# Comparison of Soil Hydraulic Properties Estimated by Steady- and Unsteady-Flow Methods in the Laboratory

^{*}

## Abstract

**:**

_{1}and C

_{2}of the 2T equation were affected by the infiltration time. The coefficient C

_{1}increased while C

_{2}decreased with increasing time when the cumulative linearization method (CL) was applied, but the change in C

_{1}tended to be smaller than that in C

_{2}. The inverse solution of the 3T equation using the Excel Solver application for β = 0.75 and β = 1.6, when positive values of K were obtained, approached better the K values estimated by the steady-flow methods compared with those estimated using β = 0.6. Regarding the estimation of S from the unsteady-flow equations (2T, 3T, Zhang), comparable S values were obtained by all equations. The differences between the S values of the various methods are smaller compared to those of K, and S is less affected than K in terms of time. The problem of negative estimates of K might be attributed to the fact that the soils used in this study are classified as soils situated in the domain of lateral capillarity or are not completely homogeneous or soil compaction is observed at some depth. In the case where the soils are not completely homogeneous, the Sequential Infiltration Analysis (SIA) method with β = 0.75 corresponding to the soil types studied was proved to be effective in estimating K values.

## 1. Introduction

_{s}(L

^{3}T

^{−1}) is the steady infiltration flux, r (L) is the radius of the disc, Κ

_{0}(L T

^{−1}) is the hydraulic conductivity at the applied pressure head, H

_{0}(L), Κ

_{i}(L T

^{−1}) is the hydraulic conductivity at the initial pressure head, H

_{i}(L), and Φ

_{0}is the matric flux potential which is defined as:

_{s}(L T

^{−1}) is the saturated hydraulic conductivity and a (L

^{−1}) is a soil texture parameter which expresses the relative importance of the gravity and capillary forces during water movement in unsaturated porous media.

_{i}is negligible compared to K

_{0}, the matric flux potential can be expressed as:

_{0}= K

_{0}/a into Equation (1), the steady-state infiltration rate under a tension disc infiltrometer can be expressed by the following equation:

_{s}(L T

^{−1}) is the steady-state infiltration rate (i

_{s}= Q

_{s}/(πr

^{2})) for the applied pressure head.

_{s}and a. Then, K at each applied pressure head is calculated from Equation (3).

_{3D}(L) is the three-dimensional cumulative infiltration, t (T) is the time, and C

_{1}(LT

^{−1/2}) and C

_{2}(LT

^{−1}) are coefficients that differ depending on the applied model.

_{1}and C

_{2}with sorptivity (S) and hydraulic conductivity (K), respectively, in a different way.

_{0}(L

^{3}L

^{−3}) is the soil water content at the applied pressure head H

_{0}(L), r (L) is the radius of the tension disc, and θ

_{i}(L

^{3}L

^{−3}) is the initial soil water content.

_{0}(L) is the radius of the tension disc, and H

_{0}(L) is the applied pressure head.

_{1}and C

_{2}of Equation (6) is of great importance for the application of non-steady-flow methods. It is usually suggested to calculate them through linearization methods of the infiltration data, because it is easy, even visually, for the researcher to detect any failure of experimental points and to decide whether Equation (6) can describe the experimental data in contrast with other methods. Two linearization methods have been proposed. The first method was proposed by Smiles and Knights [29], who suggested the linearization of Equation (6) by dividing both sides with the square root of time (Cumulative Linearization—CL method) as:

_{1}and C

_{2}can be performed by differentiating the cumulative infiltration data with respect to the square root of time (Differential Linearization—DL method):

_{1}equal to the intercept and 2C

_{2}equal to the slope of the regression line. In the case where a contact material is used to ensure the hydraulic contact between the disc and the soil, the influence of the contact material is easy to detect since it corresponds to the initially sharply decreasing part of the experimental curve of infiltration data dI/d√t vs. √t, deviating from the monotonically increasing part of the experimental curve [21].

