# Comparison of Methods Predicting Advance Time in Furrow Irrigation

^{1}

^{2}

^{*}

## Abstract

**:**

_{L}) is a key issue for improving the efficiency and uniformity of irrigation. In this study, three methods are used for estimating the t

_{L}, and their results are compared with the experimental data of fifteen different furrows from the international literature. These methods are as follows: (a) the Valiantzas equation, (b) the method presented by Walker and Skogerboe, based on solving the volume balance equation by the Newton–Raphson iterative procedure and (c) the method of Philip and Farrell. The first two methods assume that the infiltration is described by the Lewis–Kostiakov equation and the extended Lewis–Kostiakov equation, respectively, while in the case of the Philip and Farrell method, the infiltration is described by the Philip equation and the Lewis-Kostiakov equation. The results showed that in most cases of the first two methods, the absolute relative error value of the predicted time t

_{L}was less than 10%. The Philip and Farrell method using the Lewis–Kostiakov infiltration equation underestimates the time t

_{L}and fails especially in the case where the volume of the surface water is not negligible compared to the total volume of water entering the system. The Valiantzas method is recommended because it was simpler and easier to use and showed greater prediction accuracy of t

_{L}, resulting in better planning of irrigation systems and contributing to water saving, which is currently a big issue.

## 1. Introduction

_{L}) are affected by the water supply at the inlet of the system, the roughness coefficient, the longitudinal slope of the system and the infiltration rate.

_{L}, i.e., the time that the advancing water front has reached the system end, plays a decisive role, because conventionally the irrigation time is usually taken as the sum of the time t

_{L}and the time that is required to infiltrate a water depth equal to the irrigation dose at the lower end of the field (t

_{a}). Thus, in the case where a water depth equal to the net irrigation dose (Z

_{n}) has been infiltrated at the lower end of the system, the degree of storage will be E

_{s}= 1 and the degree of efficiency E

_{a}will be equal to ${\mathrm{E}}_{\mathrm{a}}=\frac{{\mathrm{Z}}_{\mathrm{n}}\mathrm{L}}{{\mathrm{q}}_{\mathrm{o}}{\mathrm{t}}_{\mathsf{\delta}}}$, where ${\mathrm{q}}_{\mathrm{o}}$ is the inlet water rate, L is the furrow length and t

_{δ}is the total irrigation time, which is equal to ${\mathrm{t}}_{\mathsf{\delta}}={\mathrm{t}}_{\mathrm{L}}+{\mathrm{t}}_{\mathrm{a}}$.

_{L}from the furrow distance. Valiantzas [16] proposed algebraic equations to calculate the advance time as a function of inflow rate, without requiring iterative calculation procedures. A disadvantage of the method is perhaps that the time of advance versus distance relationship is described by three equations of different mathematical form [16].

_{s}) and is able to give adequate infiltration and advance behavior over all time scales.

_{L}, as well as a comparison of their results with the experimental data from 15 experimental fields presented in the international literature. In the first two methods, it is assumed that the infiltration follows the Lewis–Kostiakov equation, while in the third method, both the infiltration equations described by Philip [12] and Lewis–Kostiakov are examined. The comparison of the three methods predicting the water advance in furrow irrigation could help irrigation system designers to choose the most suitable method each time. The choice of a reliable, easy and quick estimation of t

_{L}will help, among other things, to save water, which is currently a big issue.

## 2. Materials and Methods

#### 2.1. Philip and Farrell Method [8]

_{0}(L

^{3}/T) is the constant inflow rate per furrow, c (L

^{2}) is the average section area of the stream flow, x (L) is the distance water has advanced along the field at time t, t

_{s}(T) is the value of t when water has arrived at location s behind the advancing front and i (L

^{3}/L) is the infiltration volume per unit length of furrow as a function of opportunity time t − t

_{s}.

_{s}corresponding to location s.

