scholarly journals Numerical Evaluation of Fractional Vertical Soil Water Flow Equations

Water ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 511
Author(s):  
Ali Ercan ◽  
M. Levent Kavvas
Keyword(s):  
Porous Media ◽  
Soil Water ◽  
Water Flow ◽  
Water Movement ◽  
Least Square ◽  
Flow In Porous Media ◽  
Fickian Diffusion ◽  
Richards Equation ◽  
Governing Equations ◽  
Soil Water Flow

Significant deviations from standard Boltzmann scaling, which corresponds to normal or Fickian diffusion, have been observed in the literature for water movement in porous media. However, as demonstrated by various researchers, the widely used conventional Richards equation cannot mimic anomalous diffusion and ignores the features of natural soils which are heterogeneous. Within this framework, governing equations of transient water flow in porous media in fractional time and multi-dimensional fractional soil space in anisotropic media were recently introduced by the authors by coupling Brooks–Corey constitutive relationships with the fractional continuity and motion equations. In this study, instead of utilizing Brooks–Corey relationships, empirical expressions, obtained by least square fits through hydraulic measurements, were utilized to show the suitability of the proposed fractional approach with other constitutive hydraulic relations in the literature. Next, a finite difference numerical method was proposed to solve the fractional governing equations. The applicability of the proposed fractional governing equations was investigated numerically in comparison to their conventional counterparts. In practice, cumulative infiltration values are observed to deviate from conventional infiltration approximation, or the wetting front through time may not be consistent with the traditional estimates of Richards equation. In such cases, fractional governing equations may be a better alternative for mimicking the physical process as they can capture sub-, super-, and normal-diffusive soil water flow processes during infiltration.

Soil Water Flow

2003 ◽  
Author(s):  
Arthur W. Warrick
Keyword(s):  
Soil Water ◽  
Water Flow ◽  
Darcy’S Law ◽  
Soil Water Potential ◽  
Hydraulic Head ◽  
Flow Equation ◽  
Darcy's Law ◽  
Richards Equation ◽  
Flow Equations ◽  
Soil Water Flow

The definitions of hydraulic head and soil water potential in chapter 1 assumed equilibrium. However, the primary motivation was to develop a background useful to describe dynamic systems. If a system is in equilibrium, no flow will occur; otherwise, flow will occur from regions of high to low hydraulic head. The primary flow equation will be Darcy’s law. When Darcy’s law is combined with conservation of mass, the result is a continuity equation that can have several different forms. We will refer to all of those forms generically as soil water flow equations. Generally, for unsaturated conditions, the soil water flow equation is called the Richards equation. A starting point is to examine two classical relationships from fluid dynamics, the Bernoulli and the Poiseuille laws. Bernoulli’s law relates the total potential for ideal fluids and is commonly derived in introductory physics and fluid mechanics texts (see Serway, 1990). Assumptions include an ideal fluid (non-viscous), which is one that is incompressible and which exhibits steady and irrotational flow. For these conditions, the sum of gravitational, pressure, and inertial energy at positions S1and S2 are the same along any streamline For a real fluid, viscosity causes a loss of energy as friction that must be overcome. Additionally, for most problems of interest in soils, the velocity head will be negligible compared with the pressure and gravitational terms.

scholarly journals Fluid flow in porous media using image-based modelling to parametrize Richards' equation

2017 ◽  
Vol 473 (2207) ◽  
pp. 20170178 ◽  
Author(s):  
L. J. Cooper ◽  
K. R. Daly ◽  
P. D. Hallett ◽  
M. Naveed ◽  
N. Koebernick ◽  
...  
Keyword(s):  
Porous Media ◽  
Fluid Flow ◽  
Pore Structure ◽  
Soil Water ◽  
Stokes Equations ◽  
Contact Angles ◽  
Flow In Porous Media ◽  
Richards Equation ◽  
Water Retention Curve ◽  
Image Stack

The parameters in Richards' equation are usually calculated from experimentally measured values of the soil–water characteristic curve and saturated hydraulic conductivity. The complex pore structures that often occur in porous media complicate such parametrization due to hysteresis between wetting and drying and the effects of tortuosity. Rather than estimate the parameters in Richards' equation from these indirect measurements, image-based modelling is used to investigate the relationship between the pore structure and the parameters. A three-dimensional, X-ray computed tomography image stack of a soil sample with voxel resolution of 6 μm has been used to create a computational mesh. The Cahn–Hilliard–Stokes equations for two-fluid flow, in this case water and air, were applied to this mesh and solved using the finite-element method in COMSOL Multiphysics. The upscaled parameters in Richards' equation are then obtained via homogenization. The effect on the soil–water retention curve due to three different contact angles, 0°, 20° and 60°, was also investigated. The results show that the pore structure affects the properties of the flow on the large scale, and different contact angles can change the parameters for Richards' equation.

