Quantifying the role of bed surface topography in controlling sediment stability in water-worked gravel deposits

Transcription

1 WATER RESOURCES RESEARCH, VOL. 44,, doi: /2006wr005794, 2008 Quantifying the role of bed surface topography in controlling sediment stability in water-worked gravel deposits Richard Measures 1 and Simon Tait 2 Received 29 November 2006; revised 3 December 2007; accepted 17 December 2007; published 11 April [1] This study uses measured bed topography scans and fractional bed load transport rate data obtained from previous laboratory studies of gravel transport to examine the influence of the local grain-scale topography on grain entrainment and fractional transport rates. A simple steady state fluid model is developed to impose fluid forces onto the exposed areas of the bed surface. Image analysis is used to identify the surface grains and simple physically based algorithms are used to include the effects of sheltering. Comparisons of modeled and experimental fractional transport rates show that remote sheltering has a very strong influence on grain entrainment. With the effects of remote sheltering included, the model was found to be able to predict fractional transport rates well for a range of bed topographies and feed conditions. Further studies using a wider range of bed shear stresses would allow the effects of turbulence and bed re-arrangement to be included in the model. Citation: Measures, R., and S. Tait (2008), Quantifying the role of bed surface topography in controlling sediment stability in waterworked gravel deposits, Water Resour. Res., 44,, doi: /2006wr Introduction [2] Of the numerous sediment transport equations used currently in gravel bed rivers, most were formulated from concepts originally developed to deal with the movement of uniform sized sediments under steady flow conditions. The most common concept relates the rate of sediment transport to the degree by which the hydraulic conditions are above the threshold of grain motion, where the threshold of grain motion is a function of grain size. A key difficulty in the application of many bed load transport rate equations is determining an appropriate threshold for grain motion. Accurate estimation of incipient motion of grains is still a significant problem in many sediment transport studies. Many researchers still use a standard or modified form of the function empirically derived by Shields [1936]. Despite eight decades of incipient motion studies, limited progress has been made to explain satisfactorily the physical causes of the wide variation in observed threshold behavior, particularly in gravel bed rivers [Buffington and Montgomery, 1997]. [3] Bed load formulae have been extended to mixed grain size sediment deposits using representative grain size, selected on the basis of previous experience [Meyer-Peter and Muller, 1948]. This method assumes that mixed grain size beds behave as beds composed of uniformly sized sediment of this representative grain size. The concept has been further developed by sub dividing the grain population into grain size fractions and then performing transport rate calculations on each fraction separately. However, it has been observed that fractions within graded sediment can influence the mobility of one another. It was 1 Halcrow Group Ltd, Leeds, UK. 2 School of Engineering Design and Technology, University of Bradford, Bradford, UK. Copyright 2008 by the American Geophysical Union /08/2006WR noted that the larger fractions are more mobile than expected while the finer grain size fractions are much more stable than expected [Andrews, 1983]. These observations have led to the development of empirical hiding functions to account for this change in entrainment thresholds. Sutherland [1992] presented a review of the development of a large number of such hiding functions, each associated with a particular transport equation. Generally, the performance of a particular hiding function deteriorates when it is applied to data which was not used in its calibration; this implies a lack of generality in the concept. Parker et al. [1982] discussed the equal mobility hypothesis, which rests on two empirical results both derived from data obtained from Oak Creek by Milhous [1973]. These results suggest that the threshold for each size fraction is independent of grain size and that perfect similarity exists between the fractional transport rate relations. Parker et al. [1982] demonstrated that these conditions held approximately for Oak Creek and then used them to construct a general model for predicting fractional transport rates. However, Diplas [1987], in a reanalysis of the Oak Creek data, showed that important departures from equal mobility were present in the data indicating that perfect similarity of the fractional transport relations did not hold. In recognition of this, Diplas [1987] used the Oak Creek data to derive a hiding function that depended on shear stress, acknowledging that bed load composition varied strongly with shear stress. Wilcock [1992] used fractional transport rate data for a wide range of mixtures and demonstrated that there appears to be no systematic patterns linking their fractional transport relations. This strongly suggests that the derivation of a generalized empirical hiding function will not be possible. The most recent models of graded sediment transport have been based on the observed grain size composition of the sediment surface [Wilcock and Crowe, 2003; Wilcock and McArdell, 1993]. They suggested that fractional transport 1of17

