WEBVTT

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Rainfall, storage and delayed drainage. I = 2 m³/h for 2 h; k = 0.5 h⁻¹; initially empty.

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MATHEMATICS · define the relationship and the meaning of each symbol before calculating.

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S: stored volume (m³); I: inflow (m³/h); k = 0.5 h⁻¹. Initially S = 0. Inflow is 2 m³/h for two hours, then zero.

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INPUT · inflow schedule · read the axis units and the numerical values.

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I = 2 m³/h for 2 h; k = 0.5 h⁻¹; initially empty.

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At 2 h: storage = 2.528 m³; cumulative inflow = 4 m³. The balance leaves 1.472 m³ discharged.

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OUTPUT · stored water · read the axis units and the numerical values.

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At 2 h, 4 m³ entered, 2.528 remains and 1.472 left. The mass balance closes; storage then decays as inflow stops.

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CALCULATION · substitute the defined inputs; keep units through each intermediate step.

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Equilibrium during constant inflow: I/k = 2/0.5 = 4 m³

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Storage at rain end: 4 × (1 − exp(−1)) = 2.528482 m³

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Outflow volume by two hours: 2 × 2 − 2.528482 = 1.471518 m³

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Recession half-life: ln(2)/0.5 = 1.386294 hours

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SENSITIVITY · change one declared input · read the axis units and the numerical values.

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k = 0.25 /h: 3.14775 m³ stored at 2 h. Two-hour inflow of 2 m³/h; initial storage zero.

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k = 0.5 /h: 2.52848 m³ stored at 2 h. Two-hour inflow of 2 m³/h; initial storage zero.

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k = 1 /h: 1.72933 m³ stored at 2 h. Two-hour inflow of 2 m³/h; initial storage zero.

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Check: Check S(0) = 0, continuity at rainfall cessation and nonnegative storage. Integrate kS over time and close the water balance. Test the k → 0 limit during rainfall, where storage approaches It.

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ONE ROW · Two outlets can experience the same storm and tell very different stories six hours later.

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PARAMETER MAP · 14701 evaluated cells. The colour limits stay fixed across every row.

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Every row receives 2 m³/h for two hours, starts empty and obeys dS/dt = I − kS. Only k changes. The ridge occurs when inflow stops; recession retains the storage already

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accumulated. Compare the long tail at small k with the rapid emptying at large k. The integral of outlet flow plus remaining storage equals the 4 m³ input after the storm.

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VALUE LANDSCAPE · Height and colour encode Stored water (m³). This is a graph over the two input axes, not automatically a physical pavement shape.

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Lumped linear storage, not a spatial seepage solution. The coefficient is assumed; it is not inferred hydraulic conductivity, pavement saturation or a distress predictor.

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The reservoir is a mass balance, not a calibrated pavement moisture model.