Evaluation and comparison of infiltration models for estimating infiltration capacity of different textures of irrigated soils

Abstract

Accurate estimation of infiltration rates is crucial for effective irrigation system design and evaluation by optimizing irrigation scheduling, preventing soil erosion, and enhancing water use efficiency. This study evaluates and compares selected infiltration models for estimating water infiltration rates in the Shillanat-iv irrigation scheme in northern Ethiopia. Soil samples were collected to determine textural classes using hydrometer soil texture analysis and the United States Department of Agriculture (USDA) textural triangle. The soil textural map of the study was created using the inverse distance weight interpolation technique in ArcGIS version 10.4. Infiltration rates were measured using the double-ring infiltrometer for five soil textures: clay loam, loam, sandy clay loam, clay, and sandy loam. Six infiltration models (Kostiakov, Modified Kostiakov, Revised Modified Kostiakov, Philip, Horton, and Novel) were employed and evaluated using statistical parameters. Model calibration and validation were conducted using data from 38 points within the study area. The parameter values of the infiltration models were optimized using SPSS statistical software using least-squares errors. The results showed that, Revised Modified Kostiakov, Modified Kostiakov, and Novel infiltration models demonstrated superior capability in estimating infiltration rates for clay loam, loam, and sandy loam soil textures, respectively. Horton's model outperformed other models in estimating infiltration rates for both sandy clay loam and clay soil textures. The appropriately fitted infiltration models can be utilized in designing the irrigation system to estimate the infiltration rate of soil textures within the selected irrigation scheme and at similar sites with comparable soil textures.

Introduction

Infiltration is the vertical downward movement of water into soils from different surface sources such as snowfall, precipitation, flood, and irrigation (Mazighi et al. 2023; Vand et al. 2018; Bajirao and Vishnu 2023). Infiltration capacity is particularly an important parameter in various fields such as hydrology, agriculture, urban drainage, and the design and management of irrigation schemes (Gebul 2022). Knowledge of soil infiltration is required to predict the amount of water required to fill the root zone, establish the length of irrigation run, and estimate percolation losses in irrigation systems (Utin and Oguike 2018). Soil infiltration rate also indicates the composite soil's physical properties such as texture, structure, porosity, and hydraulic conductivity (Bajirao and Vishnu 2023). Lack of information about soil infiltration can be a source of various problems such as surface flooding, surface and groundwater pollution, water table dropdown, waterlogging of irrigated farms, and inappropriate application of irrigation water to crops.

Infiltration capacity is the ability of a soil layer for a particular land cover to percolate water till a constant rate is achieved over time (David et al. 2022). The prediction of soil infiltration capacity is not straightforward because of its variability over space and time and the difficulty of selecting proper methods/models and consequently determining the parameters of the models which depend on soil properties and meteorological characteristics. Sometimes, it may be necessary to use infiltration models than field measurements as field measurements are often time-consuming, expensive, and tedious (Rajkai et al. 2004). As a result, several studies (Musa et al. 2023; Amami et al. 2021; Bajirao and Vishnu 2023; Mazighi et al. 2023) use coarsely available soil databases for determining the infiltration capacity of soils of a particular area. Since the parameters used in infiltration models are highly dependent on the soil textures and surface conditions, a field test is still necessary for the determination of model parameters (Utin and Oguike 2018).

An accurate infiltration model, predicting the real infiltration capacity, is required to estimate runoff initiation time, planning of irrigation systems, and management of water resources (Mirzaee et al. 2014). Several infiltration models have been developed to evaluate the infiltration rate of soils. Infiltration models are classified into three main groups, namely physical, semi-empirical and empirical models (Bayabil et al. 2019). Empirical infiltration models are widely used for their simplicity and effectiveness in fitting observed infiltration rates (Bayabil et al. 2019). Empirical infiltration models are derived from experimental fields and laboratory measurements involving cumulative infiltration as a function of time (Gebul 2022). Several infiltration models have been developed, but their suitability for the real world remains less clear. Hence, it is not always evident as to which model better suits a particular condition (Oku and Aiyelari 2011; Mishra et al. 2003).

Many comparative studies have been done on different infiltration models for estimating infiltration capacity of soils (Amami et al. 2021; Bajirao and Vishnu 2023; Failache and Zuquette 2021; Kindo 2023; Mazighi et al. 2023; Rasool et al. 2021; Santra et al. 2021; Sihag et al. 2017; Sihag and Singh 2018). Mishra et al. (2003) analyzed fourteen different infiltration models using data collected from field and laboratory tests. Semi-empirical models (e.g., Horton, Holtan and Singh-Yu) performed better than other infiltration models. Zolfaghari et al. (2012) analyzed seven infiltration models for computing cumulative infiltration rate. It was found that the Modified Kostiakov and revised modified Kostiakov models got higher ranks than other models. Owing to their simplicity and yielding reasonably satisfactory results in most applications, the Kostiakov (KM), modified Kostiakov (MKM) and revised modified Kostiakov (RMKM) empirical infiltration models have been quite popular and frequently used in various water resource applications (Mazighi et al. 2023; Parhi 2014; Parhi et al. 2007; Ogbe et al. 2011; Rasool et al. 2021). The Modified Kostiakov, Horton, Holtan and Singh-Yu models were applied to accurately estimate infiltration rates for land use and management types used with Ferralic Arenosols and Rhodic Ferralsols (Failache and Zuquette 2021).

