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Thesis Defense Presentation-Sabarisha
- 1. STUDY ON CHEMICAL ACTIVITY OF PIERRE SHALES AND ITS EFFECT ON NEAR WELLBORE PORE PRESSURE DISTRIBUTION Sabarisha Subramaniyan July 2014
- 2. 3/2/2019 2 Objectives Shale - Chemical nature Contribution to wellbore instability Impact on mechanical response Sensitivity to other properties
- 3. 3/2/2019 3 Outline Introduction Matlab coding structure Case Study – Results Code Verification Conclusions Limitations
- 4. 3/2/2019 4 Introduction Wellbore Instability Problems are Borehole enlargement Hole shrinkage Hole fracture Hole collapse Principle of wellbore stability = Equilibrium between rock strength and in-situ stresses Petrowiki.spe.org
- 5. 3/2/2019 5 Causes for Wellbore instability Rock removal Stress redistribution Stress state > or < in-situ stresses = failure Contributors Mechanical, chemical, thermal, angular instability (McLean, 1990)
- 6. 3/2/2019 6 Clay Colloidal Chemistry (Negative charges) Isomorphic substitution If pH < or > Zero Point Charge (ZPC) Clay in electrolyte = Diffuse double layer Stern layer – equilibrium (zeta potential) Concentration decline (Colloidal chemistry department, University of Szeged)
- 7. 3/2/2019 7 Cation Exchange Capacity (CEC) 1. Measure of exchangeable ions 2. CEC ↑ = Reactivity ↑ 3. CEC ∝ pH Membrane Efficiency (ME) Restricting ions (size & charge) ME ∝ CEC ME ∝ 1/(Φ, k) (Colline et al., 2008)
- 8. 3/2/2019 8 Water Activity (aw) State of hydration aw ∝ P, T 𝑃𝑤 = 𝜎 𝑚 𝑅𝑇 𝑉 𝑤 ln 𝑎 𝑤,𝑚𝑢𝑑 𝑎 𝑤,𝑠ℎ𝑎𝑙𝑒 swelling / shrinking Transport Mechanisms in Shale Osmosis (Osmotic pressure) Chemical Potential , Membrane efficiency (ME)
