Skip to main content
×
×
Home

Mechanisms of dispersion in a porous medium

  • M. Dentz (a1), M. Icardi (a2) and J. J. Hidalgo (a1)
Abstract

This paper studies the mechanisms of dispersion in the laminar flow through the pore space of a three-dimensional porous medium. We focus on preasymptotic transport prior to the asymptotic hydrodynamic dispersion regime, in which solute motion may be described by the average flow velocity and a hydrodynamic dispersion coefficient. High-performance numerical flow and transport simulations of solute breakthrough at the outlet of a sand-like porous medium evidence marked deviations from the hydrodynamic dispersion paradigm and identify two distinct regimes. The first regime is characterised by a broad distribution of advective residence times in single pores. The second regime is characterised by diffusive mass transfer into low-velocity regions in the wake of solid grains. These mechanisms are quantified systematically in the framework of a time-domain random walk for the motion of marked elements (particles) of the transported material quantity. Particle transitions occur over the length scale imprinted in the pore structure at random times given by heterogeneous advection and diffusion. Under globally advection-dominated conditions, i.e., Péclet numbers larger than 1, particles sample the intrapore velocities by diffusion and the interpore velocities through advection. Thus, for a single transition, particle velocities are approximated by the mean pore velocity. In order to quantify this advection mechanism, we develop a model for the statistics of the Eulerian velocity magnitude based on Poiseuille’s law for flow through a single pore and for the distribution of mean pore velocities, both of which are linked to the distribution of pore diameters. Diffusion across streamlines through immobile zones in the wake of solid grains gives rise to exponentially distributed residence times that decay on the diffusion time over the pore length. The trapping rate is determined by the inverse diffusion time. This trapping mechanism is represented by a compound Poisson process conditioned on the advective residence time in the proposed time-domain random walk approach. The model is parameterised with the characteristics of the porous medium under consideration and captures both preasymptotic regimes. Macroscale transport is described by an integro-differential equation for solute concentration, whose memory kernels are given in terms of the distribution of mean pore velocities and trapping times. This approach quantifies the physical non-equilibrium caused by a broad distribution of mass transfer time scales, both advective and diffusive, on the representative elementary volume (REV). Thus, while the REV indicates the scale at which medium properties like porosity can be uniquely defined, this does not imply that transport can be characterised by hydrodynamic dispersion.

