[Physique statistique et géosciences appliquées : quelques résultats et perspectives]
Dans cette note, nous examinons quelques applications des concepts et outils de la physique statistique (PS) aux géosciences appliquées. Les applications vont de l’hydrologie au stockage de déchets nucléaires ou de CO, au stockage massif d’énergie, en passant par l’industrie pétrolière ou gazière, et enfin les applications géothermiques. Vu la complexité intrinsèque des applications de terrain, les ingénieurs s’attachent en général à optimiser un critère économique, tout en veillant à maintenir la meilleure sécurité et en minimisant l’empreinte environnementale des projets. Il s’agit donc d’employer les connaissances les plus actuelles sur les transferts en milieu poreux. On s’intéresse à différentes échelles, des applications de la PS pour formuler les lois de transport dans des milieux extrêmement confinés tels des nanopores constituant la porosité des argiles. Ensuite, on présente les applications de la PS pour modéliser les écoulements à des échelles kilométriques intéressant les ingénieurs en charge d’application. On est dans une situation typique de désordre gelé, hors d’équilibre où les temps de relaxation peuvent être très longs, de l’ordre de plusieurs siècles. Des perspectives sont proposées pour utiliser des outils issus de la théorie des matrices aléatoires afin de faciliter l’utilisation pratique en “aide à la décision” de ces modèles bien souvent sur-paramétrés par des données elles-mêmes incertaines.
In this paper, some applications of statistical physics (SP) concepts and techniques in applied geosciences are reviewed. The domain includes hydrology, oil and gas industry, nuclear or CO waste geological disposal, massive energy storage, heat recovery from geothermal formations and many other applications. Several scales are considered: we start from applications of SP at the molecular scale to understand the effect of extreme confinements concerning the fluid transport in nanopores in clay rocks. The paper ends with coarse graining techniques that are employed to build operational models relevant at the practical scale of several kilometers, including strongly fractured geological environments. Perspectives are proposed regarding some issues about the practical use of these often over-parameterized models in connection to random matrix and graph theories and the associated quenched disorder problems and “big data” issues.
Mots-clés : Physique statistique, Géosciences appliquées, Milieux poreux, Désordre, Nanopores, Changement d’échelle, Désordre gelé
Benoît Noetinger 1
CC-BY 4.0
@article{CRPHYS_2020__21_6_539_0,
author = {Beno{\^\i}t Noetinger},
title = {Statistical physics and applied geosciences: some results and perspectives},
journal = {Comptes Rendus. Physique},
pages = {539--560},
publisher = {Acad\'emie des sciences, Paris},
volume = {21},
number = {6},
year = {2020},
doi = {10.5802/crphys.40},
language = {en},
}
Benoît Noetinger. Statistical physics and applied geosciences: some results and perspectives. Comptes Rendus. Physique, Prizes of the French Academy of Sciences 2019 (continued), Volume 21 (2020) no. 6, pp. 539-560. doi : 10.5802/crphys.40. https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.40/
[1] Statistical hydrodynamics in porous media, J. Appl. Phys., Volume 25 (1954) no. 8, pp. 994-1001
[2] The physics of flow through porous media, Soil Sci., Volume 86 (1958) no. 6, p. 355
[3] A theory of dispersion in a porous medium, J. Fluid Mech., Volume 6 (1959) no. 3, pp. 321-349
[4] The statistical behavior of instabilities in displacement processes in porous media, Canad. J. Phys., Volume 39 (1961) no. 2, pp. 326-334
[5] Recent advances in scalable non-Gaussian geostatistics: the generalized sub-Gaussian model, J. Hydrol., Volume 562 (2018), pp. 685-691
[6] Trends in reservoir simulation: big models, scalable models? Will you please make up your mind?, SPE Annual Technical Conference and Exhibition, Society of Petroleum Engineers, 2001
[7] Methods for quantifying the uncertainty of production forecasts: a comparative study, Petrol. Geosci., Volume 7 (2001) no. S, p. S87-S96
[8] Earthquakes induced by the stimulation of an enhanced geothermal system below Basel (Switzerland), Seismol. Res. Lett., Volume 80 (2009) no. 5, pp. 784-798
[9] Climate-change impacts in a regional karst aquifer, Texas, USA, J. Hydrol., Volume 227 (2000) no. 1–4, pp. 173-194
[10] An experimental study of the Saffman–Taylor instability, Dynamics of Curved Fronts, Academic Press, 1988, pp. 219-234
[11] Lattice-gas models of phase separation: interfaces, phase transitions, and multiphase flow, Rev. Mod. Phys., Volume 66 (1994) no. 4, p. 1417
[12] Lattice Boltzmann methods for single-phase and solid–liquid phase-change heat transfer in porous media: a review, Intl J. Heat Mass Transfer, Volume 129 (2019), pp. 160-197
[13] 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, etc, Dalmont, Paris, 1856
[14] Eléments pour une théorie des milieux poreux, Masson, Paris, 1967
[15] Flow in porous media I: a theoretical derivation of Darcy’s law, Trans. Porous Med., Volume 1 (1986) no. 1, pp. 3-25
[16] Prolongement de la pression et homogénéisation des équations de Stokes dans un milieu poreux connexe, C. R. Acad. Sci. Paris, Volume 309 (1989), pp. 717-722
[17] One-phase Newtonian flow, Homogenization and Porous Media, Springer, New York, 1997, pp. 45-76
[18] Molecular dynamics of Poiseuille flow and moving contact lines, Phys. Rev. Lett., Volume 60 (1988) no. 13, p. 1282
[19] Steady moving contact line of water over a no-slip substrate: Challenges in benchmarking phase-field and volume-of-fluid methods against molecular dynamics simulations, Eur. Phys. J. Spec. Top., Volume 229 (2020) no. 10, pp. 1897-1921
[20] Flow boundary conditions from nano-to micro-scales, Soft Matt., Volume 3 (2007) no. 6, pp. 685-693
[21] Nanorheology: an investigation of the boundary condition at hydrophobic and hydrophilic interfaces, Eur. Phys. J. E, Volume 9 (2002) no. 1, pp. 47-53
[22] Nanofluidics, from bulk to interfaces, Chem. Soc. Rev., Volume 39 (2010) no. 3, pp. 1073-1095
[23] Diffusion under confinement: hydrodynamic finite-size effects in simulation, J. Chem. Theory Comput., Volume 13 (2017) no. 6, pp. 2881-2889
[24] Molecular dynamics simulations in the grand canonical ensemble: application to clay mineral swelling, J. Chem. Phys., Volume 111 (1999) no. 19, pp. 9025-9033
[25] Carbon dioxide in montmorillonite clay hydrates: thermodynamics, structure, and transport from molecular simulation, J. Phys. Chem. C, Volume 114 (2010) no. 35, pp. 14962-14969
[26] A new methodology for computing ionic profiles and disjoining pressure in swelling porous media, Comput. Geosci., Volume 20 (2016) no. 5, pp. 975-996
[27] A three-scale poromechanical model for swelling porous media incorporating solvation forces: application to enhanced coalbed methane recovery, Mech. Mater., Volume 131 (2019), pp. 47-60
[28] Thermal diffusion in micropores by molecular dynamics computer simulations, Phys. A, Volume 361 (2006) no. 2, pp. 494-510
[29] Slippage of binary fluid mixtures in a nanopore, Microfluid. Nanofluid., Volume 15 (2013) no. 2, pp. 183-189
[30] Mineral-and ion-specific effects at clay–water interfaces: structure, diffusion, and hydrodynamics, J. Phys. Chem. C, Volume 122 (2018) no. 32, pp. 18484-18492
[31] System-size dependence of diffusion coefficients and viscosities from molecular dynamics simulations with periodic boundary conditions, J. Phys. Chem. B, Volume 108 (2004) no. 40, pp. 15873-15879
[32] Fluid-driven metamorphism of the continental crust governed by nanoscale fluid flow, Nat. Geosci., Volume 10 (2017) no. 9, pp. 685-690
[33] Modelling water and ion diffusion in clays: a multiscale approach, C. R. Chim., Volume 10 (2007) no. 10–11, pp. 1108-1116
[34] A review of studies on CO sequestration and caprock integrity, Fuel, Volume 89 (2010) no. 10, pp. 2651-2664
[35] Phase behavior of hydrocarbons in nano-pores, Fluid Phase Equilib., Volume 497 (2019), pp. 104-121
[36] Giant osmotic energy conversion measured in a single transmembrane boron nitride nanotube, Nature, Volume 494 (2013) no. 7438, pp. 455-458
[37] Principles of geostatistics, Econ. Geol., Volume 58 (1963) no. 8, pp. 1246-1266
[38] Dealing with spatial heterogeneity, Hydrogeol. J., Volume 13 (2005) no. 1, pp. 161-183
[39] Geostatistics: Modeling Spatial Uncertainty, Vol. 497, John Wiley & Sons, New York, 2009
[40] Stochastic Subsurface Hydrology, Prentice-Hall, New York, 1993
[41] Random Fields for Spatial Data Modeling A Primer for Scientists and Engineers, Springer, Netherlands, 2020
[42] Flow and Transport in Porous Formations, Springer-Verlag GmbH & Co. KG, 1989
[43] A higher-order approximation to effective conductivity in media of anisotropic random structure, Water Resour. Res., Volume 30 (1994) no. 6, pp. 1857-1864
[44] Effective permittivity of log-normal isotropic random media, J. Phys. A, Volume 28 (1995) no. 3, p. 693
[45] Homogenization of Differential Operators and Integral Functionals, Springer Science & Business Media, Berlin, Heidelberg, 2012
[46] Quantitative Stochastic Homogenization and Large-Scale Regularity, Vol. 352, Springer, Cham, Switzerland, 2019
[47] The use of renormalization for calculating effective permeability, Trans. Porous Med., Volume 4 (1989) no. 1, pp. 37-58
[48] The effective permeability of a heterogeneous porous medium, Trans. Porous Med., Volume 15 (1994), pp. 99-127
[49] Renormalization group analysis of permeability upscaling, Stoch Environ. Res. Risk Assess., Volume 13 (1999) no. 1–2, pp. 131-161
[50] Computing the effective permeability of log-normal permeability fields using renormalization methods, C. R. Acad. Sci. - Ser. IIA - Earth Planet. Sci., Volume 331 (2000) no. 5, pp. 353-357
[51] Generalized coarse graining procedures for flow in porous media, Comput. Geosci., Volume 7 (2003) no. 4, pp. 253-273
[52] Coarse graining for upscaling of flow in heterogeneous porous media, Multiscale Model. Simul., Volume 2 (2004) no. 2, pp. 269-301
[53] Renormalization group method in the problem of the effective conductivity of a randomly heterogeneous porous medium, J. Expl Theoret. Phys., Volume 95 (2002) no. 1, pp. 67-76
[54] Effective hydraulic conductivity of a randomly heterogeneous porous medium, Water Resour. Res., Volume 39 (2003) no. 3, pp. 360-373
[55] A Treatise on Electricity and Magnetism, Vol. 1, Clarendon Press, Oxford, 1873
[56] Electrodynamics of Continuous Media, Vol. 8, Pergamon, New York, 1960, pp. 41-43
[57] A variational approach to the theory of the effective magnetic permeability of multiphase materials, J. Appl. Phys., Volume 33 (1962) no. 10, pp. 3125-3131
[58] Percolation theory and its application to groundwater hydrology, Water Resour. Res., Volume 29 (1993) no. 4, pp. 775-794
[59] Percolation Theory for Flow in Porous Media, Vol. 880, Springer, Cham, Switzerland, 2014
[60] The use of field theoretic methods for the study of flow in a heterogeneous porous medium, J. Phys. A, Volume 20 (1987) no. 12, p. 3935
