Mixing-Limited Reactions in Porous Media

  • Albert J. Valocchi
  • Diogo BolsterEmail author
  • Charles J. Werth


Mixing-driven reactions in porous media are ubiquitous and span natural and engineered environments, yet predicting where and how quickly reactions occur is immensely challenging due to the complex and nonuniform nature of porous media flows. In particular, in many instances, there is an enormous range of spatial and temporal scales over which reactants can mix. This paper aims to review factors that affect mixing-limited reactions in porous media, and approaches used to predict such processes across scales. We focus primarily on the challenges of mixing-driven reactions in porous media at pore scales to provide a concise, but comprehensive picture. We balance our discussion between state-of-the-art experiments, theory and numerical methods, introducing the reader to factors that affect mixing, focusing on the bracketing cases of transverse and longitudinal mixing. We introduce the governing equations for mixing-limited reactions and then summarize several upscaling methods that aim to account for complex pore-scale flow fields. We conclude with perspectives on where the field is going, along with other insights gleaned from this review.


Mixing Reactions Upscaling 



The authors thank Dr. Tim Ginn and Dr. Branko Bijelkic for their valuable review comments as well as Artin Laleian for assistance in preparing Figs. 1 and 9. Work by AV and CW was supported as part of the Center for Geologic Storage of \(\hbox {CO}_2\), an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Basic Energy Sciences, under Award DE-SC0C12504. DB greatly acknowledges financial support from the US National Science Foundation via Grants EAR 1351625 and EAR 1417264.


  1. Acharya, R.C., Valocchi, A.J., Werth, C.J., Willingham, T.W.: Pore-scale simulation of dispersion and reaction along a transverse mixing zone in two-dimensional porous media. Water Resour. Res. 43(10), W10435 (2007). CrossRefGoogle Scholar
  2. Alhashmi, Z., Blunt, M., Bijeljic, B.: Predictions of dynamic changes in reaction rates as a consequence of incomplete mixing using pore scale reactive transport modeling on images of porous media. J. Contam. Hydrol. 179, 171–181 (2015)CrossRefGoogle Scholar
  3. Alhashmi, Z., Blunt, M., Bijeljic, B.: The impact of pore structure heterogeneity, transport, and reaction conditions on fluid-fluid reaction rate studied on images of pore space. Transp. Porous Media 115(2), 215–237 (2016)CrossRefGoogle Scholar
  4. Alvarez, P.J., Illman, W.A.: Bioremediation and Natural Attenuation: Process Fundamentals and Mathematical Models, vol. 27. Wiley, New York (2005)CrossRefGoogle Scholar
  5. Bagtzoglou, A.C., Oates, P.M.: Chaotic advection and enhanced groundwater remediation. J. Mater. Civ. Eng. 19(1), 75–83 (2007)CrossRefGoogle Scholar
  6. Barnard, J.M.: Simulation of mixing-limited reactions using a continuum approach. Adv. Water Resour. 104, 15–22 (2017)CrossRefGoogle Scholar
  7. Battiato, I., Tartakovsky, D.M.: Applicability regimes for macroscopic models of reactive transport in porous media. J. Contam. Hydrol. 120–121, 18–26 (2011)CrossRefGoogle Scholar
  8. Battiato, I., Tartakovsky, D.M., Tartakovsky, A.M., Scheibe, T.: On breakdown of macroscopic models of mixing-controlled heterogeneous reactions in porous media. Adv. Water Resour. 32(11), 1664–1673 (2009)CrossRefGoogle Scholar
  9. Bear, J.: Dynamics of Fluids in Porous Media. Courier Corporation, Chelmsford (1972)Google Scholar
  10. Benson, D.A., Meerschaert, M.M.: Simulation of chemical reaction via particle tracking: diffusion-limited versus thermodynamic rate-limited regimes. Water Resour. Res. 44(12), W12,201 (2008)CrossRefGoogle Scholar
  11. Bethke, C.M.: Geochemical and Biogeochemical Reaction Modeling. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  12. Bjerg, P.L., Tuxen, N., Reitzel, L.A., Albrechtsen, H.J., Kjeldsen, P.: Natural attenuation processes in landfill leachate plumes at three Danish sites. Groundwater 49(5), 688–705 (2011)CrossRefGoogle Scholar
  13. Bolster, D., Dentz, M., Le Borgne, T.: Solute dispersion in channels with periodically varying apertures. Phys. Fluids 21(5), 056601 (2009)CrossRefGoogle Scholar
  14. Bolster, D., Valdés-Parada, F.J., Le Borgne, T., Dentz, M., Carrera, J.: Mixing in confined stratified aquifers. J. Contam. Hydrol. 120, 198–212 (2011)CrossRefGoogle Scholar
  15. Bolster, D., Paster, A., Benson, D.A.: A particle number conserving lagrangian method for mixing-driven reactive transport. Water Resour. Res. 52(2), 1518–1527 (2016)CrossRefGoogle Scholar
  16. Boso, F., Bellin, A., Dumbser, M.: Numerical simulations of solute transport in highly heterogeneous formations: a comparison of alternative numerical schemes. Adv. Water Resour. 52, 178–189 (2013)CrossRefGoogle Scholar
  17. Carleton, J.N., Montas, H.J.: A modeling approach for mixing and reaction in wetlands with continuously varying flow. Ecol. Eng. 29(1), 33–44 (2007)CrossRefGoogle Scholar
  18. Chiogna, G., Bellin, A.: Analytical solution for reactive solute transport considering incomplete mixing within a reference elementary volume. Water Resour. Res. 49(5), 2589–2600 (2013)CrossRefGoogle Scholar
  19. Cirpka, O.A., Chiogna, G., Rolle, M., Bellin, A.: Transverse mixing in three-dimensional nonstationary anisotropic heterogeneous porous media. Water Resour. Res. 51(1), 241–260 (2015)CrossRefGoogle Scholar
  20. Connor, J.A., Kamath, R., Walker, K.L., McHugh, T.E.: Review of quantitative surveys of the length and stability of MTBE, TBA, and benzene plumes in groundwater at UST sites. Groundwater 53(2), 195–206 (2015). CrossRefGoogle Scholar
  21. Cushman, J., Dentz, M., Daniel, T.: Diffusion in porous media: phenomena and mechanisms. Transp. Porous Media (under review) (2018)Google Scholar
  22. de Anna, P., Jimenez-Martinez, J., Tabuteau, H., Turuban, R., Le Borgne, T., Derrien, M., Mheust, Y.: Mixing and reaction kinetics in porous media: an experimental pore scale quantification. Environ. Sci. Technol. 48(1), 508–516 (2013)CrossRefGoogle Scholar
  23. de Anna, P., Dentz, M., Tartakovsky, A., Le Borgne, T.: The filamentary structure of mixing fronts and its control on reaction kinetics in porous media flows. Geophys. Res. Lett. 41(13), 4586–4593 (2014)CrossRefGoogle Scholar
  24. de Barros, F.P., Dentz, M., Koch, J., Nowak, W.: Flow topology and scalar mixing in spatially heterogeneous flow fields. Geophys. Res. Lett. 39(8), L08,404 (2012)CrossRefGoogle Scholar
  25. Dentz, M., Kinzelbach, H., Attinger, S., Kinzelbach, W.: Temporal behavior of a solute cloud in a heterogeneous porous medium: 1. Point-like injection. Water Resour. Res. 36(12), 3591–3604 (2000)CrossRefGoogle Scholar
  26. Dentz, M., Le Borgne, T., Englert, A., Bijeljic, B.: Mixing, spreading and reaction in heterogeneous media: a brief review. J. Contam. Hydrol. 120, 1–17 (2011)CrossRefGoogle Scholar
  27. Dentz, M., Le Borgne, T., Lester, D.R., de Barros, F.P.J.: Mixing in groundwater. In: Cushman, J.H., Tartakovsky, D.M. (eds.) The Handbook of Groundwater Engineering, Chap. 13, pp. 383–412. CRC Press, Boca Raton (2017)Google Scholar
  28. Di Dato, M., de Barros, F.P., Fiori, A., Bellin, A.: Improving the efficiency of 3-d hydrogeological mixers: dilution enhancement via coupled engineering-induced transient flows and spatial heterogeneity. Water Resour. Res. 54(3), 2095–2111 (2018)CrossRefGoogle Scholar
  29. Ding, D., Benson, D.A., Paster, A., Bolster, D.: Modeling bimolecular reactions and transport in porous media via particle tracking. Adv. Water Resour. 53, 56–65 (2013)CrossRefGoogle Scholar
  30. Ding, D., Benson, D.A., Fernàndez-Garcia, D., Henri, C.V., Hyndman, D.W., Phanikumar, M.S., Bolster, D.: Elimination of the reaction rate scale effect: application of the Lagrangian reactive particle-tracking method to simulate mixing-limited, field-scale biodegradation at the schoolcraft (MI, USA) site. Water Resour. Res. 53, 10411–10432 (2017). CrossRefGoogle Scholar
  31. Duplat, J., Innocenti, C., Villermaux, E.: A nonsequential turbulent mixing process. Phys. Fluids 22(3), 035,104 (2010)CrossRefGoogle Scholar
  32. Edery, Y., Scher, H., Berkowitz, B.: Modeling bimolecular reactions and transport in porous media. Geophys. Res. Lett. 36(2), L02407 (2009).
  33. Edery, Y., Scher, H., Berkowitz, B.: Particle tracking model of bimolecular reactive transport in porous media. Water Resour. Res. 46(7), W07524 (2010). CrossRefGoogle Scholar
  34. Engdahl, N.B., Benson, D.A., Bolster, D.: Predicting the enhancement of mixing-driven reactions in nonuniform flows using measures of flow topology. Phys. Rev. E 90(5), 051,001 (2014)CrossRefGoogle Scholar
  35. Freire, L., Gerken, T., Ruiz-Plancarte, J., Wei, D., Fuentes, J., Katul, G., Dias, N., Acevedo, O., Chamecki, M.: Turbulent mixing and removal of ozone within an amazon rainforest canopy. J. Geophys. Res. Atmos. 122(5), 2791–2811 (2017)CrossRefGoogle Scholar
  36. Gandhi, R.K., Hopkins, G.D., Goltz, M.N., Gorelick, S.M., McCarty, P.L.: Full-scale demonstration of in situ cometabolic biodegradation of trichloroethylene in groundwater 2 Comprehensive analysis of field data using reactive transport modeling. Water Resour. Res. 38(4) (2002).
  37. Gillespie, D.: The chemical Langevin equation. J. Chem. Phys. 113(1), 297–306 (2000)CrossRefGoogle Scholar
  38. Ginn, T.: Modeling bimolecular reactive transport with mixing-limitation: theory and application to column experiments. Water Resour. Res. 54(1), 256–270 (2018)CrossRefGoogle Scholar
  39. Golfier, F., Wood, B.D., Orgogozo, L., Quintard, M., Buès, M.: Biofilms in porous media: development of macroscopic transport equations via volume averaging with closure for local mass equilibrium conditions. Adv. Water Resour. 32(3), 463–485 (2009)CrossRefGoogle Scholar
  40. Goode, D.J., Konikow, L.F.: Apparent dispersion in transient groundwater flow. Water Resour. Res. 26(10), 2339–2351 (1990). CrossRefGoogle Scholar
  41. Gramling, C.M., Harvey, C.F., Meigs, L.C.: Reactive transport in porous media: a comparison of model prediction with laboratory visualization. Environ. Sci. Technol. 36(11), 2508–2514 (2002)CrossRefGoogle Scholar
  42. Guo, J., Quintard, M., Laouafa, F.: Dispersion in porous media with heterogeneous nonlinear reactions. Transp. Porous Media 109(3), 541–570 (2015)CrossRefGoogle Scholar
  43. Haggerty, R., Harvey, C.F., von Schwerin, C.F., Meigs, L.C.: What controls the apparent timescale of solute mass transfer in aquifers and soils? a comparison of experimental results. Water Resour. Res. 40(1), W01510 (2004). CrossRefGoogle Scholar
  44. Hering, J.G., Morel, F.M.: Humic acid complexation of calcium and copper. Environ. Sci.Technol. 22(10), 1234–1237 (1988)CrossRefGoogle Scholar
  45. Hochstetler, D.L., Kitanidis, P.K.: The behavior of effective rate constants for bimolecular reactions in an asymptotic transport regime. J. Contamin. Hydrol. 144(1), 88–98 (2013)CrossRefGoogle Scholar
  46. Jiménez-Martínez, J., de Anna, P., Tabuteau, H., Turuban, R., Borgne, T.L., Méheust, Y.: Pore-scale mechanisms for the enhancement of mixing in unsaturated porous media and implications for chemical reactions. Geophys. Res. Lett. 42(13), 5316–5324 (2015)CrossRefGoogle Scholar
  47. Jiménez-Martínez, J., Le Borgne, T., Tabuteau, H., Méheust, Y.: Impact of saturation on dispersion and mixing in porous media: photobleaching pulse injection experiments and shear-enhanced mixing model. Water Resour. Res. 53(2), 1457–1472 (2017)CrossRefGoogle Scholar
  48. Jose, S.C., Cirpka, O.A.: Measurement of mixing-controlled reactive transport in homogeneous porous media and its prediction from conservative tracer test data. Environ. Sci. Technol. 38(7), 2089–2096 (2004)CrossRefGoogle Scholar
  49. Kitanidis, P.: The concept of the dilution index. Water Resour. Res. 30(7), 2011–2026 (1994)CrossRefGoogle Scholar
  50. Kitanidis, P.K., Dykaar, B.B.: Stokes flow in a slowly varying two-dimensional periodic pore. Transp. Porous Media 26(1), 89–98 (1997)CrossRefGoogle Scholar
  51. Kitanidis, P.K., McCarty, P.L.: Delivery and Mixing in the Subsurface: Processes and Design Principles for In Situ Remediation. SERDP and ESCTP Monograph Series. Springer, New York (2012).
  52. Knutson, C., Valocchi, A., Werth, C.: Comparison of continuum and pore-scale models of nutrient biodegradation under transverse mixing conditions. Adv. Water Resour. 30(6–7), 1421–1431 (2007)CrossRefGoogle Scholar
  53. Le Borgne, T., Dentz, M., Bolster, D., Carrera, J., De Dreuzy, J., Davy, P.: Non-Fickian mixing: temporal evolution of the scalar dissipation rate in heterogeneous porous media. Adv. Water Resour. 33(12), 1468–1475 (2010)CrossRefGoogle Scholar
  54. Le Borgne, T., Dentz, M., Villermaux, E.: Stretching, coalescence, and mixing in porous media. Phys. Rev. Lett. 110(20), 204,501 (2013)CrossRefGoogle Scholar
  55. Lester, D., Rudman, M., Metcalfe, G., Trefry, M., Ord, A., Hobbs, B.: Scalar dispersion in a periodically reoriented potential flow: acceleration via Lagrangian chaos. Phys. Rev. E 81(4), 046319 (2010)CrossRefGoogle Scholar
  56. Li, D.: Encyclopedia of Microfluidics and Nanofluidics. Springer Science & Business Media, Berlin (2008)CrossRefGoogle Scholar
