ELASTIC CONSTANTS BASICS (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.
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. 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: 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. 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), Bulk Modulus
Kc, Shear Modulus
N, (also abbreviated as u (mu))
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, 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: 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:
Acoustic impedance:
Where:
ELASTIC PROPERTIES TRANSFORMS The velocity of longitudinal (compressional) waves in solids can
be predicted from the following two equations. Where: The
transverse (shear) wave velocity is defined by the following two
equations: Where: 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
YOUNG'S MODULUS Where:
LAME'S CONSTANT
POISSON'S RATIO Where:
DENSITY 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:
EFFECTS OF PRESSURE The
first and simplest relates compressibility (which is the inverse
of the bulk modulus K) and pressure: Where: 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: K
= Y / 3 * (1 - 2 * P) = 11.42*10^6 / 3 (1 - 2 * 0.32) = 10.6 *
10^6 psi
VOIGHT and REUSS METHODS 2.
REUSS Where: 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. |
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