Question 1 of 7: Hydrologic cycle processes, groundwater flow and surface runoff

Problem 1: 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.

Part (i) — Three interacting hydrologic processes (7 marks)

The hydrologic cycle is a closed mass-conservative circulation of water between the atmosphere, the land surface and the subsurface, driven by solar radiation and gravity. Of the processes that make it up, the three that dominate the engineering behaviour of a basin are precipitation, infiltration and evapotranspiration; surface runoff and groundwater flow are best understood as the consequences of how these three compete for the same water.

Precipitation is the input. It delivers water to the land surface as rain, snow or freezing rain at a rate that varies in both space and time, and it is the only term in the basin balance that the engineer cannot influence. Infiltration is the partitioning process. At any instant the soil surface can accept water only up to its infiltration capacity, which is high when the profile is dry and decays towards the saturated hydraulic conductivity as the profile wets up. Rainfall arriving faster than the soil can accept it becomes rainfall excess and runs off over the surface; water that does infiltrate either replenishes soil moisture, percolates below the root zone to recharge the water table, or moves laterally as interflow. Evapotranspiration is the principal loss. It returns water to the atmosphere from open water, wet vegetation and soil surfaces (evaporation) and through the stomata of plants drawing on soil moisture (transpiration), at a rate governed by available energy, the vapour-pressure deficit, wind and — critically — the moisture the soil still holds.

The three interact through the soil-moisture store, and the interaction is a feedback rather than a sequence. Precipitation refills the store; evapotranspiration empties it between storms; and the state of the store at the moment the next storm begins sets the infiltration capacity, which decides how that storm splits between runoff and recharge. A basin that has been drying for three weeks will absorb almost all of a 30 mm rainfall and produce a negligible hydrograph; the same 30 mm falling two days after a saturating event produces a sharp flood peak. This is why antecedent moisture conditions appear explicitly in every design-runoff procedure. Over a period long enough for the storage change to be small, the interaction is summarised by the basin water balance,

in which P is precipitation, R is surface runoff, ET is evapotranspiration, G is net groundwater outflow across the basin boundary and ΔS is the change in stored water (soil moisture, snowpack, surface detention and groundwater). All terms are expressed as an equivalent depth over the basin area, most often in millimetres per year. In a typical Canadian Shield basin roughly one third of annual precipitation leaves as runoff and two thirds as evapotranspiration; in a semi-arid prairie basin the evapotranspiration share can exceed ninety per cent.

Part (ii) — Darcy’s law and the prediction of groundwater flow (7 marks)

Darcy’s law is the constitutive relation for flow through a saturated porous medium, obtained experimentally by Henry Darcy in 1856 from sand-column tests on the water supply of Dijon. It states that the volumetric discharge through a porous medium is directly proportional to the cross-sectional area normal to the flow and to the hydraulic gradient driving it:

Here Q is the discharge (m³/s or m³/d), K is the hydraulic conductivity of the medium (m/s or m/d), A is the gross cross-sectional area of aquifer normal to the flow (m²), and Δh/ΔL is the hydraulic gradient, the head loss per unit distance along the flow path, which is dimensionless. The negative sign in the differential form records that flow proceeds from high head to low head. Two derived quantities follow at once and are routinely confused: the Darcy velocity or specific discharge q = Q/A = K i, which is a flux per unit gross area and not the speed of any water particle, and the seepage velocity v = q/n_e, the speed at which a tracer or a contaminant actually travels, where n_e is the effective porosity.

The law predicts groundwater flow because every quantity in it is measurable in the field. Hydraulic conductivity comes from a pumping test, a slug test or a grain-size correlation; the hydraulic gradient comes from water levels in a triplet of monitoring wells, which also fix the flow direction as normal to the contours of the potentiometric surface; and the area follows from the mapped saturated thickness and the width of the flow tube. This is the everyday basis of well-field yield estimates, contaminant travel-time predictions, dewatering design and the estimation of the groundwater contribution to a river’s base flow.

Given. A confined sand aquifer beneath a river valley is characterised by the following assumed, representative values.

Find. The discharge through the aquifer cross-section, and the speed at which a dissolved contaminant would migrate.

Approach. Form the flow area and the hydraulic gradient, apply Darcy’s law for the discharge, then divide by area and by effective porosity to obtain the Darcy and seepage velocities.

