pah mobility in contaminated industrial soils: a markov chain approach to the spatial variability of...

Ž .Geoderma 102 2001 371–389www.elsevier.nlrlocatergeoderma

PAH mobility in contaminated industrial soils:a Markov chain approach to the spatial variability

of soil properties and PAH levels

H. Weigand a, K.U. Totsche a, B. Huwe b, I. Kogel-Knabner a,)¨a Lehrstuhl fur Bodenkunde, Technische UniÕersitat Munchen,¨ ¨ ¨

D-85350 Freising-Weihenstephan, Germanyb UniÕersitat Bayreuth, Abteilung Bodenphysik, 95440 Bayreuth, Germany¨

Received 23 February 2000; received in revised form 22 November 2000;accepted 1 February 2001

Abstract

The consideration of spatially variable contaminant sources and sinks is crucial for thequantification of contaminant transport in industrial soils. To assess the seepage of polycyclic

Ž .aromatic hydrocarbons PAH at a former manufactured gas plant, a combined approach was usedcomprised of a field survey, stochastic representation of site heterogeneity and numericalsimulation of contaminant mobility. Based on field and laboratory data, the vertical transitionprobabilities of soil materials and PAH-contamination classes were derived and a non-stationaryMarkov chain model of site heterogeneity was developed. The model was used to generaterepresentative soil profiles by stochastic simulation. Eighty profiles covered 61% of the spatialvariability of the site in terms of soil forming materials and PAH levels. Positions and thickness of

Žhorizons agreed with the field survey. The seepage of different PAH fluoranthene, pyrene,Ž . Ž . .benzo b fluoranthene, phenathrene, and benzo a pyrene was calculated by numerical simulation

using experimentally derived isotherm data. The cumulative output for the individual PAHcovered a range of three orders of magnitude, demonstrating the effect of site heterogeneity oncontaminant transport. While calculated local maximum concentrations exceeded the criticalvalues for potable water, the weighted average of PAH concentration in seepage water was low.Stochastic generation of soil profiles based on Markov chain theory provides a powerful tool forthe consideration of soil variability at contaminated industrial sites. Total profile probabilityrelates to the area fractions represented by each profile. Therefore, contaminant seepage may be

) Corresponding author. Tel.: q49-8161-71-5174; fax: q49-8161-71-4466.Ž .E-mail address: [email protected] I. Kogel-Knabner .¨

0016-7061r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved.Ž .PII: S0016-7061 01 00043-X

( )H. Weigand et al.rGeoderma 102 2001 371–389372

estimated without a costly three-dimensional deterministic representation of the field site. q 2001Elsevier Science B.V. All rights reserved.

Keywords: Contaminant transport; Soil; Industrial waste; Aromatic hydrocarbons; Markov chainanalysis; Stochastic processes

1. Introduction

Ž .Seepage of polycyclic aromatic hydrocarbons PAH in the unsaturated zoneŽ .of contaminated sites coking plants, manufactured gas plants, etc. may endan-

ger groundwater quality. Pore water concentrations are controlled by the rate ofŽ .contaminant release from source materials e.g., coal–tar, tar–oil , the sorptionŽto organic and inorganic soil constituents Murphy et al., 1992; Totsche et al.

.1997; Kogel-Knabner and Totsche, 1998 and by biochemical transformations¨Ž .Field et al., 1992; Richnow et al., 1998 . On a field scale, the interplay betweenthese processes and, hence, the degree of contaminant export from the unsatu-rated zone, depends on the spatial distribution of contaminant sources and sinks.At industrial sites, abrupt changes in soil profile may be observed due to the cut,

Žfill and dumping of natural and technogenic materials Blume, 1989; Burghardt,.1994 . As a result, soil physical and chemical properties as well as contaminant

Ž .levels exhibit high spatial variability Luthy et al., 1994; Wiesmann, 1994 . Theconsideration of site heterogeneity by means of structure-imitating techniques is

Žtherefore mandatory for reactive contaminant transport modeling Koltermann.and Gorelick, 1996 .

When the spatial distribution of soil physical and chemical properties issufficiently continuous, maps of site heterogeneity may be obtained through

Ž .geostatistical estimation techniques e.g., Kriging . Common to these approachesis the estimation of parameter values at non-sample points based on the

Ž .auto-covariance structure of neighboring sample points Davis, 1973 . Thisapproach relies on the assumption that with increasing sampling distance,parameter values become increasingly independent from one another. A morerigorous approach to the spatial correlation of data is followed in Markov chainmodeling. Markov chains allow to express partial sequential dependencies in

Žwhich the nature of one event is dependent on the preceding event in space or. Ž .time but independent of all events previous to that Doveton, 1971 . The

likeliness of adjacent events is controlled by transition probabilities. Markovchains have been used to model spatial variability in geologic applications like

Ž . Žturbidite succession Doveton and Skipper, 1974 , sedimentation analysis Rolke,. Ž .1991 , and hydrofacies architecture Carle et al., 1998 . Applications to vegeta-

tion dynamics, humification processes, and textural changes in soil profiles haveŽ . Ž .been reported by Balzer 1998 , Kappler and Ziechmann 1969 , and Li et al.

Ž .1999 . So far, Markov chains have been used to analyze transitions of a single

( )H. Weigand et al.rGeoderma 102 2001 371–389 373

Ž .categorical random variable. Assessing contaminant seepage in soils, however,requires the consideration of soil heterogeneity with respect to both contaminantsources and sinks. At industrial sites, these need not to be correlated. Therefore,an extended modeling approach is required.

The aim of this paper is to assess PAH seepage in the unsaturated zone of aformer manufactured gas plant based on a set of characteristic soil profileswhich contain both the information on contaminant levels and soil forming

Žsubstrates. The concentrations of 16 PAH EPA PAH, priority pollutants.according to the US Environmental Protection Agency EPA and soil physico-

chemical properties were recorded in soil cores at fixed intervals and thecorresponding transition probabilities were derived. Soil profiles were generatedby stochastic simulation constrained by transition probabilities. The export ofPAH from the individual profiles was studied with a deterministic model andweighted by profile probability to determine the overall PAH seepage at the site.

2. Theoretical background

2.1. Discrete-lag MarkoÕ chains

Markov chains describe a sequence X of n observations of a random variableŽ .Z Doveton, 1994; Tuckwell, 1995

Xs Z ,Z . . . ,Z , 1Ž . Ž .1 2 n

� 4in which Z may adopt k exhaustive and mutually exclusive states z , z , . . . , z .1 2 k

X is called a stationary first order Markov chain, when the following holds:For every n)0, the probability that Z adopts any of the states z , z , . . . znq1 1 2 k

is solely dependent on the previous state in Z . This is formalized by thenŽ .one-step transition probability pr of adjacent events Clarke and Disney, 1970i j

Pr z lz� 4i j<pr sPr Z sz Z sz s i , js1, . . . ,k . 2Ž .� 4i j nq1 i n j � 4Pr z j

The one-step transition probabilities may be derived from empirical transitionŽ .frequencies recorded at discrete intervals discrete-lag model . All possible

transitions from one state to another are summarized in the transition probabilityŽ .matrix for discrete-lag T Dh .

pr pr PPP pr1,1 1,2 1,k

pr pr PPP pr2,1 2,2 2,kT Dh s . 3Ž . Ž .P P PPP P

pr pr PPP prk ,1 k ,2 k ,k


The index reading is from row to column, i.e., row 1 denotes the probabilitiesof a one step transition from state 1 to any other state 1, . . . ,k. The sum of rowentries equals unity. The probability Pr of a sequence X is given as thetot

product of the involved one-step transition probabilities:

ny1

Pr s pr 4Ž .Łtot i , jk kq1ks1

with n: number of transitions, i, js1, . . . ,k.

2.2. Continuous-lag MarkoÕ chains

Recent applications of Markov chain theory have introduced continuous-lagŽ .modeling of spatial variability Carle and Fogg, 1997; Fogg et al., 1998 . The

continuous-lag approach extends the probability of state transitions recorded atŽ .fixed intervals discrete-lag to any desired interval by considering conditional

rates of change per unit length. Thus, the transitions between adjacent events areŽnot only dependent on the previous event but also on observation length e.g.,

.horizon or layer thickness . By reducing the diagonal entries of the transitionprobability matrix, continuous-lag Markov chains allow a better representationof bed thickness and thereby avoid a major shortcoming of the constant-lag

Ž .approach Rolke, 1991; Li et al., 1999 .Application of continuous-lag Markov chains requires the knowledge of

Žlag-dependent transition probabilities. Through Sylvester’s theorem Agterberg,.1974; Carle and Fogg, 1997 these can be derived from

k

T h sexp RPh s exp l h PS 5Ž . Ž . Ž . Ž .Ý l lls1

Ž .where T h denotes the transition probability matrix for any desired lag hŽ .continuous-lag , R is the transition rate matrix, l represent the 1, . . . ,kl

eigenvalues of the transition rate matrix and S are the corresponding spectrall

component matrices. The eigenvalues of R can be obtained from the eigenvaluesŽ . Ž .v Dh of the discrete-lag transition probability matrix T Dh byl

ln v DhŽ .Ž .ll s . 6Ž .l

Dh

Ž .Correspondingly, the spectral components may be obtained from T Dh by

v PJyT DhŽ .Ž .Ł ii/lS s 7Ž .l

v Dh yv DhŽ . Ž .Ž .Ł i ii/l

where J is the identity matrix.


After calculation of the eigenvalues and spectral component matrices accord-Ž . Ž . Ž .ing to Eqs. 6 and 7 and substitution into Eq. 5 , the continuous transition

Ž .probabilities T h for any desired lag h may be obtained. Instead of asingle-transition probability matrix characteristic of discrete-lag Markov chains,continuous-lag Markov chains are expressed by a matrix of graphs. Herein, eachgraph represents the transition probabilities from one state to another as afunction of observation length while the slope of the graph corresponds to therespective transition rate.

When Markov chains are used as a model of spatial variability in stochasticsimulation, knowledge of the lag-dependence of transition probabilities may be

Ž .made use of in two ways. i By randomly choosing the lag in which successiveŽ .layers are built-up and selecting the corresponding transition probability or ii

by keeping the lag fixed in each simulation step and successively adapting thetransition probability as a function of established layer thickness.

3. Field survey

3.1. Sample site

Soil samples were taken from the ATestfeld SudB, a former manufactured gas¨Ž .plant MGP in southwest Germany. Coal gasification facilities operated at the

site from the middle of the 19th century. Since the 1970s, the plant is used tostore and distribute natural gas. A survey of aquifer and groundwater samplesindicated pollution by typical MGP contaminants such as BTEX, PAH, phenols

Ž .and cyanide Herford et al., 1998 . The unsaturated zone of the site comprises a2–4 m thick disturbed layer. Soil forming materials of this layer include clayey

Ž . Žmarlstones AGipskeuperB, upper Triassic and loamy alluvial deposits eutric.fluvisol as natural soil forming substrates along with construction debris and

coal gas production remnants.

3.2. Sampling, analysis, and experimental

Soil sampling was performed by core extraction. A non-sealed area of4 2 Ž .1.4=10 m was covered using a regular grid design. Cores I.D. 6 cm were

taken at 25 locations selected at random from the ensemble of 125 plots andsampled horizon-wise. Undisturbed samples for the characterization of soil

Ž .hydraulic properties were taken in PVC liners I.D. 4 cm .The physico-chemical characterization of soil material included the quantifi-

Ž . Ž . Ž . Ž .cation of i OC, and Fe-, Al- hydr oxides, ii bulk and substance densities, iiiŽ .soil texture and iv soil hydraulic properties. OC was determined from the

difference of total and inorganic C before and after combustion and measured onŽ .an Elementar Vario EL CN analyser. Fe-, Al- hydr oxides were determined as


Ž .the oxalate Schwertmann, 1964 and dithionitercitraterbicarbonate-extractableŽ .fractions Mehra and Jackson, 1960 . Analyses of the extracts were carried out

with a UNICAM AA939 AAS. Bulk densities were determined gravimetricallyŽ .from oven-dry 1058C liner samples. The grain size fraction )2 mm was

obtained by sieving aliquots of the freeze-dried soil. Soil hydraulic propertiesŽ y y2were determined from the breakthrough of an inert tracer Cl , present as 10

.M KCl through the liner samples. The dispersion coefficient was obtained byŽinverse modeling of the breakthrough curve using the code CXTFIT Parker and

.van Genuchten, 1984 .ŽPAH contents 16 EPA PAH identified as priority pollutants by the US

Environmental Protection Agency: naphthalene, acenaphthylene, acenaphthene,Ž .fluorene, phenanthrene, anthracene, fluoranthene, pyrene, benzo a anthracene,

Ž . Ž . Ž .chrysene, benzo b fluoranthene, benzo k fluoranthene, benzo a pyrene, indeno-Ž . Ž . Ž . .1,2,3- cd -pyrene, dibenzo a,h anthracene, benzo ghi perylene were determined

in 5 g subsamples of the homogenized -2 mm grain size fraction. SamplesŽ .were extracted according to Hartmann 1996 . For quantification, a cocktail of

eight deuterated internal standards was used. The addition of a second internalŽ .standard perylene D12 prior to analyses was used to calculate mass losses of

the internal standards during the extraction procedure. For additional qualitycontrol, repeated extractions of standard soil samples were performed. A

w x ŽGC-MS Fisons Instruments MD 800 was used for PAH measurements temper-.ature program 100–1608C at 158Crmin, 160–3008C at 58Crmin . All analyses

were run in two replicates.PAH desorption isotherms were obtained from sequential batch experiments.

Three soil samples covering the range of PAH contamination and OC contentsof the site were extracted with an artificial soil solution at a soil solution ratio of1:10. The samples were shaken horizontally at 200 rpm for 24 h. Phases wereseparated by centrifugation at 3500 rpm and the solution volume was replaced

Table 1Material classes and selected physico-chemical properties

a b c dMaterial class )2 mm IC OC CrN Fe Al Fe Aloxalate oxalate DCB DCBy1 y1 y1 y1 y1 y1w x w x w x w x w x w x w x% g kg g kg g kg g kg g kg g kg

Organic 28 33 64 30.3 4.24 1.40 10.21 0.62Debris 43 26 20 21.8 3.48 0.93 9.86 0.47Gipskeuper 16 20 23 17.3 2.23 0.85 8.78 0.47Loamy alluvial 10 8 6 10.6 1.54 0.94 8.37 0.50deposit

aIC: inorganic carbon.bOC: organic carbon.cOxalate-soluble iron and aluminum.dDCB-extractable iron and aluminum.


Table 2Ranges and average contamination of PAH classes

PAH class Range S16EPA PAHy1 y1w x w xmg kg mg kg

Low 0–10 3.1Moderate 10–20 12.4High 20–55 32.0Very high )55 206.3

twice. Solution phase PAH concentrations were determined according to Lan-Ž .drum et al. 1984 . The solidrsolution partition coefficients were calculated and

normalized to the organic carbon contents of the soil samples.

3.3. Classification of soil forming materials and PAH contamination

Prior to the stochastic representation of site heterogeneity, soil properties andPAH contamination were classified and re-expressed as a set of categorical data.A Chi2-test showed, that the PAH levels at the ATestfeld SudB are statistically¨

Fig. 1. Functional representation of the continuous-lag transition probability matrix for classes ofsoil forming materials. Row titles and column titles denote the underlying and overlying materialclasses.


independent from the physico-chemical soil properties. Therefore, soil materialsand PAH levels were classified separately.

The classes of soil forming materials and a selection of corresponding soilproperties are shown in Table 1. To allow a fast recognition in the field, theclassification is based on morphological features such as color and technogenicrnatural components in the )2 mm grain size fraction. The identification of fourdifferent material classes is supported by the corresponding physicochemicalproperties. Analysis of variance revealed significant differences among theclasses with respect to the parameters grain size fraction )2 mm, IC, OC andCrN. The classes of PAH contamination are shown in Table 2. A constant PAH

Ž .pattern contribution of individual compounds to the overall contamination wasŽ .found for the unsaturated zone Weigand et al., 1998 . Thus, the contamination

classes were defined according to total PAH levels.

4. Representation of site heterogeneity

4.1. The MarkoÕ chain model

The first step in a stochastic representation of site heterogeneity is thedevelopment of a conditioning model capable of representing the field data

Ž . Ž .Fig. 2. Measured symbols versus calculated lines transition probabilities for two selectedsuccessions. 3_3: AGipskeuperB overlying AGipskeuperB; 3_1: AOrganicB overlying AGipskeuperB.Errors are given as the standard deviations of measured transition probabilities. The squaresrepresent the discrete-lag transition probabilities used to derive the continuous-lag Markov chainmodel.


structure. For a Markov chain approach to site heterogeneity, transition probabil-ities must be examined for stationarity. Stationarity provided, a discrete-lag

ŽMarkov chain model may suffice to condition the stochastic simulation Dove-.ton, 1994 . If, however, the transition probabilities are not constant in all

positions of an observed sequence a continuous-lag Markov-chain may bemore appropriate.

ŽExamination of the ATestfeld SudB data revealed depth dependent non-sta-¨.tionary transition probabilities for both material and contamination classes. For

example, the soil forming materials AGipskeuperB and AOrganicB were preferen-tially encountered in top soil and subsoil layers, while the contamination classAvery highB was exclusively recorded in subsoil layers. Thus, a continuous-lag

Ž .representation of transition probabilities was calculated by solving Eqs. 6 and

Ž .Fig. 3. Stochastic simulations of three soil horizons i, ii, iii including information on PAH levelsand soil forming materials. j and j denote uniformly distributed random numbers that areMAT PAH

Ž . Ž .projected onto the respective cumulative transition probabilities. The virtual row vectors of T hare given on top of the panels. The probability of simulated horizons was calculated according to

Ž .Eq. 11 .


Ž . Ž .7 . The discrete-lag transition probability matrix T Dh was derived from theobserved transitions at a fixed observation interval of 20 cm. Eigenvalues and

Ž . Ž .spectral components of T Dh were replaced into Eq. 5 to yield the transitionŽ .probabilities for continuous-lag. Transition probability matrices T h for contin-

uous-lag were calculated both for PAH-levels and soil properties.Ž .A graphic representation of T h for material classes is shown in Fig. 1.

Generally, with increasing lag, the probability of transitions from one state toitself decreases in favor of transitions to a different state. The slopes of thecontinuous-lag transition probabilities relate to the average thickness of materialclasses. Material classes developed as relatively thin layers have high probabilityof transitions to another material class and, hence, yield steep negative slopes in

Ž .Fig. 4. a,b Results of stochastic simulation constrained by transition probabilities. Pr denotestotŽ .the overall probability of soil profiles obtained by applying Eq. 4 to the transition probabilities of

individual horizons.


Ž .Fig. 4 continued .

Ž .the diagonal entries of T h . Fig. 1 shows that material classes 1 and 2Ž .AOrganicB and ADebrisB form relatively fine layers. Soil profiles are domi-

Ž .nated by material 3 AGipskeuperB which has both a high chance to overlayother materials and to form thick layers within the profile. As found during the

Ž .field survey, material 4 loamy alluvial deposit never rests on any othermaterial. Most frequently, it is buried by material 3.

To validate the Markov chain model, the calculated transition probabilitiesmay be compared to transition probabilities measured at varied lag. Fig. 2 showsan example for two successions of the AGipskeuperB. The calculated andmeasured lag-dependent transition probabilities show good agreement. The

Ž .squared markers highlight the discrete-lag transition probabilities 20 cm usedto derive the continuous-lag model. Perfect agreement with the calculatedtransition probabilities demonstrates that the continuous-lag model preserves theprobability structure of the training set. Error bars indicate that the measuredtransition probabilities become increasingly uncertain with increasing lag. Thisis due to the decreasing number of observed transitions. The measured transition


probability thus strongly depends on the position within the profile. When therange of measured transition probabilities is taken into account, the continuous-lag Markov chain model seems to adequately describe the transition probabilitystructure of the field data.

4.2. Stochastic simulation of soil profiles

Stochastic simulations were performed to generate synthetic soil profiles thataccount for both the spatial variability of contaminant sources and sinks at theATestfeld SudB. Soil profiles were generated from bottom to top. In agreement¨with the average prospection depth, the maximum profile thickness was set to100 cm. Material properties and PAH levels at the profile basis were definedaccording to the empirical distribution. Stochastic simulations were carried out

w xby stepwise generation of uniformly 0,1 distributed random numbers in 5000replicates. To account for statistical independence of soil properties and PAH-loads, this was done simultaneously for PAH levels and soil material classes.The random numbers were assigned a realization of material and contaminantclasses as outlined in Fig. 3. First, the row of the transition probability matrix

Ž .corresponding to the underlying previous realization of PAH contaminationand material class was chosen. Second, the cumulative transition probabilityfrom the underlying to the subsequent state was calculated from the sum of the

Ž .row entries in T h . Third, the random numbers were projected onto thecumulative probability and translated into a realization of material and contami-nant classes. This procedure was repeated until the pre-established profilethickness was reached. For each horizon, the linkage of material classes andPAH levels were performed a posteriori as outlined in Appendix B.

