On two phase filtration under gravity and with boundary infiltration: application of a bäcklund transformation

Rogers, C.; Stallybrass, M.P.; Clements, D. L.

doi:10.1016/0362-546x(83)90034-2

Cited by 106 publications

(31 citation statements)

References 2 publications

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“…For example, (11a), sometimes known as Richard's equation, has been used to model the one-dimensional, nonhysteretic infiltration in uniform nonswelling soil (Broadbridge and White [26]) and to model two phase filtration under gravity (Rogers, Stallybrass, and Clement [27]). Furthermore, (11b)-sometimes known as the nonlinear telegraph equation-has been used to model the telegraphy of a two-conductor transmission line (Katayev [28]) and the motion of a hyperelastic homogeneous rod whose cross-sectional area varies exponentially along the rod (Jeffery [29]).…”

Section: Nonclassical Symmetriesmentioning

confidence: 99%

Nonclassical Symmetries of a Nonlinear Diffusion–Convection/Wave Equation and Equivalents Systems

et al. 2016

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Abstract:It is generally known that classical point and potential Lie symmetries of differential equations (the latter calculated as point symmetries of an equivalent system) can be different. We question whether this is true when the symmetries are extended to nonclassical symmetries. In this paper, we consider two classes of nonlinear partial differential equations; the first one is a diffusion-convection equation, the second one a wave, where we will show that the majority of the nonclassical point symmetries are included in the nonclassical potential symmetries. We highlight a special case were the opposite is true.

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Section: Nonclassical Symmetriesmentioning

confidence: 99%

Nonclassical Symmetries of a Nonlinear Diffusion–Convection/Wave Equation and Equivalents Systems

et al. 2016

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“…However, Richards' equation is strongly nonlinear and exact linearisation techniques, e.g. using B~cklund transformations, apply only for Fujita diffusivities (Rogers et al, 1983). Most systematic studies based on group classification and transformation properties of the equation (Lisle and Parlange, 1993) still lead to generalized Fujita-type diffusivities, Here on the contrary the method applies to arbitrary soil-water diffusivities, but the result is only approximate.…”

Section: Introductionmentioning

confidence: 99%

Superposition principle for short-term solutions of Richards' equation: Application to the interaction of wetting fronts with an impervious surface

et al. 1994

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Abstract. The interaction of a wetting front with an impervious layer is described by adding a reflected solution to the incoming solution for a semi-infinite medium. It is shown and checked by comparison with a numerical solution that the result is accurate during the early times of the interaction between the front and the impervious surface. This superposition principle is quite general and should prove especially useful to initiate numerical schemes by this analytical approximation as in the early times singularities are difficult to describe numerically.

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“…Exact one-dimensional solutions of Richards' equation have been derived for a few specialized forms of the constitutive relations describing the soil water retention and the unsaturated hydraulic conductivity functions (Rogers et al, 1983;Broadbridge and White, 1988;Sander et al, 1988;Barry and Sander, 1991;Barry et al, 1993;Ross and Parlange, 1994). However, these solutions are not generally applicable because either the functional forms are dissimilar from widely used constitutive relations that represent real soils, and/or the solutions impose strict requirements on the initial and boundary conditions.…”

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confidence: 99%

Numerical Solution of Richards' Equation: A Review of Advances and Challenges

Farthing

Ogden

2017

Soil Science Soc of Amer J

248

138

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The flow of water in partially saturated porous media is of importance in fields such as hydrology, agriculture, environment and waste management. It is also one of the most complex flows in nature. The Richards' equation describes the flow of water in an unsaturated porous medium due to the actions of gravity and capillarity neglecting the flow of the non-wetting phase, usually air. Analytical solutions of Richards' equation exist only for simplified cases, so most practical situations require a numerical solution in one-two-or threedimensions, depending on the problem and complexity of the flow situation. Despite the fact that the first reasonably complete conservative numerical solution method was published in the early 1990s, the numerical solution of the Richards' equation remains computationally expensive and in certain circumstances, unreliable. A universally robust and accurate solution methodology has not yet been identified that is applicable across the range of soils, initial and boundary conditions found in practice. Existing solution codes have been modified over years to attempt to increase robustness. Despite theoretical results on the existence of solutions given sufficiently regular data and constitutive relations, our numerical methods often fail to demonstrate reliable convergence behavior in practice, especially for higher-order methods. Because of robustness, the lack of higher-order accuracy and computational expense, alternative solution approaches or methods are needed. There is also a need for better documentation of improved solution methodologies and benchmark test problems to facilitate consistent advances and avoid re-inventing of the wheel.T his review paper is intended to serve as a touchstone on the state of the science of calculating the flow of water through unsaturated porous media. As active researchers we believe it is good to occasionally take stock in what has been accomplished up to date, where things may be headed, and what important issues remain. This review concentrates on computational modeling of flow through the unsaturated zone via Richards' equation.This effort can help provide some focus to the research community and serve to help propel new research forward, particularly by those just joining the field. There are many practical problems that require calculation of one-, two-, and threedimensional fluxes in the unsaturated zone. For a much broader review of modeling soil processes, we recommend the recent paper from Vereecken et al. (2016). A detailed review of mathematical models of infiltration can be found (Assouline, 2013), while Paniconi and Putti (2015) provide historical perspective and an overview of modeling Richards' equation in the context of catchment hydrology.The equation attributed to Richards (1931) that describes the flow of water through unsaturated porous media under the action of capillarity and gravity was first published by the English mathematician and physicist Lewis Fry Richardson in 1922(Richardson, 1922. Therefore, the equation would r...

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On two phase filtration under gravity and with boundary infiltration: application of a bäcklund transformation

Cited by 106 publications

References 2 publications

Nonclassical Symmetries of a Nonlinear Diffusion–Convection/Wave Equation and Equivalents Systems

Nonclassical Symmetries of a Nonlinear Diffusion–Convection/Wave Equation and Equivalents Systems

Superposition principle for short-term solutions of Richards' equation: Application to the interaction of wetting fronts with an impervious surface

Numerical Solution of Richards' Equation: A Review of Advances and Challenges

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