## 2. Materials and Methods

#### 2.1. Soil Types and Experimental Pressure Heads

_{φ}) of the soils was 1.41, 1.17, and 1.05 gcm

^{−3}, respectively, and their mechanical composition was 13.2% clay, 8% silt, and 78.8% sand for SL; 20% clay, 28% silt, and 42% sand for L; and 49.48% clay, 24.16% silt, and 26.36% sand for C. The clay soil includes only an aggregated fraction of 0.5–1.2 mm particles. After sampling, the soil samples were air-dried and passed through a 2 mm sieve. The initial water content of all soils ranged from 0.01 to 0.05 cm

^{3}cm

^{−3}, i.e., the experiments were carried out under dry initial conditions. For the experimental needs, a container with dimensions 49.5 × 46.4 × 22 cm, which ensured the unhindered three-dimensional infiltration for each type of studied soil, was used in the laboratory. Amounts of soil were gradually put in the container with simultaneous mechanical vibration of the container for achieving uniform bulk density to attain a uniform and homogeneous soil profile. Three repetitions were completed per pressure head and soil type.

#### 2.2. Laboratory Apparatus

#### 2.3. Experimental Procedure

_{i}is the experimental cumulative infiltration at time t

_{i}, and I

_{i}* is the corresponding estimated cumulative infiltration by applying the linear equation. In this way, we separate the steady-state flow from the transient flow. In the case of consecutive pressure head experiments, the sequence of pressure heads −15, −7, −3, and −1 cm was applied. The experiments started with the −15 cm pressure head, and when steady-state flow was reached the next pressure head of −7 cm was applied until steady flow was again achieved, and then the pressure heads of −3 and −1 cm were applied in the same way. The ascending (dry to wet) sequence (i.e., −15 cm → −7 cm → −3 cm → −1 cm) was selected because drainage occurs close to the disc while wetting continues at the infiltration front, reducing the hysteresis effect of the soil [19,42]. The duration of the experiments did not exceed two hours. The time circles in the initial stages of the experiments were every 10–15 s, while in the final stages (steady state) they reached up to 5 min.

#### 2.4. Methods

#### 2.4.1. Steady-State Methods

#### The Ankeny et al. [1] Method

_{1}and Q

_{2}(L

^{3}T

^{−1}) are the steady infiltration fluxes at the corresponding successive pressure heads H

_{1}and H

_{2}for which ΔH = H

_{1}− H

_{2}< 0 and r (L) is the radius of the infiltrometer tension disc.

_{1}and H

_{2}, and the Q

_{1}and Q

_{2}are calculated from the steady-flow data then the hydraulic conductivity at the corresponding pressure heads can be estimated from Equations (12) and (13). In the case where the experiments are carried out at more than two consecutive pressure heads, the value of K at the intermediate values of H will be equal to the arithmetic mean of the two individual values.

#### The Reynolds and Elrick [19] Method

_{s1,2}and a

_{1,2}are the saturated hydraulic conductivity and the a parameter, respectively, between a small range of pressure heads, H

_{1}and H

_{2}, and can be calculated from the equations:

_{1}and H

_{2}, and the K

_{s1,2}and a

_{1,2}are calculated from the steady-flow data using Equations (15) and (16) then the hydraulic conductivity at the range of corresponding pressure heads can be estimated from Equation (14). As in the case of the Ankeny et al. [1] method, if the experiments are carried out at more than two consecutive pressure heads, the value of K at the intermediate values of H will be equal to the arithmetic mean of the two individual values.

#### The Logsdon and Jaynes [20] Method

_{s}at the corresponding pressure heads. The main difference between this method and those of Ankeny et al. [1] and Reynolds and Elrick [19] is that this one utilizes the steady-flow data from all pressure heads simultaneously while the other two methods calculate the corresponding values from a pair of pressure heads. Logsdon and Jaynes [20] introduced the exponential relationship K(H) of Gardner (Equation (3)) into the Wooding equation (Equation (5)) resulting in Equation (17):

_{s}and a. The Microsoft Excel Solver tool was used to calculate these values. The fitted a parameter and K

_{s}were then used to calculate K(H) at each negative pressure head by applying Equation (3).

_{s}and a, the natural logarithm function was also applied in Equation (17) to compare the a and K

_{s}values of the two techniques:

_{s}, a = a, x = H, and b = ln[K

_{s}(1 + 4/(πra)]. So, from the diagram lni

_{s}vs. H, the slope of the line is the parameter a, and from the y-intercept of the line, which is b = ln[K

_{s}(1 + 4/(πra)], can be calculated the only unknown parameter K

_{s}. If the method of Logsdon and Jaynes [20] is used only for two successive pressure heads, it is considered similar to that of Reynolds and Elrick [19].

#### The White et al. [23] Method

_{s}is measured at one applied pressure head. It also utilizes both unsteady-flow data to estimate sorptivity, S, and steady flow.

