Secondary Shear Waves from Source Boreholes

The purpose of this paper is to synthesize the most important results of the thesis work of Meredith (1990) concerning radiation from seismic sources in boreholes. Previous studies of radiation from sources in boreholes have been far-field studies and have neglected the explicit contribution of the borehole. In general, this is fine for P-wave radiation and for S-wave radiation into high velocity rocks. However, tube waves '~eak" shear conical waves (Mach waves) which propagate when the tube wave velocity is greater than the shear wave velocity of the surrounding medium. These Mach waves are of high amplitude because of the dominance of the tube wave and radiate away from the borehole in a fixed conical shape. The shape of the cone is dependent on the shear wave velocity of the medium and the tube wave velocity. This paper defines the conditions under which these Mach waves exist and thoroughly describes them in a physical sense and less so in a mathematical sense. Finally, the relationship of Mach waves to data sets is examined and how Mach waves may be confused with receiver borehole tube waves. To keep the presentation simple, radiation from axial or torsional sources or radiation from empty boreholes is omitted in this paper but fully addressed in Meredith (1990). INTRODUCTION For the most part, analysis of cross well tomographic data has resulted in travel time picks which are inverted based on a forward velocity model. This produces a P-wave velocity image of the region between the two wells. However, there is also interest in determining amplitudes and determining S-wave velocities in addition to the P-wave velocities. 174 Meredith et al. To accomplish these expanded goals, understanding of the total wavefield radiated by a seismic source in a borehole is required. Radiation from seismic sources in a borehole has been developed through Lee and Balch's work (Heelan 1953a; Lee and Balch, 1982; Winbow, 1989) describing far-field radiation from a point source or radial source in a fluid-filled borehole. By assuming the far-field and the mathematical simplifications that implies, closed form solutions for far-field radiation can be achieved based on simple trigonometric functions. These functions can be graphed producing a radiation pattern such as presented in Figure 1 for a point source in a fluid-filled borehole. Following these theoretical developments, experiments have been run to test their validity in practice. White and Sengbush (1963) published the result of an experimentally determined radiation pattern versus the theoretical prediction. The experiment was conducted in the relatively homogeneous and massive Pierre Shale. A schematic of the experiment is outlined in Figure 2 where it can be seen that in a central shot hole drilled 600' below the surface, booster charges are fired and the resulting radiation is recorded by six 3 component geophones in receiver boreholes. The receiver boreholes were drilled in a circular pattern to represent one quadrant of a radiation pattern. The resulting data versus the radiation pattern prediction from Heelan (1953) is shown as Figure 3 (White and Sengbush, 1963). It is immediately apparent from Figure 3 that there is substantial disagreement between the theoretical prediction and the actual data. In particular, the radiation pattern prediction fails to account for the strong vertical directivity of the shear wave amplitude. White and Sengbush surmised that the difference was that Heelan's empty borehole theory could not account for the effects of tube waves present in the exPeriment. But at a later time, Fehler and Pearson (1984) performed a similar type experiment (Figure 4) in granite and found good agreement with their data and the radiation pattern prediction. Therefore, this work was carried out to resolve this type of discrepancy radiation pattern predictions work in high velocity reservoirs but seemingly fail in low velocity reservoirs for shear wave amplitude predictions. DEVELOPMENT OF A NUMERICAL SOLUTION The first effort of this work was to develop a full numerical technique to see if a numerical solution could reproduce the behavior of the data. To do so a program to calculate the radial and vertical displacements outside a radially layered medium was developed. This effort is complementary to that of Tubman (1984), and Tubman et al. (1984, 1986) who studied the pressure response inside a layered medium using a discrete wavenumber Thomson-Haskell approach. A numerical refinement to the Tubman et al. work for calculating the layer matrices (Schmitt and Bouchon, 1985) was also incorporated into this work. In essence, the work of Tubman et al. was turned inside out for a complete numerical solution and was tested against a boundary integral code (Meredith, 1990) Borehole Radiation 175 and yielded identical results. More details of the algorithm can be found in the TUbman et al. references, Meredith (1990), or Meredith et al. (1990). RESULTS OF NUMERICAL MODELLING In general, there was excellent agreement between the theoretical prediction of P-wave radiation and the numerical prediction even less than 1 m away from the borehole. Thus for P-wave radiation the far field assumptions are quite valid. There is also excellent agreement between theoretical and numerical predictions of S-wave behavior if the Swave velocity of the surrounding medium is fairly high (Vs » Vf) (Meredith, 1990). Conversely, the numerical prediction diverged from the theoretical but agreed quite well with the