196
7. Mathematical Models of Groundwater Quality
head distribution. Inserting the head distribution into Eqs. (7.1.17) and
(7.1.18), we can obtain Qij and U i for each element.
The solute mass conservation equation within element i is
At {I QijCij + NiCN, + RiCR, - p;ci}
U1
= Ui(t + At)· Ci(t + At) - Ui(t)· Ci(t).
(7.1.20)
This equation implies the assumption that water bodies with different
qualities flow into element i and mix instantaneously. In this equation, Cij can
be defined in such a way: when water flows fromj to i, that is, Qij > 0, Cij = C j
is used; conversely, when water flows from i to j, i.e., Qij < 0, then Cij = Ci
should be used. We mayaiso take Cij as the weighted mean of Ci and Cj:
(7.1.21)
The value of ~ = 0.75 is suggested for Qij > 0, and ~ = 0.25 for Qij < o. This
factor acts as an "upstream weight." Substituting Eq. (7.1.21) into Eq. (7.1.20),
we obtain a finite difference equation for the concentration. For elements
located on the boundary, we must consider boundary conditions when establishing the balance equation (7.1.20). If the values of concentration on the
left-hand side ofEq. (7.1.20) are taken at time t, the solution scheme is explicit
and we can direcdy calculate the unknown concentration Ci(t + At) on the
right-hand side. If the values of concentration on the left-hand side are taken
at time t + At, the solution scheme is implicit. In this case, we must solve
a set of equations containing all the elements to obtain the concentration
distribution at t + At. Regardless of whether the explicit or the implicit
scheme is used, numerical dispersion cannot be avoided, and a "transition
zone" will be generated. The "upstream weight" in Eq. (7.1.21) may reduce
the oscillations of the numerical solution.
Pure advection models of water quality should also be divided into the
tracer case and the general case. For the tracer case, Qij and U i can be
obtained by solving Eqs. (7.1.17) to (7.1.19) for each time step. Then, inserting
them into Eq. (7.1.20), we can obtain the concentration distribution. The flow
equation and the quality equation are solved separately. For the general case,
the flow and quality equations must be solved iteratively in combination with
the state equations (7.1.5).
7.1.4 Lumped Parameter Models
The advection-dispersion models and pure advection models mentioned
above are all distributed parameter models. Although in the pure advection
model, the problem of determining dispersion coeflicients can be a voided,
input of the values of hydraulic conductivity and porosity, input of infiltration and artificial recharge factors for different locations, input of the
locations of pollution sources and their intensities, and input of boundary
7. Mathematical Models of Groundwater Quality
head distribution. Inserting the head distribution into Eqs. (7.1.17) and
(7.1.18), we can obtain Qij and U i for each element.
The solute mass conservation equation within element i is
At {I QijCij + NiCN, + RiCR, - p;ci}
U1
= Ui(t + At)· Ci(t + At) - Ui(t)· Ci(t).
(7.1.20)
This equation implies the assumption that water bodies with different
qualities flow into element i and mix instantaneously. In this equation, Cij can
be defined in such a way: when water flows fromj to i, that is, Qij > 0, Cij = C j
is used; conversely, when water flows from i to j, i.e., Qij < 0, then Cij = Ci
should be used. We mayaiso take Cij as the weighted mean of Ci and Cj:
(7.1.21)
The value of ~ = 0.75 is suggested for Qij > 0, and ~ = 0.25 for Qij < o. This
factor acts as an "upstream weight." Substituting Eq. (7.1.21) into Eq. (7.1.20),
we obtain a finite difference equation for the concentration. For elements
located on the boundary, we must consider boundary conditions when establishing the balance equation (7.1.20). If the values of concentration on the
left-hand side ofEq. (7.1.20) are taken at time t, the solution scheme is explicit
and we can direcdy calculate the unknown concentration Ci(t + At) on the
right-hand side. If the values of concentration on the left-hand side are taken
at time t + At, the solution scheme is implicit. In this case, we must solve
a set of equations containing all the elements to obtain the concentration
distribution at t + At. Regardless of whether the explicit or the implicit
scheme is used, numerical dispersion cannot be avoided, and a "transition
zone" will be generated. The "upstream weight" in Eq. (7.1.21) may reduce
the oscillations of the numerical solution.
Pure advection models of water quality should also be divided into the
tracer case and the general case. For the tracer case, Qij and U i can be
obtained by solving Eqs. (7.1.17) to (7.1.19) for each time step. Then, inserting
them into Eq. (7.1.20), we can obtain the concentration distribution. The flow
equation and the quality equation are solved separately. For the general case,
the flow and quality equations must be solved iteratively in combination with
the state equations (7.1.5).
7.1.4 Lumped Parameter Models
The advection-dispersion models and pure advection models mentioned
above are all distributed parameter models. Although in the pure advection
model, the problem of determining dispersion coeflicients can be a voided,
input of the values of hydraulic conductivity and porosity, input of infiltration and artificial recharge factors for different locations, input of the
locations of pollution sources and their intensities, and input of boundary