### Implementation of the Green-Ampt Infiltration Model: Comparative between different numerical solutions

#### Abstract

The phenomena of infiltration and the percolation of water in the soil are of fundamental importance for the evaluation of runoff, groundwater recharge, evapotranspiration, soil erosion and transport of chemical substances in surface and groundwater. Within this context, the quantitative determination of the infiltration values is extremely important for the different areas of knowledge, in order to evaluate, mainly the surface runoff. Several types of changes in vegetation cover and topography result in significant changes in the infiltration process, making it necessary to use mathematical models to assess the consequences of these changes. Thus, this article aims to implement the Green-Ampt model using two numerical methods - Newton-Raphson method and W-Lambert function - to determine soil permeability parameters - *K* and matric potential multiplied by the difference between initial and of saturation - comparing them to the real data obtained in simulations using an automatic rainfall simulator from the Federal University of Goiás - UFG. The Green-Ampt model adjusted well to the data measured from the rain simulator, with a determination coefficient of 0.978 for the Newton-Raphson method and 0.984 for the W-Lambert function.

#### Keywords

#### Full Text:

PDF#### References

V. Chowdary, M. D. Rao, and C. Jaiswal, “Study of infiltration process under different experimental conditions,” Agricultural Water Management, vol. 83, no. 1-2, pp. 69-78, 2006.

K. T. M. Formiga, A. C. Seibt, T. Q. Castro, and R. S. Bernardes, Tópicos sobre infiltração, ch. A infiltração e o escoamento superficial. Editora UnB, 2012.

L. Mao, Y. Li, W. Hao, X. Zhou, C. Xu, and T. Lei, “A new method to estimate soil water infiltration based on a modified green-ampt model,” Soil and Tillage Research, vol. 161, pp. 31-37, 2016.

A. Jain and A. Kumar, “An evaluation of artificial neural network technique for the determination of infiltration model parameters,” Applied Soft Computing, vol. 6, no. 3, pp. 272-282, 2006.

P. K. Swamee, P. N. Rathie, and L. C. de SM Ozelim, “Explicit equations for infiltration,” Journal of hydrology, vol. 426, pp. 151-153, 2012.

G. A. Fox, D. M. Heeren, G. V. Wilson, E. J. Langendoen, A. K. Fox, and M. L. Chu-Agor, “Numerically predicting seepage gradient forces and erosion: Sensitivity to soil hydraulic properties,” Journal of hydrology, vol. 389, no. 3-4, pp. 354-362, 2010.

S. F. d. Sousa Júnior, T. A. Mendes, and E. Q. d. Siqueira, “Development and calibration of a rainfall simulator for hydrological studies,” RBRH, vol. 22,2017.

T. A. Mendes, G. d. F. N. Gitirana Jr, J. F. R. Rebolledo, E. F. Vaz, and M. P. d. Luz, “Numerical evaluation of laboratory apparatuses for the study of infiltration and runoff,” RBRH, vol. 25, 2020.

T. A. Mendes, S. A. d. S. Pereira, J. F. R. Rebolledo, G. d. F. N. Gitirana, M. T. d. S. Melo, and M. P. d. Luz, “Development of a rainfall and runoff simulator for performing hydrological and geotechnical tests,” Sustainability, vol. 13, no. 6, p. 3060, 2021.

H. Aksoy, N. E. Unal, S. Cokgor, A. Gedikli, J. Yoon, K. Koca, S. B. Inci, and E. Eris, “A rainfall simulator for laboratory-scale assessment of rainfall-runoff-sediment transport processes over a two-dimensional flume,” Catena, vol. 98, pp. 63-72, 2012.

T. A. Mendes, Modelagem física e numérica da infiltração e escoamento em superfícies não saturadas e com cobertura vegetativa. PhD thesis, Universidade de Brasília.

D. Silburn and R. Connolly, “Distributed parameter hydrology model (answers) applied to a range of catchment scales using rainfall simulator data i: Infiltration modelling and parameter measurement,” Journal of Hydrology, vol. 172, no. 1-4, pp. 87-104, 1995.

G. B. Paige, J. J. Stone, D. P. Guertin, and L. J. Lane, “A strip model approach to parameterize a coupled green-ampt kinematic wave model 1,” JAWRA Journal of the American Water Resources Association, vol. 38, no. 5, pp. 1363-1377, 2002.

T. E. Santos, E. R. d. Souza, and A. A. Montenegro, “Modeling of soil water infiltration with rainfall simulator in different agricultural systems,” Revista Brasileira de Engenharia Agrícola e Ambiental, vol. 20, no. 6, pp. 513-518, 2016.

A. Van den Putte, G. Govers, A. Leys, C. Langhans, W. Clymans, and J. Diels, “Estimating the parameters of the green-ampt infiltration equation from rainfall simulation data: Why simpler is better,” Journal of Hydrology, vol. 476, pp. 332-344, 2013.

J. Mohammadzadeh-Habili and M. Heidarpour, “Application of the green-ampt model for infiltration into layered soils,” Journal of Hydrology, vol. 527, pp. 824- 832, 2015.

S. Ali, A. Islam, P. Mishra, and A. K. Sikka, “Green-ampt approximations: A comprehensive analysis,” Journal of Hydrology, vol. 535, pp. 340-355, 2016.

P. Deng and J. Zhu, “Analysis of effective green-ampt hydraulic parameters for vertically layered soils,” Journal of Hydrology, vol. 538, pp. 705-712, 2016.

