Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Zheng, Wenbo
Doctor of Philosophy - PhD
University of British Columbia
Proppants are widely used in hydraulic fracturing for unconventional resources to increase the recovery rate of hydrocarbons in the oil and gas industry. More than 55 million tonnes of proppants were supplied for hydraulic fracturing in 2015 to fracture more than 35,000 wells. During hydraulic fracturing, proppants are transported by fracturing fluids into fractures to form a permeable proppant pack, which acts as a path for hydrocarbons to flow to wellbores. When high closure stresses exist, proppants can be crushed into smaller fragments and embed into fracture surfaces, which reduces the propped fracture conductivity by decreasing the proppant pack permeability and thickness. Choosing a suitable proppant for hydraulic fracturing is key to success in recovering hydrocarbons. Compared with time-consuming laboratory tests and simplified empirical/numerical models, the discrete element method (DEM) is a natural approach for simulating proppant crushing and embedment into fracture faces. However, current DEM simulations have not considered proppant crushing under high closure stresses. In addition, the mechanical properties of different proppants have not been fully characterised thus modelling of proppants in DEM lacks verification. Moreover, DEM simulations of proppant embedment into rock surface have not been calibrated/compared with laboratory hardness tests on rock faces. Finally, the influence of proppant crushing and embedment on proppant selection has not been thoroughly studied. With these in mind, this research presents comprehensive laboratory experiments on proppant crushing which were used to calibrate discrete element models of proppant crushing and proppant embedment. Three objectives are accomplished within this research: (1) characterize the mechanical behaviour of proppant crushing and proppant embedment into fracture surfaces; (2) model proppant crushing and proppant embedment with DEM; (3) study the influence of proppant crushing and proppant embedment on hydraulic fracture conductivity and recommend optimal proppant usage for hydraulic fracturing. The results deliver a better understanding of the crushing characteristics of different proppants, and improve DEM modelling of proppant crushing and embedment into hydraulic fractures, which provides a cost-efficient approach for evaluating the proppant performance for hydraulic fracturing without the need to conduct problematic hydraulic conductivity experiments.
LABORATORY AND DISCRETE ELEMENT STUDY OF PROPPANT CRUSHING AND EMBEDMENT AND THEIR INFLUENCE ON FRACTURE CONDUCTIVITY by Wenbo Zheng B.Eng., Tongji University, China, 2010 M.Eng., Tongji University, China, 2013 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The College of Graduate Studies (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) December 2017 © Wenbo Zheng, 2017ii The following individuals certify that they have read, and recommend to the College of Graduate Studies for acceptance, a thesis/dissertation entitled: LABORATORY AND DISCRETE ELEMENT STUDY OF PROPPANT CRUSHING AND EMBEDMENT AND THEIR INFLUENCE ON FRACTURE CONDUCTIVITY submitted by Wenbo Zheng in partial fulfilment of the requirements of the degree of Doctor of Philosophy. Dr. Dwayne D. Tannant, School of Engineering Supervisor Dr. Sumi Siddiqua, School of Engineering Supervisory Committee Member Dr. Hadi Mohammadi, School of Engineering Supervisory Committee Member Dr. Lukas Bichler, School of Engineering University Examiner Dr. Giovanni Grasselli, University of Toronto External Examiner iii Abstract Proppants are widely used in hydraulic fracturing for unconventional resources to increase the recovery rate of hydrocarbons in the oil and gas industry. More than 55 million tonnes of proppants were supplied for hydraulic fracturing in 2015 to fracture more than 35,000 wells. During hydraulic fracturing, proppants are transported by fracturing fluids into fractures to form a permeable proppant pack, which acts as a path for hydrocarbons to flow to wellbores. When high closure stresses exist, proppants can be crushed into smaller fragments and embed into fracture surfaces, which reduces the propped fracture conductivity by decreasing the proppant pack permeability and thickness. Choosing a suitable proppant for hydraulic fracturing is key to success in recovering hydrocarbons. Compared with time-consuming laboratory tests and simplified empirical/numerical models, the discrete element method (DEM) is a natural approach for simulating proppant crushing and embedment into fracture faces. However, current DEM simulations have not considered proppant crushing under high closure stresses. In addition, the mechanical properties of different proppants have not been fully characterised thus modelling of proppants in DEM lacks verification. Moreover, DEM simulations of proppant embedment into rock surface have not been calibrated/compared with laboratory hardness tests on rock faces. Finally, the influence of proppant crushing and embedment on proppant selection has not been thoroughly studied. With these in mind, this research presents comprehensive laboratory experiments on proppant crushing which were used to calibrate discrete element models of proppant crushing and proppant embedment. Three objectives are accomplished within this research: (1) characterize the mechanical behaviour of proppant crushing and proppant embedment into fracture surfaces; (2) model proppant crushing and proppant embedment with DEM; (3) study the influence of proppant crushing and proppant embedment on hydraulic fracture conductivity and recommend optimal proppant usage for hydraulic fracturing. The results deliver a better understanding of the crushing characteristics of different proppants, and improve DEM modelling of proppant crushing and embedment into hydraulic fractures, which provides a cost-efficient approach for evaluating the proppant performance for hydraulic fracturing without the need to conduct problematic hydraulic conductivity experiments. iv Preface This research was conducted under the supervision of Dr. Dwayne D. Tannant. Laboratory crush tests on proppants were performed by the author with the aid of two undergraduate students in the laboratory at School of Engineering, The University of British Columbia. Laboratory rock mechanics testing on Montney siltstone was designed by the author and was performed by the rock mechanics laboratory at Trican Well Service Inc. Ethics approval from the University of British Columbia Research Ethics Board was not required for this research. This dissertation is largely based on my work [J1, J2, J3, J4, J5, J6, C1, C2], for which I was the principal investigator and wrote the manuscripts under the supervision of Dr. Dwayne D. Tannant. Suyanne Costa Silva helped perform the crush tests on proppants in [J3]. Dr. Andre Phillion revised the content of [C1]. Dr. Melanie Grob revised the content of [C2]. The content of [J3, J6] appears in Chapter 2. The content of [J5, C1] appears in Chapter 2 and 6. The content of [J2, J4, C2] appears in Chapter 3 and 6. The content of [J1] appears in Chapter 4 and 5. Publication list Refereed Journal Articles [J1] Zheng, W., and Tannant, D.D. Laboratory tests and numerical simulations of proppant embedment into Montney Siltstone. International Journal of Rock Mechanics and Mining Sciences, in preparation. [J2] Zheng, W., and Tannant, D.D. Three-dimensional discrete element modelling of proppant crushing and influence on fracture conductivity. SPE Journal, in preparation. [J3] Zheng, W., Silva. C., and Tannant, D.D. Crushing characteristics of four different proppants and implication for fracture conductivity. Journal of Natural Gas Science and Engineering, under review. [J4] Zheng, W., and Tannant, D.D. (2018). Calibration of grain breakage criteria for discrete element models of sand crushing under one-dimensional compression. Computers and Geotechnics, 95: 231-239. [J5] Zheng, W., and Tannant, D.D. (2017). Improved estimate of the effective diameter for use in the Kozeny-Carman equation for permeability prediction. Géotechnique Letters, 7(1): 1-5. [J6] Zheng, W., and Tannant, D.D. (2016). Frac sand crushing characteristics and morphology changes under high compressive stress and implications for sand pack permeability. Canadian Geotechnical Journal, 53(9): 1412-1423. v Refereed Conference Proceedings [C1] Zheng, W., Tannant, D., and Phillion. A. (2016). Evaluation on hydraulic conductivity of frac sand pack under high compressive stress using PFC and Avizo. International Conference on Geo-Mechanics, Geo-Energy and Geo-Resources, Australia, 8 pages. [C2] Zheng, W., Tannant, D., and Grob, M. (2016). Discrete element modelling of frac sand crushing under one-dimensional compression using PFC2D. 69th Canadian Geotechnical Conference, Canada, 6 pages. vi Table of Contents Abstract ......................................................................................................................................... iii Preface ........................................................................................................................................... iv Table of Contents ......................................................................................................................... vi List of Tables ................................................................................................................................ ix List of Figures ............................................................................................................................... xi List of Abbreviations ............................................................................................................... xviii Acknowledgements .................................................................................................................... xix Dedication .....................................................................................................................................xx 1 Introduction ...............................................................................................................................1 1.1 Background and motivation .................................................................................................1 1.2 Literature review on state of the art .....................................................................................3 1.2.1 Laboratory testing ......................................................................................................3 1.2.2 Empirical and analytical analysis ..............................................................................4 1.2.3 Numerical simulation ................................................................................................4 1.3 Objectives and scope of the dissertation ..............................................................................6 2 Mechanical Properties of Proppants .....................................................................................10 2.1 Overview ............................................................................................................................10 2.2 Jordan formation sand crushing characteristics .................................................................11 2.2.1 Geology description ................................................................................................11 2.2.2 Grain size and microstructure ..................................................................................13 2.2.3 Test apparatus and procedure ..................................................................................16 2.2.4 Test results ...............................................................................................................20 2.3 Comparison with different types of proppants...................................................................28 2.3.1 Proppant types and sizes ..........................................................................................28 2.3.2 Grain shape characterization ...................................................................................30 2.3.3 Stress-strain response ..............................................................................................30 2.3.4 Post-test particle-size distribution ...........................................................................32 2.4 Implication for multilayer proppant pack permeability and fracture conductivity ............34 2.4.1 Kozeny-Carman equation for permeability estimation ...........................................34 2.4.2 Comparison between four proppants with three size ranges ...................................36 2.4.3 Evolution of permeability with compressive stresses .............................................40 vii 2.5 Summary ............................................................................................................................42 3 3D Modelling of Proppant Crushing in DEM ......................................................................45 3.1 Overview ............................................................................................................................45 3.2 Simulation of Jordan sand compaction without particle crushing .....................................46 3.3 Implementation of particle breakage model in DEM ........................................................51 3.3.1 Particle contact forces and particle stress ................................................................51 3.3.2 Particle stress indexes ..............................................................................................52 3.3.3 Particle strength .......................................................................................................53 3.3.4 Particle crushing criteria ..........................................................................................53 3.3.5 Implementation flow chart ......................................................................................54 3.4 Selection of crushing index for breakage model................................................................57 3.5 Determination of particle strength parameters for Jordan sand .........................................63 3.5.1 Preliminary determination of σ0 and m ...................................................................63 3.5.2 Final calibration of σ0 and m ...................................................................................65 3.6 Determination of particle strength parameters for three types of proppants .....................72 3.6.1 Genoa sand ..............................................................................................................72 3.6.2 Resin-coated sand ....................................................................................................75 3.6.3 Ceramic proppant ....................................................................................................77 3.7 Summary ............................................................................................................................79 4 Mechanical Properties of Montney Siltstone ........................................................................81 4.1 Overview ............................................................................................................................81 4.2 Montney siltstone core samples .........................................................................................83 4.3 Rock mechanics testing......................................................................................................84 4.3.1 Indirect tensile test ...................................................................................................84 4.3.2 Uniaxial compressive test ........................................................................................87 4.3.3 Triaxial compressive test .........................................................................................88 4.3.4 Comparison with laboratory test data on fresh cores ..............................................89 4.4 Brinell hardness test ...........................................................................................................95 4.5 Proppant embedment test .................................................................................................103 4.6 Summary ..........................................................................................................................114 5 3D Modelling of Proppant Embedment into Siltstone in DEM ........................................117 5.1 Overview ..........................................................................................................................117 5.2 DEM micro-parameters calibration of Montney siltstone ...............................................118 5.3 DEM Brinell indentation test ...........................................................................................124 viii 5.4 DEM proppant embedment test .......................................................................................130 5.5 Implication for optimal monolayer proppant ...................................................................140 5.6 Summary ..........................................................................................................................145 6 Influence of Proppant Crushing and Embedment on Fracture Conductivity and Proppant Selection ................................................................................................................147 6.1 Overview ..........................................................................................................................147 6.2 Validation of the Kozeny-Carman equation with flow simulation ..................................148 6.3 Parametric analysis via DEM...........................................................................................152 6.3.1 Simulations with proppant crushing ......................................................................152 6.3.2 Simulations with proppant embedment .................................................................164 6.3.3 Implications for proppant selection .......................................................................173 6.4 Case study ........................................................................................................................175 6.4.1 In-situ stress and hydraulic fracturing stimulation ................................................175 6.4.2 Fracture conductivity and hydrocarbon production ..............................................179 6.4.3 Optimal proppant selection and Return-On-Fracturing-Investment .....................181 6.5 Summary ..........................................................................................................................183 7 Conclusions and Recommendations ....................................................................................185 7.1 Conclusions ......................................................................................................................185 7.2 Recommendations ............................................................................................................188 Bibliography ...............................................................................................................................190 Appendices ..................................................................................................................................201 Appendix A: Micro-parameters Analysis of Crush Test in DEM ...........................................201 Appendix B: Laboratory Crush Test of Uniformly-graded Proppants ....................................207 B.1 Stress-strain response ................................................................................................207 B.2 Particle-size distribution ............................................................................................210 B.3 Grain strength follows Weibull statistic distribution? ...............................................211 Appendix C: Montney Siltstone ..............................................................................................215 Appendix D: Micro-parameters Analysis of Bonded Siltstone Models ..................................225 Appendix E: Mechanical Profile of Montney Formation ........................................................231 ix List of Tables Table 2-1. Grain size distribution of mesh 20/40 Jordan sand ..................................................... 13 Table 2-2. Relation between sieve size and mesh number ........................................................... 29 Table 2-3. Mean values of shape descriptors for proppants with different mesh sizes ................ 30 Table 2-4. Permeability estimation using Equations 2-5 (k) and 2-8 (k’) (at 40 MPa) ................. 37 Table 2-5. Parameters for permeability reduction estimation using empirical equations (D10 and Deff in unit mm) ................................................................................................. 41 Table 3-1. Value of micro-parameters for two levels ................................................................... 49 Table 3-2. Micro-parameters for DEM simulation of G600 Jordan sand ..................................... 50 Table 3-3. Micro-parameters for smaller particles to form clustered particle .............................. 57 Table 4-1. Indirect tensile test results ........................................................................................... 86 Table 4-2. Uniaxial compressive test results (orientation: horizontal) ......................................... 88 Table 4-3. Triaxial compressive test results (orientation: horizontal) .......................................... 89 Table 4-4. Uniaxial compressive test results for fresh cores (BC Oil and Gas Commission 2014a) ....................................................................................................................... 91 Table 4-5. Triaxial compressive test results for fresh cores (BC Oil and Gas Commission 2014a) ....................................................................................................................... 92 Table 4-6. Brinell indentation test results (horizontal, tested as received, 35 N) ......................... 99 Table 4-7. Average Brinell measurements of samples soaked in water/KCl ............................. 103 Table 4-8. Evolution of proppant-propped fracture aperture in embedment tests ...................... 106 Table 4-9. Permeability and fracture conductivity from proppant embedment tests at 40 MPa 108 Table 4-10. Interpretation of propped fracture aperture during embedment test (40 MPa) ....... 110 Table 4-11. Permeability and fracture conductivity based on proppant deformation and embedment at 40 MPa (Table 4-10)....................................................................... 114 Table 5-1. Value of micro-parameters for two levels ................................................................. 119 Table 5-2. Three sets of targeted macroscopic mechanical properties for DEM calibration ..... 121 Table 5-3. Summary of micro-parameters to represent three sets of macroscopic rock responses ................................................................................................................ 123 Table 5-4. Summary of proppants used in four DEM proppant embedment tests ..................... 131 Table 6-1. Comparison of permeability from flow simulation and the cubic law/Kozeny-Carman equation (L = 10 mm) ............................................................................... 152 Table 6-2. Brinell hardness values for different sphere sizes at different stress levels .............. 171 Table 6-3. Hydraulic fracturing multistage stimulation parameters ........................................... 177 x Table 6-4. Different proppant selection scenarios and resulting ROFIs after 1-year production .............................................................................................................. 182 Table A-1. Combination of different levels of micro-parameters for computation scenarios .... 201 Table A-2. Macroscopic outputs of the 32 scenarios .................................................................. 203 Table A-3. Estimated contribution of micro-parameters to outputs ........................................... 205 Table B-1. Constrained modulus of proppant packs (mean value ± standard deviation, unit: MPa) ....................................................................................................................... 208 Table B-2. Crushing percentage (cp) and survival probability (sp) of proppants under 40 MPa ................................................................................................................... 211 Table B-3. Parameters of G500 and G600 Jordan sand used in equation 10 .............................. 213 Table B-4. Parameters of uniformly-graded Jordan sand used in equation 11 (σv=40 MPa) ..... 214 Table C-1. RMT matrix for samples from archived core #1 ...................................................... 216 Table C-2. RMT matrix for samples from archived core #3 ...................................................... 217 Table C-3. RMT matrix for samples from archived core #4 ...................................................... 218 Table D-1. Combination of different levels of micro-parameters for computation scenarios .... 225 Table D-2. Results of 32 scenarios ............................................................................................. 227 Table D-3. Estimated contribution of micro-parameters to outputs ........................................... 229 xi List of Figures Figure 2-1. Frac sand mines and sandstone formations in Wisconsin and typical geological section (Modified from Wisconsin Geological and Natural History Survey 2014, and Runkel et al. 2012) .................................................................................. 12 Figure 2-2. Mesh 20/40 Jordan sand ............................................................................................. 13 Figure 2-3. SEM images of sand grains from different sieve sizes .............................................. 15 Figure 2-4. Steel cup and piston used for a one-dimensional compression test ........................... 16 Figure 2-5. Morphological properties of particle shape................................................................ 18 Figure 2-6. Shapes of sand particles (0.08-0.5 mm) along with their measured roundness and sphericity (each particle outline has been scaled to approximately the same size). ......................................................................................................................... 19 Figure 2-7. Morphological properties of particles with different shapes...................................... 19 Figure 2-8. Crushed and intact sand grains after 30 MPa (left) and 40 MPa (right) compression ............................................................................................................. 20 Figure 2-9. Grain size distributions of G500 and G600 sand under different stress levels .......... 21 Figure 2-10. Stress-strain response from five crush tests on G500 sand ...................................... 22 Figure 2-11. Compressive stress and AE data versus strain for G500 sand (Test 1) .................... 23 Figure 2-12. Stress-strain response from five crush tests on G600 sand ...................................... 24 Figure 2-13. Compressive stress and AE data versus strain for G600 sand (Test 2) .................... 25 Figure 2-14. Shapes of grains from different sieves (left to right: 0.5, 0.355, 0.25, and 0.125 mm) after a crushing test on G500 sand. ................................................................. 26 Figure 2-15. Mean values of geometry descriptors versus sieve size for G500 sand after 40 MPa compression ..................................................................................................... 27 Figure 2-16. Mean values of geometry descriptors versus sieve size for G600 sand after 40 MPa compression ..................................................................................................... 27 Figure 2-17. Four different types of proppants: Jordan sand (top left), Genoa sand (top right), resin-coated sand (bottom left) and ceramic proppant (bottom right) .......... 28 Figure 2-18. Particle-size distributions of three different sizes of four proppants ....................... 29 Figure 2-19. Stress-strain responses of four proppants with three different particle-size distributions .............................................................................................................. 31 Figure 2-20. One-dimensional constrained modulus of the proppant packs ................................ 32 Figure 2-21. Grain breakage after 40 MPa compression: Jordan sand (top left), Genoa sand (top right), resin-coated sand (bottom left) and ceramic proppant (bottom right) ... 33 Figure 2-22. Particle-size distributions of proppant packs with different mesh sizes under 40 MPa .......................................................................................................................... 34 xii Figure 2-23. Estimated permeability compared with conductivity test data (Preferred Sands 2016, Sea-Pacific Inc. 2016) .................................................................................... 38 Figure 2-24. Comparison of fracture conductivity estimation versus conductivity test data (Preferred Sands 2016, Hexion Specialty Chemicals 2010, Sea-Pacific Inc. 2016) ........................................................................................................................ 40 Figure 2-25. Permeability reduction ratio of G600 and G500 under different stresses ................ 42 Figure 3-1. Soft contact of two rigid particles (left) and rigid particle and wall (right) ............... 47 Figure 3-2. Liner contact model.................................................................................................... 47 Figure 3-3. Calculation cycle in PFC ............................................................................................ 47 Figure 3-4. Particles in cup under one-dimensional compression ................................................ 48 Figure 3-5. Stress-strain response of G600 sand compressed up to 40 MPa ................................ 51 Figure 3-6. Contact forces acting on a particle ............................................................................. 52 Figure 3-7. Replacement of an overloaded particle by smaller particles to represent crushing ... 54 Figure 3-8. Flowchart showing sample preparation and implementation of particle breakage model in DEM .......................................................................................................... 56 Figure 3-9. single particle (left), single clustered particle (middle) and contact force chain (right) between two platens in the crush test ............................................................ 58 Figure 3-10. Sudden drop in force accompanied by rapid increase in bond breakage number when a clustered particle fails under different contact number ............................... 58 Figure 3-11. single clustered particle and contact force chain using three platens (left 1 and 2) and four platens (left 3 and 4) in particle crush test............................................. 59 Figure 3-12. Schematic of clustered particle breakage caused by contacts with unbreakable particles and a wall (left) and the particle pack model in PFC3D (right) .................. 60 Figure 3-13. Typical contact force chain and bond breakage pattern for clustered particle with coordination number 3, 5, 7, and 9 (left to right) ............................................. 60 Figure 3-14. Relation between minimum and maximum principal stresses when a particle breaks from 234 simulations .................................................................................... 61 Figure 3-15. Distribution of the characteristic stress indexes for clustered particles when they broke ................................................................................................................. 62 Figure 3-16. Stress Index 1 distribution for 1554 particles tested to two different compressive stresses................................................................................................. 64 Figure 3-17. Void ratio vs compressive stress for G600 Jordan sand tested to 40 MPa .............. 65 Figure 3-18. Particle size distribution of particle pack using different σ0 (m = 3.5) after 40 MPa compression ................................................................................................ 66 Figure 3-19. Particle crushing of particle pack (m = 3.5) after 40 MPa compression with an increase of σ0: 31.5 MPa (top left), 35 MPa (top right), 37 MPa (bottom left) and 40 MPa (bottom right) ....................................................................................... 67 xiii Figure 3-20. Particle size distribution of particle pack using different m (σ0 = 35 MPa) after 40 MPa compression ................................................................................................ 68 Figure 3-21. Particle crushing of particle pack after 40 MPa compression with an increase of m: 1 (top left), 3 (top right), 5 (bottom left) and 7 (bottom right) (σ0 = 35.0 MPa) ......................................................................................................................... 69 Figure 3-22. Particle-size distribution with increase of σ0 (m = 3.5) (left) and increase of m (σ0 = 35.0 MPa) (right) ............................................................................................. 70 Figure 3-23. G600 stress-strain response with increase of σ0 (m = 3.5) (left) and increase of m (σ0 = 35.0 MPa) (right) ......................................................................................... 71 Figure 3-24. Stress-strain responses and particle-size distribution by PFC3D using σ0 = 35 MPa and m = 4 ..................................................................................................... 72 Figure 3-25. Void ratio vs compressive stress for G600 Genoa sand tested to 40 MPa ............... 73 Figure 3-26. Particle size distribution of Mesh 20/40 particle pack with different σ0 (m = 4.0) after 40 MPa compression ................................................................................ 74 Figure 3-27. Particle size distribution of Mesh 20/40 particle pack with different m (σ0 = 23 MPa) after 40 MPa compression ......................................................................... 74 Figure 3-28. Stress-strain responses of particle packs using σ0 = 23.0 MPa and m = 4.0 ............. 75 Figure 3-29. Particle size distribution of G600 particle pack using different σ0 (m = 4.0) after 40 MPa compression ................................................................................................ 76 Figure 3-30. Particle size distribution of mesh 20/40 particle pack (σ0 = 50 MPa and m = 4.0) after 40 MPa compression ................................................................................ 77 Figure 3-31. Stress-strain responses of particle packs using σ0 = 50.0 MPa and m = 4.0 ............. 77 Figure 3-32. Particle size distribution of G600 particle pack using different σ0 (m = 4.0) ........... 78 Figure 3-33. Particle size distribution of Mesh 20/40 particle pack (σ0 = 65 MPa and m = 4.0) ........................................................................................................................... 79 Figure 3-34. Stress-strain responses of particle packs using σ0 = 65.0 MPa and m = 4.0 ............. 79 Figure 4-1. Montney formation in Northeastern BC (BC Oil and Gas Commission 2012) ......... 81 Figure 4-2. Well HZ TOWN C-031-H/094-B-09 in Montney formation ..................................... 83 Figure 4-3. Loading orientation to bedding layer: perpendicular (left) and parallel (right) ......... 85 Figure 4-4. Failure pattern of sample loaded perpendicular (left) and loaded parallel to the bedding (right) .......................................................................................................... 87 Figure 4-5. Comparison of bulk density with depth between fresh cores and archived cores ..... 90 Figure 4-6. Comparison of failure patterns in uniaxial compressive tests: archived core – H (left); fresh core – TWN 4-11 H (middle); fresh core - TWN 4-1 V (right) (BC Oil and Gas Commission 2014a) ............................................................................. 93 xiv Figure 4-7. Comparison of failure patterns of samples in triaxial test with confining pressure 33 MPa: fresh core - TWN 3-13 H (left); fresh core – TWN 3-2 V (right) (BC Oil and Gas Commission 2014a) ............................................................................. 94 Figure 4-8. Plots of linear Mohr-Coulomb failure criteria in maximum - minimum principal stress space using peak strength ............................................................................... 94 Figure 4-9. Sample (left) and indenter (right) for Brinell hardness test ....................................... 95 Figure 4-10. Typical force-indentation curve from indentation tests on Montney siltstone ........ 96 Figure 4-11. Permanent indentation into surface after complete unloading ................................. 96 Figure 4-12. Correlation between Young’s modulus and Brinell hardness using indentation depth under different indentation loads.................................................................. 100 Figure 4-13. Force-indentation curves under conditions as received (top left), soaked in water (top right), and air dried (bottom left) at locations shown in (bottom right) for sample TWN2-05 ................................................................................... 101 Figure 4-14. Force-indentation curves under conditions as received (top left), soaked with 3% KCl (top right), and air dried (bottom left) at locations shown in (bottom right) for sample TWN2-07 ................................................................................... 102 Figure 4-15. Intact sample for elastic triaxial test (left); sample split to two halves for proppant embedment test (middle); monolayer of proppants between two split halves (right) .......................................................................................................... 104 Figure 4-16. Axial displacement of sample TWN2-15 with/without proppants during isostatic (left) and triaxial testing (right) ................................................................ 105 Figure 4-17. Proppant distribution of TWN2-15 (left: uneven) and TWN3-12 (right: even) .... 107 Figure 4-18. Proppant particles in four TWN samples after proppant embedment tests: G500 Jordan sand (top left), G250 Jordan sand (top right), G500 ceramic proppant (bottom left), and G250 ceramic proppant (bottom right) ..................................... 107 Figure 4-19. Reduction in propped fracture aperture caused by proppant deformation and embedment ............................................................................................................. 111 Figure 4-20. Fracture surface after proppant embedment test (TWN2-13) ................................ 112 Figure 5-1. Sample subjected to uniaxial compressive test (left), triaxial compressive test (middle), and indirect tensile test (right) ................................................................ 119 Figure 5-2. Micro-parameters calibration process: Poisson’s ratio (top left), Young’s modulus (top right), UCS (bottom left), and triaxial compressive strength (bottom right) ......................................................................................................... 122 Figure 5-3. Cuboid sample subjected to Brinell indentation test with a spherical indenter ....... 124 Figure 5-4. Indentation force-depth curve using different particle sizes .................................... 125 Figure 5-5. Correlation between indentation depth at 35 N and Young’s modulus, UCS and Poisson’s ratio from 32 numerical scenarios ......................................................... 126 Figure 5-6. Correlation between Brinell and Young’s modulus ................................................. 127 xv Figure 5-7. Indentation force-depth curve using three sets of micro-parameters ....................... 128 Figure 5-8. Parallel bond breakage pattern of set #1, #2, and #3 (left to right) .......................... 128 Figure 5-9. Brinell indentation test loaded to peak using new set # 3 (X+ direction) ................ 129 Figure 5-10. Proppants (blue colour) sandwiched between two rock blocks (green colour) in DEM proppant embedment test ............................................................................. 130 Figure 5-11. Change in sample length with axial stress for simulated tests ............................... 132 Figure 5-12. Change in fracture aperture with increase in axial stress for four tests ................. 133 Figure 5-13. Schematic view of proppant embedment in DEM proppant embedment test (top: X front view; bottom: Isometric view) .......................................................... 134 Figure 5-14. TWN 3-10: distribution of parallel bonds broke in tension (green) and shear (red) (top row); contact force chain in tension (red) (bottom row) ........................ 135 Figure 5-15. Comparison of parallel bonds broke in tension (green) and shear (red) for four samples: TWN 3-10 (top left), TWN 3-12 (top right), TWN 3-17 (bottom left) and TWN 3-19 (bottom right) ................................................................................ 136 Figure 5-16. Comparison of contact force chain in tension (red): TWN 3-10 (top left), TWN 3-12 (top right), TWN 3-17 (bottom left) and TWN 3-19 (bottom right).............. 137 Figure 5-17. Distributions of octahedral particle stress for four simulations ............................. 138 Figure 5-18. Relation between indentation force and particle stress .......................................... 139 Figure 5-19. Relation between indentation force/D2 and octahedral particle stress ................... 140 Figure 5-20. Relation between particle diameter and particle stress/strength and indentation force........................................................................................................................ 141 Figure 5-21. Proppant particles at 0 MPa (left) and 40 MPa (right) ........................................... 141 Figure 5-22. Assumed packing of monolayer proppants ............................................................ 142 Figure 5-23. Optimal particle diameter under different compressive stress for fully-filled monolayer of proppants.......................................................................................... 145 Figure 6-1. Flow boundary condition for permeability estimation ............................................. 149 Figure 6-2. Four geometries studied with flow simulation: parallel fractures (top left), monolayer (top right), three layers (bottom left), and eight layers (bottom right). 150 Figure 6-3. Flow velocity distribution of four geometries studied with flow simulation ........... 151 Figure 6-4. Particle pack loaded in an 8×8 mm rectangular-shape box .................................... 153 Figure 6-5. Results of Jordan mesh 20/40 particles loaded to 60 MPa (concentration 9.765 kg/m2) ..................................................................................................................... 154 Figure 6-6. Results of Jordan particles with different sizes loaded to 60 MPa (9.765 kg/m2) ... 155 Figure 6-7. Results of Genoa particles with different sizes loaded to 60 MPa (9.765 kg/m2) ... 157 Figure 6-8. Results of ceramic particles with different sizes loaded to 60 MPa (9.765 kg/m2) . 159 xvi Figure 6-9. Comparison of highest permeability (left) and fracture conductivity (right) for three types of proppants (9.765 kg/m2) .................................................................. 160 Figure 6-10. Results of mesh 30/50 particles with different concentrations loaded to 60 MPa . 162 Figure 6-11. Results of a blend of G250 and G500 particles with different portations loaded to 60 MPa ............................................................................................................... 163 Figure 6-12. Proppant embedment simulation setup .................................................................. 165 Figure 6-13. Proppant embedment versus axial stress for cores with different Young’s modulus .................................................................................................................. 166 Figure 6-14. Rock face topology (unit mm) indented by 0.6 mm spheres at 60 MPa with different effective particle-particle modulus .......................................................... 166 Figure 6-15. Correlation between particle embedment and rock Young’s modulus at different stresses ..................................................................................................... 168 Figure 6-16. Correlation between hardness and rock Young’s modulus at different stresses .... 168 Figure 6-17. Proppant embedment versus axial stress for cores with different sphere sizes (effective particle-particle modulus = 0.5 GPa) ..................................................... 169 Figure 6-18. Rock face topology (unit mm) at 60 MPa indented by different stiff sphere diameters with effective particle-particle modulus = 0.5 GPa ............................... 170 Figure 6-19. Values of a and b under different compressive stresses and correlations .............. 172 Figure 6-20. Relation between Brinell hardness and Young’s modulus and compressive stress ....................................................................................................................... 173 Figure 6-21. Profiles of in-situ pore pressure (blue), overburden or vertical stress (black) and the minimum horizontal stress (red) ...................................................................... 175 Figure 6-22. Layout of perforation clusters and hydraulic fracturing stages .............................. 176 Figure 6-23. Geometry of fractures and proppant packs after stimulation ................................. 178 Figure 6-24. Proppant particle size distributions and a mixture compressed between two rock faces ............................................................................................................... 180 Figure 6-25. Cumulative gas production versus production time ............................................... 181 Figure B-1. Stress-strain response of Jordan sand of single size range plotted from 0.5 MPa... 207 Figure B-2. Stress-strain response of four proppants of single size range after 40 MPa ............ 208 Figure B-3. Particle-size distribution of four proppants of single size range after 40 MPa ....... 210 Figure B-4. Weibull modulus and characteristic stress for Jordan G500 and G600 .................. 213 Figure C-1. Core box 2265.29 m to 2266.44 m (left) and 2266.44 m to 2267.55 m (right) ....... 215 Figure C-2. Axial stress difference versus radial and axial strains, measured during uniaxial compressive test (TWN1-01) ................................................................................. 219 Figure C-3. Pre test and post test of TWN1-01 .......................................................................... 219 xvii Figure C-4. Axial stress difference versus radial and axial strains, measured during uniaxial compressive test (TWN2-04) ................................................................................. 220 Figure C-5. Pre test and post test of TWN2-04 .......................................................................... 220 Figure C-6. Axial stress difference versus radial and axial strains, measured during uniaxial compressive test (TWN3-04) ................................................................................. 221 Figure C-7. Pre test and post test of TWN3-04 .......................................................................... 221 Figure C-8. Axial stress difference versus radial and axial strains, measured during triaxial compressive tests with confining stress 10 MPa (TWN1-03) and 20 MPa (TWN1-08) ............................................................................................................. 222 Figure C-9. Pre test and post test of TWN1-03 (10 MPa confinement) ..................................... 222 Figure C-10. Pre test and post test of TWN1-08 (20 MPa confinement) ................................... 222 Figure C-11. Axial stress difference versus radial and axial strains, measured during triaxial compressive tests with confining stress 10 MPa (TWN2-06) and 20 MPa (TWN2-08) ............................................................................................................. 223 Figure C-12. Pre test and post test of TWN2-06 (10 MPa confinement) ................................... 223 Figure C-13. Pre test and post test of TWN2-08 (20 MPa confinement) ................................... 223 Figure C-14. Axial stress difference versus radial and axial strains, measured during triaxial compressive tests with confining stress 10 MPa (TWN3-06) and 20 MPa (TWN3-08) ............................................................................................................. 224 Figure C-15. Pre test and post test of TWN3-06 (10 MPa confinement) ................................... 224 Figure C-16. Pre test and post test of TWN3-08 (20 MPa confinement) ................................... 224 Figure E-1. Well logs and mechanical profiling of the Montney formation from Well 28232. . 231 Figure E-2. Correlation between dynamic Young’s modulus and clay fraction ......................... 233 Figure E-3. Correlation between bulk density and static Young’s modulus (left) and Poisson’s ratio (right) ............................................................................................. 234 Figure E-4. Static Poisson’s ratio and Young’s modulus derived from bulk density log ........... 234 xviii List of Abbreviations 3D Three-Dimensional AE Acoustic Emission AR Aspect Ratio BC British Columbia DEM Discrete Element Method LBM Lattice Boltzmann Method MaxFeret Maximum Feret Diameter MD Measured Depth MinFeret Minimum Feret Diameter R Roundness RCA Routine Rock Analysis RMT Rock Mechanics Testing ROFI Return-on-Fracturing-Investment S Sphericity SEM Scanning Electron Microscope TVD True Vertical Depth UCS Uniaxial Compressive Strength xix Acknowledgements I would like to express my most sincere gratitude to Dr. Dwayne D. Tannant for providing guidance and insight throughout this research. This research would not be possible without his support and encouragement. Not only did I benefit from his professional integrity and clarity in thinking, but also his open mind has guided me to view things from different perspectives. It was a pleasure to work with such a professional and experienced advisor. I would like to thank my research committee members, Drs. Sumi Siddiqua and Hadi Mohammadi, for insights on this research. I would like to thank Drs. Giovanni Grasselli and Lukas Bichler for being willing to exam my thesis. Thanks also go to Dr. Jonathan Holzman for chairing the defense. I would also like to thank Drs. Andre Phillion and Maurice Dusseault for helpful comments. Thanks are owed to the faculty and staff at the School of Engineering for providing research spaces and access to the laboratory facilities. I also want to express my gratitude to Trican Well Service Inc. for conducting laboratory rock mechanics testing and to Itasca International Inc. for providing PFC through its educational program. Special thanks are given to Dr. Albert Cui in Trican and Dr. Melanie Grob in Itasca for their invaluable suggestions and mentorship that broadened my understanding of this research. I would also like to thank BC Oil and Gas Commission for providing Montney cores for testing and John Nurkowski for helpful discussions. The financial support from Chinese Scholarship Council, The University of British Columbia, BC Oil and Gas Commission, and Mitacs is truly appreciated. I am thankful to my group members for encouragement: Drs. Samuel Nunoo and Houquan Zhang, Cong Xu, Kai Zhang, Yi Qi, Josephine Morgenroth, Umair Asghar, Suyanne Silva, Drs. Xinli Hu, Peng Zhang, and Haizheng Wu, Weisheng Du, Dahai Wang, Kalie Smith, Bingqian Yan, Yuzhong Li, and Chuncan He. I am also thankful to fellow graduate students for sharing this journey: Xian Jin, Xuan Gao, Xuegui Song, Taylor Liu, Luanxia Yang, Liu Niu, Qi Zhang, Fang Fang, Qiang Li, Alamgir Usmani, Gyan Shrestha, Peng Zhang, Guoyu Zhang, and Ying Yi. I would also like to thank Arthur & Pearl Lo, Barb Vassilopoulos, and Maria Lenis for their friendship and kindness. Special thanks are owed to my wife, my parents, and my brothers and sisters for all their love, patience, and encouragement. Without their support, none of my achievements would be possible. xx Dedication This thesis is dedicated to my little one Claire Nan Zheng. 1 1 Introduction 1.1 Background and motivation Proppants are widely used in hydraulic fracturing for unconventional resources to increase the recovery rate of hydrocarbons in the oil and gas industry. During the hydraulic fracturing process, a mixture of water, proppants, and other chemical additives designed to enhance production is pumped under high pressure into a low-permeability formation to create fractures. Proppants are designed to stay in the newly created fractures to prop them open, which then act as a path for oil and gas to flow back to the well surface under reservoir pressure. Proppants used for hydraulic fracturing in tight shale formations mainly consist of quartz sand, resin-coated sand, or ceramic beads. More than 55 million tons of proppants were supplied for hydraulic fracturing in 2015, of which more than 90% of the proppants used originated as quartz sand (PropTester and Kelrik 2016). Hydraulic fracturing has been used on over 1 million producing wells in North America, and it is used to fracture 35,000 wells each year according to FracFocus (FracFocus 2015). In Western Canada, over 175,000 wells have been stimulated using hydraulic fracturing (Canadian Association of Petroleum Producers 2012). Typically, the usage of proppants for one well is more than 1,000,000 kg, and the loading of proppants in fracturing fluids is over 120 kg/m3 (Patel et al. 2014). Typically, hydraulic fractures in tight shale formations are created several kilometres beneath the ground surface. Fracturing fluids transport proppants to newly created fractures and a proppant pack is then formed within fractures. After hydraulic fracturing, stress from the surrounding formations will act on the hydraulic fractures. This stress is called the closure stress, which tends to reduce the aperture of fractures. A proppant pack is designed to resist the closure stress and prop open fractures, and act as paths for oil and gas to flow. The closure stress for different shale formations varies but is typically larger than 30 MPa, e.g. 60 MPa to 80 MPa for Haynesville Shale, less than 40 MPa for Fayetteville Shale, and roughly 40 MPa for Montney Siltstone (Song and Hareland 2012). The high closure stress acting on proppants within hydraulic fractures often crushes proppant grains into smaller fragments (Palisch et al. 2009). For example, crushed sand was observed in 2 flow back tests from Montney formation under a closure stress of 42.3 MPa (Romanson et al. 2010). Proppant crushing allows hydraulic fractures to close thus reducing the fracture aperture. Furthermore, the presence of small fragments of crushed sand decreases the permeability or hydraulic conductivity of the proppant pack because they can block natural gas flow paths. Coulter et al. (1972) and Lacy et al. (1998) both found that 5% fines could cause an over 50% reduction in the proppant pack conductivity, which is defined as the proppant pack permeability times the fracture aperture. A study of the Bakken Formation showed that the permeability of a ceramic proppant pack was five times higher than that of Ottawa sand under approximately 50 MPa closure stress. The higher permeability was attributed to the higher crushing resistance of the ceramic proppants (Kurz et al. 2013). Under such a high closure stress between the newly created fractures, another important issue affecting the fracture conductivity is proppant embedment into rock fracture surfaces. Laboratory conductivity testing using 20/40 Ottawa sand proppants sandwiched between Ohio sandstone (Lacy et al. 1998) shows that half of the grain diameter was embedded into the formation at a compression stress of 3.4 MPa and 3-4 grain diameters embedment occur under 34.5 MPa. Similar laboratory testing was performed using mesh 40/70 Badger sand proppants sandwiched between two Barnett shale cores (Zhang et al. 2015). They found that half of the proppant grains were embedded into each shale core under a compression stress of 27.6 MPa. Some proppants were completely buried into the fracture faces as the shale stiffness was softened by water. The embedment of proppants into the shale cores caused the fracture conductivity to reduce by 88% from 4500 µm3 (15 md-ft) to 530 µm3 (1.77 md-ft). When a monolayer proppant pack exists between the fractures because of a low proppant concentration, a small decrease in fracture aperture caused by proppant embedment can cause a large decrease in pore space, thus results in a severe conductivity reduction. A comprehensive understanding of proppant crushing and proppant embedment into hydraulic fractures and its influence on fracture conductivity will be helpful for optimising proppant usage in hydraulic fracturing, which is the motivation of this research. 3 1.2 Literature review on state of the art Proppant packs formed between hydraulic fractures are very complicated involving mechanical, chemical, biological, thermal, and hydraulic factors. This study only will only focus on the proppant crushing and embedment between fractures. The creep deformation of proppants and fracture walls are not considered in this research. The following sections summarise state of the art on proppant crushing and embedment. 1.2.1 Laboratory testing In practice, two ISO standards are widely used in the laboratory to evaluate the performance of proppants. The ISO 13503-2 standard (2006) measures the particle-size distribution, grain shape, and turbidity. This test standard also includes a crush test to evaluate the crushing resistance of proppants by measuring the percentage of fines generated under a specific compressive stress. The ISO 13502-5 standard (2006) uses a conductivity cell with proppants embedded between two fracture faces through which fluid flows to measure the permeability and fracture conductivity. Typically, this test lasts for more than 50 hours. For different formations under different closure stresses, multiple tests requiring specialised equipment are needed to select an optimal proppant (Rivers et al., 2012). Most proppant tests using these two standards have focused on measuring the percentage of fines generated (Palisch et al. 2009), and the proppant pack permeability and fracture conductivity (Cutler et al. 1985, Ye et al. 2009, Zhang et al. 2013, Zhang et al. 2015). Most of the proppant suppliers provide the data of fines and fracture conductivity under a specific stress. Details on the proppant crushing behaviours under compression such as the proppant stress-strain response and the variation in grain size distribution are rarely published. To the author’s knowledge, only Gaurav et al. (2012) compared the strength and crushing behaviour of three ultra-light-weight proppants using the crush test. A detailed characterization of proppant crushing behaviours under compression will be helpful for proppant selection for hydraulic fracturing. Large specialised testing equipment and long testing time are required for most proppant testing, which can lead to high testing costs. In addition, results generated from laboratory experiments may not be valid once the boundary conditions such as stress regime and proppant loading change. Artificial downhole fractures are quite different from laboratory experiments. For example, they 4 have non-uniform proppant distribution along fracture length and width, filter cake, and non-Darcy flow (Zhou et al. 2011). Furthermore, it is difficult to quantify experiment results and connect them to reservoir simulation making it difficult to evaluate the effect of the decrease in fracture conductivity on reservoir production. Fines migration within proppant packs depends on proppant loading, which cannot be well correlated to changes in fracture conductivity (Sondergeld et al. 2013). 1.2.2 Empirical and analytical analysis Some empirical or analytical models have been developed for the embedment of proppants into the rock formation. For these models, proppant crushing is typically not considered. An empirical expression for high strength sintered bauxite embedding into reservoir rock cores was proposed by Volk et al. (1981) based on embedment experiments with varying closure pressure, proppant size and size distribution, and proppant concentration. The proppants were assumed unbreakable because of their high crushing resistance. Guo and Liu (2012) used an analytical model for investigating the effect of rock characteristics and reservoir parameters on embedment. In their model, embedment was considered as elastic deformation and creep of the formation rock. Elastic deformation of the formation rock under proppant contact forces was calculated as distributed load over the semi-infinite solid, while a Maxwell viscoelastic model was used to describe creep deformation of the rock. Proppants were assumed as rigid and unbreakable in this model. The permeability was estimated by the Kozeny-Carman equation based on pore volume variations resulting from proppant deformation and embedment into fractures. Usually, empirical models are derived from experimental results, which may only apply to certain situations. Analytical models are theoretically valid but often include assumptions such as unbreakable proppants that are too simplified. 1.2.3 Numerical simulation Other researchers have used numerical tools to study the proppant-rock interaction. Mueller and Amro (2015) used a numerical model to study the effect of rock hardness on proppant embedment. Proppants were considered rigid, and embedment was assumed to happen in the formation only. The embedment under a certain load was linked with the contact area given by rock hardness. 5 A simple mathematical model was developed (Neto and Kotousov 2013, Neto et al. 2015) to study the performance of hydraulic fractures partially filled with a compressible proppant pack. Proppant compressibility was simplified by assuming a Terzaghi soil consolidation model, and the Kozeny-Carmen equation was used to predict the fracture permeability. A simple fluid flow model was utilised to quantify the productivity index of the well. A geomechanics model coupled with the fluid flow was constructed in finite element software Comsol (Han and Wang 2014, Han et al. 2014) to perform analysis of proppant pack failures and fracture conductivity decreases. Flow into the poroelastic proppant pack was governed by conservation of mass and momentum as well as Darcy’s law. The proppant deformation and crushing were described with the Mohr-Coulomb constitutive model, and the permeability was linked with the porosity using the Kozeny-Carmen equation. Reinicke et al. (2012) used a finite element software package to calculate the minimum principal stress at the contact between a proppant and the rock fracture surface. High tensile stress concentrations around the contact areas were observed. The stress distribution can be correlated with the breakage pattern of rock quartz grains and proppants. Likewise, ABAQUS was used to study lightweight proppants under compression and flow back (Kulkarni and Ochoa 2012). Simplifications adopted in some numerical models might not be valid such as unbreakable proppants. In addition, proppant crushing and proppant embedment into hydraulic fractures happen on a length scale of granular particles. Macroscopic continuum-based approaches with the Mohr-Coulomb constitutive model might not be adequate to simulate proppant pack failures considering proppant packs consists of only several layers of proppant grains. The discrete element method (DEM) has been widely used in simulating geomechanical problems, and it is particularly suitable for modelling granular materials. In DEM, the domain of interest is treated as an assemblage of rigid or deformable blocks/particles, and contacts between blocks/particles keep updated explicitly to represent the deformation or movement of the domain. Newton’s second law is used to determine the motion of each block/particle arising from the contact and body forces acting upon it, while the force-displacement law is used to update the contact forces arising from the relative motion at each contact. With an explicit time-stepping 6 algorithm, calculations in DEM are cycled between Newton’s second law to the blocks/particles and a force-displacement law at the contacts (Cundall and Strack 1979). DEM was applied to study the mechanical stability of proppants in hydraulic fractures during flow back, and the arch failure of proppants was observed (Asgian et al. 1995). DEM was combined with the lattice Boltzmann method (LBM) to investigate the conductivity of proppant packs under variable confining stresses (Mollanouri Shamsi et al. 2015). Proppant pack compaction was simulated with DEM, and the permeability of proppant pack was calculated from LBM. Three different proppant packs with different size gradations were used to generate loose packing of rigid spherical particles. The simulation results by this approach were in good agreement with permeability values measured from experiments and estimated from the Kozeny-Carman equation. DEM was also utilised to study the interaction between shale cores and proppants as shown in to evaluate the fracture aperture under different proppant sizes, Young’s moduli and stress levels (Deng et al. 2014). The DEM results showed that larger embedment of proppants into shale happened with softer shale and higher pressure, which lead to smaller fracture aperture and larger plasticity in the shale. Further investigation (Mattson et al. 2014) showed the result of DEM embedment modelling was consistent with experimental results. Zhang et al. (2017) integrated DEM with computational fluid dynamics to estimate fracture conductivity after proppant embedment. The simulated results were in good agreement with the laboratory conductivity tests as well as the predictions of the Kozeny-Carman equation. A series of numerical simulations were performed, and the results show that the fracture conductivity increases with the increase of proppant loading or proppant particle size, and decreases with the increase of fracture-closing stress or degree of shale hydration. 1.3 Objectives and scope of the dissertation DEM is a suitable approach for simulating proppant crushing and embedment into fracture surfaces. However, current DEM simulations have not considered proppant crushing under high compressive stresses. In addition, the mechanical properties of different proppants have not been fully characterised thus modelling of proppants in DEM lacks verification. Moreover, proppant 7 embedment into rock surface has not been calibrated/compared with experimental hardness tests on rock fractures. This research presents laboratory test results on proppant crushing. These results were used to develop and calibrate discrete element models of proppant crushing and proppant embedment. The objectives of this research are to: (1) understand the mechanical behaviours of proppant crushing and proppant embedment into hydraulic fractures, (2) model proppant crushing and proppant embedment with a discrete element method, and (3) study the influence of proppant crushing and proppant embedment on hydraulic fracture conductivity, which in turn can be used to guide the selection of appropriate proppant types and size distributions to optimize proppant usage for hydraulic fracturing. These objectives are accomplished by the following tasks: (1) characterize four different types of proppants using crush tests, (2) develop a breakage model in DEM to simulate proppant crushing and calibrate DEM micro-parameters to represent different proppants, (3) characterize a Montney siltstone against embedment using Brinell hardness tests and proppant embedment tests, (4) calibrate DEM micro-parameters to represent Montney siltstone and explore the micromechanics of Brinell hardness tests and proppant embedment tests, and (5) evaluate the influence of proppant crushing and embedment on hydraulic fracture conductivity and proppant selection. The structure of this dissertation is organised in accordance with the abovementioned tasks. An overview and a summary are included in each chapter from Chapter 2 to 6. The content of each chapter is described below. Chapter 1 presents the background and motivation of this research. Details are given on the research progress on proppant crushing and proppant embedment. The objectives and the scope of the dissertation are also clarified. Chapter 2 presents a characterization of proppant mechanical properties under compression. Jordan frac sand from Wisconsin was tested with a revised crush test standard. The grain shape, grain breakage pattern, stress-strain response of the proppant pack, and the change in particle-size distribution caused by compressive stresses of up to 40 MPa were measured. Following the same methodology, Jordan sand was further compared with three other sources/types of proppants with 8 three different size ranges that are widely used for hydraulic fracturing in North America. The proppant pack permeability and fracture conductivity interpreted from the crush test are compared. Chapter 3 presents the development of a particle breakage model in three-dimensional (3D) DEM that is capable of simulating proppant crushing under compression. Three existing crushing criteria are evaluated from a micro-scale approach using clustered balls to simulate grain breakage in DEM. A suitable crushing criterion is selected to use in the particle breakage model, and a DEM macro-scale calibration is performed to determine the elastic stiffness and strength parameters for the four proppants by matching both the grain size distribution and stress-strain response. Chapter 4 presents results from comprehensive rock mechanics tests on siltstone from the Montney formation of northeastern British Columbia. The mechanical behaviours of the siltstone were characterised using indirect tensile tests, uniaxial compressive tests, and triaxial compressive tests. The resistance of the siltstone to indentation was investigated with Brinell indentation tests, and the obtained Brinell hardness was correlated to the mechanical properties of the siltstone. A proppant embedment test was conducted to investigate the mechanical response of a monolayer proppant pack sandwiched between two siltstone cores under a triaxial loading. Chapter 5 presents a DEM modelling of proppant embedment into rock faces. A sensitivity analysis based on 2k-1 factorial design was performed to determine the micro-parameters in DEM to represent the Montney siltstone. The defined micro-parameters are used to reduplicate the Brinell hardness test in DEM, and the result is compared with the laboratory data. DEM simulations of proppant embedment tests are performed using the calibrated DEM micro-parameters for both the proppants and the Montney siltstone. Insights into micromechanics such as the particle stress distribution are gained. Comparison of the obtained particle stress with the particle strength established in Chapter 3 provides a means to optimise the proppant size used in monolayer proppant packs. Chapter 6 presents the influence of proppant crushing and embedment on the hydraulic fracture conductivity. These results can be used to guide appropriate proppant selection. The applicability of the Kozeny-Carman equation to estimate the permeability of a fracture filled with particles is evaluated against numerical flow simulations. A series of DEM simulations with various proppant types and particle sizes as well as different formation mechanical properties is conducted, and the 9 resulting fracture conductivity is compared. The knowledge gained is used to choose an optimal proppant in the case study of the investigated Montney siltstone. Chapter 7 summarizes the conclusions of this dissertation, as well as recommendations for future work. 10 2 Mechanical Properties of Proppants 2.1 Overview Frac sand is the main type of proppant used for hydraulic fracturing in the oil and gas industry. The North American market consumes more than 80% of the world’s supply of frac sand, and nearly 50% of this frac sand is mined in Wisconsin (Benson and Wilson 2015). The widely used Jordan formation sand from Wisconsin is used for this study. Sand crushing was noted by Terzaghi and Peck (1948) in one-dimensional compression tests on sands subjected to 95.6 MPa. Many studies have quantified a change in the grain size distribution due to crushing (Lade et al. 1996; Zhang and Baudet 2013) and its effect on macroscopic properties of the sand, e.g., density, void ratio, friction angle, critical state line, etc. (Chuhan et al. 2002, Luzzani and Coop 2002, Coop 2004, Ghafghazi et al. 2014). A single-particle crush test was used to measure the crushing characteristics and strength statistics of quartz, orthoclase, and plagioclase sand grains (Nakata et al. 1999). Sand crushing changes the shape of sand grains, which affects the mechanical behaviour of sand. The term ‘morphology’ of sand is used to describe the external size and shape of the grains and fragments (Blott and Pye 2008). The morphology of sand grains affects its density, void ratio, shear response, stiffness, and coefficient of lateral stress (Yimsiri and Soga 1999, Latham et al. 2002, Cho et al. 2006, Guo and Stolle 2006, Tsomokos and Georgiannou 2010). Sand crushing and the change in grain shape and grain size distribution reduce the sand pack permeability by decreasing the internal pore space and altering the pore shape. Sand with irregular-shape grains was shown to be 1.6~1.8 times less permeable than sand with round grains under gravity packing in a numerical study using finite element flow simulation through internal pore structures (Garcia et al. 2009). A crush test, or a one-dimensional compression test, has been widely used to measure the crushing resistance of proppants for hydraulic fracturing. However, most proppant testing has focused on measuring the percentage of fines generated (Palisch et al. 2009). Details on the crushing behaviour of proppants under compression are rarely published. To the author’s knowledge, only Gaurav et al. (2012) compared the strength and crushing behaviour of three ultra-light-weight proppants using the crush test. Further investigation of the crushing behaviour of different proppants is needed. 11 In this chapter, Jordan frac sand from Wisconsin was tested with a crush test apparatus. The grain shape, grain breakage pattern, stress-strain response of the proppant pack, and the change in particle-size distribution caused by compressive stresses of up to 40 MPa were measured. Following the same methodology, Jordan sand was further compared with three other sources/types of proppants with three different size ranges that are widely used for hydraulic fracturing in North America. The measurements from crush tests were used to empirically estimate and compare the permeability and fracture conductivity for different proppant packs without the need to conduct elaborate conductivity tests. 2.2 Jordan formation sand crushing characteristics 2.2.1 Geology description In Wisconsin and Minnesota, frac sand is mined from four sandstone formations, namely the Middle and Upper Ordovician St. Peter formation, the Upper Cambrian and Lower Ordovician Jordan formation, the Cambrian Upper Wonewoc formation, and the Cambrian Mount Simon formation. Figure 2-1 shows the distribution of these sandstone formations and the active/proposed frac sand mines and processing plants in Wisconsin. Among the four source formations, the Van Oser Member of Jordan formation has the highest percentage of desirable grain sizes (i.e., over 45% larger than mesh 40 by weight) and the lowest percentage of less desirable grain sizes (i.e., about 5% smaller than mesh 100 by weight) for hydraulic fracturing (Ostrom 1971). Jordan formation is a marine sandstone that was deposited 500 million years ago. Outcrops of this formation were first described in streams and quarries near Jordan City, Minnesota by Winchell (1874). This sandstone was extensively studied by Runkel et al. (1994a, 1994b, 2000, 2012). Chemical weathering is believed to have induced selective diagenetic leaching of plagioclase grains and other unstable minerals from the source rock for the sand that was eventually deposited. This created a mineralogically mature source area dominated by quartz grains (Odom 1975, 1978). The sand was then carried by wind and rivers to shorelines located in Wisconsin, Minnesota and Northern Iowa. Abrasion in marine conditions, as well as wind abrasion, jointly created the textural maturity of the sand. 12 Figure 2-1. Frac sand mines and sandstone formations in Wisconsin and typical geological section (Modified from Wisconsin Geological and Natural History Survey 2014, and Runkel et al. 2012) Two quartzose sandstone members are included in the Jordan formation. The uppermost Van Oser Member consists of fine to medium-grained, cross-stratified, well-sorted, quartzose sandstone with 9 to 15 m thickness. The underlying Norwalk Member is a fine-grained, moderately sorted feldspathic sandstone with a thickness of 15-18 m (Mudrey et al. 1987). The Van Oser is interpreted as a higher energy, marine intertidal sand while the Norwalk is as a low-energy, below wave base sand (Brown 2012). Jordan formation is poorly cemented but does have some cement-like calcite, which may locally create tightly cemented aggregates, so blasting is sometimes used to make the sandstone more amenable to excavation. After excavation, sandstone is crushed to recover sand grains, which are then washed to remove fine particles such as silts and clays. The sand is then sent to a dryer and screening plant to sort the sand into desired sizes for hydraulic fracturing (Wisconsin Department of Natural Resources 2012). Jordan sand used in this research was washed in a processing plant near Plum City, Wisconsin, and then shipped to a drying and screening plant near Seven Persons, Alberta, for final processing. CarbonateVery fine-grained, feldspathicsandstone, siltstone, shaleFine- to coarse-grained quartzosesandstone20-36 m9-24 m15-45 m30-120 mFrac sand minesand processing plantsSandstone formations0 20 40 60 kilometres 0 20 40 60 miles 13 2.2.2 Grain size and microstructure One of the most popular sizes of frac sand, mesh 20/40 as shown in Figure 2-2, is used for this study. Mesh 20 and 40 is equivalent to 0.85 mm and 0.425 mm respectively. A mesh 20/40 sand requires that more than 90% of the sand grains by weight be retained between 0.85 mm and 0.425 mm. A visual examination of the sand shows that most grains appear round and clean although some of the grain surfaces have a yellowish tone that may indicate a thin oxide coating. There are also occasional grains consisting of an agglomeration of smaller particles. Figure 2-2. Mesh 20/40 Jordan sand The particle size distribution of the 20/40 mesh size sand as received from the supplier was determined following the ASTM D6913 (2009) sieving standard. A 100 g representative sample of the sand under an air-dried condition was obtained and sieved with a 200 mm diameter metric sieve set on a shaker for 20 minutes. Table 2-1 shows the size distribution obtained by sieving. As expected for a 20/40 mesh sand, 94.2% of the sand grains by weight fall between sieve sizes of 0.425 mm and 0.85 mm. Table 2-1. Grain size distribution of mesh 20/40 Jordan sand Sieve size (mm) (Mesh No.) 1.0 (18) 0.85 (20) 0.71 (25) 0.6 (30) 0.5 (35) 0.425 (40) 0.355 (45) 0.3 (50) Weight percent retained 0 5.0 17.5 29.3 37.4 10.0 0.8 0 14 A Tescan Mira3 XMU field emission Scanning Electron Microscope (SEM) with 2 nm back-scattered electron resolution at 30 kV was used to investigate the microstructure of the natural and fractured sand grains. Sand collected from a crush test to a maximum stress of 50 MPa was used to obtain the textures of natural grain surfaces and new fracture faces. The collected sand grains were sieved to four different sizes and then were examined by a SEM. A sputtercoater was used to deposit a 10 nm thickness of a platinum-palladium alloy on the sand grains or particles to form a conductive surface. As shown in Figure 2-3, most sand grains from the 0.425 mm sieve remained unbroken although a few grains showed signs of abrasions or diametrical failures. For the 0.315 mm sieve, grains split in the middle were common with some cracks present. Grains from the 0.16 mm sieve consisted of angular elongated fragments of sand grains. The smallest size fraction (0.08 mm sieve) consisted of more elongated fragments. New fracture faces and grain surfaces are easily distinguished from each other by their texture (smooth or rough). 15 Figure 2-3. SEM images of sand grains from different sieve sizes Sand crushing characteristics affect the stress-strain response of sand and the grain size degradation under compression, which are directly related to sand pack permeability. The crushing tests presented in this section were performed on sand grains retained on 0.5 mm and 0.6 mm sieves respectively because these are the maximum size components of mesh 20/40 Jordan sand, a commonly used size range for hydraulic fracturing. In addition, testing using uniformly-graded grains facilitates the observation of evolution in grain size distribution and grain shape with the increase of applied compressive stress. It should also be noted that the crushing tests were performed on dry sand without considering fluids or pore pressure. The sand used for the crushing test that came from the 0.5 mm sieve is termed as G500 sand and G600 for sand from the 0.6 mm sieve. 0.425 mm 0.315 mm0.160 mm 0.080 mm6(b) (c)(a)(d) (e) (f)16 2.2.3 Test apparatus and procedure 2.2.3.1 1-D compression crush test A crush test involves placing sand into a steel cup and then applying a compressive stress to the top of the sand with a steel loading piston as seen in Figure 2-4. The testing apparatus was smaller than that used for the ISO 13503-2 standard to facilitate good acoustic emission monitoring while keeping the thickness of the sand in the cup similar to that in a real hydraulic fracture. The cup has a 25.4+ mm inner diameter, 50.8 mm outer diameter, and is 70.28 mm deep. The loading piston is 25.4- mm in diameter and 80 mm long. A compression force was vertically applied to the piston using an Instron 3385H testing machine with a load capacity of 250 kN, load measurement accuracy of 0.5% down to 2.5 kN, and strain measurement accuracy of 0.5%. The loading rate was set to 0.1 mm/min to maintain a stress increase at less than 0.5 MPa/s. The stress applied to the sand pack in the cup and the resulting strain was measured at a rate of 100 Hz during each test. Figure 2-4. Steel cup and piston used for a one-dimensional compression test A 27 mm diameter piezoelectric sensor with a resonant frequency of 6.3 kHz was attached to the outside of the steel cup to record the sound created by sand grain fractures throughout the compression test. The sound from the crushing sand creates a voltage in the piezoelectric sensor, which was amplified and recorded at a rate of 25 kHz using a National Instruments USB-6210 data logger. A Fast Fourier transform was performed on the acoustic emission (AE) data recorded during the crushing tests. The waveforms detected by the piezoelectric sensor were filtered to remove frequencies under 200 Hz, which were found to be associated with background noise. The grain breaking events have frequencies ranging from 400 to 8000 Hz. PistonPiezoelectric sensorCrushing cupSand17 For each crushing test, 10 ± 0.01 g of sand was placed into the cup. The cup was placed on a vibration table to level the sand surface and to achieve an approximately constant initial void ratio. The compressive stress applied to the sand in the crushing cup was up to 40 MPa to mimic the typical closure stresses in hydraulic fractures. The height of the sand pack in the steel cup was measured before the start of each crush test. These data were used to calculate the initial void ratio in the sand pack assuming that the quartz sand grains have an apparent density of 2.65 g/cm3. The tests were repeated five times to evaluate inherent variability in the test data and to obtain a sufficient mass of sand for subsequent sieving. After each set of five tests, a sieve analysis was performed to determine the particle size distribution, and the morphological properties of the crushed sand grains were analysed. 2.2.3.2 Grain shape analysis Sand crushing changes the grain shape and size, which affects the permeability and mechanical behaviour of the sand pack. A comprehensive analysis using images of the grains or fragments captured by a microscope was conducted to characterise sand particle size and shape before and after the crushing tests. An optical Zeiss Stereo Discovery.V8 microscope with a built-in camera (AxioCam ICC1) was used to examine the sand grains. The objective lens has a magnification from 25x to 200x. The sand was placed into a plastic petri dish with a black bottom while the lighting was from above. Grains were separated carefully to avoid grain overlap or touching before images were captured. A calibration slide was used to calibrate the scale of the field of view. ImageJ software (Schneider et al. 2012) was used to perform a morphology analysis of the grains. ImageJ is a Java-based image-processing program. Measurements for each individual particle were evaluated from its image and results were filtered by particle size and circularity. Several 2D particle shape descriptors are used. These include the 2D area (A) and perimeter (P) of each grain. Sphericity (S) and roundness (R) of the grain make use of the calculated area and perimeter and are defined in Equations 2-1 and 2-2 and have a maximum value of 1 for circles. For a square, R = 0.785 while S = 1. The maximum axis was found for the best-fit ellipse with the same area, orientation, and centroid to the original shape as shown in Figure 2-5. The maximum Feret diameter (MaxFeret) is the maximum distance between two parallel tangents touching the outline of a grain. The minimum Feret diameter (MinFeret) corresponds to the minimum distance that is 18 similar to a calliper diameter. The ratio of MinFeret and MaxFeret is the aspect ratio (AR). The parameter S defines the degree to which the shape of a particle approaches a sphere, while R refers to the overall angularity of a particle. The parameter R describes overall roundness while that defined by Krumbein and Sloss (1963) is more focused on local sharpness using a few sharpest corners, which is a scale-dependent parameter. The meaning of R and S values can be appreciated by reference to Figure 2-6, which is a plot of representative quartz particles retained on sieve sizes from 0.08 mm to 0.5 mm. 2PA4πR  (2-1)  2axismajor 4AS  (2-2) MaxFeretMinFeretAR  (2-3) Figure 2-5. Morphological properties of particle shape Fit ellipsePerimeterArea19 Figure 2-6. Shapes of sand particles (0.08-0.5 mm) along with their measured roundness and sphericity (each particle outline has been scaled to approximately the same size). As a comparison, images of sand particles classified into four different types of shape by Fair and Hatch (1933) were analyzed with the shape descriptors. These are shown in Figure 2-7. The values of S and R for rounded sand are 0.84 and 0.87, which are higher than those of angular sand (S = 0.68, R = 0.75). Figure 2-7. Morphological properties of particles with different shapes 111222333444444555555566666777777768888881 – 5002 – 4253 – 3554 – 3005 – 2506 – 1807 – 1258 – 80Sieve Size (μm)Angular sand S=0.68, R=0.75 Subangular sand S=0.75, R=0.79 Worn sand S=0.76, R=0.83 Rounded sand S=0.84, R=0.8720 2.2.4 Test results 2.2.4.1 Grain size distribution After a compression test, the piston was removed from the cup and the sand grains inside the cup were observed with a microscope. A typical example of sand crushing due to compression is shown in Figure 2-8. Diametrical failures in which a grain splits into two halves can be seen at locations 1 to 3. At location 4 and 5, the split grains were further broken into many pieces of finer fragments. The white coloured particles of sand are fresh fines generated due to crushing. Parab et al. (2014) used x-ray computed tomography to observe sand crushing under static loading. A sand grain was observed to break into several particles followed by cracking in a second grain once the first grain was completely pulverized. Figure 2-8. Crushed and intact sand grains after 30 MPa (left) and 40 MPa (right) compression The grain size distributions under applied stress levels of 10, 20, 30, and 40 MPa were obtained to understand the grain size variation with stress as shown in Figure 2-9. The results reveal that Jordan sand crushing becomes significant once the compressive stress exceeds approximately 30 MPa. For crushing tests on G600 sand, at maximum compressive stresses of 30 and 40 MPa, 14% and 29% of the original sand were respectively crushed to finer sizes, and 2.7% and 11% of sand by weight respectively passed through a 0.355 mm sieve size. For crushing tests on G500 sand under compressive stresses of 30 MPa and 40 MPa, 10% and 25% of the original sand were respectively crushed to finer size, and 3.2% and 12% of sand by weight respectively passed through a 0.355 mm sieve. This indicates that sand particles in a size range of 0.355 to 0.425 mm were further 3451 221 crushed into even smaller fragments. Note that G600 sand shows a lower crushing resistance compared to G500 sand because more grains are crushed at a given stress level. Figure 2-9. Grain size distributions of G500 and G600 sand under different stress levels 2.2.4.2 Stress-strain response with AE events detection The stress-strain responses from five crush tests on G500 sand are shown in Figure 2-10. The strain was plotted from a stress level of 0.5 MPa assuming the sand pack should achieve a similar sand pack height (12.35 ± 0.06 mm) under this stress level. From 0 to 10 MPa, the stress carried by the G500 sand pack increased smoothly as the grains were compacted. There is a drop in the vertical stress around 12.6 ± 1.6 MPa (mean ± standard deviation using 5 tests) as shown in the stress-strain curves due to one or more grains breaking as verified with AE detection. This is similar to the one-dimensional compression experiments by Hagerty et al. (1993) that indicated sand crushing began at an applied stress as low as 15 MPa. The stress increased with multiple small stress drops as grains were further broken especially when the stress exceeded 10 MPa, resulting in a zigzag shape for the stress-strain curve. It is also noted that the slope gradient of the stress-strain curve became shallower and each drop in stress is less significant after 30 MPa compared with stresses from 10 to 30 MPa. The average one-dimensional constrained modulus of the sand 01020304050607080901000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Percent finerParticle size (mm)40 MPa30 MPa20 MPa10 MPa40 MPa30 MPa20 MPa10 MPaMesh 20/40Compressive stressG500 sandG600 sand22 pack at stresses between 15 and 25 MPa is 485 ± 27 MPa while that between 32 and 38 MPa is 250 ± 31 MPa. The decrease in the constrained modulus may be explained by the generation of smaller and more tabular sand fragments at compressive stresses in the range of 30 to 40 MPa as verified by the grain size distributions and the microscopic morphology of the sand grains. The sand pack exhibits a less stiff response when these fragments are present and significant crushing of sand is occurring. Figure 2-10. Stress-strain response from five crush tests on G500 sand The positive values of the filtered AE voltage for Test 1 of G500 sand loaded up to 40 MPa compressive stress are presented in Figure 2-11. The voltage spikes and stress drops are coincident with each other. For example, a voltage spike larger than 3 V occurs at approximately 