1. Unit 07 : Advanced Hydrogeology
Solute Transport

2. Mass Transport Processes

3. 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.

4. 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.

5. 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.

6. Pure Advection

7. Advection in Stream Tube

8. Linear Advective Velocity
From Darcy’s Law:
v = q / ne = - (K / ne).dh/dx
where ne is the effective (or connected) porosity

9. 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.

10. 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.

11. Electrokinetic EffectsA A’
Pore
Clay
Clay Surface -
Pore +
Distance AA’
Velocity - +
Anion
Cation

12. Dispersion Concepts
Position in Pore
Velocity
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.

13. 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

14. Experimental Continuous Tracer
INFLOW A
OUTFLOW B
Time
C/Co 1
Start
Time
C/Co 1
Start A B

15. 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.

16. Continuous Tracer Transient
t = t1
t = t2
t = t3
C/Co = 0
C/Co = 1

17. Experimental Pulse Tracer
INFLOW A
OUTFLOW B
Time
C/Co 1
Start
Time
C/Co 1
Start A B

18. 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.

19. Pulse Tracer Transient
t = t1
t = t2
t = t3
C/Co = 0

20. 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

21. Transverse and Longitudinal Dispersion

22. 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]

23. 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.

24. 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.

25. 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 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

26. Dd for Common Ions Cation Dd (10-10 m2/s) Anion 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

27. 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.

28. 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.

29. 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.

30. Empirical Data on Dispersion

31. Transport Regimes For NPE < 0.02 diffusion dominates
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

32. 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.

33. 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.

34. 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.

35. 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).

36. 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 under the concentration graph (mass) lies between –2sL and +2sL.
To complete the analogy with dispersion, we find that sL = (2DLt)1/2
x/s C
Recall that concentration (C) against position (x) after time (t) for a pulse source resembles the Gaussian distribution function.

37. 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).

38. 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/Co to t1 t2

39. 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.

40. 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 variances are simply related by the advective velocity: sL2 = v2 st2
DL = sL2 / 2t = v2 st2 / 2t
0.16
0.50
0.84
C / Cmax
t16 t50 t84

41. 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 / (1D case)
For the 2D case, the peak width is divided by sqrt(2) so the standard deviation is given by:
sL = G / (2D case)
(See Robbins, 1983)
0.16
0.50
0.84
C / Cmax G

42. Fractured Media Assumptions:
Matrix
Advection
Dispersion
Fracture
Diffusion
Assumptions:
Advection and dispersion only occurs in the fracture network
Diffusion from fractures to the matrix is possible

43. 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

44. 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

45. 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

46. Spatial Data Stationary data series : mean independent of position
Data series with trend: mean changes with position

47. Autocorrelated Data Stationary autocorrelated data series
Autocorrelated data series with trend
The distance between peaks is the correlation length

48. 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 1
The chart plotting correlation coefficient against lag is called a correlogram.
Correlation l -1
Lag

49. 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.

50. Variogram Terminology
At 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
Range
Semivariance
Sill
Nugget
Lag

51. 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

52. 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

53. 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*