5. Much of the groundwater in areas with permafrost is brackish and impotable. Un-
lucky contamination of currently exploited aquifers that are potable can effectively
deprive communities of potable subterranean water supplies.
How these effects are incorporated in mathematical models of contaminant transport
can be outlined with the differential equations that form the basis of many of these models.
Water in a porous medium flows in a linearly proportional response to the hydrostatic
gradient across that medium. This observation has been formalized as Darcy's law, the
vector form of which is
k
(∇p - ρg∇z) = -K ∇h
v=-
(1)
ν
where v = Darcian flow velocity vector (m s1)
k = permeability tensor (m2)
ν = viscosity of fluid (Pa s)
p = fluid pressure (Pa)
ρ = fluid density (kg m3)
z = vertical distance above datum (m)
of m2 Pa1 s1
h = total head (Pa).
(The formulas presented here are from Mangold and Tsang [1991].) As will be discussed in
later sections, cold temperatures dramatically affect three of the parameters in eq 1: ν, p,
and K. The viscosity of the aqueous solution in freezing soils is affected directly by lower-
ing temperatures, which cause the viscosity of pure water to increase by an order of magni-
tude, and by solutes excluded from ice forming in the soil solution, which cause the viscos-
ity of the remaining liquid-water solution to be increased still further. The pressure of vici-
nal water in frozen porous media is controversial and an active area of basic research
(Zheng et al. 1991). It is clear that the liquid-pressure gradients accompanying differences
in temperature may be much more important than pressure gradients due to elevation or
water content (see, for example, Perfect et al. [1991]). Finally, as ice occupies some of the
pore space, the conductivity of the porous material is reduced.
Once the flow of water is established, the more challenging problem of solute flows
through porous media can be addressed. In many cases, the appropriate partial differential
equation to describe the transport of the solute j is
()
(
)
∂
∇ vc j - ∇ D ∇c j =
Φ c j + Qcj
(2)
∂t
= aqueous concentration of jth solute (mol m3)
where cj
= dispersivity tensor (m2 s1)
D
Φ
= porosity of porous medium (dimensionless)
= source-sink term for jth solute (mol m3 s1).
Qcj
As would be expected, most of the geochemical reactions that contribute to the terms Qcj
are affected by temperature. The most pronounced of these effects are on the chemical
thermodynamic state of the solutes and consequently on their solubility, miscibility, speci-
ation, and reactivity.
AQUATIC CHEMISTRY BELOW 0C
The transport of solutes and non-aqueous-phase liquids in any porous medium, frozen
or unfrozen, is directly affected by the physicalchemical properties of the solvents and
solutes in the system. These properties can be measured, but in many cases the careful
4
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