  We start with a general governing equation for diffusion transport, written
in a conserved form, in which the phenomenological flux laws can be constructed
in a number of alternative ways. We pay particular attention to flux laws that
can account for non-locality through space fractional derivative operators. The
available results on the well posedness of the governing equations using such
flux laws are discussed. A discrete control volume numerical solution of the
general conserved governing equation is developed and a general discrete
treatment of boundary conditions, independent of the particular choice of flux
law, is presented. We use numerical solutions of various test problems to
compare the operation and predictive ability of two discrete fractional
diffusion flux laws based on the Caputo (C) and Riemann-Liouville (RL)
derivatives respectively. When compared with the C flux-law we note that the RL
flux law includes an additional term, that, in a phenomenological sense, acts
as an apparent advection transport. Through our test solutions we show that,
when compared to the performance of the C flux-law, this extra term can lead to
RL-flux law predictions that may be physically and mathematically unsound. We
conclude, by proposing a parsimonious definition for a fractional derivative
based flux law that removes the ambiguities associated with the selection
between non-local flux laws based on the RL and C fractional derivatives.
