An important task in characterizing the flow of groundwater is to determine the properties of the aquifer, or the underground water-bearing formation. One approach is to disturb the aquifer, observe its response, and fit a theoretical formula to the observations.
For example, suppose a confined aquifer initially has no flow. In that case, the piezometric head h, or the level to which water would rise in an observation well, would be a uniform value 
. A well turned on and pumped at a rate 
will create a cone of depression; that is, it will draw down the piezometric head to a level 
, where r is the radial distance from the well. Applying conservation of mass and Darcy’s law to this situation leads to a diffusion equation whose solution for the drawdown 
as a function of distance r and time t is
. A well turned on and pumped at a rate will create a cone of depression; that is, it will draw down the piezometric head to a level , where r is the radial distance from the well. Applying conservation of mass and Darcy’s law to this situation leads to a diffusion equation whose solution for the drawdown as a function of distance r and time t is
where T is the transmissivity, S is the storativity, and
Write a function that achieves the objective of a pumping test: to determine the transmissivity and storativity from measurements of drawdown in time.
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