experimental observations when the shear wave velocity was close to or less than the tube wave velocity. For example, Figure 5 is a polar plot displaying White and Sengbush's observed data versus the numerical model predictions. The strong vertical directivity of the shear wave amplitude is very well duplicated and the numerical model represents a substantial improvement over the theoretical prediction in Figure 4. The reason for the improvement by using the numerical model is that the numerical solution includes the effects of the tube wave travelling in the borehole whereas the far-field solution does not. As predicted by White and Sengbush (1963) and de Bruin and Huizer (1989), the tube wave has a direct affect on the radiation. The mechanism describing how tube wave propagation affects the radiation into the borehole is discussed in physical and mathematical terms next. PHYSICS OF TUBE WAVE RADIATION: THE MACH WAVE Constructive and destructive interference due to superposition of wavefields is one of the fundamental principles of wave theory. It similarly plays a large role in describing how a tube wave radiates shear conical waves (Mach waves) into the surrounding medium. In cased boreholes, TUbman et al. (1984) and Winbow (1989) had shown that less than 1% of the source's energy escapes into the formation while the remainder is trapped in the tube waves. TUbe waves radiate sound into the formation as they travel through the borehole and in fact one of the conditions for their existence (Cheng and Toksoz, 1981) is that they radiate this energy. However, the energy decays away from the borehole. When the tube wave velocity exceeds the shear wave velocity the radiated noise overlaps and in fact constructively interferes (de Bruin and Huizer, 1989). This phenomenon is most closely related to that encountered in aerodynamics and is illustrated 176 Meredith et al. by Figure 6. In the sub-shear case sound emitted by the tube wave decays away from the borehole in a spherical fashion as in the top half of Figure 6. In the super-shear case, sound emitted by the tube wave does overlap causing constructive interference. When this sound overlaps, the cone of constructive interference is determined by an angle, the Mach angle, whose sine is the ratio of shear wave velocity over tube wave velocity. One note is that the Mach angle is measured from the axis of the cone. The Mach number is the ratio of tube wave velocity over the shear wave velocity and is greater than one when Mach waves exist. To illustrate this phenomenon of constructive interference, some models were run with high and low velocity media surrounding the borehole. The source was a 200 Hz Ricker wavelet. Radial and vertical displacements were calculated on a 50m by 50m grid at half meter intervals with the source location being the origin. Total displacement was derived from the radial and vertical displacements and the result was contoured. These displacement fields were then sampled at a particular time to produce a snapshot, analogous to the procedure commonly performed in finite difference calculations. The first model is the counter example, the high velocity case, which uses the velocities and densities (Table 1) of Berea Sandstone for the surrounding medium. The borehole was open and the contour plot of the displacements at 20 msecs is shown in Figure 7A. The P wave has moved off the graph and the S wave and tube wave are contoured. It can be seen that the tube wave response dies off with distance away from the borehole and there is no constructive interference of the sound emitted from the tube wave. This is the sub-shear case because the tube wave velocity at 1400 m/sec is less than the shear wave velocity of 2664 m/sec. Stepping in time to 30 msecs, Figure 7B, shows no real difference, there is still no constructive interference of the tube wave and the tube wave radiation shape is identical. The second model uses the physical properties of Pierre Shale (Table 1), a much slower medium and the medium used by White and Sengbush (1963). This is the supershear case because the tube wave velocity of 980 m/sec is greater than the shear wave velocity of 869 m/sec. The contour plot at 20 rnsecs, Figure 8A, shows a P and S wave but it can be seen that the borehole side of the S wave is due to constructive interference of the tube wave and is of high amplitude. The Mach angle here is relatively steep at 63 degrees. The third model (Figure 9) is the same as for Figure 8 but now a 0.5 em layer of casing has been inserted raising the tube wave velocity to 1280 m/sec using the formulas of Marzetta and Schoenberg (1985) and changing the Mach angle to 43 degrees. The P wave is only slightly affected by the presence of the casing. By changing the Mach angle, direct detection by a vertical array of receivers is even more likely. The geometric shape of the S-wave radiation when the Mach wave is present is due to the constructive interference and consists of the initial disturbance plus the Borehole Radiation 177 constructively interfering Mach wave. The initial disturbance will not experience any constructive or destructive interference until the tangent of the Mach cone intersects this disturbance as shown in Figure 10 at the angle tPc. The portion of the medium illuminated by the Mach wave is said to be in the Mach cone. Therefore, the shape of the radiated wave field will have an outwardly travelling spherical portion tangent to an upgoing and downgoing Mach cone. The Mach wave will directly intersect any geophone in the Mach cone and