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7. Mathematical Models of Groundwater Quality
and the state equations,
P = p(C,p), Ji = Ji(C,p),
(7.1.4)
where C denotes the solute concentration; p the pressure; P and Ji the density
and viscosity of the fluid; D the hydrodynamic dispersion coefficient; kij the
component of hydraulic conductivity tensor; n the effective porosity; Vi' V 2 ,
V 3 the three components of mean velocity V; and I the sourcejsink term. In
Eq. (7.1.3), Einstein's summation convention is used. For an incompressible
fluid of low concentration, we can use the following first-order approximation of Eq. (7.1.4):
P = Po + cx(C - Co), Ji = Jio + ß(C - Co),
(7.1.5)
where Co is a reference concentration, Po and Jio the density and viscosity at
Co, and cx and ß are constants.
Equations (7.1.1) to (7.1.4) contain seven equations and seven unknowns,
which are C, p, p, Ji, Vi' V 2 , and V 3 • With appropriate initial and boundary
conditions, those unknown functions can be uniquely determined. This set
of equations are called the hydrodynamic dispersion system or generalized
advection-dispersion model. For this model, the porous medium can be considered to be heterogeneous, anisotropic and having an arbitrary geometry.
Furthermore, the fluid may be inhomogeneous, the density and viscosity may
change with the solute concentration, and so forth. However, the flow velocity should not exceed the effective range ofDarcy's law and the fluid temperature should be approximately constant.
From both practical and computational considerations, it is very important to distinguish between the cases of a homogeneous fluid (p and Ji are
constants) and a heterogeneous fluid. If the solute concentration is extremely
low, the fluid may be regarded as homogeneous and the solute as an ideal
tracer. Therefore, the two cases are also called the tracer case and the !~eneral
case, respectively. The solution procedures for the two cases are greatly
different. It is simple for the tracer case and complex for the general case.
The Tracer Case
In this case, p and Ji are constants. Thus, the dispersion equation, continuity
equation, and kinetic equations have no effect on the state equations. The
solutions of these equations can be separated into two separate subproblems:
First, obtaining the velocity distribution from the continuity equation and
the kinetic equations, and secondly, substituting the velocity distribution into
the advection-dispersion equation to obtain the concentration distribution.
The flow chart in Figure 7.1 shows the process of computation.
The General Case
In this case, p and Ji are determined by the state equations. The changes of
the concentration may cause the values of p and Ji to change. Through the
7. Mathematical Models of Groundwater Quality
and the state equations,
P = p(C,p), Ji = Ji(C,p),
(7.1.4)
where C denotes the solute concentration; p the pressure; P and Ji the density
and viscosity of the fluid; D the hydrodynamic dispersion coefficient; kij the
component of hydraulic conductivity tensor; n the effective porosity; Vi' V 2 ,
V 3 the three components of mean velocity V; and I the sourcejsink term. In
Eq. (7.1.3), Einstein's summation convention is used. For an incompressible
fluid of low concentration, we can use the following first-order approximation of Eq. (7.1.4):
P = Po + cx(C - Co), Ji = Jio + ß(C - Co),
(7.1.5)
where Co is a reference concentration, Po and Jio the density and viscosity at
Co, and cx and ß are constants.
Equations (7.1.1) to (7.1.4) contain seven equations and seven unknowns,
which are C, p, p, Ji, Vi' V 2 , and V 3 • With appropriate initial and boundary
conditions, those unknown functions can be uniquely determined. This set
of equations are called the hydrodynamic dispersion system or generalized
advection-dispersion model. For this model, the porous medium can be considered to be heterogeneous, anisotropic and having an arbitrary geometry.
Furthermore, the fluid may be inhomogeneous, the density and viscosity may
change with the solute concentration, and so forth. However, the flow velocity should not exceed the effective range ofDarcy's law and the fluid temperature should be approximately constant.
From both practical and computational considerations, it is very important to distinguish between the cases of a homogeneous fluid (p and Ji are
constants) and a heterogeneous fluid. If the solute concentration is extremely
low, the fluid may be regarded as homogeneous and the solute as an ideal
tracer. Therefore, the two cases are also called the tracer case and the !~eneral
case, respectively. The solution procedures for the two cases are greatly
different. It is simple for the tracer case and complex for the general case.
The Tracer Case
In this case, p and Ji are constants. Thus, the dispersion equation, continuity
equation, and kinetic equations have no effect on the state equations. The
solutions of these equations can be separated into two separate subproblems:
First, obtaining the velocity distribution from the continuity equation and
the kinetic equations, and secondly, substituting the velocity distribution into
the advection-dispersion equation to obtain the concentration distribution.
The flow chart in Figure 7.1 shows the process of computation.
The General Case
In this case, p and Ji are determined by the state equations. The changes of
the concentration may cause the values of p and Ji to change. Through the