- 9. 3/2/2019 9 Osmosis (continued) Salinity differences in subsurface Total solute concentration Longevity of the Osmotic pressure Diffusion (Fick’s Law) Opposes Osmosis Concentration gradient Effective when ME ↓ and permeability ↑ (drags H2O)
- 10. 3/2/2019 10 Hydraulic Flow (Darcy’s Law) Hydraulic pressure gradient Mud Weight ↑ = Mechanical Stability Overbalanced drilling Hydration of Shales Conductivity ∝ fractures Electro-Osmosis Movement of charges – electrical potential difference
- 11. 3/2/2019 11 Year Process 1908 - Von Reuss Electro osmosis 1925, 1941 – Terzaghi, Biot Poroelastic consolidation 1965, 1969 – Young, Olsen Semi-Permeability of Shales 1967, 1993 – Katchalsky, Yeung Transport equations 1968 – Esrig 1D electro kinetic consolidation 2006 - Nguyen & Abousleiman 3D solution for electrokinetics in Biot’s model 2009, 2013 – Ghassemi, Tran et al., Thermal model, anisotropic model Chronology of Formulations
- 12. 3/2/2019 12 𝑞𝑖 = 𝐿11 𝜕(−𝑝) 𝜕𝑥𝑖 + 𝐿12 𝜕(−𝜓) 𝜕𝑥𝑖 + 𝐿13 𝑅𝑇 𝑚 𝑜 𝑎 𝜕(−𝑚 𝑎) 𝜕𝑥𝑖 + 𝐿14 𝑅𝑇 𝑚 𝑜 𝑐 𝜕(−𝑚 𝑐) 𝜕𝑥𝑖 𝐼𝑖 = 𝐿21 𝜕(−𝑝) 𝜕𝑥𝑖 + 𝐿22 𝜕(−𝜓) 𝜕𝑥𝑖 + 𝐿23 𝑅𝑇 𝑚 𝑜 𝑎 𝜕(−𝑚 𝑎) 𝜕𝑥𝑖 + 𝐿24 𝑅𝑇 𝑚 𝑜 𝑐 𝜕(−𝑚 𝑐) 𝜕𝑥𝑖 𝐽𝑖 𝑎,𝑑 = 𝐿31 𝜕(−𝑝) 𝜕𝑥𝑖 + 𝐿32 𝜕(−𝜓) 𝜕𝑥𝑖 + 𝐿33 𝑅𝑇 𝑚 𝑜 𝑎 𝜕(−𝑚 𝑎 ) 𝜕𝑥𝑖 + 𝐿34 𝑅𝑇 𝑚 𝑜 𝑐 𝜕(−𝑚 𝑐 ) 𝜕𝑥𝑖 𝐽𝑖 𝑐,𝑑 = 𝐿41 𝜕(−𝑝) 𝜕𝑥𝑖 + 𝐿42 𝜕(−𝜓) 𝜕𝑥𝑖 + 𝐿43 𝑅𝑇 𝑚 𝑜 𝑎 𝜕(−𝑚 𝑎 ) 𝜕𝑥𝑖 + 𝐿44 𝑅𝑇 𝑚 𝑜 𝑐 𝜕(−𝑚 𝑐 ) 𝜕𝑥𝑖 Coupled Flow Transport Equations Yeung & Mitchell (1993)
- 13. 3/2/2019 13 𝑑𝜎𝑖𝑗 = 2𝐺𝑑𝜀𝑖𝑗 + 2𝐺𝜈 1−2𝜈 𝑑𝜀 𝑘𝑘 𝛿𝑖𝑗 + 𝛼𝑑𝑝𝛿𝑖𝑗 𝑑𝜙 = −𝛼𝑑𝜀 𝑘𝑘 + 1 𝐾 𝜙 𝑑𝑝 𝑑𝜁 = −𝛼𝑑𝜀 𝑘𝑘 + 1 𝑀 𝑑𝑝 Constitutive Equations Coussy (2004)
- 14. 3/2/2019 14 𝜕𝜌 𝑒 𝜕𝑡 = − 𝜕𝐼 𝑖 𝜕𝑥 𝑖 = 0 Far Field Boundary Conditions 𝜎𝑥𝑥 = 𝑆 𝑥 𝜎 𝑦𝑦 = 𝑆 𝑦 𝜎𝑧𝑧 = 𝑆𝑧 𝜏 𝑥𝑦 = 𝑆 𝑥𝑦 𝜏 𝑦𝑧 = 𝑆 𝑦𝑧 𝜏 𝑥𝑧 = 𝑆 𝑥𝑧 𝑝 = 𝑝 𝑜 Sachs et al., 1987 Governing Equations
- 15. 3/2/2019 15 Near Wellbore Boundary Conditions 𝜎𝑟𝑟 = (𝜎 𝑚 + 𝜎 𝑑 cos 2 𝜃 − 𝜃𝑟 𝐻 −𝑡 + 𝑝 𝑚𝑢𝑑 𝐻 𝑡 𝜏 𝑟𝜃 = −𝜎 𝑑 sin 2 𝜃 − 𝜃𝑟 𝐻 −𝑡 𝜏 𝑟𝑧 = ( 𝑆 𝑥𝑧 cos θ + 𝑆 𝑦𝑥si n( 𝜃))𝐻(−𝑡 𝑝 = 𝑝 𝑜 𝐻 −𝑡 + 𝑝 𝑚𝑢𝑑 + ∆𝑝 𝑚𝑢𝑑−𝑠ℎ𝑎𝑙𝑒 𝐻 𝑡 Abousleiman et al., (2008)
