## Introduction

Measuring saturated soil hydraulic conductivity, *K _{s}*, directly in the field is recommended for interpreting and modelling soil hydrologic processes since disturbance of the sampled soil volume is minimised and its functional connection with the surrounding soil is maintained (Bouma, 1982). Due to the practical difficulties of fieldwork, reliable data should be collected with a reasonably simple and rapid experiment.

The single-ring pressure infiltrometer (PI) (Reynolds and Elrick, 1990) is a simple device that has frequently been applied in the field during the past 20 years (Vauclin *et al.*, 1994; Ciollaro and Lamaddalena, 1998; Bagarello and Iovino, 1999; Angulo-Jaramillo *et al.*, 2000; Bagarello *et al.*, 2000; Reynolds *et al.*, 2000; Mertens *et al.*, 2002; Bagarello and Sgroi, 2004; Gómez *et al.*, 2005; Verbist *et al.*, 2009, 2010; Bagarello *et al.*, 2013b; Verbist *et al.*, 2013; Bagarello *et al.*, 2014a; Angulo-Jaramillo *et al.*, 2016). A constant hydraulic head is established within a ring inserted to a short depth into the soil, and three-dimensional flow rate into the soil is monitored until near steady-state conditions have been reached. The steady-state methods developed by Reynolds and Elrick (1990), and particularly the one-ponding depth (OPD) and two-ponding depth (TPD) approaches, are the most commonly applied methods to analyse the PI data, although a variety of alternative methods have been considered in different investigations (Table 1). The OPD approach implies establishing a single depth of ponding on the infiltration surface but it needs an independent estimate of the so-called α* parameter, that represents the ratio of gravity to capillary forces during the infiltration process (Reynolds and Elrick, 2002a). The TPD approach yields a simultaneous estimate of *K _{s}* and α* but the experiment is more complicated since two depths of ponding have to be established in succession on the infiltration surface. The simplified falling head (SFH) technique is another ponding infiltration technique (Bagarello

*et al.*, 2004). In this case, an estimate of

*K*is obtained by a one-dimensional, transient, falling-head infiltration process (Bagarello and Sgroi, 2007; Bagarello

_{s}*et al.*, 2010; Agnese

*et al.*, 2011; Bagarello

*et al.*, 2013a). An independent estimate of α* is also necessary to analyse the infiltration data collected by the SFH run. For both infiltration techniques, there are experimental and/or analytical issues still needing clarifications and developments.

All available methods to analyse the PI data assume that the source is circular but using non-circular sources could be advisable in particular circumstances. For example, a square infiltrometer could allow, at least in theory, to sample completely an area of interest, which is less practical with a circular source. Therefore, the use of a ring infiltrometer precludes the possibility to uniformly collect data although intensively sampling soil represents an important step toward an improved interpretation and simulation of hydrological processes at the field scale (Gómez *et al.*, 2005; Bagarello *et al.*, 2013a). In principle, a square infiltrometer allows not to lose some possibly important information that is unavoidably lost with a ring infiltrometer. Employing a square infiltrometer raises a problem in the calculations of *K _{s}* since the developed equations include the ring radius that has to be replaced by a suitable alternative quantity if a square source is being used. Gómez

*et al.*(2005) used a square infiltrometer and determined

*K*by assuming that the ring radius coincided with the side length of the infiltrometer. Using numerically simulated data, Bagarello

_{s}*et al.*(2016) suggested that steady-state infiltration data collected with a square infiltrometer can be analysed with the model by Reynolds and Elrick (1990), assuming that infiltration occurs through a circular source having the same area of the square infiltrometer, which maintains congruence in flux density. However, field comparisons between PI data obtained with circular and square sources are still lacking. The shape of the source does not have any theoretical influence on the estimation of

*K*by the SFH technique since the infiltration process is one-dimensional in this case. Therefore, sources of different shape (

_{s}*e.g.*, circular, square) are expected to yield similar results. However, rings are commonly used even with the SFH technique and a field comparison between alternative shapes of the source has never been carried out.

