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Book: Scaling in Soil Physics: Principles and Applications
Published by: Soil Science Society of America



  1.  p. i-xxi
    SSSA Special Publication 25.
    Scaling in Soil Physics: Principles and Applications

    Daniel Hillel and David E. Elrick (ed.)

    ISBN: 978-0-89118-920-6

    unlockOPEN ACCESS

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Front Matter


The extreme variability of soils has made it very difficult to generalize our quantitative physical theories that are derived for ideal, homogenous systems. In hindsight, the concept of similitude seems both obvious and logical. Unfortunately, despite the current interest in geostatistical approaches to soil variability, the full power of the similarity concepts has yet to be realized in practice. The papers in this symposium should make this concept much more accessible to soil scientists, earth scientists, and engineers. Properly applied, similarity should bring the same insight and conceptual structure to soil science that dimensional analysis has brought to engineering and physics. The papers in this publication deserve careful study and analysis.

W.R. GARDNER, president

Soil Science Society of America




The theory of similitude and the attendant technique of scaling have long been used in applied physics to facilitate the analysis of varied problems. The principle is to formulate the relevant equations with the smallest possible number of variables, by clustering the variables and casting them into dimensionless form. The equations are thereby generalized and made applicable to any set of actual cases, provided the systems described are essentially similar (i.e., “scale models” of one another). The very exercise of scaling helps to reveal the fundamental relationships among operating variables.

It was only in 1955, however, that the concepts of scaling and similitude were introduced into soil physics by the brothers Ed and Bob Miller. In their twin papers, published that year in the Soil Science Society of America Proceedings, and in their subsequent paper published the following year in the Journal of Applied Physics, the Millers formulated the basic theory and defined the appropriate criteria for its application to surface tension-viscous flow phenomena in unsaturated porous media. Their seminal analysis provided new insights into the physical behavior of soil-water systems and has been applied ever since to the solution of many otherwise vexing problems, notably including the characterization of spatial variability, of hysteretic and unstable flow phenomena, and of mechanical stress distribution in unsaturated granular soils.

In a timely effort to acknowledge and highlight the landmark achievement of the Miller brothers, and to review subsequent developments in the area of similitude and scaling of soil systems, the Soil Physics Division of the Soil Science Society of America convened a special symposium as part of the Society's annual meetings in Las Vegas in October of 1989. This publication is a compendium of the invited papers presented at that symposium. As organizers of the symposium, we dedicate this volume to Ed and Bob Miller, with high regard for their inspired and inspiring contributions to soil physics. All who know the work of the Millers admire their outstanding professional qualities of originality, rigor, and insight. And all who have had the good fortune to associate with the Millers and learn from them directly cannot but feel an affection and a deep appreciation for their exemplary personal qualities of integrity, generosity, humility, friendliness, and—last but not least—perpetual good cheer.



University of Massachusetts

Amherst, Massachusetts


University of Guelph

Guelph, Ontario, Canada


R. J. Glass, Doctor, Sandia National Laboratory, Geoscience Analysis Division, Albuquerque, NM 87185

W. A. Jury, Professor of Soil Physics, Department of Soil and Environmental Sciences, University of California-Riverside, Riverside, CA 92521

Edward E. Miller, Professor Emeritus of Physics and Soil Science, Department of Physics, University of Wisconsin-Madison, Madison, WI 53706

R. D. Miller, Professor Emeritus of Soil Physics, Department of Soil, Crop, and Atmospheric Sciences, Cornell University, Ithaca, NY 14853

J.-Y. Parlange, Professor of Agricultural Engineering, Cornell University, Ithaca, NY 14853

P. A. C. Raats, Senior Research Scientist and Professor of Continuum Mechanics, Department of Mathematics, Wageningen Agricultural University; and Institute for Soil Fertility Research, 9750 RA Haren, the Netherlands

Victor A. Snyder, Associate Soil Scientist, Department of Agronomy and Soils, Agricultural Experiment Station, University of Puerto Rico, Rio Piedras, PR 00928

Garrison Sposito, Professor of Soil Physical Chemistry, Department of Soil Science, University of California-Berkeley, Berkeley, CA 94720

T. S. Steenhuis, Associate Professor, Department of Agricultural and Biological Engineering, Cornell University, Ithaca, NY 14853

