Determination of Soil Hydraulic Properties from Infiltration Data Using Various Methods

Abstract

In the present study, the determination of soil saturated hydraulic conductivity (K_s) and soil sorptivity (S) from one-dimensional vertical infiltration data of eight different soils were investigated using three methodologies. Specifically, the nonlinear optimization procedure with the help of the Excel Solver application using six different two-parameter infiltration equations, as described by Valiantzas, Haverkamp et al. (complete, two and three approximate expansions), Talsma and Parlange and Green and Ampt; the linearization method of cumulative infiltration data by Valiantzas and the method of Latorre et al. were used. The results showed that, in almost all cases, the relative errors in the prediction of S were smaller than those of K_s. The nonlinear optimization procedure using the Valiantzas equation gave the best prediction of S and K_s, with relative errors up to −12.49% and 13.61%, respectively. The two-term approximate expansion of Haverkamp gave the highest relative errors in both S and K_s. The various forms of the Haverkamp equation (complete and three approximate expansion), as well as the Latorre method, gave good predictions of S and K_s in fine-textured soils. In all forms of the Haverkamp equation, when parameter β was considered as an additional adjustment parameter, no improvement in the prediction of the S and K_s values was achieved, so the constant value β = 0.6 was proposed. The relative errors in the prediction of S and K_s resulting from the linearization method of the cumulative infiltration data were similar to those of the Valiantzas equation by the nonlinear optimization procedure. The accuracy in estimating the S and K_s parameters from each equation depends on its infiltration time validity and the soil type.

1. Introduction

The hydraulic properties of soil have been a field of intensive research for several years in many disciplines, such as hydrology, agricultural engineering and hydrogeology, because the hydraulic properties of a porous medium are necessary to simulate water flow in the soil profile by the Richards equation [1]. The determination of the hydraulic properties can be carried out either in the laboratory or in the field by various methods, depending on the specific research needs. Although many methodologies have been proposed, the issue of simple, rapid and reliable determination of the soil hydraulic properties, especially in the field, is still open [2].

Cumulative water infiltration in soil, which depends on the soil hydraulic properties, can be used to obtain information about these properties by proper analysis. Over the past decades, various theoretical, empirical and semi-empirical models have been developed to describe the phenomenon of one-dimensional vertical infiltration under certain initial and boundary conditions [3,4,5,6,7,8,9]. These models differ in terms of their complexity and assumptions, the shape of the infiltration front, initial and boundary conditions and valid time interval [10,11]. Most of them are two-parameter models with parameters of soil sorptivity (S) and saturated hydraulic conductivity (K_s). These parameters may be viewed as super-parameters of the hydraulic functions, i.e., water retention and unsaturated hydraulic conductivity functions, as referred by Vrugt and Gao [12]. The abovementioned cumulative infiltration models have the advantage of simple computations, do not require the determination of water retention and unsaturated hydraulic conductivity functions and allow for a rapid estimation of their unknown parameters. In general, two approaches have been proposed to estimate S and K_s by two-parameter cumulative infiltration models [13,14]. The first concerns linearization methods [9,13,15], and the second concerns an inverting procedure for the estimation of S and K_s using curve-fitting methods [16,17,18,19,20]. Rahmati et al. [21] proposed an alternative method, the characteristic time method, based on the concept of the characteristic time to estimate S and K_s.

Parlange et al. [6] proposed a quasi-exact implicit (QEI) solution of Richard’s equation describing cumulative vertical infiltration into a homogeneous soil at a uniform initial water content. Haverkamp et al. [7] redefined the Parlange et al. [6] model, which includes three parameters: S, K_s and the dimensionless parameter β. Parameter β is an integral shape parameter that is a function of the soil permeability, hydraulic conductivity and initial and final soil water content and is usually fixed at 0.6 [8]. Due to the complexity of the QEI model, Haverkamp et al. [8] proposed a two-term approximate expansion that is valid for short and intermediate times and has been widely used for the estimation of S and K_s by the differentiated linearization (DL) method [13], especially in the case of using infiltration data measured with a tension disc infiltrometer.

Latorre et al. [20] proposed an inverse analysis of the complete Haverkamp et al. [7] equation, known as the Numerical Solution of the Haverkamp equation (NSH) method, to estimate the soil hydraulic properties, S and K_s, from the full-time cumulative infiltration curve considering parameter β equal to 0.6. Compared to the DL method, the NSH predicts S and K_s more reliably. Due to the limitations on the validity time interval of the two-term approximate expansion and the complex resolution of the QEI analytical formulation that makes it complicated in inverting procedures, Rahmati et al. [22] proposed a three-term approximation of the QEI analytical formulation. This method is valid for longer times compared to the two-term approximate expansion, and its inverse solution gives better predictions of S and K_s. Additionally, they showed that the accuracy of the predicted S and K_s had much less dependency on the β parameter, presenting a constant value of β = 0.6 or β = 1.1, which resulted in enough accuracy in the predictions of S and K_s.

Latorre et al. [23] studied the effect of the value of the β parameter on the predictions of S and K_s. The results of their study showed that β had a small effect on the infiltration curve, with a large confidence interval of β (0.3 < β < 2.0), which demonstrated that S and K_s can be accurately predicted using a constant value of β. A similar range of β values was reported by Lassabatare et al. [19] for soils ranging from sand to silty and by Jaiswal et al. [11] for sand to clay soils. The results also showed that even small infiltration times (i.e., 100 s) are sufficient to predict S accurately, while longer times (i.e., 1000 s) are required to predict K_s and very long times (i.e., 10,000 s) for the prediction of β [23].

In addition to the various versions of the inverse estimation of S and K_s, the linearization method has also been widely used [9,13,15], where one-dimensional (1D) cumulative infiltration data are transformed into a linear form. Smiles and Knight [15] showed that, when cumulative infiltration data (i) over time (t) can be described by the two-term equation of Philip [4], the S and K_s can be determined from the y intercept and the slope of the line in the i/t^0.5 vs. t^0.5 plot, respectively. The differentiated linearization (DL) method has been applied mainly to three-dimensional (3D) cumulative infiltration data measured by disc infiltrometer, where a contact sand layer is placed between the soil surface and the basis of the disc. The S and K_s can be determined from the y intercept and the half of the slope of the line in the Δi/Δt^0.5 vs. (t_nt_n+1)^0.5 plot, respectively [13]. This method is applied for short to intermediate infiltration times, i.e., for times in which a non-steady-state flow is observed. Valiantzas [9] presented another form of linearization, where S² and, therefore, S are determined from the y intercept and K_s from the slope of the line in the i²/t vs. i plot. From the abovementioned, it is obvious that if no linear relationship is obtained by the linearization method used, then the corresponding infiltration equation cannot describe the relationship between the cumulative infiltration and time.

Clothier and Scotter [24] and Rahmati et al. [22] reported that the Data Solver application in Excel could be used to minimize the objective function between the experimental infiltration data and the predicted ones from at least a two-parameter infiltration equation and to predict the S and K_s parameters. Clothier and Scotter [24] presented some examples using various infiltration equations only in the case of sandy soil. However, a systematic study of this application using many infiltration equations and a wide range of soils to obtain safer conclusions has not been carried out, although this application is very simple to use and does not require any special skills.

In the present study, the two-parameter equations of Green and Ampt [3] and Talsma and Parlange [5], which describe two extreme infiltration behavior soils, were used. The infiltration behavior of natural soils is expected within the area defined by the abovementioned soils. These equations predict different shapes of wetting front, because in the soil described by the Green and Ampt model, the diffusivity function D(θ) is a delta function, while, in the soil of the Talsma and Parlange model, the function D(θ) and dK/dθ change rapidly and are almost proportional [6]. Additionally, the two-parameter equation of Valiantzas [9] was used, because on the one hand, it is a specific solution located approximately at the middle of the area defined by Green-Ampt and Talsma-Parlange, and on the other hand, it can be converted to a linear equation facilitating the determination of the S and K_s parameters. In addition, all versions of the equation of Haverkamp et al., i.e., the complete [7], the two-term [8] and the three-term equations [22], were used.

The main objective of the present study is the testing of the accuracy in the prediction of the hydraulic properties (S and K_s) from various infiltration equations using the nonlinear optimization method with the help of the Solver application in Excel. In particular, the equations of Talsma and Parlange [5], Green and Ampt [3] and Valiantzas [9], as well as the quasi-exact implicit (QEI) analytical formulation of Haverkamp [7] and its two-term [8] and three-term [22] expansions using the β parameter as fixed (β = 0.6) or variable, were studied. The additional objectives were: (i) comparison between the predicted S and Κ_s values by nonlinear methods using the Solver tool and those predicted by the linear form of the Valiantzas equation; (ii) comparison between the predicted S and Κ_s values by the complete, two-term and three-term equations of Haverkamp et al. and those predicted by the Latorre et al. [20] method and (iii) checking the time dependence of parameters S and Κ_s.

2. Materials and Methods

2.1. Porous Media

Eight soils, seven of them selected from the literature with known hydraulic properties, were studied for the purposes of the present work (

Table 1. Measured or numerical values of saturated hydraulic conductivity, K_s, and sorptivity, S, of the studied soils and the corresponding applied pressure heads, H.

Soil	H (cm)	S (cm min−0.5)	Ks (cm min−1)
Sand (Poulovassilis et al. [26])	0	1.375	0.3
Soil and Sand mixture (Poulovassilis et al. [26])	0	0.223	0.012
Sandy Soil—Grenoble (Haverkamp et al. [25])	2.25	1.319	0.255
Silty Soil (Poulovassilis et al. [26])	0	0.849	0.038
Yolo Light Clay (Poulovassilis et al. [26])	0	0.095	0.0007
Silty Loam GE3 (van Genuchten [27])	0	0.3162	0.0034
Guelph—loam (van Genuchten [27])	0	0.6181	0.0219
Sandy Loam	3	1.445	0.11

). One-dimensional (1D) vertical infiltration data of these soils were obtained experimentally or were generated by numerical simulations, applying a surface water ponding depth, H (H ≥ 0 cm). Specifically, for the Sandy soil—Grenoble (with H = 2.25 cm) and Silty soil (with H = 0 cm), there were experimental infiltration data reported by Haverkamp et al. [25] and Poulovassilis et al. [26], respectively. The experimental data for the Sandy Loam soil (with H = 3 cm) were obtained by experiments conducted in the Lab. of Agricultural Hydraulics (Agricultural University of Athens, Greece) in the context of the present study. All the experiments were conducted on vertical columns filled with initially air-dried soil samples by applying a constant head H of water at the soil surface. It is worth noting that S is affected by the soil hydraulic properties, initial soil water content and ponding depth [10]. However, in the cases of the Sandy soil—Grenoble and Sandy Loam soil used, small H values were applied.

Numerical infiltration data for Yolo Light Clay (with H = 0 cm), Sand (with H = 0 cm) and the Soil and Sand mixture (with H = 0 cm) were reported by Poulovassilis et al. [26].

The last two soils used were the Silty Loam GE3 and the Guelph—loam soil reported by van Genuchten [27]. The soil water retention and conductivity curves of these soils were assumed to be described by the van Genuchten [27]-Mualem [28] model [9] (with model parameter m = 1 − 1/n). Vertical infiltration was numerically simulated by means of the HYDRUS-1D Code [29], with an initial pressure head of −500 cm and surface ponding depth of 0 cm for both soils. At the lower boundary of the 1-m uniform soil profile, a zero-pressure head gradient was defined (free drainage).

The values of sorptivity, S, and saturated hydraulic conductivity, K_s, presented in Table 1 were obtained from Haverkamp et al. [25] for Sandy Soil—Grenoble and from Poulovassilis et al. [26] for Sand, Soil and Sand mixture, Silty soil and Yolo Light Clay. The values of K_s for Silty Loam GE3 and Guelph—loam were obtained from van Genuchten [27], and S values were obtained from 1D horizontal infiltration data for H = 0 by means of HYDRUS 1-D. For Sandy Loam soil, the values of K_s and S were determined using a constant head permeameter and experimental horizontal infiltration data, respectively (Table 1).

The selected soils cover a wide range of porous media, ranging from coarse-textured (sand) to very fine-textured soils (Yolo Light Clay).

2.2. Approaches to Estimate S and K_s from One-Dimensional (1D) Infiltration Experiments

2.2.1. Nonlinear Optimization Method Using the Solver Application in Excel

The first method used to estimate the S and K_s parameters is the inverse solution method using the Solver application in Excel. The Excel Solver minimizes the objective function between the measured and predicted cumulative infiltration values at given times and then predicts the S and K_s parameters. This tool was proposed by Šimůnek et al. [29], Wraith and Or [30], Clothier and Scotter [24] and Rahmati et al. [22]. Specifically, the mathematical method used by Solver is the Generalized Reduced Gradient method (GRG), which was proposed by Lasdon et al. [31] for solving problems of smooth nonlinear equations if the equations are differentiable at all points in their domain (continuous function). The optimal values computed come from the application of two mathematical techniques: the quasi-Newton method and the conjugate gradient method. Essentially, during the computation process, an alternation of these methods is observed, as the function gradient changes by changing the selected initial values of the parameters, and an optimal solution is determined when the partial derivatives are equal to zero. As the initial values of S and K_s, we consider the values ${\mathrm{S}}^{\prime }={\mathrm{i}}_1/\sqrt{{\mathrm{t}}_1}$ and ${\mathrm{K}}_{\mathrm{s}}^{\prime }=\left({\mathrm{i}}_{\mathrm{n}}-{\mathrm{i}}_{\mathrm{n}-1}\right)/\left({\mathrm{t}}_{\mathrm{n}}-{\mathrm{t}}_{\mathrm{n}-1}\right)$, where n is the last value of the data.

The infiltration equations used for the estimation of the S and K_s parameters are the following:

(1)

Green and Ampt [3]:

$$\mathrm{t}=\frac{\mathrm{i}}{{\mathrm{K}}_{\mathrm{s}}}-\frac{{\mathrm{S}}^2}{2{\mathrm{K}}_{\mathrm{s}}^2}\mathrm{l}\mathrm{n}\left(1+\frac{2{\mathrm{K}}_{\mathrm{s}}\mathrm{i}}{{\mathrm{S}}^2}\right),$$

(1)

(2)

Talsma and Parlange [5]:

$$\mathrm{t}=\frac{\mathrm{i}}{{\mathrm{K}}_{\mathrm{s}}}+\frac{{\mathrm{S}}^2}{2{\mathrm{K}}_{\mathrm{s}}^2}\left(\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{2{\mathrm{K}}_{\mathrm{s}}\mathrm{i}}{{\mathrm{S}}^2}\right)-1\right),$$

(2)

(3)

Haverkamp et al. [8] presented a quasi-exact analytical solution of the Richards [1] equation for 1D cumulative infiltration, i, which was firstly proposed by Parlange et al. [6] and redefined later by Haverkamp et al. [7]:

$$\frac{2{\left({\mathrm{K}}_{\mathrm{s}}-{\mathrm{K}}_{\mathrm{i}}\right)}^2}{{\mathrm{S}}^2}\mathrm{t}=\frac{2}{1-\mathsf{\beta }}\frac{\left({\mathrm{K}}_{\mathrm{s}}-{\mathrm{K}}_{\mathrm{i}}\right)\left(\mathrm{i}-{\mathrm{K}}_{\mathrm{i}}\mathrm{t}\right)}{{\mathrm{S}}^2}-\frac{1}{1-\mathsf{\beta }}\ln \left[\frac{1}{\mathsf{\beta }}\exp \left(\frac{2\mathsf{\beta }\left({\mathrm{K}}_{\mathrm{s}}-{\mathrm{K}}_{\mathrm{i}}\right)\left(\mathrm{i}-{\mathrm{K}}_{\mathrm{i}}\mathrm{t}\right)}{{\mathrm{S}}^2}\right)+\frac{\mathsf{\beta }-1}{\mathsf{\beta }}\right],$$

(3)

where Κ_i and Κ_s are the hydraulic conductivities corresponding to the initial water content θ_i and water content at saturation θ_s, respectively, and β is an integral shape parameter that ranges between 0.3 and 2 for sand to clay soils [11,19].

Equation (3) is applied for all infiltration times, i.e., for t → ∞. Based on the assumption that the initial hydraulic conductivity is negligible (K_i ≅ 0) (assuming initially that the soil is sufficiently dry) [8], Equation (3) is converted into the following equation, which will be referred to as the Complete Haverkamp equation:

$$\mathrm{t}=\frac{1}{1-\mathsf{\beta }}\left(\frac{\mathrm{i}}{{\mathrm{K}}_{\mathrm{s}}}-\frac{{\mathrm{S}}^2}{2{\mathrm{K}}_{\mathrm{s}}{}^2}\ln \left(\frac{1}{\mathsf{\beta }}\exp \left(\frac{2\mathsf{\beta }{\mathrm{K}}_{\mathrm{s}}\mathrm{i}}{{\mathrm{S}}^2}\right)+\frac{\mathsf{\beta }-1}{\mathsf{\beta }}\right)\right),$$

(4)

(4)

For short and intermediate infiltration times and K_i = 0, Equation (3) can be simplified as the following two-term approximate expansion [8]:

$$\mathrm{i}=\mathrm{S}\sqrt{\mathrm{t}}+\frac{2-\mathsf{\beta }}{3}{\mathrm{K}}_{\mathrm{s}}\mathrm{t},$$

(5)

(5)

Rahmati et al. [21] introduced the three-term approximate expansion of Haverkamp et al. [8]:

$$\mathrm{i}=\mathrm{S}\sqrt{\mathrm{t}}+\frac{2-\mathsf{\beta }}{3}{\mathrm{K}}_{\mathrm{s}}\mathrm{t}+\left({\mathsf{\beta }}^2-\mathsf{\beta }+1\right)\left(\frac{{\mathrm{K}}_{\mathrm{s}}^2}{9\mathrm{S}}\right){\mathrm{t}}^{\frac{3}{2}},$$

(6)

Equation (6) is applied for longer infiltration times compared to those of Equation (5).

(6)

Valiantzas [9]:

$$\mathrm{i}=0.5{\mathrm{K}}_{\mathrm{s}}\mathrm{t}+\mathrm{S}\sqrt{\mathrm{t}}{\left[1+{\left(\frac{0.5{\mathrm{K}}_{\mathrm{s}}}{\mathrm{S}}\right)}^2\mathrm{t}\right]}^{1/2},$$

(7)

Valiantzas [9] proposed the two-parameter Equation (7), which is expressed in a form explicit in i as function of t, and it is the solution of the following quadratic equation (Equation (8)).

2.2.2. Linearization Approach

In addition to the nonlinear optimization procedure, the linearization approach was used. In this study, we chose to use the linear form of the Valiantzas [9] equation:

$$\frac{{\mathrm{i}}^2}{\mathrm{t}}={\mathrm{K}}_{\mathrm{s}}·\mathrm{i}+{\mathrm{S}}^2,$$

(8)

Using Equation (8), the values of K_s and S² (and therefore S) can be estimated from the slope and the y intercept of the line in the i²/t vs. i plot, respectively. Therefore, the values of the S and K_s parameters predicted from the linearization method (Equation (8)), can be compared with those from the nonlinear optimization procedure by applying Equation (7).

2.2.3. Latorre et al. Approach (NSH Method)

Latorre et al. [20] proposed an inverse analysis of the complete Haverkamp equation by applying a numerical method, called as Numerical Solution of the Haverkamp equation (NSH), to estimate the S and K_s when considering the parameter β equal to 0.6. The method is applied for the full-time cumulative infiltration curves and, therefore, allows considering large datasets. The NSH method gives better results in the prediction of K_s compared with those from the differentiated linearization (DL) method, while both methods give similar results in the prediction of S. Latorre et al. [20], who created a webpage (http://swi.csic.es/infiltration-map/ (accessed on 2 March 2022)) to compute S and K_s from the equation of Haverkamp et al. [7] using their method.

2.3. The Effect of Infiltration Time on the Estimation of S and K_s by the Various Methods Used

The stability of the predicted values of S and K_s with the infiltration time is desirable in each calculation method. Therefore, the effect of the time duration of an infiltration experiment on the predicted values was investigated. Specifically, the values of S and K_s were calculated from each infiltration equation and method used in successive parts of the total infiltration time up to the end of the infiltration time. Depending on the soil type, a relatively short infiltration time is selected as an initial step, and the next time steps are gradually increased to the total infiltration time.

2.4. Statistical Analysis

To find the best method among those used to predict the S and K_s parameters, a comparison among the methods based on the relative errors (RE) of the predicted values using the following equation was made:

$$\mathrm{R}\mathrm{E}=\frac{\mathrm{Predicted} \mathrm{value}-\mathrm{Actual} \mathrm{value}}{\mathrm{Actual} \mathrm{value}}·100,$$

(9)

where the actual value is the measured or numerical one.

The simulation methods of Monte Carlo and Bootstrap [32] were applied to investigate the uncertainty of the predicted values of S and K_s from the nonlinear equations by means of Excel Solver. From these simulations, histograms with normal distributions were obtained for each parameter, and the average values of S and K_s were compared to the predicted ones computed by means of Solver in Excel for all the equations studied.

In the case of the linearization method (Equation (8)), the evaluation of the linearity of the infiltration data (i²/t vs. i) was made using the coefficient of determination R².