  57. Libera, A., de Barros, F.P., Guadagnini, A.: Influence of pumping operational schedule on solute concentrations at a well in randomly heterogeneous aquifers. J. Hydrol. 546, 490–502 (2017)CrossRefGoogle Scholar
  58. Mays, D.C., Neupauer, R.M.: Plume spreading in groundwater by stretching and folding. Water Resour. Res. 48(7), W07501 (2012). CrossRefGoogle Scholar
  59. McCarty, P.L., Semprini, L.: Engineering and hydrogeological problems associated with in situ treatment. Hydrol. Sci. J. 38(4), 261–272 (1993)CrossRefGoogle Scholar
  60. Mujeebu, M.A., Abdullah, M.Z., Bakar, M.A., Mohamad, A., Abdullah, M.: Applications of porous media combustion technology—a review. Appl. Energy 86(9), 1365–1375 (2009)CrossRefGoogle Scholar
  61. Muniruzzaman, M., Rolle, M.: Experimental investigation of the impact of compound-specific dispersion and electrostatic interactions on transient transport and solute breakthrough. Water Resour. Res. 53(2), 1189–1209 (2017)CrossRefGoogle Scholar
  62. Neupauer, R.M., Meiss, J.D., Mays, D.C.: Chaotic advection and reaction during engineered injection and extraction in heterogeneous porous media. Water Resour. Res. 50(2), 1433–1447 (2014)CrossRefGoogle Scholar
  63. Niemi, A., Bear, J., Bensabat, J.: Geological Storage of CO\(_2\) in Deep Saline Formations, vol. 29. Springer, Berlin (2017)CrossRefGoogle Scholar
  64. Nunes, J.P., Bijeljic, B., Blunt, M.: Time-of-flight distributions and breakthrough curves in heterogeneous porous media using a pore-scale streamline tracing algorithm. Transp. Porous Media 109(2), 317–336 (2015)CrossRefGoogle Scholar
  65. Oates, P.: Upscaling reactive transport in porous media: laboratory visualizations and stochastic models. MIT, Ph.D. Thesis (2007)Google Scholar
  66. Paster, A., Bolster, D., Benson, D.A.: Connecting the dots: semi-analytical and random walk numerical solutions of the diffusion-reaction equation with stochastic initial conditions. J. Comput. Phys. 263, 91–112 (2014a)CrossRefGoogle Scholar
  67. Paster, A., Bolster, D., Benson, D.A.: Connecting the dots: semi-analytical and random walk numerical solutions of the diffusion-reaction equation with stochastic initial conditions. J. Comput. Phys. 263, 91–112 (2014b)CrossRefGoogle Scholar
  68. Paster, A., Aquino, T., Bolster, D.: Incomplete mixing and reactions in laminar shear flow. Phys. Rev. E 92(1), 012,922 (2015)CrossRefGoogle Scholar
  69. Phillips, A.J., Gerlach, R., Lauchnor, E., Mitchell, A.C., Cunningham, A.B., Spangler, L.: Engineered applications of ureolytic biomineralization: a review. Biofouling 29(6), 715–733 (2013). CrossRefGoogle Scholar
  70. Piscopo, A.N., Kasprzyk, J.R., Neupauer, R.M.: An iterative approach to multi-objective engineering design: optimization of engineered injection and extraction for enhanced groundwater remediation. Environ. Model. Softw. 69, 253–261 (2015)CrossRefGoogle Scholar
  71. Pool, M., Dentz, M.: Effects of heterogeneity, connectivity, and density variations on mixing and chemical reactions under temporally fluctuating flow conditions and the formation of reaction patterns. Water Resour. Res. 54(1), 186–204 (2018)CrossRefGoogle Scholar
  72. Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge (2000)CrossRefGoogle Scholar
