Chapter nine Subsurface pollutant transport

CHAPTER

NINE

SUBSURFACE POLLUTANT TRANSPORT

9.1

INTRODUCTION

When a pollutant is released from a waste storage facility, it migrates downward through the unsaturated zone (i.e., vadose zone) to the water table and then laterally in the direction of the hydraulic gradient in the saturated zone. Throughout this process its fate is controlled by a myriad of physical, chemical, and biotic processes. These include the physical processes of advection, diffusion, dispersion, and capillary, and the biotic and abiotic processes of bioaccumulation, biodegradation, immobilization, retardation, and volatilization. Quantification of these various processes at a field site is very difficult. Assessment of subsurface fate and transport must address questions of source characterization (what is released, where, when, how much, etc.), vadose zone transport and processes, groundwater transport, and exposure and dose assessment.

9.2

MODELLING PROCESS

The hydrogeological modelling process starts with identifying and defining the problem and setting the modelling objectives. In polluted site assessment, the objectives often focus on determining the chances of a pollutant plume reaching a critical point, or on a comparative evaluation of various remediation strategies. The key steps are: (1) Formulation of a conceptual model: Because the real hydrological system is complex, we create an idealization of the system. The information consists of the extent, configuration, and properties of the hydrogeological units as well as the processes relevant to the main objectives. The coupling and interaction among these processes are particularly important, as are the assumptions that justify excluding certain processes and interactions. We must consider the spatial dimensions that are necessary to properly represent the selected processes in the system; (2) Model Design: On the basis of the conceptual model, we can design the physically-based mathematical model. This step involves: (a) identifying the governing physical principles and the corresponding equations that express these principles, (b) defining the boundary and initial conditions and material properties, and (c) selecting and implementing the solution method; (3) Calibration: In the cases where model parameters are unknown, calibration can be used to estimate these parameters by fitting a simulation to an observed response under controlled conditions; (4) Sensitivity: One of the most important uses of models is to determine the sensitivity of the 209

210

(5)

(6)

9.3

SUBSURFACE POLLUTANT TRANSPORT system with respect to the various parameters. A sensitivity analysis is usually done by defining a base case, and then varying the parameters (one at a time) within their respective ranges. These ranges are generally known to an experienced modeler. Sensitivity analysis provides an insight to the behaviour of the system under various conditions, and about the importance of the individual parameters and processes; Prediction: When the sensitivity of the model is sufficiently understood, the model can be used for predictive purposes. We have to bear in mind that the model is not the real system. It is only an idealization of the system and it is subject to many assumptions and simplifications; and Decision: Model prediction forms one component in the decision making process related to the problem under investigation. The decision process also involves regulatory, economic, environmental, and political components. The integration of all these aspects requires a broad understanding of the problem. A sound model can contribute to this understanding.

TRANSPORT PROCESSES IN SOILS

There are three basic physical mechanisms by which miscible and immiscible pollutants are transported in the subsurface environment: advection, diffusion, and dispersion. Emphasis in this section is placed on soluble species, and only single-phase flow is considered. 9.3.1

Advection

Advection is the process by which pollutants are transported along with the flowing fluid or solvent in response to a hydraulic gradient. Due to advection, non-reactive solutes are transported at an average rate equal to the seepage velocity of the fluid. For saturated flow, the seepage and Darcy velocity are related by

v -

v

[9.1]

n

where v, is seepage velocity, n is soil porosity, v is Darcy's water flux which is given by: Q - - k Oh = kwi~' A WOx

[9.2]

where Q is volumetric water flow rate, A is cross sectional area perpendicular to the flow direction, kw is hydraulic conductivity, h is total hydraulic head, x is the flow direction, and ih is hydraulic gradient. The advective mass flux of a particular chemical species is given by: dadvection

= VC = k i h C

=

nvsc

[9.3]

where Jauvect,o,,is the advective mass flow (mass flowing through a unit cross sectional area per unit time), and c is the concentration of the solute in the liquid phase of the porous material. Thus the

TRANSPORT PROCESSES IN SOILS

211

advective transport is proportional to the medium hydraulic conductivity, the hydraulic gradient, and the local concentration. 9.3.2

Diffusion

While advection is associated with the bulk macroscopic groundwater movement, diffusion is a molecular-based phenomenon. If we could see the individual molecules, we could note the continual movement of each molecule and of one molecule relative to the other. Hence, diffusion is caused by random thermal motion, as is the Brownian movement of colloidal particles observable under the microscope. In unstirred liquids, all molecules and ions of the solvent and solute have this random movement. As a result of this motion, which occurs randomly in all directions, irregularities in the concentration of a solution eventually disappear. Measurements of diffusion usually entail measuring the rate at which the irregularities disappear. Thus, if neighbouring volumes of solution have different concentrations, more solute molecules move into the volume of lower concentration than move out until the concentrations become equal. The rate of change of concentration depends on the initial difference of concentration between the two volumes, and on the mean distance between them. Expressed in other words, the rate of transfer by diffusion between two volume elements varies directly with the difference of concentration and inversely with the distance between them. This can be represented by Fick's first law of diffusion as: Jdiffusion

=

-

D

0__c_c 0x

[9.4]

where Jdiffusion is the rate of flow, or flux, and is the amount of solute diffusing per unit time across unit cross sectional area, D is diffusion coefficient, and x is the direction of transport. The two commonly used definitions of diffusion coefficient are the self-diffusion coefficient and the bulk diffusion coefficient. They can be understood by referring to the methods by which they are measured: (1) Self-diffusion coefficients (Do) are measured by adding a small amount of radioactive isotope to a system containing a uniform concentration of unlabeled ion or molecule throughout. The isotope equilibrates with the unlabeled ions or molecules, and the rate at which this occurs is used to measure the self-diffusion coefficient. Typical values are shown in Table 9.1 (Robinson and Stokes, 1965). (2) Bulk diffusion coefficients (D~) are measured where there is movement of a solute within a soil mass due to a concentration gradient. Solute is therefore transferred to the volume of solution at the lower concentration. The coefficient is also sometimes known as a salt diffusion coefficient. The bulk diffusion coefficient is usually much less than the selfdiffusion coefficient.

Effects of Soil Properties on the Magnitude of D~ The two soil properties that affect the diffusion coefficient in saturated soil are discussed below.

212

SUBSURFACE POLLUTANT TRANSPORT

Table 9.1: Self-diffusion coefficients for representative ions at infinite dilution in water D o * 10-1~m2/sec D o * 10 1~ m2/sec Anion 0~ 18~ 25~ Cation 0~ 18~ 25~ H+ Li + Na § K§ Mg 2+ Ca 2§ Fe z+ CO 2+ Ni 2+ Cu 2+ Zn 2+ Cd z+ Pb 2+ Cr 3+ Fe 3+ A13+

56.1 4.72 6.27 9.86 3.56 3.73 3.41 3.41 3.11 3.41 3.35 3.41 4.56

2.36

81.7 8.69 11.3 16.7 5.94 6.73 5.82 5.72 5.88 5.88 6.13 6.03 7.95 3.90 5.28 3.46

93.1 10.3 13.3 19.6 7.05 7.93 7.19 6.99 6.79 7.33 7.15 7.17 9.45 5.94 6.07 5.59

OH F CI Br I SO42 NO2

25.6

N O 3"

9.78 4.39

CO32

10.1 10.5 10.3 5.0

44.9 12.1 17.1 17.6 17.2 8.9 15.3 16.1 7.8

52.7 14.6 20.3 20.1 20.0 10.7 19.1 19.0 9.55

Soil Porosity Because we are considering diffusion in soil pore solution, the need to consider the porosity of the soil is self-evident. A cross a unit cross-sectional area of soil, the diffusion coefficient is proportional to the cross-sectional area of the liquid contained in the soil. We can write: D

= nD ~

[9.5]

Tortuosity of the Pathway of Diffusion Ions diffusing through the soil solution do not pass along straight tubes but follow irregular and winding paths. Diffusion is along these irregular paths because of the changes in soil pore size distribution (Porter et al., 1960; Olsen and kemper, 1968; Bear, 1972). The complexity of the pathways is too great to calculate or measure directly. So tortuosity is regarded as an empirical factor. We can now write: Ds " r,n D o

[9.6]

where 1: is the tortuosity factor, which needs to be measured indirectly. It varies with the moisture content of the soil because as soil becomes drier the diffusive pathway becomes more tortuous (Rowell et al., 1967). In free liquid the value of z = 1, in saturated soil 1: = 0.4 at field capacity of 0.2. Tortuosity is introduced into the diffusion equation to adjust x, the distance between two points in the soil, in order to give the true concentration gradient, d c / d x . Generally, tortuosity is expressed as:

TRANSPORT PROCESSES IN SOILS

213 [9.7]

: l/l e

where l is the macroscopic, straight line distance between two points defining the path, and l e is the actual, microscopic or effective distance of transport between the same two points. Since le is greater than l, it follows that z is less than 1.0. Typical reported values of 1: are in the range 0.01 to 0.67 (Perkins and Johnston, 1963; Freeze and Cherry, 1979; Daniel and Shackelford, 1988; Shackelford, 1989; Shackelford and Daniel, 1991). Fick's law for diffusion in soil can now be modified to include tortuosity and porosity: Oc

Jdiffusion = VoY'n -~x

[9.81

where c is the concentration of the diffusing ions in bulk soil solution.

9.3.3

Dispersion

The d i s p e r s i o n mechanism of pollutant transport is associated with bulk fluid movement in the porous medium. Fluid particles that are at one time close together tend to move apart or spread. The spreading nature of pollutant is attributed to variations in seepage velocity, which occur during pollutant transport in soils. These variations are related to the following mechanisms (Fried, 1975; Bear, 1979; Freeze and Cherry, 1979): (1) The flow velocity across any cross section within the soil will be greater in the middle than near the walls of the pore channels where there is greater frictional resistance; (2) From continuity principle within the soil, flow velocity across a smaller pore opening is greater than that across a larger pore opening; (3) Flow occurs along paths of varying tortuosity, giving different path lengths and therefore different rates of flow per unit length of soil; and (4) Due to the heterogenous nature of soils, which result in variations in hydraulic conductivities, flow velocity changes. This spreading mechanism is known as dispersion. In a very definite sense, dispersion occurs because of our inability to follow the details of groundwater movement from one pore scale to another. Statistically, advection refers to the average rate of movement while dispersion refers to the deviation from the mean. Also, statistically, dispersion is scale-dependent. The further a particle moves in the subsurface, the greater the range of heterogeneity of hydraulic conductivity it will experience. For example, consider a subsurface containing sand and clay layers, a particle may either start off in sand or in a clay. For short distances of movement, it will remain in the same type of material it started off in and the dispersion coefficient will be characteristic of that material. However, as it moves further from its initial point it may move from sand to clay to sand, etc., with each unit having its own characteristic velocity. Considering two particles, it is apparent that the expected deviation of their locations from the mean position will increase more through the actual heterogeneous system than it would through an idealized homogeneous system.

214

SUBSURFACE POLLUTANT TRANSPORT The dispersion mass flux is usually modelled as a Fickian-type process: OC Jdispersion = -Od ~

[9.91

where Jdispers,onis the dispersive flux, and Dd is the dispersion coefficient which is assumed to be a function of seepage velocity and longitudinal dispersivity and given by: [9.10]

D d = nOt,lVsf~

where ff is the longitudinal dispersivity of the porous medium in the transport direction, and 13is a constant ranging between 1.0 and 2.0 (Freeze and Cherry, 1979). In most applications, it is assumed to be 1.0. The dispersivity, trt, is scale-dependent. In laboratory experiments, a / is found to vary from 0.1 to 10 mm, as shown in Table 9.2. In the field, the dispersivities are sometimes measured through single and multiple well tracer tests. More often, however, what is usually done is that measured field concentrations are simulated with mathematical models and the coefficients adjusted to get an adequate match. The values found in this fashion are usually much larger than laboratory values. Recent literature has shown field values for at to vary from 1 to 100 m or larger. These values are larger than laboratory values by a factor of up to 105, suggesting that dispersion plays a different role in the field than in the laboratory. In practice one usually combines the coefficients of diffusion and dispersion into a single hydrodynamic dispersion coefficient.

Table 9.2: Longitudinal dispersivity values (Gillham and Cherry, 1982) Test type

a t (m)

Laboratory tests Natural gradient tracer tests Single well tests Radial and 2 well tests

9.4

0.0001- 0.01 0.01-2.0 0.03- 3.0 0.5- 15.0

TRANSPORT EQUATION

The fundamental equation of pollutant transport is the conservation o f mass equation. It states that for an arbitrary volume the net rate of mass increase within the volume is equal to the net mass flux into the volume plus any increase in mass due to reactions within the volume. The mass increase term represents the total mass per bulk volume, including both the sorbed mass and that in solution. For a saturated medium the mass density, m, which is the mass per bulk volume, may be represented as: m = nc + (1 - n)psq

[9.11]

TRANSPORT EQUATION

215

where m is the bulk concentration, 9s is the soil density, and q is the sorbed concentration (units of mass or activity sorbed per mass of soil). The product (1- n) p, is the bulk density of the soil, Pb. The net flux includes advective, diffusive, and dispersive mass transport. The mass flux vector, J, is the mass crossing a unit area per unit time. The reaction term includes radioactive decay, biodegradation of organic pollutants, precipitation and redox chemical reactions that may mobilize a pollutant, and others, and may be represented with the symbol S t . The source strength S ~ has units of mass per unit volume per unit time. For an arbitrary control volume, the conservation of mass equation takes the form

dmact - JfJf Jf m d V = -

f f J . n dA + f f f s * av

[9.12]

where the first term represents the time rate of increase in the total mass within the control volume, the second term is the net flux of mass into the volume across the control surface with n the outward unit normal vector to the control surface, and the last term is the mass increase due to sources located within the volume. Eq. [9.12] is an integral form of the continuity equation. Since the control volume is arbitrary, we may also write the continuity equation in the form

Om + ~7. J - - S + Ot

[9.13]

This is the general form of the continuity equation and serves as the starting point for most further investigations of subsurface fate and transport. For most applications we work with a form of the general continuity equation that is simplified in one fashion or another. These simplifications involve making further assumptions as to how to model the various processes that are of interest. Non-polar organic compounds in groundwater are found to be sorbed by soil organic matter present in the porous medium. This sorption is due primarily to hydrophobic interactions resulting in weak, non-specific sorption forces, as discussed in Chapter 5. When the organic compounds are present in trace concentrations, linear adsorption isotherms are often observed, as discussed in Chapter 5: q = Kac

[9.14]

where K d is the distribution or partition coefficient for the chemical species (1/kg). The distribution coefficient is found to be a function of the hydrophobic character of the organic compound and the amount of organic matter present and may be written as:

Kct = K f o c

[9.15]

where Koc is the organic carbon partition coefficient, and foe is the fraction of organic carbon within the soil matrix. Sorption partition coefficients indexed to organic carbon can be estimated from physico-chemical properties of pollutants such as octanol-water partition coefficient and solubility, as discussed in Chapters 5 and 6. Eq. [9.15] is valid only for foc greater than 0.001. Otherwise, sorption of organic compounds on non-organic solids (clays and mineral surfaces) can become

216

SUBSURFACE POLLUTANT TRANSPORT

significant. Also, the linear isotherm model is valid only if the solute concentration remains below the solubility limit of the compound. Typical calculated values for Kd are shown in Table 9.3 (Acar and Haider, 1990).

Table 9.3: Partition coefficient (Ka) and retardation coefficient (R) of selected organic pollutants (Acar and Haider, 1990). Pollutant Location and soil type Kd R foc % (cm3/~a) Acetone Benzene N-Butyl alcchol N-Butyl benzene Carbon Tetrachloride Chlorobenzene Chloroform 3,5-Dichlorobenzene O-Dichlorobenzene 1,2-Dichlorobenzene Dichloroethane 1,4-Dimethylbenzene Ethylbenzene Napthalene Nitrobenzene Quinoline 1,2,3,4-Tetrachlorobenzene Tetrachloroethylene 1,2,4-Trichlorobenzene Trichloroethylene

NA 0.0018 ...... Woodburn 0.34 --NA 0.018 ...... Glatt Valley, Switzerland 3.69 ----0.42 ...... Woodburn 0.50 --NA --1.2 Haggerstown silty loam 2.50 --Woodburn 3.45 --Willanette silty loam 3.30 --Catlin silty loam 4.10 --Glatt Valley, Switzerland 0.50 --Woodbum 0.91 --Haggerstown silty loam 8.00 --NA 0.12 1.4 Haggerstown silty loam 4.30 --Glatt Valley, Switzerland 10.48 --Catlin silty loam 17.30 --NA --7.00 Baton Rough high plasticity 8.30 4.00 Clay Toluene Glatt Valley, Switzerland 0.37 --O-Xylene Catlin silty loam 10.40 --foc = organic carbon content; NA = not applicable; a = organic matter content (%).

1.10 NA 1.10 NA 1.94 1.10 1.60 a 4.04 NA 1.10 1.94 NA 1.10 NA 4.04 NA 1.0 NA 4.04

Using the linear sorption isotherm and assuming local equilibrium, the expression for the bulk concentration can be written as:

m = ngbKaC

[9.16]

Losses due to biotic processes are often modelled as either first or zero order decay. Assuming that the source term is actually represented as a first order equation with an apparent or effective rate constant, A, then:

TRANSPORT EQUATION S + = - ~.m

217 [9.17]

The relationship given by Eq. [9.17] states that the loss rate is proportional to the total mass present. As long as linear partitioning relationships hold, then Eq. [9.17] remains true as an effective rate constant. This does not, however, actually represent the loss rate from the aqueous phase. The mass flux vector, J, in terms of advection, diffusion and dispersion, is given by: J = Jdvection + Jdiffusion + Jdispersion

[9.18]

J = nVsC - nDhd" Vc

[9.19]

where Dhd is hydrodynamic dispersion coefficient which is given by: [9.20]

D hd = T,D O+ a lYs ~

Substituting Eqs. [9.16], [9.17] and [9.19] into Eq. [9.13] gives: (n + PbKd) OC + 7 " ( n v s c ) + ~ (n + PbKd) c = V " (nDhd" 7C) Ot

[9.21]

If the porosity is constant then we introduce the retardation f a c t o r as: R=I+--K

Pb

[9.22]

d

n

The physical significance of the retardation f a c t o r is that it measures how much slower a solute migrates than water. Thus a retardation factor of ten means that the average speed of the solute is ten times slower than that of water. In addition, if the flow is steady and there are no volumetric sinks (leakage, infiltration, evaporation, etc.) then Eq. [9.21] may be written as: R Oc + v s 9 Vc + )~Rc - V ' ( D h d " Vc) Ot

[9.23]

Eq. [9.23] is the form of the pollutant transport equation that is used for most analytical models in one, two, and three dimensions with the added assumption that the velocity field is uniform. In particular, the one-dimensional form of the pollutant transport equation is: OC Ot

-

Dhd 02C R

Ox 2

-

Vs OC R

Ox

-,~c

[9.24]

218

SUBSURFACE POLLUTANT TRANSPORT

The first term gives the rate of change of concentration at a given location. The second term accounts for the effect of hydrodynamic dispersion. The third term accounts for the advection effect and the last term accounts for the sink term which is modelled as first order decay.

9.5

SOLUTE TRANSPORT MODELS

Analytical and numerical models are used to simulate the subsurface transport of chemicals. The numerical models are the most general since they may be tailored to address site specific conditions. However, these numerical models require a significant data base, which may not be available in the initial phases of a site investigation. Analytical models require simplifying assumptions, but are much more computationally efficient and need less specific data. They are specially useful for screening calculations. This section focuses on analytical models for column experiments, chemical spills, and chemical plumes from continuous releases of pollutants. 9.5.1

Conservative Tracer

A conservative tracer is a substance that moves through a porous media without interacting with the matrix or undergoing chemical transformations. Understanding the transport of conservative species is the first step toward understanding the fate and transport of hazardous and radioactive chemicals, which may be influenced by a wide range of processes, namely advection, diffusion, and dispersion. Diffusion

For cases where the seepage velocity is very low, diffusion will govern the transport process. The transport equation is given by:

de Ot

- Ds

02C OX 2

[9.25]

This equation represents the rate of change of solute concentration. The solution for this equation will depend on the initial and boundary conditions. Given a constant source concentration and the following initial and boundary conditions: at

t = O:

at x = O atx=l at x = 0

C = Co > 0 c=O

at

t>0."

c=co

the solution to Eq. [9.25] is given by Crank (1956) as:

erC/co

[9.26]

SOLUTE TRANSPORT MODELS

219

where Ds is soil diffusion coefficient.

Advection- Dispersion The advection- dispersion equation for conservative species takes the form: 0r

02e

Ot

- o hd OX 2

Table 9.4" Complementary error function

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.96 1.0

erf({)

erfc(~)

0 0.056372 0.112463 0.167996 0.222703 0.276326 0.328627 0.379382 0.428392 0.475482 0.520500 0.563323 0.603856 0.642029 0.677801 0.711156 0.742101 0.770668 0.796908 0.820891 0.842701

1.0 0.943628 0.887537 0.832004 0.777297 0.723674 0.671373 0.620618 0.571608 0.524518 0.479500 0.436677 0.396144 0.357971 0.322199 0.288844 0.257899 0.229332 0.203092 0.179109 0.157299

OC -

Vs

[9.27]

OX

(erfc(~))

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

erf(~)

erfc({)

0.880205 0.910314 0.934008 0.952285 0.966105 0.976348 0.983790 0.989091 0.992970 0.995322 0.997021 0.998137 0.998857 0.999311 0.999593 0.999764 0.999866 0.999925 0.999959 0.999978

0.119795 0.089686 0.065992 0.047715 0.033895 0.023652 0.016210 0.010909 0.007210 0.004678 0.002979 0.001863 0.001143 0.000689 0.000407 0.000236 0.000134 0.000075 0.000410 0.000022

The solution to Eq. [9.27] subject to the above stated initial and boundary conditions is given by Crank (1956) as"

Co

2

2 D ~h~)

exp ~

,erfc

2D~J)

[9.28]

220

SUBSURFACE POLLUTANT TRANSPORT

The erfc appearing in the analytical solution is the complementary error function which, for any argument ~, is given by: erfc ( - ~) = 2 - erfc (~)

erfc (~) = 1 - e r f (~);

[9.29]

where erf(~) is the error function of the argument ~ and is given by:

err (~)

= 2

,(-, (-1)" ~2,+1

,/-~ ~_- n((Zn + 1)

[9.30]

y--

Values of erf(~) and erfc (~) are shown in Table 9.4 (Freeze and Cherry, 1979).

Sample problem 1: A site investigation study has revealed that the site contains mixed chemicals buried beneath the ground surface and extending to a depth of 1 m, as shown in Figure 9.1. A thick layer of clay measuring 3 m in depth was found under the buried waste. The groundwater level was found at a depth of 0.5 m from the ground surface. At a depth of 4 m from the ground surface, the groundwater analysis has shown that one of the dominant pollutants has a concentration, Ce, of one half of the initial concentration, Co, at a depth of 1 m from the ground surface. The clay soil has a porosity of 0.5 and hydraulic conductivity of 5 x 10.9 m/sec. It is required to calculate the transient times by considering advection only, diffusion only, and advectiondispersion processes.

Figure 9.1. Schematic diagram showing site characteristics.

SOLUTE TRANSPORT MODELS Solution: hydraulic head, h, = soil layer depth, L, = hydraulic gradient, i, = Soil diffusion coefficient, D, = Dispersion coefficient, ~ =

221

0.5 m 3m

(h+L)/L = (0.5 + 3)/3 = 1.167 ~ v,. 1m

(a) Advection process

v~=ki/n t = L/vs

5• -9 x 1.167/0.5 = 1.167 • 10 .8 m/sec = (3/1.167 • 104) • (1/365 • 24 • 60 • 60) = 8.15 years =

(b) Diffusion process Using Eq. [9.26]"

c/c o X

D,.

