Investigation of hydraulic parameters influencing interrill erosion under simulated rainfall conditions

Agricultural soil is the land most affected by erosion, which is defined by the detachment of soil particles caused by water (rain, precipitation, and runoff) or by wind, subsequently transported to a place of deposit. However, repercussions of erosion on the soil and the environment are observed, such as loss of productivity (fertility), partial destruction of agricultural fields, and landslides, which are the objective of several studies dividing erosion into several forms, from which rill and interrill erosion are the most exploited. Thus, our study consists of evaluating these two types of erosion through experimental tests based on experiments by Maaliou et al. (2020), Maaliou (2018), Aziz & Liatim (2018), and Shen et al. (2018), which examined erosive agents and the effect of slope on runoff and soil loss.

Interrill erosion, which is a characteristic of soil erosion, is also the subject of several experimental studies, defined as the main cause of the degradation of agricultural land (Meyer 1981). We will improve our study by evaluating this characteristic by combining it with an artificial rill. Several researchers have studied interrill erosion (Meyer 1981; Bradford et al. 1987; Everaert 1991; Fox & Bryan 1999; Yan et al. 2008) and concluded that soil loss increases with precipitation and slope, significantly impacting channels more than interrill areas (Shen et al. 2018). They found that channel erosion increases considerably with slope compared to inter-channel erosion.

The experiments for this work were conducted in the LEGHYD laboratory using a modified rainfall simulator (Aziz & Liatim 2018; Maaliou 2018; Maaliou et al. 2020). The setup includes a mechanism for adjusting soil tray inclination to various slope values and an artificial rill placed in the middle to study the combined effect on soil erosion. This setup is complemented by lines drawn on the flume soil test to indicate different interrill slope values for adjustment.

Sadio & Faye (2023) provide a robust framework with their Indicators of Hydrologic Alteration (IHA) and Range of Variability Approach (RVA) methods, which are crucial for understanding how variations in flow regimes affect soil erosion rates. Mukta et al. (2023) highlight the significance of maintaining soil fertility and sustainable agricultural practices, which are directly aligned with our objective to mitigate soil degradation through effective erosion control. Samatar (2023) illustrates the relationship between soil health and agricultural output using the Autoregressive Distributed Lag (ARDL) model, emphasizing the importance of capturing both short-term and long-term impacts of erosion, which is vital for our temporal analysis of hydraulic effects on soil stability. The precise hydraulic modeling principles demonstrated by Mallampalli et al. (2023) inform our approach to accurately predict erosion patterns under different rainfall intensities and slope conditions. Loan et al. (2023) underscore the interconnectedness of environmental health and socio-economic conditions, reinforcing the necessity of integrated soil erosion management strategies in our study. Fuladipanah et al. (2024) showcase the potential of machine learning to enhance predictions of erosion rates, offering advanced computational techniques to improve the accuracy of our simulations. Lastly, Yusof et al. (2014) demonstrate the effectiveness of artificial neural networks in predicting soil erodibility, providing a powerful tool for refining our erosion models. Arjmand Sajjadi & Mahmoodabadi (2015) used a tray with two rainfall intensities (57 and 80 mm·h⁻¹), two soils with sand contents of 56.6 and 58.8%, and a slope gradient varying from 0.5 to 20% to study rain-induced overland flow under gentle slope conditions. This study shows that sediment concentration rises with slope gradient and that, among the analyzed hydraulic variables, overland flow velocity is the strongest predictor of sediment concentration (flow depth, shear stress, stream power, and unit stream power).

Di Stefano et al. (2021) tested the applicability of a theoretical rill flow resistance law using data on movable bed rill channels combined with measurements made in rill flumes, in which the sediment load of the flow was equal to its transport capacity. They showed that the analyzed soil's particle detachability and transportability have an impact on the Darcy–Weisbach friction factor using a database made up of samples with high slope values (17.6–84%) and a wide range of textural fractions. Collectively, these studies provide a comprehensive foundation that supports and enhances our investigation, highlighting the importance of advanced modeling techniques, sustainable practices, and integrated approaches in soil erosion research.

The purpose of this study is to analyze the effects of rill and interrill slopes on soil erosion and hydraulic parameters. The use of sandy soils in experimental studies of erosion, which have loose particles that are easily transported and deposited, enables the examination of an infinite number of detachment circumstances. Given that sandy soil does not require the management of flow detachment capability, experimental research can be conducted in a flume where the overland flow is not affected by the impact of rainfall.

