An examination of scale-dependent electricalresistivity measurements in Oracle granite.

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Authors Jones, Jay Walter, IV.

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An examination of scale-dependent electrical resistivity measurements in Oracle granite

Jones, Jay Walter, IV, Ph.D.

The University of Arizona, 1989

Copyright ®1989 by Jones, Jay Walter, IV. All rights reserved.

U·M·I 300 N. Zeeb Rd. Ann Arbor, MI 48106

AN EXAMINATION OF SCALE-DEPENDENT ELECTRICAL

RESISTIVITY MEASUREMENTS IN ORACLE GRANITE

by

Jay Walter Jones IV

Copyright ~ Jay Walter Jones IV 1989

A Dissertation Submitted to the Faculty of the

DEPARTMENT OF MINING AND GEOLOGICAL ENGINEERING

In Partial Fulfillment of the Requirements For the Degree of

DOCTOR OF PHILOSOPHY WITH A MAJOR IN GEOLOGICAL ENGINEERING

In the Graduate College

THE UNIVERSITY OF ARIZONA

1 9 8 9

THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE

2

As members of the Final Examination Committee, we certify that we have read

the dissertation prepared by ___ J __ ay~W __ l_a_t_e_r_J __ on __ e_s __________________ . ________ __

entitled An Examination of Scale-Dependent Electrical Resistivity

Measurements in Oracle Granite

and recommend that it be accepted as fulfilling the dissertation requirement

for the Degree of Ph.D. in Geological Engineering

Date

//~b1 Date

Date 1\ /3 , 1?9

//- 3,- cf / Date

11/<5/3 ( Date

Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College.

I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement.

~~ D ss rtat10n D ec or Da

STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this dissertation are allow­able without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manu­script in whole or in part may be granted by the copyright holder.

SIGNED:~E-

3

ACKNOWLEDGEMENTS

This research project was carried out without the benefit of any sort of explicit funding. I used up a lot of favors and appreciate the assistance and support that I received over the past few years. I am thankful for the support of the LASI lab which provided e~~ipment essential to the field work, data analysis, and report preparation. I also appreciate that the Hydrology Department allowed me the use of the Oracle test site. Don Pool of the USGS was instrumental in the early stages of this work by helping me to obtain the use of the USGS logging truck and by providing his time and expertise. Lastly, I would like to thank the SCS-40 minisupercomputer committee for providing me time on their machine and the CCIT for their continued support.

Throughout this process I was fortunate to have the sup­port of my advisor, Dr. Karl Glass. May you always find warm tropical waters to visit. I would also like to thank my other committee members, Dr. Ben Sternberg, Dr. Ian Farmer, Dr. Shlomo Neuman, and Dr. Harold Bentley, who were generally available for help, a kind word, or review of this effort. In addition, I am indebted to the flexi­bility afforded me in my job by the folks at CH2M-Hill.

This whole effort was made made much more pleasant by the support of my friends and family. To all a hearty thanks, and an invite to come sleep on my couch anytime.

4

5

TABLE OF CONTENTS

LIST OF ILLUSTRATIONS. •

LIST OF TABLES •

PAGE

7 . . . . . . . . . . . . . . . . . . . . . • • • 10

ABSTRACT . . .. . . . . . . . . . • 11

CHAPTER

1. INTRODUCTION . . . . . . . · . . . . . . . • 13

2. THE ORACLE TEST SITE • · . . . . . . . . • • 25

· . . • • 32 3. FIELD MEASUREMENTS AND DATA ANALYSIS •

Field Method • • • • • • 32

Data Reduction • . . . · . . . · . • • • 35

The Field Data • • • • 39

Anisotropy Analysis. • . . . . . . . • • 40

Discussion of Field Results. •

Summary. • • • • • • • •

4. DISORDERED (RANDOM) NETWORKS AND SCALING BEHAVIOR • • • •

Introduction • • • • • • •

· . . . • 52

• • • 58

• • 63

• • • 63

A Simple Random Network Model.

The Percolation Network Model.

· . · . . • • 67

· . • • • 77

Properties of Percolation Clusters (Lattice Animals) ••••••••• · . . General Percolative Model for Electrical

84

Resistivity. • • • • • • . . . . . . . • • • 94

Further Potential of the Fractal Model (Multi-fractal Scaling) •••••• 0 •• 98

Summary. • • • • • • • • . . . . • .102

6

TABLE OF CONTENTS--Continued

CHAPTER PAGE

5. THE ALPHA CENTER MODEL OF A NON-HOMOGENEOUS, ISOTROPIC MEDIUM •••••••••••••••• 104

The Alpha Center Technique • • .107

The Linear Model . . . . . . . • .110

The Exponential Model. . . . • .116

Conductivity Field Characteristics • • .119

Limiting Cases • • · . . . . . . . . . . . .124

Modeling Stategy and Parameterization •••• 125

4x4x7 Nodal Model Sensitivity Tests.

Forward Modeling Results • • • •

. . . .128

.136

The Exponential vs. Linear Model Response •• 146

A Note Regarding Parameter Estimation. .150

Summary. • •• · . . . . . . . . . . . .151

6. CONCLUDING REMARKS · . . . . . • • • • • • • 153

APPENDIX A: PLOTS OF CONTINUOUS LOGGING RESULTS, BOREHOLES M1, H2, H3, H5, AND H7 •••• 160

APPENDIX B: PLOTS AND TABULATED DATA FOR LARGE AM SINGLE-HOLE DATA, BOREHOLES Ml, H2, H3,

APPENDIX C:

APPENDIX D:

H5, H6, H7, AND H8 •••••• • •• 166

TABULATIONS OF CROSS-HOLE DATA.

FRACTAL SCALING • • • • • . . . · . . • .178

.192 · . . . APPENDIX E: EQUIVALENT CIRCUIT MODEL FOR ALPHA

CENTERS • • • • • • • • • • • • · . ••• 210

APPENDIX F: TESTS OF PARAMETER ESTIMATION •••••• 218

REFERENCES • . . . . . . . . . . . . . . . . . . ••• 242

7

LIST OF ILLUSTRATIONS

FIGURE PAGE

1. Arrangement of the boreholes •• · . . · . • • • 18

2. The pole-pole (normal) electrode array · 19

3. site Location. . . . . . . . . • · . · · 26

4. Zone of investigation. • . . . . · . · · . · 31

5 Comparison of borehole data as a function of measurement scale, boring H2. · • . · 34

6. An example of a cross-hole data set. • • • • • • 37

7. Histograms of corrected data values for a) continuous logging, b) large AM single-hole data, and c) cross-hole data. • • • • • • • 42

8. Semivariograms of corrected data values for a) continuous logging, b) large AM single-hole data, and c) cross-hole data. •• ••• 45

9. Schematic of the test for anisotropy. • • • 49

10. Plots of electrical resistivity with direction of measurement of borehole cross-sections for a) H3, H2, and M1: b) H3, H6, and H7: and c) H3, H5, and H8 •••••••••••••••• 50

11. Fracture groups and fitted hydraulic conductivi~y ellipsoid • • • • • • • · . . .

12. comparison of histograms for a) large AM single-hole measurements, and b) cross-hole

53

measurements • • • • • • • • • • • • • • • • • • 55

13. Comparison of semivariograms: large AM versus cross-hole • • • • • • • • • • • • • • • • • • • 56

14. Log-log plot of the 64-inch and 20 ft AM spacing variograms • • • • • • • • • • • • • • • • • • • 59

15. Plot of apparent resistivity with distance • • • 60

16. The five-point finite difference operator. • •• 70

17. 2D Random network model: Avg 10g10 apparent resistivity with distance, 51x51 network • • • • 73

18. 2D Random network model: Comparison of avg 10g10 apparent resistivity with network size • • • 75

19. Illustration of the AHL percolation model. • 80

8

LIST OF ILLUSTRATIONS--Continued

FIGURE PAGE

20. Sketch showing links and blobs . . . . • • • 83

21. Scaling of percolation cluster sizes • • • • 89

22. Correlation structure in the experiment of Clement et ale • • • • • • • • • • • • • • • •• 91

23. Distribution of percolation threshold for a range of sytem sizes • • • • • • • • • • • • 92

24. Examples of 3D fractal structures ••••• e •• 97

25. Example of hierarchial lattice structures. .100

26. Sketch of an individual alpha center. • •• 111

27. Sketch of the relationship among alpha centers .111

28. Conductivity around a single alpha center •••• 117

29. Conductivity for a 2x2x2 grid of alpha centers .120

30. The 4x4x7 alpha center grid used for modeling •• 126

31. Linear model scaling for random ci on the 4x4x7 grid. • • •• • • • • • • • • • .131

32. Sensitivity analysis for exponential model on the 4x4x7 model grid with 687 measurements a) scaling slope as fcn of ci and cii b) average calculated voltage c) sum of squared error. • • • • • • • .134

33. Effect of variance for ci and Cii near the feasible solution. • • • • • • • • • •

34. Cross-plot of calculated versus measured voltages for the 4x4x7 forward model • • . . .

35. Calculated voltage field around and electrode

.137

.139

for an East-West cross-section ••••••••• 140

36. Ratio of the secondary voltage field to the primary voltage field •••••••••••••• 142

37. Calculated 10g10 apparent resistivity around a single electrode ••••••••••••••• 143

38. Value of alpha for cross-section •••••••• 145

39. Calculated semivariograms for forward model ••• 147

9

LIST OF ILLUSTRATIONS--Continued

FIGURE PAGE

A.1 Short AM log: boring M1. · . · . . . · . A.2 Short AM log: boring H2 ••

A.3 Short AM log: boring H3.

• • · . • .161

.162 · . . . · . . · . . . • •• 163

A.4 Short AM log: boring H5.

A.5 Short AM log: boring H7. · . . · . . .

· . . . . . . .164

.165

B.1 Large AM log: boring Ml. · · · · · · · · · .167

B.2 Large AM log: boring H2. · · · · · · · · · · · .168

B.3 Large AM log: boring H3. · · · · · · · · · .169

B.4 Large AM log: boring H5. · · · · · · · • · · · .170

B.5 Large AM log: boring H6. · · · · · · · · · · · .171

B.6 Large AM log: boring H7. · · · · · .172

B.7 Large AM log: boring H8. · · · · · · · .173

D.l The Sierpinski gasket. . · · · · · · · .195

D.2 Examples of Fractional Brownian motion · · .199

D.3 Log-log plots of the semivariance of electrical resistivity. • • • • • • • • • • • • • • • • • .203

D.4 Test of resistivity as a Gaussian process •••• 204

F.1 Transmitter - Receiver geometry ••••••••• 228

F.2 Input variation in alpha for a) 4x4 grid: and 2) 7x7 grid ••••••••••••••••••• 229

F.3 Estimation for 4x4 grid: a) difference in alpha b) estimation variance ••••••••••••• 231

F.4 Estimation for 7x7 grid: difference in alpha •• 232

F.5 Test of objective functions in feasible range a) cii tests b) ci tests • • • • • • • • • .238

10

LIST OF TABLES

TABLE PAGE

1. Basic borehole geometry data • . . . . • • • 27

2. Summary of single-hole tests •• · . · . . • • • 36

3. Summary of cross-hole tests. · . . . . • • • 36

4. Basic data statistics (10g10 transformed values) . . . . . . . . • 41

5. Values used for anisotropy analysis. • · . • • • 51

6. 20 random network model behavior • · . . • • 72

B-1 Large AM data for boring M1. • · . • • .174

B-2 Large AM data for boring H2 •• .174

B-3 Large AM data for boring H3. · · · · · · · .175

B-4 Large AM data for boring H5. · · · · · · · .175

B-5 Large AM data for boring H6. · · · .176

B-6 Large AM data for boring H7. · · · · · · · .176

B-7 Large AM data for boring H8. · · · · · .177

C-1 Crosshole data (TX/Rx) for H2-M1 · · · · • · · .179

C-2 Crosshole data (Tx/Rx) for H3-M1 · · · · · .181

C-3 Crosshole data (TX/Rx) for M1-H3 · · · · · .183

C-4 Crosshole data (Tx/Rx) for H2-H3 · · · · · .185

C-5 Crosshole data (Tx/Rx) for H5-H3 · · · · · · · .187

C-6 Crosshole data (Tx/Rx) for H6-H3 · · · · · · · .189

C-7 Crosshole data. (Tx/Rx) for H7-H3 · · · · · .191

C-8 Crosshole data (Tx/Rx) for H8-H3 · · • · · · · .192

D-1 Fractal "measurement" techniques · · · · · .198

F-1 Exponential Model 4x4 estimation tests · · · · .235

11

ABST~~CT

Geotechnical characterization of crystalline rock is often

dependent upon the influence of the rock's fracture sys­

tem. To test ensemble fracture behavior in situ, a series

of cross-hole and single-hole electrical conductivity

measurements were made within saturated Oracle granite.

