Question 2 of 7: The Hydrologic Cycle, Stormwater Culverts and the Rational Formula

Problem 2: The Hydrologic Cycle, Stormwater Culverts and the Rational Formula (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(i) The Natural Hydrologic Cycle

Three key processes and how each depends on the others: (1) Precipitation is the cycle's input flux, condensing from atmospheric moisture that was itself supplied by evapotranspiration — it depends on the vapour delivered by process (2). (2) Evapotranspiration draws water from soil moisture (recharged by infiltration, process 3) and from open-water/interception storage (fed directly by precipitation), returning vapour to the atmosphere to form the next precipitation event — it depends on both precipitation's storage and infiltration's soil-moisture reservoir. (3) Infiltration partitions incoming precipitation between soil/groundwater recharge and surface runoff, and its rate depends on the antecedent soil moisture set by the balance of the previous two processes (a soil already wetted by earlier infiltration infiltrates more slowly, diverting more of the next storm's precipitation to runoff). Together the three processes close a loop in which each is both fed by, and feeds, its neighbours.

(ii) Stormwater Culverts

A stormwater culvert is a covered conduit — typically circular, elliptical or box-shaped — that conveys concentrated stormwater flow beneath a roadway, railway embankment or other crossing, maintaining the natural or engineered drainage path without an open-channel interruption. Design consideration 1 — hydraulic capacity/headwater: the culvert is sized so the design-storm headwater depth to diameter ratio (HW/D) does not exceed the allowable freeboard beneath the roadway or cause unacceptable upstream ponding; whether inlet control or outlet control governs determines which design chart/equation set applies. Design consideration 2 — outlet scour and debris: the outlet velocity must be checked against the receiving channel's erosion-resistant velocity (often requiring a riprap apron or energy dissipator), and a practical minimum diameter (often about 450 mm) is kept even where hydraulically unnecessary, to reduce blockage risk from debris. Primary function: to safely pass the design storm's peak flow under the crossing without overtopping the roadway or causing unacceptable upstream flooding.

(iii) The Rational Formula

Given. $Q=C\cdot I\cdot A$, the standard peak-flow estimator for small urban catchments.

Find. The meaning and consistent dimensions of each term.

$Q$ is the estimated peak runoff rate at the outlet [m³/s]; $C$ is the dimensionless runoff coefficient (0–1), the ratio of peak runoff rate to average rainfall intensity, reflecting the catchment's imperviousness and land use; $I$ is the design rainfall intensity [mm/hr], read from the IDF curve at a duration equal to the time of concentration; $A$ is the contributing drainage area [ha]. In consistent SI form, $Q\,[\text{m}^3/\text{s}]=C\times I\,[\text{mm/hr}]\times A\,[\text{ha}]/360$ — the constant 360 absorbs the mm·ha/hr→m³/s unit conversion (in US customary units, $Q\,[\text{cfs}]=C\times I\,[\text{in/hr}]\times A\,[\text{acres}]$ needs no constant, since 1 acre·in/hr is numerically almost exactly 1 cfs). The method's key assumption — a single, catchment-uniform $C$ and a storm duration set equal to the time of concentration — limits it to small catchments, typically under about 200 ha.