Modelling soil water infiltration variability using scaling

Highlights

• Cumulative infiltration-curve obtained using minimum infiltration-time data.

• Unlike previously with infiltration equation, reference curve is optional in proposed method.

• Accuracy of proposed method and infiltration scaling optimisation method were similar.

• Proposed scaling is also applicable to empirical equations (e.g. Kostiakov–Lewis).

One of the most substantial and crucial parameters for irrigation in agriculture is water infiltration into the soil. Given the variability of soil properties, the infiltration characteristics also vary. The present study aims to model soil water infiltration using scaling. In the present research, a new approach for scaling the infiltration equation was developed and evaluated for 22 infiltration tests. In the new method, the scaling factor ($F_s$) for each equation equals the depth of infiltrated water after the specified time ($t_s$) in the reference infiltration equation. This new method was applied to Philip's equation and was compared with previous Philip equation scaling methods such as sorptivity coefficient (${\alpha }_s$), transmissivity coefficient (${\alpha }_A$), the optimal factor obtained using the least squares error (${\alpha }_{opt}$), geometric, arithmetic and harmonic mean ${\alpha }_s$ and ${\alpha }_A$. Results showed that scaling using $F_s$ has the best match with measured data (R² = 0.99, MBE = 0.0006 and RMSE = 0.001) when compared to other Philip equation scaling methods. Among scaling techniques from the literature, $F_s$ had the highest correlation with ${\alpha }_{opt}$ (R² = 0.96). The proposed method applied to the Kostiakov–Lewis infiltration equation was also evaluated. The results of the scaling of the Kostiakov–Lewis equation showed that to use reference curve is optional and each of the infiltration curves can be used as reference curve. As a whole, the results of the proposed method showed that the cumulative infiltration curve can be obtained using minimum infiltration-time measurement data.

Introduction

Hydrological behaviour and transportation of sediment and water to surface and subsurface waters are affected by spatial distribution of soil physical and hydraulic properties. The infiltration rate has the most spatial variability among soil physical and hydraulic properties (Tsegaye & Hill, 1998). A large number of models have been put forward for infiltration prediction, mostly due to the importance of infiltration. Some models are empirical and based on plotting the curves of infiltration rate against time, and do not incorporate soil physical properties (Kostiakov, 1932, pp. 17–21; Horton, 1941, Holtan, 1961; SCS, 1974) while others are based on soil physical properties (Green and Ampt, 1911, Philip, 1957).

Infiltration parameters are usually determined using single-ring or double-ring experimental data (Duan, Fedler, & Borrelli, 2011). However, differences in soil properties cause temporal and spatial variations in soil infiltration capacity at farm scale (Childs et al., 1993, Guzmán-Rojo et al., 2019, Oyonarte et al., 2002, Schwankl et al., 2000) and regional scale (Machiwal et al., 2006, Zapata and Playán, 2000). Since infiltration parameters are a function of temporal and spatial factors, therefore, a relatively large number of farm experiments are needed to express the average farm conditions (Bautista & Wallender, 1985). At the same time, a major issue for modelling infiltration in catchments is how to accommodate soil spatial variability. Previous studies have shown that spatial variability of soils results in high variability of soil infiltration behaviour (Warrick, 1998, Zapata and Playán, 2000). In recent years, researchers have proposed ways to obviate the need to measure farm and regional data to express soil water dynamics. One such approach is scaling, first developed by Miller and Miller (1956), relying on the theory of similar media in water and soil science (Sadeghi, Ghahraman, Warrick, Tuller, & Jones, 2016). According to similar media theory, soils can be called similar, provided that different soils can be placed on a reference curve with ratios of a physical characteristic length, called “scaling factor".