- 16. 3/2/2019 16 Poroelastic Plane strain problem 1. Elastic radial loading 2. Diffusional loading 3. Deviatoric loading Elastic Uniaxial stress Elastic anti plane shear problem (Cui et al., 1997)
- 17. 3/2/2019 17 Input Parameters (CEC, aw ) Primary Parameters (mole fractions) Secondary Parameters (𝝈 𝒎, 𝝈 𝒅, 𝜽 𝒓) if 𝝈 𝒅, 𝜽 𝒓 = 0 Deviatoric stress loading is included Deviatoric stress loading is ignored Sum of Inverse (individual solutions) = Superposed solutions Matlab code structure No Yes
- 18. 3/2/2019 18 Case Study Properties of Pierre Shale considered Parameters Values (Salisbury et al.,1991) Shear (G), Bulk (K) modulus 600 , 4800 (MPa) Porosity (φ) 0.176 Permeability (k) 6E-21 (m2) CEC 36 meq/100 grams dry clay Pore pressure (p0) 21.4 (MPa) Sv , SH , Sh 54, 44, 44 (MPa) Membrane efficiency 0.8 (dimensionless) Temperature 82° C Depth 2200 (meters) mfc ,ma , mc 0.0719, 0.0052, 0.0771
- 19. 3/2/2019 19 Case 1: When mud activity > pore fluid activity Pore Pressure (in MPa) Vs Distance Ratio Water moves from mud → shale (hydration)
- 20. 3/2/2019 20 Effective Radial Stress (in MPa) Vs Distance Ratio Tensile Stress > Compressive Stress ( Spalling Failure)
- 21. 3/2/2019 21 Effective Tangential Stress (in MPa) Vs Distance Ratio Tensile Stress > Compressive Stress ( Spalling Failure)
- 22. 3/2/2019 22 Case 2: When mud activity < pore fluid activity Pore Pressure (in MPa) Vs Distance Ratio Water moves from shale → mud (Stability ↑ )
- 23. 3/2/2019 23 Effective Radial Stress (in MPa) Vs Distance Ratio More Compressive than Poroelastic medium (Stability ↑ )
- 24. 3/2/2019 24 Effective Tangential Stress (in MPa) Vs Distance Ratio More Compressive than Poroelastic medium (Stability ↑ )
- 25. 3/2/2019 25 Case 3: Time Propagation of the pore pressures generated (when mud activity < Pore fluid activity) Saturation of Shale ↓ with increase in time (stability ↑)
- 26. 3/2/2019 26 Case 4: Sensitivity of the pore pressures generated to mechanical properties of the formation Pore Pressure Vs distance ratio (varying G ∝ 1/ ν)
- 27. 3/2/2019 27 Pore Pressure Vs distance ratio (varying K ∝ ν) Limits for Pierre Shales = 300 – 13000 MPa
- 28. 3/2/2019 28 Pore Pressure Vs Distance ratio (varying Poisson’s ratio) The phenomena is sensitive only to differential volumetric ratio
- 29. 3/2/2019 29 Sensitivity of the Pore Pressures generated to Petrophysical and Surface Charge properties Membrane Efficiency Vs CEC (varying porosities < 30%) ME ∝ CEC/ porosity 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02 0 20 40 60 80 100 120 Reflectioncoefficient CEC (meq/100 g) por=0.0001 por=0.1 por=0.2 por=0.3
- 30. 3/2/2019 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 20 40 60 80 100 120 Reflectioncoefficient CEC (meq/100 g) por=0.4 por=0.5 por=0.6 Membrane Efficiency Vs CEC (varying porosities > 30%) Changes in ME is more sensitive for higher porosities Higher porosities affect ME even if CEC is high
- 31. 3/2/2019 31 Pore pressure Vs distance ratio (varying porosities) Osmosis is counter checked by Diffusion at higher porosities 30% porosity - threshold value for Pierre Shales
- 32. 3/2/2019 32 Pore pressure Vs distance ratio (varying Cation Exchange Capacity) Osmosis is counter checked by Diffusion at lower CEC
- 33. 3/2/2019 33 Pore pressure Vs distance ratio (varying Permeability) Low permeable formations least affected if mud activity > shale activity as pore pressure propagation is slower
- 34. 3/2/2019 34 Code Verification Type 1: Comparing the analytical solutions with results of Nguyen et al., (2008) Pore pressure Vs Distance Ratio (Poroelastic & Porochemoelastic models) The matching between the results is excellent 28 29 30 31 32 33 34 35 36 1 1.1 1.2 1.3 1.4 1.5 Porepressure(MPa) r/rw abousleiman PE model Matlab PE model abousleiman PC model Matlab PC model
- 35. 3/2/2019 35 -6 -4 -2 0 2 4 6 8 10 12 14 1 1.05 1.1 1.15 1.2 1.25 Effectiveradialstress(MPa) r/rw abousleiman PC model Matlab PC model abousleiman PE model Matlab PE model 47 48 49 50 51 52 53 54 55 56 57 1 1.05 1.1 1.15 1.2 1.25 Effectivetangentialstress(MPa) r/rw abousleiman PC model Matlab PC model abousleiman PE model Matlab PE model Comparison of the effective radial stresses obtained in Matlab and by Abousleiman et al.,(2008) Comparison of the effective tangential stresses obtained in Matlab and by Abousleiman et al.,(2008)
- 36. 3/2/2019 36 Type 2: Verifying the results of sensitivity analysis based on Jaeger’s analytical solutions for 1-D Poroelastic consolidation Berea Sand - draining starts around t = 1000 (dimensionless) 0.36 0.37 0.38 0.39 0.4 0.41 0.42 0.43 0.44 0.45 0.000001 0.0001 0.01 1 100 10000 1000000 100000000 1E+10 Verticaldisplcement(m) Dimensionless time (kt/μSh2) Berea Sand formations
- 37. 3/2/2019 37 0.633905 0.63391 0.633915 0.63392 0.633925 0.63393 0.000001 0.001 1 1000 1000000 1E+09 1E+12 1E+15 Verticaldisplacement(m) Dimensionless time (kt/μSh2) Shale formations Displacement at top of the column Vs Dimensionless time Shale - draining starts around t = 1011 (dimensionless) due to low Permeability
- 38. 3/2/2019 38 0.633918 0.63392 0.633922 0.633924 0.633926 0.633928 0.63393 0.000001 0.001 1 1000 1000000 1E+09 1E+12 1E+15 Verticaldisplacement(m) Dimensionless time (kt/μSh2) k=E-15 m2 k=E-16 m2 k=E-17 m2 k=E-18 m2 k=E-19 m2 k=E-20 m2 Displacement Vs Dimensionless time (varying permeability) Low permeable formations need more time to initiate draining
- 39. 3/2/2019 39 0.63385 0.63386 0.63387 0.63388 0.63389 0.6339 0.63391 0.63392 0.63393 0.63394 0.000001 0.001 1 1000 1000000 1E+09 1E+12 1E+15 Verticaldisplacement(m) Dimensionless time (kt/μSh2) por=0.05 por=0.1 por=0.2 por=0.3 por=0.4 Displacement Vs Dimensionless time (varying Porosities) Pore fluid drainage is rapid in less compacted formations Drainage initiation is quicker in consolidated shales - Pc
- 40. 3/2/2019 40 Displacement Vs Dimensionless time (varying Bulk Modulus) Draining = function (volumetric changes) Shales with higher bulk modulus drains less -0.00005 0 0.00005 0.0001 0.00015 0.0002 0.00025 0.000001 0.01 100 1000000 1E+10 1E+14 1E+18 1E+22 Verticaldisplacement(m) Dimensionless time (kt/μSh2) K=2528 MPa K=3550 MPa K=4800 MPa
- 41. 3/2/2019 41 Displacement Vs Dimensionless time (varying Shear Modulus) Shales with higher shear modulus drains more Poisson’s ratio – Deciding factor -0.00002 0 0.00002 0.00004 0.00006 0.00008 0.0001 0.000001 0.01 100 1000000 1E+10 1E+14 1E+18 1E+22 Verticaldisplacement(m) Dimensionless time (kt/μSh2) 600 MPa 700 MPa 800 MPa
- 42. 3/2/2019 42 Conclusions Mud activity < pore fluid activity (favorable condition) Diffusion > osmosis (salt concentration ↑ ) Sensitivity to CEC, porosity and membrane efficiency Compressibility of the medium is important (ν >> K, G) Effect of permeability
- 43. 3/2/2019 43 Limitations Valid only for isotropic Thermal effects neglected Elasticity of shales
- 44. 3/2/2019 44 Acknowledgement: • Dr. Ahmad Jamili – Advisor • Dr. Deepak Devegowda • Dr. Ben Shiau
- 46. 3/2/2019 46 Supporting Slides Conductivity coefficients Coupling Coefficients 𝐿11 = 𝑘ℎ 𝛾𝑡 𝑛 + 𝐿12 𝐿21 𝐿22 𝐿12 = 𝐿21 = 𝑘 𝑒 𝑛 𝐿22 = 𝜅 𝑛 𝐿33 = 𝐷𝑐 ∗ 𝐶𝑐 𝑅𝑇 𝐿23 = 𝐿32 = 𝐷𝑐 ∗ 𝑧 𝑐 𝐹𝐶𝑐 𝑅𝑇 𝐿44 = 𝐷 𝑎 ∗ 𝐶 𝑎 𝑅𝑇 𝐿24 = 𝐿42 = 𝐷 𝑎 ∗ 𝑧 𝑎 𝐹𝐶 𝑎 𝑅𝑇 𝐿13 = − 𝜔𝐶𝑐 𝐿11 𝐿22 − 𝐿12 𝐿21 − 𝐿12 𝐿23 𝐿22 Yeung & Mitchell (1993) 𝐿14 = − 𝜔𝐶 𝑎 𝐿11 𝐿22 − 𝐿12 𝐿21 − 𝐿12 𝐿24 𝐿22 𝐿34 = 𝐿43 = 0
- 47. 3/2/2019 47 Governing Equations Strain Displacement equation 𝜀𝑖𝑗 = 0.5 ∗ 𝜕𝑢 𝑖 𝜕𝑥 𝑗 + 𝜕𝑢 𝑗 𝜕𝑥 𝑖 Semi-static stress equilibrium equation 𝜕𝜎 𝑖𝑗 𝜕𝑥 𝑖 = 0 Katchalsky & Curran (1967)
- 48. 3/2/2019 48 Final Equations – Abousleiman et al., (2008) −𝛼 𝜕𝜀 𝑘𝑘 𝜕𝑡 + 1 𝑀 𝜕𝑝 𝜕𝑡 = 𝐷11 𝛻2 𝑝 + 𝐷12 𝛻2 𝑝 𝑎 + 𝐷13 𝛻2 𝑝 𝑐 𝑚 𝑜 𝑎 −𝛼 𝜕𝜀 𝑘𝑘 𝜕𝑡 + 1 𝑀 𝜕𝑝 𝜕𝑡 + 𝜙 𝑜 𝑉𝑜 𝑓 𝑅𝑇 𝜕𝑝 𝑎 𝜕𝑡 = 𝐷21 𝛻2 𝑝 + 𝐷22 𝛻2 𝑝 𝑎 + 𝐷23 𝛻2 𝑝 𝑐 𝑚 𝑜 𝑐 −𝛼 𝜕𝜀 𝑘𝑘 𝜕𝑡 + 1 𝑀 𝜕𝑝 𝜕𝑡 + 𝜙 𝑜 𝑉𝑜 𝑓 𝑅𝑇 𝜕𝑝 𝑐 𝜕𝑡 = 𝐷31 𝛻2 𝑝 + 𝐷32 𝛻2 𝑝 𝑎 + 𝐷33 𝛻2 𝑝 𝑐
- 49. 3/2/2019 49 D matrix with coefficients of diffusion equations – transport coefficients – Abousleiman et al., (2008) 𝐷11 𝐷12 𝐷13 𝐷21 𝐷22 𝐷23 𝐷31 𝐷32 𝐷33 = 𝜅 −𝜒𝜅 −𝜒𝜅 𝑚 𝑜 𝑎(1 − 𝜒)𝜅 𝐷𝑒𝑓𝑓 𝑎 𝑉𝑜 𝑓 𝑅𝑇 − 𝑚 𝑜 𝑎(1 − 𝜒)𝜒𝜅 −𝑚 𝑜 𝑎(1 − 𝜒)𝜒𝜅 𝑚 𝑜 𝑐 (1 − 𝜒)𝜅 −𝑚 𝑜 𝑐 (1 − 𝜒)𝜒𝜅 𝐷𝑒𝑓𝑓 𝑐 𝑉𝑜 𝑓 𝑅𝑇 − 𝑚 𝑜 𝑐 (1 − 𝜒)𝜒𝜅
- 50. 3/2/2019 50 Matrix for converting global coordinates to local coordinates 𝑆 𝑥 𝑆 𝑦 𝑆𝑧 𝑆 𝑥𝑦 𝑆 𝑦𝑧 𝑆 𝑥𝑧 = 𝑙 𝑥𝑥′ 2 𝑙 𝑦𝑥′ 2 𝑙 𝑧𝑥′ 2 𝑙 𝑥𝑥′ 𝑙 𝑦𝑥′ 𝑙 𝑦𝑥′ 𝑙 𝑧𝑥′ 𝑙 𝑧𝑥′ 𝑙 𝑥𝑥′ 𝑙 𝑥𝑦′ 2 𝑙 𝑦𝑦′ 2 𝑙 𝑧𝑦′ 2 𝑙 𝑥𝑦′ 𝑙 𝑦𝑦′ 𝑙 𝑦𝑦′ 𝑙 𝑧𝑦′ 𝑙 𝑧𝑦′ 𝑙 𝑥𝑦′ 𝑙 𝑥𝑧′ 2 𝑙 𝑦𝑧′ 2 𝑙 𝑧𝑧′ 2 𝑙 𝑥𝑧′ 𝑙 𝑧𝑧′ 𝑙 𝑦𝑧′ 𝑙 𝑧𝑧′ 𝑙 𝑧𝑧′ 𝑙 𝑥𝑧′ 𝑆 𝑥′ 𝑆 𝑦′ 𝑆𝑧′ Where 𝑙 𝑥𝑥′ 𝑙 𝑥𝑦′ 𝑙 𝑥𝑧′ 𝑙 𝑦𝑥′ 𝑙 𝑦𝑦′ 𝑙 𝑦𝑧′ 𝑙 𝑧𝑥′ 𝑙 𝑧𝑦′ 𝑙 𝑧𝑧′ = 𝑐𝑜𝑠𝜑 𝑧 𝑐𝑜𝑠𝜑 𝑦 𝑠𝑖𝑛𝜑 𝑧 𝑐𝑜𝑠 𝜑 𝑦 −𝑠𝑖𝑛𝜑 𝑦 −𝑠𝑖𝑛𝜑 𝑧 𝑐𝑜𝑠𝜑𝑧 0 𝑐𝑜𝑠𝜑 𝑧 𝑠𝑖𝑛𝜑 𝑦 𝑠𝑖𝑛𝜑 𝑧 𝑠𝑖𝑛𝜑 𝑦 𝑐𝑜𝑠𝜑 𝑦 Fjaer (2000)
- 51. 3/2/2019 51 Miscellaneous Equations – Abousleiman et al., (2008) 𝑚 𝑓𝑐 = 10−2 ∗ 𝐶𝐸𝐶 1−𝜙 𝑜 𝜌 𝑠 𝑉𝑜 𝑓 𝜙 𝑜 𝑚 𝑠ℎ𝑎𝑙𝑒 𝑎 = 0.5 −𝑚 𝑓𝑐 + (𝑚 𝑓𝑐)2 + 4(𝑚 𝑚𝑢𝑑 𝑠 )2 𝑚 𝑠ℎ𝑎𝑙𝑒 𝑐 = 0.5 𝑚 𝑓𝑐 + (𝑚 𝑓𝑐)2 + 4(𝑚 𝑚𝑢𝑑 𝑠 )2 𝑎 𝑜 𝑓 = 1 − 𝑥 + 𝑦 𝑚 𝑒𝑞 𝑠 ∆𝑝 𝑚𝑢𝑑−𝑠ℎ𝑎𝑙𝑒= 𝑅𝑇 𝑉𝑜 𝑓 ∗ (𝑚 𝑓𝑐)2 + 4(𝑚 𝑚𝑢𝑑 𝑠 )2 − 2𝑚 𝑚𝑢𝑑 𝑠 𝜂 = 𝑙𝑢𝑚𝑝𝑒𝑑 𝑝𝑜𝑟𝑜𝑒𝑙𝑎𝑠𝑡𝑖𝑐 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 = 𝛼 1−2𝜈 2 1−𝜈
- 52. 3/2/2019 52 Continued… 𝜎 𝑚 = 𝑆 𝑥+𝑆 𝑦 2 𝜎 𝑑 = 0.5 (𝑆 𝑥 − 𝑆 𝑦)2+4𝑆 𝑥𝑦 2 𝜃𝑟 = 0.5𝑡𝑎𝑛−1 2𝑆 𝑥𝑦 𝑆 𝑥−𝑆 𝑦 ∆𝑚 𝑚𝑢𝑑−𝑠ℎ𝑎𝑙𝑒 𝑎 = 𝑚 𝑠ℎ𝑎𝑙𝑒 𝑎 − 𝑚 𝑚𝑢𝑑 𝑠 ∆𝑚 𝑚𝑢𝑑−𝑠ℎ𝑎𝑙𝑒 𝑐 = 𝑚 𝑠ℎ𝑎𝑙𝑒 𝑐 − 𝑚 𝑚𝑢𝑑 𝑠
- 53. 3/2/2019 53 Continued… 𝑌 = 1 𝑀 + 𝛼𝜂 𝐺 𝑚 𝑜 𝑎 ( 1 𝑀 + 𝛼𝜂 𝐺 ) 𝑚 𝑜 𝑐 ( 1 𝑀 + 𝛼𝜂 𝐺 ) 0 𝜙 𝑂 𝑉𝑜 𝑓 𝑅𝑇 0 0 0 𝜙 𝑂 𝑉𝑜 𝑓 𝑅𝑇 𝑍 = 𝑌 −1 𝐷 Superposed solutions 𝜎𝑟𝑟 = 𝜎 𝑚 + 𝜎 𝑑 cos 2 𝜃 − 𝜃𝑟 + 𝜎𝑟𝑟 1 + 𝜎𝑟𝑟 2 + 𝜎𝑟𝑟 3 𝜏 𝑟𝜃 = −𝜎 𝑑 sin 2 𝜃 − 𝜃𝑟 + 𝜏 𝑟𝜃 3
- 54. 3/2/2019 54 𝑝 = 𝑝 𝑜 + 𝑝2 + 𝑝3 𝜎𝑟𝑟 1 = − 𝜎 𝑚 − 𝑝 𝑚𝑢𝑑 𝑅 𝑤 2 𝑟2 𝜎 𝜃𝜃 1 = 𝜎 𝑚 − 𝑝 𝑚𝑢𝑑 𝑅 𝑤 2 𝑟2 𝜎𝑧𝑧 = 𝑆𝑧 − 2𝜈𝜎 𝑚 − 𝛼 1 − 2𝜈 𝑝 𝑜 𝜏 𝑟𝑧 = 𝑆 𝑥𝑧 𝑐𝑜𝑠𝜃 + 𝑆 𝑦𝑧 𝑠𝑖𝑛𝜃 1 − 𝑅 𝑤 2 𝑟2 𝜏 𝜃𝑧 = − 𝑆 𝑥𝑧 𝑠𝑖𝑛𝜃 − 𝑆 𝑦𝑧 𝑐𝑜𝑠𝜃 1 + 𝑅 𝑤 2 𝑟2 Continued…
- 55. 3/2/2019 55 Continued… Mode 2: Diffusional loading - Detourney et al.,(1988) 𝑠 𝑝(2) = 𝑚11∆1 𝛷 𝜉1 + 𝑚12∆2 𝛷 𝜉2 + 𝑚13∆3 𝛷 𝜉3 𝑠 𝜎𝑟𝑟 (2) = −2𝜂{𝑚11∆1 𝛯 𝜉1 + 𝑚12∆2 𝛯 𝜉2 + 𝑚13∆3 𝛯 𝜉3 𝑠 𝜎 𝜃𝜃 (2) = 2𝜂{ 𝑚11∆1 𝛯 𝜉1 + 𝛷 𝜉1 + 𝑚12∆2 𝛯 𝜉2 + 𝛷 𝜉1 +
- 56. 3/2/2019 56 Continued… Mode 3: Deviatoric loading – Detourney et al., (1988) 𝑠 𝑝(3) = 𝜎 𝑑 𝑚11 𝐷1 𝐾2 𝜉1 𝑟 + 𝑚12 𝐷2 𝐾2 𝜉2 𝑟 + 𝑚13 𝐷3 𝐾2 𝜉3 𝑟 + 𝐷4 𝑓1 𝑅 𝑤 2 𝑟2 𝑐𝑜𝑠 2 𝜃 − 𝜃𝑟 𝑠 𝜎𝑟𝑟 3 = − 𝜎 𝑑 2𝜂 𝑚11 𝐷1 𝛩 𝜉1 + 𝑚12 𝐷2 𝛩 𝜉2 + 𝑚13 𝐷3 𝛩 𝜉3 − 2𝐺 ℎ + 𝛼 𝜂 𝐷4 𝑅 𝑤 2 𝑟2 −𝐷5 𝑅 𝑤 4 𝑟4 𝑐𝑜𝑠 2 𝜃 − 𝜃𝑟
- 57. 3/2/2019 57 Continued… 𝑠 𝜎 𝜃𝜃 3 = 𝜎 𝑑 2𝜂 𝑚11 𝐷1 𝛱 𝜉1 + 𝑚12 𝐷2 𝛱 𝜉2 + 𝑚13 𝐷3 𝛱 𝜉3 −
- 58. 3/2/2019 58 Inverse Laplace – Stehfest’s algorithm 𝑓𝑛𝑢𝑚 𝑡 = 𝑙𝑛2 𝑡 𝑖=1 𝑁 𝐺𝑖 𝑓(𝑖 ∗ 𝑙𝑛2 𝑡 ) 𝑁 = 𝑆𝑡𝑒ℎ𝑓𝑒𝑠𝑡 𝑛𝑢𝑚𝑏𝑒𝑟 𝑎𝑛𝑑 ℎ𝑎𝑠 𝑡𝑜 𝑏𝑒 𝑒𝑣𝑒𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 2 𝑎𝑛𝑑 20 𝐺𝑖 = (−1)𝑖+ 𝑁 2 𝑘=(𝑖+1)/2 min(𝑖, 𝑁 2 ) 𝑘 𝑁 2 2𝑘 ! 𝑁 2 −𝑘 !𝑘! 𝑘−1 ! 𝑖−𝑘 ! 2𝑘−𝑖 !