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Worksheet Grade 12 Math

Kirtlebeck Well 4 Pump Test: Drawdown per Log Cycle

A grade 12 worksheet in which students fit a semi-log drawdown model to a pumping test record, predict later drawdown, and judge an extrapolation.

Objectives

  1. Students will compute drawdown per log cycle from a pumping test record.
  2. Students will write and apply a semi-logarithmic model of drawdown against time.
  3. Students will evaluate a residual and judge whether an extrapolation is supported by the test.

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.

Also: static water level, log cycle, specific capacity, extrapolation

Teacher overview

When a well is pumped at a steady rate, the water level in it drops quickly at first and then more slowly, so drawdown plotted against clock time gives a curve with no easy slope. Plotted against the logarithm of time, the same readings fall close to a straight line, and the rise of that line across one tenfold step in time is what well operators call drawdown per log cycle. This worksheet uses a constant-rate test on Well 4 in the fictional Kirtlebeck aquifer beneath the town of Mardleton, pumped at 480 gpm. Students find the drawdown per log cycle, write the model, extend it, and then decide how far the extension can honestly be pushed.

Kirtlebeck Well 4 Pump Test: Drawdown per Log Cycle — Grade 12 Math
Name: ____________________________ Date: ______________

Student Worksheet

The fictional town of Mardleton ran a constant-rate test on Well 4, which draws from the Kirtlebeck aquifer. The pump held a steady 480 gpm for the whole test. The static water level before pumping began was 62.0 feet below grade. Drawdown means how far the water level has fallen below that static level.

Time since pumping started, t (minutes) Drawdown, s (feet)
1 4.0
10 10.0
30 13.2
100 16.0
1,000 22.0
  1. Compute the base ten logarithm of 30. Round to three decimal places.
  2. Using the readings at 10 minutes and 100 minutes, find the drawdown per log cycle in feet.
  3. Using the readings at 100 minutes and 1,000 minutes, find the drawdown per log cycle in feet.
  4. The record fits a model of the form s = a + b times the base ten logarithm of t. Report the value of a, which is the drawdown at 1 minute.
  5. Use the model from question 4 to predict the drawdown at 10,000 minutes.
  6. The static water level is 62.0 feet below grade. How far below grade is the pumping water level at 1,000 minutes?
  7. Use the model to find the time, in minutes, at which drawdown reaches 34.0 feet.
  8. Convert your answer to question 7 into days, using 1,440 minutes per day. Round to one decimal place.
  9. The model predicts 12.9 feet of drawdown at 30 minutes, and the crew recorded 13.2 feet. Find the residual, using observed minus predicted.
  10. Well 4 was pumped at 480 gpm throughout. Compute its specific capacity at 1,000 minutes, in gallons per minute per foot of drawdown. Round to one decimal place.
  11. Explain in two or three sentences why plotting drawdown against the logarithm of time gives a straight line for this well, while plotting it against time itself does not.
  12. The test ran for 1,000 minutes. Explain in two or three sentences whether you would report the answer to question 8 to the fictional town's engineer as a reliable prediction.

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: 1.477.
    The base ten logarithm of 30 is about 1.4771, which rounds to 1.477.

  2. Answer: 6.0 feet.
    Ten minutes to 100 minutes is one full log cycle, and the drawdown rises from 10.0 to 16.0 feet, a difference of 6.0 feet.

  3. Answer: 6.0 feet.
    One hundred minutes to 1,000 minutes is another full log cycle, and drawdown rises from 16.0 to 22.0 feet, again 6.0 feet. The matching value is what makes the semi-log line straight.

  4. Answer: 4.0 feet.
    At t = 1 minute the logarithm of t is 0, so s = a. The recorded drawdown at 1 minute is 4.0 feet, giving the model s = 4.0 + 6.0 times the base ten logarithm of t.

  5. Answer: 28.0 feet.
    The logarithm of 10,000 is 4, so s = 4.0 + 6.0(4) = 4.0 + 24.0 = 28.0 feet.

  6. Answer: 84.0 feet below grade.
    Drawdown at 1,000 minutes is 22.0 feet, and the water started 62.0 feet below grade, so the pumping level sits at 62.0 + 22.0 = 84.0 feet below grade.

  7. Answer: 100,000 minutes.
    Setting 4.0 + 6.0 times the logarithm of t equal to 34.0 gives 6.0 times the logarithm of t equal to 30.0, so the logarithm of t is 5 and t is 10^5, or 100,000 minutes.

  8. Answer: 69.4 days.
    Dividing 100,000 minutes by 1,440 minutes per day gives about 69.44 days, which rounds to 69.4 days.

  9. Answer: 0.3 feet.
    The residual is 13.2 minus 12.9, which is 0.3 feet. The positive sign means the crew measured slightly more drawdown than the fitted line predicts at that time.

  10. Answer: 21.8 gpm per foot.
    Specific capacity is the pumping rate divided by the drawdown, so 480 gpm divided by 22.0 feet is about 21.82, which rounds to 21.8 gpm per foot.

  11. Answer: Each tenfold step in time adds the same amount of drawdown.
    Going from 10 to 100 minutes and from 100 to 1,000 minutes both add 6.0 feet, so equal steps along a logarithmic axis produce equal rises. On a plain time axis those same two intervals have very different widths, so the same readings bend into a curve.

  12. Answer: No.
    Reaching 34.0 feet of drawdown requires two more log cycles past the end of the test, so the answer rests entirely on the assumption that the straight line continues for a hundred times longer than anyone measured. A nearby boundary, a leaky layer, or a second well starting up could change the slope well before then, so the figure should be reported as an extrapolation rather than a prediction.

Differentiation

Support

Provide these students with a semi-log grid on which the five fictional Mardleton readings are already plotted, so their work begins with reading a slope off a line rather than constructing the plot. Give a worked example of one log cycle on a neutral data set, and supply the model s = 4.0 + 6.0 times the logarithm of t as a reference for items 5 through 8.

Extension

Have students suppose that the fictional town repeats the test on Well 5, which yields drawdowns of 5.0 feet at 1 minute and 21.5 feet at 1,000 minutes. Ask them to find that well's drawdown per log cycle, compare it with Well 4's, and explain which well's water level is more sensitive to a long pumping run.

ELL supports

Pre-teach "drawdown," "static water level," "log cycle," and "extrapolation" with a labeled cross-section of a well showing the two water levels and the distance between them. Provide sentence frames for items 11 and 12, such as "The line is straight because each tenfold step in time adds ___," and allow a labeled sketch in place of prose for item 11.

Extensions

Students can repeat the analysis on the same fictional data after converting every time to hours instead of minutes, then explain why the drawdown per log cycle does not change while the value of a does. A follow-up asks students to look up how their state reports well yields in public well logs and to compare those units with the gallons per minute per foot used here.