Fluid Connectivity, Percolation Theory, and the …kellen/research/Percolation_Thesis.pdfFluid Connectivity, Percolation Theory, and the Southern ... Fault has low seismic P-wave velocity

Fluid Connectivity, Percolation Theory, and the SouthernAlps of New Zealand

by

Kellen Petersen

A Senior Honors Thesis Submitted to the Faculty ofThe University of Utah

In Partial Fulfillment of the Requirements for the

Honors Degree of Bachelor of Science

In

Mathematics

Approved:

————————————– ————————————–Kenneth M. Golden Aaron BertramSupervisor Chair, Mathematics

————————————– ————————————–Gordan Savin Martha BradleyDepartmental Honors Advisor Director, Honors Program

1

I would like to thank Professor Kenneth M. Golden of the Universityof Utah for allowing me to work on this project, for the time and sup-port he has given me throughout my research and entire undergradutecareer, and for his inspiring attitude and devotion toward mathemat-ics. I would also like to thank Dr. Julie Vry of Victoria Universityof Wellington for her suggestion to collaborate on this work and herpatience and help during the process of this study even though we areoceans apart. More generally, I would like to thank all the professorsfrom whom I have taken classes or with whom I have had any contact.They have all helped me immensely in my mathematical education,interests, and maturity.

Abstract. The results of recent studies conducted as part ofthe South Island GeopHysical Transect (SIGHT) programme haveshown that crust beneath the Southern Alps and near the AlpineFault has low seismic P-wave velocity zones and a zone of low re-sistivity coincident with that of low P-wave speeds. This region oflow resistivity has been interpreted as a region of interconnectedgrain boundaries filled with saline fluid. Understanding the inter-connected fluid of the system is a problem than can be addressedfrom the standpoint of percolation theory, where the connectivityof the microscopic elements describe the macroscopic properties ofthe system. In this paper the background of the problem will beexplained, followed by an introduction to the fundamentals of per-colation theory and its applications to hydrological systems, andwe conclude with results obtained from applying a simple cubiclattice to solve for fluid volume in the crustal layer of rock.

Contents

Chapter 1. Geological Background 2

1. Historical Setting 2

2. Geological Environment 4

3. Geophysical Evidence for Fluid Connectivity 5

4. Metamorphic Fluid Flow 6

5. A Model of Microcrack Coalescence 7

Chapter 2. Percolation Theory 8

1. Introduction 8

2. Texas Farm Example 9

3. The Percolation Problem and Universality 11

4. Fundamental Concepts 12

5. One-Dimensional Lattice 15

Chapter 3. Application to the Alpine Fault 19

1. Simple Cubic Model 19

2. Percolation Volume vs. Thermodynamic Volume 24

Bibliography 26

1

CHAPTER 1

Geological Background

This chapter introduces the the background to the problem which

initiated this topic of research, that is, the orogenesis of the South-

ern Alps of New Zealand. More specifically, terminology, geophysical

evidence, an explanation of the studies done by the South Island Geo-

Physical Transect (SIGHT) programme, and a suggested model which

explains anomalies of the data will follow.

1. Historical Setting

The history of orogenesis around the South Island of New Zealand

consists of two main periods. The first period occurred around 86 Ma

and consisted of an oblique collision of the Hikurangi Plateau with the

Gondwana margin (see FIGURE 1.1). It is believed to have ended

aroung 82 Ma when the Tasman Sea began to open up between Aus-

tralia and New Zealand.

The Alpine Fault became a through-going feature of the region

around 25 Ma. During the early part of its history its motion was dom-

inantly strike-slip as one side slides past the other in horizontal motion

versus motion in other directions such as up-down motion. This is im-

portant since the motion of the rock was only horizontal during this

time there was then no mountain formation and therefore no accom-

panying fluid production.

2

1. HISTORICAL SETTING 3

The second event occurred around 5-6 Ma during which time the

Southern Alps of New Zealand began to form. Currently the Pacific

Plate is colliding obliquely with the Australian Plate at the Alpine

Fault accompanied with mountain formation on the Pacific Plate. This

occurs since rocks are riding up the Pacific Plate towards the surface

along the Alpine Fault resulting in a thickening of the the crust of

the Pacific Plate as the tectonic plates collide. FIGURE 1.2 shows a

cross-sectional picture of the crustal region with a topological profile

in relation to the two tectonic plates.

