190 Computational Modelling in Hydraulic and Coastal Engineering
The modelled area extends from the coast to an upstream boundary
where the water table in the ground is assumed either constant (adequate
recharge from water infiltrating in the ground upstream of the hydrological basin) or of predefined slope (limited recharge from upstream). On the
coastal boundary the salt water depth is constant and the fresh water layer
becomes negligible. The two immiscible layers are characterized by their
thickness (h o and h u ) varying with time along the horizontal spatial directions (h o = h o (x,y,t)) and their constant densities ρ o and ρ u .
According to the Dupuit assumption, the specific discharges in the two
layers (of the almost horizontally flowing water) are related to the pressure
gradient along the horizontal dimensions. Gravity is also taken into consideration since the interface between the two layers and the impermeable
bottom of the aquifer are inclined (‘mild’ slopes are assumed) (Arvanitidou,
Katsifarakis and Koutitas 2012).
7.2.3.1 One-dimensional equations for saltwater intrusion
Application of the continuity equation to a control water column of base
Δx and Δy in both the upper and the lower layers lead to two PDEs relating
the time variation of the h o and h u with their spatial gradients, the aquifer
permeability K(x,y), and the soil porosity n (the void ratio).
In the simplest form, the model, in one spatial direction, reads
n
h
t
x
Kh
h h
x
q
o
o
o
u
b
wo
∂
∂
=
∂
∂
∂ + +
∂
−
(
)
ζ
(7.38)
n
h
t
x
Kh
h h
x
Kh
h
x
q
u
u
o
u
b
u
o
wu
∂
∂
=
∂
∂
∂ + +
∂
−
∂
∂
−
(
)
ζ
δ
(7.39)
where q wo and q wu are the local well discharges pumping or recharging
water from or to the upper or the lower layer, respectively, and δ is the relative density difference defined as
δ
ρ ρ
ρ
=
−
u
o
u
(7.40)
The boundary conditions at sides A and B of the control volume (Figure
7.10) are defined as
Left side boundary A: h oA = 0 and h uA = H u
Right side boundary B: h oB = H o or
∂
∂
=
h
x
defined
oB
and h uB = 0
The modelled area extends from the coast to an upstream boundary
where the water table in the ground is assumed either constant (adequate
recharge from water infiltrating in the ground upstream of the hydrological basin) or of predefined slope (limited recharge from upstream). On the
coastal boundary the salt water depth is constant and the fresh water layer
becomes negligible. The two immiscible layers are characterized by their
thickness (h o and h u ) varying with time along the horizontal spatial directions (h o = h o (x,y,t)) and their constant densities ρ o and ρ u .
According to the Dupuit assumption, the specific discharges in the two
layers (of the almost horizontally flowing water) are related to the pressure
gradient along the horizontal dimensions. Gravity is also taken into consideration since the interface between the two layers and the impermeable
bottom of the aquifer are inclined (‘mild’ slopes are assumed) (Arvanitidou,
Katsifarakis and Koutitas 2012).
7.2.3.1 One-dimensional equations for saltwater intrusion
Application of the continuity equation to a control water column of base
Δx and Δy in both the upper and the lower layers lead to two PDEs relating
the time variation of the h o and h u with their spatial gradients, the aquifer
permeability K(x,y), and the soil porosity n (the void ratio).
In the simplest form, the model, in one spatial direction, reads
n
h
t
x
Kh
h h
x
q
o
o
o
u
b
wo
∂
∂
=
∂
∂
∂ + +
∂
−
(
)
ζ
(7.38)
n
h
t
x
Kh
h h
x
Kh
h
x
q
u
u
o
u
b
u
o
wu
∂
∂
=
∂
∂
∂ + +
∂
−
∂
∂
−
(
)
ζ
δ
(7.39)
where q wo and q wu are the local well discharges pumping or recharging
water from or to the upper or the lower layer, respectively, and δ is the relative density difference defined as
δ
ρ ρ
ρ
=
−
u
o
u
(7.40)
The boundary conditions at sides A and B of the control volume (Figure
7.10) are defined as
Left side boundary A: h oA = 0 and h uA = H u
Right side boundary B: h oB = H o or
∂
∂
=
h
x
defined
oB
and h uB = 0