Roger Painter^{1*}, Tom Byl^{2}, Lonnie Sharpe^{1}, Valetta Watson^{1} and Tony Patterson^{1}  
^{1}Civil and Environmental Engineering, Tennessee State University, Nashville, TN 37209, USA  
^{2}United States Geological Survey, Nashville, TN 37211, USA  
Corresponding Author :  Roger Painter Civil and Environmental Engineering Tennessee State University Nashville, TN, 37209, USA Tel: (615)9635388 Fax: (615)9635902 Email: [email protected] 
Received December 27, 2012; Accepted June 22, 2012; Published June 25, 2012  
Citation: Painter R, Byl T, Sharpe L, Watson V, Patterson T (2012) A Residence Time Distribution Approach to Biodegradation in Fuel Impacted Karst Aquifers. J Civil Environ Eng 2:121. doi:10.4172/2165784X.1000121  
Copyright: © 2012 Painter R, et al. This is an openaccess article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. 
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It is widely perceived that karst groundwater often has insufficient residence time for significant biodegradation of contaminants to occur. It is perhaps due to these perceptions that less research has been conducted for quantitative modeling of biodegradation in karst as compared to consolidated aquifers. Modeling biodegradation in karst is in the domain of nonideal chemical reaction kinetics. The residence time distribution function (RTD) for tracer molecules in a single karst conduit or a complex system of conduits is a probability density function which can be interpreted to define the probability that contaminant molecules present at the influent at time equals zero will arrive at the effluent after a particular amount of time. To demonstrate this methodology the biodegradation rate of a contaminant (toluene) in raw karst groundwater from a BTEX impacted site in central Kentucky was quantitatively measured in batch microcosm studies and the extent of biodegradation of toluene in the same groundwater was measured for a complex flow system.The values of the pseudo first order rate constant (k’) obtained ranged from 0.017 (hr)1 to 0.0210 (hr)1 compared to 0.0186 (hr)1 for the microcosm experiments. The close agreement between the values of k’ obtained from the static microcosms and the ADE model indicate that the model adequately describes the RTD for modeling biodegradation in karst aquifers. The values of k’ obtained correspond to a halflife of less than two days for toluene and this has major
Keywords 
Bioremediation; Karst; Resident time distribution; Advection dispersion 
Introduction 
The high costs and technical difficulties associated with conventional remediation strategies make in situ biodegradation a promising approach for cleaning up contaminated aquifers [1]. The lack of studies examining biodegradation in karst aquifers may be due to the widespread perception that contaminants are rapidly flushed out of karst aquifers [2]. In highly developed and wellconnected conduit systems, the rate of contaminant migration is expected to be much faster than the rate of biodegradation [3,4]. However, the belief that contaminants are rapidly flushed out of karst aquifers is a popular misconception. Tracer studies suggest that large volumes of water may be trapped in fractures along bedding planes and vertical fractures adjacent to karst conduits [5]. In these areas isolated from the major conduit flow paths, contaminant migration may be slow enough that biodegradation could significantly reduce contaminant mass if respective microorganisms, carbon and energy sources, and geochemical conditions are present [68]. Researchers have also implied that natural bioremediation in karst or fractured rock is unlikely to occur because of the microbiological characteristics of karst aquifers; small microbial populations and low surfaceareatovolume ratio. Typical microbial numbers for material from unconsolidated aquifers have been reported to range from 1 × 10^{4} to 1 × 10^{7} cells per milliliter (cells/mL) [9]. Studies have shown that water from bedrock aquifers also may contain microbial populations within this range. For example, total microbial populations of 9.7 × 10^{5} to 8.5 × 10^{6} cells/mL and heterotrophic bacteria populations of 3.5 × 10^{3} to 5.0 × 105 cells/mL were detected in groundwater samples collected from a gasolinecontaminated karst aquifer in Missouri [10]. Greater than 70 percent of bacteria in consolidated aquifers are attached to solid surfaces. This fact may have led to the assumption that natural bioremediation in karst conduits is negligible because contact between attached bacteria and contaminants would be limited by the smaller surface area to volume ratio for karst [11]. 
