Scattering through a natural porous formation (by far the most ubiquitous example of disordered medium) represents a formidable tool to identify effective flow and transport properties. In particular, we are interested here in the scattering of a passive scalar as determined by a steady velocity field, which is generated by a line of singularity. The velocity undergoes erratic spatial variations, and concurrently, the evolution of the scattering is conveniently described within a stochastic framework that regards the conductivity of the hosting medium as a stationary, Gaussian, random field. Unlike the similar problem in uniform (in the mean) flow fields, the problem at stake results much more complex. Central to the present study is the fluctuation of the driving field that is computed in closed (analytical) form as a large time limit of the same quantity in the unsteady-state flow regime. The structure of the second-order moment Xrr, quantifying the scattering along the radial direction, is explained by the rapid change of the distance along which the velocities of two fluid particles become uncorrelated. Moreover, two approximate, analytical expressions are shown to be quite accurate in reproducing full simulations of Xrr. Finally, the same problem is encountered in other fields, belonging both to classical and to quantum physics. As such, our results lend themselves to being used within a context much wider than that exploited in the present study.