The tests were conducted with a point source and a point

reference electrode and employed electrode separations

ranging from 8 inches to over 100 feet. A volume of rock

approximately 50 x 50 x 150 feet was tested (as bounded by

the vertical test borings). Analysis of the data in terms

of an equivalent homogeneous material showed that the

effective electrical conductivity increased with electrode

separation. The cross-hole data indicate that the rock can

be treated as a non-homogeneous, isotropic material. Fur­

ther, the spatial variation of measured conductivities

along a line can be fit to a fractal model (fractional

Brownian motion), implying that the scale-dependence is a

result of a fractal process supported by the fracture sys­

tem. scale-dependence exists at the upper scale limit of

the measurements, hence a classical representative

elemental volume was not attained.

Two directions were taken to understand the scale­

dependence. The rock mass is treated in terms of a disor­

dered material, a continuum with spatially varying

conductivity. First, a percolation-based model of a

disordered material was examined that relates the conduc­

tivity pathways within the rock to the backbone of a crit-

12

ical percolation cluster. Using the field data, a fractal

dimension of 2.40 was derived for the dimensionality of

the subvolume within the rock that supports current flow.

The second approach considers an analytic solution for a

non-homogeneous, isotropic material known as the alpha

center model (Stefanescu, 1950). This model, an analytic

solution for a continuously varying conductivity in three

dimensions, is a non-linear transform to Laplace's equa­

tion. It is employed over a regular grid of support points

as an alternative to spatially discretized (piece-wise

continuous) numerical methods. The model is shown to be

capable of approximating the scale-dependent behavior of

the field tests. scaling arises as a natural consequence

of the disordered electrical structure caused by the frac­

ture system.

13

CHAPTER 1

INTRODUCTION

It is the intent of this dissertation to demonstrate a

technique to obtain and analyze measurements characteris­

tic of a crystalline rock. The physical properties of this

rock type are often dominated by the characteristics of

the internal fracture system. In situ measurements of

individual fractures are difficult to perform and do not

address the problem of the interrelationship among frac­

tures. By taking bulk measurements over a range of

scales, the basic behavior of the fracture system can be

observed from the standpoint of network behavior, rather

than trying to postulate how individual elements should

behave. In this study, the measurements are shown to be

scale-dependent and are related to a fractal-based

description of the rock mass. A non-discretized 3D model

is also presented to numerically model the observed behav­

ior. The results are this study can be used to understand

how scale-dependence can occur and how the underlying

scale-dependent processes can be analyzed using continuum

models.

The intrinsic properties of fractured rock masses have

proven to be difficult to analyze, especially when mea­

surements taken at one scale of investigation are applied

to larger or smaller scales. A great deal of effort has

been put towards the measurement of individual fractures,

yet to date there is really no satisfactory three­

dimensional description of most types of fractures. Of

special interest is the manner in which fractures interact

14

to form the bulk properties of a rock mass. Some of the

problems involved with the theoretical link between indi­

vidual crack properties and fracture system behavior are

summarized by Pai11et (1985). He makes four points

regarding the basic conceptual models often used to char­

acterize fracture systems:

1. The asperities within fractures are irregular and the overall fractures discontinuous. Many models still attempt to treat fractures as semi-infinite planar ele­ments.

2. Rock mass lithology (and structural evolution) is quite important for characterizing fractures.

3. The idea that a depth-fracture frequency relation­ship exists can now be generally disproved.

4. The relationship between fracture aperture and effective permeability is oversimplified. Field tests can­not provide more than a limited sample, and the basic geo­metric model of a flat, open-walled aperture is basically invalid. More importantly, the interconnection between fractures needs to be directly determined.

The problem of defining a "Representative Elemental Vol­

ume" is examined in some detail by de Marsi1y (1985) and

Neuman (1987). They both point out that the flow of water

is governed by the degree of intersection of fractures and

that it is very difficult to relate measured fracture

geometries to hydraulics. Rather than treat the rock mass

in terms of discrete features, a stochastic continuum is

proposed. The use of statistically generated subvo1umes

(not "representative" of any particular geometrically

defined features) is suggested as a way to study the

effect of connectivity. A fractal scaling model is further

15

suggested by deMarsily to account for the wide range of

observable fracture patterns (i.e., to allow for a wide

range of "major" and "minor" features). Basically, when

examined in relation to bulk measurements, the properties

of fractures are very difficult to quantify in any mean­

ingful way. This study demonstrates that fractal concepts

are indeed valid for the evaluation of bulk measurements.

One possible route around the quantification of discrete

cracks is to examine the bulk properties of the fracture

system. The possibility that scale-dependent values exist,

however, presents a problem in determining the scale at

which to test the rock mass. The measurements often vary

rapidly from point to point, reducing one's confidence in

the data set. Essentially a fractured rock mass appears

highly disordered.

This study approaches the properties of fractured rock, in

particular the electrical resistivity of Oracle granite,

as being described by a disordered system. From this per­

spective, the fracture system can be viewed as a statis­

tical process and treated using the ideas of percolation

theory (e.g., Stauffer (1985». In this theory one

discretizes a system into a network of lattice points that

can be scaled up or down in size. Studies of electrical

resistances on a lattice (Kirkpatrick, 1973) suggest that

the bulk properties of a network will be scale dependent.

That is to say, a scaling relationship will hold between

the elements of a network and the bulk behavior of a

larger network composed of these elements. Further results

16

of Ambegaokar, Halperin and Langer (1971) and Berman et

al. (1986) suggest that only a subset of the network will

actually control the flow of current in a disordered sys­

tem. This subset of conductors has been shown in percola­

tion models (e.g., Stauffer (1985» to have scaling

properties described by fractal geometry (discussed by

pynn and Skjeltorp (1985) and originally due to Mandelbrot

(1983». In addition, recent examination of fracture trace

lengths (Barton, 1985), fault patterns (Hirata, 1989),

fracture surfaces (Brown and Scholz, 1981), and rock frag­

mentation (Turcotte, 1985), all suggest that fractal scal­

ing properties can be observed in naturally fractured rock

systems.

Given the aforementioned suggestions that rock mass prop­

erties may involve scale-dependent phenomena, the obvious

step is to collect field data to test this hypothesis.

One approach is to take a spatial distribution of rock

properties and examine them in terms of a fractal model as

done for rock porosity measurements by Hewitt (1985).

Another approach is to take many measurements over a range

of scales or measures and directly assess whether scale

dependence can actually be observed. This approach was

taken here.

The actual field experiment is fairly simple. Electrical

resistivity measurements were made both within and between

boreholes at the Oracle test site. A pole-pole array, the

simplest of all electrode configurations, was employed.

17

Measurements were made by varying the interelectrode sepa­

ration and thus changing the scale of measurement without

changing the experimental technique. Separations of 8

inches to over 100 feet were used. (Figure 1 shows a lay­

out of the boreholes: figure 2 is a diagram of the elec­

trode array used.) The measurements, more fully described

in Chapter 3, show a clear shift in bulk rock properties

as the scale changes. Effective electrical conductivities

increase with the scale of the experiment.

There appear to be no published accounts of direct mea­

surements of the type performed here. A somewhat related

approach is the examination of scale-dependent

dispersivities in the analysis of solute transport in

porous media (e.g., Gelhar and Axness (1983): Tyler and

Wheatcraft (1988». Some of this work has used data spe­

cific to the Oracle test site (Neuman, 1987: Neuman and

Depner, 1988: Neuman et aI, 1989).

A number of publications demonstrate that scaling behavior

of measured properties occurs in fractured rock. Most

analyses of in situ measurements, however, have not

attempted to assess scale-dependent behavior. For the pur­

poses of this dissertation, it is important to briefly

discuss some of the more relevant work as it applies to

the measurement of fractured media. Here it is stressed

that the analysis of field data will be discussed. Scaling

phenomena in "theoretical media" and the application of

theoretical models will be treated in Chapter 4 in the

context of fractal geometry. The following, then, is an

18

.... 0

~b~ In

~

L:

'~ C' ....

~ 0

(/) (Y)

W -.--J D I I (\j ::: W ~ I D ....

0 (:Q C\J ... ...

LL In In D (Db C\J I C\J

r---.. 0 (Y) "::

l- I I I Z W L: "-

W £.

LJ .... C\J Z o -<[ In~ ~

'-../

~ Z <[ ~ ./1

Figure 1. Arrangement of the boreholes.

19

BOREHOLE ARRAYS I v v

40'

r -I 64°

3~' .. lD l'

20' 32° 15'

10' 16"

64" 8"

o I I

Figure 2. The pole-pole (normal) electrode array.

20

attempt to look at prior work that has had some influence

towards the interpretation and application of the I

resistivity tests performed at Oracle. To improve the

readability, the discussion is subdivided into five top-

ics.

1. Geophysical Measurements: Hole-to-hole and

hole-to-surface measurements are fairly common in the lit­

erature, but none have been tested for scale dependence.

Daniels (1983 and 1987), Snyder and Merkel (1973), and

Poirmeur and Vasseur (1988), among others, describe field

measurements that are primarily established for the iden­

tification of distinct (geometrically discrete) electrical

anomalies, rather than the examination of fracture

systems. Hewett (1986) presents an analysis of porosity

measurements in a west Texas carbonate. His measurements,

taken only for one scale of investigation, are interpreted

to have a spatial correlation structure consistent with a

fractal process and hence have a scale-dependent (fractal)

behavior. Other measurements have been made in fractured

rock to demonstrate electrical anisotropy (e.g., Leonard­

Mayer (1984», but again are restricted to one scale of

investigation.

2. Hydraulic Testing: An interesting discussion

by de Marsily (1985) examines a number of questions that

support the idea of scale-dependent transport phenomena,

but again no direct measurements are presented. The actual

process of "exhaustively" sampling a fracture system is

presented in terms of an extensive site characterization

21

study. One of his major points relevant here is that frac­

ture interconnectivity (when posed in terms of percolation

theory, discussed in Chapter 4) will show a scaling

behavior.

An extensive compilation of longitudinal dispersivity data

in Neuman et aI, (1989) indicates that scale-dependent

solute transport occurs in a wide range of geologic condi­

tions. This behavior is then linked to the spatial covari­

ance structure of the hydraulic conductivity by means of a

quasilinear theory of non-Fickian solute transport. Based

upon this model it is shown that log-transformed hydraulic

conductivity measurements can be treated as a self-similar

(fractal) process for a wide range of environments. In

general, an exponential covariance structure can be uni­

versally applied and a mean fractal dimension of 1.75

interpreted for the log hydraulic conductivity structure.

(see Appendix D for the relationship between covariance

and fractal dimension).

Often cited are the numerical simUlation results of Long

and witherspoon (1985) and Long et ale (1982) where dis­

crete fracture models are used to "test" the permeability

structure of fractured rock. A numerical test of scale was

performed by randomly generating a fracture system in

terms of discrete fracture lengths and densities. The

question they addressed was the determination of an appro­

priate scale of measurement for a particular statistical

realization of their discrete model under stationary

conditions. Therefore, their objective was to compare how

22

large a sample of discretel~ modeled fractures is required

to match the calculated permeability of an equivalent

homogeneous, anisotropic medium. With scale, the bulk

hydraulic conductivity tended asymptotically to a constant

value and no sort of scaling relationships appeared. In

comparison, this study does not assume that the fracture

process is stationary, but can be approximated by a frac­

tal process and therefore to not reach a constant variance

with sample size.

3. Rock Fragmentation: Turcotte (1985) has

examined the number-size distribution of rock fragments

and shows that there is a scaling relationship inherent to

fragmentation. His arguments are based on the observation

that self-similar or repeating patterns can be quantified

using the ideas of fractal geometry (to be discussed in

Chapter 4 and Appendix D).

4. Fracture Surfaces: Brown and Scholz (1985)

and Brown (1987) examined naturally fractured surfaces

which are shown to exhibit a statistically scaling behav­

ior, again based upon fractal arguments. They examined

the topography of the surface profiles and related the

variations in the height with distance to a random process

that has fractal (scaling) properties. The fractal proper­

ties of fracture surfaces have been used to relate flow

through single fractures by Nolte et al (1989) and by

Brown (1989). These attempt to relate the geometry of the

23

asperities within a crack to its effective transport prop­

erties, but do not extend the analysis to the properties

of an ensemble of fractures.

5. Trace Patterns of Fracture Networks: An

extensive field effort by Barton and Larsen (1985) ana­

lyzed the distribution of trace lengths from a scale of

0.20 to 25 meters. The lognormal distribution was analyzed

using a simple test of self-similarity and was found to

exhibit consistent scaling behavior. Hirata et al (1989)

and Sholz and Aviles (1986) applied box-counting algo­

rithms to demonstrate the self-similarity of large-scale

fault patterns.

Visual observations by Allegre et al. (1982) of fracture

exposures were compared with fracture patterns observed

locally and regionally. Patterns were observed to appear

to be similar over a range of scales, and therefore deemed

self-similar or fractal. The same can be inferred about

the fracture maps at Oracle when local and regional frac­

turemaps are compared, although a more formal analysis of

the fracture patterns would provide more convincing

evidence.

The overall approach of this work is as follows. First, a

brief description of the test site is given in Chapter 2

as are references to background work. Then, by chapter:

Chapter 3 is an explanation of the field method and data

reduction. The basic data are presented as are the obser­

vations of scale-dependent behavior. Briefly, there is no

24

finite volume of rock that can be described as a

representative elemental volume (REV) over the scale of

the tests. The rock is more appropriately described as a

non-homogeneous, isotropic material with point-wise mea­

surements that vary continuously over the test volume. The

variation of measured electrical resistivity with depth

can be treated as a fractal process.

Chapter 4 assesses a general model of electrical conduc­

tivity of a disordered system in terms of percolation

theory and electrical networks. The field behavior is

explained in terms of a disordered or random system with

fractal scaling properties. The electrical conductivity is

seen to be controlled by a fractal subvolume of rock

described by percolating continuum with a fractal dimen­

sion approximately equal to 2.4.

Chapter 5 presents the alpha center model, a numerical

method used in this application to model a three­

dimensional, non-homogeneous, isotropic rock mass. It is

introduced because it allows for a description of the

conductivity by a continuously varying function described

by a fixed grid of support points. The alpha center model

is shown to be capable of replicating the observed scale­

dependence.

Chapter 6 provides a summary of this research and avenues

for further work.

25

CHAPTER 2

THE ORACLE TEST SXTE

All testing performed for this study was conducted at a

test site established in 1980 by the university of Arizona

Department of Hydrology. The location was selected to

test the characteristics of saturated fractured crystal­

line rock and was originally part of a research project

funded by the u.s. Nuclear Regulatory Commission. It is

located within the Oracle granite batholith, a

1400-mi11ion-year-01d quartz monzonite, and can be found

along the northeastern flank of the Santa Catalina Moun­

tains (see Figure 3). Eight closely spaced, vertical bore­

holes have been drilled to depths of 250 to 300 feet in an

area where the depth to water is approximately 40 feet.

The layout of the borings is shown in Figure 1, and the

borehole depths and diameters are included in Table 1.

A great deal of hydraulic and geophysical testing has been

performed at the site. Tests have been performed by the

U.S.G.S. borehole geophysics research group, Lawrence Liv­

ermore National Laboratory, the Tucson Water Resources

Division of the U.S.G.S., and by the University of

Arizona.

The tests can be organized by the type of information col­

lected and analyzed. These are summarized in a rough

chrono1ogic order that lists the major summary documents.

They are as follows:

Basic site geology and geophysical testing: Jones (1983), Winstanley (1984), Jones et a1. (1985).

)),ORAClE 'V HILL

BONtT~ r -<

Z o

Figure 3. site Location.

EXPLANATION ~ ELEVA~ON OF LAND

SURFACE (FT AMSL)

,'---- MAdOR ROAD

• FIELD SITE

I I MILE

•

26

27

Table 1. Borehole Geometry-------------------------------

Borehole

Ml

H2

H3

H4

H5

H6

H7

H8

Depth

(ft)

300

300

300

288

250

250

250

250

Casing Depth

(ft)

58

59

58

43

61

63

66

59

Nominal Diam.

(in)

6.90

4.50

6.75

4.25

4.00

4.50

3.75

4.00

28

Cross-hole electromagnetic imaging: Ramirez et al. (1982), Ramirez (1986). .

Hydraulic testing: Hsieh, Neuman, and Simpson (1983), Hsieh and Neuman, Parts 1 and 2 (1985). Further analysis of the results in Neuman and Depner (1988) that discusses the problem of scale and the relation of the single-hole tests to the cross-hole tests.

Tracer testing: Cullen, stetzenbach, and Simpson (1985), Aiken (1985), Barrachman (1986).

Stochastic analysis of dispersion: Neuman and Depner (1988).

Analysis of advective heat flow: Silliman et al. (1986, unpublished), Silliman (1988), Leo(1988), Wollushun (1989).

General examination of the continuum representation of fractured rock systems, based in part upon Oracle data: Neuman (1987).

To briefly summarize, the following results of the exten­

sive test program are listed because of their application

to this study:

1. While the individual fractures could be readily

mapped and segregated into fracture sets, there was little

correlation, if any, between fracture densities and mea­

sured hydraulic conductivities. Neutron-log measurements

(a measure of total porosity) correlated well with the

measured conductivities.

2. Cross-hole hydraulic testing at a scale of about 100

feet (the single-hole tests used a 13-foot-long straddle

packer assembly) showed the rock mass to behave as a uni­

form, weakly anisotropic material. The directional conduc-

29

tivities describe an ellipsoid that appears strongly

correlated with the orientation of the major fracture set

intersections.

3. The measured single-hole hydraulic conductivities,

corrected for leakage, have a lognormal distribution and

an autocorrelation length of roughly 30 meters. This upper

correlation length is approximate as an exponential

covariance model is used to fit the data.

4. Comparison of the scalar equivalent hydraulic conduc­

tivities calculated from the single-hole tests using sto­

chastic theory (Neuman and Depner, 1988) with larger scale

cross-hole tests shows that the two data sets can both be

viewed as quantities defined over a continuum. The mea­

surements are scalar at the local scale, but define a

homogeneous, weakly anisotropic material over a large

scale. In general, the stochastic theory predicts that the

hydraulic conductivity will increase as a function of the

data variance, which in this case increases up to a scale

of approximately 30 meters.

5. An overall geologic picture has been developed that

relates fracture patterns to specific geologic events. A

major fault zone has been identified at a depth of 240 to

280 feet. There is a diabase dike to the west side of the

site with an associated hydrothermal alteration pattern

characterized by calcite-filled fractures. An extensive

surficial weathering horizon can be identified by the

associated clay content detected by natural gamma logs.

30

The resistivity measurements were taken between these fea­

tures to avoid their effects. These zones are illustrated

in figure 4.

A zone that can be seen in the cross-section between bor­

ings M3 and Ml was selected for this study based upon

prior lithologic analysis. The interval between 100 and

200 feet in depth appears to have the most consistent

properties. It lies below the near-surface weathering (de­

termined from natural gamma logs, observation of core and

chip samples, and from the interpreted matrix rock

velocity) and above the major fault zone. The diabase dike

and its associated hydrothermal alteration zone lie west

of the test zone, as indicated in Figure 4. Overall, the

zone selected for study has the most consistent lithologic

properties and is not dominated by any large-scale struc­

tures.

The following chapters are based upon measurements taken

within the aforementioned volume of rock. Geophysical

tests were conducted along and between the boreholes and

were designed to assess the effective rock properties as a

function of the scale of the test. A direct analysis of

the data is first shown, then techniques are presented to

evaluate the rock mass behavior.

ZONE OF INVESTIGA TION

H4 H3 H2 M1

FAULT

-50'-

Figure 4. Zone of investigation.

o

-100

:I I-0-W t=I

-200

-300

31

32

CHAPTER 3

FIELD MEASUREMENTS AND DATA ANALYSIS

A combination of cross-hole and single-hole electrical

resistivity measurements were made at the Oracle test

site. The purpose was to directly measure the electrical

interconnection among fractures as a function of the scale

of the test. As shown in this chapter, there is a system­

atic decrease in measured electrical resistivity as a

function of the inter-electrode separation. First, the

data collection and reduction will be discussed. Then the

data will be presented and the summary statistics pres­

ented. The survey results show that the effective electri­

cal conductivity increases with the size of the tested

sample and that the point measure of resistivity with

depth can be approximated by a 20 fractal process. The

fracture system within the rock can thus be inferred to be

self-similar over the range of the tests.

Field Method

All of the measurements used a pole-pole or normal elec­

trode array. As shown in Figure 2, the array consists of

a point current electrode and a point potential electrode.

The reference current and potential electrodes were

located 1200 feet from the test measurement. This distance

was a minimum of 10 times the interelectrode spacing for

all the measurements.

Two data collection systems were used. Short single-hole

measurements were made with a Mt. Sopris geophysical log­

ging system. This unit was provided by the U.S. Geological

33

Survey (Water Resources Division, Tucson, AZ) and operated

by Mr. Don Pool. A standard resistivity sonde was used and

data were collected for current-potential distances of 8,

16, 32, and 64 inches. The probe was modified for this

application because the return current electrode is the

outer sheathing of the wire line. For the high resistiv­

ity materials at oracle, the cable connection distance was

increased to a distance of 66 feet from the top of the

sonde to minimize the effect of the second current elec­

trode upon the array. Prior U.S.G.S. experience (Scott,

per. comm., 1987) showed that this increase in distance is

necessary in high resistivity rock.

single-hole measurements with electrode separations of 64

inches, 10, 15, 20, 30 and 40 feet were also made at ten­

foot intervals along each of the test boreholes. These

measurements were made using a standard Scintrex IPR-10

time domain reduced polarization system donated by Phil­

lips Petroleum Corporation to the Laboratory of Advanced

Subsurface Imaging, of the Department of Mining and

Geological Engineering at the University of Arizona. An

example of the single-hole data can be seen in figure 5.

A set of fixed length insulated wires were used to place

stainless steel electrodes within the boreholes. The

IPR-10 system operates with a square-wave source having a

two-second duty cycle (2 seconds on positive, off 2 sec­

onds, 2 seconds reversed current, and off again for 2 sec­

onds). An in-line resistor with a Fluke digital voltmeter

was used to measure the output current. The voltage was

H2 RESISTIVITY' LOG (corrected) 8 16 32 and 64-in 4.25 -r-----~-'--~~....;;....~-'--'--------_.,

,-.... E 4.00 I E 3.75 .c o

'-" 3.50

~ > 3.25 I-Cf) 3.00 Cf) W ~ 2.75

o 'r" 2.50 (9 o .....J 2.25

2.00 -t----r---,.-----r--r__-or---,---,.---,.-----r---\

34

100 120 140 160 180 200 220 240 260 280 300 DEPTH (ft)

LARGE AM DATA: 64-in to 40 ft 4.5 ~----------------------------~

if)

if) w n:: 0.. 4.0 0.. «

o r- 3.5 o o ---l

3.0 -t-~--r-~~~~~~~r__~~~_r_~_,._-r_,._-.___r~ 100 150 200 250 300

DEPTH (ft)

Figure 5. comparison of borehole data as a function of measurement scale, boring H2.

35

measured by the IPR-l0 receiver and checked by a second

digital voltmeter. Spontaneous potential drift was exter­

nally monitored for temporal variations by a voltmeter and

removed by the SP buckout circuit in the receiver. The

voltage values taken by the receiver are rated by the man­

ufacturer to be accurate within 3 percent with 0.1 percent

resolution.

