Volume 3, Number 2

In This Issue

Reno Office

Holiday Wishes

Welcome Aboard

Is Darcy Dead?

Fractional Calculus

Happy Holidays!
Hazlett-Kincaid would sincerely like to thank all of our clients for choosing our services in 2002. We have enjoyed working with each and every one of you and look forward to strengthening existing relationships and forging new ones. Because of you, our 2003 outlook is bright and we hope that yours is as well.  

Please continue to keep us in mind for all of your specialized geological modeling needs.

Happy Holidays and Best Wishes for a Successful New Year!!!!  §

Hazlett-Kincaid Moves West

Hazlett-Kincaid is proud to announce the opening of a new office in Reno, Nevada. We now have three locations across the U.S.; Florida (HK South), Pennsylvania (HK East), and Nevada (HK West). The new Reno office will enable us to more easily pursue modeling work on west coast projects as it is centrally located between Los Angeles and Seattle and only three hours east of San Francisco.

HK West, led by Todd Kincaid, will be taking the lead on our solids and parameter modeling and data visualization services, using our newly acquired license of EarthVision.  HK South, led by Tim Hazlett, will continue as our lead office for groundwater flow and fate and transport modeling services, as well as our corporate headquarters. HK East, led by Michael Voorhees, remains in place to support our on-going projects in the northeast and will take the lead on model optimization and NAPL volume estimation services.  Though we've spread out in terms of location, we are leveraging VPN (Virtual Private Networking) technologies to remain cohesive as a project team and we remain dedicated to our Dual Modeling Approach; providing cutting-edge technological solutions to environmental, water resource, and geotechnical problems.  §

Since this is our first news letter in several months, Hazlett-Kincaid, Inc. would like to formally welcome Mr. Kevin Day on board as a new (relatively) member of our modeling staff. Mr. Day is a graduate of Colgate University and the University of Wyoming. He is a hydrogeologist with more than 5 years experience with groundwater modeling, geophysical site characterization, and contaminant remediation projects, as well as more than a year of experience with UNIX systems administration. He is already making a significant contribution to our team through his work in all aspects of our business; including solids and parameter modeling, flow and transport modeling, and computer systems administration.

Hazlett-Kincaid Welcomes Dr. Robert Cook 

Hazlett-Kincaid would also like to formally welcome aboard Dr. Robert Cook, who has joined us to strengthen our effort to become a significant force in water resources modeling and aquifer protection.  Dr. Cook is a graduate of Kutztown University and Bryn Mawr College with more than 13 years of experience in groundwater resources characterization and modeling.  He is a professor of Environmental Resources at Keystone College in Pennsylvania and has been actively involved with groundwater resource modeling and aquifer characterization in northeastern PA.  With Hazlett-Kincaid, Dr. Cook will be providing oversight and direction for our Pennridge Aquifer Protection project in Bucks County, PA and will serve as the liaison between HK and various Pennridge authorities.  §

Please feel free to distribute this newsletter to your friends and colleagues!

Is Darcy Dead?
Yes of course. Henry Darcy died in 1858. However, Darcy's Law, the empirical porous media flow relationship underlying modern hydrogeology is alive and well. We see it in its various forms throughout many industries, including petroleum, water resources, environmental, materials science, and many others.

Over the years since Darcy's work, though Darcy's Law has provided the foundation for almost all groundwater flow models (i.e. - MODFLOW), its limitations have become more and more apparent. One fundamental assumption when using these models is that one can define an REV (Representative Elementary Volume) for the system being studied. This means, in practical terms, that the material properties of the aquifer do not change within the largest element or cell in your model. The volume-averaging effect of the REV assumption works nicely in predictable materials, such as those man made or engineered. However, in the world of geology, the underlying complexity and inherent heterogeneity of soil and rock make the definition of an REV often precarious.

When using models to make predictions of heads or even flow rates, the standard Darcian approach can be quite useful and suited to the purpose. However, when it comes to using those velocities to make predictions of the fate and transport of contaminants, the approach is often woefully inept. Volume-averaged models can't account well for pore-scale processes, such as dead-end pores, capillary effects, etc. All of the effects for which the Darcian approach cannot account compound errors in the system, producing results that never exactly match field observations. In fact, it is often observed that the flow of contaminants through porous media does not occur evenly. Contaminants tend to follow sand stringers, fractures, and other paths of least resistance: these features are not in most models made today.

Alternatives to the standard Darcian approach are on the rise. Popular among these are the models that attempt, in one form or another, to include the effects of heterogeneities, either explicitly or implicitly. Examples of the former include algorithmic models, where one might attempt to numerically construct a porous media from a physically-based mechanistic point of view.  An example of the latter, talked about more in depth in the next article, is the fractional advection-dispersion equation approach, where the intricacies of heterogeneity and its effect on solute transport are accounted for by a dispersion tensor with a fractional power. Each of these approaches has merit and can, under the right circumstances, produce strikingly good results, matching much more closely the observed phenomena than the old approach.

Until such methods have been fully developed and can be implemented for real-world problems, we must find ways to honor subsurface complexities as closely as possible. At HKI, we capitalize on as much geologic data as is available to construct 3-D models of soil and rock heterogeneities that ultimately serve as the framework for our finite element flow and transport models, resulting in more reliable model predictions.

No, Darcy is not dead. But, he and his law are on the ropes. I believe that within the next decade or so, breakthroughs in computing technology and the advancement of techniques such as those mentioned above will precipitate fundamental changes in the practice of computational hydrogeology, very much removed from Darcy's Law. As always, Hazlett-Kincaid has identified these emergent techniques and is already hard at work, adding them to our skill set and leveraging them for our clients. §

Fractional Calculus and Applications to Groundwater Flow
Fractional calculus is not a new idea, with roots tracing back to at least 1695.  The basic idea behind it is that one can write differentials or integrations, two of the most common operations performed in calculus, with fractional derivatives or integrals. Fractional derivatives can be written in terms of the gamma function (Γ): where the D indicates derivative of fractional power µ and t^λ is a function. So, when writing a finite difference equation approximation to a partial differential equation, one can write it in terms of a fractional derivative. I've plotted some fractional and integer derivatives of f(x) = x². You can view them here. What these plots illustrate is that fractional derivatives form a continuum between integer derivatives; in this case 0 and 1. Integer order derivatives act locally (remember the slope of a line at a point thing...) on a function.  Fractional derivatives have some very desirable properties, such as the fact that they do not act locally. By this, I mean that the nature of the entire function is "felt" by the fractional derivative (Blank, 1996).

You are probably asking yourself by now just exactly how or why this line of discussion is at all relevant. It is relevant because fractional derivatives can be used to write a fractional advection-dispersion equation. The advection-dispersion equation is the mathematical basis for most contaminant transport models. A fractional version of the advection-dispersion equation has some very desirable properties, due to the way in which fractional derivatives "feel" the entire function on which they're operating. In practical terms, the fractional advection-dispersion equation has both forward and backward memory such that the a fractional derivative incorporates global effects into what would ordinarily be a local derivative.

This takes us back to the discussion in the preceding article about the difficulty in defining an REV, or control volume, in heterogeneous porous media. An REV is easily defined for a perfect (isotropic, homogeneous) porous media as the entire volume. However, when your aquifer is heterogeneous, one must consider non-local effects as well and must scale parameters such as the dispersion coefficient accordingly......or do we? Eureka! (go back to the beginning and read this article again if you're not saying Eureka with me!). Recalling that fractional derivatives act non-locally, it turns out that we can write a fractional derivative for the dispersive transport term in the advection dispersion equation. For 1-D dispersion it looks like this:

In terms of time dependent probability, try to think about measuring contaminant mass at a well. The figure to the right shows in blue, the probability of a particle (contaminant) arriving at a given point at a given time (Sokolov, et. al., 2002.). For nicely behaved systems, we get a normal distribution. The rust colored curve illustrates an entirely different behavior characteristic of a fractional-ordered system. As a generalization of this figure, anyone familiar with breakthrough curves should recognize that the rust-colored curve looks a lot more like what is observed in the field than the normally-distributed data. 

In summary, the fractional advection-dispersion equation encapsulates aquifer heterogeneities into a single term, without the need for scaling the dispersion coefficient. This results in a more realistic model for advection-dispersion problems and holds a lot of promise for future advances in modeling tools. §

References & Further Reading:

Blank, L., 1996. Numerical treatment of differential equations of fractional order, Manchester Centre for Computational Mathematics Numerical Analysis Report No. 287.

Love, E. R. "Fractional Derivatives of Imaginary Order." J. London Math. Soc. 3, 241-259, 1971.

Miller, K. S. "Derivatives of Noninteger Order." Math. Mag. 68, 183-192, 1995.

Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, 1993.

Sokolov, et. al., "Fractional Kinetics", Physics Today, November 2002.

Spanier, J. and Oldham, K. B. The Fractional Calculus: Integrations and
Differentiations of Arbitrary Order. New York: Academic Press, 1974.