Theory and Applications of Macroscale Models in Porous Media

  • Ilenia BattiatoEmail author
  • Peter T. Ferrero V
  • Daniel O’ Malley
  • Cass T. Miller
  • Pawan S. Takhar
  • Francisco J. Valdés-Parada
  • Brian D. Wood


Systems dominated by heterogeneity over a multiplicity of scales, like porous media, still challenge our modeling efforts. The presence of disparate length- and time-scales that control dynamical processes in porous media hinders not only models predictive capabilities, but also their computational efficiency. Macrosopic models, i.e., averaged representations of pore-scale processes, are computationally efficient alternatives to microscale models in the study of transport phenomena in porous media at the system, field or device scale (i.e., at a scale much larger than a characteristic pore size). We present an overview of common upscaling methods used to formally derive macroscale equations from pore-scale (mass, momentum and energy) conservation laws. This review includes the volume averaging method, mixture theory, thermodynamically constrained averaging, homogenization, and renormalization group techniques. We apply these methods to a number of specific problems ranging from food processing to human bronchial system, and from diffusion to multiphase flow, to demonstrate the methods generality and flexibility in handling different applications. The primary intent of such an overview is not to provide a thorough review of all currently available upscaling techniques, nor a complete mathematical treatment of the ones presented, but rather a primer on some of the tools available for upscaling, the basic principles they are based upon, and their specific advantages and drawbacks, so to guide the reader in the choice of the most appropriate method for particular applications and of the most relevant technical literature.


Upscaling Porous media Volume averaging method Mixture theory Thermodynamically constrained averaging theory Homogenization theory Renormalization group theory 



The work of IB was fully supported by the Department of Energy under the Early Career Award DE-SC0014227953 “Multiscale dynamics of reactive fronts in the subsurface.” The work of DO was supported by the Laboratory Directed Research and Development program of Los Alamos National Laboratory under Project Number 20180481ER. The work of CTM was supported by Army Research Office Grant W911NF-14-1-02877 and National Science Foundation Grants 1619767 and 1604314. The work of PST was supported by the National Science Foundation award 0756762, and USDA-NIFA awards 2009- 35503-05279 and 2003- 3550-13963/TEXR-2006-03846. BDW was supported in part by NSF award EAR 1521441.

Author Contributions

The authors are listed in alphabetical order. IB led the effort, conceptualized the analysis, organized the entire manuscript, reviewed the homogenization method and integrated all the contributions by the other co-authors; DO reviewed renormalization group theory; CTM and FJV-P provided the overview on the thermodynamically constrained averaging theory; PST reviewed hybrid mixture theory; BDW and PTF-V provided the overview of the volume averaging method. All authors have reviewed and provided feedbacks on the manuscript structure and content.


  1. Achanta, S., Cushman, J.H., Okos, M.R.: On multicomponent, multiphase thermomechanics with interfaces. Int. J. Eng. Sci. 32(11), 1717–1738 (1994)CrossRefGoogle Scholar
  2. Achanta, S., Okos, M., Cushman, J., Kessler, D.: Moisture transport in shrinking gels during saturated drying. AICHE J. 43(8), 2112–2122 (1997)CrossRefGoogle Scholar
  3. Acharya, R.C., der Zee, S.E.A.T.M.V., Leijnse, A.: Transport modeling of nonlinearly adsorbing solutes in physically heterogeneous pore networks. Water Resour. Res. 41(W02), 020 (2005). CrossRefGoogle Scholar
  4. Adler, P.M.: Porous Media: Geometry and Transports. Butterworth-Heinemann, Oxford (1992)Google Scholar
  5. Alam, T., Takhar, P.S.: Microstructural characterization of fried potato discs using x-ray micro computed tomography. J. Food Sci. 81(3), E651–E664 (2016)CrossRefGoogle Scholar
  6. Alexander, F.J., Garcia, A.L., Tartakovsky, D.M.: Algorithm refinement for stochastic partial differential equations: 1. Linear diffusion. J. Comput. Phys. 182, 47–66 (2002)CrossRefGoogle Scholar
  7. Alexander, F.J., Garcia, A.L., Tartakovsky, D.M.: Algorithm refinement for stochastic partial differential equations: II. Correlated systems. J. Comput. Phys. 207(2), 769–787 (2005)CrossRefGoogle Scholar
  8. Anderson, T.B., Jackson, R.: Fluid mechanical description of fluidized beds. Equations of motion. Ind. Eng. Chem. Fundam. 6(4), 527–539 (1967)CrossRefGoogle Scholar
  9. Andrade Jr., J., Costa, U., Almeida, M., Makse, H., Stanley, H.: Inertial effects on fluid flow through disordered porous media. Phys. Rev. Lett. 82, 5249–5252 (1998)CrossRefGoogle Scholar
  10. Aris, R.: On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc A Math. Phys. Eng. Sci. 235(1200), 67–77 (1956)Google Scholar
  11. Arunachalam, H., Onori, S., Battiato, I.: On veracity of macroscopic Lithium-ion battery models. J. Electrochem. Soc. 162(9), A1–A12 (2015)Google Scholar
  12. Arunachalam, H., Korneev, S., Battiato, I., Onori, S.: Multiscale modeling approach to determine effective lithium-ion transport properties. In: 2017 American Control Conference, Seattle (2017).
  13. Astarita, G., Sarti, G.: A class of mathematical models for sorption of swelling solvents in glassy polymers. Polym. Eng. Sci. 18, 388–395 (1978)CrossRefGoogle Scholar
  14. Auriault, J.L.: On the domain of validity of Brinkman’s equation. Transp. Porous Med. 79, 215–223 (2009)CrossRefGoogle Scholar
  15. Auriault, J.L., Adler, P.M.: Taylor dispersion in porous media: analysis by multiple scale expansions. Adv. Water Resour. 4(18), 217–226 (1995)CrossRefGoogle Scholar
  16. Auriault, J.L., Geindreau, C., Boutin, C.: Filtration law in porous media with poor separation of scales. Transp. Porous Med. 60, 89–108 (2005)CrossRefGoogle Scholar
  17. Bakhvalov, N., Panasenko, G.: Homogenisation: Averaging Processes in Periodic Media. Kluwer Academic Publishers, Dordrecht (1989)CrossRefGoogle Scholar
  18. Bansal, H., Takhar, P.S., Maneerote, J.: Modeling multiscale transport mechanisms, phase changes and thermomechanics during frying. Food Res. Int. 62, 709–717 (2014)CrossRefGoogle Scholar
  19. Bansal, H., Takhar, P.S., Alvarado, C.Z., Thompson, L.D.: Transport mechanisms and quality changes in chicken nuggets during frying-hybrid mixture theory based multiscale modeling and experimental verification. J. Food Sci. 80(12), E2759–E2773 (2015)CrossRefGoogle Scholar
  20. Barry, D.A., Prommer, H., Miller, C.T., Engesgaard, P., Brun, A., Zheng, C.: Modelling the fate of oxidisable organic contaminants in groundwater. Adv. Water Resour. 25(8–12), 945–983 (2002)CrossRefGoogle Scholar
  21. Bassingthwaighte, J.B., Liebovitch, L.S., West, B.J.: Fractal Physiology. Springer, Berlin (2013)Google Scholar
  22. Battiato, I.: Multiscale models of flow and transport, chap 29. In: Cushman, J.H., Tartakovsky, D.M. (eds.) Handbook of Groundwater Engineering. CRC Press, Boca Raton (2016)Google Scholar
  23. Battiato, I., Rubol, S.: Single-parameter model of vegetated aquatic flows. Water Resour. Res. 50(8), 6358–6369 (2014)CrossRefGoogle Scholar
  24. 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
  25. 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, 1664–1673 (2009)CrossRefGoogle Scholar
  26. Battiato, I., Bandaru, P.R., Tartakovsky, D.M.: Elastic response of carbon nanotube forests to aerodynamic stresses. Phys. Rev. Lett. 105(144), 504 (2010). CrossRefGoogle Scholar
  27. Battiato, I., Tartakovsky, D.M., Tartakovsky, A.M., Scheibe, T.D.: Hybrid models of reactive transport in porous and fractured media. Adv. Water Resour. 34(9), 1140–1150 (2011). CrossRefGoogle Scholar
  28. Battiato, I., Tartakovsky, D.M., Cabrales, P., Intaglietta, M.: Role of glycocalyx in attenuation of shear stress on endothelial cells: from in vivo experiments to microfluidic circuits. In: IEEE Transactios on Biomedical Circuits and Systems Conference, Catania (2017)Google Scholar
  29. Baveye, P., Sposito, G.: The operational significance of the continuum hypothesis in the theory of water movement through soils and aquifers. Water Resour. Res. 20(5), 521–530 (1984)CrossRefGoogle Scholar
  30. Bear, J.: Modeling Phenomena of Flow and Transport in Porous Media. Springer, Cham (2018)CrossRefGoogle Scholar
  31. Bedford, A., Ingram, J.D.: A continuum theory of fluid saturated porous media. J. Appl. Mech. Trans. ASME 38(1), 1–7 (1971). CrossRefGoogle Scholar
  32. Bennethum, L.S., Cushman, J.H.: Multiscale, hybrid mixture theory for swelling systems.1. Balance laws. Int. J. Eng. Sci. 34(2), 125–145 (1996a)CrossRefGoogle Scholar
  33. Bennethum, L.S., Cushman, J.H.: Multiscale, hybrid mixture theory for swelling systems. 2: Constitutive theory. Int. J. Eng. Sci. 34(2), 147–169 (1996b)CrossRefGoogle Scholar
  34. Bennethum, L., Murad, M., Cushman, J.: Macroscale thermodynamics and the chemical potential for swelling porous media. Transp. Porous Med. 39(2), 187–225 (2000)CrossRefGoogle Scholar
  35. Berkowitz, B., Scher, H.: Theory of anomalous chemical transport in random fracture networks. Phys. Rev. E 57(5), 5858–5869 (1998)CrossRefGoogle Scholar
  36. Bhattad, P., Willson, C.S., Thompson, K.E.: Effect of network structure on characterization and flow modeling using x-ray micro-tomography images of granular and fibrous porous media. Transp. Porous Med. 90(2), 363–391 (2011)CrossRefGoogle Scholar
  37. Blunt, M.J.: Multiphase Flow in Permeable Media. Cambridge University Press, Cambridge (2017)CrossRefGoogle Scholar
  38. Boso, F., Battiato, I.: Homogenizability conditions of multicomponent reactive transport processes. Adv. Water Resour. 62, 254–265 (2013)CrossRefGoogle Scholar
  39. Bouchaud, J.P., Georges, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195(4–5), 127–293 (1990)CrossRefGoogle Scholar
  40. Bowen, R.: Theory of Mixtures, vol. 3. Academic Press, New York (1976)Google Scholar
  41. Brenner, H.: Transport Processes in Porous Media. McGraw-Hill, New York (1987)Google Scholar
  42. Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A1, 27–34 (1949)CrossRefGoogle Scholar
  43. Bruna, M., Chapman, S.J.: Diffusion in spatially varying porous media. SIAM J. Appl. Math. 75(4), 1648–1674 (2015)CrossRefGoogle Scholar
  44. Celia, M.A., Hassanizadeh, S.M., Dahle, H.K.: Inclusion of dynamic capillary pressure in unsaturated flow simulators. In: EOS Transactions, American Geophysical Union Fall Meeting, American Geophysical Union, San Francisco, vol. 82(47), H12F–02 (2001)Google Scholar
  45. Chen, L.Y., Goldenfeld, N., Oono, Y.: Renormalization-group theory for the modified porous-medium equation. Phys. Rev. A 44(10), 6544 (1991). CrossRefGoogle Scholar
  46. Christakos, G., Hristopulos, D.T.: Stochastic radon operators in porous media hydrodynamics. Q. Appl. Math. LV(1), 89–112 (1997)CrossRefGoogle Scholar
  47. Christakos, G., Hristopulos, D., Miller, C.: Stochastic diagrammatic analysis of groundwater flow in heterogeneous soils. Water Resour. Res. 31(7), 1687–1703 (1995)CrossRefGoogle Scholar
  48. Christensen, R.: Theory of Viscoelasticity. Academic Press, New York (1982)Google Scholar
  49. Ciucci, F., Lai, W.: Derivation of micro/macro lithium battery models from homogenization. Transp. Porous Med. 88(2), 249–270 (2011). CrossRefGoogle Scholar
  50. Crochet, M.J., Naghdi, P.M.: On constitutive equations for flow of fluid through an elastic solid. Int. J. Eng. Sci. 4(4), 383–401 (1966)CrossRefGoogle Scholar
  51. Cushman, J.H.: Proofs of the volume averaging theorems for multiphase flow. Adv. Water Resour. 5(4), 248–253 (1982)CrossRefGoogle Scholar
  52. Cushman, J.H.: Multiphase transport equations: I-general equation for macroscopic statistical, local space-time homogeneity1. Transp. Theory Stat. Phys. 12(1), 35–71 (1983)CrossRefGoogle Scholar
  53. Cushman, J.H.: On unifying the concepts of scale, instrumentation, and stochastics in the development of multiphase transport theory. Water Resour. Res. 20(11), 1668–1676 (1984)CrossRefGoogle Scholar
  54. Cushman, J.H. (ed.): Dynamics of Fluids in Hierarchical Porous Media. Academic Press, San Diego (1990)Google Scholar
  55. Cushman, J.H.: The Physics of Fluids in Hierarchical Porous Media: Angstroms to Miles. Kluwer Academic Publishers, Dordrecht (1997)CrossRefGoogle Scholar
  56. Cushman, J.H., O’Malley, D.: Fickian dispersion is anomalous. J. Hydrol. 531, 161–167 (2015). CrossRefGoogle Scholar
  57. Cushman, J.H., Bennethum, L.S., Hu, B.X.: A primer on upscaling tools for porous media. Adv. Water Resour. 25(8–12), 1043–1067 (2002)CrossRefGoogle Scholar
  58. Darcy, H.: Les fontaines publiques de la Ville de Dijon: exposition et application des principes à suivre et des formules à employer dans les questions de distribution d’eau, vol Paris. V. Dalmont (1856)Google Scholar
  59. Davit, Y., Quintard, M.: Technical notes on volume averaging in porous media I: how to choose a spatial averaging operator for periodic and quasiperiodic structures. Transp. Porous Med. 119(3), 555–584 (2017)CrossRefGoogle Scholar
  60. Davit, Y., Bell, C.G., Byrne, H.M., Chapman, L.A.C., Kimpton, L.S., Lang, G.E., Leonard, K.H.L., Oliver, J.M., Pearson, N.C., Shipley, R.J., Waters, S.L., Whiteley, J.P., Wood, B.D., Quintard, M.: Homogenization via formal multiscale asymptotics and volume averaging: How do the two techniques compare? Adv. Water Resour. 62, 178–206 (2013)CrossRefGoogle Scholar
  61. de Marsily, G.: Quantitative Hydrogeology. Academic Press, San Diego (1986)Google Scholar
  62. Deck, C., Ni, C., Vecchio, K.S., Bandaru, P.: The response of carbon nanotube ensembles to fluid flow: applications to mechanical property measurements and diagnostics. J. Appl. Phys. 106(074), 304 (2009)Google Scholar
  63. Dit-u dompo, S., Takhar, P.S.: Hybrid mixture theory based modeling of transport mechanisms and expansion-thermomechanics of starch during extrusion. Am. Inst. Chem. Eng. J. (AIChEJ) 61(12), 4517–4532 (2015)CrossRefGoogle Scholar
  64. Doyle, M., Newman, J.: The use of mathematical modeling in the design of lithium/polymer battery systems. Electrochim Acta 40(13–14), 2191–2196 (1995). CrossRefGoogle Scholar
  65. Doyle, M., Fuller, T.F., Newman, J.: Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell. J. Electrochem. Soc. 140(6), 1526–1533 (1993). CrossRefGoogle Scholar
  66. Dummit, D.S., Foote, R.M.: Abstract Algebra, vol. 3. Wiley, New York (2004)Google Scholar
  67. Durlofsky, L., Brady, J.: Analysis of the brinkman equation as a model for flow in porous media. Phys. Fluids 30(11), 3329–3341 (1987). CrossRefGoogle Scholar
  68. Durlovsky, L., Brady, J.F.: Analysis of the brinkman equation as a model for flow in porous media. Phys. Fluids 30(11), 3329–3341 (2009)CrossRefGoogle Scholar
  69. Ehlers, W., Bluhm, J.: Porous Media: Theory, Experiments and Numerical Applications. Springer, Berlin (2002)CrossRefGoogle Scholar
  70. Einstein, A.: Über die von der molekularkinetischen theorie der wärme geforderte bewegung von in ruhenden flüssigkeiten suspendierten teilchen. Ann. Phys. 4(17), 549 (1905)CrossRefGoogle Scholar
  71. Einstein, A.: Investigations on the Theory of the Brownian Movement. Dover Publications, Mineola (1956)Google Scholar
  72. Eringen, A.C.: Mechanics of Continua. R. E. Krieger Pub. Co., Huntington (1980)Google Scholar
  73. Farassat, F.: Discontinuities in aerodynamics and aeroacoustics: the concept and applications of generalized derivatives. J. Sound Vib. 55(2), 165–193 (1977)CrossRefGoogle Scholar
  74. Finden, E.: A homogenized thermal model for lithium ion batteries. Master’s thesis, Norwegian University of Life Sciences (2012)Google Scholar
  75. Forchheimer, P.: Wasserbewegung durch boden. Z. Ver. Deutsch. Ing. 45, 1782–1788 (1901)Google Scholar
  76. Freeze, R.A., Cherry, J.A.: Groundwater, vol 7632, 604. Prentice-Hall Inc., Englewood Cliffs (1979)Google Scholar
  77. Fulks, W., Guenther, R., Roetman, E.: Equations of motion and continuity for fluid flow in a porous medium. Bewegungs- und Kontinuitätsleichungen von Flüssigkeitsströmungen in einem porösen medium. Acta Mech. 12(1/2), 121 (1971)CrossRefGoogle Scholar
  78. Gell-Mann, M., Low, F.E.: Quantum electrodynamics at small distances. Phys. Rev. 95(5), 1300 (1954). CrossRefGoogle Scholar
  79. Giorgi, T.: Derivation of the forchheimer law via matched asymptotic expansions. Transp. Porous Med. 29, 191–206 (1997)CrossRefGoogle Scholar
  80. 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
  81. Goyeau, B., Benihaddadene, T., Gobin, D., Quintard, M.: Averaged momentum equation for flow through a nonhomogeneous porous structure. Transp. Porous Med. 28, 19–50 (1997)CrossRefGoogle Scholar
  82. Gray, W.G., Lee, P.: On the theorems for local volume averaging of multiphase systems. Int. J. Multiph. Flow 3(4), 333–340 (1977)CrossRefGoogle Scholar
  83. Gray, W.G., Miller, C.T.: Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 1. Motivation and overview. Adv. Water Resour. 28(2), 161–180 (2005)CrossRefGoogle Scholar
  84. Gray, W.G., Miller, C.T.: Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 3. Single-fluid-phase flow. Adv. Water Resour. 29(11), 1745–1765 (2006). CrossRefGoogle Scholar
  85. Gray, W.G., Miller, C.T.: Thermodynamically constrained averaging theory approach for heat transport in single-fluid-phase porous medium systems. J. Heat Transf. 131(10), 101,002 (2009a). CrossRefGoogle Scholar
  86. Gray, W.G., Miller, C.T.: Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 5. Single-fluid-phase transport. Adv. Water Resour. 32(5), 681–711 (2009b). CrossRefGoogle Scholar
  87. Gray, W.G., Miller, C.T.: Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 7. Single-phase megascale flow models. Adv. Water Resour. 32(8), 1121–1142 (2009c).