According to the existing literature, the PI technique is expected to yield more reliable estimates of *K _{s}* than α* (Reynolds and Elrick, 1990; Mertens

*et al.*, 2002). Notwithstanding this, a plausible α* value,

*i.e.*, falling within the realistic range of 1 to 100 m

^{–1}, together with positive estimates of both

*K*and α*, should be indicative of reliable TPD calculations on the basis of the existing guidelines (Reynolds and Elrick, 2002b). This criterion has much interest from a practical point of view since it seems to suggest that the calculated

_{s}*K*and α* values contain the necessary information to discriminate between reliable and unreliable data. To our knowledge, however, this reliability criterion has never been checked in the field.

_{s}The general objective of this investigation was to improve our ability to use ponding infiltration runs for field determination of saturated hydraulic conductivity, *K _{s}*, of a sandy-loam soil. The specific objectives were to: i) establish a comparison between circular and square sources with reference to the

*K*values obtained by both the pressure infiltrometer and the simplified falling head technique; and ii) test if the reliability criterion developed for the pressure infiltrometer and the two-ponding depth approach is enough to obtain good quality data.

_{s}## Theory

### Pressure infiltrometer

Steady, ponded infiltration from within a single ring into rigid, homogeneous, isotropic, uniformly unsaturated soil can be approximated by the following analytical expression for steady-state infiltration fluxes (Reynolds and Elrick, 1990):

where*Q*(L

_{s}^{3}T

^{–1}) is the steady-state flow rate,

*r*(L) is the ring radius,

*G*is a dimensionless shape parameter,

*K*(L T

_{s}^{–1}) is the saturated soil hydraulic conductivity,

*H*(L) is the steady depth of ponding in the ring, and φ

_{m}(L

^{2}T

^{–1}) is the matric flux potential. For practical purposes, the following estimate of

*G*can be used (Reynolds and Elrick, 1990): where

*d*(L) is the depth of ring insertion. With the TPD approach,

*K*and φ

_{s}_{m}are given by (Reynolds and Elrick, 1990): where

*H*

_{1}and

*H*

_{2}(L) are the steady depths of ponding (

*H*

_{2}>

*H*

_{1}) and

*Q*

_{s}_{1}and

*Q*

_{s}_{2}(L

^{3}T

^{–1}) are the corresponding steady flow rates. Then, the so-called α* parameter (L

^{–1}) can be calculated as:

The OPD approach implies measurement of a single *Q _{s}* value corresponding to an established ponding depth of water,

*H*, and the estimation of α* on the basis of the visually determined soil textural/structural characteristics (Elrick and Reynolds, 1992; Reynolds and Elrick, 1990; Reynolds

*et al.*, 2002). The following relationship is then used to obtain

*K*:

_{s}The α* parameter generally varies from 1 to 50 m^{–1} because of the direct and partially compensatory relationship between *K _{s}* and φ

_{m}that instead can individually range over many orders of magnitude (Reynolds and Elrick, 2002a). The reduced variability of α* and its connection to soil texture and structure make it a useful parameter in simplified single-head analyses for estimation of

*K*

_{s}.According to Bagarello *et al.* (2016), *K _{s}* can be determined by using the steady-state infiltration data collected by a square infiltrometer and the relationships by Reynolds and Elrick (1990) because it is possible to assume that infiltration occurred through a circular surface having the same area of the surface sampled by the square infiltrometer. In other terms, the equivalent radius,

*r*(L), that replaces

_{eq}*r*in Eq. (1) and the subsequent equations when steady-state flow rates are obtained from a square source, is:

*l*(L) is the side length of the square infiltrometer. In the analysis by Gómez

*et al.*(2005), also using a square infiltrometer, the following assumption was made:

### Simplified falling head technique

The SFH technique (Bagarello *et al.*, 2004) consists of quickly pouring a known volume of water, *V* (L^{3}), on the soil, generally confined by a ring inserted a fixed distance, *d* (L), into the soil, and in measuring the time, *t _{a}* (T), from the application of water to the instant in which the surface area,

*A*(L

^{2}), is no longer covered by water. To determine

*K*, the following equation, based on the analysis by Philip (1992) for falling-head one-dimensional cumulative infiltration, is applied:

_{s}^{3}L

^{–3}) is the difference between the saturated (θ

_{s}) and the initial (θ

_{i}) volumetric soil water content and

*D*=

*V/A*(L) is the depth of water corresponding to

*V.*Knowledge of Δ

_{θ}and

*d*allows to determine the volume of voids within the soil volume confined by the ring. A volume of water less than or equal to the volume of voids has to be used to assure one-dimensional flow during the experiment. Since Eq. (8) includes gravity, the only time limitation will occur if the wetting front emerges from the bottom of the ring and three-dimensional flow commences.

## Materials and methods

### Field site

The field experiment was carried out at the so-called AR site that was established at the Department of Agricultural and Forestry Sciences of Palermo University (Italy) (Table 2). A 400 m^{2} flat area of a citrus orchard, with trees spaced 4×4 m apart, was selected. The soil (Typic Rhodoxeralf) had a relatively high sand and gravel content. The soil texture of the upper part of the profile, 0.1 m thick, was sandy-loam according to the United States Department of Agriculture (USDA) classification (Alagna *et al.*, 2016a).

### Pressure infiltrometer experiment

A PI infiltration test was conducted in 39 randomly chosen locations within the experimental area. An infiltrometer consisting of a Mariotte reservoir 1.0 m high with a volume of approximately 11 L was used. All runs were carried out at the soil surface, after removing the first few centimetres of soil. A ring with an inner radius *r*=0.075 m was inserted to a depth *d*=0.03 m at 19 locations whereas a square box having a side length *l*=0.133 m with opened top and bottom ends was inserted to the same depth at other 20 locations, so that the infiltration surface was of 0.0177 m^{2} with both sources. Ring and square box insertion was conducted by gently using a rubber hammer and ensuring that the upper rim of the ring remained horizontal during insertion. A pile driver could also be used for a vertical insertion of the ring. A constant depth of ponding, *H*_{1}=0.053 m, was established on the soil surface and flow rate was monitored to detect near steady-state conditions. A constant depth of ponding, *H*_{2}=0.11 m, was then established and flow rate was monitored until another quasi steady-state condition was reached. The total duration of the run varied between 150 and 170 min, depending on the sampling point (Table 2). The rate of fall of the water level in the infiltrometer reservoir was monitored at 0.5 to 2 min time intervals. In several cases, high infiltration rates were observed and the water reservoir emptied before concluding the test. Refilling of the reservoir and changing the ponded depth of water from *H*_{1} to *H*_{2} were conducted by maintaining ponded conditions on the soil surface confined by the ring to avoid air entrapment in the sampled soil volume (Reynolds, 1993). Due to the refilling procedure, apparent steady-state flow rates (*Q _{s}*

_{1}and

*Q*

_{s}_{2}) corresponding to the two applied

*H*levels (

*H*

_{1}and

*H*

_{2}) were estimated from the flow rate

*versus*time plot. In general, a run with the ring and another run with the square box were carried out on a single day to reduce the risk to detect differences between the two sources that could be in reality expressive of differences in initial soil conditions. At an intermediate distance between the two sources, two undisturbed soil cores (0.05 m diam. by 0.05 m high) were collected at a depth of 0 to 0.05 m and 0.05 to 0.10 m, respectively, before the PI test. These cores were used to determine the bulk density, ρ

_{b}(Mg m

^{–3}), and the initial volumetric soil water content, θ

_{i}(m

^{3}m

^{–3}), that were averaged over the two sampled depths. The experiments were carried out in a period of seven months (Table 2), to explore a wide range of initial soil moisture conditions. For each two-level infiltration run,

*K*and α* were simultaneously calculated by Eqs. (3) and (4), with Eqs. (6) or (7) used in the case of a square source, and a dataset was developed for each source (circular, square). In particular, only the two-level runs simultaneously yielding positive results for the two variables and α* values ranging from 1 m

