PROCESSING
N O I T I P ATRICK ATRICK C C ONNOLLY ONNOLLY , BP Amoco, Houston, Texas, U.S. S I It is now commonplace for 3-D data sets to be processed erty of EI that the level decreases with increasing angle. as partial offset volumes to exploit the AVO information At this well, the sands are predominantly class III and U the data. However, However, there has been significant asymmeso have slightly slightly higher amplitudes at 30° than at normal Qin try in the way these volumes could be calibrated and incidence. This can be more clearly seen in Figure 2 in Cinverted. The amplitudes of near-offset, or intercept, stacks which the EI log has been scaled to have approximately A relate to changes in acoustic impedance and can be tied to the same shale baseline as the AI log. When the sands are
Elastic impedance
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s i o b m a C e m u a l l i u G y b d e t a n i d r o o C
well logs using synthetics based on acoustic impedance (AI) or inverted, to some extent, back to AI using poststack inversion algorithms. However, there have been no simple analogous processes for far-offset stacks. The symmetry can be largely restored using a function I call elastic impedance (EI). This is a generalization of acoustic impedance for variable incidence angle. EI provides a consistent and absolute framework to calibrate and invert nonzero-offset seismic data just as AI does for zero-offset data. EI, an approximation derived from a linearization of the Zoeppritz equations (Appendix, part 1), is accurate enough for widespread application. As might be expected, EI is a function of P-wave velocity, S-wave velocity, density, and incidence angle. To relate EI to seismic, the stacked data must be some form of angle stack rather than a constant range of offsets. There are several ways of constructing suitable data sets by either careful mute design or by linear combination of intercept and gradient functions. (Part 2 of the Appendix reviews these methods.) EI was initially developed developed by BP in the early 1990s to help exploration and development in the Atlantic Margins province, west of the Shetlands, where Tertiary reservoirs are typified by class II and class III AVO responses. responses. Figure 1 shows a suite of logs from the Foinaven discovery well drilled in 1992. The 30° elastic-impedance log, EI(30), is broadly similar in appearance to the acoustic-impedance aco ustic-impedance log although the absolute numbers are lower; it is a prop-
class II, a more dramatic difference is evident between the AI and EI logs. The seismic data around Foinaven suffer from very strong peg-leg multiples. Even after demultiple, the signal-to-noise ratio of the near-trace data is often poor, especially from the class II events, whereas the far-offset data are generally of good quality. EI allows the well data to be tied directly to the high-angle seismic which can then be calibrated and inverted without reference to the near offsets. Figure 3 shows part of the EI(30) log from another Foinaven well overlain on an inverted 30° angle stack. The data were inverted using a constrained sparse spike algorithm for which the EI log provided the basis for the constraints and was used to QC the result. An EI log provides an absolute frame of reference and so can also calibrate the inverted data to any desired rock property with which it correlates. In the case of Foinaven, a strong correlation was found between EI(30) and hydrocarbon pore volume, and this relationship was used to estimate the in-place volumes for the field from the inverted 30° seismic seismic vo volume. lume. Figure 4 shows a section from from the inverted 30° volume used to design the trajectory of the first high-angle development well. The oil sands correlated closely with the areas of low elastic impedance. The EI volume was used to design the trajectories of all subsequent development wells.