  88. Gray, W.G., Miller, C.T.: Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 8. Interface and common curve dynamics. Adv. Water Resour. 33(12), 1427–1443 (2010). CrossRefGoogle Scholar
  89. Gray, W.G., Miller, C.T.: TCAT analysis of capillary pressure in non-equilibrium, two-fluid-phase, porous medium systems. Adv. Water Resour. 34(6), 770–778 (2011). CrossRefGoogle Scholar
  90. Gray, W.G., Miller, C.T.: A generalization of averaging theorems for porous medium analysis. Adv. Water Resour. 62, 227–237 (2013). CrossRefGoogle Scholar
  91. Gray, W.G., Miller, C.T.: Introduction to the Thermodynamically Constrained Averaging Theory for Porous Medium Systems. Springer, Switzerland (2014). CrossRefGoogle Scholar
  92. Gray, W.G., Miller, C.T., Schrefler, B.A.: Averaging theory for description of environmental problems: What have we learned? Adv. Water Resour. 51, 123–138 (2013). CrossRefGoogle Scholar
  93. Gray, W.G., Dye, A.L., McClure, J.E., Pyrak-Nolte, L.J., Miller, C.T.: On the dynamics and kinematics of two-fluid-phase flow in porous media. Water Resour. Res. 51(7), 5365–5381 (2015). CrossRefGoogle Scholar
  94. Green, A.E., Steel, T.R.: Constitutive equations for interacting continua. Int. J. Eng. Sci. 4(4), 483–500 (1966)CrossRefGoogle Scholar
  95. Grmela, M.: Why GENERIC. J Non-Newton Fluid 165, 980–986 (2010)CrossRefGoogle Scholar
  96. Grmela, M., Grazzini, G., Lucia, U., Yahia, L.: Multiscale mesoscopic entropy of driven macroscopic systems. Entropy 15(11), 5053–5064 (2013)CrossRefGoogle Scholar
  97. Groot, J.: State-of-health estimation of li-ion batteries: cycle life test methods. PhD thesis, Chalmers University of Technology (2012)Google Scholar
  98. Hassanizadeh, S., Gray, W.: High velocity flow in porous media. Transp. Porous Med. 2, 521–531 (1987)CrossRefGoogle Scholar
  99. Hassanizadeh, S.M., Gray, W.G.: General conservation equations for multiphase systems: 1. Averaging procedure. Adv. Water Resour. 2, 131–144 (1979a)CrossRefGoogle Scholar
  100. Hassanizadeh, S.M., Gray, W.G.: General conservation equations for multiphase systems: 2. Mass, momenta, energy, and entropy equations. Adv. Water Resour. 2, 191–208 (1979b)CrossRefGoogle Scholar
  101. Hollister, S.J., Kikuchi, N.: Homogenization theory and digital imaging: a basis for studying the mechanics and design principles of bone tissue. Biotechnol. Bioeng. 43, 586–596 (1994)CrossRefGoogle Scholar
  102. Hornung, U.: Homogenization and Porous Media. Springer, New York (1997)CrossRefGoogle Scholar
  103. Howes, F.A., Whitaker, S.: The spatial averaging theorem revisited. Chem. Eng. Sci. 40(8), 1387–1392 (1985)CrossRefGoogle Scholar
  104. Hristopulos, D., Christakos, G.: Diagrammatic theory of nonlocal effective hydraulic conductivity. Stoch. Hydrol. Hydraul. 11(5), 369–395 (1997)CrossRefGoogle Scholar
  105. Hristopulos, D., Christakos, G.: Renormalization group analysis of permeability upscaling. Stoch. Environ. Res. Risk Assess. 13(1–2), 131–161 (1999). CrossRefGoogle Scholar
  106. Hristopulos, D.T.: Renormalization group methods in subsurface hydrology: overview and applications in hydraulic conductivity upscaling. Adv. Water Resour. 26(12), 1279–1308 (2003). CrossRefGoogle Scholar
  107. Hu, R., Oskay, C.: Spatial–temporal nonlocal homogenization model for transient anti-plane shear wave propagation in periodic viscoelastic composites. Comput. Methods Appl. Mech. Eng. 342, 1–31 (2018)CrossRefGoogle Scholar
  108. Hu, R., Oskay, C.: Multiscale nonlocal effective medium model for in-plane elastic wave dispersion and attenuation in periodic composites. J. Mech. Phys. Solids 124, 220–243 (2019)CrossRefGoogle Scholar
  109. Hubbert, M.K.: Darcy’s law and the field equations of the flow of underground fluids. Hydrol. Sci. J 2(1), 23–59 (1957)Google Scholar
  110. Hui, T., Oskay, C.: A nonlocal homogenization model for wave dispersion in dissipative composite materials. Int. J. Solids Struct. 50(1), 38–48 (2013)CrossRefGoogle Scholar
  111. Jackson, A., Rybak, I., Helmig, R., Gray, W., Miller, C.: Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 9. Transition region models. Adv. Water Resour. 42, 71–90 (2012). CrossRefGoogle Scholar
  112. Jackson, A.S., Miller, C.T., Gray, W.G.: Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 6. Two-fluid-phase flow. Adv. Water Resour. 32(6), 779–795 (2009). CrossRefGoogle Scholar
  113. Jones, D.S.: The Theory of Generalized Functions. Cambridge University Press, Cambridge (1982)CrossRefGoogle Scholar
  114. Kanwal, R.P.: Generalized Functions Theory and Technique: Theory and Technique. Academic Press, New York (1983)Google Scholar
  115. Karlin, S.: A First Course in Stochastic Processes. Academic press, New York (2014)Google Scholar
  116. Kechagia, P.E., Tsimpanogiannis, I.N., Yortsos, Y.C., Lichtner, P.C.: On the upscaling of reaction-transport processes in porous media with fast or finite kinetics. Chem. Eng. Sci. 57(13), 2565–2577 (2002). CrossRefGoogle Scholar
  117. Kim, D.J., Caruthers, J.M., Peppas, N.A.: Viscoelastic properties of dodecane polystyrene systems. Polymer 34(17), 3638–3647 (1993)CrossRefGoogle Scholar
  118. King, P.: The use of renormalization for calculating effective permeability. Transp. Porous Med. 4(1), 37–58 (1989)CrossRefGoogle Scholar
  119. Klika, V.: A guide through available mixture theories for applications. Crit. Rev. Solid State Mater. Sci. 39(2), 154–174 (2014)CrossRefGoogle Scholar
  120. Klika, V., Krause, A.: Beyond Onsager–Casimir relations: shared dependence of phenomenological coefficients on state variables. J. Phys. Chem. Lett. 9, 7021–7025 (2018)CrossRefGoogle Scholar
  121. Klika, V., Pavelka, M., Benziger, J.: Functional constraints on phenomenological coefficients. Phys. Rev. E 95(022), 125 (2017)Google Scholar
  122. Korneev, S., Battiato, I.: Sequential homogenization of reactive transport in porous media. Multiscale Model. Simul. 14(4), 1301–1318 (2016)CrossRefGoogle Scholar
  123. Korneev, S.V., Yang, X., Zachara, J.M., Scheibe, T.D., Battiato, I.: Downscaling-based segmentation for unresolved images of highly heterogeneous granular porous samples. Water Resour. Res. 54, 2871–2891 (2018). CrossRefGoogle Scholar
  124. Kreutzer, M.T., Kapteijn, F., Moulijn, J.A.: Shouldn’t catalysts shape up?: structured reactors in general and gas-liquid monolith reactors in particular. Catal. Today 111(1–2), 111–118 (2006)CrossRefGoogle Scholar
  125. Lai, W.M., Hou, J.S., Mow, V.C.: A triphasic theory for the swelling and deformation behaviors of articular cartilage. J. Biomech. Eng. 113(3), 245–258 (1991)CrossRefGoogle Scholar
  126. Lalam, S., Sandhu, J., Takhar, P.S., Thompson, L., Alvarado, C.: Experimental study on transport mechanisms during deep fat frying of chicken nuggets. LWT-Food Sci. Technol. 50(1), 110–119 (2013)CrossRefGoogle Scholar
  127. Lasseux, D., Valdés-Parada, F.J.: On the developments of darcy’s law to include inertial and slip effects. C. R. Méc. 345(9), 660–669 (2017)CrossRefGoogle Scholar
  128. Lasseux, D., Parada, F.J.V., Porter, M.L.: An improved macroscale model for gas slip flow in porous media. J. Fluid Mech. 805, 118–146 (2016)CrossRefGoogle Scholar
  129. Leemput, P., Vandekerckhove, C., Vanroose, W., Roose, D.: Accuracy of hybrid lattice Boltzmann/finite difference schemes for reaction diffusion systems. Multiscale Model. Simul. 6(3), 838–857 (2007)CrossRefGoogle Scholar
  130. Lévy, T.: Fluid flow through an array of fixed particles. Int. J. Eng. Sci. 21, 11–23 (1983)CrossRefGoogle Scholar
  131. Li, H., Pan, C., Miller, C.T.: Pore-scale investigation of viscous coupling effects for two-phase flow in porous media. Phys. Rev. E 72(2), 026705 (2005)CrossRefGoogle Scholar
  132. Ling, B., Tartakovsky, A.M., Battiato, I.: Dispersion controlled by permeable surfaces: surface properties and scaling. J. Fluid Mech. 801, 13–42 (2016)CrossRefGoogle Scholar
  133. Ling, B., Oostrom, M., Tartakovsky, A.M., Battiato, I.: Hydrodynamic dispersion in thin channels with micro-structured porous walls. Phys. Fluids 30, 076601 (2018). (Accepted) CrossRefGoogle Scholar
  134. Malley, D.O., Karra, S., Currier, R.P., Makedonska, N., Hyman, J.D., Viswanathan, H.S.: Where does water go during hydraulic fracturing? Groundwater 54(4), 488–497 (2016)CrossRefGoogle Scholar
  135. Marle, C.: Ecoulements monophasiques en milieu poreux. Rev. Inst. Francais du Petrole 22(10), 1471–1509 (1967)Google Scholar
  136. Marle, C.: On macroscopic equations governing multiphase flow with diffusion and chemical reactions in porous media. Int. J. Eng. Sci. 20(5), 643–662 (1982a)CrossRefGoogle Scholar
  137. Marle, C.M.: On macroscopic equations governing multiphase flow with diffusion and chemical reactions in porous media. Int. J. Eng. Sci. 20(5), 643–662 (1982b)CrossRefGoogle Scholar
  138. Marušić-Paloka, E., Mikelic, A.: The derivation of a non-linear filtration law including the inertia effects via homogenization. Nonlinear Anal. Theory Methods Appl. 42, 97–137 (2000)CrossRefGoogle Scholar
  139. Marušić-Paloka, E., Piatnitski, A.: Homogenization of a nonlinear convection–diffusion equation with rapidly oscillating coefficients and strong convection. J. Lond. Math. Soc. 2(72), 391–409 (2005)CrossRefGoogle Scholar
  140. McClure, J.E., Armstrong, R.T., Berrill, M.A., Schlüeter, S., Berg, S., Gray, W.G., Miller, C.T.: A geometric state function for two-fluid flow in porous media. Phys. Rev. Fluids 3, 084306 (2018). CrossRefGoogle Scholar
  141. Mikelic, A., Devigne, V., Van Duijn, C.J.: Rigorous upscaling of the reactive flow through a pore, under dominant Peclet and Damköhler numbers. SIAM J. Math. Anal. 38(4), 1262–1287 (2006)CrossRefGoogle Scholar
  142. Miller, C.T., Gray, W.G.: Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 2. Foundation. Adv. Water Resour. 28(2), 181–202 (2005)CrossRefGoogle Scholar
  143. Miller, C.T., Gray, W.G.: Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 4. Species transport fundamentals. Adv. Water Resour. 31(3), 577–597 (2008). CrossRefGoogle Scholar
  144. Miller, C.T., Christakos, G., Imhoff, P.T., McBride, J.F., Pedit, J.A., Trangenstein, J.A.: Multiphase flow and transport modeling in heterogeneous porous media: challenges and approaches. Adv. Water Resour. 21(2), 77–120 (1998)CrossRefGoogle Scholar
  145. Miller, C.T., Valdés-Parada, F.J., Wood, B.D.: A pedagogical approach to the thermodynamically constrained averaging theory. Transp. Porous Med. 119(3), 585–609 (2017). CrossRefGoogle Scholar
  146. Miller, C.T., Gray, W.G., Kees, C.E.: Thermodynamically constrained averaging theory: principles, model hierarchies, and deviation kinetic energy extensions. Entropy 20(4), 253 (2018a). CrossRefGoogle Scholar
  147. Miller, C.T., Valdés-Parada, F.J., Ostvar, S., Wood, B.D.: A priori parameter estimation for the thermodynamically constrained averaging theory: species transport in a saturated porous medium. Transp. Porous Med. 122(3), 611–632 (2018b). CrossRefGoogle Scholar
  148. Mls, J.: On the existence of the derivative of the volume average. Transp. Porous Med. 2(6), 615–621 (1987)CrossRefGoogle Scholar
  149. Montroll, E.W., Scher, H.: Random walks on lattices. IV. Continuous time random walks and influence of adsorbing boundaries. J. Stat. Phys. 9(2), 101–135 (1973)CrossRefGoogle Scholar
  150. Morse, J.W., Arvidson, R.S.: The dissolution kinetics of major sedimentary carbonate minerals. Earth Sci. Rev. 58, 51–84 (2002)CrossRefGoogle Scholar
  151. Neuman, S.P.: Theoretical derivation of Darcy’s law. Acta Mecanica 25, 153–170 (1977)CrossRefGoogle Scholar
  152. Neuman, S.P., Tartakovsky, D.M.: Perspective on theories of anomalous transport in heterogeneous media. Adv. Water Resour. 32(5), 670–680 (2009)CrossRefGoogle Scholar
  153. Newman, J., Tiedemann, W.: Porous-electrode theory with battery applications. AIChE J. 21(1), 25–41 (1975). CrossRefGoogle Scholar
  154. Nitsche, L.C., Brenner, H.: Eulerian kinematics of flow through spatially periodic models of porous media. Arch. Ration. Mech. Anal. 107(3), 225–295 (1989)CrossRefGoogle Scholar
  155. O’Malley, D., Cushman, J.H.: A renormalization group classification of nonstationary and/or infinite second moment diffusive processes. J. Stat. Phys. 146(5), 989–1000 (2012a). CrossRefGoogle Scholar
  156. O’Malley, D., Cushman, J.H.: Two-scale renormalization-group classification of diffusive processes. Phys. Rev. E 86(1), 011126 (2012b). CrossRefGoogle Scholar
  157. Oztop, M., Bansal, H., Takhar, P., McCarthy, M.J., McCarthy, K.L.: Using multi-slice-multi-echo images with NMR relaxometry to assess water and fat distribution in coated chicken nuggets. LWT Food Sci. Technol. 55(2), 690–694 (2014)CrossRefGoogle Scholar
  158. Pangarkar, K., Schildhauer, T.J., van Ommen, J.R., Nijenhuis, J., Kapteijn, F., Moulijn, J.A.: Structured packings for multiphase catalytic reactors. Ind. Eng. Chem. Res. 47(10), 3720–3751 (2008)CrossRefGoogle Scholar
  159. Peszynska, M., Lu, Q., Wheeler, M.: Coupling different numerical coupling different numerical algorithms for two phase fluid flow. In: Mathematics of Finite Elements and Applications X. Elsevier, pp. 205–214 (2000)Google Scholar
  160. Peter, M.A.: Homogenization in domains with evolving microstructure. C. R. Méc. 335, 357–362 (2007)CrossRefGoogle Scholar
  161. Peter, M.A.: Coupled reaction–diffusion processes inducing an evolution of the microstructure: analysis and homogenization. Nonlinear Anal. Theory Methods Appl. 70(2), 806 (2009)CrossRefGoogle Scholar
  162. Picchi, U.A.B.N.D.: Modelling of core-annular and plug flows of newtonian/non-newtonian shear-thinning fluids in pipes and capillary tubes. Int. J. Multiph. Flow 103, 43–60 (2018)CrossRefGoogle Scholar
  163. Pinson, M.B., Bazant, M.Z.: Theory of SEI formation in rechargeable batteries: capacity fade, accelerated aging and lifetime prediction. J. Electrochem. Soc. 160(2), A243–A250 (2013). CrossRefGoogle Scholar
  164. Ploehn, H.J., Ramadass, P., White, R.E.: Solvent diffusion model for aging of lithium-ion battery cells. J. Electrochem. Soc. 151(3), A456–A462 (2004). CrossRefGoogle Scholar
  165. Quintard, M., Whitaker, S.: Transport in ordered and disordered porous media ii: generalized volume averaging. Transp. Porous Med. 14(2), 179–206 (1994a)CrossRefGoogle Scholar
  166. Quintard, M., Whitaker, S.: Transport in ordered and disordered porous media iii: closure and comparison between theory and experiment. Transp. Porous Med. 15(1), 31–49 (1994b)CrossRefGoogle Scholar
  167. Quintard, M., Whitaker, S.: Transport in ordered and disordered porous media iv: computer generated porous media for three-dimensional systems. Transp. Porous Med. 15(1), 51–70 (1994c)CrossRefGoogle Scholar
  168. Quintard, M., Whitaker, S.: Transport in ordered and disordered porous media v: geometrical results for two-dimensional systems. Transp. Porous Med. 15(2), 183–196 (1994d)CrossRefGoogle Scholar
  169. Rajagopal, K.R., Tao, L.: Mechanics of Mixtures. Series on Advances in Mathematics for Applied Sciences. World Scientific, Singapore (1995)CrossRefGoogle Scholar
