Synthetic vertical seismic profiles in elastic media

Transcription

1 GEOPHYSCS VOL. 5 NO.2 (FEBRUARY 1985); P FGS. Synthetic vertical seismic profiles in elastic media Michel Dietrich* and Michel Bouchon] ABSTRACT Numerical simulations of vertical seismic profiles in flat-layered elastic media using the discrete wavenumber method are presented. The effect of source-borehole separation on recorded wave types and amplitudes is studied. For nonzero source offsets transverse and converted waves become very important and can be more energetic than the direct compressional arrivals. A systematic comparison of results from acoustic and elastic simulations shows that the acoustic approximation is quite valid for a zero source offset but becomes inadequate when the configuration of the source and vertical geophone array is two-dimensional. Recording of both pressure and displacement allows a simple separation of transverse and compressional arrivals as long as the effect of the borehole on the incoming waves can be neglected. NTRODUCTON The use of advanced synthetic seismogram techniques in exploration geophysics has been surprisingly far less extensive than in earthquake seismology. For many years the classical normal-incidence response of a horizontally stratified medium to an impulsive plane wave (Baranov and Kunetz 196; Wuenschel 196) has been widely employed and has proved to be a very helpful technique for the interpretation of seismic cross-sections. This approach has recently been used by Wyatt (1981) and Ganley (1981) for the computation of synthetic vertical seismic profiles (VSPs). n addition point source synthetic seismograms of exploration interest have been obtained by Kelly et al. (1976) and Hernon (1978) by using finitedifference schemes. The method may in theory handle the problem of laterally heterogeneous media but it entails some artifacts such as grid dispersion and spurious reflection at the edges of the model. More recently Temme and Miiller (1982) employed the reflectivity method to model the VSP technique in acoustic media and Apsel (1982)adapted the Luco and Apsel (1983) Green's function calculation method to VSP computation. n this paper we use the discrete wavenumber representation of seismic wave fields (Bouchon 1981) for generating synthetic VSP traces. mplementation of the method in the acoustic case is described in a companion paper (Dietrich and Bouchon 1985). ts formulation is extended here to elastic media and a comparative study of acoustic and elastic simulations for several source-borehole separations is performed. The results show the importance (usually underrated) of the shear and converted waves and demonstrate the inadequacy of the acoustic approximation for nonzero source offsets. CONSTRUCTON OF SYNTHETC SESMOGRAMS n the discrete wavenumber method the seismograms are first computed in the frequency domain. n essence the point source problem is numerically solved by treating a somewhat different problem where the single source is replaced by a two-dimensional (2-D) periodic array of sources. This results in an exact discretization of the elastic wave field which makes evaluation of synthetic seismograms very effective and fast. A complete description of the discrete wavenumber method is given in a series of papers by Bouchon and Aki (1977) and Bouchon ( ). ts application to the calculation of vertical seismic profiles in acoustic media was recently presented (Dietrich and Bouchon 1984). n the elastic case the coupling of P- and SV-waves increases the complexity of the calculation but the principle of discretization stays unchanged. Each phase velocity component of the discretized wave field is propagated through the stratified medium using Dunkin's (1965) formulation of the Thomson Haskell method (Thomson 195; Haskell 1953; Harkrider 1964). For computational convenience we have used the reciprocity theorem. n our scheme the receiver is located at the surface at range r from the borehole and the point source is embedded in the medium at depth z. A complete profile is then achieved by considering successive source locations along the well axis. Our VSP simulations show the dilatation vertical and horizontal displacements at interior points of the layering for a vertical point force acting on the surface. This implies computation of the vertical motion at the surface for a buried explosive point source and for buried vertical and horizontal point forces. Expressions of the source potentials for an explosive source are given by Dietrich and Bouchon (1984) while Manuscript received by the Editor April ; revised manuscript received July *Formerly Universite de Grenoble;presently Labortoirede Geophysique Marine Universite de Bretagne Occidentale Brest France. tlab de Geophysique nterne Universite Scientifique et Medicale de Grenoble BP St. Martin d'heres.france 1985 Society of Exploration Geophysicists. Allrightsreserved. 224

2 Synthetic VSP in Elastic Media 225 those for horizontal and vertical forces can be found in Bouchon (1981). TRANSVERSE AND CONVERTED WAVES OBSERVED ON VSP SECTONS When the source moves away from the borehole the amplitude of converted waves increases. P-waves impinging obliquelyon the layer boundaries produce S-waves by mode conversion and these waves may in turn generate P-waves. Thus the acoustic approximation is no longer satisfactory: mode conversion accounts for propagation anomalies (Lash ) which may lead to some interpretation errors. Nevertheless as pointed out by Helbig and Mesdag (1982) transverse and converted waves should not be regarded as undesirable events. On the contrary. they contain additional information on rock VELOC TY MS 4 LOG OFFSET. M ACOUSTC TME '51 1. i VERTCAL DSPLACEMENT 5 L- : ;:: 15 o OJ 2-25 VELOCTY LOG OFFSET. M ELASTC VERTCAL DSPLACEMENT TiME 1. ls 5 i : S s 15 l- o OJ 2 f- ~J r-r- P 25 - FG. 1. Synthetic vertical displacement seismograms for a vertical surface force in four-layer acoustic and elastic media.

3 226 Dietrich and Bouchon properties and geometry of the subsurface. n this respect it is now well established that the combined use of both compressional and shear waves (rather than the utilization of only one wave type) will be of great interest for lithological studies. Offset VSPs also allow study of lateral continuity of reflectors. When the source is located at a certain distance from the borehole an upgoing or downgoing event recorded at different depths does not correspond to a single ray. n reality the corresponding waves have traveled along adjacent raypaths. Consequently offset VSPs offer the possibility to explore reflectors over a distance which is approximately equal to half the source offset and to bring out faulted or dipping interfaces (see Wyatt and Wyatt 1982). However increasing the source offset may result in complex wave patterns. Therefore the first stage of such investigations should be a realistic modeling of vertical seismic profiles in elastic media. VELOCTY LOG OFFSET. 5 M ACOUSTC DLATATON MS 4 TME (5) '-- f- " ~ 15 o w f- - VELOCTY LOG OFFSET. 5 M ELASTC DLATATON MS 4 TME (5) " S p :r 15 f- o w L_..l--_-- FG. 2. Synthetic dilatation signals produced by a vertical surface force with an offset of 5 m in four-layer acoustic and elastic media.

4 Synthetic VSP in Elastic Media 227 n the following examples simultaneous examination of the acoustic and elastic seismograms allows the discrimination between compressional and shear converted waves. P- and S waves may also be separated by comparative analysis of the pressure vertical motion and horizontal motion recordings at depth. EXAMPLES OF SMULAnONS The first example (Figure 1) shows a zero-offset VSP computed in the acoustic and elastic cases for the four-layer model displayed on the left of the figure. The receiver array is made up of 1 vertical component geophones equally spaced between VELOCTY LOG OFFSET. 5 M ACOUSTC VERTC~L D5PL~CEMENT MS 4 TME 1. S) 5 1: ~ 15 Q. '" 2 ELASTC VERTC~L [5PL~CEMENT LME (5) : ~ 15 S Q. '" P 2 - L r L_-'-.-_--- r 3 L-~ 'L.J.._ FG. 3. Synthetic acoustic and elastic VSPs for a four-layer model and a source offset of 5 m.

