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Water lab · Learning tool

Change the inputs.
Understand the balance.

A residual can change sign when an input changes

A simple conservation-of-water exercise with transparent assumptions. All calculations stay in your browser.

Storage change ΔS = P − ET − Q+50 mm

Equivalent storage change: +500,000 m³

Positive storage change: more water enters than leaves in this simplified balance.

Precipitation800 mm
800
Evapotranspiration450 mm
450
Runoff300 mm
300

The worked example

With 800 mm of precipitation, 450 mm of evapotranspiration and 300 mm of runoff over the same period, the storage residual is 800 − 450 − 300 = 50 mm. Over 10 km², that equals 50 × 10 × 1,000 = 500,000 m³.

Why the unit conversion works

1 mm = 0.001 m and 1 km² = 1,000,000 m². Their product is 1,000 m³. Use one time period and one catchment boundary for all inputs.

Read the residual carefully

A negative balance can indicate storage depletion or incomplete inputs; it is not automatically an error. A zero residual means inputs and outputs balance in the simplified model. Water imports, abstractions, groundwater exchange and measurement uncertainty may require additional terms.

Assumptions and source

This is a closed-boundary teaching example, not a flood prediction, water availability forecast or engineering design tool. Its inputs are illustrative. The conservation of water follows the water-budget framework described by the USGS Circular 1308: Water Budgets. For a real assessment, define boundaries, measure flows and evaluate uncertainty.

Before calculating: a boundary card and four acceptance checks

The equation needs a consistent boundary before it needs a precise number. Write down the catchment, observation period and which transfers are excluded. These fictional acceptance cases let you distinguish a legitimate negative residual from an invalid input.

  1. Choose one catchment and one period for precipitation P, evapotranspiration ET and runoff Q. Use depths in millimetres and catchment area in square kilometres. Annual precipitation and monthly runoff cannot share one residual.
  2. List transfers that the three-term model omits: imports, abstractions or groundwater exchange may matter. The residual cannot identify which omitted process occurred.
  3. Run the four cases below. Positive, zero and negative residuals are all allowed. A negative precipitation or a zero area is invalid for this teaching tool; correct it before interpreting a number.
  4. Save the inputs and output together. When using real measurements, record their uncertainty and sampling period. Additional decimal places in the output do not increase the quality of the observations.
P=800, ET=450, Q=300, area=10: ΔS=+50 mm; equivalent +500,000 m³; P=800, ET=450, Q=350, area=10: ΔS=0 mm; equivalent 0 m³; P=800, ET=450, Q=400, area=10: ΔS=−50 mm; equivalent −500,000 m³
P=800, ET=450, Q=300, area=10: ΔS=+50 mm; equivalent +500,000 m³. © water-msc.org

Diagram text: P=800, ET=450, Q=300, area=10: ΔS=+50 mm; equivalent +500,000 m³; P=800, ET=450, Q=350, area=10: ΔS=0 mm; equivalent 0 m³; P=800, ET=450, Q=400, area=10: ΔS=−50 mm; equivalent −500,000 m³

FieldExample or check
P=800, ET=450, Q=300, area=10ΔS=+50 mm; equivalent +500,000 m³
P=800, ET=450, Q=350, area=10ΔS=0 mm; equivalent 0 m³
P=800, ET=450, Q=400, area=10ΔS=−50 mm; equivalent −500,000 m³
P=−1 or area=0Invalid input; no interpreted balance
Unit check1 mm × 1 km² = 1,000 m³; 50 × 10 × 1,000 = 500,000

The worked cases test arithmetic and input handling. They do not validate a real catchment, predict flooding or establish sustainable withdrawals. If your boundary contains an unmeasured transfer, record that limitation beside the residual rather than silently treating the transfer as zero.

Download the working record ↓

Source context: USGS: Water Budgets, Circular 1308

A residual can change sign when an input changes

Keep the arithmetic separate from confidence in the inputs. The calculator’s positive storage residual is a result of the numbers entered. It does not establish that a real catchment is gaining water when those numbers are uncertain or incomplete.

Try the same boundary with three runoff estimates

Retain precipitation of 800 mm, evapotranspiration of 450 mm and area of 10 km² over one common period. Use illustrative runoff estimates of 250, 300 and 400 mm. Change only that input so you can see its effect.

Runoff Q800 − 450 − QEquivalent residual over 10 km²
250 mm+100 mm+1,000,000 m³
300 mm+50 mm+500,000 m³
400 mm−50 mm−500,000 m³

Run a one-input sensitivity check

  1. Write the catchment boundary and the common observation period before entering values.
  2. Record a baseline and one defensible alternative for the input under review. These classroom alternatives are not statistical confidence limits.
  3. Recalculate, keeping every other input fixed. Report which input changed and the size of the residual change.
  4. If the sign changes, say that the conclusion depends on the selected input. Do not average away disagreement without a measurement rationale.
  5. List missing transfers and measurement uncertainties before interpreting a real residual as storage change.

Each extra 1 mm of runoff reduces the residual by 1 mm in this simplified equation. At 10 km², that is 10,000 m³. A negative result is permitted: do not clamp it to zero. The USGS discusses uncertainty arising from variability and measurement; a full assessment needs uncertainties for all terms and their relationships. This sensitivity exercise changes one number and is neither a flood forecast nor a probability interval.

Q = 250 mm; ΔS = +100 mm; Q = 300 mm; ΔS = +50 mm; Q = 400 mm; ΔS = −50 mm
Original signed-residual plot for a fictional sensitivity exercise. The horizontal zero line separates positive and negative results.

Text equivalent: Q = 250 mm → ΔS = +100 mm → Q = 300 mm → ΔS = +50 mm → Q = 400 mm → ΔS = −50 mm.

Download the working card ↓

Written by Water MSC. Source: USGS · Evaluation of methods and uncertainties in the water budget.