202
7. Mathematical Models of Groundwater Quality
eters (hydraulic conductivity and effective porosity) may propagate to the
identified dispersion parameters through the uncertainty in velocity. It is also
difficult to differentiate the effects of advection transport and macroscopic
dispersion transport. In this section, we will discuss the parameter identification problem of advection-dispersion equations by assuming that the velocity
field is given. In the next section, a general coupled groundwater flow and
mass transport inverse problem will be considered, in which flow and dispersion parameters are determined simultaneously based on both head and
concentration observations.
Assuming that:
1. We have a set of observations of concentration
C* = {Cf,C!, ... ,Cl},
(7.2.1)
which are the sampling concentrations of certain observation wells at
certain times, and which generally include measurement errors.
2. The unknown distribution parameters (e.g., the longitudinal or transverse
dispersivity) can be approximately expressed by a vector of finite dimension:
(7.2.2)
According to the prior information (such as the upper-bound or lowerbound estimations) of unknown parameters, we can define an admissible
set P ad ofp.
3. There is a simulation model M, which can output the time-space distribution of concentration for each given parameter p, including, of course, a
set of calculated concentrations corresponding to the observed concentrations (7.2.1):
C(p) = {Cl (p), C2 (p),···, Cdp)}·
(7.2.3)
Errors will inevitably exist because model M cannot depict the real physical process exactly. If M is a discrete finite difference or finite element
model, its output also contains calculation errors.
The purpose of parameter identification is to find a p* E P ad , such that the
model output C(p*) is "dosest" to the observations of C*. The simplest way
to find p* is the trial-error method. First of all, make a guess of an initial
value Po, input it into model M, and get a set of outputs C(Po). Then, analyze
the difference between the outputs and observed concentrations in Eq. (7.2.1).
An experienced modeler knows how to modify the parameter to reduce the
difference between C* and C(p) and obtain a modified parameter Pl. These
steps are repeated until no more improvement can be made. The advantage
of this process is that it is easy to operate and it does not need any programs
except the simulation model. Moreover, the physical intuition of the hydrogeologist can be fuHy utilized to ensure the reasonableness of the identified
parameters. However, this method is inefficient in computation because of
7. Mathematical Models of Groundwater Quality
eters (hydraulic conductivity and effective porosity) may propagate to the
identified dispersion parameters through the uncertainty in velocity. It is also
difficult to differentiate the effects of advection transport and macroscopic
dispersion transport. In this section, we will discuss the parameter identification problem of advection-dispersion equations by assuming that the velocity
field is given. In the next section, a general coupled groundwater flow and
mass transport inverse problem will be considered, in which flow and dispersion parameters are determined simultaneously based on both head and
concentration observations.
Assuming that:
1. We have a set of observations of concentration
C* = {Cf,C!, ... ,Cl},
(7.2.1)
which are the sampling concentrations of certain observation wells at
certain times, and which generally include measurement errors.
2. The unknown distribution parameters (e.g., the longitudinal or transverse
dispersivity) can be approximately expressed by a vector of finite dimension:
(7.2.2)
According to the prior information (such as the upper-bound or lowerbound estimations) of unknown parameters, we can define an admissible
set P ad ofp.
3. There is a simulation model M, which can output the time-space distribution of concentration for each given parameter p, including, of course, a
set of calculated concentrations corresponding to the observed concentrations (7.2.1):
C(p) = {Cl (p), C2 (p),···, Cdp)}·
(7.2.3)
Errors will inevitably exist because model M cannot depict the real physical process exactly. If M is a discrete finite difference or finite element
model, its output also contains calculation errors.
The purpose of parameter identification is to find a p* E P ad , such that the
model output C(p*) is "dosest" to the observations of C*. The simplest way
to find p* is the trial-error method. First of all, make a guess of an initial
value Po, input it into model M, and get a set of outputs C(Po). Then, analyze
the difference between the outputs and observed concentrations in Eq. (7.2.1).
An experienced modeler knows how to modify the parameter to reduce the
difference between C* and C(p) and obtain a modified parameter Pl. These
steps are repeated until no more improvement can be made. The advantage
of this process is that it is easy to operate and it does not need any programs
except the simulation model. Moreover, the physical intuition of the hydrogeologist can be fuHy utilized to ensure the reasonableness of the identified
parameters. However, this method is inefficient in computation because of