We notice that the various outcomes for the 0% slope of the rill and the interrill are not included in this paper because the values of the sediment concentrations were essentially insignificant when compared to the 3 and 5% slopes. Additionally, it should be highlighted that at slopes of 0%, there was no sediment concentration inside the rill; instead, there was clear water produced by the spray nozzles. The results presented here are those obtained for the combinations of slopes: 3 and 5% for the rill and 15 and 22.5% for the interrill under the six values of rainfall intensities.

After fixing the artificial rill, a succession of soil layers (interrill banks) is placed to ensure good distribution throughout the soil tray and uniformity. The surface layer contains the soil to be tested, which is of a variable thickness according to levels of 1 cm in height that represent inter-fill slopes of 0, 15, and 22.5% as shown in the floor pan. The first layer is made up of coarse gravel, which facilitates the infiltration of water. The second layer is formed by an inhomogeneous soil. In order to get the desired slope, the floor is modified using a flat squeegee from the top to the edge of the channel.

In this experiment, six simulated rainfall intensities were selected using a spray nozzle type (TEEJET SS 65 60) with values of 101.94, 90.39, 81.89, 69.49, 37.82, and 31.40 mm·h⁻¹. A gate valve was installed at the outlet of the pump and another in the by-pass line to vary the rainfall intensity (see Figure 1). Maaliou (2018) and Aziz & Liatim (2018) used similar equipment and procedures.

The experiments were carried out on reworked sandy agricultural soil and collected from the ITCMI (Technical Institute for Industrial Market Gardening). Physical and chemical analyses of the soil are being conducted in INRAA's Pedological Lab (National Institute of the Agronomic Research of Algeria). The results obtained are shown in Tables 1 and 2.

Over 54 tests are carried out by performing three tests for each composition of intensity, variation of interrill slope, and slope of the soil tray. The intensities are adjusted by varying the pressure through the valve of the by-pass line (Aziz & Liatim 2018; Maaliou 2018; Maaliou et al. 2020).

When the soil is saturated, we adjust the flume soil test at the desired slope value (3 and 5%). These slope values were used by different researchers according to Bryan (1979), Collinet & Valentin (1984), Abderzak et al. (2017), Aziz & Liatim (2018), Maaliou (2018), Maaliou et al. (2020). A simulated rainfall is applied when the slope is fixed at the desired value to avoid irregular soil saturation. Once the runoff is established, we proceed to collect the volume of the water–sediment mixture in a 1,000 ml beaker at the end of the soil tank (see Figure 1). This process will repeat every 4 min until the experiment ends (Fox & Bryan 1999; Aziz & Liatim 2018; Maaliou et al., 2020). The test lasts 58 min, one dose is scheduled every 4 min with a setting time of 30 s to have the final 13 samples. Four measurements are noted during the test: the speed flow, volume, and mass of sediments.

After reading the amount of mixture collected in a 1,000 ml beaker, we shake this beaker up and down to get a uniform mixture, and then take the 100 ml amount into a small beaker that has already been weighed. At the end of the experiment, the beaker filled with the water/sediment mixture is dried in an oven at 105 °C for 24 h until a constant weight is reached (Fox & Bryan 1999; Pan & Shangguan 2006; Aziz & Liatim 2018; Abderzak et al. 2019; Maaliou et al. 2020). After this time, the beaker with the dried sediment is weighed, and the difference in weight between the empty and filled beakers gives the mass of sediment in 100 ml (Pan & Shangguan 2006; Aziz & Liatim 2018).

The surface flow velocity and mean flow velocity are expressed in m·s⁻¹. The ratio 2/3 of Equation (1) can be evaluated theoretically. It was used by Maaliou et al. (2020), Abderzak et al. (2019), Aziz & Liatim (2018), Pan & Shangguan (2006), and Li et al. (1996).

The Reynolds number given by Equation (2) is a dimensionless number that represents the flow turbulence index, where the increase in erosive power is related to the increase in turbulence generated by a flow (Maaliou 2018). It is defined as being the ratio between the force of inertia and that of friction.

The overland flow power represents the energy of the water flowing over the surface of the soil; some or all of this power can remove and transport soil particles from the surface. It is defined as the best predictor of soil detachment and transport and the most efficient (Nearing et al. 1997; Zhang et al. 2009; Niu et al. 2020) and is exploded by many researchers (Hairsine & Rose 1992; Nearing et al. 1997; Zhang et al. 2009; Shi et al. 2013; Abderzak et al. 2017; Aziz & Liatim 2018; Niu et al. 2020).