_{0}:

_{i}) is negligible and the parameter b = 0.55 [22,44] yields the following equation:

_{s}= Q

_{s}/(πr

^{2}) (L T

^{−1}) is the steady-state infiltration rate.

_{s}is obtained from the steady-state flow experimental data, the values of θ

_{0}and θ

_{i}are calculated using a dielectric device, and the value of sorptivity S can be estimated from the infiltration data i(t) at the initial experimental times or from a linearization method. Due to the great uncertainty in the selection of the appropriate initial time interval for the estimation of the S parameter, its value is usually calculated with the linear fitting techniques, the so-called CL or DL (Equation (10) or Equation (11)) where C

_{1}= S. In our study, S was calculated both ways. The value of K at each pressure head is calculated from Equation (20).

#### 2.4.2. Unsteady-Flow Methods

#### The Haverkamp et al. [2] Expansion Models

_{0}(LT

^{−0.5}) and K

_{0}(LT

^{−1}) are the sorptivity and hydraulic conductivity, respectively, at the applied pressure head H

_{0}(L), K

_{i}(LT

^{−1}) is the initial hydraulic conductivity corresponding to the initial soil water content θ

_{i}(L

^{3}L

^{−3}), β (−) is a dimensionless integral shape parameter that depends on the diffusivity of the porous medium and ranges between 0.6 and 1.7 [45], and γ (−) is a dimensionless parameter equal to 0.75 [26,35].

_{0}and S [6,16,37,46]. Recently, Yilmaz et al. [39] suggested a value β = 0.9 for coarse-textured soils, such as sand and loamy sand. For the rest of the soil types, they suggested values of 0.75 for sandy loam and loam soils and 1.5 for silty loam and silty soils. The γ parameter is a dimensionless parameter for which the value 0.75 is usually used [26,35]. Lassabatere et al. [45] showed that the value of the γ parameter depends on the soil type and ranges between 0.75 and 1. Yilmaz et al. [39] suggested γ = 0.9 for sand and loamy sand soils and γ = 0.75 for the rest of the soils.

_{i}can be neglected, satisfactorily describes the water flow through a tension disc infiltrometer for a time less than or equal to t

_{grav}(T), where

_{grav}using the analytic implicit model proposed by Parlange et al. [34] valid for all times:

_{grav}is about three times larger than the classical t

_{grav}.

_{1}and C

_{2}can be related to sorptivity, hydraulic conductivity, and lateral capillary flow as follows:

_{1}and C

_{2}must first be calculated. Estimation of coefficients C

_{1}and C

_{2}can be obtained by fitting Equation (10) (CL method) or Equation (11) (DL method) to the experimental data. The adequacy of Equation (10) or Equation (11) can be checked by observing the linearity of the inputted data set.

_{0}at the applied pressure head H

_{0}can be calculated from Equation (26) and the value of S from Εquation (25). Because the accuracy of the estimating value of hydraulic conductivity from the equation of Haverkamp et al. [2] depends on the relative magnitude of the individual terms of Equation (22), the criteria of Vandervaere et al. [5] (Equation (7)) and Dohnal et al. [24] (Equation (8)) are taken into consideration.

#### The Zhang [4] Method

_{1}with the sorptivity S

_{0}and the C

_{2}with the hydraulic conductivity K

_{0}, as follows:

_{1}and A

_{2}are dimensionless parameters that depend on soil water content, soil water retention, and infiltrometer parameters. Zhang’s hypothesis that C

_{2}is correlated with gravitational forces (i.e., K

_{0}) has been doubted by several researchers [48]. For soils with the van Genuchten [28] type retention function, the mathematical expressions of these parameters were derived from numerical simulations of infiltration data [4]:

^{−1}) are the van Genuchten’s coefficients, and b (-) is a dimensionless parameter equal to 0.55. The van Genuchten equation is given as follows:

_{1}and C

_{2}can be calculated from the linearization methods (CL and DL) of Equation (6). On the other hand, the A

_{1}and A

_{2}can be calculated taking into consideration van Genuchten’s coefficients α and n as suggested by Carsel and Parrish [49] for each soil textural class (Table 1). Then, the hydraulic conductivity at the applied pressure head (K

_{0}) can be estimated from Equation (29).