Ž .Results ordered by the total probability Pr of soil profiles are shown intot

Fig. 4a and b. The sum of Pr yields 0.61. Thus, with regard to PAH levels andtot

material classes the 80 profiles presented cover 61% of the site variability. Inagreement with the field survey, Gipskeuper and construction debris are thedominant topsoil materials, indicating that these materials were deposited re-

Ž .cently e.g., during construction activities . In contrast, the loamy alluvialdeposits as the natural soil forming substrate of the site as well as organic matterrich horizons are mainly encountered in subsurface horizons due to burying bythe deposited materials. When transitions in material classes occur, the averagetopsoil layer is 45 cm thick. This agrees with the typical thickness of soil layers

Ž .formed by caterpillar leveling of deposited materials Burghardt, 1994 andindicates that Markov chain models are suitable to describe the build-up ofanthropogenic soil profiles. The simulated depth profiles of PAH-levels showsimilar agreement with the field survey. The PAH-level Avery highB is encoun-tered mainly in subsoil horizons at depths between 60 and 100 cm. All othercontamination classes are not related to a specific depth. Overall, the PAH depthprofiles show a higher variability than the soil forming materials. Results of


Žseparately generated depth profiles for PAH and soil materials Weigand et al.,.1998 corroborate this finding by a more skewed distribution of contaminant

depth profiles.

5. Numerical simulation of PAH seepage

To assess the PAH export from the unsaturated zone of the ATestfeld SudB¨and to study the effect of site heterogeneity in terms of the distribution ofcontaminant sources and sinks, numerical simulations were performed based onthe synthetic soil profiles and the experimental desorption isotherm data. The

Ždeterministic reactive transport model CARRY Knabner et al., 1996; Totsche et.al., 1996 was used. This code allows to incorporate soil horizon information

Ž .bulk density fractions of sorbents, PAH levels into sorptionrdesorption scenar-ios. To reflect a range of compounds, the five dominant PAH of the field sitewere considered. The high affinity of the contaminants to the soil constituentsand thus their low mobility was accounted for by a simulation period of 50years. As experimental investigations showed no clear dependence of soilhydraulic properties from soil forming materials, a single dispersion coefficientwas used throughout. Simulation parameters are listed in Table 3.

Fig. 5 displays the histograms for the output of the individual compounds.The probability of each output class was calculated as the sum of total profile

Ž .probabilities Pr whose simulated output lay in the corresponding range. Astot

the Pr relate to the area fractions represented by each profile, the histogramstot

can directly be interpreted as the expectancy of PAH output from a given areafraction. For example, 24% of the area represented by the synthetic profiles islikely to have a cumulative fluoranthene output between 0.05 and 0.1 mg,whereas less than 1% is likely to have a cumulative output between 10 and 50mg. Total PAH output at the sample site may be calculated in the same way.

Table 3Parameters for the numerical simulation of PAH transport

Ž . Ž .PAH Fla Pyr B b Fla Phe B a Pyr

Fraction of S16 EPA PAH 0.179 0.143 0.127 0.078 0.075y1w xmg kg

3 y1w xlog K cm g C 5.42 5.48 5.26 4.98 5.34OC3 y3w xu cm cm 0.12y1w xq cm a 12

w xl cm 15

Ž . Ž . Ž .Fla: Fluoranthene; Pyr: pyrene; B b Fla: benzo b fluoranthene; Phe: phen-anthrene; B a Pyr:Ž . Ž .benzo a pyrene; K : OC-normalised partition coefficient experimental ; q: Darcy-velocity; l:OC

Ž .longitudinal dispersivity experimental .


Fig. 5. Export of PAH from the profiles presented in Fig. 3a and b. Results of the numericalsimulation. The probabilities were calculated from the sum of Pr of profiles with an output intot

the corresponding range. Total export was calculated as the sum of PAH output weighted by Pr .tot

For all compounds the range of PAH output covers four orders of magnitude,illustrating the effect of spatially variable PAH sources and sinks. The total

Ž .output of individual compounds decreases in the order phenanthrene Phe )Ž . Ž . Ž . Ž .benz b fluornathene B b Fla ) fluoranthene Fla ) pyrene Pyr ) benzo-

Ž . Ž Ž . .a pyrene B a P . In the simulations of PAH seepage, sorption and desorption


processes were expressed by means of a linear isotherm. Thus, output ofindividual compounds is proportional to the solid phase concentration and the

Ž .K Table 1 . Due to their lower K , a higher output was therefore calculatedOC OCŽ .for Phe and B b Fla in spite of a lower solid phase concentration relative to Fla.

