© 2004 – 2009 WELLOG
WAVES IN ELASTIC MEDIA:
Elasticity is that property of a substance that causes the substance to resist deformation and to recover its original shape when the deforming forces are removed. A medium that recovers its shape completely after being deformed is considered to be elastic. The earth is considered almost elastic for small displacements. Most of the theory used in acoustic logging is described mathematically using the theory of elastic waveforms in elastic media.
A force applied to a material is called stress (s). Stress is measured in terms of force per unit area. Stress can be compressional, expansional, or shear.
s = F/A
Where: F = force
A = Area
Stress in terms of young’s modulus is expressed as:
s = E/e
e = strain
Compressional stress occurs when a force is applied over one side of a body while it is supported by an equal force on the opposite side.
Tensile stress (pulling or stretching) is considered as a negative compressional stress.
An example of compression stress is the force of a weight drop used in seismic geophysics. A 100 pound weight is dropped from a height of 10 feet. The resulting compressional stress is 1000 ft. pounds.
Most of the phenomena in acoustic logging are related to strain. When an elastic material is subjected to stress, changes in physical dimension and shape called strain, occur.
Strain (e) = Elongation = change in dimension/original dimension
Hooke’s Law states that in an elastic medium with small strains, the strain is directly proportional to the stress that caused it. Elastic materials are referred to as exhibiting “Hookean” behavior.
Volume stress is defined as:
Volume stress = DF/A
Volume strain is defined as:
Volume strain = DV/A
Where: DV = volume
A = Area
Bulk Modulus (B) is defined as:
B = - volume stress/volume strain
Note: Bulk modulus can be obtained from acoustic velocity and bulk density logs.
B = r * Vp2 – (4 * m /3)
Note: A negative sign is used to indicate a decrease in volume due to compression.
Water is compressible. When subjected to 500 atm. Water is compressed 2 to 3 percent.
The inverse of bulk modulus is compressibility.
Compressibility = 1/B
The velocity of acoustic waves in a medium are approximately related to the square root of its elastic properties and inversely related to its inertial properties.
In gases or liquids:
V (approx.) = (elastic property/inertial property)1/2
V = B/r
Where: B = Bulk Modulus (an elastic property)
r = Density (an inertial property)
Velocity of compressional (P) waves in Rock materials:
Vp = ((B + 4/3 x S)/r)1/2
Where: S = shear modulus (defined below)
B = Bulk modulus (defined above)
r = Density
Velocity of shear (s) waves:
Vs = (S/r) 1/2
The velocity of shear waves is about .7 that of compressional waves.
Velocities in various rock types will be discussed later.
BULK MODULUS VALUES x1010 dynes per square centimeter
(from Guyod, Geophysical Well Logging, 1967)
Non-porous solids: Bulk Modulus:
Water saturated 5-20% porous rocks in situ:
Elastic materials are materials in which stress and stain are proportional to each other. If the stress is doubled, the strain is doubled. The ratio of the stress and stain in an object is referred to as elastic modulus or Young’s modulus.
Young’s modulus = E = tensile or compressive stress/tensile or compressive strain
Young’s modulus can be obtained from acoustic velocity and bulk density logs.
E = 2 * r * Vs2 * ( a – 1)
Vs = velocity of the shear wave
r = bulk density
a = Poisson’s ratio
Young’s Modulus chart.
Shear stress = shear force/A
Shear strain = Ds/L
Shear Modulus ( m ) is defined as:
m = shear stress/ shear strain
Shear modulus can be obtained from acoustic velocity and bulk density.
m = r * Vs2
Shear Modulus chart.
Poisson’s ratio may be considered as a measurement of the geometric change in shape due to extensional stress.
Poisson’s ratio ( s ) is defined as the ratio of relative increase or decrease in diameter to relative compression or elongation.
s = (Dd/d) / (Dl/l)
Where: d = diameter
l = length
Rock Density: Young’s Poisson’s Vp: Vs: Vp/Vs: Vs as %Vp:
Types: Gm/cc Modulus: Ratio: m/sec m/sec
Shale (AZ) 2.67 0.120 0.040 2124 1470 1.44 69.22%
Siltstone (CO) 2.50 0.130 0.120 2319 1524 1.52 65.71%
Limestone(AZ) 2.44 0.170 0.180 2750 1718 1.60 62.47%
Schist (MA) 2.70 0.544 0.181 4680 2921 1.60 62.41%
Note: Velocities are calculated from Density, Young’s modulus, and Poisson’s ratio.
Compressional (P) wave velocities: (m/sec)
Unconsolidated: Velocity: Consolidated: Velocity:
Weathered layer: 300 - 900 Granite: 5000 - 6000
Soil: 250 - 600 Basalt: 5400 - 6400
Alluvium: 500 – 000 Metamorphic Rocks: 3500 - 7000
Unsaturated Sand: 200 – 1000 Sandstone & Shale: 2000 - 4500
Saturated Sand: 800 – 2200 Limestone: 2000 - 6000
Sand & Water: 1400 - 1600
Unsaturated Gravel: 400 – 500 Air: 331.5
Saturated Gravel: 500 - 1500
Note: This is only Partial listing
Table 1 and Table 2 From: Press, Frank
(1966), Seismic velocities, in
Using the prior tables, it is possible to distinguish velocities of dry sediments from saturated sediments. It is further possible to distinguish sediments from rocks and igneous rocks from metamorphic rocks.