# An Improved Empirical Model for Estimating the Geometry of the Soil Wetting Front with Surface Drip Irrigation

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

^{−1}, and organic matter contents from 1.7 to 2.8%, and different irrigation conditions: discharge rates of 1.44, 2.90, 3.00, 3.75, and 4.00 L h

^{−1}, initial moisture levels between permanent wilting point and field capacity, and irrigation times from 0.58 to 9.50 h. The experimental conditions and the strategy for measuring the wetting front and soil moisture are detailed so the experiment is verifiable. The proposed model performed better than five other empirical models, with average values of 3 cm for the root mean square error and 0.88 for the Nash and Sutcliffe efficiency coefficient. The generated model is efficient and simple and can be a very useful tool for the design and operation of surface drip irrigation systems in soils with conditions similar to those of this study.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Site

#### 2.2. Characteristics of the Experimental Soils

_{b}) was determined with the paraffin-coated clod method, moisture content at field capacity (θ

_{CC}) was obtained with the pressure cooker method, moisture content at permanent wilting point (θ

_{PMP}) was obtained with the pressure membrane method, and percent organic matter (OM) was measured with the Walkley and Black method.

_{r}, θ

_{s}), inverse soil air-entry pressure ratio (α), pore size distribution (n), and saturated hydraulic conductivity (K

_{s}) were obtained (Table 2).

#### 2.3. Experimental System

^{−2}) (Figure 1).

^{®}corporation), with an error of 3%, were installed under the emitter in a 4 by 4 grid on a vertical plane transversal to the observation wall. With this arrangement, the behavior of half of the wetted front was observed and it was assumed that the same occurred in the other half, approaching it mathematically as an asymmetric axis problem [14]; in effect, the wetting front showed that type of behavior in preliminary tests (Figure 2).

^{3}cm

^{−3}.

#### 2.4. Experimental Trials

^{−1}); the long irrigations were used to try to represent the sowing irrigations in an initially dry soil, with an initial moisture (θ

_{i}) close to permanent wilting point (0.13 to 0.14), while the moderate-duration irrigations were applied with an initial soil moisture close to field capacity (0.22 to 0.26), representing the supply of the daily evapotranspiration demand (Table 3). The second experimental irrigation block consisted of applying short-duration irrigations with 3.00 and 4.00 L h

^{−1}drippers in soils with an initial moisture content close to the permanent wilting point, simulating the short-period supply of the evapotranspiration demand of crops in early vegetative stages where the root system occupies a reduced volume of soil (Table 4). In both irrigation blocks, a final moisture content between field capacity and saturation was guaranteed, a situation commonly carried out by farmers. In Table 3 and Table 4 the symbol θ

_{f}represents the average soil moisture content at the end of irrigation.

#### 2.5. Soil Drying after Irrigations

^{®}, Bridgend, UK), whose error is 3%.

#### 2.6. Proposed Empirical Model

^{−1}), K

_{s}is the saturated hydraulic conductivity (cm h

^{−1}), t is the irrigation time (h), θ

_{i}is the initial volumetric moisture content (cm

^{3}cm

^{−3}), P

_{b}is the soil bulk density (g cm

^{−3}), and OM is the percent organic matter (%).

#### 2.7. Evaluated Models

_{s}, t, θ

_{i}, and P

_{b}mean the same as in Equations (1) and (2); V is the total volume of water applied (l); Δθ is the average change in water content due to irrigation (cm

^{3}cm

^{−3}) which is obtained as Δθ = θ

_{s}/2, where θ

_{s}is the water content at saturation; S is the percentage of sand (%); Si is the percentage of silt (%); C is the percentage of clay (%) and θ

_{r}is the residual soil moisture content (cm

^{3}cm

^{−3}).

#### 2.8. Evaluation of the Models

_{i}corresponds to the i-th estimated data, ${\mathrm{O}}_{\mathrm{i}}$ is the i-th observed data, and $\overline{\mathrm{O}}$ is the mean of the observed data.

## 3. Results

#### 3.1. Wetting Patterns Geometry

#### 3.2. Model Performance

#### 3.3. Effect of Input Parameters on Model Response

## 4. Discussion

#### 4.1. Wetting Patterns Geometry

^{−1}), the wetting front had greater horizontal than vertical displacement at the beginning of the irrigation because in those first moments the capillary forces dominated the movement of water in the soil, but as time passed the gravitational forces began to dominate causing an increase in the vertical advance and a decrease in the horizontal advance (irrigations 1A to 3A of Figure 4). In irrigation 4A of the first block (1.44 L h

^{−1}) (Figure 4) and in the irrigations of the second block (Figure 5), a similar displacement was observed in all directions of the wetting front.

#### 4.2. Model Performance

#### 4.3. Effect of Input Parameters on the Model Response

^{−1}, 3.24 cm h

^{−1}, 0.14 cm

^{3}cm

^{−3}, 1.31 g cm

^{−3}, 2.4%, and 3.17 h, for discharge rate (q), saturated hydraulic conductivity (K

_{s}), initial moisture content (${\mathsf{\theta}}_{\mathrm{i}}$), bulk density (P

_{b}), percent organic matter (OM), and irrigation time (t), respectively. P

_{b}had a similar impact on the radius and depth of the wetting front, while OM and K

_{s}had a greater impact on radius than on depth. P

_{b}had a direct effect (as its value increases, its effect on the magnitude of the wetting front increases) and K

_{s}and OM had an inverse effect (as its value increases, its effect on the magnitude of the wetting front decreases) in both directions of the wetting front. The q, ${\mathsf{\theta}}_{\mathrm{i}}$, and t had a direct and smaller effect than the physical properties of the soils on the displacements in both directions of the wetting front; q and t had practically identical effects on the radius, and q and ${\mathsf{\theta}}_{\mathrm{i}}$ on the depth. The ${\mathsf{\theta}}_{\mathrm{i}}$ has the least effect in both directions of the wetting front and attenuates more in depth, so Fan et al. [47] suggested removing it from the modeling; however, in this study the results showed the need for its inclusion as it considerably improved the accuracy of the estimates with respect to those that do not include it. The better predictive capacity of the proposed model to estimate the dimensions of the wetting front was due to the consideration of the parameters involved in water movement, the range of parameter values, and the care taken in the execution of the methodology.

## 5. Conclusions

_{s}), bulk density (P

_{b}), and organic matter content (OM)), initial moisture content (θ

_{i}), and irrigation operating characteristics (emitter discharge rate q and irrigation time t).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Geometry of the wetting front as a function of irrigation time in the first irrigation block.

**Figure 5.**Geometry of the wetting front as a function of irrigation time in the second irrigation block.

**Figure 8.**Temporal evolution of the width and depth of the wetting front of (

**a**,

**b**) irrigation 1B (q = 4.00 L h

^{−1}and θ

_{i}= 0.09 cm

^{3}cm

^{−3}), (

**c**,

**d**) irrigation 3B (q = 4.00 L h

^{−1}and θ

_{i}= 0.10 cm

^{3}cm

^{−3}), and (

**e**,

**f**) irrigation 5B (q = 4.00 L h

^{−1}and θ

_{i}= 0.13 cm

^{3}cm

^{−3}) [6,10,12,13,24].

**Figure 9.**Temporal evolution of the width and depth of the wetting front of (

**a**,

**b**) irrigation 2B (q = 3.00 L h

^{−1}and θ

_{i}= 0.13 cm

^{3}cm

^{−3}), (

**c**,

**d**) irrigation 4B (q = 3.00 L h

^{−1}and θ

_{i}= 0.09 cm

^{3}cm

^{−3}), and (

**e**,

**f**) irrigation 6B (q = 3.00 L h

^{−1}and θ

_{i}= 0.07 cm

^{3}cm

^{−3}) [6,10,12,13,24].

**Figure 10.**Effect of input parameters on the response of the proposed models for (

**a**) width and (

**b**) depth of the wetting front.

Soil | Sand (%) | Silt (%) | Clay (%) | P_{b}(g cm ^{−3}) | θ_{CC}(cm ^{3} cm^{−3}) | θ_{PMP}(cm ^{3} cm^{−3}) | OM (%) |
---|---|---|---|---|---|---|---|

1 | 57.50 | 22.00 | 20.50 | 1.23 | 0.26 | 0.14 | 1.70 |

2 | 57.50 | 22.00 | 20.50 | 1.38 | 0.23 | 0.13 | 1.70 |

3 | 60.50 | 18.50 | 21.00 | 1.34 | 0.22 | 0.13 | 2.60 |

4 | 61.50 | 18.00 | 20.50 | 1.31 | 0.23 | 0.14 | 2.80 |

**Table 2.**Hydraulic parameters of the experimental soils obtained with the ROSETTA version 1.2 software package [30].

Soil | θ_{r}(cm ^{3} cm^{−3}) | θ_{s}(cm ^{3} cm^{−3}) | α (cm ^{−1}) | n | K_{s}(cm h ^{−1}) |
---|---|---|---|---|---|

1 | 0.052 | 0.469 | 0.026 | 1.316 | 3.241 |

2 | 0.058 | 0.428 | 0.036 | 1.333 | 2.316 |

3 | 0.065 | 0.444 | 0.045 | 1.363 | 3.834 |

4 | 0.065 | 0.454 | 0.044 | 1.344 | 3.945 |

**Table 3.**Characteristics of the moderate- and long-duration irrigations of the first experimental block.

Irrigation Trial | Irrigation Duration | Soil | q (L h ^{−1}) | θ_{i}(cm ^{3} cm^{−3}) | θ_{f}(cm ^{3} cm^{−3}) | t (h) |
---|---|---|---|---|---|---|

1A | Long | 1 | 2.90 | 0.07 | 0.42 | 9.50 |

2A | Moderate | 1 | 2.90 | 0.23 | 0.41 | 3.00 |

3A | Moderate | 1 | 3.75 | 0.24 | 0.40 | 2.00 |

4A | Long | 1 | 1.44 | 0.14 | 0.41 | 8.00 |

Irrigation Trial | Soil | q (L h ^{−1}) | θ_{i}(cm ^{3} cm^{−3}) | θ_{f}(cm ^{3} cm^{−3}) | t (h) |
---|---|---|---|---|---|

1B | 2 | 4.00 | 0.09 | 0.39 | 0.58 |

2B | 2 | 3.00 | 0.13 | 0.35 | 0.58 |

3B | 3 | 4.00 | 0.10 | 0.42 | 0.58 |

4B | 3 | 3.00 | 0.09 | 0.39 | 0.58 |

5B | 4 | 4.00 | 0.13 | 0.45 | 0.58 |

6B | 4 | 3.00 | 0.07 | 0.43 | 0.58 |

Model | Soil Texture | q (L h ^{−1}) | K_{s}(cm h ^{−1}) | θ_{s}(cm ^{3} cm^{−3}) | P_{b}(g cm ^{−3}) | θ_{i}(cm ^{3} cm^{−3}) | θ_{r}(cm ^{3} cm^{−3}) | MO (%) |
---|---|---|---|---|---|---|---|---|

Schwartzman and Zur (1986) [24] | Silt and sandy loam. | 4.16–20.06 | 0.84–8.4 | ^{1} N.S. | N.S. | N.S. | N.S | N.S. |

Amin and Ekhmaj (2006) [6] | Silt, loam, sand, and clay loam. | 0.60–12.30 | 0.85–5.80 | 0.45–0.58 | 1.28–1.46 | 0.03–0.27 | N.S. | N.S. |

Malek and Peters (2011) [12] | Clay loam. | 2.00–6.00 | 3.66 | N.S. | 1.48 | 0.22 | N.S. | N.S. |

Al-Ogaidi et al. (2015) [13] | Sand, silt, loam, and clay loam. | 0.50–12.30 | 0.85–5.80 | 0.42–0.58 | 1.28–1.46 | 0.03–0.27 | N.S. | N.S. |

Cruz-Bautista et al. (2016) [10] | Sandy loam, clay loam, and silt loam. | 2.00–8.00 | 2.05–3.28 | 0.39–0.51 | 1.18–1.51 | 0.05–0.11 | 0.04–0.08 | N.S. |

Proposed model | Sandy clay loam. | 1.44–4.00 | 2.32–3.95 | 0.43–0.47 | 1.23–1.38 | 0.07–0.24 | 0.05–0.06 | 1.70–2.80 |

^{1}N.S.: range not specified by the authors.

**Table 6.**Statistical indicators of the empirical models for the first irrigation block for the width (d) and depth of the wetting front (z).

Irrigation Trial | Model | Statistical Indicators | |||||
---|---|---|---|---|---|---|---|

ME (cm) | RMSE (cm) | NSE | |||||

d | z | d | z | d | z | ||

1A | Schwartzman and Zur (1986) [24] | −14.78 | 22.23 | 17.63 | 25.67 | 0.32 | −6.05 |

Amin and Ekhmaj (2006) [6] | −9.14 | 7.84 | 11.17 | 8.07 | 0.73 | 0.30 | |

Malek and Peters (2011) [12] | −8.74 | 88.40 | 9.23 | 96.69 | 0.81 | −99.08 | |

Al-Ogaidi et al. (2015) [13] | −25.04 | −6.59 | 27.60 | 8.19 | −0.66 | 0.28 | |

Cruz-Bautista et al. (2016) [10] | −3.63 | 7.77 | 4.26 | 8.11 | 0.96 | 0.30 | |

Proposed model | 7.01 | −0.08 | 7.65 | 1.31 | 0.87 | 0.98 | |

2A | Schwartzman and Zur (1986) [24] | −22.92 | 7.50 | 24.79 | 8.17 | −0.73 | −0.19 |

Amin and Ekhmaj (2006) [6] | −19.94 | 3.74 | 21.37 | 4.25 | −0.29 | 0.68 | |

Malek and Peters (2011) [12] | −42.86 | −0.08 | 44.97 | 1.77 | −4.70 | 0.94 | |

Al-Ogaidi et al. (2015) [13] | −28.59 | −6.33 | 30.43 | 7.69 | −1.61 | −0.06 | |

Cruz-Bautista et al. (2016) [10] | −33.00 | 12.49 | 34.68 | 13.02 | −2.39 | −2.03 | |

Proposed model | −5.40 | 1.46 | 5.69 | 1.99 | 0.91 | 0.93 | |

3A | Schwartzman and Zur (1986) [24] | −9.48 | 1.97 | 12.46 | 2.66 | 0.42 | 0.85 |

Amin and Ekhmaj (2006) [6] | −8.37 | 1.12 | 10.75 | 2.12 | 0.57 | 0.91 | |

Malek and Peters (2011) [12] | −30.96 | −3.56 | 33.05 | 4.13 | −3.11 | 0.64 | |

Al-Ogaidi et al. (2015) [13] | −16.38 | −8.17 | 18.67 | 9.14 | −0.31 | −0.75 | |

Cruz-Bautista et al. (2016) [10] | −21.81 | 9.08 | 23.63 | 9.56 | −1.10 | −0.92 | |

Proposed model | 4.67 | −1.41 | 5.10 | 2.01 | 0.90 | 0.91 | |

4A | Schwartzman and Zur (1986) [24] | −16.65 | 15.48 | 18.77 | 18.32 | −0.18 | −4.42 |

Amin and Ekhmaj (2006) [6] | −9.23 | 2.33 | 10.74 | 2.49 | 0.62 | 0.90 | |

Malek and Peters (2011) [12] | −28.95 | 11.54 | 30.59 | 12.67 | −2.12 | −1.59 | |

Al-Ogaidi et al. (2015) [13] | −20.16 | −9.56 | 22.05 | 10.43 | −0.62 | −0.75 | |

Cruz-Bautista et al. (2016) [10] | −17.66 | 8.58 | 18.82 | 9.36 | −0.18 | −0.41 | |

Proposed model | 6.28 | −1.05 | 6.63 | 1.18 | 0.85 | 0.98 |

**Table 7.**Statistical indicators of the empirical models for the second irrigation block, with an emitter discharge rate of 4 L h

^{−1}, for the width (d) and depth of the wetting front (z).

Irrigation Trial | Model | Statistical Indicators | |||||
---|---|---|---|---|---|---|---|

ME (cm) | RMSE (cm) | NSE | |||||

d | z | d | z | d | Z | ||

1B | Schwartzman and Zur (1986) [24] | 5.69 | −4.94 | 6.09 | 5.29 | 0.53 | 0.04 |

Amin and Ekhmaj (2006) [6] | 5.02 | −1.52 | 5.45 | 1.71 | 0.62 | 0.90 | |

Malek and Peters (2011) [12] | −9.10 | 11.76 | 9.74 | 13.24 | −0.20 | −5.01 | |

Al-Ogaidi et al. (2015) [13] | 2.77 | −2.21 | 2.99 | 2.42 | 0.89 | 0.80 | |

Cruz-Bautista et al. (2016) [10] | 0.05 | −1.94 | 1.16 | 2.11 | 0.98 | 0.85 | |

Proposed model | 0.86 | −0.33 | 1.58 | 0.85 | 0.97 | 0.98 | |

3B | Schwartzman and Zur (1986) [24] | 4.44 | −3.48 | 4.88 | 3.81 | 0.70 | 0.52 |

Amin and Ekhmaj (2006) [6] | 5.01 | −0.98 | 5.42 | 1.15 | 0.63 | 0.96 | |

Malek and Peters (2011) [12] | −3.77 | 10.16 | 4.13 | 11.56 | 0.78 | −3.45 | |

Al-Ogaidi et al. (2015) [13] | −0.55 | −4.68 | 1.48 | 5.07 | 0.97 | 0.14 | |

Cruz-Bautista et al. (2016) [10] | −1.01 | −0.59 | 1.40 | 1.05 | 0.98 | 0.96 | |

Proposed model | 6.58 | 1.21 | 7.27 | 1.93 | 0.33 | 0.88 | |

5B | Schwartzman and Zur (1986) [24] | 3.42 | −3.99 | 4.30 | 4.37 | 0.81 | 0.41 |

Amin and Ekhmaj (2006) [6] | 3.77 | −1.64 | 4.36 | 1.80 | 0.80 | 0.90 | |

Malek and Peters (2011) [12] | −7.16 | 3.16 | 7.92 | 4.00 | 0.34 | 0.51 | |

Al-Ogaidi et al. (2015) [13] | −1.94 | −6.28 | 3.26 | 6.80 | 0.89 | −0.42 | |

Cruz-Bautista et al. (2016) [10] | −3.81 | 0.19 | 4.50 | 1.21 | 0.79 | 0.96 | |

Proposed model | −0.47 | −1.00 | 1.53 | 1.48 | 0.98 | 0.93 |

**Table 8.**Statistical indicators of the empirical models for the second irrigation block, with an emitter discharge rate of 3 L h

^{−1}, for the width (d) and depth of the wetting front (z).