11.5 MPa corresponding to the first large drop in stress. This shows that the stress drop was caused by grain breaking. The larger voltage spikes (>1 V) occur between compressive stresses of 10 and 30 MPa. The voltage spikes occurring when the stress was between 30 and 40 MPa are typically smaller than 1 V. The smaller AE signals that occur at higher stress levels might be explained by the evolving nature of the grain size distribution under different stresses. As shown in Figure 2-9, only 3.5% of G500 sand was crushed to finer particles under a compressive stress of 20 MPa. Most of these fragments remained on sieve sizes of 0.355 mm and 0.425 mm, indicating that the initial 0510152025303540450 0.02 0.04 0.06 0.08 0.1 0.12 0.14Compressive stress (MPa)Vertical strain4231523 sand crushing was dominated by grains being split into two halves. The creation of small sand particles as the compressive stress increased may increase the number of contact points among the grains or fragments and thus more evenly distribute the contact forces. This, in turn, may result in smaller energy releases during subsequent crushing and thus lower voltage spikes. Figure 2-11. Compressive stress and AE data versus strain for G500 sand (Test 1) For G600 sand, the sand pack height at 0.5 MPa is 12.32 ± 0.07 mm. The stress applied to the sand increased smoothly and then experienced its first drop at a compressive stress around 9.6 MPa (± std. dev. 1.7 MPa using 5 tests) as seen in Figure 2-12, which is slightly lower compared with G500 sand (12.6 ± 1.6 MPa). This may be explained by larger size grains having a lower strength than the smaller size grains since more defects such as micro-cracks are likely in a bigger sand grain. The average one-dimensional constrained modulus at stresses between 15 and 25 MPa is 467 ± 44 MPa while that between 32 and 38 MPa is 221 ± 31 MPa. The average one-dimensional constrained modulus of G600 sand under lower stress levels (15 to 25 MPa) is similar to G500 sand. This may be because the degree of sand crushing is similar for the two sand sizes at this stress level. At a higher stress the sand pack consisting of the larger G600 sand grains had a lower modulus than G500 sand. The compressive stress and AE voltage versus time for Test 2 of G600 sand are presented in Figure 2-13. A spike with a voltage larger than 2.1 V occurred at 0200400600800100012000510152025303540450 0.02 0.04 0.06 0.08 0.1 0.12 0.14AE event count &AE event rate (events/2% strain)Compressive stress (MPa) &AE voltage (5x V)Vertical strainBr=0.002Br=0.007Br=0.026Br=0.086StressEvent countEvent rateAE voltage24 approximately 9 MPa when the first large stress drop occurred. This further shows that the stress drop was caused by grain breaking. Voltage spikes that are larger than 1 V occur between compressive stresses of 9 and 25 MPa, while voltage amplitudes at stresses between 25 to 40 MPa were typically lower than 1 V. Figure 2-12. Stress-strain response from five crush tests on G600 sand An attempt was made to count the number of individual AE events that occurred during each crush test. The approach suggested by Zhao et al. (2013) was used for this purpose. AE readings with a voltage higher than a threshold of 0.02 V were first determined and then those readings that occurred within an event duration 0.03 s were counted as a single crushing event. The AE voltage and the AE event count are plotted versus the strain in Figures 2-11 and 2-13 for tests on G500 and G600 sand. For the G500 sand, the number of AE events increased slowly to approximately 300 at 25 MPa and then more quickly to just over 1000 events at 40 MPa. For G600 sand, the number of AE events increased slowly to around 360 at 25 MPa and then dramatically to 1450 at 40 MPa. The AE event rate is defined as the number of AE events per every 2% vertical strain of the sand pack. As shown in Figures 2-11 and 2-13, the AE event rate of G500 sand increases from about 50 events/2% strain at 10 MPa to about 200 events/2% strain at stresses between 25 to 40 MPa. For G600 sand, the AE event rate increases from about 50 events/2% strain near 10 MPa to 0510152025303540450 0.02 0.04 0.06 0.08 0.1 0.12 0.14Compressive stress (MPa)Vertical strain1534225 300 events/2% strain at stresses between 25 to 40 MPa. Consistent with the grain size data, the number of AE events and the AE event rate both indicate that more grains of G600 sand were broken than G500 sand under the same stress levels. Figure 2-13. Compressive stress and AE data versus strain for G600 sand (Test 2) 2.2.4.3 Grain shape evolution Two hundred grains of G500 sand were randomly selected for morphology analysis. The mean MinFeret was 0.645 ± 0.049 mm. The mean value of MinFeret is a slightly larger than the finest sieve size that the G500 sand passed through, which is 0.6 mm. This is in good agreement with results by Altuhafi et al. (2012) since the sieve size that grains can pass through depends on both the intermediate and minimum grain dimensions. The aspect ratio was 0.776 ± 0.085, the roundness was 0.848 ± 0.030, and the sphericity was 0.783 ± 0.096. The shape of G500 sand is close to that of the worn sand according to Figure 2-7. The morphology analysis was repeated on 200 randomly sampled grains of G600 sand. The MinFeret was 0.757 mm ± 0.060 mm. Again, the mean value of MinFeret is slightly larger than the finest sieve size that the G600 sand passed through, which is 0.71 mm. The aspect ratio was 0.792 ± 0.081, the roundness was 0.852 ± 0.029, and the sphericity was 0.798 ± 0.092. Compared 020040060080010001200140016000510152025303540450 0.02 0.04 0.06 0.08 0.1 0.12 0.14AE event count &AE event rate (events/2% strain)Compressive stress (MPa) &AE voltage (5x V) Vertical strainStressAE voltageEvent countEvent rateBr=0.007Br=0.010Br=0.030Br=0.09426 to G500 sand, the G600 sand grains are closer to circular shapes and are close to round sand shown in Figure 2-7, and have a slightly lower standard deviation in these three descriptors. After 40 MPa crushing tests had been performed, the sand was sieved into different size ranges. The grains retained on each sieve were used for morphology analysis. As shown in Figure 2-14, sand grains of G500 sand retained on a 0.5 mm sieve remain unbroken and have high sphericity and roundness. Sand particles from 0.425 mm, 0.355 mm and 0.3 mm sieves, typically consist of grains that have been split into two or more fragments. Elongated or tabular particles were more common for the small sand fragments collected on sieves finer than 0.3 mm in size. Figure 2-14. Shapes of grains from different sieves (left to right: 0.5, 0.355, 0.25, and 0.125 mm) after a crushing test on G500 sand. For morphology analysis, 100 grains from each sieve were randomly selected. The grains collected from the 0.5 mm sieve had a slightly higher mean roundness (0.842), sphericity (0.834) and aspect ratio (0.821) than the original G500 sand prior to the crushing test. Likewise, the grains collected from the 0.6 mm sieve had a slightly higher mean roundness (0.853), sphericity (0.811) and aspect ratio (0.803) than the original G600 sand. These results indicated that rounder and more spherical grains tend to stay unbroken under compression. For particles collected from the 0.08 mm sieve, the aspect ratio was 0.570 for G500 sand and 0.615 for G600 sand, which means the maximum Feret diameter was almost two times of the minimum Feret diameter indicating an elongated grain shape. As the sieve size reduces, the mean value of roundness, sphericity and aspect ratio also decrease as seen in Figure 2-15 and Figure 2-16. The mean value of minimum Feret diameter of 0.5 mm27 grains from each sieve has a nearly linear relation with the sieve size. This may imply that the ratio of the intermediate and minimum grain dimension remains similar for each grain size. Figure 2-15. Mean values of geometry descriptors versus sieve size for G500 sand after 40 MPa compression Figure 2-16. Mean values of geometry descriptors versus sieve size for G600 sand after 40 MPa compression 0.10.20.30.40.50.60.70.80.50.60.70.80.90 0.1 0.2 0.3 0.4 0.5 0.6MinFeret(mm)R, S, ARSieve size of collected particles (mm)SphericityMinFeretRoundnessAspectratio0.10.20.30.40.50.60.70.80.50.60.70.80.90 0.1 0.2 0.3 0.4 0.5 0.6 0.7MinFeret(mm)R, S, ARSieve size of collected particles (mm)RoundnessAspectratioSphericityMinFeret28 2.3 Comparison with different types of proppants 2.3.1 Proppant types and sizes Four different types of proppants were studied as shown in Figure 2-17. In addition to Jordan sand, the proppants included silica sand from Genoa, Nebraska (Preferred Sands 2016), resin-coated silica sand (Hexion Specialty Chemicals 2010), and a carbon ceramic polymeric proppant (Sea-Pacific Inc. 2016). A visual examination of the proppants shows that most grains appear round and clean. For the Genoa sand, there are dark grains indicating non-silica minerals. The ceramic proppant consists of nearly spherical grains, which are more round than the other proppants. Figure 2-17. Four different types of proppants: Jordan sand (top left), Genoa sand (top right), resin-coated sand (bottom left) and ceramic proppant (bottom right) The three most popular sizes of proppants used for hydraulic fracturing are mesh size ranges of 20/40, 30/50 and 40/70. These size ranges were also used in this section. Table 2-2 shows the 29 relation between sieve size and mesh number. A mesh 20/40 sand means more than 90% of sand grain by weight is retained between 0.85 mm and 0.425 mm. The particle-size distributions of the proppants were determined following the ASTM D6913 (2009) sieving standard. Figure 2-18 shows the particle-size distribution of three different size ranges for the four different proppants. For mesh 40/70, all proppants have very similar particle-size distributions, most of which were retained between 0.18 and 0.425 mm sieve. For mesh 30/50, most of the ceramic proppant grains were retained on 0.425 mm and 0.5 mm sieves, while the other three proppants were mainly retained on 0.300 mm and 0.425 mm sieves. For mesh 20/40, the Jordan sand and the resin-coated sand have a similar particle-size distribution. Most of the proppants were collected from 0.5 mm and 0.6 mm sieves. The ceramic proppant grains were retained mainly on 0.6 mm and 0.71 mm sieves. The Genoa sand has a wider distribution with grains mainly retained from 0.355 mm to 0.6 mm sieves. Table 2-2. Relation between sieve size and mesh number Sieve size (mm) 1.0 0.85 0.71 0.6 0.5 0.425 0.355 0.3 0.25 0.18 0.125 Mesh No. 18 20 25 30 35 40 45 50 60 80 120 Figure 2-18. Particle-size distributions of three different sizes of four proppants 01020304050607080901000.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Percent finerParticle size (mm)Jordan 20/40Jordan 30/50Jordan 40/70Genoa 20/40Genoa 30/50Genoa 40/70Resin 20/40Resin 30/50Resin 40/70Ceramic 20/40Ceramic 30/50Ceramic 40/7030 2.3.2 Grain shape characterization 50 grains outlines were randomly selected for grain shape analysis. As shown in Table 2-3, the value of S ranged from 0.74 to 0.95, and that of R ranged from 0.83 to 0.89. Of all proppants tested, the mesh 20/40 ceramic proppant (S=0.95, R=0.89) is closest to spherical in shape. The mesh 40/70 Jordan sand has the lowest value of S and AR (S=0.74, R=0.84). Smaller particles tended to be more angular than larger ones for each proppant type. Referring to the values of S and R in Figure 2-7 for the shape types defined by Fair and Hatch (1933), overall, the shapes of the Jordan sand, the Genoa sand, and the resin-coated sand are close to that of the worn sand. Note that the mesh 20/40 range of Jordan sand and the resin-coated sand are slightly more spherical than the worn sand while are less spherical and less round than rounded sand. The shape of the ceramic proppant is close to that of a rounded sand while having a much higher sphericity especially for the mesh 20/40 and 30/50 size ranges. Table 2-3. Mean values of shape descriptors for proppants with different mesh sizes Jordan Genoa Resin Ceramic 20/40 30/50 40/70 20/40 30/50 40/70 20/40 30/50 40/70 20/40 30/50 40/70 S 0.81 0.76 0.74 0.76 0.78 0.76 0.80 0.75 0.76 0.95 0.92 0.86 R 0.85 0.84 0.84 0.83 0.84 0.83 0.84 0.84 0.84 0.89 0.88 0.86 2.3.3 Stress-strain response For each proppant, a crush test was repeated five times yielding five stress-strain curves. The stress-strain curve that was closest to the average stress-strain response was selected as the typical stress-strain for the proppant. Figure 2-19 shows these stress-strain responses. Overall, the stress carried by the proppants increased smoothly as the grains were compacted at the initial stage and continued to stress increase almost linearly with the increase of strain. At higher stress levels, the Jordan sand showed a softening stress-strain response with stress drops caused by grain breaking as indicated in Section 2.1, while the stress-strain response of the resin-coated and ceramic proppants behaved linearly. The smallest proppants (mesh 40/70) showed the stiffest stress-strain response except for the ceramic proppant. Conversely, largest proppants (mesh 20/40) were the least stiff, within each type of proppants. 31 Figure 2-19. Stress-strain responses of four proppants with three different particle-size distributions The average one-dimensional constrained modulus of the proppant pack at stresses between 15 and 25 MPa (M1) and that between 30 and 40 MPa (M2) were calculated (Figure 2-20. One-dimensional constrained modulus of the proppant packs). The constrained modulus ranged from 192 to 774 MPa. For the Jordan sand and the Genoa sand, the decrease in the constrained modulus (M2<M1) may be explained by grain crushing and the generation of smaller and more tabular sand fragments at compressive stresses in the range of 30 to 40 MPa (Cho et al. 2006). For the resin-coated sand, the constrained modulus increased slightly (M2>M1) possibly because the grain coating holds together the fragments if grains are crushed during grain compaction. For the ceramic proppant, the value of M2 is quite close to M1 because very few proppant grains were crushed at the stresses that were used during the test (<40 MPa). It should be noted that the values of M2 for 05101520253035400 0.02 0.04 0.06 0.08 0.1 0.12 0.14Compressive stress (MPa)Vertical strainJordan 20/40Jordan 30/50Jordan 40/7005101520253035400 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2Compressive stress (MPa)Vertical strainGenoa 20/40Genoa 30/50Genoa 40/7005101520253035400 0.02 0.04 0.06 0.08 0.1 0.12Compressive stress (MPa)Vertical strainResin 20/40Resin 30/50Resin 40/7005101520253035400 0.02 0.04 0.06 0.08 0.1 0.12Compressive stress (MPa)Vertical strainCeramic 20/40Ceramic 30/50Ceramic 40/7032 the resin-coated sand and the ceramic proppant were much higher than those for the Jordan sand and the Genoa sand. For mesh 20/40 the modulus was almost twice the magnitude. Figure 2-20. One-dimensional constrained modulus of the proppant packs 2.3.4 Post-test particle-size distribution After a compression test, the piston was removed from the cup and the proppant grains inside the cup were observed with a microscope. Typical images of grain breakage for the four different proppants are shown in Figure 2-21. Diametrical failures occur when a grain splits into two halves (locations 1, 4). These fragments can be further broken into many pieces of finer fragments (locations 2, 3). Broken grains were commonly seen for Genoa sand, with less damage observed for the Jordan sand. For the resin-coated sand, some broken grains were observed (location 5) but the resin coating often held the fragments together. For the ceramic proppant, only a few split grains (location 6) or cracks (location 7) were observed. 01002003004005006007008009001 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Constrained modulus (MPa)Mesh 20/40 Mesh 30/50 Mesh 40/7001002003004005006007008009001 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Constrained modulus (MPa)1 2 3 4 5Jordan01002003004005006007008009001 3 5 7 9 11 13 15 17 19 2 23 25 27 29 31 33Constrained modulus (MPa)1 2 3 4 5Genoa01020304050607080901 3 5 7 9 13 15 17 19 1 23 25 27 29 31 3Constrained modulus (MPa)1 2 3 4 5Resin01002003004 05 06 07 08 09 01 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Constrained modulus (MPa)1 2 3 4 5CeramicM1M2M1M2 M1M233 Figure 2-21. Grain breakage after 40 MPa compression: Jordan sand (top left), Genoa sand (top right), resin-coated sand (bottom left) and ceramic proppant (bottom right) The particle-size distributions for the proppants after they were subjected to an applied stress level of 40 MPa were obtained as shown in Figure 2-22. The change in particle-size distributions is most significant for the Genoa sand, followed by the Jordan sand. The particle-size distributions of the resin-coated sand and the ceramic proppant changed only slightly. The change in the particle-size distribution was less for proppants with a smaller size range compared to larger proppants. This indicates that smaller particles have a higher crushing resistance (Nakata et al. 2001). This is in good agreement with the stress-strain responses (Figure 2-19 and Figure 2-20) where grain crushing appears to reduce the constrained modulus of the larger size ranges of proppants more significantly. 1 23456734 Figure 2-22. Particle-size distributions of proppant packs with different mesh sizes under 40 MPa 2.4 Implication for multilayer proppant pack permeability and fracture conductivity 2.4.1 Kozeny-Carman equation for permeability estimation The conductivity of a fracture propped open by a proppant pack, Cs, is an important parameter for evaluating the ability of the fracture to transmit a fluid. The fracture conductivity is defined as fs wkC  (2-4) where k and wf are the permeability and thickness of a proppant pack within a hydraulic fracture respectively. The Kozeny-Carman equation is widely used to predict the hydraulic conductivity or intrinsic permeability k of granular porous media (Kozeny 1927, Carman 1937). Chapuis (2004) gave a 01020304050607080901000 0.2 0.4 0.6 0.8 1Percent finerParticle size (mm)Jordan 20/40Jordan 20/40 40MPaJordan 30/50Jordan 30/50 40MPaJordan 40/70Jordan 40/70 40MPa01020304050607080901000 0.2 0.4 0.6 0.8 1Percent finerParticle size (mm)Genoa 20/40Genoa 20/40 40MPaGenoa 30/50Genoa 30/50 40MPaGenoa 40/70Genoa 40/70 40MPa01020304050607080901000 0.2 0.4 0.6 0.8 1Percent finerParticle size (mm)Resin 20/40Resin 20/40 40MPaResin 30/50Resin 30/50 40MPaResin 40/70Resin 40/70 40MPa01020304050607080901000 0.2 0.4 0.6 0.8 1Percent finerParticle size (mm)Ceramic 20/40Ceramic 20/40 40MPaCeramic 30/50Ceramic 30/50 40MPaCeramic 40/70Ceramic 40/70 40MPa35 comprehensive review of the applicability of the Kozeny-Carman equation for soils. The intrinsic permeability is independent on the fluid properties and reflects the ease of a porous structure to transmit a fluid. Using the Kozeny-Carman equation (Carrier 2003) the permeability can be expressed as:   2023 111SCkCK  (2-5) where CK-C is a unitless empirical coefficient related to flow tortuosity, ϕ is the porosity of the porous medium, S0 is the specific surface (area per unit volume of a particle pack). For non-uniform grain sizes, the specific surface can be expressed as: effDSFS 0 (2-6) where SF is a unitless grain shape factor, and Deff is the effective grain diameter for non-uniform grain sizes as given by Carrier (2003): isiliieffDDfD))/((1595.0404.0 (2-7) where fi is the percentage of grains retained between a larger sieve Dli and a smaller sieve Dsi. For uniform size spheres, the specific surface = area/volume = πD2/(πD3/6) = 6/D. Substitution of this expression into Equation 2-6 yields a shape factor SF for spheres equal to 6. For non-spherical grains, the recommendations by Fair and Hatch (1933) were used to estimate the shape factor SF for different grain shapes, i.e., 6.0 for spheres, 6.1 for rounded sand, 6.4 for worn sand, 7.4 for subangular sand, and 7.7 for angular sand. For comparison, a combination of the Kozeny-Carman equation and the Hazen formula (Chapuis 2004, Göktepe and Sezer 2010) was used to estimate the permeability of the proppant pack. The permeability (k’) is expressed as: 2103'1DeeCk  (2-8) 36 C is a constant related to the geometry of pore channels reflecting tortuosity, e is the proppant pack void ratio, D10 is the effective diameter at which 10% of the particles are finer by weight. Here, the value of CK-C in Equation 2-5 is assumed to be 5.0 (Carman 1956). The value of C in Equation 2-8 is determined as 1/180 by allowing permeability from Equation 2-8 equal to that of Equation 2-5 for uniform-sized spherical particles. The sand pack height wf was measured during the crush tests and was used to interpret the void ratio. The values of D10 and Deff were obtained from the post-test particle-size distributions obtained by sieve analysis. The grain shape factor in Equation 2-5 was based on a shape analysis of grains collected from the tests with reference to results by Fair and Hatch (1933) for different grain shapes. 2.4.2 Comparison between four proppants with three size ranges Based on the crush tests on four different proppants with three different sizes, the values listed in Table 2-4 were used to estimate the permeability of a proppant pack using Equations 2-5 and 2-8. The value of e was calculated from Figure 2-19, and the values of Deff and D10 in unit mm were obtained from the particle size distributions shown in Figure 2-22. The recommendations by Fair and Hatch (1933) were used to estimate the shape factor SF for different grain shapes, i.e., 6.0 for spheres, 6.1 for rounded sand, 6.4 for worn sand, 7.4 for subangular sand, and 7.7 for angular sand. Here, a shape factor of 6.1 is used for all sizes of ceramic proppant because they are close to spheres (R and S >0.86) and stay unbroken under compression. For resin-coated sand, a shape factor of 6.4 is used for all of the resin-coated sand because most sand grains are close to the shape of worn sand (R: 0.75-0.8, S=0.84) and the resin coating can bond grain fragments together after the grain is fractured. For Jordan sand and Genoa sand, the value of SF after 40 MPa varies from 6.4 to 7.7 depending on the degree of grain crushing that produces angular fragments. 37 Table 2-4. Permeability estimation using Equations 2-5 (k) and 2-8 (k’) (at 40 MPa) Type Mesh No. Parameters Permeability (µm2) Conductivity (µm3) e D10 Deff SF wf k’ k kmean Jordan 20/40 0.419 0.366 0.507 7.4 10.6 38.6 48.7 43.6 462000 30/50 0.521 0.265 0.356 7.4 11.3 36.3 43.0 39.7 448000 40/70 0.573 0.206 0.265 6.4 11.7 28.2 41.1 34.6 404000 Genoa 20/40 0.340 0.191 0.406 7.7 10.0 5.9 16.3 11.2 112000 30/50 0.413 0.191 0.340 7.4 10.5 10.1 21.0 15.6 163000 40/70 0.450 0.167 0.252 6.4 10.8 9.8 19.4 14.6 157000 Resin 20/40 0.498 0.494 0.576 6.4 11.2 111.8 133.5 122.7 1374000 30/50 0.493 0.315 0.392 6.4 11.1 44.2 60.2 52.2 579000 40/70 0.484 0.211 0.272 6.4 11.1 18.8 27.6 23.3 258000 Ceramic 20/40 0.599 0.565 0.635 6.1 11.4 238.3 291.3 264.9 3019000 30/50 0.632 0.435 0.487 6.1 11.7 162.6 197.2 179.9 2104000 40/70 0.617 0.251 0.291 6.1 11.6 50.8 66.1 58.5 678000 As shown in Table 2-4, the generalized Kozeny-Carman equation (Equation 2-5) tends to give higher permeability than that using the Kozeny-Carman equation combined with the Hazen formula (Equation 2-8). The average value kmean = (k+k’)/2 is used here for permeability comparison. The ceramic proppant has the highest predicted permeability for all proppant sizes among the four types of proppants, while the Genoa sand has the lowest permeability for all proppant size ranges. The mesh 20/40 resin-coated sand has much higher permeability than those of Jordan sand and Genoa sand. The mesh 30/50 Jordan sand has a slightly lower permeability than that of the resin-coated sand. In contrast, the mesh 40/70 Jordan sand is more permeable than the mesh 40/70 resin-coated sand. The mesh 20/40 Genoa sand has the lowest permeability among all the proppants. The average value of estimated permeability kmean was plotted against published data from conductivity tests (Preferred Sands 2016, Hexion Specialty Chemicals 2010, Sea-Pacific Inc. 2016) as shown in Figures 2-23 and 2-24. The conductivity tests followed the ISO 13502-5 standard (2006) using a conductivity cell with proppant embedded between two fracture faces (mostly Ohio Sandstone) through which 2% KCl fluid flows to measure the permeability and 38 fracture conductivity. It should be noted that the proppant loading of the laboratory conductivity tests is 9.765 kg/m2 (2 lb/ft2), while the loading of proppant for the crush tests and the subsequent conductivity estimation is 19.530 kg/m2 (4 lb/ft2). The average values of permeability based on the two empirical equations (Equations. 2-5 and 2-8) are in a similar magnitude as the measured values from the conductivity tests. The estimated permeability of ceramic proppant closely matches the measured permeability, especially for mesh 20/40 and 40/70 sizes. For the Genoa sand, the estimated permeability is lower than the measured permeability, which may result from particle-size distributions that had slightly larger grain sizes during the conductivity tests. The mean particle diameters of the three mesh sizes used for the conductivity tests are larger than those used in the crush tests; this would typically result in a higher permeability for the conductivity tests. For the conductivity tests, it is also possible that fine grains could have been transported away through the pores as observed in conductivity tests (Lacy 1998), which would result in a more permeable proppant pack. Figure 2-23. Estimated permeability compared with conductivity test data (Preferred Sands 2016, Sea-Pacific Inc. 2016) 0501001502002503003504004500 20 40 60 80Permeability (µm2)Compressive stress (MPa)Blue: mesh 30/50Red: mesh 40/700501001502002503003504004500 20 40 60 80Permeability (Darcy)Compressive stress (MPa)Genoa 30/50Genoa 40/70Ceramic 30/50Ceramic 40/70Genoa 20/40Ceramic 20/40JordanResinCeramicGenoaMesh 20/40Conductivity testEmpirical estimation0501001502002503003504004500 20 40 60 80Permeability (Darcy)Compressive stress (MPa)Genoa 30/50Genoa 40/70Ceramic 30/50Ceramic 40/70Genoa 20/40Ceramic 20/40JordanResinCeramicGenoa39 The fracture conductivity that would exist within a propped hydraulic fracture was also estimated using the proppant pack permeability and proppant pack height (Equation 2-4). The proppant pack height decreases as the applied stress increases and the height varies from 10.5 to 11.8 mm for different proppants at 40 MPa. Figure 2-24 shows the estimated fracture conductivity plotted against the measured conductivity obtained from the same laboratory conductivity tests shown in Figure 2-23. At a compression of 40 MPa, the mesh 20/40 ceramic proppant has the highest estimated fracture conductivity at approximate 3028000 µm3, while the mesh 20/40 Genoa sand has the lowest estimated conductivity at 112000 µm3. The height of the proppant packs from the conductivity test can be back-calculated using the measured conductivity and permeability (Equation 2-4). The laboratory conductivity tests at a proppant loading of 9.765 kg/m2 give a proppant pack height approximately half the height measured from the crush test (proppant loading 19.530 kg/m2). For example, at 40 MPa, the proppant height from the crush tests is 10.97 mm for the mesh 40/70 Genoa sand and is 11.71 mm for the mesh 30/50 ceramic proppant, while it is, respectively, 5.39 mm and 5.87 mm in the laboratory conductivity tests. For ceramic proppant, the estimated fracture conductivity is approximately twice the measured conductivity as the measured permeability is close to the estimated permeability. For the mesh 20/40 resin-coated sand, the estimated fracture conductivity (1374000 µm3) is about 1.7 times the measured conductivity (842000 µm3), implying the estimated permeability is about 0.85 times the measured permeability. 40 Figure 2-24. Comparison of fracture conductivity estimation versus conductivity test data (Preferred Sands 2016, Hexion Specialty Chemicals 2010, Sea-Pacific Inc. 2016) 2.4.3 Evolution of permeability with compressive stresses This subsection uses G500 and G600 Jordan sand as examples to illustrate the evolution of permeability under compression. The particle-size distributions of G500 and G600 Jordan sand under 10, 20, 30, and 40 MPa were measured. Following Equations 2-5 and 2-8, a permeability reduction factor (kr follows Equation 2-5 and kr’ follows Equation 2-8) is defined here as the ratio of permeability after and before crushing, which can be expressed as: 22221,22,231132 1)1()1(SFSFDDeeeekeffeffr  (2-9) 21,1022,10231132')1()1(DDeeeekr  (2-10) where the subscripts 1 and 2 are the value of each parameter before and after crushing respectively. At 10 MPa increments in the compressive stress, the values of the shape factor SF were estimated using the suggestion by Fair and Hatch (1933) for different grain shapes. The values of SF are 6.1 0500000100000015000002000000250000030000000 20 40 60 80Conductivity (µm3)Compressive stress (MPa)Mesh 20/40100020003000400056000700080009000100000 20 40 60 80Conductivity (md-ft)Compressive stress (MPa)Genoa 30/50Genoa 40/70Ceramic 30/50Ceramic 40/70Genoa 20/40Resin 20/40Ceramic 20/40JordanResinCeramicGenoaBlue: mesh 30/50Red: mesh 40/70Conductivity testEmpirical estimation0100020003000400050006000700080009000100000 20 40 60 80Conductivity (md-ft)Compressive stress (MPa)Genoa 30/50Genoa 40/70Ceramic 30/50Ceramic 40/70Genoa 20/40Resin 20/40Ceramic 20/40JordanResinCeramicGenoa41 for the original sand (0 MPa) as grain shapes are nearly round (R =0.84 and S = 0.83), 6.4 for sand loaded to 10 and 20 MPa because a small portion of grains are abraded and split, 7.4 for sand loaded to 30 MPa because a significant portion of the grains are split, and 7.7 for 40 MPa because many grains are split and fragments are further crushed into angular shapes. Using the empirical equations and the data in Table 2-5, the calculated permeability reduction for each stress level using Equations 2-9 and 2-10 is shown in Figure 2-25. As the stress increases from 0 to 40 MPa, the permeability reduces by up to one order of magnitude. The permeability reduction at stress levels less than 20 MPa is mainly caused by a decrease in void ratio due to the grain position realignment by compaction. The permeability reduction at compression stresses larger than roughly 20 to 30 MPa is caused by decreases in both void ratio and grain diameter as well as changes in the grain shape due to crushing. Ignoring the grain size, round grains are 1.6 times, (SFround /SFangular)2 = (7.7/6.1)2, more permeable than angular grains, which matches a numerical study using finite element flow simulation through internal pore structures (Garcia et al. 2009). Although the permeability reduction factor for a G500 sand pack is similar to G600 sand at the same stress, G600 sand is 1.4 times more permeable than G500 sand under 0 to 20 MPa compressive stresses. So while the G600 sand experiences more crushing and fines generation than the G500 sand, the larger initial grain size more than compensates for the presence of the fines in terms of the resulting permeability through the sand pack. Table 2-5. Parameters for permeability reduction estimation using empirical equations (D10 and Deff in unit mm) Stress (MPa) G500 sand G600 sand e D10 Deff SF e D10 Deff SF 0 0.658 0.60 0.539 6.1 0.655 0.71 0.643 6.1 10 0.597 0.53 0.536 6.4 0.597 0.62 0.634 6.4 20 0.561 0.515 0.530 6.4 0.561 0.61 0.628 6.4 30 0.521 0.50 0.508 7.4 0.517 0.575 0.593 7.4 40 0.454 0.30 0.443 7.7 0.444 0.325 0.516 7.7 42 Figure 2-25. Permeability reduction ratio of G600 and G500 under different stresses Permeability tests by Coulter et al. (1972) found that 5% fines can cause more than a 50% reduction in proppant pack permeability for mesh 20/40 sand under a compressive stress of 24.1 MPa. The tests give a similar result. As shown in Figure 2-32, the permeability reduces by about 50% for both G500 and G600 sand tested under 20 MPa compression. At this stress, 3.5% and 5% of the sand grains are crushed to finer particles respectively, while the sand pack height decreases by 6 to 7% due to compression. The resulting fracture conductivity that would exist in a hydraulic fracture is estimated to reduce by about 50%. 2.5 Summary For G500/G600 Jordan sand, when the effective stresses applied to a sand pack exceed roughly 10 MPa grain crushing initiates and when stresses exceed roughly 30 MPa a significant portion of the quartz grains are crushed. Acoustic emissions caused by grain breaking were directly associated with drops in the compressive stress. With further increase in compressive stress, a zigzag stress-strain response is observed as sand grains break and the sand becomes more compacted. The overall sand pack stress-strain response becomes softer as grains are crushed, especially at stress levels above roughly 30 MPa. Crushed sand particles collected from finer sieve sizes have smaller mean values of roundness, sphericity and aspect ratio as particles become smaller. The shape of 0.00.30.60.91.21.500.20.40.60.810 10 20 30 40Permeability ratio of G600 to G500Permeability reductionCompressive stress (MPa) G600 - kr' G500 - kr' G600 - kr G500 - Kr K600'/k500' K600/K500Kozeny-Carman combined with HazenG600 – krG500 – krkG600/kG500Kozeny-CarmanG600 – kr’G500 – kr’kG600’/kG500’0.00.30.60.91.21.500.20.40.60.810 10 20 30 40Permeability ratio of G600 to G500Permeability reductionCompressive stress (MPa) 6 - r' G500 - kr' G600 - kr G500 - Kr K600'/k500' K600/K5000.00.30.60.91.21.500.20.40.60.810 10 20 30 40Permeability ratio of G600 to G500Permeability reductionCompressive stress (MPa) G600 - kr' G500 - kr' G600 - kr G500 - K K6 '/k5 ' K600/K5000.00.30.60.91.21.500.20.40.60.810 10 20 30 40Permeability ratio of G600 to G500Permeability reductionCompressive stress (MPa) G600 - k '65 KK6 '/k500'/K0.00.30.60.9..00.20.40.6.0 10 20 30 40Permeability ratio of G600 to G500Permeability reductiono pressive stress ( Pa) - r' 500 - kr' G600 - kr G500 - Kr K600'/k500' K6 /K5000.00.30.60.91.21.500.20.40.60.810 10 20 30 40Permeability ratio of G600 to G500Permeability reductionCompressive stress (MPa) G600 - kr' G500 - kr' - r 500 - Kr K600'/k500' K600/K5000.0.3.6.91 2500.2.4.6810 10 20 30 40Permeability ratio of G600 to G500Permeability reductionr ssiv str ss ( ) ' k ' - k - Kr K600'/k500' K600/K500Kozeny-Carman combined with HazenKozeny-Carman43 particles shifts from nearly spherical grains to diametrically split grains and then to small, elongated and angular fragments of the smaller particle sizes. For comparison of different types/sizes of proppants, the ceramic proppant is closest to spheres with larger grains being most spherical. The mesh 40/70 Jordan sand was found to be the most angular. More angular grain shapes occur for smaller particle sizes for all proppant types. The proppants were subjected to compressive stresses of up to 40 MPa in the crush tests. For Jordan sand and Genoa sand, grain diametrical splitting and split grains further broken into finer fragments were commonly observed when the proppants were examined after the test. For the resin-coated sand, fragments from crushed grains were often held together by the resin coating. For the ceramic proppants, only a few grains with splitting/cracking were observed. The Genoa sand exhibits a more significant change in the particle-size distribution than that of the Jordan sand, while the particle-size distribution of resin-coated sand and ceramic proppant remain nearly unchanged. A softening stress-strain response with stress drops was found for Jordan sand after 30 MPa, while the resin-coated proppant and the ceramic proppant had a nearly linear stress-strain response. The constrained moduli of the resin-coated sand and the ceramic proppant are 30% higher than the Jordan sand and the Genoa sand. The Kozeny-Carman equation was used to estimate the proppant pack permeability and the resulting fracture conductivity from the crush tests. A good agreement was found between the estimated permeability/fracture conductivity and published laboratory results for fracture conductivity. This suggests that grain size data should be collected as supplemental data from a crush test. Use of the grain size and shape data provides a fast and reliable approach for evaluating proppant pack permeability and conductivity at different stress levels. The ceramic proppant and the Genoa sand are, respectively, the most and least permeable at a given mesh size. The mesh 20/40 and 30/50 ceramic proppants, and the mesh 20/40 resin-coated sand are at least two times more permeable than the other proppants at 40 MPa. The mesh 20/40 ceramic proppant has the highest fracture conductivity at approximate 3019000 µm3, while the mesh 20/40 Genoa sand has the lowest conductivity at 112000 µm3. Examination of the G500/G600 Jordan sand pack permeability at four different stress levels shows the permeability reduction that occurs as the sand is crushed is associated with three factors: 44 reduction in void ratio, reduction in particle size, and a shift away from spherical particle shapes. The permeability decreases by > 40% at 20 MPa due to a decrease in void ratio by compaction. The permeability reduces by more than 80% due to reduction in void ratio and reduction in particle size at 40 MPa when the sand is crushed. 45 3 3D Modelling of Proppant Crushing in DEM 3.1 Overview Proppant crushing reduces the fracture aperture and void ratio of proppant packs. However, most models developed for estimating proppant pack permeability only represent proppant pack compaction with Terzaghi’s soil consolidation theory (Neto et al. 2015, Chen et al. 2017) or the Mohr-Coulomb constitutive model (Han et al. 2016) and do not consider proppant crushing that alters the grain size distribution and stress-strain response. The purpose of this study is to develop numerical models to capture the change of the grain size distribution and the stress-strain response during the grain breakage process (Elghezal et al. 2013) DEM has been widely used to simulate grain breakage and comminution processes under static compression (Hosseininia and Mirghasemi 2006, Refahi et al. 2016, Jo et al. 2011). In DEM, there are two approaches to simulate grain breakage. The first approach is to represent each grain with a cluster of many smaller particles, and grain breakage is modelled as contact bond breakage within the cluster of particles (Ueda et al. 2013, Wang and Yan 2013, Cil and Alshibli 2014). The second approach is to represent each grain with a single particle in the DEM model. A breakage criterion is used to determine whether a particle breaks (Lobo-Guerrero and Vallejo 2005, Kwok and Bolton 2013, De Bono and McDowell 2014). A breakage criterion is a measure of a stress index within the particle that can be related to the particle strength obtained from experiment. When the stress acting on a particle exceeds the particle strength, the particle is replaced with smaller particles to mimic fragmentation. Due to the higher computational effort of the first method, the second approach has been widely used to simulate the crushing of sand grains. McDowell and De Bono (2013) discussed the influence of different particle splitting approaches and particle strength distributions on the void ratio-compressive stress as well as grain size distributions. De Bono and McDowell (2016) gave a comprehensive comparison of several crushing criteria using different crushing stress indexes. They found that criteria that use the maximum contact forces and octahedral shear stress match with the experimental responses. In spite of numerous DEM studies on grain breakage under one-dimensional compression, most studies have focused on exploring the effect of micro-parameters on macro-scale behaviours such as normal compression lines and critical state lines or micro-scale responses such as the evolution 46 of contact numbers and contact forces (Kevin et al. 2015). The calibration of the DEM for grain crushing with a breakage model has rarely been reported. Belheine et al. (2009) calibrated micromechanical properties of DEM particles to match the triaxial stress-strain response of Labenne sand but grain breakage was not considered. Behraftar et al. (2017) used dimensional analysis to calibrate the micro-parameters of bonded DEM elements against the laboratory data of cracked chevron notch Brazilian discs, which can be classified into the first approach to simulate grain breakage and is computational expensive when many particles are present. This chapter develops a particle breakage model in 3D DEM that is capable of simulating proppant particle crushing under compression. This is the first exploration of the validity of three different crushing indexes from a micro-scale approach using clustered balls to simulate grain breakage. Then a macro-scale DEM calibration was performed to get the strength parameters in the breakage model for Jordan sand by mainly matching the laboratory particle-size distribution. The calibration procedure was repeated to get the micro-parameters including the particle stiffness, particle strength, and Weibull modulus that respectively represent Genoa sand, the resin-coated sand, and the ceramic proppant. 3.2 Simulation of Jordan sand compaction without particle crushing A discrete element tool PFC3D (Itasca Consulting Group 2014) was used to simulate the Jordan sand compaction in a crush test without considering grain breaking. Basic elements in PFC3D are rigid spherical particles and walls. Deformation between two particles or between a particle and a wall happens in its contact. In this soft-contact approach, a rigid particle is allowed to overlap with another particle or with a rigid wall (Figure 3-1) and the magnitude of overlap is related to the contact force via a contact-displacement law such as linear contact model shown in Figure 3-2. Particles in PFC3D can be used to represent individual grains or can be glued together with bonds to mimic bulk materials like rock. The calculation cycle in PFC3D is shown in Figure 3-3. 47 Figure 3-1. Soft contact of two rigid particles (left) and rigid particle and wall (right) Figure 3-2. Liner contact model Figure 3-3. Calculation cycle in PFC A uniform distribution of particle sizes between 710 µm and 600 µm was used to create a particle assembly in PFC3D to represent G600 Jordan sand. The linear contact model shown in Figure 3-2 was used for both the particle-particle and particle-wall interactions in PFC3D as the displacement of a quartz grain typically behaves linearly with the applied force (Nakata et al. 1999). The model would require 25467 particles in PFC3D to reproduce a 10.0 g particle assembly (2.65 g/cm3) if the **Wall>0 OpenTensile11|Start of cycleAdvance time ΔtDetermine timestep Δt Law of motionForce-displacement lawContact detection48 model cup diameter was the same as the laboratory crush cup. This many particles would require excessive computational processing time, especially when particles are allowed to break into smaller particles for mimicking particle fragmentation. Therefore, to have a more manageable numerical model, a smaller cup with a diameter of 10 mm was simulated. In total 1554 spheres were created in this scaled-down oedometer (similar concept to Ciantia et al. 2015) to reproduce a 0.61 g (= 10.0×(10/25.4)3 g) particle assembly, and the particle assembly was loaded one-dimensionally as shown in Figure 3-4. Figure 3-4. Particles in cup under one-dimensional compression To analyze the influence of micro-parameters on macroscopic behaviours of a particle pack, a sensitivity analysis based on 2k-1 factorial design (Montgomery 2001) was used. Two levels of micro-parameters are shown in Table 3-1. The constrained modulus of the particle pack compressed between stresses 15 MPa and 25 MPa was used as a targeted macroscopic output result. The sand pack height and void ratio at 0.5 MPa were also used as targeted macroscopic outputs. With different combinations of the two levels for six micro-parameters, there are 32 different scenarios simulated (Table A-1). Loading platenSteel cupProppant particles49 Table 3-1. Value of micro-parameters for two levels Micro Parameter Level -1 Level +1 Particle friction coefficient 0.1 0.7 Particle-particle effective modulus (GPa) 0.8 1.6 Wall friction coefficient 0.0 0.5 Raito of particle-wall effective modulus over particle-particle effective modulus 1.0 2.0 Normal-to-shear stiffness ratio 0.5 1.5 Damping coefficient 0.1 0.7 The percentage contribution of each micro-parameter to the variability of the output results (Table A-2) is obtained as shown in Table A-3. The results show that the particle-particle effective modulus and particle friction coefficient contribute, respectively, 74% and 21% to the constrained modulus while that of other micro-parameters is less than 2%. The particle friction coefficient and wall friction coefficient contribute, respectively, 84% and 14% to the sand pack height and void ratio at 0.5 MPa. The results indicate that the particle-particle friction coefficient is the main factor controlling the initial sand pack height, and the particle-particle effective modulus is the main factor controlling the constrained modulus of the particle pack. The particle-particle effective modulus and particle-particle friction coefficient in the PFC3D simulations can be adjusted to match the laboratory stress-strain response and initial sand pack height/void ratio. The micro-parameters used in PFC3D to match the stress-strain curve of G600 sand are shown in Table 3-2 that was adapted from McDowell and de Bono (2013). The particle-particle effective modulus was adjusted to 1.1 GPa to match the stress-strain curve of G600 sand by trial and error. The wall-particle effective modulus was slightly higher than that between particle-particle to prevent particles from escaping out of the cup in the PFC3D simulation. 50 Table 3-2. Micro-parameters for DEM simulation of G600 Jordan sand Micro Parameter Value Particle diameter D (mm) 0.6-0.71 Particle density (kg/m3) 2650 Particle friction coefficient 0.5 Particle-particle effective modulus (GPa) 1.1 Wall friction coefficient 0 Particle-wall effective modulus (GPa) 1.32 Normal-to-shear stiffness ratio 1 Damping coefficient 0.4 The particle pack was initially loaded to 0.5 MPa and the particle pack is 5.31 mm, which is slightly higher than 4.85 mm, 10/25.4 of the measured height (12.32 mm) in the laboratory crush test. This results in a higher void ratio (0.812) in the particle pack in PFC3D simulation than that (0.655) in the sand pack from the laboratory test. This may be due to the spherical shape of particles in PFC3D, while in reality sand grains are with different shapes that can adjust into the void space between grains and may facilitate dense compaction. McDowell and de Bono (2013) also found a higher height and void ratio of the particle pack in PFC3D than that from laboratory tests. Based on the sensitivity analysis, the void ratio in the particle pack can be adjusted to approach the laboratory data by changing the value of the particle friction coefficient. A value of 0.3 for the particle friction coefficient would result in a particle pack height of 5.28 mm and a void ratio of 0.800 at 0.5 MPa. A value of 0.1 for the particle friction coefficient would result in a height of 5.13 mm (void ratio = 0.751). A value of 0.0 for the particle friction coefficient would give an initial height of 4.98 mm (void ratio = 0.699). However, a friction coefficient of 0.5 was used here considering the round quartz sand grains have an internal friction angle around 27-30 degree (Das 2011; Geotechdata 2017). The strain of the particle pack was calculated using the particle pack vertical displacement divided by the initial particle pack height at 0.5 MPa. As shown in Figure 3-5, the stress-strain curve given by the PFC3D simulation matches the laboratory data for axial compressive stresses less than about 30 MPa. For compressive stress higher than 30 MPa, the modelled stress continues to increase at a nearly constant gradient while the laboratory test data exhibits a non-linear and softer stress 51 increase as sand grains are crushed. To match better the strain softening as well as changes in the particle-size distribution associated with grain crushing, a particle breakage model was incorporated in PFC3D simulations of the crush test. Figure 3-5. Stress-strain response of G600 sand compressed up to 40 MPa 3.3 Implementation of particle breakage model in DEM 3.3.1 Particle contact forces and particle stress Based on the contact forces acting on a particle as shown in Figure 3-6, the ‘particle stress’, σij, within a particle (Potyondy and Cundall 2004) can be expressed as: cnc(c)j(c)i FnVD1ij2σ (3-1) where D and V are the diameter and the volume of the particle, nc is the number of contacts around the particle – also called coordination number, ni.(c) is the i.th component of the unit direction vector of contact c, and Fj.(c) is the j.th component of the force magnitude of contact c. 0102030400 0.03 0.06 0.09 0.12 0.15Compressive stress (MPa) Vertical strainLabPFC52 Figure 3-6. Contact forces acting on a particle 3.3.2 Particle stress indexes Three different definitions of particle stress indexes are presented here. De Bono and McDowell (2013) believed grain crushing is induced by deviatoric stresses and thus used the octahedral shear stress to quantify a characteristic stress. Stress Index 1 is based on the octahedral shear stress as shown in Equation 3-2.        5.0232221231)1( 31  i (3-2) In this equation, σ1 is the maximum principal stress and σ3 is the minimum principal stress within a particle calculated from the stress tensor given by Equation 3-1. Lobo-Guerrero and Vallejo (Lobo-Guerrero and Vallejo 2005) noted that particles with just a few contact points can break because forces acting on the particles are not well distributed. An analytical analysis of internal stress also showed that primarily the largest contact force controls the crushing of individual particles within a granular assembly (Russell et al. 2009). The stress index assumes that a particle breaks under tension much like an indirect tensile test. Stress Index 2 is given by: 2)2( DFmaxti  (3-3) where Fmax is the maximum normal contact force acting on the particle and D is the particle diameter. Tsoungui et al. (1999) simplified the contact forces acting on a particle by two pairs of opposing forces or stresses. This simplification is analogous to biaxial loading. Based on the stress distribution inside the particle from finite element modelling, Stress Index 3 is defined as: 53 23 31)3(i (3-4) 3.3.3 Particle strength The strength of a particle with a diameter D can be defined as (De Bono and McDowell 2014):  mDsDD/300  (3-5) where σ0 is the known strength of grain with a diameter D0, and m is a Weibull modulus of the material that is used to describe the size effect on the particle strength. Equation 3-5 is similar to the particle strength used by Lobo-Guerrero and Vallejo (2005) and Vallet and Charmet (1995). The strength σ0 can be obtained by measuring the maximum normal contact force that can be carried by a single particle of a given particle diameter, based on which the stress can be calculated (Nakata et al. 1999). It is assumed the particle strength is a single value for a specific diameter of material, as the particle strength of proppants with the same diameter seems not to follow Weibull statistic distribution (Appendix B). It should be noted that for different stress indexes the value of σ0 differs due to the different definitions of the characteristic stress. The value of D0 is set to 0.655 mm, which is the average diameter of G600 particle pack (0.6-0.71 mm). The values of σ0 of G600 grains corresponding to D0 = 0.655 mm and the Weibull modulus are determined by matching the laboratory results regarding the particle-size distribution and the stress-strain response, which will be introduced in Section 3.5. 3.3.4 Particle crushing criteria A particle crushing criterion is to judge a particle to break during the PFC3D simulations if the particle stress index exceeds the particle strength.  Dsi   (3-6) In this study, Criterion 1 corresponds to stress Index 1, Criterion 2 corresponds to stress Index 2, and Criterion 3 corresponds to stress Index 3. 54 Ben-Nun and Einav (2010) showed that final grain size distributions are not sensitive to the configuration of replaced particles for particle crushing. As shown in Figure 3-7, the overloaded particle is replaced by four smaller particles (red) in the model. Two of these particles are larger and represent larger split fragments while the two smaller particles represent smaller fragments. The orientation of the contact between the larger two particles is aligned with the direction of the maximum principal stress that was acting on the particle. To maintain the particle volume when replacing a broken particle with four smaller particles, the diameters of the four smaller replacement particles were increased by a factor (Fr) of 1.4559. Figure 3-7. Replacement of an overloaded particle by smaller particles to represent crushing 3.3.5 Implementation flow chart The procedures for sample preparation and implementation of the crushing model in PFC3D are summarized in a flowchart shown in Figure 3-8. For sample preparation of a particle pack, 0.61 g particles were generated inside the cup following a specified particle size distribution and were settled under gravity. A platen shown in Figure 3-1 was used to load the particle pack to 0.5 MPa to obtain the initial particle pack height, based on which the axial strain of the particle pack was measured with the further deformation of the particle pack. The platen continued to load the particle pack to 25 MPa and the averaged constrained stiffness was calculated based on strains at 15 and 25 MPa. The calculated constrained stiffness was compared with laboratory data from the crush test (Figure 3-5). The above-mentioned procedures were repeated until the calculated constrained stiffness was matched to the laboratory data. The matched sample loaded to 0.5 MPa was used for subsequent simulations. σ1σ1 σ3σ355 After the sample preparation, the values of particle strength and Weibull modulus shown in Equation 3-5 were set and the loading on the particle pack was continued by moving the platen downward at a constant velocity. The particle breakage model was visited every 100 cycles in PFC3D until reaching the target compressive stress on the proppant pack. In this breakage model, the comminution limit is set as 0.1 mm that means particles with a diameter smaller than 0.1 mm are deemed unbreakable. Each particle was looped to check the comminution limit against the particle size and then the particle stress against the particle strength. A broken particle was replaced with four smaller particles shown in Figure 3-7 once the failure criterion was met. 56 Figure 3-8. Flowchart showing sample preparation and implementation of particle breakage model in DEM Generate a 0.61 g particle pack with a particle size distribution and settle to a cup under gravityLoad the particle pack to 0.5 MPa, record the particle pack heightLoad the particle pack to 25 MPa, calculate the average constrained stiffness between 15 and 25 MPaStiffness match lab data?Adjust particle-particle modulusNoSave the pack at 0.5 MPa for subsequent simulations Set particle strength and Weibull modulusParticle pack preparationi = 1Particle i at comminution l imit?i = i + 1exit modelYesCalculate stress for particle iMeet failure criterion? Calculate strength for particle iNoReplace particle i with smaller particles, run 200 cyclesYesi <n?NoReach targeted stress? StopNoParticle breakage modelLoad downward platen movement(Execute every 100 cycles)YesStartYesNo57 3.4 Selection of crushing index for breakage model Three stress indexes introduced in Section 3.3.2 have been used to represent the grain breakage threshold in the previous studies. A reasonable crushing index should: (i) give a similar stress threshold when loaded with a similar contact configuration and (ii) reflect the influence of different contact configurations in the crushing stress index. To understand the applicability of the three particle stress indexes for capturing the stress threshold of proppant crushing, the three crushing stress indexes were examined from a micro-scale perspective using PFC3D. A single particle with a diameter 0.6 mm was replaced with a cluster of 3680 smaller particles joined by 12104 linear parallel bonds that act like finite-size cement. The micro-parameters (shown in Table 3-3) for the clustered particle were selected to reproduce the mechanical behaviour of a quartz particle under compression. The linear parallel bonds can transmit the contact force and moment between two contact pieces, unless the maximum stresses from the moment and force overcome the bond strength that eliminates the bond. Table 3-3. Micro-parameters for smaller particles to form clustered particle Micro Parameter Value Particle diameter Ds (mm) 0.03-0.04 Particle density (kg/m3) 2650 Particle friction angle (°) 30 Particle-particle effective modulus (GPa) 4.0 Bond friction angle (°) 30 Bond effective modulus (GPa) 4.0 Bond tensile strength (GPa) 0.1 Bond cohesion (GPa) 0.5 Normal-to-shear stiffness ratio 1 Damping coefficient 0.4 First, a crush test on a single clustered particle was performed shown in Figure 3-9. The single clustered particle was placed between two platens (two contacts) and the compressive loading on the particle was conducted by moving the upper platen downward. 58 Figure 3-9. single particle (left), single clustered particle (middle) and contact force chain (right) between two platens in the crush test The number of the parallel bonds that failed in tension or shear was tracked as the particle was loaded. The single clustered particle was judged as broken when the number of ruptured bonds suddenly increased rapidly to more than roughly 500, which was accompanied by sudden drops in contact forces with the platen (Figure 3-10). Subsequently visible particle fragmentation was observed with the further loading. The loading process was repeated to a stage just before the particle fragmentation and the contact forces acting on the particle from platens were recorded. The average stress acting on the clustered particle was estimated using Equation 3-1 and the three stress indexes were also calculated using Equations 3-2, 3-3 and 3-4. Figure 3-10. Sudden drop in force accompanied by rapid increase in bond breakage number when a clustered particle fails under different contact number 03006009001200150018000510152025300 0.02 0.04 0.06 0.08Bond breakage numberForce (N)Displacement (mm)Force-Disp. (2 contacts)Force-Disp. (3 contacts)Force-Disp. (4 contacts)Bond-Disp. (2 contacts)Bond-Disp. (3 contacts)Bond-Disp. (4 contacts)59 A peak force of 20.1 N on the upper platen was recorded during the compression. The crushing threshold stress of the particle is 49.8 MPa using Index 1, 55.7 MPa using Index 2, and 52.8 MPa using Index 3. The clustered particle has a similar peak force to a quartz grain and the resulting particle threshold stress (Index 2: 55.7 MPa) is close to a quartz grain threshold stress (Index 2: 60 MPa for a 0.85 mm quartz grain) from the laboratory crush test on single grains (Nakata 1999). The number of platens was increased to three and four to investigate the influence of the contact number on the particle threshold stresses (Figure 3-11). The peak force of the upper platen increases with an increase in the contact number (Figure 3-10). The peak force is 23.8 N for the test with three platens, and is 26.3 N with four platens. The crushing threshold stress of the particle loaded with three platens is 45.4 MPa using Index 1, 66.0 MPa using Index 2, and 53.1 MPa using Index 3. When loaded with four platens, the crushing threshold stress of the particle is 35.4 MPa using Index 1, 73.0 MPa using Index 2, and 10.3 MPa using Index 3. It can be seen that the variation of stress threshold using Index 3 is much higher than that of Index 1 and 2 for different contact numbers. The crushing index 3 may not be sufficient in describing the threshold stress when a particle breaks. Figure 3-11. single clustered particle and contact force chain using three platens (left 1 and 2) and four platens (left 3 and 4) in particle crush test The whole process was repeated on a different randomly selected single particle inside a particle pack to investigate the particle crushing stress threshold under different contact configurations as shown in Figure 3-12. In the particle pack, the single clustered particle was surrounded by unbreakable particles and/or walls. The particle pack was compressed one-dimensionally and the number of ruptured parallel bonds was tracked during the loading to determine the threshold for 60 the clustered particle crushing. In total, the above procedures were repeated for 234 different clustered particles under different contact configurations. Figure 3-12. Schematic of clustered particle breakage caused by contacts with unbreakable particles and a wall (left) and the particle pack model in PFC3D (right) The bond breakage pattern of the clustered particle differs depending on the nature of the contact with the surrounding particles and the coordination number as shown in Figure 3-13. The bonds of the clustered particle usually break in a tensile mode. The first few bonds to break under different contact patterns typically occur along the direction of the largest contact force, which agrees with the analysis by Russell et al. (2009). When the coordination number is 3 or 4, large contact force chains developed at the contact points and the particle split into two halves along a plane formed by the three strongest contacts. This finding is similar to the results from the experimental crushing of single sand grains; the sand grains fail in a tensile mode (Nakata et al. 2001). With an increase in the coordination number, the contact force chain becomes well distributed throughout the particle and splitting/spalling of a smaller portion of the particle was observed. Figure 3-13. Typical contact force chain and bond breakage pattern for clustered particle with coordination number 3, 5, 7, and 9 (left to right) WallBond breakageUnbreakable particleBond-clustered particleUnbreakable particleUnbreakable particleUnbreakable particle61 The maximum principal stress and the minimum principal stress acting on a particle at the time that it breaks can be calculated from the stress tensor (Equation 3-1). The relation between the maximum principal stress and the minimum principal stress for 234 particles under different contact configurations is plotted in Figure 3-14. Particles with three contacts have a similar maximum principal stress with a minimum principal stress of approximately 0 MPa. Particles with a contact number of 4-6 have a minimum principal stress under 20 MPa when particles crush, while those of 7-10 have a minimum principal stress larger than 24 MPa. With an increase in the number of contacts on a single particle, the minimum principal stress increases and a higher maximum principal stress occurs when the particle breaks. Some variations in the principal stresses are observed for the same contact numbers. This may result from a different pattern (such as locations) of the contacts and possibly the average nature of the stress tensor (Equation 3-1) used to represent the stress distribution inside a particle. Figure 3-14. Relation between minimum and maximum principal stresses when a particle breaks from 234 simulations The values of the three stress indexes for each clustered particle were calculated and the results are plotted in Figure 3-15. The stresses were distributed in a narrow range of 24 to 60 MPa for Stress Index 1 and 52 to 80 MPa Stress Index 2. For Stress Index 3, a wide distribution from 8 to 60 MPa was found and about 10% of the stresses were less than 0, which means these particles 0501001502002503000 20 40 60 80 100Major principal stress (MPa)Minor principal stress (MPa)3 contacts4-6 contacts7-10 contactsMin mum principal stress (MPa)Maximum principal stress (MPa)62 would not break in a crushing model if Stress Index 3 was used in the crushing criterion because the particle strength is always larger than zero. The average and the standard deviation is 43.2 ± 12.4 MPa for Stress Index 1, 67.2 ± 18.6 MPa for Stress Index 2, and 27.1 ± 20.8 MPa for Stress Index 3. Figure 3-15. Distribution of the characteristic stress indexes for clustered particles when they broke The stress to break a clustered particle under different contact numbers was calculated. For Stress Indexes 1 and 2, the average stress to break a clustered particle increases with an increase of contact number. For Stress Index 1, the average stress is 30.0 ± 2.5 MPa, 41.4 ± 10.4 MPa, and 45.7 ± 13.8 MPa when the coordination number is 3, 4-6, and 7-10 respectively. For Stress Index 2, the average stress is 44.9 ± 6.5 MPa, 63.8 ± 14.5 MPa, 71.7 ± 21.4 MPa when the coordination number is 3, 4-6, and 7-10 respectively. For Stress Index 3, the average stress is 32.5 ± 2.5 MPa, 31.7 ± 17.4 MPa, 21.8 ± 23.2 MPa when the coordination number is 3, 4-6, and 7-10 respectively. The results above show that the Stress Index 3 may not be suitable as a stress threshold for particle breakage. Stress Indexes 1 and 2 are better because they enable particles to break at a similar stress with some variability for a given loading configuration, which is desirable for a reasonable 0510152025303540-40 0 40 80 120 160FrequencyCharacteristic stress index (MPa)Criterion 1Criterion 2Criterion 363 crushing model in reflecting the variability of grain strength (Hanley et al. 2015, Ciantia et al. 2015). To reflect the fact that a higher stress is required to break a particle with a higher contact number, Stress Indexes 1 and 2 can be modified by including a coordination-number dependent reduction factor, however, this is out of the scope of this research. 3.5 Determination of particle strength parameters for Jordan sand The particle strength σ0 of G600 Jordan sand corresponding to D0 = 0.655 mm and the Weibull modulus m in Equation 3-5 were determined by matching the particle grain size distribution in the DEM crush test with the laboratory data. Criterion 1 corresponding to Index 1 was chosen here for particle crushing. A preliminary calibration was performed in PFC3D to roughly determine the beginning values of σ0 and m respectively based on the laboratory crushing percentage and the void ratio versus compressive curve of a G600 sand pack. The values of σ0 and m were further adjusted to match the laboratory particle size distribution of G600 and Mesh 20/40 proppant packs. 3.5.1 Preliminary determination of σ0 and m The first step in the calibration process was to run models to compressive stresses of 30 MPa and 40 MPa without allowing particles to split to determine the stress index for all particles. The calculated stresses were sorted by magnitude and the percentage of particles with a stress index above an arbitrary level was determined. The percentages were determined for multiple levels enabling the cumulative probability curves shown in Figure 3-16 to be drawn. The next step was to use the measured percentage of crushed sand grains that occurred in the crush tests at overall compressive stresses of 30 MPa (13.9%) and 40 MPa (29.2%). The percentage of particles with a stress exceeding a stress level corresponding to a given crushing percentage from the laboratory test was used to estimate a starting value for the particle strength σ0 as reflected in the stress index. A reasonable particle strength σ0 in the PFC3D model should also result in a similar particle crushing percentage. The modelled value of Stress Index 1 corresponding to a 13.9% crushing percentage is 31.1 MPa. Likewise, for a particle pack loaded to 40 MPa, the corresponding value of Stress Index 1 is 31.9 MPa. The average value of 31.1 and 31.9 MPa, that is 31.5 MPa, was used as the initial value of the particle strength σ0 for the next calibration. 64 Figure 3-16. Stress Index 1 distribution for 1554 particles tested to two different compressive stresses The value of m can be determined based on the gradient of void ratio (e) versus compressive stress (σ) of the sand pack after a clear yield point from the laboratory crush test. McDowell and de Bono (2013) found a power law between e and σ as: 6/me  (3-7) which gives a linear normal compression line on double logarithmic axes of e and σ after the yield point. The void ratio versus compressive stress curves for G600 Jordan sand tested to 40 MPa under laboratory one-dimensional compression test are plotted in Figure 3-17. A value of m = 3.5 gives a good linear approximation to the linear normal compression line after the onset of particle crushing at around 30 MPa. This Weibull modulus was used for the next calibration. 01020304050607080901000 10 20 30 40 50 60 70 80Percentage largerCrushing stress index (MPa)40 MPa30 MPa31.9 MPa31.1 MPa13.9% particlecrushed29.2% particle crushed65 Figure 3-17. Void ratio vs compressive stress for G600 Jordan sand tested to 40 MPa 3.5.2 Final calibration of σ0 and m The calibration process continued by allowing a particle with a diameter D to break once the particle stress index exceeds the particle strength σs(D) calculated from σ0 shown in Equation 3-6. Using the value of σ0 = 31.5 MPa obtained from the preliminary calibration, the particle crushing percentage at 40 MPa is 35.5% and the particle-size distribution is shown in Figure 3-18. As more particle crushing happens in the numerical particle pack than that of the laboratory test, the value of σ0 was increased to reduce the particle crushing while fixing m = 3.5. As shown in Figure 3-18 and Figure 3-19, the degree of particle crushing become less severe with an increase in σ0. In Figure 3-18, the blue particles are those stay unbroken, the orange ones are particles after one fragmentation, and the red ones are those after more than one fragmentation. A value of σ0 = 35 MPa in the PFC3D simulation gives a close approximation to the laboratory particle-size distribution after a 40 MPa compression stress. 0.40.5 5 50Void ratioCompressive stress (MPa)12345Linear approximationy = 3.82x-3.5/666 Figure 3-18. Particle size distribution of particle pack using different σ0 (m = 3.5) after 40 MPa compression 01020304050607080901000 0.1 0.2 0.3 0.4 0.5 0.6 0.7Percent finerParticle size (mm)Lab data31.5 MPa35 MPa37 MPa40 MPa67 Figure 3-19. Particle crushing of particle pack (m = 3.5) after 40 MPa compression with an increase of σ0: 31.5 MPa (top left), 35 MPa (top right), 37 MPa (bottom left) and 40 MPa (bottom right) The value of σ0 was fixed to 35 MPa while varying the value of m to investigate the influence of the Weibull modulus on the particle size distribution. As shown in Figure 3-20 and Figure 3-21, the percentage of G600 particles finer than 0.6 mm is about the same with an increase of m from 1 to 7. However, a more well-distributed particle size distribution was obtained with a higher value of m. With an increase in m, the strength of fragments decreases according to Equation 3-5. When m < 3, a very small percentage of particles is finer than 300 µm at 40 MPa because the fragments are more resistant to crushing. Few fragmented particles were further crushed to finer fragments as shown in Figure 3-20. For m > 3, fragments are relatively easier crushed into finer fragments thus once a particle breaks it will be quickly crushed into many smaller particles. 68 Figure 3-20. Particle size distribution of particle pack using different m (σ0 = 35 MPa) after 40 MPa compression 01020304050607080901000 0.1 0.2 0.3 0.4 0.5 0.6 0.7Percent finerParticle size (mm)Lab m=1m=2 m=3m=4 m=5m=6 m=769 Figure 3-21. Particle crushing of particle pack after 40 MPa compression with an increase of m: 1 (top left), 3 (top right), 5 (bottom left) and 7 (bottom right) (σ0 = 35.0 MPa) Repeating the abovementioned procedures on a mesh 20/40 particle pack loaded to 40 MPa, a similar trend regarding the influence of m and σ0 on the degree of particle crushing and particle-size distributions was obtained. As shown in Figure 3-22, less particle crushing occurs with an increase in σ0. The increase in the value of m prohibits crushing in larger particles and facilitates crushing in smaller particles. It can be concluded that the particle strength controls the degree of particle crushing while the Weibull modulus governs the distribution pattern of particle sizes. The values of m and σ0 in the PFC3D crushing model were adjusted to match the laboratory results. A value of Weibull modulus equal to 4.0 gives a good approximation to the laboratory particle-size distribution. 70 Figure 3-22. Particle-size distribution with increase of σ0 (m = 3.5) (left) and increase of m (σ0 = 35.0 MPa) (right) The stress-strain responses of the G600 particle pack are plotted in Figure 3-23. From those simulations using different particle strengths but with the same Weibull modulus (m = 3.5), it can be seen that the onset stress of particle crushing resulting in the softening of the stress-strain responses are affected by the particle strength, after which the stress-strain curves show a similar gradient due to the same Weibull modulus used in the simulations. Likewise, the onset stress of particle crushing is similar for particle packs using different values of m but with the same particle strength. The gradient of the stress-strain curves the onset of particle crushing in PFC3D changes with the value of m. Higher values of m tend to give a softer stress-strain response and lower stress-strain gradient according to Equation 3-5. The PFC3D simulations give a softer stress-strain response than the laboratory data, as bigger stress drops occur when particles are crushed. The stress drops are larger with a higher value of m. This may due to the spherical particle shape that facilitates fragmented particle movement in the particle pack, resulting in a void space causing loss of support to adjacent particles. In addition, particle overlap between fragmented particles exists because of particle volume conservation, which creates artificial forces to push fragmented particles apart. One solution is to use clumps to mimic the shape of fragments to avoid overlap. However, using clumps is beyond the scope of this research. 01020304050607080901000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Percent finerParticle size (mm)OriginalAfter 40 MPa (lab)35 MPa40 MPa45 MPa01020304050607080901000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Percent finerParticle size (mm)OriginalAfter 40 MPa (lab)m=2m=3m=4m=571 Figure 3-23. G600 stress-strain response with increase of σ0 (m = 3.5) (left) and increase of m (σ0 = 35.0 MPa) (right) By slightly adjusting the Weibull modulus and particle strength, a particle strength of σ0 = 35 MPa and Weibull modulus m = 4.0 gives a good approximation to the stress-strain response and particle-size distribution of G600 loaded to 30 and 40 MPa and Mesh 20/40 load to 40 MPa as shown in Figure 3-24. Compared with the laboratory stress-strain response, PFC3D simulations still give a slightly softer stress-strain response. However, this may compensate for the higher value of initial porosity of the particle pack in PFC3D than that in the laboratory, resulting in a similar porosity at 40 MPa (G600: 0.458 from PFC3D, 0.445 from lab; Mesh 20/40: 0.458 from PFC3D, 0.421 from lab). 0102030400.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24Compressive stress (MPa) Vertical strainLab31.5 MPa35 MPa37 MPa0102030400.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21Compressive stress (MPa) Vertical strainLabm=1m=2m=3m=4m=572 Figure 3-24. Stress-strain responses and particle-size distribution by PFC3D using σ0 = 35 MPa and m = 4 Based on the calibration of the particle strength parameters, the strength (unit: MPa) of a particle of Jordan formation sand with a diameter D (unit: mm) following Equation 3-5 can be defined as:  4/3655.035DDs (3-8) 3.6 Determination of particle strength parameters for three types of proppants The procedure to obtain particle strength parameters of Jordan sand in Section 3.5 was repeated to determine the particle strength parameters for the other three types of proppants, namely Genoa sand, resin-coated sand, and the ceramic proppant. 3.6.1 Genoa sand The value of particle-particle effective modulus was adjusted downward to 0.71 GPa to match the linear part of the laboratory stress-strain response of G600 Genoa sand. The particle density is 2.51 g/cm3. The initial value of σ0 is 24.3 MPa corresponding to the particle crushing percentage (47.8%) at 40 MPa. The initial value of m based on the crush test of G600 Genoa is 4.0 as shown in Figure 3-25. Simulating particle crushing in a G600 Genoa particle pack under 40 MPa would be very computationally expensive because of the large particle number. Therefore, the mesh 20/40 Genoa sand was used instead for further calibration of σ0 and m. 0102030400 0.03 0.06 0.09 0.12 0.15 0.18 0.21Compressive stress (MPa) Vertical strainLab G600PFC G600Lab Mesh 20/40PFC Mesh 20/4001020304050607080901000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Percent finerParticle size (mm)Lab 40 MPaLab 30 MPaPFC 40 MPaPFC 30 MPaLab 40 MPaPFC 40 MPaG600Mesh20/4073 Figure 3-25. Void ratio vs compressive stress for G600 Genoa sand tested to 40 MPa When allowing particles to break in PFC3D, using σ0 = 24.3 MPa and m = 4.0 gives a lower particle crushing degree than that from the laboratory tests as shown in Figure 3-26. It should be noted that grains with diameters between 0.85-1.00 mm are often dark non-silica minerals as shown in Figure 2-17 and have lower crushing resistance. These grains were crushed to finer than 0.85 mm after a 40 MPa compression as seen from the laboratory particle-size distribution in Figure 3-26. By decreasing the value of σ0 to 23 MPa, the PFC3D simulation can better match the laboratory data regarding the particle size distribution of mesh 20/40 Genoa sand (Figure 3-26). The value of m was varied as shown in Figure 3-27. These results show that m = 4.0 can generally match the particle size distributions. 0.250.5 5 50Void ratioCompressive stress (MPa)12345Linear approximationy = 3.762x-4.0/674 Figure 3-26. Particle size distribution of Mesh 20/40 particle pack with different σ0 (m = 4.0) after 40 MPa compression Figure 3-27. Particle size distribution of Mesh 20/40 particle pack with different m (σ0 = 23 MPa) after 40 MPa compression The stress-strain responses from the PFC3D models deviate from the laboratory data because excessive particle crushing occurred when loaded to 40 MPa as shown in Figure 3-28. However, the trend of the PFC3D stress-strain response matches the laboratory data; the gradient of stress-01020304050607080901000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Percent finerParticle size (mm)OriginalLab 40 MPa24.3 MPa23 MPa22 MPa01020304050607080901000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Percent finerParticle size (mm)OriginalAfter 40 MPa (lab)m=1m=3m=4m=575 strain curves first decreases with particle crushing and then increases and remains constant with further compaction of fine fragments. Again, the softer stress-strain response in PFC3D compensates for the higher value of initial porosity of the particle pack. The porosity of mesh 20/40 Genoa particle pack under 40 MPa given by PFC3D is 0.322, which is close to 0.340 measured from the laboratory crush test. Figure 3-28. Stress-strain responses of particle packs using σ0 = 23.0 MPa and m = 4.0 3.6.2 Resin-coated sand The particle-particle effective modulus was adjusted to 1.2 GPa to match the linear part of the laboratory stress-strain response of G600 resin-coated sand. The particle density is 2.63 g/cm3. The initial value of σ0 is 41.1 MPa corresponding to the particle crushing percentage (12.6%) at 40 MPa. The initial value of m could not be obtained from the void ratio versus compressive stress curve of G600 resin-coated sand using Equation 3-7 because the resin coating held the fragments together after grain crushing in the crush test. Therefore, the initial value of m was set to 4.0, a typical value of silica sand. When allowing particles to break, using σ0 = 41.1 MPa and m = 4.0 gave a higher particle crushing percentage than the laboratory data as shown in Figure 3-29. By increasing the value of σ0 to 50 MPa (m = 4.0), the PFC3D simulations gave a good match to the laboratory data regarding the 0102030400 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.3Compressive stress (MPa) Vertical strainG600 LabG600 PFC (no crushing)Mesh 20/40 LabMesh 20/40 PFC76 particle size distribution of G600 resin sand and mesh 20/40 resin sand under 40 MPa compression (Figure 3-29 and Figure 3-30). The stress-strain responses via PFC3D generally fit the laboratory data with some deviations as shown in Figure 3-31. The porosity of G600 and mesh 20/40 Genoa particle packs given by PFC3D is respectively, 0.574 and 0.586, compared with 0.474 and 0.498 measured from the laboratory crush test. Figure 3-29. Particle size distribution of G600 particle pack using different σ0 (m = 4.0) after 40 MPa compression 01020304050607080901000 0.1 0.2 0.3 0.4 0.5 0.6 0.7Percent finerParticle size (mm)Lab data41.1 MPa45 MPa47 MPa50 MPa77 Figure 3-30. Particle size distribution of mesh 20/40 particle pack (σ0 = 50 MPa and m = 4.0) after 40 MPa compression Figure 3-31. Stress-strain responses of particle packs using σ0 = 50.0 MPa and m = 4.0 3.6.3 Ceramic proppant The particle-particle effective modulus was adjusted to 2.2 GPa to match the linear part of the laboratory stress-strain response of G600 ceramic proppant. The particle density is 2.81 g/cm3. 01020304050607080901000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Percent finerParticle size (mm)OriginalAfter 40 MPa (lab)After 40 MPa (PFC)0102030400 0.03 0.06 0.09 0.12 0.15Compressive stress (MPa) Vertical strainG600 LabG600 PFC (no crushing)G600 PFC (with crushing)Mesh 20/40 LabMesh 20/40 PFC78 The initial particle strength σ0 is 47.2 MPa corresponding to the particle crushing percentage (12.8%) at 40 MPa. The initial value of m was set to 4.0. When allowing particles to break, using σ0 = 47.2 MPa and m = 4.0 gives a much higher particle crushing percentage as shown in Figure 3-32. After increasing the value of σ0 to 65 MPa (m = 4), the PFC3D simulations gave a good match to the laboratory particle size distribution of G600 ceramic proppants and mesh 20/40 ceramic proppants under 40 MPa compression (Figure 3-32 and Figure 3-33). The stress-strain responses via PFC3D generally fit the laboratory data with some deviations as shown in Figure 3-34. The porosity of G600 and mesh 20/40 ceramic particle packs given by PFC3D is respectively, 0.679 and 0.671 (more crushing should happen), which is slightly higher than 0.580 and 0.599 as measured from the laboratory crush test. Figure 3-32. Particle size distribution of G600 particle pack using different σ0 (m = 4.0) 01020304050607080901000 0.1 0.2 0.3 0.4 0.5 0.6 0.7Percent finerParticle size (mm)Lab data47.2 MPa60 MPa63 MPa65 MPa79 Figure 3-33. Particle size distribution of Mesh 20/40 particle pack (σ0 = 65 MPa and m = 4.0) Figure 3-34. Stress-strain responses of particle packs using σ0 = 65.0 MPa and m = 4.0 3.7 Summary PFC3D without a particle-crushing model can simulate the initial loading response of proppants during compaction, but is unable to capture the stiffness reduction as well as the change in the particle-size distribution in a proppant pack as grains begin to crush under high stresses. A particle 01020304050607080901000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Percent finerParticle size (mm)OriginalAfter 40 MPa (lab)After 40 MPa (PFC)0102030400 0.03 0.06 0.09 0.12Compressive stress (MPa) Vertical strainG600 LabG600 PFC (no crushing)G600 PFC (crushing)Mesh 20/40 LabMesh 20/40 PFC (crushing)80 breakage model was implemented to simulate particle crushing in PFC3D. The micro-scale examination on stress indexes shows that the octahedral shear stress or the maximum contact force can represent the threshold stress when a particle breaks, while a crushing criterion based on biaxial loading did not work well. The octahedral shear stress was selected as the particle stress index to further calibrate the particle crushing strength and Weibull modulus for four proppants. The preliminary values of the particle crushing strength and Weibull modulus were obtained from the particle crushing percentage and the compressive stress versus void ratio curve. These values were further calibrated by trial and error to match the particle-size distribution and the stress-strain response in PFC3D with the laboratory data. It was found that particle strength σ0 controls the degree of particle crushing while Weibull modulus m governs the pattern of particle size distributions. For Jordan sand, Genoa sand, resin-coated sand, and ceramic proppants, the calibrated particle-particle effective modulus is, respectively, 1.1, 0.71, 1.2, and 2.2 GPa. The particle strength is 35, 23, 50, and 65 MPa respectively, and the Weibull modulus is 4.0 for all. These values give a good approximation regarding the particle-size distributions of the laboratory data. 81 4 Mechanical Properties of Montney Siltstone 4.1 Overview The unconventional Lower Triassic Montney play is one of the biggest unconventional resources in Canada, with estimated 444,600,000,000 m3 (15.7 Tcf) recoverable unconventional natural gas across an approximate area of 29,850 km2 in Northeastern British Columbia (BC) (Figure 4-1). Over 2,344 wells have been drilled in this area since 2005 and the cumulative gas production has exceeded 65,100,000,000 m3. Based on the completion data of the Montney formation from 2010 to 2013, typically multistage hydraulic fracturing of a horizontal well contains 10 to 16 stages. For every stage, an average of 100,000 – 150,000 kg proppants and 550 - 1,190 m3 fracturing fluids were pumped to create fractures (BC Oil and Gas Commission 2013a). Figure 4-1. Montney formation in Northeastern BC (BC Oil and Gas Commission 2012) Maintaining the conductivity of hydraulic fractures after hydraulic fracturing is a key to successful exploitation of unconventional hydrocarbon resources, which requires in-depth knowledge of the reservoir and mechanical properties of the target play. Although various authors have investigated Montney PlayDry GasModerate Gas LiquidsSignificant Wet GasOilWell # 2823282 the petrophysical and geomechanical characteristics of Montney formation (Davey 2007, Cui and Brezovski 2013, Gensterblum et al. 2015, Ghanizadeh et al. 2015), few studies have been reported on proppant embedment in the Montney formation. Proppant embedment into rock formations involves the elastic and plastic deformation of local fracture surfaces, usually causing an unrecoverable indention under high compressive stress (Zhang et al. 2015). Proppant embedment into fracture surfaces is largely affected by the mechanical properties of formation rock such as Young’s modulus. Laboratory experiments on proppant embedment into rock formations show that proppant embedment increased with a decrease in Young’s modulus (Alramahi and Sundberg 2012). Large plastic strains were observed from laboratory triaxial tests on the Nordegg formation under high confining pressure and the embedment contributed by plastic deformation after reaching onset yield stress was over 100 times higher than that by elastic deformation based on finite element analysis (Cui et al. 2014). Proppant embedment was also related to the clay content in rock formations that can cause swelling under saturation of fracture fluids, which leads to a reduction in rock strength and elastic modulus (Corapcioglu et al. 2014). Some other research shows that calcite-rich formations are more sensitive to water-based fluids due to carbonate dissolution in water causing a reduction in Young’s modulus (Akrad et al. 2011). The saturated depth of fracturing fluids into rock formations typically ranges from 2.5 to 10.2 cm (McElfresh et al. 2002). Proppant embedment causing an indentation on formation fracture surfaces is fundamentally similar to indentation testing that measures the hardness of materials (Mueller and Amro 2015). In practice, indentation testing has been used to measure the hardness of rock formations, which quantifies the resistance of rock formations against the penetration of an indenter. Kumar et al. (2012) used nano-indentation tests on samples from five shale plays and found a good correlation between Young’s modulus interpreted from a nano-indentation force-displacement curve and other properties including dynamic Young’s modulus, hardness, and clay content. Changes in Young’s modulus and hardness of shale rock exposed to different fracturing fluids were quantified with the indentation tests (Yang et al. 2016). This chapter presents the results from comprehensive rock mechanics testing (RMT) on a siltstone from a well in the Montney formation. The well is HZ TOWN C-031-H/094-B-09, well # 28232 83 and its location is shown in Figure 4-1. The core samples tested were 2068-2270 m beneath the ground surface. The mechanical properties of the siltstone rock were characterized using indirect tensile tests, uniaxial compressive tests, and triaxial compressive tests. The resistance of the siltstone rock to indentation was investigated with Brinell indentation tests and the obtained Brinell hardness was correlated to the mechanical properties. Furthermore, a proppant embedment test was conducted to investigate the mechanical response of a monolayer proppant pack sandwiched between two rock surfaces under a triaxial loading. It should be noted that creep deformation of proppants and fractures is not considered in the current research. 4.2 Montney siltstone core samples The well was drilled in December 2012 and is currently active in producing natural gas. The targeted Montney formation is 2064.27 m (true vertical depth (TVD)) beneath the ground and has a thickness of about 350 m. The formation lies between the Doig formation and the Belloy formation as shown in Figure 4-2. Figure 4-2. Well HZ TOWN C-031-H/094-B-09 in Montney formation 2064 m350 m2258 mDoig formationBelloy formationMontney formation……4274 m MDWellCore depth84 Four whole cores were drilled from different depths into the Montney formation. These included a 54 m long Core #1 from the Doig and Montney formation (2068 to 2122 m measured depth (MD)), 72 m Core #2 (2122 to 2194 m MD), 81 m Core #3 (2194 to 2275 m MD), and 82 m Core #4 (2275 to 2357 m MD). The Montney formation in this location is dominated by siltstone mixed with a small portion of shale. The samples tested in this study were obtained from the archived Core #1, #3 and #4 from the core laboratory facility of BC Oil and Gas Commission as shown in Appendix C. Small diameter cores were drilled from slabs and were further sub-cored to take approximate 25.4 mm diameter by 50.8 mm long cylindrical plugs used for the uniaxial compressive tests, triaxial compressive tests, and embedment tests. Approximate 38.1 cm diameter by 19.1 cm long plugs were used for the indirect tensile tests. Approximate 25.4 mm diameter by 25.4 mm long cylindrical plugs were used for the Brinell indentation test. The RMT matrix is shown in Appendix C. In total, 57 samples were tested for these three sections, including three uniaxial compressive tests, six triaxial compressive tests, 18 indirect tensile tests, 20 Brinell hardness tests, and 10 proppant embedment tests. 4.3 Rock mechanics testing 4.3.1 Indirect tensile test Disk-shaped samples were tested as received under ambient conditions using a MetaRock testing system. The disk-shaped samples were compressed across a diameter between flat surfaces (line loading) with a loading rate of 0.22 MPa/s until failure. Two loading orientations relative to the bedding layer were used (Figure 4-3). 85 Figure 4-3. Loading orientation to bedding layer: perpendicular (left) and parallel (right) The tensile strength ft is calculated from the applied load Fpeak at failure using Equation 4-1. LDFfpeakt2 (4-1) where D and L are the diameter and the length of the sample. Table 4-1 summarizes the results of indirect tensile tests. For the first section, the tensile strength of samples at the same depth loaded perpendicular and parallel to the bedded is similar. The average strength at a depth of 2068.41 m MD is 6.57 ± 0.56 MPa. The average strength at a depth of 2068.64 m MD is 9.16 ± 0.32 MPa. For the second section, three samples were subjected to perpendicular loading while the other three samples were under parallel loading. Sample TWN2-03 failed before the test. The average tensile strength of five samples is 9.87 ± 1.11 MPa. The average tensile strength of samples loaded perpendicular to the bedding is 10.92 ± 0.59 MPa, which is slightly higher than 9.17 ± 0.68 MPa for samples loaded parallel. For the third section, the tensile strength loaded in the parallel direction (6.44 ± 0.65 MPa) is much smaller than that loaded perpendicular (13.69 ± 1.13 MPa). 