Question 2 of 5: Column Breakthrough, an Instantaneous Aquifer Spill, and the Conceptual Site Model

Question 2: Column Breakthrough, an Instantaneous Aquifer Spill, and the Conceptual Site Model (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.

Given. (a) $C_0=110\ \text{mg/L}$, column length $L=1\ \text{m}$, diameter $d=10\ \text{cm}$, $Q=6\ \text{mL/min}$, $\alpha_L=0.08\ \text{m}$, $n=0.37$, $\rho_b=1.6\ \text{g/cm}^3$, $D^{*}=1.0\times10^{-10}\ \text{m}^2/\text{s}$, $t=11\ \text{hours}$. (b) $M=1000\ \text{kg}$, aquifer thickness $b=10\ \text{m}$, Darcy velocity $q=0.05\ \text{m/day}$, $n=0.3$, $\alpha_L=\alpha_T=10\ \text{m}$.

Find. (a) Effluent chloride concentration at $t=11\ \text{h}$. (b)(i) Peak concentration after 1 year and its location. (b)(ii) Concentration 1 km downgradient after 10 years. (c) Three components of a conceptual site model.

Approach. (a) Convert the volumetric flow to a seepage velocity and evaluate the Ogata-Banks solution for a continuous, conservative source at the column outlet. (b) Build the horizontal dispersion coefficients from the seepage velocity and evaluate the 2-D instantaneous point-source (Gaussian) solution at the plume centroid and at the two specified locations/times.

1. Part (a) — column seepage velocity and dispersion coefficient. Cross-sectional area $A=\pi(0.05)^2=7.854\times10^{-3}\ \text{m}^2$. Flow $Q=6\ \text{mL/min}=1.0\times10^{-7}\ \text{m}^3/\text{s}$, so Darcy flux $q=Q/A=1.273\times10^{-5}\ \text{m/s}$ and seepage velocity $$v=\frac{q}{n}=\frac{1.273\times10^{-5}}{0.37}=3.441\times10^{-5}\ \text{m/s}\ (\boxed{2.97\ \text{m/day}}).$$ Dispersion coefficient: $D_L=\alpha_Lv+D^{*}=(0.08)(3.441\times10^{-5})+1.0\times10^{-10}\approx\boxed{2.75\times10^{-6}\ \text{m}^2/\text{s}}$ (molecular diffusion is negligible next to mechanical dispersion at this velocity).
2. Effluent concentration via Ogata-Banks. At $t=11\ \text{h}=39{,}600\ \text{s}$, the mean travel distance $vt=1.363\ \text{m}$ already exceeds the 1 m column length, so the front has broken through and the outlet concentration should be well advanced toward $C_0$: $$\frac{C}{C_0}=\tfrac12\,\mathrm{erfc}\!\left(\frac{L-vt}{2\sqrt{D_Lt}}\right)+\tfrac12\,e^{vL/D_L}\,\mathrm{erfc}\!\left(\frac{L+vt}{2\sqrt{D_Lt}}\right)=\boxed{0.838}.$$ $$C=0.838\times110=\boxed{92.1\ \text{mg/L}}.$$
3. Part (b) — horizontal dispersion coefficients. Seepage velocity $v=q/n=0.05/0.3=0.1667\ \text{m/day}$. With $\alpha_L=\alpha_T=10\ \text{m}$: $$D_x=D_y=\alpha v=(10)(0.1667)=\boxed{1.667\ \text{m}^2/\text{day}}\ \text{(equal in both directions here, since }\alpha_L=\alpha_T\text{).}$$
4. Peak concentration and location, $t=1$ year. The instantaneous-source plume is Gaussian in both horizontal directions and travels with the mean seepage velocity along the flow axis; the peak sits at the plume centroid ($y=0$): $$x_{peak}=vt=(0.1667)(365.25)=\boxed{60.9\ \text{m downgradient}}.$$ $$C_{max}=\frac{M}{4\pi t\,n\,b\sqrt{D_xD_y}}=\frac{1000}{4\pi(365.25)(0.3)(10)(1.667)}=\boxed{0.0436\ \text{kg/m}^3}\ (\boxed{43.6\ \text{mg/L}}).$$
5. Concentration 1 km downgradient, $t=10$ years. After 10 years the plume centroid has advanced to $x=vt=(0.1667)(3652.5)=608.8\ \text{m}$, so the 1 km point sits $391\ \text{m}$ ahead of the centroid, well out on the leading tail: $$C(1000,0,10\ \text{yr})=\frac{M}{4\pi t\,n\,b\sqrt{D_xD_y}}\exp\!\left[-\frac{(x-vt)^2}{4D_xt}\right]=\boxed{8.11\times10^{-6}\ \text{kg/m}^3}\ (\boxed{8.11\ \mu\text{g/L}}).$$ Even 10 years in, only a faint leading edge of the plume has reached 1 km — the peak itself is still 400 m short of that point.

Part (c) — three components of the conceptual site model (CSM). A conceptual site model integrates everything known about a contaminated site into a single working picture that drives every subsequent investigation and remediation decision; three components are always required. (1) Source characterization — the type, quantity, chemical composition, and physical form (dissolved, sorbed, free-phase/NAPL) of the contaminant(s), plus the release history (single spill vs. ongoing leak, when it began) and the source's current physical state (still present, or fully depleted). (2) Geology and hydrogeology (the pathway) — the stratigraphy, hydraulic properties (K, n, gradient), and groundwater flow directions/velocities that control how contaminants migrate from the source, including which aquifer(s) are connected, the presence of preferential pathways (fractures, sand lenses), and the vadose-zone properties governing infiltration and vapour transport. (3) Receptors and exposure pathways — the human and ecological receptors potentially affected (drinking-water wells, surface-water bodies, vapour intrusion into buildings) and the complete exposure pathway linking the source to each receptor (ingestion, inhalation, dermal contact); a pathway with no viable receptor, or a receptor with no complete pathway, is not a risk driver. All three must be integrated (not just individually characterized) into a single 3-D picture, because it is the source-pathway-receptor LINKAGE, not any one component alone, that determines whether remediation is required and what remedy is appropriate.