A solute transport equation with a fractional-order dispersion term is a model of solute movement in aquifers with very wide distributions of velocity. The equation is typically formulated in Cartesian coordinates with constant coefficients. In situations where wells may act as either sources or sinks in these models, a radial coordinate system provides a more natural framework for deriving the resulting differential equations and the associated initial and boundary conditions. We provide the fractional radial flow advection-dispersion equation with nonconstant coefficients and develop a stable numerical solution using finite differences. The hallmark of a spatially fractional-order dispersion term is the rapid transport of the leading edge of a plume compared to the classical Fickian model. The numerical solution of the fractional radial transport equation is able to reproduce the early breakthrough of bromide observed in a radial tracer test conducted in a fractured granite aquifer. The early breakthrough of bromide is underpredicted by the classical radial transport model. Another conservative, yet nonnaturally occurring solute (pentaflourobenzoate), also shows early breakthrough but does not conclusively support the bromide model due to poor detection at very low concentrations. The solution method includes, through a procedure called subordination, the effects of solute partitioning on immobile water.