8.3. Groundwater Quality Management Models
281
concentration in the river and the initial solute concentration in the aquifer
were all assumed to be 100 mg/L. For simplicity, the groundwater velocity
and the dispersion coefficient were assumed to be constants and unafTected
by pumping and injection. When sewage is injected, the solute concentration
in the aquifer will increase. The requirement of the pollution source management was to provide an optimal schedule for the sewage injection (the time
and quantity of each injection ditch) so that the total injection reaches the
maximum, and also, that the solute concentration in the source ditches
does not exceed 250 mg/L at any time. In this problem, the management
period was divided into three time intervals, each of which was 200 days, and
it was required that the management scheme should be obtained for each
time interval.
The governing equation of water quality for this problem is
oC = Do2C _ VoC + C'W
ot
ox 2
ox
bn'
(8.3.16)
where C' is the concentration of the injected water, W the injection rate per
unit area of aquifer, b the thickness of the aquifer, and n the efTective porosity.
The additional initial and boundary conditions are
C(x,O) = Co, 0 < x < L;
C(O,t) = Co, t > 0;
oC =0
0
ox
, x = L, t> .
Values of parameters used in the example are
L = 5000m,
V = 2 m/d;
D = 12m 2 /d, b = 10 m;
n = 0.2,
Co = 100 mg/L.
(8.3.17a)
(8.3.17b)
(8.3.17c)
This one-dimensional model can be solved by the FDM. The finite difTerence mesh of which is shown in Figure 8.9.
The injection rate, C' W, in the model is adecision variable to be determined. Assurne that the rates in the three injection ditches at the three time
intervals are ql,l> q2,l> ... , Q3,IIl> respectively, and are renumbered as Ql'
Q2' ... , Q9 for convenience. The objective of management is to maximize the
total injection, i.e.,
I
maxZ = L Qi'
i=l
(8.3.18)
where I = 9, which is equal to the number of injection ditches multiplied by
the number of the management intervals. The constraints are that the con-