REVIEW OF MECHANICS Schlumberger Private Cicik Sukma SW Geomechanics Engineer DCS-Jakarta Objectives Schlumberger Private Have an understanding of the types of stress, relationship to strain Need to understand relationship between different stresses Understand the unique aspects of rock as compared to other materials: porous media, effective stress, friction between grains Recognize the stages of rock deformation Outline Schlumberger Private Stress and Strain Shear and normal stresses - Mohr’s circle Effective stress Elasticity Review of mechanics: force & stress Defined by Newton's Second Law F = mg in earth's gravitational field stress = Stress in N/m2 force area Schlumberger Private F = ma Review of mechanics: stress L A A Normal stress on A is L/A Shear stress on A is L/A • In simple geometries; it looks as if only one number is needed to specify stress • In more complicated geometry - e.g., subsurface - more numbers needed • Dimensions of stress are force/area – same as pressure • Units - 106 Pascal = 1 MPa = ~145 pounds/sq. inch = 10 bar Schlumberger Private L Normal and shear stresses normal stress Maximum principal stress shear stress Minimum principal stress Schlumberger Private Normal and shear stresses are created in a rock when it is compressed Along any oblique plane, orthogonal principal stresses create a shear stress parallel to the plane and a normal stress perpendicular to the plane The relative magnitudes of shear and normal stress depend on the angle of the plane to the principal stresses When the plane, in the example, is vertical or horizontal shear stress is zero A general stress state σ xy σ yy σ zy σ xz σ yz σ zz Rotate the axes Diagonalize the tensor Principalize the stress state Normal stresses Shear stresses Schlumberger Private σ xx σ yx σ zx Principal stresses • Any state of stress can be expressed as 3 perpendicular PRINCIPAL STRESSES, plus their orientations in space. This is how stress is usually expressed. • Principal stresses are normal, not shear stresses. 0 0 σ 3 + orientation of axes Each principal stress has a magnitude, independent of the others. The orientation of the axes must also be specified. Complete specification of stress needs 6 numbers! Schlumberger Private σ1 0 0 σ 2 0 0 Strain Strain is a measure of the deformation induced in a body, by the action of stress Original length L0 New length L1 L1 − L0 εa = L0 (Simple definition of small axial strain) Schlumberger Private Strain can be normal or shear. Like stress, it can be expressed as principal quantities - three normal strains, along perpendicular axes. Stress and strain are related through material properties, such as elastic properties. Can be simple (linear isotropic elastic) or very complicated. Strain has no dimensions or units. Can be quoted as ratio, %, millistrain, microstrain…. Nomenclature Stresses: Effective stress: Schlumberger Private Pressures: σ1 = maximum compressive principal stress σ2 = intermediate compressive principal stress σ3 = minimum compressive principal stress Pp = pore pressure Pw = wellbore pressure Pc = confining pressure in rock test σ' = σ - αPp for elasticity σ' = σ - Pp for failure σ2 σ1 σ3 Calculation of normal and shear stresses σ3 τ σn = σ1 σ1 β σ3 1 (σ 1 + σ 3 ) + 1 (σ 1 − σ 3) cos 2β 2 2 1 τ = (σ 1 − σ 3 )sin 2β 2 Schlumberger Private σn Mohr’s circle A construction for illustrating the magnitudes of normal/shear stresses on a specified plane, and also for relating it to Coulomb (or other) failure criterion. σ1 Schlumberger Private τ σ3 shear stress σn τ β 2β σ3 σn σ1 normal stresses Why are we doing this? With Mohr’s circle analysis, we can predict*: when failure will occur on a predetermined plane, or even where failure will occur in an intact piece of rock (which plane will it shear on, and when?). *in simple situations Schlumberger Private Effective stress σv σH Schlumberger Private Rocks and other porous materials respond to pore pressure as well as applied stresses We can often approximate the behavior by using ‘effective stress’ in elasticity or failure calculations σh Effective normal stress = normal stress - f(pore pressure) Effective shear stress = shear stress Pp Nomenclature Stresses: Effective stress: Schlumberger Private Pressures: σ1 = maximum compressive principal stress σ2 = intermediate compressive principal stress σ3 = minimum compressive principal stress Pp = pore pressure Pw = wellbore pressure Pc = confining pressure in rock test σ' = σ - αPp for elasticity σ' = σ - Pp for failure σ2 σ1 σ3 Schlumberger Private ELASTIC PROPERTIES OF ROCKS Elasticity • Reversible behavior. Usually treated as linear, although rocks are generally non-linear • Two independent moduli for isotropic materials - Young's modulus, E, and Poisson's ratio, ν α = 1 - Kframe Ksolid • α is close to 1 for rocks with low stiffness, close to 0 for stiff rocks. Please, do not spend too much time or effort in getting accurate values! • For example of use of non-linear effects, see SPE47234 and other papers by Plona/Sinha. A lot of Sonic Scanner interpretation is based on non-linear elasticity. Schlumberger Private • For simple situations, use effective stress σ' = σ - αPp, where Elastic properties Longitudinal stress, σ Longitudinal strain, ε Young’s modulus 18 Initials 2/27/2009 Longitudinal strain, εa Poisson’ s ratio Lateral strain, εr Schlumberger Private Elastic properties of the rock help to estimate the rock strength and the in-situ earth stresses Stress and strain are related through material properties, such as elastic properties (Young’s modulus, for instance) Young’s modulus is the ratio of applied stress and the resultant strain in the same direction. It can be thought of as the stiffness of the rock Poisson’s ratio is a measure of the lateral expansion to the longitudinal contraction Elastic properties can be measure using sonic and density logs, or by testing core in the lab Typical stress/strain response: weak rock, 20 MPa confining pressure Gradient = Young’ Young’s modulus elastic region Gradient = - Poisson’ Poisson’s ratio Schlumberger Private Rock sample Static and Dynamic Moduli Static moduli are measured from slope of loading line in ‘elastic’ region: Estat = ∆σ′/∆ε Dynamic moduli are measured from acoustic wave speeds: - much smaller strains - very high rate - smaller volume sampled These differences are important in the estimation of rock strength from sonic log data Schlumberger Private Edyn = ρvp2 (3vp2-4vs2)/ (vp2-vs2) Dynamic moduli are almost always larger than static moduli:
