Unit 07 : Advanced Hydrogeology

Solute Transport

Mass Transport Processes

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.

Diffusion

• Diffusion is the process of mixing that occurs as a result of concentration gradients in porous media.

• Diffusion can occur when there is no hydraulic gradient driving flow and the pore water is static.

• Diffusion in groundwater systems is a very slow process.

Dispersion

• Dispersion is the process of mechanical mixing that takes place in porous media as a result of the movement of fluids through the pore space.

• Hydrodynamic dispersion is a term used to include both diffusion and dispersion.

Pure Advection

Advection in Stream Tube

Linear Advective Velocity

From Darcy’s Law:

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

where ne is the effective (or connected) porosity

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 there are other factors at work.

Deviations from Advective Velocity

• Electrical charges on clay mineral surfaces can force anions to the centre of pores where velocities are highest.

• Anions can then travel faster than the advective velocity.

• Cations are attracted by the clay mineral surface charge and can be retarded (travel slower than the advective velocity).

• Bi-polar water molecules can similarly be retarded giving rise to osmotic and membrane filtration effects.

Electrokinetic Effects

Distance AA’

Vel

oci

ty

A

A’

PoreClay

Clay

Clay Surface

Clay Surface

-

--- -

- - -

-

-

--- -

- - -

-

Pore

--

-

--

+

++

+

+

-

+Anion

Cation

Dispersion Concepts

• Mechanical dispersion spreads mass within a porous medium in two ways:– Velocity differences

within pores on a microscopic scale.

– Path differences due to the tortuosity of the pore network.

Position in Pore

Ve

loci

ty

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

Experimental Continuous Tracer

Time

C/C

o

0

1

Start

Time

C/C

o

0

1

Start

INFLOW A OUTFLOW B

A B

Continuous Tracer Test

• First tracer C/Co > 0.0 arrives faster than advective velocity.

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

• Last tracer C/Co = 1.0 travels slower than advective velocity.

Continuous Tracer Transient

t = t1

t = t2

t = t3

C/Co = 0C/Co = 1

Experimental Pulse Tracer

Time

C/C

o

0

1

Start

Time

C/C

o

0

1

Start

INFLOW A OUTFLOW B

A B

Pulse Tracer Test

• The “box function” of the source is both delayed and attenuated by dispersion.

• The pulse peak arrival time corresponds to the 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.

Pulse Tracer Transient

t = t1

t = t2

t = t3

C/Co = 0 C/Co = 0

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

Transverse and Longitudinal Dispersion

Diffusion Law

• Darcy’s law for relates fluid flux to hydraulic gradient:

q = -K.grad(h)

• For mass transport, there is a similar law (Fick’s law) relating solute flux to concentration 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]

Molecular Diffusion

• Molecular diffusion is mixing caused by random motion of solute molecules as a result of thermal kinetic energy.

• The diffusion coefficient in a porous medium is less than that in pure liquids because of collisions with the pore walls.

J = -Dd.[grad(nC) + / V] where V is a chemical averaging volume [moles-1L3], n is porosity and is the tortuosity of the porous medium.

Fick’s Law for Sediments• This theoretical function, for practical

applications, has been simplified to :J = -D*d.n.grad(C)

where D*d is a bulk diffusion coefficient accounting for tortuosity

• This form of the function is known as Fick’s law for diffusion in sediments often written as:

J = -D’d.grad(C) = - u.n.Dd.grad(C)

where D’d is an effective diffusion coefficient , Dd is the self diffusion coefficient of the solute ion, n is porosity and u is a dimensionless factor < unity.

Estimating D’d

• 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, D’d increases with increasing porosity and decreases with increasing tortuosity = Le/L

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- 17.3

HCO3- 11.8

Ca2+ 7.93 SO42- 10.7

Fe2+ 7.19 CO32- 9.55

Mg2+ 7.05

Fe3+ 6.07Typical factors to calculate D’d are 0.10 to 0.20 for granular materials

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

Mechanical Dispersion

• Mechanical dispersion is caused by local variations in the velocity field on scales ranging from microscopic through macroscopic to megascopic.

• Variations in hydraulic conductivity due to lithological heterogeneities are the main sources of velocity variations.

Dispersion Coefficient

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

D = D’ + D’d• 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 place normal to the flow direction.

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 diffusive transport ratio.

Empirical Data on Dispersion

Transport Regimes

• For NPE < 0.02

diffusion dominates• For 0.02 > NPE < 8

diffusion and mechanical dispersion• For NPE > 8

mechanical dispersion dominates

Some authors place the boundaries at 0.01 and 4 rather than 0.02 and 8

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) are typically around 0.1DL for NPE > 100 although values as low as 0.0 1DL have been reported.

Dispersivity

• Dispersion coefficients may be written:

DL = L.v and DT = T.v

where L and T are called the dispersivities.

• Dispersivities have units of length and are characteristic properties of porous media.

Dispersion and Scale• Most knowledge of dispersion has been gleaned

from experimental work at the microscopic scale.• A review of many dispersivity measurements

(Gelhar et al, 1992) gave values for L spanning almost six orders of magnitude.

• Microscopic scale dispersivities as a result of velocity changes on the pore scale are about two orders of magnitude smaller than macroscopic dispersivities arising from heterogeneity in hydraulic conductivity.

Fickian Model

• Hydrodynamic dispersion occurs due to a combination of molecular diffusion and mechanical dispersion.

• A Fickian dispersion model implies that mass transport is proportional to the concentration gradient and in the direction of the concentration gradient (just like Fick’s law for diffusion).

• Using such a model, we treat dispersion in a way fully analogous to diffusion (even though the processes of diffusion and dispersion are quite different).

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 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 –2L and +2L.

To complete the analogy with dispersion, we find that L = (2DLt)1/2

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

C

One-Dimensional Pulse

• The peak concentration for a pulse source travels at the advection velocity v = x / t.

DL = L2 / 2t = L

2.v / 2x

where 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 of concentration (C) with position (x).

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

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 the

centroid.

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

Breakthrough Curvet = (t84 – t16) / 2

The value t2 is a temporal

variance measured in c-t space for the breakthrough curve. Previously we recognized L

2 as a spatial variance measure in c-x space.

Fortunately the two variances are simply related by the advective velocity: L

2 = v2 t2

DL = L2 / 2t = v2 t

2 / 2t

0.16

0.50

0.84

C /

Cm

ax

t16 t50 t84

Spatial PlumeL = (x84 – x16) / 2

This may be a difficult to measure so the width of the peak at C / Cmax = 0.5 denoted by can be used.

L = / 1.665 (1D case)

For the 2D case, the peak width is divided by sqrt(2) so the standard deviation is given by:

L = / 2.345 (2D case)

(See Robbins, 1983)

0.16

0.50

0.84

C /

Cm

ax

Fractured Media

• Assumptions:– Advection and

dispersion only occurs in the fracture network

– Diffusion from fractures to the matrix is possible

Matrix

Matrix

AdvectionDispersion

FractureDiffusion

Mixing Processes in Fractures

• Mechanical mixing due to velocity variations within rough fractures

• Mixing at fracture intersections• Velocity variations between different fracture

sets• Diffusion between fractures and matrix may

be important because fractures localize mass and concentration gradients may be high

• Interactions of various processes can be complex

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

Geostatistical Parameters

• Mean (ym) measures central value:

y = yi / n

• Variance (2y) measures spread or scatter:

2y = (yi - y)2 / n

• Correlation length (y) measures spatial persistence:

y(b) = f(-|b| / y) = exp(-|b| / y)

where b is a distance sampling interval parameter called the lag

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

Autocorrelated DataStationary 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

Correlogram

• When a data series is correlated with itself for various lags, the autocorrelation eventually approaches zero after a number of lags corresponding to

• The chart plotting correlation coefficient against lag is called a correlogram.

Lag

Co

rrel

atio

n 1

0

-1

Variogram

• Geostatistical theory does not use the autocorrelation, but instead uses a related property called the semi-variance.

• The semi-variance is simply half the variance of the differences between all possible points spaced a constant distance apart.

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

• For large lags, the semi-variance approaches half the variance of the spatial dataset.

Variogram TerminologyAt lags where spatial correlations exist, the data values are similar and the semivariance is low.

A variogram is like an upside down correlogram. Special terms 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

Sem

ivar

ian

ce

Lag

Sill

Nugget

Range

Hydraulic Conductivity Fields

• Many hydrogeologic parameters, particularly hydraulic conductivity, have spatial structure

• Procedures are available for generating spatial data with a particular , and

• These measures of heterogeneity can be used to predict dispersivity

Geostatistical Estimation• Gelhar and Axness (1983) suggested:

AL = 2y / 2

where AL is called the asymptotic longitudinal dispersivity and y = ln(K) where K is hydraulic conductivity and is a flow factor (taken to be unity).

• AL accounts in a quantitative fashion for heterogeneity in the hydraulic conductivity field

Geostatistical Model of Dispersion

Dispersivity is conceptually believed to have three components: diffusive mixing, pore scale mixing and mixing through spatial heterogeneities:

AL* = AL + L + Dd

* / v

This leads to an expression for hydrodynamic dispersion coefficient with the form:

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