Weak Elastic Ani Sot Ropy Thomsen_86

8/8/2019 Weak Elastic Ani Sot Ropy Thomsen_86

1/13

GE()PHYSICS. VOL. 51. NO. IO (OCTOBER 1986); P. 19541966, 5 FIGS.. I TABLE

Weak elastic anisotropy

Leon Thomsen*ABSTRACT

Most bulk elastic media are weakly anisotropic. -Theequations governing weak anisotropy are much simplerthan those governing strong anisotropy, and they aremuch easier to grasp intuitively. These equations indi-cate that a certain anisotropic parameter (denoted 6)controls most anisotropic phenomena of importance inexploration geophysics. some of which are nonn egligibleeven when the anisotropy is weak. The critical parame-ter 6 is an awkward combination of elastic parameters,a combination which is totally independent of horizon-tal velocity and which may be either positive or nega-tive in natural contexts.

INTRODUCTIONIn most applications of elasticity theory to problems in pe-

troleum geophysics, the elastic medium is assum ed to be iso-tropic. On the other hand, most crustal rocks are found exper-imentally to be anisotropic. Further, it is known that if alayered sequen ce of different m edia (isotropic o r no t) is probedwith an elastic wave of wavelength much longer than the typi-cal layer thickness (i.e., the no rmal seismic exploration con-text). the wave propagates as though it were in a homoge-neous, but anisotropic, medium (Backus, 19 62). Hence, there isa fundamental inconsistency between practice on the one handand reality on the other.

Two major reasons for the continued existence of this in-consistency come readily to m ind:

(1) The most commonly occurring type of anisotropy(transverse isotropy) masquerades as isotropy in near-vertical reflection profiling, with the angular dependencedisguised in the uncertainty of the depth to each reflec-tor (cf., Krey an d Helbig, 1956 ).

(2) The mathematical equations for anisotropic wavepropagation are algebraically daunting, even for thissimple case.

The purpose of this paper is to point out tha t in most cases ofinterest to geophysicists the anisotropy is weak (l&20 per-cent). allowing the equations to simplify considerably. In fact,the equations become so simple that certain basic conclusions

are immediately obvious :(I) The m ost common m easure of anisotropy (con-

trasting vertical and horizontal velocities) is not v eryrelevant to problems of near-vertical P-wave propaga-tion.

(2) The most critical measure of anisotropy (denoted6) does not involve the horizontal velocity at all in itsdefinition and is often undetermined by experimentalprograms intended to measure anisotropy of rock sam-ples.

(3 ) A comm on approxim ation used to simplify theanisotropic wave-velocity equations (elliptical ani-sotropy) is usually inapprop riate and misleading for P-and SV-waves.

(4) Use of Poissons ratio, a s determined from verticalP and S velocities, to estimate h orizontal stress usua llyleads to significant error.

These co nclusions apply irrespective of the phys ical cause ofthe anisotropy. Specifically, anisotropy in sedimentary rocksequences may be caused by preferred orientation of aniso-tropic mineral grains (such as in a m assive shale formation),preferred orientation of the shapes of isotropic minerals (suchas flat-lying platelets), preferred orientation of cracks (such asparallel cracks, or vertical cracks with no preferred azimuth),or thin bedding of isotropic or anisotropic layers. The con-clusions stated here may be applied to roc ks with any or all o fthese physical attributes, with the sole restriction that the re-sulting anisotropy is weak (this condition is given precisemeaning below).

To establish these conclusions, some elementary facts abou tanisotropy are reviewed in the next section. This is followedby a presentation of the simplified angular dependence ofwave velocities appropriate for weak anisotropy. In the fo]-lowing section. the anisotropic parameters thus identified areused to analyze several common problems in petroleum geo-physics. Finally, further discussion and conclusions are pre-sented.

REVIEW OF ELASTIC ANISOTROPYA linearly elastic material is defined as one in which each

compon ent of stress oij is linearly dependent up on every com-ponent of strain &Irl Nye, 195 7). Since each directional indexmay assum e values of 1 , 2, 3 (representing directions X, JJ, ),

Manuscript eceived by the Editor September 9. 1985; revised manuscript received February 24, 1986.*Amoco Production Company. P.O. Box 3385, Tulsa, OK 74102.( 1986 Society of Exploration Geophysicists. All rights reserved.1954


2/13

Weak Elastic Anisotropy 1955there are nine such relations, each involving one com ponent ofstress and nine componen ts of strain. These nine equationsmay be written compactly as

where the 3 x 3 x 3 x 3 elastic modulus tensor Cijkr com-pletely characterizes the elasticity of the m edium. Becau se ofthe symm etry of stress (oij = ojJ, only six of these equationsare independent. Bec ause of the symm etry of strain (ckl = E&,only six of the terms on the right side of each set of equations(I) are independent.Hence, without loss of generality, the elasticity may be rep-resented more compa ctly with a chang e of indices, followingthe Voigt recipe:ij or k/ : 11 22 33 32=23 31=13 12=211 1 111 1 1 1 , (2)a P I 2 3 4 5 6

so that the 3 x 3 x 3 x 3 tensor Cipp may be represented bythe 6 x 6 matrix C,,. Each symm etry class has its own pat-tern of nonzero, independen t componen ts C,,. For example,for isotropic media the matrix assumes he simple form

where the three-direction (2) is taken as the unique axis. It issignificant that the generalization from isotropy to anisotropyintroduces three new elastic moduli, rather than just one ortwo. (If the physical cause of the anisotropy is known, e.g.,thin layering of certain isotropic media, these five moduli maynot be independent after all. However, since the physical causeis rarely determined, the general treatment is followed here.) Acompa rison of the isotropic ma trix, e quation (3), w-ith the an-isotropic matrix, equation (5), shows how the former is a de-generate sp ecial case of the latter. with

C 11-t c,, (64c hh CM

isotropy. (6b)

( 13- (3, - 2 C,, (6~)The elastic modulus matrix C,, in equation (5) may be used toreconstruct the tensor Cljkl using equation (2), so that theconstitutivc relation in equation (1) is known for the aniso-tropic medium.

