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Current Research

I study the link between solute transport in porous media and equations with fractional derivatives. With support from the National Science Foundation "Focused Research Groups in Mathematics" program, a number of researchers, post-docs, and grad students are studying fractional-order PDEs and their application to heavy tails and transport processes. Visit our Fractional Calculus Project for a first look. My students and I are focusing on transport of contaminants in porous media. You can download these files that detail some recent work in this area, much of which is supported by grants from the U.S. Dept. of Energy, Basic Energy Sciences Program (grant # DE-FG03-98ER14885 and DE-FG02-07ER15841) and the National Science Foundation (grants EAR-9980489, DMS-0139943, DMS-0706440, DMS--0417869, DMS-0417972, DMS-0139927, DMS-0539176, EAR-0748953, and EAR-0749035).

SOFTWARE

Here you'll find fortran codes and Mathcad worksheets that calculate the solutions to various fractional PDEs. These include programs to fit your data with an optimal solution. Software for soil vapor extraction simulation is found elsewhere.

Papers: Hydrology

Simulation of chemical reaction via particle tracking: diffusion-limited versus thermodynamic rate-limited regimes, Benson, D.A. and M.M. Meerschaert (2008), in press (press for preprint), Water Resour. Res.
A simple and efficient random walk solution of multi-rate mobile/immobile mass transport equations (contact me for a pre-print), D.A. Benson, M.M Meerschaert, Y. Zhang, B. Baeumer, and R. Schumer, Water Resources Research, in review.
Lagrangian simulation of multidimensional anomalous transport at the MADE site, Y. Zhang and D. Benson (2008), Geophysical Research Letters, 35, L07403, doi:10.1029/2008GL033222.
Moment analysis for spatiotemporal fractional dispersion (pdf) Y. Zhang, D.A. Benson, and B. Baeumer, Water Resources Research, 44, W04424, doi:10.1029/2007WR006291.
Comparison of instantaneous and constant-rate stream tracer experiments through non-parametric analysis of residence time distributions, R. Payn, M. Gooseff, D.A. Benson, O. Cirpka, J. Zarnetske, W. Bowden, J. McNamara, and J. Bradford (2008), Water Resour. Res. 44, W06404, doi:10.1029/2007WR006274. (AGU link)
Transport of Conservative Solutes in Simulated Fracture Networks 1. Synthetic Data Generation AND 2. Ensemble Solute Transport and the Correspondence to Operator-Stable Limit Distributions, D.M. Reeves, D.A. Benson, M.M. Meerschaert, and H.-P. Scheffler, Water Resour. Res., in press.
Ensemble Solute Transport in 2-D Operator-Scaling Random Fields (pdf file), N. Monnig, D.A. Benson, and M.M. Meerschaert (2007), Water Resour. Res., 44, W02434, doi:10.1029/2007WR005998.
Predicting the tails of breakthrough curves in regional-scale alluvial systems (pdf), Y. Zhang, D.A. Benson, and B. Baeumer (2007), Ground Water 45(4), 473–484. doi:10.1111/j.1745-6584.2007.00320.x
Recurrence of extreme events with power-law interarrival times (pdf file), Benson, D. A., R. Schumer, and M. M. Meerschaert (2007), Geophys. Res. Lett., 34, L16404, doi:10.1029/2007GL030767.
Space-fractional advection-dispersion equations with variable parameters: Diverse formulas, numerical solutions, and application to the MADE-site data, Y. Zhang, D.A. Benson, M.M. Meerschaert, and E. M. LaBolle (2007), Water Resour. Res. 43, W05439, doi:10.1029/2006WR004912 (pdf file).
Role of Volatilization in Changing TBA and MTBE Concentrations at MTBE-Contaminated Sites (click for pdf), Eweis, J. B.; LaBolle, E. M.; Benson, D. A.; Fogg, G. E., Environ. Sci. Technol., 41(19), 6822-6827, DOI:10.1021/es071250j, 2007.
Relationship between flux and resident concentrations for anomalous dispersion, by Yong Zhang, D. Benson, and B. Baeumer, Geophys. Res. Lett., 33, L18407, doi:10.1029/2006GL027251, 2006 (pdf file).
Random walk approximation of fractional-order multiscaling anomalous diffusion, by Yong Zhang, D. Benson, M. Meerschaert, E.M. LaBolle, and H.-P. Scheffler, Phys. Rev. E 74, 026706, 2006 (1Mb pdf).
On using random walks to solve the space-fractional advection-dispersion equations, by Yong Zhang, D. Benson, M. Meerschaert, and H.-P. Scheffler, J. Statistical Physics, Apr 2006, pp. 1-22, DOI 10.1007/s10955-006-9042-x, (1.5 Mb pdf).
Ensemble solute transport in 2D operator-stable random fields by N. Monnig and D.A. Benson (2006), Proceedings, MODFLOW and More 2006: Managing Ground-Water Systems, May 21-24, 2006, E. Poeter, M. Hill, and C. Zhang, eds., IGWMC, pp. 684-688. (470 kb pdf).
Aquifer Operator-Scaling and the Effect on Solute Mixing and Dispersion (with Mark M. Meerschaert, Boris Baeumer, and Hans-Peter Scheffler) (2 Mb pdf) Water Resources Research 42, W01415, doi:10.1029/2004WR003755, 2006.
Radial Fractional-Order Dispersion Through Fractured Rock (with Charles Tadjeran, Mark M. Meerschaert, Irene Farnham, and Greg Pohll) (373 Kb pdf) Water Resources Research 40, W12416, doi:10.1029/2003WR003314, 2004.
Fractal Mobile/Immobile Transport (Rina Schumer, David A. Benson, Mark M. Meerschaert, and Boris Baeumer) (1057 Kb pdf) Water Resources Research 39(10)1296, doi:10.1029/2003WR02141, 2003.
Multiscaling fractional advection-dispersion equations and their solutions (Rina Schumer, David A. Benson, Mark M. Meerschaert, and Boris Baeumer) (460 Kb pdf) Water Resources Research 39(1), 1022, doi:10.1029/2001WR001229, 2003.
Hydraulic conductivity, velocity, and the order of the fractional dispersion derivative in a highly heterogeneous system (Matt G. Herrick, David A. Benson, Mark M. Meerschaert, Katherine R. McCall, and Scott L. Painter) (800 Kb pdf) Water Resources Research 38(11), 1227, doi:10.1029/2001WR000914, 2002.
Fractional dispersion, Lévy motion, and the MADE tracer tests (222 kB .pdf) (with Rina Schumer, Stephen W. Wheatcraft, and Mark M. Meerschaert), Transp. Por. Media 42(1/2), 211-240, 2001.
Eulerian derivation of the fractional advection-dispersion equation(155 kB .pdf) (with first author Rina Schumer, Mark M. Meerschaert, and Stephen W. Wheatcraft, J. Cont. Hyd. 48(1/2), 69, 2001.
The fractional-order governing equation of Lévy motion .pfd file (with S. Wheatcraft and M. Meerschaert), Water Resour. Res. 36(6), 1413-1423, 2000.
Application of a fractional advection-dispersion equation. .pfd file (with S. Wheatcraft and M. Meerschaert), Water Resour. Res. 36(6), 1403-1412, 2000.
Subordinated advection-dispersion equation for contaminant transport. .pdf (240K) (with B. Bäumer, and M. Meerschaert), Water Resour. Res. 37(6), 1543-1550, 2001.
1998 Ph.D. thesis: The fractional advection-dispersion equation: Development and application. .pdf (1,140 K). This is a direct copy with a some mistakes that have been corrected in the papers (above). An errata sheet is coming soon.