With respect to the assessment of seepage water quality one may be moreinterested in solution phase concentrations than in total contaminant export. Inanalogy to the calculation of total output, simulated effluent concentrations wereweighted by Pr of the profiles. The weighted average concentrations summedtot

to 0.049 mg ly1 for the five PAH. Therefore, overall PAH export from theunsaturated zone is negligible in the light of a high aquifer contamination at thissite. Maximum concentrations obtained in profiles with low organic mattercontents and high PAH values exceed this value by three orders of magnitude.Thus, local effluent concentrations may be in the critical range.

Ž .Our modeling approach may be regarded as a worst-case scenario, because ithe isotherm data were obtained from batch-experiments which tend to overesti-

Ž .mate solution phase concentrations of PAH and ii PAH mass transfer is subjectŽ .to rate limitations Carmichael et al., 1997 . The latter may be of primary

importance in the presence of macropores and preferential flow paths, whichlead to a non-homogeneous flow field. Especially with regard to the calculatedmaximum effluent concentrations, results of further experimental work need to

Ž .be incorporated. Nevertheless, our modeling results highlight i the effect ofŽ .site heterogeneity on contaminant mobility and ii the need to consider spatial

variability at industrial sites in the assessment of contaminant mobility.

6. Conclusion

Stochastic simulations constrained by continuous-lag Markov chains providea powerful tool for the delineation of representative soil profiles at heteroge-neous industrial sites. When the vertical arrangement of both contaminantsources and sinks is considered, results can be directly coupled to the numericalsimulation of contaminant transport. Through their total probabilities, individualsoil profiles may be assigned area proportions. Thereby, weighting factors forthe corresponding simulated PAH outputs are provided. This renders Markovchain modeling an alternative approach to the assessment of contaminantseepage in soils without the need for a detailed, three-dimensional deterministicrepresentation of the site.

Appendix A. Linking soil properties and contaminant levels

Identity of the a priory and a posteriory approaches

Let P and M denote categorical variables defined on the sample space ofŽ . Ž .contaminant levels P and material properties M . Integration of independent


Ž .P and M into stochastic simulations may be achieved by: i an a prioriapproach which accounts for transitions between all possible combinations of M

Ž .and P formalized by a PqM=PqM transition probability matrix or ii an aposteriori approach which relies on the intersection of transition probabilities of

Ž .individual P and M. By applying Eq. 2 , the transition probabilities for P maybe expressed as

� 4Pr P lPnq1 nPpr s . 8Ž .i j � 4Pr Pn

Analogously, the transitions probabilities for M are given as

� 4Pr M lMnq1 nMpr s . 9Ž .i j � 4Pr Mn

When P and M are independent, the intersection can be calculated by theŽ .multiplication rule Ineichen, 1971 , i.e.,

� 4 � 4Pr P lP Pr M lMnq1 n nq1 nPl M P Mpr spr =pr s = . 10Ž .i j i j i j � 4 � 4Pr P Pr Mn n

Ž .Accordingly, applying Eq. 2 to the a priori intersection of P an M yieldstransition probabilities defined by

Pr P lM l P lM� 4Ž . Ž .nq1 nq1 n nPl Mpr s , 11Ž .i j � 4Pr P lMn n

which can be rearranged to yield

Pr P lP l M lM� 4Ž . Ž .nq1 n nq1 nPl Mpr s . 12Ž .i j � 4Pr P lMn n

Ž . Ž .Given the independence of P and M, the identity of Eqs. 10 and 12 isestablished.

Appendix B

List of symbolsP product operatorŽ . Ž .v Dh eigenvalues of T Dh

S sum operatorll l th eigenvalue of Rh Ž .lag observation lengthJ identity matrixk state number of random variable


Pr Ž .unconditional probabilitypri j Ž .transition probability from state j to state i conditional probabilitypr M

i j Žtransition probability from material state j to material state i condi-.tional probability

pr Pi j Žtransition probability from PAH state j to PAH state i conditional

.probabilityPrtot total probability of a random sequenceR transition rate matrixSl lth spectral component of RŽ .T Dh discrete-lag transition probability matrixŽ .T h continous-lag transition probability matrix

X random sequenceZ random variablez state of random variable

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