{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T16:30:38Z","timestamp":1770741038402,"version":"3.49.0"},"reference-count":34,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2018,4,5]],"date-time":"2018-04-05T00:00:00Z","timestamp":1522886400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The thermodynamically constrained averaging theory (TCAT) is a comprehensive theory used to formulate hierarchies of multiphase, multiscale models that are closed based upon the second law of thermodynamics. The rate of entropy production is posed in terms of the product of fluxes and forces of dissipative processes. The attractive features of TCAT include consistency across disparate length scales; thermodynamic consistency across scales; the inclusion of interfaces and common curves as well as phases; the development of kinematic equations to provide closure relations for geometric extent measures; and a structured approach to model building. The elements of the TCAT approach are shown; the ways in which each of these attractive features emerge from the TCAT approach are illustrated; and a review of the hierarchies of models that have been formulated is provided. Because the TCAT approach is mathematically involved, we illustrate how this approach can be applied by leveraging existing components of the theory that can be applied to a wide range of applications. This can result in a substantial reduction in formulation effort compared to a complete derivation while yielding identical results. Lastly, we note the previous neglect of the deviation kinetic energy, which is not important in slow porous media flows, formulate the required equations to extend the theory, and comment on applications for which the new components would be especially useful. This work should serve to make TCAT more accessible for applications, thereby enabling higher fidelity models for applications such as turbulent multiphase flows.<\/jats:p>","DOI":"10.3390\/e20040253","type":"journal-article","created":{"date-parts":[[2018,4,5]],"date-time":"2018-04-05T16:50:58Z","timestamp":1522947058000},"page":"253","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Thermodynamically Constrained Averaging Theory: Principles, Model Hierarchies, and Deviation Kinetic Energy Extensions"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6082-9273","authenticated-orcid":false,"given":"Cass T.","family":"Miller","sequence":"first","affiliation":[{"name":"Department of Environmental Sciences and Engineering, University of North Carolina, Chapel Hill, NC 27599-7431, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4045-750X","authenticated-orcid":false,"given":"William G.","family":"Gray","sequence":"additional","affiliation":[{"name":"Department of Environmental Sciences and Engineering, University of North Carolina, Chapel Hill, NC 27599-7431, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7042-6221","authenticated-orcid":false,"given":"Christopher E.","family":"Kees","sequence":"additional","affiliation":[{"name":"US Army Engineer Research and Development Center, Vicksburg, MS 39180-6199, USA"}]}],"member":"1968","published-online":{"date-parts":[[2018,4,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.advwatres.2011.12.005","article-title":"Averaging Theory for Description of Environmental Problems: What Have We Learned?","volume":"51","author":"Gray","year":"2013","journal-title":"Adv. Water Resour."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Schneider, L., and Hutter, K. (2009). Solid-Fluid Mixtures of Frictional Materials in Geophysical and Geotechnical Context, Springer.","DOI":"10.1007\/978-3-642-02968-4"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/j.advwatres.2004.09.005","article-title":"Thermodynamically Constrained Averaging Theory Approach for Modeling Flow and Transport Phenomena in Porous Medium Systems: 1. Motivation and Overview","volume":"28","author":"Gray","year":"2005","journal-title":"Adv. Water Resour."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/j.advwatres.2004.09.006","article-title":"Thermodynamically Constrained Averaging Theory Approach for Modeling Flow and Transport Phenomena in Porous Medium Systems: 2. Foundation","volume":"28","author":"Miller","year":"2005","journal-title":"Adv. Water Resour."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Gray, W.G., and Miller, C.T. (2014). Introduction to the Thermodynamically Constrained Averaging Theory for Porous Medium Systems, Springer.","DOI":"10.1007\/978-3-319-04010-3"},{"key":"ref_6","unstructured":"Gray, W.G., Leijnse, A., Kolar, R.L., and Blain, C.A. (1993). Mathematical Tools for Changing Spatial Scales in the Analysis of Physical Systems, CRC Press."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Gray, W.G., and Miller, C.T. (2007). Consistent thermodynamic formulations for multiscale hydrologic systems: Fluid pressures. Water Resour. Res., 43.","DOI":"10.1029\/2006WR005811"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/j.advwatres.2013.06.006","article-title":"A generalization of averaging theorems for porous medium analysis","volume":"62","author":"Gray","year":"2013","journal-title":"Adv. Water Resour."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1745","DOI":"10.1016\/j.advwatres.2006.03.010","article-title":"Thermodynamically Constrained Averaging Theory Approach for Modeling Flow and Transport Phenomena in Porous Medium Systems: 3. Single-Fluid-Phase Flow","volume":"29","author":"Gray","year":"2006","journal-title":"Adv. Water Resour."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1016\/j.advwatres.2008.10.013","article-title":"Thermodynamically Constrained Averaging Theory Approach for Modeling Flow and Transport Phenomena in Porous Medium Systems: 5. Single-Fluid-Phase Transport","volume":"32","author":"Gray","year":"2009","journal-title":"Adv. Water Resour."