Copyright
Corresponding author
Email address for correspondence: marco.dentz@csic.es
References
Hide All
Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions. Dover Publications.
Bear, J. 1972 Dynamics of Fluids in Porous Media. American Elsevier.
Benson, D. A. & Meerschaert, M. M. 2009 A simple and efficient random walk solution of multi-rate mobile/immobile mass transport equations. Adv. Water Resour. 32 (4), 532539.
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.
Berkowitz, B., Klafter, J., Metzler, R. & Scher, H. 2002 Physical pictures of transport in heterogeneous media: advection–dispersion, random-walk, and fractional derivative formulations. Water Resour. Res. 38 (10), 1191.
Bijeljic, B. & Blunt, M. J. 2006 Pore-scale modeling and continuous time random walk analysis of dispersion in porous media. Water Resour. Res. 42, W01202.
Bijeljic, B., Mostaghimi, P. & Blunt, M. J. 2011 Signature of non-Fickian solute transport in complex heterogeneous porous media. Phys. Rev. Lett. 107 (20), 204502.
Bijeljic, B., Muggeridge, A. H. & Blunt, M. J. 2004 Pore-scale modeling of longitudinal dispersion. Water Resour. Res. 40, W11501.
Bijeljic, B., Raeini, A., Mostaghimi, P. J. & Blunt, M. 2013 Predictions of non-Fickian solute transport in different classes of porous media using direct simulation on pore-scale images. Phys. Rev. E 87, 013011.
Carrera, J., Sánchez-Vila, X., Benet, I., Medina, A., Galarza, G. & Guimerà, J. 1998 On matrix diffusion: formulations, solution methods, and qualitative effects. J. Hydrol. 6, 178190.
Comolli, A. & Dentz, M. 2017 Anomalous dispersion in correlated porous media: a coupled continuous time random walk approach. Eur. Phys. J. B; 90.
Comolli, A., Hidalgo, J. J., Moussey, C. & Dentz, M. 2016 Non-Fickian transport under heterogeneous advection and mobile–immobile mass transfer. Trans. Porous Med. 115 (2), 265289.
Cvetkovic, V. D., Dagan, G. & Shapiro, A. M. 1991 An exact solution of solute transport by one-dimensional random velocity fields. Stochastic Hydrol. Hydraul. 5, 4554.
De Anna, P., Le Borgne, T., Dentz, M., Tartakovsky, A. M., Bolster, D. & Davy, P. 2013 Flow intermittency, dispersion, and correlated continuous time random walks in porous media. Phys. Rev. Lett. 110 (18), 184502.
De Anna, P., Quaife, B., Biros, G. & Juanes, R. 2017 Prediction of velocity distribution from pore structure in simple porous media. Phys. Rev. Fluids 2, 124103.
Delay, F., Ackerer, P. & Danquigny, C. 2005 Simulating solute transport in porous or fractured formations using random walk particle tracking. Vadose Zone J. 4, 360379.
Delay, F., Porel, G. & Sardini, P. 2002 Modelling diffusion in a heterogeneous rock matrix with a time-domain Lagrangian method and an inversion procedure. C. R. Geosci. 334, 967973.
Dentz, M. & Carrera, J. 2007 Mixing and spreading in stratified flow. Phys. Fluids 19, 017107.
Dentz, M., Cortis, A., Scher, H. & Berkowitz, B. 2004 Time behavior of solute transport in heterogeneous media: transition from anomalous to normal transport. Adv. Water Resour. 27 (2), 155173.
Dentz, M., Gouze, P., Russian, A., Dweik, J & Delay, F. 2012 Diffusion and trapping in heterogeneous media: an inhomogeneous continuous time random walk approach. Adv. Water Resour. 49, 1322.
Dentz, M., Kang, P., Comolli, A., Le Borgne, T. & Lester, D. R. 2016a Continuous time random walks for the evolution of Lagrangian velocities. Phys. Rev. Fluids 1, 074004.
Dentz, M., Le Borgne, T., Lester, D. R. & de Barros, F. P. J. 2016b The Handbook of Groundwater Engineering, 3rd edn. Mixing in Groundwater, pp. 383407. Taylor and Francis.
Feller, W. 1968 An Introduction to Probablility and Its Applications, vol. I. Wiley.
Friedman, G. M. 1962 On sorting, sorting coefficients, and the lognormality of the grain-size distribution of sandstones. J. Geology 70, 737753.
Gardiner, C. 2009 Stochastic Methods. Springer.
Gjetvaj, F., Russian, A., Gouze, P. & Dentz, M. 2015 Dual control of flow field heterogeneity and immobile porosity on non-Fickian transport in Berea Sandstone. Water Resour. Res. 51, 82738293.
Gouze, Ph., Melean, Z., 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.
Haber, S. & Mauri, R. 1988 Lagrangian approach to time-dependent laminar dispersion in rectangular conduits. Part 1. Two-dimensional flows. J. Fluid Mech. 190, 201215.
