Difference between revisions of "User:Jhurley/sandbox"

From Enviro Wiki
Jump to: navigation, search
(→‎Modeling Diffusion)
Line 1: Line 1:
−
Groundwater migrates from areas of higher [[wikipedia: Hydraulic head | hydraulic head]] toward lower hydraulic head, transporting dissolved solutes through the combined processes of [[wikipedia: Advection | advection]] and [[wikipedia: Dispersion | dispersion]].  Advection refers to the bulk movement of solutes carried by flowing groundwater. Dispersion refers to the spreading of the 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 in which dispersion coefficients can be calculated as the sum of molecular diffusion, mechanical dispersion, and macrodispersion.   
+
==Matrix Diffusion==
−
 
+
Matrix Diffusion describes the gradual transport of dissolved contaminants from higher concentration and higher hydraulic conductivity (''K'') zones of a heterogeneous aquifer into lower ''K'' and lower contaminant concentration zones by [[wikipedia:Molecular diffusion | molecular diffusion]]. Initially, the transfer of contaminant mass into the low ''K'' zones reduces the concentration in the high ''K'' zones and slows the migration of the plume. Once the contaminant source is removed and the high ''K'' zone contaminant concentration decreases, the contaminants will then diffuse back out of these low ''K'' zones. In some cases, matrix diffusion can maintain contaminant concentrations in more permeable zones at greater than target cleanup goals for decades or even centuries after the primary sources have been addressed<ref name="Chapman2005">Chapman, S.W. and Parker, B.L., 2005. Plume persistence due to aquitard back diffusion following dense nonaqueous phase liquid source removal or isolation. Water Resources Research, 41(12), Report W12411.  [https://doi.org/10.1029/2005WR004224 DOI: 10.1029/2005WR004224] [[Media:Chapman2005.pdf | Report.pdf]]  Free access article from [https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2005WR004224 American Geophysical Union]</ref>. Field and laboratory results have illustrated the importance of this process.  Analytical and numerical modeling tools are available for evaluating matrix diffusion.
 
<div style="float:right;margin:0 0 2em 2em;">__TOC__</div>
 
<div style="float:right;margin:0 0 2em 2em;">__TOC__</div>
  
 
'''Related Article(s):'''
 
'''Related Article(s):'''
  
−
*[[Dispersion and Diffusion]]
+
*[[Groundwater Flow and Solute Transport]]
 
*[[Sorption of Organic Contaminants]]
 
*[[Sorption of Organic Contaminants]]
 
*[[Plume Response Modeling]]
 
*[[Plume Response Modeling]]
−
*[[Matrix Diffusion]]
 
  
 
'''CONTRIBUTOR(S):''' [[Dr. Charles Newell, P.E.|Dr. Charles Newell]] and  [[Dr. Robert Borden, P.E.|Dr. Robert Borden]]
 
'''CONTRIBUTOR(S):''' [[Dr. Charles Newell, P.E.|Dr. Charles Newell]] and  [[Dr. Robert Borden, P.E.|Dr. Robert Borden]]
Line 14: Line 13:
 
'''Key Resource(s):'''
 
'''Key Resource(s):'''
  
−
*[http://hydrogeologistswithoutborders.org/wordpress/1979-english/ Groundwater]<ref name="FandC1979">Freeze, A., and Cherry, J., 1979. Groundwater, Prentice-Hall, Englewood Cliffs, New Jersey, 604 pages. Free download from [http://hydrogeologistswithoutborders.org/wordpress/1979-english/ Hydrogeologists Without Borders].</ref>, Freeze and Cherry, 1979.
+
*[https://www.serdp-estcp.org/content/download/23838/240653/file/ER-1740 Management of Contaminants Stored in Low Permeability Zones – A State of the Science Review]<ref name="Sale2013">Sale, T., Parker, B.L., Newell, C.J. and Devlin, J.F., 2013. Management of Contaminants Stored in Low Permeability Zones – A State of the Science Review. Strategic Environmental Research and Development Program (SERDP) Project ER-1740. [[Media: Sale2013ER-1740.pdf | Report.pdf]] Website: [https://www.serdp-estcp.org/Program-Areas/Environmental-Restoration/Contaminated-Groundwater/Persistent-Contamination/ER-1740 ER-1740]</ref>
−
*[https://gw-project.org/books/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow/ Hydrogeologic Properties of Earth Materials and Principals of Groundwater Flow]<ref name="Woessner2020">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 [https://gw-project.org/books/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow/ The Groundwater Project].</ref>, Woessner and Poeter, 2020.
 
−
 
 
−
==Groundwater Flow==
 
−
[[File:Newell-Article 1-Fig1r.JPG|thumbnail|left|400px|Figure 1. Hydraulic gradient (typically described in units of m/m or ft/ft) is the difference in hydraulic head from Point A to Point B (ΔH) divided by the distance between them (ΔL). In unconfined aquifers, the hydraulic gradient can also be described as the slope of the water table (Adapted from course notes developed by Dr. R.J. Mitchell, Western Washington University).]]
 
−
Groundwater flows from areas of higher [[wikipedia: Hydraulic head | hydraulic head]] (a measure of pressure and gravitational energy) toward areas of lower hydraulic head (Figure 1). The rate of change (slope) of the hydraulic head is known as the hydraulic gradient. If groundwater is flowing and contains dissolved contaminants it can transport the contaminants by advection from areas with high hydraulic head toward lower hydraulic head zones, or “downgradient”.
 
−
 
 
−
===Darcy's Law===
 
−
{| class="wikitable" style="float:right; margin-left:10px;text-align:center;"
 
−
|+Table 1. Representative values of total porosity (''n''), effective porosity (''n<sub>e</sub>''), and hydraulic conductivity (''K'') for different aquifer materials<ref name="D&S1997">Domenico, P.A. and Schwartz, F.W., 1997. Physical and Chemical Hydrogeology, 2nd Ed. John Wiley & Sons, 528 pgs. ISBN 978-0-471-59762-9.  Available from: [https://www.wiley.com/en-us/Physical+and+Chemical+Hydrogeology%2C+2nd+Edition-p-9780471597629 Wiley]</ref><ref>McWhorter, D.B. and Sunada, D.K., 1977. Ground-water hydrology and hydraulics. Water Resources Publications, LLC, Highlands Ranch, Colorado, 304 pgs. ISBN-13: 978-1-887201-61-2 Available from: [https://www.wrpllc.com/books/gwhh.html Water Resources Publications]</ref><ref name="FandC1979" />
 
−
|-
 
−
!Aquifer Material
 
−
!Total Porosity<br /><small>(dimensionless)</small>
 
−
!Effective Porosity<br /><small>(dimensionless)</small>
 
−
!Hydraulic Conductivity<br /><small>(meters/second)</small>
 
−
|-
 
−
| colspan="4" style="text-align: left; background-color:white;" |'''Unconsolidated'''
 
−
|-
 
−
|Gravel||0.25 - 0.44||0.13 - 0.44||3×10<sup>-4</sup> - 3×10<sup>-2</sup>
 
−
|-
 
−
|Coarse Sand||0.31 - 0.46||0.18 - 0.43||9×10<sup>-7</sup> - 6×10<sup>-3</sup>
 
−
|-
 
−
|Medium Sand||—||0.16 - 0.46||9×10<sup>-7</sup> - 5×10<sup>-4</sup>
 
−
|-
 
−
|Fine Sand||0.25 - 0.53||0.01 - 0.46||2×10<sup>-7</sup> - 2×10<sup>-4</sup>
 
−
|-
 
−
|Silt, Loess||0.35 - 0.50||0.01 - 0.39||1×10<sup>-9</sup> - 2×10<sup>-5</sup>
 
−
|-
 
−
|Clay||0.40 - 0.70||0.01 - 0.18||1×10<sup>-11</sup> - 4.7×10<sup>-9</sup>
 
−
|-
 
−
| colspan="4" style="text-align: left; background-color:white;" |'''Sedimentary and Crystalline Rocks'''
 
