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Classroom activity Grade 12 Science

Fellstone Spring: Fracture Widening and Conduit Growth

A grade 12 activity where teams measure how delivery rate controls limestone dissolution, then run three widening rounds until one fracture takes the flow.

Objectives

  1. Students will run a controlled comparison of limestone dissolution at two delivery rates and identify the variables held constant.
  2. Students will apply the cubic relationship between aperture and discharge to divide a fixed flow among three fractures.
  3. Students will explain how the feedback between flow and dissolution turns nearly identical fractures into a single conduit.
  4. Students will describe what conduit flow means for treating a karst spring used as a drinking water source.

Vocabulary

Key words: karst, aperture, conduit, swallet, saturation, positive feedback

Teacher overview

This activity pairs a short bench trial with a three-round board simulation and runs in teams of four. In the trial, teams hold the solution and the rock constant and vary only how fast the solution reaches a stick of calcium carbonate, which isolates delivery rate as a control on how much rock is removed. On the board, three fractures of nearly equal width carry water from a sinkhole to a spring in the fictional Wistow Limestone, and teams recompute the flow split and widen each fracture for three rounds. One misconception is worth planning for: the even-widening picture of cave formation, in which every opening in a limestone bed enlarges at about the same rate because water is present throughout the rock. Dissolution is instead limited by the arrival of water that is not yet saturated with calcium carbonate, so a fracture that begins slightly wider widens faster still, and the differences between the three openings grow from round to round.

Activity Directions

  1. Before class, about 15 minutes. Print one handout per student and one fracture board per team of four. Mix the trial solution, one part white vinegar to three parts water, allowing 250 mL per team. Every place, fracture, and figure in this activity is invented for the fictional Calverstock scenario; no real spring or aquifer is described.
  2. Framing, 5 minutes. Explain that rainwater picks up carbon dioxide in the soil, becomes weakly acidic, and dissolves calcium carbonate along the fractures it travels through. Tell teams that the trial substitutes diluted vinegar because carbonic acid works far too slowly to give a result in one class period, while the reaction being modeled, an acid dissolving calcium carbonate, is the same kind. Assign the four roles within each team: timekeeper, Channel A delivery, Channel B delivery, and recorder.
  3. Trial setup, 10 minutes. Teams weigh two similar sticks of calcium carbonate to 0.01 g, record both masses, and lay one stick in each groove of the tray with the tray tilted so liquid drains off rather than pooling. Teams answer handout items 1 and 2 before any solution is delivered.
  4. Run and weigh, 12 minutes. On the timer, Channel A receives 5 mL of solution each minute and Channel B receives 20 mL each minute, for eight minutes. Teams then blot both sticks the same number of times with paper towels, reweigh, and record items 3, 4, and 5. Point out that residual moisture leaves both final masses slightly high, and that the comparison between the two channels still holds because both sticks were treated identically.
  5. Board setup, 5 minutes. Give each team a fracture board and read the two model rules aloud from the handout: flow through a fracture is proportional to the cube of its aperture, and each round a fracture widens by 0.60 mm multiplied by its share of the flow. State plainly that the model holds the total flow arriving underground constant so that the fractures compete for a fixed supply.
  6. Rounds 1 through 3, 20 minutes. Teams work items 6 through 12, filling one row of the recording table per round and redrawing each fracture path on the board with a thicker line as its aperture grows. The simulation ends when every team has completed round 3 and answered item 13.
  7. Debrief, 10 minutes. Ask each team to report Fracture 1's flow in round 1 and in round 3, then to predict where Fracture 1 would stand after ten more rounds. Next ask what this model holds fixed that a real aquifer would not, and steer the discussion toward two answers: the total flow reaching the spring would itself rise as the conduit opened, and the cubic relationship stops applying once flow in the largest opening becomes turbulent. Have students write item 14 individually before the period ends, then close by asking each team to state one thing the model represents well and one thing it leaves out.
Fellstone Spring: Fracture Widening and Conduit Growth — Grade 12 Science
Name: ____________________________ Date: ______________

Student Handout

Suppose you are studying groundwater in the fictional Wistow Limestone beneath the town of Calverstock. Water sinks underground at Pendrick Swallet and returns to the surface at Fellstone Spring, 1,200 meters away. Three fractures connect the two points. All three have the same length and the same width, and all three are driven by the same difference in water level between the swallet and the spring. Every place and figure below is invented for this activity. Items 1 through 14 are the questions you answer and turn in.