The literature shows that efforts to mathematically model karst aquifers have followed two different and complementary approaches [1215]. 
1. A probabilistic approach that focuses on input events and output responses and takes into account local field observations of flow and transport processes. 
2. A determinist approach that takes into account theoretical concepts of aquifer structure, the physics of each mechanism involved in the transfer and flow of water, and from that tries to simulate the hydraulic behavior of the aquifer. 
There are four major conceptual approaches for deterministic models for karst groundwater: 
1. The equivalent porous medium (MODFLOW); 
2. The discrete fracture network; 
3. The double porosity continuum; 
4. The triple porosity approach – matrix, fracture and conduit. 
The U.S. Geological Survey (USGS) modular finitedifference groundwater flow model (MODFLOW) is a computer program that solves the groundwater flow equation. The groundwater flow equation is based on mass fluxes that are expressed in terms of hydraulic head according to Darcy’s Law. MODFLOW should be used with caution because the degree of geologic heterogeneity exhibited by karst aquifers precludes the application of Darcy’s Law and the use of average model inputs based on the assumed homogeneity of the subsurface [16]. Karst modeling is difficult due to the inability to adequately characterize the heterogeneity of the flow parameters, i.e. determine the geometry and hydraulic parameters of the highconductivity conduit network. The second problem is related to the development of a computer code based on numerical methods. Strong heterogeneity of the flow medium and different flow regimes in highconductivity channels and in the lowpermeability matrix make the development of appropriate computer modeling codes difficult. The third problem is the transfer of the simulated results to a real system. Because of the higher degree of heterogeneity, uncertainties are much higher for karst systems than for most porous aquifers [17,18]. Efforts are under way to develop more sophisticated deterministic models for karst and the triple porosity approach shows the most promise going forward since most karst aquifers have the common characteristics of storing most of the groundwater in the primary matrix and secondary fissure system and yet transmitting most of the groundwater through tertiary conduits. 
When considering the adequacy and appropriateness of a probabilistic or deterministic modeling approach, it is important to be clear about the objective of the model. Deterministic models work well if the objective is to model hydraulic heads, groundwater fluxes and spring discharge. On the other hand, existing deterministic models for karst aquifers perform poorly for the prediction of flow direction, velocity and solute destination. Consequently a probabilistic modeling approach is appropriate for modeling biodegradation of contaminants in karst aquifers. The release of a contaminant into a karst aquifer and the eventual discharge of the contaminant at a down gradient spring is analogous to the operation of a nonideal flow chemical reactor. Therefore a further justification for a probabilistic modeling approach is the fact that the state of the art chemical reaction engineering approach for modeling such a contaminant undergoing biodegradation is the application of a probabilistic biodegradation model based on an RTD derived from a field tracer study. Finally, another validation of tracer studies to provide information regarding karst flow [1921]. 
Tracer studies are an integral step in determining the RTD of contaminants in a karst system. Much work has been done to develop models for complex groundwater and surface water systems which attempt to describe the RTD in terms of mathematical functions independent of tracer study information. The numerical transient model (OTIS) and the Solute Transport and Multirate Transport – Linear Coordinates (STAMMTL) model are examples of such models [22]. However, a RTD based on tracer study data is often the single most important site specific information for modeling karst conduits. The RTD is not a complete description of a karst system but a given RTD is unique for a particular system. For large complex systems, representative tracer studies of particular conduits can be combined to provide a model of the entire complex karst system. The modeling of complex reaction systems as a number of simpler systems connected in series and parallel is an established method used for reactor design by chemical reaction engineers. In interpreting the results of a tracer study it is assumed that the contaminant experiences the same RTD as the conservative tracer except that the contaminant experiences biodegradation while in transient through the karst aquifer. 
