Page 62 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

48 3 Algorithm for Simulating Coupled Problems in Hydrothermal Systems

c 2 − b 2 ξ A
η A = . (3.58)
b 3 + b 4 ξ A
(3) a 2 = 0, a 3 = 0

2
−bb ± bb − (4aa)cc
η A = , (3.59)
2aa
c 1 − a 3 η A
ξ A = , (3.60)
a 2
where

b 4 a 3
aa = , (3.61)
a 2
b 2 a 3 − b 4 c 1
bb = − b 3 , (3.62)
a 2
−b 2 c 1
cc = . (3.63)
a 2

3.2.2.3 Case 3: a 4 = 0, b 4 = 0
This case is very similar to case 2. The corresponding equations in this case are as
follows:

a 2 ξ A + a 3 η A + a 4 ξ A η A = c 1 , (3.64)

b 2 ξ A + b 3 η A = c 2 . (3.65)
Similarly, the solution to Equations (3.64) and (3.65) can be expressed in the
following three sub-cases:

(1) b 2 = 0, b 3 = 0

c 2
η A = , (3.66)
b 3
c 1 − a 3 η A
ξ A = . (3.67)
a 2 + a 4 η A
(2) b 2 = 0, b 3 = 0

c 2
ξ A = , (3.68)
b 2
c 1 − a 2 ξ A
η A = . (3.69)
a 3 + a 4 ξ A