_{s}^{–1}to 100 m

^{–1}were included in the dataset (scenario no. 1, TPD), according to Reynolds and Elrick (2002b). Two additional scenarios were considered to develop a

*K*dataset. In the scenario no. 2 (OPD), the OPD analysis (Eq.5) was applied to both

_{s}*H*

_{1}and

*H*

_{2}values and the resulting

*K*values were averaged (Elrick and Reynolds, 1992). This approach was suggested by Reynolds and Elrick (2002b) as an alternative calculation method to be applied when the TPD approach produces negative or unrealistic results. Finally, the scenario no. 3 [OPD(

_{s}*H*

_{1})] included the

*K*data obtained by using Eq. (5) and the first ponded depth of water only (

_{s}*H*

_{1}). This last scenario was considered since a single-level run is obviously more rapid and parsimonious in terms of applied water than a two-level run. According to Elrick and Reynolds (1992) and in accordance with previous investigations at the field site (Bagarello and Sgroi, 2007), α* was set equal to 12 m

^{–1}for all OPD calculations.

### Simplified falling head experiment

An infiltration test of the SFH type was conducted in 12 randomly chosen locations within the experimental area. Applying this technique implies a relatively large insertion depth of the ring or the square box to be sure that a one-dimensional infiltration process is established during the run. However, a deep insertion increases the risk of compacting or shattering the sampled soil volume (Reynolds, 1993). To avoid this risk, a more conservative procedure (Bagarello *et al.*, 2009b) was applied to confine cylindrical and cubic soil volumes having the same infiltration surface (Figure 1), although this choice, implying a rather demanding fieldwork for preparing the soil sample, implied that a relatively small number of points were sampled. In particular, six soil cylinders (height=0.12 m, diameter=0.15 m) and six soil cubes (height=0.12 m, side length=0.135 m) were manually exposed and covered along their walls by a casing in polyurethane foam (Bagarello and Sgroi, 2008). Initially, the soil was exposed by digging a small trench. A cylindrical or cubic packing case with open ends at both the top and the bottom were placed around the soil column and a stopper in polyurethane foam, previously prepared in the laboratory, was put on the surface of the column to prevent direct contact between the expanding foam and the upper end of the sampled soil volume. The 60-70% of the space between the packing case and the soil column was filled with polyurethane foam and a tablet and a small weight of 1-2 kg were placed on the upper end of the packing case to confine foam expansion only partially. After the foam hardened, the packing case was detached along two previously realised cutting lines, and the stopper was removed to expose the soil surface for the SFH run.

Undisturbed soil cores collected two days before the infiltration run were used to determine θ_{i} and ρ_{b} and to obtain an estimate of θ_{s.} Eq. (8) with α*=12 m^{–1} was used to calculate *K _{s}.*

### Data summary

To summarise the *K _{s}* (PI, SFH) and α* (PI) data, the geometric mean and the associated coefficient of variation (

*CV*) were calculated using the appropriate lognormal equations (Lee

*et al.*, 1985) since these variables are commonly considered to be log normally distributed (Mohanty

*et al.*, 1994; Warrick, 1998). Arithmetic means and associated

*CV*s were calculated for ρ

_{b}and θ

_{i.}

## Results and discussion

### Pressure infiltrometer experiment

Regardless of the considered scenario, very similar *K _{s}* (

*i.e.*, differing by a factor of 1.05-1.16, depending on the scenario) and α* (factor of difference=1.32) values were obtained with the two sources when

*r*for the square source was defined by Eq. (6), and the differences between two

_{eq}*K*or α* datasets (one for the circular source and the other for the square source) were never statistically significant according to a two tailed t test at P=0.05 (Table 3). Using Eq. (7) to define

_{s}*r*did not produce significant differences between the two sets of α* values but the differences were statistically significant, although not substantial (

_{eq}*i.e.*, by a factor not exceeding 2.50), with reference to all

*K*calculation scenarios (Table 3). Therefore, this investigation gave experimental support to the use of Eq. (6) for analysing PI data collected with a square source.

