#### 3.1. Dynamics of the Volumetric Moisture

The dynamics of volumetric moisture in an elementary volume in the case of injection of 1500 cm

^{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].

The moisture changes in eight elementary volumes, caused by the injection of 1500 cm

^{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.

Figure 7 presents volumetric moisture distributions in two vertical planes, perpendicular to each other, containing axes 0X

^{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

The results of measurements of volumetric moisture provide a basis for the calculation of the water balance in the physical model. Each TDR sensor is representative of an elementary volume—a cube with the side of 5 cm. The sum of water volumes in the cubes is the volume of water in the entire physical model (formula 3). The initial half-hour of the measurements was the period before injection, during which no moisture changes were observed. In the course of the series of experiments, after the 30-min period, water injection took place. The effect was a jump increase of water content in the monolith—

$\Delta {V}_{c}^{{t}_{in}}$. The value of

$\Delta {V}_{c}^{{t}_{in}}$ increased with an increase in the injection dose (

Table 1). As an example, in the case of the application of 450 cm

^{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].

As can be seen from

Table 1, in the case of doses of 250 and 1500 cm

^{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.

The graphs in

Figure 8 present changes of water content in sand monolith during water injections. At the beginning, increases of water content are observable, which, then, irrespective of the passage of time, should remain at a constant level. In practice, however, already a moment after the injections, a gradual decrease of water content in the monolith is observed. What is more, the tendency resembles the runs of moisture dynamics presented in

Figure 5 and

Figure 6. After 12 h of measurements, the loss of water resulting from the water balance calculation for the doses of 250, 450, 750, 1000, 1250, and 1500 cm

^{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).

The base and the side walls of the structure of the experimental setup were impermeable barriers, and the upper surface was exposed to contact with atmospheric air (

Figure 2). Therefore, the first possible explanation for the decrease of water content is the appearance of the phenomenon of evaporation [

37]. However, the partial results of the water balance presented in

Figure 9 contradict this hypothesis. The graphs illustrate the changes in water content in division of the monolith into layers. The highest positioned first layer did not show any changes in this respect, both before and after water injection. This means that no process of evaporation took place. In fact, it could not, even if only due to the phenomenon of capillary rise in porous material of this type. In sands, the value of capillary rise after 12 h is only approximately 1.1 cm [

78]. In layers 2, 3, and 4 the content of water increased within several seconds—by 266, 251, 120 cm

^{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.

Among the characteristics presented above, the relationship shown in

Figure 9 for changes in water content in the lowest, 6th, layer of the monolith, is characterised by the smallest decrease of water content. There is a possibility that the water supplied through injection migrated, due to gravity, to the bottom of the monolith. Taking into account the kind of porous material used, which is characterised by a high filtration coefficient (10

^{−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:

where:

${V}_{pores}$—volume of pores available on the bottom of the monolith, containing a part of the injected water dose, which has not been registered by the TDR sensors (cm

^{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}.

As can be seen in

Table 2, at every injection under analysis the following relationship is noted:

${V}_{pores}>({\mathrm{V}}_{water}^{{t}_{i}}-{\mathrm{V}}_{water}^{{t}_{12h}})$. The sole exception is the injection of 750 cm

^{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.