Figure 1.1. Maps of the region near the Alpine Fault:A) Map of the South Island of New Zealand accompa-nied with important locations and regional features andB) Map of the region where the Australian and PacificPlates meet, showing where the Alpine Fault is located.The arrow indicates the direction of relative motion ofthe plates which move at a speed of 37 mm/yr

4 1. GEOLOGICAL BACKGROUND

Figure 1.2. Cross-sectional picture of the crustal re-gion located near the Alpine Fault.

2. Geological Environment

The Alpine Fault is located along the boundary of the Australian

and Pacific Plates and is on the western edge of the South Island of New

Zealand. The movement of the Pacific Plate is in the WSW direction

with a speed of approximately 37 mm/yr relative to the Australian

Plate.

From this collision there is an uptilting, rapid uplift at 8−12 mm/yr.

During this uplift, various facies of rock are exhumed which had re-

mained deep in orogen since the first event, which occured millions of

years ago. During the exhumation process, fluid is released and can

be measured. The observed amount of fluid is much more than basic

orogenetic theory predicts.

Due to the various elements of the complex geological history of the

Alpine Fault region, it is believed that the amount of fluid released dur-

ing the movement of rock during the Cenozoic evolution is less than that

of progressive regional metamorphism or simple overthrusting. There

3. GEOPHYSICAL EVIDENCE FOR FLUID CONNECTIVITY 5

must, therfore, be another process causing the increased discharge of

fluid.

The structural evolution of the rock occurs as the rocks approach

the Alpine Fault. During this time the rocks experience vertical thick-

ening and the generation of metamorphic fluids due to the devolatiza-

tion reactions. Before reaching the ramping stage, the rock undergoes

the majority of its structural deformation. Upon reaching the Alpine

Fault, the rocks undergo simple vertical shear and uplift. This is fol-

lowed by more rock continuing to approach the area and be translated

up an oblique ramp which has been created. During the devolatization

and ramping processes fluid continues to be produced and experiences

isothermal uplift and minor deformation.

3. Geophysical Evidence for Fluid Connectivity

Results of geological studies performed as part of the South Island

GeopHysical Transect (SIGHT) programme have found the crustal re-

gion beneath the Southern Alps of New Zealand to be a region of low

seismic P-wave velocity, as well as zones of low resistivity (30-100 Ω·m).

These regions are coincident and are believed to be a cause of the en-

hanced pore fluid pressure of saline fluid in the region of interconnected

grain boundaries. Further evidence suggests that metamorphic fluid

has been released in the orogenic system and has escaped vertically.

However, the region of low resistivity is not found in the same location

of metamorphic fluid production, but is offset to the west and upwards.

It is therefore believed that the fluid connectivity within the rock,

which results in regions of low resistivity, is believed to be associated

6 1. GEOLOGICAL BACKGROUND

with metamorphic fluid generation as well as the upward ramping of the

rock from the collision of the plates. As the fluid is translated along the

ramp it experiences delayed connectivity resulting in different regions

of fluid generation and connectivity.

4. Metamorphic Fluid Flow

After the generation of metamorphic fluid, we see that the fluid can

flow through rocks as the permeability of the system is increased. This

increased permeability (fluid connectivity) is reached by microcrack-

ing, followed by increased pore fluid pressures which facilitates in the

the further opening of microcracks. Therefore, fluid flow is the result

of fluid generation, the initial flow of fluid into rock fractures, and the

continued flow of fluid through fractures resulting in more microcrack-

ing.

From magnetotelluric data of the Southern Alps, we know that the

region has a high level of fluid connectivity. Additionally, the data

suggests that fluid in the region escapes upward which is consistent

with the knowledge that fluids at great depths undergo flow that is

guided by buoyancy. Basic geological knowledge asserts that fluid flows

along faults and shear zones; however, models of fluid flow along grain

boundaries are too slow and do not account for the fluid flux of this

system.

It has been suggested that this inconsistency can be accounted for

by either deformations of the rock, an influx of fluid due to metamor-

phic reactions, or dynamic permeability changes due to compaction.

5. A MODEL OF MICROCRACK COALESCENCE 7

However, they can all be shown to be inconsistent with other data ob-

tained which is related to their coupling of space and time. Therefore,

another mechanism must exist which allows the fluid to expand as it

is translated upward along the Alpine Fault.

