JOURNAL OF GEOPHYSICAL RESEARCH, VOL. tot, NO. Bt, PAGES 827-835, JANUARY t0, 1996
Fracture interface waves
BoliangGu • Departmentof Materials Scienceand Mineral Engineering,University of California, Berkeley
Kurt T. Nihei and Larry R. Myer Earth SciencesDivision, Lawrence Berkeley Laboratory, Berkeley, California
Laura J. Pyrak-Nolte Departmentof Civil Engineeringand GeologicalSciences, University of Notre Dame, Notre Dame, Indiana
Abstract. Interface waves on a single fracture in an elastic solid are investigated theoreticallyand numericallyusing plane wave analysisand a boundaryelement method. The finite mechanicalstiffness of a fracture is modeled as a displacementdiscontinuity. Analysis for inhomogeneousplane wave propagationalong a fracture yields two dispersive equationsfor symmetricand antisymmetricinterface waves. The basic form of these equationsare similar to the classicRayleigh equationfor a surfacewave on a half-space, except that the displacementsand velocities of the symmetricand antisymmetricfracture interface waves are each controlledby a normalizedfracture stiffness. For low values of the normalized fracture stiffness,the symmetric and antisymmetricinterface waves degenerate to the classicRayleigh wave on a traction-freesurface. For large values of the normalized fracture stiffness,the antisymmetricand symmetricinterface waves becomea body S wave and a body P wave, respectively,which propagateparallel to the fracture. For intermediate values of the normalizedfracture stiffness,both interface waves are dispersive. Numerical modelingperformed using a boundaryelement methoddemonstrates that a line source generatesa P-type interfacewave, in addition to the two Rayleigh-typeinterface waves. The magnitudeof the normalizedfracture stiffnessis observedto control the velocities of the interface waves and the partitioning of seismicenergy among the various waves near the fracture.
Introduction fracture were observedto be sensitiveto the magnitude of the At the microscale, a fracture in rock appears as two mechanical stiffness of the fracture or, equivalently, to the surfacesof irregular topographywhich contactto form void static stressimposed on the fracture. spacesand asperitiesof contact. A fracture with a sparse Murty [1975] examined the condition for the existence of an interface wave on a nonwelded interface with a populationof asperitiesis morecompliant than a fracturewith closely spacedasperities [Greenwood and Williamson,1966; discontinuityin shear displacementacross the thicknessof the Gangi, 1978; Brown and Scholz, 1985; Hopkinset al., 1987; interface. Pyrak-Nolte and Cook [1987] extended Murty's Cook, 1992]. It has been observed that the additional analysis to the case where both normal and shear compliance of a fracture results in a variety of interesting displacementsare discontinuousacross a nonweldedinterface. elastic wave phenomena. For example, laboratory They found that a single nonwelded interface, such as transmission measurementsof normally incident P and S fracture, can support a fast (symmetric) wave and a slow waves acrossa fractureexhibit low-passfiltering of the source (antisymmetric)wave which propagatewith phase velocities 'waveletaccompanied by a small travel time delay [ Pyrak- between the Rayleigh and shear wave velocities and are Nolte et al., 1990]. In addition, laboratory measurementsof dispersive. Recent laboratory measurementsby Pyrak-Nolte elastic wave propagationparallel to a fracture have revealed et al. [1992] have confirmed the existenceof the fast and slow that a fracture can supportdispersive interface waves [ Pyrak- interface waves in a synthetic fracture in aluminum. Very Nolte et al., 1992]. In both setsof experiments,the amplitude good agreement was found between the measured and and velocity of the waves propagating across or along a predicted group velocities of the fast and slow interface waves. In the field of nondestructive evaluation, the l Now at Earth SciencesDivision, Lawrence Berkeley Laboratory, displacement discontinuity model has been used to Berkeley,California. approximatethin elastic adhesivebonds [e.g., Rokhlin, 1984; Xuand Datta, 1990]. In these models, shear displacements. Copyright1996 by theAmerican Geophysical Union. acrossthe thin bond are discontinuousby an amount that is inverselyproportional to the shearstiffness of the bond while Papernumber 95JB02846. normal displacements are continuous, as in the case of 0148-0227/96/95JB-02846505.00 Murty's [1975] model.