For a rill slope of 3% and an interrill slope of 15%, the Reynolds number increases gradually with the rainfall intensity. When the rainfall intensity varies from 31.40 to 101.94 mm/h, the Reynolds number increases from 1,417.44 to 1,756.05, demonstrating a steady increase in the Reynolds number with rainfall intensity. This trend is accentuated when the interrill slope is increased to 22.5%. For the same intensities, the Reynolds values are higher, ranging from 1,607.08 to 1,787.54. This pattern indicates that the Reynolds number is generally higher compared to the previous scenario, with an interrill slope of 15%.

When examining a rill slope of 5%, the Reynolds numbers are significantly higher for the same rainfall intensities, demonstrating a similar trend but at elevated levels. Notably, for a rill slope of 5% and an interrill slope of 15%, with intensities ranging from 31.40 to 101.94 mm/h, the Reynolds number varies between 2,635.42 and 1,663.06. The Reynolds number is more significant when the interrill slope is increased to 22.5%. For the same rainfall intensities, the Reynolds values increase further, ranging from 2,994.46 to 1,961.94.

This increase in the Reynolds number with rainfall intensity aligns with the findings of Zhao et al. (2014), who reported that at a slope of 5° and for intensities ranging from 30 to 270 mm/h, the Reynolds number values increased from 4.17 to 122.34.

It is noteworthy that for a 5% rill slope, the Reynolds number decreases as the rainfall intensity increases. For the 15% interrill slope, at an intensity of 31.40 mm/h, the Reynolds number is highest, reaching 2,635.42. This value gradually decreases with increasing rainfall intensity, reaching 1,663.06 at an intensity of 101.94 mm/h. Similarly, for the 22.5% interrill slope, at an intensity of 31.40 mm/h, the Reynolds number is approximately 2,994.46. However, this value steadily decreases as the intensity increases, reaching 1,961.94 at an intensity of 101.94 mm/h.

This inverse relationship suggests that despite increasing rainfall intensity, the flow characteristics tend to evolve toward less turbulent regimes or become more dominated by dissipation effects at higher intensities. This trend could be attributed, on the one hand, to the fact that an increase in water flow might alter the flow structure, making it more laminar or more influenced by surface effects and stresses on the flow surface. On the other hand, the increase in slope and rainfall intensities enhance the sediment load, which implies a reduction in flow turbulence.

For a rill slope of 5%, an interrill slope of 15%, and rainfall intensities ranging from 31.40 to 101.94 mm/h, the sediment load varies between 2.94 and 5.69 kg/m³, while the Reynolds number decreases from 2,635.42 to 1,663.06. For the same rill slope and rainfall intensities but an interrill slope of 22.5%, the sediment load ranges from 3.14 to 6.99 kg/m³, and the Reynolds number decreases from 2,994.46 to 1,961.94.

The Reynolds number is strongly affected by sediment loading. As sediment load increases, the Reynolds number decreases, suggesting a reduction in flow turbulence due to increased viscosity. Flow turbulence decreases as sediment loading increases, mainly due to changes in flow viscosity with sediment loading and interactions between water molecules and sediment particles. These findings coincide with the work of Zhang et al. (2010), who indicated that the Reynolds number decreases with increasing sediment loading.

Compared to clear water, the Reynolds number in sediment-laden flow was reduced by an average of 23.0%. The inverse relationship between Reynolds number and sediment loading is consistent with the conclusions of Summer & Zhang (1998). As sediment load increases, flow turbulence decreases, primarily due to increased viscosity associated with sediment load and the interactions occurring between water molecules and sediment particles. Flow viscosity increases almost proportionally to a power function with sediment load. The higher the fluid viscosity, the less turbulent it becomes.

A steeper rill slope (5%) results in higher overland flow power across all measured rainfall intensities compared to a gentler rill slope (3%). An increase in the interrill slope (22.5%) also raises overland flow power, although this effect is more moderate than the direct impact of the rill slope.

When the rill slope is 3%, and the interrill slope is 15%, overland flow power (Ω) increases gradually with rainfall intensity. At a low rainfall intensity of 31.40 mm/h, the overland flow power is 0.120 kg/s³. As rainfall intensity rises to 101.94 mm/h, the overland flow power increases to 0.550 kg/s³. This trend indicates that even with a gentle rill slope, increased rainfall intensity leads to a continuous increase in overland flow power, suggesting that heavier rainfall boosts both the volume and velocity of surface flow.

