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Infiltration is an important component of the hydrological cycle. It provides soil moisture in the vadose zone to support plant growth. This study was conducted to compare the validity of four infiltration models with measured values from the double ring infiltrometer. The parameters of the four models compared were estimated using the linear regression analysis. The C.C was used to show the performance of the predictability of the models. The RMSE, MAE and MBE were employed to check the anomalies between the predicted and the observed values. The results showed that, average values of the C.C ranged from 0.9294 - 0.9852. The average values of the RMSE were 4.0033, −17.489, 11.2400 and 49.8448; MAE were 3.1341, 15.9802, 10.6525, and 61.4736; and MBE were 0.0786, 9.5755, 0.0007 and 47.0204 for Philip, Horton, Green Ampt and Kostiakov respectively for the wetland soils. Statistical results also from the Fisher’s multiple comparison test show that the mean infiltration rate estimated from the Green Ampt’s, Philip’s and Horton’s model was not significantly different (p > 0.05) from the observed. The results indicated that the Kostiakov’s model had the highest deviations as it overestimated the measured data in all the plots. Comparison of the statistical parameters C.C, RMSE, MAE, and MBE for the four models indicates that the Philip’s model agreed well with the measured data and therefore, performed better than the Green Ampt’s, Horton’s and Kostiakov’s models respectively in that order for Besease wetland soils. Estimation of infiltration rate by the Philip’s model is important in the design of irrigation schemes and scheduling. Therefore, in the absence of measured infiltration data, the Philip’s model could be used to produce infiltration information for inland valley bottom soils that exhibit similar characteristic as Besease wetland soils.

Infiltration is the process by which water on the ground surface enters the soil. Infiltration plays a vital role in soil and water conservation as it determines the amount of runoff over the soil surface during irrigation and precipitation. The infiltration rate of a soil, thus ability of the soil to accept heavy rainfall or irrigation depends on the characteristics of the soil [

Design, operation and management of surface irrigation system rely greatly on the infiltration behaviour or characteristics of the soil, because the infiltration behaviour of the soil directly determines the essential variables such as inflow rate, length of run, application time and depth of percolation [

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The study sought to evaluate the performance of four infiltration models (Kostiakov’s, Philips, Horton’s and Green Ampt) to determine their suitability for predicting infiltration rates for Besease wetland soils.

Besease is a predominantly farming area in the Ejisu Municipal District of the Ashanti Region in Ghana. The site lies within Latitude 1˚15'N and 1˚45'N and Longitude 6˚15'W and 7˚00'W. The study area covers about 72 ha of the valley bottom lands at Besease (^{2} [

The study area is located in the moist semi-deciduous forest zone. Grass species prominently found in the valley bottom are Santrocema trifolia, Chromolaeve ordorata, Imperata cylindrical, Mimosa pigra,Ceiba patendra, Centrosema pubescens and Mariscus flabelliformis. Plant species like Raphia hookeri (Raphia palm), Alstonia boonei, Malotus oppositifolius and Pseudospondias microcarpa extends along the margins of the Oda River. Soils of the Ejisu-Besease can be found in the soil map of Kumasi area. The study area lies in the Offin soil series which are grey to light brownish grey, poorly drained alluvial sands and clays developed within nearly flat but narrow valley bottoms along streams. The series have very slow internal drainage, very slow runoff, rapid permeability and moderate water holding capacity. The geology of the watershed is relatively heterogeneous and mainly composed of Phyllites, quartzite, shale, Tarkwain and Voltaian-sandstone and limestone. The Phyllites which underlie 59% of the area

consist of upper and lower Birimian rocks. Very few rock outcrops were encountered in the survey as the rocks are deeply weathered. The weathered phyllite is soft and easily broken, recognizable pieces and is typically found at 2 - 3 m below surface. Soils found within the Oda River catchment are grouped as those derived from granites, sandstones, alluvial materials, greenstone, andesite, schist and amphibolities. Specifically, the soils are Orthi-ferric Acrisol, Eutric Fluvisol, Gleyic Arenosols, Eutric Gleysols and Dystri-Haplic Nitisol. The Besease aquifer is composed of heterogeneous sequence of layers which is dominated by sand, clayey sand and silts. The valley bottom is developed by small holder farmers who cultivate rice in the wet season and also grow vegetables like cabbage, lettuce, bell pepper, cauliflower, cucumber and okra. Other cereals like maize are cultivated at the dry season when the water table is low.