−
|-
 
−
|Karst and Reef Limestone||0.05 - 0.50||—||1×10<sup>-6</sup> - 2×10<sup>-2</sup>
 
−
|-
 
−
|Limestone, Dolomite||0.00 - 0.20||0.01 - 0.24||1×10<sup>-9</sup> - 6×10<sup>-6</sup>
 
−
|-
 
−
|Sandstone||0.05 - 0.30||0.10 - 0.30||3×10<sup>-10</sup> - 6×10<sup>-6</sup>
 
−
|-
 
−
|Siltstone||—||0.21 - 0.41||1×10<sup>-11</sup> - 1.4×10<sup>-8</sup>
 
−
|-
 
−
|Basalt||0.05 - 0.50||—||2×10<sup>-11</sup> - 2×10<sup>-2</sup>
 
−
|-
 
−
|Fractured Crystalline Rock||0.00 - 0.10||—||8×10<sup>-9</sup> - 3×10<sup>-4</sup>
 
−
|-
 
−
|Weathered Granite||0.34 - 0.57||—||3.3×10<sup>-6</sup> - 5.2×10<sup>-5</sup>
 
−
|-
 
−
|Unfractured Crystalline Rock||0.00 - 0.05||—||3×10<sup>-14</sup> - 2×10<sup>-10</sup>
 
−
|}
 
−
In&nbsp;unconsolidated&nbsp;geologic settings (gravel, sand, silt, and clay) and highly fractured systems, the rate of groundwater movement can be expressed using [[wikipedia: Darcy's law | Darcy’s Law]]. This law is a fundamental mathematical relationship in the groundwater field and can be expressed this way:
 
−
 
 
−
[[File:Newell-Article 1-Equation 1rr.jpg|center|500px]]
 
−
 
 
−
::Where:
 
−
:::''Q'' = Flow rate (Volume of groundwater flow per time, such as m<sup>3</sup>/yr)
 
−
:::''A'' = Cross sectional area perpendicular to groundwater flow (length<sup>2</sup>, such as m<sup>2</sup>)
 
−
:::''V<sub>D</sub>'' = “Darcy Velocity”; describes groundwater flow as the volume of flow through a unit of cross-sectional area (units of length per time, such as ft/yr)
 
−
:::''K'' = Hydraulic Conductivity (sometimes called “permeability”) (length per time)
 
−
:::''ΔH'' = Difference in hydraulic head between two lateral points (length)
 
−
:::''ΔL'' = Distance between two lateral points (length)
 
−
 
 
−
[https://en.wikipedia.org/wiki/Hydraulic_conductivity Hydraulic conductivity] (Table 1 and Figure 2) is a measure of how easily groundwater flows through a porous medium, or alternatively, how much energy it takes to force water through a porous medium. For example, fine sand has smaller pores with more frictional resistance to flow, and therefore lower hydraulic conductivity compared to coarse sand, which has larger pores with less resistance to flow (Figure 2).
 
−
 
 
−
[[File:AdvectionFig2.PNG|400px|thumbnail|left|Figure 2. Hydraulic conductivity of selected rocks<ref>Heath, R.C., 1983. Basic ground-water hydrology, U.S. Geological Survey Water-Supply Paper 2220, 86 pgs. [//www.enviro.wiki/images/c/c4/Heath-1983-Basic_groundwater_hydrology_water_supply_paper.pdf Report pdf]</ref>.]]
 
−
Darcy’s Law was first described by Henry Darcy (1856)<ref>Brown, G.O., 2002. Henry Darcy and the making of a law. Water Resources Research, 38(7), p. 1106. [https://doi.org/10.1029/2001wr000727 DOI: 10.1029/2001WR000727] [//www.enviro.wiki/images/4/40/Darcy2002.pdf Report.pdf]</ref> in a report regarding a water supply system he designed for the city of Dijon, France. Based on his experiments, he concluded that the amount of water flowing through a closed tube of sand (Figure 3) depends on (a) the change in the hydraulic head between the inlet and outlet of the tube, and (b) the hydraulic conductivity of the sand in the tube. Groundwater flows rapidly in the case of higher pressure (ΔH) or more permeable materials such as gravel or coarse sand, but flows slowly when the pressure difference is lower or the material is less permeable, such as fine sand or silt.
 
−
 
 
−
[[File:Newell-Article 1-Fig3..JPG|500px|thumbnail|right|Figure 3. Conceptual explanation of Darcy’s Law based on Darcy’s experiment (Adapted from course notes developed by Dr. R.J. Mitchell, Western Washington University).]]
 
−
Since&nbsp;Darcy’s&nbsp;time,&nbsp;Darcy’s Law has been extended to develop a useful variation of Darcy's formula that calculates the actual velocity that the groundwater is moving in units such as meters traveled per year. This quantity is called “interstitial velocity” or “seepage velocity” and is calculated by dividing the Darcy Velocity (flow per unit area) by the actual open pore area where groundwater is flowing, the “effective porosity”&nbsp;(Table 1):
 
−
 
 
−
[[File:Newell-Article 1-Equation 2r.jpg|400px]]
 
  
−
:Where:
+
==Introduction==
−
::''V<sub>S</sub>'' = “interstitial velocity” or “seepage velocity” (units of length per time, such as m/sec)<br />
+
[[File:NewellMatrixDiffFig1.PNG | thumb | Figure 1.  Diffusion of a dissolved solute (chlorinated solvent) into lower ''K'' zones during loading period, followed by diffusion back out into higher ''K'' zones once the source is removed <ref name="Sale2007">Sale, T.C., Illangasekare, T.H., Zimbron, J., Rodriguez, D., Wilking, B., and Marinelli, F., 2007. AFCEE Source Zone Initiative. Air Force Center for Environmental Excellence, Brooks City-Base, San Antonio, TX. [https://www.enviro.wiki/images/0/08/AFCEE-2007-Sale.pdf Report.pdf]</ref>]]
−
::''V<sub>D</sub>'' = “Darcy Velocity”; describes groundwater flow as the volume of flow per unit area per time (also units of length per time)<br />
+
Matrix Diffusion can have major impacts on solute migration in groundwater and on cleanup time following source removal.  As a groundwater plume advances downgradient, dissolved contaminants are transported by molecular diffusion from zones with larger hydraulic conductivity (''K'') into lower ''K'' zones, slowing the rate of contaminant migration in the high ''K'' zone. However, once the contaminant source is eliminated, contaminants diffuse out of low ''K'' zones, slowing the cleanup rate in the high ''K'' zone (Figure 1).  This process, termed ‘back diffusion’, can greatly extend cleanup times.
−
::''n<sub>e</sub>'' = Effective porosity - fraction of cross section available for groundwater flow (unitless)
 
