Magnitudes of elastic constants

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Problem 2.4

To illustrate the relationships and magnitudes of the elastic constants, complete Table 2.4a. Note that these values apply to specific specimens; the elastic constants for rocks range considerably, especially as porosity and pressure change.

Solution

We use the row-column notation to designate equations from Table 2.2a.

Water

Since water is a fluid we know that . Equation (4,1) shows that also. From equation (4,5) we get Pa.

Table 2.4a. Magnitudes of elastic constants and velocities.
Constant Water Stiff mud Shale Sandstone Limestone Granite
16 54 50
2.1
0.50 0.43 0.38 0.34 0.25 0.20
1.0 1.5 1.8 1.9 2.5 2.7
1.5 1.6 3.2

Stiff mud

Because , stiff mud is equivalent to a solid, hence . From and we get using equation (9,6), while equation (9,2) expresses in terms of and , thus enabling us to find both and .

We have:


Solving the two equations, we get Pa, Pa. Using equation (6,1),


Equation (6,3) gives


Finally, to get we note that , , and are in , and so we must express in appropriate units of , i.e., . We now have


Shale

As with stiff mud, we have been given , , and , so that again and give us and gives us so that we can solve these equations for and , then find and using equations (9,3) and (9,1). Thus,


Solving the two equations gives Pa, Pa.

From equation (9,3), Pa, and equation (9,1) gives


Finally,


Sandstone

We are given the elastic constants and (plus ), so we get , , , , using equations (1,3) to (1,7) in Table 2.2a. Thus


[We could also have obtained and by using equations (9,6) and (9,7).]

Limestone

We solve in the same way as with sandstone since we are given the same constants:


Granite

Again the solution is the same as for sandstone.


Table 2.4b summarizes the results.

Table 2.4b Magnitudes of elastic constants.
Constant Water Stiffmud Shale Sandstone Limestone Granite
0 1.4 9.9 16 54 50
2.1 3.2 13 17 36 28
0 0.47 3.6 6.0 22 21
2.1 2.9 11 13 22 14
0.50 0.43 0.38 0.34 0.25 0.20
1.0 1.5 1.8 1.9 2.5 2.7
1.5 1.6 3.2 3.6 5.1 4.5
0 0.56 1.4 1.8 2.9 2.8

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