CHAPTER 20

MODELING OIL SPILLS ON RIVER SYSTEMS: EVALUATION OF AQUEOUS CONCENTRATIONS John S. Gulliver and Vaughan R. Voller Department of Civil Engineering, University of Minnesota, Minneapolis, Minnesota

David E. Hibbs Barr Engineering Company, Ann Arbor, Michigan

20.1

INTRODUCTION Accidental releases of buoyant oils, fuels, and chemicals into water bodies can be catastrophic events for the aquatic biota. Riverine spills can be particularly hazardous. Unlike spills onto oceans, in which the water column is subjected to almost an infinite dilution, rivers are shallow and confined by the river bed and banks. Consequently, aqueous concentrations resulting from riverine spills can be several orders of magnitude higher than similar spills onto oceans. Spills of any sparingly soluble buoyant compound initially form a slick on the water surface. The slick spreads across the water surface and drifts downstream as the various compounds in the slick evaporate and dissolve into the water column. To predict the impact of spills on the river biota, the concentration of compounds dissolved in the water column must be assessed. There are many models that look at the hydrodynamics of the oil slick (Yapa and Shen, 1994). These models focus on the tracking, spreading, and containment of the oil slick and are applicable to wide rivers where the two-dimensionality of the slick is important. The focus of this chapter is not on the spread of the oil slick per se, but on the effect that spread may have on the biota of the river. This requires a detailed tracking of how the aqueous concentrations develop as the slick moved downstream. Modeling the evolution of the aqueous concentrations requires an adequate accounting of the various transport coefficients that control the process. The contribution of this chapter is the identification of these parameters, citations of appropriate literature sources, and, most importantly, a sensitivity analysis that identifies the coefficients that have the greatest impact on simulation results. The model described was developed primarily for application to smaller, nonnavigable rivers where the assumptions of a cross-sectional, well-mixed water body and slick are usually valid. When the cross-sectional uniform water body and oil slick assumptions are not valid, a combination of a two-dimensional flow slick (Yapa et al., 1994; Shen et al., 20.1

20.2

CHAPTER TWENTY

1995) and the aqueous transport and transfer processes described in this chapter would be required. A spill of JP-4 jet fuel is used to illustrate the operation of the model. The rate coefficients used in this example spill are most applicable for smaller rivers, which exhibit a fair degree of meandering and are often sheltered from wind by the riverbanks and vegetation. In keeping with all environmental fate and transport modeling, the predicted aqueous concentrations have an inherent level of uncertainty due to bias in the model input parameters. Since the user-specified rate constants are seldom measured but are themselves often estimated from predictive relationships, these input parameters can often be in significant error. Depending on the particular parameter, the uncertainty associated with each input parameter can have an impact on the predicted aqueous concentrations ranging from drastic to insignificant. Within the context of a JP-4 jet fuel spill into a small river system, sensitivity analysis to identify the model parameters and / or processes that are the most important in determining aqueous concentrations of contaminants is a key component in the testing of the model.

20.2

THE DUAL-PHASE MODEL The model is established on a mixture of Eulerian and Lagrangian coordinate systems. The river is approximated as a series of completely mixed cells (typically 10–1,000 m in length) fixed in position, as shown in Fig. 20.1. The slick is approximated as a series of completely mixed cells that move across the water surface in a Lagrangian coordinate system. This treatment of the slick as a series of moving cells allows for spatial variation in the concentration of the slick. The application of the model, per se, is to situations where both flow and slick can be described as one-dimensional. This occurs when the slick is spread completely across the river, as with relatively narrow streams. The length of the river from the spill site before the one-dimensional assumption can be applied is approximately: L艑

LwUslick dl / dt

(20.1)

where Uslick is the streamwise velocity of the centroid of the slick, Lw is the largest transverse length across the river from the location of the spill, and dl / dt is the spreading velocity of the slick. If the spill occurred on one bank of the river, Lw would be the river width. All of the described processes for the evolution of aqueous concentration, however, would apply to both a one-dimensional and a two-dimensional spill model.

Slick # 3

C w,5

Slick # 2 C w,6

Slick # 1 C w,7

Cw,8

River Bottom FIGURE 20.1 Numerical approximation of river cross-section with multiple surface slicks.

MODELING OIL SPILLS ON RIVER SYSTEMS

20.2.1

20.3

Movement of the Surface Slick

The movement of the slick on the water is largely a function of the velocity of the water surface and the direction and magnitude of the wind: Uslick ⫽ ␣velU ⫹ adriftUwind where U Uwind ␣vel ␣drift

⫽ ⫽ ⫽ ⫽

the the the the

(20.2)

cross-sectional mean river velocity streamwise component of the wind speed vector velocity profile correction factor (␣vel 艑 1.1 [Addison, 1941]) wind drift coefficient (␣drift 艑 0.03–0.04 [Wu, 1983])

For smaller rivers, which are often tortuous and sheltered, the wind drift term in Eq. (20.2) is dropped. The velocities of the leading and trailing edges of the slick are: Uleading ⫽ 1.1 U ⫹ ␣driftUwind ⫹

dl dt

(20.3)

Utrailing ⫽ 1.1 U ⫹ ␣driftUwind ⫹

dl dt

(20.4)

and

where dl / dt is the spreading rate of the slick. While the oil is spilling onto the water surface, the leading edge of the slick is allowed to drift and spread downstream at a velocity determined from Eq. (20.3) and the trailing edge of the slick is assumed to be fixed in position. The oil is added to the trailing end of the overall slick, i.e., into the upstream-most individual slick. This treatment of the spilling process reasonably simulates the behavior of a slick forming on a river. During the early stages of a riverine spill, dl / dt is on the order 0.1 m / s (Fay, 1971). Thus, the spreading velocity is less than typical water surface velocities, and the slick is not likely to spread upstream from the point of spilling. Once the spilling stops, the trailing edge of the slick is released to drift and spread at a velocity determined from Eq. (20.4). The position of the leading and trailing edges of the overall slick are determined from Eqs. (20.3) and (20.4) and the boundaries of the individual slicks are interpolated from the spreading rate of the leading edges based on their distance from the midpoint of the overall slick.

20.2.2

Mass Balance for the Slick Phase

The mass of the slick decreases over time as the compounds dissolve into the water phase and evaporate into the atmosphere. A mass balance for each compound in the slick is written as: dm ⫽ dt where Cw C sat w kdis kevap m M

⫽ ⫽ ⫽ ⫽ ⫽ ⫽

the the the the the the

冕

length of slick

冋

⫺kdisW(XoC sat w ⫺ Cw) ⫺ kevapW

aqueous concentration aqueous saturation concentration dissolution rate coefficient evaporation rate coefficient mass of the compound in the slick compound’s molecular weight

冉

冊册

XoP oMi RT

dx ⫹ r

(20.5)

20.4

CHAPTER TWENTY

r Po R t T x Xo W

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

the release rate (i.e., rate at which the compound is spilled) the vapor pressure of the pure compound the universal gas constant time the absolute temperature of the interface distance downstream the mole fraction in the slick the width of the slick

This one-dimensional model formulation assumes the slick covers the water surface from bank to bank, therefore W is also the width of the river. The product XoP o represents the vapor pressure of a compound at the slick–air interface based on Raoult’s law. The mass flux rate is integrated over the length of the slick since W and Cw can vary as a function of distance. The concentration of each compound in the slick can then be expressed as a mole fraction: m(M)⫺1

冘

Xo ⫽

m(M)⫺1

(20.6)

all compounds

20.2.3

Transport Equation for Aqueous Phase

The concentrations of compounds dissolved in the water change over time and distance as the compounds dissolve from the slick into the water, volatilize from the water to the atmosphere, and disperse in the river. The one-dimensional advection-diffusion equation for each compound is written as: ⭸(ACw) ⭸(QCw) ⭸ ⫹ ⫽ ⭸t ⭸x ⭸x

冉

ADL

冊

⭸Cw ⭸x

⫹ ␾kdisW(XoC wsat ⫺ Cw) ⫺ (1 ⫺ ␾) kvolWCw

(20.7) where A DL Q kvol ␾

⫽ ⫽ ⫽ ⫽ ⫽

the cross-sectional area of river the longitudinal dispersion coefficient discharge the volatilization rate coefficient a phase marker (␾ ⫽ 1 if there is a slick at location x; ␾ ⫽ 0 otherwise)

When the surface slick is present at location x, there is an exchange between the slick and the water. When the surface slick is not present at location x, there is an exchange between the water and the atmosphere. The approximation of the river as a series of discrete well-mixed cells introduces additional dispersion into the model. Even if a value of DL ⫽ 0 is input, some dispersion will still be predicted by the model. Banks (1974) developed a mixed cell model which may be used to quantify this numerical dispersion: DL⫺num ⫽ 0.5 U ⌬x

(20.8)

where ⌬x is the length of the river control volume. Equation (20.8) is based on an infinite number of cells and is approximately valid as long as roughly 10 or more cells are used in the model (Levenspiel, 1962). Since Eq. (20.7) is often approximated as a linear partial differential equation where superposition applies, the numerical dispersion and the userspecified dispersion are nearly additive. Thus, the value of longitudinal dispersion that should

MODELING OIL SPILLS ON RIVER SYSTEMS

20.5

be specified in the model to obtain the true dispersion, DL, can be estimated from the expression: DL⫺model ⫽ DL ⫺ DL⫺num ⫽ DL ⫺ 0.5 U ⌬x

(20.9)

where DL⫺model is the longitudinal dispersion coefficient that is supplied to the model in Eq. (20.8).

20.2.4

Solution Algorithm

Equations (20.3) through (20.7) form the basis of the model. A formulation of Eq. (20.7) is needed for each compound that is to be modeled. A pair of Eqs. (20.5) and (20.6) are needed for each compound and for each individual slick used in the simulation. No direct solution for the equation set is available, so the equations are solved numerically. At each time step, the extent and location of the oil slick is determined from Eqs. (20.3) and (20.4). The mass of each compound in the oil phase is then determined by an explicit time integration of Eq. (20.5), i.e., the values of Cw and Xo are taken from the previous time step. The mole fraction of each compound in the slick is then determined from Eq. (20.6). Next, the aqueous concentration of each compound at each location in the river is determined by a fully implicit time integration of Eq. (20.7), i.e., the current values of Cw and Xo are used in the calculation.

20.3

A SAMPLE APPLICATION As an example of the capabilities of the model, consider a spill of 10,000 kg of JP-4 jet fuel released over a period of 10 minutes onto a river of the scale typical of a tributary to a navigable river. The approximate composition of the fuel, the physical properties of the compounds, the river flow data, and the model parameters for the simulation are listed in Table 20.1. Most of the jet fuel is composed of several relatively insoluble long-chain aliphatics (Burris and MacIntyre, 1987). For simplicity, these aliphatic compounds are grouped and modeled as a single compound with properties averaged from the individual aliphatic compounds. The results of the jet fuel spill simulation are shown in Figs. 20.2 through 20.5. Figure 20.2 shows the aqueous concentrations of two compounds, ethylbenzene and 1-methylnaphthalene, plotted versus distance downstream at times of 10, 20, 40, and 60 hours after the spill occurred. The concentration profiles are nearly Gaussian in shape, typical of pulse loadings in rivers. The profiles tend to broaden over time due to the streamwise mixing in the river, approximated by the longitudinal dispersion term in Eq. (20.7). Notice that the concentration profiles are slightly skewed to the downstream direction. Since the slick is drifting downstream at a rate slightly higher than the mean river velocity, the compounds in the slick are constantly dissolving into the river slightly downstream of the aqueous concentration peak, creating a skewed profile. The peak aqueous concentrations of all eight compounds in the jet fuel are plotted versus time in Fig. 20.3. The peak aqueous concentrations of the more volatile compounds (e.g., toluene and methylcyclohexane) reach maximums within the first 10 hours, then decrease as these volatile compounds rapidly evaporate from the slick and are no longer available to dissolve into the water. The aqueous concentration of each compound continues to rise until that compound has evaporated from the slick. Once a compound has been removed from the slick, the peak aqueous concentration of that compound decreases as its concentration is diluted by longitudinal dispersion and as it volatilizes from the water to the atmosphere. In this example, the least soluble compounds also are least volatile and therefore tend to remain in the slick for a longer time and continue to dissolve into the water column, resulting in

20.6

CHAPTER TWENTY

TABLE 20.1 Input Parameters Used in Sample Application Properties

Toluene

Ethylbenzene

n-Butylbenzene

Tetralin

X (⫺)a MW (g / mol) ␳ (kg / m3)b a,b C sat w (mol / L) P o (atm)b,d Dwater (ms / s)e Doil (m2 / s)f Dair (m2 / s)g

0.04331 92.1 870 10⫺2.50 10⫺1.42 9.1 ⫻ 10⫺10 1.5 ⫻ 10⫺9 8.1 ⫻ 10⫺6

0.04785 106.2 870 10⫺2.80 10⫺1.90 8.2 ⫻ 10⫺10 1.4 ⫻ 10⫺9 7.4 ⫻ 10⫺6

0.03879 134.2 860 10⫺3.97 10⫺2.86 7.0 ⫻ 10⫺10 1.1 ⫻ 10⫺9 6.4 ⫻ 10⫺6

0.02922 132.2 970 10⫺3.47 10⫺3.27 7.6 ⫻ 10⫺10 1.2 ⫻ 10⫺9 6.8 ⫻ 10⫺6

Properties

1-Methylnaphthalene

1,4-Dimethylnaphthalene

Methylcyclohexane

Aliphaticsh

X (⫺)a MW (g / mol) ␳ (kg / m3)b b,c C sat w (mol / L) P o (atm)b,d Dwater (ms / s)e Doil (m2 / s)f Dair (m2 / s)g

0.02666 142.2 1,002 10⫺3.67 10⫺4.07 7.4 ⫻ 10⫺10 1.2 ⫻ 10⫺9 6.6 ⫻ 10⫺6

0.02613 156.2 1,000 10⫺4.22 10⫺4.60 6.8 ⫻ 10⫺10 1.1 ⫻ 10⫺9 6.2 ⫻ 10⫺6

.2123 98.2 770 10⫺3.77 10⫺1.23 8.2 ⫻ 10⫺10 1.4 ⫻ 10⫺9 7.7 ⫻ 10⫺6

.5759 145.7 731 10⫺6.52 10⫺2.77 6.2 ⫻ 10⫺10 1.0 ⫻ 10⫺9 5.8 ⫻ 10⫺6

River properties ⫽ ⫽ ⫽ ⫽ Uwind ⫽ kL⫺oxygen ⫽ T⫽

Q A W DL

3 (m3 / s) 10 (m2) 10 (m) 25 (m2 / s) 2 (m / s) 2.47 ⫻ 10⫺6 (m / s) 20⬚C

Rate constantsi kevap ⫽ 5.27 ⫻ 10⫺4 (m / s) kdis ⫽ 1.50 ⫻ 10⫺6 (m / s) kvol ⫽ 1.20 ⫻ 10⫺5 (m / s) Model parameters ⌬t ⫽ 60 (sec) ⌬x ⫽ 100 (m)

a

Burris and MacIntyre, 1987. Lide, 1993. c Schwarzenbach et al., 1993. d Lyman et al., 1990. e Approximated using Hayduk and Laudie, 1974. f Approximated using Wilke and Chang, 1995. g Approximated using Fuller et al., 1966. h Average properties of the 5 aliphatic compounds in JP-4 fuel, Burris and MacIntyre, 1987. i Approximate values. Actual values determined for each compound based on wind speed, reaeration rate, and physical properties of individual compounds. b

higher aqueous concentrations. Consequently, the highest aqueous concentrations resulting from this spill are not of the most soluble compounds, but rather of some of the least soluble and least volatile compounds (e.g., 1-methylnaphthalene and 1,4-dimethylnaphthalene). Figure 20.4 shows the mass of each compound in the slick as a function of time. Figure 20.5 shows the mole fraction of each compound in the slick plotted versus time. For simplicity, this simulation was run using only one slick. The composition of the slick changes drastically with time. Notice that the evaporation and dissolution flux rates of compounds from the slick, as inferred from the slopes of the lines in Figs. 20.4 and 20.5, increase as a compound’s mole fraction in the slick increases. This behavior exemplifies the fact that the dissolution and evaporation rates are governed not only by environmental parameters but also by slick composition. Since the slick composition is constantly changing over time, the concentration of compounds in the slick must be modeled in conjunction with the aqueous concentrations.

20.7

MODELING OIL SPILLS ON RIVER SYSTEMS 1

ethylbenzene 1-methylnaphthalene

location of slick 0.8

aqueous concentration (ppm)

t = 40 hrs

0.6

0.4

t = 60 hrs

0.2 t = 20 t =10 hr

0 0

10

20

30

40 distance (km)

50

60

70

80

FIGURE 20.2 Results of jet fuel spill simulation: aqueous concentration profiles for ethylbenzene and 1-methylnaphthalene.

1

0.8 toluene ethylbenzene n-butylbenzene peak aqueous concentration (ppm)

tetralin 1-methylnaphthtalene 1,4-dimethylnaphthalene 0.6

methylcyclohexane aliphatics

0.4

0.2

0 0

10

20

30

40

50

60

70

80

90

time (hrs)

FIGURE 20.3 Results of jet fuel spill simulation: peak aqueous concentrations of compounds.

CHAPTER TWENTY 10000

mass in slick (kg)

1000

100 toluene ethylbenzene n-butylbenzene tetralin 1-methylnaphthalene 1,4-dimethylnaphthalene methylcyclohexane aliphatics

10

1 0

10

20

30

40

50

60

70

80

90

time (hrs)

FIGURE 20.4 Results of jet fuel spill simulation: mass of compounds in slick.

As shown in Fig. 20.5, the composition of the slick, and therefore the nature of the slick, can change drastically over the duration of the spill as the more volatile, more soluable, and lower molecular weight compounds evaporate and dissolve from the slick. Most notably, the specific gravity of the slick increases over time. At 30 hours, the specific gravity of the slick reaches 1.0 and the slick should break up into globules dispersed in the water column or

1

0.1

mole fraction (--)

20.8

toluene ethylbenzene n-butylbenzene tetralin 1-methylnaphthalene 1,4-dimethylnaphthalene methylcyclohexane 0.01

aliphatics

0.001 0

10

20

30

40

50

60

70

80

time (hrs)

FIGURE 20.5 Results of jet fuel spill simulation: mole fraction of compounds in slick.

90

MODELING OIL SPILLS ON RIVER SYSTEMS

20.9

sink to the bottom. Even if the slick breaks up, however, the aqueous concentration of the compounds remaining in the oil phase will continue to rise, though perhaps not precisely as shown in Fig. 20.3. If the slick degrades to small globules suspended in the water column, the aqueous concentrations of the compounds in the oil phase will increase more rapidly than predicted in Fig. 20.3 due to the increased surface area of the globules compared to the surface slick. If the globules sink, they will not be drifting downstream suspended in the water column, and therefore the peak concentrations will likely be lower than those predicted by Fig. 20.3 which assumes that the slick is drifting at roughly the same velocity as the river surface. The viscosity of the slick also increases over time, leading to an increase in resistance to transfer in the oil phase. Additionally, the surface tension and the activity coefficients of the compounds in the slick will change over time, altering the spreading rate and the slick–water and slick–air equilibrium partitioning of the compounds. Thus, the slick composition can affect the physical characteristics of the slick and, consequently, the aqueous concentrations resulting from the spill.

20.4

SENSITIVITY ANALYSIS Using the hypothetical spill of 10,000 kg of a JP-4 jet fuel, the uncertainty associated with each model parameter is estimated and the sensitivity of the predicted aqueous concentrations to variations in the input parameters over the estimated range of uncertainty is presented. As a summary of the main findings, estimates of the uncertainties in the user-supplied parameters and the resulting variations in the highest aqueous concentrations are given in Table 20.2.

20.4.1

Longitudinal Dispersion

Longitudinal dispersion accounts for the dilution of the cross-sectional average concentration of compounds dissolved in the water due to mixing in the streamwise direction. The longitudinal dispersion coefficient, DL, can be estimated as (Fischer et al., 1979): DL ⫽

0.011 U 2W 2 h 兹g h S

(20.10)

where h ⫽ the average depth of river g ⫽ acceleration due to gravity S ⫽ the slope of the water surface

TABLE 20.2 Summary of Sensitivity Analysis. Confidence Interval of Input Parameters and Resulting Range of Maximum Aqueous Concentrations are Relative to the Original Values

User-supplied parameter DL kevap kvol kdis C sat w dl / dt

Approximate uncertainty of input parameter Ⳳ Ⳳ Ⳳ Ⳳ Ⳳ Ⳳ

Factor Factor Factor Factor Factor Factor

of of of of of of

4 4 3 20 2 2

Resulting range of maximum aqueous concentration Ⳳ30% Ⳳ50% Ⳳ5% Ⳳ2000% Ⳳ200% Ⳳ10%

20.10

CHAPTER TWENTY

Fischer et al. (1979) list numerous predictive equations for longitudinal dispersion, and perhaps not inconsequentially, estimates of DL from stream parameters are notoriously inaccurate. Fischer et al. state that the recommended relationship, i.e., Eq. (20.10), will generally predict DL only within a factor of 4. Figure 20.6 shows the highest aqueous concentrations of four compounds that occur during the duration of the JP-4 jet fuel spill simulation, using the same parameters listed in Table 20.1. The highest aqueous concentrations are plotted as a function of DL over the range of uncertainty of the predicted value (i.e., from approximately 25 to 400% of the original value of DL ⫽ 25 m2 / s). Higher values of DL tend to dilute the dissolved compounds in the streamwise direction, yielding lower maximum aqueous concentrations. Lower values of DL correspond to less dilution, and therefore higher aqueous concentrations. The maximum aqueous concentration of both the soluble / volatile compounds (toluene and ethylbenzene) and the relatively insoluble / nonvolatile compounds (1-methylnaphthalene and 1,4-dimethylnaphthalene) vary by approximately 30% from the high end value of DL to the low end. Therefore, longitudinal dispersion is a fairly significant process, and whenever possible, the value of DL should be measured by performing a dye study on a river reach (Fischer et al., 1979). Dye study measurements can also be used to precisely determine the spatial mean value of U (Kilpatrick and Wilson, 1989). Evaporation Rate

Evaporation is generally described by a resistance in series model, commonly used to describe mass-transfer between phases. The overall transfer coefficient depends on the rate transfer coefficients on either side of the interface: 1 kevap

⫽

Po Mi 1 ⫹ koa R T ␳o kG

(20.11)

where koa is the oil-film transfer coefficient at the oil-air interface, kG is the gas-film transfer 1.00E+00

8.00E-01

highest aqueous concentration (ppm)

20.4.2

6.00E-01

4.00E-01 toluene ethylbenzene 1-methylnaphthalene 1,4-dimethylnaphthalene

2.00E-01

0.00E+00 0

100

200

300

400

500

% of original value

FIGURE 20.6 Sensitivity analysis: effect of longitudinal dispersion on maximum aqueous concentration.

MODELING OIL SPILLS ON RIVER SYSTEMS

20.11

coefficient, and ␳o is the density of the oil. For thick oil layers, the oil-film transport coefficient controls the evaporation of hydrocarbons due to the relatively high vapor pressures (Thibodeaux and Carver, 1997). However, in most riverine spills, the oil layer is so thin (⬍1 mm) that the resistance to transfer in the oil film is small, even if transport is assumed to be only by molecular diffusion, and kevap can be approximated by kG (Mackay and Yeun, 1983). The gas-side mass transfer coefficient for compound i can be estimated from the wind function coefficient by Ryan et al. (1974), which relates evaporative heat flux to a vapor pressure difference: ko ⫽ ƒ(wz)

冉 冊 Sci Scwater

⫺0.67

(20.12)

where kG is the gas-side mass transfer coefficient for compound i (m / s), f (wz) is the wind function (m / s). The coefficient, f (wz), is a weak function of temperature but is nearly constant for the range of temperatures typically seen in rivers. Scwater is the Schmidt number of water in air (Scwater ⫽ ␯ / D, where ␯ is kinematic viscosity of air and D is diffusivity of water in air), and Sci is the Schmidt number of compound i in air. The ratio of Schmidt numbers in Eq. (20.12) converts the gas-side mass transfer coefficient for water vapor to an equivalent coefficient for compound i. The ⫺0.67 power on the ratio of diffusivities has been used by several investigators, including Mackay and Yeun (1983), Mackay and Matsugu (1973), and Goodwin et al. (1976), and is generally associated with a ‘‘dirty’’ or fixed surface. The function f (wz) is used to describe the influence of natural convection, wind, and waves on evaporation. The two known sets of field experiments to characterize the wind function in a sheltered stream resulted in the relationships of Gulliver and Stefan (1986) and Jobson and Keefer (1979). Gulliver and Stefan’s relation is used herein: f (wz) ⫽ [8.57 U9 ⫹ 14.7 (⌬␪)1 / 3] ⫻ 10⫺4

(20.13)

where U9 is the wind speed at a height of 9 m (m / s), and ⌬␪ is the virtual temperature difference (Ryan et al., 1974) between the water surface and the air at a height of 2 m (⬚C). Many existing spill models use a relationship for kG developed by Mackay and Matsugu (1973) from pan evaporation and wind tunnel experiments. Values of kG used in this analysis are based on measurements taken in sheltered streams and are typically two to five times lower than those for spills in open areas determined by Mackay and Matsugu. Regardless of the relationship used to predict the value of kG, there is likely a large uncertainty due to its dependence on the local air turbulence level. Under similar wind velocities at a height of 10 m, the wind velocity profile over open water may be significantly different than the velocity profile over a river due to the topography and the vegetation on the river banks. Even on open lakes, the measured evaporation transfer coefficient can vary as much as Ⳳ100% for a given wind speed (Adams et al., 1990). Thus, evaporation is a complicated and site-specific process. The confidence interval of the predicted evaporation-rate constant for rivers is estimated to be plus or minus a factor of 4. The confidence interval for the value of kG is likely smaller for wide rivers in areas without significant vegetation or changes in topography. Figure 20.7 shows the highest aqueous concentrations of four compounds resulting from a series of JP-4 jet fuel spill simulations over the anticipated range of uncertainty in kG. The maximum aqueous concentrations range from 130 to 60% of the original value as the value of kG is varied from 50 to 400% of the original value. The value of kG indirectly affects the aqueous concentrations by forcing changes in the composition of the slick. At low values of kG, the compounds evaporate more slowly and remain in the slick for a longer time. In the slick, compounds continue to dissolve, increasing the aqueous concentration. At high values of kG, the compounds evaporate more quickly and are available to dissolve for less time, creating lower aqueous concentrations.

20.12

CHAPTER TWENTY sensitivity analysis effect of kevap

1.40E+00

1.20E+00

toluene ethylbenzene highest aqueous concentration (ppm)

1.00E+00

1-methylnaphthalene 1,4-dimethylnaphthalene

8.00E-01

6.00E-01

4.00E-01

2.00E-01

0.00E+00 0

100

200

300

400

500

% of original value

FIGURE 20.7 Sensitivity analysis: effect of variations in evaporation rate on maximum aqueous concentration.

20.4.3

Volatilization Rate

Volatilization from the water column (rather than from the slick) is also generally described by a two-resistance model: 1 1 RT ⫽ ⫹ kvol kL H kG

(20.14)

where kL is the liquid-side mass transfer coefficient for compound i and kG is determined from Eq. (20.12). Values of kL used in the model are adapted from the stream reaeration coefficient of Cadwallader and McDonnell (1969): kL oxygen ⫽ 2.15 ⫻ 10⫺3 (SU )0.5

(20.15)

where kL oxygen ⫽ the liquid mass transfer coefficient for oxygen at 20⬚C (m / s) S ⫽ the average slope of river U ⫽ the average streamwise velocity (m / s) Several empirical relationships for reaeration coefficients were recently reviewed and tested by Moog and Jirka (1995), who found that the form of the relationship given in Eq. (20.15) best characterized stream reaeration rates. A liquid mass transfer coefficient for each compound, kL i, can then be determined from a ratio of the Schmidt numbers (Mackay and Yven, 1983): kLi ⫽ kL oxygen

冉

冊

Sci Scoxygen

⫺0.5

(20.16)

where Scoxygen ⫽ ␯20 / Doxygen, Sci⫽ ␯ / Di, ␯ is the kinematic viscosity of water at the modeled temperature, ␯20 is the kinematic viscosity of water at 20⬚C, Doxygen is the diffusivity of oxygen in water, and Di is the diffusivity of component i in water.

20.13

MODELING OIL SPILLS ON RIVER SYSTEMS

Estimates of volatilization rates from stream parameters are also notoriously inaccurate. Moog and Jirka (1995) found that even the best empirical relationships have a 95% confidence interval of greater than a factor of 3. Values of kvol for a given river reach can be measured by conducting a gas tracer study when greater accuracy is desired (Kilpatrick et al., 1989; Hibbs et al., 1998). Figure 20.8 shows the highest aqueous concentrations resulting from a series of JP-4 jet fuel spill simulations over the anticipated range of uncertainty in the value of kvol. The highest aqueous concentrations of all the compounds are essentially unaffected by variations in the value of kvol from 50 to 300% of its original value. The insensitivity of the aqueous concentrations to variations in kvol can be explained by examining the mass balance of the compounds dissolved in the water. From Eq. (20.7), the aqueous concentration of a compound is a function of both the dissolution flux rate and the volatilization flux rate. Even though the value of kvol is typically of the same order as kdis or larger, the concentration difference driving the mass transfer is usually much greater for dissolution than it is for volatilization. Due to the extremely low concentrations of most compounds dissolved in the water after a spill, the volatilization flux is typically very small, and the aqueous concentrations are fairly insensitive to variations in the volatilization rate.

Dissolution Rate

Like evaporation and volatilization, dissolution is also commonly described by a tworesistance model: 1 1 C sat Mo ⫽ ⫹ w kdis kw ko ␳o

(20.17)

where kw is the water film transfer coefficient, ko is the oil (slick) film transfer coefficient,

sensitivity analysis effect of kvol

9.00E-01

8.00E-01

7.00E-01

toluene ethylbenzene 1-methylnaphthalene

highest aqueous concentration (ppm)

20.4.4

1,4-dimethylnaphthalene

6.00E-01

5.00E-01

4.00E-01

3.00E-01

2.00E-01

1.00E-01

0.00E+00 0

100

200

300

400

500

% of original value

FIGURE 20.8 Sensitivity analysis: effect of variations in volatilization rate on maximum aqueous concentration.

20.14

CHAPTER TWENTY

␳o is the density of the slick, and Mo is the average molecular weight of the slick. In most instances, the resistance to transfer in the oil film can be ignored due to the small value of C sat w for most hydrocarbons, such that kdis ⬇ kw. Values of ko and kw can be estimated from reaeration rates. Cohen et al. (1980, 1978) measured dissolution rates of phenol from a surface slick and volatilization rates of benzene from the water to the atmosphere in a wind tunnel at wind speeds of 0 to 9 m / s. The measured water-film volatilization rate coefficients of benzene, kL benzene, were found to be roughly 8 times the water-film dissolution rate coefficients for phenol, kw phenol, at all but the highest wind speeds when it was thought that the oil slick had a significant dampening effect on the turbulence at the oil-water interface:

kL benzene ⫽ 8 kw phenol

(20.18)

Since the molecular diffusivities of phenol and benzene in water are similar, the rate coefficients for phenol and benzene are nearly interchangeable: kL benzene ⫽ 8 kw benzene

(20.19)

Values of kw can then be estimated from values of kL oxygen by accounting for the differences in molecular diffusivities between benzene and oxygen using Eq. (20.16): kw benzene ⫽

冉

冊

kL oxygen Scbenzene 8 Scoxygen

⫺0.5

(20.20)

More generally, the water-film dissolution rate coefficient for any compound, i, can be estimated from the reaeration rate coefficient, again by correcting for the differences in molecular diffusivities: kwi ⫽

冉

冊

⫺0.5

kL oxygen Sci 8 Scoxygen

(20.21)

Cohen et al. (1980, 1978) also found the measured water-film volatilization rate coefficients of benzene to be roughly 20 times the measured water-film dissolution rate coefficients for phenol. Similarly, the oil-film dissolution rate coefficient for any compound, i, can be estimated as: koi ⫽

冉

冊

⫺0.5

kL oxygen Sci 20 Scoxygen

(20.22)

Values of kL oxygen can be obtained indirectly from stream parameters using Eq. (20.13), or more directly from volatilization measurements by conducting a gas tracer study (Kilpatrick et al., 1989; Hibbs et al., 1998). Hibbs and Gulliver (1999) recently found that when the turbulence responsible for mass transfer at the water surface is generated from the bottom, such as in a stream or in a stirred reactor, the near-surface turbulence that dominates the mass transfer process on the water side of the interface is essentially unaffected by the presence and / or the properties of an oil slick. Thus, the water-film dissolution rate coefficient can be estimated directly from the air– water liquid–film coefficient, corrected by a ratio of Schmidt number to the ⫺1⁄2 power. Kwi ⫽ KL oxygen

冉

冊

Sci Scoxygen

⫺0.5

(20.23)

Equation (20.23) was also verified with the stirred beaker data of Southworth et al. (1983). The laboratory results of Southworth et al. (1983) and Hibbs and Gulliver (1999), without wind, are much different than the wind-influenced results of Cohen et al. (1978), and thus

20.15

MODELING OIL SPILLS ON RIVER SYSTEMS 3.00E+00

highest aqueous concentration (ppm)

2.50E+00

2.00E+00

toluene ethylbenzene 1-methylnaphthalene

1.50E+00

1,4-dimethylnaphthalene

1.00E+00

5.00E-01

0.00E+00 0

100

200

300

400

500

% of original value

FIGURE 20.9 Sensitivity analysis: effect of variations in dissolution rate on maximum aqueous concentration.

there is no definitive relationship to use in the estimation of kdis. Additionally, there are no known field studies from which to estimate the uncertainty in the value of kdis on rivers. Many existing oil spill models assume constant values for the film coefficients, even though Lamont and Scott (1970) have shown that the values of the individual film coefficients depend on the level of turbulence on either side of the oil–water interface. Shen and Yapa (1988) used kdis ⫽ 1.0 cm / hr for all applications in their riverine oil spill model. Herbes and Yeh (1985) used values of ko ⫽ 0.5 cm / hr and kw ⫽ 2.0 cm / hr for navigable rivers, which for most compounds gives kdis 艑 2.0 cm / hr. The relationships used in this analysis are based on laboratory experiments relating kdis to kvol, but the dependence of kdis on kvol varies by more than a factor of 8, depending on the importance of the wind. As previously mentioned, predictive relationships for reaeration coefficients are usually only accurate to within a factor of 3. Therefore, the overall uncertainty in the estimation of kdis is a factor of 3 times a factor of 8, or roughly a whopping factor of 20 to 30. When wind is not a factor in the reaeration coefficient, the overall uncertainty of plus or minus a factor of 3 is more appropriate. Figure 20.9 shows the highest aqueous concentration resulting from a series of JP-4 jet fuel spill simulations with the value of kdis varied from 50 to 300% of its original value. The slopes of the lines in Fig. 20.9 are nearly 1:1. Thus, over the estimated range of uncertainty in the value of kdis, the maximum aqueous concentrations can be expected to range from roughly 5 to 2000% of their original values, or over several orders of magnitude. This presents a serious problem for spill modelers. 20.4.5

Saturation Concentration

Measured values of the aqueous solubility of pure compounds are available in the literature for most hydrocarbons of interest in the environment (Schwarzenbach et al., 1993). However, solubilities of compounds not available in the literature must be estimated. Depending on the compound, empirical relationships based on molecular structure can predict the aqueous

20.16

CHAPTER TWENTY 2.50E+00

highest aqueous concentration (ppm)

2.00E+00

1.50E+00

toluene ethylbenzene 1-methylnaphthalene 1,4-dimethylnaphthalene 1.00E+00

5.00E-01

0.00E+00 0

100

200

300

400

% of original value

FIGURE 20.10 Sensitivity analysis: effect of variations in saturation concentration on maximum aqueous concentration.

solubility within Ⳳ25% to Ⳳ300% (Lymann et al., 1990). Additionally, for mixtures of structurally dissimilar compounds, Burris and MacIntyre (1987) have shown that the aqueous solubilities can be 1.5 to 2.5 times higher due to increased activities in the hydrocarbon phase. In this analysis, the uncertainty in C sat w is assumed to be plus or minus a factor of 2. Figure 20.10 shows the highest aqueous concentrations resulting from a series of JP-4 jet fuel spill simulations over the anticipated range of uncertainty in the values of C sat w . The slopes of the lines in Fig. 20.10 are nearly 1:1. As the values of C sat w are varied from 50 to 200% of the original values, the resulting maximum aqueous concentrations range from roughly 50 to 200% of their original values. Thus, the aqueous concentrations of all compounds are extremely sensitive to variations in the saturation concentration.

20.4.6

Slick Spreading Velocity

Oils spread across a water surface due to differences in density and surface tension between the slick and the water. Spreading is resisted by inertial and viscous forces. Except during the initial minutes following an extremely large spill, the extent of the slick due to spreading can be determined by the larger of an expression equating the surface tension and viscous forces (Fay, 1971): l ⫽ k1t

冉 冊 ␴ 2t 3 ␳2␯

0.25

(20.24)

or an expression equating the gravitational and viscous forces (Fay, 1971): l ⫽ k1␯

冉

冊

⌬ g V 2 t1.5 ␯ 0.5

0.25

(20.25)

MODELING OIL SPILLS ON RIVER SYSTEMS

20.17

where l ⫽ the length of slick resulting from spreading in one direction k1t ⫽ the spreading law coefficient for surface tension spreading (k1t ⫽ 1.33 [Garrett and Barger, 1970]) kl␯ ⫽ the spreading law coefficient for viscous spreading (k1␯ ⫽ 1.5 [Fay, 1971]) ␴ ⫽ the net surface tension (␴ ⫽ ␴air–oil ⫹ ␴oil–water ⫺ ␴air–water) ␳ ⫽ the density of water, ␯ is the kinematic viscosity of water ⌬ ⫽ the ratio of density difference between water and oil to density of water g ⫽ gravity, t is time V ⫽ the volume of slick per unit length normal to the direction of spreading (per unit width of the river) Equations (20.24) and (20.25) were developed for spills of constant volume, constant surface tension, and low viscosity on calm water. The effects of wind and currents on spreading rates are not well studied and are difficult to estimate. Therefore, the quantifiable uncertainty in the spreading rate lies in the estimation of the parameters used in Eqs. (20.24) and (20.25). The transition from a viscous spread, i.e., Eq. (20.25) to a surface tension spread, i.e., Eq. (20.23) occurs rapidly for most spills, and the spreading rate is described by Eq. (20.24). Since the density and viscosity of water can be estimated fairly confidently, most of the uncertainty in the spreading rate lies in the estimation of the net surface tension, specifically in the estimation of the air–oil surface tension and the oil–water surface tension. There is also an uncertainty in the applications of the slick-spreading model to a crosssectional nonuniform velocity profile, where the nonuniformities would add to the spreading. In this case, the slick would experience a longitudinal dispersion in addition to the water. This phenomenon is not a component of the sensitivity analysis. There are two sources of uncertainty in estimating an interfacial surface tension for an oil slick. First, there can be a substantial error in estimating the initial value of the surface tension. For most organic compounds, the air–oil surface tension can be predicted within Ⳳ20% and the oil–water surface tension can be predicted within Ⳳ50% (Lyman et al., 1990). Second, the surface tension of a slick changes over time as the volatile and more soluble compounds selectively evaporate and dissolve from the slick. However, the range of surface tensions for most organic compounds is fairly narrow. Values for the air–oil surface tension for most organics range from approximately 0.02 to 0.04 N / m, and for oil–water surface tension from approximately 0.020 to 0.035 N / m (Lyman et al., 1990). This indicates that the uncertainty in the interfacial surface tension, and also in the overall spreading rate, is roughly a factor of 2. Figure 20.11 shows the highest aqueous concentrations resulting from a series of JP-4 jet fuel spill simulations over the anticipated range of uncertainty in the values of the spreading rate. Over the range of spreading rates simulated, the highest aqueous concentrations of all compounds are fairly insensitive to the spreading rates. Variations in the spreading rate ranging from 50 to 200% of its original value produce changes in the maximum aqueous concentration ranging from 90 to 110% of their original values. Higher spreading rates create a larger slick and more surface area for both dissolution and evaporation. A large rate of evaporation would tend to produce lower aqueous concentrations since the slick would be on the water surface for a shorter time. However, a larger dissolution rate would tend to produce higher aqueous concentrations. It is believed that effect of slick size on dissolution and evaporation counteract each other, producing little net change in the highest aqueous concentration due to variations in the spreading rate.

20.5

SPATIAL VARIATIONS IN SLICK COMPOSITION The preceding application was modeled using only one slick, and thus it was assumed that the concentration of compounds within the slick was uniform over the entire length of the

CHAPTER TWENTY 1.00E+00

8.00E-01 highest aqueous concentration (ppm)

20.18

6.00E-01 toluene ethylbenzene 1-methylnaphthalene 1,4-dimethylnaphthalene

4.00E-01

2.00E-01

0.00E+00 0

50

100

150

200

250

% of original value

FIGURE 20.11 Sensitivity analysis: effect of variations in spreading rate on maximum aqueous concentration.

slick. However, for slicks that are several kilometers in length or that are being fed slowly by a point source such as a leaky pipe, the concentration of compounds within the slick may in fact vary with distance downstream. Since the aqueous concentration is controlled largely by the concentration within the overlying slick, the assumption of a uniform concentration over the entire length of the surface slick may introduce significant errors in the prediction of the aqueous concentration. Assuming that the evaporation rate coefficient, kevap, is constant over the entire length of the slick, a streamwise variation in the concentrations of compounds within the slick can be caused in only two ways. First, a variation in slick composition could be caused by spilling a mixture of compounds into one end of a slick while compounds rapidly and selectively evaporate and dissolve from the slick. In such a case, the composition of the slick near the spilling point would resemble the composition of the spilled product, while the portions of the slick farther downstream from the spilling point could contain less of the more volatile compounds. Second, a variation in slick composition may be caused by variations in the aqueous concentration of a compound beneath the slick. If the aqueous concentration beneath the slick varies significantly from one end of the slick to the other, the dissolution rate, which is driven by the difference between the actual water concentration and the equilibrium concentration in the water, would also vary from one end of the slick to the other. Over time, the variation in dissolution rate could create a variation in slick composition. To investigate the conditions under which the composition could vary from one end of the slick to the other due to rapid evaporation, consider the following analysis of a binary spill. Let compound 1 be volatile and semisoluble, and let compound 2 be perfectly nonvolatile and insoluble. The mass balance for compound 1 in the farthest downstream end of the slick can be simplified from Eqs. (20.5) and (20.6) by assuming that the evaporative flux is much greater than the dissolution flux, and by setting M1 ⫽ M2: dm1 m1 po1M1 ⫽ ⫺kevap dt m1 ⫹ m2 R T

(20.26)

where t is the time since the product was spilled onto the water surface. Integrating Eq.

MODELING OIL SPILLS ON RIVER SYSTEMS

20.19

(20.26) from t ⫽ 0 to ts and from m1 ⫽ mo1 to m1, the time to achieve a specified reduction in the initial, or the spilled, concentration of compound 1 in the slick can be estimated as: ts ⫽

冋

册

RT m (m1o ⫺ m1) ⫺ m2 ln o1 P M1 kevap m1 o 1

(20.27)

Thus, larger variations in the concentration of a compound within a slick (as indicated by m1 / mo1) will occur when the duration of the spilling period is long, i.e., ts is large, when one of the compounds evaporates quickly, i.e., Po1 ⫻ kG is large, or when the slick is thin, i.e., m1 and m2 are small. Inserting values of mo1 ⫽ 100 g / m2 and m2 ⫽ 900 g / m2, i.e., slick thickness approx. ⫽ 1 mm, kevap ⫽ 5 ⫻ 10⫺4 m / s, Po1 ⫽ 10⫺1 atm, and M1 ⫽ 100 g / mol into Eq. (20.27), the time to achieve a 10% reduction in the concentration of compound 1 in the downstream end of the slick is on the order of ts ⫽ 10 min. The second possible cause of streamwise variations in concentrations of compounds in the slick could be the variation in dissolution rates due to streamwise gradients in the aqueous concentration beneath the slick. Hibbs et al. (1999a), however, have shown that significant streamwise concentration gradients within the slick are not likely to be caused by concentration-driven variations in dissolution fluxes. Thus, spatial variations in slick composition are likely due to the rapid evaporation of volatile compounds from the slick.

Simulation of Short-Duration Spill

The results of a series of simulations of the jet fuel spill using the multiple slicks are consistent with the above analysis. The concentrations of the more volatile compounds in the downstream end of the slick decrease only slightly during the 10 minutes that the jet fuel is being spilled onto the water. Figure 20.12 shows the maximum aqueous concentration of four compounds resulting from a 10-minute spill plotted as a function of the number of

1 maximum aqueous concentration (ppm)

20.5.1

toluene ethylbenzene 1,4-dimethylnapthalene 1-methylnaphthalene

0.8

0.6

0.4

0.2

0 0

5

10

15

20

# slicks FIGURE 20.12 Multiple-slick model: effect of the number of slicks on the maximum aqueous concentration for a release of 10,000 kg of jet fuel over 10 min.

CHAPTER TWENTY

slicks used in the model. As more slicks are used in the model, larger concentration gradients develop in the slick and the maximum aqueous concentration decreases. The variations in the slick concentration that occur while the slick is being formed affect not only the maximum highest aqueous concentrations resulting from the spill, but also the location and time at which the maximum aqueous concentration occurs after a spill. Figure 20.13 shows the model simulation results for a spill of 10,000 kg of jet fuel over a period of 10 minutes. The peak aqueous concentrations of ethylbenzene and 1-methylnaphthalene are plotted against time for simulations using the single-slick model and a 10-slick model. Similar to what was shown in Fig. 20.12, the peak concentrations of both compounds are roughly 25% lower when 10 slicks were used than when only 1 slick was used in the simulation. Also, the highest aqueous concentration occurs roughly 25% earlier when 10 slicks are used in the simulation than when only 1 slick is used. The decrease in the maximum aqueous concentration and the shift in the time of occurrence of the maximum aqueous concentration can be attributed to the selective evaporation of the more volatile compounds over the length of the slick, during the period that the fuel is being spilled onto the water surface. We were surprised that the results of the single-slick model and the multi-slick model differed for a spill of such short duration. We thus investigated the mechanisms in some detail. Using the 10-slick model for this 10-minute spill, each individual slick has fuel added to it for 60 seconds. The slick then drifts downstream while the more volatile compounds rapidly evaporate, creating a concentration gradient in the overall slick. The effective length of the overall slick for dissolution of the volatile compounds becomes shorter as the more volatile compounds evaporate from the downstream end of the slick. At roughly six hours, the effective length of the slick has become so small that the dispersion in the river dilutes any further dissolution from the slick, and the peak aqueous concentration of ethylbenzene begins to decline. In the single-slick model, as the concentration of ethylbenzene is uniform throughout the slick, it has a long effective length for dissolution. The peak aqueous concentration is able to increase until roughly eight hours, after which the dispersion in the river overpowers the declining dissolution flux. 1

peak aqueous concentration (ppm)

20.20

ethylbenzene 1-methylnaphthalene

0.8

1 slick

0.6

10 slicks 0.4

0.2 1 slick 10 slicks 0 0

10

20

30

40 time (hrs)

50

60

70

80

FIGURE 20.13 Multiple-slick model: peak aqueous concentrations resulting from a release of 10,000 kg of jet fuel over 10 min.

20.21

MODELING OIL SPILLS ON RIVER SYSTEMS

While the effective length of the slick for the volatile compounds is decreasing, the effective length for dissolution of the nonvolatile compounds is increasing. The peak aqueous concentration of 1-methylnaphthalene in the 10-slick model shown in Fig. 20.13 rises slightly faster than the peak aqueous concentration of the single-slick model. The 1-methylnaphthalene dissolves and volatilizes faster from the effectively longer 10-slick model than from the single-slick model. The 1-methylnaphthalene is thus completely gone from the slick faster in the 10-slick model than in the single slick model, resulting in a lower maximum aqueous concentration. Simulation of Long-duration Spill

The same selective evaporation phenomena illustrated in Figs. 20.12 and 20.13 can be seen in a similar plot of peak aqueous concentrations resulting from a 10-hour release of 10,000 kg of jet fuel, shown in Figs. 20.14 and 20.15. Figure 20.14 shows the maximum aqueous concentration of four compounds plotted as a function of the number of slicks used in the model. As in the 10-minute spill simulation, the highest aqueous concentration of all compounds decreases as the number of slicks in the model increases. Figure 20.15 shows peak aqueous concentrations of ethylbenzene and 1-methylnaphthalene plotted against time for simulations using the single-slick model and a 10-slick model. In the 10-minute release shown in Fig. 20.13, each slick in the 10-slick model was filled with jet fuel for 1 minute before being released. In the 10-hour spill simulation shown in Fig. 20.15, each slick is also filled with jet fuel for 1 hour before being released. During that 1-hour time period, a significant portion of the more volatile compounds evaporates from the slick. By the time the slick is released to drift downstream, most of the volatile compounds have already evaporated from the slick. Any remaining dissolution of the more volatile compounds is diluted by dispersion. Consequently, the maximum aqueous concentration of ethylbenzene occurs very early after the spill. With the single-slick model, new jet fuel is uniformly distributed over the entire slick during the 10-hour spilling period.

0.5

maximum aqueous concentration (ppm)

20.5.2

toluene ethylbenzene 1,4-dimethylnapthalene 1-methylnaphthalene

0.4

0.3

0.2

0.1

0 0

2

4

6

8

10

12

14

16

18

20

# slicks

FIGURE 20.14 Multiple-slick model: effect of the number of slicks on the maximum aqueous concentration for a release of 10,000 kg of jet fuel over 10 h.

CHAPTER TWENTY 0.5

peak aqueous concentration (ppm)

20.22

ethylbenzene 1-methylnaphthalene

0.4

0.3 1 slick 0.2

10 slicks

0.1 1 slick 10 slicks 0 0

10

20

30

40 time (hrs)

50

60

70

80

FIGURE 20.15 Multiple-slick model: peak aqueous concentrations resulting from a release of 10,000 kg of jet fuel over 10 h.

Consequently, the peak aqueous concentration of ethylbenzene continues to rise for 10 hours, after which the peak concentration is diluted by dispersion in the river. The maximum aqueous concentrations shown in Figs. 20.12 and 20.14 appear to decrease logarithmically as a function of the number of slicks used in the model, suggesting that there is a diminishing rate of return on the quality of the model prediction with the increasing number of slicks used in the model. The decrease of the maximum aqueous concentration for each compound can be described by the empirical equation: dCmax ⫽ ␤(Cmax ⫺ C⬁) dn

(20.28)

where Cmax ⫽ the maximum aqueous for a given number of slicks n ⫽ the number of slicks used in the simulation, ␤ is a fitted constant C⬁ ⫽ a fitted constant that represents the maximum aqueous concentration for an infinite number of slicks Table 20.3 shows values of ␤ and C⬁ for four compounds determined from regressions of Eq. (20.28) against the data shown in Figs. 20.12 and 20.14. Table 20.3 also shows the maximum aqueous concentrations plotted in Figs. 20.12 and 20.14 as a percentage of C⬁. For the 10-minute release simulation, the concentrations of all four compounds were within 50% of C⬁ using only 1 slick, within 10% of C⬁ using 10 slicks, and within 1% of C⬁ when using 20 slicks. For the 10-hour release simulation, the maximum aqueous concentrations of the volatile compounds varied from nearly four times the value of C⬁ using only 1 slick to within 5% using 20 slicks, while the range of variation of the maximum aqueous concentrations of the nonvolatile compounds was similar to that of the 10-minute release simulation. Thus, the number of slicks used in the simulation can have a significant impact on the predicted aqueous concentrations, especially in models of slow, continuous spills containing highly volatile compounds.

MODELING OIL SPILLS ON RIVER SYSTEMS

20.23

TABLE 20.3 Extrapolation of Cmax to an Infinite Number of Slicks. Cmax is Determined from Spill Simulations. C⬁ and ␤ are Determined from a Curve Fit of Cmax Versus the Number of Slicks Using

Eq. (20.28). Cmax as a percentage of C⬁ (i.e., Cmax / C⬁ ⫻ 100)

10-minute release Toluene Ethylbenzene 1-Methylnaphthalene 1,4-Dimethylnaphthalene 10-hour release Toluene Ethylbenzene 1-Methylnaphthalene 1,4-Dimethylnaphthalene

20.6

C⬁ (ppm)

␤

1 slick

2 slicks

5 slicks

10 slicks

20 slicks

0.103 0.081 0.585 0.603

⫺0.193 ⫺0.195 ⫺0.194 ⫺0.213

142 140 138 136

133 130 128 127

120 119 118 116

108 107 108 107

101 101 100 100

0.012 0.015 0.297 0.314

⫺0.214 ⫺0.253 ⫺0.889 ⫺0.312

386 337 134 114

325 283 113 108

224 181 105 105

141 135 100 101

105 96 97 99

CONCLUSIONS Oil spill modeling needs to be performed before a spill occurs as part of a planning process for emergency response. This avoids the ‘‘chicken-with-its-head-cut-off’’ look that is often apparent in spill response teams. There will not be sufficient time following a spill to collect appropriate data and bring a model up to speed. Oil spill models are also valuable in a forensic investigation following an oil spill event. In rivers and streams, the largest impact of an oil spill is often the aqueous concentrations that result from dissolution of the slick. The water cannot leave the slick through mixing with a surrounding water body, and aqueous concentrations of toxic compounds in the slick continue to rise. The modeling team must understand the importance of accurate interfacial chemical transfer rates between the slick and the water. This chapter discussed the best physical characterizations of chemical transfer rates and performed a sensitivity analysis on the more significant parameters. Table 20.2 indicates that aqueous concentrations resulting from spills of hydrocarbons into rivers were most sensitive to the following, listed in descending order of significance: Dissolution rate coefficients Aqueous concentration in equilibrium with the oil slick Evaporation rate coefficient of the oil slick to the atmosphere Longitudinal dispersion coefficient Spreading rate of the slick Volatilization rate between the water and the atmosphere For spills that are longer in duration and cannot be simulated by a pulse, streamwise gradients caused by the rapid evaporation of the more volatile components can have a significant impact on aqueous concentrations. A multiple-slick model can account for these streamwise gradients in the slick. For any planning of an oil spill into rivers or streams, a field investigation needs to be undertaken. This investigation involves determining the travel time of the reach, the longitudinal dispersion coefficient of the reach, and the gas transfer coefficient of the reach. An

20.24

CHAPTER TWENTY

aqueous concentration model with acceptable results can then be formulated from the relationships and references provided in this chapter.

20.7

ACKNOWLEDGMENTS This model was developed with support from a Section 104 grant distributed by the University of Minnesota Water Resources Center, a doctoral dissertation fellowship from the University of Minnesota Graduate School, and the National Science Foundation under grant No. BES-9522171.

20.8

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