Rules for Part B:

  • The total flow reaching Fellstone Spring stays at 900 gallons per minute in every round.
  • Flow through a fracture is proportional to the cube of its aperture, so a fracture's share of the total equals its aperture cubed divided by the sum of all three cubed apertures.
  • After each round, a fracture's aperture grows by 0.60 mm multiplied by its share of the flow. Round each new aperture to the nearest 0.01 mm before starting the next round.
  • Round 1 apertures: Fracture 1 is 1.00 mm, Fracture 2 is 1.10 mm, Fracture 3 is 1.20 mm.
Round Fracture 1 aperture (mm) Fracture 2 aperture (mm) Fracture 3 aperture (mm)
1 1.00 1.10 1.20
2
3
4

Part A, the dissolution trial

  1. State the one variable your team changes between Channel A and Channel B.
  2. Name one variable your team holds the same in both channels.
  3. Record the mass lost by the Channel A stick, in grams.
  4. Record the mass lost by the Channel B stick, in grams.
  5. Explain why a groove that receives more solution each minute loses more calcium carbonate each minute, in terms of how close the liquid leaving the groove is to saturation.

Part B, the fracture board

  1. Round 1: calculate the flow through Fracture 3, in gallons per minute.
  2. Round 1: calculate the flow through Fracture 1, in gallons per minute.
  3. Round 1: calculate the flow through Fracture 2, in gallons per minute.
  4. Fracture 3's aperture is 1.20 times Fracture 1's aperture in round 1, yet its flow is much more than 1.20 times as large. Explain which part of the flow rule produces that difference.
  5. Round 2, with apertures of 1.15 mm, 1.30 mm, and 1.46 mm: calculate the flow through Fracture 3, in gallons per minute.
  6. Round 3, with apertures of 1.28 mm, 1.49 mm, and 1.73 mm: calculate the flow through Fracture 1, in gallons per minute.
  7. Round 3, with apertures of 1.28 mm, 1.49 mm, and 1.73 mm: calculate Fracture 3's aperture at the start of round 4, in millimeters.
  8. Using Fracture 1's flow in round 1 and in round 3, explain why three fractures that start within 0.20 mm of one another do not stay close, and name the feedback that separates them.
  9. Fellstone Spring is the drinking water source for the fictional Calverstock system. Explain what the widening conduit means for how quickly material entering Pendrick Swallet reaches the spring, and what that implies for how the water must be treated.

