Introduction

Optimal utilization and equitable allocation of water in furrow irrigation depend on various parameters, such as inflow rate, soil texture, furrow slope, soil permeability, vegetation, roughness coefficient, and irrigation management (Amiri et al. 2016; Slatni et al. 2023). The roughness coefficient, also known as flow resistance, represents the shear stress and resistance force caused by lateral elements like soil surface roughness and vegetation. This force acts on the water at the base of the furrow, opposing its movement and contributing to roughness (Sepaskhah and Bondar 2002). To interpret the roughness, various coefficients from equations like Chezy, Manning, and Darcy can be employed (D’ippolito et al. 2021). The Manning equation is commonly chosen by irrigation engineers and researchers for estimating flow rate in open channels and surface irrigation due to its straightforward application and satisfactory level of accuracy.

The Manning roughness coefficient is a crucial factor in the design and simulation of surface irrigation systems (Seifi et al. 2023). Overestimating the roughness of the channel surface can lead to an underestimation of flow velocity, resulting in significant errors in the design and simulation of surface irrigation systems (Clemmens et al. 2001). The accurate determination of the roughness coefficient is challenging due to the spatial and temporal variability of soil and hydraulic parameters of flow. Additionally, the large variations in water depth and velocity in irrigation furrows, caused by the low flow rates, further complicate the process of determining the roughness coefficient (Li and Zhang 2001; Sedaghatdoost and Ebrahimian 2015). Several researchers have directly determined the roughness coefficient in furrow irrigation by measuring hydraulic parameters and solving the Manning equation for the coefficient (Kamali et al. 2018; Mailapalli et al. 2008; Ramesh & Ostad-Ali-Askari 2023). While this equation is primarily used to assess the accuracy of direct and indirect models and methods of roughness estimation due to the time-consuming nature of determining the various parameters involved, researchers have explored new methods for estimating the roughness coefficient. They have utilized current equations and numerical as well as inverse solutions of these equations for surface irrigation purposes. Various models have been developed by researchers to simulate flow in surface irrigation, infiltration coefficients, and Manning roughness coefficients. These models include INFILT (McClymont and Smith 1996), EVALUE (Strelkoff et al. 1999), SURDEV (Jurriens et al. 2001), SIRMOD (Walker 2003), IPARM (Gillies and Smith 2005), SIPAR_ID (Rodríguez and Martos 2008), WinSRFR (Bautista et al. 2009b), SIDES (Adamala et al. 2014), SORCOS (Burguete et al. 2014), and SISCO (Gillies and Smith 2015). The effectiveness of these models has been assessed in various studies, such as the evaluation conducted by Nie et al. (2014b, a), which examined the accuracy of the SIPAR_ID model in estimating infiltration parameters and Manning roughness coefficients in blocked-end furrow irrigation. The results indicated that this model accurately simulated infiltration parameters and roughness coefficients. Furthermore, Kamali et al. (2018) conducted a study to evaluate the Manning roughness coefficient value in furrow irrigation for maize cultivation. They also assessed the accuracy of the multilevel optimization method and SIPAR_ID in estimating the Manning roughness coefficient. The findings of the study indicated that the multilevel optimization method, with an average relative error of 5%, outperformed other methods in all irrigation events. In a related study, Weibo et al. (2012) proposed a novel approach for estimating the parameters of the infiltration equation and Manning roughness coefficient in border irrigation. This method utilized the Philip equation and volume balance and was compared against the results obtained from WinSRFR software (Bautista et al. 2009b). The authors reported the method's performance as satisfactory. Additionally, Dewedar et al. (2019) examined the efficacy of WinSRFR software in predicting the performance parameters of furrow irrigation with varying lengths and slopes in Egypt. The results demonstrated the software's effectiveness in simulation.

The roughness coefficient is commonly defined as an input parameter in various software packages, assuming constant infiltration and cross-sectional channel shape throughout the growth season (Seyedzadeh et al. 2019). However, these assumptions may lead to significant variations in advance time, infiltrated water depth, and other parameters across different field locations (Childs et al. 1993; Al-Janabi et al. 2019). Therefore, investigating the interaction between the roughness coefficient and different field parameters is crucial, as highlighted in several studies. For instance, studies by Walker (2005), Bautista et al. (2009a), Nie et al. (2014a, 2018), Sedaghatdoost and Ebrahimian (2015), Salahou et al. (2018), Smith et al. (2018), Kamali et al. (2018), Xu et al. (2019), and Seyedzadeh et al. (2019) have explored the impact of the roughness coefficient on advance time, recession time, and water infiltration in soil. Furthermore, research by Trout (1992b), Esfandiari and Maheshwari (1998), Sepaskhah and Bondar (2002), Mailapalli et al. (2008), Zheng et al. (2012), Ramezani Etedali et al. (2012), Ibrahim and Abdel-Mageed (2014), Ebrahimian (2014), Termini and Moramarco (2017), Kamali et al. (2018), Salah Abd Elmoaty (2020), and Mazarei et al. (2021) has investigated the influence of vegetation, slope, hydraulic radius, and inflow rate on the Manning roughness coefficient. Among these studies, Sepaskhah and Bondar (2002), Mailapalli et al. (2008), Ramezani Etedali et al. (2012), Kamali et al. (2018), and Mazarei et al. (2021) emphasized the impact of slope and flow rate, while Sepaskhah and Bondar (2002), Mailapalli et al. (2008), Kamali et al. (2018), and Salah Abd Elmoaty and T. A (2020) focused on vegetation. Additionally, Trout (1992b), Ibrahim and Abdel-Mageed (2014), and Termini and Moramarco (2017) highlighted the significance of hydraulic radius and flow velocity in influencing the Manning roughness coefficient. However, Zheng et al. (2012) found the relationship between roughness and slope to be insignificant, and Esfandiari and Maheshwari (1998) suggested this relationship to be negligible at flow rates exceeding 0.7 L/s.