^{2}/T

^{0.5}) is the soil sorptivity and A (L

^{2}/T) is related to saturated hydraulic conductivity Κ

_{s}(L

^{2}/T) and varied 1/3 Κ

_{s}< A < 2/3 Κ

_{s}[18], then the solution of Equation [2] for c = 0 is

_{L}can be calculated by using any generalized unconstrained technique in which f(t

_{L}) is minimized to zero. In this study, the t

_{L}was estimated using Excel Solver provided with Microsoft Excel 365 [20,21]. Excel Solver is an easy-to-use tool because it requires no programming knowledge.

_{L}. Equation (4) can be transformed into the following equation where the only unknown parameter is t

_{L}:

_{L}by using the Solver tool:

- Step 1:
- Enter the values of the parameters q
_{0}, A, S and t_{L}into an Excel worksheet. The value t_{L}= 5A_{0}L/q_{0}can be used as an initial value of t_{L}, where A_{0}is the wetted cross-sectional area of a furrow. - Step 2:
- In a new cell, calculate the f(t
_{L}) using Equation (5). - Step 3:
- Go to the tools menu and click the Solver tool.
- Step 4:
- In “set objective”, set the cell created in step 2, then set it to receive the value zero according to Equation (5), and set the cell containing the value of t
_{L}as the Solver optimization variable. GRG nonlinear is chosen as the solution method. - Step 5:
- Press OK and obtain an optimal value of t
_{L}.

^{2}/T

^{a}) and α (-) are empirical coefficients, then the following analytical solution for c = 0 is obtained:

_{L}from Equation (7) is easier compared to Equation (4) since the calculation of t

_{L}can be performed explicitly from Equation (7) if the values of q

_{0}, k and α are known.

#### 2.2. Newton–Raphson Iterative Procedure

_{L}is expressed as q

_{0}t and is equal to the sum of the water volume flowing on the furrow surface and the water volume infiltrating into the soil according to the following relationship [3,22,23]:

_{0}(L

^{3}/T) is the inflow rate, x (L) is the distance of the advance water front at each time t (T), σ

_{y}(-) is the surface profile shape factor, usually equal to 0.77 [24], σ

_{Z}(-) is the subsurface shape factor, which ranges from 0.6 to 1, and the infiltration was assumed to follow the Lewis–Kostiakov equation, which is the most commonly used infiltration equation in surface irrigation models. A

_{0}is the cross-sectional area of the inlet flow and is calculated from the Manning equation [3]:

_{0}(-) is the longitudinal slope of the system (m/m).

_{Z}is calculated by the following equation [25]:

_{1}and ρ

_{2}, the advance distance x at time t, the longitudinal slope of the system (m/m) S

_{0}, the Manning roughness coefficient n, the inflow rate q

_{0}, the coefficients k and α of the Lewis–Kostiakov equation and the two unknown parameters t

_{L}and r. The parameter r is included in the calculation equation of the parameter σ

_{z}(Equation (10)). Equation (8) can be solved using the Newton–Raphson iterative process to estimate the advance time t

_{L}, as well as the coefficient r. The steps followed are as follows:

- (i)
- First, an initial value of the parameter r is entered, which varies between 0.3 and 0.9. The value ri = 0.5 is usually chosen.
- (ii)
- Then, the value of the parameter σz is calculated, as mentioned in Equation (10). It should be noted that the value of σz is recalculated every time the value of r changes.
- (iii)
- The Newton–Raphson iterative procedure is then applied to find the advance time t
_{L}using the initial value r_{i}as follows:- An initial estimate of t
_{L0}is created. The value t_{L0}= 5A_{0}L/q_{0}is usually considered as an initial value. - A better estimate of t
_{L}is t_{L1}given by the Newton–Raphson method using the relationship$${\mathrm{t}}_{\mathrm{L}1}={\mathrm{t}}_{\mathrm{L}0}-\frac{\mathrm{f}\left({\mathrm{t}}_{\mathrm{L}0}\right)}{{\mathrm{f}}^{\prime}\left({\mathrm{t}}_{\mathrm{L}0}\right)}={\mathrm{t}}_{\mathrm{L}0}-\frac{{\mathrm{c}}_{1}{{\mathrm{t}}_{\mathrm{L}0}}^{\mathrm{a}}+{\mathrm{c}}_{2}{\mathrm{t}}_{\mathrm{L}0}+{\mathrm{c}}_{3}}{{\mathrm{c}}_{1}\mathrm{a}{{\mathrm{t}}_{\mathrm{L}0}}^{\mathrm{a}-1}+{\mathrm{c}}_{2}}$$$$\mathrm{f}\left({\mathrm{t}}_{\mathrm{L}}\right)={\mathrm{c}}_{1}{{\mathrm{t}}_{\mathrm{L}}}^{\mathrm{a}}+{\mathrm{c}}_{2}{\mathrm{t}}_{\mathrm{L}}+{\mathrm{c}}_{3}=0$$_{1}= σ_{Z}kL, c_{2}= −q_{0}and c_{3}= σ_{y}A_{0}L. - Equation (13) is a transformed form of Equation (8) when x = L and t = t
_{L}. The initial estimate t_{L0}is compared with the value t_{L1}. If the values t_{L0}and t_{L1}do not differ greatly, the next step, step 4 is applied; otherwise, step 2 is repeated and the t_{L2}value is calculated using t_{L1}. The iterative process stops when two consecutive values converge. Empirically, three to four repetitions are sufficient. - The advance time at the distance x = L/2 is calculated accordingly for the initial value r
_{i}as described in steps 2 and 3, and the volume balance equation is applied by using the value L/2 instead of L.

- (iv)
- The value r
_{i+1}is calculated using the advance times t_{L}and t_{L/2}calculated from the previous steps 3 and 4 as follows:$${\mathrm{r}}_{\mathrm{i}+1}=\frac{\mathrm{ln}\frac{\mathrm{L}}{{\mathrm{L}/}_{2}}}{\mathrm{ln}\frac{{\mathrm{t}}_{\mathrm{L}}}{{\mathrm{t}}_{\frac{\mathrm{L}}{2}}}}$$$${\mathrm{r}}_{\mathrm{i}+1}=\frac{\mathrm{ln}2}{\mathrm{ln}\left(\frac{{\mathrm{t}}_{\mathrm{L}}}{{\mathrm{t}}_{\mathrm{L}/2}}\right)}$$ - (v)
- The initial estimate r
_{i}is compared with the value r_{i+1}(Equation (15)). If the values converge, then it is assumed that the time t_{L}is the estimated one. Otherwise, steps 2 to 4 are repeated using as a new initial value the value r_{i+1}.

_{0}in the furrow is too small and the length L is too long, then there is a failure to converge the values in the iterative process in step iii. In this case, when designing furrow irrigation, either q

_{0}must be increased or the furrow length L must be reduced. Also, if the value of the coefficient r is known from some other method, then only steps ii and iii are applied to calculate t

_{L}.

#### 2.3. Valiantzas Method [17]

^{2}/T

^{a}) and α (-) are parameters of the Lewis–Kostiakov equation, A

_{0}(L

^{2}) is the inlet flow area, L (L) is the length of the furrow, q

_{0}(L

^{3}/T) is the inflow rate and σ

_{ZF}(-) is the value of parameter σ

_{Z}(subsurface shape factor) at long times proposed by Hart et al. [26]:

_{L}generally does not exceed ±7%, while in exceptional cases where the advance time is less than 30 min, the maximum error can exceed 10%.

#### 2.4. Experimental Data

_{0}of the extended Lewis–Kostiakov infiltration function:

_{0}(L

^{2}/T) is the steady infiltration rate.

^{0.5}(t

^{0.5}) is linear. In this linear relationship, the slope of the line is equal to A and the constant term is equal to S [31]. From this analysis, it appeared that only in six cases the above relationship was linear. Then, for these six cases, the values of S and A were calculated using the Solver tool. As the initial values of S and A, we considered the values S = i

_{1}/t

_{1}

^{0.5}and A = (i

_{n}− i

_{n−1})/(t

_{n}− t

_{n−1}), where n is the last value of the data. It should be noted that the values of S and A were almost the same with both the linearization method and the Excel Solver procedure.