Oku and Aiyelari (2011) deduced that Philip’s model was a better fit than the Kostiakov’s model for estimating the infiltration capacity of the soil of a humid forest in southern Nigeria. Amami et al. (2021) also analyzed three commonly used infiltration models (Kostiakov, Philip and Horton) to predict infiltration rates under tilled and untilled conditions. The Philip model was better at predicting the measured infiltration rates under all tillage treatments. Philip’s model outperformed the other four infiltration models to estimate the infiltration rate for specific land use and soil parameter models in the Chhattisgarh plains (Kindo 2023). A novel infiltration model was developed in a field based on field measurement results and its performance was the best among the other three infiltration models, namely Kostiakov, Modified Kostiakov and US Soil Conservation Service (Mazighi et al. 2023; Sihag et al. 2017). Santra et al. (2021) found that the Horton model was superior to fit measured infiltration data than the Philip model for sandy soils. Rasool et al. (2021) also found that the Horton model was the most suitable infiltration model with the highest performance followed by the Philip model for different land covers under clay and sandy clay textures. Several researches indicated that Philip’s, modified Philip’s, Kostiakov’s, modified Kostiakov’s and Horton’s infiltration models are the most commonly used methods worldwide for estimating the infiltration capacity of soils (Karahan and Pachepsky 2022; Utin and Oguike 2018).

Determining an appropriate infiltration model for a specific irrigation scheme helps to determine the optimal amount of irrigation water and irrigation time and then improve the water use efficiency of irrigation systems (Lu et al. 2019; Sun et al. 2022). Understanding the infiltration characteristics of soils during the development and management of irrigation systems is essential for the prediction of the amount of water to be applied (AL-Kayssi and Mustafa 2016). If infiltration characteristics are not determined well, irrigation performance would be overestimated (Nie et al. 2019).

Though there are several studies worldwide on infiltration models for different irrigation methods under different soil types (AL-Kayssi and Mustafa 2016; Lu et al. 2019; Sun et al. 2022), there is lack of universally accepted infiltration parameters for the infiltration models and vary across various textures of soil. Oku and Aiyelari (2011) indicated that not all models are universally applicable to all types of soils. The need for continuous in-depth and field specific study of the applicability of infiltration equations cannot be over-emphasized because models parameters and performance vary from point to point and vary with time (Farid et al. 2019). Knowing infiltration model parameters of different textures can significantly enhance existing knowledge on infiltration models and assist future designers in determining a basic infiltration rate of soil textures based solely on texture without the need of conducting infiltration tests. Thus, the evaluation of infiltration models in Northern Ethiopia can help to identify suitable infiltration models for the soils in the study area. Other arid and semi-arid regions with similar soil textures can benefit from the findings of this study and use them as a reference when selecting infiltration models for their specific contexts.

This study is selected because the region has a significant agricultural sector, with irrigation being a crucial component of agricultural practices. Understanding the infiltration capacity of different soil textures in this region can have practical implications for water management, irrigation planning, and optimizing agricultural productivity. The findings of the study can have direct implications for irrigation practices in northern Ethiopia by identifying and comparing infiltration models, providing valuable guidance for farmers, water resource managers, and policymakers in enhancing agricultural productivity, water use efficiency, and sustainable land management practices in the region. The fieldwork for finding the infiltration rate under different soil types could be time-consuming and troublesome, so it is necessary to find out the best infiltration models with due validation for the site-specific condition of the soil textures in the irrigation scheme. Therefore, the objectives of this study were; (a) to estimate model parameters of selected infiltration models for different soil textures, (b) to evaluate and compare the performance of selected infiltration models for different soil textures in an irrigated scheme.

Materials and methods

Description of the study area

The study was conducted in the Shillanat-iv irrigation scheme in the Tigray region, north of Ethiopia (Fig. 1), located between 13^o06′40.9" and 13^o07′34.4"N latitude and 39^o29′14.7" to 39^o29′50.3"E longitude. The topography of the irrigation area is almost flat, whose altitude ranges between 2060 to 2080 m above mean sea level. The source of water for this irrigation scheme is runoff water stored by an earthen dam. It has a total irrigated land of 124 ha. The dominant crops grown in the study area are sorghum, maize, wheat and some vegetables such as potatoes and tomatoes.

Fig. 1

Location map of the study area

The area has mainly a semiarid climate with a mean annual rainfall of 912 mm, and the mean daily temperature ranges between 9 °C and 23 °C (Mesfin et al. 2022). Agriculture is both rain-fed and irrigation led by small-scale farmers with an average landholding size of less than one hectare per household and it is the mainstay of the economy. Agriculture directly supports about 86.4% of the population for employment and livelihood (Ros 2013).

