Appendix C — Implicit linearization of hydraulic conductivities

An implicit linearization of hydraulic conductivities in the numerical elaboration of the Richards equation (Appendix D) requires expressions for the derivative of the conductivity to the pressure head:

Arithmetic mean:

\[ \frac{\partial K^{j+\kappa,\kappa,p}_{i-1/2}}{\partial h^{j+1,p}_{i-1}} = 0.5 \frac{\partial K^{j+1,p}_{i-1}}{\partial h^{j+1,p}_{i-1}} \tag{C.1}\]

\[ \frac{K^{j+\kappa,\kappa p}_{i-1/2}} {\partial h^{j+1,p}_i} = 0.5 \frac{\partial K^{j+1,p}_i}{\partial h^{j+1,p}_{i}}\\ \tag{C.2}\]

Weighted arithmetic mean:

\[ \frac{\partial K^{j+\kappa,\kappa,p}_{i-1/2}}{\partial h^{j+1,p}_{i-1}} = \frac{\Delta z_{i-1}}{\Delta z_{i-1}+\Delta z_i} \frac{\partial K^{j+1,p}_{i-1}}{\partial h^{j+1,p}_{i-1}} \tag{C.3}\]

\[ \frac{K^{j+\kappa,\kappa p}_{i-1/2}} {\partial h^{j+1,p}_i} = \frac{\Delta z_i}{\Delta z_{i-1}+\Delta z_i} \frac{\partial K^{j+1,p}_i}{\partial h^{j+1,p}_{i}} \tag{C.4}\]

Geometric mean:

\[ \frac{\partial K^{j+\kappa,\kappa,p}_{i-1/2}}{\partial h^{j+1,p}_{i-1}} = 0.5\left(\frac{K^{j+1,p}_i}{K^{j+1,p}_{i-1}}\right)^{0.5} \frac{\partial K^{j+1,p}_{i-1}}{\partial h^{j+1,p}_{i-1}} \tag{C.5}\] \[ \frac{\partial K^{j+\kappa,\kappa,p}_{i-1/2}}{\partial h^{j+1,p}_i} = 0.5 \left(\frac{K^{j+1,p}_{i-1}}{K^{j+1,p}_i}\right)^{0.5} \frac{\partial K^{j+1,p}_i}{\partial h^{j+1,p}_i} \tag{C.6}\]

Weighted geometric mean:

\[ \frac{\partial K^{j+\kappa,\kappa,p}_{i-1/2}}{\partial h^{j+1,p}_{i-1}} = \frac{\Delta z_{i-1}}{\Delta z_{i-1}+\Delta z_i} \left(\frac{K^{j+1,p}_i}{K^{j+1,p}_{i-1}}\right)^{\frac{\Delta z_i}{\Delta z_{i-1} + \Delta z_i}} \frac{\partial K^{j+1,p}_{i-1}}{\partial h^{j+1,p}_{i-1}} \tag{C.7}\] \[ \frac{\partial K^{j+\kappa,\kappa,p}_{i-1/2}}{\partial h^{j+1,p}_i} = \frac{\Delta z_i}{\Delta z_{i-1}+\Delta z_i} \left(\frac{K^{j+1,p}_{i-1}}{K^{j+1,p}_i}\right)^{\frac{\Delta z_{i-1}}{\Delta z_{i-1} + \Delta z_i}} \frac{\partial K^{j+1,p}_i}{\partial h^{j+1,p}_i} \tag{C.8}\]

Harmonic mean:

\[ \frac{\partial K^{j+\kappa,\kappa,p}_{i-1/2}}{\partial h^{j+1,p}_{i-1}} = \frac {0.5} {\left (\frac {0.5} {K^{j+1,p}_{i-1}} + \frac {0.5} {K^{j+1,p}_{i}} \right)^2 + \left( K^{j+1,p}_{i-1}\right )^2} \frac{\partial K^{j+1,p}_{i-1}}{\partial h^{j+1,p}_{i-1}} \tag{C.9}\]

\[ \frac{\partial K^{j+\kappa,\kappa,p}_{i-1/2}}{\partial h^{j+1,p}_i} = \frac {0.5} {\left (\frac {0.5} {K^{j+1,p}_{i-1}} + \frac {0.5} {K^{j+1,p}_{i}} \right)^2 + \left( K^{j+1,p}_{i}\right )^2} \frac{\partial K^{j+1,p}_i}{\partial h^{j+1,p}_i} \tag{C.10}\]

Weighted harmonic mean:

\[ \frac{\partial K^{j+\kappa,\kappa,p}_{i-1/2}}{\partial h^{j+1,p}_{i-1}} = \frac {\frac{\Delta z_{i-1}} {\Delta z_{i-1} + \Delta z_{i}}} {\left (\frac {\frac{\Delta z_{i-1}} {\Delta z_{i-1} + \Delta z_{i}}} {K^{j+1,p}_{i-1}} + \frac {\frac{\Delta z_{i}} {\Delta z_{i-1} + \Delta z_{i}}} {K^{j+1,p}_{i}} \right)^2 + \left( K^{j+1,p}_{i-1}\right )^2} \frac{\partial K^{j+1,p}_{i-1}}{\partial h^{j+1,p}_{i-1}} \tag{C.11}\]

\[ \frac{\partial K^{j+\kappa,\kappa,p}_{i-1/2}}{\partial h^{j+1,p}_i} = \frac {\frac{\Delta z_{i}} {\Delta z_{i-1} + \Delta z_{i}}} {\left (\frac {\frac{\Delta z_{i}} {\Delta z_{i-1} + \Delta z_{i}}} {K^{j+1,p}_{i}} + \frac {\frac{\Delta z_{i-1}} {\Delta z_{i-1} + \Delta z_{i}}} {K^{j+1,p}_{i-1}} \right)^2 + \left( K^{j+1,p}_{i}\right )^2} \frac{\partial K^{j+1,p}_i}{\partial h^{j+1,p}_i} \tag{C.12}\]

A relation for the conductivity derivative to the pressure head \(\frac{\partial K}{\partial h}\) is given by:

\[ \frac{\partial K}{\partial h} = \frac{\partial K}{\partial S_e} \frac{\partial S_e}{\partial h} \tag{C.13}\]

Where for the Mualem (1976) \(K(S_e)\) relationship:

\[ \frac{\partial K}{\partial S_e} = K_{sat}S_e^{\lambda-1}\left[1-\left(1-S_e^\frac{1}{m}\right)^m\right] \left[\lambda+\left(1-S_e^{\frac{1}{m}}\right)^{m-1}\left(\left(2+\lambda\right)S_e^{\frac{1}{m}}-\lambda\right)\right] \tag{C.14}\]

and:

\[ \frac{\partial S_e}{\partial h} = \frac{C}{\theta_{sat}-\theta_{res}} \tag{C.15}\]

The coefficients of the Jacobian are given by Equation D.1

Mualem, Y. 1976. “A New Model for Predicting the Hydraulic Conductivity of Unsaturated Porous Media.” Journal Article. Water Resources Research 12: 513–22. https://doi.org/10.1029/WR012i003p00513.