Soil samples were collected with core samplers of height 10 cm to an average depth of 100 cm (

Double ring infiltrometers, consisting of two concentric rings, were used to measure the infiltration rate. Rings were 250 mm deep and were made from 12-guage steel with sharpened bottom edges. They were driven into the ground to 50 mm depth. Grass was cut to near soil level and a pad was placed inside the inner ring to prevent puddling. The inner and outer edges were tamped to seal possible cracking. Generally, the water level was kept at or above 50 mm depth. The difference in height between the inner and outer rings was kept to a minimum. The rate of fall of water was measured in the inner ring while a pool of water was maintained at approximately the same level in the outer ring to reduce the amount of lateral flow from the inner ring. The rate of fall of the water level in the inner cylinder was measured at 2, 3, 5, 10, 15, 20, 30, 45 and 60 minutes and at 30-minute intervals thereafter. The accumulated volume of water entering the soil was converted to the infiltration rate (mm/h) and was plotted against elapsed time whereby a declining slope was obtained. Fifty-five (55) samples (replicates) were used for the measurement of soil infiltration rate. The field infiltration rate measurement was considered as observed. The aim of the measurements was to obtain a steady-state infiltration rate. This is achieved when the amount of infiltrated water was constant in time, i.e. when the infiltration curve (instantaneous infiltration against time) levels out. To estimate the infiltration rate at steady state, the terminal infiltration rate (i.e. the infiltration rate obtained at the end of the experiment in about 2 h), was used as an approximation of the steady state infiltration rate.

In this study, Kostiakov’s, Philip’s, Horton’s and Green Ampt’s infiltration models were fitted to the infiltration data.

Kostiakov’s model, an empirical model expresses cumulative infiltration equation as

F p = a t b (1)

where F p = cumulative infiltration (cm), t = time from start of infiltration (min), and a and b are constants that depends on the soil initial conditions. Where, a > 0 and 0 < b < 1.

The parameters in the Kostiakov equation are obtained from the plot of ln ( F p ) versus ln ( t ) and the best fit straight line through the plotted points gives as the intercept and b as the slope.

Philip’s physical based model expresses infiltration rate as

f p = 1 2 s t − 1 / 2 + K (2)

where f p = Infiltration capacity (cm/h) at any time t (min) from the start, S = soil water sorptivity which is a function of initial soil water content and K = Darcy’s hydraulic conductivity. The observed infiltration rate, f p values are

plotted against the reciprocal square root of time, t − 1 2 . The best fitting straight line through the plotted points gives K as the intercept and s 2 as the slope of the line.

Horton’s semi-empirical model expressed the decay of infiltration capacity with time as an exponential decay given by

f p = f c + ( f 0 − f c ) e K h t for 0 ≥ t ≤ t_{c} (3)

where f p = infiltration capacity (cm/h) at any time t (min) from the start of the rainfall

f 0 = initial infiltration capacity (cm/h) at t = 0

f c = final steady state infiltration capacity (cm/h) at t = t_{c}.

K h = Horton’s decay coefficient which depends upon soil characteristic and vegetation cover. The parameters of the Horton’s equation are determined by plotting the values of ln ( f p − f c ) against to obtain the best fit straight line through the plotted points. ln ( f 0 − f c ) depicts the intercept and the decay constant, K_{h} represent the slope.

Green Ampt proposed a model for infiltration capacity based on Darcy’s law and expresses the physical model as

f p = m + n / F p (4)

where m and n are Green Ampt’s parameters of infiltration model. Values of in-

filtration capacity, f p are plotted against 1 F P on an arithmetic graph. The in-

tercept on the ordinate axis is m and n serves as the slope when the best fit straight line is drawn through the plotted points.

Coefficient of correlation is a statistical measure that calculates the strength of the relationship between the relative movements of two variables. The coefficient of correlation is calculated as

C C = z ∑ a b − ( ∑ a ) ( ∑ b ) z ( ∑ a 2 ) − ( ∑ a ) 2 z ( ∑ b 2 ) − ( ∑ b ) 2 (5)

The root mean square error exaggerates the prediction error, thus the difference between the predicted value and the actual value. This is evaluated by

R M S E = 1 N ( ∑ i = 1 n ( a i − b i ) 2 ) (6)

where a is the calculated and b is observed values of the infiltration rate and N is the number of observations.