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: The delivery rate. Channel A receives 5 mL of solution each minute and Channel B receives 20 mL each minute, and nothing else about the two channels differs.
  2. Answer: Accept any one of the following: the solution and its strength, the starting mass and shape of the calcium carbonate stick, the tilt of the tray, the total run time, and the room temperature. Holding these constant is what allows the difference in mass loss to be attributed to delivery rate.
  3. Answer: Answers vary by team. Accept any recorded value together with the before and after masses that produced it. In a typical run the Channel A stick loses a few hundredths of a gram to about two tenths of a gram.
  4. Answer: Answers vary by team. Accept any recorded value that is larger than the Channel A value and is supported by the recorded masses. A team whose Channel B loss is not larger should check whether solution pooled in the groove or ran off the stick without contacting it.
  5. Answer: Because the liquid leaving the faster groove is further from saturation. Each volume of solution can dissolve only so much calcium carbonate before it approaches equilibrium and stops reacting appreciably, so once a small volume sits on the stick it quickly loses its capacity to dissolve more. Delivering four times as much solution per minute keeps replacing spent liquid with fresh, undersaturated liquid, and the rate of removal follows the rate of delivery rather than the amount of rock available.
  6. Answer: 383 gpm. The cubed apertures are 1.000, 1.331, and 1.728, which sum to 4.059.
    Fracture 3's share is 1.728 / 4.059 = 0.4257, and 0.4257 x 900 = 383.1 gallons per minute.
  7. Answer: 222 gpm. Fracture 1's share is 1.000 / 4.059 = 0.2464, and 0.2464 x 900 = 221.8 gallons per minute.
  8. Answer: 295 gpm. Fracture 2's share is 1.331 / 4.059 = 0.3279, and 0.3279 x 900 = 295.1 gallons per minute. The three rounded flows sum to 900 gallons per minute, which is a useful check on the arithmetic.
  9. Answer: The cube. Flow depends on the cube of the aperture rather than on the aperture itself, so an aperture ratio of 1.20 becomes a flow ratio of 1.20 cubed, or 1.728. A fracture 20 percent wider therefore carries about 73 percent more water, and that amplification is what makes small differences in aperture matter so much.
  10. Answer: 410 gpm. The cubed apertures are 1.520875, 2.197, and 3.112136, which sum to 6.830011. Fracture 3's share is 3.112136 / 6.830011 = 0.4557, and 0.4557 x 900 = 410.1 gallons per minute, up from 383 gallons per minute in round 1.
  11. Answer: 178 gpm. The cubed apertures are 2.097152, 3.307949, and 5.177717, which sum to 10.582818. Fracture 1's share is 2.097152 / 10.582818 = 0.1982, and 0.1982 x 900 = 178.4 gallons per minute, down from 222 gallons per minute in round 1.
  12. Answer: 2.02 mm. Fracture 3's share in round 3 is 5.177717 / 10.582818 = 0.4893, so it widens by 0.60 x 0.4893 = 0.2936 mm. Its aperture at the start of round 4 is 1.73 + 0.29 = 2.02 mm, rounded to the nearest 0.01 mm.
  13. Answer: A positive feedback between flow and dissolution. Fracture 1 carried 222 gallons per minute in round 1 and 178 gallons per minute in round 3, a loss of about 44 gallons per minute, while Fracture 3 gained over the same rounds. The fracture that starts widest takes the largest share of the flow, the largest share of the flow delivers the most undersaturated water, and the most undersaturated water removes the most rock, which makes that fracture wider still. Because flow depends on the cube of the aperture, each round's small advantage is amplified before it feeds back into the next round, so the three openings separate rather than converge.
  14. Answer: Material moves through fast and arrives with little treatment by the aquifer. Once one opening has grown into a conduit, water crosses the 1,200 meters from Pendrick Swallet to Fellstone Spring through what behaves like a pipe rather than through the small pore spaces that would strain particles and hold water long enough for bacteria to die off. A fuel spill or a manure application near the swallet can reach the spring in hours, and spring turbidity rises sharply after storms. For the fictional Calverstock system this means the spring cannot be treated as a protected groundwater source; it needs the filtration and disinfection a surface water source requires, along with continuous turbidity monitoring and protection of the land around the swallet.

Differentiation

Support

Give these teams a round 1 worksheet with the three cubed apertures, 1.000, 1.331, and 1.728, and their sum of 4.059 already printed, so the first round is a division and a multiplication rather than a full calculation chain. Work item 6 with them before they attempt round 2, and hand them a card with the two model rules written out in order. During the trial, assign these students the timekeeper and recorder roles first, which keeps them inside the procedure while the delivery roles demand steady hands.

Extension

Ask teams to predict what happens to the fictional Calverstock model once the largest opening carries turbulent flow, where discharge grows roughly with aperture to the 1.5 power instead of the cube. Have them recompute round 3 with the exponent changed from 3 to 1.5 and explain why the takeover slows when the exponent falls.

ELL supports

Pre-teach the four words the model depends on, aperture, saturated, conduit, and swallet, each with a labeled sketch of the fictional fracture board. Provide the sentence frame "Fracture ___ widened faster than Fracture ___ because it carried ___," and allow teams to talk through each round in their home languages before writing their answers.

Extensions

  • Have students research how their state classifies a spring that responds quickly to rainfall, including the category of groundwater under the direct influence of surface water, and report what treatment that classification requires.
  • Ask students to redesign the trial so that it measures the calcium carbonate that leaves in the liquid rather than the mass missing from the stick, and to explain which of the two measurements they would trust more and why.
  • Give teams the fictional round 4 apertures of 1.40 mm, 1.68 mm, and 2.02 mm, have them run two more rounds, and ask whether the share Fracture 3 gains each round is growing or shrinking as the gap widens. Have them also name the physical limit that would end the run in a real aquifer.