# An Improved Method to Estimate Soil Hydrodynamic and Hydraulic Roughness Parameters by Using Easily Measurable Data During Flood Irrigation Experiments and Inverse Modelling

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{0}was of the same order of magnitude of its value. All experimental datasets were simulated very well. Performance criteria were similar during both the fitting and validation stages.

## 1. Introduction

_{rl}(m

^{3}·s

^{−1}·m

^{−1}) is the surface flow rate per unit field width, f (m·s

^{−1}) is the instantaneous soil infiltration rate that represents the coupling with the infiltration process.

^{–1}) is the field slope, n (s·m

^{−1/3}) is the Manning’s roughness coefficient n = 1/k, and k (m

^{1/3}·s

^{−1}) is the Manning Strickler’s roughness coefficient. Ref. [7] proposed a modification of the Manning-Strickler’s equation by introducing a new parameter H

_{0}(m) representative for a water height that does not contribute to the surface flow: it is called the depression storage and is the water volume that has to be filled at the soil surface in the vegetation and the soil micro-topography before runoff starts [11].

_{0}accounts for the effects of vegetation, soil micro topography at small scales (due to soil roughness for instance) and large scales (due to counter slopes or local soil depressions).

_{Kos}and f

_{0}are the three empirical infiltration parameters. The Green and Ampt’s equation [15] has also been used by [7]. It can be written:

^{−1}) is the soil saturated hydraulic conductivity, ∆θ = θ

_{s}− θ

_{i}(m

^{3}·m

^{−3}) is the soil water depletion represented by the difference between the saturated water content θ

_{s}and the initial water content θ

_{i}, P

_{ini}(m) is the matric suction behind the infiltration front and F (m) is the cumulative infiltration.

## 2. Materials and Methods

#### 2.1. Model of Flood Irrigation: Streamflow Advance and Infiltration

_{i}(m

^{3}·m

^{−1}·s

^{−1}) and the field geometry (m). The main outputs are: f the instantaneous soil infiltration flow (m·s

^{−1}), F the cumulative infiltration (m), H the flow depth at the soil surface (m) and Q

_{rl}the water discharge per unit width of the field (m

^{3}·m

^{−1}·s

^{−1}). All variables are calculated at each time step ∆t along a regular grid with a ∆x space step along the streamflow propagation direction.

#### 2.2. Site of Measurements

#### 2.2.1. Field Location

#### 2.2.2. Soil Description

#### 2.3. Monitoring of the Flow Depth at the Field Surface

#### 2.4. Measurement of Soil Characteristics

^{2}= 0.96) obtained from data presented by [7]:

^{3}·m

^{−3}. This value is higher than standards for textural class of [29] but was considered to be more precise than standards obtained from pedotransfer functions not adapted to our specific stony soils.

#### 2.5. Measurement of Irrigation Characteristics

_{i}was measured with OTT C2 current-meter. Equipped with a propeller fixed in front of the water current, it measures the velocity of the water flow with an accuracy of about ± 2%. The advancing surface water front was monitored for the two irrigation events: the observed position of the stream front was marked with plastic sticks every 30 minutes by an operator walking along the field at 5 locations in the cross section of the flow direction with plastic sticks. At the end of the experiment, the position of each plastic stick was measured with a decameter and a GPS device. The precision of the measurements was evaluated at ± 2 m. Photographic monitoring was also conducted to analyze the vegetation development (data not shown). The experimental dataset of the CALHY input parameters measured during each irrigation event is presented in Table 1.

#### 2.6. Experimental Data Used for the Fitting Algorithm (“Proxy-Data”)

- T_arrive (h) is the arrival time of the surface water front at each measurement location,
- T_submersion (h) is the time interval between the arrival time and the end of the recession phase of the surface water front at a given measurement location in the field,
- H_max (mm) is the maximum measured value of the flow depth,
- H_integral (mm·h) is defined as: ${H}_{integral}={{\displaystyle \int}}_{{t}_{arrive}}^{{t}_{arrive}+{t}_{submersion}}H\left(t\right)dt$, computed directly using the trapezoidal method function.

#### 2.7. Inverse Modelling Approach

_{0}and ∆θ of the CALHY model. The data used for the fit are the proxy-variables defined above. The objective function (OF) to be minimized was based on the weighted least squares method [31], computed as follows:

_{j}

^{2}is the variance of Obs

_{j}. The values of σ

_{j}

^{2}for each proxy variable were deduced from the experimental variance. They were calculated from the three signals recorded at each position in situ for each variable of interest.

#### 2.8. Evaluation of the Estimated Parameters

#### 2.8.1. Coherence and Physical Meaning of the Fitted Parameters

#### 2.8.2. Evaluation of the Simulated Hydrograph and Surface Water Front from Irrigation Events Used for Calibration

- Estimated and the measured soil water depletion ∆θ
- Simulated and monitored time flow depth hydrograph H(t) in two cross sections of the field
- Simulated and monitored advancement of surface water front

#### 2.8.3. Evaluation of the Simulated Hydrograph and Surface Water Front for Cases Not Used for Calibration

- We simulated irrigation A4 that was not used during the calibration process. Parameters Ks, k, H
_{0}and ∆θ used for this direct simulation were derived from the parameters obtained from irrigation A5. The comparison between the simulated and measured data was performed upon the advancement of the surface water and the water depth hydrographs H(t) at the two selected sections of the field. - Two irrigation experiments referred to in [5] were obtained. They were monitored in the first experimental site and labelled irrigation A6 and A7. The first one occurred after the 1st mowing with a leaf area index (LAI) of about 1.3 and similar to those of irrigation A3 in our dataset. The second occurred before the 2nd mowing: the development of the vegetation was maximum with a LAI of about 7.7 and similar to that of irrigation A2. Therefore, we used the parameters fitted on irrigation A3 and A2 to simulate outputs of irrigation A6 and A7 respectively.
- Overall compared data for validation were also evaluated with RMSE and Nash criterion.

## 3. Results and Discussion

_{0}and ∆θ. These parameters were fitted for the irrigation events A1, A2, A3, A5, B6 and B7 with their values are listed in Table 4.

#### 3.1. Coherence and Physical Meaning of the Fitted Parameters

^{−}

^{6}and 1.5 × 10

^{−6}m·s

^{−1}. This parameter value varied within a very narrow range. An average value of 1.4 × 10

^{-6}m·s

^{−1}was calculated for this site, which is in accordance with the values obtained by [5,7] for the same plot. For the second site, the two estimations of Ks were also close to each other, with an average value of 1.1 × 10

^{−}

^{7}m·s

^{−1}. This result indicates that the soil of the second experimental site may be less permeable than the soil of the first site. The soil characteristics of the second site which is hydromorphic and has a higher clay content and a higher organic content [26].

^{1/3}·s

^{−1}. [7] calculated by inversion the value of k for the same based on six irrigations events. They found that k varied from 2.4 to 5.3 m

^{1/3}·s

^{−1}. [33] assessed the variation of parameter k in furrows under different conditions and found that k ranged between 8 and 16 m

^{1/3}·s

^{−1}. They showed that it could reduce by up to 30% when the vegetation increased by 47%. [34] suggested a value of k between 2.2 and 3.7 m

^{1/3}·s

^{−1}; [35] suggested a value of k ranging from 3.3 to 6.7 m

^{1/3}·s

^{−1}for a field planted with alfalfa and irrigated with border irrigation. [36] found that k varied from 2.8 to 5 m

^{1/3}·s

^{−1}for a field sown with perennial pasture. In conclusion, our fitted values of k agree well with the values found in the literature.

^{1/3}·s

^{−1}, which is significantly less than the values derived for the first site. From a hydraulic point of view, the effect of the hydraulic friction of on the flow advancement increases as the irrigation rate and flow rate decrease [33,37]; The field surface and vegetation stage in the second experimental site were close to those in the first one, but the applied rates during the irrigation experiments B6 and B7 were 50% lower than those applied in A1, A2 and A3 and 30% lower than in that applied A4 and A5, this justifies the lower k value.

_{0}, it is difficult to assess the coherence of the estimated results due to the difficulty in measuring this parameter. The simulated values of this parameter reported in [38,39] ranged from 0 mm to 30 mm and from 0.26 to 48.0 mm respectively. For a random soil roughness, the slope is expected to reduce the soil surface storage [40], whereas the vegetation is expected to increase this value [41]. For our experimental conditions, the soil was very flat with a very smooth microrelief, as the soil was leveled several years ago and was never tilled. Therefore, a lower value of H

_{0}is expected, which is the case.

#### 3.2. Evaluation of the Performance of the Proposed Approach for Parameter Estimation

^{3}·m

^{−3}, and 0.06–0.1 m

^{3}·m

^{−3}respectively, which is quite very low. This is due to the high content of stones with a very low porosity in the soil profile. The correlation coefficient between the two is R² = 0.85, points are distributed near the 1:1 line except one point.

#### 3.3. Evaluation of the New Parameter Estimation Approach

## 4. Conclusions

_{0}was estimated with a large uncertainty. Nevertheless, this parameter has only a slight effect on the irrigation dynamics and the water balance at the field scale during one irrigation.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Map of the Crau area showing the aquifer limits, the irrigated grassland areas including the two monitored sites and the main surface water channels.

**Figure 2.**Time evolution of water levels during an irrigation event in two locations monitored in the 1st site. The first blue curve was obtained in the upstream section of the field, and the second red curve in the downstream section of the field. Proxy-variables used in fitting process are given on this figure.

**Figure 3.**Comparison between fitted (X-axis) and measured (Y-axis) soil water depletion in the two experimental sites and for all irrigations used during the fitting process. The continuous line is the 1:1 line, black squares are for the first site, black circles are for the second site.

**Figure 4.**Comparison between simulated (blue triangles) and measured (red squares) water depth hydrographs in the upstream and downstream sections of the field. Simulations were performed with the fitted set of soil parameters. On each figure the first curve (on the left part of the X-axis) was obtained on the upstream section, whereas the second curve (on the right part of the X-axis) was obtained on the downstream section. (

**a**) is for irrigation A1; (

**b**) for irrigation A2; (

**c**) for irrigation A3, (

**d**) for irrigation A5; (

**e**) for irrigation B6 and (

**f**) for irrigation B7.

**Figure 5.**Comparison of measured and simulated advance curves of the surface water front for irrigation A5. The continuous line in red is the 1:1.

**Figure 6.**Third step of the validation of the fitting process using independent measurement. In (

**a**) comparison of simulated and measured advance curves of the stream front: data presented with square and circle are for irrigations A6 and A7 studied by [5] respectively; data presented with diamond are those of irrigation A4. Simulations in A6 and A7 were performed with the fitted set of soil parameters in A3 and A2 respectively following the vegetation conditions; (

**b**) is the comparison between simulated (blue triangle) and measured (red squares) surface water flow hydrographs in the upstream (first curve on the left) and downstream (second curve on the right) sections of the field during irrigation A4. Simulations were performed with the fitted parameters of A5.

**Table 1.**Datasets for the monitored experimental irrigations. N is the label of the irrigation, L is the field length, W is the field width, I is the field slope, Q is the irrigation discharge per unit width of the field, ∆θ is the soil water depletion, Z is the soil depth, Ti is the irrigation duration, Date is the date of the irrigation relative to the mowing date of the field.

N | L | W (m) | I (‰) | Q (L·s ^{−1}·m^{−1}) | ∆θ (m ^{3}·m^{−3}) | Z (mm) | Ti (h) | Date | |
---|---|---|---|---|---|---|---|---|---|

(m) | |||||||||

1st site | A1 | 410 | 49 | 2.8 | 2.85 | 0.070 | 450 | 7.16 | 10 days before 1st mowing |

A2 | 3 | 0.066 | 7.15 | 48 days before 2nd mowing | |||||

A3 | 3.02 | 0.072 | 6.35 | 10 days after a 2nd mowing | |||||

A4 | 2.28 | 0.072 | 9.5 | 21 days after a 2nd mowing | |||||

A5 | 2.08 | 0.081 | 9.68 | 31 days after a 2nd mowing | |||||

2nd site | B6 | 60 | 4.8 | 1.48 | 0.097 | 200 | 7.12 | 20 days after a 2nd mowing | |

B7 | 1.65 | 0.103 | 6.45 | 39 days after a 2nd mowing |

**Table 2.**Boundary values of each parameter of interest. Initial values of each parameter are drawn within these boundary values at the beginning of the fitting process.

log [K_{S} (m·s^{−1})] | k (m·s^{−1/3}) | ∆θ (m^{3}·m^{−3}) | H_{0} | |
---|---|---|---|---|

Min | −7.2 | 1.5 | 0.06 | 0 |

Max | −5.6 | 4.5 | 0.12 | 20 |

**Table 3.**The three steps of the evaluation strategy of the fitting process. Step one (“coherence”) is used to check the physical meaning of the fitted parameters; step two (“evaluation with experiments used for the inverse modelling”) is a “weak validation” performed on data which were not used in the inversing process; Step three (“evaluation with experiments had not submitted in the inverse modelling”) is the validation sensu stricto.

