Question 5 of 8: Modelling a Gasoline Release from an Underground Storage Tank

Question A-5: Modelling a Gasoline Release from an Underground Storage Tank (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.

Modelling a leaking underground storage tank (UST) is a multi-phase problem: gasoline is a light non-aqueous-phase liquid (LNAPL) that moves through the unsaturated zone under gravity, spreads and pools on the water table, dissolves a fraction into groundwater, and volatilizes a fraction into soil gas — a complete model needs a governing relationship and a parameter set for each of those four processes.

1. Release/source term. Release rate and total volume as a function of time: for a hole of area $a$ at hydrostatic head $h$ above the breach, Torricelli’s equation $Q=C_d\,a\sqrt{2gh}$ gives the leak rate, integrated over the tank-emptying period as $h$ (and hence $Q$) declines. Parameters needed: hole size/location (rarely known — back-calculated from inventory-reconciliation records if available), discharge coefficient $C_d$, and tank geometry.

2. Vadose-zone (unsaturated) transport. Multiphase (gasoline/air/water) flow governed by an extension of Darcy’s law with relative permeability, $q_{NAPL}=-\dfrac{k\,k_{rn}}{\mu_n}\nabla(P_n+\rho_n g z)$, with residual saturation $S_{nr}$ (the fraction permanently trapped in pores and unavailable for further movement) controlling how much of the release is retained before reaching the water table. Parameters: intrinsic permeability $k$, relative-permeability curves $k_{rn}(S_n)$, residual saturation $S_{nr}$, porosity, and depth to water table — relative-permeability curves in particular are rarely measured on-site and are usually taken from published soil-type correlations, a significant source of model uncertainty.

3. Dissolved-phase transport in groundwater. The advection-dispersion-reaction equation, $\dfrac{\partial C}{\partial t}=D_L\dfrac{\partial^2 C}{\partial x^2}-v_x\dfrac{\partial C}{\partial x}-\lambda C$, where $v_x=Kdh/dl/n$ is the seepage velocity (Darcy flux over porosity), $D_L$ the longitudinal dispersion coefficient, and $\lambda$ a first-order decay/attenuation rate for the dissolved BTEX plume. Retardation of individual dissolved constituents follows $R=1+(\rho_b/n)K_d$ with $K_d=f_{oc}K_{oc}$. Parameters: hydraulic conductivity $K$, gradient, porosity, dispersivity, $f_{oc}$, and a natural-attenuation decay rate — dispersivity is scale-dependent and essentially impossible to measure directly, so it is almost always estimated from an empirical correlation with plume travel distance rather than measured.

4. Volatilization / vapour-phase transport. Equilibrium partitioning to soil gas via Raoult’s law/the compound’s vapour pressure and Henry’s law constant governs the source concentration for vapour-intrusion modelling into any nearby building (relevant here at a gas bar with an on-site kiosk/store), combined with Fick’s law diffusion, $J=-D_{eff}\,dC/dz$, through the vadose zone. Parameters: pure-component vapour pressure, Henry’s constant, effective vapour diffusion coefficient (itself a function of air-filled porosity via a tortuosity correction), and building/foundation characteristics for a full vapour-intrusion model.