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Flow Analysis

Transient flow

Transient flow analysis in a partially saturated medium is driven by the solution of a general Richard’s equation (equation of continuity):




material porosity


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 [1]. 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 requires 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].


[1] 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.

[2] M. Šejnoha, Finite element analysis in geotechnical design, to appear (2015).

[3] M. Šejnoha, T. Janda, H. Pruška, M. Brouček, Metoda konečných prvků v geomechanice: Teoretické základy a inženýrské aplikace, předpokládaný rok vydání (2015).