Question 2 of 7: Hydrologic Cycle Processes, Surface Runoff and Groundwater Flow

Problem 2: Hydrologic Cycle Processes, Surface Runoff and Groundwater Flow (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 (a) — The three processes that govern hydro-electric dam design (7 marks)

A hydro-electric dam converts the potential energy of stored water into electricity, so its design is controlled by the processes that determine how much water arrives, how fast it can arrive, and how much is lost while it is held. Reading those off the cycle diagram:

1. Precipitation, and its conversion to runoff (basin yield). The firm energy output of the plant is set by the long-term mean annual runoff of the contributing basin, because energy is the product of volume and head. The designer needs the full precipitation record — mean annual depth, its seasonal distribution, and the split between rainfall and snowfall — converted to inflow through a rainfall–runoff model. In most Canadian settings the seasonal pattern matters more than the annual total: a basin whose precipitation falls as snow delivers little in January and a large fraction of its yield in a few weeks of freshet, which dictates reservoir live storage and turbine sizing. The variability of that yield, not merely its mean, fixes the reliable (firm) output.

2. Direct runoff and flood generation, which size the spillway. The same basin must be analysed for its extreme response. The inflow design flood — obtained from flood-frequency analysis of the annual maximum series, or from a probable maximum precipitation routed through a unit hydrograph for a high-consequence structure — sets the spillway capacity and the freeboard. This is the safety-critical process: under-estimating it risks overtopping, which is the dominant failure mode for embankment dams. In Canada the required design flood is tied to the dam’s consequence classification under the Canadian Dam Association Dam Safety Guidelines, rising to the probable maximum flood for very high consequence structures.

3. Evaporation from the reservoir surface, and the storage balance. Impounding a river replaces a narrow channel with a large free water surface, and the evaporative loss from that surface is a permanent debit against yield — significant on a shallow reservoir in a dry, windy or southern-interior setting. Together with infiltration and seepage through the abutments and foundation, it enters the reservoir storage balance that is routed month by month to size the live storage.

Three further processes are second-order but should be named: sediment transport, which consumes dead storage over the design life; infiltration and groundwater exchange, which governs seepage and uplift pressures on the foundation; and ice processes, which in Canada control intake design and winter operation.

Part (b) — Predicting surface runoff from a 100 km² watershed (7 marks)

Runoff prediction is the systematic removal of losses from precipitation, followed by the routing of what survives. Conceptually the chain is: gross rainfall → subtract interception, depression storage and infiltration → rainfall excess → transform by a unit hydrograph or a routing model → direct runoff hydrograph → add baseflow → total streamflow. At 100 km² the basin is large enough that a single lumped loss rate is coarse, and small enough that a lumped or semi-distributed event model is defensible; a fully distributed model is not usually justified unless land use varies sharply.

Information required for a good estimate. Physiography: basin area, drainage divide, main-channel length and slope, and the time of concentration. Land cover and soil: hydrologic soil group, land use and imperviousness, from which a curve number or a loss-rate parameter is obtained. Meteorology: the design storm, either an observed event or an IDF-derived synthetic storm of specified return period, with its temporal distribution and an areal reduction factor appropriate to 100 km². Antecedent conditions: soil moisture, water-table depth, and snowpack water equivalent if the event is a freshet or a rain-on-snow. Finally, and most important, a gauged record — concurrent rainfall and streamflow for past events — against which the model is calibrated and verified.

Given. Illustrative design case: basin area A = 100 km², design storm depth P = 80 mm, SCS curve number CN = 75 for average antecedent moisture (AMC II).

Find. The depth and volume of direct runoff produced by the storm.

1. Convert the curve number to a maximum retention. The SCS relation in SI units is $$S=\frac{25400}{\text{CN}}-254=\frac{25400}{75}-254=84.7\ \text{mm}$$ where $S$ is the potential maximum retention after runoff begins.
2. Deduct the initial abstraction. Taking the standard $I_a=0.2S$ accounts for interception, surface storage and infiltration before ponding: $$I_a=0.2\times 84.7=16.9\ \text{mm}$$ so runoff begins only once 16.9 mm has fallen.
3. Apply the runoff equation and convert to volume. For $P > I_a$, $$Q=\frac{(P-I_a)^{2}}{(P-I_a)+S}=\frac{(80-16.9)^{2}}{(80-16.9)+84.7}=26.9\ \text{mm}$$ $$\boxed{Q = 26.9\ \text{mm} \quad\Rightarrow\quad V = Q\,A = 2.69\times 10^{6}\ \text{m}^{3}}$$ Only about a third of the storm depth becomes direct runoff; the remaining two-thirds is lost to abstraction. That ratio is the single most useful sanity check on any runoff calculation.

The depth alone does not give a peak flow. To obtain the hydrograph, the excess is distributed in time using the design-storm profile and convolved with a unit hydrograph derived from gauged events (or synthesised, for example by the SCS dimensionless method using the basin lag). The peak is then read from the resulting hydrograph, and the whole model is calibrated against observed events before it is used in design.

Part (c) — Predicting the recharge potential of an aquifer supplying a municipality (6 marks)

For a municipality whose drinking water comes from an aquifer, the engineering question is whether long-term withdrawal stays below long-term recharge. The processes that must be quantified are:

1. Infiltration at the ground surface. Recharge begins with water crossing the soil surface, so infiltration capacity and its decay during a storm (Horton or Green–Ampt) control what fraction of precipitation is even available. Intense rain on a low-capacity soil runs off and never reaches the aquifer; the same depth delivered slowly, or as snowmelt over a thawing but permeable soil, largely infiltrates. This is why spring melt is the dominant recharge season across most of Canada.

2. Soil-moisture accounting and percolation through the unsaturated zone. Infiltrated water first refills the soil-moisture deficit; only the surplus percolates below the root zone. Evapotranspiration competes directly for this water and usually wins during the growing season. Quantifying recharge therefore requires a soil-moisture balance driven by precipitation and potential evapotranspiration, not merely a rainfall total.

3. Saturated groundwater flow and the aquifer’s hydraulic properties. Once at the water table, movement obeys Darcy’s law, $Q=KiA$, so the hydraulic conductivity $K$, the transmissivity $T=Kb$, the storativity $S$, and the hydraulic gradient must be measured — by pumping tests, monitoring wells and water-level mapping. These determine both how fast recharge spreads and how the water table responds to pumping.

4. Recharge-area delineation and aquifer boundaries. The recharge area is frequently not coincident with the surface watershed, and confined aquifers are recharged only where the formation outcrops. Identifying that area, and whether the aquifer is confined or unconfined, is essential for both yield and wellhead protection.

5. Discharge terms. Baseflow to streams, springs, wetlands and neighbouring pumping all remove water. Sustainable yield is recharge minus the discharge that must be preserved to maintain those ecosystems and existing users — not recharge itself.

Given. Illustrative check on the confined aquifer: hydraulic conductivity K = 25 m/d, saturated thickness b = 20 m, width of the flow section w = 1500 m, hydraulic gradient i = dH/dL = 0.004.

Find. The throughflow available to the municipal wellfield.

1. Apply Darcy’s law to the full cross-section. With the flow area $A=b\,w$, $$Q=K\,b\,w\,\frac{dH}{dL}=25\times 20\times 1500\times 0.004$$ $$\boxed{Q = 3000\ \text{m}^{3}\text{/d} = 3.0\ \text{ML/d}}$$ At a typical Canadian municipal demand of about 400 L per person per day this supports roughly 7500 people — and only if the recharge that sustains this gradient is genuinely renewed each year.