_{s}Consequently, a unique dataset was developed for the field site by pooling the data estimated with the ring and the square [Eq. (6) for *r _{eq}*] infiltrometer together.

Neither α* nor *K _{s}* were significantly correlated with ρ

_{b}and θ

_{i}, regardless of the considered scenario (Table 4). The lack of any

*K*and α*

_{s}*vs*ρ

_{b}and θ

_{i}relationship was probably due to the fact that these last two parameters did not vary very much during the experimental period. In particular, ρ

_{b}values ranging from 0.95 to 1.28 g cm-3 (mean=1.12 g cm

^{–3}) were obtained on the 20 sampling dates but the

*CV*(ρ

_{b}) was low,

*i.e.*, equal to 7.8%. On the other hand, θ

_{i}values varying between 0.07 and 0.24 m

^{3}m

^{–3}(mean=0.13 m

^{3}m

^{–3}) were measured and the corresponding

*CV*was appreciably higher,

*i.e.*, equal to 45.1%. However, most (

*i.e.*, 80%) of the θ

_{i}values were lower than 0.20 m

^{3}m

^{–}3, since relatively dry soil conditions made the access to the field easier, and the

*K*values measured at the field site were found not to depend on θ

_{s}_{i}for θ

_{i}<0.20 m

^{3}m

^{–3}(Bagarello and Sgroi, 2007). This last finding was based on a

*K*measurement campaign made with the SFH technique but Bagarello and Sgroi (2007) also showed that similar

_{s}*K*values were obtained with the PI.

_{s}Considering the developed dataset for the scenario no. 1, α* was found to significantly increase with *K _{s}* (Figure 2). This result gave additional support to previous investigations suggesting that soils with high values of the α* parameter, which is indicative of a relatively low importance of capillarity on steady flow (Reynolds

*et al.*, 1992), should be expected to also have high

*K*values (White and Sully, 1992; Yitayew

_{s}*et al.*, 1998; Bagarello

*et al.*, 2014b).

Following Reynolds and Zebchuk (1996), the Tukey’s honestly significant difference test was applied to compare the *K _{s}* values obtained with the three considered scenarios (Table 5). Significantly lower results were obtained with the TPD approach (scenario no. 1) than the two OPD approaches (scenario no. 2 and 3) that yielded statistically equivalent

*K*values. In any case, differences were not substantial since the means differed at the most by 2.05 times and an error of the estimate of

_{s}*K*by a factor of two or three can be considered acceptable for many practical purposes (Elrick and Reynolds, 1992; Reynolds and Zebchuk, 1996; Elrick

_{s}*et al.*, 2002). The coefficients of variation also were similar among the three tested approaches since they varied within the rather narrow range of 41-51%. The α* parameter was lower than expected on the basis of the textural and structural characteristics of the sampled soil (Elrick and Reynolds, 1992). However, it was very close to the mean α* parameter (3.3

*m*

^{–1}) obtained in former investigations conducted at the same field site with the PI and the TPD approach by Bagarello

*et al*(2009b, 2013b). Therefore, the two tested OPD approaches were practically equivalent but lower

*K*values, and unexpectedly low α* values, were obtained with the TPD approach.

_{s}Attempting to explain these results, a comparison was initially established between the *K _{s}* values obtained with the OPD(

*H*

_{1}) approach (scenario no. 3) and the corresponding values obtained by applying a similar approach [

*i.e.*, Eq. (5) and α*=12 m

^{–1}] with the estimated steady state flow rates for

*H=H*

_{2}(

*Q*

_{s}_{2}). This comparison was made with reference to 37 data points since only

*Q*

_{s}_{1}was measured in two infiltration runs. With a very few exceptions (two out of the 37 cases), lower

*K*values were obtained with

_{s}*H*

_{2}than

*H*

_{1}, with a mean ratio between these two estimates of 0.89 (Figure 3). According to this result, the differences between the two

*K*values were small and probably negligible form a practical point of view. However, the established comparison also suggested occurrence of underestimation of

_{s}*Q*

_{s}_{2}or overestimation of

*Q*

_{s}_{1}, or both, during the field infiltration runs.