Cross-hole resistivity measurements were also made using

the IPR-l0 equipment. Three transmitter positions of

120-, 160-, and 200-foot depths were established and

potential measurements were taken at 10-foot intervals

between 80 and 240 feet in a neighboring borehole. A wide

range of inter-electrode distances was thus derived, rang­

ing from 20 to over 100 feet. Tables 2 and 3 summarize the

data collection effort for all of the single-hole

measurements and all of the cross-hole tests, respec­

tively. An example of the cross-hole data is illustrated

in figure 6.

Data Reduction

The purpose of the measurements was to determine the

effective electrical conductivity of the rock mass. Two

factors, the effect of the fluid-filled borehole, and the

effect of the images created by the air-rock interface,

must be accounted for to properly reduce the data.

Correction for the effect of the less resistive water­

filled borehole was made using a standard technique. The

method, coded in FORTRAN by Scott (1978), treats the

Table 2. Summary of Single-hole Test Intervals

Borehole

8-inch

Ml * H2 * H3 * HS * H6 H7 * H8

continuous Record (1)

l6-inch 32-inch

* * * * * * * * No data (3) * * No data (3)

Discrete (2)

64-inch

* 100-270 * 100-270 * 100-270 * 100-240

100-240 * 100-240

100-240

36

1. continuous measurements were collected using a Mt. Sopris series II logging unit operated by the Tucson WRD of the USGS. All start at 100 ft and go to bottom of each borehole. 2. Discrete measurements were taken at 10-foot intervals. 3. Runs were attempted, but electrical leakage occurred at cablehead.

Table 3. Current Location (1) Ml

Ml H3 *

Summary of cross-hole tests. Potential Measurement Location (2) H2 H3 HS H6 H7

* *

* * * *

1. Three current electrodes used at depths of 120, 160 and 200 ft.

H8

*

2. Measurements made at 10 ft. intervals between 100 and 240 ft.

o co

ooos' ... ~ 000L3 ~

o o ... , o N ... ,

o 'Ot ... ,

o CD ... ,

o IX) ... I

o o N ,

o N N I

37

o 'Ot N ,

~100091 ~ en ... CD I")x ___ - .. -

~ EOOO~ ~ ... _-"'0:: --- ........ I ~ ~ _______ a::: _ .. !z EOOOt-l _ ~ _ ~ ........ .:::. 1.1 __ ~ _.- .. - ...

.... -....... --- ........ ~.2,OOO£ .. _ -- - • - _.x - __ c.. - ~-- a::: ~ ---~ OOOZ •

OOOB~ ~ OOOL

~~0009 -~ ffi"t) OOO~ . O:::E !z ~ OOOt

~!OOO£ 8: OOOZ < ,.

OOOL 0 co

I") ::I: (!) Z ~ 0 CD

N ::I: (!) z ~ 0 CD

0 0 ... o

N ...

~ .-~ ~.::.

* I

o CD -

DEPTH ·(ft)

~ N -.g ~.::.

* I

o IX) - o

o N

. -= I")

-.g ~~ •

o N N

, I I I I

o I")

I I I

- -" I ',' •

~ N I I I I

~ I I I I t

Figure 6. An example of a cross-hole data set. Three data sets are shown, each corresponding to the Tx electrode positions located in boring H3.

38

fluid-filled borehole and rock mass as a two-component

cylindrical system. Given a borehole filled with water

with known resistivity and diameter, the apparent

resistivity of the surrounding rock can be calculated.

wait (1987) presents a good discussion of the basic

theory. All of the single-hole data were corrected for a

previously measured fluid resistivity of 10 ohm-meters and

the borehole diameters were obtained from caliper logs.

A second significant factor that can be removed from the

data is the effect of the air/earth interface. The down­

hole current source will give rise to an image electrode,

an effect that increases closer to the ground surface.

The image plane is taken here to be the land surface. A

second image plane could be considered in the vadose zone

near the water table, but the change in resistivity

between the surface and the 40-foot-deep potentiometric

surface is probably gradational and irregular. Electrical

sounding could be used to check for a distinct contact,

but the closely spaced metal well casings create a major

interpretation problem. Therefore, no soundings were con­

ducted and only the surface interface was considered.

correction for the air/earth interface is direct and based

upon the calculation of apparent resistivity, the

resistivity of an equivalent homogeneous, isotropic earth.

In general, for the four-electrode array in a whole space:

39

~v ( Pa = 4n-I-

1 1 1 1 )-1 + --

rBN (3.1 )

rAN

where ~v is the voltage measured between M and N (Fig-

ure 2),

I is the current introduced by A and B,

and r iJ is the distance between i and j.

In the case.of the pole-pole array, all the distances

except AM are quite large and their respective terms typi­

cally neglected. When an image, AA, is located above the

conducting electrode A, the second source can be added via

superposition. The strength of the image current is also

equal to I. Then, the apparent resistivity is equal to

~V( 1 1 )-1 Pa = 4n- -- + --I rAM r AAM

(3.2)

which can now be easily corrected for the effect of the

interface. All of the cross-hole measurements and the

large-spaced (greater than 64 inches) single-hole measure­

ments were reduced to an equivalent whole-space value of

apparent resistivity. The cross-hole electrode positions

were calculated using the borehole deviation logs avail­

able conducted by LLNL within borings Ml, H2, and H3. The

remaining boreholes were assumed to be vertical.

The Field Data

The data are presented as a function of the inter­

electrope distance, AM. Three sets are shown: the con­

tinuous single-hole (USGS) data, the discrete large-spaced

single-hole data, and the cross-hole measurements.

Distance overlaps were maintained between each data set.

sixty-four-inch spacings were used by both the USGS data

sets and the discrete single-hole data. Distances of 30

and 40 feet were measured by both the larger single-hole

and shorter cross-hole measurements.

40

All the statistics are presented using loglo-transformed

values. The following statistics were generated using the

USGS STATPAC statistical analysis programs:

1. Basic descriptive statistics: mean, variance (Table 4)

2. Histograms (Figure 7)

3. semivariograms (Figure 8), (for vertical directions)

comparative plots of the single-hole resistivity logging

data are given in Appendix A. A discussion of the results

concludes the chapter.

Anisotropy Analysis

Electrical anisotropy can be measured by rotating a fixed

electrode array about a central point. Here, resistivity

measurements that share a common centerpoint can be used

to assess whether the rock is anisotropic. A schematic of

the measurement is shown in figure 9. The only available

data with such common centerpoints are the cross-hole

data; it is not possible to reliably test this particular

rock mass from the surface because of the influence of the

metal borehole casings that extend from the surface to

depths of around 40 to 50 feet. (They were cased to the

Table 4. Basic Data statistics (10g10 values)

Data set

Mt. sopris (USGS)

8-inch l6-inch 32-inch 64-inch

IPR-lO, single hole

64-inch (2) 10 ft 15 ft 20 ft 30 ft 40 ft

IPR-IO cross-hole

20-40 ft 40-60 ft 60-90 ft 90-180 ft

Mean

3.50 3.35 3.27 3.18

4.08 4.03 3.93 3.77 3.64 3.54

3.72 3.60 3.49 3.35

Variance No. Samples (1)

.0793

.0651

.0548

.0469

.0675

.0479

.0288

.0186

.0096

.0083

.0069

.0087

.0063

.0055

1660 1660 1660 1660

110 110 110 110 110 106

65 98

118 76

41

1. The Mt. Sopris data were digitized at 0.5 ft inter-vals. 2. comparison of the 64-inch data shows a difference between the response of the two measurement systems. The IPR-IO data were cross-checked during collection by exter­nal instrumentation to be correct machine values. The Mt. Sopris unit was calibrated with external resistors and checked out properly prior to use, however, an exact fac­tor of 5 separates the data sets.

~

25

20

~ 15 u 0:: I.LJ 10 0..

5

AM= 8 inches

Mean= 3.50 Var= 0.079

(1660)

25

20

~ 15 u 0:: 1.LJ10

,0..

5

AM= 16 inches

Mean= 3.35 Var= 0.065

(1660)

O~~~~~~~~~ 2.5 3.0 3.5 4.0 4.5 5.0

O~hT~~~~~~~ 2.5 3.0 3.5 4.0 4.5 5.0

~

25

20

~ 15 u 0:: I.LJ 1 0 0..

LOG RESISTIVITY

AM= 32 inches

Mean= 3.27 Var= 0.055

(1660)

25

20

~15 u 0:: ~ 10

LOG RESISTIVIlY

AM= 64 inches

Mean= 3.18 Var= 0.047

(1660)

5 5

o O~~~~~~~TM~ 2.5 3.0 3.5 4.0 4.5 5.0 2.5 3.0 3.5 4.0 4.5 5.0

LOG RESISTIVITY LOG RESISTIVITY 25 64-INCH vs 8-INCH

42

20 ~ i,

"

1a) continuous Logging:

~ 15 u 0:: I.LJ 10 0..

5

" 1'-, . , I-I

'-I , , &_-'-I ,

&-. O~~~~~~TnTM~

2.5 3.0 3.5 4.0 4.5 LOG RESISTIVITY

AM= 8",16",32",64"

Figure 7. Histograms of corrected data values for a) continuous logging, b) large AM single-hole data, and c) cross-hole data.

40

I- 30 z UJ

~ 20 UJ

0.. 10

AM - 64 in Mean- 4.08 Var-= 0.067

(110)

o~~~~~~~~~

40

I- 30 z UJ

~ 20 UJ 0.. 10

2.5 3.0 3.5 4.0 4.5 5.0 LOG RESISTIVITY

AM = 15 ft

Mean'" 3.93 Var= 0.029 (110)

O~~~~~TMTH~~

40

I- 30 z UJ u 20 ~ UJ

0.. 10

2.5 3.0 3.5 4.0 4.5 5.0 LOG RESISTIVITY

AM = 30 ft Mean= 3.64 Vara:: 0.009 (110)

o~~~~~~~~~ 2.5 3.0 3.5 4.0 4.5 5.0

LOG RESISTIVITY

40

I- 30 z UJ

~ 20 UJ 0.. 10

40

I- 30 z UJ

~ 20 UJ

0.. 10

AM - 10 ft

Mean- 4.03 Var= 0.048

(110)

43

3.0 3.5 4.0 4.5 5.0 LOG RESISTIVITY

AM = 20 ft Mean= 3.77 Var= 0.018 (110)

O~~~ThTn~~~~ 2.5

40

!z 30 UJ

~ 20 LaJ

0.. 10

3.0 3.5 4.0 4.5 5.0 LOG RESISTIVITY

AM - 40 ft

Mean= 3.54 Var= 0.008 (106)

3.0 3.5 4.0 4.5 5.0 LOG RESISTIVITY

.fb) Large AM single-hole data: AM= 64",10',15',20',30',40'

60

50

!z 40 LU

~ 30 LU

a. 20

10

AM - 20-40 ft

Mean= 3.72 Var= 0.0069 (85)

O~~nTn+~hT~~~

60

50

.- 40 z LU

~ 30 LU

a. 20

10

2.5 3.0 3.5 ~.O ~.5 5.0 LOG RESISTIVITY

AM = 60-90 ft

Mean= 3.49 Var= 0.0063

(118)

O~~n+~~~~~~

2.5 3.0 3.5 ~.O ~.5 5.0 LOG RESISTIVITY

60

50

!z 40 LU

~30 LU

a. 20

10

AM - 40-60 ft

Mean= 3.60 Var= 0.0087 (98)

44

O~nTnThTn+~~~~ 2.5

60

50

10

3.0 3.5 ~.O ~.5 5.0 LOG RESISTIVITY

AM = 90+ ft

Mean= 3.35 Var= 0.0055 (76)

O~~A+nThT~~~~ 2.5 3.0 3.5 ~.O ~.5 5.0

LOG RESISTIVITY

jc) Crosshole Data, separation classes as shown.

.q-Q)

0 or-

* r". .s:: '-' « ~ ~ « (!)

1250

1000

750

500

250

o

SEMIVARIOGRAM: LOG10 resistivity

• • 8

1/.

• •

• " • • • • ·16 • • " • • • • 32

• • " • • • • • ·64 • • • • • • • • • • • • • • • I· • • •

III • • III •

• • ~ ,. .

• • • • . ~ • .. - . i·'~ • • • t/" • -I· • .. , .. "

s: •• ' .-, ,I • . -t·1

• I o I I I

20 I I I

40 60 DISTANCE (ft)

I I , 80 100

e a) continuous Logging: AM= 8",16",32",64"

45

Figure 8. Semivariograms of corrected data values for a) continuous logging, b) large AM single-hole data, c) cross-hole data. Note that the borehole records used were generally 180 ft. A maximum lag of 90 ft is used to construct the variograms. Only vertical orientations were evaluated.

o o o

~ \0

o o 1O

0 \l) 0 - C\.J V 0 (Y)

46

0 0 .,..-

0 00

-+-" ~

w o z

~~

o N

CJ) -o

Sb) Large AM single-hole data: AM= 64" , 10 I I 15 I I 20 I , 30 I , 40 I

~ 0 ct

\

0 ~

\ \

'"

t: 0 \0

0 ~ , ,

V

.I-J 11-4

0 0\

0 0 ..-

o ro

o C\J

~------~~------~---------'---------rO o o C\J

o o ..-

t30 L * (H)'v'V'JV'fv'~

ee) Crosshole Data, separation classes as shown.

o

47

48

water table.) As shown by the previous statistics, there

is a systematic change in measured resistivities with the

scale of measurement. Because of this, directional

resistivities are segregated by electrode separation as

well as by common centerpoint.

Cross-hole measurements that share a comnon centerpoint

are graphically plotted for three cross-sectional panels

oriented along bearing of 090, 045, and 000 degrees azi­

muth. The measured apparent resistivities are plotted as a

function of the angle defined by the two test electrodes.

These plots combine borings H3, H2, and Ml in Figure lOa,

borings H3, H6, and H7 in Figure lOb, and borings H3, H5,

and H8 in Figure 10c. At each point, the directional

resistivities are typically within ten percent. Table 5

contains the values that are plotted. Overall, the avail­

able resistivity values do not exhibit strong directional

dependence and are interpreted to be point-wise isotropic.

The plots have not been combined together because of the

bias introduced by the scale-dependence.

Prior work at the site (Hsieh et al.,1985) showed that a

hydraulic conductivity tensor could be derived from cross­

hole hydraulic tests. The orientation of this tensor,

defined by an ellipsoid, is oriented along the directions

of the intersections of major fracture sets observed from

borehole data. For purposes here, the general hydraulic

conductivity along H3-H2-MI (east-west) is greater than

that observed along H3-H5-H8 (north-south) in the horizon­

tal plane. A stereonet of the major fracture sets and an

ANISOTROPY ANALYSIS

COMPARE

A c RESIS A-D

vs

RESIS B-C

B D

Figure 9. Schematic of the test for anisotropy. This measurement consists of rotating a fixed array about a central point.

49

H.3

DIRECTIONAL RESISTIVITIES

Southwest-Northe~st

H3 H6

H8~7

H3 Ml

H7 -100~~~~~~~~+-~~~~~~~

-120

~ LLI -160 Q

-180

-200 2x horiz excggenaUon __ 0_- 5000 ohm-m

South-North E~st-\Je5t

H5 H8 H3 H2

50

-1004-~~~~~--~~~~~ -1 00~~--1---I..-t--L--J.----'~'"--"'-1

-120 -120

M1

'2'-140 V v V

g-140 -:J:

~ LLI -160 V Ii:

w -160 Q Q

-180 V -180

-200 -200 2x horiz elCco9.geratlon 2x horiz exc;ger~t1on

Figure 10. Plots of the electrical resistivity with direction of measurement. Data shown for borehole cross-sections a) H3, H2, and M1 ; b) H3, H6, and H7 ; and c) H3, H5, and H8.

51

Table 5. Values used for anisotropy analysis. App. Resistivity

Panel/Point No. Fwd. Pro . Rev. Pro F R

H3-H2-M1 (east-west) 1 H3-120 H2-160 H3-160 H2-120 4816 4693 2 H3-120 H2-200 H3-200 H2-120 2891 3146

H3-160 H2-160 6027 3 H3-160 H2-200 H3-200 H2-160 5380 5348 4 H3-120 Ml-160 H3-160 Ml-120 3713 3824 5 H3-120 Ml-200 H3-200 Ml-120 2761 3033

H3-160 Ml-160 Ml-160 H3-160 4575 4363 6 H3-160 Ml-200 Ml-200 H3-160 4150 3995 7 Ml-160 H2-120 H2-120 Ml-160 4407 5090 8 Ml-200 H2-120 H2-120 Ml-200 2946 3125

Ml-160 H2-160 6754 9 Ml-160 H2-200 H2-200 Ml-160 4752 4782

H3-H6-H7 (NW-SE) 1 H3-120 H6-160 H3-160 H6-120 3588 4131 2 H3-120 H6-200 H3-200 H6-120 2623 3825

H3-160 H6-160 5086 3 H3-160 H6-200 H3-200 46-160 4232 4530 4 H3-120 H7-160 H3-160 H7-120 4022 4332 5 H3-120 H7-200 H3-200 H7-120 2779 2951

H3-160 H7-160 5207 6 H3-160 H7-200 H3-200 H7-160 4809 5300

H3-H5-H8 (north-south) 1 H3-120 H5-160 H3-160 H5-120 2495 2838 2 H3-120 H5-200 H5-200 H3-120 2351 2498

H3-160 H5-160 3261 3 H3-160 H5-200 H3-200 H5-160 3330 3388 4 H3-120 H8-160 H3-160 H8-120 2654 2754 5 H3-120 H8-200 H3-200 H8-120 2163 2346

H3-160 H8-160 3508 6 H3-160 H8-200 H3-200 H3-160 2909 3301

52

illustration of the fitted hydraulic conductivity

ellipsoid is shown in Figure 11. Comparison with the plots

of directional resistivities shows that the values

obtained in a north-south direction are less than those

obtained along the east-west profile. A loose relationship

can be inferred between the electrical and hydraulic data,

but it is not developed because a formal analysis of large

scale anisotropy has not been attempted for the electrical

data as has been done for the hydraulic data.

In general, the rock mass does not exhibit strong aniso­

tropy as measured by directional resistivities. This does

not preclude the possibility that the rock mass is

anisotropic at larger scales. At the scale of the measure­

ments, the rock is assumed to be a non-homogeneous, iso­

tropic material where the variations of the electrical

conductivity are treated as a scalar quantity that varies

continually within the rock volume.

Discussion of Field Results

A block of the subsurface sized 50 ft x 50 ft x 150 ft

between boreholes formed the basis of all the measure­

ments. Because of the orientation of the borings, only

vertically oriented statistics can be generated with

confidence, although many of the cross-hole measurements

were conducted in horizontal directions. An example of the

general behavior of the measurements is shown in Figure 5,

which is a plot of the single-hole measurements taken in

boring H2. Between electrode spacings of 64 inches and 20

to 30 feet, there is an obvious shift in the measurements.

nST 5ERIE5 .. - It l H 2 -.

I

.\~ \, i )

, I ''1-0 .

la)

53

TEST IIlIllS • TUT &EAlES C

- "3 "~- •• H2 .. 6 ' ....

Ib) Ie)

Figure 11. Fracture qroups and fitted hydraulic conductivity ellipsoid. (after Hsieh et a1 1985).

At spacings above 20 feet, the values begin to coincide,

suggesting that the scale dependence is most pronounced

for smaller rock volumes.

54

Examination of the data histograms shows the bulk rock

properties are best represented by the high-conductance

portion of the short-spaced AM histograms. Direct compari­

son of 64-inch and 40-foot AM spacing data (Figure 12a)

illustrates this relationship. Beyond a range of 20 to 40

feet, the bulk properties of the rock mass continue to

shift toward more conductive values. This is shown by com­

paring the cross-hole resistivity data in Figure 12b. The

largest distances, however, probably are most likely

influenced by lower resistivity material outside the test

zone (i.e., the near-surface weathered zone and the deeper

fault zone).

Of major concern is whether an average electrical conduc­

tivity has been reached by the large-scale measurements.

with scale, the variance of each set does decrease. Also,

the semivariograms of all the measurements using an elec­

trode separation of 30 feet or more appear similar (see

Figure 13). (The semivariograms are only for a vertical

distance and use the centerpoint between the two elec­

trodes). However, the most obvious feature of the semiva­

riograms is that no sill appears out to lags of 90 feet. A

upper limit of 90 feet was used because the maximum

distance spanned by the measurements was 180 feet (follow­

ing the general rule for the construction of variograms).

The data drift, as calculated by the semivariogram program

40

I- 30 z w ~ 20 w a. 10

40 ft vs 64 in

'i I <-- 64-in I I -I I-I

- 1--I

O-h~~MM++MM~~MM1+~~M

60

50

I- 40 z w () 30 0::: W

a. 20

10

2.5 3.0 3.5 4.0 4.5 5.0 LOG RESISTIVITY

20-40/60-90/90+

"

tIC, 9°

<-- 20 - 40 ft

O~~~~~~TMTMTM

2.5 3.0 3.5 4.0 4.5 5.0 LOG RESISTIVITY

55

Figure 12. Comparison'of histograms for a) large AM single-hole measurements, and b) cross-hole measurements.

o 'J)

I 0 __ ---------C)-------,--------------------~rO ~ or-

" " ,

o ~ I

o C\..J

" " ...... ..... ..... ..... ..... ...

0 ..... ('U

..... ..... ..... .......... ..... ..... .....

o 00

o N

~-------,--------~--------Ir--------1IO

o o C\J

o o or-

o

Figure 13. Comparison of semivariograms: large AM versus crosshole data.

56

57

GAM2D (Journe1 and Huijbregts, 1978), was always less than

10 percent of the mean value. Because of the relatively

low drift component, the experimental semivariograms which

have the form of an exponential or power model, appear to

be "real". This would imply that a constant variance is

never attained within the scale of the measurements.

Unfortunately, larger-scale measurements cannot be made to

determine the upper limits of the covariance structure due

to the borehole geometry and geologic constraints.

The exponential semivariograms can be analyzed further.

This particular covariance model has been shown theoreti­

cally (Berry and Lewis, 1980) and experimentally (Bur­

roughS, 1983; Voss, 1985; FOx, 1989) to be a measure of a

fractal process. (Refer to Appendix 0 for a more complete

description of fractal concepts.) In this particular

application, the value of measured apparent resistivity

with depth is fit to a process known as fractional Brow­

nian motion (fBm), a correlated form of a Gaussian or ran­

dom process. The slope of the data variogram is evaluated

from a log-log plot and posed in terms of the fractal

dimension of the fBm process. This has the implication

that the variation of resistivity is a measure of an

underlying scale-dependent, self-similar process.

Analysis of the semivariograms to determine a fractal

dimension is based upon a functional relationship between

the Weierstrass-Mande1brot and its Fourier power spectrum

(c.f. Berry and Lewis, 1980). The semivariogram is a spa­

tial domain equivalent to the power spectrum and thus can

58

also used to evaluate the f~acta1 function. The slope of

the log-log plot is equal to 4 - 2d l • For the relationship

to be valid, the process must be shown to be Gaussian.

This can be readily determined by examining the increments

of the process for a constant lag as done in Burroughs

(1983). The relationship is also approximate. Fox (1989)

presents a numerical analysis of the errors where he shows

that the spectral technique tend to overestimate the frac­

tal dimension for d I < 1.5 and to underestimate the dimen­

sionality for d l >I.5 for a 2D process.

Two data sets corresponding to AM distances of 64 inches

and 20 feet are examined here. Log-log plots of their

variograms are shown in figure 14. Fits to the slopes lead

to fractal dimensions of 1.75 for the 64-inch data and

1.45 for the 20 foot data. In both cases the 10g10 trans­

form of the data was used. Analysis of the increments of

the processes, included in Appendix D, confirms that the

records for AM spacings up to 20 feet can be approximated

by a Gaussian process to justify the use of the log-log

slope aanlysis.

Summary

Measurements taken between electrodes spaced from 8 inches

to 100 feet apart showed a constantly decreasing variance

and resistivity (increasing conductivity) as illustrated

in figure 15. At a scale of 30 to 40 feet, the limits of

the measurements tend to shift downward and similar exper­

imental semivariograms are derived for the data set. The

small-scale measurements (AM spacings less than 30 feet)

59

-1.0 I !

AM= 64 in. i -- i

14 I

-1.5 t>::::: 1.. I ... *' I ,-...., ... I .c

'-" I 0

I E § -2.0

I

AM= 20 ft. I Ol V i , I Ol , 0

, I

~ I -2.5 I

I

Estimate of Fractal D I I

I -3.0

1.0 1.2 1.4 1.6 1.8 2.0

log h. ft

Figure 14. Log-log plot of the 64-inch and 20 ft AM spacing variograms. The interpreted fractal dimensions are shown as a function of slope.

MEAN (OhM-M) 4.2 -..------------,

~4.0 > ~3.8 if) w 0::: 3.6

(p'/'I

30 40-60

o .- 3.4

o :3 3.2

8~ 16·~

20 ~20-40

40 60-90

90+

64

3.0 -0.5 0.5 1.5 2.5

LOG 10 DISTANCE (ft)

VARIANCE -1.0 .,.........

if)

~ W ::J -1 « o 6<; > -1.5 0 0 ~ ---1 ...........

W U -2.0 Z

~ « a:: ~

-2.5 -0.5 0.5 1.5 2.5

LOG 10 DISTANCE (ft)

60

Figure 15. Plot of apparent resistivity (and variance) with distance.

are locally influenced by individual fracture zones, as

seen by comparison of the profiles of measured apparent

resistivity with depth. This influence is marked by the

convergence of the values for a range of AM spacings at

locations within the profile. In comparison, the large­

spaced data are smoother in appearance with decreased

resistivities. This decrease is interpreted to be

influenced by the low-resistivity fracture zones.

61

Analysis of available measurements taken about common cen­

terpoints show that the electrical conductivity can be

assumed to be point-wise isotropic and treated as a scalar

process. At larger scales the data are not available to

directly assess whether large-scale anisotropy occurs, as

it does for the hydraulic conductivity measurements.

An overall assessment of the system response indicates

that as the electrode spacing increases, the electrical

conductivity is enhanced by the interconnection of frac­

tures. By increasing the spacing, the probability that a

conducting pathway forms is increased. This is in part

supported by noting that the large AM resistivities are

best represented by the lowest resistivity values measured

with the short AM arrays.

Evaluation of the correlation structure of the data shows

that no finite variance occurs over the scale of the

tests. A fractal-based analysis of the semivariograms for

64-inch and 20 foot AM data shows that the measured appar­

ent resistivity with depth can be approximated as a self­

affine process characterized by fractal dimensions of 1.75

62

and 1.45, respectively. This implies that the underlying

conductivity structure, principally determined by the

fractures within the rock mass, is fractal in nature. The

fractal analysis is limited to AM spacings less than 20 ft

for the given data set.

The next chapter will examine the electrical pathways in

the rock mass by using an argument based upon percolation

theory. The rock is taken to have an equivalent 3D circuit

such as used to model random electrical networks. The

topology of the current-carrying portions of these net­

works has been observed to follow fractal-based scaling

laws (e.g. Stauffer, 1985). These fractal networks are

shown in the next chapter to be related to the scale­

dependent resistivity measurements.

CHAPTER 4

DISORDERED (RANDOM) NETWORKS AND SCALING BEHAVIOR

Introduction

63

The electrical response of the fractured rock mass at the

Oracle test site is related in this chapter to the proper­

ties of disordered electrical networks. This approach,

which treats the rock as a continuum with rapidly varying

electrical properties, is reviewed in this chapter. A

philosophically similar approach has been taken for the

analysis of the hydraulic properties of fractured rock

systems (Neuman, 1987). Following chapters will describe a

numerical modeling approach that assumes that the field

data obtained in this study can be analyzed as a realiza­

tion of the response of a disordered material where the

scalar electrical conductivity varies continually in 3D.

Fractured rock systems are generally treated by either

discrete or continuum models. A discrete approach is based

upon a comprehensive description (size, shape, location,

intrinsic material properties, etc.) of all the fractures

that occur in a rock mass and contribute to the bulk

response of the rock. At the Oracle site the observed

fracture density, A, is 0.744 fractures per foot along

vertical scanlines and fit a negative exponential distri­

bution f(x)=Aexp(-Ax) (Jones,1983). Based upon this dis­

tribution the fracture frequency can be treated as a

random (Poisson) process as defined by Priest and Hudson

(1976). A discrete model of the rock mass requires the

numerical simulation of many hundreds of fractures. Of all

64

the measureable fracture parameters, the spacing and local

orientation data are the most reliable. The other relevant

parameters are essentially indeterminate, despite the rel­

ative abundance of data for the site. A discrete model is

exceedingly difficult to parameterize with confidence.

This is especially true of the target region chosen for

this study, because it lacks any large-scale, readily

identifiable structures such as faults, dikes or litho­

logic contacts.

The geophysical data presented in chapter 3 suggest that

the rock mass can be treated as an electrically non­

homogeneous, isotropic material. A continuum model can

therefore be used in this context to model the rock mass

behavior without the need to rely upon discrete fracture

parameters. In particular, a disordered or random network

analogy is developed to describe the rock mass. As the

name implies, a random network is the representation of a

material by a grid, lattice, or set of subvolumes that can

be assigned a wide range of electrical conductivities.

Three approaches toward the study of random networks are

distinguished here, all of which are capable of demon­

strating scaling behavior similar to that observed at the

Oracle site. The second approach is developed here for ~

direct application to the resistivity data. These include:

1) Numerical studies of random networks, typicallY by

two- or three-dimensional finite difference models, which

are used to evaluate the effective properties of various

network configurations. A two-dimensional model is used

65

here to illustrate that the measurement of effective elec­

trical resistivities about a point source can exhibit

scaling when the variance of the conductances within the

network is sufficiently large.

2) Percolation models that show how interconnected

pathways form within a random network. These models, as

explained later, can be used to intuitively understand the

behavior of disordered systems. It is the scale-dependent

structure (i.e. the fractal geometry of the pathways) that

leads to the observed network behavior. No unusual or

restrictive conditions are required to create these path­

ways within a disordered network.

3) Multi-fractal models that use a number of fractal

descriptors for a particular system. For this application,

these models examine the statistical moments of electrical

potentials calculated from random network simUlations. The

moments can then be related to a specific fractal-based

model of the current-carrying pathways that form within a

random material. These sort of hierarchical systems may

help to limit the numerical calculations necessary to

model a disordered network by focussing efforts upon the

pathways relevant to flow.

Each of these models are ultimately based upon an underly­

ing distribution of network conductances. The field data

suggest that an exponential covariance structure should be

used in this application. This particular data structure

corresponds to a system where the parameter variances

66

increase with system size (i.e. the case of infinite dis­

persion discussed in Journal and Huijbregts, 1978). It

also allows for the introduction of fractal concepts into

the description of the point measures of electrical

resistivity.

The remainder of the chapter develops an argument for

scale-dependent measurements based upon the geometry of

the current-carrying flow paths in the rock mass. First, a

simple 20 network model is presented to illustrate a Monte

Carlo technique can be applied to a disordered network. It

is included to demonstrate how the random network model

may behave around a point source. All other models

described here and elsewhere generally employ a linear

potential across the sides of a square or rectangular net­

work. Theoretical comparison of stochastic conductivity

fields under plane parallel flow conditions (Ababou, 1988)

has shown that the 20 and 30 cases can have very different

behaviors. The 20 model is presented here primarily as a

demonstration: a truly 30 finite-difference model used to

replicate the field conditions would require an extremely

large system of linear equations to properly handle the

external boundary conditions. This problem is further dis­

cussed in chapter 5 where an alternative modeling scheme

(the alpha center method) is presented and applied to the

field data.

After the 20 network model is presented, the discussion

turns toward the properties of the electrical pathways

that occur within a random network. Numerical studies of

67

both 20 and 30 random networks have shown that the geome­

try of conductivity pathways, termed percolation clusters

in this analogy, are typically fractal and hence lead to

scale-dependence of properties controlled by the clusters.

In particular, the infinite cluster that forms at the per­

colation threshold has been shown to control the observed

electrical response at and above the percolation threshold

when a lognormal distribution of conductances is intro­

duced into a network (Berman et aI, 1986). This fractal

structure is then related to the electrical resistivity

measurements to determine the fractal dimension of the

subspace that supports electrical conduction within the

rock mass. The chapter closes with a brief discussion of

the multi-fractal nature of electrical networks.

A Simple Random Network Model

The use of electrical networks to evaluate the properties

of heterogeneous systems is quite common. Early modeling

studies, for example Greenberg and Brace (1969), Shankland

and Waff (1974), and Kirkpatrick (1971), use network mod­

els to test basic constitutive relationships such as

Archie's Law (Archie, 1942) or the effective properties

predicted by binary mixing model such as proposed by

Bruggeman (1935) and summarized by Wait (1983). A great

deal of attention has been paid to the evaluation of ran­

dom networks and mixing laws at the onset of conduction at

or near a percolation threshold. The major interest has

been to apply network models to study phase transitions

and similar physical phenomena (see for example Wilson,

1979).

68

More recently the properties of hydraulically random aqui­

fers have been examined as means to test stochastic models

(e.g. Gelhar, 1986) that predict that the hydraulic

conductivity, K, ~f an aquifer will increase with the

scale of measurement. Scaling appears as a function of the

variance of the conductivity, where the effective conduc­

tivity of an aquifer increases with the variance. In this

application the point measure of K is viewed as a random

(stochastic) process. The work of Ababou et al (1988)

extensively tests the stochastic theory using 3-D finite

difference models on regular, square lattices containing

up to a million nodes (i.e. a 100x100x100 FD grid). So

far, there have been no similar approaches to examine the

scale dependence of electrical resistivities in geophysi­

cal applications. The main difference between the hydro­

logic and geophysical approaches is in the condition of

the model: the stochastic hydrology theory generally deals

with plane parallel or regional flow whereas the electri­

cal problem typically involves point sources and sinks.

Under steady state conditions, fluid flow and DC

conductivity are mathematically equivalent problems (i.e.

Laplace's equation). Stochastic theory has not been devel­

oped for a point source and is beyond the scope of this

work.

69

A random network model can be simply configured by consid~

ering a finite difference model where a randomly generated

set of conductances are input to the finite difference

grid. In this example, Monte Carlo network simulations

were performed on a regular, two-dimensional grid of

nodes. Values of internodal conductances were chosen by

sampling an arbitrary lognormal distribution. A range of

variances was used to provide a suite of statistical real­

izations.

To simulate the field experiment, the center node of the

model grid was set to a known voltage and the exterior

boundary condition was set to V = o. The output current

and the distribution of voltages were then calculated for

each simulation. This procedure was repeated 100 times to

ensure that the voltage calculated as a function of dis­

tance converged upon a mean value.

A system of linear equations is derived by using a stan­

dard five-point finite difference operator (e.g.,

Smith,1978). Kirchoff's circuit laws are satisfied at each

node, where

a IIC (V II - V c) + a NC (V N - V c) + a EC (V E - V c) + a sc (V s - V c) = / c

where aiC is the conductance between node i and the center

V iC is the voltage between node i and the center node

(4.1 )

the node index i corresponds to N,S,E,W as shown in Fig 16

70

0 OJ

> II

-.J W ~ D

U U

c U U ~ U lfl b·-> ~> ~ >

"-u u > ~

2: ~ I

W > > '-./

U Z W

~V-JTI ~ W v 0 l.L -l.L -p I---l If)

~ c 0

W 0 U

o ~ I---l \I Z I---l > l.L

\I

> €a

\I

> ~ I

(IJ

o /\

Figure 16. The five-point finite difference operator.

71

The figure shows the configuration of the standard five­

point finite difference operator and the general 20 model

conditions. Solution of the banded matrix system was done

using OMNILIB subroutines SGBCO and SGBSL, available on

the University of Arizona SCS-40x computer. The routine

used to generate normally distributed numbers was adapted

after routine GASOEV of Press et ale (1985).

