Copyright

By

Maryam Alsadat Mousavi

2010

The Dissertation Committee for Maryam Alsadat Mousavi certifies that this is the approved version of the following dissertation:

Pore scale characterization and modeling of two-phase flow in tight gas

sandstones

Committee: ____________________________________ Steven L. Bryant, (Supervisor) ____________________________________ Kitty Milliken ____________________________________ Jon E. Olson ____________________________________ Kamy Sepehrnoori ____________________________________ Sanjay Srinivasan

Pore scale characterization and modeling of two-phase flow in tight gas

sandstones

by

Maryam Alsadat Mousavi, B.S.; M.S.

Dissertation

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

The University of Texas at Austin

May 2010

Dedication

To my mother Parvaneh Eskandari for her endless support, to my husband Navid Hayeri

for his endless love and encouragement, to my lovely son Sepehr Hayeri for bringing

happiness to my life

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Acknowledgement

I would like to like to express my gratitude to all people who have supported me

throughout my PhD study. First, I would like to thank Dr. Bryant for guiding me as a

graduate student and helping me to understand the pore level petrophysics and being

available for interesting discussions. I would also like to thank Dr. Holder for lending his

time and his valuable advice on the subject of grain mechanics. To my committee

members, thank you for helpful suggestions to improve my dissertation.

I would like to acknowledge my officemates and people I had the pleasure of

working with throughout the years; especially Javad Behseresht for providing me with

codes regarding simulation of drainage capillary pressure and relative permeability and

helping me to learn how to write code in Matlab®. Without your help and kindness, I

would not finish this dissertation. I thank Elena Rodriguez for helping me with Matlab®

and her kindness whenever I needed help. I thank Jalil Varavei for helping me to learn

Transport phenomena and Kiomars Eskandari for helping me to learn petrophysics for

qualifying exam. Thank to Siavash Motealle for his helpful discussions on tight gas

sandstones properties and Masa Prodanovic for her assistant with my project and helpful

discussions. Thanks to all of my friends for helping and supporting me during my study.

All of above, I would like to thank my mother and kind sisters for their love and

support. Mom I appreciate you for being patient in these years that I was far from home. I

am also grateful to my beloved husband Navid Hayeri for being supportive during my

graduate study.

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Pore-Scale Characterization and Modeling of Two-Phase Flow in Tight

Gas Sandstones

Maryam Alsadat Mousavi, Ph.D.

The University of Texas at Austin, 2010

Supervisor: Steven L. Bryant

Unconventional natural gas resources, particularly tight gas sands, constitute a

significant percentage of the natural gas resource base and offer abundant potential for

future reserves and production. The premise of this research is that several unique

characteristics of these rocks are the consequence of post depositional diagenetic

processes including mechanical compaction, quartz and other mineral cementation, and

mineral dissolution. These processes lead to permanent alteration of the initial pore

structure causing an increase in the number of isolated and disconnected pores and thus in

the tortuosity.

The objective of this research is to develop a pore scale model of the geological

processes that create tight gas sandstones and to carry out drainage simulations in these

models. These models can be used to understand the flow connections between tight gas

sandstone matrix and the hydraulic fractures needed for commercial production rates.

We model depositional and diagenetic controls on tight gas sandstones pore

geometry such as compaction and cementation processes. The model is purely geometric

and begins by applying a cooperative rearrangement algorithm to produce dense, random

packings of spheres of different sizes. The spheres are idealized sand grains. We simulate

the evolution of these model sediments into low-porosity (3% to 10%) sandstone by

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applying different amount of ductile grains and quartz precipitation. A substantial

fraction of the original pore throats in the sediment are closed by the simulated diagenetic

alteration. Thus, the pore space in typical tight gas sandstones is poorly connected, and is

often close to being completely disconnected, with significant effect on flow properties.

The drainage curves for model rocks were computed using invasion percolation in

a network taken directly from the grain-scale geometry and topology of the model. The

drainage simulations show clear percolation behavior, but experimental data frequently

do not. This implies that either network models based on intergranular void space are not

a good tool for modeling of tight gas sandstone or the experiments are not correctly done

on tight gas samples.

In addition to reducing connectivity, the porosity-reducing mechanisms change

pore throat size distributions. These combined effects shift the drainage water relative

permeability curve toward higher values of water saturation, and gas relative permeability

shifts toward smaller values of gas. Comparison of simulations with measured relative

permeabilities shows a good match although same network fail to match drainage curves.

This could happens because the model gives the right fluid configuration but at the wrong

values of curvature and saturation.

The significance of this work is that the model correctly predicts the relative

permeabilities of tight gas sandstones by considering the microscale heterogeneity. The

porosity reduction due to ductile grain deformation is a new contribution and correctly

matches with experimental data from literature. The drainage modeling of two-phase

flow relative permeabilities shows that the notion of permeability jail, a range of

saturations over which both gas and water relative permeabilities are very small, does not

occur during drainage.

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Table of Contents

List of Tables .................................................................................................................................... x

List of Figures ................................................................................................................................. xi

Chapter 1: Introduction ................................................................................................................... 1

1.1 Description of problem ........................................................................................................... 1

1.2 Research objectives ................................................................................................................. 3

1.3 Review of chapters .................................................................................................................. 3

Chapter 2: Literature Review .......................................................................................................... 5

2.1 Introduction ............................................................................................................................. 5

2.2 What is tight gas sand? ........................................................................................................... 5

2.3 Diagenesis processes which make a rock tight ....................................................................... 6

2.4 Compaction and cementation reviews .................................................................................... 7

2.5 Pore geometries in tight gas sands .......................................................................................... 9

2.6 Ductile grains deformation ................................................................................................... 10

2.7 Multi-phase flow properties of tight gas sands ..................................................................... 13

2.8 Productivity of tight gas sands .............................................................................................. 14

2.9 Quantitative network models ................................................................................................ 15

2.10 Models of Granular Materials ............................................................................................. 16

2.11 Algorithms for computer generated packing of spheres ..................................................... 18

2.11.1 Sequential addition algorithm ................................................................................ 18

2.11.2 Cooperative rearrangement algorithms .................................................................. 19

2.11.3 Discrete element method (DEM) ........................................................................... 19

Chapter 3: Models of compaction and cementation in tight gas sandstone ................................... 20

3.1 Introduction ........................................................................................................................... 20

3.2 Compaction simulation of mixtures of rigid and ductile grains ............................................ 21

3.2.1 2D models of sand packs ......................................................................................... 26

3.2.2 Modeling hard and ductile grains ............................................................................ 29

3.3 Porosity reduction due to compaction of ductile grains ........................................................ 32

3.4 Porosity reduction due to quartz cementation ....................................................................... 38

3.5 Development of framework of rigid grains and cores during compaction ........................... 40

3.6 Role of sorting in porosity reduction .................................................................................... 42

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3.7 Pore throat size distribution .................................................................................................. 43

3.8 Effective medium: estimation of permeability...................................................................... 50

3.9 Permeability .......................................................................................................................... 53

3.10 Summary ............................................................................................................................. 54

Chapter 4: Simulation of drainage in tight gas sands .................................................................... 56

4.1 Introduction ........................................................................................................................... 56

4.2 Drainage calculations ............................................................................................................ 56

4.2.1 Effect of different grain size distribution on drainage curves .................................. 60

4.3 Drainage simulations in tight gas sandstones ....................................................................... 64

4.4 Average connectivity of pore space ...................................................................................... 68

4.5 Average connectivity of pore space for tight gas samples .................................................... 76

4.6 Percolation threshold for tight gas sand models ................................................................... 78

4.7 Effects of entry and exit pores on capillary pressure (drainage) curves ............................... 88

4.8 Comparison between drainage simulation and experimental data ........................................ 91

4.8.1 Comparison between drainage simulation from network model and experimental data ................................................................................................................................... 98

4.9 Summary ............................................................................................................................. 105

Chapter 5: Prediction of relative permeability curves in tight gas sandstones ............................ 107

5.1 Introduction ......................................................................................................................... 107

5.2 Relative permeability .......................................................................................................... 107

5.3 Relative permeability calculation ....................................................................................... 109

5.4 Relative permeability simulation in tight gas sands............................................................ 111

5.5 Relative permeability in some tight gas formations............................................................ 118

5.6 Summary ............................................................................................................................. 123

Chapter 6: Conclusions and recommendations ........................................................................... 126

6.1 Future work ......................................................................................................................... 129

Appendix A: Types of pore space ............................................................................................... 131

Appendix B: Relative permeability data from tight gas sandstone ............................................. 141

References ................................................................................................................................... 145

Vita .............................................................................................................................................. 152

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List of Tables

Table 3.1 Measurements of wet-packed sand porosity for different grain size distributions (Beard and Weyl, 1973) ........................................................................................................... 43

Table 4.1 Summary of properties of model sediments .................................................................. 61

Table 4.2 The bond-percolation threshold and connectivity of pore space (degree) for different lattices (Dean and Bird, 1967). .................................................................................. 69

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List of Figures

Figure 2.1- Photomicrograph and scanning electron microscope images illustrating slot-type pores and pore throats commonly found in low-permeability reservoirs. (A) Frontier Formation, Amoco Shute Creek 1, 10,779.8 ft (3285.6 m), 100x, plane polarized light; (B) Williams Fork Formation, MWX 3, 5830 ft (1777 m), 1400x; (C) Travis Peak Formation, SFE 2, 8275.3 ft (2522 m), 100x, plane polarized light; (D) Travis Peak Formation, SFE 2, 8275.3 ft (2522 m), 100x, fluorescent light. Figure is provided courtesy of Shanley et al., 2004. .............................................................................................. 11

Figure 3.1- Ductile grain concept. The rigid radius is a good proxy for ductility of a ductile grain.......................................................................................................................................... 21

Figure 3.2- a) Packing of a rigid and a ductile grain with overlap generation in the ductile shell. b) The overlap volume is distributed into pore space by increasing the ductile grain radius. The increment is chosen so that the ductile grain volume excluding the overlap with the hard grain is the same as the original volume of the ductile grain. This conserves mass and approximates the geometric effect of deformation of the ductile grain into pore space. ........................................................................................................................................ 25

Figure 3.3- Dense packing (porosity 21.6%) of hard and ductile grains (5000 spheres; 30% ductile matter; ductile grains have rigid core radius of 0.7R) produced by cooperative rearrangement simulator. Red spheres are ductile and gray spheres are rigid. Packing is bi-disperse (half the spheres have radius 1.5 times larger than the other half). Arrows show the big and small hard (gray) spheres and an overlap of ductile spheres (red) ............... 26

Figure 3.4- Figure shows the diagram of interactions between particles in the PFC simulations. Image courtesy from Itasca Consulting, theory and background (2004). ............ 28

Figure 3.5- Sample with random 54 ductile grains and lateral confining stress before and after compaction. The right figure was created at the end of the test. ...................................... 30

Figure 3.6- Stress-strain plot for sample with random ductile grains and initial confining stress. The vertical axis is stress (Pa) and horizontal axis is strain. Arrows show the formation of hard cages. ........................................................................................................... 31

Figure 3.7- a) There is no porosity in this thin section. The lithic grains flow into the pore space. Jurassic., Latady Fm., Antarctica. b) The pressure solution stilolite is marked by arrow. Oligocene, Frio Fm., Brazoria Co, TX. (Images courtesy from Milliken et al., 2007). ....................................................................................................................................... 32

Figure 3.8- Progressive compaction of ductile-dominated sediment. Deformation of ductile grains causes intergranular volume to drop from 40 to 0% with burial. IGV= intergranular volume; q = quartz grain; d = ductile grain (Mousavi and Bryant, 2007) ................................ 34

Figure 3.9- Simulated compaction for packings of 5000 bi-disperse spheres (half the spheres are 1.5 times larger than the other half). Each curve corresponds to a different value of the radius of the rigid core of the ductile spheres. The similar porosity trend for a wide range of rigid radii (small black arrows) is related to formation of cages (load-bearing frameworks) in the packing. The big arrow points in the direction of decreasing rigid radius, which is a proxy for increasing ductility of the lithic grains. ....................................... 35

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Figure 3.10- Porosity trends for dense packings of spheres with selected radii (0.8R and 0.9R) for the rigid cores of the ductile grains. The trends bracket the experiments of Pittman and Larese (1991) in which the ductile grains are moderately ductile fragments. ..... 36

Figure 3.11- Porosity trends for dense packings of spheres with selected radii for the rigid cores of the ductile grains. The thin soft shells yield behavior similar to the experiments of Pittman and Larese (1991) in which the ductile grains are brittle fragments. A rigid core radius of 0.92R provides a reasonable lower bound to the data. The upper bound would be grains with rigid core radius of 1R, for which no porosity reduction would occur. ........................................................................................................................................ 37

Figure 3.12- Comparison of simulated compaction with experiments using extremely ductile fragments Pittman and Larese (1991). Simulations with ductile grains having rigid radius 0.8R provide an upper bound for most of the data. When the rigid radius of ductile grains is 0.05R, the predicted porosity trend gives the correct average behavior. In contrast to Figure 3.10 and Figure 3.11, no simulation provides a lower bound for these experiments. ... 38

Figure 3.13-a) 2D thin section of a dense packing of 1000 spheres, 30% of which are ductile. The ductile grains have a rigid core of radius 0.7R. The porosity in this packing is 22%. b) The same packing with 10% of cement (0.1 of the sphere radius). The porosity in this packing is 19%. c) The same packing as with 20% cement (0.2 of the sphere radius). The porosity in this packing is 7%. d) The same packing as with 30% cement (0.3 of the sphere radius). The porosity in this packing is 3%................................................................... 39

Figure 3.14- a) Schematic of load-bearing frameworks of contacts between hard grains and/or rigid cores of ductile grains. Above a threshold size of the rigid core of a ductile grain (the value of the threshold is 70% of the original grain radius), a “cage” form when the hard spheres have point contact with rigid core of ductile spheres. b) Schematic of load-bearing frameworks of contacts between hard grains and/or rigid cores of ductile grains, below the threshold. The rigid core of ductile grain is too small to contribute to formation of hard cages. The light gray sphere is a ductile sphere with a rigid core, h=hard sphere. .......................................................................................................................... 41

Figure 3.15- Simulated compaction trends for very soft grains (rigid radius 0.05R) do not depend strongly upon the relative sphere sizes in bi-disperse packings. The spread in the experimental data is larger than the spread in the simulations. ................................................ 42

Figure 3.16- A 3D view of Delaunay cell. The middle point of pore throat (W) in face UVT can be seen in this figure. ......................................................................................................... 44

Figure 3.17- Finney pack pore throat size distributions (ф = 36%). Their “inscribed radius” characterizes the throats (see text). The x-axis is dimensionless. ............................................ 44

Figure 3.18- Comparison between Finney pack with computer generated packing of equal spheres without ductile matter (ф = 36%). The x-axis is dimensionless. ................................. 45

Figure 3.19- Comparison between Finney pack and computer generated pack with bi-dispersed spheres of 1.5 radius ratio and 50% small spheres. There is no ductile sphere (ф = 33%). The calculated throat radii from bi-dispersed packing were normalized by mean radius of mono-dispersed packing. Therefore, these two packings have the same unit for throat radii (rinscribed). The ratio of the size of small radius in bi-disperse packing to the size of spheres in mono-dispersed packing is equal to 0.77. The x-axis is dimensionless. ...... 46

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Figure 3.20- Comparison between Finney pack and mono dispersed spheres with 40% ductile matter and rigid radius of 0.9. Some of the pore throats are closed because of addition of ductile spheres to the packing (ф = 27.5%). The x-axis is dimensionless. ............ 47

Figure 3.21- Comparison between Finney pack and mono dispersed spheres with 40% ductile matter and rigid radius of 0.8 (ф = 20.7%). It is obvious that by increasing the ductility of spheres (decreasing rigid radius of spheres) we will have more closed pore throats. The x-axis is dimensionless. ........................................................................................ 47

Figure 3.22- Comparison between Finney pack and mono dispersed spheres with 40% ductile matter and rigid radius of 0.2 (ф = 15.8%). In this case, spheres are very ductile therefore, we have many closed pores (rinscribed = 0), which are clear in the plot. In addition, the pore throat sizes moved toward the smaller pore throat. The x-axis is dimensionless. .......................................................................................................................... 48

Figure 3.23- Comparison between Finney pack and mono dispersed spheres with 24% cement (0.24 of the sphere radius, ф = 4%). We have many closed pores due to cement addition. In addition, the pore throat sizes moved toward the smaller pore throat. The x-axis is dimensionless. ............................................................................................................... 49

Figure 3.24- Bidispersed packings with radius ratio of 1.5 and rigid radius of 0.7. Increasing the fraction of ductile matter in the packing causes the porosity of the packing to decrease and the number of blocked throats in the packing to increase. Thus the connectivity of the pores decreases, which reinforces the decrease in permeability caused by smaller pore throats. ...................................................................................................................................... 50

Figure 3.25- Effective conductance (estimation of permeability) of a packing with 1.5 radius ratio and 0.7 rigid radius. By increasing the ductile matter in the packing, the effective conductance decreases. In addition, it decreases by decreasing the porosity of the packing resulting from cementation. Packings with ductile grains more than 75% have very small permeability close to zero. ....................................................................................................... 51

Figure 3.26- Comparison between effective conductances predicted from mono-dispersed packing with different cementation and scaled permeability data from Fontainebleau sandstone (data courtesy from Bryant and Blunt, 1992). ......................................................... 52

Figure 3.27- Comparison between permeability calculated for mono-dispersed packing with different cementation and Fontainebleau sandstone. The dashed line is the predicted permeability values from Bryant and Blunt, 1992. The y-axis has the unit of md. ................ 53

Figure 4.1- Two views of Delaunay tessellation cell a) with four grains b) and only part of grains inside the cell. ................................................................................................................ 57

Figure 4.2- One face of Delaunay tessellation (throat) with inscribed radius shown by “rins”. ... 57

Figure 4.3- Network of sites and bonds. Sites are pores which are shown as dots and bonds are throats shown as lines connecting two pores. The image courtesy from Behseresht (2009). ...................................................................................................................................... 58

Figure 4.4- Grain size distribution for packings of Table 4.1. ...................................................... 62

Figure 4.5- Simulated drainage curves (plot of curvature versus volume fraction of wetting phase) for packings of Table 4.1. Curvature is the dimensionless capillary pressure, Eq. 4.2. ............................................................................................................................................ 63

Figure 4.6- Part of the sphere packing No. 5. The image courtesy from Behseresht (2008). ....... 63

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Figure 4.7- Connected path for packing No. 5. This shows the connected throats during drainage for that packing from one side of the packing to the other side (similar to break through). The curvature (dimensionless capillary pressure) was 0.3 at which this path was formed. ..................................................................................................................................... 64

Figure 4.8- Drainage capillary pressure curves for mono dispersed packing with fixed ductile amount (40%) and different rigid radius. Curvature is the dimensionless capillary pressure. ................................................................................................................................... 66

Figure 4.9- Drainage curves for mono dispersed packing with fixed rigid radii (0.7) and different amount of ductile matter in the packing. Curvature is the dimensionless capillary pressure. .................................................................................................................... 66

Figure 4.10- Drainage curves for mono-dispersed packing with different cement amount (different % of the sphere radius). Curvature is the dimensionless capillary pressure. ........... 67

Figure 4.11- Drainage curves for mono dispersed packing with 40% ductile matter, 0.7 rigid radius of ductile grains and different cementation (different % of the sphere radius). Curvature is the dimensionless capillary pressure. .................................................................. 68

Figure 4.12-The bond-percolation threshold for different lattices. ............................................... 69

Figure 4.13- Fraction of pores with different blocked throats status. By adding cement to the packing, the fraction of pores with 4 open throats gets smaller and fraction of pores with at least one blocked throats increases. The percentage of cement means different percentage of the sphere radius was added to the sphere radius. ............................................. 70

Figure 4.14- Average connectivity for mono dispersed packing with different amount of cement. By addition of cement, the connectivity decreases from four to 3.2. The second vertical axis shows the fraction of blocked throats versus cementation. By adding more cement, we close more throats. The values with percentage on top of the data show the porosity percentage for each packing. The percentage of cement means different percentage of the sphere radius was added to the sphere radius. ............................................. 71

Figure 4.15- Connectivity of pore space for a mono dispersed packing with fixed 40% ductile matter and different rigid radius for ductile grains. By decreasing the rigid radius, the connectivity of pore space reduces. The secondary axis shows the blocked throats versus rigid radius. It is clear that packings with 0.3 to 0.6 rigid radius have the similar blocked throats. The values with % on top of the data shows the porosity percentage for each packing. ............................................................................................................................ 73

Figure 4.16- Fraction of pores with different blocked throats status. By changing the rigid radius from 0.9 to 0.1, the fraction of pores with 4 open throats gets smaller and fraction of pores with at least one blocked throats increases. ................................................................ 74

Figure 4.17- Connectivity of pore space for a packing with fixed 0.7 rigid radius of ductile grains and different percentage of ductile matter in the whole packing. By increasing the ductile grains in the packing, the connectivity of pore space decreases. The secondary vertical axis shows the fraction of blocked throats. The values with % on top of the data shows the porosity percentage for each packing. ..................................................................... 75

Figure 4.18- Fraction of pores with different blocked throats status. By changing the amount of ductile grain from 0.1 to 0.9, the fraction of pores with four open throats gets smaller and fraction of pores with at least one blocked throat increases. ............................................. 76

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Figure 4.19- Average connectivity of pore space for packing of 40 percent ductile grains (0.7 rigid radii) and different amount of cement. By increasing the amount of cement, the connectivity of pore space decreases. The secondary vertical axis shows the fraction of blocked throats versus cement. The values with % on top of the data shows the porosity percentage for each packing. The percentage of cement means different percentage of the sphere radius was added to the sphere radius. .......................................................................... 77

Figure 4.20- Fraction of pores for mono dispersed packing with 40% ductile 0.7 rigid radius and different cementation. The percentage of cement means different percentage of the sphere radius was added to the sphere radius. .......................................................................... 78

Figure 4.21- Drainage curve for mono-dispersed packing. Drainage at 50% saturation is marked in this figure to estimate bond percolation threshold for this packing. Curvature is the dimensionless capillary pressure. ....................................................................................... 80

Figure 4.22- Inscribed radius histogram and cumulative distribution curve for mono dispersed packing. Inscribed radius at 50% saturation (determined from Fig 4.21 and Eq. 4.4) is marked in this figure to calculate bond percolation threshold. Bond percolation threshold is 1-cumulative frequency of the size of the throats invaded at 50% saturation. ..... 81

Figure 4.23- Bond percolation threshold and fraction of blocked throats for mono dispersed packing with different cementation. The values with % on top of the data shows the porosity percentage for each packing. ...................................................................................... 82

Figure 4.24- Bond percolation threshold and fraction of blocked throats for mono dispersed packing with 40% ductile and different rigid radius. The values with % on top of the data shows the porosity percentage for each packing. ..................................................................... 82

Figure 4.25- Bond percolation threshold and fraction of blocked throats for mono dispersed packing with 0.7 rigid radius (fixed ductility) and different amount of ductile grains. The values with % on top of the data shows the porosity percentage for each packing.................. 84

Figure 4.26- Bond percolation threshold and fraction of blocked throats for mono dispersed packing with fixed 0.7 rigid radius (fixed ductility), fixed 40% ductile grains and different cementation (tight gas samples). The values with % on top of the data shows the porosity percentage for each packing. ...................................................................................... 85

Figure 4.27- Bond percolation threshold and connectivity for mono-dispersed packing with different cementation. Packing with 24% cement has connectivity of 3.21 so there is no bond percolation threshold for this packing at 50% saturation. ............................................... 85

Figure 4.28- Bond percolation threshold and connectivity for mono-dispersed packing with fixed ductile matter and different rigid radius. ......................................................................... 86

Figure 4.29- Bond percolation threshold and connectivity for mono-dispersed packing with fixed 0.7 rigid radius and different ductile percentage. Packings with ductile matter more than 60% (0.6) have connectivity less than 3.21 so there is no bond percolation threshold for those packing at 50% saturation. ........................................................................................ 86

Figure 4.30- Bond percolation threshold and connectivity for mono-dispersed packing with fixed 0.7 rigid radius, fixed ductile percentage (40%) and different cementation (tight gas samples). Packings with cement more than 10% have connectivity less than 3.19 so there is no bond percolation threshold for those packing at 50% saturation. .................................... 87

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Figure 4.31- Throat size distribution for mono-dispersed packings without cement, 5% cement and 10% cement. .......................................................................................................... 88

Figure 4.32- Mono-dispersed packing with 23% cement is used with different entry and exit pores to see the influence of sample size on drainage curve. The red circle shows the difference in entry pressure for these curves. Curvature is the dimensionless capillary pressure. ................................................................................................................................... 90

Figure 4.33- Mono-dispersed packing with 23%cement is used with different entry and exit pore faces to see the influence of sample size on drainage curve. Curvature is the dimensionless capillary pressure. ............................................................................................. 90

Figure 4.34- Comparison of drainage curves between simulated packings and lab data (data courtesy from www.discovery-group.com). The dashed curve is the mono dispersed packing with 23% cement (23% of grain radius) (porosity=4.7%). The porosity for the tight gas sample is 12.1% and the mean grain radius is 175 µm. The dotted curve is a packing with lognormally distributed from sizes (mean grain radius=2.3842 R, standard deviation=0.9877R^2 and porosity=1.8). Because the lognormal distribution is not wide, the simulated drainage capillary pressure is even lower than mono-dispersed packing (explained in effect of different grain size distribution on drainage curves). The black curve is the mono-dispersed packing with 24% cement (24% of grain radius) (porosity=4%). The large difference between packing with 4.7% porosity and packing with 4% porosity is because of the amount of blocked throats (41% and 45% respectively). ............................................................................................................................ 92

Figure 4.35- The lab data from Figure 4.34 and the simulated data use the network of the packing with mono-dispersed grains with 23% cement (23% of grain radius) to which throats sizes are randomly assigned from a lognormal distribution. Each curve has different lognormal distribution with different width (mean and standard deviation). We used m and v values to calculate the mean and standard deviation as below: ���� �log �

√ ��� and standard deviation is: ��� � �log � � � 1�. ................................................ 94

Figure 4.36- The data are the same as Figure 4.35 but we kept the blocked throats of heavily cemented mono-dispersed packing and the open throats were chosen randomly from lognormal distribution. "Keeping zero" in legend means we kept the throats with zero radius (the blocked throats) from the mono-dispersed packing and then applied the lognormal throat size distribution to assign sizes to all other throats. ...................................... 95

Figure 4.37- Histogram of lognormal throat size distributions with different m and v values. a) m=1 and v=1 b) m=1, v=10 c) m=1, v=100 d) m=1, v=1000 e) m=1, v=10000. These are used in Figure 4.36. ............................................................................................................ 96

Figure 4.38- Simulated drainage curves using lognormal distribution of throat sizes assigned to open throats in a mono-dispersed packing network with 23% cement (23% of grain radius). a) Lab sample from Green River formation (depth=11605 ft) with mean grain radius of 125 µm and porosity=3.2% (Data courtesy from www.discovery-group.com). The simulation sample has m=1 and v=10000 (look at Figure 4.35 caption). We use one face of the packing for entrance and exit pores during drainage simulation. b) Lab sample from Green River formation (depth=11460 ft) with mean grain radius of 125 µm and porosity=4.4%. The simulation sample has m=1 and v=1000 (look at Figure 4.36

xvii

caption). We use one face of the packing for entrance and exit pores during drainage simulation. ................................................................................................................................ 97

Figure 4.39- The capillary pressure calculated using the real throat size distribution of field sample (Western US #2) using network modeling. It is obvious that the simulated curve does not match the field data even though the model has the same nominal distribution of throat sizes. The other curve is the simulated capillary pressure using the mono dispersed packing with heavy cementation (many blocked throats) which does not match the data either. ........................................................................................................................................ 98

Figure 4.40- Simulations with various throat size distributions (lognormal, normal and uniform distributions) with the same range of throat sizes and same mean and standard deviation as the sample (inferred from mercury intrusion). The sample is same as Figure 4.39. .......................................................................................................................................... 99

Figure 4.41- a) Throat size distribution inferred from mercury intrusion of the tight gas sample Figures 4.39 and 4.40. Notice the bimodal distribution of the sample throats. b) R-inscribed (throat sizes) distribution used in simulation. It was derived by adding two normal distribution with selected means and standard deviations. c) The throat size distribution with the same range but using reverse peaks (the same means different standard deviation). ................................................................................................................ 100

Figure 4.42- Simulated drainage capillary pressure using the range of real throat sizes but adding two normal distribution curves with the estimated means and standard deviations. The other curve (dashed line) uses the same two normal distribution but reverse the height of the peaks (look at distribution of throat sizes Figure 4.41). .................................... 101

Figure 4.43- a) A two-dimensional view of a layered packing with two grain size distributions. The ratio of big grain size to small grain size is 2.4. b) The drainage curve for the same packing. The range of capillary pressures is still narrow over wide range of saturation, and each layer percolates separately. The layer with bigger grains percolated first. Curvature is a dimensionless capillary pressure. ........................................................... 102

Figure 4.44- A schematic of a 5 layered packing. The throat size distribution is the same for the middle two gray layers with big throats. The throat size distribution for the white layer is the same with small throats. Both throat distributions are normal distributions. These distributions were imposed on the throat in corresponding spatial regions in a single network model. ............................................................................................................ 103

Figure 4.45- The drainage curve for the tight gas sample of Figure 4.30 et seq. and a simulation with heterogeneous throat size distributions. Two networks of throats were applied to a five-layered packing of Figure 4.44. The y-axis is logarithm of capillary pressure data. The irreducible water saturation is zero for experimental data but the simulated result has some trapped water as irreducible water saturation. .............................. 104

Figure 4.46- a) Intergranular pore network in highly cemented model rock with connectivity of 3.3 and porosity of 5%. The whole network of pores and throats in this sample are connected to each other, even though many individual throats are blocked. Consequently, drainage simulation show clear percolation thresholds. b) The image represents skeleton (medial axis) of the pore space in a 4% porous Western US tight gas sandstone sample. The sample was imaged at 2.3 micron resolution at Jackson School of Geosciences. Medial axis and thus the pore space are not connected across the sample in any direction. The rainbow coloring refers to the proximity of the neighboring grain (red-within one

xviii

voxel length, velvet-with 5 voxel length) (image courtesy of Dr. Maša Prodanović). There is no big connected path through this sample rock. ..................................................... 105

Figure 5.1- Typical imbibition relative permeability curves (image courtesy from Kleppe, 2009). ..................................................................................................................................... 108

Figure 5.2- Typical drainage relative permeability curves (image courtesy from Kleppe, 2009). ..................................................................................................................................... 109

Figure 5.3- a) Drainage water relative permeability for mono-dispersed packing with fixed amount of ductile grains in the packing and different ductility (rigid radius). By decreasing the rigid radius of grains from 0.9 to 0.1 (increasing ductility) water relative permeability curve shift toward right. b) The same plot with logarithmic. The curves below 0.6 rigid radius are similar as (a) because of the formation of hard cages and having similar porosity. c) Gas relative permeabilities for the same samples. By decreasing the rigid radius of grains from 0.9 to 0.1 (increasing ductility) gas relative permeability curve shift toward right. .................................................................................... 112

Figure 5.4- a) Drainage water relative permeability for mono-dispersed packing with fixed rigid radius (ductility) and different amount of ductile grains in the packing. By increasing the amount of ductile grains from 10% to 70%, water relative permeability curve shift toward right. b) The same plot with logarithmic scale for water relative permeability. c) Gas relative permeability curves for the same models. By increasing the amount of ductile grains from 10% to 70%, gas relative permeability curve shift toward right. ....................................................................................................................................... 113

Figure 5.5- a) Drainage relative permeability for mono-dispersed packing different cementation (different percentage of grain radius). By increasing the amount of cement from 0 to 24% (% of grain radius), water relative permeability curve shift toward right. b) The same plot with logarithmic water relative permeability. c) Gas relative permeability for the same model rocks. Gas relative permeability curve shifts toward right as porosity decreases. ............................................................................................................................... 115

Figure 5.6- a) Drainage water relative permeability for mono-dispersed packing with 40% ductile grains, 0.7 rigid radius of ductile grains and different cementation (different % of grain radius) (tight gas samples). By increasing the amount of cement from 0 to 15% (% of grain radius), water permeability shift toward right. b) The same plot with logarithmic scale for water relative permeability. c) Gas relative permeability curves for the same model rock. By increasing the amount of cement from 0 to 15%, gas permeability shift toward right. ........................................................................................................................... 116

Figure 5.7- Critical curvature frequency distribution of throats drained during simulation for mono-dispersed packing with 15% cement (15% of grain radius) with porosity = 12% (in Figure 5.5). The blue color shows distribution for all throats; the red color shows the distribution of throats drained when the wetting phase saturation is 50%, at which value the gas phase is assumed to reach percolation threshold. The drained throats amount to 35% of the total throats in the packing including blocked throats. ........................................ 117

Figure 5.8- Critical curvature frequency distribution of throats drained during simulation for mono-dispersed packing with 23% cement (23% of grain radius) with porosity = 4.7% (in Figure 5.5). The blue color shows distribution for all throats; the red color shows the distribution of throats drained when the wetting phase saturation is 50% and the gas phase is assumed to reach percolation threshold. The drained throats are 47% of the total

xix

throats in the packing including blocked throat. This is larger fraction than for the higher porosity packing in Figure 5.7. This explains why the gas phase relative permeability is larger in the low porosity packing, Figure 5.3, Figure 5.4, Figure 5.5, and Figure 5.6. The frequency of big throats (lower curvature values) is smaller in this case compare to Figure 5.7. The highest values of curvature (small and blocked throats) are not shown in this figure. ..................................................................................................................................... 117

Figure 5.10- Predicted and experimental drainage relative permeabilities in sandstone. The sample is from depth 3433.8 ft with porosity of 17.7%. The simulation result was used a mono-dispersed packing with 40% moderately ductile matter and a porosity of 17.3% (data courtesy of www.discovery-group.com). The sandstone relative permeability was calculated using Corey (1954) equations. .............................................................................. 120

Figure 5.11- Predicted and experimental measurements of drainage relative permeability in tight gas sandstone. The sample is from depth 8279.5 ft with porosity of 7.6%. The simulation result used a mono-dispersed packing with 40% moderately ductile matter 10% cement (10% of grain radius) and porosity of 7% (data courtesy of www.discovery-group.com). The sandstone relative permeability was calculated using Corey (1954) equations. ............................................................................................................................... 121

Figure 5.12- The relative permeability for actual tight gas sandstones, Green River basin. The relative permeabilities were calculated using Corey (1954) equation from drainage capillary pressure experiment. Data courtesy from www.discovery-group.com. .................. 122

Figure 5.13- Irreducible water saturation (Swi) increases by decreasing the porosity. This is the reason for increasing the gas relative permeability in tight gas sandstones (Figure 5.12). Swi values measured at 600 psia (4140 kPa), air-brine capillary pressure (equivalent to approximately 120 m above free-water level). Average Swi≈12% for rocks with фi>8%; average Swi≈20% with rocks with фi=6% and average Swi≈40% for rocks with фi=3%. The regression line shown represents Swi � 10��0.97 ! log10фi �2.06�(data courtesy from Castle and Byrnes, 1998). ............................................................. 124

Figure A.1- Ductile deformation of mica-rich lithic; there is no porosity in this thin section. The lithic grains flow into the pore space. Ord. Martinsburg Ss., S. Appalachian Basin. Image courtesy from Miliken et al. (2007)............................................................................. 131

Figure A.2- Compaction leads to Flowing of lithics into pore space. The pressure solution stylolites are marked by arrows. Data from Anderson well no.2, Anadarko basin. ............... 132

Figure A.3- Quartz overgrowth closed all the visible pore space in this thin section. Travis Peak formation, depth 9560.4 ft. ............................................................................................ 132

Figure A.4- Carbonate cementation close every single pore space. Travis Peak formation, depth 5974 ft. ......................................................................................................................... 133

Figure A.5- Quartz cement closed all of pore space. Frontier formation, depth 16071.8 ft. ....... 133

Figure A.6- Solution porosity shown by yellow arrows. Dolomite cement which is marked by red arrows filled the solution porosities. Travis Peak formation, depth 9753 ft. .............. 134

Figure A.7- Small pore space between quartz overgrowths marked by red arrow. Quartz overgrowth is complete in this thin section. Travis Peak formation, depth 8246.1 ft ............ 135

xx

Figure A.8- There is large pore space between quartz overgrowths. Quartz overgrowth is very thin so the pore space between grains is remained large. Travis Peak formation, depth 5973.9 ft ....................................................................................................................... 136

Figure A.9- Large pore space remained between grains with incomplete overgrowth quartz. Arrow shows the narrow pass which relates grain with pore space. Travis Peak formation, depth 6205.2 ft. ..................................................................................................... 136

Figure A.10- Rim clays prevent from forming the overgrowth cement and lead to remaining large porosity. Travis Peak formation, depth 5972.7 ft. ......................................................... 137

Figure A.11- Slot like porosity which is marked by arrows. Travis Peak formation, depth 9984 ft. ................................................................................................................................... 138

Figure A.12- Slot like porosity which is marked by red arrows. Slot pore ended to the large porosity, which was marked by yellow arrows Travis Peak formation, depth 9984 ft .......... 138

Figure A.13- Triangle porosity which is marked by red arrow. Travis Peak formation, depth 6843.7 ft ................................................................................................................................. 139

Figure A.14- Microporosity between dolomite crystals. Travis Peak formation, depth 9753 ft. ............................................................................................................................................ 140

Figure A.15- Microporosity inside dissolution feldspar (shown by arrows).Travis Peak formation, depth 10112.4 ft .................................................................................................... 140

Figure B.1- Relative permeability curves from Green River basin. The relative permeabilities were calculated using Corey (1954) equation from drainage capillary pressure experiment. By decreasing the porosity, the gas relative permeability increases. Data courtesy from www.discover-group.com. .............................................................................. 141

Figure B.2- Irreducible water saturation (Swi) versus porosity for Green River basin data. By decreasing the porosity, the Swi increases. Data courtesy from www.discover-group.com. . 142

Figure B.3- Relative permeability curves from Powder River basin. The relative permeabilities were calculated using Corey (1954) equation from drainage capillary pressure experiment. By decreasing the porosity, the gas relative permeability increases. Data courtesy from www.discover-group.com. ..................................................................... 142

Figure B.4- Irreducible water saturation (Swi) versus porosity for Powder River basin data. By decreasing the porosity, the Swi increases. Data courtesy from www.discover-group.com............................................................................................................................... 143

Figure B.5- Relative permeability curves from Uinta basin. The relative permeabilities were calculated using Corey (1954) equation from drainage capillary pressure experiment. By decreasing the porosity, the gas relative permeability increases. Data courtesy from www.discover-group.com. ..................................................................................................... 143

Figure B.6- Irreducible water saturation (Swi) versus porosity for Uinta basin data. By decreasing the porosity, the Swi increases. Data courtesy from www.discover-group.com. . 144

1

Chapter 1: Introduction

1.1.Description of problem

Unconventional natural gas resources, particularly tight gas sands, constitute a

significant percentage of the natural gas resource base and offer abundant potential for

future reserves and production. Unlike conventional oil and gas reservoirs, tight gas sands

have unique gas storage and producing characteristics. These characteristics can be

attributed to post-depositional diagenetic processes including mechanical compaction,

quartz and other mineral cementation, grain replacement and mineral dissolution. These

processes lead to permanent alteration of the initial pore structure causing an increase in

tortuosity and a subsequent increase in the number of isolated and disconnected pores.

The resulted tight gas sandstone will have low gas storage because of small pore system.

In addition, the productivity of the tight gas sandstone is reduced because digenesis

processes block most of the throats and reduce the permeability of the rock.

Diagenesis in many cases affects the connecting pore throats much more than the

main pore volume. Consequently, changes in permeability are larger than changes in

porosity. The reduction in permeability is also intensified by the presence of connate

water. Because most tight gas sands are water-wet, surface tensions cause water to

occupy the smallest pore openings such as the pore throats thus reducing the effective gas

permeability. This causes a significant reduction in ultimate gas production from a

formation. During production operations, when a pressure gradient is applied, high initial

values of connate water will exhibit limited or no mobility for gas. In fact, even small

volumes of water may have a significant and adverse impact on gas productivity. At

small porosities of tight gas sandstones, formation of pendular rings of water at grain

contacts readily coalesce in throats as wetting phase saturation increases, causing the gas

phase to snap off. This fact reduces the connectivity of gas phase and therefore reduces

the gas effective permeability (Motealleh and Bryant 2009).

Effective development of low permeability resources requires reservoir

description to identify their characteristics and quantify their impact on well productivity.

2

Core measurements are the most common direct technique for characterizing tight gas

sands on a small scale, but the low porosities and permeabilities involved often prevent

efficient and effective use of conventional measurement techniques. For example, Darcy's

law is applied in conventional absolute permeability measurements under steady-state

flow conditions, but steady–state flow is very difficult (if not impossible) to achieve in

reasonable time. In addition, most commercial laboratories use unsteady-state techniques

for relative permeability measurements, but there are questions concerning their

measurement accuracy (Rushing et al. 2004). Unsteady state relative permeabilities

usually are higher than steady state techniques measurements even when the gas slippage

and inertial effects were corrected in steady state technique. Measurements made at stress

and under two-phase (gas-water) conditions usually complicate the testing conditions and

tend to increase both the time and cost. Consequently, many operators limit the type and

number of special core analyses.

The presence of complex pore structures, such as slot-shaped pores, in tight gas

sands also makes formation evaluation and productivity predictions using indirect

measurement technique very difficult (e.g. well log analysis, pressure transient analysis

and production data analysis). From Shanley et al. (2004), a slot pore is a very narrow slit

like pores, which connect the secondary big pores. These narrow intergranular pores

control the flow of a specific phase in a rock and therefore have inverse effect on

production rate of tight gas sandstones. A specific example of indirect methods is

quantification of the productive sand thickness (i.e. net pay) which is critical for

estimating initial well productivity and ultimate gas-in-place. Most operators have

historically used porosity as a delimiter for identifying net pay from conventional well

log measurements. Unfortunately, diagenesis often not only creates isolated micro-pores

and secondary porosity, but also tends to reduce or close the pore throats connecting the

primary pores. Under these conditions, there is often no unique relationship between

porosity and permeability.

3

1.2.Research objectives

The objective of this research is to develop a robust grain-scale model to simulate

the geological processes that create tight gas sandstones (TGS). Such models would be

useful for predicting the storage and matrix flow capacities in tight gas sands under both

single and two-phase flow conditions. The matrix flow capacity can be used to

understand the flow connections between TGS matrix and the hydraulic fractures needed

for commercial production rates. The model development will include both empirical and

theoretical approaches. We will provide a detailed pore-scale description and

characterization program of core samples available from the field laboratories.

The objectives of this project include:

1- Examine previously collected data (core analyses and micro-scale imaging under

stress and two-phase (gas-water) conditions) in order to characterize typical

depositional and diagenetic controls on tight gas sandstones;

2- Develop a theoretical pore-scale model that incorporates various types of

diagenesis; and

3- Simulation of drainage and relative permeability for predictions of well

productivity in tight gas sands.

4- Validate the drainage capillary pressures simulations with tight gas sands data

from industry.

1.3.Review of chapters

There are five chapters in this dissertation. Chapter 1 is the description of problem

and objective of the dissertation. Chapter 2 (literature review) contains diagenetic

processes, quantitative grain scale model and models of granular material. In addition, an

algorithm for constructing computer-generated packings is discussed. The packings use

for drainage simulation and relative permeability models. We describe depositional and

diagenetic controls on tight gas sandstones. In this chapter, we explain how we define

tight gas sand; we describe pore geometry in those rocks; we also describe compaction

and cementation processes as characterized in the literature.

4

Chapter 3 explains compaction simulation of rigid and ductile grains based on 2D

modeling in PFC software. The objective of these simulations is to investigate whether

the pressure loaded during compaction of mixture of hard and ductile grains is supported

primarily by point contact between hard spheres. In addition, porosity reduction due to

compaction, cementation, and sorting based on simulation results is explained. We

explain throat size distribution, effects of ductile compaction and cementation on throat

size distribution, effective medium theory and effective permeability calculation of

sample rocks in this chapter. The throat size distribution of a sample is needed to describe

the geometric properties of sample rocks and is an input for drainage modeling (capillary

pressure is controlled by throat sizes).

In Chapter 4, the drainage models calculations are explained and the effects of

compaction and cementation processes on drainage curves are examined. Connectivity of

pore space and bond percolation threshold is explained in this chapter as well. Finally, we

validate our drainage simulation results with field data at the end of this chapter.

In Chapter 5, we calculate relative permeability curves in tight gas sands (water

and gas) and we examine the effect of diagenetic processes on relative permeability

curves.

5

Chapter 2: Literature Review

2.1.Introduction

The goal of this research is to model sandstones in terms of the geologic processes

that create those sandstones. Thus, it is important to review the properties of these

rocks/reservoirs. In this chapter, we first introduce a definition of tight gas sandstone

followed by a description of diagenetic processes, which make these rocks tight. Based

on Dutton et al. 1993, compaction and cementation are the most important diagenetic

processes, in reducing the porosity and permeability in rocks. Therefore, in the next

section we review the compaction and cementation process in literature. Understanding

of pore geometry of tight gas sandstone is very important in modeling of these rocks, and

this is discussed in section 2.5. Multi-phase flow properties of tight gas sandstone and the

gas productivity in these rocks are explained in the next sections. In addition, for

modeling of a rock, it is important to study the different existing modeling approaches in

literature. In the next section the quantitative network models, and models of granular

material which were applied to model the tight gas sandstone are described. In the last

section, the algorithms used in literature to model the computer generated packing of

spheres were specified.

2.2.What is tight gas sand?

In nature, the rocks ranging from 5-30% porosity and 5-1000 md permeability can

be considered as reservoirs to hold economically recoverable quantities of hydrocarbon.

Poorly permeable rocks, which have permeability less than 0.1 millidarcies, are

considered "Tight Reservoirs". There is no universal porosity cutoff, but the cutoff is

dependent on the economics of recovery pertaining to a particular field (Misra, 2007).

Tight gas reservoirs lack a formal definition, and usage of the term varies

considerably. Based on study of British Colombia report (Report of Petrel Robertson

Consulting Ltd, 2003), low-permeability (tight) reservoirs have permeabilities less than

0.1 millidarcies. Many explorationists think of tight or low-permeability reservoirs as

occurring only within basin-centered or deep basin settings. They confirmed that low-

6

permeability reservoirs characterize basin-centered gas accumulations, but can also occur

elsewhere (British Colombia report, 2003).

Low-permeability sandstone reservoirs in the United States are not dominated by

immature, muddy sandstones with large volumes of diagenetically reactive detrital clay

matrix, but rather are generally well-sorted sandstones deposited in high-energy

depositional settings whose intergranular pores have been largely occluded by authigenic

cements (mainly quartz and calcite). Post-depositional diagenetic events reduce the

effective porosity and thereby make the rock less permeable (Naik, 2007).

2.3. Diagenesis processes which make a rock tight

Tobin (1997) mentions three main possibilities for formation of a tight gas

sandstone. For the generation of tight reservoirs, either the original depositional fabric

itself was tight (Type 1), the diagenesis has so obscured the original fabric that the cause

of low porosity is uncertain (Type 2), or the original sedimentary fabric has been

subsequently reduced in porosity by post depositional diagenesis to unacceptably low

amounts (Type 3). Type 1 lithofacies include among carbonate rocks the marl, lime

mudstone or wackestone, sandy lime mudstone to micritic sandstone or fine dense

crystalline dolomite. In the clastic category, the lithofacies usually associated are facies

with turbid, low energy environments of deposition like shales and argillaceous

siltstones. Type 2 rocks include recrystallized sparry limestones, some dolomites and

some quartzose sandstones. Porosity prediction in these lithofacies is difficult hence the

exploration risks are also high. Type 3 lithofacies include grainstones, packstones and

some reefal facies among carbonates and matrix poor sandstones, arenites etc in the

clastics. Risk perception in these requires that timing and type of destructive diagenetic

processes be identified (Misra, 2007).

The reservoir burial history is of specific significance for understanding the

porosity control of a facies. Two main processes in clastics, which lead to porosity

reduction during burial, are compaction and cementation. Compaction can occur in two

modes, mechanical and chemical. Mechanical compaction predominates at shallow to

7

intermediate depths (<3 km). It involves the rearrangement (reorientation and packing) of

grains, localized brittle fracturing of rigid grains and ductile deformation of soft grains

(Berner 1980; Bloch and Helmold 1995; Paxton et al. 2002; Wilson and Stanton 1994). In

chemical compaction, rearrangement of framework grains occurs with chemical

dissolution, particularly along the regions of major contacts or stress concentrations.

Some geologists (Lundegard, 1992; Housecknecht, 1987; Pate 1989; Ehrenberg 1989)

have agreed that compaction although being generally underappreciated, is probably the

dominant mechanism of porosity loss in sandstones.

After evaluation of a large suite of sandstones, Dutton and Diggs (1992) conclude

that cementation is more important, especially in the deeper sandstones. They argued that

compaction made relatively little contribution to porosity loss once the rocks were buried

more than 3000 ft deep, the depth at which quartz cementation begins. The role of

cementation in porosity reduction is by filling the void space with authigenic mineral

precipitation.

Based on the investigation by Dutton et al., (1993) on tight gas reservoirs in the

continental United States, most tight gas reservoirs sandstones lose their original primary

porosity by both compaction and cementation.

2.4. Compaction and cementation reviews

Intergranular space decreases by increasing the packing of solid grain volume,

which makes the effect of compaction in reservoir quality, an irreversible process. In

contrast, cement phases may dissolve during burial, which makes cementation a

reversible process (Makowitz and Milliken, 2003). Porosity plotted versus depth is

usually applied to evaluate the total effect of compaction on sediments. Empirical

porosity-depth trends show a good agreement with experimental models, which indicates

porosity decreases exponentially with depth. This reduction trend in porosity models

indicates dependence on thermal maturity and first-order dependence on the age of the

sandstone, detrital quartz content, maximum burial depth, and sorting (Makowitz and

Milliken, 2003).

8

The Type of lithic material influences on the amount of compaction in sand. In

general, sedimentary lithic fragments are more ductile than metamorphic lithic material,

although there is an overlap for ductility. Volcanic lithic grains can be altered to

phyllosilicate minerals by weathering or diagenesis and may become extremely ductile

(Pittman and Larese, 1991).

Scientists have calculated porosity loss by compaction and cementation in many

ways. For example, Lundegard (1992) has determined the porosity loss due to

compaction as below:

���������� ���� ��� � ������ � �� � ��100 � ��� � ����/�100 � ���� (2. 1)

where Pi is the initial or depositional porosity and Pmc is the minus-cement porosity or

intergranular volume which is the summation of total porosity (Po) and C which is the

volume percent of pore-filling cement. He defined another parameter that determines the

porosity loss due to cementation:

���������� ���� ��� �� ������ � ��� � ����� � ��/���� (2. 2)

For comparing different data sets, he defined a parameter, which is called compaction

index (ICOMPACT) as below:

��� �!�" � ����/����� # ����� (2. 3)

When all of porosity is loss by compaction, the compaction index is equal to one; it is

equal to zero when all of porosity loss is by cementation.

Others defined porosity loss by compaction and cementation based on an

intergranular volume compaction curve, which is not relevant to this work (Ehrenberg,

1989; Houseknecht, 1987; Pate, 1989).

9

Paxton et al., (2002) have changed the minus-cement porosity (Pmc) to

intergranular volume (IGV), and used the same procedure for compaction and

cementation porosity loss. The value of IGV is 40% for well-sorted sandstone at the time

of deposition. They also considered the volume of cement in the fracture within the

grains added to the volume of cement (portion of IGV) rather than as a portion of the

grain volume.

Quartz cement is the most important and abundant form of diagenetic cement in

clastic rocks and is the main cause of porosity reduction in many deeply buried types of

sandstone (Haddad et al., 2006). There are several attempts to model quantitatively quartz

cementation and account for porosity loss, which is out of the scope of this research (e.g.,

Blatt, 1979; Bjørlykke et al., 1986; Ehrenberg, 1990; Walderhaug, 1994 and 1996).

2.5. Pore geometries in tight gas sands

Seeder and Randolph (1987) have defined the pore geometry of tight gas

sandstones into three categories: grain-supported primary porosity, narrow intergranular

slots connecting secondary solution pores, and matrix-supported grains. The grain-

supported porosity is the porosity between rounded quartz sand grains, which have point

contact to each other. They mentioned that this type of pore space is common in tight gas

sands with dry gas permeability ranging between 10 to 100 µd. This type of pore is

usually occupied by authigenic minerals, such as clay, which constrict or block the pore

throats and reduce the permeability. Shanley et al., (2004) defined the intergranular slot

pores as narrow, sheetlike slots, which connect the secondary solution-derived pores.

They are common pore geometry observed in tight gas samples with 0.1 to 10 µd

permeability (Seeder and Randolph 1987, Figure 2.1). Matrix-supported grains are a low-

porosity, very-low permeability form of tight-sandstone pore geometry. The matrix

consists of detrital clay in which quartz grains are suspended. The lack of quartz-to-

quartz grain contacts makes the rock containing these pores, to have high pore volume

compressibility. They are common in shaly sand rocks with permeability less than 0.1 µd

(Seeder and Randolph 1987; Seeder and Chowdiah 1990).

10

Appendix A contains pore geometries observed from thin section of Travis Peak

and Frontier tight gas formations.

2.6. Ductile grains deformation

Lithic grains have an important role on compactional porosity reduction. They

increase the porosity loss when they are ductile-lithic or plastically deformable grains.

(Worden et al., 2000). Study of a reservoir in South China Sea by Worden et al., (2000)

shows that the presence of ductile grains has critical role in quality of reservoir. Ductile

compaction caused a rapid reduction in porosity during burial and very rapid loss of

permeability with decreasing porosity. The porosity reduction of this reservoir had

greater rate compared to compression of rigid quartzose grains due to plastic deformation

of ductile-lithic sandstone. The presence of ductile grains has strong effects on

permeability of reservoir. Ductile grains have been squeezed between rigid sand grains,

blocked throats and lead to abundant unconnected pores (Worden et al., 2000). Ductile

grains also isolate the remaining porosity by extruding between rigid grains. Therefore,

the permeability diminishes as pores become hydrodynamically remote from one another

(Worden et. al., 2000; Worden et al., 1997).

Several scientists (Rittenhouse, 1971; Benson 1981; Pittman and Larese, 1991,

Lander and Walderhaug, 1999) modeled ductile grain deformations. Rittenhouse (1971)

has a theoretical approach to model ductile grain compaction. He has calculated the

compaction of ductile grains based on one row of ductile grain between 18 rows of hard

grains in an orthorhombic packing when ductile grains are completely compacted.

11

Figure 2.1- Photomicrograph and scanning electron microscope images illustrating slot-type pores and pore throats commonly found in low-permeability reservoirs. (A) Frontier Formation, Amoco Shute Creek 1, 10,779.8 ft (3285.6 m), 100x, plane polarized light; (B) Williams Fork Formation, MWX 3, 5830 ft (1777 m), 1400x; (C) Travis Peak Formation, SFE 2, 8275.3 ft (2522 m), 100x, plane polarized light; (D) Travis Peak Formation, SFE 2, 8275.3 ft (2522 m), 100x, fluorescent light. Figure is provided courtesy of Shanley et al., 2004.

A quantitative model of the effect of ductile grains deformation on porosity by

Benson (1981) was based on a series of sand packs with varied (5 to 50%) amount of

ductile grains between equal sized quartz grains. He compressed the mixture by 4000 to

20000 psi biaxial pressure simulating depth up to 20000 ft. By applying different tests, he

concluded that porosity decreased in a range of 1.2 to 2.1 percent per each 1000-psi

increase in pressure. He believes that porosity reduction is due to packing adjustments

and grain rotations up to 4000 psi. After that up to 20000 psi, the porosity reduction is

12

due to ductile deformations. In addition, porosity decreases in a range of 0.6 to 1.9

percent, for each additional 5% of ductile grains. Based on his results he concluded that

after 20000-psi load, two principal types of minimal porosity remains: 1) pores, which

remained separated from ductile component even after the ductile component flowed

through pore space, or 2) pores associated with ductile component, which were sheltered

by non ductile framework that did not deform. He mentioned that other factors such as

rate of loading, pore pressure, temperature duration of loading and early cementation

might have affected the ductile deformation but he did not consider those factors in his

calculation.

Pittman and Larese (1991) made more than 400 experimental compaction tests in

a high pressure, hydrothermal reactor. They studied the physical compaction behavior of

a diverse range of lithic types mixed with quartz grains. They modeled various categories

of ductile lithic materials in the experiments. The effect of the following parameters on

compaction of lithic ductile grains were studied: volume and type of lithic material,

temperature, mineralogy, fluid type, stress, cementation, overpressure, grain size, and

distribution of lithic material by grain size. They modeled preserved porosity as a result

of compaction three categories of lithic grains: moderately ductile metamorphic lithic

fragments, highly ductile shale lithic fragments and extremely ductile altered volcanic

lithic fragments. They believe the amount of compaction of lithic sand is related to

volume and type of lithic material. They conclude that temperature was not a factor in

short-term experiment. In nature, water, filling intergranular microspores in preferentially

water-wet lithic material, has an important role in ductile behavior of lithics. Early

overpressure retards compaction but late overpressure is not effective in porosity

preservation because rock has already compacted. Early cementation retards compaction

by stabilizing the sand pack and tends to preserve porosity, but late cementation does not

have this effect. They mentioned that grain size does not have any effects on compaction.

Fine-grained sand has the same compactional behavior as coarse-grained sand, although

compaction of lithic sands was influenced by grain-size distribution of lithic material. If

ductile lithic grains were concentrated at fine fraction or distributed through all sizes,

13

substantial porosity were lost by ductile deformation. Intergranular porosity will be

preserved if there is a concentration of coarse-grained lithic fragments, among fine quartz

grains. Clusters of hard grains can create "shelter porosity" during compaction that shows

the effect of mineralogy in compaction process (cf. hard grain cages explained in Chapter

3) and clusters of ductile grains lead to extensive loss of porosity. The modeled

compactional porosity loss curve by Pittman and Larese (1991) predicts the depth of

burial by knowing the intergranular volume of a lithic sandstone.

The model for compaction of ductile grains by Pittman and Larese (1991) is the

best model for porosity reduction by ductile compaction until present. Their experiments

were used to compare with our result of porosity reduction by ductile compaction in

Chapter 3.

2.7. Multi-phase flow properties of tight gas sands

Capillary pressure data are important in evaluating the producibility of tight gas

sandstones because these rocks have high surface area per unit volume. At extremely

high capillary pressure, a large amount of water will be preserved because of surface

adsorption and capillary condensation. The producibility of the tight sandstone

diminishes at high water saturation similar to conventional rocks because gas relative

permeability is greatly reduced. Thus, it takes a lot of time for water forced into the low-

permeability reservoir during drilling or fracture stimulation, to flow back to the well

(clean up). This water may never produce back in very tight reservoirs and they may

prevent the reservoir from regaining its original permeability (Dutton et al., 1993).

Effects of overburden stress and partial brine saturation are different in

conventional reservoirs and low-permeability reservoirs. Comparing unconsolidated

high-porosity, high-permeability sands with low-permeability sands shows that the

greatest response to overburden pressure is reduction in permeability in high initial

porosity-permeability sands. However, in low-permeability sands, the greatest response

to overburden stress occurs when porosity and permeability are dominated by slot pores

and pore throats (Shanley et al., 2004).

14

Geometry of the pore system, wettability and fluid saturation control relative

permeability. The flow of a phase is controlled by the distribution of that specific fluid

irrespective to any other present fluids (Almon and Thomas, 1991; Dullien, 1992).

In low-permeability reservoirs, gas relative permeability decreases rapidly at

water saturations above 40-50%. Gas and water are immobile over a wide range of water

saturations in low permeability sandstones. The saturation region across which effective

permeability to either gas or water is negligible is called "permeability jail". Shanley et

al. (2004) report the following results for multiphase flow of tight gas (low-permeability

reservoirs). 1) The lack of water production in low-permeability reservoirs does not

imply that rock is at irreducible water saturation. It only shows that water saturation is in

the region of permeability jail. 2) Significant changes in relative permeability are caused

by small changes in brine saturation. 3) Regions having more than 50% water saturation

have no significant permeability to gas. 4) The effective permeability in low-permeability

rocks is a fundamental property of reservoir. 5) Gas at high water saturation should not be

considered as a resource because there is little or no effective permeability to gas at high

water saturation in these reservoirs.

2.8. Productivity of tight gas sands

Gas productivity in tight gas sandstone reservoirs is highly controlled by

geological attributes. Based on the study of Dutton et al., (1993) on numerous tight gas

formations, most of tight gas formations in the United States are well-sorted sandstones

deposited in high-energy depositional systems, which contain large amount of authigenic

intergranular cement and are not in general, muddy sandstones with abundant detrital

clay. Production characteristics of low permeability reserves are largely controlled by

diagenesis. Important parameters, which alter the diagenetic process, are sediment

composition, depth of burial, and age of the reservoir. Other geological attributes critical

to gas production of tight gas sandstones are natural fractures and stress directions.

Knowledge of natural fracture benefits design of completion and stimulation practices

and guides drilling strategies (Dutton et al., 1993).

15

Gas production from low-permeability rocks also requires an understanding of the

petrophysical properties associated with these factors: lithofacies, facies distribution, in

situ properties, saturation, effective gas permeabilities at reservoir conditions, and the

architecture of the distribution of these properties (Naik, 2007).

Tight gas reservoir management is a challenge to petroleum engineers. Pressure

transient testing and production of these reservoirs are slowed because the low

permeability reservoir is poorly connected. Therefore, the dynamic properties and gas in

place of these reservoirs are difficult to obtain. The costs of production are high in tight

gas reservoirs based on inducing hydraulic fracture for obtaining commercial flow rates.

They are considered high economic risk reservoirs because they often show a weak

response to frac treatments which results in low production rates (Misra, 2007).

Frac treatment may cause important mechanical damage to tight gas sand

reservoirs. The most important damage is loosening and transport of fines from pore-

fillings such as clay mineral and re-deposition of them at the tight pore throats (Misra,

2007). Other common damage on productivity include problems with fluid retention,

adverse rock-fluid and fluid-fluid interactions, effects of counter current imbibitions

during under-balanced drilling, glazing and mashing, condensate dropout and

entrainment from rich gases (Bennion et al., 1995).

2.9. Quantitative network models

The geometry and topology of pore space control many macroscopic properties

and transport coefficients of engineering interest such as permeability, drainage curves,

and relative permeability. Fatt (1956) introduced the network model, which was the first

attempt to represent pore-scale geometry and topology simultaneously. Of the several

frameworks for relating macroscopic transport properties to pore-level geometry, the

network model has been most widely used, and the literature is now vast (Blunt and

Hilpert, 2001; Rajaram et al., 1997). In network model, the pore space of a porous

medium is represented by a graph of connected sites. The sites represent the pore bodies

and the bonds connecting the sites represent the pore throats of a porous medium. This

16

graph preserves the topology of the porous medium and it reflects the macroscopic

properties of a porous medium if sizes of pores and throats are assigned based on a real

porous medium. Unfortunately, measurements of the microstructure of a porous medium

are difficult to obtain and are limited to two-dimensional images of thin sections or pore

entry size of mercury injections. Therefore, the traditional network model approaches

need vast assumptions about unmeasured features of pore space. One way to escape from

making assumptions is to study topology and geometry of pore space of a porous medium

by constructing a geometric random close packing of equal spheres as a model sediment.

Of primary importance in this project is that the macroscopic consequences of

microscopic changes must be derived based on constructed geometric models of geologic

processes. Microscopic changes consist of blocking throats and reducing the connectivity

of pore space by compaction and cementation of grains, which lead to different changes

in two-phase flow properties of tight gas sandstones.

2.10. Models of granular materials

The central obstacle to quantitatively and predictively relating microscopic

structure to macroscopic properties has been the difficulty in characterizing pore space

with sufficient fidelity. One way to characterize the pore space was to construct a close

random packing of equal spheres. Finney (1968) constructed such a packing. The packing

consists of 25000 ball bearing confined with a rubber bladder, which were fixed by

means of waxes. Finney measured the spatial coordinates of the centers of some 8000

spheres in the core of his packing (by special machines), which determine both solid and

void space within the packing. An alternative to measuring grain location is to simulate a

real random packing process on a computer.

Many researchers have tried to construct models of sphere packings with different

algorithms (next section) to make packings more realistic to actual sediment. For

example, Powell (1979) used random packing of spheres using sequential deposition

algorithm (spheres from any given particle size distribution). He produced simulated

packings for equal sized spheres and spheres with lognormal size distributions. Rodriguez

17

et al., (1986) utilized a computer method for random packing of unequal sized spheres

built under gravity, particle by particle. With this method, they can build sphere packing

with a high diameter ratio (e.g., 5). The work of Soppe (1990), which was the

combination of the Rain model and the Monte Carlo technique, appears to be an effective

method for simulating loose random sphere packing. This method is used to study the

effect of particle size distribution on the structure of hard sphere sediments in three

dimensions. The problem with this method is that it is less effective for dense random

packing.

Several scientists used sphere packing to model microscopic and macroscopic

properties of porous media. Mason (1972) showed that Finney’s measurement of grain

locations in a dense random packing of equal spheres could be used to extract pore throat

sizes, which in turn could be used to estimate capillary pressure curves (phase volume

fraction versus applied capillary pressure) in an unconsolidated granular material.

The most important feature of packing of spheres is the randomness in locations

of grain pore throats. This model (sphere packing) has been shown to enable

extraordinarily accurate predictions of macroscopic properties as a function of the extent

of various geological processes, such as quartz cementation. For example, Roberts and

Schwartz (1985) used Finney’s data to estimate electrical conductivity in the packing of

non-conducting grains filled with a conducting fluid. Bryant et al., (1993) adapted the

approach of Mellor (1989) to predict permeability in sandstones. Starting with the

original Finney packing, the porosity was reduced by increasing the radius of the spheres,

without changing the sphere locations. Increasing sphere radii modeled the growth of

quartz cement. The key advantage of this model is that the geometry of pore space in the

model rocks is completely determined because the grain space is completely determined;

all sphere locations are known, as are their radii. After that, others used sphere packing to

model transport properties of porous medium. Mason and Mellor (1995) used Finney

packing data converted a network model to simulate drainage and imbibitions in a porous

medium. Sphere packing was used to model compaction and cementation of a

conventional sandstone (e.g. Jin et al., 2006). We used this approach to model tight gas

18

sandstones with low porosity and permeability based on compaction of ductile grains and

cementation. To do that, we adapted the approach of Thane (2006) to construct a sphere

packing considering ductile deformation of grains using soft shell model. To construct a

compacted sphere packing Thane (2006) adapted cooperative rearrangement algorithm,

which is one of the used algorithms in building a sphere packing (next section).

2.11. Algorithms for computer generated packing of spheres

The key advantage of a simulated packing over network model is that each

particle location is known and geometry of pore space between the grains can be

computed. Two geometry-based packing algorithms are the sequential addition algorithm

(Rodríguez et al., 1986; Visscher and Bolstereli, 1972) and the cooperative rearrangement

algorithm (Clarke and Wiley, 1987). Other algorithms, such as the Discrete Element

Method (DEM), incorporate some of the physics of the packing process by using friction,

contact forces, and particle-particle interaction (Dobry and NG, 1992; Jin et al., 2003).

2.11.1. Sequential addition algorithm

The sequential addition algorithm reproduces loose random packings and the

packing solid fraction is in the range of 0.58. To make a dense random packing using

sequential addition algorithm, the shaking steps have to be added to the simulation, which

increases the solid fraction packing to 0.6 (Visscher, 1972).

The sequential addition algorithm has one hard boundary corresponding to the

floor of the bin; this causes ordering of spheres near the boundary (Thane 2006). The

sequential algorithm has three steps, which are sphere generation, free fall under gravity,

and final positioning.

In the first step, an initial position and a radius of spheres are assigned and the

sphere is positioned above the other spheres already in place. The number of assigned

spheres can change the properties of the packing (Visscher, 1972). In the second step, the

sphere will fall with downward movement due to gravity until the sphere contacts the

floor or another sphere. Then the last step starts when the sphere has contact with the

19

floor or other sphere and the position is not stable so the sphere rolls along the floor or

contacting sphere until it contacts another sphere or the floor. If the new position is

stable, the new sphere will be generated. If the position is not stable, it rolls again and

iteration continues until the position with the floor or other sphere in contact is stable

(Thane, 2006).

2.11.2. Cooperative rearrangement algorithms

The cooperative rearrangement algorithm, described in details in Chapter 3, yields

packing fractions between 0.637 and 0.649 for mono-disperse spheres (Thane, 2006), and

has closely approximated the packing fractions of experimental random dense packings

of mono-disperse spheres, such as the Finney packing. The algorithm consists of three

basic steps, which are initial point generation, sphere growth or shrinkage, and overlap

check and removal. The second and third steps are iterated until no further increase in

sphere size is possible (Thane, 2006).

2.11.3. Discrete element method (DEM)

This algorithm is one of the best algorithms in making sphere packings because it

uses the effect of physical characteristic of granular matter on the produced packing. It

accounts for the van der Waals interactions between grains, contact forces between

particles, friction forces and gravity influence. The DEM is expensive to compute

(Thane, 2006).

There are many techniques to model the discrete element algorithm. A first step in

typical approaches is to initiate the number of particles (spheres) into the domain of

interest, with the diameters much smaller than their final size. In the next step, the

particle diameter increases to fill the domain until dense packing is reached. Another

approach is to fill a large domain with particles with final diameter, then to slowly move

the wall inward to reach to the dense packing. Another way is to use particles in their

final diameters and simulate gravitational deposition. This calculation needs a large

amount of computation for the motion of particles (Bagi, 2005).

20

Chapter 3: Models of compaction and cementation in tight gas sandstone

3.1. Introduction

We have seen that several geologic processes are responsible for porosity

reduction in tight gas sandstones. The goal of this chapter is to present a pore-scale model

of these processes. The model attempts to capture the geometric changes caused by these

processes. The geometry can then be used to compute the flow properties, which can be

compared to measurements on samples of tight gas sandstone. These models of rocks can

then be used to model tight gas sand two-phase flow properties. The chapter starts with

modeling of ductile grain compaction. It explains the method using cooperative

rearrangement algorithm and soft shell model for ductile grains. Then, the visualization

of ductile compaction is done using particle flow code (PFC2D). The hypothesis

investigated here is that the pressure loaded during compaction of mixture of hard and

ductile grains is supported primarily by point contact between hard spheres. This means

hard spheres make a framework of cages that prevent further compaction even though

some ductile grains may remain undeformed. The next step is to use the soft shell model

for compacting ductile grains and uniform quartz cementation to reduce the porosity of

the grain model and simulate tight gas sandstone. The hard cages form during modeling

of ductile grains using soft shell model. In addition, the compaction of ductile grains,

using soft shell model, results in formation of slit like pores similar to tight gas

sandstones. Modeling of compaction and cementation can yield accurate predictions of

porosity-permeability in subsurface sandstones (Lander and Walderhaug, 1999) and the

compaction and cementation models provided in this research, helps to refine this

capability. At the end of this chapter, we calculate the one-phase flow permeability of

modeled rock and compare it with real rock data to validate the model works before

proceeding to two-phase modeling.

3.2. Compaction simulation of mixtures of rigid and ductile grai

A simple ductile grain model implemented by Thane (2006)

effect of ductile grains on the packing porosity. In this model, ductile grains differ from

rigid grains by having a soft outer shell and a rigid central core.

this model of a ductile grain. Before explaining the mixture of ductile and rigid grains,

we need to explain the cooperative rearrangement algorithm in details as implemented by

Thane (2006).

Figure 3.1- Ductile grain concept.

As we mentioned before (in

consists of three basic steps, which are initial point generation, sphere growth or

shrinkage, and overlap check and removal.

Initial point generation for the packing is a process that requires three general

inputs: (1) the number of spheres in the packing, (2) the size of the box containing the

spheres, (3) the sphere radii distribution. The number of spheres within the packing

arbitrary, but in practice is limited by time constraints.

with 5000 spheres it takes about half an hour for a computer with dual core processor 2

GHz and 8 GB of RAM.

computationally expensive the simulations are. The number of spheres chosen for the

packing will have an effect on the choice of the box size of the packing. An appropriate

21

Compaction simulation of mixtures of rigid and ductile grains

A simple ductile grain model implemented by Thane (2006) was used to study the

effect of ductile grains on the packing porosity. In this model, ductile grains differ from

rigid grains by having a soft outer shell and a rigid central core. Figure 3.1

this model of a ductile grain. Before explaining the mixture of ductile and rigid grains,

we need to explain the cooperative rearrangement algorithm in details as implemented by

The rigid radius is a good proxy for ductility of a ductile grain.

before (in Chapter 2), the cooperative rearrangement algorithm

consists of three basic steps, which are initial point generation, sphere growth or

shrinkage, and overlap check and removal.

Initial point generation for the packing is a process that requires three general

inputs: (1) the number of spheres in the packing, (2) the size of the box containing the

spheres, (3) the sphere radii distribution. The number of spheres within the packing

arbitrary, but in practice is limited by time constraints. For example for making a packing

with 5000 spheres it takes about half an hour for a computer with dual core processor 2

GHz and 8 GB of RAM. The more spheres the packing contains, the more

putationally expensive the simulations are. The number of spheres chosen for the

packing will have an effect on the choice of the box size of the packing. An appropriate

was used to study the

effect of ductile grains on the packing porosity. In this model, ductile grains differ from

demonstrates

this model of a ductile grain. Before explaining the mixture of ductile and rigid grains,

we need to explain the cooperative rearrangement algorithm in details as implemented by

is a good proxy for ductility of a ductile grain.

the cooperative rearrangement algorithm

consists of three basic steps, which are initial point generation, sphere growth or

Initial point generation for the packing is a process that requires three general

inputs: (1) the number of spheres in the packing, (2) the size of the box containing the

spheres, (3) the sphere radii distribution. The number of spheres within the packing is

For example for making a packing

with 5000 spheres it takes about half an hour for a computer with dual core processor 2

The more spheres the packing contains, the more

putationally expensive the simulations are. The number of spheres chosen for the

packing will have an effect on the choice of the box size of the packing. An appropriate

22

box size is determined from experience, but there is normally a large range of suitable

box sizes. In the all packings studied here, the box is periodic. This eliminates artifacts in

the packing structure that hard walls would cause. Finally, the sphere radius distribution

function is decided. Once the number of spheres, the box size, and the distribution type

have been selected a number of points equal to the number of spheres is generated by

randomly choosing x, y, and z coordinates within the box. These points are effectively

spheres with a zero radius. At this stage, the first step of initial point generation is

complete (Mousavi and Bryant, 2007; Thane, 2006).

In the first step, initial positions of sphere centers are assigned randomly in a unit

volume of space. The points initially have radius 0. To produce packings with different

grain sizes, each point is assigned a “radius ratio.” This parameter determines the relative

size to which the sphere will grow.

The next step in the algorithm involves growing the spheres in the box to begin

filling the packing space. The first increment size for the sphere radius is arbitrary, but

choosing a larger increment size will save some time. If the increment is too small, the

spheres may need to be grown repeatedly before spheres begin encountering each other.

Overlaps can occur as the spheres begin to grow in size; overlaps represent a physically

impossible situation of two separate particles sharing the same volume of space.

Overlaps, in this algorithm, refer to two spheres overlapping by over 1x10-6 absolute

distance units. For a mono-dispersed packing this represents 0.000046% of a sphere

radius. The size of a box containing spheres is fixed (70 in x direction, 70 in y direction

and 70 in z direction). The size of the spheres will change based on the number of spheres

assigned to the box. Therefore, the sphere overlapping amount depends on the number of

spheres in the box. At each size change of the sphere radii, the packing has to be checked

for overlapping spheres.

In the next step, the size of all the spheres in the packing is changed. The

increment applied to each sphere is proportional to its radius ratio. Increasing the sphere

sizes makes the packing denser. If spheres are already touching, the increment also

causes overlaps between spheres to form.

23

As soon as overlaps are found within the packing, they are removed. In order to

remove the overlap between two spheres, they are moved apart an equal distance along

the line joining the spheres. At the end of the move, the two spheres are in point contact.

This process is repeated for all spheres in the packing. A loop through the entire packing

is considered one iteration. The packing must be checked at least twice in order to

determine if removal of overlaps in the iteration created new overlaps. However, as

packings get denser it may not be possible to remove all of the overlaps. In this study, the

algorithm goes through 500 iterations before the overlaps are considered to be impossible

to remove (Mousavi and Bryant, 2007; Thane, 2006).

When the overlaps in the packing can no longer be resolved, the radii of the

spheres are decreased so that the spheres can fit into the packing space. The size of the

radius decrement, in this algorithm, is half the size of the last radius increment applied.

Sphere radii are decremented until all overlaps can be removed, at which point the

spheres enter a growth stage again. The size of the increment is a little over half the size

of the last decrement and the increment is applied until overlaps are again impossible to

remove. At this point, spheres begin a shrinking stage again.

The increment and decrement steps in the algorithm continue to get smaller as the

packing reaches its final packing fraction. The simulation finishes when the packing

fraction no longer changes between the overlap-free packing stages.

Overlaps must be removed to have a physically feasible packing. Thus in the third

step, overlap checking is applied. In some algorithms, including the method used here,

the overlap between two spheres is removed by moving the spheres apart along the line

joining the center of the two spheres until they are in point contact. Other algorithms

remove overlaps by moving spheres along the vector sum of all overlaps.

At higher packing fractions, new overlaps with nearby spheres often are the

consequence of removing the overlap between two spheres. If even after a large number

of iterations the overlap between spheres cannot be removed, then the sphere sizes will be

reduced slightly to eliminate the overlaps in the packing. The cooperative rearrangement

simulation ends at a densely packed, overlap-free stage when no further increase in

24

sphere size can be accommodated by rearrangement (Thane, 2006; Mousavi and Bryant,

2007).

The presence of ductile grains i.e. spheres, with penetrable outer shells will cause

a few changes in the basic cooperative rearrangement algorithm. There are three steps in

creating a packing containing both rigid and ductile spheres: (1) assignment of ductile

characteristics, (2) cooperative rearrangement of the mixture of rigid and ductile spheres,

and (3) volume conservation. The following paragraphs describe the implementation of

these steps in a code written by Thane (2006).

In the first step, the desired amount of ductile matter is specified in terms of two

variables: the volume percent of ductile grains, and the size of each ductile grain's hard

core. In the results reported here, spheres are chosen randomly to be ductile (although in

some of rocks there may be a spatial arrangement of ductile grains). Thus, in the case of

bi-modal packings, large and small spheres have an equal probability of being assigned as

a ductile sphere. The size of each ductile grain’s hard core is specified as a fixed fraction

of the grain’s initial radius. We show below that the radius of the hard core can be

correlated to the degree of ductility. Thin soft shells (rigid core radius 0.92 to 0.98 of the

grain radius) yield porosity trends similar to those of brittle grains, while thick shells

(rigid radii 0.05 to 0.8 of grain radius) correspond to extremely ductile grains.

The extended cooperative rearrangement algorithm packs all spheres based on

their rigid cores, with the understanding that the “core” of a hard grain is the grain itself.

The consequence of this method of packing is that the ductile shell overlaps with other

grains in the packing. The overlaps represent pressing of hard grains into ductile grains

during compaction. It is also possible for two ductile grains to overlap. Only the ductile

shells overlap and there is no overlap between rigid portions of the grains. Figure 3.2

shows the way in which a rigid sphere and a ductile sphere might pack together through

the cooperative rearrangement.

The existence of overlap violates the conservation of mass of original grains

because some portion of volume will be lost in the overlap. To correct this problem, a

volume conservation step is applied after each iteration of cooperative rearrangement.

25

The initial volume of grains should be equal to the final volume to satisfy volume

conservation. A simple way to achieve this is to increase the overall radius of a grain

when overlap has occurred. This increment increases the overlap still more, but away

from grain contacts, it moves the boundary of the grain farther into open pore space. The

net effect is to increase the volume occupied by the grain. In this sense the model is

analogous to what happens when a ductile grain undergoes deformation when pressed by

adjacent hard grains. The final grain radius rf is chosen so that the apparent grain volume

4/3πrf3 less the total volume of overlap will be equal to the initial grain volume (Thane,

2006; Mousavi and Bryant 2007).

Figure 3.2- a) Packing of a rigid and a ductile grain with overlap generation in the ductile shell. b) The overlap volume is distributed into pore space by increasing the ductile grain radius. The increment is chosen so that the ductile grain volume excluding the overlap with the hard grain is the same as the original volume of the ductile grain. This conserves mass and approximates the geometric effect of deformation of the ductile grain into pore space.

As an example of the algorithm output, Figure 3.3 is a dense random packing of

5000 spheres. The packing has porosity 21.6%. 30% of the grain volume is ductile, and

each ductile grain has a rigid radius 75% of the initial grain radius. Thus in each ductile

grain, the original ductile shell thickness is 2

spheres are ductile, as is evident from their bigger sizes (even bigger than the biggest

rigid spheres) and also by their overlap with each other. The packing was initialized with

a bi-modal distribution of sphere si

small. The radius ratio of big spheres to small spheres is 1.5.

Figure 3.3- Dense packing (porosity 21.6%) of hard and ductile grains (5000 spheres; 30% ductile matter; ductile grains have rigid core radius of 0.7spheres are ductile and gray spheres are rigid. Packing is bilarger than the other half). Arrows show the big and small hard (gray) spheres and an overlap of ductile spheres (red)

3.2.1. 2D models of sand packs

The hypothesis investigated here is that the

mixture of hard and ductile grains is supported primarily by point contact between hard

spheres. This means hard spheres make a framework of load

further compaction even though some ductile grains may

to test this hypothesis, we used a commercial simulator, Particle Flow Code (PFC) in 2

dimensions created by Itasca Consulting Group. We load

26

each ductile grain has a rigid radius 75% of the initial grain radius. Thus in each ductile

original ductile shell thickness is 25% of the initial grain radius. The red

spheres are ductile, as is evident from their bigger sizes (even bigger than the biggest

rigid spheres) and also by their overlap with each other. The packing was initialized with

modal distribution of sphere sizes, half the spheres being large and the other half

small. The radius ratio of big spheres to small spheres is 1.5.

Dense packing (porosity 21.6%) of hard and ductile grains (5000 spheres; 30% ductile matter;

ductile grains have rigid core radius of 0.7R) produced by cooperative rearrangement simulator. Red spheres are ductile and gray spheres are rigid. Packing is bi-disperse (half the spheres have radius 1.5 times larger than the other half). Arrows show the big and small hard (gray) spheres and an overlap of ductile

D models of sand packs

The hypothesis investigated here is that the mechanical load during compaction of

mixture of hard and ductile grains is supported primarily by point contact between hard

spheres. This means hard spheres make a framework of load-bearing cages that prevent

further compaction even though some ductile grains may remain not deformed. In order

to test this hypothesis, we used a commercial simulator, Particle Flow Code (PFC) in 2

imensions created by Itasca Consulting Group. We loaded a packing of 115 disks with

each ductile grain has a rigid radius 75% of the initial grain radius. Thus in each ductile

5% of the initial grain radius. The red

spheres are ductile, as is evident from their bigger sizes (even bigger than the biggest

rigid spheres) and also by their overlap with each other. The packing was initialized with

zes, half the spheres being large and the other half

Dense packing (porosity 21.6%) of hard and ductile grains (5000 spheres; 30% ductile matter; ) produced by cooperative rearrangement simulator. Red

disperse (half the spheres have radius 1.5 times larger than the other half). Arrows show the big and small hard (gray) spheres and an overlap of ductile

during compaction of

mixture of hard and ductile grains is supported primarily by point contact between hard

bearing cages that prevent

remain not deformed. In order

to test this hypothesis, we used a commercial simulator, Particle Flow Code (PFC) in 2

a packing of 115 disks with

27

mixture of hard and ductile grains in PFC2D. The tested packing contains disks with

hexagonal orientation, which is the most dense 2D packing.

