GEOPHYSICS, VOL. 60, NO. 6 (NOVEMBER-DECEMBER 1995); P. 1933-1935, 3 FIGS.

Short Note Simplified anisotropy parameters for transversely isotropic sedimentary rocks

Colin M. Sayers* INTRODUCTION

PVNMO(P) = C44 + (C13 + C44) 2 /(C33 - C44),

Sedimentary rocks frequently possess an anisotropic structure resulting, for example, from fine scale layering, the

(2)

where p is the density of the medium. Defining a parameter 6 by

presence of oriented microcracks or fractures, or the preferred orientation of nonspherical grains or anisotropic minerals. For many rocks the anisotropy may be described, to a good approximation, as being transversely isotropic. The purpose of this note is to present simplified anisotropy parameters for these rocks that are valid when the P-wave normal moveout (NMO) and vertical velocities differ by less than 25%. This condition appears reasonable since depths calculated from P-wave stacking velocities are often within 10% of actual depths (Winterstein, 1986). It is found that when this condition is satisfied the elastic constants c 13 and c 44 affect the P-wave NMO velocity and anellipticity only through the combination c 13 + 2c 44 , a combination of elastic constants that can be determined using walkaway VSP data (Miller et al., 1993). The anellipticity quantifies the deviation of the P-phase slowness from an ellipse and also determines the difference between the vertical and NMO velocities for SV-waves. Helbig (1983) has shown that a time-migrated section for which elliptical anisotropy has been taken into account is identical to one that has been determined under the assumption of isotropy. The anellipticity is therefore the important anisotropy parameter for anisotropic time migration. The results given are of interest for anisotropic velocity analysis, time migration, and timeto-depth conversion.

s =

(Thomsen, 1986), the NMO velocity may be written in the following simple form: VNMO(P) = vv(P)V1 + 28.

(4)

Banik (1987) and Thomsen (1993) have shown that 8 also adequately describes the variation of the P-P reflection coefficient with offset at small offsets. It appears from equation (3) that c 13 and c 44 play an independent role in determining the value of 8. However, 8 may be written in the form:

fi=X+

XZ (5) 2(1 - C441C33)

where (C13 + 2 C44 - C33)

x=

C33

(6)

Figure 1 compares 8 with X using the elastic constants for the sedimentary rocks listed in Thomsen (1986). Also shown is the prediction of equation (5) for the average value c 44 1c 33 = 0.3514 calculated for these rocks. It is seen that 6 is determined, to a good approximation, by the much simpler anisotropy parameter (c 13 + 2c 44 — c 33 ), appropriately normalized. Thus, 8 depends principally on the values of c 33 and the combined parameter c 13 + 2 c44•

In terms of the elastic stiffnesses c,^ of a transversely isotropic medium, the vertical velocity, v v (P), and the NMO velocity, v NMO (P), for P-waves are given by (

(3) 2 C33(C33 - C44)

P- WAVE ANISOTROPY AND NORMAL MOVEOUT

pv . P) = C33,

(C13 + C44) 2 - (C33 - C44) 2

(1)

Manuscript received by the Editor April 18, 1994; revised manuscript received December 27, 1994. *Schlumberger Cambridge Research, Madingley Rd., Cambridge CB3 OEL, United Kingdom. © 1995 Society of Exploration Geophysicists. All rights reserved. 1933 Downloaded 25 Dec 2010 to 199.6.131.16. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/

1934



Sayers SV-WAVE ANISOTROPY AND NORMAL MOVEOUT

[C11 + C33 — 2(c13 +2C44)]13

The combination of stiffnesses c 13 + 2c 44 is also useful for SV-wave studies. For SV-waves the vertical, v v(SV), and normal moveout, V NMO (SV), velocities are given by pv 2 (SV) = C44,

(7)

PVNMO(SV) = C11 — ( C13 + C44) 2/(C33 — C44)•

(8)

( )

2c44

Figure 2 compares or defined by equation (9) with Qapprox given by equation (13) for the sedimentary rocks listed in Thomsen (1986) having 8 in the range — 0.25 <— 8 <— 0.25 (see Figure 1). It is seen that Q is determined, to a good approximation, by the much simpler anisotropy parameter [C11 +

C33 — 2(c13 + 2C44)]•

Defining a parameter Q by F."L DIJ101W11[SYM'1

A (7=

2 C44(C33 — C44)

(9)

where A = ( c11 — C44)(C33 — C44) — (C13 + C44) 2 ,

( 10)

the NMO velocity may be written in the following simple form: (11)

VNMO(SV) = vv(SV)V1 + 2Q

(Tsvankin and Thomsen, 1994). The parameter A defined in equation (10) may be written in the form A = ( C33 — C44)[C11 + C33 — 2(c13 + 2C44)]