Figure 1. Comparison of an AI curve with a 30° EI curve for the Foinaven discovery well 204/24a-2. 438
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The EI formula is an approximation and may not be applicable in all circumstances; however, however, the loss of accuracy is easy to calculate and minimize (Appendix, part 3). In most situations, more general seismic data quality issues and particularly uncertainty in the estimation of incidence i ncidence angle are probably larger than errors in the implied reflectivity from the EI values. Estimating Poisson’s ratio from seismic data has prompted much comment in the literature and was the sub ject of a workshop at SEG’s 1998 Annual Meeting. One approach is to invert a 90° angle stack (see part 2 of the Appendix) which, in theory, has amplitudes that are
between. This allows the the use userr to construct as high high an angle stack as is stable and then to calibrate or invert it using the equivalent EI log. Because of the difficulties and uncertainties of constructing angle stacks, a method of quality-controlling the results using available well data is important. EI provides a simple mechanism to produce synthetic seismograms for variable incidence angle. A V p term can be factored out of the EI expression and the remaining angle-dependent expression can be used in place of the density log in conventional synthetics software (equation 1.3). The V p log is then calibrated with a time-depth relationship in the usual
approximately in Poisson’s ratio. However, the proportional constructiontoofchanges quantitatively accurate Poisson’s ratio stacks is notoriously difficult difficul t because of sensitivity to residual moveout and bandwidth variations. EI can provide an optimum compromise. Part 4 of the Appendix shows how one variant of EI has values equal to AI at normal incidence and to (V p/Vs)2 at 90° (this b being eing closely related to Poisson’s ratio) with a smooth transition
way.Figure 5 shows near- and far-offset ties to a west of Shetlands well. There is much variation of amplitudes
Figure 2. Detail from Figure 1, but with the EI(30) curve scaled so that the shale baseline is approximately the same as the AI curve. This shows the percentage decrease in impedance at the oil-sand interface is greater than 30° at normal incidence, incidence, consistent with the class III response of these sands.
Figure 3. Part of an EI(30) log overlain on the inverted 30° angle stack. The log log was used to const constrain rain a conventional poststack sparse spike inversion and to QC the result.
Figure 4. A section through th the e inverted 30° volume, showing tthe he path of the fi first rst development wel well. l. The location location of oil-bearing sands encountered by the well correlates with the areas of low elastic impedance (yellow). 440
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with offset in this area, and the two angle stacks are quite different. Despite this, both well ties are of good quality. A principal principal benefit benefit of EI within within BP has been been its valu valuee as a communication and integration tool. EI allows AVO information to be displayed in a way that can be understood more intuitively by nongeophysical specialists. It is easily incorporated into petrophysical systems allowing AVO information to be communicated throughout the earth-science community. community. EI can be used to display rockproperty data, either from wireline or core measurements, in a way that can be directly related to far-offset stacks. Shear-wave data are now recorded routinely in many
of an AI and EI curve is often simpler to relate to the seismic response than, th an, say, Vs or Poisson’s ratio logs. An example is shown in Figure 6 that is a standard BPpetrophysical BP petrophysical display from the Gulf of Mexico. With this type of data established within a petrophysical database the EI concept can help with more general rock-property studies. Figure 7 shows AI/EI crossplots of data from 19 Gulf of Mexico wells for shales, brine, and oil sands. By measuring average impedance values we can quickly estimate the AVO response of various lithology combinations. This particular data set, for example, shows that the percentage increase in amplitude from 0-
wells so the calculation of, say, an EI(30)The logcombination within any petrophysical package is straightforward.
30° a shale/brine sand interface (~18%) is(~17%). almost exactly the for same as for a shale/oil sand interface So, for
Figure 5. Low- and high-angle synthetic ties for a west of Shetlands well. The left side of the display is a conventional AI synthetic match match to a 10° angle stack. The right si side de is a 30° EI-based synthetic tied to a 30° angle stack.
Figure 6. Standard petrophysical display for a Gulf of Mexico well (MC619-1). The two right tracks show the AI and EI(30) curves. In this example, the upper sand would be expected to generate little response at normal incidence and a tough-peak pair at far offsets. 442
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these data, AVO AVO gradient would be a poor fluid fl uid indicator. However, looking at average values only tells part of the story. Figure 8 shows simplified, Gaussian frequency curves of the AI and EI distributions of the three lithologies. / g r o . g e s . y r a r b i l / / : p t t h t a e s U f o s m r e T e e s ; t h g i r y p o c r o e s n e c i l G E S o t t c e j b u s n o i t u b i r t s i d e R . 4 3 2 . 0 8 1 . 0 0 1 . 2 9 1 o t 4 1 / 1 1 / 2 1 d e d a o l n w o D
In each case the normalized standard deviations of the EI(30) data are less than those of the AI data. The area of overlap between the the oil and brine-sand values at 30° is less than half that at normal incidence. I find that many data sets have this characteristic (i.e., that EI values are more
Figure 7. AI against EI(30) crossplots of data from 10 Gulf of Mexico wells for shales (left), brine sands (center), and oil sands (right). Average AI and EI values can be read from the histograms and from these reflection coefficients for any lithology combination combination at normal and 30° incidence. These data show almost tthe he same percentage increase in amplitude for a shale/brine-sand interface as a shale/oil-sand interface.
Figure 8. Gaussian curves equivalent to the histograms of Figure 7 showing the distribution of AI and EI(30) values for the three lithologies. The normalized standard deviations for the shale, brine-sand, and oil-sand AI data are 0.15, 0.12, and 0.11. They are 0.10, 0.08, and 0.09 for the EI data.
Figure 9. The relationship between oil saturation and AI (left) and EI(30) (right) from core sample measurements from the Foinanven Field. 444
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uniform than AI values for a given lithology). litholo gy). This implies that most forms of amplitude analysis would be less ambiguous at higher incidence angles than lower. Finally, an example of using EI to display the results of core measurement data. The Foinaven Field is the sub ject of a 4-D, 4- D, time-lapse seismic experiment and various core sample measurements have been made to calibrate the results. Figure 9 shows the relationship between oil saturation, AI, and EI(30). Clearly, Cl early, the far offsets are more sensitive to changing saturation than the near ones. In summary EI is pragmatic technology. It allows at least first-order AVO effects to be incorporated routinely
tion, using no specialist software and with minimal increase in effort. This allows for routine extraction of quantitative AVO information from large 3-D volumes. LE Acknowledgments: Th Acknowledgments: Thee examp examples les ar aree the work of many people. people. I’d especially acknowledge and thank current and former BP Atlantic Margins colleagues Mike Cooper, Dave Cowper, Robert Hanna, Mike Currie, and Dave Lynch and our joint venture partners Shell UK for support and encouragement. Also Ed Meanley, Sue Raikes, Terry Redshaw, Stan Davis, and Wayne Wendt for additional help and both Shell and BP Amoco for permission permission to publish this paper paper.. Corresponding Correspon ding author: Patrick Connolly,
[email protected]
into seismic and rock-property analysis and interpretaAppendix 1) Derivation. Equation 1.1, a well known linearization of the Zoeppritz equations for P-wave reflectivity, reflectivity, is accurate for small changes of elastic parameters for subcritical angles.
substituting K for Vs2/V p2 and rearranging
(1.1) where
but sin2θtan2θ = tan2θ – sin2θ, so
Note that had we used only the first two terms of (1.1), then the above and following expressions differ only by changing the tan2θ to sin2θ. We substitute again ∆lnx for ∆x/x;
and where
now if we make K a constant we can take all terms inside the ∆s;
and similarly for the other variables (NB. For ease of notation, the “bars” will be omitted from the averaged Vs2/V p2 ratios.) We require a function f(t) which has properties analogous to acoustic impedance, such that reflectivity can be derived from the formula given below for any incidence angle
and finally we integrate and exponentiate (i.e., remove the differential and logarithmic terms on both sides), setting the integration constant to zero: (1.2)
Call this function EI (elastic impedance), and use the alternative log derivation for reflectivity which is accurate for small to moderate changes in impedance;
An alternative form, with a V p term factored out, can be used for generating synthetics (see main text). (1.3) 2) Angle stacks. The ideal angle stack has amplitudes that relate to a specific incidence angle over a long time win-
and so,
dow and has enhanced signal to In aother words it should approximate as closely as noise. possible band-limited constant angle reflectivity sequence. The construction of an angle stack requires knowledge of the relationships between offset and incidence angle and between angle 446
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and amplitude. These are both potentially complex areas but if we restrict ourselves to first-order approximations then angle stacks can be calculated with little extra effort beyond conventional stacking. These approximations limit applicability to data for which the two-term moveout and Dix equations equatio ns are valid and for which amplitudes are proportional to sin2, (where is incidence angle). This effectively means layer-cake geometry, offset less than depth and incidence angle less than 30-35°. We are also limiting ourselves to an isotropic medium. This is probably the most severe limitation; increasing anisotropy will distort raypaths and alter AVO AVO behavior. Correctly balanced, “true” prestack amplitudes behavior. are also assumed. The following expression relates incidence angle to offset given the above constraints. Figure A1. Finite angle stack weighting functions.
(2.1) where
An intercept stack is seen to be similar to a near-offset stack and a midangle stack is similar to a far-offset stack. Higher angle stacks are projections beyond the range of recorded data which must exploit the difference between near and far offsets. This will inevitably be very sensitive to residual moveout and phase and bandwidth variations which explains some of the difficulty in trying to estimate Poisson’s ratio values from P-wave seismic data. The second way to produce an angle stack is to design
= incidence angle, x = offset, t0 = zero offset two way time vi = interval velocity vr = rms velocity
θ
There are essentially two methods of constructing angle stacks; the first uses linear combinations of intercept and gradient, and the secondThe usesformer averaging between appropriate muting functions. can also be reformulated as a weighted stack. As is well known from Shuey’s classic paper, amplitudes are linear with sin 2 up to about 30-35°. Therefore, using (2.1) we can estimate sin 2 for each sample through a CDP time slice and fit a regression regression line through through the amplitude values up to the maximum appropriate angle. Numerous AVO AVO attribute values can then be constructed from this line: the intercept, the gradient, Poisson’s ratio stack (the value at 0=90°) or the value for any intermediate angle (called Finite Angle Stack within BP). This entire process can be efficiently programmed to run almost as quickly as a conventional stack. Perhaps a more intuitive way to look at this process is to rearrange it as a weighted stack. The simple linear regression formula for intercept and gradient formula can
an appropriate muting function. Equation 2.1 can be integrated with respect to x to give the average value of sin 2θ for a range of offsets stacked between x1 and x2.
(2.2) Because amplitude is linear with sin 2, the average amplitude of the stack will also correspond to this angle. Either the outer or inner mute can be fixed and then its pair calculated. For example, for a high-angle stack the outer mute could be a 35° mute calculated from equation 2.1 possibly combined with the maximum offset-dependent upon acquisition geometry geometry.. The inner mute to provide some target angle could then be calculated from equation 2.2. In practice a fixed angle can usually only be achieved over some limited window. An example of this process is shown in Figure A2. (Note that the final muting functions should be smoothed to some extent.) This second method of angle stack construction has the advantage that only conventional stacking software is required. Its disadvantages are that it assumes regular geometry and is only capable of producing stacks at a limited range of angles. The regression approach is far more flexible. Of all the errors implicit in the EI/angle-stack approach almost certainly the largest is the estimate of incidence angle. Every effort should be made to minimize this and
be rearranged as weighting functions as follows, where X is sin2, Y is the equivalent amplitude value, and N is stack fold.
A linear combination of these will provide a weighting function for any desired angle stack A().
Figure A1slice shows a typical suite of for one time to output a range of weighting finite anglefunctions stacks from 0° (intercept) (intercept) tto o 90° (Poisson’s (Poisson’s ra ratio). tio). This This mor moree closely closely shows the similarity between this process and partial stacking. 448
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certainly this should include angles at each velocity control point whenreestimating constructingthe angle stacks. 3) Accuracy. The derivation of the EI formula (Appendix, part 1) requires the Vs/V p ratio in the exponentials be kept
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2 0 0 0
4 0 0 0
6 0 0 0
8 0 0 0
1 0 0 0 0
1 2 0 0 0
1 4 0 0 0
1 6 0 0 0
0
) s m (
5 0 0 0
1 0 0 0 0
1 5 0 0 0
2 0 0 0 0
2 5 0 0 0
3 0 0 0 0
3 5 0 0 0
4 0 0 0 0
) s m (
e m i t y
e m i t y a w o w t
a w o w t
Figure A2. A typical stacking velocity function and its Dix interval equivalen equivalent. t. From these and equation 2.1, an outer 35° mute to exclude the nonlinear AVO AVO is derived. Then, from equation 2.2, an inner-trace mute is calculated to give an average stack angle of 25°.
constant for the entire time series (or constant within any system in which absolute comparisons are being made). This reduces the accuracy of the derived reflectivity compared with that obtained directly from equation 1.1 for which the Vs/V p ratio can be set to be the average across each interface. Again substituting K for Vs2/V p2, if ∆K is the difference between the true local value and the constant value, then from equation 1.1 the error in the reflection coefficient is
This expression can be used to calculate the rms level of the error for the entire log. As an example, the error was calculated for the 204/24a-2 data and is displayed disp layed in Figure A3 as signal-to-noise for a range of K values. This provides a mechanism to obtain the optimum K value for any data set. In this example the signal-to-noise of 12 should be more than adequate for most purposes. For comparison the signal-to-noise from ignoring AVO and using an AI approximation is 1.3 which would be unacceptable for quantitative analysis. In general errors introduced by the EI approximation will probably be much smaller than those arising from the estimation of incidence angle. It is possible to derive a correction to an EI log such that the derived reflectivity is accurate for a locally averaged K value rather than the constant value. The correction is, however however,, recursive, dependent on the overburden and hence loses the prime advantage of the EI function— that it depends only on instantaneous local properties. If K is the locally averaged value and K is the constant value the following expression is calculated for each sample.
(3.1)
(3.2) R is the ratio of the correction co rrection for the (i-1)th sample to the ith sample so the correction works by calculating calc ulating the running sum of this expression and then multiplying the EI values by this factor (Figure A4). Figure A3. The ratio of the rms level of the R(30) reflectivity from (1.1) divided by the EI error term from (3.1) for a range of K values for the 204/24a-2 data. The maximum signal-to-noise is about 12 and corresponds to a K value of about 0.21. 450
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4) High-angle inversions. Both methods constructing angle stacks (outlined in Appendix, part for 2) are based on the first order AVO equation whereas the EI derivation (Appendix, part 1) is based on the second-order equation. Below 30°, this makes little difference, but if we wish to
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Figure A4. An example of the application of the EI correction (3.2) to the 204/24a-2 data. The effect of the correction is small for this example.
calibrate higher-angle stacks constructed using the regression projectionwhich method then we should use the first orderline EI1 variation I’ll denote with the subscript. As noted in Appendix 1, this is the same as the second-order version but with the tangent in the V p exponential term changing to a sine.
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(4.1) I’ve already shown that EI(0) = AI and from 4.1 we can now see that if we let K = 0.25 then EI1(90) = (V p/Vs)2. The absolute levels of EI1(90) will depend on exactly which value for K is being used but relative amplitudes should always be approximately proportional to (V p/Vs)2. And of course (V p/Vs)2 can be easily transformed into Poisson’s ratio which in appearance is a very similar function. Figure A5 shows a suite of EI 1 functions for the 204/24a-2 well with the gamma and log resistivity curves 2
for reference and showing comparisons with (V p/Vs) and Poisson’s ratio curves. The curve values have all been normalized. The optimized K value of 0.21 was used for the EI calculations, but the EI 1(90) is almost identical to the (V p/Vs)2 curve. EI 1 curves therefore provide a smooth transition between AI and (V p/Vs)2 functions. In principle a corresponding angle stack could be constructed and inverted for any desired angle but in practice these will become increasingly unreliable at higher angles. The EI concept, however,, allows an optimum balance to be chosen. however
1 curves with Figure of a suite EIthe (V /Vs)2A5. andComparison a Poisson’s ratio curveoffor 204/24a-2 p well. Gamma and log resistivity included for reference. The EI1 curves form a continuum between an AI and a (V p /Vs)2 curve. 452
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