Problem 58966. Compute the head for steady 1D flow in a homogeneous aquifer
Write a function that computes the head h (i.e., piezometric head for a confined aquifer or water table elevation above the aquifer bottom for an unconfined aquifer) at specified locations x. The input will include a Boolean variable that is true for confined aquifers and false for unconfined aquifers as well as the length L of the aquifer, hydraulic conductivity K, aquifer thickness b (set to  for unconfined aquifers), width w, and a vector BC, which contains two of three of the head at , the head at , and the flow Q. The function should also return the flow.
The movement of groundwater is governed by conservation of mass and (if the flow is slow enough) Darcy’s law. The latter relates the flow of water to the gradient in piezometric head by
where A is the flow area. For a confined aquifer, the flow area is , whereas for an unconfined aquifer .
The spatial variation of the head can be computed with these two principles. For steady, one-dimensional flow, conservation of mass says that the flow must by constant (i.e., the same at any cross section). Then Darcy’s law provides a differential equation that can be integrated using two of the three boundary conditions listed above
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