unit07-soltran

7/29/2019 Unit07-soltran

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Unit 07 : Advanced Hydrogeology

Solute Transport


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Mass Transport Processes


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Advection

Advection is mass transport due simply

to the flow of the water in which the

mass is carried. The direction and rate of transport

coincide with that of the groundwater

flow.


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Diffusion

Diffusion is the process of mixing thatoccurs as a result of concentration

gradients in porous media. Diffusion can occur when there is no

hydraulic gradient driving flow and thepore water is static.

Diffusion in groundwater systems is avery slow process.


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Dispersion

Dispersion is the process of

mechanical mixing that takes place in

porous media as a result of themovement of fluids through the pore

space.

Hydrodynamic dispersion is a termused to include both diffusion and

dispersion.


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Pure Advection


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Advection in Stream Tube


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Linear Advective Velocity

From Darcys Law:

v = q / ne = - (K / ne).dh/dx

where ne is the effective (or connected)porosity


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Fractured Rocks and Clays

In fractured rocks, the effective porosity

(ne) can be very small implying relatively

high advective velocities. In clays and shales, effective porosity

can also be very low and high advective

velocities might be expected but thereare other factors at work.


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Deviations from Advective Velocity

Electrical charges on clay mineral surfacescan force anions to the centre of pores wherevelocities are highest.

Anions can then travel faster than theadvective velocity.

Cations are attracted by the clay mineralsurface charge and can be retarded (travel

slower than the advective velocity). Bi-polar water molecules can similarly be

retarded giving rise to osmotic andmembrane filtration effects.


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Electrokinetic Effects

Distance AA

Veloci

ty

A

A

PoreClay

Clay

Clay Surface

Clay Surface

-

-- --

- --

-

-

-- -

-- -

-

-

Pore

--

-

-

-+

+ ++

+

-

+

Anion

Cation


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Dispersion Concepts

Mechanical dispersionspreads mass within aporous medium in twoways: Velocity differences

within pores on amicroscopic scale.

Path differences due tothe tortuosity of thepore network.

Position in Pore

Velocity


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Macroscopic Dispersion

Random variations in velocity and tortuous

paths through flow systems are created on a

larger scale by lithological heterogeneity. Heterogeneity is responsible for macroscopic

dispersion in flow systems


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Experimental Continuous Tracer

Time

C/C

o

0

1

Start

Time

C/C

o

0

1

Start

INFLOW A OUTFLOW B

A B


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Continuous Tracer Test

First tracer C/Co > 0.0 arrives faster

than advective velocity.

Mean tracer arrival time C/Co = 0.5corresponds to advective velocity.

Last tracer C/Co = 1.0 travels slower

than advective velocity.


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Continuous Tracer Transient

t = t1

t = t2

t = t3

C/Co = 0C/Co = 1


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Experimental Pulse Tracer

Time

C/C

o

0

1

Start

Time

C/C

o

0

1

Start

INFLOW A OUTFLOW B

A B


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Pulse Tracer Test

The box function of the source is both

delayed and attenuated by dispersion.

The pulse peak arrival time corresponds tothe advective velocity.

The peak concentration C/Co is less than 1.0

The breadth and height of the peak

characterize the dispersivity of the porous

medium.


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Pulse Tracer Transient

t = t1

t = t2

t = t3

C/Co = 0 C/Co = 0


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Pulse Zone of Dispersion

The zone of dispersion broadens and

the peak concentration C/Co reduces as

it moves through the porous medium. Ahead of the zone C/Co = 0

Behind the zone C/Co =0


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Transverse and Longitudinal Dispersion


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Diffusion Law

Darcys law for relates fluid flux to hydraulicgradient:

q = -K.grad(h)

For mass transport, there is a similar law(Ficks law) relating solute flux toconcentration gradient in a pure liquid:

J = -Dd.grad(C)

where J is the chemical mass flux [moles. L-2T-1]

C is concentration [moles.L-3]

Dd is the diffusion coefficient [L2T-1]


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Molecular Diffusion

Molecular diffusion is mixing caused byrandom motion of solute molecules as aresult of thermal kinetic energy.

The diffusion coefficient in a porous mediumis less than that in pure liquids because ofcollisions with the pore walls.

J = -Dd

.[grad(nC) + t / V]

where V is a chemical averaging volume [moles-1L3],

n is porosity and

t is the tortuosity of the porous medium.


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Ficks Law for Sediments

This theoretical function, for practicalapplications, has been simplified to :

J = -D*d.n.grad(C)

where D*d is a bulk diffusion coefficient accountingfor tortuosity

This form of the function is known as Fickslaw for diffusion in sediments often

written as:J = -Dd.grad(C) = - u.n.Dd.grad(C)

where Dd is an effective diffusion coefficient , Dd isthe self diffusion coefficient of the solute ion, n isporosity and u is a dimensionless factor < unity.


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Estimating Dd

The factor u depends on the tortuosity of the

medium and empirical values (Hellferich,

1966) lie between 0.25 and 0.50 Bear (1972) suggest values between 0.56

and 0.80 based on a theoretical evaluation of

granular media.

Whatever the factor used, Dd increases withincreasing porosity and decreases with

increasing tortuosity t = Le/L


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Dd for Common Ions

Cation Dd (10-10 m2/s) Anion Dd (10

-10 m2/s)

H+ 93.1 OH- 52.7

K+ 19.6 Cl- 20.3

Na+

13.3 HS-

17.3HCO3

- 11.8

Ca2+ 7.93 SO42- 10.7

Fe2+ 7.19 CO32- 9.55

Mg2+

7.05Fe3+ 6.07

Typical factors to calculate Dd are 0.10 to 0.20 for granular materials

Notice that diffusion coefficients are smaller the higher the charge on the ion


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Mechanical Dispersion

Mechanical dispersion is caused by

local variations in the velocity field on

scales ranging from microscopicthrough macroscopic to megascopic.

Variations in hydraulic conductivity due

to lithological heterogeneities are themain sources of velocity variations.


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Dispersion Coefficient

The hydrodynamic dispersion coefficient (D)is a combination of mechanical dispersion(D) and bulk diffusion (Dd):

D = D + Dd The advective flow velocity (v) and mean

grain diameter (dm) have been shown to be

the main controls on longitudinal dispersion(DL) parallel to the flow direction.

Transverse dispersion (DT) also takes placenormal to the flow direction.


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Peclet Number

D/Dd is a convenient ratio that

normalizes dispersion coefficients by

dividing by the diffusion coefficient. v.dm /Dd is called the Peclet Number

(NPE) a dimensionless number that

expresses the advective to diffusivetransport ratio.


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Empirical Data on Dispersion


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Transport Regimes

For NPE < 0.02

diffusion dominates

For 0.02> NPE < 8diffusion and mechanical dispersion

For NPE > 8

mechanical dispersion dominatesSome authors place the boundaries at 0.01 and 4

rather than 0.02 and 8


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Velocity Proportionality

For values of NPE > 8 the longitudinal (and

transverse) dispersion coefficient (DL) is

proportional to the advective velocity (v). This result has been generalized to describe

dispersion both on microscopic and

megascopic scales.

Tranverse dispersion coefficients (DT) aretypically around 0.1DL for NPE > 100 although

values as low as 0.0 1DL have been reported.


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Dispersivity

Dispersion coefficients may be written:

DL = aL.v and DT = aT.v

where aL and aT are called the

dispersivities.

Dispersivities have units of length and

are characteristic properties of porousmedia.


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Dispersion and Scale

Most knowledge of dispersion has beengleaned from experimental work at themicroscopic scale.

A review of many dispersivity measurements(Gelhar et al, 1992) gave values foraLspanning almost six orders of magnitude.

Microscopic scale dispersivities as a result of

velocity changes on the pore scale are abouttwo orders of magnitude smaller thanmacroscopic dispersivities arising fromheterogeneity in hydraulic conductivity.


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Fickian Model

Hydrodynamic dispersion occurs due to acombination of molecular diffusion andmechanical dispersion.

A Fickian dispersion model implies that masstransport is proportional to the concentrationgradient and in the direction of the concentrationgradient (just like Ficks law for diffusion).

Using such a model, we treat dispersion in away fully analogous to diffusion (even thoughthe processes of diffusion and dispersion arequite different).


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Quantifying Dispersion

Recall that concentration (C)

against position (x) after time(t) for a pulse source

resembles the Gaussian

distribution function.

For the Gaussian (normal)

distribution s is the

standard deviation and

measures the spread about

the mean value.

About 95.4% of the area

under the concentration

graph (mass) lies between

2sL and +2sL.

To complete the analogy

with dispersion, we find that

sL = (2DLt)1/2

-3 -2 -1 0 1 2 3 x/s

C


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One-Dimensional Pulse

The peak concentration for a pulse source

travels at the advection velocity v = x / t.

DL = sL2

/ 2t = sL2

.v / 2xwhere v is the advective velocity and x is the

distance travelled by the peak at time t.

This provides a means to estimate DL from

field or laboratory measurements ofconcentration (C) with position (x).


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Two-Dimensional Pulse

Two-dimensional spread of a pulse tracer in a

unidirectional flow field results in an elliptically shaped

concentration plume with a Gaussian mass distribution.

C/C

o

to t2t1


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Three-Dimensional Pulse

3D plumes are generally cigar-shaped

Typically, vertical transverse dispersion is

small and plumes have a surfboard shape

Pulse source plumes are symmetric about thecentroid.

Continuous source plumes are assymmetric,broadening in direction of flow.


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Breakthrough Curve

st = (t84 t16) / 2

The value st2 is a temporal

variance measured in c-t space

for the breakthrough curve.Previously we recognized sL2 as

a spatial variance measure in c-x

space.

Fortunately the two variancesare simply related by the

advective velocity: sL2 = v2 st

2

DL = sL2 / 2t = v2 st

2 / 2t

0.16

0.50

0.84

C/C

max

t16 t50 t84


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Spatial Plume

sL = (x84 x16) / 2

This may be a difficult to measure

so the width of the peak at C / Cmax

= 0.5 denoted by G can be used.sL = G / 1.665 (1D case)

For the 2D case, the peak width is

divided by sqrt(2) so the standard

deviation is given by:

sL = G / 2.345 (2D case)

(See Robbins, 1983)

0.16

0.50

0.84

C/C

max

G


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Fractured Media

Assumptions:

Advection and

dispersion onlyoccurs in the

fracture network

Diffusion from

fractures to the

matrix is possible

Matrix

Matrix

Advection

Dispersion

Fracture

Diffusion


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Mixing Processes in Fractures

Mechanical mixing due to velocity variationswithin rough fractures

Mixing at fracture intersections

Velocity variations between different fracturesets

Diffusion between fractures and matrix may

be important because fractures localize massand concentration gradients may be high

Interactions of various processes can becomplex


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Geostatistics

Geostatistics allow spatial variability to

be included in the analysis of flow and

transport in porous media Important because heterogeneity is the

at the root of macroscopic dispersion

We use three statistical parameters:mean, variance and correlation length


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Geostatistical Parameters

Mean (ym) measures central value:

my = S yi / n

Variance (s2

y) measures spread or scatter:s2y = S (yi - my)

2 / n

Correlation length (ly) measures spatial

persistence:

ry(b) = f(-|b| / ly) = exp(-|b| / ly)

where b is a distance sampling interval parameter

called the lag


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Spatial Data

-3

0

3

0 50

-3

0

3

0 50

Stationary data series : mean independent of position

Data series with trend: mean changes with position


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Autocorrelated Data

Stationary autocorrelated data series

Autocorrelated data series with trend

-3

0

3

0 50

-3

0

3

0 50

The distance between peaks is the correlation length


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Correlogram

When a data series is correlated with itself for various

lags, the autocorrelation eventually approaches zero

after a number of lags corresponding to l

The chart plotting correlationcoefficient against lag is called

a correlogram.

Lag

Correlati

on 1

0

-1

l


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Variogram

Geostatistical theory does not use theautocorrelation, but instead uses a relatedproperty called the semi-variance.

The semi-variance is simply half the varianceof the differences between all possible pointsspaced a constant distance apart.

For a lag of zero, the semi-variance is thuszero.

For large lags, the semi-variance approacheshalf the variance of the spatial dataset.


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Variogram Terminology

At lags where spatial correlations exist, the data valuesare similar and the semivariance is low.

A variogram is like an upside

down correlogram. Specialterms describe the function:

sill corresponds to the

semivariance of the dataset

range is a distance parameter

similar to correlation length

nugget is the projected

intercept on the semivariance

axis for experimental data

Semivariance

Lag

Sill

Nugget

Range


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Hydraulic Conductivity Fields

Many hydrogeologic parameters,

particularly hydraulic conductivity, have

spatial structure Procedures are available for generating

spatial data with a particularm, s and l

These measures of heterogeneity canbe used to predict dispersivity


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Geostatistical Estimation

Gelhar and Axness (1983) suggested:

AL = s2y l / g

2

where AL is called the asymptotic longitudinal

dispersivity and y = ln(K) where K is hydraulicconductivity and g is a flow factor (taken to beunity).

AL accounts in a quantitative fashion for

heterogeneity in the hydraulic conductivityfield


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Geostatistical Model of Dispersion

Dispersivity is conceptually believed to have

three components: diffusive mixing, pore

scale mixing and mixing through spatial

heterogeneities:

AL* = AL + aL + Dd

* / v

This leads to an expression for hydrodynamic

dispersion coefficient with the form:

DL = (AL + aL).v + Dd*

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