3. Results and Discussion

3.1. Results from Nonlinear Optimization for Estimation of S and K_s Parameters

As regards the nonlinear parameter estimations of S and K_s from cumulative infiltration measurements by means of Solver, six two-parameter equations (Equations (1), (2) and (4)–(7)) were used, and the corresponding results are presented in

Table 2. Predicted values of S and K_s from various infiltration equations using different methods, as well as the corresponding actual values (measured or numerical) for all soils studied.

Soil	Equation (1) with Solver		Equation (2) with Solver		Equation (4) with Solver
	S (cm min−0.5)	Ks (cm min−1)	S (cm min−0.5)	Ks (cm min−1)	S (cm min−0.5)	Ks (cm min−1)
Sand	1.254	0.315	1.515	0.363	1.393	0.358
Soil and Sand mixture	0.197	0.011	0.226	0.0137	0.211	0.0133
Sandy Soil—Grenoble	1.177	0.255	1.301	0.33	1.229	0.313
Silty Soil	0.691	0.037	0.787	0.045	0.735	0.044
Yolo Light Clay	0.09	0.00057	0.096	0.00078	0.092	0.00072
Silty Loam GE3	0.314	0.0023	0.319	0.0038	0.315	0.0031
Guelph—loam	0.593	0.0162	0.624	0.0231	0.6034	0.0209
Sandy Loam	1.484	0.082	1.518	0.129	1.494	0.11
Soil	Equation (5) with Solver		Equation (6) with Solver		Equation (7) with Solver
	S (cm min−0.5)	Ks (cm min−1)	S (cm min−0.5)	Ks (cm min−1)	S (cm min−0.5)	Ks (cm min−1)
Sand	1.154	0.568	1.374	0.37	1.359	0.341
Soil and Sand mixture	0.191	0.0191	0.203	0.0109	0.21	0.0124
Sandy Soil—Grenoble	1.149	0.422	1.228	0.318	1.233	0.287
Silty Soil	0.678	0.061	0.74	0.044	0.743	0.04
Yolo Light Clay	0.0894	0.00089	0.0926	0.00072	0.093	0.00065
Silty Loam GE3	0.311	0.0036	0.313	0.0032	0.314	0.003
Guelph—loam	0.587	0.0256	0.603	0.0211	0.605	0.0191
Sandy Loam	1.498	0.119	1.511	0.106	1.514	0.096
Soil	Equation (8) (Linearization)		Latorre et al. Approach		Actual Data
	S (cm min−0.5)	Ks (cm min−1)	S (cm min−0.5)	Ks (cm min−1)	S (cm min−0.5)	Ks (cm min−1)
Sand	1.373	0.3383	1.946	0.2874	1.375 2	0.3 2
Soil and Sand mixture	0.215	0.0121	0.214	0.0131	0.223 2	0.012 2
Sandy Soil—Grenoble	1.248	0.282	1.225	0.315	1.319 1	0.255 1
Silty Soil	0.782	0.037	0.7186	0.0445	0.849 1	0.038 1
Yolo Light Clay	0.0949	0.0006	0.093	0.0007	0.095 2	0.0007 2
Silty Loam GE3	0.31	0.0032	0.3122	0.0033	0.316 2	0.0034 2
Guelph—loam	0.606	0.019	0.6042	0.0208	0.618 2	0.0219 2
Sandy Loam	1.537	0.089	1.481	0.115	1.445 1	0.11 1

¹ Experimental values. ² Numerical values.

. In the case of the three-parameter equations, i.e., Equations (4)–(6), they were used as two-parameter ones by considering the β parameter equal to 0.6.

The comparison of the results showed that the best prediction of the S parameter was made by Equation (2), giving the lower value of the maximum relative error (RE = 10.15%) among those calculated from the other equations used (

Table 3. Relative errors (RE%) of the predicted values of S and K_s obtained from various methods for all soils studied. The underlined values are the maximum relative errors calculated from each method.

	RE (%)
Soil	Equation (1) with Solver		Equation (2) with Solver		Equation (4) with Solver
	S (cm min−0.5)	Ks (cm min−1)	S (cm min−0.5)	Ks (cm min−1)	S (cm min−0.5)	Ks (cm min−1)
Sand	−8.77	5.12	10.15	20.98	1.32	19.27
Soil and Sand mixture	−11.46	−6.53	1.52	14.43	−5.39	11.00
Sandy Soil—Grenoble	−10.79	−0.05	−1.35	29.23	−6.86	22.87
Silty Soil	−18.55	−3.89	−7.25	19.00	−13.46	15.08
Yolo Light Clay	−5.46	−19.27	1.00	11.22	−3.13	2.75
Silty Loam GE3	−0.63	−32.35	0.82	11.76	−0.32	−8.82
Guelph—loam	−4.08	−26.03	0.94	5.48	−2.38	−4.57
Sandy Loam	2.71	−25.29	5.07	17.32	3.36	0.40
Soil	Equation (5) with Solver		Equation (6) with Solver		Equation (7) with Solver
	S (cmmin−0.5)	Ks (cmmin−1)	S (cmmin−0.5)	Ks (cmmin−1)	S (cmmin−0.5)	Ks (cmmin−1)
Sand	−16.07	89.46	−0.04	23.33	−1.15	13.61
Soil and Sand mixture	−14.52	59.29	−8.84	−9.36	−5.63	3.02
Sandy Soil—Grenoble	−12.91	65.65	−6.91	24.75	−6.55	12.50
Silty Soil	−20.10	60.01	−12.80	16.32	−12.49	5.05
Yolo Light Clay	−5.91	27.60	−2.54	2.62	−2.14	−7.51
Silty Loam GE3	−1.58	5.88	−0.95	−5.88	−0.70	−11.76
Guelph—loam	−4.98	16.89	−2.44	−3.65	−2.10	−12.79
Sandy Loam	3.67	8.20	4.60	−3.93	4.79	−12.67
Soil	Equation (8) (Linearization)		Latorre et al. Approach
	S (cm min−0.5)	Ks (cmmin−1)	S (cm min−0.5)	Ks (cm min−1)
Sand	−0.15	12.77	41.53	−4.20
Soil and Sand mixture	−3.59	0.83	−4.04	9.50
Sandy Soil—Grenoble	−5.38	10.59	−7.13	23.53
Silty Soil	−7.89	−2.63	−15.36	17.00
Yolo Light Clay	−0.11	−14.29	−2.11	0.29
Silty Loam GE3	−1.90	−5.88	−1.20	−2.94
Guelph—loam	−2.04	−13.24	−2.25	−5.02
Sandy Loam	6.37	−19.09	2.49	4.55

