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Purpose In this paper, a deduced flow resistance equation for open-channel flow was tested using measurements carried out in mobile bed rills with sediment-laden flows and fixed bed rills. The main aims were to (i) assess the effect of sediment transport on rill flow resistance, and (ii) test the slope-flow velocity relationship in fixed bed rills. Methods The following analysis was developed: (i) a relationship between the Γ function of the velocity profile, the rill slope and the Froude number was calibrated using measurements carried out on fixed bed rills; (ii) the component of Darcy-Weisbach friction factor due to sediment transport was deduced using the corresponding measurements carried out on mobile bed rills (grain resistance and sediment transport) and the values estimated by flow resistance equation (grain resistance) for fixed bed rills in the same slope and hydraulic conditions; (iii) the Γ function relationship was calibrated using measurements carried out on mobile bed rills and the data of Jiang et al. (2018). Results This analysis demonstrated that the effect of sediment transport on rill flow resistance law is appreciable only for 7.7% of the examined cases and that the theoretical approach allows for an accurate estimate of the Darcy-Weisbach friction factor. Furthermore, for both fixed and mobile beds, the mean flow velocity was independent of channel slope, as suggested by Govers (1992) for mobile bed rills. Conclusions The investigation highlighted that the effect of sediment transport on rill flow resistance is almost negligible for most of the cases and that the experimental procedure for fixing rills caused the unexpected slope independence of flow velocity.

Rills caused by runoff concentration on erodible hillslopes generally have very irregular longitudinal profiles and cross-section shapes. Rill erosion directly depends on the hydraulics of flow within the rills which may differ greatly from that in larger and regular channels like streams or rivers. At first, in this paper, a review of the two different approaches to estimate rill flow velocity, based on flow regime and flow resistance laws, is presented. Recent advances in measurements of rill geometry by the three-dimensional photo-reconstruction (3D-PR) technique, which allows one to obtain a Digital Terrain Model (DTM) by low-altitude aerial and terrestrial photogrammetry, are also discussed. Indeed, information on rill geometry is also necessary to estimate flow velocity and depth using flow resistance equations. Then, the results of related laboratory and field investigations carried out from 1984 to 2022 are reported. In particular, the review was developed considering the studies conducted investigating the different components of rill flow resistance due to grain resistance, morphological resistance (step-pool), and sediment transport. Moreover, a brief mention of the slope threshold, equal to 18%, determining differences in hydraulic and sediment transport variables, is done. Also, an analysis is developed to demonstrate that a theoretical rill flow resistance law has been erroneously presented as an example of spurious correlation. Finally, some research needs on rill flow hydraulics are highlighted.

The main purpose of this article is to give a general idea of the scientific activity that was carried out starting from the 2000s on the basis of the data collected in the plots installed at the Sparacia experimental station for soil erosion measurement in Sicily, South Italy. The paper includes a presentation of the experimental site, a description of the methods and procedures for measuring soil erosion processes both available in the literature and applied at the Sparacia station (sediment sampling and water level reading in the storage tanks for total erosion measurements; profilometer, and Structure from Motion technique for rill erosion measurements), and the main results obtained in the monitoring period in the experimental site. The latter concern the effects of plot size and steepness on soil loss, the measurement variability, the frequency analysis of soil loss, the rill erosion characterization, and the comparison between rill and interrill erosion rates. Each of these topics is addressed with multi-temporal analyses performed with increasing size of the available database, which allowed to draw robust conclusions. Soil loss did not vary appreciably with plot length in contrast with the assumption made in the USLE/RUSLE. The variability of the measurements of soil loss, runoff volume, and sediment concentration at the event scale in replicated plots decreased as the mean measured value increased. The normalized event soil loss was distributed according to a two-component distribution. A power relationship between rill volumes and lengths was established. The measurements also confirmed the morphological similarity between the channels of the rills and ephemeral gullies described by a power dimensionless relationship. Rill erodibility of the sampled clay soil varied over time, maintaining relatively low values. Finally, rill erosion was dominant relative to interrill erosion, and a more efficient sediment transport system through the rill network occurred as plot steepness increased.

