Modeling Storage and Flow of Fluids in Shale Reservoirs

University of Calgary

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Modeling Storage and Flow of Fluids in Shale

Reservoirs

Haghshenas, Behjat

Haghshenas, B. (2017). Modeling Storage and Flow of Fluids in Shale Reservoirs (Unpublished

doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/26951

http://hdl.handle.net/11023/3840

doctoral thesis

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UNIVERSITY OF CALGARY

Modeling Storage and Flow of Fluids in Shale Reservoirs

by

Behjat Haghshenas

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

GRADUATE PROGRAM IN CHEMICAL AND PETROLEUM ENGINEERING

CALGARY, ALBERTA

MAY, 2017

© Behjat Haghshenas 2017

I

Abstract

Recent development of shale gas reservoirs has led to a revolution in the global energy market.

The shale gas industry has expanded rapidly through the application of new drilling and

completion technologies, particularly horizontal wells completed in multiple hydraulic fracture

stages. While these technologies play a critical role enabling economical production from these

resources, uncertainty in the understanding of basic shale gas reservoir properties, and methods

used to characterize them, has led to inefficiencies in shale gas resource development.

This thesis addresses uncertainties in the characterization of fluid storage and transport

mechanisms in shales, and uses new methods for characterization in exploring enhanced

recovery options for shale-hosted hydrocarbons. The primary gas storage mechanisms in

organic-rich shales are free gas storage and adsorption; however, there is a significant amount of

uncertainty in modeling these storage mechanisms in shale. Of the adsorption models tested, the

simplified local density (SLD) model was found to be the most useful for shale gas storage

estimation. The model was used not only for adsorption modeling, but also to rigorously correct

free gas storage calculations for the presence of adsorbed phase volume. Further, the SLD model

was used to predict changes in fluid behaviour within the confined pore space of shale reservoirs.

An important contribution of this thesis is the estimation of gas storage and transport parameters

from shale reservoir drill cuttings. Low-pressure (N2 and CO2) adsorption data was collected on

“artificial” (crushed rock) shale cuttings, and used, in combination with the SLD model to

predict high-pressure/high-temperature adsorption of hydrocarbons. Further, adsorption rate

data, collected on small masses of artificial cuttings, combined with sophisticated numerical

modeling which takes into account the physics of gas storage and transport through shale, was

used to estimate shale gas diffusivity/permeability.

Finally, the importance of hydrocarbon adsorption and diffusivity for predicting hydrocarbon

liquid recovery after CO2 injection was investigated. A history-matched (flowback data) multi-

fractured horizontal well completed in a tight liquid-rich reservoir was used as a starting point

for sensitivities using CO2 injection. Unique to this study, fluid compositions as a function of

depth in the reservoir were available. Simulation of CO2 huff-n-puff schemes using this

II

calibrated model demonstrated that inclusion of adsorption/diffusion effects has an important

impact on hydrocarbon liquid recovery.

III

Acknowledgements

I am so grateful to my supervisor, Dr. Nancy (Shengnan) Chen for her invaluable technical

suggestions, encouragement and support. Her great comments and advices helped me in different

steps of my thesis completion.

I would like to extend my sincere gratitude and appreciation to my co-supervisor, Dr.

Christopher R. Clarkson. His phenomenal research ideas, scholarly advices, and constant

encouragement and support helped me to accomplish this work. I learned from him not only how

to perform a research, but also, learned how to think and behave like a professional.

I would like to acknowledge Dr. Josephine Mary Hill and Dr. Robert A Marriott for serving on

my supervisory committee and Dr. Tatyana Plaksina and Dr. Laurence Robert Bentley for

serving on my examining committee and also Dr. Farzam G Javadpour for serving on my

external examiner.

My thanks also go to the sponsors of Tight Oil Consortium (TOC) and Department of Chemical

and Petroleum Engineering at University of Calgary.

Constant encouragement and support from my fellow researchers, Farhad Qanbari, Amin

Ghanizadeh, Atena Vahedian, Samuel Aquino are sincerely appreciated.

Journal of Natural Gas Science and Engineering (Elsevier) and Society of Petroleum Engineering

(SPE) are acknowledged for copyright permission for the published papers.

I am also so grateful to my family and friends for their support and encouragement.

IV

Table of Contents

Abstract .......................................................................................................................................................... i

Acknowledgements ...................................................................................................................................... iii

Table of Contents ......................................................................................................................................... iv

List of Tables ............................................................................................................................................. viii

List of Figures .............................................................................................................................................. ix

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

1.1 Background ................................................................................................................................... 1

1.2 Problem Statement and Motivations ............................................................................................. 3

1.3 Outlines of Thesis ......................................................................................................................... 5

1.4 References ..................................................................................................................................... 6

Chapter 2 Multiple Reservoir Engineering Applications of the Simplified Local Density Model for

Shale Gas and Liquid Rich Shale Reservoirs ................................................................................................ 8

2.1 Abstract ......................................................................................................................................... 8

2.2 Introduction ................................................................................................................................... 8

2.3 Theory and Methods ..................................................................................................................... 9

2.3.1 Single-component modeling with SLD model ...................................................................... 9

2.3.2 Multi-component modeling with the SLD model ............................................................... 15

2.3.3 Predicting high-pressure/temperature adsorption from low-pressure/temperature adsorption

using the SLD model .......................................................................................................................... 17

2.3.4 Fluid property modeling in confined pore spaces using the SLD model ............................ 18

2.4 Applications ................................................................................................................................ 20

2.4.1 Predicting phase density profile .......................................................................................... 20

2.4.2 Matching single-component adsorption data and predicting high-pressure adsorption (New

Albany Shale) ...................................................................................................................................... 21

2.4.3 Predicting high-pressure methane adsorption from low-pressure adsorption (Duvernay

Shale) 23

2.4.4 Predicting high-pressure methane adsorption from low-pressure adsorption (Montney

Shale) 30

2.4.5 Fluid property modeling ...................................................................................................... 34

2.5 Conclusions ................................................................................................................................. 36

2.6 Nomenclature .............................................................................................................................. 36

V

2.7 References ................................................................................................................................... 38

Chapter 3 Characterization of Multi-Fractured Horizontal Shale Wells using Drill Cuttings:

Permeability/Diffusivity Estimation ........................................................................................................... 41

3.1 Abstract ....................................................................................................................................... 41

3.2 Introduction ................................................................................................................................. 42

3.2.1 Diffusivity/permeability studies performed for coal ........................................................... 43

3.2.2 Diffusivity/permeability studies performed for shale ......................................................... 44

3.3 Model summary and new model development ........................................................................... 46

3.3.1 Conventional bidisperse model for coal .............................................................................. 46

3.3.2 Modified bidisperse model development for shale ............................................................. 48

3.3.3 Modified unipore model development for shale ................................................................. 52

3.3.4 Model fit to data .................................................................................................................. 52

3.3.5 Calculating apparent permeability ...................................................................................... 53

3.4 Experimental procedure .............................................................. Error! Bookmark not defined.

3.5 Results ......................................................................................................................................... 56

3.5.1 Application of unipore model to Duvernay low-pressure adsorption rate data .................. 57

3.5.2 Application of conventional bidisperse model (constant coefficients) to Duvernay low-

pressure adsorption rate data ............................................................................................................... 58

3.5.3 Application of modified bidisperse model (variable coefficients) to Duvernay low-pressure

adsorption rate data ............................................................................................................................. 59

3.5.4 Extension of the model to Montney samples ...................................................................... 62

3.6 Discussion ................................................................................................................................... 64

3.7 Conclusions ................................................................................................................................. 66

3.8 Nomenclature .............................................................................................................................. 67

3.9 Acknowledgements ..................................................................................................................... 69

3.10 References ................................................................................................................................... 69

Chapter 4 New Models for Reserve Estimation and Non-Darcy Gas Flow in Shale Gas Reservoirs ..... 74

4.1 Abstract ....................................................................................................................................... 74

4.2 Introduction ................................................................................................................................. 75

4.2.1 Gas-In-Place Calculations for Shale ................................................................................... 76

4.2.2 Material Balance Calculations for Shale ............................................................................. 77

4.2.3 Permeability Modeling in Shale .......................................................................................... 78

VI

4.3 New Approach to Shale Gas-In-Place and Material Balance Calculations, and Numerical

Simulation ............................................................................................................................................... 80

4.3.1 Gas in Place Calculation ..................................................................................................... 81

4.3.2 Material Balance Equation .................................................................................................. 82

4.3.3 Porosity Correction ............................................................................................................. 83

4.3.4 Pressure-Dependent Sorbed Phase Density ........................................................................ 83

4.4 Results ......................................................................................................................................... 86

4.4.1 Volumetric Gas-in-Place Calculation Results ..................................................................... 86

4.4.2 Material Balance Results .................................................................................................... 90

4.4.3 Porosity Correction and Production Results ....................................................................... 92

4.5 Discussion ................................................................................................................................... 95

4.6 Conclusions ................................................................................................................................. 97

4.7 Nomenclature .............................................................................................................................. 98

4.8 Acknowledgement .................................................................................................................... 100

4.9 References ................................................................................................................................. 101

Appendix A: Proof of Eqs. 1 and 2 ....................................................................................................... 103

Appendix B – Derivation of Eq. 30 ...................................................................................................... 107

Chapter 5 Modeling PVT Behavior of Gas-Condensate System under Pore Confinement Effects:

Implications for Rate-Transient Analysis of Gas-Condensate Shale Plays .............................................. 111

5.1 Abstract ..................................................................................................................................... 111

5.2 Introduction ............................................................................................................................... 112

5.3 Theory ....................................................................................................................................... 113

5.3.1 Thermophysical Properties of Fluids under Confinement in Nanopores .......................... 113

5.3.2 Use of SLD Model for Estimating Gas Properties under Pore Confinement .................... 115

5.3.3 Non-Darcy flow Calculations, Taking into Account Adsorbed Layer Thickness Changes,

Diffusivity and Slippage Effects ....................................................................................................... 118

5.4 Application ................................................................................................................................ 119

5.4.1 Confined Fluid Property Estimation ................................................................................. 119

5.4.2 RTA of Numerical Simulation Results ............................................................................. 122

5.5 Discussion ................................................................................................................................. 128

5.6 Conclusions ............................................................................................................................... 129

5.7 Nomenclature ............................................................................................................................ 130

5.8 Acknowledgements ................................................................................................................... 131

VII

5.9 References ................................................................................................................................. 131

Chapter 6 Simulation of Enhanced Recovery using CO2 in a Liquid-Rich Western Canadian

Unconventional Reservoir: Accounting for Reservoir Fluid Adsorption and Compositional Heterogeneity

135

6.1 Abstract ..................................................................................................................................... 135

6.2 Introduction ............................................................................................................................... 136

6.3 Theory and Methods ................................................................................................................. 140

6.3.1 Diffusion ........................................................................................................................... 140

6.3.2 Miscibility ......................................................................................................................... 141

6.3.3 Adsorption ......................................................................................................................... 142

6.4 Simulation Model Setup ............................................................................................................ 147

6.4.1 Reservoir Model ................................................................................................................ 148

6.4.2 Fluid Properties ................................................................................................................. 150

6.4.3 Rock Properties ................................................................................................................. 151

6.4.4 Huff-n-Puff Operating Conditions .................................................................................... 152

6.5 Results ....................................................................................................................................... 153

6.5.1 Effect of Injection Time Period ........................................................................................ 154

6.5.2 Effect of Soaking Time Period .......................................................................................... 155

6.5.3 Effect of Reservoir Fluid Composition and Reservoir Fluid Heterogeneity ..................... 157

6.6 Discussion ................................................................................................................................. 158

6.7 Conclusions ............................................................................................................................... 160

6.8 Appendix C ............................................................................................................................... 160

6.9 Nomenclature ............................................................................................................................ 161

6.10 Acknowledgements ................................................................................................................... 161

6.11 References. ................................................................................................................................ 162

Chapter 7 Conclusions and Future Work ............................................................................................... 168

7.1 Conclusions ............................................................................................................................... 168

7.2 Future Work .............................................................................................................................. 169

Copyright Permissions .............................................................................................................................. 171

List of Tables

Table 2-1 Constants for Lee’s viscosity correlations. ................................................................. 20

Table 3-1 — Summary of crushed-rock sample properties (Duvernay formation). ..................... 54

Table 3-2 — Unipore and bidisperse numerical model parameters obtained from Duvernay shale

adsorption rate data (Duvernay formation). .................................................................................. 61

Table 3-3 — Summary of crushed-rock sample properties (Montney). ....................................... 62

Table 3-4 — Bidisperse numerical model parameters obtained from Montney adsorption rate

data. ............................................................................................................................................... 62

Table 4-1 — PVT and reservoir input parameters for volumetric OGIP calculation for shale A-C

(modified after Williams-Kovacs, 2012). ..................................................................................... 87

Table 4-2 — Volumetric OGIP results for shale A-C using conventional and corrected gas

storage models. ............................................................................................................................. 88

Table 4-3 — Summery of case studies investigated in this paper. ............................................... 93

Table 5-1 Constants for Lee’s viscosity correlations. ............................................................... 117

Table 5-2 Bulk fluid (gas) composition. ................................................................................... 120

Table 5-3 Numerical model inputs used in the generation of synthetic cases. ......................... 124

Table 5-4 — Results of RTA with and without pore confinement effects. ................................ 126

Table 6-1 — Apparent diffusivity of different components used in this study. ......................... 141

Table 6-2 — Final value of tuned hydraulic fracture parameters at the end of history match

performed by Clarkson et al. (2016a). ........................................................................................ 152

List of Figures

Figure 2-1 — Schematic of a slit-shaped pore model showing the variables used to calculate

distances in the SLD approach. Modified from Zuo, 2015........................................................... 10

Figure 2-2 — Use of the SLD model to estimate gas density profiles (red solid line) in organic

matrix-pores in (a) a 2 nm diameter pore and (b) a 10 nm diameter pore. L is the width of the

pore, while z is the distance from one of the pore walls. The approximate adsorbed layer

thickness in both cases is highlighted with purple shading. The bulk phase density near the

center of the pore is shown with a dashed red line. The adsorbed phase is everything above the

dashed line. From Clarkson et al., 2016........................................................................................ 21

Figure 2-3 — Match of SLD model to high pressure CH4 and CO2 Gibbs (excess) adsorption

data. Data from Chareonsuppanimit et al. (2012) for the New Albany shale. From Clarkson et

al., 2016. ........................................................................................................................................ 22

Figure 2-4 — Predictions of SLD model for high pressure CH4 and CO2 mixture adsorption. The

2D EOS predictions are also given for comparison. From Clarkson et al., 2016. ........................ 23

Figure 2-5 — (a) N2 adsorption/desorption isotherms (hysteresis loop) and (b) CO2 adsorption

isotherms collected for 4 Duvernay shale artificial cuttings samples. From Clarkson et al., 2016.

....................................................................................................................................................... 24

Figure 2-6 — SLD model match to low-pressure (a) N2 isotherms and (b) CO2 isotherms. The

SLD model was fitted to the adsorption branch of both isotherm datasets. The relative pressure

range of around 0.05-0.2 was selected for nitrogen because this is the pressure range used for

BET model analysis. p0 is vapor pressure of gas at experimental temperature, i.e., 77 K for N2

and 273 K for CO2. From Clarkson et al., 2016. .......................................................................... 25

Figure 2-7 — Specific surface areas calculated from all models from (a) low-pressure N2

adsorption data and (b) low-pressure CO2 adsorption data. From Clarkson et al., 2016. ............. 25

Figure 2-8 — Pore size distributions obtained from N2 adsorption data (using BJH and DFT

models) and from CO2 adsorption data (using the DFT model). From Clarkson et al., 2016. ..... 27

X

Figure 2-9 — High-pressure excess and absolute adsorption isotherms predicted from the SLD

model for (a) N2 and (b) CO2 for 4 Duvernay shale artificial cuttings samples. From Clarkson et

al., 2016. ........................................................................................................................................ 28

Figure 2-10 — Fit of the SLD model to high-pressure/temperature (383.15 K) CH4 isotherms

measured on 2 Duvernay samples taken from the same interval as the artificial cuttings samples.

Solid lines are SLD model fit to the experimental data. From Clarkson et al., 2016. .................. 29

Figure 2-11 — Predicted high-pressure, high-temperature methane isotherms for 4 Duvernay

artificial cuttings samples. From Clarkson et al., 2016. ................................................................ 30




BET model analysis. ..................................................................................................................... 31

Figure 2-13 — High-pressure absolute adsorption isotherms predicted from the SLD model for

C1, C2, C3, C4+ and CO2 on Montney artificial cuttings sample#14. ............................................ 32

Figure 2-14 — a) Pore size distributions obtained from N2 adsorption data (using BJH model)

and from CO2 adsorption data (using the DFT model). Two modal pore sizes are around 0.55 and

3 nm for the 7G artificial cuttings sample#14. b) The ratio between the pore volume occupied by

adsorbed phase and the total hydrocarbon for each pore size. ...................................................... 33

Figure 2-15 — (a) Long-term constant GOR observed for producing liquids-rich shale wells

(Altman et al., 2014) and (b) use of the SLD model to predict the phase envelope for a gas

condensate system as a function of pore size, from 300 nm to 2 nm (Clarkson and Haghshenas,

2016). From Clarkson et al., 2016. ............................................................................................... 35

Figure 3-1 — Conceptual schematic of a) unipore and b) bidisperse model (after Clarkson and

Bustin, 1999). ................................................................................................................................ 47

Figure 3-2 — Experimental data obtained from SMP-200 and bidisperse model match for the

two crushed Duvernay shale samples a) sample #5, b) sample #8. The low precision of pressure

data is evident, leading to lower confidence in extracted permeability values. ............................ 57

XI

Figure 3-3 — Experimental data (for one crushed shale sample) obtained from 3Flex and unipore

model match for two pressure steps for Duvernay sample #5. ..................................................... 58

Figure 3-4 — Experimental data (for crushed Duvernay shale sample #5) obtained from the

3Flex device and conventional (constant coefficient) bidisperse model match for two pressure

steps. The bidisperse model match to the fast decay portion is better than the slow decay portion.

....................................................................................................................................................... 59

Figure 3-5 — Experimental data (for crushed shale sample #5) obtained from the 3Flex device

and new (variable coefficient) bidisperse model match for two pressure steps. The new

bidisperse model with variable coefficients is successful in matching both fast and slow decay

portions of the data........................................................................................................................ 60

Figure 3-6 — Apparent permeability trend with pressure for crushed shale sample #5. Apparent

permeability decreases as pressure increases. ............................................................................... 61

Figure 3-7 — Experimental data (for crushed Montney sample #14) obtained from the 3Flex

device and new (variable coefficient) bidisperse model match for a) N2 and b) CO2. The


portions of the data........................................................................................................................ 63


crushed Montney sample # 14. The low precision of pressure data is evident, leading to lower

confidence in extracted permeability values. ................................................................................ 64

Figure 3-9 — Pore size distributions obtained for the two studied Duvernay shale samples.

Modified from Clarkson and Haghshenas (2016). ........................................................................ 66

Figure 4-1 — Petrophysical model showing volumetric constituents of gas-shale matrix. a)

conventional model, b) modified after Ambrose (2012), c) new model. ...................................... 80

Figure 4-2 — Langmuir adsorption isotherms for three shale samples. ....................................... 88

Figure 4-3 — Partitioning coefficients of free and adsorbed gas by a) Conventional method, b)

Ambrose method, c) New method. ............................................................................................... 90

XII

Figure 4-4 — Material-Balance plots for shale B in the case of constant sorbed phase density

versus pressure. Plots created by a) uncorrected Clarkson and McGovern equation b) Clarkson

and McGovern equation corrected with Ambrose Method c) Clarkson and McGovern equation

corrected with New Method. ......................................................................................................... 91

Figure 4-5 — Material-Balance plots for shale B in the case of variable sorbed phase density



corrected with New Method. ......................................................................................................... 91

Figure 4-6 — Plot of porosity correction factor vs cumulative production. Considering variable

sorbed phase density with production causes the correction factor to be closer to one, especially

at higher production levels (correction factor equal to one means whole pore volume is available

for free gas and sorbed gas has zero volume. As pressure decreases during production, the sorbed

phase evaporates and allows more of the pore volume to be occupied with free gas. .................. 93

Figure 4-7 — Plot of gas recovery factor vs. time for the case with constant permeability. ........ 94

Figure 4-8 — Plot of gas recovery factor vs. time for the case with variable permeability. ........ 95

Figure 4-9 — Illustration of how to use new straight line method for calculation of free- and

total-gas-in-place. Modified from Haghshenas and Clarkson (2016b, in preparation). .............. 97

Figure 5-1 — Snapshot of methane (small blue spheres) and ethane (grey spheres) molecule

distribution in slit-shaped carbon pore using Monte Carlo simulations. Note the layers of

molecules parallel to the upper and lower organic walls. From Rahmani et al. (2013). ........... 115

Figure 5-2 — Apparent gas permeability calculations for a 5 nm pore radius, accounting for non-

Darcy flow and adsorbed layer thickness changes with pressure. .............................................. 119

Figure 5-3 — Comparison of gas compressibility in the bulk state versus within a 5 nm slit-pore.

..................................................................................................................................................... 120

Figure 5-4 — Comparison of gas formation volume factor in the bulk state versus within a 5 nm

slit-pore. ...................................................................................................................................... 121

Figure 5-5 — Comparison of gas viscosity in the bulk state versus within a 5 nm slit-pore. .... 121

XIII

Figure 5-6 — Comparison of gas phase behavior in the bulk state versus within a 5 nm slit-pore.

..................................................................................................................................................... 122

Figure 5-7 — Base geometry for the synthetic cases. From Clarkson and Qanbari (2015). An

element of symmetry is used to reduce the computation time. ................................................... 123

Figure 5-8 — Case 1: (a) synthetic production data, (b) log-log diagnostic plot (Transient linear

flow is evident from half-slope behavior on the log-log plot). ................................................... 125

Figure 5-9 — Square-root of time plot for Case 1 with and without correction for fluid property

changes. ....................................................................................................................................... 125


flow is evident from half-slope behavior on the log-log plot). ................................................... 127

Figure 5-11 — Square-root of time plot for Case 2 with and without correction for gas apparent

permeability and fluid property changes. .................................................................................... 127

Figure 5-12 — Cumulative condensate production from bulk and pore-confined PVT properties.

The general properties of both of the cases are listed in Table 7.3. ............................................ 129

Figure 6-1 — Separation factor (selectivity) calculations for binary mixtures of CH4 and heavier

hydrocarbons and CO2 using the EL model. In the EL model, the separation factor is assumed

not to be a function of pressure or composition. Modified from Ambrose et al. (2011). ........... 143

Figure 6-2 — Comparison of separation factor (selectivity) calculations for a binary mixture of

CO2-butane+ and CH4-butane+ using the EL model. Modified from Clarkson and Haghshenas

(2013). ......................................................................................................................................... 144




BET model analysis. ................................................................................................................... 146

Figure 6-4 — High-pressure absolute adsorption isotherms predicted from the SLD model for C1,

C2, C3, C4+ and CO2. ................................................................................................................... 147

XIV

Figure 6-5 — Cross-section (created in Petrel™) showing subject well horizontal lateral

trajectory with respect to upper and lower zones (identified during geologic characterization), the

location of 9 successful hydraulic fracturing stages (near heel of the well), and the location of the

cuttings samples analyzed for gas composition (and used in fluid property modeling). Also

projected are the initial designed hydraulic fracturing stages (not yet implemented). A gamma

ray log is provided to illustrate lithology/reservoir quality variation along the lateral. Note: Cross

section depth scale is in true vertical depth subsea (TVDSS). Note 1 m = 3.28 ft. Modified from

Clarkson et al., 2016a. ................................................................................................................. 148

Figure 6-6 — Illustration of the use of local grid refinement (with logarithmic spacing) to

represent propped and un-propped regions of the hydraulic fracture. Note that the fracture height

extends into the upper and lower zones illustrated in Figure 6.5. Modified from Clarkson et al.,

2016a. .......................................................................................................................................... 149

Figure 6-7 — Phase envelopes of the highest and lowest saturation pressure layers. Only the

layers intersected by the hydraulic fracture are shown; the pressures in non-intersected layers

won't change even after long production times. Modified from Clarkson et al. (2016a). ......... 151

Figure 6-8 — Comparison of primary recovery and CO2 huff-n-puff responses for cases

including and neglecting adsorption and diffusion effects. Note that CO2 huff-n-puff only

appears to be beneficial for the case where adsorption and diffusion are included. ................... 153

Figure 6-9 — Effect of injection time on the performance of CO2 huff-n-puff (for the case

accounting for adsorption and diffusion effects). ....................................................................... 154

Figure 6-10 — Effect of soaking time on the performance of CO2 huff-n-puff (for the case

accounting for adsorption and diffusion effects). ....................................................................... 156

Figure 6-11 — Effect of soaking time on the performance of CO2 huff-n-puff (for the case

without adsorption and diffusion effects included). ................................................................... 156

Figure 6-12 — Effect of reservoir fluid composition and heterogeneity on the performance of

CO2 huff-n-puff (for the case accounting for adsorption and diffusion effects). The pink lines

correspond to primary and huff-n-puff cases run assuming a uniform, low saturation pressure

fluid (see Figure 6.7), and the green lines were run assuming a uniform, high saturation pressure

XV

fluid. The black line correspond to the primary production in the case where layer-variable fluid

compositions is included in the model. The huff-n-puff production of variable composition case

is what used in previous sensitivity cases. .................................................................................. 158

1

Chapter 1 Introduction

1.1 Background

Economical gas production from the Barnett Shale in the late 1990’s changed the future of the

North America natural gas industry. In recent years shale gas plays have proven to be extensive,

exploitable resources for fossil fuels. Technologies such as horizontal/lateral drilling and multi-

fractured well completions have made commercial production from shale gas reservoirs possible.

Advances in reservoir characterization and simulation of unconventional gas reservoirs have

provided an approach to study production mechanisms and improve production forecasts. While

much work has been focused on the effect of hydraulic fractures on shale gas production profiles

(Cipolla, 2009; Cipolla, 2010; Rutledge, 2003), there is still a need to evaluate the impact of the

unique reservoir properties of shale on production forecasting associated with both primary and

enhanced recovery. The objective of the current thesis therefore is to address the effect of shale

matrix storage and transport properties, and the communication of the matrix-fracture, on

ultimate production from shale plays, and to develop methods for characterization of key shale

matrix properties.

In the following, a short summary of the main shale features that are considered in this thesis,

and that are important for shale reservoir characterization and simulation, is provided.

Organic matter. Organic matter is the source of gas generation in shale reservoirs. The organic

matter, which has been buried along with the inorganic matter, may consist of kerogen, bitumen

and mobile hydrocarbons and is typically reported as total organic carbon (Sondergeld, 2010). It

is generally believed that kerogen is the major component that contains porosity and gas in the

adsorbed and solution states. Organic matter has a great impact on shale properties, e.g. it lowers

the density, increases porosity, provides the source of the gas, imparts anisotropy, alters

wettability, and causes adsorption. Therefore; the distribution, behavior and concentration of

organic matter is important in economic assessments (Sondergeld, 2010).

Pore size. The shale matrix has pore throat radii in the range of 1 to 200 nanometers, which is

much smaller than that of conventional sandstone and carbonate reservoirs, having pore sizes in

2

the range of 1 to 100 micrometers (Cipolla et al., 2009). Curtis et al., (2010) suggests that, in the

shale matrix, small pores (radii in 3 to 6 nm) dominate the total pore volume. Sondergeld (2010)

also demonstrated that the shale pore volume consists mostly of pores with characteristic

dimensions between 13 nm and 20 nm.

Porosity. The pore space in shales can be classified into three main categories: porous organic

matter, interparticle and intraparticle pore system in the inorganic matrix, and fractures (induced

by hydraulic fracture stimulation and natural fractures) (Loucks et al., 2012).

Gas storage. Organic-rich shale gas reservoirs are collectively referred to as “sorbed gas”

reservoirs because a significant amount of gas storage occurs through physical adsorption onto

the internal surface area of the organic matter and clays or through absorption (solution) within

organic matter. The adsorbed gas portion is reported to be as high as 85% in some shale plays

(Lewis and Antrim Shale) and is dependent on a variety of geologic and geochemical

properties (Canadian Discovery, 2006; Drake, 2007). Strictly speaking, there are multiple

mechanisms for gas storage in coals and organic-rich shales including (Clarkson and

Haghshenas, 2013): 1) Physical adsorption upon internal surface area 2) Conventional

(compressed free gas) storage in natural and hydraulic (induced) fractures 3) Conventional

storage in matrix porosity (organic and inorganic) 4) Solution in formation water 5) Absorption

(solution) in organic matter

Heterogeneity. Compositional variation (organic matter content, mineral composition) makes

shale reservoirs highly heterogeneous. Sondergeld (2010) provided examples of shale texture and

reported that porosity occurs within organics, pyrite framboids, fossils, minerals, between grains,

and in the form of microcracks. Therefore, advanced reservoir engineering methods are required

for modeling gas storage in shale gas reservoirs. Furthermore, the effects of different

configurations of these pore systems on production profiles needs to be quantified.

Gas flow. Transport of gas molecules through tight porous media may occur by various

mechanisms controlled by pressure, temperature, gas properties and pore size, including

Knudsen diffusion, transitional flow, slip flow, viscous flow, adsorbed phase diffusion, and

liquid viscous or condensate flow (Civan, 2013). Therefore, pores in the shale matrix that are

small relative to the mean free path of the gas molecules cause the Darcy equation to be

3

inaccurate for modeling the gas flow in shale gas reservoirs. Researchers (e.g. Javadpour, 2009)

suggest defining an apparent permeability which is different from the absolute rock permeability

which is used in Darcy’s equation. Most researchers emphasize that any petrophysical study only

samples an extremely small portion of a reservoir and generalizations are tempting, and must be

supported with statistical studies to establish the universal applicability of such observations

(Sondergeld, 2010). Therefore, multiple experiments are necessary to determine the apparent

permeability for shale samples of interest.

Pore Configuration and Connectivity. Typically, shale gas reservoirs have been represented by

dual-porosity (dual-porosity, single-permeability) or dual permeability models. The dual-porosity

model assumes the shale matrix is a storage grid and intersecting fracture networks are flow

conduits that convey flow into the well. The dual-permeability model is an extended version of

the dual-porosity model and also accounts for flow through matrix, with matrix and fractures

contributing to well production. Storage and transport mechanisms in shale formations are

different from conventional, naturally-fractured reservoirs; therefore, conventional dual-porosity

and dual-permeability models cannot be directly used for shales and need to be adapted for

specific pore configuration and connectivity in shale rocks.

1.2 Problem Statement and Motivations

1. Under certain P-T conditions, much more gas is stored in the adsorbed state than in the

compressed gas state. This is because of the relatively high density of gases in the adsorbed

state. The mechanism of adsorption can therefore contribute significantly to total gas storage

in organic-rich reservoirs. This storage mechanism also significantly influences the

production mechanism of the gas, and has led to enhanced recovery strategies in organic-rich

reservoirs that involve the use of inert gases (mostly N2 or CO2 or a mixture). Indeed, it is

important to understand how much adsorbed gas is left over at abandonment of primary

depletion operations to understand what resource is left over and if enhanced recovery

operations are viable. However, there is experimental evidence that commonly-used simple

isotherm adsorption models often cannot accurately fit the data and therefore more accurate

adsorption models are required.

4

2. Historically, the focus of adsorption modeling has been on single- and multi-component dry

gases (low molecular weight hydrocarbon gases and inert gases and their mixtures) in

support of primary (coal and shale) and enhanced recovery studies (coal); of these studies,

much of the focus has been on CH4, CO2 and N2 adsorption on coal because of the relevance

to enhanced recovery in coal and CO2 sequestration. With recent interest in liquid-rich shales

(Whitson and Sunjerga, 2012), there is increased interest in the sorption properties of heavier

hydrocarbons (C2+) with a view towards understanding enhanced recovery process in those

systems using lighter hydrocarbons and inert gases as injectants. To date there have been

relatively few studies focused on the prediction of binary gas adsorption or heavy

hydrocarbon adsorption on shales, and therefore additional study is required.

3. For free gas estimation, conventional volumetric approaches assign all available ‘effective

gas pore volume’ to the free gas; however, in order to properly account for the total and free

gas in place, the volume occupied by the adsorbed gas phase must be determined and

subtracted from the free-gas calculation. Volumetric and material-balance calculations should

then be corrected for this effect. Commercial simulators need also to be trained for

accounting the free gas volume variation (increase) due to desorption of adsorbed gas.

4. Transport properties also determine whether or not production from a reservoir is

economically profitable. Although commercial simulators are properly prepared for modeling

Darcy flow in either single or dual-porosity mediums, they cannot account for permeability

variation (increase) due to molecular slippage on pore walls and diffusion through nano-scale

pores of the shale matrix. Therefore, a permeability correction factor must be defined and

used in simulating gas production with commercial simulators.

5. The effect of adsorption on hydrocarbon liquids recovery has not been sufficiently studied.

For hydrocarbon components, adsorption increases strongly with molecular weight. This is

an important observation because, in liquid-rich shales, affinity for adsorption on organic

matter competes with the commercial goal of producing heavier fractions. Ignoring

adsorption and diffusion effects while simulating the enhanced recovery processes in shale

reservoirs could lead to unfavorable results either with flooding or in huff-and-puff

processes.

5

6. The conventional methods proposed previously for shale sample evaluation, are typically

performed on core plug or crushed rock samples obtained from cores. However, sample sizes

typically obtained from MFHWs are small quantities of drill cuttings, which present

additional challenges for characterization. Converting the experimentally-measured data into

porosity, pore geometric parameters, high pressure adsorption prediction, and

permeability/diffusivity requires the application of an appropriate model and assumptions.

7. At small (nano) scales the confined hydrocarbon phase behavior deviates from bulk

measurements due to the effect of pore wall/gas molecules interactions. Without considering

this effect a conventional reservoir simulator will likely not be able to explain the

inconsistent produced GOR observed in the field compared to simulated results.

1.3 Outlines of Thesis

In the first research chapter (Chapter 2) of this thesis, the simplified local density (SLD) model is

used to model supercritical fluid adsorption, characterize pore structure, and estimate high-

pressure adsorption on organic rich shales. Because the only source of rock samples typically

available from multi-fractured horizontal wells (MFHWs) are small amounts (< 2-3 g) of rock

(drill) cuttings, an innovative method to estimate high-pressure/temperature adsorption of

hydrocarbons using low-pressure adsorption (LPA) of N2 and CO2 (which can be used to

measure adsorption on small amounts of sample) modeled with SLD is proposed. Finally, using

pore structure information extracted from LPA, the SLD model is used to predict hydrocarbon

fluid phase behavior in the confined space of shale nanopores.

In the second research chapter (Chapter 3), laboratory and modeling procedures for extracting

permeability and diffusivity from drill cuttings is discussed. As with Chapter 2, low-pressure

adsorption (LPA) is again used due to the availability of only small sample amounts from

cuttings – this time the rate-of-adsorption data is simulated using a new bidisperse pore structure

numerical model, that accounts for the physics of gas storage and transport through the shale

pore structure, for the purpose of extracting permeability and diffusivity.

In the third research chapter (Chapter 4), using adsorption modeling insight provided in Chapter

2, the SLD model is used for improving reserve estimation in shale reservoirs with high

adsorption affinity. The free gas porosity is formulated as a function of pressure (accounting for

6

the volume of the adsorbed phase) and applied in a numerical simulation to understand its effect

on recovery predictions.

In the fourth research chapter (Chapter 5), the understanding of shale fluid flow and storage

mechanisms gained from Chapters 2 and 3 is used to improve rate-transient analysis (RTA) for

liquid-rich shale reservoirs. Specifically, the impact of pore confinement on RTA is investigated

by including the change in fluid PVT behavior (using SLD model) and non-Darcy flow.

In the fifth research chapter (Chapter 6), the simultaneous effect of adsorption and condensation

on enhanced production from liquid-rich shale, while considering diffusion effects, is studied

using a real field case.

Note on authorship. This thesis is paper-based. I am the first author on the papers that were

modified to become Chapters 3-6, meaning that I was involved in the development of the

research idea, performed the majority of the technical work, and drafted the first versions of the

papers. Chapter 2 is unique in that I have taken my original contributions to previously-

published papers (ex. Clarkson and Haghshenas 2013; Clarkson and Haghshenas 2016) and

assembled them into a new chapter.

1.4 References

1. Canadian Discovery. 2006. Shale Gas in North America. Canadian Discovery Digest, 6 pp.

B1-B41.

2. Cipolla, C.L., Lolon, E., Erdle, J. and Tathed, V.S., 2009, January. Modeling well

performance in shale-gas reservoirs. In SPE/EAGE Reservoir characterization and simulation

conference. Society of Petroleum Engineers.

3. Cipolla, C.L., Lolon, E.P., Erdle, J.C. and Rubin, B., 2010. Reservoir modeling in shale-gas

reservoirs. SPE reservoir evaluation & engineering, 13(04), pp.638-653.

4. Civan, F. 2013. Impact of Fluid Behavior Modification under Elevated Pressure and

Temperature Conditions on Shale-Gas/Condensate Reservoir Engineering & Production

Analysis. Society of Petroleum Engineers. doi:10.2118/167186-MS

5. Clarkson, C.R. and Haghshenas, B. 2013. Modeling of Supercritical Fluid Adsorption on

Organic-Rich Shales and Coal. Paper SPE 154532 presented at the SPE Unconventional

7

Resources Conference-USA, The Woodlands, Texas, USA.

http://dx.doi.org/10.2118/164532-MS.

6. Clarkson, C.R. and Haghshenas, B. 2016. Characterization of multi-fractured horizontal shale

wells using drill cuttings: 1. Fluid-in-place estimation. Journal of Natural Gas Science and

Engineering, 32, pp.574-585.

7. Curtis, M.E., Ambrose, R.J., Sondergeld, C.H., et al. 2010. Structural Characterization of Gas

Shales on the Micro- and Nano-Scales. Paper SPE 137693 presented at the Canadian

Unconventional Resources and International Petroleum Conference held in Calgary, Alberta,

Canada, 19-21 October. http://dx.doi.org/10.2118/137693-MS.

8. Haghshenas, B., Clarkson, C. R., & Chen, S. 2013, November. Multi-porosity multi-

permeability models for shale gas reservoirs. In SPE Unconventional Resources Conference

Canada. Society of Petroleum Engineers

9. Javadpour, F. 2009. Nanopores and Apparent Permeability of Gas Flow in Mudrocks (Shales

and Siltstone). J. Cdn. Pet. Tech. 48 (8): 16-21. http://dx.doi.org/10.2118/09-08-16-DA.

10. Loucks, R.G., Reed, R.M., Ruppel, S.C. and Hammes, U., 2012. Spectrum of pore types and

networks in mudrocks and a descriptive classification for matrix-related mudrock

pores. AAPG bulletin, 96(6), pp.1071-1098.

11. Sondergeld, C., Ambrose, R., Rai, C., et al. 2010. Micro-Structural Studies of Gas Shales.

Paper SPE 131771 presented at the SPE Unconventional Gas Conference, Pittsburgh,

Pennsylvania, USA, 23-25 February. http://dx.doi.org/10.2118/131771-MS.

12. Whitson, C.H. and Sunjerga, S. 2012. PVT in Liquid-Rich Shale Reservoirs. Paper SPE

155499 presented at the SPE Annual Technical Conference and Exhibition held in San

Antonio, Texas, 8-10 October.

http://dx.doi.org/10.2118/164532-MS


http://dx.doi.org/10.2118/09-08-16-DA

8

Chapter 2 Multiple Reservoir Engineering Applications of the Simplified Local Density

Model for Shale Gas and Liquid Rich Shale Reservoirs

2.1 Abstract

The petroleum industry is in need of simple-rigorous methods to measure and model fluid

storage in shale reservoirs. Commonly an indirect approach is taken for estimating fluid storage

in the lab: adsorption isotherm measurements to assess adsorbed gas storage and porosity

measurements to evaluate free-gas storage. A complication is that often the only samples

available from horizontal wells is small amounts of drill cuttings, which cannot be used for high-

temperature/pressure adsorption measurements.

Complications for storage modeling are caused by the multi-pore nature of shale; different

storage mechanisms become dominant in certain pore size ranges, and mixed storage

mechanisms may also occur. Fluid properties may be significantly altered in the confined pore

space of shales. Most current adsorption/fluid storage models do not properly handle the physics

of fluid storage in these various pore size ranges.

To address these challenges, the simplified local density (SLD) model is used to model various

aspects of fluid storage. The model is flexible enough to model fluid density gradients across the

pore width, allowing adsorbed fluid density and volume to be assessed. Pore confinement effects

on fluid phase behaviour may be determined, in combination with an equation of state. Finally,

due to the inability to measure high-pressure/temperature hydrocarbon adsorption on small

sample amounts such as drill cuttings, a novel procedure is used to predict this using low-

pressure adsorption (LPA) data in combination with the SLD model.

2.2 Introduction

Adsorption is an important gas storage mechanism in organic-rich shale and coal. There are

several theoretical and empirical methods for analyzing adsorption isotherms, which were

reviewed by Clarkson and Haghshenas (2013). The SLD model, not used in that study, has some

advantages over the above mentioned models, an important one being that it directly uses pore

structure information (surface area, A, and pore width, L) to predict adsorption. In addition, the

9

SLD model can be used to predict fluid density gradients in the pore space by considering the

pore wall-fluid interaction as a function of position. The SLD model also appears to be viable for

correlating isotherms past the Gibbs excess maximum, and, by providing the adsorbed phase

density, it can estimate absolute adsorption from the Gibbs excess adsorption.

In this chapter, the practical use of the SLD adsorption model for various engineering

calculations related to fluid storage estimation in shales is demonstrated. Specifically it is used

to 1) accurately correlate single and binary-component data 2) extract pore structure information

(surface area and pore width) 3) estimate fluid density profiles for assessing adsorbed phase

volumes (to correct free gas calculations and for calculating absolute adsorption) 4) predict high-

pressure/temperature adsorption from low-pressure adsorption data for estimating in-situ gas

adsorption using small amounts of sample (e.g. for analyzing cuttings) 5) evaluating pore

confinement effects on fluid properties.

2.3 Theory and Methods

The simplified local-density (SLD) model describes adsorption behavior using fluid-fluid and

fluid-solid interactions. The model is actually a simplified combination of the Peng–Robinson

(PR) equation of state (EOS) to provide the fluid properties, and the Lennard–Jones potential to

represent interactions between gas and solid molecules.

2.3.1 Single-component modeling with SLD model

Rangarajan (1995) originally articulated the physical premises and assumptions of SLD theory as

used in this work. The fundamental principle used is the equality of chemical potential of phases.

The basic assumption of the model is that the chemical potential of the fluid at any point near the

adsorbent surface (adsorbed phase) is equal to the bulk-phase chemical potential [i.e. b

z ]. The

chemical potential at any point above the surface is then defined as the sum of the fluid-fluid and

fluid-solid interactions:

bfsffzzz )()()( (1)

10

where μb is chemical potential of the bulk-phase, μff is chemical potential of the fluid-fluid and μfs

is chemical potential of the fluid-solid. Therefore, at equilibrium, there will be no chemical

potential gradient from the surface of the solid to the bulk fluid outside (Chen et al., 1997).

The pore geometry most widely assumed with SLD for carbon adsorbents is a two-surface slit,

with a specified distance (width) L, between which the fluid molecules reside. L is defined as the

distance between the two orthogonal planes that are tangential to the surfaces of the first graphite

planes on opposing sides of the slit. This geometry is used herein, and therefore it is assumed

that nanopores in shale have this shape. Figure 2-1 illustrates the schematic of pore model used

in SLD model. It is important to note that the model is regressed on adsorption data to give one

average pore size (L) for the sample (irrespective of organic/inorganic nature of the pores) and

assumes this average L is equivalent to the diameter of any available cylindrical pores. In the

cases where pore size distribution (PSD) exhibits two modes, the model again assumes one

average mode for all pores.

Figure 2-1 — Schematic of a slit-shaped pore model showing the variables used to calculate

distances in the SLD approach. Modified from Zuo, 2015.

Adsorbent Molecules Adsorbate Molecules

11

A molecule residing within a slit has fluid–solid interactions with both surfaces at distances z and

L – z. The equilibrium criterion for chemical potentials is summarized as (Fitzgerald et al.,

2006):

bfsfsffzLzz )()()( (2)

This equation indicates that the chemical potential of the adsorbed fluid reflects the proximity of

the fluid to the molecular wall of the adsorbent. Thus, the SLD model considers inhomogeneity

of the adsorbed phase in describing the molecular interactions of the adsorbed fluid with the

adsorbent (Chen et al., 1997). In the above equation, the fluid-solid chemical potential is given

as:

)]([)( zzfsAfs

N

(3)

where NA is Avogadro's number and Ψfs is the fluid-solid potential function, typically described

by an integrated potential function such as the 10-4 Lennard-Jones model. Although the fluid

may reside anywhere, the density is negligibly small in the distance from the wall to about σff /2.

Consequently, for this work, the potential has been set to infinity for positions less than σff /2

from the slit wall:

2,

2......................................................................................)(

22......).........

))1((2

1

5(4)(

4

1

4

4

10

10

2

fsfs

fs

fsfs

i ss

fsfs

fsfsatomsfs

lzzz

Lzizz

z

(4)

with:

2

sszz

2

ffss

fs

ffssfs

12

where σss is the carbon interplanar distance, σff is molecular diameter of the adsorbate, εff is the

fluid–fluid interaction energy parameter, εss is the solid–solid interaction energy parameter, εfs is

the fluid–solid interaction energy parameter, and carbon atom density (ρatoms)=0.382 atoms/Å2.

The carbon interplanar distance was adopted to be 0.335 nm (Subramanian et al., 1995) and fluid

diameters are 0.3758, 0.3798, and 0.3941 nm for methane, nitrogen, and CO2, respectively (Reid

et al., 1987). Please note that the carbon atom density and carbon interplanar distance were taken

to be that of graphite (Subramanian et al., 1995).

Calculating the fluid-fluid chemical potential, however, is a bit more challenging. As discussed

above, the chemical potential, µ, provides the fundamental criterion for phase equilibria; that is,

the chemical potential of each component must be equal in all existing phases at equilibrium.

However, it exhibits characteristics which discourage its use. The Gibbs energy, hence µ , is

defined in relation to the internal energy and entropy, both thermodynamic quantities for which

absolute values are difficult to be characterized. As a result, calculating the absolute values for µ

is also difficult. While these characteristics do not preclude the use of chemical potential, the

application of equilibrium criteria is facilitated by introduction of fugacity, a quantity that takes

the place of µ but which does not exhibit its less desirable characteristics (Smith et al., 1987).

The chemical potential of a bulk fluid can be expressed in terms of fugacity as follows:

)ln()(

0

0f

fRTT

b

b

(5)

Similarly, the chemical potential of fluid–fluid interactions can be given as:

)ln()()(

0

)(

0f

fRTTZ

Zff

ff

(6)

In the above equations T is temperature, fb refers to bulk fugacity, fff (z) is fluid fugacity at

position z and f0 refers to the same arbitrary reference state for both equations.

Substituting for μff, μfs and μb in Eq. 2 results in:

]exp[)()(

)(Tk

ff

B

fsfs

bzff

zLz

(7)

13

which gives the local adsorbed-phase fugacity at each position z. In the above equation, kB is the

Boltzmann constant (1.3806488 × 10-23

m2 kg s

-2 K

-1).

In this study, the Peng-Robinson (PR) EOS (1976) was used to calculate the bulk density, ρbulk,

as follows:

))21(1)()21(1(1

1 )(

bb

bT

bb bbRT

a

bRT

p

(8)

where:

),(45724.0)(

22

r

c

cT

P

TRTa

c

c

P

RTb 077796.0

22/1)]1(1[),(

rrTkT

226992.05422.137464.0 k

The bulk fugacity equation, also calculated with PR-EOS, is as follows:

]

)21(1

)21(1ln[

22

]1

ln[

)21(1ln

)(

22

)(

b

bT

b

b

bb

bT

b

b

b

b

b

bRT

a

RT

b

bbRT

a

b

bf

(9)

where a and b are PR-EOS constants.

Now, substituting the fff(z) from Eq. 7, the PR-EOS is again employed to calculate the local

density of the adsorbed phase, ρ(z):

])21(1

)21(1ln[

22

]1

ln[)21(1

ln

)(

)()(

)(

)(

2

)(

2

)(

)(),(

)(

)(

)(

z

zz

z

z

zz

zzTads

z

z

zff

b

b

bRT

a

RT

b

bbRT

a

b

bf

(10)

Using this equation, the density profile can be calculated across the pore width.

Finally, Gibbs (excess) adsorption is calculated from the following equation:

14

dzA

nffL

ff

bz

Slit

Gibbs

2/

2/)(

][2

(11)

Where, for a slit geometry, the lower limit of integration σff /2 is the location of the center of an

adsorbed molecule in contact with the left planar surface, and the upper limit, L- σff /2, is the

location of the center of an adsorbed molecule in contact with the right plane surface (Figure 2-

1). The three physical parameters: pore width (L), surface area (ASlit) (it is the total surface area

of both sides of the slit, that is why the area is divided by two for slit volume calculation in Eq.

11) and fluid–solid interaction energy parameter (εfs/kB), are obtainable through regression of

experimental data using the SLD model.

Traditionally adsorption models predict absolute adsorption, not excess adsorption, as given with

Eq. 11. The absolute amount adsorbed can be estimated from following equation:

aaabsVn (12)

Or more practically from:

)(

ba

a

Gibbsabsnn

(13)

In the above equation, average adsorbed density, a

, can be calculated as:

ads

h

z

a

h

dz

ads

ff

2/

)(][

(14)

where hads is the thickness of adsorbed layer. The adsorbed-phase thickness (volume) is where

fluid-solid interactions are present nearby the solid surface. For conditions where the gas density

is the same as the average adsorbed density, the excess adsorption is zero. At very high gas

densities, the excess adsorption can become negative and this can be interpreted physically to

indicate that the adsorbed phase is nearly incompressible due to fluid-solid interactions, whereas

the gas density, lacking the fluid-solid interactions, is more compressible, and thus can surpass

the average adsorbed phase density (Fitzgerald, 2005).

The absolute amount adsorbed can be significantly different from the experimentally inferred

Gibbs excess adsorption, especially past the Gibbs excess maximum. As noted in the literature,

15

for some supercritical fluids (e.g., CO2) adsorbed at high pressure, the absolute amount adsorbed

approaches a saturation value but the excess amount adsorbed reaches a maximum plateau and

then begins to decrease with increasing pressure. This has created some uncertainty as to the

utility of the excess function formalism of adsorption thermodynamics for high pressure

adsorption (Myers and Monson, 2002). The SLD model appears to be viable for correlating

isotherms past the Gibbs excess maximum and in estimating absolute adsorption from the Gibbs

excess adsorption (Fitzgerald, 2006).

2.3.2 Multi-component modeling with the SLD model

For pure gas or mixture adsorption, the local density, ),(,

)()(

ˆziz

xffif

(also is shown as ))(,)((,

ˆzixzadsi

f

(Fitzgerald et al., 2006)), is a function of position. In addition, in mixture adsorption, the

composition, xi, changes with position. Thus, the excess adsorption on the surface, A, may be

described by

dzyzxA

nffL

ff

ibiz

Slit

Gibbsi

2/

2/)(,

])([2

(15)

For pure gas adsorption, x = y = 1. The local adsorbed fugacity of the components at each

position z can be calculated by a local equilibrium relationship

NciTkf

f

B

fsifsi

bi

zixzffi zLz:1].......[

ˆ

ˆ

ln)()(

,,

,

))(,)((,

(16)

In the adsorbed phase, the fugacity of component i in a mixture, ),(,

)()(

ˆziz

xffif

, is a function of the

local composition, local density, pressure, and temperature. In the bulk phase, the fugacity of

component i in a mixture, bulki

f,

ˆ , is solved at the bulk density, pressure, and temperature (not a

function of position). The fluid-solid potential of component i in a mixture, fsi ,

, is a function of

the slit geometry and position (not a function of composition). Following Chen et al. (1997) a

partially integrated 10-4 Lennard-Jones potential was used to describe the fluid-solid interactions

for each component, which is a truncated version of Steele’s 10-4-3 potential function.

16

ffissfsi

fsifsi

fsi

fsifsi

i ss

fsifsi

fsifsiatomsfsi

Lzzz

Lzizz

z

,,

,,

,

,,

4

1

4

4

,

10

10

,2

,,,

2,

2......................................................................................)(

22..).........

))1((2

1

5(4)(

(17)

In the bulk phase, a linear mixing rule for b and a quadratic mixing rule for a is used. The

fugacity for the bulk phase using the PR-EOS is

jTiTijijT

iii

i jijTji

b

bjijj

ii

i

bi

aaCa

byb

ayya

b

b

a

ay

b

b

bRT

a

RT

bpZZ

b

b

py

f

)()()1()(

)(

]

)21(1

)21(1ln[)

2

(

22

]ln[)1(

ˆ

ln

)()()(

)(

,

(18)

In the adsorbed phase, quadratic mixing rules are used for both the co-volume b and the

attraction constant a. The quadratic mixing rules were included for b in the adsorbed phase

because they provided a marginally better fit of the adsorption data. Using these rules, the

fugacity in the adsorbed phase is

)2

()(

)1()()()(

)(

)(

]

)21(1

)21(1

ln[))(

)()(2)(2

(

22

)(

]ln[)1(

)(2

)(

)(ˆ

ln

),(),(),(

),(

)(

)(

)()(

,

ji

ija

ijjzTaizTaijzTa

i jijaji

i jijzTaji

za

zajijj

jijj

zaza

jijj

i

ai

bbb

Caaa

bxxb

axxa

b

b

za

zazx

b

bbzx

bRT

za

RT

bp

RT

p

RT

p

b

bbzx

pzx

zf

(19)

where, Cij is binary interaction parameter (BIP) for asymmetric mixtures.

For the adsorbed amount calculation, the pore volume is taken to include the entire volume

covered by the slit, (AL)/2. Because the profile of the local-adsorbed density and compositions

are symmetric about the pore midpoint, Eq. 15 is evaluated for one side of the slit only, and the

17

result is multiplied by 2, as shown in Eq. 20. Also, because the fluid-solid potential was assumed

to be infinite at distances of less than σff /2 from the wall surface, the third integral in the

equation below is zero. The second integral is not zero to accommodate the convenient definition

of the pore volume as being A(L/2), rather than the fluid-specific definition of A(L - σi,ff )/2.

dzzxAdzyAdzyzxAnff

izSlit

ff

ibSlit

L

ff

ikizSlitGibbsi

2/

0)(

2/

0

2/

2/)(,

)]([][])([

(20)

In the SLD model evaluation of gas mixtures, all binary interaction parameters were set to zero

and no parameter regressions were performed; therefore, the regression parameters obtained

from matching the pure-gas SLD model were applied directly to model multicomponent gas

adsorption with the selected mixing rules.

In this study, the SLD model predictions are limited to binary gas mixtures.

2.3.3 Predicting high-pressure/temperature adsorption from low-pressure/temperature

adsorption using the SLD model

The small amount of drill cuttings typically available from horizontal wells cannot be used for

conventional volumetric high-pressure adsorption measurements (Clarkson and Hagshenas

2016). The SLD model, however, can be used to predict these data using low-pressure adsorption

(LPA) because it uses pore structure information (surface area, A, and pore width, L) extracted

from LPA. The SLD model can be calibrated to low-pressure adsorption data through adjustment

of A and L, then, because these parameters are assumed constant (not functions of pressure),

high-pressure adsorption can be predicted using this information. Note that for shale gas content

determination, the model must be calibrated using some high-pressure adsorption data.

The following workflow was developed by Clarkson and Haghshenas (2016) for Duvernay shale

artificial cuttings samples to enable prediction of high-pressure hydrocarbon adsorption from the

SLD model using low-pressure, non-hydrocarbon adsorption:

1. Measure low-pressure adsorption of CO2 and N2 on artificial drill cuttings

2. Estimate surface area and pore size distributions using traditionally-applied models

(Langmuir, D-R, D-A, BET, DFT and BJH)

18

3. Using the previous step as an initial guess, fit the SLD model to CO2 and N2

adsorption data by adjusting A and L and εfs/kB

4. Compare the SLD-derived pore structure and surface area information with

traditionally-applied models for calibration and as a consistency check

5. Predict high-pressure CO2 and N2 adsorption using the SLD model

Because low-pressure adsorption data for hydrocarbon gases is not available at the present time,

in order to predict high-pressure hydrocarbon adsorption, the εfs/kB for the hydrocarbon fluid

must be determined. For the Duvernay reservoir studied, however, some high-pressure

hydrocarbon adsorption data were available for larger samples with similar depth, TOC content

and thermal maturity. This information was used to calibrate the SLD model and predict

hydrocarbon adsorption for the rest of the artificial cuttings samples as follows:

6. Adjust εfs/kB in SLD model to match actual measured high-pressure hydrocarbon

adsorption

7. Use SLD model to predict high-pressure hydrocarbon adsorption using εfs/kB obtained

from step 6, and A and L obtained from step 3

2.3.4 Fluid property modeling in confined pore spaces using the SLD model

The fluid-solid interactions cause a non-uniform distribution of gas molecules in nanopores; that

is, the fluid density is lower in the central portion of the pore space. The change in gas density

subsequently alters many related fluid parameters such as critical temperature and pressure,

compressibility factor and viscosity. The SLD model is capable of predicting the density profile

within the pore, therefore, the model is used here to predict vapor-liquid equilibria and critical

properties of confined fluids in nanopores. An important starting point for estimating gas

property alteration is Eq. 1 below:

)()( ]exp[

)()(

zff

B

fsfs

bzfff

Tkff

zLz

(21)

which gives the local adsorbed-phase fugacity at each position z. In the above equation, T is

temperature, fbulk refers to bulk fugacity, fff (z) is fluid fugacity at position z, Ψfs is the fluid-solid

potential function and kB is the Boltzmann constant.

19

Using the fluid fugacity from Eq. 1, the compressibility factor, Z, can be then derived from

following expression:

ZBZ

BZ

B

ABZZ

p

f

)

414.0

414.2ln(

22

)ln(1ln (22)

where:

22TR

apA

RT

apB

),(45724.0)(

22

r

c

cT

P

TRTa

c

c

P

RTb 077796.0

22/1)]1(1[),(

rrTkT

226992.05422.137464.0 k

Because the formation volume factor and viscosity are gas parameters that are directly

influenced by z-factor (Z), these parameters are also altered by pore confinement. The formation

volume factor is calculated as:

g

sc

sc

gB

pT

ZTPB

(23)

and viscosity is calculated by Lee et al. (1966):

g

Y

gXK

)exp(10

4 (24)

where:

RT

Mw

Z

p

g

MwxT

xxX

3

2

1

20

XyyY21

TMwkk

TMwkkK

k

54

321

)(

Eq. 24 is solved using constants from Table 2-1 (the significant digits are rounded to be

compatible with the original paper by Lee et al. 1966, however, inclusion of additional

significant figures does not make a big difference in viscosity estimation).

The corrected critical pressure, Pc, and critical temperature, Tc, are calculated as follows. Altered

z-factor distributions are calculated at a set of pressure points, using Eq. 22 and, at each pressure,

the calculated z-factor is averaged over pore width (zavg.Eq.2). Then, z-factor from PR-EOS is

fitted to the resulting zavg.Eq.2 vs. pressure points by setting Tc and Pc as the regression parameters.

Table 2-1 Constants for Lee’s viscosity correlations.

Parameter Value

k1 9.4

k2 0.02

k3 1.5

k4 209

K5 19

x1 3.5

x2 986

x3 0.01

y1 2.4

y2 0.2

2.4 Applications

2.4.1 Predicting phase density profile

An example application of the SLD model for predicting the gas density profile across a pore

width is shown in Figure 2-1. It is shown than the fluid density under confinement effects is no

21

longer homogeneous and varies across the pore width, with higher density at pore walls and

lower density at the center of the pore. The density profile, i-ndeed, follows the solid-gas

interaction potential profile, where its quantity is higher at positions close to the solid surface.

Also, it is illustrated that the thickness of the adsorbed layer is a function of pore width; while in

10 nm pore, the main part of the pore is occupied with the bulk phase, in 2 nm pore, the relative

thickness of adsorbed layer to the pore width is considerable.

Figure 2-2 — Use of the SLD model to estimate gas density profiles (red solid line) in organic

matrix-pores in (a) a 2 nm diameter pore and (b) a 10 nm diameter pore. L is the width of the

pore, while z is the distance from one of the pore walls. The approximate adsorbed layer

thickness in both cases is highlighted with purple shading. The bulk phase density near the

center of the pore is shown with a dashed red line. The adsorbed phase is associated with

densities above the dashed line. From Clarkson et al., 2016.

2.4.2 Matching single-component adsorption data and predicting high-pressure

adsorption (New Albany Shale)

An example application of the SLD model for correlating New Albany Shale CH4 and CO2 data

collected by Chareonsuppanimit et al. (2012) is shown in Figures 2-3. In the validation step, the

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.2 0.4 0.6 0.8 1

Ad

so

rbe

d P

ha

se

De

ns

ity

(g/c

m3

)

z/L

Pore diameter=2nm , P=50MPa

adsorbed phase density bulk phase density

a)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.2 0.4 0.6 0.8 1

Ad

so

rbe

d P

ha

se

De

ns

ity

(g/c

m3

)

z/L

Pore diameter=10nm , P=50MPa

adsorbed phase density bulk phase density

b)

22

SLD model was utilized to match the pure CO2 and CH4 data. It is shown that the SLD model is

appropriately capable to match the excess adsorption data, especially at pressures higher than the

maximum excess adsorption pressure (Figures 2-3).

The model is also utilized to predict the amount of adsorbed gas in binary gas mixtures as a

function of pressure and composition. The predictions based on two-dimensional equation of

state (2D-EOS) model are also given for comparison. It is clear that there are differences in the

quantitative predictions of two models, it may come from the different routes the models pass to

predict the adsorption amount and more importantly, the different regression parameters that

each model uses. However, phenomenological models like the SLD and 2D-EOS are in general

better suited for predictions of gas adsorption over wide ranges of pressures and compositions.

Still, because the SLD predictions are obtained through the use of both pore characterization and

molecular properties of the adsorbates, this model is believed to give more reliable results.

Nevertheless, the binary-mixture prediction results give some preliminary insight for simulations

of enhanced shale recovery process.

Figure 2-3 — Match of SLD model to high pressure CH4 and CO2 Gibbs (excess) adsorption

data at 370 K. Data from Chareonsuppanimit et al. (2012) for the New Albany shale.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 2 4 6 8 10 12 14

To

tal

Am

ou

nt

Ad

so

rbe

d,

mm

ol/g

p, MPa

CO2 experimental dataCH4 experimental dataSLD modelSLD model

23

Figure 2-4 — Predictions of SLD model for high pressure CH4 and CO2 mixture adsorption at

370 K. The 2D EOS predictions are also given for comparison.

2.4.3 Predicting high-pressure methane adsorption from low-pressure adsorption

(Duvernay Shale)

Low-pressure N2 and CO2 adsorption modeling. In the following, steps 3-4 of the workflow

described above are illustrated for Duvernay shale samples – steps 1-2 are covered

comprehensively by Clarkson and Haghshenas (2016).

0

0.02

0.04

0.06

0.08

0.1

0.12

0 5 10 15

To

tal

Am

ou

nt

Ad

so

rbe

d,

mm

ol/g

p, MPa

yCH4=50% 2DEOSyCH4=90% 2DEOSyCH4=50% SLDyCH4=90% SLD

24

Figure 2-5 — (a) N2 adsorption/desorption isotherms (hysteresis loop) at 77 K and (b) CO2

adsorption isotherms at 273 K, collected for 4 Duvernay shale artificial cuttings samples. From

Clarkson et al., 2016.

Following the step 3 of the workflow, the fit of SLD model to the adsorption branch of N2 and

CO2 isotherm data (Figure 2.6) is used to provide surface area and pore size distribution

estimates. To obtain an initial guess for the parameters of interest, BET, Langmuir, BJH and

DFT models were formerly used to interpret the N2 data, while BET, Langmuir, D-R, D-A, and

DFT models were used for CO2 analysis (see Clarkson and Haghsehnas 2016). Figure 2.7

compares surface area calculations for all models. Generally, surface areas estimated from all

models are in good agreement (Figure 2.7), except for the D-A model for CO2. This provides

important validation for the SLD model, which has not historically been applied to low-pressure

adsorption data.

Sample CC2 which has the highest adsorption (Figure 2-6), also has the highest surface area,

whereas CC4 sample has the lowest. It is worth noting surface area estimates obtained from N2

and CO2 isotherms are not the same, with surface areas from N2 generally being much larger.

For example, CC2 surface area obtained from N2 adsorption is in the range of 35-52 (m2 /g) as

compared to 9-24 m2/g obtained from CO2 adsorption; CC4 surface area ranges from 11-12 m

2 /g

obtained from N2 adsorption and 5-10 m2/g from CO2 adsorption. Please note that the surface

area used in the model is not actually a physical parameter which remains unchanged with the

different pressure-temperature conditions. The parameter can be better described as an

adsorption surface area (model parameter rather than physical parameter) which indeed is a

function of gas component, pressure, temperature and solid properties. For high pressure

0.E+00

1.E-04

2.E-04

3.E-04

4.E-04

5.E-04

6.E-04

7.E-04

8.E-04

0 40000 80000 120000

Qu

an

tity

ad

so

rbe

d (

gm

ole

/g S

TP

)

Absolute pressure (Pa)

N2

CC2CC4CC5CC8

0.E+00

1.E-05

2.E-05

3.E-05

4.E-05

5.E-05

6.E-05

7.E-05

8.E-05

9.E-05

0 40000 80000 120000

Qu

an

tity

ad

so

rbed

(g

mo

le/g

ST

P)

Absolute pressure (Pa)

CO2

CC2CC4CC5CC8

a) b)

25

adsorption predictions based on low pressure data, the surface area is assumed constant with

respect to the pressure to minimize the number of regression parameters.

Figure 2-6 — SLD model match to low-pressure (a) N2 isotherms at 77 K and (b) CO2 isotherms

at 273 K. The SLD model was fitted to the adsorption branch of both isotherm datasets. The

relative pressure range of around 0.05-0.2 was selected for nitrogen because this is the pressure

range used for BET model analysis (BET constant (C) = 22.3). p0 is vapor pressure of gas at

experimental temperature, i.e., 77 K for N2 and 273 K for CO2. From Clarkson et al., 2016.

Figure 2-7 — Specific surface areas calculated from all models from (a) low-pressure N2

adsorption data and (b) low-pressure CO2 adsorption data. From Clarkson et al., 2016.

0.E+00

5.E-05

1.E-04

2.E-04

2.E-04

3.E-04

3.E-04

4.E-04

4.E-04

5.E-04

5.E-04

0 0.1 0.2 0.3

Qu

an

tity

ad

so

rbed

(g

mo

le/g

ST

P)

Relative pressure (p/p0)

CC2CC4CC5CC8

0.E+00

1.E-05

2.E-05

3.E-05

4.E-05

5.E-05

6.E-05

7.E-05

8.E-05

9.E-05

0 0.01 0.02 0.03 0.04

Qu

an

tity

ad

so

rbed

(g

mo

le/g

ST

P)

Relative pressure (p/p0)

CC2

CC4

CC5

CC8

a) b)

0

5

10

15

20

25

CC2 CC4 CC5 CC8

Sp

ecif

ic S

urf

ace A

rea (

m2/g

)

CO2 BET

Langmuir

D-R

D-A

SLD

0

10

20

30

40

50

60

CC2 CC4 CC5 CC8

Sp

ecif

ic S

urf

ace A

rea (

m2/g

)

N2 BET

Langmuir

BJH Ads

Total Area DFT

SLD

a) b)

26

Pore size distributions (PSDs) for the artificial cuttings samples obtained from analyzing N2 data

with the BJH and DFT models, and from CO2 data with the DFT model, are shown in Figure 2.8.

Although some portions of the PSDs are missing due to the lack of overlap between CO2 and N2

data, all samples appear to exhibit a multi-modal pore structure with peaks in the 0.4-0.6 nm

range (micropores), 1-2 nm range (micropores inferred, data missing), and 2-4 nm range

(mesopores). These values are used as an initial guess for the SLD match.

The average pore size obtained from the SLD model match to the CO2 isotherms and N2

isotherms was 1.1 nm (micropores) and 2.9 nm (mesopores), respectively. These results are

within acceptable range of those obtained from the DFT and BJH models. It is worth noting that

the SLD model normally uses one gas-specific parameter (fluid-solid interaction parameter) plus

two adsorbent-specific parameters (surface area and pore width) that are independent of the

adsorbing gas species. However, with the large difference observed between CO2 and N2

estimations of pore size and surface area, it appears that the adsorbent parameters are not

independent of the adsorbing gas species. This is likely due to the difference in pore

accessibility of the two gases at the different temperatures. Therefore, in this study, the model

was fitted specifically to each gas dataset to give the best estimates of parameters of interest.

The fluid–solid interaction energy parameter (εfs/kB) extracted using the SLD model for N2 and

CO2, respectively, is 21 (K) and 50 (K).

27

Figure 2-8 — Pore size distributions obtained from N2 adsorption data (using BJH and DFT

models) and from CO2 adsorption data (using the DFT model). From Clarkson et al., 2016.

Importantly, the broad pore size range observed in Figure 2.8 suggests that the storage

mechanisms would be expected to differ substantially in the pore structure of these shales.

Referring to the SLD simulations in Figure 2.2, pores in the < 2 nm range would be expected to

have limited to no free-gas storage and strong gas density gradients from pore wall to center,

while pores in the 10 nm range or greater have a distinct bulk phase at the center of the pore.

The SLD model is therefore capable of accurately accounting for the different storage

mechanisms and can be used for high-pressure adsorption prediction, as will be discussed in the

next section.

High-pressure N2 and CO2 adsorption prediction. In this section, step 5 of the workflow is

illustrated.

As noted previously, because there often is not enough sample mass from cuttings to perform the

high-pressure adsorption measurements directly, an accurate prediction method needs to be

developed. Using pore-structure and solid-gas interaction information obtained from the match

of the SLD model to the low-pressure adsorption data (previous section), the SLD model can

0

0.002

0.004

0.006

0.008

0.01

0.012

0.2 2 20 200

dV

/d(W

) P

ore

Vo

lum

e (

cm

³/g

.nm

)

Pore Width (nm)

CC2 CO2 DFTCC4 CO2 DFTCC5 CO2 DFTCC8 CO2 DFTCC2 N2 DFTCC4 N2 DFTCC5 N2 DFTCC8 N2 DFTCC2 N2 BJHCC4 N2 BJHCC5 N2 BJHCC8 N2 BJH

28

now be used to predict high-pressure adsorption (excess and absolute) for these same gases

(Figure 2.9) on the artificial cuttings samples. The temperature and pressure ranges were

selected to be consistent with those expected for the Duvernay shale in the study area. At low

pressure (sub-atmospheric), the difference between absolute and excess adsorption is negligible.

However, for higher pressures, when adsorbed layer is occupying a significant volume of the

pore space, the difference is more substantial as shown in Figure 2.9.

Figure 2-9 — High-pressure excess and absolute adsorption isotherms predicted from the SLD

model for (a) N2 and (b) CO2 at 383.15 K for 4 Duvernay shale artificial cuttings samples. From

Clarkson et al., 2016.

High-pressure CH4 adsorption prediction. In this section, use of the SLD model for predicting

high-pressure and temperature hydrocarbon adsorption is demonstrated. Unfortunately, no

methane adsorption isotherms (low- or high-pressure) could be measured on the 4 artificial

cuttings samples, however these measurements were made on larger samples in the same depth

interval by a commercial laboratory. Using the pore structure parameters (surface area and pore

width) derived from low-pressure adsorption measured on artificial cuttings discussed in

previous steps, the SLD model is fit to the high-pressure methane adsorption data through

adjustment of the hydrocarbon-solid interaction parameter (steps 6 and 7 of workflow). Methane

adsorption is predicted in this case because it is the major component of natural gas and gas

condensates in shales.

0.E+00

1.E-04

2.E-04

3.E-04

0 5 10 15 20 25 30 35 40 45

Qu

an

tity

ad

so

rbed

(g

mo

le/g

ST

P)

Absolute pressure (MPa)

N2CC2 - absolute

CC2 - excess

CC4 - absolute

CC4 - excess

CC5 - absolute

CC5 - excess

CC8 - absolute

CC8 - excess

0.E+00

1.E-04

2.E-04

3.E-04

0 5 10 15 20 25 30 35 40 45

Qu

an

tity

ad

so

rbed

(g

mo

le/g

ST

P)


CO2CC2 - absolute

CC2 - excess

CC4 - absolute

CC4 - excess

CC5 - absolute

CC5 - excess

CC8 - absolute

CC8 - excess

a) b)

29

Two complications arise for predicting CH4 adsorption with the SLD model for the artificial

cuttings: 1) A and L values obtained from low-pressure adsorption are different for N2 and CO2,

and one set of parameters must be selected to make the prediction and 2) the fluid-solid

interaction parameter for CH4 is not known, a-priori. To resolve both complications, two sets of

matchings are performed: one with A/L for N2 and another one with A/L for CO2. For both of

these matches, the fluid-solid interaction parameter (εfs/kB) was adjusted to match the high-

pressure CH4 isotherms collected for the larger samples in the interval (Figure 2.10). In this

way, the εfs/kB value is "calibrated" for the artificial cuttings samples. The match (A/L

combination) which results in a fluid-solid interaction value most consistent with literature

values (Chareonsuppanimit, 2012; Clarkson and Haghshenas, 2013) was chosen. The resulting

fluid–solid interaction energy parameter for CH4 is 31 K (fluid–solid interaction energy

parameter for CO2 and N2 were 100 K and 19K, respectively).

Figure 2-10 — Fit of the SLD model to high-pressure/temperature (383.15 K) CH4 isotherms

measured on 2 Duvernay samples taken from the same interval as the artificial cuttings samples.

Solid lines are SLD model fit to the experimental data. From Clarkson et al., 2016.

The high-pressure methane adsorption predictions for the four Duvernay shale artificial cuttings

samples, using low-pressure N2 adsorption-derived A/L and the adjusted fluid-solid interaction

parameter, are given in Figure 2.11. As expected from the shale adsorption literature (Hartman et

0.E+00

1.E-04

2.E-04

3.E-04

0 5 10 15 20 25 30 35 40 45

Qu

an

tity

ad

so

rbed

(g

mo

le/g

ST

P)


3263.4 m, TOC 3.93%

3268.92 m , TOC 2.7%

3263.4 m, TOC 3.93%

3268.92 m , TOC 2.7%

30

al., 2011), the high pressure adsorption capacity of methane lies between nitrogen and carbon

dioxide curves (compare Figure 2.11 with Figure 2.9).

Figure 2-11 — Predicted high-pressure, high-temperature methane isotherms at 383.15 K for 4

Duvernay artificial cuttings samples. From Clarkson et al., 2016.

2.4.4 Predicting high-pressure methane adsorption from low-pressure adsorption

(Montney Shale)

To further demonstrate application of the SLD model for this purpose, high-pressure/temperature

hydrocarbon adsorption for the Montney siltstone reservoir was also predicted. The primary

difference between this example and the previous example for Duvernay is that heavier

hydrocarbon (C1 – C4+) was also predicted for this hydrocarbon liquid-rich reservoir.

Following the same workflow as demonstrated in the previous section, the SLD model is first

used to match LPA results of N2 and CO2 (Figs. 2-12a and 2-12b, respectively). Figure 2-13

provides the SLD model predictions for all components of the reservoir fluid + CO2. To derive

the results shown in Figure 2-13, the prediction for CH4 is calibrated to high-pressure CH4

adsorption on Montney samples available in the literature (Beaton et al., 2010) (compatible with

our samples of interest). For heavier components there are no measurements currently available

for the Montney – as a result, data for these components estimated for the Barnett shale

0.E+00

1.E-04

2.E-04

3.E-04

0 5 10 15 20 25 30 35 40 45

Qu

an

tity

ad

so

rbed

(g

mo

le/g

ST

P)


CH4CC2 - absolute

CC2 - excess

CC4 - absolute

CC4 - excess

CC5 - absolute

CC5 - excess

CC8 - absolute

CC8 - excess

31

(Ambrose et al., 2012) are used for calibration. For this purpose, the ratio of Montney CH4

adsorption to CH4 adsorption obtained by Ambrose et al. is first calculated; this ratio is then

multiplied by the heavier component adsorption isotherms in the Ambrose et al. dataset to

predict heavy component adsorption (C2, C3, C4+) for Montney samples. These results will be

used for CO2-EOR studies presented in Chapter 6. It is important to note that, owing to the lack

of adsorption measurements of components heavier than butane, the adsorption of these

components was assumed to be similar to butane.

Figure 2-12 — SLD model match to low-pressure (a) N2 isotherms at 77 K and (b) CO2

isotherms at 273 K. The SLD model was fitted to the adsorption branch of both isotherm

datasets. The relative pressure range of around 0.05-0.2 was selected for nitrogen because this is

the pressure range used for BET model analysis.

0

10

20

30

40

50

60

70

80

0 0.1 0.2 0.3 0.4

Qu

an

tity

Ad

so

rbed

(s

cf/

ton

)

Relative Pressure (p/p0)

Data

Model

(a)

0

2

4

6

8

10

12

14

16

0 0.02 0.04

Qu

an

tity

Ad

so

rbed

(scf/

ton

)


Data

Model

(b)

32

Figure 2-13 — High-pressure absolute adsorption isotherms predicted from the SLD model for

C1, C2, C3, C4+ and CO2 on Montney artificial cuttings sample#14 at 360 K.

Figure 2-14 provides a pore size distribution (PSD) for a Montney artificial cuttings sample,

along with the proportionality of free and adsorbed hydrocarbon. Two dominant peaks at around

0.55 nm and 3 nm are visible in PSD graph. Further, it is shown that almost all of the volume of

0.55 nm pores is occupied by adsorbed phase and at larger pores, i.e., 3 nm, the proportionality

of adsorbed volume over total hydrocarbon volume is still not negligible (around 40%).

0

50

100

0 2000 4000 6000 8000

Qu

an

tity

Ad

so

rbed

(scf/

ton

)

Absolute Pressure (psi)

CH4 C2

C3 CO2

C4+

33

Figure 2-14 — a) Pore size distributions obtained from N2 adsorption data (using BJH model)

and from CO2 adsorption data (using the DFT model). Two modal pore sizes are around 0.55 and

3 nm for the 7G artificial cuttings sample#14. b) The ratio between the pore volume occupied by

adsorbed phase and the total hydrocarbon for each pore size.

The following procedure is employed to use the above information for estimating hydrocarbon

content of the Montney artificial cuttings samples. Assuming the entire pore volume is filled

with hydrocarbon (neglecting water saturation):

1. The ratio in Fig. 2-14b provides the relative amount of absorbed over total hydrocarbon

in place at in-situ pressure and temperature conditions obtained from the SLD model.

2. The total (in-situ) hydrocarbon in place (cm3/g) = the pore volume of 0.55 nm pores

(cm3/g) + the pore volume of 3 nm pores (cm

3/g).

3. The absorbed amount (cm3/g) = ratio for 0.55 nm pores (from step 1) times the

hydrocarbon in place of 0.55 nm pores + ratio for 3 nm pores (from step 1) times the

hydrocarbon in place of 3 nm pores.

4. The free hydrocarbon amount is the total hydrocarbon in place (from step 2) minus

absorbed amount (from step 3).

