Retention and Transfer of Pollution by Hydrocarbons: Mechanisms and Numerical Modeling - Environmental Geomechanics - page 397

14.3.4. Pollutant transported by water or air in the soil pores

For the case where hydrocarbons are transported in the soil pores by water
and/or air we have to take into account the following equations:

− a linear momentum balance equation of the mixture of the [14.11] type;

− a continuity equation for water [14.12] and for air of the [14.14] type;

− an energy balance equation;

− a solute conservation equation in water;

− a solute conservation equation in air.

This is the most complete model for coupled thermal and hygral transport of
soluble contaminants in a partially saturated porous medium. Simplifications are
always possible, as in the case of a rigid porous medium where equation [14.11] can
be neglected, or in the isothermal case where the energy balance equation is not
necessary. We have introduced this last equation because thermal variations may
produce fluid movements in the soil and these movements produce contaminant
transport, as will be shown in the last example.

The energy balance equation is hence written as:

∂ T

( ) ρC − div λ grad T

p eff

∂ t

rw rg

⎡ k k k k ⎤

( )

w
p

w g c w g

p

gw g

− C ρ grad p − grad p − p g + C ρ grad p ⋅ grad T

⎢ ⎣ ⎥ ⎦ w g

µ µ

⎡ ⎤

⎛

rw

⎞

∂ k k

( ) ( ) ⎥ ⎦

= ∆h nS ρ

w

+ S ρ

w

div v

s

− div

⎜ ⎜
⎝

ρ

w

grad p

g

− grad p

c

− p

w

g

⎟ ⎟
⎠

⎢ ⎣ vap w w

∂ t µ

w

[14.17]

where:

( ) s

w w

p

g g

p

s

ρC = n S ρ C + n S ρ C + 1 − n ρ C

p w g

[14.18]

is the heat capacity of the mixture, λ eff its thermal conductivity, T the temperature,
∆h vap the enthalpy of evaporation, and g the gravity acceleration. The solute
conservation equation in water is: