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Final Report: Superior Subsurface Characterization Using Fractal-Based Hydraulic Conductivity Distributions
EPA Grant Number: R826171Title: Superior Subsurface Characterization Using Fractal-Based Hydraulic Conductivity Distributions
Investigators: Molz, Fred J. , Lu, Silong
Institution: Clemson University
EPA Project Officer: Krishnan, Bala S.
Project Period: November 17, 1997 through November 16, 2000 (Extended to November 16, 2001)
Project Amount: $218,961
RFA: Exploratory Research - Environmental Engineering (1997)
Research Category: Engineering and Environmental Chemistry
Description:
Objective:The objectives of the research project were to: (1) identify fractal structure in existing hydraulic conductivity data sets; (2) develop and evaluate a general scheme for generating fractal property realizations conditioned on the data; and (3) incorporate the methodology into a series of computer programs for general purpose use. Summary/Accomplishments (Outputs/Outcomes):
During the period of this research, there was a great deal of change in our fundamental understanding involving fractal models of natural heterogeneity. First Gaussian-based models, then Levy-based models, and finally multifractal models were introduced. Initially, each new introduction seemed to offer distinct advantages, but upon further examination, inconsistencies emerged. For some time, it was not clear how one should proceed. During this time period, many different thoughts, observations, and concepts were presented at national meetings. There also was a lot of healthy controversy at the various meetings. Eight presentations with published abstracts were made between May 1998 and December 2000. The understanding that finally emerged is presented in the last three publications listed at the end of the executive summary, and will be outlined below. The first publication was in the form of a comment on a previously published paper by V. Di Federico and S.P. Neuman, wherein we attempted to clarify the meaning of a parameter that we considered vague.
When one is confused within a scientific problem, a good rule is to begin by carefully studying available data. Results of a detailed data study were reported in the publication by Lu and Molz (2001), entitled "How well are hydraulic conductivity variations approximated by additive stable processes?" This study was based on two data sets: intrinsic permeability (k) from an eolian sandstone, and hydraulic conductivity (K) from unconsolidated sands and clays deposited in a fluvial environment. Both data sets exhibited fractal-like scaling over a significant range of scales.
For the past 10 years, hydrologists and petroleum scientists have explored
the use of nonstationary stochastic processes with stationary increments (which
also may be viewed as stochastic fractals) as models for log hydraulic conductivity
distributions—the so-called scaling fractal models. In the ongoing effort
to arrive at the most practical and realistic model for K or log(K) increments,
we performed a careful analysis of the tail behavior of K and log(K) data sets.
Analysis of the higher statistical moments of both data sets led to the conclusion
that the increments of the data and the logs of the data are not governed in
general by Levy-stable or Gaussian distributions. The distribution tails appear
to display a Pareto-like power-law decay 
,
with 
values averaging 
3 for the data sets studied. Unlike the calculations of Liu and Molz (1997),
values were largely independent of lag size. These results suggest that the
Levy model does not fit the tail behavior of the data well even prior to the
need for truncation to keep the statistical moments of simulated K distributions
from becoming unrealistically large. However, the Levy model does fit the central
portion of the data quite well. Thus, careful data analysis led us to the conclusion
that K distributions from natural sediments do not exhibit purely Gaussian behavior
nor purely Levy-stable behavior.
