Dispersion/Diffusion Fundamentals, Isotope Hydrology, and Effective Diffusion Coefficient: 18-Geol-B1 Contaminant Hydrogeology, May 2017, Q1

Question 1: Dispersion/Diffusion Fundamentals, Isotope Hydrology, and Effective Diffusion Coefficient (equal value)

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) — True/False. Each statement is judged against the governing physics of dispersion, unsaturated flow, and remediation, with a one-line justification:

Question 1(a) — True/False answers
#	Statement (abridged)	Answer	Reason
1	No mechanical dispersion in stationary water	True	Mechanical dispersion is caused by variations in the velocity field; with zero bulk flow there is no advective spreading, so only molecular diffusion operates.
2	Hydrodynamic dispersion coefficient increases with dispersivity/diffusion coeff.	True	$D_L=\alpha_L v+D^{}$ — both terms add linearly, so a larger $\alpha_L$ or $D^{}$ directly raises $D_L$.
3	Darcy's Law cannot be used for unsaturated flow	False	The Buckingham-Darcy law extends Darcy's law to unsaturated media by replacing $K$ with the saturation-dependent $K(\theta)$ or $K(\psi)$.
4	Coarse-grained soils act as flow barriers	False	Coarse soils (sand/gravel) have high hydraulic conductivity and are preferential flow paths, not barriers; fine-grained clays/silts are the barriers.
5	Diffusion always insignificant vs. mechanical dispersion	False	At very low seepage velocities (e.g. clay aquitards), $D^{*}$ can dominate over $\alpha_L v$; diffusion is only negligible at typical aquifer velocities.
6	Lower interfacial tension lowers capillary pressure at fixed saturation	True	Young-Laplace: $P_c=2\sigma\cos\theta/r$ — $P_c$ is directly proportional to interfacial tension $\sigma$ at a given pore radius/saturation.
7	Geothermal good for heating, impractical for cooling in North America	False	Ground-source heat pumps use the same shallow-earth loop for both heating (winter) and cooling (summer) — that reversibility is the technology's main advantage.
8	Air relative permeability rises as air saturation falls	False	Relative permeability of a phase increases with that phase's OWN saturation; if $S_{air}$ decreases, $k_{r,air}$ decreases (while $k_{r,water}$ rises).
9	Excavation/disposal best for any situation	False	Excavation is limited by depth, footprint, cost, and disposal liability; deep or widespread plumes need in-situ methods (pump-and-treat, bioremediation, SVE).
10	Hydrocarbon biodegradation always needs O₂ as electron acceptor	False	Anaerobic pathways (nitrate-, iron-, sulfate-reducing, methanogenic) also degrade hydrocarbons, just more slowly — oxygen is the acceptor, not always required, and the terminology in the question ("electron donor") is itself reversed: hydrocarbons are the donor, O₂ the acceptor.
11	Biopiles always use nutrients, air, and acclimated microbes	False	Biopiles rely primarily on aeration and nutrient amendment of the native (indigenous) population; bioaugmentation with acclimated/exogenous microbes is an optional enhancement, not a universal requirement.
12	Water pressure always less than air pressure in unsaturated soil	True	By definition, matric suction $\psi=P_{air}-P_{water}>0$ in the vadose zone, so pore-water pressure is always below the air-phase pressure.

Part (b) — tritium and the meteoric water line. Tritium (³H, a radioactive hydrogen isotope with a 12.32-year half-life) enters groundwater as tritiated water molecules formed in the atmosphere by cosmic-ray spallation and, historically, by above-ground nuclear weapons testing (a large pulse peaking around 1963). Because it decays with a known half-life and its atmospheric input history is documented, tritium is used as an age tracer: a sample with tritium activity near modern atmospheric levels (or bearing the 1960s "bomb pulse" signature) contains water recharged within the last ~50-60 years, while tritium-dead water (below detection) indicates recharge that pre-dates the 1950s nuclear era. It is therefore widely used to distinguish young, recently-recharged groundwater (vulnerable to surface contamination) from old, pre-bomb water, and to estimate mean residence times in shallow flow systems when combined with a lumped-parameter age model. The stable isotopes oxygen-18 and deuterium (²H) are used differently: because both undergo similar equilibrium and kinetic fractionation during evaporation and condensation, a plot of $\delta^2\text{H}$ against $\delta^{18}\text{O}$ for global precipitation falls close to a straight line, $\delta^2\text{H}\approx 8\,\delta^{18}\text{O}+10$ (Craig, 1961), known as the Global Meteoric Water Line (GMWL). Precipitation and groundwater recharged directly from precipitation plot on or near this line; samples that plot to the right of/below the line (a lower slope, an "evaporation trend") have undergone evaporative enrichment of the heavy isotopes before or during infiltration — e.g. water that stood in a surface reservoir, wetland, or was affected by evapotranspiration in the unsaturated zone. Comparing a groundwater sample's isotopic composition against the local/global meteoric water line is therefore a standard tool for identifying the recharge source and whether evaporation has modified the water before it entered the aquifer.

Given. Part (c): free-water molecular diffusion coefficient $D_0=2\times10^{-5}\ \text{cm}^2/\text{s}$ for Cl⁻, porosity $n=0.3$, tortuosity $\tau=1.22$.

Find. The effective diffusion coefficient $D_e$ of Cl⁻ in the saturated porous medium.

Approach. Convert the free-solution diffusion coefficient to a porous-medium value using the standard tortuosity correction, which scales $D_0$ down by the medium's porosity and up by the reciprocal of its tortuosity (the actual diffusion path is longer and partly obstructed by solids).

Effective diffusion coefficient. $$D_e=\frac{D_0\,n}{\tau}=\frac{(2\times10^{-5})(0.3)}{1.22}=\boxed{4.92\times10^{-6}\ \text{cm}^2/\text{s}}.$$ The porous medium diffuses Cl⁻ roughly four times slower than free water, because only 30% of the cross-section is available for transport and the tortuous path is about 1.22× longer than a straight line.

Question 1 — Final Results
Item	Result
1(a) True/False (12 items)	T,T,F,F,F,T,F,F,F,F,F,T (see table above)
1(b)	Tritium = radioactive age tracer (bomb-pulse/decay); meteoric water line = $\delta^2\text{H}=8\delta^{18}\text{O}+10$, the precipitation baseline for spotting evaporation
1(c) Effective diffusion coefficient $D_e$	$4.92\times10^{-6}\ \text{cm}^2/\text{s}$

Concept & Theory

Topic. Hydrodynamic dispersion, isotope hydrology, and molecular diffusion in porous media.

Key relations. $D_L=\alpha_L v+D^{*}$ (longitudinal hydrodynamic dispersion); $D_e=D_0\,n/\tau$ (effective porous-medium diffusion coefficient); $\delta^2\text{H}=8\,\delta^{18}\text{O}+10$ (Global Meteoric Water Line); $P_c=2\sigma\cos\theta/r$ (Young-Laplace capillary pressure).

Why this works. Hydrodynamic dispersion is the sum of two physically distinct processes — mechanical mixing driven by pore-scale velocity variability (which vanishes at zero flow) and molecular diffusion (which persists regardless of flow) — so the total coefficient is additive in $v$. Stable-isotope fractionation during evaporation preferentially removes the lighter isotopologues from the vapour phase, so any water that has evaporated plots off the meteoric line with a shallower slope, which is why the GMWL is the diagnostic baseline. Diffusion in a porous medium is throttled by the reduced cross-sectional area (porosity) and the lengthened, winding path solute molecules must actually travel (tortuosity), so both enter the effective coefficient.

Common pitfall. Confusing tortuosity conventions: some texts define $\tau$ as a fraction $<1$ (so $D_e=\tau D_0$ directly) while others (as used here, with $\tau=1.22>1$, the ratio of actual to straight-line path length) require dividing by $\tau$ and multiplying by porosity, i.e. $D_e=D_0 n/\tau$ — always check which convention a given tortuosity value implies before applying it. A second pitfall is treating relative permeability as increasing with the OTHER phase's saturation (item 8) rather than its own.

Source. Fetter, Contaminant Hydrogeology Ch.2 (diffusion, tortuosity); Freeze & Cherry, Groundwater Ch.9 (isotope hydrology, meteoric water line) & Ch.2 (Buckingham-Darcy law, capillary pressure).

Question 1 of 5: Dispersion/Diffusion Fundamentals, Isotope Hydrology, and Effective Diffusion Coefficient

Question 1: Dispersion/Diffusion Fundamentals, Isotope Hydrology, and Effective Diffusion Coefficient (equal value)