Transient flow analysis in a partially saturated medium is driven by the solution of a general Richard’s equation (equation of continuity):
rate of change of degree of saturation
coefficient of relative permeability
permeability matrix of fully saturated soil
gradient of total head
Time discretization of Richard’s equation is based on a fully explicit Picard’s iteration scheme . This corresponds to a hybrid formulation ensuring conservation of mass. Owing to the solution of a generally nonlienear problem, the analysis is performed incrementally. Standard Newton-Raphson iteration scheme is used to satisfy equilibrium conditions.
Note that speed and stability of the iteration process is influenced to a large extent by the choice of the material model (the way of calculating the coefficient of relative permeability Kr, degree of saturation S and the approximation of capacity term C = dS / dhp) in relation to the nonlinear properties of a given soil. A significantly nonlinear behavior is for example typical of sands where improperly prescribed initial conditions may cause numerical problems. Details can be found in [2,3].
Steady state flow
The steady state analysis assumes no change of the degree of saturation in time. The governing equation then reduces to:
Unlike transient flow, the analysis is therefore time independent and the introduction of the flow boundary conditions only. However, it is still a generally nonlinear problem (e.g. unconfined flow analysis) calling for the application of the Newton-Raphson iteration method. Details can be found for example in [2,3].
 M. A. Celia and E. T. Bouloutas, A general mass-conservative numerical solutionfor the unsaturated flow equation, Water Resources Research 26 (1990), no. 7, 1483-1496.
 M. Šejnoha, Finite element analysis in geotechnical design, to appear (2015).
 M. Šejnoha, T. Janda, H. Pruška, M. Brouček, Metoda konečných prvků v geomechanice: Teoretické základy a inženýrské aplikace, předpokládaný rok vydání (2015).