86 Table 4-1. Indirect tensile test results It can be seen from Figure 4-4 that multiple cracks were formed along the interfaces of the bedding layers. These crack occurred before reaching the peak stress for those samples loaded parallel to the bedding. For those samples loaded perpendicularly, one or two maximum cracks were formed along the loading points by penetrating the bedding layers with some cracks formed along the interface of the bedding layers. Sample ID Depth MD (m) Loading direction Bulk density (g/cc) Tensile strength (MPa) TWN1-5 2068.41 Parallel 2.52 6.77 TWN1-6 2068.41 Parallel 2.51 5.93 TWN1-7 2068.41 Perpendicular 2.49 7.00 TWN1-10 2068.64 Parallel 2.59 9.24 TWN1-11 2068.64 Perpendicular 2.55 9.43 TWN1-12 2068.64 Perpendicular 2.57 8.81 TWN2-01 2265.35 Perpendicular 2.57 10.50 TWN2-02 2265.35 Perpendicular 2.56 11.34 TWN2-03 2265.35 Perpendicular Failed pre-test Failed pre-test TWN2-10 2265.89 Parallel 2.57 8.41 TWN2-11 2265.89 Parallel 2.55 9.39 TWN2-12 2265.89 Parallel 2.58 9.71 TWN3-1 2335.94 Parallel 2.53 7.39 TWN3-2 2335.94 Parallel 2.54 6.19 TWN3-3 2335.94 Parallel 2.60 6.20 TWN3-14 2336.9 Parallel 2.61 5.96 TWN3-15 2336.9 Perpendicular 2.59 14.49 TWN3-16 2336.9 Perpendicular 2.61 12.89 87 Figure 4-4. Failure pattern of sample loaded perpendicular (left) and loaded parallel to the bedding (right) 4.3.2 Uniaxial compressive test Cylindrical-shaped samples were tested as received under ambient conditions using a GCTX RTX-3000 testing system. Samples were mounted in the testing vessel and the axial and radial deformation was measured, respectively, with two LDVTs and one LDVT. Samples were loaded axially at a displacement rate of 1×10-6 m/s until failure. Based on the peak load measured at failure, the uniaxial compressive strength (UCS) was determined. Young’s modulus and Poisson’s ratio were obtained using the tangential gradient of the axial stress-strain curve and the ratio between the radial strain and the axial strain at a stress of 50% of UCS. One sample from each section was tested under uniaxial compression and the results are summarized in Table 4-2. The stress-strain responses are shown in Appendix C. Three samples have a similar Young’s modulus at approximately 29 GPa while their UCS values differ. The TWN1-01 sample has a UCS of 127.0 MPa, which is >10 times higher than the tensile strength. The TWN2-04 sample has a UCS of 98.2 MPa, which is about 10 times of the average tensile strength. The TWN3-04 sample has a UCS of 63.1 MPa, which is about 10 times of the tensile strength loaded in parallel. Using the ratio of the gradient of axial strain-axial stress over the ratio of the gradient of radial strain-axial stress, the Poisson’s ratio is 0.146 for TWN1-01, 0.440 for TWN2-04, and 0.396 for TWN 3-04. It should be noted that the Poisson’s ratio of both samples 88 TWN2-04 and TWN3-04 is slightly higher than the typical Poisson’s ratio of the Montney siltstone, which is about 0.20-0.25. This could be caused by core drying which facilitates splitting along the laminate layers as shown in the failure patterns (Cui et al. 2014). Table 4-2. Uniaxial compressive test results (orientation: horizontal) Sample ID Depth MD (m) Bulk density (g/cc) Peak strength (MPa) Young’s modulus (GPa) Poisson’s ratio TWN1-01 2068.12 2.60 127.0 27.7 0.146 TWN2-04 2265.60 2.57 98.2 29.4 0.440 TWN3-04 2336.04 2.59 63.1 30.1 0.396 4.3.3 Triaxial compressive test Samples were tested using the same loading frame as was used for the uniaxial compressive tests. A small confining pressure was applied to samples to seat the jacket and end-caps. The confining pressure was increased at a loading rate of 0.007 MPa/s, until it reached a prescribed value, after which the confining pressure was maintained constant. Strain equilibration was allowed to occur before any additional loading. An axial stress difference was then applied, at a controlled axial strain rate of 5×10-7 m/s. Loading continued until a peak stress was reached and then beyond this to determine a residual strength value. Two samples from each section were tested with confining pressures of 10 and 20 MPa and the results are summarized in Table 4-3. The stress-strain responses are shown in Appendix C. The section #1 has a compressive strength (peak deviatoric stress + confining pressure) of 170.5 MPa at a 10 MPa confining stress, and a strength of 266.0 MPa at a 20 MPa confining stress. The section #2 has a compressive strength of 211.0 MPa at 10 MPa and a compressive strength of 282.9 MPa at 20 MPa. The section #3 has a compressive strength of 124.7 MPa at 10 MPa and a compressive strength of 210.2 MPa. The Young’s modulus from triaxial tests (confining pressure 10 and 20 MPa) ranges from 40.5 to 45.0 GPa. These values are slightly higher than those from the uniaxial compressive tests. The Poisson’s ratio from the triaxial compressive tests ranges from 0.25-0.3, which were still slightly higher than the typical value of the Montney siltstone (about 0.20-0.25). 89 The samples behave elastically and show little plasticity under compression. The samples confined with 20 MPa show slightly more plasticity than the samples confined with 10 MPa when approaching the vicinity of the peak stress. After passing the peak stress, the deviator stress of both two samples decreases dramatically. The post-failure patterns of the samples shift from splitting fracture when confined at 10 MPa to shear fracture when confined at 20 MPa. Table 4-3. Triaxial compressive test results (orientation: horizontal) Sample ID Depth MD (m) Bulk density (g/cc) Confining pressure (MPa) Peak strength (MPa) Young’s modulus (GPa) Poisson’s ratio TWN1-03 2068.29 2.52 10 170.5 40.5 0.268 TWN1-08 2068.55 2.62 20 266.0 43.5 0.301 TWN2-06 2265.73 2.59 10 211.0 44.9 0.242 TWN2-08 2265.85 2.59 20 282.9 41.3 0.308 TWN3-06 2336.08 2.57 10 124.7 41.6 0.259 TWN3-08 2336.21 2.61 20 210.2 43.5 0.244 4.3.4 Comparison with laboratory test data on fresh cores The rock testing data presented in the previous sections was from archived cores, which may be dried out resulting in different mechanical properties from preserved fresh cores. The test data from the archived cores was compared with a rich data set of RMT on relatively fresh cores from the same well location (BC Oil and Gas Commission 2014a). The RMT on the fresh cores were performed mostly in 2013 and early 2014. Data of the routine rock analysis (RCA) of the fresh cores was also included here. The comparison of bulk density between fresh cores and archived cores was plotted in Figure 4-5. It should be noted that the bulk density from RCA was measured by the mercury immersion method, while that in RMT was measured by the volumetric method using the known measures of core diameter and length. It can be seen that the bulk density of fresh cores from RCA is consistent with that from RMT. The bulk density of the archived cores is slightly lower than that of fresh cores, which may result from the drying of the core samples. The depth shown in Figure 4-5 approximately covers the Montney formation and it shows that the 90 Montney formation in this location has a bulk density mostly in the range of 2500 to 2650 kg/m3. The upper part of the Montney formation has a slightly lower bulk density than that of the lower part, which may reflect the heterogeneity of the Montney formation. Figure 4-5. Comparison of bulk density with depth between fresh cores and archived cores The results of uniaxial compressive and triaxial compressive tests on the fresh cores are shown in Table 4-4 and Table 4-5. Note that the orientation of the archived cores is horizontal. The UCS of the fresh cores is very close to that of the archived cores, while the uniaxial Young’s modulus of the fresh core (2335.8 m MD) is much higher than that of the archived core (2336.04 m MD). This may result from the drying out of samples causing splitting along the laminate layers as shown in the failure patterns (Figure 4-6). The peak strength of the fresh cores from the triaxial tests is generally higher than that of the archived cores due to a higher confining pressure. The Young’s modulus from the triaxial tests is similar between the fresh cores and the archived cores at around 40 GPa. This is also close to the average Young’s modulus interpreted from the Brinell hardness tests that will be presented in Section 4.4. The Poisson’s ratio of the archived rock from triaxial tests (around 0.25) is slightly higher than that of the fresh cores (around 0.20). The Young’s modulus of the archived rock is closer to that of fresh cores in the triaxial tests than the uniaxial 205021002150220022502300235024002450 2500 2550 2600 2650 2700 2750 2800Depth (m)Bulk density (kg/m3)RCA (fresh)RMT (fresh)RMT (archived)91 compressive tests, which may be due to the confining pressure that can counteract the splitting along the laminate layers. Table 4-4. Uniaxial compressive test results for fresh cores (BC Oil and Gas Commission 2014a) Sample ID Orientation Depth MD (m) Bulk density (g/cc) Peak strength (MPa) Young’s modulus (GPa) Poisson’s ratio TWN1-13 H 2090.56 2.531 108.35 26.20 0.31 TWN2-11 H 2116.57 2.499 107.35 38.95 0.2 TWN3-11 H 2313.56 2.604 98.91 45.48 0.2 TWN4-11 H 2335.8 2.593 63.12 41.36 0.38 TWN1-7 V 2090.71 2.548 135.68 20.40 0.25 TWN2-3 V 2116.45 2.589 141.74 22.03 0.2 TWN3-1 V 2313.35 2.608 184.01 29.27 0.16 TWN4-1 V 2335.67 2.622 157.40 28.26 0.19 92 Table 4-5. Triaxial compressive test results for fresh cores (BC Oil and Gas Commission 2014a) Sample ID Orientation Depth MD (m) Bulk density (g/cc) Confining pressure (MPa) Peak strength (MPa) Young’s modulus (GPa) Poisson’s ratio TWN1-11 H 2090.54 2.544 33 277 34.93 0.19 TWN1-12 H 2090.55 2.535 33 276 37.30 0.18 TWN1-9 H 2090.51 2.532 40 304 33.03 0.17 TWN2-8 H 2116.53 2.576 33 270 41.70 0.20 TWN2-9 H 2116.45 2.577 40 293 41.56 0.20 TWN3-13 H 2313.60 2.611 33 293 47.55 0.22 TWN3-9 H 2313.54 2.606 40 302 46.96 0.22 TWN4-12 H 2335.83 2.613 33 254 45.31 0.20 TWN4-13 H 2335.83 2.607 33 255 43.30 0.20 TWN1-4 V 2090.71 2.580 33 270 24.82 0.11 TWN1-5 V 2090.71 2.556 33 269 26.64 0.20 TWN1-2 V 2090.71 2.555 40 292 25.33 0.17 TWN2-5 V 2116.45 2.578 33 255 26.84 0.20 TWN2-4 V 2116.45 2.570 40 271 25.33 0.19 TWN3-2 V 2313.35 2.608 33 285 32.56 0.19 TWN3-6 V 2313.35 2.613 40 298 32.99 0.24 TWN4-3 V 2335.67 2.618 33 252 30.87 0.20 TWN4-5 V 2335.67 2.618 33 258 35.64 0.19 93 Figure 4-6. Comparison of failure patterns in uniaxial compressive tests: archived core – H (left); fresh core – TWN 4-11 H (middle); fresh core - TWN 4-1 V (right) (BC Oil and Gas Commission 2014a) The anisotropy of Montney formation was also obtained by comparing the mechanical parameters measured from tests in the horizontal and vertical orientations. It can be seen from Table 4-4 that the Young’s modulus along the horizontal orientation is much higher than that along the vertical orientation, while the peak strength is lower than that along the vertical orientation. The Poisson’s ratio along the horizontal orientation is higher than that along the vertical orientation, which may be associated with the shear failure mode in samples cored vertically compared with splitting failure mode in samples cored horizontally (Figure 4-6). From the triaxial tests shown in Table 4-5, the Young’s modulus in the horizontal orientation is lower than that in the vertical orientation, while the peak strength and the Poisson’s ratio is similar to each other with a similar shear failure mode shown in Figure 4-7. 94 Figure 4-7. Comparison of failure patterns of samples in triaxial test with confining pressure 33 MPa: fresh core - TWN 3-13 H (left); fresh core – TWN 3-2 V (right) (BC Oil and Gas Commission 2014a) The peak strengths corresponding to different confining stresses are plotted in Figure 4-8 and each best-fitted line was used to derive the cohesion and the friction angle assuming a linear Mohr-Coulomb failure criteria. It can be seen that the cohesion and the friction angle of the fresh cores along the horizontal orientation are similar to those of the archived cores, both of which have a higher friction angle and a lower cohesion than the cores along the vertical orientation. Figure 4-8. Plots of linear Mohr-Coulomb failure criteria in maximum - minimum principal stress space using peak strength 0501001502002503003500 10 20 30 40 50Compressive strength (MPa)Confining stress (MPa)Fresh H 2116 mFresh H 2313 mFresh V 2116 mFresh V 2313 mArchived H 2068 mArchived H 2265 mArchived H 2336 mφ (°) c (MPa)414332294854452522395422171495 4.4 Brinell hardness test Disk-shaped samples were indented with a Tungsten carbide indenter with a spherical 1 mm diameter tip using a MetaRock testing system (Figure 4-9). The samples were mold with epoxy in proper position and were polished to have a smooth surface. All the samples were first tested as received under ambient conditions. The indenter was perpendicularly brought into contact with the sample surface under a contact load of 20 mN. The force was increased up to 35 N and then unloaded to 0 N under a constant loading rate of 17.5 N/min. The force and the indentation depth were recorded during the test. Nine locations (three by three) were indented for each sample as shown in Figure 4-9. To assess the influence of saturation on the fracture surface, two samples were selected for re-resting after testing all the samples as received. One sample, TWN2-05, was soaked in tap water and the other one, TWN2-07, was soaked in 3% KCl fluid, both for one week. Then Brinell hardness tests were performed again. After that, these two samples were air dried for one week and were tested for the third time. Figure 4-9. Sample (left) and indenter (right) for Brinell hardness test A typical force-indentation curve for a tested sample is shown in Figure 4-10. The test force and the gradient of test force-indentation depth increase with an increase in the indentation depth. The force-indentation depth curve during the unloading process shows a similar trend to that of the 3 by 3 Indentation points 96 loading process, except that the gradient of the force-indentation depth is steeper upon unloading and left an unrecoverable permanent indentation after complete unloading as shown in Figure 4-11. Figure 4-10. Typical force-indentation curve from indentation tests on Montney siltstone Figure 4-11. Permanent indentation into surface after complete unloading 05101520253035400 5 10 15 20Test Force (N)Indentation depth (µm)hmPmhcS=(dP/dh)hmaxɛPmSIndentation97 The true contact area Ac (in unit mm2) at the peak force Pm (in unit N) with the known indenter diameter Di (in unit mm) and the measured indentation depth hm (in unit mm) is (ASTM E10-17 2017): imc DhA  (4-2) Brinell hardness (HB, in unit HBW) is calculated as: cmAPHB80665.9 (4-3) Young’s modulus was also obtained following the method by Zeng and Chiu (2001). A power law function with constants k and h0 was used to fit the upper 80% of the unloading data shown in Figure 4-10. kmmhhhhPP 00 (4-4) The contact stiffness S was given by the derivative of P on h at the maximum indentation depth hm.   10 hhmPS mm (4-5) The test result can also be linked to the Young’s modulus of the test sample. The apparent Young’s modulus EA between the indentation tip and the test sample is: cAASE2 (4-6) The Young’s modulus of the test material, E, can be calculated using the following equation: EEE iiA22 111   (4-7) where Ei and vi are the Young’s modulus and the Poisson’s ratio of the Tungsten carbide indenter, and v is the Poisson’s ratio of the test material. In Equation 4-7, the first term on the right side of the equation is ignored considering Ei is much higher than E. A value of 0.3 is used for v based on the previous testing on the Montney siltstone. 98 The Brinell hardness and the corresponding Young’s modulus of all the samples tested are listed in Table 4-6. It can be seen that the Hardness and Young’s modulus varies greatly across the span of the sample depth from 2068.12 to 2337.39 m MD. The maximum indentation depth at a 35 N indentation force ranges from 9.7 to 19.2 µm. The value of Brinell hardness ranges from 59 to 117 HBW. The Young’s modulus interpreted from the Brinell hardness tests ranges from 17.6 to 78.1 GPa, with an average of 42.0 GPa and a standard deviation of 14.8 GPa. The average value of Young’s modulus is close to those from the triaxial compressive tests in Table 4-3. The relatively large deviation in Young’s modulus and Brinell hardness may be due to the inhomogeneity of rock along different depths and under a small scale. 99 Table 4-6. Brinell indentation test results (horizontal, tested as received, 35 N) Sample ID Depth MD (m) hm at 35 N (µm) Hardness (HBW) Young’s Modulus (GPa) TWN1-02 2068.12 15.0 75.8 36.8 TWN1-04 2068.29 17.0 66.9 34.0 TWN1-09 2068.55 15.2 74.7 32.7 TWN1-14 2068.76 14.2 80.1 58.1 TWN1-16 2068.90 14.3 79.3 46.4 TWN1-18 2069.05 16.0 71.0 35.1 TWN2-05 2265.60 18.8 60.5 28.9 TWN2-07 2265.73 16.3 69.9 32.7 TWN2-09 2265.85 18.5 61.5 17.6 TWN2-14 2266.05 14.3 79.7 31.9 TWN2-16 2266.21 14.8 76.6 32.1 TWN2-18 2266.50 12.6 90.4 57.0 TWN2-20 2266.99 14.3 79.6 41.3 TWN2-22 2267.08 9.7 117.0 69.9 TWN3-05 2336.04 15.0 75.8 58.4 TWN3-07 2336.08 11.7 97.0 78.1 TWN3-09 2336.21 16.6 68.4 48.5 TWN3-11 2336.37 12.3 92.1 48.0 TWN3-13 2336.57 18.2 62.3 25.6 TWN3-18 2337.16 19.2 59.0 30.9 TWN3-20 2337.22 15.4 73.7 42.1 TWN3-22 2337.39 14.3 79.3 37.9 100 Figure 4-12 shows the correlations between the Brinell hardness values and the Young’s modulus values calculated from the hardness tests for different indentation forces. Note that the Brinell hardness determined from Equation 4-3 changes with the indentation force, as the relation between indentation force and depth is not linear, especially at the initial loading stage with low indentation forces. The correlations between the Brinell hardness values and the Young’s modulus values are more positive at higher indentation forces compared with those at lower indentation forces. Figure 4-12. Correlation between Young’s modulus and Brinell hardness using indentation depth under different indentation loads The force-indentation curves of two samples exposed to water/KCl fluid are shown in Figure 4-13 and Figure 4-14. The force-indentation curves of the two samples tested as received are very similar. The indentation depth was around 15 µm when the test force reached 35 N. A slightly deeper indentation was observed for the sample TWN2-05 than that of the sample TWN2-07. Deviations in the force-indentation depth curves (10-20 µm at 35 N) were observed due to the inhomogeneity of the samples. After saturated with water, the force-indentation depth curves for the sample TWN2-05 show little difference from those tested as received and tested under air-dry y = 19.43x0.37R² = 0.62y = 18.69x0.34R² = 0.57y = 18.65x0.28R² = 0.42y = 20.91x0.14R² = 0.09025507510012510 30 50 70 90Brinell hardness (HB)Young's modulus (GPa)35 N25 N15 N5 N101 conditions. For the sample TWN2-07 saturated with 3% KCl, a slightly smaller indentation (about 3.5 µm) was observed compared to those tested as received. The indentation depth at 35 N after air drying was the smallest. Figure 4-13. Force-indentation curves under conditions as received (top left), soaked in water (top right), and air dried (bottom left) at locations shown in (bottom right) for sample TWN2-05 05101520253035400 5 10 15 20 25Test Force (N)Indentation depth (µm)12345678905101520253035400 5 10 15 20 25Test Force (N)Indentation depth (µm)12345678905101520253035400 5 10 15 20 25Test Force (N)Indentation depth (µm)123456789102 Figure 4-14. Force-indentation curves under conditions as received (top left), soaked with 3% KCl (top right), and air dried (bottom left) at locations shown in (bottom right) for sample TWN2-07 The hardness and Young’s moduli are summarized in Table 4-7. For the sample TWN2-05 soaked in water, the values of hardness and Young’s modulus are similar during the three indentation tests. For the sample soaked in 3% KCl, the hardness and Young’s modulus from the 2nd test and 3rd test increased greatly compared with that tested as received. Surface softening of siltstone in hydraulic fracturing is usually caused by water migration to clay minerals in siltstone resulting in clay expansion and reduction in clay bond strength. Zhang et al. (2015) also showed that Berea sandstone with a lower clay composition has much lower proppant embedment than Barnett shale. KCl fluids seem to increase the hardness of the tested Montney sample as KCl fluid can help 05101520253035400 5 10 15 20 25Test Force (N)Indentation depth (µm)12345678905101520253035400 5 10 15 20 25Test Force (N)Indentation depth (µm)12345678905101520253035400 5 10 15 20 25Test Force (N)Indentation depth (µm)13456789103 control clay swelling and increase the bonding between cement and formation (Kaufman et al. 2008). In addition, the Montney siltstone samples selected for testing are from preferably competent and thus stronger zones, which usually have a very small clay content. This may explain the observed little softening due to exposure to water/KCL fluid. Table 4-7. Average Brinell measurements of samples soaked in water/KCl Sample ID Orientation and depth (m) Test conditions hm at 35 N (µm) Hardness (HBW) Young’s modulus (GPa) TWN2-05 H 2265.60 As received 18.8 60.5 28.9 Soaked in water 17.6 64.5 35.8 Air dried 17.6 64.4 27.3 TWN2-07 H 2265.73 As received 16.3 69.9 32.7 Soaked in 3% KCl 12.7 89.2 36.4 Air dried 12.4 91.9 50.9 4.5 Proppant embedment test Cylindrical-shaped samples were tested as received under ambient condition using a GCTX RTX-3000 testing system. The testing procedure is shown in Figure 4-15. First, a triaxial test was performed on a sample with an axial stress up to 40 MPa under a confining stress of 20 MPa to get the elastic properties of the sample. Second, the sample was split along the cross-section into two smaller cylindrical cores (each approximately 25.4 mm in diameter with a 25.4 mm length) for use in a proppant embedment test. One layer of proppants (monolayer) was placed between the two split cylindrical cores to form a proppant pack and then the whole sandwiched-sample was subjected to triaxial testing with an axial stress of 40 MPa and a confining pressure of 20 MPa. The total displacement of the proppant pack resulting from proppant compaction, proppant crushing, and proppant embedment was estimated by subtracting the elastic deformation of the intact rock sample measured during the initial triaxial test from the whole sandwiched-sample displacement in the embedment test. Two narrow size ranges, namely 0.5-0.6 mm and 0.25-0.3 mm, of Jordan sand and the ceramic proppant were used in the proppant embedment tests to see how the proppant types and sizes affect the embedment into the Montney formation. 104 Figure 4-15. Intact sample for elastic triaxial test (left); sample split to two halves for proppant embedment test (middle); monolayer of proppants between two split halves (right) After placing a proppant pack between two rock cores, the initial fracture aperture prior to loading was determined as the difference between the total length of a sample with proppants and the length of the intact sample without proppants. As shown in Figure 4-16, typically the axial displacement of intact samples without proppants and samples sandwiched with proppants increase close to linearly with an increase in confining pressure and axial stress. The fracture aperture reduction due to isostatic confining was calculated by subtracting the axial deformation of the intact sample during isostatic confining from the axial deformation of the sample sandwiched with proppants during isostatic confining. Likewise, the fracture aperture reduction due to axial loading was calculated in a similar manner. 105 Figure 4-16. Axial displacement of sample TWN2-15 with/without proppants during isostatic (left) and triaxial testing (right) The measured propped fracture apertures during the proppant embedment tests are summarized in Table 4-8. The Young’s modulus was obtained from the axial stress-axial strain curves at a stress of about 40 MPa. The mass of proppants ranges from 0.173 to 0.422 g, which corresponds a proppant pack loading ranging from 0.342 to 0.830 kg/m2 over a circular cross-section with a diameter of 2.54 mm. A monolayer of G250 proppants has a mass of approximately 0.18 g, while that using G500 proppants typically has a proppant mass of 0.3~0.422 g depending on how dense of the monolayer proppant pack is. The initial fracture aperture has a thickness ranging from 0.927 to 0.338 mm. Proppants with G500 size typically result in a higher fracture aperture than that with G250 proppants. For proppant packs using G500 size, large variation in the initial fracture aperture was observed and the initial fracture aperture was > 0.65 mm. For proppant packs using G250 size, the initial fracture aperture was around 0.37 mm except for sample TWN2-15. This sample (TWN2-15) had an initial fracture aperture of 0.645 mm that is likely due to the uneven proppant distribution forming a multi-layers proppant pack between the core surfaces as shown in Figure 4-17. Note that for the sample TWN1-13 the membrane was found broken during the embedment triaxial test thus this set of data was excluded in the following analysis. The reduction in the fracture aperture by an isostatic confining stress of 20 MPa ranges from 0.045 to 0.174 mm, while the aperture reduction by an axial stress of 40 MPa (20 MPa deviatoric stress + 20 MPa confining pressure) ranges from 0.018 to 0.119 mm. 05101520250 0.05 0.1 0.15Confining stress (MPa)Axial displacement (mm)Elastic isotropicEmbedment isotropicσ3σ3σ3σ3σ3σ3σ3σ3El staticnt static05101520250 0.02 0.04 0.06 0.08Deviatoric stress (MPa)Axial displacement (mm)Elastic triaxialEmbedment triaxialσ1σ1σ3σ3σ1σ1σ3σ3106 The final remaining fracture aperture ranges from 0.433 to 0.605 mm with G500 proppants, and ranges from 0.192 to 0.263 mm with G250 proppants. Most proppant remained unbroken after the proppant embedment test (Figure 4-18). Table 4-8. Evolution of proppant-propped fracture aperture in embedment tests Sample ID Siltstone Young’s modulus (GPA) Proppant Fracture aperture (mm) Type Mass (g) F0 F1 F2 F3 TWN1-13 62.5 G500 Jordan 0.364 0.927 0.377 0.031 0.519 TWN1-15 43.1 G500 ceramic 0.363 0.711 0.115 0.019 0.577 TWN2-13 41.1 G500 Jordan 0.422 0.650 0.099 0.119 0.433 TWN2-15 40.0 G250 Jordan 0.185 0.645 0.052 0.041 0.552 TWN2-17 38.0 G500 ceramic 0.302 0.754 0.087 0.063 0.605 TWN2-19 36.1 G250 ceramic 0.173 0.338 0.079 0.067 0.192 TWN3-10 42.2 G500 Jordan 0.390 0.658 0.045 0.090 0.523 TWN3-12 45.4 G250 Jordan 0.187 0.352 0.114 0.018 0.220 TWN3-17 38.9 G500 ceramic 0.408 0.681 0.174 0.025 0.482 TWN3-19 39.8 G250 ceramic 0.173 0.389 0.062 0.064 0.263 F0 – initial fracture aperture F1 – fracture aperture reduction by isostatic confining pressure F2 - fracture aperture reduction by deviator stress F3 – remain fracture aperture 107 Figure 4-17. Proppant distribution of TWN2-15 (left: uneven) and TWN3-12 (right: even) Figure 4-18. Proppant particles in four TWN samples after proppant embedment tests: G500 Jordan sand (top left), G250 Jordan sand (top right), G500 ceramic proppant (bottom left), and G250 ceramic proppant (bottom right) 108 The permeability of a monolayer of proppant between two core faces is determined using Equation 2-5 and the results are shown in Table 4-9. The porosity of the monolayer of proppant within the fracture was calculated using the known proppant mass, and fracture aperture and cross-sectional area. The effective size of particles in proppant was determined based on Equation 2-7 assuming no proppant particles were broken. The fracture conductivity is also calculated using Equation 2-4. Table 4-9. Permeability and fracture conductivity from proppant embedment tests at 40 MPa Sample ID Proppant mass Fracture aperture (mm) e Deff (mm) SF k (µm2) Cs (µm3) Type Loading (kg/m2) TWN1-13 G500 Jordan 0.718 0.519 0.915 0.539 6.4 565.5 293000 TWN1-15 G500 ceramic 0.718 0.577 1.263 0.539 6.1 1388.6 801000 TWN2-13 G500 Jordan 0.835 0.433 0.378 0.539 6.4 55.3 24000 TWN2-15 G250 Jordan 0.366 0.552 3.007 0.269 6.4 2405.1 1327000 TWN2-17 G500 ceramic 0.596 0.605 1.852 0.539 6.1 3474.9 2102000 TWN2-19 G250 ceramic 0.342 0.192 0.580 0.269 6.1 48.4 9000 TWN3-10 G500 Jordan 0.767 0.523 0.801 0.539 6.4 403.6 211000 TWN3-12 G250 Jordan 0.371 0.22 0.580 0.269 6.4 43.4 9000 TWN3-17 G500 ceramic 0.806 0.482 0.682 0.539 6.1 294.1 141000 TWN3-19 G250 ceramic 0.342 0.263 1.165 0.269 6.1 285.2 75000 The estimated permeability ranges from 43.4 to 3474.9 µm2 and the fracture conductivity ranges from 9000 to 2102000 µm3. The void ratio of TWN 1-15, 2-15, 2-17, and 3-19 is higher than 1, resulting in a permeability and fracture conductivity that is much higher than the laboratory conductivity data (BC Oil and Gas Commission 2013b). A mesh 20/40 proppant pack with the effective particle size similar to G500 has a permeability of approximately several hundred µm2, and a mesh 40/70 proppant pack with the effective particle size similar to G250 has a permeability about several tens of µm2 at about 40 MPa (BC Oil and Gas Commission 2013b). The unusual 109 high permeability and fracture conductivity in some of the proppant embedment tests results from the unusually high void ratio that is probably due to the artifacts in the laboratory tests. The estimated permeability and fracture conductivity of the rest of proppant embedment tests are in good agreement with those in the laboratory conductivity tests (BC Oil and Gas Commission 2013b), with a permeability of 43.4-48.4 µm2 for G250 proppants and 294.1-565.5 µm2 for G500 proppants. It can be seen from Table 4-9 that a monolayer of G500 proppant has a hydraulic conductivity almost one magnitude higher than G250 proppant. As most proppants remained intact after the proppant embedment tests, the fracture aperture reduction under axial compression during the tests should mainly result from proppant deformation and proppant embedment into the rock surface without considering other factors. Here the proppant particle number between the two fractures for each embedment test was estimated based on the proppant mass and density, based on which the average force acting on each particle was calculated as shown in Table 4-10. The particle force for G500 proppant is from 11.1 to 16.4 N, while that for G250 is from 3.1 to 3.6 N. This average particle force was used to estimate the proppant embedment and proppant deformation between the fractures as shown in Figure 4-19. 110 Table 4-10. Interpretation of propped fracture aperture during embedment test (40 MPa) Sample ID Proppant HB (HBW) Fracture aperture (mm) Type Particle number Particle force/N P0 P1 P2 P3 F3 (lab) P2/D (%) TWN1-13 G500 Jordan 1577 12.9 57.8 0.55 0.027 0.026 0.497 0.519 4.7 TWN1-15 G500 ceramic 1483 13.7 51.7 0.55 0.014 0.031 0.504 0.577 5.6 TWN2-13 G500 Jordan 1828 11.1 46.5 0.55 0.023 0.028 0.498 0.433 5.1 TWN2-15 G250 Jordan 6411 3.2 35.7 0.275 0.013 0.021 0.241 0.552 7.6 TWN2-17 G500 ceramic 1234 16.4 53.6 0.55 0.017 0.036 0.496 0.605 6.5 TWN2-19 G250 ceramic 5654 3.6 34.9 0.275 0.008 0.024 0.243 0.192 8.7 TWN3-10 G500 Jordan 1689 12.0 48.4 0.55 0.025 0.029 0.495 0.523 5.3 TWN3-12 G250 Jordan 6480 3.1 36.8 0.275 0.013 0.020 0.242 0.220 7.3 TWN3-17 G500 ceramic 1667 12.2 47.5 0.55 0.013 0.030 0.507 0.482 5.5 TWN3-19 G250 ceramic 5654 3.6 35.7 0.275 0.008 0.024 0.244 0.263 8.7 P0 – predicted initial fracture aperture P1 – predicted proppant deformation P2 – predicted proppant embedment P3 – predicted remaining fracture aperture 111 Figure 4-19. Reduction in propped fracture aperture caused by proppant deformation and embedment The proppant deformation between two rock faces can be expressed as (Cundall and Strack 1979): REFkFPpnp221  (4-8) where Fp is the particle force, kn is the particle stiffness that is related to the particle radius, R, and the particle-particle effective modulus, E, which was obtained in Chapter 3. The embedment depth of proppants into fracture surfaces during a proppant embedment test can be roughly estimated using Equation 4-9 based on the known Brinell hardness, which is correlated from the measured Young’s modulus using the equations in Figure 4-12. HBDFPip80665.92 (4-9) The results of the predicted final fracture aperture after proppant deformation and embedment at 40 MPa are compared against the laboratory results shown in Table 4-10. It seems that the predicted fracture apertures are consistent with the measurements from the proppant embedment tests excluded those possibly inaccurate measurements for samples TWN1-15, TWN2-15, TWN2-17, and TWN3-19. At 40 MPa, the proppant deformation is 0.023-0.027 mm for G500 Jordan sand, 0.013-0.017 mm for G500 ceramic proppant, 0.013 mm for G250 Jordan sand, and 0.008 mm for G250 ceramic proppant. The proppant deformation of G500 is about twice of that of G250 for the same type of proppants, while the proppant deformation of Jordan sand is about twice of that of ceramic for the same proppant size. For proppant embedment, a lower Brinell hardness value was used for rock samples with G250 size proppant particles than G500 size because of smaller particle = +112 forces acting on rock surfaces. The proppant embedment of different samples ranges from 0.020 to 0.036 mm. Except for G500 Jordan sand, the proppant embedment is typically larger than the proppant deformation at 40 MPa, especially for G250 proppants. The embedment percentage was also calculated as the total embedment into two core faces over the average proppant particle diameter. Higher embedment percentage is observed in smaller size proppants, which is in good agreement with experimental work on proppant embedment tests (BC Oil and Gas Commission 2013b). Slightly higher embedment is found in ceramic proppant than Jordan sand due to larger forces acting on ceramic proppant particles. The estimated embedment percentage of G500 Jordan sand ranges from 5.1% to 5.3% at 40 MPa, which is comparable to 5.8% of mesh 20/40 sand at 34.5 MPa from the previous data (BC Oil and Gas Commission 2013b). The estimated embedment percentage of G250 Jordan sand is around 7.3%-7.6% at 40 MPa, which is comparable to 7.4%-12% of mesh 40/70 sand at 34.5 MPa. The rock samples in the previous tests have a Young’s modulus of about 40 GPa, which is close to the Young’s modulus of samples in our tests. The proppant embedment results in a ‘moon-face-like’ surface shown in Figure 4-20. The total reduction in fracture aperture is about 10% of the initial fracture aperture under a 40 MPa compression. Figure 4-20. Fracture surface after proppant embedment test (TWN2-13) 113 Table 4-11 shows the fracture aperture based on proppant embedment and proppant deformation between two fractures. The estimate fracture aperture was used to calculate the void ratio of the monolayer of proppant between the two core faces. The permeability and the fracture conductivity results are shown in Table 4-11. The fracture aperture is similar if using the same proppant type and particle size. The fracture aperture when using ceramic proppant is consistently higher than that of Jordan sand with same particle size and thus has a higher void ratio. Jordan sand has a permeability of about 80 µm2 for G250 size and about 177-440 µm2 for G500 size depending on proppant loading. Ceramic proppant has a permeability of about 19 µm2 for G250 size and about 395-1585 µm2 for G500 size. This demonstrates the fracture aperture estimated based on proppant embedment and proppant deformation can give a reasonable prediction of residual fracture aperture. Based on Table 4-10, the reduction in fracture conductivity caused by proppant embedment is equally important to that by proppant deformation for G500 Jordan sand. For ceramic proppant, the decrease in the fracture conductivity caused by proppant embedment is more significant than that caused by proppant deformation for fractures with a monolayer of proppant. Compared with multilayer proppant packs in Chapter 2, monolayer proppant packs tend to have a higher void ratio thus a higher permeability when no large embedment is occur in a stiff formation. 114 Table 4-11. Permeability and fracture conductivity based on proppant deformation and embedment at 40 MPa (Table 4-10) Sample ID Proppant mass Fracture aperture (mm) e Deff SF k (µm2) Cs (µm3) Type Loading (kg/m2) TWN1-13 G500 Jordan 0.718 0.497 0.832 0.539 6.4 440.2 218000 TWN1-15 G500 ceramic 0.718 0.504 0.979 0.539 6.1 728.3 367000 TWN2-13 G500 Jordan 0.835 0.498 0.586 0.539 6.4 177.6 88000 TWN2-15 G250 Jordan 0.366 0.241 0.748 0.269 6.4 83.9 20000 TWN2-17 G500 ceramic 0.596 0.496 1.341 0.539 6.1 1585.0 786000 TWN2-19 G250 ceramic 0.342 0.243 1.002 0.269 6.1 193.4 47000 TWN3-10 G500 Jordan 0.767 0.495 0.706 0.539 6.4 288.2 142000 TWN3-12 G250 Jordan 0.371 0.242 0.736 0.269 6.4 79.9 19000 TWN3-17 G500 ceramic 0.806 0.507 0.769 0.539 6.1 395.7 200000 TWN3-19 G250 ceramic 0.342 0.244 1.006 0.269 6.1 195.4 47000 4.6 Summary Archived cores from three sections of a Montney siltstone were subjected to RMT. The results show that the mechanical properties in the horizontal direction vary with different depths. The indirect tensile strengths were found to be 6.6, 9.8, and 6.4 MPa, respectively, in the three test sections. The UCS values were 127, 98, and 63 MPa. All the sections have a Young’s modulus of around 29 GPa from uniaxial compressive tests and a Young’s modulus of around 42 GPa from triaxial compressive tests. The lower Young’s modulus from uniaxial compressive tests may due to core drying out facilitating splitting along lamination. The Poisson’s ratio is 0.24-0.3 in triaxial compressive tests and varies greatly from 0.14 to 0.44 in uniaxial compressive tests. The friction angle and cohesion based on linear Mohr-Coulomb failure criteria are 48° and 22 MPa for first TWN section, 54° and 17 MPa for second TWN section, and 45° and 14 MPa for third TWN section. Compared with fresh cores, the bulk density of the archived cores is slightly lower likely 115 due to core drying out, and the uniaxial Young’s modulus is moderately lower, while the UCS and triaxial Young’s modulus and peak strength are similar to each other. The anisotropy in Montney siltstone mechanical properties is noticeable as the Young’s modulus in the horizontal orientation is much higher than that in the vertical orientation. The cohesion and the friction angle of the fresh cores along the horizontal orientation are similar to those of the archived cores, both of which have a higher friction angle and a lower cohesion than the cores along the vertical orientation. Brinell hardness tests on archived Montney siltstone show that the maximum indentation depth at a 35 N indentation force ranges from 9.7 to 19.2 µm, averaging at 15.2 ± 2.4 µm. The value of Brinell hardness ranges from 59 to 117 HBW, averaging at 76.8 ± 13.4 HBW. The Young’s modulus interpreted from the Brinell hardness tests ranges from 17.6 to 78.1 GPa, averaging at 42.0 ± 14.8 GPa. Saturation of the test samples in 3% KCl fluid increases the hardness of the tested Montney siltstone while saturation in water shows little influence. Proppant embedment tests on archived Montney siltstone show that the initial fracture aperture propped by a monolayer of proppant is highly variable with G500 proppant (>0.65 mm) and is around 0.37 mm with G250 proppant. The reduction in the fracture aperture by an isostatic confining stress of 20 MPa ranges from 0.045 to 0.174 mm, while the aperture reduction cause by adding an additional 20 MPa of axial stress ranges from 0.018 to 0.119 mm. The final fracture aperture ranges from 0.433 to 0.605 mm with G500 proppants, and ranges from 0.192 to 0.263 mm with G250 proppants. Most of the proppant particles remained unbroken at 40 MPa thus the decrease in fracture aperture mainly results from proppant deformation and proppant embedment into the rock faces. The average particle force acting on a proppant particle was used to calculate the proppant deformation using the particle stiffness. The average particle force acting on a proppant particle was also used to calculate the proppant embedment using the rock Brinell hardness estimated from Young’s modulus. The predicted fracture apertures at 40 MPa are consistent with the laboratory results. The proppant pack permeability and fracture conductivity were calculated. The results show that the reduction in fracture conductivity caused by proppant embedment is equally important to that by proppant deformation for G500 Jordan sand. For ceramic proppants, the fracture conductivity 116 decrease caused by proppant embedment is most significant. A monolayer proppant pack tends to have a higher void ratio thus a higher permeability than multilayer proppant packs when placed into a hard rock formation where little embedment occurs. 117 5 3D Modelling of Proppant Embedment into Siltstone in DEM 5.1 Overview The finite element method has been widely used to study indentations and many constitutive models have been developed based on the experiment results (Jackson and Green 2005, Kucharski and Mróz 2007, Liao et al. 2009, Kot et al. 2013). Compared with the finite element approach, DEM does not require complex constitutive models and is capable of capturing fracture mechanisms during indentation with appropriate contact models and micro-parameters. Some successful examples include Huang and Detournay (2013) that reproduced fracture mechanisms during normal wedge indentation. Underneath the indenter, a damage zone was developed and it leads to the development of brittle fractures. A pressure-densification model was developed in a coupled DEM and the constrained natural element method, and was applied to study the Vickers indentation of silica glass (Jebahi et al. 2013). André et al. (2013) further developed a 3D beam model and modified the failure criterion of the contact bond to fit the indentation experiment results. Zhang et al. (2011) used PFC3D to study the fracturing process of rock with the face-centered cubic crystal structure under indentation. The hard rock material was represented with a parallel-bond contact model, while the spherical indenter was simulated with stiff walls with a stiffness two-order higher than rock. A similar approach was used to capture the embedment of proppant into fracture faces and was linked to the decrease in the fracture conductivity (Zhang et al. 2017). The permeability predictions from the DEM simulations were in good agreement with those from the fracture conductivity experiment and the Kozeny-Carman equation. An elastic-plastic contact model was developed in PFC3D and predicted proppant embedment into fracture faces that are close to experiment results (Deng et al. 2014, Mattson et al. 2014). In all the simulations, the micro-parameters for proppant particles were not calibrated and proppants were assumed unbreakable. Stresses on proppant particles were typically not examined. This chapter presents the DEM modelling of proppant embedment into rock faces of Montney siltstone. The micro-parameters in the DEM for Montney siltstone were first determined with the aid of a sensitivity analysis based on a 2k-1 factorial design. The calibrated micro-parameters were used to replicate Brinell hardness tests in DEM and the results were compared with the laboratory 118 data. Numerical proppant embedment tests were performed using the calibrated DEM micro-parameters for both proppants and Montney siltstone. Insight into the micromechanics during proppant embedment tests including the nature of proppant embedment and the particle stresses are gained through the DEM simulations. Comparison of the obtained particle stresses with the particle strengths established in Chapter 3 was used to evaluate the influence of the proppant size used in monolayer proppant packs. 