## 3. Results and Discussion

#### 3.1. Comparison of the Results of Steady-Flow Methods

^{2}greater than 0.97 (Figure 1). From the coefficients of the linear equations, the values of the parameters K

_{s}and a can be derived for each soil, and then the values of K at each pressure head can be calculated from Equation (3). The results showed similar K values between the different approaches for the Logsdon and Jaynes [20] method (Table 3).

#### 3.2. Unsteady-Flow Methods

_{0}) referred to the unsteady flow were calculated (Table 4). In Figure 2, the I(t) diagrams and the separation of unsteady from steady flow (given by arrows) are depicted. These t

_{0}times are the same order of magnitude as the transient flow times reported by Latorre et al. [6] in 266 field infiltration experiments at pressure head H = 0 cm (t

_{0}ranged from 8 to 15 min) with the exception of loam soil at H = −15 cm where t

_{0}= 47 min. The hydraulic parameter estimations of the non-steady-flow equations were evaluated based on the aforementioned t

_{0}times at each pressure head.

#### 3.2.1. The Two-Term Equation (2T) of Haverkamp et al. [2]

_{1}and C

_{2}are given in Table 5. As shown in Figure 3, in all soils and pressure heads, Equations (10) and (11) had high R

^{2}values ranging from 0.8772 to 0.988 for the CL method and from 0.7202 to 0.9746 for the DL method with the exception of the clay soil at the pressure head −15 cm where R

^{2}= 0.3435. Comparing the two linearization methods, the CL method gave higher R

^{2}values than the DL method in all soils and pressure heads. The same results were reported by Moret-Fernadez et al. [51] in field experiments. Although the DL method had a smaller R

^{2}, its values were much higher than those presented by Latorre et al. [6] in 266 field infiltration experiments where only in 21 (8%) infiltration experiments was the R

^{2}≥ 0.6. This difference can probably be attributed to the fact that our experiments were conducted on disturbed soil samples with homogeneous initial soil moisture and the absence of bio-pores and cracks, in contrast to the field experiments of Latorre et al. [6]. In the latter case, abnormal changes in I(t) are very probable to occur, which, due to the derivation process in the DL method, will lead to small R

^{2}values.

_{1}and C

_{2}. The results showed that the two methods at each pressure head and soil gave quite different values of C

_{1}and C

_{2}even though they were applied to the same infiltration times. The DL method gave larger C

_{1}values and smaller C

_{2}values compared to the corresponding values of the CL method (Table 5). Specifically, the DL method compared to the CL method gave higher values of up to 41.1% for the coefficient C

_{1}, while correspondingly it gave lower values of up to 52.5% for the coefficient C

_{2}. From these results, it appears that the DL method will probably estimate values of K that are not acceptable since smaller values of C

_{2}and larger values of C

_{1}will lead to estimating negative values of K through Equation (26). Similar results were presented by Bagarello et al. [52] for the soils classified in the lateral capillarity domain after numerical infiltration experiments in sandy loam and clay soils applying the DL method to initially dry soils, where an underestimation of K was observed as C

_{2}was underestimated while C

_{1}, and thus S, was overestimated. For these cases, it is recommended to conduct experiments with higher initial soil moisture and relatively long experimental duration [52]. In contrast, for soils that were situated in the domain of gravity, Bagarello and Iovino [32] showed that the coefficients C

_{1}and C

_{2}calculated from the DL method, as well as the K estimations, were affected by the duration of the experiment and the time intervals of the water level drop measurements in the water reservoir of TI. In these soil types, the change in K depends mainly on the change in C

_{2}, while the change in C

_{1}, which is larger, has no significant effect on the K estimation.

_{1}and C

_{2}, these were calculated in the cases of the loam soil at a pressure head of −15 cm and for the sandy loam soil at −7 cm applying the CL linearization method. Two experimental times were selected for each soil: for the loam soil, the times t = 10 min (≈ t

_{0}/5) and t = t

_{0,}and for the sandy loam soil, the times t = 5 min (≈ t

_{0}/3) and the t = t

_{0}(Table 6). The results showed that the duration of the experiment significantly affected the values of C

_{1}and C

_{2}. Specifically, it was found that the value of C

_{1}decreased while the value of C

_{2}increased with the decrease in time, and the change in C

_{2}was much greater than that in C

_{1}for both CL and DL methods. These results are contrary to those of Bagarello and Iovino [32], who found a greater change in the coefficient C

_{1}. Factors that may contribute to this difference are whether the soils are classified in the lateral capillary or gravity domain, and also the initial soil water content. It is obvious that increasing C

_{2}values and decreasing C

_{1}values with decreasing infiltration time will affect the estimation of K and S values when the 2T equation of Haverkamp et al. [2] is used. Overall, the K estimations obtained from the non-steady methods varied with the duration of the experiment in a rather complex way [52].