may indirectly intersect other geophones through reflections of Mach waves. The spherical portion inside the Mach cone will be destroyed by destructive interference. When an observation point is inside the Mach cone, the arrival time of the Mach wave will be a hybrid travel time as shown in Figure 11. The travel time will consist of the time to travel up the borehole as a tube wave and time to travel through the medium at the shear wave velocity. The hybrid travel time is given by the formula analogous to that from earthquake seismology (Ben-Menahem and Singh, 1987). What this means is that for a vertical array of receivers the moveout will be controlled by the tube wave velocity exclusively and so will be linear. Therefore, a Mach wave may look like a high amplitude receiver borehole tube wave though it is not. MATHEMATICS OF TUBE WAVE RADIATION Another way to visualize this is mathematically. The radiation integrals for an infinite half-space surrounding a borehole are typically evaluated by the method of stationary phase (Lee and Balch, 1982; Winbow, 1989; Meredith, 1990). Using the method of stationary phase implies that the most significant contribution to the integral occurs in the neighborhood of a stationary point where the integral is slowly varying. Away from this point the integral is highly oscillatory and positive and negative oscillations cancel each other. The integrals of concern here have a pole at aT and a branch point at ;;where v is either "', P-wave velocity or {3, S-wave velocity. A commonly used path of integration for evaluating these stationary phase integrals is displayed in Figure 12 for the P-wave case. In Figure 12, the stationary point ranges from z/R=O to z/R=1 and consequently the stationary point ranges from 0 to ;;:. The stationary point does not come too close to the tube wave pole in a normal velocity setting. However, if the tube wave velocity is close to the P-wave velocity or greater than the tube wave velocity the pole will have an effect. This situation would be rare with P-wave velocities but in the case of shear 178 Meredith et al. waves it is much more common that the shear wave velocity is less than the tube wave velocity as illustrated in Figure 13. The sub-shear case, the top half of Figure 13, is very similar to the P-wave case. If f3 > CT no Mach wave will be produced though as f3 approaches CT the shear wavefield will be more substantially affected (Meredith, 1990). The super-shear case involves the presence of Mach waves. Here the stationary point will intersect the pole location at Cr = ~ for some value of fl. This will make the stationary phase assumptions no longer valid for evaluation of the integral. The mathematics is echoing the point of the physics that when CT > f3 the borehole can no longer be ignored and the far-field assumptions fail. MACH WAVES IN DATA SETS The presence of Mach waves and their effects has already been demonstrated for the White and Sengbush (1963) data set and their effects can also be seen on the data of de Bruin and Huizer (1989) which is shown in Meredith (1990). One final figure will show the pervasiveness of the Mach waves. Figure 14 shows data from a recent group shoot taken at Texaco's Humble test site. The vertical component of shot gathers with identical geometries but different sources are shown. Nice P-wave hyperbolas are evident but none are seen for the S waves. Strong linear events are seen for both the dynamite and airgun experiments. It might first be thought these are receiver borehole tube waves but because the receiver array was cemented and there is no receiver borehole this is impossible. Instead these are Mach waves, and as mentioned before the moveout is linear at the tube wave velocity. Since the dynamite forces the formation nonlinearly, it puts more energy into the formation and comparatively less into the tube wave. Therefore, the Mach wave response for the dynamite is less. CONCLUSIONS In some cases, the energetic tube waves in the borehole lead to direct radiation of shear corncal waves or Mach waves into the surrounding formation. However, P-wave radiation is not substantially affected by the presence of the borehole. The existence of these Mach waves is dependent on the shear wave velocity of the medium being less than the tube wave velocity. The Mach waves will be controlled by a Mach cone and an angle whose sine is the ratio of the shear wave velocity over tube wave velocity. The travel time to a receiver in the Mach cone will be a hybrid travel time and will include travel time as a tube wave and as a shear wave in the formation. The distinction between an observation point being inside the Mach cone or outside depends on the tube wave and shear wave velocity. Borehole Radiation 179 The Mach wave phenomenon has been noticed in data sets such as White and Sengbush (1963), de Bruin and Huizer (1989), and in a recent group shoot at the Texaco Humble test site. Included in these data sets have been buried receiver arrays which preclude the presence of receiver borehole tube waves thus removing ambiguity. ACKNOWLEDGMENTS We would like to thank Ari Ben-Menahem, Michel Bouchon, Luc Boelle, Benoit pa,ternoster, and Roger Turpening for valuable discussions and Jim DiSiena and Benoit Paternoster for suggesting the topic. Tills work was supported by the Full Waveform Acoustic Logging Consortium at M.LT. Table 1. Physical properties of lithologies and steel casing used in tills paper. Velocities in m/sec, densities in kg/m3 . Lithology/Property Vp Vs P CT Berea Sandstone 4206 2664 214