scholarly journals A Solution of Richards’ Equation by Generalized Finite Differences for Stationary Flow in a Dam

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1604
Author(s):  
Carlos Chávez-Negrete ◽  
Daniel Santana-Quinteros ◽  
Francisco Domínguez-Mota
Keyword(s):  
Finite Differences ◽  
Test Problem ◽  
Soil Mechanics ◽  
Stationary Flow ◽  
Standard Test ◽  
Richards Equation ◽  
Saturated Soils ◽  
Transient Case ◽  
Comparative Results ◽  
Stationary Form

The accurate description of the flow of water in porous media is of the greatest importance due to its numerous applications in several areas (groundwater, soil mechanics, etc.). The nonlinear Richards equation is often used as the governing equation that describes this phenomenon and a large number of research studies aimed to solve it numerically. However, due to the nonlinearity of the constitutive expressions for permeability, it remains a challenging modeling problem. In this paper, the stationary form of Richards’ equation used in saturated soils is solved by two numerical methods: generalized finite differences, an emerging method that has been successfully applied to the transient case, and a finite element method, for benchmarking. The nonlinearity of the solution in both cases is handled using a Newtonian iteration. The comparative results show that a generalized finite difference iteration yields satisfactory results in a standard test problem with a singularity at the boundary.

Numerical solution of Richards’ equation of water flow by generalized finite differences

2018 ◽  
Vol 101 ◽  
pp. 168-175 ◽  
Author(s):  
C. Chávez-Negrete ◽  
F.J. Domínguez-Mota ◽  
D. Santana-Quinteros
Keyword(s):  
Numerical Solution ◽  
Water Flow ◽  
Finite Differences ◽  
Richards Equation

scholarly journals Modeling Soil Water Redistribution under Gravity Irrigation with the Richards Equation

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1581 ◽  
Author(s):  
Sebastián Fuentes ◽  
Josué Trejo-Alonso ◽  
Antonio Quevedo ◽  
Carlos Fuentes ◽  
Carlos Chávez
Keyword(s):  
Finite Difference ◽  
Numerical Solution ◽  
Soil Water ◽  
Water Movement ◽  
Soil Mechanics ◽  
Water Infiltration ◽  
Richards Equation ◽  
Water Redistribution ◽  
Irrigation Drainage ◽  
Transfer Problems

Soil water movement is important in fields such as soil mechanics, irrigation, drainage, hydrology, and agriculture. The Richards equation describes the flow of water in an unsaturated porous medium, and analytical solutions exist only for simplified cases. However, numerous practical situations require a numerical solution (1D, 2D, or 3D) depending on the complexity of the studied problem. In this paper, numerical solution of the equation describing water infiltration into soil using the finite difference method is studied. The finite difference solution is made via iterative schemes of local balance, including explicit, implicit, and intermediate methods; as a special case, the Laasonen method is shown. The found solution is applied to water transfer problems during and after gravity irrigation to observe phenomena of infiltration, evaporation, transpiration, and percolation.

Hydrology of swelling soils: a review

Soil Research ◽  
2000 ◽  
Vol 38 (3) ◽  
pp. 501 ◽  
Author(s):  
D. E. Smiles
Keyword(s):  
Water Flow ◽  
Liquid Flow ◽  
Soil Mechanics ◽  
Material Balance ◽  
Flow Equation ◽  
Richards Equation ◽  
Gravitational Potential Energy ◽  
Future Research ◽  
Swelling Soils ◽  
Wide Range