_{2}, i.e., gravity plays a decisive role, whereas in the opposite case where lateral capillary forces play a dominant role, the estimated values of K are not reliable.

#### 3.2.2. The Three-Term Equation (3T) of Haverkamp et al. [2]

^{−1}, respectively, while for β = 0.75, they are 0.0923 and 0.1405 cm min

^{−1}, respectively, which are correlated with the K values estimated by the steady-state equations as well as Zhang’s equation. Similar values of K, at the corresponding pressure heads, are also obtained by applying the 3T equation instead of the 4T for the various values of β. Thus, the SIA method can also be applied using the 3T equation for laboratory experiments with these disturbed soils. However, considering the results presented in Figure 5, it is not always easy to select the appropriate infiltration time for the calculation of parameters K and S, because there is no clear descending and ascending part of the curve as observed in the diagrams of Moret-Fernadez et al. [38]. Such situations may appear in soils where there is no clear soil stratification, e.g., in soils where there is little soil compaction at some depth. In these cases, it is recommended that the measurements be very frequent in the initial stages of the infiltration so that the appropriate selection of the first part of the curve can be made easier. However, the short time intervals make it difficult to apply the method, especially in the case of visual readings.

#### 3.2.3. The Zhang [4] Method

_{1}and C

_{2}, and positive values of K were estimated in all soils and pressure heads studied (Table 7). In this respect, although it has been criticized for its physical adequacy, it is a very useful method. The estimated K values from the CL method were in almost all cases greater than those estimated from the DL method. The method of Zhang, when the CL method was used to calculate C

_{1}and C

_{2}, gave comparable values of K to those of the methods of Ankeny et al. [1] and Reynolds and Elrick [19] for all soils and pressure heads, except for the clay soil at pressure head −15 cm. The Zhang CL method compared with the Logsdon and Jaynes [20] method presented differences that are mainly focused on pressure head −1 cm for all soils and on −15 cm for clay soil (Figure 6). The differences vary by a factor of between 0.4 and 2 and may be considered acceptable for practical purposes [52]. These differences can be attributed to the fact that the Zhang model assumes that the hydraulic properties are described by the Mualem [27]-van Genuchten [28] model, while the Logsdon and Jaynes [20] method assumes that the K(H) relationship is described by the Gardner [18] equation. It is worth noting that the van Genuchten’s coefficients (α and n) used in the Zhang method were related to the soil texture class and were not obtained by fitting laboratory-determined soil water retention data.

_{2}parameter of Dohnal et al. [24] and the CL linearization method to calculate the C

_{2}parameter were also examined.

#### 3.2.4. The White et al. [23] Method (Steady-State Single Test Method—SST Method)

^{2}/(r

_{0}Δθ) in their equations [24]. This was especially observed in the case of the sandy loam soil where negative K values were obtained at pressure heads −1 and −7 cm.

#### 3.3. Estimation of Soil Sorptivity from Unsteady-Flow Data

^{0.5}) using the early-time infiltration data. In the Zhang method [4] and the 2T equation of Haverkamp et al. [2], both linearization methods (CL and DL) were used to calculate the coefficients C

_{1}and C

_{2}. Note that in the method of Haverkamp et al. S = C

_{1}, while in Zhang’s method S = C

_{1}/A

_{1}.

^{2}is included. Additionally, for the safe estimation of the linearity of the relationship I(t

^{0.5}) in the early times a sufficient amount of data is required that allows taking measurements in short time intervals, which is not always possible, especially in the field. Thus, the application of linearization methods over the entire range of measurements to calculate S is preferred.

_{1}is smaller than that on C

_{2}, it could be said that S is less affected than K in terms of time [6].

_{1}or inverse solution method of the 3T equation are preferred due to both the simplicity of the method and because no additional soil data are required, as in the case of the Zhang method, for the calculation of S.

