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1. Rock Elasticity

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1. Rock Elasticity

1. 1. James A. Craig Omega 2011
2. 2.  Linear Elasticity  Stress  Strain  Elastic moduli  Non-Linear Elasticity  Poroelasticity
3. 3.  Ability of materials to resist and recover from deformations produced by forces.  Applied stress leads to a strain, which is reversible when the stress is removed.  The relationship between stress and strain is linear; only when changes in the forces are sufficiently small.  Most rock mechanics applications are considered linear.  Linear elasticity is simple  Parameters needed can be estimated from log data & lab tests.  Most sedimentary rocks exhibit non-linear behaviour, plasticity, and even time-dependent deformation (creep).
4. 4. F = force exerted Fn = force exerted normal to surface Fp = force exerted parallel to surface A = cross-sectional area Normal Stress Shear Stress Fn  A  Fp A
5. 5.  Sign convention:  Compressive stress = positive (+) sign  Tensile stress = negative (-) sign  Stress is frequently measured in:  Pascal, Pa (1 Pa = 1 N/m2)  Bar  Atmosphere  Pounds per squared inch, psi (lb/in2)
6. 6. The Stress Tensor  Identifying the stresses related to surfaces oriented in 3 orthogonal directions.
7. 7.  Stress tensor =   x  xy  xz      yx  y  yz    zy  z  zx    Mean normal stress,   x  y  z   3  For theoretical calculations, both normal & shear stresses can be denoted by σij:  “i” identifies the axis normal to the actual surface  “j” identifies the direction of the force  ij   x ,  y ,  z ; xy , yz , xz   11  12  13   Stress tensor :       21  22  23      31  32  33 
8. 8.  Principal Stresses  Normal & shear stresses at a surface oriented normal to a general direction θ in the xy-plane. The triangle is at rest. No net forces act on it.
9. 9.    x cos2    y sin 2   2 xy sin  cos 1    y   x  sin 2   xy cos 2 2  Choosing θ such that τ = 0 tan 2   2 xy x  y   θ has 2 solutions (θ1 & θ2), corresponding to 2 directions for which shear stress vanishes (τ = 0).  The 2 directions are called the principal axes of stress.  The corresponding normal stresses (σ1 & σ2)are called the principal stresses.
10. 10. 2 1 1 2  1   x   y    xy   x   y  2 4 2 1 1 2  2   x   y    xy   x   y  2 4  The principal stresses can be ordered so that σ1 > σ2 > σ3.  The principal axes are orthogonal.
11. 11.  Mohr’s Stress Circle
12. 12. 1 1    1   2    1   2  cos 2 2 2  Radius of the circle: 1     1   2  sin 2 2  1   2  2  Center of the circle on σ-axis:  1   2   Maximum absolute shear stress: 2  max  1   2   2  …occurs when θ = π/4 (= 45o) and θ = 3π/4 (= 135o).
13. 13. Normal strain (elongation) L  L L   L L Elongation is positive (+) for contraction.
14. 14. Shear strain 1   tan  2 Change of the angle ψ between two initially orthogonal directions.
15. 15.  The Strain Tensor  x    yx   zx  xy y  zy  xz    yz  z    Volumetric Strain  vol   x   y   z  Relative decrease in volume
16. 16.  Principal Strains tan 2   2 xy x y   In 2-D, there are 2 orthogonal directions for which the shear strain vanishes (Γ = 0).  The directions are called the principal axes of strain.  The elongations in the directions of the principal axes of strain are called the principal strains.
17. 17.  A group of coefficients.  They have the same units as stress (Pa, bar, atm or psi).  For small changes in stress, most rocks may normally be described by linear relations between applied stresses and resulting strains.     E  Hooke’s law.  E is called Young’s modulus or the E-modulus.  A measure of the sample’s stiffness (resistance against compression by uniaxial stress).
18. 18.  Poisson’s ratio.  A measure of lateral expansion relative to longitudinal contraction. y   x  σx ≠ 0, σy = σz = 0.  Isotropic materials  Response is independent of the orientation of the applied stress.  Principal axes of stress and the principal axes of strain always coincide.
19. 19.  General relations between stresses and strains for isotropic materials:  x     2G   x   y   z  y   x     2G   y   z  z   x   y     2G   z  xy  2G xy  xz  2G xz  yz  2G yz  λ and G are called Lamé’s parameters.  G is called shear modulus or modulus of rigidity.  G is a measure of the sample’s resistance against shear deformation.
20. 20.  Bulk modulus.  A measure of the sample’s resistance against hydrostatic compression.  The ratio of hydrostatic stress relative to volumetric strain. p K  vol  2 If σp = σ1 = σ2 = σ3 while τxy = τxz = τyz = 0: K    G 3  Reciprocal of K (i.e. 1/K) is called compressibility.
21. 21.  In a uniaxial test, i.e. σx ≠ 0; σz = σy = τxy = τxz = τyz = 0: x 3  2G E G x  G y      x 2  G  If any 2 of the moduli are known, the rest can be determined.
22. 22.  Some relations between elastic moduli: E  3K 1  2  E  2G 1   9 KG E 3K  G 3  2G E  G  E  1  1  2    1    K 3 2G 1    K 3 1  2  2 K  G 3 3K  2G v 2  3K  G   2  G 1  2
23. 23. 2    G G 1  2   G   2G 2 1     G 3  2G 2 1     G 3  4G 2  2     G H    2G 4 H K G 3 1   H E 1  1  2  G  E  4G  H  E  3G  H  2G  2H  G  H is called Plane wave modulus or uniaxial compaction modulus.
