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The finite water-content vadose zone flux methodTalbot, C.A., and F. L. Ogden (2008), A method for computing infiltration and redistribution in a discretized moisture content domain, ''Water Resour. Res.'', 44(8), doi: 10.1029/2008WR006815.Ogden, F. L., W. Lai, R. C. Steinke, J. Zhu, C. A. Talbot, and J. L. Wilson (2015), A new general 1-D vadose zone solution method, ''Water Resour.Res.'', 51, doi:10.1002/2015WR017126. represents a one-dimensional alternative to the numerical solution of
Richards' equation The Richards equation represents the movement of water in unsaturated soils, and is attributed to Lorenzo A. Richards who published the equation in 1931. It is a quasilinear partial differential equation; its analytical solution is often limited ...
for simulating the movement of water in unsaturated soils. The finite water-content method solves the advection-like term of the Soil Moisture Velocity Equation, which is an
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
alternative to the Richards
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
. The Richards equation is difficult to approximate in general because it does not have a closed-form analytical solution except in a few cases.Ross, P.J., and J.-Y. Parlange (1994). Comparing exact and numerical solutions of Richards equation for one-dimensional infiltration and drainage. Soil Sci. Vol 1557, No. 6, pp. 341-345. The finite water-content method, is perhaps the first generic replacement for the numerical solution of the
Richards' equation The Richards equation represents the movement of water in unsaturated soils, and is attributed to Lorenzo A. Richards who published the equation in 1931. It is a quasilinear partial differential equation; its analytical solution is often limited ...
. The finite water-content solution has several advantages over the
Richards equation The Richards equation represents the movement of water in unsaturated soils, and is attributed to Lorenzo A. Richards who published the equation in 1931. It is a quasilinear partial differential equation; its analytical solution is often limited ...
solution. First, as an
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
it is explicit, guaranteed to converge Yu, H., C. C. Douglas, and F. L. Ogden, (2012), A new application of dynamic data driven system in the Talbot–Ogden model for groundwater infiltration, ''Procedia Computer Science'', 9, 1073–1080. and computationally inexpensive to solve. Second, using a finite volume solution methodology it is guaranteed to conserve mass. The finite water content method readily simulates sharp wetting fronts, something that the Richards solution struggles with. The main limiting assumption required to use the finite water-content method is that the soil be homogeneous in layers. The finite water-content vadose zone flux method is derived from the same starting point as the derivation of
Richards' equation The Richards equation represents the movement of water in unsaturated soils, and is attributed to Lorenzo A. Richards who published the equation in 1931. It is a quasilinear partial differential equation; its analytical solution is often limited ...
. However, the derivation employs a hodograph transformation to produce an advection solution that does not include soil water diffusivity, wherein z becomes the dependent variable and \theta becomes an independent variable: :\left(\frac\right)_\theta = \frac \left 1- \left (\frac\right) \right where: :K is the unsaturated
hydraulic conductivity Hydraulic conductivity, symbolically represented as (unit: m/s), is a property of porous materials, soils and rocks, that describes the ease with which a fluid (usually water) can move through the pore space, or fractures network. It depends on th ...
T−1 :\psi is the capillary
pressure head In fluid mechanics, pressure head is the height of a liquid column that corresponds to a particular pressure exerted by the liquid column on the base of its container. It may also be called static pressure head or simply static head (but not ''sta ...
(negative for unsaturated soil), :z is the vertical coordinate (positive downward), :\theta is the
water content Water content or moisture content is the quantity of water contained in a material, such as soil (called soil moisture), rock, ceramics, crops, or wood. Water content is used in a wide range of scientific and technical areas, and is expressed as ...
, (−) and :t is
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, t ...
This equation was converted into a set of three
ordinary differential equations In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...
(ODEs) using the Method of Lines to convert the
partial derivatives In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Par ...
on the right-hand side of the equation into appropriate
finite difference A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for th ...
forms. These three ODEs represent the dynamics of infiltrating water, falling slugs, and capillary groundwater, respectively.


Derivation

A superior derivation was published in 2017, showing that this equation is a diffusion-free version of the Soil Moisture Velocity Equation. One way to solve this equation is to solve it for q(\theta,t) and z(\theta,t) by integration: : \int \frac \, d\theta = \int \frac \, d\theta Instead, a finite water-content discretization is used and the integrals are replaced with summations: : \sum_^N \left frac\rightj \Delta \theta = \sum_^N \left frac\rightj \Delta \theta where N is the total number of finite water content bins. Using this approach, the conservation equation for each bin is: : \left frac\rightj = \left frac\rightj. The method of lines is used to replace the partial differential forms on the right-hand side into appropriate finite-difference forms. This process results in a set of three ordinary differential equations that describe the dynamics of infiltration fronts, falling slugs, and groundwater capillary fronts using a finite water-content discretization.


