Three-dimensional analytical solutions of Richards’ equation for a box-shaped soil sample with piecewise-constant head boundary conditions on the top

Tracy, Fred T.

doi:10.1016/j.jhydrol.2007.01.011

Cited by 45 publications

(29 citation statements)

References 12 publications

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“…The challenge is finding a form of relative hydraulic conductivity such that the nonlinear Richards' equation can be converted from a nonlinear to a linear form. The derivations presented here are mirrored after those presented earlier (Tracy, 2006(Tracy, , 2007 because they lend themselves to onedimensional (1-D), 2-D, and three-dimensional (3-D) solutions. First, 1-D and 2-D analytical solutions will be derived, and then numerical finite element solutions highlighting accuracy for different representations of relative hydraulic conductivity will be investigated.…”

Section: Analytical Solutionsmentioning

confidence: 53%

Analytical and Numerical Solutions of Richards' Equation with Discussions on Relative Hydraulic Conductivity

Tracy¹

2011

Hydraulic Conductivity - Issues, Determination and Applications

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Section: Analytical Solutionsmentioning

confidence: 53%

Analytical and Numerical Solutions of Richards' Equation with Discussions on Relative Hydraulic Conductivity

Tracy¹

2011

Hydraulic Conductivity - Issues, Determination and Applications

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“…As shown below in Equation (31), P is related to the van Genuchten parameter m. Assuming P = 1, the RE is reduced to a linearized form for which analytical solutions to various flow processes exist [16][17][18]. However, the assumption P = 1 is rarely met under realistic conditions; most soils exhibit a P significantly larger than 1 [33].…”

Section: New Solution To Richards' Equationmentioning

confidence: 99%

“…Combining Equations (10), (11), (14), (18) and (19), and denoting the constants b 1 , b 2 , and b 3 , a simple closed-form solution is obtained:…”

Section: New Solution To Richards' Equationmentioning

confidence: 99%

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Advancing NASA’s AirMOSS P-Band Radar Root Zone Soil Moisture Retrieval Algorithm via Incorporation of Richards’ Equation

et al. 2016

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P-band radar remote sensing applied during the Airborne Microwave Observatory of Subcanopy and Subsurface (AirMOSS) mission has shown great potential for estimation of root zone soil moisture. When retrieving the soil moisture profile (SMP) from P-band radar observations, a mathematical function describing the vertical moisture distribution is required. Because only a limited number of observations are available, the number of free parameters of the mathematical model must not exceed the number of observed data. For this reason, an empirical quadratic function (second order polynomial) is currently applied in the AirMOSS inversion algorithm to retrieve the SMP. The three free parameters of the polynomial are retrieved for each AirMOSS pixel using three backscatter observations (i.e., one frequency at three polarizations of Horizontal-Horizontal, Vertical-Vertical and Horizontal-Vertical). In this paper, a more realistic, physically-based SMP model containing three free parameters is derived, based on a solution to Richards' equation for unsaturated flow in soils. Evaluation of the new SMP model based on both numerical simulations and measured data revealed that it exhibits greater flexibility for fitting measured and simulated SMPs than the currently applied polynomial. It is also demonstrated that the new SMP model can be reduced to a second order polynomial at the expense of fitting accuracy.

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“…In addition, Tracy [16] obtained threedimensional analytical solutions to Richards' equation when a box-shaped soil sample with piecewise constant head boundary conditions on the top is utilized.…”

Section: Introductionmentioning

confidence: 99%

Innovative Analytical Solutions to 1, 2 and 3D Water Infiltration into Unsaturated Soils for Initial-Boundary Value Problems

2017

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Abstract. Fluid in ltration into unsaturated soil is of vital signi cance from many perspectives. Mathematically, such transient in ltrations are described by Richards' equation: a nonlinear parabolic Partial Di erential Equation (PDE) with limited analytical solutions in the literature. The current study uses separation of variables and Fourier series expansion techniques and presents new analytical solutions to the equation in one, two, and three dimensions subject to various boundary and initial conditions. Solutions for 1D horizontal and vertical water in ltration are derived and compared to numerical nite-di erence method solutions, whereby both solutions are shown to coincide well with one another. Solutions to 2 and 3D vertical water in ltration are derived for constant, no-ow, and sinusoidal boundary and initial conditions. The presented analytical solutions are such that both steady and unsteady solutions may be obtained from a single closed form solution. The solutions may be utilized to test numerical models that use di erent computational techniques.

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Three-dimensional analytical solutions of Richards’ equation for a box-shaped soil sample with piecewise-constant head boundary conditions on the top

Cited by 45 publications

References 12 publications

Analytical and Numerical Solutions of Richards' Equation with Discussions on Relative Hydraulic Conductivity

Analytical and Numerical Solutions of Richards' Equation with Discussions on Relative Hydraulic Conductivity

Advancing NASA’s AirMOSS P-Band Radar Root Zone Soil Moisture Retrieval Algorithm via Incorporation of Richards’ Equation

Innovative Analytical Solutions to 1, 2 and 3D Water Infiltration into Unsaturated Soils for Initial-Boundary Value Problems

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