The scattering analog for infiltration in porous media

This review takes the form of a set of Chinese boxes. The outermost box gives a brief general account of modem developments in the mathematical physics of unsaturated flow in soils and porous media. This provides the necessary foundations for the second box, which describes the quasi-linear analysis of steady multidimensional unsaturated flow, which is an essential prerequisite to the analog. Only then can we proceed to the innermost box, devoted to our major theme. An exact analog exists between steady quasi-linear flow in unsaturated soils and porous media and the scattering of plane pulses, and the analog carries over to the scattering of plane harmonic waves. Numerous established results, and powerful techniques such as Watson transforms, far-field scattering functions, and optical theorems, become available for the solution and understanding of problems of multidimensional infiltration. These are needed, in particular, to provide the asymptotics of the physically interesting and practically important limit of flows strongly dominated by gravity, with capillary effects weak but nonzero. This is the limit of large s, where s is a characteristic length of the water supply surface normalized with respect to the sorptive length of the soil. These problems are singular in the sense that ignoring capillarity gives a totally incorrect picture of the wetted region. In terms of the optical analog, neglecting capillarity is equivalent to using geometrical optics, with coherent shadows projected to infinity. When exact solutions involve exotic functions, difficulties of both analysis and series summation may be avoided through use of small-s and large-s expansions provided by the analog. Numerous examples are given of solutions obtained through the analog. The scope for extending the application to flows from surface sources, to anisotropic and heterogeneous media, to unsteady flows, and to linear convection-diffusion processes in general is described briefly.

[1]  A. Sommerfeld.  Partial Differential Equations in Physics , 1949 .

[2]  M. Reiner.  Ueber die Strömung einer elastischen Flüssigkeit durch eine Kapillare , 1926 .

[3]  A. Pullan,et al.  Two‐ and three‐dimensional steady quasi‐linear infiltration from buried and surface cavities using boundary element techniques , 1987 .

[4]  P. Polubarinova-Kochina,et al.  Theory of ground water movement , 1962 .

[5]  P. Raats.  Steady Infiltration from Line Sources and Furrows , 1970 .

[6]  Ross E. Moore,et al.  Water conduction from shallow water tables , 1939 .

[7]  J. Burke.  Long‐Wavelength Scattering by Hard Spheroids , 1966 .

[8]  Parry Moon,et al.  Field Theory Handbook , 1961 .

[9]  T. Senior,et al.  Electromagnetic and Acoustic Scattering by Simple Shapes , 1969 .

[10]  J. Philip.  Linearized unsteady multidimensional infiltration , 1986 .

[11]  Richard James Duffin,et al.  Yukawan potential theory , 1971 .

[12]  P. Raats,et al.  Steady Infiltration from Point Sources, Cavities, and Basins 1 , 1971 .

[13]  O. M. R. S. L. R. Pres..  XXX. On the light dispersed from fine lines ruled upon reflecting surfaces or transmitted by very narrow slits , 1907 .

[14]  M. J. Sully,et al.  Macroscopic and microscopic capillary length and time scales from field infiltration , 1987 .

[15]  G. Kirchhoff,et al.  Vorlesungen über die Theorie der Wärme , 1894 .

[16]  B. Friedman.  Principles and Techniques of Applied Mathematics , 1956 .

[17]  D. Jones.  High-frequency scattering of electromagnetic waves , 1957, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[18]  Herman Bouwer,et al.  Unsaturated flow in ground-water hydraulics , 1964 .

[19]  J. Philip.  THE THEORY OF INFILTRATION: 1. THE INFILTRATION EQUATION AND ITS SOLUTION , 1957 .

[20]  H. Hasimoto.  Steady longitudinal motion of a cylinder in a conducting fluid , 1960, Journal of Fluid Mechanics.

[21]  Herman Bouwer,et al.  Rapid field measurement of air entry value and hydraulic conductivity of soil as significant parameters in flow system analysis , 1966 .

[22]  G. Watson.  The Transmission of Electric Waves Round the Earth , 1919 .

[23]  J. Philip,et al.  APPROXIMATE ANALYSIS OF THE BOREHOLE PERMEAMETER IN UNSATURED SOIL , 1985 .

[24]  J. Philip.  Steady Infiltration From Buried Discs and Other Sources , 1986 .

[25]  L. A. Richards.  Capillary conduction of liquids through porous mediums , 1931 .

[26]  I. White,et al.  Measurement of Soil Physical Properties in the Field , 1988 .

[27]  J. Philip.  Steady infiltration from spheroidal cavities in isotropic and anisotropic soils , 1986 .