The relation may be used in the equation of motion (e.g.,Dairy and Hron. 1977; Keith and Crampin, 1977a, b, c), yield-ing a wave equation. There are three independent solutions--

c,, =L

-c,,C,? - X4,) (C,~_ 2.x,,)c 3 (c.33 2Cw)C 3 I isotropy. (3 )c44 I _C 4 c 4Only the nonzero components in the upper triangle areshown: the lower triangle is symmetrical. These componentsare related to the Lame parameters J. and u and to the bulkmodulus K by

an d (4 )C,, = u.

The simplest anisotropic case of broad geophysical applica-bility has one distinct direction (usually, but not always, verti-cal), while the other two directions are equivalent to eachother. This case- called transverse isotropy, or hexago nalsymm etry -is the only one considered explicitly h ere (al-though the present approac h is useful for any sy mme try).Hence, subs equent use of the term anisotropy refers only tothis particular case.

The elastic modulus matrix has five independent compo-nents among twelve nonzero components, giving the elasticmodulus matrix the form

C,,, =

one quasi-longitudinal, one transverse, and on e quasi-transverse for each direction of propagation. The three are@ariLe d in mutu ally orthogonal directions. The exactlytransverse wave has a polarization vector with no compo nentin the three-direction. It is denoted by SH: the other vector isdenoted by Sk Daley and Hron (1977) give a clear derivationof the directional d ependence ofthe three phase velocities:

CA.3 c,, + (C, , ~ C,,) sin 0 -t O(0)1 (7a)c,,c,, C, C,,)in 0 - O (Q )I ; (7b)an dpr


3/13

1956 Thomsen

Phase (Wavefront) Angle 0and Group (Ray) Angle 4

wave vector Z I

FIG. 1. This figure graph ically indicates the definitions ofphase (wavefront) angle and group (ray) angle.

D(O) is compact notation for the quadratic combination:D(H) = c,, - c&J2i +44)2-(C33-C44)(C11Cc,,-2C,,) sin2 B

(C,,+c,-7(,)2~4(C,3+C44)2(7d)

(note misprint in the corresponding expression in Daley andHron, 197 7). It is the algebraic complexity of D which is aprimary obstacle to use of anisotropic models in analyzingseismic exploration data.

It is useful to recast equations (7aH 7d) (involving five elas-tic mod uli) u sing notation involving only two e lastic mod uli(or equivalently, vertical P- and S-wave velocities) plus threemeasu res of anisotropy. Th ese three anisotropies should beappropriate combinations of elastic mod uli w hich (1) simplifyequations (7); (2) are nondimensional, so that one may speakof X percent P an isotropy, etc.; and (3) reduce to zero in thedegenerate case of isotropy, a s indicated by relations (6), sothat materials with small values (+ 1)of anisotropy may bedenoted weakly anisotropic.

Some suitab le combinations a re suggcstcd by the form ofequations (7):

Cl36 c44? ?C :44an d

3-

(W

2C44)1(W

The utility of the factors of two in definitions (8aW8d) will beevident shortly. The definition of equation (8~) is not u nique,and it m ay be justified only as in the case considered next,where it leads eventually to simplification. The vertical soundspeeds for P - and S-waves are, respectively.

an dC-W

Then, equations (7) become (exactly)

1 + c sin 0 + D*(8)1 (lOa)*3c,&(O) = pi [ 1 + $ E sin 0 - 5 o*(O) I ;0 0 (lob)*

a:,(H)=P~ [1+Zrsine].with

4(1 - @;/Cl:,+ E)E+ (1 - Pb4)2 (lOd)*

Before considering the case of wea k anisotrop y, it is impor-tant to clarify the distinction between the ph ase angle 0 andthe ray angle C$ along wh ich ene rgy propagates). Referring toFigure 1, the wavefront is locally perpe ndicular to the propa-gation vector k. since k points in the direction of maximumrate of increa se in p hase. The phase velocity r(0) is also calledthe wavefront velocity, since it me asures the velocity of ad-vance of the wavefront along k(B). Since the wavefront is non-spherical, it is clear that 0 (also called the wavefront-normalangle) is different from 4, the ray angle from the source pointto the wavefront. Following Berrym an (19 79), these relation-ships may be stated (for each wave type) in terms of the wavevector

k = k,% + k,i, (ll)*where the components are clearly

k, = k(0) sin 0; (I la)*k, = k(B) cos 0; (1 b)*

an dk,=O;

and the scalar length isk(B) = jm =w/v(e), (I lc)*

where w is angular frequency. The ray velocity V is then givenThis and other expressions below which are marked with an asteriskare valid for arbitrary (not just weak) anisotropy.


4/13

Weak Elastic Anisotropy 1957

by(la*

where the partial derivatives are taken with the other compo-nent of k held constant. Becaus e of the similarity in formbetween equation (12) and the usual expression for group ve-locity in a dispersive medium , V is also called the group veloc-ity and G$s the group angle. From Figure 1, 4 is given by

(13a)*

Berryman (1979) also shows that the scalar magnitude V ofthe group velocity is given in terms of the phase velocitymagnitude 1by

(14)*At 8 = 0 degrees and 0 = 90 degrees, the second term in equa-tion (14) van ishes. so that for these extreme angles (verticaland horizontal propaga tion, respectively), group velocityequals phase velocity.

WEAK ANISOTROPYAll the results of the previous section are exac t, given equa-

tions (1) and (5). However, the algebraic complexity of equa-tions (IO) impedes a clear understanding of their physical con-tent. Progress may bc m ade. however, by observing that mostrocks are only weakly anisotropic. even though many of theirconstituent minerals are highly anisotropic.

