  A function for the bed-load sediment transport rate is derived. This is
achieved from the first principle by using the entrainment probabilities of the
sediment grains by rolling and lifting, and by introducing two travel lengths,
respectively, for the first time. The predictions from the new bed-load
function agree well with the experimental results over the entire experimental
range and show significant improvement over the commonly used formula for
bed-load transport rate. The new function shows that, in terms of contributing
to the bed-load transport rate, the total entrainment probability of the
sediment grains is a weighted summation of those by the lifted and rolling
grains, rather than a simple addition of the two. The function has also been
used to predict the total entrainment probability, saltation length and the bed
layer thickness at high bed-load transport rate. These predictions all agree
well with the experimental results. It is found that, on average, the travel
length for the rolling sand grains is about one order of magnitude less than
that for the lifted ones.
