shear-wave elastic impedance

7/30/2019 Shear-Wave Elastic Impedance

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It is well known that if we are able to estimate acousticimpedance (AI) and a parameter related to shear-wavevelocity from seismic data, our ability to discriminate

between different lithologies and fluid phases will increase.Prestack inversion on individual CDP gathers and invert-ing directly for VP, VS, and density have been tested in sev-eral ways, but the estimated parameters are often poorlydetermined.

A more robust approach is to apply poststack inver-sion on partial stacks. For inversion of the near-offset stack,AI can be calculated directly from well logs. However, forthe far-offset stack, we need to derive an equivalent of theacoustic impedance that can be used to calibrate the non-zero-offset seismic reflectivity. Connolly (1998, 1999)derived such an equivalentelastic impedance (EI)anddemonstrated how, by using elastic impedance logs (which

requires shear-wave logs), he was able to perform well cal-ibration and inversion of far-offset data.

This article describes a new functionshear-wave elas-tic impedance (SEI)for linking converted-wave stacks towells using a linearization of the Zoeppritz equations. SEIis similar to EI but adapted to P-S converted seismic data.It can be computed from acoustic log data (P- and S-wavevelocities and density) and used for well calibration,wavelet estimation, and inversion ofP-S reflectivity dataleading to improved interpretability of converted wavedata. (Equation 4 in Appendix 1 is the key mathematicalformula in this approach to SEI).

Example of SEI. To test the SEI technique, we used a 3-Dfour-component (4-C) ocean-bottom cable (OBC) surveyacquired in 1997 that covers approximately 10 km2 of theStatfjord Field (Rogn et al., 1999). Both P-wave and con-verted-wave data were 3-D prestack time-migrated. Thehydrophone (P) and the vertical geophone (Z) componentwere summed to attenuate receiver side water-layer rever-

berations and both data sets (PZ and P-S) were decimated.The summed P and Z data are hereafter denoted PZ data.

1222 T HE LEADING EDGE NOVEMBER 2000 NOVEMBER 2000 T HE LEADING EDGE 00

Shear-wave elastic impedanceKENNETH DUFFAUT, TRINEALSOS, Statoil, Trondheim, NorwayMARTINLANDR, Norwegian University of Science and Technology, Trondheim, NorwayHEGEROGN, NAZIHF AL-NAJJAR, Statoil, Trondheim, Norway

Figure 1. Logs from well A, including (left to right)gamma log superimposed on net sand; EI; Zs; SEI; andEI (blue) and scaled SEI (orange).

Figure 2. The elastic impedance well tie in P-Ptime(left) and the shear elastic impedance well tie in P-Stime (right).

Figure 3. Estimated wavelets for the PZ(green) and P-S (red) volumes on the left and corresponding amplitudespectra on the right.


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The area used in the poststack inversion consists of 70 in-lines and 110 cross-lines with a common bin size of 25 25 m. The estimated angle range for the individual stacksis 5-25 for the PZ volume and 7-47 for the P-S volume.

Figure 1 shows several logs from well A. The two mainreservoir units, the Brent Group and the StatfjordFormation, are mainly sandstone and shales. The CookFormation is the only prolific unit within the Dunlin Group

composed of sand interbedded with shales.The EI and SEI logs were computed by integrating

over the given angle ranges. In computing the SEI log, weestimated an average K (=VS/VP) value of 0.52 and an aver-age incidence angle of about 27. This explains the simi-larity between the shear-impedance (Zs) and the SEI login Figure 1. The EI and the SEI logs, scaled with an aver-age value for the reservoir interval, are broadly similar


Figure 4. Cross-sections of PZ(left) and P-S data (right) through Statfjord Field. Data in both cross-sections havethe same polarity.

Figure 6. Elastic impedance plotted against shear elastic impedance log (left). The EI (blue) and scaled SEI (orange)logs on the right show an opposite impedance contrast near the Top Cook interface.

Figure 5. Cross-sections through inverted PZ(left) and P-S seismic volumes with superimposed impedance logsfrom well A. The sections display absolute elastic and shear elastic impedance obtained from combining invertedseismic and low-frequency trends from log data.


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except near Top Drake, Top Cook, and Top Nansen.Figure 2 shows examples ofPZ and P-S seismic stacks

tied to elastic and shear elastic impedance logs in well A.The central part of the log interval (Top Dunlin [Drake] toTop Statfjord [Nansen]) has a good fit between syntheticseismograms and stacks. The discrepancy between thesynthetic and real data near the top and bottom of the reser-voir interval may be caused by the high deviation angleof the well (54) and variation in data quality. The syntheticseismograms for EI and SEI are computed using 1-D con-volution.

Wavelet analysis was done separately for the PZ andP-S volumes. In each case, data were matched to the com-

puted elastic impedance and the shear elastic impedancelogs at the wells. The PZ (green) and P-S (red) waveletsare close to zero phase (left side of Figure 3). The twowavelets have similar phase properties; however, thereare significant frequency bandwidth differences. Their cor-responding amplitude spectra are on the right of Figure 3.The passband of the PZ wavelet is centered at 20 Hz, andthe P-S wavelet is centered at 10 Hz.

