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Geological Storage of CO2 in Deep Saline Formations pp 187–248Cite as

Upscaling and Scale Effects

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Part of the book series: Theory and Applications of Transport in Porous Media ((TATP,volume 29))

Abstract

Chapters 3 and 4 address the mathematical and numerical modeling of CO2 geological storage. This chapter, in turn, focuses on a specific important aspect of modeling, namely that of scale effects and upscaling . The geological systems are heterogeneous, with heterogeneity occurring at various scales. This gives rise to what is commonly named the “scale effect”. Certain process are critical at the scale of pores, while some of the effects of CO2 injection may have an effect and need to be modeled at the scale of tens and even hundreds of kilometers. Furthermore, various processes may be important at different scales. This requires understanding and methods of linking processes over a span of the scales. This is the topic of the current chapter.

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References

  • Acharya RC, Valocchi AJ, Werth CJ, Willingham TW (2007) Pore-scale simulation of dispersion and reaction along a transverse mixing zone in two-dimensional porous media. Water Resour Res 43:W10435

    Article  Google Scholar 

  • Adams EE, Gelhar LW (1992) Field study of dispersion in a heterogeneous aquifer: 2. Spatial moments analysis. Water Resour Res 28:3293–3307

    Article  Google Scholar 

  • Andres JTH, Cardoso SSS (2011) Onset of convection in a porous medium in the presence of chemical reaction. Phys Rev E 83:46312

    Article  Google Scholar 

  • Aris R (1956) On the dispersion of a solute in a fluid flowing through a tube. Proc R Soc Lond Math Phys Eng Sci 235:67–77

    Article  Google Scholar 

  • Backhaus S, Turitsyn K, Ecke RE (2011) Convective instability and mass transport of diffusion layers in a Hele-Shaw geometry. Phys Rev Lett 106:104501

    Article  Google Scholar 

  • Barenblatt GI, Zheltov IP, Kochina IN (1960) Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata]. J Appl Math Mech 24:1286–1303

    Article  Google Scholar 

  • Batchelor G (1949) Diffusion in a field of homogeneous turbulence. I. Eulerian analysis. Aust J Chem 2:437–450

    Article  Google Scholar 

  • Batchelor GK (1952) Diffusion in a field of homogeneous turbulence. Math Proc Camb Philos Soc 48:345–362

    Article  Google Scholar 

  • Battiato I, Tartakovsky DM, Tartakovsky AM, Scheibe T (2009) On breakdown of macroscopic models of mixing-controlled heterogeneous reactions in porous media. Adv Water Resour 32:1664–1673

    Article  Google Scholar 

  • Battiato I, Tartakovsky DM, Tartakovsky AM, Scheibe TD (2011) Hybrid models of reactive transport in porous and fractured media. Adv Water Resour New Comput Meth Softw Tools 34:1140–1150

    Article  Google Scholar 

  • Bear J (1972) Dynamics of fluids in porous media. Elsevier, New York

    Google Scholar 

  • Beckie R, Aldama AA, Wood EF (1996) Modeling the large-scale dynamics of saturated groundwater flow using spatial-filtering theory: 1. Theoretical development. Water Resour Res 32:1269–1280

    Article  Google Scholar 

  • Ben Y, Demekhin EA, Chang H-C (2002) A spectral theory for small-amplitude miscible fingering. Phys Fluids (1994-Present) 14:999–1010

    Article  Google Scholar 

  • Benson DA, Wheatcraft SW, Meerschaert MM (2000) Application of a fractional advection-dispersion equation. Water Resour Res 36:1403–1412

    Article  Google Scholar 

  • Berkowitz B, Scher H (1997) Anomalous transport in random fracture networks. Phys Rev Lett 79:4038–4041

    Article  Google Scholar 

  • Berkowitz B, Cortis A, Dentz M, Scher H (2006) Modeling non-Fickian transport in geological formations as a continuous time random walk. Rev Geophys 44:RG2003. doi:10.1029/2005RG000178

    Article  Google Scholar 

  • Bijeljic B, Blunt MJ (2006) Pore-scale modeling and continuous time random walk analysis of dispersion in porous media. Water Resour Res 42:W01202

    Article  Google Scholar 

  • Bolster D, Dentz M, Carrera J (2009) Effective two-phase flow in heterogeneous media under temporal pressure fluctuations. Water Resour Res 45:W05408

    Google Scholar 

  • Bolster D, Neuweiler I, Dentz M, Carrera J (2011) The impact of buoyancy on front spreading in heterogeneous porous media in two-phase immiscible flow. Water Resour Res 47:W02508

    Google Scholar 

  • Bourgeat A (1997) Two-Phase Flow. In: Hornung U (ed) Homogenization and porous media, interdisciplinary applied mathematics. Springer, New York, pp 95–127