## 3. Results and Discussion

_{L}values in Equations (7) and (16), as well as in the iterative procedure. Also, Table 2 presents the values of the parameters S and A for six cases of data sets (i.e., Flowell wheel, Flowell non-wheel, Kimberly wheel, Kimberly non-wheel, Matchett M-1 and Printz P-1). In these data sets, the relationship i/t

^{0.5}(t

^{0.5}) is strongly linear, and the calculated values of S and A from Solver are positive. In the remaining cases, the relationship was not linear while the Solver calculated values A = 0, which have no physical meaning. Thus, the remaining cases were not studied further by applying Equation (4).

_{L}from the three methods studied, as well as the measured values for all experimental furrows. The absolute values of relative errors (|RE|%) of the predicted values of t

_{L}with respect to the measured ones are presented in Table 5.

_{L}in most cases. In more than half of the experimental furrows, the relative error is less than 10%, indicating that both methods predict the time t

_{L}fairly accurately. In general, the two methods converge on the t

_{L}value, and thus their deviation from the measured t

_{L}is approximately the same. The biggest differences in the two methods compared with the experimental data are observed in the Matchett data sets, where in all three cases (M1, M2 and M3) the RE values range from 11.96% to 25.81%. These deviations may be due to experimental errors related to inflow rate, roughness or furrow shape parameters. Possible problems with experimental measurements in these furrows were also reported by Valiantzas et al. [7].

_{L}. To explain these findings, it was investigated whether the amount of surface water is negligible. For this purpose, the index V

_{s}, which is equal to the ratio of the amount of surface water at the end of the advance phase to the total amount of water applied, was estimated in the 15 data sets.

_{s}values showed that the surface water was 5.1%, 3.6%, 9.7% and 1.4% of the total application water for the Flowell wheel, Flowell non-wheel, Kimberly wheel and Kimberly non-wheel furrows, respectively. That is, the surface water amounts are very small, with the exception of the Kimberly wheel case (9.7%) where the corresponding RE value is 16.55%. In the rest of the data sets, the values of the V

_{s}index are quite large and range from 7.1% to 40.9%. The highest RE values are observed in the cases of Greeley wheel and Cordoba, 59.24% and 51.59%, respectively, where the corresponding V

_{s}index values are 0.409 and 0.355. It may be assumed that the same causes play a role in the failure of the Philip and Farrell [8] method in combination with the Philip [12] infiltration equation.

_{L}(t

_{L,EXP}) and the predicted ones (t

_{L,PRED}) for the three methods. As can be seen, these relationships are linear with a very high value of determination coefficient (R

^{2}> 0.981). The Valiantzas [17] method gave the best results, since the slope of the linear relationship had the higher value (0.98), i.e., it is closer to the 1:1 line. In addition, it should be mentioned that this method is easy and simple to use, since the time t

_{L}is directly calculated from the furrow length L.

_{L}, RMSE (Root Mean Square Error) values for the three methods were also calculated. The RMSE values for the Valiantzas [17] and Newton–Raphson iterative procedure [3] methods were 22.14 min and 26.66 min, respectively, while for the Philip and Farrell [8] method in combination with the Lewis–Kostiakov equation the value was 43.19 min, which is almost twice the value of the other two methods.

## 4. Conclusions

_{L}is important. Especially currently, where water saving is a vital issue and the rational design of irrigation systems is required, the reliable and fast prediction of t

_{L}under various irrigation scenarios can help to optimize irrigation design.

_{L}, using experimental data of 15 different furrows from the international literature.

_{L}and failed especially in the case where the volume of surface water is not negligible. This is expected since the corresponding equation is obtained by considering the amount of surface water as negligible. These results can be very beneficial for irrigation system designers to study the irrigation performance of systems that are already working and to propose optimal solutions for each area.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Comparative presentation between the measured values of advance time, t

_{L,EXP}, and the predicted ones, t

_{L,PRED}, obtained from the (a) Valiantzas method [17], (b) Newton–Raphson iterative procedure [3] and (c) Philip and Farrell [8] method using the Lewis–Kostiakov infiltration equation in all experimental furrows studied.

