# Difference between revisions of "Dispersion and Diffusion"

Dispersion of solutes in flowing groundwater results in the spreading of a contaminant plume from highly concentrated areas to less concentrated areas. In many groundwater transport models, solute transport is described by the advection-dispersion-reaction equation. The dispersion coefficient in this equation is the sum of the molecular diffusion coefficient, the mechanical dispersion coefficient and the macrodispersion effect.

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## Molecular Diffusion

Figure 1. Conceptual depiction of diffusion of a dissolved chemical recently placed in a container at Time 1 (left panel) and then distributed throughout the container (right panel) at Time 2.
Figure 2. Conceptual depiction of mechanical dispersion (adapted from ITRC (2011)[3]).

Molecular diffusion is the result of the thermal motion of individual molecules which causes a flux of dissolved solutes from areas of higher concentration to areas of lower concentration (Figure 1). The diffusion coefficient is a proportionality constant between the molar flux due to molecular diffusion and the concentration gradient and is a function of the temperature and molecular weight. In locations where advective flux is low (clayey aquitards and sedimentary rock), diffusion is often the dominant transport mechanism.

The diffusive flux J (M/L2/T) in groundwater is calculated using Fick’s Law:

Equation 1:
J = -De dC/dx
Where De is the effective diffusion coefficient and

The effective diffusion coefficient for transport through the porous media, De, is estimated as:

Equation 2:
De = Dm ne δ/Τ
Where Dm is the diffusion coefficient of the solute in water,
ne is the effective porosity (dimensionless),
δ is the constrictivity (dimensionless) which reflects the restricted motion of particles in narrow pores[4], and
Τ is the tortuosity (dimensionless) which reflects the longer diffusion path in porous media around sediment particles[5].

Dm is a function of the temperature, fluid viscosity and molecular weight. Values of Dm for common groundwater solutes are shown in Table 1.

Table 1. Diffusion Coefficients (Dm) for Common Groundwater Solutes.
Aqueous Diffusion Coefficient Temperature
(°C)
Dm
(cm2/s)
Reference
Acetone 25   1.16x10-5   Cussler 1997
Benzene 20 1.02x10-5 Bonoli and Witherspoon 1968
Carbon dioxide 25 1.92x10-5 Cussler 1997
Carbon tetrachloride 25 9.55x10-6 Yaws 1995
Chloroform 25 1.08x10-5 Yaws 1995
Dichloroethene 25 1.12x10-5 Yaws 1995
1,4-Dioxane 25 1.02x10-5 Yaws 1995
Ethane 25 1.52x10-5 Witherspoon and Saraf 1965
Ethylbenzene 20 8.10x10-6 Bonoli and Witherspoon 1968
Ethene 25 1.87x10-5 Cussler 1997
Helium 25 6.28x10-5 Cussler 1997
Hydrogen 25 4.50x10-5 Cussler 1997
Methane 25 1.88x10-5 Witherspoon and Saraf 1965
Nitrogen 25 1.88x10-5 Cussler 1997
Oxygen 25 2.10x10-5 Cussler 1997
Perfluorooctanoic acid (PFOA) 20 4.80x10-6 Schaefer et al. 2019
Perfluorooctane sulfonic acid (PFOS) 20 5.40x10-6 Schaefer et al. 2019
Tetrachloroethene 25 8.99x10-6 Yaws 1995
Toluene 20 8.50x10-6 Bonoli and Witherspoon 1968
Trichloroethene 25 8.16x10-6 Rossi et al. 2015
Vinyl chloride 25 1.34x10-5 Yaws 1995

## Mechanical Dispersion

Mechanical dispersion (hydrodynamic dispersion) results from groundwater moving at rates both greater and less than the average linear velocity. This is due to: 1) fluids moving faster through the center of the pores than along the edges, 2) fluids traveling shorter pathways and/or splitting or branching to the sides, and 3) fluids traveling faster through larger pores than through smaller pores[6]. Because the invading solute-containing water does not travel at the same velocity everywhere, mixing occurs along flow paths. This mixing is called mechanical dispersion and results in distribution of the solute at the advancing edge of flow. The mixing that occurs in the direction of flow is called longitudinal dispersion. Spreading normal to the direction of flow from splitting and branching out to the sides is called transverse dispersion (Figure 2).