As indicated, the roughness coefficient, a key soil property characteristic, is subject to influence by various factors including vegetation, slope, hydraulic radius, and inflow rate (Harun-ur-Rashid 1990; Trout 1992a; Kamali et al. 2018; Tahmid et al. 2021; Takata et al. 2024). Altering any of these parameters leads to changes in flow conditions and subsequently affects the roughness coefficient. Therefore, it is crucial to investigate the temporal and spatial variations and the interactive impact of the Manning roughness coefficient with different field parameters. While many studies have focused on temporal changes (variations in roughness coefficient during different irrigation events) of the Manning roughness coefficient (Sepaskhah and Bondar 2002; Sepaskhah and Shaabani 2007; Mailapalli et al. 2008; Alkassem Alosman et al. 2018; Guzmán-Rojo et al. 2019; Seyedzadeh et al. 2019; Mazarei et al. 2021; Ramesh and Ostad-Ali-Askari 2023), they have overlooked the examination of alterations in the Manning roughness coefficient and its values during various phases of irrigation within each irrigation event. Therefore, this study investigated the Manning roughness coefficient in bare furrows during the advance, storage, and total irrigation phases across three irrigation events. Furthermore, this research explored the impact of inflow rate and furrow slope on the Manning roughness coefficient. Additionally, the study analyzed the relationship between Manning roughness coefficient, flow cross-sectional area, and advance time during different irrigation phases in three irrigation events to explore the interactions among these parameters.

Materials and methods

Field experiments

Field experiments were conducted at the experimental station of the College of Agriculture and Natural Resources, University of Tehran, located in Karaj, Iran, at a longitude of 50°57′30.2 and a latitude of 35°48′18.5 N. Two fields with distinct soil textures were chosen for the study. The first field, denoted as F, had a history of long-term cultivation, while the second field, named E, had not been cultivated for several years. The soil characteristics of these fields are detailed in Table 1. In this study, we investigated the effects of inflow rate, initial soil moisture, different irrigation events, and phases on the roughness coefficient. Therefore, six inflow rates (corresponding to the conditions of the furrow and maximum non-erosive inflow rate), three irrigation events (first to third irrigation), advance (the advance phase starts from the moment water enters the field and ends when the water reaches the end of the field) and storage (the period of time between the end of the advance phase and the cut-off of the inflow) phases, two irrigation intervals (5 and 10 days, the values of soil moisture for each interval are presented in Tables 2 and 3), and two types of soil texture were studied. Hence, in field E, three low inflow rates (with an average value of 0.27 L/s) and three high inflow rates (with an average value of 0.57 L/s) were considered in each of the 5 and 10-irrigation intervals, in consecutive irrigation events (first to third irrigation event). Similar treatments were also investigated in field F. To achieve a more accurate and comprehensive analysis, additional data were required. Therefore, the 10-day irrigation interval in field F considered six low inflow rates (with an average of 0.27 L/s) and six high inflow rates (with an average of 0.57 L/s), instead of the previous three low and three high inflow rates. Given the significant time and cost involved in data acquisition, the increase in the number of data was not applied in field E. As a result, a total of 90 furrows were examined in three consecutive events, with 18 furrows in field F and 12 furrows in field E. Noting that in each of these experiments, the advance and storage phases of irrigation were investigated separately because of their significance and impact on the Manning roughness coefficient.

Table 1 Soil properties of the experimental fields
Full size table
Table 2 Data summary of the experimental furrows for the field E
Full size table
Table 3 Data summary of the experimental furrows for the field F
Full size table

A 2000-L tank was located to maintain consistent inflow rates, and water was conveyed to the furrows through polyethylene pipes. In both fields, furrows measuring 75 cm in width and 20 m in length were established. Within these furrows, 9 nails were strategically placed at 2 m intervals in the soil to monitor the advance and recession times. To assess the inflow and outflow rates, Washington State College (WSC) flumes type 2 were set up at the commencement and end of each furrow. The water depth and subsequent flow rate in these flumes were recorded at 5-min intervals during the first hour (when the soil nearly reaches a constant infiltration rate) and at 10-min intervals thereafter until the end of the irrigation event. Additionally, four rulers spaced 4 m apart were installed along the furrows to gage water depth throughout irrigation, with readings taken concurrently with those from the flumes. The irrigation duration for all furrows was approximately 180 min. The slope of the furrow base was assessed prior to the first irrigation event.

Soil moisture was determined before each irrigation event by oven drying at 105 °C for 24 h. Three samples were extracted from the soil surface layer (0–10 cm) within each furrow, and the average moisture content was reported in this study.

