UNIVERSITY OF LONDON

DEPARTMENT OF EARTH SCIENCE AND ENGINEERING

PREDICTING ABSOLUTE AND RELATIVE PERMEABILITIES

OF CARBONATE ROCKS USING IMAGE ANALYSIS AND

EFFECTIVE MEDIUM THEORY

by

Mathieu Jurgawczynski

A thesis submitted in fulfilment of the requirements for

the degree of Doctor of Philosophy of the University of London

and the Diploma of Imperial College

February 2007

2

ABSTRACT

Relating the transport properties of a reservoir rock to its pore structure is of great

relevance to many scientific, environmental or industrial problems, such as extraction of

hydrocarbons from oil and gas reservoirs. Traditionally, most studies of rock physics for

petroleum applications have focused on sandstones. Recently, attention has turned to

carbonates, which are the host rocks of most remaining oil and gas reserves. The

complex and heterogeneous pore structure of carbonates poses many new difficulties for

petrophysical modelling.

In this thesis, a combination of image analysis and effective medium theory is used

to develop a method for estimating the permeability of reservoir rocks from two-

dimensional pore images, without resorting to computationally intensive image analysis

or network calculation procedures. Areas and perimeters of individual pores are first

measured from scanning electron micrographs of thin sections. The hydraulic

conductance of each pore is then inferred using various stereological corrections and

hydrodynamic approximations. Finally, Kirkpatrick’s effective medium theory is used

to infer an effective pore conductance, which leads to a permeability estimate.

This methodology was applied to twelve carbonate samples from hydrocarbon

basins from various parts of the world. In most cases the method accurately predicts the

permeability within a factor of two, for a set of samples with permeabilities ranging

from 1-1000 milliDarcies. The few exceptions were rocks for which the small available

image size does not fully capture a sufficient population of pores.

The method was also extended to the problem of two-phase relative permeability.

This was tested for both sandstones and carbonate rocks. By assuming a water-wet rock

and a pore occupancy based on the pore size, it is possible to build phase-specific

conductance distributions. The same procedure as is applied to the single-phase case is

then applied to each phase separately. After several preliminary tests, a probabilistic

accessibility factor is introduced to modify the pore occupancies. In general, the correct

behaviour of the two relative permeability curves, as functions of phase saturation, was

obtained, although neither the curves nor the end points were perfectly matched.

3

CONTENTS

ABSTRACT.………..………………………………………..…………………….….... 2

TABLE OF CONTENTS…………………………………..…………………….….….. 3

LIST OF FIGURES………………………………………………..………….….……... 8

LIST OF TABLES………………………………………………………….…..………. 14

NOMENCLATURE…………………………………………………………………….. 15

ACKNOWLEDGEMENTS…………………………………………….…….………… 17

1 INTRODUCTION…………………………………….…………..……….…………. 18

2 PREVIOUS PORE-SCALE MODELS FOR PERMEABILITY……….….….……... 21

2.1 Permeability and Darcy’s law.………………………………….…….…………. 21

2.2 Hagen-Poiseuille equation.…………………………………….…….………..…. 22

2.3 Bundle of identical tubes.……………………………………………..……....…. 24

2.4 Kozeny-Carman equation and the hydraulic radius approximation………..….… 25

2.4.1 Kozeny Carman equation……..…………………………………..………... 25

2.4.2 Hydraulic radius approximation………………………………..………..…. 26

2.5 Permeability prediction using network models……..………………..………….. 28

2.5.1 Adler et al. …………………………………………....…….….…..……..... 29

2.5.2 Reconstruction methods………………………………………………….… 30

2.5.3 Okabe and Blunt…………………………………….………………............ 31

2.5.4 The NETSIM code………………………………………………...…..…..... 32

2.6 Image analysis and permeability prediction………..….………….…..….……… 35

2.6.1 Berryman and Blair.…………...………………..……………….................. 35

2.6.2 Cerepi et al. ……………………………...…..……………………...…….... 37

3 REVIEW OF THE METHOD OF SCHLUETER AND LOCK……………….…….. 39

4

3.1 Image analysis…………………………..………………………....………..…… 39

3.2 Effective medium theory……..………………………………….………….…… 40

3.3 The model of Koplik et al. …………………..………………….………….…… 42

3.4 The model of Schlueter………………………………………….……………..... 44

3.5 The model of Lock……..…………………………………..…….……………… 49

3.5.1 Stereological factors...………………………………....…….….………….. 49

3.5.2 Areal thresholding…………………………………………….…...…..…… 52

3.5.3 Pore number density…………………………………………….………..… 53

3.5.4 Automated image analysis…………………………………………..…..….. 53

3.5.5 Summary of Lock’s computational procedure…………………….…..…… 54

3.5.6 Results…………………………………………………………………..….. 55

4 EFFECTIVE MEDIUM THEORY….…………………………..…………….……… 57

4.1 Description of the theory………………………………………….……………... 57

4.2 Integral form of the EMA and some closed-form solutions………....…..………. 60

4.3 Special examples with binary conductances.…………………….……………… 61

4.4 Discrete form of Kirkpatrick's equation…………….…………………………… 63

4.5 Limitations of Kirkpatrick’s EMA…………......................................................... 65

4.5.1 Limitations near the percolation threshold……..…….………..…..……..… 65

4.5.2 Limitations with the input data.………………………...…………..………. 75

5 REVIEW OF DATA SET AND SAMPLES.………………………………………… 80

5.1 Carbonates vs. sandstones…………………...………………………..…………. 80

5.2 Data acquisition………………………………………………………….………. 82

5.2.1 Core sampling and sample preparation……………………..…………....… 82

5

5.2.2 Scanning electron microscopy………………………………..……...……... 83

5.2.3 Data presentation………..…………………………………………...……... 83

5.3 Outcrop samples: Southeast France (SEF)………………………......…………... 84

5.3.1 Orgon and Belvedere samples (SEF-1 & SEF-2)…………………………... 85

5.3.2 Rustrel sample (SEF-3)……………………………….……………....….… 86

5.4 Middle East field 1 (ME1)………………….…………………….……………... 88

5.4.1 Geology, texture and diagenesis.…………….……………...……...…….… 88

5.4.2 Pore system and petrophysical properties…………….…………..……...… 91

5.5 Middle East field 2 (ME2)………………….………..………………………….. 94

5.5.1 Geology, texture and diagenesis.…………….………………………...…… 94

5.5.2 Pore system and petrophysical properties………….….……………….…... 95

6 SINGLE PHASE PERMEABILITY PREDICTION………………….…………....… 97

6.1 Image analysis.……………………..……………………….…………………… 97

6.1.1 Image analysis software…………….……………………..……..….……… 97

6.1.2 Grey-level histogram and image segmentation…………...…….….…….… 98

6.2 Data acquisition and manipulation…………….………………………....... 104

6.2.1 Data collection…………….………………………………….....…….…. 105

6.2.2 Data manipulation and the elimination of micro-porosity…………..….… 106

6.3 Computational procedure……………...……………………….……………..... 108

6.4 Absolute permeability predictions……………………………………………... 109

6.4.1 Results using a co-ordination number of 6……………………………….…109

6

6.4.2 Introduction of a varying co-ordination number………………......…...…. 113

6.4.3 Updating the cubic lattice……………………………………………..…... 116

6.4.4 New results……………………………………………………………..…. 119

6.4.5 Special case: non-touching vugs…………………………………….……. 121

7 RELATIVE PERMEABILITY PREDICTIONS……………….……..…………….. 123

7.1 Definition………………………………………………………..……………... 123

7.2 Relative permeability measurements…………………….……………..……… 125

7.2.1 Steady state flow method…………………………………………...…..… 125

7.2.2 Unsteady state flow method……………………………………….…….... 127

7.3 Network modelling of relative permeability………………………………….... 127

7.3.1 History of network modelling……………………………..…….………... 128

7.3.2 Wettability changes…………………………………………….………..... 129

7.3.3 Computation of relative permeability…………………………….…….… 129

7.4 The model of Levine and Cuthiell………………………………………....…... 133

7.5 Preliminary test of two-phase flow model………………………………..……. 134

7.5.1 Constant pore-size distribution……………………………………...….…. 134

7.5.2 Uniform pore-size distribution……………………………………..…..…. 136

7.5.3 Log-normal pore-size distribution………………………………..……….. 138

7.6 Relative permeability of Berea sandstone……………………………………… 139

7.6.1 Schlueter’s two-phase work…………………………………….……...…. 139

7.6.2 Preliminary results for Berea…………………………………..………….. 140

7.6.3 Introduction of the GPA………………………………………..…………. 142

7

7.6.4 Probabilistic pore wettability…………………………………...……….… 144

7.6.5 Introduction of the irreducible water saturation……………………...…… 149

7.6.6 Final Berea relative permeability curves…………………………..…...…. 150

7.7 Relative permeability of carbonate rock ME2-2…………………………….…. 152

8 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK…………….…… 155

REFERENCES…………………………………………………………………...…… 160

APPENDICES: CODE LISTING…………………………………………………...… 170

A Effective-Medium Approximation (Newton-Raphson method)………..………..… 171

B Effective-Medium Approximation (Bisection method)…………….……………… 173

C SEM images of the rocks used during the research………………………………... 175

D Excel template for 2-phase relative permeability computation…...……………...… 181

8

LIST OF FIGURES

Figure 2.1 Cylindrical element of fluid during laminar flow through a

circular pipe. Force balance on the element yields eq. (2.5).

23

Figure 2.2 A plot of absolute difference between the outputs obtained for

simulations on ISONETSIM and Kirkpatrick’s equation as a

function of lattice size at 20-30 realisations.

35

Figure 3.1 An illustration showing the replacement of a discrete network of

conductances by a homogeneous network having the same

topology.

41

Figure 3.2 Illustration showing how the area of a slice through a single pore

at some arbitrary angle θ will generally be larger than the true

cross-sectional area.

44

Figure 3.3 Constriction factor for hydraulic conductivity as function of the

ratio of the minimum pore radius to the maximum pore radius of

an individual pore.

47

Figure 3.4 Constriction factor f as a function of amin/amax for saw-tooth and

sinusoidal profiles.

52

Figure 3.5 A two-dimensional illustration showing the number of additional

pores bisected when a slice is taken at an angle θ that is not

perpendicular to a lattice direction.

53

Figure 3.6 Measured permeabilities plotted against predicted values for both

UKCS and St. Bees data. The upper and lower lines correspond to

errors of a factor of two in either direction.

56

Figure 4.1 Construction used in calculating the “pressure” induced across one

conductance, Cm, surrounded by a uniform medium (after

Kirkpatrick, 1973).

58

Figure 4.2 Kirkpatrick’s function for the twenty-one conductance values used 64

9

by Priest (1992), plotted for the case 6=z . Note that Priest’s

definition of C has the factor of μL incorporated into it.

Figure 4.3 Sensitivity of Ceff (mm2 s-1) to co-ordination number, using

Priest’s data set.

64

Figure 4.4 Bond percolation on a square network at bond occupancies of

p=1/3 and p=2/3. Note that there is no continuous path spanning

the entire region in the former case, but there is in the latter case.

The percolation limit, pc, for the square network is exactly 0.5

(Sahimi, 1995).

66

Figure 4.5 Comparison of normalised permeabilities at co-ordination number

ranging from 1.5 to 6 for both NETSIM and the EMA, both

obtained while artificially lowering the co-ordination number

using zero conductances.

69

Figure 4.6 Comparison of normalised permeabilities at co-ordination number

ranging from 2 to 6, obtained with the EMA. In one method we

add zero conductance to the data set, and in the other we lower the

co-ordination inside the equation, and use eq. (4.29).

70

Figure 4.7 Comparison between the NETSIM prediction and the scaling law,

near the percolation threshold, presented here with three different

critical exponents t.

73

Figure 4.8 Comparison between the NETSIM predictions and the scaling

law, over the entire range of bond probabilities, for three different

critical exponents, t.

74

Figure 4.9 Comparison of NETSIM, Kirkpatrick’s EMT and the GPA for

log-normal conductivity distributions (after Lock et al., 2004).

78

Figure 5.1 Illustration of the Dunham classification (Dunham, 1962). 81

Figure 5.2 Low magnification image from the Belvedere sample (SEF-2), a

very fine peloidal grainstone. The presence of a slightly leached

benthic foraminifer is indicated on the plate (B).

86

10

Figure 5.3 Figure 5.3. Low magnification image from the Rustrel sample

(SEF-3). The presence of the gastropod is illustrated on the plate

with G, while D illustrates a severely leached orbitolinid and O

shows an oversized pore created after leaching occurred in the

sample.

87

Figure 5.4 (i) General view of sample ME1-1, at ×45 magnification. The

presence of a partially dissolved benthic foraminifer is highlighted

at the top of the image (B). Note the high porosity (in blue) due to

the lack of inter-granular cement.

90

Figure 5.4 (ii) A closer view of the sample ME1-1 (×113 magnification) with the

presence of oversized pores (O) confirming the effects of leaching

on the sample.

90

Figure 5.5 (i) Overall view of sample ME1-4 (×70 magnification) with the

presence of partially dissolved benthic foraminifers (B) and

dissolution vugs (V), due to leaching, as well as calcite-cemented

areas (C).

93

Figure 5.5 (ii) Closer view (×280 magnification) of the macro-pores system with

the presence of an oversized pore (O) following the leaching

process.

93

Figure 5.5 (iii) Illustration of more calcite-cemented areas (C) and oversized

pores (O), at the same magnification of the previous plate (×280).

This also illustrates the fairly good connectivity of the porosity

system.

93

Figure 5.5 (iv) Illustration of the irregular-shaped faces of calcite crystals

surrounding a peloidal grain. This probably indicates re-

crystallisation process, which could have enhanced microporosity

networking.

93

Figure 6.1 Illustration of the thresholding procedure on sample ME1-5, from

our second data set. In (a), the darkest regions represent the pore

space while the remainder is the grain area. In (b), the pores are

99

11

now in black and the remaining features mapped to a binary

output of zero, hence providing us a binary representation of the

image.

Figure 6.2 Illustration of the grey-level histogram for sample ME1-5, where

the abscissa is the grey-level values (0-255), and the ordinate is

frequency of occurrence.

100

Figure 6.3 Illustration of the histogram generated for sample SEF-1, with

cemented grains and micro-porosity. The histogram presents only

one clear peak for the grains and then a slow decrease towards the

final black value of zero, where a peak appears.

103

Figure 6.4 Illustration of the variation of the image porosity as a function of

the segmentation value. The helium porosity value is circled in

black on the graph, and the corresponding grey-level threshold

value was used for the segmentation process.

104

Figure 6.5 Predicted permeabilities plotted against measured values for the

three data sets. The upper and lower lines correspond to errors of a

factor of two in either direction.

112

Figure 6.6 Correlation between the co-ordination number and the pore

volume for Fontainebleau sandstone (after Sok et al., 2002).

114

Figure 6.7 Illustration of the derived correlation between the co-ordination

number and the equivalent spherical radius.

115

Figure 6.8 Illustration of the derived correlation between the permeability

and the co-ordination number and the equivalent spherical.

116

Figure 6.9 Illustration of the cubic lattice with a maximum of four added

opposite branches, thereby yielding a co-ordination number of 14.

Please note the branches are not intended to intersect each other in

the middle of the lattice.

117

Figure 6.10 Predicted permeabilities plotted against measured values for the

carbonate rocks data using the varying co-ordination number.

119

12

Figure 6.11 Predicted permeabilities plotted against measured values for

Lock’s sandstone data. Both the original and new predictions are

presented here.

120

Figure 7.1 Illustration of typical relative permeability curves for different

cores of Berea sandstone. The data presented here is taken from

Oak et al. (1990).

125

Figure 7.2 Predicted and measured oil/water relative permeabilities as

obtained by Blunt et al. (2002). The points are from the

experimental data of Oak (1990), while the solid lines are

predictions using their network model previously described. (a)

Primary drainage: In the network model a receding contact angle

of 0º is assumed. (b) Imbibition: Here a distribution of advancing

contact angles, randomly assigned to pores and throats is assumed.

Contact angles range from 30º to 90º, with a mean of 60º.

132

Figure 7.3 Relative permeability curves obtained for the two-phase model

computed using a delta-function pore-size distribution.

136

Figure 7.4 Relative permeability curves obtained for the two-phase model

computed using a uniform pore-size distribution.

137

Figure 7.5 Relative permeability curves obtained for the two-phase model

computed using a log-normal pore-size distribution, with a mean

of 1 and a standard deviation of 0.5.

139

Figure 7.6 Prediction of the relative permeability curves for Berea sandstone

using the model described previously and the EMA, along with

values measured by Oak (1990)

141

Figure 7.7 Prediction of the relative permeability curves for Berea sandstone

using the first model and the GPA in combination with the scaling

law, for two critical exponents.

144

Figure 7.8 Illustration of the phase pore size distribution from a Berea sample

impregnated with both non-wetting and wetting phase (after

145

13

Schlueter, 1995).

Figure 7.9 Wetting phase relative permeability curves obtained with a range

of probability thresholds, and a critical exponent of t = 1.6.

146

Figure 7.10 Wetting phase relative permeability curves obtained with a range

of probability thresholds, and a critical exponent of t = 2.

147

Figure 7.11 Non-wetting phase relative permeability curves obtained with a

range of probability thresholds, and a critical exponent of t = 1.6.

147

Figure 7.12 Non-wetting phase relative permeability curves obtained with a

range of probability thresholds, and a critical exponent of t = 2.

148

Figure 7.13 Comparison between the final relative permeability curves for

Berea, presented for both critical exponents, and the experimental

data from Oak (1990).

151

Figure 7.14 Comparison between the experimental data of sample ME2-2 and

the final relative permeability curves, shown here for both critical

exponents.

153

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LIST OF TABLES

Table 2.1 Accuracy of the hydraulic approximation (Schlueter, 1995) 27

Table 2.2 Porosity and permeability values estimated by Berryman and Blair

(1987), compared with laboratory values.

37

Table 3.1 Permeability predictions from Schlueter (1995). 48

Table 3.2 Comparison of stereological factors used by Schlueter and Lock. 51

Table 4.1 Numerical estimates of bond percolation threshold, site percolation

threshold for four common three-dimensional networks (after

Sahimi, 1995)

67

Table 5.1 Basic petrophysical properties for the SEF data set. 87

Table 5.2 Basic petrophysical properties for the ME1 data set 92

Table 5.3 Basic petrophysical properties for the ME2 data set. 96

Table 6.1 Illustration of the correspondence between the grey-scale values and

some mineral phases found in rocks, taken from standard grey-level

calibration standards, originally supplied by Applied Reservoir

Technologies, Ltd.

101

Table 6.2 Absolute permeability predictions using 6=z . 110

15

NOMENCLATURE

a mean pore radius [m]

a radius of pore tube [m]

A cross-sectional area [m2]

Apore surface area of pore [m2]

Bc co-ordination number at bond percolation threshold

c product of the tortuosity factor and the Kozeny constant

C conductance [m4]

C0 net conductance [m4]

C[1] effective conductance when all conductances are set to unity [m4]

Ceff effective conductance [m4]

Ceff-w effective conductance of the wetting phase [m4]

Ceff-nw effective conductance of the non-wetting phase [m4]

Ceq equivalent conductance [m4]

Ch hydraulic conductance [m4]

d diameter [m]

g acceleration due to gravity [m/s2]

Itot total current [A]

k permeability [mD]

kg geometric mean [mD]

kri relative permeability of phase i

L tube length [m]

n number of nodes in the lattice

p probability

pc percolation threshold

pcb bond percolation threshold

pcs site percolation threshold

pth probability threshold

P pressure [Pa]

P∇ average pressure gradient [Pa/m]

Q flowrate [m3/s]

Qtm total flow rate for the multi-phase configuration [m3/s]

Qts total flow in single-phase configuration [m3/s]

16

r radius [m]

rh hydraulic radius [m]

R(u) correlation function

S specific surface area

S2(r) two-point correlation function

t critical exponent

u Darcy velocity [m/s]

Vbulk bulk volume [m3]

Vip volume of phase p in element i [m4]

V0 upper row potential [V]

V∞ lower row potential [V]

z co-ordination number

Z(x) phase function

Greek letters

λ radius variation wavelength

ξ roughness magitude

μ viscosity [Pa s]

φ porosity

ρ density [kg/m3]

Π pore perimeter [m]

τ shear stress [Pa]

Subscripts

actual actual value

apparent apparent value

eff effective value

eff-w effective value for wetting phase

eff-nw effective value for non-wetting phase

i phase i

measured measured value

r relative

w wetting phase

nw non-wetting phase

17

ACKNOWLEDGEMENTS

I would like to express my most sincere gratitude to my supervisor, Prof. Robert

Zimmerman, for giving me the opportunity to carry out this project, as well as for his

knowledge, advice and guidance. I would also like to thank Dr. Xudong Jing for his

assistance throughout the project, as well as Dr. Carlos Grattoni, who spent many hours

helping and supporting me along the way.

Many thanks are due to Shell for providing both the financial support as well as

most of the data used during this research. The carbonate team was a pleasure to work

with, in particular Paul Wagner, who spent much time with me during my visit to

Rijswijk. Special thanks also go to Fons Marcelis, who provided support with the data,

and acted as a vital link between Shell and me.

There are many people at Imperial College whose help has been very much

appreciated throughout the years; Prof. Martin Blunt, for his help with the two-phase

work, Dr. Mike Ala and Dr. Matthew Jackson for giving me the opportunity to help

with various courses in the department, as well as Dr. Chris Harris for his comments

and suggestions. Also, special thanks to Dr. Peter Lock for providing some support at

the start of the thesis that helped me hit the ground running.

Special thanks must go out to my friends here at Imperial, including, Ruben Rubio,

Igor Siveroni, Greg Lusted, Karl Charvin, Romain Guises, Hu Dong, Hiroshi Okabe,

Ernesto Addiego-Guevara, Martin Putz, Matthew Rhodes, Malin Kalynder, and John

Stephenson, for their friendship and help over the past three years.

Also big thanks to all my friends outside of Imperial College. It would take too

long to mention everyone here, but special mention must be made to Colin, Hector, Ian,

Johanna, Lidwine, Bola, Sam, Yan, Vero, Erika, Luke, Christophe, Julien, Ludovic,

Pablo, Jon, Jerome and Cecile for their friendship.

Finally, my greatest thanks go to my parents and my sister, without whom this

thesis would not have been possible.

18

1 INTRODUCTION Being able to relate the transport properties of rocks to their internal pore structure has

long been of great interest to hydrologists, earth scientists and petroleum engineers.

This problem can be addressed with various degrees of detail, with the required amount

of micro-structural data varying depending on the model. At one end of the spectrum,

empirical permeability models such as that of Kozeny-Carman predict values of the

permeability using knowledge only of porosity and a mean pore diameter or mean grain

size. The Kozeny-Carman approach therefore requires some means of estimating the

specific surface, which can be problematic (Berryman and Blair, 1986). It is also known

that this model, although fairly accurate for unconsolidated sands, tends to become

unreliable for consolidated sandstones.

At the other end of the spectrum are models that attempt to reconstruct the

complete 3-D micro-structure of the rock, which can then be used either to compute the

properties directly using the Navier-Stokes equations (Adler et al., 1990), or as a

starting point for the development of network models (Blunt, 2001). These network

models can then be used to compute various flow properties, such as absolute or relative

permeabilities (Blunt et al., 2002), as well as to gain insight into the relationship

between pore geometry and petrophysical properties (Arns et al., 2004). It is also

possible to reconstruct the pore space without having to resort to the use of imaging

techniques. For example, Øren and Bakke (2000, 2002) constructed an artificial porous

medium using sedimentation algorithms, and used this medium as a basis for their

permeability calculations. Okabe and Blunt (2004) used multiple point statistics on a

series of 2-D images to reconstruct the pore structure of a 3-D sample. The network

permeability was finally computed using the lattice-Boltzmann method, with good

results. However, although these methods can provide insight into the relation between

permeability and pore geometry, they do not yet appear to lead to the prediction of the

permeability of specific rocks.

Lock (2001) developed a method for predicting single-phase permeabilities of

sandstones, using only a small number of SEM images, with a minimum of

computational effort. His approach was a refinement of the method developed by

Schlueter (1995), which in turn was an extension of the method used by Koplik et al.

19

(1984). In Lock’s approach, the hydraulic radius approximation is used to compute the

individual conductivities of the pores, stereological correction factors are applied to

determine the true cross-sectional shapes from the images, and a constriction factor is

applied to account for the effect of the variation of the cross-sectional are along the tube

length. The effective medium approximation of Kirkpatrick (1973) is then used to

determine an effective pore conductance from the measured distribution of conductances.

The procedure was tested on several sandstones with permeabilities ranging from 10 to

1000 mD, and the permeabilities were typically predicted within a factor of two of the

measured values.

In the present research, we attempt to extend the methodology developed by Lock

for sandstones to the prediction of the permeability of carbonate rocks. In recent years,

attention has turned towards carbonates within the oil industry, because most oil

reserves are found in carbonate reservoirs. These are, however, notoriously more

difficult to produce than are sandstone reservoirs. This is due to the fact that, in

carbonate rocks, the processes of sedimentation and diagenesis produce a wide range of

pore size distributions, resulting in a complex internal pore structure. The methodology

is therefore of particular interest for carbonates because, while tomographic imaging can

be used for sandstones, that technique is still being refined for carbonates. Thus, in

those cases where a good quality core is not available, a methodology such as Lock’s

can provide a quick means for the prediction of permeability from drill cuttings.

Because of the more complex pore structure found in carbonates, the methodology

set up by Lock for sandstones had to be entirely reviewed. Firstly, the image analysis

process had to be adjusted in some cases because of the difficulty in identifying a

thresholding point using Lock’s method. We then present a throughout study of

Kirkpatrick’s effective medium approximation, and assess the suitability of the method

with regards to carbonate rocks. The second, areal, thresholding process is also adjusted

to account for the wide pore size distributions encountered in carbonates. Finally, in

order to improve the methodology, a varying co-ordination number is introduced to

compensate for an over/under-prediction that was noticed in the preliminary results. The

method is then applied to twelve carbonate samples, with permeabilities ranging from

0.5-1000 milliDarcies, and the permeabilities are in most cases predicted within a factor

of two.

In the final chapter, the method is extended to the prediction of relative

permeabilities, which are of particular importance in the oil industry. Indeed, in most

cases, a combination of fluids will be found in the pore space. The fractional saturation

20

of each phase will influence the phase permeabilities, thereby leading to the concept of

relative permeability. The extensive use of reservoir simulation in the evaluation,

development and management of oil fields is placing increased importance on data such

as relative permeability. The extended model for the relative permeability prediction is

first tested on Berea sandstone, before being applied to a carbonate sample. In both

cases, the behaviour of the two relative permeability curves is correctly predicted, as

functions of phase saturation, although it proves more problematic to accurately predict

the end-points as well as the precise relative permeability values. In the final chapter,

some suggestions are given for future work.

21

2 PREVIOUS PORE-SCALE MODELS FOR PERMEABILITY 2.1 Permeability and Darcy’s law

Consider a porous rock filled with a pore fluid such as water. Whenever there exists a

spatial gradient in the hydraulic potential of the pore fluid, the fluid will flow through

the rock in response to this gradient. The precise relationship between the potential

gradient and the flowrate is given by Darcy’s law (de Marsily, 1986). The

“permeability” is the constitutive coefficient that relates the flowrate to the potential

gradient in Darcy’s law. Consider first the simplest case of a horizontally oriented,

isotropic rock core subjected to a pressure gradient. The flowrate will be given by

dxdPkAQ

μ−= , (2.1)

where Q is the volumetric flowrate, A is the area of the core normal to the flow, μ is the

viscosity of the pore fluid, dP/dx is the pressure gradient, and k is the permeability.

This equation can be generalised to a three-dimensional case if the flow is not

necessarily horizontal. First, we define a flow velocity (i.e., volumetric flow per unit

area) as AQq /= , which must be interpreted as a vector. Next, we note that if the fluid

is stagnant, the pressure will vary with depth as gzPP o ρ+= , where oP is some

constant, ρ is the fluid density, and the z co-ordinate points downwards. Hence, a

pressure gradient of ),0,0( gP ρ=∇ will not give rise to any flow. For laboratory

purposes, we can consider both g and ρ to be constant. So, it seems that we should

subtract this pressure gradient from the pressure gradient in eq. (2.1), to yield

)( gzPk ρμ

−∇−=q , (2.2)

where gzP ρ− is the hydraulic potential. Finally, we can generalise this to the case of

an anisotropic rock by taking k to be a second-order tensor, in which case (2.2) becomes

)( gzP ρμ

−∇−=kq . (2.3)

22

However, in this thesis we will only consider isotropic rocks, and always take k to be a

scalar. The permeability has dimensions of length squared (L2), suggesting that its

numerical value should be given in units of m2. However, in the petroleum industry the

“Darcy unit” is commonly used, defined by

212 m10987.0Darcy1 −×= . (2.4)

Finally, it should be mentioned that Darcy’s law is only valid for low flowrates (Bear,

1972). At higher flowrates, defined as those for which the Reynolds number is greater

than unity, the pressure gradient is generally found to be a quadratic rather than a linear

function of the flowrate. But such high flowrates are the exception rather than the rule in

petroleum engineering, and non-Darcy effects will not be considered in the present

work.

The influence of the pore fluid on the flowrate is contained solely in the viscosity

term. Hence, the permeability coefficient is a property of the rock, not the fluid, and

therefore depends on the pore geometry of the rock. We now review some of the models

that have been developed to relate pore geometry to permeability. Further details can be

found in the monographs by Scheidegger (1974) and Adler (1992).

2.2 Hagen-Poiseuille equation

The simplest models start from the basic idealisation of pores as being conduits of

circular cross-section. Fluid flow through such a tube is governed by the famous Hagen-

Poiseuille law, which was first discovered empirically in 1840. Due to its central role in

permeability modelling, we will begin with a derivation of this law from first principles.

Consider a straight pipe with a constant radius a through which fluid is flowing

under laminar conditions. We can consider a cylinder of fluid of radius r moving from

left to right, with the axial velocity u that varies with r. This cylindrical element is acted

on by pressure forces at its two ends, and by a shear force along its curved face that

resists the flow. A force balance on this cylindrical element, in the x-direction, gives

02)( 22 =++− xrrPPrP δπτπδπ , (2.5)

which gives, upon passing to the limit as 0→xδ ,

rdxdP τ2

= . (2.6)

23

For a Newtonian viscous fluid, the shear stress is related to the velocity gradient through

the relation

ru∂∂

= μτ . (2.7)

The shear stress arises from the fact that the inner cylinder is moving through the tube

more rapidly than is the outer annular cylinder that surrounds it. Combining eqs. (2.6)

and (2.7) yields

drdu

rdxdP μ2

= . (2.8)

In steady-state flow, the pressure drop will be a constant, and u will vary with r, but not

with x, so we can use a total derivative on the right side of eq. (2.8). If we separate

variables in (2.8) and integrate once, we find

DdxdPrru +=

μ4)(

2

, (2.9)

where D is a constant of integration. The usual boundary condition imposed is that of

“no-slip” at the pipe wall, which implies that .0)( =au This leads to

)/)(4/( 2 dxdPaD μ−= , in which case the velocity profile is found to be

dxdParru

μ4)()(

22 −= . (2.10)

Note that if fluid flows left to right, u will be a positive number, and the pressure will

decrease from left to right, showing that dP/dx is negative, as expected.