  
−
Effective porosity is smaller than total porosity. The difference is that total porosity includes some dead-end pores that do not support groundwater. Typical values for total and effective porosity are&nbsp;shown&nbsp;in&nbsp;Table&nbsp;1.
+
The impacts of back diffusion on aquifer cleanup have been examined in controlled laboratory experiments by several investigators<ref name="Doner2008">Doner, L.A., 2008. Tools to resolve water quality benefits of upgradient contaminant flux reduction. Master’s Thesis, Department of Civil and Environmental Engineering, Colorado State University.</ref><ref name="Yang2015">Yang, M., Annable, M.D. and Jawitz, J.W., 2015. Back Diffusion from Thin Low Permeability Zones. Environmental Science and Technology, 49(1), pp. 415-422.  [https://doi.org/10.1021/es5045634 DOI: 10.1021/es5045634] Free download available from: [https://www.researchgate.net/publication/269189924_Back_Diffusion_from_Thin_Low_Permeability_Zones ResearchGate]</ref><ref name= "Yang2016">Yang, M., Annable, M.D. and Jawitz, J.W., 2016. Solute source depletion control of forward and back diffusion through low-permeability zones. Journal of Contaminant Hydrology, 193, pp. 54-62. [https://doi.org/10.1016/j.jconhyd.2016.09.004 DOI: 10.1016/j.jconhyd.2016.09.004] Free download available from: [https://www.researchgate.net/profile/Minjune_Yang/publication/308004091_Solute_source_depletion_control_of_forward_and_back_diffusion_through_low-permeability_zones/links/5a2ed2c44585155b6179f489/Solute-source-depletion-control-of-forward-and-back-diffusion-through-low-permeability-zones.pdf ResearchGate]</ref><ref name="Tatti2018">Tatti, F., Papini, M.P., Sappa, G., Raboni, M., Arjmand, F., and Viotti, P., 2018. Contaminant back-diffusion from low-permeability layers as affected by groundwater velocity: A laboratory investigation by box model and image analysis. Science of The Total Environment, 622, pp. 164-171. [https://doi.org/10.1016/j.scitotenv.2017.11.347 DOI: 10.1016/j.scitotenv.2017.11.347]</ref>.  The video in Figure 2 shows the results of a 122-day tracer test in a laboratory flow cell (sand box)<ref name="Doner2008"/>. The flow cell contained several clay zones (''K'' = 10<sup>-8</sup> cm/s) surrounded by sand (''K'' = 0.02 cm/s).  During the loading period, water containing a green fluorescent tracer migrates from left to right with the water flowing through the flow cell, while diffusing into the clay.  After 22 days, the fluorescent tracer is eliminated from the feed, and most of the green tracer is quickly flushed from the tank’s sandy zones. However, small amounts of tracer continue to diffuse out of the clay layers for over 100 days.  This illustrates how back diffusion of contaminants out of low ''K'' zones can maintain low contaminant concentrations long after the contaminant source as been eliminated.
  
−
[[File:Newell-Article 1-Fig4.JPG|500px|thumbnail|left|Figure 4.  Difference between Darcy Velocity (also called Specific Discharge) and Seepage Velocity (also called Interstitial Velocity).]]
+
[[File:www.enviro.wiki/images/e/e0/GreenTank.mp4 | thumb | Figure 2. Video of dye tank simulation of matrix diffusion]]
 +
In some cases, matrix diffusion can maintain contaminant concentrations in more permeable zones above target cleanup goals for decades or even centuries after the primary sources have been addressed.  At a site impacted by Dense Non-Aqueous Phase Liquids (DNAPL), [[Chlorinated Solvents | trichloroethene (TCE)]] concentrations in downgradient wells declined by roughly an order-of-magnitude (OoM), when the upgradient source area was isolated with sheet piling. However, after this initial decline, TCE concentrations appeared to plateau or decline more slowly, consistent with back diffusion from an underlying aquitard.  Numerical simulations indicated that back diffusion would cause TCE concentrations in downgradient wells at the site to remain above target cleanup levels for centuries<ref name="Chapman2005"/>.
  
−
===Darcy Velocity and Seepage Velocity===
+
One other implication of matrix diffusion is that plume migration is attenuated by the loss of contaminants into low permeability zones, leading to slower plume migration compared to a case where no matrix diffusion occurs.  This phenomena was observed as far back as 1985 when Sudicky et al. observed that “A second consequence of the solute-storage effect offered by transverse diffusion into low-permeability layers is a rate of migration of the frontal portion of a contaminant in the permeable layers that is less than the groundwater velocity.”<ref name="Sudicky1985"> Sudicky, E.A., Gillham, R.W., and Frind, E.O., 1985. Experimental Investigation of Solute Transport in Stratified Porous Media: 1. The Nonreactive Case. Water Resources Research, 21(7), pp. 1035-1041. [https://doi.org/10.1029/WR021i007p01035 DOI: 10.1029/WR021i007p01035]</ref> In cases where there is an attenuating source, matrix diffusion can also reduce the peak concentrations observed in downgradient monitoring wells.  The attenuation caused by matrix diffusion may be particularly important for implementing [[Monitored Natural Attenuation (MNA)]] for contaminants that do not completely degrade, such as [[Metal and Metalloid Contaminants | heavy metals]] and [[Perfluoroalkyl_and_Polyfluoroalkyl_Substances_(PFAS) | PFAS]].  
−
In&nbsp;groundwater&nbsp;calculations, Darcy Velocity and Seepage Velocity are used for different purposes. For any calculation where the actual flow rate in units of volume per time (such as liters per day or gallons per minute) is involved, the original Darcy Equation should be used (calculate ''V<sub>D</sub>'', Equation 1) without using effective porosity. When calculating solute travel time however, the seepage velocity calculation (''V<sub>S</sub>'', Equation 2) must be used and an estimate of effective porosity is required. Figure 4 illustrates the differences between Darcy Velocity and&nbsp;Seepage&nbsp;Velocity.
 
  
−
===Mobile Porosity===
+
==SERPD/ESTCP Research==
−
{| class="wikitable" style="float:right; margin-left:10px; text-align:center;"
+
{|  
−
|+Table 2.  Mobile porosity estimates from 15 tracer tests<ref name="Payne2008">Payne, F.C., Quinnan, J.A. and Potter, S.T., 2008. Remediation Hydraulics. CRC Press. ISBN 9780849372490  Available from: [https://www.routledge.com/Remediation-Hydraulics/Payne-Quinnan-Potter/p/book/9780849372490 CRC Press]</ref>
+
The SERDP/ESTCP programs have funded several projects focusing on how matrix diffusion can impede progress towards reaching site closure, including:
 
|-
 
|-
−
!Aquifer Material
+
|
−
!Mobile Porosity<br /><small>(volume fraction)</small>
+
*[https://www.serdp-estcp.org/Program-Areas/Environmental-Restoration/Contaminated-Groundwater/Persistent-Contamination/ER-1740 SERDP Management of Contaminants Stored in Low Permeability Zones, A State-of-the-Science Review] <ref name="Sale2013"/>
−
|-
 
−
|Poorly sorted sand and gravel||0.085
 
−
|-
 
−
|Poorly sorted sand and gravel||0.04 - 0.07
 
−
|-
 
−
|Poorly sorted sand and gravel||0.09
 
−
|-
 
−
|Fractured sandstone||0.001 - 0.007
 
−
|-
 
−
|Alluvial formation||0.102
 
−
|-
 
−
|Glacial outwash||0.145
 
−
|-
 
−
|Weathered mudstone regolith||0.07 - 0.10
 
−
|-
 
−
|Alluvial formation||0.07
 
−
|-
 
−
|Alluvial formation||0.07
 
−
|-
 
−
|Silty sand||0.05
 
−
|-
 
−
|Fractured sandstone||0.0008 - 0.001
 
−
|-
 
−
|Alluvium, sand and gravel||0.017
 
−
|-
 
−
|Alluvium, poorly sorted sand and gravel||0.003 - 0.017
 
−
|-
 
−
|Alluvium, sand and gravel||0.11 - 0.18
 
−
|-
 
−
|Siltstone, sandstone, mudstone||0.01 - 0.05
 
−
|}
 
−
 
 
−
Payne&nbsp;et&nbsp;al.&nbsp;(2008)&nbsp;reported the results from multiple short-term tracer tests conducted to aid the design of amendment injection systems<ref name="Payne2008" />. In these tests, the dissolved solutes were observed to migrate more rapidly through the aquifer than could be explained with typically reported values of ''n<sub>e</sub>''. They concluded that the heterogeneity of unconsolidated formations results in a relatively small area of an aquifer cross section carrying most of the water, and therefore solutes migrate more rapidly than expected. Based on these results, they recommend that a quantity called “mobile porosity” should be used in place of ''n<sub>e</sub>'' in equation 2 for calculating solute transport velocities. Based on 15 different tracer tests, typical values of mobile porosity range from 0.02 to 0.10 (Table 2). 
 
