Question 2 of 7: Hydrologic cycle processes, groundwater flow and surface runoff

Question 2: Hydrologic cycle processes, groundwater flow and surface 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.

(i) Precipitation, infiltration and surface runoff (10 marks)

The three processes are not three separate phenomena but one partition problem observed at three places. Precipitation supplies water to the land surface; infiltration is the rate at which the surface accepts it; and surface runoff is whatever the surface will not accept. Over any control volume of the basin, and over any period, they are tied by the continuity statement drawn at the foot of the schematic,

$$P = R + F + \mathrm{ET} + \Delta S$$

in which $P$ is precipitation, $R$ surface runoff, $F$ infiltration, ET evapotranspiration and $\Delta S$ the change in surface and soil storage. Every one of the three named components appears in that identity, and changing one necessarily changes another.

Precipitation is the only input to the land phase and therefore fixes the ceiling on everything else. Its importance is not only in depth but in rate and phase. Rate matters because the partition between infiltration and runoff is decided by comparing the rainfall intensity $i$ with the current infiltration capacity $f$: while $i < f$ everything infiltrates and there is no Hortonian runoff; once $i$ exceeds $f$ the excess ponds and then runs off. Phase matters because snow places the input into seasonal storage and releases it weeks later as melt, which is why the annual peak on most Canadian rivers is a spring freshet rather than a rainfall event. Without precipitation there is no cycle at all: it is the term that recharges soil moisture, groundwater, lakes and channels alike.

Infiltration is the gate between the surface and the subsurface, and it is the process that decides how much of a storm the basin keeps. Water entering the soil first satisfies soil-moisture deficit in the unsaturated zone, where it is available to vegetation and returns to the atmosphere as transpiration; the surplus percolates to the water table and becomes groundwater recharge. Infiltration capacity is not constant — it falls during a storm from an initial capacity $f_0$ towards a final capacity $f_c$ as the surface layer wets up and pore space fills, which is exactly what the Horton relation describes. Its importance to the cycle is twofold: it is the sole mechanism of natural groundwater recharge, and it is the mechanism that sustains streamflow between storms. Groundwater discharged slowly back to the channel is the baseflow that keeps rivers running through a dry August, and that is the same water that infiltrated months earlier.

Surface runoff is the residual, and it is the fast pathway. It is generated in two distinct ways — the infiltration-excess (Hortonian) mechanism just described, which dominates on urban, compacted and arid surfaces, and the saturation-excess mechanism, in which rain falls on ground whose profile is already saturated from below, typically in valley bottoms and near the channel. Its importance is that it carries the flood peak: it is the component with the shortest travel time, so it controls the magnitude and timing of the hydrograph and therefore controls the sizing of every drainage structure. It is also the component that carries sediment and surface contaminants to the receiving water.

The interrelation is best seen by disturbing one term. Urbanising a catchment replaces pervious soil with roofs and pavement; infiltration capacity collapses, so for the same precipitation $F$ falls and $R$ rises sharply — peak flows increase, the hydrograph becomes shorter and steeper, and downstream erosion increases. At the same time the reduced recharge lowers the water table, so baseflow in the dry season falls and the stream may go from perennial to intermittent. One partition, changed at the surface, has altered both the flood regime and the low-flow regime. That coupling is why stormwater management in Canada is now written in terms of maintaining pre-development infiltration volumes, not merely capping peak discharge.

(ii) Peak runoff from the 10 ha development by the Rational Method (10 marks)

Given. A new suburban development is to be drained by a storm sewer, and the sub-watershed data supplied by the question are collected below.

Find. The peak surface runoff Q leaving the development, in m³/s, by the Rational Method, with every assumption stated and justified.

Approach. Choose the critical storm duration (the time of concentration), read the design intensity off the supplied IDF relation at that duration, select a runoff coefficient appropriate to suburban residential development, and combine them in the Rational formula in its metric form.

1. State the four assumptions the Rational Method requires, and why each is acceptable here. (a) The storm duration equals the time of concentration. For durations shorter than $t_c$ only part of the catchment contributes at any instant; for longer durations the whole area contributes but the intensity has fallen. With a monotonically decreasing IDF relation the product is largest at $t_d=t_c$, so the critical duration is 90 minutes. (b) Rainfall is uniform over the area and constant in intensity over the duration. Over 10 ha this is entirely reasonable — the area is far smaller than any convective cell. (c) The runoff coefficient is constant and independent of storm magnitude and antecedent conditions. This is the method's crudest assumption; it is accepted because the catchment is small, largely impervious and sewer-drained, which is the case the method was calibrated for. (d) There is no significant storage in the system — no ponds, no on-site detention — so the sewer is sized for the undamped peak. All four are the standard qualifications on the Rational Method and all four hold for a 10 ha serviced development.
2. Select the runoff coefficient for suburban residential land use. Canadian municipal practice (TAC Drainage Manual, and essentially every municipal design standard) gives $C$ in the range 0.30 to 0.50 for detached suburban residential development, the value rising with lot coverage and with the fraction of roof and driveway drained directly to the sewer. Taking the midpoint of that range, $$C = 0.40$$ which corresponds to roughly 35 to 45 per cent effective impervious cover. The sensitivity of the answer to this choice is reported in the final results table, because $C$ is the largest single source of uncertainty in the calculation.
3. Read the design intensity from the IDF relation at the critical duration. With $t_d = t_c = 1.5$ h, $$i = 7.0 - 0.2\,t_d = 7.0 - 0.2(1.5) = \boxed{6.7\ \text{mm/h}}$$ The relation is linear and therefore only a local approximation valid over the range of durations for which it was fitted; it becomes negative beyond $t_d = 35$ h and must not be used there. The units are not printed on the question; mm/h is adopted, for the reasons set out in the callout at the head of this paper.
4. Apply the Rational formula in metric form. The formula is $Q = CiA$ in consistent units; with $i$ in mm/h and $A$ in hectares the convenient working form is $$Q=\frac{C\,i\,A}{360}$$ where the 360 converts mm/h × ha into m³/s. Substituting, $$Q=\frac{0.40 \times 6.7 \times 10}{360}=\frac{26.8}{360}=\boxed{0.0744\ \text{m}^3/\text{s}}$$ that is 74.4 L/s, or about 268 m³ of water per hour at the peak.
5. Check the result for reasonableness against the catchment. Expressed as a unit rate the answer is $0.0744/0.10 = 0.744$ m³/s per km², and as a depth-equivalent it is $0.40 \times 6.7 = 2.68$ mm/h of net rainfall — both entirely ordinary for a small suburban catchment. A 300 mm diameter concrete sewer at 1 per cent grade carries 0.0968 m³/s flowing full by Manning with $n = 0.013$, so a 300 mm pipe is the sensible first trial size and the answer is of a magnitude the drainage system can actually accept.
6. Report the sensitivity to the two open choices. Varying the runoff coefficient across its full published band gives $Q = 0.0558$ m³/s at $C = 0.30$ and $Q = 0.0931$ m³/s at $C = 0.50$, a spread of roughly $\pm 25$ per cent about the adopted value — small enough that the pipe size is unlikely to change. Reading the IDF relation in in/h instead of mm/h would give $i = 170.2$ mm/h and $Q = 1.891$ m³/s, twenty-five times larger and far outside anything a 10 ha suburban catchment produces in Canada, which settles the units question on physical grounds.