Using this procedure for the artificial cuttings sample in Fig. 2-14, micro- and mesopore

volume measurements from low pressure adsorption data, and assuming the entire pore volume

is made up of 0.55 nm pores and 3 nm pores, the relative proportion of pore volume in the 0.55

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.2 2 20

dV

/dlo

g(W

) p

ore

vo

lum

e (

cm

3/g

)

Pore width (nm)

7G#14-N2

7G#14-CO2

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.55 3

Ad

so

rbed

Vo

lum

e O

ver

To

tal In

-Pla

ce

Pore Width (nm)

(b)

34

nm range and 3 nm range are evaluated. Approximately 8.4% of the total pore volume is

estimated to be in the 0.55 nm range, while 91.6% is estimated to be in the 3 nm range. The total

pore volume (from low-pressure adsorption data) for the sample is 0.01 cm3/g –0.000914 cm3/g

in the 0.55 nm range and 0.009 cm3/g in the 3 nm range. Using Fig. 2-14b, and steps 2 and 3

above, the amount of adsorbed and free hydrocarbon can be estimated to be 0.005 cm3/g and

0.006 cm3/g, respectively.

Limitations of this methodology are listed below:

1. The entire pore volume is not sampled with low-pressure adsorption data.

2. Attributing the entire mesopore volume to be associated with 3 nm pores may cause an

overestimation of the adsorbed phase volume; the ratio of the adsorbed phase thickness to

the effective pore width sharply decreases in pores larger than 2-3 nm, as can be inferred

from Fig. 2-14.

To evaluate the second possible error, another scenario is performed assuming the entire meso-

pore volume is made up from 6.5 nm pores. On this basis, the amount of adsorbed and free

hydrocarbon content is calculated as 0.0014 cm3/g and 0.009 cm

3/g, respectively, which is

significantly different than that found assuming 3 nm pores.

2.4.5 Fluid property modeling

It has been suggested that fluid property changes due to pore confinement result in anomalous

fluid production behavior in liquid-rich shale plays - an example of anomalous fluid production

is long periods of constant (and small) condensate-gas ratios for wells (or conversely, high gas-

oil ratios, see Fig. 2-15a) as given by Altman et al. (2014). This phenomenon has been referred

to as “dew point suppression”, and inferred to be caused by fluid property changes in the

confined nanopore space of shales.

The SLD model is used to predict fluid property changes in a shale gas condensate system. The

results of PVT calculations for such a system subject to confinement effects is shown in Fig. 2-

15b. As shown in the figure, the SLD model is capable of predicting the phase envelope for a

gas condensate system as a function of pore size, from 300 nm to 2 nm. It is found that the phase

enveloped shifts downward (from higher saturation pressures/temperatures to lower values) and

correspondingly, predicts a later onset for condensate dropout in shale reservoirs than for bulk

35

systems. This result is consistent with the reported dewpoint suppression behavior that others

have reported for liquid-rich shale systems (Singh, 2011; Devegowda, 2012; Ma and Jamili,

2014; Didar and Akkutlu, 2015; Pitakbunkate, 2015).

Figure 2-15 — (a) Long-term constant GOR observed for producing liquids-rich shale wells

(Altman et al., 2014) and (b) use of the SLD model to predict the phase envelope for a gas

condensate system as a function of pore size, from 300 nm to 2 nm. From Clarkson et al., 2016.

0

10

20

30

40

50

60

70

80

90

0 5 10 15 20 25 30 35 40 45 50 55 60

GO

R (

MS

CF

/ST

B)

time (months)

Monthly GOR behavior of gas condensate wells in Eagle Ford shale. Modified after Altman et al., 2014

Well 27 Well 28 Well 29 Well 30

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

-200 0 200 400 600 800

pre

ssu

re (

psia

)

temperature (oF)

phase envelop of a gas-condensate fluid under confinement

pore width=300nm pore width=10nm

pore width=5nm pore width=2nm

a)

b)

36

2.5 Conclusions

The simplified local density (SLD) model is a powerful tool for modeling various aspects of

fluid storage in shale. Specifically, in this work the SLD model has proved useful for:

1. Correlating high-pressure adsorption data of shales

2. Predicting phase density profiles across a nanopore

3. Predicting high-pressure/temperature hydrocarbon adsorption from low-pressure non-

hydrocarbon adsorption (which is useful for estimating hydrocarbon storage from small

amounts of cuttings)

4. Predicting fluid property changes in the confined nanopore spaces of shales

Application 2. will be used as the basis for estimating adsorbed phase volume to correct free-gas

storage and material calculations in shale, as demonstrated in Chapter 4. Application 3. is used

for primary and enhanced recovery compositional numerical simulations performed in Chapter 6.

Finally, application 4. is used for rate-transient analysis studies in Chapter 5.

2.6 Nomenclature

Field Variables

A slit total surface area, m3

f fugacity, Pa

hads adsorbed phase thickness, m

ka apparent permeability (m2)

kB Boltzmann’s constant, R/NA=1.3806488×10-23

m2 kg s

-2 K

-1

L slit width, m

Mw molar mass (kg/kgmol) or (lb/lbmol)

NA Avogadro’s number, 6.02214129×1023

mol−1

n molar number

nGibbs excess number of sorbed phase moles, gmol

37

nabs absolute number of sorbed phase moles, gmol

p pressure, psia

R gas constant (J/mol/K) or (psi.ft3/lbmol/R)

r pore radius (m)

T temperature, K

z distance from the surface of the wall

Z gas compressibility factor

Greek Symbols

εff energy parameter of fluid-fluid molecular interaction

εfs energy parameter of fluid-solid molecular interaction

μb chemical potential of bulk gas

μff chemical potential of fluid-fluid interaction

μfs chemical potential of fluid-solid interaction

μg viscosity

ρ molar density

ρa adsorbed phase molar density

ρb bulk fluid molar density

ρbulk gas bulk density, g/cm3

ρgr grain density, g/cm3

σff molecular diameter of the adsorbate

σss carbon molecules interplanar distance

Ψfs fluid-solid potential

ϕ total porosity fraction, dimensionless

Subscript

38

ff fluid-fluid

fs fluid-solid

i component identifier

j component identifier

ss solid-solid

st standard condition

2.7 References

Altman, R. M., Fan, L., Sinha, S., Stukan, M., and Viswanathan, A., 2014. Understanding

Mechanisms for Liquid Dropout from Horizontal Shale Gas Condensate Wells. Paper SPE

170983 presented at eh SPE Annual Technical Conference and Exhibition held in Amsterdam,

The Netherlands, 27-29 October.

Ambrose, R. J., Hartman, R. C., Diaz-Campos, M., Akkutlu, Y., and Sondergeld, G. H., 2012.

Shale Gas-in-Place Calculations Part I: New Pore-Scale Considerations. SPE Journal 17 (01),

219-229.

Beaton A.P., Pawlowicz J.G., Anderson S.D.A., Berhane H. and Rokosh C.D., 2010. Rock

Eval™, total organic carbon and adsorption isotherms of the Montney Formation in Alberta:

shale gas data release. Energy Resources Conservation Board, ERCB/AGS Open File Report, 5,

p.30.

Chen, J.H., Wong, D.S.H., Tan, C.S., Subramanian, R., Lira, C.T. and Orth, M., 1997.

Adsorption and Desorption of Carbon Dioxide onto and from Activated Carbon at High

Pressures. Industrial & Engineering Chemistry Research 36 (7), 2808-2815.

Clarkson, C.R. and Haghshenas, B., 2013. Modeling of Supercritical Fluid Adsorption on

Organic-Rich Shales and Coal. Paper SPE 154532 presented at the SPE Unconventional

Resources Conference-USA, The Woodlands, Texas.

39

Clarkson, C.R. and Haghshenas, B., 2016. Characterization of multi-fractured horizontal shale

wells using drill cuttings: 1. Fluid-in-place estimation. Journal of Natural Gas Science and

Engineering, 32, pp.574-585

Devegowda, D., Sapmanee, K., Civan, F., Sigal, R.F., 2012. Phase Behavior of Gas Condensates

in Shales Due to Pore Proximity Effects: Implications for Transport, Reserves and Well

Productivity. Paper SPE 160099 presented at the SPE Annual Technical Conference and

Exhibition held in San Antonio, Texas, 8-10 October.

Didar, B.R., and Akkutlu, I.Y., 2015. Confinement Effects on Hydrocarbon Mixture Phase

Behavior in Organic Nanopore. Paper SPE 178519 presented at the Unconventional Resources

Technology Conference held in San Antonio, Texas, 20-22 July.

Fan, L. et al., 2005. Understanding Gas Condensate Reservoirs. Schlumberger Oilfield Review.

Fitzgerald, J.E., 2005. Adsorption of pure and multi-component gases of importance to enhanced

coalbed methane recovery: measurements and simplified local density modeling (Doctoral

dissertation, Oklahoma State University.

Fitzgerald, J.E., Robinson, R.L. and Gasem, K.A., 2006. Modeling high-pressure adsorption of

gas mixtures on activated carbon and coal using a simplified local-density

model. Langmuir, 22(23), pp.9610-9618.

Hamada, Y., Koga, K. and Tanaka, H., 2007. Phase Equilibria and Interfacial Tension of Fluids

Confined in Narrow Pores. The Journal of Chemical Physics 127 (8), 084908.

Hartman, R.C., Ambrose, R.J., Akkutlu, I.Y., and Clarkson, C.R. 2011. Shale Gas-in-Place

Calculations Part II – Multi-component Gas Adsorption Effects. Paper SPE 144097, presented

at the SPE Unconventional Gas Conference held in Woodlands, TX, 14-16 June, 2011.

Jiang, J., Sandler, S.I., Schenk, M. and Smit, B., 2005. Adsorption and Separation of Linear and

Branched Alkanes on Carbon Nanotube Bundles from Configurational-Bias Monte Carlo

Simulation. Physical Review B 72 (4), 045447.

Ma, Y., and Jamili, A., 2014. Using Simplified Local Density/Peng-Robinson Equation of State

to Study the Effects of Confinement in Shale Formations on Phase Behavior. Paper SPE 168986

40

presented at the SPE Unconventional Resources Conference held in the Woodlands, Texas, 1-3

April.

Mohammad, S. A., Chen, J. S., Robinson Jr, R. L., and Gasem, K. A., 2009. Generalized

Simplified Local-Density/Peng− Robinson Model for Adsorption of Pure and Mixed Gases on

Coals. Energy & Fuels 23 (12), 6259-6271.

Pitakbunkate, T., Balbuena, P. B., Moridis, G. J., and Blasingame, T. A., 2015. Effect of

Confinement on Pressure/Volume/Temperature Properties of Hydrocarbons in Shale Reservoirs.

SPE Journal in press.

Rangarajan, B., Lira, C. T., and Subramanian, R. (1995). Simplified Local Density model for

Adsorption over Large Pressure Ranges. AIChE Journal 41(4), 838-845.

Reid, R. C.; Prausnitz, J. M.; Poling, B. E, 1987. The Properties of Gases and Liquids. McGraw-

Hill: New York.

Singh, S. K., and. Singh J. K., 2011. Effect of Pore Morphology on Vapor–Liquid Phase

Transition and Crossover Behavior of Critical Properties from 3D to 2D. Fluid Phase Equilibria

300 (1–2), 182-187.

Singh, S.K., Sinha, A., Deo, G. and Singh, J.K., 2009. Vapor− Liquid Phase Coexistence,

Critical Properties, and Surface Tension of Confined Alkanes. The Journal of Physical Chemistry

C 113 (17), 7170-7180.

Subramanian, R.; Pyada, H.; Lira, C. T., 1995. Ind. Eng. Chem. Res., 34, 3830-3837.

Zarragoicoechea, G.J. and Kuz, V.A., 2002. van der Waals Equation of State for a Fluid in a

Nanopore. Physical Review E 65 (2), 021110.

Zarragoicoechea, G.J. and Kuz, V.A., 2004. Critical Shift of a Confined Fluid in a

Nanopore. Fluid Phase Equilibria 220 (1), 7-9.

Zuo, L., 2015. A New Method to Calculate the Absolute Amount of High-Pressure Adsorption of

Supercritical Fluid. Iranian Journal of Chemistry and Chemical Engineering (IJCCE), 34(2),

pp.61-71.

41

Chapter 3 Characterization of Multi-Fractured Horizontal Shale Wells using Drill

Cuttings: Permeability/Diffusivity Estimation1

3.1 Abstract

There is considerable research interest in the transport properties of shales to assist in their

evaluation as reservoirs for natural gas and oil. However, shales have proven difficult to

characterize, in part because of the challenges of obtaining viable reservoir samples from multi-

fractured horizontal wells used to produce from them. Often the only reservoir samples available

from horizontal wells are drill cuttings – the sample sizes obtained from cuttings are typically too

small for quantitative analysis using conventional techniques. Therefore, new, high-precision

methods are required to analyze the smaller cuttings samples. Further, the physics of gas storage

and transport through the multi-model pore structure of shale is complex, requiring rigorous

modeling approaches to extract parameters of interest such as permeability/diffusivity.

In this chapter, the use of a high-precision, low-pressure adsorption device is explored for

extracting permeability/diffusivity parameters from small amounts (1-2 g) of artificial (crushed

core sample) drill cuttings of Duvernay shale. The model is also developed to match Montney

samples - although the models was not found significantly different than the ones used for

Duvernay. In order to extract the transport parameters, gas flow through the complex,

heterogeneous matrix pore structure of the shale has been approximated using a general dual

porosity numerical model which assumes that (1) gas flows through macropores by continuum

viscous flow (2) gas flows through meso and micropores by Knudsen diffusion and molecular

slippage on pore walls and (3) adsorption occurs in meso and micropores. The model can be

simplified into two sub-models, a macro/micropore system or meso/micropore system,

depending on the measured pore size distribution of the samples of interest.

1 This chapter is a modified version of a paper published by Journal of Natural Gas Science and Engineering as:

Haghshenas, B., Clarkson, C.R., Aquino S., and Chen S. 2016. Characterization of Multi-Fractured Horizontal Shale

Wells using Drill Cuttings: 2. Permeability/Diffusivity Estimation. Journal of Natural Gas Science and

Engineering,32, pp.586-596. Copyright approval has been obtained from Elsevier (see “Copyright Permissions”

section of this thesis).

42

The new multi-pore (bidisperse) numerical model is applied to low-pressure adsorption rate data

obtained from the crushed Duvernay and Montney shale core samples, and apparent permeability

for each gas/sample group is calculated at different pressure steps. The low-pressure adsorption

device yields pressure-time data that is of much better quality than a commercial crushed rock

permeability device that requires larger sample sizes. The new bidisperse pore structure

numerical model, which allows permeability to vary (at each pressure step) due to gas slippage

effects, properly describes the entire adsorption rate history of the samples studied. For

Duvernay samples, mesopore apparent permeabilities measured using CO2 range from 1E-2

-1E-3

mD and micropore apparent diffusivities are in the 1E-7

mD range. The calculated apparent

diffusivities obtained from modeling adsorption rate data change with pressure. For Montney

samples, mesopore apparent permeabilities obtained from the low pressure adsorption data range

from 1×10-4

mD using CO2 to 1×10-3

mD using N2, while micropore apparent diffusivities are in

the 1×10-7

mD range. The mesopore apparent permeability calculated from a commercial

permeameter using larger sample masses 2×10-3

mD.

The results of this study have important implications for shale matrix transport characterization.

The resulting data can be used for making completions decisions and in reservoir models which

capture reservoir property changes along a horizontal lateral.

3.2 Introduction

Although static volumetric calculations often indicate large in-place hydrocarbon volumes for

shale (adsorbed plus free gas), it is the rate of desorption (in organic-rich shales) and

diffusion/flow that dictate the timescales needed to produce hydrocarbon gas through primary

production, or inject CO2 for enhanced recovery of hydrocarbons and storage in shale. Therefore,

a good understanding of the transport properties of the shale matrix and fracture system is

required for accurate production predictions. Matrix permeability, which is the subject of the

current study, is a particularly important control on long term fluid flow in unconventional

reservoirs.

However, shale matrix permeability is challenging to measure in the laboratory. The various

techniques used for this purpose operate on different physical principals and utilize samples of

different sizes and geometries, subjected to contrasting measurement conditions (Ghanizadeh et

43

al., 2015a). Crushed rock permeability measurements are often performed to obtain a “true”

measurement of matrix permeability (Handwerger et al., 2011). Although commercial labs

routinely perform these analyses, the procedures and algorithms used for analysis are not always

disclosed. Further, commercial equipment often gives one value (average) value for

permeability. The experimental and modeling attempts for describing diffusion/flow mechanism

of coal reservoirs have a long history; however, these techniques are still being evolved for

shales. In the following, a brief summary of attempts to extract coal diffusivity values is

provided, followed by a summary of methods for shale matrix permeability calculation.

3.2.1 Diffusivity/permeability studies performed for coal

Some researchers suggest that a single coefficient is sufficient for describing matrix transport

through coal (Charrière et al., 2010; Ciembroniewicz and Marecka, 1993; Jian et al., 2012; Pone

et al., 2009; Švábová et al., 2012), while others apply a more general two coefficient model

(Busch et al., 2004; Clarkson and Bustin, 1999; Cui et al., 2004; Shi and Durucan, 2003;

Siemons et al., 2007) to describe diffusion in samples with a relatively wide pore size

distribution. Proponents of the “bidisperse” pore structure approach suggest that one single

average value for pore size may no longer represent the whole sample.

In order to determine the desorption/diffusion behaviour coal, experiments can be designed to

directly measure sorption kinetics (Charrière et al., 2010; Gruszkiewicz et al., 2009; Shi and

Durucan, 2003).

A subject of debate in the experimental estimation of coal diffusion coefficients is whether

diffusion coefficients increase or decrease with an increase in pressure. The dominant trend in

diffusivity/permeability with pressure has implications for modeling both primary and enhanced

recovery/CO2 storage in unconventional reservoirs. Even with similar models, some authors

have found that diffusion coefficients increase with increasing pressure (Charrière et al., 2010;

Ciembroniewicz and Marecka, 1993; Jian et al., 2012), while others have found that they

decrease (Busch et al., 2004; Cui et al., 2004; Pone et al., 2009; Shi and Durucan, 2003; Siemons

et al., 2007). Still others have found that different models may give different pressure trends

depending on the model chosen, even when using the same data (Clarkson and Bustin, 1999;

Staib et al., 2013). Staib et al. (Staib et al., 2013) recently summarized that the lack of

44

consistency in the deduced effects of pressure could be due to: (i) choice of model, (ii) choice of

experimental conditions, and (iii) choice of coal sample.

3.2.2 Diffusivity/permeability studies performed for shale

Matrix transport mechanisms of shale are likely quite different from coal, in part due to the

difference in pore structure of the matrix, and also different pore associations within organic and

inorganic matter, the latter of which is typically not as important for coal. There appear to be

very few studies performed that account for specific transport physics while estimating the

permeability of shale samples. Recently, Heller et al. (2014) used helium at relatively high

pressure (~ 1 MPa) as the test gas and measured permeability of some crushed shale samples.

Helium was used to avoid the effects of adsorption and/or associated swelling that might impact

permeability. Those authors then applied the model suggested by Cui et al. (2009), while

neglecting the Klinkenberg slippage effect, to analyze pressure vs. time data of crushed shale

samples. Because of low accuracy of the experimental data, Heller et al. (2014) were only able to

fit their data with lower and upper bound curves, as opposed to a single curve. For shales,

particularly when adsorption gases are used for measurement, it is important to properly capture

the potentially significant effects of adsorption, slippage and diffusion.

Researchers such as Ertekin et al. (1986), Javadpour (2009), and Civan (2010) have evaluated

permeability coefficients in the shale matrix and concluded that, while gas flows through nano-

scale pores at low pressures, the mean-free path of gas molecules is comparable to the average

effective rock pore throat radius causing the gas molecules to “slip” along pore surfaces (as

noted by Kinkenberg, 1941) – this slip-flow creates an additional flux mechanism which may be

additive to viscous flow and diffusion flow, causing a higher apparent permeability. In the

Javadpour model (Javadpour, 2009), pressure-driven flow of shale nano-pores was modeled

using Darcy’s Law corrected for slippage, while concentration-driven flow was modeled with

Fick’s Law.

These complexities in matrix transport property determination for shale make it difficult for

reservoir engineers to obtain representative values for use in shale reservoir simulation. Further,

use of reservoir simulation to study the effects of fluid storage and transport mechanisms on

primary and enhanced shale gas recovery requires a relatively large dataset. Although some of

45

these data are available for well-developed shale reservoirs, they are limited for other

unconventional reservoirs such as the liquid-rich portions of the Duvernay and Montney

formations in Western Canada, which have recently received a great deal of attention and are in

early stages of development (Ghanizadeh et al., 2015a,b,c).

Rock samples of these reservoirs are required as a source of reservoir property information;

however, typically the only source of rock samples from horizontal wells used to develop these

unconventional reservoirs are rock (drill) cuttings. Because rock properties and reservoir quality

are expected to vary significantly along the length of a horizontal well (Clarkson and

Haghshenas, 2016), it is critical to assess these properties quantitatively from cuttings. However,

quantitative analysis procedures for drill cuttings are in their infancy (Ortega and Aguilera, 2013;

2014). The conventional methods proposed in the literature for shale sample permeability

evaluation require a large quantity of sample (i.e. cores or core plugs) that are not typically

available for horizontal laterals, but rather from offset (and rare) vertical wells. Matrix

permeability is then typically measured using 30 g (or more) of crushed rock samples obtained

from cores extracted from the vertical wells.

In this chapter, experimental procedures and modeling techniques are developed to allow the

extraction of permeability/diffusivity from drill cuttings collected at multiple intervals along a

horizontal well, which in turn enables the evaluation of reservoir heterogeneity. Drill cuttings

obtained from horizontal wells present challenges for characterization due to small sample sizes

(typically < 2-3 g). Therefore, in the current paper, laboratory and modeling procedures for

extracting critical reservoir properties (e.g. permeability or diffusivity) from small sizes of

crushed samples are developed. "Artificial" cuttings derived from previously-analyzed core plug

samples (Ghanizadeh et al., 2015a) are used to develop the procedures and allow for comparison

of the results with larger-scale samples. The experimental procedures used historically for

analyzing coal (Busch et al., 2004; Ciembroniewicz and Marecka, 1993; Clarkson and Bustin,

1999; Cui et al., 2004; Jian et al., 2012; Pan et al., 2010; Pone et al., 2009; Siemons et al., 2007;

Švábová et al., 2012) are adapted for estimating diffusivity/permeability in shales. For this

purpose, the pressure-time data recorded during low-pressure CO2 and N2 adsorption isotherm

collection (volumetric method) is used to extract the parameters of interest from two different

shale sample suites.

46

A new model developed herein is applied to convert the kinetic data to diffusion coefficients. In

addition to applying the previous unipore and bidisperse models, a new, more general numerical

model is developed which utilizes partial differential equations that consider the effects of

adsorption, slippage and diffusion in two pore size domains for shale. This model was used to

evaluate the effects of gas pressure on extracted permeability values. Also, it is proved that this

model works well on the artificial cuttings; our long term goal is then to extend the application of

this model to actual drill cuttings for the purpose of getting permeability estimates along a well.

3.3 Model summary and new model development

The diffusion coefficient for crushed samples is commonly calculated from the solution of the

conservation equation in combination with Fick’s second law for spherically symmetric flow.

Two pore structural models are commonly assumed: unipore and bidisperse.

3.3.1 Conventional bidisperse model for coal

The conventional unipore model (Crank, 1975; Sevenster, 1959; Crosdale and Beamish, 1995;

Nandi and Walker, 1970; Smith and Williams, 1984) assumes spherical particles, where internal

porosity is in the form of spheres of a certain radius ( r ), with a single diffusion coefficient

(Figure 3.1a). The general form of the equation for concentration-dependent (pressure-

dependent) diffusivity is written as:

t

C)r

r

CD(

rrr

2

2

1

(1)

Where r is radius, C is the adsorbate concentration, D is the diffusion coefficient, and t is time.

Presented in terms of fractional uptake, the solution to Eq. 1 for constant concentration of the

diffusing species (or concentration-independent diffusivity) may be expressed as follows:

)exp(16

12

22

1

22t

r

Dn

nM

M

n

t

(2)

Where Mtis the amount of gas undergoing diffusion and adsorption in the pores, expressed as a

function of the final adsorbed amount at equilibrium M∞. The term D/r2, when the pore radius is

47

unknown, is expressed as De, referred to as the effective diffusivity, and therefore remains

insensitive to the value used for the estimate of the mean particle radius. If experimental data are

presented in terms of fractional uptake, then Eq. 2 can be curve-fitted to experimental data by

optimizing De .

Although fractures and microfractures may be removed during the crushing process

(Handwerger et al., 2011), pore size distribution measurements for some coals and shales still

suggest a dual porosity structure for the resulting particles. In these cases, a bidisperse model

(e.g. Ruckenstein et al., 1971) is preferred for retrieving diffusivities from adsorption rate data.

The bidisperse model (Figure 3.1b) assumes spherical particles (macro or mesosphere) which are

filled with an assemblage of non-overlapping microporous2 microspheres of uniform size, with

the space between microspheres making up the macro/mesoporosity (macro/mesoporosity is the

void volume unoccupied by the microspheres). The internal porosity is therefore distributed

between two discrete pore sizes with one diffusion coefficient for each pore size.

Figure ‎3-1 — Conceptual schematic of a) unipore and b) bidisperse model (after Clarkson and

Bustin, 1999).

2 The IUPAC classification of pore sizes (IUPAC, 1994) is used herein: micropores (pore width < 2 nm), mesopores

(pore width between 2–50 nm) and macropores (pore width > 50 nm)

macrosphere macropore

Ra

microsphere

macrosphere macropore

micropore

Ra

Ri

a) b)

48

Following the work of previous researchers (Ruckenstein et al., 1971; Clarkson and Bustin,

1999; Cui et al., 2004), the diffusion–adsorption of gas in macropores and micropores is

described by Eqs. 3 and 4, respectively:

t

)(

rD

R

)()r

rD(

rr

aa

Rri

i

i

i

ia

aa

ra

a

a

aaiia

131 2

2

(3)

t

))(C(

t

)()r

rD(

rr

iiii

ii

ri

i

i

iii

11 2

2

(4)

The above equations assume (i) the transport mechanism is diffusion in both macro- and

micropores and (ii) no adsorption is occurring in the macroporosity.

The derivation and application of a new bidisperse model, specifically adapted for the physics of

storage and flow in shales, is given in the following section.

3.3.2 Modified bidisperse model development for shale

The assumptions for the model introduced in the current study may be summarized as follows:

(i) Isothermal system.

(ii) Applicable transport equations are Fick’s first law (r

DJdiff

) and slippage-

viscous flow ( r

pFkJ

D

visslip

); where J is molar flux (mole/(m

2.s)), Dis the

diffusion coefficient (m2/s), kD is Darcy’s permeability (m

2), Fis the slippage factor,

φ is porosity, ρ is gas phase density (mole/m3), p is the gas pressure (Pa), and μ is

gas phase viscosity (Pa.s).

(iii) Gas phase densities can be expressed using the Real Gas Law:zRT

p , where p is

the gas pressure (Pa), Zis the compressibility factor, Tis temperature (K), and R is

the universal gas constant (J/(mole.K)).

(iv) Void volume is constant with time; pores are incompressible and no correction is

made for void volume shrinkage during adsorption of gas.

(v) The gas phase is mobile whereas the adsorbed phase is immobile.

49

(vi) Particles (crushed samples) are assumed to be perfect spheres with equal radius.

(vii) Adsorption is active in both meso and microporosity and can be modeled using the

Langmuir isotherm.

(viii) Boundary pressure is time-varying.

Following the above assumptions, a more general form of the diffusion-adsorption equations

including the effects of adsorption, diffusion and slippage-viscous flow for larger (Eq. 5) and

smaller (Eq. 6) pores is introduced as follows:

t

C

t

rD

r

pFk

Rr

rD

r

pFk

rr

aaaa

iRiri

ii

i

i

ii

iDi

i

iaaa

ara

aa

a

a

aa

aDa

aa

))1.(().(

,)1(3).

,(

1 2

2

(5)

t

C

tr

rD

r

pFk

rr

iiii

ii

ri

i

i

i

i

ii

iD

i

iii

))1.(().().

,(

1 2

2

(6)

where gas slippage is expressed as a function of pressure as below (Brown et al. 1946; Javadpour

2009):

)12

(.

)8

(15.0

aa

a

arpM

RTF and )1

2(

.)

8(1

5.0

ii

i

irpM

RTF (7)

and C is the adsorbate concentration determined by the Langmuir isotherm:

and ii

iiim

ipb

pbCC

1

, (8)

The parameter which is used in the material balance calculation is the absolute mass of adsorbed

gas. Although the Langmuir model assumes single-layer adsorption even when modeling the

multi-layer adsorption data, the model is capable of correlating absolute adsorption amounts.

Therefore, either the Langmuir model or a table of absolute adsorption versus pressure can be

used in material-balance calculations.

It should be noted that in the previous models for coal, the adjusted parameters (Kdiff,a, Kdiff,i,)

were assumed to be constant values at each pressure step. However, in the new model that

considers the effect of gas slippage on the pore walls, as suggested by several researchers

(Klinkenberg, 1941; Ertekin et al., 1986; Javadpour, 2009; Civan, 2010), the adjusted parameters

50

(Kvis,a, Kdiff,a, Kvis,i, Kdiff,i) depend on pressure even at each pressure step. Langmuir constants

should be determined experimentally for each gas/sample set.

The model has the following initial conditions:

aaiiaaRrrr at ),0(),0(

0 (9)

),0(0vaa

R (10)

and boundary conditions:

at(t,0) 0 and 0

00

iri

i

ara

a

rr

(11)

),(),(aaii

rtRt (12)

),

(3

aaa r

a

a

a

a

a

aa

aD

aa

a

p

Rr

a

vr

Dr

pFk

R

V

tV

(13)

To represent the equations in dimensionless form, the following variables and parameters are

introduced:

a

a

aR

r ,

i

i

iR

r

a

a

ap

,

i

i

ip

,

a

a

ap

Cq

,i

i

ip

Cq

2,

,

aa

aD

avis

R

FkK

, 2,

,

ii

iD

iivis

R

FkK

, 2,

a

aa

adiff

R

DK

,

2,

i

ii

idiff

R

DK

Using the differential chain rule, Eqs. (5) and (6) can be written in terms of pressure, p:

0))1(())((1

,,

2

2

t

pqgKK

pa

aaaaiaadiffaavis

a

a

a

aa

(14)

0))1(())((1

,,

2

2

t

pqKK

pi

iiiiiidiffiivis

i

i

i

ii

(15)

Where

51

1

1,,

)1(3

i

ii

i

iidiffiivisai

pKKg

(16)

with initial conditions:

1at ),0(),0(0

aiiaa

ppp (17)

)1,0(0va

pp (18)


at(t,0) 0 and 0

00

ii

i

aa

app

(19)

),()1,(aaii

tptp (20)

1

1,,

1

3)(

a

a

aa

a

aadiffaavisp

a

av

pKKV

t

pV

(21)

where Vvis measured and ))1(/(tgraingrainp

mV . At t=0, the gas pressure is assumed to be

equal to p0in the meso and microspheres (Eq. 16). Pressure at the boundary of the mesospheres is

equal to cell pressure (Eq. 17). A no (free gas) flow internal boundary condition is used for the

mesospheres and microspheres (Eq. 18). Eq. 19 states that the gas pressure at the microsphere

boundary is equal to the gas pressure in the mesoporosity at ra. Eq. 20 is a mass balance

statement which expresses that the change in mass of gas stored in the cell void space (including

interparticle void space but not internal void space of the particles) is equal to the mass flux of

gas across all particle boundaries for t>0.

The coupled Eqs. 14 and 15 are nonlinear because the coefficients and q are dependent on

the gas phase pressure p. Thus the above equations are solved numerically. A computer code was

developed using a discretization algorithm similar to that used in previous studies (e.g. Clarkson

and Bustin, 1999).

52

3.3.3 Modified unipore model development for shale

A simplified version of the bidiperse model, which assumes a unimodal pore volume

distribution, was also applied to the adsorption rate data. The model equation and boundary

conditions are expressed as follows:

0))1(())((1

,,

2

2

t

pqKK

pa

aaaaaadiffaavis

a

a

a

aa

(22)

with initial conditions:

1at ),0(0

aaa

pp (23)

)1,0(0va

pp (24)


at(t,0) 0

0

a

a

ap

(25)

1

1,,

1

3)(

a

a

aa

a

aadiffaavisp

a

av

pKKV

t

pV

(26)

3.3.4 Model fit to data

Numerical solutions for ),(aa

Rtp (or )1,( aa

tp ) are compared with the experimental pressure

data at different time steps. The permeability coefficients (Kvis,a, Kdiff,a, Kvis,i, Kdiff,i) of each gas

are adjusted to minimize the least-squares function:

N

j

jcaljppL

1

2

,exp,)(

2

1

where jp

exp, and jcalp

, are the experimental measured and numerically computed gas pressures

in the domain vV external to shale particles, respectively, at different time steps j during one

adsorption step.

53

3.3.5 Calculating apparent permeability

To derive the new bidisperse model equations, the format of previous bidisperse models

(Ruckenstein et al., 1971; Clarkson and Bustin, 1999; Cui et al., 2004) is used to allow the reader

to easily track the model improvements over previous formulations.

For ease of comparison with previous studies, the term “apparent permeability” is defined for

meso and micropores (Eq. 27 and Eq. 28) to be in common Darcy permeability units (m2) or (D).

This allows the derived apparent permeabilities to be compared with other tests or modeling

approaches and to be used in reservoir simulators or flow analysis equations. As an additional

advantage, the apparent permeability term reflects the combined effects of all flow mechanisms

in a single parameter.

Apparent permeability of shales is defined as (Swami and Settari, 2012):

,

,, aaga

a

aD

aappDc

Fkk

and

,,

, iigi

i

iD

iappDc

Fkk

note that:

aapp

a

a

a

a

aadiffaavisk

R

KK ,2,,

and

iapp

i

i

i

i

iidiffiivisk

R

KK ,2.,

kapp,a and kapp,i can then be extracted by adjusting parameters Kvis,a, Kdiff,a, Kvis,i, Kdiff,i and

calculated parameters (aa

, and a

) at each pressure step:

a

a

a

a

aadiffaavis

aapp

R

KKk

2

,,,

(27)

i

i

i

i

iidiffiivis

iapp

R

KKk

2

,,,

(28)

Please note that the number of regression parameters can be decreased from four (Kvis,a, Kdiff,a,

Kvis,i, Kdiff,i) to two (kapp,a and kapp,i), if there is no need to discretize the viscous and diffusion

permeability. For this, Eqs 14 and 15 need to be rewritten based on Eqs. 27 and 28.

54

3.4 Experimental procedure

In order to simulate drill cuttings, crushed core samples were used. This was done to allow for a

greater amount of sample to be used for conventional permeability testing and to allow for

comparison between conventional permeability tests and the procedures described herein.

The analyzed samples from the Duvernay shale differ in total organic carbon (TOC) content,

pore network characteristics (porosity, pore size distribution), pore-fluid content (‘‘as-received’’

and cleaned/dried) and mineralogy. Ghanizadeh et al. (2015a) provided details of the sample

properties. Table 3.1 summarizes the relevant properties used in this study.

Table ‎3-1 — Summary of crushed-rock sample properties (Duvernay formation).

Sample

No.

Depth [m]

TOC

[wt %]

Cleaned/dried1

Grain density

[g/cm3]

Total porosity

[vol%]

Microporosity2

[vol%]

mp

[g]

Vp

[cm3]

Langmuir constants

CL

[mole/m3]

b

[Pa-1]

5 3251.49 3.6 2.61 5.7 0.52 2 0.813 242.73 0.0166e-3

8 3257.00 4.5 2.58 5.2 0.57 2 0.818 198.66 0.0157e-3

1 Dean-Stark extraction was performed using Toluene-Methanol (10 days) to remove residual fluid, and the samples were dried in a vacuum

oven at 110oC (10 days).

2 Microporosity obtained from low pressure CO2 adsorption.

1.5 inch diameter core plugs were crushed and sieved between 20 to 35 US Mesh screens to yield

particle sizes with a diameter between 0.5 mm to 0.853 mm. Crushed-rock permeability

measurements were then performed on these larger mass samples (~30 g) in the cleaned/dried

state using a commercial permeameter (SMP-200™, Corelab). The crushed samples were then

placed into the test chamber of the SMP-200, and helium, at approximately 200 psig, expanded

into the test chamber. The pressure decline curve was recorded for up to 2,000 seconds. A

simulator history match of the pressure decline curve yields the matrix permeability of the shale

sample. Typical results are given in Figure 3.2. The SMP-200 results are insufficient for the

55

current use because 1) precision of the pressure data is poor, leading to errors in parameter

extraction and 2) helium is a non-adsorptive gas, providing only a diffusion rate estimate rather

than a diffusion/adsorption rate estimate. Further, the SMP-200 requires a relatively large

amount of sample, which is not usually available when the sample source is drill cuttings.

To improve data quality for permeability estimation, a low-pressure automatic adsorption

apparatus (a Micromeritics 3Flex™) was used to obtain adsorption rate data using CO2 for the

two crushed Duvernay shale samples; the higher precision of the measurements and use of a

sorptive gas met our criteria for analysis. The instrument has three analysis ports for higher

throughput and is capable of recording the rate of adsorption with high accuracy and resolution.

Only ~ 1-2 grams of crushed and sieved sample (splits of the larger crushed rock samples used

for SMP-200 device) are required for analysis in each port, which is much smaller than that

required for the SMP-200. Hence the instrument is useful if only small amounts of cuttings are

available, which is typically the case. Pressure was applied to the samples incrementally, from

vacuum to a maximum pressure (a bit less than atmospheric pressure). During an adsorption step,

gas was dosed into the reference cell, and a few minutes allowed for thermal equilibration. The

gas was then dosed into the sample cell for a few seconds and then the dosing was stopped. The

pressure in the sample cell was monitored using two, fast response high precision (1e-6

torr)

pressure transducers. For computer-controlled adsorption analysis, pressure data in the sample

cell was collected every 0.5 second until equilibrium was achieved. Time intervals for data

collection were evenly spaced, resulting in uniform pressure points taken either at early time or

later time. The uniform spacing of data points lowers the risk of biased curve fits toward a

specific section of time data. The cell was then pressurized to the next pressure step

(programmed before the experiment) and the above procedure repeated. The experimental

pressure–time (p-t) data recorded during adsorption experiments at different pressure stages were

then modeled to obtain permeabilities of the gases in shale particles using the derived bidisperse

gas transport model (Section 2).

The equilibrium adsorption isotherm data are best-fitted using the Langmuir isotherm (Langmuir

parameters are listed in Table 6.1). The equation may be expressed for plotting purposes as:

L

L

LaC

p

C

p

C

p

56

A linear regression is performed for a

Cp / vs. p plots to obtain the Langmuir constants, Lp (

b/1 ) and LC .

3.5 Results

In the following sections, the two pore structure models – unipore and bidisperse – were first

applied to compare their applicability in determining permeability coefficients in the Duvernay

shale artificial cuttings samples (Sections 3.5.1 – 3.5.3). The bidisperse model was then extended

to apply to Montney samples (Section 3.5.4).

Input parameters for the Duvernay samples (Table 3.1) included meso and microporosity

(assumed to be constant), cell void volume (v

V ) and the total shale particle volume (p

V ) (both

volumes assumed to be constant), Langmuir constants (L

C and b ), initial pressure within the

sample particles ( 0p ) and in the cell void volume (

0vp ). Cell void volume was measured to be

23.0748 cm3 and 23.0243 cm

3 for samples #5 and #8, respectively. Micropore volume must also

be provided as input; this is estimated from CO2 low-pressure adsorption data and application of

the Dubinin-Rudushkevich (D-R) model (Dubinin and Astakhov, 1971).

Table 3.2 summarizes the best-fit apparent mesopore and micropore permeabilities (kapp,a and

kapp,i, respectively) at different pressure steps.

Before discussing the analysis of the more accurate low-pressure adsorption (3Flex) data, the

limitations of the SMP-200 results are discussed. Figure 3.2 provides the pressure decay data

measured by SMP-200 instrument for the Duvernay samples. As can be seen, the low resolution

and sampling frequency of pressure transducers limit the ability to precisely constrain the shape

of the pressure- decay curve. Further, the analysis gas is helium, which has insignificant

adsorption affinity. Finally, as mentioned previously, a relatively large amount of crushed

sample (~ 30 g) is required for this instrument and measurement technique. All these limitations

are removed by using the 3Flex low-pressure adsorption instrument, which will be the focus of

the rest of this study.

