPENDIX A: METHOD OF CHARACTERISTICS TO
SOLVE SYSTEM OF QUASILINEAR FIRST-ORDER
PARTIAL DIFFERENTIAL EQUATIONS (PDES)
Consider a general system of quasilinear first-order partial differen-
tial equations (PDEs) for two dependent variables (Rhee et al., 2001), C 1
and C 2 , with two independent variables, x D and t D :

@C 1 @C 1 @C 2 @C 2
L 1 5 A 1 1 B 1 1 C 1 1 D 1 1 E 1 5 0
@x D @t D @x D @t D
(A.1)
@C 1 @C 1 @C 2 @C 2
L 2 5 A 2 1 B 2 1 C 2 1 D 2 1 E 2 5 0
@x D @t D @x D @t D
where A 1 ; B 1 ; C 1 ; D 1 ; E 1 ; A 2 ; B 2 ; C 2 ; D 2 ; E 2 are given continuous func-
tions of C 1 ; C 2 , x D , and t D .
The above PDEs are homogeneous if E 1 ; E 2 5 0. They are also called
reducible although they are homogeneous and the coefficients are only func-
tions of C 1 and C 2 . For different cases of multiphase multicomponent flow
with adsorption and chemical reactions, the functions of E 1 ; E 2 depend on
the assumption of equilibrium or nonequilibrium reaction theory.
To ensure the characteristic lines of both variables C 1 and C 2 , along
the same direction, a linear combination of the above two equations is
defined, as L 5 λ 1 L 1 1 λ 2 L 2 to yield:

@C 1 @C 1 @C 2
ð
ð
L 5 λ 1 A 1 1 λ 2 A 2 Þ 1 λ 1 B 1 1 λ 2 B 2 Þ 1 λ 1 C 1 1 λ 2 C 2 Þ
ð
@x D @t D @x D
@C 2
ð
ð
1 λ 1 D 1 1 λ 2 D 2 Þ 1 λ 1 E 1 1 λ 2 E 2 Þ 5 0
@t D
(A.2)