Experimental analysis of size and distance of travel

Transcription

1 WATER RESOURCES RESEARCH, VOL. 34, NO. 9, PAGES , SEPTEMBER 1998 Experimental analysis of size and distance of travel of unconstrained particles in interrill flow Anthony J. Parsons and Simon G. L. Stromberg Department of Geography, University of Leicester, University Road, Leicester, England, United Kingdom Abstract. The travel distances of particles ranging in size from 2.88 mm to mm were investigated in!.?.boratory simulations of interrill overland flow. Using travel distances scaled for differences,: among the experiments in flow and rainfall energy, a relationship between distance traveled and particle size is obtained that shows a steep reduction in travel distance with increase in particle size. Travel distance is the outcome of two probabilities: that of moving and that of coming to rest. In interrill flow, the former is controlled by rainfall energy, but the latter is controlled by flow energy. Analysis of subsets of the data in which only rainfall or flow energy varied shows that the steep reduction in travel distance with particle size is primarily due to sensitivity to flow energy. Although particle movement (entrainment) by rainfall energy does vary with particle size, the sensitivity is less. 1. Introduction ML = (Er El) (1) Process-based predictions of sediment removal from hillslopes in interrill overland flow require information on the rate and timing of sediment detachment, rate of transport, and distances traveled by particles. Although a substantialiterature exists on sediment detachment in, and transport capacity in which M is particle mass (grams), L is distance moved in unit time (cm min- ), Er is rainfall energy (J m -2 S -1) and Ef is flow energy (J m -2 s-1). This relationship indicates that smaller particles are transported farther than larger particles and that effective particle transport is achieved in interrill flow of, interrill overland flow, very few studies of transport distance only when both rainfall and flow energy are significant because of particles have been undertaken. Wainwright [1991] exam- the combined effect of rainfall energy and flow energy on ined the transport of ceramic and flint fragments in laboratory particle transport is multiplicative. While this study demonexperiments, and Parsons et al. [1993] studied the movement of strates the importance of both rainfall and flow energy to fine-sand-sized magnetite under simulated rainfall on a large particle transport, it describes only the relationship between rate of transport and total energy available. No data have been runoff plot. The results of these empirical studies seem to presented that clearly demonstrate the relationship between accord with more theoretical considerations [Kirkby, 1991] particle size and transport distance. which indicate that transport distances for particles in interrill For rivers a number of studies exist that characterize the flow have a gamma distribution, with most sediment traveling movement of individual particles [e.g., Einstein, 1937; only a short distance before coming to rest. These studies, Takayama, 1965; Ashworth and Ferguson, 1989; Hassan et al., however, have not yielded information on the relationship 1991]. Church and Hassan [1992] used the results of several of between transport distance and particle size that can be incor- these studies to determine the relationship between sediment porated into process-based. predictions of sediment removal in size and mean distance of travel. These authors took data from interrill overland flow.!.n,vainwright's [1991] study the particles were too large to ½ detached by raindrop impact, in contrasto most sedimem transported in interrill flow, and the data were obtained from only one rainfall intensity. In Parsons et al.'s [1993] experiments, even though the particles were small enough to be detached by raindrop impact, they were all of similar size, and again, all the data were obtained under one rainfall intensity. As a consequence, transport distance to date a wide range of hydrological regimes and a wide range of bed structures. To eliminate variability introduced by variable event size and different bed particle sizes, the data were scaled, enabling a simple size-distance relationship to be determined. For each particle size group in each data set, the mean travel distance was standardized by Li/[L], where [L] is the reference travel distance for the event (a measure of event size) and L i is the mean distance of travel of particles in the size group has mainly been expressed in terms of the total percentage of i. The travel distances were also normalized to eliminate variparticles moved [Kirkby, 1991; Parsons et al., 1993] and the ability in range of particle size and differences in median size probability of a particle of a given size traveling a given dis- of surface and subsurface bed material. Analysis of the scaled tance [Kirkby, 1991; Wainwright, 1991]. data indicated that the travel distance is not simply propor- More recently, laboratory experiments have been conducted tional to l/d, where D is the grain size, as might have been that examine the relationships that exist among transport disexpected. For scaled particle sizes greater than 2 the average tance, particle mass, flow energy and rainfall energy [Parsons et relationship was shown to be L -- D -2'25, where L = Li/{LD50surf} and D = Di/DSOsu b are the scaled values of al., 1998]. This study yielded the relationship travel distance and particle size, respectively. Where the scaled Copyright 1998 by the American Geophysical Union. Paper number 98WR /98/98WR particle size was less than 2, the scaled distance was shown to change only gradually with size and L -- D -ø'8. Church and Hassan interpreted their results to indicate that particles finer 2377