=0.5 =3m = 1 x 1.167 x 1 0 4 = 1.167 • 10 .8 m2/sec =x/2VDst=3/(2 v / 1 . 1 6 7 x 1 0 .8 x 365 x 24 x 60 x 60 t) = 2.4726/ V t

The transient time calculations are shown in Table 9.5. By plotting the relationship between

CJCo and t, the transient time at CJCo =0.5 found to be approximately 28 years.

Table 9.5" Transient time calculations for diffusion process t (years)

~

erfc (~)

c~/c~

5 10 20 30

1.105 0.78 0.55 0.451

0.1198 0.2889 0.4366 0.5245

0.1198 0.2889 0.4366 0.5245

(c) Advection-dispersion processes By using Eq.[9.28]

[9.28]

222

SUBSURFACE POLLUTANT TRANSPORT x vst ~

v~ t/D~ ~2

=3m = 1.167 • 10 .8 • 365 • 24 • 60 • 60 t = 0.368 t = (x- v, t) / 2 v/ D,. t = (3- 0.368 t )/(2 v/ 0.368 t) = ( 3 - 0.368 t) / 1.2133 ~/t = 1.167 x 10 .8 • 3/1.167 x 10-8= 3 = (x + vs t) / 2 ~/Ds t = (3 + 0.368 t )/(2 v/ 0.368 t) = ( 3 + 0.368 t) / 1.2133 v/t

The transient time calculations for advection- dispersion process are shown in Table 9.6. By plotting the relationship between C/Co and t, the transient time at c/co =0.5 found to be approximately 7 years.

Table 9.6: Transient time calculations for advection- dispersion process

9.5.2

t (years)

~l

erfc (~l)

~z

erfc (~z)

c,/c o

5 7 8 10

0.427 0.424 0.0163 -0.177

0.5716 0.5716 0.943 1.168

1.784 1.737 1.732 1.741

0.0109 0.0162 0.0162 0.0162

0.395 0.448 0.634 0.747

Reactive Chemical Species

For reactive chemical species with retardation factor, R, the equivalent forms of Eqs. [9.26] and [9.28] are:

Co and Co

er cl I 2V/(

/R)t

l(e lX Vsl tl+ IvJ

2

2~/(D~a/R)---~tj

exp ~ h a

,erfc

[9.311

-2v/(Dhd/R)t) -

I

[9.32]

respectively.

Sample P r o b l e m 2" For the same information given in sample problem 1, what are the transient times if R - 2.

SOLUTE TRANSPORT MODELS

223

Solution: By following the same procedures as described previously, the transient times will be doubled. For diffusion only, t = 56 years, and for advection dispersion, t = 14 years. 9.5.3

Spill of Pollutants

Pollutant releases to the subsurface may occur over short periods of time, or there may be a slow continual release over a long time period. The first of these cases may be modelled by assuming that the release is instantaneous, while the second leads to plume models. If a mass of pollutant is released to the water table over a short duration, the chemical slug is subsequently transported downgradient as it spreads in the direction of flow, transverse horizontally and transverse across the thickness of the aquifer. If the flow is uniform in the x-direction at a seepage velocity, vs, then the concentrations in one, two, and three dimensions are given by (Hunt, 1978; Wilson and Miller, 1978):

(x-(v]R)t)2) 4(Dx/R)t

Mexp(-M) exp[-

c(x,t) =

[9.33]

nR~(4nD~~)

(x -(v]R)t) 2 M exp(-M) exp

c(x,y,t)

-

4~Dx~

ye

)

- 4(/~-)R)t

=

[9.34]

nR

~(4~~-~)(4r~_Dff_[)

and M c(x,y,z,t)

( exp(-M) exp [ -

-

(x

y2

(v]R)t) 2

4(D]R)t

-

_

4(D~/R)t

z 2

)

4(D=/R)t [9.35]

nR

d(4~-~)(4~-D~

f ) ( 4 n ( D ; t)

where c is the concentration (M/L3), M is the mass of spilled pollutant which varies with the number of dimensions: (a) in one dimension, it is an instantaneous plane source (M/L2), (b) in two dimensions, it is an instantaneous line source (M/L), and (c) in three dimensions, it is an instantaneouspoint source (M), x is distance measured in the direction of down-gradient from the point of spill (L), D~, D~, and D= are hydrodynamic dispersion coefficients in x, y, and z directions respectively (L2/T), n is the porosity (L3/L3), t is time (T), R is retardation parameter, and v~, is

224

SUBSURFACE POLLUTANT TRANSPORT

seepage velocity (L/T). According to Eqs. [9.33] to [9.35] the maximum concentration occurs at the location x = vs t/R and y = z = 0, and is given by: M exp(-~t) C m a X --

nR

~(4~-Dff f ) ( 4 ~ _ D ~ ) ( 4 ~

(Dzt)

[9.36]

The main feature of interest in Eq. [9.36] is that near the source, the maximum concentration decreases as:

Cmax

N

1

t3/2

or since the distance of migration of the centroid of the polluted mass is given by concentration decreases a s L -3/2.

[9.37]

L = vs t/R, the

Sample Problem 3: An instantaneous release of 5 kg of a pollutant due to an accident took place on a surface of a soil deposit. The soil was characterized by having a porosity of 0.3 and seepage velocity of 0.2 m/day. Assume a dispersion coefficient of 1 m and decay constant of 0 day -1 (no decay). It is required to calculate the following, considering one-dimensional analysis: (1) The concentration at a distance of 1 m from the source after 1 day, and (2) The maximum concentration and it is location.

Solution: We use Eq. [9.33 ]" Mexp(-~,t) exp (-(x-(vs/R)t)2)4-(D--~

c(x,t)

=

[9.33]

nRI(4~D~~)

where: M

D~ R A n x

t

=5kg = 0.2 m/day = err v,.= 1 • 0.2 = 0.2 m2/day =1 = 0 day l =0.3 =lm =lday

SOLUTE TRANSPORT MODELS

225

yielding the concentration at x = 1 m and t = 1 day: c(1,1) = { 5 e x p [ - ( 1 - 0 . 2 x l ) 2/(4x0.20]}/{0.3xl~/4x• = 4.72 kg/m 3 The maximum concentration is located at a distance x = Cmax = {5}/ {0.3 X 1 X/4 X • 0.2 t} = 10.513 kg/m 3

9.5.4

Vs t/R

= 0.2 m.

Pollutant Plume

If a pollutant is released at a constant rate from the source for a long enough period of time, a concentration distribution in the shape of a pollutant plume develops. The steady plume models are useful in conservative predictions because they provide the maximum likely concentration that may be observed at a given location. The concentration distribution in one, two, and three dimensions are given by (Hunt, 1978; Wilson and Miller, 1978):

c(x,t)

= Am exp (-~,t)

2n(vJR)

erfc ( x - (Vs/R)t I ( -V/4(D-~t )

[9.381

where A m is continuous plane source (ML2T~). The concentration in two dimensions is given by:

c(x,y,t)

-

Am exp (-~,t)

exp

(x

2n X/4X(Vs/R)(D~/R)r

r)vs 2D

erfc - - ~ ~/4(Dxx/R)t )

[9.39]

where A m is continuous line source (MLITI), and r=

X2 +

D,y

y

The concentration in three dimensions is given by:

c(x,y,z,t) -

Am exp (-Xt)

exp

8nxr ~/(D /R)(DJR)

rvsI i r vs ,l

219=

erfc - V/4(Dxx/R)t )

where d m is continuous point source (MT~), and r = I x 2 + D yy xx

2 +D= Z 2

Dzz

[9.40]

226

SUBSURFACE POLLUTANT TRANSPORT

Sample P r o b l e m 4: A continuous release of 5 mg/day of a pollutant due to leakage is taking place on a surface of a soil deposit. The soil was characterized by a porosity of 0.3 and seepage velocity of 0.2 m/day. Assume a dispersion coefficient of 1 m and a decay constant of 0 day 1. It is required to calculate the concentration of pollutant as a function of distance after one year, using onedimensional analysis. Solution: We use Eq. [9.38]

c(x,t) =

Am exp (-~,t)

erfc

2n(vs/R)

x - (v/R)t]

[9.38]

(4(D J R ) t )

where: Am

= 5 mg/day = 0.2 m/day = a'r v,.= 1 • 0.2 = 0.2 m2/day =1 = 0 day l =0.3 = 365 days

Ys

D~x R 2 n t

and obtain:

c(x,

365 days) = {5 / (2 • 0.3 • 0.2)}

erfc

[(x- 0.2 • 365)/( v/4 • 0.2 • 365)]

The concentration profile is tabulated in Table 9.7.

Table 9.7: Concentration variations with distance after 1 year for a pollutant source with a constant rate of release x (m)

~

erfc (~)

c (mg/m 3)

1 10 20 30 40 50 60 70 80 90 100

-4.21 -3.686 -3.10 -2.516 -1.931 -1.346 -0.761 -0.176 0.409 0.995 1.58

2 2 2 1.999 1.993 1.934 1.711 1.223 0.572 0.157 0.024

83.33 83.33 83.33 83.29 83.03 80.58 71.29 50.96 23.83 6.55 0.987

METHODS FOR CALCULATING TRANSPORT PARAMETERS

9.6

METHODS FOR CALCULATING TRANSPORT PARAMETERS

9.6.1

Laboratory Methods for Hydraulic Conductivity Testing

227

Notable differences are observed between hydraulic conductivities measured in the laboratory and in the field. The ratio between field and laboratory hydraulic conductivities may be as large as 1000. The difference may be attributed to the fact that the laboratory apparatus is capable of testing only a small volume of soil. Therefore, the sample may be too homogeneous, and not representative of actual conditions. In the field, soils tend to be heterogeneous, with cracks, fissures, roots, and animal burrows. These factors will all affect soil hydraulic conductivity. Also, construction procedures may not be highly controlled, creating variation in water content and compaction. For these reasons, extreme care must be taken when testing soils for hydraulic conductivity in the laboratory. Due to time constraint and expense of field testing, laboratory measurements will almost always be required. There are at least two basic types of apparatus used for laboratory testing: the rigid wall permeameter and the flexible wall permeameter. Considerable discussion on the relative merits of these permeameters exists (Daniel et al., 1985; Bowders et al., 1986; Madsen and Mitchell, 1989). Each has its advantages and disadvantages and a discussion of these considerations is presented here.

(a) Compaction mould permeameter

Figure 9.2. Compaction mould permeameters.

(b) Double-ring compaction mould permeameter

228

SUBSURFACE POLLUTANT TRANSPORT

Rigid Wall Permeameter The are two principal types of rigid or fixed wall permeameters: the modified compaction mould, and the consolidation cell. The compaction mould permeameter is the same as that used in ASTM D698 for determining moisture-density relations for soils. It consists of two end plates, a cylindrical compaction mould, and a collar, as shown in Figure 9.2(a). The soil is compacted directly into the mould, trimmed, and permeated with the liquid stored in the collar place above the soil specimen. Back pressure saturation is not applied, and the hydraulic pressure gradient is achieved by pressurizing the permeant with compressed air. The inflow and outflow quantities are measured to determine the rate of flow for a period of time. Darcy's Law is used to calculate the hydraulic conductivity (Bowders et al., 1986). The main advantages of this cell, as reported by Madsen and Mitchell (1989), are the low cost of test sampling, and the simplicity of operation. Other strong points include the ease in testing compacted samples, and the use of reasonable confining pressures. However, the compaction mould has several serious drawbacks. The first is that since no back pressure is applied, complete saturation cannot be achieved. Also, there is no means of controlling the applied stresses, since the applied vertical stress is zero. This is inconsistent with field conditions where an overburden creates vertical stress. The greatest disadvantage is the potential for sidewall leakage. This is a major concem when permeating the soil with organic chemicals, which have a tendency to reduce the diffuse double layer and cause shrinkage. Daniel et al. (1985) reported tests where visible gaps were seen between the soil and the cell wall following permeation of kaolinite with heptane. A double-ring compaction mould consists of a compaction mould and a ring built into the base plate, as shown in Figure 9.2(b). The primary function of the ring is to separate the outflow that occurs through the central portion of the soil from the outflow that occurs near or along the sidewall. If there is significant sidewall leakage, the rate of flow into the outer collection ring will be much greater than the rate of flow into the inner ring. If the rates of flow, adjusted to take into account the differences in cross-sectional area of the test specimen that is intercepted, are unequal, one could reject the test and setup a new one. Although at present there is relatively little experience with this device, the cell has worked well in a few tests and shows excellent promise. The disadvantages are the same as those indicated previously for the compaction mould permeameter, except that one has some indication of the magnitude of sidewall leakage. For hydraulic conductivity testing, the standard consolidation procedure is followed, whereby the sample is trimmed into a ring (typical height of 13 to 25 mm and typical diameter of 50 to 80 mm) and then clamped into the base. The reservoir surrounding the ring is filled with water, and the soil is consolidated by applying the desired vertical stress, with deformation measured by a dial gauge. Permeant is then introduced at the base of the sample, with the leachate flowing upward. A schematic presentation of the consolidation cell is shown in Figure 9.3. A modification may be made whereby a thicker walled cell is used, so that compaction takes place directly in the ring. Consolidation and permeation follow the same procedures as for the standard consolidation cell. Since the sample is consolidated, the applied vertical stress can be of the same magnitude as that found in the field. The applied stress tends to push the soil against the sidewalls, helping to minimize leakage. For the modified cell, compacting directly in the mould ensures a seal between the material and the cell walls. The dial gauge provides for the determination of vertical deformation. Other advantages include cost efficiency and the fact that the tests do not take as long to conduct since the samples are relatively thin. The upward flow of the permeant helps saturation. Although the potential for sidewall leakage is reduced, it can never be eliminated for a rigid wall permeameter.

METHODS FOR CALCULATING TRANSPORT PARAMETERS

229

Another drawback is that complete saturation cannot be ensured. The application of back pressure may expand the ring, thereby creating sidewall leakage. Finally, the thin samples, though they shorten testing time, may not be representative of field conditions.

Figure 9.3. A schematic of the consolidation cell.

Flexible Wall Permeameter

Triaxial cells are used to perform flexible wall hydraulic conductivity tests. The specimen is confined in a membrane which is pressurized to keep it in contact with the soil specimen. This virtually eliminates sidewall leakage. A variety of specimen heights and diameters may be used, depending on the size of the apparatus. A pressure transducer measures pressure drop across the sample, and double drainage lines help to clear air bubbles. Back pressure is generally applied to fully saturate the soil. All stresses, horizontal and vertical, may be controlled, and specimen deformation may be measured. A schematic presentation of the flexible-wall permeameter is shown in Figure 9.4. A drawback of this device is the possibility of chemical pollution of the membrane. Some chemicals react with latex, thereby destroying the membrane. To alleviate this problem, the specimen may be wrapped with Teflon tape prior to placement of the membrane, or, for short term tests, the membrane could be placed in the chemical solution before starting the test to determine if chemical attack will occur. The high confining pressure applied to the specimen tends to close any cracks in the soil. Since a field soil is not homogeneous, this situation does not replicate actual conditions. The laboratory hydraulic conductivity in this case will be lower than the true value. Other disadvantages include higher costs and increased testing time due to the use of larger samples.

230

SUBSURFACE POLLUTANT TRANSPORT

Figure 9.4. A schematic presentation of the flexible-wall permeameter.