This is the average difference between the predicted values and the observed values of the infiltration models. The mean bias error is estimated by

M B E = 1 N ∑ i = 1 n ( a i − b i ) 2 (7)

where a is the calculated and b is observed values of the infiltration rate and N is the number of observations.

The absolute error is the absolute value of the difference between the predicted value and the observed value. The absolute error is estimated by

M A E = 1 N ∑ i = 1 n | a i − b i | (8)

where a is the calculated and b is observed values of the infiltration rate and N is the number of observations.

Kostiakov’s, Philip’s, Horton’s and Green Ampt’s infiltration models were used to predict soil infiltration rate using the 55 (replicates) observed field infiltration rate. The replicates were used to compute means and standard deviation for each model and the observed in IBM SPSS version 23. One-way analysis of variance (ANOVA) was used to determine whether there are any statistically significant differences in the infiltration rates among the four different infiltration rate models and the observed at α = 5% significance level. Fisher Multiple comparison post hoc test was used to separate the means.

Results from

A comparison between the measured and estimated infiltration rates as calculated from the four models is shown in

Site | Soil type | Initial infil. rate (cm/h) | Final infil. rate (cm/h) | Moisture Cont. (%) |
---|---|---|---|---|

P7 - P8 | silt loam | 270 | 37.5 | 18.3 |

P6 - P9 | silt loam | 36 | 2.8 | 17.5 |

P1 - P9 | sandy loam | 24 | 3.3 | 16 |

P13 - P14 | sandy loam | 45 | 9 | 11.5 |

P1 - P2 | sandy loam | 24 | 1.2 | 11.8 |

P10 - P1 | silt loam | 270 | 42 | 20 |

Test Site | Philip’s model | Green Ampt’s model | Kostiakov’s model | Horton’s model | |||||
---|---|---|---|---|---|---|---|---|---|

s | k | m | n | b | a | k | f_{0} | f_{c} | |

Site: P7 - P8 | 103.92 | −23.68 | 40.76 | 305.1 | 0.84 | 205.02 | 2.42 | 231.68 | 37.5 |

Site: P6 - P9 | 14.97 | −9.26 | −2.43 | 47.68 | 0.67 | 15.08 | 5.65 | 37.67 | 2.4 |

Site: P1 - P9 | 10.94 | −5.7 | 1.97 | 20.02 | 0.8 | 16.35 | 3.28 | 21.57 | 2.2 |

Site: P13 - P4 | 15.16 | −0.27 | 8.88 | 57.43 | 0.92 | 40.84 | 2.03 | 33.15 | 9 |

Site: P1 - P2 | 10.03 | −6.9 | −4.21 | 23.17 | 0.55 | 6.68 | 6.68 | 23.06 | 1.2 |

Site: P10 - P11 | 104.78 | −10.63 | 59.43 | 2217.8 | 0.85 | 224.17 | 2.16 | 212.85 | 46.8 |

Test Site | Philip’s model | Horton’s model | Green-Ampt’s model | Kostiakov’s model |
---|---|---|---|---|

coefficient of correlation (C.C) | ||||

P10 - P11 | 0.993 | 0.9477 | 0.8986 | 0.9838 |

P1 - P2 | 0.9663 | 0.9079 | 0.9833 | 0.9255 |

P13 - P4 | 0.9863 | 0.8941 | 0.985 | 0.9242 |

P1 - P9 | 0.9921 | 0.9597 | 0.9139 | SS0.9789 |

P6 - P9 | 0.9783 | 0.9289 | 0.9833 | 0.9789 |

P7 - P8 | 0.9955 | 0.9378 | 0.9536 | 0.9746 |

Average | 0.9852 | 0.9294 | 0.9529 | 0.9522 |

Root means square error (RMSE) | ||||

P10 - P11 | 9.5465 | 39.8549 | 35.3858 | 127.1988 |

P1 - P2 | 1.9409 | 3.5617 | 1.3734 | 4.3387 |

P13 - P4 | 1.91 | 7.3604 | 1.9929 | 25.0485 |

P1 - P9 | 0.967 | 4.354 | 3.1239 | 10.687 |

P6 - P9 | 2.215 | 5.7666 | 1.942 | 9.6733 |

P7 - P8 | 7.4405 | 44.0367 | 23.6221 | 122.1226 |

Average | 4.0033 | 1.7489 | 11.24 | 49.8448 |

Mean absolute error (MAE) | ||||

P10 - P11 | 7.1456 | 35.2567 | 32.7114 | 121.52 |

P1 - P2 | 1.6478 | 2.7967 | 1.1234 | 3.9711 |

P13 - P4 | 1.3022 | 6.6778 | 3.9716 | 23.7778 |

P1 - P9 | 0.7733 | 4.0167 | 2.8773 | 9.9289 |

P6 - P9 | 1.746 | 5.041 | 1.5755 | 93.573 |

P7 - P8 | 6.19 | 42.0922 | 21.6556 | 116.0711 |

Average | 3.1341 | 15.9802 | 10.6525 | 61.4736 |

Mean bias error (MBE) | ||||

P10 - P11 | 0.1944 | 14.6522 | −0.0034 | 121.52 |

P1 - P2 | 0.0189 | 1.6278 | 0.0033 | 2.3867 |

P13 - P4 | 0.0267 | 3.6978 | 0.0015 | 23.7778 |

P1 - P9 | 0.02 | 3.0344 | −0.003 | 9.9289 |

P6 - P9 | 0.026 | 4.175 | −0.003 | 8.438 |

P7 - P8 | 0.1856 | 30.2656 | 0 | 116.0711 |

Average | 0.0786 | 9.5755 | −0.0007 | 47.0204 |

infiltration rate estimated by the Philip’s model was the most successful in predicting fitting measured experimental data.

Statistical results from

The average values of the RMSE were 4.0033, 17.489, 11.2400 and 49.8448, MAE were 3.1341, 15.9802, 10.6525, and 61.4736, and MBE were 0.0786, 9.5755, −0.0007 and 47.0204 for Philip, Horton, Green Ampt and Kostiakov respectively for the entire study area (

Methods | Mean Infiltration rate (cm/h) | Std. Error of Mean (cm/h) |
---|---|---|

Observed | 44.44b | 9.12 |

Green Ampt | 44.44b | 8.81 |

Horton | 53.92b | 8.90 |

Kostiakov | 90.76a | 13.95 |

Philip | 44.52b | 9.11 |

F-ratio | 3.89 | |

df | 4 | |

P-value | 0.004 |

Means that do not share a letter are significantly different by Fisher’s multiple comparison test.

measured data and therefore, performed better than the Green Ampt’s, Horton’s and Kostiakov’s models respectively in that order for Besease wetland soils. This result corroborates with the findings of [

The prediction accuracy of four infiltration models was validated with measured values using the double ring infiltrometer. Comparison of the field and predicted infiltration rate indicated that the infiltration rate predicted by the Philip’s model was much closer to the observed data. The statistical results of C.C show that infiltration rate can be predicted by the Philip, Green Ampt, Horton and Kostiakov models, respectively.

Statistical results also from the Fisher’s multiple comparison test show that the mean infiltration rate estimated from the Green Ampt’s, Philip’s and Horton’s model was not significantly different (p > 0.05) from the observed.

Based on the mean values of RMSE, MAE and MBE values, the Philip’s model provided the lowest values and could be deduced that infiltration rate was well described by this model. Quantification of infiltration rate by this model will be of importance in the design of irrigation schemes and scheduling of irrigation. Therefore, in the absence of measured infiltration data, the Philip’s model could be employed to generate infiltration information for inland valley bottom soils that exhibit similar characteristics of Besease wetland soils.

The authors acknowledged the Ministry of Food and Agriculture (MoFA) for providing monetary support to this research work and Mr. Frank Boakye of Grains Development Board, a subsidiary of MoFA for assisting in conducting field work.

The studies reported in this publication, were supported by a grant from the Ministry of Food and Agriculture (MoFA), Ghana. The author is a lecturer at the University of Energy and natural Resources, Ghana. The terms of this arrangement have been reviewed and approved by the University of Energy and natural Resources at Sunyani in accordance with its policy on objectivity in research.

Thomas, A.-D., Ofosu, A.E., Emmanuel, A., De-Graft, A.J., Ayine, A.G., Asare, A. and Alexander,^{ }A. (2020) Comparison and Estimation of Four Infiltration Models. Open Journal of Soil Science, 10, 45-57. https://doi.org/10.4236/ojss.2020.102003