Problem 59002. Compute piezometric head in a leaky confined aquifer

ChrisR

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Problem statement
Write a function to compute the piezometric head h in a leaky confined aquifer and the position of a groundwater divide  given the boundary heads  at  and  at  and head H (assumed constant) in the unconfined aquifer above the leaky confining layer. The hydraulic conductivity and thickness of the aquifer are K and b, and the hydraulic conductivity and thickness of the leaking confining layer are  and . If there is no groundwater divide, set xd to the empty set []. 
Background
As in other problems in this series, the movement of groundwater is governed by conservation of mass and (if the flow is slow enough) Darcy’s law. For steady, one-dimensional flow, conservation of mass applied to the control volume indicated by the dashed line says that the flow into the volume equals the flow out of the volume. If v is the specific discharge in the aquifer, or the flow per unit area, the flow into the left side is , and the flow out of the right side is . If  is the specific discharge through the leaky confining layer, the flow into the volume through the top is . Then the mass (or volume) balance is
Rearranging, dividing by , and taking the limit as  goes to zero gives
Darcy’s law allows the specific discharge through the aquifer to be written as  and the specific discharge through the leaky confining layer to be approximated as . Then 
One can solve this second-order linear ordinary differential equation using the boundary heads to get the piezometric head in the aquifer. 