Haggerty, R. & Gorelick, S. M. 1995 Multiple-rate mass transfer for modeling diffusion and surface reactions in media with pore-scale heterogeneity. Water Resour. Res. 31 (10), 23832400.
Holzner, M., Morales, V. L., Willmann, M. & Dentz, M. 2015 Intermittent Lagrangian velocities and accelerations in three-dimensional porous medium flow. Phys. Rev. E 92, 013015.
Hornung, U. 1997 Homogenization and Porous Media. Springer.
Icardi, M., Boccardo, G., Marchisio, D. L., Tosco, T. & Sethi, R. 2014 Pore-scale simulation of fluid flow and solute dispersion in three-dimensional porous media. Phys. Rev. E 90 (1), 013032.
Jin, C., Langston, P. A., Pavlovskaya, G. E., Hall, M. R. & Rigby, S. P. 2016 Statistics of highly heterogeneous flow fields confined to three-dimensional random porous media. Phys. Rev. E 93, 013122.
de Josselin de Jong, G. 1958 Longitudinal and transverse diffusion in granular deposits. Trans. Amer. Geophys. Un. 39, 6774.
Kang, P. K., de Anna, P., Nunes, J. P., Bijeljic, B., Blunt, M. J. & Juanes, R. 2014 Pore-scale intermittent velocity structure underpinning anomalous transport through 3-d porous media. Geophys. Res. Lett. 41 (17), 61846190.
Kenkre, V. M., Montroll, E. W. & Shlesinger, M. F. 1973 Generalized master equations for continuous-time random walks. J. Stat. Phys. 9 (1), 4550.
Koponen, A., Kataja, M. & Timonen, J. 1996 Tortuous flow in porous media. Phys. Rev. E 54 (1), 406.
Kreft, A. & Zuber, A. 1978 On the physical meaning of the dispersion equation and its solutions for different initial and boundary conditions. Chem. Engng Sci. 33, 14711480.
Le Borgne, T., Bolster, D., Dentz, M., de Anna, P. & Tartakovsky, A. 2011 Effective pore scale dispersion upscaling with a correlated CTRW approach. Water Resour. Res. 47, W12538.
Lester, D. R., Metcalfe, G. & Trefry, M. G. 2013 Is chaotic advection inherent to porous media flow? Phys. Rev. Lett. 111 (17), 174101.
Lester, D. R., Metcalf, G. & Trefry, M. G. 2014 Anomalous transport and chaotic advection in homogeneous porous media. Phys. Rev. E 90, 063012.
Liu, Y. & Kitanidis, P. K. 2012 Applicability of the dual-domain model to nonaggregated porous media. Groundwater 50 (6), 927934.
Margolin, G., Dentz, M. & Berkowitz, B. 2003 Continuous time random walk and multirate mass transfer modeling of sorption. Chem. Phys. 295, 7180.
Matyka, M., Golembiewski, J. & Koza, Z. 2016 Power-exponential velocity distributions in disordered porous media. Phys. Rev. E 93, 013110.
Montroll, E. W. & Weiss, G. H. 1965 Random walks on lattices, 2. J. Math. Phys. 6 (2), 167.
Noetinger, B., Roubinet, D., Russian, A., Le Borgne, T., Delay, F., Dentz, M., De Dreuzy, J.-R. & Gouze, P. 2016 Random walk methods for modeling hydrodynamic transport in porous and fractured media from pore to reservoir scale. Trans. Porous Med. 141.
Painter, S. & Cvetkovic, V. 2005 Upscaling discrete fracture network simulations: an alternative to continuum transport models. Water Resour. Res. 41, W02002.
Pfannkuch, H. O. 1963 Contribution a l’étude des déplacements de fluides miscibles dans un milieux poreux. Rev. Inst. Fr. Petr. 18, 215270.
Redner, S. 2001 A Guide to First-Passage Processes. Cambridge University Press.
Risken, H. 1996 The Fokker–Planck Equation. Springer.
Russian, A., Dentz, M. & Gouze, P. 2016 Time domain random walks for hydrodynamic transport in heterogeneous media. Water Resour. Res. 52, doi:10.1002/2015WR018511.
Saffman, P. G. 1959 A theory of dispersion in a porous medium. J. Fluid Mech. 6 (03), 321349.
Scher, H. & Lax, M. 1973 Stochastic transport in a disordered solid. I. Theory. Phys. Rev. B 7 (1), 44914502.
Scheven, U. M. 2013 Pore-scale mixing and transverse dispersivity of randomly packed monodisperse spheres. Phys. Rev. Lett. 110 (21), 214504.
Scheven, U. M., Harris, R. & Johns, M. L. 2007 Intrinsic dispersivity of randomly packed monodisperse spheres. Phys. Rev. Lett. 99 (5), 054502.
Siena, M., Riva, M., Hyman, J. D., Winter, C. L. & Guadagnini, A. 2014 Relationship between pore size and velocity probability distributions in stochastically generated porous media. Phys. Rev. E 89 (1), 013018.
Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219, 186203.
Whitaker, S. 1999 The Method of Volume Averaging. Kluwer Academic Publishers.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

JFM classification

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed