Environmental Fluid Mechanics - Chapter 10 doc

This will be seen in the present chapter, in which the classic tion–diffusion equation is derived to express a mass balance statement for adissolved chemical species distributed in a flui

of interest This will be seen in the present chapter, in which the classic tion–diffusion equation is derived to express a mass balance statement for adissolved chemical species distributed in a fluid flow field In later chaptersthis idea is expanded to include transport of suspended sediment particles,and several important classes of environmental flows are discussed Typicalparameters of interest might include concentrations of dissolved gases (partic-ularly oxygen), nutrients such as phosphorus or nitrogen, various chemicalcontaminants, both organic and inorganic, salinity, suspended solids, temper-ature, biological species, and others In order to fully describe the fate andtransport of a particular species, a knowledge of specific source and sink terms,including interactions with other species, must be incorporated in the generalconservation equation The present text, with several exceptions, generallydoes not deal directly with these terms, but rather concentrates on the physicaltransport mechanisms.

advec-10.1.1 Water, Heat, and Solute Transport

Earlier we saw that certain quantities control the rate at which different erties of a flow are transported, either by mean motions or by diffusion Forexample, kinematic viscosity may be thought of as a molecular diffusivity formomentum Thermal diffusivity represents a similar transport term for heatenergy, and solute diffusivity represents a corresponding transport mechanism

prop-for a dissolved species Mean motions generally carry all properties of a flow atthe same rate, but molecular diffusivities can vary widely For instance, kine-matic viscosity of water is about 10
2 cm2/s, thermal diffusivity is of order

10
3 cm2/s, and salt diffusivity is of order 10
5 cm2/s These differences areassociated with molecular activity, as shown below, and can be related byvalues of the Prandtl and Schmidt numbers,

the same rate (this is generally referred to as the Reynolds analogy — see

Chap 5).Of course, the net transport depends on mean gradients, as describedpreviously, but the diffusion coefficients or diffusivities are the same This

implies that the turbulent Prandtl and Schmidt numbers (defined similar to

Eqn 10.1.1, but using turbulent or eddy diffusivities) are both around 1, which

is a basic result of the Reynolds analogy

Another transport term of interest is that of dispersion Many authors

have used the terms diffusion and dispersion interchangeably, since the netresults of these processes are similar in causing spreading or mixing of materialfluid properties In fact, both terms are often represented mathematically in thesame way in the conservation equations However, dispersion arises from acompletely different process than diffusion, as described hereinafter The effect

of dispersion on transport of different properties of a flow is similar to that

of turbulent diffusion, in the sense that dispersion causes mixing of a fluidproperty about a mean position, and dispersion coefficients for momentum,heat and mass all tend to be similar

Various transport processes of interest may be summarized as follows:

Advection These motions are associated with mean flow or currents,

such as rivers, streams, or tidal motions They are normally driven by gravity

or pressure forces and are usually thought of as primarily horizontal motions

Convection This term usually refers to vertical motions induced by

hydrostatic instability, i.e., they are buoyancy driven Examples of this type

of motion include heating a pot of water on a stove, or fall and spring lakeoverturns occurring when the surface temperature on a lake passes through

4°C (temperature of maximum density)

Molecular diffusion Molecules of a fluid are naturally in random motion,

relative to other molecules (Brownian motion), and this leads to a mixing orspreading of fluid properties, consistent with the second law of thermody-namics

Turbulent diffusion This is a type of mixing similar to molecular

diffu-sion but with a much stronger effect Mixing in this case derives from thelarger scale movement of packets of fluid (rather than individual molecules)

by turbulent eddies

Shear Shear exists when there is a variation of advection (mean flow

velocity) at different locations in a flow field, so that a gradient exists forflow velocity This produces a variation in the rate of advective transport of afluid property, with associated spreading of the average concentration of thatproperty, as illustrated inFig 10.1

Dispersion This is spreading of a fluid property by the combined effects

of shear and transverse diffusion

The main distinction between advection (or convection) and diffusion ordispersion is that advection represents a net movement of the center of mass

of a packet of fluid or fluid property, while diffusion and dispersion represent

a spreading about the center of mass This is illustrated inFig 10.2 As will

be seen later, dispersion is normally included in the conservation equations in

a similar manner as diffusion, but the effects of dispersion are normally muchgreater than those of diffusion In this text the term mixing will refer to eitherdiffusion or dispersion

Figure 10.1 Effect of shear on spreading of mean (here, depth-averaged) concentration.

Figure 10.2 Illustration of advective and diffusive transport; advection moves the center of mass while diffusion spreads mass relative to the center, whether it is moving

or not.

The basic balance equation for dissolved mass may be derived followingprocedures similar to those used in the derivations for water mass, momentum,and energy balances fromChap 2.However, it will first be important to definethe various transport properties involved, especially diffusion, and this will

lead to a derivation of the classic advection–diffusion equation.

It will be important to keep in mind the definition of a flux, which is

the transport of a given property across a surface, per unit time and per unit

area of that surface, and the definition of mass concentration, or density, C

or
1 (see below), which is the amount of mass of dissolved solids per unit

volume of fluid In the following discussion, it is also helpful to define a pure concentration, CŁ, which is a dimensionless ratio of dissolved solid mass in

a given volume of fluid to total mass in that volume C is related to CŁby

CŁ1 and
1 turn out to be the same, since the density of water is typically

2 ¾D 1 gm/cm3D 106 mg/L Thus for chemical concentrations on the order

of 1 mg/L, both CŁ1 and
1 would have values of about 10
6 (1 mg/L or

1 ppm)

In addition, we will be concerned primarily with binary systems, i.e.,

water plus one other component (the extension to conditions with more thanone additional component is straightforward) Thus let
1 D mass of species

1 per unit volume of solution and
2D mass of species 2 per unit volume ofsolution The total density of the solution is
D
1C
2 Normally there is arelatively small amount of one species compared with the other, such as with

a pollutant dissolved in water If species 2 is water, then
2 ×
1 and
¾D
2

In other words, the addition of the pollutant, or tracer, does not significantlyaffect the water (solution) density The (nondimensional) concentration ofspecies 1 is

10.3.1 Molecular Diffusion, Fick’s Law

The basic form of diffusion is molecular diffusion, which is due to the random

motions all molecules undergo (Brownian motion) From considerations ofnonequilibrium thermodynamics, the simplest assumption about diffusion ofmass that is consistent with the second law of thermodynamics (increasingentropy), is that this diffusion is proportional to the gradient of the chemicalpotential of the system (c) This statement is analogous to Fourier’s law forheat conduction and can be written as

The main contribution to the flux is the middle term on the right-hand

side The first term is called the Soret effect and indicates the possible diffusion

of mass due to a temperature gradient This term may become important

under some circumstances, usually involving high salinity concentrations, but

is ignored in most applications Not much is known about the third term, but it

is assumed to be negligible Considering only the density gradient term, then,

where k1 is defined as the molecular diffusivity for species 1 This result,

extended to all three directions, becomes



F1D
k1



which is known as Fick’s law or Fickean diffusion.

The concept of diffusivity can also be derived using mixing length

theory, illustrated in Fig 10.3 for a simple one-dimensional stratification (zdirection) of species 1 On the left-hand plot is shown a general density profilefor species 1, while the right-hand plot is a magnified view of a small part ofthe profile The average velocity of molecules in the z-direction, due to inherentBrownian motion, is w1, and lm is the mixing length, which is assumed to berelated to the average distance the molecules travel before colliding with othermolecules (molecular free path) Considering a small “window” in the fluid,perpendicular to the z axis, the flux of species 1 through the window in thepositive z-direction is 
1w11, and in the negative z-direction it is

1w12,where the subscripts outside the parentheses indicate the levels at which theflux terms are evaluated If w1 is assumed constant, the net flux across thiswindow is

where the minus sign is used since (∂
1/∂z) is negative Substituting Eq (10.3.6)into Eq (10.3.5),

Fz 1 D
w1lm

∂
1

The product of mean molecular velocity and mean free path (w1lm)

is a property of the system and is given the symbol k1 and defined as the

diffusivity for species 1, similar to the term in Eq (10.3.3) Similar analyses

may be applied for the x and y directions, to arrive at the same result as before(Eq 10.3.4) In general, k1 is a function of temperature and density (
1), andpossibly pressure It is interesting to note that k1 is defined here as the product

of a characteristic length and a characteristic velocity scale — this idea also

is applied when discussing turbulent diffusion, though of course the velocityand length scales are different

10.3.2 Turbulent Diffusion

The concept of turbulent diffusive transport is analogous to the turbulent port of momentum, discussed inChap 5.In particular, the Reynolds transportterms are derived in exactly the same way as the Reynolds transport terms formomentum, (i.e., Eqs 5.4.7 and 5.4.8)

trans-Here, however, fluctuations of concentration are used instead of a secondvelocity component In other words, the turbulent or Reynolds transport ofdissolved mass in the i direction is

Ex, Ey, and Ez could all have different values (anisotropic) and they might

all be functions of (x, y, z) (inhomogeneous) Often turbulent diffusivities are

not very well known and must be estimated, unless direct measurements ofthe terms in Eq (10.3.8) are available Instrumentation must be capable ofmeasuring the fluctuating quantities making up the Reynolds transport term(left-hand side), and mean concentration gradient must be measured In manycases, the turbulent diffusivities are treated as fitting parameters, chosen tooptimize the results of a particular model, compared to observations

10.3.3 Statistical Theory of Diffusion

One example of the use of observations to estimate diffusive-type spreadingcharacteristics is illustrated by the use of dye release experiments in a naturalsystem This approach is based on the observation that average dye concen-tration distributions are often closely approximated by Gaussian profiles (i.e.,they are normally distributed) Figure 10.4shows a conceptual sketch of theouter extent of spreading of dye in the surface layer of a lake, following

a concentrated “instantaneous” release over a small area The dye patch isshown at three different times after this release For simplicity, mean velocity

is assumed to be in the x-direction.Figure 10.5shows corresponding tration profiles measured across the patch at one of these times The ensembleaverage profile is approximately normally distributed At earlier times it would

concen-be more peaked, and at later times it would concen-be more flattened In other words,the variance of the distribution increases with time

Assuming the Gaussian distribution is appropriate, the concentrationcan be described in terms of the variances in each of the three coordinate

Figure 10.4 Spreading of a dye patch at three times: t 3 > t2> t1.

Figure 10.5 Measurements of concentration across the patch at one time, with semble average.

en-directions as

CŁD Me
Kt

23/2

x y zexp

-
x
Ut2

2
y2

2
z22 z

The variances are measures of the degree of spreading along each of thecoordinate directions For example, variance in the x-direction is calculatedfrom

Having developed the basic transport terms, we are now in position to write

a conservation equation for dissolved mass of a tracer in an aqueous system.First, a total flux of dissolved mass of species 1 is defined as the sum of theadvective (Eq 10.2.2) and diffusive terms (Eq 10.3.4),

diffu-Sometimes it is convenient to define a representative velocity for species

1,q1, such that the total flux may be written as



Although this velocity cannot be measured directly, it is defined by equatingEqs (10.4.1) and (10.4.2) Ifq2is a representative velocity for species 2, then

the total momentum per unit volume is

F2D0 ), which is consistent with the idea that diffusion causesspreading about the center of mass and not net transport of the center of mass.The mass balance for species 1 may be formulated in several ways, butthe most direct way is to consider a differential fluid element and incorporatethe transport of mass of species 1 across the boundaries of the element Theapproach is similar to the development leading to the continuity equation(conservation of fluid mass — see Chap 2),except diffusive flux must also

be included here A general statement of dissolved mass conservation for thefluid element (volume 8) sketched inFig 10.6is

[rate of change of mass in 8 per unit time] D[rate at which mass moves across the boundaries by net flux]

š [rate at which mass is produced C or consumed 
 bychemical and biological reactions]

For simplicity, only the fluxes in the x-direction are shown inFig 10.6,but similar fluxes may be defined for the y- and z-directions Using a truncated

Figure 10.6 Fluid element, showing fluxes of species 1 in the x-direction.

Taylor series to evaluate differences in flux values across dx, dy, and dz, the

general statement above is expressed mathematically by

in the other species, so R1 D
R2 Adding Eqs (10.4.6) and (10.4.7),

of mass

A more common form of Eq (10.4.6) can be derived by substituting

Eq (10.4.1) for total flux and dividing by total density (assuming
Dconstant),

Ł 1

∂t C CŁ

1r ÐV CV ÐrCŁ

V D0 and, if k1D constant, Eq (10.4.9) becomes

∂y C w∂CŁ1

∂z

D k1r2CŁ1C R1

which is commonly known as the advection–diffusion equation for

incom-pressible, dilute (and laminar) flow As noted earlier, turbulent transport can

be incorporated by defining a turbulent diffusivity as appropriate If V D0and R1 D 0 (in which case the material is known as a conservative substance),

Eq (10.4.10) reduces to

∂CŁ1

This result is a simple diffusion equation and is known as Fick’s second law.

It is analogous to Fourier’s law of heat conduction

For some problems it is useful to apply the advection–diffusion equation

in cylindrical coordinates The development of this equation is not presentedhere, since it follows the same general procedure as described above, but thefinal result is

coor-10.4.1 Source and Sink Reaction Terms

Reactions leading to increases or decreases in species 1 are classified into twocategories, depending on whether those reactions occur uniformly throughoutthe volume or at specific locations within the system, usually at a boundary

The former are called homogeneous reactions and are usually incorporated in

the governing equation through the source/sink term R1 The latter are called

heterogeneous reactions; these are more appropriately included as boundary

conditions In some cases, usually depending on the number of physicaldimensions being modeled, reactions that might appear as homogeneous inone situation may appear as an internal or boundary source in another Forexample, gas transfer across an air/water interface is normally incorporated as

a boundary condition when gas concentrations in the vertical direction are ofinterest, such as in lakes In this case the advection–diffusion equation would

be solved explicitly for the vertical direction (horizontal directions might also

be modeled), and the air/water interface would represent a boundary alongthat direction However, when considering gas modeling in a river, a common

approach (seeChap 12)is to use a one-dimensional (longitudinal) model, inwhich case any flux across the air/water surface would be considered to beinstantaneously mixed over the entire depth In other words, the model wouldhave no capability for simulating a vertical distribution of concentration, andtherefore the air/water surface has no meaning as a boundary condition Inthis case, fluxes across the surface would be considered as an internal sourceterm for the model.

Internal (homogeneous) reactions may be specified in a number of ways,and a full description of all possibilities, for all potential parameters of interest,would require a separate text It is worthwhile to note here, at least, twocommon classes of reaction terms In general, reaction terms may be groupedaccording to the assumed dependence of the reaction on concentration ofthe chemical species of interest The most common reaction terms are eitherzero order or first order A zero-order source/sink term does not depend onconcentration at all but would be a constant (possibly time-varying) termadded to the right-hand side of Eq (10.4.10) An example of this type ofreaction is a municipal waste stream discharging into a river The loading

of a contaminant of interest to the river through this waste stream wouldnot depend on concentrations in the river, and would depend only on thecharacteristics of the discharge A first-order reaction is one that dependslinearly on concentration In this case, a first-order reaction rate K would bedefined so that

R1

where either plus (C) or minus (
) is used depending on whether concentration

is growing or decaying This form of reaction term is useful for many naturalprocesses and also allows analytical solutions for the advection–diffusionequation, as discussed further in Section 10.7

10.4.2 Boundary and Initial Conditions

Both boundary conditions and initial conditions are needed to obtain solutions

to any differential equation, and the advection–diffusion equation is no tion Boundary conditions apply to specific locations in the modeled physicaldomain, as noted above, and are usually specified in one of three ways (seealso Sec 4.6):

excep-1 Specify concentration (e.g., C D C0at x D 0), possibly dent; in combination with velocity, this gives advective flux

time-depen-2 Specify gradient (also possibly time-dependent), which, in nation with the diffusivity, gives diffusive flux — this is useful,

combi-for example, at impermeable surfaces, where velocities go to zero;zero gradient implies a “perfectly insulating surface” (using heatconduction analogy).

3 Specify total flux, as a (linear) combination of both diffusive andadvective fluxes

Boundary conditions play an important role in determining the behavior

of a particular solution, and care must be taken in specifying the correctconditions for any given problem This is particularly true when solutionsare desired near one of the boundaries of the system domain In some cases,where numerical solutions are applied, it is useful to define additional grids ornodes outside of the actual system being modeled, and to apply the boundaryconditions at the limits of these additional grids That way, the boundarycondition itself does not directly affect as strongly the solution at the point ofinterest This and other considerations for numerical solutions are discussed

in more detail in Sec 10.8

Initial conditions also are needed for time-dependent problems (i.e.,

∂C/∂t 6D0) These are usually specified by setting all values of C throughoutthe domain to known values for t D 0 A useful function for specifying initial

conditions, which allows analytical solutions, is the Dirac delta function, υ,

used to indicate an instantaneous input of mass at a point, along a line or across

a surface (corresponding to a three-dimensional, two-dimensional, or dimensional problem, respectively) For example, for a planar source parallel

one-to the y –z plane and passing through x D x1, the initial concentration can bedescribed by Cxi,0dx D Mυx
x1, where M D input mass per unit area,

dx is the thickness of the source and

υx
x1 D0 if x 6D x1υx
x1 D1 if x D x1

8

Application of this function to solve a simple example problem showshow it can be used For this example, consider an infinitely long cylinder filledwith quiescent water, as shown inFig 10.7.At time t D 0, an infinitely thincylinder of dye with mass M is introduced at x D 0 The problem is then tocalculate the concentration at any x and any t > 0 For this problem, note that

C is constant across any cross section, so there is no diffusion in the radial(y –z) plane The initial concentration distribution is shown inFig 10.8,alongwith several distributions at later times Note that C ! 1 at x D 0, since

a finite amount of mass (M) is injected into an infinitely small volume (if

dx !0)

Figure 10.7 Infinite cylinder with initial injection of dye at x D 0.

The full advection–diffusion equation is given by Eq (10.4.9) (the script 1 is dropped here and in most of the text to follow, since it will beunderstood that species 1 is being analyzed) Since the water is quiescent,

sub-u DvD w D 0, and CŁ is a function of x only The dye also is assumed

to be conservative (R D 0), so the governing equation is the simple diffusionequation (10.4.11) The initial condition (t D 0) states that CŁD 0 when x 6D 0and CŁ! 1 when x D 0 (i.e., υx D 0 would be used to specify the initialinjection of mass) The boundary conditions are CŁ! 0 when x is very large,i.e., x ! š1 The general solution is

CŁD pB

texp



x24kt

texp



x24kt



dx D
ApB

t 2

pkt

Note that M is independent of time, as it should be for a conservative dye.Rearranging this last result to solve for B, and substituting into Eq (10.4.15),the final solution is



10.4.18

Figure 10.8 Concentration distributions at three different times t 1 < t2.

This solution is sketched schematically inFig 10.8for two times, t > 0 Notethat the solution is in the form of a normal distribution, symmetric and centeredaround the dye center of mass at x D 0 This is consistent with the statisticalapproach to diffusion as discussed previously in Sec 10.3.3

character-diffusivities, for example Dispersion, however, results from spatial averaging.

In many ways, the idea of dispersion is analogous to that of turbulent diffusion.For instance, it would not be necessary to define turbulent closure schemes(Chap 5)if models were solved with time and spatial steps small enough toresolve directly the turbulent fluctuations In most applications this is imprac-tical, so that turbulent diffusivities, or eddy viscosities, in the case of themomentum equations, are defined to account for processes occurring on timescales much less than the model time step

Similarly, simplifications in spatial representation require that scale processes be represented in some way The most common application

smaller-of dispersion is when the spatial domain, particularly for the velocity field, issimplified by integrating (averaging) over one or more coordinate directions.For example, one-dimensional river models are by definition averaged overdepth and width, and thus require a longitudinal dispersion term to accountfor variations or processes occurring in those directions, which are not directly

included in the model formulation Dispersion is often included in such modelsusing a Fickean diffusion term, i.e., dispersive transport may be defined by

the product of a dispersion coefficient, or dispersivity, with a mean gradient.

This dispersion coefficient is in most cases much larger than the turbulentdiffusivity, so that diffusion terms are often neglected when dispersion isincluded In many applications dispersion is used as a sort of bulk adjustment

to the model, to account for any processes (known or unknown) which arenot directly represented In fact, the advection–diffusion equation might bemore aptly called the advection–dispersion equation in these cases As long asthe dispersion coefficient can be estimated, this usually allows for significantsimplifications in model approach and structure, by reducing the number ofspatial dimensions that must be considered

of x, t, and r D radial position, so CŁD CŁx, r, t The dependence on r

arises because the velocity is a function of r The governing equation for

this problem is a two-dimensional (radially symmetric) advection–diffusionequation, written in cylindrical coordinates as

∂CŁ

∂t C ur∂CŁ

∂x D 1r

Figure 10.9 Problem definition for Taylor analysis of dispersion in a tube.

Although the velocity distribution is not necessarily known, it is assumedthat the Reynolds analogy holds, and Er is the same as the eddy viscosity inthe radial direction, so

Err D


where  D shear stress, given by

 D 0r

and 0D wall shear stress and R D tube radius Further substituting uŁ D

0/
1/2D friction velocity, Eq (10.5.2) is rewritten as

Err D
r

R

u2Ł

Also, following Taylor, the longitudinal diffusion term is neglected (this should

be valid after a sufficient time has elapsed following the injection)

In principle, once u(r) and Err are specified, the originalequation (10.5.1) may be integrated directly to obtain a solution However,this information is not always available, and the integration may be difficult,possibly requiring a numerical solution A simpler alternative approach wassuggested by Taylor, involving use of a one-dimensional transport equation,

is the mean velocity, Q D flow rate, A D cross-sectional area, and EL is a

longitudinal dispersion coefficient Note that both CŁ and U are no longer

functions of r The solution to this problem is already known; it is the same

as Eq (10.3.9), rewritten here in terms of the present variables,



10.5.6where M is the original mass of dye injected Taylor showed that this solutionwas a good approximation to the exact solution, using

ELD 10.1RuŁ

10.5.7This value may be compared with the radial or longitudinal diffusivities,estimated from

Er ¾D E

x¾D 0.07RuŁ

10.5.8which are several orders of magnitude smaller

The different effects of diffusion and dispersion are illustrated inFig 10.10, for an instantaneous injection of dye at time t D 0 and x D 0

Figure 10.10 Effect of velocity shear on the spreading of dye, with a longitudinal dispersion approach: (a) spreading due to diffusion only; (b) spreading due to diffusion and velocity shear; and (c) resulting cross-sectional average concentration profiles, according to the dispersion model.

For purposes of demonstration, the dye is assumed to be conservative InFig 10.10aa uniform velocity is assumed, so that only longitudinal diffusionoccurs (because there are no gradients in the radial direction, radial diffusion

is not important) In this case the dye spreads symmetrically upstream anddownstream about the center of mass being advected downstream at velocity

U.Figure 10.10bshows a more realistic situation that accounts for the no-slip

conditions at the tube wall, although the mean velocity is unchanged Thevariations in velocity cause a stretching of the dye distribution, effectivelyspreading the concentration more quickly than by diffusion alone, as shown

inFig 10.10c.In this case, radial diffusion will occur because this stretchingcreates radial gradients For the longitudinal dispersion approach, the samevelocity field as in part (a) is assumed (i.e., uniform), but EL is defined toaccount for the effects of nonuniform velocity, so that a better fit with thecross-sectional average concentration (CŁ) is obtained, as seen inFig 10.10c.Although not shown here, this produces a response similar to that shown inpart (a), but with a faster spreading rate

10.5.2 Longitudinal Dispersion in Rivers

As previously noted, dispersion is commonly used when a model is simplified

in the number of spatial dimensions or coordinate directions that it includesexplicitly In general, since almost all real processes are three-dimensional innature, this implies that any model that does not solve for all three coordinatedirections should include a dispersion term to account for those processes notdirectly included In order to illustrate this procedure, and also to show how adispersion coefficient may be estimated, the problem of two-dimensional openchannel flow is considered here for modeling in a one-dimensional (longitu-dinal) framework Two-dimensional flow is a common assumption for openchannel modeling (see Chap 7)and in principle should include a dispersionterm to account for the fact that any processes occurring in the lateral direction(the y-direction, in the following discussion) are already averaged The focushere is on the further simplifying step of going to a one-dimensional model

A two-dimensional flow and concentration profile for a tracer, assumed

to be conservative for simplicity of discussion, are shown inFig 10.11,wherearbitrary distributions are assumed The important point to note is the decom-position of both velocity u and concentration c into mean and fluctuating

Figure 10.11 Two-dimensional velocity and concentration profiles, with raged values and spatial deviations shown.

depth-ave-components, i.e.,

(note that a lower case c is used to denote a local value, while the upper caseindicates an average, as defined below; also, the discussion here applies equally

to a dimensional and a nondimensional concentration) The decomposition in

Eq (10.5.9) is similar to the Reynolds decomposition discussed in Sec 5.3,except that the means and fluctuations here are defined over space instead oftime, so

U D 1hh 0

hh 0

and by definition,

h 0

u00dz D

h 0

In general, u and c also may be functions of x and/or time t For the present

discussion it is assumed that the profiles are already averaged over a timeperiod for which a dispersion coefficient value is desired Similar calcula-tions as shown below could be performed for other time periods or for otherlongitudinal positions, as needed

The governing equation for transport of a conservative tracer in dimensional flow is

Eq (10.5.9) into Eq (10.5.12),

right-hand side (RHS) of Eq (10.5.14) is the difference between the verticaltransport at the upper and lower boundaries and, as long as there is no loss

or gain of material at these boundaries (i.e., zero-flux boundary conditions),both of these transport rates are zero

The third term on the left-hand side (LHS) of Eq (10.5.14) is similar to

a Reynolds stress term It is rewritten as

u00∂c00

While it is possible to calculate terms such as u00c00directly, as long assufficient data are available, an alternate approach may be used that relies only

on velocity data and an estimate for Ez For this approach, the original dimensional equation, neglecting longitudinal diffusion, is used as a startingpoint,

The coordinate system is then transformed by considering a new coordinate

, which is moving with mean velocity U, so

This equation may be simplified for the case of steady state, also assuming

c00− C and noting that C is not a function of z,

u00∂C

∂ dz

0D z0

1

Ez z 0

1

Ez z 0

u00dz0 dz00C u00

c00 0

H) u00c00D 1

hh 0

u00c00dz D 1

hh 0



u00∂C

∂

z 0

1

Ez z 0

u00dz0 dz00



dz

C1hh 0

u00c00 0 dz

Or, since the last term on the right-hand side is zero because c00(0) is treatedlike a constant, the final result is

u00c00D 1hh 0



u00∂C

∂

z 0

1

Ez z 0

may be taken one step further, to develop an expression for the longitudinaldispersivity, by substituting Eq (10.5.17),

ELD
1

hh 0



u00z 0

1

Ezz 0

et al (1979) show that ELis two to four orders of magnitude larger than tudinal turbulent diffusivity, Ex, and may be approximated by EL¾D 20.2 huŁ.For practical applications, ELis normally chosen as a fitting parameter, since

longi-it is difficult to know longi-its precise value wlongi-ithout detailed veloclongi-ity profile mation, which normally is not available Statistical approaches can be used, asdiscussed previously in the context of turbulent diffusivities, or values can bechosen to allow the best fit of a particular model The final value also can takeinto account numerical dispersion resulting from the solution procedure used tosolve the partial differential equation of mass transport (advection–dispersionequation), as discussed further in Sec 10.8.1

Experiments of solute diffusion in the stagnant fluid phase that saturates ments have indicated that the molecular diffusive transport in such a domain

sedi-is subject to attenuation due to several factors, including electrical effects and

tortuosity Electrical effects originate from the gradients of other ions, which

may be present in the solution or sorbed onto the particles However, theelectric effects are usually much smaller than the effect of tortuosity.Tortuosity is associated with the ratio of the actual path of ions as theymove around sediment particles to the straight distance of that path Basically,

it may be assumed that the tortuosity should depend on the porosity of theporous medium and a characteristic length of the sediment particles It iscommon to define the tortuosity, , by

D0D Dm

where Dm is the coefficient of molecular diffusion of the contaminant in freesolution and D0 is the diffusion coefficient in the fluid phase which saturatesthe porous medium In the case of lake sediments of similar characteristicparticle size, the tortuosity can be expressed as a function of the porosity,and experiments have indicated that the following approximation can often be

where  is the porosity (see Sec 4.3) In the case of a heterogeneous porousmedium with an axis of symmetry, the tortuosity may be considered as asecond-order tensor depending on the vector, which indicates the direction ofpreferred diffusion Then the diffusion coefficient also should be represented

by a tensor

Generally, the fluid phase that saturates the porous medium is subject

to flow Then the solute, advected by the flowing fluid particles, follows thecurved paths of the fluid particles, and some mixing between flow lines isinevitable, even though the general macroscale fluid motion occurs alongstraight lines Again, the difference between straight-line advection and advec-tion in curved lines through the porous medium is associated with the tortousity

of the porous medium Section 4.4 shows how the laminar flow through porousmedia can be represented by a model of flow through small capillaries Then,

the average flow rate per unit area, namely the specific discharge, which is a

macroscale (scale much larger than the characteristic pore size) parameter, isshown to be proportional to the hydraulic gradient Therefore, though the flow

is basically laminar, its macroscale characteristics can be modeled and lated by methods applied to inviscid flows Hence, the specific discharge isshown to originate from a potential function that is proportional to the piezo-metric head The macroscale average interstitial velocity through the porousmedium is equal to the specific discharge divided by the porosity of the porousmedium However, contaminant advection in the domain is accomplished bythe microscale flow of the fluid particles The microscale flow velocity can beexpressed, in Eulerian terms, by

simu-

where EVis the average macroscale local flow velocity, which originates fromthe gradient of the potential function, and EV0 is the local deviation of themicroscale velocity, relative to the macroscale local flow velocity The expres-sion given by Eq (10.6.3) is very similar to the expression represented by

Eq (5.2.1), with regard to turbulent flow However, in turbulent flow, understeady-state conditions, the value of the local velocity deviation from theaverage value is still a time-dependent quantity With regard to flow throughporous media, under steady-state conditions, the local deviation from themacroscale velocity is a space-dependent variable This, then, is more consis-tent with the definition of a dispersive transport, as defined in the previoussection It should be noted that by the employment of the Lagrangian approach,the deviation of the fluid particle velocity from the local macroscale velocity

is always a time-dependent variable

As with the discussion of dispersion in open channel flow in the previoussection, the deviation of the microscale velocity from the macroscale velocity

is associated with the dispersion of contaminant in the porous medium domain

On the other hand, the macroscale velocity is considered as the only parameterleading to contaminant advection in the domain In addition, there are twomajor differences between dispersion in surface waters and dispersion in flowthrough porous media: (a) dispersion in free turbulent flow often tends to benearly isotropic, while dispersion in flow through a porous medium, even in

an homogeneous and isotropic porous medium, is a nonisotropic phenomenon,provided that the Peclet number is high (the definition of Peclet number isgiven hereinafter); and (b) for large-size domains the dispersion coefficients offlow through a porous medium are larger than for smaller size domains — inlarge domains, there is likely to be a greater degree of inhomogeneity in theproperties of the porous medium, which intensifies the effect of contaminantdispersion, as discussed below