In order to relate the results of the 20 network model to

the geophysical methods used in this study, it is neces­

sary to translate the 20 simulation results from voltages

to measures of apparent resistivity. This is done through

the evaluation of the geometric factor used for a particu­

lar electrode array. The geometric factor, GF, relates the

calculated resistance (VII) determined from the

measurement of potential at a given distance and known

input current to the resistivity of an equivalent homoge­

neous, isotropic material. In general, Pa=(VIJ) GF , so

for known resistivity, current, and potential, the

geometric factor can be directly calculated from the

results of a network model under homogeneous conditions.

The two-dimensional apparent resistivities were thus cal­

culated from the voltages by (1) establishing a constant

resistivity for the 20 network by setting all the

internodal conductances equal to a constant value (10 mhos

(or siemens», (2) solving for the voltages on the homoge­

neous network, and (3) calculating the GF value at every

node in the network given the resistivity, voltage, and

input current. These GF values allow for the imposed

boundary conditions so that the calculated apparent

72

resistivities are constant ~or all points in the grid. In

the case of the homogeneous network, the apparent and true

resistivities are the same. For the non-homogeneous case,

the geometric factors are used to calculate an equivalent

homogeneous conductivity (or resistivity) for a given

electrode separation, voltage measurement, and input cur­

rent. Here, the GF values were directly calculated from

the homogeneous model.

Square grids were set up for the calculation of apparent

resistivity with distance. Here 51 x 51, 71 x 71, and 81 x

81 grids were established for the disordered network mod­

els. The randomly generated network values were input

using lognormally distributed values with a mean 10g10

value of zero and log transformed variances ranging up to

1. To complete the analog to the field test, the apparent

2D resistivities were sorted by distance to derive an

average apparent resisitivity as a function of the dis­

tance between the current source and potential measurement

point. Each simulation was repeated 100 times to get a

suite of statistical realizations that converged upon a

mean resistivity value as a function of distance. The con­

vergence behavior is summarized in Table 6, which shows

that the mean was fairly stable after approximately 50

iterations of the 51x51 network model.

With increasing variance, a pronounced change in calcu­

lated apparent resistivity can be seen in figure 17.

Decreases in average resistivity occur with distance as

the variance of the input network values increases. For

Table 6. 2D Random Network Model Behavior

A listing of averaged log10 resistivities per number of simulations. Shown are values for var= 0.1 and var = 1.0. Values for 51 x 51 network (figure 17)

variance= 0.01

No. Sim.

10 30 50 70 90

100

Data Pts

5

-.0007 -.0008 -.0005 -.0006 -.0005 -.0005

per 100 sim.: 2000

Variance= 1.0

No. Sim.

10 30 50 70 90

100

Data Pts

5

-.0257 -.0165 -.0177 -.0172 -.0176 -.0174

per 100 sim.: 2000

10

-.0014 -.0007 -.0008 -.0008 -.0007 -.0007

2000

10

-.0404 -.0244 -.0284 -.0257 -.0251 -.0252

2000

distance 15

-.0012 -.0010 -.0012 -.0012 -.0011 -.0012

4440

distance 15

-.0431 -.0400 -.0433 -.0415 -.0410 -.0410

4440

20

-.0017 -.0006 -.0012 -.0014 -.0015 -.0015

5200

20

-.0705 -.0673 -.0741 -.0725 -.0752 -.0753

5200

72

73

0.0200 -------------------

0.0000 .---. . • • • ••••••• 1411 .... var= 0.01 0_ .. -

>-. -.-4J .-> .-4J .~ -0.0200 en <l.) var= 0.25 L-

a .... -0.0400 tJ) 0 var= 0.50

en ~ -0.0600 \

\ var= 1.0

-O.OBOO 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

log10 distance

Figure 17. 20 random network model: Average 10g10 apparent resistivity with distance on a 51x51 network. Shown are results for input conductances with zero 10g10 mean and variance = 0.01, 0.25, 0.50, and 1.0. (100 simulations).

74

the lowest variance case, the 2D model behaves nearly as a

homogeneous material with a slight shift in apparent

resistivity. The scale-dependent behavior becomes more

obvious with increasing variance. Note that the mean con­

ductance increases due to the skew introduced by the log­

normal distribution.

In order to evaluate the effect of the size of the finite­

sized system upon the scale-dependent behavior, a series

of simulations were done for 71x71 and 81x81 grid sizes

(refer to figure 18). For In[O,l] distributions, there is

a marked decrease in resistivity near the periphery of the

network. This behavior does not occur for the low variance

case and is more pronounced as the variance is increased

for any grid size. The shift occurs at roughly half the

distance between the center 'electrode' node and the

boundary. The geometry of the model appears to affect the

calculated scaling behavior. Note, however, that the

external voltage boundary conditions are accounted for by

the geometric factors calculated using the homogeneous

case.

The scaling behavior of the 2D model is influenced by the

location of the external boundary. Under homogeneous con­

ditions the boundary is not seen, as illustrated by the

low-variance simulation data of Figure 17. To fully

understand the disordered system behavior at the equipo­

tential boundary, it appears necessary to examine the cur­

rent distribution in detail to determine how conditions

vary from the middle to the periphery of the grid. This

0.0000

~ -0.0200 ...... . s: :;:;

(fJ

(fJ

~ -0.0400

o r-01 o tJ) -0.0600 > o

71 x 71

(bdy= 1.23)

• -0.0800 -+----r--,...-~-_.--r_-r_-._____,

0.00 0.25 0.50 0.75 1.00 1.25 1.50

log 1 0 distance

75

Figure 18. 2D random network model: Comparison of average 10g10 apparent resistivity for SlxS1, 71x71, and 81x81 networks with 1n[0,1] conductances in the finite difference grid.

76

has not been done and is beyond the immediate scope of

this project because the objective of the 20 model was to

simply establish whether scaling could occur on a disor­

dered lattice.

Overall, the 20 finite difference model roughly imitates

the field data behavior and establishes that a disordered

network will give rise to scale-dependent measurements. At

distances more than halfway to the boundary, the scale­

dependence appears exaggerated along the outer portion of

the grid. Extremely large finite differenc~ grids could be

employed, but the problem would not realistically simulate

the 30 field conditions. An alternative 30 model is pres­

ented in Chapter 5 as a means to avoid massive computa­

tional problems. The 20 model shown here is useful as an

illustration of a random network model that demonstrates

scale-dependence, but is not intended as a direct analog

to the field conditions.

Most of the use of random conductance networks such as

described here has been restricted to (stochastic) hydrol­

ogy and statistical physics. The study. of disordered sys­

tems in physics, for example the properties of doped

semi-conductors or magnetic spin-systems near the Curie

point (e.g. Wilson, 1979), show that scaling behavior

arises as a function of disorder. in a material. These stu­

dies, however, emphasize transitional properties that

occur at or near percolation thresholds. Applications in

geophysical prospecting have not occurred because the

exploration targets are viewed as discrete bodies, not a

77

disordered assemblage, and ~hould not be expected to dis­

play any sort of scale-dependent measurements. In this

particular study, the random network model is applicable

on the basis that no defined target was examined. The

field response, as measured by apparent resistivities,

displays strong scale-dependent behavior. The following

section will develop the use of the disordered network

model to help understand the behavior of the fractured

rock system.

The Percolation Network Model

By far, the widest application of random network modeling

has been to study percolation theory. Percolation theory,

introduced as a distinct methodology by Broadbent and Ham­

mersley (1957), was originally concerned with the flow of

fluid through a random maze or network. Consider a

regular, square, two-dimensional mesh. Either the inter­

sections (the nodes) or the mesh connections (the bonds)

can be viewed as present or vacant. There is a minimum

percentage of interconnected nodes (or bonds) that must be

present so that a continuous pathway can be traced across

the mesh. This minimum percentage is termed the percola­

tion threshold. Percolation models generally examine con­

stitutive properties of a mesh as a function of the

properties of the statisties of the nodes (or bonds) such

as correlation length, variance, and probability of resi­

dence at a site on a lattice. continuum models can also be

considered where spatial rather that point measures are

defined. The continuum problem is generally regarded as

78

being equivalent to the lattice approach. Some of the bet­

ter references include Stauffer (1985), Shklovskii and

Efros (1984), and Pynn and Skeltjorp (1985).

The importance of percolation theory to the electrical

properties of the fractured granite becomes evident

through the treatment of random systems via percolation

theory. Conductance in rock sees no minimum threshold and

a measure of conductivity can almost always be made. Per­

colation comes into play when the conductive pathways are

examined in detail as an analogy can be made between, for

example, the pore space in a rock and a random network. In

rock that demonstrates basic properties (conductivity,

strength, etc.) with high variances, the percolating net­

work is of direct consequence because it is the conductive

pathways that should dominate the behavior of the rock

mass.

This percolation-based picture of a material dominated by

an internal network of conductive pathways is especially

appealing to the study of fracture systems. In this

instance, an exact analogy is not being proposed where

individual fractures can be mapped onto a percolation net­

work. Instead, the fracture system is seen as a process

that leads to a random conductivity field which can be

evaluated at a variety of scales without regard to the

discrete fracture network.

Ambegoakar, Halperin and Langer (1971) (hereafter referred

to as AHL) proposed a model in which the critical pathway

is used to examine the bulk resistance of a random

79

resistor network. Their model was originally used to

understand the highly variable, microscopic conductivity

variations in semiconductors (hopping conductivity). Their

model is created as follows. Consider an empty resistor

network. For a given distribution of conductances, fill

the network randomly with resistors, starting with the

largest conductance values and working downward. At the

point that conductances span the network (i.e., current

can pass across the network), a critical path is created.

This path dominates the behavior of the system for a dis­

tribution with a large variance, such as a lognormal data

set. The addition of the remaining, lower conductances

make less important contributions because they are effec­

tively shorted out by the spanning electrical pathway. An

illustration of the sequential process of "filling" a

random network is shown in figure 19.

Tests of the AHL model by Kirkpatrick (1973) and Webman,

Jortner, and Cohen (1975) showed that the critical path

model did not work very well as a general model. Their

early emphasis was to test effective medium properties

through numerical modeling studies of binary mixtures.

More recent numerical tests by Berman, Orr, Jaeger, and

Goldman (1985) showed that networks formed by skewed dis­

tributions of conductances, such as lognormal, could be

approximated fairly well by the AHL model. Further worle by

Giordano (1988) suggests that the variance of conductance

can be related to effective conductivities for a wide

range of two- and three- dimensional random networks.

80

CONDUCTIVITY

CONDUCTIVITY'

CONDUCTIVITY

Figure 19. Illustration of the AHL model.

81

Increasing variances correlate well with increasing effec­

tive conductivities. The point to be made is that the

critical conductance pathway, previously seen to be

readily described by fractal-based scaling properties in

percolation models, is of importance in examining disor­

dered networks in some instances. In turn, the internal

network geometry should relate to the bulk scaling

behavior.

Percolation models of the effective electrical conductiv­

ity of a heterogeneous material usually express the con­

ductivity of a lattice with a binary mix of conducting and

non-conducting bonds as

Jorx>xc (4.2)

where Xc is the critical fraction or percolation thresh­

old. Most studies, such as Kirkpatrick (1973) rely upon

finite difference calculations to estimate a(x) for a

series of non-conditional simulations. For a binary mix­

ture of conductances of unit strength, the exponent ~= 1.3

in 2D and ~= 1.9 in 3D (Stauffer, 1985).

To date, there has been a great deal of interest in deter­

mining the critical exponent ~, as well as many other

critical exponents that describe the behavior of discrete

and continuum models near the percolation threshold. Pynn

and Skjeltorp (1985) provide a good overview of these

efforts. Description of rock is often related to a nodes,

82

links, and blobs model that can be posed in terms of per­

colation theory. The pore space is broken down into por­

tions corresponding to features of a percolating fluid.

The infinite cluster is the association of pores (nodes in

the lattice) that are interconnected and span the network

at the percolation threshold. Some of the pores along the

infinite cluster are dangling channels that are wetted,

but are dead ends and do not contribute to the conductiv­

ity. A link is the branch of the cluster that connect

nodes. The links contribute to flow if they are not part

of dangling clusters. Multiply connected links are termed

blobs. These do not necessarily contribute to flow,

depending on the relative importance of the links within

the blob. A sketch of these features is shown in figure

20.

