Finite water-content vadose zone flow method

Infiltration fronts in finite water-content domain.

With reference to Figure 1, water infiltrating the land surface can flow through the pore space between ${\displaystyle \theta _{d}}$ and ${\displaystyle \theta _{i}}$. In the context of the method of lines, the partial derivative terms are replaced with:

${\displaystyle {\frac {\partial K(\theta )}{\partial \theta }}={\frac {K(\theta _{d})-K(\theta _{i})}{\theta _{d}-\theta _{i}}}.}$

Given that any ponded depth of water on the land surface is ${\displaystyle h_{p}}$, the Green and Ampt (1911)[12] assumption is employed,

${\displaystyle {\frac {\partial \psi (\theta )}{\partial z}}={\frac {|\psi (\theta _{d})|+h_{p}}{z_{j}}},}$

represents the capillary head gradient that is driving the flow. Therefore the finite water-content equation in the case of infiltration fronts is:

${\displaystyle \left({\frac {dz}{dt}}\right)_{j}={\frac {K(\theta _{d})-K(\theta _{i})}{\theta _{d}-\theta _{i}}}\left({\frac {|\psi (\theta _{d})|+h_{p}}{z_{j}}}+1\right).}$

Falling slugs in the finite water-content domain. The water in each bin is considered a separate slug.

After rainfall stops and all surface water infiltrates, water in bins that contains infiltration fronts detaches from the land surface. Assuming that the capillarity at leading and trailing edges of this 'falling slug' of water is balanced, then the water falls through the media at the incremental conductivity associated with the ${\displaystyle j^{\text{th}}\ \Delta \theta }$ bin:

${\displaystyle \left({\frac {dz}{dt}}\right)_{j}={\frac {K(\theta _{j})-K(\theta _{j-1})}{\theta _{j}-\theta _{j-1}}}}$

Groundwater capillary fronts in finite water-content domain.

In this case, the flux of water to the ${\displaystyle j^{\text{th}}}$ bin occurs between bin j and i. Therefore in the context of the method of lines:

${\displaystyle {\frac {\partial K(\theta )}{\partial \theta }}={\frac {K(\theta _{j})-K(\theta _{i})}{\theta _{j}-\theta _{i}}},}$

and,

${\displaystyle {\frac {\partial \psi (\theta )}{\partial z}}={\frac {|\psi (\theta _{j})|}{H_{j}}}}$

which yields:

${\displaystyle \left({\frac {dz}{dt}}\right)_{j}={\frac {K(\theta _{j})-K(\theta _{i})}{\theta _{j}-\theta _{i}}}\left({\frac {|\psi (\theta _{j})|}{H_{j}}}-1\right).}$

The performance of this equation was verified for cases where the groundwater table velocity was less than 0.92 ${\displaystyle K_{s}}$,[13] using a column experiment fashioned after that by Childs and Poulovassilis (1962).[14] Results of that validation showed that the finite water-content vadose zone flux calculation method performed comparably to the numerical solution of Richards' equation.

Because the hydraulic conductivity rapidly increases as the water content moves towards saturation, with reference to Fig.1, right-most bins in both capillary groundwater fronts and infiltration fronts can "out-run" their neighbors to the left. In the finite water content discretization, these shocks[15] are dissipated by the process of capillary relaxation, which represents a pore-scale free-energy minimization process that produces no advection beyond the REV scale[11] Numerically, this process is a numerical sort that places the fronts in monotonically-decreasing magnitude from left-right.