university of calgary drill cuttings, petrophysical, … · methodology developed in this thesis...
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UNIVERSITY OF CALGARY
Drill Cuttings, Petrophysical, and Geomechanical Models for Evaluation of Conventional
and Unconventional Petroleum Reservoirs
by
Bukola Korede Olusola
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL
FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF
SCIENCE
DEPARTMENT OF CHEMICAL AND PETROLEUM ENGINEERING
CALGARY, ALBERTA
SEPTEMBER, 2013
© Bukola Korede Olusola 2013
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ABSTRACT
This thesis concentrates on some of the aspects of a ‘Total Petroleum System’ including
natural gas and oil stored in conventional and unconventional reservoirs.
The original contributions of this thesis include:
1) The use of electromagnetic mixing rules for construction of dual and triple
porosity models with a view to quantifying matrix, fracture and non-effective porosity
and the porosity exponent (m) of naturally fractured reservoirs.
2) The use of electromagnetic mixing rules for construction of dual and triple
porosity models with a view to quantify the water saturation exponent (n) and to estimate
the wettability of reservoir rocks in naturally fractured reservoirs.
3) Measurements of porosity and permeability in drill cuttings collected directly in a
horizontal well. Although these measurements have been carried out previously in drill
cuttings collected in vertical and deviated wells, this is the first time that they are
performed in horizontal well drill cuttings.
The models developed in Items 1 and 2 are compared successfully against core laboratory
data. Water and/or oil stored in each of the porous media considered in the models,
affects rock wettability and consequently the values of n. Robustness of the models is
important because, in practice, while logging a well in a naturally fractured reservoir, the
tool will probably go through some intervals with only matrix porosity; some intervals
with matrix porosity and fractures, some with matrix porosity and isolated porosity; and
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some intervals with matrix, fractures and non-connected porosities. As there are
variations in the contribution of each porosity system with depth, there are also variations
in m with depth that have to be taken into account.
The laboratory measurements of porosity and permeability from drill cuttings mentioned
above in Item 3 were conducted at the University of Calgary. Based on a thorough review
of literature this is the first time that this measurements are conducted in drill cuttings
samples collected in horizontal wells.
Starting with only drill cuttings measurements of porosity and permeability, the
methodology developed in this thesis allows for complete formation evaluation and
geomechanical analysis through the use of a successive approach for determination of
several parameters of interest including pore throat aperture radius (rp35), water
saturation, porosity exponent (m), true formation resistivity, capillary pressure, Knudsen
number, depth to the water contact (if present), construction of Pickett plots, Young’s
Modulus, Poisson’s ratio and brittleness index throughout the horizontal length of the
well, and for locating the best hydraulic fracture initiation points during multi-stage
fracturing jobs.
It is concluded that the use of electromagnetic mixing rules and drill cuttings provide a
valuable and practical addition for quantitative characterization of conventional and
unconventional petroleum reservoirs.
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ACKNOWLEDGEMENT
I will like to specially thank Dr. Roberto Aguilera for giving me an opportunity to join
the GFREE (Geoscience, Formation Evaluation, Reservoir Drilling, Completion and
Stimulation, Reservoir Engineering, Economics and Externalities) research team at the
University of Calgary. I sincerely appreciate all his efforts in guiding and supervising my
work and also ensuring the successful completion of my thesis. I feel privileged to have
had Dr. Roberto Aguilera as my supervisor on this research.
Parts of this work were funded by the Natural Sciences and Engineering Council of
Canada (NSERC agreement 347825-06), ConocoPhillips (agreement 4204638), Alberta
Innovates Energy and Environmental Solutions (AERI agreement 1711), the Schulich
School of Engineering at the University of Calgary and Servipetrol Ltd. Darcylog
equipment for measuring permeabilities from drill cuttings was provided by Roland
Lenormand of Cydarex in Paris. GOHFER software was provided by Barree &
Associate. Their contributions are gratefully acknowledged. Also, special thanks to past
and present GFREE research team members at the University of Calgary for their
consistent academic support and advice.
I also specially thank my wife, daughter and family for their moral support.
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DEDICATION
I dedicate this achievement to my family in Calgary and Nigeria especially to my wife,
daughter, parents and sisters.
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TABLE OF CONTENTS
Abstract ............................................................................................................................... ii
Acknowledgement ............................................................................................................. iv
Dedication ............................................................................................................................v
Table of Contents ............................................................................................................... vi
List of Tables ..................................................................................................................... ix
List of Figures and Illustrations ......................................................................................... xi
List of Symbols, Abbreviations and Nomenclature………………………………….….xix
Chapter One: INTRODUCTION .........................................................................................1
1.1 JUSTIFICATION ..........................................................................................................1
1.2 OBJECTIVES ................................................................................................................5
1.3 GEOLOGICAL BACKGROUND OF STUDY AREA ................................................5
1.3.1 Study Area ..............................................................................................................8
1.4 RESEARCH APPROACH ............................................................................................9 1.4.1 Laboratory Work .....................................................................................................9
1.4.2 Analytical Model Development ..............................................................................9 1.4.3 Petrophysical and Geomechanical Evaluation ......................................................10
1.4.4 Hydraulic Fracturing and Simulation ...................................................................10
1.5 TECHNICAL PUBLICATIONS .................................................................................12
Chapter Two: LITERATURE REVIEW ...........................................................................14
2.1 OVERVIEW ................................................................................................................14
2.1.1 Drill Cuttings ........................................................................................................14 2.1.2 Porosity Exponent (m) ..........................................................................................19 2.1.3 Water Saturation Exponent (n) .............................................................................24 2.1.4 Petrophysical and Geomechanical Evaluation ......................................................28 2.1.5 Hydraulic Fracturing .............................................................................................32
Chapter Three: METHODOLOGY ...................................................................................39
3.1 LABORATORY WORK .............................................................................................39
3.1.1 Sample Collection .................................................................................................40 3.1.2 Microscopic Analysis ...........................................................................................41 3.1.3 Measurable Sample Collection .............................................................................43 3.1.4 Cleaning, Drying, and Weighting of Sample ........................................................45 3.1.5 Porosity Measurement ..........................................................................................46
3.1.6 Screening of Samples Prior to Permeability Measurement ..................................53 3.1.7 Permeability Measurement ...................................................................................53
Chapter Four: POROSITY EXPONENT...........................................................................58
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4.1 THE CONCERNS .......................................................................................................58
4.2 USE OF ELECTROMAGNETIC UNIFIED MIXING RULE FOR BUILDING
DUAL AND TRIPLE POROSITY MODELS .........................................................59 4.2.1 Maxwell Garnett Electromagnetic Mixing Rule ...................................................61 4.2.2 Bruggeman Electromagnetic Mixing Rule ...........................................................66
4.2.3 Coherent Potential Electromagnetic Mixing Rule ................................................68
4.3 MODEL DEVELOPMENT .........................................................................................69 4.3.1 Derivation of Maxwell Garnett Mixing Rule Extension to Matrix and Non-
Connected Vugs .....................................................................................................69 4.3.2 Maxwell Garnett Mixing Rule Extension to Matrix and Fractures ......................71
4.3.3 Bruggeman Mixing Rule Extension to Matrix and Non-Connected Vugs ...........72 4.3.4 Coherent Potential Mixing Rule Extension to Matrix and Non-Connected
Vugs .......................................................................................................................74
4.4 MODEL VALIDATION .............................................................................................75
4.4.1 Comparison of Available Models .........................................................................75 4.4.2 Comparison with Core Data .................................................................................83
Chapter Five: WATER SATURATION EXPONENT(n) .................................................91
5.1 THE CONCERNS .......................................................................................................91
5.2 THEORETICAL MODELS.........................................................................................91
5.2.1 Dual Porosity (Matrix and Isolated Porosity) .......................................................93
5.2.2 Dual Porosity (Matrix and Fracture Porosity) ......................................................97 5.2.3 Triple Porosity (Matrix, Isolated and Fracture Porosity) ....................................100
5.3 MODEL VALIDATION ...........................................................................................101
5.3.1 First Step: Single Porosity Reservoirs (matrix porosity) ....................................102 5.3.2 Second Step: Dual Porosity Reservoirs (Matrix and Isolated Porosity or
Matrix and Fracture Porosity) ..............................................................................111 5.3.3 Third Step: Triple Porosity Reservoirs (Matrix, Isolated and Fracture
Porosities) ............................................................................................................118 ..........................................................................................................................................121 Chapter Six: PETROPHYSICAL AND GEOMECHANICAL EVALUATION ............122
6.1 OVERVIEW ..............................................................................................................122
6.2 PETROPHYSICAL EVALUATION BASED ON DRILL CUTTINGS ..................122
6.2.1 Case Study: Drill Cuttings Collected in Horizontal Well ...................................122 6.2.2 Porosity ...............................................................................................................124 6.2.3 Permeability ........................................................................................................125 6.2.4 Pore Throat Aperture Radii .................................................................................125
6.2.5 Porosity Exponent ( ) ........................................................................................127 6.2.6 Irreducible Water Saturation ...............................................................................128
6.2.7 True Formation Resistivity .................................................................................131
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6.2.8 Capillary Pressure ...............................................................................................132 6.2.9 Distinguishing Between Viscous and Diffusion-Like Flow ...............................133 6.2.10 Location of Water Contact ................................................................................135 6.2.11 Flow (or Hydraulic) Units .................................................................................136 6.2.12 Construction of Pickett Plots ............................................................................137
6.3 GEOMECHANICS BASED ON DRILL CUTTINGS .............................................145 6.3.1 Brittleness Index .................................................................................................146
Chapter Seven: HYDRAULIC FRACTURE DESIGN OPTIMIZATION USING DRILL
CUTTINGS ......................................................................................................................157
7.1 HYDRAULIC FRACTURING OF TIGHT GAS RESERVOIRS ............................157
7.1.1 Hydraulic Fracture Design Using Drill Cuttings ................................................158
7.2 MODEL DEVELOPMENT .......................................................................................161 7.2.1 Data Input and Processing ..................................................................................161 7.2.2 Calibration of GOHFER Generated Data with Drill Cuttings Data ...................163
7.2.3 Multi-Stage Hydraulic Fracture Treatment Design ............................................169
7.3 RESULTS ..................................................................................................................171
Chapter Eight: CONCLUSION AND RECOMMENDATIONS ....................................175
8.1 DRILL CUTTINGS ...................................................................................................175
8.2 POROSITY EXPONENT (M) ...................................................................................175
8.3 WATER SATURATION EXPONENT (N) ..............................................................177
8.4 PETROPHYSICAL AND GEOMECHANICAL EVALUATION ...........................178
8.5 HYDRAULIC FRACTURING AND MODELING .................................................179 References ........................................................................................................................181
APPENDIX A: SCREEN SHOTS OF PERMEABILITY MEASUREMENTS USING
DARCYLOG ...................................................................................................................196
APPENDIX B: RELEVANT EQUATIONS ……………………….……………...…..225
APPENDIX C: ANGLE BETWEEN FRACTURES AND DIRECTION OF CURRENT
FLOW………………………………………………………………………..…….…...226
APPENDIX D: WELL CONFIGURATION FOR INITIAL DESIGN………………...227
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LIST OF TABLES
Table 3.1— FRAC VALUE ESTIMATION AT A GIVEN DEPTH FOLLOWING
MICROSCOPIC ANALYSIS OF DRILL CUTTINGS (Adapted from Ortega
and Aguilera, 2012) .................................................................................................. 42
Table 3.2— RESULTS OF LABORATORY WORK ON DRILL CUTTINGS
(DETERMINATION OF POROSITY) .................................................................... 52
Table 3.3— RESULTS OF LABORATORY WORK ON DRILL CUTTINGS
(DETERMINATION OF PERMEABILITY) .......................................................... 57
Table 5.1—RESULTS FOR SINGLE POROSITY RESERVOIR WHEN ISOLATED
POROSITY IS EQUAL TO ZERO AND MATRIX POROSITY IS EQUAL TO
THE TOTAL POROSITY ...................................................................................... 104
Table 5.2—RESULTS FOR SINGLE POROSITY RESERVOIR WHEN
FRACTURE POROSITY IS EQUAL TO ZERO AND MATRIX POROSITY IS
EQUAL TO THE TOTAL POROSITY ................................................................. 105
Table 5.3—RESULTS FOR SINGLE POROSITY RESERVOIR WHEN ISOLATED
POROSITY IS EQUAL TO ZERO AND MATRIX POROSITY IS EQUAL TO
THE TOTAL POROSITY ...................................................................................... 107
Table 5.4— RESULTS FOR SINGLE POROSITY RESERVOIR WHEN
FRACTURE POROSITY IS EQUAL TO ZERO AND MATRIX POROSITY IS
EQUAL TO THE TOTAL POROSITY ................................................................. 110
Table 5.5— RESULTS FOR DUAL POROSITY RESERVOIR WITH MATRIX
AND ISOLATED POROSITY ............................................................................... 113
Table 5.6 — RESULTS FOR DUAL POROSITY RESERVOIR WITH MATRIX
AND ISOLATED POROSITY ............................................................................... 114
Table 6.1— PETROPHYSICAL DATA FOR WESTERN CANADA
SEDIMENTARY BASIN TIGHT GAS SANDSTONE. THE HORIZONTAL
WELL POROSITY AND PERMEABILITY DATA FROM DRILL CUTTINGS
(COLUMN 2 & 3) ARE OBTAINED FROM LABORATORY WORK AND IT
IS USED AS A STARTING POINT IN DETERMINING OTHER
PETROPHYSICAL DATA (COLUMN 5 TO 11) USING EMPIRICAL
EQUATIONS. ......................................................................................................... 123
Table 6.2— GEOMECHANICAL DATA FOR WESTERN CANADA
SEDIMENTARY BASIN TIGHT GAS SANDSTONE. THE HORIZONTAL
WELL POROSITY AND PERMEABILITY DATA FROM DRILL CUTTINGS
(COLUMN 3) ARE OBTAINED FROM LABORATORY WORK AND IT IS
USED AS A STARTING POINT IN DETERMINING OTHER
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PETROPHYSICAL DATA (COLUMN 4 TO 12) USING EMPIRICAL
EQUATIONS. ......................................................................................................... 148
Table 7.1—BOTTOM HOLE PUMP SCHEDULE ....................................................... 169
Table 7.2— SIMULATION OUTPUT FROM INITIAL MODEL DESIGN BASED
ON SYMMETRICAL DISTANCES FOR HYDRAULIC FRACTURE
INITIATION. .......................................................................................................... 173
Table 7.3— SIMULATION OUTPUT FROM RE-DESIGN MODEL BASED ON
HYDRAULIC FRACTURE INITIATIONS FROM THE CUTTINGS-BASED
CUT-LOG. .............................................................................................................. 174
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LIST OF FIGURES
Figure 1.1— Distribution of world proved oil reserves in 1992, 2002, and 2012
(Adapted from BP Statistical Review of World Energy. June, 2013). ....................... 1
Figure 1.2 — Distribution of world proved natural gas reserves in 1992, 2002, and
2012 (Adapted from BP Statistical Review of World Energy. June, 2013). .............. 2
Figure 1.3—Location of western Canada sedimentary basin and other major
sedimentary basins in North America (Zaitlin and Moslow, 2006). ........................... 6
Figure 1.4—Map of western Canada sedimentary basin (Adapted from Moslow,
University of Calgary Presentation, 2013). ................................................................. 6
Figure 1.5— Stratigraphic framework of the Nikanassin group (Adapted from Stott,
1998; Miles, 2010; and Solano, 2010). ....................................................................... 7
Figure 1.6—Typical wireline log section through the Nikanassin group in the study
area 66-0W6 (Miles, 2010). ........................................................................................ 8
Figure 1.7— Location of Study Area (Red Rock Field) shaded in Light Blue (Adapted
from Solano, 2010). .................................................................................................... 9
Figure 2.1— Example of the use of drill cuttings to analyse shale caving types and
wellbore stability prediction (source: www.geomi.com). ......................................... 16
Figure 2.2—Example of Cut-Log showing the brittleness index, permeability and
Frac-Value (Adapted from Ortega and Aguilera, 2012). .......................................... 18
Figure 2.3—Main sources of data concerning sub-surface rocks (Source: Dr. Paul
Glover, Petrophysics MSc course notes, University of Laval, Quebec, Canada,
2013) ......................................................................................................................... 29
Figure 2.4 —Example of a non-fractured reservoir (upper schematic) and a fractured
reservoir (lower schematic). The upper schematic represents radial fluid flow
(red arrows) into the well (circle) from a smaller area of the reservoir. The lower
schematic represents a hydraulically fractured reservoir that allows linear fluid
flow (red arrows) from a larger area of the reservoir into the fracture and finally
into the wellbore. This leads to an increase in recovery (API, 2009). ...................... 33
Figure 2.5—Example of an open rock with well-rounded ceramic proppants keeping
the fracture open (Adapted from thorsoil.com). ....................................................... 34
Figure 2.6—Example of a multistage hydraulic fracturing operation. Halliburton’s
Swell packer systems isolating various zones of a horizontal wellbore (open
hole) that will be stimulated (Adapted from www.egyptoil-gas.com). .................... 35
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Figure 2.7‒ Example of horizontal well contacting a large area of the reservoir layer.
The vertical well in the right can only contact a smaller area of the reservoir
(Adapted from API, 2009). ....................................................................................... 36
Figure 3.1—Example of drill cuttings under a 10x microscope (Source:
http://en.wikipedia.org/wiki/Drill_cuttings, 2013 ). ................................................. 39
Figure 3.2—Labeled (well name and measured depth) vials containing drill cutting
samples. ..................................................................................................................... 40
Figure 3.3— Identifying structural features through drill cuttings. (a)The upper right
corner shows a planar (flat surface), and unmineralized drill cuttings sample. (b)
The lower right shows drill cutting samples with mineral infill or linings. (c)
Upper left shows loose, clear, euhedral quartz crystal in sandstone. (d) Lower
left shows a fracture set with mineral infill (Adapted from Hews, 2011). ............... 41
Figure 3.4—Chart for visual estimation of percentages of minerals in rock sections.
The interpretation is extended in this work to represent sizes of drill cutting
samples (Terry and Chillingar, 1955). ...................................................................... 43
Figure 3.5—Material safety data sheet for Toluene ......................................................... 46
Figure 3.6—Apparatus for measuring saturated weight of drill cuttings (Adapted from
Ortega, 2012). ........................................................................................................... 47
Figure 3.7—Apparatus for measuring immersed weight based on Archimedes
principle. The Immersed weight is used to determine the bulk volume of the drill
cuttings using Eq. 3.2 (Adapted from Ortega, 2012). ............................................... 50
Figure 3.8— The left side shows the diagram of the spring and bellow system while
the right side shows the Darcylog Equipment (Adapted from Lenormand and
Fonta, 2007). ............................................................................................................. 54
Figure 4.1—Schematic of mixture for spherical inclusions with permittivity εi that
occupy random positions in a host environment of permittivity εe. The mixture
effective permittivity is εeff (Sihvola, 1999). ............................................................. 59
Figure 4.2— Chart for determining m as a function of non-connected vug porosity
(nc) or fracture porosity (2) for the case in which mb= 2.0. Petrophysical model
is developed on the basis of the Electromagnetic Mixing Rule (Maxwell Garnett,
vs = 0). ....................................................................................................................... 64
Figure 4.3— Chart for determining m as a function of non-connected vug
porosity(nc) and fracture porosity(2) for the case in which mb = 2.0.
Petrophysical model developed on the basis of the Unified Electromagnetic
Mixing Rule (Maxwell Garnett, vs = 0). ................................................................... 66
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Figure 4.4— Chart for determining m as a function of non-connected vug
porosity(nc) for the case in which mb = 2.0. Petrophysical model developed on
the basis of the Unified Electromagnetic Mixing Rule (Bruggeman, vs = 2). .......... 67
Figure 4.5—Chart for determining m as a function of non-connected vug porosity(nc)
for the case in which mb = 2.0. Petrophysical model developed on the basis of
the Unified Electromagnetic Mixing Rule (Coherent Potential, vs = 3). .................. 69
Figure 4.6—Comparison chart for determining m as a function of non-connected vug
porosity (nc) for the case in which mb = 2.0. Petrophysical model developed on
the basis of the Unified Electromagnetic Mixing Rule -Maxwell Garnett( vs = 0),
Bruggeman (vs = 2) and Coherent Potential (vs = 3). ................................................ 76
Figure 4.7— Comparison of dual porosity models made out of matrix and fractures or
matrix and non-connected vugs from various theories show good agreement.
Porosity exponent of the matrix mb = 2.0. Maxwell Garnet and Aguilera’s
equations for calculating m are explicit. Berg’s solution uses an iteration
procedure and for the vugs case assumes an infinite value of mv (in reality it
assumes mv = 1E35 in a spread sheet for the above curves in the right hand side
of the graph). ............................................................................................................. 78
Figure 4.8—Comparison of models based on the Unified Electromagnetic Mixing
Rule (Maxwell Garnett, vs = 0) represented by the solid lines and Berg effective
medium theory (x symbol). Comparison is made for naturally fractured
reservoirs and reservoirs with vugs (mb = 2.0, mv =1.5, mf =1.0). According to
Berg, small values of mv might be indicative of connected (or partially
connected) vugs. ....................................................................................................... 79
Figure 4.9—Comparison of models based on the Electromagnetic Mixing Rule, vs = 0
(solid lines) and Aguilera fracture dip model, θ = 90°- fracture dip (x symbol).
Comparison is made for naturally fractured reservoirs (mb = 2.0, fracture dip,
0°). ............................................................................................................................. 80
Figure 4.10—Comparison of triple porosity models based on the Electromagnetic
Mixing Rule, vs = 0 (solid lines) and Berg’s effective medium theory (x symbol).
Comparison is good (mb = 2.0, mv =1.5, mf =1.0, θ’ = 90°). ..................................... 81
Figure 4.11—Comparison of models based on the Electromagnetic Mixing Rule, vs =
2 (solid lines) and Berg’s effective medium theory (x symbol). Comparison is
for reservoirs with non-connected vugs (mb = 2.0, mv =1.5). ................................... 82
Figure 4.12—Comparison of models based on the Electromagnetic Mixing Rule, vs =
3 (solid lines) and Berg’s effective medium theory (x symbol). Comparison is
for reservoirs with non-connected vugs (mb = 2.0, mv =1.5). ................................... 83
xiv
Figure 4.13—Comparison of limestone core data from well A (Ragland, 2002) and
dual porosity models made out of matrix and fractures developed by Aguilera
and Aguilera (2003), Berg (2006) and this study (Maxwell Garnett, vs = 0).The
open red circles show the absolute error as compared with core data. The mean
absolute percentage error (MAPE) is 1.84%. ............................................................ 85
Figure 4.14—Comparison of limestone core data from well B (Ragland, 2002) and
dual porosity models made out of matrix and fractures developed by Aguilera
and Aguilera (2003), Berg (2006) and this study(Maxwell Garnett, vs = 0). The
blue data point enclosed in a red square corresponds to a sample where
approximately 64% of the pore system is dominated by non-connected moldic
pores (Ragland 2002). The MAPE is 7.15% ............................................................. 86
Figure 4.15—Comparison of limestone core data from well J (Ragland, 2002) and
dual porosity models made out of matrix and fractures developed by Aguilera
and Aguilera (2003), Berg (2006) and this study(Maxwell Garnett, vs = 0). The
blue data point enclosed in a red square with m = 1.63 shows an error in the
order of 20%. This sample is characterized by a connected moldic pore system
that reduces significantly the value of m (Ragland 2002). The MAPE is equal
7.40%. ....................................................................................................................... 87
Figure 4.16—Comparison of dolomite core data from well C (Ragland, 2002) and
dual porosity models made out of matrix and fractures developed by Aguilera
and Aguilera (2003), Berg (2006) and this study(Maxwell Garnett, vs = 0). The
error is generally less than 10%. The MAPE is 4.91%. ............................................ 88
Figure 4.17—Comparison of dolomite core data from well E (Ragland, 2002) and
dual porosity models made out of matrix and fractures developed by Aguilera
and Aguilera (2003), Berg (2006) and this study (Maxwell Garnett, vs = 0). The
largest errors correspond to data points enclosed in red squares. These are
dominated by either connected and interparticle pore system (lower values of m)
or non-connected moldic pore systems (larger values of m) (Ragland 2002). The
MAPE is equal to 7.67%. .......................................................................................... 89
Figure 4.18—Comparison of tight gas sandstone core data from wells in the
Mesaverde formation (Byrnes et al., 2006) and triple porosity models made out
of matrix, fractures and slots, and non-connected porosity developed by Al-
Ghamdi et al. (2011), Berg (2006) and this study(Maxwell Garnett, vs = 0). The
error is generally less than 10%. The MAPE is equal to 3.58% .............................. 90
Figure 5.1— Comparison of Eq. 5.19 for matrix and isolated porosity using nc = 0
and preferentially oil-wet core data published by Sweeney and Jennings (1960).
The blue diamonds represent core data, the black open diamonds represent other
data from the dual porosity model used to match the core data. All the core data
points are matched. The red squares correspond to data calculated for illustration
purposes in Table 5.3 for samples 1 to 5. ............................................................... 108
xv
Figure 5.2— Comparison of Eq. 5.33 for matrix and fractures using 2 = 0 and
preferentially water-wet core data published by Sweeney and Jennings (1960).
The blue diamond’s represent core data, the black open diamonds represent other
data from the dual porosity model used to match the core data. All the core data
points are matched. The red squares correspond to data calculated for illustration
purposes in Table 5.4 for samples 1 to 5. ............................................................... 111
Figure 5.3— Graph of versus developed with the use of the dual porosity
model for matrix and isolated porosity (Eq. 5.17). The chart shows the effects of
increasing isolated porosity (PHINC) on while keeping matrix porosity
constant at b = 0.315 Matrix and composite system (vugs+matrix) are oil wet. .. 113
Figure 5.4— Graph of versus developed with the use of the dual porosity
model for matrix and fractures (Eq. 5.32). The chart shows the effects of
increasing fracture porosity (PHI2) on while keeping matrix porosity constant
at b = 0.315 Matrix and composite system (matrix plus fractures) are water wet. 115
Figure 5.5— Plot of versus for a dual porosity model made up of matrix and
isolated porosity (red squares) and comparison with core data (single porosity
model). For the same resistivity index, water saturations are smaller in the case
of the dual porosity model. Matrix and the composite system (matrix + vugs) are
oil wet. ..................................................................................................................... 116
Figure 5.6— Plot of versus for a dual porosity model made up of matrix and
fracture porosity (red squares) and comparison against core data (single porosity
model). For the same resistivity index, water saturations are larger in the case of
the dual porosity model for the example at hand. The slope values indicate that
the matrix and the composite system are preferentially water wet. ........................ 118
Figure 5.7— Graph of versus using Eqs. 5.34 and 5.35 for the case of a triple
porosity model. The chart shows the combined effects of isolated porosity and
fracture porosity on Matrix porosity (b) is maintained constant at 0.315 in
this case. .................................................................................................................. 120
Figure 5.8— Graph of versus calculated for a triple porosity model (red
squares). The calculated results are compared against core data represented by
black squares (single porosity model). For the same resistivity index the water
saturation of the triple porosity model is generally larger than Sw from cores
(single porosity). ..................................................................................................... 121
Figure 6.1—Plot of porosity exponent ( versus porosity ( . The porosity
exponent was determined using Byrnes empirical correlation (Eq. 6.2). ............... 128
Figure 6.2— Buckles plot. The red circles represent porosity and permeability data
from drill cuttings. Lines of constant permeability are represented by solid lines.
The red circles and solid lines are determined using Eq. 6.4. The dashed lines
xvi
represent the constant values of the product of porosity and irreducible water
saturation also known as Buckles number. All water saturations are irreducible
since the wells have not produced any water for several years. ............................. 130
Figure 6.3— Plot of capillary pressure (Pc) vs. irreducible water saturation (Swi). Pc
was determined using Eq. 6.7 and Swi was obtained using Eq. 6.4 based on
knowledge of porosity and permeability from drill cuttings................................... 133
Figure 6.4—Chart for estimating pore throat apertures on the basis of permeability
and porosity. The green triangular symbols represent data obtained from drill
cuttings. Nikanassin flow units are dominated by microports. Source of template:
Aguilera (2003). ...................................................................................................... 137
Figure 6.5— Pickett plot including lines of constant water saturation. The black
circles represent drill cuttings data from a horizontal well drilled in the
Nikanassin group of the WCSB. ............................................................................. 141
Figure 6.6— Pickett plot including lines of constant water saturation and constant
permeability. The black circles represent drill cuttings data from a horizontal
well drilled in the Nikanassin group of the WCSB. ................................................ 141
Figure 6.7—Pickett plot including lines of constant water saturation, constant
permeability, and constant capillary pressure. The black circles represent the
drill cuttings data from a horizontal well drilled in the Nikanassin Group of the
WCSB. .................................................................................................................... 142
Figure 6.8—Pickett plot including lines of constant water saturation, constant
permeability, and constant pore throat radius. The black circles represent drill
cuttings data from a horizontal well drilled in the Nikanassin group of the
WCSB. .................................................................................................................... 142
Figure 6.9—Pickett plot including lines of constant water saturation, constant
permeability and constant height above the water contact. The black circles
represent drill cuttings data from a horizontal well drilled in the Nikanassin
group of the WCSB. ................................................................................................ 144
Figure 6.10—Pickett plot including constant lines of water saturation and constant
Knudsen number. The black circles represent drill cuttings data from a
horizontal well drilled in the Nikanassin group of the WCSB. .............................. 145
Figure 6.11—Plot of ( ) versus ( ) for the Nikanassin formation using data
from an offset vertical well (Ortega and Aguilera, 2012). ...................................... 149
Figure 6.12—Cross plot of Young Modulus (YM) versus Poisson’s ratio (PR). The
porosity and bulk density values from drill cuttings (WCSB) were used as input
data in calculating the YM and PR using Eq.6.22 and Eq. 6.27 respectively. ........ 152
xvii
Figure 6.13—Empirical log-log cross plot of porosity from drill cuttings versus
Poisson’s ratio (PR) results in a nearly straight line (R2 = 0.9999). Thus porosity
can be used for obtaining a reasonable estimate of Poisson’s ratio in those cases
where compressional and shear velocities are not available. .................................. 155
Figure 7.1— Example of fracture network created by hydraulic fracturing operation
(Source: fracfocus.org, 2013).................................................................................. 157
Figure 7.2— Track 2 shows a good comparison between drill cuttings data (DTC-
Lab-Dark blue line between the depth interval of 3095-3205m MD) and DTC-
Sonic Log(Green).The track scale is 120-420 us/m.(Adapted from Ortega et al.
2012). ...................................................................................................................... 159
Figure 7.3 — Wellbore survey for horizontal Well A 3953 MD / 3115m TVD and
reference Well B 3240m MD / 3192m TVD. ......................................................... 163
Figure 7.4 — Calibration of compressional travel time DTCGR from GOHFER
(orange Line) with DTCLab extracted from drill cuttings (blue line). The 2
curves are shown in Track 2. .................................................................................. 165
Figure 7.5 — Calibration of Poisson’s ratio PRGR from GOHFER (orange Line) with
PRLab (blue line) extracted from drill cuttings. The 2 curves are shown on Track
2. .............................................................................................................................. 166
Figure 7.6 — Calibration of dynamic Young Modulus YMEGR from GOHFER
(Orange Line) with YMELab (pink line). The 2 curves are shown on Track 2...... 167
Figure 7.7— Example of grid setup showing grid properties such as total closure
stress, brittleness factor, permeability and static young modulus . The reference
well (well B) data was used to populate the grid properties from surface to
3225m MD while the remaining depth interval to 3950m MD that represents the
horizontal section was populated using the horizontal well (well A) data. ............ 168
Figure 7.8 — Cut-Log with three parameter tracks ranging from 3200 to 3900m MD.