Scaling research on infiltration consists of three categories. The first category involves scaling the Richards equation for the water infiltration process in homogeneous soils (Kutilek et al., 1991, Sadeghi et al., 2012, Warrick and Hussen, 1993). Warrick and Hussen (1993) used the scaling technique to scale the Richards equation for similar soils and infiltration process. Sadeghi, Ghahraman, Davary, Hasheminia, and Reichardt (2011) proposed a method for scaling the Richards equation in redistribution conditions so that this method is not restricted to specific hydraulic functions and it is possible to use all models presented for hydraulic functions in solving the Richards equation. Sadeghi et al. (2012) scaled the Richards equation for different soils and, after solving the scaled Richard's equation, proposed an infiltration equation similar to Philip's three-term infiltration equation for the soil water infiltration process. The results showed that the proposed infiltration equation predicts the water infiltration process with good accuracy. These approaches are mostly used for laboratory conditions and less for farm conditions.

The second category involves the application of scaling to the infiltration equations used in surface irrigation. Rasoulzadeh and Sepaskhah (2003), using dimensional analysis, scaling, and infiltration data from six soil series, provided a general equation for furrow irrigation. The resulting equation was a function of the wet medium and the volume of water entering the farm. Evaluation of the scaled infiltration equation showed that this equation would be applicable to other furrows with different textures and hydraulic conditions, and estimated the infiltration very accurately. The studies of Rasoulzadeh and Sepaskhah (2003) on infiltration scaling have a great deal of complexity to determine infiltration coefficients and require a great deal of preliminary data. Consequently Khatri and Smith (2006) proposed a method of estimating infiltration coefficients using a single advance point and a reference infiltration curve (sample). They defined a scale factor for each field and obtained the parameters of the Kostiakov–Lewis infiltration equation. This is an easy-to-use method at the field level that only needs to advance at one point plus the flow rate and the cross-sectional area at the furrow entrance. Koech, Smith, and Gillies (2014) used the scaling method proposed by Khatri and Smith (2006) to determine real-time flow cut off time in automatic furrow irrigation. According to this method, the parameters of infiltration in furrow irrigation were determined according to the time of water arrival in the middle of the field and then the optimum cut off time was determined. Nie et al. (2018), using scaling process, presented a general equation similar to the Lewis-Kostiakov infiltration equation. Given this method, a pedotransfer function was introduced to determine the scaling factor, which depended on the clay percentage, the sand percentage, the bulk density, and initial moisture content. On the other hand, in many cases water advance data are not available in the strip or furrow and the parameters of the infiltration equation are present in the strip or furrow prior to irrigation, so this method cannot be used for scaling.

The third category involves modelling the spatial variability of infiltration in catchments and agricultural lands. In these methods, the reference curve can be determined by averaging all the infiltration curves and then the scaling factor of each infiltration curve is determined such that each scaled curve is placed on the reference curve with the least error. Sharma, Gander, and Hunt (1980) adopted the scaling method to determine the unit infiltration equation at the basin level and proposed a method for calculating Philip's two-term infiltration equation parameters. Machiwal et al. (2006), by referring to Sharma et al. (1980), introduced Philip's two-term infiltration equation model as a suitable model for the process of scaling and infiltration variability analysis. In the study by Machiwal et al. (2006), a total of 24 infiltration tests were conducted on a systematic squared grid pattern over the study area using double-ring infiltrometers in a basin with 66.2 ha. In that research the scaling factor for the sorptivity and transmissivity were computed. The results showed that the scaling factor based on the transmissivity factor gave better results. Babaei, Zolfaghari, Yazdani, and Sadeghipour (2018) used scaling to model spatial variability of infiltration in farmland in an arid region in Iran. In this method, the reference curve was calculated as the arithmetic mean of the transmissivity and the sorptivity for 60 infiltration measurement points, and the scale factor was then obtained for each soil using six different methods. In the methods outlined above, reference and scaling curves cannot be used if the soil is outside the range of the original data. Also the methods above are applicable only to Philip's two-term infiltration equation and cannot be used for other widely used infiltration models such as Kostiakov–Lewis and others.

The objective of this research is to present a new method for scaling spatial variability of infiltration so that it can be applied to all infiltration models and as well as the selection of the reference curve.