9.6.2

Laboratory Methods for Adsorption Characteristics

In order to study the adsorption characteristics of soils, two experimental techniques are generally used in the laboratory: (1) batch equilibrium test, and (2) soil column leaching test. These techniques are described below:

Batch Equilibrium Test The main objectives of this test are to: (I) study soil attenuation of pollutants at equilibrium, (2) estimate the number of pore volumes required to achieve breakthrough of a constituent into the effluent liquid, and (3) calculate the retardation parameter required in the pollutant transport equation. Test procedures The clay is air-dried and ground with a mortar and pestle until a uniform powdery texture is obtained. A fixed amount of soil, say 4 g, is placed in glass sample bottles. Various concentrations of leachate are prepared, say 60, 250, 500, 1000, 1500, 2000 ppm. A constant volume of each leachate concentrations, say 40 ml, is then added to each bottle which is then capped tightly. A solution with a soil ratio of 10:1 was recommended by the United States Environmental Protection Agency (EPA, 1987) for estimating soil attenuation of chemicals from batch adsorption tests. Triplicates are prepared for each concentration to ensure accuracy. The bottles are then placed in a rotary shaker and kept at a constant temperature of 20 ~C. The bottles are shaken for at least 24 hours to ensure equilibrium. At the end of the shaking period, the samples are centrifuged to separate the clay from the liquid. One of the bottles, however, contains just the leachate and no clay (blank). The supernatant liquid from the bottles is filtered, and the equilibrium concentration in the liquid phase

METHODS FOR CALCULATING TRANSPORT PARAMETERS

231

of a constituent of interest (c, expressed in units of mass of constituent per unit volume of liquid) is measured using the appropriate analytical method. The concentration of the constituent in the leachate itself is determined from the bottle with no soil and is denoted Co. Analysis The adsorption mass ratio, q, is computed for each bottle as follows: q :

(Co- C) V

M

[9.41]

where V is the volume of liquid in a bottle (40 m/), and M is the mass of soil in the bottle (4 g). The numerator in Eq. [9.41 ] represents the mass of constituent adsorbed onto the solid phase, and it is divided by the mass of the soil to obtain a measure of the relative mass of the constituent adsorbed on the solid phase. The values of q are plotted as a function of the equilibrium concentration. For constituents at low or moderate concentrations, the relationship between q and c can be expressed as: q = kdc b

[9.42]

where kd and b are coefficients that depend on the constituents, nature of the porous material and the interaction mechanism between it and the constituents. Eq. [9.42] is known as the Freundlich isotherm. If b = 1, then q versus c data will be a straight line. Such an isotherm is termed linear, and Eq. [9.42] with b = 1 reduces to: dq _ kd dc

[9.43]

where kd, known as the distribution coefficient, is used for pollutant partitioning between liquid and solids only if the reactions that cause the partitioning are fast and reversible and only if the isotherm is linear. For cases where the partitioning of the contaminants can be adequately described by the distribution coefficient (i.e. fast and reversible adsorption, with linear isotherm), the retardation factor, R, can be expressed as: R = 1 + Pd kd

[9.44]

n

where Pd is the dry mass density (mass of dry solids divided by the total volume of the soil) of the test specimen, and n is the porosity of the test specimen. The retardation parameter also can be expressed as the ratio of the breakthrough times of an adsorbed chemical relative to that of a nonadsorbed tracer.

232

SUBSURFACE POLLUTANT TRANSPORT

Leaching Column Test The main objectives of leaching column test are to: (1) study pollutant migration and attenuation by soils, and (2) estimate the transport parameters which control the migration of pollutants through soils.

Test procedures In the leaching column test, rigid-wall or flexible-wall permeameter is used. The soil is compacted into four replicate soil columns, and the leaching cells are then assembled. First, steadystate fluid flow is established through the soil specimen. After steady state has been established, the fluid in the effluent reservoir (usually water) is changed to a solution with known and constant concentration (Co) of particular chemical species. The concentration, c e, of a chemical species appearing in the effluent reservoir is measured over time and the results are plotted in the form of solute breakthrough curves, or relative concentration, CJCo, versus time or pore volumes of flow (PV). A pore volume of flow for a saturated soil is the cumulative volume of flow through the soil divided by the volume of the void space in the soil. Chemical Analysis Soluble ions: A known amount of soils, 30 g, is taken from each layer. The soil is placed in two plastic containers, to which distilled water is added in two stages. The soil/liquid ratio is 1:10, which is the same ratio used for the adsorption isotherm. The mixture is shaken for approximately 3 hours, poured into small tubes, and centrifuged. The supernatant is separated from the clay and stored in two glass bottles. This washing procedure removes pore liquid, which may contain some of the pollutants, from the clay. The soil remaining after washing is placed in an oven and allowed to dry overnight. The supernatant, washes, and the leachate are analysed with an atomic absorption spectrophotometer. The reading from the machine minus the concentration found in the background solution is the true quantity of the required cation in ppm. The conversion to meq/100g from ppm requires the following formula meq/lOOg = ppm x 2.5 EWC

[9.45]

where EWC is equivalent weight of cation. The amount of specific constituent in the pore solution is determined as follows: m =

[C] x WDS • 400 ml WDS., x 1000 ml

[9.46]

where mc is mass of constituents (g), [C] is the concentration of specific constituent in the wash in (g/l), WDS is the weight of dry soil for each layer (g), 400 ml is the volume of distilled water used in washing, WDSw = 30 g/moisture content at end of test, and 1000 ml is the amount of ml in one litre. The total mass of specific constituent is the sum of the mass of constituents per layer.

METHODS FOR CALCULATING TRANSPORT PARAMETERS

233

Exchangeable ions: Exchangeable ions can be determined from the Silver-thiourea method (Chhabra et al., 1975) method in which ammonium acetate is prepared under a fume hood by adding 57 ml of concentrated acetic acid and 68 ml of concentrated ammonium hydroxide to 700 ml of distilled water. The pH of the solution can be adjusted as required. Once the required pH is reached, distilled water is added to the flask to make one litre. Four grams of the dry soil are carefully weighed into a plastic centrifuge tube. Triplicates are desired to ensure accuracy, so a total of 12 g of dry soil is needed. To each of the three centrifuge tubes, 33 ml of ammonium acetate is added. The tubes are then placed in a mechanical shaker for approximately 2 hours. Following shaking, the tubes are placed in a high speed centrifuge for approximately 10 minutes. The supernatant is then decanted into a 100 ml container. The whole process is repeated two more times by adding 33 ml and 34 ml of ammonium acetate, respectively. The supernatant is, then, analysed using double beam atomic absorption spectrophotometer. The analysis involves measuring various ions in the supernatant, as well as any background levels in the ammonium acetate, with the final concentration equal to the difference between the two. To convert from p p m to meq/100g, the following formula is used m e q / l OOg -

[C] • 100 ml • 100 g x 1000 1000 ml • 4 g • E W C

[9.47]

where 100 ml is the amount of ammonium acetate, 1000 ml is the amount of ml in one litre, 4 g is the amount of dry soil, and 1000 is used to convert to meq. The amount of specific constituent in the soil is found using the expression: _ [C] x W D S x 100 ml 4 g x 1000 ml

m -

[9.48]

where [C] is the average concentration of specific constituent in soil in (g/I), WDS is the weight of dry soil for the layer, 100 ml is the amount of ammonium acetate added, 1000 ml is the amount of ml in one litre, and 4 g is the weight of the dry soil used.

Mass Balance: To determine the total volume input, the total number of pore volumes collected is multiplied by the size of one pore volume, in ml. The final output component is the leachate. For each pore volume, the exact volume of solution which passed through the sample is determined. The amount of specific constituents in g/ml is multiplied by the volume of leachate, thereby giving the amount of specific constituent in a particular pore volume in grams. The procedure is repeated for all the pore volumes. The final mass balance equation for specific constituent is M e : M L + Msl + MEt 4-sink~source

[9.491

where M I is the mass introduced, M E is the mass leached, MsI is the mass of soluble ions (in pores), and MEI is the mass of exchangeable ions. The sink term is due to microbial action while the source term is due to desorption of ions.

234

SUBSURFACE POLLUTANT TRANSPORT

Data Extracted From Leaching Column Tests Four kinds of data can be obtained from leaching column test. These are: (1) breakthrough curve (effluent concentration versus pore volume), (2) migration profiles (concentration versus depth) of soluble ions, (3) exchangeable profiles (concentration versus depth) of adsorbed constituents by soils, and (4) adsorption isotherm (q versus c). To highlight these kinds of data, a specific example is discussed (Mohamed et al., 1994). Material and Methods: The natural micaceous soil used in this investigation consists of illite, phlogopite, hydrobiotite, and vermiculite as basic elements. Specific surface area determined by using the Ethylene Glycol Monoethyl Ether (EGME) adsorption method (Carter et al., 1965) = 206 m2/g. Cation exchange capacity of soil was determined by using two methods" (1) batch equilibrium test - ASTM D 4319 (1984), CEC = 14.89 meq/100g, and (2) the silver-thiourea method (Chhabra et al. 1975), CEC = 13.2 meq/100g. The low CEC values indicated that the soil is mainly illitic and mica in composition. The engineering properties of this soil are" (1) maximum dry density = 1.81 Mg/m 3, (2) optimum moisture content = 16.1%, and (3) hydraulic conductivity, using rigid wall permeameter = 2.3 x 10.9 m/sec.

Test Procedure: A leaching column test was conducted as described above. Soil was compacted at its optimum moisture content and maximum dry density using the static compaction method. Leaching was carried out using municipal solid waste leachate spiked with heavy metals (Pb 2+, Zn2+) and cations (Na +, K § Mg 2+, Ca 2+) in the form of chlorides. The pH of the reconstituted leachate was adjusted to pH of 1.33 by adding concentrated hydrochloric acid. The chemical composition of the reconstituted leachate is: (1) heavy metal concentrations (ppm): Pb 2+= 1372.2, and Zn 2§ = 1141.6, and (2) cation concentrations (ppm): Na 2+ =346, K + = 164.8, Mg 2§ = 43.8, and Ca 2+= 94.5.

Figure 9.5(a). Variation of heavy metal relative concentration with number of pore volumes.

METHODS FOR CALCULATING TRANSPORT PARAMETERS

235

Leaching was carried out under a constant pressure of 103.5 kPa, resulting in a hydraulic gradient of 87.2. During the leaching process, the effluent was collected as a function of time and analysed to determine its chemical composition. At the end of the test, the specimen was extruded, cut into 10 mm thick slices, which were, then, analysed for pore fluid contents (soluble ions) and exchangeable ions.

Migration of Heavy Metals: In a clay soil system, heavy metal may (1) occur in ion exchange sites, (2) be incorporated into or on the surface of crystalline or non-crystalline compounds, or (3) be in the soil pore solution. Most investigations have recognized that heavy metals occur predominantly in a sorbed state. Because of their low solubility, movement of heavy metals in soil has generally been considered to be minimal. Figure 9.5(a) shows the effluent relative concentration, C/Co, in the leachate, for lead, Pb 2+, and Zinc, Zn 2+, as a function of pore volume passage. Ceis the concentration of the ion concerned in the effluent, and Co is the original concentration of the ion concerned in the influent. From the diagram, it is evident, therefore, that most of the Pb > and Zn > are retained in the soils. The results show that a significant amount of heavy metals was retained in the top portion of the soil samples, as seen in the concentration profiles depicted in Figure 9.5(b). Due to lower pH in the influent leachate, it is expected that the retention capacity of the soil in the top part of the column is reduced. However, this depends on the buffering capacity of the soil to any change in pH. It is known that heavy metals would generally precipitate out of the solutions if the solution's pH is high (e.g. Pb > precipitates at pH >5). Since soil pH was initially about 6.5, Pb> precipitates in the soil at the start of leaching. Further leaching decreases soil pH and enhances the mobility ofPb > in solution. After 5 pore volumes, the top 25% of the soil column has pH values ranging from 1.33 to 5, thus enhances the mobility of Pb 2§ in this part of the soil column. Pb > was retained by cation exchange replaceability, as shown in Figure 9.5(c). For the rest of the soil column, which has pH greater than 5, Pb 2+retention mechanism will be due to precipitation in various forms.

Figure 9.5(b). Variation of heavy metal pore fluid concentration with distance.

236

SUBSURFACE POLLUTANT TRANSPORT

Figure 9.5(d). Heavy metal adsorption isotherm; Y(1) and Y(2) are the left and the right vertical axes, respectively.

It can be seen also from Figure 9.5(b) that the amount of Zn 2§ retained by the soil column is less than the amount of Pb 2§ retained. This can be explained by the ease of exchange or the strength with which cations of equal charge are held, which is generally inversely proportion to the hydrated radii or proportional to the unhydrated radii. Therefore, the predicted order of soil retention based on: (1) unhydrated radii is Pb 2§ (0.120 nm) > Zn 2+ (0.074 nm) which agrees with the experimental

METHODS FOR CALCULATING TRANSPORT PARAMETERS

237

data, and (2) metal ion softness, which is a function of ionization potential, change of metal ion and ionic radius, is Pb 2+ (3.58) > Zn 2+ (2.34). In order to investigate the adsorption isotherm of Pb 2+ and Zn 2+, chemical test data are presented in terms of exchangeable cations and equilibrium solution concentration in the pore fluid, as shown in Figure 9.5(d). It can be seen from Figure 9.5(d) that exchangeable Pb 2§ and Zn 2§ increased for lower pHs up to pH = 5, hence the exchangeable cations decreased with further decrease in pH.

Migration of Cations: Figure 9.6(a) shows the effluent relative concentration, C/Co, in the leachate for cations (Na+, K § Mg 2+, and Ca2+). With increasing number of pore volumes permeated, the relative concentration of cations increased. For Na § Mg 2§ and Ca 2§ cations, the relative concentration exceeds 1.0. This can be attributed to the elution of cations from the solid particles. The high relative concentrations of Ca2+and Mg 2+in the effluent collected can be attributed to cation exchange or replacement by Pb 2§ and Zn 2+. Low concentration of K § in the effluent leachate is due to the fact that K § is often adsorbed and incorporated into the interlayer lattice of micaceous soils. Since the soil tested is basically micaceous, there is greater affinity for K § hence adsorption and incorporation into the inter-layer lattice of the soils of the K § ions occurred during the leaching process. Also, it can be noticed that Na + relative concentration reached a steady state approximately after 4 pore volumes while for K § Mg 2+, and Ca 2+, steady state conditions arrived after approximately 2 pore volumes.

Figure 9.6(a). Variation of metal ion relative concentration with number of pore volumes.

The migration profiles of pore fluid cation Concentration versus depth of the soil column are shown in Figure 9.6(b). The migration profiles depict how a particular cationic species migrate or move through the soil column with increasing permeation by the leachate. The initial concentration ofNa + in the influent leachate was 345 ppm while the measured concentration in the pore fluid was

238

SUBSURFACE POLLUTANT TRANSPORT

greater than 500 ppm, indicating Na § desorption. Similar results are obtained for K § Mg 2+ and Ca 2+. As discussed, the increase in concentration of the pore fluid as a function of time is attributed to cation exchange or replacement by A13+, Pb 2+, and Zn 2§ in the top part of the soil column while in the bottom part, Na § and K § are exchanged by Ca 2+ and Mg z+. This is due to the higher valence of Ca 2+ and Mg 2+ compared to Na § and K § and, hence higher replacing power, as shown in Figure 9.6(c).

Figure 9.6(c). Variation of adsorbed metal ion concentration with distance.

METHODS FOR CALCULATING TRANSPORT PARAMETERS

239

The adsorption isotherms of various cations are shown in Figure 9.6(d). Generally, it can be seen that maximum exchange capacity of various cations were obtained at lower pH values due to replacement by heavy metals. For higher pH values, cation exchange decreased.

Figure 9.6(d). Metal ion adsorption isotherm.

9.6.3

Estimation of Transport Parameters

In order to predict the transport and the fate of various pollutant species in the subsurface environment, the transport parameters involved in the goveming set of equations that describe the transport process need to be accurately defined. The laboratory methods which can be used to estimate the transport parameters of chemical species diffusing through waste containment barriers are discussed. In general, methods used to calculate the transport parameters fall into two broad categories-- steady and transient states. This section describes some of the more common procedures which have been used to calculate the transport parameters.