2 2378 PARSONS AND STROMBERG: PARTICLE size AND TRAVEL DISTANCE Table 1. Median Travel Distances of Particles Rainfall Flow Median Travel Distance mm energy energy (J m -2 s -1) (J m -2 s -1) i = 2.88 i = 5.04 i = 5.25 i = 5.98 i = 7.38 i = 8.41 i = 9.5 i = * * * * ii 85 i: '5' Here i is particle size (millimeters). *In these experiments, some particles were transported out of the flume, so median distance could not be calculated. than the median size of the bed surface material all have a Flow was introduced into the flume from a header tank at rates relatively high chance of being trapped in their journey downstream. Thus whereas the movement of small stones appeared to depend mainly on the relative trapping efficiency in the bed, the movement of larger stones depended mainly on their size. Recently, Wilcock [1997] has argued that the different relationships of grain size to travel distance for small and large grains reflect differing mobilities of the two classes of particles. The strong dependence of travel distance on particle size is charranging from 0 to s-1. Experiments were conducted gradients between 3.5 ø and 10 ø. Once the flow and rainfall for a particular experiment were at equilibrium, particles were introduced into the flow by placing them onto the bed using tweezers over a period of 1 min, and the experiment was run for a further 14 min, after which time the distances traveled by the particles were measured. In all, 224 experiments were conducted in which rainfall kinetic energy was varied between 0.00 acteristic of partial transport, whereas the weaker dependence and 0.85 J m -2 S -1 and flow energy was varied between is indicative of fully mobilized transport. In this paper we undertake an analysis for rain-impacted interrill flow similar to that presented by Church and Hassan [1992], using data given by Parsons et al. [1998]. The Church and Hassan methodology is adopted, allowing data from a wide range of flow and rainfall regimes to be included while enabling a single relationship to be established. and J m -2 s -. For each experiment, 10 or more particles of a particular size were introduced into the flow. A summary of the results is given in Table 1. Of these experiments, 220 yielded data that could be used in the present analysis. Using methods similar to those of Church and Hassan [1992], the data have been scaled to remove variability introduced by variable event size, which in the case of interrill flow 2. Data is a function of rainfall and flow energy (equation (1)). Whereas Church and Hassan [1992] utilize the mean for each size class of particles, we have chosen to use the median for Parsons et al. [1998] measured median transport distances in a laboratory flume of particles ranging from 2.88 mm to mm. The full details of these experiments and their results are given by Parsons et al. [1998], and for the purposes of this paper it is sufficiento note only the following. The experiments were expressing transport distance. Because transport distances of particles in interrill flow are described by a gamma distribution [Kirkby, 1991], the median is a better descriptor of central tendency than is the mean. On the basis of (1), the data given in Table 1 were scaled using; conducted on a fixed bed consisting of silica sand that had particle diameters of between 1 and 2 mm with a median diameter of 1.5 mm. Rainfall was supplied to the flume from an Li = (ErEf) (2) overhead sprin er system hat could deliver intensities varyigg between 51 and 138 mm h -, with most raindrops attaining where Li is the scaled transport distance (cm m 4 s 2 j-2) of terminal velocity within the available fall height to the flume. particles of size i and li = median transport distance (centi-

3 PARSONS AND STROMBERG: PARTICLE SIZE AND TRAVEL DISTANCE 2379 meters) of 10 or more particles of size i. Implicit in this scaling is the view that travel distance is a function of the separate, independent interactions of rainfall and flow energy with the particles. The rationale for this view is examined later in this paper. Unlike the river data used by Church and Hassan, our experimental data were obtained from experiments performed using a fixed bed material size. Thus there is no need to scale the data to take into account bed material variability Results and Discussion The results of scaling the data for event variability (equation (2)) are given in Figure 1. As was the case for Church and Hassan's data, regression is the appropriate procedure to estimate the functional relationship between the observed variables. A power law regression fitted to the data yielded the equation L = 9970D -3'44 (3) in which L is the scaled median transport distance and D is the particle diameter (centimeters) for which r 2 = Thus, just as Church and Hassan found for rivers, for particles that are coarser than the bed material, the travel distance declines with increasing particle size more rapidly than might be expected. As Church and Hassan noted for rivers, the smaller particles outrun the larger ones. For rivers this relationship breaks down where particles approach the median bed size. In the case of the interrill flow experiments the data do not make it possible to determine whether the relationship breaks down in a similar way. However, for natural overland flow it does seem intuitively likely that smaller particles will either become trapped by more pronounced surface irregularities and larger immobile particles (the explanation proposed by Church and Hassan [1992]), or be fully mobile (the explanation proposed by Wilcock [1997]. It is therefore probable that the relationship will be similar to that determined for rivers, in which the travel distances of particles smaller than the median diameter of the bed vary less with particle size than do those of larger particles. The transport distance-particle size relationships developed here and by Church and Hassan are based upon aggregations, over time, of several smaller, individual movements. Particles, whether in rivers or interrill overland flow, move in a series of steps separated by rest periods. Kirkby [1991] has argued that such particle movement can be modeled as the outcome of two probabilities: that of moving and that of coming to rest. In the case of rain-impacted overland flow the probability of moving (the probability of being detached) is principally a function of rainfall energy (as was shown, for example, by the work of Borst and Woodbum [1942] and Ellison [1945]), with the frequency of detachment being directly related to rainfall intensity. In contrast, the probability of coming to rest is primarily a function of flow energy [Young and Wiersma, 1973; Morgan, 1980]. Consequently, rainfall and flow energy act upon particles more or less independently. Because travel distance is the outcome of two independent probabilities and each of these probabilities is associated with one of the energy sources, the travel distance is a function of the product of the two probabilities and hence of the two energy values. Insight into the separate roles of rainfall energy and flow energy in controlling the transport distance-particle size relationship might be gained by examining scaled transport distances under conditions of constant rainfall and flow energy, in which cases (2) reduces to :1: - -I. -I, ß,, Particle Size, D (mm) Figure 1. Relationship between scaled travel distance and particle size. li Li=krEf (4) Li-- kfer (5) respectively, where kr and kf denote the energy associated with the constant rainfall and flow, respectively. To some extent, the data obtained by Parsons et al. [1998] provide an opportunity for such analyses. A range of experiments was conducted in which the rainfall energy was held constant at J m -2 s - but the flow energy varied from to J m -2 s -. Unfortunately, a comparable set of experiments in which flow energy remained constant was not conducted, but a small set of data can be extracted in which flow energy varied only from

4 2380 PARSONS AND STROMBERG: PARTICLE SIZE AND TRAVEL DISTANCE to J m -2 S -1 but rainfall energy varied from 0.20 and 0.85 J m -2 s -1. in which h is flow depth (meters) and b is an exponent [Morgan et al., 1992]. The results of the analyses of the two subsets of data yield the regression equations with D, imply that flow energy is more important than settling velocity in controlling travel distance of rain-detached particles. These subsets of the data do not provide exact equivalents for values of L i to those of (4) and (5) because the rainfall energy available to detach particles is neither necessarily equal to the energy supplied from the sprinkler system, nor independent of the flow energy. Where particles are submerged by the 4. Conclusion In this study we have analyzed travel distances of particles of flow, some of the rainfall energy striking the surface of the flow will be absorbed by the water and therefore will not be available for particle detachment. In the present case, depths of flow in the flume were generally between 2 and 4 mm, so that this effect would be minimal and apply only to the smallest particles studied. In consequence, it may be expected that only a range of sizes in laboratory simulations of interrill overland flow encompassing a range of overland flow discharges and rainfall intensities. The results are similar to the findings of Church and Hassan [1992] for travel distances of coarse particles in rivers, namely, that distance of travel declines more sharply than in proportion to the inverse of particle diameter. the travel distances of the smallest particles might be slightly Because these travel distances are in fact summations of sevunderestimated at higher flow rates (depths), leading to a reduction in the estimate of the steepness of the relationship between D and L determined from (4) and an increase in the effect of the variable flow energy on the data used to estimate the relationship between D and L determined from (5). More generally, in applying this type of analysis to particle travel distances in interrill flow it would be appropriate for deeper flows to use effective rainfall energy E;, where eral small steps, they can be regarded as the joint outcome of the probability of moving and of coming to rest. For interrill overland flow, the former probability