Track 1 is the brittleness index, track 2 is permeability and track 3 is the Frac-
Value. Higher values of all three parameters are desired but two high values are
still acceptable. The blue color represents the zones suitable for optimum
fracture initiation and the color intensity is intentionally related to better
conditions. ............................................................................................................... 170
Figure 7.9—Comparison of cumulative gas production profile between the Initial
Design and the Re-design case using Cut-Log. The two cases involved a seven
stage fracture treatment but the fracture initiation zones for the two cases are
different. The Re-design case shows a better performance for the 364 days of
production forecast; these performance shows that selecting symmetrical
distances for hydraulic fracture initiation, as done commonly in the oil and gas
industry due to data scarcity, is not optimum. Better performance is obtained
xviii
selecting those zones with high brittleness index, permeability and Frac-Value in
the Cut-Log, parameters obtained from drill cuttings in this study. ....................... 172
xix
LIST OF ABBREVIATIONS, NOMENCLATURE, AND SYMBOLS
Nomenclature
A = Empirical parameter function of saturation (Kwon et al 1975)
a = Constant in formation factor equation, dimensionless
A= Function of water saturation in Pickett and Kwon empirical equation
B = Empirical parameter in permeability correlation
BRITi = Brittleness Index, dimensionless
BV = Bulk Volume, cubic centimeter
C = Degrees Celsius
C = Parameter function of type of fluid (Morris et al. 1967)
constant1= Empirical constant (A determination)
constant2 = Empirical constant (A determination)
DT= Wave traveling delta time, (μs/m) and (μs/ft)
DTC =Compressional wave slowness (μs/m) and (μs/ft)
DTCGR =Compressional wave slowness from gamma ray data (μs/m) and (μs/ft)
DTCLab =Compressional wave slowness from drill cuttings (μs/m) and (μs/ft)
DTf = Fluid Compressional slowness, (μs/m) and (μs/ft)
DTma = Matrix Compressional slowness (μs/m) and (μs/ft)
DTS= Shear wave slowness, (μs/m) and (μs/ft)
F = Formation factor, dimensionless
f = Volume fraction of the spherical inclusion
G =Rigidity Modulus, (psi) and (GPa)
xx
I = Resistivity Index, dimensionless
k = Absolute permeability, md
kx= Absolute permeability in x direction, md
ky= Absolute permeability in y direction, md
kz= Absolute permeability in z direction, md
Kn =Knudsen number, dimensionless
L = Characteristic length (Knudsen number calculation), m
m = Triple/dual porosity exponent for the reservoir, dimensionless
mb= Matrix block porosity exponent attached only to the matrix system, dimensionless
mf = Fracture block porosity exponent attached only to the fracture system, dimensionless
n = Water saturation exponent, dimensionless
nb = Water saturation exponent attached only to the matrix system, dimensionless
nb’ = Water saturation exponent in electromagnetic mixing rule triple porosity formula
NA= Avogadro's constant, 1/mol
Pc = Capillary pressure, psi
PHI = Porosity, fraction
PR = Poisson's Ratio, fraction
PRGR = Poisson's Ratio from gamma ray data, fraction
PRLab = Poisson's Ratio from drill cuttings, fraction
PRbrit = Poisson’s Ratio brittleness term, fraction
r = Radius of a capillary tube, μm
r35 = Winland's average pore throat radius at 35th percentile mercury saturation, μm
xxi
Rg= Universal gas constant, Pa.m3/mol.K
rp35= Average pore throat radius at 35% mercury saturation, microns
Rt = True resistivity of a porous media at a brine saturation ( , (Ω.m)
= Resistivity of the matrix system at reservoir temperature when it’s 100% saturated
with water of resistivity , (Ω.m)
= Water resistivity at reservoir temperature, (Ω.m)
= Resistivity of the spherical inclusions (water phase)
= Resistivity of the composite system at reservoir temperature when it’s 100%
saturated with water of resistivity , (Ω.m)
= Resistivity of the composite system (matrix and fractures) at θ= 0 when it’s 100%
saturated with water of resistivity , (Ω.m)
= Resistivity of the composite system (matrix and fractures) at θ= 90 when it’s
100% saturated with water of resistivity , (Ω.m)
Rw = Water resistivity, ohm.m
Sirr = Irreducible Water Saturation, fraction
Sw = Water Saturation, fraction
Swi = Irreducible Water Saturation, fraction
Swirr= Irreducible Water Saturation, fraction
Sw = Water saturation, fraction
Swb = Water saturation attached only to the matrix system, fraction
Swnc = Water saturation attached only to the isolated porosity system, fraction
Sw2 = Water saturation attached only to the fracture porosity system, fraction
xxii
Swb’ = Water saturation in electromagnetic mixing rule triple porosity formula; the water
saturation of nb’, fraction
T = Temperature, K
V = Sonic wave velocity, ft/s
Vc = Compressional wave velocity, ft/s
Vs = Shear wave velocity, ft/s
v = partitioning coefficient, fraction
vnc= Lucia’s isolated porosity ratio, fraction
vs = Sihvola dimensionless parameter in unified mixing formula
YM = Young's Modulus, (psi) and (GPa)
YMbrit =Young’s Modulus Brittleness term, fraction
YMd= Dynamic Young’s Modulus, (psi) and (GPa)
YMEGR= Dynamic Young’s Modulus from gamma ray data, (psi) and (GPa)
YMELab= Dynamic Young’s Modulus from drill cuttings, (psi) and (GPa)
YMs= Static Young’s Modulus, (psi) and (GPa)
YMs_max= Maximum Static Young’s Modulus, (psi) and (GPa)
YMs_min = Minimum Static Young’s Modulus, (psi) and (GPa)
Greek Symbols
= Permittivity
εi = Permittivity for spherical inclusions
εe = Permittivity for background material (host)
xxiii
εeff =Effective permittivity of the mixtures
= Real part of the dielectric permittivity
θ = Aguilera’s, angle between fracture and current flow direction
= Bulk density, g/cm3
= Conductivity
= Conductivity for spherical inclusions
= Conductivity for background material (host)
= Conductivity of mixtures (composite system)
= Conductivity of water phase
= Conductivity of the composite system (matrix and fractures) in parallel at θ= 0
= Conductivity of the composite system (matrix and fractures) in series at θ= 90
σ = Interfacial tension, dyne/cm
σh = Minimum in-situ horizontal stress, psi
σH = Maximum in-situ horizontal stress, psi
σv = Vertical stress or overburden stress, psi
δ = Collision diameter, m
θ = Interface contact angle, degrees
θ’ = Berg’s angle that the current makes with the normal to a fracture
λ = Molecular mean free path, m
π= Pi number, dimensionless
μ = Viscosity, mPa.s
= Porosity from laboratory test on drill cuttings, fraction
ϕ = Total porosity, fraction
xxiv
ϕb =Matrix block porosity attached to the bulk volume of the matrix system, fraction
ϕ2 = Fracture block porosity attached to the bulk volume of the composite system,
fraction
ϕm = Matrix block porosity attached to the bulk volume of the composite system, fraction
ϕnc = Isolated block porosity attached to the bulk volume of the composite system,
fraction
= Angular frequency
Abbreviations
API= American Petroleum Institute
ASCII= American Standard Code for Information Interchange
GFREE=Geoscience, Formation Evaluation, Reservoir Drilling, Completion and
Stimulation, Reservoir Engineering, Economics and Externalities
LHS= Left Hand Side
RHS= Right Hand Side
PHINC= Isolated Porosity
PHI2= Fracture Porosity
WCSB= Western Canada Sedimentary Basin
1
Chapter One: INTRODUCTION
1.1 Justification
As at the end of 2012 the total world proved oil reserves stood at 256.33 billion cubic
metres (Fig. 1.1) and the total world proved natural gas reserves at 187.3 trillion cubic
meters (Fig. 1.2). This proved reserves of oil and natural gas represents the quantities
found both in conventional and unconventional petroleum reservoirs. Proved reserves of
oil and natural gas are generally defined as those quantities that geological and
engineering information indicate with reasonable certainty can be recovered in the future
from known reservoirs under existing economic and operating conditions (BP Statistical
Review of World Energy, June, 2013).
Figure 1.1— Distribution of world proved oil reserves in 1992, 2002, and 2012
(Adapted from BP Statistical Review of World Energy. June, 2013).
2
Figure 1.2 — Distribution of world proved natural gas reserves in 1992, 2002, and
2012 (Adapted from BP Statistical Review of World Energy. June, 2013).
As evidenced in Fig. 1.1 and Fig. 1.2 the quantity of proved oil and natural gas reserves
continues to increase from year to year due to technological advancements. However,
significant challenges remain especially in quantitative evaluation of conventional and
unconventional petroleum reservoirs. Some of the questions this research addresses for
contributing to the solution of these challenges include: (1) Is there an alternate and
reliable direct source of information, apart from well logs, seismic data, or core data,
which can be added for quantitative formation evaluation?, (2) how does the porosity
exponent (m) change with depth in reservoirs with complex pore structures?, (3) what is
the water saturation exponent (n) in heterogeneous reservoirs with mixed wettability?,
and (4) in the absence of well logs and/or core data, is it still possible to design an
3
hydraulic fracture model that will improve the success probabilities of the stimulation
job?.
The answers to these questions will help to increase the accuracy of reservoir properties
required for determining petroleum reserves in conventional and unconventional
reservoirs. This will also serve as a means of improving recovery from challenging
reservoirs, hence reducing the number of unproven reserves.
Therefore, the justification for this thesis is to address these challenges (question 1 to 4)
and offer possible solutions. It does so by considering that all hydrocarbons and reservoir
types can be integrated under the umbrella of ‘Total Petroleum Systems’. That is the
premise for being able to integrate in this thesis conventional and unconventional
reservoirs. Aguilera (2010, 2013) set up the stage for this integration by considering flow
units: from conventional to tight gas to shale gas to tight oil to shale oil reservoirs, as a
means for characterizing these types of reservoirs and estimating potential production
rates.
An excellent explanation of the Petroleum System has been presented by Magoon and
Beaumont (1999). “The Petroleum System is a unifying concept that encompasses all of
the disparate elements and processes of petroleum geology including a pod of active
source rock and all genetically related oil and gas accumulations.”
The Petroleum System includes all the geologic elements and processes required for an
oil and gas accumulation to exist. The word ‘petroleum’ includes high concentrations of
4
any of the following substances: (1) Thermal and biological hydrocarbon gas found in
conventional reservoirs as well as in unconventional reservoirs (gas hydrates, tight
reservoirs, fractured shale, and coal). (2) Condensates. (3) Crude oils. (4) Natural
bitumen. The word ‘system’ describes the interdependent elements and processes that
form the functional unit that creates hydrocarbon accumulations.
Magoon and Beaumont (1999) indicate that the essential elements of a Petroleum System
include the following: (1) Source rock. (2) Reservoir rock. (3) Seal rock. (4) Overburden
rock. The Petroleum System includes two processes: (1) Trap formation. (2) Generation–
migration–accumulation of hydrocarbons. These essential elements and processes must
be correctly placed in time and space so that organic matter included in a source rock can
be converted into a petroleum accumulation. A Petroleum System exists wherever all
these essential elements and processes are known to occur or are thought to have a
reasonable chance or probability to occur (Aguilera, 2013).
The segments of the Total Petroleum System described above, dealing with conventional
oil in carbonate reservoirs and natural gas in tight reservoirs are the primary concern of
this thesis. The significant paradigm shift is that tight rocks that could not produce any
petroleum in the past or were nearly impermeable ‘seals’ are now economic reservoir
rocks (Aguilera, 2013).
5
1.2 Objectives
Part of the objectives of the GFREE (Geoscience, Formation Evaluation, Reservoir
Drilling, Completion and Stimulation, Reservoir Engineering, Economics and
Externalities) research group at the University of Calgary is to understand the reservoir
rocks through their characterization, and develop new petrophysical models and concepts
to facilitate optimal recovery of petroleum from challenging reservoir rocks.
As continuation of the GFREE research program, the primary objectives of this research
are to:
Extend the use of drill cuttings-based petrophysical and geomechanical evaluation
methods to a horizontal well that penetrates a tight gas reservoir.
Develop new petrophysical models for calculating the porosity exponent (m) in
dual and triple porosity reservoirs.
Develop new petrophysical models for calculating the water saturation exponent
(n) in dual and triple porosity reservoirs with mixed wettability.
Design a multi-stage hydraulic fracture model using drill cuttings data to improve
recovery from the Nikanassin formation
1.3 Geological Background of Study Area
Zaitlin and Moslow (2006) reviewed the deep basin gas reservoirs of the western Canada
sedimentary basin (WCSB) as shown in Fig. 1.3 and Fig.1.4. The western Canada deep
basin is mapped northward from the United States-Montana border into the northeastern
part of British Columbia, Canada. This is a regional extensive hydrocarbon saturated
6
area, abnormally pressured, thermally matured, composed of Mesozoic to Paleozoic
clastics rocks with low permeability and little or no water production.
Figure 1.3—Location of western Canada sedimentary basin and other major
sedimentary basins in North America (Zaitlin and Moslow, 2006).
Figure 1.4—Map of western Canada sedimentary basin (Adapted from Moslow,
University of Calgary Presentation, 2013).
7
The Nikanassin rocks in the WCSB are dominated by a sandstone sequence. The
Nikanassin spreads above the Fernie shale and below the Cadomin conglomerate
(Mackay, 1930; Stott, 1998). The Nikanassin Group represents a significant tight gas
resource and is composed by the following three formations: Monteith, Beattie Peaks,
and Monach (Fig. 1.5 and Fig. 1.6). The horizontal well evaluated in this study
penetrates the Monteith formation and the collected drill cutting samples are also from
this same formation. The Monteith formation is the oldest of the three and generally
decreases in thickness from west to east. It is made up of a succession of very fine to fine
grain sandstones but has variable amounts of coarser grained quartzose sandstone,
siltstone, mudstone and carbonaceous rocks (Stott, 1998; Solano, 2010). Detailed
description of this formation was done by Solano (2010).
Figure 1.5— Stratigraphic framework of the Nikanassin group (Adapted from Stott,
1998; Miles, 2010; and Solano, 2010).
8
Figure 1.6—Typical wireline log section through the Nikanassin group in the study
area 66-0W6 (Miles, 2010).
1.3.1 Study Area
With the aid of stratigraphic correlation and mapping of three major coarsening-up,
siliciclastic sequences, the Monteith formation was divided into Red rock, Wapiti and
Knopcik field (Miles, 2009, 2010; Solano, 2010). The study area for the present work is
in the Red Rock Field in Alberta (Canada) where drill cuttings were collected at different
depths in the horizontal well (referred as well A in Chapters 3 and 7 ) (Fig. 1.7).
9
Figure 1.7— Location of Study Area (Red Rock Field) shaded in Light Blue
(Adapted from Solano, 2010).
1.4 Research Approach
The research work in this thesis can be summarised under four main components listed
below:
1.4.1 Laboratory Work
Carry out laboratory measurement on drill cuttings to determine porosity and
permeability.
1.4.2 Analytical Model Development
Develop new petrophysical models to estimate the porosity exponent (m) in dual
and triple porosity reservoirs using electromagnetic mixing rules.
10
Develop new petrophysical models to estimate the water saturation exponent (n)
in dual and triple porosity reservoirs with mixed wettability using electromagnetic
mixing rules.
Both m and n exponents in Archie’s equation used for determining the formation factor
and water saturation.
1.4.3 Petrophysical and Geomechanical Evaluation
Use drill cuttings data to perform complete petrophysical and geomechanical
evaluations of a horizontal well that penetrates a tight gas Nikanassin reservoir.
1.4.4 Hydraulic Fracturing and Simulation
Use drill cuttings data to design a multi-stage hydraulic fracture model to
optimize production from a tight gas reservoir and simulate production for 364
days.
The above topics are discussed in this thesis throughout seven chapters. Chapter One (this
chapter) presents an introduction to drill cuttings, petrophysical, and geomechanical
models for evaluation of conventional and unconventional petroleum reservoirs.
Chapter Two discusses an overview of the work done by several researchers and can be
broadly grouped into five categories: (1) Drill cuttings, (2) Porosity exponent, (3) Water
11
saturation exponent, (4) Petrophysical and geomechanical evaluation, and (5) Hydraulic
fracturing and simulation
Chapter Three discusses the methodology used in the laboratory at the University of
Calgary to determine porosity and permeability from drill cuttings samples. Results are
also presented in this chapter.
Chapter Four develops new petrophysical methods with the use of electromagnetic
mixing rules for determining the porosity exponent (m) in reservoirs represented by dual
(matrix and non-connected vugs or matrix and fractures) and /or triple porosity models
(matrix, fractures and non-connected vugs). This chapter also includes the models
validation with core data from carbonate and tight gas reservoirs.
Chapter Five also uses electromagnetic mixing rules to develop new petrophysical
models capable of estimating the water saturation exponent ( ) in heterogeneous
reservoirs with mixed wettability. The reservoirs are represented by dual porosity (matrix
and fractures or matrix and isolated porosity) and triple porosity (matrix, fractures and
isolated porosity) models. This chapter also include the model validation with core data
from preferentially oil-wet and water-wet reservoirs.
Chapter Six presents a case study on petrophysical and geomechanical evaluation of
horizontal wells in the tight gas Nikanassin Group of the WCSB using drill cuttings.
Their utilization as an aid in performing complete quantitative petrophysical and
12
geomechanical evaluations is also discussed. This includes measurements in the
laboratory of porosity and permeability, which allow determining pore throat apertures,
capillary pressures, irreducible water saturation, porosity exponent (m), true formation
resistivity, location of water contact, Knudsen’s number, Young Modulus, Poisson’s
ratio, and brittleness index.
Chapter Seven discusses the application of hydraulic fracturing as a means of optimizing
production from tight gas reservoirs using drill cuttings. Two cases are compared: One
with and one without the use of drill cuttings for selecting fracture initiation zones in a
horizontal well drilled in the lower Nikanassin Group of the western Canada sedimentary
basin.
Chapter Eight contains a summary of the thesis’s finding, conclusions and
recommendations.
1.5 Technical Publications
Parts of this research work have been accepted for presentation at the following
international conferences:
Olusola, B. K., Yu, G., Aguilera, R. 2012. The Use of Electromagnetic Mixing
Rules for Petrophysical Evaluation of Dual and Triple Porosity Reservoirs. Paper
SPE 162772-PP presented at the SPE Canadian Unconventional Resources
Conference., Calgary, Canada. Oct - Nov. In press: SPE Reservoir Evaluation &
Engineering-Formation Evaluation
13
Olusola, B. K, Aguilera, R. 2013. How to Estimate Water Saturation Exponent in
Dual and Triple Porosity Reservoirs with Mixed Wettability. Paper SPE 167213-
MS prepared for SPE Canadian Unconventional Resources Conference., Calgary,
Canada. Oct - Nov.
Olusola, B. K, Aguilera, R. 2013. A Case Study on Formation Evaluation of
Horizontal Wells in the Tight Gas Nikanassin Group of the Western Canada
Sedimentary Basin Using Drill Cuttings. Paper SPE 167214-MS prepared for SPE
Canadian Unconventional Resources Conference., Calgary, Canada. Oct - Nov.
14
Chapter Two: LITERATURE REVIEW
2.1 Overview
This chapter presents an overview of the work done by several researchers and can be
broadly grouped into five categories: (1) Drill cuttings, (2) Porosity exponent, (3) Water
saturation exponent, (4) Petrophysical and geomechanical evaluation, and (5) Hydraulic
fracturing and simulation. In reality, there are interdependencies between each of these
five categories but for the purpose of this thesis, each category is treated as a distinct
topic in order to provide the reader with a basic overview of the impact of each category
on petroleum reservoirs.
2.1.1 Drill Cuttings
Drill cuttings are produced as the rock is broken by the action of the drilling bit
advancing through rock or soil. The cuttings are usually carried to the surface by drilling
fluid circulating up from the drill bit to the surface. Drill cuttings are separated from the
drilling fluid at the surface using shale shakers, de-sanders and de-silters.
At the rig site, drill cuttings are commonly monitored and examined by mud loggers, mud
engineers, pore pressure specialists and other on-site personnel. These professionals make
a record (a well log) of the formations penetrated at various depths; and various
properties including among others drill-cuttings composition, size, shape, color, texture,
and hydrocarbon content (http://www.glossary.oilfield.slb.com, 2013).
15
In the petroleum industry, this record is often called a mud log. The quality of mud
logging data is highly dependent on the well site personnel technical skills and
experience. The quality of the mud log data can be improved through detailed cutting
selection and analysis. Many visual inspections are focused on the large cutting samples
because they are the easiest to manipulate with tweezers and picks but in reality, the
larger drill cuttings samples may not be representative of the interval of interest due to
many factors including lack of structural features that indicates the presence of vugs or
fractures, this structural features will be explained further in chapter two (Georgi et al.,
1993). Based on experiments performed by GFREE members at the University of
Calgary, drill cuttings samples equal to or greater than 1mm can provide representative
values of porosity and permeability for the intervals of interest.
Mud log data provides direct source of information about the formation and helps to
answer certain questions that well logs data cannot explain during formation evaluation.
Fig. 2.1 shows how shapes of drill cutting samples are used to predict the sub-surface
stress and recommends solutions to avoid wellbore stability problems during drilling
operations, for example platy or tabular caving samples at the shale shakers may indicate
the drill bit has penetrated a weakly bedded or fissile subsurface rock
16
Figure 2.1— Example of the use of drill cuttings to analyse shale caving types and
wellbore stability prediction (source: www.geomi.com).
2.1.1.1 Drill Cuttings based Formation Evaluation
High quality formation evaluation data is critical for successful development of oil and
gas fields especially in tight gas reservoirs where this information is required as input
data for building hydraulic fracturing models. Formation evaluation data assists in
identifying zones that are brittle, permeable and suitable for stimulation in order to
improve oil and gas recovery.
Historically, the methods used to obtain petrophysical formation evaluation data has been
limited to well logs and cores but recently, advances in technology have made the use of
17
drill cuttings possible for obtaining formation evaluation data. An example is provided by
the Darcylog; an equipment designed by the French Institute of Petroleum (IFP) that uses
the liquid pressure pulse method for determining permeability from drill cuttings.
The Darcylog equipment ensures effective flow inside the cuttings by compression of the
residual gas that the drill cuttings contain. The equipment is especially suited for
measuring permeability in reservoir rocks that cannot be measured by gas pressure test
(Lenormand and Fonta, 2007). The Darcylog, explained in chapter three is used for the
experimental work carried out in this thesis.
Talabani and Thamir (2004) designed equipment (Portable Permeameter) that allows
measuring vector permeability in a piece of drill cutting. The recommended average size
of drill cutting samples is 0.32 x 0.5 x 0.5 cm. According to Talabani and Thamir (2004),
three points are chosen and marked perpendicular to each other to measure permeability
of the drill cutting sample. The three points represents permeability in three directions
(kx, ky, and kz). Once the permeability measurement is completed after injecting gas
through a probe at certain pressure, two permeability measurement will be close or
identical (plane k: kx and ky) while the last permeability measurement will be different
(kz).
Drill cuttings can also detect micro fractures in some cases but most of the in-situ micro
fractures and slot porosities are destroyed by the action of the bit during drilling
operations (Ortega and Aguilera, 2012). Micro fractures, when present in large quantities,
can significantly improve fluid flow into the wellbore from the reservoir. Hews (2012)
recommends a staining technique such as the use of Rhodamine B dye to identify micro
18
fracture porosity in drill cuttings. Hews (2012) also presents procedures to evaluate drill
cuttings and identify structural features using a standard binocular microscope. Some of
the features that can be identified include fracture sets with mineral lining, fracture sets
with planar, unmineralized surfaces, brecciation, micro faulting, slickensides,
crenulations in shales, and loose mineral crystals.
Ortega and Aguilera (2012) adapted the drill cuttings evaluation procedure of Hews
(2012) to develop the Cut-Log. The Cut-Log is made of three main parameters
(brittleness Index, permeability and Frac Value) that can be used as a guide in selecting
optimum locations for initiating hydraulic fractures in vertical and horizontal wells. An
example of the Cut-Log is presented in Fig 2.2. The shades of green show the optimum
locations for performing multi-stage fracturing in a Nikanassin horizontal well.
Figure 2.2—Example of Cut-Log showing the brittleness index, permeability and
Frac-Value (Adapted from Ortega and Aguilera, 2012).
19
Egermann et al. (2004), Lenormand and Fonta (2007); Solano (2010); Ortega (2012); and
Zambrano (2013) have used drill cuttings to effectively characterize tight gas sandstone
and carbonate reservoirs. In a recent work by Ortega (2012) he concluded that in the
absence of well logs and cores; quantitative petrophysical evaluation can presently be
considered a possibility owing to the fact that permeability and porosity can now be
measured directly from drill cuttings. However, his study as well as works by Solano
(2010), and Zambrano (2013) were performed on drill cuttings from vertical and deviated
wells drilled in the tight gas Nikanassin group of the western Canada sedimentary basin.
In the present study the methodology is extended to drill cuttings collected in a horizontal
well drilled in the same formation. The benefit of this work is that it provides direct
information about reservoir properties of the horizontal well to be stimulated using
hydraulic fracturing technology.
2.1.2 Porosity Exponent (m)
Understanding of the role played by vuggy and naturally fractured reservoirs on
hydrocarbon recovery has changed over the past few years as a thorough petrophysical
knowledge of this reservoir properties has helped to improve estimates of hydrocarbons-
in-place and recoveries. This has happened for example in the case of ‘unconventional’
reservoirs.
When conducting petrophysical evaluation in a tight gas reservoir, an accurate value of
the porosity exponent (m) is important because variations in m change significantly the
water saturation values and therefore affect the hydrocarbons-in-place estimates. In
20
general, decrease in m tends to reduce the computed water saturation values. It has been
demonstrated in the literature (Al-Ghamdi et al., 2012) that petrophysical evaluation
techniques that uses a constant porosity exponent for all depths will likely magnify the
errors in the calculated water saturation value especially in tight reservoirs with low
porosity. Laboratory studies by Byrnes et al. (2006) show that m tends to become lower
as porosity decreases in the Mesaverde formation of the United Sates.
Archie (1942) studied a group of core samples and observed that m ranged between 1.8
and 2.0 for consolidated sandstones and that for loosely or partly unconsolidated
sandstone the value of m was as low as 1.3. Archie expressed the formation factor, F, as
follows (Eq. 2.1):
…………………………………………………………..….Eq. 2.1
Towle (1962) investigated the porosity exponent and discovered that one of the reasons m
varies is because of changes on the type of pore system, e.g. inter-granular, inter-
crystalline, vuggy or fractured. Towle’s investigation of reservoirs containing vugs
showed that m values were larger than usual ranging from 2.67 to > 7.3. He concluded
that the vuggier the rock, the higher the value of m.
Lucia (1983, 1992) showed that the porosity exponent of carbonate rocks was related to
particle size, amount of inter-particle porosity, amount of non-connected vug porosity,
and the presence or absence of connected vugs. Lucia defined vuggy porosity as that pore
space larger than or within the particles of rock. A common characteristic was the
presence of leached particles, fractures, and large irregular cavities.
21
Lucia described the interconnectivity of vugs in two general ways: (1) Vugs connected
only through the interparticle/matrix pore network (isolated, non-connected or non-
touching vugs) and (2) Vugs connected to each other (connected or touching vugs).
Values of the porosity exponent were related to a vug porosity ratio (vnc) which he
defined as non-connected vug porosity (non-touching vugs) divided by total porosity. He
further observed that m became larger as the vug porosity ratio increased. This
observation was based on data from the Smackover dolomite in the Bryans Mill field,
Texas; Mississippian dolomite of the Harmattan field, Alberta; Magnolia field and
Quitman field in Texas, and Snipe Lake field in Canada.
In Lucia’s work, the term non-connected vugs refers to vugs that are not touching each
other; but each vug is connected to the matrix pore system, hence making it possible for
the vuggy space to be filled with water or hydrocarbon during migration and
accumulation processes. Non-connected vugs are usually formed through diagenetic
processes by which selective fabric (carbonates and evaporite minerals) is dissolved and
removed, thus creating and modifying pore spaces in reservoir rocks (Lucia, 1992).
Aguilera and Aguilera (2004) developed a triple porosity model for evaluation of
naturally fractured reservoirs. They assumed that the matrix and fractures have
conductivities that are connected in parallel and the combined matrix and fractures are
connected in series with the non-connected vugs. An improved triple porosity model
using the same assumptions was developed by Al-Ghamdi et al. (2011). Values obtained
22
from this model matched reasonably well petrographic work and core data and indicated
that m can be smaller than, equal to or larger than the porosity exponent of only the
matrix system, mb, depending on the contribution of non-connected vugs and fractures to
the total porosity system of the reservoir.
Aguilera and Aguilera (2009, 2010) developed a model for petrophysical evaluation of
dual-porosity naturally fractured reservoirs that included the angle between the fracture
and the direction of current flow, θ, between 0o and 90
o. They used the Maxwell Garnett
mixing formula for calculating effective permittivity of a system with aligned ellipsoids
and depolarization factors equal to 0 and 1 that led to the parallel and series resistance
networks required to establish the matrix and fracture relationships (Eq. 2.2 to Eq. 2.4).
They concluded that the change in angle θ between the fracture and the direction of
current flow had significant impact on the value of m. Their theoretical model was used
as part of the validation of the dual porosity (matrix and fractures) model developed in
this paper.
[
]
( ………………………………………...……………….Eq. 2.2
(
……………………………………………………...Eq. 2.3
( ………....…..…………………..……………Eq. 2.4
Berg (2006) developed an equation that calculates the effects of fractures or vugs on the
total porosity exponent (m) of the composite system for a dual porosity reservoirs using
effective medium theory. He further extended the method to calculate the triple porosity
23
exponent, m. The model works by calculating first the dual porosity exponent, m, for the
combination of matrix (bulk) porosity and vugs using Eq. 2.5 and then using the
calculated dual porosity exponent, m as mb, and the total porosity of the dual porosity
model ϕ as ϕb along with a porosity exponent of only the fractures, mf, to calculate the
triple porosity exponent, m, with Eq. 2.6. Berg’s model has been used as part of the
validation of the dual and triple porosity models based on electromagnetic mixing rules
developed in this thesis.
( (
)
……………………………………….…….Eq. 2.5
……………..……………………………………….Eq. 2.6
Berg’s model is very useful as it allows introduction of the porosity exponents for the
fractures (mf) and vugs (mv). Most generally mf is assumed to be equal to 1.0 but it can
reach as high as 1.2 or 1.3 depending on the amount of fracture tortuosity in the fractures.
When mv is infinite (actually in an excel sheet a very large value, for example mv = 1E35)
Berg’s model provides approximately the same results as other models discussed in this
thesis.
Berg suggests that small values of mv might be indicative of connected vugs where
electric current has to contend with tortuosity of the vugs until the vugs approach the
shape of smooth tubes in which case the porosity exponent would be 1.0 as in the case of
smooth fractures. The problem is that the starting point of effective medium theory
assumes that the inclusions in the host material are not connected. So thinking in terms of
24
connected ‘inclusion’ vugs using the effective medium theory appears contradictory. A
further problem from a practical point of view is thinking in terms of mv for a vug. How
to determine it? What does it mean? The assumption of mv being infinite is intuitively
correct for the unlikely case of truly isolated vugs (how do they form and fill with
hydrocarbons if they are truly isolated? So this is arguably an abstract case. But
additional work in the subject is warranted.
There are also excellent contributions along the same lines by Sen et al. (1981) for
various types of sedimentary rocks and fused glass beads; Rasmus and Kenyon for
ooilitic reservoirs (1985); Focke and Munn (1987) for limestones and dolomites; Lucia
(1983) for carbonates; and Kennedy and Herrick (2004) for non-shaly sandstones. The
new method developed in this thesis for petrophysical analysis can handle with the use of
one single equation the individual mixing rules of Maxwell, Bruggeman and Coherent
potential. The unified equation can further handle matrix, fracture and non-connected vug
porosity as well as calculate the porosity exponent (m) of the total porosity system
represented by dual or triple porosity reservoir systems.
2.1.3 Water Saturation Exponent (n)
One of the key petrophysical parameters when evaluating reserves estimates is water
saturation; water saturation relies on the water saturation exponent (n) and resistivity
index (I) using Archie’s equation (Eq. 2.7)
(
)
…………………………………………………………………Eq. 2.7
25
where is the water saturation, is the resistivity of the porous media when
completely saturated with brine, is the true resistivity of a porous media at a brine
saturation ( , is the resistivity index, and is the water saturation exponent; a
dimensionless empirical parameter (Archie, 1942). The resistivity index ( ) is readily
available from resistivity well logs but the water saturation exponent can only be
correctly determined through laboratory work. The water saturation exponent ( appears
equal to two (2) for clean, consolidated, and water-wet sandstones (Archie, 1942) and has
been assumed equal to two (2) in many petrophysical evaluation methods but in reality n
is not constant but varies as a function of different factors including fluid distribution in
the pore spaces and rock wettability.
Many researchers have conducted experimental work to show that the water saturation
exponent truly varies as a function of the preferential wetting of the porous rock to water
or oil. Donaldson and Siddiqui (1989) performed some experiments on some Berea and
Elgin core samples and found that the water saturation exponent range from less than two
(2) to eight (8). They later concluded that the relationship between wettability and n is
linear and that oil/water/ rock systems have more preference for water as the temperature
is increased.
Sweeney and Jennings (1960) carried out resistivity measurements on some porous
carbonate rocks by varying the amounts of electrolyte and aliphatic hydrocarbon; this was
done to make the carbonate surface either preferentially water wet or oil wet. They
observed that the resistivity data obtained from preferentially water-wet samples were
26
considerably different from the resistivity data obtained from preferentially oil-wet
samples; thereby leading to different n values. The data from of Sweeney and Jennings
(1960) laboratory work have been used to validate the models developed in this thesis.
Morgan and Pirson (1964) examined the effect of fractional wettability on ; they
prepared unconsolidated sand packs in capillary pressure resistivity cells using mixtures
covering the entire wettability spectrum. They saturated each pack with brine and then
displace the packs with oil at progressive stages; at each stage the water saturation value
was measured and the n value for each pack was determined. They recorded n values
ranging from 2.48 to 25.17 at different wettability condition. Therefore, wettability is a
major controlling factor on n.
Wettability is defined as the tendency of a particular fluid to spread on or adhere to a
solid surface in the presence of other immiscible fluids. In a rock/oil/brine system; when
the rock is water-wet; there is a tendency for water to occupy the small pores and also
make contact with most of the rock surfaces. For an oil-wet rock the reverse is true.