USEPA’s QTRACER model is a probabilistic model that uses the one dimensional advection dispersion equation (ADE) to model the RTD for karst. In this context, it is a natural extension of chemical reaction engineering to use field tracer data to construct an RTD based on the ADE and couple the RTD to a chemical kinetic model to describe biodegradation in the karst aquifers. It is important to stress that the ADE is not being used here to model the physical characteristics of the aquifer flow. The physical characteristics of the aquifer are subsumed into the RTD and the ADE is modeling the RTD. The goal of this study was to investigate and access the ability of the one dimensional ADE to adequately model biodegradation in karst groundwater. The appropriateness of this approach is evident in the fit of the model RTD to the experimental RTD derived from the tracer study data. 
Materials and Methods 
Laboratory flow through karst microcosms shown in Figure 1 were constructed using a 20liter glass reservoir and twelve 1liter volumetric flasks arranged in three replicate systems of four flasks connected in series. Water maintained at a temperature of 2325 degrees Celsius was pumped into all three systems at a constant flow rate of 3.0 milliliters per minute by using a highperformance peristaltic pump. A stirred injection cell (10 mL volume) was placed at the entrance of each replicate system for the injection of tracer or toluene. During the conservative tracer studies Rhodamine dye was simultaneously injected into the stirred injection cells. The Rhodamine concentration at the discharge port was monitored through time by collecting samples at 12 hour time intervals over a 4 day period. A fluorometer was used to quantify the Rhodamine in the water samples. The lower detection limit on the fluorometer was established at 100 parts per trillion. 
Toluene was selected as the experimental contaminant because it is a component in most fuels and because previouswork indicated pseudomonad bacteria, which are heterotrophic aerobic bacteria (HAB), from the Kentucky site could grow using toluene as a food source. The biodegradation experiments used water containing live bacteria collected from a 120footdeep well completed in a karst aquifer in southcentral Kentucky. Bacteria counts ranged from approximately 7x10^{5} bacteria per milliliter to 1.2x10^{6} million depending on the well and sample collection time. These bacteria counts were derived using the BART growth test. Additionally, bacteria isolated from fuel contaminated groundwater samples readily grew with dissolved gasoline as the only source of food. Testtube (batchwise) microcosms set up using aerated raw karst water and spiked with Toluene at 1 mg/L established a pseudo first order biodegradation rate constant of 0.0186 hr^{−1}. Sterilized control microcosms containing distilled water had less than 10% toluene loss over the same time period. 
Before the tracer study was initiated, the experimental systems were sterilized with bleach. The bleach was neutralized with sterile sodium thiosulfate. During the conservative tracer study, a constant flow rate of approximately 3 milliliters per minute (mL/min) was established for both systems. The water flowing through the abiotic system was sterile water. At the beginning of the tracer study, 300 micrograms (μg) of Rhodamine dye (1,500 μg of 20% wt/wt solution) was injected into each stirred injection cell. The Rhodamine concentration at the discharge port was monitored through time by collecting 134 samples over a 4day period. 
Before the toluene biodegradation study was initiated toluene was injected into an abiotic system to investigate loss of toluene to vaporization and or adsorption. The experimental systems were sterilized with bleach. Filtersterilized toluene (87 138 μg) dissolved in 100 micro liters (μL) of methanol was delivered into the injection cell and pushed through the abiotic system with sterile water. Toluene concentration was monitored at the discharge port over the next 5 days. Water samples were collected in clean 40mL volatile organic compound (VOC) vials every 1 to 4 hours. Twenty ml alaquots were immediately analyzed on a gas chromatograph (GC) equipped with a purgeandtrap system, silicafilm capillary column, argoncarrying gas, and temperature program: 5°C/min from 50°C to 100°C, and microargon ionization detector. 