When the rill slope remains at 3% but the interrill slope increases to 22.5%, the overland flow power values are generally higher for each rainfall intensity compared to the 15% interrill slope. For example, at a rainfall intensity of 31.40 mm/h, the overland flow power is 0.187 kg/s³ (compared to 0.120 kg/s³ previously). At a rainfall intensity of 101.94 mm/h, the overland flow power remains at 0.550 kg/s³, similar to the lower interrill slope scenario. This increase with the steeper interrill slope suggests that greater inclination facilitates more efficient water drainage even with a constant rill slope.

With the rill slope increased to 5%, the overland flow power is significantly higher than with a 3% rill slope. At a rainfall intensity of 31.40 mm/h, overland flow power rises to 0.344 kg/s³ (compared to 0.120 kg/s³ for a 3% rill slope and 15% interrill slope). This phenomenon intensifies with increasing rainfall intensity, with overland flow power reaching 1.010 kg/s³ at 101.94 mm/h. This marked increase demonstrates that steeper rill slopes promote greater and faster flow, likely due to enhanced drainage facilitation and more efficient rainwater management.

For a rill slope of 5% combined with an interrill slope of 22.5%, overland flow power is higher overall but follows a similar trend to that observed with a 15% interrill slope. The overland flow power values increase significantly with rainfall intensity, reaching 0.312 kg/s³ at 31.40 mm/h and rising to 0.875 kg/s³ at 101.94 mm/h. These results confirm that increasing the rill slope leads to higher overland flow power, regardless of the interrill inclination.

The relationship between overland flow power and rainfall intensity is represented by a linear function for a 3% rill slope and an exponential function for a 5% rill slope, with coefficients of determination ranging from 0.96 to 0.97. These coefficients demonstrate that overland flow power is closely related to rainfall intensity, evolving through linear and exponential functions.

A steeper rill slope results in greater flow depths, indicating an enhanced capacity to drain rainfall. Similarly, a higher interrill slope also contributes to increased flow depth, although the effect is more pronounced with changes in the rill slope.

For a rill slope of 3% and an interrill slope of 15%, flow depth (h) increases progressively with rainfall intensity. At a low rainfall intensity of 31.40 mm/h, the flow depth is 0.65 cm. As rainfall intensity rises, the flow depth also increases, reaching 1.13 cm at a rainfall intensity of 101.94 mm/h. These values demonstrate a direct relationship between rainfall intensity and flow depth, indicating that more intense rainfall increases water volume.

When the interrill slope is increased to 22.5% while keeping the rill slope at 3%, flow depths are slightly higher for each rainfall intensity level. At 31.40 mm/h, the flow depth is 0.67 cm (compared to 0.65 cm previously), and at 101.94 mm/h, it is 1.39 cm. This increase suggests that steeper interrill slopes facilitate more significant runoff and contribute to slightly higher flow depths, even if the rill slope remains constant.

They increase the rill slope to 5% while maintaining an interrill slope of 15%, resulting in generally higher flow depths than a 3% rill slope. At a rainfall intensity of 31.40 mm/h, the flow depth rises to 0.85 cm (compared to 0.65 cm with a 3% rill slope and a 15% interrill slope), and at 101.94 mm/h, it reaches 1.15 cm. These values indicate that steeper rill slopes increase flow depth at each rainfall intensity.

For a 5% rill slope combined with a 22.5% interrill slope, flow depths follow a similar trend but remain higher overall. At 31.40 mm/h, the flow depth is 0.68 cm (compared to 0.85 cm with a 15% interrill slope); at 101.94 mm/h, it reaches 1.35 cm. These results show that increasing the rill slope, along with a steeper interrill slope, contributes to higher flow depths at every level of rainfall intensity.

The observed increase in flow depth with rainfall intensity aligns with the findings of Cochrane & Flanagan (2006), who reported that flow depth increases with higher rainfall intensity, rising from 0.59 cm at 80 mm/h to 0.73 cm at 160 mm/h.

For a rill slope of 3% and an interrill slope of 15%, there is a progressive increase in flow depth with increasing rainfall intensity. At a rainfall intensity of 31.40 mm/h, the flow depth is 0.65 cm, corresponding to an overland flow power of 0.120 kg/s³. As the rainfall intensity rises to 101.94 mm/h, the flow depth increases to 1.13 cm, with an overland flow power of 0.550 kg/s³. This trend indicates that more intense rainfall boosts both flow depth and overland flow power, suggesting that drainage systems must be capable of managing these increasing volumes of water.

When the interrill slope is increased to 22.5% while maintaining a 3% rill slope, flow depths are slightly higher at each level of rainfall intensity. At 31.40 mm/h, the flow depth is 0.67 cm, and the overland flow power is 0.187 kg/s³, while at 101.94 mm/h, it reaches 1.39 cm with an overland flow power of 0.550 kg/s³. This increase in interrill slope facilitates more effective water runoff, resulting in higher flow depths and powers, indicating that steeper interrill slopes enhance drainage capacity.

Increasing the rill slope to 5% with an interrill slope of 15% leads to generally higher flow depths and overland flow power. At 31.40 mm/h, the flow depth reaches 0.85 cm with an overland flow power of 0.344 kg/s³, and at 101.94 mm/h, it is 1.15 cm with an overland flow power of 1.010 kg/s³. These values demonstrate that steeper rill slopes significantly increase both flow depth and overland flow power, reflecting an improved capacity to manage higher water volumes.

When both the rill and interrill slopes are increased to 5 and 22.5%, respectively, flow depths remain high, ranging from 0.68 to 1.35 cm, and overland flow powers range from 0.312 to 0.875 kg/s³. This configuration, combining steeper slopes, results in more significant flows, demonstrating the cumulative impact of both types of slopes on runoff.

For a rill slope of 3% with an interrill slope of 15%, the theoretical flow rate (Q_th) gradually increases with increasing flow depth. Flow rates range from 0.18 l/min at a depth of 0.65 cm–0.38 l/min at a depth of 1.13 cm. This gradual increase aligns with hydraulic theory, which suggests that greater water depth allows for higher flow due to the increased capacity of the rill to carry water. However, the 3% slope is relatively gentle, which limits flow acceleration because gentler slopes lead to higher resistance against water flow.

In comparison, when the interrill slope is increased to 22.5% while maintaining a 3% rill slope, observed flows increase more markedly. For instance, at a flow rate of 0.53 l/min, the depth reaches 1.39 cm. These observations demonstrate that increasing the interrill slope improves hydraulic performance by increasing flow even when the rill slope remains constant.

When the rill slope is increased to 5% with an interrill slope of 15%, the results show a noticeable increase in flow depth compared to a 3% slope. For a flow rate ranging from 0.16 to 0.50 l/min, the flow depth increases from 0.85 to 1.15 cm. This increase results from the steeper slope, which reduces flow resistance and facilitates faster flow. The 5% slope provides a better longitudinal gradient, allowing for improved water transport within the gully.

The impact of the rill slope becomes even more pronounced when the interrill slope is increased to 22.5%. Flow rates are the highest observed in the dataset, ranging from 0.24 to 0.54 l/min for depths ranging from 0.68 to 1.35 cm. This arrangement leverages the benefits of both rill and interrill slopes, resulting in optimal flow conditions and increased flow rates. The interaction between a higher rill slope and a steeper interrill slope creates ideal conditions for maximized flow, reducing pressure drops and enhancing flow efficiency.

For a 3% rill slope and a 15% interrill slope, the overland flow power also rises as the theoretical flow discharge increases progressively. The theoretical flow discharge Q_th values increase from 0.18 to 0.38 l/min, while the overland flow power Ω rises from 0.120 to 0.550 kg/s³. This trend indicates that as the theoretical flow discharge increases, the overland flow power also increases, reflecting greater energy and water transport capacity.

Both measurements show increases when the interrill slope increases to 22.5% for the same 3% rill slope. The theoretical flow discharge rises from 0.21 to 0.53 l/min, and the overland flow power from 0.187 to 0.550 kg/s³. Compared to the 15% interrill slope results, the overland flow power is higher for each theoretical flow discharge value. These findings suggest that steeper slopes increase flow discharge and enhance the sediment concentration and soil particle transport capacity of the water, thereby increasing the overland flow power.

Considering a 5% rill slope, the observed trends are similar, but the theoretical flow discharge and overland flow power values are overall higher than those recorded with a 3% rill slope. For example, for a 15% interrill slope, the theoretical flow discharge varies from 0.16 to 0.50 l/min, and the overland flow power increases from 0.344 to 1.010 kg/s³. The values increase even more for an interrill slope of 22.5%, with a theoretical flow discharge rising from 0.24 to 0.54 l/min and an overland flow power from 0.312 to 0.875 kg/s³. This observation confirms steeper rill slopes intensify flow discharge and overland flow power.

Based on previous studies of the relationship between sediment concentration and overland flow power, this is the relationship that provides the best prediction of soil particle detachment (Maaliou 2018); however, this parameter can be used for the evaluation of the interrill erosion.