The measurement of the furrow cross section (Af) was conducted using a profile meter (Walker 1987) at three specific points: the upstream section (2 m from the beginning of the furrow), the middle section (10 m from the beginning of the furrow), and the downstream section (18 m from the beginning of the furrow) both before and after each irrigation event. The flow cross section (Aflow) and the wetted perimeter (Pw) of the flow within each furrow were determined by measuring the flow depth using rulers and a profile meter at various time points (Fig. 1). The average length of the nails on each profile meter was calculated for the upstream, middle, and downstream sections of the furrow before irrigation, and a similar procedure was repeated for the post-irrigation measurements. The following equations were utilized to calculate the flow cross section (Aflow) and the wetted perimeter (Pw):

$$a_{i} = ({{\text{Max}}({l_{1} ,l_{2} ,l_{3} \ldots l_{n} })}) - L$$
(1)

where ai represents the length of the profile meter nail that extends above the water surface (cm), L denotes the depth of water in the furrow (cm), and li indicates the length of each profile meter nail within the furrow (cm).

$$b_{i} = l_{i} - a_{i}$$
(2)
Fig. 1
figure 1

Schematic view of furrow profile

Full size image

Utilizing Eq. (2), bi was computed for all profile meter nails both pre- and post-irrigation. Subsequently, the area between the two nails was determined by applying the trapezoid area formula and denoted as Ai. Finally, the cross-sectional area for flow was ascertained by employing Eq. (3).

$$A_{{{\text{flow}}}} = \mathop \sum \limits_{i = 1}^{n} A_{i}$$
(3)

where Aflow represents the flow cross section (cm2) and Ai denotes the area between the two profile meter nails (cm2).

The wetted perimeter of the flow was determined by calculating the total length of the larger base of each trapezoid (Eq. 4).

$$P_{w} = \mathop \sum \limits_{i = 1}^{n} p_{i}$$
(4)

where Pw, is the wetted perimeter of the furrow (cm) and pi is the length of the large edge of each trapezoid (cm).

Upon establishing the cross-sectional area and wetted perimeter of the flow both before and after irrigation, the average values were computed and reported (Tables 2 and 3).

Manning roughness coefficient determination

Manning equation and similar equations have primarily been developed for fully turbulent and deep flows. Consequently, the utilization of these equations in surface irrigation, where flow depth is shallow and flow characteristics undergo significant alterations due to rough elements, is questionable (Maheshwari 1992). Nevertheless, due to their user-friendly nature, researchers employ the Manning equation to model flow in surface irrigation. In this equation, the Manning roughness coefficient (n) serves as the water resistance factor and varies based on factors such as soil type, aggregates, flow depth, hydraulic radius, among others. Consequently, its value is anticipated to fluctuate across different locations and time periods. Clemmens et al. (2001) noted that the Manning roughness coefficient diminishes over time due to aggregate softening and gap filling in the furrow bed, with the roughness coefficient typically being higher during the advance phase compared to other irrigation phases. Assuming a constant roughness coefficient for determining the advance and recession phases can lead to numerous inaccuracies in parameter estimation. Therefore, this study separately determined the Manning roughness coefficient in the advance, storage, and total irrigation phases. The method of determining the roughness coefficient in each phase is elaborated upon.

Advance phase (n a)

According to the assumptions of the Manning equation, its application in irrigation is limited due to infiltration and the resulting nonuniformity of flow. Despite this limitation, researchers often resort to using this equation to characterize the hydraulic behavior of flow in surface irrigation after the stabilization of water infiltration in the soil (reaching base infiltration). During the advance phase, the fluctuating and decreasing rate of water infiltration in the soil makes it impractical to determine the flow depth, cross-sectional area, and wetted perimeter accurately. Consequently, direct application of the Manning equation to describe the hydraulics of flow may introduce significant errors. In this context, the SIPAR_ID model is proposed as an alternative to the Manning equation for determining the Manning roughness coefficient during the advance phase. By utilizing data from the advance phase, including inflow rate and flow depth at various time points (Rodríguez and Martos 2010), this model offers a more suitable approach for estimating the Manning roughness coefficient in the advance phase.

The SIPAR_ID model

Rodríguez and Martos (2010) introduced a multi-objective inverse optimization method to estimate the parameters of the Kostiakov infiltration equation and the Manning roughness coefficient. This method integrates a combined model of volume balance, artificial neural network, and differential evolution optimization algorithm. To facilitate this process, they developed the SIPAR_ID software. The model is designed to estimate the infiltration parameters and Manning roughness by minimizing the sum of the squares of the differences between the observed and simulated flow advancement values. Apart from the inflow rate and advance data, the model necessitates information on cross-sectional geometry and flow depth at a specific location. The objective functions incorporated in this model are as follows:

$${\text{OF}}_{1} = {\text{Min}}. {\text{SSE}}_{{\text{Advance Distance}}} = \mathop \sum \limits_{i = 1}^{{N_{s} }} \left( {X_{i} - \overline{X}_{i} } \right)^{2}$$
(5)
$${\text{OF}}_{2} = {\text{Min}}. {\text{SSE}}_{{\text{Flow Depth}}} = \mathop \sum \limits_{t = 1}^{{N_{t} }} \left( {Y_{t} - \overline{Y}_{t} } \right)^{2}$$
(6)

where SSE is the sum of the squares of error, Xi and are the measured and estimated advance of the model, Yt and Y are measured and estimated flow depth of the model at a specific station at different times, Ns represents the number of flow advance measurement stations, while Nt indicates the number of flow depth measurements at different times for a specific station. The final objective function in the model is presented in Eq. (7).

$${\text{OF}}_{{{\text{Aggre}}{.}}} = w\left( {{\text{OF}}_{1} } \right) + \left( {1 - w} \right)\left( {{\text{OF}}_{2} } \right)$$
(7)

where w is the weight parameter. The SIPAR_ID model, apart from necessitating inflow rate and advance data, requires cross-sectional geometry and flow depth data from a specific station. The optimization model’s decision variables encompass the parameters of the Kostiakov infiltration equation and the Manning roughness coefficient. Furthermore, a defined range is established for both the variables and constraints within the optimization model.