Steady State Methods Decreasing Source Concentration: A schematic diagram illustrating the steady state method is shown in Figure 9.7. The soil is contained between two reservoirs -- a source reservoir containing the solution of interest and a collection reservoir from which samples are drawn for specified chemical analysis. The concentration of the chemical species of interest is higher in the source reservoir than it is in the collection reservoir, hence a concentration gradient is established across the soil sample. Once the steady state condition has been reached, Fick's first law for diffusion can be applied and the diffusion parameter can be calculated as follows:

240

SUBSURFACE POLLUTANT TRANSPORT J=-D

a___c_c

[9.50]

8x The diffusion coefficient can then be calculated (Shackelford, 1991; Yong, Mohamed, and Warkentin, 1992) using: [9.51]

where Js is the mass flux, D is the diffusion coefficient, L and A are the length and cross sectional area of the soil sample respectively, and Ac is the change in mass of the chemical species in an increment of time, At. Since the quantity (L/A Ac) in Eq. [9.51] can be measured or set independently of the test, only the change in mass with respect to time, (Am~At), is measured during the test.

Figure 9.7. Steady state method with decreasing source concentration.

At steady state,

Ami At

-

Am2 At

-

Am At

[9.52]

where Am~ is the decrease in mass of the chemical species in the source reservoir, and Am2 is the increase in mass of the chemical species in the collection reservoir. The use of the difference operator in Eq. [9.52] implies that the concentration gradient across the sample is linear. However,

METHODS FOR CALCULATING TRANSPORT PARAMETERS

241

due to coupled flow processes, the concentration gradient within the soil sample is non-linear. As a result, the calculated diffusion parameter using the external (across the clay) concentration gradient may not be the same as that determined using the internal (within the clay) distribution of concentration. In order to apply this technique, the following conditions have to be satisfied: (1) The concentration of the chemical species of interest should be higher in the source reservoir than in the collection reservoir; (2) The chemical species diffusing from the source reservoir must be continuously replenished while the mass of the chemical species diffusing into the collection reservoir needs to be continuously removed in order to maintain constant concentration difference Ac across the sample; (3) The concentration of the chemical species of interest at steady state in the soil at x = 0 is higher than it is at x = L. In other words, the concentration profile within the soil should be inward concave, as shown in Figure 9.7(c); and (4) The solution pH in the source reservoir as well as in the soil column should be greater than 5. This is due to the fact that when pH < 5, heavy metals interact with clay through exchangeable cations, hence most cations (Na +, K +, Mg 2+, Ca 2+) are desorbed from clay resulting in outward concave profiles in the soil. Time-Lag Method: This method is commonly used to obtain the diffusion coefficient through porous membrane (Jost, 1960; Crank, 1975). A schematic diagram illustrating the time- lag method is shown in Figure 9.8. The soil is contained between two reservoirs -- a source reservoir containing the solution of interest and a collection reservoir. This soil is initially at zero concentration and the concentration at the interface with the collection reservoir is maintained effectively at zero concentration. In this case, the total amount of diffusing substance per crosssectional area, Q,, which has passed through the soil approaches a steady state value as t -~oogiven by (Jost, 1960; Crank, 1975):

[9.53]

where Qt = f0tJ~9 Cl is the concentration in the source reservoir, which is maintained at a constant value with time. Eq. [9.53] is the equation of a straight line on a plot of Qt versus t, as shown in Figure 9.8(c). The intercept on the time axis is the time lag, TL, which is given by: L2

TL- 6D

[9.54]

Therefore, D can be calculated using Eq. [9.54] by plotting Qt versus time and determining the value for the intercept TL. The time lag method requires less control of the test conditions than the decreasing source concentration method since steady state condition has to be established, not maintained. Furthermore, the time required to establish steady state condition can be excessive, especially for relative thick samples.

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SUBSURFACE POLLUTANT TRANSPORT

Figure 9.8. Time lag method

Root Time Method: This method was developed by Mohamed and Yong (1991). The approach is based on an analytical solution of the differential equation of solute transport in clay barrier. The equation is first cast in a non-dimensional form and the Fourier Series is used to solve the differential equation for specified initial an boundary conditions. The concentration, in a nondimensional form, is given by:

c*(~,1:) = (1-~)- 1 exp(_nzz) sin(g~)

[9.55]

and the relative change in concentration is given by: CRC

= exp (-11:21:)

[9.56]

where: Dt

c - c2

x

L 2'

c1 -

L

c2

and 1: is non-dimensional time factor, { is non-dimensional distance, c" is non-dimensional concentration, c is the concentration at specific time and distance, Cl is the concentration at x = 0, c 2 is the concentration at x = L, and L is the length of soil specimen. The theoretical relationship between the non-dimensional relative concentration Cec* and the root time factor, z, is shown in Figure 9.9(c). The theoretical curve is linear up to relative

METHODS FOR CALCULATING TRANSPORT PARAMETERS

243

concentration of 0.2 (80% equilibrium). At a relative concentration of 0.1 (90% equilibrium), the abscissa (AC) is 1.055 times the abscissa (AB). This characteristic is used to determine the point on the experimental curve corresponding to relative concentration of 0.1 (i.e. 90% of the steady state equilibrium time).

Figure 9.9. Root time method

The experimental data reduced in terms of relative change in concentration of specific ion in the collected effluent versus root time generally consists of a short curve representing initial increase in concentration (in the effluent), a linear part and a second curve as shown in Figure 9.9(d). The point (D) shown in Figure 9.9(d) corresponding to the initial condition is obtained by projecting the linear part of the curve until it meets the vertical axis at zero time. A straight line(DE) is then drawn having abscissa 1.055 times the corresponding abscissa on the linear part of the experimental part. The intersection of the line (DE) with the experimental curve locates the point (ago) corresponding to relative concentration of 0.1. The corresponding value t9o can then be obtained. The value of 1: corresponding to CRC*is 0.2436 and the diffusion coefficient, D, is given by: 0.2436 L 2 O

__

%

[9.57]

The method is applicable for the case of adsorption and desorption. It is also applicable for various values of pH in the influent solution (i.e., acid and alkaline conditions). It should be noted that in the case of low pH in the influent, some of the cations are desorbed from the clay, yielding

244

SUBSURFACE POLLUTANT TRANSPORT

c 2 > Cl, hence the experimental relative change in concentration is negative. Transient M e t h o d s

Several different transient methods have been used to calculate transport parameters. The column method is described. The soil column test, traditionally known as leaching column test, has been used to study adsorption and migration of contaminants through clay barriers. First, steady state flow is established through the soil sample by using distilled water in the source reservoir. After steady-state fluid has been established, the fluid in the influent reservoir is changed to a solution with known and constant concentration (Co'S) of particular chemical constituents. The concentration, c e, in the effluent reservoir is measured as a function of time. The data are reduced in the form of breakthrough curves, of relative concentration, C/Co, versus time or pore volumes of flow. Breakthrough curves are modelled using either one of the analytical solutions given by Eqs. [9.26], [9.28],[ 9.31 ], and [9.32]. The diffusion coefficient can then be calculated once, ce, c o, v, L, and t are known.

9.7

S U M M A R Y AND CONCLUDING REMARKS

The manner in which the transport parameters are determined for use in modelling is important. At least five options are available: (1) Using values and relationships reported in various publications for similar situations and circumstances; (2) Using the infinite solution diffusion coefficient as the starting point for the determination of the diffusion-dispersion coefficients and also using adsorption isotherms and adsorption characteristics, with modifications for tortuosity and advective velocity effects in regard to diffusion-dispersion phenomenon; (3) Using experimentally determined values from laboratory experiments with soil from the site and representative leachates; (4) Using information from monitoring wells and chemical analysis from the samples recovered, to back-calculate the transport and adsorption parameters; and (5) A combination of any of the preceding options. Where possible, it should be standard practice to match numerical solutions with established analytical solutions, to have full confidence of the model developed. In the final analysis, a combination of field, laboratory, and analytical studies is the best approach for evaluating pollutant transport in subsurface.