−
 
 
−
A data mining analysis of 43 sites in California by Kulkarni et al. (2020) showed that on average 90% of the groundwater flow occurred in about 30% of cross sectional area perpendicular to groundwater flow.  These data provided “moderate (but not confirmatory) support for the&nbsp;mobile&nbsp;porosity&nbsp;concept.”<ref name="Kulkarni2020">Kulkarni, P.R., Godwin, W.R., Long, J.A., Newell, R.C., Newell, C.J., 2020. How much heterogeneity? Flow versus area from a big data perspective. Remediation 30(2), pp. 15-23. [https://doi.org/10.1002/rem.21639 DOI: 10.1002/rem.21639]  [//www.enviro.wiki/images/9/9b/Kulkarni2020.pdf Report.pdf]</ref>
 
−
 
 
−
==Advection-Dispersion-Reaction Equation==
 
−
The transport of dissolved solutes in groundwater is often modeled using the Advection-Dispersion-Reaction (ADR) equation. As shown below (Equation 3), the ADR equation describes the change in dissolved solute concentration (''C'') over time (''t'') where groundwater flow is oriented along the ''x'' direction.
 
−
 
 
−
{|
 
−
| || [[File:AdvectionEq3r.PNG|center|635px]]
 
−
|-
 
−
| Where: ||
 
−
|-
 
−
|
 
−
:''D<sub>x</sub>, D<sub>y</sub>, and D<sub>z</sub>''&nbsp;&nbsp;
 
−
| are hydrodynamic dispersion coefficients in the ''x, y'' and ''z'' directions (L<sup>2</sup>/T),
 
−
|-
 
−
|
 
−
:''v''
 
−
| is the advective transport or seepage velocity in the ''x'' direction (L/T), and
 
−
|-
 
−
|
 
−
:''λ''
 
−
| is an effective first order decay rate due to combined biotic and abiotic processes (1/T).
 
−
|-
 
−
|
 
−
:''R''
 
−
| is the linear, equilibrium retardation factor (see [[Sorption of Organic Contaminants]]), 
 
−
|}
 
−
 
 
−
The term on the left side of the equation is the rate of mass change per unit volume.  On the right side are terms representing the solute flux due to dispersion in the ''x, y'', and ''z'' directions, the advective flux in the ''x'' direction, and the first order decay due to biotic and abiotic processes. Dispersion coefficients (''D<sub>x,y,z</sub>'') are commonly estimated using the following relationships (Equation 4):
 
−
 
 
−
{|
 
−
| || [[File:AdvectionEq4.PNG|center|360px]]
 
−
|-
 
−
| Where: ||
 
 
|-
 
|-
 
|  
 
|  
−
:''D<sub>m</sub>''
+
*[https://www.serdp-estcp.org/Tools-and-Training/Environmental-Restoration/Groundwater-Plume-Treatment/Matrix-Diffusion-Tool-Kit ESTCP Matrix Diffusion Toolkit]<ref name="Farhat2012">Farhat, S.K., Newell, C.J., Seyedabbasi, M.A., McDade, J.M., Mahler, N.T., Sale, T.C., Dandy, D.S. and Wahlberg, J.J., 2012. Matrix Diffusion Toolkit. Environmental Security Technology Certification Program (ESTCP) Project ER-201126.  [[Media:Farhat2012ER-201126UsersManual.pdf | User’s Manual.pdf]]  Website: [https://www.serdp-estcp.org/Tools-and-Training/Environmental-Restoration/Groundwater-Plume-Treatment/Matrix-Diffusion-Tool-Kit ER-201126]</ref>
−
| is the molecular diffusion coefficient (L<sup>2</sup>/T), and
 
 
|-
 
|-
 
|  
 
|  
−
:''&alpha;<sub>L</sub>, &alpha;<sub>T</sub>'', and ''&alpha;<sub>V</sub>''&nbsp;&nbsp;
+
*[https://www.serdp-estcp.org/Program-Areas/Environmental-Restoration/Contaminated-Groundwater/Persistent-Contamination/ER-200530 ESTCP Decision Guide]<ref>Sale, T. and Newell, C., 2011. A Guide for Selecting Remedies for Subsurface Releases of Chlorinated Solvents. Environmental Security Technology Certification Program (ESTCP) Project ER-200530. [[Media: Sale2011ER-200530.pdf | Report.pdf]]  Website: [https://www.serdp-estcp.org/Program-Areas/Environmental-Restoration/Contaminated-Groundwater/Persistent-Contamination/ER-200530 ER-200530]</ref>
−
| are the longitudinal, transverse and vertical dispersivities (L), respectively.
 
−
|}
 
−
 
 
−
===ADR Applications===
 
−
[[File:AdvectionFig5.png | thumb | right | 350px | Figure 5. Curves of concentration versus distance (a) and concentration versus time (b) generated by solving the ADR equation for a continuous source of a non-reactive tracer with ''R'' = 1, λ = 0, ''v'' = 5 m/yr, and ''D<sub>x</sub>'' = 10 m<sup>2</sup>/yr.]]
 
−
The ADR equation can be solved to find the spatial and temporal distribution of solutes using a variety of analytical and numerical approaches.  The design tools [https://www.epa.gov/water-research/bioscreen-natural-attenuation-decision-support-system BIOSCREEN]<ref name="Newell1996">Newell, C.J., McLeod, R.K. and Gonzales, J.R., 1996. BIOSCREEN: Natural Attenuation Decision Support System - User's Manual, Version 1.3. US Environmental Protection Agency, EPA/600/R-96/087. [https://www.enviro.wiki/index.php?title=File:Newell-1996-Bioscreen_Natural_Attenuation_Decision_Support_System.pdf Report.pdf]  [https://www.epa.gov/water-research/bioscreen-natural-attenuation-decision-support-system BIOSCREEN website]</ref>, [https://www.epa.gov/water-research/biochlor-natural-attenuation-decision-support-system BIOCHLOR]<ref name="Aziz2000">Aziz, C.E., Newell, C.J., Gonzales, J.R., Haas, P.E., Clement, T.P. and Sun, Y., 2000. BIOCHLOR Natural Attenuation Decision Support System. User’s Manual - Version 1.0. US Environmental Protection Agency, EPA/600/R-00/008.  [https://www.enviro.wiki/index.php?title=File:Aziz-2000-BIOCHLOR-Natural_Attenuation_Dec_Support.pdf Report.pdf]  [https://www.epa.gov/water-research/biochlor-natural-attenuation-decision-support-system BIOCHLOR website]</ref>, and [https://www.epa.gov/water-research/remediation-evaluation-model-chlorinated-solvents-remchlor REMChlor]<ref name="Falta2007">Falta, R.W., Stacy, M.B., Ahsanuzzaman, A.N.M., Wang, M. and Earle, R.C., 2007. REMChlor Remediation Evaluation Model for Chlorinated Solvents - User’s Manual, Version 1.0. US Environmental Protection Agency. Center for Subsurface Modeling Support, Ada, OK.  [[Media:REMChlorUserManual.pdf | Report.pdf]]  [https://www.epa.gov/water-research/remediation-evaluation-model-chlorinated-solvents-remchlor REMChlor website]</ref> (see also [[REMChlor - MD]]) employ an analytical solution of the ADR equation.  [https://www.usgs.gov/software/mt3d-usgs-groundwater-solute-transport-simulator-modflow MT3DMS]<ref name="Zheng1999">Zheng, C. and Wang, P.P., 1999. MT3DMS: A Modular Three-Dimensional Multispecies Transport Model for Simulation of Advection, Dispersion, and Chemical Reactions of Contaminants in Groundwater Systems; Documentation and User’s Guide. Contract Report SERDP-99-1 U.S. Army Engineer Research and Development Center, Vicksburg, MS. [[Media:Mt3dmanual.pdf | Report.pdf]]  [https://www.usgs.gov/software/mt3d-usgs-groundwater-solute-transport-simulator-modflow MT3DMS website]</ref> uses a numerical method to solve the ADR equation using the head distribution generated by the groundwater flow model MODFLOW<ref name="McDonald1988">McDonald, M.G. and Harbaugh, A.W., 1988. A Modular Three-dimensional Finite-difference Ground-water Flow Model, Techniques of Water-Resources Investigations, Book 6, Modeling Techniques. U.S. Geological Survey, 586 pages. [https://doi.org/10.3133/twri06A1  DOI: 10.3133/twri06A1]  [[Media: McDonald1988.pdf | Report.pdf]]  Free MODFLOW download from: [https://www.usgs.gov/mission-areas/water-resources/science/modflow-and-related-programs?qt-science_center_objects=0#qt-science_center_objects USGS]</ref>.
 
−
 
 
−
Figures 5a and 5b were generated using a numerical solution of the ADR equation for a non-reactive tracer (''R'' = 1; λ = 0) with ''v'' = 5 m/yr and ''D<sub>x</sub>'' = 10 m<sup>2</sup>/yr.  Figure 5a shows the predicted change in concentration of the tracer, chloride, versus distance downgradient from the continuous contaminant source at different times (0, 1, 2, and 4 years).  Figure 5b shows the change in concentration versus time (commonly referred to as the breakthrough curve or BTC) at different downgradient distances (10, 20, 30 and 40 m).  At 2 years, the mid-point of the concentration versus distance curve (Figure 5a) is located 10 m downgradient (x = 5 m/yr * 2 yr).  At 20 m downgradient, the mid-point of the concentration versus time curves (Figure 5b) occurs at 4 years (t = 20 m / 5 m/yr).
 
−
 
 
−
===Modeling Dispersion===
 
−
Mechanical&nbsp;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<ref>Fetter, C.W., 1994. Applied Hydrogeology: Macmillan College Publishing Company. New York New York. ISBN-13:978-0130882394</ref>. 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 6).  Typical values of the mechanical dispersivity measured in laboratory column tests are on the order of 0.01 to 1 cm<ref name="Anderson1979">Anderson, M.P. and Cherry, J.A., 1979. Using models to simulate the movement of contaminants through groundwater flow systems. Critical Reviews in Environmental Science and Technology, 9(2), pp.97-156.  [https://doi.org/10.1080/10643387909381669 DOI: 10.1080/10643387909381669]</ref>.
 
−
[[File:Fig2 dispanddiff.JPG|thumbnail|left|400px|Figure 6. Conceptual depiction of mechanical dispersion (adapted from ITRC (2011)<ref name="ITRC2011">ITRC Integrated DNAPL Site Strategy Team, 2011. Integrated DNAPL Site Strategy. Technical/Regulatory Guidance Document, 209 pgs. [//www.enviro.wiki/images/d/d9/ITRC-2011-Integrated_DNAPL.pdf Report pdf]</ref>).]]
 
−
 
 
−
The dispersion coefficient in the ADR equation accounts for the combined effects of mechanical dispersion and molecular diffusion, both of which cause spreading of the contaminant plume from highly concentrated areas toward less concentrated areas.  [[wikipedia:Molecular diffusion | 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.
 
−
 
 
−
===Modeling Diffusion===
 
−
[[File:Fig1 dispanddiff.JPG|thumbnail|right|400px|Figure 7. 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.]]
 
−
[[wikipedia: Molecular diffusion | Molecular&nbsp;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 7). 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&nbsp;diffusive&nbsp;flux&nbsp;''J'' (M/L<sup>2</sup>/T) in groundwater is calculated using [[wikipedia:Fick's laws of diffusion | Fick’s Law]]:
 
−
 
 
−
{|
 
−
|
 
−
|<big>'''''J&nbsp;=&nbsp;-D<sub>e</sub>&nbsp;<sup>dC</sup>/<sub>dx</sub>'''''</big>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(Equation&nbsp;5)
 
−
|-
 
−
|Where:||
 
−
|-
 
−
|
 
−
:''D<sub>e</sub>''
 
−
|is the effective diffusion coefficient and
 
−
|-
 
−
|
 
−
:''dC/dx''&nbsp;&nbsp;&nbsp;&nbsp;
 
−
|is the concentration gradient.
 
−
|}
 
−
The effective diffusion coefficient for transport through the porous media, ''D<sub>e</sub>, is estimated as:''
 
−
{|
 
−
|
 
−
|<big>'''''D<sub>e</sub>&nbsp;=&nbsp;D<sub>m</sub>&nbsp;n<sub>e</sub>&nbsp;<sup>&delta;</sup>/<sub>&Tau;</sub>'''''</big>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(Equation&nbsp;6)
 
−
|-
 
−
|Where:||
 
−
|-
 
−
|
 
−
:''D<sub>m</sub>''&nbsp;&nbsp;&nbsp;&nbsp;
 
−
|is the [[wikipedia:Mass diffusivity | diffusion coefficient]] of the solute in water,
 
−
|-
 
−
|
 
−
:''n<sub>e</sub>''
 
−
|is the effective porosity (dimensionless),
 
−
|-
 
−
|
 
−
:''&delta;''
 
−
|is the constrictivity (dimensionless) which reflects the restricted motion of particles in narrow pores<ref name="Grathwohl1998">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: [https://link.springer.com/book/10.1007/978-1-4615-5683-1 Springer.com]</ref>, and
 
 
|-
 
|-
 
|
 
|
−
:''&Tau;''
+
*[https://www.serdp-estcp.org/Program-Areas/Environmental-Restoration/Contaminated-Groundwater/Persistent-Contamination/ER-201426 ESTCP REMChlor-MD: the USEPA’s REMChlor model with a new matrix diffusion term for the plume]<ref name="Farhat2018">Farhat, S. K., Newell, C. J., Falta, R. W., and Lynch, K., 2018. A Practical Approach for Modeling Matrix Diffusion Effects in REMChlor. Environmental Security Technology Certification Program (ESTCP) Project ER-201426.  [https://enviro.wiki/images/0/0b/2018-Falta-REMChlor_Modeling_Matrix_Diffusion_Effects.pdf  User’s Manual.pdf] Website: [https://www.serdp-estcp.org/Program-Areas/Environmental-Restoration/Contaminated-Groundwater/Persistent-Contamination/ER-201426 ER-201426]</ref>
−
|is the [[wikipedia:Tortuosity | tortuosity]] (dimensionless) which reflects the longer diffusion path in porous media around sediment particles<ref name="Carey2016">Carey, G.R., McBean, E.A. and Feenstra, S., 2016. Estimating Tortuosity Coefficients Based on Hydraulic Conductivity. Groundwater, 54(4), pp.476-487.  [https://doi.org/10.1111/gwat.12406 DOI:10.1111/gwat.12406] Available from: [https://ngwa.onlinelibrary.wiley.com/doi/abs/10.1111/gwat.12406 NGWA]</ref>.
 
