An Excel^®-based visualization tool of 2-D soil gas concentration profiles in petroleum vapor intrusion

Abstract

In this study we present a petroleum vapor intrusion tool implemented in Microsoft^® Excel^® using Visual Basic for Applications (VBA) and integrated within a graphical interface. The latter helps users easily visualize two-dimensional soil gas concentration profiles and indoor concentrations as a function of site-specific conditions such as source strength and depth, biodegradation reaction rate constant, soil characteristics and building features. This tool is based on a two-dimensional explicit analytical model that combines steady-state diffusion-dominated vapor transport in a homogeneous soil with a piecewise first-order aerobic biodegradation model, in which rate is limited by oxygen availability. As recommended in the recently released United States Environmental Protection Agency's final Petroleum Vapor Intrusion guidance, a sensitivity analysis and a simplified Monte Carlo uncertainty analysis are also included in the spreadsheet.

1 Introduction

In the recent guidelines on petroleum vapor intrusion (PVI) released by Interstate Technology & Regulatory Council (ITRC, 2014) and by United States Environmental Protection Agency (U.S. EPA, 2015), it is suggested that mathematical models be used as one line of evidence in determining whether a building is likely to be impacted by petroleum chemical vapors migrating through the subsurface from contaminated soils or groundwater. The use of PVI modeling is recommended to help understand the observed behavior and to provide information useful for developing and visualizing a conceptual site model (CSM; Lahvis et al. 2013). In recent decades, there have been proposed numerous 1-D analytical models (e.g. DeVaull, 2007; Davis et al. 2009; Yao et al. 2014; Verginelli and Baciocchi, 2014) and multi-dimensional numerical models (e.g. Abreu and Johnson, 2006; Abreu et al. 2009; Hers et al. 2014) for estimating the migration of subsurface petroleum vapors into potentially impacted buildings. These models differ in their underlying assumptions and the conditions under which they may be applied. Namely, 1-D analytical models are widely used since they can be easily implemented in Excel spreadsheets but they are, of course, incapable of comprehensively predicting multi-dimensional subsurface soil gas concentration profiles, which may sometimes be critical to understanding particular situations. Conversely, multi-dimensional numerical models are quite powerful in visualizing comprehensive subslab soil gas concentration profiles of hydrocarbon and oxygen but they require significant computational effort and greater expertise with advanced numerical techniques. This has limited their use as quick and easy to understand screening tools. In this work we present a new tool (Petroleum Vapor Intrusion, Two-Dimensional - PVI2D) that represents a compromise between the need for an easy-to-use tool and the need for a more comprehensive and realistic representation of subslab hydrocarbon and oxygen soil gas concentration profiles. Specifically, the developed tool incorporates a 2-D analytical model that is based on a basic conceptual model similar to the one employed by Abreu and Johnson (2006) and Abreu et al. (2009) in their numerical 3-D modeling of PVI with biodegradation. The use of a 2-D approach adds the ability to represent many important subsurface profile phenomena, but avoids most of the additional mathematical complexity associated with the full 3-D representation.

PVI2D was developed in Microsoft^® Excel^® using Visual Basic for Applications (VBA) and integrated within a graphical interface that helps practitioners easily generate two-dimensional soil gas concentration profiles, and from these, indoor concentrations, as a function of the site-specific conditions such as source strength and depth, biodegradation reaction rate constant, soil characteristics and building features. Furthermore, a one-factor-at-a-time (OAT) Sensitivity Analysis and a simplified Monte Carlo uncertainty analysis are also included in the spreadsheet to permit assessing the sensitivity of results to any assumptions regarding input parameters.

2 Description of the tool

2.1 Model development

The developed tool implements a two-dimensional explicit analytical model that combines a steady-state diffusion-dominated vapor transport model in a homogeneous soil with a piecewise first-order aerobic biodegradation model, assumed to be limited by oxygen availability. The governing equations and the boundary conditions adopted for the model derivation are shown in Table 1. The underlying analytical solution was derived by coupling a 1-D analytical solution to the diffusion-reaction problem with the Schwarz-Christoffel mapping method (Verginelli et al. 2016). Table 2 reports a summary of the derived equations implemented in the tool. The full description and derivation of the analytical solution, together with a validation of the model is reported in Yao et al. (2016).

Table 1.

Governing equations and boundary conditions of the 2-D analytical solution.

Governing Equations and Boundary Conditions
Governing Equations
Hydrocarbons	Oxygen
Di∇2ci = Ri	Di∇2co = Σ;δiRi
Reaction Rate (R)
$\begin{array}{cc}{R}_i=\frac{{\lambda }_i{\theta }_w}{H_i}{c}_i& c_0>c_o^{\mathit{min}}\\ R_i=0& c_o\le c_o^{\mathit{min}}\end{array}$
Boundary conditions
Hydrocarbons	Oxygen
Foundation subslab: ∇ci·n⃗ = 0	Foundation subslab: ∇co·n⃗ = 0
$\text{Source depth}:c_i=c_i^s$	$\text{Open ground}:c_o=c_o^{\mathit{\text{atm}}}$

Table 2.

Main equations implemented in the tool.

Equations implemented in the developed tool
Iso-concentration contour curve position
$\begin{array}{c}\begin{array}{ccc}z(x)=\frac{L}{\pi }\mathit{\text{arccos}}(\frac{a}{b}+i\sqrt{\frac{a^2}{b^2}+\frac{c}{b}})& \text{with}& i=\{\begin{array}{cc}1& \mathit{\text{for}}w(x,z)\le \frac{1}{2}\\ -1& \mathit{\text{for}}w(x,z)>\frac{1}{2}\end{array}\end{array}\\ \text{where}\\ \begin{array}{ccc}a=\frac{\Omega (\Omega -1)\mathit{cos}h\left(\frac{\pi x}{L}\right)}{{\mathit{cos}}^2(\pi w(x,z))};& b={\left(\frac{\Omega \mathit{cos}h\left(\frac{\pi x}{L}\right)}{\mathit{cos}(\pi w(x,z))}\right)}^2+{\left(\frac{\Omega \mathit{sin}h\left(\frac{\pi x}{L}\right)}{\mathit{sin}(\pi w(x,z))}\right)}^2;& c=1+{\left(\frac{\Omega \mathit{sin}h\left(\frac{\pi x}{L}\right)}{\mathit{sin}(\pi w(x,z))}\right)}^2-\frac{{(\Omega -1)}^2}{{\mathit{cos}}^2(\pi w(x,z))}\end{array}\\ \text{with}\Omega =\frac{1}{\cos {\mathrm{h}}^2\left(\pi \frac{L_{\mathit{\text{slab}}}}{4\mathit{L}}\right)}\end{array}$
Oxygen iso-concentration contour curve
$w(x,z)=w_a+\frac{(c_o(x,z)-c_o^{\mathit{min}})}{c_o^{\mathit{\text{atm}}}-c_o^{\mathit{min}}}(1-w_a)$
Hydrocarbons iso-concentration contour curve
$w(x,z)=\{\begin{array}{cc}1-\frac{1}{k_{i}L}ln\left(2\frac{c_i(x,z)}{c_i^s}\left[(1+k_i{L}_b)cosh\left[k_{i}L(1-w_a)\right]\right]\right)& w(x,z)>z_a\\ (\frac{1+k_i{L}_b}{k_i{L}_b})(1-\frac{c_i(x,z)}{c_i^s})w_a& w(x,z)\le w_a\end{array}$
Slab-on-grade foundation	Basement foundation
$w_a=\frac{\mathrm{\sum }{\delta }_i{D}_i{c}_i^s\frac{k_i{L}_b(x=+\infty )}{1+k_i{L}_b(x=+\infty )}}{\sum {\delta }_i{D}_i{c}_i^s+D_o(c_o^{\mathit{\text{atm}}}-c_o^{\mathit{min}})}$	$w_a=\frac{\mathrm{\sum }{\delta }_i{D}_i{c}_i^s\frac{k_i{L}_b(x=+\infty )}{1+k_i{L}_b(x=+\infty )}}{D_o\left(c_o\right(x>\frac{L_{\mathit{\text{slab}}}}{2},L)-c_o^{\mathit{min}})+\sum {\delta }_i{D}_i{c}_i^s}$
Thickness of the anaerobic zone in the subsurface
$\begin{array}{ccc}{L}_b(x=+\infty )=\frac{L}{1+\frac{D_o(c_o^{\mathit{\text{atm}}}-c_o^{\mathit{min}})}{\mathrm{\sum }{\delta }_i{D}_i{c}_i^s}}& \text{with}& k_i=\sqrt{\frac{{\lambda }_i{\theta }_w}{H_i{D}_i}}\end{array}$
The diffusion coefficients Do and Di can be calculated with the Millington and Quirk (1961) expression:
$D=D_a\cdot {\theta }_a^{10/3}/{\theta }_t^2$
Subslab concentration of vapors in correspondence of the position of the cracks
$\frac{c_i(x_{\mathit{\text{ck}}},L)}{c_i^s}=\{\begin{array}{cc}\frac{1}{1+k_i{L}_b(x=+\infty )}\frac{\mathit{cos}h[k_{i}L(1-w(x_{\mathit{\text{ck}}},L))]}{\mathit{cos}h[k_{i}L(1-w_a)]}& w(x_{\mathit{\text{ck}}},L)>w_a\\ 1-\frac{w(x_{\mathit{\text{ck}}},L)}{w_a}\frac{k_i{L}_b(x=+\infty )}{1+k_i{L}_b(x=+\infty )}& w(x_{\mathit{\text{ck}}},L)\ge w_a\end{array}$
Indoor Concentration
$\begin{array}{c}{c}_i^{\text{indoor}}=c_i(x_{\mathit{\text{ck}}},L)\left(\frac{R_{\mathit{mix}}}{R_{\mathit{\text{crack}}}+R_{\mathit{mix}}}\right)\\ \text{with}{R}_{\mathit{mix}}=\frac{1}{L_{\mathit{mix}}\cdot \mathit{\text{ER}}};R_{\mathit{\text{crack}}}=(R_{\mathit{mix}}-\frac{A_b}{Q_s})[\mathit{exp}(-\frac{Q_s}{A_b}\cdot \frac{L_{\mathit{\text{crack}}}}{D_i^{\mathit{\text{crack}}}\cdot \eta })-1]\end{array}$
The convective flow rate Qs from the soil into the building can be calculated as follows:
$\begin{array}{ccc}{Q}_s=\frac{2\pi \cdot k_{\nu }\cdot \mathrm{\Delta }p\cdot X_{\mathit{\text{crack}}}}{\mu ln\left(\frac{2d_f}{r_{\mathit{\text{crack}}}}\right)}& \text{with}& r_{\mathit{\text{crack}}}=\frac{\eta \cdot A_b}{X_{\mathit{\text{crack}}}}\end{array}$

2.2 Tool development

An illustration of the main screen of the spreadsheet tool, which is automatically opened at startup, is shown in Figure 1. Here the user can select the compounds of concern (by clicking on the compound of concern and then clicking the button “Insert ≫”) and specify their vapor source concentrations and the relevant biodegradation rate constants. Note that by simultaneously adding more compounds (up to 15) the tool accounts for the total oxygen demand required to sustain the aerobic biodegradation of the selected chemicals. On this screen, the user specifies the building features and the vadose zone parameters. Here, some of the input parameters required in the tool (such as soil permeability to vapor flow or the airflow rate through foundation breaches) can be directly specified or automatically calculated by the tool (by activating the check box “Calculated”), as a function of the other input parameters selected by user.

Figure 1.

Main screen of the developed tool.

To review the chemical properties of the compounds of concern, the user can enter the Database screen by clicking on the “Database” button. On this screen, the user can edit the chemical properties of petroleum compounds and oxygen or add new chemicals to the system database. Here, some parameters such as the effective diffusion coefficient and the stoichiometric mass of oxygen consumed per mass of hydrocarbon can be automatically calculated by the tool (based on the other chemical properties set by the user) or directly specified by the user. In this model, the assumption is made that the compounds of concern are fully degraded to carbon dioxide and water.

The model results can be accessed by clicking on the “Results” button reported in the Main screen. On this screen (see Figure 2) the model results for each compound added in the Main screen (which can be selected by the user from the drop-down box present in this screen) are represented as 2-D normalized hydrocarbon iso-concentration curves and normalized oxygen iso-concentration contour curves beneath and beyond building footprint. The hydrocarbons and oxygen soil-gas concentrations are normalized to source concentration and atmosphere concentration, respectively. On the hydrocarbons graph are also reported the source to indoor air attenuation factor $(\alpha ={\mathrm{c}}_{\mathrm{i}}^{\text{indoor}}/{\mathrm{c}}_{\mathrm{i}}^{\mathrm{s}})$ and the indoor concentration and the source to outdoor air attenuation factor $(\alpha ={\mathrm{c}}_{\mathrm{i}}^{\text{outdoor}}/{\mathrm{c}}_{\mathrm{i}}^{\mathrm{s}})$ calculated by the model. Here the user can specify an empirical subslab to indoor air attenuation factor as an alternative to that automatically calculated by the tool. The other options present in this screen refer to the total length of x domain and to the boundary conditions at the bottom of the foundation slab, i.e., the choice of a strictly impervious slab or pseudo-impervious slab (for more details see Yao et al. 2016).

Figure 2.

Results screen of the developed tool.

A sensitivity analysis for some of the key model input parameters can be performed by accessing the sensitivity analysis screen (see Figure 3) through the “Sensitivity Analysis” button. Here the user can select from a drop-down box the compound of concern and the minimum and maximum parameter values for a one-factor-at-a-time (OAT) sensitivity analysis. By clicking the button “(Re)Run Analysis” shown in this screen, the tool calculates the source to indoor air attenuation factor calculated as function of the main parameter selected by the user for the sensitivity analysis. Furthermore, in this screen the user can select a secondary parameter and its corresponding input values for which to perform the sensitivity analysis. For instance, in Figure 3 the primary parameter selected for the sensitivity analysis is the vapor source concentration while the secondary parameter is the source depth below ground surface. Hence, in this illustrative example the source to indoor air attenuation factor as a function of vapor source concentration can be easily evaluated for different vapor source depths. Other combinations of parameters can be easily explored using this feature.

Figure 3.

Sensitivity analysis screen of the developed tool.

Finally, an uncertainty analysis for some of the key input parameters required in the model can be performed by accessing the Uncertainty analysis screen (see Figure 4) through the “Monte Carlo Analysis” button. Here the user selects the compound of concern and the input parameter for which to perform the uncertainty analysis (activating the appropriate check box “Include in the MC analysis”). Based on the ranges specified by user, the tool performs a simplified Monte Carlo analysis and reports the frequency and cumulative frequency of the expected source to indoor (or outdoor) air attenuation factor. Specifically, by clicking on the “Run Monte Carlo Analysis” button reported in this screen, the tool analyzes the attenuation factors obtained by performing up to 1000 simulations randomly varying the selected site parameter with a uniform distribution (i.e. each value is equally likely) in the ranges specified by the user.

Figure 4.

Monte Carlo analysis screen of the developed tool.

3 Demonstration of the tool application

Figure 5 shows for a benzene vapor source of 200 g/m³ located at a depth of 5 m bgs and a first-order biodegradation rate of λ_i= 0.18 hr^-1, the soil gas concentration profiles for oxygen and hydrocarbons obtained by Abreu and Johnson (2006) from 3-D numerical simulations and those generated by the Excel-based tool presented in this work. The other model input parameters used in generating this figure are reported in Table 3. It is worth noting that for this comparison, Abreu and Johnson (2006) in their simulation assumed benzene to be the sole vapor component for a gasoline source. In practice, modeling of PVI should simultaneously consider all the aerobically degradable petroleum vapor components.

Figure 5.

Comparison between the results provided by the 3-D numerical model of Abreu and Johnson (2006) and by the tool presented in this work (PVI2D). In this scenario a benzene vapor source concentrations of 200 mg/L and a biodegradation rate λ = 018 h^-1 are assumed. The hydrocarbons and oxygen soil-gas concentrations are normalized to source concentration and atmosphere concentration, respectively. (Figure adapted from Abreu and Johnson, 2006).

Table 3.

Models input parameters used for Figure 5.