Soil sampling

The most commonly used method for soil sampling is a grid method (Hardy 2014). A grid size of 1 hectare and a depth of 30 cm (0 to 15 cm and 15 to 30 cm) is commonly used for cultivated lands (Soil Survey staff 2009). As recommended by Soil Survey staff (2009), grid sizes of 1 hectare were used for randomly sampling soils from the study area. The irrigation scheme has a border that separates the irrigated land into three zones. Thus, before sampling, the study area was classified into three zones based on visual observation of the borders of the irrigation scheme. Then, a total of 32 sampling points (Fig. 2) were randomly selected from the irrigation scheme, in which 10 points were from Zone 1, 8 points were from Zone 2 and 14 points were from Zone 3.

Fig. 2

Soil sampling points in Shillanat-iv irrigation area

Determination of soil properties

Texture of soil can be determined by estimating the three basic components of soil such as sand, silt and clay (Gangwar et al. 2019). Soil texture of the irrigation scheme was determined from the sampling points indicated in (Fig. 2) using sampling auger. All the soil samples were dried at room temperature to remove the water content and particles more than 2 mm were separated using sieve analysis method (Fig. 3b). The physical proportions of three primary sizes of soil particles such as sand, silt and clay have been estimated by their settling rates in an aqueous solution with the help of soil hydrometer as per ASTM-D422 (Fig. 3c). As per USDA specifications, the size of sand ranging from 2 to 0.050 mm, silt 0.050 to 0.002 mm and clay less than 0.002 mm has been considered for their particle size analysis (Huluka and Miller. 2014; Gangwar et al. 2019). Finally, the soil texture was determined using the USDA textural triangle having the percentages of sand, silt and clay soils (Groenendyk et al. 2015).

Fig. 3

Soil and infiltration testing

Undisturbed soil samples were also collected from the irrigation scheme near the infiltration measurement site using core sampler of 5 cm height and 5 cm diameter (Fig. 3a) to determine the soil properties such as saturated hydraulic conductivity (Ks), initial moisture contents (MC), soil bulk density (ρb), and total porosity.

The gravimetric method as described by Wilke ( 2005) was used to determine initial moisture content, porosity and bulk density for each selected sites of the irrigation scheme. For the determination of water content, soil samples were collected with standard samplers (core method) and taken to the laboratory while kept in plastic bags to avoid evaporation loss. Initially, the wet weights of the samples were determined by weighing. Then the samples were oven-dried at 105°C for 24 h and then the weights were measured to determine the dry weight of the samples. In the process, the weight of water lost is determined. The initial moisture contents (MC), soil bulk density (ρ_b), and total porosity (P) were then determined from Eq. (1), (2) and (3) respectively.

$$MC =\,\ \frac{{M_{w} }}{{M_{d} }}$$

(1)

$${\rho }_{b}=\frac{{M}_{s}}{{V}_{T}}$$

(2)

$$P=1-\frac{{\rho }_{b}}{{\rho }_{s}}$$

(3)

where; M_w = mass of water (g), M_s = mass of dry soil (g), V_T = total volume of soil in the core sampler (cm³) and ρ_s is particle density of the soil (ρ_s = 2.65g.cm⁻³), recommended by studies (Santos et al. 2022).

The initial soil moisture content can affect the infiltration rate and cumulative infiltration capacity of soils and affects parameters of the infiltration models (Zheng and Gao 2020; Ketsela et al. 2023). The influence of initial moisture content on some infiltration model parameters such as the Kostiakove, Horton, and Philips model parameters was investigated by Zheng and Gao (2020). The study showed that there were some changes in parameters as the initial moisture content was modified. In the three models, most of the affected parameters decreased with increasing initial water content.

From the undisturbed soil samples, the saturated hydraulic conductivity (Ks) of the soil samples were also determined by constant head permeameter method in the laboratory. With careful inspections for loosening of soils during core sampling, only the intact soil samples were fully saturated in the laboratory. Thereafter, the saturated soil samples were imposed to a hydraulic-head difference and the resulting flux of water was measured until a steady flow rate was obtained. The saturated hydraulic conductivity of the soil samples was measured under a constant head (Mahapatra et al. 2020).

Soil texture mapping

The soil texture map of the irrigation scheme was prepared using an inverse distance weighting (IDW) method in ArcGIS version 10.4. The IDW method is the most frequently used spatial interpolation method in environmental studies (Li and Heap 2011). The mapped soil texture was validated by dividing the total 32 soil sampling points into two data sets; training data sets and testing data sets. During soil texture mapping, 70% of the total data sets were used during training and the remaining 30% of the data sets were used for testing the soil texture map. Thus, in this study 23 and 9 of the total sampling points were used for training and validation of the soil texture map, respectively.

Infiltration measurement

The infiltration capacity of the soil in the study area was measured using a double-ring infiltrometer for selected sites in the irrigation scheme. As shown in Fig. 3d, the double ring infiltrometer has two concentric rings having diameters of 28 cm and 55 cm for the inner and outer rings, respectively with 25 cm depth. Both rings were driven at about 10 cm deep into the soil by a hammer without disturbing the soil surface. First, the annular space was filled with water, followed by the inner space and the initial reading of the water level in the inner space was noted. The water level in the inner space was noted at regular intervals until the rate of infiltration became steady (Vand et al. 2018).