Storage phase (n s)

The storage phase commences once the water reaches the end of the furrow and persists until the inflow ceases. Observations indicate that approximately one hour after the initiation of irrigation, the rate of water infiltration into the soil stabilizes, while the outflow rate from the furrows (maintaining a constant inflow) remains consistent. Subsequently, it can be inferred that a state of uniform flow in the furrow is established, and the Manning roughness coefficient was determined using the Manning equation (Eq. 8) as a representation of the storage phase (from achieving the basic infiltration rate until the irrigation cutoff time).

$$Q = \frac{A}{n} \times R^{\frac{2}{3}} \times S_{0}^{\frac{1}{2}}$$
(8)

where Q represents the flow rate (m3/s), A denotes the flow cross-sectional area (m2), n stands for the Manning roughness coefficient (s/m1/3), R signifies the hydraulic radius (m), and S0 represents the bottom slope (or hydraulic gradient) of the furrow (m/m).

To enhance the accuracy in estimating the Manning roughness coefficient during this phase, the roughness coefficient at the beginning and the end of the furrow was computed separately. For this purpose, the cross-sectional area and wetted perimeter at the upstream location were determined utilizing a profile meter and Eqs. (1–4). Subsequently, the Manning roughness coefficient was determined at the upstream position at 10-min intervals until the end of irrigation (cutoff time). The average roughness coefficient within these time intervals was regarded as the roughness of the storage phase at the upstream section. Likewise, by analyzing the cross section of the downstream area and its outflow, the roughness coefficient was evaluated at various time intervals, and its average value was calculated. Ultimately, the average roughness coefficient at the beginning and the end of irrigation was reported.

The whole irrigation event (n t)

The Manning roughness coefficient for the whole irrigation event was determined utilizing the WinSRFR software. To achieve this, all necessary input data, including the parameters of the Kostiakov-Lewis infiltration equation (Eq. 9) (a, k, and f0), as well as the Manning roughness coefficient, were provided to the software as initial values for a, k, f0, and n. Subsequent simulations were conducted. To assess the software's performance, the simulated values of advance and recession times, and the volume of infiltrated water, were compared with the measured values to ascertain the extent of simulation error. If the simulation error exceeded 5%, the software underwent calibration by adjusting the values of the infiltration parameters of the Kostiakov-Lewis equation and the Manning roughness coefficient to minimize errors in estimating advance and recession times, as well as the volume of infiltrated water.

$$Z = kt^{a} + f_{0} t$$
(9)

where Z represents the cumulative infiltration (mm), t denotes the infiltration time (min), k (mm/mina) and a (dimensionless) are constants, and f0 (mm/min) stands for the basic infiltration rate.

The WinSRFR software

The WinSRFR software is a one-dimensional mathematical model utilized for simulating, evaluating, and designing various surface irrigation methods and regimes (Bautista et al. 2009b). The software is founded on the numerical solution of the equations of Saint–Venant. This software has the ability to design, simulate and evaluate different irrigation regimes and comprises four different components as follows: (1) an unsteady flow simulation model for predicting surface and subsurface water flow under various boundary conditions, (2) a tool for assessing irrigation system performance and estimating infiltration and roughness based on field measurements, (3) a tool for designing diverse surface irrigation systems, and (4) a tool for optimizing irrigation system performance (Bautista et al. 2009b).

Evaluation

Several indices, such as the coefficient of determination (R2) (Eq. 10), root mean square error (RMSE) (Eq. 11), normalized root mean square error (NRMSE) (Eq. 12), and relative error (RE) (Eq. 13), were employed to determine the accuracy of estimating the Manning roughness coefficient.

$$R^{2} = \left( {\frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {X_{i} \times Y_{i} } \right) - N \times \overline{{X_{i} }} \times \overline{{Y_{i} }} }}{{\sqrt {\left( {\mathop \sum \nolimits_{i = 1}^{n} X_{i}^{2} - N \times \overline{X}^{2} } \right)(} \mathop \sum \nolimits_{i = 1}^{n} Y_{i}^{2} - N \times \overline{Y}^{2} )}}} \right)$$
(10)
$${\text{RMSE}} = \sqrt {\frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {X_{i} - Y_{i} } \right)^{2} }}{N}}$$
(11)
$${\text{NRMSE}} = \frac{{\sqrt {\frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {X_{i} - Y_{i} } \right)^{2} }}{N}} }}{{\overline{X}}}$$
(12)
$${\text{RE}} = \frac{{Y_{i} - X_{i} }}{{X_{i} }}$$
(13)

where xi is the measured data, Yi is estimated data, N is total number of experimental data, X is average measured data, and Y is average estimated data.

During the storage phase, it was presumed that the measurement error associated with field data was minimal. Consequently, the Manning equation was applied directly, and the values derived from this equation were adopted as the Manning roughness coefficient for this phase.