Newton’s second law was used for interactions between spherical grains of sand

using discrete element model (DEM). Normal compressive and shear particle stiffness

give rise to forces between particles of the packing when the particles deform at contacts.

This interaction is a linear force-distance relationship. Pairs of particles can also be

subject to compressive and tensile forces from a cement bond between particles, which

acts in parallel with the contact forces. This feature of the code is not used in this study.

Inter-particle interactions can also be described using a Hertz-Mindlin contact model,

which is a non-linear contact model, although this interaction cannot be used in parallel

with the cement bond (Itasca 2004). We used Hertz-Mindlin model in one of our

packings to compare the results with linear model. The stress test conducted here is based

on a uni-axial test although we used hydrostatic pressure as an initial condition in the

packing to make it more realistic.

Figure 3.4 shows the nature of the interactions between particles in the DEM

simulation. In this model forces are defined by linear spring constants kn and ks which

refer to normal and shear forces respectively. The normal force Fn between two

uncemented particles i and j is defined as the amount of overlap between spheres times

the normal stiffness of the particles:

�� � �� � ��� � � �� (3.1)

where dn is distance between centers of two particles and Ri and Rj are radii. This force

acts only at a point, at the intersection of the line connecting the centers of the particles

with the surface of the sphere. Linear-spring model is the default stiffness model in PFC

and we used that in our PFC test. When two balls (or a ball and a wall) are in contact, an

effective contact-stiffness is calculated by assuming that the springs act in series. The

units for ball stiffness are [force/displacement] (e.g., N/m). If not assigned, stiffness

values are set to zero by default (Itasca 2004).

28

To make a packing in PFC, the first step is to create a packing of disks, the second

step is to initiate stress state, and the last step is to do elastic loading. The packing

contains both hard grains (disks) and ductile grains in a closed hexagonal packing. The

soft shell model of a ductile grain is a geometric model and is not used in PFC software.

Ductility is represented in PFC by assigning grains a smaller value of stiffness.

Parallel Bonds(Cement)

• Initial forces = 0• Forces increment by spacing• Moment increments by beam bending• All forces zero if bond exceeded

• kn, ks: ball stiffness

Contact Forces

δ

• Normal force = kn * δ• Shear force = ks * γγγγ

γγγγ

Figure 3.4- Figure shows the diagram of interactions between particles in the PFC simulations. Image courtesy from Itasca Consulting, theory and background (2004).

Rittenhouse (1971) believed that the hexagonal packing arrangement is applicable

to well-sorted and rounded natural sands, which tolerated extensive compaction. He used

a packing with 115 spheres with hexagonal (orthorhombic) arrangement of 19 rows hard

sphere with one row of ductile spheres in the middle. The porosity was calculated

knowing the radius of spheres, height, width and thickness of the sphere pack. He simply

removed the ductile row in the packing as a consequence of compaction. Then he

reported the new porosity and height reduction of the pack. He determined that when the

percentage of ductile grains are small in the packing the formation of load bearing hard

29

cages prevent further compaction and the ductile grain may remain not deformed.

Rittenhouse believed the orthorhombic (hexagonal) packing is the closest packing and is

appropriate for modeling of well-sorted and rounded sand grains. The Rittenhouse

compaction model is very simple and it does not account for grain contact mechanism.

3.2.2. Modeling hard and ductile grains

We modeled the packing of Rittenhouse in PFC2D program with different amount

of ductile and different positions for ductile grains to find the behavior of ductile grains

under the actual load (initial configuration was made geometrically and do not consider

actual load). First, we defined a packing of 115 disks (2D packing is enough for our

calculation) with 54 ductile grains that distributed randomly in the packing. The

sequences of steps to make this test are as below:

1. Create a hexagonal array

2. Apply an isotropic initial stress in all direction equal to -106 Pa as a hydrostatic

pressure.

3. Move the top wall in negative direction and keep the lateral forces constant. This

is an axial load with constant strain rate.

4. Plot the stress-strain data.

Contact model was chosen to be stiffness model for walls and balls in the

packing. The inputs for the linear-spring model (stiffness model) are normal and shear

stiffness for walls of the packing and balls. The normal stiffness for the wall was 3×1014

N/m and the shear stiffness chosen to be zero. The normal and shear stiffness for balls

were 2×1010 N/m and 2×109 N/m respectively. The walls are fixed in x direction and the

top wall moves at the constant velocity of 0.4 after increasing the velocity in 80 chunks

of 2000 steps. These boundary and initial conditions were chosen after several trial and

error tests to get the best options for the compaction of ductile grains under load.

The stiffness values for quartz and ductile grains were chosen based on the

approach of Holder (2005). A lithic grain is an aggregate of other minerals such as

quartz, feldspar, clay etc, which represents a fragment of an older rock that served as a

sediment source. Ductility of a lithic fragment varies with rock type. For this type of test

30

in PFC we chose the lithic fragment to be very ductile as might be true of a lithic

fragment composed of clay particles. We explain the steps to get the best stiffness for

quartz and clay briefly as below:

The quartz shear stiffness and Poisson’s ratio were assigned as inputs for the

Hertz-Mindlin contact model in PFC from Park (2006) (G=44 GPa and ν=0.1

respectively). Those values for clay grain were found to be G=6 GPa and ν=0.48). Then

the Hertz-Mindlin contact model was applied to the packing of only quartz grains and

another packing of only clay grains to derive the macroscopic Young’s modulus of quartz

and clay. The slopes of the axial stress–strain were computed to give a macroscopic

Young’s modulus, E for both packings of quartz and clay grains. Next, a linear force-

displacement (linear-spring model) contact model was applied to the same packings. We

varied the values of normal stiffness to get the slopes of the axial stress–strain plot to be

equal to the ones in Hertz-Mindlin contact model. Then those values of stiffness were

chosen for the compaction of ductile and rigid packing with a linear contact model. Based

on this approach, the values for normal stiffness of quartz and clay are 2×1010 N/m and

2×109 N/m respectively.

Figure 3.5- Sample with random 54 ductile grains and lateral confining stress before and after compaction. The right figure was created at the end of the test.

31

Figure 3.5, shows the sample before and after compaction. The quartz grains

(white) carry the most of the load (thick lines) and make a framework of rigid grains (a

"hard cage" that surrounds ductile grains). No further compaction is possible even though

some ductile grains may remain not deformed. The stress-strain path shows different

irregularities (marked by arrows in the Figure 3.6), which are related to formation of hard

cages. When the stress-strain path decreases, it means that substantial deformation of a

soft grain occurs. Because ductile grains are softer than rigid grains, they are the first

grains that reach to this peak point (yield point). Decreasing in the stress-strain path,

shows that the ability of one region of rock (consisting of ductile grains) to support an

additional load is decreasing and it is compensated by transferring of the some of the load

to the adjacent grains. Therefore, the load finds a path through the hard grains after

deformation of ductile grains. Then, the stress-strain path continues to increase again

until it finds the next concentration of ductile grains. This process happens several times

in the packing for different regions consist of ductile grains and it is responsible for the

formation of "hard cages" in the packing under the load. This result supports the

hypothesis of point contact between hard grains during compaction of mixture of ductile

and hard grains.

Figure 3.6- Stress-strain plot for sample with random ductile grains and initial confining stress. The vertical axis is stress (Pa) and horizontal axis is strain. Arrows show the formation of hard cages.

0.00E+00

2.00E+08

4.00E+08

6.00E+08

8.00E+08

1.00E+09

1.20E+09

1.40E+09

0 0.1 0.2 0.3 0.4 0.5 0.6

Str

ess

Strain

32

3.3. Porosity reduction due to compaction of ductile grains

The structure of granular packings is complex and very sensitive to boundary

conditions, filling procedures, and grain-grain interactions (Uri et al., 2004). In nature,

plastic deformation of lithic grains is very important after the initial mechanical

compaction. Thus, a comparison of the structure of ductile granular material is of major

interest in the understanding of the later stages of compaction.

Figure 3.7- a) There is no porosity in this thin section. The lithic grains flow into the pore space. Jurassic., Latady Fm., Antarctica. b) The pressure solution stilolite is marked by arrow. Oligocene, Frio Fm., Brazoria Co, TX. (Images courtesy from Milliken et al., 2007).

Deformable grains in sediment change the properties of porous media under load.

Unlike rigid grains, ductile grains can deform and flow into the nearby pore space (Figure

3.7). After compaction, the porosity of a packing containing ductile grains will be less

than that of a packing containing only rigid grains. Ductile matter also has an important

33

role in reduction of permeability in porous media. The other importance of deformable

material is their role in mechanical properties of sediments (Persson and Göransson,

2005).

Sediment compaction reduces the pore volume under a load and is defined by two

processes: shallow and deep compaction. Mechanical compaction takes place in a

shallow or intermediate depth (<3 km) and consists of rearrangement (reorientation and

repacking) of grains, localized brittle fracturing of rigid grains, and ductile deformation

of soft grains (Makowitz and Milliken, 2003; Paxton et al., 2002).

Intergranular volume is a good indicator of mechanical and chemical (pressure

solution) compaction because it reflects the grain framework (Paxton et al., 2002).

Intergranular volume is the sum of intergranular pore space, intergranular cement, and

depositional matrix (small solid particles) which can be measured by point counting of

petrographic thin sections. When there is no cement or matrix, the intergranular volume

equals the maximum intergranular porosity at that depth. For the purpose of IGV

calculation of the authigenic phases, the replace grains are tabulated as part of the grain

volume. The amount of porosity reduction not attributable to intergranular cement or

depositional matrix is the indicator of compaction extent (Mousavi and Bryant, 2007).

Compaction of a ductile-dominated sandstone can be seen schematically in Figure

3.8. Because of the ductility and high initial volume of ductile grains, the intergranular

volume was completely closed after compaction. One of the primary goals of this work is

to develop a simple model of this mode of reduction of intergranular volume.

One goal of this study is to determine whether the purely geometric algorithm for

creating dense packings with soft-shell spheres yields porosities comparable to buried

sediments containing ductile grains. We compare the simulations to the data of Pittman

and Larese (1991), comprising more than 400 experiments in a high pressure

hydrothermal reactor to simulate the compaction process. They tested three models of

lithic content in sandstones: moderately ductile metamorphic lithic fragments, highly

ductile shale lithic fragments and extremely ductile altered volcanic lithic fragments. The

models contained from 5% to 95% lithics and were loaded to a maximum effective stress

of 7500 psi (51.7 MPa).

Figure 3.8- Progressive compaction of ductileintergranular volume to drop from 40 to 0% with burial. IGV= intergranular volume; q = quartz grain; d = ductile grain (Mousavi and Bryant, 2007

We generated a large number of dense packings

algorithm (section 2 of this chapter), varying the rigid radius of soft grains from zero to

100 percent and the total amount of ductile matter from zero to 100 percent. We used

5000 spheres in each packing which proved to b

reproducible results. Figure 3.9

plot shows the trend in porosity reduction as the ductile fraction increases and the rigid

radius of ductile spheres decreases.

modeled by soft shell model of ductile

penetrates through the ductile grain to reach to the rigid core of ductile grain. Then the

amount of overlap will be calc

34

models contained from 5% to 95% lithics and were loaded to a maximum effective stress

Progressive compaction of ductile-dominated sediment. Deformation of ductile grains causes intergranular volume to drop from 40 to 0% with burial. IGV= intergranular volume; q = quartz grain; d =

(Mousavi and Bryant, 2007)

We generated a large number of dense packings using cooperative rearrangement

algorithm (section 2 of this chapter), varying the rigid radius of soft grains from zero to

100 percent and the total amount of ductile matter from zero to 100 percent. We used

5000 spheres in each packing which proved to be a sufficient number of grains for

3.9 summarizes more than 190 numerical experiments. This

plot shows the trend in porosity reduction as the ductile fraction increases and the rigid

radius of ductile spheres decreases. The flow of ductile matter in the pore space is

soft shell model of ductile grains. When a hard grain is near a ductile grain, it

through the ductile grain to reach to the rigid core of ductile grain. Then the

amount of overlap will be calculated and added to the outer shell of ductile grain.

models contained from 5% to 95% lithics and were loaded to a maximum effective stress

minated sediment. Deformation of ductile grains causes intergranular volume to drop from 40 to 0% with burial. IGV= intergranular volume; q = quartz grain; d =

using cooperative rearrangement

algorithm (section 2 of this chapter), varying the rigid radius of soft grains from zero to

100 percent and the total amount of ductile matter from zero to 100 percent. We used

e a sufficient number of grains for

summarizes more than 190 numerical experiments. This

plot shows the trend in porosity reduction as the ductile fraction increases and the rigid

ow of ductile matter in the pore space is

When a hard grain is near a ductile grain, it

through the ductile grain to reach to the rigid core of ductile grain. Then the

ulated and added to the outer shell of ductile grain.

Therefore, ductile grains are bigger than hard grains

reduced and the size of ductile grain

of the throats. The result will lead to

packings.

Figure 3.9- Simulated compaction for packings of 5000 bilarger than the other half). Each curve corresponds to a different value ductile spheres. The similar porosity trend for a wide range of rigid radii (small black arrows)formation of cages (load-bearing frameworks) in the packing. The big arrow points in the direction of decreasing rigid radius, which is a proxy for increasing ductility of the lithic grains.

A plot of porosity versus percentage of ductile matter was constructed for each

lithic type used in the Pittman and Larese experiments (

It is remarkable that the radius of the rigid core of the ductile grains proves to be a good

proxy for the type of ductile material

between our simulations with soft spheres of rigid radius 0.8

varied the amount of ductile matter from 0 to 100 percent and observed same trend as

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20

(Po

rosi

ty /

Init

ial P

oro

sity

)*10

0

35

Therefore, ductile grains are bigger than hard grains. The pore space and throats

the size of ductile grain increased. In addition, this process will block some

t will lead to porosity and permeability reduction in simulated

Simulated compaction for packings of 5000 bi-disperse spheres (half the spheres are 1.5 times larger than the other half). Each curve corresponds to a different value of the radius of the rigid core of the ductile spheres. The similar porosity trend for a wide range of rigid radii (small black arrows)

bearing frameworks) in the packing. The big arrow points in the direction of easing rigid radius, which is a proxy for increasing ductility of the lithic grains.

A plot of porosity versus percentage of ductile matter was constructed for each

lithic type used in the Pittman and Larese experiments (Figure 3.10 through

It is remarkable that the radius of the rigid core of the ductile grains proves to be a good

proxy for the type of ductile material. The experiments with moderately ductile grains fall

between our simulations with soft spheres of rigid radius 0.8R to 0.9R, Figure

varied the amount of ductile matter from 0 to 100 percent and observed same trend as

40 60 80 100

Ductile %

and throats are

. In addition, this process will block some

reduction in simulated

disperse spheres (half the spheres are 1.5 times

of the radius of the rigid core of the ductile spheres. The similar porosity trend for a wide range of rigid radii (small black arrows) is related to

bearing frameworks) in the packing. The big arrow points in the direction of

A plot of porosity versus percentage of ductile matter was constructed for each

Figure 3.12).

It is remarkable that the radius of the rigid core of the ductile grains proves to be a good

The experiments with moderately ductile grains fall

Figure 3.10. We

varied the amount of ductile matter from 0 to 100 percent and observed same trend as

0.8 Rigid R

0.85 Rigid R

0.9 Rigid R

0.92 Rigid R

0.94 Rigid R

0.96 Rigid R

0.98 Rigid R

0.99 Rigid R

36

seen in Pitman and Larese paper, notably the fact that porosity does not go to zero even

when all the grains are ductile.

Figure 3.11 shows the comparison between the brittle grain experiments of

Pitman and Larese and our simulations with when the rigid radius of soft spheres is 0.92R

and 0.98R. Although our simulations do not account for breakage of brittle grains, they

nevertheless capture the observed trends.

Figure 3.10- Porosity trends for dense packings of spheres with selected radii (0.8R and 0.9R) for the rigid cores of the ductile grains. The trends bracket the experiments of Pittman and Larese (1991) in which the ductile grains are moderately ductile fragments.

The experiments using extremely soft ductile grains show a porosity trend

bounded above by our simulations with rigid radius equal to 0.8R, Figure 3.12. The trend

for packings with soft spheres of rigid radius less than 0.7R correctly predicts that all

porosity is lost when the ductile fraction increases to about 70%. In contrast to the other

types of ductile grains, these experiments in many cases fall below the lower limit

predicted by the simulations, especially when ductiles comprise 20% to 60% of the

grains. This discrepancy is discussed in the context of the framework of load-bearing

grains that emerges during the compaction simulation (section 5 of this Chapter).

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

( P

oro

sity

/Init

ial P

oro

sity

)*10

0

Ductile (%)

Green Shale

Phyllite

Black Shale

Mica Schist

0.9 Rigid R

0.8 Rigid R

37

Figure 3.11- Porosity trends for dense packings of spheres with selected radii for the rigid cores of the ductile grains. The thin soft shells yield behavior similar to the experiments of Pittman and Larese (1991) in which the ductile grains are brittle fragments. A rigid core radius of 0.92R provides a reasonable lower bound to the data. The upper bound would be grains with rigid core radius of 1R, for which no porosity reduction would occur.

Modeling the compaction of sediments containing extremely ductile grains with

the soft-shell model captures the average behavior observed experimentally, but does not

establish a lower bound on porosity loss. However, in the case of moderately ductile and

brittle fragments, simulations with a narrow range of rigid core sizes accurately predict

the upper and lower limits on porosity loss.

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

(Po

rosi

ty/In

itia

l Po

rosi

ty)*

100

Ductile %

Ooids

Rhyolite

Red Slate

Basalt

0.92 Rigid R

0.98 Rigid R

38

Figure 3.12- Comparison of simulated compaction with experiments using extremely ductile fragments Pittman and Larese (1991). Simulations with ductile grains having rigid radius 0.8R provide an upper bound for most of the data. When the rigid radius of ductile grains is 0.05R, the predicted porosity trend gives the correct average behavior. In contrast to Figure 3.10 and Figure 3.11, no simulation provides a lower bound for these experiments.

3.4. Porosity reduction due to quartz cementation

From Figure 3.9 and Figure 3.12, we can see that some of the porosity is

preserved in packings containing 30% to 40% ductile matter even when ductile grains

have rigid radius = 0.05 R (very soft grains). Typical porosities after simulated

compaction are 10% to 25%. Many tight gas sandstones contain up to one third ductile

grains, like these packings, but exhibit much smaller porosities (less than 10%). We

conclude that extensive cementation is also needed to reduce the IGV in these compacted

rocks. We model this process by adding quartz overgrowth in the shape of uniform rim

cement to each sphere. This model is purely geometric and it is not deposited under the

overburden load. Figure 3.13 shows the 2D thin section of 1000 spheres, 30% of which

are ductile having rigid radius 70% of original grain radius, for several different

thicknesses of cement. Cement addition was applied as a percent of the sphere radius. For

example “10% cement” means that a layer of thickness equal to 10% of the sphere radius

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100 120

(Po

rosi

ty/In

itia

l Po

rosi

ty)*

100

Ductile %

Glauconite

Weathered BasaltBlack Shale

Bentonite

Red Shale

Green Shale

0.8 Rigid R

0.05 Rigid R

39

has been added onto each sphere. The thickness of cement is thus different for different

grain sizes.

Figure 3.13-a) 2D thin section of a dense packing of 1000 spheres, 30% of which are ductile. The ductile grains have a rigid core of radius 0.7R. The porosity in this packing is 22%. b) The same packing with 10% of cement (0.1 of the sphere radius). The porosity in this packing is 19%. c) The same packing as with 20% cement (0.2 of the sphere radius). The porosity in this packing is 7%. d) The same packing as with 30% cement (0.3 of the sphere radius). The porosity in this packing is 3%.

Based on Houseknecht (1984) a large volume of quartz cement appears in

samples of larger mean grain size whereas larger dissolution of quartz occurs in the case

of smaller mean grain size. Fine-grain sandstones have more contact per grain than coarse

grain sandstones. Therefore, higher pore volumes remain in the coarser-grained

sandstones and they have more available surface area for precipitation of cement. Based

40

on Houseknecht (1984), we modeled cementation unequally (based on grain radius) in bi-

dispersed packing of spheres. It means the bigger the grain, the more the cement. This

model of cementation is plausible for porosity reduction in sphere packings based on the

Houseknecht statement.

3.5.Development of framework of rigid grains and cores during compaction

Figure 3.9 shows that the trend of porosity reduction with increasing ductile

content is dependent on rigid core radius of ductile grains, when that radius exceeds the

threshold value of 70% of the grain radius. When the rigid cores are smaller than the

threshold, the porosity trends are independent of the rigid core radius. The reason for this

behavior is the emergence of load-bearing frameworks of rigid grains. The frameworks

may include the rigid cores of ductile grains. Recall that in cooperative rearrangement, a

hard sphere cannot penetrate the rigid core of a ductile grain (Figure 3.2). As the

algorithm proceeds, the hard spheres surrounding a ductile grain can penetrate it to form

a “cage” in which each hard sphere contacts the rigid core of the ductile sphere (Figure

3.14). When a cage forms, no further compaction of the ductile grain is possible. When

enough cages form in the whole packing, a rigid lattice or framework is established and

the spheres can no longer move (The same sequence of events establishes a framework of

point contacts when none of the grains are ductile, occurring at a porosity of about 36%

for mono-disperse spheres). Therefore, the cooperative rearrangement algorithm will stop

and no further compaction is possible.

In our simulations, the emergence of a framework of cages involving the cores of

ductile grains only occurs above a threshold size of the cores, namely 70% of the initial

grain size. It follows that porosity can be preserved during compaction in these packings,

as there is still a large amount of space between the hard spheres. Above the threshold

rigid core size, the hard core of a ductile grain contributes to the formation of cages.

Below the threshold, we find that cages still form, but they do not involve the hard core

of ductile grains. Instead, they result from hard grains penetrating the ductile grain until

they contact each other (Figure 3.14b). Compaction still stops, even though the hard core

41

of the ductile sphere does not touch the hard spheres. This is the reason why below the

threshold, all the packings with different ductile rigid cores have very similar trends in

porosity reduction.

Figure 3.14- a) Schematic of load-bearing frameworks of contacts between hard grains and/or rigid cores of ductile grains. Above a threshold size of the rigid core of a ductile grain (the value of the threshold is 70% of the original grain radius), a “cage” form when the hard spheres have point contact with rigid core of ductile spheres. b) Schematic of load-bearing frameworks of contacts between hard grains and/or rigid cores of ductile grains, below the threshold. The rigid core of ductile grain is too small to contribute to formation of hard cages. The light gray sphere is a ductile sphere with a rigid core, h=hard sphere.

A limitation of our simulation is the behavior when the ductile grains are very

soft. In Figure 3.12, many experimental data fall below the steepest predicted trend of

porosity loss. Broadening the very narrow distribution of sphere sizes in the simulations

does not change the trend. We repeated the simulations with smallest rigid radius (0.05R)

with different ratios of large sphere radius to small sphere radius. Figure 3.15 shows the

result for radius ratios of 1, 1.5, 3 and 5. The lower boundary is almost the same in the

case of 1, 1.5 and 3 radius ratio but for radius ratio of 5 it moves upward. This range of

sphere sizes is still quite narrow compared to natural sediments, and in any case, the grain

size distributions are bi-disperse. We speculate that broader, continuous distributions of

42

grain size may be needed to explain this observation, or that the soft-shell model

implemented here is less applicable for extremely ductile grains.

Figure 3.15- Simulated compaction trends for very soft grains (rigid radius 0.05R) do not depend strongly upon the relative sphere sizes in bi-disperse packings. The spread in the experimental data is larger than the spread in the simulations. 3.6. Role of sorting in porosity reduction

Based on experiments of Beard and Weyl (1973) the porosity decreases as sorting

index increases (Table 3.1). The sorting index used in their paper was calculated based on

Trask sorting coefficient (square root of larger quartile, Q1, of particle to smaller quartile,

Q3 (Scherer 1987)). We calculated the porosity for some packings with different sorting

index and our result shows the same trend as Beard and Weyl (1973) experiment (Table

4.1, Chapter 4). The absolute values of porosity are about 5 percentage points smaller

than the sand pack data. This is consistent with the fact that our packings are maximally

dense and do not account for the effects of angular grains, which hinder settling into

dense packings.

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

(Po

rosi

ty/In

itia

l Po

rosi

ty)*

100

Ductile %

Glauconite

Weathered BasaltBlack Shale

Bentonite

Red Shale

Green Shale

Radius ratio = 1

Radius ratio = 1.5Radius Ratio = 3

Radius ratio = 5

43

Table 3.1. Measurements of wet-packed sand porosity for different grain size distributions (Beard and Weyl, 1973).

Sorting Index Porosity 1.05 0.42 1.15 0.41 1.3 0.39 1.7 0.34 2.35 0.30 4.2 0.27

3.7. Pore throat size distribution

Pore space is often characterized in terms of a network model. The network model

represents pore space as a graph of interconnected sites. The nodes of the graph represent

pore bodies, while the edges connecting nodes indicate the pore throats, which connect

pore bodies. Once the network model has been established it can be used to predict the

permeability of the medium if the geometry of the throats and connectivity of the pores

can be determined a priori (Bryant et al., 1993).

We used Delaunay tessellation to derive the pore throat size distribution of our

packing. Delaunay tessellation finds the nearest neighbor spheres in the packing and

groups them in tetrahedron. It identifies pore bodies and pore throats in the packing. A

pore body is the empty space between four spheres in tetrahedron, and a pore throat is

empty space between three spheres of each face. Therefore, each tetrahedron has four

pore throats and one pore body (Figure 3.16).

44

Figure 3.16- A 3D view of Delaunay cell. The middle point of pore throat (W) in face UVT can be seen in this figure.

Figure 3.17- Finney pack pore throat size distributions (ф = 36%). Their “inscribed radius” characterizes the throats (see text). The x-axis is dimensionless.

We characterize each pore throat by its “inscribed radius.” The inscribed radius is

the radius of the inscribed circle between three grains in each face of Delaunay cell. For

45

reference, Figure 3.17 shows the pore throat size distribution in the Finney pack of equal

spheres (Finney 1968).

Figure 3.18- Comparison between Finney pack with computer generated packing of equal spheres without ductile matter (ф = 36%). The x-axis is dimensionless.

The pore throat size distribution in a mono-dispersed packing of 5000 hard

spheres (no ductile grains) (Figure 3.18) matches very well with the Finney data.

Changing the sphere size distribution from mono-dispersed to bidispersed (radius

ratio=1.5, 50% big spheres, no ductile grains), the peaks move toward the left (more,

smaller pore throats), Figure 3.19. The smaller pore throats correspond to the formation

of groups of the smaller spheres. Then we considered the case of mono-dispersed packing

with 40% ductile grains, each ductile sphere having rigid radius of 0.9 (corresponding to

moderately ductile grains). This shifted the distribution toward the left relative to mono

disperse hard spheres, Figure 3.20. The shift is greater than that produced by adding

small spheres, Figure 3.19. To illustrate how ductility affects the pore throat size

distribution, we used two other packings with the same ductile fraction (40%) but

different rigid radii, Figure 3.21 and Figure 3.22. Decreasing the rigid radius of spheres

46

(making them more ductile) shifts the distribution to smaller sizes. The shift is nonlinear

with the rigid core size. Of particular note is the large fraction of the pore throats that are

completely closed during compaction when the rigid cores are small; see the peak in the

histogram at zero throat size in Figure 3.22.

Figure 3.19- Comparison between Finney pack and computer generated pack with bi-dispersed spheres of 1.5 radius ratio and 50% small spheres. There is no ductile sphere (ф = 33%). The calculated throat radii from bi-dispersed packing were normalized by mean radius of mono-dispersed packing. Therefore, these two packings have the same unit for throat radii (rinscribed). The ratio of the size of small radius in bi-disperse packing to the size of spheres in mono-dispersed packing is equal to 0.77. The x-axis is dimensionless.

47

Figure 3.20- Comparison between Finney pack and mono dispersed spheres with 40% ductile matter and rigid radius of 0.9. Some of the pore throats are closed because of addition of ductile spheres to the packing (ф = 27.5%). The x-axis is dimensionless.

