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Near pumping wells or near/beneath a stream, the groundwater flow is not totally horizontal and the vertical component of groundwater movement at these points cannot be ignored in reality. Often, groundwater flow equations are derived while assuming that the flow in largely horizontal which is called Duipuit-Forscheimer assumption.

I want to know if we consider the vertical flow near pumping wells and streams, what form will the groundwater flow equations have?

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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.

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