## 4. Conclusions

_{1}and C

_{2}of the 2T infiltration equation are affected by the duration of the infiltration experiment. The coefficient C

_{1}increases while C

_{2}decreases with increasing time. From the experimental data, for the category of soils studied, the change in C

_{2}is more important than that of C

_{1}for the CL method. Comparing the two linearization methods, the CL method gave smaller C

_{1}values and larger C

_{2}values than those calculated from the DL method, as well as higher R

^{2}values in all soils and pressure heads. Therefore, in this category of soils, when no contact material is used, the CL method is considered more appropriate due to its simplicity.

^{2}/(r

_{0}Δθ) in their equations. The problem of negative K values of the equation of White et al. and the 2T equation of Haverkamp et al. will increase as the disc radius of the infiltrometer decreases and the lateral flow increases.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Presentation of the linear equations of the Logsdon and Jaynes [20] method (Equation (18)) for loam, sandy loam, and clay soils at pressure heads of −15, −7, −3, and −1 cm.

**Figure 2.**Diagrams of cumulative infiltration versus time, I(t), for loam (

**a**), sandy loam (

**b**), and clay (

**c**) soils at pressure heads of −15, −7, and −1 cm. The lines show that the water flow is steady, and the arrows show the time (t

_{0}) where the unsteady flow prevails.

**Figure 3.**Applying the CL and DL linearization methods for loam, sandy loam, and clay soils, at pressure heads of −15, −7, and −1 cm at corresponding times t

_{0}.

Soil | Van Genuchten Parameters | |
---|---|---|

α (cm^{−1}) | n (-) | |

Loam | 0.036 | 1.56 |

Sandy Loam | 0.075 | 1.89 |

Clay | 0.008 | 1.09 |

H (cm) | K (cm min^{−1}) | |||
---|---|---|---|---|

White et al. [23] | Logsdon and Jaynes [20] | Ankeny et al. [1] | Reynolds and Elrick [19] | |

Loam Soil | ||||

−15 | 0.0289 | 0.0089 | 0.0099 | 0.0097 |

−7 | 0.0369 | 0.0139 | 0.0129 | 0.0126 |

−3 | (no data) | 0.0175 | 0.0151 | 0.0145 |

−1 | 0.0631 | 0.0195 | 0.0255 | 0.0247 |

Sandy loam soil | ||||

−15 | 0.0496 | 0.0469 | 0.0428 | 0.0436 |

−7 | <0 | 0.1282 | 0.1289 | 0.1292 |

−3 | (no data) | 0.2119 | 0.2041 | 0.1998 |

−1 | <0 | 0.2724 | 0.2220 | 0.2155 |

Clay soil | ||||

−15 | 0.0152 | 0.0116 | 0.0131 | 0.0131 |

−7 | 0.0434 | 0.0411 | 0.0349 | 0.0348 |

−3 | (no data) | 0.0771 | 0.0783 | 0.0773 |

−1 | 0.0547 | 0.1057 | 0.1087 | 0.1066 |

**Table 3.**Hydraulic conductivity (K) values estimated by the Logsdon and Jaynes [20] method using the Solver tool and the linearization/logarithmization method for loam, sandy loam, and clay soils at pressure heads of −15, −7, −3, and −1 cm.

H (cm) | K (cm min^{−1}) | |||||
---|---|---|---|---|---|---|

Linearization/Logarithmization Method | Solver Tool | |||||

Loam | Sandy Loam | Clay | Loam | Sandy Loam | Clay | |

−15 | 0.0090 | 0.0446 | 0.0142 | 0.0089 | 0.0469 | 0.0116 |

−7 | 0.0144 | 0.1277 | 0.0412 | 0.0139 | 0.1282 | 0.0411 |

−3 | 0.0183 | 0.2160 | 0.0700 | 0.0180 | 0.2119 | 0.0771 |

−1 | 0.0206 | 0.2809 | 0.0912 | 0.0195 | 0.2724 | 0.1057 |

**Table 4.**Experimental times and unsteady-flow times according to the method of Angulo-Jaramillo et al. [21].

H (cm) | Experimental Time (t_{exp})(min) | Unsteady-Flow Time (t_{0})(min) |
---|---|---|

Loam Soil | ||

−15 | 103.0 | 47.00 |

−7 | 40.00 | 19.00 |

−1 | 19.50 | 11.00 |

Sandy loam soil | ||

−15 | 29.00 | 12.00 |

−7 | 52.00 | 16.00 |

−1 | 29.00 | 14.00 |

Clay soil | ||

−15 | 29.00 | 11.00 |

−7 | 33.00 | 11.00 |

−1 | 37.00 | 14.00 |

**Table 5.**The values of coefficients C

_{1}and C

_{2}using the cumulative linearization (CL) and differential linearization (DL) methods for loam, sandy loam, and clay soils, at pressure heads of −15, −7, and −1 cm at corresponding times t

_{0}.

