Hydraulic models
Hydraulic models in numgeo define the constitutive relationship known as the Soil-Water Retention Curve (SWRC). This relationship governs the amount of water a porous medium can retain at a given level of suction. It fundamentally links the degree of saturation (\(S^w\)) to the capillary pressure (\(p^c\)) and the soil's structural state, typically represented by the void ratio (\(e\)) or volumetric strain (\(\varepsilon_{vol}\)).
The general form of this dependency is:
It is important to note that many of the implemented models are non-hysteretic. They represent a single path (either the main drying or main wetting curve) and do not, by default, capture the hysteretic phenomena observed during cyclic wetting and drying processes.
Available hydraulic models in numgeo are:
| Model | Dependence | Hysteresis |
|---|---|---|
| Van Genuchten | capillary pressure \(p^c\) | no |
| Brooks & Corey | capillary pressure \(p^c\) | no |
| Fredlund & Xing | capillary pressure \(p^c\) | no |
| Exponential model | capillary pressure \(p^c\) | no |
| Karlsruhe Model | capillary pressure \(p^c\), void ratio \(e\) | yes |
Effective Degree of Saturation
The effective degree of saturation (\(S^e\)) is a normalized, dimensionless quantity central to the mechanics of unsaturated soils. It quantifies the portion of the pore space occupied by mobile water, effectively scaling the saturation between its irreducible minimum and full saturation.
Its significance extends beyond the SWRC, as \(S^e\) serves as a critical state variable for other constitutive models. For instance, it governs:
- The relative permeability of the water and gas phases, which dictates multi-phase flow behavior.
- The Bishop's effective stress variable (\(\chi\)), which modulates the contribution of suction to the soil's mechanical strength and stiffness.
The effective saturation is computed from the total saturation (\(S^w\)) and the residual water saturation (\(S^{wr}\)), which is the fraction of water considered immobile due to strong adsorptive forces:
Numerical Implementation: Jacobian Contributions
For the implicit, fully-coupled numerical schemes employed in numgeo, the solver requires the partial derivatives of the constitutive relationships. The partial derivatives of the degree of saturation with respect to suction and void ratio, \(\frac{\partial S^w}{\partial s}\) and \(\frac{\partial S^w}{\partial e}\), are critical terms in the Jacobian matrix that influences the stability and convergence rate of the numerical simulation. numgeo provides several methods for its evaluation.
To specify which method to use, add the following command line to the material definition below the main model parameters:
*Jacobian = <method>
Where <method> is one of the options described below. If this line is omitted, the Smoothed method is used by default.
-
Analytical
This method uses the direct, analytical derivative of the selected SWRC model evaluated at the current suction, \(s\).
\[ \frac{\partial S^w}{\partial s} = \frac{d}{ds} f(s, e) \] -
Smoothed (Default)
This method provides a numerically smoothed derivative to enhance solution stability, especially where the SWRC is highly nonlinear. The analytical derivative is evaluated over 10 substeps within the current suction increment and then averaged.
-
Chord-Slope
This method approximates the derivative using a first-order finite difference (the chord slope) between the state at the beginning (\(t_0\)) and the end (\(t\)) of the current time step.
\[ \frac{\partial S^w}{\partial s} \approx \frac{\Delta S^w}{\Delta s} = \frac{S^w_{t} - S^w_{t_0}}{s_{t} - s_{t_0}} \] -
Time-Step-Centered
This method computes the derivative by averaging the analytical derivatives evaluated at the current suction (\(s_t\)) and the suction from the previous time step (\(s_{t_0}\)).
\[ \frac{\partial S^w}{\partial s} \approx \frac{1}{2} \left( \left. \frac{\partial S^w}{\partial s} \right|_{t} + \left. \frac{\partial S^w}{\partial s} \right|_{t_0} \right) \]