[61] Use of the Fourier–Laplace transform and of diagrammatical methods to interpret pumping tests in heterogeneous reservoirs, Adv. Water Resour., Volume 21 (1998) no. 7, pp. 581-590
[62] Spin Glass Theory and Beyond: An Introduction to the Replica Method and its Applications, Vol. 9, World Scientific Publishing Company, Singapore, 1987
[63] On the multiscale characterization of effective hydraulic conductivity in random heterogeneous media: a historical survey and some new perspectives, Adv. Water Resour. (2020), 103594
[64] Calculating equivalent permeability: a review, Adv. Water Resour., Volume 20 (1997) no. 5, pp. 253-278
[65] Higher-order correction of effective permeability of heterogeneous isotropic formations of lognormal conductivity distribution, Trans. Porous Med., Volume 12 (1993) no. 3, pp. 279-290
[66] Correlation structure dependence of the effective permeability of heterogeneous porous media, Phys. Fluids, Volume 7 (1995) no. 11, pp. 2553-2562
[67] Prediction of steady state flow in nonuniform geologic media by conditional moments: exact nonlocal formalism, effective conductivities, and weak approximation, Water Resour. Res., Volume 29 (1993) no. 2, pp. 341-364
[68] Field Theory, the Renormalization Group, and Critical Phenomena: Graphs to Computers, World Scientific Publishing Company, Singapore, 2005
[69] Permeability of a random array of fractures of widely varying apertures, Trans. Porous Med., Volume 2 (1987) no. 1, pp. 31-43
[70] Introduction to Percolation Theory, Taylor & Francis, London, Philadelphia, 2014
[71] Crack instabilities of a heated glass strip, Phys. Rev. E, Volume 52 (1995) no. 4, p. 4105
[72] Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata], J. Appl. Math. Mech., Volume 24 (1960) no. 5, pp. 1286-1303
[73] Fractured reservoir simulation: a challenging and rewarding issue, Oil Gas Sci. Technol.–Revue de l’Institut Français du Pétrole, Volume 65 (2010) no. 2, pp. 227-238
[74] Connectivity of random fault networks following a power law fault length distribution, Water Resour. Res., Volume 33 (1997) no. 7, pp. 1567-1583
[75] Hydraulic properties of two-dimensional random fracture networks following power law distributions of length and aperture, Water Resour. Res., Volume 38 (2002) no. 12, p. 12-1
[76] Connectivity, permeability, and channeling in randomly distributed and kinematically defined discrete fracture network models, Water Resour. Res., Volume 52 (2016) no. 11, pp. 8526-8545
[77] Fractures and Fracture Networks, Vol. 15, Springer Science & Business Media, Dordrecht, 1999
[78] Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches, John Wiley & Sons, Germany, 2011
[79] Flow, transport, and reaction in porous media: percolation scaling, critical-path analysis, and effective medium approximation, Rev. Geophys., Volume 55 (2017) no. 4, pp. 993-1078
[80] Application of fractal geometry to the study of networks of fractures and their pressure transient, Water Resour. Res., Volume 31 (1995) no. 3, pp. 527-540
[81] A quasi steady state method for solving transient Darcy flow in complex 3D fractured networks, J. Comput. Phys., Volume 231 (2012) no. 1, pp. 23-38
[82] A quasi steady state method for solving transient Darcy flow in complex 3D fractured networks accounting for matrix to fracture flow, J. Comput. Phys., Volume 283 (2015), pp. 205-223
[83] The Laplacian spectrum of graphs, Graph Theory, Combin. Appl., Volume 2 (1991) no. 871–898, p. 12
[84] Some applications of Laplace eigenvalues of graphs, Graph Symmetry, Springer, Dordrecht, 1997, pp. 225-275
[85] Core percolation in random graphs: a critical phenomena analysis, Eur. Phys. J. B, Volume 24 (2001) no. 3, pp. 339-352
[86] Random incidence matrices: moments of the spectral density, J. Stat. Phys., Volume 103 (2001) no. 1–2, pp. 301-337
[87] Computation of the equivalent macroscopic permeability tensor of discrete networks with heterogeneous segment length, J. Hydraul. Eng., Volume 134 (2008) no. 6, pp. 784-793
[89] Sparse random matrices: the eigenvalue spectrum revisited, J. Phys. A, Volume 35 (2002) no. 23, p. 4837
[90] Percolation on sparse networks, Phys. Rev. Lett., Volume 113 (2014) no. 20, 208702
[91] Modeling flow and transport in fracture networks using graphs, Phys. Rev. E, Volume 97 (2018) no. 3, 033304
[92] Predictions of first passage times in sparse discrete fracture networks using graph-based reductions, Phys. Rev. E, Volume 96 (2017) no. 1, 013304
[93] Emergence of stable laws for first passage times in three-dimensional random fracture networks, Phys. Rev. Lett., Volume 123 (2019) no. 24, 248501
[94] Quasi-steady two-equation models for diffusive transport in fractured porous media: large-scale properties for densely fractured systems, Adv. Water Resour., Volume 24 (2001) no. 8, pp. 863-876
[95] Multiple-rate mass transfer for modeling diffusion and surface reactions in media with pore-scale heterogeneity, Water Resour. Res., Volume 31 (1995) no. 10, pp. 2383-2400
[96] Multi-rate mass transfer (MRMT) models for general diffusive porosity structures, Adv. Water Resour., Volume 76 (2015), pp. 146-156
[97] Up-scaling of double porosity fractured media using continuous-time random walks methods, Trans. Porous Med., Volume 39 (2000) no. 3, pp. 315-337
[98] A direct determination of the transient exchange term of fractured media using a continuous time random walk method, Trans. Porous Med., Volume 44 (2001) no. 3, pp. 539-557
[99] Random walk methods for modeling hydrodynamic transport in porous and fractured media from pore to reservoir scale, Trans. Porous Med., Volume 115 (2016) no. 2, pp. 345-385
[100] Random Matrices, Elsevier, Netherlands, 2004
[101] Cavity approach to the spectral density of sparse symmetric random matrices, Phys. Rev. E, Volume 78 (2008) no. 3, 031116
[102] Recent progress on the random conductance model, Probab. Surv., Volume 8 (2011), pp. 294-373
[103] A First Course in Random Matrix Theory, Cambridge University Press, 2019 (in press)
[104] Machine learning for graph-based representations of three-dimensional discrete fracture networks, Comput. Geosci., Volume 22 (2018) no. 3, pp. 695-710
[105] Efficient Monte Carlo with graph-based subsurface flow and transport models, Water Resour. Res., Volume 54 (2018) no. 5, pp. 3758-3766
[106] Calculation of internodal transmissivities in finite difference models of flow in heterogeneous porous media, Water Resour. Res., Volume 31 (1995) no. 4, pp. 943-959
[107] Extreme value problems in random matrix theory and other disordered systems, J. Statist. Mech.: Theory Exp., Volume 2007 (2007) no. 07, 07019
[108] Distribution of eigenvalues for some sets of random matrices, Mat. Sborn., Volume 114 (1967) no. 4, pp. 507-536
[109] A random matrix approach to neural networks, Ann. Appl. Probab., Volume 28 (2018) no. 2, pp. 1190-1248
[110] Classification spectrale par la laplacienne déformée dans des graphes réalistes, XXVII colloque GRETSI (GRETSI 2019), Aug 2019, lille, France, 2019 (hal-02153901)
[111] Dispersion in fixed beds, J. Fluid Mech., Volume 154 (1985), pp. 399-427
[112] Anomalous diffusion in heterogeneous porous media, Phys. Fluids, Volume 31 (1988) no. 5, pp. 965-973
[113] Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular diffusion, Proc. R. Soc. Lond. A. Math. Phys. Sci., Volume 225 (1954) no. 1163, pp. 473-477
[114] Dispersion due to molecular diffusion and macroscopic mixing in flow through a network of capillaries, J. Fluid Mech., Volume 7 (1960) no. 2, pp. 194-208
[115] Dispersion dans un écoulement de Stokes, J. Phys. Lett., Volume 46 (1985) no. 21, pp. 991-998
[116] Hydrodynamic dispersion at stagnation points: simulations and experiments, Phys. Rev. E, Volume 52 (1995) no. 5, p. 4952
[117] Dual control of flow field heterogeneity and immobile porosity on non-Fickian transport in Berea sandstone, Water Resour. Res., Volume 51 (2015) no. 10, pp. 8273-8293
[118] Is transport in porous media always diffusive? A counterexample, Water Resour. Res., Volume 16 (1980) no. 5, pp. 901-917
[119] Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications, Phys. Rep., Volume 195 (1990) no. 4–5, pp. 127-293
[120] Superdiffusion in random velocity fields, Phys. Rev. Lett., Volume 64 (1990) no. 21, p. 2503
[121] Renormalization group analysis of macrodispersion in a directed random flow, Water Resour. Res., Volume 33 (1997) no. 10, pp. 2287-2299
[122] Probability of return to the origin at short times: a probe of microstructure in porous media, Phys. Rev. B, Volume 51 (1995) no. 21, p. 14936
[123] Space-fractional advection–diffusion and reflective boundary condition, Phys. Rev. E, Volume 73 (2006) no. 2, 021104
[124] Continuous-time random-walk model of transport in variably saturated heterogeneous porous media, Phys. Rev. E, Volume 81 (2010) no. 3, 031104
[125] Superdispersion in homogeneous unsaturated porous media using NMR propagators, Phys. Rev. E, Volume 87 (2013) no. 4, 043007
[126] Model to interpret pulsed-field-gradient NMR data including memory and superdispersion effects, Phys. Rev. E, Volume 89 (2014) no. 6, 062121
[127] Mixing, spreading and reaction in heterogeneous media: a brief review, J. Contam. Hydrol., Volume 120 (2011), pp. 1-17
[128] Non-Fickian mixing: temporal evolution of the scalar dissipation rate in heterogeneous porous media, Adv. Water Resour., Volume 33 (2010) no. 12, pp. 1468-1475
[129] Lagrangian statistical model for transport in highly heterogeneous velocity fields, Phys. Rev. Lett., Volume 101 (2008) no. 9, 090601
[130] Continuous time random walks for the evolution of Lagrangian velocities, Phys. Rev. Fluids, Volume 1 (2016) no. 7, 074004
[131] Scalar gradients in stirred mixtures and the deconstruction of random fields, J. Fluid Mech., Volume 812 (2017), pp. 578-610
[132] The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid, Proc. R. Soc. Lond. A, Volume 245 (1958) no. 1242, pp. 312-329
[133] Viscous fingering in porous media, Annu. Rev. Fluid Mech., Volume 19 (1987) no. 1, pp. 271-311
[134] Diffusion-limited aggregation and the Saffman–Taylor problem, Phys. Rev. A, Volume 31 (1985) no. 3, p. 1977
[135] Velocity selection and the Saffman–Taylor problem, Phys. Rev. Lett., Volume 56 (1986) no. 19, p. 2028
[136] Dendrites, viscous fingers, and the theory of pattern formation, Science, Volume 243 (1989) no. 4895, pp. 1150-1156
[137] Why waterflood works: a linearized stability analysis, SPE Annual Technical Conference and Exhibition, Society of Petroleum Engineers, 1989
[138] Viscous fingering in periodically heterogeneous porous media. I. Formulation and linear instability, J. Chem. Phys., Volume 107 (1997) no. 22, pp. 9609-9618
[139] Viscous fingering in periodically heterogeneous porous media. II. Numerical simulations, J. Chem. Phys., Volume 107 (1997) no. 22, pp. 9619-9628
[140] Dynamics of the water–oil front for two-phase, immiscible flow in heterogeneous porous media. 1–stratified media, Trans. Porous Med., Volume 56 (2004) no. 3, pp. 283-303
[141] Dynamics of the water–oil front for two-phase, immiscible flow in heterogeneous porous media. 2–isotropic media, Trans. Porous Med., Volume 56 (2004) no. 3, pp. 305-328
[142] Up-scaling two-phase flow in heterogeneous reservoirs: current trends, Oil Gas Sci. Technol., Volume 59 (2004) no. 2, pp. 185-195
[143] Steady-state two-phase flow in porous media: statistics and transport properties, Phys. Rev. Lett., Volume 102 (2009) no. 7, 074502
[144] Two-phase flow: structure, upscaling, and consequences for macroscopic transport properties, Vadose Zone J., Volume 11 (2012) no. 3, 2011.0123 | DOI
[145] A method for predicting the performance of unstable miscible displacement in heterogeneous media, Soc. Petrol. Eng. J., Volume 3 (1963) no. 02, pp. 145-154
[146] The development, testing, and application of a numerical simulator for predicting miscible flood performance, J. Petrol. Tech., Volume 24 (1972) no. 07, pp. 874-882
[147] A theoretical analysis of vertical flow equilibrium, Trans. Porous Med., Volume 18 (1995) no. 2, pp. 107-129
[148] How to predict viscous fingering in three component flow, Trans. Porous Med., Volume 12 (1993) no. 3, pp. 207-236
[149] Linear viscous fingering: new experimental results, direct simulation and the evaluation of averaged models, Chem. Eng. Sci., Volume 50 (1995) no. 4, pp. 601-616
[150] Diffusion-limited aggregation, a kinetic critical phenomenon, Phys. Rev. Lett., Volume 47 (1981) no. 19, p. 1400
[151] Invasion percolation: a new form of percolation theory, J. Phys. A, Volume 16 (1983) no. 14, p. 3365
[152] Diffusion-limited aggregation and two-fluid displacements in porous media, Phys. Rev. Lett., Volume 52 (1984) no. 18, p. 1621
[153] A diffusion-limited aggregation model for the evolution of drainage networks, Earth Planet. Sci. Lett., Volume 119 (1993) no. 3, pp. 379-386
[154] Phase diagram of sustained wave fronts opposing the flow in disordered porous media, Europhys. Lett., Volume 101 (2013) no. 3, p. 38003
[155] Experimental evidence for three universality classes for reaction fronts in disordered flows, Phys. Rev. Lett., Volume 114 (2015) no. 23, 234502
[156] Dynamic scaling of growing interfaces, Phys. Rev. Lett., Volume 56 (1986) no. 9, p. 889
[157] Multiscale description and upscaling of fluid flow in subsurface reservoirs, Oil Gas Sci. Technol., Volume 59 (2004) no. 2, pp. 119-139
[158] From pore scale to wellbore scale: impact of geometry on wormhole growth in carbonate acidization, Chem. Eng. Sci., Volume 63 (2008) no. 12, pp. 3088-3099
[159] Chemo-hydrodynamic patterns in porous media, Phil. Trans. R. Soc. A, Volume 374 (2016) no. 2078, 20150419
[160] A random fuse model for breaking processes, J. Phys. Lett., Volume 46 (1985) no. 13, pp. 585-590
[161] Scaling and multiscaling laws in random fuse networks, Phys. Rev. B, Volume 39 (1989) no. 4, p. 2678
[162] Connectivity of growing random networks, Phys. Rev. Lett., Volume 85 (2000) no. 21, p. 4629
[163] Scaling, universality, and geomorphology, Annu. Rev. Earth Planet. Sci., Volume 28 (2000) no. 1, pp. 571-610
[164] Self-organized complexity in geomorphology: observations and models, Geomorphology, Volume 91 (2007) no. 3–4, pp. 302-310
[165] Climate change and geomorphological hazards in the eastern European Alps, Phil. Trans. R. Soc. A, Volume 368 (2010) no. 1919, pp. 2461-2479
[166] Inverse Problem Theory and Methods for Model Parameter Estimation, Vol. 89, SIAM, Philadelphia, 2005
[167] Pilot point methodology for automated calibration of an ensemble of conditionally simulated transmissivity fields: 2. Application, Water Resour. Res., Volume 31 (1995) no. 3, pp. 495-516
[168] Optimizing subsurface field data acquisition using information theory, Math. Geosci., Volume 42 (2010) no. 6, pp. 603-630
[169] Statistical physics of inference: thresholds and algorithms, Adv. Phys., Volume 65 (2016) no. 5, pp. 453-552
[170] Phase transitions in the coloring of random graphs, Phys. Rev. E, Volume 76 (2007) no. 3, 031131
Cité par Sources :
Commentaires - Politique