P235 Modelling Anisotropy for Improved Velocities, Synthetics and Well Ties P.W. Wild* (Ikon Science Ltd), M. Kemper (Ikon Science Ltd), L. Lu (Ikon Science Ltd) & C.D. MacBeth (Heriot Watt University) SUMMARY The construction of a new-work flow for deriving well ties from deviated well log data is comprised of three main stages. These are the building of an anisotropic model of the sub-surface; the verticalisation of well velocities measured in deviated wells; and the replacement of isotropic formulations for computing PP and PS reflectivities with anisotropic counterparts, which are then used in extracting wavelets from seismic data and in computing synthetic traces. The development of this work-flow is described, together with field data that underlines its strength in producing well-ties that provide a better fit with observed data than those produced using a conventional, isotropic work-flow.
Conventional work-flows for deriving synthetic seismic traces from bore-hole velocity data fail to take into account the velocity variations caused by the anisotropic nature of the rock through which the well trajectory passes. Errors, caused by assuming the vertical velocity is equivalent to well trajectory velocities, are compounded by using isotropic formulations for the reflectivity coefficient calculations. To limit the effects of these shortcomings, a new work-flow is presented for computing synthetic seismograms from log velocity and density data. Firstly, an anisotropic model is constructed by apportioning different anisotropy, such as layering or fracture induced, to the lithological intervals. Secondly, the bore-hole P- and S-velocities, which represent the velocities recorded in the direction of the well trajectory, are corrected to the vertical velocity using the anisotropic model. Finally, the standard approach of using isotropic formulations for PP and PS reflectivity computations, such as Zoeppritz (1919) or Aki and Richards (1980), are replaced by anisotropic counterparts. Where the model contains layers which are assumed to be anisotropic, but where the strength is unknown, empirical and computational methods are used to assign anisotropic parameters. As these methods require knowledge of the vertical velocities, and since in many instances only velocities in the bore-hole direction are known, the anisotropy is initially estimated using the log velocities and then updated iteratively as the vertical velocities are derived. Parameterising Anisotropy Our approach to parameterise the anisotropy remains faithful to modelling theories, while at the same time is accessible to as wide a user base as possible. To accomplish this, we set intervals in the model to isotropic, VTI, HTI or orthorhombic (Figure 1), which may be rotated about horizontal or vertical axes using Bond Transformations (MacBeth 2001). Transverse Isotropy with a Vertical axis of symmetry (VTI) shale related, e.g. from fine layering. Transverse Isotropy with a Horizontal axis of symmetry (HTI) sand related, e.g. from vertical fracturing. Orthorhombic mixture of VTI and HTI, for example the postdepositional fracturing of a finelayered sequence. Figure 1. Side view representation of the anisotropies supported by the modelling: horizontal layering (VTI), vertical fracturing and cracking (HTI) and orthorhombic. VTI and HTI anisotropy is quantified using Thomsen s (1986) parameters of epsilon, gamma and delta, plus the P-wave (Vp) and shear-wave velocities (Vs). However, once the model is rotated, or to consider orthorhombic anisotropy, the full set of elastic constants is required. At the front end of the model builder we use the Thomsen parameters to define the anisotropy, including orthorhombic which we build from VTI and HTI components. The Thomsen parameters are then converted to the full set of elastic constants to allow the trajectory and model rotations and the computations of orthorhombic material to be computed, before being converted back to effective Thomsen parameters at the required observational azimuth and passed to reflectivity computations.
Building an Anisotropic Model Where the anisotropy is known, the Thomsen parameters are set to constant values throughout the interval. Where the anisotropy is unknown (the general case), empirical or computational methods are used to define the anisotropy based on velocities or fracture compliances. For VTI, Ryan-Grigor (1997) presents a set of empirical formulations to give the three Thomsen parameters in terms of the Vp and Vs values. However, this presents a problem because we need the vertical Vp and Vs values (assuming horizontal layering) as inputs, but only log based values will be available, typically measured from non-vertical well paths. The problem is overcome in a two stage solution by computing and then applying factors to the log Vp and Vs values to correct for the well s deviation. Regression factors are computed by fitting the input log velocities within each interval to the velocities given after deriving the anisotropy and rotating the resultant elastic constant matrix by the well s incidence angle. These factors are then used to modify the log Vp and Vs at each sample, which are input back into the empirical relationship to derive a new, final set of elastic constants which contain corrections to verticalise the well and are used in our model. For HTI, Hudson (1981) and Schoenberg and Sayers (1995) present methods for calculating elastic constants from crack parameters or fracture compliance values, respectively. These methods assume vertical velocities, but also require a dry frame as input to their algorithms, requiring an additional step to our work-flow where any fluid is removed, the crack or fracture computation made, and then the original fluid re-inserted. Like the VTI approach, we compute the HTI parameters is two steps. Firstly, the log Vp values and fluid values are applied to Gassmann (1951) to yield the dry Vp values. With the above cracking or fracturing applied, the model is now anisotropic and an anisotropic implementation of Gassmann is applied to reinsert the original fluid (Gurevich, 2003). New Vp values are computed from the fractured material for the log deviation and fracture azimuth and fitted against the original log Vp values to obtain a regression factor. In the second step, the original log Vp values are taken and modified by the regression factor; these are used as input to the fracturing/cracking steps to yield a set of elastic constants for the HTI interval, containing corrections made for the well s trajectory. The procedure for the derivation of an orthorhombic material proceeds along similar lines as to the HTI method, the main difference being that a VTI framework is fractured rather than an isotropic one. The proportion of VTI material in the model is defined by using the shale volume to reduce the effective VTI anisotropy at each sample. Cracks and fractures are assumed to be post-depositional Verticalisation of Velocities Having arrived at an anisotropic model for each interval, we take the non-vertical log velocities and use the model to back-out velocities for the equivalent vertical propagation. The methodology is demonstrated using data from fields where logs are available for both vertical and deviated wells. The left of Figure 2 shows two separate clusters of points corresponding to Vp against Vs values for measurements made using deviated and vertical logs within the same sand formation. In the right plot, velocities in the deviated log have been verticalised using a model containing HTI anisotropy, where the HTI is parameterised using our implementation of Schoenberg and Sayers method for computing elastic constants for fractured material using normal and tangential compliances values. The compliance values were selected to be in a range typically expected in the field (Worthington and Lubbe, 2007) and the fracture azimuth was set using knowledge of the regional stress direction.
