8.2. Seawater Intrusion
273
In order to use the Galerkin method, the four equations are expressed as:
o
0
L 2 (C,p, Vx, v,,) == o)pVx) + oz(pv,,) = 0,
k op
L 3 (C,p, Vx , v,,) == Vx + - ~ = 0,
Jm ux
L 4 (C,p, Vx,vz) == v" + :n (~~ + pg) = o.
We mayadopt the following approximate solutions:
N
C(x, z, t) = L Cj(t)(MX, z),
j=l
N
p(x, z, t) = L pj(t)(MX, z),
j=l
N
Vx(x, z, t) = L VX,j(t)/h(x, z),
j=l
N
~(x,z,t) = L v"jt)#x,z),
j=l
(8.2.44a)
(8.2.44b)
(8.2.44c)
(8.2.44d)
(8.2.45a)
(8.2.45b)
(8.2.45c)
(8.2.45d)
where {(Mx, z), i = 1,2, ... , N} are basis functions. When the basis functions
are chosen, the problem of solving approximate solutions will become
a problem of determining 4 x N coefficients: {Ci(t), Pi(t), v",;(t), v",;(t)}, (i =
1,2, ... , N). According to the Galerkin method, these coefficients may be
determined by solving the 4 x N equations as follows
f r L1(C,p, v", ~)(/J;dxdz = 0,
J(Ql
f r L 2 (C,p, V x , Vz)(/J;dxdz = 0,
Jm
f r L 3 (C,p, V x , ~)(/J;dxdz = 0,
JQl
f r L 4 (C,p, v", ~)(/J;dxdz = O.
JQl
(i = 1,2, ... ,N)
(8.2.46a)
(8.2.46b)
(8.2.46c)
(8.2.46d)