Irrigation Trial | Model | Statistical Indicators | |||||
---|---|---|---|---|---|---|---|

ME (cm) | RMSE (cm) | NSE | |||||

d | z | d | z | d | Z | ||

2B | Schwartzman and Zur (1986) [24] | 4.31 | −4.61 | 4.63 | 4.97 | 0.70 | 0.15 |

Amin and Ekhmaj (2006) [6] | 4.56 | −1.60 | 4.93 | 1.99 | 0.66 | 0.86 | |

Malek and Peters (2011) [12] | −11.55 | 2.55 | 12.45 | 2.91 | −1.20 | 0.71 | |

Al-Ogaidi et al. (2015) [13] | 3.30 | −2.25 | 3.56 | 2.61 | 0.82 | 0.77 | |

Cruz-Bautista et al. (2016) [10] | −2.88 | −0.82 | 3.12 | 0.98 | 0.86 | 0.97 | |

Proposed model | 1.15 | 0.50 | 1.61 | 0.76 | 0.96 | 0.98 | |

4B | Schwartzman and Zur (1986) [24] | 0.74 | −1.61 | 1.14 | 2.15 | 0.98 | 0.76 |

Amin and Ekhmaj (2006) [6] | 2.08 | 0.50 | 2.41 | 0.79 | 0.93 | 0.97 | |

Malek and Peters (2011) [12] | −5.46 | 14.87 | 5.91 | 16.77 | 0.56 | −13.89 | |

Al-Ogaidi et al. (2015) [13] | −3.44 | −3.12 | 3.79 | 3.35 | 0.82 | 0.41 | |

Cruz-Bautista et al. (2016) [10] | −2.74 | 0.02 | 3.09 | 0.98 | 0.88 | 0.95 | |

Proposed model | 2.94 | 1.86 | 3.72 | 2.56 | 0.82 | 0.65 | |

6B | Schwartzman and Zur (1986) [24] | 0.23 | −1.27 | 1.43 | 1.68 | 0.98 | 0.86 |

Amin and Ekhmaj (2006) [6] | 1.36 | 0.69 | 1.69 | 0.79 | 0.97 | 0.97 | |

Malek and Peters (2011) [12] | −2.56 | 24.21 | 2.95 | 26.97 | 0.90 | −35.64 | |

Al-Ogaidi et al. (2015) [13] | −5.28 | −3.94 | 5.91 | 4.29 | 0.60 | 0.07 | |

Cruz-Bautista et al. (2016) [10] | −1.31 | −0.60 | 1.64 | 0.76 | 0.97 | 0.97 | |

Proposed model | −3.89 | −0.72 | 4.25 | 0.91 | 0.79 | 0.96 |

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Cristóbal-Muñoz, I.; Prado-Hernández, J.V.; Martínez-Ruiz, A.; Pascual-Ramírez, F.; Cristóbal-Acevedo, D.; Cristóbal-Muñoz, D.
An Improved Empirical Model for Estimating the Geometry of the Soil Wetting Front with Surface Drip Irrigation. *Water* **2022**, *14*, 1827.
https://doi.org/10.3390/w14111827

**AMA Style**

Cristóbal-Muñoz I, Prado-Hernández JV, Martínez-Ruiz A, Pascual-Ramírez F, Cristóbal-Acevedo D, Cristóbal-Muñoz D.
An Improved Empirical Model for Estimating the Geometry of the Soil Wetting Front with Surface Drip Irrigation. *Water*. 2022; 14(11):1827.
https://doi.org/10.3390/w14111827

**Chicago/Turabian Style**

Cristóbal-Muñoz, Irouri, Jorge Víctor Prado-Hernández, Antonio Martínez-Ruiz, Fermín Pascual-Ramírez, David Cristóbal-Acevedo, and David Cristóbal-Muñoz.
2022. "An Improved Empirical Model for Estimating the Geometry of the Soil Wetting Front with Surface Drip Irrigation" *Water* 14, no. 11: 1827.
https://doi.org/10.3390/w14111827