# 14.1: Introduction

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In a certain sense, this is the most significant chapter in Part 2 of these course notes—inasmuch as virtually all natural sediments comprise a range of particle sizes, not just a single size. Most of what was said in earlier chapters, on threshold, transport mode, and transport rate, involve an implicit assumption that the sediment is effectively of a single size (hence the term “unisize” sediment), in the sense that the effect of the spread of sizes around the mean (that is, the **sorting**) is sufficiently small that it can be ignored, at least for very well sorted sediments. All sedimentationists know, however, that such an assumption cannot be valid even for moderately sorted sediments, to say nothing of poorly sorted sediments, with a wide spread of particle sizes, like the sand–gravel mixtures that are so common in rivers.

I will probably be insulting your intelligence when I explain the meaning of “size fraction”. A **size fraction** in a natural sediment or an artificial mixture of sediments is a specified range of sizes within the size distribution of the sediment. Such size fractions are usually perceived or chosen to be very narrow relative to the overall range of sizes in the sediment. The choice of lower and upper size limits of the fraction are basically arbitrary—in practice, usually governed by the subdivisions of the conventional powers-of-two grade scale for sediment size. Keep in mind, however, that the size varies, perhaps non-negligibly, even within a narrowly defined size fraction. A size fraction is *not* a single size.

If, for definiteness, we assume a certain definite size-distribution shape for mixed-size sediments, like a log-normal distribution, then the relative size of a given size fraction is specified by three things: the sorting of the distribution, the mean or median size of the distribution, and the position of the given size fraction within the distribution (which is most naturally described by \(D_{i}/D_{m}\), where \(D_{m}\) is the mean or median size and \(D_{i}\) is the size of the given fraction). Beyond this, of course, matters become much more complex (hopelessly so?) when we allow the shape of the size distribution to vary, as it does greatly, even to the point of bimodal and trimodal distributions, in natural sediments. (A great many natural sediments, particularly sand–gravel mixtures, are strongly bimodal.) You can see that the task of addressing the problem of threshold and transport of mixed-size sediments is a daunting one.

A final note seems in order here. The focus of this chapter is on sediment size. As you saw in Chapter 8, sediments in general have a joint frequency distribution of size, shape, and density. The study of mixed-shape and mixed-density sediment has not progressed as far as study of mixed-size sediment. It seems fair to say that the effect of mixed shapes is not as significant as the effect of mixed sizes—except, perhaps, for uncommon sediments with extremely nonspherical shapes. The effect of mixed-density sediments is important, for example, in understanding the development of placers. For completeness, these notes should have additional sections on mixed-shape and mixed-density sediments.

## A Useful Thought Experiment

To get your thinking started, imagine a planar bed of mixed-size sediment, with a wide range of sizes from sand to gravel, over which a uniform flow is arranged to be passed. Assume that the particle-size distribution is unimodal. Suppose that the flow extends uniformly so far upstream and downstream as to be effectively infinite in extent. Clearly this is an idealization of the flow in real streams and rivers—but there is an essential element of reality to it, inasmuch as during a period of strong flow in a river the flow for the most part works on a bed of sediment that was lying there, waiting to be worked on before the event, and the flow picks up and moves what it wants to, without an externally constrained supply of sediment. A sediment-recirculating flume (see Chapter 8) works the same way, and in that sense is a good model for fluvial sediment transport.

You could attempt to measure three significant aspects of the transport of the mixed-size sediment in such an experiment. One is the relationship among the load (the sediment in transport at a given time), the bed surface (the sediment that is exposed to the flow at any given time), and the substrate (the bulk sediment from which the flow entrains, transports, and deposits sediment particles of various sizes). A second question has to do with movement thresholds: how do the thresholds for the various size fractions in the sediment mixture differ from one another? A third aspect is the relationship among the rates of transport of the various size fractions (usually called fractional transport rates; see below) of the sediment mixture. These three aspects are considered in some detail in the following sections.