Figure 3.21- Comparison between Finney pack and mono dispersed spheres with 40% ductile matter and rigid radius of 0.8 (ф = 20.7%). It is obvious that by increasing the ductility of spheres (decreasing rigid radius of spheres) we will have more closed pore throats. The x-axis is dimensionless.

48

Figure 3.22- Comparison between Finney pack and mono dispersed spheres with 40% ductile matter and rigid radius of 0.2 (ф = 15.8%). In this case, spheres are very ductile therefore, we have many closed pores (rinscribed = 0), which are clear in the plot. In addition, the pore throat sizes moved toward the smaller pore throat. The x-axis is dimensionless.

In the last step, we add cement to our model rocks. Cement makes the spheres

bigger so ductile spheres will have bigger overlap. Also the hard spheres, which were in

point contact with each other, will have overlap after cement addition Therefore the pore

throat size distribution will move again toward the left (Figure 3.23).

49

Figure 3.23- Comparison between Finney pack and mono dispersed spheres with 24% cement (0.24 of the sphere radius, ф = 4%). We have many closed pores due to cement addition. In addition, the pore throat sizes moved toward the smaller pore throat. The x-axis is dimensionless.

Increasing the ductile matter in the model rocks and increasing the ductility of

grains (decreasing the rigid radius of ductile spheres) cause the porosity and permeability

(as indicated by the pore throat size distributions) to decrease. Like ductile grains in

sandstones, which deform into open pore space, possibly closing pores and their

associated throats, the soft-shell model for ductile grains causes closure of pores and pore

throats (Figure 3.24). A closed pore throat reduces the connectivity of pore space, and

this strongly affects many fluid/rock transport properties (we will discuss this matter in

Chapter 4).

50

Figure 3.24- Bidispersed packings with radius ratio of 1.5 and rigid radius of 0.7. Increasing the fraction of ductile matter in the packing causes the porosity of the packing to decrease and the number of blocked throats in the packing to increase. Thus the connectivity of the pores decreases, which reinforces the decrease in permeability caused by smaller pore throats.

3.8. Effective medium: estimation of permeability

Effective medium theory (EMT) is used to calculate the effective properties of

random fields. In this theory an inhomogeneous medium is replaced by an equivalent

homogenous medium until the fluctuations induced by restoring the heterogeneity

average to zero (Kilpatrick, 1973). The effective medium theory determines a single

value of conductance that gives the same response as a distribution of conductances. With

EMT one can upscale parameters such as permeability and use it in reservoir simulation

studies.

To calculate an effective medium conductance (Fokker, 2001; Hansen and Muller,

1992; Garboczy, 1991), we used a network with tetrahedral lattice, which has

connectivity of 4. Then we derived the conductance of each bond and as a result the

effective medium (gm) of our network. We computed the effective conductance from the

equation below:

020406080100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4

Ductile (%)

Blo

cked

th

roat

s (f

ract

ion

)

Porosity (fraction)

Porosity

Ductile

51

�� � ��� ��� � �� � �

������ � �

(3.2)

where: gm: effective medium conductance g: conductance of each bond (here we assume g = rins

4) z: connectivity of lattice, which is 4 in our case (using tetrahedral lattice) f(g)dg: frequency distribution of bond conductance (throats)

Figure 3.25- Effective conductance (estimation of permeability) of a packing with 1.5 radius ratio and 0.7 rigid radius. By increasing the ductile matter in the packing, the effective conductance decreases. In addition, it decreases by decreasing the porosity of the packing resulting from cementation. Packings with ductile grains more than 75% have very small permeability close to zero.

The presence of ductile grains in our packing leads to closed pore throats, which

have zero conductances. We include these throats in the frequency distribution in order to

quantify the effect of the reduced connectivity on gm. By increasing the amount of ductile

grains, we increase the number of zero conductance values or decrease the connectivity.

This reduces the effective conductance of our packing. Figure 3.25 show the effective

conductance of a bi-dispersed packing with radius ratio of 1.5 and rigid radius of 0.7. It

shows that by increasing the ductile matter in the medium the porosity of the medium

decreases and the number of closed pores (Figure 3.24) increases. Increasing, the number

020406080

1.00E-06

1.00E-05

1.00E-04

1.00E-03

0 0.1 0.2 0.3

Ductile (%)

Eff

ecti

ve c

on

du

ctan

ce

Porosity (fraction)

Porosity

Ductile

52

of closed pores causes the effective conductance and therefore the permeability of the

packing to decrease.

To compare the calculated effective conductance of each packing with

experimental data we need to calculate permeability from equation below:

� � �� ���

(3.3)

where K is permeability (md), gm is the effective conductance values and N/A is number

of throats per unit cross section. Figure 3.26 shows the effective medium conductance for

mono-dispersed packing with different cementation and Fontainebleau sandstone

permeability scaled by a constant (data courtesy from Bryant and Blunt, 1992). The

trends are similar, as expected since N/A is constant as cement is added to model.

Figure 3.26- Comparison between effective conductances predicted from mono-dispersed packing with different cementation and scaled permeability data from Fontainebleau sandstone (data courtesy from Bryant and Blunt, 1992).

0.01

0.1

1

10

100

1000

10000

100000

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Sca

led

eff

ecti

ve c

on

du

ctan

ce (

arb

itra

ry u

nit

s)

Porosity

prediction

Fontainebleau sandstone

53

3.9. Permeability

The permeability of the medium was also directly determined from the network

flow calculation in the sphere packing. The detailed procedure can be found in Bryant

and Blunt (1992). The calculated permeability for a packing for mono-dispersed sphere

pack with different addition of cement were compared with permeability data from

Fontainebleau sandstone (Figure 3.27). The predicted values of permeability are in good

agreement with real data and predicted permeability values of Bryant and Blunt (1992).

The only difference in two predictions is that Bryant and Blunt used reffective

(rinscribed+requivalent/2), which is the mean value of equivalent and inscribed radii in their

conductance calculation (equivalent radius is the radius of a circle whose area is the same

as the minimum cross-section of the pore throat). We only used the inscribed radius (rins)

in our conductance calculation. This is the reason for difference between our calculation

and Bryant and Blunt prediction.

Figure 3.27- Comparison between permeability calculated for mono-dispersed packing with different cementation and Fontainebleau sandstone. The dashed line is the predicted permeability values from Bryant and Blunt, 1992. The y-axis has the unit of md.

0.01

0.1

1

10

100

1000

10000

100000

0 0.1 0.2 0.3 0.4

Per

mea

bili

ty (

md

)

Porosity fraction

Fontainebleau sandstone

prediction (this work)

Bryant_Blunt 1992

54

3.10.Summary

In this Chapter, the compaction modeling of mixture of ductile and hard grains

was explained using soft shell model. A compaction test was done in PFC2D software on

a mixture of ductile and hard grains. An initial hydrostatic pressure was applied to all

four walls of the packing for simulating the hydrostatic pressure in subsurface rocks. In

addition, a one-dimensional axial load was applied as an overburden pressure to the

packing. The load finds a path through the hard grains after deformation of ductile grains.

By increasing the load, the ability of soft grains to support the load decreases, and the

load transfers to the adjacent hard grains. This makes a cage of hard grains or hard grains

and rigid core of ductile grains, with point contact to each other. When enough cages

form in the whole packing, a rigid lattice or framework is established and compaction

stops.

A large set of computer generated sphere packings (using cooperative

rearrangement algorithm) was used to model porosity loss during compaction of

sediments containing ductile grains. To model a ductile grain we used a sphere with a

hard core and soft (penetrable) shell. Our model yields good agreement with

hydrothermal compaction experiments reported in the literature, when the radius of the

hard core in the ductile grains is chosen in an appropriate range. With different, mutually

exclusive ranges for the hard-core radius, this simple geometric model correctly predicts

trends in porosity during compaction for brittle, moderately ductile, and very soft lithic

grains. The model shows the emergence of “cages” of contacts between hard grains

and/or rigid cores of ductile grains. These cages form a load-bearing framework around

ductiles, prevent further porosity loss during burial. Thus the model correctly predicts the

preservation of porosity even in sediments containing 95% ductile matter. The model also

quantifies the change in pore throat size distribution resulting from deformation of ductile

grains and from overgrowth cement. It also identifies the closure of pore throats. The

closed pore throats give a direct measure of the reduction of connectivity of pore space,

and reduction of the effective conductance, which is a good proxy of permeability

reduction in tight gas sands.

55

The actual effective permeability was calculated from the network model of pore

space in the modeled rocks. There is good agreement between predicted permeability

from our model and Fontainebleau sandstone. The predicted result is similar to predicted

effective permeability from Bryant and Blunt, (1992).

56

Chapter 4: Simulation of drainage in tight gas sands

4.1.Introduction

Tight gas sandstones are characterized by very high capillary pressure. Accurate

description of reservoir parameters such as capillary pressure data is required in effective

exploitation of tight gas reservoirs especially for calculation of vertical water saturation

distribution and resource in place (Newsham et al., 2003). The goal of this chapter is to

examine the effect of diagenesis processes on drainage capillary curves in tight gas

sandstones. In this chapter, we first explain the drainage model calculations briefly. Then

we model the effect of ductile compaction and cementation on drainage curves in tight

gas samples. Connectivity of pore space and the bond percolation threshold of the model

rock are two concepts important in interpreting the capillary pressure curves. The

connectivity is the number of open throats in each pore. Percolation threshold is the

critical fraction of lattice points that must be filled to create a continuous path of nearest

neighbors from one side to another. We derived the connectivity of pore space and bond

percolation threshold and their changes with diagenetic process in the modeled tight gas

sandstones in this chapter. Finally, the modeled drainage capillary pressure curves were

compared with experimental data from tight gas samples for validation.

4.2.Drainage calculations

Recall that to model pore space in tight gas sands we used a packing of spheres

with ductile grains and quartz cementation similar to one we modeled for diagenesis in

Chapter 3 with mixture of hard and ductile grains. This packing data is an input for a

network modeling of drainage in tight gas sands. To calculate drainage we used methods

of Behseresht (2008) and his code in Matlab software. I explain his methods briefly here.

To obtain a network of pores and throats from a grain scale model we need a tool

to change the grain scale data (radius and center of spheres) to void space of bonds and

sites in a network model (bonds represent pore throats and sites represent pore spaces of a

real rock). The tool we apply is Delaunay tessellation. Delaunay tessellation finds the

57

nearest neighbor spheres and joins the centers of those spheres in a tetrahedron. We

tessellate our packing with the Delaunay tessellation tool in Matlab®. Each tetrahedron

has four spheres with void space inside which we call pore space or porosity. Each face

of this tetrahedron has three spheres with an open space in between which we call pore

throat. Therefore, each tetrahedron has one pore and four pore throats (Figure 4.1). When

all of pores are open, the connectivity is four.

Figure 4.1- Two views of Delaunay tessellation cell a) with four grains b) and only part of grains inside the cell.

Figure 4.2- One face of Delaunay tessellation (throat) with inscribed radius shown by “rins”.

We quantify the pore space between spheres by

between four spheres in each Delaunay cell

inscribed between three spheres in each face of Delaunay cell. The radius of sphere

throat is called inscribed radius, which has

computer generated packings (

In drainage modeling we simulate the capillary displacement of wetting phase by a non

wetting phase in our network model of porous media. We quantify the pores and throats

as described above and extract the network of sites (pores) and bonds (throats) (

4.3).

Figure 4.3- Network of sites and bonds. Sites are pores which are shown as dots and bonds are throats shown as lines connecting two pores.

We need the critical drainage

network. This critical drainage

58

We quantify the pore space between spheres by considering a sphere inscribed

between four spheres in each Delaunay cell. We quantify a pore throat by a sphere

inscribed between three spheres in each face of Delaunay cell. The radius of sphere

throat is called inscribed radius, which has a crucial role in simulation of flow in the

computer generated packings (Figure 4.2).

In drainage modeling we simulate the capillary displacement of wetting phase by a non

wetting phase in our network model of porous media. We quantify the pores and throats

escribed above and extract the network of sites (pores) and bonds (throats) (

Network of sites and bonds. Sites are pores which are shown as dots and bonds are throats shown as lines connecting two pores. The image courtesy from Behseresht (2009).

We need the critical drainage curvature of the throats to simulate drainage

. This critical drainage curvature is the minimum curvature for non-wetting phase

considering a sphere inscribed

pore throat by a sphere

inscribed between three spheres in each face of Delaunay cell. The radius of sphere in the

al role in simulation of flow in the

In drainage modeling we simulate the capillary displacement of wetting phase by a non-

wetting phase in our network model of porous media. We quantify the pores and throats

escribed above and extract the network of sites (pores) and bonds (throats) (Figure

Network of sites and bonds. Sites are pores which are shown as dots and bonds are throats

of the throats to simulate drainage in this

wetting phase

59

to pass through a throat. To estimate this drainage curvature we used the Haines criterion

(insphere approximation) (Haines, 1930) with Mason and Mellor (1995) modification.

Mason and Mellor subtract 1.6 from Haines criterion:

� � ��������� � . � (4.1)

where rinscribed is the radius of inscribed sphere between three spheres in each face of

Delaunay cell (Figure 4.2).

Following Behseresht (2008), we considered pendular rings and liquid bridges

when we determine the connectivity of wetting phase in drainage calculation. Pendular

ring is a volume of wetting phase between two grains that are in point contact. Liquid

bridges are bridges of wetting phase between two grains having a small gap between

them. Liquid bridges disappear when the drainage exceeds the threshold drainage for that

specific throat whereas pendular rings are always there regardless of drainage

(Behseresht, 2008; Gladkikh and Bryant 2005; Fisher, 1926)

We used drainage code made by Behseresht (2008), which uses percolation theory

to determine the trapping of wetting phase. We use Mason and Mellor’s (1995) method to

model drainage as invasion percolation. The method amounts to a set of rules to

determine which candidates for drainage are invaded by nonwetting phase. In this rule a

pore with wetting phase and at least one neighbor pore empty of wetting phase is a

candidate for drainage. This candidate will be drained when the applied curvature

exceeds the critical curvature of the throats connecting this pore and its neighbor and

when there is a connected path of wetting phase through this pore to the outlet.

The simulation starts from few pores or all pores on the one face of the packing

(in our case) in contact with non-wetting phase. The applied curvature is increased as

small increments. In each step of increment all of the pores in the path from source,

which have critical curvature less than or equal to the applied curvature, will be invaded

until no more candidates can be found. The drainage will be completed when all of the

open pores have been drained. The simulated drainage curves here show dimensionless

capillary pressure, which is defined as below:

60

� � �� � ������

(4.2)

where C is drainage curvature, Pc is drainage capillary pressure, Rmean is average grain

radius before cement or compaction, and σ is the interfacial tension between gas and

water (or wetting and non-wetting fluids used in experimental data).

4.2.1. Effect of different grain size distribution on drainage curves

We ran the drainage code for different grain size distribution to see how the

capillary pressure (drainage) curve changes by changing the grain distribution and

sorting. The sorting index that we used was calculated based on Behseresht (2008) as

below:

�� � ������� (4.3)

where d75 is the grain size that is larger than 75% of all grains and d25 is the grain size

larger than 25% of all grains. The index can be correlated as below:

Extremely well sorted: 1.0 ≤ sorting index ≤ 1.1 Very well sorted: 1.1 ≤ sorting index ≤ 1.2 Well sorted: 1.2 ≤ sorting index ≤ 1.4

Moderately sorted: 1.4 ≤ sorting index ≤ 2.0

61

Table 4.1. Summary of Properties of Model Sediments

Grain sizes Porosity (fraction)

Sorting Index distribution

Packing No.

Minimum Radius

Maximum radius

Mean radius

Standard deviation

Number fraction

basis

Volume or weight fraction

basis

1 0.32 2.58 2.18 0.11 0.37 1.04 1.04 Log-normal distribution

2 1.47 3.06 2.16 0.22 0.37 1.07 1.07 Log-normal distribution

3 0.93 4.64 2.12 0.43 0.34 1.14 1.14 Log-normal distribution

4 3.44E-03 6.44 1.90 0.79 0.35 1.31 1.31 Log-normal distribution

5 2.39E-03 10.16 1.31 1.17 0.25 1.72 1.54 Log-normal distribution;

truncated

6 1.97 2.40 2.19 0.07 0.34 1.02 1.02 Normal

distribution

7 2.19 2.19 2.19 - 0.36 - - Mono-

dispersed 8 1.69 2.54 2.12 - 0.35 - - Bi-dispersed

The packings have 5000 spheres with different distributions as mono-dispersed,

bi-dispersed, normal and lognormal distribution with different mean and standard

deviation to see how drainage differs with different distribution and sorting index. The

packings have no cement so they are model sediments. The table above shows the

summary of used packings (all of the packings except the last two are from Behseresht

2008).

The bi-dispersed packing is a packing with 50% large spheres and the radius ratio of

large grains to small grains is 1.5 (meaning the large grains are 1.5 bigger than the

smaller grains). The grain size distribution and drainage curve for packings of Table 4.1

are shown in Figure 4.4 and Figure 4.5 respectively.

Figure 4.4- Grain size distribution for packings of T

62

ution for packings of Table 4.1.

63

Figure 4.5- Simulated drainage curves (plot of curvature versus volume fraction of wetting phase) for packings of Table 4.1. Curvature is the dimensionless capillary pressure, Eq. 4.2.

Figure 4.6- Part of the sphere packing No. 5. The image courtesy from Behseresht (2008).

64

The drainage plot shows that grain size distribution does not change the drainage curve

except when we have a much-skewed distribution with high standard deviation (Packing

No. 5). In the case of having big difference in size between grains (Figure 4.6), the pores

and throats in the big grain area are big and the drainage can find a path through big pores

and never invade the small throats and pores. Therefore the drainage capillary pressure

for that big path (connected path) (Figure 4.7) of big pores and throats is lower than when

we have mono-disperse spheres.

Figure 4.7- Connected path for packing No. 5. This shows the connected throats during drainage for that packing from one side of the packing to the other side (similar to break through). The curvature (dimensionless capillary pressure) was 0.3 at which this path was formed.

4.3.Drainage simulations in tight gas sandstones

The drainage curve was made for different models of rock in the previous section.

To model tight gas rocks we first construct the tight gas sand packs by inserting ductile

grains and cement to the mono dispersed packing. Ductile grains have 0.7 rigid radius as

discussed in Chapter 4 and the amount of ductile grain in the tight gas sands was chosen

65

to be 30% ductile (most tight gas sands have close to 30% ductile grains). We simulated

drainage in two kinds of different packings. In one set, we change the rigid radius of

ductile grains and hold a fixed amount of ductile matter in the packing. In the other, we

fix the rigid radius and change the amount of ductile matter in the packing.

Figure 4.8 shows the drainage (capillary pressure) curves for mono dispersed

packing with fixed amount of ductile grains and different rigid radius for ductile grains.

By decreasing the rigid core of ductile grains, we increase the ductility of grains and

therefore increase the pressing of ductile grains to other grains. The result of this process

is that we increase the amount of blocked throats in the packing, and thereby increase the

percolation threshold and decrease the connectivity of pore space (which is explained in

the next sections). This fact leads to shifting the curves toward larger irreducible water

saturation. Decreasing the rigid radius also increases the drainage capillary pressure

slightly but the increase in the irreducible water saturation is more obvious. There are

some exceptions. The packing with 0.9 rigid radii has higher irreducible water saturation

than packing with 0.8 rigid radius and also the packing with 0.2 rigid radius has higher

irreducible water saturation than packing with 0.1 rigid radius. This is explained in the

next sections.

Figure 4.9 shows the drainage curves for packings with fixed rigid radii of ductile grains

and different amount of ductile matter in the packing. Increasing the percentage of ductile

grains in the packing causes the irreducible water saturation and drainage curves both to

increase but the rate of increasing is greater than previous case (changing the rigid radius

and fixing the percentage of ductile grains, Figure 4.8). This means that increasing the

amount of ductile matter in the packing has more effect on reducing the connectivity of

pore space and therefore has greater effect on the drainage curves. We still have one

exception in the trend of irreducible water saturation increasing which is explained in the

next sections.

66

Figure 4.8- Drainage capillary pressure curves for mono dispersed packing with fixed ductile amount (40%) and different rigid radius. Curvature is the dimensionless capillary pressure.

Figure 4.9- Drainage curves for mono dispersed packing with fixed rigid radii (0.7) and different amount of ductile matter in the packing. Curvature is the dimensionless capillary pressure.

67

Next, we added only cement to mono dispersed packing and changed the amount

of cement to see the effect of cement on the drainage curves. The cement was added

uniformly to all of grains after the mono dispersed packing was made. This means we

hold the center of grains fixed and then increase the radius of grains uniformly by certain

amount, which is a fraction of radius of grain as explained in the cementation section in

Chapter 3. Figure 4.10 shows the drainage curves for mono dispersed packing with

different cement amount. Increasing the amount of cement causes the drainage and the

irreducible water saturation to increase rapidly but the rate of increasing the drainage is

higher than when we have addition of ductile matter (Figure 4.8 and Figure 4.9). To

explain this fact we have to look at the pore scale in these different packings in the next

section.

Figure 4.10- Drainage curves for mono-dispersed packing with different cement amount (different % of the sphere radius). Curvature is the dimensionless capillary pressure.

Figure 4.11 shows the drainage curves for mono dispersed packing with 0.7 rigid

radius of ductile grains, 40% ductile matter in the packing and different cementation.

This is an example of a tight gas sands. In these packings both effects of ductile addition

68

(increasing the irreducible water saturation) and cement (increasing the drainage) come

into account. By addition of 22% cement we have no percolation (the drainage curve

shifts toward the right side of the plot) while in the case of only cement addition (Figure

4.10), by adding 24% cement, we do not reach to that point. It means having ductile

grains and cement together make the packing much tighter than when we have only one

of those processes. By looking at the connectivity and percolation threshold in the next

section, we explain this matter in more detail.

Figure 4.11- Drainage curves for mono dispersed packing with 40% ductile matter, 0.7 rigid radius of ductile grains and different cementation (different % of the sphere radius). Curvature is the dimensionless capillary pressure.

4.4.Average connectivity of pore space

Delaunay tessellation was used to derive the pore throat size distribution of the

packing. Average connectivity of the pore space is the number of the open throats in each

pore in the packing. Cementation and ductile deformation will close some of the pore

space and related pore throats in rocks. Therefore, the average connectivity should be

below four for packings containing ductile grains and cement. Average connectivity of

69

pore space is equal to four for a mono dispersed packing of spheres such as Finney’s

packing with 36% porosity.

Table 2 and Figure 4.12 show the connectivity of pore space for different lattices.

Mono dispersed packing has all pores with four open throats in Figure 4.13. By adding

different amount of cement to this packing, the number of pores with four open throats

decreases and the number of pores with three, two and one open throats increases. We

considered throats with inscribed radii twenty orders of magnetude (10-20) smaller than

average throat radius to be closed.

Table 4.2- The bond-percolation threshold and connectivity of pore space (degree) for different lattices (Dean and Bird, 1967).

Lattice Degree Bond threshold

hexagonal 3 0.65 square 4 0.5

tetrahedral 4 0.39 triangular 6 0.347

simple cubic 6 0.257

face-centered cubic 12 0.125

Voronoi 16.27 0.0822

Figure 4.12-The bond-percolation threshold for different lattices.

3

4

46

6

1216.27

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20

Bo

nd

per

cola

tio

n t

hre

sho

ld

Connectivity of the Lattice

Figure 4.13- Fraction of pores with different blocked fraction of pores with 4 open throats gets smaller and fraction of pores with at least one blocked throats increases. The percentage of cement means different percentage sphere radius.

Figure 4.14 shows the average connectivity of pore space for mono dispersed

packing when we add different cementation to this packing.

the mono dispersed packing, the connectivity of pore space will be reduced

adding 5 to 10% cement to the mono dispersed packing, the connectivity of pore space

does not change which means there is no closed throats in those packings (

The cement is not thick enough to close any throats. Addition of more ceme

will reduce the connectivity of pore space to be less than four.

axis in this figure shows the fraction of blocked throats for the mentioned packings. It is

clear that by addition of cement, the number of blocked throat

of the packing decreases (values on top of the data)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Four blocked throats

Three blocked throats

Fra

ctio

n o

f p

ore

s

70

Fraction of pores with different blocked throats status. By adding cement to the packing, the fraction of pores with 4 open throats gets smaller and fraction of pores with at least one blocked throats

The percentage of cement means different percentage of the sphere radius was added to

shows the average connectivity of pore space for mono dispersed

packing when we add different cementation to this packing. By adding more cement to

the mono dispersed packing, the connectivity of pore space will be reduced

adding 5 to 10% cement to the mono dispersed packing, the connectivity of pore space

t change which means there is no closed throats in those packings (

The cement is not thick enough to close any throats. Addition of more cement, 15 to 24%

will reduce the connectivity of pore space to be less than four. The secondary vertical

the fraction of blocked throats for the mentioned packings. It is

clear that by addition of cement, the number of blocked throats increases and the porosity

(values on top of the data).

Three blocked throats

Two blocked throats

One blocked throats

Four open

throats

mono_dis

mono-dis-

mono-dis-

mono-dis-

mono-dis-

mono-dis-

mono-dis-

mono-dis-

mono-dis-

throats status. By adding cement to the packing, the fraction of pores with 4 open throats gets smaller and fraction of pores with at least one blocked throats

was added to the

shows the average connectivity of pore space for mono dispersed

y adding more cement to

the mono dispersed packing, the connectivity of pore space will be reduced more. By

adding 5 to 10% cement to the mono dispersed packing, the connectivity of pore space

t change which means there is no closed throats in those packings (Figure 4.13).

nt, 15 to 24%

The secondary vertical

the fraction of blocked throats for the mentioned packings. It is

s increases and the porosity

mono_dis- no cement

-5%cement

-10%cement

-15%cement

-20%cement

-21%cement

-22%cement

-23%cement

-24%cement

71

Figure 4.14- Average connectivity for mono dispersed packing with different amount of cement. By addition of cement, the connectivity decreases from four to 3.2. The second vertical axis shows the fraction of blocked throats versus cementation. By adding more cement, we close more throats. The values with percentage on top of the data show the porosity percentage for each packing. The percentage of cement means different percentage of the sphere radius was added to the sphere radius.

Addition of ductile grains alone (without any cement content) to the mono

dispersed packing will reduce the connectivity of pore space differently. There are two

ways to add ductile grains to the packing, either to keep the rigid radius of ductile sphere

constant and increase the ductile matter in the packing, or keep the ductile matter

constant and decrease the rigid radius of ductile grains in the packing. By decreasing the

rigid radius of ductile grains, we increase the ductility of the grains. Therefore, they can

press more into other grains (either hard grains or other ductile grains) and close some of

the throats.

To make a packing with ductile grains we first fixed the amount of ductile matter

in the packing and used different rigid radii to see how the drainage curves (Figure 4.8)

and connectivity of pore space change (Figure 4.15). To do this we used a mono

dispersed packing with fixed 40% ductile matter (most of tight gas sands containing lithic

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

0 5 10 15 20 25

Blo

cked

th

roat

s

Ave

rag

e co

nn

ecti

vity

Cement (% of grain radius)

Average connectivity

Blocked throats

36%

4%

4.7%5.3%

6%

7%

11.5%

18%26%

72

grains have around 30 to 40% ductile grains based on study of Dutton et al., (1993) and

different rigid radii for ductile grains. By decreasing the rigid radius of ductile grains, the

connectivity of pore space decreases because some of the throats will be closed by

pressing ductile grains into other grains. Small rigid radius means the grains are very soft

so they can be deformed into the pore space and block the throats easily. High rigid

radius (packings with 0.9 to 1 rigid radius) means grains are rigid similar to quartz and

feldspar, so they do not deform easily in the pore space. The connectivity is almost the

same for packings with 0.3 to 0.6 rigid radii. The secondary vertical axis in the same

figure shows the fraction of blocked throats versus rigid radius. It is obvious that

packings with rigid radius from 0.3 to 0.6 have similar blocked throats fraction. This is

the reason for similarity in connectivity for those packings. Although they have different

pore throat closure status (Figure 4.16), having the same fraction of blocked throats

makes no difference in packing’s properties. The porosity values of those packings

confirm that all of those packings have similar porosity (Figure 4.15). This figure shows

that packings with rigid radius below 0.6 have the same porosity but blocked throats are

different for packing with 0.3, 0.2 and 0.1 rigid radius. Only packings with 0.4, 0.5 and

0.6 rigid radii have similar blocked throats. We expect these to have similar connectivity

too.