Infiltration measurements were carried out in the field to deal with spatial variability of infiltration rate for the five soil textures of the irrigation scheme considering soil texture variability (Table 1). A total of 38 infiltration tests were conducted in the irrigation scheme, 10 tests were for clay soil, 8 tests were for each of loam and sandy clay loam, and 6 tests were for each of clay sandy loam soils. As Vand et al. (2018) indicated, the total measurement of infiltration data was divided into two data sets. Thus, 70% of the data was used for calibration of the infiltration models and 30% of the data was used for validation of the infiltration models.

Table 1 Number of infiltration tests for calibration and validation of the infiltration models

Infiltration Models

Referring to previous studies (Mishra et al. 2003; Zolfaghari et al. 2012; Mirzaee et al. 2014; Parhi 2014; Mazighi et al. 2023; Rasool et al. 2021; Amami et al. 2021; Failache and Zuquette 2021; Karahan and Pachepsky 2022; Santra et al. 2021), six infiltration models namely, Kostiakov model, Modified Kostiakov model, Revised Modified Kostiakov model, Horton’s model, Novels model, and Philip Model were found to be a good fit for estimating the infiltration capacity of soil textures.. These models, are described in Table 2.

Table 2 Description of the infiltration models used in this study and their parameters

Estimation of the infiltration model parameters: calibration process

Infiltration model parameters were estimated using the method of least squares, the best fit is obtained by minimizing the sum of the squares of differences between the observed and the model estimated values (Parhi 2014; Zakwan 2020). The most commonly used method of computing nonlinear least squares estimator, the Levenberg–Marquardt algorism in the SPSS software package was used (Prajneshu 2011), See Eq. 1.

$$MinSSE=\sum_{i=1}^{N}{\left[{f}_{obs}\left(i\right)-{f}_{com}\left(i\right) \right]}^{2}$$

(4)

where, MinSSE is the minimization of sum of squares of errors, f_obs is the observed infiltration rate at the i^th time, N is the number of observations and f_com is the estimated infiltration rate at i^th time.

Evaluation of the infiltration model performances: validation process

The most commonly used statistical measures, namely the Coefficient of determination (R²), Maximum absolute error (MAE), Bias, Root mean square error (RMSE) and Percentage average error (PAE) were used for evaluating the performance of the infiltration models (Table 3). The model with higher values of R² and correspondingly lower values of MAE, Bias, RMSE and PAE was considered to give a good fit of the field-measured data to model estimations (Vand et al. 2018).

Table 3 Statistical measures used for evaluating performance of infiltration models

The overall data acquisition, data processing, selection and evaluation of the infiltration models for the study area is summarized in Fig. 4.

Fig. 4

Overall methodological framework for data collection, processing and evaluation of infiltration models

Results and discussion

Soil textures of the study area

Five soil textures namely clay, clay loam, sandy clay loam and sandy loam were found in the Shillanat-IV irrigation scheme. Out of the total number of soil samples, the numbers of samples obtained as clay, clay loam, loam, sandy clay loam, and sandy loam soil textures were 2, 12, 9, 7, and 2, respectively. The soil textures were similar at the depths of 0 to 15 cm and 15 to 30 cm below the ground surface, showing no variation in soil texture with depth. The soil texture variability, and percentages of sand, silt and clay are summarized in Table 4.

Table 4 Soil samples and textures of the study area

The minimum and maximum percentages of each soil texture in the irrigation scheme were obtained as Table 4. The percentages of sand, silt and clay for all soil textures ranged from 25 to 65%, 17 to 25% and 18 to 19% respectively. Thus the area has less percentages of clay and high percentages of sand.

Soil texture mapping

As shown in Fig. 5, the soil texture map of the irrigation scheme has five soil textures namely clay, sandy loam, loam, sandy clay loam and clay loam with area coverage of 1.1, 1.5, 26.9, 38.1 and 32.4%, respectively. The majority of the soil textures in the selected study area were clay loam soils followed by sandy clay loam and loam soils. Clay and sandy loam soil textures cover small areas of the irrigation scheme.

Fig. 5

Soil textural map of Shillanat-iv irrigation scheme

Validation of soil textural mapping

Out of the total number of soil sampling points, 30% of the data’s (9 samples) were selected for validating the soil textural map as shown in Fig. 6. From the total sampling points, 8 points were overlaid within corresponding soil textural maps and 1 testing point was almost near to the corresponding soil texture map. Therefore, the soil texture mapping created using IDW was good, because 88.9% of the soil texture testing points were overplayed on their corresponding soil textural maps.

