Abstract
Classical stochastic theories for transport in subsurface are revisited and transport models are formulated as stochastic processes. The process of diffusion with space variable drift coefficients is proposed as a general frame for stochastic modeling in subsurface hydrology. Stochastic homogeneity properties, first order approximations, and the occurrence of anomalous diffusion, ergodic, and self-averaging properties are presented.
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References
Aït-Sahalia, Y.: Telling from discrete data whether the underlying continuous-time model is a diffusion. J. Financ. 57, 2075–2112 (2002)
Arnold, V.I.: Ordinary Differential Equations. Springer, Berlin (1992)
Attinger, S., Dentz, M., Kinzelbach, H., Kinzelbach, W.: Temporal behavior of a solute cloud in a chemically heterogeneous porous medium. J. Fluid Mech. 386, 77–104 (1999)
Avellaneda, M., Majda, M.: Stieltjes integral representation and effective diffusivity bounds for turbulent diffusion. Phys. Rev. Lett. 62(7), 753–755 (1989)
Avellaneda, M., Majda, M.: Superdiffusion in nearly stratified flows. J. Stat. Phys. 69(3/4), 689–729 (1992)
Avellaneda, M., Elliot, F. Jr., Apelian, C.: Trapping, percolation and anomalous diffusion of particles in a two-dimensional random field. J. Stat. Phys. 72(5/6), 1227–1304 (1993)
Balescu, R.: Transport Processes in Plasmas. North-Holland, Amsterdam (1988)
Balescu, R., Wang, H-D., Misguich, J.H.: Langevin equation versus kinetic equation: subdiffusive behavior of charged particles in a stochastic magnetic field. Phys. Plasmas 1(12), 3826–3842 (1994)
Bear, J.: On the tensor form of dispersion in porous media. J. Geophys. Res. 66(4), 1185–1197 (1961)
Bhattacharya, R.N., Gupta, V.K.: A theoretical explanation of solute dispersion in saturated porous media at the Darcy scale. Water Resour. Res. 19, 934–944 (1983)
Bouchaud, J.-P., Georges, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293 (1990)
Bouchaud, J.-P., Georges, A., Koplik, J., Provata, A., Redner, S.: Superdiffusion in random velocity fields. Phys. Rev. Lett. 64, 2503–2506 (1990)
Chilès, J.P., Delfiner, P.: Geostatistics: Modeling Spatial Uncertainty. Wiley, New York (1999)
Clincy, M., Kinzelbach, H.: Stratified disordered media: exact solutions for transport parameters and their self-averaging properties. J. Phys. A. Math. Gen. 34, 7142–7152 (2001)
Colucci, P.J., Jaberi, F.A., Givi, P.: Filtered density function for large eddy simulation of turbulent reacting flows. Phys. Fluids 10(2), 499–515 (1998)
Dagan, G.: Solute transport in heterogeneous porous formations. J. Fluid Mech. 145, 151–177 (1984)
Dagan, G.: Theory of solute transport by groundwater. Annu. Rev. Fluid Mech. 19, 183–215 (1987)
Dagan, G.: Time-dependent macrodispersion for solute transport in anisotropic heterogeneous aquifers. Water Resour. Res. 24, 1491–1500 (1988)
Dagan, G.: Flow and Transport in Porous Formations. Springer, Berlin (1989)
Dagan, G.: Transport in heterogeneous porous formations: spatial moments, ergodicity, and effective dispersion. Water Resour. Res. 26(6), 1281–1290 (1990)
Deng, W., Barkai, E.: Ergodic properties of fractional Brownian-Langevin motion. Phys. Rev. E 79, 011112 (2009)
Deng, F.W., Cushman, J.H.: On higher-order corrections to the flow velocity covariance tensor. Water Resour. Res. 31(7), 1659–1672 (1995)
Dentz, M., de Barros, F.P.J.: Dispersion variance for transport in heterogeneous porous media. Water Resour. Res. 49, 3443–3461 (2013)
Dentz, M., Kinzelbach, H., Attinger, S., Kinzelbach, W.: Temporal behavior of a solute cloud in a heterogeneous porous medium 1. Point-like injection. Water Resour. Res. 36, 3591–3604 (2000)
Dentz, M., Kinzelbach, H., Attinger, S., Kinzelbach, W.: Temporal behavior of a solute cloud in a heterogeneous porous medium 2. Spatially extended injection. Water Resour. Res. 36, 3605–3614 (2000)
Di Federico, V., Neuman, S.P.: Scaling of random fields by means of truncated power variograms and associated spectra. Water Resour. Res. 33, 1075–1085 (1997)
Doob, J.L.: Stochastic Processes. Wiley, New York (1990)
Dybiec, B., Gudowska-Nowak, E.: Discriminating between normal and anomalous random walks. Phys. Rev. E 80, 061122 (2009)
Eberhard, J.: Approximations for transport parameters and self-averaging properties for point-like injections in heterogeneous media. J. Phys. A. Math. Gen. 37, 2549–2571 (2004)
Eberhard, J., Suciu, N., Vamos, C.: On the self-averaging of dispersion for transport in quasi-periodic random media. J. Phys. A: Math. Theor. 40, 597–610 (2007)
Fannjiang, A., Komorowski, T.: Diffusive and nondiffusive limits of transport in nonmixing flows. SIAM J. Appl. Math. 62, 909–923 (2002)
Fiori, A.: Finite Peclet extensions of Dagan’s solutions to transport in anisotropic heterogeneous formations. Water Resour. Res. 32, 193–198 (1996)
Fiori, A.: On the influence of local dispersion in solute transport through formations with evolving scales of heterogeneity. Water Resour. Res. 37(2), 235–242 (2001)
Fiori, A., Dagan, G.: Concentration fluctuations in aquifer transport: a rigorous first-order solution and applications. J. Contam. Hydrol. 45, 139–163 (2000)
Fried, J.J.: Groundwater Pollution. Elsevier, New York (1975)
Gardiner, C.W.: Stochastic Methods. Springer, Berlin (2009)
Gelhar, L.W.: Stochastic subsurface hydrology from theory to applications. Water Resour. Res. 22(9S), 135S–145S (1986)
Gelhar, L.W., Axness, C.: Three-dimensional stochastic analysis of macrodispersion in aquifers. Water Resour. Res. 19(1), 161–180 (1983)
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Elsevier, Amsterdam (2007)
Honkonen, G.: Stochastic processes with stable distributions in random environments. Phys. Rev. E 53(1), 327–331 (1996)
Isichenko, M.B.: Percolation, Statistical Topography, and Transport in Random Media. Rev. Mod. Phys. 64, 961 (1992)
Jaekel, U., Vereecken, H.: Renormalization group analysis of macrodispersion in a directed random flow. Water Resour. Res. 33, 2287–2299 (1997)
Jeon, J.-H., Metzler, R.: Fractional Brownian motion and motion governed by the fractional Langevin equation in confined geometries. Phys. Rev. E 81, 021103 (2010)
Jury, W.A., Sposito, G.: Field calibration and validation of solute transport models for the unsaturated zone. Soil Sci. Soc. Am. J. 49, 1331–1241 (1985)
Kesten, H., Papanicolaou, G.C.: A limit theorem for turbulent diffusion. Commun. Math. Phys. 65, 97–128 (1979)
Kitanidis, P.K.: Prediction by the method of moments of transport in a heterogeneous formation. J. Hydrol. 102, 453–473 (1988)
Kloeden, P.E., Platen, E.: Numerical Solutions of Stochastic Differential Equations. Springer, Berlin (1999)
Kolmogorov, A.N.: Grundbegriffe der Warscheinlichkeitsrechnung. Springer, Berlin (1933)
Le Doussal, P., Machta, J.: Annealed versus quenched diffusion coefficient in random media. Phys. Rev. B 40(12), 9427–9430 (1989)
Lumley, J.L.: The mathematical nature of the problem of relating Lagrangian and Eulerian statistical functions in turbulence. In: Mécanique de la Turbulence. Coll. Intern. du CNRS à Marseille (Ed. CNRS, Paris, 1962), pp. 17–26
Majda, A.J., Kramer, P.R.: Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena. Phys. Rep. 14, 237–574 (1999)
Majumdar, S.N.: Persistence of a particle in the Matheron–de Marsily velocity field. Phys. Rev. E 68, 050101(R) (2003)
Matheron, G., de Marsily, G.: Is transport in porous media always diffusive? Water Resour. Res. 16, 901–917 (1980)
Monin, A.S., Yaglom, A.M.: Statistical Fluid Mechanics, Volume II: Mechanics of Turbulence. MIT Press, Cambridge (1975)
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 (2012)
O’Malley, D., Cushman, J.H.: Two scale renormalization group classification of diffusive processes. Phys. Rev. E 86(1), 011126 (2012)
Papoulis, A., Pillai, S. U.: Probability, Random Variables and Stochastic Processes. McGraw-Hill, Singapore (2009)
Pope, S.B.: PDF methods for turbulent reactive flows. Prog. Energy Combust. Sci. 11(2), 119–192 (1885)
Port, S.C., Stone, C.J.: Random measures and their application to motion in an incompressible fluid. J. Appl. Prob. 13, 498–506 (1976)
Reuss, J.-D., Misguish, J.H.: Low frequency percolation scaling for particle diffusion in electrostatic turbulence. Phys. Rev. E 54(2), 1857–1869 (1996)
Ross, K., Attinger, S.: Temporal behaviour of a solute cloud in a fractal heterogeneous porous medium at different scales. Geophys. Res. Abstr. 12, EGU2010-10921-2 (2010)