## Macrodispersion

Figure 3. Matrix diffusion processes and their effects on plume persistence and attenuation.

Macrodispersion is the name given to the plume spreading caused by large-scale heterogeneities and associated spatial variations in advective transport velocity. In some groundwater modeling projects, large values of the macrodispersion coefficient are used as an adjustment factor to help match the apparent large-scale spreading of the plume[3]. However, there is limited theoretical support for using large mechanical dispersion coefficients[7][8]. In transmissive zones, macrodispersion coefficients are often orders of magnitude greater than molecular diffusion coefficients, leading some to conclude that molecular diffusion can be ignored.

Figure 4. Comparison of tracer breakthrough (upper graph) and cleanup curves from advection-dispersion based (gray lines) and advection-diffusion based (black lines) solute transport[3] (adapted from Payne et al. (2008)[7]).

Recently, an alternate conceptual model for describing large-scale plume spreading in heterogeneous soils has been proposed[7][3][8]. In this approach, solute transport in the transmissive zones is reasonably well described by the advection-dispersion equation using relatively small dispersion coefficients representing mechanical dispersion. However, overtime, molecular diffusion slowly transports solutes into lower permeability zones (Figure 3). As the transmissive zones are remediated, these solutes slowly diffuse back out, causing a long extended tail to the flushout curve. This process is controlled by diffusion and the presence of geologic heterogeneity with sharp contrasts between transmissive and low permeability media[9] as discussed in the video shown in Figure 3.

## References

1. ^ Freeze, A., and Cherry, J., 1979. Groundwater, Prentice-Hall, Englewood Cliffs, New Jersey, 604 pages. Free download from Hydrogeologists Without Borders.
2. ^ Woessner, W.W., and Poeter, E.P., 2020. Properties of Earth Materials and Principals of Groundwater Flow, The Groundwater Project, Guelph, Ontario, 207 pages. Free download from The Groundwater Project.
3. ^ 3.0 3.1 3.2 3.3 ITRC Integrated DNAPL Site Strategy Team, 2011. Integrated DNAPL Site Strategy. Technical/Regulatory Guidance Document, 209 pgs. Report pdf
4. ^ Grathwohl, P., 1998. Diffusion in Natural Porous Media: Contaminant Transport, Sorption/Desorption and Dissolution Kinetics. Kluwer Academic Publishers, Boston. DOI: 10.1007/978-1-4615-5683-1 Available from: Springer.com
5. ^ Carey, G.R., McBean, E.A. and Feenstra, S., 2016. Estimating Tortuosity Coefficients Based on Hydraulic Conductivity. Groundwater, 54(4), pp.476-487. DOI:10.1111/gwat.12406 Available from: NGWA
6. ^ Fetter, C.W., 1994. Applied Hydrogeology: Macmillan College Publishing Company. New York New York. ISBN-13:978-0130882394
7. ^ 7.0 7.1 7.2 Payne, F.C., Quinnan, J.A. and Potter, S.T., 2008. Remediation hydraulics. CRC Press. ISBN:978-1-4200-0684-1
8. ^ 8.0 8.1 Hadley, P.W. and Newell, C., 2014. The new potential for understanding groundwater contaminant transport. Groundwater, 52(2), pp.174-186. doi:10.1111/gwat.12135
9. ^ Sale, T.C., Illangasekare, T., Zimbron, J., Rodriguez, D., Wilkins, B. and Marinelli, F., 2007. AFCEE source zone initiative. Report Prepared for the Air Force Center for Environmental Excellence by Colorado State University and Colorado School of Mines. Report pdf