    Chapter  Google Scholar 

  • Brenner H (1980) Dispersion resulting from flow through spatially periodic porous media. Philos Trans R Soc Lond 297(1430):81–133

    Article  Google Scholar 

  • Brinkman HC (1949) A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl Sci Res 1:27–34

    Article  Google Scholar 

  • Bronstert A, Carrera J, Leavesley G, Mölders N (2005) Scale issues. In: Bronstert A, Carrera J, Kabat P, Lütkemeier S (eds) Coupled models for the hydrological cycle integrating atmosphere, biosphere and pedosphere. Springer, pp 21–43

    Google Scholar 

  • Caltagirone J-P (1980) Stability of a saturated porous layer subject to a sudden rise in surface temperature: comparison between the linear and energy methods. Q J Mech Appl Math 33:47–58

    Article  Google Scholar 

  • Carrera J (1993) Chemistry and migration of actinides and fission products an overview of uncertainties in modelling groundwater solute transport. J Contam Hydrol 13:23–48

    Article  Google Scholar 

  • Carrera J, Sánchez-Vila X, Benet I, Medina A, Galarza G, Guimerà J (1998) On matrix diffusion: formulations, solution methods and qualitative effects. Hydrogeol J 6:178–190. doi:10.1007/s100400050143

    Article  Google Scholar 

  • Cirpka OA, Kitanidis PK (2000) An advective-dispersive stream tube approach for the transfer of conservative-tracer data to reactive transport. Water Resour Res 36:1209–1220. doi:10.1029/1999WR900355

    Article  Google Scholar 

  • Cueto-Felgueroso L, Juanes R (2012) Macroscopic phase-field model of partial wetting: bubbles in a capillary tube. Phys Rev Lett 108:144502

    Article  Google Scholar 

  • Cunningham JA, Werth CJ, Reinhard M, Roberts PV (1997) Effects of grain-scale mass transfer on the transport of volatile organics through sediments: 1. Model development. Water Resour Res 33:2713–2726

    Article  Google Scholar 

  • Cushman JH, Ginn TR (1993) Nonlocal dispersion in media with continuously evolving scales of heterogeneity. Transp Porous Media 13:123–138

    Article  Google Scholar 

  • Cushman JH, Ginn TR (2000) Fractional advection-dispersion equation: a classical mass balance with convolution-Fickian Flux. Water Resour Res 36:3763–3766

    Article  Google Scholar 

  • Cushman JH, Bennethum LS, Hu BX (2002) A primer on upscaling tools for porous media. Adv Water Resour 25:1043–1067

    Article  Google Scholar 

  • Cvetkovic V, Dagan G (1996) Reactive transport and immiscible flow in geological media. ii. applications. Proc R Soc Lond A, 452:202–328

    Google Scholar 

  • Dagan G (1984) Solute transport in heterogeneous porous formations. J Fluid Mech 145:151–177

    Article  Google Scholar 

  • Dagan G (1986) Statistical theory of groundwater flow and transport: pore to laboratory, laboratory to formation, and formation to regional scale. Water Resour Res 22:120S–134S

    Article  Google Scholar 

  • Dagan G (1988) Time-dependent macrodispersion for solute transport in anisotropic heterogeneous aquifers. Water Resour Res 24:1491–1500

    Article  Google Scholar 

  • Dagan G (1989) Flow and transport in porous formations. Springer

    Book  Google Scholar 

  • Dagan G, Bresler E (1979) Solute dispersion in unsaturated heterogeneous soil at field scale: I. Theory1. Soil Sci Soc Am J 43:461

    Article  Google Scholar 

  • Dagan G, Cvetkovic V (1996) Reactive transport and immiscible flow in geological media. I. General theory. Proc R Soc Lond Math Phys Eng Sci 452:285–301

    Article  Google Scholar 

  • Dahle HK, Celia MA, Hassanizadeh SM (2005) Bundle-of-tubes model for calculating dynamic effects in the capillary-pressure–saturation relationship. Transp Porous Media 58:5–22

    Article  Google Scholar 

  • Daniel D, Tilton N, Riaz A (2013) Optimal perturbations of gravitationally unstable, transient boundary layers in porous media. J Fluid Mech 727:456–487

    Article  Google Scholar 

  • Das DB, Hassanizadeh SM (2005) Upscaling multiphase flow in porous media. Springer, Berlin

    Book  Google Scholar 

  • Das DB, Mirzaei M (2012) Dynamic effects in capillary pressure relationships for two-phase flow in porous media: experiments and numerical analyses. AIChE J 58:3891–3903

    Article  Google Scholar 

  • de Dreuzy J-R, Carrera J, Dentz M, Le Borgne T (2012) Time evolution of mixing in heterogeneous porous media. Water Resour Res 48:W06511

    Google Scholar 

  • De Simoni M, Carrera J, Sánchez-Vila X, Guadagnini A (2005) A procedure for the solution of multicomponent reactive transport problems. Water Resour Res 41:W11410

    Article  Google Scholar 

  • De Simoni M, Sanchez-Vila X, Carrera J, Saaltink MW (2007) A mixing ratios-based formulation for multicomponent reactive transport. Water Resour Res 43:W07419

    Article  Google Scholar 

  • Dean DS, Drummond IT, Horgan RR (2007) Effective transport properties for diffusion in random media. J Stat Mech Theory Exp 2007:P07013

    Article  Google Scholar 

  • Delay F, Ackerer P, Danquigny C (2005) Simulating solute transport in porous or fractured formations using random walk particle tracking. Vadose Zone J 4:360

    Article  Google Scholar 

  • Dentz M, Berkowitz B (2003) Transport behavior of a passive solute in continuous time random walks and multirate mass transfer. Water Resour Res, 39(5):1111

    Google Scholar 

  • Dentz M, Berkowitz B (2005) Exact effective transport dynamics in a one-dimensional random environment. Phys Rev E 72:31110

    Article  Google Scholar 

  • Dentz M, Castro A (2009) Effective transport dynamics in porous media with heterogeneous retardation properties. Geophys Res Lett 36:L03403

    Article  Google Scholar 

  • Dentz M, Kinzelbach H, Attinger S, Kinzelbach W (2000) Temporal behavior of a solute cloud in a heterogeneous porous medium: 1. Point-like injection. Water Resour Res 36:3591–3604

    Article  Google Scholar 

  • Dentz M, Gouze P, Carrera J (2011a) Effective non-local reaction kinetics for transport in physically and chemically heterogeneous media. J Contam Hydrol 120–121:222–236

    Article  Google Scholar 

  • Dentz M, Le Borgne T, Englert A, Bijeljic B (2011b) Mixing, spreading and reaction in heterogeneous media: a brief review. J Contam Hydrol 120–121:1–17

    Article  Google Scholar 

  • Di Donato G, Lu H, Tavassoli Z, Blunt MJ (2007) Multirate-transfer dual-porosity modeling of gravity drainage and imbibition. SPE J. 12:77–88

    Article  Google Scholar 

  • Donado LD, Sanchez-Vila X, Dentz M, Carrera J, Bolster D (2009) Multicomponent reactive transport in multicontinuum media. Water Resour Res 45:W11402

    Article  Google Scholar 

  • Edwards DA, Shapiro M, Brenner H (1993) Dispersion and reaction in two-dimensional model porous media. Phys Fluids A, 5:837–848

    Google Scholar 

  • Elenius MT, Gasda SE (2013) Convective mixing in formations with horizontal barriers. Adv Water Resour 62(Part C):499–510

    Article  Google Scholar 

  • Ennis-King J, Paterson L (2005) Role of convective mixing in the long-term storage of carbon dioxide in deep saline formations. Soc Petrol Eng 10(03):349–356

    Google Scholar 

  • Espinoza C, Valocchi AJ (1997) Stochastic analysis of one-dimensional transport of kinetically adsorbing solutes in chemically heterogeneous aquifers. Water Resour Res 33:2429–2445

    Article  Google Scholar 

  • Foster TD (1965) Stability of a homogeneous fluid cooled uniformly from above. Phys Fluids 1958–1988(8):1249–1257

    Article  Google Scholar 

  • Fu X, Cueto-Felgueroso L, Juanes R (2013) Pattern formation and coarsening dynamics in three-dimensional convective mixing in porous media. Philos Trans R Soc Lond Math Phys Eng Sci 371:20120355

    Article  Google Scholar 

  • Geiger S, Dentz M, Neuweiler I (2013) A novel multi-rate dual-porosity model for improved simulation of fractured and multiporosity reservoirs. SPE J 18:670–684

    Article  Google Scholar 

  • Gelhar LW (1993) Stochastic subsurface hydrology. Prentice-Hall, Upper Saddle River

    Google Scholar 

  • Gelhar LW, Axness CL (1983) Three-dimensional stochastic analysis of macrodispersion in aquifers. Water Resour Res 19:161–180

    Article  Google Scholar 

  • Gelhar LW, Welty C, Rehfeldt KR (1992) A critical review of data on field-scale dispersion in aquifers. Water Resour Res 28:1955–1974

    Article  Google Scholar 

  • Gerke HH (2006) Preferential flow descriptions for structured soils. J Plant Nutr Soil Sci 169:382–400

    Article  Google Scholar 

  • Ghesmat K, Hassanzadeh H, Abedi J (2011) The impact of geochemistry on convective mixing in a gravitationally unstable diffusive boundary layer in porous media: CO2 storage in saline aquifers. J Fluid Mech 673:480–512

    Article  Google Scholar 

  • Ginn TR (2001) Stochastic–convective transport with nonlinear reactions and mixing: finite streamtube ensemble formulation for multicomponent reaction systems with intra-streamtube dispersion. J Contam Hydrol 47:1–28

    Article  Google Scholar 

  • Ginn TR, Simmons CS, Wood BD (1995) Stochastic-convective transport with nonlinear reaction: biodegradation with microbial growth. Water Resour Res 31:2689–2700

    Article  Google Scholar 

  • Gouze P, Melean Y, Le Borgne T, Dentz M, Carrera J (2008) Non-Fickian dispersion in porous media explained by heterogeneous microscale matrix diffusion. Water Resour Res 44:W11416

    Google Scholar 

  • Gramling CM, Harvey CF, Meigs LC (2002) Reactive transport in porous media: a comparison of model prediction with laboratory visualization. Environ Sci Technol 36:2508–2514

    Article  Google Scholar 

  • Guimerà J, Carrera J (2000) A comparison of hydraulic and transport parameters measured in low-permeability fractured media. J Contam Hydrol 41:261–281

    Article  Google Scholar 

  • Gutjahr AL, Gelhar LW, Bakr AA, MacMillan JR (1978) Stochastic analysis of spatial variability in subsurface flows: 2. Evaluation and application. Water Resour Res 14:953–959

    Article  Google Scholar 

  • Haggerty R, Gorelick SM (1995) Multiple-rate mass transfer for modeling diffusion and surface reactions in media with pore-scale heterogeneity. Water Resour Res 31:2383–2400

    Article  Google Scholar 

  • Hänggi P, Talkner P, Borkovec M (1990) Reaction-rate theory: fifty years after Kramers. Rev Mod Phys 62:251–341

    Article  Google Scholar 

  • Hassanizadeh SM, Gray WG (1990) Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv Water Resour 13:169–186

    Article  Google Scholar 

  • Hassanzadeh H, Pooladi-Darvish M, Keith DW (2007) Scaling behavior of convective mixing, with application to geological storage of CO2. AIChE J 53:1121–1131

    Article  Google Scholar 

  • Hewitt DR, Neufeld JA, Lister JR (2013) Convective shutdown in a porous medium at high Rayleigh number. J Fluid Mech 719:551–586

    Article  Google Scholar 

  • Hidalgo JJ, Carrera J (2009) Effect of dispersion on the onset of convection during CO2 sequestration. J Fluid Mech 640:441–452

    Article  Google Scholar 

  • Hidalgo JJ, Fe J, Cueto-Felgueroso L, Juanes R (2012) Scaling of convective mixing in porous media. Phys Rev Lett 109:264503

    Article  Google Scholar 

  • Hidalgo JJ, MacMinn CW, Juanes R (2013) Dynamics of convective dissolution from a migrating current of carbon dioxide. Adv Water Resour 62(Part C):511–519

    Article  Google Scholar 

  • Hornung U (1997) Homogenization and porous media. Springer, New York

    Book  Google Scholar 

  • Kang Q, Lichtner PC, Zhang D (2006) Lattice Boltzmann pore-scale model for multicomponent reactive transport in porous media. J Geophys Res Solid Earth 111:B05203

    Google Scholar 

  • Kapoor V, Kitanidis PK (1998) Concentration fluctuations and dilution in aquifers. Water Resour Res 34:1181–1193

    Article  Google Scholar 

  • Kapoor V, Gelhar LW, Miralles-Wilhelm F (1997) Bimolecular second-order reactions in spatially varying flows: segregation induced scale-dependent transformation rates. Water Resour Res 33:527–536

    Article  Google Scholar 

  • Kazemi H, Merrill LS, Porterfield KL, Zeman PR (1976) Numerical simulation of water–oil flow in naturally fractured reservoirs. Soc Pet Eng J. 16:317–326

    Article  Google Scholar 

  • Kechagia PE, Tsimpanogiannis IN, Yortsos YC, Lichtner PC (2002) On the upscaling of reaction-transport processes in porous media with fast or finite kinetics. Chem Eng Sci 57:2565–2577

    Article  Google Scholar 

  • Keller JB (1964) A theorem on the conductivity of a composite medium. J Math Phys 5:548–549

    Article  Google Scholar 

  • Kinzelbach W (1988) The random walk method in pollutant transport simulation. In: Custodio E, Gurgui A, Ferreira JPL (eds) Groundwater flow and quality modelling, NATO ASI series. Springer, Amsterdam, pp 227–245

    Chapter  Google Scholar 

  • Kitanidis PK (1988) Hydrologic research: The U.S.—Japan experience prediction by the method of moments of transport in a heterogeneous formation. J Hydrol 102:453–473

    Article  Google Scholar 

  • Kitanidis PK (1994) The concept of the dilution Index. Water Resour Res 30:2011–2026

    Article  Google Scholar 

  • Knabner P, van Duijn CJ, Hengst S (1995) An analysis of crystal dissolution fronts in flows through porous media. Part 1: compatible boundary conditions. Adv Water Resour 18:171–185

    Article  Google Scholar 

  • Kneafsey TJ, Pruess K (2009) Laboratory flow experiments for visualizing carbon dioxide-induced, density-driven brine convection. Transp Porous Media 82:123–139. doi:10.1007/s11242-009-9482-2

    Article  Google Scholar 

  • LaBolle EM, Fogg GE, Tompson AFB (1996) Random-walk simulation of transport in heterogeneous porous media: local mass-conservation problem and implementation methods. Water Resour Res 32:583–593

    Article  Google Scholar 

  • Lallemand-Barres A, Peaudecerf P (1978) Recherche des relations entre la valeur de la dispersivité macroscopique d’un milieu aquifer, ses autres caracteristiques et les conditions de mesure. Bull BRGM 3:227–287

    Google Scholar 

  • Langlo P, Espedal MS (1994) Macrodispersion for two-phase, immiscible flow in porous media. Adv Water Resour 17:297–316

    Article  Google Scholar 

  • Langlo P, Espedal M (1995) Macrodispersion for two-phase, immiscible flow in porous media. Adv Water Resour 17:297–316

    Google Scholar 

  • Le Borgne T, Dentz M, Carrera J (2008) Lagrangian statistical model for transport in highly heterogeneous velocity fields. Phys Rev Lett 101:90601

    Article  Google Scholar 

  • Le Borgne T, Dentz M, Davy P, Bolster D, Carrera J, de Dreuzy J-R, Bour O (2011) Persistence of incomplete mixing: a key to anomalous transport. Phys Rev E 84:15301

    Article  Google Scholar 

  • Li L, Peters CA, Celia MA (2006) Upscaling geochemical reaction rates using pore-scale network modeling. Adv Water Resour 29:1351–1370

    Article  Google Scholar 

  • Li L, Steefel CI, Yang L (2008) Scale dependence of mineral dissolution rates within single pores and fractures. Geochim Cosmochim Acta 72:360–377

    Article  Google Scholar 

  • Lichtner PC (1985) Continuum model for simultaneous chemical reactions and mass transport in hydrothermal systems. Geochim Cosmochim Acta 49:779–800

    Article  Google Scholar 

  • Lichtner PC, Kang Q (2007) Upscaling pore-scale reactive transport equations using a multiscale continuum formulation. Water Resour Res 43:W12S15

    Google Scholar 

  • Lichtner PC, Tartakovsky DM (2003) Stochastic analysis of effective rate constant for heterogeneous reactions. Stoch Environ Res Risk Assess 17:419–429

    Article  Google Scholar 

  • Liu C, Zachara JM, Qafoku NP, Wang Z (2008) Scale-dependent desorption of uranium from contaminated subsurface sediments. Water Resour Res 44:W08413

    Google Scholar 

  • MacMinn CW, Juanes R (2013) Buoyant currents arrested by convective dissolution. Geophys Res Lett 40:2017–2022. doi:10.1002/grl.50473

    Article  Google Scholar 

  • MacQuarrie KTB, Sudicky EA (1990) Simulation of biodegradable organic contaminants in groundwater: 2. Plume behavior in uniform and random flow fields. Water Resour Res 26:223–239

    Google Scholar 

  • Marle CM (1981) Multiphase flow in porous media. Gulf Publishing Co, Houston

    Google Scholar 

  • Matheron G (1967) Composition des perméabilités en milieu poreux héterogène. Méthode de Schwydler et règles de pondération. Revue de l’Institute Français du Petrole 443–466

    Google Scholar 

  • Matheron G, De Marsily G (1980) Is transport in porous media always diffusive? A counterexample. Water Resour Res 16:901–917

    Article  Google Scholar 

  • McWhorter DB, Sunada DK (1990) Exact integral solutions for two-phase flow. Water Resour Res 26:399–413

    Article  Google Scholar 

  • Meakin P, Tartakovsky AM (2009) Modeling and simulation of pore-scale multiphase fluid flow and reactive transport in fractured and porous media. Rev Geophys 47:RG3002

    Article  Google Scholar 

  • Meerschaert MM, Benson DA, Scheffler H-P, Becker-Kern P (2002) Governing equations and solutions of anomalous random walk limits. Phys Rev E 66:60102

    Article  Google Scholar 

  • Meile C, Tuncay K (2006) Scale dependence of reaction rates in porous media. Adv Water Resour 29:62–71

    Article  Google Scholar 

  • Metzler R, Klafter J (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339:1–77

    Article  Google Scholar 

  • Mikelić A, Devigne V, van Duijn C (2006) Rigorous upscaling of the reactive flow through a pore, under dominant Peclet and Damkohler numbers. SIAM J Math Anal 38:1262–1287

    Article  Google Scholar 

  • Molz FJ, Widdowson MA (1988) Internal inconsistencies in dispersion-dominated models that incorporate chemical and microbial kinetics. Water Resour Res 24:615–619

    Article  Google Scholar 

  • Morales-Casique E, Neuman SP, Guadagnini A (2006) Non-local and localized analyses of non-reactive solute transport in bounded randomly heterogeneous porous media: theoretical framework. Adv Water Resour 29:1238–1255

    Article  Google Scholar 

  • Muskat M (1937) The flow of homogeneous fluids through porous media: soil science. McGraw-Hill, New York

    Google Scholar 

  • Neufeld JA, Hesse MA, Riaz A, Hallworth MA, Tchelepi HA, Huppert HE (2010) Convective dissolution of carbon dioxide in saline aquifers. Geophys Res Lett 37:L22404. doi:10.1029/2010GL044728

    Article  Google Scholar 

  • Neuman SP, Orr S (1993) Prediction of steady state flow in nonuniform geologic media by conditional moments: exact nonlocal formalism, effective conductivities, and weak approximation. Water Resour Res 29:341–364

    Article  Google Scholar 

  • Neuman SP, Tartakovsky DM (2009) Perspective on theories of non-Fickian transport in heterogeneous media. Adv Water Resour 32:670–680

    Article  Google Scholar 

  • Neuman SP, Zhang Y-K (1990) A quasi-linear theory of non-Fickian and Fickian subsurface dispersion: 1. Theoretical analysis with application to isotropic media. Water Resour Res 26:887–902

    Google Scholar 

  • Neuman SP, Winter CL, Newman CM (1987) Stochastic theory of field-scale fickian dispersion in anisotropic porous media. Water Resour Res 23:453–466

    Article  Google Scholar 

  • Neuweiler I, Attinger S, Kinzelbach W, King P (2003) Large scale mixing for immiscible displacement in heterogeneous porous media. Transp Porous Media 51:287–314

    Article  Google Scholar 

  • Noetinger B, Artus V, Ricard L (2004) Dynamics of the water–oil front for two-phase, immiscible flow in heterogeneous porous media. 2: Isotropic media. Transp Porous Media 56:305–328

    Article  Google Scholar 

  • Nordbotten JM, Celia MA (2012) Geological storage of CO2: modeling approaches for large-scale simulation. Wiley, New York

    Google Scholar 

  • Panfilow M, Floriat S (2004) Nonlinear two-phase mixing in heterogeneous porous media. Transp Porous Media 57:347–375

    Article  Google Scholar 

  • Pau GSH, Bell JB, Pruess K, Almgren AS, Lijewski MJ, Zhang K (2010) High-resolution simulation and characterization of density-driven flow in CO2 storage in saline aquifers. Adv Water Resour 33:443–455. doi:10.1016/j.advwatres.2010.01.009

    Article  Google Scholar 

  • Pfannkuch H-O (1963) Contribution a l’etude des d placement de fluides miscible dans un milieu poreux. Revue de l’Institut français du pétrole 18:215

    Google Scholar 

  • Pope SB (2001) Turbulent flows. Meas Sci Technol 12:2020

    Article  Google Scholar 

  • Quintard M, Whitaker S (1994) Convection, dispersion, and interfacial transport of contaminants: homogeneous porous media. Adv Water Resour 17:221–239

    Article  Google Scholar 

  • Rapaka S, Pawar RJ, Stauffer PH, Zhang D, Chen S (2009) Onset of convection over a transient base-state in anisotropic and layered porous media. J Fluid Mech 641:227–244

    Article  Google Scholar 

  • Rees DAS, Selim A, Ennis-King JP (2008) The instability of unsteady boundary layers in porous media. In: Vadász P (ed) Emerging topics in heat and mass transfer in porous media, theory and applications of transport in porous media. Springer, Amsterdam, pp 85–110

    Chapter  Google Scholar 

  • Renard P, de Marsily G (1997) Calculating equivalent permeability: a review. Adv Water Resour 20:253–278. doi:10.1016/S0309-1708(96)00050-4

    Article  Google Scholar 

  • Rezaei M, Sanz E, Raeisi E, Ayora C, Vázquez-Suñé E, Carrera J (2005) Reactive transport modeling of calcite dissolution in the fresh-salt water mixing zone. J Hydrol 311:282–298

    Article  Google Scholar 

  • Riaz A, Hesse M, Tchelepi HA, Orr FM (2006) Onset of convection in a gravitationally unstable diffusive boundary layer in porous media. J Fluid Mech 548:87–111. doi:10.1017/S0022112005007494

    Article  Google Scholar 

  • Risken PDH (1984) Fokker-Planck equation. In: The Fokker-Planck equation, Springer Series in Synergetics. Springer, Berlin, pp 63–95

    Chapter  Google Scholar 

  • Risken H (1996) The Fokker-planck equation. Springer, Heidelberg, New York

    Google Scholar 

  • Robinson BA, Viswanathan HS (2003) Application of the theory of micromixing to groundwater reactive transport models. Water Resour Res 39:1313

    Article  Google Scholar 

  • Rubin Y (2003) Applied stochastic hydrogeology. Oxford University Press, New York

    Google Scholar 

  • Rubin Y, Sun A, Maxwell R, Bellin A (1999) The concept of block-effective macrodispersivity and a unified approach for grid-scale- and plume-scale-dependent transport. J Fluid Mech 395:161–180

    Article  Google Scholar 

  • Saaltink MW, Ayora C, Carrera J (1998) A mathematical formulation for reactive transport that eliminates mineral concentrations. Water Resour Res 34:1649–1656

    Article  Google Scholar 

  • Sahimi M (1995) Flow and transport in porous media and fractured rock. VCH, Weinheim

    Google Scholar 

  • Sahimi M (2011) Flow and transport in porous media and fractured rock: from classical methods to modern approaches. Wiley, New York

    Book  Google Scholar 

  • Salamon P, Fernàndez-Garcia D, Gómez-Hernández JJ (2006) A review and numerical assessment of the random walk particle tracking method. J Contam Hydrol 87:277–305

    Article  Google Scholar 

  • Sanchez-Vila X, Guadagnini A, Carrera J (2006) Representative hydraulic conductivities in saturated groundwater flow. Rev Geophys 44:RG3002

    Article  Google Scholar 

  • Sanchez-Vila X, Dentz M, Donado LD (2007) Transport-controlled reaction rates under local non-equilibrium conditions. Geophys Res Lett 34:L10404

    Article  Google Scholar 

  • Schmid KS, Geiger S (2012) Universal scaling of spontaneous imbibition for water-wet systems. Water Resour Res 48:W03507

    Article  Google Scholar 

  • Schumer R, Benson DA, Meerschaert MM, Baeumer B (2003) Fractal mobile/immobile solute transport. Water Resour Res 39:1296

    Article  Google Scholar 

  • Seeboonruang U, Ginn TR (2006a) Upscaling heterogeneity in aquifer reactivity via exposure-time concept: forward model. J Contam Hydrol 84:127–154

    Article  Google Scholar 

  • Seeboonruang U, Ginn TR (2006b) Upscaling heterogeneity in aquifer reactivity via the exposure-time concept: inverse model. J Contam Hydrol 84:155–177. doi:10.1016/j.jconhyd.2005.12.010

    Article  Google Scholar 

  • Šimůnek J, Jarvis NJ, van Genuchten MT, Gärdenäs A (2003) Review and comparison of models for describing non-equilibrium and preferential flow and transport in the vadose zone. J Hydrol 272:14–35

    Article  Google Scholar 

  • Slim AC, Ramakrishnan TS (2010) Onset and cessation of time-dependent, dissolution-driven convection in porous media. Phys Fluids 22:124103

    Article  Google Scholar 

  • Slim AC, Bandi MM, Miller JC, Mahadevan L (2013) Dissolution-driven convection in a Hele–Shaw cell. Phys Fluids (1994-Present) 25:24101

    Article  Google Scholar 

  • Steefel CI, MacQuarrie KTB (1996) Approaches to modeling of reactive transport in porous media. Rev Miner Geochem 34:85–129

    Google Scholar 

  • Steefel CI, DePaolo DJ, Lichtner PC (2005) Reactive transport modeling: an essential tool and a new research approach for the Earth sciences. Earth Planet Sci Lett 240:539–558

    Article  Google Scholar 

  • Szulczewski ML, Hesse MA, Juanes R (2013) Carbon dioxide dissolution in structural and stratigraphic traps. J Fluid Mech 736:287–315. doi:10.1017/jfm.2013.511

    Article  Google Scholar 

  • Tartakovsky AM, Meakin P, Scheibe TD, Eichler West RM (2007) Simulations of reactive transport and precipitation with smoothed particle hydrodynamics. J Comput Phys 222:654–672

    Article  Google Scholar 

  • Tartakovsky AM, Redden G, Lichtner PC, Scheibe TD, Meakin P (2008) Mixing-induced precipitation: experimental study and multiscale numerical analysis. Water Resour Res 44:W06S04

    Article  Google Scholar 

  • Tartakovsky AM, Tartakovsky GD, Scheibe TD (2009) Effects of incomplete mixing on multicomponent reactive transport. Adv Water Resour 32:1674–1679

    Article  Google Scholar 

  • Tecklenburg J, Neuweiler I, Dentz M, Carrera J, Geiger S, Abramowski C, Silva O (2013) A non-local two-phase flow model for immiscible displacement in highly heterogeneous porous media and its parametrization. Adv Water Resour 62(Part C):475–487

    Article  Google Scholar 

  • Tilton N, Daniel D, Riaz A (2013) The initial transient period of gravitationally unstable diffusive boundary layers developing in porous media. Phys Fluids (1994-Present) 25:92107

    Article  Google Scholar 

  • Villermaux J, Devillon JC (1972) Représentation de la coalescence et de la redispersion des domaines de ségrégation dans un fluide par un modèle d’interaction phénom. In: Proceedings of 2nd international symposium chemistry reaction engineering. Elsevier, New York

    Google Scholar 

  • Wen X-H, Gómez-Hernández JJ (1996) Upscaling hydraulic conductivities in heterogeneous media: an overview. J Hydrol 183:9–32

    Article  Google Scholar 

  • West B, Bologna M, Grigolini P (2003) Physics of fractal operators. Springer, Berlin

    Book  Google Scholar 

  • Whitaker S (1986) Flow in porous media I: a theoretical derivation of Darcy’s law. Transp Porous Media 1:3–25

    Article  Google Scholar 

  • Whitaker S (1999) The method of volume averaging. Kluwer, Berlin

    Book  Google Scholar 

  • White A, Peterson M (1990) Role of reactive-surface-area characterization in geochemical kinetic models. In: Chemical modeling of aqueous systems II. American chemical society, pp 461–475

    Google Scholar 

  • Willingham TW, Werth CJ, Valocchi AJ (2008) Evaluation of the effects of porous media structure on mixing-controlled reactions using pore-scale modeling and micromodel experiments. Environ Sci Technol 42:3185–3193

    Article  Google Scholar 

  • Willmann M, Carrera J, Sánchez-Vila X (2008) Transport upscaling in heterogeneous aquifers: what physical parameters control memory functions? Water Resour Res 44:W12437

    Article  Google Scholar 

  • Willmann M, Carrera J, Sanchez-Vila X, Silva O, Dentz M (2010) Coupling of mass transfer and reactive transport for nonlinear reactions in heterogeneous media. Water Resour Res 46:W07512

    Article  Google Scholar 

  • Winter CL, Tartakovsky DM, Guadagnini A (2003) Moment differential equations for flow in highly heterogeneous porous media. Surv Geophys 24:81–106

    Article  Google Scholar 

  • Wyckoff RD, Botset HG (1936) The flow of gas-liquid mixtures through unconsolidated sands. J Appl Phys 7:325–345

    Google Scholar 

  • Xu X, Chen S, Zhang D (2006) Convective stability analysis of the long-term storage of carbon dioxide in deep saline aquifers. Adv Water Resour 29:397–407. doi:10.1016/j.advwatres.2005.05.008

    Article  Google Scholar 

  • Yang Z, Niemi A, Fagerlund F, Illangasekare T (2013) Two-phase flow in rough-walled fractures: comparison of continuum and invasion-percolation models. Water Resour Res 49:993–1002

    Article  Google Scholar 

  • Zhang D, Tchelepi H (1999) Stochastic analysis of immiscible two-phase flow in heterogeneous media. SPE J 4:380–388

    Article  Google Scholar 

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Authors and Affiliations

  1. Institute of Environmental Assessment and Water Research (IDAEA), Spanish National Research Council (CSIC), Barcelona, Spain

    Marco Dentz, Jesus Carrera & Juan Hidalgo

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  1. Marco Dentz
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  2. Jesus Carrera
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  3. Juan Hidalgo
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Correspondence to Marco Dentz .

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Editors and Affiliations

  1. Uppsala University Dept. Earth Sciences, Uppsala, Sweden

    Auli Niemi

  2. Engineering, Technion-Israel Institute of Technology Dept. Civil & Environmental, Haifa, Israel

    Jacob Bear

  3. Haifa, Israel

    Jacob Bensabat

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Dentz, M., Carrera, J., Hidalgo, J. (2017). Upscaling and Scale Effects. In: Niemi, A., Bear, J., Bensabat, J. (eds) Geological Storage of CO2 in Deep Saline Formations. Theory and Applications of Transport in Porous Media, vol 29. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-0996-3_5

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