# fronts.D.van_genuchten¶

fronts.D.van_genuchten(n=None, m=None, l=0.5, alpha=1.0, Ks=None, k=None, nu=1e-06, g=9.81, theta_range=0.0, 1.0)

Return a Van Genuchten moisture diffusivity function.

Given the saturated hydraulic conductivity $$K_S$$ and parameters $$\alpha$$, m, l, $$\theta_r$$ and $$\theta_s$$, the Van Genuchten moisture diffusivity function D is defined as:

$D(\theta)=\frac{(1-m)K_S}{\alpha m (\theta_s-\theta_r)} S_e^{l-\frac{1}{m}}\left((1-S_e^\frac{1}{m})^{-m} + (1-S_e^\frac{1}{m})^m - 2 \right)$

where:

$S_e = \frac{\theta-\theta_r}{\theta_s-\theta_r}$

and $$\theta$$ is water content.

In common usage, the m parameter is replaced with an n parameter so that $$m=1-1/n$$. This function supports either parameter.

Parameters
• n (float, optional) – n parameter in the Van Genuchten model. Must be >1. Either n or m must be given (but not both).

• m (float, optional) – m parameter in the Van Genuchten model. Must be strictly between 0 and 1. Either n or m must be given (but not both).

• l (float, optional) – Pore connectivity parameter. The default is 0.5.

• alpha (float, optional) – $$\alpha$$ parameter of the Van Genuchten model. The default is 1. Must be positive.

• Ks (None or float, optional) – $$K_S$$, the saturated hydraulic conductivity. Must be positive. If neither Ks nor k are given, the saturated hydraulic conductivity is assumed to be 1.

• k (None or float, optional) – Intrinsic permeability of the porous medium. Can be given in place of Ks, which results in the saturated hydraulic conductivity being computed using $$K_S = kg/\nu$$. Must be positive.

• nu (float, optional) – $$\nu$$, the kinematic viscosity of the wetting fluid. Only used if k is passed instead of Ks. Must be positive. Defaults to 1e-6, approximately the kinematic viscosity of water at 20°C in SI units.

• g (float, optional) – Magnitude of the gravitational acceleration. Only used if k is passed instead of Ks. Must be positive. Defaults to 9.81, the gravity of Earth in SI units.

• theta_range (sequence of two floats, optional) – ($$\theta_r$$, $$\theta_s$$), where $$\theta_r$$ is the minimum (also known as residual) water content and $$\theta_s$$ is the maximum water content. The default is (0, 1). $$\theta_s$$ must be greater than $$\theta_r$$.

Returns

D

Function to evaluate $$D$$ and its derivatives:

• D(theta) evaluates and returns $$D$$ at theta

• D(theta, 1) returns both the value of $$D$$ and its first derivative at theta

• D(theta, 2) returns the value of $$D$$, its first derivative, and its second derivative at theta

In all cases, the argument theta may be a single float or a NumPy array.

Return type

callable

Notes

The expression used is the one found in Van Genuchten’s original paper [1], but with the addition of the optional l parameter.

References

[1] VAN GENUCHTEN, M. Th. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Science Society of America Journal, 1980, vol. 44, no 5, p. 892-898.