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"779","DOI":"10.1016\/j.advwatres.2008.11.010","article-title":"Thermodynamically Constrained Averaging Theory Approach for Modeling Flow and Transport Phenomena in Porous Medium Systems: 6. Two-Fluid-Phase Flow","volume":"32","author":"Jackson","year":"2009","journal-title":"Adv. Water Resour."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.advwatres.2012.01.006","article-title":"Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 9. Transition region models","volume":"42","author":"Jackson","year":"2012","journal-title":"Adv. Water Resour."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1016\/j.jhydrol.2014.11.051","article-title":"Modeling two-fluid-phase flow and species transport in porous media","volume":"521","author":"Rybak","year":"2015","journal-title":"J. Hydrol."},{"key":"ref_14","first-page":"222","article-title":"Darcy\u2019s law and the field equations of the flow of underground fluids","volume":"207","author":"Hubbert","year":"1956","journal-title":"Trans. Am. Inst. Min. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1021\/ie50720a004","article-title":"Advances in theory of fluid motion in porous media","volume":"61","author":"Whitaker","year":"1969","journal-title":"Ind. Eng. Chem."},{"key":"ref_16","unstructured":"Bear, J. (1972). Dynamics of Fluids in Porous Media, Elsevier."},{"key":"ref_17","unstructured":"Boruvka, L. (1975). An Extension to Classical Theory of Capillarity. [Master\u2019s Thesis, University of Toronto]."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Courant, R., and Hilbert, D. (1989). Methods of Mathematical Physics, Wiley.","DOI":"10.1002\/9783527617210"},{"key":"ref_19","unstructured":"Gelfand, I.M., and Fomin, S.V. (2000). Calculus of Variations, Dover."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Whitaker, S. (1999). The Method of Volume Averaging, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-017-3389-2"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5365","DOI":"10.1002\/2015WR016921","article-title":"On the dynamics and kinematics of two-fluid-phase flow in porous media","volume":"51","author":"Gray","year":"2015","journal-title":"Water Resour. Res."},{"key":"ref_22","unstructured":"Thorne, K.S., and Blandford, R.D. (2017). Modern Classical Physics Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics, Princeton University Press."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1063","DOI":"10.5194\/hess-21-1063-2017","article-title":"On the Concistency of Scale Among Experiments, Theory, and Simulation","volume":"21","author":"McClure","year":"2017","journal-title":"Hydrol. Earth Syst. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1427","DOI":"10.1016\/j.advwatres.2010.07.002","article-title":"Thermodynamically Constrained Averaging Theory Approach for Modeling Flow and Transport Phenomena in Porous Medium Systems: 8. Interface and Common Curve Dynamics","volume":"33","author":"Gray","year":"2010","journal-title":"Adv. Water Resour."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1007\/s11242-017-0900-6","article-title":"A Pedagogical Approach to the Thermodynamically Constrained Averaging Theory","volume":"119","author":"Miller","year":"2017","journal-title":"Transp. Porous Meda"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Davidson, P.A. (2015). Turbulence: An introduction for Scientists and Engineers, Oxford University Press. [2nd ed.].","DOI":"10.1093\/acprof:oso\/9780198722588.001.0001"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"873","DOI":"10.1061\/(ASCE)0733-9429(2007)133:8(873)","article-title":"Double-averaging concept for rough-bed open-channel and overland flows: Theoretical background","volume":"133","author":"Nikora","year":"2007","journal-title":"J. Hydraul. Eng."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.coastaleng.2016.08.007","article-title":"SedFoam: A multi-dimensional Eulerian two-phase model for sediment transport and its application to momentary bed failure","volume":"119","author":"Cheng","year":"2017","journal-title":"Coast. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1016\/j.ces.2011.09.050","article-title":"Mechanics and thermodynamics of diffusion","volume":"68","author":"Whitaker","year":"2012","journal-title":"Chem. Eng. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3769","DOI":"10.3390\/e16073769","article-title":"On Conservation Equation Combinations and Closure Relations","volume":"16","author":"Gray","year":"2014","journal-title":"Entropy"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1016\/0301-9322(90)90077-V","article-title":"An assessment of multiphase flow models using the second law of thermodynamics","volume":"16","author":"Arnold","year":"1990","journal-title":"Int. J. Multiph. Flow"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/S0921-4526(96)00338-9","article-title":"Is turbulent motion chaos or order? Is the hydrodynamic or the kinetic descriptin of turbulent motion more natural","volume":"228","author":"Klimontovich","year":"1996","journal-title":"Phys. B"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1016\/j.crma.2008.05.013","article-title":"Entropy-based nonlinear viscosity for Fourier approximations of conservation laws","volume":"346","author":"Guermond","year":"2008","journal-title":"C. R. Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"4248","DOI":"10.1016\/j.jcp.2010.11.043","article-title":"Entropy viscosity method for nonlinear conservation laws","volume":"230","author":"Guermond","year":"2011","journal-title":"J. Comput. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/4\/253\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:59:46Z","timestamp":1760194786000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/4\/253"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,4,5]]},"references-count":34,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2018,4]]}},"alternative-id":["e20040253"],"URL":"https:\/\/doi.org\/10.3390\/e20040253","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,4,5]]}}}