Publications

A guide to relevant Continuous Time Random Walk (CTRW) papers

We provide here a brief guide to our publications that are relevant to theoretical, numerical, and applications aspects of CTRW, particularly in the context of transport in porous media. While the CTRW software provided on on this website can be used independently, we encourage users to familiarize themselves with at least some of the papers mentioned here. Other papers that deal with CTRW theory and applications are cited within the publications listed below.

CTRW theory was first applied to calculate impurity conduction in semiconductors (Scher and Lax, 1973a,b) and to analyze properties of amorphous semiconductors (Montroll and Scher, 1973; Scher and Montroll, 1975). An excellent pedagogical treatment of these analyses is given in Scher et al. (1991).

In the context of geological materials, CTRW theory has been developed and applied extensively. For an extensive review, see: B. Berkowitz, A. Cortis, M. Dentz and H. Scher, Modeling non-Fickian transport in geological formations as a continuous time random walk, Reviews of Geophysics, 44, RG2003, doi:10.1029/2005RG000178, 2006. A related extensive review is given in: B. Berkowitz, I. Dror, S.K. Hansen and H. Scher, Measurements and models of reactive transport in geological media, Reviews of Geophysics, 54, 930-986, doi:10.1002/2016RG000524, 2016.

A full listing of our CTRW-related papers:

  • Hansen, S.K. and B. Berkowitz (2020) Aurora: A non-Fickian (and Fickian) particle tracking package for modeling groundwater contaminant transport with MODFLOW, Environmental Modelling and Software, 134, 104871, doi:10.1016/j.envsoft.2020.104871.

  • Hansen, S.K. and B. Berkowitz (2020) Modeling non‐Fickian solute transport due to mass transfer and physical heterogeneity on arbitrary groundwater velocity fields, Water Resources Research, 56(10), e2019WR026868, doi:10.1029/2019WR026868.
  • Goeppert, N., N. Goldscheider and B. Berkowitz (2020) Experimental and modeling evidence of kilometer-scale anomalous tracer transport in an alpine karst aquifer, Water Research, 178, 115755, doi:10.1016/j.watres.2020.115755.
  • Berkowitz, B. and E. Zehe (2020) Surface water and groundwater: unifying conceptualization and quantification of the two “water worlds”, Hydrology and Earth System Sciences, 24, 1831-1858, doi:10.5194/hess-24-1831-2020.
  • Ben-Zvi, R., H. Scher and B. Berkowitz (2019) Bimolecular reactive transport in a two-dimensional velocity field in disordered media, Journal of Physics A, 52, 424005, doi:10.1088/1751-8121/ab4077.
  • Ben-Zvi, R., S. Jiang, H. Scher and B. Berkowitz (2019) Finite Element Method solution of non-Fickian transport in porous media: The CTRW-FEM Package, Groundwater, 57(3), 479-484, doi:10.1111/gwat.12813.
  • Nissan, A. and B. Berkowitz (2019) Anomalous transport dependence on Péclet number, porous medium heterogeneity, and a temporally-varying velocity field, Physical Review E, 99, 033108, doi:10.1103/PhysRevE.99.033108.
  • Nissan, A. and B. Berkowitz (2018) Inertial effects on flow and transport in heterogeneous porous media, Physical Review Letters, 120, 054504, doi:10.1103/PhysRevLett.120.054504.
  • Ben-Zvi, R., A. Nissan, H. Scher and B. Berkowitz (2018) A continuous time random walk (CTRW) integro-differential equation with chemical interaction, European Journal of Physics B, 91, 15, doi.org/10.1140/epjb/e2017-80417-8.
  • Nissan, A. and B. Berkowitz (2018) Inertial effects on flow and transport in heterogeneous porous media, Physical Review Letters, 120, 054504, doi:10.1103/PhysRevLett.120.054504.
  • Nissan, A., I. Dror and B. Berkowitz (2017) Time-dependent velocity-field controls on anomalous chemical transport in porous media, Water Resources Research, 54, doi:10.1002/2016WR020143.
  • Ben-Zvi, R., H. Scher and B. Berkowitz (2017) Two-dimensional Finite Element Method solution of a class of integro-differential equations: Application to non-Fickian transport in disordered media, International Journal for Numerical Methods in Engineering, doi:10.1002/nme.5524.
  • Hansen, S. K., B. Berkowitz, V. V. Vesselinov, D. O'Malley and S. Karra (2016) Push-pull tracer tests: Their information content and use for characterizing non-Fickian, mobile-immobile behavior, Water Resources Research, 52, 9565-9585, doi:10.1002/2016WR018769.
  • Berkowitz, B., I. Dror, S.K. Hansen and H. Scher (2016) Measurements and models of reactive transport in geological media, Reviews of Geophysics, 54, 930-986, doi:10.1002/2016RG000524.
  • Edery, Y., S. Geiger and B. Berkowitz (2016) Structural controls on anomalous transport in fractured porous rock, Water Resources Research, 52, 5634-5643, doi:10.1002/2016WR018942.
  • Naftaly, A., I. Dror and B. Berkowitz (2016) Measurement and modeling of engineered nanoparticle transport and aging dynamics in a reactive porous medium, Water Resources Research, 52, 5473-5491, doi:10.1002/2016WR018780.
  • Ben-Zvi, R., H. Scher, S. Jiang and B. Berkowitz (2016) One-dimensional finite element method solution of a class of integro-differential equations: Application to non-Fickian transport in disordered media, Transport in Porous Media, 239-263, doi:10.1007/s11242-016-0712-0.
  • Edery, Y., G. M. Porta, A. Guadagnini, H. Scher and B. Berkowitz (2016) Characterization of bimolecular reactive transport in heterogeneous porous media, Transport in Porous Media, 291-310, doi:10.1007/s11242-016-0684-0.
  • Raveh-Rubin, S., Y. Edery, I. Dror and B. Berkowitz (2015) Nickel migration and retention dynamics in natural soil columns, Water Resources Research, 51, 7702-7722, doi:10.1002/2015WR016913.
  • Naftaly, A., Y. Edery, I. Dror and B. Berkowitz (2015) Visualization and analysis of nanoparticle transport and ageing in reactive porous media, Journal of Hazardous Materials, 299, 513-519, doi:10.1016/j.jhazmat.2015.07.043.
  • Edery, Y., I. Dror, H. Scher and B. Berkowitz (2015) Anomalous reactive transport in porous media: Experiments and modeling, Physical Review E, 91, 052130, doi:10.1103/PhysRevE.91.052130.
  • Ciriello, V., Y. Edery, A. Guadagnini and B. Berkowitz (2015) Multimodel framework for characterization of transport in porous media, Water Resources Research, 51, 3384-3402, doi:10.1002/2015WR017047.
  • Hansen, S. K. and B. Berkowitz (2015) Integrodifferential formulations of the continuous-time random walk for solute transport subject to bimolecular A + B → 0 reactions: From micro- to mesoscopic, Physical Review E, 91, 032113, doi:10.1103/PhysRevE.91.032113.
  • Hansen, S. K. and B. Berkowitz (2014) Interpretation and nonuniqueness of CTRW transition distributions: Insights from an alternative solute transport formulation, Advances in Water Resources, 74, 54-63, doi:10.1016/j.advwatres.2014.07.011.
  • Hansen, S. K., H. Scher and B. Berkowitz (2014) First-principles derivation of reactive transport modeling parameters for particle tracking and PDE approaches, Advances in Water Resources, 66, 146-158, doi:10.1016/j.advwatres.2014.04.007.
  • Edery, Y., A. Guadagnini, H. Scher and B. Berkowitz (2014) Origins of anomalous transport in disordered media: Structural and dynamic controls, Water Resources Research, 50, 1490-1505, doi:10.1002/2013WR015111.
  • Ciriello, V., A. Guadagnini, V. Di Federico, Y. Edery and B. Berkowitz (2013) Comparative analysis of formulations for conservative transport in porous media through sensitivity-based parameter calibration, Water Resources Research, 49, 5206-5220, doi:10.1002/wrcr.20395.
  • Nowamooz, A., G. Radilla, M. Fourar and B. Berkowitz (2013) Non-Fickian transport in transparent replicas of rough-walled rock fractures, Transport in Porous Media, 98, 651-682, doi:10.1007/s11242-013-0165-7.
  • Y. Berkowitz, Y. Edery, H. Scher and B. Berkowitz. (2013) Fickian and non-Fickian diffusion with bimolecular reactions, Physical Review E, 87, 032812, doi:10.1103/PhysRevE.87.032812.
  • Y. Edery, A. Guadagnini, H. Scher and B. Berkowitz. (2012) Reactive transport in disordered media: Role of fluctuations in interpretation of laboratory experiments. Advances in Water Resources, doi:10.1016/j.advwatres.2011.12.008.
  • S. Rubin, I. Dror and B. Berkowitz. (2012) Experimental and modeling analysis of coupled non-Fickian transport and sorption in natural soils. Journal of Contaminant Hydrology, 132, 28-36, doi:10.1016/j.jconhyd.2012.02.005.
  • B. Bijeljic, S. Rubin, H. Scher and B. Berkowitz. (2011) Non-Fickian transport in porous media with bimodal structural heterogeneity. Journal of Contaminant Hydrology, 120/121, 213-221, doi:10.1016/j.jconhyd.2010.05.007.
  • Y. Edery, H. Scher and B. Berkowitz. (2011) Dissolution and precipitation dynamics during dedolomitization, Water Resources Research, 47, W08535, doi:10.1029/2011WR010551.
  • B.W. Kuntz, S. Rubin, B. Berkowitz and K. Singha. (2011) Quantifying solute transport at the Shale Hills Critical Zone Observatory. Vadose Zone Journal, 10, 843-857, doi:10.2136/vzj2010.0130.
  • M. Dentz and B. Berkowitz. (2010) Exact effective transport dynamics in a one-dimensional random environment. Physical Review E, 72, 031110, doi:10.1103/PhysRevE.72.031110, 2005. (Erratum: 81, 059901(E).)
  • Y. Edery, H. Scher and B. Berkowitz. (2010) Particle tracking model of bimolecular reactive transport in porous media. Water Resources Research, 46, W07524, doi:10.1029/2009WR009017.
  • G. Srinivasan, D. M. Tartakovsky, M. Dentz, H. Viswanathan, B. Berkowitz and B. A. Robinson. (2010) "Random walk particle tracking simulations of non-Fickian transport in heterogeneous media." Journal of Computational Physics, 229, 4304-4314, doi:10.1016/j.jcp.2010.02.014.
  • H. Scher, K. Willbrand and B. Berkowitz. (2010) Transport equation evalution of coupled continuous time random walks. Journal of Statistical Physics, 141, 1093-1103, doi:10.1007/s10955-010-0088-4.
  • H. Scher, K. Willbrand and B. Berkowitz. (2010) Transport in disordered media with spatially nonuniform fields. Physical Review E, 81, 031102, doi:10.1103/PhysRevE.81.031102.
  • B. Berkowitz and H. Scher. (2010) Anomalous transport in correlated velocity fields. Physical Review E, 81, 011128, doi:10.1103/PhysRevE.81.011128.
  • B. Berkowitz and H. Scher. (2009) Exploring the nature of non-Fickian transport in laboratory experiments. Advances in Water Resources, 32, 750-755, doi:10.1016/j.advwatres.2008.05.004.
  • Y. Edery, H. Scher and B. Berkowitz. (2009) Modeling bimolecular reactions and transport in porous media. Geophysical Research Letters, 36, L02407, doi:10.1029/2008GL036381.
  • B. Berkowitz, A. Cortis, I. Dror and H. Scher. (2009) Laboratory experiments on dispersive transport across interfaces: The role of flow direction. Water Resources Research, 45, W02201, doi:10.1029/2008WR007342.
  • B. Berkowitz, S. Emmanuel and H. Scher. (2008) Non-Fickian transport and multiple-rate mass transfer in porous media. Water Resources Research, 44, W03402, doi:10.1029/2007WR005906.
  • M. Dentz, H. Scher, D. Holder and B. Berkowitz. (2008) Transport behavior of coupled continuous-time random walks. Physical Review E, 78, 041110, doi:10.1103/PhysRevE.78.041110.
  • S. Emmanuel and B. Berkowitz. (2007) Continuous time random walks and heat transfer in porous media. Transport in Porous Media, 67, 413-430, doi:10.1007/s11242-006-9033-z.