Method essentials

The finite water-content vadose zone flux calculation method replaces the
Richards' equation The Richards equation represents the movement of water in unsaturated soils, and is attributed to Lorenzo A. Richards who published the equation in 1931. It is a quasilinear partial differential equation; its analytical solution is often limited ...
PDE with a set of three
ordinary differential equations In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...
(ODEs). These three ODEs are developed in the following sections. Furthermore, because the finite water-content method does not explicitly include soil water diffusivity, it necessitates a separate capillary relaxation step. Capillary relaxation Moebius, F., D. Canone, and D. Or (2012), Characteristics of acoustic emissions induced by fluid front displacement in porous media, ''Water Resour. Res.'', 48(11), W11507, doi:10.1029/2012WR012525. represents a free-energy minimization process at the pore scale that produces no advection beyond the REV scale.


Infiltration fronts

With reference to Figure 1, water infiltrating the land surface can flow through the pore space between \theta_d and \theta_i. In the context of the method of lines, the partial derivative terms are replaced with: : \frac=\frac. Given that any ponded depth of water on the land surface is h_p, the Green and Ampt (1911) assumption is employed, : \frac=\frac, represents the capillary head gradient that is driving the flow. Therefore the finite water-content equation in the case of infiltration fronts is: : \left(\frac\right)_j= \frac \left(\frac+1\right).


Falling slugs

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 j-th \Delta\theta bin: : \left(\frac\right)_j = \frac


Capillary groundwater fronts

In this case, the flux of water to the j^\text bin occurs between bin ''j'' and ''i''. Therefore in the context of the method of lines: : \frac= \frac, and, : \frac = \frac which yields: : \left(\frac\right)_j= \frac \left(\frac-1\right). The performance of this equation was verified for cases where the groundwater table velocity was less than 0.92 K_s,Ogden, F. L., W. Lai, R. C. Steinke, and J. Zhu (2015b), Validation of finite water-content vadose zone dynamics method using column experiments with a moving water table and applied surface flux, ''Water Resour. Res.'', 10.1002/2014WR016454. using a column experiment fashioned after that by Childs and Poulovassilis (1962). Results of that validation showed that the finite water-content vadose zone flux calculation method performed comparably to the numerical solution of Richards' equation.


Capillary relaxation

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 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 Numerically, this process is a numerical sort that places the fronts in monotonically-decreasing magnitude from left-right.


Constitutive relations

The finite water content vadose zone flux method works with any monotonic
water retention curve Water retention curve is the relationship between the water content, θ, and the soil water potential, ψ. This curve is characteristic for different types of soil, and is also called the soil moisture characteristic. It is used to predict the ...
/unsaturated hydraulic conductivity relations such as Brooks and Corey Clapp and Hornberger and van Genuchten-Mualem.van Genuchten, M. Th. (1980). "A Closed-Form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils" (PDF). ''Soil Sci. Soc. Am. J.'', 44 (5): 892-898. doi:10.2136/sssaj1980.03615995004400050002x The method might work with hysteretic water retention relations- these have not yet been tested.


Limitations

The finite water content method lacks the effect of soil water diffusion. This omission does not affect the accuracy of flux calculations using the method because the mean of the diffusive flux is small. Practically, this means that the shape of the wetting front plays no role in driving the infiltration. The method is thus far limited to 1-D in practical applications. The infiltration equation was extended to 2- and quasi-3 dimensions. More work remains in extending the entire method into more than one dimension.


Awards

The paper describing this method was selected by the Early Career Hydrogeologists Network of the International Association of Hydrogeologists to receive the "Coolest paper Published in 2015" award in recognition of the potential impact of the publication on the future of hydrogeology.


See also

*
Richards' equation The Richards equation represents the movement of water in unsaturated soils, and is attributed to Lorenzo A. Richards who published the equation in 1931. It is a quasilinear partial differential equation; its analytical solution is often limited ...
*
Infiltration (hydrology) Infiltration is the process by which water on the ground surface enters the soil. It is commonly used in both hydrology and soil sciences. The infiltration capacity is defined as the maximum rate of infiltration. It is most often measured in meter ...
* Soil Moisture Velocity Equation


References

{{reflist, 2 Soil physics Hydrology Partial differential equations Ordinary differential equations