H (cm) | CL Method | DL Method | ||
---|---|---|---|---|

C_{1} (cm min^{−0.5}) | C_{2} (cm min^{−1}) | C_{1} (cm min^{−0.5}) | C_{2} (cm min^{−1}) | |

Loam Soil | ||||

−15 | 0.2752 | 0.0424 | 0.3855 | 0.0265 |

−7 | 0.3537 | 0.0524 | 0.4698 | 0.0298 |

−1 | 0.4850 | 0.0388 | 0.5288 | 0.02845 |

Sandy loam soil | ||||

−15 | 0.3258 | 0.0605 | 0.4216 | 0.03875 |

−7 | 0.8826 | 0.1398 | 1.0259 | 0.10425 |

−1 | 1.3566 | 0.2058 | 1.5994 | 0.1463 |

Clay soil | ||||

−15 | 0.1943 | 0.0085 | 0.1955 | 0.0098 |

−7 | 0.2402 | 0.0429 | 0.3347 | 0.0204 |

−1 | 0.5793 | 0.1020 | 0.8175 | 0.04875 |

**Table 6.**The values of coefficients C

_{1}and C

_{2}using the CL linearization method at different times.

Coefficients C _{1}, C_{2} | Loam, H = −15 cm | Sandy Loam, H = −7 cm | ||
---|---|---|---|---|

t = 10 min (≈ t_{0}/5) | t_{0} = 47 min | t = 5 min (≈ t_{0}/3) | t_{0} = 16 min | |

C_{1} (cm min^{−0.5}) | 0.2752 | 0.0424 | 0.3855 | 0.0265 |

C_{2} (cm min^{−1}) | 0.4850 | 0.0388 | 0.5288 | 0.02845 |

**Table 7.**Estimated hydraulic conductivity values (K) from unsteady-flow methods for loam, sandy loam, and clay soils, at pressure heads of −15, −7, and −1 cm. In the case of the Haverkamp et al. [2] method, the K values were obtained considering the constant value β = 0.6.

H (cm) | K (cm min^{−1}) | ||||
---|---|---|---|---|---|

Haverkamp (2T) CL Method | Haverkamp (2T) DL Method | Haverkamp (3T) Using Solver Tool | Zhang CL Method | Zhang DL Method | |

Loam Soil | |||||

−15 | 0.0569 | <0 | 0.0288 | 0.0080 | 0.0050 |

−7 | 0.0581 | <0 | 0.0369 | 0.0205 | 0.0117 |

−1 | <0 | <0 | 0 | 0.0264 | 0.0193 |

Sandy Loam soil | |||||

−15 | 0.0710 | <0 | 0.0509 | 0.0559 | 0.0358 |

−7 | <0 | <0 | 0 | 0.1351 | 0.1007 |

−1 | <0 | <0 | 0 | 0.2057 | 0.1462 |

Clay soil | |||||

−15 | 0.0037 | 0.0063 | 0.0052 | 0.0041 | 0.0047 |

−7 | 0.0724 | 0.0057 | 0.0539 | 0.0304 | 0.0145 |

−1 | 0.1116 | <0 | 0.0752 | 0.0968 | 0.0454 |

**Table 8.**Application of the criteria of Vandervaere et al. [5] and Dohnal et al. [24] for loam, sandy loam, and clay soils at pressure heads of −15, −7, and −1 cm. The values of C

_{1}and C

_{2}used were derived from the CL and DL linearization methods. The symbols Y and N are used when the criteria are satisfied and not satisfied, respectively.