The applicability of a theoretical rill flow resistance equation, based on the integration of a power velocity distribution, was tested using measurements carried out in mobile and fixed bed rills, shaped on plots having different slopes (9, 14, 15, 18, 22, 24, 25 and 26%) and soil textures (clay fractions ranging from 32.7% to 73% and silt of 19.9–30.9%), and flume measurements available in the literature. The Darcy–Weisbach friction factor resulted dependent on the slope, Froude number, Reynolds number and clay and silt percentages, used as variables representative of soil transportability and detachability, respectively. This theoretical approach was applied to two different databases distinguished by the slope threshold of 18%. The results showed that for both steep (≥18%) and gentle (

Flow velocity is one of the most important hydrodynamic variables for both channelized (rill and gullies) and interrill erosive phenomena. The dye tracer technique to measure surface flow velocity Vs is based on the measurement of the travel time of a tracer needed to cover a known distance. The measured Vs must be corrected to obtain the mean flow velocity V using a factor αv=V/Vs which is generally empirically deduced. The Vs measurement can be influenced by the method applied to time the travel of the dye-tracer and αv can vary in different flow conditions. Experiments were performed by a fixed bed small flume simulating a rill channel for two roughness conditions (sieved soil, gravel). The comparison between a chronometer-based (CB) and video-based (VB) technique to measure Vs was carried out. For each slope-discharge combination, 20 measurements of Vs, characterized by a sample mean Vm, were carried out. For both techniques, the frequency distributions of Vs/Vm resulted independent of slope and discharge. For a given technique, all measurements resulted normally distributed, with a mean equal to one, and featured by a low variability. Therefore, Vm was considered representative of surface flow velocity. Regardless of roughness, the Vm values obtained by the two techniques were very close and characterized by a good measurement precision. The developed analysis on αv highlighted that it is not correlated with Reynolds number for turbulent flow regime. Moreover, αv is correlated neither with the Froude number nor with channel slope. However, the analysis of the empirical frequency distributions of the correction factor demonstrated a slope effect. For each technique (CB, VB)-roughness (soil, gravel) combination, a constant correction factor was statistically representative even if resulted in less accurate V estimations compared to those yielded by the slope-specific correction factor.

Rills caused by run-off concentration on erodible hillslopes have very irregular profiles and cross-section shapes. Rill erosion directly depends on the hydraulics of flow in the rills, which may differ greatly from hydraulics of flow in larger and regular channels. In this paper, a recently theoretically deduced rill flow resistance equation, based on a power–velocity profile, was tested experimentally on plots of varying slopes (ranging from 9% to 26%) in which mobile and fixed bed rills were incised. Initially, measurements of flow velocity, water depth, cross-section area, wetted perimeter, and bed slope, carried out in 320 reaches of mobile bed rills and in 165 reaches of fixed rills, were used for calibrating the theoretical flow resistance equation. Then the relationship between the velocity profile parameter Γ, the channel slope, and the flow Froude number was separately calibrated for the mobile bed rills and for the fixed ones. The measurements carried out in both conditions (fixed and mobile bed rills) confirmed that the Darcy–Weisbach friction factor can be accurately estimated using the proposed theoretical approach. For mobile bed rills, the data were supportive of the slope independence hypothesis of velocity, due to the feedback mechanism, stated by Govers. The feedback mechanism was able to produce quasicritical flow conditions. For fixed bed rills, obtained by fixing the rill channel, by a glue, at the end of the experimental run with a mobile bed rill, the slope independence of the flow velocity measurements was also detected. Therefore, an experimental run carried out by a rill bed fixed after modelling flow action is useful to detect the feedback mechanism. Finally, the analysis showed that, for the investigated conditions, the effect of sediment transport on the flow resistance law can be considered negligible respect to the grain roughness effect.

In this paper, the morphology of step–pool features is analysed using rill measurements and literature data for streams. Close-range photogrammetry was used to carry out ground measurements on rills with step–pool units, shaped on a plot having slope equal to 14, 15, 22, 24 and 26%. Data were used to compare the relationships between H/L, in which H is the step height and L is the step length, and the mean gradient of the step–pool sequence, Sm, for streams or the slope of the step–pool unit, S, for rills. The relationship of H/L against Sm is widely used to test the occurrence of the maximum flow resistance condition in streams, which is associated with the range 1 ≤ (H/L)/Sm ≤ 2. Further analyses were carried out to compare both the formation process and the profile of the pool in rills with those related to streams. Moreover, for a single rill channel, an analysis of flow characteristics expressed in terms of Darcy–Weisbach friction factor and Froude number was developed. The results allowed us to state: (i) the relationships of H/L versus Sm and S are quite similar and the steepness ratio for streams, (H/L)/Sm, and for rills, (H/L)/S, generally ranges from 1 to 2; (ii) the formation process and the profile of the pool in rills are not consistent with those occurring in streams; (iii) in the rills, the longitudinal size of the pool is dominant with respect to the maximum scour depth; (iv) the presence of a sequence of step–pool units within a rill segment noticeably increases flow resistance compared to segments with a flat bed; (v) the Froude number of the flow over the sequence of step–pool units in rills is slightly below the range of 0.8–1 corresponding to the maximum flow resistance in step–pool units.