Table I presents data o n anisotropy for a num ber of sedi-mentary rocks. The original d ata con sist (in the labo ratorycases) of ultrason ic velocity me asuremen ts or (in the fieldcases)of seismic-band velocity mea suremen ts. These data wereinterpreted by the original investigators in terms of the fiveelastic mod uli of transverse isotropy. In T able 1, these modu liare recast in terms of the vertical velocities IX,,, and PO, andthe three anisotropies E, 15*, and y defined above. Also, afourth measure of anisotropy (6, defined below as a moreuseful alternative to 6*) is shown. As seen in the table, thesequantities provide an immediate estimate of the three types ofanisotropy that are unavailable by simple inspection of themodu li themselves. Table I co nfirms that m ost of these rockshave anisotropy in the weak-to-moderate range (i.e., ~0.2). asexpected. The table also shows data for some common crys-talline minerals (which in some cases are strongly anisotropic)and for some layered composites.

The listing of measurements of sedimentary rock anisotrop yin Table I from the literature is nearly exhaustive. Howev er,perhaps twice as many partial studies of anisotropy (me a-suring vertical a nd horizontal P an d S velocities) have ap-peared. It is clear that the requisite five parameters may not be

obtained from four measurements (at least one measurementat an oblique angle, preferably 45 degrees, is required). As isshown below, this omitted da tum is the most important onefor most a pplications in petroleum geophysics. Hence, thesepartial s tudies are omitted from Table 1.

In addition to intrinsic anisotropy, one must consider ex-trinsic anisotropy, for example, due to fine layering of iso-tropic beds. Many examples could be listed, but it is not clearhow to pick representative examples. This table has been lim-ited to the particu lar exam ples defined by Levin (1 979 ) (thesechoices are discussed urther below).

With Table I as justification of the approximation of weakanisotropy. it now make s sense to expand equations (10) in aTaylor series in the small parameters E. 6*, and y at fixed 0.Retaining on ly terms linear in these small param eters, thequad ratic D* is approxim ately

s in 0 co? 0 + E sin4 0. (15)Substituting this expression into equations (1O a) and (lob) andfurther linearizing yields a final set of phase velocities that isvalid for weak anisotropy:

L;~6) = a, (1 + 6 sin 8 co? 0 + E sin4 O), (16a)v,,(B) = POI d+ -5 (E - 6) s in 6 cos2 000 1 WW

an dcSH 9) = PO I + y sin 0) . (16~)

Equations (16) display the required simplicity of form. Theyhave been arranged so that, for s mall angles 8, each successiveterm contributes to the total by successivelysmaller orders ofmagnitude. This relation leads to replacement of the (initiallydefined) anisotropy pa rameter 6* in equation (8~ ) by

(Cl, + G.J2 cc,, cd= 2C,,(C,, - C,,) (17)$The parameter 6* is not required further.

From Table 1, note that a ll three anisotropies E, 6, and y areusually of the sam e order of m agnitude. Becau se of this, it isclear from equation (16a) that at small angles 0 w heresin cos is not nearly as small as sin4, the second term (in F) isnot nearly as small as the last term (in E). It is in this regime(small 0) where most reflection profiling takes place. Hencethis 6 term will domina te m ost anisotropic effects in this con-text, unlessE $ 6 in some particular case.

The trigonometric factor cos 0 in the second term of equa-tion (16a) rather than (1 - sin 0) appears by design. Thisfactor e nsures that the angu lar depend ence of t?,(e), at n ear-horizontal propaga tion, is clearly dominated by E (sincecos n,2 = O ), just as the n ear-vertical p ropagation is domi-nated by 6. In fact. at horizontal incidence,

U,(K/2) = Cl,(l + E). (184Since a0 is the vertical P velocity, it is now abundantly clear


5/13

1958 ThomsenTable 1. Measured anisotropy in sedimentary rocks. This table comp iles and condense s irtually all published data o nanisotropy of sedim entary rocks, plus some related materials.

Sample Conditions

TaylorsandstoneMesaverde (4903j2mudshale :nD = 27.58 MPa, undrndMesaverde (4912j2immature sandstone 1%

= 27.58 MPa, undrnd

Mesaverde (494612immature sandstone kE 6= 27.58 MPa

, undrndMesaverde (5469.5)silty limestone kt ;

= 27.58 MPa, undrnd

Mesaverde (5481.3)immature sandstone 1% = 27.58 MPa, undrndMesaverde (5501)clayshale LB = 27.58 MPa, undrndMesaverde (5555.5)immature sandstone k

= 21.58 MPa, undrnd

Mesaverde (5566.3j2laminated siltstone L& 6 = 27.58 MPa, undrndMesaverde (5837.5)immature sandstone P&E = 27.58 MPa, undrnd

Mesaverde (5858.6)clayshale = 27.58 MPa 12 448 6 804 0.189, undrnd 3 794 2 074Mesaverde (6423.6)calcareous sandstone ~cE~,=u~~~~~ MpaMesaverde (6455.1) P

s% = 27.58 MPaimmature sandstone , undrndMesaverde (6542.6) P

sEEi= 27.58 MPaimmature sandstone , undrnd

Mesaverde (6563.7) PS% i

= 27.58 MPamudshale , undrndHesaverde (7888.4j2 P

&d= 27.58 MP~sands tone , undrnd

Mesaverde (7939.5) Ps,eD

= 27.58 MPamudshale , undrnd

v (f/s)m/s) v (f/s) Em/s)11 050 6 000 0.110

3 368 1 a2914 860 8869 0.0344 529 2 70314 684 9 232 0.097

4 476 2 81413 449 7 696 0.077

4 099 2 34616 312 9 512 0.056

4 972 2 89914 2704 349 a 434 0.0912 57112 887 6 742 0.3343 928 2 05514 891 8 877 0.060

4 539 2 70614 596 8 482 0.0914 449 2 58515 327 9 294 0.0234 672 2 833

17 914 10 560 0.0005 460 3 21914 496 a 487 0.0534 418 2 58714 451 8 339 0.080

4 405 2 54216 644 9 a37 0.0105 073 2 99815 973 9 549 0.0334 869 2 91114 096 8 106 0.081

