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ELASTIC CONSTANTS BASICS (ROCK PHYSICS)
This page discusses how well logs are 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".

The best known elastic constants are the bulk modulus of compressibility, 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 (pressure) applied per unit area, 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.

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

Young's Modulus,
       1: Y = (F/A) / (dL/L)

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

Shear Modulus,
       3: N = (F/A) / tanX

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

Another important elastic constant, called Poisson's Ratio, 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.

 

LAB MEASUREMENT PROCEDURES

The general procedures for triaxial compressive test are:
    1. A right cylindrical plug is cut from the sample core and their ends ground parallel according to International Society for Rock Mechanics (ISRM) and American Society for Testing and Materials (ASTM) standards.  A length to diameter ratio of 2:1 is recommended to obtain representative mechanical properties of the sample, which is also recommended by ASTM and ISRM.  Physical dimensions and weight of the specimen are recorded and the specimen is saturated with simulated formation brine.

   2. The specimen is then placed between two plates and a heat-shrink jacket is placed over the specimen.

   3. Axial strain and radial strain devices are mounted in the endcaps and on the lateral surface of the specimen, respectively.

    4. The specimen assembly is placed into the pressure vessel and the pressure vessel is filled with hydraulic oil.

   5. Confining pressure is increased to the desired hydrostatic testing pressure.

   6. Measure ultrasonic velocities at the hydrostatic confining pressure.

   7. Specimen assembly is brought into contact with a loading piston that allows application of axial load.

   8. Increase axial load at a constant rate until the specimen fails or axial strain reaches a desired amount of strain while confining pressure is held constant.

   9. Reduce axial stress to the initial hydrostatic condition after sample fails or reaches a desired axial strain.

   10. Reduce confining pressure to zero and disassemble sample.

Dynamic elastic properties measured with ultrasonic impulse in the lab. Note differences between static and dynamic values. Elastic properties from log analysis models match lab dynamic data better than static data.
 

Dynamic elastic properties calculated from density and sonic log data, showing close match to dynamic data from lab measurements (coloured dots). Lab data is from table shown above. Note synthetic sonic and density plotted next to measured log curves (Tracks 2 and 3), showing reasonably small differences due to minor borehole effects. Synthetic curves can repair worse logs or even replace missing curves.

ELASTIC CONSTANTS THEORY
The velocity of sound in a rock is related to the elastic properties of the rock/fluid mixture and its density, according to the Wood, Biot, and Gassmann equations.

The composite compressional bulk modulus of fluid in the pores (inverse of fluid compressibility) is:   ____   
      1:  Kf = 1/Cf = Sw / Cwtr + (1 - Sw) / Coil
OR 1a: Kf = 1/Cf = Sw / Cwtr + (1 - Sw) / Cgas

The pore space bulk modulus (Kp) is derived from the porosity, fluid, and matrix rock properties:
      2: ALPHA = 1 - Kb / Km
        3: Kp = ALPHA^2 / ((ALPHA - PHIt) / PHIt / Kf )

The composite rock/fluid compressional bulk modulus is:
      4: Kc = Kp + Kb + 4/3 * N

Compressional and shear velocity (or travel time) depend on density and on the elastic properties, so we need a density value that reflects the actual composition of the rock fluid mixture:
      5:  DENS = (1 - Vsh) * (PHIe * Sw * DENSW + PHIe * (1 - Sw) * DENSHY + (1 - PHIe) * DENSMA)
                        + Vsh * DENSSH 

Compressional velocity (Vp) and shear velocity (Vs) are defined as:
      6: Vp = KS4 * (Kc / DENS) ^ 0.5
      7: Vs = KS4 * (N / DENS) ^ 0.5

Although it is not a precise solution, we often invert equations 5 and 6 to solve for Kb and N from sonic log compressional and shear travel time values.

WHERE:
  ALPHA = Biot's elastic parameter (fractional)
  Cgas = gas compressibility
  Coil = oil compressibility
  Cwtr = water compressibility
  DENS = rock density (Kg/m3 or g/cc)
  DENSW = density of fluid in the pores (Kg/m3 or g/cc)
  Kb = compressional bulk modulus of empty rock frame
  Kc = compressional bulk modulus of porous rock
  Kf = compressional bulk modulus of fluid in the pores
  Km = compressional bulk modulus of rock grains
  Kp = compressional bulk modulus of pore space
  N = shear modulus of empty rock frame
  PHIt = total porosity of the rock (fractional)
  Sw = water saturation (fractional)
  Vp = compressional wave velocity (m/sec or ft/sec)
  Vs = shear wave velocity (m/sec or ft/sec)
  Vp = Stoneley wave velocity (m/sec or ft/sec)
  KS4 = 68.4 for English units
  KS4 = 1.00 for Metric units

The Biot-Gassmann approach looks deceptively simple. However, the major drawback to this approach is the difficulty in determining the bulk moduli, particularly those of the empty rock frame (Kb and N), which cannot be derived from log data. Murphy (1991) provided equations for sandstone rocks (PHIe < 0.35) that predict Kb and N from porosity:
      8: Kb = 38.18 * (1 - 3.39 * PHIe + 1.95 * PHIe^2)
      9: N   = 42.65 * (1 - 3.48 * PHIe + 2.19 * PHIe^2)

Examples of Mechanical Properties Logs
The above equations can be computed continuously and presented as logs. The format and curve complement vary widely between service companies and age of log. Some logs have Metric depths but the moduli are in English units. Some are vice versa. Here are some examples.

Mechanical properties log with lithology/porosity track at the right.

Another mechanical properties log
 

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