Physics of fractional PDEs:

Advection and Dispersion in Time and Space (pdf format) (Baeumer, Benson, and Meerschaert), Physica A, 350 (2-4): 245-262 May 15 2005.
Governing equations and solutions of anomalous random walk limits, Physical Review E 66(6), #060102(R), 2003, (60 Kb .pdf) (Mark M. Meerschaert, David A. Benson, Peter Becker-Kern and Hans-Peter Scheffler, Department of Mathematics, University of Dortmund, Germany)
Stochastic solution of space-time fractional diffusion equations, (pdf format) (with Mark M. Meerschaert, Hans-Peter Scheffler, and Boris Baeumer), Phys Rev. E. 65, 041103, 2002.
Operator Lévy motion and multiscaling anomalous diffusion..pdf (240K) (with M. Meerschaert and B. Bäumer), Phys. Rev. E. 63(2), 021112, 2001.
Multidimensional advection and fractional dispersion. .pdf (123K) (with M. Meerschaert and B. Bäumer), Phys. Rev. E, 59(5), 5026-5028, 1999.

Unsaturated flow and transport:

Simulating scale-dependent solute transport in soils with the fractional advective-dispersive equation, with 1st author Y. Pachepsky and W. Rawls, Soil Sci. Soc. Am. J. 64(4), 1234, 2000.
A model of water streaking down a wall (460 kB .pdf), Water Resour. Res. 37(2), 427, 2001.

David Benson

Colorado School of Mines
Hydrologic Sciences and Engineering Program
1500 Illinois St
Golden, CO 80401
(303) 273-3806