5. A Model of Microcrack Coalescence

As mentioned above, there must exist some mechanism that causes

fluid to expand while being translated upward. The metamorphic con-

nectivity found in regions of low resistivity must then be the result of

fluid expansion experienced during exhumation. As the fluid moves

upward and is exhumed, there is a decrease in surrounding pressure.

Therefore, the small droplets of fluid found throughout the rock are

able to expand. This expansion allows fluid to fill the grain boundaries

and to create “spiderlegs” which cause microcracking in the rock and

allow tubules of fluid to connect with one another. This expansion is

a function of the depth of the fluid in the rock since pressure depends

of the depth. There is a critical depth where the pressure allows fluid

to escape and creates a connected system of fluid. This corresponds to

the percolation threshold which is discussed in the next section.

CHAPTER 2

Percolation Theory

1. Introduction

Initially introduced as a model for understanding the flow of carbon

particles through a gas mask follwed by applications as a stochastic

method of studying the effects of polymerization, percolation theory

has developed into a fundamental branch of statistical physics. Cur-

rently, many problems from diverse fields are studied in the context of

percolation because of its simple description of random behavior and

the universal laws that are obtained from such simple models. Over

the past fifty years, a large amount of literature has been written about

percolation, resulting in a vast assortment of applications of the theory

to a variety of fields of study.

Successful early applications of the theory include the design of elec-

tronic and magnetic materials, the conceptualization of geometrical and

topological characteristics of porous media, and the understanding of

miscible and immiscible displacements in porous media. Use of perco-

lation theory has proven to be successful in many applications and this

is due to its ability to provide from simple models the universal laws

which determine the geometrical and physical properties of the system.

Imagine a very large array of squares where the lines cross perpen-

dicularly at fixed intervals. This array is known as a lattice and can

8

2. TEXAS FARM EXAMPLE 9

be considered large enough such that boundary of the lattice has a

neglible effect on the interior of the lattice.

We allow the distribution of the occupation of squares to be ran-

dom, that is, their occupation state is independent of the the state of

other squares. We call p the probability of whether or not a square is

occupied. Therefore, each square has probability p to be occupied, or

open to flow, and probability 1− p to be unoccupied, or closed to flow.

If the lattice is an NxN lattice, then there are pN2 squares occupied

and (1 − p)N2 unoccupied. We then define a cluster to be a group

of neighboring occupied sites. For a square lattice, a cluster is formed

wherever the sides of two occupied squares touch. The two squares are

then called nearest neighbors. If two corners touch then the squares

are called next nearest neighbors. It follows that a cluster consists

of an unbroken chain of nearest neighbor squares.

Percolation theory is the study of these clusters, their size, prop-

erties, and characteristics, particularly in very large (nearly infinite)

lattices. Of special interest is the phenomena of percolation near spe-

cial values of p = pc where the first percolating cluster of the system is

formed. This value pc is known as the percolation threshold.

2. Texas Farm Example

Many fundamental concepts of percolation theory can be developed

by considering this simple Texas Farm Irrigation system. Suppose there

is an “almost infinite” irrigation system with a square network of pipes.

In such an arrangement the pipes meet perpendicularly and make an

interconnect. Suppose also that there is a large river located at the

10 2. PERCOLATION THEORY

northen edge of the farm from which the irrigation receives a constant

flow of water. From our own intuition and experience we know that

if one of the pipes becomes clogged then the flow of water will be

interrupted and diverted through another route of pipes to escape from

the pipes at the southern end of the farm.

The key question asked is: Can the number of clogged pipes can

be determined from the flow of water that escpaes the pipes? That is,

can the connectivity of the system be understood by analyzing external

observation and measurement? If the number of unclogged segments

of pipes is denoted by N , the number of clogged segments at the per-

colation threshold pc is Nc, and Q is the volumetric water flow, then

there is a power law which describes the fluid flow. With an exponent κ

which is determined by theoretical or computational means, this power

law takes the form

Q ∝ (N − Nc)κ(2.1)

This simple law holds for N close to Nc. Other power laws, such

as the infinite cluster density, can be proved when N is very large

because of various assumptions that can be considered. However, as

stated earlier, most of our attention will be directed to phenomena that

occurs near the percolation threshold, thus falling into the situation

where the above expression applies. While this power may appear

to be appropriate for only a small number of applications, it actually

has broad application and historical success in accurately describing

3. THE PERCOLATION PROBLEM AND UNIVERSALITY 11

natural phenomena such as water flow through porous rock, the flow of

electricity through a network, termite penetration through wood, and

wild fire spread through epidemics.