Vs Vs Deviated Vertical 1 2 3 4 1 2 3 4 G/C SAND LINE Vp G/C SAND LINE Vp Figure 2. Cross plots of Vp and Vs log data from vertical (3 & 4) and deviated wells (1 & 2) through a sand formation from a North Sea field. On the left the raw log velocity values are plotted against each other, whereas the right shows the velocities once the data from the deviated wells has been verticalised based on an HTI model. Well ties With the anisotropy set in each interval, and the vertical velocities computed, Thomsen parameters are now available for every log sample and are passed to algorithms for computing reflectivity coefficients on a sample by sample basis. Formulations from Varycuik and Pecik (1998) and Jilek (2000) are used to compute PP and PS coefficients, respectively, which are convolved with wavelets extracted from supporting seismic data using wavelet extraction (White, 1980) with the anisotropic reflectivity spike series. The benefit of using corrected vertical velocities and the anisotropic reflectivity calculations is highlighted in Figure 3, where two synthetic gathers have been computed using the conventional log derived velocity method (left) and the new work-flow (right). The conventional work-flow predicts a low impedance sand interval with no phase reversal in the Amplitude Verses Offset (AVO) response (see box in track g). On the other hand, the new work-flow predicts a class II AVO response (box in track k); that is phase reversal from near to far offsets. Seismic data extracted from near and far stacks confirm the class II AVO response we predict. Where access is available to both vertical and deviated logs passing through the same formation, a useful reorganisation of this new work-flow enables the anisotropy to be parameterised from log velocity data. For this reorganised work-flow we input the velocities from the vertical logs and find the best anisotropic model that fits them with verticalised velocities from the deviated data. It is a valuable quality control step to compare the log derived anisotropic model with the anisotropic model derived using the empirical or computation methods described above. A further extension of this work-flow is that compliance values can be calculated for fractured intervals from the log based anisotropic model by backing them out from the elastic constant matrix. Summary The new work-flow provides a method to build a model incorporating several anisotropic types and uses it to compute vertical velocities from deviated log data. These, together with reflectivity values computed using the anisotropic information, provide more robust well-ties than those computed using a conventional, isotropic work-flow.
(a (b) (c (d) (e) (f) (g) (i) (j) (l) (k Figure 3. Two synthetic traces are displayed alongside corresponding composite traces extracted along the deviated well-bore path from a seismic data volume. The synthetic gather in track (g) has been computed using raw log velocities and is poorly correlated with the near stack trace (track f); whereas the synthetic in track (k) has been computed using verticalised velocities and an anisotropic model and correlates well with the near (j) and far stack traces (l). The red curves in (c) and (d) show the verticalised Vp and Vs, which for vertical fractured media will tend to have higher values than their deviated counterparts shown in black. Other tracks show shale volume (a), porosity (b) and density (e). References Aki, K., and Richards, P.G., 1980. Quantitative Seismology: W.H. Freeman & Co. Gassmann, F., 1951. Elastic waves through a packing of spheres. Geophysics, 16, 673-685. Gurevich, B., 2003. Elastic properties of saturated porous rocks with aligned fractures. Journal of Applied Geophysics, 54, 203-218. Hudson, J.A., 1981. Wave speeds and attenuation of seismic waves in materialc containing cracks. Geohysical Journal of the Royal Astronomical Society, 64, 133-150. Jilek, P., 2000. Approximate reflection coefficients of PS-waves in anisotropic media, 70 th SEG Conference and Exhibition, Expanded Abstracts, 19, 182. MacBeth, C., 2001. Multicomponent VSP analysis for applied seismic anisotropy. Seismic Exploration Series, Volume 26. Pergamon. Ryan-Grigor, S., 1997. Empirical relationships between transverse isotropy parameters and Vp/Vs: Implications for AVO. Geophysics, 62, 1359-1364. Schoenberg, M., and Sayers, C.M., 1995. Seismic anisotropy of fractured rock: Geophysics, 60, 204-211. Thomsen, L., 1986. Weak elastic anisotropy. Geophysics, 51, 1954-1966. White, R.E., 1980. Partial coherence matching of synthetic seismograms with seismic traces, Vavrycuk, V., and Psencik, I., 1998. PP-wave reflection coefficients in weakly anisotropic elastic media. Geophysics, 63., 2129-2141. Geophysical Prospecting, 28, 333-358. Worthington, M.H., and Lubbe, R., 2007. The scaling of fracture compliance. In: Lonergan, L., Jolly, R.J.H., Rawnsley, K., and Sanderson, D.J. (Eds.) Fractured Reservoirs. Geological Society, London, Special Publications, 270, 73-82. Zoeppritz, K., 1919. Erdbebenwellen, on the reflection and penetration of seismic waves through unstable layers. Gottinger Nachrichten, 1(VIIIB), 66-84.