Flow in porous media 191
The last boundary condition implies that the toe of the saline wedge, propagating under the fresh water layer, does not reach the end of the modelled
control volume area.
7.2.3.2 Computational scheme of the governing equations
The form of the governing Equations 7.38 and 7.39 allows the decoupling
of the solution for the two basic variables h o and h u and the utilization of
the time derivatives of h o and h u appearing on the left-hand side of the two
equations for the explicit propagation of the solution from a time level (n) to
the next (n + 1). After discretization of the domain, the numerical solution
is accomplished on a staggered grid, where the seepage specific discharges
Q, are computed on the sides of a grid cell and the layers thickness h o , h u at
the cell’s centre. The numerical scheme first estimates the fluxes Q o and Q u
from known data at time n, and then computes the layers thickness h o and
h u at the new time step n + 1. The explicit FTCS (forward in time, central
in space) finite differences scheme is given by Equations 7.41 through 7.46:
Q
K
h
h
h
h
o i
o i
n
u i
n
b i
o i
n
u i
n
b i
,
,
,
,
,
,
,
= −
+
+
−
−
−
(
−
−
−
ζ
ζ
1
1
1 ) ) ⋅
+
(
)
−
∆x
h
h
o i
n
o i
n
,
, 1
2
(7.41)
Q
K
h
h
h
h
o i
o i
n
u i
n
b i
o i
n
u i
n
b
,
,
,
,
,
,
,
+
+
+
+
= −
+
+
−
−
−
1
1
1
1
ζ
ζ i i
oi
n
o i
n
x
h
h
(
) ⋅
+
(
)
+
∆
,
, 1
2
(7.42)
h
h
n
t
x
Q
Q
q
o i
n
o i
n
o i
o i
wo
,
,
,
,
(
)
+
+
= −
−
−
+
1
1
1 ∆
∆
(7.43)
Q
K
h
h
h
h
u i
o i
n
u i
n
b i
o i
n
u i
n
b i
,
,
,
,
,
,
,
= −
+
+
−
−
−
−
−
−
−
ζ
ζ
1
1
1
δ δ h
h
x
h
h
o i
n
o i
n
u i
n
u i
n
,
,
,
,
−
(
)
(
⋅
+
(
)
−
−
1
1
2
∆
(7.44)
Q
K
h
h
h
h
u i
o i
n
u i
n
b i
o i
n
u i
n
b
,
,
,
,
,
,
,
+
+
+
+
= −
+
+
−
−
−
1
1
1
1
ζ
ζ i i
o i
n
o i
n
u i
n
u i
n
h
h
x
h
h
−
−
(
)
(
⋅
+
(
)
+
+
δ ,
,
,
,
1
1
2
∆
(7.45)
h
h
n
t
x
Q
Q
q
u i
n
u i
n
u i
u i
wu
,
,
,
,
(
)
+
+
= −
−
−
+
1
1
1 ∆
∆
(7.46)
The last boundary condition implies that the toe of the saline wedge, propagating under the fresh water layer, does not reach the end of the modelled
control volume area.
7.2.3.2 Computational scheme of the governing equations
The form of the governing Equations 7.38 and 7.39 allows the decoupling
of the solution for the two basic variables h o and h u and the utilization of
the time derivatives of h o and h u appearing on the left-hand side of the two
equations for the explicit propagation of the solution from a time level (n) to
the next (n + 1). After discretization of the domain, the numerical solution
is accomplished on a staggered grid, where the seepage specific discharges
Q, are computed on the sides of a grid cell and the layers thickness h o , h u at
the cell’s centre. The numerical scheme first estimates the fluxes Q o and Q u
from known data at time n, and then computes the layers thickness h o and
h u at the new time step n + 1. The explicit FTCS (forward in time, central
in space) finite differences scheme is given by Equations 7.41 through 7.46:
Q
K
h
h
h
h
o i
o i
n
u i
n
b i
o i
n
u i
n
b i
,
,
,
,
,
,
,
= −
+
+
−
−
−
(
−
−
−
ζ
ζ
1
1
1 ) ) ⋅
+
(
)
−
∆x
h
h
o i
n
o i
n
,
, 1
2
(7.41)
Q
K
h
h
h
h
o i
o i
n
u i
n
b i
o i
n
u i
n
b
,
,
,
,
,
,
,
+
+
+
+
= −
+
+
−
−
−
1
1
1
1
ζ
ζ i i
oi
n
o i
n
x
h
h
(
) ⋅
+
(
)
+
∆
,
, 1
2
(7.42)
h
h
n
t
x
Q
Q
q
o i
n
o i
n
o i
o i
wo
,
,
,
,
(
)
+
+
= −
−
−
+
1
1
1 ∆
∆
(7.43)
Q
K
h
h
h
h
u i
o i
n
u i
n
b i
o i
n
u i
n
b i
,
,
,
,
,
,
,
= −
+
+
−
−
−
−
−
−
−
ζ
ζ
1
1
1
δ δ h
h
x
h
h
o i
n
o i
n
u i
n
u i
n
,
,
,
,
−
(
)
(
⋅
+
(
)
−
−
1
1
2
∆
(7.44)
Q
K
h
h
h
h
u i
o i
n
u i
n
b i
o i
n
u i
n
b
,
,
,
,
,
,
,
+
+
+
+
= −
+
+
−
−
−
1
1
1
1
ζ
ζ i i
o i
n
o i
n
u i
n
u i
n
h
h
x
h
h
−
−
(
)
(
⋅
+
(
)
+
+
δ ,
,
,
,
1
1
2
∆
(7.45)
h
h
n
t
x
Q
Q
q
u i
n
u i
n
u i
u i
wu
,
,
,
,
(
)
+
+
= −
−
−
+
1
1
1 ∆
∆
(7.46)