Non-saturated analysis

with diagrams

You can perform a non-saturated analysis by making the permeability dependent on the groundwater total pressure (= pore pressure) by a dependency diagram. The diagram accounts for high permeability at saturation and low permeability at non-saturation. For example, do something like:

...
dependency_item 10 -group_groundflow_permeability 0 -to_pres 4
dependency_diagram 10 -100. 0.0 0.05 100.
1.e-2 1.e-2 1.e-8 1.e-8
1.e-2 1.e-2 1.e-8 1.e-8
...
...

The atmospheric air pressure is 0 in tochnog, so that is where the permeability starts changing it's value in the table. You can also specify a table for group_groundflow_capacity to model non-saturated capacity.

van Genuchten

As an alternative to specifying diagrams you can use the specific van-Genuchten model for non-saturated ground water flow. The pore-pressure head is defined by

$$\phi_p = - \frac{p}{\rho g}$$

with $p$ the pore pressure (= total pressure), $\rho$ the ground water density and $g$ is the absolute value of the gravity acceleration (typically 9.81). De degree of saturation is a function of the pore-pressure head

$$S = S( \phi_p )$$

The total capacity is the sum of the saturated capacity and a non-saturated part:

$$c = c_{\rm sat} + n \frac{d S(\phi_p)}{d \phi_p}$$

where where $c_{\rm sat}$ is the saturated groundflow capacity as specified by group_groundflow_capacity and $n$ is the porosity specified by group_porosity. The total permeabilities $k_i$ are written as a relative factor of the saturated permeabilities

$$k_i = k_{\rm rel}(S) k_{sat,i}$$

where $k_i$ is the total permeability in direction $i$, $k_{\rm rel}(S)$ is a factor dependent on the saturation $S$ and $k_{sat,i}$ is the saturated permeability specified by group_groundflow_permeability.

Now for the van-Genuchten model we have

$$S(\phi_p) = S_{\rm residu} + (S_{\rm sat} - S_{\rm residu}) \left( 1 + (g_a \vert \phi_p \vert )^{g_n} \right) ^ {(1-g_n)/g_n}$$

which has the following model parameters: $S_{\rm residu}$ is the residual saturation, $S_{\rm sat}$ normally is 1.0 but may be less than 1.0 if in case of trapped air, and $g_a$ and $g_n$ are constants to be determined experimentally. The derivative of this law defines the additional non-saturated capacity as defined above. After definition of the effective saturation $S_e$

$$S_e = \frac{S - S_{\rm residu}}{S_{\rm sat} - S_{\rm residu}}$$

the relative permeability factor is defined as

$$k_{\rm rel}(S) = (S_e)^{g_l} \left( 1 - ( 1 - S_e^{g_n/(g_n-1)} ) ^ { (g_n - 1)/g_n } \right) ^ 2$$

To use the model you need to specify the saturated parameters group_groundflow_capacity and group_groundflow_permeability as usual, specify the porosity in group_porosity, specify specific van-Genuchten parameters in group_groundflow_nonsaturated_vangenuchten and initialise groundflow_saturation in the initialisation part.

Since the model is strongly linear it might be needed to specify a relaxation of, say, 0.1 with control_relaxation to obtain convergence.