258
8. Applications of Groundwater Quality Models
better reflect the properties of fractured porous media, such as heterogeneity
and anisotropy. The dis advantages of the equivalent method are evident:
even if the details of the fracture structures are known, we still do not know
how to obtain the equivalent parameters, and moreover, when the effect of
fractures is very significant, a fractured medium may not be equivalent to a
porous medium at all.
2. Double-Medium Method
The fracture-pore medium composed of fractures and pores may be regarded
as a double-medium, which means that two kinds of continuum, the fractured
medium and the porous medium, exist simultaneously in the same spatial
region. Both have their own physical parameters. The fractured medium
has significant hydraulic conductivity and weak storage, and hence, large
velocities. On the contrary, the porous medium has weak conductivity and
significant storage, and hence, small velocities. Therefore, there are two
groups of quite different parameters related to the fractured and the porous
media at each spatial point. Meanwhile, there are exchanges of water and
solute between the two media.
It is easy to derive a mathematical expression for such an idealized physical model. The governing equation for solute transport in the fractured
medium is still the advection-dispersion equation:
oC
0 (OC)
OC
(1 -8)
at = OX i DijOXj - Vi OXi + I + - 8 - r, (i =j = 1,2,3) (8.1.12)
where Dij and V; are the components of the hydrodynamic dispersion coefficient and the average velocity, respectively, and Cis the concentration distribution, all of which are macroscopic average parameters; I is the source and
sink term, which may contain the pumping and injection of water, radioactive decay, chemical reactions, and adsorption. The last term on the righthand side ofEq. (8.1.12) denotes the exchange ofsolute between fractures and
pores, where 8 is the fracture porosity, i.e., the volume occupied by fractures
in per unit volume of aquifer; and r represents the rate of solute transport
from the fractured medium to the porous medium per unit volume of aquifer.
Thus, the dimensions of rare M/L 3 T.
In order to solve for r, Bibby (1981), Huyakorn et al. (1983) made an
idealized treatment to the geometrie structures of fractured media. They
assumed a structure of porous blocks and fractures like the one in Figure 8.2,
in which the flows in the porous blocks are one-dimensional, and the solute
enters into the fractures through the interfaces of the porous blocks. The
concentration distributions in the porous blocks are expressed as C'. From
the middle plane of each porous block, a local one-dimensional coordinate Z'
can be established, as shown in Figure 8.3. Then, from the definition of r, we
have
r = (V1C' - D10C:)1 ~ = !(V1C' - D 10C :) I ,(8.1.13)
oz z'=a2a'['d a
oz z'=a
8. Applications of Groundwater Quality Models
better reflect the properties of fractured porous media, such as heterogeneity
and anisotropy. The dis advantages of the equivalent method are evident:
even if the details of the fracture structures are known, we still do not know
how to obtain the equivalent parameters, and moreover, when the effect of
fractures is very significant, a fractured medium may not be equivalent to a
porous medium at all.
2. Double-Medium Method
The fracture-pore medium composed of fractures and pores may be regarded
as a double-medium, which means that two kinds of continuum, the fractured
medium and the porous medium, exist simultaneously in the same spatial
region. Both have their own physical parameters. The fractured medium
has significant hydraulic conductivity and weak storage, and hence, large
velocities. On the contrary, the porous medium has weak conductivity and
significant storage, and hence, small velocities. Therefore, there are two
groups of quite different parameters related to the fractured and the porous
media at each spatial point. Meanwhile, there are exchanges of water and
solute between the two media.
It is easy to derive a mathematical expression for such an idealized physical model. The governing equation for solute transport in the fractured
medium is still the advection-dispersion equation:
oC
0 (OC)
OC
(1 -8)
at = OX i DijOXj - Vi OXi + I + - 8 - r, (i =j = 1,2,3) (8.1.12)
where Dij and V; are the components of the hydrodynamic dispersion coefficient and the average velocity, respectively, and Cis the concentration distribution, all of which are macroscopic average parameters; I is the source and
sink term, which may contain the pumping and injection of water, radioactive decay, chemical reactions, and adsorption. The last term on the righthand side ofEq. (8.1.12) denotes the exchange ofsolute between fractures and
pores, where 8 is the fracture porosity, i.e., the volume occupied by fractures
in per unit volume of aquifer; and r represents the rate of solute transport
from the fractured medium to the porous medium per unit volume of aquifer.
Thus, the dimensions of rare M/L 3 T.
In order to solve for r, Bibby (1981), Huyakorn et al. (1983) made an
idealized treatment to the geometrie structures of fractured media. They
assumed a structure of porous blocks and fractures like the one in Figure 8.2,
in which the flows in the porous blocks are one-dimensional, and the solute
enters into the fractures through the interfaces of the porous blocks. The
concentration distributions in the porous blocks are expressed as C'. From
the middle plane of each porous block, a local one-dimensional coordinate Z'
can be established, as shown in Figure 8.3. Then, from the definition of r, we
have
r = (V1C' - D10C:)1 ~ = !(V1C' - D 10C :) I ,(8.1.13)
oz z'=a2a'['d a
oz z'=a