2 MEASURES AND TAIT: BED TOPOGRAPHY AND SEDIMENT STABILITY rates for finer grain sizes are a function of the surface grain size distribution and that for coarser sizes there was a dependence on the proportion of that grain size that remained immobile at a particular flow condition. In practical situations a limitation of such models is that the surface composition is often a consequence of transport and as such is often unknown and is a desired prediction rather than a known variable. [4] Field observations by Buffington et al. [1992] and Church et al. [1998] suggested that the resistance of grains to entrainment from naturally formed gravel beds is not related simply to grain weight as described by the surface grain size distribution but to some other property of the bed, possibly the arrangement of the surface grains and topography of the bed surface. These field observations suggest that the use of models that link transport simply to surface grain size distribution may over predict sediment transport because they ignore the contribution to bed stability caused by different patterns of grain topography observed in waterworked deposits. Church et al. [1998] suggested that entrainment thresholds may double because of the development of grain-scale structures under certain transport conditions in gravel bed rivers. The concept of textural influences supports previous laboratory work by Kirchner et al. [1990] which indicated that the range and size of grain fiction angles within grain size fractions depended on local grain topography as well as grain size. Detailed examination of bed load transport rates and compositions by Hassan and Church [2000] suggested that over static armored gravel beds formed in the laboratory, up to 50% of the bed shear stress could be carried by small, well developed grain scale structures. [5] It is likely that empirical hiding functions, derived from observations of average boundary shear stress and bed load, carry information on the stabilizing effects of grain scale organization and its interaction with the near bed turbulent flow. However, their empirical nature means it is difficult to examine systematically the contribution to grain stability from the individual processes. More detailed laboratory studies on grain entrainment have provided evidence of the processes which may enhance grain stability. Observations have indicated that the enhanced stability may be caused by the sheltering of a grain in which either the upstream projected area is reduced or the near bed flow is adjusted by adjacent upstream grains [Schmeeckle and Nelson, 2003] or by the development of grain geometries which result in larger fluid forces being required to entrain grains [Hassan and Church, 2000]. [6] Detailed modeling studies using discrete particle models (DPM) based on spherical, or spherically based particles have suggested that grain geometries can have a significant impact on the conditions under which grains are entrained [McEwan and Heald, 2001; McEwan et al., 2004; Heald et al., 2004; Calatroni et al., 2004]. Other studies have attempted to relate the patterns of near bed flow velocity to the presence of large exposed grains creating zones of reduced velocity in their lee in which other grains are less likely to be entrained [Schmeeckle and Nelson, 2003]. In either case the reduction in the likelihood of entrainment is related to the relative geometries of neighboring particles. In the above DPM studies the geometries of the sediment surfaces have been conceptual in that they involved the modeling or measurement in situations using spherical particles that have not been arranged in the surface patterns typically seen in water worked sediment deposits. Several researchers now routinely measure the grain scale topography of water worked sediment beds both in the laboratory and the field [e.g., Willetts et al., 1998; Nikora et al., 1998]. Nikora et al. [1998] used a one dimensional second order structure function to investigate the structure of streamwise transects. This was extended to two dimensions by Goring et al. [1999] for application to bed areas rather than line transects. Similar higher order statistical functions have been used to describe objectively the surface geometries of water worked gravel beds [Marion et al., 2003; Nikora and Walsh, 2004]. These statistical functions have been used to identify bed forms of different scale, to discover whether deposits are characteristic of the upstream sediment boundary conditions and even the direction of the mobilizing flows. However, no studies have examined the link between surface topography, grain entrainment and bed load transport explicitly. [7] The aim of this study is to use existing laboratory data to examine the influence of the local grain-scale topography on grain entrainment and estimated fractional transport rates. The study examines the relative importance of adjusted physical exposure and fluid sheltering independently. 2. Experimental Data Sources [8] It is relatively rare for both the bed surface topography and grain entrainment thresholds of the gravel deposits to be measured in the same study. This paper uses data from tests reported by Willetts et al. [1998] in which the bed surface topography, bed load composition and transport rate were measured. [9] The data set used was collected during a large research program, led by the Universities of Aberdeen and Glasgow at HR Wallingford Ltd s facilities in the UK. These tests were designed to investigate the development of armored surface layers under different conditions of flow and upstream sediment feeding. During the tests, the flow rates were steady, while the upstream sediment conditions were varied by use of a conveyor belt at the upstream end of the flume. Experiments with no upstream sediment feed were also conducted. The sediment used was a washed, bimodal natural river gravel, with D 16 = 1 mm, D 50 = 4 mm, D 84 = 8 mm and a standard deviation of approximately 2.8. The channel was 800 mm wide and 18 m long with a flow depth of mm. Initial bed slope was set at in all the tests. [10] In the tests the upstream feed conditions were varied from zero to 5.0 g/s. In all tests the fed material composition corresponded to the initial mixture with the coarsest fractions removed. In the first two tests (f and g) the mixture was placed in the flume and a steady flow applied with zero sediment feed until a static armor developed. In the other tests (h, i, l, m and n) the sediment input rate was fixed using a conveyor belt system and feed continued until a state of equilibrium was achieved. This was estimated by weighing the collected bed load samples. A dynamic armor was now formed in equilibrium with the imposed flow and sediment input conditions. Once this had happened the sediment feed was stopped and the bed degraded creating a static armor layer that progressively moved down the flume. Further details of the changes in bed load transport 2of17

