200
7. Mathematical Models of Groundwater Quality
The third consideration in model selection is the computation efl'ort. A
complex distributed parameter model requires solving a large system of
equations and thus needs much more computational effort than a lumped
parameter model does. Of course, with the modern computer technology, the
problem of calculation will no longer be the main factor for consideration.
Based on the above considerations, we often use the distributed parameter
model to study groundwater quality problems. If there is sufficient data for
model calibration, the distributed parameter model can produce an exact
distribution of solute concentration. Next, let us turn to the issue of wh at
kind of distributed parameter model should be used.
The first consideration is still the purpose of using the model. The
advection-dispersion model may depict the transition zone accurately because it involves the effects of hydrodynamic dispersion and it conforms
better to reality than the pure advection model does. For problems which
require high accurate solutions, e.g., to determine the concentration of a
poisonous substance in a weIl, we should not neglect the effect of the transition zone. In this ca se, we have to use the advection-dispersion model. However, if the ratio of the width of transition zone to the whole region is quite
smaIl, the effect of the transition zone on the whole concentration distribution is negligible, and thus the pure advection model may be used.
The second factor to be considered is still the data available. To build an
advection-dispersion model, we need extra data to determine the values of
dispersion coefficients. A transition zone calculated with incorrect dispersion
coefficients will not be reliable.
The third consideration is the computational effort. Compared with the
methods of characteristics and Random-Walk, the pure advection model
requires a lesser amount of calculation, because it only needs to solve the
advection part of water flow equation rat her than both the advection and
dispersion parts. However, the accuracy of the pure advection model is relatively low. When using the finite difference method, the computational effort
required to solve the pure advection model, which is a flow equation coupled
with an advection equation, is almost the same as that for the adve:ctiondispersion model. The numerical solution of a pure advection model wJill also
suffer from "numerical dispersion" and "overshoot."
The fourth factor we should consider is the degree of difficulty in solving
the inverse problem. Using observed data to calibrate the conceptual model
is a necessary step for building any applicable model. Observation of the
concentration is the main basis for calibrating a water quality model. When
a pure advection model is used, it is difficult to fit the observed concentration
values only by adjusting the advection parameters. On the other hand, when
an advection-dispersion model is used, the model output may fit the concentration observation quite weIl by adjusting both advection and dispersion
parameters.
With the above considerations, we conclude that generally we should
7. Mathematical Models of Groundwater Quality
The third consideration in model selection is the computation efl'ort. A
complex distributed parameter model requires solving a large system of
equations and thus needs much more computational effort than a lumped
parameter model does. Of course, with the modern computer technology, the
problem of calculation will no longer be the main factor for consideration.
Based on the above considerations, we often use the distributed parameter
model to study groundwater quality problems. If there is sufficient data for
model calibration, the distributed parameter model can produce an exact
distribution of solute concentration. Next, let us turn to the issue of wh at
kind of distributed parameter model should be used.
The first consideration is still the purpose of using the model. The
advection-dispersion model may depict the transition zone accurately because it involves the effects of hydrodynamic dispersion and it conforms
better to reality than the pure advection model does. For problems which
require high accurate solutions, e.g., to determine the concentration of a
poisonous substance in a weIl, we should not neglect the effect of the transition zone. In this ca se, we have to use the advection-dispersion model. However, if the ratio of the width of transition zone to the whole region is quite
smaIl, the effect of the transition zone on the whole concentration distribution is negligible, and thus the pure advection model may be used.
The second factor to be considered is still the data available. To build an
advection-dispersion model, we need extra data to determine the values of
dispersion coefficients. A transition zone calculated with incorrect dispersion
coefficients will not be reliable.
The third consideration is the computational effort. Compared with the
methods of characteristics and Random-Walk, the pure advection model
requires a lesser amount of calculation, because it only needs to solve the
advection part of water flow equation rat her than both the advection and
dispersion parts. However, the accuracy of the pure advection model is relatively low. When using the finite difference method, the computational effort
required to solve the pure advection model, which is a flow equation coupled
with an advection equation, is almost the same as that for the adve:ctiondispersion model. The numerical solution of a pure advection model wJill also
suffer from "numerical dispersion" and "overshoot."
The fourth factor we should consider is the degree of difficulty in solving
the inverse problem. Using observed data to calibrate the conceptual model
is a necessary step for building any applicable model. Observation of the
concentration is the main basis for calibrating a water quality model. When
a pure advection model is used, it is difficult to fit the observed concentration
values only by adjusting the advection parameters. On the other hand, when
an advection-dispersion model is used, the model output may fit the concentration observation quite weIl by adjusting both advection and dispersion
parameters.
With the above considerations, we conclude that generally we should