Rudzki (1911) and Gassmann (1964) have shown that the deviation of the P-phase slowness from an ellipse is determined by the anellipticity parameter A defined by equation (10). When A = 0, the P-phase slowness and group velocity are elliptical. v is therefore a normalized anellipticity parameter. For P-waves, the normalization given in equation (9) is inconvenient since the elastic constant c44 cannot be obtained from P-wave data alone. A more convenient normalized anellipticity parameter for P-waves is a defined by A =1—

a— c11(c33 — c44)

pvH(P)

The last term in this equation is seen to be of second order in Thomsen's 6 parameter [see equations (5) and (6)]. For small 8, it then follows from equations (9) and (12) that

0.8

1

0.6



VH(P)

where v H (P) is the horizontal P-wave velocity given by (12)

— (C33 — C13 — 2C44)2.

VNMO(P) 2 (14)

=

(15)

c11•

For an elliptically anisotropic medium vNMO(P) = v H (P) and the anellipticity parameter a vanishes. Alkhalifah and Tsvankin (1994) have recently introduced a parameter q defined by

•

1.6 6approx

1.2 0.4

x

0.8 0.2

Equation (5) 0.4

0

•

:7' —0.2

0

•

—0.4 • 0.2

0

0.4

0.6

0.8

s FIG. 1. Comparison of S defined by equation (3) with X given by equation (6) for the sedimentary rocks listed in Thomsen (1986). The curve is the prediction of equation (5) for the average value c441c33 = 0.3514.

—0.4

0

0.4

0.8

1.2

1.6

6 FIG. 2. Comparison of o defined by equation (9) with vap rox given by equation (13) for the sedimentary rocks listed in Thomsen (1986) having S in the range — 0.25 <— S < 0.25.

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1935

Anisotropy Parameters for Rocks 1 f v(P) 1 _ — z— 1 .

Figure 3 shows a comparison of a defined by equation (14) with a approx given by equation (18) for the sedimentary rocks listed in Thomsen (1986) having S in the range —0.25 < S < — S < 0.25, a is seen to be determined by the 0.25. For —0.25< simpler anisotropy parameter [c 11 + c 33 — 2(c 13 + 2c44)], appropriately normalized.

(16)

2 vNMO(P) This may be written in terms of the anellipticity parameter a as follows: a

(17)

2(1—a)



Since S is usually small for sedimentary rocks, it follows from equations (12) and (14) that c11 + C33

— 2(c13 + 2C44)

a =

(18) C11

0.8 a

0.6

approx

CONCLUSIONS

In conclusion, the accuracy of equations (13) and (18) as demonstrated in Figures 2 and 3 occurs because the last term in equation (12) is of second order in Thomsen's S parameter, which is usually small for sedimentary rocks (see Figure 1). It is for this reason that depths calculated from P-wave stacking velocities are often within 10% of actual depths. It follows from this work that if v NMO (P) does not differ from the vertical P-wave velocity by more than 25%, the elastic constants c i3 and c 44 affect the anellipticity and the P and S V-wave NMO velocities only through the combination c 13 + 2c 44 . This combination of elastic constants can be determined using walkaway VSP data (Miller et al., 1993). REFERENCES

0.4 0.2

0

—0.2

—0.4" —0.4

•

—0.2

0.2

0

0.4

0.6

0.8

a FIG. 3. Comparison of a defined by equation (14) with aap roX

defined by equation (18) for the sedimentary rocks listedr in Thomsen (1986) having S in the range —0.25 <— S <— 0.25.

Alkhalifah, T. A., and Tsvankin, I., 1994, Velocity analysis for transversely isotropic media: 64th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1000-1003. Banik, N. C., 1987, An effective anisotropy parameter in transversely isotropic media: Geophysics, 52, 1654-1664. Gassmann, F., 1964, Introduction to seismic traveltime methods in anisotropic media: Pure Appl. Geophys., 58, 53-112. Helbig, K., 1983, Elliptical anisotropy—Its significance and meaning: Geophysics, 48, 825-832. Miller, D. E., Leaney, S., and Borland, W., 1993, An in-situ estimation of anisotropic elastic moduli for a submarine shale: 55th Ann. Mtg., Eur. Assn. Expl. Geophys., Expanded Abstracts, Expanded Abstract CO29. Rudzki, M. P., 1911, Parametrische Darstellung der elastichen Welle in anisotropen Medien: Anzeiger der Akademie der Wissenschaften Krakau, 503-536. Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954-1966. 1993, Weak anisotropic reflections, in Castagna, J. P., and Backus, M., Eds., Offset dependent reflectivity: Soc. Expl. Geophys. Tsvankin, I., and Thomsen, L., 1994, Nonhyperbolic reflection moveout in anisotropic media: Geophysics, 59, 1290-1304. Winterstein, D. F., 1986, Anisotropy effects in P-wave and SHwave stacking velocities contain information on lithology: Geophysics, 51, 661-672.

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