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
|
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|
|
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
|
Continue reading
Also in this chapter
External links
find literature about Magnitudes of elastic constants
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