  • B. Berkowitz, A. Cortis, M. Dentz and H. Scher. (2006) Modeling non-Fickian transport in geological formations as a continuous time random walk. Reviews of Geophysics, 44, RG2003, doi:10.1029/2005RG000178.
  • G. Hornung, B. Berkowitz and N. Barkai. (2005) Morphogen gradient formation in a complex environment: An anomalous diffusion model. Physical Review E, 72, 041916, doi:10.1103/PhysRevE.72.041916.
    [Also featured in: Virtual Journal of Biological Physics Research, 1 November 2005]
  • A. Cortis and B. Berkowitz. (2005) Computing ''anomalous'' contaminant transport in porous media: The CTRW Matlab toolbox. Ground Water, 43(6), 947-950, doi:10.1111/j.1745-6584.2005.00045.x.
  • A. Cortis and B. Berkowitz. (2005) Anomalous transport in ''classical'' soil and sand columns. Soil Science Society of America Journal, 68, 1539-1548. (Erratum: 69, 285, 2005)
  • A. Cortis, C. Gallo, H. Scher and B. Berkowitz. (2004) Numerical simulation of non-Fickian transport in geological formations with multiple-scale heterogeneities. Water Resources Research, 40, W04209, doi:10.1029/2003WR002750.
  • A. Cortis, Y. Chen, H. Scher and B. Berkowitz.(2004) Quantitative characterization of pore-scale disorder effects on transport in "homogeneous" granular media. Physical Review E, 70, 041108, doi:10.1103/PhysRevE.70.041108.
  • G. Margolin and B. Berkowitz. (2004) Continuous time random walks revisited: First passage time and spatial distributions. Physica A, 334, 46-66.
  • M. Dentz, A. Cortis, H. Scher and B. Berkowitz. (2004) Time behavior of solute transport in heterogeneous media: Transition from anomalous to normal transport. Advances in Water Resources, 27(2), 155-173.
  • M. Dentz and B. Berkowitz. (2003) Transport behavior of a passive solute in continuous time random walks and multirate mass transfer. Water Resources Research, 39(5), 1111, doi:10.1029/2001WR001163.
  • M. Levy and B. Berkowitz. (2003) Measurement and analysis of non-Fickian dispersion in heterogeneous porous media. Journal of Contaminant Hydrology, 64(3-4), 203-226.
  • G. Margolin, M. Dentz and B. Berkowitz. (2003) Continuous time random walk and multirate mass transfer modeling of sorption. Chemical Physics, 295(1), 71-80.
  • G. Margolin and B. Berkowitz. (2002) Spatial behavior of anomalous transport. Physical Review E, 65, 031101, 1-11.
  • H. Scher, G. Margolin and B. Berkowitz. (2002) Towards a unified framework for anomalous transport in heterogeneous media. Chemical Physics, 284, 349-359.
  • H. Scher, G. Margolin, R. Metzler, J. Klafter and B. Berkowitz. (2002) The dynamical foundation of fractal stream chemistry: The origin of extremely long retention times. Geophysical Research Letters, 29(5), doi:10.1029/2001GL014123.
  • B. Berkowitz, J. Klafter, R. Metzler, and H. Scher. (2002) Physical pictures of transport in heterogeneous media: Advection-dispersion, random-walk and fractional derivative formulations. Water Resources Research, 38(10), 1191, doi:10.1029/2001WR001030.
  • B. Berkowitz and H. Scher. (2001) The role of probabilistic approaches to transport theory in heterogeneous media. Transport in Porous Media, 42(1-2), 241-263.
  • B. Berkowitz, G. Kosakowski, G. Margolin and H. Scher. (2001) Application of continuous time random walk theory to tracer test measurements in fractured and heterogeneous porous media. Ground Water, 39(4), 593-604.
  • G. Kosakowski, B. Berkowitz and H. Scher. (2001) Analysis of field observations of tracer transport in a fractured till. Journal of Contaminant Hydrology, 47(1), 29-51.
  • G. Margolin and B. Berkowitz. (2000) Application of continuous time random walks to transport in porous media. Journal of Physical Chemistry B, 104(16), 3942-3947. (Erratum: 104(36), 8762, 2000)
  • B. Berkowitz, H. Scher and S. E. Silliman. (2000) Anomalous transport in laboratory-scale, heterogeneous porous media. Water Resources Research, 36(1), 149-158. (Erratum: 36(5), 1371, 2000)
  • B. Berkowitz and H. Scher. (1998) Theory of anomalous chemical transport in fracture networks. Physical Review E, 57(5), 5858-5869.
  • B. Berkowitz and H. Scher. (1997) Anomalous transport in random fracture networks. Physical Review Letters, 79(20), 4038-4041.
  • B. Berkowitz and H. Scher. (1995) On characterization of anomalous dispersion in porous and fractured media. Water Resources Research, 31(6), 1461-1466.
  • H. Scher, M. F. Shlesinger, and J. T. Bendler. (1991) Time-scale invariance in transport and relaxation. Physics Today, January, 26-34.
  • H. Scher and E. W. Montroll. (1975) Anomalous transit-time dispersion in amorphous solids. Physical Review B, 12(6), 2455-2477.
  • H. Scher and M. Lax. (1973) Stochastic transport in a disordered solid, I. Theory, Physical Review B, 7(10), 4491-4502.
  • H. Scher and M. Lax. (1973) Stochastic transport in a disordered solid, II. Impurity conduction. Physical Review B, 7(10), 4502-4519.