The relationship between manning roughness and field parameters

The Manning roughness coefficient interacts with various hydraulic and field parameters. Changes in flow rate, slope, and hydraulic radius can influence the Manning roughness coefficient. Conversely, alterations in the Manning roughness coefficient can impact advance and recession times, infiltrated water volume, among other factors. This study investigates the correlation between the roughness coefficient and these parameters during different phases of irrigation. The Manning roughness coefficient was determined in the advance, storage, and whole irrigation phases. The study examined the correlation between the Manning roughness coefficient and experimental parameters such as flow rate (Q), initial soil moisture (M), irrigation event number (IE), slope (S0), furrow cross section (Af), flow cross-sectional area (Aflow), wetted perimeter (P), basic infiltration rate (f0), advance time (Tadv), and recession time (Trec). SPSS (IBM SPSS Statistics for Windows, Version 22.0) was utilized to analyze the relationship between these parameters and the roughness coefficient in the advance (na), storage (ns), and whole irrigation (nt) phases. The relation between the Manning roughness coefficient and irrigation events was assessed using the Kendall test (Kendall 1990) due to their rank nature, while other quantitative parameters were evaluated using the Pearson test (Kirch 2008).

Results and discussion

Manning roughness coefficient values

Advance phase

Table 4 shows the values of the evaluation indices for estimating the advance curve in fields E and F. The mean values of the R2, RMSE, and RE indices in both fields were 0.995, 0.363 min, and 4.24%, respectively, indicating the SIPAR_ID model's high accuracy in estimating the advance curve. Previous studies by Ramezani Etedali et al. (2011), Nie et al. (2014b, 2018), and Mazarei et al. (2021) have also confirmed the acceptable accuracy of the SIPAR_ID model in estimating advance time.

Table 4 Evaluation indices of SIPAR_ID model in estimating flow advance curve in the first to third irrigations in the fields E and F
Full size table

Table 5 presents the minimum, maximum, and average Manning roughness coefficients during various phases and irrigation events in fields E and F. It was observed that the Manning roughness coefficient during the advance phase in field F was lower than that in field E, potentially attributed to the heavier soil texture in field F. Ramezani Etedali et al. (2012) highlighted the significant impact of soil texture on the Manning roughness coefficient, indicating that light soils tend to have a higher roughness coefficient compared to heavy soils. Moreover, the number of aggregates formed in the field was noted to influence the Manning roughness coefficient, particularly during the advance phase. In field E, where perennial cultivation was absent, a greater number and size of aggregates were created compared to the other field, potentially contributing to the higher roughness coefficient in field E relative to field F. The reduction in the average inflow rate in field E compared to field F led to an increase in the average Manning roughness coefficient for field E (Table 5). Sepaskhah and Bondar (2002), Mailapalli et al. (2008), Kamali et al. (2018), and Yilmaz et al. (2023) also underscored the decrease in inflow rate as a significant factor influencing the increase in the Manning roughness coefficient.

Table 5 Maximum, minimum and average values of Manning roughness coefficient in advance phase, storage phase, and whole irrigation event
Full size table

Manning roughness coefficient during the advance phase exhibited a decrease as the number of irrigation events increased, as shown in Table 5. This decline can be attributed to the wetting and drying cycle of the soil between successive irrigation applications, leading to the dissolution of clods. Additionally, the eroded and dissolved soil tends to settle in the cracks on the soil surface, causing the surface layer to become wetter and softer after each irrigation. Consequently, this process results in a decrease in the roughness coefficient. Similar explanations for the reduction in roughness coefficient during various irrigation events have been noted by Harun-ur-Rashid (1990), Clemmens et al. (2001), Kassem and Ghonimy (2011), and Amiri et al. (2016).

It is noteworthy that in field F, the average Manning roughness coefficient during the second irrigation exceeded that of the third irrigation, potentially attributed to variations in the furrow inflow rates. As depicted in Table 5, the average inflow during the second irrigation decreased in comparison to the third irrigation in field F, likely resulting in a higher Manning roughness coefficient during the third irrigation than the second irrigation.

The storage phase

The mean Manning roughness coefficient of the storage phase was 0.073 for field E and 0.041 for field F (Table 5). Similar to the advance phase, a higher roughness coefficient was noted in the storage phase of field E compared to field F.

Figure 2 illustrates the trend of the Manning roughness coefficient during the storage phase in field E. The coefficient decreased as the frequency of irrigation events increased. This trend can be attributed to the wetting cycle of the furrow caused by irrigation, similar to the observations in the advance phase. Comparable results were also noted in field F.

Fig. 2
figure 2

Average of Manning roughness coefficient in all irrigation phases at three irrigation events (first to third) in the fields E and F

Full size image

Whole irrigation event

The WinSRFR software's accuracy in estimating the Manning roughness coefficient throughout the whole irrigation event was assessed by comparing the error percentages of advance and recession times and the infiltrated water volume. Table 6 presents the average error percentages of these parameters in two fields (E and F) over three consecutive irrigation events. The determination of advance and recession phases and infiltrated water volume in both fields yielded average values of RE, RMSE, and NRMSE as follows: 0.5%, 0.1 min, 1% for Field E and 1.2%, 3.5 min, 1.8% for Field F, and 0.6%, 5.4 L, 0.5% for both fields, respectively. These results indicate the software's effective performance in estimating the Manning n coefficient. Mazarei et al. (2020), Fadul et al. (2020), and Mehri et al. (2023) also reported the satisfactory performance of the WinSRFR software in estimating infiltration and roughness parameters and simulating water flow in surface irrigation.

Table 6 Evaluation indices of the WinSRFR software in estimating advance and recession times, and the total infiltrated volume
Full size table

The findings indicated that the Manning roughness coefficient was lower throughout the whole irrigation event as in the advance and storage phases in field F. Conversely, in field E, the Manning roughness coefficient was highest during the initial irrigation, decreasing as the number of irrigation events increased. However, in field F, this trend of changes resembled that of the advance and storage phases.