Data Series | Walker and Busman [27] | Camacho et al. [29] | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Flowell Wheel | Flowell Non-Wheel | Kimberly Wheel | Kimberly Non-Wheel | Greeley Wheel | Cordoba | |||||||||

Inflow rate | q_{0} (m^{3}/min) | 0.12 | 0.12 | 0.09 | 0.048 | 0.114 | 0.09 | |||||||

Furrow slope | S_{0}(m/m) | 0.008 | 0.008 | 0.0104 | 0.0104 | 0.008 | 0.003 | |||||||

Manning roughness coefficient | n | 0.04 | 0.04 | 0.04 | 0.04 | 0.04 | 0.04 | |||||||

Furrow shape parameters | ρ_{1} | 0.3269 | 0.3269 | 0.6644 | 0.6644 | 0.369 | 0.39 | |||||||

ρ_{2} | 2.734 | 2.734 | 2.8787 | 2.8787 | 2.81 | 2.797 | ||||||||

Furrow length | L (m) = x_{2} | 360 | 274 | 360 | 112 | 411 | 200 | |||||||

Advance time | t_{L} (min) = t_{2} | 400 | 432 | 208 | 560 | 63 | 51.5 | |||||||

Advance distance and corresponding time | x_{1} (m) | 180 | 140 | 160 | 60 | 205.5 | 100 | |||||||

t_{1} (min) | 41 | 40 | 48 | 120 | 26 | 20.25 | ||||||||

Surface profile shape factor | σ_{y} | 0.77 | 0.77 | 0.77 | 0.77 | 0.77 | 0.77 | |||||||

Extended Lewis–Kostiakov parameters | α | 0.534 | 0.673 | 0.212 | 0.533 | 0.45 | 0.4550 | |||||||

k (m^{2}/min^{α}) | 0.0028 | 0.0022 | 0.0088 | 0.007 | 0.0021 | 0.0033 | ||||||||

f_{0} (m^{2}/min) | 0.00022 | 0.00022 | 0.00017 | 0.00017 | 0.0000 | 0.0000 | ||||||||

Data Series | Wilson and Elliot [28] | |||||||||||||

Benson B-1 | Benson B-2 | Benson B-3 | Matchett M-1 | Matchett M-2 | Matchett M-3 | Printz P-1 | Printz P-2 | Printz P-3 | ||||||

Inflow rate | q_{0} (m^{3}/min) | 0.1668 | 0.0684 | 0.0702 | 0.051 | 0.0552 | 0.0264 | 0.2886 | 0.2094 | 0.1662 | ||||

Furrow slope | S_{0} | 0.0044 | 0.0044 | 0.0044 | 0.0092 | 0.0095 | 0.0095 | 0.0023 | 0.0025 | 0.0025 | ||||

Manning roughness coefficient | n | 0.03 | 0.02 | 0.02 | 0.03 | 0.02 | 0.02 | 0.03 | 0.02 | 0.02 | ||||

Furrow shape parameters | ρ_{1} | 0.46 | 0.58 | 0.34 | 0.3 | 1.35 | 2.12 | 0.92 | 0.615 | 0.73 | ||||

ρ_{2} | 2.86 | 2.91 | 2.84 | 2.73 | 3 | 3.15 | 2.91 | 2.924 | 2.98 | |||||

Furrow length | L (m) = x_{2} | 500 | 500 | 500 | 400 | 400 | 400 | 200 | 300 | 300 | ||||

Advance time | t_{L} (min) = t_{2} | 175 | 344.5 | 247 | 124.3 | 232.2 | 213 | 178 | 45.5 | 73 | ||||

