I. Groundwater flow equation
The text derives this equation based on a conservation of fluid mass. To simplify the derivation for class it was presented in terms of conservation of fluid volume. This is not strictly correct since volume may change as a function of temperature or pressure, but the result is the same in the end. Considering an REV such as that shown in the text we can express the difference between fluid entering and exiting through opposite sides of a cube in the x direction as
Further, assuming that the specific discharge, q, varies linearly with distance across the REV
Similar equations can be derived for flow in and out in the y and z directions. Putting these together and using the definition of specific storage, our conservation equation becomes
Dividing both sides by the REV volume and substituting Darcy's law for the q's yields
Which is a general equation for 3D transient flow in a heterogeneous, anisotropic aquifer. If the aquifer is homogeneous and isotropic, the above equation becomes
The text provides a number of variant on this for different assumptions.
For an unconfined homogeneous aquifer employing the Dupuit assumptions the governing equation becomes
The unconfined, homogeneous aquifer equation can also be written as
where the dependent variable is h-squared rather than h. If the aquifer is very thick or the slope of the water table is small, you can replace Kh in the above equation by an approximately constant T (transmissivity). If the 2D equation for a confined aquifer will yield a good solution for an unconfined aquifer as well.