EEI_2002

GEOPHYSICS, VOL. 67, NO. 1 (JANUARY-FEBRUARY 2002); P. 63–67, 6 FIGS.10.1190/1.1451337

Extended elastic impedance for fluid and lithology prediction

David N. Whitcombe∗, Patrick A. Connolly‡, Roger L. Reagan∗∗,and Terry C. Redshaw§

ABSTRACT

Constant angle projections of seismic sections can bedesigned to provide maximum discrimination betweenfluids or lithologies. The optimum projection for a noise-free, isotropic environment can be obtained using an ex-tension to the elastic impedance concept, which itselfis an extension of acoustic impedance (AI) to nonzeroangles of incidence. To achieve this, we modify the def-inition of elastic impedance (EI) beyond the range ofphysically meaningful angles by substituting tanχ forsin2θ in the two-term reflectivity equation. The primaryvariable now becomes χ rather than θ . We allow it tovary between −90◦ and +90◦, which gives an extensionof EI for any combination of intercept and gradient. Werefer to this form of elastic impedance as extended elasticimpedance (EEI).

In this paper we demonstrate that EEI can be tuned us-ing different χ values to be approximately proportionalto a number of elastic parameters, and we give EEI ex-pressions for shear impedance (SI), bulk modulus, shearmodulus, Lame’s parameter, and Vp/Vs. This leads to theidentification of different areas of EEI space that tendto be optimum for fluid and lithology imaging. Havingidentified an appropriate χ value, the equivalent seismicsection can be obtained from combinations of interceptand gradient stacks from routine AVO processing.

INTRODUCTION

The present work builds on several key pieces of work. Smithand Gidlow (1987) first showed that prestack data could bestacked with different weights to produce their fluid factor andpseudo-Poisson’s ratio sections for predicting fluids and lithol-ogy. Later, Dong (1996) showed how changes in bulk modulus

Presented at the 70th Annual Meeting, Society of Exploration Geophysicists. Manuscript received by the Editor December 5, 2000; revisedmanuscript received May 30, 2001.∗BP, GFU Business Unit, Burnside Road, Farburn Industrial Estate, Dyce, Aberdeen AB21 7PB, U.K. E-mail: [email protected].‡BP, Angola Business Unit, Compass Point, 79-87 Kingston Road, Knowle Green, Staines, Middlesex TW18 1DY, U.K. E-mail: [email protected].∗∗BP, 501 Westlake Park Boulevard, Houston, Texas, 77079-2696. E-mail: [email protected].§BP Research Centre, Chertsey Road, Sunbury Upon Thames, Middlesex TW16 7LN, U.K. E-mail: [email protected]© 2002 Society of Exploration Geophysicists. All rights reserved.

κ , shear modulus µ, and density ρ could be expressed in termsof the AVO parameters A, B, and C as defined by Aki andRichards (1980). Goodway et al. (1997) highlighted the use ofshear modulus and Lame’s constant, λ (or, more strictly, µρand λρ) as optimum lithology and fluid indicators.

These previous works were all formulated in the reflectivitydomain. Connolly (1999) introduced elastic impedance (EI) asa generalization of acoustic impedance (AI) for nonnormal in-cidence angle, enabling the benefits of inversion to be exploitedfor AVO data. He described derivations from both the two- andthree-term Zoeppritz linearizations (Aki and Richards, 1980)using the commonly used A, B, and C parameters:

R(θ) = A+ B sin2 θ + C sin2 θ tan2 θ, (1)

where θ is the average of the incidence and transmission anglesat a plane reflecting interface.

Connolly also showed examples of its use to provide a frame-work for calibrating and inverting angle stacks. He showed thatthe two-term formulation of EI could be expressed as a simplefunction of Vp,Vs, and density (α, β, and ρ):

EI(θ) = αaβbρc, (2)

where

a = (1+ sin2 θ),

b = −8K sin2 θ, (3)

c = (1− 4K sin2 θ),

and where K is a constant, usually set to the average value of(β/α)2 over the log interval of interest. This technology wasdeveloped and applied as a fluid imaging tool in the appraisaland development of the Foinaven field, West of Shetlands.

The EI function [equation (2)] was recently modified(Whitcombe, 2002) by introducing reference constants αo,βo, and ρo, which remove the variable dimensionality of

63

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64 Whitcombe et al.

equation (2) and provide an EI function which returns nor-malized impedance values for all angles θ :

EI(θ) = αoρo

[(α

αo

)a(β

βo

)b(ρ

ρo

)c]. (4)

The present work integrates the previous reflectivity domainformulations with EI concepts and enables EI to be extendedto identify both fluids and lithology.

THEORY

Dong (1996) showed that the linearized Zoeppritz equationslead to the following approximation:

1κ = (3A+ B+ 2C)α2ρ

1.5, (5)

where 1κ is the change in bulk modulus across the interface.Various authors have noted the difficulty of extracting the C

term from real seismic data. Shuey (1985) examined the ratioC/A and noted that this parameter tended to lie between 0and 1. We define C/A as f . The value of f = 0.8 describesrocks that follow Gardner et al.’s (1974) relationship. Ratherthan determine C directly, we replace it with f A and select anf value appropriate to the rock properties in the area.

