Tube-wave relationships

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Problem 2.16a

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.

Figure 2.15a  Response of vertical and horizontal geophones to different waves. (a) Direct P-wave, (b) reflected P-wave, (c) converted S-wave, (d) Rayleigh wave, (e) Love wave. U, D = up, down; A, T = away (from), toward (source); L, R = left, right.

Background

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)]


(2.16a)

being the density and the bulk modulus of the fluid while is the rigidity modulus of the wall material.

Solution

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

Problem 2.16b

Repeat for and 1300 m/s. What do you conclude about the accuracy of this method for determining ?

Solution

When ,

When ,

Summarizing the results for versus , we get the following:

1.05 3.44
1.20 6.80 +14 +156
1.30 35.7 +24 +938

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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Directional geophone responses to different waves Relation between nepers and decibels
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Introduction Partitioning at an interface

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Tube-wave relationships
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