ELASTIC PROPERTIES TRANSFORMS
 
VELOCITY
Velocity of sound, density, and elastic properties of rocks are
intimately connected by a series of transforms. Knowledge of any two
of these properties means all the others can be calculated.
The velocity of longitudinal (compressional) waves in solids can
be predicted from the following two equations.
1: Vp = 68.4 * (((K + 4/3 * N) / DENS) ^ 1/2)
OR: 1A: Vp = 68.4 * (((Y * (1 - N) / (DENS * (1 - 2 * N) * (1 - N))
^ 1/2)
WHERE:
K = bulk modulus of elasticity (psi)
DENS = density (lb/cuft)
N = shear modulus or modulus or rigidity (psi)
Vp = compressional velocity (ft/sec)
Y = Young's modulus (psi)
The
transverse (shear) wave velocity is defined by the following two
equations:
2: Vs = 68.4 * ((N / DENS) ^ 1/2)
OR 2A: Vs = 68.4 * (((Y / DENS) / 2 * (1 + PR)) ^ 1/2)
WHERE:
DENS = density (lb/cuft)
N = shear modulus or modulus or rigidity (psi)
PR = Poisson's ratio (unitless)
Vs = shear wave velocity (ft/sec)
To
translate these formulae into metric, convert density into gm/cc,
velocity to Km/sec and the various moduli to megabars, and change
the constant terms to 1.0. To
convert moduli in megabars to psi, multiply by 6.89 * 10^-6. To
convert megabars to Kilopascals, multiply by 10^4.
The
elastic constants K, N, Y and PR are often known, and many values
are listed in handbooks. Identities exist which show that knowledge
of any two constants infers knowledge of the other two. This in
turn, infers knowledge of velocity. These identities follow.
BULK MODULUS
Bulk
modulus (K) can be calculated from any of the following six equations
depending on which parameters are known about a rock:
3: K = L + 2 * N / 3
4: K = Y * N / (3 * (3 * N - Y))
5: K = L * (1 + PR) / (3 * PR)
6: K = S * (2 * (1 + PR)) / (3 * (1 - 2 * PR))
7: K = Y / (3 * (1 - 2 * PR))
8: K = DENS * (Vp ^ 2 - 4 / 3 * Vs ^ 2)
YOUNG'S MODULUS
Young's
modulus (Y) is related to the other properties by:
9: Y = N * (3 * L + 2 * N) / (L + N)
10: Y = 9 * K * (K - L) / (3 * K - L)
11: Y = 9 * K * L / (3 * K + L)
12: Y = L * (1 + PR) * (1 - 2 * PR) / PR
13: Y = 2 * N * (1 + PR)
14: Y = 3 * K * (1 - 2 * PR)
15: Y = ((9 * DENS * R3 ^ 2 * R2 ^ 2) / (3* R2 ^ 2 + 1))
WHERE:
16: R2 = (K / (DENS * (Vs ^ 2))) ^ (1 / 2)
17: R3 = (K / (DENS * (Vp ^ 2))) ^ (1 / 2)
LAME'S CONSTANT
Lame's
constant (L) is found from:
18: L = K - 2 * N / 3
19: L = N * (Y - 2 * N) / (3 * N - Y)
20: L = 3 * K * (3 * K - Y) / (N * K - Y)
21: L = N * (2 * PR / (1 - 2 * PR))
22: L = 3 * K * (PR / (1 - PR))
23: L = Y * PR / ((1 + PR) * (1 - 2 * PR))
24: L = DENS * (Vp^2 - 2 * Vs ^ 2)
POISSON'S RATIO
Poisson's
ratio (P) is related by:
25: PR = L / 2 * (L + N)
26: PR = L / (3 * K - L)
27: PR = (3 * K - 2 * N) / (2 * (3 * K + N))
28: PR = (Y / (2 * N)) - 1
29: PR = (3 * K - Y) / (6 * K)
30: PR = ((R1^2 - 2) / (R1^2 - 1) / 2)
31: PR = ((3 * (R2^2) - 2) / (3 * (R2^2) + 1) / (3 * (R3^2) + 1)
/ 2)
WHERE:
32: R1 = Vp / Vs
R2 and R3 are as defined before.
DENSITY
By
rearranging all of the above, density can be found in a large
variety of circumstances.
