Soil moisture velocity equation
The first "advection-like" term was developed to simulate surface infiltration [4] and was extended to the water table,[5] which was verified using data collected in a column experimental that was patterned after the famous experiment by Childs & Poulovassilis (1962)[6] and against exact solutions.
The advection-like term of the Soil Moisture Velocity Equation is particularly useful for calculating the advance of wetting fronts for a liquid invading an unsaturated porous medium under the combined action of gravity and capillarity because it is convertible to an ordinary differential equation by neglecting the diffusion-like term.
[5] and it avoids the problem of representative elementary volume by use of a fine water-content discretization and solution method.
These three ODEs represent the dynamics of infiltrating water, falling slugs, and capillary groundwater, respectively.
of water in the vadose zone starts with conservation of mass for an unsaturated porous medium without sources or sinks: We next insert the unsaturated Buckingham–Darcy flux:[9] yielding Richards' equation[2] in mixed form because it includes both the water content
: Applying the chain rule of differentiation to the right-hand side of Richards' equation: Assuming that the constitutive relations for unsaturated hydraulic conductivity and soil capillarity are solely functions of the water content,
Employing the Implicit function theorem, which by the cyclic rule required dividing both sides of this equation by
Written in moisture content form, 1-D Richards' equation is[10] Where D(θ) [L2/T] is 'the soil water diffusivity' as previously defined.
as the dependent variable, physical interpretation is difficult because all the factors that affect the divergence of the flux are wrapped up in the soil moisture diffusivity term
Analytical and experimental results show that these assumptions are acceptable under most conditions in natural soils.
The first term on the right-hand side of the SMVE represents the two scalar drivers of flow, gravity and the integrated capillarity of the wetting front.
This term is responsible for the true advection of water through the soil under the combined influences of gravity and capillarity.
Neglecting gravity and the scalar wetting front capillarity, we can consider only the second term on the right-hand side of the SMVE.
In this case the Soil Moisture Velocity Equation becomes: This term is strikingly similar to Fick's second law of diffusion.
This is a physically realistic result because an equilibrium hydrostatic moisture profile is known to not produce fluxes.
Notably, sharp wetting fronts are notoriously difficult to resolve and accurately solve with traditional numerical Richards' equation solvers.
Comparing against exact solutions of Richards' equation for infiltration into idealized soils developed by Ross & Parlange (1994)[12] revealed[1] that indeed, neglecting the diffusion-like term resulted in accuracy >99% in calculated cumulative infiltration.
This result indicates that the advection-like term of the SMVE, converted into an ordinary differential equation using the method of lines, is an accurate ODE solution of the infiltration problem.
The advection-like term of the SMVE can be solved using the method of lines and a finite moisture content discretization.
These three ODEs are: With reference to Figure 1, water infiltrating the land surface can flow through the pore space between
Using the method of lines to convert the SMVE advection-like term into an ODE: Given that any ponded depth of water on the land surface is
, the Green and Ampt (1911)[13] assumption is employed, represents the capillary head gradient that is driving the flow in the
bin: This approach to solving the capillary-free solution is very similar to the kinematic wave approximation.
Therefore, in the context of the method of lines: and which yields: Note the "-1" in parentheses, representing the fact that gravity and capillarity are acting in opposite directions.
[6] Results of that validation showed that the finite water-content vadose zone flux calculation method performed comparably to the numerical solution of Richards' equation.
It is noteworthy that the SMVE advection-like term solved using the finite moisture-content method completely avoids the need to estimate the specific yield.
Calculating the specific yield as the water table nears the land surface is made cumbersome my non-linearities.
However, the SMVE solved using a finite moisture-content discretization essentially does this automatically in the case of a dynamic near-surface water table.
The paper on the Soil Moisture Velocity Equation was highlighted by the editor in the issue of J. Adv.
The paper describing the finite moisture-content solution of the advection-like term of the Soil Moisture Velocity Equation was selected to receive the 2015 Coolest Paper Award by the early career members of the International Association of Hydrogeologists.