4 296 2 471

6 Y p(g/cm3>

-0.127 -0.035 0.255 2.500

0.250 0.211 0.046 2.520

0.051

-0.039

-0.041

0.134

0.818

0.147

0.688

-0.013

0.154-0.345

0.173

-0.057

0.009

0.030

0.118

0.091 0.051 2.500

0.010 0.066 2.450

-0.003 0.067 2.630

0.148 0.105 2.460

0.730 0.575 2.590

0.143 0.045 2.480

0.565 0.046 2.570

0.002 0.013 2.470

0.204 0.175 2.560-0.264 -0.007 2.690

0.158

-0.003

0.012

0.040

0.129

0.133 2.450

0.093 2.510

-0.005 2.680

-0.019 2.500

0.048 2.660

Rai and Frisillo, 1982Kelley, 1983 (number in parentheses is depth label)


6/13

Weak Elastic AnisotropyTable I. Continued

1959

Sample

Mesaverdeshale (350j3

Pd:;f

= 20.00 MPa 11 100 8000 0.0653 383 2 438

Mesaverdesandstone (1582j3 Pd:;f = 20.00 MPa 12 1003 688 9 100 0.0812 774Mesaverdeshale (1599j3

Pdf;f

= 20.00 MPa 12 800 8 800 0.1373 901 2 682

Mesaverdesandstone (1958)

Pd:Ff

= 50.00 MPa 13 900 9 900 0.0364 237 3 018

Mesaverdeshale (1968)

Pd:Ff

= 50.00 MPa 15 900 10400 0.0634 846 3 170

Mesaverdesandstone (3512)

Pd;if

= 50.00 MPa 15 200 10600 -0.0264 633 3 231

Mesaverdeshale (3511)

Pd$f

= 50.00 MPa 14 300 10 000 0.1724 359 3 048

Mesaverdesandstone (3805)"

Pd:cf

= 20.00 MPa 13 000 9 600 0.0553 962 2 926

Mesaverdeshale (3883)"

Pd:;f

= 50.00 MPa 12 300 8 600 0.1283 749 2 621

Dog Creek4 in situ, 6 150 2 710 0.225shale 2 = 143.3 m (430 ft) 1 875 82 6Wills Point in situ, 3 470 1270 0.215shale 2 = 58.3 m (175 ft) 1 058 38 7

Cotton Valleyshale

= 111.70 MPa 15 490 9480 0.135, undrnd 4 721 2 890

Pierre in situ, 6 804 2 850 0.110shale z = 450 m 2 074 86 9

Conditions

in situ, 6 910 2 910 0.195z = 650 m 2 106 807in situ, 7 224 3 180 0.0152 = 950 m 2 202 96 9P = 3~, :o 300sitd, undrnd 3 048

v (f/s)m/s>

v (f/s) t:m/s 1

13 550 7 810 0.0854 130 2 380

fj Y p(g /cm )

-0.003 0.059 0.071 2.35

0.010 0.057 0.000 2.73

-0.078 -0.012 0.026 2.64

-0.037 -0.039 0.030 2.69

-0.031 0.008 0.028 2.69

-0.004 -0.033 0.035 2.71

-0.088 0.000 0.157 2.81

-0.066 -0.089 0.041 2.87

-0.025 0.078 0.100 2.92

-0.020 0.100 0.345 2 .ooo

0.359 0.315 0.280 1.800

0.104 0.120 0.185 2.640

0.172 0.205 0.180 2.640

0.058 0.090 0.165 2.25?

0.128 0.175 0.300

0.030

0.480

2.25?

0.085

-0.2~70

0.060 2.25?

-0.050 2.420

Lin, 1985 (number in parentheses is depth label)4Robertson and Corrigan, 1983Tosay a, 198 2GWhite, et al., 19827Jones and Wang, 1981 (depth of core sho wn)


7/13

ThomsenTable 1. Continued

1960

Sample

Oil Shale*

Green River9shale

Berea'csandstoneBandera'osandstoneGreen River" P = 202.71 MPa, 10 800 5 800 0.195shale a?r dry 3 292 1768Lance" P = 202.71 Mpa, 16 500 9800 -0.005sandstone a?r dry 5 029 2 987Ft. Union" P = 202.71 Mpa, 16 000 9650 0.045siltstone a?r dry 4 877 2 941Timber Mtn" P = 202.71 Mpa, 15 900 6090 0.020tuff afr dry 4 846 1856Muscovite"crystalQuartz crystall(hexag. approx.)Calcite crystal'2(hexag. approx.)Biotite crystall

Apatite crystal"

Ice I crystal"

Aluminum-lucite':j clamped; oil 9 410 4430 0.97composite between layers 2 868 1350

Conditions v (f/s)'(m/s)

P = 101.36 MPa 11080s&d, undrnd 3377unknown 13880

4231p 0, 13670s&d, undrnd 4167P = 68.95 MPa, 14450s&d, undrnd 4404;gEsr;ti;.95 Mpa, 12;;;

;sEir;t8i.95 Mpa, 12 5003 810

P =oc

Peff = 0

P eff = 0

Peff = 0

Peff =0

P. =o,eff 4OF

7 770 0.0302 368

14 500 6860 1.124420 2091

20000 14700 -0.0966 096 4481

17 500 11000 0.3695 334 3 353

13 3004 054

4400 1.2221341

20 800 14400 0.0976 340 4 389

11900 5 500 -0.0383627 1676

v (f/s) 6'(m/s)

4890 0.20014908330 0.2002 5397 980 0.0402 4328470 0.0252 5828 740 0.0022 664

-0.282

0.000

-0.013

0.056

0.023

0.037

-0.45

-0.032

-0.071 -0.045 0.040 2.600

-0.003 -0.030 0.105 2.330

-1.23 -0.235 2.28 2.79

0.169 0.273 ,0.159 2.65

0.127 0.579 0.169 2.71

-1.437 -0.388 6.12 3.05

0.257 0.586 3.218

-0.10 -0.164

0.079

0.031

1.30

1.064

-0.89 -0.09 1.86

6

-0.075

0.100

0.010

0.055

0.020

0.045

-0.220

-0.015

d p(glcm3)

0.510 2.420

0.145 2.370

0.030 2.310

0.020 2.310

0.005 2.140

0.030 2.160

0.180 2.075

0.005 2.430

'Kaarsberg, 1968'Podio et al., 1968

"King, 1964"Schock et al., 1974'*Simmons and Wang, 1971


8/13

Weak Elastic Anisotropy 1961

Table 1. Continued

P(g/cm3)ample Conditions 6--

-0.010

v (f/s)p(m/s) v (f/s)S(m/s)Sandstone- hypothetical 9 871 5 426shaleIll 50-50 3 009 1 654

r.