3 MEASURES AND TAIT: BED TOPOGRAPHY AND SEDIMENT STABILITY rate with time are given by Marion et al. [2003]. Details of the experimental conditions are given in Table 1. [11] Bed load was measured 16.7 m downstream of the upstream feed conveyor. Bed load samples were collected using a slot trap with collection boxes which were changed at frequent intervals during each test. The bed topography was measured using a computer controlled laser displacement sensor. It was moved over the bed surface in a grid pattern, with vertical displacement measurements being obtained every 0.5 mm. The bed elevations were measured using a Keyence LAS8010 laser displacement sensor. The sensor could measure to a precision of ±0.1 mm in the vertical with this measurement being integrated over an area of 50 mm in diameter. The vertical elevations were measured relative to an arbitrary datum position that was dependent on the vertical position of the laser displacement sensor. It had a valid measurement range of ±8 mm relative to this zero position. Given the grain size distribution of the sediment deposits the vast majority of the point elevation measurements are within this range. However, due to occasional large variations in bed elevation a small number of measurements were outside this range. [12] The area of bed measured was 192 mm by 192 mm in area and was located on the centerline of the flume 1.5 m upstream of the bed load trap. It was believed that the resolution of the sensor and the measurement grid produced bed surface topography data that was capable of describing the key features of the bed surface arrangement at a grain scale. 3. Estimation of Entrainment Thresholds and Influence of Bed Surface Topography [13] The study aimed to quantify the influence of grain scale topography on grain entrainment. It was believed that there may be two types of topographical effect present; direct sheltering, in which the physical position and the geometry of the adjacent grains influence entrainment; and remote sheltering, in which a more remote part of the bed surface changes the local flow conditions and alters the probability of a grain s entrainment. Work was then carried out to develop an approach so that a quantitative assessment could be made of the relative importance of direct and remote sheltering on gravel beds. [14] This was accomplished by first assuming a particular type of vertical velocity profile close to the bed following the work of Nikora et al. [2004]. This velocity distribution was then applied to the measured surface topography. An appropriate momentum balance was used to estimate the fluid velocities to be applied to elemental areas of each exposed grain, thus allowing an estimate of applied fluid force on identified individual grains (including local sheltering). The probability of entrainment of each grain has been estimated based on the ratio of an estimated grain weight and applied fluid force. This allows the number of grains susceptible to be entrained for different bed topographies to be estimated for different levels of bed shear stress. By calculating the displacement of entrained grains using the relationship proposed by Wilcock [1997] it was possible to estimate fractional bed load transport rates. [15] The effect of remote sheltering was then incorporated separately by adjusting the near bed velocities based on the experimental observations reported by Schmeeckle and Nelson [2003] of the mean flow velocity adjustment pattern behind a spherical grain. [16] Forty seven bed surface topography data sets were investigated in this study, representing data collected at different times during seven different armoring experiments. In order to apply the analysis approach to all of these data sets it was necessary to develop a number of programming tools to: Identify grains, their upstream projected areas and weights; apply and modify the near bed flow field; estimate entrainment thresholds; estimate fractional bed load transport rates; and then compare these with observed bed load compositions. The computational tools to carry out the analysis were developed in Matlab. The structure of these computational tools is show in Figure 1. [17] The analysis process involves six stages: [18] 1. Filtering and detrending the bed surface topography data. [19] 2. Identifying grains on the bed surface using automated image analysis techniques. [20] 3. Calculating the small areas of grain exposed to streamwise flow (upstream projected area) [21] 4. Applying a velocity distribution to the upstream projected area of each grain and using this to estimate fluid drag forces acting on the grains, while maintaining an overall momentum balance. [22] 5. Linking applied fluid forces to grains and extracting information about the properties of each grain s weight to estimate susceptibility to entrainment. [23] 6. Integrating all the grains susceptible to entrainment and then estimating potential fractional transport rates for comparison with measured bed load data. [24] The results were then used to examine the relative influences of direct and remote sheltering for different water-worked gravel bed topographies, formed under static and dynamic armoring conditions Filtering and Detrending Bed Surface Topography Data [25] In the forty seven topography data sets used in this study, the typical number of out of range readings was Table 1. Summary of Test Conditions Experiment Discharge, l/s Upstream Sediment Feed Rate, g/s Feed Duration, min Bed Slope Uniform Water Depth, mm Flow Condition Number of Texture Surveys f inbank 4 g overbank 7 h inbank 10 i inbank 9 l overbank 5 m inbank 5 n overbank 7 3of17

