Question 6 of 7: Hydrologic Equation, Energy Budget and Infiltration Simulation

Question 6: Hydrologic Equation, Energy Budget and Infiltration Simulation (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) Annual evapotranspiration from the basic equation of hydrology (8 marks)

Given. An annual water balance for a single watershed, with the data exactly as printed in the question.

Find. The annual evapotranspiration depth ET = E + T in mm, with every assumption stated and justified — which, as the arithmetic below shows, must include an assumption about the printed precipitation figure.

Approach. Convert the measured river discharge into an equivalent runoff depth over the watershed, apply the basic equation of hydrology with justified assumptions on G and ΔS, and test the result for physical admissibility before reporting it.

1. State and justify the two closure assumptions. Over one complete hydrologic year, beginning and ending at the same point in the annual cycle, the soil-moisture, snowpack and groundwater storage return to comparable states, so $\Delta S \approx 0$. Deep groundwater outflow bypassing the gauge, G, is small compared with the surface outflow in a basin of this size whose divide is topographic, so $G \approx 0$. Both are the standard assumptions for an annual basin-scale balance, and both are conservative here because either one, if non-zero and positive, would only make the residual ET smaller. With them the balance reduces to $$ET = E + T = P - R$$
2. Convert the river discharge into a runoff depth. The annual runoff volume is the mean flowrate multiplied by the length of the year, $$V_R = Q_r\,\Delta t = 200\ \tfrac{\text{m}^3}{\text{s}} \times 31\,536\,000\ \text{s} = 6.3072 \times 10^{9}\ \text{m}^3$$ and dividing by the watershed area converts it to the depth that is directly comparable with a rainfall depth: $$\boxed{R = \frac{V_R}{A} = \frac{6.3072 \times 10^{9}\ \text{m}^3}{1.0 \times 10^{10}\ \text{m}^2} = 0.63072\ \text{m} = 630.72\ \text{mm}}$$ This number is firm: it follows from two unambiguously printed values and nothing else. As a cross-check, the specific discharge is $200/10\,000 = 0.020$ m³/s per km², or 20 L/s per km², an entirely ordinary yield for an eastern Canadian basin.
3. Substitute the printed precipitation and test the result. With $P = 40$ mm as printed, $$ET = P - R = 40 - 630.72 = -590.72\ \text{mm}$$ A negative evapotranspiration is physically impossible — it would require the atmosphere to deliver water to the watershed by some route other than precipitation. The failure is not marginal: the runoff depth exceeds the stated rainfall by $$\frac{R}{P} = \frac{630.72}{40} = 15.8$$ and no choice of G or ΔS rescues the data, because both of those terms can only make the residual ET smaller, never larger.
4. Identify which datum must be wrong, and correct it explicitly. Conservation requires $P \ge R$ for any watershed over a long averaging period. The runoff depth is fixed by the area and the discharge, and the area is stated unambiguously, so the printed precipitation depth is the term in error — almost certainly a lost digit. The minimum admissible value is $P \ge 631$ mm, which would imply zero evapotranspiration; a basin yielding 630.72 mm of runoff at 20 L/s per km² is characteristic of the boreal Shield of eastern Canada, where annual precipitation is of the order of 900–1100 mm. Adopting $P = 1000$ mm/a as representative of such a basin: $$\boxed{ET = P - R = 1000 - 630.72 = 369.28\ \text{mm/a}}$$ which corresponds to a runoff coefficient of $R/P = 0.631$ and an annual evapotranspiration volume of $0.36928 \times 1.0 \times 10^{10} = 3.6928 \times 10^{9}$ m³. An ET near 370 mm/a is exactly what is measured on cool, forested Shield catchments, where evapotranspiration is limited by available energy rather than by available water.
5. State the method independently of the numbers. The general result — and the one the question is testing — is that annual evapotranspiration on a gauged basin is obtained as the residual of the water balance, $$ET = P - \frac{Q_r\,\Delta t}{A} - G - \Delta S$$ Because ET is the residual, it inherits the errors of every other term: a 10 percent error in the areal precipitation propagates to roughly a 27 percent error in ET at this runoff coefficient. That is the principal weakness of the residual method and the reason an independent estimate — Penman–Monteith, Thornthwaite, or an eddy-covariance measurement — is always used as a cross-check.

(ii) Using the energy budget to predict drought conditions (5 marks)

The surface energy budget states that the net radiation arriving at a surface must be partitioned among the ground heat flux, the sensible heat that warms the air, and the latent heat that evaporates water:

where $R_n$ is net radiation, $G_0$ the soil heat flux, H the sensible heat flux and $\lambda E$ the latent heat flux, all in W/m², with $\lambda = 2.45$ MJ/kg the latent heat of vaporisation. The budget is useful for drought prediction because it sets an upper bound on evaporation from energy alone and then, compared with the actual evaporation, reveals when water rather than energy has become the limiting factor.

Worked example. Consider an agricultural basin in mid-July under a persistent ridge with no rainfall for six weeks. A flux station measures $R_n = 150$ W/m² as a daily mean and $G_0 = 10$ W/m², so the available energy is 140 W/m². If all of that energy went into evaporation — the potential rate —

Now bring in the “other information” the question refers to: soil-moisture probes, or a soil-water accounting model driven by the rainfall record, show that the root zone has been drawn down to near the wilting point, and the actual evapotranspiration measured by eddy covariance or estimated from the crop coefficient has fallen to $ET_a = 1.2$ mm/d. In energy terms,

The Bowen ratio has risen from a well-watered value below 0.5 to 3.11: nearly all the available energy is now heating the air instead of evaporating water. That is the diagnostic. Three consequences follow directly and are what makes the calculation a prediction rather than a description. The evaporative fraction $\lambda E/(R_n - G_0) = 0.24$ is a normalised drought index that can be mapped from thermal satellite imagery over a whole province. The cumulative deficit $\sum (ET_p - ET_a) = 3.74$ mm/d of unmet demand accumulates into the irrigation requirement and, if unmet, into crop loss. And the feedback is self-reinforcing: with the latent flux suppressed, surface and air temperatures rise, the vapour-pressure deficit grows, and the atmospheric demand increases further, which is why droughts, once established, persist until a synoptic change breaks them. Coupling the energy budget with soil-moisture accounting therefore gives a lead time of weeks — the deficit is visible in the flux partition before it is visible in the crop or in the streamflow.

(iii) Modelling infiltration to predict groundwater recharge, with two simplifying assumptions (7 marks)

Recharge to a drinking-water aquifer is the fraction of infiltrated water that passes below the root zone and reaches the water table. It cannot be measured directly over an area, so it is modelled — and the model has two linked parts: an infiltration model that computes how much water enters the soil surface, and a soil-water accounting model that decides how much of it survives the root zone.

Infiltration at the surface. Two standard formulations are used. Horton’s empirical equation, $f(t) = f_c + (f_0 - f_c)e^{-kt}$, treats the capacity as decaying exponentially from a dry value to a saturated one. For a silt-loam with $f_0 = 75$ mm/h, $f_c = 12$ mm/h and $k = 2.5$ h⁻¹, after one hour of ponded rain the capacity has fallen to

and the cumulative infiltration is

The Green–Ampt model, $f = K\left(1 + \dfrac{\psi\,\Delta\theta}{F}\right)$, reaches the same behaviour from physics rather than curve-fitting, with parameters — hydraulic conductivity K, wetting-front suction ψ, and moisture deficit Δθ — that can be measured or estimated from soil texture, which is why it is preferred when the model must be transferred to an ungauged area.

From infiltration to recharge. The infiltrated depth is then routed through a daily soil-moisture budget: water entering the root zone first satisfies the soil-moisture deficit, evapotranspiration draws the store down at the actual rate computed from the energy budget of part (ii), and only the surplus above field capacity percolates to the water table. Over a year the recharge is therefore a small residual of large fluxes — commonly 10–25 percent of precipitation in southern Canadian settings — and it is concentrated in spring snowmelt and late-autumn rain, when evapotranspiration is small and the profile is wet. Models of this class (HELP, SWAT, or a distributed MODFLOW recharge package) are calibrated against baseflow separation and against observed water-table hydrographs.

Assumption 1 — one-dimensional vertical flow through a homogeneous soil profile with a sharp wetting front. Both Horton and Green–Ampt treat the soil as a uniform column and ignore lateral flow, macropores, root channels, cracks and frozen layers. The engineering consequence for sustainable yield is that the model tends to under-predict the speed of recharge and, more seriously, under-predict the speed at which surface contamination reaches a supply well: preferential flow through macropores or karst can deliver a pathogen or a nitrate pulse to a well in days when the modelled travel time is years. Wellhead protection areas must therefore be based on tracer or particle-tracking evidence, not on the one-dimensional model alone.

Assumption 2 — a spatially uniform, stationary recharge rate, usually applied as a long-term annual average. Recharge in the model is a single value applied over an area and over a year, whereas the real process is patchy (recharge concentrates on permeable outwash and is near zero on clay till) and strongly episodic between years. The engineering consequence for sustainable yield is that a permitted water-taking rate set equal to the average annual recharge will fail in a run of dry years, because the model contains no information about that variability. Sustainable yield must be set below the average recharge, and must additionally respect the flow needed to sustain baseflow to streams and wetlands and to keep neighbouring wells and any saline or contaminated interface from being drawn in. In practice this means a permit conditioned on observed water levels rather than a single fixed rate: a monitoring network with defined trigger levels, mandatory reduction when levels fall, and periodic re-analysis as the record lengthens.