0.013

6

-0.001

Y__-

0.035 2.34

0.059 -0.042 -0.001 0.163 2.34

0.134 -0.094 0.000 0.156 2.44

0.169 -0.123 0.000 0.271 2.44

0.103 -0.073

-0.002

-1.075

-0.001 0.345 2.34

0.022 0.018 0.004 2.03

1.161 -0.140 2.781 2.35

SS-a nisotro pic hypothetical 9 871 5 426shale 50-50 3 009 1 654Limestone- hypothetical 10 845 5 968shale 50-50 3 306 1819LS-anisotropic hypothetical 10 845 5 968shale 4 50-50 3 306 1819Ani sot ropic hypothetical 9 005 4 949shale 50-50 2 745 1 508Gas sand- hypothetical 4 624 2 560wate r sand14 50-50 1 409 780Gypsum -weathered hypothetical 6 270 2 609ma teria l 50-50 1911 79 5

Dalke, 198314Levin, 1979

cusps or triplications are present in the limit of weak ani-sotropy. (The term I,, in these figures is discussed in thenext section.)

At this point, where the linearization procedure h as identi-tied F as the crucial anisotropic parameter for near-verticalf-wave propaga tion, it is appropriate to discuss a special caseof transvcrsc isotropy which has received much attention: el-liptical anisotropy. An elliptically anisotropic medium ischaracterized by elliptical P wavefronts emanating from apoint so urce. It is defined (cf.. Daley and H ron, 197 9) by thecondition

6=c elliptical anisotropy.Notable for its algebraic simp licity. this special case, is, ofcourse. detincd by a mathetnatical restriction of the parame-ters which has no physical justification. Accordingly, one maycxpcct the occurrence of such a cast in nature to be van-ishingly r;lrc_ Ins act,~ able 1~ hows !hat 6 and _c are not evenwell-correlated (being frequently of opposite sign), so that theassum ption of their equality m ay lead to serious error. Thispoint is rcinforccd by F igure 4 , which plots 6 versus c for therocks of Table I and shows for comparison the elliptic con-dition dcftncd above. The inadequacy of the elliptic assump-tion is immed iately ob vious.

These results are for intrinsic anisotropy. B erryman (197 9)and Helbig (1979) show that if anisotropy is caused by tinelayering of isotropic materials, then strictly

8 0 (Table 1). How-ever, this is of little use in understan ding anisotropy in near-vertical reflection problems, b ecause E is multiplied by sin4 8in equation (16a). The near-vertical anisotropic response isdominated by the 6 term, and few can claim m yth intuitivefamiliarity with this comb ination of parame ters [equation(17)]. In fact, Table 1 shows a substantial fraction of caseswith negative 6.

Figures 2 and 3 show P wavefronts radiating from a pointsource into two uniform half-spaces, each with p ositive E butdilTerent values of 6. one positive and one negative. These arejust plots of I/,($) in polar coordinates. It is clear from thefigures that quite complicated wavefronts may occur. Similarcomplications arise with SL wavefronts. although no actual


9/13

1962 ThomsenWAVEFRONTS__._~~~._~~~

I

6 = 0.20- NM0

FIG. 2. This figure indicates an elliptical wavefront (6 = E). Th ecurve marke d V,,, is a segment of the wavefront that wouldbe inferred from isotropic mov eout analysis of reflectedenergy. VkMo> V,,,,

WAVEFRONTS

ANISOTROPIC:E = 0.20

FIG. 3. This figure indicates a plausible anisotropic wavefront(6 = -E). The curve marked VNNMos a segment of the wave-front that would be inferred from isotropic moveout analysisof reflected energy. V,,, < V>_,since 6 < 0.

As a final remark. note that. for small 0 the last term inequation (16a). E sin 0. might be comparable to a neglectedquadratic term in 6 sin 0, or 6c sin 8. However, all neglect-ed terms quadratic in anisotropy are multiplied by trigono-metric terms of order sin 0 co? 0 or smaller. and hence are infact negligible, even for small 0.

Rerryman (19 79) writes a perturbation approximation toequation (10) in which the small parameter is a combinationof anisotropic param eters and trigonometric functions. Hisderivation, which is also vfalid for strong anisotropy at smallangles. reduces to the present equations (16 a) and (16b ) forweak aniso tropy (at any angle). His a pproxima tion is less re-strictive than the present one, but it yields formulas which areless simple (and w hich do not readily d isclose the crucial roleof the parameter 6, or contrast it with c ). It is therefore anapproxim ation intermediate between the exact expressions(IO) and the intuitively accessible approxim ation (16).

Backus (1965) treats the case of weak anisotropy of arbi-trary sym metry, detining anisotropy difterently than is donehere, M Gthou t imple,menting criteria ( !) an d (2) wh ich followsequation (7d).

Consttleration of the linearized SV result equation (16b)immed iately confirms the well-known special result that ellip-tical P wa vefronts (6 = c) imply spherical SV wav,efronts (no 0dependence). However, equ ation (Ibb) shows that the moregeneral case of weakly a nisotropic but none lliptical med ia isstill algebraically tractable.

For completeness, note from equa tion (16 ~) thatls11 71/2) BoY= so

so that y corresponds to the conventional meaning of SHanisotropy. Also note that in the elliptical case 6 = E, thefunctional form of equation (16a) becomes the same as that ofequation (16~).This demonstrates that SH wavefronts are el-liptical in the general case; this is true even for strong ani-sotropy.