24. 24.  The stress-strain relations for isotropic materials can be rewritten in alternative forms: E x   x     y   z  E y   y     x   z  E z   z     x   y  1    1  xy   xy  xy 2G E 1    1  xz   xz  xz 2G E 1    1  yz   yz  yz 2G E
25. 25.  Strain Energy  Potential energy may be released during unloading by a strained body.  For a cube with sides a, the work done by increasing the stress from 0 to σ1 is: Work = force × distance 1 1 1 d  a 2    a  d     a 3   d   a 3   Work    d  E 0 0 0 1 3  12 1 3 2 Work  a  a E 1 2 E 2 1 3 Work  a  11 2
26. 26.  When the other 2 principal stresses are non-zero, corresponding terms will add to the expression for the work.  Work (= potential energy) per unit volume is: 1 W   11   2 2   3 3  2  W is called the strain energy.  It can also be expressed as: 1 2 2 W     2G  12   2   3   2 1 2  1 3   2 3  2
27. 27.  Any material not following a linear stress-strain relation.  It is complicated mathematically.   E1  E2 2  E3 3   Types of non-linear elasticity:  Perfectly elastic  Elastic with hysteresis  Permanent deformation
28. 28.  Perfectly Elastic  Ratio of stress to strain is not the same for all stresses.  The relation is identical for both the loading and unloading processes.
30. 30.  Permanent Deformation  It occurs in many rocks for sufficiently large stresses.  The material is still able to resist loading (slope of the stress-strain curve is still positive), i.e. ductile.  Transition from elastic to ductile is called the yield point.
31. 31.  Sedimentary rocks are porous & permeable.  The elastic response of rocks depend largely on the non-solid part of the materials.  The elastic behaviour of porous media is described by poroelastic theory.  Maurice A. Biot was the prime developer of the theory.  We account for the 2 material phases (solid & fluid).  There are 2 stresses involved:  External (or total) stress, σij  Internal stress (pore pressure), Pf
32. 32.  There are 2 strains involved:  Bulk strain – associated with the solid “framework” of the rock. The framework is the “construction” of grains cemented together with a certain texture.  vol   V   s  u V  Zeta (ζ) parameter – increment of fluid, i.e. the relative amount of fluid displaced as a result of stress change.     Vp Pf  Vp  V f    us  u f       V V Kf  p  
33. 33.  The simplest linear form of stress-strain relationship is:   K  vol  C Pf  C vol  M   This is Biot-Hooke’s law for isotropic stress conditions.  C and M are poroelastic coefficients. They are moduli.  C → couples the solid and fluid deformation.  M → characterizes the elastic properties of the pore fluid.
34. 34.  Drained Loading (Jacketed Test)  A porous medium is confined within an impermeable “jacket.”  It is subjected to an external hydrostatic pressure σp.  Pore fluid allowed to escape during loading → pore pressure is kept constant.  Stress is entirely carried by the framework. 
35. 35. Pf  0 0  C vol  M  C vol   M   K  vol  C vol C  M  C2      vol  K fr  vol   K fr  K  M   vol  There are no shear forces associated with the fluid.  Shear modulus of the porous system is that of the framework. G  G fr
36. 36.  Drained Loading (Unjacketed Test)  A porous medium is embedded in a fluid.  Pore fluid is kept within the sample with no possibility to escape.  Hydrostatic pressure on sample is balanced by the pore pressure.
37. 37.  The following equations are combined to give the elastic constants K, C and M in terms of the elastic moduli of the constituents of rock (Ks & Kf) plus porosity φ and Kfr: p C2 K  K fr  vol M K fr Ks  C 1 M  1 1  CK    K K  K M  f  fr  s
38. 38. K fr K 1 Kf   K s  K K s  K fr  K s  K f  Or, 2  K fr  1   Kf Ks   K  K fr  Kf  K fr   1 1      Ks  Ks   This is known as Biot-Gassmann equation. Biot hypothesized that the shear modulus is not influenced by the presence of the pore fluid, i.e.: Gundrained  Gdrained  G fr
39. 39.  K fr  C  1  M Ks   CK s M K s  K fr C M Kf  1 K fr Ks Kf  K fr  1 1      Ks  Ks  Kf 1  Kf  Kf  1  1     Ks  Ks 
40. 40.  Limit 1 – Stiff frame (e.g. hard rock)  Frame is incompressible compared to the fluid: K fr , G fr , K s  K f  Finite porosity (porosity not too small):  Kf     2   K s  K fr   Ks   Then: K  K fr K f  K fr  C 1     Ks  M  Kf 
41. 41.  Limit 2 – Weak frame  For bulk modulus: For porosity: K fr , G fr , K f  K s  Then: K  K fr   Kf  Kf Ks CM  Kf   K is influenced by both rock stiffness and Kf.  In a limiting case when Kfr → 0 (e.g. suspension): K = C= M (≈ Kf /φ) are all given mainly by fluid properties.  For practical calculations, complete K, C and M expressions are used.
42. 42.  Undrained Test (Effective Stress Principle)  Jacketed test with the pore fluid shut in.  No fluid flow in or out of the rock sample.  Increase in external hydrostatic load (compression) will cause an increase in the pore pressure.  No relative displacement between pore fluid and solid during the test.  0   K  vol Pf  C vol C   K
43. 43. C   Pf  K fr  vol M    Pf     σp = total stress σ’p = effective stress  The solid framework carries the part σ’p of σp, while the fluid carries the remaining part αPf. This is called the Effective stress concept (Terzaghi, 1923). K fr C   1 M Ks  α is called Biot constant.  φ < α ≤ 1.  In unconsolidated or weak rocks, α is close to 1.  Upper limit for Kfr is (1- φ)Ks. The lower limit is zer0.