There are lots of expensive ...

There are lots of expensive ways to get at this answer that involves modeling and geochemistry.  However, the simplest way may be to measure the depth to water at the well compared with the average surface elevation of the water at the river.  If the static elevation of water in the well is significantly above the surface elevation of the river, then you are likely drawing from groundwater that flows to the river.  If there is only a slight difference above or if the static elevation Is below the surface of the river, you are likely drawing water from the river.  Without further investigation, that's the best I can come up with.