Returning to the central point of 6 as the crucial parameterin near-vertical anisotropic P-wave propaga tion, some dis-cussion is necessary regarding the reliability of its measu re-ment. It is clear from equations (16a) and (18b) that 6 may befound directly (in the case of weak aniso tropy) from a singleset of measurements at 0 = 0.45 and 90 degrees:

6 = 4FP(rti4),VP(0) - 1 - VP(7t~2)WP(0) II[ IBecau se of the factor of 4. errors in VP(x/4)/Vr,(0) propagateinto F with con siderable magnification. In fact, if the relativestandard error in each velocity is 2 percent. then the (indepen-dently) propagated absolute standard error in 6 is of order .12,which is of the same order as 6 itself (Table 1). The propaga -tion of this error through equation (17) implies that the rela-tive error in C,, is even larger. To reduce these errors towithin acceptable limits requires many redundant experiments,of both V, (fl) and V&. 0). Since the measuremen t at 45 degreesmay involve cutting a separate core, questions of sample het-erogeneity (as distinct from anisotropy) naturally arise. Thedata of Table 1 should be viewed with appropriate caution.

SOME APPLICATIONS OF WEAK ANISOTROPYGroup velocity

For the quasi-P-wave, the derivative in equation (14) isgiven for the case of weak anisotropy [equation (I 6a)] by

u; (0) sin Clcos 0 6i;o + 2(E - 6) sin 0np 1 (19)i.e., it is linear in an isotropy. Therefore, the group velocity[equation (14)] expanded in such terms,

is quadratic in anisotropy. Therefore, this term is neglected inthe linear approximationV,(4) = n,(e). (20aj

Similarly for the other wave types,V& (4) = 2s (8); (20bj

an dVW (4) = t.SH(@). (2Oc)

Note that equations (20) do notsay that group velocity equalsphase velocity (or equivalently, that ray velocity equals wave-front velocity). These equations do say that at a given ray


10/13

Weak Elastic Anisotropy 1963(group) angle +, if the corresponding wavefront normal(phase) angle 0 is calculated using equations (22) below, thenequations (16) and (20) may be used to find the ray (group)velocity.Group angle

The relationship (13) between group angle C$ and phaseangle 0 is, in the linear approximation.

I 1 dc--sm 8 cos 0 ~(0) dO1 (21)For P-waves, use of equation (19) fully linearized) in equation(21) leads to

tan $P = tan 8, 1 + 26 + 4(& - 6) sin 8,L I . (224Similarly, for SV-waves,tan +sy = tan El,, L

4l+?p(E-&)(I--2sin*(1,,) ;I CW0and for SH-waves,

tan & = tan &(l + 2~). (22c)These expressions, along w ith equations (16 ) and (20), definethe group velocity, at any angle, for each wave type.

Note that inclusion of the anisotropy terms in the angles[equations (22)], when used in conjunction with the phase-

velocity formulas [equation (16)], does not constitute a viola-tion o f the linearization process, even though p roducts ofsmall quantities implicitly appear. In linearizing equa tions (10)in terms of anisotropy, the angle 0 was held constant, i.e., wasnot pa rt of the linearization process. The linear dependen ce of0 on anisotropy. at fixed $, is then given by equations (22).Polarization angle

The particle motion of a quasi-P-wave is polarized in thedirection of the eigenvector g, (Daley and Hron , 1977 ), where

g,(B) = FP sin O,% + mP cos 0,&. *Since this is not parallel to the propagation vector k, [equa-tion (1 )], the wave is said to be quasi-longitudinal, ratherthan strictly longitudinal; similar remarks apply to the quasi-SV-wave. The ang le j, between k, and g,is given by

cos 6, = & (k,, g,) = & VP sin 8, + inP co? 0,). *PC PExpressions for the sc alars /,, and m P are given by Daley andHron (197 7); in the case of weak an isotropy, these expressionsreduce lo

an d/@ = I + A/,;

Comparison of P-Anisotropies0.4

0.8

0.4

0. 2

0.0

-0.2

-0.4 _

I-

I-

I--

aData onCrustal Rocks(Lab & Field)

I I I I I I-0.2 0.0 0.2 0.4 0.8 0.8Anisotropy Parameter EFIG. 4. This figure indicates the noncorrelation of the two anisotropy parame ters 6 and E for the data in Table 1.


11/13

1964 Thomsen

- --

HomogeneousAnisotropicElasticMedium

FIG. 5. A cartoon showing a simple reflection experimentthrough a homogeneous anisotropic medium.

where A/,, and Am, are linear in anisotropies E and 6. Itfollows directly thatcos & = 1.

i.e., C,,= 0, and in the linear approxim ation the departure ofthe polarization direction g, from the wave vector k, is negli-giblc. Of course, g, deviates from the ray by the amount de-fined in equation (2 2a). This c onclusion appears to disagreewith a result by Backus (1965).

Correspond ing rem arks apply to the shear polarization vec-tors: they are each transverse to the corresponding k, in thecase of weak an isotropy. The polarization of each S-wave de-viates from the normal to the ray direction by the amountdefined in equation (22b).Moveout velocity

Consider a homog eneous anisotropic layer through w hich aconventional reflection survey is performed (Figure 5). Theraypath to any offset x(+) consists of two straight segmen ts, asshown in the figure. The traveltime r is given (trivially) as

*

where r is the vertical traveltime. So lving for 1

Becau se of the 4 dependence, the function in eq uation (2 3)plots along a curved line (instead of a straight line) in theI* - .x plane. The slope of this line is

fir* 1 f f/V2_=~d?? V2(4J) v ds

Norm al-moveou t velocity is defined using the initial slope of

this line:&=liyI($)=&[l -&*I,. (25)*

(This is, of course, the short-spread I/NM0 ) The second term onthe right is generally not zero. Hence it is clear from equation(25) that, even in the limit of small s offsets (i.e., with all wavespropaga ting nearly vertically), with all velocities near V (O), theresulting moveout velocity is not the vertical velocity V(0).Carrying out the derivative [using equation (13)] is alge-braically tedious. but straightforward. For P-waves, the deri-vation is

L,,(P) = a,, J-TZ, (26a)*independent of c. The value of this velocity is indicated inFigures 2 and 3 by a short arc below the origin. This linerepresents a segment of that wavefront which w ould beinferred by an isotropic analysis of a surface reflection experi-ment such as that depicted in Figure 5.

