194
7. Mathematical Models of Groundwater Quality
mate location of the front at t + fl.t can be depicted. In fact, this procedure
has been used in the method 01 characteristics (MOC).
It should be noted that the location obtained by this method is only the
mean location of the front, and not the real spreading area of the tracer. The
real spreading area is always larger than the area defined by the front.
Consequently, the travel time of the pollutants from the source to a certain
place calculated by the location of the front only represents the mean travel
time. The estimate of the travel time is very useful in practice. For instance,
Cherry et al. (1973) studied the movement of a radioactive substance using
this method to see if the substance could pass through an aquifer and enter a
river from its disposal site. They obtained a velocity field based on the field
studies and the finite element analysis. There is a fault zone between the
disposal site and the river. Their calculation results show that the passing
time is about 100 years. Even if the uncertainty of the estimated transmissivity is taken into consideration, the travel time is stilliarger than 20 years.
Because of adsorption, the practical transport velocity of the radioactive
substance is only about 1/10 of that of the groundwater flow. Therefore, the
real traveI time should be at least 200 years. During this period, the concentration of the radioactive substance ( 90 Sr and 137CS) should have been reduced to a very low level by natural decay.
With respect to the study of travel time, the research by Kirkham and
Stotres (1978) and Cushman and Kirkham (1978) should be mentioned. They
studied the travel time of a solute from a single or multi-Iayer aquifer to a
weIl, where the flow field is defined by an analytic solution. Nelson (1978) also
adopted the concept of travel time in the study of regional groundwater
pollution problems. The transient flow fieId that he used was obtained by the
finite difference method.
2. The Method of Coupling the Flow and Water Quality Equations
To explain this method, let us consider the groundwater pollution problem
in a two-dimensional aquifer. Partition the flow region into several elements,
the shapes of which are usually rectangles or other polygons. Figure 7.3
shows a polygonal element i, where j is an neighboring element of it. To
obtain the water mass balance and solute mass balance equations with respect
to element i, the following balance factors are usually involved:
1. The flow rate Qij between element i and its adjacent elementj. It is positive
for inflow and negative for outflow. The solute concentration of Qij is
denoted by Cij;
2. The flow rate Ni from the unsaturated zone. The solute concentration of
Ni is denoted by CN .;
3. The flow rate R i oi artificial recharge. The solute concentration of Ri is
denoted by CR,;
4. The drainage rate P; from wells or springs in the element, the solute
concentration of which is just the concentration of the element, Ci.
7. Mathematical Models of Groundwater Quality
mate location of the front at t + fl.t can be depicted. In fact, this procedure
has been used in the method 01 characteristics (MOC).
It should be noted that the location obtained by this method is only the
mean location of the front, and not the real spreading area of the tracer. The
real spreading area is always larger than the area defined by the front.
Consequently, the travel time of the pollutants from the source to a certain
place calculated by the location of the front only represents the mean travel
time. The estimate of the travel time is very useful in practice. For instance,
Cherry et al. (1973) studied the movement of a radioactive substance using
this method to see if the substance could pass through an aquifer and enter a
river from its disposal site. They obtained a velocity field based on the field
studies and the finite element analysis. There is a fault zone between the
disposal site and the river. Their calculation results show that the passing
time is about 100 years. Even if the uncertainty of the estimated transmissivity is taken into consideration, the travel time is stilliarger than 20 years.
Because of adsorption, the practical transport velocity of the radioactive
substance is only about 1/10 of that of the groundwater flow. Therefore, the
real traveI time should be at least 200 years. During this period, the concentration of the radioactive substance ( 90 Sr and 137CS) should have been reduced to a very low level by natural decay.
With respect to the study of travel time, the research by Kirkham and
Stotres (1978) and Cushman and Kirkham (1978) should be mentioned. They
studied the travel time of a solute from a single or multi-Iayer aquifer to a
weIl, where the flow field is defined by an analytic solution. Nelson (1978) also
adopted the concept of travel time in the study of regional groundwater
pollution problems. The transient flow fieId that he used was obtained by the
finite difference method.
2. The Method of Coupling the Flow and Water Quality Equations
To explain this method, let us consider the groundwater pollution problem
in a two-dimensional aquifer. Partition the flow region into several elements,
the shapes of which are usually rectangles or other polygons. Figure 7.3
shows a polygonal element i, where j is an neighboring element of it. To
obtain the water mass balance and solute mass balance equations with respect
to element i, the following balance factors are usually involved:
1. The flow rate Qij between element i and its adjacent elementj. It is positive
for inflow and negative for outflow. The solute concentration of Qij is
denoted by Cij;
2. The flow rate Ni from the unsaturated zone. The solute concentration of
Ni is denoted by CN .;
3. The flow rate R i oi artificial recharge. The solute concentration of Ri is
denoted by CR,;
4. The drainage rate P; from wells or springs in the element, the solute
concentration of which is just the concentration of the element, Ci.