Fig. 6

Validation of the soil texture map of the study area with point observed data

Similar studies were also conducted near the study area in the south-eastern of Tigray, in the Chelokot area (Gebre et al. 2015). Thus, the majority of soil textures in Chelokot were similar to the findings of this study, which were loam, clay loam, and sandy loam textures. Soil survey assessment was also conducted by Berhane et al. (2024), to characterize and map the distribution of the physical characteristics of irrigated soils from irrigated areas in Raya Azebo, northern Ethiopia. The results of their study were most of the soil texture in the irrigation schemes were clay and clay loam soils. Similarly, Seifu et al. (2023) were found Clay, sandy loam, loam soil textures are the majority of textures in Ayiba watershed, northern Ethiopia. Most of the soil textures found by the previous studies were similar to this study.

Soil physical properties

The soil physical properties of the study area showed little variability within the soil textures, but there is some variation from one soil texture to another (Table 5). For instance, loam had the highest average bulk density (1.46 g/cm³) followed by sandy loam (1.29 g/cm³), clay (1.30 g/cm³), sandy clay loam (1.26 g/cm³), and then the clay loam (1.12 g/cm³) in that order. Clay loam had the lowest average bulk density probably because of its high clay content and low sand fraction. The relatively lower average bulk density values obtained for clay may be attributed to the structural aggregation of the soils due to relatively high organic matter content (Seifu et al. 2023). Moisture contents of the soil textures also varied from 6.1 to 20.5% among the different soil textures. Clay loam had the highest moisture content (20.5%) and loam had the least moisture content (6.85%).

Table 5 Average soil physical properties of the irrigation scheme

There were also slight differences in average saturated hydraulic conductivity within the soil textures, with loam soil having the highest (0.28 cm/sec) and clay loam soil having the lowest (0.09 cm/sec) value. The average porosity among the soil textures varied from 44.90% (loam soil) to 57.74% (clay loam soil). Overall analysis indicated that the irrigation area has heterogeneous soil physical properties.

Similar studies indicated that the bulk density of clay, loam and sandy loam soil textures were 1.26–1.36 g/cm³, 1.43 g/cm³ and 0.86–3.58 g/cm³ respectively. The soil horizons bulk densities and available moisture contents were found to increase with depth (Mebrate et al. 2022; Berhane et al. 2024). The subsurface soil bulk density was higher than the surface layer, indicating the tendency of bulk density to increase as soil depth increases, due to the effects of the weight of the overlying soil and the corresponding decrease in soil organic matter content and clay content (Mebrate et al. 2022). Yimer et al. (2007) also reported that the application of organic materials from the plant system leads to a decrease in the surface soil bulk density than the layer below.

Infiltration rate and capacity of the soil in the irrigation scheme

Field infiltration tests for each soil textures of the irrigation scheme were carried out as shown in Table 1 and the average measured infiltration rates and capacities of each soil texture of the irrigation scheme was computed as shown in Table 6.

Table 6 Average measured infiltration rate and capacity soil textures in the irrigation scheme

The infiltration rate of soils has a general shape, starting from very high values, reducing sharply to almost half the initial values within the first 30 min, and continued to decrease gradually until a constant value is reached. Most of the runs attained this constant value within the first 1.67 h, in this time the infiltration capacity of the different soil textures was determined.

The variation of infiltration rate in the study area within soil textures was high. The initial infiltration rate in the first 2 min varied from 2.31 to 11.57, 2.93 to 8, 3 to 5.4, 3 to 4.8 and 2.55 to 3.6 cm/hr for clay loam, loam, sandy clay loam, clay and sandy loam soils, respectively. There was also variation in the steady infiltration rate within the soil textures; it varied from 0.6 to 2.4, 1.26 to 2.4, 1.5 to 1.86, 0.66 to 2.4 and 1.16 to 2.10 cm/hr for clay loam, loam, sandy clay loam, clay and sandy loam soils, respectively. The variations in infiltrations within the soil textures could be due to differences in soil physical properties such as moisture content, bulk density, saturated hydraulic conductivity and porosity. According to Gebre et al. (2015), the infiltration rate depends on soil texture. In sandy soils, the infiltration rate is higher than in silty soils and other fine-textured soils. In clay soil, the infiltration rate may be initially high (for heavy black clay with cracking), but it becomes low when the soil is moist to wet. Research findings demonstrated that clay has the lowest water storage capability, whereas sand could store a comparatively large amount of rainwater. In contrast, sand has the highest infiltration rate, followed by sandy loam, loam, and clay loam. Clay has the lowest infiltration rate (Kuok et al. 2023).

Determination of infiltration model parameters

A curve fitting between observed infiltration data and the infiltration model prediction was shown to determine the parameters of the infiltration models for each soil texture of the irrigation scheme. The fitting of infiltration models to measured soil infiltration data is presented in Fig. 7, for each soil texture.

Fig. 7

Curve fitting of the infiltration models for different soil textures of the irrigation scheme

Parameters of the infiltration models were also determined by fitting the measured infiltration data to the infiltration models as shown in Fig. 7. Thus, the average values of the parameters for the six selected infiltration models and the soil textures of the irrigation scheme is also in Table 7.

