# Dynamics of Volumetric Moisture in Sand Caused by Injection Irrigation—Physical Model

^{1}

^{2}

^{*}

## Abstract

**:**

^{3}under pressure (4 bar). At the same time, measurements of volumetric moisture were conducted with the use of TDR sensors, which were positioned within the soil in a regular grid pattern. It was demonstrated that the primary cause of water movement at the moment of injection is the pressure potential gradient of water molecules. The spatial reach of moisture change in relation to the injected water dose was also defined. It was also observed that in the course of water injection there is a risk of disturbing the structure of the porous material. The correctness of the adopted method was verified through the calculation of the water balance.

## 1. Introduction

^{3}, 16.7% of which is used by industry, 15.6% is household consumption, and as much as 67.7% is used in agriculture [2]. Therefore, special attention should be paid to the segment related directly to food production [3]. Water consumption in agriculture relates a.o. to the use of irrigation systems, and a measurable effect of their use is an increase in the level of yields [4,5,6]. For this reason, in open field cultivations there is an extensive application of such irrigation systems as mobile or fixed sprinkler systems, which are used globally on an area of approximately 35 million ha [7]. An increasing popularity is also observed in the case of water-saving drip lines, for which the area of use in 2030 will be doubled relative to that of 2000 [8]. Selected features of those irrigation systems became the base point for a concept of an injection irrigation system for crop plants, which is being realized within the framework of the project “Mobile system of injection irrigation and fertilisation meeting the individual requirements of the plant” by the consortium Mobile Irrigation and Fertilisation (MSINiN) [9,10,11]. The concept consists in subsurface application of a specific dose of water, within soil space containing plant root systems, which is characteristic of the subsurface drip line. In addition, the application of water is to proceed in a mobile manner, and that will allow the use of the system on the largest possible area, which is an advantage of mobile sprinkler irrigation systems (Figure 1) [12]. According to this idea, a one-time injection for one plant should last several seconds at maximum. Such an approach to irrigation is in conformance with the idea and solutions of “precision agriculture” [13,14]. One of the issues that should be addressed during the research and development work on this concept is to determine the form of the body of moisture that is formed during pointwise application of water to soil under high pressure of the liquid.

#### 1.1. Distribution of Soil Moisture Dynamics during Injection Irrigation

^{3}of water within a few seconds). Therefore, the dynamics of volumetric moisture in the soil monolith is not determined by the gravity gradient and the pressure equivalent, but solely by the hydraulic pressure of the injected liquid [17]. After the process of injection, the hydraulic pressure disappears and water movement takes place as a result of the difference in the total potential composed mainly of the gravity potential and the pressure equivalent.

#### 1.2. Standards of Injection Irrigation of Field Cultivations

#### 1.2.1. Plant Spacing in Cultivations Dedicated for Injection Irrigation

#### 1.2.2. Irrigation Dose in the Case of Injection Irrigation

^{3}), $d$—water dose determined as liquid column height in the case of irrigation using a sprinkler system or a drip line (mm H

_{2}O), $A$—surface area per a single plant (cm

^{2}), 10—unit correction number (−).

^{3}, respectively.

#### 1.2.3. Depth of Water Injection Relative to Ground Surface

#### 1.3. Research Objective

## 2. Materials and Methods

#### 2.1. Scope and Run of the Experiment

^{−3}, and porosity was 0.425 (–) [41].

^{3}∙s

^{−1}(±10%) and under pressure of 4 bar, which is in conformance with the concept of injection irrigation presented in the introduction.

^{3}. The dimensions of the monolith, injection depth, and the volume of the applied water doses with the required outflow intensity were determined on the basis of the theoretical considerations presented in the introduction and verified in pilot experiments.

#### 2.2. Monitoring of Volumetric Water Content

^{3}of water to the soil profile lasts about 7 s. In addition, the filtration coefficient of sand is in the range from 10

^{−6}to 10

^{−3}m∙s

^{−1}[56]. For this reason it was decided that the measurements would be taken at the highest possible frequency for apparatus calibrated for those parameters, i.e., at 30-s intervals.

^{i}—the origin of the X

^{i}Y

^{i}Z

^{i}system, which is the location of the tip of the injector nozzle oriented along the 0Y

^{i}axis. The fundamental condition is that the moisture sensors cover the entire space XYZ, dividing it into equal elementary volumes. Thus, the space was divided into 5 cm cubes. This creates the possibility of analysing the volumetric moisture in every direction (not only in directions 0

^{i}X

^{i}, 0

^{i}Y

^{i}, and 0

^{i}Z

^{i}), and also allows the construction of water balance which verifies the correctness of the adopted method. In relation to the above, in the experiment conducted within the scope of this study the LP/ms sensors were distributed in a regular grid. The adopted solution can be compared to the grid of the calculation area in the finite difference method. It is a numerical method used for the solution of differential equations [63], such as the Richards equation, used for the description of soil water movement [64].

^{3}), ${\theta}_{i}$—moisture of the solid after injection, moisture in the saturated zone for coarse-grained sand is ${\theta}_{s}$ = 0.425 cm

^{3}∙cm

^{−3}[65], ${\theta}_{in}$—initial moisture, ${\theta}_{in}$ = 0.1 cm

^{3}∙cm

^{−3}.

^{3}, the radius of the sphere is 5.7 cm, while for the maximal dose of 1500 cm

^{3}, the radius is 10.3 cm. In relation to the above it was assumed that, for the analysed water doses, changes in moisture should be expected at distances of approximately 5–15 cm from the injection point. For this reason, in the adopted distribution of the TDR sensors the distances between neighbouring sensors were 5 cm.