F-or S V-waves,

1Ii2 ; (26b)*

for SH-waves (which have elliptical wavefronts)VW, (SW = so &=Y

= L$,l(X/2) = V,,(n/2). (26c)*The last result (for SH) does not require the limit x + 0. i.e..the I - x2 graph is a straight line, as for isotropic media. Thisis a well-known result for elliptical wavefronts (Van der Stoep,1966 ), as is the fact that the resulting V,,, is identical to thehorizontal velocity, even though all ra ys are near-vertical.

For weak anisotropy, equations (26) reduce toV,,,(P) = a,(1 + 6); (27a)vN,,(sv~=P,[l+q (274

an dV,,oW) = Po(l + Y). (27c)

Comparison of equations (27a) and (18a) shows that, for P-waves, the moveout velocity is equal to neither the verticalvelocity u,, nor the horizontal velocity a,(1 + E). Neither isthe moveou t velocity nece ssarily intermediate between thesetwo values, since F and c m ay be of opposite sign (Table 1L

Equation (26a) is equivalent to expression (3) of Helbig(198 3). In the present version, the departure of V NMo /u, fromunity is clearly related to the same anisotropic parameter 6which app ears so prominently in other applications.Horizontal stress

One way to estimate the horizontal stress in the sedi-mentary crust of the Earth is to describe an element of rock atdepth as a linearly elastic medium in un iaxial strain (Hubertand W illis, 1957 ). Despite the obvious shortcom ings of thisapproxim ation (e.g., the difference between static and dynamicmod uli, L in, 198 3, it remains widely used as a means to esti-mate ho rizontal stress for hydrofracture control, etc.


12/13

Weak Elastic Anisotropy 1965The analysis is normally done in terms of isotropic med ia; it

is instructive to consider the same problem in anisotropicmedia. Here, anisotropy still is taken to mean transverseisotropy with symmetry axis vertical, even though this sort ofanalysis is usually done in a context of preferred horizontalstress direction, resulting in an o riented h ydrofracture. T hismay often be justified, since the resulting azimutha l anisotropyis usually of the order of 1-2 percent, whereas the convention-al anisotropy is frequently 10-20 percent.The vertical stress crj3 and the ho rizontal stress ol, are,from equation (l), related by

an d0 33 = C31c11+ c32 E22 + c33 E33 GW

(311 = C,,a,, + C,z%* + C13c33. (28b)The other s train terms vanish because of null va lues of C,,.(In the application to hydrofracturing, these stresses are re-placed by effective stresses; however, the following argum entis not affected.)

In the case of uniaxial strain, &t, = cJ2 = 0, so that thehorizontal stress s

cIs II =(J33 13C 33The vertical stress s due to gravity:

033 = - Lw?

where y is the acceleration due to gravity and z is the depth.In the isotropic case, the ratio of elastic modu li [equation (4)]may be expressed n sereral equivalent ways:

(JII c i_=!.A \ K -f~ b203.1 c =-zz-=-= ] -2-J. + 2u (30)33 l-v K+jp a2where u and 8 are the velocities of P-waves and S-waves.respectively, and v is Poissons ratio. Hence, a and fi could bemeas ured in situ and, given the assu mptions just~stated, cri icould be estimated using equations (29)(30).

In the anisotropic case, the corresponding exp ression is,from equation (17)

For weak anisotropy. equation (31) reduces to011 C_=_!2=033 C 33 ( >-2@ +&a; (32)

Comparison with the last formulation of equation (30) showsthat the anisotropic correction is given simply by the ani-sotropy parame ter 6. In a typical case. P,/a, z 0.5, so that thefirst term in equation (32) is also ~0.5. Table I shows that 6 isoften not negligible in comparison to 0.5; the correction maybe either positive or negative. Therefore, use of the isotropicmodel, equation (3 0) may lead to serious overestimates orunderestima tes of horizontal stress [equation (29)]. Th ese

errors may reinforce or reduce errors due to failure of otheraspects of the mode l of elastic uniaxial strain.

DISCUSSIONThe ca sual term the anisotropy of a rock us ually means

the quan tity E, calculated using equation (18b ). It is oftenimplied that, if E and the vertical velocity a, are known, thevelocity at o blique angles is calculable simply b y using sometrigonometric relations. Of course, this assumption is not true;the P velocity at o blique ang les requires knowledge of a thirdphysical parameter, in addition to the trigonometric functions.Equation (16a)shows that, for we akly anisotropic m edia, therelevant third parameter is the anisotropic parameter 6. Theequation further shows that, for near-vertical P-wave propa-gation, the 6 contribution completely dom inates the E contri-bution. Becau se of this, 6 (rather than E) controls the aniso-tropic features of most situations in exploration geophysics,including the relationships among ray ang les, wavefrontangles. and p olarization angles and the move out ve locity forP-waves, and the horizontal stress-overburden ratio.

With todays compu ters. there is little excuse for using thelinearized equations (16) for compu tational purposes, evenwhen the anisotropy is so small that their numerical accuracyis high. All program s should be written using the exact equa-tions (IO) or (7). The linearized equations are useful becausetheir simplicity of form aids in the understanding of the phys-ics. For example, in a forward modeling program, a routinemay b e able to predict data for comparison with real datathat seem to call for an anisotropic interpretation. A primaryobstacle is that few geophysicists are prep ared to propose~rea-sonable values for the five different C,, required by the pro-gram, or to adjust values iteratively to m atch the real data.however most interpreters can propose reasonable values ofvertical velocities a0 and b. from direct experience with iso-tropic ideas. Further. most are prepared to estimate the valuesof anisotropies E (and */ if needed); the sign (+) and gen eralmagnitude (020 percent) are commonplace. That leaves onlyd (for a P or SC problem), and the linearized equations (16)imply that determination of 6 is where iterative adjustmentsshould be concentrated. since the value of 6 is probably themost crucial. Table 1 illustrates its range of values, extendinginto both the positive and negative ranges.