Table 7 Statistics of optimized parameters of the infiltration models for different soil textures

Referring to Table 7, the Kostikov’s infiltration model’s parameter α value was estimated to be in the range of 0.88 to 2.77. The infiltration decay constant β value was in the range of 0.12 to 0.55 considering all soil textures. The values of the infiltration decay constant β are described to be positive and always less than one. It has also been reported that most values of these parameters lie between 0.2 and 0.9 (Ogbe et al. 2011; Blair and Reddell 1983).

For the Modified Kostiakov’s model parameters α₁ and β₁ were estimated to be in the range of 0.12 to 0.84 and 0.52 to 0.89, respectively. These parameters are approximately similar to other studies (Mirzaee et al. 2014). For the Revised Modified Kostiakov model parameters α₂, β₂, α₃ and β₃ were estimated to be in the range of 0.04 to 3.81, − 0.51 to 3.02, -2.90 to 2.13 and − 10.66 to 6.46 respectively. Philip’s infiltration model parameters such as sorptivity ‘S’ values were found in the range of 0.53 to 4.59 and conductivity constant ‘Ko’ values were obtained as –0.32 to 2.08 cm/min considering all the soil textures. The estimated values for sorptivity (S) were in close agreement to the values reported by Mirzaee et al. (2014) and Vand et al. (2018). Similarly, Horton’s model parameters such as constant ‘β₄’ were determined in the range of 2.68 to 8.86. The values of initial infiltration capacity ‘fo’ were obtained to range from 2.45 to 13.23 cm/hr considering all the soil textures. While the values of model infiltration parameters have no physical meaning, these values reflected the effects of soil physical properties on infiltration as well as surface conditions and soil moisture contents (Oyedele et al. 2019; Zerihum and Sanchez 2003). Similarly, Novel’s model parameters such as constants α₄, β₅ and p were estimated in the range of − 11.94 to 20.87, − 0.14 to 0.68, and − 13.55 to 10.22, respectively.

The statistics of statistical parameters during the calibration of the infiltration models (curve fitting) are summarized in Table 8 for all soil textures.

Table 8 Performance evaluation of infiltration models during calibration

Considering all soil textures, performance of the Kostiakov model with R², SSE and RMSE values ranged 0.87 to 1.0, 0.01 to 3.03 and 0.03 to 0.50, respectively. For the Modified Kostiakov’s model, the statistical measures ranged 0.79 to 098, 0.03 to 5.44, and 0.09 to 0.67, respectively. For the Revised Modified Kostiakov model and Philips model the values of R², SSE and RMSE ranged 0.95 to 1.0, 0.01 to 1.27 and 0.02 to 0.36 and 0.82 to 0.99, 0.09 to 2.66 and 0.09 to 0.47 respectively. Similarly, for Horton and Novel model performances ranged 0.95 to 1.0, 0.01 to 1.91 and 0.02 to 0.38 and 0.96 to 1.0, 0.01 to 1.21 and 0.03 to 0.33, respectively.

The performance of the infiltration models during determination of parameters of the models can be observed from the curve fittings in Fig. 8. From this, we can observe that for clay loam soil texture, MKM and HM had less fits than the other remaining infiltration models with the measured infiltration data. RMKM and NM had better fits with the measured infiltration data than the other four infiltration models for loam and sandy clay loam soils. Sihag et al. 2017 assessed the suitability of the Novel model for different soil textures. Thus, they found that the Novel model was suitable for sandy loam, loam and clayey loam. In this regard, it may be inferred that the Novel model may accurately predict the infiltration rate in loamy soils (Zakwan 2020).

Fig. 8

Infiltration model’s parameters comparison for studied textures from literature and present study

For clay soil texture, HM had better fits with measured infiltration data than the other infiltration models. Horton’s model was a best-fitting model with measured values of infiltration rates for clay soils (Dagadu and Nimbalkar 2012). Similarly, for sandy loam soil texture, all models had a good correlation with measured infiltration data.

Almost all the R² values of the infiltration models were close to 1 and the SSE and RMSE of the infiltration models were minimum, near to 0. This indicates that the calibration of infiltration models was good for all the soil textures of the irrigation scheme because of higher values of R² and lower values of SSE and RMSE.

Comparison framework of infiltration model parameters

An exhaustive literature review was conducted by referring to previous studies (Mishra et al. 2003; Mirzaee et al. 2014; Zolfaghari et al. 2012; Failache and Zuquette 2021; Igbadunh et al. 2016; Kindo et al. 2024; Ketsela et al. 2023; Farid et al. 2019; Faridah et al. 2023; Thomas et al. 2020) to determine the ranges of different infiltration model parameters. A comparison framework was formulated for the parameters of infiltration models obtained from literature and from present the study, considering each texture of the soil as shown in Fig. 8.

The results of the present study and literature allow us to estimate the ranges of the model parameters for different soil textures. For the Kostiakov model, the 'α' parameter ranges from 0.76 to 7.12, 0.76 to 29.92, 1.68 to 18.64, 0.12 to 32.25, and 1.23 to 16.35 for clay loam, loam, sandy clay loam, clay, and sandy loam soil textures, respectively. Additionally, the 'β' parameter of the Kostiakov model ranges from 0.10 to 0.67, 0.22 to 1.0, 0.14 to 0.7, 0.20 to 0.78, and 0.12 to 0.80 for clay loam, loam, sandy clay loam, clay, and sandy loam soil textures, respectively.

