Thermal conduction exchanges (Chapter 4) in solids and diffusion of contaminants (Chapter 6) are modelled by similar diffusion equations.
The balance equation is written:
дг/г + Q — S (11.6)
where / represents a flux of fluid or heat, Q represents a sink term and S represents the storage of fluid or of heat. When modelling a diffusion problem with the finite element method, the most often used formulation is based on fluid pore pressure, u, or on temperature, T.
Then the Darcy’s law for fluid flow in porous media gives the fluid flux (this equation has been presented in a slightly different form in Chapter 1 (Eq. 1.2) and in Chapter 2, Eqs. 2.15 and 2.16):
/і — — (дги + дг pgz) (11.7)
д
with the intrinsic permeability K (possibly depending on the saturation degree), the dynamic viscosity, д, the density, p, the fluid pressure, u, the altitude, z, and the gravitational acceleration, g. The fluid storage term, S, depends on the saturation degree, Sr, and on the fluid pressure (see Chapter 2, Section 2.7):
S — /n(u, Sr)
For thermal conduction one obtains Fourier’s law — see Eq. 4.1, rewritten here as:
fi = — kdiT (11.9)
with the conductivity coefficient, X. The heat storage (enthalpy) term depends on the temperature, T (Chapter 4, Section 4.4):
£ = fn(T) (11.10)
Diffusion of contaminant follows a similar law (Chapter 6, Section 6.3.1). The diffusion problem is non-linear when:
• the permeability depends (directly or indirectly) on the fluid pore pressure;
• the fluid storage is a non-linear function of the pore pressure;
• partial saturation occurs;
• the conductivity coefficient depends on the temperature; and
• the enthalpy is a non-linear function of the temperature.
When the storage term is considered, the time dimension of the problem has to be addressed.