[28]  A. Klute,et al.  A NUMERICAL METHOD FOR SOLVING THE FLOW EQUATION FOR WATER IN UNSATURATED MATERIALS , 1952 .

[29]  R. Wooding,et al.  Steady Infiltration from a Shallow Circular Pond , 1968 .

[30]  Use and Hydrological Robustness of Time-to-Incipient-Ponding , 1989 .

[31]  J. Philip,et al.  Steady Two- and Three-Dimensional Flows in Unsaturated Soil: The Scattering Analog , 1985 .

[32]  Rayleigh Lord,et al.  V. On the incidence of aerial and electric waves upon small obstacles in the form of ellipsoids or elliptic cylinders, and on the passage of electric waves through a circular aperture in a conducting screen , 1897 .

[33]  Joseph B. Keller,et al.  Rays, waves and asymptotics , 1978 .

[34]  R. Hurd.  A note on the diffraction of a scalar wave by a small circular aperture , 1961 .

[35]  J. Philip.  Steady Infiltration from Circular Cylindrical Cavities , 1984 .

[36]  D. S. Jones.  CIX. On the scattering gross section of an obstacle , 1955 .

[37]  R. T. Waechter.  Steady longitudinal motion of an insulating cylinder in a conducting fluid , 1968, Mathematical Proceedings of the Cambridge Philosophical Society.

[38]  D. Jones.  Approximate methods in high-frequency scattering , 1957, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[39]  J. Philip,et al.  Steady Infiltration From Buried Point Sources and Spherical Cavities , 1968 .

[40]  J. R. Philip,et al.  Reply To “Comments on Steady Infiltration from Spherical Cavities , 1985 .

[41]  I. White,et al.  Designs for disc permeameters , 1988 .

[42]  H. C. van de Hulst,et al.  On the attenuation of plane waves by obstacles of arbitrary size and form , 1949 .

[43]  John Knight,et al.  The seepage exclusion problem for spherical cavities , 1989 .

[44]  J. R. Philip.  RECENT PROGRESS IN THE SOLUTION OF NONLINEAR DIFFUSION EQUATIONS , 1974 .

[45]  J. R. Philip.  Flow in porous media , 1970 .

[46]  M. Reiner.  Ueber die Strömung einer elastischen Flüssigkeit durch eine Kapillare , 1926 .

[47]  J. Philip,et al.  THE THEORY OF INFILTRATION: 2. THE PROFILE OF INFINITY , 1957 .

[48]  W. R. Gardner.  SOME STEADY‐STATE SOLUTIONS OF THE UNSATURATED MOISTURE FLOW EQUATION WITH APPLICATION TO EVAPORATION FROM A WATER TABLE , 1958 .

[49]  E. Buckingham.  On Physically Similar Systems; Illustrations of the Use of Dimensional Equations , 1914 .

[50]  S. P. Neuman,et al.  Vadose Zone Permeability Tests: Summary , 1982 .

[51]  Unsaturated seepage and subterranean holes: Conspectus, and exclusion problem for circular cylindrical cavities , 1989 .

[52]  Garrison Sposito,et al.  The “physics” of soil water physics , 1986 .

[53]  J. R. Philip.  AN ANALOG FOR INFILTRATION AND UNSATURATED SEEPAGE , 1987 .

[54]  J. Philip,et al.  Multidimensional steady infiltration to a water table , 1989 .

[55]  J. Philip.  Quasianalytic and Analytic Approaches to Unsaturated Flow , 1988 .

[56]  J. Philip.  General Theorem on Steady Infiltration from Surface Sources, With Application to Point and Line Sources , 1971 .

[57]  E. Feenberg.  The Scattering of Slow Electrons by Neutral Atoms , 1932 .

[58]  R J Duffin,et al.  Hilbert transforms in yukawan potential theory. , 1972, Proceedings of the National Academy of Sciences of the United States of America.

[59]  W. H. Gardner.  Early Soil Physics into the Mid-20th Century , 1986 .

[60]  T. Talsma,et al.  Hydraulic conductivity measurement of forest catchments , 1980 .

[61]  J. Philip.  Scattering Functions and Infiltration , 1985 .

[62]  J. Philip,et al.  Theory of Infiltration , 1969 .

[63]  J. Philip.  Nonuniform Leaching from Nonuniform Steady Infiltration1 , 1984 .

[64]  E. C. Childs,et al.  The permeability of porous materials , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[65]  Emil Wolf,et al.  Principles of Optics: Contents , 1999 .

[66]  W. R. Gardner,et al.  Comparison of Empirical Relationships between Pressure Head and Hydraulic Conductivity and Some Observations on Radially Symmetric Flow , 1971 .

[67]  T. Wu.  High-Frequency Scattering , 1956 .