scholarly journals Governing equations of transient soil water flow and soil water flux in multi – dimensional fractional anisotropic media and fractional time

2016 ◽  
Author(s):  
M. Levent Kavvas ◽  
Ali Ercan ◽  
James Polsinelli
Keyword(s):  
Fractional Derivative ◽  
Soil Water ◽  
Water Flow ◽  
Continuity Equation ◽  
Water Flux ◽  
Anisotropic Media ◽  
Governing Equations ◽  
Soil Matrix ◽  
Soil Water Flow ◽  
Soil Water Flux

Abstract. In this study dimensionally-consistent governing equations of continuity and motion for transient soil water flow and soil water flux in fractional time and in fractional multiple space dimensions in anisotropic media are developed. Due to the anisotropy in the hydraulic conductivities of natural soils, the soil medium within which the soil water flow occurs is essentially anisotropic. Accordingly, in this study the fractional dimensions in two horizontal and one vertical directions are considered to be different, resulting in multi-fractional multi-dimensional soil space within which the flow takes place. Toward the development of the fractional governing equations, first a dimensionally-consistent continuity equation for soil water flow in multi-dimensional fractional soil space and fractional time is developed. It is shown that the fractional soil water flow continuity equation approaches the conventional integer form of the continuity equation as the fractional derivative powers approach integer values. For the motion equation of soil water flow, or the equation of water flux within the soil matrix in multi-dimensional fractional soil space and fractional time, a dimensionally consistent equation is also developed. Again, it is shown that this fractional water flux equation approaches the conventional Darcy's equation as the fractional derivative powers approach integer values. From the combination of the fractional continuity and motion equations, the governing equation of transient soil water flow in multi-dimensional fractional soil space and fractional time is obtained. It is shown that this equation approaches the conventional Richards equation as the fractional derivative powers approach integer values. Then by the introduction of the Brooks-Corey constitutive relationships for soil water into the fractional transient soil water flow equation, an explicit form of the equation is obtained in multi-dimensional fractional soil space and fractional time. The governing fractional equation is then specialized to the case of only vertical soil water flow and of only horizontal soil water flow in fractional time-space. It is shown that the developed governing equations, in their fractional time but integer space forms, show behavior consistent with the previous experimental observations concerning the diffusive behavior of soil water flow.

scholarly journals Governing equations of transient soil water flow and soil water flux in multi-dimensional fractional anisotropic media and fractional time

2017 ◽  
Vol 21 (3) ◽  
pp. 1547-1557 ◽  
Author(s):  
M. Levent Kavvas ◽  
Ali Ercan ◽  
James Polsinelli
Keyword(s):  
Fractional Derivative ◽  
Soil Water ◽  
Water Flow ◽  
Continuity Equation ◽  
Water Flux ◽  
Anisotropic Media ◽  
Governing Equations ◽  
Soil Matrix ◽  
Soil Water Flow ◽  
Soil Water Flux

Abstract. In this study dimensionally consistent governing equations of continuity and motion for transient soil water flow and soil water flux in fractional time and in fractional multiple space dimensions in anisotropic media are developed. Due to the anisotropy in the hydraulic conductivities of natural soils, the soil medium within which the soil water flow occurs is essentially anisotropic. Accordingly, in this study the fractional dimensions in two horizontal and one vertical directions are considered to be different, resulting in multi-fractional multi-dimensional soil space within which the flow takes place. Toward the development of the fractional governing equations, first a dimensionally consistent continuity equation for soil water flow in multi-dimensional fractional soil space and fractional time is developed. It is shown that the fractional soil water flow continuity equation approaches the conventional integer form of the continuity equation as the fractional derivative powers approach integer values. For the motion equation of soil water flow, or the equation of water flux within the soil matrix in multi-dimensional fractional soil space and fractional time, a dimensionally consistent equation is also developed. Again, it is shown that this fractional water flux equation approaches the conventional Darcy equation as the fractional derivative powers approach integer values. From the combination of the fractional continuity and motion equations, the governing equation of transient soil water flow in multi-dimensional fractional soil space and fractional time is obtained. It is shown that this equation approaches the conventional Richards equation as the fractional derivative powers approach integer values. Then by the introduction of the Brooks–Corey constitutive relationships for soil water into the fractional transient soil water flow equation, an explicit form of the equation is obtained in multi-dimensional fractional soil space and fractional time. The governing fractional equation is then specialized to the case of only vertical soil water flow and of only horizontal soil water flow in fractional time–space. It is shown that the developed governing equations, in their fractional time but integer space forms, show behavior consistent with the previous experimental observations concerning the diffusive behavior of soil water flow.

scholarly journals Soil water flow due to surface topography in research drainage plots

age ◽  
2020 ◽  
Vol 3 (1) ◽  
Author(s):  
Sally Logsdon ◽  
Cindy Cambardella
Keyword(s):  
Surface Topography ◽  
Soil Water ◽  
Water Flow ◽  
Soil Water Flow
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