Table I also provides a guide for construction, for mo delingpurposes. of an equivalent anisotropic med ium from finelylayered isotropic media. A tempting simplification is toassum e a con stant P oissons ratio amon g these isotropiclayers (Lcvin, 197 9). It is easy to show analytically (Back us,1962 ). as verified num erically in the corresponding entries ofTable 1. that assum ption of constant Poissons ratio leads to6 = 0. While this value is indeed plausible, nonzero v alues ofeither sign are also plausible. This particular choice happensto minim ize the resultant anisotropic effects for P-waves. Theassum ption of constant Poissons ratio is, therefore, a danger-ous one. The hypothetical sequencesshown in Table 1 shouldnot be taken as representative for any actual area withoutcareful justification. (These comments are entirely consistentwith those of Thomas and Lucas, 1977, and of course with thecalculations of Levin, 197 9.)

A second point that deserves further em phasis is thatweak anisotropy (defined as E, 6, y CC ), by definition, leads


13/13

1966 Thomsento second-order corrections whenever the anisotropy is com-pared to unity, as in equa tions (16). However, the anisotropysometimes occurs in a context where it is comparable, not tounity, but to another small number [e.g., equation (32)]. Inthis case, the anisotropy makes a first-order contribution,rather than a second-order correction (even though it is de-fined as weak), and sh ould therefore not be neglected. Othercomm on contexts of interest in exploration geophysics wherethe anisotropy appears in this way as a first-order effect willbe the subject of future contributions.

ACKNOWLEDGMENTSI thank C. S. Rai, A. L. Frisillo, J. A. Kelley (of Amoco

Production Company), and W. Lin (of Lawrence LivermoreLabora tory) for permission to publish their data (Table 1) inadvance of its public appearance under their own names. Ithank A. S eriff (of Shell) for useful comm ents.

REFERENCESBackus, G. E.. 1962, Long-wave elastic anisotropy produced by hori-

zontal layering: J. Geophys. Res.: 7. 442744 40.~ 1965, Possibleorms of seismic anisotropy of the uppermostmantle under oceans: J. Geophys. Res., 70. 3429.Berryman, J. G:, lY79, Long-wave elastic anisotropy in transverselyisotropic media: Geophysics, 44, 8969 17.Dale!, P. F., and Hran, F.. 1977, Reflection and transmission coef-&tents for transversely isotropic media: Bull., Seis. Sot. Am., 67.h61-675.___ 1979, Reflection and transmission coefficients for seismicwaves in ellipsoidally anisotropic media: Geophysics, 44, 27738.Dalke, R. A., 1983, A model study: wave propagation in a trans-versely isotropic medium: MSc. thesis, Colorado School of Mines.Helbtg, K1, lY79. Disucssion on The reflection, refraction and dif-fraction of waves in media with elliptical velocity dependence byF. K. Levitt: geophysics 44. 987-990__ 1983, Ellipi&tl anisotropy--its significance and meaning:Geophysics. 48, 825 X32.

Hubbert, M. K.. and Willis, D. B., 1957, Mechanics of hydraulicfracture: Trans., Am. Inst. Min. Metallurg. Eng., 210, 1533166.Jones, E. A., and Wang, H. F., 1981. Ultrasonic velocities in Cre-taceous shales rom the Williston basin: Geophysics, 46, 288297.Kaarsberg, E. A., 1968, Elastic studies of isotropic and anisotropicrock samples: Trans., Am. Inst. Min. Metallurg. Eng., 241,47&475.Keith. C. M.. and Cramnin. S.. l977a. Seismic body waves in aniso-tropic media: Reflection and refraction at a plane interface: Geo-phys. J. Roy. Astr. Sot.. 49, 181-208.. 1977b. Seismic body waves in anisotropic media: Propagationthrough a layer: Geophys. J. Roy. Astr. Sot., 49, 209-224.1977~. Seismic body waves in anisotropic media: Syntheticseismograms: Geophys. J. Roy. Astr. Sot., 49, 225243.King. M. S., 1964, Wave velocities and dynamic elastic moduli ofsedimentary rocks: Ph.D. thesis. Univ. of California at Berkeley.Kellcy, J. A., 1983, Amoco Production Company: Private communi-cation (Table 1).Krey, T. H.. and Helbig, K., 1956. A theorem concerning anisotropyof stratified media and its significance for reflection seismics:Geo-phys. Prosp.. 4,294302.Levin. F. K.. 1979, Seismic vzelocities n transversely isotropic media:Geophysics, 44,91&936.Lin. W., 1985, Ultrasonic velocities and dynamic elastic moduli ofMesaverde rock :Lawrence Livermore Nat. Lab. Rep. 20273, rev. 1.Nye. I . F., 1957, Physical properties of crystals: Oxford Press.Podio, A. L., Gregory, A. R., and Gray, M. E., 1968, Dynamic proper-ties of dry and water-saturated Green River shale under stress: Sot.Petr. Eng. J., 8, 389-404.Rai. c S., and Frisillo, A. L., 1982. Amoco Production Company:Private communication (Table 1).

Robertson J. D., and Corrigan. D., 1983. Radiation patterns of ashear-wave vibrator in near-surface shale: Geophysics, 48, 19-26.Schock, R. N., Banner. B. P., and Louis, H., 1974, Collection ofultrasonic velocity data as a function of pressure for polycrystallinesolids: Lawrence Livermore Nat. Lab. Rep. UCRL-51508.Simmons, G.. and Wang, H., 1971, Single crystal elastic constants andcalculated aggregate properties: A handbook: Mass. Inst. Tech.Press.Thomas, J. H., and Lucas. A. L., 1977, The effects of velocity ani-sotropv on stacking velocities and time-to-depth conversion: Geo-phys..Prosp,. 25. 58j.Toaava. C.. 1982. Acoustical moperties of clav-bearing rocks: Ph.D..thesis,Sianf~rrd~irtiv. .

Van dcr Steep, D. M.. 1966, Velocity anisotropy measurements inwells: Geophysics, 31. 9t%916.White. J. E.. Martineau-Nicoletis. l., and Monash, C. 1982, Measuredanisotropy in Pierre shale: Geophys. Prosp., 31, 709-725.

Documents

Weak Elastic Ani Sot Ropy Thomsen_86