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Lesson plan Grade 12 Science

The Freshwater Lens Under Selvey Point

A 55-minute grade 12 lesson building a spreadsheet model of a barrier island's freshwater lens and testing which withdrawal rates keep seawater below the well screen.

Objectives

  1. Students will derive the Ghyben-Herzberg ratio from the densities of fresh water and seawater.
  2. Students will build a spreadsheet model relating withdrawal rate, water-table height, and interface depth.
  3. Students will express the well-screen depth limit as an inequality and interpret its solution as a withdrawal limit.

Vocabulary

Recharge
Recharge is water soaking down through the ground to refill an aquifer, the way rain refills an underground sponge. Aquifers stay healthy only if recharge keeps up with pumping.

Also: freshwater lens, interface, Ghyben-Herzberg ratio, hydrostatic balance, skimming well

Teacher overview

Selvey Point is a fictional barrier island town whose entire drinking water supply is a lens of fresh water floating on seawater inside the island's sand. This lesson has students derive why the lens is so much thicker than it looks from the surface, build a small spreadsheet model of it, and find the withdrawal rate at which seawater reaches the bottom of the town's well screen. The misconception the lesson targets is that a small change in the water table is a small change underground. Because fresh water is only about two-and-a-half percent less dense than seawater, one foot of lost water table lifts the salt water forty feet, so a lens can fail long before anything noticeable happens at the surface.

Procedure

Hook (5 min): Show a sketch of an island in cross-section with a lens of fresh water floating in the sand. Ask students to guess how deep the fresh water extends if the water table at the island's center stands 3 feet above sea level. Write the guesses on the board and hold them until the derivation.

Deriving the ratio (10 min): Set up the hydrostatic balance. At the depth where fresh water meets salt water, the weight of the column of fresh water above that point must equal the weight of the column of seawater beside it. Writing the fresh column as the height above sea level h plus the depth below sea level z, and the seawater column as z, gives fresh density times the quantity h plus z equal to salt density times z. Solving for z gives z equals fresh density times h divided by the difference in densities. With fresh water at 1.000 grams per cubic centimeter and seawater at 1.025 grams per cubic centimeter, that difference is 0.025, so z equals 40 times h. Return to the hook guesses and compare.

Guided worked example (8 min): Give the class the fictional calibration for Selvey Point in an average recharge year, h equals 4.5 minus 5.0 times P, where h is the water table height above sea level in feet and P is the average daily withdrawal in million gallons per day. Work one case with the class at P equal to 0.25 million gallons per day. Then h equals 4.5 minus 1.25, which is 3.25 feet, and the interface sits 40 times 3.25, or 130 feet, below sea level. Emphasize that the model gives a steady-state position, not the day the salt arrives.

Building the model (12 min): Pairs set up a spreadsheet with one column for withdrawal rate from 0.20 to 0.60 million gallons per day in steps of 0.05, a second column computing h from the average-year relation, a third computing interface depth as 40 times h, and a fourth flagging whether the interface is above the bottom of the well screen at 100 feet below sea level. Circulate and check that pairs have written the interface formula as a cell reference rather than typing in values.

Independent practice (15 min): Students complete the handout, which repeats the model for a drought year and evaluates a shallow well field. Circulate and watch for two errors: multiplying by 40 before subtracting, and treating a shallower screen as though it required a thicker lens.

Closure and exit ticket (5 min): As the formative check, ask every student to answer one question on a slip: the island's water table drops by half a foot, so how far does the interface rise, and why does that number worry a hydrogeologist? Collect the slips and scan for the value 20 feet before dismissal.

The Freshwater Lens Under Selvey Point — Grade 12 Science
Name: ____________________________ Date: ______________

Student Handout

The fictional town of Selvey Point sits on a barrier island. Its only supply is a lens of fresh water floating on seawater inside the island's sand aquifer. Fresh water has a density of 1.000 grams per cubic centimeter and the surrounding seawater has a density of 1.025 grams per cubic centimeter. The town's production well is screened from 70 feet to 100 feet below sea level.

The island's calibrated relations are as follows. In an average recharge year, h equals 4.5 minus 5.0 times P. In a drought year, h equals 3.6 minus 5.0 times P. In both, h is the water table height above sea level in feet at the island center, and P is the average daily withdrawal in million gallons per day.

  1. Using the two densities above, calculate the Ghyben-Herzberg ratio: the number of feet the interface lies below sea level for each foot the water table stands above sea level.

  2. In an average recharge year with a withdrawal of 0.30 million gallons per day, find the water table height h above sea level, in feet.