−
|}
 
−
''D<sub>m</sub>'' is a function of the temperature, fluid viscosity and molecular weight.  Values of ''D<sub>m</sub>'' for common groundwater solutes are shown in Table 3.
 
−
 
 
−
{| class="wikitable" style="float:left; margin-right:20px; text-align:center;"
 
−
|+Table 3. Diffusion Coefficients (''D<sub>m</sub>'') for Common Groundwater Solutes.
 
−
|-
 
−
!Aqueous Diffusion Coefficient
 
−
!Temperature<br /><small>(&deg;C)</small>
 
−
!''D<sub>m</sub>''<br /><small>(cm<sup>2</sup>/s)</small>
 
−
!Reference
 
−
|-
 
−
|Acetone||25||&nbsp;&nbsp;1.16x10<sup>-5</sup>&nbsp;&nbsp;||Cussler 1997 <ref name="Cussler1997">Cussler, E.L., 1997. Diffusion: Mass Transfer in Fluid Systems, Cambridge University Press, New York, 580 pages.  ISBN: 9780521450782</ref>
 
−
|-
 
−
|Benzene||20||1.02x10<sup>-5</sup>||Bonoli and Witherspoon 1968 <ref name="Bonoli1968">Bonoli, L. and Witherspoon, P.A., 1968. Diffusion of Aromatic and Cycloparaffin Hydrocarbons in Water from 2 to 60 deg. The Journal of Physical Chemistry, 72(7), pp.2532-2534.  [https://doi.org/10.1021/j100853a045 DOI: 10.1021/j100853a045]</ref>
 
−
|-
 
−
|Carbon dioxide||25||1.92x10<sup>-5</sup>||Cussler 1997 <ref name="Cussler1997"/>
 
−
|-
 
−
|Carbon tetrachloride||25||9.55x10<sup>-6</sup>||Yaws 1995 <ref name="Yaws1995">Yaws, C.L., 1995. Handbook of Transport Property Data: Viscosity, Thermal Conductivity and Diffusion Coefficients of Liquids and Gases, Gulf Publishing Company, Houston, TX. ISBN: 0884153924</ref>
 
−
|-
 
−
|Chloroform||25||1.08x10<sup>-5</sup>||Yaws 1995 <ref name="Yaws1995"/>
 
−
|-
 
−
|Dichloroethene||25||1.12x10<sup>-5</sup>||Yaws 1995 <ref name="Yaws1995"/>
 
−
|-
 
−
|1,4-Dioxane||25||1.02x10<sup>-5</sup>||Yaws 1995 <ref name="Yaws1995"/>
 
−
|-
 
−
|Ethane||25||1.52x10<sup>-5</sup>||Witherspoon and Saraf 1965 <ref name="Witherspoon1965">Witherspoon, P.A. and Saraf, D.N., 1965. Diffusion of Methane, Ethane, Propane, and n-Butane in Water from 25 to 43&deg;. The Journal of Physical Chemistry, 69(11), pp. 3752-3755.  [https://doi.org/10.1021/j100895a017 DOI: 10.1021/j100895a017]</ref>
 
−
|-
 
−
|Ethylbenzene||20||8.10x10<sup>-6</sup>||Bonoli and Witherspoon 1968 <ref name="Bonoli1968"/>
 
−
|-
 
−
|Ethene||25||1.87x10<sup>-5</sup>||Cussler 1997 <ref name="Cussler1997"/>
 
−
|-
 
−
|Helium||25||6.28x10<sup>-5</sup>||Cussler 1997 <ref name="Cussler1997"/>
 
−
|-
 
−
|Hydrogen||25||4.50x10<sup>-5</sup>||Cussler 1997 <ref name="Cussler1997"/>
 
−
|-
 
−
|Methane||25||1.88x10<sup>-5</sup>||Witherspoon and Saraf 1965 <ref name="Witherspoon1965"/>
 
−
|-
 
−
|Nitrogen||25||1.88x10<sup>-5</sup>||Cussler 1997 <ref name="Cussler1997"/>
 
−
|-
 
−
|Oxygen||25||2.10x10<sup>-5</sup>||Cussler 1997 <ref name="Cussler1997"/>
 
−
|-
 
−
|Perfluorooctanoic acid (PFOA)||20||4.80x10<sup>-6</sup>||Schaefer et al. 2019 <ref name="Schaefer2019">Schaefer, C.E., Drennan, D.M., Tran, D.N., Garcia, R., Christie, E., Higgins, C.P. and Field, J.A., 2019. Measurement of Aqueous Diffusivities for Perfluoroalkyl Acids. Journal of Environmental Engineering, 145(11).  [https://doi.org/10.1061/(ASCE)EE.1943-7870.0001585 DOI: 10.1061/(ASCE)EE.1943-7870.0001585]</ref>
 
−
|-
 
−
|Perfluorooctane sulfonic acid (PFOS)||20||5.40x10<sup>-6</sup>||Schaefer et al. 2019 <ref name="Schaefer2019"/>
 
−
|-
 
−
|Tetrachloroethene||25||8.99x10<sup>-6</sup>||Yaws 1995 <ref name="Yaws1995"/>
 
−
|-
 
−
|Toluene||20||8.50x10<sup>-6</sup>||Bonoli and Witherspoon 1968 <ref name="Bonoli1968"/>
 
−
|-
 
−
|Trichloroethene||25||8.16x10<sup>-6</sup>||Rossi et al. 2015 <ref name="Rossi2015">Rossi, F., Cucciniello, R., Intiso, A., Proto, A., Motta, O. and Marchettini, N., 2015. Determination of the Trichloroethylene Diffusion Coefficient in Water. American Institute of Chemical Engineers Journal, 61(10), pp.3511-3515.  [https://doi.org/10.1002/aic.14861 DOI: 10.1002/aic.14861]</ref>
 
−
|-
 
−
|Vinyl chloride||25||1.34x10<sup>-5</sup>||Yaws 1995 <ref name="Yaws1995"/>
 
 
|}
 
|}
−
</br>
 
  
−
===Macrodispersion===
+
===Impacts on Breakthrough Curves===
−
[[File:ADRFig2.PNG | thumb | right | 350px | Figure 8. Predicted variation in macrodispersivity with distance for varying ''σ<sup>2</sup>Y'' and correlation length = 3 m.]]
+
[[File:ADRFig3.png | thumb| 350px| Figure 3. Comparison of tracer breakthrough (upper graph) and cleanup curves (lower graph) from advection-dispersion based (gray lines) and advection-diffusion based (black lines) solute transport<ref name="ITRC2011">Interstate Technology and Regulatory Council (ITRC), 2011. Integrated DNAPL Site Strategy (IDSS-1),  Integrated DNAPL Site Strategy Team, ITRC, Washington, DC. [https://www.enviro.wiki/images/d/d9/ITRC-2011-Integrated_DNAPL.pdf Report.pdf] Free download from: [https://itrcweb.org/GuidanceDocuments/IntegratedDNAPLStrategy_IDSSDoc/IDSS-1.pdf ITRC]</ref>.]]
−
[[File:NewThinkingAboutDispersion.mp4 |thumbnail|right|500px|Figure 9. Matrix diffusion processes and their effects on plume persistence and attenuation.]]
+
The impacts of matrix diffusion on the initial breakthrough of the solute plume and on later cleanup are illustrated in Figure 3<ref name="ITRC2011"/>. Using a traditional advection-dispersion model, the breakthrough curve for a pulse tracer injection appears as a bell-shaped ([[wikipedia:Gaussian function |Gaussian]]) curve (gray line on the right side of the upper graph) where the peak arrival time corresponds to the average groundwater velocity.  Using an advection-diffusion approach, the breakthrough curve for a pulse injection is asymmetric (solid black line) with the peak tracer concentration arriving earlier than would be expected based on the average groundwater velocity, but with a long extended tail to the flushout curve.
−
Spatial variations in hydraulic conductivity can increase the apparent spreading of solute plumes observed in groundwater monitoring wells. For example, in an aquifer composed of alternating layers of lower hydraulic conductivity (''K'') silty sand and higher ''K'' sandy gravel layers, the dissolved solute rapidly migrates downgradient through the sandy gravel layers resulting in relatively high concentration fingers surrounded by relatively uncontaminated material. Over time, contaminants in lower ''K'' layers eventually breakthrough at the monitoring well, causing a more gradual further increase in measured concentrations.  This rapid breakthrough followed by gradual increases in solute concentrations gives the appearance of a plume with a very large dispersion coefficient. This spreading of the solute caused by large-scale heterogeneities in the aquifer and the associated spatial variations in advective transport velocity is referred to as macrodispersion.
 
  
−
In some groundwater modeling projects, large values of the dispersion coefficient are used as an adjustment factor to better represent the observed large-scale spreading of plumes<ref name="ITRC2011"/>. Theoretical studies suggest that macrodispersivity will increase with distance near the source, and then increase more slowly farther downgradient, eventually approaching an asymptotic value<ref name="Gelhar1979">Gelhar, L.W., Gutjahr, A.L. and Naff, R.L., 1979. Stochastic analysis of macrodispersion in a stratified aquifer. Water Resources Research, 15(6), pp.1387-1397.  [https://doi.org/10.1029/WR015i006p01387 DOI:10.1029/WR015i006p01387]</ref><ref name="Gelhar1983">Gelhar, L.W. and Axness, C.L., 1983. Three‐dimensional stochastic analysis of macrodispersion in aquifers. Water Resources Research, 19(1), pp.161-180.  [https://doi.org/10.1029/WR019i001p00161 DOI:10.1029/WR019i001p00161]</ref><ref name="Dagan1988">Dagan, G., 1988. Time‐dependent macrodispersion for solute transport in anisotropic heterogeneous aquifers. Water Resources Research, 24(9), pp.1491-1500.  [https://doi.org/10.1029/WR024i009p01491 DOI:10.1029/WR024i009p01491]</ref>.  Figure 8 shows values of macrodispersivity calculated using the theory of Dagan<ref name="Dagan1988"/> with an autocorrelation length of 3 m and several different values of the variance of ''Y'' (σ<small><sup>2</sup><sub>''Y''</sub></small>) where ''Y'' = Log ''K''. The calculated macrodispersivity increases more rapidly and approaches higher asymptotic values for more heterogeneous aquifers with greater variations in ''K'' (larger σ<small><sup>2</sup><sub>''Y''</sub></small>).  The maximum predicted dispersivity values were in the range of 0.5 to 5 m.  Zech, et al. (2015)<ref>Zech, A., Attinger, S., Cvetkovic, V., Dagan, G., Dietrich, P., Fiori, A., Rubin, Y. and Teutsch, G., 2015. Is unique scaling of aquifer macrodispersivity supported by field data? Water Resources Research, 51(9), pp.7662-7679.  [https://doi.org/10.1002/2015WR017220 DOI: 10.1002/2015WR017220]&nbsp;&nbsp; Free access article from [https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1002/2015WR017220 American Geophysical Union]</ref> presented moderate and high reliability measurements of longitudinal macrodispersivity versus distance. Typical values of the longitudinal macrodispersivity varied from 0.1 to 10 m, with much lower values for transverse and vertical dispersivities.  
+
The lower graph shows the predicted cleanup concentration profiles following complete elimination of a source area. The advection-dispersion model (gray line) predicts a clean-water front arriving at a time corresponding to the average groundwater velocity.  The advection-diffusion model (black line) predicts that concentrations will start to decline more rapidly than expected (based on the average groundwater velocity) as clean water rapidly migrates through the highest-permeability strata. However, low but significant contaminant concentrations linger much longer (tailing) due to diffusive contaminant mass exchange between zones of high and low permeability.
  
−
===Matrix Diffusion===
 
−
Recently, an alternate conceptual model for describing large-scale plume spreading in heterogeneous soils has been proposed<ref name="ITRC2011" /><ref name="Payne2008"/><ref name="Hadley2014">Hadley, P.W. and Newell, C., 2014. The new potential for understanding groundwater contaminant transport. Groundwater, 52(2), pp.174-186. [http://dx.doi.org/10.1111/gwat.12135 doi:10.1111/gwat.12135]</ref>. 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, over time, molecular diffusion slowly transports solutes into lower permeability zones. As the transmissive zones are remediated, these solutes slowly diffuse back out, causing a long extended tail to the flushout curve. This process, referred to as [[Matrix Diffusion |matrix diffusion]], is controlled by [[wikipedia: Molecular diffusion | molecular diffusion]] and the presence of geologic heterogeneity with sharp contrasts between transmissive and low permeability media<ref>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. [//www.enviro.wiki/images/0/08/AFCEE-2007-Sale.pdf Report pdf]</ref> as discussed in the [//www.enviro.wiki/images/8/8a/NewThinkingAboutDispersion.mp4 video] shown in Figure 9. In some cases, matrix diffusion can maintain contaminant concentrations in more permeable zones at greater than target cleanup goals for decades or even centuries after the primary sources have been addressed<ref>Chapman, S.W. and Parker, B.L., 2005. Plume persistence due to aquitard back diffusion following dense nonaqueous phase liquid source removal or isolation. Water Resources Research, 41(12): W12411.  [https://doi.org/10.1029/2005WR004224 DOI: 10.1029/2005WR004224] &nbsp;&nbsp; Free access article from [https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/2005WR004224 American Geophysical Union]</ref>.
 
 
<br clear="left" />
 
<br clear="left" />
  

Revision as of 23:08, 21 December 2020

Matrix Diffusion

Matrix Diffusion describes the gradual transport of dissolved contaminants from higher concentration and higher hydraulic conductivity (K) zones of a heterogeneous aquifer into lower K and lower contaminant concentration zones by molecular diffusion. Initially, the transfer of contaminant mass into the low K zones reduces the concentration in the high K zones and slows the migration of the plume. Once the contaminant source is removed and the high K zone contaminant concentration decreases, the contaminants will then diffuse back out of these low K zones. In some cases, matrix diffusion can maintain contaminant concentrations in more permeable zones at greater than target cleanup goals for decades or even centuries after the primary sources have been addressed[1]. Field and laboratory results have illustrated the importance of this process. Analytical and numerical modeling tools are available for evaluating matrix diffusion.

Related Article(s):

CONTRIBUTOR(S): Dr. Charles Newell and Dr. Robert Borden

Key Resource(s):

Introduction

Figure 1. Diffusion of a dissolved solute (chlorinated solvent) into lower K zones during loading period, followed by diffusion back out into higher K zones once the source is removed [3]

Matrix Diffusion can have major impacts on solute migration in groundwater and on cleanup time following source removal. As a groundwater plume advances downgradient, dissolved contaminants are transported by molecular diffusion from zones with larger hydraulic conductivity (K) into lower K zones, slowing the rate of contaminant migration in the high K zone. However, once the contaminant source is eliminated, contaminants diffuse out of low K zones, slowing the cleanup rate in the high K zone (Figure 1). This process, termed ‘back diffusion’, can greatly extend cleanup times.

