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8. Applications of Groundwater Quality Models
of the fresh water aquifer was the intrusion of marine water from adjacent
aquifers, in addition to the direct seawater intrusion.
Owing to the remarkable difference between the density of the seawater
and that of the fresh water, significant calculation errors would occur if the
seawater was considered as a tracer. Strictly speaking, the problem of seawater intrusion should refer to the general case mentioned in Section 7.1, i.e.,
the fluid flow may be affected by the change of fluid density. Therefore, the
water flow equation and water quality equation cannot be solved separately.
Instead, they must be solved iteratively through the state equations. The
Galerkin FEM for solving this kind of problem will be introduced below.
Let us consider a two-dimensional seawater intrusion problem on the xz
plane, where the x axis is horizontal and the z axis is vertically upward. For
the sake of simplicity, the media of aquifers are assumed to be isotropie. This
problem is governed by:
1. Dispersion equation
oe 0 (oe
oe) 0 (oe
oe)
7ft = ox D xx ox + D xz oz + oz D zx ox + D zz oz
o
0
- ox (vxc) - oz (~C),
(8.2.39)
2. Continuity equation of an incompressible fluid
[
O(P VJ o(p ~)J -
-----ax + ----az - 0,
3. Two movement equations
k op
V x = - - - ,
Iln OX
~ = _~(op + pg).
Iln OZ
4. State equation
p = Po + Ee,
(8.2.40)
(8.2.41)
(8.2.42)
(8.2.43)
where Po is the density of fresh water; E can be taken as 0.7; eis the salt
concentration determined by Eq. (8.2.39); and viscosity 11 is assumed to be
a constant.
The general procedure for solving these equations has been presented in
Section 7.1. Assurne that the solution at time t is known. First, estimate the
distribution of density p at time t + M, then simultaneously solve for the
distributions of p, Vx , ~ and e at time t + M. Finally, use Eq. (8.2.43) to
update the distribution of p. This procedure is repeated until convergence is
achieved. Thus, the major problem is how to simultaneously solve Eqs.
(8.2.39) to (8.2.42).
8. Applications of Groundwater Quality Models
of the fresh water aquifer was the intrusion of marine water from adjacent
aquifers, in addition to the direct seawater intrusion.
Owing to the remarkable difference between the density of the seawater
and that of the fresh water, significant calculation errors would occur if the
seawater was considered as a tracer. Strictly speaking, the problem of seawater intrusion should refer to the general case mentioned in Section 7.1, i.e.,
the fluid flow may be affected by the change of fluid density. Therefore, the
water flow equation and water quality equation cannot be solved separately.
Instead, they must be solved iteratively through the state equations. The
Galerkin FEM for solving this kind of problem will be introduced below.
Let us consider a two-dimensional seawater intrusion problem on the xz
plane, where the x axis is horizontal and the z axis is vertically upward. For
the sake of simplicity, the media of aquifers are assumed to be isotropie. This
problem is governed by:
1. Dispersion equation
oe 0 (oe
oe) 0 (oe
oe)
7ft = ox D xx ox + D xz oz + oz D zx ox + D zz oz
o
0
- ox (vxc) - oz (~C),
(8.2.39)
2. Continuity equation of an incompressible fluid
[
O(P VJ o(p ~)J -
-----ax + ----az - 0,
3. Two movement equations
k op
V x = - - - ,
Iln OX
~ = _~(op + pg).
Iln OZ
4. State equation
p = Po + Ee,
(8.2.40)
(8.2.41)
(8.2.42)
(8.2.43)
where Po is the density of fresh water; E can be taken as 0.7; eis the salt
concentration determined by Eq. (8.2.39); and viscosity 11 is assumed to be
a constant.
The general procedure for solving these equations has been presented in
Section 7.1. Assurne that the solution at time t is known. First, estimate the
distribution of density p at time t + M, then simultaneously solve for the
distributions of p, Vx , ~ and e at time t + M. Finally, use Eq. (8.2.43) to
update the distribution of p. This procedure is repeated until convergence is
achieved. Thus, the major problem is how to simultaneously solve Eqs.
(8.2.39) to (8.2.42).