4 MEASURES AND TAIT: BED TOPOGRAPHY AND SEDIMENT STABILITY The magnitude of the change of elevation gradient at each point (see Figure 2b) is then calculated by equation (1). d 2 z dx; ð y Þ 2 ðx;yþ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2! u 2 d ¼ t 2 z þ d2 z dxð 2 x;yþ dyð 2 x;yþ ð1þ Figure 1. Sketch showing the structure of the grain entrainment analysis. approximately equal to 0.04% of the total data points, with the poorest data set having 0.57% of the measurement locations with out of range values. In order to facilitate matrix handling operations, these out of range values were replaced with the average elevation of the adjacent eight measured bed elevations. The data was then detrended in both the lateral and streamwise directions. This was done to remove any bias in the data, caused either by long wavelength bed features or by the measurement frame not being parallel with the water worked bed. Detrending was carried out by performing a simple linear regression to calculate the gradient of any underlying trend in both streamwise and lateral directions and then removing that trend from the raw bed topography data. Part of a typical bed elevation data set following filtering and detrending is shown in Figure 2a Analysis to Identify Individual Grains-Estimation of Upstream Projected Area and Weight [26] Several previous studies, for example, McEwan et al. [2000] and Sime and Ferguson [2003] have developed Automated Image Analysis (AIA) as a tool for extracting grain size information from images of gravel sediment deposits. The AIA technique used in this paper broadly follows the processes reported in these papers but particular attention had to be given to edge detection as the resolution of data available for this study was lower (in comparison to the grain size) than in these papers. This lower resolution also necessitates a different approach to small grains. [27] Edge detection of individual grains was carried out using the Canny method [Canny, 1986] followed by dilation and watershed erosion. The various stages of the image analysis process are shown in Figure 2. [28] First the elevation gradient between each measurement point and the points surrounding it are calculated. Following this the change in elevation gradient at each point is calculated in the streamwise and transverse directions. [29] In order to ensure grain edges are only one pixel wide potential edge points are identified using non-maximal suppression. This is carried out by disregarding any points which are not local maxima in either the streamwise or transverse direction (see Figure 2c). [30] Thresholding is then carried out with an upper and lower hysteresis threshold. Edges are seeded from potential edge points with elevation gradients greater than an upper threshold (see Figure 2d). In order to reduce erroneous edges isolated seed pixels are removed; this process is known as edge tidying (see Figure 2e). The seeds are then extended to surrounding potential edge points with values greater than the lower threshold (see Figure 2f) in order to produce more complete edges (see Figure 2g). [31] In order to improve edge connectivity watershed erosion is then carried out. This involves dilating the identified edges several pixels (see Figure 2h) then eroding the dilated image while maintaining its connectivity (see Figure 2i). [32] The closed regions made by the identified edges represent the grains. These regions are numbered and geometrical information about each grain is extracted. [33] It was found that this image identification process was sensitive to the thresholds selected and amount of dilation carried out during watershed erosion. An iterative approach was used to find appropriate thresholds. An assessment of the effectiveness of the grain identification process was made visually by comparing images of bed elevations and identified grains (such as the ones shown in Figures 2a and 2i). The assessment showed that once calibrated the automated image analysis correctly identified most of the medium to large surface grains within all the measured bed topography data. Consistent thresholds were used for the analysis of all the beds within the experimental data set. [34] Once individual grains have been identified then the upstream projected area of each grain, A dx.dy, is calculated by using the height differences between adjacent measurements to estimate the area of bed surface perpendicular to the flow for each grain (Figure 3). The calculation of projected area does not take into account remote sheltering as it merely investigates the geometry of the exposed part of each grain. [35] The upstream projected area of each grain and the midpoint elevation of the projected area, z A are calculated using equations (2) and (3). A dx:dy ðx; yþ ¼ ðzx; ð yþ zx ð dx; yþþ:dy ð2þ where dx and dy represent the difference between consecutive elevation measurement points in the streamwise (x) and lateral (y) directions. z dx;dy ðx; y Þ ¼ zx; ð y Þþzx ð dx; y Þ 2 [36] Data about each grain is then extracted from the labeled image obtained from the AIA process. This data ð3þ 4of17

5 MEASURES AND TAIT: BED TOPOGRAPHY AND SEDIMENT STABILITY Figure 2. Stages of the automated grain identification process. includes the long axis dimension of the grain (on plan), a, taken to be the distance between the two points on the surface of the grain which are furthest separated, and the short axis dimension of the grain (on plan), b, taken as the maximum dimension of the grain perpendicular to the long axis on plan. Grain volume is then calculated by assuming that each grain is an ellipsoid with its long axis equal to the long axis dimension of the grain and both other dimensions equal to the short axis dimension of the grain. Grain mass, m g, is then calculated: m g;n ¼ 4p: ð a n=2þ: ðb n =2Þ 2 3 r s ¼ p:a n:b 2 n r 6 s where r s = grain density and n denotes a specific grain. [37] Once information on the size and mass of each grain was generated the grain size distribution could be found by sieving the identified grains according to their short axis dimension. Both McEwan et al. [2000] and Sime and Ferguson [2003] showed that grain size distributions obtained using AIA compare well to data obtained by ð4þ sieving. Attempts were made to compare the grain size distribution (GSD) of identified grains with physically sieved, experimental grain size distribution data. During the experimental program bulk samples of bed material as well as wax samples of the initial and final surface grain size distributions were taken for each test. Further details of the sampling and the processing of the raw areal data to provide equivalent bulk volumetric samples are described by Marion [1996]. [38] Initially this comparison revealed that the automated image analysis was poor at identifying small grains. It was found that fines, especially those with short axis dimension less than 2 pixels (1 mm) were impossible to identify reliably. The comparison was also hampered by difficulties in comparing samples obtained by different sampling techniques, particularly when considering surface grain size distributions. This problem is well documented and tools for converting areal to volumetric samples have been developed by others [e.g., Marion and Fraccarollo, 1998]. [39] It was possible to convert the area by weight GSD obtained through image analysis of a non-water worked bed to a volume by weight GSD for comparisons with the bulk 5of17