73

Figure 4.15- Connectivity of pore space for a mono dispersed packing with fixed 40% ductile matter and different rigid radius for ductile grains. By decreasing the rigid radius, the connectivity of pore space reduces. The secondary axis shows the blocked throats versus rigid radius. It is clear that packings with 0.3 to 0.6 rigid radius have the similar blocked throats. The values with % on top of the data shows the porosity percentage for each packing.

To study the effect of ductile grains on porosity reduction we perform one more test. This

time we fixed the rigid radius of ductile grains (use ductile grains with the same ductility)

and changed the amount of ductile grains in the packing each time (Figure 4.17).

Therefore, we used packings with fix 0.7 rigid radius, which is ductile grain with

moderate ductility (Mousavi and Bryant, 2007). By increasing the amount of ductile

matter in the packing the connectivity of pore space decreases due to closure of some

throats (Figure 4.17). Figure 4.18 shows the status of pores by number of blocked throats.

By increasing the amount of ductile grains in the packing the fraction of pores with four

open throats decreases and the fraction of throats with at least one blocked throats

increases. This figure is different from Figure 4.15 that had fixed amount of ductile grains

and different rigid radius. In fact changing the amount of ductile grain is a different

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

0.1 0.3 0.5 0.7 0.9

Blo

cked

th

roat

s

Ave

rag

e co

nn

ecti

vity

Rigid radius

Average connectivity

Blocked throats

14.9%

27.5%

20.7%

17.3%

15.4%15.3%

15.2%

15.1%

15%

35.7%

process than changing the ductility (rigid

grains. Therefore, we expect to have different status of blocked throats. It seems by

increasing the amount of ductile grains the number of pores with two blocked throats

increases more rapidly than other pore status. In the case of changing the

grains, the fraction of pores with at least one blocked throats increases more rapidly than

the other pores. In this case, the fraction of blocked throats is almost similar for packing

with 0.4, 0.5 and 0.6 rigid radi

changing the percentage of ductile matter.

Figure 4.16- Fraction of pores with different blocked throats status. By changing the rigid radius from 0.9 to 0.1, the fraction of pores with 4 open throats gets smaller and fthroats increases.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

four closed throats

three closed throats

Fra

ctio

n o

f p

ore

s

74

process than changing the ductility (rigid radius) of grains by fixing the amount

grains. Therefore, we expect to have different status of blocked throats. It seems by

increasing the amount of ductile grains the number of pores with two blocked throats

increases more rapidly than other pore status. In the case of changing the

grains, the fraction of pores with at least one blocked throats increases more rapidly than

the other pores. In this case, the fraction of blocked throats is almost similar for packing

radius whereas we do not have similar situation in the case of

changing the percentage of ductile matter.

Fraction of pores with different blocked throats status. By changing the rigid radius from 0.9 to 0.1, the fraction of pores with 4 open throats gets smaller and fraction of pores with at least one blocked

two closed throats

one closed throats

four open throats

40% duc_0.9rr40% duc_0.8rr40% duc_0.7rr40% duc_0.6rr40% duc_0.5rr

) of grains by fixing the amount of ductile

grains. Therefore, we expect to have different status of blocked throats. It seems by

increasing the amount of ductile grains the number of pores with two blocked throats

increases more rapidly than other pore status. In the case of changing the ductility of

grains, the fraction of pores with at least one blocked throats increases more rapidly than

the other pores. In this case, the fraction of blocked throats is almost similar for packing

similar situation in the case of

Fraction of pores with different blocked throats status. By changing the rigid radius from 0.9 raction of pores with at least one blocked

75

Figure 4.17- Connectivity of pore space for a packing with fixed 0.7 rigid radius of ductile grains and different percentage of ductile matter in the whole packing. By increasing the ductile grains in the packing, the connectivity of pore space decreases. The secondary vertical axis shows the fraction of blocked throats. The values with % on top of the data shows the porosity percentage for each packing.

Figure 4.17 also shows the porosity percentage of packings with different percentage of

ductile grains. Because the rigid radius is fixed, there is no ranged ductile fraction

showing similar porosity values for these packings. As we said above, the rigid radius has

contribution in formation of cages and load bearing formations, which cause similar

porosity value for packings with different rigid radius of grains.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Blo

cked

th

roat

s

Ave

rag

e co

nn

ecti

vity

Ductile (fraction)

Average connectivity

Blocked throats

36%

0.26%

1.5%

4.3%

8%

12%

17%22%

27%

31.5%

Figure 4.18- Fraction of pores with different blocked throats status. By changing the amount of ductile grain from 0.1 to 0.9, the fraction of pores with least one blocked throat increases.

4.5.Average connectivity of

We also calculated the average connectivity of pore space for tight gas samples

with 40 percent ductile grains (0.7 rigid

(Figure 4.19). The 40 percent ductile matter is a

tight gas sandstones. We vary

ductility of the grain. The 0.7 rigid

fragments (Mousavi and Bryant

cement in the packing, the average connectivity of the pore space in the packing

decreases more rapidly than when we have only ductile grains. These results show that by

adding cement and ductile grains to the p

pore throats therefore the average connectivity of the whole packing will be decreased.

This can be also seen from blocked throats and porosity

cement (Figure 4.19) which shows by inc

blocked throats increases and porosity decreases.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

four closed throats

three closed throats

Fra

ctio

n o

f p

ore

s

76

on of pores with different blocked throats status. By changing the amount of ductile , the fraction of pores with four open throats gets smaller and fraction of pores w

Average connectivity of pore space for tight gas samples

We also calculated the average connectivity of pore space for tight gas samples

with 40 percent ductile grains (0.7 rigid radius) and different amount of cementation

The 40 percent ductile matter is a reasonable amount of ductile grains for

vary the rigid radius of each ductile grain to account for

ductility of the grain. The 0.7 rigid radius is a good proxy for ductile grains such as lithic

fragments (Mousavi and Bryant, 2007). We can see that by increasing the amount of

cement in the packing, the average connectivity of the pore space in the packing

decreases more rapidly than when we have only ductile grains. These results show that by

adding cement and ductile grains to the packing, we close more pore space and related

pore throats therefore the average connectivity of the whole packing will be decreased.

This can be also seen from blocked throats and porosity values versus percentage of

) which shows by increasing the amount of cement, the fraction of

blocked throats increases and porosity decreases. Figure 4.20 shows the pores status with

three closed throats

two closed throats

one closed throats

four open throats

0.1duc_0.7rr0.2duc_0.7rr0.3duc_0.7rr0.4duc_0.7rr0.5duc_0.7rr0.6duc_0.7rr0.7duc_0.7rr0.8duc_0.7rr0.9duc_0.7rr

on of pores with different blocked throats status. By changing the amount of ductile open throats gets smaller and fraction of pores with at

We also calculated the average connectivity of pore space for tight gas samples

) and different amount of cementation

amount of ductile grains for

the rigid radius of each ductile grain to account for

is a good proxy for ductile grains such as lithic

We can see that by increasing the amount of

cement in the packing, the average connectivity of the pore space in the packing

decreases more rapidly than when we have only ductile grains. These results show that by

acking, we close more pore space and related

pore throats therefore the average connectivity of the whole packing will be decreased.

versus percentage of

reasing the amount of cement, the fraction of

the pores status with

0.1duc_0.7rr0.2duc_0.7rr0.3duc_0.7rr0.4duc_0.7rr0.5duc_0.7rr0.6duc_0.7rr0.7duc_0.7rr0.8duc_0.7rr0.9duc_0.7rr

77

different blocked throats. By adding more cement from 3% to 22%, the fraction of pores

with four open throats decreases and fraction of other pores increases especially pores

with two and three open throats. From this plot, it is obvious that fraction of pores with

four open throats are much smaller than when we have only cement or only ductile grains

but the status shape is almost similar to packing with different ductile matter. It means the

pores with at least two blocked throats are increasing more rapidly than the other pores

similar to packing with different ductile percentage.

Figure 4.19- Average connectivity of pore space for packing of 40 percent ductile grains (0.7 rigid radii) and different amount of cement. By increasing the amount of cement, the connectivity of pore space decreases. The secondary vertical axis shows the fraction of blocked throats versus cement. The values with % on top of the data shows the porosity percentage for each packing. The percentage of cement means different percentage of the sphere radius was added to the sphere radius.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

2.5

2.7

2.9

3.1

3.3

3.5

3.7

0 5 10 15 20 25

Blo

cked

th

roat

s

Ave

rag

e co

nn

ecti

vity

Cement (% of grain radius)

Average connectivity

Blocked throats

17.3%

3.6%

6.7%

9%

11%13.5%

1.3%

Figure 4.20- Fraction of pores for mono dispersed packing cementation. The percentage of cement means different percentage sphere radius.

Tight gas sands have many close

grains and cementation. These results show that these rocks should have a very poor

connectivity. This connectivity controls the flow properties of these tight reservoirs.

4.6.Percolation threshold for tight gas sand models

Percolation threshold is the critical frac

create a continuous path of nearest neighbors from one side to another. In the other word,

percolation threshold of wetting and non

the pore space. The fluid can b

clusters in the pore space.

To calculate the bond percolation threshold, first we need the capillary pressure

curve (or drainage) to get the drainage at 50% saturation of wetting phase for each

packing. We choose 50% saturation because in this point half of the packing was drained

and it is reasonable to expect a

from one side to another. If we have percolation,

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

four closed throats throats

Fra

ctio

n o

f p

ore

s

78

raction of pores for mono dispersed packing with 40% ductile 0.7 rigid radius and different The percentage of cement means different percentage of the sphere radius was added to the

Tight gas sands have many closed pores and throats due to compaction of ductile

d cementation. These results show that these rocks should have a very poor

connectivity. This connectivity controls the flow properties of these tight reservoirs.

Percolation threshold for tight gas sand models

Percolation threshold is the critical fraction of lattice points that must be filled to

create a continuous path of nearest neighbors from one side to another. In the other word,

wetting and non-wetting fluids depend on fluid connectivity in

the pore space. The fluid can be totally connected or broken into independent drop

To calculate the bond percolation threshold, first we need the capillary pressure

curve (or drainage) to get the drainage at 50% saturation of wetting phase for each

ng. We choose 50% saturation because in this point half of the packing was drained

reasonable to expect a continuous path of non-wetting phase (breakthrough)

from one side to another. If we have percolation, the gas phase permeability

three closed throats

two closed throats

one closed throats

four open throats

no cement3% cement5% cement7% cement10%cement15% cement22% cement

with 40% ductile 0.7 rigid radius and different was added to the

compaction of ductile

d cementation. These results show that these rocks should have a very poor

connectivity. This connectivity controls the flow properties of these tight reservoirs.

tion of lattice points that must be filled to

create a continuous path of nearest neighbors from one side to another. In the other word,

wetting fluids depend on fluid connectivity in

e totally connected or broken into independent drops or

To calculate the bond percolation threshold, first we need the capillary pressure

curve (or drainage) to get the drainage at 50% saturation of wetting phase for each

ng. We choose 50% saturation because in this point half of the packing was drained

(breakthrough)

the gas phase permeability in this

no cement3% cement5% cement7% cement10%cement15% cement22% cement

79

packing (which is equivalent to a specific rock sample) should be significant. If there is

no percolation at 50% wetting phase saturation, it may not be economical to produce

from that reservoir. Choosing the exact point to calculate the percolation is depend upon

the economics, oil and gas price, production price and other prices for extracting from the

rock. By calculating percolation threshold for each rock and knowledge of different

production prices, we can define if that specific rock is reservoir or not.

When we have simulated drainage to 50% saturation (Figure 4.21), we then

calculate the inscribed radius equivalent to the curvature needed to reach that saturation

from equation given below (Haines criteria):

������� � ! ��"�#$%"� & . �' � ($� (4.4)

where:

Rinscribed = inscribed radius

Rmean = mean radius of hard grains

Curvature = curvature at 50% saturation

We use mean radius of hard grains to make the inscribed radius dimensional because the

curvature is dimensionless (Eq. 4.2). The next step is to plot the inscribed radius

histogram and cumulative frequency distribution to find the calculated inscribed radius

on that plot. This provides a convenient estimate of how much of the throats (throats

frequency) have been drained (Figure 4.22). That is, if all throats had access to non-

wetting phase then all throats larger than the calculated inscribed radius would drain.

Thus, one minus the cumulative throat frequency at the calculated inscribed radius is an

estimate of the bond percolation threshold for that packing.

Figure 4.21- Drainage curve for monofigure to estimate bond percolation threshold for this pressure.

Figure 4.23 shows the bond percolation threshold for mono dispersed packing with

different cementation. When the packing does not have any cement and all of the throats

are open, the bond percolation of t

be drained to have continuous path from one side to another. Adding cement reduces the

open throats (increases blocked throats) in the packing and increases the percolation

threshold. Addition of 5 to 10% cement to this packing do

threshold because the fraction of blocked throats does not change and we do not have any

blocked throats for that amount of cementation. After that by adding more cement to the

packing we get some fraction of blocked throat therefore we have higher percolation

threshold than a packing without cement or few cement (5 to 10% cementation).

Figure 4.10, there is no percolation at 50% saturation for packing with 24% cement.

Therefore, we cannot calculate the percolation at 50

packing. It means there is no percolation for this packing with 41% blocked throats.

80

for mono-dispersed packing. Drainage at 50% saturation is marked in this bond percolation threshold for this packing. Curvature is the dimensionless capillary

shows the bond percolation threshold for mono dispersed packing with

different cementation. When the packing does not have any cement and all of the throats

are open, the bond percolation of this packing is 55%. It means 55% of the throats should

be drained to have continuous path from one side to another. Adding cement reduces the

open throats (increases blocked throats) in the packing and increases the percolation

10% cement to this packing does not increase the percolation

because the fraction of blocked throats does not change and we do not have any

blocked throats for that amount of cementation. After that by adding more cement to the

e fraction of blocked throat therefore we have higher percolation

threshold than a packing without cement or few cement (5 to 10% cementation).

there is no percolation at 50% saturation for packing with 24% cement.

lculate the percolation at 50% saturation of wetting phase for this

packing. It means there is no percolation for this packing with 41% blocked throats.

dispersed packing. Drainage at 50% saturation is marked in this Curvature is the dimensionless capillary

shows the bond percolation threshold for mono dispersed packing with

different cementation. When the packing does not have any cement and all of the throats

his packing is 55%. It means 55% of the throats should

be drained to have continuous path from one side to another. Adding cement reduces the

open throats (increases blocked throats) in the packing and increases the percolation

not increase the percolation

because the fraction of blocked throats does not change and we do not have any

blocked throats for that amount of cementation. After that by adding more cement to the

e fraction of blocked throat therefore we have higher percolation

threshold than a packing without cement or few cement (5 to 10% cementation). From

there is no percolation at 50% saturation for packing with 24% cement.

saturation of wetting phase for this

packing. It means there is no percolation for this packing with 41% blocked throats.

Figure 4.22- Inscribed radius histogram and cumulative distribution curve for mono dispersed packinInscribed radius at 50% saturationcalculate bond percolation threshold. Bond percolation threshold is 1the throats invaded at 50% saturation.

We also calculated the bond percolation threshold for packings with ductile

grains. First, we consider packings with fix

and change the rigid radius of the ductile grains (changing ductility) to see this effect on

the percolation threshold.

81

Inscribed radius histogram and cumulative distribution curve for mono dispersed packinInscribed radius at 50% saturation (determined from Fig 4.21 and Eq. 4.4) is marked in this figure to calculate bond percolation threshold. Bond percolation threshold is 1-cumulative frequencythe throats invaded at 50% saturation.

so calculated the bond percolation threshold for packings with ductile

grains. First, we consider packings with fixed 40% ductile matter in the entire packing

and change the rigid radius of the ductile grains (changing ductility) to see this effect on

Inscribed radius histogram and cumulative distribution curve for mono dispersed packing. is marked in this figure to

frequency of the size of

so calculated the bond percolation threshold for packings with ductile

40% ductile matter in the entire packing

and change the rigid radius of the ductile grains (changing ductility) to see this effect on

82

Figure 4.23- Bond percolation threshold and fraction of blocked throats for mono dispersed packing with different cementation. The values with % on top of the data shows the porosity percentage for each packing.

Figure 4.24- Bond percolation threshold and fraction of blocked throats for mono dispersed packing with 40% ductile and different rigid radius. The values with % on top of the data shows the porosity percentage for each packing.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0 5 10 15 20 25

Blo

cked

th

roat

s

Bo

nd

per

cola

tio

n t

hre

sho

ld

Cement (% of grain radius)

Bond percolation

Blocked throats

36%

4.7%

5.3%

6%

7%

11.5%

18%26%

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.55

0.6

0.65

0.7

0.75

0.8

0.1 0.3 0.5 0.7 0.9

Blo

cked

th

roat

s

Bo

nd

per

cola

tio

n t

hre

sho

ld

Rigid radius

Bond percolationBlocked throats

14.9%

27.5%20.7%

17.3%15.4%

15.3%15.2%

15.1%

15%

35.7%

83

Figure 4.24 shows bond percolation threshold for a mono dispersed packing with

40% ductile and different rigid radii. The total trend shows that by decreasing the

ductility of ductile grains (rigid radius from 1 to 0.1) the percolation threshold increases.

The only exception is the percolation of packings with 0.4 to 0.6 rigid radius. Similar to

connectivity for those packings, their percolation threshold are similar. We explained

why this happens in connectivity part but this may related to another reason, which is the

random nature of the grains. All of grains, both ductile and hard grains, were randomly

distributed in the packings so the fraction of blocked throats may change in different

realization of the same packing (different position of random grains). If we use another

set of packings we may have different response and we may have different percolation

(increasing percolation by decreasing the rigid radius) for those packings.

In the case of packings with fixed ductility (0.7 rigid radii) and different amount

of ductile matter, Figure 4.25, the bond percolation threshold increases by increasing the

ductile percentage. Addition of more ductile grains to the packing has more effect on

percolation result than decreasing rigid radius with a fixed ductile percentage. Because

from Figure 4.9, the packings with more than 60% ductile matter did not drain to 50%

saturation, for the range of curvatures imposed we could not calculate their percolation

threshold. This happens because many throats are closed in packing with more than 60%

ductile grains (more than 40% blocked throats).

We calculated the percolation threshold for sample tight gas sands, which are

mono dispersed packings with 40% ductile matter, 0.7 rigid radius of ductile grains and

different percentage of cement (Figure 4.26). By adding cement (up to 10% cementation)

to this packing we increase the bond percolation threshold increases from 0.63 to 0.85.

By adding more cement, we close many throats therefore the packing does not drain to

50% saturation of wetting phase (Figure 4.11) anymore.

84

Figure 4.25- Bond percolation threshold and fraction of blocked throats for mono dispersed packing with 0.7 rigid radius (fixed ductility) and different amount of ductile grains. The values with % on top of the data shows the porosity percentage for each packing.

Therefore, we can increase the bond percolation threshold in different ways: 1)

addition of only cement to the packing, 2) addition of only ductile grains to the packing

3) addition of ductile grains and cement together. To add ductile grains we can either

decrease the rigid radius of ductile grains and fix the ductile percentage in the packing or

increase the percentage of ductile grains and fix the rigid radius of ductile grains. The

later has more effect on increasing the bond percolation threshold.

By looking at blocked throats and bond percolation thresholds for different

packings (Figure 4.23 through Figure 4.26) it is obvious that having blocked throats more

than 40% makes the percolation threshold approach unity. It means that nearly all the

throats must be drained for gas to flow, or that there is no connected path (breakthrough)

for those samples.

By looking at the connectivity of pore space for those packings (Figure 4.27 through

Figure 4.30), we can recognize that there is no percolation for packings with connectivity

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.55

0.6

0.65

0.7

0.75

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Blo

cked

th

roat

s

Bo

nd

per

cola

tio

n t

hre

sho

ld

Ductile (fraction)

Bond percolation

Blocked throats

36%

0.26%

1.5%

4.3%

8%

12%17%

22%27%

31.5%

of pore space less than 3.20 at 50% saturation of wetting phase because they have more

than 40% blocked throats or small throats.

Figure 4.26- Bond percolation threshold and fraction of blocked throats for mono dispersed packing with fixed 0.7 rigid radius (fixed ductility), fixsamples). The values with % on top of the data shows the poro

Figure 4.27- Bond percolation threshold and connectivity for monocementation. Packing with 24% cement has connectivity of 3.21 so there is no bond percolation threshold for this packing at 50% saturation.

0.62

0.67

0.72

0.77

0.82

0.87

0

Bo

nd

per

cola

tio

n t

hre

sho

ld

17.3%

11%

13.5%

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0 5

Bo

nd

per

cola

tio

n t

hre

sho

ld

Bond percolation

Average connectivity

85

of pore space less than 3.20 at 50% saturation of wetting phase because they have more

than 40% blocked throats or small throats.

ond percolation threshold and fraction of blocked throats for mono dispersed packing with ductility), fixed 40% ductile grains and different cementation (tight gas

The values with % on top of the data shows the porosity percentage for each packing.

Bond percolation threshold and connectivity for mono-dispersed packing with different cementation. Packing with 24% cement has connectivity of 3.21 so there is no bond percolation threshold

0.15

0.25

0.35

0.45

0.55

0.65

0.75

0.85

5 10 15 20 25

Cement (% of grain radius)

Bond percolation

Blocked throats

3.6%

6.7%

9%11%

13.5%

1.3%

3

3.2

3.4

3.6

3.8

4

4.2

10 15 20 25 30

Ave

rag

e co

nn

ecti

vity

Cement (% of grain radius)

Bond percolation

Average connectivity

of pore space less than 3.20 at 50% saturation of wetting phase because they have more

ond percolation threshold and fraction of blocked throats for mono dispersed packing with 40% ductile grains and different cementation (tight gas

sity percentage for each packing.

dispersed packing with different

cementation. Packing with 24% cement has connectivity of 3.21 so there is no bond percolation threshold

Blo

cked

th

roat

s

86

Figure 4.28- Bond percolation threshold and connectivity for mono-dispersed packing with fixed ductile matter and different rigid radius.

Figure 4.29- Bond percolation threshold and connectivity for mono-dispersed packing with fixed 0.7 rigid radius and different ductile percentage. Packings with ductile matter more than 60% (0.6) have connectivity less than 3.21 so there is no bond percolation threshold for those packing at 50% saturation.

3.29

3.39

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Figure 4.30- Bond percolation threshold and connectivity for mono-dispersed packing with fixed 0.7 rigid radius, fixed ductile percentage (40%) and different cementation (tight gas samples). Packings with cement more than 10% have connectivity less than 3.19 so there is no bond percolation threshold for those packing at 50% saturation.

Porosity values on Figure 4.23, through Figure 4.26 show that packings with

porosity less than 4.5% do not have percolation at 50% saturation. It is related to

connectivity of pore space, which is below 3.20, and percentage of blocked throats,

which are higher than 40% of the throats.

In the Figure 4.27, a mono dispersed packing with 15% cement has connectivity

of 3.71 and bond percolation value is 0.66. There are three more packings with the same

percolation values but different connectivity which are: mono dispersed packing with

40% ductile and 0.4 rigid radius (Figure 4.28, connectivity=3.6), mono dispersed packing

with 50% ductile and 0.7 rigid radius (Figure 4.29, connectivity=3.49) and mono

dispersed packing with 40% ductile, 0.7 rigid radius and 3% cement (Figure 4.30,

connectivity=3.58). Packings with the same percolation threshold may have different

connectivity values. This means addition of cement and ductile grains to the packing has

different effect on the connectivity of throats even if we get the same percolation

threshold.

The other issue is packings with the same connectivity may have different bond

percolation thresholds (Figure 4.27). This is correct only if we do not block any throat in

2.5

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88

those packings. For example, in by addition of 5 to 10% cement to the mono dispersed

packing, there is no blocked throats for these packings and connectivity is the same but

percolation threshold is different. This may is the result of different effects of

cementation on throats sizes (Figure 4.31). Because the cement is not enough to block

any throats (zero R_inscribed) and the shape of the throats are almost similar (similar

distribution) but addition of cement makes the throats smaller (they shift toward left).

This may is the reason for having different percolation threshold in those packings. The

connectivity is the same because there are no blocked throats in those packings.

Figure 4.31- Throat size distribution for mono-dispersed packings without cement, 5% cement and 10% cement.

4.7.Effects of entry and exit pores on capillary pressure (drainage) curves

Although geometry and wetting properties of the pores of a porous sample have

strong relations with the shape of capillary pressure curves, the accessibility of those

pores from the surface of the sample changes the shape of the curves too. The relation of

accessibility of pores to the sample surface with capillary pressure curves show that,

those curves are sensitive to sample size. Pores lying in the interior of the sample are less

accessible than pores close to the surface of the sample. The reason is that to have

89

connection between surface with pores in interior, the pores and pore throats, which

connect the interior pores to the surface must be connected first. On the other hand,

invasion can occur when groups of connected pores are accessible by each small

increasing of pressure. Generally, small throats control the access to larger throats.

Therefore, the larger throats are invaded in high capillary pressure and the throats size

distribution derived from capillary pressure is not a good representative of porous

medium. This means, the larger throats are assigned to high capillary pressure part of the

throat size distribution (Larson and Morrow, 1981).

To find the relation of sample size to the capillary pressure curves, I have used

one of the sphere packing samples and modeled the drainage curve for that sample at

different number of entrance and exit pores. By changing this boundary condition each

time, I changed the accessibility of interior pores to the surface of the sample. Figure

4.32, shows the effect of sample size on drainage curves. By changing the entry pores

from 1 pore to one entire face of the sample, the drainage curve is pretty similar in all

cases. When we have only 1 pore or few pores as entrance, the first part of curve is

sharper than when we use more pores for entrance. On the other hand the entry pressure

in samples with few entry pores is sharper than for samples with higher entry pores

(showed with red circle in the picture).

Exit pores do not have as strong influence on drainage as entry pores. Also by

changing the entry and exit faces of a sample, the drainage curve does not change a lot.

Figure 4.33 shows the same sample of Figure 4.32, with different entry and exit faces of

the packing (each packing has 6 faces, we use different entrance and exit pore faces in

this case) to see the influence of different entry pore faces on drainage curves. It is

obvious from the Figure 4.33 that changing the entrance face or exit face does not change

the shape of drainage curve. Also by changing the only the exit pores with similar

entrance pores, the drainage curve does not change.

90

Figure 4.32- Mono-dispersed packing with 23% cement is used with different entry and exit pores to see the influence of sample size on drainage curve. The red circle shows the difference in entry pressure for these curves. Curvature is the dimensionless capillary pressure.

Figure 4.33- Mono-dispersed packing with 23%cement is used with different entry and exit pore faces to see the influence of sample size on drainage curve. Curvature is the dimensionless capillary pressure.

91

4.8.Comparison between drainage simulation and experimental data

To compare our simulation data with experimental data, first we have to change

the drainage curves (dimensionless curvatures) to capillary pressure curves. All of the

simulated drainage curves in this dissertation are dimensionless curvature and for

comparing with experimental data, they have to be converted to capillary pressure. To do

that, we need to know the mean grain radius of the sample and the value for interfacial

tension. Then we have to change the simulated drainage data to capillary pressure data by

the following given equation:

)� � � � * � +,- ($�

(4.5)

where Pc is capillary pressure (KPa), σ is interfacial tension between air and mercury

(which is 484 dynes/cm or mN/m), Rmean is the mean radius of the sample in (m), and C is

the curvature which is derived from simulation. Therefore, the equation should be written

as below:

)� � � � -.- ($� (4.6)

We are simulating tight gas sands therefore our simulated samples should have a lot of

cement and low porosity. We chose a mono dispersed packing with 23% cement (4.7%

porosity) to compare with lab data. For comparison, we need to have at least one

parameter similar in both simulation data and lab data. Therefore, we decided to have

similar mean radius of grains for comparison. By choosing the similar mean grain radius

(175 µm), we did not find a very good match between our simulations and lab data

(Figure 4.34). The difference is in the value of irreducible water saturation, which is

smaller in simulation result than the actual sample. In addition, the capillary pressure is

lower in our simulation. Neither the curve nor the porosity values (12.1%) in the last

sample are similar. Comparing the same sample with another simulation in a model rock

with 24% cement (4% porosity), the curve is more similar to the sample curve than the

92

previous simulation curve. In this case, the entry pressure is different from the actual

sample (Figure 4.34). However, the porosity in the model is much smaller than the

sample.

