Chapter 22, Problem 36TAS

Darcy’s law gives us the volume flow rate—volume per time (for example, cubic meters per day, m³/day). Another way to express volume is on a per-unit-area basis. For example, if we have a 1-m³ cube, we know its base is I m²—the cross-sectional area of the cube—and its height is 1 m. If we fill the cube with water, we can say we have 1 m of water per unit area. Darcy’s law can be rearranged to calculate the volume flow rate per unit area. This is called the specific discharge, which has units of length per time (for example, meters per day, m/day). Completing parts (a) through (c) will illustrate how specific discharge is related to the volume flow rate. (Hint: Assume that the units for hydraulic conductivity are m/day and the units for cross-sectional area are m².)

1. (a) Suppose 1 m³ of water is pumped from a well into an empty cylindrical tank. If the water level is 2 m above the base, what is the cross-sectional area of the tank? (Hint: Volume of a cylinder = πr²b = area of base X height.)
2. (b) If it takes half a day to pump the 1 m³ of water into the tank, what is the flow rate in terms of both volume per unit time and specific discharge?
3. (c) Write Darcy’s law so that it calculates the specific discharge.