Of importance here is the infinite cluster, the intercon­

nected set of nodes and links that first appears at the

percolation threshold. within the cluster is a backbone

structure that carries the majority of flow and is the

cluster stripped of dangling channels and unnecessary

blobs. According to the AHL conjecture, the backbone

should govern the behavior of the network. The conductance

of the backbone determined at the percolation threshold

can then be used to approximate the overall system

response, even above the percolation threshold. On the

other hand, the binary mixing model (eqn 4.2) allows for

increasing conductivity as the volume fraction of conduc­

tors is increased. This behavior is consistent with most

effective medium approximations (EMA). It is important to

o Node

\ Blob \ ''-----

Figure 20. Sketch showing links and blobs (percolation network)

83

84

note that the EMA typically apply to systems with short­

range correlation structures and stationary input conduc­

tances. The fractal behavior observed over the scales of

the field data leads to long-range correlations.

In order to offer an explanation for this apparent dis­

crepancy between the two percolation models, a basic argu­

ment will be developed to relate the properties of the

infinite cluster at the percolation threshold (i.e. the

structure that controls the AHL model behavior) to the

experimentally measured conductivities at Oracle. First,

the infinite cluster will be examined in terms of fractal

measures. After that the geometry of the conducting path­

ways will be related to the volumetric measure of apparent

resistivity.

properties of the percolation clusters (lattice animals)

The interconnected structures that form within a percola­

tion lattice as the sites become occupied, sometimes

termed lattice animals (e.g. stauffer,1985), have received

widespread attention in statistical physics. Applications

of the study of these structures range from viscous fin­

gering (stanley et aI, 1985), to diffusion processes (Red­

ner et aI, 1987), polymer studies (Meakin et aI, 1984),

semiconductors (Shklovskii and Efros, 1984), colloids

(Schaefer et aI, 1984), and the study of pores in sand­

stone (Krohn and Thompson, 1986). Many other applications

exist. Perhaps the prime motivation behind the recent

resurgence in interest in percolative processes is the

observation that the lattice animals invariably display

85

fractal scaling behavior. This behavior, as explained in

Appendix D, leads to convenient geometric or statistical

descriptions of the various natural processes that can be

posed in terms of percolation processes.

Description of a lattice animal begins with statistics

regarding the fraction of sites belonging to clusters (in­

terconnected nodes) of a given size, as well as mean clus­

ter sizes and site-site spatial correlations. Following

the general notation of stauffer (1985) and Shklovskii and

Efros, 1984), let Ns be the number of clusters containing

s sites or nodes. The percolating cluster is excluded

since it is defined for an infinite-sized system and thus

contains an infinite number of sites (hence the term infi­

nite cluster or IC). The sum Lsn s is then equal to the

total number of occupied lattice points belonging to

finite clusters and nss is the probability that an arbi­

trary site belongs to an s-sized cluster. The fraction of

sites in the IC is then

P(x) == x- L sns jorx> Xc (4.3) s

The mean cluster size Sex), a second measure of a lattice

animal, is based on the probability, W s , that a particular

occupied site is contained in a site of size s, where

(4.4)

the mean cluster size is

86

Ln S2 S(X)=LW S= s

S LSn S

(4.5)

A number of types of averages can be defined, here the

definition is based upon the fact that the Ie is excluded.

This is done to evaluate the behavior as x ~ x c, often the

point of observable system transitions. For example, as

the percolation threshold is approached, the mean cluster

size will diverge as

(4.6)

where y is a fixed exponent.

The statistics of the lattice animals can then be used to

evaluate numerical simUlations of systems described by

percolation processes.

A third measure of interest is the spatial correlation

among points belonging to the same finite cluster. It can

be used, for example, to evaluate the statistical radius

of a cluster. Define a function g(r,r+h)=l if two sites at

separation h belong to the same cluster and set g=O if

they don't. Averaging over all lattice sites

G(r,x)= G(h,x)- <g(r,r+h» (4.7)

where lim G (r I x) = O. The percolation connectedness correla-"'-tCD

tion function is important for describing the fractal

87

geometry of the clusters in a network or lattice. The

fractal dimension of a cluster can be shown to be related

to the site correlation function (Kapitulnik et aI, 1984).

The basic percolation statistics have been used success­

fully in the studies cited here to demonstrate that the

intrinsic geometry of lattice animals is fractal. This

implies that

1) the structures are self-similar, and

2) a fractal dimension can be derived that helps to predict the size, shape, mass, or other geometrically­defined properties as a function of the scale of measure­ment.

These basic observations were noted in a general way in

percolation theory but not termed fractal until Mandel­

brot's efforts (e.g. Mandelbrot, 1983). Self-similarity

implies that large and small-scale magnifications of an

image of the network will appear alike, over the range of

scales that self-similarity holds. The existence of self­

similarity implies that geometrically-defined properties

of a network will be proportional to Ld

, where L is a

characteristic length measure of a system and df is the

Hausdorf-Besikovitch dimension (usually referred to as the

fractal dimension). For a fractal structure in a three­

dimensional space, 2< Df <3 • A fractional geometric

dimension is defined which lies between the classical

Euclidean dimensional limits. Two examples follow.

Monte Carlo methods can be used to generate realizations

of a percolating network to test various system behaviors.

88

The cluster statistics are then generated to examine the

system. Figure 21 shows a plot used to test cluster sizes

as a function of scale for classical percolation on a 2D

triangular lattice at the percolation threshold. The log­

log plot tests the relationship between the size of the

infinite cluster and the length scale of the percolation

network, S(x)«L, where the slope shows that the fractal

dimension of the cluster size, d, is equal to 1.895. The

clusters do not quite span a complete 2D space. The back­

bone of the IC can also be examined using a method known

as "burning" to remove dangling clusters and dead-ends

(Herrman et AI, 1984). This and other approaches have dem­

onstrated that the lattice animals and in particular clus­

ter backbones are consistently fractal.

A more physically-based test of percolation was performed

by Clement et al (1987) to analyze the structure of a

fluid invasion front percolating into a porous material. A

liquid metal alloy (Wood's metal) was injected into a col­

umn. The metal was cooled and the sample cut into slices

to examine the metal distribution that was the invasion

front. Digitized images were produced to quantitatively

analyze the spatial distribution of the metal and a point

correlation function was generated from the images where

C(r) = 0 or 1 in the absence/presence of the invading

fluid. A fractal dimension could then be calculated by

plotting the log of c(r) versus log r (similar in theory

to evaluating the log-log plot of the semivariogram as

explained in Appendix D).

The plots are normalized to the mean pore size, R. The

89

fJ8

t Sa, fJ8 ••

(at Pc) I ••

0 . 1 I-' ,:::: r rn rt CD ~. 11 , fI) , .... , N , CD ••

/ 10 I • , , 10

Lattice Size

Figure 21. Scaling of percolation cluster sizes on a triangular lattice (after Stauffer, 1985). The slope of the line gives a fractal dimension of 1.895.

90

results shown in figure 22 show definite scaling behavior

up to a limit, beyond which the correlation function

becomes constant.

Finally, a point should be made regarding the character­

ization of lattice or network properties. The value of the

percolation threshold can only be regarded as a constant

for infinite systems. In "real" or finite-sized systems,

a number of realizations are required to determine the

expected value of the threshold value xc' For a system of

dimension l,

Xc= iim<x c' > 1-+00

(4.8)

where, as shown by numerical experiments (Levinshtein et

aI, 1975) the distribution of the values of Xci are Gaus­

sian, as illustrated in figure 23, and the mean of Xci ten­

ded to shift slightly towards higher values of xc with

increasing system size. A variance was found for their

experiment where

= (4.9)

for a constant Band v= 0.9 in 3~. The shift in the

threshold as a function of sample size was found to be

<x >=X+AC 11h cl c (4.10)

91

, d =-2. 0 1 ~

'L:'- v '-' 0.1 (.)

0.01 1 10 100

r/R

..... ::s 0 11 , d~ :. C!.2. 1-(1) I» (II

~ ..... ::s 'L:'-\Q 0.1 '-' --N (II

(.)

I» rt c

0.01 11 I»

1 10 100 rt ..... r/R 0

::s ~I ~ I a v iA , ~ .t'.t'~ \~ ~t:~.z dr- e·40 ~ 'L:'-

'-' 0.1 (.)

0.01 1 10 100

rlR Note: the correlation structures show

the transition from fractal to statistical homogeneity

Figure 22. Correlation structure in the experiment of Clement et ale

-P c OJ U L OJ

a.. 8 2

~~--~~;-~~~

0.4 0.5 0.6 0.7 0.8

Percola.tion Threshold

Figure 23. Distribution of percolation threshold for a range of system sizes. (after Shlovskii and Efros, 1985)

92

93

for a constant A and ~ = 1 in 3D. The point made here is

that the threshold value is a random value for finite­

sized systems and that the variance of the threshold is

also scale-dependent.

94

General Percolative Model For Electrical Resistivity

The studies of random networks and percolation models were

presented to show that:

1) Random network models of binary mixtures agree with effective medium theories for random systems. The system behavior follows 0 (x) = (x - xc) 't, where x is the fraction of occupied sites. No scaling is implied or generally observed in such systems and the input conductances of the network are spatially uncorrelated.

2) Random systems with high variability, for example of the form 0 = 0 0 exp(-;), where ~ is a random variable, can be observed to have effective properties governed by the critical backbone conductance gc defined as the conduc­tance of the spanning interconnected cluster at percola­tion in accordance with the AHL conjecture. The electrical conductivity of the systems at and above the percolation threshold can be approximated by the critical conductance.

3) Analysis of percolation processes show the percolating backbone to be fractal. This backbone, the current­carrying portion of the network, will have a mass that scales proportional to L

d" where 2 < Of < 3 for 30

problems. Under the AHL condition, this backbone will determine the system behavior. Given that this backbone is fractal, the properties of the system should be expected to be scale-dependent and scale according to a power-law relationship.

In order to apply the per~olation concepts to the

resistivity measurements it is necessary to relate the

measure of electrical resistivity to the sub-volume of

rock that carries current. Electrical Resistivity, p, is

defined for a block of cross-section A and length L as

RA L

(Ohms L) (4.11 )

95

where R is the linear resistance of the homogeneous mate­

rial and current flows evenly through the cube of cross­

section A. Under homogeneous conditions, the current flow

completely spans the rock. The previous AHL model of the

percolating cluster control upon the network properties of

a disordered material states that only a portion of the

network supports the bulk of the current flow. To allow

for this condition, a term is introduced into the

resistivity equation for the effective porosity,~, which

is equal to 1 when the current spans the rock mass. Then

writing (4.11) in terms of the geometric factor:

P a = GF R <I> (4.12)

Under homogeneous conditions the apparent resistivity is

scale-independent. That is to say that the product (GF)(R)

is constant in a homogeneous material. The key to scale­

dependence is the effective porosity term. It can be

determined for a fractal material by evaluating the mass

of the percolating backbone. The mass of the backbone

varies as a function of the fractal dimension of the

structure, d / • The effective porosity for this structure is

then:

<I> == mass of percolating network =

mass of the total network

( 4.13)

The behavior of the electrical resistivity as a function

of length is:

96

pel) ex [GF R(L)] L d,-3 (4.14)

where R(L) is the resistance measured for an interelec­

trode separation L. Again, the product (GF R) is scale

invariant.

A measure of the fractal dimension of the current-carrying

pathway can be simply derived from a log-log plot of equa­

tion 4.14. The slope calculated from the field data is

roughly equal to -0.6. (A hand-fitted line appears to vary

between -.55 and -.65.) This leads to a fractal dimension

of approximately 2.4 which corresponds to a structure that

is self-similar and does not completely fill a 3D space.

For illustration, figure 24 shows a few structures with

fractal dimensions similar to those proposed here.

It is interesting to note that for d/~3, the current­

carrying pathway will completely span the material. The

resistivity will no longer be scale-dependent. This will

be the case when the upper correlation length scale for

fractal behavior is less than the test length or for the

case of a homogeneous material. The analysis of Clement et

al previously described shows the relation between corre­

lation lengths and fractal or scale-dependent behavior. A

sandstone, for example, could have a fractal pore

structure, but if the measurement is made at scales larger

than the characteristic fractal length scales, no scaling

would be expected. Thus, this fractal-based definition of