Roko Andričević

University of SplitCivil and Architectural Engineering Faculty

Ò A finite quantity M is released (naturally or accidently) into the environmental medium (air, water, soil) and one wants to know the contaminant concentration f ield at some location of interest

Ò The host medium and contaminant are misci ble and regarded as continua

Ò All variables will be assumed to have continuum scale resolution yielding concentration def ined as:

c = limv →L3

Mv

L is the length scale containing the volume v and M is the mass contained within v

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River Po discharging sediment plume into the Adri atic Sea

River Rio Plata discharging the plume into the Atl antic ocean

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Mixing in rivers; wastewater efluent discharged in the river from the point source

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Borden site 1983 (sandy aquifer), Ontario Canada

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Vertical cross sections through the wastewater (treated sewage) plume on Cape Cod, MA, showing the distribution of dissolved ammonium (NH4+ ) in micromoles per liter (µmol/L) during 1994.

Cape COD, MA, treated wastewater disposal beds, which created a large subsurface plume of contaminated ground water

Winter-time view of multilevel well sampling array. There are over 10,000 subsurface sampling ports

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Ò In environmental f low the concentration f ield is governed by the advective diffusive equation:

Ò where d is the molecular di ffusion coefficient and u is the turbulent velocity f ield governed by the Navie r-Stokes equations

Ò The incompressibility c ondition of fluids, , is assume d in nature

Ò This is fundamental description how any scalar quantity is transported through a velocit y field in environmental medium (air, water, subsurf ace)

∂c∂t

+ ∇ ⋅ uc( )= d∇2c

∇ ⋅ u = 0

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Ò Consider incompressible and steady groundwater flow taking placethrough a heterogeneous aquifer

Ò Groundwater and contaminants are flowing through the openings (empty space)

Ò In practice it is almost impossible to obtain measurements at smaller than Darcy scale – ”sampling averaging”

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REV or Darcy scale

Ò Mass conversation requires resident concentration to satisf y

Ò Where c(x,t) is the mass per aqui fer volume (of constant porosity) surrounding x at time t, u(x,t) is the groundwater velocity defined on Darcy scale and D=dI+ αu(x,t) is hydrodynamic dispersion, where α is a constant dispersivit y tensor, generally of fourth rank (81 components) !

Ò Thus, the above equation is up -scaled previous general advective diff usive equation at the Darcy scal e (or REV)

∂c(x, t)∂t

+ ∇ ⋅ uc(x, t)[ ]= ∇ ⋅ D∇c(x, t)

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Ò A common approach in accounting for local dispersion D is to replace velocity u(x,t) with the uniform ensemble mean U

Ò or to set D as a constant

Ò which is followed in this presentation

∂c(x, t)∂t

+ u ⋅ ∇c(x, t) = (d + αLU) ∂2

∂x 2 c(x, t) + (d + αTU) ∂ 2

∂y 2 c(x, t) +∂2

∂z2 c(x, t)

∂c(x, t)∂t

+ u ⋅ ∇c(x,t) = D∇2c(x, t)

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Ò In practice the sampling volume creates a strong difference in concentration fluctuations which results in great effort to describe transport theoretically

Ò The heterogeneity of porous media creates highly complex velocity field at pore, outcrop and regional scale yielding fluctuating velocities around the average

Ò The variations in velocity field create additional process called mechanical dispersion

Ò The tortuous and unpredictable pathways of groundwater flow resulting from geologic heterogeneity yield random concentration field which is common in all environmental flows; surface waters, atmosphere and groundwater

Ò Due to this randomness, a single realization of concentration field is meaningless and prevents one from using any deterministic approach in concentration prediction

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Ò To achieve a solution with any meaning repeated measurements (or time average f rom continuous record) must occur to approximate ensemble averages using ergodic theorem

Ò If one wished to solve Navier -Stokes or groundwater f low and advective diffusive equation numerically one would require more computing power than is currently a vailable

Ò The inherent di fficulties in attempting to solve advective diffusive equations are illustrated using a Raynolds decomposition

c = c + ′ c , ′ c = 0

u = u + ′ u , ′ u = 014

Ò Substituting decomposed velocity and concentration into mean and f luctuating part in advective diff usive equation one gets:

Ò A new additional term , , appears resulting in more unknowns than equati ons. By trying to find the ensemble of the square of concentrations yields:

Ò The term is present but more unknowns appear, creating a well kno wn ”non closure ” problem

∂c ∂t

+ ∇ ⋅ u c + ′ u ′ c ( )= d∇2c

∂ ′ c 2

∂t+ ∇ ⋅ u ′ c 2 + 2 ′ u ′ c c + ′ u ′ c 2( )= d∇2 ′ c 2 − 2d ∇ ′ c ( )2

′ u ′ c

′ u ′ c 15

Ò A natural approach to describe the evolution of a contaminant concentration field in a subsurface transport problem is to use the probability density function (pdf) defined with p(c;x,t)dc=prob(c≤c(x,t)≤c+dc)

Ò The concentration field is bounded between 0 and C0 which is the released concentration

Ò Since c is the continuous random variable, there is a non-negative pdf such that:

Ò where P is distribution functionÒ Since c is bounded above and below, p is ”truncated pdf”Ò only defined in the range (0,C0) such that

Pr c ≤ C*( )= p(c)dc = P(C*)0

C *

∫

p(c)dc = 10

C0∫ 16

Ò If the groundwater concentration above some threshold C* is considered unhealthy for drinking, then the probability of polluted groundwater is:

Ò From the pdf one can define an expectation, E, of f, where f is a continuous function of c, as

Ò This linear operator, as it names implies, provides the value off(c) that one expects to observe, on average, over an infinite set of repeated realizations

p(c)dc = P(C0C *

C0∫ ) − P(C*)

E( f ) = f (c)p(c)dc0

C0∫

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Ò An important set of expectations in statistics i s the set of absolute moments

Ò Setting n=1 the mean of concentration f ield isobtained which is relativel y insensitive to temporal and spatial experimental resolution and requires fewest realizations (or length of time series record)

Ò Each higher order moment is generally expe cted to require approximately an order of magnitude more reali zations (or longer record length f or each increasing order)

Ò In prediction of mean concentration the di ffusion term or local dispersion in the subsurf ace has negligible influence

mn = c n0

C0∫ p(c)dc

m1 = c(x,t)

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Ò The central moments of the pdf

Ò The central and absolute moments are related to each other

μ

µn = c − m1( )0

C0∫np(c)dc

µn =nk

k =0

n

∑ −1( )n −k mkµ1n −k

mn =nk

k =0

n

∑ −1( )n −k µkm1n −k

with m0=1

with μ0=1 and μ1=0

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Ò The skewness and kurtosis are standardized moments describing degree of asymmetry and peakedness or flatness

Ò If the Laplace transform of the pdf is f inite around the origin then the set of uniquel y determines the pdf

Ò Mean and variance are location and size pdf parameters while all higher moments are called shape parameters

S =µ3

µ23 2 K =

µ4

µ22

mi[ ]i=1

∞

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Ò The local dispersion term is the mechanism which enables a continuous range of values to occur in the concentration field

Ò Advection process works on the fast scale compared to the slow local dispersion process

Ò The balance is expected to occur at some scale/time during the transport

Ò Withouth local dispersion the advection pulls and conveys the contaminant into layers and fingers of higher permeability and if one takes the probe the measurements will be either 0 or C0

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Ò For zero local dispersion case and for uniform initial concentration c(0,0)=C0, the concentration field is a two-state process with pdf:

Ò where f(x,t) is the probability of being within the contaminant at point (x,t) and is related to the concentration mean

Ò The central moments are given as

Ò Although zero local dispersion is unrealistic in the environment, near source or at small times since release this approximation is reasonable, particularly for the subsurface where solid phase is further preventing local dispersion to develop compared to other environmental media

p(c;x,t) = 1− f (x, t)[ ]δ(c) + f (x,t)δ(c − C0)

f (x,t) =m1(x, t)

C0

µn =1

C0

m1(C0 − m1)n + −1( )n (C0 − m1)m1

n[ ]

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Ò By multiplying starting equation with cn(x,t) and taking the ensemble average, the concentration mass balance in terms of absolute moments can be written

Ò The local dispersion term is much smaller than the advective term and will be neglected

Ò Since the local dispersion acts very slowly compared to the heterogeneous advective velocity we estimate advective term for D=0

Ò The exact absolute moments for D=0 case are and when used above we estimate advective term as:

∂∂t

mn +1 + ∇ ⋅ ucn +1 = D∇2mn +1 − n(n +1)Dcn−1(∇c)2

D∇2mn +1

mn +1(x,t) = C0nm1(x,t)

∇ ⋅ ucn +1 = −C0n ∂

∂tm1

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Ò Using estimated advective term in the starting equation the concentration mass balance in terms of absolute moments becomes:

Ò Reduction of the release concentration C0 occurs only through D>0 such that advection and local dispersion balance each other out at some effective scale λc which at large time corresponds to the actual size of developed fingers or layers

∂∂t

mn +1 − C0n ∂c

∂t= −n(n +1)Dc n−1(∇c)2

λc24

Ò The geologic characteristics of the aquifer and its connectivity structure will affect the size of λc

Ò Using this scale parameter the concentration gradient can be approximated

Ò Where c*(t) is a background threshold concentration such that c*(t) 0 as t 0 and c*(t) as t

Ò Using results in a suitable form of the governing equation for the concentration absolute moments with n≥1:

Ò Where is dimensionless time, Pe=UI/D is the Peclet number, and is the scale factor between concentration scale and log-conductivity correlation scale

∇c(x, t) =c(x,t) − c*(t)

λc (t)→ →

→c * ( t ) = c ( x , t )

→ ∞

∂∂τ

mn +1 +n(n +1)

γ 2Pemn +1 − 2c mn + c 2mn−1[ ]= C0

n ∂c ∂τ

τ = tUI

γ = λcI

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c (x,t)

Ò This moment equation is a first-order linear homogeneous ordinary differential equation which has suitable solution in the recursive form:

Ò where k=n(n-1) and is dimensionless parameter that represents the ratio between the local dispersion and flow condition described with concentration scale λc

Ò For infinite Peclet number (e.g., zero local dispersion) a equals zero and above Eq. reduces to ex act expression for D=0 case

Ò The magnitude of a determines the strength of local dispersion process

∂∂τ

mn +1 +n(n +1)

γ 2Pemn +1 − 2c mn + c 2mn−1[ ]= C0

n ∂c ∂τ

mn +1(x, t) = C0nc (x, t) + kaexp(−kaτ ) c (x,ζ )

0

τ∫ ⋅ exp(kaζ ) ⋅

2mn (x,ζ ) − c (x,ζ )mn−1(x,ζ ) − C0n[ ]dζ

a = γ 2Pe( )−1= DI Uλc

2

mn +1(x, t) = C0nc (x, t)

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Ò Pulse injection over a finite volume (all distances scaled by log-K correlation scale

Ò Mean concentration assumed to follow the Gaussian distribution

Ò where , Yi is the particle displacement relative to the center of mass and Yii denotes the relative displacement variance

hx

h y

c (x,τ ) =MV

C0(τ )exp −Y1

2

2Y11

−Y2

2

2Y22

−Y3

2

2Y33

V0

∫V∫ dadb

C0(τ) = 2π( )3 / 2 Y11Y22Y33( )1/ 2[ ]−1

c (x, t)

dxdy

y

x

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Comparison between the Cape Cod field experimental and theoretical plume average st. deviation for several values of transversal local dispersivities

Comparison between the Borden Site field experimental and theoretical plume average st. deviation for several values of transversal local dispersivities

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Normalized concentration variance, third and fourth moment along the plume center line compared to upper limit for two levels of local dispersion strength

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Second, third and fourth concentration central moments in the absolute frame of reference as a plume passes two fixed control points CP(8,0) and CP(8,0.5) with local dispersion parameter a=0.05

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Ò Consider that at each time the contaminant concentration field at position c(x,t) has a position vector of the plume center denoted as

where

Ò At each time t, the absolute frame of reference is x=xr+R such that

R(t) = R1 = X1C ,R2 = X2

C ,R3 = X3C[ ]

X1C , X2

C , X3C( )= M−1 x1x2x3c(x,t)dx1dx2dx3ℜ3∫∫∫

c(xr,t) =c(x −R,t)

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Ò The absolute frame one-point concentration pdf p(c;x,t) can be related to the relative frame pdf pr(c;xr,t) through the spatial convolution

Ò where pR denotes the pdf of plume center positionÒ From above the relationship between concentration moments in

absolute frame of reference and relative moments is

Ò where are relative frame of reference concentration moments evaluated as presented earlier

p(c;x,t) = pr (c;xr,t)pR (x − x r;t)dx1rdx2

rdx3r

ℜ3∫∫∫

mns (x, t) = mn

r (x r,t)pR (x − x r;t)dx1rdx2

rdx3r

ℜ3∫∫∫mn

r (x, t)

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Numerical Example

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Pe=10000

Pe=100

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Numerički model zajedni čkog rada podmorski h ispusta na području Splitskog kanala

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Ò Simple, yet theoretical sound and consistent method of evaluating any number of concentration moments in absolute and relative frame of reference for a plume subject to heterogeneous velocity field in the subsurface is presented

Ò The local dispersion appears to have significant influence on the magnitude and shape of the concentration higher moments

Ò Applying the spatial convolution principle a simple relationship between the relative and absolute framework concentration moments is presented

Ò Both relative and absolute frameworks are providing crucial information needed in practice since concentration fluctuations observed at a fixed point in space is combined result of internal mixing and plume meandering

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Ò Deterministic ApproachÉ Risk Quotient Approach

ÐBased on point estimates of exposure and effectÐe.g. expected environmental concentration

LD50

“low” number is OK, “high” number is bad

UncertaintyUncertainty Natural VariationNatural Variation

How can we account for this?How can we account for this?

Ò Probabilistic ApproachÉ General Strategy

ÐUse information about variation and uncertainty in our estimates of exposure and effects to determine the probability that individuals or species will be impa cted.

ÐThat is, replace point estimates of exposure and effects with information about the distribution of possible exposures and possible effects

Area = probabilityArea = probability

Total Area = 1.0Total Area = 1.0

F(x) = Pr(x≤xi)

xi

xi

So, 40% of all of the possible So, 40% of all of the possible values for x lie below Xvalues for x lie below Xii..

XXii is the 40is the 40thth percentile for percentile for these data!these data!

xi

Prob = Pr(x≥xi)

So, 60% of all of the possible So, 60% of all of the possible values for x lie above Xvalues for x lie above Xii..

Pr (exceeding XPr (exceeding Xii) = .60) = .60

xi

Ò General Strategy

Concentration

Prob

abilit

y

Magnitude of Effect

Prob

abilit

y

RiskRisk

Ò Method 1É Derive distribution for exposureÉ Compare exposure distribution to a point estimate of toxicityÉ Risk = Pr (exceeding point estimate)

RiskRisk

xobsxobs-4 -2 0 2 4

0.0

0.1

0.2

0.3

0.4

ExposureDensity Function

Concentration

f(x)

ÒMethod 1

xobsxobs

Exposure Exceedence Function(EXF)

Concentration

-4 -2 0 2 4

0.0

0.2

0.4

0.6

0.8

1.0

Prob

abili

tyRiskRisk

ÒMethod 2É Derive distribution for exposure (ECD)É Derive distribution for toxicity (TD)É Compare distributions using convolutionÉ Risk = Pr (exposure) * Pr (toxic effect)

(i.e., overlap between distributions)

ÒMethod 2

-4 -2 0 2 4

0.0

0.2

0.4

0.6

0.8

1.0

Concentration

Prob

abili

ty

Exposure Concentration Distribution (ECD)Exposure Concentration Distribution (ECD)(as EXF)(as EXF)

TD TD (as PDF)(as PDF)

Risk = ECD * T DRisk = ECD * T D

R = f (Cexp )Ctox

∞

∫0

∞

∫ g(Ctox )dCexpdCtox

ÒMethod 2

-4 -2 0 2 4

0.0

0.2

0.4

0.6

0.8

1.0

Exceedence Function and TD

Concentration

Prob

abili

ty

Toxicity EffectToxicity Effect

ExposureExposure

ÒMethod 3

-4 -2 0 2 4

0.0

0.2

0.4

0.6

0.8

1.0

Exceedence Function and TD

Concentration

Prob

abili

ty

0.0 0.2 0.4 0.6 0.8 1.00.

00.

20.

40.

60.

81.

0

Joint Probability Curve

TD

EXF

ÒMethod 3

-4 -2 0 2 4

0.0

0.2

0.4

0.6

0.8

1.0

Exceedence Function and TD

Concentration

Prob

abili

ty

0.0 0.2 0.4 0.6 0.8 1.00.

00.

20.

40.

60.

81.

0

Joint Probability Curve

TD

EXF

(Per

cent

Exc

eedi

ng)

Increasin

g Risk

Increasin

g Risk

Increasing ConcentrationIncreasing Concentration

Ò What is a probabilistic risk approachÉ Attempt to move beyond characterizing risk in terms of a number (e.g.,

some quotient)É Characterize risk in terms of a distribution of possible outcomes (can

estimate the probability of effect)Ò Why?

É Allows us to incorporate both uncertainty and variabilityÉ More realistic assessment of risk

Ò How to do it?É Replace “exposure” and/or “toxicity” with probability distributions

(empirical or theoretical)É Details of how you do this vary greatly!