PREDICTING ABSOLUTE AND RELATIVE PERMEABILITIES OF

182 Pages·2007·6.63 MB·English

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
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.

2
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
37
2.6.2 Cerepi et al. ……………………………...…..……………………...……....
3 REVIEW OF THE METHOD OF SCHLUETER AND LOCK……………….…….. 39
3
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
4
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
5
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
6
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

7
LIST OF FIGURES

Figure 2.1 Cylindrical element of fluid during laminar flow through a 23
circular pipe. Force balance on the element yields eq. (2.5).
Figure 2.2 A plot of absolute difference between the outputs obtained for 35
simulations on ISONETSIM and Kirkpatrick’s equation as a
function of lattice size at 20-30 realisations.
Figure 3.1 An illustration showing the replacement of a discrete network of 41
conductances by a homogeneous network having the same
topology.
Figure 3.2 Illustration showing how the area of a slice through a single pore 44
at some arbitrary angle θ will generally be larger than the true
cross-sectional area.
Figure 3.3 Constriction factor for hydraulic conductivity as function of the 47
ratio of the minimum pore radius to the maximum pore radius of
an individual pore.
Figure 3.4 Constriction factor f as a function of a /a for saw-tooth and 52
min max
sinusoidal profiles.
Figure 3.5 A two-dimensional illustration showing the number of additional 53
pores bisected when a slice is taken at an angle θ that is not
perpendicular to a lattice direction.
Figure 3.6 Measured permeabilities plotted against predicted values for both 56
UKCS and St. Bees data. The upper and lower lines correspond to
errors of a factor of two in either direction.
Figure 4.1 Construction used in calculating the “pressure” induced across one 58
conductance, C , surrounded by a uniform medium (after
m
Kirkpatrick, 1973).
Figure 4.2 Kirkpatrick’s function for the twenty-one conductance values used 64
8
by Priest (1992), plotted for the case z =6. Note that Priest’s
definition of C has the factor of μL incorporated into it.
Figure 4.3 Sensitivity of C (mm2 s-1) to co-ordination number, using 64
eff
Priest’s data set.
Figure 4.4 Bond percolation on a square network at bond occupancies of 66
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, p , for the square network is exactly 0.5
c
(Sahimi, 1995).
Figure 4.5 Comparison of normalised permeabilities at co-ordination number 69
ranging from 1.5 to 6 for both NETSIM and the EMA, both
obtained while artificially lowering the co-ordination number
using zero conductances.
Figure 4.6 Comparison of normalised permeabilities at co-ordination number 70
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).
Figure 4.7 Comparison between the NETSIM prediction and the scaling law, 73
near the percolation threshold, presented here with three different
critical exponents t.
Figure 4.8 Comparison between the NETSIM predictions and the scaling 74
law, over the entire range of bond probabilities, for three different
critical exponents, t.
Figure 4.9 Comparison of NETSIM, Kirkpatrick’s EMT and the GPA for 78
log-normal conductivity distributions (after Lock et al., 2004).
Figure 5.1 Illustration of the Dunham classification (Dunham, 1962). 81
Figure 5.2 Low magnification image from the Belvedere sample (SEF-2), a 86
very fine peloidal grainstone. The presence of a slightly leached
benthic foraminifer is indicated on the plate (B).
9
Figure 5.3 Figure 5.3. Low magnification image from the Rustrel sample 87
(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.
Figure 5.4 (i) General view of sample ME1-1, at ×45 magnification. The 90
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.
Figure 5.4 (ii) A closer view of the sample ME1-1 (×113 magnification) with the 90
presence of oversized pores (O) confirming the effects of leaching
on the sample.
Figure 5.5 (i) Overall view of sample ME1-4 (×70 magnification) with the 93
presence of partially dissolved benthic foraminifers (B) and
dissolution vugs (V), due to leaching, as well as calcite-cemented
areas (C).
Figure 5.5 (ii) Closer view (×280 magnification) of the macro-pores system with 93
the presence of an oversized pore (O) following the leaching
process.
Figure 5.5 (iii) Illustration of more calcite-cemented areas (C) and oversized 93
pores (O), at the same magnification of the previous plate (×280).
This also illustrates the fairly good connectivity of the porosity
system.
Figure 5.5 (iv) Illustration of the irregular-shaped faces of calcite crystals 93
surrounding a peloidal grain. This probably indicates re-
crystallisation process, which could have enhanced microporosity
networking.
Figure 6.1 Illustration of the thresholding procedure on sample ME1-5, from 99
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
10