Comparing the manning coefficient across various irrigation phases

The roughness coefficient during the advance phase exhibited the highest value compared to the other two phases in both fields. This difference could be attributed to factors such as the size and number of clods, lower soil moisture levels, and higher infiltration rates during the advance phase in contrast to the storage phase and the whole irrigation event (which encompasses all three phases). The findings indicated that the advance phase had the highest roughness values. The roughness coefficient values throughout the entire irrigation event fell between those of the advance and storage phases. However, due to the limited contribution of the advance phase to the total irrigation event, the roughness values for the entire event closely resembled those of the storage phase. Given the short length of the furrows, the advance-to-total irrigation time ratio was small, resulting in a minimal impact of the roughness coefficient of the advance phase on the whole irrigation event. It can be inferred that when the advance time-to-whole irrigation time ratio is low, the roughness coefficient of the storage phase or the whole irrigation event can be reliably used as the roughness coefficient for the furrow during the irrigation event. However, Mailapalli et al. (2008) have shown that the Manning roughness coefficient tends to decrease over time due to clod softening and crack filling in the furrow bed. Typically, the roughness coefficient is higher during the advance phase compared to the other irrigation phases. Therefore, assuming a constant roughness coefficient for determining the advance and recession phases may lead to inaccuracies in estimating advance time and water distribution during irrigation.

Hence, in extended furrows where the proportion of advance time to the total irrigation duration is significant, it is recommended to determine the roughness of distinct irrigation phases individually. Subsequently, employing the weighted average of roughness values from various irrigation phases is suggested for estimating the overall roughness of the whole irrigation event.

There were elevated values of the roughness coefficient observed during the initial irrigation. For instance, the roughness coefficient of E5-1-1 was estimated as 0.636 during the advance phase, which was deemed unexpected. This particular furrow exhibited numerous aggregates, a byproduct of tillage operations, based on the field parameters measured. Additionally, the low inflow rate (0.11 L/s) resulted in reduced flow velocity, leading to an escalation in the Manning roughness coefficient (Abdullah et al. 2023). This underscores the significant influence of inflow and aggregates on the Manning roughness coefficient. However, in several studies (Harun-ur-Rashid 1990; Nie et al. 2014a, 2018), aggregates are manually removed without addressing their impact. It is crucial to acknowledge that the determination of the Manning roughness coefficient during the advance phase and the whole irrigation event is determined by inverse solution and model utilization, which may introduce uncertainties in estimation. The estimation method allows for the possibility of multiple parameter combinations yielding the same roughness coefficient. Therefore, to explain the unexpected values, the influence of these uncertainties can be considered. The presence of flow impediments (such as large aggregates in the furrow) results in backwater conditions and nonuniform flow (Fenton 2008). Manning's Equation is applicable under conditions of uniform flow (Mohamoud 1992; Nie et al. 2014b). This discrepancy may explain why the roughness coefficients derived during the advance phase of the initial irrigation sometimes exhibit unreasonably high values, surpassing the Manning's n values reported in the literature.

The relationship between the manning roughness coefficient and various parameters

Values of the correlation coefficient

To explore the relationship between the measured parameters and the Manning roughness coefficient, Pearson and Kendall statistical tests were employed. Pearson and Kendall tests were utilized to examine the relationship between quantitative and ordinal data, respectively (Tables 7 and 8). The findings indicated that the roughness of the whole irrigation event (nt) exhibited correlations with the parameters of inflow and outflow rates, advance time, initial soil moisture, flow cross-sectional area, and the number of irrigation events at a significance level of 1%. Additionally, it showed a correlation with the slope of the furrow bottom at a significance level of 5%. According to Pallant (2010), when the numerical correlation coefficient falls within the ranges of 0.1 to 0.29, 0.3 to 0.49, or 0.5 to 1, the correlation between the experimental parameters is considered weak, moderate, or strong, respectively. Applying this criterion, advance time demonstrated a strong correlation, while slope and initial soil moisture exhibited weak correlations with the Manning roughness coefficient data (nt); the remaining parameters showed moderate correlations.

Table 7 Correlation analysis of Manning roughness coefficient and different parameters
Full size table
Table 8 Correlation analysis of manning roughness coefficient and irrigation event (IE)
Full size table

The Manning roughness coefficient during the advance phase (na) was highly correlated with the advance time (Tables 7 and 8). Manning roughness coefficient during this phase showed a moderate correlation with both inflow and outflow rates, as well as the number of irrigations.

The correlation coefficients between the parameters and the Manning roughness coefficient of the storage phase (ns) varied, with the highest correlation observed for advance time (0.65) and the lowest for inflow rate (-0.31). These values suggest a strong positive correlation between advance time and roughness coefficient, while indicating a weak negative correlation between inflow rate and roughness coefficient. Further analysis is warranted to explore the relationship between the Manning roughness coefficient and other significant parameters.

Flow rate

The relationship between the inflow and outflow rates with the Manning roughness coefficient during the initial irrigation event and various phases is illustrated in Fig. 3. To establish the correlation between the roughness and flow rate values, an exponential equation was deemed suitable. While some graphs may exhibit a more fitting relationship with a higher coefficient of determination between the roughness coefficient and flow rate values, the roughness is expected to have a positive value at zero velocity and gradually approach the base value asymmetrically as velocity increases. Therefore, an exponential equation could be employed under such circumstances to yield more realistic outcomes. Trout (1992b) suggested that, apart from the exponential equation, the power equation is also suitable for representing the connection between flow velocity and roughness, albeit not effective for low velocities.

