Mass-time and space-time fractional partial differential equations of water movement in soils: Theoretical framework and application to infiltration

Su, Ninghu

doi:10.1016/j.jhydrol.2014.09.021

Cited by 35 publications

(40 citation statements)

References 46 publications

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“…The purpose of this paper is to investigate the movement of water in soils using a newly developed fPDE [1]. The results of this investigation in this paper include the following:…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…In a comprehensive development [1], we have introduced a set of fPDEs for water flow in both swelling and non-swelling soils. For swelling soils, with the material diffusivity given by where D 0 , b, K 0 and k are constants, the equation governing the movement of water in swelling soils is given as b 1…”

Section: The Ctrw Theory and Its Connection With The Fpde For Water Mmentioning

confidence: 99%

“…Due to the fact that the diffusivity, D(θ), and the hydraulic conductivity, K(θ), are related functions [1], the fractional connections between D(θ) and K(θ) are more complex which need to be explored. In the present case, a constant diffusivity and hydraulic conductivity are used and, with functions for D(θ) and K(θ), the fPDE with a fractional term…”

Section: ∂K(θ) ∂|M|mentioning

confidence: 99%

“…The fPDE discussed in this paper is one special form of the fPDEs presented earlier [1] following the extensive analysis and parameterisation of the flow processes in soils. The fPDEs incorporate both the material coordinate for water movement in swelling-shrinking soils and the Cartesian coordinates in non-swelling soils.…”

Section: Introductionmentioning

confidence: 99%

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Exact and Approximate Solutions of Fractional Partial Differential Equations for Water Movement in Soils

2017

Hydrology

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This paper presents solutions of the fractional partial differential equation (fPDE) for analysing water movement in soils. The fPDE explains processes equivalent to the concept of symmetrical fractional derivatives (SFDs) which have two components: the forward fractional derivative (FFD) and backward fractional derivative (BFD) of water movement in soils with the BFD representing the micro-scale backwater effect in porous media. The distributed-order time-space fPDE represents water movement in both swelling and non-swelling soils with mobile and immobile zones with the backwater effect operating at two time scales in large and small pores. The concept of flux-concentration relation is now updated to account for the relative fractional flux of water movement in soils.

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“…The purpose of this paper is to investigate the movement of water in soils using a newly developed fPDE [1]. The results of this investigation in this paper include the following:…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Section: The Ctrw Theory and Its Connection With The Fpde For Water Mmentioning

confidence: 99%

Section: ∂K(θ) ∂|M|mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Exact and Approximate Solutions of Fractional Partial Differential Equations for Water Movement in Soils

2017

Hydrology

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“…This makes A proportional to x 0 2 (or K S ) but S proportional to x 0 1.53 , or K S 0.77 , suggesting a different proportionality S ≈ A 0.77 , than that in Philip's theory, S ≈ A 0. 5 .…”

Section: Relevance Of Percolation Scaling To Infiltrationmentioning

confidence: 99%

A Percolation‐Based Approach to Scaling Infiltration and Evapotranspiration

Hunt

Holtzman

Ghanbarian

2017

Water

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Abstract:Optimal flow paths obtained from percolation theory provide a powerful tool that can be used to characterize properties associated with flow such as soil hydraulic conductivity, as well as other properties influenced by flow connectivity and topology. A recently proposed scaling theory for vegetation growth appeals to the tortuosity of optimal paths from percolation theory to define the spatio-temporal scaling of the root radial extent (or, equivalently, plant height). Root radial extent measures the maximum horizontal distance between a plant shoot and the root tips. We apply here the same scaling relationship to unsteady (horizontal) flow associated with plant transpiration. The pore-scale travel time is generated from the maximum flow rate under saturated conditions and a typical pore size. At the field-scale, the characteristic time is interpreted as the growing season duration, and the characteristic length is derived from the measured evapotranspiration in that period. We show that the two scaling results are equivalent, and they are each in accord with observed vegetation growth limits, as well as with actual limiting transpiration values. While the conceptual approach addresses transpiration, most accessed data are for evapotranspiration. The equivalence of the two scaling approaches suggests that, if horizontal flow is the dominant pathway in plant transpiration, horizontal unsteady flow follows the same scaling relationship as root growth. Then, we propose a corresponding scaling relationship to vertical infiltration, a hypothesis which is amenable to testing using infiltration results of Sharma and co-authors. This alternate treatment of unsteady vertical flow may be an effective alternative to the commonly applied method based on the diffusion of water over a continuum as governed by Richards' equation.

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Random fractional partial differential equations and solutions for water movement in soils: Theory and applications

2023

Hydrological Processes

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This paper analyses a set of random fractional partial differential equations (rfPDEs) for water movement in soils. The rfPDEs for both rigid and swelling soils are solved for both a random flux boundary condition (BC), and random concentration BC. Solutions from a random flux BC are presented for the large-time and small-time situations with the large-time solution as a very simple method for determining the flux through the surface of the soil. The equation of cumulative infiltration is presented with random parameters of the rfPDE subject to a random concentration BC. The simulations using the results of the rfPDE for the two types of BCs yielded encouraging and stable results based on two sets of field data: the first set of the data was measurements at a single site while the second set was from 26 measurements in a small catchment. The results suggest that the presented procedures are very useful methods for the interpolation, extrapolation, and prediction of hydrological variables and parameters such as water content, hydraulic conductivity or the flux through the surface of the soil. The methodologies presented in this paper are able to reveal and reproduce the realistic hydrological processes in nature which are often stochastic and random.

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Mass-time and space-time fractional partial differential equations of water movement in soils: Theoretical framework and application to infiltration

Cited by 35 publications

References 46 publications

Exact and Approximate Solutions of Fractional Partial Differential Equations for Water Movement in Soils

Exact and Approximate Solutions of Fractional Partial Differential Equations for Water Movement in Soils

A Percolation‐Based Approach to Scaling Infiltration and Evapotranspiration

Random fractional partial differential equations and solutions for water movement in soils: Theory and applications

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