Scott W. Tyler, Assistant Research Soil Scientist, Water Resources Center, Desert Research Institute, Reno, NV 89506

A. W. Warrick, Professor of Soil Physics, Soil and Water Science Department, University of Arizona, Tucson, AZ 85721

Stephen W. Wheatcraft, Associate Research Professor of Hydrogeology, Desert Research Institute, University of Nevada System, Reno, NV 89506

E. G. Youngs, Visiting Professor of Soil Physics, Silsoe College, Cranfield Institute of Technology, Silsoe, Bedford MK45 4DT, England


Given the nature of this symposium, and the passage of a full third of a century since my brother, Bob, and I submitted our three papers on scaling, I have permitted both personal and historical material to flavor this submission quite freely, even to the extent of detailing how a personal hobby helped open the road to development. Readers preferring the filtered language of today's scientific discourse are cordially invited to thumb forward to more conventional entries.

Lesson from a Hobby. In the early 1950s, bows for hunting were often made of alloy aluminum. It was my conceit that, being a physicist, I might design an aluminum bow somewhat superior to those commercially available. To facilitate design experimentation, I wanted to use quarter-scale models, using 1/16 inch sheet stock to model the 1/4 inch sheet stock that I planned to use for my actual bow. The question was, “Can full-scale performance be deduced from small-scale models, and if so, how?” Mulling this over, I happened to invent the basic idea of similitude analysis, which was entirely unknown to me at the time.

Those aspects of physics that involve materials are normally expressed as general differential equations, and these are then integrated or solved to compute the behavior of any given macroscopic system. It dawned on me that simply by doing my scale-modeling at the level of the differential equations, I could assure that the resulting scaling of any derived systems would also hold good, even though working out the solution, mathematically, might be far too horrendous to contemplate for the given system. Such scaling of systems would be exactly as applicable as the differential equations themselves.

For my bow-and-arrow scaling, I would employ the same alloy for the full and small scale models, prestressing the bow limbs through exactly the same cycles of strain. In a super nutshell (leaving details for the well-known student), the differential equations for this problem were (i) elastic—generically, stress/strain = modulus, (S/s = M), and (ii) Newtonian—generically, df = dm a. Applying these conditions to arbitrary differential elements dX,dY,dZ of corresponding position and size in the two models (their respective scales being referenced to some characteristic length, L, such as the draw distance), I obtained generic results of the form dS/dX = pd2x/dt2 (x = a positional coordinate, X = a material coordinate). Because dS and d(X/L) are the same for the two models, dS/d(X/L), L(d2x/dt2), and thus L2d2(x/L)/d(t)2, are also the same. Therefore, d2(x/L)/d(t/L)2 is the same for both systems, meaning that time is scaled in exactly the same way as space. Consequently, mass /L3, energy /L3, force /L2, velocity /L0, and acceleration /L−1 are the same for both the bow and its scale model.

With this result, I could now construct a miniature ballistic pendulum and use other comparisons for evaluating various designs accurately. (Incidentally, a miniature arrow flies just as fast as a full-sized arrow; when I shot one of those toy arrows straight up, it just plain vanished into thin air—never saw it again.) After three models, the bottom line was that my final bow performed very well and actually put venison on our table. If the steel handle hadn't been so darn heavy, I might still be carrying it.

Step-by-Step Development of Soil-Moisture Scaling. A year or two after I finished that bow, Bob and I got together at the Penn State meetings where he showed me his elegant infiltration data taken at Berkeley in the wee hours, using his touchy needle-and-thread tensiometer sensor. We discussed the big differences between his sets of data for different soils. Bob pointed out that the early stages of one of the coarse soils resembled the later-stage development of one of the fine soils. We wondered whether a major portion of these differences might be simply a reflection of the average pore size of each soil. Could the differences be usefully narrowed if the complex soil moisture behavior could somehow be scale-modeled?

Back in Madison, WI, I thought about this, and played around with dimensional analysis without much luck. Eventually I tried letting gravity be zero. With this simplification, I found that infiltration distances should simply scale as the square root of time. Discussing this by phone with Bob, I learned of the Kirkham and Feng paper, which pulled together the horizontal infiltration data from many different investigators and fitted them all reasonably well to the square root of time. Showing that this result agreed with theory for such linear systems as heat flow, they remarked that because soil moisture behavior was tremendously nonlinear, there must be something interesting here that we don't yet understand. This excited me, because, of course, the result that I had deduced from dimensional analysis did not depend upon linearity. Although Arnold Klute soon published a nonlinear solution and showed that Boltzmann had also done it long before in a different context, the hook of soil physics was now set into me solidly. I never recovered from that first spurt of excitement. (Obviously, my incredible luck at having Champ Tanner dropped in my lap when he was a graduate student was also a major factor.)

