## 1. Introduction

[2] The wide range of grain sizes found in most gravel bed rivers poses a difficult and incompletely solved problem for the successful modeling of transport rate. The influence of grain size on transport rate can be described qualitatively as a competition between absolute and relative grain size effects. The absolute size effect causes the inherent mobility of sediment grains to decrease with increasing grain size. A given flow will transport finer sediment more rapidly than coarser sediment. When different sizes are placed in a mixture, the relative size effect tends to increase the transport rate of larger grains and decrease the transport rate of smaller grains. The magnitude of the relative size effect and, therefore, the transport rate of individual sizes within a mixture, will be sensitive to the composition of the mixture, which can change during transport and in response to variation in flow and sediment supply. A quantitative model for mixed-size transport must account for the distribution of grain sizes available for transport.

[3] Early transport models avoided some, but not all, of the difficulties associated with specifying size distribution by predicting the total transport rate as a function of a single representative grain size [e.g., *Meyer-Peter and Müller*, 1948; *Engelund and Hansen*, 1972]. This approach is relatively practical because the only sediment information required is the representative size, but it is unable to predict changes in grain size and it is also likely to underpredict the transport rate of the finer fractions, which may be much larger than that of the coarser fractions [*Leopold*, 1992; *Lisle*, 1995]. To account for size-dependent variations in transport, transport models can be formulated for many finely divided size fractions. This approach is able to capture variation in transport rate among different sizes, as well as interactions among different sizes. This detail comes at the expense of greater computational effort and, more critically, requires specification of the full size distribution of the bed sediment. The grain size distribution in a reach is generally not accurately known, is subject to variation during transport, and is sensitive to the history of flow and sediment supply. At present, these data constraints make many-fraction transport models useful primarily for simulation, rather than for predicting the transport at a particular location, for which specific grain size information is required [*Wilcock*, 2001b].

[4] This paper presents a transport model in which the bed size distribution is divided into two fractions, sand and gravel. The effort required to determine the proportion of sand and gravel in a reach is comparable to that required to determine a representative grain size (and much less than that required to determine the full size-distribution), so a two-fraction model retains much of the practicality of a single-fraction estimate, while permitting variation in bed grain size through changes in the relative proportion of sand and gravel. This provides a means of predicting the variation in the fines content of the bed, which may often be more variable than that of the coarse fraction, and whose passage, intrusion, or removal may be a specific environmental or engineering objective. A two-fraction transport relation also admits relatively large sand transport rates at low to moderate flows that transport little gravel.

[5] That a two-fraction approximation of widely sorted sediment might capture mixed-size transport dynamics of practical significance is suggested by a number of observations, including differences in the behavior of the sand and gravel, similarity of transport rates of different sizes within the two fractions, consistent variation of sand and gravel behavior with bed sand content *f*_{s}, and the fact that the fines content of a river bed tends to be more transient than the gravel/cobble framework of the bed. In this paper, we outline previous work that indicates why a two-fraction approach might be effective and why the particular model proposed here is consistent with (in fact, could be deduced from) previous observations.

[6] In addition to practical considerations, the binary nature of a two-fraction model has the particular advantage that it supports a simple description of the interaction between the two fractions. Because the proportions of sand and gravel sum to one, the effect on transport of variations in bed composition may be represented as a simple function of the proportion of either fraction. We find that sand and gravel transport rates depend on *f*_{s} not only as it specifies the amount of each fraction available for transport, but also through an additional nonlinear effect on the mobility of each fraction. With a two-fraction model, we can efficiently and directly ask: how does *f*_{s} affect gravel transport rates? Earlier work indicates that this effect is quite pronounced and not adequately accounted for in current transport models [*Jackson and Beschta*, 1984; *Ikeda and Iseya*, 1988; *Wilcock et al.*, 2001].

[7] The two-fraction model presented here was partially described by *Wilcock* [1998]. This treatment differs in several respects. First, it presents a complete model, including a function for transport rate as well as incipient motion. Second, it incorporates new data from flume experiments with four sediments specifically designed to define the two-fraction model in its most sensitive range. Third, it explicitly develops versions of the model referenced to the grain size of either the bed surface or subsurface.