In this paper a recently theoretically deduced rill flow resistance equation, based on a power-velocity profile, was tested experimentally on plots of varying slopes and soil texture in which mobile bed rills are incised. Measurements of flow velocity, water depth, cross section area, wetted perimeter and bed slope conducted in rill reaches incised on experimental plots, having different slope values (9, 14, 22, 24 and 26%) and soil texture (clay fraction ranging from 42 to 73%), and literature data were used to calibrate the flow resistance equation. In particular, the relationship between the velocity profile parameter Γ, the channel slope, the flow Froude number and texture fractions was firstly calibrated using 147 rill reach data. Then this relationship was tested using 126 measurements carried out with soils having different texture (percentage of clay ranging from 9.6 to 73%) and slopes (6.9%–26%). The measurements allowed to establish that a) the Darcy-Weisbach friction factor can be accurately estimated using the proposed theoretical approach, and b) the data were supportive of the soil texture influence on rill velocity and flow resistance.

In this paper the results of a plot experiment on rill erosion are reported. The rill network, manually incised on the soil and further shaped by a clear inflow discharge, was surveyed using the three-dimensional photo-reconstruction (3D-PR) technique which allows to obtain a digital terrain model (DTM) by a large series of oblique images of the channel from consumer un-calibrated and non-metric cameras. The three-dimensional (3D) DTM and the quasi-tridimensional (2.5D) model were generated by Agisoft Photoscan software. For a single rill channel, the reliability of the 3D image-based ground measurements of morphological and hydraulic variables was positively tested by the corresponding measurements carried out on the rill gypsum cast. Moreover, the morphological and hydraulic variables were measured at 11 transects of the plot by a profilometer P, the 2.5D and 3D models. Using the 3D model as reference, the analysis showed that the reliability of the measurements by P and 2.5D methods was comparable and the error was generally lower than 15%. The rill measurements carried out by the three methods agreed with the available literature measurements and supported the applicability of both the empirical relationship between rill length and its eroded volume and the theoretical dimensionless relationship among the morphological variables describing the channelized erosion process. The analysis also showed that the effect of the distance between two consecutive cross sections on the rill volume measurement by 3D model is negligible. Finally, the shaping effect of different flow discharges on three straight rills incised into the plot was tested. In particular, three increasing flow discharges (0.15, 0.35 and 0.5 l s − 1 ) were used sequentially in two rills while the maximum discharge of the sequence (0.5 l s − 1 ) was used into the third rill. Both the grain-size distribution of the eroded sediment and the total amount of soil loss were affected by the discharge sequence.

In this paper, a new flow resistance equation for rill flow was deduced applying dimensional analysis and self‐similarity theory. At first, the incomplete self‐similarity hypothesis was used for establishing the flow velocity distribution whose integration gives the theoretical expression of the Darcy–Weisbach friction factor. Then the deduced theoretical resistance equation was tested by some measurements of flow velocity, water depth, cross section area, wetted perimeter, and bed slope carried out in 106 reaches of some rills shaped on an experimental plot. A relationship between the velocity profile, the channel slope, and the flow Froude number was also established. The analysis showed that the Darcy–Weisbach friction factor can be accurately estimated by the proposed theoretical approach based on a power–velocity profile.