, underlined values) for all soils studied. On the other hand, the best prediction of the K_s parameter was made by Equation (7) with a maximum relative error of 13.61% (Table 3). The higher relative errors were obtained by Equation (5) for both S and K_s (−20.10% and 89.46%, respectively).

The comparison between the two-term (Equation (5)) and three-term (Equation (6)) approximation expansions of the QEI model showed that the estimations of S and K_s were markedly improved by using Equation (6), and the corresponding relative errors were significantly reduced (Table 2 and Table 3). Specifically, in all studied soils, a notable reduction of the relative errors of S, and even greater reduction in the case of K_s, were observed (Table 3). However, no significant differences were observed between the results obtained by Equations (4) and (6), as the difference between the relevant errors for both S and K_s did not exceed 4% (Table 3). Therefore, the three-term equation of Haverkamp et al. (Equation (6)), which is simpler than the Complete Haverkamp equation (Equation (4)), can be used instead of that for the estimation of the S and K_s parameters.

It is worth noting that the lower values of the relative error were observed in the prediction of S compared to K_s, with a few exceptions relating to Equation (1), mainly in coarse-textured soils.

Overall, it can be argued that the best results in the simultaneous determination of parameters S and K_s for the examined soils are given by Equation (7), with absolute relative errors less than 14% for both parameters. However, there are equations that resulted in slightly lower RE values, especially in the prediction of S, such as Equation (2), but they led to much higher RE values in the prediction of Κ_s. It is also worth pointing out that the predicted values of S and K_s from Equation (7) are almost the semi-sum of the corresponding values obtained from Equations (1) and (2). This can be attributed to the fact that the Valiantzas [9] equation (Equation (7)) is a solution that is approximately located at the middle of the area defined by the Green-Ampt (Equation (1)) and Talsma-Parlange (Equation (2)) infiltration equations for real soils.

The accuracy of each equation used seems to depend on the soil type and the infiltration time, the role of which will be examined in detail below. The equations of Talsma and Parlange (Equation (2)), complete Haverkamp (Equation (4)) and three-term approximation (Equation (6)) demonstrated particularly good results in soils with very large ratio S/K_s, i.e., in fine-textured soils [10]. Specifically, for these soils, the values of the ratio were 135.7, 93 and 28.22 for Yolo Light Clay; Silty Loam GE3 and Guelph—loam, respectively. The corresponding absolute RE values ranged from 0.32% to 3.13% for S and 2.62% to 11.76% for K_s.

The reliability of the estimated values of S and K_s given by the Solver application was evaluated using the Monte Carlo and Bootstrap simulation methods, as suggested by Hu et al. [32]. These methods rely on repeating the experiment under the same conditions with randomly generated numbers to evaluate the variations of the parameters (initial values of S and K_s used in all iterations are the values estimated by Solver, and the values of cumulative infiltration are randomly generated based on the estimated ones). The main differences between the two methods are the way the random numbers are generated and the number of iterations required for the results to be considered reliable. In this study, we applied both methods in 200 simulations. Histograms with a normal distribution of the parameters S and K_s obtained from both methods for Sandy Soil—Grenoble using Equation (7) are presented in Figure 1. It appears that both S and K_s, in both methods, tend to have a normal distribution. The average values of the parameter S from the Monte Carlo and Bootstrap methods were similar (1.2353 and 1.2356 cm min^−0.5, respectively) and very close to the predicted value (1.233 cm min^−0.5) from Solver using Equation (7). Similarly, the average values of the parameter K_s from the Monte Carlo and Bootstrap methods were similar (0.2859 and 0.2862 cm min⁻¹, respectively) and very close to the predicted value (0.287 cm min⁻¹). Similar results were obtained for all the studied soils.

Based on the abovementioned results, it can be said that the Excel Solver application is a reliable tool for estimating the S and K_s parameters from two-parameter infiltration equations, such as the studied Equations (1)–(7).

3.2. Results from the Latorre et al. Approach for S and K_s Estimation

The comparison between the predicted values of S and K_s obtained by the Latorre et al. [20] method and those obtained from the other methods used, as well as the actual values, showed that the method of Latorre et al. can satisfactorily estimate these parameters, since the absolute RE values for S and K_s do not exceed 16% and 24%, respectively, for all soils, except for Sand (RE = 41.53% for S) (Table 2 and Table 3). The RE values range from similar levels to those of Equations (4) and (6), as the differences of the relative errors between them do not exceed 5%. The method of Latorre et al. [20] showed particularly good results in soils where the ratio S/K_s had high values. These results were expected, since the method of Latorre et al. [20] uses the equation of Haverkamp et al. [7].

