Question 2 of 7: Runoff hydrographs, unit hydrographs and conceptual models of runoff

Problem 2: Runoff hydrographs, unit hydrographs and conceptual models of runoff (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.

Part (i) — How a runoff hydrograph is generated (7 marks)

A runoff hydrograph is the record of discharge against time at a specified point on a stream, and it is generated by the basin acting on a storm in three stages: production, translation and attenuation.

Production of rainfall excess. Rain falling on the basin is first depleted by interception on vegetation and structures, by depression storage in surface hollows, and above all by infiltration. What remains is the rainfall excess, or effective rainfall, and it alone generates direct runoff. Because infiltration capacity decays through the storm as the soil profile wets, the excess is concentrated in the later part of a long storm and the loss rate is commonly represented as a constant φ-index or by an infiltration model such as Horton or Green–Ampt.

Translation. The excess moves as thin overland flow to rills, then to channels, and down the channel network to the gauge. Water generated close to the outlet arrives first; water from the hydraulically most remote part of the basin arrives after one time of concentration. The superposition of arrivals from progressively more distant contributing areas builds the rising limb.

Attenuation and recession. Storage in the channels, the floodplain and the surface itself delays and flattens the response, so the peak lags the centroid of the rainfall excess and is lower than a simple arrival calculation would predict. Once rainfall stops, the stored water drains out and the hydrograph falls along a recession that is characteristic of the basin rather than of the storm, approaching the base flow contributed by groundwater discharge to the channel.

The three elements labelled on the schematic are the ones an examiner expects: the peak discharge with its time to peak, which governs the sizing of conveyance structures; the rising limb, whose steepness reflects the storm intensity and the basin’s response speed; and the recession (falling) limb, which reflects storage drainage and is nearly independent of the storm that produced it. The dashed line separating base flow from direct runoff is the fourth feature always drawn, and the area between it and the hydrograph is the direct-runoff volume.

Two important limitations of the hydrograph method. First, a measured hydrograph is specific to the storm that produced it as well as to the basin. Its shape embodies that storm’s duration, its spatial pattern over the basin, its movement relative to the drainage network and the antecedent moisture at the time. A storm advancing down-basin produces a markedly higher and earlier peak than the same rainfall advancing up-basin. Transferring a single observed hydrograph to a different event, or to an ungauged basin, therefore carries a large and largely unquantified error, which is precisely why the unit hydrograph was invented as a normalising device. Second, separating base flow from direct runoff is arbitrary. The straight-line, fixed-base and master-recession methods in common use give different separations of the same record, and every subsequent quantity — direct-runoff volume, the derived unit hydrograph, the φ-index inferred from it — inherits that arbitrariness. A related practical limitation is that the high-flow end of the hydrograph is usually not measured at all but extrapolated from a rating curve beyond its highest calibrated gauging, so the peak itself may carry a larger uncertainty than any other point on the curve.

Part (ii) — The unit hydrograph and its time characteristics (7 marks)

A unit hydrograph for a given watershed is the direct-runoff hydrograph produced by one unit depth of rainfall excess — conventionally 1 cm in SI practice — falling uniformly over the whole basin at a uniform rate during a specified duration D. It is written as a D-hour unit hydrograph, and its ordinates have units of discharge per unit depth, m³/s per cm. Base flow has been removed, so it describes direct runoff only. It is a property of the basin, obtained either by analysing a suitable isolated single-peaked storm on a gauged basin or, on an ungauged basin, from a synthetic method such as the SCS dimensionless or triangular unit hydrograph, or Snyder’s method.

The device rests on three assumptions: the basin behaves as a linear system, so doubling the excess doubles every ordinate; it is time-invariant, so the same excess always produces the same shape irrespective of when it occurs; and responses to successive blocks of excess superpose. Together these permit the direct-runoff hydrograph for any storm to be built by convolution of its excess blocks with the unit hydrograph.

Given. An ungauged watershed is characterised by the following assumed, representative values, and a two-hour unit hydrograph is required.

Find. The peak ordinate and time base of the two-hour unit hydrograph, and the peak of the direct-runoff hydrograph produced by a storm delivering 1.5 cm of excess in the first two-hour block and 0.8 cm in the second.

Approach. Use the SCS triangular unit hydrograph — lag time from the time of concentration, time to peak from the lag and the duration, peak ordinate from the requirement that the triangle contain one unit depth — then convolve the ordinates with the two blocks of excess and add base flow.

