Question 6 of 7: Section B — One of Two Questions

Question B-1: Pump-and-Treat Remediation Time for a TCE-Contaminated Aquifer (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.

Given.

Find. (a) the theoretical time to remediate the site per well; (b) whether that time is a reasonable/realistic estimate.

Approach. The extraction rate a single well can sustain is set by steady-state radial flow (the Thiem equation) between the well and its radius of influence; the total TCE mass held within that same zone of influence — both dissolved in the pore water and sorbed to the soil — is found from the equilibrium concentration and the linear-sorption retardation factor. Since the extracted water is always at the equilibrium concentration $C_e$ throughout pumping, the clean-up time is simply that total mass divided by the constant mass-removal rate, which reduces to the retardation factor times one pore-volume-flush time.

1. Unit conversion and retardation factor. $K_d=0.00016$ L/mg $=160$ L/kg $=0.16\ \text{m}^3/\text{kg}$ (multiplying by $10^6$ mg/kg then by $10^{-3}$ m³/L). Retardation factor: $R=1+\dfrac{\rho_b}{n}K_d=1+\dfrac{1700}{0.40}\times0.16=1+680=\boxed{681}$.
2. Extraction rate (Thiem equation). Steady radial flow to a fully penetrating well, drawdown $s_w$ at $r_w$ and zero drawdown at $R_i$: $Q=\dfrac{2\pi K b\,s_w}{\ln(R_i/r_w)}=\dfrac{2\pi(2\times10^{-3})(3)(0.3)}{\ln(25/0.05)}=\dfrac{0.01131}{6.215}=\boxed{1.82\times10^{-3}\ \text{m}^3/\text{s}}$ ($\approx$1.82 L/s, or 157.2 m³/day).
3. Pore volume within the well's zone of influence. Treating the zone of influence as a cylinder of radius $R_i$ and thickness $b$: $V=\pi R_i^2 b=\pi(25)^2(3)=5890\ \text{m}^3$; pore (water-filled) volume $V_p=nV=0.40\times5890=\boxed{2356\ \text{m}^3}$.
4. Theoretical remediation time. Because the extracted water is always at $C_e$, the total dissolved-plus-sorbed mass in the zone is $M=C_e\,V_p\,R$, and it is removed at a constant rate $C_e\,Q$, so $C_e$ cancels: $t=\dfrac{R\,V_p}{Q}=\dfrac{681\times2356}{1.82\times10^{-3}}=8.82\times10^{8}\ \text{s}=\boxed{\approx10{,}200\ \text{days}\ (\approx27.9\ \text{years})}$.

Part (b) — does the remediation time make sense? A ~28-year theoretical clean-up time does make physical sense given the inputs, and is not an arithmetic anomaly: the retardation factor of 681 means that, at equilibrium, more than 99.8% of the TCE mass in the zone of influence is sorbed onto the soil rather than dissolved in the pumped water at any instant ($1/R\approx0.15\%$ of the total mass is in the mobile, extractable phase). Pump-and-treat can only remove the dissolved fraction directly; the very slow overall clean-up time is the mathematical expression of the well-known real-world observation that pump-and-treat performs poorly wherever sorption (or a persistent DNAPL source) continually re-supplies the dissolved phase. If anything, the ~28-year estimate is optimistic: it assumes the extracted concentration stays at a constant 150 mg/L for the entire period, whereas real systems show pronounced concentration "tailing" as the most accessible sorbed mass is depleted first and desorption/back-diffusion from lower-permeability zones becomes rate-limiting, meaning the true time to reach a low residual concentration is typically longer still. The practical conclusion is that pump-and-treat alone is not an appropriate stand-alone remedy for this site; it should be paired with (or replaced by) a technology that removes or destroys the sorbed/source mass directly — in-situ chemical oxidation, enhanced anaerobic bioremediation (reductive dechlorination), or thermal treatment of the source zone — with pump-and-treat retained, if at all, mainly for hydraulic containment of the dissolved plume.