57

Figure ‎3-2 — Experimental data obtained from SMP-200 and bidisperse model match for the

two crushed Duvernay shale samples a) sample #5, b) sample #8. The low precision of pressure

data is evident, leading to lower confidence in extracted permeability values.

3.5.1 Application of unipore model to Duvernay low-pressure adsorption rate data

Figure 3.3 provides the 3Flex pressure-decay measurements and the results of the unipore

numerical model matches to pressure-decay data (using CO2) for the Duvernay artificial cuttings

samples (1-2 g splits taken from the larger crushed rock samples analyzed in previous section).

Calculated permeability is approximately proportional to the rate of pressure-decay, or, in other

words, to the amount of time required to reach equilibrium (when all intraparticle porosity has

been filled). The differences in resolution of experimental data obtained from SMP-200 and

3Flex are clearly observable (compare Figure 3.3 with Figure 3.2), with the 3Flex providing

higher precision data. From the modeling perspective, it appears that the unipore model does not

provide an adequate fit to CO2 adsorption rate data over the entire time scale and significantly

underestimates the time required to reach equilibrium.

664040

664060

664080

664100

664120

664140

664160

664180

664200

664220

664240

0 100 200 300

Pre

ssu

re (

Pa

)

Time (s)

experimental data

model

666000

666050

666100

666150

666200

666250

666300

0 100 200 300

Pre

ssu

re (

Pa

)

Time (s)

experimental data

model

a) b)

58

Figure ‎3-3 — Experimental data (for one crushed shale sample) obtained from 3Flex and unipore

model match for two pressure steps for Duvernay sample #5.

3.5.2 Application of conventional bidisperse model (constant coefficients) to Duvernay

low-pressure adsorption rate data

Figure 3.4 provides the same 3Flex pressure-decay measurement as in Figure 3.3, but with the

results of the conventional (constant coefficients) bidisperse numerical model matches. From the

modeling perspective, it is clear that, for this set of samples in the given pressure range, the

conventional bidisperse model (Eqs. 3 and 4) gives much better matches to pressure-time data

than the unipore model. This result suggests that the pressure-decay data obeys a two-stage

sorption process, with the first fast stage modeled with kapp.a and the slower second stage

modeled with kapp.i. Pore size distribution plots (Figure 3.7), discussed further later and in

Clarkson and Haghshenas (2016), also support the interpretation of a bimodal pore size

distribution. A comprehensive shale matrix gas transport model must therefore describe

transport in several pore systems (see Discussion section).

23500

23550

23600

23650

23700

23750

23800

23850

23900

23950

24000

24050

0 100

Pre

ssu

re (

pa

)

Time (s^0.5)

2nd dose

experimental data

model

35800

35850

35900

35950

36000

36050

36100

36150

36200

0 100

Pre

ssu

re (

Pa

)

Time (s^0.5)

3rd dose

experimental data

model

59

Figure ‎3-4 — Experimental data (for crushed Duvernay shale sample #5) obtained from the

3Flex device and conventional (constant coefficient) bidisperse model match for two pressure

steps. The bidisperse model match to the fast decay portion is better than the slow decay portion.

From Figure 3.4, it can also be seen that the conventional bidisperse model is more successful in

matching the fast decay portion of the data than the slow decay portion. The new (modified),

variable coefficient model is therefore applied below to improve the late-time match.

3.5.3 Application of modified bidisperse model (variable coefficients) to Duvernay low-

pressure adsorption rate data

To obtain the matches observed in Figures 3.2-4, the permeability coefficients were assumed to

be constant for all pressures in a pressure step. Therefore, the coefficients represent an average

value of apparent permeability for a specific pressure range. Figure 3.5, however, demonstrates

application of the modified bidisperse model accounting for pressure-dependence of the

coefficients (Eqs. 14 and 15) for the same set of data. Applying the modified model, the

limitation of constant permeability at each pressure step is removed, and the micropore

permeability is allowed to change due to gas slippage. Because a reasonable match of the fast

decay portion of the pressure-decay data was already achieved with the conventional bidisperse

model, only kapp.i is set as a variable function of pressure (time). It appears that the pressure-

dependent micropore permeability greatly improves the match quality, especially for the slow

decay portion.

23500

23600

23700

23800

23900

24000

24100

0 100

Pre

ssu

re (

pa

)

Time (s^0.5)

2nd dose

experimental data

model

35800

35850

35900

35950

36000

36050

36100

36150

36200

0 100

Pre

ssu

re (

Pa

)

Time (s^0.5)

3rd dose

experimental data

model

60

The better match can be explained by the theory of gas slippage; that is, when a gas is flowing

along a solid wall, if the wall has a zero velocity, then the velocity of the gas layer in the

immediate vicinity of the wall has a finite value. As a consequence, the quantity of gas flowing

through a capillary is larger than would be expected from Poiseuille’s formula (which assumes

zero velocity for fluids next to the solid surface) (Klinkenberg, 1941). In other words, at low

pressures, the effect of gas slippage on micropore walls is not negligible. Most of the previous

studies using adsorption rate data to extract permeability were performed at high pressure and

therefore neglected the effect of slippage.

Figure ‎3-5 — Experimental data (for crushed shale sample #5) obtained from the 3Flex device

and new (variable coefficient) bidisperse model match for two pressure steps. The new


portions of the data.

Figure 3.6 illustrates the trend of apparent permeability variation with pressure. In general, it

appears that both meso and micropore permeabilities decrease as pressure increases.

23500

23550

23600

23650

23700

23750

23800

23850

23900

23950

24000

24050

0 100

Pre

ssu

re (

pa

)

Time (s^0.5)

2nd dose

experimental data

model

35800

35850

35900

35950

36000

36050

36100

36150

36200

0 100

Pre

ssu

re (

Pa

)

Time (s^0.5)

3rd dose

experimental data

model

61

Figure ‎3-6 — Apparent permeability trend with pressure for crushed shale sample #5. Apparent

permeability decreases as pressure increases.

Table 3-2 — Unipore and bidisperse numerical model parameters obtained from Duvernay shale

adsorption rate data (Duvernay formation).

Sample

No.

Pressure

][ Pa

Unipore model Conventional bidisperse model Modified bidisperse model

aappk , ][ mD

aappk , ][ mD

iappk , ][ mD

aappk , ][ mD

iappk , ][ mD

At equilibrium

pressure

5

24000 1.4×10-02 1.1×10-02 3.3×10-07 1.2×10-02 3.0×10-07

36169 6.6×10-03 6.2×10-03 2.1×10-07 6.3×10-03 1.9×10-07

48449 5.1×10-03 4.8×10-03 2.3×10-07 4.9×10-03 2.0×10-07

60770 3.8×10-03 3.4×10-03 1.1×10-07 3.4×10-03 1.0×10-07

8

24020 0.8×10-02 1.0×10-02 3.4×10-07 1.0×10-02 3.1×10-07

36171 4.0×10-03 5.9×10-03 2.2×10-07 5.9×10-03 2.1×10-07

48453 3.1×10-03 4.6×10-03 2.3×10-07 4.6×10-03 2.1×10-07

60776 2.1×10-03 3.2×10-03 1.3×10-07 3.2×10-03 1.1×10-07

0.0E+00

5.0E-08

1.0E-07

1.5E-07

2.0E-07

2.5E-07

3.0E-07

3.5E-07

4.0E-07

0.0E+00

2.0E-03

4.0E-03

6.0E-03

8.0E-03

1.0E-02

1.2E-02

1.4E-02

0 20000 40000 60000 80000

ka

pp

,i(m

d)

ka

pp

,a(m

d)

Pressure (Pa)

kapp,a

kapp,i

62

3.5.4 Extension of the model to Montney samples

The new multi-pore (bidisperse) numerical model was also applied to carbon dioxide and

nitrogen low-pressure adsorption rate data obtained from the crushed Montney samples. Table 3-

3 provides a summary of sample properties. The low-pressure adsorption device (Fig. 3-7) yields

pressure-time data that is of much better quality than the commercial (SMP-200) crushed rock

permeability device (Fig. 3-8) that requires larger sample sizes. Mesopore apparent

permeabilities obtained from the low pressure adsorption data range from 1×10-4 mD for CO2 to

1×10-3 mD for N2, while micropore apparent diffusivities are in the 1×10-7 mD range (see table

3-4). The mesopore apparent permeability calculated from SMP data is 2×10-3 mD.

Table ‎3-3 — Summary of crushed-rock sample properties (Montney).

Sample

No.

Depth [m]

TOC

[wt %]

Cleaned/dried1

Grain density

[g/cm3]

Total porosity

[vol%]

Microporosity2

[vol%]

mp

[g]

Vp

[cm3]

Langmuir constants

CL

[mole/m3]

b

[Pa-1]

14 3145.02 0.55 2.71 2.1 0.3 2 0.7 113 0.015e-3

1 Dean-Stark extraction was performed using Toluene-Methanol (10 days) to remove residual fluid, and the samples were dried in a vacuum

oven at 110oC (10 days).

2 Microporosity obtained from low pressure CO2 adsorption.

Table 3-4 — Bidisperse numerical model parameters obtained from Montney adsorption rate

data.

Sample

No.

Pressure

][ Pa

Modified bidisperse model (CO2) Modified bidisperse model(N2)

aappk , ][ mD

iappk , ][ mD

aappk , ][ mD

iappk , ][ mD

At equilibrium

pressure

14 24000 1×10-04 2×10-07 1×10-03 1×10-07

63

Figure 3-7 — Experimental data (for crushed Montney sample #14) obtained from the 3Flex

device and new (variable coefficient) bidisperse model match for a) N2 and b) CO2. The


portions of the data.

64


crushed Montney sample # 14. The low precision of pressure data is evident, leading to lower

confidence in extracted permeability values.

3.6 Discussion

The disadvantages of performing measurements on crushed samples include potential

elimination of heterogeneities (such as bedding and laminations), and the inability to investigate

sensitivity of permeability to stress (or effective confining pressure). In addition, straightforward

and established methods for interpreting data are mostly developed for samples of large (and

perhaps more representative) volumes of rock. However, crushed rock analysis has some

important advantages, the main one being that small sample (particle) sizes allow matrix

diffusivity/permeability to be obtained quickly, even for nanodarcy samples. Further, although

this study has been focused on artificial drill cuttings (crushed rock), the method could

potentially be used for real drill cuttings samples, as will be explored in future work.

Nonetheless, although significant progress in permeability modeling of shale crushed samples

has been made in recent years, there are still apparent mismatches obtained in the modeling of

pressure-decay data. Cui et al. (2004) noted that small mismatches between the experimental

data and numerical solution are likely related to the assumptions of the model:

94.85

94.86

94.87

94.88

94.89

0 20 40 60 80 100 120 140 160

Pre

ss

ure

(p

si)

Time (s)

Data

Model

65

(i) Uniform particle sizes and spherical shape of all macro/meso and micro particles

(Figure 3.1). Generally smaller particles have faster adsorption than larger particles, and

the shale particles have different geometries. Average geometry effects are, however,

only considered in the model.

(ii) Constant porosity and void volume. During the adsorption process, with increase of gas

loading, the porosity and void volume of the system may change due to adsorption

swelling adding of adsorbed gases and compression (stress) effects.

(iii) Langmuir isotherm. The best-fit Langmuir parameters generally have several percent

errors, which may also contribute to the mismatch.

Although the new bidisperse diffusion model developed in this work also makes a number of

simplifying assumptions, it does a reasonable job of matching the experimental adsorption rate

data for the shale samples studied to date. The better match of the bidisperse model compared to

the unipore model for the Duvernay samples, for example, can be explained in the context of the

pore structure of the samples. Figure 3.7 illustrates pore size distributions for the Duvernay

samples studied in this work, obtained from the combination of low pressure carbon dioxide (273

K) and low pressure nitrogen (77 K) isotherm data and application of density functional theory

(DFT) (Do and Do, 2003; Adesida et al., 2011) – these results are explained comprehensively in

the accompanying study (Clarkson and Haghshenas, 2016). Some general observations can be

made regarding the pore structure of the Duvernay shale samples, despite the difficulties of

obtaining a quantitative evaluation of pore volume distribution of shale from gas adsorption data.

Both the studied samples are multimodal with respect to pore volume distributions, but have

varying proportions of micro and mesoporosity. The pore volume distributions have one peak at

around 0.6 nm (micropore range) and an additional peak at around 3-4 nm (mesopore range).

Appearance of more than one large peak in the volume distribution suggests heterogeneity in the

pore structure, and provides evidence that model with at least a bimodal pore structure is

required. A similar conclusion can be drawn for Montney samples (see Chapter 2 for a PSD for a

Montney sample).

66

Figure 3-9 — Pore size distributions obtained for the two studied Duvernay shale samples.

Modified from Clarkson and Haghshenas (2016).

As a last point of discussion, it should be emphasized that the apparent permeability variation

with pressure shown in Figure 3.6 suggests that it is necessary to specify the specific pressure

range used when reporting permeability data, when using experimental data as an input for

simulation (primary or enhanced recovery in shales) or flow analysis purposes.

3.7 Conclusions

Typically, the only reservoir samples obtained along the length of a horizontal well drilled in a

shale reservoir are drill cuttings – it is desirable to use these samples in a quantitative way to

evaluate reservoir quality along the length of the lateral to 1) assist with stimulation optimization

and 2) to help calibrate reservoir models used in forecasting the well production. Shale matrix

diffusivity/permeability is an important rock property that, along with other reservoir and rock

mechanical properties, can be used for this purpose. Conventional methods for shale sample

diffusivity/permeability evaluation, however, usually require larger sample sizes such as core

plugs that are not typically available for horizontal laterals.

Drill cuttings, which typically are available, present additional challenges for characterization

due to small sample sizes (typically < 2-3 g). Therefore, in the current chapter, laboratory and

modeling procedures for extracting diffusivity/permeability from small volumes of crushed

0

0.001

0.002

0.003

0.004

0.005

0.006

0.2 2 20 200

dV

/dW

Po

re V

olu

me

(cm

³/g

·nm

)

Pore Width (nm)

CC5

CC8

67

samples (synthetic cuttings) were pursued. The low pressure adsorption apparatus used in this

study to collect CO2 rate-of-adsorption data for two Duvernay shale samples, and CO2 and N2

ROA data for a Montney sample, provided a high degree of precision and resolution necessary

for extracting diffusivity/permeability coefficients. A commercial apparatus used for crushed

rock permeability required more sample (more than what is typically available from cuttings),

and did not have the data precision necessary for accurate modeling of pressure-time data.

In order to extract diffusivity/permeability parameters from the low pressure adsorption rate data,

a new numerical model incorporating the appropriate physics was developed. Using this model,

which assumes a bidisperse pore structure, and the simpler unipore model, it is generally

concluded that: (i) the bidisperse diffusion model better describes the adsorption rate curve than

the unipore model for the shale samples studied; and (ii) a modified bidisperse model that

accounts for gas slippage effects better describes the entire adsorption rate curve than the

bidisperse model with constant permeability. This is because at low pressures, for each pressure

step, micropore permeability varies as a function of gas slippage and is no longer a constant

value (as was assumed with the original bidisperse model). The effective permeability of the

rock is enhanced at very low pore pressures because of molecular slippage. Finally, it is observed

that meso and micropermeability show a decreasing trend with increasing pressure.

In future work, the experimental and modeling procedures used herein will be expanded to

different sets of shale samples/diffusing gases to further investigate the effect of rock

petrophysical and geological properties (pore structure, rock composition, grain size, etc) on

permeability coefficients. Continued comparisons with more conventional permeability testing

will also be performed. Finally, an actual drill cuttings dataset from a horizontal well completed

in a Canadian shale reservoir will be analyzed with the new procedures to quantify the reservoir

quality variability along the lateral.

3.8 Nomenclature

C sorbate concentration (mole/m3)

Cg gas isothermal compressibility (1/Pa)

D diffusion coefficient (m2/s)

68

De effective diffusivity coefficient (m2/s)

Da mesopore diffusivity coefficient (m2/s)

Di micropore diffusivity coefficient (m2/s)

Mt fractional amount of gas diffused into and adsorbed on the pores

M gas molecular weight (g/mole)

mp mass of particles (g)

N number of shale particles (macro/meso spheres) in system

kD Darcy permeability

kapp apparent permeability (m2)

Kvis apparent viscous permeability (1/(Pa.s.m2))

Kdiff apparent diffusion permeability (m2/s)

F gas slippage factor

P average reservoir pressure (Pa)

r radius (m)

rp diffusion path length (m)

Ra macro/mesosphere or particle radius (m)

Ri microsphere radius (m)

S pore surface area per unit bulk volume of coal (m2/m

3)

t time (s)

Vp bulk volume of particles (cm3)

fraction of molecules striking pore wall which are diffusely reflected, and is set to

the value 0.8.

μ gas viscosity (Pa.s)

Subscripts

69

a macro/meso pore

i micropore

L Langmuir

v voids

Greek symbols

γa dimensionless macro/mesosphere radial position

γi dimensionless microsphere radial position

porosity

ρ gas density (kg/m3)

β gas density derivative (mole/m3/Pa)

q sorbed phase concentration derivative (mole/m

3/Pa)

3.9 Acknowledgements

Chris Clarkson would like to acknowledge Shell, Encana and Alberta Innovates Technologies

Futures (AITF) for support of his Chair position in Unconventional Gas and Light Oil Research

at the University of Calgary, Department of Geoscience. The authors thank the sponsors of Tight

Oil Consortium (TOC), hosted at the University of Calgary, for their support of this research.

Partial funding for this study was provided through an NSERC Discovery grant to Clarkson.

3.10 References

1. Adesida, A.G., Akkutlu, I.Y., Resasco, D.E, et al. 2011. Characterization of Kerogen

Pore Size Distribution of Barnett Shale using DFT Analysis and Monte Carlo

Simulations. Paper SPE 147397 presented at the SPE Annual Technical Conference and

Exhibition held in Denver, Colorado, 30 October - 2 November.

2. Brown, G. P., DiNardo, A., Cheng, G. K., & Sherwood, T. K., 1946. The flow of gases in

pipes at low pressures. Journal of Applied Physics, 17(10), 802-813.

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3. Busch, A., Gensterblum, Y., Krooss, B.M., Littke, R., 2004. Methane and carbon dioxide

adsorption-diffusion experiments on coal: upscaling and modeling. International Journal

of Coal Geology 60, 151–168.

4. Charrière, D., Pokryszka, Z., Behra, P., 2010. Effect of pressure and temperature on

diffusion of CO2 and CH4 into coal from the Lorraine basin (France). International

Journal of Coal Geology 81, 373–380.

5. Ciembroniewicz, A., Marecka, A., 1993. Kinetics of CO2 sorption for two Polish hard

coals. Fuel 72, 405–408.

6. Civan, F., 2010. Effective correlation of apparent gas permeability in tight porous

media. Transport in porous media, 82(2), 375-384.

7. Clarkson, C., Bustin, R., 1999. The effect of pore structure and gas pressure upon the

transport properties of coal: a laboratory andmodeling study. 2. Adsorption ratemodeling.

Fuel 78, 1345–1362.

8. Crank, J., 1953. A theoretical investigation of the influence of molecular relaxation and

internal stress on diffusion in polymers. Journal of Polymer Science 11 (2), 151–168.

9. Crank, J., 1975. The mathematics of diffusion. Clarendon Press.

10. Crosdale, P. J., & Beamish, B. B., 1995. Methane diffusivity at South Bulli (NSW) and

Central (QLD) collieries in relation to coal maceral composition. InInternat. Symp.

Mangt. and control of High Gas Emissions and outbursts in U/G mines, at

Wollongong (pp. 363-367).

11. Cui, X., Bustin, R.M., Dipple, G., 2004. Selective transport of CO2, CH4, and N2 in

coals: insights from modeling of experimental gas adsorption data. Fuel 83, 293–303.

12. Cui, X., Bustin, A. M. M., & Bustin, R. M., 2009. Measurements of gas permeability and

diffusivity of tight reservoir rocks: different approaches and their

applications. Geofluids, 9(3), 208-223.

13. Do, D.D., and DO, H.D., 2003. Pore Characterization of Carbonaceous Adsorbents by

DFT and GCMC Simulations: A Review. Adsorption Science and Technology 21 (5),

381-423.

71

14. Dubinin, M.M., Astakhov, V.A., 1971. Description of adsorption equilibria of vapors on

zeolites over wide ranges of temperature and pressure. In: Advances in Chemistry Series,

No. 102. American Chemical Society Publications, Washington, DC, pp. 69–85.

15. Ertekin, T., King, G. A., & Schwerer, F. C., 1986. Dynamic gas slippage: a unique dual-

mechanism approach to the flow of gas in tight formations. SPE formation

evaluation, 1(01), 43-52.

16. Handwerger, D. A., Suarez-Rivera, R., Vaughn, K. I., Keller, J. F., 2011, October.

Improved petrophysical measurements on tight shale reservoirs using retort and crushed

samples. In SPE 147456 presented at the SPE Annual Technical Conference and

Exhibition, Denver, Colorado, USA (Vol. 30).

17. Heller, R., Vermylen, J., & Zoback, M., 2014. Experimental investigation of matrix

permeability of gas shales. AAPG bulletin, 98(5), 975-995.

18. Ghanizadeh, A., Bhowmik, S., Haeri-Ardakani, O., Sanei, H., Clarkson, C. R., 2015a. A

comparison of shale permeability coefficients derived using multiple non-steady-state

measurement techniques: Examples from the Duvernay Formation, Alberta

(Canada). Fuel, 140, 371-387.

19. Ghanizadeh, A., Clarkson, C.R., Aquino, S. Ardakani, O.H., Sanei, H., 2015b.

Petrophysical and Geomechanical Characteristics of Canadian Tight Oil and Liquid-Rich

Gas Reservoirs: I Pore Network and Permeability Characterization. Fuel 153, 664-681.

20. Ghanizadeh, A., Clarkson, C.R., Aquino, S. Ardakani, O.H., Sanei, H., 2015c.

Petrophysical and Geomechanical Characteristics of Canadian Tight Oil and Liquid-Rich

Gas Reservoirs: II Geomechanical Property Estimation. Fuel 153, 682-691.

21. Gruszkiewicz, M., Naney,M., Blencoe, J., Cole, D., Pashin, J., Carroll, R., 2009.

Adsorption kinetics of CO2, CH4, and their equimolarmixture on coal from the

BlackWarrior Basin, West-Central Alabama. International Journal of Coal Geology 77,

23–33.

22. International Union of Pure and Applied Chemistry Physical Chemistry Division,

Commission on Colloid and Surface Chemistry, Subcommittee on Characterization of

Porous Solids, 1994. Recommendations for the characterization of porous solids

(Technical Report), Pure Appl. Chem. 66(8), 1739–1758.

72

23. Javadpour, F., 2009. Nanopores and apparent permeability of gas flow in mudrocks

(shales and siltstone). Journal of Canadian Petroleum Technology,48(08), 16-21.

24. Jian, X., Guan, P., Zhang, W., 2012. Carbon dioxide sorption and diffusion in coals:

experimental investigation and modeling. Science China Earth Sciences 55, 633–643.

25. Klinkenberg, L. J., 1941, January. The permeability of porous media to liquids and gases.

In Drilling and production practice. American Petroleum Institute.

26. Nandi, S. P., Walker, P. L., 1970. Activated diffusion of methane in coal. Fuel, 49(3),

309-323.

27. Ortega, C., Aguilera, R., 2013. A Complete Petrophysical-Evaluation Method for Tight

Formations from Drill Cuttings Only in the Absence of Well Logs. SPE Journal in-press.

28. Ortega, C., Aguilera, R., 2014. Quantitative Properties from Drill Cuttings to Improve

the Design of Hydraulic-Fracturing Jobs in Horizontal Wells. Journal of Canadian

Petroleum Technology 53 (01): 55-68.

29. Pan, Z., Connell, L.D., Camilleri, M., Connelly, L., 2010. Effects ofmatrixmoisture on

gas diffusion and flow in coal. Fuel 89, 3207–3217.

30. Pone, J.D.N., Halleck, P.M., Mathews, J.P., 2009. Sorption capacity and sorption kinetic

measurements of CO2 and CH4 in confined and unconfined bituminous coal. Energy &

Fuels 23, 4688–4695.

31. Ruckenstein, E., Vaidyanathan, A. S., & Youngquist, G. R., 1971. Sorption by solids

with bidisperse pore structures. Chemical Engineering Science, 26(9), 1305-1318.

32. Sevenster, P. G., 1959. Diffusion of gases through coal. Fuel, 38(4), 403-418.

33. Shi, J., Durucan, S., 2003. A bidisperse pore diffusion model formethane displacement

desorption in coal by CO2 injection. Fuel 82, 1219–1229.

34. Siemons, N., Wolf, K.H.A.A., Bruining, J., 2007. Interpretation of carbon dioxide

diffusion behavior in coals. International Journal of Coal Geology 72, 315–324.

35. Smith, D. M., Williams, F. L., 1984. Diffusional effects in the recovery of methane from

coalbeds. Society of Petroleum Engineers Journal, 24(05), 529-535.

36. Staib, G., Sakurovs, R., Gray, E. M. A. 2013. A pressure and concentration dependence

of CO 2 diffusion in two Australian bituminous coals. International Journal of Coal

Geology, 116, 106-116.

73

37. Švábová, M.,Weishauptová, Z., Přibyl, O., 2012. The effect ofmoisture on the sorption

process of CO2 on coal. Fuel 92, 187–196.

38. Swami, V., Settari, A., 2012, January. A pore scale gas flow model for shale gas

reservoir. In SPE Americas Unconventional Resources Conference. Society of Petroleum

Engineers.

74

Chapter 4 New Models for Reserve Estimation and Non-Darcy Gas Flow in Shale Gas

Reservoirs3

4.1 Abstract

Organic-rich shale gas reservoirs have various complexities related to the physics of gas storage

and transport. Traditionally, the original gas in place (OGIP) in shales has been calculated as the

sum of the adsorbed gas and the free gas, using CBM reservoirs as an analog. However, as

recently noted in the literature, the free gas volume must be corrected for presence of adsorbed

gas, assuming all gas storage occurs in kerogen. Even with these corrections in place, shales are

also complex reservoirs in terms of flow characteristics. The contribution of viscous, diffusive,

and slip forces in nano-scale conduits cause the permeability calculated from Darcy’s Law to be

higher than the value for liquids.

In this chapter, a new model is developed to address the effect of the adsorbed gas volume on the

nanopore storage capacity. The relative fraction of adsorbed gas volume is treated as a sorbed-

phase saturation. The initial free gas volume is then calculated by subtracting any non-free gas

saturation from the effective void volume. This concept is extended to a gas material balance

equation through which the free gas volume is dynamically adjusted during depletion. The

Simplified Local Density (SLD) adsorption model is used to evaluate sorbed-phase density and

volume. To address the complexity in gas flow, permeability of the reservoir model is assumed

to be a function of pressure in order to determine the impact of advection, slippage and diffusion

mechanisms. The permeability is calculated via a multi-mechanism flow model. Finally, the

dynamically-corrected permeability is used in parallel with dynamically-corrected porosity to

simulate the primary recovery of a shale gas reservoir.

3 This chapter is a modified version of a paper presented at the SPE/EAGE European Unconventional Conference

and Exhibition held in Vienna, Austria, 25-27 February 2014 as: Haghshenas, B. , Clarkson, C.R. and Chen, S.

2014. New Models for Reserve Estimation and Non-Darcy Gas Flow in Shale Gas Reservoirs. In SPE/EAGE

European Unconventional Conference and Exhibition. Society of Petroleum Engineers. Copyright approval has been

obtained from SPE (see “Copyright Permissions” section of this thesis).

75

The new models successfully describe the unique characteristics of shale reservoirs and correct

the conventional methods for overestimation of reserves and underestimation of permeability.

The format of the final material balance equation and flow model used here preserves the

conventional reservoir engineering framework, but with some important modifications.

4.2 Introduction

Shale gas reservoirs are organic-rich, fine grained reservoirs in which the pore space can be

classified into three main categories: porous organic matter, interparticle and intraparticle pore

system in the inorganic matrix, and fractures (induced by hydraulic fracture stimulation and

natural fractures) (see Loucks et al. 2012). Natural gas (mainly methane) in shale gas reservoirs

is generally believed to be stored as either free or adsorbed gas (Faraj et al., 2004; Hamblin,

2006; Bustin et al., 2008), although solution gas within pore fluids and bitumen may also be

important. The adsorbed gas portion is reported to be as high as 85% in some shale plays (Lewis

and Antrim Shale) and is dependent on a variety of geologic and geochemical properties

(Canadian Discovery, 2006; Drake, 2007). Strictly speaking, there are multiple mechanisms for

gas storage in coals and organic-rich shales including (Clarkson and Haghshenas, 2013):

1) Adsorption upon internal surface area

2) Conventional (compressed gas) storage in natural and hydraulic (induced) fractures

3) Conventional storage in matrix porosity (organic and inorganic)

4) Solution in formation water

5) Absorption (solution) in organic matter

In this chapter, mechanisms 1, 2 and 3 are the main focus. Therefore, both free and sorbed gas

need to be considered in estimating the storage capacity of a shale gas reservoir.

In addition to complex gas storage mechanisms, shales may also exhibit complex fluid transport

mechanisms. Transport of gas, for example, through the shale organic and inorganic matrix may

be complicated by non-Darcy flow through nanopores (slip flow and diffusion), which is

controlled by pressure, temperature, gas properties and pore size.

In the following, the current approaches for calculating gas-in-place and material balance are

summarized, and modified approaches are introduced. Javadpour’s method (2009) for

76

accounting for non-Darcy flow is summarized and a discussion provided on how it is

implemented in this work to model shale gas transport.

4.2.1 Gas-In-Place Calculations for Shale

For free gas estimation, current volumetric approaches use conventional gas-in-place equations

and assign all available ‘effective gas pore volume’ to the free gas phase. These methods are

appropriate for single-phase gas reservoirs in which the reservoir fluid (i.e. the natural gas)

remains as unassociated gas during the entire producing life of the reservoir. In order to capture

the extra hydrocarbon-in-place volume associated with sorption within organic matter, the

current volumetric approaches use the Langmuir adsorption isotherm. Under these assumptions,

the following two equations are used to calculate free and sorbed phase volumes in field units:

gi

wi

fB

SAhG

)1(43560

(1)

iL

iL

baPP

PVAhG

7.1359 (2)

In these equations, Gf is free gas-in-place (scf), Ga is sorbed gas-in-place (scf), A is the drainage

area (ft2), h is the net pay thickness (ft), ϕ is the reservoir porosity (fraction), Swi is initial water

saturation (fraction), Bgi is initial gas formation volume factor (res. ft3/scf), ρb is shale bulk

density (gr/cm3), VL is Langmuir volume (scf/ton), PL is Langmuir pressure (psi), and Pi is initial

reservoir pressure (psi). The total gas-in-place is consequently the summation of the free and

sorbed gas:

aftGGG (3)

Eq. 3 assumes that free gas is calculated completely independent of the sorbed gas volume.

However, as noted by Ambrose et al. (2012) the sorbed phase volume has to be considered in

calculation of the free gas storage capacity. In Ambrose et al.’s work, the contribution of free-gas

storage in the inorganic fraction was ignored, and assumed to be completely associated with

organic matter, which may not be true in all cases.

77

The adsorbed gas volume correction proposed by Ambrose et al. (2012) is not included in

commonly used commercial simulators. Ambrose et al. demonstrated that failure to correct free-

gas storage for the volume occupied by sorbed gas (i.e. use of Eq. 3) results in an over-estimate

of free-gas volume. They therefore modified Eq. 1 to account for the pore space occupied by the

sorbed phase, using the Langmuir equation to calculate the sorbed phase amount and converting

this to a volume assuming a sorbed-phase density. Although the approach first assumed single-

component adsorption, it was extended to multi-component adsorption by Hartman et al. (2011).

A complication is that an accurate adsorption model and sorbed phase density calculations are

required to implement these corrections.

In this chapter, a more tradition reservoir engineering approach is used through introduction of a

new parameter: average sorbed fluid saturation, Sa. This parameter is then applied to the

conventional gas-in-pace (GIP) equation to account for the sorbed phase fraction. This method

facilitates the inclusion of a ‘partitioning coefficient’ for free and sorbed gas components of

‘storage capacity’ in a numerical simulator. Specifically speaking, in the new method, the gas

pore volume is related to the bulk productive volume by not only the average porosity, ϕ, and the

average connate water saturation, Sw, but also by the average sorbed fluid saturation, Sa. This

provides a more accurate estimate of the hydrocarbons-in-place, from which ultimate recovery

can be estimated by using an appropriate recovery factor.

4.2.2 Material Balance Calculations for Shale

When conducting a field study, particularly during the early development period, the bulk

volume is not accurately known. In this situation, one option is to use material balance

calculations, provided shut-in pressures are available. The general (static) material balance

equation in either water-drive or volumetric gas reservoirs could be generated by applying the

law of conservation of mass to the reservoir and associated production:

pwgpe

wi

fwiw

gigigWBBGWp

S

cScGBBBG

1)( (4)

When there is neither water encroachment into, nor water production from, a reservoir of

interest, the gas reservoir is said to be volumetric (Craft, 1990). For reservoirs under volumetric

78

control, there is no change in the interstitial water saturation, so the reservoir gas volume remains

the same. Also, for most gas reservoirs, the gas compressibility term is much greater than the

formation and water compressibility and so the second term on the left hand side of the Eq. 4 can

be neglected. Further, since production is an isothermal process, the reservoir temperature is

assumed constant in calculating gas volume factor, Bg. Williams-Kovacs et al. (2012) derived a

new material balance equation (MBE) that dynamically adjusts free-gas storage volume during

depletion according to the amount of volume occupied by sorbed gas, as suggested by

Ambrose et al. (2012) for volumetric gas-in-place determination. However, they considered

sorbed gas density to be a constant value throughout the calculations. With the aforementioned

common assumptions for gas reservoirs (volumetric depletion and negligible water and

formation volume factors) and application of a new remaining GIP formula, a simplified form of

the material balance equation is created that can be used in graphical form for finding original

gas-in-place (OGIP). Unlike previous studies, pressure-dependent sorbed gas density in the MBE

is accounted for by using the simplified local density (SLD) method.

4.2.3 Permeability Modeling in Shale

Shales are known to exhibit ultra-low matrix permeability, on the order of 10-4

to 10-8

mD (Civan

2008). In some cases, diffusion and gas slippage help to explain the higher-than-expected gas

permeability in nano-capillaries of shale matrix. Javadpour (2009) summarized the simultaneous

contribution of viscous, diffusive and slippage effects on flow of gas molecules in nanoscale

pores, and introduced an apparent gas permeability. His method preserves the format of Darcy’s

equation, and hence can easily be incorporated into commercial simulators:

pk

ua

a

(5)

The apparent permeability proposed by Javadpour, 2009 is as follows:

D

avg

k

aFk

RT

MDk

(6)

where ua is apparent velocity (m/s), p is pressure gradient (Pa/m), ka is apparent permeability

(m2), Dk is Knudsen diffusion constant (m

2/s), μ is gas viscosity (Pa.s), M is gas molecular mass

79

(kg/mol), R is universal gas constant (j/mol/K), T is temperature (K), ρ is gas density (kg/m3), kD

is Darcy permeability (m2), and F is the gas slippage factor (fraction) defined by:

)12

(8

1

PrM

RTF (7)

Where P (Pa) is average reservoir pressure, r (m) is pore radius, and is the fraction of

molecules striking pore wall which are diffusely reflected, and is set to the value 0.8.

Substituting for slippage factor in Eq. 6, the apparent permeability is given as:

8)1

2(

83

282

r

PRT

M

M

RTrk

avg

a

(8)

This corrected pressure-permeability function (Eqs. 6-7) is used, as well as a new pressure-

porosity function derived based on the gas-in-place corrected concepts, to predict the production

performance of a sample shale gas play.

80

4.3 New Approach to Shale Gas-In-Place and Material Balance Calculations, and

Numerical Simulation

The conceptual models representing the volumetric constituents of the shale gas matrix

according to the conventional model, the Ambrose model and the new model provided in this

work, are compared in Figure 4.1. Although the new model still lacks the complexity of a real

reservoir rock, it allows the concept of different fluid storage mechanisms in shale formations to

be introduced. This schematic emphasizes that, from a mathematical point of view, there is a

distinct difference between the three models; that is, the three models treat the nature and

position of the space occupied by the sorbed phase differently.

Figure 4-1 — Petrophysical model showing volumetric constituents of gas-shale matrix. a)

conventional model, b) modified after Ambrose (2012), c) new model.

Conventional Model Ambrose Model New Model

Non-Clay Grain Volume



Dry Clay Volume Dry Clay Volume Dry Clay Volume

Bound (Clay) Water

Volume Bound (Clay) Water

Volume Bound (Clay) Water

Volume

Organic Content+

Sorbed-phase Volume Organic Content

Organic Content

Connected Pore Volume

Containing Free Gas,Oil,

and Water

Sorbed-phase Pore

Volume Connected Pore Volume

Containing Free

Gas,Oil, and

Water+Sorbed Phase

volume

Connected Pore Volume

Containing Free Gas,Oil,

and Water

Isolated Pore Volume Isolated Pore Volume Isolated Pore Volume

81

The following observations can be made from Figure 4-1: the conventional model does not

account for volume occupied by the sorbed phase within the connected pore space; Ambrose’s

model considers a part of pore space to be taken up by sorbed-phase pore volume; and the new

model considers the sorbed fluid to occupy a fraction of connected pore volume. In the following

sections the influence of these different suppositions on reservoir calculations is discussed.

4.3.1 Gas in Place Calculation

Ambrose (2012) treated sorbed phase volume similarly to a non-effective porosity (i.e. a fraction

of bulk volume) and derived the following formula for calculating standard cubic feet of in-place

free gas:

)(10318.1)1(43560

6

L

L

s

b

W

g

fPP

PVMS

B

AhG

(9)

where M is apparent molecular weight (lb/lbmol) of the sorbed phase and ρs is sorbed phase

density (gr/cm3). Note that the second term within the brackets is the sorbed phase porosity

fraction that is subtracted from the free gas porosity fraction to provide a corrected free gas

volume.

However, because the sorbed phase is a fluid phase, not a solid grain or void space, it would be

more realistic to treat its volume like a fluid saturation (i.e., a fraction of pore volume). In other

words, the sorbed phase saturation occupies some portions of the available void volume in

exactly the same way as the other fluid components within the void space (i.e. water, gas etc.).

This new approach has an additional advantage that it will be more familiar to reservoir

engineers. Therefore, the conceptual model that is used in this work is that shown on the right

side of Figure 4.1 and the free gas-in-place expression is corrected as given below:

g

aw

B

SSAhGf

)1(43560

(10)

s

b

L

L

a

M

PP

PVS

)(10*318.1

6

(11)

82

Where, Sa is introduced here as sorbed phase saturation (fraction). The derivation of saturation

formula is given in appendix A.

4.3.2 Material Balance Equation

The main concept in generating the material balance equation is simply a volumetric balance,

which states that (with all parameters at standard conditions, 14.7 psia and 60 oF) the algebraic

sum of volume changes of gas in reservoir and the gas produced must be zero. Here, it is

assumed that water and formation compressibility factors are negligible compared to gas

compressibility and further that the reservoir is a volumetric gas reservoir (no water influx or

production), and that the pores are only occupied by water and gas (no oil saturation).

The volumetric material balance equation proposed by Clarkson and McGovern (2001),

assuming negligible water production from the matrix, is used as a starting point:

)1(

037.327355.0)1(

037.32

wi

LbLLb

p

wi

LbgL

SVBgPPi

Pi

VAh

GS

VBPP

P

(12)

The modification of Clarkson and McGovern’s equation, using Ambrose’s approach for

correcting for sorbed phase volume, is given by Williams-Kovacs et al. (2012):

Li

i

aLb

wi

LLb

p

LaLb

wi

gL

PP

PM

V

S

BgPPi

Pi

VAh

G

PP

PM

V

S

BPP

P

6

6

10318.1)1(037.327355.0

10318.1)1(037.32

(13)

The new equations for calculating OGIP (Eqs. 10 and 11) are used to rewrite Clarkson and

McGovern’s material balance equation as below:

)1(

037.327355.0)1(

037.32

awi

LLb

p

awi

LbgL

SSBgPPi

Pi

VAh

GSS

VBPP

P

(14)

Plotting )1(037.32

awi

LbgL

SSVBPP

P

vs. p

Gx results in a straight line with slopeLb

p

VAh

G

7355.0,

from which the reservoir bulk volume could be calculated. Having reservoir thickness (h) from

83

well logs, Langmuir Isotherm parameters and rock density, one may easily calculate the drainage

area (A). Using this plot, OGIP can be calculated from x-intercept.

4.3.3 Porosity Correction

In the new model derived in this chapter, the effective pore volume within the rock matrix is

not only occupied by water and free gas but also by the adsorbed gas. This means that, the

free gas volume in the new model is less than the case in which the sorbed phase volume is

ignored. As pressure decreases the sorbed phase vaporizes and creates a vacant space that is

instantly occupied by free gas. Numerical simulators are not able to distinguish this change in

free gas void space; the porosity needs to be corrected for these effects. For this purpose, we

introduce the porosity correction factor (or multiplier) to the simulator through porosity-pressure

look-up tables. The pore volume available for free gas is allowed to increase as pore pressure

decreases.

Previously, Williams-Kovacs et. al. (2012) presented a porosity correction factor based on

Ambrose’ free-gas correction formula as provided below:

)1(

10318.1)1(6

wi

L

L

a

b

wi

c

S

PP

PVMS

(15)

In this work, the new free gas accessible volume provided in Eqs. 10-11 is used and the porosity

correction factor developed from that. Therefore, porosity can be altered as a function of pressure

to calculate corrected free gas. Because sorbed phase saturation (Sa) is a positive value, the

porosity multiplier is always a quantity smaller than one. Also, as Sa gets larger (e.g. at higher

pressures or heavier gas molecules), the porosity multiplier shifts more from unity.

)1(

)1(

wi

awic

S

SS

(16)

4.3.4 Pressure-Dependent Sorbed Phase Density

As shown in Eqs. 9-10, for calculating sorbed phase saturation, the sorbed phase density is

needed. The simplified local density approach is used to determine sorbed phase density versus

84

pressure in this work. Rangarajan (1995) originally articulated the physical premises and

assumptions of SLD theory as presented in this work. The model assumes that (1) the chemical

potential at any point near the adsorbent surface is equal to the bulk-phase chemical potential

[i.e. b

z )( ] and (2) the chemical potential at any point above the surface is the sum of the

fluid-fluid and fluid-solid interactions [i.e.

)()()( zzfsff

z ]. Accordingly, the equilibrium

chemical potential is calculated as:

)()()( zzfsffb

z (17)

Here, b is chemical potential of the bulk-phase, ff

is chemical potential of the fluid-fluid and

fs is chemical potential of the fluid-solid. Therefore, at equilibrium, there will be no chemical

potential gradient from the surface of the solid to the bulk fluid outside (Chen, 1997).

The pore geometry most widely used in a local-density model for carbon adsorbents is a slit with

a specified distance (width) L. In this work, the slit width L is defined as the distance between the

two orthogonal planes that are tangential to the surfaces of the first graphite planes on opposing

sides of the slit. Note that we are assuming that nanpores in kerogen are analogous to these slit

pores between graphite planes.

For a slit of width L, the chemical potential is written as (Fitzgerald, 2006):

bulkfsfsffzLzzz )()()()( (18)

Where the subscript “bulk” refers to the bulk fluid, “ff” refers to fluid-fluid interactions, and “fs”

refers to the fluid-solid interactions. The position within a slit is z, where z is orthogonal to the

plane of the solid phase defined as a flat surface formed by the peripheral carbon atoms. A

molecule within a slit has fluid-solid interactions with both slit surfaces at distances z and L – z.

In this study, the PR-EOS was used to calculate the bulk density:

bulk

bulk

bulk

bulk

bulkbulk

bulk

bulk

bulk

bulk

RT

a

RT

b

bbRT

a

b

bf

)21(1

)21(1ln

22

1ln

)21(1ln

22 (19)

85

Where bulk

f fugacity of the bulk-phase is,

bulk is density of the bulk-phase and a and b are PR-

EOS constants.

The local adsorbed fugacity at each position z can be calculated by a local equilibrium

relationship:

Tkff

B

fs

zl

fs

z

bulkzffexp

)( (20)

Where, )( zfff is the fugacity of the fluid-phase that is a function of position, k is Boltzmann

constant (1.3806488 × 10-23

m2 kg s

-2 K

-1), T is temperature, and Ψ

fs is fluid-solid potential

function and typically is described by an integrated potential function such as the 10-4 Lenard-

Jones model:

)

))1((2

1

5

(4

4

14

4

10

10

2

i ss

fsfs

fsfsatoms

fs

z

izz

(21)

2

sszz

(22)

2

ffss

fs

(23)

ffssfs

(24)

Here, σss is carbon interplanar distances, σff is molecular diameter of the adsorbate, εff is fluid–

fluid interaction energy parameter, εss is solid –solid interaction energy parameter, εfs is fluid–

solid interaction energy parameter and ρatoms=0.382 atoms / Å2.

Now, substituting the )( zfff , the PR-EOS is again employed to calculate the local density of

sorbed phase, )( z :

)(

)()(

)(

)(

2

)(

2

)(

)()(

)(

)(

)(

)21(1

)21(1ln

22

1ln

)21(1ln

z

zz

z

z

zz

zz

z

z

zff

RT

a

RT

b

bbRT

a

b

bf

(25)

86

In the above equation, average adsorbed density, a

, can be calculated as:

ads

h

z

a

h

dz

ads

ff

2/

)(][

(26)

where hads is the thickness of adsorbed layer (the distance where fluid density is more than 1.2

times bulk fluid density).

4.4 Results

4.4.1 Volumetric Gas-in-Place Calculation Results

In order to quantify the impact of the free gas correction (for sorbed phase volume) on OGIP

estimation, the shale gas data provided by Williams-Kovacs et al. (2012) was used. The

parameters for three shale gas examples with varying sorption capacity are given below in Table

3.1. Shale A has the lowest TOC content and sorption capacity (VL) and expectedly is the

reservoir with a small adsorbed gas fraction; Shale B represents a reservoir with an intermediate

adsorbed gas fraction; and Shale C has the highest TOC content and the greatest sorption

affinity. The Langmuir adsorption isotherms for the three shale examples are shown in Figure

4.2.

87

Table ‎4-1 — PVT and reservoir input parameters for volumetric OGIP calculation for shale A-C

(modified after Williams-Kovacs, 2012).

Inputs Shale A (Low Sorption) Shale B (Medium Sorption) Shale C (High Sorption)

PVT Inputs

T (°F) 200 200 200

γg (air = 1) 0.69 0.69 0.69

N2 (%) 0 0 0

CO2 (%) 0 0 0

H2S (%) 0 0 0

cw (psi-1) 3.0 * 10-6 3.0 * 10-6 3.0 * 10-6

cf (psi-1) 4.0 * 10-12 4.0 * 10-12 4.0 * 10-12

Bgi (rcf/scf) 0.00478 0.00478 0.00478

Reservoir Inputs

pi (psi) 3500 3500 3500

Net Pay (ft) 100 100 100

Porosity (%) 10 10 10

Sw (%) 0 0 0

TOC (%) 4.00 4.26 11.26

VL (scf/ton) 45 88.5 145.6

PL (psia) 720 535.5 749.3

ρB (g/cm3) 2.47 2.47 2.47

ρs (g/cm3) 0.37 0.37 0.37

Mw (lb/lb-mol) 20 20 20

A (acres) 57 57 57

88

Figure 4-2 — Langmuir adsorption isotherms for three shale samples.

In Table 3.2, the volumetric OGIP estimations by the conventional method, Ambrose’s equation

and our new equation, are summarized for the three shale gas examples at 365 K. The

partitioning coefficient defined in this table is the fraction of the total gas content which is

occupied by each phase (free gas phase or sorbed gas phase):

Partitioning Coefficient of Free Gas = Free Gas Volume (Gf)/ Total Gas Volume (Gf) (27)

Partitioning Coefficient of Sorbed Gas = Sorbed Gas Volume (Gf)/ Total Gas Volume (Gf) (28)

Percent Difference= 100×(GIP from corrected method-GIP from conventional method)/ GIP

from conventional method (29)

Table ‎4-2 — Volumetric OGIP results for shale A-C using conventional and corrected gas

storage models.

Gas-in-Place (scf) Shale A Shale B Shale C

Conventional Conventional Conventional Conventional

Gf 5.2E+09 5.2e+09 5.2E+09

Ga 7.1E+08 1.5e+09 2.3E+09

G 5.9E+09 6.6e+09 7.5E+09

0

20

40

60

80

100

120

140

0 500 1000 1500 2000 2500 3000 3500 4000

sorp

tio

n c

apac

ity

(S

CF

/to

n)

pressure (psi)

Sored Phase Volume per Unit Mass of Rock

Shale A Shale B Shale C

89

Corrected by Ambrose Corrected Corrected Corrected

Gf 4.8E+09 4.5e+09 4.1E+09

Ga 7.1E+08 1.5e+09 2.3E+09

G 5.6E+09 5.9e+09 6.4E+09

Corrected by Eq. 10 Corrected Corrected Corrected

Gf 4.8E+09 4.5e+09 4.1E+09

Ga 7.1E+08 1.5e+09 2.3E+09

G 5.6E+09 5.9e+09 6.4E+09

Partitioning coefficient Conventional Conventional Conventional

(fraction)free 0.88 0.78 0.69

(fraction)sorbed 0.12 0.22 0.31

Corrected by Ambrose Corrected by Ambrose Corrected by Ambrose

(fraction)free 0.87 0.75 0.64


Corrected by Eq. 10 Corrected by Eq. 10 Corrected by Eq. 10

(fraction)free 0.87 0.7534 0.64


Percent Difference Corrected by Ambrose Corrected by Ambrose Corrected by Ambrose

(%Difference)free -7.69% -13.46% -21.15%

(%Difference)total -5.08% -10.61% -14.67%

Corrected by Eq. 10 Corrected by Eq. 10 Corrected by Eq. 10

(%Difference)free -7.69% -13.46% -21.15%

(%Difference)total -5.08% -10.61% -14.67%

Table 3.2 indicates that the conventional gas-in-place calculation method results in equal values

of Gf (Gf =271.35) for all three shale cases, regardless of whether there is high or low sorption

capacity. The new approach for free-gas storage correction (Eq. 10) and Ambrose’s approach

provide exactly the same results - this is because Eq. 10 is developed to rewrite Ambrose’s

90

equation in terms of the more standard definition of reservoir parameters (sorbed fluid saturation

instead of sorbed fluid porosity).

The partitioning coefficients of free and adsorbed gas are illustrated in Figure 4.2. The

conventional method predicts that sorbed gas is a lower fraction of the total gas content while

free gas is a larger fraction, compared to the Ambrose and new methods. The error caused by

using the conventional approach increases as the TOC content (adsorption capacity) of the shale

increases.

Figure 4-3 — Partitioning coefficients of free and adsorbed gas by a) Conventional method, b)

Ambrose method, c) New method.

4.4.2 Material Balance Results

Figure 4.4a-c show the results of the material balance calculations using the uncorrected

Clarkson and McGovern equation, and the corrected equation using the Ambrose approach, and

new approach. For the Ambrose and new approach, the results overlap, confirming the

robustness of the new approach for correcting for sorbed phase volume. The slope of the line for

the uncorrected Clarkson and McGovern equation is smaller than that for the corrected version.

Because the estimated OGIP is obtained from the inverse of this slope, the OGIP calculated by

the uncorrected Clarkson and McGovern equation is also lower than for the corrected methods.

free 88%

sorbed 12%

Shale A

Partitioning Coeffitiont Conventional

Method

free 78%

sorbed 22%

Shale B

Partitioning Coeffitiont Conventional Method

free 69%

sorbed 31%

Shale C

Partitioning Coeffitiont Conventional Method

free 87%

sorbed 13%

Shale A

Partitioning Coeffitiont Corrected by

Ambros or Equation (6)

free 75%

sorbed 25%

Shale B

Partitioning Coeffitiont Corrected by

Ambrose or Equation (6)

free 64%

sorbed 36%

Shale C

Partitioning Coeffitiont Corrected by Ambrose

or Equation (6)

91

The values calculated by uncorrected and corrected methods are respectively given as -

5.8426×10-10

and -5.1048×10-10

for slopes and 6.7277×10+9

(SCF) and 6.8895×10+9

(SCF) for

OGIP, which is a difference of 2%.

Figure 4-4 — Material-Balance plots for shale B in the case of constant sorbed phase density



corrected with New Method.

In order to generate Figure 4.4a-c, the sorbed phase density was considered to remain constant

during depletion, but this would not be the case in reality. In Figure 4.5b-c the MB plots for the

case that sorbed fluid density decreases as average reservoir pressure decreases are provided. The

sorbed phase density is calculated as a function of pressure using simplified local density model

as described previously.

Figure 4-5 — Material-Balance plots for shale B in the case of variable sorbed phase density


0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 2000 4000 6000

y b

y c

on

ven

tio

nal

met

ho

d

Gp (MMSCF)(a)

S=-5.8426

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1000 2000 3000 4000 5000 6000

y b

y A

mb

rose

met

ho

d

Gp (MMSCF)(b)

S=-5.1048

0

0.5

1

1.5

2

2.5

3

3.5

4

0 2000 4000 6000

y b

y n

ew m

eth

od

Gp (MMSCF)(c)

S=-5.1048

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 2000 4000 6000

y b

y c

on

ven

tio

nal

met

ho

d

Gp (MMSCF)(a)

S=-5.8426

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1000 2000 3000 4000 5000 6000

y b

y A

mb

rose

met

ho

d

Gp (MMSCF)(b)

S=-5.0978

0

0.5

1

1.5

2

2.5

3

3.5

4

0 2000 4000 6000

y b

y n

ew m

eth

od

Gp (MMSCF)(c)

S=-5.0978

92


corrected with New Method.

Because the uncorrected Clarkson and McGovern equation does not account for changes in

sorbed phase density, the predicted OGIP (6.7277×10+9

) is unchanged. However the corrected

equation (using the Ambrose and new approaches), now including the change in sorbed phase

density, results in a higher OGIP than before (6.8990×10+9

versus 6.8895×10+9

SCF (1%

difference).

4.4.3 Porosity Correction and Production Results

To look at the impact of the models on porosity correction and production from the numerical

simulator, permeability is first assumed to be constant (case 1), and then the pressure-dependent

porosity model is compared with the constant porosity model for three scenarios: 1) porosity

varies, but sorbed phase density remains invariant with respect to pressure 2) porosity and sorbed

phase density remain invariant with respect to pressure and 3) porosity and sorbed phase density

are dynamically adjusted during pressure depletion. In the next step, same cases are run but

assuming variable permeability (case 2). Table 3.3 summarizes the different cases investigated

in this study. The results are illustrated in Figures 4-6 to 4-8. For all cases, flow is assumed to be

from organic pores to inorganic pores to natural fractures (as was done in the work by

Haghshenas and Clarkson, 2013).

93

Table ‎4-3 — Summery of case studies investigated in this paper.

Case Number Case1: permeability is invariant with

pressure

Case2: permeability is variable with

pressure

Scenario Scenario:1 Scenario:2 Scenario:3 Scenario:1 Scenario:2 Scenario:3

porosity is variable with

pressure

porosity is invariant with

pressure

sorbed phase density is variable

with pressure

sorbed phase density is

invariant with pressure

Figure 4-6 — Plot of porosity correction factor vs cumulative production. Considering variable

sorbed phase density with production causes the correction factor to be closer to one, especially

at higher production levels (correction factor equal to one means whole pore volume is available

0.86

0.87

0.88

0.89

0.9

0.91

0.92

0.93

0.94

0 1000 2000 3000 4000 5000 6000

poro

sity

corr

ecti

on f

acto

r

Gp (MMSCF)

Porosity Correction Factor- Constant ρs vs. Variable ρs

porosity correction factor_New formula (constant ρs)

porosity correction factor_New formula (variable ρs)

94

for free gas and sorbed gas has zero volume. As pressure decreases during production, the sorbed

phase evaporates and allows more of the pore volume to be occupied with free gas.

Figure 4.7 illustrates that when permeability does not change with pressure (case 1), the recovery

factor for the scenarios that porosity changes with pressure (scenarios 2 and 3) is around 10 %

lower than the scenario where porosity is assumed to be constant (scenario 1). Further, the

recovery factor for the scenario with variable sorbed phase density (scenario 3) is lower than that

with constant sorbed phase density (scenario 2).

Figure 4-7 — Plot of gas recovery factor vs. time for the case with constant permeability.

For case 2 (permeability changes with pressure), Figure 4.8 illustrates that the difference

between recovery factor for case 2- scenario 3 (porosity and sorbed phase density change with

pressure) and case 2- scenario 1 (porosity is assumed to be constant) is even greater

(approximately 10 percent) than this difference between case 1- scenario 3 and case 1- scenario

1. Therefore, when permeability is considered to be sensitive to pressure depletion, the role of

porosity correction is more important. Further, as before, the recovery factor of the case with

variable sorbed phase density is lower than the case with constant sorbed phase density.

0

10

20

30

40

50

60

70

0 2000 4000 6000 8000 10000 12000

RF ( permeability constant- porosity constant)RF ( permeability constant- porosity changes- constant ρs)RF ( permeability constant- porosity changes- variable ρs)

40

42

44

46

48

50

7500 8500 9500 10500 11500

95

Figure 4-8 — Plot of gas recovery factor vs. time for the case with variable permeability.

4.5 Discussion

From the above, it is observed that, with the same OGIP and average reservoir pressure drop, the

corrected model results in lower recovery factor than the uncorrected model. This is because, in

order to keep the total original gas in place constant, with equal effective porosity the

uncorrected model contains more free gas and less adsorbed gas than corrected model.

Consequently, for an equal time scale the uncorrected model produces more free gas than

corrected model whereas the corrected model produces more sorbed gas than uncorrected model.

The recovery of adsorbed gas needs a higher pressure drop than the equivalent standard cubic

feet of free gas. This can be explained by the physical mechanism which accounts for the

occurrence of the adsorption phenomenon; the intermolecular attractive forces between the

adsorbent (with high specific surface area) and natural gas is greater than those between bulk gas

molecules themselves (bulk gas is defined as the gas phase sufficiently far from the adsorbent

surface where gas almost is not under the influences of attractive surface forces). Therefore gas

molecules tend to remain condensed on the surface of the solid (organic materials and certain

clay particles). Therefore, production of sorbed gas requires a higher pressure drop to overcome

this additional attractive energy.

0

10

20

30

40

50

60

70

0 2000 4000 6000 8000 10000 12000

RF ( permeability changes- porosity constant)RF ( permeability changes- porosity changes- constant ρs)RF ( permeability changes- porosity changes- variable ρs)

50

55

60

65

7500 8500 9500 10500 11500

96

The material balance approach presented above has also been extended to non-volumetric

reservoirs. While Eq. 14 accounts for free-gas storage volume changes due to adsorption, and

desorption, it assumes a volumetric reservoir (no change in fracture porosity, no water influx

etc.). Recently, a general equation Eq. 30 was developed by the Haghshenas and Clarkson

(2016b, in preparation) that can be applied to both dual porosity coalbed methane and shale

reservoirs. The new equation accounts for water encroachment/production, expansion of

formation and residual liquids in overpressured reservoirs in the matrix and fractures, gas

desorption and matrix shrinkage and swelling effects, the latter being important for some CBM

reservoirs. As with Eq. 14, the non-zero volume of the adsorbed phase in the matrix pore

volume is accounted for with an adsorption phase saturation term. Variation in adsorbed phase

thickness is accounted for by introducing adsorbed phase compressibility into the material

balance equation. An important aspect of the new equation is that it is presented in a simple and

familiar P/Z form and the straight line plot simultaneously gives i,f

G and itG

, .

GGZ/P

Z/P

p

i,f

*

ii

11

(30)

Where

i,fg

wpe

L

L

iL

iL

gii

gi

bieq,feq,m

ii

*

iiGB

.)BWW(

PP

PV

PP

PV

S

BP]CC)[(

Z/P

Z/P

Z/P

Z/P 615511

(31)

All of these terms are defined in the nomenclature section. The full derivation is provided in

Appendix B.

Fig. 4-9 provides and illustration of how to apply the new material balance equation to

production data.

97

Figure 4-9 — Illustration of how to use new straight line method for calculation of free- and

total-gas-in-place. Modified from Haghshenas and Clarkson (2016b, in preparation).

4.6 Conclusions

In this chapter, new corrections for gas-in-place and material balance calculations which account

for the volume occupied by sorbed gas in the pore network are provided. A rigorous approach

for accounting for sorbed phase density change with pressure is also provided, which was not

demonstrated previously. The impact of these new corrections was demonstrated. It is observed

that the correction for sorbed phase volume leads to a smaller void space available for free gas,

which in turn leads to lower initial gas in place estimates using both volumetric and material-

balance methods.

The corrected pore volume is a dynamic function of pressure (free gas pore volume increases as

pressure decreases). Therefore, one needs to define a porosity correction factor to use in a

numerical simulator in order to predict the production performance of shale reservoirs. The

calculated recovery factor using this porosity correction is lower than the case in which the

simulator assumes a constant free gas volume [ϕ(1-Swi)].

-3

-2

-1

0

1

2

0 1 2 3 4 5 6

[(p

/z)/

(pi/z

i)]*

Gp

Fitted line Production data

[(p/z)/(pi/zi)]*min

Gfree,i Gt,i

98

Finally, the permeability of shale gas reservoirs is not constant, but may increase during pressure

depletion due to non-Darcy flow mechanisms. Therefore, in order to improve predictions of

shale gas production using a conventional numerical simulator, the model must include a

porosity correction factor as well as a permeability correction factor.

4.7 Nomenclature

Field Variables

A areal extent of gas reservoir, acres

Bg gas formation volume factor, rcf/SCF

Bgi initial Gas formation volume factor, bbl/SCF

Bw water formation volume factor, bbl/STB

cf formation isothermal compressibility, psi-

cw water isothermal compressibility, psi-

Dk Knudsen diffusion constant (m2/s)

F theoretical slippage dimensionless coefficient

f fugacity, Pa

Gf free-gas storage capacity, scf

Gp Cumulative Gas Produced, MMscf

h net formation thickness, ft

ka apparent permeability (m2)

kB Boltzmann’s constant, R/NA=1.3806488×10-23

m2 kg s

-2 K

-1

kD Darcy’s permeability (m2)

L slit width, nm

M molar mass (kg/mol) or (lb/lbmol)

ms sorbed phase mass, ton

99

NA Avogadro’s number, 6.02214129×1023

mol−1

n molar number

ns number of sorbed phase moles, lbmol

P average reservoir pressure, Pa

p pressure, psia

pL Langmuir pressure, psia

pst standard pressure, psia

R gas constant (J/mol/K) or (psi.ft3/lbmol/R)

r pore radius (m)

Sf free gas saturation, dimensionless

Sw water saturation, dimensionless

Sa adsorbed phase saturation, dimensionless

T temperature, K

Tst standard temperature, °R

ua apparent velocity (m)

Vb bulk volume, ft3

VL Langmuir volume (scf/ton)

Vp pore (void) volume, ft3

Vs sorbed phase volume, ft3

Vst standard volume, scf

We water influx into reservoir, bbl

Wp cumulative produced water, STB

z distance from the surface of the wall

zst standard gas compressibility, dimensionless

100

Greek Symbols

α tangential momentum accommodation coefficient, fraction

p change in average reservoir pressure, psi

εff energy parameter of fluid-fluid molecular interaction

εfs energy parameter of fluid-solid molecular interaction

μ viscosity, Pa.s

μb chemical potential of bulk gas

μff chemical potential of fluid-fluid interaction

μfs chemical potential of fluid-solid interaction

ρ molar density, kgmol/m3

ρa adsorbed phase density, g/cm3

ρb reservoir bulk density, g/cm3

ρbulk gas bulk density, g/cm3

ρgr grain density, g/cm3

σff molecular diameter of the adsorbate

σss carbon molecules interplanar distance

Ψ fluid-solid potential

ϕ total porosity fraction, dimensionless

4.8 Acknowledgement

The authors would like to thank Shell Canada for their support of this research. Clarkson would

like to thank Encana and Alberta Innovates Technology Futures for their contributions to his

Chair in Unconventional Gas and Light Oil research at the University of Calgary, Department of

Geoscience.

101

4.9 References

1. Ambrose, R., Hartman, R., Diaz-Campos, M., Akkutlu, I. Y., & Sondergeld, C. 2012.

Shale Gas-in-Place Calculations Part I: New Pore-Scale Considerations. SPE Journal,

17(1), 219-229.

2. Anderson, D., Nobakht, M., Moghadam, S., & Mattar, L. 2010, February. Analysis of

production data from fractured shale gas wells. In SPE unconventional gas conference.

3. Bustin, M., Bustin, A., Ross, D., Chalmers, G., Murthy, V., Laxmi, C. et al. 2008. Shale

Gas Opportunities and Challenges.AAPG Annual Convention. San Antonio, TX:

American Association of Petroleum Geologists.

4. Canadian Discovery. 2006. Shale Gas in North America. Canadian Discovery Digest, 6

pp. B1-B41.

5. Chen, J.H., Wong, D.S.H., Tan, C.S., Subram anian, R., Lira, C.T., Orth, M. 1997.

Adsorption and desorption of carbon dioxide onto and from activated carbon at high

pressures. Industrial and Engineering Chemistry Research 36: 2808– 2815.

6. Clarkson, C.R. and McGovern, J.M. 2001. Study of the Potential Impact of Matrix Free

Gas Storage Upon Coalbed Gas Reserves and Production Using a New Material Balance

Equation. Proceedings of the 2001 International Coalbed Methane Symposium, The

University of Alabama, Tuscaloosa, Alabama, p. 133-149.

7. Clarkson, C.R. and Haghshenas, B. 2013a. Modeling of Supercritical Fluid Adsorption

on Organic-Rich Shales and Coal. Paper SPE 154532 presented at the SPE

Unconventional Resources Conference-USA, The Woodlands, Texas, USA.

http://dx.doi.org/10.2118/164532-MS.

8. Craft, B. C., Hawkins, M. F., & Terry, R. E. 1991. Applied petroleum reservoir

engineering.

9. Curtis, M.E., Ambrose, R.J., Sondergeld, C.H., et al. 2010. Structural Characterization of

Gas Shales on the Micro- and Nano-Scales. Paper SPE 137693 presented at the Canadian

Unconventional Resources and International Petroleum Conference held in Calgary,

Alberta, Canada, 19-21 October. http://dx.doi.org/10.2118/137693-MS.

102

10. Drake, S. 2007. Unconventional Gas Plays. Southwest Land Institute Presentation.

American Association of Professional Landmen.

11. Ertekin, T., King, G.R., and Schwerer, F.C. 1986. Dynamic Gas Slippage: a Unique

Dual-Mechanism Approach to the Flow of Gas in Tight Formations. SPE Form Eval 1(1):

43-52. Paper SPE 12045-PA. http://dx.doi.org/10.2118/12045-PA.

12. Faraj, B., Williams, H., Addison, G. and McKinstry, B. 2004, Winter. Gas Potential of

Selected Shale Formations in the Western Canadian Sedementary Basin. Gas TIPS, pp.

21-25.

13. Fitzgerald, J.E., Robinson Jr., R.L., and Gasem, K.A.M. 2006. Modeling of High-

Pressure Adsorption of Gas Mixtures on Activated Carbon and Coal using a Simplified

Local Density Model. Langmuir 22: 9610-9618.

14. Haghshenas, B, Clarkson, C. 2013. Multi-Porosity, Multi-Permeability Models for Shale

Gas Reservoirs. In presented at SPE Canadian Unconventional Resources Conference

held in Calgary, AB, Canada, November.

15. Hamblin, A. P. 2006. The “Shale Gas” Concept in Canada: A Preliminary Inventory of

Possibilities. Ottawa, ON: Geological Survey of Canada.

16. Hartman, R.C., Ambrose, R.J., Yucel Akkuutlu, I. and Clarkson, C.R. 2011. Shale Gas-

in-Place Calculations Part II – Multi-component Gas Adsorption Effects. Paper SPE

144097, presented at the SPE Unconventional Gas Conference held in Woodlands, TX,

14-16 June.

17. Javadpour, F. 2009. Nanopores and Apparent Permeability of Gas Flow in Mudrocks

(Shales and Siltstone). J. Cdn. Pet. Tech. 48 (8): 16-21. http://dx.doi.org/10.2118/09-08-

16-DA.

18. Loucks, R.G., Reed, R.M., Ruppel, S.C., et al. 2009. Morphology, Genesis, and

Distribution of Nanometer-Scale Pores in Siliceous Mudstones of the Mississippian

Barnett Shale. Journal of Sedimentary Research 79 (12): 848-861.

http://dx.doi.org/10.2110/jsr.2009.092.

19. Passey, Q.R., Bohacs, K.M., Esch, et al. 2010. From the Oil-Prone Source Rock to Gas-

Producing Shale Reservoir – Geological and Petrophysical Characterization of

Unconventional Shale Gas Reservoirs. Paper SPE 131350 presented at the International

http://dx.doi.org/10.2118/12045-PA

103

Oil and Gas Conference and Exhibition, Beijing, China, 8-10 June.

http://dx.doi.org/10.2118/131350-MS.

20. Sondergeld, C., Ambrose, R., Rai, C., et al. 2010. Micro-Structural Studies of Gas Shales.

Paper SPE 131771 presented at the SPE Unconventional Gas Conference, Pittsburgh,

Pennsylvania, USA, 23-25 February. http://dx.doi.org/10.2118/131771-MS.

21. Rangarajan, B., Lira, C.T., Subramanian, R. 1995. Simplified local density model for

adsorption over large pressure ranges. AICHE Journal 41: 838– 845.

22. Williams-kovacs, J., Clarkson, C., & Nobakht, M. 2012, October. Impact of Material

Balance Equation Selection on Rate-Transient Analysis of Shale Gas. In SPE Annual

Technical Conference and Exhibition.

Appendix A: Proof of Eqs. 1 and 2

gB

aSWSAh

gB

aSWSpV

gB

fSpV

fG)1(43560)1.(.

(A4-1)

iL

iL

b

iL

iLb

aPP

PVAh

PP

PVAhG

7.1359

037.3243560 (A4-2)

Sorbed Phase Saturation. If total porosity and total saturation is used, and mgr=mb , then

bb

V

bm

bV

mgr

:

104

tot

Mb

PL

P

PL

V

tot

Mb

PL

P

PL

V

tot

Mb

PL

P

PL

V

tot

Mb

PL

P

PL

V

stRT

stP

tot

M

bV

bm

PL

P

PL

V

stRT

stP

Sb

mgrmassume

tot

M

bV

grm

PL

P

PL

V

RTst

Pst

totbV

M

grmP

LP

PL

V

stRTstz

stVstP

S

stRTstz

stVstPn

totbV

Mn

totbV

m

totbV

m

pV

VS

aaa

aa

a

aa

aa

a

a

a

a

a

a

a

a

)(

610*318.1

037.32

2000037.3248.379

1

037.32

2000037.3252073.10

7.14

037.32

2000037.32

037.32

2000

037.32

2000

037.32

2000

037.32

2000

037.32

037.32

(A4-3)

If total porosity and total saturation is used with ρgr:

tot

M

PL

P

PL

V

tot

M

PL

P

PL

V

tot

M

PL

P

PL

V

tot

M

V

m

PL

P

PL

V

stRT

stP

S

totbV

grV

totbVVp

tot

M

bV

grm

PL

P

PL

V

RTst

Pst

totbV

M

grmP

LP

PL

V

stRTstz

stVstP

S

stRTstz

stVstPn

totbV

Mn

totbV

m

totbV

m

pV

VS

a

totgr

a

tot

gr

a

tot

gr

a

tot

gr

gr

a

aa

aa

a

a

a

a

a

a

a

a

)1(

)(

610*318.1

037.32

2000)1(

037.3248.379

1

037.32

2000)1(

037.3252073.10

7.14

037.32

2000

1

)1(

037.32

2000

037.32

2000

037.32

2000

037.32

037.32

(A4.4)

If effective porosity and effective saturation is used and mgr=mb , thenb

bV

bm

bV

mgr

:

105

effa

eff

a

eff

a

eff

a

eff

a

b

b

eff

a

eff

a

aa

eff

a

a

eff

a

a

eff

a

a

a

a

Mb

PL

P

PL

V

Mb

PL

P

PL

VMb

PL

P

PL

V

Mb

PL

P

PL

V

RTst

PstM

bV

m

PL

P

PL

V

RTst

Pst

mgrmassume

M

bV

grm

PL

P

PL

V

RTst

Pst

bV

M

grm

PL

P

PL

V

stRTstz

stVstP

S

stRTstz

stVstPn

bV

Mn

bV

m

bV

m

effpV

V

effS

)(

610*318.1

037.32

2000037.3248.379

1

037.32

2000037.3252073.10

7.14

037.32

2000037.32

037.32

2000

037.32

2000

037.32

2000

037.32

2000

037.32

037.32

(A4.5)

If effective porosity and effective saturation is used with ρgr:

106

effa

Mgr

PL

P

PL

V

eff

a

Mgr

PL

P

PL

V

eff

a

Mgr

PL

P

PL

V

eff

a

Mgr

PL

P

PL

V

RTst

Pst

eff

a

M

grV

grm

PL

P

PL

V

RTst

Pst

aS

totbV

grV

totbV

pV

eff

a

M

bV

grm

PL

P

PL

V

RTst

Pst

effbV

a

M

grm

PL

P

PL

V

stRTstz

stVstP

aS

stRTstz

stVstP

an

effbV

a

M

an

effbV

a

am

effbV

a

am

effpV

aV

effaS

)1(

)(

610*318.1

037.32

2000

)1(

037.3248.379

1

037.32

2000

)1(

037.3252073.10

7.14

037.32

2000

)1(

037.32

037.32

2000

1

)1(

037.32

2000

037.32

2000

037.32

2000

037.32

037.32

(A4.6)

107

Appendix B – Derivation of Eq. 30

The general material balance equation, with consideration of water encroachment/production,

differences in formation and residual fluid expansion in matrix and fractures, gas desorption as a

source term, matrix shrinkage and adsorbed gas layer thickness change is given as below. It

starts with a general material balance equation for naturally fractured reservoirs and then is

extended to CBM reservoirs.

A material balance equation for naturally fractured gas reservoirs (with no adsorption and no

water encroachment/production) is written as follows (Aguilera, 2008):

pBS

CCSCSBBG

pBS

CCSCSBBGBG

gi

gfi

pfwwfioofi

ggiifreef

gi

gmi

pmwwmioomi

ggiifreemgp

,,

,,

(B-1)

where p

G (MMSCF) is produced gas volume at standard conditions, m

G (MMSCF) is initial

gas-in-place in matrix at standard conditions, f

G (MMSCF) is initial gas-in-place in fracture at

standard conditions, g

B (RCF/SCF) is gas formation volume factor, w

C (psi-1) ,pm

C (psi-1) and

pfC (psi-1) are water, matrix and fracture compressibilities, respectively,

wmiS and

wfiS are

initial water saturation in matrix and fracture, respectively, and p (psi) is current average

reservoir pressure minus initial reservoir pressure ( )(i

ppp ). For CBM reservoirs with

water encroachment/production, adsorbed gas storage, and adsorbed phase compressibility, we

modify the above equation as below:

108

615.5)(

,

,,

,,

wpeg

L

L

iL

iL

gii

giifree

Bi

gi

gfi

pfwwfioofi

ggiifreef

gi

gmi

pmaamwwmioomi

ggiifreemgp

BWWBPP

PV

PP

PV

S

BG

pBS

CCSCSBBG

pBS

CCSCSCSBBGBG

(B4-2)

where ifreem

G,,

is initial free gas-in-place in matrix, ifreef

G,,

is initial free gas-in-place in fracture.

Dividing both sides of previous equation byifree

G,

, it changes to:

615.5)(

,

,

,,

,

,,

,

ifree

wpe

g

L

L

iL

iL

gii

gi

Bi

gi

gfi

pfwwfioofi

ggi

ifree

ifreef

gi

gmi

pmaamwwmioomi

ggi

ifree

ifreem

g

ifree

p

G

BWWB

PP

PV

PP

PV

S

B

pBS

CCSCSBB

G

G

pBS

CCSCSCSBB

G

GB

G

G

(B4-3)

For simplification, the ratio of free gas-in-place in fracture divided by total free gas-in-place is

defined as (Aguilera, 2008):

ifreemifreef

ifreef

GG

G

,,,,

,,

(B4-4)

Note that it is not an easy task to calculate the individual terms of the numerator and

denominator of Eq. (A1-4), however, the ratio ( ) can still be calculated based on measurable

data as follows:

gmiimgfiif

gfiif

SS

S

,,

,

(B4-5)

Substituting for , Eq. (A1-3) is simplified to:

109

615.5)(

11)1(

,

,,

,

ifreeg

wpe

L

L

iL

iL

gii

gi

Bi

g

gi

eqf

g

gi

g

gi

eqm

g

gi

ifree

p

GB

BWW

PP

PV

PP

PV

S

B

pB

BC

B

Bp

B

BC

B

B

G

G

(B4-6)

where eqm

C,

and eqf

C,

are defined as:

gmi

pmaamwwmioomi

eqmS

CCSCSCSC

,

(B4-7)

and

gfi

pfwwfioofi

eqfS

CCSCSC

,

(B4-8)

Finally, using the definition of gas formation volume factor:

p

zTB

g0282793.0

(B4-9)

Eq. (37) changes to:

615.5)(

])1[(1/

/1

,

,,

,

ifreeg

wpe

L

L

iL

iL

gii

gi

Bieqfeqm

iiifree

p

GB

BWW

PP

PV

PP

PV

S

BpCC

zp

zp

G

G

(B4-10)

For plotting purposes, similar to that is commonly used for conventional gas reservoir (i.e.

)/

/(

iizp

zpversus

pG plot, we derive the following expression:

110

ifreeg

wpe

L

L

iL

iL

gii

gi

Bieqfeqm

ii

ii

GB

BWW

PP

PV

PP

PV

S

BpCC

zp

zp

zp

zp

,

,,*

615.5)(

])1[(1/

/

/

/

(B4-11)

Hence Eq. (A1-10) simplifies to:

*

,/

/1

iiifree

p

zp

zp

G

G

(B4-12)

Rewriting Eq. (A1-12) to be equivalent to conventional volumetric equation, results in:

1

1/

/

,

*

p

ifreeii

GGzp

zp

(B4-13)

Where

*

/

/

iizp

zpdecreases from 1 at

ipp , to its minimum value of *

min

)/

/(

iizp

zpat 0p :

iL

iL

gii

gi

Bi

iiPP

PV

S

B

zp

zp

*

min/

/

(B4-14)

Based on Eq. (A1-13), for a plot of

*

/

/

iizp

zpversus

pG , where 0

/

/*

iizp

zp,

pG reflects

ifreeG

, and where *

*

min

)/

/(

/

/

iiiizp

zp

zp

zp

,

pG reflects

itG

,.

111

Chapter 5 Modeling PVT Behavior of Gas-Condensate System under Pore Confinement

Effects: Implications for Rate-Transient Analysis of Gas-Condensate Shale Plays4

5.1 Abstract

Rate-transient analysis (RTA) is a robust technique for evaluating reservoir/stimulation

properties and for forecasting production from shale reservoirs. However, knowledge of fluid

storage and flow mechanisms, and controlling rock and fluid parameters, is critical for obtaining

meaningful information from RTA.