  73. Porta, G.M., Riva, M., Guadagnini, A.: Upscaling solute transport in porous media in the presence of an irreversible bimolecular reaction. Adv. Water Resour. 35, 151–162 (2012)CrossRefGoogle Scholar
  74. Porta, G.M., Chaynikov, S., Thovert, J.F., Riva, M., Guadagnini, A., Adler, P.M.: Numerical investigation of pore and continuum scale formulations of bimolecular reactive transport in porous media. Adv. Water Resour. 62, 243–253 (2013)CrossRefGoogle Scholar
  75. Porta, G.M., Ceriotti, G., Thovert, J.F.: Comparative assessment of continuum-scale models of bimolecular reactive transport in porous media under pre-asymptotic conditions. J. Contam. Hydrol. 185, 1–13 (2016)CrossRefGoogle Scholar
  76. Raje, D.S., Kapoor, V.: Experimental study of bimolecular reaction kinetics in porous media. Environ. Sci. Technol. 34(7), 1234–1239 (2000)CrossRefGoogle Scholar
  77. Rasa, E., Bekins, B.A., Mackay, D.M., Sieyes, N.R., Wilson, J.T., Feris, K.P., Wood, I.A., Scow, K.M.: Impacts of an ethanol-blended fuel release on groundwater and fate of produced methane: simulation of field observations. Water Resour. Res. 49(8), 4907–4926 (2013)CrossRefGoogle Scholar
  78. Rehfeldt, K.R., Gelhar, L.W.: Stochastic analysis of dispersion in unsteady flow in heterogeneous aquifers. Water Resour. Res. 28(8), 2085–2099 (1992). CrossRefGoogle Scholar
  79. Rolle, M., Eberhardt, C., Chiogna, G., Cirpka, O.A., Grathwohl, P.: Enhancement of dilution and transverse reactive mixing in porous media: experiments and model-based interpretation. J. Contam. Hydrol. 110(3–4), 130–142 (2009)CrossRefGoogle Scholar
  80. Rolle, M., Hochstetler, D., Chiogna, G., Kitanidis, P.K., Grathwohl, P.: Experimental investigation and pore-scale modeling interpretation of compound-specific transverse dispersion in porous media. Transp. Porous Media 93(3), 347–362 (2012)CrossRefGoogle Scholar
  81. Rubio, A., Zalts, A., El Hasi, C.: Numerical solution of the advection-reaction-diffusion equation at different scales. Environ. Model. Softw. 23(1), 90–95 (2008)CrossRefGoogle Scholar
  82. Sanchez-Vila, X., Fernàndez-Garcia, D., Guadagnini, A.: Interpretation of column experiments of transport of solutes undergoing an irreversible bimolecular reaction using a continuum approximation. Water Resour. Res. 46(12) (2010).
  83. Scheibe, T.D., Schuchardt, K., Agarwal, K., Chase, J., Yang, X., Palmer, B.J., Tartakovsky, A.M., Elsethagen, T., Redden, G.: Hybrid multiscale simulation of a mixing-controlled reaction. Adv. Water Resour. 83, 228–239 (2015)CrossRefGoogle Scholar
  84. Semprini, L., Roberts, P.V., Hopkins, G.D., McCarty, P.L.: A field evaluation of in-situ biodegradation of chlorinated ethenes: part 2, results of biostimulation and biotransformation experiments. Groundwater 28(5), 715–727 (1990)CrossRefGoogle Scholar
  85. Shapiro, M., Brenner, H.: Dispersion of a chemically reactive solute in a spatially periodic model of a porous medium. Chem. Eng. Sci. 43(3), 551–571 (1988)CrossRefGoogle Scholar
  86. Sheng, J.: Modern Chemical Enhanced Oil Recovery: Theory and Practice. Gulf Professional Publishing, Houston (2010)Google Scholar
  87. Siuliukina, N., Tartakovsky, D.M.: A hybrid multiscale model of miscible reactive fronts. Water Resour. Res. 54, 61–71 (2018). CrossRefGoogle Scholar
  88. Souzy, M., Zaier, I., Lhuissier, H., Le Borgne, T., Metzger, B.: Mixing lamellae in a shear flow. J. Fluid Mech. 838, R3 (2018).