Criteria for CL Method | ||||||

H (cm) | Loam Soil | Sandy Loam Soil | Clay Soil | |||

Vandervaere et al. [5] | Dohnal et al. [24] | Vandervaere et al. [5] | Dohnal et al. [24] | Vandervaere et al. [5] | Dohnal et al. [24] | |

−15 | Y | Y | Y | Y | N | Y |

−7 | N | Y | N | N | Y | Y |

−1 | N | N | N | N | N | Y |

Criteria for DL method | ||||||

H (cm) | Loam soil | Sandy Loam soil | Clay soil | |||

Vandervaere et al. [5] | Dohnal et al. [24] | Vandervaere et al. [5] | Dohnal et al. [24] | Vandervaere et al. [5] | Dohnal et al. [24] | |

−15 | N | N | N | N | Y | Y |

−7 | N | N | N | N | N | Y |

−1 | N | N | N | N | N | N |

**Table 9.**Estimated hydraulic conductivity values from the inverse solution method and Equation (27) for loam, sandy loam, and clay soils, at pressure heads of −15, −7, and −1 cm considering the values of β = 0.6 and β = 1.6, and for the loam and sandy loam soils the value of β = 0.75 [39].

H (cm) | K (cm min^{−1}) | ||
---|---|---|---|

β = 0.6 | β = 0.75 | β = 1.6 | |

Loam Soil | |||

−15 | 0.0288 | 0.0307 | 0.0354 |

−7 | 0.0369 | 0.0398 | 0.0518 |

−1 | 0 | 0 | 0 |

Sandy Loam soil | |||

−15 | 0.0509 | 0.0547 | 0.0685 |

−7 | 0 | 0 | 0 |

−1 | 0 | 0 | 0 |

Clay soil | |||

−15 | 0.0052 | - | 0.0126 |

−7 | 0.0539 | - | 0.0692 |

−1 | 0.0752 | - | 0.1025 |

H (cm) | Clay Soil | ||
---|---|---|---|

K (cm min^{−1}) | |||

Zhang [4] CL Method | Zhang [4] DL Method | Dohnal et al. [24] | |

−15 | 0.0041 | 0.0047 | 0.0022 |

−7 | 0.0304 | 0.0145 | 0.0120 |

−1 | 0.0968 | 0.0454 | 0.0305 |

**Table 11.**Values of soil sorptivity (S) estimated from unsteady-flow methods for loam, sandy loam, and clay soils at −15, −7, and −1 cm pressure heads. In the case of the Haverkamp et al. [2] method using the Solver tool, the S values were estimated considering the constant value β = 0.6.

S (cm min^{−0.5}) | ||||||
---|---|---|---|---|---|---|

H (cm) | Early Stage | Haverkamp (2T) CL Method | Haverkamp (2T) DL Method | Haverkamp (3T) Using Solver Tool | Zhang CL Method | Zhang DL Method |

Loam soil | ||||||

−15 | 0.2825 | 0.2752 | 0.3855 | 0.3126 | 0.1695 | 0.2374 |

−7 | 0.3812 | 0.3537 | 0.4698 | 0.3732 | 0.2896 | 0.3847 |

−1 | 0.4750 | 0.4850 | 0.5288 | 0.4822 | 0.4743 | 0.5172 |

Sandy Loam soil | ||||||

−15 | 0.3790 | 0.3258 | 0.4216 | 0.3394 | 0.3956 | 0.5119 |

−7 | 1.0013 | 0.8826 | 1.0259 | 0.8415 | 1.0420 | 1.2111 |

−1 | 1.4964 | 1.3566 | 1.5994 | 1.1436 | 1.6176 | 1.9071 |

Clay soil | ||||||

−15 | 0.2054 | 0.1943 | 0.1955 | 0.1930 | 0.1190 | 0.1198 |

−7 | 0.2731 | 0.2402 | 0.3347 | 0.2530 | 0.1666 | 0.2321 |

−1 | 0.6523 | 0.5793 | 0.8178 | 0.6070 | 0.4446 | 0.6277 |

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**MDPI and ACS Style**

Koka, D.; Kargas, G.; Londra, P.A.
Comparison of Soil Hydraulic Properties Estimated by Steady- and Unsteady-Flow Methods in the Laboratory. *Water* **2023**, *15*, 3554.
https://doi.org/10.3390/w15203554

**AMA Style**

Koka D, Kargas G, Londra PA.
Comparison of Soil Hydraulic Properties Estimated by Steady- and Unsteady-Flow Methods in the Laboratory. *Water*. 2023; 15(20):3554.
https://doi.org/10.3390/w15203554

**Chicago/Turabian Style**

Koka, Dimitrios, George Kargas, and Paraskevi A. Londra.
2023. "Comparison of Soil Hydraulic Properties Estimated by Steady- and Unsteady-Flow Methods in the Laboratory" *Water* 15, no. 20: 3554.
https://doi.org/10.3390/w15203554