The karst groundwater used in the study was collected from a site located at an airfield in southern Kentucky. Ten gallon samples were collected from several contaminated monitoring wells that had varying concentrations of dissolved fuel the karst groundwater was then pumped through the system for four days to establish a biofilm on the glass surfaces. An 87μg aliquot of toluene was then injected into the biotic system in the same manner as for the abiotic test. In order to document the presence of attached bacteria, glass slides were suspended in the flow systems. The suspended slides were removed prior to and at the end of the experiments and viewed using an epifluorescent microscope and the direct count method [23]. Fields were examined for CTCstained bacteria as well as DAPIstained bacteria. Because CTC absorbs the fluorescent energy emitted by DAPI, total counts were obtained by adding the CTC and DAPI counts together. Figures 2 and 3 shown below are representative of the photographs of the glass surfaces. 
(1) 
where: 
C_{A} = Concentration of Contaminant; 
C_{B} = Concentration of Microbes or limiting electron acceptors; 
k = Biodegradation Rate Constant; 
and; 
t = Time 
But in a case where the microbes are acclimated and at steady state and electron acceptors are not limiting theconcentration C_{B} may not change appreciably while C_{A} does. Treating C_{B} like a constant: 
(2) 
k’ is the pseudo first order rate constant and the biodegradation is modeled as pseudofirst order. 
This does not however require that the mechanism of the microscopic chemical reaction be monomolecular. Biodegradation of organic contaminants in ground water is accomplished by biochemical reactions where one compound (electron donor) loses electrons and is oxidized and the other compound (electron acceptor) receives electrons and is reduced. If the organic contaminant is oxidized, some other compound must be reduced. The compound that is reduced (receives or gains electrons) is termed a Terminal Electron Acceptor (TEA). Oxygen, nitrate, sulfate, and ferric iron (Fe+3) minerals in the aquifer can serve as TEAs. The actual biodegradation rate is variable order and is a function of electron donor (carbon substrate) and microbial population. However, it is a common practice in chemical engineering to model complex reactions with pseudo first order kinetics. This is a practical concern as complex reaction mechanisms are routinely modeled with zero to second order equations. The first order decay rate has the added benefit of being linear but nonlinear decay rate can be used in the methodology below by resorting to a numerical solution in lieu of the exact solution presented here. The following application of chemical reaction engineering theory to is strictly valid if the chemical biodegradation kinetics information is linear with respect to intensive fluid properties. These assumptions facilitated the development of a simple mathematical model based on the ADE that does not address the physical chemistry mechanism for the reaction kinetics but addresses the practical question of the rate of decay of toluene in karst conditions. 
The residence time distribution function (E(t)) for contaminant molecules in a single karst conduit or a complex system of conduits is a probability density function (pdf) which can be interpreted to define the probability that contaminant molecules present at the influent at time equals zero will arrive at the effluent after a particular amount of time. The RTD is depicted as a plot of E(t) versus time as time goes from zero to infinity. 
The mathematical approach presented below involves coupling biodegradation kinetics to the RTD by application of the ADE. The transient form of the ADE was solved and fitted to the tracer study data to estimate the systems Peclet number. This Peclet number was incorporated into a second solution of the ADE with a chemical reaction term that was fitted to the toluene degradation data to estimate the pseudo first order rate constant. 
E(t) was determined by injecting a pulse of a conservative tracer (rhodamine) into the system shown in Figure 2 at time (t) = 0 and then measuring the tracer concentration in the effluent as a function of time. At some time after injection of the tracer, consider an increment of time dt small enough that the tracer concentration exiting the reactor between t and t + dt is essentially constant. The amount of tracer leaving the reactor between t and t + dt is then in differential form [24]. 
