T. Sinclair says the Batman Su river was likely named after a nearby settlement.
T. Sinclair, "The Site of Tigranocerta. II," Review of Armenian Studies 26, 1996–1997, 58
For Sitae a site by a river crossing is most likely, as Miller and Dillemann observed, and the most important river crossing between Bismil and the vicinity of Tilmis is that of the Batman Su. There is no settlement now bearing a suitable name on the relevant sector of the Batman Su, but the river was known in Arabic аз Sātīdamā, of which “Šit‘ma” is an Armenianised version, and the name Sitae is no doubt connected. One or other of the two settlement mounds to the west of the Batman Su will be the site. Strictly speaking it is much more likely that the river was named after the settlement, as the Batman Su is now.