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

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

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

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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

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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

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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

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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

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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

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SINTEF Petroleum Research 9 Observations Hydrostatic test

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SINTEF Petroleum Research 10 Observations Hydrostatic test

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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

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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

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SINTEF Petroleum Research 13 Observations Uniaxial loading test

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SINTEF Petroleum Research 14 Observations Uniaxial loading test

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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

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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

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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

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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

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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

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

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

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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?

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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

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

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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 11 22 33 The intermediate principal stress ( 2 ) has no impact Stress symmetry is not important

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

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

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

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

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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

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SINTEF Petroleum Research 31 How can stress symmetry affect the strength? - It’s because it affects the probability for failure! 11 22 33

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SINTEF Petroleum Research 32 Classical picture 11 22 33 Probability for failure 0 1 mm

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SINTEF Petroleum Research 33 Classical picture 11 22 33 Probability for failure 0 1 mm

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SINTEF Petroleum Research 34 Classical picture 11 22 33 Probability for failure 0 1 mm

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SINTEF Petroleum Research 35 Classical picture 11 22 33 Probability for failure 0 1 mm

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SINTEF Petroleum Research 36 Classical picture 11 22 33 Probability for failure 0 1 mm

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SINTEF Petroleum Research 37 Classical picture 11 22 33 Probability for failure 0 1 mm

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SINTEF Petroleum Research 38 Classical picture 11 22 33 Probability for failure 0 1 mm

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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 +

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SINTEF Petroleum Research 40 Classical picture 11 2 33 All planes oriented at an angle relative to the 1 axis 22 Many potential failure planes in a critical state High probability for failure

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SINTEF Petroleum Research 41 Classical picture 11 22 33 Only planes oriented at an angle relative to the 1 axis, and parallel to the 2 axis 22 Few potential failure planes in a critical state Low probability for failure

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SINTEF Petroleum Research 42 Classical picture 33 2 11 All planes oriented at an angle /2 - relative to the 3 axis 22 Many potential failure planes in a critical state High probability for failure

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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:

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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:

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SINTEF Petroleum Research 45 Mathematical model

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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

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

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

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

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SINTEF Petroleum Research Borehole breakouts in a non-fractured rock

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

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

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SINTEF Petroleum Research 53 Simple example No fractures

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

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

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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.

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SINTEF Petroleum Research Other possible failure modes – bedding plane splitting

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SINTEF Petroleum Research Other possible failure modes – bedding plane splitting

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SINTEF Petroleum Research Other possible failure modes – bedding plane splitting

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SINTEF Petroleum Research Other possible failure modes – bedding plane splitting

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SINTEF Petroleum Research Other possible failure modes – bedding plane splitting

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SINTEF Petroleum Research Other possible failure modes – bedding plane splitting

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SINTEF Petroleum Research Other possible failure modes – bedding plane splitting

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SINTEF Petroleum Research Økland and Cook 1998

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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

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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)

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SINTEF Petroleum Research 67 Geologocal Strength Index - GSI

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

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

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

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

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

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SINTEF Petroleum Research 73 Failure probability model

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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

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