Wanting to go beyond gravity-free scaling, but getting no further with dimensional analysis, one day I happened to recall my adventures with the bow and arrow, and began wondering what would happen if I tried adapting the similitude analysis idea to soils. Maybe, with similitude, I could carry scaling into the real world of gravity systems.

Fitted in between lots of other work—considerable other research, teaching, and even rebuilding part of my house at home—the work moved ahead in jerks, requiring altogether about a year. Certainly it attracted more of my interest than anything else during that time. It also led to several happy excuses for visiting Bob (who was the real soil physicist, and who was by this time located in Ithaca).

I began with the unrealistic idea of perfectly similar media, just to see where it would lead. From the beginning, I used surface tension and viscous flow (STVF) to generate the basic differential equations for similitude, accepting the consequent limitations on generality imposed by this STVF assumption.

Surface tension, with its multiple, history-dependent options of interface shape led immediately to reduce potential as {pλ/σ} and to the time-scale invariance property of hysteresis. This was extended easily enough to the averaging of water content over large numbers of pores, yielding water content, θH{pλ/σ}, thereupon defined as a scaled hysteresis function.

We next considered the viscous flow boundary condition for the air-water interfaces—slip, nonslip, or something in between? A lot of study, literature searching, discussion, and argument went into that phase. (Incidentally, not all of our conclusions from that aspect appeared in the final paper, having been deleted in response to the editor's request for shortening.) Basically we accepted the no-slip assumption as being simplest, and also because it evidences itself in the rising-bubble paradox, the dynamics of soap bubbles and antibubbles, and in other experimental observations. Adopting this boundary condition, and then averaging the net viscous flow components over many pores yielded the reduced Darcy and Richards Equations—hysteretic and in scaled form. The reduced {ηK/λ2} was no surprise. Petroleum people always separate viscosity from their “permeability.”

These averaging processes wiped away the need for “exact” similitude. If you start with one medium and then imagine shaking up a medium exactly similar to it, then repacking it to the original bulk density, it is no longer exactly similar. Yet its macroscopic properties, the θ and K hysteresis functions, are exactly the same as before the shakeup.

The final stage—using these midscale differential equations for spring-boarding one more step to reach the scaling of behavior for whole systems (columns, profiles)—all came tumbling out in a heap; took about an hour. It was an exciting and interesting finish with some unexpected implications. The reduced form of the gravity force required that finer-pored media must scale into larger-sized systems. Counter-intuitive.

When you mention this scale inversion effect to most nonsoil physicists, you sow the seeds of skepticism, glazing their eyeballs. (However, this glazing is also seen when you mention water under negative absolute pressure. Most of them have forgotten about Glaser's Nobel Prize for the bubble chamber, and they have never heard of Lyman Briggs.).

I remember Bob saying to me at the end of all this, “Boy, if we've slipped up on this somewhere, our necks will be out a mile.” After considerable digging, we located some relevant preexisting data that supported our ideas. These went into the initial SSSA papers of 1955. Before long, Klute and Wilkinson published their spectacular confirming experiments involving both gravity and hysteresis aspects simultaneously. Then Dave Elrick and others at Wisconsin published further confirmations, and they also tested and confirmed our assumption that departures would begin to show up under conditions for which the STVF differential equations began to be inadequate for particular soils.

Insufficiency of Dimensionless Coordinates Alone. I have tried to outline how similitude analysis is based on scaling of underlying differential equations (along with their boundary conditions of course). As illustrated by our similar-medium scaling, similitude analysis is always at least as powerful as dimensional analysis and is usually more powerful. Take, for example, our micro and macro lengths, λ and L. Dimensional analysis provides no handle whatever for sorting out one of these from the other. We can take advantage of this example to show that dimensionless coordinates, alone, need not be equivalent to scaling. Just interchanging these two lengths everywhere they occur in our list of reduced coordinates yields complete garbage. What is the meaning of {pL/σ}, {ηK/L2}, or {λ,div}? It is as meaningful as a “purple smell” or a “bitter color.” Whenever I see a paper purporting to scale some system purely in terms of cooking up dimensionless variables with no indication of their source or meaning, I wince. I have seen some of these.