Russo, D.: On the velocity covariance and transport modeling in heterogeneous anisotropic porous formations. Water Resour. Res. 31(1), 129–137 (1995)
Russo, D.: A note on ergodic transport of a passive solute in partially saturated porous formations. Water Resour. Res. 32(12), 3623–3628 (1996)
Saffman, P.G.: Application of the Wiener-Hermite expansion to the diffusion of a passive scalar in a homogeneous turbulent flow. Phys. Fluids 12(9), 1786–1798 (1969)
Scheidegger, A.E.: Statistical hydrodynamics in porous media. J. Appl. Phys. 25(8), 994–1001 (1954)
Scheidegger, A.E.: General theory of dispersion in porous media. J. Geophys. Res. 66(10), 3273–3278 (1961)
Schwarze, H., Jaekel, U., Vereecken, H.: Estimation of macrodispersivity by different approximation methods for flow and transport in randomly heterogeneous media. Transp. Porous Media 43, 265–287 (2001)
Shapiro, A.M., Cvetkovic, V.D.: Stochastic analysis of solute travel time in heterogeneous Porous media. Water Resour. Res. 24(10), 1711–1718 (1988)
Suciu, N.: Spatially inhomogeneous transition probabilities as memory effects for diffusion in statistically homogeneous random velocity fields. Phys. Rev. E 81, 056301 (2010)
Suciu, N.: Diffusion in random velocity fields with applications to contaminant transport in groundwater. Adv. Water Resour. 69, 114–133 (2014)
Suciu, N., Vamoş, C.: Comment on “Nonstationary flow and nonergodic transport in random porous media” by G. Darvini and P. Salandin. Water Resour. Res. 43, W12601 (2007)
Suciu, N., Vamoş, C.: Ergodic estimations of upscaled coefficients for diffusion in random velocity fields. In: L’Ecuyér, P., Owen, A.B. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2008, pp. 617–626. Springer, Berlin (2009)
Suciu, N., Vamoş, C., Vanderborght, J., Hardelauf, H., Vereecken, H.: Numerical investigations on ergodicity of solute transport in heterogeneous aquifers. Water Resour. Res. 42, W04409 (2006)
Suciu, N., Vamoş, C., Eberhard, J.: Evaluation of the first-order approximations for transport in heterogeneous media. Water Resour. Res. 42, W11504 (2006)
Suciu, N., Vamos, C., Vereecken, H., Sabelfeld, K., Knabner, P.: Memory effects induced by dependence on initial conditions and ergodicity of transport in heterogeneous media. Water Resour. Res. 44, W08501 (2008)
Suciu, N., Vamos, C., Vereecken, H., Sabelfeld, K., Knabner, P.: Ito equation model for dispersion of solutes in heterogeneous media. Rev. Anal. Num. Theor. Approx. 37, 221–238 (2008)
Suciu, N., Vamos, C., Radu, F.A., Vereecken, H., Knabner, P.: Persistent memory of diffusing particles. Phys. Rev. E 80, 061134 (2009)
Suciu, N., Attinger, S., Radu, F.A., Vamoş, C., Vanderborght, J., Vereecken, H., Knabner, P.: Solute transport in aquifers with evolving scale heterogeneity. Print No. 346, Mathematics Department—Friedrich-Alexander University Erlangen-Nuremberg (2011)
Suciu, N., Attinger, S., Radu, F.A., Vamoş, C., Vanderborght, J., Vereecken, H. Knabner, P.: Solute transport in aquifers with evolving scale heterogeneity. Analele Universitatii “Ovidius” Constanta-Seria Matematica 23(3), 167–186 (2015)
Suciu, N., Schüler, L., Attinger, S., Knabner, P.: Towards a filtered density function approach for reactive transport in groundwater. Adv. Water Resour. 90, 83–98 (2016)
Taylor, G.I.: Diffusion by continuous movements. Proc. Lond. Math. Soc. 2(20), 196–212 (1921)
Trefry, M.G., Ruan, F.P., McLaughlin, D.: Numerical simulations of preasymptotic transport in heterogeneous porous media: departures from the Gaussian limit. Water Resour. Res. 39(3), 1063 (2003)
Vanderborght, J.: Concentration variance and spatial covariance in second order stationary heterogeneous conductivity fields. Water Resour. Res. 37(7), 1893–1912 (2001)
Yaglom, A.M.: Correlation Theory of Stationary and Related Random Functions, Volume I: Basic Results. Springer, New York (1987)
Zirbel, C.L.: Lagrangian observations of homogeneous random environments. Adv. Appl. Prob. 33, 810–835 (2001)
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Suciu, N. (2019). Diffusion in Random Velocity Fields. In: Diffusion in Random Fields . Geosystems Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-15081-5_4
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