Solution to Exercise 12

William W. Woessner; Eileen P. Poeter

Solution to Exercise 12

12) What conditions does the following governing equation represent? Explain the features of the terms of the equation that indicate each condition.

$\displaystyle S_{s}\frac{\partial h}{\partial t}=\frac{\partial }{\partial x}\left ( K_{x}\frac{\partial h}{\partial x} \right )+\frac{\partial }{\partial y}\left ( K_{y}\frac{\partial h}{\partial y} \right )+\frac{\partial }{\partial z}\left ( K_{z}\frac{\partial h}{\partial z} \right )$

This is Equation 67 of this book. It represents transient, confined, groundwater flow where conditions are anisotropic and heterogeneous. The K values are within the differential on the right-hand side of the equation signifying they are variables and can be different values everywhere in the domain (x,y,z), thus the equation is applicable to a heterogeneous and anisotropic system. The left-hand side has a time derivative and thus, this equation represents transient conditions, that is, the heads change with time and water can go into, or come out of, storage. If the left-hand side was set to 0, the equation would represent steady state conditions. If the equation represented unconfined conditions, there would be an h inside the derivative, and S_y would be used instead of S_s as follows (as shown in Equation 74 of this book):

$\displaystyle S_y\frac{\partial h}{\partial t}\ =\frac{\partial}{\partial x}K_x\left(h\frac{\partial h}{\partial x}\right)+\frac{\partial}{\partial y}K_y\left(h\frac{\partial h}{\partial y}\right)$

Return to Exercise 12

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