To test what happens with the TPD approach in these cases, the three representative sand (*K _{s}*=1×10

^{–4}m s

^{–1}, α*=36 m

^{–1}), loam (

*K*=1×10-6 m s

_{s}^{–1}, α*=12

*m*

^{–1}) and clay (

*K*=1×10

_{s}^{–8}m s

^{–1}, α*=4 m

^{–1}) soils according to Reynolds and Elrick (1990) were considered and the true

*Q*

_{s}_{1}and

*Q*

_{s}_{2}values were calculated by Eqs. (1) and (2). In other terms, the calculated steady-state flow rates for

*H*

_{1}=0.053 m and

*H*

_{2}=0.11 m, respectively, were those expected for the three theoretical soils according to the model by Reynolds and Elrick (1990). Then,

*K*and α* were calculated by Eqs. (2), (3) and (4),

_{s}*i.e.*, the TPD approach, considering a 5% and a 10% error in the estimation of

*Q*

_{s}_{1}(true value + error) and/or

*Q*

_{s}_{2}(true value – error). Only an overestimation was considered for the lower ponded level (

*H*

_{1}) since field runs have unavoidably a limited duration for practical reasons and the decrease of flow rates to steady-state conditions can be long, especially in fine textured and initially dry soils (Reynolds, 1993; Reynolds and Elrick, 2002b). Prolonged wetting is expected to weaken the soil aggregates because it lowers their cohesiveness, soften the cements and causes clay particle swelling (Morgan, 2005). All these phenomena are expected to induce a decrease in flow rates into the soil and, for this reason, only an underestimation was considered for the higher ponded level (

*H*

_{2}). With a single exception (clay soil, 10% overestimation of

*Q*

_{s}_{1}and 10% underestimation of

*Q*

_{s}_{2}), simultaneously positive

*K*and α* values were obtained and α* varied between 1.03 and 21.7 m

_{s}^{–1}, which means that there were signs of a successful two-level run. However, both

*K*and α* were systematically underestimated (Figure 4). A single error (overestimation of

_{s}*Q*

_{s}_{1}or underestimation of

*Q*

_{s}_{2}) was enough to determine erroneous

*K*and α* predictions and the absolute value of the prediction error was highest when overestimation of

_{s}*Q*

_{s}_{1}and underestimation of

*Q*

_{s}_{2}occurred simultaneously. Larger errors in

*Q*implied larger differences between the estimated and the true soil hydraulic parameters. Considering the scenario no. 2 (OPD) with the true α* parameter and the same error levels for

_{s}*Q*

_{s}_{1}and

*Q*

_{s}_{2}, the error of the estimated

*K*varied from –5% to +5%. With reference to the scenario no. 3 [OPD(

_{s}*H*

_{1}), only the error in

*Q*

_{s}_{1}was considered], the error of the

*K*prediction did not exceed 10%. These last results were expected taking into account that

_{s}*K*is directly proportional to

_{s}*Q*according to Eq. (5).

_{s}Therefore, this analysis showed that, with the TPD approach, excessively low *K _{s}* and α* values have to be expected as a consequence of a small overestimation of

*Q*

_{s}_{1}, a small underestimation of

*Q*

_{s}_{2}or both. Taking into account that, in practice, the errors in the calculated soil hydraulic parameters could not be detectable, since physically possible and also plausible results are obtained by the two-level analysis, the conclusion should be that the OPD approach has to be preferred in general to the TPD approach. This suggestion is based on the circumstance that single-level calculations appear to be less sensitive to small uncertainties in the estimated steady-state flow rates than two-level calculations, but also on the premise that no uncertainties affect the estimate of α*, which cannot be always true. Fortunately, a reduced variability of α* has to be expected (Reynolds and Elrick, 2002a).