3. The Percolation Problem and Universality

The problem that is fundamental to the theory of percolation con-

sists of three basic questions. First, what is the geometrical or physical

property relevant to the connectivity of the system under investiga-

tion? In physical situations this may be fluid flow or penetration of

some object through a system. Second, what is the percolation thresh-

old? This important threshold depends very much on the system that

is considered and it is around this point that many physical systems ex-

hibit interesting behavior. Third, what is the exponent that describes

the behavior of the property of the system that the theory is being

applied to? In the above example this exponent was κ, but many other

exponents describe other physical phenomena.

One advantage of percolation theory is that many different systems

have the same exponent that descibes the behavior of the properties

of the system. This principle is known as universality and is a cen-

tral principle of the theory. Its usefulness lies in the fact that one can

create a very simple model of a system and solve this system, either

exactly or approximately, and then apply the results to more compli-

cated and complex systems. Often it occurs that a realistic model is

unsolvable and approximations of the solutions can only be obtained

through computational analysis, and even so, the time of computation

may be so large that it is not realistic to tackle such a problem. In this

12 2. PERCOLATION THEORY

respect, simple models are invaluable and the principle of universality

essential.

The Texas Farm model, described above, is an intuitive way of

introducing some concepts of percolation. However, we now transition

to the more general lattice model by changing the “pipes” from the

farm to “lattice sites.” If we let the occupied sites, represented by

large black dots, be those that are open to flow and the unoccupied

site, small black dots, be those closed to flow, then the two systems are

equivalent. An example of such a lattice model is shown in FIGURE

2.1. We say that two nearest neighbors are bound if they are both

occupied. Additionally, we say that two sites are connected if there

is a continuous path of bounded sites between them. A connected

path of lattice sites allows flow of fluid or electricity (or whatever other

property is being studied) from one end of the system to the other. The

importance of percolation theory is that the connectivity of microscopic

elements is directly related to the physical properties of the macroscopic

system.

4. Fundamental Concepts

Consider a finite square lattice as shown in FIGURE 2.1. Each

lattice point is either occupied or unoccupied and the represenation of

its occupation state is the same as described above. It is an important

assumption that we each lattice’s occupation state be random. This

means that the state of occupation of one site is completely independent

of the state of occupation of any other site. Then, for the finite square

lattice, we know that the probability that any given site is occupied

4. FUNDAMENTAL CONCEPTS 13

Figure 2.1. Example of a square lattice with p > pc

such that there is a cluster from one end of the lattice tothe other.

is p = N/N0, where N is the number of occupied sites and N0 is the

total number of sites in the lattice. This probability of occupation is

well defined as the lattice becomes infinite, that is, as N0 approaches

infinity.

A main focus of percolation theory is the study of clusters of the

lattice and their properties such as shape and size. Different lattices

will produce different clusters and, therefore, it is important that the

statistics of the clusters are studied. Before mentioning how these

clusters are studied it is appropriate to discuss the variety of well-

known and well-studied lattice types.

The most simple lattice is the one-dimensional lattice with lattice

points along a line where lattice sites are a fixed distance from each

other. This simple model can be solved exactly and will be discussed

more thoroughly in a later section. The next natural model to consider

is the simple two-dimensional square lattice, such as we discussed in the

14 2. PERCOLATION THEORY

Texas Farm Irrigation example. Other two-dimensional cases that have

been studied are the triangular lattice and the honeycomb lattice. The

triangular lattice is obtained when the sites of the lattice are placed

on the crossing points of a lattice model with diagonally directioned

lines. The honeycomb model is similarly obtained, but site locations

are placed in the centers of the triangles rather than the crossing points.

The simple square lattice can naturally be extended to the three-

dimensional cubic lattice model. Model modifications of this cubic

model exist such as the body-centered cubic (bcc) lattice, the face-

centered cubic (fcc) lattice, and the diamond lattice.

