ELASTIC CONSTANTS BASICS (ROCK PHYSICS)
Well logs are often used to determine the mechanical properties of rocks. These properties are often called the elastic properties or elastic constants of rocks. The subject matter and practice of calculating these rock properties is often called "rock physics".

Mechanical properties are used to design hydraulic fracture stimulation programs in oil and gas wells, and in the design of mines and gas storage caverns. In this situation, the mechanical properties are derived in the laboratory or from well log analysis, calibrated to the lab results.

In seismic petrophysics, these same mechanical properties are called seismic attributes. They are derived by inversion of time-domain seismic data, calibrated to results from well log analysis, which in turn were calibrated to the lab data. The vertical resolution of seismic data is far less than that of well logs, so some filtering and up-scaling issues have to be addressed to make the comparisons meaningful.

The main purpose for finding  these attributes is to distinguish reservoir quality rock from non-reservoir. The ultimate goal is to determine porosity, lithology, and fluid type by "reverse-engineering" the seismic attributes. The process is sometimes called "quantitative seismic interpretation". In high porosity areas such as the tar sands, and in high contrast areas such as gas filled carbonates,, modest success has been achieved, usually after several iterative calibrations to log and lab data. Something can be determined in almost all reservoirs, but how "quantitative" it is may not be known.

There are many other types of seismic attributes related to the signal frequency, amplitude, and phase, as well as spatial attributes that infer geological structure and stratigraphy, such as dip angle, dip azimuth, continuity, thickness, and a hundred other factors. While logs may be used to calibrate or interpret some of these attributes, they are not discussed further here.

The best known elastic constants are the bulk modulus of compressibility, shear modulus, Young's Modulus (elastic modulus), and Poisson's Ratio. The dynamic elastic constants can be derived with appropriate equations, using sonic log compressional and shear travel time along with density log data.

Dynamic elastic constants can also be determined in the laboratory using high frequency acoustic pulses on core samples. Static elastic constants are derived in the laboratory from tri-axial stress-strain measurements (non-destructive) or the chevron notch test (destructive).

Elastic constants are needed by five distinct disciplines in the petroleum industry:
        1. geophysicists interested in using logs to improve synthetic seismograms, seismic models, and interpretation of seismic attributes, seismic inversion, and processed seismic sections.
        2. production or completion engineers who want to determine if sanding or fines migration might be possible, requiring special completion operations, such as gravel packs
        3. hydraulic fracture design engineers, who need to know rock strength and pressure environments to optimize fracture treatments
        4. geologists and engineers interested in in-situ stress regimes in naturally fractured reservoirs
        5. drilling engineers who wish to prevent accidentally fracturing a reservoir with too high a mud weight, or who wish to predict overpressured formations to reduce the risk of a blowout.

The elastic constants of rocks are defined by the Wood-Biot-Gassmann Equations. The equations can be transformed to derive rock properties from log data. If crossed dipole sonic data is available, anisotropic stress can be noticed by differences in the X and Y axis displays of both the compressional and shear travel times. When this occurs, all the elastic constants can be computed for both the minimum and maximum stress directions. This requires the original log to be correctly oriented with directional information, and may require extra processing in the service company computer center.

Elasticity is a property of matter, which causes it to resist deformation in volume or shape. Hooke's Law, describing the behavior of elastic materials, states that within elastic limits, the resulting strain is proportional to the applied stress. Stress is the external force applied per unit area (pressure), and strain is the fractional distortion which results because of the acting force.

The modulus of elasticity is the ratio of stress to strain:
      0: M = Pressure / Change in Length =  {F/A} / (dL/L)

This is identical to the definition of Young's Modulus. Both names are used in the literature so terminology can be a bit confusing.

Different types of deformation can result, depending upon the mode of the acting force. The three elastic moduli are:

Young's Modulus Y (also abbreviated E in various literature),
       1: Y = (F/A) / (dL/L)

Bulk Modulus Kc,
       2: Kc = (F/A) / (dV/V)

Shear Modulus N, (also abbreviated as u (mu))
       3: N = (F/A) / (dX/L) = (F/A) / tanX

Where F/A is the force per unit area and dL/L, dV/V, and (dX/L) = tanX are the fractional strains of length, volume, and shape, respectively.

Poisson's Ratio PR (also abbreviated v (nu)), is defined as the ratio of strain in a perpendicular direction to the strain in the direction of extensional force,
       4: PR = (dX/X) / (dY/Y)

Where X and Y are the original dimensions, and dX and dY are the changes in x and y directions respectively, as the deforming stress acts in y direction.

Young's Modulus vs Poison's Ratio: Brittleness increases toward top left, density increases toward top right, porosity plus organic content and depth decrease toward bottom left. PR values less than 0.17 indicate gas or organic content or both. (image courtesy Canadian Discovery Ltd)

All of these moduli can be derived directly from well logs and indirectly from seismic attributes:
      5: N = KS5 * DENS / (DTS ^ 2)
      6: R = DTS / DTC
      7: PR = (0.5 * R^2 - 1) / (R^2 - 1)
      8: Kb = KS5 * DENS * (1 / (DTC^2) - 4/3 * (1 / (DTS^2)))
      9: Y = 2 * N * (1 + PR)

Lame's Constant Lambda, (also abbreviated λ) is a measure of a rocks brittleness, which is a function of both Young's Modulus and Poisson's Ratio:
      10: Lambda = Y * PR / ((1 + PR) * (1 - 2 * PR))
OR 10A: Lambda = DENS * (Vp^2 - 2 * Vs ^ 2)

Some people prefer different abbreviations: Mu or u for shear modulus, Nu or v for Poisson's Ratio, and E for Young's Modulus. The abbreviations used above are used consistently trough these training materials.

In the seismic industry, it is common to think in terms of velocity and acoustic impedance in addition to the more classical mechanical properties described above.

The compressional to shear velocity ratio is a good lithology indicator:
      11. R = Vp / Vs = DTS / DTC

Acoustic impedance:
      12: Zp = DENS / DTC
      13: Zs = DENS / DTS

Where:
  DTC = compressional sonic travel time
  DTS = shear sonic travel time
  DENS = bulk density
  KS5 = 1000 for metric units

ELASTIC PROPERTIES TRANSFORMS

VELOCITY OF SOUND
Velocity of sound, density, and elastic properties of rocks are intimately connected by a series of transforms. Knowledge of any two of these properties means all the others can be calculated.

The velocity of longitudinal (compressional) waves in solids can be predicted from the following two equations.
         1: Vp = 68.4 * (((K + 4/3 * N) / DENS) ^ 1/2)
OR: 1A: Vp = 68.4 * (((Y * (1 - N) / (DENS * (1 - 2 * N) * (1 - N)) ^ 1/2)

Where:
  K = bulk modulus of elasticity (psi)
  DENS = density (lb/cuft)
  N = shear modulus or modulus or rigidity (psi)
  Vp = compressional velocity (ft/sec)
  Y = Young's modulus (psi)

The transverse (shear) wave velocity is defined by the following two equations:
        2: Vs = 68.4 * ((N / DENS) ^ 1/2)
OR 2A: Vs = 68.4 * (((Y / DENS) / 2 * (1 + PR)) ^ 1/2)

Where:
  DENS = density (lb/cuft)
  N = shear modulus or modulus or rigidity (psi)
  PR = Poisson's ratio (unitless)
  Vs = shear wave velocity (ft/sec)

To translate these formulae into metric, convert density into gm/cc, velocity to Km/sec and the various moduli to megabars, and change the constant terms to 1.0. To convert moduli in megabars to psi, multiply by 6.89 * 10^-6. To convert megabars to Kilopascals, multiply by 10^4.

The elastic constants K, N, Y and PR are often known, and many values are listed in handbooks. Identities exist which show that knowledge of any two constants infers knowledge of the other two. This in turn, infers knowledge of velocity. These identities follow.

  BULK MODULUS
Bulk modulus (K) can be calculated from any of the following six equations depending on which parameters are known about a rock:
    3: K = L + 2 * N / 3
    4: K = Y * N / (3 * (3 * N - Y))
    5: K = L * (1 + PR) / (3 * PR)
    6: K = S * (2 * (1 + PR)) / (3 * (1 - 2 * PR))
    7: K = Y / (3 * (1 - 2 * PR))
    8: K = DENS * (Vp ^ 2 - 4 / 3 * Vs ^ 2)

  YOUNG'S MODULUS
Young's modulus (Y) is related to the other properties by:
    9:  Y = N * (3 * L + 2 * N) / (L + N)
    10: Y = 9 * K * (K - L) / (3 * K - L)
    11: Y = 9 * K * L / (3 * K + L)
    12: Y = L * (1 + PR) * (1 - 2 * PR) / PR
    13: Y = 2 * N * (1 + PR)
    14: Y = 3 * K * (1 - 2 * PR)
    15: Y = ((9 * DENS * R3 ^ 2 * R2 ^ 2) / (3* R2 ^ 2 + 1))

Where:
    16: R2 = (K / (DENS * (Vs ^ 2))) ^ (1 / 2)
    17: R3 = (K / (DENS * (Vp ^ 2))) ^ (1 / 2)

  LAME'S CONSTANT
Lame's constant (L) is found from:
    18: L = K - 2 * N / 3
    19: L = N * (Y - 2 * N) / (3 * N - Y)
    20: L = 3 * K * (3 * K - Y) / (N * K - Y)
    21: L = N * (2 * PR / (1 - 2 * PR))
    22: L = 3 * K * (PR / (1 - PR))
    23: L = Y * PR / ((1 + PR) * (1 - 2 * PR))
    24: L = DENS * (Vp^2 - 2 * Vs ^ 2)

  POISSON'S RATIO
Poisson's ratio (PR) is related by:
    25: PR = L / 2 * (L + N)
    26: PR = L / (3 * K - L)
    27: PR = (3 * K - 2 * N) / (2 * (3 * K + N))
    28: PR = (Y / (2 * N)) - 1
    29: PR = (3 * K - Y) / (6 * K)
    30: PR = ((R1^2 - 2) / (R1^2 - 1) / 2)
    31: PR = ((3 * (R2^2) - 2) / (3 * (R2^2) + 1) / (3 * (R3^2) + 1) / 2)

Where:
    32: R1 = Vp / Vs
          R2 and R3 are as defined before.

 DENSITY
By rearranging all of the above, density can be found in a large variety of circumstances.
    33: DENS = (L + 2 * N) / (Vp ^ 2)
    34: DENS = (3 * K - 2) / (Vp ^ 2)
    35: DENS = (K + 4 * N / 3) / (Vp ^ 2)
    36: DENS = N * (4 * N - Y) / (3 * N - Y) / (Vp ^ 2)
    37: DENS = 3 * K * (3 * K + Y) / (9 * K - Y) / (Vp ^ 2)
    38: DENS = L * ((1 - PR) / PR) / (Vp ^ 2)
    39: DENS = N * (2 - 2 * PR) / (1 - 2 * PR) / (Vp ^ 2)
    40: DENS = 3 * K * (1 - PR) / (1 + PR) / (Vp ^ 2)
    41: DENS = Y * (1 - PR) / ((1 + PR) * ( 1 - 2 * PR)) / (Vp ^ 2)
    42: DENS = 3 * ( K - L) / 2 / (Vs ^ 2)
    43: DENS = 3 * K * Y / (9 * K - Y) / (Vs ^ 2)
    44: DENS = L * ((1 - 2 * PR) / (2 * PR) / Vs ^ 2)
    45: DENS = 3 * K * (1 - 2 * PR) / (2 + 2 * PR) / (Vs ^ 2)
    46: DENS = Y / (2 + 2 * PR) / (Vs ^ 2)

Such relationships are used to reconstruct density logs in bad hole conditions by using sonic log data and assumed values for Poisson's ratio. PR is often a function of shale volume and lithology, which can be determined in zones where hole condition is good.

