# Hydrodynamics of the Vadose Zone of a Layered Soil Column

^{*}

## Abstract

**:**

## 1. Introduction

**Κ**is the hydraulic conductivity that corresponds to the initial soil moisture

_{i}**θ**, and

_{i}**S**is a series of coefficients that are calculated as functions of the soil characteristics and the initial and boundary conditions of infiltration. For

_{m}**m**= 1, coefficient

**S**is called sorptivity [29,32], and it is valid for early times. Sorptivity is also given by Equation (2) which came out from soil moisture profiles, where

_{m}**K**is saturated hydraulic conductivity,

_{s}**θ**is initial water content,

_{i}**θ**is the boundary condition applied,

_{s}**H**

_{0}is the load on the soil surface, and

**H**is the suction at the wet front.

_{f}**θ**is soil moisture, while

**D**(

**θ**) is diffusivity. Diffusivity is a hydraulic parameter given below:

**C**(

**θ**) is the specific water capacity given by the derivative:

**H**pressure head.

_{f}**K**is the saturated hydraulic conductivity,

_{s}**S**is the sorptivity,

**I**is cumulative infiltration, and

**t**is time. Parlange [35,36,37,38,39] also presented a model (P model) for the prediction of infiltration (Equation (10)) from Richard’s Equation (9) [40], valid for one-dimensional water movement:

_{r}is the residual water content, and θ

_{s}is the saturated water content. The hydraulic parameters a, m, and n indicate the shape and curvature of the SWCC curve and m and n are related to each other as follows:

## 2. Materials and Methods

**K**) was measured for each layer using the constant head method experimental setup. The first drainage was achieved gradually by placing the Mariotte bottle in four positions (stages). At the beginning (1st stage), the Mariotte bottle was placed 25 cm from the soil surface, where it remained for 20,715 min. Then, in the 2nd stage, we put the Mariotte bottle at the interface of the layers (44.5 cm from the soil surface) for 22,318 min. In the 3rd stage, the Mariotte bottle was placed 79.5 cm from the soil surface for 8768 min, and during the 4th stage, it was placed 84.5 cm from the soil surface for 9686 min. The experiment lasted a total of 61,487 min (1025 h). After drainage, the 2nd imbibition followed, which was achieved in 4 stages. in the 1st stage, the Mariotte bottle was placed −21 cm from the 4th CC; in the 2nd stage, it was placed at the interface (−8 cm from the 3rd CC); in the 3rd stage it was placed at the height of the 1st CC (−11.5 cm from the 2nd CC), and at the 4th stage it was placed at a height equal to the soil surface (−25 cm from the 1st CC).

_{s}## 3. Results

**K**and

_{s}**θ**, respectively) for each layer were found:

_{s}**K**= 0.170 cm min

_{s}^{−1},

**θ**= 0.38

_{s}**K**= 1.490 cm/min

_{s}^{−1},

**θ**= 0.26

_{s}#### 3.1. Infiltration under Ponded Conditions

**θ(z)**is presented.

**m**= 1 and found:

**S**= 1.0607 cm min

_{LS}^{−0.5}

**dI/dt**was calculated, and the result was:

**dI/dt**= 0.177 cm min

^{−1}.

#### 3.2. Pressure Head–Soil Moisture Research

#### 3.2.1. First Drainage

#### 3.2.2. Second Imbibition

#### 3.2.3. Characteristic Curves of the Two Layers

**D(θ)**was obtained from Equations (4)–(7) and simulated in Figure 13. The fitting Equation is given below:

**D(θ)**into three parts, found their fitting Equations and calculated the separated integrals. Thus,

**S**was found as a sum of the separated integrals (Table 2).

^{2}## 4. Discussion

**dI/dt**, which expresses the infiltration rate, decreases. The decrease in infiltration rate over time is due to several factors, such as the deformation of the soil structure, soil pore clogging, and the possibility of trapped air into the soil pores, but mainly due to the reduction of the hydraulic gradient [21,30,34,35,36,45]. Specifically, during the vertical movement of water into completely dry soil, the ponding conditions at the soil surface cause immediate saturation at the soil surface. Hence, the hydraulic gradient at the soil layer very close to the surface is steep, but over time, as water moves towards greater depths, the gradient decreases [46]. The continuous reduction of the hydraulic gradient near the soil surface results in a continuous reduction of the infiltration rate, which eventually stabilizes at a certain value. This constant value of the infiltration rate is practically equal or tends toward the saturated hydraulic conductivity (

**K**) [21,30,31,47,48]. The values of the measured saturated hydraulic conductivity tend to the gradient

_{s}**dI/dt**at late infiltration times (relative mse = 0.002), indicating good validation of the experimental process. Infiltration experimental data of the upper layer were approximated with the GA and P models, and results showed relative mse 0.11 and 0.04, respectively, in agreement with previous infiltration experiments under ponded conditions [21,23].

**I(t)**data,

**θ(z)**curves can lead to the prediction of cumulative infiltrate, indicating that the soil moisture profiles curve is a useful tool to simulate infiltration over time.

**dθ**/

**dt**is significant, as expected [40,44,46]. During the fourth stage of drainage, a decrease in soil moisture at all three sensors of the upper layer was observed but lesser than that during the third stage. At the end of the stage, the experimental water content values were stabilized; specifically, the first sensor showed the lowest soil moisture value (0.17 cm

^{3}/cm

^{3}) as expected, while the second sensor was stable at 0.18 cm

^{3}/cm

^{3}and the third sensor at 0.20 cm

^{3}/cm

^{3}. Figure 8b shows the reduction of water content at the S-layer during the four stages of the 1st drainage. It is noticed that at the first two stages of the drainage, we observed absolutely no change in the layer’s water content. This was expected during the first two stages, and the drainage bottle was placed above the two TDR sensors (Z4 and Z5); thus, no drainage could occur. During the third stage, when the drainage container was located −21 cm from the fourth CC, therefore below the two soil moisture sensors, a significant water content decrease was observed. Moreover, the

**dθ**/

**dt**gradient was steep at the beginning of the stage, indicating low holding capacity [45,46]. During the fourth stage, when the drainage container was placed −26 cm from the fourth CC, we observed a further decrease and, afterward, the stabilization of water content. The Z4 sensor stabilized at 0.14 cm

^{3}/cm

^{3}and the Z5 at 0.15 cm

^{3}/cm

^{3}.

**h**(

**t**) curve was significant and gradually decreased at the end of the stage and finally stabilized at the 4th stage, where equilibrium occurred, and gradient tended toward zero. The final value of tension recorded by the first tension sensor (T1) was −64 cm, while for the second sensor (T2) was −53 cm, which is justified by the fact that the T2 sensor was placed at −11.5 cm lower than the T1 sensor, indicating that the stabilized values of tension tend to the external loads applied [22].

**h**(

**t**) curve is steep, while later, the tension stabilizes, as can be seen in Figure 9b. Finally, at the fourth (final) stage, we see that equilibrium has been achieved, and sensors T3 and T4 stabilize at −23 cm and −22 cm, respectively, almost equal to the external loads applied [41,42,49].

**θ**.

_{s}**h**(

**θ**) was extracted for each layer (Figure 12a,b). The hydraulic hysteresis phenomenon is significant at both layers indicating that the pressure head-soil moisture function is not a 1-1 function [22,46]. The S-layer shows a greater hysteresis loop than the LS-layer, which can be explained by the nature, structure and physical properties of the sandy soil, which has low holding capacity along with lower values of saturated soil moisture (

**θ**) and residual soil moisture (

_{s}**θ**). During the draining and wetting circles of the soil column, we obtained differences between the soil moisture values of the two layers. On the contrary, the pressure head changes show similar behavior in both soils. This is in agreement with the theory of hydraulic hysteresis [49,52,53,54], where it is indicated that the area under the drainage curve expresses the work required to lead to the complete drainage of a unit volume of saturated soil. The same goes for imbibition. Hence, the difference between the above areas is equal to what is called hydraulic hysteresis work [$\frac{\mathit{W}}{\mathit{\gamma}}={{\displaystyle \int}}_{\mathbf{0}}^{{\mathit{\theta}}_{\mathit{s}}}\mathit{h}\left(\mathit{\theta}\right)\mathit{d}\mathit{\theta}$], where

_{r}**γ**is the specific weight of water. Hydraulic hysteresis work is greater as the soil texture becomes lighter [22,52,54].

**m**= 1, Equation (1) can lead to the estimation of sorptivity. We can also lead to sorptivity estimation through calculations of hydraulic capacity and diffusivity using Equations (3)–(7). Due to the impossibility of the analytical solution of the Integral (13), we proceeded to a graphical solution of the integral by dividing it into three parts and approximating each part with linear Equations. Summing the results of the three integrals led to the estimation of the sorptivity of the S-layer. Sorptivity is a crucial hydraulic parameter that has a great impact on water movement through porous media. As it is a component of the flow process, it needs to be incorporated in applications where the adsorption or desorption of water from a porous media is occurring [22,24,55,56,57,58,59]. The results of the current study suggest deep research into the vadose zone of the soils under research, using methods to estimate crucial hydraulic parameters that are involved in the soil-water motion, constituting useful tools for further investigation of runoff, aquifer horizon recharge, rational water management and water saving.

## 5. Conclusions

**θ**(

**z**) led to a good approximation of the infiltration process; hence, soil moisture profiles can simulate infiltration over time. Green and Ampt and Parlange infiltration models were used to predict infiltration for the upper layer with good results. Thus, using early time data is a quick and easy way to predict water movement under ponded conditions. Drainage and imbibition experimental circles were used to extract the Soil Water Characteristic Curves of each layer, indicating the hydraulic hysteresis effect during the drainage and imbibition processes. Simulation of Soil Water Characteristic Curves led to van Genuchten’s hydraulic parameters assessment, and further mathematical modeling of water motion led to the estimation of specific water capacity and diffusivity. To avoid the impossibility of an analytical solution of the integral that provides sorptivity, a graphical solution was used. The results indicate that the methods used in the current study are useful for the simulation of vertical water motion through the under-research layered soils.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

a | Hydraulic parameter in Van Genuchten’s model |

θ | soil moisture |

θ_{i} | initial water content |

θ_{s} | soil moisture at saturation |

C | Specific water capacity |

CC | Ceramic capsule(s) |

D | Diffusivity |

h | Suction |

H_{f} | Suction at the wet front |

H_{0} | Pressure head at the soil surface |

I | Cumulative infiltration |

GA | Green and Ampt |

K_{i} | Hydraulic conductivity |

K_{s} | Hydraulic conductivity at saturation |

LS-layer | Loamy sand layer |

m | Hydraulic parameter in Van Genuchten’s model |

n | Hydraulic parameter in Van Genuchten’s model |

P | Parlange |

PT | Pressure transducer(s) |

S | Sorptivity |

S-layer | Sand layer |

S_{m} | A series of coefficients in Philip’s Equation |

SWCC | Soil water characteristic curve(s) |

t | time |

TDR | Time domain reflectometry |

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**Figure 7.**Approximation of the infiltration models with the experimental points for the upper layer (LS-layer).

Hydraulic Parameter | θ_{s} | θ_{r} | α | n | m |
---|---|---|---|---|---|

RETC value | 0.24937 | 0.16 | 0.0797 | 6.47629 | 0.845591 |

1st part: (θ: 0.15–0.18) | D_{1} = 84.632θ − 9.5346(R ^{2} = 1) | ${S}_{1}^{2}=2\int \left(84.632{\theta}^{2}-9.5346\theta \right)d\theta =0.4420$ |

2nd part: (θ: 0.18–0.21) | D_{2} = 92.099θ − 10.879(R ^{2} = 1) | ${S}_{2}^{2}=2\int \left(92.099{\theta}^{2}-10.879\theta \right)d\theta =0.0830$ |

3rd part: (θ: 0.21–0.24) | D_{3} = 6085.9θ − 1269.6(R ^{2} = 1) | ${S}_{3}^{2}=2\int \left(6085.9{\theta}^{2}-1269.6\theta \right)d\theta =1.3737$ |

S^{2} = 1.5012 cm·min^{−0.5} | ||

S= 1.2252 cm min^{−0.5} |

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**MDPI and ACS Style**

Batsilas, I.; Angelaki, A.; Chalkidis, I. Hydrodynamics of the Vadose Zone of a Layered Soil Column. *Water* **2023**, *15*, 221.
https://doi.org/10.3390/w15020221

**AMA Style**

Batsilas I, Angelaki A, Chalkidis I. Hydrodynamics of the Vadose Zone of a Layered Soil Column. *Water*. 2023; 15(2):221.
https://doi.org/10.3390/w15020221

**Chicago/Turabian Style**

Batsilas, Ioannis, Anastasia Angelaki, and Iraklis Chalkidis. 2023. "Hydrodynamics of the Vadose Zone of a Layered Soil Column" *Water* 15, no. 2: 221.
https://doi.org/10.3390/w15020221