4. Two-phase medium and flow

The coupled problem of the equilibrium of a soil in presence of transient underground flow conditions is solved using a mixed formulation in which displacement and total water pressure are the main variables. The definition of effective stresses (Eq.4.1) is modified to account for non saturated media:

^(4.1)

where the saturation ratio S is a function of the water pressure.

The two-phase coupled equilibrium equations are:

The first equation expresses equilibrium of the two-phase medium in total stresses and b_i = g_i, with = nS _w +(1-n) _s and n, the porosity; the second equation expresses transient continuity in the pore fluid and accounts for consolidation and flow. _kk is the volumetric strain time derivative in the skeleton, q the averaged (Darcy) fluid velocity and c the specific storage coefficient.

An appropriate set of boundary conditions must be prescribed. These include imposed surface loads in total stress t and imposed displacements (as on ₁). In addition, constant pressure values are prescribed where pressure is constant (as on ₂), flow across the boundary can be prescribed where necessary ( = 0 on ₁) and seepage boundary conditions are imposed on ₃ and ₄ to account for the fact that the free surface and H2 are functions of time. This means that seepage elements are introduced along these boundaries and external, time-dependent pressures are prescribed. This type of boundary condition will switch automatically to the appropriate one depending on the saturation in the medium.

Appropriate initial conditions must also be prescribed for the fluid in case of transient flow. In the situation described above, these correspond to the hydrostatic pressure in the medium, i.e to a water table following the upper surface of the medium.