High School Geology

ABSTRACT OF DISSERTATION


Michael C. Sukop






The Graduate School
University of Kentucky
2001POROSITY, PERCOLATION THRESHOLDS, AND WATER RETENTION
BEHAVIOR OF RANDOM FRACTAL POROUS MEDIA


____________________________________________
ABSTRACT OF DISSERTATION

A dissertation submitted in partial fulfillment of the
Requirements for the Degree of Doctor of Philosophy
at the University of Kentucky


By
Michael C. Sukop
Co-Directors: Dr. Edmund Perfect, Assistant Professor of Agronomy
and Dr. John Grove, Associate Professor of Agronomy
Lexington, Kentucky
2001ABSTRACT OF DISSERTATION


POROSITY, PERCOLATION THRESHOLDS, AND WATER RETENTION
BEHAVIOR OF RANDOM FRACTAL POROUS MEDIA

Fractals are a relatively recent development in mathematics that show promise as a
foundation for models of complex systems like natural porous media. One important
issue that has not been thoroughly explored is the affect of different algorithms
commonly used to generate random fractal porous media on their properties and
processes within them. The heterogeneous method can lead to large, uncontrolled
variations in porosity. It is proposed that use of the homogeneous algorithm might lead to
more reproducible applications. Computer codes that will make it easier for researchers
to experiment with fractal models are provided.

In Chapter 2, the application of percolation theory and fractal modeling to porous media
are combined to investigate percolation in prefractal porous media. Percolation thresholds
are estimated for the pore space of homogeneous random 2-dimensional prefractals as a
function of the fractal scale invariance ratio b and iteration level i. Percolation in
prefractals occurs through large pores connected by small pores. The thresholds increased
beyond the 0.5927… porosity expected in Bernoulli (uncorrelated) networks. The
thresholds increase with both b (a finite size effect) and i. The results allow the prediction
of the onset of percolation in models of prefractal porous media. Only a limited range of
parameters has been explored, but extrapolations allow the critical fractal dimension to be
estimated for many b and i values. Extrapolation to infinite iterations suggests there may
be a critical fractal dimension of the solid at which the pore space percolates. The
extrapolated value is close to 1.89 -- the well-known fractal dimension of percolation
clusters in 2-dimensional Bernoulli networks.

The results of Chapters 1 and 2 are synthesized in an application to soil water retention in
Chapter 3.


Keywords: Fractals, Porous Media, Percolation, Soil Water Retention
Michael C. Sukop
March 21, 2001

POROSITY, PERCOLATION THRESHOLDS,
AND WATER RETENTION BEHAVIOR
OF RANDOM FRACTAL POROUS MEDIA

By
Michael C. Sukop











Dr. Edmund Perfect
Co-Director of Dissertation

Dr. John Grove
Co-Director of Dissertation

Dr. John Grove
Director of Graduate Studies

March 21, 2001

RULES FOR THE USE OF DISSERTATIONS


Unpublished dissertations submitted for the Doctor's degree and deposited in the
University of Kentucky Library are as a rule open for inspection, but are to be used only
with due regard to the rights of the authors. Bibliographical references may be noted, but
quotations or summaries of parts may be published only with the permission of the
author, and with the usual scholarly acknowledgements.


Extensive copying or publication of the dissertation in whole or part also requires the
consent of the Dean of the Graduate School of the University of Kentucky.



Name Date

______________________________________________________________________
______________________________________________________________________
DISSERTATION





Michael C. Sukop



The Graduate School
University of Kentucky
2001
POROSITY, PERCOLATION THRESHOLDS,
AND WATER RETENTION BEHAVIOR
OF RANDOM FRACTAL POROUS MEDIA




____________________________________________

DISSERTATION

A dissertation submitted in partial fulfillment of the
Requirements for the Degree of Doctor of Philosophy
at the University of Kentucky


By

Michael C. Sukop

Lexington, Kentucky

Co-Directors: Dr. Edmund Perfect, Assistant Professor of Agronomy
and Dr. John Grove, Associate Profegy


Lexington, Kentucky

2001

Copyright by

Michael C. Sukop

2001