Problem 59002. Compute piezometric head in a leaky confined aquifer

Problem statement
Write a function to compute the piezometric head h in a leaky confined aquifer and the position of a groundwater divide xd given the boundary heads h0 at x = 0 and hL at x = L 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 K' and b'. 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 vbw, and the flow out of the right side is (v+Deltav)bw. If v' is the specific discharge through the leaky confining layer, the flow into the volume through the top is v'wDeltax. Then the mass (or volume) balance is
0 = vbw - (v+Deltav)bw + v'wDeltax = -bwDeltav + v'wDeltax
Rearranging, dividing by , and taking the limit as goes to zero gives
dv/dx - v'/b = 0
Darcy’s law allows the specific discharge through the aquifer to be written as v = -K(dh/dx) and the specific discharge through the leaky confining layer to be approximated as v' = K'(H-h)/b'. Then
K d2h/dx2 + (K'/(bb'))(H-h) = 0
One can solve this second-order linear ordinary differential equation using the boundary heads to get the piezometric head in the aquifer.
leaky confined aquifer

Solution Stats

100.0% Correct | 0.0% Incorrect
Last Solution submitted on Sep 28, 2023

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