# Implications of Hysteresis on the Horizontal Soil Water Redistribution after Infiltration

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Theoretical Analysis

_{i1}, θ

_{i2}, …, θ

_{in}are the water content values during reversing the process from drying to wetting and vice-versa until reaching the value of θ.

_{0}, and where no water evaporation is allowed, the horizontal infiltration process is taking place, with the water front reaching a distance X. After the cessation of the infiltration, the redistribution process inevitably follows.

#### 2.2. Scenarios Examined

^{3}cm

^{−3}), infiltration durations (6, 12, and 25 min), and original infiltration depths (5.34 to 12.3 cm) for the case that hysteresis is considered or is neglected and for the cases that the initial water content varies following either the boundary wetting or the boundary drying curve of the hysteresis loop. After detailed examination of the obtained results for the above combinations, the following representative scenarios were analyzed to highlight the implications of hysteresis on horizontal infiltration and soil water redistribution.

_{0}= 0.12 cm

^{3}cm

^{−3}(before infiltration), without considering hysteresis and using the boundary wetting curve were compared with the profiles obtained in the presence of hysteresis when θ

_{0}was lying on the boundary wetting curve. The same comparison was made for the case when θ

_{0}was lying on the boundary drying curve.

_{0}= 0.12 cm

^{3}cm

^{−3}, (ii) Ι = 10.90 cm and θ

_{0}= 0.12 cm

^{3}cm

^{−3}, and (iii) I = 12.30 cm and θ

_{0}= 0.055 cm

^{3}cm

^{−3}. In all cases of the second scenario, the initial soil water content was assumed to lie on the boundary wetting curve.

_{0}= 0.12 cm

^{3}cm

^{−3}is the same. For the two cases of the third scenario, the initial soil water content was also assumed to lie on the boundary wetting curve.

_{0}= 0.12 cm

^{3}cm

^{−3}, before the beginning of the original infiltration, lied on the boundary drying and on the boundary wetting curves. The time duration of the original infiltration was 25 min for both cases.

_{0}(0.055, 0.12, and 0.2 cm

^{3}cm

^{−3}) on the redistribution rate was examined when the time duration of the original infiltration (25 min) was the same for both cases.

#### 2.3. Experimental Determination of the Boundary Wetting and Drying Curves of the Hysteresis Loop

^{−1}), was independently determined by the constant-head method [26].

_{s}is the volumetric soil water content at saturation; θ

_{r}is the residual volumetric soil water content; m = 1−(1/n); and Se = (θ−θ

_{r})/(θ

_{s}−θ

_{r}) (degree of saturation or effective saturation).

_{r}, α, and n. The values of the parameters m and p were taken as m = 1−(1/n) and p = 0.5, a value widely used [28]. The fitted curves of the Mualem–van-Genuchten model to the experimental data are also presented in Figure 1.

#### 2.4. Hydrus 1-D

_{r}were obtained from experimental data using the RETC program (Table 1) as described above.

_{0}, x > 0

_{s}(saturation, as H = 0)

## 3. Results

#### 3.1. Comparison of the Redistribution Profiles with and without Hysteresis When the Initial Water Content Lies on the Boundary Wetting or the Boundary Drying Curve

_{0}= 0.12 cm

^{3}cm

^{−3}(H = −49.9 cm), the infiltration duration was T = 25 min, and the infiltration depth was 10.863 cm in both cases. The H–θ relationship (for both cases) during the infiltration from the state B caused an increase in θ and H following the boundary wetting curve (Figure 2a). During the redistribution process for the WH case, again, the H–θ relationship in every position in the soil column was following the boundary wetting curve, while, for the HY in every position in the soil column, the H–θ relationship would follow, depending on the θ value attained in the specific soil position, an appropriate drying scanning curve (e.g., DE drying scanning curve, Figure 2a), or the boundary drying curve as such.

