Keep a dated log of every flow measurement with the rain before it, and fit the dry-season readings to the exponential recession Q = Q0 × e^(−k×t), which turns a few readings into a forecast of how low the spring will fall: readings dropping from 12 L/min to 7.7 L/min over 60 days give k = 0.0074 per day, a half-life of 94 days and about 3.2 L/min at day 180. Use it to size storage for a dry season longer than the one you saw, and never use readings taken just after rain, which belong to a different curve.
Part of Spring Capture
Flow records are the written series of a spring’s measured flows, with the conditions they were taken in. Besides showing the minimum, a good record lets you predict the dry-season decline and the water stored behind the spring.
A Spring That Drains Its Store Falls By A Fixed Fraction Each Day
When rain stops, a spring is fed only by the water stored in the ground above its outlet. The simplest model of that store is a single reservoir that drains in proportion to how full it is. The solution is the exponential recession curve, attributed to Maillet (1905) and given in standard hydrogeology texts (recalled from them here, not derived from this guide’s sources):
Q(t) = Q0 × e^(−k×t)
where Q0 is the flow at the start of the recession, t is the time in days, and k is the recession coefficient in 1/day. Taking logarithms, ln Q = ln Q0 − k × t, a straight line: so plot ln Q against t and the slope is −k.
Two useful results follow. The half-life, the time for the flow to halve, is ln 2 ÷ k = 0.693 ÷ k. The water still stored above the outlet when the flow is Q is S = Q ÷ k.
Worked Example: Fitting And Forecasting
Five dry-season readings, with no rain in the days before each (illustrative values):
| Day | 0 | 15 | 30 | 45 | 60 |
|---|---|---|---|---|---|
| Q (L/min) | 12.0 | 10.7 | 9.6 | 8.6 | 7.7 |
| ln Q | 2.485 | 2.370 | 2.262 | 2.152 | 2.041 |
Test the fit. The fall in ln Q per 15 days is 0.115, 0.108, 0.110, 0.111: nearly constant, so the points lie close to a straight line and the model is reasonable. If the steps shrink steadily, the spring has a fast component and a slow one and a single exponential is optimistic early and pessimistic late.
Fit. k = (ln 12.0 − ln 7.7) ÷ 60 = (2.485 − 2.041) ÷ 60 = 0.0074 per day.
- Half-life = 0.693 ÷ 0.0074 = 94 days
- Q(120) = 12 × e^(−0.0074 × 120) = 12 × e^(−0.888) = 12 × 0.412 = 4.9 L/min
- Q(150) = 12 × e^(−1.11) = 12 × 0.330 = 4.0 L/min
- Q(180) = 12 × e^(−1.332) = 12 × 0.264 = 3.2 L/min
Water stored. At day 0, Q0 = 12 L/min = 12 × 1,440 = 17,280 L/day, so S = 17,280 ÷ 0.0074 = 2.34×10⁶ L, about 2,300 m³ of water draining slowly from the hillside. At day 60 the store left is 7.7 × 1,440 ÷ 0.0074 = 1.5×10⁶ L.
Storage to bridge a longer dry season. Suppose the settlement uses D = 7,200 L/day (illustrative). The spring falls below it when 17,280 × e^(−0.0074 × t) = 7,200, so e^(−0.0074 t) = 7,200 ÷ 17,280 = 0.4167 and t = −ln 0.4167 ÷ 0.0074 = 118 days. If the dry spell goes on to day 180, the shortfall is the demand minus the supply summed from day 118 to day 180:
deficit = D × (180 − 118.3) − (Q0 ÷ k) × (e^(−k × 118.3) − e^(−k × 180)) = 7,200 × 61.7 − 2.335×10⁶ × (0.4167 − 0.2639) = 444,200 − 356,700 = 87,500 L, about 88 m³.
A tank holding that, filled while the spring is strong, carries the settlement to day 180. A spell to day 150 would need only about 25 m³. The tank is the cost of the unknown length of the dry season (Storage Tanks).
How Far To Trust The Fit
- Two readings cannot show curvature. Use at least four or five over the recession.
- Small reading errors change the forecast. If the day-60 reading were 8.0 instead of 7.7, k = ln(12 ÷ 8.0) ÷ 60 = 0.0068 and Q(120) = 5.3 L/min, 8% higher.
- Rain resets the curve. Fit only readings with no significant rain in the preceding days.
- A year does not contain every drought. The fit extrapolates your dry season, not the worst one (Seasonal Monitoring).
What You Need
- A bound notebook — numbered pages, ink, no torn-out sheets
- The measuring set-up — bucket, watch, channel (Bucket Timing)
- A rain gauge (Rainfall Data)
- Two copies of the record, kept in different places
- Log-paper or a ruler and a table of natural logarithms — for plotting ln Q
Method: Log, Plot, Fit, Forecast
Step 1: Log every visit with these columns: date, time, who measured, fill volume, fill time, Q in L/min, rain in the last 7 days and today, water clarity, notes (changes at the spring).
Step 2: Keep copies. Copy the log to a second sheet after each visit and store it away from the first.
Step 3: Pick the recession. Mark the run of readings after the last significant rain.
Step 4: Plot ln Q against day and judge whether the points lie on a line.
Step 5: Fit the line, compute k, half-life and the forecast, and note the uncertainty.
Step 6: Use the forecast to set the storage and the supply plan, and re-fit as new readings come in.
Common Mistakes
- Fitting readings that include rain. They sit on a different curve.
- Extrapolating far beyond the data. A forecast to day 180 from 60 days of readings carries real uncertainty.
- Changing the way the flow is gathered. Steps in the record that are not steps in the spring.
- Losing the book. The record is the only evidence of what the spring did.
Safety
A forecast is not a guarantee
A drought can be longer than any in the record, and a tank sized for the forecast can still run dry. Keep the community’s use of the spring flexible, hold reserves, and treat the water: a spring that has been low may also be contaminated more easily when it recovers (Community Water Testing).
What’s Next
- Learn next: Storage Tanks — holding the shortfall
- Combine with: Spring Box Construction — building to the measured flows