A possible solution to the problem of Levy versus Gaussian fractal structure is to hypothesize that sedimentary deposits often are characterized by various distinct zones or facies that may change abruptly in both the vertical and horizontal directions, with facies structure relating to the depositional and post-depositional environments. A physical parameter such as hydraulic conductivity, K, varies within each facies, and mean values in one facies may be several orders of magnitude larger or smaller than those in another facies. In many situations, it is possible, perhaps even probable, that the statistical properties of K variations within a facies are very different from those between facies. In such a situation, it may not make sense to perform a single statistical analysis on permeability values taken from a mix of distinct facies. As an alternative, we employed the transition probability Markov approach with indicator Kriging (Carle and Fogg, 1996, 1997) to simulate large-scale facies distributions, in which a single mean K value was assigned to each facies. To further represent the natural heterogeneity of sedimentary deposits on an intra-facies scale, we used a monofractal Gaussian model, fractional Brownian motion (fBm), to represent the log(K) variations about the mean inside each facies, as suggested by the data analyses mentioned previously. Further study shows that the simulated log(K) distributions for the entire multifacies domain produce non-Gaussian log(K) increment distributions similar to those observed experimentally, which we called "Levy-like." It turns out that a superposition of several Gaussian distributions having the same mean of zero and different variances produces a distribution that is Levy-like in the region surrounding the mean, but non-Levy in the tail regions. This is what we saw in the careful data analysis discussed above. Further study is needed to determine how general this result will be. However, many geologists feel intuitively that the model makes sense.
The facies/fractal idea was first presented at the 2000 Fall Annual Meeting of the American Geophysical Union, and a detailed manuscript, entitled "Combining stochastic facies and fractal models for representing natural heterogeneity," was submitted to the Hydrogeology Journal. We believe that the fractal/facies research went beyond the scope of our original proposal, but it was necessary to motivate selection of the appropriate fractal generation procedures for programming. Ultimately, we selected both Gaussian and Levy fractal generation procedures based on a three-dimensional successive random additions (SRA) approach. This resulted in a manuscript, entitled "An efficient, three-dimensional, anisotropic, fractional Brownian motion and truncated fractional Levy motion simulation algorithm based on successive random additions," which has been submitted to the journal Computers and Geosciences. Along the way, we found an improved methodology for detecting fractal scaling in data sets called "dispersion analysis." Therefore, a FORTRAN code for dispersion analysis is included along with the SRA code. Originally, we submitted our SRA manuscript to Mathematical Geology, where it was accepted tentatively for publication. However, the editor told us that Mathematical Geology does not emphasize computer work, but that Computers and Geosciences does. In fact, that journal maintains a Web site where computer codes submitted with manuscripts accepted for publication are stored and made available for download by any interested reader, which seems ideal for our purposes. Therefore, we elected to withdraw from Mathematical Geology and submit to Computers and Geosciences.
Partly as a result of the EPA STAR research reported herein, Dr. Molz has been invited to author a review article for the prestigious journal Reviews of Geophysics, which is published quarterly by the American Geophysical Union. This article is tentatively titled "Fractal-based concepts in subsurface hydrology: origins, applications, limitations, and future research questions." It is scheduled to be submitted during January 2002.
Journal Articles on this Report: 4 Displayed | Download in RIS Format
| Other project views: | All 9 publications | 4 publications in selected types | All 4 journal articles |
| Type | Citation | ||
|---|---|---|---|
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Liu HH, Molz FJ. Comment on . Water Resources Research. 1998;34(11):3207-3208. |
R826171 (1998) R826171 (Final) R826765 (Final) |
not available |
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|
Lu S, Molz FJ. An efficient, three-dimensional, anisotropic, Fractional Brownian motion and truncated fractional Levy motion simulation method. Journal of Mathematical Geology 2000. |
R826171 (2000) R826171 (Final) |
not available |
|
|
Lu S, Molz FJ, Boufadel MC. Numerical studies of flow and solute transport in three-dimensional, anisotropic, fractal porous media. Journal of Contaminant Transport 2000. |
R826171 (2000) R826171 (Final) |
not available |
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Lu S, Molz FJ. How well are hydraulic conductivity variations approximated by additive stable processes? Advances in Environmental Research 2000, Volume 5, Issue 1, February 2001, Pages 39-45. |
R826171 (2000) R826171 (Final) |
not available |
porous media, transport, characterization. , Ecosystem Protection/Environmental Exposure & Risk, Scientific Discipline, Waste, Environmental Engineering, Fate & Transport, Environmental Chemistry, risk assessment, contaminant transport models, natural attenuation, subsurface characterization, computer science, porous media, groundwater, fractal properties
Progress and Final Reports:
1998 Progress Report
2000 Progress Report
Original Abstract