6 MEASURES AND TAIT: BED TOPOGRAPHY AND SEDIMENT STABILITY Figure 3. Estimation of projected grain areas using bed topography data. GSD obtained by sieving (Figure 4, top panel). It was also possible to compare the image analysis GSD with that from the wax sampling (Figure 4, bottom panel). In these comparisons only grains with a short axis dimension greater than 2 pixels have been considered in order to allow for the fact that the image identification is poor at identifying small grains. [40] Figure 4 shows that, after these adjustments, there is a reasonable fit between the two grain size distributions. Differences between the two can be attributable to difficulties of comparing grain size distributions obtained by different sampling methodologies and inaccuracies in the image identification and the assumption used in the grain weight calculation. [41] It was necessary to account for the fine grains not identified during the image analysis process. To do this it was assumed that all areas not identified as being part of a grain with short axis dimension greater than 2 pixels (1 mm) were made up of fines. The weight of fines in these areas was then calculated by assuming that the surface grains in these areas made up a layer which was on average 0.5 mm thick. 0.5 mm was selected as representative of the mean grain size of fines Application of Fluid Force to Individual Grains [42] The estimation of the likelihood of grain entrainment requires the ratio of grain weight to applied fluid force to be estimated. In order to apply forces onto the projected areas of the individual grains it is necessary to estimate a velocity distribution within the near bed region. Such velocity data was not collected in the experimental program; however Nikora et al. [2004] investigated detailed experimental velocity measurements within the near bed region over rough surfaces including gravel. They proposed three different velocity distributions in the near bed layer for flow over rough surfaces. Each distribution was associated with a physically different type of roughness. The three different distributions were; a constant velocity model, a linear vertical distribution of velocity and an exponential vertical distribution of velocity. A constant velocity model assumes wake turbulence dominates and that the turbulent shear stress is approximately constant with depth. This is suitable in environments in which the roughness elements are tall and regular such as partially submerged vegetation in streams or on an inundated floodplain. The linear vertical distribution of velocity is suitable for a range of roughness types in which the proportion of area of the flow occupied by roughness elements at a given elevation increases monotonically from zero at the elevation of the highest roughness elements to one at the elevation of the lowest roughness trough. This type of distribution seems appropriate for describing flows over rough gravel beds. The third type of distribution suggested was an exponential distribution and is the most suitable velocity distribution for well submerged roughness elements where the change in the area of the roughness element with height is negligible, as in flows over submerged vegetation. Given the nature of the bed being investigated, it was clear that the linear velocity distribution would be the most appropriate to describe the velocity field that the grains would be likely to experience. [43] In order to calculate this linear velocity distribution it is assumed that the velocity and the velocity gradient are consistent at the boundary between the near bed roughness sublayer and the logarithmic layer (see Figure 5). The velocity in the logarithmic layer is given by: uz ðþ ¼ 1 u * k ln z ð5þ z 0 p u * ¼ ffiffiffiffiffiffiffiffiffi t 0 =r where z o equals the elevation where there is zero velocity and r equals water density [44] The elevation of the top boundary (z c ) of the roughness sublayer is given by Nikora et al. [2004] as being the elevation of the highest roughness crest. However, with natural sediment beds it is difficult to find the maximum and minimum bed elevations [Nikora et al., 2002]. For this analysis z c is defined as being four times the standard deviation of bed elevations (the 95th percentile elevation relative to the 5th percentile elevation assuming a normal distribution of bed elevations). This gives a more consistent and reliable approach to defining these elevations from each of the bed surface topography data sets. This gives: z c ¼ 4s z where s z = standard deviation of bed elevations [45] Based on this assumption it is possible to calculate the gradient at the base of the logarithmic layer (z = z c ): dhu i dz ¼ u * ð8þ k:z c pffiffiffiffiffiffiffiffiffi dhu i dz ¼ t 0 =r k:4s z [46] Using this velocity gradient it is possible to define the linear velocity distribution in the interfacial sublayer as: hui ðþ¼ z ðz z o Þ dhui dz ð6þ ð7þ ð9þ ð10þ [47] Rather than assume a value for z o it is obtained by equating the sum of forces on the projected grain areas, 6of17

7 MEASURES AND TAIT: BED TOPOGRAPHY AND SEDIMENT STABILITY Figure 4. Comparison of numerically sieved GSDs from image analysis and physically sieved experimental samples. F dx.dy, and the resultant force on the bed from mean bed shear force, t o. t o was obtained during the laboratory tests from measurements of the flow depth, d, and channel slope, S (flow depth was uniform in all tests): t 0 ¼ r:g:d:s ð11þ [48] The force exerted on each grains projected area, F dx.dy, is calculated as being equal to the drag force exerted on the water. This has been calculated with the assumption that there is zero velocity on the downstream side of the grains. The coefficient of drag, C d has been taken to be unity, given the shape of the grain projected areas and range of experimental Reynolds numbers. F dx:dy ðx; yþ ¼ C d 2 r:a 2 dx:dyðx; yþ: hui:z dx:dy ðx; yþ ð12þ [49] Based on the equations (10), (11), and (12) it is possible to iteratively find the value of z 0 that satisfies: X N i¼1 X M j¼1 F dx:dy x i ; y j ¼ t0 :N:dx:M:dy ð13þ where N and M are the number of measurement points in the x and y directions respectively. [50] Using this method, the bed elevation at which the flow velocity equals zero was found to be close to the minimum bed elevation in all the topography data sets examined. It was therefore believed that this selection of near bed flow velocity profile was reasonable. [51] This estimation of vertical velocity profile and bed shear force allows the calculation of force acting on each individual grain, F g. F g;n ¼ X i X j x i ; y j ð14þ where x i, y j represent the coordinates of the bed areas identified as being within grain n Definition of Grain Entrainment [52] The previous sections have described how the grain weight and fluid force on individual grains have been estimated using the measured bed topography and experimental hydraulic data. Using this information it is possible to define those grains which are susceptible to being entrained. This relates closely to the concept of partial transport which is explored in detail by Wilcock and McArdell [1993, 1997]. They demonstrate that during partial transport some of the grains within each size fraction 7of17