For an erosion event (October 2016) occurred at the Sparacia experimental area (Southern Italy), both terrestrial and low-altitude aerial surveys were carried out by consumer grade camera and quadcopter (low-cost unmanned aerial vehicle [UAV]) to measure rill erosion on two plots with steepness of 22% and 26%. Applying the structure from motion (SfM) technique, the three-dimensional digital terrain models (3D-DTMs) and the quasi three-dimensional models (2.5D-digital elevation model [DEM]) were obtained by the two surveys. Furthermore, 3D-DTM and DEM were built using the available aerial photographs (166) and adding 40 terrestrial photographs. For the first time, the convergence index was applied to high-resolution rill data for extracting the rill network, and a subsequent separation into contributing and non-contributing rills was carried out. The comparison among the three surveys (terrestrial, UAV, and UAV+terrestrial) was developed using two morphometric parameters of the rill network (drainage density and drainage frequency). Moreover, using as reference the weight of sediment stored on the tanks located downstream of the plots, the reliability of soil loss measurement by 3D models was tested. For both contributing and non-contributing rills, the morphometric parameters were higher for the terrestrial than for UAV and UAV+terrestrial surveys. For both plots, SfM always provided reliable soil loss measurements, which were affected by errors ranging from −8% to 13%. Although the applied technique used a low-cost UAV and a consumer grade camera, the obtained results demonstrated that a reliable estimate of rill erosion can be obtained in an area of interest.

This paper focuses on the dissipative similarity of step-pool units at rill, flume and stream scale. This investigation is carried out using recent advances in open channel flow resistance, applications of close-range photogrammetry to rill erosion, available published data on step-pool features in flumes and streams and a new dataset of measurements in fixed bed step-pool rills. A theoretically-based equation for calculating the Darcy-Weisbach friction factor obtained by integration of a power velocity profile is presented. The scale factor Γ of this power velocity profile, which is included in the flow resistance equation, was previously calibrated (Eq. 10) for mobile bed rills with step-pool units. At first, in this investigation an additional test of this flow resistance equation is developed using measurements carried out in fixed bed rills with step-pool features. In particular, the proposed Γ function is tested using the measurements carried out in 63 fixed bed rill reaches incised on a 24 and 26% sloped plot. This test shows that the effect of sediment transport on the flow resistance law can be considered negligible as compared to the effect of the form-induced dissipative mechanisms due to the presence of step-pools structures. Then, using measurements carried out in 61 flume experimental runs and 109 reaches of step-pool streams, this investigation demonstrates that the Γ function for flume and stream conditions can be obtained scaling that (Eq. 10) determined for step-pool rills. The Darcy-Weisbach friction factor values measured in flumes and streams are, on average, higher than those related to rills. Finally, the application of the theoretical equation for calculating the Darcy-Weisbach friction factor for flume and stream conditions requires Γ, estimated by Eq. (10), to be multiplied by a specific scale factor.

Flow velocity controls hillslope soil erosion and is a key hydrodynamic variable involved in sediment transport and deposition processes. The dye-tracer technique is one of the most applied methods for measuring velocity of shallow interrill and rill flow. The technique is based on the injection of a tracer in a specific point and the measurement of its speed to travel the known distance from the injection point to a given channel section. The dye-tracer technique requires that the measured surface flow velocity has to be corrected to obtain the mean flow velocity using a correction factor which is generally empirically deduced. The technique has two sources of uncertainties: i) the method applied for measuring the travel time of the dye-tracer and ii) the estimate of the correction factor, which is the ratio between the mean flow velocity and the surface velocity, in different flow conditions. In this paper the results of a wide experimental investigation, carried out using a fixed bed small flume simulating a rill channel, are presented. At first, the comparison between a chronometer-based (CB) and video-based (VB) technique was carried out for establishing the influence of the travel time measuring technique. For each experimental run, which was characterized by a sample of 20 measurements carried out with the same values of slope and discharge, the developed analysis showed that the empirical frequency distribution of the ratio between the single measurement and the sample mean (i.e., the average of 20 measurements) is more uniform for the VB technique than for the CB one. In any case, this sample mean was representative of surface flow velocity for both the CB and the VB technique. Furthermore, the mean value obtained by the CB measurements lightly underestimated (−1.7%) the corresponding mean obtained by the VB technique. Finally, the effects of slope (0.1–8.7%), flow Reynolds number (3462–10040), Froude number (1.44–5.17) on the correction factor are presented. The analysis demonstrated that the correction factor is independent of flow Reynolds number while a relationship with a Froude number, obtained by surface velocity measurement, or channel slope can be established.