is controlled by the rainfall energy, and the latter is controlled by the flow energy. Separate analyses of variation in these two energy sources indicate that the probability of coming to rest declines much more sharply with particle size than does the probability of moving, so that it is the former that largely explains the steeper -bh than expected decrease in travel distance with particle size. It E'r = E e would be interesting to know if these differences in detachment and coming to rest are also responsible for the relationship between size and travel distance of coarse particles in rivers identified by Church and Hassan [1992]. Acknowledgments. We thank Mark Greener for his assistance in L = 15,744D -4'ø6 (6) collecting the data on which this paper is based, and Athol Abrahams and Ian Prosser for their reviews of the manuscript. This research was L = 421D -2'5 (7) supported by a grant from the Natural Environment Research Council (grant GR3/8809). with r 2 of 0.83 and 0.51 for the cases of constant rainfall energy and near-constant flow energy, respectively. Thus it would seem that in interrill overland flow the smaller particles outrun References the larger ones mainly because of the strength of the inverse Ashworth, P. J., and R. I. Ferguson, Size-selectiv entrainment of bed load in gravel bed streams, Water Resour. Res., 25, , relationship between particle size and the probability of com- Borst, H. L., and R. Woodburn, The effect of mulching and methods ing to rest. The probability of being detached, on the other of cultivation on run-off and erosion from Muskingum silt loam, hand, appears to be less strongly dependent on particle size. Agtic. Eng., 23, 19-22, Indeed, it may seem surprising that there is an inverse rela- Church, M., and M. A. Hassan, Size and distance of travel of unconstrained clasts on a streambed, Water Resour. Res., 28, , tionship between particle size and the probability of being detached, at all. For much smaller (fine sand sized) particles, Einstein, H. A., Bedload transport as a probability problem (in Ger- Parsons et al. [1993] found a positive relationship between man), Ph.D. thesis, Eidgenoess. Tech. Hochsch, Zurich, Switzerland, splash transport and particle size that was in proportion to the surface areas of the particles. A probable explanation for the (English translation by W. W. Sayre in Sedimentation, edited by H. W. Shen, Appendix C, H. W. Shen, Fort Collins, Colo., 1972.) Ellison, W. D., Some effects of raindrops and surface-flow on soil difference between that result and the present one may lie in erosion and infiltration, Eos Trans. AGU, 26, , our observations during experiments that raindrop impact did Hairsine, P. B., and C. W. Rose, Rainfall detachment and deposition: not always result in particle movement. Larger particles move Sediment transport in the absence of flow-driven processes, Soil Sci. only when impacted by large raindrops, whereas smaller par- Soc. Am. J., 55, , Hassan, M. A., M. Church, and A. P. Schick, Distance of movement of ticles move when impacted by both large and small raindrops. coarse particles in gravel bed streams, Water Resour. Res., 27, 503- This effect may well counteract that of the smaller target that 511, the smaller particles provide. Kirkby, M. J., Sediment travel distance as an experimental and model In modeling interrill soil erosion, few studies have explicity recognized this process as one in which particles move in a variable in particulate movement, Catena Suppl., 19, , Morgan, R. P. C., Field studies of sedimentransport by overland flow, Earth Surf. Proc., 5, , series of steps. A significant exception is the model presented Morgan, R. P. C., J. N. Quinton, and R. J. Rickson, Eurosem docuby Hairsine and Rose [199!]. This model assumes that detach- mentation manual, 34 pp., Silsoe Coll., Bedfordshire, England, ment is wholly in response to rainfall energy and that the only Parsons, A. J., J. Wainwright, and A.D. Abrahams, Tracing sediment role of overland flow is to translate the detached sediment movement in interrill overland flow on a semi-arid grassland hilldownslope as it settles back to the surface. The model assumes slope using magnetic susceptibility, Earth Surf. Processes Landforms, 18, , that travel distance will be an inverse function of settling ve- 1/2 Parsons, A. J., S. G, L. Stromberg, and M. Greener, Sediment- 1ocity (and hence, for coarse particles; D ), our results, transport competence of rain-impacted interrill overland flow, Earth which show a much greater rate of decrease of travel distance Surf. Processes Landforms, 23, , 1998.

5 PARSONS AND STROMBERG: PARTICLE SIZE AND TRAVEL DISTANCE 2381 Takayama, S., Bedload movement in torrential mountain streams (in Japanese), Tokyo Geogr. Pap., 9, , Wainwright, J., Erosion of semi-arid archaeological sites: A study in natural formation processes, Ph.D. thesis, Univ. of Bristol, Bristol, England, Wilcock, P. R., Entrainment, displacement and transport of tracer gravels, Earth Surf. Processes Landforms, 22, , Young, R. A., and J. L. Wiersma, The role of rainfall impact in soil detachment and transport, Water Resour. Res., 9, , A. J. Parsons and S. G. L. Stromberg, Department of Geography, University of Leicester, University Road, Leicester LE1 7RH, England, United Kingdom. ( ajp16@le.ac.uk) (Received August 19, 1997; revised April 28, 1998; accepted April 29, 1998.)