Everything defined so far falls under the category of site percola-

tion. However, there exists another fundamental type of percolation

model known as bond percolation. Site percolation is exactly what

we have considered thus far, the occupation state is referring to the

site location. In bond percolation, we let there be a line, or bond,

connecting every two sites. Then the probability that this connecting

bond is open is p and the probability that it is closed is (1− p). In this

case, a cluster is a group of connected open bonds. Historically, bond

percolation preceded site percolation, but it is now more traditional to

introduce the basics of percolation theory using site percolation because

the counting of clusters sizes is more straightforward.

The following table shows the various percolation thresholds for the

site and bond percolation cases for each of the lattice models discussed,

including some higher-dimensional lattices.

For the rest of our discussion we consider only the site percolation case.

5. ONE-DIMENSIONAL LATTICE 15

Site BondHoneycomb 0.6962 0.65271

Square 0.592746 0.50000Triangular 0.500000 0.34729Diamond 0.43 0.388

Simple Cubic 0.3116 0.2488Cubic (Body-centered) 0.246 0.1803Cubic (Face-centered) 0.198 0.119

d = 4 hypercubic 0.197 0.1601d = 5 hypercubic 0.141 0.1182d = 6 hypercubic 0.107 0.0941d = 7 hypercubic 0.089 0.0787

Table 1. Percolation thresholds for a variety of latticesfor both site and bond percolation

Continuing our study if the clusters of a lattice, by analyzing the

number of clusters with s sites per lattice site and defining this to be

ns(p). We begin with the one-dimensional lattice which shows the fun-

damental concepts without the complications of more complex models

and the advantage of this lattice is that it can be solved exactly.

5. One-Dimensional Lattice

Here we consider the site percolation problem along an infinitely

long linear chain where the site locations are separated by a fixed dis-

tance throughout the chain. The occupation of each site is random

with probability p that it is occupied. Since the occupation of each

site is random it follows that the probability of any two given sites are

occupied is p2. It follows by the independence of site occupation that

the probability that any s sites are occupied is ps.

We next ask how many clusters of length s are there along the chain

if the length of the chain is L, where L → ∞ and is much larger than

16 2. PERCOLATION THEORY

s? Notice that each site on the chain has probability ps(1 − p)2. This

is because ps is the probability that any s sites are occupied and then

since the two sites on the end must be unoccupied, which occurs with

probability (1 − p)2, it then follows that ps(1 − p)2 is the probability

that a site is the left end of a cluster of size s. Then the total number of

s-sized clusters is Lps(1−p)2. This is the reason that we talk about the

number of clusters per lattice site, which is the total number divided

by L and therefore independent of the lattice size. Since there are s

sites in each s-cluster, it follows that the probability that a site is part

of an s-cluster instead of just then left end is sps(1 − p)2.

We see that the percolation threshold must be pc = 1. If p were less

than 1 then the the number of unoccupied sites along the chain would

be L(1 − p) and as L goes to infinity it guarentees that there will be

at least one unoccupied site which will divide the chain to at least two

clusters. And recall that the percolation threshold occurs when there

is a continuous cluster from one end of the lattice to the other.

As mentioned above, the probability that a site belongs to an s-

cluster is sns(p). We also note that every site must belong to some

cluster. Even if a site is isoloted it still belongs to a cluster, that is,

a cluster of size 1. This is called the unity cluster. Therefore, by

summing over all the possible cluster sizes we find that

∞∑

s=1

nss = p (p < pc)(5.1)

5. ONE-DIMENSIONAL LATTICE 17

This says that the probability of that a site belongs to some cluster (of

any size) is equal to the probability that it is occupied.

Using the fact that p < 1, we are able to express the summation

as a derivtive and verify the above statemetn by using the formula for

geometric series which says∑

∞

s=1 ps = p1−p

.

∞∑

s=1

ps(1 − p)2s = (1 − p)2

∞∑

s=1

pd(ps)

dp

= (1 − p)2pd

∑∞

s=1 ps

dp

= (1 − p)2pd(p/(1 − p))

dp

= p(5.2)

Additionally, we may want to know the average cluster size of the

lattice system. Recall that nss is the probability that any given site

belongs to an s-cluster and∑

s nss is the probability that it belongs to

any finite cluster. We now define

ws =nss∑s nss

(5.3)

where ws is interpreted as being the probability that the cluster to

which an arbitrary occupied site belongs is an s-cluster. We can say

that the average cluster size S is

18 2. PERCOLATION THEORY

S =∑

s

wss(5.4)