A generally accepted theory of liquid flow in rigid systems has been used in soil science for more than 50 years. Liquid flow in systems that change volume with liquid content is not so well described and remains a major challenge to soil scientists, although its application in chemical and mining engineering and soil mechanics is increasingly accepted. Theory of water flow in swelling soils must satisfy material continuity. It must also account for changes in the gravitational potential energy of the system during swelling and for anisotropic stresses that constrain the soil laterally but permit vertical movement. A macroscopic and phenomenological analysis based on material balance and Darcy’s law is the most useful first approach to water flow and volume change in such soils. Use of a material coordinate based on the solid distribution results in a flow equation analogous to that L. A. Richards enunciated for non-swelling soils. This framework is strain-independent and solutions to the flow equation exist for a wide range of practically important conditions. The approach has been well tested in clay suspensions and saturated systems such as mine tailings and sediments. It is also applied in soil mechanics. This paper reviews central elements in application of the analysis to swelling soils. It argues that, as with use of the Richards’ equation in rigid soils, complexities are evident, but the approach remains the most coherent and profitable to support current need and future research. The use of material coordinates, to ensure material balance is assessed correctly, is simple.

Corrigendum to: Hydrology of swelling soils: a review

Soil Research ◽  
2001 ◽  
Vol 39 (6) ◽  
pp. 1467
Author(s):  
D. E. Smiles
Keyword(s):  
Water Flow ◽  
Liquid Flow ◽  
Soil Mechanics ◽  
Material Balance ◽  
Flow Equation ◽  
Richards Equation ◽  
Gravitational Potential Energy ◽  
Future Research ◽  
Swelling Soils ◽  
Wide Range

A generally accepted theory of liquid flow in rigid systems has been used in soil science for more than 50 years. Liquid flow in systems that change volume with liquid content is not so well described and remains a major challenge to soil scientists, although its application in chemical and mining engineering and soil mechanics is increasingly accepted. Theory of water flow in swelling soils must satisfy material continuity. It must also account for changes in the gravitational potential energy of the system during swelling and for anisotropic stresses that constrain the soil laterally but permit vertical movement. A macroscopic and phenomenological analysis based on material balance and Darcy’s law is the most useful first approach to water flow and volume change in such soils. Use of a material coordinate based on the solid distribution results in a flow equation analogous to that L. A. Richards enunciated for non-swelling soils. This framework is strain-independent and solutions to the flow equation exist for a wide range of practically important conditions. The approach has been well tested in clay suspensions and saturated systems such as mine tailings and sediments. It is also applied in soil mechanics. This paper reviews central elements in application of the analysis to swelling soils. It argues that, as with use of the Richards’ equation in rigid soils, complexities are evident, but the approach remains the most coherent and profitable to support current need and future research. The use of material coordinates, to ensure material balance is assessed correctly, is simple.

One-Dimensional Infiltration and Vertical Flow

2003 ◽  
Author(s):  
Arthur W. Warrick
Keyword(s):  
Steady State ◽  
Numerical Methods ◽  
Analytical Solutions ◽  
Finite Differences ◽  
Numerical Solutions ◽  
Water Dynamics ◽  
Richards Equation ◽  
Vertical Flow ◽  
One Dimensional ◽  
The One

This chapter addresses one-dimensional infiltration and vertical flow problems. Traditionally, infiltration has received more attention than other unsaturated flow procedures, both for empirical formulations and for applications of Richards’ equation. Rarely is infiltration the only process of interest, and from an overall point of view it is only one example of soil water dynamics. Here, we will first emphasize systems for which analytical (or quasi-analytical) solutions can be found. These include the Green and Ampt solution (1911), which adds gravity to the simplified analysis discussed in chapter 4. Then a linearized form of Richards’ equation will be examined, followed by the perturbation of the horizontal problem of Philip leading to his famous series solution. Although the closed-form and quasi-analytical solutions are convenient for calculations and discussing the physical principles, generally, the nonlinearity of Richards’ equation precludes such convenient forms. However, numerical approximations can be used. The conventional numerical methods applied in water and solute transport are based on finite differences and finite elements. Because of its greater simplicity, we will emphasize finite differences and build on the methodology from the saturated-flow example in chapter 3. Richards’ equation is a parabolic partial differential equation reducing to an elliptical form for steady-state cases. The analyses and methods parallel developments for techniques developed primarily for the linear diffusion equation. Many texts exist for numerical methods; one to which we refer is by Smith (1985). Ideally, numerical methods give solutions that are as accurate as the input warrants or as necessary for application. In some cases, results may be easier or more accurate than the evaluation of a complex analytical expression. Clearly, infiltration is of limited duration, with drainage and redistribution occurring over much longer time frames. We will visit briefly some steady-state examples, including layered profile and upward flow from a shallow water table. Other examples include modeling plant water uptake from the profile and drainage of initially wet profiles. The rapid increase in computational power and availability of computers make solutions feasible and routine for problems that were very tedious or time consuming only a few years ago. This is particularly true of the one-dimensional numerical solutions.