^{3}[66]. The adoption of sand monolith dimensions of 40 cm × 45 cm × 30 cm and the division of the model into elementary volumes in the form of cubes with dimensions of 5 cm × 5 cm × 5 cm resulted in 432 sensors that should be placed within the model. The resultant total volume of the sensors relative to the entire volume of the model would be 0.026%. However, symmetry of moisture distribution in planes normal in relation to axis 0Y

^{i}was assumed, due to which the number of sensors was reduced. Such an assumption is typical in the study of the spatial and temporal distribution of moisture during the operation of drip lines [21]. The distribution of the sensors was therefore simplified, which is presented in Figure 4. The figure presents also the division of the monolith into columns (along axis 0X), rows (along axis 0Y), and layers (along axis 0Z). The localization of individual spaces in relation to the assigned axis is very important. As an example, the spaces situated along a horizontal line identical with the direction of water injection (axis 0Y

^{i}) were also presented. The centre points of volumes h, g, f, e are situated at distances of 17.5, 12.5, 7.5, and 2.5 cm from the point of injection, and distributed in the direction of water outflow from the injector, and points d, c, b, a at distances of 2.5, 7.5, 12.5, 17.5 cm, respectively, and oriented in the opposite direction.

^{®}10 [25], on the basis of 48, 54 (vertical planes), and 72 (horizontal plane) measurement points. Between the indicated points, the values of volumetric moisture were interpolated with the kriging method. It is a geostatistical method of interpolation which was applied a.o. for the presentation of changes of volumetric moisture in selected sections with the use of a drip line [67]. Kriging is also used in the monitoring of moisture changes caused by the operation of sprinkler irrigation systems [68].

#### 2.3. Water Balance

^{3}), ${V}_{i}$—elementary i-th volume, representative for TDR sensor type LP/ms (cm

^{3}), ${\theta}_{i}^{t}$—volumetric moisture of i-th volume at moment $t$ (cm

^{3}∙cm

^{−3}), $n$—number of elementary volumes of the monolith (−).

^{3}), ${\theta}_{i}^{t-1}$—volumetric moisture in i-th volume at moment $t-1$ (cm

^{3}∙cm

^{−3}), ${\theta}_{i}^{t+1}$—volumetric moisture in i-th volume at moment $t+1$ (cm

^{3}∙cm

^{−3}), remaining symbols as in the formula above.

^{3}), ${\theta}_{i}^{{t}^{in+1}}$—volumetric moisture in i-th volume immediately after injection, ${t}^{in+1}$ (cm

^{3}∙cm

^{−3}), ${\theta}_{i}^{t+1}$—volumetric moisture in i-th volume immediately prior to injection, ${t}^{in-1}$ (cm

^{3}∙cm

^{−3}). The value of $\Delta {V}_{c}^{{t}^{in}}$ can then be compared with the actual dose of water applied by the injector.

## 3. Results and Discussion

#### 3.1. Dynamics of the Volumetric Moisture

^{3}of water is presented in Figure 5. This is the dynamics of moisture for a point which is 2.5 cm distant from the injector nozzle and which is situated in accordance with the orientation and sense of injection. In other words, this is the first encountered moisture sensor on the path of the water outflow from the injector nozzle. Prior to the start of water application under pressure, in a 30-min period lasting from moment t

_{1}to t

_{2}, the volumetric moisture was constant at ${\theta}_{in}$ = 0.11 cm

^{3}∙cm

^{−3}. In the period from moment t

_{2}to t

_{3}, lasting for 12 s, the injector delivered to the physical model 1500 cm

^{3}of water under high pressure. This caused a jump increase of volumetric moisture, from ${\theta}_{in}$ = 0.11 cm

^{3∙}cm

^{−3}to ${\theta}_{max}$ = 0.47 cm

^{3}∙cm

^{−3}. The mean rate of moisture increase was as much as 18 cm

^{3}∙cm

^{−3}∙min

^{−1}. Such an intensive increase of volumetric moisture is not observed in the case of other irrigation systems. As an example, Mmolawa and Or [74] studied the changes in water content in soil during the operation of a surface drip line. For a point situated 5 cm beneath the emitter, an increase of moisture from 0.25 cm

^{3}∙cm

^{−3}to 0.42 cm

^{3}∙cm

^{−3}was noted. That moisture increase was observed as late as after about 3 h (the intensity of water outflow from the emitter was 1.6 L∙h

^{−1}). To elucidate such a jump in soil moisture, we need to reach for the theory of soil water movement. Water movement in soil is due to differences in the total potential which is the sum of the matrix, gravity, osmotic, and pressure potentials [16]. In the course of injection, the cause of the jump increase of moisture is the gradient of the pressure potential resulting from the pressure of the liquid applied. The remaining gradients can be left out. However, this is a hypothesis which needs to be verified. The maximum measured volumetric moisture of ${\theta}_{max}$ = 0.47 cm

^{3}∙cm

^{−3}is higher than the value of ${\theta}_{S}$ = 0.425 cm

^{3}∙cm

^{−3}defined as moisture at the state of full saturation in the case of sand [65]. This is evidence of a disturbance of the structure of the porous medium, caused by the high pressure of liquid from the injector nozzle. This is supported by the fact that during injection the surface of the monolith building the model became deformed—the surface in the area of the fixed injector moved up. In addition, after the injection was completed, the surface settled down. The observed phenomenon of ground settlement is a known process which takes place at the moment of saturation of pores with water to the level of ${\theta}_{S}$ and subsequent decrease of volumetric moisture [75]. The period from moment t

_{3}to t

_{4}, lasting for 3 min, is a period of rapid decrease of moisture. During that time the volumetric moisture dropped from 0.47 cm

^{3}∙cm

^{−3}to 0.22 cm

^{3}∙cm

^{−3}, i.e., by 0.25 cm

^{3}∙cm

^{−3}, at an average rate of 0.083 cm

^{3}∙cm

^{−3}∙min

^{−1}. Therefore, the rate of volumetric moisture decrease in that stage is over 20-fold lower than the rate of moisture increase in the period from moment t

_{2}to t

_{3}. In the next 3-min period, from moment t

_{4}to t

_{5}, the rate of moisture decrease is lower still and amounts to only 0.007 cm

^{3}∙cm

^{−3}∙min

^{−1}. The process of slow decrease of volumetric moisture lasts till the end of the experiment (12 h), and it is now caused by a difference of total potential which is the sum of the gravity potential and the pressure equivalent [15].

^{3}of water, is presented in Figure 6. The elementary spaces were described and visualized in Figure 4. Prior to the application of water, the volumetric moisture in those elementary volumes was in the range of 0.10–0.12 cm

^{3}∙cm

^{−3}. In volumes d, e, and f the run of changes of volumetric moisture was similar in character to that presented in Figure 5. In elementary volume g there was a jump increase of moisture, but not to the level of full saturation of pores with water. The maximum moisture in that volume was ca. 0.25 cm

^{3}∙cm

^{−3}. After the injection, no dynamic decrease of water content was noted but only a slow decrease of volumetric moisture. In volumes a, b, c, and h the content of water did not change in the course of the experiment. Therefore, the range of the effect of injection along the analysed axis 0Y

^{i}is 12.5 cm in the direction of water outflow, and only 2.5 cm in the opposite direction. The results of measurements for the analysed elementary volumes indicate that volumetric moisture changes caused by injection irrigation are not homogeneous, and their intensity decreases with increasing distance from the injector nozzle. A similar character of moisture change is noted also in the case of other irrigation systems. Naglić et al. studied the propagation of water in a sandy soil (with moisture of 0.12 cm

^{3}∙cm

^{−3}) irrigated with the use of a surface drip line [21]. The emitters of the line dosed water with the intensity of 2 L∙h

^{−1}, which resulted in an increase of the volumetric moisture of the soil. The character of the increase just under the soil surface was as follows: at the distance of 10 cm from the emitter the moisture was 0.38 cm

^{3}∙cm

^{−3}, at 20 cm it was 0.28 cm

^{3}∙cm

^{−3}, and at 30 cm it was only 0.16 cm

^{3}∙cm

^{−3}. Badr and Abuarab determined soil moisture changes for a subsurface drip line, installed in a sandy soil at the depth of 30 cm [25]. The results had an identical character—the rate of soil moisture increase decreased with increasing distance from water emitters. Also, a similar character of the volumetric moisture changes was obtained during measuring the intensity of evaporation caused by induced irrigation [11]. In this work, the moisture content of the top layer of the monolith during a water injection was measured. For example, when 1000 cm

^{3}of water was implemented into sand, the sensor located 8 cm from the injection site showed an increase in volumetric moisture by 0.07 cm

^{3}∙cm

^{−3}at the time of injection.

^{i}and 0Y

^{i}, and in a horizontal plane containing axis 0Z

^{i}. The visualisations have been prepared on the basis of data from measurements taken at the moment of injection termination (t

_{3}). Based on the visualisations of the shape and extent of the moisture front in the figure below, one can conclude that the range of changes of the volumetric moisture is proportional to the volume of the applied water doses. As an example, in the case of the injection of 250 cm

^{3}of water, the maximum volumetric moisture was noted at 3.5 cm from the injector nozzle and it amounted to 0.47 cm

^{3}∙cm

^{−3}. In turn, for the dose of 1500 cm

^{3}, the maximum moisture was as high as 0.65 cm

^{3}∙cm

^{−3}and it was recorded at the distance of 12.8 cm from the injector nozzle. The recorded maximum moisture content was higher than the initial porosity of the monolith due to the disturbance of the porous structure by the water jet injected at high pressure. In every case, the maximum values of moisture at the moment of injection were noted in the direction of water outflow from the injector nozzle. The proportion between the intensity of changes in water content and the volume of water applied to the monolith is visible also in the analysis of the dynamics of moisture caused by the operation of other irrigation systems. An example can be the spatial reach of moisture changes in the case of irrigation with the use of a subsurface drip line [76]. When the water dose applied to the soil profile was 1 dm

^{3}, the front of soil moisture with the value of 0.33 cm

^{3}∙cm

^{−3}, in the horizontal direction, could have the reach of 6.7 cm, while for the dose of 5 dm

^{3}it was as much as 16 cm, and for 10 dm

^{3}—even 21 cm. In the cited study, the determination of soil moisture dynamics was conducted with the use of TDR probes. Numerical models (Hydrus software) also confirm that in a sandy soil formation the spatial extent of changes in volumetric moisture depends on the volume of water supplied to the porous medium [77].

#### 3.2. Water Balance in the Physical Model

^{3}of water the increase was 435 cm

^{3}, and for the dose of 1250 cm

^{3}–1200 cm

^{3}. The $\Delta {V}_{c}^{{t}_{in}}$ increments should be comparable with the volumes of water planned for injection—${\mathrm{V}}_{i}$. However, according to the instructions from the manufacturer of the water injection device, the accuracy of the liquid dose applied is ± 10% [43].

^{3}the 10% relative error between the water volume calculated on the basis of the water balance and the planned water dose was exceeded slightly. The causes of that can be potential error in the amount of injected water and the change of the structure of the porous medium mentioned earlier.

^{3}was 110, 209, 412, 357, 495, and 346 cm

^{3}, respectively. Assuming the correctness of the measurements and the calculations, the results clearly indicate an outflow of water from the analysed space. To localise the flow boundary, one should analyse the values of changes in the water balance in time, for the individual layers of the monolith, and also for the rows and columns (Figure 4). Taking into account the greatest changes in water content, the analysis was performed for the results of water balance calculations for the injection of 750 cm

^{3}of water (Figure 9).

^{3}, respectively—but after 4 h there was a decrease of water content, by 232, 218, and 98 cm

^{3}, respectively. This indicates a nearly total evacuation of water from those volumes. In layer 5 no major changes were recorded in water content. Whereas, in the deepest situated, 6th layer, an increase of water content was noted, but it was much less violent than in the higher parts of the sand profile. Within 8 min the increase amounted to 200 cm

^{3}. What is more, the subsequent decrease of water content was very slight. After 4 h a decrease by 22 cm

^{3}was noted. Changes of water content over time, but in a division into columns and rows, were also analysed. Changes in water content were noted only in the central rows and columns, i.e., in the immediate vicinity of the point of injection. The run of the changes can be compared to the moisture dynamics noted in layers 2, 3, and 4. In the extreme rows and columns no changes in water content were observed. This indicates that there was no water escape through the extreme vertical barriers of the monolith.

^{−6}–10

^{−3}m∙s

^{−1}[56]), high porosity (0.425 (−) [65]), low maximum capillary rise (3 cm [78]), and small retention capacity [79], it is certain that the orientation and sense of water flow was identical to the orientation and sense of the force of gravity. In the 6th layer, the TDR sensors were situated 2.5 cm above the impermeable bottom. The zone of sensitivity of TDR sensors type LP/ms in saturated soil has the form of a cylinder with dimensions of 5.5 cm along the sensor rods and 0.5 and 0.8 cm in the cross-section of the rods [80]. This means that above the vertically installed sensors there is a layer with a maximum thickness of 2.25 cm, which is included in the volumes represented by the LP/ms sensors, but in which moisture changes are not recorded in reality. Taking into account the section of the volumes in the plan view (5 cm × 5 cm) and 2.25 cm of thickness, the volume isolated for a single elementary volume, in which the moisture sensors do not register moisture dynamics, amounts to as much as 56.25 cm

^{3}. What is more, the results of measurements demonstrated that, depending on the injected water dose, the moisture jump in the bottom, 6th, layer was noted in varying numbers of volumes. Based on this, it is possible to calculate the volume of pores in which there could have been water not registered by the TDR sensors:

^{3}), $L$—number of elementary volumes in 6th layer, where an increase of volumetric moisture was noted during the 12 h period of the experiment (−), $56.25$—part of the volume of the isolated space, for which no volumetric moisture was recorded (cm

^{3}), ${\theta}_{s}$—volumetric moisture in the saturated zone, 0.425 cm

^{3}∙cm

^{−3}[65], ${\theta}_{in}$—initial volumetric moisture immediately before injection, 0.10 cm

^{3}∙cm

^{−3}.

^{3}. This means that in the bottom layer, with a thickness of 2.25 cm, beneath the sensors which registered an increase of volumetric moisture, there is a sufficient volume in the form of free pores than can accommodate water migrating down under gravity from the upper layers of the monolith. This is the basis for the acceptance of the method employed in the experiment as correct for the determination of the dynamics of the moisture front caused by injection irrigation. There is a possibility of independent verification of the correctness of the measurements—with the use of mathematical modelling [19,81]. Even conducting an identical experiment with the use of a soil formation with a percentage of silt and clay particles should ensure the maintenance of water balance at the level from the moment of injection. Heavier soil formations are characterised by a notably lower filtration coefficient, lower porosity, and higher capillary rise. The above hypotheses will be the subject of future research.

## 4. Conclusions

_{2}–t

_{3}). After the process of injection (maximum several seconds), the hydraulic pressure ceases and there takes place a decrease of moisture, the intensity of which reduces with the passage of time (period t

_{3}–t

_{4}–t

_{5}). Within the scope of the experiment, the spatial reach of the changes was also determined. As an example, in the case of the injection of 250 cm

^{3}of water, the maximum volumetric moisture was noted at 3.5 cm from the injector nozzle, and it amounted to 0.47 cm

^{3}∙cm

^{−3}. For the dose of 1500 cm

^{3}, the maximum moisture was as high as 0.65 cm

^{3}∙cm

^{−3}and it was registered at the distance of 12.8 cm from the injector nozzle (Figure 7). It was also observed that during water injection there is a risk of disturbing the sand structure.

_{3}), the calculation of the water balance showed that the amounts of water applied to the monolith were comparable to the planned water doses (with the accuracy of ±10%). The water balance calculations revealed, however, that the content of water decreased with the passage of time (Figure 8). There was no possibility of water outflow from the monolith, as it was enclosed in impermeable barriers. Also, no evaporation from the surface was noted (Figure 9). The cause of the decrease of the content of water was its migration, conforming with the orientation and sense of the force of gravity, below the zone of sensitivity of the lowest positioned layers of moisture sensors (Table 2). In future research, it is planned to focus on the determination of moisture dynamics through mathematical modelling, and to extend the scope of the research into other types of soil.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Jackson, R.B.; Carpenter, S.R.; Dahm, C.N.; McKnight, D.M.; Naiman, R.J.; Postel, S.L.; Running, S.W. Issues in Ecology. WATER A Chang. WORLD
**2001**, 11, 1027–1045. [Google Scholar] - Wada, Y.; Bierkens, M.F.P. Sustainability of global water use: Past reconstruction and future projections. Environ. Res. Lett.
**2014**, 9, 104003–104020. [Google Scholar] [CrossRef] - Walczak, A. The Use of World Water Resources in the Irrigation of Field Cultivations. J. Ecol. Eng.
**2021**, 22, 186–206. [Google Scholar] [CrossRef] - Ünlü, M.; Kanber, R.; Şenyigit, U.; Onaran, H.; Diker, K. Trickle and sprinkler irrigation of potato (Solanum tuberosum L.) in the Middle Anatolian Region in Turkey. Agric. Water Manag.
**2006**, 79, 43–71. [Google Scholar] [CrossRef] - Kang, Y.; Wang, F.X.; Liu, H.J.; Yuan, B.Z. Potato evapotranspiration and yield under different drip irrigation regimes. Irrig. Sci.
**2004**, 23, 133–143. [Google Scholar] [CrossRef] - Dyśko, J.; Kaniszewski, S. Effect of drip irrigation, n-fertigation and cultivation methods on the yield and quality of carrot. Veg. Crop. Res. Bull.
**2007**, 67, 25–33. [Google Scholar] [CrossRef][Green Version] - Kulkarni, S. Innovative Technologies for Water Saving in Irrigated Agriculture. Int. J. Water Resour. Arid Environ.
**2011**, 1, 226–231. [Google Scholar] - Sauer, T.; Havlík, P.; Schneider, U.A.; Schmid, E.; Kindermann, G.; Obersteiner, M. Agriculture and resource availability in a changing world: The role of irrigation. Water Resour. Res.
**2010**, 46, 1–12. [Google Scholar] [CrossRef][Green Version] - Janik, G.; Walczak, A.; Dawid, M.; Pokładek, R.; Adamczewska-Sowińska, K.; Wolski, K.; Sowiński, J.; Gronostajski, Z.; Reiner, J.; Kaszuba, M.; et al. Innovative Conception of Irrigation and Fertilization—Section in Monography; Agricultural Advisory Center in Brwinów: Brwinów, Poland, 2013; ISBN 978-83-88082-18-4. [Google Scholar]
- Janik, G.; Kłosowicz, I.; Walczak, A.; Adamczewska-Sowińska, K.; Jama-Rodzeńska, A.; Sowiński, J. Application of the TDR technique for the determination of the dynamics of the spatial and temporal distribution of water uptake by plant roots during injection irrigation. Agric. Water Manag.
**2021**, 252, 106911–106923. [Google Scholar] [CrossRef] - Walczak, A.; Lipiński, M.; Janik, G. Application of the tdr sensor and the parameters of injection irrigation for the estimation of soil evaporation intensity. Sensors
**2021**, 21, 2309. [Google Scholar] [CrossRef] - Kaczyński, P.; Kaszuba, M.; Dworzak, Ł. Obrotowy Rozdzielacz Hydrauliczny Obrotowy do Iniekcyjnego Nawadniania Gruntów. Polish Patent P.436556, 2021. [Google Scholar]
- McBratney, A.; Whelan, B.; Ancev, T.; Bouma, J. T. Future Directions of Precision Agriculture. Precis. Agric.
**2005**, 6, 7–23. [Google Scholar] [CrossRef] - Gebbers, R.; Adamchuk, V.I. Precision Agriculture and Food Security Published by: American Association for the Advancement. Precis. Agric. Food Secur.
**2017**, 327, 828–831. [Google Scholar] - Shock, C.C.; Wang, F.X. Soil water tension, a powerful measurement for productivity and stewardship. HortScience
**2011**, 46, 178–185. [Google Scholar] [CrossRef][Green Version] - Bittelli, M. Measuring soil water potential forwater management in agriculture: A review. Sustainability
**2010**, 2, 1226–1251. [Google Scholar] [CrossRef][Green Version] - King, B.A.; Bjorneberg, D.L. Characterizing droplet kinetic energy applied by moving spray-plate center-pivot irrigation sprinklers. Trans. ASABE
**2010**, 53, 137–145. [Google Scholar] [CrossRef] - Richards, L.A. A pressure-membrane extraction apparatus for soil solution. Soil Sci.
**1941**, 51, 377–386. [Google Scholar] [CrossRef] - Šimůnek, J.; Van Genuchten, M.T.; Šejna, M. Hydrus: Model use, calibration, and validation. Trans. ASABE
**2012**, 55, 1261–1274. [Google Scholar] - List, F.; Radu, F.A. A study on iterative methods for solving Richards’ equation. Comput. Geosci.
**2016**, 20, 341–353. [Google Scholar] [CrossRef][Green Version] - Naglič, B.; Kechavarzi, C.; Coulon, F.; Pintar, M. Numerical investigation of the influence of texture, surface drip emitter discharge rate and initial soil moisture condition on wetting pattern size. Irrig. Sci.
**2014**, 32, 421–436. [Google Scholar] [CrossRef] - Reynolds, W.D.; Bowman, B.T.; Drury, C.F.; Tan, C.S.; Lu, X. Indicators of good soil physical quality: Density and storage parameters. Geoderma
**2002**, 110, 131–146. [Google Scholar] [CrossRef] - Van Genuchten, M.T. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J.
**1980**, 44, 892–898. [Google Scholar] [CrossRef][Green Version] - Puértolas, J.; Alcobendas, R.; Alarcón, J.J.; Dodd, I.C. Long-distance abscisic acid signalling under different vertical soil moisture gradients depends on bulk root water potential and average soil water content in the root zone. Plant Cell Environ.
**2013**, 36, 1465–1475. [Google Scholar] [CrossRef] [PubMed] - Badr, A.E.; Abuarab, M.E. Soil moisture distribution patterns under surface and subsurface drip irrigation systems in sandy soil using neutron scattering technique. Irrig. Sci.
**2013**, 31, 317–332. [Google Scholar] [CrossRef] - Elfstrand, S.; Båth, B.; Mårtensson, A. Influence of various forms of green manure amendment on soil microbial community composition, enzyme activity and nutrient levels in leek. Appl. Soil Ecol.
**2007**, 36, 70–82. [Google Scholar] [CrossRef] - Baumann, D.T.; Bastiaans, L.; Kropff, M.J. Competition and crop performance in a leek-celery intercropping system. Crop Sci.
**2001**, 41, 764–774. [Google Scholar] [CrossRef] - Smilde, D. Phytophthora Porri in Leek: Epidemiology and Resistance; Wageningen University & Research: Wageningen, Netherlands, 1996; ISBN 9789054855071. [Google Scholar]
- Borowy, A. Effect of Celeriac-Leek Intercropping on Weeds, Insects and Plant Growth. In Second European Allelopathy Symposium “Allelopathy—from understanding to application”; Institute of Soil Science and Plant Cultivation State Research Institute: Puławy, Poland, 2003; pp. 119–120. [Google Scholar]
- Niemiec, M.; Cupiał, M.; Szeląg-Sikora, A. Efficiency of Celeriac Fertilization with Phosphorus and Potassium Under Conditions of Integrated Plant Production. Agric. Agric. Sci. Procedia
**2015**, 7, 184–191. [Google Scholar] [CrossRef] - Sogut, T.; Arioglu, H. Plant density and sowing date effects on sugarbeet yield and quality. J. Agron.
**2004**, 3, 215–218. [Google Scholar] [CrossRef][Green Version] - Yang, X.; You, X. Estimating parameters of van genuchten model for soil water retention curve by intelligent algorithms. Appl. Math. Inf. Sci.
**2013**, 7, 1977–1983. [Google Scholar] [CrossRef][Green Version] - Elmaloglou, S.; Diamantopoulos, E. Soil water dynamics under surface trickle irrigation as affected by soil hydraulic properties, discharge rate, dripper spacing and irrigation duration. Irrig. Drain.
**2010**, 59, 254–263. [Google Scholar] [CrossRef] - Arbat, G.; Cufí, S.; Duran-Ros, M.; Pinsach, J.; Puig-Bargués, J.; Pujol, J.; de Cartagena, F.R. Modeling approaches for determining dripline depth and irrigation frequency of subsurface drip irrigated rice on different soil textures. Water
**2020**, 12, 1724. [Google Scholar] [CrossRef] - Li, J.; Liu, Y. Water and nitrate distributions as affected by layered-textural soil and buried dripline depth under subsurface drip fertigation. Irrig. Sci.
**2011**, 29, 469–478. [Google Scholar] [CrossRef] - Mali, S.S.; Jha, B.K.; Singh, R.; Meena, M. Bitter Gourd Response to Surface and Subsurface Drip Irrigation under Different Fertigation Levels. Irrig. Drain.
**2017**, 66, 615–625. [Google Scholar] [CrossRef] - Gliński, J.; Horabik, J.L.J. Encyclopedia of Agrophyscis; Springer: Berlin, Germany, 2011. [Google Scholar]
- Smit, A.L.; Zuin, A. Root growth dynamics of Brussels sprouts (Brassica olearacea van gemmifera ) and leeks (Allium porrum L.) as reflected by root length, root colour and UV fluorescence. Plant Soil
**2021**, 185, 271–280. [Google Scholar] [CrossRef] - Hossne, A.J.; Jesus Méndez, N.; Leonett, F.A.; Meneses, J.E.; Gil, J.A. Maize root growth under regular water content, subjected to compaction, irrigation frequencies, and shear stress. Rev. Fac. Nac. Agron. Medellin
**2016**, 69, 7867–7880. [Google Scholar] [CrossRef] - García-Gaines, R.A.; Frankenstein, S. USCS and the USDA Soil Classification System: Development of a Mapping Scheme. United States Army Corps of Engineers: Washington DC, USA, 2015; 37. [Google Scholar]
- Sulewska, M.J. Neural modelling of compactibility characteristics of cohesionless soil. Comput. Assist. Mech. Eng. Sci.
**2010**, 17, 27–40. [Google Scholar] - Kaczyński, P.J.; Kaszuba, M.D.; Dworzak, Ł.J.; Hawryluk, M. Iniektor do Nawadniania Gruntu 2019. nr W.128720, Polish Patent Office: Warszawa, Poland. Available online: https://ewyszukiwarka.pue.uprp.gov.pl/search/pwp-details/W.128720?lng=en.
- Isweek.com. YF-B Series Water Flow Sensor Description. Available online: https://www.isweek.com/product/water-flow-sensor-yf-b1-b2-b3-b7_2430.html (accessed on 5 June 2021).
- Bianchi, A.; Masseroni, D.; Thalheimer, M.; de Medici, L.O.; Facchi, A. Field irrigation management through soil water potential measurements: A review. Ital. J. Agrometeorol.
**2017**, 22, 25–38. [Google Scholar] - Evett, S. Coaxial Multiplexer for time domain rflectometry measurement of soil water content and bulk electrical conductivity. Encylopedia Water Sci.
**2003**, 41, 181–190. [Google Scholar] - Jones, S.B.; Sheng, W.; Xu, J.; Robinson, D.A. Electromagnetic Sensors for Water Content: The Need for International Testing Standards. In Proceedings of the 2018 12th International Conference on Electromagnetic Wave Interaction with Water and Moist Substances (ISEMA), Lublin, Poland, 4–7 June 2018. [Google Scholar]
- Šarec, P.; Šarec, O.; Prošek, V.; Dobek, T.K. Description of new design of an instrument measuring soil humidity. Inżynieria Rol.
**2008**, 11, 219–225. [Google Scholar] - Susha Lekshmi, S.U.; Singh, D.N.; Shojaei Baghini, M. A critical review of soil moisture measurement. Meas. J. Int. Meas. Confed.
**2014**, 54, 92–105. [Google Scholar] - Malicki, M.A.; Plagge, R.; Roth, C.H. Improving the calibration of dielectric TDR soil moisture determination taking into account the solid soil. Eur. J. Soil Sci.
**1996**, 47, 357–366. [Google Scholar] [CrossRef] - Skierucha, W. Accuracy of soil moisture measurement by TDR technique. Int. Agrophys.
**2000**, 14, 417–426. [Google Scholar] - Skierucha, W.; Wilczek, A.; Alokhina, O. Calibration of a TDR probe for low soil water content measurements. Sens. Actuators A Phys.
**2008**, 147, 544–552. [Google Scholar] [CrossRef] - Skierucha, W.; Wilczek, A.; Szypłowska, A.; Sławiński, C.; Lamorski, K. A TDR-based soil moisture monitoring system with simultaneous measurement of soil temperature and electrical conductivity. Sensors
**2012**, 12, 13545–13566. [Google Scholar] [CrossRef] - Manoel, C.; Vaz, P.; Hopmans, J.W. with a Combined Penetrometer—TDR Moisture Probe. Soil Sci. Soc. Am. J.
**2001**, 65, 4–12. [Google Scholar] - Gong, Y.; Cao, Q.; Sun, Z. The effects of soil bulk density, clay content and temperature on soil water content measurement using time-domain reflectometry. Hydrol. Process.
**2003**, 17, 3601–3614. [Google Scholar] [CrossRef] - Chen, R.P.; Chen, Y.M.; Xu, W.; Yu, X. Measurement of electrical conductivity of pore water in saturated sandy soils using time domain reflectometry (TDR) measurements. Can. Geotech. J.
**2012**, 47, 197–206. [Google Scholar] [CrossRef] - Telichenko, V.; Rimshin, V.; Eremeev, V.; Kurbatov, V. Mathematical modeling of groundwaters pressure distribution in the underground structures by cylindrical form zone. MATEC Web Conf. 2018, 196, 02025. Available online: https://www.matec-conferences.org/articles/matecconf/abs/2018/55/matecconf_rsp2018_02025/matecconf_rsp2018_02025.html (accessed on 6 January 2021).
- Karlberg, L.; Jansson, P.E.; Gustafsson, D. Model-based evaluation of low-cost drip-irrigation systems and management strategies using saline water. Irrig. Sci.
**2007**, 25, 387–399. [Google Scholar] [CrossRef] - Koumanov, K.S.; Hopmans, J.W.; Schwankl, L.W. Soil Water Dynamics in the Root Zone of a Micro-Sprikler Irrigated Almond Tree. Available online: https://www.ishs.org/ishs-article/664_46 (accessed on 6 June 2021).
- Alomran, A.M. Management of Irrigation Water Salinity in Greenhouse Tomato Production under Calcareous Sandy Soil and Drip Irrigation. J. Agric. Sci. Technol.
**2012**, 14, 939–950. [Google Scholar] - Simionesei, L.; Ramos, T.B.; Brito, D.; Jauch, E.; Leitão, P.C.; Almeida, C.; Neves, R. Numerical Simulation of Soil Water Dynamics Under Stationary Sprinkler Irrigation With Mohid-Land. Irrig. Drain.
**2016**, 65, 98–111. [Google Scholar] [CrossRef] - Elnesr, M.N.; Alazba, A.A. Computational evaluations of HYDRUS simulations of drip irrigation in 2D and 3D domains (i-Surface drippers). Comput. Electron. Agric.
**2019**, 162, 189–205. [Google Scholar] [CrossRef] - Shan, G.; Sun, Y.; Zhou, H.; Schulze Lammers, P.; Grantz, D.A.; Xue, X.; Wang, Z. A horizontal mobile dielectric sensor to assess dynamic soil water content and flows: Direct measurements under drip irrigation compared with HYDRUS-2D model simulation. Biosyst. Eng.
**2019**, 179, 13–21. [Google Scholar] [CrossRef] - Lipnikov, K.; Manzini, G.; Shashkov, M. Mimetic finite difference method. J. Comput. Phys.
**2014**, 257, 1163–1227. [Google Scholar] [CrossRef] - Chávez-Negrete, C.; Domínguez-Mota, F.J.; Santana-Quinteros, D. Numerical solution of Richards’ equation of water flow by generalized finite differences. Comput. Geotech.
**2018**, 101, 168–175. [Google Scholar] [CrossRef] - Curry, C.W.; Bennett, R.H.; Hulbert, M.H.; Curry, K.J.; Faas, R.W. Comparative study of sand porosity and a technique for determing porosity of undisturbed marine sediment. Mar. Georesources Geotechnol.
**2004**, 22, 231–252. [Google Scholar] [CrossRef] - E-TEST Sp. z o.o. Innovative Measurement Techniques for Environment and Soil Science. Available online: https://www.e-test.eu/ (accessed on 20 January 2021).
- Douh, B.; Boujelben, A.; Khila, S.; Bel Haj Mguidiche, A. Effect of Subsurface Drip Irrigation System Depth on Soil Water Content Distribution At Different Depths and Different Times After Irrigation. Larhyss. J.
**2013**, 13, 7–16. [Google Scholar] - Barker, J.B.; Franz, T.E.; Heeren, D.M.; Neale, C.M.U.; Luck, J.D. Soil water content monitoring for irrigation management: A geostatistical analysis. Agric. Water Manag.
**2017**, 188, 36–49. [Google Scholar] [CrossRef][Green Version] - Bear, J. The General Conservation Principle. In Dynamics of fluids in porous media; Courier Corporation: Chelmsford, MA, USA, 2013; pp. 74–113. [Google Scholar]
- Sahimi, M. The Mass Conservation Equation. In Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches; John Wiley and Sons: Weinheim, Germany, 2011. [Google Scholar]
- Quinones, H.; Ruelle, P.; Nemeth, I. Comparison of three calibration procedures for tdr soil moisture sensors. Irrig. Drain.
**2003**, 52, 203–217. [Google Scholar] [CrossRef] - Pfletschinger, H.; Engelhardt, I.; Piepenbrink, M.; Königer, F.; Schuhmann, R.; Kallioras, A.; Schüth, C. Soil column experiments to quantify vadose zone water fluxes in arid settings. Environ. Earth Sci.
**2012**, 65, 1523–1533. [Google Scholar] [CrossRef] - Janik, G.; Wolski, K.; Daniel, A.; Albert, M.; Skierucha, W.; Wilczek, A.; Szyszkowski, P.; Walczak, A. TDR technique for estimating the intensity of evapotranspiration of turfgrasses. Sci. World J. Hindawi
**2015**, 2015, 626545. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mmolawa, K.; Or, D. Root zone solute dynamics under drip irrigation: A review. Plant Soil
**2000**, 222, 163–190. [Google Scholar] [CrossRef] - Janik, G.; Dawid, M.; Walczak, A.; Słowińska-Osypiuk, J.; Skierucha, W.; Wilczek, A.; Daniel, A. Application of the TDR technique for the detection of changes in the internal structure of an earthen flood levee. J. Geophys. Eng.
**2017**, 14, 292–302. [Google Scholar] [CrossRef][Green Version] - Elaiuy, M.L.C.; Dos Santos, L.N.S.; De Sousa, A.C.M.; Souza, C.F.; Matsura, E.E. Wet bulbs from the subsurface drip irrigation with water supply and treated sewage effluent. Eng. Agric.
**2015**, 35, 242–253. [Google Scholar] [CrossRef][Green Version] - Saefuddin, R.; Saito, H.; Šimůnek, J. Experimental and numerical evaluation of a ring-shaped emitter for subsurface irrigation. Agric. Water Manag.
**2019**, 211, 111–122. [Google Scholar] [CrossRef] - Wysocka, M.; Szypcio, Z.; Tymosiak, D. The speed of capillary raise in granular soils. Civ. Environ. Eng.
**2013**, 4, 167–172. [Google Scholar] - Hewelke, P.; Gnatowski, T.; Hewelke, E.; Tyszka, J.; Żakowicz, S. Analysis of water retention capacity for select forest soils in Poland. Polish J. Environ. Stud.
**2015**, 24, 1013–1019. [Google Scholar] [CrossRef] - Janik, G.; Szpila, M.; Slowinska, J.; Brej, G.; Turkiewicz, M.; Skierucha, W.; Pastuszka, T. Method for the determination of the sensitivity zone of the probe. Acta Agrophys.
**2011**, 18, 269–286. [Google Scholar] - Subbaiah, R. A review of models for predicting soil water dynamics during trickle irrigation. Irrig. Sci.
**2013**, 31, 225–258. [Google Scholar] [CrossRef]

