What? I thought this was going to be about layered aquifers, but I don't see anything about layers here - wait - are you referring to the river as a separate layer?
Rivers can cut through an aquifer and there will invariably be some kind of interaction between the surface and ground waters; actually these relationships are very complex and I am not an expert so I will avoid the complexities. In simple terms the aquifer will either feed the river, or the river will feed the aquifer - this relationship will vary along the river (at the source you would imagine the aquifer to feed the river but at the sink you would expect the opposite) and this relationship will vary with time (you might expect a different relationship in the dry season than in the wet season).
Don't think of the ground water as flowing through some kind of underground river, it's a common misconception, only in very specific geology will this happen, like karstic geology where you can get cave systems developing and the water can flow through the caves. The groundwater flows through the pore spaces and fracture planes in the aquifer, the aquifer simply being a rock unit that yields water at economic rates, in other words just a layer of rock with water moving through it. It is quite possible, indeed it is quite likely, for the groundwater to be flowing perpendicular to the river, I've certainly come across case studies where this is what happens.