When the rock has no strong affinity for either water or oil; the system is defined as
having neutral (or intermediate) wettability (Anderson, 1986).
Apart from oil-wet, water-wet, and neutral-wet systems; there is also another type of
wettability known as fractional wettability in which some parts of the rock have affinity
for water while other parts have affinity for oil (Morgan and Pirson, 1964). The idea of
fractional wettability (also called heterogeneous, spotted, or Dalmatian wettability) was
27
also proposed by Brown and Fatt (1956). Fractional wettability is conceptually different
from intermediate wettability, which assumes that all parts of the rock surface have slight
but equal affinity for either water or oil (Anderson, 1986).
Salathiel (1973) introduced the concept of mixed wettability as a special type of
fractional wettability in which the oil-wet surfaces form continuous paths through the
larger pores. In this condition, the fine pores and grains contacts would be water-wet and
contain no oil while the surfaces of the larger pores would be strongly oil-wet. Mixed
wettability is different from fractional wettability because fractional wettability does not
imply specific locations for the oil-wet surfaces nor continuous oil-wet paths but that a
portion of the rock is strongly water-wet while the rest is strongly oil-wet. Salathiel
(1973) explained that the generation of mixed wettability conditions in the reservoir may
result from the following process: originally water is present in the larger pores, small
capillaries and at grain contacts and the reservoir is water-wet, but as oil accumulates in
the reservoir, it displaces the water from the larger pores while the water in the small
capillaries and at grain contacts are retained due to capillary forces.
As this fluid migration and displacement process continues for a long period of time,
some organic material from the oil will deposit on the rock surfaces in direct contact with
the oil and the larger pores will become strongly oil-wet. This last process results in a
mixed wettability condition. Melrose (1982) and Hall et al. (1983) further describe that
the water films in the originally water-wet reservoir becomes thinner and thinner as more
oil accumulates in the reservoir. At a time when the water film thickness is reduced to a
28
critical thickness, the water film become unstable in the larger pores (only part of the
total pore surface area is affected) after which it will rupture and give way for the
displacing oil to contact the rock.
However, most studies carried out on reservoirs with mixed wettability condition are
laboratory studies with limited work done on estimating using analytical method. To
contribute to the solution of this concern, we utilize electromagnetic mixing rules to
develop new petrophysical models capable of estimating the water saturation exponent
( ) in heterogeneous reservoirs with mixed wettability. The reservoirs are represented by
dual porosity (matrix and fractures or matrix and isolated porosity) and triple porosity
(matrix, fractures and isolated porosity) models. In our petrophysical model, we follow
the concept of mixed wettability proposed by Salathiel, 1973; Melrose, 1982; and Hall et
al., 1983. We assume that the larger pores represent the isolated porosity (non-connected
vugs or isolated porosity) and the small capillaries and grain contacts represent fractures
and matrix porosity respectively.
2.1.4 Petrophysical and Geomechanical Evaluation
2.1.4.1 Petrophysical Evaluation
Petrophysical evaluation can be described as the process of sub-surface data acquisition,
processing and interpretation to detect and quantify oil and gas reserves in the rock
adjacent to a well (Wikipedia, 2013). Petrophysical evaluation techniques are used to
answer basic questions, such as:
What does the reservoir contain: water or hydrocarbons?
29
If hydrocarbons, oil or gas?
How much is there?
Where is it? And how to get it out?
What type of rock is there and what are its properties?
Fig. 2.3 shows the main sources of data used in the oil and gas industry for petrophysical
evaluation. Generally, the more sources of data are available for a particular reservoir the
more accurate and reliable the petrophysical evaluation results.
Figure 2.3—Main sources of data concerning sub-surface rocks (Source: Dr. Paul
Glover, Petrophysics MSc course notes, University of Laval, Quebec, Canada, 2013)
2.1.4.1.1 Archie’s Equation
Archie (1942) introduced empirical equations that became the keystone of log analysis
(See Eq. 2.8 to Eq. 2.10). The equations are still used till date to determine water
saturation in hydrocarbon reservoirs.
30
(
)
…………………………..……………………………………………Eq. 2.8
…………………………………………………………………………...…Eq. 2.9
(
)
…………………………………….………………………………….Eq. 2.10
Archie’s equations produce acceptable results in “clean” (shale free) formations but they
produces questionable results in heterogeneous and complex reservoirs. The application
of Archie’s equation requires knowledge of the parameters in Eqs. 2.8 to 2.10. Most of
the parameters especially the resistivity parameters can be determined from field
measurements and /or log analysis estimations but the porosity exponent (m) and
particularly the water saturation exponent (n) are difficult to estimate in tight formations.
As a result assumed values are assigned based on experience of the petrophysicist or
geoscientist. The problem is that and values can change continuously with depth
especially in heterogeneous and complex reservoirs. Thus the assigned constant value for
the whole reservoir is misleading since it either overestimates or underestimates the water
saturation value. This affects the reserves estimates in conventional and unconventional
petroleum reservoirs.
To accommodate this deficiency, a new petrophysical model is developed in this thesis
using electromagnetic mixing rules proposed by Sihvola (1999). The petrophysical model
allows the estimation of and in reservoirs made up of dual porosity (matrix and
fractures or matrix and isolated porosity) and triple porosity (matrix, fractures and
isolated porosity) systems. These petrophysical models are validated with core data with
31
a good level of accuracy. The models development is explained in detail in chapters four
and five.
2.1.4.1.2 Pickett Plot
Pickett (1973) devised a formation evaluation interpretive technique using cross plots of
log responses. In his approach resistivity is plotted against porosity values on log-log
coordinates. The result is straight lines for zones with constant water saturation (for
example Sw = 100%). Hydrocarbon-bearing intervals fall to the right of the 100% water
saturation line. Pickett’s plot is based on Archie’s (1942) equations (Eq. 2.8 to Eq. 2.10),
which can be rearranged to obtain:
( ………………………………………..Eq. 2.11
The conventional construction of a Pickett plot requires the availability of porosity and
resistivity logs. In this thesis, these logs are not available for the horizontal well under
consideration, and the Pickett plot is built on the basis of data extracted from drill
cuttings. Use of Pickett plot is illustrated in chapter six of this thesis.
2.1.4.2 Geomechanical Evaluation
Geomechanical evaluation of conventional and unconventional reservoirs is used to
predict important rock mechanics parameters, such as in-situ rock stresses, Young
Modulus, Poisson’s ratio and brittleness index. These parameters are useful for designing
hydraulic fracture treatments. Some important work along these lines has been
contributed by Barree et al. (2009) for stress and rock property profiling for
unconventional reservoir simulation; Mullen et al. (2007) for determination of
32
mechanical rock properties for stimulation design in the absence of sonic logs; Aoudia
and Miskimins (2010) for statistical analysis on the effects of mineralogy on rock
mechanical properties of the Woodford shale; and Ortega and Aguilera (2012) for
evaluating the impact of drill cuttings on the design of multi-stage hydraulic fracturing
jobs in tight gas formations. Chapter six of this thesis shows an application of drill
cuttings based geomechanical evaluation to characterize the tight gas reservoir considered
in this study.
2.1.5 Hydraulic Fracturing
The use of hydraulic fracturing technology started in the 1940s (King, 2012). Without
hydraulic fracturing it would have been impossible to recover hydrocarbons such as oil
and gas from tight reservoirs with permeabilities equal to or smaller than 0.1 md (Center
for Energy, 2013). Permeability is a rock property and defines the ability of a fluid to
flow through porous media. This porous media must have interconnected pores for the
fluids to travel through a tortuous path to reach the wellbore. Tight gas reservoirs have
low permeabilities. Without using hydraulic fracturing technology, it is impossible to
recover economically gas or oil from these types of reservoirs (API, 2009).
Hydraulic fracturing removes the skin effect of a wellbore (Economides et. al. 1994),
creating a high conductivity path that covers a long distance and extends from the
wellbore to the hydrocarbon reservoir. This fractured path allows reservoir fluids to flow
more easily from the reservoir, into the fracture, and finally into the wellbore. Fig. 2.4
33
shows a comparison of fluid flow between a well completed without hydraulic fracturing
and a well completed using hydraulic fracturing technology (API, 2009).
Figure 2.4 —Example of a non-fractured reservoir (upper schematic) and a
fractured reservoir (lower schematic). The upper schematic represents radial fluid
flow (red arrows) into the well (circle) from a smaller area of the reservoir. The
lower schematic represents a hydraulically fractured reservoir that allows linear
fluid flow (red arrows) from a larger area of the reservoir into the fracture and
finally into the wellbore. This leads to an increase in recovery (API, 2009).
2.1.5.1.1 Hydraulic Fracturing Process
The hydraulic fracturing process involves injecting specialized fluids down the wellbore
and through perforations in the casing (or open hole) to the reservoir. The fluid is injected
at pressures high enough to cause tensile failure of the rock (Economides et al., 1994).
This process is known as “breaking down” the formation. As additional fluids are
injected at the propagating pressure, the rock continues to open and the fracture continues
to propagate into the reservoir. During this process, measured amounts of proppants, such
as sand, are added into the injected fluid. When the fluid injection is stopped and the
excess pressure is removed, the proppants keep the fractured path open (Fig. 2.5),
34
allowing the oil or gas (also water) to easily flow through the conductive (fractured) path
(API, 2009).
Figure 2.5—Example of an open rock with well-rounded ceramic proppants keeping
the fracture open (Adapted from thorsoil.com).
2.1.5.1.2 Multistage Hydraulic Fracturing
Multistage hydraulic fracturing is shown in Fig. 2.6 and refers to the process whereby
multiple fractures are created along the horizontal section of the wellbore in a
consecutive manner. The operation is carried out by first isolating and fracturing the
deepest segment of the horizontal wellbore, after which the depth before the deepest
segment is treated; this process continues upwards until the last segment (shallowest
depth) is treated (ERCB, 2011).
35
Figure 2.6—Example of a multistage hydraulic fracturing operation. Halliburton’s
Swell packer systems isolating various zones of a horizontal wellbore (open hole)
that will be stimulated (Adapted from www.egyptoil-gas.com).
Ortega and Aguilera (2012) developed a new methodology to improve multistage
hydraulic fracturing design using drill cuttings. This is valuable particularly in those
cases where log and core data are scarce or unavailable. The methodology selects the
optimum locations to initiate each hydraulic fracturing stage.
2.1.5.1.3 Horizontal Wells
Many tight gas reservoirs that are candidates for hydraulic fracturing are drilled using
horizontal well technology. This allows optimum formation area penetration and
maximum gas and oil recovery. The use of horizontal well technology started in the
1930s (King, 2012). Horizontal well technology with directional survey technology
allows drilling multiple horizontal wells from a single surface location, thereby reducing
36
time spent on moving rigs to different surface locations and reducing the footprint and
environmental problems.
Tight gas reservoirs are candidates for horizontal well technology because horizontal
wells allow a large area of the reservoir to be placed in contact with the wellbore (Fig.
2.7); horizontal wells make it possible to create a complex fracture network in the tight
reservoir using multistage hydraulic fracturing technology.
Figure 2.7‒ Example of horizontal well contacting a large area of the reservoir
layer. The vertical well in the right can only contact a smaller area of the reservoir
(Adapted from API, 2009).
37
2.1.5.1.4 3D Hydraulic Fracture Simulation Software
Robust hydraulic fracture simulation software requires a model that can be used to
analyze, target and optimize hydraulic fracturing design, and predict well production.
Several years ago, simple two-dimensional (2D) models were used in the petroleum
industry to simulate hydraulic fracturing in unconventional reservoirs. The 2D model is
based on the concept that the fluid pumped into the rock is directly related to the fracture
geometry. The limitation of the 2D model is that it imposes unrealistic height restrictions
that are often not representative of the formation.
This limitation led to the development of pseudo three-dimensional (P3D) and fully
three-dimensional (3D) models. The 3D models require more reservoir input data and
greater computational time (Christopher et al. 2007). There are many commercial 3D
models for hydraulic fracturing including Stimplan, Fracpro, and GOHFER (Grid-
Oriented Hydraulic Fracture Extension Replicator). GOHFER is used in this thesis.
GOHFER is a well-known robust simulator used to model complex hydraulic fractures in
tight gas and oil reservoirs. GOHFER is a planar 3D geometry, finite difference hydraulic
fracturing modeling software and has a fully coupled fluid/solid transport simulator.
GOHFER was developed by Dr. Bob Barree & Associates in association with Stim-Lab,
a division of Core Laboratories. GOHFER uses a regular grid structure to describe the
reservoir; this grid structure permits vertical and lateral variations, multiple perforated
intervals as well as single and bi-wing asymmetric fractures to model the most complex
reservoirs. The software is capable of modeling multiple fracture initiation sites
38
simultaneously and shows diversion between perforations. GOHFER handles elastic rock
displacement calculations as well as a planar finite difference grid for the fluid flow
calculations. At each grid cell variables such as pressure, width, fluid composition,
proppant concentration, shear rate, fluid age, viscosity, velocity, and proppant
concentration are defined. The robustness of GOHFER to handle hydraulic fracturing
design, modeling and production forecast makes it a valuable tool in this research
(GOHFER user manual version 8.1.5, 2013).
39
Chapter Three: METHODOLOGY
3.1 Laboratory Work
Due to technology advancements, drill cuttings are gaining recognition and acceptance as
a direct source of information for quantitative petrophysical analysis. Petrophysical data
such as porosity and permeability are now measurable in the laboratory by the GFREE
team at the University of Calgary with minimal error. Fig. 3.1 shows an example of drill
cuttings examined under a microscope that include red, brown and gray shales, limestone,
sand grain, consolidated sand and limestone conglomerate with imbedded glauconitic.
Figure 3.1—Example of drill cuttings under a 10x microscope (Source:
http://en.wikipedia.org/wiki/Drill_cuttings, 2013 ).
This chapter discussed the following laboratory steps leading to the evaluation of drill
cuttings (Adapted from Ortega, 2012):
40
Sample collection
Microscopic analysis
Measurable sample selection
Cleaning and drying of sample
Porosity measurement
Screening of samples prior to permeability measurement
Permeability measurement
Also, good practice to follow while doing this laboratory work was recommended.
3.1.1 Sample Collection
During drilling operations at the rig site; the mud logger picks samples of drill cuttings at
regular intervals for lithology identification and, generally verifies his/her interpretation
with available well site data such as a gamma ray log. These samples are usually taken at
5m intervals after which they are washed and preserved in vials. To preserve the identity
of the samples, each vial is labeled. The common label information on the vial includes
well name and measured depth (Fig. 3.2).
Figure 3.2—Labeled (well name and measured depth) vials containing drill cutting
samples.
41
3.1.2 Microscopic Analysis
Hews (2009-2012) showed that microscopic analysis of drill cuttings provides
information that permits identification of many structural features including fractures,
vugs, lost circulating materials, and shale beddings .
Figure 3.3— Identifying structural features through drill cuttings. (a)The upper
right corner shows a planar (flat surface), and unmineralized drill cuttings sample.
(b) The lower right shows drill cutting samples with mineral infill or linings. (c)
Upper left shows loose, clear, euhedral quartz crystal in sandstone. (d) Lower left
shows a fracture set with mineral infill (Adapted from Hews, 2011).
Ortega and Aguilera (2012) developed the concept of ‘Frac Value’ using drill cuttings.
The Frac Value represents the mean value of three features obtained through visual
inspection of drill cuttings: (1) Fracture sets with mineral lining. (2) Fracture sets with
planar; unmineralized surfaces. (3) Loose mineral crystals (Table 3.1 and Fig. 3.3).
The Frac Value is expressed in percentage using as a base a chart by Terry and Chillingar
(1955) presented in Fig. 3.4. This allows visual estimation of percentages of drill cutting
42
chips in laboratory. Therefore, microscopic analysis is carried out for each drill cuttings
sample to obtain its Frac Value. This information is compared with a brittleness index
and permeability and the integrated information is useful for identifying fracture
initiation zones during hydraulic fracturing design job. Ortega and Aguilera (2012) used
this concept to show how production can be improved from the Nikanassin formation by
pointing to optimum locations to initiate multi-stage hydraulic fractures in horizontal
wells.
Petrographic work involving thin section image analysis of drill cuttings can also serve as
a means of identifying isolated porosity from drill cuttings.
Table 3.1— FRAC VALUE ESTIMATION AT A GIVEN DEPTH FOLLOWING
MICROSCOPIC ANALYSIS OF DRILL CUTTINGS (Adapted from Ortega and
Aguilera, 2012)
SAMPLE None 1-5% 6-10% 11-15% 15%+
0 1 2 3 4
FEATURE A
FEATURE B
FEATURE B
FRAC_VALUE Mean Value
43
Figure 3.4—Chart for visual estimation of percentages of minerals in rock sections.
The interpretation is extended in this work to represent sizes of drill cutting samples
(Terry and Chillingar, 1955).
3.1.3 Measurable Sample Collection
Obtaining an amount of sample that meets the criteria for porosity and permeability
measurement is important. This step is actually tedious and time consuming but the work
is important as it simplifies other steps that follows in the laboratory procedure by
providing only drill cuttings that can be used for both porosity and permeability
measurements. This laboratory step is divided into two stages:
44
3.1.3.1 Stage 1:
This involves sieving the drill cuttings to remove samples with sizes greater than 5mm
and lower than 1mm from the rest of the samples. To save time, it is important to start
with this sieving process first, previous to the other steps explained below as this helps to
concentrate on samples that meet the criteria for porosity and permeability measurements.
Drill cuttings samples less than 1mm are difficult to handle when removing the excess
brine from the surfaces of the saturated drill cuttings as the small samples tend to enter
into the material (soft paper or sponge) used in removing the excess water. Also, smaller
samples tend to be unstable and float while measuring the immersed weight of the
drilling cuttings using Archimedes principle, hence introducing errors into the
measurement. Larger samples (more than 5mm) are acceptable for porosity measurement
but it is beyond the domain of application of Darcylog for measuring permeability.
Therefore, since managing time is essential to the success of this laboratory work,
isolating samples with sizes outside the range of application is important.
3.1.3.2 Stage 2:
Sandstone is the reservoir rock in the Nikanassin Group. This stage involves the picking
and isolation of non-reservoir (non-sandstone) rocks and foreign materials (for example
metals from the drilling string) from the entire sample of drill cuttings. Non-reservoir
rocks such as shale, quartz, lost circulation materials (L.C.M.) are removed from the
sample of drill cuttings to end up with only samples that are sandstone chips between
1mm to 5mm in size.
45
3.1.4 Cleaning, Drying, and Weighting of Sample
Cleaning the drill cutting samples is essential as this helps to remove remaining fluids
and any foreign dirt from the pore space of the drill cuttings. According to Gant and
Anderson (1988), mixtures or series of solvents are generally more effective than using a
single solvent. In their work, they orderly ranked the effectiveness of solvents after
performing a 12-hour solvent evaluation of dolomite cores using Dean Stark extraction
method and demonstrated that a 50% toluene and 50% methanol (mixture) is highly
effective in cleaning core samples. They reached their conclusion because toluene
removed the hydrocarbons and some weakly polar compounds, and methanol removed
the dissolved precipitated salt and polar compounds.
Therefore, in this work we utilize a 50% - 50% mixture of toluene and methanol to clean
the drill cutting samples. The process involves soaking the drill cuttings for at least 30
minutes in this cleaning solvent and then gently stirring the drill cuttings inside the glass
bottle. Once the cleaning solvent appears dirty, the solvent is poured into a waste
container and a new cleaning solvent is used for rinsing the drill cuttings. This process is
repeated continuously until the rinsing solvent is clean.
In the case of drill cuttings used in this work, most of the cleaning had been done at the
rig site and so the cleaning done at the laboratory was not overly time consuming. After
cleaning, the drill cuttings are poured into a glass bottle, which is placed in the oven for
about 3 hours at 110oC. The drying temperature is selected using the boiling point of the
46
highest solvent in the mixture as a guide (the toluene used in this experiment has a
boiling point of 110.60 oC). Fig. 3.5 shows the material safety data sheet for toluene
where the boiling point is stated. After drying the drill cuttings, they are weighted and the
dry weight is recorded. This measured weight is later used for calculating porosity.
Figure 3.5—Material safety data sheet for Toluene
(Adapted from http://kni.caltech.edu/facilities/msds/toluene.pdf ).
3.1.5 Porosity Measurement
The laboratory procedure outlined in the American Petroleum Institute recommended
practice 40 (API RP-40) for core analysis was adapted to determine porosity in the drill
cuttings. According to API RP-40, porosity is defined as the ratio of void space volume
to the bulk volume of the whole rock. Porosity measurement is important since it is used
in estimating the initial hydrocarbons-in-place using volumetric method. Eqs. 3.1 to 3.2
were used to determine from drill cuttings pore volume, bulk volume and porosity,
respectively.
(
…………………….….…….. Eq. 3.1
47
(
…………………........…………Eq. 3.2
(
………………..………………………………Eq. 3.3
Prior to the calculation using Eq. 3.1 to Eq. 3.3, the saturated weight followed by the
immersed weight was measured as explained next.
3.1.5.1 Saturated Weight
3.1.5.1.1 Apparatus
Analytical mass balance accurate to 1 milligram
Beaker that can hold de-aerated liquid under vacuum
Test tbe with tap (125 milli-litre)
Vacuum pump
Brine composed of 20 grams of sodium chloride (NaCl) in one litre of distilled
water
Drill cutting basket for holding cuttings in the beaker
Figure 3.6—Apparatus for measuring saturated weight of drill cuttings (Adapted
from Ortega, 2012).
48
3.1.5.1.2 Procedure
Re-measure and record the weight of the dry and cleaned drill cuttings (Column 3
of Table 3.2)
Pour the drill cuttings inside the cuttings basket
Place the drill cutting basket inside the beaker, the beaker can contain five (5) drill
cutting basket at a time
Fill the test tube with 125 milliliter of brine of known density (1.023g/cc)
Place the test tube on top of the beaker such that it seals the beaker
Apply a high vacuum on the beaker for about 15 minutes
Open the tap on the test tube to allow the brine to flow into the beaker containing
the drill cuttings until the drill cuttings have been completely submerged in the
brine
The drill cuttings are allowed to saturate for at least 45 minutes
The drill cuttings basket is removed from the beaker
The drill cuttings are placed on a paper sheet to remove excess water on the
surface of the drill cuttings
The saturated cuttings are placed on the mass balance and the saturated weight is
recorded (column 4 of Table 3.2)
The difference in weight between the saturated weight and dry weight is divided
by the density of the saturant to obtain the pore volume (column 6 of Table 3.2)
49
3.1.5.1.3 Recommendation
The following recommendation will minimize potential sources of error:
Distilled water must be used in preparing the brine since un-distilled water may
introduce impurities that may plug the pores of the drill cuttings
Extreme care should be taken to prevent the brine from contacting air in the
beaker as this will impair the saturation process
The excess brine on the saturated drill cuttings must be removed in order to
minimize error when measuring the saturated weight
3.1.5.2 Immersed Weight
3.1.5.2.1 Apparatus
Analytical mass balance accurate to one milligram
Transparent liquid container
Drill cuttings basket for holding drill cuttings during suspension
Brine made up of 20 grams of sodium chloride (NaCl) in one litre of distilled
water
50
Figure 3.7—Apparatus for measuring immersed weight based on Archimedes
principle. The Immersed weight is used to determine the bulk volume of the drill
cuttings using Eq. 3.2 (Adapted from Ortega, 2012).
3.1.5.2.2 Procedure
Fill the liquid container with brine of known density (1.023g/cc)
Suspend the empty drill cutting basket with a fine wire and lower it into the liquid
container
Tare the weight of the balance until it reads zero
Remove the empty drill cuttings basket from the liquid container and pour the
drill cuttings inside it
Re-suspend the drill cuttings basket containing the drill cuttings using the fine
wire and lower it into the liquid container
After the weight is stabilized, record the immersed weight (column 5 of Table
3.2)
The immersed weight is divided by the density of immersion liquid to get the bulk
volume (column 7 of Table 3.2)
51
3.1.5.3 Porosity Calculation
The ratio of pore volume to bulk volume is determined to yield the porosity of the
drill cuttings sample. Columns 8 or 9 of Table 3.2 show the determined porosity
values.
3.1.5.4 Recommendation
Ensure the analytical mass balance is tared with empty drill cuttings basket inside
the liquid container so that accurate immersed weight can be obtained
Fill the liquid container to a level that allows it to have some room to
accommodate increase in liquid level in the container due to buoyancy effects
3.1.5.5 Results
Results of the experimental work on drill cuttings up to the determination of porosity are
presented in Table 3.2. The table includes the identification of the samples in column 1,
bottom depth of sample collection in column 2, column 3 shows the measured dry
weight, column 4 contains the measured saturated weight, column 5 has the measured
immersed weight, column 6 has the calculated pore volume, column 7 shows the
calculated bulk volume, and columns 8 and 9 shows the determined porosity quantified in
fraction and percentage, respectively.
52
Table 3.2— RESULTS OF LABORATORY WORK ON DRILL CUTTINGS
(DETERMINATION OF POROSITY)
1 2 3 4 5 6 7 8 9
Sample
No.
Bottom
Depth
MD (m)
Dry Weight (g)Saturated Weight
(g)
Immersed Weight
(g)
Pore Volume(PV)
(cc)
Bulk
Volume(BV)
(cc)
Porosity=
(PV/BV)
(fraction)
Porosity (%)
1 3185.0 1.594 1.643 0.743 0.048 0.734 0.066 6.595
2 3190.0 1.981 2.030 0.773 0.048 0.764 0.063 6.339
3 3195.0 3.113 3.198 1.307 0.084 1.291 0.065 6.503
4 3200.0 1.841 1.883 0.799 0.041 0.789 0.053 5.257
5 3202.5 2.084 2.143 0.901 0.058 0.890 0.065 6.548
6 3205.0 1.642 1.711 0.653 0.068 0.645 0.106 10.567
7 3207.5 1.616 1.675 0.681 0.058 0.673 0.087 8.664
8 3210.0 1.415 1.469 0.599 0.053 0.592 0.090 9.015
9 3217.5 1.061 1.095 0.496 0.034 0.490 0.069 6.855
10 3220.0 1.075 1.121 0.470 0.045 0.464 0.098 9.787
11 3225.0 1.710 1.780 0.708 0.069 0.699 0.099 9.887
12 3235.0 1.154 1.191 0.567 0.037 0.560 0.065 6.526
13 3265.0 2.257 2.334 0.891 0.076 0.880 0.086 8.642
14 3285.0 2.878 3.048 1.137 0.168 1.123 0.150 14.952
15 3290.0 3.634 3.755 1.446 0.120 1.428 0.084 8.368
16 3295.0 3.933 4.023 1.625 0.089 1.605 0.055 5.538
17 3300.0 4.391 4.550 1.751 0.157 1.730 0.091 9.081
18 3305.0 5.009 5.145 1.998 0.134 1.974 0.068 6.807
19 3310.0 5.063 5.174 2.040 0.110 2.015 0.054 5.441
20 3315.0 3.329 3.411 1.316 0.081 1.300 0.062 6.231
21 3320.0 2.833 2.954 1.154 0.120 1.140 0.105 10.485
22 3325.0 1.200 1.245 0.469 0.044 0.463 0.096 9.595
23 3345.0 1.307 1.378 0.585 0.070 0.578 0.121 12.137
24 3355.0 2.085 2.148 0.819 0.062 0.809 0.077 7.692
25 3435.0 1.992 2.045 0.874 0.052 0.863 0.061 6.064
26 3440.0 2.100 2.156 0.850 0.055 0.840 0.066 6.588
27 3445.0 1.595 1.650 0.650 0.054 0.642 0.085 8.462
28 3450.0 9.469 9.762 3.812 0.289 3.766 0.077 7.686
29 3455.0 7.040 7.166 2.808 0.124 2.774 0.045 4.487
30 3460.0 1.664 1.699 0.671 0.035 0.663 0.052 5.216
31 3665.0 1.449 1.475 0.641 0.026 0.633 0.041 4.056
32 3690.0 1.329 1.379 0.721 0.049 0.712 0.069 6.935
33 3710.0 9.801 9.994 3.913 0.191 3.865 0.049 4.932
34 3715.0 7.958 8.105 3.188 0.145 3.149 0.046 4.611
35 3720.0 9.529 9.728 3.871 0.197 3.824 0.051 5.141
36 3725.0 8.613 8.845 3.537 0.229 3.494 0.066 6.559
37 3730.0 6.447 6.648 2.579 0.199 2.548 0.078 7.794
38 3735.0 5.751 5.872 2.273 0.120 2.245 0.053 5.323
39 3740.0 5.973 6.133 2.367 0.158 2.338 0.068 6.760
40 3745.0 2.120 2.176 0.879 0.055 0.868 0.064 6.371
41 3750.0 6.534 6.624 2.570 0.089 2.539 0.035 3.502
42 3755.0 7.127 7.250 2.799 0.122 2.765 0.044 4.394
43 3760.0 4.595 4.687 1.792 0.091 1.770 0.051 5.134
44 3765.0 4.398 4.482 1.744 0.083 1.723 0.048 4.817
45 3770.0 6.724 6.857 2.683 0.131 2.650 0.050 4.957
46 3775.0 3.663 3.760 1.435 0.096 1.418 0.068 6.760
47 3785.0 4.243 4.359 1.762 0.115 1.741 0.066 6.583
48 3790.0 2.986 3.031 1.239 0.044 1.224 0.036 3.632
49 3800.0 1.170 1.198 0.567 0.028 0.560 0.049 4.938
50 3820.0 1.889 1.933 0.765 0.043 0.756 0.058 5.752
51 3835.0 2.943 3.028 1.279 0.084 1.263 0.066 6.646
52 3845.0 1.048 1.070 0.619 0.022 0.611 0.036 3.554
53 3850.0 5.572 5.710 2.257 0.136 2.230 0.061 6.114
54 3855.0 4.369 4.517 1.765 0.146 1.744 0.084 8.385
55 3860.0 3.367 3.452 1.385 0.084 1.368 0.061 6.137
56 3865.0 2.333 2.403 0.958 0.069 0.946 0.073 7.307
57 3880.0 2.771 2.862 1.082 0.090 1.069 0.084 8.410
58 3885.0 2.979 3.043 1.174 0.063 1.160 0.055 5.451
59 3890.0 2.626 2.709 1.018 0.082 1.006 0.082 8.153
60 3895.0 2.607 2.685 1.067 0.077 1.054 0.073 7.310
61 3900.0 2.203 2.268 0.871 0.064 0.860 0.075 7.463
62 3930.0 3.907 4.030 1.621 0.122 1.601 0.076 7.588
63 3935.0 3.145 3.223 1.273 0.077 1.258 0.061 6.127
64 3940.0 5.075 5.214 2.057 0.137 2.032 0.068 6.757
65 3945.0 3.006 3.102 1.262 0.095 1.247 0.076 7.607
66 AVERAGE 3.574 3.668 1.455 0.093 1.438 0.069 6.892
TABLE 3.2 —RESULTS OF LABORATORY WORK ON DRILL CUTTINGS
53
3.1.6 Screening of Samples Prior to Permeability Measurement
After going through the previous procedures and calculating the porosity of each drill
cutting sample, we are now equipped with all required information necessary to
determine if some of the drill cuttings meet the criteria to be used for measuring
permeability using Darcylog equipment. In this step, drill cutting samples with porosity
less than 5% and occupying less than half of the drill cutting cell (basket) are not
included for permeability measurement due to Darcylog limitations (www.cydarex.fr,
2013).
3.1.7 Permeability Measurement
Darcylog equipment for measuring permeability in drill cuttings was designed and is
patented by the French Petroleum Institute, and is built by Cydarex (Paris, France) (Fig.
3.8). According to Lenormand and Fonta (2007), the concept of liquid pressure pulse is
used in Darcylog equipment in such a way that an effective flow inside the pores of the
drill cuttings by compression of residual gas inside the cuttings can be achieved. A
viscous liquid (in this work a mixture of 90% glycerol and 10% distilled water) is used as
displacing fluid in such a way that the pressure decrease in the rock is slowed down.
During the displacement process the pressure versus time is recorded by the equipment
and the volume of oil invading into the pores of the cuttings is derived from the
calibration of the spring/bellow system. After the displacement process, a numerical
simulation model based on equations describing the flow of a viscous liquid in a
compressible medium of spherical geometry is used to calculate the permeability by
54
matching the simulated model data with the experimental model data. A rule of thumb is
to use the beginning of the curve corresponding to more than 1/3 of the total curve. For
more details please refer to Lenormand and Fonta (2007).
Figure 3.8— The left side shows the diagram of the spring and bellow system while
the right side shows the Darcylog Equipment (Adapted from Lenormand and Fonta,
2007).