3.3. Results from the Linearization Method with Valiantzas Equation

The linearization method of Valiantzas [9] using Equation (8) was also applied. As shown in Figure 2, the transformed infiltration data i²/t with the cumulative infiltration i gave a strong linear correlation, with a coefficient of determination R² > 0.96 for all the soils studied. The results of the linearization method can be considered very satisfactory (Table 2). The absolute RE values did not exceed 8% and 19% for S and K_s, respectively. This method can be considered as equivalent to the nonlinear optimization procedure with Equation (7) by means of Solver, as they gave almost the same results for predicting S and K_s (Table 2), thus enhancing the reliability of the method with Solver. Only in the cases of Yolo Light Clay and Sandy Loam soil were remarkable differences observed in the estimation of K_s, where the linearization method gave RE −14.29% and −19.09%, respectively, while the Solver method gave −7.51% and −12.67%, respectively.

3.4. Fixed β = 0.6 vs. Variable β Generated from Nonlinear Optimization Equations

Parameter β is usually fixed at 0.6 [8] or 1.1 [23]. However, various researchers have reported values ranging from 0.3 to 2 [11,19].

Τhe case where β = 0.6 has been already investigated and presented (Equations (4)–(6), Table 2 and Table 3). Then, we will investigate parameter β as an additional adjustment parameter, such as S and K_s, in the three-parameter Equations (4)–(6) of the nonlinear optimization procedure. As shown in

Table 4. Predicted values of S and K_s from the complete Haverkamp (Equation (4)), two-term (Equation (5)) and three-term (Equation (6)) approximate expansions using the Solver application and considering β as an additional fitting parameter.

Soil	Equation (4) with Solver			Equation (5) with Solver			Equation (6) with Solver
	β	S (cm min−0.5)	Ks (cm min−1)	β	S (cm min−0.5)	Ks (cm min−1)	β	S (cm min−0.5)	Ks (cm min−1)
Sand	0.808	1.458	0.361	0.439	1.154	0.51	0.914	1.446	0.395
Soil and Sand mixture	1.215	0.234	0.0139	0.508	0.191	0.0179	1.286	0.232	0.0153
Sandy Soil—Grenoble	1.146	1.329	0.333	0.504	1.149	0.395	1.321	1.337	0.368
Silty Soil	4.39	1.116	0.047	0.542	0.678	0.058	2.532	0.98	0.042
Yolo Light Clay	1.401	0.1	0.00081	0.566	0.0894	0.00087	1.340	0.0984	0.00088
Silty Loam GE3	0.013	0.314	0.0023	0.591	0.311	0.0036	0.0138	0.3126	0.0024
Guelph—loam	0.764	0.6107	0.022	0.590	0.5873	0.0254	0.851	0.6096	0.0235
Sandy Loam	1.502	1.571	0.146	0.583	1.498	0.118	1.680	1.586	0.158

, the results differ among the equations. Using Equation (5), the value of β ranged from 0.439 to 0.591, with an average value of 0.54, which is close to the constant value 0.6 [8]. The reliability of the estimation of parameter K_s was slightly improved, as the maximum RE value decreased (from 89.46% to 69.89% for Sand), while no differences were observed in the predicted values of S compared to those computed using β = 0.6 (Table 2 and Table 4). Therefore, since the predicted values of S and K_s are nearly independent of parameter β, it is recommended to use the constant value β = 0.6. These results are the same to those of Rahmati et al. [22] and Latorre et al. [23] regarding the effect of parameter β on the reliability of the predictions of S and K_s. Additionally, in both cases (β = 0.6 and β ≠ 0.6), it seems that Equation (5) underestimates the S and overestimates the K_s values.

As regards Equation (6), a greater variation of the β values was observed in relation to Equation (5). Specifically, they ranged from 0.0138 to 2.532 (Table 4). However, it was found that the estimations of S and K_s were not improved compared to the case where β was 0.6 (Table 2 and Table 4). In fact, the RE values increased in some cases, such as in Yolo Light Clay and Sandy Loam soil. Regarding the S prediction, the RE values slightly increased: from −2.54% to −3.5% for Yolo Light Clay and from 4.6% to 9.7% for Sandy Loam soil. On the other hand, in the case of the K_s prediction, the RE values considerably increased: from 2.62% to 25.7% for Yolo Light clay and from −3.93% to −43.3% for Sandy Loam soil. In other words, it seems that the values of S and K_s, in most cases, are nearly independent from the β value. Therefore, in this case, the use of the fixed value 0.6 can be suggested to predict S and K_s as well.

Equation (4) also showed similar characteristics to those of Equation (6) in terms of estimating the parameters S and K_s using β as an adjustment parameter. In this case, the values of β ranged from 0.013 to 4.39 (Table 4). These values are similar to those from Equation (6) in all the studied soils, except Silty soil, and the corresponding RE values are similar as well. However, it is worth noting that Equation (6) is simpler than Equation (4), which has an implicit QEI formulation and may be complicated to use. Therefore, it seems that using parameter β as an adjustment parameter in Equation (4) has no improvement on the predictions of S and K_s. Therefore, even in this case, we can consider as the best scenario for the estimation of S and K_s the fixed value of β (β = 0.6).