Figure 4.34- Comparison of drainage curves between simulated packings and lab data (data courtesy from www.discovery-group.com). The dashed curve is the mono dispersed packing with 23% cement (23% of grain radius) (porosity=4.7%). The porosity for the tight gas sample is 12.1% and the mean grain radius is 175 µm. The dotted curve is a packing with lognormally distributed from sizes (mean grain radius=2.3842 R, standard deviation=0.9877R^2 and porosity=1.8). Because the lognormal distribution is not wide, the simulated drainage capillary pressure is even lower than mono-dispersed packing (explained in effect of different grain size distribution on drainage curves). The black curve is the mono-dispersed packing with 24% cement (24% of grain radius) (porosity=4%). The large difference between packing with 4.7% porosity and packing with 4% porosity is because of the amount of blocked throats (41% and 45% respectively).

We used mono-dispersed packing to compare with lab data. This is not correct

because field samples have usually lognormal grain size distribution rather than mono-

sized grains. Therefore, we decided to use a packing with lognormal distribution to see if

we can capture any lab data. We report the drainage curve for a lognormal distribution

sphere packing with cement in Figure 4.34. Because the lognormal distribution is not

wide enough, the drainage curve is even lower than mono-dispersed packing (this was

explained earlier as an effect of different grain size distribution on drainage curves). The

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93

code that we are using to make the sphere packing, does not support lognormal

distributions wider than the one used above. We would need to use a very large number

of spheres to make lognormal packings with wide distribution. It needs large cpu for a

computer to handle that much data. Therefore, we decided to do a shortcut toward more

realistic packings. We assigned the throat sizes to the network of mono-dispersed model

sandstone from a lognormal distribution. We used different lognormal distributions of

throat sizes, with different mean and standard deviation (changing the width of the

distribution curve by changing the mean and standard deviation of lognormal

distribution) (Figure 4.35). Using different throats size distribution did not help in

improving the simulated curve to match with lab data. The lognormally distributed

throats, which we used in this figure, did not have a good match with lab sample. The

reason is that none of the simulated curves with lognormal distribution of throat sizes has

closed throats. Therefore, most of those curves are below the lab sample curve, which is

heavily cemented and has many blocked throats.

To make correction for this problem, we assigned sizes from a lognormal

distribution to the open throats and we kept the blocked throats from the heavily

cemented mono-dispersed network. By keeping the blocked throats of mono-dispersed

network, the capillary curves move to higher values (Figure 4.36). We did not have any

good match between simulated curves and lab curve. Figure 4.37 is the histogram of

lognormal throat sizes used in Figure 4.35 and Figure 4.36.

94

Figure 4.35- The lab data from Figure 4.34 and the simulated data use the network of the packing with mono-dispersed grains with 23% cement (23% of grain radius) to which throats sizes are randomly assigned from a lognormal distribution. Each curve has different lognormal distribution with different width (mean and standard deviation). We used m and v values to calculate the mean and standard deviation

as below: /012 � log ! �6789�6' and standard deviation is: :;< � =log > 8�6 & 1@.

The sample used in Figure 4.34 through Figure 4.36 has a porosity value different from

the model rocks used although the grain sizes were similar. We decided to compare

samples and simulations with the same porosity values instead of same grain sizes for the

next test.

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lognorm-throats, m=1, v=1000

lognorm-throats, m=1, v=10000

95

Figure 4.36- The data are the same as Figure 4.35 but we kept the blocked throats of heavily cemented mono-dispersed packing and the open throats were chosen randomly from lognormal distribution. "Keeping zero" in legend means we kept the throats with zero radius (the blocked throats) from the mono-dispersed packing and then applied the lognormal throat size distribution to assign sizes to all other throats.

Figure 4.38 shows a good match with our simulation and lab data. The simulation

samples have lognormal distribution of throats sizes with different mean and standard

deviation, using mono-dispersed packing network. Part a of this figure is a sample from

Green River formation depth 11605 ft with mean grain radius of 125µm and porosity of

3.2%. The lognormal throat size distribution with mean of -4.6052 (log value) and

standard deviation of 3.0349 using mono-dispersed packing network (porosity=4.7%),

matched very well with the lab sample. The simulated packing has the blocked throats of

mono-dispersed sample with heavy cementation. In part b of this figure, a sample from

the same formation and different depth (11460 ft) with the same mean grain radius and

porosity of 4.4% was matched well with our simulation sample (porosity=4.7%). The

simulated sample has lognormal distribution of throat sizes with mean of -3.45 (log

value) and standard deviation of 2.6 using mono-dispersed packing network having

blocked throats of mono-dispersed packing with 23% cement.

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lognorm-throats, m=1, v=1 keeping zeros

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lognorm-throats, m=1, v=100 keeping zeros

lognorm-throats, m=1, v=1000 keeping zeros

lognorm-throats, m=1, v=10000 keeping zeros

Figure 4.37- Histogram of lognormal throat size distributions with different m and v values. a) m=1v=1 b) m=1, v=10 c) m=1, v=100 d) m=1, v=1000 e) m=1, v=10000.

96

Histogram of lognormal throat size distributions with different m and v values. a) m=1v=1 b) m=1, v=10 c) m=1, v=100 d) m=1, v=1000 e) m=1, v=10000. These are used in Figure 4.36.

Histogram of lognormal throat size distributions with different m and v values. a) m=1 and These are used in Figure 4.36.

97

Figure 4.38- Simulated drainage curves using lognormal distribution of throat sizes assigned to open throats in a mono-dispersed packing network with 23% cement (23% of grain radius). a) Lab sample from Green River formation (depth=11605 ft) with mean grain radius of 125 µm and porosity=3.2% (Data courtesy from www.discovery-group.com). The simulation sample has m=1 and v=10000 (look at Figure 4.35 caption). We use one face of the packing for entrance and exit pores during drainage simulation. b) Lab sample from Green River formation (depth=11460 ft) with mean grain radius of 125 µm and porosity=4.4%. The simulation sample has m=1 and v=1000 (look at Figure 4.36 caption). We use one face of the packing for entrance and exit pores during drainage simulation.

98

4.8.1. Comparison between drainage simulation from network model and

experimental data

We repeated the modeling approach with a different sample, this time with the

drainage curve converted to gas-water:

)� � * � �� (4.7)

where Pc is a capillary pressure, σ is the interfacial tension of gas-water (72 dynes/cm), C

is dimensionless curvature and r is the mean radius of the field sample data. The result of

applying different distribution for throat size data is shown in Figure 4.39. This sample is

from a western US tight gas formation. There are two curves beside the experimental

capillary pressure curve. One is the mono-dispersed packing with heavy cementation,

which shows no match and is below the sample capillary pressure curve. The other curve

is the simulate using throat size distribution of the field sample (inferred from the

mercury intrusion capillary pressure curve). The curve shows higher capillary pressure

than the sample and the shape of the curve does not match either.

Figure 4.39- The capillary pressure calculated using the real throat size distribution of field sample (Western US #2) using network modeling. It is obvious that the simulated curve does not match the field data even though the model has the same nominal distribution of throat sizes. The other curve is the simulated capillary pressure using the mono dispersed packing with heavy cementation (many blocked throats) which does not match the data either.

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mono-dis-23%cement

99

To seek a better match with experimental data we changed the throat size

distribution of the real sample and used a lognormal, normal or uniform distribution in

the same range of throat sizes (the mean and standard deviation of real sample was used

in the case of lognormal and normal distributions). The simulated drainage curves with

the new distributions are shown in Figure 4.40. The lognormal distribution curve with the

same mean and standard deviation of the sample (throat size distribution of the sample)

does not match the data at all. The normal and uniform distribution (with the same mean

and standard deviation of the real data and the same range of data in the case of uniform

distribution) are close to the real data but does not match very well. We always have a

clear percolation threshold in network calculation whereas the data show a gradual

decrease in water saturation as capillary pressure increases.

Figure 4.40- Simulations with various throat size distributions (lognormal, normal and uniform distributions) with the same range of throat sizes and same mean and standard deviation as the sample (inferred from mercury intrusion). The sample is same as Figure 4.39.

We next used the same data range of throat sizes (Figure 4.41) but we tried to

build the curve (and the corresponding two peaks) using a bi-normal distribution of data

with two means and two standard deviations approximated from the sample throat size

distribution curve (Figure 4.41, part a). Then the two normal curves were added up to

make a bi-normal distribution of data (Figure 4.41, part b). The simulated drainage curve

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100

still does not match the real data and it shows percolation (Figure 4.42). The throat size

distribution inferred for the sample comes from a mercury intrusion capillary pressure

experiment via bundle of tubes model. The bundle of tube model does not consider the

volume of tubes in the calculation, only the number of tubes. In the throat size

distribution computed this way (Figure 4.41, part a) the bigger throats have higher

frequency than the smaller one. If we consider the volume of tubes in the calculation,

which seems to be the correct way, the smaller throats will have the higher frequency.

Therefore, in the next step we used the same two normal distributions (Figure 4.41, part

c) for throat sizes but reversed the peaks (changed the standard deviation and kept the

means the same). The simulated capillary pressure curve for that throat size distribution

did not match with the real data either and still shows percolation.

Figure 4.41- a) Throat size distribution inferred from mercury intrusion of the tight gas sample Figures 4.39 and 4.40. Notice the bimodal distribution of the sample throats. b) R-inscribed (throat sizes) distribution used in simulation. It was derived by adding two normal distribution with selected means and standard deviations. c) The throat size distribution with the same range but using reverse peaks (the same means different standard deviation).

101

Figure 4.42- Simulated drainage capillary pressure using the range of real throat sizes but adding two normal distribution curves with the estimated means and standard deviations. The other curve (dashed line) uses the same two normal distribution but reverse the height of the peaks (look at distribution of throat sizes Figure 4.41).

In all of the previous cases, we got a percolation in our simulation, which is

barely seen in experimental data. Percolation behavior happens when the throat sizes are

distributed randomly upon a network. To weaken the percolation behavior, we decided to

use a grain packing, which is layered. Each layer should have different grain sizes. This

idea comes from one of the mercury experiments on tight gas sandstone, which has a

bimodal distribution of throats. For this test, we used a grain packing with three layers

(Figure 4.43). The middle layer has big grains (mono-dispersed) and the upper and lower

layers have the same small grain size distribution (mono-dispersed) (the ratio of radius of

big grains to radius of small grains is 2.4). The simulated drainage curve still has the

percolating behavior, though a gradual decrease in water saturation as capillary pressure

increases is more evident.

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Figure 4.43- a) A two-dimensional view of a layered packing with two grain size distributions. The ratio of big grain size to small grain size is 2.4. b) The drainage curve for the same papressures is still narrow over wide range of saturationbigger grains percolated first. Curvature is a dimensionless capillary pressure.

In many previous simulations, we u

dispersed grain packing. In the next simulation, we decided to use two different networks

within a single packing. To do that, we used a layered throat size distribution for the

packing. We created two sets of throat s

throat sizes in the range of

result with that sample. The two normal distributions were allowed to overlap to simulate

the shape of the throat size distrib

normal distribution of throats with bigger mean value was

middle layers of the network. T

applied to throats in the rest

102

dimensional view of a layered packing with two grain size distributions. The ratio of to small grain size is 2.4. b) The drainage curve for the same packing. The range of

wide range of saturation, and each layer percolates separately. The layer with Curvature is a dimensionless capillary pressure.

previous simulations, we used one network coming from the mono

dispersed grain packing. In the next simulation, we decided to use two different networks

packing. To do that, we used a layered throat size distribution for the

packing. We created two sets of throat sizes using normal distributions. We used the

throat sizes in the range of a Western US tight gas sample to compare the simulation

result with that sample. The two normal distributions were allowed to overlap to simulate

the shape of the throat size distribution in that tight gas sample (Figure 4.41

normal distribution of throats with bigger mean value was assigned to throats in

of the network. The normal distribution with smaller mean value was

the rest of the layers. We also used three layers of big throats in the

dimensional view of a layered packing with two grain size distributions. The ratio of range of capillary

separately. The layer with

sed one network coming from the mono-

dispersed grain packing. In the next simulation, we decided to use two different networks

packing. To do that, we used a layered throat size distribution for the

izes using normal distributions. We used the

to compare the simulation

result with that sample. The two normal distributions were allowed to overlap to simulate

4.41 part a). The

throats in two

he normal distribution with smaller mean value was

used three layers of big throats in the

103

middle to get the better transition (not percolating behavior) in drainage curve. By using

two separate distributions for throat sizes, we effectively create two networks within the

same packing. A schematic figure of this layered throat arrangement is shown in Figure

4.44. We compared the simulated data from this test with the actual data of a tight gas

sample (Figure 4.45). The result shows a better match with experimental data than other

simulations although we still can see some percolation. Each layer percolated separately;

the bigger throats percolated first and it followed by percolating the small throats. By

changing the distribution of throat sizes (using trial and error) for big and small layer of

throats as an input to the simulator, we may capture the experimental data.

Figure 4.44- A schematic of a 5 layered packing. The throat size distribution is the same for the middle two gray layers with big throats. The throat size distribution for the white layer is the same with small throats. Both throat distributions are normal distributions. These distributions were imposed on the throat in corresponding spatial regions in a single network model.

Most of the experimental data for tight gas sands show a gradual drainage

capillary pressure curve and do not show percolation over a wide range of saturation. In

contrast most network simulations show sharp percolation, spanning a wide range of

saturations within a narrow range of capillary pressure. It seems that the intergranular

pore network in actual samples (Figure 4.46) is very different from the ones from the

network simulations. The intergranular pore network in real samples does not show any

104

connected structure whereas all of a network model results shows a connected path. The

nature of the network model is based on percolation in the packings. The simulated

results show that either the network models fail to capture the topology of pores in tight

gas sandstones, or mercury intrusion experiments in tight gas sands are altering the

intergranular void space in the samples.

Figure 4.45- The drainage curve for the tight gas sample of Figure 4.30 et seq. and a simulation with heterogeneous throat size distributions. Two networks of throats were applied to a five-layered packing of Figure 4.44. The y-axis is logarithm of capillary pressure data. The irreducible water saturation is zero for experimental data but the simulated result has some trapped water as irreducible water saturation.

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105

Figure 4.46- a) Intergranular pore network in highly cemented model rock with connectivity of 3.3 and porosity of 5%. The whole network of pores and throats in this sample are connected to each other, even though many individual throats are blocked. Consequently, drainage simulation show clear percolation thresholds. b) The image represents skeleton (medial axis) of the pore space in a 4% porous Western US tight gas sandstone sample. The sample was imaged at 2.3 micron resolution at Jackson School of Geosciences. Medial axis and thus the pore space are not connected across the sample in any direction. The rainbow coloring refers to the proximity of the neighboring grain (red-within one voxel length, velvet-with 5 voxel length) (image courtesy of Dr. Maša Prodanović). There is no big connected path through this sample rock.

4.9. Summary

Grain scale simulation of porosity-reducing mechanisms in tight gas sandstones

(ductile grain deformation, cement precipitation) shows that pore space is poorly

connected in these rocks. The ductile grains content and cement reduces the connectivity

rapidly due to changing the pore size distribution. The calculated percolation threshold at

50% saturation of wetting phase for these low porosity packings is very high. It means a

high amount of throats and pores should be drained to get the 50% saturation of wetting

phase. This is a result of decreasing the size of pores and throats due to compaction of

106

grains and cementation. The drainage curves are sensitive to the amount of ductile grains,

the ductility of ductile grains (changing the rigid radius of ductile grains) and the amount

of cementation. These processes change the shape of drainage curves differently based on

their influence on throat size distributions but they all increase the amount of irreducible

water saturation and entry pressure.

By comparing the simulation result with experimental data of tight gas sandstone,

we realized that most of the experimental data does not show the percolation behavior,

yet percolation is usually observed in network simulation results. We tried different

procedure to decrease the sharpness of percolation in simulated drainage curves, but most

procedures failed. It seems that the intergranular pore networks in actual samples are very

different from the ones in the network simulations. The intergranular pore network in real

samples exhibit even less connected structure, than in the poorly connected network

models.

The drainage capillary pressure simulation based on network modeling exhibits

some difficulty in agreement with experimental data. The following suggestions are

recommended for future simulation of capillary pressure curves. 1) The mercury intrusion

experiments in tight gas sands are altering the intergranular void space in the samples and

the experiment may not be appropriate for this type of rocks. 2) The mono-dispersed

packing of spheres may not be appropriate as an input for network modeling of capillary

pressure simulation in tight gas sandstones. The solution for the later problem is to use a

wide distribution of sphere sizes (normal or lognormal distribution) as an input for a

network model because tight gas sandstones seem to have distributions of grain sizes. An

alternative solution is that to use two or three network of throat sizes as an input for

network modeling of tight gas sandstones to remove the percolation behavior in

simulation drainage curves. Another way is to modify the throat size distribution of the

actual rock by iteration and use the new distribution of throat sizes as an input to the

network model. The literature shows that network model works well for convensional

rocks, which are not as heterogeneous as tight gas sandstones and show percolation

behavior in even experimental data.

107

Chapter 5: Prediction of relative permeability curves in tight gas sandstones

5.1.Introduction

Relative permeabilities are one of the important parameters for estimating tight

gas reservoir performance. To understand gas production from low permeability tight gas

sandstone we need to understand the petrophysical properties of the gas reservoir

including gas-water relative permeability. Relative permeability is a function of pore

geometry. It is a goal of this chapter to examine how different geologic processes

(discussed in Chapters 2) that control the pore geometry can affect relative permeability

of tight gas sandstones. In this chapter, we first introduce the relative permeability curves

and the two methods obtaining the laboratory measurements are briefly explained. Then

we discuss the details of calculation of relative permeability model used in this research.

Finally, the effects of diagenesis and pore geometry on relative permeability curves are

discussed.

5.2.Relative permeability

Conductivity of a phase with respect to another phase as a function of saturation is

called relative permeability. The geometry of the pore space, viscous forces and surface

tension are the controlling factors on displacing one fluid by another. For two-phase

system, wetting phase permeability and non-wetting phase permeability are fractions of

total permeability. The effective permeability of each phase is a product of relative

permeability of that phase and total permeability. Usually an empirical model, which was

fitted to the experimental data, is used to obtain relative permeability curves as a function

of saturation. There are two methods used by industry to measure relative permeability:

steady state and unsteady state methods. Steady state method is easier to calculate from

data but lab measurements are difficult and the unsteady state is vice versa.

To measure steady state relative permeability we have to inject fluids to the core

at fixed ratio until pressure and saturation equilibrium are reached. Relative permeability

curves are calculated from pressure drop and flow rate data. To calculate relative

108

permeability at other saturation we have to fix the total injection rate but change the fluid

ratio.

��� � ���

∆���

(5.1)

where krj is the relative permeability of the phase j, qj is the flux, L is the length the

pressure drop is taking over, ∆P is the pressure drop, A is the cross sectional area to flow,

µj is the viscosity of the phase j, and k is the absolute permeability of the porous medium.

In the unsteady state method a single fluid will be injected to the core initially

saturated with the other fluid phase in either constant rate or constant pressure. Then the

amount of displaced fluid and the pressure drop or rate are measured with time. The

relative permeability should be calculated using an empirical model such as Johnson et

al., (1959), Archer and Wong (1973) or Sigmund and McCaffery (1979).

Figure 5.1- Typical imbibition relative permeability curves (image courtesy from Kleppe, 2009).

When the relative permeability of wetting phase is zero there is irreducible water

saturation (Swir) and in the same position the relative permeability of the non-wetting

phase is at its maximum end point value (knwr). In imbibition curves when the relative

permeability of the wetting phase is at maximum value and relative permeability of non-

wetting phase is zero, there is a residual non-wetting phase saturation (Snwr). And the two

phase flow only occurs at the saturation of Swir<Sw<(1-Snwr). The imbibition curves

109

(Figure 5.1) are used when water displaces oil or gas in a water wet reservoir in

waterflood calculation, water displaces oil or gas in a water wet reservoir in natural water

influx, oil displaces gas in when oil is forced into a gas cap (Peters, 2009).

In the typical drainage curve (Figure 5.2), the wetting phase starts from saturation

of 1 and non-wetting phase starts at saturation of zero. The permeability of the wetting

phase at the beginning is the absolute permeability of the wetting phase. Drainage relative

permeability curves are used as when gas replaces oil in solution gas drive calculation,

gravity drainage calculation when gas replaces drained oil, when gas displaces oil or

water in gas drive calculation and in tertiary recovery process when oil or gas displace

water (Peters, 2009).

Figure 5.2- Typical drainage relative permeability curves (image courtesy from Kleppe, 2009). 5.3.Relative permeability calculation

For predicting the two-phase flow permeabilities, the network models are useful.

We consider a specific network of throats and pore bodies for each phase, which

correspond to a particular set of grains in our grain packing. We define pore bodies and

throats by Delaunay tessellation of the network and calculate drainage to reach to

irreducible water saturation. We used the drainage capillary pressure simulation, which

was calculated in Chapter 4 to determine the configuration of each phase i.e. which pores

110

and pore throats, contain wetting phase, and which contain non wetting phase. We then

compute the drainage relative permeability curves assuming the fluid configuration does

not change during steady flow.

We need to define the hydraulic conductance and saturation to calculate the

relative permeability of the system (Bryant and Blunt 1992). The drainage of non-wetting

fluid is controlled by the throats shape, which contains wetting pendular rings and bridges

(calculated in drainage code, Chapter 4). Behseresht 2008 implemented the relative

permeability simulator, which is briefly explained here. The conductance of the throat is

the flow rate divided by pressure difference in Hagen-Poisueille equation and is

proportional to fourth power of the radius of the throat:

�∆� � ���� � ������

���

(5.2)

where q is the flow rate, µ is the viscosity of displaced fluid, ∆P is the pressure

difference between two neighbor pores, ghydr is the hydraulic conductance of the throat, L

is the distance between center of the two neighbors and r ins is the inscribed radius of the

throat connects two neighbor pores.

The inscribed radius of the throat, is the radius of the biggest circle that can be

inscribed into the throat. The hydraulic conductance of the throat reduces because of the

presence of the pendular rings, which reduce the void area of the throat for non-wetting

fluid to pass through.

The relative permeability was calculated similar to the approach of Bryant and

Blunt (1992). It was assumed to have steady state flow, incompressible fluids and a

constant pressure drop across the wetting and non-wetting phases. The net flow in or out

of each pore should be zero based on Kirchoff’s law:

� �� � ��

���

(5.3)

�� � �,!"#$�,�%$�� & �!"#$�,�%%

(5.4) where z is the number of neighbor pores, qk is the flow rate, gi,NBR(I,k) is hydraulic

conductance of the throat which connects pore i to its kth neighbor pore, Pi is the pressure

at pore i and PNBR(i,k) is the pressure at the kth neighbor of pore i (Motealleh, 2009).

111

� �,!"#$�,�%�!"#$�,�% & � �,!"#$�,�%�� � ��

���

�

���

(5.5)

Equations for pressure in each pore body will produce the system of simultaneous

equations of all pores in the porous medium. The overall flow rate (calculated from value

of pressures in pore bodies) for an imposed pressure drop across the network gives the

permeability to the non-wetting phase. This value will be normalized by the single phase

permeability of the medium and reported as relative permeability of that phase.

Behseresht (2008) in his drainage code calculated the geometry of pendular ring and

throat.

5.4. Relative permeability simulation in tight gas sands

We modeled drainage relative permeability for rocks with ductile grains, cement

or both together. The model rocks are the same as ones in Chapter 4. First, we modeled

rocks with only ductile grains. We first model rocks with fixed amount of ductile grains

and change the ductility of soft grains to see how it change the relative permeability

curve of the simulated rock (Figure 5.3). By decreasing the rigid radius of ductile grains

from 0.9 to 0.1 (increasing the ductility of grains), both water and gas permeability

curves shift toward right. The packings having 0.1 to 0.6 rigid radii have similar relative

permeabilities. This is due to similar connectivity of pore space and fraction of blocked

throats (Chapter 4) which itself is related to formation of load-bearing frameworks of

rigid grains (Mousavi and Bryant 2007). The reason the curves shift toward the right will

be explained at the end of this section.

The next step is computing relative permeability curves for model rocks with

ductile grains with fixed ductility and different amount of ductile grains in the packing

(Figure 5.4). Increasing the amount of ductile grains has similar effect on relative

permeability curves as when we change the rigid radius of ductile grains (both water and

gas curves shift toward right). In this case, the first three packings with 10% and 20%

ductile grains have almost the same relative permeability curves. This is because of their

close similarity of fraction of blocked throats and average connectivity of pore space

(Chapter 4).

112

Figure 5.3- a) Drainage water relative permeability for mono-dispersed packing with fixed amount of ductile grains in the packing and different ductility (rigid radius). By decreasing the rigid radius of grains from 0.9 to 0.1 (increasing ductility) water relative permeability curve shift toward right. b) The same plot as (a) with logarithmic scale. The curves below 0.6 rigid radius are similar because of the formation of hard cages and having similar porosity. c) Gas relative permeabilities for the same samples. By decreasing the rigid radius of grains from 0.9 to 0.1 (increasing ductility) gas relative permeability curve shift toward right.

(a)

(b)

(c)

113

Figure 5.4- a) Drainage water relative permeability for mono-dispersed packing with fixed rigid radius (ductility) and different amount of ductile grains in the packing. By increasing the amount of ductile grains from 10% to 70%, water relative permeability curve shift toward right. b) The same plot with logarithmic scale for water relative permeability. c) Gas relative permeability curves for the same models. By increasing the amount of ductile grains from 10% to 70%, gas relative permeability curve shift toward right.

(a)

(b)

(c)

114

By adding cement to a mono dispersed packing (Figure 5.5), we close many small

throats therefore we reduce the connectivity of pore space (Chapter 4). Packings with no

cement and 5% to 10% cement have similar relative permeability (especially gas

permeability) curves. This is because those packings have no blocked throats yet. The

cement has not grown enough to close any throats.

The last step was to model tight gas sand with both ductile grains and cement

together. We used a packing with 40% ductile grains and 0.7 rigid radii of ductile grains

and different amount of cementation to model tight gas sands (Figure 5.6). Increasing

cement causes both water and gas relative permeability curves shift toward the right.

The reason for this behavior in all of the relative permeability samples is that by

increasing the cement, the fraction of ductile grains or both, the connectivity of pore

space decreases. The water permeability shifts toward right because throats containing

water become disproportionately smaller by increasing cement or ductile grains. The gas

permeability shows higher relative values as cement (or ductile grains or both) increases.

This counterintuitive result is a consequence of the connectivity of pore space, as

explained below.

In the case of low porosity packings, the fraction of the drained throats is larger at

50% volume fraction of wetting phase (saturation) compare to higher porosity packings.

Packings with lower porosity are poorly connected and a larger fraction of throats needs

to be drained to get the same volume fraction of wetting phase (50% saturation) compare

to packings with higher porosity (Figure 5.7 and Figure 5.8).

115

Figure 5.5- a) Drainage relative permeability for mono-dispersed packing different cementation (different percentage of grain radius). By increasing the amount of cement from 0 to 24% (% of grain radius), water relative permeability curve shift toward right. b) The same plot with logarithmic scale for water relative permeability. c) Gas relative permeability for the same model rocks. Gas relative permeability curve shifts toward right as porosity decreases.

116

Figure 5.6- a) Drainage water relative permeability for mono-dispersed packing with 40% ductile grains, 0.7 rigid radius of ductile grains and different cementation (different % of grain radius) (tight gas samples). By increasing the amount of cement from 0 to 15% (% of grain radius), water permeability shift toward right. b) The same plot with logarithmic scale for water relative permeability. c) Gas relative permeability curves for the same model rock. By increasing the amount of cement from 0 to 15%, gas permeability shift toward right.