Water flow on hillslope soil surface supplies energy which is required to detach soil particles, to transport and deposit sediments, therefore flow velocity is a key variable related to hillslope hydrodinamics of soil erosion processes. Among the different methods available for measuring velocity of shallow interrill and rill flow, the trace technique is widely used. Trace technique is applied by adding a material (salt, magnetic material, water isotope, floating object) and then measuring the speed of the material to travel a known distance from the injection point. When flow velocity is measured using a dye-tracing method, the mean velocity is calculated by multiplying the measured surface velocity of the leading edge of the tracer plume by a correction factor which was generally empirically deduced. The main uncertainty of the dye-tracing technique stands in the estimate of this correction factor, which is the ratio between the mean flow velocity and the surface velocity. The main goal of this paper is establishing a theoretical relationship for calculating the correction factor by using a power velocity profile. At first, the developed analysis demonstrated that the correction factor only depends on the exponent of the power velocity distribution (Eq. (5) ). Then, this theoretical expression of the correction factor was applied using the velocity profiles measured by Baiamonte and Ferro for the condition of sediment-free flow in motion on a rough bed and by Coleman for a sediment-laden flow moving on a smooth flume. Using the correction factor values calculated by the velocity measurements carried out for the sediment-free flow in motion on a rough bed, the relationship between the correction factor and roughness height was established. In agreement with a previous study by Ali et al., for a sediment-free flow the correction factor increases with roughness height. Finally, the velocity profile measurements carried out for a sediment-laden flow in motion on a smooth bed were used to state the effect of sediment load on the correction factor. This last analysis allowed to conclude, in agreement with Li and Abrahams and Zhang et al., that the correction factor decreases when the sediment load increases.

In this paper, 1017 rainfall events from 2008 to 2017 are used to identify the rainfall threshold that produces upland erosion at the Masse (central Italy) and Sparacia (southern Italy) experimental stations. The rainfall events are classified into three classes: non-erosive, interrill-only and rill. The threshold values for separating as correctly as possible the erosive rains (case I) and the rill rains (case II) are derived solely from the hyetograph. Each threshold value is obtained by imposing that the long-term erosivity of the events above the threshold is equal to the long-term erosivity of all erosive events (case I) or only rill events (case II). The performances of selective criteria based on 31 threshold variables are compared, and those most effective in separating erosive and rill events are identified. The identification of the best criterion depends on the aim of the analysis. It could be required to provide the greatest accuracy for separating erosive and rill events or the lowest error in the prediction of long-term erosivity. In general, the results clearly show that the best variables are those that quantify the characteristics of rainfall patterns, such as rain showers (periods of continuous rain) and the deviations in the rain records over a truncation level. These results are especially significant for the operational estimation of rainfall erosivity and for identifying the trigger of the erosion process and rill development by using only a hyetograph. This is obtained by relatively simple field measurements and is also widely available on a global scale. The most effective variables are potentially usable in water erosion prediction models as proxies of variables that are more rarely available and/or more difficult to measure.

Water erosion is one of the most important soil degradation processes and rill erosion contribution to total soil loss is usually dominant as compared to interrill erosion. Rill erosion modelling requires that rill flow has to be adequately modelled. Flow depths in rills are typically of the order of millimeters to several centimeters and bed topography, characterized by steep slope values, significantly affects flow hydraulics. In this paper, a new technique for measuring the water depth inside a rill channel is proposed and the effects on flow resistance estimate are examined. This technique couples an accurate ground survey of the rill channel, obtained by close-range photogrammetry, with the survey of the water tracks inside the channel. The comparison between the water depth measurements carried out by a micro-hydrometer (local technique) and those obtained by the water tracks (volumetric technique) demonstrated that the data pairs are characterized by a wide scattering respect to the line of perfect-agreement. Furthermore, the hydraulic radius values obtained by the proposed volumetric technique are always greater than those corresponding to the local technique. The developed analysis showed that the volumetric technique is affected by the adopted cross-section spacing even if reliable results can be also obtained using a distance interval which is equal to ten times the best investigated reference condition, which corresponds to the minimum tested distance interval, having the same order of magnitude of the mean 3D-DTM resolution. Using of the volumetric measurement technique for the water depth improved the agreement between the measured Darcy-Weisbach friction factor values and those calculated by the theoretical flow resistance law based on a power-velocity profile. Finally, a constant scale factor (equal to 0.8257) was proposed to convert the Γ function of the power velocity profile obtained by local measurements into that corresponding to the volumetric technique.