Fading and Revival. The scaling theory and its initial experimental confirmations were completed by the late 1950s. Then a decade-plus passed without much being heard about scaling, though a few people were quietly using it on occasion. Eventually, the famous crew of activists at Davis began struggling with the disconcerting degree of field heterogeneity that was turning up in what seemed to me the world's most ambitious field experiments in soil physics. Claus Reichardt looked into how well such heterogeneity might be described in terms of patches of similar media with the heterogeneity residing solely in the micro-length λ. Don Nielsen claims that in the discussion following the presentation of his thesis work at one of these annual meetings, Claus found himself answering penetrating questions from two fellows in the front row. It suddenly dawned on him that these two must be Miller and Miller. It shook him up; he had assumed they were dead. Well, Don tells great stories. I won't vouch for this one, but he still sticks to it.

Abruptly, a lot of such scaling-related activity was springing up everywhere, as heterogeneity became the hot subject of the time (it still is, of course). The irony was that now I was overhearing people voicing their assumption that tests of the validity of this separate hypothesis (i.e., that heterogeneity could be described as patches of similar media) was the same thing as testing the basic validity of the scaling itself. What was being tested, of course, was whether the processes of soil genesis tend to generate patches of similar media. The testing of scaling itself that would prove useful (and has been almost entirely neglected) is the mapping out, over many practical soils, of the degree of limitation on the usefulness of scaling that is imposed by the original STVF assumption. Within realms of behavior that are controlled by larger pores, the STVF assumption should be valid. But just where, how, and to what degree will the limitations imposed by the very small pores of clay fractions become important?

General Perspectives. For a number of years, a large fraction of the experiments in our Soil Physics Laboratory course have been devoted to mapping out, qualitatively, the strange behavior of soils—hysteresis, infiltration, redistribution, drainage, 3D behavior, and so on. For this we have used compact, quick experiments with cheap, coarse sands. I initiated this strategy purely on the basis of scaling, since macro-size varies as the reciprocal of micro-size, and the duration of a macro-process varies as the reciprocal cube of the micro-size. (The rest of our semester largely explores practical instrumentation for finer soils and for plants, and the one-shot measurements that can be made with them in the limitations of an afternoon lab.) It works out very efficiently and I recommend it.

Go back now to our initial 1950s idea, when we were first seeking a scaling method for soils. Recall that we hoped to cut down the range of variation of soil-moisture characteristics and of flow behavior by scaling away the major effect of texture. If results are plotted in reduced coordinates, infiltration and other classes of behavior look much the same for sands as for silts, provided, of course, that everything is scaled—fluxes, total infiltration, and so on. (For practical purposes, one would also like a chance to stare at the physical coordinates; no reason that both can't be shown on all plotted data.)

In using this approach, generally, we must deal with all kinds of soils, hence with dissimilar soils. Plotting such data with reduced coordinates shows up the degree of dissimilitude, and this can be instructive. The way in which such differences are exhibited depends on the selection of a definition of micro-length λ—whether in terms of saturated conductivity, satiated (or rewetted) conductivity, gravity-free sorptivity, air entry value, or whatever. If we compare two similar media in terms of the ratio of their microlengths, we can choose any one of these possible definitions and the ratio of micro-lengths will always come out the same. For dissimilar media this will not be the case, and here lies a problem that generally requires further attention. The choice of definition for λ will depend on what soil property is considered most central for a given purpose. I believe that we should be thinking about such choices more than we do.

One of my relevant unfulfilled interests has been the development of better parameterized models for hysteretic soil moisture characteristics. I suggest that these parameters should, if possible, be so designed that one of them serves the role of representing microlength, somehow defined. Then, on this model, all similar media would differ only by this one parameter.

Summary. In this space I have tried to give some feeling for a small bit of history, an idea of what is—and what isn't—similitude analysis … how it works … why it is powerful. I have outlined some of the ways in which it has been useful, and a few possible ways in which it may be useful in future.

As for me, I hope it has come through how interesting this has all been. And how much fun.


University of Wisconsin

Madison, Wisconsin



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