5 228 Dietrich and Bouchon depths of 2 and 2 7 m. The seismograms are normalized with respect to the peak amplitude of each trace. The signal is computed for a time duration of 2 s and a frequency cut-off of 64 Hz. For all the examples considered in this paper the source time dependence is a Ricker wavelet defined by its spectrum as for a peak frequency F(oo) = 2 exp (-2t~) 2n =-. to The first P-wave arrival has the largest amplitude. The direct P pulse and the subsequent reflected and refracted waves are continuously trackable along the entire vertical profile and the seismograms display a certain symmetry between the upward and downward traveling waves. Let us now compare the acoustic and elastic VSP. The seismograms are almost identical except for the presence of additional events in the elastic simulation. There in the upper layer we observe a small secondary arrival coming from the source. This wave follows the direct P-wave but propagates at the shear-wave velocity. After reaching the bottom of the layer the wave refracts into the second layer where it progressively decays. t also gives rise to reflected and converted waves in the surface layer. n addition to the waves mentioned above the downgoing source pulse generates a converted PS-wave of very small amplitude at the base of the upper layer. The presence of these events is due to the existence of a longitudinal component of S-wave in the vicinity of the source (Aki and Richards 198). This usually neglected contribution is present in our solution because the method of calculation takes into account the complete wave field including the near-field terms. The combined use of the dilatation vertical and horizontal displacements is illustrated in Figures 2 3 and 4 in the case of a source offset of 5 m. Hydrophone receivers only record compressional waves; consequently comparison of the pressure vertical and horizontal motion will allow separation of the wave modes. n practice this is true only as long as the effect of the borehole can be neglected. Figure 2 shows the dilatation. t is seen that the upgoing and downgoing waves have a hyperbolic shape and are no longer symmetric. n the elastic case new phases appear on the seismograms. The most energetic ones correspond to the conversion of the direct S-wave into P-wave energy. n the upper layer the reflected SP-wave energy gives rise to an additional system of P-wave multiples. The corresponding vertical displacement is displayed in Figure 3. The upper part of the figure shows the close harmony between the acoustic vertical displacement and the acoustic dilatation. On the other hand the differences in the elastic case are much more important than the similarities. The direct shear wave dominates the seismogram between depths of 2 and 7 m. Wave conversions are clearly observed in the vicinity of the upper boundary for the direct P- and S-waves. Both phases generate a reflected P-wave a reflected S-wave a transmitted P-wave and a transmitted S-wave. At greater depths the S-wave travels almost vertically and its motion is then mostly horizontal. Thus shear waves are best displayed on the horizontal component of displacement as shown in Figure 4. On this figure transverse and converted waves are distinctly visible over the entire extent of the record. t should be emphasized VELOCTY LOG OFFSET: 5 M ELASTC HORZONTAL DSPLACEMENT MS 4 TME 8.. ~_~ ~ ~_~ :..;-::--_~_~ ~_~ -= a r S) l c n wo 2 is r~ r-r-: - p a : FG. 4. Horizontal displacement produced by a vertical surface force with a 5 m offset.

6 Synthetic VSP in Elastic Media 229 that converted PS- or SP-waves are generated at all boundaries including the deepest ones where the waves are impinging at a near vertical incidence. However double mode conversion such as PSP- or SPS-waves could hardly be detected on the profile. These waves have in general very small amplitudes if the source offset is not too large. Another feature observed on Figure 4 is the third arrival appearing between the direct P- and S-waves in the first layer. Examination of the dilatation indicates that the corresponding wave is a shear one. n reality this phase propagates along the surface as a P-wave (surface P-wave) and is refracted as an S-wave in the lower medium. This arrival rapidly merges with the direct S-wave. Figure 5 shows the dilatation for a very large source offset. On this example the source is located at 2 m from the well axis. We first note the discontinuity of the direct downgoing P-wave on crossing the first interface. The small refracted arrival in the upper layer is continuous in time with the direct arrival in the second layer. n the acoustic case a series of P-wave VELOCTY MS LOG O"FSET' 2 M ACOUSTC TME is DL~T~TDN 4 1. i ' r;! Q. "" 15J 2 "' 25 r 3 z; VELOCTY MS 4 :L + l :! 1' Q. u > 2lt 2'O~ r i ~ 3 1 r- 1 :5 p U LOG '---r ; OFFSET' 2 M ELASTC T ME FG. 5. Same as Figure 2 for a source offset of 2 m. 1. [S) OL~HTON 2.

7 23 Dietrich and Bouchon multiple reflections is clearly visible in the first layer. The angle of incidence of these waves decreases with the order of the multiple and corresponds to decreasing apparent vertical velocity. All reflections are postcritical and we observe that the amplitudes of the multiples may exceed those of singly reflected waves. n the elastic case however the P-wave impinging on the free surface around 1.2 s is almost entirely converted into S-wave so that the P-wave multiples in the first layer are no longer seen. The corresponding synthetic seismograms for the horizontal motion displayed in Figure 6 illustrate the increasing complexity of the wave pattern. The direct P-wave has significant horizontal motion when propagating horizontally and is therefore easily observed. n contrast the direct S-wave in the first layer has a rather small amplitude since its motion is principally vertical. t should also be noted that converted waves are easily identifiable at all layer boundaries. We next consider a more complicated velocity model depicted on the left of Figure 7 which consists of 47 layers. The S-wave velocities and densities of the model are inferred from the P-wave velocities by assuming a Poisson's ratio of.25 and by using the following relation (Gardner et al. 1974): with Po = 2.6 a ).25 P = Po( - ao ao = 4 mis and (2) Offset 5 m: The main difference between the two figures is the presence in Figure 9 of a downward traveling S-wave train which is easily tracked down to a depth of about 1 9 m. The observed S-wave train results from a complex superposition of the S-wave radiated by the source PS converted waves and shear wave multiples generated in the nearwhere P denotes the density and a the compressional velocity. A cut-off frequency of 1 Hz was used in the calculation. The theoretical seismograms displayed in Figure 7 correspond to the elastic horizontal displacement for a 1 m offset and they show the great complexity of the wave pattern. The results which are very similar to an actual VSP reveal the predominance of downgoing events although some intense reflections on the group of reflectors extending between the depths of 1 24 and 1 42 m are also present. Of particular importance is the strong and stable downgoing PS phase originating at the same level which dominates the lower part of the section. Converted waves of high amplitude and stability were also reported by Gal'perin (1974) who emphasized their importance on VSP sections even for small source offsets. The last two figures show the evolution of the wave patterns in both acoustic (Figure 8) and elastic (Figure 9) cases with increasing source offsets. The following observations can be made from comparison of the acoustic and elastic seismograms. (1) Offset m: A visual inspection of the two profiles shows that for the zero-offset case the acoustic and elastic vertical displacement responses are virtually identical. As a consequence the seismograms computed for vertically traveling waves in the acoustic case are a very good approximation of the elastic wave field. surface layers. n addition some downgoing converted waves originating from the first arrival curve are also discernible. (3) Offset 1 m: The acoustic and elastic seismograms are now very different. The enhancement of the "source S wave" and the occurrence of converted waves at all layer boundaries are the main features which can be noted on Figure 9. (4) Offset 2 m: n the depth range m the wave refracted at the base of the layer at 1 8 m depth is the first arrival. Except for this common feature the wave patterns in the acoustic and elastic cases do not present any similarity. The elastic seismograms are entirely dominated by transverse waves which are either pure S-waves or converted PS phases. CONCLUSON Application of the discrete wavenumber method to the calculation of vertical seismic profiles in flat layered media has been presented. Acoustic and elastic simulations have been considered. The method can handle large numbers of layers and includes all existing arrivals and wave types. The discrete wavenumber technique may be readily implemented and is therefore well suited for the computation of synthetic seismograms of exploration interest. t allows fine-scalestudies of seismic reflection data and should prove useful to the seismic interpreter. Examples of simulations done for several source-borehole separations explicitly show the inadequacy of the acoustic approximation for nonzero source offsets; shear and converted waves are shown to be prevalent in some situations even for small source offsets. ACKNOWLEDGMENTS Most of the calculations presented in this paper were performed at the Centre de Calcul Vectoriel pour la Recherche. This work was also supported by the ATP Geophysique Appliquee ofthe C.N.R.S. REFERENCES Aki K. and Richards P. G. 198 Quantitative seismology: Theory and Methods 1: W. H. Freeman and Co. Apsel R VESPA: Viscoelastic seismic profile algorithm for complete synthetic seismograms in layered media: Sierra Geophysics tech. rep. SG-R Baranov V. and Kunetz G. 196 Film synthetique avec reflexions multiples. Theorie et calcul rapide: Geophys. Prosp Bouchon M Discrete wavenumber representation of elastic wave fields in three space dimensions: J. Geophys. Res ~ A simple method to calculate Green's functions for elastic layered media: Bull.Seis. Soc. Am Bouchon M. and Aki K Discrete wavenumber representation of seismic source wave fields: Bull.Seis. Soc. Am Dietrich M. and Bouchon M Measurements of attenuation from vertical seismic profiles by iterative modeling: submitted to Geophysics. Dunkin J. W Computation of modal solutions in layered elastic media at high frequencies: Bull.Seis. Soc. Am Gal'perin E. L 1974 Vertical seismic profiling: Soc. of Expl. Geophys. Ganley D. C 1981 A method for calculating synthetic seismograms which includes the effects of absorption and dispersion: Geophysics Gardner G. H. F. Gardner L. W. and Gregory A. R Formation velocity and density: the diagnostic basis for stratigraphic traps: Geophysics

8 Synthetic VSP in Elastic Media 231 VELOCTY LOG OFFSET: 2 M ELASTC HORZONTAL DSPLACEMENT MS 4 8 -"O----~-. r ME 1. ls r 15. o W o is P l- i r-r-: L~ :! FG. 6. Same as Figure 4 for a source offset of 2 1l1. VELOCTY LOG OFFSET M ELASTC HORZONTAL DSPLACEMENT MS TME S).J r~""""----'-"-r-:::-~._-=8-;ooo i-"--_~_~ ~_---'-i--=---- LO -;2. i 5 2: :r: 15.- o W 2 P 25 3 '---_~--'--.L.L-_J FG. 7. Horizontal displacement calculated for a 2-D VSP configuration.

9 232 Dietrich and Bouchon TME (s> i J: :t: 15 a.. 2 \ ~ OFFSET. ( 25 r :t: 15 a.. '''t 25 OFFSET. :l: :t: o w i5 2 OFFSET. 25 :l: ::.5 c w 2 < ~ 9 ~ ~ O"FSET 25 - FG. 8. Effect of the source offset on the vertical displacement for a multilayered acoustic medium.

10 Synthetic VSP in Elastic Media 233 -~ >-. 1: 15..! ljj -..i{.. :S p OFFSET: 2 4" _.J r- TME (S) 1: 15 >-.. ljj 2 p - OFFSET. 25 '"1 z; rs: --; 15 >-... ljj -..i{ 'l :S p OFFSET: 2 4" 25 _.J.. - r- iooo 1: l >-.. ljj 15r f "oo~ P f OFFSET: 25 FG. 9. Effect of the source offset on the vertical displacement for a multilayered elastic medium.

11 234 Dietrich and Bouchon Harkrider D. G Surface waves in multilayered elastic media.. Rayleigh and Love waves from buried sources in a multilayered clastic half-space: Bull. Seis. Soc. Am Haskell N. A The dispersion of surface waves on multilayered media: Bull. Seis. Soc. Am Helbig K. and Mesdag C S The potential of shear-wave observations: Geophys. Prosp Hernon C 1978 Equations d'onde et modeles: Geophys. Prosp Kelly K. R. Ward R. W. Treitel S. and Alford R. M Svnthetic seismograms: A finite-difference approach: Geophysics 41 i 27. Lash C. C 198 Shear waves multiple reflections and converted waves found by a deep vertical wave test (Vertical seismic profiling): Geophysics nvestigation of multiple reflections and wave conversion by means of a vertical wave test (Vertical seismic profiling). Geophysics Luco J. E. and Apsel R On the Green's functions for a layered half-space. Part : Bull. Seis. Soc. Arn Temme P. and MiJller G Numerical simulation of vertical seismic profiling: 1. Geophys. SO Thomson W. T. 195 Transmission of elastic waves through a stratified solid medium: J. Appl. Phys Wuenschel P. C 196 Seismogram synthesis including multiples and transmission coefficients: Geophysics Wyatt K. D Synthetic vertical seismic profiles: Geophysics Wyatt K. D. and Wyatt S. B. \982 The determination of subsurface structural information using the vertical seismic profile: Geophysics