W. Green, “Ampt. ga: Studies on soil physics. 1. flow of air and water through soils,” J. Agr. Sci, vol. 4, pp. 1-24, 1911.

Q. Shao and T. Baumgartl, “Estimating input parameters for four infiltration models from basic soil, vegetation, and rainfall properties,” Soil Science Society of America Journal, vol. 78, no. 5, pp. 1507-1521, 2014.

J. Enciso-Medina, D. Martin, and D. Eisenhauer, “Infiltration model for furrow irrigation,” Journal of irrigation and drainage engineering, vol. 124, no. 2, pp. 73-80, 1998.

D. Barry, P. Culligan-Hensley, and S. Barry, “Real values of the w-function,” ACM Transactions on Mathematical Software (TOMS), vol. 21, no. 2, pp. 161-171, 1995.

J.-Y. Parlange, D. Barry, and R. Haverkamp, “Explicit infiltration equations and the lambert w-function,” Advances in water resources, vol. 25, no. 8-12, pp. 1119-1124, 2002.

A. d. Costa and L. Prado, “Espacialização de chuvas intensas para o estado de goiás e o sul de tocantins,” Revista de Engenharia Agrícola, vol. 23, no. 2, pp. 268-276, 2003.

T. Steihaug, “Computational science in the eighteenth century. test cases for the methods of newton, raphson, and halley: 1685 to 1745,” Numerical Algorithms, vol. 83, no. 4, pp. 1259-1275, 2020.

T. Talsma and J. Parlange, “One dimensional vertical infiltration,” Soil Research, vol. 10, no. 2, pp. 143-150, 1972.

J.-Y. Parlange, I. Lisle, R. Braddock, and R. Smith, “The three-parameter infiltration equation,” Soil Science, vol. 133, no. 6, pp. 337-341, 1982.

P. K. Swamee, P. N. Rathie, L. C. de SM Ozelim, and A. L. Cavalcante, “Recent advances on solving the three-parameter infiltration equation,” Journal of Hydrology, vol. 509, pp. 188-192, 2014.

D. Barry, J.-Y. Parlange, L. Li, D.-S. Jeng, and M. Crapper, “Green-ampt approximations,” Advances in Water Resources, vol. 28, no. 10, pp. 1003-1009, 2005.

A. Q. d. Almeida, A. Ribeiro, Y. G. Paiva, N. Rascon Jr, E. P. Lima, et al., “Geoestatística no estudo de modelagem temporal da precipitação,” Revista Brasileira de Engenharia Agrícola e Ambiental, vol. 15, no. 4, pp. 354-358, 2011.

B. B. d. Silva, A. C. Braga, C. C. Braga, L. M. M. d. Oliveira, J. D. Galvíncio, and S. M. G. L. Montenegro, “Evapotranspiração e estimativa da água consumida em perímetro irrigado do semiárido brasileiro por sensoriamento remoto,” Pesquisa Agropecuária Brasileira, vol. 47, no. 9, pp. 1218-1226, 2012.

B. D. Pimenta, A. D. Robaina, M. X. Peiter, A. C. Pereira, S. A. Rodrigues, and M. V. Loregian, “Desempenho e precisão de equaçõe explícitas do coeficiente de perda de carga em regime de fluxo turbulento,” Revista Brasileira de Agricultura Irrigada, vol. 12, no. 2, p. 2443, 2018.

S. Ali and A. Islam, “Solution to green-ampt infiltration model using a two-step curve-fitting approach,” Environmental earth sciences, vol. 77, no. 7, pp. 1-9, 2018.

D. N. Moriasi, J. G. Arnold, M. W. Van Liew, R. L. Bingner, R. D. Harmel, and T. L. Veith, “Model evaluation guidelines for systematic quantification of accuracy in watershed simulations,” Transactions of the ASABE, vol. 50, no. 3, pp. 885-900, 2007.

N. Suryoputro, W. Soetopo, E. S. Suhartanto, L. M. Limantara, et al., “Evaluation of infiltration models for mineral soils with different land uses in the tropics,” Journal of Water and Land Development, vol. 37, no. 1, pp. 153-160, 2018.

C. J. Willmott, “On the validation of models,” Physical geography, vol. 2, no. 2, pp. 184-194, 1981.

A. L. Fernandes, E. F. Fraga Júnior, and B. Y. Takay, “Avaliação do Método penman-piche para a estimativa da evapotranspiração de referência em uberaba, mg,” Revista Brasileira de Engenharia Agrícola e Ambiental, vol. 15, no. 3, pp. 270-276, 2011.

H. Xiang-Wei, S. Ming-An, and R. Horton, “Estimating van genuchten model parameters of undisturbed soils using an integral method,” Pedosphere, vol. 20, no. 1, pp. 55-62, 2010.

R. Duan, C. B. Fedler, and J. Borrelli, “Field evaluation of infiltration models in lawn soils,” Irrigation Science, vol. 29, no. 5, pp. 379-389, 2011.

N. Romano and P. Nasta, “How effective is bimodal soil hydraulic characterization? functional evaluations for predictions of soil water balance,” European Journal of Soil Science, vol. 67, no. 4, pp. 523-535, 2016.

G. Gitirana Jr and D. Fredlund, “Statistical assessment of hydraulic properties of unsaturated soils,” Soils and Rocks, p. 81, 2016.

#### Article Metrics

_{Metrics powered by PLOS ALM}

### Refbacks

- There are currently no refbacks.

**Trends in Computational and Applied Mathematics**

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)

Indexed in: