Question 3 of 7: Unit Hydrographs, Runoff Hydrographs and Conceptual Models

Problem 3: Unit Hydrographs, Runoff Hydrographs and Conceptual Models (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) — Significance of three unit-hydrograph terms for design (7 marks)

1. The duration of excess precipitation, t_e. A unit hydrograph is not a property of the basin alone: it is the response to one unit depth of excess rainfall applied uniformly over a stated duration. The duration is part of the definition, and it is the term most often mishandled in practice. It matters for design because convolution is only valid when the design storm is broken into blocks of exactly that duration; a 1-hour unit hydrograph cannot be applied to 3-hour rainfall blocks without first being converted, by the S-curve method or by lagging and averaging. Choosing a duration that is too long smooths the storm and under-predicts the peak — the classic cause of an undersized culvert. A useful rule is to take the duration at roughly one quarter of the time to peak.

2. The peak of the quick-response runoff and the time to peak, t_p. The peak ordinate is the design number: it sizes culverts, storm sewers, bridge openings and spillways, all of which are capacity-governed rather than volume-governed. The time to peak carries the operational information — it is effectively the flood warning time available between the causative rainfall and the arrival of the flood, and it governs how quickly gates must be operated. Urbanisation attacks both terms at once: impervious cover and piped drainage shorten $t_p$ and raise the peak, which is precisely the effect stormwater management is designed to reverse. Because the unit hydrograph is linear, the peak also scales directly with the excess depth, so a doubling of excess rainfall doubles the direct-runoff peak.

3. The baseflow separation line and the inflection point. A unit hydrograph describes direct runoff only, so baseflow — the groundwater contribution that persists between storms — must be removed before the unit hydrograph is derived, and added back after the design hydrograph is computed. The inflection point on the recession marks where surface inflow to the channel has essentially ceased and the falling limb becomes pure storage depletion; it is the conventional and physically defensible end point for the separation. Getting this wrong distorts the whole answer: too much flow assigned to baseflow shrinks the unit hydrograph and under-predicts design floods, while too little inflates it. For design purposes on a large river, baseflow may be a substantial fraction of the total flood peak and cannot simply be neglected.

Part (ii) — Generating the runoff hydrograph at the outfall of four subwatersheds (8 marks)

The problem is solved by discretise, transform, route, and superpose. Each subwatershed is treated as its own rainfall–runoff system, the resulting hydrographs are routed through the connecting ditches, and they are summed at the junction to give the outfall hydrograph. The steps are:

1. Characterise each subwatershed. For each of the four, determine area $A_k$, imperviousness, slope, and the time of concentration $t_{c,k}$. In an urban catchment $t_c$ is the sum of overland-flow, gutter and pipe travel times, and it differs markedly between the four; that difference is what makes simple summation of peaks wrong.
2. Compute rainfall excess. Apply the design storm — typically a Chicago or SCS-type distribution built from the local IDF curves for the chosen return period — and remove losses. In urban catchments the common practice is to treat directly connected impervious area as producing nearly 100 % runoff and to apply an SCS curve number or a Horton infiltration rate to the pervious fraction.
3. Transform excess into a hydrograph by convolution. With a unit hydrograph of ordinates $U_j$ for the subwatershed and excess-rainfall blocks $P_i$, the direct-runoff ordinates follow from the discrete convolution $$Q_n=\sum_{i=1}^{n} P_i\,U_{n-i+1}$$ Worked for one subwatershed with a 1-hour unit hydrograph $U=\{0,\,5,\,15,\,10,\,4,\,0\}$ m³/s per cm and excess blocks of 2 cm then 1 cm:

The peak is 35 m³/s. The volume check confirms the arithmetic: the unit-hydrograph ordinates sum to 34 m³/s over 1-hour steps, which is 1.224 × 10⁵ m³ per centimetre of excess, so 3 cm of excess must deliver 3.672 × 10⁵ m³ — and the summed direct-runoff ordinates give exactly that. Conservation of volume is the check to run every time.

4. Route each hydrograph down its ditch. The ditches translate the hydrograph in time and attenuate its peak. A hydrologic (Muskingum or Muskingum–Cunge) routing suffices for design; a hydraulic (St Venant) solution is used where backwater at the junction or the outfall matters. Routing is what makes the analysis honest — it is why four peaks of 10 m³/s do not produce 40 m³/s at the outfall.
5. Superpose at the junction and route to the outfall. Add the four routed hydrographs ordinate by ordinate at matching clock times, never peak-to-peak, then route the combined hydrograph the last reach to the outfall: $$\boxed{Q_{\text{outfall}}(t)=\sum_{k=1}^{4} Q_k\!\left(t-\tau_k\right)}$$ where $\tau_k$ is the travel time from subwatershed $k$ to the junction. Where storage ponds exist, apply level-pool (storage-indication) routing at each before summation.
6. Calibrate and check. Compare the computed hydrograph against gauged events at the outfall, and confirm that the timing of the combined peak is plausible — the critical case is often not the largest storm but the one whose duration makes two subwatershed peaks coincide.

Part (iii) — Three differences between conceptual and empirical runoff models (5 marks)

1. Basis of the formulation. A conceptual model represents the physical processes explicitly, as interconnected stores and fluxes — interception, soil moisture, infiltration, groundwater — each governed by a continuity equation, so its internal states correspond to real quantities. An empirical model is a fitted input–output relation with no internal process representation: the Rational Method, $Q=CiA$, and a regression of peak flow on drainage area are the archetypes. The conceptual model explains; the empirical model only correlates.

2. Parameters and data demand. Conceptual model parameters have physical meaning and units (storage capacities, conductivities, lag times), can be constrained by measurement, but are numerous, and the model needs detailed spatial and temporal data to run. Empirical parameters (a runoff coefficient, a regression exponent) are calibration constants with no independent physical measurement, but there are few of them and the data demand is light — which is exactly why the Rational Method survives for small culverts.

3. Transferability and behaviour outside the calibration range. This is the practically decisive difference. Because it encodes process, a conceptual model can be applied to an ungauged basin by parameter estimation from physical attributes, and can be used to predict the effect of a change the data never contained — new development, a climate-shifted storm, a proposed detention pond. An empirical model is valid only within the range of catchment size, land use and storm magnitude from which it was fitted; extrapolating it beyond that range is unsupported. The corresponding costs are that conceptual models are far more expensive to build and are vulnerable to over-parameterisation, where several different parameter sets reproduce the calibration record equally well but diverge in prediction.