260
8. Applications of Groundwater Quality Models
Huyakom et al. (1983) adopted the upstream weighted FEM (see Section
6.2). Eq. (8.1.15) can be solved by either FEM or FDM.
The whole problem can be solved by an iteration procedure. Assume that
the solutions at time n are {C'}n (the concentration in porous blocks) and
{C} n (the concentration in fractures), and the solutions at time n + 1 are to be
solved. For this purpose, we first extrapolate an estimation {C'}n+l' and
substitute it into Eq. (8.1.13) to obtain r. Then, r is substituted into Eq.
(8.1.12) to yield {C}n+l. {C}n+l is taken as the concentration at the interface
and used as the boundary condition (8.1.16) of Eq. (8.1.15). The first iteration
is completed when we solve Eq. (8.1.15) to obtain {C'}n+1. For the next
iteration, the original estimation of {C'}n+l is replaced by the calculated
values. The whole iteration procedure is repeated until convergence is
reached.
Bibby (1981) made an assumption that the velocities in the porous blocks
may be neglected, i.e., let V' = 0, so that D' in Eqs. (8.1.13) and (8.1.15) can be
simplified to the molecular diffusion coefficient ofthe porous medium. Under
this condition, he obtained an analytic solution for C', and then coupled the
analytic solution for concentration in the blocks with the two-dimensional
form of Eq. (8.1.12) for mass transport in fractures. To solve the two-dimensional equation, he used the common Galerkin FEM. Bibby (1981) gave an
example of using the method in a limes tone aquifer located in the eastem
Kent district of England.
The dis advantage of the double-medium method is its undetermination in
physics. When we measure the concentration at a point, we can only obtain
one value and we do not know whether this value represents the concentration in the fractured medium, or in the porous medium. Thus, it is difficult to
solve the inverse problem with the double-medium model. The advantage of
the double-medium method is that the knowledge of the fracture structure is
not required.
3. Discrete Fracture Method
If there are some large fractures and solution grooves in the aquifer, the
assumption of continuum will not be valid. In that case, we should use the
discrete fracture method to consider the effects caused by these large fractures. The discrete fracture method is a numerical method in which mass
conservation equations are buHt up for fractured elements and porous elements, respectively. For instance, in Figure 8.4, the elements marked with
oblique lines are fractured elements, and the others are porous elements.
The former have large transmissivities and small storage coefficients, and the
latter have the opposite properties. Transport through fractured elements is
advection-dominated, while in porous elements it is dispersion-dominated.
The exchanges of water and solute between the two obey Darcy's law and
Fick's law. Therefore, the general goveming equation is still the advectiondispersion equation, differing only in that different parameters are taken
for different elements. As long as the fractures are divided into elements
8. Applications of Groundwater Quality Models
Huyakom et al. (1983) adopted the upstream weighted FEM (see Section
6.2). Eq. (8.1.15) can be solved by either FEM or FDM.
The whole problem can be solved by an iteration procedure. Assume that
the solutions at time n are {C'}n (the concentration in porous blocks) and
{C} n (the concentration in fractures), and the solutions at time n + 1 are to be
solved. For this purpose, we first extrapolate an estimation {C'}n+l' and
substitute it into Eq. (8.1.13) to obtain r. Then, r is substituted into Eq.
(8.1.12) to yield {C}n+l. {C}n+l is taken as the concentration at the interface
and used as the boundary condition (8.1.16) of Eq. (8.1.15). The first iteration
is completed when we solve Eq. (8.1.15) to obtain {C'}n+1. For the next
iteration, the original estimation of {C'}n+l is replaced by the calculated
values. The whole iteration procedure is repeated until convergence is
reached.
Bibby (1981) made an assumption that the velocities in the porous blocks
may be neglected, i.e., let V' = 0, so that D' in Eqs. (8.1.13) and (8.1.15) can be
simplified to the molecular diffusion coefficient ofthe porous medium. Under
this condition, he obtained an analytic solution for C', and then coupled the
analytic solution for concentration in the blocks with the two-dimensional
form of Eq. (8.1.12) for mass transport in fractures. To solve the two-dimensional equation, he used the common Galerkin FEM. Bibby (1981) gave an
example of using the method in a limes tone aquifer located in the eastem
Kent district of England.
The dis advantage of the double-medium method is its undetermination in
physics. When we measure the concentration at a point, we can only obtain
one value and we do not know whether this value represents the concentration in the fractured medium, or in the porous medium. Thus, it is difficult to
solve the inverse problem with the double-medium model. The advantage of
the double-medium method is that the knowledge of the fracture structure is
not required.
3. Discrete Fracture Method
If there are some large fractures and solution grooves in the aquifer, the
assumption of continuum will not be valid. In that case, we should use the
discrete fracture method to consider the effects caused by these large fractures. The discrete fracture method is a numerical method in which mass
conservation equations are buHt up for fractured elements and porous elements, respectively. For instance, in Figure 8.4, the elements marked with
oblique lines are fractured elements, and the others are porous elements.
The former have large transmissivities and small storage coefficients, and the
latter have the opposite properties. Transport through fractured elements is
advection-dominated, while in porous elements it is dispersion-dominated.
The exchanges of water and solute between the two obey Darcy's law and
Fick's law. Therefore, the general goveming equation is still the advectiondispersion equation, differing only in that different parameters are taken
for different elements. As long as the fractures are divided into elements