Dispersion and Diffusion

5.5.4 Dispersion and Diffusion

Dispersion is three-dimensional spreading of fluid particles as they flow through porous media. On the microscopic scale, dispersion is caused by deviation in velocity of the fluid particles. Particles within an individual pore will have different velocities; the ones in the center of the pore will travel the fastest, whereas those next to the pore walls will hardly move. Flow directions and velocities will also change as the particles navigate through tortuous travel paths around the individual grains of the porous material.

As discussed by Franke et al. (1990), on a larger (macroscopic) scale, local hetero- geneity in the aquifer causes both the magnitude and the direction of velocity to vary as the flow concentrates along zones of greater permeability or diverges around pockets of lesser permeability. The term “macroscopic heterogeneity” is used to suggest variations in features large enough to be readily discernible in surface exposures or test wells, but too small to map (or to represent in a mathematical model) at a working scale. For example, in

a typical problem involving transport away from a landfill or waste lagoon, macroscopic heterogeneities might range from the size of a baseball to the size of a building. Although the phenomenon of dispersivity has a physical explanation that is relatively easy to understand, the process itself cannot be feasibly measured in the field. There are no widely accepted or routinely applied methods of quantifying field-scale dispersivity, and there are still very few credible large-scale field experiments that can help in better understanding dispersivity in heterogeneous porous media. It has been argued that defining the actual field distribution of hydraulic conductivity and its anisotropy to a satisfactory level of detail would eliminate the need for quantifying yet another uncertain parameter such as dispersivity. However, it is apparent that in many cases it would also not be feasible to determine distribution of the hydraulic conductivity and effective

404 ChapterFive

porosity (and therefore the velocity field) with very fine resolution, particularly in cases of large travel distances. It is for this reason that the dispersivity is explicitly included, through surrogate parameters, in common equations of contaminant fate and transport, in an attempt to somehow account for deviations from the average linear (advective) flow velocity.

The key assumption in deriving a term to represent dispersion is that dispersion can

be represented by an expression analogous to Fick’s second law of diffusion (Anderson, 1983):

Mass flux due to dispersion = ∂

ij ∂

(5.17) x j

where c = concentration and D ∗ ij = the coefficient of dispersion (the i, j indices refer to cartesian coordinates). The coefficient of dispersion is assumed to be a second-rank tensor, where

ij = D ij + D d (5.18)

D ij = coefficient of mechanical dispersion and D d = coefficient of molecular diffusion (a scalar). Molecular diffusion is a microscale process (it happens at the molecular level), which causes movement of a solute in water from the area of its higher concentration to the area of its lower concentration. An effective diffusion coefficient is generally as- sumed to be equal to the diffusion coefficient of the introduced liquid (or ion) in water times a tortuosity (convoluted travel paths between solids) factor, which accounts for the obstructing effects of solids and tortuous paths of the fluid particles. Effective diffu-

sion coefficients are generally around 10 − 6 cm 2 /s, which means that, except for systems in which groundwater velocities are very low, the coefficient of mechanical dispersion

will be one or more orders of magnitude larger than D d . Therefore, in many practical applications, the effects of molecular diffusion may be neglected and the coefficient of dispersion assumed to be equal to the coefficient of mechanical dispersion (Anderson, 1979, 1983). An additional parameter called dispersivity (α), which has units of length, relates the coefficient of mechanical dispersion (or just dispersion) to the average linear velocity in the main direction of groundwater flow (v):

α= D ij

The coefficient of mechanical dispersion and dispersivity are commonly expressed with three components: longitudinal (in the main direction of groundwater flow), trans- verse (perpendicular to the main direction in the horizontal plane), and vertical (perpen- dicular to the main direction in the vertical plane): D x ,D y , and D z , respectively, and α x , α y , and α z , respectively.

A number of researchers have questioned the validity of the quantitative parameters of dispersion described above and their inclusion in equations of contaminant fate and transport. For example, the basic assumption of Fick’s law is that the driving force for the diffusion is the concentration gradient. It is then assumed that Fick’s law is applicable to hydrodynamic dispersion as shown with Eq. (5.17) and that coefficient of disper- sion includes a coefficient of molecular diffusion—Eq. (5.18). However, the molecular

GroundwaterQuality

diffusion is then ignored for all practical purposes. At the end, the pore channel veloci- ties and their variations are “left” to be driven by concentration gradients even though the molecular diffusion is excluded altogether (Knox et al., 1993).

Studies by various researchers have shown non-fickian behavior of dispersion in porous media flow, arguing that use of the coefficient of mechanical dispersion and dispersivity in fate and transport equations has to be reevaluated, including finding

a better mathematical formulation for all time and space scales, possibly in favor of stochastic (probabilistic) approaches (e.g., Matheron and de Marsily, 1980; Smith and Schwartz, 1980; Pickens and Grisak, 1981a, 1981b; Gelhar and Axness, 1983; Dagan, 1982, 1984, 1986, 1988). The main issues when assuming a fickian behavior of dispersion and selecting a value for D or a (in any direction) are as follows (Anderson, 1983; Knox et al., 1993):

r The approach to fickian flow is asymptotic in many cases, resulting in significant non-fickian transport early in the process (dispersivity steadily increases with

distance before it reaches an asymptotic value after a long time). r During development of the dispersion process, there are significant departures

from the classical normal concentration distribution associated with fickian pro- cess. Concentration-time curves are typically skewed on the right, except for long times or large distances from the source.

r There may be hydrogeologic settings where macroscopic dispersion never be-

comes a fickian process. Unfortunately, dispersion remains one of the more uncertain and unquantifiable fate

and transport processes, posing a significant challenge to both the practitioners and the legal community (courts), in trying to find the most appropriate ways of incorporating it into predictive fate and transport models. The rule of thumb, suggested by the USEPA, is that longitudinal dispersivity in most cases could be initially estimated from the plume length as being 10 times smaller (Wiedemeier, 1998; Aziz et al., 2000). This means that, for example, if the plume length is 300 ft, the initial estimate of the longitudinal dispersivity is about 30 ft. However, recognizing the limitations of the available and reliable field- scale data on dispersivity, the agency also suggests that the final values of dispersivity used in fate and transport calculations should be based on calibration to the site-specific (field) concentration data.

The main reason why very few (if any) projects for practical groundwater remedi- ation consider field determinations of dispersivity is that they would require a large number of monitoring wells and the application of large-scale tracer tests. Such studies are expensive and are usually not feasible due to the generally slow movement of tracers in intergranular porous media over long distances. Two of the most widely analyzed and reported large-scale, controlled tracer field tests of dispersion, the Borden Landfill test (Sudicky et al., 1983; Mackay et al., 1986; Freyberg, 1986) and the Cape Cod test (Garabedian et al., 1991; LeBlanc et al., 1991), show similarly small macrodispersivity values compared to the overall travel distance of the injected tracers. At the Borden land- fill site, the dispersivities showed possible scale- and time-dependent behavior, with the approximated asymptotic value of 0.43 m for the longitudinal dispersivity and the travel distance of 65 m after 647 days. The transverse dispersivity was about 11 times smaller than the longitudinal. At the Cape Cod site, the plume did not show time dependence:

406 ChapterFive

a constant longitudinal dispersivity of 0.96 m developed shortly after the tracer injec- tion and a short period of nonlinear growth. The transverse and vertical dispersivities were 0.018 and 0.0015 m, respectively, or 53 and 640 times smaller than the longitudinal dispersivity. The travel distance of the plume was about 230 m after 461 days.

Xu and Eckstein (1995) authored one of the most widely cited practical studies relat- ing the scale effects of dispersion and the selection of appropriate values of dispersivity. The authors conclude that “. . . the rate of increase of dispersivity declines as the scale in- creases. Theoretically, the rate will asymptotically approach zero as the scale approaches infinity. However, our analysis shows that the rate of increase is very small, and the curve of dispersivity versus scale is almost horizontal (with a slope angle of 0.24 ◦ ) when the scale of flow exceeds 1 km. The increase in longitudinal dispersivity at that scale is so small that it can be practically ignored without causing significant error.”

In their analysis, Xu and Eckstein used field data and model-calibration data re- ported by Gelhar et al. (1992) and shown here in Fig. 5.18. Based on the weighted least- squares data-fitting technique, which addresses the reliability of data and the nonlinear

d ispers Longitudinal Dispersivity

10 (scale)] 2.414 itud

inal 10 0 = 0.83.[log

(Xu and Eckstein, 1995) Long

RELIABILITY

10 -1

High Intermediate

10 -2

Low

Data Source: Gelhar et al.,1992

Scale (m)

F IGURE 5.18 Longitudinal dispersivity versus scale data reported by Gelhar et al. (1992). Size of circle represents general reliability of dispersivity estimates. High reliability data are considered to be accurate within a factor of about 2 or 3. (Graph modified from Aziz et al., 2000.)

GroundwaterQuality

characteristics of the correlation between longitudinal dispersivity and the scale of ob- servations, they proposed the following two equations for estimating the longitudinal dispersivity:

α L = 0.94(log 10 L) 2.693 for 1:1.5:2 scheme

(5.21) where α L = longitudinal dispersivity (in units of length), L = scale (length of observa-

α L = 0.83(log 10 L) 2.414 for 1:2:3 scheme

tion), and the first, second, and thirds numbers in the scheme correspond to the weighting factors for low, medium, and high reliability data, respectively.