8 MEASURES AND TAIT: BED TOPOGRAPHY AND SEDIMENT STABILITY Figure 5. Assumed near bed velocity profile, after Nikora et al. [2004]. on the bed surface remain immobile while others are available for transport. It is this proportion of grains that are available for transport within each size fraction that is investigated here using the detailed grain scale information we have developed. [53] For the purposes of this study the likelihood of entrainment for each of these grains has been assumed to be related to the ratio of fluid drag forces to submerged grain weight. It should be noted that unlike many previous studies using ratios of fluid drag and weight this study uses grain specific fluid forces rather than average forces acting on the bed. This ratio (t* g ) is given by the equation below for each grain n: t* g;n ¼ m g;n : F g;n r s r r s :g ð15þ [54] In order to asses the likelihood of entrainment of a grain on the bed surface the value of t g * acting on each grain is compared to a critical dimensionless shear stress, t c *. If it exceeds this critical value then that grain is considered susceptible to entrainment. By examining all the grains in the bed in this way it is possible to find the proportion of grains (by mass) in a particular size fraction that are susceptible to entrainment at a particular mean bed shear stress. [55] This critical dimensionless shear stress approach has been widely used for estimating the threshold of motion at scales larger than the grain scale approach described here. Buffington and Montgomery [1997] discuss and reanalyze results from many studies carried out to quantify the critical dimensionless shear stress. They show that a range values from to may be appropriate in different circumstances. For this study a value of 0.06 was selected. Sensitivity analysis for the range of critical dimensionless shear stress from to indicated that the value selected did not significantly influence the results below. The reason for this is that grain specific non-dimensional shear stresses vary widely within each size fraction and only a small proportion of grains have non-dimensional stresses in the range to Also, while the critical shear stress affects the mass of grains susceptible to entrainment it has less of an effect on the proportion of susceptible grains within each size fraction as it affects all size fractions. [56] Further investigation into more advanced representations of entrainment considering the contact forces between grains were hampered by the resolution of the data set. It would be very difficult to deduce any grain specific contact forces from the available bed topography data. This is a subject where further research may refine the model Mathematical Description of Sheltering [57] Grain sheltering can be considered as two separate mechanisms: [58]. direct sheltering This occurs when a sediment particle is embedded within a water worked bed so that a lower area of the grain is exposed to the flow. [59]. remote sheltering This occurs when a sediment particle or particles protrude into the flow and affect flow velocities downstream of them. This can reduce forces on grains located downstream. [60] By calculating the upstream projected area of grains from measurements of topography, this study inherently incorporates the effects of direct sheltering. The effects of remote sheltering are more difficult to include. Some of the effects of remote sheltering are intrinsically incorporated by using a vertically varying near bed velocity profile. However, this does not account for any spatial variation in velocity in the streamwise or transverse directions due to sheltering caused by the interference of upstream grains on the near bed flow field. [61] To account for remote sheltering additional algorithms are required to simulate the effect of a remote upstream grain surface on the fluid velocity impacting a grain that is being examined. [62] Schmeeckle and Nelson [2003] used velocity measurements downstream of an exposed spherical particle in an artificial bed to investigate the effects of remote sheltering. They derived an empirical relationship for the ratio of velocity downstream of an exposed particle to the velocity in the same location if the sheltering particle was removed. A similar form of relationship is used here so that the remote sheltering factor (Sr) is defined as: l d h p S r ¼ erf 5h p ð16þ where h p = height of projection of exposed particle, l d = distance downstream of sheltering particle. [63] Spatial variations in remote sheltering were incorporated into the calculation of velocity by analyzing the topography data and identifying local peaks in the streamwise direction as projections into the flow which cause remote sheltering. The height of projection was defined as the elevation of the local peak minus the elevation of the closest local bed elevation trough upstream of the peak. When incorporating remote sheltering it was necessary to discount the most upstream 20 mm of our measurement area for further stages of investigation as we had insufficient information about the upstream topography sheltering this area. [64] In order to maintain the linear velocity profile it is necessary to impose a condition on the sheltering that forces the near bed velocity profiles to maintain their equilibrium 8of17