3.1.7.1 Apparatus
Darcylog
Pump
Viscous liquid (mixture of 90% glycerol and 10% distilled water) suitable for
tight formation since the rate of invasion also depends on fluid viscosity
(Lenormand and Fonta, 2007)
Glass bottle for holding drill cuttings
55
3.1.7.2 Procedure
Pour some quantity of the viscous liquid inside the glass bottle containing the drill
cuttings and stir gently to remove excess air in the pores of the drill cuttings
Open the cover of the drill cutting cell and open all valves i.e. valve 1 to 3
After 30 minutes pour the drill cutting samples inside the drill cuttings basket and
place it in the drill cutting cell of the Darcylog and cover it but not tightly
Using Darcylog software, click start data display from the data acquisition
window to monitor the real-time pressure and temperature
Input required drill cuttings information (well information, porosity, dry weight,
range of cuttings diameter, grain density, temperature, and viscosity) into the
software, and save this information because it will be used for the simulation run
Apply some pressure via the pump to the Darcylog equipment to remove any
trapped air in the circulating system of the Darcy Log
Stop the pump, close the cover of the cutting cell completely
With valve 1 remaining open ,close valves 2 and 3
Apply pressure between 9.5 to10 bars to the pressure cell using the pump
Close valve 1 and wait for few minutes to allow the pressure in the cell to
stabilized
Start recording data from the Darcylog software and open only valve 2
The apparatus will record the measured pressure versus time
Stop the recording after about 100 seconds and match the simulated data with the
experimental data to determine the permeability
56
Remove the drill cuttings basket from the cuttings cell and repeat the same step to
determine the permeability of the next sample
3.1.7.3 Recommendation
1. Ensure the temperature in the input data window is updated to the present liquid
temperature as the experiment continues, as this is used to determine the liquid
viscosity
2. Ensure to gently stir the drill cuttings in the container after pouring the liquid into
it to remove excess air to avoid crushing the drill cuttings samples
3. Ensure the Darcylog equipment is not pressurized for more than 10 bars to avoid
liquid leakage and rupturing of the cylinders in the equipment
3.1.7.4 Results
Results of the experimental work on drill cuttings up to the determination of permeability
are presented in Table 3.3. The table includes the identification of the samples in column
1, bottom depth of sample collection in column 2, column 3 shows the viscosity of the
viscous liquid measured by Darcylog, column 4 contains the temperature of the viscous
liquid measured by Darcylog, column 5 has the outside gas volume which represents the
volume of gas in the outer side of the cuttings fragments or gas trapped between the
rubber seals and Darcylog valves, column 6 contains the initial gas saturation, this values
refers to the gas trapped inside the drill cutting fragments after the spontaneous
imbibition process, column 7 has the porosity values mentioned in previous section and
column 8 contains the determined permeability from drill cuttings. Appendix A shows
screenshots of the best fit between simulated and experimental data.
57
Table 3.3— RESULTS OF LABORATORY WORK ON DRILL CUTTINGS
(DETERMINATION OF PERMEABILITY)
1 2 3 4 5 6 7 8
Sample
No.
Bottom Depth
MD (m)Viscosity (cp) Temp.(deg C)
Outside Gas
Volume (mm3)
Initial Gas
Saturation
(fraction)
Porosity (fraction) K (md)
1 3185.0 154.4 23.8 8.8 0.075 0.066 0.031
2 3190.0 152.2 24.0 11.5 0.067 0.063 0.022
3 3195.0 151.1 24.1 15.0 0.076 0.065 0.030
4 3200.0 151.1 24.1 10.4 0.098 0.053 0.038
5 3202.5 151.1 24.1 106.5 0.085 0.065 0.056
6 3205.0 152.2 24.0 56.0 0.062 0.106 0.088
7 3207.5 151.1 24.1 23.0 0.067 0.087 0.079
8 3210.0 158.8 23.4 139.2 0.097 0.090 0.216
9 3217.5 159.9 23.3 16.2 0.050 0.069 0.025
10 3220.0 159.9 23.3 43.5 0.071 0.098 0.017
11 3225.0 162.2 23.1 13.4 0.042 0.099 0.017
12 3235.0 161.1 23.2 7.0 0.120 0.065 0.069
13 3265.0 156.6 23.6 14.0 0.082 0.086 0.058
14 3285.0 156.6 23.6 11.9 0.060 0.150 0.064
15 3290.0 161.1 23.2 80.0 0.108 0.084 0.078
16 3295.0 159.9 23.3 10.0 0.136 0.055 0.032
17 3300.0 158.8 23.4 135.0 0.125 0.091 0.190
18 3305.0 158.8 23.4 30.4 0.141 0.068 0.109
19 3310.0 158.8 23.4 12.0 0.150 0.054 0.055
20 3315.0 159.9 23.3 8.6 0.148 0.062 0.062
21 3320.0 158.8 23.4 38.0 0.085 0.105 0.069
22 3325.0 157.7 23.5 7.5 0.082 0.096 0.046
23 3345.0 157.7 23.5 95.0 0.102 0.121 0.150
24 3355.0 158.8 23.4 9.6 0.040 0.077 0.025
25 3435.0 157.7 23.5 8.3 0.117 0.061 0.065
26 3440.0 157.7 23.5 8.7 0.105 0.066 0.048
27 3445.0 157.7 23.5 6.8 0.074 0.085 0.051
28 3450.0 157.7 23.5 8.6 0.029 0.077 0.010
29 3460.0 157.7 23.5 9.8 0.066 0.052 0.020
30 3690.0 151.1 24.1 7.7 0.058 0.069 0.023
31 3710.0 157.7 23.5 11.8 0.088 0.05 0.030
32 3715.0 156.6 23.6 75.5 0.078 0.05 0.029
33 3720.0 156.6 23.6 7.9 0.080 0.051 0.025
34 3725.0 156.6 23.6 68.0 0.065 0.066 0.028
35 3730.0 156.6 23.6 41.0 0.058 0.078 0.036
36 3735.0 155.5 23.7 8.7 0.059 0.053 0.010
37 3740.0 153.3 23.9 26.8 0.044 0.068 0.018
38 3745.0 153.3 23.9 120.0 0.100 0.064 0.066
39 3760.0 154.4 23.8 114.0 0.082 0.051 0.057
40 3765.0 153.3 23.9 12.0 0.076 0.048 0.018
41 3770.0 153.3 23.9 57.5 0.077 0.050 0.012
42 3775.0 153.3 23.9 12.0 0.050 0.068 0.015
43 3785.0 153.3 23.9 11.5 0.055 0.066 0.016
44 3820.0 157.7 23.5 12.8 0.052 0.058 0.019
45 3835.0 155.5 23.7 12.2 0.015 0.066 0.011
46 3855.0 153.3 23.9 125.0 0.091 0.084 0.184
47 3860.0 153.3 23.9 18.2 0.069 0.061 0.020
48 3865.0 153.3 23.9 14.2 0.051 0.073 0.015
49 3880.0 153.3 23.9 8.8 0.037 0.084 0.012
50 3885.0 153.3 23.9 9.0 0.072 0.055 0.020
51 3890.0 153.3 23.9 8.0 0.049 0.082 0.017
52 3895.0 154.4 23.8 8.8 0.046 0.073 0.015
53 3900.0 154.4 23.8 10.2 0.053 0.075 0.016
54 3930.0 154.4 23.8 7.0 0.035 0.076 0.010
55 3935.0 154.4 23.8 9.2 0.048 0.061 0.014
56 3940.0 153.3 23.9 28.6 0.045 0.068 0.020
57 3945.0 153.3 23.9 88.5 0.058 0.076 0.080
AVERAGE 155.8 23.7 33.0 0.075 0.047
TABLE 3.3—RESULTS OF LABORATORY WORK ON DRILL CUTTINGS
58
Chapter Four: POROSITY EXPONENT
4.1 The Concerns
The understanding of the role played by vuggy and naturally fractured reservoirs in
hydrocarbon recovery has improved significantly over the past few years as thorough
petrophysical knowledge of reservoir properties has helped to improve estimates of
petroleum-in-place and recoveries. This has happened for example in the case of
‘unconventional’ reservoirs. When conducting petrophysical evaluation in a tight gas
reservoir, an accurate value of the porosity exponent (m) is very important because
variations in this exponent will change the calculated water saturation and therefore affect
the petroleum-in-place estimates. In general, a decrease in m tends to reduce the
computed water saturation values.
A detailed literature review (Chapter 2) demonstrated that petrophysical evaluation
techniques that uses a constant porosity exponent for all depths will likely magnify the
errors in the calculated water saturations especially in tight reservoirs with low porosity.
To address this concern, we develop as part of the present research petrophysical models
that takes into consideration the individual porosity components in heterogeneous
reservoirs for estimating the value of m to be used in log interpretation.
This chapter dicusses the development of the petrophysical model for determining m in a
reservoir represented by dual (matrix and non-connected vugs or matrix and fractures)
59
and /or triple porosity reservoir (matrix, fractures and non-connected vugs/or isolated
porosity).
4.2 Use of Electromagnetic Unified Mixing Rule for Building Dual and Triple
Porosity Models
The dielectric mixing rules of Maxwell Garnett rule, Bruggeman formula and Coherent
potential approximation are unified into one family by Sihvola (1999). The unified
mixing formula assumed a spherical geometry in three dimensions in which isotropic
spherical inclusions with permittivity ԑi occupy random positions in an isotropic host
environment with permittivity ԑe (Fig. 4.1).
Figure 4.1—Schematic of mixture for spherical inclusions with permittivity εi that
occupy random positions in a host environment of permittivity εe. The mixture
effective permittivity is εeff (Sihvola, 1999).
The unified formula is expressed as:
(
( )…………………………………Eq. 4.1
εe
εi
60
Where f is the volume fraction occupied by a spherical inclusion, and vs is a Sihvola
dimensionless parameter. Using different value of the individual mixing rules can be
recovered; for example when vs = 0 in Eq.4.1 gives the Maxwell Garnett rule, vs = 2 leads
to the Bruggeman equation, and vs = 3 simplifies to the Coherent potential
approximation.
In the present work, the term permittivity relates to a material’s capacity to allow (or
“permit’) an electric field. In order to replace the permittivity term with conductivity
equivalences, we use the complex dielectric permittivity which is defined as (Seybold,
2005):
(
) ……….…………………………………………………………..Eq.
4.2
where is the real part of the dielectric permittivity, =√-1, is the conductivity, and
is the angular frequency. At low frequencies ( approaches zero), the conductivity term
in Eq. 4.2 becomes large, and dominates the complex dielectric permittivity ( ). Based
on these observations Sihvola’s unified mixing formula can be recast for the special case
of low frequency where it becomes defined in terms of the porosity exponent (m) that is
familiar to petrophysics practitioners. As a result can be replaced with their
conductivity terms at low frequencies in Eq. 4.1. The same type of reasoning has been
presented by Rasmus and Kenyon (1985).
61
Also, to convert the unified Eq. 4.1 to standard petroleum nomenclature, we assigned
Sihvola’s permittivity notation the following petrophysical equivalences: host
environment permittivity (ԑe) to be equivalent to the conductivity of the matrix porosity;
the spherical inclusions permittivity (ԑi ) corresponds to the water phase, the volume
fraction ( f ) is equal to non-connected vugs or fracture porosity depending on the model,
and the effective permittivity (ԑeff ), which is the summation of the host and the inclusion
is equivalent to the conductivity of the total porosity of the composite system. Since
resistivity is the inverse of conductivity, the permittivity terms in Eq. 4.1 are re-written
with their resistivity value equivalent during the petrophysical model development in
order to relate it to Archie’s empirical equation.
A good match between the theoretical and laboratory values of porosity exponent
presented later in the section dealing with ‘comparison with core data’ lends empirical
evidence that this substitution is valid. Development of the petrophysical equations are
presented in detail later in this chapter (section 4.5). The equivalences between individual
mixing rules, dual and triple porosity models are presented next.
4.2.1 Maxwell Garnett Electromagnetic Mixing Rule
Sihvola (1999) Unified Mixing Rule, when in Eq. 4.1, leads to the Maxwell
Garnett equation:
(
( ………………………………………………………Eq. 4.3
62
Maxwell Garnett developed his famous equation by considering the effective dielectric
constant of a medium where metal spheres occupy a given fraction of the host medium
(Maxwell, 1904). The positions of the spheres are random. The same randomness
assumption is used in the case of the non-connected vugs considered in this chapter.
4.2.1.1 Dual Porosity (Matrix and Non-Connected Vugs)
This model represents the matrix and any type of non-connected porosity that can occur
in a given reservoir. But for simplicity we refer to non-connected vugs throughout this
chapter. For example in the case of carbonate rocks; intragranular, ooilitic, moldic, and/
or fenestral porosity are the mathematical equivalent of non-connected (or isolated) vugs.
In this case the summation of matrix and non-connected vugs is equal to the total porosity
(ϕ) of the system. In the case of tight gas formations isolated non-effective porosity is the
mathematical equivalent of non-connected vugs. When the permittivity terms in Eq. 4.3
are replaced by their respective conductivity and resistivity terms (inverse of
conductivity), all resistivities cancel out as shown in Section 4.5.1 and we end up with
Eq. 4.4, which is a function of porosities at different scales and porosity exponents. The
equation allows calculating m of the composite system of matrix and non-connected vugs
from the equation:
[
((
(
( ( )]
( ………………………………………...Eq. 4.4
where ϕnc is the non-connected vug porosity, ϕb is matrix porosity attached (or scaled) to
the bulk volume of only the matrix block and mb is the porosity exponent of the matrix.
63
The right hand side of Fig. 4.2 shows values of m calculated with the use of Eq. 4.4 for
various values of total and non-connected porosities. The consistency of the model is
demonstrated by the fact that all lines become horizontal when the value of non-
connected porosity becomes equal to the total porosity (matrix porosity is zero). This is
an indication that the model predicts a value of infinity for m of non-connected vugs.
4.2.1.2 Dual Porosity (Matrix and Fractures)
Eq. 4.5 is an adaptation of Eq. 4.3 developed in this paper to construct a model that
incorporates fracture and matrix porosities:
( (
( …………………………………………….Eq. 4.5
where αt is a dimensionless parameter representing the volume fraction of the host and
the inclusion phase, expressed by the empirical equation:
……………………………………….Eq. 4.6
The equation was developed as part of this research by determining the αt that gave a
fracture porosity equal to the total porosity (matrix porosity is zero); then the respective
fracture porosity was correlated with αt using a polynomial regression that led to Eq. 4.6.
Eq. 4.6 is valid for ϕ2 smaller than 0.1 which covers most cases of practical importance.
When the permittivity terms in Eq. 4.6 are replaced by their respective conductivities
(and resistivities), all resistivities cancel out as shown in Section 4.5.2. This leads to Eq.
4.7 that calculates the dual porosity exponent (m) in reservoirs made out of matrix and
fracture porosity.
64
[ ( ] (
[ ( ] (
( ……………………………………….…Eq. 4.7
Results from Eq. 4.7 are presented in the left hand side of Fig. 4.2. Values of m are
shown as a function of total and fracture porosities. Note that m = 1.0 when the fracture
porosity is equal to total porosity. Thus, in this model the porosity exponent of the
fractures is equal to 1.0. This equation was compared with Aguilera’s (2006, 2009)
petrophysical model that considers the effect of the angle between the fracture and the
direction of current flow on the porosity exponent, and Berg’s (2006) effective medium
petrophysical model that includes the angle that the normal to the fracture makes with the
current flow. It is concluded that the porosity exponent obtained with the petrophysical
model represented by Eq. 4.7 calculates about the same m when the angle, θ = 0°, using
Aguilera (mf =1.0) (2009, 2010) and Berg petrophysical model (θ’ = 90°, mf =1.0).
Figure 4.2— Chart for determining m as a function of non-connected vug porosity
(nc) or fracture porosity (2) for the case in which mb= 2.0. Petrophysical model is
0.0001
0.001
0.01
0.1
1
1.0 1.5 2.0 2.5 3.0
Tota
l Po
rosi
ty, φ
Dual Porosity Exponent, m
ɸ2=0.01
ɸ2=0.015
ɸ2=0.001
ɸ2=0.0001
ɸnc=0.050
ɸnc=0.010
ɸnc=0.0010
ɸnc=0.0001
65
developed on the basis of the Electromagnetic Mixing Rule (Maxwell Garnett, vs =
0).
4.2.1.3 Triple Porosity (Matrix, Non Connected Vugs and Fractures)
Aguilera and Aguilera (2004) developed a triple porosity model for evaluation of
naturally fractured reservoirs, which was subsequently improved by Al-Ghamdi et al.
(2011). The basic assumptions are that the matrix and fractures have conductivities that
are connected in parallel; and the combined matrix and fractures are connected in series
with the non-connected vugs. Berg (2006) provided another way of quantifying the same
triple porosity model parameters with the use of Eqs. 2.5 and 2.6 as discussed in Chapter
two. This chapter uses the same methodology proposed by Berg as follows: (1) calculate
a porosity exponent mb’ using Eq. 4.8; note that Eq. 4.8 has the same form as Eq. 4.4 but
ϕ and m are replaced with ϕb’ and mb’. (2) Transfer the parameter mb’ and ϕb’ into Eq. 4.7
to replace mb and ϕb, respectively. This results in Eq. 4.9 from which the triple porosity
exponent, m is calculated.
[
((
(
( ( )]
(
…………………………………….Eq. 4.8
[ ( ] (
[ ( ] (
( ……………………………………….Eq. 4.9
Fig. 4.3 shows an example of triple porosity results under the assumption that vs is equal
to zero (Maxwell Garnett). Note that the value of m for the triple porosity model can be
bigger, equal to, or smaller than mb depending on the contribution of matrix, non-
connected vugs and fractures to the total porosity system.
66
Figure 4.3— Chart for determining m as a function of non-connected vug
porosity(nc) and fracture porosity(2) for the case in which mb = 2.0. Petrophysical
model developed on the basis of the Unified Electromagnetic Mixing Rule (Maxwell
Garnett, vs = 0).
4.2.2 Bruggeman Electromagnetic Mixing Rule
Use of the Unified Mixing Rule, when in Eq. 4.1 results in the Bruggeman
equation:
(
( )…………………………………..Eq. 4.10
When the permittivity terms in Eq. 4.10 are replaced by the corresponding conductivity
and resistivity terms, all the resistivities cancel out as shown in Section 4.5.3 and we
obtain Eq. 4.11 that allows calculation of the dual porosity exponent (m) for reservoirs
with matrix and non-connected vug porosities:
0.01
0.1
1
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0To
tal P
oro
sity
, φ
Triple Porosity Exponent, m
ɸnc=0.1,ɸ2=0.01
ɸnc=0.075,ɸ2=0.01
ɸnc=0.05,ɸ2=0.01
ɸnc=0.01,ɸ2=0.01
ɸnc=0.02,ɸ2=0.01
67
(
)………………………...………………..Eq. 4.11
This equation cannot be solved explicitly to estimate the porosity exponent (m) as in the
case of the Maxwell Garnett approach. An iterative approach has to be used by assuming
m in the right hand side and calculating m in the left hand side until a minimum
acceptable error is accepted. Microsoft Excel Solver was utilized for this purpose. Fig.
4.4 shows an example of calculated results using Eq. 4.11.
Figure 4.4— Chart for determining m as a function of non-connected vug
porosity(nc) for the case in which mb = 2.0. Petrophysical model developed on the
basis of the Unified Electromagnetic Mixing Rule (Bruggeman, vs = 2).
0.01
0.1
1
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0
Tota
l Por
osit
y, ɸ
Dual Porosity Exponent,m
ɸnc=0.125
ɸnc=0.010
ɸnc=0.050
ɸnc=0.075
ɸnc=0.100
68
4.2.3 Coherent Potential Electromagnetic Mixing Rule
When vs = 3 the unified mixing formula (Eq. 4.1) leads to the Coherent potential mixing
rule:
(
( )…………………………………..Eq. 4.12
Development of the above equation is presented in section 4.5.4. As in the previous
cases, when the permittivity terms in Eq. 4.12 are replaced by conductivity and resistivity
terms, the resistivities cancel out. The result is Eq. 4.13 that allows calculating the dual
porosity exponent (m) for reservoirs represented by matrix and non-connected vug
porosities:
[
(
)]
(
)………………….......…………………..Eq. 4.13
As in the case of the Bruggeman model this equation cannot be solved directly to
estimate the porosity exponent (m). An iterative method is required. Microsoft Excel
Solver has been utilized in this thesis. Fig. 4.5 shows an example of calculated results
using Eq. 4.13.
69
Figure 4.5—Chart for determining m as a function of non-connected vug
porosity(nc) for the case in which mb = 2.0. Petrophysical model developed on the
basis of the Unified Electromagnetic Mixing Rule (Coherent Potential, vs = 3).
4.3 Model Development
4.3.1 Derivation of Maxwell Garnett Mixing Rule Extension to Matrix and Non-
Connected Vugs
The unified mixing formula according to Sihvola (1999) is stated as follows:
(
( )……………………...………….Eq. 4.14
When, the unified mixing formula (Eq. 4.14) leads to the Maxwell Garnett mixing
rule as follows:
(
( …………………………………………………...….Eq. 4.15
0.01
0.1
1
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0
Tota
l Por
osit
y, φ
Dual Porosity Exponent,m
ɸnc=0.125
ɸnc=0.010
ɸnc=0.050
ɸnc=0.075
ɸnc=0.100
70
By expansion of Eq. 4.15 we obtain:
( )( ( )( …………….…………Eq. 4.16
Factorization of Eq. 4.16 leads to:
[ ( ( ] [ ( ] ( …………….Eq. 4.17
[ ( (
( ( ]………………………………………….……Eq. 4.18
which is the Maxwell Garnett’s permittivity of the host is assumed in this chapter to
be equal to the conductivity of the rock matrix while is assumed to be equivalent to
the conductivity of the composite system. Therefore the basic Archie’s equations for only
matrix and composite system can be written respectively as follows:
…………………………………………..………………..Eq. 4.19
……………………………………..………………………Eq. 4.20
………………………………………………………...…Eq. 4.21
…………………….……………………………………..Eq. 4.22
Permittivity of the inclusion phase, which has a volume fraction , is assumed to
represent the water phase in the non-connected vugs and is written as:
……………………………………………………………………..Eq. 4.23
Replacing the permittivity with conductivity and using standard petrophysical resistivity
notation (the inverse of conductivity); and replacing with nc transforms equations (Eq.
4.18) to (Eq. 4.23) into:
71
[
(
(
(
(
]…………………………………….……Eq. 4.24
[
(
(
(
(
] ……………………………………..Eq. 4.25
leading to:
[
(
(
( ( ]……………...………………………..Eq. 4.26
[
(
(
( ( ]
( …………………………………………..Eq. 4.27
This is the dual porosity exponent equation for the non-connected vugs and matrix
porosity presented in Eq. 4.4 of this chapter.
4.3.2 Maxwell Garnett Mixing Rule Extension to Matrix and Fractures
Maxwell Garnett equation (Eq. 4.18), used for development of the dual porosity exponent
for matrix and non-connected vugs (Eq. 4.27), has been extended in this section to
provide an equation for a dual porosity model made out of matrix and fractures as
follows:
(
………………………………………….………Eq. 4.28
Eq. 4.28 can be expanded:
( )( ( ( )( ………………….Eq. 4.29
72
( ( ( ) ( ( ))
( ( ( ) ( ( ))………..……………………..……Eq. 4.30
Replacing permittivity with conductivity, using standard petrophysical resistivity notation
(the inverse of conductivity) and replacing with 2 transforms Eq. 4.30 to Eq. 4.31:
(
(
( )
(
(
( )
…………………………..…...Eq. 4.31
[
(
( ]
(
(
…………………….……….Eq. 4.32
leading to:
[ ( ] (
( ( ) ( ………...……………………………….Eq. 4.33
[ ( ] (
[ ( ] (
( …………………………...……………..Eq. 4.34
where (for 0.0001 ≤ ϕ2 ≤ 0.1 )…………...……….Eq. 4.35
The extension of the Maxwell Garnett mixing formula represented by Eq. 4.35 is the
same as Eq. 4.7 in this chapter for the case of a dual porosity model represented by
matrix and fractures.
4.3.3 Bruggeman Mixing Rule Extension to Matrix and Non-Connected Vugs
When, the Sihvola’s (1999) unified mixing formula, Eq. 4.14 leads to the
Bruggeman mixing rule as follows:
73
(
( )…………………………………...Eq. 4.36
………………………………………….……………..Eq. 4.37
By factorization we obtain:
( …………………………………..…………..Eq. 4.38
Replacing permittivity with conductivity, using standard petrophysical resistivity notation
(the inverse of conductivity) and replacing with nc transforms Eq. 4.38 to Eq. 4.39:
(
)……………………………………...…....Eq. 4.39
(
)
………………………………………..Eq. 4.40
[
(
)]…………………………………..Eq. 4.41
leading to:
(
)………………...………………………..Eq. 4.42
This is the dual porosity exponent equation presented in Eq. 4.11 for the non-connected
vugs and matrix porosity system developed starting with the Bruggeman mixing formula.
The equation has to be solved by iterations to obtain the value of m for the composite
system of matrix and non-connected vugs.
74
4.3.4 Coherent Potential Mixing Rule Extension to Matrix and Non-Connected Vugs
When, the unified mixing formula, Eq. 4.14 results to the Coherent potential
mixing rule as follows:
(
( )……………………………….…..Eq. 4.43
……………………………………………..……….Eq. 4.44
[ ( ] [ ( ]……....Eq. 4.45
Replacing permittivity with conductivity, using standard petrophysical resistivity notation
(the inverse of conductivity) and replacing with nc transforms Eq. 4.45 to Eq. 4.46.
[
(
)]
[
(
)]…………………………………………..………….Eq. 4.46
[
(
)]
[
(
)]………………………………………….….Eq. 4.47
(
)
[ (
)]
(
)
[
( )]……………………….………………Eq. 4.48
(
)
and we obtain:
75
[ (
)]
[
( )]………………………………………...……….Eq. 4.49
[
(
)]
(
)………………………...………………..Eq. 4.50
This is the dual porosity exponent equation for the non-connected vugs and matrix
porosity system presented in section 4.2.3 as Eq. 4.13.
4.4 Model Validation
4.4.1 Comparison of Available Models
Sihvola’s (1999) unified equation for handling the three electromagnetic mixing rules
was used to develop petrophysical means of estimating the porosity exponent of dual and
triple porosity models made out of matrix, non-connected vugs and natural fractures.
Only the Maxwell Garnett mixing rule has been extended to determine the porosity
exponent in reservoirs made out of matrix and fractures but the three electromagnetic
mixing rules have been used to determine the porosity exponent in reservoirs made out of
matrix and non-connected vugs. The three rules provide consistent results but the
difference between the three models gets bigger as the value of the non-connected
porosity gets bigger than 0.1. This is shown in Fig. 4.6.
76
Figure 4.6—Comparison chart for determining m as a function of non-connected
vug porosity (nc) for the case in which mb = 2.0. Petrophysical model developed on
the basis of the Unified Electromagnetic Mixing Rule -Maxwell Garnett( vs = 0),
Bruggeman (vs = 2) and Coherent Potential (vs = 3).
However, models based on the assumption of parallel and series resistors and the
Maxwell Garnett mixing rule are easier to run as the calculations for determining m are
explicit. On the other hand the Bruggeman, Berg and Coherent potential models require
iterative procedures for calculating m. The porosity exponent of the matrix (mb) is
assumed to be equal to 2.0 in all these comparisons. Key equations that link matrix,
fracture and non-connected porosity are presented in APPENDIX B.
0.01
0.1
1
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0
Tota
l P
oro
sity
, Ф
Comparison of Dual Porosity Exponent,m for Different EMR Models
Vs=0,PHINC=0.05
Vs=0,PHINC=0.010
Vs=0,PHINC=0.10
Vs=0,PHINC=0.075
Vs=2.0,PHINC=0.10
vs=2.0,PHINC=0.050
Vs=2.0,PHINC=0.010
Vs=3.0,PHINC=0.10
Vs=3.0,PHINC=0.050
Vs=3.0,PHINC=0.010
Vs=3.0,PHINC=0.075
Vs=2.0,PHINC=0.075
ɸnc=0.100
ɸnc=0.050
ɸnc=0.010
ɸnc=0.075
77
Fig. 4.5 to 4.6 discussed previously show charts for evaluating the dual and triple
porosity exponent for reservoirs represented by matrix, vugs and /or fracture porosity.
Fig. 4.7 shows a comparison of dual porosity models based on three different theories:
Maxwell Garnett (this study, Eq. 4.4 and Eq. 4.7); Aguilera and Aguilera (2003) and
Berg (2006). In all cases the porosity exponent of the matrix (mb) is equal to 2.0. All the
models give essentially the same values of m for the case of fractures and matrix as
shown in the left hand side of the graph. The first two methods allow calculation of m
explicitly. Berg’s m values are calculated by an iteration procedure. In the case of matrix
and non-connected vugs Berg’s iteration procedure uses an infinite value of the porosity
exponent of the vugs (actually mv = 1E35 in an Excel spread sheet). There are some
differences in the case of the dual porosity models made out of matrix and vugs. Aguilera
and Aguilera (2003) and Berg (2006) provide the same results for the cases involving
matrix and fractures. Eq. 4.4 based on Maxwell Garnett (vs = 0) for matrix and non-
connected vugs gives m values that are somewhat smaller at large total porosities. These
smaller values can be matched with Berg’s model when mv is made equal to 1.5.
78
Figure 4.7— Comparison of dual porosity models made out of matrix and fractures
or matrix and non-connected vugs from various theories show good agreement.
Porosity exponent of the matrix mb = 2.0. Maxwell Garnet and Aguilera’s equations
for calculating m are explicit. Berg’s solution uses an iteration procedure and for the
vugs case assumes an infinite value of mv (in reality it assumes mv = 1E35 in a spread
sheet for the above curves in the right hand side of the graph).
Fig. 4.8 shows a good comparison of models based on the Unified Electromagnetic
Mixing Rule (Maxwell Garnett, vs = 0) represented by the solid lines and Berg’s effective
medium theory (X symbol). The comparison is made for naturally fractured reservoirs
and reservoirs with vugs (mb = 2.0, mv =1.5, mf =1.0). Berg suggests that values such as
mv =1.5 could be indicative of connected vugs (as opposed to mv being infinite). It could
be, but the problem from a practical point of view is the quantification of the porosity
exponent (mv) for vugs.
0.0001
0.001
0.01
0.1
1
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Tota
l Po
rosi
ty, Ф
Maxwell PHINC=0.10
Maxwell PHINC=0.05
Maxwell PHINC=0.010
Maxwell PHI2=0.01
Maxwell PHI2=0.001
Maxwell PHI2=0.0001
Aguil_PHINC=0.10
Aguil_PHINC=0.05
Aguil_PHINC=0.01
AGUIL PHI2=0.01
AGUIL PHI2=0.001
AGUIL Phi2=0.0001
Berg PHIV=0.10
Berg PHIV=0.05
Berg PHIV=0.010
Berg PHI2 =0.01
Berg Phi2=0.001
Berg PHI2=0.0001
ɸnc=0.100
ɸnc=0.050
ɸnc=0.010
ɸ2=0.01
ɸ2=0.001
ɸ2=0.0001
Dual Porosity Exponent, m
79
Figure 4.8—Comparison of models based on the Unified Electromagnetic Mixing
Rule (Maxwell Garnett, vs = 0) represented by the solid lines and Berg effective
medium theory (x symbol). Comparison is made for naturally fractured reservoirs
and reservoirs with vugs (mb = 2.0, mv =1.5, mf =1.0). According to Berg, small values
of mv might be indicative of connected (or partially connected) vugs.
Fig. 4.9 compares results obtained with the model developed in this paper (Eq. 4.7) for
dual porosity (matrix and fractures) and Aguilera’s model with a fracture angle between
the fracture and the direction of current flow equal to 0°. Fracture porosities in the graph
are equal to 0.0150, 0.010, 0.0010, and 0.0001.
80
Figure 4.9—Comparison of models based on the Electromagnetic Mixing Rule, vs =
0 (solid lines) and Aguilera fracture dip model, θ = 90°- fracture dip (x symbol).
Comparison is made for naturally fractured reservoirs (mb = 2.0, fracture dip, 0°).
Fig. 4.10 presents a comparison of triple porosity models based on the electromagnetic
mixing rule, vs = 0 (solid lines), and Berg’s effective medium theory (X symbol). The
comparison is very good as results are approximately the same. The graph is developed
for mb = 2.0, mv =1.5, mf =1.0, θ = 90°.
81
Figure 4.10—Comparison of triple porosity models based on the Electromagnetic
Mixing Rule, vs = 0 (solid lines) and Berg’s effective medium theory (x symbol).
Comparison is good (mb = 2.0, mv =1.5, mf =1.0, θ’ = 90°).
Fig. 4.11 is a comparison of models based on the Unified Electromagnetic Mixing Rule,
Bruggeman’s equation (vs = 2) and Berg’s effective medium theory (X symbol).
Comparison is for reservoirs with matrix and non-connected vugs (mb = 2.0, mv =1.5).
82
Figure 4.11—Comparison of models based on the Electromagnetic Mixing Rule, vs
= 2 (solid lines) and Berg’s effective medium theory (x symbol). Comparison is for
reservoirs with non-connected vugs (mb = 2.0, mv =1.5).
Fig. 4.12 shows a comparison of models based on the Electromagnetic Mixing Rule for
the coherence potential approximation ( vs = 3) and Berg’s effective medium theory (X
symbol). The comparison is good for the assumed values of mb = 2.0 and mv =1.5.
0.010
0.100
1.000
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0
Tota
l P
oro
sity
Ф
Dual Porosity Exponent,m
ɸnc=0.010
ɸnc=0.125
ɸnc=0.100
ɸnc=0.075
ɸnc=0.050
83
Figure 4.12—Comparison of models based on the Electromagnetic Mixing Rule, vs
= 3 (solid lines) and Berg’s effective medium theory (x symbol). Comparison is for
reservoirs with non-connected vugs (mb = 2.0, mv =1.5).
4.4.2 Comparison with Core Data
A key component of this thesis is the comparison of the model developed in this study
and previous models with core data. An excellent data bank for carbonate reservoirs has
been presented by Ragland (2002). In her work values of the porosity exponent were
calculated from laboratory resistivity and porosity data and compared to thin section
analyses of the same pore systems. Ragland’s data base for limestone and dolomite
reservoirs are used for comparison with the theoretical models presented in this chapter.
Equally outstanding is a data bank presented by Byrnes et al. (2006) for the case of tight
gas sandstones in the Mesaverde formation of the United States. The data of Byrnes et al.
(2006) are also used for comparison purposes in this section. All comparisons show that
0.010
0.100
1.000
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0
Tota
l Po
rosi
ty Ф
Dual Porosity Exponent,m
ɸnc=0.100
ɸnc=0.075
ɸnc=0.050
ɸnc=0.010
84
the theoretical methods provide good results that match the core data. The advantage of
the explicit methods (Aguilera and Aguilera, 2003; this study) is that they are easier to
use. Berg’s porosity exponent for the fractures is assumed to be equal to 1.0. Berg’s
porosity exponent for the vugs is assumed to be 1.5 in the case of the triple porosity
example.
The difference between the porosity exponent from core data and the model introduced in
this study is generally small as shown by the absolute error scale (%) on the secondary
vertical axis and the open red circles at the bottom of Fig. 4.13 to Fig. 4.18. Some data
points with higher error represent pore systems with different characteristics such as non-
connected moldic pore systems, partially connected moldic and interparticle pore systems
discussed by Ragland (2002). The error value is calculated by considering that m from
core data is the correct value.
4.4.2.1 Limestone Reservoirs
Wells A, B, and J of Ragland’s (2002) data base are used in this case. The comparison of
laboratory data and three different theoretical models are presented in Fig. 4.13, Fig. 4.14
and Fig. 4.15. . The partitioning coefficient (v) mentioned in the title of these figures is
equal to fracture porosity divided by the total porosity. The values of v and mb are
selected in such a way that they provide a good fit of the core data. The theoretical
models were developed by Aguilera and Aguilera (2003), Berg (2006) and in this study
using the unified mixing model (Sihvola, 1999) and vs = 0, which leads to the Maxwell
Garnett equation. The Maxwell Garnett and Aguilera’s models are explicit and permit
85
determination of the porosity exponent (m) without any trial and error. Berg’s curve was
developed with an iteration procedure.
All the theoretical models do a good job in well A (Fig. 4.13). The difference between the
m values from core data and the model introduced in this study (Maxwell Garnett with vs
= 0) is less than 10% as shown by the error scale. The mean absolute percentage error
(MAPE) for this case is 1.84%.
Figure 4.13—Comparison of limestone core data from well A (Ragland, 2002) and
dual porosity models made out of matrix and fractures developed by Aguilera and
Aguilera (2003), Berg (2006) and this study (Maxwell Garnett, vs = 0).The open red
circles show the absolute error as compared with core data. The mean absolute
percentage error (MAPE) is 1.84%.
Fig. 4.14 shows the difference between m from core data and the theoretical model for
well B. The absolute error is generally less than 10%. The data point enclosed in the red
square has an m from cores equal to 2.40. In his case the error is slightly larger than 10%.
The reason for the difference is that about 64% of the pore system in this sample is
0
10
20
30
40
50
60
70
80
90
100
1.0
1.5
2.0
2.5
3.0
0.14 0.16 0.18 0.20 0.22 0.24 0.26
Err
or,
%
Du
al P
oro
sit
y E
xp
on
en
t,m
Total Porosity
Well A Limestone,mb =2.15,v=0.025
Maxwell Vs=0
Ragland's m,cores,2002
Aguilera &Aguilera,2003
Berg's 2006
Error: Core data-Maxwell
86
dominated by a non-connected moldic pore system (Ragland 2002). The MAPE in this
well is 7.15%
Figure 4.14—Comparison of limestone core data from well B (Ragland, 2002) and
dual porosity models made out of matrix and fractures developed by Aguilera and
Aguilera (2003), Berg (2006) and this study(Maxwell Garnett, vs = 0). The blue data
point enclosed in a red square corresponds to a sample where approximately 64% of
the pore system is dominated by non-connected moldic pores (Ragland 2002). The
MAPE is 7.15%
The difference between m from core data and the model introduced in this study
corresponds to an error smaller than 10% for well J as shown in Fig. 4.15. The data point
enclosed in the red square shows an m equal to 1.63 and an errror that is in the order of
20%. This data point corresponds to a sample characterized by a connected moldic pore
system (Ragland 2002) which reduces significantly the core value of m. The MAPE for
the sample population of this well is 7.40%.
87
Figure 4.15—Comparison of limestone core data from well J (Ragland, 2002) and
dual porosity models made out of matrix and fractures developed by Aguilera and
Aguilera (2003), Berg (2006) and this study(Maxwell Garnett, vs = 0). The blue data
point enclosed in a red square with m = 1.63 shows an error in the order of 20%.
This sample is characterized by a connected moldic pore system that reduces
significantly the value of m (Ragland 2002). The MAPE is equal 7.40%.
4.4.2.2 Dolomite Reservoirs
Wells C and E of Ragland’s (2002) data are used in this case. The comparison of m
values from core data and the same theoretical models mentioned above is good as shown
on Fig. 4.16 and Fig. 4.17. Another example using a triple porosity model has been
presented by Al-Ghamdi et al. (2011).
Fig. 4.16 shows the difference between m from core data and the Maxwell Garnett
model. The comparison is good with an error that is generally less than 10%. The MAPE
is 4.91%.
88
Figure 4.16—Comparison of dolomite core data from well C (Ragland, 2002) and
dual porosity models made out of matrix and fractures developed by Aguilera and
Aguilera (2003), Berg (2006) and this study(Maxwell Garnett, vs = 0). The error is
generally less than 10%. The MAPE is 4.91%.
The errors stemming from differences between m from core data and the model
introduced in this study are generally less than 10% as shown in Fig. 4.17. The data
points enclosed in red squares correspond to m values equal to 2.26, 2.91, 2.81 and 2.21.
In these cases the errors are above 10%. These data points correspond to samples
dominated by connected and interparticle pore systems or moldic pore systems (Ragland
2002). The MAPE for this case including all samples is 7.67%.
0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
90.00
100.00
1.0
1.5
2.0
2.5
3.0
0.00 0.02 0.04 0.06 0.08 0.10
Err
or,
%
Du
al
Po
rosit
y E
xp
on
en
t,m
Total Porosity
Well C Dolomite,mb =3.0,v=0.016
Maxwell Vs=0
Ragland's m,cores,2002
Aguilera &Aguilera,2003
Berg's 2006
Error Value: Core data-Maxwell
89
Figure 4.17—Comparison of dolomite core data from well E (Ragland, 2002) and
dual porosity models made out of matrix and fractures developed by Aguilera and
Aguilera (2003), Berg (2006) and this study (Maxwell Garnett, vs = 0). The largest
errors correspond to data points enclosed in red squares. These are dominated by
either connected and interparticle pore system (lower values of m) or non-
connected moldic pore systems (larger values of m) (Ragland 2002). The MAPE is
equal to 7.67%.
4.4.2.3 Tight Gas Sandstone
In this case m data based on core analysis published by Byrnes et al. (2006) are compared
against theoretical models developed by Al-Ghamdi et al. (2011), Berg (2006) and in this
study using the unified mixing model (Sihvola, 1999) and vs = 0, which leads to the
Maxwell Garnett equation but now considering a triple porosity model. The comparison
with all three models is good as shown on Fig. 4.18. The error is generally less than 10%.
The MAPE is 3.58%.
APPENDIX B shows relevant equations that (Aguilera and Aguilera, 2003, 2004) are useful
during the calculations to maintain consistency in the scaling of porosity.
0
10
20
30
40
50
60
70
80
90
100
1.0
1.5
2.0
2.5
3.0
0.10 0.12 0.14 0.16 0.18
Err
or,
%
Du
al P
oro
sit
y E
xp
on
en
t,m
Total Porosity
Well E Dolomite,mb =2.7,v=0.015
Maxwell Vs=0
Ragland's m,cores,2002
Aguilera &Aguilera,2003
Berg's 2006
Error: Core data-Maxwell
90
Figure 4.18—Comparison of tight gas sandstone core data from wells in the
Mesaverde formation (Byrnes et al., 2006) and triple porosity models made out of
matrix, fractures and slots, and non-connected porosity developed by Al-Ghamdi et
al. (2011), Berg (2006) and this study(Maxwell Garnett, vs = 0). The error is
generally less than 10%. The MAPE is equal to 3.58%
0
10
20
30
40
50
60
70
80
90
100
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
0.00 0.05 0.10 0.15
Err
or,
%
Tri
ple
Po
rosit
y E
xp
on
en
t,m
Total Porosity
Comparison of Petrophysical Model with Tight Sandstone Reservoir Core Data
Maxwell Vs=0
Mesaverde m,core data
Al-Ghamdi 2011
Berg's 2006
Error: Core data-Maxwell
91
Chapter Five: WATER SATURATION EXPONENT
5.1 The Concerns
A detailed literature review (Chapter 2) demonstrated that the water saturation exponent
( ) is not constant in petroleum reservoirs but varies as a function of different factors
including fluid distribution in the pore spaces and rock wettability. However, most
studies carried out on petroleum reservoirs with mixed wettability condition are
laboratory studies with limited work done for estimating . To contribute to the solution
of this concern, we present in this chapter how we used electromagnetic mixing rules to
develop a new petrophysical model capable of estimating in heterogeneous reservoirs
with mixed wettability. The reservoirs are represented by dual porosity (matrix and
fractures or matrix and isolated porosity) and triple porosity (matrix, fractures and
isolated porosity) models. In our petrophysical model, we follow the concept of mixed
wettability proposed by Salathiel, 1973; Melrose, 1982; and Hall et al., 1983. This
concept was explained in Chapter 2.
5.2 Theoretical Models
The three models developed in this chapter are as follows: (1) Petrophysical model for
calculating values of n in reservoirs made up of matrix and isolated porosity, (2)
Petrophysical model for calculating values of n in reservoirs made up of matrix and
fracture porosities, and (3) Petrophysical model for calculating n in reservoirs made up of
matrix, isolated and fracture porosities.
92
To use the three petrophysical models proposed in this study, the porosity exponent (m)
representing these multi-porosity reservoirs is required as input data. To accomplish this
task; we used two recent petrophysical models (Eq. 5.1 and Eq. 5.2) that can determine
m for dual and triple porosity reservoirs. These models have been validated with core
data and proven to estimate with reasonable certainty the multi-porosity values of m
(Aguilera, 2009, 2010; Olusola et al., 2013). Eq. 5.1 was developed using the
electromagnetic mixing formula of Maxwell Garnett (Olusola et al., 2012). It calculates
m for a dual porosity system made up of matrix and any type of non-connected or isolated
porosity. For example in the case of carbonate rocks, intragranular, ooilitic, moldic, and/
or fenestral porosity are the mathematical equivalent of isolated porosity. In the case of
this dual porosity model the summation of matrix and isolated porosity is equal to the
total porosity ϕ of the system.
[
((
(
( ( )]
( ……………...…………………………………Eq. 5.1
Eq. 5.2 is used to determine the m value of a dual-porosity model made up of matrix and
fractures (Aguilera and Aguilera, 2009, 2010). This case includes an angle θ between the
fracture and the direction of current flow (0o to 90
o).
[
]
( ……………………………………………...………………….Eq. 5.2
where,
(
………………………………………………………….……Eq. 5.3
( ………...……………………………….……………Eq. 5.4
93
Values of m from the above equations are used in the determination of n. The next
section explains the development of the petrophysical model that determines n in dual
and triple porosity reservoirs.
5.2.1 Dual Porosity (Matrix and Isolated Porosity)
The dielectric mixing rules of Maxwell Garnett, Bruggeman and Coherent Potential
approximation were unified into one family (Eq. 5.5) by Sihvola (1999). The unified
mixing formula assumes a spherical geometry in three dimensions in which isotropic
spherical inclusions with permittivity ԑi occupy random positions in an isotropic host
environment with permittivity ԑe . In this present work, the term permittivity relates to a
material’s capacity to allow (or “permit’) an electric field (Olusola et al. 2013).
( )
( ) …………………………...……………….Eq. 5.5
where f is the volume fraction occupied by a spherical inclusion, is the effective
permittivity, and vs is a Sihvola dimensionless parameter. Using different value of the
individual mixing rules can be recovered; for example when vs = 0 in Eq. 5.5 it gives the
Maxwell Garnett formula (Eq. 5.6):
[ ( (
( ( ]……………………………………………..…………Eq. 5.6
Recently, Olusola et al. (2013) used Eq. 5.6 to develop a petrophysical model that
determines the porosity exponent m in dual and triple porosity reservoirs. The model
compares well with core data and petrophysical models developed by Berg (2006),
94
Aguilera and Aguilera (2003, 2009) and Al-Ghamdi et al. (2011). To convert Eq. 5.6 to
oil and gas notation, we assigned Sihvola’s notation the following petrophysical
equivalences: host environment permittivity (ԑe) to be equivalent to the conductivity of
the matrix porosity; the spherical inclusions permittivity (ԑi ) corresponds to the
conductivity of the inclusion phase, the volume fraction f is equal to isolated porosity
( , and the effective permittivity (ԑeff) is equivalent to the conductivity of the total
porosity system. In order to replace the permittivity term with conductivity equivalences,
we use the complex dielectric permittivity which is defined as (Seybold, 2005):
(
) …………………..…………………………………..…..Eq. 5.7
where is the real part of the dielectric permittivity, =√-1, is the conductivity, and
is the angular frequency. At low frequencies ( approaches zero), the conductivity term
in Eq. 5.7 becomes large, and dominates the complex dielectric permittivity ( ). Based
on these observations, Eq. 5.6 can be recast for the special case of low frequency where it
becomes defined in terms familiar to petrophysics practitioners. As a result permittivity
terms can be replaced with their conductivity terms at low frequencies in Eq. 5.6 to yield
Eq. 5.8 (Olusola et al., 2013). The same type of reasoning using the complex dielectric
permittivity has been presented by Rasmus and Kenyon (1985), and Sihvola (1999).
[ ( (
( ( ]……….…………………………………………Eq. 5.8
Now, to use Eq. 5.8 to develop a model that estimates the water saturation exponent in a
dual porosity reservoir with mixed wettability; it is important to note that the term
wettability in this work refers to (1) the wetting preference of the rock and (2) the fluid in
contact with the rock at any given time. Also, except when stated otherwise; the concept
95
of mixed wettability as used in this chapter refers to a reservoir condition in which the
matrix porosity is water wet, the isolated porosity is oil wet and that the inclusion phase
is completely filled with oil. The conductivity terms in Eq. 5.8 can also be written using
standard petrophysical notation in terms of resistivity, which is the inverse of
conductivity. This allows expressing Archie’s-types relationships in standard form as
follows (Eq. 5.9 and 5.10):
For the composite system:
……………………………………………………………Eq. 5.9
For the host environment (matrix):
………………………………………………………....Eq. 5.10
and for the inclusion phase:
………………………………………….……………………Eq. 5.11
Also, an averaging equation for water saturation in a dual porosity reservoir made up of
matrix and isolated porosity can be written as,
( …………………………………..................……..Eq. 5.12
where is the water saturation in the isolated porosity and is the isolated porosity
ratio (ratio of the isolated porosity to the total porosity of the composite system). Eq. 5.12
is written such that if the average equation will represent that of a single
porosity (matrix porosity) reservoir, i.e.
…………………….……..……………..……………………………..Eq. 5.13
If (i.e. isolated pores filled completely with oil), Eq. 5.12 can be re-arranged to
yield:
96
(
( )…………….………….….…………………….……………….Eq. 5.14
Eq. 5.14 can be related to the reservoir condition in which the conductivity of the
inclusion phase is very low ( ≈ 0). Inserting Eqs. 5.9, 5.10, and 5.11 into Eq. 5.8 leads
to:
[
(
(
(
(
]……….……Eq. 5.15
cancels out of Eq. 5.15 leading to:
[ (
(
( (
]………………….……Eq. 5. 16
Re-arranging Eq. 5.16 results in:
[
[
]
(
(
( ( ]
[ ( ] ………………..……………………………………..…Eq. 5.17
Eq. 5.16 can be re-written in terms of the resistivity index as expressed in Eq. 5.18 to
give Eq. 5.19
…………………………………………………………………………Eq. 5.18
[ (
(
( (
] ( …………………..……Eq. 5.19
Eqs. 5.17 and 5.19 represent the petrophysical model used to determine the water
saturation exponent n in a dual porosity reservoir represented by matrix and isolated
porosity with mixed wettability. The two equations (Eqs. 5.17 and Eq. 5.19) give the
same results. Availability of data will direct which model to use for determining the
97
water saturation exponent n. The method on how to use the models will be explained
further in the model validation section of this chapter.
5.2.2 Dual Porosity (Matrix and Fracture Porosity)
To develop a model that determines the water saturation exponent (n) in a dual porosity
reservoir made up of matrix and fractures, we start with Eqs. 5.20 and 5.21 which
represent the effective permittivities of mixtures. From Maxwell Garnett formulas with
aligned ellipsoids, the depolarization factor is equal to zero in Eq. 5.20 and equal to 1.0 in
Eq. 5.21 (Sihvola, 1999):
( …………………………………………………..…….Eq. 5.20
( ………………………………………………...………......…Eq. 5.21
Based on the complex dielectric permittivity reasoning stated above and expressed in Eq.
5.7, the permittivity terms in Eq. 5.20 and 5.21 are replaced with their conductivity terms
at low frequencies but written in their equivalent resistivity terms in Eqs. 5.22 and 5.23:
(
……………………………..……..…..…….Eq. 5.22
(
…………………….………………..………...…..Eq. 5.23
and
………………………………………………………………...... Eq. 5.24
Eqs. 5.22 and 5.23 can be combined to give the total conductivity (inverse of resistivity)
of the system for current flowing at any angle ( with respect to the fractures (Eq. 5.25);
98
except when stated otherwise the angle ( between the fracture and the direction of
current flow has been taken to be 0o throughout this chapter so that the tortuosity is equal
to one (1.0) and the fracture porosity exponent, mf =1.0. Appendix C shows a schematic
of the angle between fractures and the direction of current flow.
(
) (
) ………………………………...…………Eq. 5.25
Therefore;
(
(
) (
(
) ………...…………Eq. 5.26
An averaging equation for water saturation in a dual porosity reservoir made up of matrix
and fracture porosity can be written as
( ……….…………………………………………...........Eq. 5.27
where is the water saturation in fractures and is the fracture porosity ratio (ratio of
the fracture porosity to the total porosity of the composite system) . If in Eq. 5.27
the average equation will represent that of a single porosity (matrix porosity) reservoir,
i.e.,
…………………….…………………….…………..…………………Eq. 5.28
Also, if (i.e. fractures are filled completely with water), Eq. 5.28 can be re-
arranged such that,
(
( ……………………..………………………………………………Eq. 5.29
Inserting Eqs. 5.9, 5.10 and 5.24 into Eq. 5.26 leads to:
(
(
)
99
(
(
) .………………………...……Eq. 5.30
cancels out of Eq. 5.30 resulting in:
[( (
)
(
(
) ]…......................................................Eq. 5.31
To obtain the water saturation exponent (n) in dual porosity system represented by matrix
and fracture porosities, Eq. 5.31 is re-arranged to yield Eq. 5.32:
[
([( ⟨(
⟩) ]
[
(
) (
] )]
[ ( ] ……………………………….………………….…Eq. 5.32
To include the resistivity index, Eq. 5.31 can be re-written as expressed in Eq. 5.33
using :
[( (
)
(
(
) ] ( ……………………………………………Eq. 5.33
100
Eqs. 5.32 and 5.33 represent the petrophysical model used to determine n in a dual
porosity reservoir made up of matrix and fracture porosities with mixed wettability. Eqs.
5.32 and 5.33 give the same results. Availability of data will direct which model to use
for determining the water saturation exponent. This will be explained further in the
modeling section of this chapter.
5.2.3 Triple Porosity (Matrix, Isolated and Fracture Porosity)
To determine the triple water saturation exponent n in a reservoir with mixed wettability;
the following steps are taken:
(1) Calculate a water saturation exponent nb’ using Eq. 5.34; note that Eq. 5.34 has the
same form as Eq. 5.17 but and n are replaced with ’ and nb’,
(2) Transfer the parameters ’ and nb’ into Eq. 5.32 to replace Swb and nb, respectively.
This results in Eq. 5.35 from which the triple porosity water saturation exponent (n) is
calculated. Berg (2006) and Olusola et al. (2013) followed a similar approach to
determine the triple porosity exponent (m) by using two dual porosity models.
[
[
]
(
(
( ( ] [ ( ]
………
……………………………………………………………………………….....….Eq. 5.34
101
[
([( ⟨(
⟩) ]
[
(
) (
] )] [ ( ] ………………….……Eq. 5.35
5.3 Model Validation
An essential part of this chapter is the validation of the models and their application to
mixed wettability reservoirs using core data published by Sweeney and Jennings (1960).
The intergranular core plugs studied by Sweeney and Jennings (1960) include twenty five
one-inch diameter core plugs, three-inches long with porosity values in the range of 11.8
to 31.5%. Throughout this chapter we demonstrate the use of the single, dual and triple
porosity models using a matrix porosity ( of 31.5%. Note that is scaled to the bulk
volume of only the matrix block.
Crossplots of resistivity index ( ) versus water saturation ( ) are used to validate the
models developed in this chapter. Since the available core data represent a single porosity
reservoir (Sweeney and Jennings do not mention fractures or isolated porosity); we
validate the models by converting the dual and triple porosity equations into a single
porosity model by making This corroborates that the dual and triple
porosity models developed in this study also apply to the case where there is only matrix
porosity. The single porosity model can next be extended to reservoirs made up of dual or
102
triple porosity systems. Three steps, one for each type of porosity, and various methods
are shown next for validation purposes.
5.3.1 First Step: Single Porosity Reservoirs (matrix porosity)
The first validation step of the methodology presented in this chapter is explained using
two methods for determining n; the first method uses water saturation while the second
method uses the resistivity index. Both methods are for single porosity reservoirs.
5.3.1.1 First Method: Use of Sw for Determining n
As indicated previously the available core data and wettability studies are for a single
porosity reservoir; therefore the core data water saturation ( is equal to the saturation
of the matrix block (Swb) and the matrix block porosity (b) can be handled with a dual
porosity model assuming that the isolated and fracture porosities are equal to zero (nc =
2 = 0).
5.3.1.1.1 Dual Porosity Model for matrix and isolated porosity system:
To convert the dual porosity model representing matrix and isolated porosity (Eq. 5.17)
to a single porosity model, we assume in Eq. 5.17. Columns 1 to 12 in Table
5.1 show results of the various petrophysical parameters determined for 5 samples.
Colum 1 presents the core porosities that correspond to matrix porosities scaled to the
103
bulk volume of only the matrix blocks ( b). For example, for sample No. 1,
Since the calculations are for only the matrix, vnc and as shown in
columns 2 and 3. matrix porosity scaled to the bulk volume of the composite system of
matrix and isolated porosity (m) is calculated with the use of equation B-2 in the
Appendix, and is shown in column 4. Total porosity scaled to the bulk volume of the
composite system of matrix and isolated porosity () is calculated with the use of
equation B-3 in the Appendix, and is shown in column 5. The porosity exponent of only
the matrix (mb = 2.01) is obtained from core data and is shown in column 6. The dual
porosity exponent (m) in column 5 is determined using Eq. 5.1. In this case, m= mb
because we are dealing with a single porosity reservoir.
The results presented on columns 9 to 12 of Table 5.1 show that and
indicating that the dual porosity model is robust and estimates the water saturation and
water saturation exponent correctly in the case of a single porosity reservoir. Since the
isolated porosity is zero in this dual porosity case, a value of Swnc is mathematically
needed. In this case, as shown in column 8. can be any other number
depending on the wettability conditions of the rocks . Water saturation in the matrix
block (Swb) is taken from the core data and is presented in column 9. The matrix block
water saturation exponent ( ) in column 10 is equal to the water saturation exponent
from core data and is also the same as calculated using Eq. 5.18. The values of nb range
between 3.92 and 4.50 suggesting the presence of a rock that is preferentially oil wet.
Water saturation ( ) in column 11 is calculated from Eq. 5.12 and in column 12 is
determined using Eq. 5.17.
104
Table 5.1—RESULTS FOR SINGLE POROSITY RESERVOIR WHEN
ISOLATED POROSITY IS EQUAL TO ZERO AND MATRIX POROSITY IS
EQUAL TO THE TOTAL POROSITY
5.3.1.1.2 Dual Porosity Model for matrix and fracture porosity system:
To use the dual porosity model representing matrix and fracture porosity (Eq. 5.32) in a
conventional single porosity reservoir, we assume in Eq. 5.32. Table 5.2
(columns 1 to 12) shows results for petrophysical parameters determined in the following
manner: starting with column 1 assume that the porosity attached to the bulk volume of
only the matrix system is the core porosity ( as shown on column 2. For a
single porosity reservoir (column 3), it can also be determined using equation B-4
in the Appendix. and in columns 4 and 5 are determined from equations B-5 and
B-6 in the Appendix, respectively. Column 6 is the matrix porosity exponent attached to
only the matrix system (mb = 1.57). The dual porosity exponent (m) in column 7 is
determined using Eq. 5.2. Since this is a single porosity system, m= mb.
Since this is a dual porosity model where 2 is zero there is a mathematical need for
water saturation in the fractures. In this case is assumed equal to 1.0 as shown in
column 8. However, can be any other number depending on wettability conditions
Sample No. 1 2 3 4 5 6 7 8 9 10 11 12
ɸb Vnc ɸnc ɸm ɸ mb m Swnc Swb nb Sw n
1 0.315 0 0 0.315 0.315 2.01 2.01 0 0.22 4.41 0.22 4.41
2 0.315 0 0 0.315 0.315 2.01 2.01 0 0.23 4.50 0.23 4.50
3 0.315 0 0 0.315 0.315 2.01 2.01 0 0.25 4.25 0.25 4.25
4 0.315 0 0 0.315 0.315 2.01 2.01 0 0.26 3.92 0.26 3.92
5 0.315 0 0 0.315 0.315 2.01 2.01 0 0.26 4.40 0.26 4.40
105
and the angle between fractures and direction of flow ( ). Water saturation in the matrix
( ) in column 9 is given by the core data water saturation. The matrix block water
saturation exponent ( ) in column 10 is equal to the water saturation exponent from core
data. Water saturation ( ) in column 11 is calculated from Eq. 5.27 and in column 12
is determined using Eq. 5.32. The results of Table 5.2 show that and
indicating that the dual porosity model is robust and calculates Sw and n correctly in the
case of a single porosity reservoir.
Table 5.2—RESULTS FOR SINGLE POROSITY RESERVOIR WHEN
FRACTURE POROSITY IS EQUAL TO ZERO AND MATRIX POROSITY IS
EQUAL TO THE TOTAL POROSITY
5.3.1.2 Second Method: Use of I for Determining n
5.3.1.2.1 Dual Porosity Model for matrix and isolated porosity system:
This second method also uses a dual porosity model for determination of n in a single
porosity reservoir. In this case the resistivity index ( is known and some of the
calculations require an iteration procedure. To convert the dual matrix and isolated
porosity model to a single porosity model is made equal to zero in Eq. 5.19. Table
5.3 (columns 1 to 15) shows results of the calculated petrophysical parameters for 5
Sample No. 1 2 3 4 5 6 7 8 9 10 11 12
ɸb V ɸ2 ɸm ɸ mb m Sw2 Swb nb Sw n
1 0.315 0 0 0.315 0.315 1.57 1.57 1 0.69 2.40 0.69 2.40
2 0.315 0 0 0.315 0.315 1.57 1.57 1 0.70 2.38 0.70 2.38
3 0.315 0 0 0.315 0.315 1.57 1.57 1 0.72 2.15 0.72 2.15
4 0.315 0 0 0.315 0.315 1.57 1.57 1 0.84 2.59 0.84 2.59
5 0.315 0 0 0.315 0.315 1.57 1.57 1 0.95 3.46 0.95 3.46
106
samples determined in the following manner: A value of from core data is
introduced in column 1.
Since there is no isolated porosity, as shown in columns 2 and 3 (see also
equation B-1 in the Appendix). Porosities and in columns 4 and 5 are determined
from equations B-2 and B-3 in the Appendix. Column 6 represents the matrix porosity
exponent attached only to the matrix system (mb = 2.01) determined from core data. The
dual porosity exponent (m) shown in column 7 is determined using Eq. 5.1. The result is
that m equals mb validating the use of the dual porosity equation for calculating m values
in single porosity systems.
The dual porosity model requires mathematically water saturation in the isolated porosity.
For this case it assumed that as shown in column 8. However, can be any
other value depending on the rock wettability conditions. Water saturations in the matrix
blocks ( ) in column 9 are obtained via an iteration procedure. The water saturation
exponent of the matrix blocks ( ) shown in column 10 are taken from cores. They range
between 3.92 and 4.50 suggesting that the rock is preferentially oil wet.
The resistivity index (I) in column 11 is also taken from the core data. Columns 12 and
13 are determined using Eq. 5.19. To use Eq. 5.19 iterations are run to determine the
unknown value of that makes the right hand side (RHS) of Eq. 5.19 equal to the left
hand side (LHS). In this work we used Microsoft Excel Solver package. To use this
package we followed these steps:
107
(1) Input available data into Eq. 5.19 and assume any reasonable value greater than zero
for .
(2) Set up as a constraint: LHS = RHS. (3) Run the iterations until reaching convergence;
at convergence LHS =RHS. The result from Eq. 5.19 is unique since it depends on a
constant resistivity index at each depth in the well.
Water saturation ( ) in column 14 is calculated from Eq. 5.12 and in column 15 is
determined with the use of Eq. 5.18. The results of Table 5.3 show that and
indicating that the matrix-isolated dual porosity model is robust and capable of
calculating correct values of n for single porosity reservoirs.
Table 5.3—RESULTS FOR SINGLE POROSITY RESERVOIR WHEN
ISOLATED POROSITY IS EQUAL TO ZERO AND MATRIX POROSITY IS
EQUAL TO THE TOTAL POROSITY
Fig. 5.1 shows a comparison of results from this dual porosity model (with nc = 0) and
core data (single porosity) published by Sweeney and Jennings (1960) for preferentially
oil-wet rocks. The blue diamonds in the graph correspond to core data, the black open
diamonds represent calculated parameters to match the core data, and the numbered red
square symbols represent the calculated values shown in Table 5.3 (samples 1 to 5) for
Sample No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
ɸb Vnc ɸnc ɸm ɸ mb m Swnc Swb nb I LHS RHS Sw n
1 0.315 0 0 0.315 0.315 2.01 2.01 0 0.22 4.41 805.85 0.10 0.10 0.22 4.41
2 0.315 0 0 0.315 0.315 2.01 2.01 0 0.23 4.50 753.78 0.10 0.10 0.23 4.50
3 0.315 0 0 0.315 0.315 2.01 2.01 0 0.25 4.25 349.40 0.10 0.10 0.25 4.25
4 0.315 0 0 0.315 0.315 2.01 2.01 0 0.26 3.92 192.88 0.10 0.10 0.26 3.92
5 0.315 0 0 0.315 0.315 2.01 2.01 0 0.26 4.40 354.43 0.10 0.10 0.26 4.40
108
illustration purposes. The red squares make it easier for the readers to compare the
calculations presented in Table 5.3 with the graph shown in Fig. 5.1.
Figure 5.1— Comparison of Eq. 5.19 for matrix and isolated porosity using nc = 0
and preferentially oil-wet core data published by Sweeney and Jennings (1960). The
blue diamonds represent core data, the black open diamonds represent other data
from the dual porosity model used to match the core data. All the core data points
are matched. The red squares correspond to data calculated for illustration
purposes in Table 5.3 for samples 1 to 5.
5.3.1.2.2 Dual Porosity Model for matrix and fracture porosity system:
To convert this dual porosity model into a single porosity petrophysical model we assume
in Eq. 5.33. Table 5.4 (columns 1 to 15) shows petrophysical parameters
determined for this case in the following manner: The porosity attached only to the
matrix system ( shown in column 1 is from unfractured cores. Column 2 is
determined using equation B-4 in the Appendix. For a single porosity reservoir as
1
10
100
1000
0.1 1
Re
sist
ivit
y R
ati
o
Fractional Water Saturation(Sw)
1 253
4
109
shown in column 3. Porosities and in column 4 and 5 are determined from equation
B-5 and B-6 (Appendix), respectively. Column 6 is the matrix porosity exponent attached
only to the matrix system (mb = 1.57). The dual porosity exponent (m) in column 7 is
determined using Eq. 5.2. For this case, m= mb because we are dealing with a single
porosity system.
Since this is a dual porosity model where 2 is zero there is a mathematical need for
water saturation in the fractures. In this case is assumed equal to 1.0 as shown in
column 8 indicating thus water wet fractures. However, can be any other value
depending on wettability conditions of the fractures and the angle between fractures and
direction of flow ( ). Water saturation in the matrix ( ) in column 9 is obtained via
iteration and in column 10 is the water saturation exponent from core data. The small
values of nb (1.37 to 1.53) indicate that the matrix is preferentially water wet. The
resistivity index (I) in column 11 is taken from core data, columns 12 and 13 are
determined using Eq. 5.33. This involves iterations to determine the unknown value
( ) that will make the RHS = LHS in Eq. 5.33. In this work we used Microsoft Excel
Solver package and the following steps: (1) Input available data into Eq. 5.33 and assume
any reasonable value greater than zero for . (2) Set up a constraint: LHS = RHS. (3)
Run iterations until attaining convergence at LHS = RHS. The iteration result from Eq.
5.33 is unique since it depends on a constant resistivity index at each depth in the well.
Water saturation ( ) in column 14 is calculated from Eq. 5.27 and in column 15 is
determined using Eq. 5.18. The results in Table 5.4 show that and
indicating that the dual porosity model is capable of calculating water saturation and the
110
water saturation exponent correctly in single porosity reservoirs. For this example, =
90o
in Eq. 5.33. The model (Eq. 5.33) is capable of running iterations without any
convergence problem with .
Table 5.4— RESULTS FOR SINGLE POROSITY RESERVOIR WHEN
FRACTURE POROSITY IS EQUAL TO ZERO AND MATRIX POROSITY IS
EQUAL TO THE TOTAL POROSITY
Fig. 5.2 shows a comparison of results from this dual porosity model (with 2 = 0) and
core data (single porosity) published by Sweeney and Jennings (1960) for preferentially
water-wet rocks. The blue diamonds in the graph correspond to core data, the black open
diamonds represent the calculated parameters that match the core data, and the numbered
red square symbols represent the calculated values shown in Table 5.4 (samples 1 to 5)
for illustration purposes. The red squares make it easier for the readers to compare the
calculations presented in Table 5.4 with the graph shown in Fig. 5.2.
Sample No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
ɸb V ɸ2 ɸm ɸ mb m Sw2 Swb nb I LHS RHS Sw n
1 0.315 0 0 0.315 0.315 1.57 1.57 1 0.15 1.37 13.32 0.16 0.16 0.15 1.37
2 0.315 0 0 0.315 0.315 1.57 1.57 1 0.15 1.54 18.18 0.16 0.16 0.15 1.54
3 0.315 0 0 0.315 0.315 1.57 1.57 1 0.16 1.56 16.79 0.16 0.16 0.16 1.56
4 0.315 0 0 0.315 0.315 1.57 1.57 1 0.19 1.40 9.94 0.16 0.16 0.19 1.40
5 0.315 0 0 0.315 0.315 1.57 1.57 1 0.20 1.53 12.17 0.16 0.16 0.20 1.53
111
Figure 5.2— Comparison of Eq. 5.33 for matrix and fractures using 2 = 0 and
preferentially water-wet core data published by Sweeney and Jennings (1960). The
blue diamond’s represent core data, the black open diamonds represent other data
from the dual porosity model used to match the core data. All the core data points
are matched. The red squares correspond to data calculated for illustration
purposes in Table 5.4 for samples 1 to 5.
5.3.2 Second Step: Dual Porosity Reservoirs (Matrix and Isolated Porosity or Matrix
and Fracture Porosity)
The second validation step of the dual porosity models developed in this chapter is
explained using two methods; the first method shows how to determine the water
saturation exponent (n) using water saturation values while the second step shows how to
use resistivity index values to determine the water saturation exponent (n).
5.3.2.1 First Method: Use of Sw for Determining n
As indicated previously the available core data and wettability studies are for a single
porosity reservoir; therefore the core water saturations are equal to the saturation of the
1
10
100
1000
0.1 1
Resi
stiv
ity R
atio
Fractional Water Saturation(Sw)
5
4
3
1
2
112
matrix blocks (Swb) and the porosity of cores is equal to the matrix block porosity (b).
Furthermore it was shown previously that single porosity can be handled with a dual
porosity model assuming that the isolated and fracture porosities are equal to zero (nc =
2 = 0). In this second step we investigate the effects of isolated and fracture porosities
greater than zero on and .
5.3.2.1.1 Dual Porosity Model for matrix and isolated porosity system:
In this example we assume in the dual porosity model made out of matrix
and isolated pores (Eq. 5.17). Each data value in Table 5.5 (columns 1 to 12) is
calculated using the first method explained previously in the section “First Step: Single
Porosity Reservoirs (matrix porosity).” In this case Sw is known. The only difference is
that now for illustration purposes (rather than zero) in Table 5.5. The water
saturation in the isolated porosity (Swnc) shown in column 8 is equal to zero. Thus the
isolated porosity is full of oil and is oil-wet. Also oil-wet is the matrix as the values of nb
in column 10 are large (3.92 to 4.50). The effects of increasing isolated porosity (PHINC)
on the water saturation exponent are shown on Fig. 5.3. The plot was constructed using
data developed with the use of the dual porosity model for matrix and isolated porosity
represented by Eq. 5.17 under the assumption that nb is equal to 4.0.
113
Table 5.5— RESULTS FOR DUAL POROSITY RESERVOIR WITH MATRIX
AND ISOLATED POROSITY
Figure 5.3— Graph of versus developed with the use of the dual porosity
model for matrix and isolated porosity (Eq. 5.17). The chart shows the effects of
increasing isolated porosity (PHINC) on while keeping matrix porosity constant at
b = 0.315 Matrix and composite system (vugs+matrix) are oil wet.
5.3.2.1.2 Dual Porosity Model for matrix and fracture porosity system:
This case is handled assuming in the dual porosity model made out of matrix
and fractures (Eq. 5.32). Each data value in Table 5.6 (columns 1 to 12) is determined
using the first method explained previously in the section “First Step: Single Porosity
Reservoirs (matrix porosity).” The only difference is that now for illustration purposes
(rather than zero) in Table 5.6.
Sample No. 1 2 3 4 5 6 7 8 9 10 11 12
ɸb Vnc ɸnc ɸm ɸ mb m Swnc Swb nb Sw n
1 0.315 0.143 0.050 0.299 0.349 2.01 2.06 0 0.22 4.41 0.19 4.14
2 0.315 0.143 0.050 0.299 0.349 2.01 2.06 0 0.23 4.50 0.20 4.21
3 0.315 0.143 0.050 0.299 0.349 2.01 2.06 0 0.25 4.25 0.22 3.97
4 0.315 0.143 0.050 0.299 0.349 2.01 2.06 0 0.26 3.92 0.22 3.67
5 0.315 0.143 0.050 0.299 0.349 2.01 2.06 0 0.26 4.40 0.23 4.10
1.5
2.5
3.5
4.5
0.0 0.2 0.4 0.6 0.8 1.0
Wat
er S
atur
atio
n Ex
pone
nt,n
Water Saturation
PHINC=0.025
PHINC=0.050
PHINC=0.075
PHINC=0.100
nb = 4.0
114
The water saturation in the fracture porosity (Sw2) shown in column 8 is equal to 1. Thus
the fracture porosity is full of water and is water-wet. The matrix has intermediate
wettability as the values of nb in column 10 are moderate (2.15 to 3.46).
The effects of increasing fracture porosity (PHI2) on the water saturation exponent are
shown on Fig. 5.4. The plot was constructed using data developed with the use of the
dual porosity model for matrix and fractures represented by Eq. 5.32 under the
assumption that nb is equal to 4.0. The value of n for the composite system shown in
column 12 ranges between 1.75 and 2.92 suggesting a system that is preferentially water
wet with the possible exception of sample 5. Fig. 5. 4 (matrix and fracture porosity) has
an opposite slope compared to Fig. 5.3 (matrix and isolated porosity) because the Sw2 is
equal to one in Fig. 5.4 while Swnc is equal to zero in Fig. 5.3.
Table 5.6 — RESULTS FOR DUAL POROSITY RESERVOIR WITH MATRIX
AND ISOLATED POROSITY
Sample No. 1 2 3 4 5 6 7 8 9 10 11 12
ɸb V ɸ2 ɸm ɸ mb m Sw2 Swb nb Sw n
1 0.315 0.143 0.050 0.299 0.349 1.57 1.51 1 0.69 2.40 0.73 1.90
2 0.315 0.143 0.050 0.299 0.349 1.57 1.51 1 0.70 2.38 0.74 1.90
3 0.315 0.143 0.050 0.299 0.349 1.57 1.51 1 0.72 2.15 0.76 1.75
4 0.315 0.143 0.050 0.299 0.349 1.57 1.51 1 0.84 2.59 0.87 2.16
5 0.315 0.143 0.050 0.299 0.349 1.57 1.51 1 0.95 3.46 0.96 2.92
115
Figure 5.4— Graph of versus developed with the use of the dual porosity
model for matrix and fractures (Eq. 5.32). The chart shows the effects of increasing
fracture porosity (PHI2) on while keeping matrix porosity constant at b = 0.315 Matrix and composite system (matrix plus fractures) are water wet.
5.3.2.2 Second Method: Use of I for Determining n
5.3.2.2.1 Dual Porosity for matrix and isolated porosity system:
In this section, we assume in the dual porosity model made out of matrix and
isolated pores (Eq. 5.19). The matrix porosity (b) is maintained constant at 0.315. We
follow similar calculation steps as shown previously in the second method in the section
“First Step: Single Porosity Reservoirs (matrix porosity).” In this method the resistivity
index (I) is known.
0.0
1.0
2.0
3.0
4.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Wate
r S
atu
rati
on
Exp
on
en
t,n
Water Saturation
PHI2=0.025
PHI2=0.050
PHI2=0.075
PHI2=0.100
nb = 4.0
116
The red squares in Fig. 5.5 show a cross plot of the calculated water saturation against the
resistivity index (red squares). The isolated porosity is 100% saturated with oil. Also
shown for comparison are the oil wet data (black diamonds) extracted from core analysis
(single porosity model) by Sweeney and Jennings (1960). These core data was also
shown on Fig. 5.1. In this case, for the same resistivity index the values of water
saturation are lower in the case of the dual porosity model.
In both cases the slopes of the linear trends for matrix and composite systems are steep (n
> 7) indicating that both the matrix and composite system (matrix and isolated pores) are
preferentially oil wet.
Figure 5.5— Plot of versus for a dual porosity model made up of matrix and
isolated porosity (red squares) and comparison with core data (single porosity
model). For the same resistivity index, water saturations are smaller in the case of
the dual porosity model. Matrix and the composite system (matrix + vugs) are oil
wet.
1
10
100
1000
0.1000 1.0000
Res
isti
vity
Rat
io
Fractional Water Saturation(Sw)
Dual Porosity Model
Single Porosity Model
117
5.3.2.2.2 Dual Porosity for matrix and fracture porosity system:
In this section, we assume in the dual porosity system made out of matrix and
fractures (Eq. 5.33). We follow similar calculation steps as shown previously in the
second method in the section “First Step: Single Porosity Reservoirs (matrix porosity).”
In this method the resistivity index (I) is known.
The red squares in Fig. 5.6 show a cross plot of the calculated water saturation values vs.
resistivity index. Also shown for comparison (black diamonds) are the data extracted
from core analysis (single porosity model) published by Sweeney and Jennings (1960).
As opposed to the previous dual porosity model (oil wet matrix and isolated pores), for
the same resistivity index the values of water saturation are larger in the case of the dual
porosity model made out of matrix and fractures. This is because the angle between the
fractures and the direction of flow was taken as 90o for this case.
In the case of core experiments (single porosity model), also shown in Fig. 5.2, the
average value of n is 1.666 (R2 = 0.86) indicating a system that is preferentially water
wet. In the case of the dual porosity model the slope is steeper with an average value of n
equal to 2.22 (R2 = 0.896) indicating that both the matrix and composite system (matrix
and fractures combined) are preferentially water wet.
118
Figure 5.6— Plot of versus for a dual porosity model made up of matrix and
fracture porosity (red squares) and comparison against core data (single porosity
model). For the same resistivity index, water saturations are larger in the case of the
dual porosity model for the example at hand. The slope values indicate that the
matrix and the composite system are preferentially water wet.
5.3.3 Third Step: Triple Porosity Reservoirs (Matrix, Isolated and Fracture Porosities)
The third step is treated using two methods. The first method shows how to determine the
water saturation exponent (n) using water saturation values (Sw) while the second method
shows how to use resistivity index values (I) to determine the water saturation exponent
(n). Both methods are for a triple porosity reservoir system that include matrix, isolated
and fracture porosities.
y = x-2.222
R² = 0.8964
y = x-1.666
R² = 0.8633
1
10
100
0.1 1.0
Re
sist
ivit
y In
dex
(I)
Fractional Water Saturation (Sw)
Dual Porosity Model
Single Porosity Model
119
5.3.3.1 First Method: Use of Sw for Determining n
In this section, to determine the triple water saturation exponent in a reservoir with mixed
wettability. The approach involves the following steps:
(1) Calculate an intermediate water saturation exponent nb’ using Eq. 5.34. Note that Eq.
5.34 has the same form as Eq. 5.17 and all the petrophysical parameters are determined in
the same way as done previously for the dual porosity model represented by isolated and
matrix porosities (first method).
(2) Transfer the parameter ’ and nb’ from Eq. 5.34 into Eq. 5.35 making ’ =
and nb’ = n. Use Eq. 5.35 to calculate the triple porosity water saturation exponent (n).
Note that Eq. 5.35 has the same form as Eq. 5.32. All the other petrophysical parameters
beyond Swb and nb are determined in the same way as was done with the dual porosity
model represented by fractures and matrix porosity (first method).
(3) Prepare a cross plot of water saturation exponent (n) versus water saturation as shown
on Fig. 5.7. This provides a good visualization of the calculations. The matrix porosity
(b) in the graph is 0.315, fracture porosity (2 ) is 0.01 and the isolated porosities (nc )
are 0.025, 0.05, 0.075, and 0.1.
120
Figure 5.7— Graph of versus using Eqs. 5.34 and 5.35 for the case of a triple
porosity model. The chart shows the combined effects of isolated porosity and
fracture porosity on Matrix porosity (b) is maintained constant at 0.315 in this
case.
5.3.3.2 Second Method: Use of I for Determining n
In this example matrix porosity (b) is equal to 0.315 and . The matrix
system (core) is oil wet as well as the isolated vugs which have a water saturation of zero
(100% oil). The fractures are water wet with a water saturation of 1. The calculations are
as in the second method for the single porosity reservoir case but
in Eq. 5.34
and Eq. 5.35. The calculated results are plotted as red squares in Fig. 5.8. A comparison
is made with core data from an oil wet system (black squares), which corresponds to only
matrix (core, single porosity model). Contrary to the case shown on Fig. 5.5, for the same
1.5
2.5
3.5
0.0 0.2 0.4 0.6 0.8
Wa
ter
Satu
rati
on
Exp
on
en
t,n
Water Saturation
PHINC=0.025,PHI2=0.01
PHINC=0.050,PHI2=0.01
PHINC=0.075,PHI2=0.01
PHINC=0.100,PHI2=0.01
nb = 4.0
121
resistivity index, water saturations are higher in the triple porosity than in the single
porosity model. The overall triple porosity system is oil wet with values of n greater than
7.
Figure 5.8— Graph of versus calculated for a triple porosity model (red
squares). The calculated results are compared against core data represented by
black squares (single porosity model). For the same resistivity index the water
saturation of the triple porosity model is generally larger than Sw from cores (single
porosity).
1.0000
10.0000
100.0000
1000.0000
0.1000 1.0000
Re
sist
ivit
y R
atio
Fractional Water Saturation(Sw)
Triple Porosity Model
Single Porosity Model
122
Chapter Six: PETROPHYSICAL AND GEOMECHANICAL EVALUATION
6.1 Overview
This chapter presents a case study on petrophysical and geomechanical evaluation of
horizontal wells in the tight gas Nikanassin Group of the WCSB using drill cuttings. The
advantages of measuring porosity and permeability from drill cuttings has also been
discussed as they provide an aid in performing a complete quantitative petrophysical and
geomechanical evaluation of reservoirs in those cases where well logs and/or core data
are not available. Several empirical equations using porosity and permeability data for
calculating capillary pressure, irreducible water saturation, porosity exponent (m), pore
throat aperture radius, true formation resistivity, location of water contact, Knudsen’s
number, Young Modulus, Poisson’s ratio, and brittleness index have been used to
characterize this tight gas formation
6.2 Petrophysical Evaluation Based on Drill Cuttings
6.2.1 Case Study: Drill Cuttings Collected in Horizontal Well
Column 1 of Table 6.1 identifies 57 drill cutting samples considered in this study. The
measured depth in Column 2 represents the bottom depth where the drill cutting samples
were collected in the horizontal well.
123
Table 6.1— PETROPHYSICAL DATA FOR WESTERN CANADA
SEDIMENTARY BASIN TIGHT GAS SANDSTONE. THE HORIZONTAL
WELL POROSITY AND PERMEABILITY DATA FROM DRILL CUTTINGS
(COLUMN 2 & 3) ARE OBTAINED FROM LABORATORY WORK AND IT IS
USED AS A STARTING POINT IN DETERMINING OTHER PETROPHYSICAL
DATA (COLUMN 5 TO 11) USING EMPIRICAL EQUATIONS.
1 2 3 4 5 6 7 8 9 10 11
Sample
No.
BOTDepth
MD (m)
Ф (Drill
Cuttings)
K (md)
(Drill
Cuttings)
rp35
(μm)m
Swi
(fraction)F
Rt
(ohm.m)
Pc (Hg-Air)
(Psi)Kn
1 3185.0 0.066 0.031 0.239 1.827 0.129 143.61 231 7101 0.00052
2 3190.0 0.063 0.022 0.208 1.823 0.136 152.61 221 7442 0.00060
3 3195.0 0.065 0.030 0.237 1.825 0.125 146.74 247 7478 0.00053
4 3200.0 0.053 0.038 0.290 1.801 0.059 201.16 1254 22116 0.00043
5 3202.5 0.065 0.056 0.313 1.826 0.094 145.19 416 9297 0.00040
6 3205.0 0.106 0.088 0.309 1.866 0.314 66.22 22 1204 0.00041
7 3207.5 0.087 0.079 0.322 1.852 0.183 92.65 82 2903 0.00039
8 3210.0 0.090 0.216 0.497 1.855 0.125 86.71 157 3608 0.00025
9 3217.5 0.069 0.025 0.213 1.831 0.161 135.22 146 5443 0.00059
10 3220.0 0.098 0.017 0.153 1.861 0.568 75.49 8 891 0.00082
11 3225.0 0.099 0.017 0.152 1.861 0.586 74.20 8 850 0.00082
12 3235.0 0.065 0.069 0.344 1.826 0.084 145.97 515 10271 0.00036
13 3265.0 0.086 0.058 0.280 1.851 0.212 93.04 63 2595 0.00045
14 3285.0 0.150 0.064 0.229 1.885 1.044 35.92 1 211 0.00055
15 3290.0 0.084 0.078 0.325 1.849 0.166 98.13 103 3394 0.00039
16 3295.0 0.055 0.032 0.262 1.807 0.075 186.63 765 16195 0.00048
17 3300.0 0.091 0.190 0.468 1.855 0.136 85.67 132 3314 0.00027
18 3305.0 0.068 0.109 0.415 1.830 0.075 136.72 588 10136 0.00030
19 3310.0 0.054 0.055 0.337 1.805 0.054 191.47 1400 21839 0.00037
20 3315.0 0.062 0.062 0.335 1.821 0.077 156.66 638 12199 0.00037
21 3320.0 0.105 0.069 0.278 1.865 0.347 67.11 18 1132 0.00045
22 3325.0 0.096 0.046 0.241 1.859 0.325 78.08 24 1454 0.00052
23 3345.0 0.121 0.150 0.369 1.874 0.365 52.04 13 782 0.00034
24 3355.0 0.077 0.025 0.202 1.842 0.227 112.57 65 3185 0.00062
25 3435.0 0.061 0.065 0.346 1.818 0.069 163.22 798 14105 0.00036
26 3440.0 0.066 0.048 0.291 1.827 0.103 143.83 347 8497 0.00043
27 3445.0 0.085 0.051 0.267 1.850 0.212 96.34 65 2719 0.00047
28 3450.0 0.077 0.010 0.134 1.842 0.359 112.72 28 2216 0.00093
29 3460.0 0.052 0.020 0.218 1.800 0.079 203.38 740 17735 0.00057
30 3690.0 0.069 0.023 0.204 1.832 0.174 132.79 125 4988 0.00061
31 3710.0 0.049 0.030 0.268 1.792 0.055 219.90 1525 27055 0.00047
32 3715.0 0.046 0.029 0.272 1.783 0.045 240.89 2260 36506 0.00046
33 3720.0 0.051 0.025 0.243 1.798 0.068 207.59 994 20747 0.00052
34 3725.0 0.066 0.028 0.229 1.826 0.133 144.82 218 6991 0.00055
35 3730.0 0.078 0.036 0.237 1.843 0.197 110.21 84 3467 0.00053
36 3735.0 0.053 0.010 0.158 1.802 0.119 197.58 347 12227 0.00079
37 3740.0 0.068 0.018 0.185 1.829 0.182 138.21 119 5094 0.00068
38 3745.0 0.064 0.066 0.341 1.823 0.080 151.45 582 11281 0.00037
39 3760.0 0.051 0.057 0.352 1.798 0.045 207.98 2102 29029 0.00036
40 3765.0 0.048 0.018 0.215 1.789 0.066 227.17 1122 24630 0.00058
41 3770.0 0.050 0.012 0.177 1.793 0.088 218.38 650 18320 0.00071
42 3775.0 0.068 0.015 0.170 1.829 0.199 138.21 100 4736 0.00073
43 3785.0 0.066 0.016 0.178 1.827 0.178 143.99 128 5494 0.00071
44 3820.0 0.058 0.019 0.204 1.812 0.109 176.59 372 11029 0.00061
45 3835.0 0.066 0.011 0.149 1.828 0.221 141.91 85 4527 0.00084
46 3855.0 0.084 0.184 0.478 1.849 0.109 97.79 225 4739 0.00026
47 3860.0 0.061 0.020 0.203 1.819 0.129 160.30 252 8326 0.00062
48 3865.0 0.073 0.015 0.165 1.837 0.252 122.25 59 3297 0.00076
49 3880.0 0.084 0.012 0.140 1.849 0.429 97.31 18 1568 0.00090
50 3885.0 0.055 0.020 0.214 1.805 0.090 190.95 555 14444 0.00059
51 3890.0 0.082 0.017 0.166 1.847 0.328 102.41 30 2082 0.00076
52 3895.0 0.073 0.015 0.165 1.837 0.252 122.16 58 3290 0.00076
53 3900.0 0.075 0.016 0.168 1.839 0.260 118.20 54 3067 0.00075
54 3930.0 0.076 0.010 0.135 1.840 0.345 115.08 31 2352 0.00093
55 3935.0 0.061 0.014 0.173 1.819 0.154 160.69 184 7273 0.00073
56 3940.0 0.068 0.020 0.194 1.829 0.172 138.28 131 5321 0.00065
57 3945.0 0.076 0.080 0.343 1.841 0.123 114.62 206 5342 0.00036
58 AVERAGE 0.073 0.047 0.253 1.831 0.195 137.14 381 8553 0.00055
124
6.2.2 Porosity
Porosity which represents the amount of void space in a bulk volume of rock is
considered as a measure of the storage capacity of the rock since it is capable of holding
fluids (Ahmed 2006). Porosity is an important rock property, cheaper to measure and
most readily used to correlate with other rock properties such as permeability and pore
throat aperture that are more expensive to obtain.
In this case study, we provide support to the conclusion that measuring porosity from
drill cuttings is an important step to complete petrophysical evaluation of reservoirs in the
absence of well logs or cores. The porosity values obtained from drill cuttings closely
match those from well log data (Ortega and Aguilera, 2012); therefore we anticipate that
future quantitative formation evaluation work will also include drill cuttings. Care will
always have to be exercised in tight reservoirs because in all probability some of the slot
porosity and micro-fractures will be destroyed by the action of the drilling bit, but drill
cuttings can still provide relevant quantitative information as shown in this case study
using drill cuttings collected in a horizontal well. Furthermore they can provide structural
information (Hews, 2012) that is very relevant for estimating the Frac Value (Ortega and
Aguilera, 2012). The third column in Table 6.1 shows the laboratory porosity values
from drill cuttings.
125
6.2.3 Permeability
Permeabilities were determined using Darcylog equipment patented by the Institut
Français du Pétrole (IFP) and methodologies presented by Egermann et al. (2002) and
Lenormand et al. (2007). Nikanassin data (Solano, 2010; Ortega, 2012) confirmed the
reliability of permeability measurements from drill cuttings. These permeabilities are
most likely associated with the matrix system since micro fractures are probably not
preserved in drill cuttings. Permeability is a function of many petrophysical variables
including grain size, pore-throat aperture, porosity, pore architecture and irreducible
water saturation. Among these independent variables porosity represents the most easily
measured variable used for predicting permeability (Byrnes, 2005). Column 4 in Table
6.1 show the laboratory permeabilities from drill cuttings determined in this study.
6.2.4 Pore Throat Aperture Radii
Pore throat size represents one of the dominant variables that control permeability in low
permeability rocks (Byrnes, 2005). Pore throat aperture is generally estimated from
mercury injection test but this test is expensive to acquire and the chemical element in
mercury is not environmentally friendly due to its toxic nature. Therefore, empirical
equations relating pore throat aperture to less expensive parameters such as permeability
and porosity are generally used to estimate the pore throat aperture (Pittman, 1992).
126
The Washburn equation shows a relationship that determines the pore aperture radius in
porous rocks after conducting mercury injection test in the laboratory (Washburn, 1921).
H. D. Winland (1980s) of Amoco Research Department (Tulsa, Oklahoma) used
Washburn equation and performed mercury injection tests on sandstones from 14
formations. These 14 formations which age range from Ordovician to Tertiary include
Simpson, Delaware, Tensleep, Nugget, Cotton Valley, Muddy, Meserverde, Terry, First
Wall Creek, Second Wall Creek, Frontier, Montrose, Vicksburg, and Frio Sandstones.
From this mercury injection test, Winland established an empirical correlation that relates
porosity and permeability to pore throat size. 202 porosity (data range from 3.3% to 28%)
and uncorrected air permeability (data range from 0.05 to 998md) data sets was used in
his analysis (Pittman, 1992).
A similar equation (Eq. 6.1) that relates porosity and permeability to pore throat aperture
was developed by Aguilera (2002):
[
]
.......................................................................................... Eq. 6.1
This equation is rigorously valid between 30 to 90% water saturation, but in practice we
have been able to extend it beyond those limits with reasonable results for the tight
formations we are studying in the GFREE research group (Aguilera 2002 and Ortega et al
2013). The porosity and permeability obtained from drill cuttings were used to estimate
the pore throat radii (rp35) shown in Column 5 of Table 6.1.
127
6.2.5 Porosity Exponent ( )
Data from the Mesaverde tight gas formation in the United States was used by Byrnes et
al. (2006) to develop the following empirical correlation for calculating the porosity
exponent ( ):.
…………………………………..…………………………….… Eq. 6.2
A crossplot of Eq. 6.2 is shown in Fig. 6.1. Given that porosity exponent ( ) data are not
available in the study area and that there are some similarities between the Nikanassin
and Mesaverde continuous accumulations, the above equation was used to generate
porosity the exponent ( data shown in column 6 of Table 6.1.
128
Figure 6.1—Plot of porosity exponent ( versus porosity ( . The porosity
exponent was determined using Byrnes empirical correlation (Eq. 6.2).
6.2.6 Irreducible Water Saturation
Low permeability gas-production sandstone reservoirs are typically known for high
capillary pressure due to the small pore throat size of the reservoir rocks. Since
irreducible water saturation affects the gas recovery from low permeability reservoirs; it
is essential to account for the presence of this water. In conventional reservoirs
irreducible water saturation is defined as the saturation at which a further increase in
capillary pressure does not significantly decrease the water saturation (Byrnes, 2005). In
129
the case of tight gas sandstones it is possible to have from very small to nearly 100%
water saturation at irreducible conditions.
Morris and Biggs (1997) developed an empirical equation to estimate permeability for
formations at irreducible condition. They expressed the correlation of log derived values
of water saturation (Sw) versus porosity ( ) as:
…………………………………………………………………………Eq. 6.3
In Eq. 6.3, k = formation permeability, milidarcies, Swi = water saturation at irreducible
saturation, i.e., it corresponds to the beginning of krw equal to zero and c equal to a
constant whose value depends on the density of the type of hydrocarbon occupying the
formation. For the case of medium gravity oil (≈25o API), c = 250, and for dry gas at
shallow depth, c = 79. Since the formation under study is a tight gas formation, a constant
value of c = 79 was used in this work.
This correlation is applicable in this case study because for several years of gas
production from the Nikannasin tight gas sandstone, the water production has been zero.
This indicates that the formation has immobile water even in zones with high values of
water saturation (Solano el al., 2011 ; Ortega and Aguilera, 2013). Therefore, Eq. 6.3 can
be written to calculate the irreducible water saturation from:
130
………………………………………………………………………. Eq. 6.4
The calculated values of irreducible water saturation are presented in column 7 of Table
6.1.
Figure 6.2— Buckles plot. The red circles represent porosity and permeability data
from drill cuttings. Lines of constant permeability are represented by solid lines.
The red circles and solid lines are determined using Eq. 6.4. The dashed lines
represent the constant values of the product of porosity and irreducible water
saturation also known as Buckles number. All water saturations are irreducible
since the wells have not produced any water for several years.
Morris and Biggs (1967) also corroborated Buckles’ (1965) observation that the product
of porosity and water saturation was approximately constant for intervals at irreducible
water saturation. Since porosity and permeability (columns 3 and 4 in Table 6.1) are
131
obtained from drill cuttings, and the tight gas formation under consideration does not
produce any water, it is possible to estimate irreducible water saturation with the use of
Eq. 6.4. Porosity, permeability and Buckles number are introduced in Fig. 6.2; the
increasing Buckles number is indicative of increasing reservoir heterogeneity (not
moveable water) since the Nikanassin tight gas formation is known for zero water
production (Solano et al., 2011).
6.2.7 True Formation Resistivity
True formation resistivity is an important parameter in petrophysical evaluation of
hydrocarbon reservoirs (Archie, 1942). It is also important in the construction of Pickett
plots.
Archie established that the ratio of resistivity of a completely brine saturated rock ( ) to
the resistivity of the brine ( saturating the rock is equal to the formation factor ( ).
He also observed that has a linear relationship with porosity:
..............................................................................................................Eq. 6.5
Eq. 6.5 was used for calculating the values of F shown in column 8 of Table 6.1. Next,
values of were calculated as shown in column 9 using irreducible water saturations
values from column 7, m values from column 6, formation water resistivity Rw = 0.038
ohm-m (at 100o C of temperature) and constant a=1, with the use of the equation (Archie,
1942):
132
…………………………..……………………………………Eq. 6.6
where is the water saturation exponent (assumed equal to in this study). Eq. 6.6 can
be applied in this study because the drill cuttings used in the laboratory for determination
of porosity and permeability are clean sandstone samples.
6.2.8 Capillary Pressure
A discontinuity in pressure exists between two immiscible fluids that are in contact. This
pressure difference is generally known as capillary pressure and commonly defined as the
difference in pressure across the interface between two fluids that are not miscible
(Ahmed, 2006). Capillary pressure shown in column 10 of Table 6.1 was calculated from
Eq. 6.7 (Aguilera, 2002):
[ ] ( ⁄ )
…………………………………..…………….Eq. 6.7
where Sw is water saturation (fraction). In order to extend Eq. 6.7 to this study, we
replaced the water saturation (Sw) with irreducible water saturation (Swi). This is valid
because several years of gas production from the Nikannasin formation have resulted in
zero water production, which indicates that the formation is at irreducible water
saturation (Solano el al., 2011; Ortega and Aguilera, 2013).
133
The trend line fit between the capillary pressure and irreducible water saturation was
performed using the power regression option in excel (Fig. 6.3). Strictly, however,
separate capillary pressures have to be developed for each value of porosity and
permeability. For the case at hand, the results are similar to the regression line.
Figure 6.3— Plot of capillary pressure (Pc) vs. irreducible water saturation (Swi). Pc
was determined using Eq. 6.7 and Swi was obtained using Eq. 6.4 based on
knowledge of porosity and permeability from drill cuttings.
6.2.9 Distinguishing Between Viscous and Diffusion-Like Flow
A thorough understanding of gas flow through micro or nano scale pore space (tight
porous media) is important for improving techniques that will ensure economic flow rates
and recoveries from tight formations (Rahmanian et al., 2013). Knudsen number has been
recognized as a flow regime indicator that helps to understand the flow regime (viscous
or diffusion-like flow or maybe a combination of the two) in tight formations. Knudsen
134
number is defined in gas dynamics as the ratio of the molecular mean free path ( ) to
some characteristics length ( ). The molecular mean free path ( ) is the average
distance covered by a moving molecule between successive collisions that modify its
direction or energy or its other properties. The characteristics length ( ) depends on the
type of problem under consideration and the flow geometry (Knudsen, 1909, after
Kennard, 1938, Klinkenberg, 1941, Rahmanian et al., 2013).
Four different flow regimes have been recognized in gas dynamics of porous media and
the value of Knudsen number (Kn) helps to identify these flow regimes: Continuum flow
(Kn<0.01), slip flow (0.01<Kn<0.1), transition flow (0.1<Kn<10), and free molecular flow
(Kn>10).
The basic data and the assumed gas mixture parameters used in this work were published
originally by Javadpour et al. (2007) and have been used by Ortega and Aguilera (2013).
Several researchers including Winland (in Kolodzie, 1980), Pittman (1992), Aguilera
(2002), Byrnes (2005), and Ortega and Aguilera (2013) have shown the relation between
porosity, permeability, and pore throat radii. The pore throat radius (rp35) is used to
calculate Knudsen number in this study. Hence, another benefit of measuring porosity
and permeability from drill cuttings in the laboratory is that it serves as an aid for
determining the dominant flow regime in a reservoir especially for areas where well logs
or core data is scarce. For the tight gas sandstone considered in this study, continuum-
135
flow condition is the dominant flow regime as evidenced by the Kn values calculated
using Eq. 6.7 and 6.8, and shown in Table 6.1, column 11.
…………………………………………………………………...…Eq. 6.7
( ) is expressed as
√ ……………………………………………… ……..……… Eq. 6.8
where is the universal gas constant (Pa.m3/mol.K), is temperature
(K), is Avogadro’s number, is the
average collision diameter of the gas molecules (gm), and is pressure in
the porous media (Pa).
6.2.10 Location of Water Contact
Water contact refers to the elevation above which a gas-water contact or oil-water contact
can be found in a reservoir. However, the Nikanassin formation is a tight gas continuous
accumulation characterized by lack of a water leg. In the Nikanassin formation water is
found above gas due to very low permeability and capillary pressure effects (Masters,
1979; Ramirez and Aguilera, 2011). However, for reservoirs with water below the gas
bearing zone it is possible to make an estimate of the water contact depth with the use of
136
Eq. 6.9 (Aguilera and Ortega, 2013). Note that this estimate starts with the evaluation of
drill cuttings in the laboratory.
……………………………………………..…………………………Eq. 6.9
where h is water contact depth in feet and Pc is capillary pressure in psi.
6.2.11 Flow (or Hydraulic) Units
Flow unit is a reservoir subdivision defined on the basis of similar pore type (Hartmann
and Beaumont, 1999). Data from carbonate and sandstone reservoirs was originally used
by H.D. Winland of Amoco (Kolodzie, 1980) to develop flow units on the basis of pore
throat apertures. Following Winland’s format Aguilera (2003) developed the template
presented in Fig. 6.4, which includes pore throat apertures ( ) as a function of porosity
and permeability. The triangles in the graph correspond to porosities and permeabilities
from the drill cutting samples evaluated in this study. The crossplot indicates that
Nikanassin flow units are dominated by micro pore throats (microports). In this case,
hydraulic fracturing stimulation is required to attain commercial production.
137
Figure 6.4—Chart for estimating pore throat apertures on the basis of permeability
and porosity. The green triangular symbols represent data obtained from drill
cuttings. Nikanassin flow units are dominated by microports. Source of template:
Aguilera (2003).
6.2.12 Construction of Pickett Plots
Pickett (1973) devised a formation evaluation interpretation technique using crossplots of
log responses. His approach indicates that a log-log crossplot of Rt vs. porosity should
result in a straight line with a negative slope equal to the porosity exponent ( ) for
intervals with constant water saturation. Pickett’s plot is based on Archie’s (1942)
equations:
0.001
0.010
0.100
1.000
10.000
100.000
1000.000
10000.000
0.000 5.000 10.000 15.000 20.000 25.000 30.000
Pe
rme
ab
ilit
y, k
ma
x (
mD
)
Porosity (%)
© Servipetrol, 2003
CHART FOR ESTIMATING PORE THROAT APERTURE (Source: Aguilera, CSEG Recorder, Feb 2003)
rp35
20
10
4
2
1
0.5
0.2
me
ga
po
rtsm
acro
po
rtsm
eso
po
rtsm
icrop
orts
0.1
138
…………………………………………………………………….……Eq. 6.10
( ……………………………………………………….................Eq. 6.11
………………………………………………………………………Eq. 6.12
Combining and rearranging Eqs. 6.10 to 6.12 yield:
( ……………………………………..…Eq. 6.13
The conventional construction of a Pickett plot requires the availability of porosity and
resistivity logs. In the present case study, these logs are not available and the Pickett plot
is built on the basis of data extracted from drill cuttings. The 100% water saturation
straight line is drawn by knowing the value of the porosity exponent m, the formation
water resistivity and the value of a (used in Eq. 6.12). For the purpose of the present work
a is assumed to be equal to 1.0, n is assumed equal to m = 1.83 (column 6, Table 6.1).
The water resistivity Rw is equal to 0.038 ohm-m at reservoir temperature (100 C). Fig.
6.5 shows the drill cuttings-based Pickett Plot for the tight gas formation under
consideration built with the porosity and true resistivity data shown in Table 6.1.
In order to construct lines of constant permeability on a Pickett plot, Aguilera (1990)
derived an equation (Eq. 6.14) that indicates that a cross-plot of Rt versus on log-log
139
coordinates should result in a straight line with a slope equal to -3n-m for intervals at
irreducible water saturation with constant k and constant aRw.
( [ (
⁄ ]
………………...……………Eq. 6.14
Fig. 6.6 shows the same Pickett plot presented in Fig. 6.5 but now including lines of
constant permeability. The black circles are drill cuttings data and follow same trend as
some of the lines of constant permeability. Constant permeability values equal to 0.005,
0.05, 0.5 and 5md are plotted on the Pickett plot, the cuttings data fall between the lines
of constant permeability ranging between 0.005 and 0.5 md. The interpretation of Pickett
plot is facilitated by these lines of constant permeability. For instance, note the red square
enclosing a black circle (drill cuttings sample No. 39 in Table 6.1) in Fig. 6.6. From the
graph the interpreter can assign a value slightly larger than 0.05md to this sample. The
actual value is 0.057md as shown in column 4 of Table 6.1.
The construction of the Pickett plot is extended next to include lines of constant capillary
pressure using Eq. 6.15 (Aguilera, 2002). Results are shown in Fig. 6.7 where the black
dot inside the red square corresponds to Sample No. 6 (Table 6.1, column 10). Sample 6
has a porosity of 0.106, a permeability of 0.088 md, a true formation resistivity of 22
ohm-m and a mercury /air capillary pressure of 1204 psi. The same values can be
observed in the Pickett plot shown in Fig. 6.7.
140
( ([ ][ ]
) ……….…Eq. 6.15
Lines of constant pore throat radius (r, microns) can also be included in the Pickett plot
as shown on Fig. 6.8 Eq. 6.15 that was used for constructing lines of constant capillary
pressure can be modified by introducing the Washburn (1921) pore throat equation. This
modification results in the following equation:
( ([ ] [ (
)
]
) ..……Eq. 6.16
For intervals with constant pore throat radius and constant aRw, the crossplot of Rt versus
on log-log coordinates results in a straight line with a slope equal to ( )
(Ortega and Aguilera, 2013). These lines were constructed with the use of Eq. 6.16 based
on the assumption that the mercury/air interfacial tension (IFT) is 480 dyne/cm, and the
mercury/air contact angle is 140o.
141
Figure 6.5— Pickett plot including lines of constant water saturation. The black
circles represent drill cuttings data from a horizontal well drilled in the Nikanassin
group of the WCSB.
Figure 6.6— Pickett plot including lines of constant water saturation and constant
permeability. The black circles represent drill cuttings data from a horizontal well
drilled in the Nikanassin group of the WCSB.
0.01
0.10
1.00
0.01 0.10 1.00 10.00 100.00 1,000.00 10,000.00
Ф
Rt, ohm*m
Cuttings Based Pickett Plot (WCSB)
Cuttings Data
Sw=100%
Sw=50%
Sw=25%
Sw=12.5%
Sw=5%
aRw=0.038
aRw=0.038ohm-m
142
Figure 6.7—Pickett plot including lines of constant water saturation, constant
permeability, and constant capillary pressure. The black circles represent the drill
cuttings data from a horizontal well drilled in the Nikanassin Group of the WCSB.
Figure 6.8—Pickett plot including lines of constant water saturation, constant
permeability, and constant pore throat radius. The black circles represent drill
cuttings data from a horizontal well drilled in the Nikanassin group of the WCSB.
143
To construct constant lines of constant height above the water table in a Pickett plot in
gas reservoirs where an aquifer is present ( ), the capillary pressure in Eq. 6.15 is
substituted with
to give Eq. 6.17
( ([ ] [ (
)
]
) ……..Eq. 6.17
where represents the height in feet above the free water level when capillary pressure is
equal to zero and is mercury-air capillary pressure (psi). The example shown in Fig.
6.9, generated with the use of Eq. 6.17, is presented for illustration purposes only.
However, as mentioned previously in this study one of the characteristics of the
Nikanassin continuous gas accumulation is the lack of a water leg. This has been
corroborated as water-free throughout several years of production from a Nikanassin area
covering more than 15,000 km2 (Solano et al., 2011).
Lines of constant Knudsen number can also be constructed on a Pickett plot with the use
of the equation (Ortega and Aguilera, 2013):
(
([ ][ ] [ ( ( ⁄ ( )⁄ ]
)………………Eq. 6.18
144
The equation indicates that a crossplot of Rt versus on log-log coordinates should result
in a straight line with a slope equal to ( ) (Ortega and Aguilera, 2013) for
intervals with constant aRw. Fig. 6.10 shows the Pickett plot constructed with lines of
constant Knudsen’s number using Eq. 6.18. The drill cuttings data fall between the lines
of Kn equal to 1E-4 and 1E-2. Thus for the tight gas sandstone considered in this study,
continuum-flow is the dominant regime as evidenced in the Pickett plot shown in Fig.
6.10.
Figure 6.9—Pickett plot including lines of constant water saturation, constant
permeability and constant height above the water contact. The black circles
represent drill cuttings data from a horizontal well drilled in the Nikanassin group
of the WCSB.
0.01
0.10
1.00
0.01 0.10 1.00 10.00 100.00 1,000.00 10,000.00
Ф
Rt, ohm*m
Cuttings Based Pickett Plot (WCSB)
Cuttings Data
Sw=100%
Sw=50%
Sw=25%
Sw=12.5%
Sw=5%
k=5md
k=0.5md
k=0.05md
k=0.005md
aRw=0.038
h1=1600 ft
h2 = 800ft
h3 = 400 ft
h4 = 200 ft
h5 = 100 ft
h6 = 50 ft
h7 = 25 ft
h8 = 10 ft
aRw=0.038ohm-m
145
Figure 6.10—Pickett plot including constant lines of water saturation and constant
Knudsen number. The black circles represent drill cuttings data from a horizontal
well drilled in the Nikanassin group of the WCSB.
6.3 Geomechanics Based on Drill Cuttings
Column 1 of Table 6.2 identifies 65 drill cutting samples from a horizontal well used in
this study. Column 2 shows the collection measured depth. As in the case of the
formation evaluation presented above, the drill-cuttings based methodology is not meant
to replace detailed geomechanical studies but to provide estimates of basic parameters for
designing hydraulic fracturing jobs, particularly in those situations where core and log
data are scarce such as the case of some horizontal wells. This facilitates the success of
the operations by pointing to ideal locations for fracture initiation (Ortega and Aguilera,
2012).
146
6.3.1 Brittleness Index
Brittleness index serves as a good indicator for identifying lithological zones suitable for
well stimulation (hydraulic fracturing). Poisson’s ratio and Young’s Modulus are key
mechanical properties that are combined to determine the brittleness index. Poisson’s
ratio indicates the tendency of the rock to fail under stress while Young’s Modulus
indicates the resistance of the rock to deformation which helps to maintain a fracture after
well stimulation (Rick et al., 2008). Therefore, for optimal hydraulic fracturing design in
tight formations the intervals with high brittleness index (high Young’s Modulus and low
Poisson’s ratio) must be selected as zones of interest for initiation of hydraulic fractures.
Furthermore, the more brittle the tight formation, the more likely for these intervals to be
naturally fractured (Ortega and Aguilera, 2012).
The porosity data obtained from laboratory work (column 3 in Table 6.2) performed on
drill cuttings can serve as raw data for calculation of the compressional travel time ( )
shown in column 4 of Table 6.2. In this study DTC is calculated using the sonic log
equation published by Raymer et al. (1980):
(
……………………….…………………………Eq. 6.19
The Raymer et al.(1980) equation has been shown previously to work well for the lower
Nikanassin formation (Ortega and Aguilera, 2012). Also needed is the shear wave
traveling time ( ). In the absence of shear wave information, the Nikanassin empirical
147
correlation shown on Fig. 6.11 can be used as a good approximation as the coefficient of
determination R2 is equal to 0.89 (Ortega and Aguilera, 2012). The linear trend in Fig.
6.11 is represented by the equation:
………………………………………………..…Eq. 6.20
148
Table 6.2— GEOMECHANICAL DATA FOR WESTERN CANADA
SEDIMENTARY BASIN TIGHT GAS SANDSTONE. THE HORIZONTAL
WELL POROSITY AND PERMEABILITY DATA FROM DRILL CUTTINGS
(COLUMN 3) ARE OBTAINED FROM LABORATORY WORK AND IT IS
USED AS A STARTING POINT IN DETERMINING OTHER PETROPHYSICAL
DATA (COLUMN 4 TO 12) USING EMPIRICAL EQUATIONS.
1 2 3 4 5 6 7 8 9 10 11 12
Sample
No.
Bottom
Depth
MD (m)
PHILab
(fraction)
DTC
(us/ft)
DTS
(us/ft)PR G (E6 psi)
YM
Dynamic
(E6 psi)
YM Static
Eissa
(E6psi)
PR-BRIT
(%)
YM -BRIT
(%)
BRIT-
Index(%)
1 3185.0 0.07 60.12 95.04 0.17 3.65 8.52 5.91 64.46 68.48 66.47
2 3190.0 0.06 59.84 94.33 0.16 3.71 8.62 5.98 67.09 70.98 69.03
3 3195.0 0.07 60.02 94.79 0.17 3.67 8.55 5.93 65.39 69.37 67.38
4 3200.0 0.05 58.65 91.34 0.15 3.95 9.08 6.30 78.78 81.75 80.27
5 3202.5 0.07 60.07 94.91 0.17 3.66 8.54 5.92 64.93 68.94 66.93
6 3205.0 0.11 64.77 106.80 0.21 2.89 6.99 4.85 29.35 32.83 31.09
7 3207.5 0.09 62.48 101.02 0.19 3.23 7.69 5.34 44.95 49.20 47.07
8 3210.0 0.09 62.90 102.07 0.19 3.17 7.56 5.24 41.91 46.08 44.00
9 3217.5 0.07 60.41 95.78 0.17 3.59 8.41 5.83 61.84 65.98 63.91
10 3220.0 0.10 63.82 104.40 0.20 3.03 7.27 5.04 35.49 39.38 37.44
11 3225.0 0.10 63.94 104.70 0.20 3.01 7.24 5.02 34.69 38.53 36.61
12 3235.0 0.07 60.04 94.85 0.17 3.67 8.55 5.93 65.17 69.16 67.16
13 3265.0 0.09 62.46 100.95 0.19 3.24 7.70 5.34 45.14 49.40 47.27
14 3285.0 0.15 70.48 121.21 0.24 2.24 5.59 3.88 0.00 0.00 0.00
15 3290.0 0.08 62.14 100.15 0.19 3.29 7.81 5.41 47.56 51.87 49.71
16 3295.0 0.06 58.96 92.11 0.15 3.89 8.96 6.22 75.64 78.91 77.27
17 3300.0 0.09 62.98 102.26 0.19 3.15 7.53 5.23 41.35 45.50 43.43
18 3305.0 0.07 60.36 95.64 0.17 3.61 8.43 5.85 62.32 66.44 64.38
19 3310.0 0.05 58.85 91.84 0.15 3.91 9.00 6.25 76.72 79.89 78.30
20 3315.0 0.06 59.72 94.02 0.16 3.73 8.67 6.01 68.21 72.03 70.12
21 3320.0 0.10 64.67 106.54 0.21 2.91 7.02 4.87 29.97 33.51 31.74
22 3325.0 0.10 63.59 103.81 0.20 3.06 7.34 5.09 37.06 41.03 39.04
23 3345.0 0.12 66.74 111.78 0.22 2.64 6.46 4.48 17.89 20.31 19.10
24 3355.0 0.08 61.36 98.18 0.18 3.42 8.07 5.60 53.74 58.07 55.90
25 3435.0 0.06 59.53 93.56 0.16 3.77 8.74 6.06 69.96 73.68 71.82
26 3440.0 0.07 60.11 95.03 0.17 3.65 8.52 5.91 64.53 68.55 66.54
27 3445.0 0.08 62.25 100.42 0.19 3.27 7.77 5.39 46.73 51.02 48.87
28 3450.0 0.08 61.35 98.16 0.18 3.42 8.07 5.60 53.80 58.12 55.96
29 3455.0 0.04 57.83 89.26 0.14 4.14 9.42 6.54 87.72 89.64 88.68
30 3460.0 0.05 58.61 91.23 0.15 3.96 9.10 6.31 79.24 82.16 80.70
31 3665.0 0.04 57.38 88.12 0.13 4.25 9.61 6.67 92.97 94.14 93.55
32 3690.0 0.07 60.50 96.00 0.17 3.58 8.38 5.81 61.05 65.21 63.13
33 3710.0 0.05 58.31 90.46 0.14 4.03 9.23 6.40 82.48 85.06 83.77
34 3715.0 0.05 57.96 89.60 0.14 4.11 9.37 6.50 86.24 88.36 87.30
35 3720.0 0.05 58.53 91.02 0.15 3.98 9.13 6.34 80.09 82.93 81.51
36 3725.0 0.07 60.08 94.94 0.17 3.66 8.53 5.92 64.82 68.83 66.83
37 3730.0 0.08 61.48 98.47 0.18 3.40 8.03 5.57 52.80 57.12 54.96
38 3735.0 0.05 58.73 91.52 0.15 3.94 9.06 6.28 78.03 81.08 79.55
39 3740.0 0.07 60.30 95.51 0.17 3.62 8.45 5.86 62.79 66.89 64.84
40 3745.0 0.06 59.87 94.42 0.16 3.70 8.61 5.97 66.76 70.66 68.71
41 3750.0 0.04 56.80 86.66 0.12 4.39 9.87 6.84 100.00 100.00 100.00
42 3755.0 0.04 57.73 89.02 0.14 4.16 9.46 6.56 88.83 90.61 89.72
43 3760.0 0.05 58.52 91.01 0.15 3.98 9.14 6.34 80.17 83.00 81.58
44 3765.0 0.05 58.18 90.15 0.14 4.06 9.28 6.43 83.82 86.24 85.03
45 3770.0 0.05 58.33 90.53 0.15 4.02 9.21 6.39 82.19 84.80 83.50
46 3775.0 0.07 60.30 95.51 0.17 3.62 8.45 5.86 62.79 66.89 64.84
47 3785.0 0.07 60.11 95.01 0.17 3.65 8.52 5.91 64.58 68.60 66.59
48 3790.0 0.04 56.94 87.00 0.13 4.36 9.81 6.80 98.32 98.62 98.47
49 3800.0 0.05 58.31 90.48 0.14 4.03 9.22 6.40 82.41 85.00 83.70
50 3820.0 0.06 59.19 92.70 0.16 3.84 8.87 6.15 73.31 76.78 75.04
51 3835.0 0.07 60.18 95.19 0.17 3.64 8.50 5.89 63.94 67.99 65.97
52 3845.0 0.04 56.86 86.80 0.12 4.38 9.84 6.83 99.32 99.45 99.38
53 3850.0 0.06 59.59 93.70 0.16 3.76 8.72 6.05 69.43 73.18 71.31
54 3855.0 0.08 62.16 100.20 0.19 3.28 7.80 5.41 47.41 51.71 49.56
55 3860.0 0.06 59.61 93.76 0.16 3.75 8.71 6.04 69.19 72.95 71.07
56 3865.0 0.07 60.92 97.07 0.18 3.50 8.23 5.71 57.41 61.68 59.54
57 3880.0 0.08 62.19 100.27 0.19 3.28 7.79 5.40 47.18 51.48 49.33
58 3885.0 0.05 58.86 91.87 0.15 3.91 9.00 6.24 76.60 79.78 78.19
59 3890.0 0.08 61.89 99.52 0.18 3.33 7.89 5.47 49.49 53.82 51.66
60 3895.0 0.07 60.92 97.08 0.18 3.50 8.22 5.70 57.38 61.64 59.51
61 3900.0 0.07 61.10 97.51 0.18 3.47 8.16 5.66 55.92 60.21 58.06
62 3930.0 0.08 61.24 97.87 0.18 3.44 8.11 5.63 54.73 59.04 56.88
63 3935.0 0.06 59.60 93.74 0.16 3.75 8.71 6.04 69.30 73.05 71.17
64 3940.0 0.07 60.30 95.50 0.17 3.62 8.45 5.86 62.82 66.91 64.87
65 3945.0 0.08 61.26 97.93 0.18 3.44 8.11 5.62 54.55 58.86 56.70
149
Strictly, if the formation of interest is anisotropic to velocity, then the relationship of
vertical slowness to porosity is not applicable in the horizontal well. However, previous
Nikanassin work (Ortega and Aguilera, 2012) suggests that the empirical correlations
presented here provide a good approximation to DTC and DTS.
Figure 6.11—Plot of ( ) versus ( ) for the Nikanassin formation using data
from an offset vertical well (Ortega and Aguilera, 2012).
The correlation presented in Fig. 6.11 makes the estimation of ( ) possible (Column 5,
Table 6.2). Calculation of compressional wave velocities ( and shear wave
velocities ( are obtained from the following equation:
150
(μsec/ft) and
(μsec/ft) ……………………………………Eq. 6.21
Using (ft/sec) and (ft/sec) from Eq. 6.21, it is possible to estimate values of
Poisson’s ratio (column 6 in Table 6.2) from the equation:
………………………………………… …………………..…Eq. 6.22
The bulk density ( is calculated from:
……………………………………………………….Eq. 6.23
The dry weight and bulk volume parameters in Eq. 6.23 are measured in the laboratory
using drill cuttings. Shear or Rigidity Modulus, G (column 7, Table 6.2) is calculated
from (Barree, 2011):
………………..…………………………………………Eq. 6.24
The dynamic Young’s Modulus, YMd (column 8, Table 6.2) is determined from:
( …………………………………………………………Eq. 6.25
151
Mullen et al. (2007) worked on determining the mechanical rock properties for
stimulation design in the absence of sonic log and presented a derived correlation for
estimating the static Young Modulus, YMs (Eq. 6.26) that can be applied to tight gas
sands, coals and shales. Similarly, he developed for the same purpose a modified form of
the Eissa and Kazi (1988) empirical equation (Eq. 6.27). This modified form of Eissa
and Kazi correlation was used in this study to convert the calculated dynamic Young’s
Modulus to the static Young’s Modulus. Results are shown in column 9, Table 6.2.
( ………………………………………………………… Eq. 6.26
( ( …………………………...………………Eq. 6.27
Fig. 6.12 shows a plot of Poisson’s ratio versus Eissa and Kazi static Young Modulus
developed from drill cuttings data, and a relative indication of the change in brittleness
and ductility. The data are presented in columns 6 and 9 of Table 6.2. For comparison
purposes, results from the Mullen et al. correlation (Eq. 6.26) are also included in Fig.
6.12. Aoudia et al. (2010) show a similar plot on the effects of mineralogy on rock
mechanical properties.
152
Figure 6.12—Cross plot of Young Modulus (YM) versus Poisson’s ratio (PR). The
porosity and bulk density values from drill cuttings (WCSB) were used as input
data in calculating the YM and PR using Eq.6.22 and Eq. 6.27 respectively.
Rickman et al. (2008) have developed a unitized Poisson’s ratio (PRbrit) and unitized
Young’s Modulus (YMbrit) on the basis of the maximum and minimum values of these
parameters with the use of Eq. 6.28 and Eq. 6.29. Since Poisson’s ratio and Young’s
Modulus are mathematically expressed in different units, it is important to unitize these
two mechanical properties. Results are presented in columns 10 and 11 of Table 6.2.
y = -3.0899E+07x + 1.1126E+07R² = 9.9902E-01
y = -2.3870E+07x + 9.8627E+06R² = 9.9713E-01
3.E+06
4.E+06
5.E+06
6.E+06
7.E+06
8.E+06
0.12 0.14 0.16 0.18 0.20 0.22 0.24
Sta
tic Y
ou
ng
Mo
du
lus
(E6
psi
)
Poisson's Ratio
Mullen
Eissa and Kazi
Linear (Mullen)
Linear (Eissa and Kazi)
Increasingbrittleness
Increasingductility
153
(
) …………………………………………………………Eq. 6.28
(
) ……………………………………………………Eq. 6.29
The arithmetic average of (column 10, Table 6.2) and (column 11, Table
6.2) yield the brittleness index ( ) expressed by:
……………………………….………………………Eq. 6.30
The brittleness index value obtained using Eq. 6.30 is a fraction and can be expressed as
a percentage if desired. Column 12 in Table 6.2 shows the brittleness Index values. A
ductile rock will have a brittleness index of zero or close to zero while a brittle rock will
have a brittleness index of one or close to one. In the case of the lower Nikanassin
formation considered in this study, the zones with high brittleness index correspond to
zones that may contain micro fractures and slot porosities. This might be supported by
fracture indicators such as the presence of lost circulation materials, loose crystals and
planar shape features in the corresponding drill cuttings samples.
In those cases where compressional and shear velocities are not available it is still
possible to obtain reasonable estimates of Poisson’s ratio and static Young Modulus
solely on the basis of porosity. This is demonstrated with the use of Fig. 6.13 that shows
154
a log-log crossplot of drill-cuttings porosity versus Poisson’s ratio (Data are extracted
from columns 3 and 6 of Table 6.2). The crossplot results in a nearly straight line with a
coefficient of determination (R2) equal to 0.9999. Thus based on this result, Poisson’s
ratio can be calculated for the study area considered in this paper from the best fit
equation obtained in Fig. 6.13:
…………...………………………………………………Eq. 6.31
where porosity is a fraction. Fig. 6.13 shows that a cross plot of static Young Modulus
versus Poisson’s ratio results in approximate straight line for each, Mullen et al. (2007)
and Eissa and Kazi (1988) correlations. In both cases R2 is greater than 0.99. Based on
these results, static Young Modulus can be calculated for the study area from the best fit
equations shown on Fig. 6.12:
( …………………..……………Eq. 6.32
( …………..…………Eq. 6.33
Having these results, the values of PRbrit, YMbrit and brittleness index are calculated with
the use of Eq. 6.28, 6.29 and 6.30. Results are not shown but they are very close to the
ones generated in Table 6.2. The bottom line of this approach is that important
geomechanical parameters can be estimated when only drill cuttings porosities (and for
that matter porosities from cores and/or logs) are available.
155
Figure 6.13—Empirical log-log cross plot of porosity from drill cuttings versus
Poisson’s ratio (PR) results in a nearly straight line (R2 = 0.9999). Thus porosity can
be used for obtaining a reasonable estimate of Poisson’s ratio in those cases where
compressional and shear velocities are not available.
Although the results are reasonable, it must be emphasized that the methods presented in
this chapter are not meant to replace detailed petrophysical and geomechanical studies
but to provide useful information when core and log data are scarce. To make the
methods presented in this chapter more valuable, even when core, log data and detailed
y = 0.6139x0.4798
R² = 0.9999
0.10
1.00
0.01 0.10 1.00
Po
isso
n's
Ra
tio
Porosity
Drill cuttings
Power (Drill cuttings)
156
studies are available, the methods presented above should still be implemented with a
view to calibrate the equations presented in this chapter for use in wells short of data.
157
Chapter Seven: HYDRAULIC FRACTURE DESIGN OPTIMIZATION USING
DRILL CUTTINGS
7.1 Hydraulic Fracturing of Tight Gas Reservoirs
A high performance reservoir can be described as a reservoir that can produce large
volumes of hydrocarbons at economic rates without performing some extraordinary
completion, stimulation and development practices. Unfortunately, tight gas reservoirs
cannot be described as a high performance reservoir because fluid flow from pore spaces
to wellbore is restricted; their low permeability nature typically equal to or smaller than
0.1 md is responsible for this restriction (Center for Energy, 2013). One of the well
stimulation methods recognized to optimize gas recovery from tight gas reservoirs is
hydraulic fracturing; this method involves the continuous injection of fluids containing
proppants and other chemical additives at a high pressure into tight rock; until it causes
“tensile failure” of the rock and creates a good fracture network (Economides et al.,
1994). This is illustrated in Fig. 7.1.
Figure 7.1— Example of fracture network created by hydraulic fracturing
operation (Source: fracfocus.org, 2013).
158
The starting point for designing a hydraulic fracture treatment is an understanding of the
in-situ stress profile. To determine the in-situ stress profile the mechanical rock
properties and the pore pressure variations throughout the wellbore are required (Mullen,
2007).
In this chapter, we discuss the application of drill cuttings in constructing a hydraulic
fracturing model for optimizing production from a tight gas reservoir. A comparison is
made of two cases: One with drill cuttings for estimating geomechanical properties to
select fracture initiation zones, and one without drill cuttings. The geomechanical data
used in this chapter are taken from Chapter six.
7.1.1 Hydraulic Fracture Design Using Drill Cuttings
A methodology for improving design of multi-stage hydraulic fracturing jobs in
horizontal wells using drill cuttings from vertical wells was developed by Ortega et al.
(2012). This methodology started with porosity data determined from drill cuttings to
estimate shear slowness (DTS) based on the knowledge of compressional wave travel
time (DTC). These data allowed estimating critical geomechanical parameters such as
Poisson’s ratio, Young’s Modulus, and Brittleness Index (see Chapter six for a complete
workflow of how the geomechanical properties are computed). Fig. 7.2, track 2 shows a
good comparison between the compressional wave travel time (DTC) data obtained from
a sonic well log and drill cuttings. This shows that drill cuttings data are reliable in
designing hydraulic fracturing jobs since subsequent geomechanical parameters used for
identifying fracture initiation zones depends on DTC value.
159
Figure 7.2— Track 2 shows a good comparison between drill cuttings data (DTC-
Lab-Dark blue line between the depth interval of 3095-3205m MD) and DTC-Sonic
Log(Green).The track scale is 120-420 us/m.(Adapted from Ortega et al. 2012).
2 1
160
Another important use of drill cuttings data is the generation of the Frac Value (Chapter
three), which is an aid in selecting the selecting the optimum points for fracture initiation
(Ortega et al., 2012). The Frac Value brings microscopic observations of structural
features on drill cuttings into an understandable and readable approach.
The Frac Value, brittleness index, and permeability from drill cuttings are incorporated
into a qualitative model called Cut-Log. The advantages of the Cut-Log as stated by
Ortega et al. (2012) are as follows:
Open-Hole Completions:
Optimized the fracture spacing between packers and number of fracturing stages
in the reservoir
Identify where to initiate the stimulation process using as a guide the brittleness
index, friction losses optimization and tortuosity reductions related to the initial
axial growth length (commonly occur in horizontal wells with vertical hydraulic
fractures; but transverse fractures are also desired and expected, Daneshy, 2011)
Identify where to increase volume of injected fracturing fluid and proppant
concentration
Cemented Casing/Liner Completions:
Maximize the success probability of the fracturing job by identifying where to
perforate
Optimize the number, length and spacing of perforation clusters (Beard, 2011) for
each stage
161
The next section shows how drill cuttings data was used to optimize hydraulic fracturing
design from a horizontal well that penetrates a tight gas reservoir in the Nikanassin Group
of the western Canada sedimentary basin.
7.2 Model Development
GOHFER simulator was used to build the hydraulic fracturing model and to generate
production forecasts. A significant amount of input data are required to build an accurate
model that describes properly the reservoir to be stimulated. Unfortunately, but typical of
way too many cases in the oil and gas industry, the horizontal well (well A) considered in
this study only has a gamma ray log. But fortunately there are drill cuttings (refer to
Table 6.2 in Chapter six). The cuttings data were used to populate the grid cells of
GOHFER to create a hydraulic fracturing model. The availability of this geomechanical
data sourced from drill cuttings in the absence of complete standard well logs and cores
serve as a means of emphasizing another benefit of measuring porosity and permeability
from drill cuttings.