(3) 
where: 
Q = volumetric flow rate; 
C(t) = concentration at time t 
and; 
N = Mass 
Where dN denotes the amount of material that has spent an amount of time between t and t + dt and Q is the volumetric flow rate which is taken to be constant. If N_{0} is the amount of tracer injected at t = 0 then the fraction of material leaving the reactor between t and t + dt is 
(4) 
And for a pulse input the residence time distribution function is defined as 
(5) 
So that 
(6) 
Noting that N_{0} is the sum of all the tracer leaving the system 
(7) 
And combining Equations 3, 6 and 7 gives 
(8) 
E(t) is the pdf for the random variable C(t) and the first two moments of E(t), define the mean residence time and variance of the distribution. The mean residence time (tm) is: 
(9) 
And the variance (σ2) is:’ 
(10) 
For the nonideal flow characteristics of karst E(t) and its moments are obtained by numerical integration of the tracer breakthrough data 
The advection, dispersion equation for the conservative tracer is: 
(11) 
where 
Da = dispersion coefficient; 
z = distance 
and; 
U = velocity. 
Da is a measure of the degree of mixing in the fluid due to molecular and eddy diffusion as well an indication of the nature of the fluids velocity profile due to viscous effects. This equation effectively superimposes mass transfer of the tracer/contaminant due to axial onto the convective mass transfer. It may appear that this model is capable of accounting only for axial mixing but it can be shown that the model accounts for radial mixing as well. 
Equation 11 can be put in dimensionless form to facilitate its solution as: 
(12) 
where, 
After a narrow pulse of tracer is injected, molecular diffusion and eddy diffusion cause the pulse to widen as the tracer molecules disperse randomly in all directions. Consequently Da and hence Pe are directly related to the resulting mean residence time and variance of the RTD obtained from the tracer concentration versus time data. Consequently the solution of Equation 12 for Dankwerts boundary conditions for an openopen system gives the following relationship between the Peclet number (Pe) for the nonideal flow system, the mean residence time (tm), and the variance (σ2) [25,26]. 
(13) 
(14) 
where, 
and V is the volume of the flowing groundwater. 
(15) 
Values of σ2 and t_{m} from the tracer study are used to estimate P_{ep} and τ and these parameters are used to estimate the extent to which the toluene flowing through the system is biodegraded. The experimental value for the fraction of toluene biodegraded was obtained by numerically integrating the toluene concentration versus time data. The conversion as a function of time (X(t)) is: 
(16) 
X(t) is the fraction of the initial contaminant biodegraded after some time (t) and since the RTD function (E(t)) is the probability density function (p.d.f.) for C(t), the mean or expected value of X(t) is: 
(17) 
And noting that the rate of change of X(t) is independent of concentration and hence independent of the degree of mixing it follows that for first order kinetics: 
(18) 
And the mean conversion is obtained by combining equations 17 and 18 [27]. 
(19) 
(20) 
where; 
rA = volumetric reaction rate. 
This equation is linear for zero or first order kinetics and thus can be solved analytically. For higher order kinetics the equation must be solved numerically. As with the advection dispersion equation for the conservative tracer the equation for toluene undergoing degradation can be written in dimensionless terms as: 
(21) 
Where D is the Damkohler number defined as: 
and for first order reactions (or assuming the reaction can be modeled by pseudofirst order kinetics): 
Thus Equation 17 couples information from the RTD obtained from the tracer study to the chemical reaction kinetics term. In steady state mass transfer, the transient terms in the “OpenOpen System” boundary conditions disappear and reduce to the Dankwert’s “Closed Closed” Boundary Condition. Furthermore, this equation is linear for first order kinetics and thus can be solved analytically. The solution for assumed pseudo first order biodegradation kinetics is [28,29]. 
(22) 
where; 
Xm = the fraction of toluene biodegraded; 
and; 
Results 
The experimental values for the conversion (Xm) of toluene from the flow microcosms and the Peclet numbers from the conservative tracer studies allows for the calculation of the values for k’. The results from numerical integration of the concentration versus time data for the three tracer studies conducted are shown in Table 1. 
The toluene data shown in Figure 5 was numerically integrated to determine the experimental values for Xm and the values of k’ shown in Table 2 were calculated with Equation 22. 
To facilitate the comparison of the experimental RTD to the model RTD a dimensionless RTD function was calculated as: 
(23) 
The experimental dimensionless RTDs are compared to the model RTD in Figure 6. 
Discussion and Conclusions 
The values obtained for k’ correspond to a halflife of less than two days for toluene. This result has major implications for issues regarding the natural attenuation of fuelimpacted sites. This result indicates that the natural aerobic biodegradation of toluene will proceed to a significant degree in karst aquifers despite shorter residence times. This methodology is an adequate and appropriate approach for modeling biodegradation of contaminants traversing any karst system that is characterized by tracer study breakthrough data. The literature reports values of Pe ranging from 6.3 to 22 and in this study Pe ranged from 12.01 to 14.80. The appropriateness of Equation 12 for Pe in this range is directly verifiable by comparing the RTD obtained from the ADE to the experimental RTD from the tracer study data as shown in Figure 6. 
The scope of the results of this research is not limited to model development; it also addresses the fundamental issue of whether natural aerobic biodegradation occurs to a significant degree in karst [30]. The perception that contaminants are quickly flushed out of karst aquifers is largely incorrect [31]. Tracer studies for karst aquifers often indicate that these aquifers are characterized by diverse flow regimes and storage capabilities [32,33]. The range of flow velocities between the least permeable and the most permeable portions of the aquifer is typically over 6 to 10 orders of magnitude [34]. 
An RTD approach for modeling biodegradation in karst aquifers is validated by the wide use of quantitative field tracer tests to provide information regarding karst flow. It is a natural extension of chemical reaction engineering to use field tracer data to construct an RTD and to model biodegradation in the system. The biodegradation of toluene in groundwater from a karst aquifer in central Kentucky was adequately described by pseudo firstorder reaction kinetics. The values of k’ obtained from the model ranged from 0.017 (hr)^{1} to 0.0210 (hr)^{1}. The close agreement between the values of k’ obtained from the static microcosms and the ADE model indicate that the ADE does an adequate job of modeling the RTD with regard to the area beneath the RTD curve. This area is proportional to the mass of the contaminant in the effluent. 
Equations 14 and 15 often appear in the literature as equalities, in reality, the equations result from truncation of infinite series and are strictly valid only for large Pe. This is of practical importance since the whole concept of an apparent diffusivity in a dispersionlike equation breaks down for very low values of Pe where the flow approaches mixed flow. Mixed flow on the macroscopic scale is not achieved in typical karst. Fortunately Pe values for karst aquifers are relatively large due to their long effective path lengths. The Peclet number is a measure of the relative contribution ofmechanical dispersion and diffusion to solute transport. It relates the effectiveness of mass transport by advection to the effectiveness of mass transport by either dispersion or diffusion. Peclet numbers below 0 4 indicate diffusion control; 0.4 – 6.0 suggest that diffusion and advection are in transition and thus approximately equal to each other; and greater than 6 0 indicates 359 advection control. The Peclet numbers for karst are typically many times greater than 6.0. [35,36]. In this study, Pe 360 ranged from 12.01 to 14.80. 
Comparison of the experimental RTDs to the RTD from the ADE in Figure 6 reveal that the solution of the ADE tends to be more symmetric than typical tracer RTD and fails to reflect the long upper tail that is typical of tracer study data. Multi parameter models have been demonstrated that address this deficiency, however the simplicity of the ADE and its apparent relationship to the ADE of the physical system via the Peclet number are compelling reasons using this approach [37]. It is important to stress again that the ADE solution as presented here is not intended as a deterministic model of the karst system. 
Methanol was used to promote dissolution of toluene to allow the delivery of a true concentration impulse input to the system. This poses a concern as the presence of methanol as a energy source could augment biodegradation of toluene artificially. The close agreement of the reaction rate constants for the flow microcosm with the batch microcosms which did not incorporate methanol addresses this concern to some extent. However, this is a legitimate concern and future work may address this concern by conducting the experiment for a stepwise input instead of an impulse. This experimental approach would not require the injection of methanol. 
References 

Table 1  Table 2 
Figure 1  Figure 2  Figure 3 
Figure 4  Figure 5  Figure 6 
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