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
Pore
- +
- +
+
+
- + - Clay Surface
- -
Clay Surface
Clay
A
Pore
Velocity
A’
Clay
Distance AA’
- Anion
+ Cation
• 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.
Velocity
Dispersion Concepts
Position in Pore
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
INFLOW A
1
1
C/Co
C/Co
OUTFLOW B
0
0
Time
Start
A
Time
Start
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
C/Co = 1
t = t3
C/Co = 0
Experimental Pulse Tracer
INFLOW A
1
1
C/Co
C/Co
OUTFLOW B
0
0
Time
Start
A
Time
Start
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
C/Co = 0
t = t3
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) + t / V]
where V is a chemical averaging volume [moles-1L3],
n is porosity and
t 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 t = 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+
6.07
Typical 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 = aL.v
and
DT = aT.v
where aL and aT 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 aL
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
For the Gaussian (normal)
distribution s is the
standard deviation and
measures the spread about
the mean value.
About 95.4% of the area
-3 -2 -1
0 1
2 3 x/s
under the concentration
Recall that concentration (C)
graph (mass) lies between
against position (x) after time –2sL and +2sL.
(t) for a pulse source
To complete the analogy
resembles the Gaussian
with dispersion, we find that
distribution function.
sL = (2DLt)1/2
C
One-Dimensional Pulse
• The peak concentration for a pulse source
travels at the advection velocity v = x / t.
DL = sL2 / 2t = sL2.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
C/Co
• Two-dimensional spread of a pulse tracer in a
unidirectional flow field results in an elliptically shaped
concentration plume with a Gaussian mass distribution.
to
t1
t2
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 Curve
st = (t84 – t16) / 2
C / Cmax
0.84
0.50
0.16
t16 t50 t84
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 variances
are simply related by the
advective velocity: sL2 = v2 st2
DL = sL2 / 2t = v2 st2 / 2t
Spatial Plume
sL = (x84 – x16) / 2
C / Cmax
0.84
This may be a difficult to measure
so the width of the peak at C / Cmax
= 0.5 denoted by G can be used.
0.50
sL = G / 1.665 (1D case)
0.16
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)
G
(See Robbins, 1983)
Fractured Media
• Assumptions:
Matrix
Diffusion
Fracture
Matrix
Advection
Dispersion
– Advection and
dispersion only
occurs in the
fracture network
– Diffusion from
fractures to the
matrix is possible
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:
my = S yi / n
• Variance (s2y) 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
Spatial Data
Stationary data series : mean independent of position
3
0
0
50
-3
Data series with trend: mean changes with position
3
0
0
-3
50
Autocorrelated Data
Stationary autocorrelated data series
3
0
0
50
-3
Autocorrelated data series with trend
3
0
0
50
-3
The distance between peaks is the correlation length
Correlogram
Correlation
• When a data series is correlated with itself for various
lags, the autocorrelation eventually approaches zero
after a number of lags corresponding to l
1
• The chart plotting correlation
coefficient against lag is called
0
a correlogram.
l
-1
Lag
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 Terminology
Semivariance
At lags where spatial correlations exist, the data values
are similar and the semivariance is low.
Range
Sill
Nugget
Lag
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
Hydraulic Conductivity Fields
• Many hydrogeologic parameters,
particularly hydraulic conductivity, have
spatial structure
• Procedures are available for generating
spatial data with a particular m, s and l
• These measures of heterogeneity can
be used to predict dispersivity
Geostatistical Estimation
• Gelhar and Axness (1983) suggested:
AL = s2y l / g2
where AL is called the asymptotic longitudinal
dispersivity and y = ln(K) where K is hydraulic
conductivity and g 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 + aL + Dd* / v
This leads to an expression for hydrodynamic
dispersion coefficient with the form:
DL = (AL + aL).v + Dd*