1. SINTEF Petroleum Research The strength of fractured rock Erling Fjær SINTEF Petroleum Research 1

2. 2 Porosity, Density, Sonic,.... Challenge: Estimation of rock strength from log data Strength Available Wanted Traditional approach: correlations

3. SINTEF Petroleum Research 3 Porosity, Density, Sonic,.... Challenge: Estimation of rock strength from log data Strength Available Wanted Brandås et al. (2012)

4. SINTEF Petroleum Research 4 Alternative approach: 1.Establish a constitutive model for static and dynamic moduli of rocks 2.Use the measured dynamic moduli (i.e. velocities) to calibrate the model 3.Use the calibrated model to simulate a test where strength can be measured

5. SINTEF Petroleum Research 5 static moduli vs dynamic moduli Rock mechanical test including acoustic measurements on a dry sandstone static moduli  dynamic moduli The differences changes with stress and strain Dry, weak sandstone

6. SINTEF Petroleum Research 6 static moduli vs dynamic moduli Rock mechanical test including acoustic measurements on a dry sandstone static moduli  dynamic moduli The differences changes with stress and strain Dry, weak sandstone We are seeking mathematical relations between the static and the dynamic moduli

7. SINTEF Petroleum Research 7 We introduce a parameter P, defined as: P is a measure of the inelastic part of the deformation caused by a compressive hydrostatic stress increment. Building relations  v - total volumetric strain Hydrostatic test - elastic strain

8. SINTEF Petroleum Research 8 We introduce a parameter P, defined as: P is a measure of the inelastic part of the deformation caused by a compressive hydrostatic stress increment. Building relations  v - total volumetric strain Hydrostatic test - elastic strain  K = Static bulk modulus K e = Dynamic bulk modulus

9. SINTEF Petroleum Research 9 Observations Hydrostatic test

10. SINTEF Petroleum Research 10 Observations Hydrostatic test

11. SINTEF Petroleum Research 11 We introduce a parameter F, defined as: F is a measure of the inelastic part of the deformation caused by a shear stress increment. Building relations  z - total axial strain Uniaxial loading test - elastic strain

12. SINTEF Petroleum Research 12 We introduce a parameter F, defined as: F is a measure of the inelastic part of the deformation caused by a shear stress increment. Building relations  z - total axial strain Uniaxial loading test - elastic strain  E = Static Young’s modulus E e = Dynamic Young’s modulus

13. SINTEF Petroleum Research 13 Observations Uniaxial loading test

14. SINTEF Petroleum Research 14 Observations Uniaxial loading test

15. SINTEF Petroleum Research 15 Discussion: the F - parameter Since E  (1 - F)  when F =1 then E = 0  peak stress Note: F = 1  rock strength

16. SINTEF Petroleum Research 16 Griffith’s failure criterion: If we can assume that: (  1 -  3 )  (  1 -  3 ) then we could state that F = 1  Fulfilment of the Griffith criterion Our model: Discussion: the F - parameter

17. SINTEF Petroleum Research 17 (  1 -  3 )  (  1 -  3 ) ? OK for a purely elastic material Also OK at the intact parts of the material even after local failure has occurred elsewhere Local (  1 -  3 )  Global (  1 -  3 ) ! Discussion: the F - parameter

18. SINTEF Petroleum Research 18 The development of F can be seen as a gradual fulfillment of the Griffith criterion May be associated with local failure at various places in the rock, triggered at different stress levels due to variable local strength Discussion: the F - parameter

19. SINTEF Petroleum Research 19 We have a set of equations…… These represent a constitutive model for the rock We may use it to predict rock behavior, and thereby derive mechanical properties for the rock

20. SINTEF Petroleum Research 20 Porosity, Density, Sonic,.... Constitutive model Application for logging purposes Simulates rock mechanical test on fictitious core Strength

21. SINTEF Petroleum Research 21 Courtesy of Statoil Prediction from logs Core measurements … an example:

22. SINTEF Petroleum Research 22 In the lab  2 =  3  1   2   3 in general In the field Challenge: What is the impact of the intermediate principal stress on rock strength?

23. SINTEF Petroleum Research Most convenient description:  -plane cross sections (planes normal to the hydrostatic axis)  -plane Hydrostatic axis Projections of the principal axes Cross section of the failure surface

24. SINTEF Petroleum Research 24 Failure criteria (  -plane): Assumption: Rotational symmetry in  -plane (No physical argument) No impact of the intermediate stress Empirical

25. SINTEF Petroleum Research 25 Basic theory on shear failure: Shear failure occurs when the shear stress over some plane within the rock exceeds the shear strength of the rock 11 22 33 The intermediate principal stress (  2 ) has no impact Stress symmetry is not important 

26. SINTEF Petroleum Research 26 Experimental observations: No impact of intermediate stress

27. SINTEF Petroleum Research 27 Experimental observations: Takahashi & Koide (1989)

28. SINTEF Petroleum Research 28 Numerical simulations: Fjær & Ruistuen (2002)

29. SINTEF Petroleum Research 29 Experimental observations:  -plane Mohr- Coulomb Drucker- Prager

30. SINTEF Petroleum Research 30 Question: What is similar when  2 =  3 and  2 =  1 but different when  1 >  2 >  3 ? It’s the stress symmetry! Tetragonal Orthorhombic

31. SINTEF Petroleum Research 31 How can stress symmetry affect the strength? - It’s because it affects the probability for failure! 11 22 33 

32. SINTEF Petroleum Research 32 Classical picture  11 22 33  Probability for failure   0 1 mm

33. SINTEF Petroleum Research 33 Classical picture  11 22 33  Probability for failure   0 1 mm

34. SINTEF Petroleum Research 34 Classical picture  11 22 33  Probability for failure   0 1 mm

35. SINTEF Petroleum Research 35 Classical picture  11 22 33  Probability for failure   0 1 mm

36. SINTEF Petroleum Research 36 Classical picture  11 22 33  Probability for failure  0 1 mm 

37. SINTEF Petroleum Research 37 Classical picture  11 22 33  Probability for failure   0 1 mm

38. SINTEF Petroleum Research 38 Classical picture  11 22 33  Probability for failure   0 1 mm

39. SINTEF Petroleum Research 39 Classical picture Probability for failure  0 1 Classical picture: Failure occurs if the shear stress across any plane in the rock sample exceeds S o +  – otherwise not. Introducing fluctuations: The shear strength varies from plane to plane. The rock fails when  exceeds the shear strength for one of them. The probability for failure increases when   S o +  S o + 

40. SINTEF Petroleum Research 40 Classical picture  11  2 33 All planes oriented at an angle  relative to the  1 axis  22 Many potential failure planes in a critical state  High probability for failure

41. SINTEF Petroleum Research 41 Classical picture  11 22 33 Only planes oriented at an angle  relative to the  1 axis, and parallel to the  2 axis  22 Few potential failure planes in a critical state  Low probability for failure

42. SINTEF Petroleum Research 42 Classical picture  33  2 11 All planes oriented at an angle  /2 -  relative to the  3 axis  22 Many potential failure planes in a critical state  High probability for failure

43. SINTEF Petroleum Research 43 Mathematical model Probability for failure of a plane with orientation specified by ( ,  ): ( n    classical Mohr-Coulomb ) Overall probability for failure: Expected strength of the material:

44. SINTEF Petroleum Research 44 Mathematical model Probability for failure of a plane with orientation specified by ( ,  ): ( n    classical Mohr-Coulomb ) Overall probability for failure: Expected strength of the material:

45. SINTEF Petroleum Research 45 Mathematical model

46. SINTEF Petroleum Research 46 Mathematical model The impact of the intermediate principal stress is directly linked to the non-sharpness of the failure criterion (represented by 1/ n ) i.e. to the rock heterogeneity

47. SINTEF Petroleum Research 47 Comparing model and observations Takahashi and Koide, 1989 n = 30

48. SINTEF Petroleum Research 48 Comparing model and observations Numerical model n = 25

49. SINTEF Petroleum Research Outcrop from a Marcellus shale formation Han, 2011 Fractures are planes with largely reduced or no strength

50. SINTEF Petroleum Research Borehole breakouts in a non-fractured rock

51. SINTEF Petroleum Research Shear failure planes Borehole breakouts in a non-fractured rock

52. SINTEF Petroleum Research Borehole breakouts in a non-fractured rock Shear failure planes

53. SINTEF Petroleum Research 53 Simple example No fractures

54. SINTEF Petroleum Research 54 Simple example No fractures Sealed fractures || borehole

55. SINTEF Petroleum Research 55 Simple example No fractures Sealed fractures || borehole Open fractures || borehole

56. SINTEF Petroleum Research Several fracture sets complicates the situation. Blocks may become detached at washed away by the circulating mud.  More fractures will be exposed to the drilling fluid.

57. SINTEF Petroleum Research Other possible failure modes – bedding plane splitting

58. SINTEF Petroleum Research Other possible failure modes – bedding plane splitting

59. SINTEF Petroleum Research Other possible failure modes – bedding plane splitting

60. SINTEF Petroleum Research Other possible failure modes – bedding plane splitting

61. SINTEF Petroleum Research Other possible failure modes – bedding plane splitting

62. SINTEF Petroleum Research Other possible failure modes – bedding plane splitting

63. SINTEF Petroleum Research Other possible failure modes – bedding plane splitting

64. SINTEF Petroleum Research Økland and Cook 1998

65. SINTEF Petroleum Research Økland and Cook 1998 To avoid the problem: The “angle of attack” between the well and the bedding plane should be at least 20 . Well

66. SINTEF Petroleum Research 66 Challenge: What is the strength of a fractured rock (if we consider it as homogeneous)? Available alternative: Hoek-Brown Purely empirical criterion Hoek & Brown (1980)

67. SINTEF Petroleum Research 67 Geologocal Strength Index - GSI

68. SINTEF Petroleum Research 68 Rocks are heterogeneous – treating them as homogeneous comes at a price…..

69. SINTEF Petroleum Research 69 Hoek & Brown (1980) The strength of a homogeneous material is size invariant. Rocks, on the other hand, -

70. SINTEF Petroleum Research 70 Current work: Relate the failure probability model to Hoek-Brown

71. SINTEF Petroleum Research 71 Data from Hoek; Kaiser (2008) Challenge: Match with observations Failure probability model

72. SINTEF Petroleum Research 72 Kaiser (2008) Consideravble scatter in measured strength

73. SINTEF Petroleum Research 73 Failure probability model

74. SINTEF Petroleum Research 74 Conclusions: Physics helps us to make better tools for rock mechanics applications There is still room for more physics in rock mechanics