Where:
  K = bulk modulus (megabars)
  DENS = density (gm/cc)
  L = Lame's constant (unitless)
  PR = Poisson's ratio (unitless)
  N = shear modulus (megabars)
  Vs = shear wave velocity (km/sec)
  Vp = compressional wave velocity (km/sec)
  Y = Young's modulus

EFFECTS OF PRESSURE
Considerable data is available on elastic constants versus pressure. Three methods are available for tabulation of results and are covered in the Handbook of Physical Constants.

The first and simplest relates compressibility (which is the inverse of the bulk modulus K) and pressure:
    47: Ce = 1 / K = (6.89*10^-8) * a + (47.5*10^-16) * b * Pf

Where:
  a = constant (psi^-1)
  K = bulk modulus (psi)
  b = constant (psi^-2)
  Ce = compressibility (psi^-1)
  Pf = formation pressure (psi)

The constants a and b, for particular solids can be found in the Handbook of Physical Constants.

For example assume the following measured values on a limestone sample:
    DENS = 2.712 gm/cc = 170.0 lb/cuft
    Y = 0.789 mb = 11.42*10^6 psi
    N = 0.229 mb = 4.35*10^6 psi
    PR = 0.32

    K = Y / 3 * (1 - 2 * P) = 11.42*10^6 / 3 (1 - 2 * 0.32) = 10.6 * 10^6 psi
    Vp = 68.4 ((10.6*10^6 + (4 / 3) * 4.35*10^6) / 170)) ^ 1 / 2 = 21,200 ft/sec
    DTC = (10^6) / 21200 = 47.4 usec/ft

VOIGHT and REUSS METHODS
The other two methods are termed the Voight and Reuss schemes for obtaining the elastic constants of aggregates. They lead to the following relationships:
  1. VOIGHT
      48: a = (C11 + C22 + C33) * 4.83*10^6
      49: b = (C23 + C31 + C12) * 4.83*10^6
      50: c = (C44 + C55 + C66) * 4.83*10^6
      51: K = (a + 2 * b) / 3
      52: N = (a - b + 3 * c) / 5

  2. REUSS
      53: a = (S11 + S22 + S33) * 2.29 * 10^-8
      54: b = (S23 + S31 + S12) * 2.29 * 10^-8
      55: c = (S44 + S55 + S66) * 2.29 * 10^-8
      56: K = 1 / (3 * a + 6 * b)
      57: N = 5 / (4 * a - 4 * b + 3 * c)

Where:
  a,b,c = intermediate terms (psi^-1)
  K = bulk modulus (psi)
  Cij = compressibility constants for the Voight method (psi^-1)
  N = shear modulus (psi)
  Sij = shear constants for the Reuss method (psi^-1)

The Cij and Sij values are obtained from the tables in The Handbook of Physical Constants. Other coefficients for the aggregate may be obtained from K and N, by use of the relationships between the various elastic constants given earlier. Examples of these two methods are also shown in the Handbook of Physical Constants.

For many rocks, elastic constants are known, whereas velocity is unknown. This is especially true when the effects of pressure and temperature are being considered. It is also clear that given a reasonable set of elastic constants and either a velocity or density log, the other log can be constructed with confidence. This is particularly useful in seismography. Note that the sonic velocity log as a rule, measures the travel time associated with the longitudinal or compressional wave. Therefore, the appropriate equations should be used for log interpretation work.