1. Obtain the basin lag from the time of concentration. The SCS relation puts the lag, measured from the centroid of the excess to the peak, at six tenths of the time of concentration: $$t_L = 0.6\,t_c = 0.6(3.0\ \text{h}) = 1.8\ \text{h}$$
2. Locate the peak in time. The centroid of a D-hour block of excess sits at D/2, so $$T_p = \frac{D}{2} + t_L = \frac{2.0}{2} + 1.8 = 2.8\ \text{h}, \qquad T_b = 2.67\,T_p = 7.48\ \text{h}$$
3. Fix the peak ordinate from the unit volume. Requiring the triangle of base T_b to contain 1 cm of runoff over A square kilometres gives the standard SCS constant, $$u_p = \frac{2.08\,A}{T_p} = \frac{2.08(25)}{2.8} = \boxed{18.6\ \text{m}^3/\text{s per cm}}$$ The volume check confirms it: ½(18.57)(7.476)(3600) = 2.499 × 10⁵ m³, against 0.01 m × 25 × 10⁶ m² = 2.50 × 10⁵ m³, the small difference being the rounding built into the constant 2.08.
4. Convolve the excess with the unit hydrograph. Reading the triangle at two-hour intervals gives ordinates u = 0, 6, 15, 10, 4, 0 m³/s per cm. Applying the convolution equation $$Q_n = \sum_{m} P_m\,u_{\,n-m+1}$$ with P = 1.5 cm and 0.8 cm in successive blocks gives direct-runoff ordinates of 0, 9.0, 27.3, 27.0, 14.0, 3.2 and 0 m³/s. The volume checks: their sum, 80.5, equals the sum of the unit ordinates (35) times the total excess (2.3 cm).
5. Add base flow to recover the stream hydrograph. $$Q_{\text{peak}} = 27.3 + 3.0 = \boxed{30.3\ \text{m}^3/\text{s}}$$

The four features the question asks to be identified are marked on the schematic. Lag time t_L is measured from the centroid of the rainfall excess to the peak of the unit hydrograph and is the basin’s characteristic response delay. Time of concentration t_c is the travel time from the hydraulically most remote point of the basin to the outlet, and on a hydrograph it is conventionally taken from the end of the rainfall excess to the point of inflection on the recession, the moment at which the last of the direct runoff from that remote point has arrived. The recession curve is the falling limb, controlled by drainage of channel and surface storage and, once direct runoff has ceased, by groundwater depletion. Base flow is the groundwater contribution to the channel, present before and after the storm; it is removed before a unit hydrograph is derived and added back when one is applied.

Part (iii) — Conceptual runoff models: method, strengths, best use and a limitation (6 marks)

Method. A conceptual runoff model represents the basin as a small set of interconnected storage elements — typically an interception or canopy store, an upper soil-moisture store, a lower groundwater store and sometimes a snowpack store — each governed by a simple water balance and drained by empirical flux relations such as a linear-reservoir outflow Q = S/k, a threshold-controlled percolation, or a power-law drainage. Rainfall and potential evapotranspiration are the driving inputs; the model steps continuously through time, updating each store, and routes the generated runoff to the outlet through a linear-reservoir cascade or a unit hydrograph. The internal structure is a caricature of the real processes rather than a solution of their governing equations, and the parameters — store capacities, drainage coefficients, threshold levels — are not directly measurable and must be obtained by calibration against an observed flow record. The Sacramento Soil Moisture Accounting model, the Scandinavian HBV model, GR4J and the soil-moisture-accounting option in HEC-HMS are the familiar examples; in Canada, WATFLOOD and the MESH modelling system occupy the same niche with a distributed grid.

Strengths. The structure is parsimonious, so the model calibrates and runs quickly and can be applied for very long simulation periods at low computational cost. Because it carries a soil-moisture state forward continuously, it represents antecedent conditions automatically, which an event-based method such as the unit hydrograph cannot do — and antecedent moisture is often the difference between a nuisance flow and a flood. Its data requirements are modest: a precipitation and temperature record and an estimate of potential evapotranspiration are usually enough. It preserves the water balance by construction, and on a well-gauged basin a calibrated conceptual model routinely reproduces both the flood peaks and the low-flow recessions with a single parameter set.

Best use. Continuous long-term simulation on a gauged or regionally transferable basin: reservoir yield and operating-rule studies, water-supply and drought analysis, the generation of long synthetic flow series for frequency analysis, real-time flood forecasting with state updating, and the assessment of land-use or climate-change scenarios where decades of continuous record must be simulated.

A limitation. The parameters have no directly measurable physical counterpart, so the model must be calibrated — and calibration is not unique. Many different parameter sets reproduce the calibration record almost equally well, a condition known as equifinality, and these sets diverge when the model is used outside the range of conditions it was fitted to, which is exactly the extrapolation that design flood estimation demands. The practical consequences are that a conceptual model cannot be applied with confidence to an ungauged basin without regionalising its parameters, and that its performance degrades when the basin changes physically — through urbanisation, forest harvest or a shifting climate — because the calibrated parameters silently encode the conditions of the calibration period.