Fig. 3
figure 3

The relationship between Manning roughness coefficient (na, nt and ns), inflow (Qin) and outflow (Qout) rates for different phases of first irrigation in the fields E and F

Full size image

The relationship between flow rate and roughness coefficient was observed to be inversely proportional, indicating that an increase in flow rate led to a decrease in the roughness coefficient, as depicted in Figs. 3. This finding is consistent with previous studies by Ramezani Etedali et al. (2012), Sepaskhah and Bondar (2002), Ebrahimian (2014), Ibrahim and Abdel-Mageed (2014), and Salah Abd Elmoaty and T. A (2020). Additionally, an increase in flow rate resulted in reduced scattering of the intersection in the graphs, possibly attributed to the diminishing impact of initial flow conditions with higher flow rates (Trout 1992b). During the first irrigation compared to the second and third, variations in inflow and outflow rates significantly influenced the Manning roughness coefficient across all experimental phases. These variations exhibited a declining trend in subsequent irrigations, to the extent that by the third irrigation, the roughness coefficient and flow rate outcomes became nearly independent of each other. Notably, in the first irrigation, the presence of numerous clods in the furrow led to increased flow resistance at low flow rates, as the flow closely interacted with the soil surface, consequently elevating the roughness coefficient.

As the flow rate and water depth increase during the initial irrigation, the clods and soil surface generate less friction in the water flow path, resulting in a reduction in the roughness coefficient. With an increase in the number of irrigations, clods and irregularities in the furrow are eradicated through erosion and sedimentation, leading to a smoother furrow surface and a decrease in flow resistance. It is important to note that in furrow irrigation, the flow resistance is primarily attributed to the soil surface, and this resistance does not vary significantly with increasing depth (Esfandiari and Maheshwari 1998). Consequently, as the furrow bed becomes smoother, the coefficient of determination (R2) in any given equation will approach zero in the second and third irrigations.

The analysis presented in Fig. 3 indicates that during the initial irrigation phase, the correlation between flow rate and Manning roughness coefficient is more pronounced. Specifically, the outflow rate exhibited a stronger association with the Manning roughness coefficient compared to the inflow rate. It is important to highlight that, in most instances, the correlation between flow rate and Manning roughness coefficient did not surpass 0.6. Consequently, establishing a definitive relationship between inflow or outflow rate and the Manning roughness coefficient across all phases was not feasible. Trout (1992) conducted a study aiming to establish a connection between Manning roughness coefficient, flow velocity, and hydraulic radius through linear regression analysis, yielding a modest correlation (R2 = 0.6). Similarly, Sepaskhah & Bonder (2002), Mailapalli et al. (2008), and Kamali et al. (2018) characterized the relationship between flow rate and roughness coefficient as weak and inverse. In contrast, Abdullah et al. (2023) and Mohammadpour et al. (2020) identified a robust and inverse correlation between flow rate and Manning roughness coefficient.

Esfandiari and Maheshwari (1998) concluded that the hydraulic resistance to water flow in irrigation furrows arises from intricate interactions between water and the soil surface. They argued that attempting to establish a direct relationship between the Manning roughness coefficient, flow velocity, and hydraulic radius is futile as these variables lack a physical connection. Any observed systematic relationship between these parameters in field data should be viewed as a secondary effect. Consequently, any correlation established between these parameters is likely to be weak.

Slope

The slope is a significant parameter affecting the Manning roughness coefficient in furrow irrigation (Sepaskhah and Bondar 2002) due to the direct relationship between slope and flow velocity. In fields E and F, the minimum, maximum, and average slopes were measured as 0.0050, 0.0076, and 0.0066, and 0.0140, 0.0095, respectively, and 0.0065, respectively.

The correlation between slope and the Manning roughness coefficient throughout the whole irrigation event was found to be statistically significant at the 5% level. However, this relationship was not significant during both the advance and storage phases (Table 7). The Pearson correlation coefficient indicated a weak and positive association between slope and the roughness of the entire irrigation event. Similar findings have been reported by various researchers (Sepaskhah and Bondar 2002; Mailapalli et al., 2008a; Ramezani Etedali et al. 2012; Ebrahimian 2014). The lack of significance of the slope and roughness coefficient during the advance and storage phases, as well as the weak correlation between the roughness of the whole irrigation event and slope, could be attributed to the assumption of a constant slope in all three irrigation events. However, it is important to note that erosion and sedimentation in the furrow can alter its slope. Studies have indicated that despite the increasing trend of roughness and slope variations (Zhang et al. 2020), establishing a logical relationship between roughness and slope data in both field and irrigation settings across all three phases may be challenging due to the impact of slope on parameters such as flow velocity, erosion levels, and clod size. Furthermore, Zheng et al. (2012) highlighted that even though flow velocity tends to increase with steeper and eroded paths like irrigation furrows, the average flow velocity does not necessarily rise in tandem with the slope due to heightened surface roughness and soil erosion. Consequently, an increase in slope may not lead to a corresponding increase in the roughness coefficient and could potentially remain unchanged.

