Fractional Vertical Infiltration
The infiltration phenomena has been studied by several authors for decades, and numerical and approximate results have been shown through the asymptotic solution in short and long times. In particular, it is worth highlighting the works of Philip and Parlange, who used time and volumetric content as...
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MDPI AG
2021-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/4/383 |
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| author | Carlos Fuentes Fernando Alcántara-López Antonio Quevedo Carlos Chávez |
| author_facet | Carlos Fuentes Fernando Alcántara-López Antonio Quevedo Carlos Chávez |
| author_sort | Carlos Fuentes |
| collection | DOAJ |
| description | The infiltration phenomena has been studied by several authors for decades, and numerical and approximate results have been shown through the asymptotic solution in short and long times. In particular, it is worth highlighting the works of Philip and Parlange, who used time and volumetric content as independent variables and space as a dependent variable, and found the solution as a power series in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>t</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></semantics></math></inline-formula> that is valid for short times. However, several studies show that these models are not applicable to anomalous flows, in which case the application of fractional calculus is needed. In this work, a fractional time derivative of a Caputo type is applied to model anomalous infiltration phenomena. Fractional horizontal infiltration phenomena are studied, and the fractional Boltzmann transform is defined. To study fractional vertical infiltration phenomena, the asymptotic behavior is described for short and long times considering an arbitrary diffusivity and hydraulic conductivity. Finally, considering a constant flux-dependent relation and a relation between diffusivity and hydraulic conductivity, a fractional cumulative infiltration model applicable to various types of soil is built; its solution is expressed as a power series in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>t</mi><mrow><mi>ν</mi><mo>/</mo><mn>2</mn></mrow></msup></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ν</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> is the order of the fractional derivative. The results show the effect of superdiffusive and subdiffusive flows in different types of soil. |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T00:52:53Z |
| publishDate | 2021-02-01 |
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| spelling | doaj.art-bc276b1a34e5417eac0a0ed4f8b90c1c2023-12-11T17:06:24ZengMDPI AGMathematics2227-73902021-02-019438310.3390/math9040383Fractional Vertical InfiltrationCarlos Fuentes0Fernando Alcántara-López1Antonio Quevedo2Carlos Chávez3Mexican Institute of Water Technology, Paseo Cuauhnáhuac Núm. 8532, Jiutepec 62550, Morelos, MexicoDepartment of Mathematics, Faculty of Science, National Autonomous University of México, Av. Universidad 3000, Circuito Exterior S/N, Delegación Coyoacán 04510, Ciudad de México, MexicoMexican Institute of Water Technology, Paseo Cuauhnáhuac Núm. 8532, Jiutepec 62550, Morelos, MexicoWater Research Center, Department of Irrigation and Drainage Engineering, Autonomous University of Querétaro, Cerro de las Campanas SN, Col. Las Campanas 76010, Querétaro, MexicoThe infiltration phenomena has been studied by several authors for decades, and numerical and approximate results have been shown through the asymptotic solution in short and long times. In particular, it is worth highlighting the works of Philip and Parlange, who used time and volumetric content as independent variables and space as a dependent variable, and found the solution as a power series in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>t</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></semantics></math></inline-formula> that is valid for short times. However, several studies show that these models are not applicable to anomalous flows, in which case the application of fractional calculus is needed. In this work, a fractional time derivative of a Caputo type is applied to model anomalous infiltration phenomena. Fractional horizontal infiltration phenomena are studied, and the fractional Boltzmann transform is defined. To study fractional vertical infiltration phenomena, the asymptotic behavior is described for short and long times considering an arbitrary diffusivity and hydraulic conductivity. Finally, considering a constant flux-dependent relation and a relation between diffusivity and hydraulic conductivity, a fractional cumulative infiltration model applicable to various types of soil is built; its solution is expressed as a power series in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>t</mi><mrow><mi>ν</mi><mo>/</mo><mn>2</mn></mrow></msup></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ν</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> is the order of the fractional derivative. The results show the effect of superdiffusive and subdiffusive flows in different types of soil.https://www.mdpi.com/2227-7390/9/4/383asymptotic solutionparlange equationsDarcy’s lawfractional Caputo derivative |
| spellingShingle | Carlos Fuentes Fernando Alcántara-López Antonio Quevedo Carlos Chávez Fractional Vertical Infiltration Mathematics asymptotic solution parlange equations Darcy’s law fractional Caputo derivative |
| title | Fractional Vertical Infiltration |
| title_full | Fractional Vertical Infiltration |
| title_fullStr | Fractional Vertical Infiltration |
| title_full_unstemmed | Fractional Vertical Infiltration |
| title_short | Fractional Vertical Infiltration |
| title_sort | fractional vertical infiltration |
| topic | asymptotic solution parlange equations Darcy’s law fractional Caputo derivative |
| url | https://www.mdpi.com/2227-7390/9/4/383 |
| work_keys_str_mv | AT carlosfuentes fractionalverticalinfiltration AT fernandoalcantaralopez fractionalverticalinfiltration AT antonioquevedo fractionalverticalinfiltration AT carloschavez fractionalverticalinfiltration |