Advance distance and corresponding time | x_{1} (m) | 300 | 300 | 300 | 200 | 200 | 200 | 100 | 100 | 200 | ||||

t_{1} (min) | 84.5 | 159 | 123.5 | 38 | 70.5 | 88.2 | 13.5 | 15 | 43 | |||||

Surface profile shape factor | σ_{y} | 0.77 | 0.77 | 0.77 | 0.77 | 0.77 | 0.77 | 0.77 | 0.77 | 0.77 | ||||

Extended Lewis–Kostiakov parameters | α | 0.02 | 0.02 | 0.01 | 0.48 | 0.4 | 0.16 | 0.4 | 0.02 | 0.02 | ||||

k (m^{2}/min^{α}) | 0.0252 | 0.018 | 0.0173 | 0.0011 | 0.0033 | 0.0039 | 0.0078 | 0.013 | 0.0161 | |||||

f_{0} (m^{2}/min) | 0.00023 | 0.0001 | 0.00008 | 0.00003 | 0.00003 | 0.00002 | 0.00141 | 0.00049 | 0.0004 |

**Table 2.**Parameters of Lewis–Kostiakov equation (α and k) and Philip [12] equation (S and A) calculated with Solver tool.

Parameters | Walker and Busman [27] | Camacho et al. [29] | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Flowell Wheel | Flowell Non-Wheel | Kimberly Wheel | Kimberly Non-Wheel | Greeley Wheel | Cordoba | ||||||

α | 0.7534 | 0.7916 | 0.5143 | 0.6458 | 0.4500 | 0.4550 | |||||

k (m^{2}/min^{α}) | 0.0017 | 0.0018 | 0.0039 | 0.0050 | 0.0021 | 0.0033 | |||||

S (m^{2}/min^{0.5}) | 0.003105 | 0.003547 | 0.003916 | 0.007826 | - | - | |||||

A (m^{2}/min) | 0.000237 | 0.000355 | 0.000022 | 0.000205 | - | - | |||||

Parameters | Wilson and Elliot [28] | ||||||||||

Benson B-1 | Benson B-2 | Benson B-3 | Matchett M-1 | Matchett M-2 | Matchett M-3 | Printz P-1 | Printz P-2 | Printz P-3 | |||

α | 0.3533 | 0.3952 | 0.2767 | 0.5720 | 0.4731 | 0.2966 | 0.758 | 0.2929 | 0.3894 | ||

k (m^{2}/min^{α}) | 0.0103 | 0.0051 | 0.0077 | 0.0009 | 0.0027 | 0.0027 | 0.00576 | 0.0100 | 0.0083 | ||

S (m^{2}/min^{0.5}) | - | - | - | 0.5720 | - | - | 0.006103 | - | - | ||

A (m^{2}/min) | - | - | - | 0.000025 | - | - | 0.001296 | - | - |

**Table 3.**Calculated values of coefficient r from Equation (11) and the Newton–Raphson iterative procedure [3].

Coefficient r | ||
---|---|---|

Experimental Furrows | Equation (11) | Newton–Raphson Iterative Procedure |

Flowell wheel | 0.304 | 0.354 |

Flowell non-wheel | 0.282 | 0.297 |

Kimberly wheel | 0.553 | 0.574 |

Kimberly non-wheel | 0.405 | 0.382 |

Greeley Wheel | 0.783 | 0.763 |

Cordoba | 0.743 | 0.734 |

Benson B-1 | 0.702 | 0.701 |

Benson B-2 | 0.661 | 0.665 |

Benson B-3 | 0.737 | 0.764 |

Matchett M-1 | 0.585 | 0.606 |

Matchett M-2 | 0.582 | 0.582 |

Matchett M-3 | 0.786 | 0.749 |

Printz P-1 | 0.269 | 0.309 |

Printz P-2 | 0.990 | 0.735 |

Printz P-3 | 0.766 | 0.691 |

Advance Time t_{L} (min) | |||||
---|---|---|---|---|---|

Experimental Furrows | Measured Values | Valiantzas Method | Newton–Raphson Iterative Procedure | Philip and Farrell Method | |

