**REVISED ****11-15-2009**

**© 2004 – 2009 WELLOG**

**WAVES IN
ELASTIC MEDIA:**

**Elastic
waveforms:**

**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.**

**Stress:**

**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.**

**Example:**

**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.**

**Strain:**

**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:**

**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:**

**Volume stress is
defined as:**

** Volume
stress = ****D****F/A**

**Volume strain:**

**Volume strain is
defined as:**

** Volume
strain = ****D****V/A**

**Where: ****D****V = volume**

** A = Area**

**Bulk modulus:**

**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 * Vp**^{2}
– (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 x10 ^{10} dynes
per square centimeter**

**===============================================**

**(from Guyod, Geophysical
Well Logging, 1967)**

**Non-porous solids: Bulk Modulus:**

**Anhydrite 62**

**Dolomite 83**

**Limestone 69**

**Salt 31**

**Steel 173**

**Water saturated 5-20% porous rocks in
situ:**

**Dolomites 62-34**

**Limestones 54-23**

**Sandstones 32-18**

**Young’s Modulus:**

**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**** * Vs**^{2}** * ( ****a**** – 1)**

**Where:**

** Vs
= velocity of the shear wave**

** ****r ****= bulk density**

** ****a**** = Poisson’s
ratio**

**Young’s Modulus chart.**

**Shear Modulus:**

**Shear stress:**

** Shear
stress = shear force/A**

**Shear strain:**

** Shear
strain = ****D****s/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**** * Vs ^{2}**

**Shear Modulus chart.**

**Poisson’s Ratio:**

**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 = (D****d/d) /**** (D****l/l)**

**Where: d
= diameter**

** l
= length**

**Table 1:**

**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%**

**Partial listing**

**Note: Velocities are calculated from
Density, Young’s modulus, and Poisson’s ratio.**

**Table 2:**

**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 ****Clark****, S. P. Jr.
ed., Handbook of physical constants, revised edition, Geological Society of ****America**** Memoir 97,
p. 97-173.**

**Conclusion:**

**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.**