1. Form the cross-sectional area normal to flow. For a confined aquifer the area is the saturated thickness times the width of the flow tube, $$A = b\,w = (15\ \text{m})(400\ \text{m}) = 6000\ \text{m}^2$$
2. Compute the hydraulic gradient. The gradient is the head drop divided by the distance along the flow path, $$i = \frac{\Delta h}{\Delta L} = \frac{4.5\ \text{m}}{2500\ \text{m}} = 0.0018$$ a value entirely typical of a regional aquifer, where gradients of order 10⁻³ are the rule.
3. Apply Darcy’s law. Substituting into Q = K A i, $$Q = (30\ \text{m/d})(6000\ \text{m}^2)(0.0018) = \boxed{324\ \text{m}^3/\text{d} \;=\; 3.75\times10^{-3}\ \text{m}^3/\text{s}}$$
4. Separate the two velocities. The Darcy velocity is the flux per unit gross area, whereas the seepage velocity divides that flux over the pore area actually available for flow: $$q = K i = (30)(0.0018) = 0.054\ \text{m/d}, \qquad v = \frac{q}{n_e} = \frac{0.054}{0.22} = 0.245\ \text{m/d}$$ so a conservative contaminant advances about 90 m per year — four and a half times faster than the Darcy velocity would suggest, which is exactly why the distinction matters in a travel-time assessment.

Two assumptions the law makes. First, the flow is laminar. Darcy’s law is linear in velocity, which holds only while viscous forces dominate inertial ones, conventionally for a pore Reynolds number below about 1 to 10. It therefore fails in coarse gravel or fractured rock near a heavily pumped well, and in the immediate vicinity of a well screen, where the head loss grows faster than the first power of the discharge. Second, the medium is fully saturated and behaves as a continuum that is homogeneous and isotropic at the scale of the calculation, with a rigid matrix and a fluid of constant density and viscosity. A single scalar K is only meaningful under that assumption; real stratified deposits are anisotropic, with horizontal conductivity commonly one to two orders of magnitude greater than vertical, and require a conductivity tensor or an equivalent layered treatment.

Part (iii) — Three watershed factors controlling runoff from a large developed watershed (6 marks)

In a developed watershed the factors below are ranked from the greatest to the least influence on the peak surface runoff. The ranking is for peak discharge specifically; a ranking for runoff volume would weight land use even more heavily and drainage efficiency hardly at all.

1. Land use and the directly connected impervious area (greatest impact). Development replaces soil and vegetation with roofs, roads and parking areas that have essentially zero infiltration and negligible depression storage. Every millimetre falling on a directly connected impervious surface reaches the drainage system, so both the runoff coefficient and the runoff volume rise steeply with imperviousness. A rural catchment with a rational-method runoff coefficient C of about 0.35 becomes, at typical suburban densities, a catchment with C near 0.75, so for the same design rainfall intensity the peak discharge rises by the ratio 0.75/0.35 = 2.14, before any change in timing is considered. What makes this the dominant factor is that it acts on the runoff volume and the peak together.

2. Drainage-network efficiency, basin slope and shape — that is, the time of concentration (second). Urban development replaces slow, rough overland and channel paths with gutters, storm sewers and lined channels, and it shortens flow paths by adding inlets. The time of concentration falls, sometimes by half; because the design intensity is read from an intensity–duration–frequency curve at a duration equal to t_c, and IDF intensity rises sharply as duration falls, a shorter t_c raises the design intensity as well as compressing the same runoff volume into a narrower hydrograph. Basin shape reinforces this: an elongated basin delivers a flat, late peak, whereas a compact or fan-shaped basin concentrates the contributions and produces a sharp early one. Slope acts through the same mechanism, by controlling flow velocity.

3. Soil type, storage and antecedent moisture conditions (third). On the pervious remainder of a developed basin, the hydrologic soil group and the moisture already in the profile decide how much of the rainfall infiltrates. Group A sands may absorb most of a design storm while Group D clays behave almost like pavement. Depression storage, lakes and wetlands, and any engineered detention add further attenuation. This factor is ranked last for a large developed watershed for two reasons: the pervious fraction over which it acts has been reduced by development, and its influence is greatest in small and moderate events, whereas at the extreme return periods that govern design the soil approaches saturation and the peak becomes relatively insensitive to it.