For a 3% rill slope and a 15% interrill slope, the overland flow power Ω varies from 0.120 to 0.550 kg/s³. It is observed that the sediment concentration C_s increases significantly with the increase in the overland flow power Ω. The sediment concentrations C_s increased from 2.42 to 5.76 kg/m³. This trend indicates that as the overland flow power increases, the capacity of the flow to transport sediment also increases. The direct relationship between overland flow power and sediment concentration can be explained by higher flow powers associated with higher velocities, leading to increased soil erosion and greater sediment mobilization.

When the interrill slope is increased to 22.5% while maintaining a rill slope of 3%, the sediment concentration follows a similar trend but with slightly higher values. For overland flow power ranging from 0.187 to 0.550 kg/s³, the sediment concentration varies from 2.94 to 5.69 kg/m³. This phenomenon can be attributed to the combined effect of higher slopes, which increase water velocity and enhance its ability to entrain and transport more soil particles.

Considering a 5% rill slope and a 15% interrill slope, the overland flow power values are higher, ranging from 0.344 to 1.010 kg/s³, and the sediment concentration also increases significantly from 2.38 to 5.88 kg/m³. This more pronounced increase in sediment concentration, compared to a 3% rill slope, indicates that steeper slopes lead to increased flow discharge, overland flow power, and greater soil erosion, resulting in higher sediment concentrations in the water.

Finally, the sediment concentration is even higher for a 22.5% interrill slope with a 5% rill slope for the corresponding overland flow power values. With overland flow powers ranging from 0.312 to 0.875 kg/s³, sediment concentrations range from 3.14 to 6.99 kg/m³. This observation confirms that steeper slopes intensify sediment transport. The steeper slope promotes higher flow velocities and greater erosion, increasing sediment concentration in the water flow.

It is worth noting that the relationship between overland flow power and sediment concentration is represented by an exponential function with highly significant coefficients of determination ranging from 0.94 to 0.96.

Wang et al. (2016) examined how hydraulic parameters affect the modeling of soil detachment capacity through rill flow. They proposed a power–function relationship between soil detachment capacity and flow power for slopes ranging from 15.8 to 38.4%. The coefficients of determination for their model ranged from 0.865 to 0.987. Similarly, Ali et al. (2012), in their investigation of the effect of hydraulic parameters on sediment transport capacity in surface flows over erodible beds, also identified a power–function relationship between sediment transport capacity and flow power, reporting a coefficient of determination of 0.87. Shen et al. (2021) found that the relationship between soil detachment capacity and flow power can be described by a linear function with coefficients of determination between 0.95 and 0.98 for various soil types. Tian et al. (2020) proposed a linear relationship between the rill erosion rate and flow power for slopes of 25.9, 42.3, and 57.4%, with coefficients of determination ranging from 0.97 to 0.98.

Similarly, Zhang et al. (2010) established a linear relationship between sediment load and flow power. Zhu et al. (2019) presented a linear relationship between soil detachment capacity and flow power, with a coefficient of determination of 0.83. They also reported that flow power (ω) is the optimal parameter for predicting the soil detachment capacity (Dc) of shallow surface runoff, and the linear power equation Dc = 0.070 + 0.062ω was the optimal model for predicting detachment capacity of shallow surface runoff with an Nash-Sutcliffe Efficiencys (NSE) of 0.83. Moussouni et al. (2013) proposed a polynomial relationship between soil erodibility and flow power, with a coefficient of determination of 0.96. Finally, Nearing et al. (1997) modeled the relationship between flow power and sediment concentration using a logarithmic distribution, achieving a correlation coefficient of 0.93.

Researchers found that for a rill slope of 3% and an interrill slope of 15%, soil erodibility K gradually decreases from 9.52 × 10⁶ to 4.90 × 10⁶ kg·s/m⁴ as the interrill slope increases. This decrease can be explained by better stabilization of the soil at lower slopes, thus reducing its susceptibility to erosion. At the same time, the sediment concentration C_s increases significantly from 2.42 to 5.76 kg/m³, indicating a corresponding increase in the amount of sediment transported, probably due to greater mobility of soil particles due to runoff.

For a rill slope of 3% and an interrill slope of 22.5%, soil erodibility K starts at 8.05 × 10⁶ kg·s/m⁴ and decreases to 4.14 × 10⁶ kg·s/m⁴. This significant reduction in K with an increase in the interrill slope indicates the increased vulnerability of the soil to erosion. As a result, the sediment concentration shows a moderate increase from 2.94 to 5.69 kg/m³, supporting the idea that higher slopes promote more significant sediment mobilization.

In comparison, for a 5% rill slope with the same 15% interrill slope, the initial values of soil erodibility K are slightly lower than those observed with a 3% rill slope, indicating a similar but less pronounced trend. On the other hand, the sediment concentration shows a similar increasing trend with higher values at steeper rill slopes, reaching up to 5.88 kg/m³.

Finally, for a rill slope of 5% and an interrill slope of 22.5%, the values of K start at 8.85 × 10⁶ kg·s/m⁴ and decrease to 6.45 × 10⁶ kg·s/m⁴, again showing a progressive reduction in erodibility with an increase in the integral slope. The sediment concentration reaches high values of up to 6.99 kg/m³, confirming that steeper slope conditions lead to substantial sediment mobilization.

These findings demonstrate that rill and interrill slopes play a crucial role in the dynamics of water erosion. Higher slopes increase soil erodibility and sediment concentration, requiring special attention in soil management and conservation, especially in areas with intense rainfall or intensive land use. This understanding is essential for developing sustainable agricultural practices and land use planning to minimize the adverse effects of erosion on ecosystems and water resources.

The relationship between sediment concentration and soil erodibility is described by a power function, with coefficients of determination varying between 0.94 and 0.97. These coefficients indicate a strong correlation between sediment concentration and soil erodibility. They demonstrate a substantial and statistically significant correlation between sediment concentration and soil erodibility. In other words, they indicate that the amount of sediment transported is closely related to the susceptibility of the soil to erosion, which is crucial for understanding and predicting water erosion processes.

Researchers have found that soil erodibility increases with increasing slope and decreases with increasing rainfall intensity. For a rill slope of 3% and a rainfall intensity of 31.40 mm/h, the soil erodibility is 9.52 × 10⁶kg·s/m⁴. On the other hand, for a rainfall intensity of 101.94 mm/h, the soil erodibility is 4.9 × 10⁶4 kg·s/m⁴. For a slope of 5% and a rainfall intensity of 31.40 mm/h, the soil erodibility is 8.85 × 10⁶ kg·s/m⁴, while for a rainfall intensity of 101.94 mm/h, the soil erodibility is 6.99 × 10⁶ kg·s/m⁴.

These results coincide with those of Fang et al. (2015), who conducted a rainfall simulation study on soil erodibility and nutrient losses in China, showing that soil loss increased more with increasing slope than precipitation intensity. They observed that total soil loss with a rainfall intensity of 60 mm/h and a slope of 20% exceeded that of a rainfall intensity of 120 mm/h and a slope of 10% and even approached it in the treatments with a rainfall intensity of 120 mm/h and a slope of 20%. They indicated that increasing the slope from 10 to 20% had a more substantial influence on increasing soil loss than increasing the erosive force of precipitation (from 60 to 120 mm/h).

For a rill slope of 3% and an interrill slope of 15%, the erodibility values vary from 9.52 × 10⁶ to 4.90 × 10⁶ kg·s/m⁴, showing a gradual decrease with an increase in the Froude number from 0.35 to 0.74. This decrease in K indicates a reduction in soil erodibility as the flow dynamics, represented by Fr, increase. This trend can be attributed to better soil stability in the face of less turbulent flows and a reduction in the carrying capacity of soil particles.

In contrast, for a 3% rill slope with an interrill slope of 22.5%, the values of K start at a slightly lower level of 8.05 × 10⁶ kg·s/m⁴ but also decrease as Fr increases to 0.95. The data suggest that, even with a constant rill slope, an increase in the interrill slope can enhance flow dynamics, reducing soil erodibility.

For a rill slope of 5% with an interrill slope of 15%, the values of k show a less pronounced decrease from 8.35 × 10⁶ to 5.21 × 10⁶ kg·s/m⁴ with an increase in Fr from 0.42 to 0.79. This observation suggests that steeper slopes may slightly increase soil erodibility despite relatively stable flow conditions, possibly due to greater soil exposure to water erosion.

Finally, for a rill slope of 5% with an interrill slope of 22.5%, the values of k decrease from 8.85 × 10⁶ to 6.45 × 10⁶ kg·s/m⁴, with Fr increasing from 0.62 to 0.95. This substantial decrease in k indicates significant soil stabilization despite more dynamic flow conditions, highlighting the critical importance of topography in regulating water erosion.

The relationship between the Froude number and soil erodibility is represented by a power function characterized by high coefficients of determination between 0.8 and 0.96. These coefficients indicate a significant correlation between the Froude number and soil erodibility.