^{3}cm

^{−3}from θ

_{s}= 0.38 cm

^{3}cm

^{−3}(data not shown), while, for t = 975 min, it fell to a value of θ = 0.288 cm

^{3}cm

^{−3}. The HY redistribution soil water content profiles after the time t = 75 min almost coincided with the one obtained for WH at an earlier time t = 25 min. This confirms that the inclusion of hysteresis causes a large delay in the redistribution process and consequently to soil water removal from the soil root zone where it was originally (with the initial infiltration) stored. One other observation was that the soil water content at the soil surface at a time t = 975 min was θ = 0.288 cm

^{3}cm

^{−3}, while in the absence of hysteresis it was 0.215 cm

^{3}cm

^{−3}.

_{0}= 0.12 cm

^{3}cm

^{−3}) before the initiation of the horizontal infiltration lies on the boundary drying curve (state B, Figure 4a). The infiltration depth in this case was I = 12.5 cm for the HY case and I = 15 cm for WH.

_{0}= 0.12 cm

^{3}cm

^{−3}at H = −64.1 cm lying on the boundary drying curve (state B, Figure 4a), following a first-order wetting scanning curve BDA (Figure 4a). When hysteresis was not considered, the θ–H relation was single-valued and followed the boundary drying curve. In HY, during the redistribution process in the soil zone, where soil water content reduction starts after the cessation of infiltration (the one closer to the soil surface, which is wetter than the rest), the θ–H relation in every position x will follow either the respective 2nd order drying scanning curve DB (Figure 4a) or the boundary drying curve depending on the values of θ and H attained during the original infiltration. From Figure 4b, one may observe that redistribution at early times, after the cessation of infiltration, was faster for both cases (HY or WH), and this appeared to be more obvious for the HY case. In this case, at t = 75 min, θ = 0.334 cm

^{3}cm

^{−3}from θ

_{s}= 0.38 cm

^{3}cm

^{−3}while, at t = 975 min, θ = 0.276 cm

^{3}cm

^{−3}, and, for the WH case, θ = 0.221 cm

^{3}cm

^{−3}. Comparing the above with the case of using the boundary wetting curve, the values of the soil water content at the soil surface appear rather close to the respective values obtained with or without the inclusion of hysteresis. Nonetheless significant differences were observed at the soil at a distance x = 200 cm, where θ for both cases (HY and WH), when it followed the boundary drying curve, was larger than the initial one, θ = 0.12 cm

^{3}cm

^{−3}. This phenomenon was more pronounced when hysteresis was not included, since, in this distance (x = 200 cm), θ was even larger (θ = 0.176 cm

^{3}cm

^{−3}). We should mention that for the case of using the boundary wetting curve soil water redistribution profile development was not advanced up to that distance (x = 200 cm) for both cases (HY or WH).

#### 3.2. Comparison of the Redistribution Profiles Forms in the Presence of Hysteresis for Different Infiltration Depths and Initial Water Content Values

_{0}, were compared for the case that the initial water content values lie on the boundary wetting soil water content retention curve (Figure 6).

#### 3.3. Investigation of the Redistribution Rates for Substantially Different Infiltration Durations and Depths

_{0}= 0.12 cm

^{3}cm

^{−3}). In this case, we examined, as sub-cases, the ones with small and large time durations of redistribution.

^{3}cm

^{−3}, while, for the case of the larger original infiltration depth, it was 0.373 cm

^{3}cm

^{−3}. For the case of the larger infiltration depth, the slope $\frac{d{\theta}_{i}}{dx}$near the soil surface will have a smaller magnitude, and it will tend to zero as the initial infiltration depth increases compared to the respective slope, which will be attained in the case of a smaller depth being originally infiltrated, always assuming that the above refers to the same porous medium and the same initial soil water content before the initiation of the original infiltration process. From the above one could argue that the redistribution rate will be larger in the second case due to the larger magnitude of the soil water pressure head gradient (Equation (3)), and therefore the reduction of the soil surface water content will be larger too.

^{3}cm

^{−3}(from 0.373 cm

^{3}cm

^{−3}at the beginning of redistribution) for the larger infiltration depth and equal to 0.266 cm

^{3}cm

^{−3}(from 0.362 cm

^{3}cm

^{−3}) for the lower infiltration depth (Figure 8).

#### 3.4. Comparison of the Redistribution Rates for the Cases That Initial Water Content Values Lie on the Boundary Drying and the Boundary Wetting Curves

_{0}= 0.12 cm

^{3}cm

^{−3}) but where it was taken lying on the boundary drying and the boundary wetting curves, respectively.

_{0}lies on the boundary drying curve then, during the infiltration process, θ and H will increase following the appropriate scanning wetting curve, starting at θ

_{0}, while the drying process during redistribution in every soil position will result in reduction in θ and H following the appropriate second-order drying–scanning curve each time. The larger the θ

_{0}value the narrower is the width of the hysteresis loop, which participates during the infiltration–redistribution process.

_{0}increases, which means that the redistribution rate increases with the increase in the value of the initial water content θ

_{0}.

^{3}cm

^{−3}. It could be argued that this small difference can be attributed to the fact that the term $\frac{d{\theta}_{i}}{dx}$ was not the same in these two cases, and therefore the effect of the narrower width of the soil water content loop for this occasion was annihilated when the initial water content lies on the boundary drying curve.

#### 3.5. Comparison of the Redistribution Rates for Different Initial Water Content Values Lying on the Boundary Drying and the Boundary Wetting Curves

_{0}= 0.055, θ

_{0}= 0.12, and θ

_{0}= 0.2 cm

^{3}cm

^{−3}) lying on the boundary wetting or boundary drying curve on the redistribution rate was examined when the time duration of the original infiltration was equal to 25 min; it was the same for both cases. The corresponding pressure heads (H) to these initial water content values for the case of the boundary wetting curve were −96.3, −49.9, and −30.3 cm, and for the boundary drying curve, they were −97.2, −64.1, and −49.6 cm.

_{0}lying at the boundary wetting curve. As can be seen in this figure, smaller values of θ

_{0}correspond to a larger redistribution rate. In other words, the redistribution rate was inversely related to θ

_{0}in the case of boundary wetting curves of the hysteresis loop.

_{0}corresponded to larger redistribution rates.

_{0}tended to keep larger values of Hx (Hx being the value of the soil water pressure head at the transition plane separating the drying zone from the wetting one during the redistribution process) and therefore maintained larger values of the pressure head (H) (smaller negative ones) in the drying zone. Therefore, it is anticipated that the redistribution rate decreases with the increase in the values of the initial water content [7].

#### 3.6. General Considerations

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The experimental data for the boundary wetting and drying curves of the hysteretic loop (points) and the fitted curves according to the Mualem–van-Genuchten model (VG). The corresponding values of parameters of the Mualem–van-Genuchten model are presented in Table 1.

**Figure 2.**Schematic representation of the boundary drainage and wetting curves and the pathways followed during infiltration and redistribution in the case where the initial water content lies on the boundary wetting curve (

**a**). The θ(x) relationship for the horizontal infiltration (INF) with initial soil water content θ

_{0}= 0.12 cm

^{3}cm

^{−3}and for T = 25 min and for the redistribution (RED) when hysteresis was considered (HY) and without hysteresis (WH), making use of the boundary wetting H–θ curve for redistribution times: 25 min (WH) and 75 min (HY) and 975 min for (WH) and (HY) (

**b**).

**Figure 3.**The relation of the soil water pressure head H with the horizontal distance x, H(x), at the end of the horizontal infiltration (INF) with duration T = 25 min and during the redistribution (RED) with hysteresis being included (HY) at times t = 1, 5, and 25 min, after the cessation of the infiltration making use of the boundary wetting H–θ curve. The initial soil water content before the commencement of the infiltration was θ

_{0}= 0.12 cm

^{3}cm

^{−3}(H = −49.9 cm).