Gelhar et al. (1992) include 106 data classified as of high, low, or intermediate relia- bility and state that the high reliability data are considered to be accurate within a factor of about 2 or 3. Their dataset includes model-calibrated longitudinal dispersivities, some of which are with low reliability and are at the same time extremely high (e.g., one data point has longitudinal dispersivity of 20 km for a “modeled” migration scale of 100 km?!). Neuman (1990) excludes available model-calibrated values of α L in a similar analysis and proposes the following equation for longitudinal dispersivity when L ≥ 100 m:

α L = 0.32L 0.83 (5.22) Selecting a “representative” value of longitudinal dispersivity and then applying it

to predictive models of contaminant fate and transport is a rather subjective process. Modeling should, therefore, include a thorough sensitivity analysis of this parameter, with the understanding that “more emphasis should be placed on field study and the ac- curate determination of hydraulic conductivity variations and other non-homogeneities and less on incorporating somewhat “arbitrary” dispersion coefficients into complex mathematical models” (Molz et al., 1983).

When the groundwater velocity becomes very low, due to small pore sizes and very convoluted pore-scale pathways, diffusion may become an important fate and transport process. Porosity that does not readily allow advective groundwater flow (flow under the influence of gravity), but does allow movement of the contaminant due to diffusion, is sometimes called diffusive porosity. Dual-porosity medium has one type of porosity that allows preferable advective transport through it; it also has another type of porosity where free gravity flow is significantly smaller than the flow taking place through the higher effective (advective) porosity. Examples of dual-porosity media include fractured rock, where advective flow takes place preferably through fractures, while the advective flow rate through the rest of the rock mass, or rock matrix, is comparably lower, is much lower, or does not exist for all practical purposes. This gradation depends on the nature of matrix porosity; in some rocks such as sandstones and young limestones, matrix porosity may be fairly high and it may allow a very significant rate of advective flow, often as high as or higher than through the fractures. In most hard rocks, matrix porosity is usually low, less than 5 to 10 percent, and it does not provide for significant advective flow. Other examples of dual-porosity media include fractured clay and residuum sediments. In some cases, various discontinuities and fractures in such media may serve as pathways for advective contaminant transport, while the bulk of the sediments may have a high overall matrix porosity and low effective porosity where advective transport is slow.

408 ChapterFive

Flow of solutes with high concentration through the fractures may result in the solute diffusion into the surrounding matrix.

Diffusion is movement of a contaminant from higher concentration toward lower con- centration solely due to concentration gradients; it does not involve “bulk,” free-gravity movement of water particles (as in case of advection and dispersion). The contaminant will move as long as there is a concentration gradient, including when this gradient reverses, such as when fractures are flushed out by clean groundwater and the contam- inant starts to move back from the invaded rock matrix into the fractures (the so-called back-diffusion).

The rate of diffusion for different chemicals (solutes) in water depends on the con- centration gradient and the coefficient of diffusion, which is solute specific (different solutes have different coefficients of diffusion). The diffusion coefficients for electrolytes,

such as major ions in groundwater (Na − ,K , Mg 2+ , Ca 2+ , Cl − , HCO

3 , and SO 2− 4 ) range

C (Robinson and Stokes, 1965; from Freeze and Cherry, 1979). Coefficient of diffusion is temperature dependent and decreases with the decreasing temperature (e.g., at 5 ◦

between 1 × 10 − 9 and 2 × 10 − 9 m 2 /s at 25 ◦

C these coefficients are about 50 percent smaller than at 25 ◦ C).

Flux (F ) of a contaminant moving due to diffusion in a porous medium is described by Fick’s first law:

F = −D e (5.23)

The second Fick’s law describes the change in concentration of a nonsorbing contam- inant due to diffusion:

(5.24) and if the contaminant is also subject to sorption as it moves through the porous media:

dx 2

= D e ∂ · 2 C (5.25)

R dx 2

where D e = effective coefficient of diffusion in the porous medium R = coefficient of retardation (see next section on sorption and retardation)

C = contaminant concentration in groundwater Because of tortuosity, effective diffusion coefficients in the subsurface are smaller than

in free water. The effective diffusion coefficient (D e ) can be determined by using the known (experimentally determined) tortuosity of the porous media or by multiplying the aqueous diffusion coefficient (D 0 ) with an empirical coefficient, called apparent tortuosity factor (τ ), which can range between 0 and 1. This empirical coefficient is related to the aqueous (D 0 ) and effective (D e ) diffusions, and the rock matrix porosity (θ m ) through the following expression (Parker et al.; from Pankow and Cherry, 1996):

D e = τ∼ = θ p

GroundwaterQuality

where the exponent p varies between 1.3 and 5.4, depending on the type of porous geologic medium. Low porosity values result in small τ values and low D e values. Laboratory studies of nonadsorbing solutes show that apparent tortuosity usually has values between 0.5 and 0.01. For example, for generic clay τ is estimated at 0.33, for shale/sandstone it is 0.10, and for granite it is quite small: 0.06 (Parker et al.; from Pankow and Cherry, 1996).

Concentration profile of a nonsorbing solute in subsurface, moving only due to dif- fusion and in one direction (x) from the high- into the zero-concentration layer, can be analytically calculated for various times (t) based on Fick’s second law, using Crank’s equation (Freeze and Cherry, 1979):

i (x, t) = C 0 erfc ·

· (D e t)

2 where C 0 = initial concentration on the high-concentration side of the contact between

two layers and erfc = complimentary error function.