7.2.1 Data Input and Processing
Ideally, reservoir data such as rock mechanical properties are derived from cores and
openhole well logs, and the treatment data such as fluid properties, pumping rates,
proppant concentrations and quantity are provided by the service company.
162
GOHFER is built using log derived input data to generate rock elastic properties, porosity
and lithology. For optimum hydraulic fracturing design, the following data are required :
gamma ray (GR), neutron porosity (NPHI), bulk density (RHOB) , deep resistivity (ILD),
caliper (CAL), compressional travel time (DTC) and shear travel time (DTS) but in the
absence of standard log suite, the optimum minimum log suite required are gamma ray
(GR), neutron porosity (NPHI), and bulk density (RHOB).
Since these logs are not available for the hydraulic fracturing design in the horizontal
well considered in this study (Well A), a reference well (Well B) in the same geological
area was used to generate the grid properties in the vertical direction of the model while
the horizontal well data was superimposed along the bottom of the vertical grid.
Reference well B has gamma ray (GR), neutron porosity (NPHI), bulk density (RHOB),
deep resistivity (ILD), caliper (CAL), compressional travel time (DTC) and shear travel
time (DTS) data. This reference well data correspond to true vertical depth (TVD).
Porosity and permeability from drill cuttings, and geomechanical data extracted from
these properties were used for horizontal well A. Both Wells A and B are drilled
vertically from surface before they start deviating at 1710 m TVD and 1803 m TVD,
respectively. It is assumed that both wells have the same ‘zero’ level for TVD. Fig. 7.3
shows the wellbore survey for wells A and B.
163
Figure 7.3 — Wellbore survey for horizontal Well A 3953 MD / 3115m TVD and
reference Well B 3240m MD / 3192m TVD.
7.2.2 Calibration of GOHFER Generated Data with Drill Cuttings Data
The basic assumption is that data extracted from drill cuttings data represents the tight
gas reservoir accurately. As mentioned above, horizontal well A only has a gamma ray
(GR) log. The GR log was used by GOHFER to generate rock mechanical properties
using in-built correlations. The coefficients of the GR empirical equations for generating
rock mechanical properties such as Poisson’s ratio (PRGR), compressional travel time
(DTCGR), and dynamic Young Modulus (YMEGR) were calibrated with the
geomechanical data extracted from drill cuttings measurements of porosity and
164
permeability in the laboratory (Poisson’s ratio, PRLab; compressional travel time,
DTCLab; and dynamic Young Modulus, YMELab). Fig. 7.4 to 7.6 shows matches
between the GR GOHFER correlated curves and geomechanical data from the drill
cuttings.
Where in track 2 of Fig. 7.4,
DTCRESIST = DTC from resistivity, DTCPHIN = DTC from Neutron Porosity,
DTCACT = Actual DTC from Log, DTC PHIA = DTC from average porosity, DTCGR =
DTC from gamma ray, and DTCLab = DTC from drill cuttings. All these parameters are
expressed in microseconds per meter (µSec/m).
In track 2 of Fig. 7.5,
PRRESIST = PR from resistivity, PRPHIA = PR from average porosity, PRDTC = PR
from DTC Log, PRACT = Actual PR from Log, PRGR = PR from gamma ray, and
PRLab = PR from drill cuttings. All these parameters are expressed in fractions.
And in track 2 of Fig. 7.6,
YMERESIST = YME from resistivity, YMEPHIA = YME from average porosity,
YMEDTC = YME from DTC log, YMEACT = Actual YME from Log, YMEGR = YME
from gamma ray, and YMELab = YME from drill cuttings. All these parameters are
expressed in Giga Pascals (GPa).
165
Figure 7.4 — Calibration of compressional travel time DTCGR from GOHFER
(orange Line) with DTCLab extracted from drill cuttings (blue line). The 2 curves
are shown in Track 2.
1 2
166
Figure 7.5 — Calibration of Poisson’s ratio PRGR from GOHFER (orange Line)
with PRLab (blue line) extracted from drill cuttings. The 2 curves are shown on
Track 2.
1 2
167
Figure 7.6 — Calibration of dynamic Young Modulus YMEGR from GOHFER
(Orange Line) with YMELab (pink line). The 2 curves are shown on Track 2.
1 2
168
With the data input, and calibration process completed, the grid properties are populated
in the model. Fig. 7.7 show examples of the model populated with four grid properties.
Figure 7.7— Example of grid setup showing grid properties such as total closure
stress, brittleness factor, permeability and static young modulus . The reference well
(well B) data was used to populate the grid properties from surface to 3225m MD
while the remaining depth interval to 3950m MD that represents the horizontal
section was populated using the horizontal well (well A) data.
169
7.2.3 Multi-Stage Hydraulic Fracture Treatment Design
With the construction phase completed, the next step is to design the treatment
configuration. Two seven-stage hydraulic fracturing design cases were considered in this
work, Case I involves initiating fractures in the zones that were actually treated in
horizontal well A (shown in Appendix D). Case II involves using the Cut-Log (Fig. 7.8)
to determine zones where fractures can be initiated to improve gas recovery (i.e. re-
design case using drill cuttings data).
To emphasize the importance on production performance of fracture initiation points, a
constant pump schedule was used for the seven stages (Table 7.1) and the only variable
was the treated zones.
Table 7.1—BOTTOM HOLE PUMP SCHEDULE
1 2 3 4 5 6
FLUID FLUID CUM PROP PROP TOTAL B.H.
TYPE VOLUME FLUID TYPE CONC INJ. RATE
(m3) (m3)
Kg/m3 (m3/min)
PAD 34.4 34.4 None 0.0 9
2 58.5 92.9 100 MESH 50.0 9
3 31.7 124.6 None 0.0 9
4 75.6 200.2 40/70 SAND 25.0 9
5 83.5 283.7 40/70 SAND 50.0 9
6 68.9 352.6 None 0.0 9
7 80.1 432.7 40/70 SAND 25.0 9
8 79.1 511.8 40/70 SAND 50.0 9
9 34.6 546.3 40/70 SAND 75.0 9
10 62.2 608.6 None 0.0 9
11 62.4 670.9 40/70 SAND 50.0 9
12 55.7 726.7 40/70 SAND 75.0 9
13 44.6 771.3 40/70 SAND 100.0 9
SPACER 12.2 783.5 None 0.0 9
FLUSH 66.7 850.2 None 0.0 9
In Table 7.1, column 1 shows the sequence of operation, columns 2 and 3 show the
injected fluid volume, column 4 the type of proppant, column 5 the proppant
concentration, and column 6 shows the total bottom hole injection rate. Fig. 7.8 shows the
Cut-Log that was used as a guide in selecting zones with potential for high gas recovery.
170
Three parameter tracks ranging in depth from 3200m MD to 3900m MD are displayed in
Fig. 7.8. Track 1 is the brittleness index, track 2 is the permeability and track 3 is the Frac
Value. Higher values of all three parameters are desired but two high values are still
acceptable. The blue color represents the zones suitable for optimum fracture initiation
and the color higher intensity is intentionally related to better conditions.
Figure 7.8 — Cut-Log with three parameter tracks ranging from 3200 to 3900m
MD. Track 1 is the brittleness index, track 2 is permeability and track 3 is the Frac-
Value. Higher values of all three parameters are desired but two high values are still
acceptable. The blue color represents the zones suitable for optimum fracture
initiation and the color intensity is intentionally related to better conditions.
171
7.3 Results
The economic recovery of gas from tight gas reservoirs depends on successful
stimulation of wells using hydraulic fracturing techniques. The number of treatment
stages, the location, spacing, and number of hydraulic fractures created per stage; all
affect gas recovery. Therefore accurate modeling of hydraulic fractures is required to
predict production rates and to improve future stimulation strategies. Production forecasts
were carried out for the hydraulic fracture designs with and without the inclusion of drill
cuttings data. The results (shown for 364 days) highlight the importance of using drill
cuttings for selecting fracture initiation zones as opposed to using symmetric intervals for
multi-stage hydraulic fracturing, a common practice in the oil and gas industry due to
data scarcity.
Fig. 7.9 shows a comparison of results. The Re-design case based on drill cuttings shows
a better performance for the 364 days of production forecast Tables 7.2 and 7.3 shows
results for each hydraulic fracturing stage. Columns 2, 4, 6, 8, 10, 12, and 14 show the
cumulative gas production per stage, while columns 3, 5, 7, 9, 11, 13 and 15 show the
daily gas production rate for each fracture stage. Columns 16 and 17 show the total
cumulative gas production and the gas rate, respectively; these totals were determined by
adding the individual fracture stage cumulative gas production and rates.
172
Figure 7.9—Comparison of cumulative gas production profile between the Initial
Design and the Re-design case using Cut-Log. The two cases involved a seven stage
fracture treatment but the fracture initiation zones for the two cases are different.
The Re-design case shows a better performance for the 364 days of production
forecast; these performance shows that selecting symmetrical distances for
hydraulic fracture initiation, as done commonly in the oil and gas industry due to
data scarcity, is not optimum. Better performance is obtained selecting those zones
with high brittleness index, permeability and Frac-Value in the Cut-Log,
parameters obtained from drill cuttings in this study.
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
0.0 100.0 200.0 300.0 400.0
Cu
mu
lati
ve
Ga
s P
rod
uct
ion
, E
6 m
3
Time, days
Re-Design
Initial Design
173
Table 7.2— SIMULATION OUTPUT FROM INITIAL MODEL DESIGN BASED
ON SYMMETRICAL DISTANCES FOR HYDRAULIC FRACTURE
INITIATION.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
TimeCumGas
(e6m3)
Gas Rate
(e3m3/day)
CumGas
(e6m3)
Gas Rate
(e3m3/day)
CumGas
(e6m3)
Gas Rate
(e3m3/day)
CumGas
(e6m3)
Gas Rate
(e3m3/day)
CumGas
(e6m3)
Gas Rate
(e3m3/day)
CumGas
(e6m3)
Gas Rate
(e3m3/day)
CumGas
(e6m3)
Gas Rate
(e3m3/day)
CumGas
(e6m3)
Gas Rate
(e3m3/day)
0.000 0.004 4.318 0.004 4.318 0.004 4.317 0.026 25.848 0.026 25.939 0.003 3.074 0.002 1.527 0.069 69.341
1.000 0.008 3.963 0.008 3.963 0.008 3.963 0.049 23.450 0.050 23.533 0.006 2.618 0.003 1.313 0.132 62.803
2.100 0.012 3.669 0.012 3.669 0.012 3.669 0.073 21.412 0.073 21.441 0.008 2.270 0.004 1.169 0.195 57.299
3.310 0.017 3.454 0.017 3.454 0.017 3.483 0.097 19.969 0.097 20.039 0.011 2.004 0.005 1.078 0.260 53.481
4.641 0.021 3.319 0.021 3.319 0.021 3.347 0.122 18.936 0.123 18.962 0.013 1.863 0.007 1.027 0.328 50.773
6.105 0.026 3.190 0.026 3.190 0.026 3.216 0.149 18.067 0.149 18.129 0.016 1.732 0.008 0.979 0.399 48.503
7.716 0.031 3.115 0.031 3.115 0.031 3.090 0.177 17.378 0.177 17.403 0.018 1.651 0.010 0.933 0.474 46.685
9.487 0.036 2.993 0.036 2.993 0.036 3.018 0.206 16.784 0.207 16.808 0.021 1.574 0.011 0.889 0.553 45.059
11.436 0.042 2.923 0.042 2.923 0.042 2.947 0.238 16.244 0.239 16.267 0.024 1.500 0.013 0.862 0.639 43.666
13.579 0.048 2.855 0.048 2.855 0.048 2.878 0.272 15.720 0.272 15.742 0.027 1.430 0.015 0.835 0.729 42.315
15.937 0.054 2.788 0.054 2.788 0.054 2.766 0.308 15.244 0.308 15.266 0.030 1.407 0.017 0.809 0.826 41.068
18.531 0.061 2.722 0.061 2.722 0.062 2.701 0.346 14.783 0.347 14.804 0.034 1.341 0.019 0.783 0.930 39.856
21.384 0.069 2.659 0.069 2.659 0.069 2.637 0.387 14.335 0.388 14.355 0.038 1.320 0.021 0.759 1.040 38.724
24.523 0.077 2.596 0.077 2.596 0.077 2.576 0.431 13.915 0.431 13.935 0.042 1.258 0.023 0.735 1.158 37.611
27.975 0.086 2.535 0.086 2.535 0.086 2.515 0.477 13.453 0.478 13.472 0.046 1.239 0.026 0.712 1.284 36.461
31.772 0.095 2.436 0.095 2.436 0.095 2.456 0.527 13.059 0.528 13.078 0.051 1.219 0.028 0.729 1.419 35.413
35.950 0.105 2.379 0.105 2.379 0.105 2.399 0.579 12.625 0.580 12.643 0.055 1.162 0.031 0.707 1.561 34.294
40.545 0.116 2.323 0.116 2.323 0.116 2.343 0.636 12.256 0.637 12.273 0.061 1.144 0.035 0.685 1.716 33.347
45.599 0.127 2.269 0.127 2.269 0.127 2.251 0.696 11.849 0.697 11.865 0.066 1.126 0.038 0.663 1.878 32.292
51.159 0.139 2.216 0.139 2.216 0.139 2.198 0.759 11.455 0.761 11.471 0.073 1.109 0.042 0.680 2.051 31.345
57.275 0.153 2.129 0.153 2.129 0.153 2.147 0.827 11.075 0.828 11.091 0.079 1.057 0.046 0.659 2.239 30.287
64.003 0.167 2.079 0.167 2.079 0.167 2.096 0.899 10.707 0.901 10.722 0.086 1.040 0.050 0.639 2.437 29.362
71.403 0.182 2.031 0.182 2.031 0.182 2.015 0.976 10.352 0.977 10.366 0.094 1.024 0.055 0.619 2.647 28.438
79.543 0.197 1.951 0.197 1.951 0.198 1.967 1.057 10.008 1.059 10.022 0.102 1.008 0.059 0.599 2.869 27.506
88.497 0.214 1.906 0.214 1.906 0.215 1.921 1.144 9.676 1.146 9.689 0.110 0.961 0.065 0.616 3.108 26.675
98.347 0.233 1.861 0.233 1.861 0.233 1.846 1.236 9.317 1.237 9.330 0.120 0.946 0.071 0.597 3.363 25.758
109.182 0.252 1.788 0.252 1.788 0.253 1.803 1.333 9.007 1.335 9.020 0.130 0.931 0.077 0.579 3.632 24.916
121.100 0.273 1.760 0.273 1.760 0.273 1.718 1.436 8.673 1.439 8.685 0.140 0.887 0.084 0.560 3.918 24.043
134.210 0.295 1.678 0.295 1.678 0.295 1.692 1.546 8.351 1.548 8.363 0.152 0.859 0.091 0.543 4.222 23.164
148.631 0.319 1.652 0.319 1.652 0.318 1.612 1.662 8.041 1.664 8.053 0.164 0.833 0.099 0.526 4.545 22.369
164.494 0.344 1.574 0.344 1.574 0.344 1.587 1.785 7.743 1.787 7.754 0.176 0.807 0.107 0.543 4.887 21.582
181.943 0.370 1.500 0.370 1.500 0.370 1.513 1.914 7.425 1.917 7.436 0.191 0.824 0.116 0.526 5.248 20.724
201.138 0.398 1.477 0.398 1.477 0.399 1.489 2.051 7.121 2.054 7.131 0.206 0.798 0.126 0.510 5.632 20.003
222.252 0.428 1.408 0.428 1.408 0.429 1.419 2.195 6.815 2.198 6.824 0.222 0.773 0.136 0.494 6.036 19.141
245.477 0.459 1.342 0.459 1.342 0.460 1.353 2.346 6.522 2.350 6.531 0.240 0.749 0.148 0.478 6.462 18.317
271.024 0.492 1.279 0.492 1.279 0.493 1.289 2.504 6.191 2.508 6.199 0.258 0.726 0.159 0.464 6.906 17.427
299.127 0.526 1.219 0.526 1.219 0.527 1.229 2.670 5.876 2.674 5.885 0.278 0.703 0.172 0.449 7.373 16.580
330.039 0.562 1.162 0.562 1.162 0.564 1.171 2.842 5.578 2.846 5.586 0.299 0.681 0.185 0.421 7.860 15.761
364.000 0.600 1.107 0.600 1.107 0.602 1.116 3.020 5.251 3.025 5.259 0.321 0.660 0.200 0.438 8.368 14.938
STAGE 6 (3419)STAGE 1 (3876) STAGE 2 (3814) STAGE 3 (3708) STAGE 4 (3616) STAGE 5 (3511)
BASECASE-INITIAL DESIGN
STAGE 7 (3314) TOTAL
174
Table 7.3— SIMULATION OUTPUT FROM RE-DESIGN MODEL BASED ON
HYDRAULIC FRACTURE INITIATIONS FROM THE CUTTINGS-BASED
CUT-LOG.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
TimeCumGas
(e6m3)
Gas Rate
(e3m3/day)
CumGas
(e6m3)
Gas Rate
(e3m3/day)
CumGas
(e6m3)
Gas Rate
(e3m3/day)
CumGas
(e6m3)
Gas Rate
(e3m3/day)
CumGas
(e6m3)
Gas Rate
(e3m3/day)
CumGas
(e6m3)
Gas Rate
(e3m3/day)
CumGas
(e6m3)
Gas Rate
(e3m3/day)
CumGas
(e6m3)
Gas Rate
(e3m3/day)
0.000 0.023 22.690 0.021 20.606 0.026 26.052 0.002 1.527 0.002 1.527 0.002 1.527 0.002 1.527 0.076 75.456
1.000 0.043 20.585 0.039 18.694 0.050 23.635 0.003 1.313 0.003 1.313 0.003 1.313 0.003 1.313 0.144 68.166
2.100 0.064 18.876 0.058 17.179 0.073 21.535 0.004 1.169 0.004 1.169 0.004 1.169 0.004 1.169 0.212 62.266
3.310 0.086 17.715 0.078 16.089 0.098 20.126 0.005 1.078 0.005 1.078 0.005 1.078 0.005 1.078 0.283 58.242
4.641 0.108 16.798 0.098 15.256 0.123 19.045 0.007 1.027 0.007 1.027 0.007 1.027 0.007 1.027 0.356 55.207
6.105 0.131 16.126 0.119 14.645 0.150 18.171 0.008 0.979 0.008 0.979 0.008 0.979 0.008 0.979 0.433 52.858
7.716 0.156 15.511 0.142 14.087 0.178 17.479 0.010 0.933 0.010 0.933 0.010 0.933 0.010 0.933 0.515 50.809
9.487 0.183 15.012 0.166 13.620 0.208 16.881 0.011 0.889 0.011 0.889 0.011 0.889 0.011 0.889 0.602 49.069
11.436 0.211 14.557 0.192 13.221 0.240 16.304 0.013 0.862 0.013 0.862 0.013 0.862 0.013 0.862 0.695 47.530
13.579 0.242 14.116 0.219 12.833 0.273 15.811 0.015 0.835 0.015 0.835 0.015 0.835 0.015 0.835 0.793 46.100
15.937 0.274 13.703 0.249 12.457 0.309 15.301 0.017 0.809 0.017 0.809 0.017 0.809 0.017 0.809 0.899 44.697
18.531 0.309 13.355 0.280 12.092 0.348 14.838 0.019 0.783 0.019 0.783 0.019 0.783 0.019 0.783 1.012 43.417
21.384 0.346 12.964 0.314 11.785 0.389 14.389 0.021 0.759 0.021 0.759 0.021 0.759 0.021 0.759 1.133 42.174
24.523 0.385 12.584 0.350 11.440 0.433 13.967 0.023 0.735 0.023 0.735 0.023 0.735 0.023 0.735 1.261 40.931
27.975 0.427 12.215 0.388 11.105 0.479 13.503 0.026 0.712 0.026 0.712 0.026 0.712 0.026 0.712 1.396 39.671
31.772 0.472 11.857 0.429 10.779 0.529 13.108 0.028 0.729 0.028 0.729 0.028 0.729 0.028 0.729 1.544 38.660
35.950 0.520 11.510 0.473 10.464 0.582 12.673 0.031 0.707 0.031 0.707 0.031 0.707 0.031 0.707 1.701 37.475
40.545 0.572 11.173 0.519 10.157 0.639 12.301 0.035 0.685 0.035 0.685 0.035 0.685 0.035 0.685 1.868 36.371
45.599 0.627 10.845 0.569 9.860 0.699 11.893 0.038 0.663 0.038 0.663 0.038 0.663 0.038 0.663 2.047 35.250
51.159 0.685 10.528 0.622 9.571 0.763 11.498 0.041 0.642 0.042 0.680 0.042 0.680 0.042 0.680 2.236 34.279
57.275 0.747 10.178 0.679 9.253 0.831 11.116 0.046 0.660 0.046 0.659 0.046 0.659 0.046 0.659 2.440 33.184
64.003 0.814 9.880 0.739 8.982 0.903 10.747 0.050 0.639 0.050 0.639 0.050 0.639 0.050 0.639 2.656 32.165
71.403 0.884 9.552 0.804 8.684 0.980 10.390 0.054 0.619 0.055 0.619 0.055 0.619 0.055 0.619 2.886 31.102
79.543 0.960 9.235 0.872 8.395 1.062 10.045 0.059 0.600 0.059 0.599 0.059 0.599 0.059 0.599 3.131 30.072
88.497 1.040 8.928 0.945 8.149 1.148 9.673 0.065 0.617 0.065 0.616 0.065 0.616 0.065 0.616 3.393 29.215
98.347 1.125 8.632 1.023 7.879 1.240 9.351 0.071 0.598 0.071 0.597 0.071 0.597 0.071 0.597 3.671 28.251
109.182 1.215 8.345 1.105 7.617 1.338 9.004 0.077 0.579 0.077 0.579 0.077 0.579 0.077 0.579 3.966 27.282
121.100 1.311 8.068 1.193 7.334 1.442 8.705 0.084 0.561 0.084 0.560 0.084 0.560 0.084 0.560 4.281 26.348
134.210 1.413 7.800 1.286 7.091 1.552 8.382 0.091 0.543 0.091 0.543 0.091 0.543 0.091 0.543 4.614 25.445
148.631 1.522 7.511 1.384 6.842 1.668 8.071 0.098 0.526 0.099 0.526 0.099 0.526 0.099 0.526 4.968 24.528
164.494 1.636 7.232 1.489 6.601 1.791 7.740 0.107 0.544 0.107 0.543 0.107 0.543 0.107 0.543 5.344 23.746
181.943 1.758 6.978 1.599 6.317 1.921 7.453 0.116 0.527 0.116 0.526 0.116 0.526 0.116 0.526 5.742 22.853
201.138 1.886 6.678 1.716 6.095 2.057 7.118 0.126 0.510 0.126 0.510 0.126 0.510 0.126 0.510 6.163 21.931
222.252 2.021 6.391 1.839 5.833 2.201 6.812 0.136 0.494 0.136 0.494 0.136 0.494 0.136 0.494 6.605 21.012
245.477 2.163 6.116 1.969 5.583 2.353 6.520 0.147 0.479 0.148 0.478 0.148 0.478 0.148 0.478 7.076 20.132
271.024 2.313 5.854 2.106 5.343 2.511 6.189 0.159 0.464 0.159 0.464 0.159 0.464 0.159 0.464 7.566 19.242
299.127 2.469 5.556 2.248 5.072 2.676 5.874 0.172 0.449 0.172 0.449 0.172 0.449 0.172 0.449 8.081 18.298
330.039 2.632 5.274 2.397 4.814 2.848 5.576 0.185 0.421 0.185 0.421 0.185 0.421 0.185 0.421 8.617 17.348
364.000 2.802 5.006 2.552 4.570 3.027 5.249 0.200 0.439 0.200 0.438 0.200 0.438 0.200 0.438 9.181 16.578
STAGE 6 (3394) STAGE 7 (3299)
REDESIGN AFTER CUT-LOG
TOTALSTAGE 1 (3854) STAGE 2 (3746) STAGE 3 (3645) STAGE 4 (3446) STAGE 5 (3342)
175
Chapter Eight: CONCLUSION AND RECOMMENDATIONS
Conclusions stemming from this thesis are presented below separately for drill cuttings,
porosity exponent (m), water saturation exponent (n), petrophysics and geomechanical
evaluations, and finally hydraulic fracturing and modelling. This is followed by some
recommendations.
8.1 Drill Cuttings
1. Drill cuttings are powerful direct sources of information that can be used in the
laboratory for determining porosities and permeabilities of tight formations
penetrated by horizontal wells. This is of significant practical value particularly
because of the usual data scarcity in horizontal wells.
2. The data allow quantitative formation evaluation of horizontal wells even in the
absence of well logs. This includes the determination of pore throat aperture
radius (rp35), water saturation, porosity exponent (m), true formation resistivity,
capillary pressure, Knudsen number, depth to the water contact (if present) and
construction of Pickett plots.
3. The data also allow estimation of geomechanical properties including Poisson’s
ratio, Young’s Modulus, Brittleness Index, and a Cut-Log useful for determining
the optimum locations for initiating hydraulic fractures in multi-stage fracturing
jobs.
8.2 Porosity Exponent (m)
1. Sihvola’s unified electromagnetic mixing rules are shown to be applicable for
176
handling at once Maxwell Garnett, Bruggeman and the Coherent Potential
formulas.
2. These formulas have been used for developing new dual and triple porosity
models for calculation of the porosity exponent m in naturally fractured
reservoirs.
3. The models have been compared successfully with core data from limestones,
dolomites and tight sandstones. When applicable the models have also been
compared successfully with other theoretical models developed previously in the
literature.
4. The models are robust and can be used for any combination of porosities
(including one or two porosities equal to zero). This makes the models powerful
as they can be run throughout heterogeneous naturally fractured reservoirs
without a need for changing equations with depth.
5. The advantage of the petrophysical models based on the Maxwell Garnett
equation is that the calculation of m can be done explicitly without trial and error.
The Bruggeman and coherent potential formulas require an iteration procedure for
calculating m.
177
8.3 Water Saturation Exponent (n)
1. The same electromagnetic mixing rule mentioned above has been used for
developing new theoretical models for determining the water saturation exponent
(n) and water saturation (Sw) in dual and triple porosity reservoirs with mixed
wettability.
2. The dual and triple porosity models are robust and work correctly in the case of
single porosity (matrix) reservoirs when isolated and fracture porosities are equal
to zero. The single porosity model has been validated with the use of
preferentially oil wet and also preferentially water wet core data.
3. The models require water saturation of the matrix blocks or resistivity index as
input data. The value of n for dual and triple porosity reservoirs can be calculated
explicitly when water saturation of the matrix block (Swb) is known. However, an
iterative procedure is required when resistivity index of the matrix block or whole
block is used as input data.
4. Use of the models requires knowledge of the porosity exponent of the matrix (mb)
as well as the porosity exponent of the composite system (m). These parameters
are determined from petrophysical models for dual and triple porosity reservoirs
developed in this thesis with the use of the same electromagnetic mixing rules
mentioned above.
178
8.4 Petrophysical and Geomechanical Evaluation
1. Porosity and permeability data determined from drill cuttings of a horizontal well
have been used for complete petrophysical evaluation of a tight gas sandstone in
the Nikanassin Group.
2. Porosity data from drill cuttings can serve as input data for geomechanical
calculations and also serve as an aid in identifying zones with high brittleness
index. This increases the success probabilities of stimulation operations by
pointing to ideal locations for fracture initiation.
3. Starting with only drill cuttings, Pickett plots that include water saturation,
permeability, pore throat radii, capillary pressure, height above the water table,
and Knudsen numbers have been constructed.
4. For those cases where wells are short of data, a new method has been developed
for estimating Poisson’s ratio, static Young Modulus and brittleness index based
on knowledge of only porosity.
5. For the tight gas sandstone considered in this study, the dominant flow regime is
continuum-flow as evidenced by calculated Knudsen numbers.
6. The proposed drill cuttings-based petrophysical and geomechanical evaluations
are not meant to replace detailed studies when complete data sets are available.
179
But drill cuttings can provide good estimates of several important reservoir
parameters particularly in those cases where core and log data are scarce.
8.5 Hydraulic Fracturing and Modeling
1. A multi-stage hydraulically fractured horizontal well has been modeled
successfully using GOHFER. Drill cuttings data were used to calibrate
geomechanical correlations built in the simulator.
2. Grid properties in the hydraulic fracturing model were populated using data from
drill cuttings.
3. Design of hydraulic fracturing models using drill cuttings data can serve as an aid
in identifying optimum fracture initiation zones during multi-stage hydraulic
fracturing jobs. These zones can be at any distance to each other if wellbore
configuration allows.
4. Comparison of cumulative gas production profiles indicate that better
performance is obtained selecting those zones with high brittleness index,
permeability and Frac-Value in the Cut Log developed from drill cuttings. The
performance selecting symmetrical distances for hydraulic fracture initiation, as
done commonly in the oil and gas industry due to data scarcity, is not optimum.
Recommendations are as follows:
1. Extend the evaluation of drill cuttings to the case of petroleum shale reservoirs.
180
2. Conduct the evaluations at various confining pressures.
3. Use the results to develop petrophysical models for evaluation of shale petroleum
reservoirs.
4. Demonstrate that the drill cuttings based methodologies presented in this thesis
are applicable in the case of conventional reservoirs.
5. Extend the drill cuttings and petrophysical methodologies to the case of oil sands.
181
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APPENDIX A: SCREEN SHOTS OF PERMEABILITY MEASUREMENTS
USING DARCYLOG
Sample 1_ 3185m MD
Sample 2_ 3190m MD
197
Sample 3_ 3195m MD
Sample 4_ 3200m MD
198
Sample 5_ 3205.5m MD
Sample 6_ 3205m MD
199
Sample 7_ 3207.5m MD
Sample 8_ 3210m MD
200
Sample 9_ 3217.5m MD
Sample 10_ 3220.0 MD
201
Sample 11_ 3225MD
Sample 12_ 3235MD
202
Sample 13_ 3265MD
Sample 14_ 3285MD
203
Sample 15_ 3290MD
Sample 16_ 3295MD
204
Sample 17_ 3300MD
Sample 18_ 3305MD
205
Sample 19_ 3310MD
Sample 20_ 3315MD
206
Sample 21_ 3320MD
Sample 22_ 3325MD
207
Sample 23_ 3345MD
Sample 24_ 3355MD
208
Sample 25_ 3435MD
Sample 26_ 3440MD
209
Sample 27_ 3445MD
Sample 28_ 3450MD
210
Sample 29_ 3460MD
Sample 30_ 3690MD
211
Sample 31_ 3710MD
Sample 32_ 3715MD
Sample 33_ 3720MD
212
Sample 33_ 3720MD
Sample 34_ 3725MD
213
Sample 35_ 3730MD
Sample 36_ 3735MD
214
Sample 37_ 3740MD
Sample 38_ 3745MD
215
Sample 39_ 3760MD
Sample 40_ 3765MD
216
Sample 41_ 3770MD
Sample 42_ 3775MD
Sample 43_ 3785MD
217
Sample 43_ 3785MD
Sample 44_ 3820MD
218
Sample 45_ 3835MD
Sample 46_ 3855MD
219
Sample 47_ 3860MD
Sample 48_ 3865M
220
Sample 49_ 3880MD
Sample 50_ 3885MD
221
Sample 51_ 3890MD
Sample 52_ 3895MD
222
Sample 53_ 3900MD
Sample 54_ 3930MD
223
Sample 55_ 3935MD
Sample 56_ 3940MD
224
Sample 57_ 3945MD
225
APPENDIX B : RELEVANT EQUATIONS
The following equations (Aguilera and Aguilera, 2003, 2004) are useful for petrophysical
calculations in order to maintain consistency in the scaling of porosity.
For dual porosity models:
……….……………………………...………………………………………….(B-1)
( ………………..…………………...………………………..……………..(B-2)
( ………………………………………...……….……...(B-3)
………..…………….………………………………………………………….….(B-4)
( …………………………………………………………………….………..(B-5)
( ……………………………………………………...….....(B-6)
For triple porosity models:
( ……………………………….……..(B-7)
226
APPENDIX C : ANGLE BETWEEN THE FRACTURES AND DIRECTION OF
CURRENT FLOW
Schematics assumed that current direction is horizontal in all cases, and the angle θ in the
schematic corresponds to fracture dip. Case A shows the horizontal fracture with no
tortuosity (mf = 1.0), Case B shows horizontal fracture with tortuosity that leads to a
porosity exponent of the fractures (mf) equal to 1.3, Case C shows the non-horizontal
fracture (θ = 50°) with no tortuosity (mf = 1.0); the 50° angle leads to a pseudo fracture
porosity exponent (mfp) equal to 1.19, and Case D shows the non-horizontal fracture (θ =
50°) with tortuosity (mf = 1.3). The 50° angle leads to a pseudo fracture porosity
exponent (mfp) equal to 1.49. If the flow of current is vertical, the angle corresponds to
1.0 minus fracture dip (Adapted from Aguilera and Aguilera, 2006).
227
APPENDIX D : WELL CONFIGURATION FOR INITIAL DESIGN
Well A