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Hydraulic - Hydraulic constitutive model

*Material, ...
*Hydraulic = <hydraulic model> [,Swr=residual water saturation]
*<parameter 1>, ... , <parameter n>
...

Set this parameter equal to the hydraulic model to be used for the calculation of the degree of saturation. The subsequent lines \texttt{, ... , } depend on the model chosen. This parameter is only allowed for materials with more than one phase.

Optionally, a residual water saturation \(S^{wr}\) can be prescribed by adding \texttt{[,Swr=]} after the model specification. \(S^{wr}\) is given as fraction of 1, e.g. \(S^{wr}=0.32\). If this parameter is omitted, \(S^{wr}=0\) is assumed.

The different hydraulic models available in numgeo are described in the subsections accessible through the navigation bar on the left.

For the evaluation of the partial derivative \(d(S^w)/d(s)\) with respect to the suction three methods are available. The default setting is the usage of the smoothed derivative. To specify the method a second input line is added after the material parameters. The input takes the following form:

*Material, ...
*Hydraulic = <hydraulic model> [, Swr=<residual water saturation>] 
<parameter 1>, ... , <parameter n>
*Jacobian = 
...

The methods available in numgeo are:

  • Analytical derivative: The analytical derivative of the respective constitutive model for the current suction \(s\) is used. *Jacobian = Analytical

  • Smoothed representative of the analytical derivative: The analytical derivation is smoothed by evaluating the Jacobian over 10 substeps. This is the default option. *Jacobian = Smoothed

  • Chord-Slope approximation: A chord-slope approximation of the Jacobian for the current values of \(d (S^w)\) and \(d(s)\) is used. *Jacobian = Chord-Slope

  • Averaged analytical derivative: The analytical derivation is averaged for the current suction \(s\) and previous suction \(s_0\). *Jacobian = Time-Step-Centered

Theory

The saturation-suction relation interrelates the degree of saturation \(S^w\) with the capillary pressure \(p^c\) (also suction \(s\)) and in the simplest case links the degree of saturation via an injective function with the suction, i.e. \(S^w(s)\). Due to their simplicity most models are not capable of reproducing the hysteretic behaviour observed during wetting and drying cycles and have to be adjusted to either match the main drying or main wetting curve. In general, the degree of saturation \(S^w\) at time \(t + \Delta t\) may be expressed as a function of the degree of saturation \(S^{w,t}\) at time \(t\), the rate of capillary pressure \(\dot{p}^{c,t}\), the void ratio \(e\) and other state variables \(sv\).

\[\begin{equation*} S^w = S^w(S^{w,t},\dot{p}^{c,t},e,sv) \end{equation*}\]

The Jacobian tensor of the degree of saturation with respect to the solid strain \(J^{S,\varepsilon}_{ij}\) and the suction \(J^{S,pc}\) read:

\[\begin{equation*} \label{eq:swrc_deriv} J^{S,\varepsilon}_{ij} = \frac{\partial S^w}{\partial \varepsilon_{jk}} ~~~;~~~ J^{S,pc} = \frac{\partial S^w}{\partial p^c} \end{equation*}\]

Note that to account for the presence of residual water (\(wr\)) the effective degree of saturation \(S^e\)and its time derivative are evaluated and passed on by the hydraulic model. \(S^e\) is calculated via:

\[\begin{equation*} S^e = \frac{S^w-S^{wr}}{1-S^{wr}} \end{equation*}\]