Overall, in all Haverkamp expressions (complete, two-term and three-term approximate expansions), the use of parameter β as an additional adjustment parameter during the computation procedure does not improve the prediction of S and K_s. Therefore, we suggest using a constant value of β to predict the S and K_s parameters. In addition, it seems that Equation (6), due to its advantages (better prediction of the values of S and K_s compared to Equation (5), and simpler form compared to Equation (4)) could be used for the direct and inverse modeling of water infiltration into soils.

Rahmati et al. [22] reported that the parameters S and K_s are nearly independent from parameter β in the three-term expansion equation of Haverkamp and that the estimation of the former parameters is best with a fixed value of the latter one. Additionally, Latorre et al. [23] found that S and K_s in Equation (4) can be reliably calculated by considering a value of β fixed at 1.1 from 1D and 3D infiltration experiments with a 50-mm radius disc infiltrometer for times up to 1000 s.

3.5. Evaluation of S and K_s through Time from Nonlinear Optimization and Linear Equation of Valiantzas

In the present study, the accuracy of the predicted S and K_s from all studied equations with respect to the infiltration time was also examined (Figure 3). In these equations, the parameter β was fixed at 0.6.

Regarding the different versions of the Haverkamp equation (complete, two-term and three-term expansions), Equation (5) of the two-term, after an infiltration time, leads to a severe deviation of the S estimation while generally overestimating the K_s. In other words, it seems that the infiltration time significantly affects the reliability of the predictions of S and K_s, which is expected, since the validity time of the equation is generally limited [22]. On the contrary, the reliability of the prediction using Equations (4) and (6) generally increases with time, and both equations give almost the same results in all soils for all infiltration times. In some soils (i.e., sand, sandy soil, Guelph loam, silty loam and sandy loam), an overestimation of K_s is observed in the early times, but the accuracy of the estimation becomes very good in later times (Figure 3).

Rahmati et al. [22] investigated the accuracy of the predicted S and K_s with respect to the infiltration time for Equations (5) and (6) and found that Equation (6) behaves better in the estimation of S and K_s and is valid for longer times than Equation (5). Indeed, in Figure 3, we observe, in almost all soils, that the estimations of S and K_s from Equation (6) are closer to the actual values for all the infiltration times compared to Equation (5). It seems that the third term in Equation (6) increases the accuracy of the equation and decreases the variability of the parameters with time.

Equation (7) gave similar results with those of Equations (4) and (6) for all infiltration times.

The S estimations from Equation (1) (Green-Ampt equation) seem to deviate after an infiltration time, while the values of K_s are relatively stable with time, with a tendency toward underestimation. Equation (2) (Talsma-Parlange) generally gives stable values with a good prediction of S and reliable prediction of K_s over time. It is worth mentioning that the S and K_s predictions from Equations (4), (6) and (7), for all infiltration times, are located between the S and K_s values predicted from Equations (1) and (2), which define the limits of the infiltration areas of real soils [9].

Additionally, the linear and nonlinear optimization methods of Valiantzas (Equations (7) and (8), respectively) have similar behaviors in terms of the parameter estimations with time. In both methods, the S and K_s estimations are similar, thus enhancing the validity of the nonlinear optimization method. For later times, the values present a similar stability and satisfactorily approach the actual values. The overestimations of K_s or the underestimations of S at early times are improved at later times. Therefore, the infiltration of Equation (7) has a large validity time interval.

Overall, from the results presented in Figure 3, it is clear that longer infiltration measurements were needed to predict K_s more accurately for medium (e.g., t > 30 min for Guelph loam soil) and fine (e.g., t > 500 min for Yolo Light Clay)-textured soils than coarse-textured ones (e.g., t > 10 min for Sandy soil—Grenoble). After these times, it seems that almost all the equations used were less time-dependent.

Additionally, another common feature observed in all the soils is that parameters S and K_s behave as inversely proportional amounts, i.e., when one parameter increases, the other decreases and vice versa. This may be due to the fact that capillary forces (S) dominate the soil water infiltration at early times, whereas, at later times, the effect of the capillary forces decreases, and the gravitational force (K_s) becomes dominant [21].

4. Conclusions

There are many methods and equations applied to estimate parameters S and K_s from cumulative infiltration data. Our aim was to select the method that will predict the required parameters with accuracy and be relatively easy, simple and rapid to use. Overall, from the studied equations, the Valiantzas equation with the nonlinear optimization method (Equation (7)) gave better results, with the smallest relative errors for both parameters. The method of Latorre et al. [20] and the nonlinear optimization method with complete and three-term approximate expansions (Equations (4) and (6)) gave very satisfactory predictions for both parameters in the soils with a large value of the S/K_s ratio, i.e., in fine-textured soils. The linearization method gave almost the same results as the nonlinear optimization method. The only equation that seemed to seriously fail, especially in estimating K_s, was the two-term approximate expansion (Equation (5)).

In the three-parameter infiltration equations, the use of parameter β as a fitting parameter did not lead to an improvement in the predictions of parameters S and K_s. For this reason, in all three parameter equations, the use of a constant value of β was proposed. The three-term approximate expansion is a reliable equation for predicting both i(t) and the S and K_s parameters.

The infiltration time seemed to play a serious role in estimating the S and K_s parameters from the equations used. The application of the equations outside of their validity time intervals may corrupt the accuracy of these equations to predict the parameters. Thus, the accuracy of each equation in estimating the S and K_s parameters depends on its theoretical assumptions, the soil type and the infiltration time.