827 828 GU ET AL.' FRACTURE INTERFACE WAVES
medium 1 1.02 •( kz)a fracture
X 0.98 medium 2 0.96 Figure 1. Problem geometry used in the derivation of OO•"0.94 2.45 interface wave equations. 1.87 antisymmetric 0.92 1.63 interface wave 1.50 This paperinvestigates interface waves on a planar fracture symmetric in an elastic solid using plane wave and numerical analysis. interface wave The fracture is modeled as a displacementdiscontinuity 0.88 10-3 10-2 lfi 1 10ø 101 102 boundarycondition in both shear and normal displacements. ix and [z Closed-form equations for fracture interface waves are developed. The characteristicsof the dispersionand particle Figure 3. Normalized phasevelocities of the symmetricand motions of trapped and leaky interface waves are examined. antisymmetricinterface waves as a function of the normalized The partitioning of seismic energy among waves near the fracture stiffnessfor a rangeof C•,/Cs ratios. The numbersof fracture is also evaluated. 1.50, 1.63, 1.87, and 2.45 labeledon the curvesare the C•,/Cs ratios, which are equivalent to Poisson'sratios of 0.1, 0.2, 0.3 Plane Wave Analysis and0.4, correspondingly. Symbol(•z)cf is thecut-off normalized fracture stiffness above which the symmetric The seismic behavior of a thin fracture compared to the interface wave ceases to exist in the normal mode. wavelength can be well described by the displacement discontinuity model. Across such a discontinuity, seismic stresses are continuous and particle displacements are discontinuousby an amount which is determinedby the ratio on the fracturesurfaces, kx and kz arethe x andz components• of the stresson the fracture surface to the fracture specific of the fracturespecific stiffness, and subscripts 1 and 2 referto stiffness[ Kendall and Tabor, 1971; Schoenberg, 1980; Rokhlin the mediaabove and below the fracture,respectively. and Wang, 1991; Cook, 1992]. This boundary condition The generalized potentials for an inhomogeneousplane degenerates to that for a welded fracture as the fracture wave propagatingin the x direction with exponentially specific stiffnessapproaches infinity and, for two traction-free decayingamplitude in the z directioncan be expressedas surfaces,as the fracture specific stiffnessreaches zero. The displacement discontinuity model was found to accurately •l=Ale-PøøZeiøø(x/C-t)' z->O (2) predictthe frequency-dependenttransmission of a plane ,wave •2= A2ePøøZeiøø(x/C-t),z_
1.6 respectto the fracture;and when B2= B1 and A2 =-A1, the wave field is antisymmetricwith respectto the fracture. The C•,/Cs leakysymmetric ' 1.4 interfacewave •' symmetricand antisymmetricwave fields are schematically shownin Figure 2. SubstitutingB2 = -B1 and A2 = A1 into 1.2 normalsymmetric / (9) resultsin two linearly independentequations, 1.0 interfacewv• CAl+ +q2Bl=O ' \ antisymmetric 0.8 (10) interface wave i i i i 10-a 10-2 10-1 10ø 10• 102 C2sC2 ]go)AI +'•'(•'•+q BI=0. •z and • a) a symmetricinterface wave Figure 4. Normalized phasevelocities of the symmetricand antisymmetricinterface waves, including the leaky mode of the symmetricinterface wave, as a function of the normalized fracture stiffnesswhen C•,/Cs=1.53. lowersurface propagation ofa fracture for the uppermedium and
u2= O) '•A2ePeøZ-q B2 O. 11)•sm Ii 1 eqøazleiøa(x/C-t), (7) w2=o)PA2e•'ø•z +i• B2eqø•z eiOa(x/C-t), for the lower medium. Seismic stressesin the upper and lower media obtained 0.24)•sm from (6) and (7) and Hooke's Law are
Tzxl=(,02 _2ipkt c Ale_vO•z_k t +p2 Ble-qO•z eiOa(x/C-t), O. 47)•sm
- A1 B1e-qm z eim(x/C-t) 0.96)•sm0 •,=•2• • •) C 1 ' (s) z 1 ß•2=•'L•A2eP• + B2e• ei•(x/C-t) b) an antisymmetricinterface wave
w •2=•2F(2L••-• 1• A2e• z+2i•c B2e•Zei•(x/C-•)' lower surface of a fracture where X •d • are the Lame'sconstants. By substituting(6) through(8) into the boundaryconditions propagation• (equations (1)), the following four homogeneous,linear equationsare obtained, i(kx + 2/zo)p) O. 11)•asm A• C II,C2 to)s 21,to)C2 qkx)B1--•-a2+qkxB2=O, ik x,, (9a) 0.22)•asm C2s C2 Pkz A1 i(kz+ 21'ttøq) i kz,• (9b) + C BI-PkzA2-'•-a2=0, O. 43)•asm 2ip• (A1 +A2) + •'7 + q2 (B1-B2)=O, (9c) 0.85)•asm0 (9d) -'• (A1-A2) + (B1+B2) =0. Figure 5. Particlemotions of a symmetricand antisymmetric interfacewave versusdepth from a fracturewhen Cp/Cs=l.53 A comparisonof displacementsfor the uppermedium in (6) and the normalizedfracture stiffness is assigneda value of 0.1. and those for the lower medium in (7) shows that when Symbols 3,smand /•asmare the wavelengthsof the symmetric B2=-B1 and A2 =A1, the wave field is symmetric with and antisymmetricinterface waves, respectively. 830 GU ET AL.' FRACTURE INTERFACE WAVES
5.0 ß 0.8 , . 0.8
4.0 0.6 0.6 • antisymmetric/ • • 3.0 insymmetric 0.4 0.4 • • 2.0 interfacey__•.symmetric • 1.0 nterfacewave 0.2 0.2 o 0.0 i i i antisymmetric interface wave 10-3 10'2 10-1 10ø 101 0.0 , , , •0 kz and kx 10-3 10'2 104 10ø 10 kx and kz Figure 7. Radius ratios of the particle motion ellipse on the surfaces of a fracture as a function of the normalized fracture Figure 6. Reversaldepths of the particle motion direction as stiffnesswhen Ce/Cs=l.53. Symbolswpp and upp are the a functionof the normalizedfracture stiffness for Ce/Cs=l.53. peak-peak amplitudes of the vertical and horizontal Symbols •sm and •asm are the wavelengthsof the symmetric components,respectively, of the displacementon the fracture and antisymmetricinterface waves, respectively. surface.
For a nontrivial solution of (10) to exist, the coefficient it can be shownthat the Ce/Cs valuesof 1.50, 1.63, 1.87, and matrix of the two constants A1 and B1 must vanish, which 2.45 correspondto the Poisson'sratios of 0.1, 0.2, 0.3, and 0.4, yields an equationfor the symmetricinterface wave: correspondingly. Thus Figure 3 also shows that with increasingPoisson's ratio, the phasevelocities of the interface waves increase relative to the S wave velocity for a given value of the normalized fracture stiffness. where• = Cs/C, •' = Cs/Ce, and •z = kz/røZs ( Zs= PCs is The explanation 'for the increase of C/Cs with Poisson's the S wave impedance). ratio may be as follows. As Poisson'sratio increases, the Similarly, substituting B2 = B• and A2=-A1 into (9), an velocity of the ordinary Rayleigh wave increases. Therefore equationfor the antisymmetricinterface wave is obtained, the phase velocities of the interface waves accordingly increasesbecause the interfacewaves are formed of coupling of two ordinary Rayleigh waves by the fracture stiffness. where•x = kx/OZs. It should be pointed out that the normalized phase velocities of the interface waves depicted in Figure 3 are Whenthe normalized fracture stiffnesses, /•z and •x, are obtainedby searchingfor roots of (11) and (12) in the real zero, both symmetric and antisymmetric interface wave domain. When the solutionrange is extendedto the complex equations (11) and (12) degenerateto the ordinary Rayleigh domain,a complexroot of (11), that is, the leaky mode of the equation for a traction-free surface. However, when the normalized fracture stiffnessesare finite, (11) and (12), unlike symmetricinterface wave, isobtained for•z >(•z)cf. The the Rayleigh equation,contain wave frequency. Therefore the normalized phase velocities of the symmetric and interface waves propagating along a fracture with a finite antisymmetricinterface waves, including the leaky mode of normalized stiffness are dispersive. The respective the symmetric interface wave, are shown in Figure 4 for appearanceof /•z and •x in (11) and(12) indicatesthat the C•,/Cs=1.53. symmetric and antisymmetric interface waves are supported By setting • = 1 in (11), the normalizedfracture stiffness by the normal and tangentialcoupling between the surfacesof which defines the boundarybetween the normal and leaky the fracture, respectively. modesofthe symmetric interface wave, (•z)cf, can be fobriff Using (11) and (12), the normalized phase velocities, with the form C/Cs, of the symmetricand antisymmetricinterface waves are calculatedas a functionof •z and /7:xfor a rangeof Ce/Csratios and displayed in Figure3. As •z and /•x a non-welded fracture increase, the phase velocities of the two interface waves increasefrom the Rayleigh wave velocities to the shear wave source 45.79 m velocity, with the symmetricinterface wave propagatingfaster than the antisymmetricinterface wave. Examination of the Cp= 5800m/s normalized fracture stiffness terms reveals that an increase in Cs= 3800m/s /7:z and /7:xcan be inducedby a decreaseeither in w or Zs P= 2600kg/m3 or by an increase in kz and kx, respectively. Hence the increase of the normalized phase velocities of the interface waves with the normalized fracture stiffness can be a result of Figure 8. Simulation geometryused to generate symmetric an increasein w or Zs or of an increasein kz and kx. By and antisymmetric interface waves. The source line is useofv=0.5(1-2C2)/(1-C 2)where v is the Poisson's ratio,perpendicular to the plane of the paper. GU ET AL.' FRACTUREINTERFACE WAVES 831
a) horizontalsource, kz = 5 x 109Pa/m
DISTRNCE FROI1 THE SOURCE (11) HORIZONTRL DI SPLRCEI1ENT VERTI CRL DI SPLRCEI1ENT 3. 69 15.8 28.8 48.1 52.3 64.43.69 15.8 28.8 48.1 52.3 6t•.4 , , , , ,, , , , , , 8. 88
2. 81
4.82
6.83
8. 84
18.8.
12.6
14.6
16.8
18.8
28.1 ti me ms)
b) verticalsource, kx = 5 x 109 Pa/m
DISTRNCE FROI'I THE SOURCE (1'1) HORIZONTRL OI SPLRCEI'IENT VERTICRL OlSPLRCEI'IENT 3.69 15. 8 28. 8 t•8.1 52.3 64.4 3. 6g 15.8 28. 8 t•8.1 52.3 6t•.t, 8. 88
.
8•
. ti me ms)
Figure 9. Seismicsections recorded along the lower surfaceof the fracture.
1 for thewhole range of normalizedfracture stiffnesses (see also Figures3 and4). (•Z)cf:241_•.2 ' (13)Particle motions of the two interface waves for a or normalized fracture stiffness of 0.1 and C•,/Cs=1.53 are plottedin Figure 5. The particle motionsof both the (•'Z)cf:O.542(1-v), (14) symmetricand antisymmetric interface waves, like thosein by substituting•.2= (1- 2v)/(2- 2v) into(13). Equationsthe Rayleighwave field on a traction-freesurface, exhibit (13) and(14) give an expressionfor the cut-offnormalized elliptic trajectoriesand they are retrogradenear the fracture fracturestiffness for the symmetricinterface wave. However, andreverse to progradeat a certaindepth. The reversaldepth no cut-off normalized fracture stiffness is found to exist for the of particlemotion direction is calculatedas a functionof the antisymmetricinterface wave from (12). This demonstratesnormalized fracture stiffnessesand is displayed in Figure 6. onceagain that the antisymmetricinterface wave is nonleaky With increasingnormalized fracture stiffness, the reversal 832 GU ET AL.' FRACTURE INTERFACE WAVES
a) horizontal source a result, as the normalized fracture stiffness increases, the particle motion ellipse of the antisymmetric interface wave 20.0 becomesmore verticallypolarized because it is compressed only in the tangential direction and not in the normal 13.3 direction. 3 6.7 PIW Figures3, 4, and7 showthat as •'x--->oo,the p/irticle 0.0 motion ellipse of the antisymmetric interface wave approachesan asymptotedefined by the particle polarization -6.7 of a body S wave propagatingparallel to the fracture,and the velocity of the antisymmetric interface wave reaches the -13.3 , I I I I I I I I velocityof the S wave. With (6), (7), (10), and (11), it can 8.7 10.4 12.2 13.9 15.7 be demonstratedthat as •'x -• oo,the antisymmetricinterface time (msec) wavedegenerates to a body S wave which propagatesparallel to the fracture with the form b) vertical source Ul = U2= 0,