It is common practice to use PVT data measured in laboratories (i.e. bulk fluid properties) for

reservoir modeling and production data analysis purposes. These measurement techniques were

developed for conventional reservoirs and cannot explain some of the anomalous fluid

production behaviors observed for shale gas-condensate wells, such as long-term constant gas/oil

ratio (GOR) trends. One explanation for this behavior is that the PVT properties of fluids are

affected by confinement in nano-scale pores, and hence deviate from bulk fluid properties. On

the other hand, it is also addressed very well in the recent literature that, the apparent

permeability of gas in nanopores is different from Darcy (liquid) permeability.

In order to study the effects of pore confinement on fluid properties in shales, the simplified local

density (SLD) model is used. The SLD model can be used to estimate fluid density gradients

from pore wall to pore center, and therefore explicitly considers pore geometry in adsorption

modeling. This model can also be used to adjust the confined fluid critical properties, phase

envelope and viscosity. Significant shifts in phase envelope and fluid properties due to pore

confinement are observed in this work. Importantly, the corrected equation-of-state predicts a

later onset for condensate dropout in shale reservoirs than for bulk systems. The SLD model is

4 This chapter is a slightly modified version of a paper presented at the SPE Low Perm Symposium held in Denver,

Colorado, 05-06 May 2016 as: Haghshenas, B., Qanbari, F., Clarkson, C.R., and Chen S. 2016. Modeling PVT

Behavior of Gas-Condensate System under Pore Confinement Effects: Implications for Rate-Transient Analysis of

Gas-Condensate Shale Plays. In SPE Low Perm Symposium. Society of Petroleum Engineers. Copyright approval

has been obtained from SPE (see “Copyright Permissions” section of this thesis).

112

also used to estimate adsorbed layer thickness, and then the corrected pore radius along with

diffusion and slippage effects where used to modify permeability calculations.

The corrections for fluid properties, adsorbed layer thickness and non-Darcy flow are then

analytically incorporated into transient linear flow analysis of nanoporous shale gas-condensate

wells. Analysis of simulated cases using the “corrected” (for pore confinement effects) and

“uncorrected” RTA is performed to quantify errors associated with the latter.

This study demonstrates that failure to account for pore confinement effects on fluid properties

and fluid flow results in errors in linear flow parameter estimation using RTA, but the error

depends on the fluid composition, pore size, permeability and pressure. The effects of pore

confinement should therefore be considered for proper evaluation of shale gas-condensate

reservoirs using analytical or numerical methods.

5.2 Introduction

The focus of shale gas activity in recent years has shifted from dry shale gas to liquid-rich shale

reservoirs that produce more profitable condensate and oil (Rahmani and Akkutlu 2015). Multi-

fractured horizontal wells (MFHWs) are currently the most popular method for exploiting these

low-permeability liquid-rich shale (LRS) systems. The combination of complex fracture,

reservoir and fluid properties encountered in these systems has necessitated the modification of

conventional reservoir engineering approaches for quantitative analysis. Among different

available engineering approaches, production data analysis is an applicable engineering tool for

analyzing low-permeability reservoirs to obtain reserves estimates, hydraulic fracture and

reservoir properties, and for development planning.

For performing production data analysis, the first step is to recognize the flow regime. The most

common transient flow regime encountered in the analysis of MFHWs completed in shale

reservoirs is transient linear flow. This flow regime is characterized by a straight line on a

square-root-time plot, which is a plot of rate-normalized pseudo pressure (for gas reservoirs)

against square-root of time (Wattenbarger et al. 1998). The slope of the square-root-time plot is

used to estimate the linear flow parameter, ickA , provided that certain reservoir and fluid

properties are known. However, square-root of time analysis historically has assumed that bulk

113

fluid properties are applicable and that Darcy’s Law is valid; both fluid properties and transport

mechanisms may deviate away from these basic analysis assumptions for shales due to

confinement effects in nano-scale pores. In nanopores, the mean-free path of gas molecules is

comparable to or larger than the average effective rock pore radius; thus, the molecule-pore wall

interactions may become significant. These molecule-pore wall interactions may be significant

enough to change the fluid and rock characteristics such as gas critical properties, z-factor, gas

formation volume factor, viscosity, effective permeability, effective porosity and adsorption

affinity (Rahmani and Akkutlu 2013; Devegowda et al. 2012; Ma et al. 2013). Therefore, the

slope of the square-time-plot needs to be adjusted to account for these effects in certain

instances.

In this study, the effect of alteration of shale gas condensate fluid phase behavior and transport

properties due to pore confinement is investigated using numerical simulation. The simplified

local density model is applied to investigate bulk gas-adsorbed phase coexistence, z-factor and

critical properties under confinement. The transport properties are adjusted through diffusion and

slippage effects and adsorbed layer thickness.

5.3 Theory

5.3.1 Thermophysical Properties of Fluids under Confinement in Nanopores

The small scale of the pore structure in shales causes a shift in the fluid thermophysical

properties, in particular, the onset point of condensate dropout in the reservoir. Measurement of

fluid phase behavior in nanoscale porous media is a difficult task. Ball and Evans (1989),

Morishige et al. (1997) and Morishige and Shikimi (1998) demonstrated experimentally that

critical temperatures of bulk fluid decrease substantially with decreased pore size. More recently,

Parsa et al. (2015) established experimentally that the point of phase transition obtained from

bulk PVT experiments does not represent the phase behavior of hydrocarbon fluids confined in

nano-pores and that confinement leads to a reduction in vapor pressure.

Due to the lack of experimental data on fluid-phase confinement, theoretical approaches can be

helpful. Zarragoicoechea and Kuz (2002, 2004) used the van der Waals equation-of-state and

Lennard-Jones potential equations to predict vapor-liquid equilibria and critical properties of

114

confined fluids in nanopores and found good agreement with Morishige et al.’s (1997)

experimental results. They presented empirical equations for critical temperature and pressure

adjustment. Other semi-empirical methods, such as the simplified local density (SLD) model

(Rangarajan, 1995; Fitzgerald, 2006) and molecular simulation, have also found use in

describing fluid properties under pore confinement. Jiang et al. (2005) investigated the phase

coexistence of n-alkanes in single-wall carbon nanotubes using grand

canonical ensemble Monte Carlo simulation (GCEMC) and observed the critical temperature in

the confined state to be lower than that of the bulk state. Singh et al (2009) used configurational-

bias grand-canonical transition-matrix Monte Carlo simulations to model critical property

alterations for alkanes and observed that the critical temperature decreases as pore size decreases,

although this decrease occurs below a certain pore size. Ma et al. (2013) used correlations for

critical property alteration suggested by Singh et al. (2009). Rahmani et al. (2013, 2015) also

used the GCEMC technique and reported that the liquid production from nanoporous rocks is

enhanced due to a significant decrease in the bubble point and dew point pressures. Other

theoretical efforts using the Monte Carlo simulation approach include the works of Nagy and

Siemek (2014), Pitakbunkate et al. (2015), Haider (2015), Allawe et al. (2015), and Sandoval

(2015). For illustration purposes, a typical snapshot of methane-ethane within a slit-shape nano-

pore is shown in Figure 5.1 (Rahmani et al. 2013). Parallel alignment of the ethane molecules to

the pore walls is caused by physical adsorption. It is observable that methane develops a density

profile within the pore. This non-uniform distribution of gas molecules causes the fluid to be less

dense in the central portion of the pore space. The change in gas density subsequently alters

many related fluid parameters such as critical temperature and pressure, compressibility factor

and viscosity.

115

Figure 5-1 — Snapshot of methane (small blue spheres) and ethane (grey spheres) molecule

distribution in slit-shaped carbon pore using Monte Carlo simulations. Note the layers of

molecules parallel to the upper and lower organic walls. From Rahmani et al. (2013).

Although molecular simulation is able to describe the pore confinement phenomenon in detail,

the computational demand of this approach makes it relatively impractical. Hence, a

mathematical model, such as the SLD model that can reasonably reproduce the results of

molecular simulation, yet yields computational savings, is desirable (Ma and Jamili 2014). Ma et

al. (2013, 2014) used a conventional bulk-fluid equation-of-state and altered the attraction term

using the SLD model.

In the current study, the SLD model was used for calculating equilibrium adsorption of gas onto

slit-pore walls as well as gas fugacity and compressibility factors. All parameters of the

equation-of-state (not just the attraction term) were altered using the SLD model. The use of the

SLD model for this purpose is presented in the following section.

5.3.2 Use of SLD Model for Estimating Gas Properties under Pore Confinement

The SLD model theory and use in calculating phase envelope shifts of gas condensate in

nanopores was recently discussed in Clarkson and Haghshenas (2016). An important starting

point for estimating gas property alteration is Eq. 1 below:

116

)()( ]exp[

)()(

zff

B

fsfs

bulkzfff

Tkff

zLz

(1)

which gives the local adsorbed-phase fugacity at each position z. In the above equation, T is

temperature, fbulk refers to bulk fugacity, fff (z) is fluid fugacity at position z, Ψfs is the fluid-solid

potential function and kB is the Boltzmann constant.

Using the fluid fugacity from Eq. 1, the compressibility factor, Z, can be then derived from

following expression:

)( )414.0

414.2ln(

22)ln(1ln zZ

BZ

BZ

B

ABZZ

p

f

(2)

where:

22TR

apA

RT

apB

),(45724.0)(

22

r

c

cT

P

TRTa

c

c

P

RTb 077796.0

22/1)]1(1[),(

rrTkT

226992.05422.137464.0 k

And the averaged compressibility factor is defined as:

ads

adsh

ff

z

h

dzZ

Z

2/

)( ][

(3)

Because the formation volume factor and viscosity are gas parameters that are directly

influenced by z-factor (Z), these parameters are also altered by pore confinement. The formation

volume factor is calculated as:

117

g

sc

sc

gB

pT

ZTPB

(4)

and viscosity is calculated by Lee et al. (1966):

g

Y

gXK

)exp(10

4 (5)

where:

RT

Mw

Z

p

g

MwxT

xxX

3

2

1

XyyY21

TMwkk

TMwkkK

k

54

321

)(

Eq. 4 is solved using constants from Table 5-1. Please note that the coefficients in Table 5-1 are

rounded to represent the same significant numbers provided in original paper (Lee et al., 1966).

The results, however, are not considerably sensitive to the number of significant digits.

The corrected critical pressure, Pc, and critical temperature, Tc, are calculated as follows. Altered

z-factor distributions are calculated at a set of pressure points, using Eq. 2 and, at each pressure,

the calculated z-factor is averaged over pore width (zavg.Eq.2). Then, z-factor from PR-EOS is

fitted to the resulting zavg.Eq.2 vs. pressure points by setting Tc and Pc as the regression parameters.

Table ‎5-1 Constants for Lee’s viscosity correlations.

Parameter Value

k1 9.4

k2 0.02

118

k3 1.5

k4 209

K5 19

x1 3.5

x2 986

x3 0.01

y1 2.4

y2 0.2

5.3.3 Non-Darcy flow Calculations, Taking into Account Adsorbed Layer Thickness

Changes, Diffusivity and Slippage Effects

Rock absolute (liquid-equivalent) permeability decreases with a decrease in pore size; however,

for gases, non-Darcy flow effects cause apparent gas permeability to be larger than the liquid-

equivalent value. The reason is that the assumption of zero gas velocity at the pore wall in

continuum theory is violated in nanopores at low pressure values; that is, gas molecules slide (or

“slip”) along the pore walls. To describe this behavior, an apparent permeability, which is a

corrected form of Darcy permeability for the positive impact of gas slippage and diffusion, is

introduced (Javadpour 2009). However, conventional applications of apparent permeability

models that account for non-Darcy flow (slippage/diffusion) assume a constant pore radius,

neglecting the effect of adsorbed layer thickness changes with pressure. Haghshenas et al. (2014)

used the SLD model to calculate the thickness of the adsorbed layer, which in turn was used to

derive the effective radius for gas flow (pore radius minus the adsorbed layer thickness) in

apparent gas permeability models. Figure 5.2 illustrates the ratio of apparent to absolute

permeability versus pressure for the gas composition that is used in the “Application” section,

and includes adsorbed layer thickness changes, diffusion and slippage for a 5 nm pore. The

apparent gas permeability increases dramatically with decreased pressure.

119

Figure 5-2 — Apparent gas permeability calculations for a 5 nm pore radius, accounting for non-

Darcy flow and adsorbed layer thickness changes with pressure.

5.4 Application

In this section, the impact of altered fluid properties and non-Darcy flow caused by pore

confinement on the results of RTA is demonstrated. First, the SLD model is used to derive

modified fluid properties, and then numerical simulation is used to demonstrate the impact of

pore confinement on RTA.

5.4.1 Confined Fluid Property Estimation

Starting with the bulk fluid compositions provided in Table 7.2, the SLD model is used to predict

fluid properties in a 5 nm pore. Figure 5.3-5 compare the bulk and altered gas properties

obtained from the relevant correlations for gas compressibility factor (Eq. 2), formation volume

factor (Eq. 3), and viscosity (Eq. 4). The results demonstrate the pore size-dependence of

thermophysical properties - as pore size decreases, z-factor (and therefore formation volume

factor) increases, whereas viscosity decreases.

1

10

100

1 10 100 1000 10000

Pe

rme

ab

ility

ra

tio

Pressure, psia

kapp/kD

120

Table ‎5-2 Bulk fluid (gas) composition.

Gas Component Percent

CH4 0.7

C2H6 0.08

C3H8 0.08

NC4 0.08

NC5 0.06

Figure 5-3 — Comparison of gas compressibility in the bulk state versus within a 5 nm slit-pore.

0

0.5

1

1.5

2

2.5

3

0 2000 4000 6000 8000 10000

z-f

acto

r

Pressure, psia

z bulk

z confined

121

Figure 5-4 — Comparison of gas formation volume factor in the bulk state versus within a 5 nm

slit-pore.

Figure 5-5 — Comparison of gas viscosity in the bulk state versus within a 5 nm slit-pore.

Figure 5.6 illustrates a comparison between the bulk fluid phase envelope and that corrected for

the effect of pore proximity. For plotting this figure, the modified phase envelopes were

0.0025

0.0035

0.0045

0.0055

0.0065

0.0075

0.0085

0.0095

0 2000 4000 6000 8000 10000

Bg

, b

bl/scf

Pressure, psia

Bg bulk

Bg confined

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 2000 4000 6000 8000 10000

µ, cp

Pressure, psia

µ bulk

µ confined

122

calculated using the Peng-Robinson EOS by inputting the critical point data for the confined

fluid into a commercial PVT package. The two-phase envelope shrinks as a result of pore

confinement, and therefore can potentially reduce condensate drop-out and near wellbore

permeability impairment. This shift in phase envelope can explain the commonly observed “dew

point suppression” phenomenon discussed in the literature in association with LRS reservoirs

(Rahmani and Akkutlu 2013; Devegowda et al. 2012; Ma et al. 2013).

Figure 5-6 — Comparison of gas phase behavior in the bulk state versus within a 5 nm slit-pore.

5.4.2 RTA of Numerical Simulation Results

In order to demonstrate the impact of pore confinement two synthetic production data sets are

generated using numerical simulation. Fluid properties used in numerical simulation of both

cases are those altered by pore confinement (Figures 5-3 to 5-6 in the previous section). The only

difference between simulation cases is that, in Case 1, the impact of pore confinement on gas

apparent permeability is neglected whereas in Case 2, the apparent permeability profile that

includes pore confinement effects (Figure 5-2) is used. The numerical model for the synthetic

cases is similar to that used previously by Clarkson and Qanbari (2015) in which identical

0

500

1000

1500

2000

2500

3000

0 50 100 150 200 250 300

Pre

ssu

re, p

sia

Temperature, oF

2-Phase boundary2-Phase boundary confinedReservoir pressure pathCritical pointCritical point

123

hydraulic fractures (Figure 5-7) are assumed and discretization is performed perpendicular to

fracture direction with nth grid block size calculated from:

grids

an

nNneyy ,...,2,1,0 ,)()(

0 (6)

where 0

)( y is the size of fracture grid, grids

N is total number of matrix grid blocks, and a is a

factor controlling the size of the grids, which can be calculated by the equation for fracture half-

distance, e

y :

gridsN

n neyy

0)( (7)

Figure 5-7 — Base geometry for the synthetic cases. From Clarkson and Qanbari (2015). An

element of symmetry is used to reduce the computation time.

The simulation runs are performed under constant flowing pressure constraint for 200 days.

Inputs for the numerical model are provided in Table 5-3.

For each set of synthetic data, two RTA studies were performed, one with the confinement effect

and the other one without this effect. This was to illustrate that, for a given production data set,

how much error can be encountered into calculations if an uncorrected RTA approach is used.

The synthetic production data for Case 1 is presented in Figure 5-8. For analyzing the

production data, the first step is to identify the flow regime. Transient linear flow regime is

xe

ye

xf

Element of

symmetry

124

evident from half-slope behavior on the log-log diagnostic plot (Figure 5-8b). The production

data is therefore analyzed using the square-root of time plot (linear flow plot) (Figure 5-9), which

is corrected for drawdown and associated nonlinearities (pressure-dependent fluid properties and

condensate dropout) using the methods of Qanbari and Clarkson (2013).

In Figure 5-9, the linear flow parameter, ickA , is determined from the slope of the line passing

through the data. In order to evaluate the impact of altered fluid properties on RTA, two linear

flow plots are presented in Figure 5-9; one with correction for pore-confined fluid PVT (red

circles) and the other one with bulk fluid PVT (black diamonds). Linear flow plots (Figure 5-9)

yield values of total ickA = 4890 ft

2md

0.5 (2.2 % lower than the input value) using confined

PVT versus ickA = 4260 ft

2md

0.5 (14.8 % lower than the input value) using bulk PVT. These

results are provided in Table 7.4. In this example, therefore, use of bulk PVT fluid properties

does not introduce large errors in the analysis. However, the sources of any input or calculation

error should be identified and removed (if possible) to prevent accumulation of errors and

unreasonable characterization of the reservoir/completion system. It is important to note that the

error could be higher for a case with higher condensate content – this is discussed further in the

“Discussion” section.

Table ‎5-3 Numerical model inputs used in the generation of synthetic cases.

Parameter Case 1 Case 2

Fluid Properties Figures. 5.3-6 Figures 5.3-6

Initial pressure (psi) 5000 5000

Temperature (°F) 150 150

Flowing pressure (psi) 1000 1000

Porosity (%) 7 7

Rock compressibility (psi-1) 7E-6 7E-6

Initial gas saturation (%) 1 1

Input total Ac√ki (ft2md0.5) 5000 5000

Apparent permeability due to slippage, diffusion, and

adsorption layer

Not modeled Figure 5.2

125


flow is evident from half-slope behavior on the log-log plot).

Figure 5-9 — Square-root of time plot for Case 1 with and without correction for fluid property

changes.

0

5

10

15

20

25

30

35

40

0

200

400

600

800

1000

1200

1400

0 50 100 150 200

Co

nd

en

sa

te R

ate

(S

TB

/da

y)

Ga

s R

ate

(M

scf/

da

y)

Time (days)

Gas Rate

Flowing Bottomhole Pressure

Condensate Rate

(a)

100

1000

10000

100000

1 10 100 1000

Co

rre

cte

d n

orm

aliz

ed

Ga

s R

ate

(1

06p

si2

/cp

/MM

scf)

Time (days)

(b)

1/2 slope line

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 5 10 15

Corr

ecte

d n

orm

aliz

ed

Ga

s R

ate

(1

06p

si2

/cp

/MM

scf)

√t (√days)

Linear Flow Plot - Confined Properties

Linear Flow Plot - Bulk Properties

126

Table ‎5-4 — Results of RTA with and without pore confinement effects.

Parameter Case 1 Case 2

Total Ac√ki from confined PVT (ft2md0.5) 4890 (2.2 % error)

Total Ac√ki from bulk PVT (ft2md0.5) 4260 (14.8 % error)

Total Ac√ki with apparent permeability and confined PVT

(ft2md0.5)

5240 (4.8 % error)

Total Ac√ki without apparent permeability and bulk PVT

(ft2md0.5)

10400 (108 %

error)

In Case 2, the apparent gas permeability profile of Figure 5.2 (accounting for the combination of

slippage, diffusion, and adsorption layer changes) is used in generating production data.

Production data and log-log diagnostic plots for Case 2 are shown in Figures 5.10a and 5.10b,

respectively.

In order to evaluate the summed impact of altered fluid properties and non-Darcy permeability

on RTA, two linear flow plots are presented in Figure 5-11; one with correction for apparent

permeability and fluid property changes (red circles) and the other one without these corrections

(black diamonds). In the former case, the permeability was corrected for non-Darcy flow using

the method of Qanbari et al. (2014). There is considerable discrepancy between the plots

implying that neglecting the impact of diffusion, slippage, and adsorption layer changes results

in overestimation of total ickA (10400 ft

2md

0.5 vs. the input value of 5000 ft

2md

0.5 as listed in

Table 5-4).

127


flow is evident from half-slope behavior on the log-log plot).

Figure 5-11 — Square-root of time plot for Case 2 with and without correction for gas apparent

permeability and fluid property changes.

0

5

10

15

20

25

30

35

40

0

200

400

600

800

1000

1200

1400

0 50 100 150 200

Co

nd

en

sa

te R

ate

(S

TB

/da

y)

Ga

s R

ate

(M

scf/

da

y)

Time (days)

Gas Rate

Flowing Bottomhole Pressure

Condensate Rate

(a)

100

1000

10000

100000

1 10 100 1000

Co

rre

cte

d n

orm

aliz

ed

Ga

s R

ate

(1

06p

si2

/cp

/MM

scf)

Time (days)

(b)

1/2 slope line

0

1000

2000

3000

4000

5000

6000

7000

8000

0 5 10 15

Corr

ecte

d n

orm

aliz

ed

Ga

s R

ate

(1

06p

si2

/cp

/MM

scf)

√t (√days)

Linear Flow Plot - With Apparent Permeability

Linear Flow Plot - Without Apparent Permeability

128

5.5 Discussion

Conventional black oil and compositional simulators lack the necessary physics to account for

pore confinement effects on fluid behavior. In this study, the SLD model is used to account for

change in PVT properties under confinement. The resulting PVT properties can be utilized in

numerical and analytical reservoir models.

As demonstrated in the previous section, neglecting pore-confined PVT properties introduces an

error in RTA results. For Case 1, where only the impact of altered fluid properties was examined,

the error of not accounting for pore confinement effects on RTA results was not substantial.

However, depending on fluid properties, the error could be much more significant. For example,

in this study, a lean gas condensate fluid was assumed, resulting in only a modest shift in the

phase envelope (Figure 5-6). For rich-gas condensate fluids, the shift in phase envelope caused

by pore confinement could be much more dramatic. The importance of this from an RTA

perspective is that, using bulk fluid properties to guide analysis would require condensate

dropout and relative permeability effects to be accounted for in the analysis. However, if the

phase envelope shift due to pore confinement is so significant that little to no condensate dropout

occurs, then a single-phase RTA analysis would suffice – the difference in the results of RTA

using bulk and confined fluid properties could therefore be much more significant in richer-gas

cases. Therefore, in future work, the impact of using bulk PVT should be tested for cases with

higher levels of CGR than what was assumed in this study.

Further, from the production forecasting perspective, the impact of confinement on producing

CGR and condensate production (as the most valuable commodity) also needs to be considered.

To illustrate, cumulative condensate production for forecasts assuming bulk and pore-confined

fluid properties (Case 1, provided in the previous section) are compared (Figure 5.12). The

results show that the case with confined PVT yields higher CGR than the case with bulk PVT

(37.3 vs. 32.7 STB/MMscf, respectively), resulting in higher cumulative condensate production.

This is because the reservoir fluid in nanopores exhibits a behavior similar to a leaner gas-

condensate system, thereby reducing the condensate dropout or at least delaying it.

129

Figure 5-12 — Cumulative condensate production from bulk and pore-confined PVT properties.

The general properties of both of the cases are listed in Table 7.3.

5.6 Conclusions

The simplified local density model is used to correct gas compressibility, viscosity and phase

behavior for the effects of pore confinement in liquid-rich shales. These changes in gas

properties due to pore confinement, as well as the effects of non-Darcy flow and adsorbed layer

thickness changes (also calculated with use of the SLD model) are incorporated into transient

linear flow analysis of nanoporous shale gas condensate reservoirs. It is concluded that the effect

of pore confinement on fluid properties (in this study, a 5 nm pore size was assumed) can be

significant. For example, for the gas condensate fluid modeled in this study, the dewpoint

pressure was altered from 1800 psi to 1500 psi (a drop of 300 psi) at reservoir temperature of

150oF. This is consistent with the phenomenon of “dewpoint suppression” which has been noted

previously for liquid-rich shale reservoirs.

This study also demonstrates, using numerical simulation, that pore confinement can

significantly affect the results of rate-transient analysis. For example, neglecting changes due to

pore confinement results in underestimation of the linear flow parameter derived from linear

20

22

24

26

28

30

32

34

36

38

40

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 50 100 150 200

Pro

du

cin

g C

GR

(S

TB

/MM

scf)

Cu

mu

lative

Co

nd

en

sa

te (

MS

TB

)

Time (days)

Cumulative Condensate - Confined PVT

Cumulative Condensate - Bulk PVT

Producing CGR - Confined PVT

Producing CGR - Bulk PVT

130

flow analysis. Conversely, neglecting non-Darcy flow effect results in overestimation of the

linear flow parameter. Therefore, the effects of pore confinement on fluid properties and fluid

flow should be considered for production forecasting, field planning, well placement, and

completion design and facility management.

5.7 Nomenclature

Field Variables

Bg Gas formation volume factor (RCF/SCF)

fbulk Bulk fugacity (psia)

fff Fluid fugacity at position z (psia)

kB Boltzmann constant

Mw Gas molecular weight (lb/lbmole)

p Pressure (psia)

R Universal gas constant (psia.ft3.lbmole

-1.R

-1)

T Temperature (K)

z Real gas compressibility factor (fraction)

Greek Letters

ρ Density (lb/ft3)

ω Acentric factor

μ Gas viscosity at T and p (cp)

Ψfs Fluid-solid potential function

Subscripts

c Critical

g Gas

r Reduced

131

sc Standard condition







5.9 References

1. Allawe, E. M., Stockdale, A. P., Aminiaa, K., & Ameri, S. 2015. Impact of Nanoscale

Pore Confinement on Estimation of Gas Resources and Gas/Condensate Behavior in

Shale Reservoirs. Paper SPE 177285 presented at the SPE Eastern Regional Meeting held

in Morgantown, West Virginia, USA, 13-15 October 2015.

2. Ball, P.C. and Evans, R. 1989. Temperature Dependence of Gas Adsorption on a

Mesoporous Solid: Capillary Criticality and Hysteresis. Langmuir 5(3): 714-723.

3. Clarkson, C.R., and Haghshenas, B. 2016. Characterization of Multi-Fractured Horizontal

Shale Wells using Drill Cuttings: 1. Fluid-in-Place Estimation Journal of Natural Gas

Science & Engineering. Journal of Natural Gas Science and Engineering, in press.

4. Clarkson, C.R. and Qanbari, F. 2015. An Approximate Semianalytical Multiphase

Forecasting Method for Multifractured Tight Light-Oil Wells With Complex Fracture

Geometry. J of Can Pet Technol 54 (06): 489-508.

http://dx.doi.org.ezproxy.lib.ucalgary.ca/10.2118/178665-PA.

5. Devegowda, D., Sapmanee, K., Civan, F. and Sigal, R.F. 2012. Phase Behavior of Gas

Condensates in Shales Due to Pore Proximity Effects: Implications for Transport,

Reserves and Well Productivity. Paper SPE 160099 presented at the SPE Annual

Technical Conference and Exhibition held in San Antonio, Texas, USA, 8-10 October

2012.

132

6. Fitzgerald, J.E., Robinson, R.L. and Gasem, K.A. 2006. Modeling High-Pressure

Adsorption of Gas Mixtures on Activated Carbon and Coal Using A Simplified Local-

Density Model. Langmuir 22(23): 9610-9618.

7. Haghshenas, B., Clarkson, C. R., Bergerson, J., Chen, S., & Pan, Z. 2014. Improvement

in Permeability Models for Unconventional Gas Reservoirs and Model Selection Using

Statistical Analysis. Paper SPE 171641 presented at the SPE/CSUR Unconventional

Resources Conference held in Calgary, Alberta, Canada, 30 September–2 October 2014.

8. Haider, B.A. 2015. Impact of Capillary Pressure and Critical Properties Shift Due to

Confinement on Hydrocarbon Production from Shale Reservoirs. PhD diss., Stanford

University, 2015.

9. Javadpour, F. 2009. Nanopores and Apparent Permeability of Gas Flow in Mudrocks

(Shales and Siltstone). JCPT 48(08): 16-21.

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and Branched Alkanes on Carbon Nanotube Bundles from Configurational-Bias Monte

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11. Ma, Y., Jin, L. and Jamili, A. 2013. Modifying van der Waals Equation of State to

Consider Influence of Confinement on Phase Behavior. Paper SPE 166476 presented at

SPE Annual Technical Conference and Exhibition held in New Orleans, Louisiana, USA,

30 September-2 October 2013.

12. Ma, Y. and Jamili, A. 2014. Using Simplified Local Density/Peng-Robinson Equation of

State to Study the Effects of Confinement in Shale Formations on Phase Behavior. Paper

SPE 168986 presented at SPE Unconventional Resources Conference, The Woodlands,

Texas, USA, 1-3 April 2014.

13. Morishige, K., Fujii, H., Uga, M., and Kinukawa, D. 1997. Capillary Critical Point of

Argon, Nitrogen, Oxygen, Ethylene, and Carbon Dioxide in MCM-41. Langmuir 13(13):

3494–3498.

14. Morishige, K. and Shikimi, M. 1998. Adsorption Hysteresis and Pore Critical

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133

15. Nagy, S. and Siemek, J. 2014. Confined Phase Envelope of Gas-Condensate Systems in

Shale Rocks. Archives of Mining Sciences 59(4): 1005-1022.

16. Parsa, E., Yin, X. and Ozkan, E. 2015, September. Direct Observation of the Impact of

Nanopore Confinement on Petroleum Gas Condensation. Paper SPE 175118 presented at

SPE Annual Technical Conference and Exhibition held in Houston, Texas, USA, 28-30

September 2015.

17. Peng, D.Y. and Robinson, D.B. 1976. A New Two-Constant Equation of State. Industrial

& Engineering Chemistry Fundamentals 15(1): 59-64.

18. Pitakbunkate, T., Balbuena, P.B., Moridis, G.J. and Blasingame, T.A. 2015. Effect of

Confinement on Pressure/Volume/Temperature Properties of Hydrocarbons in Shale

Reservoirs. SPE Journal August 2015.

19. Qanbari, F., Haghshenas, B., and Clarkson C.R. 2014. Effects of Pore Confinement on

Rate-Transient Analysis of Shale Gas Reservoirs. Paper SPE 171637 presented at

SPE/CSUR Unconventional Resources Conference held in Calgary, Alberta, Canada, 30

September–2 October 2014.

20. Qanbari, F., and Clarkson C.R. 2013. A New Method for Production Data Analysis of

Tight and Shale Gas Reservoirs during Transient Linear Flow Period. Journal of Natural

Gas Science and Engineering 14: 55-65.

21. Qanbari, F., and Clarkson C.R. 2013. A New Method for Production Data Analysis of

Tight and Shale Gas Reservoirs during Transient Linear Flow Period. Journal of Natural

Gas Science and Engineering 14: 55-65.

22. Rangarajan, B., Lira, C.T. and Subramanian, R. 1995. Simplified Local Density Model

for Adsorption Over Large Pressure Ranges. AIChE Journal 41(4), pp.838-845.

23. Rahmani, B. and Akkutlu, I.Y. 2013. Pore-Size Dependence of Fluid Phase Behavior and

the Impact on Shale Gas Reserves. Paper SPE 168939 presented at Unconventional

Resources Technology Conference held in Denver, Colorado, USA, 12-14 August 2013.

24. Rahmani, B. and Akkutlu, I.Y. 2013. Pore-Size Dependence of Fluid Phase Behavior and

Properties in Organic-Rich Shale Reservoirs. Paper SPE 164099 presented at SPE

International Symposium on Oilfield Chemistry held in The Woodlands, Texas, USA, 8-

10 April 2013.

134

25. Rahmani, B. and Akkutlu, Y.I. 2015. Confinement Effects on Hydrocarbon Mixture

Phase Behavior in Organic Nanopore. Paper SPE 178519 presented at Unconventional

Resources Technology Conference held in San Antonio, Texas, USA, 20-22 July 2015.

26. Sandoval, D., Yan, W., Michelsen, M.L. and Stenby, E.H. 2015. Phase Envelope

Calculations for Reservoir Fluids in the Presence of Capillary Pressure. Paper SPE

175110 presented at SPE Annual Technical Conference and Exhibition held in Houston,

Texas, USA, 28-30 September 2015.

27. Singh, S.K., Sinha, A., Deo, G., and Singh, J.K. 2009. Vapor- Liquid Phase Coexistence,

Critical Properties, and Surface Tension of Confined Alkanes. The Journal of Physical

Chemistry C 113(17): 7170–7180. http://dx.doi.org/10.1021/jp8073915.

28. Smith, J.M., Abbott, M.M. and Van Ness, H.C. 1987. Introduction to chemical

engineering thermodynamics. McGraw-Hill.

29. Wattenbarger, R.A., El-Banbi, A.H., Villegas, M.E., and Maggard, J.B. 1998. Paper SPE

39931 presented at Rocky Mountain Regional Meeting/Low Permeability Reservoirs

Symposium, Denver, Colorado, USA, 5-8 April. http://dx.doi.org/10.2118/39931-MS.

30. Zarragoicoechea, G.J., & Kuz, V.A. 2002. van der Waals Equation of State for a Fluid in

a Nanopore. Physical Review E 65(2): 021110.

http://dx.doi.org/10.1103/PhysRevE.65.021110.

31. Zarragoicoechea, G.J., & Kuz, V.A. 2004. Critical Shift of a Confined Fluid in a

Nanopore. Fluid Phase Equilibria 220(1): 7–9.

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135

Chapter 6 Simulation of Enhanced Recovery using CO2 in a Liquid-Rich Western

Canadian Unconventional Reservoir: Accounting for Reservoir Fluid Adsorption and

Compositional Heterogeneity5

6.1 Abstract

Liquid-rich unconventional reservoirs are currently popular targets for development by the

industry. However, hydrocarbon liquid recovery in unconventional reservoirs can be very low,

primarily due to low permeability, but also partly due to adsorption of heavier hydrocarbon

components. Previous studies have demonstrated that the heaviest components (butane+) are the

most strongly adsorbed while being the most valuable commodity. Therefore, the development

of methods to enhance recovery of these strongly-adsorbed components is very appealing to

operators. The purpose of this chapter is therefore to investigate the possibility of incremental

recovery of oil in a low-permeability reservoir by injecting a non-hydrocarbon gas (CO2) into the

reservoir using a huff-n-puff procedure.

A feasibility study of CO2-enhanced production in a liquid-rich (volatile oil) low-permeability

(tight) reservoir in Western Canada is conducted using rigorous compositional simulation

combined with multi-component adsorption modelling. The simulation model used for a

sensitivity analysis was previously calibrated using flowback data obtained from a multi-

fractured horizontal well (Clarkson et al., 2016a). A unique aspect of that study was that multi-

layer PVT and fluid properties in the reservoir were estimated using a novel procedure; however,

adsorption of the reservoir fluids was ignored. In the current study, an innovative approach

developed by Clarkson and Haghshenas (2016) was applied for estimating high

pressure/temperature (in-situ) adsorption of reservoir fluid components and CO2 using a

combination of low pressure adsorption data and the simplified local density model. This

approach was required because, typically, the only reservoir samples available along horizontal

5 This chapter is a slightly modified version of a paper presented at SPE Canada Unconventional Resources held in

Calgary, Alberta, Canada, 15-16 February 2017 as: Haghshenas, B., Qanbari, F.,and Clarkson, C.R., 2017.

Simulation of Enhanced Recovery using CO2 in a Liquid-Rich Western Canadian Unconventional Reservoir:

Accounting for Reservoir Fluid Adsorption and Compositional Heterogeneity. In SPE Canada Unconventional

Resources. Copyright approval has been obtained from SPE (see “Copyright Permissions” section of this thesis).

136

wells are cuttings, which are not available in sufficient quantities for direct high pressure

adsorption measurements. A general equation was also developed for defining the diffusivity

coefficient in nanopores which can be directly applied in a commercial numerical simulator.

Sensitivity studies were then performed for different huff-n-puff operating conditions, and for

the range in different reservoir fluids obtained by Clarkson et al. (2016a).

The huff-n-puff sensitivity study demonstrates that, for the operating conditions applied, results

of CO2 injection are positive (incremental recovery over primary production) only when

adsorption/diffusion effects are included in the model. Further, for the 1000 day evaluation

period, the combination of shorter injection times (40 days) and longer soak periods (60 days)

are required to yield incremental recovery. When uniform in-situ fluid compositions are

assumed, lower saturation pressure fluids are more amenable to the CO2 huff-n-puff procedure

than higher bubble point fluids. However, when fluid compositions vary by geologic horizon, as

they do in this study, this heterogeneity must be considered in the analysis for an accurate

assessment of CO2 EOR.

For the first time, reservoir fluid component adsorption and reservoir fluid property variability by

layer in an unconventional reservoir has been considered while planning for CO2-enhanced

liquid recovery. This study provides some insight into the selection of optimal well operating

conditions for CO2 injection while considering the effects of adsorption selectivity, pore wall-

fluid molecular interaction, and thermodynamic behavior of the fluid.

6.2 Introduction

The current common practice for producing liquid-rich unconventional reservoirs is through

primary depletion using multi-fractured horizontal wells (MFHWs). However, due to the low

permeability of these reservoirs, drainage is typically restricted to small regions around the

well/fractures. As a result, even with the combination of long horizontal wells, multiple

hydraulic fractures, and massive hydraulic fracturing treatments, oil recoveries may be very low.

For example, some researchers (e.g. Christensen et al., 1998; Hoffman et al., 2012) have

estimated primary oil recovery factors in shale reservoirs to be only 5-10% of original oil in

place. Such low recovery factors have motivated operators to consider enhanced oil recovery

(EOR) schemes. An additional motivation is that in-place hydrocarbon volumes of

137

unconventional reservoirs may be massive; therefore, even a minor increase in oil recovery

factor could yield several billion barrels of additional oil (Hawthorne et al., 2013). It is of

practical importance to understand the underlying storage and flow mechanisms in tight

formations and use this information to evaluate EOR processes.

Water flooding and CO2/natural gas flooding are the most common EOR processes in

conventional reservoirs (Jarrell et al., 2002). However, the success of any well-to-well flooding

process relies on sweep efficiency, which in turn requires reasonable reservoir permeability

levels. As a result, conventional water flooding is often not considered suitable for enhancing

recovery in tight formations mainly due to extremely low injectivity, poor sweep efficiency, and

clay swelling problems (Yu et al., 2015). There are exceptions, however, as demonstrated by

Ghaderi et al. (2017) for a waterflood pilot in the Viewfield Bakken in Saskatchewan. Gas

injection (CO2, N2, natural gas or a mixture) is believed to be less challenging because of the

lower viscosity and density and also because of the ability of gas to achieve first- or multiple-

contact miscibility. However, because of the ultra-low permeability of some unconventional

reservoirs, it takes a long time for pressure to propagate from injection to production wells,

which may limit the success of gas flooding in these scenarios (Chen et al., 2014). An additional

issue is that hydraulic fracturing, required for economical production from tight formations and

providing a route for the injected fluid into the reservoir, may result in undesirable early

breakthrough during flooding processes. Although some researchers have proposed gas flooding

as an EOR method in shale reservoirs (Shoaib and Hoffman, 2009; Wang et al., 2010; Ghedan,

2009), the huff-n-puff gas injection scheme is an alternative method that may avoid some of the

limitations of gas flood scenarios (Kanfar and Clarkson, 2017).