**Figure 1.**Schematic presentation of the concept of a platform for injection irrigation with a possibility of fertilization [9].

**Figure 4.**Schematic of the sensor distribution within the physical model and the division of the sand monolith into columns, rows, and layers. Dimensions in centimetres.

**Figure 5.**Dynamics of volumetric moisture in a selected elementary volume during injection of 1500 cm

^{3}of water.

**Figure 6.**Dynamics of volumetric moisture in eight elementary volumes with centre points situated on the line of water outflow from the injector—axis 0Y

^{i}(dose of 1500 cm

^{3}). Elements (

**a**–

**h**) were described in Figure 4.

**Figure 7.**Distributions of moisture in selected sections for doses of 250 (

**a**), 750 (

**b**), and 1500 (

**c**) cm

^{3}.

**Figure 8.**Changes of water content in sand monolith during the injection of 250 (

**a**), 450 (

**b**), 750 (

**c**), 1000 (

**d**), 1250 (

**e**), and 1500 (

**f**) cm

^{3}of water, in a 12-h time interval.

**Figure 9.**Changes of water content in sand monolith during the injection of 750 cm

^{3}of water, in a 4-h time interval, in the individual layers: (

**a**)—layer 1, (

**b**)—layer 2, (

**c**)—layer 3, (

**d**)—layer 4, (

**e**)—layer 5, (

**f**)—layer 6. The layers were presented in Figure 4.

Planned Water Dose ${\mathbf{V}}_{\mathit{i}}\left(\mathbf{c}{\mathbf{m}}^{3}\right)$ | Calculated Water Dose ${\mathit{V}}_{\mathit{c}}^{{\mathit{t}}_{\mathit{i}}}\left(\mathbf{c}{\mathbf{m}}^{3}\right)$ | Relative Error $\frac{\left|{\mathbf{V}}_{\mathit{i}}-{\mathit{V}}_{\mathit{c}}^{{\mathit{t}}_{\mathit{i}}}\right|}{{\mathbf{V}}_{\mathit{i}}}(\%)$ |
---|---|---|

250 | 285 | 14 |

450 | 435 | 3 |

750 | 675 | 10 |

1000 | 950 | 5 |

1250 | 1200 | 4 |

1500 | 1315 | 12 |

**Table 2.**Difference between the volume of pores available on the bottom of the sand monolith and the decrease of water content over time, for various water injection doses.

Planned Water Dose $\text{}{\mathbf{V}}_{\mathit{i}}\left(\mathbf{c}{\mathbf{m}}^{3}\right)$ | Calculated Water Dose at Injection Moment ${\mathbf{V}}_{\mathit{w}\mathit{a}\mathit{t}\mathit{e}\mathit{r}}^{{\mathit{t}}_{\mathit{i}}}\left(\mathbf{c}{\mathbf{m}}^{3}\right)$ | Calculated Water Content after 12 h ${\mathbf{V}}_{\mathit{w}\mathit{a}\mathit{t}\mathit{e}\mathit{r}}^{{\mathit{t}}_{12\mathit{h}}}\left(\mathbf{c}{\mathbf{m}}^{3}\right)$ | ${\mathbf{V}}_{\mathit{w}\mathit{a}\mathit{t}\mathit{e}\mathit{r}}^{{\mathit{t}}_{\mathit{i}}}-{\mathbf{V}}_{\mathit{w}\mathit{a}\mathit{t}\mathit{e}\mathit{r}}^{{\mathit{t}}_{12\mathit{h}}}$ $\left(\mathbf{c}{\mathbf{m}}^{3}\right)$ | $\mathit{L}$ $(-)$ | ${\mathbf{V}}_{\mathit{p}\mathit{o}\mathit{r}\mathit{e}\mathit{s}}$$\left(\mathbf{c}{\mathbf{m}}^{3}\right)$ |
---|---|---|---|---|---|

250 | 285 | 175 | 110 | 15 | 274 |

450 | 435 | 226 | 209 | 17 | 311 |

750 | 675 | 222 | 453 | 22 | 402 |

1000 | 950 | 589 | 361 | 25 | 457 |

1250 | 1200 | 705 | 495 | 27 | 493 |

1500 | 1315 | 969 | 346 | 33 | 603 |

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**MDPI and ACS Style**

Walczak, A.; Szypłowska, A.; Janik, G.; Pęczkowski, G.
Dynamics of Volumetric Moisture in Sand Caused by Injection Irrigation—Physical Model. *Water* **2021**, *13*, 1603.
https://doi.org/10.3390/w13111603

**AMA Style**

Walczak A, Szypłowska A, Janik G, Pęczkowski G.
Dynamics of Volumetric Moisture in Sand Caused by Injection Irrigation—Physical Model. *Water*. 2021; 13(11):1603.
https://doi.org/10.3390/w13111603

**Chicago/Turabian Style**

Walczak, Amadeusz, Agnieszka Szypłowska, Grzegorz Janik, and Grzegorz Pęczkowski.
2021. "Dynamics of Volumetric Moisture in Sand Caused by Injection Irrigation—Physical Model" *Water* 13, no. 11: 1603.
https://doi.org/10.3390/w13111603