The impacts of back diffusion on aquifer cleanup have been examined in controlled laboratory experiments by several investigators[4][5][6][7]. The video in Figure 2 shows the results of a 122-day tracer test in a laboratory flow cell (sand box)[4]. The flow cell contained several clay zones (K = 10-8 cm/s) surrounded by sand (K = 0.02 cm/s). During the loading period, water containing a green fluorescent tracer migrates from left to right with the water flowing through the flow cell, while diffusing into the clay. After 22 days, the fluorescent tracer is eliminated from the feed, and most of the green tracer is quickly flushed from the tank’s sandy zones. However, small amounts of tracer continue to diffuse out of the clay layers for over 100 days. This illustrates how back diffusion of contaminants out of low K zones can maintain low contaminant concentrations long after the contaminant source as been eliminated.

File:Www.enviro.wiki/images/e/e0/GreenTank.mp4
Figure 2. Video of dye tank simulation of matrix diffusion

In some cases, matrix diffusion can maintain contaminant concentrations in more permeable zones above target cleanup goals for decades or even centuries after the primary sources have been addressed. At a site impacted by Dense Non-Aqueous Phase Liquids (DNAPL), trichloroethene (TCE) concentrations in downgradient wells declined by roughly an order-of-magnitude (OoM), when the upgradient source area was isolated with sheet piling. However, after this initial decline, TCE concentrations appeared to plateau or decline more slowly, consistent with back diffusion from an underlying aquitard. Numerical simulations indicated that back diffusion would cause TCE concentrations in downgradient wells at the site to remain above target cleanup levels for centuries[1].

One other implication of matrix diffusion is that plume migration is attenuated by the loss of contaminants into low permeability zones, leading to slower plume migration compared to a case where no matrix diffusion occurs. This phenomena was observed as far back as 1985 when Sudicky et al. observed that “A second consequence of the solute-storage effect offered by transverse diffusion into low-permeability layers is a rate of migration of the frontal portion of a contaminant in the permeable layers that is less than the groundwater velocity.”[8] In cases where there is an attenuating source, matrix diffusion can also reduce the peak concentrations observed in downgradient monitoring wells. The attenuation caused by matrix diffusion may be particularly important for implementing Monitored Natural Attenuation (MNA) for contaminants that do not completely degrade, such as heavy metals and PFAS.

SERPD/ESTCP Research

The SERDP/ESTCP programs have funded several projects focusing on how matrix diffusion can impede progress towards reaching site closure, including:

Impacts on Breakthrough Curves

Figure 3. Comparison of tracer breakthrough (upper graph) and cleanup curves (lower graph) from advection-dispersion based (gray lines) and advection-diffusion based (black lines) solute transport[12].

The impacts of matrix diffusion on the initial breakthrough of the solute plume and on later cleanup are illustrated in Figure 3[12]. Using a traditional advection-dispersion model, the breakthrough curve for a pulse tracer injection appears as a bell-shaped (Gaussian) curve (gray line on the right side of the upper graph) where the peak arrival time corresponds to the average groundwater velocity. Using an advection-diffusion approach, the breakthrough curve for a pulse injection is asymmetric (solid black line) with the peak tracer concentration arriving earlier than would be expected based on the average groundwater velocity, but with a long extended tail to the flushout curve.

The lower graph shows the predicted cleanup concentration profiles following complete elimination of a source area. The advection-dispersion model (gray line) predicts a clean-water front arriving at a time corresponding to the average groundwater velocity. The advection-diffusion model (black line) predicts that concentrations will start to decline more rapidly than expected (based on the average groundwater velocity) as clean water rapidly migrates through the highest-permeability strata. However, low but significant contaminant concentrations linger much longer (tailing) due to diffusive contaminant mass exchange between zones of high and low permeability.


References

  1. ^ 1.0 1.1 Chapman, S.W. and Parker, B.L., 2005. Plume persistence due to aquitard back diffusion following dense nonaqueous phase liquid source removal or isolation. Water Resources Research, 41(12), Report W12411. DOI: 10.1029/2005WR004224 Report.pdf Free access article from American Geophysical Union
  2. ^ 2.0 2.1 Sale, T., Parker, B.L., Newell, C.J. and Devlin, J.F., 2013. Management of Contaminants Stored in Low Permeability Zones – A State of the Science Review. Strategic Environmental Research and Development Program (SERDP) Project ER-1740. Report.pdf Website: ER-1740
  3. ^ Sale, T.C., Illangasekare, T.H., Zimbron, J., Rodriguez, D., Wilking, B., and Marinelli, F., 2007. AFCEE Source Zone Initiative. Air Force Center for Environmental Excellence, Brooks City-Base, San Antonio, TX. Report.pdf
  4. ^ 4.0 4.1 Doner, L.A., 2008. Tools to resolve water quality benefits of upgradient contaminant flux reduction. Master’s Thesis, Department of Civil and Environmental Engineering, Colorado State University.
  5. ^ Yang, M., Annable, M.D. and Jawitz, J.W., 2015. Back Diffusion from Thin Low Permeability Zones. Environmental Science and Technology, 49(1), pp. 415-422. DOI: 10.1021/es5045634 Free download available from: ResearchGate
  6. ^ Yang, M., Annable, M.D. and Jawitz, J.W., 2016. Solute source depletion control of forward and back diffusion through low-permeability zones. Journal of Contaminant Hydrology, 193, pp. 54-62. DOI: 10.1016/j.jconhyd.2016.09.004 Free download available from: ResearchGate
  7. ^ Tatti, F., Papini, M.P., Sappa, G., Raboni, M., Arjmand, F., and Viotti, P., 2018. Contaminant back-diffusion from low-permeability layers as affected by groundwater velocity: A laboratory investigation by box model and image analysis. Science of The Total Environment, 622, pp. 164-171. DOI: 10.1016/j.scitotenv.2017.11.347
  8. ^ Sudicky, E.A., Gillham, R.W., and Frind, E.O., 1985. Experimental Investigation of Solute Transport in Stratified Porous Media: 1. The Nonreactive Case. Water Resources Research, 21(7), pp. 1035-1041. DOI: 10.1029/WR021i007p01035
  9. ^ Farhat, S.K., Newell, C.J., Seyedabbasi, M.A., McDade, J.M., Mahler, N.T., Sale, T.C., Dandy, D.S. and Wahlberg, J.J., 2012. Matrix Diffusion Toolkit. Environmental Security Technology Certification Program (ESTCP) Project ER-201126. User’s Manual.pdf Website: ER-201126
  10. ^ Sale, T. and Newell, C., 2011. A Guide for Selecting Remedies for Subsurface Releases of Chlorinated Solvents. Environmental Security Technology Certification Program (ESTCP) Project ER-200530. Report.pdf Website: ER-200530
  11. ^ Farhat, S. K., Newell, C. J., Falta, R. W., and Lynch, K., 2018. A Practical Approach for Modeling Matrix Diffusion Effects in REMChlor. Environmental Security Technology Certification Program (ESTCP) Project ER-201426. User’s Manual.pdf Website: ER-201426
  12. ^ 12.0 12.1 Interstate Technology and Regulatory Council (ITRC), 2011. Integrated DNAPL Site Strategy (IDSS-1), Integrated DNAPL Site Strategy Team, ITRC, Washington, DC. Report.pdf Free download from: ITRC

See Also

Retrieved from "https://www.enviro.wiki/index.php?title=User:Jhurley/sandbox&oldid=14223"