Among the different gases used for injection, CO2 has received much attention for EOR,

because: 1) under supercritical conditions6 (likely the case for high pressure and temperature

conditions of many unconventional reservoirs), CO2 has a kinematic viscosity and liquid-like

density of approximately 0.1–0.25 and 70% that of water, respectively, thus, allowing for better

injectivity (Chen and Zhang, 2010); 2) CO2 has a considerably lower minimum miscibility

pressure (MMP) than other gases such as N2 and CH4 (Stalkup, 1987; Holm, 1987); 3) CO2 has

6 Temperatures higher than 31°C= 304 K= 87.5°F

138

higher adsorption affinity than N2 and CH4 and can assist with the desorption of valuable heavier

components (Clarkson and Haghshenas, 2013) and; 4) injecting CO2 for EOR purposes may have

the additional benefit of greenhouse gas sequestration (Ghaderi et al., 2017).

The CO2 huff-n-puff process has become a focus of a number of research studies in recent years.

For example, Liu et al. (2005) performed simulation of the CO2 huff-n-puff process and noted

that the process is still favorable in oil reservoirs with relatively small pore sizes and poor well

performance. Wan et al. (2013a,b,c), Sheng and Chen (2014) and Gamadi et al. (2013)

extensively evaluated cyclic gas injection using numerical simulation, and demonstrated that

cyclic gas injection could be a viable method to improve oil recovery in shale oil reservoirs.

However, in those studies, a black oil model was used without detailed analysis of the

thermodynamic properties of the reservoir fluid and injected gas. Vega et al. (2010) performed

an experimental study to investigate oil recovery using CO2 injection into a 1.3 mD siliceous

shale reservoir and demonstrated positive results for recovery. Their compositional simulation

results, however, could not reproduce their experimental data. Chen et al. (2014) used an EOS-

based compositional reservoir simulator to evaluate the impact of reservoir heterogeneity on CO2

huff-n-puff efficiency. Their study interestingly found that the final recovery factor achieved

with huff-n-puff is lower than for primary production - even the use of a longer shut-in time did

not help increase recovery during the production stage. Those authors also found that greater

reservoir heterogeneity resulted in a reduced final recovery factor. Chen et al. (2014) suggested

that the negative huff-n-puff results are due to limited CO2 migration into the shale matrix.

Recently, Yu et al. (2015) evaluated CO2 huff-n-puff performance using diffusion coefficients as

high as 0.1 cm2/s (for CO2 in oil). They concluded that the CO2 huff-n-puff process is favorable

for increasing oil recovery in the Bakken Formation. However, in that study, the simulation

model was not validated and tuned with the real production data, and adsorption effects were

ignored. There are also several other laboratory and simulation studies that have suggested that

CO2 huff-n-puff is an effective method for enhancing recovery in tight oil formations (Pu et al.,

2016; Wilson, 2015; Liu, 2005; Kong et al., 2016; Song and Yang, 2017). Successful field

implementation of CO2 injection into shale reservoirs has yet to be reported; however, a pilot

field test of CO2 huff-n-puff in a low permeability offshore reservoir showed a favorable

response (Konishi et al., 2013).

139

As a result of the lack of reported field tests for CO2 huff-n-puff in unconventional reservoirs,

researchers have relied heavily on the results of lab or numerical simulation studies to evaluate

the efficiency of this EOR scheme – some of these studies were reported on above. Kanfar and

Clarkson (2017), however, noted that previous simulation studies of the CO2 huff-n-puff process

in hydraulically-fractured tight reservoirs yielded contradictory results, and investigated some of

the possible sources of these discrepancies. Those authors found that grid refinement and

fracture pseudo width representations used in the simulation model can greatly affect the results;

for example, coarsely gridded models can lead to falsely optimistic results for the CO2 huff-n-

puff scheme. They further suggested that higher permeability, fine fracture spacing and complex

fractures can lead to improved CO2 huff-n-puff efficiency.

The current study differs from previous numerical simulation studies of the CO2 huff-n-puff

process in liquid-rich unconventional reservoirs in several important respects. First, although no

actual CO2 huff-n-puff data were available to simulate, an element of realism is provided in this

study by using, as the basis for the sensitivity study, a compositional numerical model that was

calibrated by history-matching flowback fluid production rates and pressures obtained from an

actual producing MFHW completed in a liquid-rich tight reservoir in Western Canada (Clarkson

et al., 2016). The calibrated model, which was used to estimate fracture height growth, is an

ideal starting model because 1) a detailed petrophysical and geologic model was used in its

development; 2) PVT and fluid properties were estimated with depth in the reservoir and

bounding horizons using a novel procedure; 3) propped and unpropped sections of the hydraulic

fracture were included. However, CO2 EOR using the huff-n-puff procedure was not considered

in the Clarkson et al. (2016) study, nor was the adsorption of reservoir fluids. A second

important aspect of the current study is therefore the consideration of hydrocarbon and CO2

adsorption; for this purpose, an innovative approach for estimating in-situ hydrocarbon

component and CO2 adsorption using low-pressure adsorption data, combined with the

simplified local density model, was adapted from Clarkson and Haghshenas (2016). The base

model, with and without adsorption and diffusion accounted for, was used to perform a

sensitivity study to the effect of reservoir fluid composition (using the range of fluid

compositions obtained from Clarkson et al., 2016) and various aspects of CO2 huff-n-puff

operations, such as the length of the injection and soak periods.

140

6.3 Theory and Methods

In this section, several parameters that have an impact on oil recovery through the cyclic CO2

injection process are first discussed, as well as how they are implemented in the reservoir model.

A separate section is dedicated to simulation model description and setup.

6.3.1 Diffusion

Typically, the primary mechanisms for gas-EOR in naturally fractured reservoirs include viscous

forces, gravity drainage, and molecular diffusion (Hoteit, 2013). However, in low permeability

reservoirs, the viscous forces and gravity drainage become less important, while molecular

diffusion will be dominant (Hoteit and Firoozabadi, 2006). The CO2 diffusion coefficient in oil

and gas phases can be determined based on published laboratory measurements (Grogan et al.,

1988; Renner, 1988). Grogan et al. (1988) conducted experimental measurements of CO2

diffusion coefficients in pentane, decane, and hexadecane at a temperature of 77 oF (25

oC) and

pressures up to 870 psi (6000 kPa) and reported CO2 diffusion coefficients in the range of

1.80×105 cm

2/s–7.59×10

5 cm

2/s. Renner (1988) measured CO2 diffusion coefficients in decane at

a temperature of 100 oF (38

oC) and pressures up to 850 psi (5860 kPa) and reported CO2

diffusion coefficients in the range of 1.97 ×105 cm

2/s to 12.6 ×10

5 cm

2/s. Based on the work of

Denoyelle and Bardon (1983), CO2 diffusion coefficients in the oil phase at reservoir conditions

are 5–10 times higher than those measured at ambient conditions. More importantly, the

diffusion coefficient for super critical CO2 is 10–100 times of that for liquid (Kumar et al.,

1999).

Herein, a new definition for apparent diffusivity is introduced which takes into account the

effects of surface interactions on fluid flow. The new definition for diffusivity coefficient

explains the high diffusivity values measured in the above experiments and can be directly used

to model diffusion of different components in a numerical simulator. This approach can be used

to define the flow properties of each component; previous methods (e.g. Javadpour, 2009)

provided an equation in the form of Darcy permeability, which is input as a rock property in

commercial simulators and, therefore, assumes the same value for all components. In this work,

141

an apparent diffusivity of 0.001 cm2/s (Yu et al., 2015) is used for CO2 and the apparent

diffusivities of other components are calculated based on Eq. 1, which correlates the Da of each

component to Da,CO2 using the reverse ratio of molecular weights (Table 4.1). The derivation of

the new equation (Eq. 1) can be found in appendix C.

DpFk

DD

a

(1)

where Dk (m

2) is Darcy permeability, D (m

2/s) is Fick’s diffusion constant, (Pa.s) is gas

viscosity, , (kg/m3) is gas density, is porosity, p (Pa) is pore pressure, and F is slippage

factor. Da values are input into the commercial simulator as diffusivity coefficients, which are

coded as component properties, i.e., specific for each component.

Table 6-1 — Apparent diffusivity of different components used in this study.

Component CO2 N2 X1+ C2 C3+ C7+ C8+ C11+ C15+ C19+ C23+ C27+

Da (cm2/s) 0.001 0.0012 0.0016 0.0012 0.0009 0.0007 0.0006 0.0005 0.0004 0.0004 0.0004 0.0003

6.3.2 Miscibility

Once crude oil makes contact with CO2 at pressures above the minimum miscibility pressure

(MMP), CO2 will dissolve in the oil to swell its volume, decrease its viscosity, reduce its

interfacial tension, and extract its light-components. This in turn results in the oil and CO2

phase, the latter of which contains some extracted hydrocarbon components, flowing together

more easily through the porous media (Yang et al., 2005; Li et al., 2013; Taber and Martin, 1983.

Lambert et al., 1996; Martin and Taber, 1992). However, although the reduction in interfacial

tension and capillary pressure are critical for miscible processes in heavy oils, Fai-Yengo et al.

(2014) suggested that the effect of capillary pressure has a negligible effect on oil recovery in

shale reservoirs. The extraction of hydrocarbons is highly dependent on the density of the CO2,

and the CO2 will extract more and heavier hydrocarbons with the increasing CO2 density (Holm

and Josendal, 1982; Orr et al., 1983; Sigmund et al., 1984). The CO2 density varies from 0.1 to

142

0.8 g/cm3 at pressures from 1000 to 4000 psi when the temperature is above its critical

temperature of 87.9 oF (Holm and Josendal, 1982). Holm and Josendal (1982) found that

sufficient hydrocarbon extraction occurs when the CO2 density is about 0.42 g/cm3, which is

close to the CO2 critical density of 0.468 g/cm3. As noted above, under typical pressure and

temperatures of shale reservoirs, the injected CO2 is actually at super critical conditions. The

density of the super critical CO2 is more like a liquid, but the viscosity is like a gas. Therefore,

the miscible process is very likely to occur under these conditions (Lambert et al., 1996).

6.3.3 Adsorption

Accurate adsorption modeling for sorbed gas reservoirs is important for resource and gas

recovery estimation during both primary and enhanced recovery operations. Although the

permeability of unconventional reservoirs such as shales is low and abandonment pressures

typically high, wells completed in these reservoirs are often operated at high drawdowns, with

pressures near the wellbore or fracture face being < 1000 psi, or possibly less than 500 psia. In

such scenarios, desorbed gas near the wellbore could impact production and influence

evaluations of enhanced recovery methods in shale gas and oil reservoirs. In liquid-rich shales

with high organic matter content, adsorption of heavier hydrocarbon components can be

significant. Based on the preliminary investigations of light and heavy hydrocarbon, N2, and CO2

adsorption on shale (Ambrose et al., 2011), it was concluded that heavier hydrocarbon

components are more strongly adsorbed on shale than light hydrocarbons, and CO2 is more

strongly adsorbed than ethane and methane. The heaviest components (e.g butane +) are the most

strongly adsorbed, and also the most valuable commodity. It is logical therefore to investigate

methods to enhance recovery of these strongly-adsorbed components. In a previous study by the

authors (Clarkson and Haghshenas, 2013), binary gas phase diagrams were generated using the

extended Langmuir (EL) method which demonstrated that butane+ has a much higher selectivity

over methane than the other components (Figure 6.1). Those authors then investigated the

influence of CO2 by replacing methane with CO2 in the free-gas phase of the binary system and

recalculating the binary gas phase diagram (Figure 6.2). It was observed that the impact of

replacing methane with CO2 in the free-gas phase is a reduction in the selectivity of butane+,

suggesting that CO2 could be used to improve recovery of the adsorbed heavy hydrocarbon

143

components. This analysis was over-simplified due to the assumption of equilibrium conditions

in the analysis (kinetics ignored), and the assumption that the EL model is valid in such systems.

Figure ‎6-1 — Separation factor (selectivity) calculations for binary mixtures of CH4 and heavier

hydrocarbons and CO2 using the EL model. In the EL model, the separation factor is assumed

not to be a function of pressure or composition. Modified from Ambrose et al. (2011).

0

2

4

6

8

10

12

14

16

18

20

CH4+Butane+ CH4+Propane CH4+CO2 CH4+Ethane

Se

pa

ratio

n fa

cto

r (r

ela

tive

to

CH

4)

Binary mixture

144

Figure ‎6-2 — Comparison of separation factor (selectivity) calculations for a binary mixture of

CO2-butane+ and CH4-butane+ using the EL model. Modified from Clarkson and Haghshenas

(2013).

In the current work, rigorous compositional simulation combined with multi-component

adsorption modeling is used to model the system more accurately.

Importantly, a method is required to estimate high-pressure/temperature (in-situ) adsorption of

each component in the fluid system. The problem is that, for multi-fractured horizontal wells,

typically the only reservoir samples available are small masses of drill cuttings along the lateral,

which are usually insufficient for high pressure adsorption measurements which often require >

150-200g of sample. A solution, provided by Clarkson and Haghshenas (2016), is to use low-

pressure adsorption equipment (which only requires small amounts of sample), combined with

the simplified local density model (SLD, calibrated to the low-pressure adsorption data) to

predict adsorption of the hydrocarbon components (as well as CO2 and N2) under in-situ

conditions. The SLD model is a rigorous method for representing the adsorbed phase volume

and density, and was chosen due to its capability in matching experimental data, as well as its

ability to predict the fluid density distribution within nanopores (Clarkson et al., 2016b).

0

2

4

6

8

10

12

14

16

18

20

CO2+Butane+ CH4+Butane+

C4H

10

se

pa

ratio

n fa

cto

r (r

ela

tive

to

CH

4&

CO

2)

Binary mixture

145

The procedure described by Clarkson and Haghshenas (2016) was used to estimate component

adsorption under in-situ conditions for the subject liquid-rich tight reservoir. Figure 6.3

illustrates the match of low pressure adsorption results for N2 (Figure 6.3a) and CO2 (Figure

6.3b) using the SLD model. Figure 6.4 provides high pressure adsorption SLD model

predictions, after calibration to the low pressure adsorption data, for the components of the

reservoir fluid and CO2. The prediction for CH4 is calibrated using available data for high

pressure CH4 adsorption on samples from the same reservoir (Beaton et al., 2010). These

samples were not obtained from the study area and are assumed to be compatible with the

samples of interest. For heavier components, no actual adsorption measurements for the reservoir

of interest are available; hence literature data for shale samples (Ambrose et al., 2011) were used

to estimate heavy component adsorption for the subject reservoir. For this purpose, ratios of

heavy component adsorption to methane were obtained from the Ambrose et al. (2011) study and

used to predict heavy component adsorption (C2, C3, C4+) for the studied reservoir. It is

important to note that, owing to the lack of adsorption measurements of components heavier than

butane, the adsorption of these components was assumed to be similar to butane. We realize that

this approach for predicting heavy hydrocarbon adsorption is likely in error due to 1) the

assumption that the available CH4 adsorption isotherm for the reservoir of interest, although

obtained from different locations, depths and samples of dissimilar composition, is applicable to

the study area reservoir; 2) adsorption will occur in a similar ratio to the Ambrose et al. (2011)

samples; and 3) all components heavier than butane have a similar adsorption capacity to butane.

However, these assumptions should serve as a reasonable starting point for evaluating the impact

of adsorption on CO2 huff-n-puff results; once actual adsorption measurements are acquired for

reservoir samples in the study area, the sensitivities will be re-run.

146

Figure ‎6-3 — SLD model match to low-pressure (a) N2 isotherms and (b) CO2 isotherms. The



BET model analysis.

0

10

20

30

40

50

60

70

80

0 0.1 0.2 0.3 0.4

Qu

an

tity

Ad

so

rbe

d (

scf/

ton

)


Data

Model

(a)

0

2

4

6

8

10

12

14

16

0 0.01 0.02 0.03 0.04

Qu

an

tity

Ad

so

rbe

d (

scf/

ton

)


Data

Model

(b)

147

Figure ‎6-4 — High-pressure absolute adsorption isotherms predicted from the SLD model for C1,

C2, C3, C4+ and CO2.

6.4 Simulation Model Setup

In the following, the methods used for numerical model setup and implementation are provided.

The rock and fluid properties, and reservoir model, were adapted from previous work (Clarkson

et al., 2016a). In that previous study, a compositional numerical model was used to history-

match flowback fluid production rates and pressures of a multi-fractured horizontal well

completed in a liquid-rich tight reservoir in Western Canada (Clarkson et al., 2016a). The

purpose of that study was to estimate fracture height growth using reservoir interval fluid

compositions (as assessed from drill cuttings data) and flowback fluid compositions as a

constraint. The model serves as an ideal starting point for the current study because it captures

reservoir rock and fluid property heterogeneities, the latter of which, to our knowledge, has never

been considered in a CO2 huff-n-puff study in tight reservoirs.

The studied horizontal well was drilled to a measured depth of 18,110 ft with a kickoff point of

9905 ft. The lateral section was directionally drilled from 10,545 ft to TD (Figure 6.5). During

0

10

20

30

40

50

60

0 2000 4000 6000 8000

Qu

an

tity

Ad

so

rbe

d (

scf/

ton

)

Absolute Pressure (psi)

CH4 C2

C3 CO2

C4+

148

the openhole completion, mechanical difficulties resulted in the implementation of only 9 of the

planned 28 stages, with the successful stages located in the heel of the well. Additional details of

the well drilling and completion can be found in the original paper (Clarkson et al., 2016a).

Figure ‎6-5 — Cross-section (created in Petrel™) showing subject well horizontal lateral

trajectory with respect to upper and lower zones (identified during geologic characterization), the

location of 9 successful hydraulic fracturing stages (near heel of the well), and the location of the

cuttings samples analyzed for gas composition (and used in fluid property modeling). Also

projected are the initial designed hydraulic fracturing stages (not yet implemented). A gamma

ray log is provided to illustrate lithology/reservoir quality variation along the lateral. Note: Cross

section depth scale is in true vertical depth subsea (TVDSS). Note 1 m = 3.28 ft. Modified from

Clarkson et al., 2016a.

6.4.1 Reservoir Model

Using an element of symmetry, one effective hydraulic fracture (HF) within one stage

(representative of the successful heel stages shown in Figure 6.5) was used to save computation

time. Grid blocks around the fracture plane are locally refined using logarithmically spacing,

such that very fine grid blocks are placed in the vicinity of the fracture, and larger grid blocks

Samples

Frac stagesUpper Zone

Lower Zone

9 Successful stages

Vertical exaggeration: 3x

Frac stage 1

GR Log

Heel

Toe (TD)

NW SE

149

away from the fracture plane. Fine gridding is essential to avoid numerical dispersion caused by

large pressure or property changes near fractures, which are characteristics of tight/shale

reservoirs operated under large drawdown pressures and exhibiting large permeability

differences between fracture and matrix. Moreover, logarithmic spacing preserves the total

number of simulation grids at a manageable level.

The following assumptions were made in the construction of the reservoir model by Clarkson et

al. (2016a): 1) two distinct types of interacting porous media are present: the reservoir and the

HF; 2) the HF is made up of “un-propped” and “propped” sections which have differing fluid

storage and conductivity characteristics. Figure 6.6 displays the location of fracture plane and

logarithmic refinement in both propped and unpropped zones within the fracture plane.

The relative permeability curves used herein (water–oil relative permeability and liquid–gas

relative permeability) are from the original work (Clarkson et al., 2016a). Other details of

reservoir model setup are provided in that study.

Figure ‎6-6 — Illustration of the use of local grid refinement (with logarithmic spacing) to

represent propped and un-propped regions of the hydraulic fracture. Note that the fracture height

extends into the upper and lower zones illustrated in Figure 6.5. Modified from Clarkson et al.,

2016a.

150

6.4.2 Fluid Properties

Clarkson et al. (2016a) used an innovative approach for estimating the in-situ fluid composition

of this liquid-rich reservoir as a function of depth. Cuttings samples were collected from the

vertical, bend and lateral sections of the well shown in Figure 6.5 (see sample locations) and

placed in isojars®. Gas compositions were then determined in the lab and recombined with oil

compositions obtained from the separator to calculate the recombined in-situ fluid composition

as a function of depth (at cuttings locations). The detailed calculation procedure for

determination of in-situ composition is presented by Clarkson et al. (2016a).

Figure 6.7 presents the phase envelope of the fluids (volatile oil) with the highest and lowest

saturation pressures obtained using the procedure described above. These phase envelopes

illustrate the range of variability encountered during production – however, as noted by Clarkson

et al. (2016a), the phase envelopes and in-situ fluid properties differ layer-by-layer in the model

because of the differing cuttings gas compositions.

As noted in the Theory and Methods section, layer-by-layer adsorption of each component was

also accounted for in the current work, but was not considered in the Clarkson et al. (2016a)

study.

151

Figure ‎6-7 — Phase envelopes of the highest and lowest saturation pressure layers. Only the

layers intersected by the hydraulic fracture are shown; the pressures in non-intersected layers

won't change even after long production times. Modified from Clarkson et al. (2016a).

6.4.3 Rock Properties

Permeability and porosity of the fracture is assumed to be homogeneous; however, different

values of these properties were assigned to propped and unpropped regions. Fracture (half) width

was also set to 0.03 ft with 100% porosity (as starting point) for the unpropped region. The

porosity of the propped region is used, combined with the mass of injected proppant, to calculate

initial propped fracture volume, as described by Clarkson et al. (2016a). Note the HF fracture

height, half-length and permeability (propped and unpropped) were derived from history-

matching the flowback data of the subject well, as described by Clarkson et al. (2016a). The

resulting fracture properties used in the current model are provided in Table 4.2.

The geomodel used to populate matrix properties of the simulation model (after upscaling) was

generated based on a wide range of measured data that captures reservoir rock and fluid

heterogeneity (Clarkson et al., 2016a). The stochastic geologic model was constructed using the

0

1000

2000

3000

4000

5000

6000

-200 0 200 400 600 800

Pre

ssu

re (

psia

)

Temperature (F)

Layer 12

Layer 13

Critical point layer 12

Critical point layer 13

Reservoir initial condition

152

Petrel™ software and the numerical simulation model was constructed in Eclipse 300™. Other

details of rock property inputs are provided in that study.

As noted in the Theory and Methods section, component diffusivity was also accounted for in the

current work, which was not considered in the Clarkson et al. (2016a) study.

Table 6-2 — Final value of tuned hydraulic fracture parameters at the end of history match

performed by Clarkson et al. (2016a).

Matching parameters Unit Value

HF (half) width ft 0.03

HF (total) half-length ft 1640

HF (propped) half-length ft 350

HF (total) height (ft) ft 230

HF (propped) height (ft) ft 175

HF (propped) permeability (Darcy) ft 56.2

HF (unpropped) permeability (Darcy) ft 18.5

6.4.4 Huff-n-Puff Operating Conditions

Using the base simulation model tuned by history-matching flowback data, and modified to

account for component adsorption and diffusivity, forecasts were then performed using the CO2

huff-n-puff procedure. For forecasting this process, the horizontal well initially produces

(primary production) for 100 days under a constant flowing bottomhole pressure of 1000 psi.

The well is then converted to a CO2 injector with the following constraints: maximum injection

rate of 500 MSCF/day and maximum bottomhole pressure of 6000 psi (average formation

fracture pressure estimated to be 7000 psi). For the optimized case (see sensitivity study to

operating conditions below), after 40 days of CO2 injection, the well is shut-in and allowed to

soak for 60 days. Finally, the well is put back on production. This is one cycle of the CO2 huff-n-

puff process. The case with diffusion and adsorption is then compared with the base case with no

diffusion and adsorption.

153

6.5 Results

Figure 6.8 compares the following results: primary recovery oil production (1000 days), with and

without adsorption and diffusion, and CO2 huff-n-puff, with and without adsorption and

diffusion. Primary recovery oil production (at 1000 days of production) is greater with

adsorption and diffusion taken into account than the base case without. This result was expected.

However, interestingly, CO2 huff-n-puff (even after optimization – see sensitivities below), for

the case where adsorption and diffusion are neglected, does not result in incremental oil recovery

over primary production, whereas it does when these effects are included. These results point to

the possible importance of including adsorption and diffusion in simulation of CO2 huff-n-puff

for low permeability reservoirs where surface attractions are significant.

Figure ‎6-8 — Comparison of primary recovery and CO2 huff-n-puff responses for cases

including and neglecting adsorption and diffusion effects. Note that CO2 huff-n-puff only

appears to be beneficial for the case where adsorption and diffusion are included.

0

5

10

15

20

0 200 400 600 800 1000

Cu

mu

lative

Oil

Pro

du

ctio

n, 1

0×

Mstb

Time, day

Primary production—with ads./diff.

Optimized huff/puff—with ads./diff.

Primary production—without ads./diff.

Optimized huff/puff—without ads./diff.

154

In the following, the impact of operational parameters and in-situ fluid composition on CO2 huff-

n-puff results is explored.

6.5.1 Effect of Injection Time Period

The effect of the injection time period on CO2 huff-n-puff performance is shown in Figure 6.9 –

all other parameters, including adsorption and diffusion, are held constant. The sum of injection

and soaking times (the time when production well is shut down) is the same for all cases. As

illustrated, longer injection time has either positive or negative effect on incremental oil

production, the optimum injection time is determined as 55 days here. The injected CO2 should

be high enough to provide the required driving force for huff-n-puff process.

Figure ‎6-9 — Effect of injection time on the performance of CO2 huff-n-puff (for the case

accounting for adsorption and diffusion effects).

0

5

10

15

20

0 200 400 600 800 1000

Cu

mu

lative

Oil

Pro

du

ctio

n, 1

0×

Mstb

Time, day


45 days injection huff/puff—with ads./diff.



155

6.5.2 Effect of Soaking Time Period

Figure 6.10 illustrates the impact of the soaking time period on well performance, using the

optimized injection time (55 days) from the previous sensitivity – all other parameters are kept

the same for all cases. It can be seen that the parameter needs to be optimized to achieve

maximum recovery. The longer soaking time provides more time for the CO2 molecules to

penetrate into the low permeability matrix and helps with desorption of heavier components, but,

the optimum soaking time is not necessarily the longest time. A soaking time of 20 days is found

the best here. Without a soaking period, most of the injected gas accumulates near the wellbore

and is produced back during the production period. As a result of the near well gas

accumulation, oil recovery is reduced because of unfavorable relative permeability.

Figure 6.11 illustrates, however, that increasing the soaking time does not lead to incremental

recovery over primary recovery when adsorption and diffusion effects are neglected.

Summarizing the results of these two sensitivities, the combination of injection time of 55 days

and soak period of 20 days, referred to here as the “optimized” case, will lead to incremental

recovery for CO2 huff-n-puff over primary recovery for the scenarios run, but only if adsorption

and diffusion effects are included. All scenarios were run using the layer fluid compositions

derived from the previous study (Clarkson et al., 2016a). In order to isolate the impact of in-situ

fluid composition, an additional set of simulation runs were performed as illustrated in the next

section.

156

Figure ‎6-10 — Effect of soaking time on the performance of CO2 huff-n-puff (for the case

accounting for adsorption and diffusion effects).

Figure ‎6-11 — Effect of soaking time on the performance of CO2 huff-n-puff (for the case

without adsorption and diffusion effects included).

0

5

10

15

20

0 200 400 600 800 1000

Cu

mu

lative

Oil

Pro

du

ctio

n, 1

0×

Mstb

Time, day


10 days soaking huff/puff—with ads./diff.



0

5

10

15

20

0 200 400 600 800 1000

Cu

mu

lative

Oil

Pro

du

ctio

n, 1

0×

Mstb

Time, day

Primary production—without ads./diff.

10 days soaking huff/puff—without ads./diff.



157

6.5.3 Effect of Reservoir Fluid Composition and Reservoir Fluid Heterogeneity

To isolate the effect of reservoir fluid composition, CO2 was injected into the reservoir assuming

uniform fluid composition, as opposed to the layer-variable fluid compositions used in the

previous sections. The fluid compositions shown in Figure 6.7 were used for this purpose to

provide a range according to saturation pressure.

The results provided in Figure 6.12 suggest that, for the operating conditions used in this study,

results of huff-n-puff are positive only for the lower saturation pressure fluid. For the higher

saturation pressure fluid, there is no incremental recovery over primary production. Further, the

results assuming uniform fluid composition differ significantly from those with layer-variable

fluid composition (see black line in Figure 6.12 which shows layer-variable fluid composition

case). Therefore, the inclusion of geologic interval fluid compositions has an important effect on

the performance of the huff-n-puff process and needs to be characterized and considered in field

studies.

158

Figure ‎6-12 — Effect of reservoir fluid composition and heterogeneity on the performance of

CO2 huff-n-puff (for the case accounting for adsorption and diffusion effects). The pink lines

correspond to primary and huff-n-puff cases run assuming a uniform, low saturation pressure

fluid (see Figure 6.7), and the green lines were run assuming a uniform, high saturation pressure

fluid. The black line correspond to the primary production in the case where layer-variable fluid

compositions is included in the model. The huff-n-puff production of variable composition case

is what used in previous sensitivity cases.

6.6 Discussion

For the sensitivities performed, the simulation results indicate that for optimized conditions

(injection and soaking time), some incremental oil recovery could be achieved for the CO2 huff-

n-puff process over primary recovery for a 1000 day simulation forecast period (single huff-n-

puff cycle). However, the effects of reservoir fluid component adsorption and diffusion were

required to be included in order to achieve positive results for huff-n-puff. Further, it was

demonstrated that lower saturation pressure fluids are more amenable to the huff-n-puff success,

0

5

10

15

20

0 200 400 600 800 1000

Cum

ula

tive

Oil

Pro

du

ctio

n, 1

0×

Mstb

Time, day

Primary production—with ads./diff.Optimized huff/puff—with ads./diff.Primary production, high Ps—with ads./diff.Optimized huff/puff, high Ps—with ads./diff.Primary production, low Ps—with ads./diff.Optimized huff/puff, low Ps—with ads./diff.

159

and that assuming reservoir fluid homogeneity when it varies vertically in the reservoir can lead

to errors in huff-n-puff evaluation.

Many of the simulation runs used in this study point to negative results for CO2 huff-n-puff. In a

parallel study by our research group (Kanfar and Clarkson, 2017), the causes of contradictory

results reported for the success of CO2 huff-n-puff using numerical simulation were investigated.

Those authors found that when fine gridding is used, and the well is still in transient linear flow,

CO2 huff-n-puff results are mostly negative. This finding is consistent with the current study for

the cases where adsorption/diffusion are ignored. Kanfar and Clarkson (2017) also found,

however, that CO2 huff-n-puff may yield positive results after fracture interference, and during

pseudodepletion periods, when mixing is improved. In the current study, because an element of

symmetry was used, and fracture interference ignored, the bulk of the production was in the

transient flow period, and hence, without pseudodepletion effects, the results are mostly

negative. However, as also demonstrated in this work, even with fine gridding, and in the

absence of depletion, huff-n-puff results could be positive, if adsorption/diffusion is considered,

and operating conditions for the scheme are optimized. In future work, additional huff-n-puff

cycles will be run to evaluate their impact on huff-n-puff success.

Finally, although the results of adsorption were considered in this work, there was no actual high

pressure/temperature (in-situ) adsorption data available for methane, CO2 and heavier

hydrocarbon adsorption for the reservoir of interest in the study area. The combination of low-

pressure adsorption and the simplified local density model was used to predict component

adsorption under in-situ conditions using methane data from the reservoir in a different location

(Beaton et al., 2010), and heavy component adsorption ratios derived from the data of Ambrose

et al. (2011). The resulting adsorption isotherms calculated from this approach (Figure 6.4) yield

adsorption amounts that are less for each component than those reported by Ambrose et al.

(2011). This result makes sense because the studied reservoir is a low organic matter content

siltstone, not an organic rich shale as studied by Ambrose et al. Even with the relatively smaller

adsorption amounts for the studied reservoir, the impact of adsorption on huff-n-puff results is

significant.

160

6.7 Conclusions

In this study, a compositional simulation model, tuned to match flowback data of a producing

multi-fractured horizontal well completed in a liquid-rich (volatile oil) tight reservoir, was used

to study the effects of fluid component adsorption and diffusion, operational parameters, and

reservoir fluid composition on CO2 huff-n-puff performance. An innovative method for

estimating reservoir fluid component adsorption at in-situ conditions by using low-pressure

adsorption combined with the simplified local density model was adopted from Clarkson and

Haghshenas (2016). Further, a method for calculating component diffusivity was introduced. To

our knowledge, the effect of component adsorption and diffusivity on the results of CO2 huff-n-

puff has not been studied in detail until now. Another unique aspect of this study is that layer-

by-layer fluid compositions were taken into effect, and results were compared with the more

common assumption of uniform fluid composition.

The following are the primary conclusions derived from this study:

1. For certain huff-n-puff operating conditions, CO2 huff-n-puff results (in terms of

incremental recovery over primary production) are positive only when component

adsorption/diffusion effects are considered.

2. The combination of injection time of 55 days and soak period of 20 days yield the

most favorable huff-n-puff results for the studied 1000 day period.

3. Assuming uniform reservoir fluid composition, the lower saturation pressure fluid

yielded better results for huff-n-puff than the higher saturation pressure fluid.

4. The assumption of a uniform reservoir fluid composition, when in-situ fluid

compositions are variable by reservoir layer, can lead to errors in the assessment of

huff-n-puff efficiency.

In future work, additional sensitivities to huff-n-puff operations will be performed, such as the

effect of additional cycles.

6.8 Appendix C

Writing the material balance equation in terms of diffusivity (rather than common writing based

on permeability) gives the definition for apparent diffusivity:

161

t

p

p

C

t

p

pr

rD

r

pFk

rr

a

a

a

a

a

a

a

aaa

ra

a

a

a

a

a

a

aa

aD

a

aaa

)1().,

(1 2

2

(C-1)

t

p

p

C

pr

rD

pFk

rr

a

a

a

a

a

a

aaa

ra

a

r

a

a

a

aa

aD

a

aaaa

)1().

,(

1 2

2

(C-2)

As an analogue to Fick’s law, in the above equation, Da, is the coefficient in the bracket

multiplied by density gradient.

DpFk

DD

a

(C-3)

6.9 Nomenclature

D Fick’s diffusion constant, (m2/s)

aD Apparent Fick’s diffusion constant, (m2/s)

F Slippage factor

Dk Darcy permeability, (m2)

p Pore pressure, (Pa)

Gas viscosity, (Pa.s)

Gas density, (kg/m3)

Porosity







162

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168

Chapter 7 Conclusions and Future Work

7.1 Conclusions

The current thesis addresses several of the key properties of shale reservoirs listed in the

introduction section through a combination of numerical reservoir simulation, analytical

modeling and experimental approaches.

The following is a summary and primary conclusions from this thesis:

1. The Simplified Local Density (SLD) model is developed based on rigorous

thermodynamic fundamentals and can be used to 1) identify the adsorbed layer thickness

and give the ratio of adsorbed gas versus total gas in-place with respect to the pore size;

2) represent adsorbed phase density which is necessary for converting excess adsorption

quantity to absolute adsorption; 3) extract pore width (L) and surface area (ASlit) by

matching to the low pressure adsorption data; 4) predict fluid property alteration from

pore confinement, which leads to a drop in dew point pressures of gas condensate

mixture; and 5) predict high-pressure hydrocarbon gas adsorption from low-pressure non-

hydrocarbon adsorption data collected for small amounts of cuttings samples.

2. New laboratory and modeling procedures for extracting permeability/diffusivity from

drill cuttings are also pursued. A binary pore structure model is developed for the nano-

pore-scale molecular interactions (slippage and diffusion) and is fitted to several sets of

experimental data. The results show that, compared to the unipore model or conventional

bidisperse model (accounts only for diffusion but not for the slippage), the new bidisperse

mode can provide a better match to the both steep and shallow portions of pressure decay

data and provides a better representation for shale samples which typically exhibit a

bimodal pore size distribution (PSD).

3. Corrections for gas-in-place and material balance calculations are provided, which

account for the volume occupied by adsorbed gas. It is observed that such correction

leads to a smaller void space available for free gas, which in turn leads to lower initial gas

in place estimates using both volumetric and material-balance methods. In addition, the

corrected pore volume is a function of pressure (free gas pore volume increases as

169

pressure decreases) because the adsorbed layer thickness decreases through desorption

process. Thus, the calculated recovery factor using this porosity correction is lower than

the case in which the simulator assumes a constant free gas volume [ϕ(1-Swi)]. One needs

to define a porosity correction factor in a numerical simulator in order to predict the

production performance of shale reservoirs.

4. Changes in gas properties due to pore confinement, as well as the effects of non-Darcy

flow and adsorbed layer thickness changes (also calculated with use of the SLD model),

are incorporated into transient linear flow analysis of nanoporous shale gas condensate

reservoirs. Simulation results demonstrate that pore confinement can significantly affect

the results of rate-transient analysis. For example, neglecting changes due to pore

confinement results in underestimation of the linear flow parameter derived from linear

flow analysis. Conversely, neglecting non-Darcy flow effects results in the

overestimation of the linear flow parameter.

5. A sensitivity study of primary production and CO2 huff-n-puff is performed in a tight

liquid rich gas reservoir in Western Canada, where the in-situ fluid composition

variability and adsorption/transport processes are considered. The results demonstrate

that the adsorption and diffusion effects increase the ultimate recovery of the reservoir.

Inclusion of adsorption increases the initial hydrocarbon in-place and also allows for

adsorption of CO2 which in turn, enhances the production of adsorbed hydrocarbons.

Also, neglecting the diffusion effect results in slow transportation of either injected gas

(into the reservoir) or reservoir fluid (towards the well) and therefore, results in lower

recovery estimates.

7.2 Future Work

The following topics are recommended for future work:

1. Predictions of heavy hydrocarbon adsorption provided in this thesis, have not been

validated by measured data. Adsorption models for predicting heavy component

adsorption in shales need to be developed and calibrated.

2. The adsorption/diffusion model developed in this thesis is applied to a limited set of

experimental data collected from two reservoirs. Reservoir-specific adsorption/diffusion

170

calculations and diffusivity trends need to be conducted as the estimated

diffusivity/permeability values are quite different for different reservoirs.

3. Well communication (possibly through natural fracture system, hydraulic fractures,

faults, etc.) may affect the ultimate recovery of the reservoir and therefore it is

recommended to evaluate field scale cases with multiple wells and more cycles for CO2

huff-n-puff process.

4. The rate transient study performed in this thesis provides sensitivity analysis outcomes

based on synthetic production data. It is recommended to analyze field production data

while incorporating combined effects of confinement, adsorption layer and diffusion-

slippage on rate transient analysis.

5. This study provides advanced understanding of the thermodynamic and kinetic

characteristics of shale reservoirs, however, other characteristics of shale plays such as

geomechanical effects need to be investigated to offer a comprehensive workflow for

characterizing a shale reservoir and optimizing its primary and EOR production.

171

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GENERAL TERMS2. Elsevier hereby grants you permission to reproduce the aforementioned material subject tothe terms and conditions indicated.3. Acknowledgement: If any part of the material to be used (for example, figures) hasappeared in our publication with credit or acknowledgement to another source, permissionmust also be sought from that source. If such permission is not obtained then that materialmay not be included in your publication/copies. Suitable acknowledgement to the sourcemust be made, either as a footnote or in a reference list at the end of your publication, asfollows:"Reprinted from Publication title, Vol /edition number, Author(s), Title of article / title ofchapter, Pages No., Copyright (Year), with permission from Elsevier [OR APPLICABLESOCIETY COPYRIGHT OWNER]." Also Lancet special credit "Reprinted from TheLancet, Vol. number, Author(s), Title of article, Pages No., Copyright (Year), withpermission from Elsevier."4. Reproduction of this material is confined to the purpose and/or media for whichpermission is hereby given.5. Altering/Modifying Material: Not Permitted. However figures and illustrations may bealtered/adapted minimally to serve your work. Any other abbreviations, additions, deletionsand/or any other alterations shall be made only with prior written authorization of ElsevierLtd. (Please contact Elsevier at [email protected]). No modifications can be madeto any Lancet figures/tables and they must be reproduced in full.6. If the permission fee for the requested use of our material is waived in this instance,please be advised that your future requests for Elsevier materials may attract a fee.7. Reservation of Rights: Publisher reserves all rights not specifically granted in thecombination of (i) the license details provided by you and accepted in the course of thislicensing transaction, (ii) these terms and conditions and (iii) CCC's Billing and Paymentterms and conditions.8. License Contingent Upon Payment: While you may exercise the rights licensedimmediately upon issuance of the license at the end of the licensing process for thetransaction, provided that you have disclosed complete and accurate details of your proposeduse, no license is finally effective unless and until full payment is received from you (eitherby publisher or by CCC) as provided in CCC's Billing and Payment terms and conditions. Iffull payment is not received on a timely basis, then any license preliminarily granted shall bedeemed automatically revoked and shall be void as if never granted. Further, in the eventthat you breach any of these terms and conditions or any of CCC's Billing and Paymentterms and conditions, the license is automatically revoked and shall be void as if nevergranted. Use of materials as described in a revoked license, as well as any use of thematerials beyond the scope of an unrevoked license, may constitute copyright infringementand publisher reserves the right to take any and all action to protect its copyright in thematerials.9. Warranties: Publisher makes no representations or warranties with respect to the licensedmaterial.10. Indemnity: You hereby indemnify and agree to hold harmless publisher and CCC, andtheir respective officers, directors, employees and agents, from and against any and allclaims arising out of your use of the licensed material other than as specifically authorizedpursuant to this license.11. No Transfer of License: This license is personal to you and may not be sublicensed,assigned, or transferred by you to any other person without publisher's written permission.