The 'α₁' parameter for the modified Kostiakov’s model ranges from 0.37 to 12.01, 0.068 to 2.58, 0.12 to 12.01, 0.24 to 18.65, and 0.20 to 12.74 for clay loam, loam, sandy clay loam, clay, and sandy loam soil textures, respectively. Furthermore, the 'β₁' parameter of the modified Kostiakov’s model ranges from − 0.05 to 0.87, 0.47 to 0.87, − 0.05 to 0.87, 0.46 to 0.98, and 0.1 to 1.25 for clay loam, loam, sandy clay loam, clay, and sandy loam soil textures, respectively. Additionally, the 'f_c' parameter of the modified Kostiakov’s model ranges from 0.25 to 3.1, 0.59 to 5.4, 0.25 to 3.1, 0.11 to 1.36, and 0.55 to 2.36 for clay loam, loam, sandy clay loam, clay, and sandy loam soil textures, respectively.

For Horton's model, the ‘f_o’ parameter ranges from 2.58 to 13.22, 3.11 to 7.15, 5.38 to 21.85, 1.9 to 7.5, and 2.4 to 29.52 for clay loam, loam, sandy clay loam, clay, and sandy loam soil textures, respectively. Moreover, the ‘β₄’ parameter of Horton's model ranges from 0.04 to 6.67, 0.11 to 7.88, − 1.18 to 3.82, 0.03 to 3.03, and 2 to 6.68 for clay loam, loam, sandy clay loam, clay, and sandy loam soil textures, respectively. Additionally, the 'fc' parameter of Horton's model ranges from 1.01 to 7.03, 1.48 to 7.88, 2.64 to 16.34, 0.22 to 3.23, and 1.2 to 10.36 for clay loam, loam, sandy clay loam, clay, and sandy loam soil textures, respectively.

For Philip's model, the ‘S’ parameter ranges from 0.15 to 2.69, 0.08 to 3.13, 0.58 to 8.73, 1.56 to 7.06, and 0.53 to 13.73 for clay loam, loam, sandy clay loam, clay, and sandy loam soil textures, respectively. Furthermore, the ‘K_o’ parameter of Philip's model ranges from 0.08 to 2.07, 0.15 to 1.9, 0.39 to 3.24, 0.01 to 3.28, and 0.29 to 7.13 for clay loam, loam, sandy clay loam, clay, and sandy loam soil textures, respectively.

By estimating the appropriate parameter values within the provided ranges for specific soil textures, researchers and practitioners can improve their hydrological models and optimize water management strategies. These findings contribute to a better understanding of the variability of model parameters in different soil textures, providing essential information for accurate modeling and analysis in the field of soil science and hydrology.

Comparison of infiltration models with validation data

Predicted infiltration rate for different soil textures

After employing the average parameter values obtained using curve fitting, Table 7, the ability of the infiltration models to simulate the infiltration process was validated by comparing the measured infiltration rate and predicted infiltration rate of different soil textures (Fig. 9).

Fig. 9

Infiltration rates of soil textures predicted by models compared with the measured value

Performance evaluation of the infiltration models

To compare the selected infiltration models statistical measures, namely the Coefficient of determination (R²), Root mean square error (RMSE), Maximum absolute error (MAE), Bias, and Percentage average error (PAE) were calculated and their statistics are summarized in Table 9.

Table 9 Values of statistically measures for each model in different soil textures during validation

Referring to Table 9, the ability of infiltration models during validation was evaluated based on the value of R², RMSE, MAE, Bias, and PAE. Considering all soil textures, performances of R², RMSE, MAE, Bias, and PAE values for Kostiakov’s infiltration model ranged 0.90 to 0.99, 0.18 to 1.46, 0.35 to 3.74, − 1.16 to 0.86 and − 42.06 to 32.65, respectively. For the Modified Kostiakov’s model, the performance parameters R², RMSE, MAE, Bias, and PAE ranged from 0.76 to 0.99, 0.24 to 1.43, 0.34 to 3.74, − 1.21 to 0.43 and − 45.46 to 14.29 respectively. For the Revised Modified Kostiakov’s infiltration model the performances ranged from 0.57 to 0.99, 0.35 to 1.91, 0.56 to 4.49, − 0.54 to 1.46 and − 27.44 to 233.40, respectively.

For the Philip’s model, the value of the statistical parameters R², RMSE, MAE, Bias, and PAE ranged 0.84 to 0.99, 0.21 to 1.46, 0.33 to 3.90, − 1.14 to 0.84 and − 41.67 to 32.86 respectively. Finally, it was observed that for Horton’s and Novel’s model the statistical parameters R², RMSE, MAE, Bias, and PAE ranged 0.93 to 0.99, 0.17 to 1.39, 0.30 to 3.27, − 1.06 to 0.91 and − 38.43 to 36.64; and 0.94 to 1.0, 0.28 to 2.07, 0.43 to 6.02, − 0.79 to 0.66 and − 34.48 to 55.88, respectively.

The infiltration model performance for validation was compared using the average values of these statistical parameters on different data sets for each soil texture. To select a better model for the estimation of infiltration rate, ranks of the average values of the statistical parameters were summarized in Table 10 and the model with good rank was a better estimator of infiltration rate of soils. The acceptable ranges of model performance indicators were evaluated for the infiltration models and ranked for each soil texture. The results of ranking models according to statistical measures of the models, the goodness of infiltration rate can be estimated (Vand et al. 2018; Mirzaee et al. 2014).

Table 10 Ranks of the infiltration models in the soil textures using statistical measures

Referring to Table 10, the Revised Modified Kostiakov’s gave better performance followed by Novel’s, Horton’s, Kostiakov’s, Philip’s, and Modified Kostiakov’s Models for estimating infiltration rates in clay loam soil. The results of Mirzaee et al. (2014) indicate that the Revised Modified Kostiakov’s model was the best-fit infiltration model for clay loam soils for quantifying the infiltration process compared to the other infiltration models. Rasool et al. (2021); Mazighi et al. (2023); Zolfaghari et al. (2012); Parhi et al. (2007) and Parhi (2014) also reported similar performance of the Revised Modified Kostiakov’s model for estimating infiltration rate in clay loam soils. For loam soil texture, the Modified Kostiakov’s model was the best performing model followed by Horton’s, Kostiakov’s, Philip’s, Novels, and Revised Modified Kostiakov’s models. Results of Igbadun and Idris (2008); Zolfaghari et al (2012); Mirzaee et al. (2014) indicate that the Modified Kostiakov’s model is good for estimating infiltration of soils compared to other models.

For sandy clay loam, Horton’s model had the best performance followed by Kostiakov’s, Philip’s, Novel’s, Revised Modified Kostiakov’s models and then Modified Kostiakov’s. For clay soils, Horton’s model had the best performance followed by Kostiakov’s, Novel’s, Modified Kostiakov’s, Philip’s, and then Revised Modified Kostiakov’s infiltration models. Santra et al. (2021) found that the Horton model was superior to the Philip model for estimating infiltration rates in sandy soils. Skaggs et al. (1978) also found that Horton’s model was the best-fit model for estimating infiltration rates of soils. Dagadu and Nimbalkar (2012) also found that for clay soils, Horton’s model was the most suitable model for the prediction of infiltration.

For sandy loam soils, the Novels infiltration model gave better performance followed by Horton’s, Kostiakov’s, Modified Kostiakov’s Philip’s, and then Revised modified Kostiakov’s models for estimating infiltration rates. Mazighi et al. (2023) and Sihag et al. (2017) also reported that Novel’s infiltration model was the most suited as compared to others for estimating infiltration rates in sandy load soils. Thus the results found by other similar studies were similar to the results found in this study. Considering all soil textures of the study area the Modified Kostiakov’s, Revised Modified Kostiakov’s, Hortons’s, and Novel’s infiltration models were best fits for the simulation of the infiltration rate of the soil textures because they are based on empirical parameters. Empirical models are generally preferred over theoretical models because they reflect in-situ conditions (Mazighi et al. 2023; Bharali 2019).

Conclusions

This study investigated the capability of selected infiltration models namely Kostiakov’s Model (KM), Modified Kostiakov’s Model (MKM), Revised Modified Kostiakov’s Model (RMKM), Philip’s Model (PM), Horton’s Model (HM) and Novel’s Model (NM), for estimating the infiltration rate of different soil textures from actual field data of an irrigation scheme in Northern Ethiopia. To evaluate the performance of the infiltration models, 38 infiltration measurements were made in the Shillant-IV irrigation scheme, out of which 70% of the data was used for calibration of the model parameters and the remaining 30% of the data was used for validation of the model prediction results. Among the six selected infiltration models, the infiltration rates of clay loam soil were better estimated by RMKM than the other five infiltration models with a final infiltration rate of 1.74 cm/hr. The infiltration rate of loam soil was better estimated by MKM than the other five infiltration models with a final infiltration rate of 2.01 cm/hr. For sandy clay loam and clay soils, the infiltration rate was better estimated by HM than the other five infiltration models with final infiltration rates of 1.78 cm/hr and 1.14 cm/hr, respectively. Similarly, for sandy loam soil texture, the NM had better infiltration prediction capability than the other models with a final infiltration rate of 1.83 cm/hr. This study showed that the infiltration prediction capabilities of models varied among soil texture classes. Generally, the MKM, RMKM, HM, and NM models are considered more suitable than the KM and PM models for estimating soil infiltration rates, primarily due to their simplicity and ability to accurately. The best fitted infiltration models under verified field conditions can simulate the infiltration rate of soils and can be used to adjust stream discharge, slope, and field geometry based on the varying field infiltration rate in the surface irrigation system. This study covered only five soil textures of the irrigation scheme. It would be better to study other soil texture sites to evaluate infiltration models beyond this study site so as represent a wide range of measured infiltration data.