Underestimation of *K _{s}* and α* by the TPD approach is likely a consequence of overestimation of

*Q*

_{s}_{1}and underestimation of

*Q*

_{s}_{2}. Steady-state conditions are slowly approached for the first infiltration run, particularly in medium- to fine-textured soils, which can imply an erroneous estimate of steady-state flow rate (

*e.g.*, Bagarello

*et al.*, 1999; Reynolds

*et al.*, 2000; Reynolds and Elrick, 2002b). However, with reference to this experimental investigation, overestimation of

*Q*

_{s}_{1}was considered a less likely cause of the low

*K*and α* values by the TPD approach since convincing near steady-state conditions were detected for

_{s}*H=H*

_{1}in all cases (Figure 5). Conversely, the low

*K*and α* values by the TPD approach were more likely attributable to a systematic underestimation of

_{s}*Q*

_{s}_{2}. The theoretical model by Reynolds and Elrick (1990),

*i.e.*, Eq. (1), was developed under the hypothesis of an initially uniformly unsaturated soil. When the first ponding depth of water (

*H*

_{1}) is established on the infiltration surface, the hypothesis of a uniform θ

_{i}can perhaps be considered plausible. However, when the subsequent ponding depth of water (

*H*

_{2}>

*H*

_{1}) is established on the infiltration surface, it is certain that the soil is initially wet (saturated, θ

_{s}) below the infiltration surface, and relatively dry (antecedent soil water content, θ

_{i}) outside the wetting front formed at the end of the run with the

*H*

_{1}level. Therefore, the uniform θ

_{i}hypothesis of the wetted soil at the beginning of the run with the

*H*

_{2}level is no longer valid. The TPD approach assumes that φ

_{m}does not vary in the passage from

*H*

_{1}to

*H*

_{2}but a higher initial soil water content implies smaller values of φ

_{m}in Eq. (1) and hence smaller steady-state flow rates. Therefore, the measured

*Q*

_{s}_{2}value could be expected to be lower than the one that would theoretically allow use of Eqs. (3) for calculation of soil hydraulic parameters. However, a check of this reasoning appears necessary since, at steady-state, the wetted soil volume with

*H=H*

_{2}envelopes that corresponding to

*H=H*

_{1}. Therefore, θ is equal to θ

_{i}outside the wetting front for both

*H=H*

_{1}and

*H=H*

_{2}, and the θ

_{i}uniformity hypothesis could also be valid for the analysis of steady-state flow rates. Other reasons for lower than expected

*Q*

_{s}_{2}values are of practical nature. As time passes, swelling phenomena, reducing macropore volume, have more opportunity to occur, also in soils with a low clay content (Bagarello and Sgroi, 2007), and obstruction of macropores become more likely since particle bonds are weakened by prolonged wetting. Moreover, some turbulence at the infiltration surface can occur in the passage from

*H*

_{1}to

*H*

_{2}. Finally, low

*Q*

_{s}_{2}values could also be due to a low permeability layer close to the bottom edge of the ring.

In conclusion, the estimate of *Q _{s}*

_{1}appeared more reliable than that of

*Q*

_{s}_{2}and therefore the

*K*values obtained with the scenario no. 3 were used for the subsequent comparison with the SFH technique. In other terms, the OPD approach appears to be less sensitive to

_{s}*Q*approximations than the TPD approach and, in particular, using

_{s}*Q*

_{s}_{1}is better than considering both

*Q*

_{s}_{1}and

*Q*

_{s}_{2}(scenario no. 2) since only one source of errors is included in the former approach.

### Simplified falling head experiment

Very similar *K _{s}* values (

*i.e.*, differing by a factor of 1.19) were obtained with the two sources (circular, square) by the transient SFH experiment and the differences between the two

*K*datasets were not statistically significant according to a two-tailed t test at P=0.05 (Table 6). Therefore, this investigation gave experimental support to the theoretically expected equivalence of the two sources since a one-dimensional infiltration process is established with the SFH technique.