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12. No Amendment Except in Writing: This license may not be amended except in a writingsigned by both parties (or, in the case of publisher, by CCC on publisher's behalf).13. Objection to Contrary Terms: Publisher hereby objects to any terms contained in anypurchase order, acknowledgment, check endorsement or other writing prepared by you,which terms are inconsistent with these terms and conditions or CCC's Billing and Paymentterms and conditions. These terms and conditions, together with CCC's Billing and Paymentterms and conditions (which are incorporated herein), comprise the entire agreementbetween you and publisher (and CCC) concerning this licensing transaction. In the event ofany conflict between your obligations established by these terms and conditions and thoseestablished by CCC's Billing and Payment terms and conditions, these terms and conditionsshall control.14. Revocation: Elsevier or Copyright Clearance Center may deny the permissions describedin this License at their sole discretion, for any reason or no reason, with a full refund payableto you. Notice of such denial will be made using the contact information provided by you. Failure to receive such notice will not alter or invalidate the denial. In no event will Elsevieror Copyright Clearance Center be responsible or liable for any costs, expenses or damageincurred by you as a result of a denial of your permission request, other than a refund of theamount(s) paid by you to Elsevier and/or Copyright Clearance Center for deniedpermissions.

LIMITED LICENSEThe following terms and conditions apply only to specific license types:15. Translation: This permission is granted for nonexclusive world English rights onlyunless your license was granted for translation rights. If you licensed translation rights youmay only translate this content into the languages you requested. A professional translatormust perform all translations and reproduce the content word for word preserving theintegrity of the article.16. Posting licensed content on any Website: The following terms and conditions apply asfollows: Licensing material from an Elsevier journal: All content posted to the web site mustmaintain the copyright information line on the bottom of each image; A hypertext must beincluded to the Homepage of the journal from which you are licensing athttp://www.sciencedirect.com/science/journal/xxxxx or the Elsevier homepage for books athttp://www.elsevier.com; Central Storage: This license does not include permission for ascanned version of the material to be stored in a central repository such as that provided byHeron/XanEdu.Licensing material from an Elsevier book: A hypertext link must be included to the Elsevierhomepage at http://www.elsevier.com . All content posted to the web site must maintain thecopyright information line on the bottom of each image.

Posting licensed content on Electronic reserve: In addition to the above the followingclauses are applicable: The web site must be passwordprotected and made available only tobona fide students registered on a relevant course. This permission is granted for 1 year only.You may obtain a new license for future website posting.17. For journal authors: the following clauses are applicable in addition to the above:Preprints:A preprint is an author's own writeup of research results and analysis, it has not been peerreviewed, nor has it had any other value added to it by a publisher (such as formatting,copyright, technical enhancement etc.).Authors can share their preprints anywhere at any time. Preprints should not be added to orenhanced in any way in order to appear more like, or to substitute for, the final versions ofarticles however authors can update their preprints on arXiv or RePEc with their AcceptedAuthor Manuscript (see below).If accepted for publication, we encourage authors to link from the preprint to their formalpublication via its DOI. Millions of researchers have access to the formal publications onScienceDirect, and so links will help users to find, access, cite and use the best available

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version. Please note that Cell Press, The Lancet and some societyowned have differentpreprint policies. Information on these policies is available on the journal homepage.Accepted Author Manuscripts: An accepted author manuscript is the manuscript of anarticle that has been accepted for publication and which typically includes authorincorporated changes suggested during submission, peer review and editorauthorcommunications.Authors can share their accepted author manuscript:

immediatelyvia their noncommercial person homepage or blogby updating a preprint in arXiv or RePEc with the accepted manuscriptvia their research institute or institutional repository for internal institutionaluses or as part of an invitationonly research collaboration workgroupdirectly by providing copies to their students or to research collaborators fortheir personal usefor private scholarly sharing as part of an invitationonly work group oncommercial sites with which Elsevier has an agreement

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link to the formal publication via its DOIbear a CCBYNCND license this is easy to doif aggregated with other manuscripts, for example in a repository or other site, beshared in alignment with our hosting policy not be added to or enhanced in any way toappear more like, or to substitute for, the published journal article.

Published journal article (JPA): A published journal article (PJA) is the definitive finalrecord of published research that appears or will appear in the journal and embodies allvalueadding publishing activities including peer review coordination, copyediting,formatting, (if relevant) pagination and online enrichment.Policies for sharing publishing journal articles differ for subscription and gold open accessarticles:Subscription Articles: If you are an author, please share a link to your article rather than thefulltext. Millions of researchers have access to the formal publications on ScienceDirect,and so links will help your users to find, access, cite, and use the best available version.Theses and dissertations which contain embedded PJAs as part of the formal submission canbe posted publicly by the awarding institution with DOI links back to the formalpublications on ScienceDirect.If you are affiliated with a library that subscribes to ScienceDirect you have additionalprivate sharing rights for others' research accessed under that agreement. This includes usefor classroom teaching and internal training at the institution (including use in course packsand courseware programs), and inclusion of the article for grant funding purposes.Gold Open Access Articles: May be shared according to the authorselected enduserlicense and should contain a CrossMark logo, the end user license, and a DOI link to theformal publication on ScienceDirect.Please refer to Elsevier's posting policy for further information.18. For book authors the following clauses are applicable in addition to the above: Authors are permitted to place a brief summary of their work online only. You are notallowed to download and post the published electronic version of your chapter, nor may youscan the printed edition to create an electronic version. Posting to a repository: Authors arepermitted to post a summary of their chapter only in their institution's repository.

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19. Thesis/Dissertation: If your license is for use in a thesis/dissertation your thesis may besubmitted to your institution in either print or electronic form. Should your thesis bepublished commercially, please reapply for permission. These requirements includepermission for the Library and Archives of Canada to supply single copies, on demand, ofthe complete thesis and include permission for Proquest/UMI to supply single copies, ondemand, of the complete thesis. Should your thesis be published commercially, pleasereapply for permission. Theses and dissertations which contain embedded PJAs as part ofthe formal submission can be posted publicly by the awarding institution with DOI linksback to the formal publications on ScienceDirect. Elsevier Open Access Terms and ConditionsYou can publish open access with Elsevier in hundreds of open access journals or in nearly2000 established subscription journals that support open access publishing. Permitted thirdparty reuse of these open access articles is defined by the author's choice of CreativeCommons user license. See our open access license policy for more information.Terms & Conditions applicable to all Open Access articles published with Elsevier:Any reuse of the article must not represent the author as endorsing the adaptation of thearticle nor should the article be modified in such a way as to damage the author's honour orreputation. If any changes have been made, such changes must be clearly indicated.The author(s) must be appropriately credited and we ask that you include the end userlicense and a DOI link to the formal publication on ScienceDirect.If any part of the material to be used (for example, figures) has appeared in our publicationwith credit or acknowledgement to another source it is the responsibility of the user toensure their reuse complies with the terms and conditions determined by the rights holder.Additional Terms & Conditions applicable to each Creative Commons user license:CC BY: The CCBY license allows users to copy, to create extracts, abstracts and newworks from the Article, to alter and revise the Article and to make commercial use of theArticle (including reuse and/or resale of the Article by commercial entities), provided theuser gives appropriate credit (with a link to the formal publication through the relevantDOI), provides a link to the license, indicates if changes were made and the licensor is notrepresented as endorsing the use made of the work. The full details of the license areavailable at http://creativecommons.org/licenses/by/4.0.CC BY NC SA: The CC BYNCSA license allows users to copy, to create extracts,abstracts and new works from the Article, to alter and revise the Article, provided this is notdone for commercial purposes, and that the user gives appropriate credit (with a link to theformal publication through the relevant DOI), provides a link to the license, indicates ifchanges were made and the licensor is not represented as endorsing the use made of thework. Further, any new works must be made available on the same conditions. The fulldetails of the license are available at http://creativecommons.org/licenses/byncsa/4.0.CC BY NC ND: The CC BYNCND license allows users to copy and distribute the Article,provided this is not done for commercial purposes and further does not permit distribution ofthe Article if it is changed or edited in any way, and provided the user gives appropriatecredit (with a link to the formal publication through the relevant DOI), provides a link to thelicense, and that the licensor is not represented as endorsing the use made of the work. Thefull details of the license are available at http://creativecommons.org/licenses/byncnd/4.0.Any commercial reuse of Open Access articles published with a CC BY NC SA or CC BYNC ND license requires permission from Elsevier and will be subject to a fee.Commercial reuse includes:

Associating advertising with the full text of the ArticleCharging fees for document delivery or accessArticle aggregationSystematic distribution via email lists or share buttons

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ELSEVIER LICENSETERMS AND CONDITIONS

Apr 07, 2017

This Agreement between Behjat Haghshenas ("You") and Elsevier ("Elsevier") consists ofyour license details and the terms and conditions provided by Elsevier and CopyrightClearance Center.



Licensed Content Publisher Elsevier

Licensed Content Publication Journal of Natural Gas Science and Engineering

Licensed Content Title Characterization of multifractured horizontal shale wells using drillcuttings: 2. Permeability/diffusivity estimation

Licensed Content Author B. Haghshenas,C.R. Clarkson,S.D. Aquino,S. Chen

Licensed Content Date May 2016

Licensed Content Volume 32

Licensed Content Issue n/a

Licensed Content Pages 11

Start Page 586

End Page 596

Type of Use reuse in a thesis/dissertation

Intended publisher of newwork

other


Format both print and electronic

Are you the author of thisElsevier article?

Yes

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Title of yourthesis/dissertation




200

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INTRODUCTION1. The publisher for this copyrighted material is Elsevier. By clicking "accept" in connectionwith completing this licensing transaction, you agree that the following terms and conditionsapply to this transaction (along with the Billing and Payment terms and conditionsestablished by Copyright Clearance Center, Inc. ("CCC"), at the time that you opened yourRightslink account and that are available at any time at http://myaccount.copyright.com).

GENERAL TERMS2. Elsevier hereby grants you permission to reproduce the aforementioned material subject tothe terms and conditions indicated.3. Acknowledgement: If any part of the material to be used (for example, figures) hasappeared in our publication with credit or acknowledgement to another source, permissionmust also be sought from that source. If such permission is not obtained then that materialmay not be included in your publication/copies. Suitable acknowledgement to the sourcemust be made, either as a footnote or in a reference list at the end of your publication, asfollows:"Reprinted from Publication title, Vol /edition number, Author(s), Title of article / title ofchapter, Pages No., Copyright (Year), with permission from Elsevier [OR APPLICABLESOCIETY COPYRIGHT OWNER]." Also Lancet special credit "Reprinted from TheLancet, Vol. number, Author(s), Title of article, Pages No., Copyright (Year), withpermission from Elsevier."4. Reproduction of this material is confined to the purpose and/or media for whichpermission is hereby given.5. Altering/Modifying Material: Not Permitted. However figures and illustrations may bealtered/adapted minimally to serve your work. Any other abbreviations, additions, deletionsand/or any other alterations shall be made only with prior written authorization of ElsevierLtd. (Please contact Elsevier at [email protected]). No modifications can be madeto any Lancet figures/tables and they must be reproduced in full.6. If the permission fee for the requested use of our material is waived in this instance,please be advised that your future requests for Elsevier materials may attract a fee.7. Reservation of Rights: Publisher reserves all rights not specifically granted in thecombination of (i) the license details provided by you and accepted in the course of thislicensing transaction, (ii) these terms and conditions and (iii) CCC's Billing and Paymentterms and conditions.8. License Contingent Upon Payment: While you may exercise the rights licensedimmediately upon issuance of the license at the end of the licensing process for thetransaction, provided that you have disclosed complete and accurate details of your proposeduse, no license is finally effective unless and until full payment is received from you (eitherby publisher or by CCC) as provided in CCC's Billing and Payment terms and conditions. Iffull payment is not received on a timely basis, then any license preliminarily granted shall bedeemed automatically revoked and shall be void as if never granted. Further, in the eventthat you breach any of these terms and conditions or any of CCC's Billing and Paymentterms and conditions, the license is automatically revoked and shall be void as if nevergranted. Use of materials as described in a revoked license, as well as any use of thematerials beyond the scope of an unrevoked license, may constitute copyright infringementand publisher reserves the right to take any and all action to protect its copyright in thematerials.9. Warranties: Publisher makes no representations or warranties with respect to the licensedmaterial.10. Indemnity: You hereby indemnify and agree to hold harmless publisher and CCC, andtheir respective officers, directors, employees and agents, from and against any and allclaims arising out of your use of the licensed material other than as specifically authorizedpursuant to this license.11. No Transfer of License: This license is personal to you and may not be sublicensed,assigned, or transferred by you to any other person without publisher's written permission.

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12. No Amendment Except in Writing: This license may not be amended except in a writingsigned by both parties (or, in the case of publisher, by CCC on publisher's behalf).13. Objection to Contrary Terms: Publisher hereby objects to any terms contained in anypurchase order, acknowledgment, check endorsement or other writing prepared by you,which terms are inconsistent with these terms and conditions or CCC's Billing and Paymentterms and conditions. These terms and conditions, together with CCC's Billing and Paymentterms and conditions (which are incorporated herein), comprise the entire agreementbetween you and publisher (and CCC) concerning this licensing transaction. In the event ofany conflict between your obligations established by these terms and conditions and thoseestablished by CCC's Billing and Payment terms and conditions, these terms and conditionsshall control.14. Revocation: Elsevier or Copyright Clearance Center may deny the permissions describedin this License at their sole discretion, for any reason or no reason, with a full refund payableto you. Notice of such denial will be made using the contact information provided by you. Failure to receive such notice will not alter or invalidate the denial. In no event will Elsevieror Copyright Clearance Center be responsible or liable for any costs, expenses or damageincurred by you as a result of a denial of your permission request, other than a refund of theamount(s) paid by you to Elsevier and/or Copyright Clearance Center for deniedpermissions.

LIMITED LICENSEThe following terms and conditions apply only to specific license types:15. Translation: This permission is granted for nonexclusive world English rights onlyunless your license was granted for translation rights. If you licensed translation rights youmay only translate this content into the languages you requested. A professional translatormust perform all translations and reproduce the content word for word preserving theintegrity of the article.16. Posting licensed content on any Website: The following terms and conditions apply asfollows: Licensing material from an Elsevier journal: All content posted to the web site mustmaintain the copyright information line on the bottom of each image; A hypertext must beincluded to the Homepage of the journal from which you are licensing athttp://www.sciencedirect.com/science/journal/xxxxx or the Elsevier homepage for books athttp://www.elsevier.com; Central Storage: This license does not include permission for ascanned version of the material to be stored in a central repository such as that provided byHeron/XanEdu.Licensing material from an Elsevier book: A hypertext link must be included to the Elsevierhomepage at http://www.elsevier.com . All content posted to the web site must maintain thecopyright information line on the bottom of each image.

Posting licensed content on Electronic reserve: In addition to the above the followingclauses are applicable: The web site must be passwordprotected and made available only tobona fide students registered on a relevant course. This permission is granted for 1 year only.You may obtain a new license for future website posting.17. For journal authors: the following clauses are applicable in addition to the above:Preprints:A preprint is an author's own writeup of research results and analysis, it has not been peerreviewed, nor has it had any other value added to it by a publisher (such as formatting,copyright, technical enhancement etc.).Authors can share their preprints anywhere at any time. Preprints should not be added to orenhanced in any way in order to appear more like, or to substitute for, the final versions ofarticles however authors can update their preprints on arXiv or RePEc with their AcceptedAuthor Manuscript (see below).If accepted for publication, we encourage authors to link from the preprint to their formalpublication via its DOI. Millions of researchers have access to the formal publications onScienceDirect, and so links will help users to find, access, cite and use the best available

http://www.sciencedirect.com/science/journal/xxxxx





version. Please note that Cell Press, The Lancet and some societyowned have differentpreprint policies. Information on these policies is available on the journal homepage.Accepted Author Manuscripts: An accepted author manuscript is the manuscript of anarticle that has been accepted for publication and which typically includes authorincorporated changes suggested during submission, peer review and editorauthorcommunications.Authors can share their accepted author manuscript:

immediatelyvia their noncommercial person homepage or blogby updating a preprint in arXiv or RePEc with the accepted manuscriptvia their research institute or institutional repository for internal institutionaluses or as part of an invitationonly research collaboration workgroupdirectly by providing copies to their students or to research collaborators fortheir personal usefor private scholarly sharing as part of an invitationonly work group oncommercial sites with which Elsevier has an agreement

After the embargo periodvia noncommercial hosting platforms such as their institutional repositoryvia commercial sites with which Elsevier has an agreement

In all cases accepted manuscripts should:

link to the formal publication via its DOIbear a CCBYNCND license this is easy to doif aggregated with other manuscripts, for example in a repository or other site, beshared in alignment with our hosting policy not be added to or enhanced in any way toappear more like, or to substitute for, the published journal article.

Published journal article (JPA): A published journal article (PJA) is the definitive finalrecord of published research that appears or will appear in the journal and embodies allvalueadding publishing activities including peer review coordination, copyediting,formatting, (if relevant) pagination and online enrichment.Policies for sharing publishing journal articles differ for subscription and gold open accessarticles:Subscription Articles: If you are an author, please share a link to your article rather than thefulltext. Millions of researchers have access to the formal publications on ScienceDirect,and so links will help your users to find, access, cite, and use the best available version.Theses and dissertations which contain embedded PJAs as part of the formal submission canbe posted publicly by the awarding institution with DOI links back to the formalpublications on ScienceDirect.If you are affiliated with a library that subscribes to ScienceDirect you have additionalprivate sharing rights for others' research accessed under that agreement. This includes usefor classroom teaching and internal training at the institution (including use in course packsand courseware programs), and inclusion of the article for grant funding purposes.Gold Open Access Articles: May be shared according to the authorselected enduserlicense and should contain a CrossMark logo, the end user license, and a DOI link to theformal publication on ScienceDirect.Please refer to Elsevier's posting policy for further information.18. For book authors the following clauses are applicable in addition to the above: Authors are permitted to place a brief summary of their work online only. You are notallowed to download and post the published electronic version of your chapter, nor may youscan the printed edition to create an electronic version. Posting to a repository: Authors arepermitted to post a summary of their chapter only in their institution's repository.

http://www.crossref.org/crossmark/index.html

http://www.elsevier.com/about/open-access/open-access-policies/article-posting-policy



19. Thesis/Dissertation: If your license is for use in a thesis/dissertation your thesis may besubmitted to your institution in either print or electronic form. Should your thesis bepublished commercially, please reapply for permission. These requirements includepermission for the Library and Archives of Canada to supply single copies, on demand, ofthe complete thesis and include permission for Proquest/UMI to supply single copies, ondemand, of the complete thesis. Should your thesis be published commercially, pleasereapply for permission. Theses and dissertations which contain embedded PJAs as part ofthe formal submission can be posted publicly by the awarding institution with DOI linksback to the formal publications on ScienceDirect. Elsevier Open Access Terms and ConditionsYou can publish open access with Elsevier in hundreds of open access journals or in nearly2000 established subscription journals that support open access publishing. Permitted thirdparty reuse of these open access articles is defined by the author's choice of CreativeCommons user license. See our open access license policy for more information.Terms & Conditions applicable to all Open Access articles published with Elsevier:Any reuse of the article must not represent the author as endorsing the adaptation of thearticle nor should the article be modified in such a way as to damage the author's honour orreputation. If any changes have been made, such changes must be clearly indicated.The author(s) must be appropriately credited and we ask that you include the end userlicense and a DOI link to the formal publication on ScienceDirect.If any part of the material to be used (for example, figures) has appeared in our publicationwith credit or acknowledgement to another source it is the responsibility of the user toensure their reuse complies with the terms and conditions determined by the rights holder.Additional Terms & Conditions applicable to each Creative Commons user license:CC BY: The CCBY license allows users to copy, to create extracts, abstracts and newworks from the Article, to alter and revise the Article and to make commercial use of theArticle (including reuse and/or resale of the Article by commercial entities), provided theuser gives appropriate credit (with a link to the formal publication through the relevantDOI), provides a link to the license, indicates if changes were made and the licensor is notrepresented as endorsing the use made of the work. The full details of the license areavailable at http://creativecommons.org/licenses/by/4.0.CC BY NC SA: The CC BYNCSA license allows users to copy, to create extracts,abstracts and new works from the Article, to alter and revise the Article, provided this is notdone for commercial purposes, and that the user gives appropriate credit (with a link to theformal publication through the relevant DOI), provides a link to the license, indicates ifchanges were made and the licensor is not represented as endorsing the use made of thework. Further, any new works must be made available on the same conditions. The fulldetails of the license are available at http://creativecommons.org/licenses/byncsa/4.0.CC BY NC ND: The CC BYNCND license allows users to copy and distribute the Article,provided this is not done for commercial purposes and further does not permit distribution ofthe Article if it is changed or edited in any way, and provided the user gives appropriatecredit (with a link to the formal publication through the relevant DOI), provides a link to thelicense, and that the licensor is not represented as endorsing the use made of the work. Thefull details of the license are available at http://creativecommons.org/licenses/byncnd/4.0.Any commercial reuse of Open Access articles published with a CC BY NC SA or CC BYNC ND license requires permission from Elsevier and will be subject to a fee.Commercial reuse includes:

Associating advertising with the full text of the ArticleCharging fees for document delivery or accessArticle aggregationSystematic distribution via email lists or share buttons

http://www.elsevier.com/about/open-access/open-access-policies/oa-license-policy

http://creativecommons.org/licenses/by/4.0

http://creativecommons.org/licenses/by-nc-sa/4.0

http://creativecommons.org/licenses/by-nc-nd/4.0



Posting or linking by commercial companies for use by customers of those companies. 20. Other Conditions: v1.9Questions? [email protected] or +18552393415 (toll free in the US) or+19786462777.




SOCIETY OF PETROLEUM ENGINEERS LICENSETERMS AND CONDITIONS

Apr 06, 2017

This Agreement between Behjat Haghshenas ("You") and Society of Petroleum Engineers("Society of Petroleum Engineers") consists of your license details and the terms andconditions provided by Society of Petroleum Engineers and Copyright Clearance Center.



Licensed Content Publisher Society of Petroleum Engineers

Licensed Content Publication SPE Proceedings

Licensed Content Title New Models for Reserve Estimation and NonDarcy Gas Flow in ShaleGas Reservoirs

Licensed Content Author B. Haghshenas, University of Calgary;C. R. Clarkson, University ofCalgary;S. Chen, University of Calgary et al

Licensed Content Date 2014

Type of Use Thesis/Dissertation

Requestor type author of the original work

SPE member yes

SPE member number 4105753

Format print and electronic


Will you be translating? no

Distribution 10


Title of your thesis /dissertation




200



Billing Type Invoice

Billing Address Behjat Haghshenas152 Strathbury Circle, SW


Total 0.00 CAD

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STANDARD TERMS AND CONDITIONS FOR REPRODUCTION OF MATERIAL1. The Society of Petroleum Engineers, Inc. (“SPE”) holds the copyright for this material.By clicking "accept" in connection with completing this licensing transaction, you agree thatthe following terms and conditions apply to this transaction (along with the Billing andPayment terms and conditions established by Copyright Clearance Center, Inc. ("CCC"), atthe time that you opened your RightsLink account and that are available at any time at ).2. SPE hereby grants to you a nonexclusive license to use this material. Licenses are foronetime use only with a maximum distribution equal to the number that you identified inthe licensing process; any form of republication must be completed within six months fromthe date hereof (although copies prepared before then may be distributed thereafter); and anyelectronic posting is limited to the period identified in the licensing process.3. You may not alter or modify the material in any manner (except that you may use, withinthe scope of the license granted, one or more excerpts from the copyrighted material,provided that the process of excerpting does not alter the meaning of the material or in anyway reflect negatively on SPE or any writer of the material or their employer), nor may youtranslate the material into another language.4. Total excerpts from the license material may not exceed thirty percent (30%) of the totaltext. Not more than five (5) excerpts, figures, tables, or images may be used from any givenpaper. Multiple permission requests may not be used to exceed these limits.5. SPE reserves all rights not specifically granted in the combination of (i) the license detailsprovided by you and accepted in the course of this licensing transaction, (ii) these terms andconditions and (iii) CCC's Billing and Payment terms and conditions.6. While you may exercise the rights licensed immediately upon issuance of the license atthe end of the licensing process for the transaction, provided that you have disclosedcomplete and accurate details of your proposed use, no license is finally effective unless anduntil full payment is received from you (either by SPE or by CCC) as provided in CCC'sBilling and Payment terms and conditions. If full payment is not received on a timely basis,then any license preliminarily granted shall be deemed automatically revoked and shall bevoid as if never granted. Further, in the event that you breach any of these terms andconditions or any of CCC's Billing and Payment terms and conditions, the license isautomatically revoked and shall be void as if never granted. Use of materials as described ina revoked license, as well as any use of the materials beyond the scope of an unrevokedlicense, may constitute copyright infringement and SPE reserves the right to take any and allaction to protect its copyright in the materials7. You must include the appropriate copyright and permission notice and disclaimer inconnection with any reproduction of the licensed material.The copyright information isfound on the front page of the paper immediately under the title and author. This statementwill then be followed with the disclaimer, “Further reproduction prohibited withoutpermission.” Examples:1) Copyright 1990, Society of Petroleum Engineers Inc.Copyright1990, SPE. Reproduced with permission of SPE. Further reproduction prohibited withoutpermission.2) Copyright 2010, IADC/SPE Drilling Conference and ExhibitionCopyright2010, IADC/SPE Drilling Conference and Exhibition. Reproduced with permission of SPE.Further reproduction prohibited without permission.3) Copyright 2008, Offshore TechnologyConferenceCopyright 2008, Offshore Technology Conference. Reproduced with permissionof OTC. Further reproduction prohibited without permission.4) Copyright 2005,International Petroleum Technology ConferenceCopyright 2005, International PetroleumTechnology Conference. Reproduced with permission of IPTC. Further reproductionprohibited without permission.If for any reason, the copyright on the paper is missing orunclear, please follow Example 1 above, using SPE as the default copyright holder. SPEadministers copyright for OTC, IPTC and other joint events on behalf of all parties in thoseevents.8. SPE makes no representations or warranties with respect to the licensed material andadopts on its own behalf the limitations and disclaimers established by CCC on its behalf in



its Billing and Payment terms and conditions for this licensing transaction.9. You hereby indemnify and agree to hold harmless SPE and CCC, and their respectiveofficers, directors, employees and agents, from and against any and all claims arising out ofyour use of the licensed material other than as specifically authorized pursuant to thislicense.10. This license is personal to you, but may be assigned or transferred by you to a businessassociate (or to your employer) if you give prompt written notice of the assignment ortransfer to SPE. No such assignment or transfer shall relieve you of the obligation to pay thedesignated license fee on a timely basis (although payment by the identified assignee canfulfill your obligation).11. This license may not be amended except in a writing signed by both parties (or, in thecase of SPE, by CCC on SPE's behalf).12. SPE hereby objects to any terms contained in any purchase order, acknowledgment,check endorsement or other writing prepared by you, which terms are inconsistent with theseterms and conditions or CCC's Billing and Payment terms and conditions. These terms andconditions, together with CCC's Billing and Payment terms and conditions (which areincorporated herein), comprise the entire agreement between you and SPE (and CCC)concerning this licensing transaction. In the event of any conflict between your obligationsestablished by these terms and conditions and those established by CCC's Billing andPayment terms and conditions, these terms and conditions shall control.13. This Agreement shall be governed and interpreted by the laws of the State of Texas,United States of America. Regardless of the place of performance or otherwise, theAgreement, and all schedules, amendments, modifications, alterations, or supplementsthereto, will be governed by the laws of the State of Texas, United States of America. If anyprovisions of the Agreement are unenforceable under applicable law, the remainingprovisions shall continue in full force and effect.Other Terms and Conditions:v1.1Questions? [email protected] or +18552393415 (toll free in the US) or+19786462777.





Apr 06, 2017






Licensed Content Title Modeling PVT Behavior of GasCondensate System Under PoreConfinement Effects: Implications for RateTransient Analysis of GasCondensate Shale Plays

Licensed Content Author B. Haghshenas, University of Calgary;F. Qanbari, University ofCalgary;C. R. Clarkson, University of Calgary et al




SPE member yes





Distribution 10






200






Behjat

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Total 0.00 CAD


STANDARD TERMS AND CONDITIONS FOR REPRODUCTION OF MATERIAL1. The Society of Petroleum Engineers, Inc. (“SPE”) holds the copyright for this material.By clicking "accept" in connection with completing this licensing transaction, you agree thatthe following terms and conditions apply to this transaction (along with the Billing andPayment terms and conditions established by Copyright Clearance Center, Inc. ("CCC"), atthe time that you opened your RightsLink account and that are available at any time at ).2. SPE hereby grants to you a nonexclusive license to use this material. Licenses are foronetime use only with a maximum distribution equal to the number that you identified inthe licensing process; any form of republication must be completed within six months fromthe date hereof (although copies prepared before then may be distributed thereafter); and anyelectronic posting is limited to the period identified in the licensing process.3. You may not alter or modify the material in any manner (except that you may use, withinthe scope of the license granted, one or more excerpts from the copyrighted material,provided that the process of excerpting does not alter the meaning of the material or in anyway reflect negatively on SPE or any writer of the material or their employer), nor may youtranslate the material into another language.4. Total excerpts from the license material may not exceed thirty percent (30%) of the totaltext. Not more than five (5) excerpts, figures, tables, or images may be used from any givenpaper. Multiple permission requests may not be used to exceed these limits.5. SPE reserves all rights not specifically granted in the combination of (i) the license detailsprovided by you and accepted in the course of this licensing transaction, (ii) these terms andconditions and (iii) CCC's Billing and Payment terms and conditions.6. While you may exercise the rights licensed immediately upon issuance of the license atthe end of the licensing process for the transaction, provided that you have disclosedcomplete and accurate details of your proposed use, no license is finally effective unless anduntil full payment is received from you (either by SPE or by CCC) as provided in CCC'sBilling and Payment terms and conditions. If full payment is not received on a timely basis,then any license preliminarily granted shall be deemed automatically revoked and shall bevoid as if never granted. Further, in the event that you breach any of these terms andconditions or any of CCC's Billing and Payment terms and conditions, the license isautomatically revoked and shall be void as if never granted. Use of materials as described ina revoked license, as well as any use of the materials beyond the scope of an unrevokedlicense, may constitute copyright infringement and SPE reserves the right to take any and allaction to protect its copyright in the materials7. You must include the appropriate copyright and permission notice and disclaimer inconnection with any reproduction of the licensed material.The copyright information isfound on the front page of the paper immediately under the title and author. This statementwill then be followed with the disclaimer, “Further reproduction prohibited withoutpermission.” Examples:1) Copyright 1990, Society of Petroleum Engineers Inc.Copyright1990, SPE. Reproduced with permission of SPE. Further reproduction prohibited withoutpermission.2) Copyright 2010, IADC/SPE Drilling Conference and ExhibitionCopyright2010, IADC/SPE Drilling Conference and Exhibition. Reproduced with permission of SPE.Further reproduction prohibited without permission.3) Copyright 2008, Offshore TechnologyConferenceCopyright 2008, Offshore Technology Conference. Reproduced with permissionof OTC. Further reproduction prohibited without permission.4) Copyright 2005,International Petroleum Technology ConferenceCopyright 2005, International PetroleumTechnology Conference. Reproduced with permission of IPTC. Further reproductionprohibited without permission.If for any reason, the copyright on the paper is missing orunclear, please follow Example 1 above, using SPE as the default copyright holder. SPEadministers copyright for OTC, IPTC and other joint events on behalf of all parties in thoseevents.



8. SPE makes no representations or warranties with respect to the licensed material andadopts on its own behalf the limitations and disclaimers established by CCC on its behalf inits Billing and Payment terms and conditions for this licensing transaction.9. You hereby indemnify and agree to hold harmless SPE and CCC, and their respectiveofficers, directors, employees and agents, from and against any and all claims arising out ofyour use of the licensed material other than as specifically authorized pursuant to thislicense.10. This license is personal to you, but may be assigned or transferred by you to a businessassociate (or to your employer) if you give prompt written notice of the assignment ortransfer to SPE. No such assignment or transfer shall relieve you of the obligation to pay thedesignated license fee on a timely basis (although payment by the identified assignee canfulfill your obligation).11. This license may not be amended except in a writing signed by both parties (or, in thecase of SPE, by CCC on SPE's behalf).12. SPE hereby objects to any terms contained in any purchase order, acknowledgment,check endorsement or other writing prepared by you, which terms are inconsistent with theseterms and conditions or CCC's Billing and Payment terms and conditions. These terms andconditions, together with CCC's Billing and Payment terms and conditions (which areincorporated herein), comprise the entire agreement between you and SPE (and CCC)concerning this licensing transaction. In the event of any conflict between your obligationsestablished by these terms and conditions and those established by CCC's Billing andPayment terms and conditions, these terms and conditions shall control.13. This Agreement shall be governed and interpreted by the laws of the State of Texas,United States of America. Regardless of the place of performance or otherwise, theAgreement, and all schedules, amendments, modifications, alterations, or supplementsthereto, will be governed by the laws of the State of Texas, United States of America. If anyprovisions of the Agreement are unenforceable under applicable law, the remainingprovisions shall continue in full force and effect.Other Terms and Conditions:v1.1Questions? [email protected] or +18552393415 (toll free in the US) or+19786462777.





Apr 06, 2017






Licensed Content Title Simulation of Enhanced Recovery using CO2 in a LiquidRich WesternCanadian Unconventional Reservoir: Accounting for Reservoir FluidAdsorption and Compositional Heterogeneity

Licensed Content Author B. Haghshenas, University of Calgary;F. Qanbari, University ofCalgary;C. R. Clarkson, University of Calgary et al




SPE member yes





Distribution 10






200






Behjat

Highlight



Total 0.00 CAD


STANDARD TERMS AND CONDITIONS FOR REPRODUCTION OF MATERIAL1. The Society of Petroleum Engineers, Inc. (“SPE”) holds the copyright for this material.By clicking "accept" in connection with completing this licensing transaction, you agree thatthe following terms and conditions apply to this transaction (along with the Billing andPayment terms and conditions established by Copyright Clearance Center, Inc. ("CCC"), atthe time that you opened your RightsLink account and that are available at any time at ).2. SPE hereby grants to you a nonexclusive license to use this material. Licenses are foronetime use only with a maximum distribution equal to the number that you identified inthe licensing process; any form of republication must be completed within six months fromthe date hereof (although copies prepared before then may be distributed thereafter); and anyelectronic posting is limited to the period identified in the licensing process.3. You may not alter or modify the material in any manner (except that you may use, withinthe scope of the license granted, one or more excerpts from the copyrighted material,provided that the process of excerpting does not alter the meaning of the material or in anyway reflect negatively on SPE or any writer of the material or their employer), nor may youtranslate the material into another language.4. Total excerpts from the license material may not exceed thirty percent (30%) of the totaltext. Not more than five (5) excerpts, figures, tables, or images may be used from any givenpaper. Multiple permission requests may not be used to exceed these limits.5. SPE reserves all rights not specifically granted in the combination of (i) the license detailsprovided by you and accepted in the course of this licensing transaction, (ii) these terms andconditions and (iii) CCC's Billing and Payment terms and conditions.6. While you may exercise the rights licensed immediately upon issuance of the license atthe end of the licensing process for the transaction, provided that you have disclosedcomplete and accurate details of your proposed use, no license is finally effective unless anduntil full payment is received from you (either by SPE or by CCC) as provided in CCC'sBilling and Payment terms and conditions. If full payment is not received on a timely basis,then any license preliminarily granted shall be deemed automatically revoked and shall bevoid as if never granted. Further, in the event that you breach any of these terms andconditions or any of CCC's Billing and Payment terms and conditions, the license isautomatically revoked and shall be void as if never granted. Use of materials as described ina revoked license, as well as any use of the materials beyond the scope of an unrevokedlicense, may constitute copyright infringement and SPE reserves the right to take any and allaction to protect its copyright in the materials7. You must include the appropriate copyright and permission notice and disclaimer inconnection with any reproduction of the licensed material.The copyright information isfound on the front page of the paper immediately under the title and author. This statementwill then be followed with the disclaimer, “Further reproduction prohibited withoutpermission.” Examples:1) Copyright 1990, Society of Petroleum Engineers Inc.Copyright1990, SPE. Reproduced with permission of SPE. Further reproduction prohibited withoutpermission.2) Copyright 2010, IADC/SPE Drilling Conference and ExhibitionCopyright2010, IADC/SPE Drilling Conference and Exhibition. Reproduced with permission of SPE.Further reproduction prohibited without permission.3) Copyright 2008, Offshore TechnologyConferenceCopyright 2008, Offshore Technology Conference. Reproduced with permissionof OTC. Further reproduction prohibited without permission.4) Copyright 2005,International Petroleum Technology ConferenceCopyright 2005, International PetroleumTechnology Conference. Reproduced with permission of IPTC. Further reproductionprohibited without permission.If for any reason, the copyright on the paper is missing orunclear, please follow Example 1 above, using SPE as the default copyright holder. SPEadministers copyright for OTC, IPTC and other joint events on behalf of all parties in thoseevents.



8. SPE makes no representations or warranties with respect to the licensed material andadopts on its own behalf the limitations and disclaimers established by CCC on its behalf inits Billing and Payment terms and conditions for this licensing transaction.9. You hereby indemnify and agree to hold harmless SPE and CCC, and their respectiveofficers, directors, employees and agents, from and against any and all claims arising out ofyour use of the licensed material other than as specifically authorized pursuant to thislicense.10. This license is personal to you, but may be assigned or transferred by you to a businessassociate (or to your employer) if you give prompt written notice of the assignment ortransfer to SPE. No such assignment or transfer shall relieve you of the obligation to pay thedesignated license fee on a timely basis (although payment by the identified assignee canfulfill your obligation).11. This license may not be amended except in a writing signed by both parties (or, in thecase of SPE, by CCC on SPE's behalf).12. SPE hereby objects to any terms contained in any purchase order, acknowledgment,check endorsement or other writing prepared by you, which terms are inconsistent with theseterms and conditions or CCC's Billing and Payment terms and conditions. These terms andconditions, together with CCC's Billing and Payment terms and conditions (which areincorporated herein), comprise the entire agreement between you and SPE (and CCC)concerning this licensing transaction. In the event of any conflict between your obligationsestablished by these terms and conditions and those established by CCC's Billing andPayment terms and conditions, these terms and conditions shall control.13. This Agreement shall be governed and interpreted by the laws of the State of Texas,United States of America. Regardless of the place of performance or otherwise, theAgreement, and all schedules, amendments, modifications, alterations, or supplementsthereto, will be governed by the laws of the State of Texas, United States of America. If anyprovisions of the Agreement are unenforceable under applicable law, the remainingprovisions shall continue in full force and effect.Other Terms and Conditions:v1.1Questions? [email protected] or +18552393415 (toll free in the US) or+19786462777.


Documents

Modeling Storage and Flow of Fluids in Shale Reservoirs