scholarly journals Test Problem Construction for Single-Objective Bilevel Optimization

2014 ◽  
Vol 22 (3) ◽  
pp. 439-477 ◽  
Author(s):  
Ankur Sinha ◽  
Pekka Malo ◽  
Kalyanmoy Deb
Keyword(s):  
Evolutionary Algorithm ◽  
Optimization Algorithm ◽  
Test Problem ◽  
Bilevel Optimization ◽  
Standard Test ◽  
Test Problems ◽  
Problem Construction ◽  
Constrained Problems ◽  
Construction Procedure ◽  
Single Objective

In this paper, we propose a procedure for designing controlled test problems for single-objective bilevel optimization. The construction procedure is flexible and allows its user to control the different complexities that are to be included in the test problems independently of each other. In addition to properties that control the difficulty in convergence, the procedure also allows the user to introduce difficulties caused by interaction of the two levels. As a companion to the test problem construction framework, the paper presents a standard test suite of 12 problems, which includes eight unconstrained and four constrained problems. Most of the problems are scalable in terms of variables and constraints. To provide baseline results, we have solved the proposed test problems using a nested bilevel evolutionary algorithm. The results can be used for comparison, while evaluating the performance of any other bilevel optimization algorithm. The code related to the paper may be accessed from the website http://bilevel.org .

A Transformed Pressure Head-Based Approach to Solve Richards' Equation for Variably Saturated Soils

1995 ◽  
Vol 31 (4) ◽  
pp. 925-931 ◽  
Author(s):  
Lehua Pan ◽  
Peter J. Wierenga
Keyword(s):  
Pressure Head ◽  
Richards Equation ◽  
Saturated Soils

Algorithms for solving Richards' equation for variably saturated soils

1992 ◽  
Vol 28 (8) ◽  
pp. 2049-2058 ◽  
Author(s):  
M. R. Kirkland ◽  
R. G. Hills ◽  
P. J. Wierenga
Keyword(s):  
Richards Equation ◽  
Saturated Soils

scholarly journals BEARING CAPACITY OF DEEP PILE FOUNDATION FOR HIGH-RISE FACILITY ON WEAK SOILS: COMPARING OF ANALYSIS RESULTS AND EXPERIMENTAL DATA

2019 ◽  
Vol 15 (1) ◽  
pp. 90-97
Author(s):  
Rashid Mangushev ◽  
Nadezhda Nikitina
Keyword(s):  
Bearing Capacity ◽  
Pile Foundation ◽  
Standard Test ◽  
Strain Gauges ◽  
Construction Site ◽  
Static Testing ◽  
High Rise ◽  
Weak Soils ◽  
Comparative Results ◽  
Pile Length

The results of static testing of the pile and comparative results of analytical and numerical calculations for the experimental deep pile (length 65 m, diameter 1.2 m) under the high-rise building, designed in the area of a large thickness of weak soils, are presented in the paper. At the same construction site, an experimental barrette pile of rectangular crosssection with a size of 3.3 x 1.1 and a length of 65 m with the location of the base in solid Proterozoic clays was made. This pile was tested with the use of Osterberg cells, for which strain gauges were mounted in its reinforcement cage at 9 levels. In the first stage, a standard test of the entire experimental barrette pile in the top-down direction was conducted; in the second, after reaching the maximum possible load, the tests were carried out using the “O-cells” located at a depth of 50 m in the thickness of solid clays and transmitting the load in two directions (up and down). A General assessment of the bearing capacity of the barrette pile obtained by three methods is given.

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