Groundwater Flow and Solute Transport - Revision history

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2022-04-28T03:24:27Z

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Admin at 21:37, 26 April 2022

2022-04-26T21:37:31Z

Jhurley: /* Matrix Diffusion */

2020-12-29T15:49:09Z

‎Matrix Diffusion

Jhurley: /* Macrodispersion */

2020-12-28T16:43:06Z

‎Macrodispersion

Jhurley: /* Matrix Diffusion */

2020-12-28T16:41:38Z

‎Matrix Diffusion

Jhurley: /* Modeling Dispersion */

2020-12-28T14:10:04Z

‎Modeling Dispersion

Jhurley at 19:05, 22 December 2020

2020-12-22T19:05:20Z

Jhurley: /* ADR Applications */

2020-12-22T18:51:39Z

‎ADR Applications

Jhurley at 22:19, 21 December 2020

2020-12-21T22:19:29Z

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 *[[Plume Response Modeling]]
 *[[Sorption of Organic Contaminants]]
 '''Contributor(s):''' [[Dr. Charles Newell, P.E.|Dr. Charles Newell]] and  [[Dr. Robert Borden, P.E.|Dr. Robert Borden]]
 '''Key Resource(s):'''

@@ Line 9: / Line 9: @@
 *[[Matrix Diffusion]]
-'''CONTRIBUTOR(S):''' [[Dr. Charles Newell, P.E.|Dr. Charles Newell]] and  [[Dr. Robert Borden, P.E.|Dr. Robert Borden]]
 '''Key Resource(s):'''

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 ===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>.
+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 potentially 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="right" />

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 [[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:NewThinkingAboutDispersion.mp4 |thumbnail|right|500px|Figure 9. Matrix diffusion processes and their effects on plume persistence and attenuation.]]
-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.
+Spatial&nbsp;variations&nbsp;in&nbsp;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.
+In&nbsp;some&nbsp;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.
 ===Matrix Diffusion===

@@ Line 296: / Line 296: @@
 ===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="right" />
 ==References==

@@ Line 185: / Line 185: @@
 ===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>.
+Mechanical&nbsp;dispersion ([https://en.wikipedia.org/wiki/Hydrogeology#Hydrodynamic_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===

@@ Line 14: / Line 14: @@
 *[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://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.
+*[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. Hydrogeologic 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==

@@ Line 180: / Line 180: @@
 ===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>.
+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://xmswiki.com/wiki/GMS:MT3DMS 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://xmswiki.com/wiki/GMS:MT3DMS 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).