  3. Using your answer to item 2, find the depth of the freshwater-saltwater interface below sea level, in feet.

  4. Extend your spreadsheet across withdrawals of 0.20, 0.30, 0.40, 0.50, and 0.60 million gallons per day in an average recharge year. Report the interface depth below sea level at a withdrawal of 0.60 million gallons per day.

  5. The bottom of the well screen is 100 feet below sea level. In an average recharge year, find the largest average daily withdrawal, in million gallons per day, that still keeps the interface at or below the bottom of the screen.

  6. In a drought year the town keeps withdrawing 0.40 million gallons per day. Find the interface depth below sea level, in feet.

  7. In a drought year, find the largest average daily withdrawal, in million gallons per day, that keeps the interface at or below 100 feet below sea level.

  8. The town proposes replacing the deep well with three shallow skimming wells, none screened deeper than 60 feet below sea level. In a drought year, find the largest average daily withdrawal, in million gallons per day, that keeps the interface at or below 60 feet.

  9. Subtract your answer to item 7 from your answer to item 8 to find how much additional withdrawal, in million gallons per day, the shallow well field allows in a drought year.

  10. Suppose the island's water table falls by exactly 1 foot. Using the ratio from item 1, find how far the interface rises, in feet.

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: 40 feet per foot.
    At the interface the fresh column and the salt column balance, so 1.000 times the quantity h plus z equals 1.025 times z. Rearranging gives 1.000h equals 0.025z, so z equals h divided by 0.025, which is 40h.

  2. Answer: 3.0 feet. h equals 4.5 minus 5.0 times 0.30, which is 4.5 minus 1.5, or 3.0 feet above sea level.

  3. Answer: 120 feet.
    Interface depth equals 40 times h, so 40 times 3.0 equals 120 feet below sea level.

  4. Answer: 60 feet.
    At 0.60 million gallons per day, h equals 4.5 minus 3.0, or 1.5 feet, and the interface sits 40 times 1.5, or 60 feet, below sea level. The full column runs 140, 120, 100, 80, and 60 feet across the five withdrawal rates.

  5. Answer: 0.40 million gallons per day.
    The constraint is 40h at least 100, so h must be at least 2.5 feet. Setting 4.5 minus 5.0P at least 2.5 gives 5.0P at most 2.0, so P is at most 0.40 million gallons per day.

  6. Answer: 64 feet.
    In a drought year h equals 3.6 minus 5.0 times 0.40, which is 3.6 minus 2.0, or 1.6 feet. The interface sits 40 times 1.6, or 64 feet, below sea level, which is above the bottom of the screen.

  7. Answer: 0.22 million gallons per day.
    The constraint is again h at least 2.5 feet. Setting 3.6 minus 5.0P at least 2.5 gives 5.0P at most 1.1, so P is at most 0.22 million gallons per day.

  8. Answer: 0.42 million gallons per day.
    A screen no deeper than 60 feet requires 40h at least 60, so h must be at least 1.5 feet. Setting 3.6 minus 5.0P at least 1.5 gives 5.0P at most 2.1, so P is at most 0.42 million gallons per day.

  9. Answer: 0.20 million gallons per day.
    0.42 minus 0.22 equals 0.20 million gallons per day of additional withdrawal in a drought year. The shallow wells help because they sit farther above the interface, not because they draw from a thicker lens.

  10. Answer: 40 feet.
    Each foot of water table above sea level supports 40 feet of fresh water below it, so losing 1 foot at the surface lifts the interface 40 feet.

Exit ticket answer: A half-foot drop lifts the interface 20 feet, which is a large share of the distance between the interface and the bottom of the screen, and it can happen without any visible change at the ground surface.

Differentiation

Support

Provide the spreadsheet with the first two columns already filled and the interface formula written in the first row, so students extend the pattern rather than build it. Give a worked template for item 5 that shows the inequality set up and asks only for the algebra.

Extension

Ask students to add a recharge column to the model and derive the intercept themselves, assuming the intercept falls in direct proportion to annual recharge and that the drought year received 80 percent of average recharge. Have them state one physical reason the real relation would not stay perfectly linear at very high withdrawal rates.

ELL supports

Pre-teach lens, interface, and recharge with a labeled cross-section of the island showing the water table above sea level, the interface below it, and the well screen marked as a segment. Provide the handout with each relation written once in symbols and once in words, and allow students to answer item 10 orally before writing.

Extensions

Have students find a coastal or island community that monitors chloride in its supply wells and compare the monitoring thresholds it uses with the interface depths in this model. Ask them to explain why a utility would rather watch chloride climb slowly in a monitoring well than wait for it to appear in the production well.