_{s}A unique dataset was developed with all the SFH data (Table 6) and a comparison with the PI results (Table 5, scenario no. 3) was established by a two-tailed t test performed on ln(*K _{s}*). The difference between the mean values of

*K*(139.7 mm h

_{s}^{–1}for the SFH technique and 239.4 mm h

^{–1}for the PI) was statistically significant (P=0.05) but not substantial (means differing by 1.71 times). Moreover, all

*K*values obtained with the SFH technique fell within the range of the

_{s}*K*values obtained with the PI. Therefore, it seems plausible to suggest that detecting a statistical significance of the difference between the means was a consequence of the reduced number of data collected with the SFH technique. In other words, larger sample sizes for the transient technique should be expected to increase the probability to detect clearer similarities with the steady-state technique,

_{s}*i.e.*, even from a statistical point of view.

### Comparison with previous experiments

The mean *K _{s}* values obtained in this investigation with the PI and SFH runs were in line with the saturated conductivity previously measured at the same field site with the same techniques and also with other techniques, such as the so called BEST procedure of soil hydraulic characterisation by Lassabatere

*et al.*(2006) (Figure 6) (Bagarello and Sgroi, 2007; Bagarello

*et al.*, 2014b; Alagna

*et al.*, 2016a). In particular, it was confirmed that high

*K*values have to be expected at the field site when the soil is initially relatively dry.

_{s}## Conclusions

According to this investigation, the classical steady-state analysis of PI data is usable if a square infiltrometer is employed in the field. In this case, however, an equivalent radius has to be used in the calculations. In particular, it is possible to assume that infiltration occurs through a circular surface having the same area of the square infiltrometer. Assuming that the ring radius coincides with the side length of the infiltrometer is not recommended.

A plausible α* value, *i.e.*, falling within the realistic range of 1 to 100 m^{–1}, together with positive estimates of both *K _{s}* and α*, is not always indicative of reliable TPD calculations. In addition, excessively low

*K*and α* values have to be expected with the TPD approach as a consequence of a small overestimation of

_{s}*Q*

_{s}_{1}, a small underestimation of

*Q*

_{s}_{2}or both. Overestimating

*Q*

_{s}_{1}and underestimating

*Q*

_{s}_{2}is a practical possibility during application of the PI in the field for different reasons of experimental nature, such as a too short duration of the first phase of the run (

*H*=

*H*

_{1}) or the weakening of particle bonds by prolonged wetting (during the

*H*=

*H*

_{2}phase). Reasons of theoretical nature could also be suggested, taking into account that the soil is wetter at the beginning of the second phase of the two-level run than the first one. This circumstance could imply that the measured

*Q*

_{s}_{2}value is lower than the one that would theoretically allow use of the analytical model for calculation of soil hydraulic parameters. However, this reasoning is not free from doubts since, at steady state, a higher

*H*value determines wetting of larger soil volumes.

This is one of the reasons why the OPD approach with steady-state flow rate estimated under homogeneous initial moisture conditions (*i.e., Q _{s}*

_{1}) appears to be the most appropriate way to analyse the PI data. An additional reason is that estimating α* on the basis of the soil texture and structure characteristics appears a relatively straightforward task since a very limited number of soil categories have been defined. Clearly, the conclusion that the simplest PI run yields the most reliable

*K*data needs additional support.

_{s}Finally, despite limited to a single soil, this investigation confirmed that, as predicted by theory, the SFH technique can be applied with both circular and square sources since the shape of the infiltration surface does not influence the measured *K _{s}* values.

Theoretical developments on source shape effects for ponding infiltration experiments are advisable since the currently used equations for *K _{s}* estimation were derived for a circular infiltration source, which implies radial symmetry of the flux under and outside the ring. Radial symmetry cannot be assumed if the infiltration source has a square shape and the use of an equivalent radius represents a practical means to estimate

*K*from infiltration data collected by square sources. A topic to be developed experimentally is the comparison between the PI and SFH techniques. In particular, establishing in detail what are the sample size effects on the results of a comparison between these two techniques appear advisable. Even if some error is surely included in the

_{s}*K*estimation by using the equations developed for the circular source, in this application such error turned out to be negligible compared to the spatial and temporal variability (

_{s}*i.e.*, the mean values were not significantly different) and, thus, square infiltration appears to be a practical way to characterise the soil.