Experiment | Coherence of Fitted Parameters | Evaluation with Data Didn’t Used in Fitting Process | Evaluation Using Data from Independent Experiments |
---|---|---|---|

A1, A2, A3, A5, B6, B7 | - Temporal Stability of Ks | - Measured/fitted ∆θ. | |

- f (Hay development, k). | - Measured/simulated H(t) | ||

A5 | - Measured/simulated streamflow advance curve. | ||

A4, A6, A7 [5] | - H(t) in A4 simulated with fitted parameters from A5 - Streamflow advance curve in A4, A6 and A7 using fitted parameters from A5, A3 and A2 respectively. |

**Table 4.**Irrigation parameters estimated for the different irrigations. Fitted values +/− one square root of the variance are given in brackets.

A1 | A2 | A3 | A5 | B6 | B7 | |
---|---|---|---|---|---|---|

Ks (m·s^{−1}) | 1.5 × 10^{−6}(1.08–2.07) × 10 ^{−6} | 1.34 × 10^{−6}(0.9–1.99) × 10 ^{−6} | 1.36× 10^{−6}(0.98–1.88) × 10 ^{−6} | 1.41× 10^{−6}(0.94–2.1) × 10 ^{−6} | 1.15× 10^{−7}(0.83–1.59) × 10 ^{−7} | 1.11× 10^{−7}(0.8–1.53) × 10 ^{−7} |

k (m·s^{−1/3}) | 2.94 (2.43–3.44) | 2.61 (2.15–3.06) | 3.54 (3.15–3.92) | 2.53 (1.99–3.06) | 1.69 (1.58–1.79) | 1.65 (1.45–1.85) |

∆θ (m^{3}·m^{−3}) | 0.071 (0.05–0.09) | 0.06 (0.04–0.08) | 0.078 (0.06–0.09) | 0.07 (0.05–0.09) | 0.1 (0.08–0.12) | 0.092 (0.07–0.11) |

H_{0} (mm) | 0.3 (0–6.3) | 1.9 (0–5.6) | 2.6 (0–6.13) | 4.2 (0–10.2) | 3.3 (0–7.1) | 2.6 (0–4.8) |

**Table 5.**Values of the performance criteria calculated during the verification step of the fitting process. CNASH is the Nash criterion. RMSE is the Root Mean Square Error between simulated and measured data.

Experiments | A1 | A2 | A3 | A5 | B6 | B7 | |
---|---|---|---|---|---|---|---|

Cnash (−) | Upstream | 0.99 | 0.85 | 0.86 | 0.89 | 0.85 | 0.96 |

Downstream | 0.97 | 1.00 | 0.98 | 0.94 | 0.98 | 0.94 | |

RMSE (mm) | Upstream | 3.63 | 10.12 | 8.18 | 10.29 | 8.61 | 4.13 |

Downstream | 3.78 | 1.57 | 3.41 | 4.63 | 3.55 | 4.74 |

**Table 6.**Values of the performance criteria calculated during the validation step of the fitting process. CNASH is the Nash criterion. RMSE is the Root Mean Square Error between simulated and measured data. RMSE units are m for the advance curve and mm for the hydrograph.

Streamflow Advanced Curve | Streamflow Hydrograph | ||||
---|---|---|---|---|---|

Experiments | A4 | A6 | A7 | Upstream | Downstream |

CNASH (−) | 1.00 | 1.00 | 1.00 | 0.90 | 0.95 |

RMSE (m) | 5.10 | 5.38 | 7.32 | 8.96 | 5.86 |

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## Share and Cite

**MDPI and ACS Style**

Alkassem Alosman, M.; Ruy, S.; Buis, S.; Lecharpentier, P.; Bader, J.C.; Charron, F.; Olioso, A.
An Improved Method to Estimate Soil Hydrodynamic and Hydraulic Roughness Parameters by Using Easily Measurable Data During Flood Irrigation Experiments and Inverse Modelling. *Water* **2018**, *10*, 1581.
https://doi.org/10.3390/w10111581

**AMA Style**

Alkassem Alosman M, Ruy S, Buis S, Lecharpentier P, Bader JC, Charron F, Olioso A.
An Improved Method to Estimate Soil Hydrodynamic and Hydraulic Roughness Parameters by Using Easily Measurable Data During Flood Irrigation Experiments and Inverse Modelling. *Water*. 2018; 10(11):1581.
https://doi.org/10.3390/w10111581

**Chicago/Turabian Style**

Alkassem Alosman, Mohamed, Stéphane Ruy, Samuel Buis, Patrice Lecharpentier, Jean Claude Bader, François Charron, and Albert Olioso.
2018. "An Improved Method to Estimate Soil Hydrodynamic and Hydraulic Roughness Parameters by Using Easily Measurable Data During Flood Irrigation Experiments and Inverse Modelling" *Water* 10, no. 11: 1581.
https://doi.org/10.3390/w10111581