**Figure 4.**Schematic representation of the boundary drainage and wetting curves and the pathways followed during infiltration and redistribution in the case that the initial water content lies on the boundary drying curve (

**a**). The θ(x) relation for the case of the initial horizontal infiltration for T = 25 min (INF) with initial soil water content θ

_{0}= 0.12 cm

^{3}cm

^{−3}and the case of water content redistribution with the inclusion of hysteresis (HY) and without hysteresis (WH) making use of the boundary drying curve and initial soil water content θ

_{0}= 0.12 cm

^{3}cm

^{−3}for two times of redistribution: 75 and 975 min (

**b**).

**Figure 5.**The relation between soil water pressure head H and horizontal distance x, H(x), after a horizontal infiltration (INF) with duration T = 25 min and during the redistribution (RED) process with the inclusion of hysteresis for time intervals t = 1, 5, and 75 min with initial water content θ

_{0}= 0.12 cm

^{3}cm

^{−3}(H = −64.1 cm) and using the boundary drying curve.

**Figure 6.**The soil water content profiles of the initial infiltration (INF) and the redistribution (RED) water content profiles for several cases of infiltration water depths, I, and different values of the initial water content. (

**a**) The profiles for θ

_{0}= 0.12 cm

^{3}cm

^{−3}and duration of infiltration T = 6 min, corresponding to I = 5.34 cm; (

**b**) the profiles for θ

_{0}= 0.12 cm

^{3}cm

^{−3}and duration of infiltration T = 25 min, corresponding to I = 10.9 cm; and (

**c**) the profiles for θ

_{0}= 0.055 cm

^{3}cm

^{−3}and duration of infiltration T = 25 min, corresponding to I = 12.3 cm.

**Figure 7.**The water content profiles of the initial infiltration (INF) for two different infiltration durations (T = 6 min and T = 25 min corresponding to I = 5.34 cm and I = 10.90 cm) and for water content redistribution (RED) for small times (t = 2min).

**Figure 8.**The soil water content profiles after the initial horizontal infiltration (INF) for two infiltration durations (T = 6 min and T = 25 min corresponding to I = 5.34 cm and I = 10.90 cm) and their subsequent profiles of redistribution (RED) for large time durations, t = 994 min (for I = 5.34 cm) and t = 975 min (for I = 10.90 cm).

**Figure 9.**The horizontal soil water content redistribution (RED) profiles after infiltration with duration (T = 25 min) for times, t = 25 min and t = 275 min, for the boundary wetting (WETTING) and the boundary drying (DRYING) θ–H curves when the initial water content (θ

_{0}= 0.12 cm

^{3}cm

^{−3}) lies on the respective boundary wetting and drying curves.

**Figure 10.**The soil water content profiles of the initial horizontal infiltration (INF) with duration T = 25 min and of the redistribution (RED) for time t = 75 min after the cessation of the infiltration for different values of initial water content lying at the boundary wetting curve before the commencement of the initial infiltration (θ

_{0}= 0.055, θ

_{0}= 0.12, and θ

_{0}= 0.2 cm

^{3}cm

^{−3}).

**Table 1.**The values of the parameters of the Mualem–van-Genuchten model obtained according to a curve-fitting procedure (RETC program) on experimental measurements and the measured Κs.

Boundary Wetting | Boundary Drying | |
---|---|---|

θ_{r} (cm^{3}cm^{−3}) | 0.0309 | 0.0309 |

θ_{s} (cm^{3}cm^{−3}) | 0.38 | 0.38 |

a (cm^{−1}) | 0.0364 | 0.02227 |

n | 3.12 | 4.427 |

M = 1−(1/n) | 0.679487 | 0.774113 |

Ks (cm/h) | 33.61 |

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**MDPI and ACS Style**

Kargas, G.; Soulis, K.X.; Kerkides, P. Implications of Hysteresis on the Horizontal Soil Water Redistribution after Infiltration. *Water* **2021**, *13*, 2773.
https://doi.org/10.3390/w13192773

**AMA Style**

Kargas G, Soulis KX, Kerkides P. Implications of Hysteresis on the Horizontal Soil Water Redistribution after Infiltration. *Water*. 2021; 13(19):2773.
https://doi.org/10.3390/w13192773

**Chicago/Turabian Style**

Kargas, George, Konstantinos X. Soulis, and Petros Kerkides. 2021. "Implications of Hysteresis on the Horizontal Soil Water Redistribution after Infiltration" *Water* 13, no. 19: 2773.
https://doi.org/10.3390/w13192773