9 MEASURES AND TAIT: BED TOPOGRAPHY AND SEDIMENT STABILITY with the applied averaged boundary shear stress. This allows the calculated sheltering to introduce spatial variation in velocity without affecting the velocity profile averaged over the whole measurement area. The calculation of the effects of spatial variations in remote sheltering is shown below: uðx; y; zþ hu iðþ z ¼ S r ðx; y Þ P M P N i¼1 j¼1 1 S r x i ; y j ð17þ [65] No vertical variation in sheltering was considered using this approach as the use of the vertical velocity profile described in equation (10) already accounts for some of these effects and it is difficult to incorporate any spatially varying variations in vertical velocity profile without significantly more experimental data on velocity variations within the near bed region over a water worked gravel surface. 4. Model Validation Estimation of Fractional Transport Rates [66] The authors are aware of no published studies in which spatially distributed measurements of near bed flow velocities have been collected over measured water worked sediment beds. It is therefore difficult to compare the estimated fluid drag forces on identified grains with measured data. However, the model predictions of the number of grains susceptible to entrainment can be used to make comparisons with measured data on fractional transport rates. This data allows us to compare the predictive performance of the analysis against such measured bed load data. [67] By assuming a constant rate of entrainment for susceptible grains, k, (units s 1 ) it is possible to use the model to estimate fractional transport rates (q bi ) from the fractional mass entrainment rate and the mean grain displacement length for each size fraction (L 1i ). [68] The fractional mass entrainment rate per unit area, f i, for each size fraction can be calculated by: f i ¼ m ei k ð18þ where m ei equals mass of grains per unit area in size fraction i that are susceptible to entrainment P mgi where t* g > t* c m ei ¼ ð19þ MdxNdy [69] Wilcock [1997] developed dimensionless equations for calculating mean displacement length for different sizes of entrained sediment under different shear stress conditions based on experimental observations: where L 1i t* 0:5 50 ¼ 15 for D i > D 50 ð20þ D i 0:093 L 1i ¼ 15 D x i t* 0:5 50 for D i < D 50 ð21þ D i 0:093 D 50 t * 0 ¼ ð22þ ðr s rþgd 50 t 50 x = calibration parameter ( 1 x 0), x = 0.3 used in this study. [70] Using these equations an expression for the fractional transport rate is developed: q bi ¼ f i L 1i ð23þ [71] By algebraic manipulation it is possible to generate an expression for p i (the proportion of bed load within a particular size fraction by mass) which does not involve the unknown constant k: p i ¼ q bi q b ¼ p i ¼ p i f i ¼ P n i¼1 f i P n P n i¼1 f il 1i m eikl 1i f i L 1i m ei kl 1i m eil 1i i¼1 m ei L 1i ð24þ ð25þ ð26þ where f i = proportion of grains on the surface within a particular size fraction (by mass). [72] This allows comparison of the modeled and the experimentally observed (p i /f i ) ratio for different grain size fractions to be performed. Experimental values of p i were calculated from observed values of q bi and values of f i obtained from the results of the image analysis. 5. Application to Bed Load and Bed Topography Data Sets for Different Upstream Sediment Feed Conditions [73] All the topographical data sets were examined so that there was a range of upstream sediment feed conditions and as was shown by Marion et al. [2003] a typical range of surface topographies. The aim of the examination was to observe the impact of direct sheltering and then the combination of both direct and remote sheltering. Thence it could be determined whether the selected remote sheltering algorithm was appropriate under a range of upstream sediment feed conditions. The results from two tests are shown; one from a test in which there was no feed rate and the other from a test with a steady feed rate throughout the test. Figure 6 shows the result of the comparison between direct and remote sheltering for beds measured during a test with no upstream sediment feed. It shows the comparison at two different times, one in which a static armor is developing and the other when it has developed and the transport rate has dropped to a low stable value. Figure 7 shows the result between direct and remote sheltering for a fed bed but after a mobile armor layer had developed. Both figures clearly show that without the effects of remote sheltering there is 9of17

10 MEASURES AND TAIT: BED TOPOGRAPHY AND SEDIMENT STABILITY Figure 6. Comparison of p i /f i for model results with and without remote sheltering flow algorithm during test F (no upstream feed). poor correspondence between the observed and predicted values of p i /f i especially for the finer grain size fractions. With the remote sheltering algorithm included more of the fluid forces are transferred onto the larger, more exposed grains and they become much more susceptible to entrainment and transport. The predicted and observed values of p i /f i correspond reasonably well over the whole grain range, except for values of grain size below 3.0 mm. The reason for the poor prediction of transport of fine grain sizes is because of inaccuracies in identification of the smaller grains using the automated image analysis process. Any inaccuracies in identification of the grains will result in the forces not being applied correctly as well as the effects on surface grain size distribution (f i ) that have already been discussed. Model results compare well both before and after bed armoring has taken place indicating that the algorithms used to describe direct and remote sheltering work well on static armored and mobile armored beds. This comparison clearly shows that remote sheltering has a very significant impact on fractional transport rates. [74] Using the information available, it is possible to calculate the sum of fluid forces acting on each grain size fraction, F g,i : F g;i ¼ Xni F g;n n¼1 ð27þ where n i = the number of grains in size fraction i. [75] This allows the ratio of fluid force acting on each grain size fraction, F R,i to be investigated: F R;i ¼ F g;i P N F g;n n¼1 ð28þ where N = total number of grains on bed. [76] Plotting F R,i against D i shows how the fluid forces are distributed on the different grain size fractions on the different water worked bed surfaces. Figure 8 shows this distribution for 3 different beds created under different feed Figure 7. Comparison of p i /f i for model results with and without remote sheltering flow algorithm during test h (steady feed rate of 2.5 g/s) after a mobile armor layer had developed. 10 of 17