  89. Sposito, G.: Chaotic solute advection by unsteady groundwater flow. Water Resour. Res. 42(6), W06D03 (2006).
  90. Steefel, C., Appelo, C., Arora, B., Jacques, D., Kalbacher, T., Kolditz, O., Lagneau, V., Lichtner, P., Mayer, K.U., Meeussen, J., et al.: Reactive transport codes for subsurface environmental simulation. Comput. Geosci. 19(3), 445–478 (2015)CrossRefGoogle Scholar
  91. Stephenson, T., Brindle, K., Judd, S., Jefferson, B.: Membrane Bioreactors for Wastewater Treatment. IWA Publishing, Bloomberg (2000)Google Scholar
  92. Sund, N., Porta, G., Bolster, D., Parashar, R.: A Lagrangian transport Eulerian reaction spatial (laters) Markov model for prediction of effective bimolecular reactive transport. Water Resour. Res. 53(11), 9040–9058 (2017a)CrossRefGoogle Scholar
  93. Sund, N.L., Porta, G.M., Bolster, D.: Upscaling of dilution and mixing using a trajectory based spatial Markov random walk model in a periodic flow domain. Adv. Water Resour. 103, 76–85 (2017b)CrossRefGoogle Scholar
  94. Tang, Y., Valocchi, A.J., Werth, C.J.: A hybrid porescale and continuumscale model for solute diffusion, reaction, and biofilm development in porous media. Water Resour. Res. 51(3), 1846–1859 (2015). CrossRefGoogle Scholar
  95. Tartakovsky, A.M., Tartakovsky, G.D., Scheibe, T.D.: Effects of incomplete mixing on multicomponent reactive transport. Adv. Water Resour. 32(11), 1674–1679 (2009)CrossRefGoogle Scholar
  96. Trefry, M.G., Lester, D.R., Metcalfe, G., Ord, A., Regenauer-Lieb, K.: Toward enhanced subsurface intervention methods using chaotic advection. J. Contam. Hydrol. 127(1–4), 15–29 (2012)CrossRefGoogle Scholar
  97. Vafai, K.: Porous Media: Applications in Biological Systems and Biotechnology. CRC Press, Boca Raton (2010)CrossRefGoogle Scholar
  98. Werth, C.J., Cirpka, O.A., Grathwohl, P.: Enhanced mixing and reaction through flow focusing in heterogeneous porous media. Water Resour. Res. 42(12), W12414 (2006). CrossRefGoogle Scholar
  99. Willingham, T., Zhang, C., Werth, C.J., Valocchi, A.J., Oostrom, M., Wietsma, T.W.: Using dispersivity values to quantify the effects of pore-scale flow focusing on enhanced reaction along a transverse mixing zone. Adv. Water Resour. 33(4), 525–535 (2010)CrossRefGoogle Scholar
  100. Willingham, T.W., Werth, C.J., Valocchi, A.J.: Evaluation of the effects of porous media structure on mixing-controlled reactions using pore-scale modeling and micromodel experiments. Environ. Sci. Technol. 42(9), 3185–3193 (2008)CrossRefGoogle Scholar
  101. Wise, D.L.: Remediation Engineering of Contaminated Soils. CRC Press, Boca Raton (2000)CrossRefGoogle Scholar
  102. Wright, E.E., Richter, D.H., Bolster, D.: Effects of incomplete mixing on reactive transport in flows through heterogeneous porous media. Phys. Rev. Fluids 2(11), 114,501 (2017)CrossRefGoogle Scholar
  103. Ye, Y., Chiogna, G., Cirpka, O.A., Grathwohl, P., Rolle, M.: Experimental investigation of transverse mixing in porous media under helical flow conditions. Phys. Rev. E 94(1), 013,113 (2016)CrossRefGoogle Scholar
  104. Zhang, Y., Papelis, C.: Particle-tracking simulation of fractional diffusion-reaction processes. Phys. Rev. E 84(6), 066704 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringUniversity of IllinoisUrbanaUSA
  2. 2.Department of Civil and Environmental Engineering and Earth SciencesUniversity of Notre DameNotre DameUSA
  3. 3.Department of Civil, Architectural and Environmental EngineeringUniversity of Texas at AustinAustinUSA

Personalised recommendations