The results from this study clearly show that as soil erodibility increases, the Froude number decreases, corroborating the findings of Zhang et al. (2010). They pointed out that the Froude number decreased as the sediment load increased, and the effect of the sediment load on the Froude number depended on the flow conditions. Tian et al. (2020) reported that for slopes ranging from 26 to 57%, the Froude number ranged from 1.9 to 3.1. These values are more significant than 1, indicating that the rill flows were in supercritical conditions. This observation is also supported by Mirzaee & Ghorbani (2018), who noted similar situations due to significant slopes ranging from 26 to 57%. In contrast, the regime in our study is fluvial due to slopes ranging from 3 to 5%.

Shen et al. (2016) and Zhang et al. (2017) indicated that the slope gradient is an essential factor affecting rill erosion processes. Zhao et al. (2014) also pointed out that Froude numbers were more significant than 1, indicating that surface flows on all slopes were supercritical. They added that Froude numbers decreased with increasing slope gradients under low precipitation intensities. These changes can be explained by the reduction in flow depth and the increase in velocity with increasing slope gradients, leading to a rise in Froude numbers. However, under moderate and high precipitation intensities, the relationship between Froude numbers and slope gradients was relatively complex due to the combined effects of precipitation intensity and slope gradient on flow depth and velocity.

Our investigation into the hydraulic parameters influencing interrill erosion under simulated rainfall conditions provides a comprehensive understanding of the complex interplay between slope, rainfall intensity, and sediment transport. The experimental setup incorporated variable slope conditions and rainfall intensities, enabling a detailed analysis of rill and interrill erosion dynamics and revealing several critical insights.

The study demonstrated that the Reynolds number (Re), a measure of flow turbulence, generally increases with rainfall intensity. For a rill slope of 3% and interrill slopes of 15 and 22.5%, the Reynolds number increased progressively with rainfall intensity, highlighting a positive correlation between these factors. The Reynolds number values were significantly elevated at a higher % rill slope of 5%, indicating enhanced turbulence due to steeper slopes. However, an inverse relationship emerged at high rainfall intensities, suggesting a shift toward less turbulent flow regimes, potentially due to increased sediment load and altered flow structures.

Sediment load played a crucial role in modulating flow characteristics. Higher sediment loads were associated with decreased Reynolds numbers, indicating reduced flow turbulence. This relationship underscores the importance of sediment transport dynamics in influencing hydraulic parameters. The study found that sediment concentration rises with increasing slope gradients and rainfall intensities. This trend emphasizes the critical role of slope and precipitation in driving soil erosion processes.

Overland flow power (Ω), an indicator of surface runoff's erosive potential, increased with rainfall intensity and slope. Steeper rill slopes (5%) and higher interrill slopes (22.5%) significantly enhanced overland flow power, suggesting a direct relationship between slope steepness, rainfall intensity, and erosive force. Flow depth (h) also showed a positive correlation with rainfall intensity. Steeper rill slopes and higher interrill slopes resulted in greater flow depths, reflecting improved drainage and runoff efficiency.

The interrelationship between flow depth and overland flow power demonstrated that both parameters are closely linked to slope and rainfall intensity. Steeper slopes facilitated greater flow depths and higher overland flow power, highlighting the compounded effect of slope and precipitation on erosion dynamics. Our results extend existing models by providing empirical data under controlled experimental conditions.

The findings from this study are highly relevant for managing soil erosion, especially in agricultural environments. Effective slope management strategies, including contour plowing and terracing, can mitigate the impact of slope erosion. By reducing slope gradients, the erosive potential of surface runoff can be minimized, preserving soil fertility and preventing land degradation. While controlling natural rainfall is impractical, adopting soil conservation practices such as mulching and cover cropping can help mitigate the effects of heavy rain on soil erosion. These practices enhance soil structure and increase infiltration, reducing surface runoff and sediment transport. Managing sediment load through vegetative barriers and sediment traps can facilitate the transport of soil particles, minimizing the impact on downstream water bodies. Such measures are crucial for maintaining water quality and preventing sedimentation in rivers and reservoirs.

In conclusion, our study highlights the intricate relationships between hydraulic parameters, slope, and rainfall intensity in driving interrill erosion. The findings underscore the need for integrated soil erosion management strategies that consider the combined effects of these factors to mitigate soil degradation and promote sustainable agricultural practices effectively.

A. M. conceptualized and investigated the whole process, rendered support in data curation, wrote the original draft, wrote the review and edited the article; A. M. rendered support in funding acquisition, wrote the original draft, wrote the review and edited the article; D. Z. investigated the study and rendered support in data curation.

Special thanks are addressed to Mr Cherif DRAIFI, technician at the hydraulics laboratory of the University of Science and Technology Houari Boumediene for his strong mobilization and contribution to carry out this study.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.