Problem 59029. Compute the water table in a layered unconfined aquifer

Write a function to compute the flow per unit width and the elevation of the water table above the bottom of unconfined aquifer with two soil layers. The input to the function will be the positions where the water table elevation (WTE) is requested, the hydraulic conductivities of the two layers, the thickness of the bottom layer, the WTEs at x = 0 and x = L, and the length L of the aquifer.
Background
Cody Problem 52070 dealt with the effective conductivity of a heterogeneous confined aquifer. In that case, the effective conductivity Keff for the case of flow through soil units in parallel could be computed by realizing that the head difference is same for the two soils and the total flow is the sum of the flows through the soils. The effective conductivity then is the weighted average of the individual conductivities, using the layer thicknesses as weights.
For an unconfined aquifer, the effective conductivity is computed in a similar way, but because the WTE h varies with x, so does Keff:
Keff = (1/h)[(K2-K1) b2 + K1(h-b2)]
Then using Darcy’s law as in Cody Problem 58966, one can write the flow Q through the aquifer as
Q = -Keff(dh/dx)A = -Keff(dh/dx)hw
Conservation of mass says that in steady one-dimensional flow, the flow Q past any section is constant. Then the WTE can be computed by inserting the effective conductivity, integrating, and using the boundary conditions h(0) = h0 and h(L) = hL to determine the flow Q and the constant of integration.
unconfined aquifer with soils in parallel
Solve

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100.0% Correct | 0.0% Incorrect
Last Solution submitted on Oct 01, 2023

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