  170. Rubol, S., Battiato, I., de Barros, F.P.J.: Vertical dispersion in vegetated shear flows. Water Resour. Res. 52, 8066–8080 (2016). CrossRefGoogle Scholar
  171. Rubol, S., Ling, B., Battiato, I.: Universal scaling-law for flow resistance over canopies with complex morphology. Sci. Rep. 8, 4430 (2018). CrossRefGoogle Scholar
  172. Rybak, I.V., Gray, W.G., Miller, C.T.: Modeling two-fluid-phase flow and species transport in porous media. J. Hydrol. 521, 565–581 (2015). CrossRefGoogle Scholar
  173. Salvadori, A., Bosco, E., Grazioli, D.: A computational homogenization approach for Li-ion battery cells: part 1—formulation. J. Mech. Phys. Solids 65, 114–137 (2014). CrossRefGoogle Scholar
  174. Sanchez-Palencia, E.E.: Solutions périodiques par rapport aux variables d’espaces et applications. C. R. Acad. Sci. Paris Sér. A-B 271(A), 1129–1132 (1970)Google Scholar
  175. Scher, H., Montroll, E.W.: Anomalous transit time dispersion in amorphous solids. Phys. Rev. B 12(6), 2455–2477 (1975)CrossRefGoogle Scholar
  176. Schreyer-Bennethum, L.: Theory of flow and deformation of swelling porous materials at the macroscale. Comput. Geotech. 34(4), 267–278 (2007)CrossRefGoogle Scholar
  177. Schwartz, L.: Théorie des Distributions. Hermann, Paris (1966)Google Scholar
  178. Sciumé, G., Shelton, S.E., Gray, W.G., Miller, C.T., Hussain, F., Ferrari, M., Decuzzi, P., Schrefler, B.A.: Tumor growth modeling from the perspective of multiphase porous media mechanics. Mol. Cell. Biomech. 202(1), 1–20 (2012)Google Scholar
  179. Sciumé, G., Shelton, S., Gray, W.G., Miller, C.T., Hussain, F., Ferrari, M., Decuzzi, P., Schrefler, B.A.: A multiphase model for three-dimensional tumor growth. New J. Phys. 15, 015005 (2013). CrossRefGoogle Scholar
  180. Semenenko, M.G.: Application effective medium approximation approach for economic researching. Phys. A Stat. Mech. Appl. 329(1–2), 264–272 (2003)CrossRefGoogle Scholar
  181. Shapiro, M., Brenner, H.: Taylor dispersion of chemically reactive species: irreversible first-order reactions in bulk and on boundaries. Chem. Eng. Sci. 41(6), 1417–1433 (1986)CrossRefGoogle Scholar
  182. Shapiro, M., Brenner, M.: 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
  183. Shapiro, M., Fedou, R., Thovert, J., Adler, P.M.: Coupled transport and dispersion of multicomponent reactive solutes in rectilinear flows. Chem. Eng. Sci. 51(22), 5017–5041 (1996). CrossRefGoogle Scholar
  184. Shilov, G.E.: Generalized Functions and Partial Differential Equations. Gordon and Breach, London (1968)Google Scholar
  185. Singh, P.P., Cushman, J.H., Bennethum, L.S., Maier, D.: Thermomechanics of swelling biopolymeric systems. Transp. Porous Med. 53(1), 1–24 (2003a)CrossRefGoogle Scholar
  186. Singh, P.P., Cushman, J.H., Maier, D.: Multiscale fluid transport theory for swelling biopolymers. Chem. Eng. Sci. 58(11), 2409–2419 (2003b)CrossRefGoogle Scholar
  187. Singh, P.P., Cushman, J.H., Maier, D.: Three scale thermomechanical theory for swelling biopolymeric systems. Chem. Eng. Sci. 58(17), 4017–4035 (2003c)CrossRefGoogle Scholar
  188. Singh, P.P., Maier, D., Cushman, J.H., Haghighi, K., Corvalan, C.: Effect of viscoelastic relaxation on moisture transport in foods. Part i: solution of general transport equation. J. Math. Biol. 49(1), 1–19 (2004)Google Scholar
  189. Slattery, J.C.: Flow of viscoelastic fluids through porous media. AIChE J. 13(6), 1066–1071 (1967)CrossRefGoogle Scholar
  190. Slattery, J.C.: Advanced Transport Phenomena. Cambridge University Press, Cambridge (1999)CrossRefGoogle Scholar
  191. Smith, A.J., Burns, A.C., Zhao, X., Xiong, D., Dahn, J.R.: A high precision coulometry study of the SEI growth in Li/graphite cells. J. Electrochem. Soc. 158(5), A447–A452 (2011). CrossRefGoogle Scholar
  192. Song, H., Litchfield, J., Morris, H.D.: Three-dimensional microscopic MRI of maize kernels during drying. J. Agric. Eng. Res. 53, 51–69 (1992)CrossRefGoogle Scholar
  193. Spagnolo, S.: Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Scuola. Norm. Sup. Pisa. 3(22), 571–597 (1968)Google Scholar
  194. Sun, T.Y., Mehmani, Y., Balhoff, M.T.: Hybrid multiscale modeling through direct numerical substitution of pore-scale models into near-well reservoir simulators. Energy Fuels 26, 5828–5836 (2012)CrossRefGoogle Scholar
  195. Takhar, P.S.: Hybrid mixture theory based moisture transport and stress development in corn kernels during drying: coupled fluid transport and stress equations. J. Food Eng. 105(4), 663–670 (2011)CrossRefGoogle Scholar
  196. Takhar, P.S.: Unsaturated fluid transport in swelling poroviscoelastic biopolymers. Chem. Eng. Sci. 109, 98–110 (2014)CrossRefGoogle Scholar
  197. Takhar, P.S., Maier, D.E., Campanella, O., Chen, G.: Hybrid mixture theory based moisture transport and stress development in corn kernels during drying: validation and simulation results. J. Food Eng. 106, 275–282 (2011)CrossRefGoogle Scholar
  198. Tartakovsky, D.M., Dentz, M.: Diffusion in porous media: phenomena and mechanisms. Transp. Porous Med. 1–23, (2019).
  199. Tartakovsky, D.M., Neuman, S.P.: Transient flow in bounded randomly heterogeneous domains. 1. Exact conditional moment equations and recursive approximation. Water Resour. Res. 34(1), 1–12 (1998)CrossRefGoogle Scholar
  200. Tartakovsky, D.M., SP, S.P.N., Lu, Z.: Conditional stochastic averaging of steady state unsaturated flow by means of kirchoff transformation. Water Resour. Res. 35(3), 731–745 (1999)CrossRefGoogle Scholar
  201. Tartakovsky, A.M., Meakin, P., Scheibe, T.D., West, R.M.E.: Simulation of reactive transport and precipitation with smoothed particle hydrodynamics. J. Comput. Phys. 222, 654–672 (2007)CrossRefGoogle Scholar
  202. Taverniers, S., Alexander, F.J., Tartakovsky, D.M.: Noise propagation in hybrid models of nonlinear systems: the ginzburg-landau equation. J. Comput. Phys. 262, 313–324 (2014)CrossRefGoogle Scholar
  203. Taylor, G.I.: Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. A 219(1137), 186–203 (1953)Google Scholar
  204. Thomas, K.E., Newman, J.: Thermal modeling of porous insertion electrodes. J. Electrochem. Soc. 150(2), A176–A192 (2003). CrossRefGoogle Scholar
  205. Thomas, N.L., Windle, A.H.: A deformation model for case-ii diffusion. Polymer 21(6), 613–619 (1980)CrossRefGoogle Scholar
  206. Thomas, N.L., Windle, A.H.: A theory of case-ii diffusion. Polymer 23(4), 529–542 (1982)CrossRefGoogle Scholar
  207. Um, W., Serne, R.J., Yabusaki, S.B., Owen, A.T.: Enhanced radionuclide immobilization and flow path modifications by dissolution and secondary precipitates. J. Environ. Qual. 34, 1404–1414 (2005)CrossRefGoogle Scholar
  208. Valdés-Parada, F.J., Alvarez-Ramírez, J.: A volume averaging approach for asymmetric diffusion in porous media. J. Chem. Phys. 134(20), 204709 (2011)CrossRefGoogle Scholar
  209. Valdés-Parada, F.J., Romero-Paredes, H., Espinosa Paredes, G.: Numerical simulation of a tubular solar reactor for methane cracking. Int. J. Hydrogen Energy 36(5), 3354–3363 (2011)CrossRefGoogle Scholar
  210. Valdés-Parada, F.J., Varela, J.R., Alvarez-Ramirez, J.: Francisco j. valdés-parada and juan r. varela and josé alvarez-ramirez. Phys. A Stat. Mech. Appl. 391(3), 606–615 (2012)CrossRefGoogle Scholar
  211. Vervloet, D., Kapteijn, F., Nijenhuis, J., van Ommen, J.R.: Process intensification of tubular reactors: considerations on catalyst hold-up of structured packings. Catal. Today 216, 111–116 (2013)CrossRefGoogle Scholar
  212. Veverka, V.: Theorem for the local volume average of a gradient revised. Chem. Eng. Sci. 36(5), 833–838 (1981)CrossRefGoogle Scholar
  213. Weibel, E.R.: The pathway for oxygen: structure and function in the mammalian respiratory system. Harvard University Press, Cambridge (1984)Google Scholar
  214. Whitaker, S.: Diffusion and dispersion in porous media. AIChE J. 13(3), 420–427 (1967)CrossRefGoogle Scholar
  215. Whitaker, S.: Simultaneous heat, mass, and momentum transfer in porous media: a theory of drying. Adv. Heat Transf. 13, 119–203 (1977)CrossRefGoogle Scholar
  216. Whitaker, S.: A simple geometrical derivation of the spatial averaging theorem. Chem. Eng. Educ. 19(1), 18–21 (1985)Google Scholar
  217. Whitaker, S.: Flow in porous media I: a theoretical derivation of darcy’s law. Transp. Porous Med. 1(1), 3–25 (1986). CrossRefGoogle Scholar
  218. Whitaker, S.: The forchheimer equation: a theoretical development. Transp. Porous Med. 25, 27–61 (1996)CrossRefGoogle Scholar
  219. Whitaker, S.: The method of volume averaging, vol. 13. Springer, Berlin (1999)Google Scholar
  220. Wilson, K.G.: The renormalization group and critical phenomena. Rev. Mod. Phys. 55(3), 583 (1983). CrossRefGoogle Scholar
  221. Wilson, K.G., Kogut, J.: The renormalization group and the \(\epsilon \) expansion. Phys. Rep. 12(2), 75–199 (1974). CrossRefGoogle Scholar
  222. Wojciechowski, K.J.: (2011) Analysis and numerical solution of nonlinear volterra partial integrodifferential equations modeling swelling porous materials. ThesisGoogle Scholar
  223. Wojciechowski, K.J., Chen, J., Schreyer-Bennethum, L., Sandberg, K.: Well-posedness and numerical solution of a nonlinear volterra partial integro-differential equation modeling a swelling porous material. J. Porous Med. 17(9), 763–784 (2014)CrossRefGoogle Scholar
  224. Wood, B.D.: Review of upscaling methods for describing unsaturated flow. Tech. Rep. 13325, Pacific Northwest National Lab., Richland (2000)Google Scholar
  225. Wood, S.N.: Generalized Additive Models. An Introduction with R. Chapman & Hall/CRC, Boca Raton (2006)CrossRefGoogle Scholar
  226. Wood, B.D.: The role of scaling laws in upscaling. Adv. Water Resour. 32(5), 723–736 (2009)CrossRefGoogle Scholar
  227. Wood, B.D., Valdés-Parada, F.J.: Volume averaging: local and nonlocal closures using a green’s function approach. Adv. Water Resour. 51, 139–167 (2013)CrossRefGoogle Scholar
  228. Wood, B.D., Quintard, M., Whitaker, S.: Calculation of effective diffusivities for biofilms and tissues. Biotechnol. Bioeng. 77(5), 495–516 (2002)CrossRefGoogle Scholar
  229. Wood, B., Apte, S., Liburdy, J., Ziazi, R., He, X., Finn, J., Patil, V.: A comparison of measured and modeled velocity fields for a laminar flow in a porous medium. Adv. Water Resour. 85, 45–63 (2015)CrossRefGoogle Scholar
  230. Wood, B.D., He, B., Apte, S.V.: (2020) Modeling turbulent flows in porous media. Ann. Rev. Fluid Mech. 52. (in production) Google Scholar
  231. Xing, H., Takhar, P., Helms, G., He, B.: Nmr imaging of continuous and intermittent drying of pasta. J. Food Eng. 78, 61–68 (2007)CrossRefGoogle Scholar
  232. Yakhot, V., Orszag, S.A.: Renormalization group analysis of turbulence. I. Basic theory. J. Sci. Comput. 1(1), 3–51 (1986). CrossRefGoogle Scholar
  233. Youzefzadeh, M., Battiato, I.: Nonintrusive hybrid models of reactive transport in fractured media: an immersed boundary method approach. J. Comput. Phys. 244, 320–338 (2017)CrossRefGoogle Scholar
  234. Zhang, X., Tartakovsky, D.M.: Effective ion diffusion in charged nanoporous materials. J. Electrochem. Soc. 164(4), E53–E61 (2017a)CrossRefGoogle Scholar
  235. Zhang, X., Tartakovsky, D.M.: Optimal design of nanoporous materials for electrochemical devices. Appl. Phys. Lett. 110(14), 143,103 (2017b)CrossRefGoogle Scholar
  236. Zhao, Y., Takhar, P.: Freezing of foods: mathematical and experimental aspects. Food. Eng. Rev. (2017). CrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Energy Resources EngineeringStanford UniversityStanfordUSA
  2. 2.Chemical, Biological, and Environmental EngineeringOregon State UniversityCorvallisUSA
  3. 3.Computational Earth Science GroupLos Alamos National LaboratoryLos AlamosMexico
  4. 4.Department of Computer Science and Electrical EngineeringUniversity of MarylandBaltimore County, BaltimoreUSA
  5. 5.Environmental Sciences and EngineeringUniversity of North CarolinaChapel HillUSA
  6. 6.Food Science and Human NutritionUniversity of IllinoisUrbana-ChampaignUSA
  7. 7.Departamento de Ingeniería de Procesos e HidráulicaMetropolitan Autonomous UniversityMexico CityMexico

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