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Math problem set Grade 8 Science

Pinehurst Well Drawdown: Linear Functions for Pumping

Grade 8 students model a fictional well's drawdown and pumping level as linear functions of pumping rate, then check finite aquifer storage with scientific notation.

Objectives

  1. Students will construct linear functions for well drawdown and pumping water level.
  2. Students will solve linear equations to find safe pumping rates and use scientific notation for aquifer storage.

Vocabulary

Drawdown
Drawdown is how far the water level in a well drops when a pump pulls water out. The harder and longer you pump, the bigger the drawdown gets.
Aquifer
An aquifer is an underground layer of rock, sand, or gravel that holds water in the tiny spaces between its grains. Wells reach down into aquifers to bring water up to the surface.

Also: static water level, specific capacity

Teacher overview

When a pump pulls water from a well, the water level inside the well drops, and the harder it pumps, the more the level falls. That drop is called drawdown, and it can be modeled with a linear function. This set uses the fictional town of Pinehurst to build those functions and find safe pumping limits. A common misconception is to mix up the static water level (how far down the water sits before pumping) with the drawdown (the extra drop caused by pumping); this set keeps the two apart.

Pinehurst Well Drawdown: Linear Functions for Pumping — Grade 8 Science
Name: ____________________________ Date: ______________

Problems

Pinehurst (a fictional town) has a groundwater well. Before pumping, the static water level sits 40 feet below the ground. Tests show that for every 100 gpm (gallons per minute) pumped, the water level in the well drops another 10 feet. Drawdown is the pumping water level minus the static water level.

  1. Write a linear function for the drawdown D, in feet, in terms of the pumping rate Q in gpm, using the rate of 10 feet of drawdown for every 100 gpm.
  2. What is the drawdown when the well is pumped at 200 gpm?
  3. At 200 gpm, how far below the ground is the pumping water level? Add the static level and the drawdown.
  4. Write a linear function for the pumping water level L, in feet below the ground, in terms of Q.
  5. The pump intake sits 90 feet below the ground. If the water level falls to the intake, the pump breaks suction. At what pumping rate would the water level reach 90 feet? Solve 40 + 0.1Q = 90.
  6. To stay safe, operators keep the water level at least 10 feet above the intake, so no deeper than 80 feet. What is the maximum safe pumping rate? Solve 40 + 0.1Q = 80.
  7. Specific capacity is the pumping rate divided by the drawdown. At 200 gpm with 20 feet of drawdown, what is this well's specific capacity in gpm per foot?
  8. A second well nearby has a specific capacity of 4 gpm per foot. Compared with Pinehurst's well, which well gives more water for the same drawdown, and why?
  9. The aquifer under Pinehurst holds about 8.0 × 10^8 gallons of usable water. If the town withdraws 4.0 × 10^5 gallons per day, about how many days of supply is that? Write your answer in scientific notation.
  10. Using your drawdown function, explain why pumping the well at a higher rate is not always a good idea.

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key
  1. Answer: D = 0.1Q.
    Ten feet per 100 gpm is a rate of 10 ÷ 100 = 0.1 feet per gpm, so D = 0.1Q.
  2. Answer: 20 feet.
    D = 0.1 × 200 = 20 feet of drawdown.
  3. Answer: 60 feet.
    Pumping level = static level + drawdown = 40 + 20 = 60 feet below the ground.
  4. Answer: L = 40 + 0.1Q.
    The static level of 40 feet is the initial value, and the drawdown rate 0.1 is the slope.
  5. Answer: 500 gpm.
    Solve 40 + 0.1Q = 90, so 0.1Q = 50 and Q = 500 gpm.
  6. Answer: 400 gpm.
    Solve 40 + 0.1Q = 80, so 0.1Q = 40 and Q = 400 gpm as the maximum safe rate.
  7. Answer: 10 gpm per foot.
    Specific capacity = 200 ÷ 20 = 10 gpm per foot of drawdown.
  8. Answer: Pinehurst's well, because it gives more water per foot of drawdown.
    Pinehurst's 10 gpm per foot is higher than the second well's 4 gpm per foot, so it yields more flow for the same drop. Accept any answer that compares 10 to 4.
  9. Answer: 2.0 × 10^3 days.
    8.0 × 10^8 ÷ (4.0 × 10^5) = 2.0 × 10^3 days, which is about 2,000 days.
  10. Answer: Higher pumping means more drawdown and more strain on the aquifer.
    A larger Q raises the drawdown, which can pull the water level down to the intake and break suction, and it draws harder on a groundwater supply that refills slowly. Accept any answer connecting higher Q to more drawdown or aquifer strain.

Differentiation

Support

Provide a table (Q, drawdown) with the values for 100 and 200 gpm filled in so students can see the constant rate before writing the function. Give the frame "L = ___ + ___ Q" with the blanks labeled static level and drawdown rate.

Extension

Ask students to graph both D and L against Q on the same axes for pumping rates from 0 to 500 gallons per minute and mark where L reaches the 90-foot intake. Then have them find the pumping rate that would leave exactly 25 feet of drawdown and explain their method.

ELL supports

Pre-teach "static," "drawdown," and "aquifer" with a labeled cross-section of a well showing the water level before and during pumping. Provide the sentence frame "Drawdown equals ___ times ___." Allow students to show the equation steps rather than writing explanations for the calculation items.