We can also calculate the mean cluster size Sm which is

Sm =(1 − p)2

∑∞

s=1 s2ps

∑∞

s=1 nss

=(1 − p2)

∑∞

s=1 s2ps

p

=(1 − p)2

p(p

d

dp)2

∞∑

s=1

ps

=1 + p

1 − p(p < pc)(5.5)

CHAPTER 3

Application to the Alpine Fault

1. Simple Cubic Model

We consider a simple 3D cubic lattice to determine fluid volume in

a bond percolated system. This lattice type has been well studied and

it is known that the percolation threshold is pc = 0.2488, the critical

probability that a site is occupied resulting in an infinite cluster in the

lattice. Below this critical value the system is not percolated; however,

at this value the fluid percolates through the system. We note that

in relation to conductivity, infinite lattices with values p < pc have no

infinite cluster and therefore no conductvity.

First consider a two-dimensional lattice of points that have pipes

with interconnects. Then, for a lattice of one segment along its edge

(one pipe along its edge and therefore four interconnects) there are

A1 = 4 (where A is the total number of pipes for the two-dimensional

lattice and the subscript is the number of segments along the edge the

lattice). Similarly, we find

A2 = 4(1 + 2) = 12

A3 = 4(1 + 2 + 3) = 24

A4 = 4(1 + 2 + 3 + 4) = 40

19

20 3. APPLICATION TO THE ALPINE FAULT

Continuing we obtain the general two-dimensional result for a lat-

tice of s segments

As = 4s∑

i=1

i(1.1)

A similar process for a three-dimensional cubic lattice leads to the

following:

T1 = 2(4) +2

2(4) = 4[(1 + 1) +

1 + 1

2] = 12

T2 = 3(12) +3

2(12) = 12[(2 + 1) +

2 + 1

2] = 54

T3 = 4(24) +4

2(24) = 24[(3 + 1) +

3 + 1

2] = 144

where T is the total number of pipes for the three-dimensional lattice

and the subscript is the number of segments along a given edge of the

lattice, as for the two-dimensional case.

From this we obtain the general three-dimensional result for a lat-

tice of s segments

Ts =3

2(s + 1)[4(1 + 2 + 3 + ...)]

=3

2(s + 1)(4)

s∑

i=1

i

= 6(s + 1)s∑

i=1

i(1.2)

1. SIMPLE CUBIC MODEL 21

Now consider a large cube with total length L and pipe lengths l

such that there are s = L/l pipes along any given edge of the cube.

Then there are

T = 6(L

l+ 1)

L/l∑

i=1

i

pipes with volume v = πr2l, where r and l are the radius and lengths of

the individual pipes, respectively. Then it follows that the total volume

in this system of pipes is

Vtotal = Tv

= 6πr2l(L

l+ 1)

L/l∑

i=1

i

= 6πr2(L + l)

L/l∑

i=1

i(1.3)

Note that∑n

i=1 i = n(n+1)2

. Using this, we get for the total volume

22 3. APPLICATION TO THE ALPINE FAULT

Figure 3.1. Scanning Electron Microscopy (SEM) im-age of a sample of rock from the region near the AlpineFault. The image gives approximate sizes of droplets andtubes in the rock and along grain boundaries.

Vtotal = 6πr2(L + l)

L/l∑

i=1

i

= 3πr2(L + l)L

l(L

l+ 1)

= 3πr2(L + l)L2 + Ll

l2

= 3πr2 L

l2(L + l)2

= 3πr2L(1 +L

l)2(1.4)

The percolation model used in this calculation of critical fluid vol-

ume in the rock is a simple cubic bond lattice which has a percolation

threshold pc = 0.2488 (see TABLE 1).