I ß = 4.7 •o 2.3 PIW • Wl=W2 = --B1 Cs ('osin•( t, •- Cs t). (15) a. 0.0 With increasing•'z, the particlemotion of the symmetric interface wave becomesmore horizontallypolarized (see -2.3 4 3 Figure7). As •'z--->•,, the phasevelocity of the symmetric o -4.7 interfacewave approaches the P wavevelocity (see Figure 4). 'oc -7.0 From (6), (7), (9), and (12), it has been derived that as ß I I I I I I I I gz--> •,, thedisplacements for the symmetricinterface wave 8.7 10.4 12.2 13.9 15.7 become time (msec) Figure 10. Waveforms recorded on the lower fracture surface,at distance45.79 m from source. Fracturespecific stillnesses are (1) kz=109, (2) kz=Sx109, (3) a) PIW interface wave kz= 2.5x10lø, and(4) kz= l0ll PaJmin thehorizontal source case,and (1) kx=Sx109, (2) kx=10lø, (3) kx=2.Sx10lø and(4) kx= l0 ll PaJmin the verticalsource case. 'horizontal source)
u Pay(vertical source) depth for the symmetricinterface wave greatly increases while that for the antisymmetricinterface wave slightly decreases. Figure5 alsoshows that the particlemotion ellipse on the fracture surface is more vertically polarized for the a, 0 antisymmetricinterface wave than the one for the symmetric 107 108 109 101ø 1011 10t2 1013 interfacewave. The ratio of the ellipseradii of the particle fracture specific stiffness(Pa/m) motion,or equivalently,the ratio of the peak-peakamplitude of the vertical displacement to that of the horizontal b) RIW interface wave displacement,on the fracture surfacehas been evaluated for a range of normalizedfracture stiffnessesand displayedin Figure 7. The two curvesin Figure 7 indicate that, with W RIW increasingthe normalized fracture stiffness, the particle (horizontal source) motion ellipse on the fracture surface becomes more horizontallypolarized for the symmetricinterface wave and Ua•w (vertical source) more vertically polarized for the antisymmetric interface wave. This dependenceof particlemotion polarization on the symmetryof interface wave fields may be explained as 6 follows. The symmetricinterface wave is supportedby only 0 the normalcomponent and not the tangentialcomponent of the normalized fracture stiffness (see (11)). Thus, as the 107 108 10ø 101ø 1011 iOn IOn normalized fracture stiffness increases, the symmetric fracture specific stiffness(Pa/m) interfacewave is compressedin the normal directionand not in the tangentialdirection, which leadsto the particlemotion Figure 11. Peak-peak amplitudes of the PIW and R/W of the symmetricinterface wave becomingmore horizontally interface waves as a function of fracture specific stiffness. polarized. For the antisymmetric interface wave, the The rangeof the fracturespecific stiffness from 107 to 1013 tangentialcomponent and not the normal componentof the Pa/m in this figure correspondsto that of the normalized normalizedfracture stiffness has an effect on it (see (12)). As fracturestiffness from 0.2 x 10-3 to 0.2 x 103 ß GU ET AL.: FRACTURE INTERFACE WAVES 833
3900 numericalsimulations is not long enoughfor the two waves to separatefrom each other. Clearly, there exist on the surface S-wave velocity 3800 of the fractureonly uv in the horizontalsource case and ws __antisymmetric, analytical•'- •--••"•- in the vertical source case when the fracture is completely • symmetric,analytical / o/' welded. Hence, w row, umw and wmw in the horizontal . antisymmetric,numerical/ •ø/ sourcecase and urow, wmw, and umw in the vertical source 3600 O symmetric,n case are generated due to the finite stiffness of the fracture and are interface waves. 3500 To examine the effects of the fracture specific stiffnesson Rayleighwave velocity__ the interface waves, displacements recorded on the lower 3400 fracture surface,45.79 m from the source,are displayedin Figure 10 for a range of fracturespecific stiffnesses. Both the 3300 I I I I I 107 108 109 10lø 10n 10• lOb shape and amplitude of the waveforms vary with fracture specificstiffness. fracture specific stiffness(Pa/m)
Figure 12. Phase velocities of the R/W interface waves as a a) symmetric interface wave function of fracture specific stiffness. The range of the fracturespecific stiffness from 107 to 1013Pa/m in thisfigure correspondsto that of the normalized fracture stiffness from propagation 0.2x10 -3 to 0.2x10 3. upper surface P+PIW