Figure 2.1. Cylindrical element of fluid during laminar flow through a circular pipe. Force balance on the element yields eq. (2.5).

δr

P + δP P

r u + δu

uR

δx

24

The velocity profile is parabolic, and the velocity reaches a maximum at the centre

of the pipe, where 0=r . The total flowrate through the pipe is found by integrating

(2.10) across a face of the pipe:

,8244

2

2)(412)()(

4440

22

0

⎟⎠⎞

⎜⎝⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

−=== ∫∫∫

dxdPaaa

dxdP

rdrardxdPrdrrudAruQ

aa

A

μπ

μπ

πμ

π (2.11)

which is known as the Hagen-Poiseuille equation. For the case of a pressure drop ΔP

over a length of pipe L, we have

LaPQ

μπ

8

4Δ= . (2.12)

This solution could also be obtained in a less ad hoc manner by solving the full three-

dimensional Navier-Stokes equations for a straight, circular tube. In this particular case,

the nonlinear terms usually present in the Navier-Stokes equations are identically zero,

and one eventually arrives at eq. (2.12).

2.3 Bundle of identical tubes

Having derived the flow rate equation for a straight circular tube, we can then use this

result to develop a simple model for the permeability of a rock, by assuming a

collection of such tubes passing through a cubical piece of rock of length L. If we have

n tubes of length L and radii a, passing directly through the sample at right angles to one

pair of faces, then the porosity φ will be given by

2

2

Lna πφ = . (2.13)

The total volumetric flow through the medium is given by Qtotal = nQ, with Q given by

eq. (2.12):

LPaLQ Δ

=μ

φ8

22. (2.14)

Comparison of eq. (2.14) with Darcy’s law, eq. (2.1), gives us the following expression

for the permeability of this simple one-dimensional model:

25

8

2ak φ= . (2.15)

A pseudo three-dimensional tube model could be obtained by arranging the n capillaries

in such a way that n/3 capillaries are parallel to the x-axis, n/3 parallel to the y-axis, and

the final n/3 parallel to the z-axis. In this more realistic, isotropic model, the

permeability for the same porosity φ is one-third of the permeability of the 1-D model,

and so

9624

22 dak φφ== . (2.16)

This three-dimensional model can also be thought of in terms of a tortuosity factor,

usually denoted as τ, which is sometimes interpreted as an average path length

parameter. In the context of a three-dimensional model, only one-third of the pores

tubes will be aligned in one given direction. It is worth noting that this same result, eq.

(2.16), can be obtained by assuming that the pores are randomly oriented in all

directions.

2.4 Kozeny-Carman equation and the hydraulic radius approximation

The model developed in the previous section assumes that all of the pores have circular

cross-sections. There are two ways to generalise this approach so as to be applicable to

pores with irregular cross-sections.

2.4.1 Kozeny-Carman equation

The Kozeny-Carman generalisation of the tube model assumes that resistance to flow

arises from viscous drag along the pore walls, and so permeability should show a

negative correlation with the amount of surface area per unit volume. We define the

specific surface S as the internal surface area per unit bulk volume,

bulk

pore

VAS = , (2.17)

which for the circular tube model takes the form

32

LaLnS π

= . (2.18)

The porosity for this pore geometry can be expressed as

26

3

2

LLan

VV

bulk

pore πφ == , (2.19)

which shows that

aS φ2= . (2.20)

Inserting this expression into eq. (2.16) gives the “Kozeny-Carman” equation, in which

the permeability is expressed in terms of φ and S:

2

3

6Sk φ= . (2.21)

Since the “tortuosity” of the tube model presented in §2.3 was equal to 3, the factor 6 in

the numerator of eq. (2.21) is often replaced by the more general term “2τ”.

2.4.2 Hydraulic radius approximation

The Kozeny-Carman equation generalises the parallel–tube model in a manner that

treats the entire pore space as a single entity, characterised by a single value of S.

Another way to generalise the parallel circular tube model is to use it to develop an

expression for the hydraulic conductivity of a single pore of non-circular cross-section.

This is done by noting that the pore radius a that appears in eq. (2.16) can also be

interpreted as

Π==⎟⎟

⎠

⎞⎜⎜⎝

⎛=

APerimeter

Areaa

aa 222

22

ππ , (2.22)

where we denote the perimeter by Π to avoid confusion with the pressure. If we express

the hydraulic conductivity of a given pore, as given by the Hagen-Poiseuille law in eq.

(2.12), in terms of the area-perimeter ratio, we arrive at

⎟⎟⎠

⎞⎜⎜⎝

⎛Π

Δ=⎟

⎠⎞

⎜⎝⎛Π

Δ=

Δ=

Δ= 2

32224

22

8)(

88A

LPAA

LPaa

LPa

LPQ

μμπ

μπ

μ. (2.23)

The bracketed term on the right can be defined as the hydraulic conductance of the

pore,

2

3

2Π=

ACh , (2.24)

27

in which case we have the simple relationship LPCQ h μ/Δ= for an individual pore.

This is equivalent to using the Hagen-Poiseuille law for all pores, and replacing the

radius a with the “hydraulic radius”, defined as

PerimeterArearh 2≡ . (2.25)

The foregoing “derivation” of course provides no guarantee that approximation (2.24) is

valid for non-circular pores. Schlueter (1995) examined the error incurred by this

approximation for different pore shapes. Her results are presented in Table 2.1.

Table 2.1. Accuracy of the hydraulic approximation (Schlueter, 1995)

Cross-section kexact kHR Error (%)

Circle, radius a a2/8 a2/8 0%

Equilateral Triangle, side a a2/80 a2/96 -20%

Square, side a a2/28 a2/32 -11%

Slit, aperture h h2/12 h2/8 50%

a:b = 2:1 a2/68 a2/88 14% Ellipse

a:b = 10:1 a2/403 a2/324 -21%

These results for geometrically regular pores unfortunately provide little

confidence in the accuracy of the hydraulic radius approximation for actual rock pores.

Sisavath et al. (2000) tested the hydraulic approximation against the boundary element

method (BEM) for a set of pores observed in SEM images of Berea and Massilon

sandstones. They also compared this approximation against other approximate

expressions, such as those of Saint-Venant (1879) and Aissen (1951). Surprisingly, it

was found that the overall error was the smallest when using the hydraulic radius

approximation. The range of errors for these “real” pores was found to be between 1-

15%, for each sample, showing that, to some extent, the errors made by the hydraulic

radius approximation in estimating the conductances of the individual pores partially

cancel out.

28

2.5 Permeability prediction using network models

In the field of fluid flow in porous media and predictive modelling, network modelling

has long been used, to allow for a more realistic representation of the pore space, with

regards to both pore shape and topology. Network modelling was first introduced by

Fatt (1956), who adopted an electrical network analogy that he used to predict the

capillary pressure characteristics of porous media. Today, pore network modelling is

used to investigate a wide range of properties, from capillary pressure characteristics to

interfacial area, as well as absolute and relative permeabilities (Blunt et al., 2001; Blunt,

2002; Arns et al., 2003). There are several ways one can build a network model. We

will not review all of them here, but explain a few of them and describe the main steps

and developments that are involved in the construction of such networks.

In principle, any type of process that can be modelled at the pore scale can be

incorporated in a network to compute properties at a mesoscopic scale. One approach

that can be adopted to generate a network is to use synchrotron or X-ray micro-

tomography to directly image the three dimensional pore structure of the rock at very

high resolution, so as to capture as many details as possible. These generated images

can then be used to generate a network. This approach has been used, for example, by

Ferréol and Rothman (1995), Lindquist et al. (2000) and Arns et al. (2001, 2003). This

is a direct and accurate method, and generally leads to results that are in reasonable

agreement with experiments. However, even with the recent advances of computer

technology, this is a time-consuming process, and is not likely to become a routine

process for core analysis in the immediate future. In practice, 2-D images of the pore

space are more readily available. Another approach to generate a network is to use a

“reconstruction” method, which simulates the geological and sedimentary processes to

recreate the pore space of the rock. These have led to interesting results, especially

when used in conjunction with 2-D information (Øren and Bakke, 2002, 2003). One of

the drawbacks of this method is that it can be very difficult to replicate the actual

geological processes that occur to create certain rocks. This is particularly true for

carbonate rocks, which go through many sedimentary processes before reaching their

final form and composition. We will look in more detail at this method later.

Meanwhile, as stated before, two-dimensional images are often readily available. These

can then be the starting point for the measurements of porosity and the two-point

correlation function, to finally generate a three dimensional image with the same

statistical properties. This technique is quite general and has been used for example by

29

Ioannidis et al. (1996) and Liang et al. (2000). An equivalent network can then be

reconstructed and the flow equations can then be solved on the image of the pore space.

A similar technique has also been used by Talukdar et al. (2002, 2004) where the

porous media was stochastically reconstructed using a simulated annealing technique.

The technique was applied to vuggy chalk samples (Talukdar et al., 2004) with good

results.

2.5.1 Adler et al.

Adler et al. (1990, 1992) used 2-D pore space images and statistical techniques to

model a Fontainebleau sandstone and generate an equivalent three-dimensional pore

space that has the same properties as the original images. They then successfully

predicted the electrical properties and the absolute permeability using the 3-D network,

at the expense of extensive data collection and computation, since they had to solve the

Navier-Stokes equation numerically in the pore space. More recently, Moctezuma-

Berthier et al. (2002, 2004) have been investigating reconstructed vugular porous media.

A detailed survey of this technique can be found in Adler and Thovert (1998). We will

not here go into too much detail, which can be found in the aforementioned papers; only

a brief description will be given in order to introduce the reconstruction of bimodal

media. This method starts by defining a phase function Z(x) at each point x within a

sample of a real porous medium, such that

Z(x) = 1, if x belongs to the pore space, (2.26)

Z(x) = 0, otherwise. (2.27)

This phase function can be partly characterized in a statistical sense by the two average

properties:

φ = Z(x), (2.28)

( )[ ] ( )[ ]2)(

φφφφ

−−+⋅−

=uxZxZuR , (2.29)

where φ is the porosity, R(u) is the second moment of the phase function (which is also

called the correlation function), and u is the magnitude ||u|| of the translation vector u.

Generally speaking, the correlation function and the magnitude of the translation vector

can be measured from the binarised thin section, if the medium is isotropic. However, in

30

their most recent work, Moctezuma-Berthier et al. have assumed the correlation to be of

a certain form. This actually leads to slightly “unrealistic” looking vugular porous

media. They finally used the lattice Boltzmann calculation to compute permeabilities.

2.5.2 Reconstruction methods

As mentioned earlier, another way to construct a realistic network model is to use the

so-called “reconstruction” approach. It is based on modelling the geological processes

by which the porous medium was made. This was introduced and pioneered by Bryant

and Blunt (1992). This idea has come from the fact that long-range connectivity cannot

be taken into account when using two point correlation statistics alone. Bryant and

Blunt based their models on a random close packing of equally sized spheres. They then

took into account the different geological processes involved. Diagenesis was

represented by the swelling of the spheres and allowing them to overlap. Compaction

was modelled by moving the centre of the spheres closer together in the vertical

direction, again by allowing overlap. The co-ordination number from the resulting

network was found to be around four. They then assigned equivalent radii and hydraulic

conductivities to the throats, and volumes to the pores. They could compute the flow in

each cylinder, as well as apply the mass conservation principle at each pore. This leads

to set of linear equations, which were solved using Gauss-Seidel iteration. Finally, by

conveniently applying an approximately spherically symmetric pressure gradient, the

permeability of the network k is defined as

⎥⎦

⎤⎢⎣

⎡−

Δ=

outin

tot

rrPQ

k 114π

μ, (2.30)

where totQ is the total flow, inr is the radius of the inlet, and outr is the radius of the

outlet boundary. The results obtained compared well with data available from random

sphere packs composed of beads of different materials and sizes. However, this

approach could only be applied to porous media resembling a pack of spherical grains

of the same size. Their approach was subsequently extended by Øren et al. (1998) and

Øren and Bakke (1997, 2002, 2003) who developed a reconstruction method in which

spheres of different sizes were used. The grain size distribution and other petrographical

data were directly derived from 2-D images of the rock of interest. Geological processes

were modelled in the same way as was done by Bryant and Blunt. It was shown that the

connectivity of this reconstructed network was comparable to that of a real rock. They

31

finally used the geological reconstructions to construct a topologically equivalent

network, which they then used to predict relative permeabilities for a variety of water-

wet sandstones, showing promising results.

However promising the results, there are several issues of concern regarding this

type of method. First, the reconstruction process is based on explicit simulation of the

geological processes involved in the formation of the rock. Clearly, this is likely to be a

problem for more challenging rocks, which might involve, for example, microporosity,

the presence of clay, as well as others complex sedimentary processes. This includes, in

particular, carbonate rocks, which are the focus of the present work. It must also be

pointed out that this method is rather time consuming, and a vast amount of work is

required. Indeed, heavy data requirements and manipulation of thin sections are needed

to build a network that is both geologically and topologically realistic. This is certainly

why, so far, this research has concentrated mainly on three different sandstones, namely,

Bentheimer, Berea and Fontainebleau sandstones. Finally, the characterisation of the

pore shape and the distribution of fluids within the pore space are not completely

understood, and are still subject to ongoing research (Blunt et al., 2002).

2.5.3 Okabe and Blunt

There are other more general tools available, such as statistical reconstruction, which

require only two-dimensional images as input. Even though their application has so far

been limited, they are likely to become more important as research in this field expands.

For example, Okabe and Blunt (2004, 2005) recently used multiple point statistics to

reconstruct a carbonate rock model from 2-D images. It is easy to realise that, in many

cases the submicron structures observed in many rocks, particularly carbonates (Lucia,

1999), preclude direct imaging, and the geological history is too complex for process-

based reconstruction to be applied easily. Hence, this has led to the method developed

by Okabe and Blunt. In order to able to reproduce the pore space of such rocks, they

decided used the multi-point statistics technique, which was originally developed in the

field of geostatistics by Strebelle et al. (2003). This was chosen mainly for two reasons.

Firstly, multi-point statistics require little input data and can therefore be used with 2-D

images, which are easily available. More importantly, it preserves higher-order

information, describing the statistical relation between multiple spatial locations.

Therefore, the patterns of the void space seen in the 2-D images are preserved, and this

enables the construction of realistic networks. It is also worth noting, that while

32

Strebelle et al. used multi-point statistics to generate 2-D realisations, Okabe and Blunt

expanded this by using it to generate 3-D images from 2-D images.

The extraction of multiple-point statistics from the training image, and their

reproduction in a stochastic model mainly consists of two steps: (1) extracting multiple-

point statistics from training images and (2) pattern reproduction. It is not our intent

here to give a detailed description of the method, which can be found in Okabe’s papers.

In his work, Okabe firstly used a training image of a well-known rock in order to test

his method, before attempting to reconstruct a carbonate rock network from 2-D images.

The training was done on a Berea sandstone, known for its homogeneity, and for which

a network is already available. Okabe was then able to compare the properties of the

reconstructed network with the one obtained from micro-CT scanning, and also

compare the permeabilities obtained from the two different networks. It appeared that

the reconstructed network gave very good results, within 20% of the laboratory

measurements, and very close to the computed permeability on the scanned network. He

then reconstructed a network for a carbonate rock. The final step of the method is to

compute the permeability. This was done using the lattice-Boltzmann method, which

provides a good approximation to the Navier-Stokes by using an efficient algorithm that

deals with complex boundaries such as are found in porous media.

Their method was tested on this single carbonate, and they obtained reasonable

results, predicting the permeability within a factor of three. Obviously, the method they

developed is still at an early stage, and it is likely that increases in computing power

will allow the construction of more realistic networks, once they are not limited by the

size of the template images used, which are currently quite small (9×9 template).

However, it is also worth noting that the method is still quite time consuming, and that

certain types of carbonates cannot be reconstructed due to their very heterogeneous

nature.

2.5.4 The NETSIM code

NETSIM (Jing, 1990) is a network simulation code that essentially performs an exact

network calculation by solving the flow equations in the entire networks of tubes. As

such, it can be used to model various phenomena within a porous material, such as the

electrical resistivity and the permeability. It is based on a cubic lattice of pore tubes, and

was originally written by Jing (1990) based on previous work by Yale (1984).

33

The original NETSIM code can be described as consisting of five major blocks:

1. A cubic lattice is specified in terms of nodes that connect bonds, and the latter are

subsequently allocated with conductances from an external file in a random manner.

These conductances can be drawn from an idealised distribution (i.e., normal,

lognormal, etc.), or from a distribution estimated from the SEM images.

2. The boundary conditions are then specified, with a fixed pressure gradient in one

direction only, and no-flow boundaries along the remaining two sets of orthogonal

faces.

3. The equation )/)(/( dxdPCQ μ−= is used for each tube to relate the flux to the

pressure drop, and conservation of mass is applied at each node, to yield a set of

linear algebraic equations, which are solved using the successive over-relaxation

technique of Young (1971).

4. The total flowrate is found by summing up the fluxes through each of the

conductors that exit the cube at the low-pressure face.

5. Finally, the permeability is calculated from the total flowrate and the overall

pressure drop, using Darcy’s law.

Sisavath (2000) modified the original NETSIM code so that the simulations only

require the main block to specify the cubic lattice size, impose the boundary conditions

and calculate the total conductance and flowrate. Lock (2001) subsequently modified

the code again. First, in order to accommodate anisotropy studies, Lock made it possible

for each of the three lattice directions to be decorated with a different conductance

distribution, which may be read from an external file. Moreover, it was made possible to

use NETSIM to compute an effective conductance. David et al. (1990) showed that a

conductance Ceq adopted by all bonds in a regular homogenised network is given by the

following expression

[ ]1o

CC

Ceq = , (2.31)

where Ceq is the equivalent conductance, Co the net conductance and C[1] is defined as

the overall conductance of the equivalent homogeneous network with the same

topology, when the individual conductances are set to unity. Co can be calculated by

34

dividing the net flow rate coming out of the cubic network by the total potential. For a

cubic lattice, C[1] is expressed as follows (David et al., 1990):

1]1[

2

+=

nnC cubic , (2.32)

where n is the number of nodes of the lattice. The values of Ceq calculated thusly allow

validation of the values effC obtained using the effective medium technique of

Kirkpatrick (Chapter 4), since we can directly assess the difference between Ceq and Ceff

obtained from the two simulations.

The development of consistent means for pre-processing the input data, and the

subsequent running of each simulation, are necessary in order to form a methodology

that can be applied confidently to any dataset. Indeed, the data available are rarely

sufficient to decorate the entire cubic lattice. It is therefore necessary to randomise the

data and then to expand it. When the code was originally developed by Jing, the lattice

size was restricted by the computing capabilities of the time, and Jing used a lattice of

15 nodes. However, Lock (2001) investigated the issue of lattice size versus the quality

of prediction by testing NETSIM with lattices ranging from 5 to 40 nodes, increasing in

increments of 5, running 30 realisations in each instance. To monitor the fluctuations in

the output of the network code, an average percent difference was used, by comparing

the effective-medium solution provided by Kirkpatrick’s equation for the same set of

conductances. It was found that the average levelled out to a constant value after the

twentieth realisation for all the lattice sizes, with the overall error at this realisation

being 10% on a 5×5×5 lattice. This difference gradually fell with increasing lattice size,

until it reached a consistent 2% error on the 30×30×30 lattice, with virtually no

improvement for simulations run on a lattice containing 403 nodes. Therefore, Lock

decided to use a lattice of 30×30×30 nodes, using the average permeability computed

over twenty realisations. This is also what we will be using for our NETSIM

simulations.

35

Figure 2.2. A plot of absolute difference between the outputs obtained for simulations on ISONETSIM and Kirkpatrick’s equation as a function of lattice size at 20-30 realisations.

As mentioned before, NETSIM can be used for a variety of modelling applications.

Jing (1990) originally developed the code so that the mechanisms of the effects of

pressure and temperature on the electrical and hydraulic properties could be understood

better, as the experimental data was not completely understood at the time. But it can

also simply predict the properties of the medium such as porosity, formation resistivity

factor or permeability. Lock et al. (2004) used NETSIM to compare the predictions of

their effective medium theory model against an exact network calculation. The method

they used will be reviewed in the next chapter. NETSIM will also be used throughout

this thesis in order to assess the behaviour of Lock’s method with different parameters.

One of the main limitations of NETSIM, at least within the context of this thesis, is the

impossibility to change the co-ordination number of the lattice. Indeed, in the present

form of the code, only a cubic lattice can be used, therefore limiting the applicability of

NETSIM. This will be discussed in more detail in Chapter 4.

2.6 Image analysis and permeability prediction

2.6.1 Berryman and Blair

Berryman and Blair (1986, 1987) used image analysis to estimate the parameters

appearing in the Kozeny–Carman equation. They showed that specific surface

estimation could be derived from the tangent of the two-point correlation function at the

|CN

ETS

IM-C

Kirk

patri

ck/C

NE

TSIM

|

Lattice Size (Nodes)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0 5 10 15 20 25 30 35 40 45

20 REALISATIONS

25 REALISATIONS

30 REALISATIONS

36

origin, S2(r = 0). The two-point correlation function represents the probability that two

points that are a distance r apart are either both in the pore space, or both in the mineral

phase. For statistically isotropic materials, Sn is independent of direction and will only

exhibit a dependency on the absolute value of the distances between the n points. If we

then take the two point correlation function under the condition r = |r|, the following

relationships can be derived (Debye et al., 1957; Berryman and Blair, 1986)

φ=)0(2S , (2.33)

22 )(lim φ=∞→

rSr

, (2.34)

4)0(2 S

drdS −

= . (2.35)

Berryman and Blair concluded that, if the magnification is chosen in an appropriate

way, a typical correlation length estimate for a thin section would then be approximately

100 times larger than the size of a pixel, while the measurement of the specific surface

area (as a pore structure parameter) will become acceptable at a given pixel size. The

magnification required was defined by Berryman and Blair (1987), who proposed that

the pixel size, h, be approximately 1% of the size of an average pore radius, such that

100/crh ≅ . (2.36)

The above criterion restricts the measurement error in the average pore radius, due

to surface roughness and pixel quantization, to be of the order of 1%. The magnification

cannot be increased to the point where an average pore diameter is larger than the width

of the digitised image. Consequently, we may conclude that there is little advantage in

increasing the image magnification past the point where h violates these limits. More

importantly, it has also been shown by Berryman and Blair (1987), and later by Sisavath

et al. (2000), that small-scale roughness, here defined as any corrugation of amplitude

05.0/ ≤cRε , is not actually relevant to permeability computation, as its effect on the

predictions is minimal. Indeed, the error induced by roughness on the hydraulic

conductance for real pore shapes can be neglected. Furthermore, these arguments about

the resolution and pixel sizes have also in a way become outdated, as the digital

imaging technologies have greatly improved over the last ten years, and have given

researchers access to large high-quality 2-D images.

37

Table 2.2. Porosity and permeability values estimated by Berryman and Blair (1987), compared with laboratory values.

Sample Magnif’n S (μm-1) φIMAGE φMEAS kIMAGE (D) kMEAS (D)

50 0.0241 0.35 0.3 9.7 8.7 Glass beads (55μm)

78 0.0281 0.43 0.3 10.7 8.7

Berea 100 0.0281 0.17 0.15-0.18 0.312 0.023

Berea 200 0.0354 0.18 0.15-0.18 0.197 0.023

Berea 490 0.1109 0.233 0.15-0.18 0.021 0.023

Berea 1000 0.1231 0.393 0.15-0.18 0.016 0.023

As can be seen from these results, the predictions are accurate, but only at certain

resolutions. This is due to the approximations the method introduces into the estimation

of specific surface. These arise from the resolution constraints dictated by the image

digitisation. It is the major drawback of their method, in that it allows accurate

prediction at a suitably chosen magnification, which is not known a priori.

2.6.2 Cerepi et al.

Cerepi et al. (2001) used what they call “Petrographic Image Analysis” (PIA) to predict

the petrophysical properties of a set of twenty-five carbonate rocks. They used images

from both an optical microscope (OM) and from a scanning electron microscope (SEM)

to capture the different porosity systems displayed by carbonate rocks. Specifically, the

optical microscope provided them with images with lower resolution so as to study the

macroporosity, and the SEM images provided them with the higher resolution, to enable

them to study the microporosity. This allowed them to describe the pore space from the

submicron to a millimetre scale. Using the information collected from the images, such

as pore area, specific surface area, average pore diameter and pore shape factor, they

then modelled petrophysical properties such as porosity, capillary pressure and

permeability. For modelling the permeability, they used two different approaches, a

bundle of capillary tubes model as well as a Kozeny-Carman model. The Kozeny-

Carman model they used is given by

( ) 2

22

2

3

161 LS

ck π

φφ−

= , (2.37)

38

where φ is the porosity, S the pore area, c is a constant related to the tortuosity and the

pore shape, and L is the total pore perimeter. According to Carman, the best value of c

to fit most experimental data on packed beds is 5, which is the value chosen here.

Cerepi et al. (2001) also used a bundle of capillary tubes, expressed as follows:

∫∞

=0

2 )(8

drrrk ατφ , (2.38)

where τ is the tortuosity factor, and )(rα is the pore size distribution function.

Using this model they predicted the permeability of the samples, and compared the

results to the standard petrophysical analysis they also carried out. They found that the

Kozeny-Carman model gave better predictions than did the parallel tube bundle model.

The values of permeability predicted were higher in the grain-dominated texture than in

mud-dominated texture. It was also found that 5 was not the best value for 2τ, but rather

the “best” value seemed to vary from sample to sample.

39

3 REVIEW OF THE METHOD OF SCHLUETER AND LOCK In the previous chapter, we reviewed some basic permeability concepts and some of the

available methods for permeability modelling and prediction. As the work presented in

this thesis follows from the methodology developed in two previous PhD projects, by

Schlueter (1995) and Lock (2001), we now review their work in more detail, including

earlier works that helped to form the basis of their approach. This approach attempts to

minimise the computational effort involved in the permeability computation by using

effective medium theory to approximately solve the network equations, and tries to limit

the amount of data collection and manipulation required by using only a handful of

SEM images.

We start by briefly reviewing image analysis, as this forms the basis of the data

acquisition and manipulation, before quickly introducing the effective medium theory

(EMT). The work of Koplik et al. (1984) will then be reviewed, before presenting the

algorithm developed by Lock. We then finish by presenting some of Lock’s results for

sandstones.

3.1 Image analysis

Image analysis may be regarded as a technique for characterising and classifying

images, and features within images. However, the purpose of using image analysis, in

the context of Lock’s and our work, is to obtain quantitative information concerning the

size and shapes of the pores in a given rock sample, such as the areas and perimeters of

the pores. In the case of permeability modelling, image analysis has been used in several

different ways to estimate different parameters (Koplik et al., 1984; Berryman and

Blair, 1986; Cerepi et al., 2001).

The procedure needed to obtain information from the 2-D images will typically

involve the use of an electronic microscope, together with a video scanner or camera, a

digital converter, and image processing software. In order to have a successful analysis

of the data, we need a method to consistently process the images, whereby the regions

of interest can be identified and segmented from the remainder of the image. Once

stored in digital format, the image can be accessed by use of suitable image processing

40

software that will allow the editing, enhancement and subsequent analysis of digital

images. In order to extract useful information from the images, one is likely to use a

process called segmentation. Segmentation is the assignment of a grey-level or energy

threshold with which to convert the digitised image into a binary signal (Castleman,

1996). The term “thresholding” in this context then refers to the mapping of all

“points”, or pixels, with energy levels up to and including a certain predetermined

threshold value to a binary output of zero, with all higher levels mapped to a binary

output of 1. In the context of deriving fluid transport parameters, a threshold is sought to

allow the separation of the pore space from the remainder of the mineral phases present

in the 2-D section. It can also be used to separate different fluid phases within the pore

space, when dealing with two-phase data, as we will see in Chapter 7. The thresholding

level is usually determined with the help of the histogram, usually chosen at the lowest

point of the latter. This section on image analysis will expanded and explained more

specifically in Chapter 6, where we will present in detail the computational procedure

and the single-phase results for carbonates.

3.2 Effective medium theory

Numerous methods have been proposed over the years to estimate an “effective”

macroscopic value of a given physical property that is heterogeneously distributed on

some smaller scale. In different scientific and technological fields, these methods are

known as homogenisation, upscaling, or effective medium methods. These methods can

apply to elastic properties, thermal properties, magnetic properties, electrical properties,

fluid transport properties, etc. This field is much too broad and varied to allow even a

brief overview in the context of this thesis. For reviews of this field, the interested

reader can consult the monographs by Sahimi (1995), Choy (1999), or Torquato (2002).

Although there is some overlap, the various methods can be grouped according to

whether they are intended to apply to continuous media or discrete networks. One very

widely used such approach is the effective medium equation that was originally

developed by Kirkpatrick (1973) in the context of a disordered electrical resistance

network. This equation will be used in the present work, and so we now briefly review

the developments of the method, its underlying theory, and its justification.

Landauer (1952) attempted to describe the resistance of binary mixtures of

conducting materials using simple mixing laws. However, these laws were found to be

inadequate when dealing with insulating-conducting mixtures, a very common case in

41

rock physics. Kirkpatrick (1971, 1973) presented a solution to this problem, the so-

called effective medium approximation (EMA). By incorporating the EMA within our

method, it means that we can infer an effective pore conductance from a distribution of

discrete conductances of pores arranged on a lattice. This means that we are able to use

the data provided by the image analysis on a pore basis, and “average” the conductances

so that we can then compute a permeability value using the effective conductance. In

other words, the EMA is an ingenious way of transforming a heterogeneous medium

into a homogeneous one, as illustrated in Fig. 3.1.

Figure 3.1. An illustration showing the replacement of a discrete network of conductances by a homogeneous network having the same topology.

The self-consistent procedure of Kirkpatrick’s EMA leads to the construction of an

effective conductance by the superposition principle, in which the replacement of each

conductor Ci by a trial conductance is applied to each bond in the network, such that the

resulting fluctuations in the potentials at the nearby nodes are minimised. This

procedure is repeated until the successive updating of the trial conductance eventually

causes the fluctuations to average out to zero. The trial conductance has then reached

the “effective” value throughout the network. This procedure leads to the following

equation that implicitly defines the effective conductance:

( )[ ]∑=

=+−

−=

N

i ieff

ieffeff CCz

CCCf

10

12/)( , (3.1)

where N is the number of conductors and z is the “co-ordination number” of the

network, which represents the number of conductors that meet at each node. By

multiplying through by each denominator, this equation can be transformed into an Nth-

order polynomial, which can be solved by any standard root-finding algorithm. Lock

wrote two simple FORTRAN codes to solve this equation, using the bisection algorithm

C5

C1 C2 C3 C4

C8 C6 C7

C9 C11 C12

C10

Ceff

Ceff Ceff Ceff Ceff

Ceff Ceff Ceff

Ceff Ceff Ceff

Ceff

42

and the Newton-Raphson procedure. If bisection is used, one must begin with upper and

lower bounds for the root of eq. (3.1); these can be provided by the arithmetic and

harmonic means of the individual conductances. If Newton-Raphson is used, an initial

guess is required that is close to the actual root; this can be provided by the geometric

mean of the conductivity distribution.

Kirkpatrick derived his equation using the assumption that the co-ordination

number at each node had to be identical, but it was later shown by Koplik (1982) that

this equation could also be used for topologically inhomogeneous networks, with z then

being the mean co-ordination for all the nodes in the network. The EMA is a vast

subject in itself and we will, in the next chapter, investigate it in more detail, as well as

discussing the limitations of this approximation, and the solutions.

3.3 The model of Koplik et al.

Koplik et al. (1984) were the first to combine image analysis with effective medium

theory in order to compute rock properties. In retrospect, this paper stands out as

groundbreaking work, even though the results they obtained were not very accurate.

Using their techniques, they calculated both the electrical conductivity and the

permeability of a Massilon sandstone. We will now review their work.

Their first step was to conceptualise the pore space with a standard “ball and

sticks” network model. They assumed simple spherical pores, connected by cylindrical

throats of constant cross-section. The flow in the cylindrical throats is described by

pL

CQ Δ=μ

, (3.2)

where μ is the viscosity and C the conductance, which is known for standard

geometrical shapes. They assumed the throats to be of elliptical cross-section, in which

case the conductance is given by (White, 1974):

)(4 22

33

bahbaC+

=π , (3.3)

where a and b are the major and minor semi-axes of the ellipse, respectively. The term h

here represents the length of the pore. They neglected the pressure drops in the pore

bodies, on the assumption that the pores are large and compact, while the throats are

long and narrow. It was then assumed that the net flux at the pore bodies has to be equal

43

to zero, as there is no accumulation of fluid. These assumptions are standard

assumptions when dealing with this type of network problem. They then used the EMA

to define the effective conductance Ceff, and to finally calculate permeability by

assuming a continuous medium. By fixing the pressures at the ends of a network of

conductors, they created an average pressure gradient P∇ . They finally expressed the

total flux through any plane perpendicular to P∇ as

bPb

efftot LP

CQ ∇∑=

∩ μ, (3.4)

in which we take the sum of the individual fluxes Qb through the bonds b intersecting P,

which is a plane perpendicular to the pressure gradient. Finally, dividing eq. (3.4) by the

area of the plane, they obtain the velocity average, and comparing this with Darcy’s

law, they identify the permeability as

tCk eff= , (3.5)

where t is defined by

( ) ∑∩⋅=

Pbb nL

PAt ˆ1 . (3.6)

where n̂ is a unit vector along the pressure gradient. However, if the porous material is

statistically isotropic and sufficiently homogeneous, then any unit vector will suffice.

Physically, t represents the number of pores flowing per planar section perpendicular to

the direction of the pressure gradient.

Having performed a detailed section analysis on the forty-one slices of a Massilon

sandstone that were available, they reconstructed the pore space, with a total volume of

3.3 mm3. The average co-ordination number for the network was found to be 3.49. This

was estimated from the rock slices used in the reconstruction process. Using this, they

typically over-predicted the permeability by about a factor of ten. This is a rather

substantial error, but it has to be remembered that it was the first time that the problem

was tackled this way, so this might have been expected. There are several reasons that

can explain the errors, but according to Lock et al. (2004) these are mainly the lack of

appropriate stereological corrections applied to the pore images, and the lack of

consideration of the converging-diverging nature of the pore throats. These factors will

be discussed in more detail below.

44

3.4 The model of Schlueter

The model of Schlueter (1995) is conceptually based on the work of Koplik et al., who

seemed to have abandoned this approach after they failed to get accurate predictions

from their model. In her 1995 thesis, Schlueter used the same overall approach,

although with some modifications, to predict the permeability of four sandstones: Berea,

Boise, Massilon and St. Giles.

The main difference with the model of Koplik et al. was that stereological and

constriction factors were introduced. “Stereology” is the name given to the technique

used to reconstruct the characteristics of a three-dimensional object from randomly

orientated planar slices. If one wants to be able to predict the permeability of a three-

dimensional porous material of rocks using 2-D images, these corrections are necessary

and must be implemented in the procedure. It is easy to see why there is a need for

stereological factors. In general, the areas and perimeters of the individual pores, as

measured from the SEM images, will be larger than the actual values for the pore cross-

sections. Let us consider a tubular pore of uniform cross-section, with radius a. If,

during the sample preparation, the pore happens to be sliced perpendicular to its axis,

then the exposed pore would indeed be a circle of radius a. In general, however, the

pore will be intersected at some arbitrary angle relative to its axis. In this case, it would

appear on the SEM image as an ellipse, with a semi-minor axis equal to a, but with a

semi-major axis of θcos/ar = that is typically larger than a. Consequently, both the

measured area and the perimeter of the image would overestimate the actual value.

Figure 3.2. Illustration showing how the area of a slice through a single pore at some arbitrary angle θ will generally be larger than the true cross-sectional area.

θcos/ar =

θ

a

45

Following this argument, Schlueter estimated the different corrections that need to

be made for each of the parameters. These can be estimated with the combined use of

geometry and trigonometry. We will not here go through all the calculations, but merely

describe the work involved and the actual values obtained. It also worth noting that in

order to compute the value of the correction factor for the conductances, it is first

necessary to evaluate it for the area, perimeter, and hydraulic radius.

Considering a cylinder of radius a sliced at an angle θ, as in Fig. 3.2, the true cross-

sectional area will be 2aπ , whereas the measured area will be θπ cos/2a . Hence, for

this pore we would have θcos/actmeas AA = . If we assume that the rock contains a large

number of pores of radius a, that are randomly oriented with respect to the slicing plane,

and then take averages of both sides of this relation, we find actmeas AA θcos/1= .

Note that this ratio is independent of the radius a, and hence holds for a distribution of

radii. So, the actual area is therefore given by

measuredactual AA1

cos1 −

=θ

, (3.7)

where

1cosθ

=

1cosθ

sinθdθdΦ0θmax∫0

π∫

sinθdθdΦ0θmax∫0

π∫. (3.8)

According to this logic, it is not possible to consider the entire range of angles, because

the integral would diverge if taken up to 2/max πθ = . This corresponds to the fact that

if an infinitely–long cylinder was sliced at this angle, the semi-major axis of the

resulting ellipse would be of infinite extent. In reality, each pore is of finite length, and

so there is indeed a maximum possible value of θ, defined by )/arctan(max DL=θ ,

where DL / is the maximum ratio of pore length to diameter. Schlueter took the ratio of

DL / to be 5, based on observation of the BSE images. Using this value, integral (3.8)

can be evaluated to yield:

measuredactual AA 61.0= . (3.9)

The perimeter Π of an ellipse is given exactly in terms of an elliptic integral, but in

order to avoid integrating such a function, Schlueter used for the following

approximation (Beyer, 1987), where 2r is the major axis, and 2a the minor one:

Πmeasured ≈ π 2(a2 + r 2 ) = π r 2(1+ cos2θ) . (3.10)

46

Using this approximation, along with the previous calculation for the area, she was able

to evaluate the correction for the hydraulic radius, RH. The evaluation of the integral

was done again for θ over the interval ),0( maxθ . The result is

HmeasuredHactual RR 85.0= . (3.11)

Finally, Schlueter computed the correction for the hydraulic conductance:

(RH

2 A)actual =12

1cosθ(1+ cos2θ )

−1

, (3.12)

where

1cosθ(1+ cos2θ)

=

1cosθ(1+ cos2θ )

sinθdθdΦ0θmax∫0

π∫

sinθdθdΦ0θmax∫0

π∫. (3.13)

Again using 69.785arctanmax ==θ , Schlueter obtained:

Cactual = 0.4Cmeasured. (3.14)

Another factor that was introduced by Schlueter is a constriction factor for the

throats. It is fair to assume that the radii along the pores will not be uniform. As a

consequence of the continuously varying cross-section, there will be an excess pressure

drop associated with the constrictions, as the fluid passes through the pore channel. This

can be quantified by the hydraulic constriction factor, f, defined such that the

conductance of the tube is given by 8/4 faC π= . For a uniform pore, 1=f , and

aa = , and the conductance reduces to that given by the Hagen-Poiseuille equation.

Following the analogous derivation given by Zimmerman et al. (1991) for a rough-

walled fracture, Schlueter assumed a sinusoidal variation of radius along the length of

the pore,

( )[ ]λπξ /2sin1)( xaxa += , (3.15)

where λ is the wavelength of the radius variations, ξ is the magnitude of the

“roughness” and a is the mean radius of the pore. If we plug eq. (3.15) into the

integral form of the Hagen-Poiseuille equation, and integrate over one wavelength λ, we

find the following expression for the constriction factor:

47

,)5335()(1

256)sinusoidal( 234

7/2

++++=

ρρρρρf (3.16)

where ( ) ( ) maxmin /1/1 aa=+−= ξξρ . The constriction factor is plotted as a function of

ρ in Fig. 3.3. Schlueter used different numerical values for the constriction factors,

depending on the rock. For example, she used a factor of 0.26 for the St. Giles

sandstones, estimated from a value of 33.0=ρ . A value of 50.0=ρ was used for

Berea sandstone, which leads to a constriction factor of 0.55.

Figure 3.3. Constriction factor for hydraulic conductivity as function of the ratio of the minimum pore radius to the maximum pore radius of an individual pore.

These stereological and constriction factors are then used to compute the

conductances values using the data extracted from the BSE images. These conductances

are then input into Kirkpatrick’s EMT equation to obtain an effective conductance Ceff,

from which the permeability can finally be calculated. It must be noted that, after the

study of BSE images of Berea sandstone, it was found by Schlueter that the pore

structure was statistically isotropic. This led to the idealisation that the pores and throats

could be arranged on a cubic lattice, so that the co-ordination number is six. Hence, a

value of 6=z was used all EMT calculations. We will now briefly review the

procedure for the permeability calculation, and then review the results.

We start by recalling that the hydraulic conductance per unit length of each tube is

given by

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Con

stric

tion

fact

or, f

amin /amax

48

ihii ARC 2

21

= . (3.17)

According to the Hagen-Poiseuille equation, the volumetric flux of fluid through one

tube is given by

PAR

PAC

q ihiii ∇=∇=

μμ 2

2

. (3.18)

Over the whole cubic lattice, the total flow rate in the direction parallel to the pressure

gradient is given by

PNC

QQ effN

ii ∇== ∑

= μ1, (3.19)

where Qeff is the flow rate through an effective conductor, and N is the total number of

pores aligned in the lattice direction. We now recall Darcy’s law, which expresses the

total flow rate through the porous medium as

PAkQ total∇=μ

, (3.20)

where k is the permeability and Atotal is the total area of the porous medium under

consideration. By combining the two previous equations, we finally get the permeability

as a function of known factors, in the form of

total

eff

ANCk = . (3.21)

Having now discussed the method, we will present and briefly discuss the results.

Table 3.1. Permeability predictions from Schlueter (1995).

Rock Type kmeasured (D) kpredicted (D) Error (%)

Berea Sandstone 0.48 0.56 +17

Boise Sandstone 1.30 1.59 +22

Massilon Sandstone 2.50 3.65 +46

Saint Gilles Sandstone 0.17 0.34 +100

49

The predictions of the permeability all fell within a factor of two of the measured

values, with three of them falling below the fifty percent error line. It is also worth

pointing out that all the predictions are actually higher than the measurements. This

must be considered as successful, given the discrepancy sometimes seen in laboratory

permeability measurements. It is therefore a major improvement over the results of

Koplik et al. (1984), which were too high by about a factor of ten. This is in part due to

the stereological correction added to the method. Indeed, when put together, the two

factors account for at least a seventy five percent reduction in the conductance values,

which will decrease the predictions. Also, Koplik et al. used an “equivalent ellipse” for

estimating the conductance of the individual pores, as opposed to the hydraulic radius

approximation used by Schlueter. Comparisons with the boundary element calculations

show that this is likely to account for a 10-50% error in the estimated conductances

(Sisavath et al., 2001). All of these factors contribute to the fact Schlueter’s predictions

are in much better agreement with the measured values than were those of Koplik et al.

(1984).

3.5 The model of Lock

The work of Lock (2001) and Lock et al. (2002) was a refinement of the model

developed by Schlueter. It is based on the same method and concepts, but a few changes

and improvements were introduced. Furthermore, it was tested against a much larger

database of rocks, albeit only sandstones. We will first review the changes and

improvements before presenting the results obtained.

3.5.1 Stereological factors

Lock (2001) retained the stereological factors introduced by Schlueter, but re-

evaluated the logic behind their calculation. He started from the relation

θcosmeasact AA = , and averaged both sides, holding Ameas constant, to obtain

∫ ∫∫ ∫== 2/

0 0

2/

0 0

sin

sincoscos π π

π π

θφθ

θφθθθ

dd

dd

AA

measured

actual

( )21

)2/(cos1cos14/1

sin

sincos2/

0

2/

0 =−

−==

∫∫

πθπ

θθ

θθθπ

π

d

d. (3.22)

50

This value is lower than that obtained by Schlueter, which was 0.61.

We also need to make a similar correction for the perimeter, hydraulic radius, etc.

The actual perimeter of a pore is given by aactual πΠ 2= , while the measured perimeter

can be approximated by the following expression, which is accurate to within 10%

(Beyer, 1987):

)cos1(2)(2 222 θππΠ +=+≈ rrameas . (3.23)

The expectation value of the actual perimeter is given by

59.0cos1

cos2)cos1(2

222

=+

=+

=θ

θθπ

πΠΠ

ra

measured

actual . (3.24)

Similarly, the measured hydraulic radius can be expressed as

)cos1(222][

2θππ

Π +==

rraAR

measured

measuredmeasuredh . (3.25)

Recalling that [RH ]actual = a for a circular pore, the expectation value of the actual

hydraulic radius is given by

81.0cos12

1][][ 2 =+= θmeasuredh

actualh

RR

. (3.26)

Finally, we derive an approximate expression for a stereological correction factor that

converts the “measured” values of the hydraulic conductance, into “actual” values. To

simplify the calculation, we again assume a circular pore for which the conductance is

given by

22

2

2

82

82 hh RAAAAAC =⎥⎦⎤

⎢⎣⎡==ΠΠ

. (3.27)

The corresponding expression for the actual conductance is:

822

8][

8

4222 aa

aaRA

C actualhactual

actualπ

πππ

=⎥⎦

⎤⎢⎣

⎡== . (3.28)

Using eq. (3.25), the measured conductance is given by

51

)cos1(cos2

81][

8 2

4

θθπ+

==aR

AC measuredh

measuredmeasured . (3.29)

Thus,

375.0)cos1(cos21 2 =+= θθ

measured

actual

CC

. (3.30)

We therefore have to multiply the conductances that are estimated from planar sections

by 0.375 to calculate the true hydraulic conductance of the pores, i.e.,

measuredactual CC 375.0=>< . (3.31)

The approaches used by Schlueter and Lock differ in that, whereas Schlueter chose to

hold Aactual constant, Lock held Ameasured constant, leading to a slight difference between

the expectation values obtained after the averaging process. It is interesting that the

values obtained by the two methods are reasonably close (around 6%), and will not

make a great difference in the final calculation. We will, in this thesis, use the approach

chosen by Lock because it seems difficult to choose a value )/( DL simply using the

SEM images. The value of 5 chosen by Schlueter appears slightly arbitrary, and one can

also expect this value likely to be different for carbonate rocks. The next table

summarises the different values for the stereological factors.

Table 3.2. Comparison of stereological factors used by Schlueter and Lock.

Correction Factor Schlueter Lock

Area 0.61 0.50

Perimeter 0.69 0.59

Hydraulic Radius 0.85 0.81

Conductance 0.40 0.375

In Lock’s work, the constriction factor was evaluated in two different ways, using

both a sinusoidal (as in Schlueter’s work) and a saw-tooth variation for the throats

radius. The results are shown in Fig. 3.4.

52

Figure 3.4. Constriction factor f as a function of amin/amax for saw-tooth and sinusoidal profiles.

The saw-tooth variation gives a higher constriction factor, therefore reducing the

conductances slightly more than the sinusoidal variation. It must also be noted that these

constriction factors are only correct for circular cross-section. It is generally difficult to

arrive at good estimates of maxmin aa / , as has been discussed in the literature (Cargill,

1984; Bourbié et al., 1987). It is also difficult to estimate an appropriate “average”

value, as the value is likely to change for different rocks. Schlueter, for example,

estimated the ratio from the pore casts. As Lock et al. (2002) wanted to develop a model

which uses no other information than from 2-D images, they decided to use the average

value between the simple cubic and hexagonal packing of 0.43, which then gives

hydraulic constriction factors of 0.44 and 0.57.

3.5.2 Areal thresholding

Lock (2001) considered that, as the procedure of collecting measurements of the pores

involves a wide range of scales, it was conceivable that the measurements recorded for

many of the single and small pixel cluster features corresponded to artefacts from the

thresholding procedure. They therefore defined an additional thresholding procedure

that eliminates these smaller non-conducting features that are still present in the image

after a grey-level thresholding. They decided to use an areal cut-off based on some

percentage of the area of the largest conducting feature, such that we do not lose more

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

sinusoid

Saw-tooth

Con

stric

tion

fact

or, f

amin /amax

53

than 3% of the “total” hydraulic conductivity, calculated as if all the pores were in

parallel. In their case, they decided to use a 1% areal cut-off. The value shown here was

chosen for sandstones, and this is one of the parameters we will be reviewing when

using the method with carbonate rocks.

3.5.3 Pore number density

Another consideration that was added by Lock is a stereological correction factor for the

pore number density. Indeed, an arbitrary two-dimensional slice probably will not lie in

a plane perpendicular to a lattice direction, therefore increasing the areal number

density. Assuming a cubic lattice, a slice taken perpendicular to a given lattice direction

will only intersect with the pores that lie in that direction. If, on the other hand, the

slicing plane is not normal to the lattice direction, it will also intersect some pores that

are also orthogonal to that first lattice direction (Fig. 3.5), leading to an overestimation

in the number density of pores.

Using this argument, Lock (2001) derived the expectation value for a number

density. We will not here include the original calculation, but only include the results. It

was found that for a three-dimensional cubic lattice that the number of pores found on

the images had to be divided by 1.47 to arrive at the actual number of pores, i.e.,

47.1apparent

actualNN = . (3.32)

Figure 3.5. A two-dimensional illustration showing the number of additional pores bisected when a slice is taken at an angle θ that is not perpendicular to a lattice direction.

3.5.4 Automated image analysis

The above-described changes and improvements were combined with a much more

automated approach to the image analysis procedure. Whereas Schlueter calculated the

perimeters and areas manually, Lock used an image analysis program called Scion

Y

X

θ

54

Image Analysis (http://www.scioncorp.com/frames/fr_scion_products.htm). Several

features of the program were therefore used to automate the method. The software

allows one to “threshold” the BSE images using the grey-level histogram, a process also

known as segmentation. This process then converts a digitised image into a binary

signal, i.e., a black and white image. This is necessary to be able to separate the porous

phase from the solid phase, and therefore provides us with the necessary measurements

about the porous phase, such as the perimeter and area of each pore. This, in turn,

allows the computation of the hydraulic conductances. The corrected conductances are

then entered into Kirkpatrick’s equation, which is solved using a FORTRAN program.

3.5.5 Summary of Lock’s computational procedure

The final steps of the calculation are similar to those in Schlueter’s work, but for clarity,

we will quickly go through them again. We must here recall that the basis of the

modelling is done using a cubic lattice, so we consider a plane that slices the lattice

perpendicular to one of the principal lattice directions, containing N pores in a region of

cross-sectional area A. The total flow rate through this region will be given by

LPNCQ eff

μΔ

= . (3.33)

Darcy’s law, on the other hand, expresses the flow rate as

LPkAQ

μΔ

= . (3.34)

Equating (3.33) and (3.34) then gives the permeability as

total

eff

ANC

k = . (3.35)

The permeability is therefore computed as follows:

• Take BSEI photographs of polished sections.

• Digitise pore images.

• Apply grey-level thresholding procedure to identify the “pores”.

• Compute perimeter and area of each pore with image analyser.

55

• Apply stereological correction and hydraulic constriction factors to estimate

the hydraulic conductivity Ci of the individual pores.

• Employ areal thresholding procedure to truncate the data set.

• Obtain Ceff using the effective-medium approximation.

• Compute the areal density of pores inside the image, using eq. (3.32).

• Assuming a cubic lattice, calculate k using k = NCeff/Atotal.

3.5.6 Results

Lock’s study involved two independent data sets. The first consisted of a collection of

backscattered electron micrographs that were prepared from the core extractions of a

commercial well that lies inside the UK Continental Shelf (UKCS), and was supplied by

Enterprise Oil, now part of Shell. The second dataset was prepared by coring a sample

of St. Bees sandstone from a quarry in Cumbria, UK. These were sent to the laboratory

for permeability measurements, the end trims being taken to prepare a collection of

backscattered electron micrographs.

As can be seen from Fig. 3.6, the predictions are within a factor of two of the

measured values in almost every case. There seems to be a slight trend of over-

prediction in the low permeability range, and under-prediction in the high permeability

range, although this bias may be due to the small number of samples. If absolute values

are considered, the average error is 48%.

56

Figure 3.6. Measured permeabilities plotted against predicted values for both UKCS and St. Bees data. The upper and lower lines correspond to errors of a factor of two in either direction.

10

100

1000

104

10 100 1000 104

Sandstonesy = xy = 2xy = 0.5x

K p

redi

cted

(mD

)

K measured (mD)

57

4 EFFECTIVE MEDIUM THEORY In the previous chapter, we briefly introduced the effective medium approximation. This

method allows the replacement, in a consistent manner, of a given disordered medium

with a hypothetical homogeneous one. By doing so, computation of the effective

macroscopic properties of the homogeneous medium is then much simpler than for the

original problem. Considering the central role of this theory in the work presented here,

it is now described in more detail. In this chapter, the methodology will be reviewed,

starting with the basic concepts, before presented some special cases. Later, the

limitations of the theory and possible solutions are discussed in the context of our

prediction model.

4.1 Description of the theory

Consider a heterogeneous continuum in which we define a local conductive property.

We may approximate, using a finite difference representation, the conducting medium

by a network with a regular topology, where each bond is occupied by a given

conductance Ci. To account for possible heterogeneity, the set of all Ci follows some

probability density function. Our intention is to build up a network with the same

topology, but in which all the conductances have a single effective value, Ceff. The

appropriate value of Ceff is found as follows. First, a single conductance Cm is inserted

into the homogeneous system. The inclusion of Cm in the effective medium locally

disturbs the uniform field. However, if this procedure is repeated over all local

conductances, the disturbing effects of embedding Cm into the voltage distribution

should eventually average out to zero.

58

Figure 4.1. Construction used in calculating the “pressure” induced across one conductance, Cm, surrounded by a uniform medium (after Kirkpatrick, 1973).

In the self-consistent approximation, it is assumed that the homogeneous network

is made up of a set of equal conductances, Ceff, connected to each other on a cubic

lattice. The single conductances are then calculated assuming that each bond is subject

to the local potential, as well as an external potential; the external potential in this case

is taken to be a continuous flowrate. The overall effect of the local potential should

average to zero over a sufficiently large region. If we then consider one conductance,

having the value Cm, and orientated along the external pressure field surrounded by the

effective medium (Fig. 4.1), then the self-consistent solution in the presence of Cm is

constructed by adding the effects of a fictitious current or flowrate, qm, introduced at A

and extracted at B, to the uniform field. Far from Cm, the perturbation will be small and

the solution will correspond to the uniform field. Across A and B however, the uniform

solution fails to satisfy flow conservation, and so the magnitude of qm is chosen to

account for this:

meffeff qCCP =− )( m . (4.1)

The extra pressure, Pm, induced between A and B, can be calculated if we know the

conductance GAB' of the network between points A and B when the perturbation is

absent. Then:

)( 'ABm

mm CC

qP

+= . (4.2)

At this point, we need to calculate 'ABC before going any further. First, we express the

conductance CAB in the uniform effective medium as

(a) (b)

Ceff

Ceff

Ceff Ceff Cm

Ceff

Ceff

Ceff

Ceff

Cm

qm qm

qm qm

59

effABAB CCC += / . (4.3)

The total flowrate through each of the z equivalent bonds at the points where the fluid

enters and leaves is partitioned as zqm/ , so that a total flowrate of zqm/2 passes through

the AB bond. This leads us to

effAB CzC )2/(= . (4.4)

Combining eqs. (4.3) and (4.4) gives us

effAB CzC )12/(/ −= . (4.5)

By substituting eqs. (4.1) and (4.5) into (4.2), we obtain

meff

meffeffm CCz

CCPP

+−

−=

]1)2/[(. (4.6)

Eq. (4.6) must be solved to give a new set of discrete conductances that are then used to

construct new potentials. This iterative process is continued until no further significant

changes occur, at which point the system is said to be self-consistent. The requirement

that the average of mP vanishes gives

0]1)2/[(

=+−

−

ieff

ieff

CCzCC

, (4.7)

where the angle brackets corresponds to an averaging procedure with respect to the

probability density function of the Ci. Consequently, the relation above represents the

minimisation of the discrepancies between Cm and Ci, normalised to the macroscopic

conductance of the homogeneous network model. Furthermore, we ignore any possible

spatial correlation between the magnitudes of the local conductances. The co-ordination

number z can be defined locally as the number of conducting bonds reaching a node in

the network, which implies that, in Kirkpatrick’s approach, it is lattice-dependent. This

is the assumption under which Kirkpatrick (1973) originally derived his effective

medium theory. However, Koplik (1981) later showed through numerical simulations

that the EMA provides good accuracy for topologically disordered media, where z is

then the mean co-ordination number. As stated before in the previous chapter, both

Lock (2001) and Schlueter (1995) used a hypothetical isotropic cubic lattice, thereby

assuming 6=z . Doyen (1988) also used a mean co-ordination number of six for a

60

Fontainebleau sandstone under the assumption that the pore-space connectivity was

similar to that of a close random packing of grains. Koplik et al. (1984) used a co-

ordination number 49.3=z , based on a connectivity analysis of ten sectionals of

Massilon sandstone. Jerauld and Salter (1990) reviewed the experimental evidence on

co-ordination numbers and concluded that z typically ranged from four to eight. More

recent work by Lindquist et al. (2000) and Arns et al. (2004) gave average co-

ordination numbers ranging from 3.3-3.9 for Fontainebleau sandstone using three-

dimensional tomographic techniques. Our aim in this study is to incorporate a varying

co-ordination number, using recent work (Sok et al., 2002, Seright et al., 2003) that

suggests a trend between the absolute permeability and the co-ordination number of

rock samples. This will be presented in Chapter 6. There are also others issues that arise

when the co-ordination number approaches the percolation threshold. These will be

investigated later in this chapter.

4.2 Integral form of the EMA and some closed-form solutions

It is possible to express the EMA as an integral. To do so, we need to be able to

represent the values of Ci as a continuous probability density function. If this is defined

over the interval ],[ 21 CC as p(C), then the integral form is given by

0)(]1)2/[(

2

1

=+−

−∫ dCCp

CCzCCC

Ceff

eff . (4.8)

For example, if C is uniformly distributed between two values 1C and 2C , i.e.,

21

1)(CC

Cp−

= , (4.9)

then eq. (4.8) can be evaluated analytically to yield the following explicit equation for

Ceff (David et al., 1990):

121

12

]1)2/[(1ln

2CC

CCzCC

Cz

effeff −=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+−−

+ . (4.10)

In general, eq. (4.10) is difficult to interpret, and must be solved numerically. But in the

limiting case of a very narrow distribution, we can put CCC Δ+= 12 , and expand eq.

(4.10) in a power series to eventually find

61

2/)(2/ 2112 CCCCC +=Δ+≈ , (4.11)

which is to say, the effective conductivity approaches the arithmetic mean conductivity

as the distribution becomes narrower.

Schlueter (1995) considered a log-uniform distribution that has the form f(C) =

(2ClnCg)-1 for ],/[ αα gg CCC ∈ , where gC is the geometric mean of the distribution,

and α is some dimensionless factor >1. Schlueter solved eq. (4.8) for co-ordination

numbers corresponding to a series arrangement of conductors, ( 2=z ), a parallel

arrangement of conductors, ( ∞=z ), and for the isotropic cubic lattice, ( 6=z ). The

results were:

2=z : 1

ln22 −

=α

ααgeff

CC , (4.12)

∞=z : αα

αln2

)1( 2 −= g

eff

CC , (4.13)

6=z : )(2)1(

3/1

3/4

ααα−

−= g

eff

CC . (4.14)

The case of 6=z is of particular interest to us, as this is the one that applies to a cubic

lattice. For narrow distributions, setting εα += 1 and expanding eq. (4.14) for small ε

shows that geff CC → , i.e., the effective conductance approaches the geometric mean.

For broad distributions, which is to say large values of α, the effective conductance

given by eq. (4.14) can be approximated as

geff CC2

3/1α≈ . (4.15)

However, quite broad distributions are required for eq. (4.15) to hold, as the error is still

4% when 100=α .

4.3 Special examples with binary conductances

We now consider the case where we have conductances distributed according to some

distribution function, f(C); the self-consistent condition then gives

0]1)2/[(

)( =⎥⎥⎦

⎤

⎢⎢⎣

⎡

−+

−∫ dC

CzCCC

Cfeff

eff . (4.16)

62

If we assume a binary distribution of conductances, in which two strictly positive

conductances C1 and C2 occur with respective probabilities f and 1-f, eq. (4.16) reduces

to

01

2

)1(1

2 2

2

1

1 =⎟⎠⎞

⎜⎝⎛ −+

−−+

⎟⎠⎞

⎜⎝⎛ −+

−

eff

eff

eff

eff

CzC

CCf

CzC

CCf . (4.17)

The following quadratic equation on Ceff is thereby obtained:

( ) 02

1112

12 2112

2 =−⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ +−+⎟

⎠⎞

⎜⎝⎛ − CCfzCfzCCCz

effeff , (4.18)

for which we can find the following solution:

( )

22

1112 12

−

⎥⎦⎤

⎢⎣⎡ −−⎥⎦

⎤⎢⎣⎡ +−−

=z

fzCfzCCeff .

( )

2

12

42

1112 2112

−

⎟⎠⎞

⎜⎝⎛ −−⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦⎤

⎢⎣⎡ −−⎥⎦

⎤⎢⎣⎡ +−

±z

CCzfzCfzC. (4.19)

Now, if we let 02 →C , we obtain the following expression for Ceff:

22

12

1 11

−

⎥⎦⎤

⎢⎣⎡ −±⎥⎦

⎤⎢⎣⎡ −−

=z

fzCfzCCeff , (4.20)

with solutions

01 =effC , (4.21)

and

Ceff 2 =

2C1z2

f −1⎡

⎣ ⎢

⎤

⎦ ⎥

z − 2. (4.22)

If the fraction of conducting bonds, f, is greater than 2/z, then the conductance given by

eq. (4.22) is positive, and represents the physically meaningful root. If the fraction of

conducting bonds falls below 2/z, then solution (4.22) is not physically plausible, and

the effective conductance will be given by eq. (4.21), i.e., it will be zero. Hence,

63

Kirkpatrick’s method predicts that percolation (see §4.5.1 below) will occur when the

fraction of conducting bonds reaches at least 2/z.

For the case of a cubic lattice, where 6=z , the solution (4.22) becomes

( )132

1 −= fCCeff , (4.23)

i.e.,

21

23

1

−= fC

Ceff . (4.24)

In this case, the effective conductance will vanish for f < 1/3.

4.4 Discrete form of Kirkpatrick's equation

It may not be possible to always fit the distribution of the local conductance with a

continuous distribution function. In such cases, we can replace the integral in eq. (4.8)

by a summation:

0]1)2/[(1

=+−

−∑

=

N

i ieff

ieff

CCzCC

, (4.25)

where N is the total number of conductors. As Ceff is a monotonically increasing

function of z, the two cases 2=z and ∞=z provide bounds on Ceff that hold for all

values of z. In the case 2=z , which corresponds physically to a series arrangement of

conductors, eq. (4.25) yields the harmonic mean, whereas as ∞→z , which corresponds

to a parallel arrangement, it yields the arithmetic mean. Hence, we have the following

bounds:

( ) ∑∑=

−

=

≤≤⎥⎦

⎤⎢⎣

⎡ N

iieff

N

i i

CN

zCCN 1

1

1

111 . (4.26)

An example of how the effective conductivity is bounded by these two values is shown

in Fig. 4.2, for a set of hydraulic conductances of fracture segments taken from a

fracture network discussed by Priest (1992). In the figure we have plotted the left-hand

side of eq. (4.25), which we call the Kirkpatrick function, f, as a function of Ceff. The

root is seen to be bracketed from below by the harmonic mean, corresponding to the

limiting case of a serially connected network, 2=z , and from above by the arithmetic

mean, corresponding to the network being connected in parallel, ∞=z . It is worth

64

noting also that the actual root of eq. (4.25) does not lie very far from the geometric

mean. The variation of the effective conductance, as a function of the co-ordination

number, is shown in Fig. 4.3.

Figure 4.2. Kirkpatrick’s function for the twenty-one conductance values used by Priest (1992), plotted for the case 6=z . Note that Priest’s definition of C has the factor of μL incorporated into it.

Figure 4.3. Sensitivity of Ceff (mm2 s-1) to co-ordination number, using Priest’s data set.

Geometric Mean

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1 10 100 1000

Cef

f

z

65

Another issue that arises is whether or not Kirkpatrick’s equation will have one real

solution. The physically meaningful solution to the conductance problem requires that

the desired root lies between the bounds provided by the harmonic and arithmetic mean

of the conductances, and so it is only necessary to consider the uniqueness of the

solution inside this interval. Kirkpatrick’s function is continuous at every point in the

closed interval defined by the harmonic and arithmetic means of the hydraulic

conductances. Fig. 4.2 shows that f(Ceff) is negative at the lower bound and positive at

the upper bound. This is true in general, as examination of eq. (4.25) reveals that

Nf −= when 0=effC , and ]1)2//[( −= zNf as ∞→effC . Hence, by the

Intermediate Value Theorem, there must exist at least one point where the function

crosses the Ceff axis; indeed, this is the basis of the bisection algorithm. We also

recognise that this only occurs only once inside this interval, since the first derivative of

the Kirkpatrick function is positive everywhere for all values of z inside this interval,

i.e.,

f '(Ceff ) =(z /2)Ci

[(z /2) −1]Ceff + Ci{ }2 > 0i=1

N

∑ . (4.27)

Consequently, we can expect only one real root within this interval.

4.5 Limitations of Kirkpatrick’s EMA

Since the publication of the article by Kirkpatrick (1973) on the effective medium

approximation, there have been numerous studies on the reliability of his EMA, and its

limitations (David et al., 1990). It is not possible to cover all the literature available

within the scope of this thesis, and this is not our aim. However, we will explore the

issues that relate directly to our work.

4.5.1 Limitations near the percolation threshold

Before describing the limitations of Kirkpatrick’s EMA, we first briefly introduce

percolation theory and define the percolation threshold. Percolation theory was first

introduced for the flow of fluids in porous media by Broadbent and Hammersley

(1957). For many years subsequent to its introduction, it was used mainly in various

areas of physics, and only in the past two decades has it featured more prominently in

the field of porous media, with contributions from Berkowitz (1993, 1998) and Shante

and Kirkpatrick (1971), as well as in monographs by Stauffer and Aharony (1992),

66

Adler (1992) and Sahimi (1995). One of the specific questions relating to percolation is,

in fact, a very simple one. On a given lattice, if we start removing bonds (bond

percolation) or sites (site percolation) with a given probability p−1 , then the

probability p is the probability for a bond or site to be occupied. It is found, both

empirically and via computer simulations, that there exists a threshold pc for which, if

cpp > , an infinite cluster will span the lattice, whereas if cpp < , no such infinite

cluster will exist. Hence, when cpp < , the macroscopic conductivity will be zero. In

other words, the percolation threshold may be defined as the fraction of conducting sites

that need to be occupied for conduction to occur across the network. In practice,

calculating pc is difficult and, so far, has only been solved for a few lattices, including,

for example, square lattices.

Figure 4.4. Bond percolation on a square network at bond occupancies of p=1/3 and p=2/3. Note that there is no continuous path spanning the entire region in the former case, but there is in the latter case. The percolation limit, pc, for the square network is exactly 0.5 (Sahimi, 1995).

Some numerically computed percolation parameters (Sahimi, 1995) for several

three-dimensional lattices are shown in Table 4.1, where pcb is the bond percolation

threshold, which is defined as the largest fraction of occupied bonds below which there

is no sample-spanning cluster of occupied bonds. Similarly, pcs is the site percolation

threshold; above which an infinite cluster of occupied site spans the network. Bc is

defined as the mean co-ordination number at the bond percolation threshold, i.e.,

cbpc zB = .

67

Table 4.1. Numerical estimates of bond percolation threshold, site percolation threshold for four common three-dimensional networks (after Sahimi, 1995)

Lattice z pcb Bc pcs

Diamond 4 0.389 1.55 0.430

Simple cubic 6 0.249 1.49 0.312

Body-centred cubic 8 0.179 1.44 0.264

Face-centred cubic 12 0.119 1.43 0.199

The parameters obtained for the simple cubic lattice are of most interest to us, as

this is the lattice used by Lock et al. (2002), and which will serve as the basis of our

permeability computations. It is interesting to compare the percolation value for the

simple cubic lattice listed in the above table with the one predicted by the EMA. Indeed,

as mentioned before, the lower limit of the co-ordination number in the EMA is

obtained when 2=z , corresponding to a bond percolation threshold of 3/1=cbp . This

is higher than the value found for the simple cubic lattice, estimated to be around 0.25.

The difference between the two values has several implications. Firstly, it means that

the EMA is unable to handle co-ordination numbers lower than two, while we know that

in theory, 5.1=cB . This problem has been well documented in the literature

(Kirkpatrick, 1973) and several remedies have been proposed (Sahimi et al., 1983;

Zhang and Seaton, 1992).

The other implication of this problem is that it is expected that the predictions of

the EMA will start to diverge from the actual values at some sufficient low co-

ordination number. It is one of our objectives to be able to incorporate a varying co-

ordination number in our permeability predictions. However, in order to do so, one must

know the value at which the predictions between the EMA and an exact network

calculation start diverging. Lock et al. (2004) compared the performance of the EMA to

the results of explicit network calculations, using the code NETSIM (Jing, 1990), at a

fixed co-ordination number. As their calculation was based on a cubic lattice, the co-

ordination number used was fixed and equal to 6. They investigated the effect of a

spread in the conductance data, for the case of a lognormal distribution, and compared

the results of Kirkpatrick’s EMA, NETSIM and another approximate theory known as

the General Perturbation Ansatz (Gelhar and Axness, 1983). It was found that, in

general, all of these methods agree within a few percent for log-variances less than 5. In

the present work, we would like to extend this type of comparison to networks having

68

co-ordination numbers that are less than 6. Indeed, as mentioned before, Koplik (1981)

showed that the EMA can be used for topologically disordered media, where z is then

interpreted as the mean co-ordination number. We feel that, in the scope of this thesis,

verification of Koplik’s results is important; therefore, this will be presented next. We

will also introduce and use two independent methods for the lowering of the co-

ordination number, and compare the results. We will also compare the results obtained

with these methods with the scaling laws derived from percolation theory.

There are several ways in which one can lower the co-ordination number within

Kirkpatrick’s EMA. One approach would be to directly lower the co-ordination number

in Kirkpatrick’s equation. However, one has to recall that NETSIM, being based on a

cubic lattice, has a fixed co-ordination number of 6, and does not allow us to directly

change that value. There is however another way to “artificially” reduce the co-

ordination number in NETSIM. We can directly input some zero conductances within

the conductance distribution. By inputting a number of zero-conductances proportional

to the desired co-ordination number, one can recreate a lower co-ordination number,

with the zero-conductances simply blocking some of the nodes. This enables us to use

the same conductance files for both the methods, while having a co-ordination lower

than 6.

These two approaches have been investigated with a conductance file taken from

an image from one of our data sets. The sample from which the conductance file is

taken is sample ME2-2. More information on the sample can be found in Chapter 5,

where the data sets used in this thesis are presented. When using this with the EMA, the

co-ordination number within the equation was fixed to 6, in the same way that it is fixed

for NETSIM. Lock (2001) investigated the issues of lattice size and number of

realisations for NETSIM. In order to do so, a conductance file was used with lattice

ranging in size from 53 to 403 nodes, sizes increasing by increments of 5. These lattices

were then used to run with the conductance file for 20, 25 and 30 realisations,

respectively. Following this, Lock concluded that taking the average values from 20

realisations on a 303 lattice provided sufficient accuracy, as a larger lattice and more

realisations provided virtually no improvement. We will therefore follow these

guidelines when running our NETSIM simulations. The conductance data is taken from

sample ME2-2. We will present the normalised permeability as a function of the co-

ordination number. Before doing so, we here recall how the permeability is computed

from the effective conductance. Eq. (3.35) in Chapter 3 gives

69

k =NCeff

Atotal

, (4.28)

where A is the cross sectional area of the image, N is a number of pores that are normal

to the plane of this image, and Ceff is the effective conductance, as obtained from the

effective medium approximation.

Figure 4.5. Comparison of normalised permeabilities at co-ordination number ranging from 1.5 to 6 for both NETSIM and the EMA, both obtained while artificially lowering the co-ordination number using zero conductances.

It is seen that the results obtained for both NETSIM and the EMA are in many

regards very similar. However, they differ in their predicted value of the percolation

threshold. While the EMA effective conductivity vanishes at z = 2, as expected from eq.

(4.24), corresponding to a bond percolation threshold of 31=cbp , the NETSIM values

do not reduce to zero until the co-ordination number falls to about 5.1=z ,

corresponding to a value of pcb of around 0.25. This percolation limit agrees with the

one presented in Table 4.1 for a cubic lattice. We also notice that for 4≥z , the values

obtained by the EMA are marginally higher than the ones obtained from NETSIM. This

trend is reversed when z is lower than 4, and persists until the percolation threshold is

reached. Overall, the EMA seems to be handling the presence of zero conductances very

well, except very close to the percolation limit.

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6

Normalised k (NETSIM)Normalised k (EMA)

Nor

mal

ised

k

Co-ordination Number

70

As mentioned earlier, there is another way to lower the co-ordination number in the

EMA. Indeed, it is possible to vary that co-ordination number directly when computing

the effective conductance. However, when doing so, one must remember that a

correction is needed when converting the effective conductance into a macroscopic

permeability, as we must take into account the reduced numbers of pores in the lattice.

This was done by using a simple linear relationship, running from 6=z to 2=z , so

that the permeability agrees with the cubic lattice for 6=z , and vanishes at the

predicted percolation limit of 2=z . Therefore, the final computation to obtain the

permeability when lowering the co-ordination directly is expressed as follows:

k =z4

−12

⎛ ⎝ ⎜

⎞ ⎠ ⎟

NCeff

Atotal

⎛

⎝ ⎜

⎞

⎠ ⎟ , (4.29)

where N, Atotal, and Ceff are as defined previously. The results obtained using this

method are presented below in Fig. 4.6, where they are compared to some of the earlier

results.

Figure 4.6. Comparison of normalised permeabilities obtained with the EMA. In one method we add zero conductances to the data set, and in the other we lower the co-ordination inside the equation, and use eq. (4.29).

It is clear from Fig. 4.6 that the two different methods yield the same results, as the

data points obtained using a decreasing co-ordination number agree perfectly with the

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6

Normalised k (EMA with added zero conductances) Normalised k (Decreasing z in the EMA)

Nor

mal

ised

k

Co-ordination Number

71

curve obtained using the EMA with added zero conductances. This has several

implications for our work. First of all, it means that we can use the more direct approach

of lowering the co-ordination number inside the EMA equation, as we know that this

compares very well with the values computed with NETSIM. Clearly, the difference

between NETSIM and the EMA was minimal (Fig. 4.5), and we can safely say that,

down to a co-ordination number of three, the EMA is quite accurate. While the

percolation thresholds obtained with the EMA ( 2=z ) and NETSIM ( 5.1=z ) are

different, this does not have an impact on our predictions of absolute permeability.

Indeed, the co-ordination number of rocks will very rarely be lower than three. Recent

tomographic studies of rocks have shown that an average co-ordination number for

sandstone rocks is usually between 3 and 4 (Lindquist et al., 2000; Seright et al., 2003).

Koplik (1984) analysed forty-one SEM slices of Massilon sandstone and found the co-

ordination number to be 3.49.

Such studies have yet to be carried out for carbonate rocks, as the technology

currently does not provide high enough resolution for the 3-D imaging of the pore

structure. This is mainly due to the fact that most carbonate rocks exhibit micro-

porosity, with some pores being in the μm or sub-µm range. However, one can safely

assume that the co-ordination of these rocks will still be higher than 3. Another

implication of these results is that it gives us the opportunity to consider co-ordination

number higher than 6. Indeed, while it is possible to artificially reduce the co-ordination

number by adding some zero conductances to the conductance distribution, it is not

possible to increase the co-ordination number in this way. However, since the two

methods tested with the EMT give us the same results, we can confidently vary the co-

ordination number inside the EMA equation, if we need to account for a higher co-

ordination number. This opens up the possibility of implementing a varying co-

ordination number for our predictions, which is one of our objectives. For example, as

our model is based on a cubic lattice, it would be possible to add extra branches within

the cubic lattice to increase the co-ordination number. Each extra branch going from

opposite corners of the lattice will increase the co-ordination number of the lattice by

two, so adding the maximum of four branches could provide us with a maximum co-

ordination number of 14. In reality, such a high co-ordination number is very unlikely.

Nevertheless, it allows us to implement co-ordination numbers slightly higher than 6, a

possibility when dealing with high-permeability rocks. This, however, still leaves us

with the problem of devising a method for estimating the co-ordination number for our

72

samples. This issue and the expanded cubic lattice will be investigated in more details in

Chapter 6, when presenting the single-phase results.

In Chapter 7, we will attempt to predict two-phase relative permeability curves for

a few samples. However, the use of Kirkpatrick’s EMA will be compromised in our

calculations, due to the over-estimated percolation threshold one obtains when using the

EMA. This will be explained in more detail in the relevant chapter. However, this can

be avoided by using the Generalised Perturbation Ansatz (GPA) in conjunction with

percolation theory, which enables us to predict the correct percolation threshold.

Percolation theory, which was briefly introduced earlier, also provides us with a

powerful way to model percolation processes. Indeed, one of the most important

characteristics of percolation processes is the universal scaling laws they obey. The

behaviour of many percolation quantities near the percolation threshold is insensitive to

the network structure, and to whether the percolation process is a site or bond

percolation process. The quantitative statement of this apparent universality is that

many percolation properties obey scaling laws in the vicinity the percolation threshold.

These are characterised by the critical exponents and the Euclidian dimensionality of the

system, d. Scaling laws exist for several parameters such as the percolation probability,

the correlation length and the effective conductivity. Similarly, the permeability of a

given system can be defined as follows

tcppk )( −∝ , (4.30)

where pc is the percolation threshold and t is the critical exponent. As mentioned above,

the value of the critical exponent t depends on the dimension of the system. While the

value for two-dimensional system is now commonly accepted ( 3.1≈t ), there has been,

over the years, a debate over the value of t in 3-D. Using computer simulations,

Kirkpatrick (1973) obtained a value of 1.06.1 ±=t , whereas Adler and Brenner (1988)

found a value of 1.87. Sahimi (1995) and Gingold and Lobb (1990) computed numerical

estimates for t and found a value of around 2. O’Carroll and Sorbie (1992) also found a

value of 1.6 ± 0.1 for the critical exponent for randomly-filled networks. They assumed

this to be true in the vicinity of the percolation threshold and, as well as over the whole

range of probabilities p, with 1≤≤ ppc . It is not our aim to resolve the discrepancy

between the values presented here, but merely to be aware of the range of exponent

values that have been reported in the literature. Here, we primarily wish to investigate

how well the scaling law and the network prediction agree with each other. It is easy

73

enough to do so to assume that a co-ordination number of 6 corresponds to a probability

1=p . We also wish to determine which value of the critical exponent can be used over

the whole range of probabilities, therefore enabling us to use percolation theory for our

relative permeability predictions. We will therefore plot the different curves obtained

using different critical exponents, respectively 1.6, 1.8 and 2, both near the percolation

threshold and over the entire range of probabilities.

Figure 4.7. Comparison between the NETSIM prediction and the scaling law, near the percolation threshold, presented here with three different critical exponents t.

Figure 4.7 shows the curves obtained for the normalised permeability using the

scaling law with a range of critical exponents, as well as the curve originally computed

with NETSIM near the percolation threshold. We here focus on the critical region so as

to analyse the behaviour of the network predictions and that of the scaling law, with

precision. The NETSIM curve presented is again the one obtained with the same sample

used previously, ME2-2. It is interesting to note that 0.2=t seems to work well very

close to pc, in the critical region. However, when 35.0≥p , the NETSIM curve increases

0

0.02

0.04

0.06

0.08

0.1

0

0.02

0.04

0.06

0.08

0.1

0.2 0.25 0.3 0.35 0.4

1.2 1.5 1.8 2.1 2.4

Scaling Law - t = 2Scaling Law - t = 1.8 Scaling Law - t = 1.6

Normalised k (NETSIM)

Nor

mal

ised

k

Probability p

Co-ordination Number

74

more rapidly compared to the curves 0.2=t and 8.1=t . The NETSIM curve appears

to get closer to the curve obtained with 6.1=t . This confirms the values quoted in the

literature for the value of the critical exponent near the critical region (Sahimi, 1995,

1998; Gingold and Lobb, 1990), which appear to be close to 2.

Figure 4.8. Comparison between the NETSIM predictions and the scaling law, over the entire range of bond probabilities, for three different critical exponents, t.

Figure 4.8 shows the same comparison as in Fig. 4.7, but for the entire range of

values of the bond probability. Away from the percolation threshold, the value 6.1=t

gives the best agreement with the NETSIM values, confirming the assertion of

O’Carroll and Sorbie (1993). As seen in Fig. 4.8, and in agreement with previous results

(Sahimi, 1995), only very close to the percolation threshold does the critical exponent of

2 gives a good match for the normalised permeability. However, it is very unlikely that

we would need to go to such low probabilities (or co-ordination numbers) - certainly not

for the single-phase predictions. However, we will use a percolation-type model for our

two-phase predictions (Chapter 7), so we will revisit this issue.

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

1.2 2.4 3.6 4.8 6

Scaling Law - t = 2Scaling Law - t = 1.8Scaling Law - t = 1.6

Normalised k (NETSIM)

Nor

mal

ised

k

Probability p

Co-ordination Number

75

Overall, the use of the EMA at lower co-ordination numbers is here justified. Other

studies (David et al., 1990) also investigated the effect of different co-ordinations

numbers by using different lattices, such as hexagonal, square and triangular, with

respective co-ordination number of 3, 4 and 6. It was found that, independently of the

distribution function used, the behaviour of the EMA was not influenced by the co-

ordination, at least down to a value of 3=z , in accordance with the results presented

above.

4.5.2 Limitations with the input data

We have reviewed the limitations of the EMA near the percolation threshold, as well as

the implementation and the validity of the use of the EMA over a range of possible co-

ordination numbers. There have also been several articles (David et al., 1990; Zhang

and Seaton, 1992; Adler, 2000; Lock et al., 2004) that have investigated the accuracy of

the EMA when used with certain types of distributions, such as uniform, log-normal or

exponential. We will briefly review some of this work, with particular attention to the

implications for the present work of computing effective conductivities of a porous

medium.

David et al. (1990) investigated the numerical accuracy of the EMA when used

with various distributions on 2-D lattices. In order to do so, they compared the effective

values obtained with EMA to the values obtained from a standard network calculation.

The calculation of the conducting properties for regular networks is a known problem;

therefore we will only describe the main steps. Consider a regular network of nodes and

bonds and apply “voltage” on nodes and “current” on bonds. Kirchhoff’s law of

conservation tells us that, at each node, the sum of all currents will be zero. Following

that, we can express the problem using a matrix formulation. As David et al. used

networks of finite size, they had to define boundary conditions; the nodes on the upper

rows of the network have the same potential V0, whereas the nodes on the lower row

have a different potential V∞. The potential gradient present therefore drives the flow.

After solving the matrix using the Choleski method, they finally arrived at the following

expression for the effective conductance:

( ) [ ]10 CVVI

C toteff −

=∞

, (4.31)

76

where Itot is the total current, V∞ the lower row potential, V0 the upper row boundary

condition potential, and C[1] is the overall conductance of the homogeneous network

with the same topology, in which all the local elements are set to unity. Therefore, in

this case, the effective conductance is determined numerically without any

approximation, by solving the flow equations. Their study was based on three different

2-D networks, which included hexagonal, square and triangular networks, with co-

ordination numbers of 3, 4 and 6, respectively. It is interesting to note that they only

focused on 2-D networks, and did not investigate the accuracy of the EMA for 3-D

networks. They also used three types of distributions, which included uniform, peak-

like, and decreasing exponential distributions. All three distributions were tested for

several parameters, over a standard test interval of [0,50]. The conductivity files were

then populated with a collection of 1000 conductances taken from the tested

distributions, as it was concluded by the authors that a minimum of 500 conductances

were required to eliminate statistical variation from the results.

In general, they found that the EMA worked very well with uniform and peak-like

distributions, with errors in the region of only a few percent. However, when used with

decreasing exponential distributions, the EMA gave much larger errors, under-

predicting by about 40-50%, when compared to the network computations. It was also

concluded that the error increased as the ratio of probability between the extreme values

increased. It is interesting to notice that in the case of the decreasing exponential

distribution, the EMA predictions consistently under-predict when compared to the

network values. Decreasing exponential distributions are similar to the distributions

found in rocks, so that the errors generated by the EMA when these distributions could

potentially pose a problem for our work.

However, other studies of the validity of the EMA (Adler and Berkowitz, 2000;

Lock et al., 2004) with 2-D and 3-D lattices did not lead to such alarming conclusions.

Adler and Berkowitz (2000) compared the effective conductance values obtained from

randomly generated 2-D and 3-D lattices with the effective values computed using the

EMA. A complete description of the numerical simulations can be found in Margolin et

al. (1998). Their study was undertaken using log-normal distributions over a range of

probabilities and log standard deviations. In 2-D, they found that the predictions of the

EMA agreed very well with the numerical results from the simulations, somewhat

differing from the results of David et al. In 3-D, they concluded that the EMA provided

good prediction of the overall conductivity when the log standard deviation, b, was

equal to 0; i.e., when the lattice was reduced to a binary system. However, as b

77

increased, the error between the effective values from the simulations and from the

EMA also increased, with the errors also becoming more significant as the probability

was lowered, a result already acknowledged earlier in the chapter. Adler and Berkowitz

concluded that, when the log standard deviation was greater than 2, the EMT should not

be used. This criterion seems a little conservative, as Lock et al. (2004) concluded that a

log standard deviation of up to 5 could be used. It was also noticed that, as b was

increased, the values obtained with the EMA were too low, compared to the network

values. A similar trend was also noticed by David et al. This can be explained by flow

channelling, as has been demonstrated using flow simulations under these conditions

(Margolin et al., 1998). Indeed, the effective medium theory, when used with a

decreasing conductance distribution, will tend to give preference to the smaller values

of the distribution, simply because these are more numerous than the larger ones. In

effect, the effective values from the EMT will be shifted to the lower range, as the

theory does not take into account what the current or flow pattern might be in such a

heterogeneous medium. On the other hand, the flow in a given network will not be

constricted so much by the lower values, as there might be a path connecting the more

conductive elements, therefore creating flow channelling, hence explaining the

difference between the network and EMA values.

Lock et al. (2004) investigated different methods to upscale the permeability from

the pore scale to the core scale. They compared different solutions to do so, including an

exact network solution (NETSIM), Kirkpatrick’s EMA and the so-called Generalised

Perturbation Ansatz (GPA), which in three dimensions predicts that 6/2σekk geff = ,

where gk is the geometric mean of the permeability distribution, and σ is the variance.

Prior to testing these methods with “real” conductivity data, they first investigated the

accuracy of the EMA for a log-normal conductivity distribution. They investigated the

accuracy of both the GPA and the EMA against a NETSIM network calculation on a

cubic lattice filled with conductors. The results of their work are shown in Fig. 4.10, for

log-variances up to about 6. Both approximations are reasonably accurate over this

range of variances, with the error of the EMA never exceeding 10%. More specifically,

the GPA’s predictions are within 8% of the network’s values, for log-variances up to 5,

and just below that for higher log-variances. This is similar to the errors previously

found by Neuman et al. (1992), using a continuum model. Kirkpatrick’s EMA is

slightly more accurate than the GPA for log-variances up to about 4, with the accuracy

decreasing slightly as σ increases. This gives us confidence for the use of the EMA in

our work, as Lock et al. carried this study out for a 3-D cubic lattice rather than a

78

random one. Also, the calculations we carried earlier on in the chapter proved that the

EMA can handle real-rock data, as this was carried out one of our samples. Statistical

studies on our data set revealed that our mean σ was around 3.1, with a minimum of

0.64 and a maximum of 5.1, for a vuggy carbonate. Apart from the log-variance found

in a vuggy carbonate, the range of expected σ is contained within the interval

recommended by Lock et al. (2004).

1.0

1.5

2.0

2.5

3.0

0 1 2 3 4 5 6

GPANETSIMKirkpatrick

k eff

/ kge

o

Variance (lnC)

Figure 4.9: Comparison of NETSIM, Kirkpatrick’s EMT and the GPA for log-normal conductivity distributions (after Lock et al., 2004).

One might think that such good agreement between the EMA and the GPA would

advocate for the use of the somewhat simpler GPA in our work. Indeed, if we were to

restrict our predictions to a co-ordination number of 6, the use of the GPA for our

single-phase predictions could be considered. However, as it is our aim to try to

implement a varying co-ordination numbers in our single-phase predictions, this rules

out the use of the GPA at this stage. Its use will, however, be justified in the case of

two-phase relative permeability predictions (Chapter 7).

We have, throughout this chapter, reviewed the basic mechanisms behind the

effective medium theory, and studied some specific cases, as well as potential problems

faced when using this theory for our work. The literature on the subject is very

extensive and it would not have been possible for us to review in its entirety, nor was it

one of our aims. We have however highlighted the mains problems of the EMA. Firstly,

the theory was found to be inadequate to deal with low co-ordination numbers, which,

in turn, leads to the inability of the EMA to correctly predict the percolation thresholds.

79

Several solutions have been proposed in the literature (Sahimi et al., 1983; Hollewand

and Gladden, 1992; Mukhopadhyay and Sahimi, 1994; Zhang and Seaton, 1992) to

minimise these effects, with various level of success. It was also found that certain types

of distributions (decreasing exponential, log-normal) would not necessarily work with

the EMA, when subject to extreme parameters (high log-variance or log standard

deviation). However, the distributions present in our dataset are not so extreme (Chapter

6), therefore allowing us to use the EMA in the context of this work.

80

5 REVIEW OF DATA SET AND SAMPLES In this chapter, the data sets and samples that have been used during the course of this

research will be described in detail. Several independent data sets have been used in this

study. As the main focus of this project has been to extend the methodology developed

by Lock to the prediction of absolute permeability of carbonate rocks (Chapter 6), most

of the data are on carbonate samples. However, another aim of this work has been to

extend the methodology to the prediction of two-phase relative permeabilities. For this

purpose, it was decided to first work with a rock having a simpler pore structure, and so

Berea sandstone was chosen for this purpose (Chapter 7). We will briefly describe the

main differences between sandstones and carbonates, before presenting the data in more

detail, both in geological and petrophysical terms.

5.1 Carbonates vs sandstones

Carbonate rocks make up only 20% of the sedimentary rock record, yet they account for

more than 60% of the world’s proven hydrocarbon resources, and presently provide

around 40% of the world oil production (Akbar et al., 1995). This is expected to last

well into the present century, due to several giant carbonate fields in the Middle East.

However, while sandstones or silici-clastics are reasonably well known and defined in

terms of their pore-structural controls on petrophysical properties, the same cannot be

said of carbonate rocks. Indeed, several post-depositional processes can completely alter

their micro-structure and properties. It is therefore important to have a good

understanding of these rocks in order to be able to produce oil from them efficiently.

Sandstones are composed of a variety of silica-based grains that have generally

travelled away from their original depositional location. These rocks generally stand up

to the rigor of geologic time, and therefore undergo only minor modifications during

diagenesis. The depositional record is usually preserved, and correlation between wells

is usually done fairly easily. The grains present within the rocks are regularly shaped,

and the pore space, while possibly complicated, remains inter-granular. On the other

hand, carbonates consist of mainly two minerals, calcite and dolomite, as well as a

limited numbers of accompanying minerals. They usually remain near their point of

origin and form in shallow and deep marine settings, evaporitic, basins lakes and windy

81

deserts. Most of the older carbonates have been formed in shallow marine origins, but

the most widespread type of modern carbonates are those formed in deep water. Unlike

sandstones, carbonates are chemically unstable and can undergo substantial alteration

after deposition. Such alteration processes include cementation, mineral dissolution, and

re-crystallisation (Akbar et al., 1995). There are several other such processes that will

not be discussed here. Interested readers can consult the monographs by Scholle and

Ulmer-Scholle (2003) and Lucia (1999), which are authoritative references on carbonate

diagenesis and petrography.

Diagenesis can either reduce or increase both porosity and permeability. In general,

however, the trend is towards a progressive loss of both porosity and permeability with

increased time and depth of burial, with the shift possibly quite substantial. There are

two common methods to describe and classify carbonates that have been developed,

namely by Dunham (1962), and by Folk (1959, 1962). The Dunham classification is

based on the characterisation of the support framework of the rock or sediment, while

the Folk classification uses multiple descriptive terms, where the fundamental name is

based on the four grain types as well as the relative abundance of grains, matrix and

cement or pore spaces. For example, Dunham’s classification firstly differentiates

between mud-supported or grain-supported structure. Then, the amount of mud in the

samples is determined, hence leading to the four main categories: grainstone, packstone,

wackestone and mudstone. These generic names are then combined with the main type

of grains found to give rock names such as “oolitic grainstone”, for a grainstone made

up mainly of ooids.

Figure 5.1. Illustration of the Dunham classification (Dunham, 1962).

82

It is recommended to refer to Scholle and Ulmer-Scholle (2003) or Lucia (1995,

1999) for a detailed overview of the two classification methods, as well as their

advantages and drawbacks. Lucia has also devised another classification process in his

1999 monograph, which is also widely used. For the purpose of this study however, we

will use the Dunham classification to describe the samples presented here.

The data used for this thesis consist of either petrographic (or thin section) images

and Back-Scattering Electron Microscopy (BSEM) images. While the former are used

for general rock description, the latter form the basis of our permeability prediction

model. All data used herein have been supplied by Shell International Exploration &

Production (SIEP), Rijswijk, The Netherlands. The database of samples used here

consists of three independent data sets. One data set consists of images taken from three

outcrop locations in Southeast France. These will be presented first. More BSEM

images make up the other two data sets. These are taken from two carbonate fields in

the Middle East. Before describing the data, we will first summarise the different steps

involved in the acquisition of the data.

5.2 Data acquisition

5.2.1 Core sampling and sample preparation

Core samples provide valuable data source for understanding reservoir quality and

performance, as well as for investigating geological heterogeneity. In our case, this is

the starting point of our data acquisition process. In carbonate formations, core recovery

and quality are often poor, as these are often more fragile formations, which can lead to

lost or damaged core. Special attention is therefore needed in all mechanical aspects of

the coring process, such as retrieval, handling, washing, etc. In practice, rock cores,

approximately four inches in diameter, are taken perpendicular to the bedding plane in

the course of drilling. A one inch circular section of material, known as a core plug, is

then taken parallel to the bedding laminations, and all trim material is carefully retained

as it can later be used for the Scanning Electron Microscopy (SEM) imaging process.

From here the process is then two-fold. Firstly, the core plugs are used for the thin

sections preparation. In order to help with the identification of porosity, the plugs are

impregnated with blue-dyed epoxy resin. The thin sections are also usually stained to

help distinguish between different minerals. For example, Alizarin Red S and Potassium

Ferricyanide can help distinguishing between calcite and dolomite, while sodium

cobaltinitrite is used to identify feldspars. In the meantime, the trim material retained

83

from the core plug process is used for the preparation of the SEM samples. The rough

cubes, approximately 125 mm3 in volume, are highly polished and then coated with

carbon using a Biorad Sputter coater, while the part of the block surrounding the rock is

coated with gold. An energy dispersive analysing system (EDAX) was used to identify

minerals observed while the BSEM images were acquired using a Philips XL-40

scanning electron microscope, which was equipped with a BSE detector (Foglietta,

2002).

5.2.2 Scanning electron microscopy

Essentially, SEM and BSEM are the same type of scanning processes. The electron

beam is focused by one or two condenser lenses into a beam with a very fine focal spot,

sized 1 nm to 5 nm. The beam passes through pairs of scanning coils in the objective

lens, which deflect the beam in a raster fashion over a rectangular area of the sample

surface. As the primary electrons strike the surface they are inelastically scattered by

atoms in the sample. Through these scattering events, the primary electron beam

effectively spreads and fills a teardrop-shaped volume, known as the interaction

volume, extending from less than 100 nm to around 5 µm into the surface. Interactions

in this region lead to the subsequent emission of electrons, which are then detected to

produce an image (Joyce and Loebl, 1985).

The difference between SEM and BSEM comes from the detection of

backscattered electrons in the case of BSEM. In the context of rock imaging,

backscattered electrons can be used to detect contrast between areas with different

chemical compositions. These can be observed especially when the average atomic

number of the various regions is different. Indeed, since heavy atoms with a high atomic

number are stronger scatterers than are light ones, images with back-scattered electrons

(BSE) can contain compositional information.

5.2.3 Data presentation

Thin section and BSEM imaging allow us to generate different kind of images, and at

several resolutions. The thin sections allow us to generate images over large areas,

generally one or several millimetres. These images generally do not show enough detail

of the porosity system for them to be successfully used for direct prediction of rock

properties. However, these can still be analysed under the microscope to provide

textural and depositional information. Firstly, incident light microscopy can be used at

84

low magnification to establish macroscopic textural characteristics of the samples.

Secondly, transmitted light microscopy can be used at higher magnifications to evaluate

depositional and diagenetic textural and compositional characteristics.

As mentioned earlier, BSEM images were, for this project, the main source of data.

The samples were examined using the following settings: electron beam voltage, 15 kV;

working distance, 10 μm; spot size 4. For the Southeast France samples, the electron

beam voltage was 25 kV. Four or more greyscale images were taken of each sample.

Different magnifications were used in order to give the best possible view of the pore

system. The pore space in these pictures is shown in black. Generally, a magnification

ranging between ×25 to ×70 was used for a general view of the rock, while for

analysing macro- and micro-pore system inter-relations, a magnification ranging from

×140 to ×280 was used. For a better view of the micro-porosity, a higher resolution

ranging from ×1180 to ×1400 was used. Generally, the settings used are specified on the

images. On these images, the porosity is shown in black, and the grains and minerals are

shown in different shades of grey. This will be investigated in more detail in the next

chapter, when the image analysis process is presented.

All BSEM and petrographic images were received from SIEP in an uncompressed

*.TIF format, at a pixel definition of 1424×968. Depending on the magnification used,

the field and pixel sizes varied accordingly. All images were then cropped to remove the

scale-bar and other extraneous information, and finally re-sized to a more manageable

size of 700×400 pixels.

5.3 Outcrop samples: Southeast France (SEF)

The first part of our data set is composed of three limestone samples from Southeast

France. Three main locations were investigated: Belvedere, with very fine grainstone,

Orgon, which also shows some very fine grainstone, and Rustrel, with a vuggy coarse

grainstone. Plug samples were drilled in freshly slabbed quarries to avoid blasting

fractures and weathering transformations.

These outcrops from Provence were specifically selected as stratigraphical and

petrophysical analogues to Lower Cretaceous carbonate reservoirs in the Middle East.

The petrographical analyses also revealed significant pore type similarities with Middle

East carbonate reservoirs, especially with the presence of microporosity associated with

grainstone textures. These outcrops were originally collected by Jean Borgomano, of

Marseille University, while working for SIEP. However, probably because these are not

85

reservoir rocks, a full geological and petrophysical study does not appear to have been

performed, and information for these samples is generally scarce. Most of the

information presented below has been compiled by Borgomano, and has not been

published.

5.3.1 Orgon and Belvedere samples (SEF-1 & SEF-2)

These predominant rock fabric observed in these two samples is a slightly compacted,

very fine grainstone, most likely from the Lower Cretaceous geological era. The

depositional bodies correspond to laterally extensive massive sand sheets, and sand

waves with cross-bedding. These samples are characterised by textural homogeneity

(100% grain-supported) and grain size variability (very fine to gravel). A majority of the

rock fabric is composed of peloids in various proportions. Other components are also

found within the grainstone facies, such as benthic foraminifers and orbitolinids, which

are often leached (Fig. 5.2). These features are also present in sample SEF-1, which can

be viewed in Appendix C (Fig. C.1), where pictures from all the samples used in this

thesis are presented. The depositional environment suggested here by the rock fabric is

that of a shallow-water, low-energy setting.

The textures and pore types of these grainstones are characterised by intra-granular

micro-porosity and, in lesser extent, inter-granular porosity. The combination (amount

and distribution) of these two end members is most likely one of the primary controls on

the petrophysical properties in the Lower Cretaceous Middle East reservoirs. The

diagenetic transformations of these rocks have not been studied specifically here;

however, they seem to be homogeneous and conformed to strata, and consist mainly of

calcite cementations (marine and freshwater) and leaching/micritisation. In particular,

leaching can easily be identified by of the intra-granular micro-porosity present in the

rocks, where it looks as if unstable material has been replaced. The basic petrophysical

properties of these rocks are presented at the end of the next section, in Table 5.1. As

mentioned earlier, Fig. 5.2 shows a plate for sample SEF-2, whereas a plate for sample

SEF-1 is attached in the appendix (Fig. C.1).

86

Figure 5.2. Low magnification image from the Belvedere sample (SEF-2), a very fine peloidal grainstone. The presence of a slightly leached benthic foraminifer is indicated on the plate by B.

5.3.2 Rustrel sample (SEF-3)

The third sample from this data set is a vuggy coarse grainstone. It is the only vuggy

sample present in this study, and it is also from the Lower Cretaceous time. There are

two types of vuggy rock fabric, in which the vugs are either non-touching or touching,

both of which will have different implications for the petrophysical properties of the

rocks. Non-touching, or separate, vugs are not connected to each other directly, but are

connected via the inter-particle pore space. Generally, the presence of non-touching

vugs increases the porosity but does not significantly improve the permeability (Lucia,

1983). In fact, the permeability of a moldic grainstone is less than would be expected if

all the porosity were inter-particle, and at constant porosity, permeability increases with

decreasing separated vug porosity (Lucia and Conti, 1987). This of course is likely to

have some impact on the permeability prediction of our model, and will be discussed in

more detail in Chapter 6.

This sample is composed mainly of coarse grains as well as peloids. Most of the

image of the sample is taken by a gastropod of more than 2 mm across (Fig 5.3). As

with the two other samples, benthic foraminifers and orbitolinids are also present. There

B

87

are evident signs of heavy leaching everywhere across the sample, leading to the

formation of vugs and oversized pores. This is noticeable within the gastropod as well

as with the dissolved orbitolinids found at the left of the gastropod. These oversized

pores, formed by leaching, would most likely not be connected to each other, but only

via the inter-granular pore space (Wagner, pers. comm., 2005) This will have

implications for the permeability predictions.

Figure 5.3. Low magnification image from the Rustrel sample (SEF-3). The presence of the gastropod is illustrated on the plate by G, while D illustrates a severely leached orbitolinid, and O shows an oversized pore created after leaching occurred.

Table 5.1. Basic petrophysical properties for the SEF data set.

Sample no.

Rock Classification (Dunham)

Core depth (m)

He porosity (%)*

Air perm. (mD)

Grain density (g/cm3)

SEF-1 Grainstone n/a 13.4 0.5 2.70

SEF-2 Grainstone n/a 16.7 1.1 2.70

SEF-3 Grainstone n/a 21.7 25.2 2.69

G

O

D

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5.4 Middle East field 1 (ME1)

The second data set used in this thesis consists of samples originating from a Middle

East oil field. There are six samples available, and they are all taken from the same well.

Most samples in this data set are relatively similar in terms of geology and textural

properties, while presenting some marked differences in terms of petrophysical

properties. We will first describe the geology, texture and diagenesis of the samples

before describing the pore system and petrophysical properties. Most of the information

presented in the following sections is taken from the SIEP internal report G434

(Foglietta, 2002).

5.4.1 Geology, texture and diagenesis

The six samples present in this data set can all be described as grainstones. Of the

samples, two can be described as peloidal grainstones (ME1-4 and ME1-5), while

another two can be described as peloidal and intraclastic grainstones (ME1-1 and ME1-

2). In these four samples, the most abundant granular components of these samples are

peloids, followed by benthonic foraminifers (including orbitolinids), intraclasts, and

echinoderm fragments (Fig. 5.4). The last two remaining samples, ME1-3 and ME1-6,

are more correctly described as intraclastic and peloidal grainstones, therefore having

intraclasts as their main granular component, especially in the case of sample ME1-6,

while still retaining some peloids and other components. Almost all of these samples are

poorly sorted, apart from ME1-1. The plates for all these samples can be found in

Appendix C, Figs. C.4-C.9.

All samples exhibit early diagenetic products in the form of both finely bored

bioclasts (by micro-organisms such as algae, fungi, etc.) and micritic envelopes. In

addition, bioclasts are often abraded, indicating that they have been subjected to

mechanical erosion, most probably by current action. Almost all samples have been

exposed to evaporitic conditions, as indicated by the presence of dolomite cement. The

most important diagenetic process in these samples is, supposedly, carbonate

dissolution. Although difficult to prove and, even more so to quantify, carbonate

dissolution is here suggested by a number of indications, both direct and indirect. One

indication is the presence, on virtually every sample, of partially leached dolomite

crystals. This is clear on samples ME1-3 to ME1-6, which all display various degrees of

leaching, but it is even more evident on samples ME1-5 and ME1-6, which have heavy

leaching showing within the rock (Figs. C.8 and C.9). However, calcite dissolution can

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only be inferred indirectly. Indirect evidence of carbonate dissolution includes, amongst

other things, the presence of cemented areas, fossil moulds, and oversized pores.

Further to the petrographic imaging, BSEM images confirm the presence of dolomite

crystals, albeit in a heavily dissolved form, especially in the lower samples.

90

(i) Magnification ×45

(ii) Magnification ×113

Figure 5.4. (i) General view of sample ME1-1, at ×45 magnification. The presence of a partially dissolved benthic foraminifer is highlighted at the top of the image (B). Note the high porosity (in blue) due to the lack of inter-granular cement. (ii) A closer view of the sample ME1-1 (×113 magnification) with the presence of oversized pores (O) confirming the effects of leaching on the sample.

O

B

91

5.4.2 Pore system and petrophysical properties

Porosity in these samples is strongly affected by the texture of the rock. Grainstone

fabrics are generally characterised by well to fairly well connected inter-particle pores,

followed by subordinate sparse to very sparse, usually isolated, intra-particle porosity.

The sorting of the grains within the rock fabric will also influence the porosity, as well

as the connectivity between the pores. This is indeed confirmed when looking at the

BSEM images. The porosity measurements are also constant through the studied part of

the well, with a porosity reading of around 30% and no obvious correlation with well

depth. However, it is clear that the porosity of these samples is mainly controlled by

rock texture. Indeed, it is known that deeper samples in this well are mud-supported,

and the porosity and permeability are much lower than in this batch of samples.

The main diagenetic controls on porosity and permeability for these samples relate

to the effects of exposure to meteoric diagenesis, especially re-crystallisation and

dissolution. All of these samples have almost certainly been affected by dissolution

processes, as highlighted in sample ME1-4 (Fig. 5.5). The dissolution of unstable

mineral phases was probably quite effective in the grain-supported facies studied here,

likely to be richer in marine cement phases. This, most likely, resulted in the

enlargement of the pore-to-pore conduits and, consequently, in the enhancement of both

porosity and permeability.

In addition, it was observed that in the grain-supported fabrics, most of the calcite

crystals that constitute the micritic grains (e.g., peloids) have irregular crystal

boundaries. This feature is typical of re-crystallisation processes. This observation

probably has a major impact on the pore connectivity, in that pore throats are never

completely blocked by dense cement phases; instead they are always connected through

a maze of micro-pores, as in samples ME1-4 (Fig. 5.5 (iv)) and ME1-5. This is

especially noticeable when viewing the high resolution BSEM images. This effect,

however, is likely to have less impact in grain-supported fabric than in the mud-

supported facies.

Sorting is also a major textural control on permeability. Clearly, better-sorted

samples will have a better-connected and more homogeneous pore system, therefore

resulting in higher permeability performances. This is certainly noticeable in these

samples, as the better-sorted samples have the higher permeabilities, such as sample

ME1-1, shown in Fig. 5.2. BSEM images also reveal the presence of oversized pores

(Fig. 5.5, plate (ii) & (iii)), as well as micro-fractures which are also locally present,

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which very likely increase the permeability of the samples. All samples here display an

appreciable amount of permeability (Table 5.1), certainly much more than the two other

data sets. This was in fact the reason why this data set was chosen, so that the method

could be tested with rocks that had permeability ranging over several orders of

magnitude.

Table 5.2. Basic petrophysical properties for the ME1 data set

Sample no. Rock Classification (Dunham, 1962)

Core depth (ft)

Porosity* (%)

Air permeability (mD)

Grain density g/cm3

ME1-1 Peloidal grainstone d 32.5 400 2.708

ME1-2 Peloidal grainstone d + 0.4 29.3 56 2.706

ME1-3 Peloidal grainstone d + 26.7 27.4 260 2.708

ME1-4 Peloidal grainstone d + 28.8 30.8 73 2.709

ME1-5 Peloidal grainstone d + 29.4 31.4 103 2.707

ME1-6 Intraclastic-peloidal grainstone d + 33.3 29.8 1010 2.709

*Helium porosity

Permeability readings do not appear to display a particular trend with well depth.

However, we know that permeability is, in fact, correlated with rock texture (Dullien,

1992). One can, nevertheless, notice the range of permeability values offered by a depth

range of not more than ten metres. This shows clearly the problems faced with

permeability modelling, when working on carbonate reservoir characterisation.

Nevertheless, one must be sceptical of the permeability value shown for sample

ME1-6. This value is high and certainly not completely unrealistic, as it is possible for

carbonates to have very high permeability. However, because the sample has been very

heavily leached, and therefore lacking cement between the grains, it could have easily

been damaged during preparation if it was not handled carefully. The presence of a

fracture across the sample, as mentioned in Shell internal report G434, would certainly

have had an influence in the permeability measurement. It is not necessarily so clear on

the image, because the sample has been infiltrated with mud, but it is certainly possible

that the permeability measurement may have been affected by this.

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Figure 5.5. BSEM images of sample ME1-4, at different magnifications. (i) Overall view of the rock (×70 magnification) with the presence of partially dissolved benthic foraminifers (B) and dissolution vugs (V), due to leaching, as well as calcite-cemented areas (C). (ii) Closer view (×280 magnification) of the macro-pores system with the presence of an oversized pore (O) following the leaching process. (iii) Illustration of more calcite-cemented areas (C) and oversized pores (O), at the same magnification of the previous plate (×280). This also illustrates the fairly good connectivity of the porosity system. Plate (iv) shows the irregular-shaped faces of calcite crystals surrounding a peloidal grain. This probably indicates re-crystallisation process, which could have enhanced microporosity networking.

(ii) Magnification ×280

(iii) Magnification ×280 (iv) Magnification ×1400

(i) Magnification ×70

O

C

O

C

C

V

B

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5.5 Middle East field 2 (ME2)

The second data set is composed of three samples, all originating from a Middle East

gas field. All the samples are from a single well and are vertically separated from one

another by about 10 metres. They will therefore present some similarities. We will

describe these samples briefly, both in terms of geological processes and petrophysical

properties. We will outline the common features between them, as well as point out the

main differences observed. The plates for these 3 samples are presented in Appendix C

(Figs. C.10-C.12).

5.5.1 Geology, texture and diagenesis

The predominant rock fabric observed in these three samples is a moderately- to fairly-

compacted peloidal grainstone, more or less enriched in biostatic components. In two of

these samples (ME2-1 and ME2-2) a bindstone fabric has also been observed, locally

occupying a rather large portion of these samples. Although peloids are the main

constituents of the grainstone rock-fabrics, they are not always easily recognisable

because of compactional processes that tend to amass them together. However, it is

always possible to single them out in areas that have been partially protected by

compaction (because of the presence of rigid components) (Fig. C.9). Other components

are also found within the grainstone facies, such as orbitolinids, echinoderms. The

bindstone rock fabric, when present, occurs as an irregular micritic framework, either

empty, or locally filled with homogeneous micrite, calcite cement, or grains from the

surrounding grainstone facies. The depositional environment suggested here by the rock

fabric is that of a shallow-water, low-energy setting. Little water circulation throughout

the sediment is suggested by presence of thin rim cements around peloids.

The pore system within grainstones is dominated by chalk-type micro-porosity.

This has probably developed in consequence of the selective leaching of micritic

components (mostly peloids and benthonic micritic forms). Also present within the

grainstones, but volumetrically less important, are mouldic, intra-particle, and inter-

particle pores. The bindstone fabric, on the other hand, often exhibits large pores within

its growth-framework. In general, pore interconnection appears to be rather poor both

within the grainstone facies (owing to micro-porosity and inter-granular cement) and in

the bindstone (due to its micritic framework).

95

5.5.2 Pore system and petrophysical properties

With regard to the pore system, back scattering analysis confirmed the observations

from the thin sections. In particular, the pore system of sample ME2-1 is highly

heterogeneous, showing both macro and micro-porosity (Fig. C-10), while the other two

samples, ME2-2 and ME2-3 are characterised by micro-porosity only (Figs. C.11-C.12).

This is mostly fabric-dependent, as already observed from petrographic results: the

presence of large pores in sample ME2-1 being related to the presence of an

encrusting/binding organism.

BSEM images were also very helpful in analysing the sub-microscopic pore

system, or micro-porosity. Sub-microscopic pores are similar both in shape and,

apparently, in origin in all three samples. BSEM images show that the micro-porous

system is most likely product of a dissolution/re-crystallisation process, leading to

porosity “reversal”. This means that formerly non-porous areas, i.e., peloid grains,

orbitolinids, have been turned into micro-porous zones, while original primary porosity

(i.e., inter-granular space, foraminifer) are now cemented (Fig. C.12). Dissolution/re-

crystallisation is also indicated by irregular-shaped, mud-sized crystals in the “leached”

areas. These micro-pores appear to be rather isolated due to intervening cemented areas.

Based on their morphological properties, no apparent differences were found in cement

types present within the three samples.

Generally speaking, the pore-size distribution appears to be the only discriminating

factor between the three samples, with sample ME2-1 showing a highly heterogeneous

pore space, exhibiting both macro- and micro-porosity. It is therefore expected that

petrophysical properties may vary among the three samples.

Clearly, both porosity and permeability would be expected to be higher in sample

ME2-1, because of the presence of macro-porosity and this is indeed confirmed when

looking at the properties table of the samples (Table 5.3). While the porosity stays more

or less constant for the three samples (ranging from 24-30%), the highest value of

permeability is to be found in sample ME2-1. Furthermore, there also appears to be a

correlation between burial depth and the rock petrophysical properties, especially in the

case of permeability. This is also confirmed by the petrographic images, as it appears

that compaction seems higher in the deeper samples. Overall, while the displayed

porosity is fairly high, ranging from 24-29%, the permeability values are generally in

the lower end of the common permeability range, ranging from 2-12 mD. This is most

likely due to secondary processes that have occurred, such as the dissolution/re-

96

crystallisation process. The original porosity having disappeared, it seems that the

secondary porosity is less connected because it has been produced in the former non-

porous areas, which are not so likely to be connected to the original pore network,

resulting in permeability loss. Similarly, when observing the petrographic images of the

samples, it can be inferred that permeability is primarily dependent on dissolution. The

shallower samples (in terms of core depth) seem to have experienced more intense

dissolution than the deeper ones. We therefore here suggest that dissolution probably

derived from a major depositional and diagenetic event occurring somewhere above in

the rock sequence (i.e., an exposure surface).

Table 5.3. Basic petrophysical properties for the ME2 data set.

Sample no.

Rock Classification (Dunham, 1962)

Core depth (ft)

Porosity* (%)

Air permeability (mD)

Grain density (g/cm3)

ME2-1 Grainstone/Bindstone d 29.3 13.0 2.69

ME2-2 Grainstone/Bindstone d + 9.2 26.1 2.5 2.68

ME2-3 Grainstone d + 18.7 23.9 1.9 2.69

The combination of thin section petrography and SEM techniques is a very

valuable tool for determining the nature and abundance of matrix porosity. This

combination will allow us to fully characterising the micro- and meso-matrix porosity

of a rock sample. In addition, by coupling thin section analysis with SEM techniques it

is possible to unravel complicate diagenetic problems otherwise impossible to solve by

applying only one of the above techniques.

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6 SINGLE-PHASE PERMEABILITY PREDICTIONS In the previous chapters, we reviewed the models of Schlueter (1995) and Lock (2001),

presented Kirkpatrick’s effective medium approximation and its range of validity, and

introduced the data sets used during the course of this thesis. In this chapter, we will

summarise the computational procedure in detail before presenting the results obtained

for the carbonate single-phase permeability predictions. Before doing so, we will

discuss in more detail the image analysis process, which was briefly mentioned in

Chapter 3. The image analysis process is the first part of the computational procedure

and will be presented in the next section.

6.1 Image analysis

In the present context, image analysis can be regarded as a technique for extracting

quantitative geometric information about the pore space of a rock, with the aim of using

this data as an input to our permeability model. The procedure of obtaining information

from thin sections will typically involve the use of a scanning electron microscope,

together with a digital camera and image processing software. The acquisition process

and the equipment necessary for the collection of images from the samples using

scanning electron microscopy (SEM) were previously described in Chapter 5. In order

to successfully analyse the digitised images, one requires the development of consistent

processing methods with which regions of interest can be identified and segmented

from the rest of the image. We now discuss this issue in more detail.

6.1.1 Image analysis software

All the images available for this thesis were delivered in *.TIF digital format. These

images can be accessed by various imaging software. In order to carry out the image

analysis process, one needs to use suitable image processing software that will allow the

editing, enhancement and subsequent analysis of these digital images. The software

used in this thesis was the “Image J” package for Windows (Version 1.37, 2006,

http://rsb.info.nih.gov/ij/). Image J is an image processing and analysis software

package for the PC. It operates as a menu-driven interface that has the capacity to

98

provide the conventional image processing operations such as smoothing, sharpening,

edge detection, and median filtering. Its main feature is the ability to perform automated

feature analysis while spatial calibration is supported and provides real-world area and

length/perimeter measurements. Collected results can be printed or exported to a text

file for further study.

There are of course other programs available for image analysis, such as Scion

Image (http://www.scioncorp.com/) or UTHSCSA Image Tool

(http://ddsdx.uthscsa.edu/). Lock (2001) used Scion Image for his image analysis.

However, it was decided to switch to Image J after a few months of testing, as it was

found that Scion Image rendered the Windows operating system unstable. However,

Image J and Scion Image are very similar in structure, and tests carried out by using

each program with the same image indeed generated very similar results.

6.1.2 Grey-level histogram and image segmentation

Segmentation is the assignment of a grey-level or energy threshold with which to

convert the digitised image into a binary signal, therefore partitioning the image into

non-overlapping regions (Joyce-Loebl, 1985; Castleman, 1996). Thus, the term

“thresholding” refers in this context to the mapping of all “points”, or pixels, with

energy levels up to and including a certain predetermined threshold value to a binary

output of zero, with all higher levels mapped to a binary output of 1. In the context of

deriving fluid transport parameters, a threshold is sought that allows separation of the 2-

D representation of the pore space from the remainder of the mineral phases present in

the 2-D section. As a result, the procedure of image analysis is performed only on this

“meaningful” representation of the conducting phase of the porous material.

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(a) BSEM image of sample ME1-5 before the thresholding procedure

(b) BSEM image of sample ME1-5 after the thresholding procedure

Figure 6.1. Illustration of the thresholding procedure on sample ME1-5, from our second data set. In (a), the darkest regions represent the pore space while the remainder is the grain area. In (b), the pores are now in black and the remaining features mapped to a binary output of zero, hence providing us a binary representation of the image.

The determination of the correct image segmentation and thresholding is a key part

of the image analysis process, as it will influence the measurements, and therefore the

subsequent permeability predictions. One important visual aid to help with the choice of

the segmentation level is the histogram window available in the program. As mentioned

in Chapter 5, the images provided by SIEP are displayed using the grey-scale colour

mode. In this mode, the difference between the pixels is represented using up to 256

shades of grey. Therefore, every pixel is assigned a grey-scale, or brightness, value that

ranges between 0 (black) to 255 (white). The grey-level histogram is a display device

100

that, for a given image, computes the number of pixels that are attributable to a given

value of grey from the greyscale, therefore providing a visual summary of grey-level

content of an image.

Figure 6.2. Illustration of the grey-level histogram for sample ME1-5, where the abscissa is the grey-level values (0-255), and the ordinate is frequency of occurrence.

Figure 6.2 shows a typical histogram for a carbonate rock. Whereas the sandstone

histograms display a three-peak type grey-level histogram (Lock, 2001), carbonate

rocks usually display a two-peak histogram. This is easily explained, as grey-scale

values can also be correlated to the different mineral phases that can be found within a

given rock. Table 6.1 below shows grey-level standard values against minerals found in

rocks, as supplied by Applied Reservoir Technologies Ltd, which are specific to the

SEM instrument used. Different microscopes will generate slightly different values for

the same minerals, but this can be used as a rough guide. The fact that sandstones

display a three-Gaussian type curve is due to the presence of a few carbonate grains,

after the peak generated by the quartz/feldspar minerals. Carbonate rocks, on the other

hand, simply display one peak for the carbonate grains, usually having a steeper

increase leading to that peak, due to the presence of clay minerals. The first peak, in

both cases, represents the pore space. However, for the purpose of image segmentation,

these standard grey-level value tables cannot be relied upon. Indeed, it is desirable to

0

0.005

0.01

0.015

0.02

0.025

0.03

0 50 100 150 200 250 300

Nor

mal

ised

Fre

quen

cy

Grey-Level Value

101

develop a set of segmentation criteria that remain unaffected by the characteristics of a

given SEM. Consequently, in this thesis, these standard tables will not be used for the

image segmentation process.

Table 6.1. Illustration of the correspondence between the grey-scale values and some mineral phases found in rocks, taken from standard grey-level calibration standards, originally supplied by Applied Reservoir Technologies, Ltd.

Greyscale Value Corresponding Mineral

0 – 35 Pore Space 35 – 40 Iron 40 – 170 Clay 171 – 202 Quartz 203 – 220 Feldspar 221 – 239 Carbonate 240 – 255 Anhydrite

Ideally, for a given a set of mineral phases, the grey-level histogram would be

composed of a set of delta-type functions with heights proportional to the relative

proportions of the phase in the field of view of the image. However, there are several

reasons why this is not the case when working with SEM images of rocks. Firstly, as the

pixel size is finite, a given pixel may contain more than one mineral. This will result in

the broadening of the signals into Gaussian-type functions. Moreover, there are

problems regarding the dispersion and reflection of the energy, as a grain sitting at the

surface of the sample may not be sitting at a right angle, while the surface of the sample

may also not be flat. It can therefore become problematic to define the appropriate

segmentation threshold. Certain convolution algorithms can be used to separate the

various peaks (Lu et al., 1994), but this is not considered to be ideal.

Weszka and Rosenfeld (1979) argued that a reasonable methodology for choosing

the threshold in a multi-modal histogram is to locate the deepest point or valley between

two overlapping Gaussian peaks. The idea behind this method is that in the vicinity of

this dip, the histogram takes on relatively small values, implying that the area function

changes slowly with threshold grey-level. Therefore, if we place the threshold grey-

level at the dip, we minimise its effect on the boundary (Castleman, 1996), because of

the relatively low number of pixels associated with these boundary features.

102

Generally, if the background grey-level is reasonably constant and the region(s) of

interest has a consistent range of grey-levels, then the assignment of a threshold,

applicable to the whole field of view of the image, may be applied. This method of

segmentation is known as global thresholding, and was applied in the analysis of all

BSEI used in this study.

There are, however, a few cases in this study where the choice of image threshold

was not as straightforward as detecting the lowest point of the valley, as suggested by

Weszka and Rosenfeld (1979). Indeed, the lack of macro-porosity coupled with the

BSEM meant that the grey-level histogram was not as useful for selecting a

thresholding point. The ability to see “through” the pores meant that the grey-level

values for the pores were constantly decreasing, rather than forming a peak. An example

of this is shown in the next figure when displaying the grey-level threshold for sample

SEF-1.

103

Figure 6.3. Illustration of the histogram generated for sample SEF-1, with cemented grains and micro-porosity. The histogram presents only one clear peak for the grains, and then a very gradual decrease towards the final black value of zero, where a peak appears.

In this case, the segmentation value had to be determined in a different manner.

Because the pore space has been filled with epoxy prior to the image acquisition

process, it is represented in the lower values of the grey level table. Consequently, we

expect the optimum segmentation value is likely to be found between 45 and 100. It was

therefore decided to use the “Preview” function in Image J to compare the thresholded

image to the original BSEM, as well as generating several thresholded images. Using

these thresholded images, it was then possible to generate measurements for each of the

(a)

(b)

0

0.01

0.02

0.03

0.04

0.05

0 50 100 150 200 250 300

Nor

mal

ised

Fre

quen

cy

Grey-Level Value

104

images, and therefore calculate the porosity for each image. The segmentation value can

then be chosen when the image porosity matches up with the core helium porosity.

Figure 6.3 below shows such a plot where the image porosity is expressed as a function

of the segmentation value. The change in porosity from the image is quite noticeable,

but this is considered to be a good way to find a realistic threshold point. The original

and thresholded images can then be easily compared to check the segmentation value.

Figure 6.4. Illustration of the variation of the image porosity as a function of the segmentation value. The helium porosity value is circled in black on the graph, and the corresponding grey-level threshold value was used for the segmentation process.

This technique had to be used for four samples (SEF1, SEF2, ME2-2, and ME2-3),

each of which displayed a fair amount of micro-porosity. It is clear that the presence of

micro-porosity contributes to the removal of the first peak in the grey-level histogram,

normally describing the pore space of the sample.

6.2 Data acquisition and manipulation

This section describes the various steps involved in the automated collection of the

quantitative measurement of the properties of the pore space, specifically the areas and

perimeters of the pores, which will subsequently be used for the computation of the

hydraulic conductances of the pores. However, prior to any analysis, it was thought to

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105

be necessary to run various features analyses with known geometrical shapes and shades

of grey, so as to test of the software’s ability to analyse the images. Using Adobe

Photoshop 7.0, it was possible to build a collection of images and then test them in

Image J. This led to the conclusion that, not only did the software return the true level of

grey value, but the correct perimeters and areas were also computed.

6.2.1 Data collection

The following outline lists the various stages involved in collecting the areas and

corresponding perimeter measurements of the pore space, starting with image display

and finishing with a final text file that can be exported for subsequent calculation of the

individual pore conductances in Microsoft Excel.

1. Open image by going to the “File” tab and then select the “Open” menu. The

individual co-ordinates and grey-level value of each pixel are displayed on the main

window of the program, while the information about the image is displayed on top.

2. Go to the “Analyse” tab and choose the “Set Scale” menu. This allows for the

spatial calibration of the image, including the pixel aspect ratio if necessary. The

“global” option enables the recall of the calibration parameters for each newly

opened image. If this is not ticked, then the spatial calibration has to be set each time

a new image is opened.

3. The next step is to display the grey-level histogram. To do so, again, go to the

“Analyse” tab and select the “Histogram” option. The grey-level histogram then

displays as a new window. Various pieces of information are displayed on that

window such, as the total pixel-count, the mean, and the mode. By scrolling the

mouse over the histogram, it is the possible to view the pixel-count for each

individual grey-level value. It is possible to display the values for the histogram, as

well as copying these values in the clipboard for subsequent exporting in other

programs.

4. The next step is to apply the thresholding process to segment the image, and allow

for the separation of the pore space and the grains. To open the threshold window, go

to the “Image” tab, choosing the “Display” menu, and then selecting the “Threshold”

sub-menu. We can select the threshold value using Weszka’s methodology and the

previously opened histogram so as to separate the pore space and the grains. If, as

illustrated previously in Fig. 6.3, the histogram generated by the image fails to

106

display a clear peak for the pore space, the threshold value can then be found by

matching the image porosity to the core porosity. The thresholding procedure alters

the digital signal of the image, so that the grey-level value of each pixel is converted

to a binary value, which facilitates the measurements.

5. The next step is to select the type of measurements that need to be retrieved from

the thresholded image. Again, go to the “Analyse” tab and select the “Set

Measurements” menu. A wide range of measurements is available from this menu;

however for our work, only areas and perimeters need to be selected. The numbers of

decimals can also be selected from this menu.

6. At this point, everything is set for the analysis of the pore space. In order to

proceed with the measurements, one needs to select the “Analyse Particles” from the

“Analyse” menu. Several windows are created after the measurements have been run,

with a window outlining the features detected, a summary window, as well as a new

window displaying the results. These results can then either be saved as a *.txt file,

or exported directly in Microsoft Excel for additional computations.

6.2.2 Data manipulation and the elimination of micro-porosity

Lock (2001) noted that micro-porosity might not be completely removed from the

region of interest following the segmentation procedure. Although this did not prevent

his method from providing reliable estimates of the effective transport properties of his

sandstones, it is conceivable that measurements recorded for many of the single and

small pixel cluster features corresponding to artefacts from the thresholding procedure

will overlap with the micro-porosity component of the image. Lock therefore performed

an additional post-image-analysis thresholding step to eliminate these smaller features,

and effectively exclude the contribution of the small void spaces. The criterion chosen

by Lock was to have an areal cut-off based on some percentage of the area of the largest

conducting feature, in a way such that not more than 3% of the total hydraulic

conductance was lost. The justification for choosing an areal cut-off was based on the

fact that the hydraulic radius of individual pores correlates more strongly with area than

with, say, perimeter or some other geometrical attribute (Lock, 2001). After having

applied this areal cut-off to several images, it was discovered that removal of any

feature whose area is less than 1% of the area of the largest feature accounts for the

107

removal of not more than 3% of the total hydraulic conductance. This was later kept as

a general criterion for the areal thresholding procedure.

This step of a second, post-image analysis, threshold will also be applied to our

carbonate predictions, although the specific criterion chosen for sandstones may not be

appropriate when used with carbonate rocks. Indeed, the pore systems described in

Chapter 5 are quite different from the ones obtained from sandstones. While sandstones

contain simple, inter-granular pore systems, carbonate rocks display a wide range of

porosity systems, which have very often been subjected to diagenesis, resulting in

dramatic petrophysical changes (Lucia, 1999). Thus, both pore size and hydraulic

conductance distributions from carbonate rocks differ quite significantly from the ones

generated for sandstones. This is especially true for most of the porosity systems

displayed in the carbonates studied here. Most samples display both micro- and macro-

porosity, making the criterion used by Lock difficult to apply. Indeed, in the case of

grainstones, where only micro-porosity is present, the use of an areal thresholding as

used by Lock would lead to the removal of too many features that should be included in

the analysis. This is easily confirmed when computing the amount of hydraulic

conductance being removed when using this criterion. In extreme cases, for example

samples SEF-1 and SEF-2, the removal of the features less than 1% of the largest ones

led to the removal of around 10% of the total hydraulic conductance. This was not

thought to be acceptable in the context of our permeability predictions. Rather than

employing a direct areal thresholding, it was decided to implement a conditional

thresholding. Since we are primarily concerned with the loss of conductance, it was

decided to first set a threshold of 3% of the total hydraulic conductance. Then, having

set this criterion, it is possible to evaluate the percentage of the area of the largest

feature to which this corresponds. If it is less than 3%, then it is decided to keep this

thresholded conductance file for the prediction. If, however, this corresponds to more

than 3% of the largest feature, then it is recommended to use this new criterion for the

prediction. In most cases, the removal of 3% of the overall conductance rarely

corresponds to the removal of features that are 3% the size of the largest feature. Of the

twelve samples used for this thesis, there was only one instance of this occurring, so this

certainly is the exception rather than the norm. Clearly, the criterion for this second

thresholding needed to be adjusted, as the pore size distributions found in carbonate

rocks are much more varied and heterogeneous than those found in sandstones.

108

6.3 Computational procedure

The various stages of the prediction method have thus far been presented and discussed

in detail. We will now collect and summarise the steps presented up to now, and explain

how to eventually estimate the permeability of the continuous medium from the

effective conductance of the individual hydraulic conductances. The single-phase

predictions for the samples will then be presented.

As all the simulations are based on flow through a cubic lattice, we consider a

plane that slices the lattice perpendicular to one of the principal lattice directions,

containing N pores in a region of cross-sectional area A. The total flow-rate going

through this region will be given by

LpNC

Q eff

μΔ

= , (6.1)

where μ is the viscosity of the fluid, and effC the effective conductance of the pores.

Darcy’s law, on the other hand, expresses the flow-rate as

LPkAQ

μΔ

= . (6.2)

Equating (6.1) and (6.2) then gives the permeability as

total

eff

ANC

k = . (6.3)

After the images have been cropped and resized, the first step of the analysis is to

process them so as to separate the pore space and estimate the necessary parameters,

namely the area and perimeter of each pore. These measurements can then be exported

to Microsoft Excel for further manipulations. The next step is the application of the

stereological factors. The first factor that needs to be applied is the one that yields the

actual area and perimeter values, 3/8 (eq. 3.31), from which the hydraulic conductance

can be calculated using the hydraulic radius approximation. The sinusoidal hydraulic

constriction factor of 0.44 (§3.5.1) is then applied so as to account for the variations in

pore radius along the pore length. The set of conductances thus obtained is then

truncated, using the criteria presented earlier in this chapter, to remove artefacts or

micro-pores that do not effectively contribute to the flow. This new conductance

distribution file is then used as an input for Kirkpatrick’s effective medium

109

approximation, so as to compute an effective conductance value. This effective

conductance value is then populated on a cubic lattice from which the permeability can

be calculated. This final permeability is computed using another stereological correction

to convert the apparent number density of pores in the field of the image to the actual

density perpendicular to the flow direction by dividing equation (6.3) by the areal

density factor 1.47. The entire procedure can be summarised as follows:

• Generate digital BSE images of polished rock thin sections

• Import BSE images into image analysis software

• Apply grey-level thresholding procedure as to separate the pore space from the

grains

• Generate measurements of area and perimeter for the detected pores

• Apply stereological correction and hydraulic constriction factor to estimate the

hydraulic conductance Ci of each individual pore

• Employ second thresholding procedure to truncate the data set

• Obtain Ceff using Kirkpatrick’s effective-medium approximation

• Compute the areal density of pores inside the image

• Assuming a cubic lattice, calculate k using

total

effeff

ACN

k = (6.4)

6.4 Absolute permeability predictions

In this section, we will present the single-phase, absolute permeability predictions for

our carbonate samples. These will first be computed with the use of a common co-

ordination number of six for all samples. We will analyse these results, and then try to

improve the predictions by introducing a variable co-ordination number into the

procedure.

6.4.1 Results using a co-ordination number of 6

The preliminary single-phase permeability predictions for the carbonate samples were

computed using the procedure outlined above. Table 6.2 shows lists the predicted

110

permeabilities for the samples, as well as the values measured in the laboratory and

provided by Shell International Exploration & Production (SIEP).

Table 6.2. Absolute permeability predictions using 6=z .

Core Ceff (µm4)

Total area (µm2)

kpred (mD)

kmeasured (mD)

kpred/kmeas

SEF-1 1.8 9.82×105 1.1 0.5 2.20

SEF-2 1.5 9.82×105 2.2 1.1 2.00

ME1-1 7646 1.97×106 243 400 0.61

ME1-2 935 7.97×106 53 56 0.95

ME1-3 2086 1.97×106 39 260 0.15

ME1-4 294 1.97×106 26 73 0.36

ME1-5 2492 1.97×106 133 104 1.28

ME1-6 9852 1.97×106 145 1010 0.15

ME2-1 688 1.97×106 20 13 1.54

ME2-2 0.4 9.82×105 1.8 2.5 0.72

ME2-3 1.0 1.97×106 1.8 1.9 0.95

This method, when used by Lock (2001), proved to be successful in predicting the

absolute permeability of sandstones within a factor of two, over a range of

permeabilities from about 10-1000 mD. The predictions for our carbonate data set are

shown in Table 6.2. In this case, the accuracy varies considerably from specimen to

specimen, and it is not the case that all the predictions fall within a factor of two of the

measured values. Nevertheless, out of the eleven samples analysed here, in eight cases

the permeability is predicted within or very close to a factor of two, while in the other

cases the permeability is under-predicted by a substantial amount. This is especially true

for samples ME1-3 and ME1-6, which are both under-predicted by a factor of nearly 7.

The permeability of sample ME1-4 is under-predicted by a factor of 3, which given the

wide range of pore space encountered, can certainly be considered a reasonable

prediction.

111

There are however a few reasons that might explain the errors in predicting the

absolute permeability. It is worth remembering that the core of sample ME1-6 was

described in the previous chapter as having an apparent fracture, which was probably

due to sample preparation. If this happened prior to the laboratory measurements, which

is very likely, it would have increased the permeability of the sample quite dramatically.

By looking at the stratigraphic column and the various properties above and below this

sample, it appears that the permeability of sample ME1-6 is abnormally high when

compared to the other surrounding samples for which petrophysical data is available.

This is not justification in itself, merely clues as a possible way of explaining the under-

predicting of the absolute permeability. As far as sample ME1-3 is concerned, there

does not appear to be any obvious reason to explain the difference between the

measured and predicted permeability. It is possible that some of the more conducting

features of the pore space were not captured during the imaging process, which brings

us to the issue of heterogeneity and Representative Elementary Volume (REV). Indeed,

our model is based on the implicit assumption that the image used for the permeability

prediction contains a representative sample of the pore population. Moreover, this issue

has to be balanced with that of image resolution, since the bigger the image, the lower

will be the resolution. However, there are several reasons why, in this particular case, it

is difficult to assess whether or not a REV was captured. Firstly, we only have one low-

resolution image of the sample providing a fairly large view, making it impossible to

analyse other regions of the sample. Secondly, the author has not had the chance to

personally examine the sample. Consequently, one can only make hypothetical guesses

as to why the sample’s permeability is so under-predicted by the method.

112

Figure 6.5. Predicted permeabilities plotted against measured values for the three data sets. The upper and lower lines correspond to errors of a factor of two in either direction.

An interesting trend observed in Fig. 6.5, which was also noticed by Lock (2001),

is the tendency of the method to over-predict the permeability of low-permeability

rocks, and over-predict the permeability of high-permeability rocks. At the time, Lock

suggested that this bias might have been an artefact of having a very small number of

samples, or may indeed have indicated a problem with the method. As this same trend is

now observed for the carbonates, it seems likely that it is a real property of the

prediction method.

One possible explanation for this bias may lie in our use of a co-ordination number

of six for all samples. Assuming that any errors due to other aspects of our prediction

method show no preference for (or against) high (or low) permeability rocks, the trend

can be qualitatively explained by assuming that the actual co-ordination number is

higher than 6 for the “high perm” samples, and lower than 6 for the “low perm”

samples. This assumption is consistent with recent work on the co-ordination number

(Sok et al., 2002; Arns et al., 2003, 2004; Seright et al., 2003). Having carried out

three-dimensional tomographic analysis of a number of Fontainebleau sandstone

samples, they found a positive correlation between permeability and mean co-ordination

number. Therefore, in the high permeability range, we might expect the rocks to have a

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Carbonatesy = xy = 2x y = 0.5x

K p

redi

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(mD

)

K measured (mD)

113

co-ordination number higher than six, thereby explaining the under-prediction of the

model. Conversely, in the low permeability range, it is expected to find lower co-

ordination numbers, thus explaining the over-predictions generated by the method.

Following this argument, it would therefore be desirable to implement a varying co-

ordination in the method to try to eliminate this trend.

There are several steps one needs to implement in order to introduce a varying co-

ordination number in the prediction model. Firstly, it is necessary to relate the co-

ordination number to some known characteristic of the rock. Furthermore, the

permeability estimation method itself needs to be modified to be able to incorporate

different co-ordination numbers in a rational way.

6.4.2 Introduction of a varying co-ordination number

As mentioned in Chapter 2, the tomographic imaging technique is a very useful tool for

the three-dimensional study of rocks (Lindquist et al., 2000; Arns et al., 2003). The data

generated from such techniques can be presented in a various formats, and be used to

analyse various geometric and topological properties of the pore space (Arns et al.,

2004). For some researchers, this data is often used of the primary step of network

building (Blunt, 2001; Blunt et al., 2002). Ideally, one would like to find a direct

correlation between the co-ordination number and some simple parameters directly

measurable from the 2-D images, such as the mean pore area or radius. This would

allow us to employ a priori a suitable co-ordination number in our effective medium

calculation. However, it turned out not to be possible to do so, nor has it been possible

to relate the prediction error to some other measurable parameters of the samples.

Indeed, no clear relationship could be found between the error generated and the

samples’ parameters, probably because of the wide range of carbonates used here.

Tomographic imaging for carbonate rocks could be helpful in generating a relationship

between pore size and co-ordination number, but, although this technique is fairly well

established for sandstones, it is still being perfected for carbonate rocks (Coenen et al.,

2004; Arns et al., 2005). Indeed, the high resolution needed to capture the smaller

features can only be achieved when scanning a very small part of the rock. The process

is therefore very computationally intensive and time-consuming.

However, tomographic studies have been carried out for sandstones, and the

correlations between co-ordination number and various parameters were obtained. Sok

et al. (2002) and Seright et al. (2003) presented this kind of data for Fontainebleau and

114

Berea sandstones, respectively. Studying the tomographic data generated for these

samples, they were able to relate the pore volume with the co-ordination number. An

example of this kind of data is shown below in Fig. 6.6.

Figure 6.6. Correlation between the co-ordination number and the pore volume for Fontainebleau sandstone (after Sok et al., 2002).

Our aim is to use the data available in the literature, as presented in Fig. 6.6, to

generate a correlation between the co-ordination number and the permeability of the

rock, which would allow us to correct the co-ordination number after the initial

permeability prediction. This would then enable us to iterate the computational process

and make a new prediction. Because our model is based on 2-D images, it would be

desirable to be able to relate the co-ordination number with a parameter obtainable from

such images, rather than a 3-D parameter, or one obtained from 3-D imaging

techniques. If one assumes the pores present in the rock to be of spherical shape, it is

then easy to convert the volume of the spheres into an equivalent radius. One can

therefore convert the data from Sok et al. (2002) presented in Fig. 6.6 into a correlation

between the average pore radius and the mean co-ordination number of the pore space.

Once this has been done, it is then possible to calibrate this data with some rock samples

for which a range of permeabilities and pore sizes are available. In order to do so, it was

decided to use Lock’s data set for sandstones, which provides us with a wide range of

measured permeabilities. By analysing the pore space measurements acquired by Lock

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Co-ordination Number

115

(2001), we are able to extract the radius information, once again simply assuming the

pores to be of circular shape. Once this is done, we can then use the relationship

highlighted in Fig. 6.7, and generate a correlation between the co-ordination number

and permeability of the rock, shown in Fig. 6.8 on the next page.

Figure 6.7. Illustration of the derived correlation between the co-ordination number and the equivalent spherical radius.

2

4

6

8

10

12

15 20 25 30 35 40 45

y = 1.86*10 -2 * x 1.722

Co-

ordi

natio

n N

umbe

r

Equivalent Radius (µm )

116

Figure 6.8. Illustration of the derived correlation between the permeability and the co-ordination number and the equivalent spherical radius.

Using the correlation presented in Fig. 6.8, it is therefore possible to approximately

relate the permeability and co-ordination number of sedimentary rocks. The data

presented here is based solely on the data previously published by Sok et al. (2002) for

Fontainebleau sandstone. It would certainly be desirable to have the same kind of data

for carbonate rocks. However, as we mentioned earlier, such data it is not available yet,

due to technical constraints. Meanwhile, we will use the correlation derived from

sandstones for our carbonate samples.

6.4.3 Updating the cubic lattice

Before we can introduce a varying co-ordination number to rectify the over/under-

predicting trend noticed in the predictions, we need to modify our conceptual model of

the cubic lattice so that it can handle co-ordination numbers other than 6. We have seen

in Chapter 4 that we can use the EMA when z represents the mean co-ordination of the

rock. Firstly, it is easy enough to make the co-ordination number lower than 6 by simply

removing one branch of the lattice within each unit cell; this results in the co-ordination

being lowered from 6 to 4. One could also remove one branch every other cubic lattice

to obtain an average co-ordination of 5. As we have seen in Chapter 4, when using a

lower co-ordination number, one must also take this into account when converting the

2

4

6

8

10

12

1 10 100 1000 104

z = 3.348 * k 0.132

Co-

ordi

natio

n N

umbe

r

k (mD)

117

effective conductance into a macroscopic permeability (§4.5.1). We showed then that

the permeability could be computed as follows:

k =z4

−12

⎛ ⎝ ⎜

⎞ ⎠ ⎟

NCeff

Atotal

⎛

⎝ ⎜

⎞

⎠ ⎟ , (6.5)

where z ≤ 6 is the co-ordination number.

On the other hand, it is also possible to add branches to the cubic lattice in order to

increase the co-ordination number. As all the adjacent corners are connected to each

other, the only way to add branches is to join up opposite corners of the lattice. If one

adds a branch going from the front top left-hand corner to the back bottom right-hand

corner, then the co-ordination number will be increased by two, from 6 to 8. By adding

yet another similar branch, that would take the co-ordination to 10. One could repeat the

same process a maximum of two more times to make it a maximum of 14=z . In

practice, it is very unlikely that such a high co-ordination will be observed in rocks and

this is confirmed when looking at the correlation presented on Fig. 6.8. Indeed, the

highest co-ordination number obtained using this correlation for Lock’s data set is

around 9.

Figure 6.9. Illustration of the cubic lattice with a maximum of four added opposite branches, thereby yielding a co-ordination number of 14. Please note the branches are not intended to intersect each other in the middle of the lattice.

We must point out that this conceptual model is a heuristic device that is intended

merely to give us a way to rigorously calculate the relation between Ceff and the

macroscopic permeability k, for some specific network with a given co-ordination

number greater than 6, in the same way that our cubic lattice was used for this purpose

when 6=z . We do not necessarily claim that the structure shown in Fig. 6.9 represents

the actual pore space topology of any of our rocks.

118

Again, as with a co-ordination lower than 6, it is also possible to add branches

every other unit cell to yield co-ordination numbers such as such as 7, etc. However, we

first need to verify if eq. (6.5) is still valid for the computation the macroscopic

permeability when z is greater than 6. For example, consider the case 8=z , in which

the cubic lattice is augmented by one additional diagonal bond. If we consider a cube

with an edge of length L, the length of this extra branch would be L3 , whereas the

pressure drop over each bond is ΔP. The total flow through a unit cell would then be

given by the flow through the one bond per unit cell that is aligned with the pressure

drop, plus the flow through this new diagonal bond, i.e.,

⎟⎟⎠

⎞⎜⎜⎝

⎛ +Δ=⎟

⎠⎞

⎜⎝⎛ Δ

+Δ

=3

313 L

PCL

PLPC

Q effeff

μμ. (6.6)

If we compare this with Darcy’s law, LPkAQ μ/Δ= , we find

cellunit

eff

cellunit

eff

AC

AC

k58.1

313

=⎟⎟⎠

⎞⎜⎜⎝

⎛ += . (6.7)

The permeability generated by this new lattice would therefore be 58% greater than that

of the cubic lattice before the addition of the extra bond. On the other hand, if we used a

co-ordination number of 8 in eq. (6.5), we would predict an increase of almost the same

amount, specifically 50%, despite the fact that eq. (6.5) was derived only for z ≤ 6. A

similar analysis for other values of z > 6 would show the same permeability increase of

29% per unit increase in z, as opposed to the 25% increase predicted by eq. (6.5).

Hence, we conclude that it is in fact permissible to use eq. (6.5) over the entire range of

co-ordination numbers.

The correlation between the co-ordination number and the permeability presented

in Fig. 6.8 can therefore be used to generate a new co-ordination number following an

initial permeability prediction, using a co-ordination number of 6. The two parameters

are related by the following equation, as taken from the curve fit on the data compiled:

132.035.3 kz = , (6.8)

where k must have units of mD, and z is dimensionless. Eq. (6.8) is therefore used after

an initial permeability prediction to generate a new co-ordination number. This new

value of z is then used to compute a new effective hydraulic conductance, and a new

permeability is generated. The results obtained are presented in the next section.

119

6.4.4 New results

As well as refining the predictions for the carbonate rocks, we also decided, as implied

in the previous paragraph, to apply the varying co-ordination number to the sandstone

samples used by Lock (2001) for his studies. This was thought to be an interesting test

to check whether or not any improvement could be made on Lock’s results using this

technique. The new predictions will be compared to the previous ones, and conclusions

will be drawn.

Figure 6.10. Predicted permeabilities plotted against measured values for the carbonate rocks data, using a variable co-ordination number.

0.1

1

10

100

1000

104

0.1 1 10 100 1000 104

Carbonates (Old)Carbonates (New)y = xy = 2xy = 0.5x

K p

redi

cted

(mD

)

K measured (mD)

120

Figure 6.11. Predicted permeabilities plotted against measured values for Lock’s sandstone data, using a variable co-ordination number.

The above figures show the new permeability results obtained with a varying co-

ordination number. This appears to improve the results for both the sandstones and the

carbonate rocks. In the case of sandstones, the improvement is particularly evident,

especially in the higher range of the permeability, where a tendency towards under-

prediction had been visible. Indeed, most data points were either falling on the xy 5.0=

line, or below. Using a refined co-ordination number distinctly improves the

permeability predictions. In the lower range, the results are also improved because of

the use of a lower co-ordination. There are only a handful of samples for which the

permeability is not improved using the varying co-ordination number, which

corresponds to cases in which the permeability was originally over-predicted.

In the case of the carbonate samples, the predictions are also improved, although

the overall effect is certainly debatable. In the high permeability range, one of the

predictions (ME1-1) is improved, while the other two samples barely show any

improvement, because the original permeability prediction was not high enough to

subsequently yield a higher co-ordination number when applying the correlation. We

have, however, explained the possible reasons why these samples are so under-

predicted, when we first presented the results (§6.4.1). In the very low permeability

10

100

1000

104

10 100 1000 104

Sandstones (Old) Sandstones (New) y = xy = 2xy = 0.5x

K pr

edic

ted

(mD

)

K measured (mD)

121

range, the permeability estimates were lowered by this procedure. However, in some

cases the new predictions for the low-k samples are now too low. It must be

remembered, however, that the correlation was generated using sandstone data, so it is

hardly surprising that it leads to better results with sandstones than with carbonate

rocks. When considering the absolute error, the mean error of the predicted

permeabilities is now 48% while, before the correction made for the co-ordination

number, it was 56%. Similarly, for Lock’s sandstones, while the original error was 48%,

the new mean error is now 42%. Consequently, the introduction of a varying co-

ordination number can be considered an improvement of the method, both for

sandstones and carbonate rocks.

6.4.5 Special case: non-touching vugs

There is one more sample that has not been analysed in the previous sections − sample

SEF-3, described previously as a vuggy grainstone. We mentioned in the previous

chapter that this sample features the presence of non-touching vugs. Non-touching, or

separated, vugs are not connected to each other directly, but are possibly linked together

via the inter-particle pore space. It was mentioned earlier that these types of vugs, even

though they generally increase the porosity, do not significantly improve the

permeability (Lucia, 1983). An SEM of sample was SEF-3 was shown in the previous

chapter, and the number of vugs and oversized pores was quite apparent. However, it is

clear that this pore will not directly participate in conducting fluid through the rock

(Wagner, pers. comm.). However, these pores will be included during the data

acquisition process, and they will therefore influence the computation of the effective

pore conductance. Because these pores are generally quite large, up to a millimetre

across, they will generally have quite a dramatic impact on the computed effective

conductance. Furthermore, because of their size, they will also have an impact on the

thresholding process, resulting in the discarding of many of the smaller features present

in the rock.

In order to develop a systematic method for somehow excluding such pores from

our calculations, we start by carrying out a permeability prediction taking into account

all the pores, using our usual procedure for the permeability computation. The

permeability thus predicted for this sample was around 900 mD, while the measured

permeability (Table 5.1) was only 25 mD – an over-prediction of a factor of almost 40.

As expected, because of the presence of vugs and oversized pores, the thresholding

122

process eliminated most of the smaller pores, therefore yielding a very high value of the

predicted permeability. Because these large pores are almost certainly not connected to

each other, we felt it would be worthwhile trying to run other predictions while

excluding some of the larger features present in the picture. We discarded from the

image the whole gastropod, the leached orbitolinid, and the oversized pore present at the

bottom of the picture. The acquisition process was then run again, and the permeability

was predicted once more. The permeability thus obtained was much lower, with the new

prediction giving us an absolute permeability of 48 mD, only a factor of two above the

measured permeability.

This example illustrates the need to exclude isolated vugs from our calculation

procedure. However, the decision as to which pores to “ignore” was based on expert

advice from a carbonate sedimentologist. If this method is ever to be used in a routine

way, by technicians not highly trained in carbonate sedimentology, for example, it

would be necessary to develop a systematic, “objective” way to decide which pores

should be eliminated form the prediction process.

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7 RELATIVE PERMEABILITY PREDICTIONS In the previous chapter, we extended the model developed by Schlueter (1995) and

Lock (2001) so that it can be used for the prediction of the absolute permeability of

carbonate rocks. The results of these predictions were subsequently presented and

discussed. Another key objective of this research has been to extend the methodology to

the prediction of two-phase relative permeabilities. We will first define the concept of

relative permeability, before briefly reviewing the literature and extending our

computational method as to allow for the prediction of relative permeabilities. The

method will be tested on Berea sandstone, before being applied on carbonate sample

ME2-2. Conclusions and suggestions will then be made at the end of the chapter.

7.1 Definition

Although we have attempted to predict the absolute (single-phase) permeability of rocks

in the previous chapters, the situation that is more relevant to the oil and gas industry is

that in which there are two, and sometimes three, fluids in the pore space – for example,

oil, water and gas. This is especially relevant to situations in which water or gas is

injected into oil reservoirs during the later stages of the field life, so as to enhance oil

production. It is clear that the relative permeability of each phase will change as the

saturation of the each phase varies. Knowing how these relative permeabilities will vary

with saturation is of paramount importance in reservoir engineering.

The flow of immiscible fluids in porous media can be divided into two categories:

steady state, where all the macroscopic properties of the system are invariant at all

times, and unsteady state, where properties change with time. Steady flow occurs when

there is no displacement of any fluid by any of the other fluids present in the pore space,

while in unsteady state, the saturation of a given location in the system will generally

change. This means that immiscible displacements will fall into this category. While

these will not discussed in detail here, a good review of these phenomena can be found

in Dullien (1992).

In this chapter, we will mainly concentrate on the modelling of the relative

permeability curves, and the various approaches to predict these curves. This will

generally occur under steady-state conditions. Theoretically, the problem of two-phase

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flow can be treated from the pore level by solving the Navier-Stokes equations. This

process would need to be combined with the equations of state for both fluids, and the

equations of mass conservation. However, one requires a great deal more information,

such as pore morphology, type of flow, and fluid distribution within the pores, to

entirely solve the two-phase problem (Dullien, 1992).

In practice, macroscopic equations can be used to describe two-phase flow in

porous media, which are generalisations of Darcy’s law for single-phase flow:

dxdPAkQ i

i

ii μ

−= )2,1( =i , (7.1)

where Qi is the flow rate of phase i, dxdPi / is the pressure gradient in this phase, iμ is

the viscosity of this phase, and A is the macroscopic area normal to the flow. The

parameter ki is referred to as the effective permeability of the porous medium to fluid i.

One usually non-dimensionalises this term by defining the relative permeability as the

ratio of the effective permeability to the absolute permeability, i.e., kkk iri /≡ , after

which eq. (7.1) can be rewritten as:

dxdPAkkq i

i

rii μ

−= )2,1( =i . (7.2)

If one assumes water to be the wetting phase, the relative permeabilities curves are

generally plotted against water saturation. In most cases, the relative permeability of the

wetting phase will display an increasing trend with increasing wetting phase saturation,

while the relative permeability of the non-wetting phase will decrease. Relative

permeability can therefore be simply described as the specific permeability of a porous

medium to a particular phase measured at a specific fluid saturation, usually expressed

as a fraction of the absolute permeability.

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Figure 7.1. Illustration of typical relative permeability curves for different cores of Berea sandstone. The data presented here is taken from Oak et al. (1990).

7.2 Relative permeability measurements

In general, relative permeability data are obtained in the laboratory by testing

cylindrical rock cores. There are two methods for measuring permeability at various

saturation states to obtain relative permeability measurements: steady-state and

unsteady-state flow tests, which were defined earlier. A significant problem that arises

in measuring relative permeability in the laboratory is the restoration of reservoir

conditions. Pore surfaces, especially in carbonate rocks, are reactive to changes of

fluids, and these reactions can alter the wettability state. Elaborate methods have been

devised to preserve the original wettability state of the core material, and the accuracy

of any relative permeability data is dependent upon these methods. Many carbonate

reservoirs are considered to have mixed wettability at present, with some pore walls

being water-wet and some others being oil-wet. However, it is very likely that the

reservoirs were water-wet at the time of oil migration (Lucia, 1999).

7.2.1 Steady state flow method

Generally speaking, the steady state method is the most accurate, but it is time

consuming and expensive. The criterion of steady state is determined by the condition

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Kr-w

Kr-nw

Rel

ativ

e P

erm

eabi

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Kr

Wetting Phase Saturation (%)

126

that the inflows equal the outflows and/or that constant pressure drop has been reached

across the sample. The attainment of steady state conditions may take from two to forty

hours, sometimes even longer, depending on the sample permeability and the method

used.

If the porous medium is initially saturated with the wetting fluid, and subsequently

a non-wetting fluid is pumped into the sample either a fixed flow-rate, or at a fixed

pressure, there will be non-steady flow as long as there is displacement, and the

saturation keeps changing. This is similar to the primary drainage process that occurs in

a reservoir. At the end of this displacement process, a certain amount of residual wetting

fluid remains in the sample, known as the “irreducible” wetting fluid saturation. This

saturation can generally be lowered quite significantly in the laboratory over a long

period of time (Dullien, 1992). However, in the context of reservoir engineering, this

saturation is often found to be between 10 and 25%. When there is no additional

displacement, we have attained steady state, single-phase flow of the non-wetting fluid.

Subsequent displacement of the non-wetting fluid by the wetting fluid, pumped into the

sample at the residual wetting phase saturation, runs a similar course, and the process

ends with stationary residual non-wetting fluid remaining in the sample. This process is

known as “imbibition”. This residual non-wetting phase saturation depends on many

parameters, such the pore structure and the capillary number, which is defined as the

ratio of the viscous forces to that of the capillary ones, acting across an interface

between two immiscible liquids.

The pressure drop is usually measured in the fluid under the higher pressure, i.e.,

the non-wetting fluid, and it is assumed that the pressure drop in the wetting phase is

identical to the pressure drop in the non-wetting phase, since it is assumed that the

saturation is the same everywhere under steady flow conditions. There are several

methods available for the determination of saturations, such as direct weighing or X-ray

absorption. Most of the available methods have been presented and discussed by

Scheidegger (1974). More recently, nuclear magnetic measurement (NMR) and X-ray

tomography have been used to determine the relative saturations. Using the saturation

information, the flow rates and the dimensions of the core, the relative permeabilities

can be calculated using eq. (7.2).

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7.2.2 Unsteady state flow method

In the unsteady-state, or external drive, method, one phase is displaced from a core by

pumping in the other phase, and the relative permeability ratio is calculated from the

produced fluid ratios. The unsteady-state method presents several advantages over the

steady state method, including the speed at which the experiments can be carried out, as

well as the fact that the relative permeabilities measured this way will exactly reproduce

the experimental oil recovery data.

There has been a great deal of work to investigate whether the measurements

generated by the unsteady state method are comparable to those generated with the

steady state flow technique. In general, the two methods usually give the same results.

But there have been numerous exceptions to this rule, as reported in the literature

(Schneider and Owens, 1970; Archer and Wong, 1971). However, the use of more

modern techniques such as sonic and computed tomographic scanning now allow for in

situ saturation measurements, thereby allowing unsteady state relative permeabilities to

be measured to be measured inside the sample as a function of position and time.

7.3 Network modelling of relative permeability

There are various ways one can model the relative permeability of rocks. Capillary

bundle-of-tubes models have been developed in the past, but because they ignore the

inter-connected nature of the pores, they cannot give a realistic description of two-phase

flow phenomena. Empirical models are practical and useful ways of modelling the

relative permeabilities. These various methods have been reviewed by Dullien (1992),

and we refer the interested reader to this monograph.

In recent years, networks models have superseded the other methods, due to their

ability to describe quantitatively the two-phase flow in porous media, as well as

allowing for a better understanding of the flow phenomena. This has been possible

because networks models can take into account the inter-connectedness of the pore

structure, as well as the microscopic flow mechanisms. Therefore, most recent methods

of predicting relative permeability have been based on network models. We have, in

Chapter 2, briefly reviewed several methods to generate networks, and presented how

they can be used to predict the permeability. Here, we will briefly review the history of

network modelling, by recalling some of the main papers, and show how it is used for

the predictions of two-phase relative permeability.

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7.3.1 History of network modelling

Modelling flow behaviour using network models was pioneered by Fatt in the 1950s

(Fatt, 1956). By distributing the pores and throats on a regular 2-D lattice, and filling

them according to randomly assigned radii using the Young-Laplace equation, he was

able to produce capillary pressure and relative permeability curves for drainage that had

the same characteristics as those obtained experimentally. He investigated the effects of

different network lattices, and concluded that the results were closer to experimental

measurements than those found using the bundle of parallel tubes model. The pore-size

distribution was found to have more influence on the results than did the choice of

lattice. Fatt computed the flow properties using an equivalent physical network of

electrical resistors, since, at that time, numerical solution of the flow/conductance

equations was non-trivial.

Further advances in network modelling did not occur until the late 1970s, when

computer processing power became more readily available. Chatzis and Dullien (1977)

built on the pioneering work of Fatt. They noted that 2-D networks could not reliably

predict 3-D behaviour, contrary to what had until then been commonly assumed. Their

networks consisted of both pores and throats, each of which had assigned volumes,

while the invading fluid was injected from the inlet face. They also allowed clusters of

defending fluid to become surrounded by the invader and become trapped. Chatzis and

Dullien (1981), and later Lenormand et al. (1983), developed experimental micro-

models of drainage and imbibition that allowed the pore-scale physics of displacement

to be observed and described. These observations of Lenormand et al. (1983) remain the

foundation of network flow modelling. Since the mid-1990s, there has been an ever-

greater trend towards realistic networks generation. The various methods to generate

networks include, amongst others, X-ray micro-tomography, process-based

reconstruction, and numerical reconstruction. These were briefly reviewed in Chapter 2.

Other important parameters of these network models are the characterisation of the

pore shape and wettability, because these can have a significant influence when

predicting transport properties. Most early work in pore-scale modelling assumed that

the throats were cylinders with circular cross sections, while pore bodies were assumed

to be spherical or cylindrical in shape. While it is certainly very difficult to represent

every pore directly, many researchers have tried to implement more realistic looking

pores that can accommodate wetting layers, such as square or triangular shapes. Øren et

al. (1998) defined a shape factor, G, as the ratio of the cross-sectional area over the

129

perimeter squared. They then modelled the elements as having a circular, square or

scalene triangular cross section, so that the shape factors was matched to those of their

reconstructed pore space. In most cases, the pores and throats had a triangular cross-

section.

7.3.2 Wettability changes

Another important factor that needs to be accounted for when modelling multi-phase

flow using network models is the definition of pore wettability. The term “wetting”

means that a given liquid will spread over the solid surface, while “non-wetting” means

that the liquid tends to ball up and run off from the surface. Wettability can be estimated

by determining the contact angle, or by calculating the so-called spreading coefficient.

Because of the prolonged contact of oil with the solid surface, which influences the

wettability, few hydrocarbons reservoir are strongly water-wet. For strongly water-wet

media, the multi-phase flow displacement mechanisms are now well understood from

micro-model experiments (Lenormand et al., 1983). Scenarios for media of arbitrary

wettability are harder to recreate experimentally, but Kovscek et al. (1993) proposed a

pore-level model for wettability change and fluid distribution. In this model, the

reservoir is considered to be initially completely saturated with water, and to be water-

wet. During the oil migration stage, the oil invades the pore space. If the oil directly

contacts the solid phase, the wettability is then changed. For the region of the pore space

where a thick wetting film is present, the surface remains water-wet, as do the corners

of the pores where water still resides, as well as the pores that remain completely water

filled. The alteration of the wettability depends on many parameters, such as the

composition of fluids, mineralogy of the surface, and capillary pressure. Oil is then

displaced during the imbibition process, during which it is assumed that the wettability

does not change. Kovscek et al. then examined the various pore-scale configurations for

the two fluids for a grain boundary shape. Additional reviews of two-phase network

flow modelling have been given by Jackson et al. (2003), Blunt et al. (2002), and Piri

and Blunt (2005).

7.3.3 Computation of relative permeability

Once the physics of the model have been established, one can then proceed to compute

the relevant parameters of the model. Firstly, the capillary pressures need to be

calculated using the Young-Laplace equation, assuming a strongly water-wet system.

130

The pores are then filled in order of capillary pressure. The oil invasion continues until

some specified maximum capillary pressure, or minimum water saturation, is reached.

The saturation of phase p can then be calculated using the following equation:

∑ ∑∑

== =

=p e

e

ni

ni ip

ni ip

pV

VS

1 1

1 , (7.3)

where ipV is the volume of phase p in element i, np is the number of phases, and ne is the

total number of pores and throats. The relative permeabilities are then calculated using

some appropriate variant of Poiseuille’s equation. Because the pores are assumed to

have a triangular geometry, if two or more phases are present, the expressions for the

conductance (as for capillary pressure) are more complicated. Normally, exact analytic

results are not possible, and empirical expressions are derived from solutions of the

Stokes’ equation for flow in pores of different geometries and for different fluid

configurations (Blunt et al., 2002).

At this point, the pressure can be computed everywhere in the network, therefore

allowing the computation of both absolute and relative permeability. Constant pressures

are assigned at the inlet and outlet and, assuming that all oil-water interfaces are frozen

in place, the pressure of each phase is computed separately. Conservation of mass is

used at each pore, which is equivalent to conservation of volume if it is assumed that the

fluids are incompressible. A set of simultaneous equations yield equations that allow the

pressures in the pores to be solved for using standard matrix techniques. The pressure is

normally computed when the network is entirely saturated with wetting phase. Then the

absolute permeability of the network is found from Darcy’s law,

)( outletinlet

tsp

PPALQ

k−

=μ

, (7.4)

where Qts is the total flow rate across the network for single-phase flow, summed over

all throats connected to the inlet, A is the cross-sectional area of the network model

normal to the flow direction, and L is the length of the model in the flow direction.

When multiple phases are present in the network, if flow rates are computed using the

same pressure drop as for single-phase flow, the relative permeability is simply

computed as follows:

ts

tmrp Q

Qk = , (7.5)

131

where Qtm is the total flow rate for the multi-phase situation.

Øren and Bakke (1998, 2002) used a disordered network of pores connected by

throats of triangular shapes, and a wettability model as described by Blunt et al. (2002),

and concluded that the assumptions were sufficient to make relatively good predictions

of relative permeability for water-wet sandstones. Blunt and co-workers (Valvatne and

Blunt, 2004; Piri and Blunt, 2005) have published several papers discussing predictions

for two and three-phase flow in various wettability settings. Overall, they managed to

match the experimental data relatively well. However, one of the problems they faced

when dealing with reservoir rocks is the lack of data for the wettability. Indeed, data for

the characterisation of the contact angles is scarce in the literature, therefore making

predictive modelling still relatively difficult. For example, Øren and Bakke (1998) had

to tune the fraction of oil-wet pores to match the residual oil saturation, and from there

were able to predict reasonably well the relative permeability of oil a mixed-wet

reservoir sample.

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Figure 7.2. Predicted and measured oil/water relative permeabilities as obtained by Blunt et al. (2002). The points are from the experimental data of Oak (1990), while the solid lines are predictions using their network model previously described. (a) Primary drainage: In the network model a receding contact angle of 0º is assumed. (b) Imbibition: Here a distribution of advancing contact angles, randomly assigned to pores and throats is assumed. Contact angles range from 30º to 90º, with a mean of 60º.

Figure 7.2 shows a typical relative permeability prediction obtained using network

modelling. It is clear that pore-scale modelling combined with geologically realistic

networks can reliably predict two-phase relative permeabilities for porous media,

especially water-wet media. However, it is certainly a time-consuming process that

requires a large amount of computational power. Another downside of the method is

that there are currently very few rocks for which a network has been fully extracted, and

133

in particular, this technique has not yet been applied to carbonate rocks. If the pore

structure is not known a priori, one can adjust the pore size distribution to match

capillary pressure data, while retaining the connectivity of the network. This can

subsequently be used this to make predictions of relative permeability. However, these

predictions might not always be reliable, especially in the case where experimental data

is scarce or not available. While network modelling does provide a very powerful tool

for relative permeability modelling, and is capable of good predictions, it is both a time-

and energy-consuming process that it is still far from being a standard industry practice.

One of the objectives of this research is to extend the methodology presented in the

previous chapter to the prediction of two-phase flow, using 2-D SEM images. In a

sense, this is the natural extension of the single-phase model. Although over the last two

decades many researchers have used the effective medium approximation to predict

transport properties, very few have used this approximation to predict two-phase flow

relative permeabilities. Amongst those, Levine and Cuthiell (1986) developed an

effective-medium model of two-phase flow based on a cubic lattice. We will here

briefly review their model and discuss their results.

7.4 The model of Levine and Cuthiell

Levine and Cuthiell (1986) assumed that the pore-size distribution of the rock could be

described by an idealised probability function, which was partitioned so that pores

below a certain size, or radius, would only allow the presence of a water phase. On the

other hand, above this threshold, the pores will only admit the oil phase. Hence, their

model seeks to obtain two-phase relative permeabilities by separating the flow tubes

according to a pre-defined size criterion to account for wettability. They used

Poiseuille’s equation to compute the conductances of each phase, and then assumed the

relative permeabilities to be proportional to those conductances. Finally, by assuming

various pore-size distributions, they solved the equations to generate the relative

permeabilities curves.

The relative permeability curves generated in this way did resemble those obtained

experimentally. However, it was noticed that the shape of the non-wetting phase curve

was concave towards the Sw axis in the neighbourhood of the flow cut-off, whereas most

experimental data show this part of the curve to be convex. Furthermore, it also

appeared that the predicted end-points of the curves were quite low, with a reported

average Sw at 0=−nwrk of just under 0.4. This certainly seems lower than most reported

134

experimental values. These two problems, as the authors noted, could well have been

due to the fact that the EMA does not work well near the percolation threshold, thereby

lowering that saturation, as well as changing the shape of the curve. They also extended

their model so that they could account for simultaneous flow of water and oil, by having

the oil flow on top of a pre-existing water film, and included the possible dependence of

the relative permeabilities on the ratio of the viscosity of fluids. Unfortunately, they did

not present any numerical results for this extended model in their paper.

The other drawback of their model was that the distribution of pore radii was

drawn from theoretical distributions, and not from real porous media. This certainly did

not contribute to the generation of realistic relative permeability curves. It is therefore

our aim to combine the use of two-dimensional images with some simple wettability

rule to generate a simple two-phase relative permeability model.

7.5 Preliminary test of two-phase flow model

Conceptually, the model we wish to develop here is, to start with, similar to that of

Levine and Cuthiell (1986) described previously, with the notable difference that we

will be using direct measurements of the pore space, as described in the Chapter 6, to

use an input for the model. However, before doing so, we decided to carry out

preliminary predictions with pre-determined pore size distributions under the

assumption of a completely water wet rock, to assess the viability of our model. These

preliminary tests will be described in the next sections.

7.5.1 Constant pore-size distribution

The first test we decided to run was to simply have a near-constant pore size

distribution, ranging from 1.00001 to 1.01, with increments of 5101 −× , therefore

making a distribution of 1000 pores, each of which were assumed to have a circular

cross-section. If we assume the rock to be water-wet, we know from the micro-model

experiments of Lenormand et al. (1983) that the wetting phase will fill the smallest

pores first, while the non-wetting phase will primarily reside in the larger pores. In a

way, this is similar to the approach used by Levine and Cuthiell, in that the occupancy

of the pore will be determined by the size of the pore. This is also the first assumption

under which we will develop our model.

To generate the relative permeability curves, we will make up a series of

conductance distributions at several saturation points, using the aforementioned pore-

135

occupancy model. The EMA will then be used to generate an effective conductance at

each saturation level. For example, for the wetting phase permeability, we first assume

that all the pores are filled with the wetting phase, before inputting zero-conductances

by saturation increment so as to simulate the advancement of the non-wetting phase in

the pore space. This is done by increments of 5 or 10% of the pore volume, and the

filling-up process is starting from the largest pore to the smaller ones, i.e., equivalent to

a “top-down” approach. As hinted earlier, the EMA is then used to compute an effective

conductance at each of the various saturation points. The wetting-phase relative

permeability is then calculated using eq. (6.4), solely applied to the wetting phase,

which gives us

total

weffweffwr A

CNk −−

− = . (7.6)

This calculation is repeated at the various saturation levels until the percolation

threshold of the EMA is reached, when two-thirds of the pores are “filled” with the non-

wetting phase. This process is comparable to a primary drainage process that occurs

when the non-wetting phase is forced into a porous rock. For the computation of the

non-wetting phase permeability, a similar approach is used, by firstly assuming the pore

space to be fully saturated with the non-wetting phase, and progressively filling it with

the wetting phase by saturation increments, starting from the smallest pore to the larger

ones, i.e., in a “bottom-up” approach. The effective permeability of the non-wetting

phase is then calculated in the same way as that for the wetting phase:

total

nweffnweffnwr A

CNk −−

− = . (7.7)

In the present case of a pore-size distribution that is essentially a delta function, we

expect the relative permeability to simply plot as a straight line, going from a

normalised relative permeability of 1 to 0 when two-thirds of the saturation of each

respective phase is blocked off – 2/3rds being the percolation threshold generated by the

EMA, as presented in Chapter 4.

136

Figure 7.3. Relative permeability curves obtained for the two-phase model computed using a delta-function pore-size distribution.

Figure 7.3 shows the relative permeability curves obtained using a delta-function

pore-size distribution. Reassuringly, the shape of the curves obtained is what was

expected from this very simple model. Because of the binary nature of the conductance

distribution used in this case, coupled with the use of the EMA and its percolation

threshold, we obtain very simple behaviour from these assumptions. However simple

these assumptions, we feel that this gives us a good basis to work from, as the behaviour

is akin to that which is observed experimentally.

7.5.2 Uniform pore-size distribution

The second test we wish to carry out is by using a pore-size distribution that is uniform

between two values. This is a closer approximation to actual conditions, and will allow

us to observe the effects of the width of the distribution on the relative permeability

curves. We use a uniform distribution with a pore-size (in arbitrary units) ranging from

1-11, with increments of 2101 −× , hence making again an artificial distribution with

1000 pores. Again, the pores are assumed to be of circular cross-section, and the

hydraulic radius approximation is used to generate the hydraulic conductances. The

effective permeabilities are then calculated using eqs. (7.6) and (7.7).

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Kr-w

Kr-nw

Rel

ativ

e P

erm

eabi

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Kr

Wetting Phase Saturation (%)

137

Figure 7.4. Relative permeability curves obtained for the two-phase model computed using a uniform pore-size distribution.

When using a uniform pore-size distribution as shown on Fig. 7.4, we notice some

changes in the shapes of the relative permeability curves obtained, compared to the

previous case of a very narrow distribution. First, the saturation at which the wetting

phase permeability vanishes has been shifted to the left, down to a value of about 15%.

This is easily explained from the pore-occupancy assumption that the wetting phase will

fill the smaller pores first. Indeed, while more than a third of the pores have to be filled

to allow the EMA to generate a positive conductance, this condition occurs at a lower

saturation than in the previous case, simply because, by definition, this one-third of the

pores are smaller than the other two-thirds, therefore yielding a percolating saturation

less than 33%. Similarly, the end-point for the non-wetting phase relative permeability

curve has also been shifted to the left, to a value of 48%. Again, this is due to the pore-

wettability conditions that we assume for the model. Because the non-wetting phase fills

up the bigger pores first, the smaller pores are assumed to be filled with the wetting

phase, and therefore have a hydraulic conductance (to the non-wetting phase) equal to

zero. This means that the saturation at which the non-wetting phase curve vanishes is

lowered, compared to the previous case. The other trend that is noticed is that the non-

wetting phase relative permeability curve is concave towards the Sw axis in the

neighbourhood of the flow cut-off, while it is generally found that experimental data

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Kr-w

Kr-nw

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e P

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display a convex shape. Levine and Cuthiell (1986) noticed the same trend in their

results. It must be pointed out that the results obtained here with the linear pore-size

distribution are very similar to those obtained by Levine and Cuthiell.

7.5.3 Log-normal pore-size distribution

Before applying the methodology to actual data collected from 2-D SEM images, it was

thought to be a worthwhile test to run the method with a log-normal pore-size

distribution. The relative permeability curves obtained with this distribution are shown

in Figure 7.5, assuming a mean radius of 1 and a standard deviation of 0.5. The trends

that were previously noticed when using a linear distribution are here more pronounced,

due to the fact that the log-normal distribution has a much longer “tail” at the low end of

radii. The starting point of the wetting-phase relative permeability curve is again

slightly shifted to the left, down to a value of 5%, again for the same reasons as

discussed above. Similarly, the end point of the non-wetting phase curve is even more

shifted to the left as well, dropping to value a little less than 30%, which seems

unrealistic when compared with experimental values (Oak, 1990). The concave shape of

the curve is also accentuated by the use of the log-normal pore-size distribution.

Furthermore, a slight change in the shape of the wetting phase relative permeability

curve is also noticed, with the curve taking on an S-shape rather than a convex shape, as

observed in the previous case.

Although these preliminary tests are based on theoretical distributions that may not

as realistic as an actual rock pore-size distribution, it is encouraging to see the curves

created are not too dissimilar to those measured in the laboratory. In a way, these results

are similar to those of Levine and Cuthiell (1986), and display the same limitations.

However, as mentioned earlier, Levine and Cuthiell did not apply their model to a pore-

size distribution extracted from a rock. In the next section, we will try to generate the

relative permeability curves for Berea sandstone, while in the last part of this chapter,

our model will be applied to a carbonate sample used in the previous chapter.

139

Figure 7.5. Relative permeability curves obtained for the two-phase model computed using a log-normal pore-size distribution, with a mean of 1 and a standard deviation of 0.5.

7.6 Relative permeability of Berea sandstone

Before predicting the relative permeability of any carbonate rocks, it was considered

important to test the model on sandstone, since the homogeneity of the pore space will

not be as much of a problematic issue as it is with carbonate rocks. There are several

sandstones that could have been used to test the method, such as Fontainebleau or

Massilon. However, for several reasons, we decided to use Berea sandstone. Firstly,

Berea sandstone is a well-studied rock for which relative permeability data are readily

available in the literature. Oak et al. (1990) has carried out experimental measurements

for two- and three-phase flow with several Berea samples, while Jadhunandan and

Morrow (1995) have published a thorough study of the wettability of Berea sandstone.

Some of Oak’s data was shown in Fig. 7.1 earlier in this chapter. Secondly, Schlueter

(1995) also studied Berea sandstone in her thesis.

7.6.1 Schlueter’s two-phase work

We have reviewed in Chapter 2 the single-phase permeability model of Schlueter.

Although she did not explicitly compute two-phase relative permeability from 2-D

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images, she ran a series of primary drainage experiments to visualise directly the 2-D

pore structure and wettabilities of the samples for the full range of saturations, as well

as to evaluate relative permeabilities in terms of the microphysics and microchemistry

of the process involved. These experiments involved preparing various Berea samples

over the whole range of saturations, so as to generate 2-D images of the fluid

distribution at these various saturations. In order to do so, she oven-dried the samples

before filling them with paraffin wax at various saturations. Finally, the samples were

partially saturated with distilled water. Thus, she was able to observe the fluid

placement within the pore space of the rock. By doing so, she constructed a library of 2-

D images at various saturations. Although the resolution of these images was generally

too low for the direct computation of relative permeabilities, it allowed for a statistical

study of the pore occupancy, which we will use later.

It was originally intended for this research to use a similar approach, i.e.,

experimentally saturate rock samples at various saturations with different coloured

epoxies, before taking SEM images of the pore space. Having these images, it would

have been possible to generate measurements for each phase, and directly compute

relative permeability from this data. However, for various reasons, it has not been

possible to carry out these experiments, hence the motivation to devise the conceptual

model discussed above. Nevertheless, although these laboratory measurements would

have been interesting, we note that our ultimate aim is to devise a prediction method

that does not require such complex laboratory procedures.

7.6.2 Preliminary results for Berea

Using one of Schlueter’s 2-D Berea images, we were able to extract the necessary

parameters, i.e., pore area and perimeter, as explained in the previous chapter. Using

this data, we first ran our model exactly as described in section 7.5. The resulting

relative permeability curves obtained are shown in Fig. 7.6. These predictions are made

using the model described earlier using the data taken from an SEM image of Berea.

The simple model used here for the prediction of two-phase data more or less recreates

the overall shape of the relative permeability curves, as measured by Oak (1990).

141

Figure 7.6. Prediction of the relative permeability curves for Berea sandstone using the model described previously and the EMA, along with values measured by Oak (1990).

However, there are two problems with the computed curves. First, the starting

points of the computed curves clearly do not match those of the experimental data.

There are several reasons for this, as will be addressed later. The other problem that we

notice in these curves, which was highlighted earlier, is the impossibility for the non-

wetting relative permeability curve to be extended in the higher range of the wetting

phase saturation. Indeed, the non-wetting phase reaches zero permeability at a wetting

phase saturation of 31.5%, which appears to be unrealistic, especially when compared

with the experimental data from Oak, also presented on Fig. 7.6. This trend is partly due

to the pore-wettability conditions assumed for the model, i.e., the non-wetting phase

fills the larger pores first, but it is also related to the percolation threshold generated by

the EMA.

These appear to be the two mains reasons that prevent our model from generating

more realistic relative permeability curves. We have highlighted in Chapter 4 how the

EMA and percolation theory yield different percolation thresholds (§4.5.1). We also

pointed out then that a main characteristic of percolation processes is that they obey

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specific scaling laws. In particular, it was noted that the permeability of a given system

obeys the following law:

tcppk )( −∝ , (7.8)

where pc is the percolation threshold, and t is the critical exponent. We then investigated

the behaviour of eq. (7.8), both near the percolation threshold and over the entire

saturation range, and concluded that, although a value of 2=t was most appropriate in

the vicinity of the percolation threshold, a value of 6.1=t gave the best fit over the

entire range of p. This was similar to the findings of O’Carroll and Sorbie (1993), who

also used the scaling law in combination with Poiseuille’s law to generate a simple two-

phase flow model. They used the scaling law to compute the relative permeability down

to the percolation threshold. We wish here to use a similar technique and combine it

with the Generalised Perturbation Ansatz (GPA), developed by Gelhar and Axness

(1983), which was briefly introduced in Chapter 4.

7.6.3 Introduction of the GPA

As implied earlier, one of the main problems of using the EMA for the prediction of

two-phase permeability is the percolation threshold that it generates. Indeed, we know

that this threshold is not accurately predicted by the EMA, and this has direct

implications for the shapes of the relative permeability curves, as seen on Figs. (7.4)-

(7.6). By using the so-called GPA in conjunction with the scaling law, we will be able

to use the correct percolation threshold for a cubic lattice, which presumably will

improve the predictions. Gelhar and Axness (1983) claimed that, by using the GPA, the

effective permeability of a conducting medium in three dimensions could be computed

as follows:

)6/exp( 2ln kgeff kk σ= , (7.9)

where gk is the geometric mean of the conductivity distribution, and 2ln kσ denotes the

log permeability variance. Although no rigorous proof for its validity is available,

Dagan (1989, 1993) has derived a perturbation expansion that agrees with the Taylor

series expansion of eq. (7.9) up to the fourth order in klnσ . In the context of our relative

permeability predictions, eq. (7.9) can be used to estimate an effective pore-level

conductance from the hydraulic conductances. Having computed the effective

conductance in this manner, the permeability is then computed as usual:

143

total

eff

ANC

k = . (7.10)

Using this, in conjunction with the scaling law in manner similar to that used by

O’Carroll and Sorbie (1993), we arrive at

t

c

c

total

eff

ppp

ANC

k ⎥⎦

⎤⎢⎣

⎡−−

=1

, (7.11)

where p is the fraction of pores filled by a given phase, pc is the percolation threshold,

and t is the critical exponent. As stated earlier, the obvious advantage of using this is the

ability to choose the percolation threshold. Rather than being limited to the percolation

threshold generated by the EMA, it is now possible to use the correct value for this

parameter, i.e., 0.25 for a 3-D cubic lattice. When inserting this value in eq. (7.11), we

find the following equation for the permeability:

t

pANC

ktotal

eff⎥⎦⎤

⎢⎣⎡ −=

31

34 . (7.12)

By introducing the correct percolation threshold for a 3-D cubic lattice, we hope to

displace the ending and starting points of the relative permeability curves in the right

direction, as well as to lower the wetting phase relative permeability curve. We will

present results with two critical exponents 6.1=t and 2=t , although we anticipate, as

highlighted in Chapter 4, that a value of 1.6 will work better over the full range of

saturation. At this point, apart from the actual relative permeability calculation, nothing

has been changed to the model. The results are presented in Fig. 7.7.

Figure 7.7 shows the relative permeability curves obtained using the GPA in

conjunction with the scaling law. We notice that the end-point of the non-wetting phase

curve has been shifted to the right, with the point 0=−nwrk now reached at a saturation

of 42%, in contrast of an end-point saturation of 31.5% when using the EMA. This

represents a substantial improvement. Although we lose the hump that is present at low

wetting phase saturation when the relative permeability is computed using the EMA, the

overall shape of the non-wetting phase curve is certainly more realistic, and closer to the

original data than was that computed using the EMA. Also, it is clear that a critical

exponent of 2 lowers the curves too much, this being especially noticeable on the non-

wetting phase curve, which becomes too convex.

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Figure 7.7. Prediction of the relative permeability curves for Berea sandstone using the first model and the GPA in combination with the scaling law, for two critical exponents.

7.6.4 Probabilistic pore occupancy

Thus far, our pore occupancy model, in which we assume that a given pore can only

contain one phase, simply states that the wetting phase will fill the smaller pores first,

whilst the non-wetting phase will normally fill the larger pores. Under the assumption

of a water-wet porous medium, this very simple rule has been verified experimentally

using micro-models (Lenormand et al., 1983). However, in practice, it is very unlikely

that this will always be the case. In theory, the primary drainage process is driven by

capillary forces, with the non-wetting phase permeating the pore or throat with the

lowest capillary entry pressure, in a process akin to an invasion percolation process

(Wilkinson and Willemsen, 1983; Dias and Wilkinson, 1986). One might expect that

this process would eventually lead to the pore occupancy scenario described above.

However, there is an accessibility issue, caused by the fact that a given “large” pore

may be, for example, surrounded by smaller pores that the non-wetting phase cannot

enter. In this case, this given “large” pore will not be filled by the non-wetting phase.

We mentioned earlier in §7.6.1 that Schlueter ran primary drainage experiments in

which she impregnated several Berea samples with different phases at different

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saturations. Although most of the SEM images shown in her thesis were not usable in

the present work, because of the low resolution of the images, one of those images had a

good enough resolution to enable us to analyse the pore-occupancy in relation with the

pore sizes, using Image J. In this case we were able to analyse how the two phases

position themselves within the pore space. The results are displayed in Fig. 7.8.

Figure 7.8. Illustration of the phase pore size distribution from a Berea sample impregnated with both non-wetting and wetting phase fluids (after Schlueter, 1995).

It is clear from Fig. 7.8 that, although the phase distribution is generally as had

been expected from the simple considerations outlined above, we nevertheless find

some smaller pores filled with the non-wetting phase, while the wetting phase has

occupied one of the biggest pores. This scenario seems compatible with the percolation

invasion process and the accessibility problem highlighted above. A network model

would be helpful in simulating this process, but in the present context we would prefer

some simpler method to make our pore occupancy assumptions more realistic. One

simple way to do so is to assign a random probability between 0 and 1 to each pore.

This probability can then be used in combination with a probability threshold, pth, above

which the pore occupancy would be “reversed”. Thus, if a given pore has thpp ≤ , then

the pore occupancy is kept unchanged, with small pores occupied by the wetting phase,

whereas if thpp > , the pore occupancy is reversed. The model we have been using until

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146

now therefore corresponds to a probability threshold of 1, i.e., the pore-occupancy is

never reversed. We then let the threshold take on values from 1 down to 0.5. It is hoped

that the introduction of the probabilistic component to the assignment of pore

occupancy will lead to more realistic relative permeability curves, by re-adjusting the

shape and shifting the end-points in the right directions. The introduction of the pore-

occupancy probability does not change the first part of the model, i.e., when 1=thp .

The points of each relative permeability curve are found by first assuming a theoretical

water saturation, generally in increments of 5% or 10%. Probabilities are subsequently

assigned to the pores, and the water saturation is re-calculated accordingly to the chosen

probability threshold. The random probabilities were generated using the software

Crystal Ball (http://www.decisioneering.com). This process has been implemented for

all probability thresholds from 1 to 0.5, in decrements of 0.1. The results are presented

in the figures below.

Figure 7.9. Wetting phase relative permeability curves obtained with a range of probability thresholds, and a critical exponent of t = 1.6.

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Figure 7.10. Wetting phase relative permeability curves obtained with a range of probability thresholds, and a critical exponent of t = 2.

Figure 7.11. Non-wetting phase relative permeability curves obtained with a range of probability thresholds, and a critical exponent of t = 1.6.

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Figure 7.12. Non-wetting phase relative permeability curves obtained with a range of probability thresholds, and a critical exponent of t = 2.

Figures 7.9-7.12 show the results obtained for Berea sandstone when using this

probabilistic pore occupancy model. In each case the results are presented for five

values of the probability threshold, as well as for two different critical exponents. The

introduction of the probability factor has a clear effect on the relative permeability

curves. The wetting phase relative permeability curves are progressively shifted to the

right as the probability threshold is lowered, and the overall shape of the curve seems to

match the data better as the probability threshold decreases. A probability threshold of

0.6 seems to give the best match to the shape of the curve.

As far as the non-wetting phase relative permeability is concerned, the effect of the

probability threshold is also noticeable. Again, as expected, the curves are shifted to the

right, closer to the actual data points, as the probability threshold is lowered. A critical

exponent of 2=t seems to generate curves with a slightly more realistic shape, but

lowers the relative permeability values too much. As with the wetting phase curves, a

probability threshold of 0.6 seems to work best in matching the experimental data. This

is equivalent to what can be computed from the relative pore size distribution presented

in Fig. 7.8. Overall, as mentioned earlier, it appears that a critical exponent of 6.1=t

works better over the whole water saturation range, whereas 2=t appears marginally

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better in the higher water saturation range, again for the same reasons as mentioned in

§4.5.1.

7.6.5 Introduction of the irreducible water saturation

There is one salient difference between our predicted curves and the experimental data

presented in the earlier figures, and that is the gap between the ending and starting

points of the curves. Whereas, due to the assumptions of our models, the relative

permeability curves normally start, or finish, at a water saturation of 0%, it is easily

noticed that the experimental curves either start or finish at a water saturation wS of

22%. This saturation is known as the irreducible water saturation, and was introduced

earlier in §7.2.1. It is defined as the point where the wetting phase saturation appears to

be independent of any further increases in the externally measured capillary pressure

(Dullien, 1992). In the case of a water-wet medium, it will be the lowest water

saturation obtained after a non-wetting fluid has flowed through the porous medium.

This saturation is usually not very well defined, apart for media such as beds of smooth

glass beads. In the case of Berea sandstone, this saturation is quite high, and its value is

around 22%, as reported by Oak (1990), as well as by Jadhunandan and Morrow (1995).

The irreducible saturation is unusually high in the case of Berea, mostly due to the

presence of water in the clays contained within the rock. It is possible in the laboratory

to lower this irreducible water saturation to create a large surface area for wettability

alteration, as was reported by Jadhunandan and Morrow (1995). The lowest Swirr they

obtained was 8%. This issue is discussed by Jackson et al. (2002), who also explained

how this value is implemented in their network modelling.

While we have not incorporated this immobile water volume in our model, because

it is impossible to determine this value a priori, it is possible, however, to normalise the

water saturations using the value reported by Oak (1990). If we consider the various

volumes present in the total pore space, tV , we have the total mobile pore volume, tmV ,

the mobile volume mV , and the immobile (water saturated volume), imV . The irreducible

water saturation can be expressed as

t

imwirr V

VS = . (7.13)

We can also express the total pore volume, tV , as

imtmt VVV += , (7.14)

150

where

t

imwi V

VS = . (7.15)

Thus, we obtain

wirr

tmt S

VV

−=

1. (7.16)

We then finally compute the normalised water saturation, Snor, as

wit

m

t

imnor S

VV

VVV

S +=+

= , (7.17)

with the mobile water saturation expressed as

tm

mm V

VS = , (7.18)

and thereby obtain

wirrmwirrnor SSSS +×−= )1( . (7.19)

Sm, the mobile saturation, here corresponds to the saturation used in the previous

figures. Using eq. (7.19), it is therefore possible to convert the saturations used

previously, and start our relative permeability curves at the experimentally reported

irreducible water saturation (Blunt, pers. comm.).

7.6.6 Final Berea relative permeability curves

Having introduced the pore wettability probability and the irreducible water saturation,

we now present our final relative permeability curves for Berea sandstone. We stated

previously that a probability threshold of 0.6 appeared to yield the best results for the

prediction of the relative permeabilities. However, it is also worth noting that the first

value obtained for each probability threshold, down to 6.0=thp , appears to fit the

relative permeability curves. This is particularly true for the wetting phase relative

permeability curve, as seen in Figs. 7.9-7.10. This allows us to have relative

permeability data points that lie between a water saturation of 60% and 100% for the

wetting phase. Below that, the data points obtained from 6.0=thp are used for

generating the relative permeability curves. The final results are presented in Fig. (7.13).

151

Figure 7.13. Comparison between the final relative permeability curves for Berea, presented for both critical exponents, and the experimental data from Oak (1990).

Figure 7.13 shows the final relative permeability curves obtained for a probability

threshold of 0.6, presented for two values of the critical exponent, as well as the original

data from Oak (1990). The introduction of the normalised water saturation brings the

previously obtained Snor curves more in line with the experimental values. This is

particularly noticeable, as the value for the irreducible water saturation is quite high.

Overall, the match obtained between the curves is good, considering the simplicity of

the model presented here. The non-wetting phase relative permeability curve is indeed

matched very well when using a critical exponent of 6.1=t , if only for the starting

point of the curve. In our case, this point occurs at a water saturation of around 80%,

while the experimental data curve starts at 100%. Unfortunately, this value cannot be

increased with our model, as this corresponds to the percolation threshold generated by

the extended GPA formula. This percolation is reached when 3/4ths of the conductances

are zero, which corresponds to this saturation.

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The wetting phase relative permeability is unfortunately not matched as well as the

non-wetting phase, when using 6.1=t , as the predicted curves tend to over-predict the

relative permeability values, although the shape of the curve is broadly predicted. While

the use of 2=t does correct that trend, we believe that a choice of 6.1=t is more

appropriate in relation to the work presented in §4.5.1. The over-prediction may be due

to the presence of clays in the rock, as well as numerous small pores present in the

Berea sample, both of which were not necessarily captured properly during the image

analysis process, due to the low resolution of the images. These two factors, as

mentioned earlier, are also partly responsible for the high irreducible water saturation

Swirr found in Berea. Overall, we believe that the results obtained for the Berea sample

are good, especially after introducing the probabilistic pore occupancy assignment. This

renders both the curves more realistic, as well as making the ending and starting points

shift in the right direction in each case.

7.7 Relative permeability of carbonate rock ME2-2

Having verified that our relative permeability model gives reasonable results for Berea

sandstone, it was decided to test the method with one of the carbonate rocks that was

used in the previous chapter for the prediction of absolute permeability. The only

carbonate sample for which we had access to experimental two-phase relative

permeability measurements was sample ME2-2, so this was the only carbonate sample

on which we could test our relative permeability model. It is reassuring that the absolute

permeability predicted in Chapter 6 was very close to the experimental value.

All the curves presented earlier for the Berea sample have also been generated for

sample ME2-2. However, we feel it is redundant to present yet again several figures of

two-phase relative permeability curves. The exact same processes presented for Berea

have been used in the computation of the properties. The probability threshold was used

in decrements of 0.1, from 1 to 0.5. Again, it appeared that a probability threshold of 0.6

gave us the best fit for the curves, while it was again possible to use the initial values

obtained with higher probability thresholds to complete the relative permeability curves.

It is also interesting to notice that we were also able to do this for the non-wetting phase

curve. An irreducible water saturation of Swirr = 12.1% was implemented, in accordance

with the experimental data available. As with the Berea predictions, this saturation was

not computed using the model, but was simply input into the model after the simulations

were run. Finally, the non-wetting phase relative permeability was also re-scaled to

153

match the starting point of the curve. Using our model, the first point of the non-wetting

phase relative permeability curve will always have a value of 1, as we simply assume

that all the pores present in the rocks are filled with oil and therefore oil-conducting.

However, in this case, the presence of the irreducible water saturation caused the sample

to display a value of 595.0=−nwrk at Swirr. It was therefore decided to re-scale the non-

wetting phase curves to match this value. The relative permeability curves obtained for

sample ME2-2 are presented in Fig. 7.14.

Figure 7.14. Comparison between the experimental data of sample ME2-2 and the final relative permeability curves, shown here for both critical exponents.

Figure 7.14 shows the relative permeability curve obtained for sample ME2-2.

Overall, the match between the experimental data and the predicted curves is relatively

good. There are several trends that are similar to those observed with Berea. Generally,

the shape of the wetting phase curve is matched well, although again the model tends to

over-predict the actual values. Similarly, a critical exponent 2=t appears to give a

better match for the wetting phase curve, while the values are over-predicted by a

maximum factor of two for Sw around 60% to 70% when a critical exponent 6.1=t is

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used. On the other hand, a critical exponent of 1.6 appears to give us a better match for

the non-wetting phase, with the values slightly under-predicting for values of Sw around

50% to 60%. This is similar to the trends noticed when working with the Berea sample.

Generally, both the shape and the values of the non-wetting phase curve are matched

very well with our model.

As mentioned earlier, it is interesting to notice that the relative permeability curves

can be computed easily, since one can simply take the overall pore conductance, and

apply a decreasing probability threshold down to 0.6, to generate the first few data

points of the curve, while thereafter one needs to alter the pore conductance distribution

as shown in §7.5.1. The overall predictions, while not perfect, are quite reasonable, with

an accuracy that is similar to that obtained in our predictions of absolute permeability.

Of course, the irreducible water saturation needs to be input in the model after having

computed the curves, but this is not dissimilar to what other researchers have done

(Jackson et al., 2002; Blunt et al., 2002). More problematic are situations in which the

maximum non-wetting phase relative permeability does not reach a value near 1. This

problem cannot be solved with the current model, but there are several possible ways

that this could be improved; these will be discussed in Chapter 8.

155

8 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK In this thesis, we have presented a methodology for the prediction of the hydraulic

permeability of carbonate rocks, based solely on 2-D SEM images. This method is

applied to both the absolute and relative permeabilities. Because the internal pore

structure displayed by carbonate rocks is comparatively more complex than that of

sandstones, the method, originally developed by Lock (2001), had to be entirely

reviewed to accommodate its application with carbonates. Firstly, the image analysis

process had to be updated for several samples because of the impossibility of picking a

thresholding point using Lock’s criterion. This was due the fact that, in those samples, a

significant amount of micro-porosity was present. This problem was solved by inferring

a value for this parameter by matching the image porosity to the core porosity.

Kirkpatrick’s effective medium approximation was also reviewed in order to assess its

applicability to the conductance distributions generated from our carbonate samples. We

concluded that it was permissible to use the EMA with such distributions, although the

more extreme distributions are close to the permitted bound. It also turned out that the

areal post-image analysis thresholding had to be adjusted for carbonate rocks; this was

hardly surprising, because of the wide range of pore-size distributions displayed by

carbonates.

Another refinement to the method was the introduction of a varying co-ordination

number. Lock had assumed a cubic lattice, and a co-ordination number of six for each

sample. However, we noticed that this method leads to a tendency to underpredict the

permeability of high-permeability samples, and overpredict the permeability of low-

permeability samples. It seemed plausible that this error was caused by the fact that the

actual co-ordination number tends to correlate positively with the permeability. We

therefore investigated several possible ways to relate the permeability to the co-

ordination number. Because no experimental data was directly for carbonate rocks, we

based our correlation on data available in the literature for Fontainebleau sandstone

(Sok et al., 2002). Using this data, we generated a correlation between the permeability

and the co-ordination number. We then used the following procedure to estimate the

permeability. First, the method was used, with a co-ordination number of six, to predict

156

the permeability. The correlation between k and z was then used to estimate a new co-

ordination number, after which this new value of z was input to our prediction

procedure to yield a new prediction. Although this procedure did help to improve the

predictions for some of the samples, and gave a reduction in the mean error for the

entire data set, it did not yield much improvement for those samples that were

significantly under-predicted to start with. This can be explained as follows. If the initial

permeability prediction for a high-k sample is, say, slightly low, our correction

procedure will indeed give a “new” co-ordination number greater than 6, after which the

refined permeability prediction will be increased, as it should be. But if initial

permeability prediction for a high-k sample is much too low, our correction procedure

will only give a slight increase in z (or even a decrease, in extreme cases), and so the

second prediction of the permeability will again be lower than the actual value. Other

possible ways to estimate a value for z based solely on our two-dimensional images

were investigated, but none yielded any substantial further improvement in the

permeability predictions. This is certainly an issue that could benefit from further

investigation.

For most of the rocks analysed during this research, the permeability was predicted

within a factor of two. However, for two samples, the permeability was substantially

under-predicted, in some cases by as much as a factor of seven. One of these two

samples (ME1-6) appeared not to have been carefully handled during the laboratory

procedures, resulting in the formation of fractures within the rock, which very likely

caused an increase in the laboratory permeability. In the other case, we have found no

obvious reason for the sample to be so under-predicted by the method, which brings up

the issue of representative elementary volume (REV). Indeed, it must be remembered

that carbonate rocks are much more heterogeneous than sandstones, therefore it is quite

possible that the small area of our image did not capture the full pore size distribution

that existed in the core. And, as highlighted earlier, there was no opportunity to directly

examine the sample, which means to we can only speculate as to why the sample’s

permeability is so greatly under-predicted. In these two cases, the varying co-ordination

number did not substantially improve the original predictions.

Overall, the results obtained are good, considering the range of rocks analysed for

this study. The permeabilities are predicted within a factor of two in most cases, as was

achieved with the sandstone samples. This must be put in perspective with the errors

inherent in the measured permeability values. McPhee and Arthur (1994) reported the

results of a series of gas permeability measurements on a Clashach quarried sandstone,

157

with an average permeability of 693 mD, and found an “error” of 32%, based on the

standard deviation of all the measurements. Although these values are not directly

comparable, the mean (absolute) relative error in our measurements seems to be

consistent with the error expected when carrying out experimental permeability

measurements. The introduction of a varying co-ordination number, while refining

some of the predictions, especially for sandstones, did not appear to yield a systematic

improvement for the carbonate samples. It must be mentioned, however, that the

relative computational simplicity of the model is due to the numerous “physical”

approximations made during the procedure.

While it would obviously be desirable to test the method with more carbonate

samples, it may well turn out “± a factor of two” may represent the inherent limitation

of the method. To test this hypothesis, we calculated, for each sample, the value of z

that would be needed for our predicted permeabilities to exactly match the measured

values. In most cases, the required value of z was between 4 and 7. Such values of the

co-ordination number are quite plausible, and so these cases could be interpreted as

implying that all that is needed to “perfect” the present method is a better way to

estimate the co-ordination number. On the other hand, in order to rectify the error in the

case of samples ME1-3 and ME1-6, we would need to use co-ordination numbers of

around 16, which seem to be unrealistic. These two cases seem to imply that either (a)

the problem lies in our original image not capturing the full REV scale, or (b) this error

reflects an inherent limitation in our prediction methodology.

A tomographic study of carbonate rocks would be helpful in generating a direct

relation between the permeability and the co-ordination number specific to carbonates.

This could potentially improve the predictions, as it clearly did for the sandstone data

set. However, while these techniques exist, they are still being perfected for carbonate

rocks, because of the technical constraints when dealing with such rocks. Thus, it may

take some time before seeing such studies.

In Chapter 7, the methodology was extended to the prediction of relative

permeability. In a sense, the work presented is an extension of some earlier work by

Levine and Cuthiell (1986). However, whereas they used idealised pore size

distributions, we used conductance distributions inferred for the individual rocks from

analysis of our SEM images. By assuming a water-wet rock, we partition the pore-size

distribution so that pores below a certain size only allow the presence of the wetting-

phase (water), while above this size point the pores will only admit the non-wetting

phase (oil). This simple model was based on the micro-model experiments of

158

Lenormand et al. (1983). Using this partitioning system of the pore size distribution, we

were able to generate the conductance distributions corresponding to the various

wetting-phase saturation points. Using these, the relative permeability as a function of

the wetting phase saturation was computed. The preliminary tests using Kirkpatrick’s

effective medium approximation led to the conclusion that, although the general shape

of the relative permeability curves was fairly realistic, the match was not good

quantitatively, and the end-points of the curves were not correctly predicted. This was

due to the fact that the EMA does not predict the correct percolation threshold, as

highlighted in Chapter 4.

To improve the results, we used a combination of the GPA and the percolation

scaling law for permeability. Percolation theory allows us to define the behaviour of

many quantities near the percolation threshold, independently of the network structure.

In the case of permeability, we have tcppk )(~ − , where pc is the percolation threshold

and t is the critical exponent. By using this result, we were able to use the correct

percolation threshold, thus improving our predictions. In order to generate a more

realistic pore occupancy distribution, we associated a random variable f to each pore,

uniformly distributed between 0 and 1. A probability threshold pth was then introduced,

such that the pore occupancy of each pore was kept as described above if thpf ≤ , but

the phase occupancy was reversed (i.e., wetting phase in the larger pores, etc.) if

thpf > . This led to a more accurate prediction of the end points, as well as more

accurate prediction of the actual values of the curves. A range of thresholds was tested,

and it appeared that a value of 6.0=thp yielded the best results. Finally, the presence of

the “irreducible” water saturation was taken into account. Indeed, there is always some

amount of water trapped in the pore space that does not necessarily contribute to the

flow, meaning that the lowest Sw experimental value will not be zero. We simply

accounted for that volume by normalising our water saturation to match our zero-point

to the experimental “irreducible” saturation.

This methodology was firstly tested with Berea sandstone, before being applied to

one of the carbonate sample (ME2-2). The results obtained for the relative permeability

predictions in both cases are very encouraging. The shape of the curves is predicted

fairly well, although it appears harder to match the wetting phase relative permeability

curve. A critical exponent of 6.1=t appear to give the best results for the non-wetting

phase, while, for the wetting phase, a value of 2=t gives the best fit to the

experimental data. It is obviously difficult to draw clear conclusions from this model,

because it was only tested on two samples. Therefore, a more thorough study with a

159

larger database is recommended, so that the method, as well as the values for both the

probability threshold and the critical exponent, can be tested further.

At present, because of the assumptions used for the model, it is only possible to

simulate the primary drainage relative permeability curves. Of course, it would be more

valuable to have the primary imbibition relative permeability curves, because this

corresponds to the case where the water moves back into the pore space after some of

the oil has been produced; this is generally the kind of data that is needed in reservoir

simulation. A more sophisticated water-wet model, which allows for the presence of a

water film in all the pores, regardless of their size, might improve the results presented

here. It could also help simulate the imbibition process, and possibly generate the

relative permeability curves for this process. It would, however, be necessary to work

out how to determine the volume of oil left behind. The inclusion of more data, such as

wettability data and capillary curves, could help to do so. However, in order to keep the

computational procedure down to a reasonable time, a correct balance will have be

found between the need for more data, and any incremental improvement in the final

results.

160

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APPENDICES: CODE LISTING

A Effective-Medium Approximation (Newton-Raphson), modified from Lock (2001)

B Effective-Medium Approximation (Bisection), modified from Lock (2001)

C SEM images of the rocks used during the research

D Excel template for 2-phase relative permeability computation

Programs A and B were written and compiled using DIGITAL visual Fortran for

Windows, version 5a.

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A Effective-Medium Approximation (Newton-Raphson)

PROGRAM NEWTONKIRKPATRICK C FINDS THE ROOT OF KIRKPATRICK'S EQUATION USING THE NEWTON- C RAPHSON PROCEDURE C COMPUTES THE REQUIRED DERIVATIVE ANALYTICALLY C USES DOUBLE PRECISION ARITHMETIC TO ALLOW MORE STRINGENT AND C CONSISTENT CONVERGENCE CRITERION

IMPLICIT NONE REAL G, L, Z, ALPHA, GMAX, RMAX, GMIN DOUBLE PRECISION A, B, SQ, SER, DER, FX, RATIO, SUM, GGEO, X DIMENSION G(10000) INTEGER I, J, K PARAMETER(L=7397.) PARAMETER(Z=6) K=INT(L)

C OPEN EXTERNAL DATAFILE AND READ IN THE CONDUCTIVITIES

OPEN(UNIT=3,FILE='1YCOMPILED.txt', STATUS='OLD') REWIND(3) DO 10 I=1, K READ(3,*) G(I)

10 END DO C CALCULATE BOUNDS FOR GEFF, ALONG WITH THE GEOMETRIC MEAN, Ggeo

Ggeo = 1.0 Gmax = 0.0 Rmax = 0.0 DO 20 I=1, K Ggeo = Ggeo*(G(I))**(1/L) Gmax = Gmax + G(I) Rmax = Rmax + 1.0/G(I)

20 CONTINUE Gmax = Gmax/L Rmax = Rmax/L Gmin = 1.0/Rmax

PRINT*, 'NO OF INPUT CONDUCTANCES:' , K PRINT*,'Gmin=', Gmin PRINT*,'Ggeo=', Ggeo PRINT*,'Gmax=', Gmax

ALPHA=(Z/2)-1

C INITIALISE OUR GUESS FOR X X = GGEO C DO LOOP FOR N-R ITERATION

DO 30 J=1, 200 C INITIALISE FUNCTION VALUES

FX = 0.0 SUM= 0.0

C DO LOOP FOR CALCULATING FUNCTION VALUES DO 40 I=1, K

FX = FX + (X-G(I))/(ALPHA*X+G(I)) C COMPUTE DERIVATIVE USING QUOTIENT RULE AND SUM

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A=(ALPHA*X+G(I)) B=ALPHA*(X-G(I)) SER=A-B SQ=(ALPHA*X+G(I))**2 DER=SER/SQ SUM=SUM+DER

40 CONTINUE RATIO=FX/SUM

C UPDATE SOLUTION FOR ROOT X = X – RATIO

C TEST TO SEE IF X HAS JUMPED OUTSIDE OF RANGE IF((GMIN-X)*(X-GMAX) .LT.0) THEN PAUSE 'X JUMPED OUT OF BRACKETS' ENDIF C TEST(S) FOR CONVERGENCE

IF(ABS(RATIO) .LT. 0.0000001 .AND. ABS(FX) .LT. 0.0000001)THEN

C COMPUTES PERMEABILITY (micron2) AND ACCOUNTS FOR THE PORE NUMBER DENSITY P = (L*X)/(1.47*S)

C CONVERTS K INTO mD P = P*1.013171226*1000 PRINT*,'N-R HAS CONVERGED TO', X PRINT*, 'Permeability = ', P,'mD' STOP ENDIF

30 CONTINUE END

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B Effective-Medium Approximation (Bisection) PROGRAM ROOTKIRK C FINDS THE ROOT TO KIRKPATRICK'S EQUATION USING THE BISECTION C PROCEDURE C USES THE HARMONIC AND ARITHMETIC MEANS AS BOUNDS TO BRACKET C THE ROOT IMPLICIT REAL (A-H, M-Z) DIMENSION G(10000) PARAMETER (L=7397) PARAMETER (S=2541) C OPEN EXTERNAL DATAFILE AND READ IN THE CONDUCTIVITIES OPEN(UNIT=5,FILE='1ycompiled30.txt', STATUS='old') REWIND(5) DO 10 I=1, L READ(5,*) G(I) 10 END DO C CALCULATE BOUNDS FOR GEFF Gmax = 0.0 Rmax = 0.0 DO 20 i=1, L Gmax = Gmax + G(i) Rmax = Rmax + 1.0/G(i) 20 CONTINUE Gmax = Gmax/L Rmax = Rmax/L Gmin = 1.0/Rmax C BISECTION fmid = 0.0 DO 30 i=1,L sum = sum + (Gmax-G(i))/(2*Gmax+G(i)) 30 CONTINUE fmid=sum root = Gmin dx = Gmax-Gmin C NOW DO THE BISECTION LOOP Kmax = 50 DO 40 k=1,Kmax dx = dx*0.5 Gmid = root + dx fmid = 0.0 DO 50 I=1,L fmid = fmid + (Gmid-G(i))/(2*Gmid+G(i)) 50 END DO IF(fmid .LE. 0.0) root = Gmid 40 END DO Ceff = ROOT C COMPUTES PERMEABILITY (microns2) AND ACCOUNTS FOR THE PORE NUMBER

DENSITY P= (L*Ceff)/(1.47*S) C CONVERTS K IN mD

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P= P*1.013171226*1000 PRINT *, 'Ceff =', Ceff PRINT*,Gmin,Gmax PRINT*,'Permeability = ', P,'mD' END

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C SEM images of the rocks used during the research

Figure C.1. Low magnification SEM image of the Orgon sample (SEF-1).

Figure C.2. Low magnification SEM image of the Belvedere sample (SEF-2).

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Figure C.3. Low magnification SEM image of the Rustrel sample (SEF-3).

Figure C.4. Low magnification SEM image of sample ME1-1.

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Figure C.5. Low magnification SEM image of sample ME1-2.

Figure C.6. Low magnification SEM image of sample ME1-3.

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Figure C.7. Low magnification SEM image of sample ME1-4.

Figure C.8. Low magnification SEM image of sample ME1-5.

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Figure C.9. Low magnification SEM image of sample ME1-6.

Figure C.10. Low magnification SEM image of sample ME2-1.

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Figure C.11. Intermediate magnification SEM image of sample ME2-2.

Figure C.12: Intermediate magnification SEM image of sample ME2-3.

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D Excel template for two-phase relative permeability computation

The CD contains an example of the Excel program used for the generation of the

relative permeability curves in Chapter 7. The only input data needed are the areas and

perimeters of the pores, which can be extracted using image analysis, and the total area

of the image. At the various saturations level (say, every 5 or 10%), the user has to

manually create the appropriate pore conductance distributions by taking the original

distribution, i.e., with all the pores, and “blocking” the necessary pores. This is done by

inputting zero conductances in place of the original values. For example, at a water

saturation of 90%, the pores that represent 10% of the area should be replaced by zeros.

Once this is done, the probability threshold formulas need to be matched to the new

conductance distribution. This is because there are two different formulas depending on

the phase, one for the wetting-phase and one for the non-wetting phase. These formulae

are in the Excel sheet provided, and simply need to be copied and pasted to correspond

to the new distribution. Finally, the user needs to input the correct size of the image (in

μm2) in the cell where the relative permeability is computed (G5801, M5801, T5801,

etc., on the provided file). The normalised relative permeabilities are finally computed

at the top far right end of the sheet.

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