Advance time

The findings indicated a direct correlation between the Manning roughness coefficient and the advance time have a direct relationship (Table 7). Therefore, if the inflow rate is held constant, an increase in the number of irrigation events leads to a decrease in the Manning roughness coefficient. This decrease is attributed to the stabilization of the bed, resulting in a reduction in the advance time. Similar results were reported by Mwendera and Feyen (1992), Bautista et al. (2009a), and Xu et al. (2019).

The Manning roughness coefficient in the phases under investigation was unexpectedly not found to exhibit a positive correlation with the advance time at certain locations, indicating that as the Manning roughness decreased, the advance time increased (Fig. 4). This observation may be attributed to the impact of flow rate and other influential factors, such as hydraulic radius, clod size, and a combination of these variables. Additionally, this discrepancy could be a result of the unrealistic estimation of the Manning roughness coefficient, as the inverse solution of equations may yield several estimates with varying parameter values that lead to the same outcome. The findings also revealed a relatively strong quadratic relationship, with R2 values of 0.94, 0.89, and 0.92 for the advance, storage, and whole irrigation phases, respectively. This can be explained by the fact that as the advance phase concludes upon the flow reaching the end of the furrow, the roughness coefficient becomes more reliant on the advance time data.

Fig. 4
figure 4

The trend of changes in Manning roughness coefficient of advance phase and advance time in the experimental furrows in the field E

Full size image

The relationship between advance time and roughness coefficient during the storage phase, which exhibits the least reliance on advance time data, demonstrates a lower R2 value. The relationship between roughness coefficient and advance time in all three experimental phases was more correlated in first irrigation and declined with subsequent irrigations. Previous studies by Mwendera and Feyen (1992), Zerihun et al. (1996), Bautista et al. (2009a), and Xu et al. (2019) have acknowledged the significance of the relationship between roughness coefficient and advance time. In contrast, Nie et al. (2014a, 2018), Salahou et al. (2018), and Smith et al. (2018) have reported varying results, indicating that the impact of roughness coefficient variations on advance time estimation is ineffective.

Flow cross-sectional area

The relationship between the cross-sectional area of the flow and the furrow's cross-sectional area with the roughness coefficient during the irrigation event was examined. The results indicated a moderate positive correlation between the flow cross-sectional area and the roughness coefficient during the advance phase, storage phase, and the entire irrigation event, with correlation coefficients of 0.344, 0.344, and 0.313, respectively (Table 7). Conversely, the shape and size of the furrow's cross section did not show a significant correlation with the roughness coefficient in all three phases studied. Previous research by Chow (1959) suggested that the size and shape of a channel may not be a crucial factor affecting the roughness coefficient. However, the current study's findings align with those of Song et al. (2017) and Zhu et al. (2020), indicating a significant relationship between the roughness coefficient and the flow cross section and hydraulic radius. The interaction between roughness and flow cross-sectional area is intricate. Factors that impede flow or reduce flow velocity lead to an increase in water depth, subsequently affecting the cross-sectional area and roughness (Mera et al. 2017). Additionally, roughness changes in response to variations in flow rate due to inflow rate fluctuations, flow path obstructions, alterations in bed type, and the presence of vegetation. Consequently, the mutual influence of flow cross section on roughness does not yield definitive outcomes due to the impact of other variables. This complexity is evident in Fig. 5, which depicts the changes in flow cross-sectional area and roughness coefficient during the storage phase of three irrigation events in field F. The figure illustrates a decrease in the flow cross section from the first to the third irrigation event, despite no significant change in the inflow rate. This reduction can be attributed to the disintegration of clods, the redistribution of soil particles in cracks, the smoothing of the soil surface, and consequently, a decrease in the roughness coefficient. Predicting the general trend of changes in roughness coefficient and flow cross section is a challenging task, primarily due to the impact of various parameters like flow rate, clod size, and slope.

Fig. 5
figure 5

The trend of changes in Manning roughness coefficient of storage phase (ns) and flow cross section in three irrigation events

Full size image

Conclusions

In this study, the Manning roughness coefficient was determined for furrow irrigation during the advance, storage, and entire irrigation phases across three irrigation events. Additionally, the relationship between the Manning roughness coefficient and factors such as flow rate, slope, advance time, and flow cross-sectional area was examined for different irrigation phases.

The results indicated that the SIPAR_ID model and WinSRFR software effectively estimated the Manning roughness coefficient during the advance phase and the entire irrigation event, respectively. Manning’s n values ranged from 0.017 to 0.636 in the advance phase, 0.015 to 0.317 in the storage phase, and 0.015 to 0.34 for the entire irrigation event, with mean values of 0.083, 0.054, and 0.055, respectively.

The study found that the Manning roughness coefficient decreased across irrigation phases as the number of irrigation events increased, with flow rate having a significant influence on this reduction. A strong correlation was observed between the roughness coefficient and advance time during the whole irrigation event, while a weak correlation was noted with the furrow slope. Moderate correlations were found with inflow and outflow rates, flow cross-sectional area, and the number of irrigation events. In the advance phase, the roughness coefficient showed a strong correlation with advance time, a weak correlation with outflow, and moderate correlations with other parameters. During the storage phase, the highest correlation was with advance time, while the lowest was with inflow rate.

In conclusion, when the advance time is relatively short compared to the overall irrigation duration, it is feasible to use a single Manning roughness coefficient for the entire irrigation event without needing to separate it for the advance and storage phases. The variation in roughness coefficient values between these phases has minimal impact on simulation accuracy. However, due to the significant differences in Manning roughness values across the phases, improving simulation models to account for distinct roughness coefficients in different phases is recommended to enhance accuracy.