ul=u2 =-Al Cv•øsin•ol'-•-x •.Cv- t i, (16) w1= w2 = 0. This indicatesthat the symmetricinterface wave degenerates RIW to a body P wave that propagatesparallel to the fractureas propagatio•P+PIWL •ower surface
Line Source Analysis Interface waves along a fracture generatedby a line source at the fracture are investigated using a boundary element method (BEM). In the BEM scheme, the upper and lower fracture surfacesare divided into quadraticboundary elements. b) antisymmetricinterface wave The time variable is discretized using an implicit time- steppingalgorithm. The displacementdiscontinuity boundary conditions given in (1) are applied between the upper and lower surfacesof a fracture. A system of linear equations is propagation• then formed, which is used to solve for unknown displacementsand stresseson the surfaces of the fracture. The displacements in the two half-spaces are computed afterwardby direct use of the dynamic integral representation. =•.•• upp• Details of the BEM for a fractured medium have been given by Gu et el. [1994]. In the following, numerical experimentsare conducted for the model geometry shown in Figure 8. The two elastic half- spacesare assignedP and S wave velocities and a density, Cp= 5800 m/s, Cs= 3800 m/s and p=2600 kg/m3, respectively. The sourceis a three-lobeRicker wavelet with a centralfrequency of 800 Hz. Figure 9 displays two seismic sections recorded on the lower surface of the fracture. The four arrivals observed are propagation ! • labeled up+mw, wrow, umw and wmw in the horizontal sourcecase and umw wrow, umw and ws+•w in the vertical source case. Here superscripts PIW and RIW indicate, respectively, P-type and Rayleigh-type interface waves, P+PIW denotes a mixture of a body P wave and a P IW interface wave, and S+RIW refers to a mixture of a body S wave and a R/W interface wave. The mixing of the interface Figure 13. Particle motions on the upper and lower surfaces waves and the body waves is becausethe fracture used in the of the fracture, at distance 45.79 m from source. 834 GU ET AL.: FRACTURE INTERFACE WAVES
Figure 11 shows that the peak-peak amplitudes of the displacementsfor the PIW and RIW interface waves decrease P+PIW• RIW as the fracture specific stiffness increases, which indicates that less seismicenergy is partitioned from body waves into the interface waves with increasingfracture specific stiffness. propagation• Figure 12 displaysthe phasevelocities of the RIW interface waves versusfracture specific stiffness. The analytical phase S+RIW velocities are calculated using the symmetric and antisymmetric interface equations (11) and (12). The O.06X,, numerical phase velocities of the symmetric and 0.09)% antisymmetricinterface waves are obtained by calculating 14.82w/AO for a line sourceat the fracture,where AO is the phase difference of a Fourier component of the interface S+RIW waves recorded on the fracture surface at distance 29.83 m and 44.65 m from the sourcesand the angularfrequency of the Fourier component w = 2n:'690 Hz. The numerical and 0.11X,, O.17Xs analytical velocities are basically consistentwith each other P+PIW although there is some discrepancy between them. The S+RIW discrepancyis induced, most probably, by the use of not-fine- enoughgrids and time intervalsfor the numericalsimulations. Figure 13 shows particle motions of the symmetric and 0.17)% 0.25)% antisymmetric interface waves observed on the upper and P+PIW lower fracture surfaces, 45.79 m from the source, for the geometry shown in Figure 8. The symmetric and antisymmetricinterface waves are generatedby the horizontal and vertical sources,respectively. The fracture is assigneda +RIW specificstiffness of kz= 5x109Pa/m in the symmetriccase and kx= 5x109pa/min the antisymmetriccase. The particle motions of the RIW interface waves on both the upper and 0.22•,p+p•• -- 0.42•s lower surfacesof the fracture are retrograde. Figure 14 shows particle motions recorded along the vertical receiverprofile in the lower mediumfor the horizontal •S-wave sourcecase shownin Figure 8. Th.eP+PIW wave tracesout largely horizontally polarized particle motions, and the 0.77)•pP-wa• 1.18)% S+RIW and RIW waves trace out more vertically polarized particle motions. The particle motion of the S+RIW wave reverses from retrograde to prograde at depth 0.17 Xs (3' wavelength) from the fracture. The particle motion of the Figure 14. Particle motions observed along the vertical P+PIW wave, opposite in particle motion direction to the profile for the horizontally polarized source(see Figure 8). The fractureis assigneda specificstiffness of kz=5xlO 9 S+RIW wave, changes from prograde near the fracture to Pa/m. The depth is normalizedby the P and S wavelengths, retrogradeat depth 0.17 Xv (P wavelength)from the fracture. •,•, and •,s. Now, the vertical componentof the P+PIW wave, insteadof the horizontal component of the 3'+RIW wave, goes through zero at depth 0.17Xv from the fracture. The vertical In the plane wave analysis, the displacementdiscontinuity displacementof the P+PIW wave reachesa maximum, not on boundarycondition for a fracturewas applied to the potentials the fracturesurface, but at a slight depth, around0.06 Xv away for the displacement field of inhomogeneousplane waves. from the fracture. Deeper than 1.18 X s from the fracture, the This directly resultsin a 4 by 4 matrix equationfor interface PIW and RIW interface waves disappearand the body waves waves. By consideringthe symmetryof the wave field with dominate. respect to the fracture, the 4 by 4 matrix equation is decomposed into two 2 by 2 matrices which give the symmetricand antisymmetricinterface wave equations. The Summary and Conclusions normalized fracture stiffness (which is the ratio of the fracture An extensive fracture in a solid can be modeled for seismic specific stiffness to the product of the wave's angular waves as a displacement discontinuity boundary condition frequencyand the S wave impedanceof the two half-spaces) between the surfaces of the fracture. Across the fracture, controls the behavior of the interface waves. The normal and seismicstresses are continuousand particle displacementsare tangential componentsof the normalized fracture stiffness discontinuousby an amount which is proportional to the supportthe symmetric and antisymmetric interface waves, stressesand inversely proportionalto the mechanicalspecific respectively. As the normalized fracture stiffness increases stiffness of the fracture. Based on the fracture model, from lower to higher values, the phase velocities of the interface waves along a single, extensivefracture in an elastic symmetric and antisymmetricinterface waves increase from solid were investigated using plane wave analysis and a the Rayleigh wave velocity to the velocities of the body P and boundaryelement method. S waves, respectively. For values of the normalized fracture GU ET AL.: FRACTURE INTERFACE WAVES 835
Gangi, A.F., Variation of whole- and fractured-porous-rock stiffnessabove 0.5•/2(1-v) (v is Poisson'sratio), the permeabilitywith confiningpressure, lnt. J. RockMech. Min. Sci. symmetricinterface wave existsin a leaky mode. Geomech.Abstr., 57, 249-257, 1978. In the wave field near a fracturegenerated by a line source, Greenwood,J.A., and J.B.P. Williamson, Contact of nominally flat a P-type interfacewave is observedto exist in additionto the surfaces,Proc. R. Soc.London., A, 295, 300-318, 1966. Gu, B., N.G.W. Cook, K.T. Nihei, and L.R. Myer, Modeling elastic Rayleigh-typeinterface waves predicted by the plane analysis. wavepropagation in a mediumwith non-weldedinterfaces using The particlemotion of the RIW interfacewave reversesfrom theboundary integral equation method, in ComputerMethods and retrogradenear the fracture to progradeat a certain depth. Advancesin Geomechanics,edited by H.J. Siriwardane,and M.M. The particle motion direction of the PIW interface wave, Zaman,pp. 387-392,A.A. Balkema,Brookfield, Vt., 1994. Hopkins,D.L., N.G.W. Cook,and L.R. Myer, Fracturestiffness and oppositeto the particlemotion direction of the RIW interface apertureas a functionof appliedstress and contactgeometry, wave, changesfrom progradenear the fractureto retrogradeat Proc. U.S. Syrup.Rock Mech., 29th,673-680, 1987. a certain depth. With decreasingfracture specific stiffness, Kendall,K., andD. Tabor, An ultrasonicstudy of the area of contact more seismicenergy is partitionedfrom body waves into the betweenstationary and slidingsurfaces, Proc. R. Soc.London, A, interface waves. 1323, 321-340, 1971. Murty,G.S., A theoreticalmodel for the attenuationand dispersion of These results may find direct application to seismic Stoneleywaves at the looselybonded interface of elastichalf- detection and characterization of fractures in the field. space,Phys. Earth Planet Inter., 11, 65-79,1975. Interface waves are characterized by localization of the Myer,L.R., D.L. Hopkins,and N.G.W. Cook, Effects of contactarea of seismicenergy in the neighborhoodof the fractureand hence an interface on acoustic wave transmission characteristics, Pro. U.S.Rock Mech. Syrup., 26th, 565-572, 1985. travel with lessloss in amplitudealong the fracturethan body Pyrak-Nolte,L.J., and N.G.W. Cook, Elasticinterface waves along a waveswhich spreadin three dimensions.In addition,fracture fracture,Geophys. Res. Lett., 14, 1107-1110,1987. interface waves directly sample the mechanicalproperties of Pyrak-Nolte,L.J., L.R. Myer, and N.G.W. Cook, Transmissionof fractures. Thereforeinterface wave techniquesmay become a seismicwaves across natural fractures, J. Geophys.Res., 95, 8617- quantitative diagnostic tool for evaluating the physical 8638, 1990. Pyrak-Nolte,L.J., J. Xu, and G.M. Haley, Elastic interfacewaves propertiesof fracturesin geoengineeringand the strengthof propagatingin a fracture,Phys. Rev. Lett., 68, 3650-3653,1992. welding, bonds,and adhesivesin nondestructivetesting. For Rokhlin,S., Adhesivejoint characterizationby ultrasonicsurface and example, the leaky mode of the symmetricinterface wave interfacewave, in AdhesiveJoints, edited by K.L. Mittal, Plenum, allows fracturesto be detected by receivers located off the New York, 307-345, 1984. Rokhlin,S.I., and Y.J. Wang, Analysisof boundaryconditions for fracture. The dependenceof the interface waves on the elastic wave interaction with an interface between two solids, J componentsof fracturestiffness may allow separateestimates Acoust.Soc. Am., 89(2), 503-515, 1991. of the horizontaland verticalcomponents of fracturestiffness. Schoenberg,M., Elasticwave behavior across linear slip interfaces,J. Acoust.Soc. Am., 68, 1516-1521, 1980. Acknowledgments. The work was supportedby the Director, Xu, P.C., and S.K. Datta, Guided waves in a bonded plate: A Office of Energy Research,Office of Basic Energy Sciences under parametricstudy, J. Appl. Phys., 67, 6779-6789,1990. U.S. Departmentof Energy ContractsDE-AC03-76SF00098 and DE- F602-93ER14391. B. Gu, L.R. Myer, and K.T. Nihei, Earth SciencesDivision, LawrenceBerkeley Laboratory, Bldg 50E, 1 CyclotronRoad, Berkeley,CA 94720.(e-mail: boliang@jard:lbl.gov) References L.J. Pyrak-Nolte,Depm•tment of Civil Engineeringand Geological Brown, S.R., and C.H. Scholz, Closure of random surfacesin contact, Sciences,University of Notre Dame, Notre Dame, IN 46556. J. Geophys.Res., 90, 5531-5545, 1985. Cook, N.G.W., Natural joints in rock: Mechanical hydraulic and seismic behavior and properties under normal stress, Int. J. Rock (ReceivedFebruary 6, 1995; revisedSeptember 5, 1995; Mech. Min. Sci. Geomech. Abstr., 29, 188-223, 1992. acceptedSeptember 13, 1995)