11 MEASURES AND TAIT: BED TOPOGRAPHY AND SEDIMENT STABILITY Figure 8. Distribution of fluid force on bed surface (shown with measured experimental fractional transport rate for comparison). 11 of 17

12 MEASURES AND TAIT: BED TOPOGRAPHY AND SEDIMENT STABILITY Figure 9. Measured transport rate and feed rate during test n, from Willetts et al. [1998]. conditions; no feed static armor and then two mobile armors with different feed rates. [77] Figure 8 shows that with different surface topographies the fluid forces are distributed differently between the grain size fractions on the surface of the bed. It suggests that static armored beds have more of the force applied to the larger grains compared to mobile armored beds. It also shows that the mobile armor layer which develops under conditions of higher feed and transport rate has more of the forces distributed onto the smaller grain size fractions. This suggests that in mobile armors the stability of the grain size fractions is strongly related to the distribution of forces onto the bed surface. Marion et al. [2003] use data from the same experimental program as used in this paper to show that tests with static and mobile armored beds had distinctly different surface topography arrangements. In well developed static armors the surface topography developed slowly and demonstrated correlations in both the streamwise and lateral directions while, in mobile armors, which developed more quickly there was only a strong correlation in the streamwise direction. This suggested that the surface topography pattern can be linked with the distribution of fluid Figure 10. than 3 mm. Modeled and experimental p i /f i for test n, R2 values shown are calculated for grains larger 12 of 17

13 MEASURES AND TAIT: BED TOPOGRAPHY AND SEDIMENT STABILITY Figure 11. Comparison experimental and modeled fractional transport rate for tests with static and mobile armors. k values have been optimized for each test to generate the comparisons shown. forces and thus the stability of different grain size fractions. It should be noted that the conclusions drawn here can only relate to the condition of partial transport as the experiments were conducted over a relatively narrow range of bed shear stresses close to the threshold of motion. [78] Having established that the model reliably predicts fractional transport of the larger grain sizes well for beds with constant sediment feed rates an investigation of tests with time varying feed rates was made. Figure 9 shows a measured transport and feed rates for a test where the upstream sediment feed was stopped after 33 h. During the first part of the test a dynamic armor layer was formed at the upstream end of the flume and propagated down the flume over a number of hours. After approximately 20 h the whole length of the flume was uniformly armored in this way and a steady rate of sediment transport was reached... When the sediment feed was stopped the dynamic armor layer started to be eroded at the upstream end of the flume to leave a slowly developing coarser static armor layer to form. This layer propagated down the flume until the whole bed was once again uniformly armored and a steady rate of sediment transport was reached was reached as the test finished. Figure 10 shows that the model reliably predicts values of p i /f i for most of the bed surface topography data sets collected during test except those gathered at 15 h and 56 h. At these times the change in bed armor was propagating past the bed measurement area and toward the sediment slot trap (which was located 1.5 m downstream of the measurement area). This meant conditions at the measurement area and at the location supplying sediment to the slot trap were not the same causing inaccurate model predictions of sediment transport at the slot trap. 6. Prediction of Fractional Transport Rates Based on Bed Topography Data [79] Given that the model can predict bed load composition adequately for a range of bed topographies and sediment feed regimes, once remote sheltering is accounted for, it was decided to attempt to predict fractional transport rates 13 of 17

14 MEASURES AND TAIT: BED TOPOGRAPHY AND SEDIMENT STABILITY Figure 12. Plot of W i * against t o * generated from predicted transport rates (test n, k = s 1 ). following the methods described by Wilcock [1988]. To estimate fractional transport rates it is necessary to find an appropriate value for k in equation (18). This value can be found by comparing transport rate predicted by the model and transport rate measured in the physical experiments for each of the topographic data sets and simultaneous measurements of bed load. Once a relationship for k has been found fractional transport rates can be calculated. [80] Using the validated model with the remote sheltering algorithms included it was possible to investigate any variation in the constant, k. This was done by optimizing k for each test to match modeled and experimental bed transport rates, q b. It should be noted that k was kept constant for all size fractions within any given test. This process was carried out for all the tests in which the model was representing fractional transport well. It showed k values to be in the range s 1 to s 1. This shows that k is reasonably constant for all the tests examined as experimental transport rate is unsteady and exhibits variations with a similar order of magnitude to the range of k values calculated (see Figure 11). Values of k were found to be similar for feed and no feed beds. Examples of comparisons between different observed and predicted fractional transport rates for tests with different feed conditions are shown in Figure 11. They show that the model can produce fractional transport rates which compare well across the full range of grain size fractions despite the fact that the small grains are not well identified in the AIA process. The likely reason for this is that the small grains have relatively little impact on overall bed load transport rate due to their low mass. Figure 13. Modified plot of W i * against t o * using varying k values according to equation (29) (test n, a =5 and k = s t o =1.6Nm 2 ). 14 of 17