33: DENS = (L + 2 * N) / (Vp ^ 2)
34: DENS = (3 * K - 2) / (Vp ^ 2)
35: DENS = (K + 4 * N / 3) / (Vp ^ 2)
36: DENS = N * (4 * N - Y) / (3 * N - Y) / (Vp ^ 2)
37: DENS = 3 * K * (3 * K + Y) / (9 * K - Y) / (Vp ^ 2)
38: DENS = L * ((1 - PR) / PR) / (Vp ^ 2)
39: DENS = N * (2 - 2 * PR) / (1 - 2 * PR) / (Vp ^ 2)
40: DENS = 3 * K * (1 - PR) / (1 + PR) / (Vp ^ 2)
41: DENS = Y * (1 - PR) / ((1 + PR) * ( 1 - 2 * PR)) / (Vp ^ 2)
42: DENS = 3 * ( K - L) / 2 / (Vs ^ 2)
43: DENS = 3 * K * Y / (9 * K - Y) / (Vs ^ 2)
44: DENS = L * ((1 - 2 * PR) / (2 * PR) / Vs ^ 2)
45: DENS = 3 * K * (1 - 2 * PR) / (2 + 2 * PR) / (Vs ^ 2)
46: DENS = Y / (2 + 2 * PR) / (Vs ^ 2)
Such
relationships are used to reconstruct density logs in bad hole
conditions by using sonic log data and assumed values for Poisson's
ratio. PR is often a function of shale volume and lithology, which
can be determined in zones where hole condition is good.
WHERE:
K = bulk modulus (megabars)
DENS = density (gm/cc)
L = Lame's constant (unitless)
PR = Poisson's ratio (unitless)
N = shear modulus (megabars)
Vs = shear wave velocity (km/sec)
Vp = compressional wave velocity (km/sec)
Y = Young's modulus
EFFECTS OF PRESSURE
Considerable
data is available on elastic constants versus pressure. Three
methods are available for tabulation of results and are covered
in the Handbook of Physical Constants.
The
first and simplest relates compressibility (which is the inverse
of the bulk modulus K) and pressure:
47: Ce = 1 / K = (6.89*10^-8) * a + (47.5*10^-16) * b * Pf
WHERE:
a = constant (psi^-1)
K = bulk modulus (psi)
b = constant (psi^-2)
Ce = compressibility (psi^-1)
Pf = formation pressure (psi)
The
constants a and b, for particular solids can be found in the Handbook
of Physical Constants.
For
example assume the following measured values on a limestone sample:
DENS = 2.712 gm/cc = 170.0 lb/cuft
Y = 0.789 mb = 11.42*10^6 psi
N = 0.229 mb = 4.35*10^6 psi
PR = 0.32
K
= Y / 3 * (1 - 2 * P) = 11.42*10^6 / 3 (1 - 2 * 0.32) = 10.6 *
10^6 psi
Vp = 68.4 ((10.6*10^6 + (4 / 3) * 4.35*10^6) / 170)) ^ 1 / 2 =
21,200 ft/sec
DTC = (10^6) / 21200 = 47.4 usec/ft
VOIGHT and REUSS METHODS
The
other two methods are termed the Voight and Reuss schemes for
obtaining
the elastic constants of aggregates. They lead to the following
relationships:
1. VOIGHT
48: a = (C11 + C22 + C33) * 4.83*10^6
49: b = (C23 + C31 + C12) * 4.83*10^6
50: c = (C44 + C55 + C66) * 4.83*10^6
51: K = (a + 2 * b) / 3
52: N = (a - b + 3 * c) / 5
2.
REUSS
53: a = (S11 + S22 + S33) * 2.29 * 10^-8
54: b = (S23 + S31 + S12) * 2.29 * 10^-8
55: c = (S44 + S55 + S66) * 2.29 * 10^-8
56: K = 1 / (3 * a + 6 * b)
57: N = 5 / (4 * a - 4 * b + 3 * c)
WHERE:
a,b,c = intermediate terms (psi^-1)
K = bulk modulus (psi)
Cij = compressibility constants for the Voight method (psi^-1)
N = shear modulus (psi)
Sij = shear constants for the Reuss method (psi^-1)
The
Cij and Sij values are obtained from the tables in The Handbook
of Physical Constants. Other coefficients for the aggregate may
be obtained from K and N, by use of the relationships between
the various elastic constants given earlier. Examples of these
two methods are also shown in the Handbook of Physical Constants.
For
many rocks, elastic constants are known, whereas velocity is unknown.
This is especially true when the effects of pressure and temperature
are being considered. It is also clear that given a reasonable
set of elastic constants and either a velocity or density log,
the other log can be constructed with confidence. This is particularly
useful in seismography. Note that the sonic velocity log as a
rule, measures the travel time associated with the longitudinal
or compressional wave. Therefore, the appropriate equations should
be used for log interpretation work.
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