Parameter	Symbol	Unit	Value
Concentration of benzene at vapor source	$c_i^s$	g/m3	200
Biodegradation rate constant of benzene in water phase	λi	hr-1	0.18
Stoichiometric mass of oxygen consumed per mass of benzene	δi	g/g	3
Oxygen atmospheric concentration	$c_o^{\mathit{\text{atm}}}$	g/m3	279
Minimum oxygen concentration to sustain biodegradation	$c_o^{\mathit{min}}$	g/m3	13.7
Depth of contaminant source below ground surface	ds	m	5
Water-filled porosity of the soil	θw	m3/m3	0.07
Total porosity of the soil	θt	m3/m3	0.35
Soil permeability to vapor flow	kν	m2	10-11
Depth of the building foundation below ground surface	df	m	0.2
Width of the building slab (foundation)	Lslap	m	10
Pressure difference between the soil and the building	Δp	Pa	5
Building air exchange rate	ER	hr-1	0.5
Thickness of the building foundation slab	Lcrack	m	0.15
Foundation cracks area fraction	η	m2cracks/m2tot	0.00039
Enclosed space volume/infiltration area ratio	Lmix	m	1.74

As shown in Figure 5, there is a good agreement between the soil-gas concentration profiles obtained by 3-D numerical simulation and those returned by the tool presented in this work. Slight differences observed can be in part attributed to the simplifying assumptions employed in the analytical model implemented in the Excel-based tool but also to the difficulty of the numerical model of simulating strictly non-flux boundary conditions for a foundation slab (Yao et al. 2016).

Regardless of the small differences the results shown in Figure 5 clearly establish that the tool presented in this work can be used as a valid alternative to more sophisticated 3-D numerical models for cases involving diffusion dominated soil gas transport with biodegradation under steady-state conditions and homogenous source and soil, The application of PVI2D at sites involving more complicated (e.g.. asymmetric) building foundations or high methane concentrations (involving gaseous-phase advection - see Yao et al., 2015) should be approached with caution, as it has not been developed for such applications.

Nomenclature
δi	Stoichiometric mass of oxygen consumed per mass of hydrocarbon	g/g
Δp	Pressure difference between the soil and the building	Pa
θt	Total porosity of the soil	m3/m3
θw	Water-filled porosity of the soil	m3/m3
λi	Biodegradation rate of hydrocarbon in water phase	Hr−1
μ	Vapor viscosity	g/(m – hr)
η	Foundation cracks area fraction	m2 cracks/m2 tot
Ab	Foundation footprint area	m2
ci	Concentration of hydrocarbon in the soil-gas phase	g/m3
$c_i^{\mathit{\text{indoor}}}$	Concentration of hydrocarbon in indoor air	g/m3
$c_i^s$	Concentration of hydrocarbon at vapor source	g/m3
co	Concentration of oxygen in the soil-gas phase	g/m3
$c_o^{\mathit{\text{atm}}}$	Oxygen concentration in the atmosphere	g/m3
$c_o^{\mathit{min}}$	Minimum oxygen concentration to sustain biodegradation	g/m3
df	Depth of the building foundation slab below ground surface	m
ds	Depth of contaminant source below ground surface	m
Di	Effective porous medium diffusion coefficient of hydrocarbon	m2/hr
D0	Effective porous medium diffusion coefficient of oxygen	m2/hr
ER	Building air exchange rate	hr–1
kν	Soil permeability to vapor flow	m2
Hi	Henry's law constant for hydrocarbon	dimensionless
L	Vertical distance from source to the the bottom of the foundation	m
La	Thickness of the aerobic zone in the subsurface	m
Lb	Thickness of the anaerobic zone in the subsurface	m
Lcrack	Thickness of the building foundation slab	m
Lmix	Enclosed space volume/infiltration area ratio	m
Lslab	Width of the building slab (foundation)	m
W	Dimensionless oxygen or hydrocarbon concentration	dimensionless
Wa	Dimensionless oxygen or hydrocarbon variable at the interface	dimensionless
x	Coordinate in the horizontal direction	m
Xcrack	Perimeter of the building foundation	m
xck	Distance of the slab entry cracks from the building center	m
Z	Coordinate in the vertical direction	m