A tube wave has a velocity of 1050 m/s. The fluid in the borehole has a bulk modulus of Pa and a density of . The wall rock has and density . Calculate and for the wall rock.
Several types of tube waves exist (Sheriff and Geldart, 1995, Section 2.5.5). The classical type consists of a P-wave traveling in a fluid within a tubular cavity (such as a borehole) in a solid medium, the wall of the tube expanding and contracting as the pressure wave passes. Because the wall material interacts with the fluid, the tube-wave velocity depends upon the properties of both the wall material and the fluid. The formula for the tube-wave velocity is [see Sheriff and Geldart, 1995, equation (2.97)]
being the density and the bulk modulus of the fluid while is the rigidity modulus of the wall material.
Assuming the wall material to be rock, we write , , , , and for the rock, and for the fluid. We solve equation (2.16a) for but first we note that and are in units of , so we express as . Then,
Using equation (5,5) of Table 2.2a, we have
To get , we have
Repeat for and 1300 m/s. What do you conclude about the accuracy of this method for determining ?
Summarizing the results for versus , we get the following:
Since the relative change in is very much larger than the relative change in , the method is very sensitive to changes in , hence the accuracy is very poor.
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Also in this chapter
- The basic elastic constants
- Interrelationships among elastic constants
- Magnitude of disturbance from a seismic source
- Magnitudes of elastic constants
- General solutions of the wave equation
- Wave equation in cylindrical and spherical coordinates
- Sum of waves of different frequencies and group velocity
- Magnitudes of seismic wave parameters
- Potential functions used to solve wave equations
- Boundary conditions at different types of interfaces
- Boundary conditions in terms of potential functions
- Disturbance produced by a point source
- Far- and near-field effects for a point source
- Rayleigh-wave relationships
- Directional geophone responses to different waves
- Relation between nepers and decibels
- Attenuation calculations
- Diffraction from a half-plane