5.2 DEM micro-parameters calibration of Montney siltstone The linear parallel bond model has been widely used to simulate rock-like materials (Wang et al. 2003, Potyondy and Cundall 2004, Park and Song 2009, Duan et al. 2015), which is adopted here to simulate Montney siltstone. The linear parallel bond model can be used to attach DEM particles together to create objects with specified shapes. A parallel bond acts like a finite-size cement on the circular contact plane of two contact pieces (particles/walls) that can transmit force and moment. The linear parallel bond can act together with the linear contact model, unless the maximum stresses from the moment and force overcome the bond strength that cancels the bonding and leaves only the linear contact model between two contact pieces. Loading on bonded samples is via moving platens that are simulated as rigid walls in DEM. Sample deformation as well as platen forces are recorded during the loading process. Typically, micro-parameters used in DEM simulations cannot be physically measured directly. This presents challenges to simulate targeted mechanical properties as most micro-parameters are determined by trial and error, which is time-consuming and it is sometimes difficult to duplicate several macroscopic mechanical properties simultaneously. The design of experiment approach has been widely used to evaluate the influence of micro-parameters on macroscopic mechanical properties. Yoon (2007) used the Plackett–Burman design to select the key micro-parameters affecting each mechanical property. The selected micro-parameters were used in a response surface analysis via the central composite design method to estimate the non-linear relations between the key micro-parameters and the mechanical properties. A similar approach was used by Hanley et al (2011), Kazerani (2013), Sun et al. (2013), and Chen (2017). In this study, a calibration procedure is used to enable micro-parameters in DEM simulations to reproduce macroscopic rock mechanics properties measured in the laboratory. The influence of 119 micro-parameters on macroscopic properties of rock was studied through a sensitivity analysis based on 2k-1 factorial design. Two levels of micro-parameters are shown in Table 3-1. It is assumed that the particle-particle effective modulus is equal to the bond effective modulus and the particle normal-to shear stiffness ratio is equal to the bond normal-to-shear stiffness ratio. Bonded samples using the micro-parameters shown in Table 5-1 were used in DEM simulations of uniaxial compressive tests, triaxial compressive tests, and indirect tensile tests as shown in Figure 5-1. Table 5-1. Value of micro-parameters for two levels Micro Parameter Level -1 Level +1 Particle-particle effective modulus (bond effective modulus) (GPa) 15 30 Particle normal-to-shear stiffness ratio (bond normal-to-shear stiffness ratio) 1.0 4.0 Bond tensile strength (MPa) 5 15 Bond cohesion (MPa) 25 75 Bond friction angle (°) 0 30 Particle friction coefficient 0.1 0.7 Figure 5-1. Sample subjected to uniaxial compressive test (left), triaxial compressive test (middle), and indirect tensile test (right) 120 UCS, Young’s modulus, and Poisson’s ratio from uniaxial compressive tests, peak strength from triaxial tests under 10 and 20 MPa confining pressure, and indirect tensile strength from indirect tensile tests, were used as the macroscopic outputs. There are 32 different scenarios of simulations based on the different combinations of the six micro-parameters with two levels (Appendix D). The percentage contribution of each micro-parameter to the variability of the output results is obtained as shown in Appendix D. The results show that the particle-particle effective modulus and the particle normal-to-shear stiffness ratio contribute, respectively, 75% and 20% to the Young’s modulus while that of other factors is less than 3%. The particle normal-to-shear stiffness ratio contributes 93% to the Poisson’s ratio and the particle friction coefficient contributes 6% while the contribution of the other micro-parameters is under 1%. For the UCS, the bond tensile strength and the particle normal-to-shear stiffness ratio have maximum influence with a contribution of 49% and 25%, respectively. For the triaxial peak strength under a confining pressure of 20 MPa, the bond cohesion and the particle friction coefficient have maximum influence with a contribution percentage of 34% and 30%. The particle normal-to-shear stiffness ratio also has a positive influence on the triaxial peak strength with an individual contribution of 8% and a contribution of 15% when combined with the bond cohesion. For the indirect tensile strength, the bond tensile strength and the particle normal-to-shear stiffness ratio have maximum influence with a contribution of 61% and 26%, respectively. Based on the sensitivity analysis above, a procedure to determine the micro-parameters to match the macroscopic rock mechanics parameters is established as follows: (1) the particle normal-to-shear stiffness ratio in uniaxial compressive tests was adjusted to give a Poisson’s ratio equal to laboratory measurements; (2) the particle-particle effective modulus in uniaxial compressive tests was adjusted to give a Young’s modulus equal to laboratory measurements; (3) the bond tensile strength in uniaxial compressive tests was adjusted to give UCS equal to laboratory measurements; (4) the bond cohesion and the particle friction coefficient were adjusted simultaneously to give a triaxial peak strength equal to the laboratory measurements. Three sets of the micro-parameters were calibrated to match the rock mechanics properties (shown Table 5-2) of the archived cores at three different depths presented in Chapter 4. In the following, the macroscopic mechanical parameters at a depth around 2336 m are used as an example to 121 demonstrate the calibration process. It should be noted that the Young’s modulus values and the Poisson’s ratio values in Table 5-2 are from the laboratory triaxial compressive tests as the laboratory uniaxial compressive tests give high Poisson’s ratio values that might be artificial. The indirect tensile strengths are from the indirect tests loaded perpendicular to the bedding layers. Table 5-2. Three sets of targeted macroscopic mechanical properties for DEM calibration Set Depth MD (m) UCS (MPa) Young’s modulus (GPa) Poisson’s ratio Peak strength (MPa) at 10 MPa confining pressure Peak strength (MPa) at 20 MPa confining pressure Indirect tensile strength (MPa) # 1 ~2068 127.0 42.0 0.28 170.5 266.0 8.4 # 2 ~2265 98.2 43.1 0.28 211.0 282.9 10.9 # 3 ~2336 63.1 42.5 0.25 124.7 210.1 13.7 When the values of six micro-parameters are all at level +1 shown in Table 5-1, the DEM sample yields a UCS of 64.5 MPa, a Young’s modulus of 50.4 GPa, and a Poisson’s ratio of 0.236. As shown in Figure 5-2, Firstly, the particle normal-to-shear stiffness ratio (kratio) was adjusted to 4.6 to give a Poisson’s ratio close to 0.25. Next, the particle-particle effective modulus (Emod) was adjusted to 26 GPa to give a Young’s modulus close to 42.5 GPa with kratio fixed to 4.6. Thirdly, the bond tensile strength was adjusted to 17 MPa to give a UCS close to 63.1 MPa while fixing all the other micro-parameters. It was found that changing the bond cohesion and the bond friction angle does not affect the compressive strength or the indirect tensile strength. Thus in the last step of the calibration process the particle friction coefficient was adjusted to 0.6 to give peak strengths of 136 MPa and 207 MPa, respectively, at 10 and 20 MPa confining pressures. Eventually the micro-parameters used to reproduce the three sets of macroscopic rock mechanics properties are shown in Table 5-3. 122 Figure 5-2. Micro-parameters calibration process: Poisson’s ratio (top left), Young’s modulus (top right), UCS (bottom left), and triaxial compressive strength (bottom right) 010203040506070-0.002 -0.001 0 0.001 0.002Axial compressive stress (MPa)Radial strain (left) and axial strain (right)kratio=4.0kratio=4.6kratio=8.0Kratio=4.0Kratio=8.0kratio=4.6Poisson ratio0.2360.2560.2660102030405060-0.001 0 0.001 0.002Axial compressive stress (MPa)Radial strain (left) and axial strain (right)Emod26Emod20Emod30Emod=30 GPaEmod=26 GPaEmod=20 GPaYoung's Modulus(GPa)49.042.432.801020304050607080-0.001 0 0.001 0.002Axial compressive stress (MPa)Radial strain (left) and axial strain (right)TenBond15TenBond17TenBond20pb_ten=15 MPapb_ten=17 MPapb_ten=20 MPaUCS strength (MPa)74.863.656.10501001502002503000 0.002 0.004 0.006 0.008 0.01Axial compressive stress (MPa)Axial strain (right)2.0 2.00.7 0.70.6 0.60.1 0.1Friction coeffiction(10 MPa) (20 MPa)123 Table 5-3. Summary of micro-parameters to represent three sets of macroscopic rock responses Micro Parameter Value for set 1 Value for set 2 Value for set 3 Particle-particle effective modulus (bond effective modulus) (GPa) 27.5 27.5 26 Particle normal-to-shear stiffness ratio (bond normal-to-shear stiffness ratio) 6.4 6.4 4.6 Bond tensile strength (MPa) 40 30 17 Bond cohesion (MPa) 75 75 75 Bond friction angle 30 30 30 Particle friction coefficient 0.7 50 0.6 Resulting macroscopic responses Value for set 1 Value for set 2 Value for set 3 UCS (MPa) 124.0 97.4 62.8 Young’s (GPa) 42.1 42.4 42.3 Poisson’s ratio 0.276 0.273 0.258 Peak strength (MPa) with 10 MPa confining pressure 175.8 184.0 136.3 Peak strength (MPa) with 20 MPa confining pressure 246.4 282.5 207.3 Indirect tensile strength (MPa) 25.2 22.3 13.3 It should be noted that the simulated indirect tensile strength of set #1 and #2 are higher than those from the laboratory tests. The ratio of the UCS over the indirect tensile strength is 4.9, 4.4, and 4.7, respectively for the 3 sets of micro-parameters used here. From the 32 different scenarios simulated in the sensitivity analysis, the ratio of the UCS over the indirect tensile strength falls into a narrow range from 4.0 to 6.0 with an average value of 4.9. Similar results were found by Yoon (2007) and Zhang et al. (2011) and this may result from the simplification of the parallel bond model that may not fully represent the mechanical behaviours of rocks. Further investigation on particle contact models is required but it is out of the scope of this research. 124 5.3 DEM Brinell indentation test As shown in Figure 5-3, a DEM sample in cuboid shape (1.44 mm×1.44 mm×0.96 mm) was generated with particles using the parallel bond contact model. The DEM sample was confined with rigid walls to mimic laboratory samples placed in a mold with epoxy. The indenter was simulated by a 1 mm diameter sphere that consists of many triangular rigid walls. The sphere was brought into perpendicular contact with the sample surface with an indentation force of up to 1 N. The force was increased to a load to 35 N and then unloaded to 0 N. The indentation depth of the indenter into the sample was recorded during the test. Figure 5-3. Cuboid sample subjected to Brinell indentation test with a spherical indenter Typically, the size of mineral grains in Montney siltstone core samples range from 20 to 50 μm (BC Oil and Gas Commission 2014b). The first set of the micro-parameters shown in Table 5-3 was used to generate a DEM sample for the Brinell indentation test and the particle size was varied to investigate its influence on indentation results. Five locations on the sample were selected for the indentation test. The resulting average indentation depth corresponding to a 35 N indentation force was, respectively, 10.9 ± 3.4 μm for size 100-150 μm, 10.0 ± 1.4 μm for size 80-120 μm, 10.3 ± 2.1 μm for size 60-90 μm, 10.2 ± 0.4 for size 40-60 μm, and 10.5 ± 0.3 for size 30-45 μm. The average indentation depths with different particle sizes are about the same, while a smaller 125 particle size results in a lower deviation in the indentation depth, especially for the sizes 40-60 μm and 30-45 μm. This may be due to the relatively more homogenous structure when using a smaller particle size. Typical curves of the indentation force and the indentation depth are shown in Figure 5-4. Similar to the laboratory indentation test, the test force and the gradient of test force-indentation depth increase with an increase in the indentation depth. The force-indentation depth curve during the unloading process shows a similar trend to that of the loading process, except the gradient of the force-indentation depth curve is steeper upon unloading and left an unrecoverable permanent indentation after complete unloading. Figure 5-4. Indentation force-depth curve using different particle sizes The particle size of DEM samples for the Brinell indentation test was set to a range between 40 and 60 μm for computational efficiency. Using the micro-parameters for the 32 scenarios presented in the micro-parameter sensitivity analysis in Section 5.2, the Brinell indentation depth corresponding to a 35 N indentation force at each scenario was determined. The detailed results can be found in Appendix D. Note that in some scenarios the sample failed before reaching the targeted indentation force 35 N. The macroscopic rock mechanics properties including the UCS, Young’s modulus, Poisson’s ratio for each scenario are shown in Appendix D. A positive correlation was found between indentation depths and Young’s modulus, while the correlations between UCS and Poisson’s ratio are negative (Figure 5-5). Using the power regression between 0102030400 3 6 9 12Indentation force (N)Indentation depth (μm)30-45 nm40-60 nm80-120 nm126 indentation depths and Young’s modulus, the indentation corresponding to a Young’s modulus of about 42 GPa is around 13.4 μm, which is slightly lower than 15.2 ± 2.4 µm from laboratory Brinell indentation tests. Figure 5-5. Correlation between indentation depth at 35 N and Young’s modulus, UCS and Poisson’s ratio from 32 numerical scenarios The Brinell hardness value corresponding to a 35 N indentation force was calculated using Equation 4-3 and is plotted against the Young’s modulus shown in Figure 5-6. A positive power correlation was found between Brinell hardness and Young’s modulus from DEM simulations, which is similar to the results of the laboratory Brinell indentation depth with similar regression coefficients. y = 59.88x-0.40R² = 0.7481114172020 40 60 80Indentation depth (μm)Young's modulus (GPa)y = -3.2175x + 106.68R² = 0.0465y = 0.0023x + 0.1223R² = 0.004600.050.10.150.20.250.3040801201608 11 14 17 20Poisson's ratioUCS (MPa)Indentation depth (μm)127 Figure 5-6. Correlation between Brinell and Young’s modulus It should be noted that the DEM simulations with the same Young’s modulus as the laboratory tend to give a slightly higher Brinell hardness thus a lower indentation depth under the same indentation force. For example, using the power regression relations in Figure 5-6, the Brinell hardness based on DEM simulations corresponding to a Young’s modulus of around 42 GPa is 84.6 HBW, which is slightly higher than that from the laboratory tests (77.5 HBW). Three sets of micro-parameters shown in Table 5-3 were used for the Brinell indentation test simulations and the results are shown in Figure 5-7. The third set of micro-parameters gives a slightly higher indentation depth (11.2 μm) than the other two sets (around 10.6 μm). Compared with the laboratory Brinell indentation tests on Montney samples, the numerical tests give a smaller indentation depth for a 35 N indentation force. y = 18.97x0.40R² = 0.74 y = 19.43x0.37R² = 0.6250709011013010 30 50 70 90Brinell Hardness (HBW)Young's modulus (GPa)DEM simulationLab data128 Figure 5-7. Indentation force-depth curve using three sets of micro-parameters Figure 5-8. Parallel bond breakage pattern of set #1, #2, and #3 (left to right) To better represent the surface deformation of Montney siltstone under indentation in DEM simulations, a lower Young’s modulus was used for the Montney siltstone to achieve a similar Brinell hardness to the laboratory measurements. Using the third set as an example, the value of the particle-particle effective modulus was reduced from 26 GPa to 20 GPa, which gives a Young’s modulus of 32.5 GPa while the other macroscopic rock mechanics properties remained approximately the same. Based on the power regressions in Figure 5-6, the Young’s modulus of 32.5 GPa in DEM gives a Brinell hardness of 76.3 HBW, which is close to that of 77.7 HBW for a Young’s modulus of 42.4 GPa measured in the laboratory. 0102030400 3 6 9 12 15 18Indentation force (N)Indentation depth (μm)Set 1Set 2Set 3LabSet 3 - new129 The DEM indentation result using the new set of micro-parameters is shown in Figure 5-7. The indentation force-depth curve given by the numerical simulation is similar to the laboratory result. This demonstrates that the discrete element model with the chosen micro-parameters is capable of capturing the local deformation of rock surfaces under an indentation force although the indentation result is sensitive to the Young’s modulus, which should be carefully selected. The micro-parameters of the linear parallel bond calibrated from the laboratory compression tests tend to give higher hardness and lower indentation depth than the laboratory. It should also be noted that the laboratory indentation tests leave a larger permanent indentation upon completed unloading than that of numerical simulations. A more sophisticated bond model can be developed to capture this feature, which, however, is out of the scope of this research. Using the new set of micro-parameters, the indentation into the DEM sample was continued until a peak indentation force was reached. As shown in Figure 5-9, at 35 N the indentation depth is 14.3 μm and the crack number is 3012. Most of the cracks are caused by failure in the tensile mode (green colour). When the indentation force increases to 100 N, the crack number is 11470 and cracks expand to deeper locations in the sample. With the increase of the indentation depth to 35.4 μm, the indentation reaches the peak load 161 N with a crack number of 21134. Particle spalling resulting from local failures of bonded particles was found near the indentation area. The parallel bond breakage is mainly distributed underneath the indentation tip within a depth of the indenter diameter. Figure 5-9. Brinell indentation test loaded to peak using new set # 3 (X+ direction) 35 NCrack: 3012 Crack: 11470100 N 161 NCrack: 21134Spalling14.3 μm 25.3 μm 35.4 μm130 5.4 DEM proppant embedment test To verify the interaction between proppants and rock via DEM as well as to gain insight to the micromechanics of proppant embedment tests that can not be observed directly in the laboratory, DEM simulations of proppant embedment tests were performed. The numerical simulation setup for a proppant embedment test is shown in Figure 5-10. Two cuboid rock samples with a dimension of 6.35×6.35×3.25 mm (length × width × height) were generated and the corrected third set of DEM micro-parameters corresponding to cores from a depth around 2336 m (MD) was used. Proppant particles were placed between two rock samples and were subjected to a triaxial loading with an axial stress of 40 MPa and a confining pressure of 20 MPa to mimic the laboratory triaxial settings in Chapter 4. The total DEM particle number for this model is 51556. A fine particle size (120 μm) was used for the end of the rock sample that is in direct contact with proppants in order to better capture the local interaction with proppants. Four proppant embedment tests consisting of two different types of proppants and two different proppant sizes were simulated in DEM. The detailed information is listed in Table 5-4. Figure 5-10. Proppants (blue colour) sandwiched between two rock blocks (green colour) in DEM proppant embedment test 131 Table 5-4. Summary of proppants used in four DEM proppant embedment tests Sample ID Proppant Parameters for proppants in DEM simulation type loading (g/cm2) size (mm) particle number particle-particle modulus (GPa) particle strength at 0.655 mm Weibull modulus TWN3-10 G500 Jordan sand 0.0769 0.55 134 1.1 35 MPa 4 TWN3-12 G250 Jordan sand 0.0369 0.275 515 1.1 35 MPa 4 TWN3-17 G500 ceramic 0.0804 0.55 132 2.2 65 MPa 4 TWN3-19 G250 ceramic 0.0342 0.275 450 2.2 65 MPa 4 Using the micro-parameters in Table 5-4, the proppants sandwiched between the two rock cores were subjected to a 20 MPa isostatic compression and then a 20 MPa deviatoric stress was added to generate a 40 MPa axial stress. The decrease in the propped fracture aperture during the deviatoric stress loading stage was determined based on the difference of the sample length with and without proppants as shown in Figure 5-11. 132 Figure 5-11. Change in sample length with axial stress for simulated tests The change in fracture aperture with axial stress for the four simulated tests is shown in Figure 5-12. At an axial stress of around 20 MPa, the fracture aperture from four DEM simulations is 0.514 mm for TWN 3-10, 0.252 mm for TWN 3-12, 0.524 mm for TWN 3-17, and 0.257 mm for TWN 3-19. The fracture apertures given by the G500 proppants are much higher than those with G250 proppants. With the increase in axial stress, the fracture aperture decreases linearly. The decrease for simulated Jordan sand is slightly faster than that of ceramic proppant. The ceramic proppants consistently give a larger fracture aperture than the Jordan sand for each size (G500 and G250). At an axial stress of 40 MPa, the fracture aperture from DEM simulations is, respectively, 0.499, 0.243, 0.514, and 0.250 mm for TWN 3-10, TWN 3-12, TWN 3-17 and TWN 3-19, which is a good match to laboratory measurements of 0.523, 0.220, 0.482 and 0.263 mm and empirical predictions of 0.495, 0.242, 0.507, and 0.244 mm (Table 4-10). 6.36.46.56.66.76.86.920 25 30 35 40Sample length (mm)Axial stress (MPa)TWN 3-10TWN 3-12TWN 3-17TWN 3-19Without proppants133 Figure 5-12. Change in fracture aperture with increase in axial stress for four tests Proppant embedment into two rock surfaces is shown in Figure 5-13 using the known rock surfaces (planes in blue colour). The embedment of every individual proppant into the two rock surfaces was measured using the known particle location and diameter, as well as the locations of the two planes representing the two rock surfaces. It should be noted that rigid particles are used in DEM and particle deformation is simulated as particle overlap. The magnitude of the simulated proppant embedment in DEM includes deformation of each proppant particle. The results show that the embedment of proppant particles into the upper rock surface and the lower rock surface is slightly different but the total embedment is almost identical due to the uniform size proppant particles used for each simulation. The G500 Jordan sand particles in TWN 3-10 have an embedment depth of 50.8 μm. The G250 Jordan sand particles in TWN 3-12 have an embedment depth of 32.5 μm. The G500 ceramic proppants in TWN 3-17 have an embedment depth of 35.9 μm. The G250 ceramic proppants in TWN 3-19 have an embedment depth ranging from 24.0 μm. As evidenced by the fracture aperture after a 40 MPa axial stress, the embedment depth of Jordan sand is consistently higher than that of the ceramic proppants, and the G250 proppants have a smaller embedment depth than the G500 proppants. 0.20.250.30.350.40.450.50.5520 25 30 35 40Fracture aperture (mm)Axial stress (MPa)TWN 3-10 TWN 3-12TWN 3-17 TWN 3-19134 Figure 5-13. Schematic view of proppant embedment in DEM proppant embedment test (top: X front view; bottom: Isometric view) The spatial distribution of the linear parallel bonds that broke in tension/shear for the TWN 3-10 sample is shown in Figure 5-14. The total number of broken bonds is 7325 and the bonds mainly broke in tension. Most of the bond breakage occurs near the surfaces of the two fracture walls, especially near the contact areas between proppant particles and rock particles. The contact force chain also shows that most of the bonds in tension are near the fracture surfaces. The total number of the bonds in tension is 10845 and these bonds are distributed throughout a thicker zone than that of the parallel bond breakage. 135 Figure 5-14. TWN 3-10: distribution of parallel bonds broke in tension (green) and shear (red) (top row); contact force chain in tension (red) (bottom row) A comparison of the broken linear parallel bonds for four samples is shown in Figure 5-15. The total number of broken bonds is 7325 for TWN 3-10, 1582 for TWN 3-12, 7833 for TWN 3-17, and 2478 for TWN 3-19. For each proppant type, G250 proppant particles caused a less dense pattern of broken bonds that are distributed closer to the fracture wall surfaces than that of G500 proppant particles. Most of the broken bonds occur within a depth of half the diameter of the proppant particles from the fracture wall surfaces. For the same proppant size, ceramic proppant particles cause more broken bonds than that of Jordan sand particles, especially for size G250. 136 Similar findings regarding the contact force chain in tension are found as shown in Figure 5-16. The total number of broken bonds is 10845 for TWN 3-10, 3097 for TWN 3-12, 11542 for TWN 3-17, and 6114 for TWN 3-19. Most of the bonds in tension occur at a deeper depth than the broken parallel bonds but are within a depth of the diameter of the proppant particles from the fracture wall surfaces. Figure 5-15. Comparison of parallel bonds broke in tension (green) and shear (red) for four samples: TWN 3-10 (top left), TWN 3-12 (top right), TWN 3-17 (bottom left) and TWN 3-19 (bottom right) 137 Figure 5-16. Comparison of contact force chain in tension (red): TWN 3-10 (top left), TWN 3-12 (top right), TWN 3-17 (bottom left) and TWN 3-19 (bottom right) Based on the contact forces acting on proppant particles in DEM simulations, the particle octahedral shear stress for each proppant particle was calculated using Equation 3-2. The particle strength for each particle was determined using Equation 3-5. The distributions of particle stresses from four DEM simulations are plotted in Figure 5-17. Both for Jordan sand and ceramic proppant, a wider range of particle stress is found for G250 proppants than for G500 proppants and the average particle stress of G250 is slightly higher. For each particle size, the particle stress of the ceramic particles has a wider range and higher average stress than that of Jordan sand particles. 138 The maximum particle stress is, respectively, 37.9, 49.6, 42.5, and 40.7 MPa, which is smaller than the particle strength of 39.9, 67.1, 74.1, and 124.6 MPa. The particle stress for all the particles is smaller than the corresponding particle strength, which is consistent with observations that most proppants remain intact after the laboratory proppant embedment tests shown in Figure 4-18. Figure 5-17. Distributions of octahedral particle stress for four simulations The indentation force between each proppant particle and the rock surface was also determined using the sum of vertical contact forces between proppant particles and bonded particles representing rock. For the G500 Jordan sand particles in TWN 3-10, the indentation force ranges from 8.5 to 13.6 N, while that of the G250 Jordan sand particles in TWN 3-12 ranges from 0.4 to 05101520253023.6 26.4 29.3 32.2 35.0 37.9FrequencyOctahedral particle stress (MPa)TWN 3-100204060801001201401601804.8 13.7 22.7 31.7 40.7 49.6FrequencyOctahedral particle stress (MPa)TWN 3-12051015202519.4 24.0 28.6 33.2 37.9 42.5FrequencyOctahedral particle stress (MPa)TWN 3-1701020304050607080900.0 13.7 27.4 41.1 54.8 68.4FrequencyOctahedral particle stress (MPa)TWN 3-19139 4.6 N. Slightly larger indentation force is found in the ceramic proppant particles. The indentation force of the G500 ceramic proppant particles ranges from 7.2 to 15.0 N, while that of the G250 ceramic proppant particles ranges from 0 to 6.2 N. The total sum of particle indentation force for each TWN sample is 1612.7 N, which is equivalent to the axial force to produce a 40 MPa compression stress on a 6.35*6.35 mm cross-section area. The indentation force of each particle is plotted against its octahedral particle stress as shown in Figure 5-18. Linear relations between indentation forces and octahedral particle stresses were found for all the DEM simulations. Higher indentation forces with rock surfaces in turn result in higher particle octahedral shear stresses. A similar linear gradient was found between TWN 3-10 and TWN 3-17, which is steeper than that of TWN 3-12 and TWN 3-19. Figure 5-18. Relation between indentation force and particle stress According to McDowell and De Bono (2013), the octahedral particle stress σ is proportional to F/D2 where a particle with the diameter D is subjected to an axial force F. The data shown in Figure 5-18 were re-plotted as F/D2 assuming F equals the indentation force acting on each proppant particle. As shown in Figure 5-19, a linear relation was found between σ and F/D2. A coefficient 010203040506070800 5 10 15 20Octehedral particle stress (MPa)Indentation force (N)TWN 3-10 TWN 3-12TWN 3-17 TWN 3-19140 δ is found from Figure 5-19 to be used in Equation 5-1, which is 0.8475. This equation will be used in Section 5.5. 2DF  (5-1) Figure 5-19. Relation between indentation force/D2 and octahedral particle stress 5.5 Implication for optimal monolayer proppant From the above analysis of a monolayer of proppant particles with different particle sizes between two rock faces, larger particles, which have lower particle strength, are subjected to higher indentation force and have higher particle stress. Conversely, smaller particles, with high particle strength, are subjected to smaller indentation force and the particle stress is much lower than particle strength. If a mixture of larger size and smaller size particles is used for monolayer proppant, the larger size particles with lower strength are likely to carry more load and are thus likely to break, while the smaller size particles with higher strength carry less load. A simple example with a size range from 0.5 to 0.6 mm of Jordan sand particles in a proppant embedment simulation is used here for illustration. The proppant particles first were assumed unbreakable and the particle stress for each proppant particle was determined as shown in Figure 5-20. It can be seen that larger particles with lower particle strengths tend to have higher particle stresses and 010203040506070800 20 40 60 80 100Octehedral particle stress (MPa)Indentation force/D2 (MPa)TWN 3-10TWN 3-12TWN 3-17TWN 3-19Linear regressiony=0.8475xR2=0.9938141 higher indentation forces. For this simulation, 39 particles have a particle stress larger than its particle strength, which means these particles would have broken if the particle breakage model was turned on. Some small size particles with diameter near 0.5 mm have an indentation force close to 0 N. When the particles are allowed to break with the implementation of the particle breakage model, the fracture aperture under 40 MPa compression is 0.464 mm and all 132 original particles were crushed into 1425 smaller fragment particles as shown in Figure 5-21. Figure 5-20. Relation between particle diameter and particle stress/strength and indentation force Figure 5-21. Proppant particles at 0 MPa (left) and 40 MPa (right) To fully utilize proppants to maintain propped fractures without excessive proppant breakage, uniform size proppants are preferred for fractures propped with monolayer proppants. In the 0714212802550751000.5 0.55 0.6Indentation force (N)Particle stress and strength (MPa)Particle diameter (mm)Particle stressParticle strengthIndentation force142 following, a simple model was used to optimize the particle size of proppants used to form a monolayer proppant pack between fracture faces under a target stress. Using Figure 5-22 for illustration, assuming the applied axial stress is σv (Pa) on a cross-section with an area of A (m2). The proppants used have a concentration or loading of lp, a density of ρ, and a uniform diameter of D. It is assumed the proppant strength follows the size hardening law shown in Equation 3-4 with a Weibull modulus of m and a known strength σ0 at a known particle diameter D0. Figure 5-22. Assumed packing of monolayer proppants The number of proppants N between the cross-section area is calculated as: 33661 DAlDlANpp (5-2) Assuming the possible maximum axial force Fm acting on a proppant particle is 1.1 times the average force acting on a proppant: pvpvvmlDlDNAF61.161.11.133  (5-3) According to Equation 5-1, the particle stress σ is proportional to Fm /D2 when a particle is subjected to an axial force F. The particle stress should not exceed the particle strength as expressed below: 143  mDsDD/300  (5-4) Combining Equations 5-2, 5-3, 5-4, the particle size D follows the equation below:  mvmp DlD313001.16 (5-5) It should be noted that lp has an upper bound that is related to the particle diameter corresponding to a fully-filled monolayer. Following Figure 5-22, the maximum value of lp is expressed as: 633619 3DDDDl p  (5-6) The maximum proppant loadings for G500 Jordan sand and ceramic proppants are 0.763 and 0.809 kg/m2 respectively. These values are close to 0.769 and 0.804 kg/m2 used in the laboratory proppant embedment tests. For G250 Jordan sand and ceramic proppants, the maximum proppant loadings are 0.381 and 0.405 kg/m2 compared to 0.369 and 0.342 kg/m2 used in the laboratory proppant embedment tests. Using Equation 5-6 in Equation 5-5, yields:    mvmmvmpDDDlD31300313001.1661.16  (5-7) The above equation can be expressed as: mvmDD33001.1  (5-8) Equation 5-5 was used to calculate an optimal value of D respectively for Jordan sand and ceramic proppants at 40 MPa using the parameters in Table 5-4 without changing the original proppant loading. The value of D was determined to be 0.57 mm for Jordan sand and 0.81 mm for ceramic proppants. As a demonstration, ceramic proppants with a uniform diameter 0.80 mm result in a fracture aperture of 0.742 mm, which is 44% higher than the fracture aperture of 0.514 mm when 144 using a 0.55 mm particle size. All proppant particles have a particle stress lower than the particle strength. Equation 5-8 was used to determine an optimal value of D when the proppant loading can be varied to achieve a fully filled monolayer proppant pack. The optimal D is 0.60 mm for Jordan sand under a 40 MPa compression and lp is 0.0833 g/cm2. For ceramic proppants, the optimal D is 1.37 mm and lp is 0.2016 g/cm2. As a demonstration, Jordan sand with a uniform diameter of 0.60 mm and a lp of 0.0833 g/cm2 was used. The resulting fracture aperture is 0.546 mm, which is a 9% improvement compared with the original fracture aperture of 0.499 mm. Ceramic proppant with a uniform diameter of 1.30 mm and a lp of 0.1913 g/cm2 was used. The resulting fracture aperture is 1.235 mm, which is a 140% improvement compared with the original fracture aperture of 0.514 mm. All the proppant particles have a particle stress lower than the particle strength. Using Equation 5-8, the value of the optimal D used for different compression stresses is plotted in Figure 5-23. Genoa sand is also included using the strength parameters from Chapter 4. The optimal particle diameter decreases with the increase in the compressive stress acting on monolayer proppants. For a 40 MPa compression, the optimal diameter is 0.60 mm if using Jordan sand, 0.38 mm if using Genoa sand, and 1.37 mm if using ceramic proppants. For a 60 MPa compression, the optimal diameter is 0.35 mm for Jordan sand, 0.20 mm for Genoa sand, and 0.80 mm for ceramic proppants. A larger size proppant would likely result in a higher fracture aperture and a higher fracture conductivity. 145 Figure 5-23. Optimal particle diameter under different compressive stress for fully-filled monolayer of proppants 5.6 Summary The linear parallel bond model was used to simulate Montney siltstone and the micro-parameters representing macroscopic mechanical properties of Montney siltstone were calibrated with the aid of design of experiment. Thirty-two scenarios of different combinations of micro-parameters were used for DEM simulations of uniaxial compressive tests, indirect tensile tests, and triaxial compressive tests. The contribution of each micro-parameter to the macroscopic rock mechanics properties was determined, based on which the micro-parameters were calibrated to give a Young’ modulus, Poisson’s ratio, UCS and triaxial compressive strength matched to the laboratory data of Montney siltstone. The calibrated micro-parameters were used for subsequent simulations of Brinell indentation tests and proppant embedment tests. The 32 scenarios of different combinations of micro-parameters were also used in the DEM simulations of Brinell indentation tests. Similar to the laboratory indentation tests, the obtained 00.511.522.533.540 20 40 60 80 100Particle diameter (mm)Compressive stress (MPa)Jordan sandCeramic proppantsGenoa sand146 Brinell indentation depth/Brinell hardness at 35 N shows a power regression with Young’s modulus from DEM uniaxial compressive simulations. A DEM Brinell indentation test using the same Young’s modulus as the laboratory tends to give a slightly higher Brinell hardness thus a lower indentation depth than the laboratory data. A lower particle-particle effective modulus yielding a lower Young’s modulus was used to represent Montney siltstone to better capture the interaction between proppant packs and rock surfaces. Using the adjusted micro-parameters gave a prediction of the indentation depth-force curve similar to laboratory data. At a peak load of 161 N particle spalling occurs near the indentation tip and local failures of bonded particles are mainly distributed within a depth of half the indenter diameter beneath the tip. Four proppant embedment tests with two different proppant types and two different particle sizes were modelled in DEM. The fracture aperture reduces nearly linearly under axial stress and the final fracture aperture in DEM simulation of these four tests is, respectively, 0.499, 0.243, 0.514, and 0.250 mm, which is close to laboratory measurements of 0.523, 0.220, 0.482, and 0.263 mm and analytical predictions of 0.495, 0.242, 0.507, and 0.244 mm. Insights into the micromechanics of the proppant embedment tests were gained. The total embedment depth for the four tests is, respectively, 50.8, 32.5, 35.9, and 24.0 μm. Most of the broken bonds between rock particles occur within a depth of half the diameter of the proppant particles from the fracture wall surfaces. More bond breakage was observed in simulations using larger size proppant particles. The particle octahedral shear stress for each proppant particle was calculated and no proppant particle stress exceeds the particle strength, which is in good agreement with observations of the intact particles in the laboratory tests. Larger particles with lower particle strength are subjected to higher indentation force and have higher particle stress. The particle stress shows a linear relation with the value of the indentation force divided by the square of particle diameter. Uniform-size proppants are preferred for fractures propped with monolayer proppants. An equation for choosing an optimal size of uniform-sized proppant particles under different compressive stresses was developed based on the principle that the particle stress should not overcome the particle strength. Based on the model, for a 40 MPa compression, the optimal diameter is 0.60 mm if using Jordan sand, 0.38 mm if using Genoa sand, and 1.37 mm if using ceramic proppants. For a 60 MPa compression, the optimal diameter is 0.35 mm for Jordan sand, 0.20 mm for Genoa sand, and 0.80 mm for ceramic proppants. 147 6 Influence of Proppant Crushing and Embedment on Fracture Conductivity and Proppant Selection 6.1 Overview As proppant crushing and embedment can greatly decrease the fracture conductivity, a proppant pack should resist the fracture closure stress and maintain fracture conductivity. While the traditional way to select a proppant is based on the proppant crushing percentage and the propped fracture conductivity under a targeted closure stress using two ISO testing standards, a methodology to optimize the proppant usage in hydraulic fracturing has shown great potential (Gu 2013, Gu and Mohanty 2014, Gu et al. 2015). This optimization methodology can link proppant selection to the fracture conductivity, the hydrocarbon production, and the Return-on-Fracturing-Investment (ROFI). Several components are included in this methodology. First, a fracture propagation model such as those based on the Perkins-Kern-Nordgren model is used to predict the fracture geometry created by pressure pumping. A proppant transport model is also incorporated into the fracture propagation model to obtain the distribution of proppants inside the fracture geometry. Second, based on the proppant distribution, the fracture conductivity distribution is estimated using either laboratory conductivity test data or numerical flow simulations such as those by lattice Boltzmann method (Gu et al. 2017). Next, the fracture conductivity distribution is used in reservoir simulations to predict the fracture well productivity. The stimulated hydrocarbon production can be used to compare the ROFI by subtracting the capital value of the cumulative production after a certain production time by hydraulic fracturing treatment cost. Applying the methodology to different proppant selection scenarios can yield a direct comparison of the ROFI values thus an optimal proppant can be chosen. This methodology has been successfully applied to compare different fracturing fluid systems with different proppant sizes and concentrations. However, converting the proppant distribution to the fracture conductivity could be challenging due to the lack of time-consuming laboratory conductivity data for a certain proppant particle size. In addition, the laboratory conductivity measured under specified boundary conditions might not represent the real conditions such as different proppant concentrations or several proppants being used together. Furthermore, the influence of proppant embedment into the formation rock on the fracture conductivity is difficult to quantify in the conductivity test. The discrete element method combined with the lattice Boltzmann method (Gu et al. 2017, Mollanouri Shamsi et al. 2015) has 148 been used to simulate the interactions between proppant particles and predict the proppant pack permeability under varying particle sizes and concentrations. However, these proppant particle simulations have not been calibrated and the proppant crushing was not considered in previous studies. In this research, with the developed proppant breakage model and the calibrated proppant-rock interaction in DEM, the proppant crushing and the proppant embedment can be fully considered and can be easily converted to fracture conductivity. This chapter shows how to use the knowledge gained from proppant crushing and embedment on fracture conductivity to optimize the proppant usage in hydraulic fracturing treatments. First, the applicability of the Kozeny-Carman equation to estimate the permeability of a fracture filled with particles is validated against numerical flow simulations. A parametric analysis on the influence of proppant crushing and proppant embedment on fracture conductivity is performed in DEM. The knowledge gained is used to guide and optimize proppant selection in a case study for the same well (#28232) where the tested Montney siltstone came from. 6.2 Validation of the Kozeny-Carman equation with flow simulation As an alternative to permeameter testing, the permeability of a porous media can be determined numerically by solving the Navier-Stokes equations to yield the pressure and fluid velocity throughout the pore space, from which the permeability can be estimated with Darcy’s law (Keehm et al. 2004). Several numerical studies have been successfully conducted to investigate the influence of grain shapes (Garcia et al. 2009), tortuosity (Ghassemi and Pak 2011) and sphere arrangements (Blazejewski and Murat-Blazejewska 2014) on the permeability. The particle packs with different particle size distributions were also studied (Zheng and Tannant 2017). Comsol Multiphysics was used here to conduct flow simulations to verify the Kozeny-Carman equation for the prediction of particle pack permeability. A particle assembly geometry was generated by specifying the particle diameter and coordinate for each particle, which can be obtained from DEM simulations. In Comsol Multiphysics the particle assembly was trimmed to a smaller size (10 mm length) and non-permeable boundary walls were applied. 149 It is assumed that the fluid passing through the porous medium (pore structure) is incompressible and follows Newtonian and steady-state flow. The Kozeny-Carman equation is applicable to the same flow conditions. A finite element method is used to discretize the Stoke equations (Equation 6-1) on the pore structure. 0.02VPV (6-1) where µ is the dynamic viscosity of a fluid, ?⃗? and P are the velocity and pressure of the fluid respectively. ∇2 is the Laplacian operator, ∇⃗ is the gradient operator, and ∇⃗ . is the divergence operator. As shown in Figure 6-1, the flow boundary conditions are (1) no slip occurs at the fluid-solid interface, (2) non-slip walls are added to the faces that are parallel to the main flow direction and no fluid flows out of the system through these walls, and (3) uniform velocity on the input face and uniform pressure on the output face of the porous medium. Figure 6-1. Flow boundary condition for permeability estimation The permeability of the porous medium is calculated by Darcy’s law using the pressure and velocity fields determined from the flow simulation using PLvk   (6-2) Non-permeable wallNon-permeable wall OutputPoreLInputface face150 where v is the mean fluid flow velocity through the medium, L is the length of the porous medium between input and output faces as shown in Figure 6-1, and ΔP is the pressure difference between the input and output faces. Several cases are studied with flow simulation as shown in Figure 6-2, including flow through two parallel fractures, fractures with monolayer spherical particles, fractures with three and eight layers spherical particles. The length of the flow model is 10 mm, which is 10 times the spherical particle diameter (1 mm) used in the simulations. The inlet velocity was set to 1e-20 m/s and the outlet pressure was set to 0 Pa. The distribution of flow velocity for each case is shown in Figure 6-3. Figure 6-2. Four geometries studied with flow simulation: parallel fractures (top left), monolayer (top right), three layers (bottom left), and eight layers (bottom right) 151 Figure 6-3. Flow velocity distribution of four geometries studied with flow simulation The average flow pressure at the inlet was calculated in Comsol via an integration function on the inlet faces. Based on that, the permeability for each flow simulation was calculated using Equation 6-2. The simulated permeability values are compared with the predictions using the Kozeny-Carman equation shown in Table 6-1. It can be seen that the permeability via flow simulation is very close to the predictions by the empirical equations. The permeability of two parallel fractures via flow simulation is in good agreement with that by the cubic law, and the permeability of monolayer/multilayer particle packs between fractures generally follows the Kozeny-Carman equation. 152 Table 6-1. Comparison of permeability from flow simulation and the cubic law/Kozeny-Carman equation (L = 10 mm) Scenario Fracture aperture (mm) k – flow simulation (µm2) e Deff (mm) SF k – empirical equations (µm2) Parallel fractures 0.1 828.0 - - - 832.9* Monolayer 1 958.3 0.654 1.0 6.0 939.5 Three layers 3 1688.6 0.821 1.0 6.0 1689.6 Eight layers 8 1622.5 0.818 1.0 6.0 1673.8 * The cubic law was used, while the Kozeny-Carman equation was used for the other scenarios 6.3 Parametric analysis via DEM 6.3.1 Simulations with proppant crushing A series of DEM simulations were performed to investigate the influence of proppant types, particle sizes, and concentrations on the permeability and fracture conductivity of the particle pack. Proppant embedment into rock surfaces was not considered in the simulations. Each particle pack sample was generated in a 8×8 mm (length × width) rectangular-shape box and was loaded axially with a top platen moving downward (Figure 6-4). The particle pack was loaded to 60 MPa and the void ratio and the height of the particle pack were recorded during the loading process. The particle size distribution was recorded from 10 MPa at an interval of 5 MPa. 153 Figure 6-4. Particle pack loaded in an 8×8 mm rectangular-shape box Use DEM simulations of Jordan mesh 20/40 with a concentration of 9.765 kg/m2 (2 lb/ft2) as an example, the evolution of the particle size distribution, effective particle size, void ratio, particle pack height, and the resulting proppant pack permeability and conductivity are plotted in Figure 6-5. Note that the effective particle size was calculated using Equation 2-7. As seen from Figure 6-5, with an increase in compressive stress, the particles were crushed and the effective particle size decreased. An obvious decrease in the effective particle size begins from 25 MPa and becomes significant after 35 MPa. The particle pack height and void ratio decrease close to linearly with the stress. The resulting particle pack permeability and fracture conductivity decreases dramatically from 394.8 µm2 and 2796000 µm3 at 10 MPa to 90.8 µm2 and 495000 µm3 at 40 MPa, then slowly decreases to 24.7 µm2 and 124000 µm3 at 60 MPa. Using the same approach above, the simulation results of Jordan sand using different particle sizes are shown in Figure 6-6. 154 Figure 6-5. Results of Jordan mesh 20/40 particles loaded to 60 MPa (concentration 9.765 kg/m2) 01020304050607080901000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Percent finerParticle size (mm)10 MPa20 MPa30 MPa40 MPa50 MPa60 MPa0.40.450.50.550.60.650 10 20 30 40 50 60 70Effective particle diameter (mm)Axial stress (MPa)Effetive particle size4.555.566.570 10 20 30 40 50 60 70Particle pack height (mm)Axial stress (MPa)Particle pack height0.30.40.50.60.70.80 10 20 30 40 50 60 70Void ratioAxial stress (MPa)Void ratio01002003004005000 10 20 30 40 50 60 70Permeability (µm2)Axial stress (MPa)Permeability0500000100000015000002000000250000030000000 10 20 30 40 50 60 70Fracture conductivity (µm3 )Axial stress (MPa)Fracture conductivity155 Figure 6-6. Results of Jordan particles with different sizes loaded to 60 MPa (9.765 kg/m2) 0.20.30.40.50.60.70.80.9110 20 30 40 50 60Effective particle diameter (mm)Axial stress (MPa)Mesh 20/40 Mesh 30/50Mesh 40/70 G850G600 G500G300 G2504.555.566.5710 20 30 40 50 60Particle pack height (mm)Axial stress (MPa)Mesh 20/40 Mesh 30/50Mesh 40/70 G850G600 G500G300 G2500.20.30.40.50.60.70.80.910 20 30 40 50 60Void ratioAxial stress (MPa)Mesh 20/40 Mesh 30/50Mesh 40/70 G850G600 G500G300 G2501010010001000010 20 30 40 50 60Permeability (µm2)Axial stress (MPa)Mesh 20/40 Mesh 30/50Mesh 40/70 G850G600 G500G300 G2501000010000010000001000000010 20 30 40 50 60Fracture conductivity (µm3)Axial stress (MPa)Mesh 20/40 Mesh 30/50Mesh 40/70 G850G600 G500G300 G250156 The G850 particle pack has the largest effective particle size at around 0.9 mm at 10 MPa while the G250 particle pack has the smallest size at around 0.27 mm. The mesh 20/40 particle pack, which mainly consisted of G500 and G600 particles, has an effective particle size at around 0.54 mm, between that of the G500 and G600 particle packs. The mesh 30/50 particle pack has an effective particle size of 0.4 mm and that of the mesh 40/70 is about 0.28 mm. With an increase of compressive stress acting on the particle pack, an obvious decrease in effective particle size occurs in particle packs with larger particle diameters as more crushing happens due to smaller particle strength. The significant decrease in the effective particle size of the G850 particle pack initiates at 15 MPa while that of G500/G600/mesh 2040 particles starts at around 25 to 30 MPa, which is in good agreement to the yield strength of G500/G600 Jordan sand. After 40 MPa, the G850 particle pack has an effective particle size similar to that of the mesh 20/40 particle pack, which is slightly higher than that of the G500 and G600 particle packs. A very small change in the effective particle size was observed in the rest of the particle packs when loaded to 60 MPa. The particle packs with a poly-disperse particle size distribution (mesh 20/40, 30/50 and 40/70) have a smaller sample height and void ratio at 10 MPa than that of mono-disperse size particle packs such as G850. Among the poly-disperse size particle packs, the mesh 40/70 has the smallest height of 6.21 mm and void ratio of 0.685, while the mesh 20/40 has the largest height of 6.34 mm and void ratio of 0.722. For mono-disperse particle sizes, particle packs with a higher effective particle size show a larger particle pack height and a void ratio at 10 MPa. The G850 particle pack has a height of 6.70 mm and a void ratio of 0.818, while the G250 particle pack has a height of 6.34 mm and a void ratio of 0.722. The particle pack height and void ratio show a similar response with the increase of compressive stresses. A more rapid decrease in the particle pack height and void ratio is found in the particle packs with larger effective particle size. After 35 MPa, the G250 and G300 particle packs have a larger height and void ratio than that of mesh 30/50 and 40/70, which are >G500>mesh 20/40 > G600 > G850. For the particle pack permeability, the G850 particle pack is the most permeable for compressive stresses less than 20 MPa due to its high effective particle size. For stress between 20 and 40 MPa, the G500/G600/mesh 20/40 particle packs are higher than the rest because of the relatively high values in both the void ratio and effective particle size. For stress greater than 40 MPa, the permeability of all the particle packs becomes similar to each other with the G300/G250/mesh 157 30/50 slightly more permeable than the rest. The highest permeability at 60 MPa is 37.0 µm2 for the G250 particle pack. Similar results are found in the fracture conductivity. The highest fracture conductivity is 204000 µm3 of the G250 particle pack at 60 MPa. Using the same approach above, the simulation results of Genoa sand using different particle sizes are shown in Figure 6-7. Figure 6-7. Results of Genoa particles with different sizes loaded to 60 MPa (9.765 kg/m2) As Genoa sand has lower particle strength than Jordan sand, the decrease in effective particle diameter initiates at a lower compressive stress. The decrease in the G850 particle pack starts at 10 MPa and that of G600/G500/mesh 20/40 begins at around 20 MPa. At 60 MPa, the mesh 20/40, mesh 30/50, G850, G600 and G500 particle packs have an effective particle size at around 0.30-0.32 mm. The effective particle size of G300 decreases moderately from around 35 MPa and is 0.20.30.40.50.60.70.80.9110 20 30 40 50 60Effective particle diameter (mm)Axial stress (MPa)Mesh 20/40 Mesh 30/50Mesh 40/70 G850G600 G500G300 G2500.10.20.30.40.50.60.70.810 20 30 40 50 60Void ratioAxial stress (MPa)Mesh 20/40 Mesh 30/50Mesh 40/70 G850G600 G500G300 G250110100100010 20 30 40 50 60Permeability (µm2)Axial stress (MPa)Mesh 20/40 Mesh 30/50Mesh 40/70 G850G600 G500G300 G25010001000010000010000001000000010 20 30 40 50 60Fracture conductivity (µm3)Axial stress (MPa)G850 G600G500 G300G250 Mesh 20/40Mesh 30/50 Mesh 40/70158 around 0.265 mm at 60 MPa. The effective particle size of G250 and mesh 20/40 changes very slightly with compressive stresses and is around 0.248 mm at 60 MPa. The void ratio and particle pack height has the similar response to compressive stress thus only the results of the void ratio are plotted here. Likewise, the mesh 20/40, 30/50, and 40/70 particle packs have a smaller void ratio than the other particle packs at 10 MPa. The G850 has the lowest void ratio after 15 MPa due to extensive particle crushing while G250 has the highest void ratio after 25 MPa. The G850 particle pack is the most permeable for compressive stresses less than 15 MPa due to its high effective particle size. For stress between 15 and 30 MPa, the G500/G600/mesh 20/40 particle packs are higher than the rest because of the relatively high values in both the void ratio and effective particle size. For stress between 30 and 45 MPa, G300 is more permeable than the other particle packs. For stress between 45 and 60 MPa, G250 is the most permeable with a permeability of 11.0 µm2 and the most conductive with a fracture conductivity of 58000 µm3 at 60 MPa. The simulation results of ceramic proppant using different particle sizes are shown in Figure 6-8. 159 Figure 6-8. Results of ceramic particles with different sizes loaded to 60 MPa (9.765 kg/m2) Ceramic proppant has much higher particle strength than Jordan sand and Genoa sand, thus the decrease in effective particle diameter starts at a much higher compressive stress. The decrease in the effective particle size of the G850 particle pack initiates at 25 MPa. Note that the mesh 20/40 proppant particles mainly consist of G600 particles. The G600 and mesh 20/40 particle packs have a similar response to compressive stresses in terms of the effective particle size/void ratio/permeability and fracture conductivity. A noticeable decrease in the G600/mesh 20/40 particle packs begins at around 40 MPa, while that of G500/G300/G250/ mesh 20/40 changes very slightly with compressive stresses. At 60 MPa, the effective size is 0.58 mm for G850, around 0.54 mm for G600/mesh 20/40, 0.497 mm for G500, 0.481 mm for mesh 30/50, 0.321 mm for G300, 0.291 mm for mesh 40/70, and 0.269 mm for G250. 0.20.30.40.50.60.70.80.9110 20 30 40 50 60Effective particle diameter (mm)Axial stress (MPa)Mesh 20/40 Mesh 30/50 Mesh 40/70 G850G600 G500 G300 G2500.40.450.50.550.60.650.70.750.80.850.910 20 30 40 50 60Void ratioAxial stress (MPa)Mesh 20/40 Mesh 30/50Mesh 40/70 G850G600 G500G300 G2501010010001000010 20 30 40 50 60Permeability (µm2)Axial stress (MPa)Mesh 20/40 Mesh 30/50Mesh 40/70 G850G600 G500G300 G25010000010000001000000010000000010 20 30 40 50 60Fracture conductivity (µm3)Axial stress (MPa)Mesh 20/40 Mesh 30/50Mesh 40/70 G850G600 G500G300 G250160 The void ratio is slightly higher than those of Jordan sand and Genoa sand at 10 MPa due to higher ceramic particle stiffness. The mesh 20/40 particle pack has the lowest void ratio at 10 MPa while that of mesh 20/40 and 30/50 mainly made of one or two particle sizes is higher than some mono-disperse size particles packs. The G850 has the lowest void ratio after 25 MPa due to particle crushing. G500 has the highest void ratio between 30 and 40 MPa, while G250/G300 are the highest between 40 and 60 MPa. Due to the high particle strength, the G850 particle pack is the most permeable for compressive stresses less than 40 MPa because of its high effective particle size. For stress between 40 and 60 MPa, the G600/mesh 20/40 particle packs are higher than the rest because of the relatively high values in both the void ratio and effective particle size. At 60 MPa the highest permeability is 166.8 µm2 and the highest fracture conductivity is 935000 µm3. The highest permeability of each stress level for the three types of proppants is plotted in Figure 6-9 for comparison. As expected, much higher permeability and fracture conductivity are found in ceramic particles than that of Jordan sand and Genoa sand. At 40 MPa, the ratio of permeability and fracture conductivity, respectively, is 1:3:11.4 and 1:2.9:11.6 (Genoa:Jordan:ceramic). At 60 MPa, the ratio of permeability and fracture conductivity, respectively, is 1:3.4:15.3 and 1:3.5:16.1. Figure 6-9. Comparison of highest permeability (left) and fracture conductivity (right) for three types of proppants (9.765 kg/m2) From the analysis above, it can be concluded the proppant size and proppant type have important influences on the final particle pack permeability and fracture conductivity. The particle size should be tailored to the expected compressive stresses to achieve the desired conductivity of the particle pack. 1010010001000010 20 30 40 50 60Permeability (µm2)Axial stress (MPa)Jordan Genoa Ceramic1000010000010000001000000010000000010 20 30 40 50 60Fracture conductivity (µm3)Axial stress (MPa)Jordan Genoa Ceramic161 The influence of the proppant loading on the permeability and fracture conductivity is also investigated with the Jordan mesh 30/50 particle pack. Three different proppant loadings are used, namely 4.882 kg/m2, 7.324 kg/m2 and 9.765 kg/m2. The results are shown in Figure 6-10. Initially a particle pack with different proppant loadings has the same effective particle size at 10 MPa. With an increase in compressive stress, slightly more particle crushing happens in particle packs with lower proppant loadings thus causing a slightly lower effective particle size under the same compressive stress. The final effective particle size at 60 MPa is 0.363 mm with 9.765 kg/m2, which is slightly higher than 0.346 mm with 4.882 kg/m2. Regarding the void ratio, a particle pack with a lower proppant loading has a slightly higher void ratio but with an increase in the compressive stress the void ratio is close to each other. Similar results are found for permeability and it seems particle packs with different proppant loadings have a similar permeability. The particle pack height is linearly proportional to the proppant loading. The fracture conductivity using 9.765 kg/m2 is about two times of that using 4.882 kg/m2. 162 Figure 6-10. Results of mesh 30/50 particles with different concentrations loaded to 60 MPa In the oil and gas industry, different sizes of proppants are commonly used together for hydraulic fracturing treatments for the same well. Here a mixture of Jordan particles with three different percentages of G250 and G500 (50/50, 30/70, and 70/30) was used to study the influence of different size mixtures on the particle pack permeability and fracture conductivity. The results are shown in Figure 6-11. 0.340.350.360.370.380.390.410 20 30 40 50 60Effective particle diameter (mm)Axial stress (MPa)9.7657.3244.882Proppant loading (kg/m2)0.40.50.60.70.810 20 30 40 50 60Void ratioAxial stress (MPa)9.7657.3244.882Proppant loading (kg/m2)05010015020025010 20 30 40 50 60Permeability (µm2)Axial stress (MPa)9.7657.3244.882Proppant loading (kg/m2)02000004000006000008000001000000120000010 20 30 40 50 60Fracture conductivity (µm3)Axial stress (MPa)9.7657.3244.882Proppant loading (kg/m2)163 Figure 6-11. Results of a blend of G250 and G500 particles with different portations loaded to 60 MPa As expected, the initial effective particle size of the three mixtures at 10 MPa is equal to the weighted effective particle size based on the percentage of G250/G500 and the effective particle size of G250/G500. With an increase of compressive stress, the effective particle size of the mixtures is consistently higher than the weighted value, which means less particle crushing happens in the mixtures than using a single mono-disperse size particles. At 60 MPa, the effective particle size is 0.343 mm for 50% G250 + 50% G500, 0.373 mm for 30% G250 + 70%, and 0.309 mm for 70% G250 + 30%, compared with weighted values of 0.316 mm, 0.343 mm, and 0.294 mm. On the other hand, the void ratio of the mixtures at 10 MPa is lower than particle packs with single mono-disperse particles due to a wider distribution of particle sizes. With an increase of compressive stress, the void ratio of mixtures is still consistently lower than that with single mono-disperse particles. The weighted permeability and fracture conductivity for each particle mixture 0.10.20.30.40.50.610 20 30 40 50 60Effective particle diameter (mm)Axial stress (MPa)50/50 50/50 (weight)30/70 30/70 (weight)70/30 70/30 (weight)100/0 0/100Percentage of G250 and G5000.10.20.30.40.50.60.70.810 20 30 40 50 60Void ratioAxial stress (MPa)50/50 50/50 (weight)30/70 30/70 (weight)70/30 70/30 (weight)100/0 0/100Percentage of G250 and G500010020030040010 20 30 40 50 60Permeability (µm2)Axial stress (MPa)0/10030/7030/70 (weight)50/5050/50 (weight)70/3070/30 (weight)100/0Percentage of G250+G50005000001000000150000020000002500000300000010 20 30 40 50 60Fracture conductivity (µm3)Axial stress (MPa)0/10030/7030/70 (weight)50/5050/50 (weight)70/3070/30 (weight)100/0Percentage of G250+G500164 are calculated based on the weighted effective particle size, void ratio, and particle height. When compressive stress is less than 45 MPa, the permeability and fracture conductivity of each particle mixture are much lower than the weighted permeability and fracture conductivity of single mono-disperse particles due to the much lower void ratio in the particle pack. Between 45 and 60 MPa, the permeability and fracture conductivity of each particle mixture are close to the weighted permeability and fracture conductivity due to a larger effective particle size in the particle mixture. It can be concluded that while mixing the particle pack with smaller particles reduces particle crushing, it may cause a significant decrease in permeability and fracture conductivity due to a loss in void ratio. 6.3.2 Simulations with proppant embedment The laboratory and DEM indentation tests in previous chapters show that the proppant embedment is mainly related to the Young’s modulus of the host rock. Proppant embedment into rock faces with different Young’s modulus is studied with DEM simulations. The simulation setup is shown in Figure 6-12, which is similar to the experimental setup by Alramahi and Sundberg (2012). In DEM a 6.35×6.35×3.175 mm cuboid rock sample was generated and bonded with the linear parallel bonds. A monolayer of uniform-sized stiff spheres was then placed on the rock faces. A platen (rigid wall) was placed on the top of the spheres and the model was loaded to 60 MPa. The displacement of the platen was recorded during the increment of axial stress. The embedment of the spheres into the rock face under a specific axial stress was determined as the difference between the platen displacement and the rock model deformation and sphere diameter. 165 Figure 6-12. Proppant embedment simulation setup First, the diameter of the stiff spheres was set to 0.6 mm and the Young’s modulus was varied by changing the effective particle-particle modulus while using the other DEM micro-parameters in Table 5-4. The total number of the spheres on the 6.35×6.35 mm cross-section is 112. The results are shown in Figure 6-13. The embedment-stress curves are similar to the experimental results by Alramahi and Sundberg (2012). A larger gradient of embedment-stress is found for all the curves at the beginning of the loading process, which is in good agreement to the force-indentation curve in the Brinell indentation tests (Figure 4-10). With the further loading, the embedment increases linearly. When Young’s modulus is 32.6 GPa, the embedment of the 0.6 mm spheres into the core face at 40 MPa is 7.3 μm and is 8.5 μm at 60 MPa. The Young’s modulus has great influence on the degree of embedment. For example, when Young’s modulus decreases to 3.3 GPa, the embedment at 60 MPa is 24.9 μm. The distribution and extent of particle embedment into the core face with different Young’s modulus can be appreciated in Figure 6-14. Deeper indentation into the core face is seen when the core Young’s modulus is lower. 166 Figure 6-13. Proppant embedment versus axial stress for cores with different Young’s modulus Figure 6-14. Rock face topology (unit mm) indented by 0.6 mm spheres at 60 MPa with different effective particle-particle modulus 01020304050600 10 20 30 40 50 60Embedment (µm)Axial stress (MPa)20 GPa10 GPa8 GPa5 GPa2 GPa1 GPa0.5 GPaEffective particle-particle modulusYoung's modulus (GPA)32.616.413.08.23.31.60.8effective particle-particle modulus: 20 GPaeffective particle-particle modulus: 5 GPaeffective particle-particle modulus: 1 GPaeffective particle-particle modulus: 0.5 GPa167 The embedment into rock faces with different Young’s modulus at five different axial stress levels is plotted against Young’s modulus as shown in Figure 6-15. These results are similar to the experimental results by Alramahi and Sundberg (2012). A power correlation was found between the embedment and Young’s modulus with a power coefficient from -0.494 to -0.418. At a higher axial stress level, a higher constant factor and a lower power factor exist. The embedment increases sharply for Young’s modulus less than 5 MPa at all the axial stress levels. The embedment at each stress level is used to calculate the Brinell hardness using Equations 4-2 and 4-3 from Chapter 4. The average force acting on each sphere is calculated as the total axial force divided by the total number of spheres. The Brinell hardness is plotted against the Young’s modulus as shown in Figure 6-16. A power correlation exists between Young’s modulus and Brinell hardness for a wide range of Young’s modulus; this is similar to the power correlation in Figure 4-12. A higher constant factor and a higher power factor exist for a higher compressive stress level. Note that the absolute values of power factors in Figure 6-17 are the same those in Figure 6-16 except they have positive sign because of the reciprocal relationship between of the embedment and Brinell hardness (Equation 4-3). The correlations can be used to estimate the Brinell hardness from the known Young’s modulus, and then the Brinell hardness can be used to back calculate the approximate embedment under a target stress level. 168 Figure 6-15. Correlation between particle embedment and rock Young’s modulus at different stresses Figure 6-16. Correlation between hardness and rock Young’s modulus at different stresses The above analysis uses a constant sphere size for embedment while varying the value of Young’s modulus. The influence of sphere size on embedment was also investigated while keeping the value of Young’s modulus at 0.8 GPa (DEM effective particle-particle modulus = 0.5 GPa). In y = 44.346x-0.494R² = 0.9968y = 39.101x-0.486R² = 0.9937y = 33.793x-0.472R² = 0.9891y = 27.716x-0.449R² = 0.9852y = 21.501x-0.418R² = 0.9717010203040500 5 10 15 20 25 30 35Embedment (µm)Young's modulus (GPa)60 MPa50 MPa40 MPa30 MPa20 MPa乘幂(60 MPa)乘幂(50 MPa)乘幂(40 MPa)乘幂(30 MPa)乘幂(20 MPa)Axial stress levelsy = 26.351x0.4936R² = 0.9968y = 24.906x0.4855R² = 0.9937y = 23.053x0.4723R² = 0.9891y = 21.081x0.4486R² = 0.9852y = 18.116x0.4183R² = 0.97170204060801001201401600 5 10 15 20 25 30 35Brinell hardness (HBW)Young's modulus (GPa)60 MPa 50 MPa40 MPa 30 MPa20 MPaAxialstress levels169 addition to 0.6 mm spheres, four other sphere sizes, namely 0.5, 0.425, 0.3, and 0.25 mm, were used for embedment analysis. The resulting number of spheres on the 6.35×6.35 mm cross-section is 112 for 0.6 mm size, 161 for 0.5 mm size, 223 for 0.425 mm size, 448 for 0.3 mm size, and 645 for 0.25 mm size. The results of embedment versus applied stress are shown in Figure 6-17. As expected, smaller embedment occurs when using smaller size spheres, because more spheres exist per unit area and thus each sphere carries less force from the axial stress. The embedment at 60 MPa is 49.3 μm for 0.6 mm spheres and is 39.2 μm for 0.5 mm spheres, which are more than two times higher than that of 0.3 mm spheres (23.1 μm) and 0.25 spheres (17.3 μm), respectively. The core face profile after embedment using different size spheres is shown in Figure 6-18. Shallower particle indentation occurs when using smaller size spheres. Figure 6-17. Proppant embedment versus axial stress for cores with different sphere sizes (effective particle-particle modulus = 0.5 GPa) 01020304050600 10 20 30 40 50 60Embedment (µm)Axial stress (MPa)G600G500G425G300G250Stiff sphere sizes170 Figure 6-18. Rock face topology (unit mm) at 60 MPa indented by different stiff sphere diameters with effective particle-particle modulus = 0.5 GPa The embedment results using different sphere sizes at five different stress levels are summarized in Table 6-2. These embedment values were used to calculate the Brinell hardness at each stress levels using Equations 4-4 and 4-5. A higher Brinell hardness is found at a higher stress level. The Brinell hardness of the five different sphere sizes has a mean value of 14.9 ± 1.3 HBW at 20 MPa, 18.2 ± 0.7 HBW at 30 MPa, 21.3 ± 1.1 HBW at 40 MPa, 23.7 ± 1.3 HBW at 50 MPa, 25.5 ± 1.6 HBW at 60 MPa. For the same stress level, the Brinell hardness using different sphere sizes is almost the same. This knowledge can be combined with the correlation between Young’s modulus and Brinell harness to estimate the embedment of a specific size sphere into a rock surface with a specific Young’s modulus under a specified stress. Stiff sphere diameter: 0.5 mmStiff sphere diameter: 0.425 mmStiff sphere diameter: 0.3 mmStiff sphere diameter: 0.25 mm171 Table 6-2. Brinell hardness values for different sphere sizes at different stress levels Sphere size (mm) Embedment (μm) at stress levels (MPa) Brinell hardness at stress levels (MPa) 20 30 40 50 60 20 30 40 50 60 0.6 25.9 31.9 38.4 44.2 49.3 15.0 18.3 20.3 22.0 23.7 0.5 21.2 26.5 30.1 34.1 39.2 15.3 18.4 21.6 23.9 24.9 0.425 16.9 23.3 27.1 30.1 32.7 16.4 17.8 20.4 22.9 25.3 0.3 13.3 15.1 17.0 19.6 23.1 14.6 19.3 22.9 24.8 25.3 0.25 12.6 14.0 15.1 16.3 17.3 12.9 17.4 21.5 25.0 28.2 Based on Figure 6-16 and Table 6-2, the Brinell hardness of rock is mainly influenced by its Young’s modulus and the applied stress acting on rock faces through a proppant particle pack, regardless of particle sizes. An equation to predict Brinell hardness of rock is derived from the results of the laboratory tests and DEM simulations. First, a power relation (like those in Figure 6-16) was used to correlate the Young’s modulus (E, unit GPa) and Brinell hardness (HB, unit HBW) in Figure 5-6, resulting in Equation 6-3 (from the laboratory data, R2 = 0.62) and Equation 6-4 (from the simulation data, R2 = 0.74). 37.043.19 labEHB  (6-3) 40.094.18 DEMEHB  (6-4) By setting Equation 6-3 equal to Equation 6-4 (same value of Brinell hardness), the Young’s modulus values simulated in DEM can be transformed to represent values in the laboratory. The Young’s modulus values Figure 6-13, namely 32.6, 16.4, 13.0, 8.2, 3.3, 1.6, and 0.8 GPa, are equivalent to 40.7, 19.3, 15.1, 9.1, 3.4, 1.6, 0.8 GPa in the laboratory. Using the laboratory values of Young’s modulus, the relationship between Young’s modulus and Brinell hardness shown in Figure 6-16 can be expressed in a general form as Equation 6-5. blabEaHB  (6-5) 172 where a and b are coefficients that are related to applied compressive stresses on a particle pack as plotted in Figure 6-19. Figure 6-19. Values of a and b under different compressive stresses and correlations Using the correlations between a and b and compressive stresses (σ, unit MPa) in Figure 6-19, Equation 6-5 can be expressed as Equation 6-6:   1926.00652.03409.07511.6 InlabEHB (6-6) Equation 6-6 can be used to estimate the Brinell hardness from known Young’s modulus (via compression tests, Brinell indentation, or log data), which can be appreciated in Figure 6-20, which is a plot with the data from Figure 6-16. The estimated values of Brinell hardness can be used to estimate proppant embedment under a target stress level using Equation 4-9. y = 6.7511x0.3409R² = 0.9985y = 0.0652ln(x) + 0.1926R² = 0.99220.380.40.420.440.46152025300 20 40 60 80Value of bValue of aCompressive stress (MPa)Value of aValue of b乘幂(Value of a)对数 (Value of b)173 Figure 6-20. Relation between Brinell hardness and Young’s modulus and compressive stress 6.3.3 Implications for proppant selection Selection of proppants for hydraulic fracturing involves choosing the proppant type, the proppant particle size, and the proppant concentration. The selection of a proppant should also consider the proppant cost, the formation closure stress and mechanical properties, as well as the targeted fracture conductivity based on the hydraulic fracturing system design. To illustrate the proppant selection process, a typical example is presented. It is assumed that a fully-filled monolayer proppant pack is selected for a formation with a closure stress of 40 MPa and a Young’s modulus (after fracturing fluid treatment) of 10 GPa. Based on the analysis in Section 5.5, a uniform proppant size is preferred and the proppant size to accommodate a 40 MPa compression is 0.6 mm if using Jordan sand, 0.38 mm if using Genoa sand, and 1.37 mm if using ceramic proppant. It is reasonable to assume that the initial fracture aperture is equal to the proppant size, and that this aperture is reduced by the proppant deformation and proppant embedment. According to Equations 5-2, 5-3, and 5-6, the force acting on each particle is expressed as: 2DF vp   (6-7) Using Equation 6-7, the particle force is 14.4 N for a 0.6 mm Jordan particle, 5.8 N for a 0.38 mm Genoa particle, and 75.1 N for a 1.37 mm ceramic particle. Using Equation 4-8, the proppant deformation is 0.014 mm for Jordan particles, 0.014 mm for Genoa particles, and 0.016 mm for ceramic particles. For a rock with a Young’s modulus of 10 GPa under 40 MPa compression, the 174 Brinell hardness is about 64 HBW using Equation 6-6. The proppant embedment into a rock face can be calculated using Equation 4-9 and the total embedment to both side of a rock fracture is 0.024 mm for a Jordan particle, 0.015 mm for a Genoa particle, and 0.055 mm for a ceramic particle. The residual fracture aperture considering proppant deformation and proppant embedment is 0.562 mm using Jordan particles, 0.351 mm using Genoa particles, and 1.299 mm using ceramic particles. The resulting void ratio of the monolayer particle pack is 0.789, 0.764, and 1.299, respectively. Using Equation 2-5, the predicted permeability is 540.8, 200.3, and 3027.8 µm2, respectively for Jordan, Genoa and ceramic particles, and the fracture conductivity is 308026, 71291, and 3985000 µm3, respectively. Compared with the multi-layer proppant packs in Section 6.5.1, the permeability of monolayer proppant packs are higher due to the higher proppant pack void ratio at 40 MPa. The fracture conductivity of a monolayer of Jordan particles (0.830 kg/m2) is equivalent to a G500 Jordan particle pack with 4.882 kg/m2 proppant concentration, while that of a monolayer of Genoa particles (0.586 kg/m2) is equivalent to a G300 Genoa particle pack with 3.271 kg/m2 proppant concentration. The fracture conductivity of a monolayer of ceramic particles (2.0 kg/m2) is equivalent to a G600 ceramic particle pack with 16.6 kg/m2 proppant concentration. The effectiveness of monolayer proppant packs decreases greatly in soft formations. To illustrate this effect the previous example was compared with a rock with a Young’s modulus of 1.0 GPa after fracturing fluid treatment (Josh et al. 2012, Zhang et al. 2017). Using the procedure above, the residual fracture aperture at 40 MPa is 0.520, 0.325, and 1.204 mm, respectively, for Jordan, Genoa and ceramic particles. The resulting void ratio is 0.657, 0.632, and 0.679, respectively. The particle pack permeability is 342.5, 124.3, 1943.2 µm2 and the fracture conductivity is 178078, 40308, and 2340000 µm3, respectively. The decrease in the fracture conductivity is more than 40% compared with that with a Young’s modulus of 10 GPa. If a multilayer Jordan sand pack is chosen for a formation with a closure stress of 50 MPa and a Young’s modulus of 1.0 GPa, the G300 particle pack should be selected as it provides the best fracture conductivity at 50 MPa compared with other sizes (Figure 6-6). Assuming the proppant concentration is 4.882 kg/m2, the G300 particle pack has a permeability of 55.3 µm2. The thickness of the G300 particle pack at 4.882 kg/m2 is around 2.83 mm. Using the effective particle size of the G300 particle pack at a 50 MPa compression, 0.313 mm, the average force acting on the core face from each particle is about 4.9 N. The Brinell hardness for a 1.0 GPa Young’s modulus at a 175 50 MPa compression is 25.6 HBW. The total embedment into the two rock faces is 0.040 mm, which is negligible compared to the particle pack thickness. The resulting residual fracture aperture is 2.790 mm and the estimated fracture conductivity is 156000 µm3. This example shows that a multilayer proppant pack can mitigate the effect of proppant embedment on the fracture conductivity when a soft formation is encountered. 6.4 Case study 6.4.1 In-situ stress and hydraulic fracturing stimulation Based on the log-derived density profile (Appendix E), the vertical stress overburden was calculated. The reservoir pressure was measured as 15.688 MPa at a vertical depth of 2246 m (TVD) from the diagnostic fracture injection test, based on which a pressure gradient of 6.98 kPa/m was derived. The obtained profiles of vertical stress and reservoir pressure were used together with the profile of Poisson’s ratio to obtain the minimum horizontal stress (Zoback 2010) as plotted in Figure 6-21. The resulting minimum horizontal stress is around 27.5 MPa at a depth of around 2246 m, resulting in a minimum horizontal stress gradient of around 12.2 kPa/m. Figure 6-21. Profiles of in-situ pore pressure (blue), overburden or vertical stress (black) and the minimum horizontal stress (red) 2100212021402160218022002220224022602280230010 20 30 40 50 60Depth (m)In-situ stress profile (MPa)Vertical stressReservoir pressureMinimum horizontal stress176 A horizontal well was drilled at a depth of around 2258 m (TVD) and the total length is about 2 km. Hydraulic fracturing was conducted with 35 perforation clusters along the horizontal portion of the well as shown in Figure 6-22. The first perforation cluster near the toe of the horizontal well is 72 m away from the second perforation cluster, while the others have a spacing of 50 m. The length of the first perforation cluster is 1 m while that of the other perforation clusters (3 holes per cluster) is 0.15 m. Figure 6-22. Layout of perforation clusters and hydraulic fracturing stages The hydraulic fracturing treatments are divided into nine stages and the stimulation parameters are shown in Table 6-3. The breakdown pressure at different stages varies from 40.2 to 64.7 MPa, while the instantaneous shut-in pressure is relatively constant at around 33.5 MPa, which is slightly larger than the minimum horizontal stress. The first stage only contains one perforation cluster and was pumped at a relatively high injection rate of 4.4 m3/min. Each stage from #2 to #7 have a total injection rate of 5.7-8.1 m3/min and cover four perforation clusters per stage. Five perforation clusters were covered in stages # 8 and # 9 and the average pumping rate was 8.4 m3/min and 5 m3/min respectively. #1#2#4#6#8#9 #3#5#7PerforationFracturing stage177 Table 6-3. Hydraulic fracturing multistage stimulation parameters Stage Perforation cluster number Breakdown pressure (MPa) Instantaneous shut-in pressure (MPa) Fluid pumped (m3) Proppant usage * (tonne) Average injection rate (m3/min) 1 1 40.2 32.1 441.4 1+ 20 + 3 4.4 2 4 49.1 33 1196.3 4 + 100 + 20 8.1 3 4 64.7 33.6 1113.9 4 + 100 + 20 8.1 4 4 55.5 32.8 1136.9 4 + 100 + 20 7.2 5 4 57.2 33.3 1206.7 4 + 100 + 20 6.3 6 4 51.8 33.6 1075 4 + 100 + 20 7.2 7 4 57.2 33.7 1031 4 + 100 + 20 5.7 8 5 58.2 33.6 1121 5 + 125 + 25 8.4 9 5 52.4 35.6 1401.3 4.3 + 127.7 + 25.7 5 * Mass of Proppant 1 + mass of Proppant 2 + mass of Proppant 3. Proppant 1 is a mesh 50/140 sand, Proppant 2 is a mesh 40/70 sand, and Proppant 3 is a mesh 20/40 resin-coated sand (SB Excel from Hexion) Three types of proppant were used at each stimulation stage. Proppant 1 is a mesh 50/140 sand mainly for fluid leak off control, Proppant 2 is a mesh 40/70 sand, and Proppant 3 is a mesh 20/40 resin-coated sand (SB Excel from Hexion) that can bond the proppant pack to control proppant flowback. The mass ratio of Proppant 1 to Proppant 2 to Proppant is 1:25:5 for stages from #2 to #9. In total, 9723.5 m3 of slickwater and 1080.7 tonnes of proppants were pumped in the stimulation. The volume concentration of proppants in the fracturing fluid is 0.042 assuming the apparent proppant density is 2650 kg/m3. The average injection rate at each perforation is 1.73 m3/min. 178 Assuming an identical vertical bi-wing fracture was created in each perforation cluster location, a total 35 fractures were produced. The fracture geometry is assumed to follow the Perkins–Kern–Nordgren model with a constant height and an elliptical cross-section as shown in Figure 6-23. The net fluid pressure inside the fracture at the perforation locations was estimated to be 6 MPa by subtracting the minimum horizontal closure stress from the instantaneous shut-in pressure. Referring to a case study by Gu et al. (2015) with a similar set of stimulation parameters to our case study, a fracture height of h = 60 m was assumed. Figure 6-23. Geometry of fractures and proppant packs after stimulation The fracture width at the perforation point (the centre of the bi-wing fracture) during the instantaneous shut-in was estimated to be 3.79 mm using Equation 6-8 (Gidley et al. 1989, Gu et al. 2015).  min1hPhGw  (6-8) where v and G are the Poisson’s ratio and the shear modulus of the formation, P is the fluid pressure, σhmin is the minimum horizontal stress. WellFracturesProppant packs (ellipsoidal shape )179 The half-length of the fractures (L) was estimated to be around 660 m using Equation 6-9.   fQGwL1814 (6-9) where Q is the average injection rate (0.865 m3/min) in each wing (two wings for one single fracture), and μf is the dynamic viscosity of the fracturing fluid, which is assumed to be 6.37 cp for slickwater (Gu et al. 2015). The obtained fracture geometry is similar to those from numerical simulations (Gu 2013, Gu et al. 2015). In addition, those simulations considered the proppant transport and found the proppant distribution in hydraulic fractures after hydraulic fracturing occurs in an approximately ellipsoidal shape (Figure 6-23). Here, an ellipsoidal shape was adopted for the proppant distribution. The principal axis half-length in the vertical direction was assumed 30 m and the half-length in the fracture horizontal propagation direction was 300 m. The ellipsoid thickness is equal to the thickness of proppant pack and is oriented in the minimum horizontal stress direction. The averaged proppant bulk volume for each fracture is 19.3 m3 using the total proppant mass of 1080 tonnes for the 35 fractures with an assumed proppant bulk density of 1600 kg/m3. Using the averaged bulk volume of proppants in a single bi-wing fracture, the maximum thickness of the proppant pack was estimated as 1.02 mm (19.3/3.1416/30/300/4×3×2 m), which is a reasonable value for a proppant pack thickness between hydraulic fractures. 6.4.2 Fracture conductivity and hydrocarbon production To calculate the average fracture conductivity provided by the proppant pack, the ellipsoidal proppant pack was simplified as a proppant pack with a uniform thickness/concentration on the elliptical cross-section perpendicular to the minimum horizontal stress. The average proppant concentration after simplification is 1.092 kg/m2 (=1080.7×1000/35/300/30/π). The fracture conductivity provided by the average proppant concentration was determined from a DEM simulation. Assuming the mechanical properties of Proppant 1 and Proppant 2 are similar to Jordan sand, and those of Proppant 3 are similar to the resin-coated sand presented in Chapter 2 and 3. A mixture of Proppant 1, Proppant 2, and Proppant 3 was generated following a mass ratio of 1:25:5. The particle size distributions of each proppant and the resulting mixture are shown in Figure 6-24. 180 The proppant mixture was placed between two rock faces and compressed to 27.5 MPa. The micro-parameters for rock in Section 5.4 were used for the rock samples. Figure 6-24. Proppant particle size distributions and a mixture compressed between two rock faces The resulting fracturing aperture is 0.689 mm at 27.5 MPa and the proppant particle-size distribution at 27.5 MPa is plotted in Figure 6-24. It can be seen only a small portion of Proppant 3 was crushed to smaller fragments. The effective particle size is 0.300 mm and the void ratio is 0.674. Using the Kozeny-Carman equation, the predicted permeability is 90.8 µm2 for the proppant mixture and the fracture conductivity is 63000 µm3. The formation has a permeability of 6.9e-8 to 2.6e-7 µm2 (0.07 to 0.26 μd) based on a gas permeability determination using a pressure decay system on crushed cores (BC Oil and Gas Commission 2014b). The relation between the actual cumulative gas volume and the production time (BC Oil and Gas Commission 2017) is shown in Figure 6-25. The actual cumulative production after 1 year is 11.8 million m3. 01020304050607080901000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Percent finerParticle size (mm)Proppant 1Proppant 2Proppant 3MixtureMixture (27.5 MPa)181 Figure 6-25. Cumulative gas production versus production time 6.4.3 Optimal proppant selection and Return-On-Fracturing-Investment Several scenarios of proppant selection were simulated in DEM and the resulting proppant pack permeability and fracture conductivity are summarized in Table 6-4. In Scenario 1, G600 proppant was used to replace the mesh 40/70 sand as G600 sand can give a better fracture conductivity at 27.5 MPa referring to Figure 6-9. As expected, a portion of G600 sand was crushed at 27.5 MPa but overall Scenario 1 gives a higher proppant pack permeability (121.4 µm2) and fracture conductivity (80000 µm3) compared with the base case due to a larger effective particle size. In the second scenario, the G600 sand was replaced with G300 sand to avoid the sand crushing. The resulting proppant pack conductivity is 149 µm2 and the fracture conductivity is 108000 µm3, which are higher than those of the base case and Scenario 1 due to relatively high values of the void ratio and the effective particle size. It should be noted that some large resin-coated sand particles were crushed at 27.5 MPa in Scenario 2. Thus, in Scenario 3, the mesh 20/40 resin-coated sand was replaced with the mesh 30/50 resin-coated sand. The resulting proppant pack conductivity is 150 µm2 and the fracture conductivity is 110000 µm3. In Scenario 4, no resin-coated sand was used and only the mesh 50/140 and G300 sand was utilized. The resulting proppant pack conductivity is 152 µm2 and the fracture conductivity is 114000 µm3, which is the highest for all the proppant selection scenarios and is a 70% improvement compared with the base case. 0200040006000800010000120001400016000180000 200 400 600 800Cumulative gas (103 ×m3)Production time (day)1 year production182 Table 6-4. Different proppant selection scenarios and resulting ROFIs after 1-year production Scenario Base 1 2 3 4 Proppant type 50/140 sand, 40/70 sand, 20/40 resin-coated sand 50/140 sand, G600 sand, 20/40 resin-coated sand 50/140 sand, G300 sand, 20/40 resin-coated sand 50/140 sand, G300 sand, 30/50 resin-coated sand 50/140 sand, G300 sand Total proppant usage in tonne (mass ratio) 1080 (1:25:5) 1080 (1:25:5) 1080 (1:25:5) 1080 (1:25:5) 1080 (1:30) fluid (m3) 9723.5 9723.5 9723.5 9723.5 9723.5 Fracture aperture (mm) 0.689 0.654 0.718 0.727 0.738 Void ratio 0.674 0.591 0.744 0.765 0.792 Effective diameter (mm) 0.300 0.410 0.336 0.326 0.314 Permeability (µm2) 90.8 121.4 149.0 150.0 152.0 Fracture conductivity (µm3) 63000 80000 108000 110000 114000 Production after 1 year (million m3) 11.8 13.3 15.4 15.5 15.8 Production value (million $) 1.84 2.07 2.40 2.42 2.46 Proppant cost (million $) 0.26 0.26 0.26 0.26 0.16 Non-proppant cost (million $) 1.56 1.56 1.56 1.56 1.56 ROFI (million $) 0.03 0.25 0.59 0.61 0.74 183 To evaluate the impact of proppant selection in hydraulic fracturing treatments on the economics of the well production, the Return-On-Fracturing-Investment (ROFI) after a 1-year production was calculated as the total gas production value after 1-year production subtracted by the proppant cost and non-proppant hydraulic fracturing cost. The gas production after one year was assumed proportional to the square root of the propped fracture conductivity. The gas price is assumed to be $25/boe (boe = ‘barrel of oil equivalent’, 1 boe = 160.2 m3). The average well completion cost for Montney gas plays in 2014-2015 was $2.7 million dollars per well with an average proppant usage of 2020 tonnes and an average pumped fluid of 14595 m3 (Fockler and McKenzie 2016). Referring to an analysis of frac sand prices (PacWest Consulting Partners 2014), the wholesale frac sand price in Montney area ranges from $120 to $180 per tonne due to a high transport cost. Here assuming the proppant price is $180/tonne, the non-proppant completion cost per well was calculated as $2.34 million dollars (= 2.7-180*2020/1e6) and the non-proppant completion cost per pumped volume was calculated as $160/m3 (= 2.34*1e6/14595). The values of the proppant cost and non-proppant cost were used to calculate the completion cost of the investigated well. The price of the resin-coated sand is assumed to be $540/tonne, 3 times of that of regular frac sand. The calculated ROFI for the base scenario is 0.03 million dollars. The results show that Scenario 2, 3 and 4 can significantly improve the hydrocarbon production and thus the ROFIs at 1 year. Scenario 4 gives the highest ROFI of 0.74 million dollars due to the lowest cost for the proppant when not using the expensive resin-coated sand. When proppant flow back is considered, Scenario 3 with the resulting ROFI 0.61 million dollars should be used. In addition, the mesh 30/50 resin-coated sand in Scenario 3 has better transport capacity than the larger mesh size 20/40 used in the base case. 6.5 Summary The flow simulation results show that the Kozeny-Carman equation can be useful for predicting the permeability for monolayer and multilayer proppant packs. A DEM parametric analysis on proppant crushing shows that the decrease of the particle pack permeability and fracture conductivity by compressive stresses is largely affected by the proppant sizes and proppant types. An optimal particle size to give a better permeability and fracture conductivity exists, which should be tailored to a specific stress expected in the field. Proppants 184 with uniform/narrow particle size distribution tend to have a higher conductivity than those with a wide particle size distribution. Multi-layer proppant packs with different proppant concentrations have a similar proppant pack permeability and the fracture conductivity is proportional to proppant concentrations. Adding smaller particles with a higher crushing strength to the particle pack can alleviate overall particle crushing; however, this may also cause a significant decrease in permeability and fracture conductivity due to a reduction in the void ratio. A DEM parametric analysis on proppant embedment shows that the proppant embedment into a rock face increases significantly for rock with a Young’s modulus less than 5 MPa at all the stress levels. A power relationship was found between the embedment and Young’s modulus and a power relationship exists between Young’s modulus and Brinell hardness over a wide range of Young’s modulus. In addition, using smaller particle sizes can result in shallower particle indentation into the rock face. When the particle indentation is converted to the Brinell hardness of the simulated formation, the Brinell hardness increases with the applied compressive stress but is not sensitive to proppant particle sizes under the same compressive stress. An equation to derive Brinell hardness from Young’s modulus under a specific stress was given, from which the embedment of a specific size sphere into a rock surface can be determined. The gained knowledge on proppant crushing and proppant embedment can be used to guide the selection of a monolayer or multilayer proppant pack with a proppant size that is compatible with the formation closure stress and mechanical properties. A more comprehensive comparison of the resulting fracture conductivity between different proppant selections can be conducted using the DEM simulation of proppants embedded between two rock faces. The usefulness of the DEM simulation of proppant crushing and embedment is demonstrated in the case study of proppant selection for a Montney well. Four different proppant scenarios were studied and the results show that the fracture conductivity could have been improved by over 70% with a different proppant size compared with the base scenario. The resulting ROFI could have been 0.74 million dollars after 1-year production instead of 0.03 million dollars. 185 7 Conclusions and Recommendations 7.1 Conclusions This research presents a comprehensive characterization of proppant crushing and embedment in fracture faces in a host rock and improves the DEM modelling of the proppant-rock interaction within fractures. Detailed research outcomes are concluded below. A laboratory crush test can be used to characterize the stiffness and particle strength of proppants with different types and sizes by measuring stress-strain responses and the change in particle-size distributions. A crush test can be improved to provide a fast and reliable approach for evaluating proppant pack permeability and conductivity at different stress levels by collecting supplemental data such as proppant pack height, grain size distributions, and grain shape. The permeability/fracture conductivity reduction with increased stresses is associated with three factors: reduction in void ratio/proppant pack height, reduction in particle size, and a shift away from spherical particle shapes. The values of these three factors change dramatically after the onset of proppant crushing. The stress level at which a proppant begins to experience significant crushing is related to proppant type and particle sizes. Four different types of proppants were tested: Jordan sand, Genoa sand, a resin-coated sand, and ceramic proppants. To simulate proppant crushing, a particle breakage model was developed in DEM. A proppant particle breaks into smaller fragments when the particle stress exceeds the particle strength. The stress-strain responses measured from crush tests were used to calibrate the stiffness of proppant particles in DEM, and the change in grain size distributions was utilized to determine particle strength and Weibull modulus. The particle strength in the breakage model controls the onset stress for particle crushing while Weibull modulus affects the particle size distributions. Four different sets of DEM micro-parameters were successfully obtained to represent four different types of proppants. The procedures presented here can be used to characterize and model other proppant types. Through a DEM parametric analysis with the developed DEM model and micro-parameters, it is found that proppants with a uniform/narrow particle size distribution tends to give a better conductivity than those with a wide range of particle sizes. For an expected compressive stress in 186 a fracture in the field, an optimal particle size for a given proppant type exists, which gives the best proppant pack permeability and fracture conductivity. Multilayer proppant packs with different proppant concentrations tend to have a similar proppant pack permeability and the fracture conductivity is proportional to the proppant concentrations as the proppant pack thickness is almost linearly related to the proppant concentration. Adding smaller particles, which tend to have a higher crushing strength, to the particle pack can alleviate overall particle crushing but may cause a significant decrease in permeability and fracture conductivity due to a loss in void ratio. RMT on cores from Montney siltstone shows that the mechanical properties in the horizontal direction vary with depth. Compared with fresh cores, the bulk density and uniaxial Young’s modulus of the archived cores is slightly lower likely due to the core drying out. The uniaxial and triaxial Young’s modulus and peak strength are similar for the fresh and archived cores. The anisotropy in Montney siltstone mechanical properties is noticeable, for example, the Young’s modulus in the horizontal orientation is much higher than that in the vertical orientation. The cohesion and the friction angle of the fresh cores along the horizontal orientation are similar to those of the archived cores, both of which have a higher friction angle and a lower cohesion than the cores along the vertical orientation. Brinell indentation tests on archived Montney siltstone show that Brinell hardness has a power regression with Young’s modulus. The tested Montney siltstone has a hardness ranging from 59 to 117 HBW at a 35 N indentation force. Saturation with 3% KCl fluid increases the hardness of the tested Montney siltstone while saturation with water shows little influence. Proppant embedment tests show that most of the proppant particles remained unbroken up to a 40 MPa compression and the propped fracture aperture reduction mainly results from proppant deformation and proppant embedment into rock faces. The average particle force of each proppant particle under specific compressive stresses can be used to calculate proppant particle deformation with a known particle stiffness and the proppant embedment with the known Brinell hardness correlated from Young’s modulus. The predicted fracture apertures are consistent with the laboratory results and previous data. Compared with multiple proppant packs, a monolayer proppant pack tends to have a higher void ratio thus a higher permeability when used in a formation where particle embedment is not significant. Based on sensitivity analysis, a procedure to calibrate the micro-parameters of a linear parallel bond model for rock-like materials in DEM was successfully applied to reproduce the macroscopic 187 mechanical properties of Montney siltstone. DEM simulations of Brinell indentation tests also show a power regression between Brinell hardness with Young’s modulus. Brinell hardness was not sensitive to Poisson’s ratio and uniaxial compressive strength. The calibrated micro-parameters from laboratory compression tests tend to give a lower indentation compared with the laboratory Brinell indentation results, which can be compensated by using a lower Young’s modulus in DEM with a lower particle-particle effective modulus. The calibrated DEM micro-parameters for both Montney siltstone and the proppant particles was used to simulate proppant embedment into rock fracture. The modelling results were consistent with laboratory measurements. Smaller size proppant particles have a smaller indentation and cause fewer broken bonds than larger particles. Particle stress has a linear relationship with the particle indentation force divided by the square of the particle diameter. When loading a proppant pack with a range of particle sizes, the larger particles carry larger forces but these particles also have lower strength. Proppants with a uniform-size are preferred for fractures propped with a monolayer of proppants. An equation is proposed for choosing an optimal proppant particle size for a given expected compressive stress to ensure that the particle stress remains smaller than the particle strength. A DEM parametric analysis shows that the proppant embedment increases sharply when the Young’s modulus is less than 5 MPa. A power relationship was found between the embedment and the Young’s modulus and a power relationship exists between Young’s modulus and Brinell hardness over a wide range of Young’s modulus. The Brinell hardness increases with the applied compressive stress but is not sensitive to proppant particle sizes under the same compressive stress. This knowledge can be used together with the Young’s modulus - Brinell harness correlation to estimate the embedment of a proppant pack into the surface of rock with a specific Young’s modulus under a specific stress. The improved characterization and DEM modelling provides a cost-efficient approach to evaluate the performance of proppants in hydraulic fractures without carrying out complicated hydraulic conductivity experiments. The research also shows that proppant crushing and embedment can greatly affect the fracture conductivity and the selection of an optimal proppant to achieve a desired fracture conductivity should consider the formation closure stress and mechanical properties, as well as the proppant strength, which in turn is related to the proppant type and particle sizes. 188 The results from this research can be used to guide the choice of a monolayer or multilayer proppant pack with a proppant size that is compatible with the formation closure stress and mechanical properties. A more comprehensive comparison of the resulting fracture conductivity between different proppant selections can be conducted using DEM simulations of proppants embedded between two rock faces. The usefulness of the developed DEM modelling of proppant crushing and embedment was demonstrated in a case study of proppant selection for a Montney well. Four different proppant scenarios were studied and the results show that the fracture conductivity could have been improved by over 70% with a different proppant size compared with the base case. The resulting ROFI could have been 0.74 million dollars after 1-year production instead of 0.03 million dollars. 7.2 Recommendations Although most of the research objectives were achieved, some limitations are present in this study and a few recommendations could be implemented in the future to improve the characterization and modelling of proppant crushing and embedment. In the current study, proppants were tested dry in crush tests while in real conditions proppants are mixed with fracturing fluids. Further studies should test proppants in wet conditions. Furthermore, different fracturing fluids are composed of different chemical contents and the chemical effect of fracturing fluids on proppants should be considered. In addition, the creep deformation of proppants and formation fractures and the influence of cyclic stresses on proppant crushing behaviours need to be further studied. Current DEM simulations simplify non-spherical proppant grains as spherical particles, which likely results in an overestimated void ratio. When replacing a broken particle with smaller particles as fragments, the overlap of spherical particles creates a pulse to neighbour particles thus causes an unrealistic change to the particle contact configuration. The smaller spherical particles (fragments) are easier to be pushed away causing a bigger drop in stress-strain responses compared with real conditions. To overcome these shortcomings, proppant grains can be modelled as clumps in future studies, which requires a much higher computational expense and a more comprehensive particle stress index considering different contact force configurations on a particle with different shapes. 189 The current study examines the proppant-rock interaction from a relatively small scale, in which fracture roughness is not considered and the proppant distribution between fractures is assumed uniform. Future studies should consider the created fracture geometry as well as proppant transport dynamics and the resulting distribution pattern inside the fracture geometry. In addition, hydrocarbon flow through proppant packs may not consist of laminar flow, thus non Darcy-flow effect on proppant pack permeability needs to be considered. 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Combination of different levels of micro-parameters for computation scenarios Scenario Particle-particle effective modulus A Particle friction coefficient B Raito of particle-wall effective modulus over particle-particle effective modulus C Wall friction coefficient D Normal-to-shear stiffness ratio E Damping coefficient F 1 -1 -1 -1 -1 -1 -1 2 1 -1 -1 -1 -1 1 3 -1 1 -1 -1 -1 1 4 1 1 -1 -1 -1 -1 5 -1 -1 1 -1 -1 1 6 1 -1 1 -1 -1 -1 7 -1 1 1 -1 -1 -1 8 1 1 1 -1 -1 1 9 -1 -1 -1 1 -1 1 10 1 -1 -1 1 -1 -1 11 -1 1 -1 1 -1 -1 12 1 1 -1 1 -1 1 13 -1 -1 1 1 -1 -1 14 1 -1 1 1 -1 1 15 -1 1 1 1 -1 1 16 1 1 1 1 -1 -1 17 -1 -1 -1 -1 1 1 18 1 -1 -1 -1 1 -1 202 Scenario Particle-particle effective modulus A Particle friction coefficient B Raito of particle-wall effective modulus over particle-particle effective modulus C Wall friction coefficient D Normal-to-shear stiffness ratio E Damping coefficient F 19 -1 1 -1 -1 1 -1 20 1 1 -1 -1 1 1 21 -1 -1 1 -1 1 -1 22 1 -1 1 -1 1 1 23 -1 1 1 -1 1 1 24 1 1 1 -1 1 -1 25 -1 -1 -1 1 1 -1 26 1 -1 -1 1 1 1 27 -1 1 -1 1 1 1 28 1 1 -1 1 1 -1 29 -1 -1 1 1 1 1 30 1 -1 1 1 1 -1 31 -1 1 1 1 1 -1 32 1 1 1 1 1 1 203 Table A-2. Macroscopic outputs of the 32 scenarios Scenario Constrained modulus 15-25 MPa (MPa) Particle pack height at 0.5 MPa (mm) Particle pack void ratio at 0.5 MPa 95% Stress index 1 95% Stress index 2 95% Stress index 3 1 279.7 6.42 0.722 28.2 49.5 119.7 2 504.0 6.46 0.735 34.0 55.9 138.8 3 372.4 6.70 0.799 37.6 54.6 138.7 4 625.4 6.73 0.807 43.3 60.6 159.7 5 318.5 6.44 0.731 29.6 51.1 124.0 6 509.6 6.47 0.736 34.7 57.7 141.3 7 400.3 6.71 0.802 37.6 55.6 139.5 8 693.6 6.70 0.800 43.8 61.1 160.8 9 300.8 6.54 0.755 28.5 48.0 118.6 10 530.4 6.53 0.754 34.0 55.0 137.3 11 400.3 6.88 0.846 38.6 54.9 141.9 12 669.2 6.87 0.844 44.2 61.8 160.4 13 325.2 6.52 0.751 29.1 48.6 119.7 14 535.0 6.54 0.755 34.1 55.4 139.5 15 438.1 6.82 0.832 38.1 52.7 139.1 16 705.0 6.89 0.850 45.4 62.4 165.6 17 293.8 6.42 0.724 28.8 49.9 121.1 18 452.6 6.43 0.728 35.5 56.6 145.0 19 364.6 6.70 0.799 35.9 54.0 134.3 20 620.2 6.69 0.795 41.5 59.2 154.0 21 312.5 6.43 0.727 29.3 51.0 123.8 22 458.2 6.46 0.735 35.5 58.3 146.8 204 Scenario Constrained modulus 15-25 MPa (MPa) Particle pack height at 0.5 MPa (mm) Particle pack void ratio at 0.5 MPa 95% Stress index 1 95% Stress index 2 95% Stress index 3 23 377.4 6.71 0.801 36.2 54.2 135.0 24 642.7 6.71 0.802 42.2 61.1 156.3 25 309.5 6.51 0.748 28.3 49.0 117.9 26 527.5 6.54 0.756 34.1 55.1 139.1 27 395.8 6.83 0.834 36.6 53.2 135.5 28 648.4 6.88 0.847 43.4 60.2 157.6 29 360.4 6.54 0.757 29.0 49.1 119.8 30 499.3 6.53 0.754 35.4 57.6 142.8 31 414.6 6.85 0.841 37.3 54.4 136.7 32 660.1 6.84 0.838 45.2 61.8 165.7 205 Table A-3. Estimated contribution of micro-parameters to outputs Micro-parameters and its combination Constrained modulus 15-25 MPa (MPa) Particle pack height at 0.5 MPa (mm) Particle pack void ratio at 0.5 MPa 95% Stress index 1 95% Stress index 2 95% Stress index 3 - - - - - - - A 73.6 0.2 0.2 32.1 64.8 56.8 B 20.6 84.1 84.0 65.2 29.5 38.8 AB 1.9 0.0 0.0 0.1 0.0 0.1 C 0.7 0.0 0.0 0.4 1.1 0.6 AC 0.1 0.0 0.0 0.0 0.3 0.1 BC 0.1 0.1 0.0 0.0 0.1 0.0 ABC 0.2 0.0 0.0 0.1 0.0 0.1 D 1.4 14.0 14.0 0.2 0.7 0.0 AD 0.0 0.0 0.0 0.0 0.4 0.1 BD 0.0 0.9 0.9 0.7 0.9 1.2 ABD 0.1 0.1 0.1 0.1 0.2 0.1 CD 0.0 0.1 0.1 0.0 0.1 0.0 ACD 0.1 0.0 0.0 0.0 0.1 0.2 BCD 0.0 0.0 0.0 0.0 0.0 0.0 ABCD 0.0 0.0 0.0 0.0 0.0 0.0 E 0.4 0.1 0.1 0.1 0.0 0.1 AE 0.4 0.0 0.0 0.1 0.0 0.2 BE 0.0 0.0 0.0 0.7 0.7 1.1 ABE 0.1 0.0 0.0 0.0 0.1 0.1 CE 0.1 0.0 0.0 0.0 0.2 0.0 ACE 0.0 0.0 0.0 0.0 0.0 0.0 206 Micro-parameters and its combination Constrained modulus 15-25 MPa (MPa) Particle pack height at 0.5 MPa (mm) Particle pack void ratio at 0.5 MPa 95% Stress index 1 95% Stress index 2 95% Stress index 3 BCE 0.0 0.0 0.0 0.0 0.1 0.0 ABCE 0.0 0.0 0.0 0.0 0.1 0.0 DE 0.0 0.0 0.0 0.0 0.1 0.0 ADE 0.0 0.0 0.0 0.0 0.0 0.0 BDE 0.1 0.0 0.0 0.1 0.0 0.1 ABDE 0.0 0.0 0.0 0.0 0.0 0.1 CDE 0.0 0.0 0.0 0.1 0.2 0.1 ACDE 0.0 0.3 0.3 0.0 0.0 0.0 BCDE 0.0 0.0 0.0 0.0 0.0 0.0 ABCDE 0.1 0.0 0.0 0.0 0.2 0.0 207 Appendix B: Laboratory Crush Test of Uniformly-graded Proppants It is widely acknowledged that sand grains with different sizes might have different crushing strength (Nakata et al. 2001). To design a sand pack with higher crushing resistance and permeability, it is important to understand the particle strength of different proppant sizes and types in order to optimize the grain size distribution of a proppant pack to resist different closure stresses. This appendix presents crush test results on uniformly graded proppant grains that were retained on an individual sieve with sizes including 0.25, 0.3, 0.355, 0.425, 0.5, 0.6, and 0.71 mm. The proppant grains from these individual sieves were term as G250, G300, G355, G425, G500, G600, and G710. Note that G500 and G600 Jordan sand presented in section 2.1 were included here. B.1 Stress-strain response Crush tests were loaded up to a compressive stress of 40 MPa. The initial stage of stress-strain responses from 0 MPa to 5 MPa for different uniformly graded sizes shows deviation as seen in Figure B-1 for Jordan sand, after which the stress-strain curves have a similar linear response for different sizes. The stress-strain responses were plotted beginning from 5 MPa as shown in Figure B-2 to facilitate comparison. The average one-dimensional constrained modulus of the proppant pack at stresses between 15 and 25 MPa (M1) and that between 30 and 40 MPa (M2) were calculated as shown in Table B-. Figure B-1. Stress-strain response of Jordan sand of single size range plotted from 0.5 MPa 05101520253035400 0.02 0.04 0.06 0.08 0.1 0.12 0.14Compressive stress (MPa)Vertical strainG250-1G300-4G355-5G425-1G500-4G600-4G710-4G710-5208 Figure B-2. Stress-strain response of four proppants of single size range after 40 MPa Table B-1. Constrained modulus of proppant packs (mean value ± standard deviation, unit: MPa) Size Jordan Genoa Resin Ceramic M15-25 M30-40 M15-25 M30-40 M15-25 M30-40 M15-25 M30-40 G710 407 ± 57 185 ± 32 184 ± 23 238 ± 10 425± 83 442 ± 54 728 ± 36 541 ± 32 G600 457 ± 43 223 ± 30 239 ± 17 234 ± 7 438± 40 502 ± 35 785 ± 25 727 ± 55 G500 474 ± 27 242 ± 23 305 ± 25 246 ± 18 466± 15 536 ± 24 566 ± 31 618 ± 42 G425 460 ± 9 329 ± 17 326 ± 35 249 ± 31 473± 19 554 ± 20 555 ± 28 658 ± 59 G355 462 ± 26 353 ± 21 399 ± 31 327 ± 27 480± 20 577 ± 29 634 ± 14 684 ± 26 G300 474 ± 18 379 ± 11 429 ± 36 324 ± 27 497± 6 609 ± 8 681 ± 19 731 ± 31 G250 481 ± 11 413 ± 7 438 ± 25 374 ± 23 471± 15 588 ± 23 709 ± 42 768 ± 30 05101520253035400 0.02 0.04 0.06 0.08 0.1 0.12 0.14Compressive stress (MPa)Vertical strainG250-1G300-1G355-5G425-2G500-4G600-4G710-505101520253035400 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18Compressive stress (MPa)Vertical strainG250-5G300-4G355-1G425-5G500-5G600-5G710-205101520253035400 0.02 0.04 0.06 0.08 0.1Compressive stress (MPa)Vertical strainG250-4G300-4G355-1G425-4G500-5G600-2G710-405101520253035400 0.02 0.04 0.06 0.08Compressive stress (MPa)Vertical strainG250-3G300-1G355-1G425-5G500-1G600-3G710-4209 For all the uniformly graded sizes of Jordan sand, the stress increases linearly with the increase of the strain up to 25 MPa. The average one-dimensional constrained modulus M1 is around 460 MPa. After 25 MPa, sand packs including G500, G600, and G710 show less stiff response due to the generation of fragments under compression shown in the particle-size distributions (Figure 2-26). The average one-dimensional constrained modulus M2 for G500, G600, and G710 is much smaller than M1 and is 242, 223 and 185 MPa, respectively. For G250, G300, G355, and G425, the stress increases close to linear with the increase of strain and the generation of fragments by weight is smaller than 20%. The average one-dimensional constrained modulus M2 for G250, G300, G355, and G425 is moderately smaller than M1 and is 413, 379, 353, and 329 MPa, respectively. For Genoa sand, the sand pack with smaller uniformly graded sizes has higher constrained modulus than that with higher particle sizes at the stress level under 25 MPa. The value of average one-dimensional constrained modulus M1 for G250 and G710 is 438 MPa and 184 MPa, respectively. With the further increase of stress, the stress-strain response becomes less stiff. However, it should be noted that the stress-strain response becomes more stiff for G710 and G600 sand after 30 MPa. The value of M2 is 238 MPa for G710 sand compared to the value of M1 at 184 MPa. As shown in the particle-size distribution of G600 and G710 sand, more than 45% of sand grains were crushed into finer fragments. The increase of the stiffness in stress-strain response may result from the further compaction of induced fragments. For resin-coated sand, the stress-strain response for different uniformly graded sizes has a similar constrained modulus around 450 MPa for M1. With the increase of stress, the stress-strain behaves almost linearly and the constrained modulus increases slightly to around 550 MPa for M2. For ceramic proppants, the constrained modulus for different uniformly graded sizes is quite different. G250, G300 and G355 have a similar value of M1 ranging from 634 to 709 MPa, which is larger than that of G425 and G500 at around 560 MPa. G600 and G710 have the highest value of M1 ranging from 785 to 728 MPa. The constrained modulus for uniformly grades sizes is consistent with that of three mesh sizes, namely mesh 20/40, 30/50, and 40/70, shown in Figure 2-22. The mesh 20/40, 30/50, and 40/70 ceramic proppant, respectively, mainly consists of G600, G500 and G425, and G250, G300 and G355. The mesh 20/40 has the highest value of M1, while that of mesh 30/50 is the lowest. 210 B.2 Particle-size distribution The particle-size distributions were determined by sieve analysis after the crush test. The results are shown in Figure B-3. With the increase of particle size, the percentage of particles retained on its original sieve decreases implying more grain crushing happens on grains with larger sizes. For example, G710 Jordan sand has a 33.3% of sand passing through the 0.710 mm sieve after the crush test, while G250 Jordan sand has 9.0% of sand passing through the 0.25 mm sieve. For the same grain size tested, Genoa sand exhibits the largest change in particle-size distribution among the four proppant types, while Jordan sand is the second largest. The resin-coated sand and ceramic have much smaller change regarding the particle-size distribution than the other two proppants. Figure B-3. Particle-size distribution of four proppants of single size range after 40 MPa As shown in Figures 2-14 to 2-16, for uniformly-graded grains originally collected from a single size range, grains crushed to finer sizes have a distinct lower value regarding the roundness, sphericity, aspect ratio, and minferet. The particles passing through the original sieve can be 01020304050607080901000 0.2 0.4 0.6 0.8 1Percent finerParticle size (mm)G250G300G355G425G500G600G71001020304050607080901000 0.2 0.4 0.6 0.8 1Percent finerParticle size (mm)G250G300G355G425G500G600G71001020304050607080901000 0.2 0.4 0.6 0.8 1Percent finerParticle size (mm)G250G300G355G425G500G600G71001020304050607080901000 0.2 0.4 0.6 0.8 1Percent finerParticle size (mm)G250G300G355G425G500G600G710211 regarded as crushed fragments from the original particles. Based on this, the crushing percentage and the survival probability of four proppants under 40 MPa can be determined as shown in Table B-. It should be noted that for resin-coated sand the coating can hold the fragments once the grains are crushed, thus the particle size distribution may not validly reflect the crushing percentage and the survival probability. Table B-2. Crushing percentage (cp) and survival probability (sp) of proppants under 40 MPa Size Jordan Genoa Resin Ceramic cp sp cp sp cp sp cp sp G710 33.3 66.7 56.0 44.0 15.7 84.3 18.8 81.2 G600 29.2 70.8 47.8 52.2 12.6 87.4 12.8 87.2 G500 25.3 74.7 40.4 59.6 9.9 90.1 11.4 88.6 G425 19.2 80.8 36.5 63.5 8.8 91.2 10.7 89.3 G355 15.1 84.9 27.3 72.7 6.1 93.9 5.9 94.1 G300 13.2 86.8 26.3 73.7 5.8 94.2 5.8 94.2 G250 9.0 81.0 26.3 73.7 4.4 95.6 1.8 98.2 B.3 Grain strength follows Weibull statistic distribution? It can be seen from the particle-size distributions and stress-strain responses that the particle size might have an influence on the grain strength. This section discusses the grain strength using a similar approach by Nataka et al. (1999 and 2001). Experimental studies of the crush test on a single silica grain have shown that the particle force F increases proportionally to the displacement, followed by a sudden decrease as the particle is fractured (Nataka et al. 1999). Based on the peak force Fpeak, the characteristic stress σ when a particle with a diameter d breaks, which is termed as particle strength, can be determined as: 2dFpeak (B-1) 212 By repeating the single grain crush test on different single silica grains, it was shown that the particle strength is not a single value and its distribution follows the Weibull statistics. The probability of survival of a particle of size d subjected to a given stress can be given by:  msddtestedparticlesofnumbertotalstressatcrushingparticlesofnumberdP030exp, (B-2) where σ0 is the characteristic stress at which 37% of particles of size d0 survive, and m is the Weibull modulus. The value of Weibull modulus for silica sand is around 3 from experimental data (Nakata et al. 1999). However, there are some discrepancies regarding the value of m. For example, it seems value of Weibull modulus for different grain sizes are different. A value of 3.04 was reported for silica with a grain size range from 1.4 to 1.7 mm, while the value is 2.17 and 1.82, respectively, for a size range of 0.6-0.71 mm and 0.25-0.3 mm. For this study, no crush test on single proppant particles was performed thus the distribution of single grain strength for different proppants is unknown. However, the survival probability of proppants from crush test can be used to investigate whether the strength of particle follows the Weibull distribution. This approach was adopted from Nakata et al. (1999) that defines an average characteristic tensile stress σ that is linked to the void ratio e of the proppant pack and the applied vertical stress σv on the proppant pack.  232 61 edFv (B-3) For particles with the same diameter d, the equation B-2 can be presented as:  mnsP exp (B-4) where σn is the characteristic stress at which 37% of particles of size d survive. Equation B-4 can be rewritten as:      ns mP  loglogloglog  (B-5) 213 For both G500 and G600 Jordan sand, the void ratio and the probability of survival under four applied vertical stress (10, 20, 30, and 40 MPa) are known as shown in Table B-. The data can be plotted according to equation 10 to get the slope gradient as the Weibull modulus as shown in Figure B-4. It can be seen that the Weibull modulus is 1.3782 for G600 and is 2.2499 for G500. The characteristic stress can be calculated from the y incept, which is 94.1 MPa for G600 and is 65.6 MPa for G500. The Weibull modulus seems to be in the range of the typical values for silica sand. The characterises stress of G600 Jordan sand is of the same magnitude to that of a G600 silica sand, which is 72.87 MPa with a Weibull modulus 2.17 using single particle crush test (Nakata et al. 2001). However, the characteristic stress of G500 sand smaller than G600 sand contradicts with the experimental results showing that the characteristic stress increases with the decrease of particle size for silica sand. Table B-3. Parameters of G500 and G600 Jordan sand used in equation 10 Stress level σv (MPa) G500 G600 e σ Ps e σ Ps 10 0.599 8.9 0.986 0.596 8.9 0.950 20 0.565 17.5 0.965 0.561 17.5 0.944 30 0.526 25.8 0.897 0.517 25.7 0.862 40 0.451 33.3 0.747 0.442 33.2 0.708 Figure B-4. Weibull modulus and characteristic stress for Jordan G500 and G600 y = 1.3782x - 6.2625R² = 0.7803y = 2.2499x - 9.4116R² = 0.9463-5-4-3-2-102 2.5 3 3.5 4log(-log(Ps))log(σ) (MPa)Jordan G600Jordan G500Linear (Jordan G600)Linear (Jordan G500)214 The applicability of Weibull distribution to the particle strength can be further explored with the probability of survival of different uniformly-graded proppant packs under a 40 MPa compression. The equation B-2 can be rewritten as:     00 loglog3loglog mddPs (B-6) The parameters used for equation B-6 are shown in Table B-. Here, the values of m were calculated by using d0 and σ0 from G500 (d0=0.550 mm, σ0=65.6 MPa) and G600 (d0=0.655 mm, σ0=94.1 MPa). The results show that the value of Weibull modulus is not a constant. The value of m decreases from 2.81 to 0.54 for G600 and decreases from 1.35 to -0.15 for G500 when the data used in equation 11 is from a smaller size of uniformly graded particles. The single particle crushing test results (Nakata et al. 2001) also show that the Weibull modulus decreases from 3.04 for 1.4-1.7 mm silica particles to 1.82 for 0.25-0.3 mm silica particles, which means more variability in particle strength for smaller particles. However, the value of Weibull modulus may not be valid as the value becomes close to zero or negative when using the data from G250 and G300. It can be concluded that the particle strength becomes stronger with the decrease of particle size as the survival of probability increases. However, the distribution of particle strength may not fully follow the Weibull statistic and the Weibull modulus to reflect the Weibull statistic may not be a constant value for different particle sizes. Table B-4. Parameters of uniformly-graded Jordan sand used in equation 11 (σv=40 MPa) Size e σ Ps d (mm) mlog(σ/σ0)G500 mG500 mlog(σ/σ0)G600 mG600 G710 0.414 32.7 0.667 0.780 -1.95 2.81 -1.43 1.35 G600 0.415 32.7 0.708 0.655 -1.59 2.28 -1.06 1.01 G500 0.398 32.5 0.747 0.550 -1.23 1.75 -0.71 0.66 G425 0.398 32.5 0.808 0.463 -1.03 1.46 -0.50 0.47 G355 0.486 33.8 0.849 0.390 -0.78 1.17 -0.25 0.25 G300 0.492 33.9 0.868 0.328 -0.40 0.61 0.12 -0.12 G250 0.428 33.0 0.917 0.275 -0.37 0.54 0.16 -0.15 215 Appendix C: Montney Siltstone Figure C-1. Core box 2265.29 m to 2266.44 m (left) and 2266.44 m to 2267.55 m (right) 216 Table C-1. RMT matrix for samples from archived core #1 Sample ID Depth MD (m) Bulk density (g/cm3) Test type Lithology description TWN1-01 2068.12 2.60 Uniaxial “Shale: 100%, medium to dark gray, black in part, recomposed rock flour, amorphous to sub blocky, earthy and granular, soft to friable, firm in part, very fine texture, traces grading to shaley Siltstone, occasional Siltstone stringers, calcareous, argillaceous, carbonaceous in part, local softly dolomitic in part, tight, no shows, rare loose very fine quartz sand grains, poor to no porosity or shows” (BC Oil and Gas Commission 2013b) TWN1-02 2068.12 - Brinell TWN1-03 2068.29 2.52 Triaxial TWN1-04 2068.29 - Brinell TWN1-05 2068.41 - Tensile TWN1-06 2068.41 - Tensile TWN1-07 2068.41 - Tensile TWN1-08 2068.55 2.62 Triaxial TWN1-09 2068.55 - Brinell TWN1-10 2068.64 - Tensile TWN1-11 2068.64 - Tensile TWN1-12 2068.64 - Tensile TWN1-13 2068.76 2.59 Embedment TWN1-14 2068.76 - Brinell TWN1-15 2068.90 2.62 Embedment TWN1-16 2068.90 - Brinell 217 Table C-2. RMT matrix for samples from archived core #3 Sample ID Depth MD (m) Bulk density (g/cm3) Test type Lithology description TWN2-01 2265.35 - Tensile “Siltstone, medium to light gray, dark gray, firm to friable in part, quartzose, well sorted, well consolidated, very argillaceous cement to matrix, dolomitic locally becoming slightly calcareous, scattered carbonaceous plant remains, even weak to spotty fair, 2-5% earthy porosity, faint brownish to tan flourescence with even medium to bright yellow, moderately fast ribbon cut, occasional laminae very fine silt to silty Shale, medium gray, blocky to amorphous, locally carbonaceous, locally very silty grading to very fine siltstone.” (BC Oil and Gas Commission 2013b) TWN2-02 2265.35 - Tensile TWN2-03 2265.35 - Tensile TWN2-04 2265.60 2.57 Uniaxial TWN2-05 2265.60 - Brinell TWN2-06 2265.73 2.59 Triaxial TWN2-07 2265.73 - Brinell TWN2-08 2265.85 2.59 Triaxial TWN2-09 2265.85 - Brinell TWN2-10 2265.89 - Tensile TWN2-11 2265.89 - Tensile TWN2-12 2265.89 - Tensile TWN2-13 2266.05 2.57 Embedment TWN2-14 2266.05 - Brinell TWN2-15 2266.21 2.57 Embedment TWN2-16 2266.21 - Brinell TWN2-17 2266.50 2.56 Embedment TWN2-18 2266.50 - Brinell TWN2-19 2266.99 2.58 Embedment TWN2-20 2266.99 - Brinell TWN2-22 2267.08 - Brinell 218 Table C-3. RMT matrix for samples from archived core #4 Sample ID Depth MD (m) Bulk density (g/cm3) Test type Lithology description TWN3-01 2335.94 - Tensile “Siltstone: abundant pulverized chalky and recomposed cuttings, medium to light gray, stringers dark gray, scattered firm to friable, quartzose, well sorted, well consolidated, very argillaceous cement to matrix, calcareous, locally slightly dolomitic, scattered carbonaceous plant remains, traces finely disseminated pyrite, even weak to spotty fair, 2-5% earthy porosity, faint brownish to tan flourescence with even medium to bright yellow, moderately fast ribbon to massive cut, traces interludes very fine silty Shale, medium gray, blocky to amorphous, locally carbonaceous, silty grading to very fine” (BC Oil and Gas Commission 2013b) TWN3-02 2335.94 - Tensile TWN3-03 2335.94 - Tensile TWN3-04 2336.04 2.59 Uniaxial TWN3-05 2336.04 - Brinell TWN3-06 2336.08 2.57 Triaxial TWN3-07 2336.08 - Brinell TWN3-08 2336.21 2.61 Triaxial TWN3-09 2336.21 - Brinell TWN3-10 2336.37 2.62 Embedment TWN3-11 2336.37 - Brinell TWN3-12 2336.57 2.61 Embedment TWN3-13 2336.57 - Brinell TWN3-14 2336.87 - Tensile TWN3-15 2336.87 - Tensile TWN3-16 2336.87 - Tensile TWN3-17 2337.16 2.56 Embedment TWN3-18 2337.16 - Brinell TWN3-19 2337.22 2.59 Embedment TWN3-20 2337.22 - Brinell 219 Figure C-2. Axial stress difference versus radial and axial strains, measured during uniaxial compressive test (TWN1-01) Figure C-3. Pre test and post test of TWN1-01 020406080100120140-0.75 -0.25 0.25 0.75Axial compressive stress (MPa)Radial strain (left) and Axial strain (right) in %Axial strain - LVDT1Axial strain - LVDT2Radial strain - LVDT350% peakstrength220 Figure C-4. Axial stress difference versus radial and axial strains, measured during uniaxial compressive test (TWN2-04) Figure C-5. Pre test and post test of TWN2-04 020406080100120-1 -0.5 0 0.5 1Axial compressive stress (MPa)Radial strain (left) and Axial strain (right) in %Axial strain - LVDT1Axial strain - LVDT2Radial strain - LVDT350% peakstrength221 Figure C-6. Axial stress difference versus radial and axial strains, measured during uniaxial compressive test (TWN3-04) Figure C-7. Pre test and post test of TWN3-04 010203040506070-1 -0.5 0 0.5Axial compressive stress (MPa)Radial strain (left) and Axial strain (right) in %Axial strain - LVDT1Axial strain - LVDT2Radial strain - LVDT350% peakstrength222 Figure C-8. Axial stress difference versus radial and axial strains, measured during triaxial compressive tests with confining stress 10 MPa (TWN1-03) and 20 MPa (TWN1-08) Figure C-9. Pre test and post test of TWN1-03 (10 MPa confinement) Figure C-10. Pre test and post test of TWN1-08 (20 MPa confinement) 0306090120150180210240270-1 0 1 2Deviator stress (MPa)Radial strain (left) and Axial strain (right) in %Axial strain - 10 MPaAxial strain - 20 MPaRadial strain - 10 MPaRadial strain - 20 MPa223 Figure C-11. Axial stress difference versus radial and axial strains, measured during triaxial compressive tests with confining stress 10 MPa (TWN2-06) and 20 MPa (TWN2-08) Figure C-12. Pre test and post test of TWN2-06 (10 MPa confinement) Figure C-13. Pre test and post test of TWN2-08 (20 MPa confinement) 0306090120150180210240270300-1 0 1 2Deviator stress (MPa)Radial strain (left) and Axial strain (right) in %Axial strain - 10 MPaAxial strain - 20 MPaRadial strain - 10 MPaRadial strain - 20 MPa50% (10 MPa)50% (20 MPa)224 Figure C-14. Axial stress difference versus radial and axial strains, measured during triaxial compressive tests with confining stress 10 MPa (TWN3-06) and 20 MPa (TWN3-08) Figure C-15. Pre test and post test of TWN3-06 (10 MPa confinement) Figure C-16. Pre test and post test of TWN3-08 (20 MPa confinement) 0306090120150180210-1 0 1 2Deviator stress (MPa)Radial strain (left) and Axial strain (right) in %Axial strain - 10 MPaAxial strain - 20 MPaRadial strain - 10 MPaRadial strain - 20 MPa225 Appendix D: Micro-parameters Analysis of Bonded Siltstone Models Table D-1. Combination of different levels of micro-parameters for computation scenarios Scenario Particle-particle effective modulus A Particle normal-to-shear stiffness B Bond tensile strength C Bond cohesion D Bond friction angle E Particle friction coefficient F 1 -1 -1 -1 -1 -1 -1 2 1 -1 -1 -1 -1 1 3 -1 1 -1 -1 -1 1 4 1 1 -1 -1 -1 -1 5 -1 -1 1 -1 -1 1 6 1 -1 1 -1 -1 -1 7 -1 1 1 -1 -1 -1 8 1 1 1 -1 -1 1 9 -1 -1 -1 1 -1 1 10 1 -1 -1 1 -1 -1 11 -1 1 -1 1 -1 -1 12 1 1 -1 1 -1 1 13 -1 -1 1 1 -1 -1 14 1 -1 1 1 -1 1 15 -1 1 1 1 -1 1 16 1 1 1 1 -1 -1 17 -1 -1 -1 -1 1 1 18 1 -1 -1 -1 1 -1 19 -1 1 -1 -1 1 -1 20 1 1 -1 -1 1 1 226 Scenario Particle-particle effective modulus A Particle normal-to-shear stiffness B Bond tensile strength C Bond cohesion D Bond friction angle E Particle friction coefficient F 21 -1 -1 1 -1 1 -1 22 1 -1 1 -1 1 1 23 -1 1 1 -1 1 1 24 1 1 1 -1 1 -1 25 -1 -1 -1 1 1 -1 26 1 -1 -1 1 1 1 27 -1 1 -1 1 1 1 28 1 1 -1 1 1 -1 29 -1 -1 1 1 1 1 30 1 -1 1 1 1 -1 31 -1 1 1 1 1 -1 32 1 1 1 1 1 1 227 Table D-2. Results of 32 scenarios Scenario UCS (MPa) Young’s modulus (GPa) Poisson’s ratio Indirect tensile strength (MPa) Triaxial strength (20 MPa) Indentation depth at 35 N (µm) 1 36.5 32.7 0.13 7.75 87.3 - 2 47.5 72.9 0.07 9.21 157 11.6 3 22.0 24.7 0.24 4.56 151 17.8 4 17.8 45.0 0.27 3.89 111 - 5 73.7 36.7 0.07 18.30 154 17.2 6 60.2 65.5 0.12 14.60 89.2 - 7 51.5 23.5 0.26 11.00 126 - 8 64.5 50.4 0.24 12.80 165 12.1 9 54.6 33.7 0.07 9.06 241 16.4 10 36.3 64.6 0.12 8.13 183 - 11 17.5 23.5 0.26 3.95 127 - 12 22.2 50.2 0.24 4.59 187 12.1 13 113.0 32.2 0.13 22.00 206 - 14 145.0 69.7 0.07 25.60 270 10.9 15 63.8 25.2 0.24 12.70 211 15.2 16 51.0 47.1 0.27 11.30 160 - 17 52.0 38.8 0.07 9.06 181 16.2 18 36.0 65.2 0.12 8.11 119 - 19 17.5 23.5 0.26 3.95 125 - 20 22.2 50.2 0.24 4.59 177 11.8 21 70.8 32.7 0.12 17.00 126 - 22 89.6 72.6 0.07 21.00 194 10.7 228 Scenario UCS (MPa) Young’s modulus (GPa) Poisson’s ratio Indirect tensile strength (MPa) Triaxial strength (20 MPa) Indentation depth at 35 N (µm) 23 63.8 25.2 0.24 12.70 196 14.6 24 51.0 47.1 0.27 11.30 149 - 25 38.5 32.5 0.12 7.75 196 - 26 51.2 72.1 0.07 9.22 278 10.5 27 22.0 24.7 0.24 4.56 188 15.6 28 17.8 45.0 0.27 3.89 123 - 29 149.0 34.7 0.07 26.00 319 14.2 30 112.0 65.6 0.12 22.40 251 - 31 51.8 23.4 0.26 11.00 162 - 32 64.5 50.4 0.24 12.80 211 10.5 229 Table D-3. Estimated contribution of micro-parameters to outputs Micro-parameters and combination UCS (MPa) Young’s modulus (GPa) Poisson’s ratio Indirect tensile strength (MPa) Triaxial strength (20 MPa) - - - - - - A 0.0 75.2 0.0 0.0 0.0 B 25.0 20.5 93.4 26.5 7.8 AB 0.0 2.4 0.1 0.0 0.0 C 49.2 0.0 0.0 61.0 4.3 AC 0.0 0.0 0.0 0.0 0.1 BC 2.1 0.0 0.0 3.2 0.0 ABC 0.0 0.0 0.0 0.0 0.0 D 4.6 0.1 0.0 1.5 34.0 AD 0.0 0.0 0.0 0.0 0.0 BD 4.6 0.0 0.0 1.5 15.0 ABD 0.0 0.0 0.0 0.0 0.0 CD 3.9 0.0 0.0 1.5 1.0 ACD 0.0 0.0 0.0 0.0 0.0 BCD 0.0 0.0 0.0 0.0 0.0 ABCD 0.0 0.0 0.0 0.0 0.1 E 0.1 0.0 0.0 0.1 4.6 AE 0.1 0.0 0.0 0.0 0.0 BE 0.1 0.0 0.0 0.1 1.1 ABE 0.0 0.0 0.0 0.0 0.0 CE 0.1 0.0 0.0 0.1 0.2 ACE 0.0 0.0 0.0 0.0 0.0 BCE 0.0 0.0 0.0 0.0 0.0 ABCE 0.1 0.0 0.0 0.0 0.1 DE 0.1 0.0 0.0 0.1 0.2 ADE 0.0 0.0 0.0 0.0 0.0 BDE 0.0 0.0 0.0 0.0 0.0 230 Micro-parameters and combination UCS (MPa) Young’s modulus (GPa) Poisson’s ratio Indirect tensile strength (MPa) Triaxial strength (20 MPa) ABDE 0.5 0.0 0.0 0.5 0.0 CDE 0.0 0.0 0.0 0.0 0.0 ACDE 0.7 0.1 0.7 0.3 0.6 BCDE 0.0 0.2 0.0 0.0 0.0 ABCDE 4.4 1.4 5.7 2.0 29.7 231 Appendix E: Mechanical Profile of Montney Formation A set of geophysical log data from the Well 28232 in the Montney formation was retrieved and processed as plotted in Figure E-. The curves are either direct log data, or interpreted from logs, and the symbols represent laboratory measurements and are explained by the legend in each plot. Laboratory core gamma scan data was not available thus no depth shift was applied to the gamma ray well log. Figure E- shows the log data generally matches well with the laboratory data at the same depth thus no depth adjustment should not affect the results of this study. Figure E-1. Well logs and mechanical profiling of the Montney formation from Well 28232. 205021002150220022502300235024002450 2550 2650 2750Depth (m)Bulk density (kg/m3)RCA (fresh)RMT (fresh)RMT (archived)Calibrated log205021002150220022502300235024000 1Depth (m)Quartz+Feldspar+MicaLogXRD205021002150220022502300235024000 1Depth (m)Carbonate205021002150220022502300235024000 200 400Depth (m)DT (μs/m)DTC - logDTS - logDTS -ultrasoundDTC -ultrasound205021002150220022502300235024000 0.1 0.2 0.3Depth (m)Poisson ratioLogUltrasound freshStatic freshStatic archived2050210021502200225023002350240020 40 60 80Depth (m)Young's modulus (GPa)LogUltrasound freshStatic freshStatic archived232 The bulk density was measured with different methods on fresh and archived cores. The value of logged bulk density was reduced by 30 kg/m3 in the calibration process to better match the laboratory measurements from the rock core analysis (RCA) and rock mechanics testing (RMT). The calibrated log of the bulk density is consistent with the data from RCA using the mercury immersion method. The calibrated log shows that the upper part (2050-2150 m) of the Montney formation has a slightly lower bulk density at around 2550 kg/m3, and the middle part (2150-2350 m) has a density at around 2600 kg/m3, while the lower part has a density higher than 2600 kg/m3. The mineralogy log also shows the laboratory measurements determined from the X-ray diffraction method as the dry weight fraction. It can see the log data matches well with the laboratory X-ray diffraction results. It is evident that the lithology shifts from the Doig formation to the Montney formation at a depth of 2068 m MD with a compositional change in the carbonate and the quartz+feldspar+mica. In the Montney formation at a depth from 2100 to 2400 m, the lithology is consistently quartz-rich with roughly a 0.5 fraction of quartz+feldspar+mica, 0.2 fraction of carbonate, and 0.3 fraction of clay. The total organic content from the Source Rock Analyzer ranges from 0.18 to 4.29 vol.% with an average of 1.15 ± 0.60 vol.%. The P- and S- wave travel time from the sonic log are plotted along with the laboratory ultrasound measurements (horizontal orientation). The laboratory ultrasound measurements were acquired during triaxial compressive tests on samples with a confining pressure ranging from 20 to 40 MPa. Only the data with an axial stress close to the in-situ overburden stress of ~ 50 MPa are plotted here. It can be seen that the sonic log P- and S- wave velocities are consistently slightly lower than the ultrasound measurements. The difference may result from the different frequencies used for the sonic wave (tens kilohertz) and ultrasound wave (several megahertz). The dynamic Poisson’s ratios determined with sonic velocities and ultrasound velocities are plotted along with the static ones from triaxial compression tests (horizontal orientation). The ultrasound Poisson’s ratios scatter around the log-derived dynamic Poisson’s ratios, while the static Poisson’s ratios from fresh cores are close to the log-derived dynamic Poisson’s ratios and the static Poisson’s ratios from archived cores are consistently higher than the log-derived Poisson’s ratios. Based on the log-derived dynamic Poisson’s ratios, the upper part of the Montney 233 formation (2050-2150 m) has a dynamic Poisson’s ratio at around 0.20, which is slightly lower than that of the rest of the Montney formation at around 0.22. The dynamic Young’s moduli derived from the sonic log data and laboratory ultrasound data are plotted along with the static ones from triaxial compression tests. The ultrasound Young’s moduli are consistently higher than the log-derived ones, while some of the static measurements are close to the log-derived dynamic ones and some are smaller. From the log-derived data, the dynamic Young’s moduli range from 40 to 57 GPa with an average value of 49 ± 3 GPA. The log-derived dynamic Young’s modulus changes with the core depth and a positive relation with the fraction of clay was found as shown in Figure E-. A larger clay fraction would likely result in a lower dynamic Young’s modulus with is consistent with the conclusion by Alramahi and Sundberg (2012). Figure E-2. Correlation between dynamic Young’s modulus and clay fraction The correlations between static Young's modulus and Poisson's ratios with bulk density from triaxial compression tests on fresh cores are shown in Figure E-. Positive correlations between the bulk density and the static Young’s modulus and the Poisson’s ratios were observed. The linear regression relations from this figure were used to derive the static Poisson’s ratios and Young’s moduli using the calibrated bulk density log. The results plotted in Figure E- show that the density y = 39.187x-0.183R² = 0.5112304050607080900 0.2 0.4 0.6Young's modulus (GPa)Clay fraction234 derived logs can generally match the logs derived from sonic wave velocities except for the depth from 2250 to 2320 m. Figure E-3. Correlation between bulk density and static Young’s modulus (left) and Poisson’s ratio (right) Figure E-4. Static Poisson’s ratio and Young’s modulus derived from bulk density log A comparison of the field log and the laboratory data show that the log data generally matches the laboratory measured from rock taken at the same depth. The Montney siltstone at a depth from y = 0.1466x - 336.74R² = 0.903630354045502520 2540 2560 2580 2600 2620Young's modulus (GPa)Bulk density (kg/m3)y = 0.0004x - 0.8828R² = 0.74120.150.170.190.210.230.252520 2540 2560 2580 2600 2620Poisson ratioBulk density (kg/m3)205021002150220022502300235024000.1 0.2 0.3Depth (m)Poisson ratio2050210021502200225023002350240020 40 60 80Depth (m)Young's modulus (GPa)LogUltrasound freshStatic freshStatic archivedDensity derived log235 2100 to 2400 m is consistently quartz-rich with roughly a 0.5 fraction of quartz+feldspar+mica, 0.2 fraction of feldspar, and 0.3 fraction of clay. The dynamic Young’s modulus decreases with an increase in clay composition. Positive correlations between the bulk density and the static Young’s modulus and the Poisson’s ratios were observed, based on which the static Young’s modulus along the depth was derived.
Thesis/Dissertation
eng
Civil Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Laboratory and discrete element study of proppant crushing and embedment and their influence on fracture conductivity
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