Figure 4 shows final processed PZ and P-S stacks forin-line 430. Note the significant difference in the frequencycontent. This makes time interpretation challenging

because the seismic response of the reflecting interfaceswithin the PZ and P-S data appears quite different. Thevertical resolution is, however, quite similar for the PZ and

P-S volumes. There is a significant amplitude differencebetween the P-S and PZ synthetic seismograms (Figure 2)for the Cook interval. This amplitude difference is alsoapparent on the real seismic sections (Figure 4). Figure 6,a plot of EI versus SEI from well A, indicates that the sandyCook interval will be easier to detect in the inverted P-Svolume than the inverted PZ volume.

The corresponding inverted sections are shown (Figure5) with overlays of EI and SEI logs from well A. The datawere inverted using a sparse spike algorithm with con-straints from the EI and SEI logs. In the SEI section, a largecontinuous impedance contrast separates the CookFormation into low and high SEI layers. This impedanceincrease correlates with the porosity decrease through the

Cook interval. Asimilar impedance boundary is less appar-ent in the inverted PZ volume. Another interesting featureis a continuous low impedance layer below Top Statfjordin the EI section. A similar feature is less apparent in theSEI section, which is expected due to a smaller SEI con-trast across the Top Statfjord interface (Figures 2 and 3).This feature corresponds to the position of the gas-fluidcontact in the upper part of the Statfjord Formation. Figure7 compares seismically derived EI and SEI traces (blue) andtheir corresponding log impedance (red) at two blind wells.The seismically derived EI trace and the log-derived EImatch very well especially within the reservoir below theTop Viking. A good match is also obtained between theseismically derived SEI trace and the log-derived SEI inthe lowermost part of the reservoir interval.

Estimation of VP/VS from inverted seismic data. Twocubes are output from the poststack inversion of the OBCdata: an elastic impedance cube and a shear elastic imped-ance cube. These were combined to compute an instanta-neous VP/VS volume. Figure 8 shows a VP/VS cross-sectionof in-line 430 (position of line shown in Figure 9). ProperP-P and P-S event correlation is crucial because it has con-siderable effect on the outcome of this sample-by-sample

based VP/VS computation. Prior to VP/VS computation, theSEI volume is transformed to P-P time using interval VP/VSratios derived from the event correlation. In addition, toperform the computation, we need to choose a mean valuefor K (VS/VP = 0.5) and density (2.24 g/cm

3).To reduce the impact of minor mismatches in the event

correlation, average interval VP/VS maps can be computedfrom average impedance maps. Figure 9 shows attributemaps extracted over the upper part of the StatfjordFormation.

A band of low VP/VS trending nearly N-S appears inthe eastern part of the map. This coincides with the posi-tion of the gas-fluid contact within the formation. A sim-ilar feature is seen as a low-impedance boundary on theEI map to the left but not on the SEI map (middle).

Conclusions. A new quantity derived from acoustic welllog measurements, denoted shear-wave elastic impedance,is presented. Computing shear-wave elastic impedance inaddition to acoustic and elastic impedance logs makes itpossible to compare the relative strength between P-P andP-S events and to predict which interfaces might givestrong P to S conversions. When VS/VP is close to 0.5 atincidence P-wave angles around 30, the SEI log is approx-imately equal to the shear impedance log.

We have shown, with real data from the Statfjord Field,how SEI links converted-wave data to well logs. SEI can

be used for P-S data as EI is used for P-P data. The tech-nique is easy to implement in conventional interpretation

packages where the SEI log can constrain the poststackinversion. The output from such an inversion could bedirectly compared with the shear-wave elastic impedancecurve derived from acoustic well logs. Finally, elasticimpedance and shear-wave elastic impedance can be com-

bined to compute instantaneous VP/VS. The VP/VS estimateis based upon amplitude and may therefore serve as com-plementary (and more detailed) information for travel-time estimates ofVP/VS.

Suggested reading. SumicA new strategic tool for explo-ration and reservoir mapping by Berg et al. (EAEG Expanded

Abstracts, 1994). Calibration and inversion of nonzero-offsetseismic by Connolly (SEG 1998 Expanded Abstracts). Elastic


Figure 7. Blind well comparisons for EI (left) and SEI(right) for two wells show that the inversion (blue)corresponds fairly well with the log-derived imped-ance (red).


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impedance by Connolly (TLE, 1999). Well calibration ofseabed seismic data by Landr et al. (SEG 1999 Expanded

Abstracts). The Alba Field ocean bottom cable seismic sur-vey: Impact on development by MacLeod et al. (TLE, 1999).The Statfjord 3-D, 4-C OBC survey by Rogn et al. (TLE,1999). Elastic impedance inversion by VerWest and Masters(SEG 2000 Expanded Abstracts). Converted-wave imaging ofValhall Reservoir by Thomsen et al. (EAGE 1997 Expanded

Abstracts). LE

Acknowledgments: We thank Statoil and the Statfjord license partnersfor permission to present the sea bed seismic data.

Corresponding author: K. Duffaut, [email protected]

Appendix 1

Connolly (1998, 1999) derived the angle or offset equiv-alent of acoustic impedance, and used the term elasticimpedance (EI). According to Connolly, EI is defined as:

(1)

where EI() is the elastic impedance log at incidence P-wave angle ; VP, VS, and denote the P-wave and S-wavevelocity and density, respectively; and K is the averageVS/VP.

The basic assumption about elastic impedance is thatit should play the same role for angle-dependent P-wavereflectivity as AI does for zero-offset reflectivity. If that con-dition is fulfilled, one can apply poststack inversionschemes on partial stacks. For an incidence angle of zero,EI equals AI. However, note that the magnitude and unitsof EI vary with angle. VerWest and Masters (2000) pre-sented a dimensionless version of this equation. EI logsare now being used as a quick way of assessing AVO char-acteristics expected from a given well, and as a tool for cal-ibration and inversion of partial stack volumes.

An approximate equation (assuming weak contrastsand small angles) for P to S reflectivity, RPS(), as a func-tion of incidence P-wave angle at a plane interface is:

(2)


Figure 9. Average EI (left), SEI (middle), and VP/VS maps computed within the upper part of the StatfjordFormation. White line shows position of cross-section.

Figure 8. Cross-section through VP/VS volume with VP/VS logs superimposed (left) and a wiggle trace comparisonbetween seismically derived VP/VS (blue) and log-derived VP/VS (orange) at well A (right).

EI Vp1 tan

2

Vs8K

2sin

24K

2sin

2

( )

pp p p

=+ ( ) ( ) ( )1

Rps1

21

Vs

Vssin( ) ( )

p K K p + + +( )

2 4

Vs

Vssin

3

K K

Kp+ +

( )

1

2

2

4


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where

and

denote the fractional changes in density and S-wave veloc-ity across an interface and the bar denotes the average.

To relate SEI to P-S reflectivity in the same way that AIrelates to P-P reflectivity, the following equation must befulfilled:

(3)

Subscripts 1 and 2 refer to the layer above and below theinterface, respectively. To derive SEI with a similar depthtrend as acoustic, elastic, and shear impedance, we chooseto use the negative offsets of CCP (common conversionpoint) gathers. A one-term version of SEI (using only thefirst term in equation 2 in the derivation) is given byLandr et al. (1999). A higher-order approximation isobtained by combining equations 2 and 3

(4)

where

K is assumed constant in the derivation of equation 4. Foran incidence angle of zero, SEI(p=0) equals 1 (no con-trasts), corresponding to no P-to-S conversion (as expected).Figure A1 compares P-S reflectivity using the one-term SEIequation, the two-term SEI equation, and Zoeppritz equa-tion for a two-layer model. For incidence P-wave anglesabove 25, the one-term SEI equation is probably not suf-ficiently accurate. One should bear in mind that most con-verted waves are generated at higher incidence angles,hence we should use the two-term SEI equation (4).

Asimpler P-S reflectivity function using only the shearimpedance contrast can be written

(5)

where

denotes the shear impedance contrast and

is an angle dependent scalar.It is assumed in this derivation that the density con-

trast is small compared with the shear impedance contrast,and that the average VS/VP ratio equals 0.5. Figure A2 illus-trates the P-S reflectivity error between equations 5 and 3at incidence angles of 10, 30, and 40. The error curvesare computed by varying only the VP. Given the model inFigure A1 (VS/VP = 0.6), the error is approximately -15%for all three angles. For an incidence angle of 30 and Kequal to 0.5, SEI is approximately equal to the shear imped-ance. For other incidence angles and K values different from0.5, the error in P-S reflectivity increases.


Figure A1. Comparison of P-S reflectivity versus inci-dence P-wave angle for one- and two-term SEI equa-tions and Zoeppritz equation. Layer properties aregiven in the table.

Figure A2. Estimated P-S reflectivity error plotted

against VS/VP between the simplified express (equation5) and the two-term SEI equation (equations 3 and 4).

Vs

Vs

Rps Rps = =

+( ) ( )

( ) ( )

( ) ( )

p p

SEI p SEI p

SEI p SEI p

2 1

2 1

SEI Vsm n

( ), ,

pK p K p

=( ) ( )

m K p p p, ( ) ( ) ( ) ( )

= +4Ksin 11

21 2K sin

2

n K p p p, ( ) ( ) ( ) ( )

= +

+

+

( )1 2K sin 1

K 13

2K

1 2Ksin

2

RpsZs

Zs

( ) p Sp

( )

Zs

Zs

S p p p ( ) = sin cos2

shear-wave elastic impedance

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