(a)

(b)

(c)

Figure 5.7- Critical curvature frequency dispersed packing with 15% cementcolor shows distribution for all thrwetting phase saturation is 50%, at which value The drained throats amount to 35% of the total throats in the packing including

Figure 5.8- Critical curvature frequency dispersed packing with 23% cementcolor shows distribution for all throats; the red color shows the distribution of throats drained when the wetting phase saturation is 50% and the gas phase is assumed to reach percolation threshold. The drained throats are 47% of the total throats in the packing including blocked throatthe higher porosity packing in Figure the low porosity packing, Figure 5.3(lower curvature values) is smaller in this case compare(small and blocked throats) are not shown in this In Chapter 2, we explained that g

immobile over a wide range of water saturations in low permeability sandstones. The

117

frequency distribution of throats drained during simulation

dispersed packing with 15% cement (15% of grain radius) with porosity = 12% (in Figure color shows distribution for all throats; the red color shows the distribution of throats drained when the

, at which value the gas phase is assumed to reach percolation threshold. The drained throats amount to 35% of the total throats in the packing including blocked throats.

frequency distribution of throats drained during simulation for mono

dispersed packing with 23% cement (23% of grain radius) with porosity = 4.7% (in Figure throats; the red color shows the distribution of throats drained when the

wetting phase saturation is 50% and the gas phase is assumed to reach percolation threshold. The drained throats are 47% of the total throats in the packing including blocked throat. This is larger fraction than for

Figure 5.7. This explains why the gas phase relative permeability is larger in 5.3, Figure 5.4, Figure 5.5, and Figure 5.6. The frequency of

(lower curvature values) is smaller in this case compared to Figure 5.7. The highest values of curvature (small and blocked throats) are not shown in this figure.

Chapter 2, we explained that gas and water are commonly believed to be

ide range of water saturations in low permeability sandstones. The

of throats drained during simulation for mono-Figure 5.5). The blue

oats; the red color shows the distribution of throats drained when the the gas phase is assumed to reach percolation threshold.

blocked throats.

distribution of throats drained during simulation for mono-Figure 5.5). The blue

throats; the red color shows the distribution of throats drained when the wetting phase saturation is 50% and the gas phase is assumed to reach percolation threshold. The drained

. This is larger fraction than for . This explains why the gas phase relative permeability is larger in

frequency of big throats values of curvature

commonly believed to be

ide range of water saturations in low permeability sandstones. The

saturation region across which effective permeability to

is called "permeability jail" (Shanley

sandstone, we realized that we do not see that behavior in the simulated relative

permeability curves for tight gas sandstone (Figure 5.6).

permeability sketched by Shanley

region of the plot can be obtained from drainage simulation. In

water relative permeability moves toward low values of permeability

water saturation, which is similar to right region of

al., figure (Figure 5.9), the gas relative permeability increase

which was explained quantitatively

sketched in Figure 5.9 may be created by imbibition process of tight gas s

is a good subject for future work.

Figure 5.9- Schematic figure of relative permeability curves in low permeability sandstone. Image courtesy

5.5.Relative permeability in some tight gas formations

To make sure that the relative permeability calculations work well, we needed to

compare the result with experimental data.

permeability of real tight gas

118

saturation region across which effective permeability to both gas and water is negligible

(Shanley et al., 2004). After drainage simulation of tight gas

one, we realized that we do not see that behavior in the simulated relative

permeability curves for tight gas sandstone (Figure 5.6). If we divide the relative

permeability sketched by Shanley et al., (2004) (Figure 5.9) to two regions

the plot can be obtained from drainage simulation. In Figure 5.6 part a, the

water relative permeability moves toward low values of permeability at high values of

which is similar to right region of Figure 5.9. But unlike the

the gas relative permeability increases at larger water saturation

quantitatively in the last paragraph. The gas relative permeability

may be created by imbibition process of tight gas sandstone

future work.

Schematic figure of relative permeability curves in low permeability sandstone. Image courtesy

from Shanley et al., (2004),

Relative permeability in some tight gas formations

that the relative permeability calculations work well, we needed to

compare the result with experimental data. We compared our simulations with relative

gas samples from discovery-group website (www.discovery

water is negligible

2004). After drainage simulation of tight gas

one, we realized that we do not see that behavior in the simulated relative

we divide the relative

regions, the right

igure 5.6 part a, the

high values of

unlike the Shanley et

at larger water saturation,

The gas relative permeability

andstone. This

Schematic figure of relative permeability curves in low permeability sandstone. Image courtesy

that the relative permeability calculations work well, we needed to

compared our simulations with relative

(www.discovery-

119

group.com). The curves are not actual measurements of relative permeability; the relative

permeabilities were calculated using Corey (1954) equations following the Byrnes et al.

(2009) conclusion that many measurements can be fit well by this form. The Corey

equations for effective water saturation, relative permeability to water and relative

permeability to gas are given as below:

'() � $'( & '(*��%$1 & '(*��%

(5.6)

��( � '(), (5.7)

��- � $1 & '()%. / $1 & '().% (5.8)

where, Swe is the effective water saturation, Sw is fractional water saturation, Swirr is

irreducible water saturation, krw is the relative permeability to water and krg is the relative

permeability to gas. Byrne et al. (1979) modified Corey equation to predict gas relative

permeability for low-permeability sandstones:

0�- � 11 & $'( & '(2,-%$1 & '-2 & '(2,-%3

4/ 11 & 5'( & '(2,-1 & '(2,- 6

73 (5.9)

where, Sgc is the fractional critical gas saturation, Swc,g is the fractional critical water

saturation relevant to the gas phase, and p and q are exponents expressing pore size

distribution influence (Byrnes et al., 2009).

Figure 5.10, shows a comparison between one of the simulated drainage relative

permeability result with a measurements on a conventional sandstone. The porosity

values are close (actual sample porosity=17.7% and simulation porosity=17.3%). The

predicted relative permeability curves match the experiment. The only difference is that

the gas-relative permeability for the actual data is less than the predicted one. This may

be because in this simulation result, we did not consider pendular rings (explained in

Chapter 4) in the water saturation calculations. Therefore, the irreducible water saturation

in our simulation result is smaller than the actual water saturation. The simulation works

well for packings with high porosity.

120

Figure 5.10- Predicted and experimental drainage relative permeabilities in sandstone. The sample is from depth 3433.8 ft with porosity of 17.7%. The simulation result was used a mono-dispersed packing with 40% moderately ductile matter and a porosity of 17.3% (data courtesy of www.discovery-group.com). The sandstone relative permeability was calculated using Corey (1954) equations.

We needed to compare the simulation result with tight gas sandstone with smaller

porosity values. Therefore, we compared the simulation result of a packing with 40%

moderately ductile matter, 10% cement and porosity of 7% to a tight gas sample (Figure

5.11). The results show that the relative water saturation curve matches well with the

experiment but the gas-relative permeability is a little different and it shows different

shape than the experiment result. In overall, the simulation results match well with

experimental data.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Kr

SW

Green_River 3433.8ft Mono-dispersed

121

Figure 5.11- Predicted and experimental measurements of drainage relative permeability in tight gas sandstone. The sample is from depth 8279.5 ft with porosity of 7.6%. The simulation result used a mono-dispersed packing with 40% moderately ductile matter 10% cement (10% of grain radius) and porosity of 7% (data courtesy of www.discovery-group.com). The sandstone relative permeability was calculated using Corey (1954) equations.

By looking at the drainage relative permeability from actual tight gas samples

(Figure 5.12), we can see that by decreasing the porosity and permeability, the gas

relative permeability shows higher values. The water relative permeability shows lower

values by decreasing the porosity and permeability. This trend is seen in other tight gas

basins such as Powder River, and Uinta (Appendix B). The reason for this behavior is

that by decreasing the porosity and permeability, the irreducible water saturation of the

sample increases (Figure 5.13) which is the reason for increasing the gas relative

permeability of the sample.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Kr

SW

Uinta 8279.5 ft

mono-dispersed 40%ductile, 0.7 rr, 10% cement

122

Figure 5.12- The relative permeability for actual tight gas sandstones, Green River basin. The relative permeabilities were calculated using Corey (1954) equation from drainage capillary pressure experiment. Data courtesy from www.discovery-group.com.

These data confirm our simulation of relative permeability for tight gas

sandstones, which shows decreasing in water relative permeability and increasing in gas

relative permeability as cementation and ductile compaction increase. In addition, the

irreducible water saturation increases as the cement/ductile compaction reduce

connectivity (Chapter 4). The tight gas relative permeability data do not show the

permeability jail trend from Shanley et al., (2004).

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

100.0

0.0 20.0 40.0 60.0 80.0 100.0

Krg

SW%

12553.7 ft

13672.5 ft

12520.9 ft

11587.2 ft

11548 ft

11956.1 ft

3433.8 ft

2717.1 ft

ф = 1.3%

ф = 20%

ф = 17.7%

ф = 9.1%

ф = 6%

ф = 4.4%

ф = 3.4%

ф = 2.8%

123

Figure 5.13- Irreducible water saturation (Swi) increases by decreasing the porosity. This is the reason for increasing the gas relative permeability in tight gas sandstones (Figure 5.12). Swi values measured at 600 psia (4140 kPa), air-brine capillary pressure (equivalent to approximately 120 m above free-water level). Average Swi≈12% for rocks with фi>8%; average Swi≈20% with rocks with фi=6% and average Swi≈40% for rocks with фi=3%. The regression line shown represents '(* � 10$:;.=>/?@-ABфDE..;F%(data courtesy from Castle and Byrnes, 1998). 5.6. Summary

Pore geometry has a large effect on gas-water relative permeability curves. In

general, rocks with low amounts of ductile grains and cement have low irreducible water

saturation and relatively have a large amount of pore space available for flow. This

condition is due to high connectivity of pore space, which leads to large changes in

saturation during two-phase flow. Samples with high amount of ductile grains, cement or

both have high amount of irreducible water saturation and little room for fluid to flow.

The low connectivity of pore space due to increasing the blocked throats is the reason for

this behavior.

124

The porosity-reducing mechanisms change pore throat size distributions (in

Chapter 3). This affects the phase relative permeabilities. Simulation of drainage in

model tight gas sands shows that by decreasing the connectivity of pore space due to

diagenesis (compaction of ductile grains and cementation), the water relative

permeability curves shift toward right. The gas relative permeability curves move to the

higher values of gas relative permeability at a given water saturation. The consequence of

this process is the increasing of irreducible water saturation. Simulated relative

permeability curves show good agreement with curves predicted from drainage capillary

pressure experiment using an empirical correlation.

The comparison of relative permeability with experimental data suggests that we

could predict the major flow channels accurately. The invasion percolation model of

drainage relative permeability correctly predicts the fluid configuration in the tight gas

sandstone although the predicted capillary pressure curves do not show good agreement

with experimental data (Chapter 4). It seems that the fluid configuration is predicted

correctly but in the wrong values of curvatures (capillary pressure) and saturations. A

possible explanation is that the values of inscribe radii (throat sizes) used in calculation of

hydraulic conductance are correct but the distribution of those values is wrong. The

capillary pressure is very sensitive to throat size distribution but the relative permeability

simulation shows a relative configuration of two phases together and it does not depend

on the exact position of the throat values. Thus, the relative permeability simulation could

be correct even though the fluid configurations were obtained at saturations and

curvatures inconsistent with capillary pressure measurements.

It seems that the permeability jail from Shanley et al., 2004 paper for two-phase

flow relative permeability of tight gas sandstone does not occur during a drainage-only

displacement. Even relative permeability data from actual tight gas samples from

different basins do not show permeability jail. The gas relative permeability always

increases by decreasing the porosity of the sample. Only, the water relative permeability

part of the Shanley et al., plot is consistent with drainage simulation.

125

We have quantified the sensitivity of relative permeability to rock structure. In

addition, we have predicted the changes in relative permeability due to compaction of

ductile grains and cementation. The models provided in this work in turn can provide

insight into the flow properties of tight gas reservoirs, particularly the sensitivity of gas

permeability to water saturation. Such models are useful for predicting the storage and

matrix flow capacities in tight gas sands under both single and two-phase flow

conditions.

126

Chapter 6: Conclusions and recommendations We modeled two of the main porosity reducing mechanisms in tight gas

sandstones; cementation and deformation of ductile grains during compaction. The effect

of these mechanisms on the geometry of intergranular pore space was studied in model

sediments (packing of spheres).

A compaction test was done in PFC2D software on a mixture of ductile and hard

grains. An initial hydrostatic pressure was applied to all four walls of the packing for

simulating the hydrostatic pressure in subsurface rocks. In addition, a one-dimensional

axial load was applied as an overburden pressure to the packing. The load finds a path

through the hard grains after deformation of ductile grains. By increasing the load, the

ability of soft grains to support the load decreases, and the load transfers to the adjacent

hard grains. This makes a cage of hard grains, each in point contact with the others.

Ductile grains are suspended within the cages. When enough cages form in the whole

packing, a rigid lattice or framework is established and compaction stops.

A large set of computer generated sphere packings was used to model porosity

loss during compaction of sediments containing ductile grains. The penetrable sphere

model of lithic grains deformation predicts trends of porosity reduction during

compaction of brittle, moderately ductile, and very soft grains. The model yields good

agreement with hydrothermal compaction experiments reported in the literature, when the

radius of the hard core in the ductile grains is chosen in an appropriate range. With

different, mutually exclusive ranges for the hard core radius, this simple geometric model

correctly predicts trends in porosity during compaction for brittle, moderately ductile, and

very soft grains. The model shows the emergence of “cages” of contacts between hard

grains and/or rigid cores of ductile grains. These cages form a load-bearing framework

around ductile grains prevent further porosity loss during burial. Therefore, the model

correctly predicts the preservation of porosity even in sediments containing 95% ductile

matter.

The model also quantifies the changes in throat size distribution resulting from

deformation of ductile grains and from overgrowth cement. It also identifies the closure

127

of pore throats. The closed pore throats give a direct measure of the reduction of

connectivity of pore space and reduction of permeability in tight gas sands. The predicted

effective permeability has a good agreement with effective permeability in quartz-

cemented sandstone.

The percolation threshold was calculated from drainage simulations for low

porosity packings. The threshold increases as porosity decreases. This is a result of

decreasing the size of pores and throats and closure of some throats due to compaction of

grains and cementation.

The simulated drainage curves are sensitive to the amount of ductile grains, the

ductility of ductile grains (changing the rigid radius of ductile grains) and the amount of

cementation. The shape of drainage curves changes differently depending on the

influence of ductile grain compaction and cementation on throats size distributions. All of

the drainage curves show increases in the amount of irreducible water saturation and

entry pressure as the degree of compaction or cementation increases.

By comparing the simulation result with mercury intrusion measurements on tight

gas sandstone, we realized that few of the experimental data exhibits the same percolation

behavior observed in the network simulation results. Various numerical experiments,

such as replacing throat sizes in the network with values taken directly from mercury

experiments, did not improve the agreement. It seems that the intergranular pore

networks in many samples are very different from the ones in the network simulations.

The intergranular pore network in a sample from the Western US, obtained from x-ray

tomography, does not show any connected structure. In contrast the network of

intergranular voids in the compacted/cemented packings remains connected at small

porosity. This difference is an important topic for future work.

Pore geometry has a large effect on gas-water relative permeability curves. In

general, rocks with small amounts of ductile grains and cement have low irreducible

water saturation and relatively have a large amount of pore space available for flow. This

condition is due to high connectivity of pore space, allowing a wide range of saturations

during two-phase flow. Samples with large amounts of ductile grains, cement or both

128

have large irreducible water saturation and little room for fluid to flow. The low

connectivity of pore space due to throats blocked by cement or ductile grain deformation

is the reason for this behavior.

The porosity-reducing mechanisms change pore throat size distributions. This

affects the phase relative permeabilities. Simulation of drainage in model tight gas sands

shows that by decreasing the connectivity of pore space due to diagenesis (compaction of

ductile grains and cementation), the water relative permeability curves shift toward right

(i.e. smaller values of relative permeability at a given saturation). The gas relative

permeability curves move to the higher values of gas relative permeability at a given

saturation. The consequence of this process is the increasing of irreducible water

saturation. Simulated relative permeability curves show a good agreement with trends of

relative permeabilities inferred from drainage experiments using Corey's formula.

The comparison of relative permeability with experimental data suggests that the

model rocks predict the major flow channels accurately. The invasion percolation model

of drainage relative permeability appears to predict the fluid configuration that controls

relative permeability despite the fact that the predicted capillary pressure curves do not

show good agreement with experimental data. A possible explanation is that the values of

inscribed radii (throat sizes) used in calculation of hydraulic conductance are correct but

the distribution of those values is wrong. The capillary pressure is very sensitive to throat

size distribution but the relative permeability simulation shows a relative configuration of

two phases together and it does not depend on the exact position of the throat values.

Thus, the relative permeability simulation could be correct even though the fluid

configurations were obtained at saturations and curvatures inconsistent with capillary

pressure measurements.

Permeability jail (Shanley et al., 2004) for gas-water relative permeability in tight

gas sandstones does not occur during a drainage-only displacement. Relative

permeability data reported in literature from actual tight gas samples from different

basins do not show permeability jail. In the models studied here, the gas relative

permeability always increases when the porosity decreases due to cementation or ductile

129

grain deformation. Only the water relative permeability part of the Shanley et al., plot is

consistent with our drainage simulations.

6.1. Future work

We have quantified the sensitivity of relative permeability to rock structure. In

addition, we have predicted the changes in relative permeability due to compaction of

ductile grains and cementation. The models provided in this work in turn can provide

insight into the flow properties of tight gas reservoirs, particularly the sensitivity of gas

permeability to water saturation. Such models are useful for predicting the storage and

matrix flow capacities in tight gas sands under both single and two-phase flow

conditions.

The drainage capillary pressure simulation result of tight gas sandstone based on

network modeling exhibits some difficulty in agreement with experimental data. The

literature shows that network model approach used in this research works well for

conventional rocks, which are not as heterogeneous as tight gas sandstones and show

percolation behavior in even experimental data. Thus, the following suggestions are

recommended for future simulation of capillary pressure data in tight gas sandstones. 1)

The experiment processes used to measure the capillary pressure data in tight gas

sandstone samples may not be appropriate for this type of rocks. The very large pressures

(exceeding >1000 psi with mercury, and often exceeding 10,000 psi) may alter the rock

structure. 2) The intergranular pore space in the models used here (dense, disordered

packings of penetrable spheres whose radii are increased without moving their centers)

may not be appropriate as an input for network modeling of capillary pressure simulation

in tight gas sandstones. A potential solution for the latter problem is to use a wide

distribution of sphere sizes (normal or lognormal distribution) as an input for a network

model because tight gas sandstones seem to have distributions of grain sizes. An

alternative solution is to use two or three networks of different throat sizes as an input for

network modeling of tight gas sandstones to broaden the percolation region in simulated

drainage curves. Another way is to modify the throat size distribution of the actual rock

by iteration and use the new distribution of throat sizes as an input to the network model.

130

The models provided in this work can provide insight into the flow properties of

tight gas reservoirs, particularly the sensitivity of gas permeability to water saturation.

For future work, it is important to simulate the imbibition process in tight gas sands to

determine whether the gas relative permeability is consistent with the permeability jail of

Shanley et al. (2004). Finally, the models also can be used to understand the flow

connections between tight gas sandstone matrix and the hydraulic fractures needed for

commercial production rates.

131

Appendix A: Types of Pore Space

Thin sections from Travis Peak and Frontier tight gas formations show different

types of pores. Some of them have no porosity which may relate to different reasons for

example: a) the rock had considerable lithics or other soft grains which lead to porosity

loss during compaction by flowing into the pore space. Pressure solution is common in

this kind of thin sections (Figure A.1and Figure A.2). b) Cementation of quartz

overgrowths or other cements were so high, which lead to closing the pore space (Figure

A.3, Figure A.4 and Figure A.5).

Figure A.1- Ductile deformation of mica-rich lithic; there is no porosity in this thin section. The lithic grains flow into the pore space. Ord. Martinsburg Ss., S. Appalachian Basin. Image courtesy from Miliken et al. (2007).

132

Figure A.2- Compaction leads to Flowing of lithics into pore space. The pressure solution stylolites are marked by arrows. Data from Anderson well no.2, Anadarko basin.

Figure A.3- Quartz overgrowth closed all the visible pore space in this thin section. Travis Peak formation, depth 9560.4 ft.

133

Figure A.4- Carbonate cementation close every single pore space. Travis Peak formation, depth 5974 ft.

Figure A.5- Quartz cement closed all of pore space. Frontier formation, depth 16071.8 ft.

134

Secondary pores are most abundant macroporosity (pores having pore-aperture

radii >0.5 µm) in tight gas sands, which formed by dissolution of detrital grains,

particularly feldspar (Spencer, 1989). Figure A.6 shows this kind of porosity in Travis

peak formation. Dolomite crystals filled some of these pores.

Figure A.6- Solution porosity shown by yellow arrows. Dolomite cement which is marked by red arrows filled the solution porosities. Travis Peak formation, depth 9753 ft.

Another type of pore space is the porosity between quartz overgrowths. The most

observed porosity in thin sections occurs when the quartz cement has growth large

enough and the remaining pore space between the grains becomes small (Figure A.7).

Sometimes quartz overgrowth is so minor that visible pore space remains between grains

(Figure A.8). This can be seen when a thin layer of clay minerals surrounds the quartz

grain and there is no overgrowth cement (Dewers and Ortoleva, 1991). When rim clay is

not complete overall the quartz, there is some narrow pass which relate quartz grain with

fluids in pore space and the overgrowth quartz is formed near these passes (Figure A.9).

Large opening porosity is formed when the rim clay is complete and the grain doesn’t

135

have any relation with fluid in pore space (Figure A.10). This kind of porosity will form

when there is early precipitation of cement (clay, carbonate…) in the diagenetic process.

Figure A.7- Small pore space between quartz overgrowths marked by red arrow. Quartz overgrowth is complete in this thin section. Travis Peak formation, depth 8246.1 ft

136

Figure A.8- There is large pore space between quartz overgrowths. Quartz overgrowth is very thin so the pore space between grains is remained large. Travis Peak formation, depth 5973.9 ft

Figure A.9- Large pore space remained between grains with incomplete overgrowth quartz. Arrow shows the narrow pass which relates grain with pore space. Travis Peak formation, depth 6205.2 ft.

137

Figure A.10- Rim clays prevent from forming the overgrowth cement and lead to remaining large porosity. Travis Peak formation, depth 5972.7 ft. Many of geologists believe that slots which connect secondary pores are the most

common porosity in tight gas sandstones. These narrow slots which are formed by

precipitation of authigenic cements in intergranular pore network are easily closed by

increasing pressure. Soeder and Randolph (1987) believe that the slot pores are partially

filled or lined with authigenic minerals and mostly consist of illite and mixed-layer clays.

However that slot must have been connected to the flow paths of pore fluids carrying

those minerals in solution. Ostensen (1983) refers to the irregularities in the quartz walls

that prop the slot pore open under reservoir stress as “asperities”. However, there were a

few visible slots like porosity in thin sections of Travis Peak and Frontier formation

(Figure A.11and Figure A.12).

138

Figure A.11- Slot like porosity which is marked by arrows. Travis Peak formation, depth 9984 ft.

Figure A.12- Slot like porosity which is marked by red arrows. Slot pore ended to the large porosity, which was marked by yellow arrows Travis Peak formation, depth 9984 ft

139

Triangle pore was another pore type that was seen in thin sections. This kind of

porosity is formed between the quartz overgrowths (Figure A.7 and Figure A.13).

Figure A.13- Triangle porosity which is marked by red arrow. Travis Peak formation, depth 6843.7 ft

The last type of porosity which can be seen in Travis Peak thin sections is

microporosity. This porosity is available between dolomite crystals and also inside the

alteration minerals (Figure A.14 and Figure A.15).

140

Figure A.14- Microporosity between dolomite crystals. Travis Peak formation, depth 9753 ft.

Figure A.15- Microporosity inside dissolution feldspar (shown by arrows).Travis Peak formation, depth 10112.4 ft

141

Appendix B: Relative permeability curves for tight gas sandstone

The relative permeability curves in this appendix come from www.discovery-

group.com. Relative permeability curves were calculated using Corey (1954) equation

from actual drainage experiment. This correlative approach is based on Byrnes et al.

(2009). The curves show by decreasing the porosity and permeability, the irreducible

water saturation increases and gas relative permeability shifts to the right (they show the

same trend as our simulation result, Chapter 5). The curves are for different basins such

as Green River, Powder River and Uinta.

Figure B.1- Relative permeability curves from Green River basin. The relative permeabilities were calculated using Corey (1954) equation from drainage capillary pressure experiment. By decreasing the porosity, the gas relative permeability increases. Data courtesy from www.discovery-group.com.

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

100.0

0.0 20.0 40.0 60.0 80.0 100.0

Krg

(%

)

SW%

12553.7 ft

13672.5 ft

12520.9 ft

11587.2 ft

11548 ft

11956.1 ft

3433.8 ft

2717.1 ft

ф = 1.3%

ф = 20%

ф = 17.7%

ф = 9.1%

ф = 6%

ф = 4.4%

ф = 3.4%

ф = 2.8%

142

Figure B.2- Irreducible water saturation (Swi) versus porosity for Green River basin data. By decreasing the porosity, the Swi increases. Data courtesy from www.discovery-group.com.

Figure B.3- Relative permeability curves from Powder River basin. The relative permeabilities were calculated using Corey (1954) equation from drainage capillary pressure experiment. By decreasing the porosity, the gas relative permeability increases. Data courtesy from www.discovery-group.com.

0

10

20

30

40

50

60

70

0 5 10 15 20 25

SW

I%

Porosity %

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

100.0

0.0 20.0 40.0 60.0 80.0 100.0

Krg

SW%

7550.1 ft

6996 ft

7538 ft

7053 ft

ф = 23.8%

ф = 16.7%

ф = 7% ф = 4%

143

Figure B.4- Irreducible water saturation (Swi) versus porosity for Powder River basin data. By decreasing the porosity, the Swi increases. Data courtesy from www.discovery-group.com.

Figure B.5- Relative permeability curves from Uinta basin. The relative permeabilities were calculated using Corey (1954) equation from drainage capillary pressure experiment. By decreasing the porosity, the gas relative permeability increases. Data courtesy from www.discovery-group.com.

0

10

20

30

40

50

60

0 5 10 15 20 25

SW

I%

Porosity %

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

100.0

0.0 20.0 40.0 60.0 80.0 100.0

Krg

SW%

6650.5 ft

6482 ft

7825.5 ft

7279.9 ft

6530.3 ft

6515.6 ft

ф = 1.3%

ф = 13.8%

ф = 9.5%

ф = 7%

ф = 3.7%ф = 4.7%

144

Figure B.6- Irreducible water saturation (Swi) versus porosity for Uinta basin data. By decreasing the porosity, the Swi increases. Data courtesy from www.discovery-group.com.

0

10

20

30

40

50

60

70

80

90

100

0 5 10 15

SW

I%

Porosity %

145

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VITA

Maryam Mousavi received her B.Sc. degree in geology as a first ranked student from

University of Tehran, Iran on August 2000. She pursued her M.Sc. degree in petroleum

geology from Tehran University, where she graduated as a first ranked student in 2003.

She worked as a teaching assistant for two courses in University of Tehran. From June

2003 to August 2004, she worked at National Iranian Oil Company as a researcher

working on petrophysical modeling of two gas reservoirs in Persian Gulf. In September

2004, she started her PhD degree in petroleum engineering in The University of Texas at

Austin. She worked as a teaching assistant in several courses during her PhD study in

petroleum engineering department. She began working with Dr. Bryant as a research

assistant in September 2005 where she worked on grain based modeling of two-phase

flow in tight gas sandstones. Maryam worked as a researcher intern for Baker Hughes in

summer 2008. She is a recipient of 2006 "The University of Texas continuing Doctoral"

fellowship. She presented technical articles in several oil and gas symposiums in the

United States. Her research interests include petrophysics, log analysis, reservoir

characterization of carbonates and sandstones, and pore-scale petrophysics.

Permanent address: 3457 Lake Austin Blvd. #A, Austin, Texas, 78703

This dissertation was typed by Maryam Mousavi