Therefore, the threshold volume is

1. SIMPLE CUBIC MODEL 23

Vc = pcVtotal

= 3πr2pcL(1 +L

l)2(1.5)

For one cubic meter of medium (L = 1)the critical volume reduces

to

Vc = 3πr2pc(1 +1

l)2(1.6)

Measurements from Scanning Eletron Microscope (SEM) images

show that the grainsize of these of these pipes lie between 10-50 µm

with an average grainsize of about 35 µm. Similarly, it has been found

that the droplet radii vary between 0.05 and 2.5 µm with a typical

radius of 0.1 and 0.5 µm. FIGURE 3.1 shows a Scanning Electron

Microscope (SEM) image of a sample of rock.

r=0.125 µm r=0.25 µm r=0.5 µm r=0.75 µm r=1.0 µml = 20 µm 91.6 366.4 1465.6 3297.6 5862.4l = 25 µm 58.6 234.5 938.0 2110.5 3752.0l = 30 µm 40.7 162.8 651.3 1465.6 2605.5l = 35 µm 29.9 119.6 478.6 1076.8 1914.3l = 40 µm 22.9 91.6 366.4 824.4 1465.7l = 45 µm 18.1 72.4 289.5 651.4 1158.0

Table 1. Critical fluid volume values (cm3) for variouspipe radii and lengths near typical values in one cubicmeter of rock. Average values for the pipe parameters arel = 30µm and r = 0.5µm. The estimated fluid volumeobtained by other means is 917 cm3.

24 3. APPLICATION TO THE ALPINE FAULT

Figure 3.2. Fluid volume versus length for a simplethree-dimensional cubic lattice. We consider a bond per-colation model with a percolation threshold pc = 0.2488.The various curves correspond to different radii of thelattice pipes ranging from 0.0125 to 3.0 µm. SEM imag-ing shows the average length to be near 30 µm and theaverage radius to be around 0.5 µm. The black horizon-tal line is the fluid volume calculated by thermodynamicconsiderations.

2. Percolation Volume vs. Thermodynamic Volume

Dr. Julie Vry of Victoria University of Wellington in New Zealand

has calculated the estimated fluid volume of in one cubic meter of rock.

Her methods, which will not be explained here, were accomplished

through thermodynamic considerations. She considered two cases: one

with no fluid escape through the millions of years of plate motion and

another case where all the fluid was assumed to have escaped. The

2. PERCOLATION VOLUME VS. THERMODYNAMIC VOLUME 25

second case is believed more plausible ny geologists and is the case we

compare to the percolation prediction.

She has found the saline fluid volume in on cubic meter of rock to

be 917 cm3. Referring TABLE 2 and FIGURE 3.2 we see that this

estimated fluid volume falls directly within the possible range for the

various volumes calculated over different pipe lengths. We then see

that the two predictions appear to be in agreement.

The fundamental problem of trying to understand the anomalous

behavior of fluid in crustal rocks of the Southern Alps of New Zealand

has been investigated. We have found percolation theory to be success-

ful in accurately predicting a critical depth which corresponds to a crit-

ical fluid volume in the rock. This predicted fluid volume is consistent

with the fluid volume predicted by thermodynamic calculations. We

see that percolation theory demonstrates how to upscale information

about the microstructure to understand macroscopic fluid transport

properties.

Bibliography

[1] D. Stauffer and A. Aharony Introduction to Percolation Theory, Taylor and

Francis, London, ed. 2, 1992.

[2] J. Vry Development of Fluid Connectivity in Collisional Orogenesis: Southern

Alps, New Zealand, (Pre-print 2006).

[3] B. Berkowitz and I. Balberg Percolation Theory and Its Application to Ground-

water Hydrology, Water Resources Research, Vol. 29, No. 4, Pages 775-594,

April 1993.

[4] P. Wannamaker, et al. Fluid generation and pathways beneath an active com-

pressional orogen, the New Zealand Southern Alps, inferred from magnetotel-

luric data, J. Geophys. Res., 107(B6), 2117, doi:10.1029/2001JB000186

[5] FIGURE 1.1 Maps obtained through correspondence with Dr. Julie Vry of

Victoria University at Wellington in New Zealand

[6] FIGURE 1.2 Picture obtained through correspondence with Dr. Julie Vry of

Victoria University at Wellington in New Zealand

[7] FIGURE 2.1 Picture obtain from Introduction to Percolation Theory by D.

Stauffer and A. Aharony, page 7, Taylor and Francis, London, ed. 2, 1992.

[8] FIGURE 3.1 SEM image obtained through correspondence with Dr. Julie Vry

of Victoria University at Wellington in New Zealand

[9] FIGURE 3.2 Graph made using MATLAB, a numerical computing program

made by The Mathworks

[10] TABLE 1 Data obtain from Introduction to Percolation Theory by D. Stauffer

and A. Aharony, page 17, Taylor and Francis, London, ed. 2, 1992.

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