The dispersion in flow through a porous medium depends on the erties of the porous medium and on the magnitude of the flow velocity Forlarger macroscale velocity, the deviations of the microscale velocities fromthe macroscale velocity also become larger Therefore the dispersion coef-ficient value increases with an increase of the macroscale velocity If thefluid that saturates the porous medium is flowing, and the porous medium

prop-is prop-isotropic, then dprop-ispersion coefficients are larger in the direction of themacroscale velocity vector

Assuming that the porous medium is isotropic, the dispersion coefficientshould depend on scalar properties of the porous medium, as well as thevelocity vector and its invariant, namely, its absolute value or magnitude.Therefore the dispersion coefficient should be a second-rank tensor, whichcan be represented using a series approximation,

DijD D0υijC a1VυijC a2ViVj

V C a3V2υijC a4ViVj 10.6.4where D0is the molecular diffusivity affected by the tortuosity of the domain,

aii D1 4 are coefficients with constant values, V is the absolute value

of the velocity, Vi is the ith component of the velocity vector, and υij is theKronecker delta

The coefficients aiof Eq (10.6.4) depend on the structure of the porousmedium, which can be represented by a characteristic advection length Inmost cases, not all terms of the series given by Eq (10.6.4) are considered instudies of contaminant dispersion in a porous medium Basically, the number

of significant terms of Eq (10.6.4) depends on Peclet number, defined as

Pe D Vd

Dm

10.6.5

where V is the large-scale interstitial flow velocity, d is the mean grain size

or any other characteristic length of contaminant advection in the domain, and

Dm is molecular diffusivity of the contaminant in the fluid phase The Pecletnumber, in this case, represents the ratio between contaminant advection andmolecular diffusion

If Pe is extremely small, then only the first term on the right-hand side of

Eq (10.6.4) should be considered If Pe is of order O(1), then the first threeterms should be taken into account In most cases relevant to contaminanttransport in aquifers, Pe is quite high Then the first and two last right-hand-side terms of Eq (10.6.4) can be neglected, and an approximate relation isobtained,

DijD aTVυijC aL
aTViVj

where aTis called the transverse dispersivity, and aLis called the longitudinal dispersivity The longitudinal dispersivity is normally about 20 times larger

than the transverse dispersivity

As indicated by Eq (10.6.6), the principal directions of the sion tensor are parallel and perpendicular to the macroscale flow direction.Therefore by adopting a coordinate system with a coordinate parallel to themacroscale velocity, we obtain a matrix of dispersion coefficients whoseentries are zero, except for those occupying the major diagonal The values ofthe major diagonal dispersion coefficients are given by

where DLis the dispersion coefficient in the longitudinal direction (the tion parallel to the velocity vector) and DTis the dispersion coefficient in anarbitrary direction perpendicular to the velocity vector

direc-It was noted previously that in large domains dispersion is usually moresignificant than in smaller domains This phenomenon is commonly referred

to as the scale effect It is generally connected with some heterogeneity that

characterizes common large-size porous domains To exemplify the possibleeffect of the domain heterogeneity, consider the conceptual model shown inFig 10.12 The large-size domain shown in this figure incorporates porousblocks, which are permeable In these blocks, two sets of equidistant fracturesare embedded There is laminar flow through the small-aperture fractures.Therefore these fractures may be considered as another type of porous medium.Through the large-size domain of Fig 10.12, two types of flow are avail-able for contaminant advection, the porous block flow and the fracture flow.Contaminant disposed at a certain point of the domain is subject to advection

by both of these flows However, the flow through the fractures is usuallymuch quicker than that through the porous blocks Therefore the fracture

Figure 10.12 A conceptual model of a fractured permeable medium: (a) porous blocks embedding small aperture fractures; (b) mixing in the elementary fracture volume.

flow conveys contaminant into regions in which the porous block flow is notcontaminated Furthermore, there is mixing between the porous block flow andthe fracture flow The incorporation of contaminant advection in the blocks andthe fractures and the mixing between the fracture flow and the porous blockflow can be represented as advection and dispersion, respectively, in a homo-geneous domain However, the presentation of the domain ofFig 10.12as ahomogeneous continuum porous matrix requires consideration of a sufficientlylarge volume of the blocks with the embedded fractures Such a volume can betermed a representative elementary volume (REV) with regard to contaminanttransport in the domain

InFig 10.12ba schematic description of the mixing between the porousblock flow and the fracture flow is given under steady-state flow The mixing isassumed to take place in the elementary volume of the fracture As an example,consider the case when there is complete mixing between the fracture flow and

porous block flow in the elementary fracture volume, shown in Fig 10.12b.

By referring to the Figure we obtain

As shown in the figure, initially (at t D 0), the value of Cb is zerofor the entire domain, and the value of Cf is C0 at the entrance of the firstfracture segment (at x D 0) Therefore direct integration of Eq (10.6.8) yieldsthe initial distribution of Cf as

CfD C0exp



qb tan

Qfx



10.6.9Equation (10.6.9) indicates that, due to the mixing between the fractureflow and the porous block flow, the contaminant distribution in the domain

is subject to variations The contaminant distribution in the porous blocks isnot uniform downstream of the fracture segment Therefore, in the segmentfollowing, the mixing between the porous block flow and the fracture flowproduces another type of distribution of contaminant in the domain By numer-ical experiments, it is possible to show that contaminant transport in thedomain of Fig 10.12 can be simulated as a combination of advection anddispersion in a domain composed of a continuum porous medium, providedthat the simulated domain is sufficiently large

ADVECTION – DIFFUSION EQUATION

In this section we consider analytical solutions for the advection–diffusionequation that have been developed for certain simplified conditions Thesesolutions can be applied directly to predict the behavior of a system underthe stated conditions, and they are often useful for checking the results of anumerical solution that might be developed for more complicated situations

In other words, numerical model solutions are often used for simulations ofmore realistic conditions than are usually assumed for the analytical solutionspresented here In many cases the problems introduced in real applications areassociated with the specification of initial and/or boundary conditions Thenumerical model might be run under simplified conditions to compare with

an analytical solution, as a verification that the model is properly formulated.Numerical modeling considerations are discussed further in Sec 10.8

Consider first the general three-dimensional form of the sion equation, with first-order reaction,

where, as before, C is concentration (CŁ could be used just as well), (x,

y, z) are the three spatial coordinates, (u, v, w) are the three corresponding

velocity components, (Ex, Ey, Ez) are the corresponding diffusion coefficients,and K is the first-order decay constant Specific solutions to Eq (10.7.1) areoutlined below for different domains, boundary conditions, and source condi-tions, which are commonly introduced through the boundary conditions

10.7.1 Point Sources

Instantaneous Point Source

The instantaneous point source is perhaps the most fundamental situation toconsider, since it forms a basis for most of the other solutions presented inthis section An instantaneous point source is one where a finite amount ofmass is injected instantaneously at an infinitesimally small point A solution isobtained first for the following conditions: (1) infinitely large domain with asource located at (x1, y1, z1) — seeFig 10.13;(2) homogeneous, anisotropicturbulence, constant in time; and (3) uniform and steady velocity field in thex-direction only

The governing equation is Eq (10.7.1), without the advection terms for

vand w A change of variables is used to eliminate the advection and decayterms, by defining

Figure 10.13 Instantaneous point sources in an infinite domain.

š1 The solution, in terms of the original variables, is

u D u0t C∂u

∂yy C

∂u

∂zz D u0t C yy C zz and D w D 0 10.7.7For simplicity in notation, a parameter ϕ also is defined as

ϕ2D 112



2yEy

Ex

C 2 z

Ez

Ex



10.7.8The inverse of ϕ is thought of as a time scale for the importance of velocityshear in causing mixing of concentration The general solution under these

conditions is

4t3/2ExEyEz1/2 1 C ϕ2t21/2 10.7.9where M is again the mass injected, but ˇ is defined here as

Continuous Point Source

In general, to develop solutions for continuous sources, the basic procedure is

to integrate the corresponding instantaneous source solution over time That

is, a continuous source is considered to be a series of instantaneous sourcesacting over a given time interval As will be seen, for continuous sources theconcept of the steady state becomes of interest, when the source is acting over

a very long time

For a continuous point source the same basic assumptions are used asfor the instantaneous source, except in this case the source occurs over a timeinterval t1 The velocity field is steady and uniform and in the x-direction only,and the source is located at position (x1, y1, z1), as before Each instantaneoussource (with solution given by Eq 10.7.5) is associated with an amount ofmass (q dt), where q D dM/dt D mass injection rate The total concentrationresponse is then obtained by integrating over time t1,

C Dd

p

2p

afe2 p

ab[erfF1C F2
erfF3C F4]

C e
2pab[erfF1
F2
erfF3
F4]g 10.7.11

a Dx
x1

24Ex C y
y12

4Ey Cz
z12

b D U2

and erf is the error function Values of the error function can be found in

various statistic texts or from various web sites A useful calculator may befound at http://ourworld.compuserve.com/homepages/MTE/gerr
e.htm

For a continuous injection, the actual time is set equal to the injection

time, t D t1 In this case it should be noted that F1 becomes unbounded(approaches 1), but F2 D 0 and the error function is still defined For a

steady-state solution, we let t ! 1, and Eq (10.7.11) becomes

C Dd

pp

port neglected This assumption is sometimes referred to as a boundary layer approximation, since longitudinal gradients are often neglected in boundary

layer analyses, relative to transverse gradients Figure 10.14 illustrates thissituation, and the solution (for a source at x1D y1 D z1D 0) is

4EyEz1/2x exp



y2U4xEy

z2U4xEz

KxU



10.7.17

... normally chosen as a fitting parameter, since

longi-it is difficult to know longi-its precise value wlongi-ithout detailed veloclongi-ity profile mation, which normally is not available Statistical... coef-ficient value increases with an increase of the macroscale velocity If thefluid that saturates the porous medium is flowing, and the porous medium

prop-is prop-isotropic, then dprop-ispersion... through the small-aperture fractures.Therefore these fractures may be considered as another type of porous medium.Through the large-size domain of Fig 10. 12, two types of flow are avail-able for contaminant