What forms do groundwater flow equations have when Dupuit supposition is not considered?
If the pumping well is partially penetrating in the aquifer or the source above the aquifer (e.g., unconfined aquifer, leaky aquifer, under the stream), the vertical flow should be accounted. 

The groundwater equation like Thies' solution is not considering the vertical flow. Incorporating the vertical flow would change the governing equation (P.D.E.) form including the vertical hydraulic conductivity. Lots of papers have shown the solution form for considering that.