Lewis–Kostiakov Equation | Philip Equation | ||||

Flowell wheel | 400 | 386.70 | 340.53 | 357.10 | 397.80 |

Flowell non-wheel | 432 | 479.51 | 404.48 | 456.86 | 616.97 |

Kimberly wheel | 208 | 201.87 | 198.13 | 173.57 | 175.31 |

Kimberly non-wheel | 560 | 558.46 | 534.69 | 547.60 | 577.78 |

Greeley Wheel | 63 | 61.47 | 61.04 | 25.67 | - |

Cordoba | 51.5 | 50.33 | 50.40 | 24.93 | - |

Benson B-1 | 175 | 181.86 | 180.44 | 143.07 | - |

Benson B-2 | 344.5 | 310.00 | 305.30 | 271.13 | - |

Benson B-3 | 247 | 235.54 | 231.84 | 195.37 | - |

Matchett M-1 | 124.3 | 92.22 | 94.11 | 60.41 | 59.44 |

Matchett M-2 | 232.2 | 204.44 | 202.08 | 181.33 | - |

Matchett M-3 | 213 | 176.85 | 174.30 | 143.13 | - |

Printz P-1 | 178 | 160.31 | 156.95 | 145.62 | 533.71 |

Printz P-2 | 45.5 | 52.22 | 51.25 | 32.53 | - |

Printz P-3 | 73 | 79.35 | 79.08 | 57.29 | - |

**Table 5.**Absolute values of relative error (|RE|) between the measured and predicted values of advance time t

_{L}from the (a) Valiantzas method [17], (b) Newton-Raphson iterative procedure [3] and (c) Philip and Farrell [8] method using the Lewis–Kostiakov and Philip [12] infiltration equations in all experimental furrows studied.

|RE| (%) | ||||
---|---|---|---|---|

Experimental Furrows | Valiantzas Method | Newton–Raphson Iterative Procedure | Philip and Farrell Method | |

Lewis–Kostiakov Equation | Philip Equation | |||

Flowell wheel | 3.33 | 14.87 | 10.72 | 0.55 |

Flowell non-wheel | 11.00 | 6.37 | 5.76 | 42.82 |

Kimberly wheel | 2.95 | 4.74 | 16.55 | 15.72 |

Kimberly non-wheel | 0.27 | 4.52 | 2.21 | 3.17 |

Greeley wheel | 2.42 | 3.11 | 59.24 | - |

Cordoba | 2.27 | 2.13 | 51.59 | - |

Benson B-1 | 3.92 | 3.11 | 18.25 | - |

Benson B-2 | 10.01 | 11.38 | 21.30 | - |

Benson B-3 | 4.64 | 6.14 | 20.90 | - |

Matchett M-1 | 25.81 | 24.29 | 51.39 | 30.83 |

Matchett M-2 | 11.96 | 12.97 | 21.91 | - |

Matchett M-3 | 16.97 | 18.17 | 32.80 | - |

Printz P-1 | 9.94 | 11.83 | 18.19 | 199.84 |

Printz P-2 | 14.76 | 12.63 | 28.50 | - |

Printz P-3 | 8.70 | 8.33 | 21.51 | - |

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**MDPI and ACS Style**

Kargas, G.; Koka, D.; Londra, P.A.; Mindrinos, L.
Comparison of Methods Predicting Advance Time in Furrow Irrigation. *Water* **2024**, *16*, 1105.
https://doi.org/10.3390/w16081105

**AMA Style**

Kargas G, Koka D, Londra PA, Mindrinos L.
Comparison of Methods Predicting Advance Time in Furrow Irrigation. *Water*. 2024; 16(8):1105.
https://doi.org/10.3390/w16081105

**Chicago/Turabian Style**

Kargas, George, Dimitrios Koka, Paraskevi A. Londra, and Leonidas Mindrinos.
2024. "Comparison of Methods Predicting Advance Time in Furrow Irrigation" *Water* 16, no. 8: 1105.
https://doi.org/10.3390/w16081105