Using this substitution and dividing both sides of equa-tion (5) by κ , the average bulk modulus across the interface,allows us to determine bulk modulus reflectivity Rκ :

Rκ =(1κ

2κ

)=(

A+ B

3+ 2 f

)(3+ 2 f

3− 4K

). (6)

The first bracketed term can be considered in the form of afirst-order AVO equation, namely

R(θ) = A+ B sin2 θ. (7)

Therefore,

sin2 θκ = 13+ 2 f

. (8)

Setting f = 0.8 (Gardner’s relationship) results in sin2 θκ =0.22, or θκ = 28◦. The value Rκ is therefore approximately pro-portional to R(28◦). Note that f = 0.0 and f = 1.0 providesin2

θκ values of 0.33 and 0.20 (equivalent θκ of 35◦ and 26◦,respectively).

The second bracketed term in equation (6) can be consid-ered a scaling factor. If K and f are constant over the intervalof interest, this term will be a constant. For typical values off = 0.8 and K = 0.25, this scaling factor equals 2.3.

Thus, the reflectivity associated with a bulk modulus log canbe considered as a scaled AVO projection.

A similar treatment (C. Sondergeld, 1999, personal commu-nication) for Lame’s constant λ yields

1λ = (2A+ B+ C)α2ρ. (9)

Dividing both sides of this equation by λ, the average Lame’sconstant across the interface, allows us to determine Lame’sconstant reflectivity Rλ:

Rλ =(1λ

2λ

)=(

A+ B

2+ f

)(2+ f

2− 4K

). (10)

Again, the first bracketed term can be considered in the form ofa first-order AVO equation with a projection angle θλ definedby

sin2 θλ = 12+ f

. (11)

Using Gardner’s relationship gives sin2 θλ= 0.36 or θλ= 37◦.Thus, Rλ is approximately proportional to R(37◦).

The second bracketed term in equation (10) can be consid-ered a scaling factor. If K and f are constant over the intervalof interest, this term will be a constant. For typical values off = 0.8 and K = 0.25, this scaling factor will equal 2.8.

These observations are consistent with the observation thatfluids often better image at far offsets, and they imply thatthe elastic impedance equation can deliver impedance propor-tional to bulk modulus and Lame’s constant.

For lithological imaging we look to shear modulus µ. Thisis attractive because, for rocks that follow Gassmann’s (1951)assumptions, shear modulus will be independent of fluid fill.Following Dong (1996),

1µ = (C − B)α2ρ

2. (12)

Dividing both sides of equation (12) by µ, the average shearmodulus across the interface, allows us to determine shearmodulus reflectivity Rµ:

Rµ =(1µ

2µ

)=(

A− B

f

)(f

4K

). (13)

Thus, as for bulk modulus and Lame’s parameter, the re-flectivity associated with a shear modulus log can be consid-ered a scaled AVO projection. For typical values of f = 0.8and K = 0.25, the scaling factor will equal 0.8.

The projection angle θµ is defined by

sin2 θµ = −1f. (14)

For f = 0.8, a value for sin2 θµ of−1.25 is obtained. In this casethere is no real solution for θµ, though we can clearly combineA and B together with any weights we choose. The two-termAVO linearization defines a straight line for reflectivity againstsin2(θ). This line can be extended infinitely in either direction,as indicated in Figure 1. It only approximates physical realityfrom zero to about 30◦, as outlined by Shuey (1985), and has

FIG. 1. Prestack amplitude observations are fit with thetwo-term AVO linearization. The linear model can be extrapo-lated beyond the range of measured data and beyond the rangeover which sin2θ is physically definable.


Extended Elastic Impedance 65

an equivalent angle for values of sin2θ between 0 and 1. Forvalues of sin2 θ less than zero or greater than unity, there is noequivalent real angle. But we can measure A and B from realdata between zero and 30◦ and hence construct reflectivity byextrapolation for any value of sin2 θ that we choose.

EXTENDED ELASTIC IMPEDANCE

Our objective is to express the preceding reflectivity equa-tions in terms of the corresponding impedance relationships.There are two difficulties with using the current EI definition.There is the requirement for |sin2

θ | to exceed unity, and re-flectivity values may exceed unity as sin2 θ increases; clearly, noimpedance contrast can give rise to a reflectivity value greaterthan unity (unless we allow negative impedance). In practice,this will mean that as |sin2

θ | approaches and passes unity,the EI log, by its current definition, will become increasinglyinaccurate.

To compensate for these difficulties, we make two changes tothe current definition of EI. First, we replace sin2

θ by tanχ sothe equation is defined between ±∞ rather than the 0–1 limitimposed by sin2

θ . We also define a scaled version of reflectivityto be normal reflectivity multiplied by cosχ , which ensures thatreflectivity never exceeds unity.

The first substitution in the two-term linearized Zoeppritzequation gives

R= A+ B tanχ, (15)

from which we derive

R= (Acosχ + B sinχ)cosχ

. (16)

Now introducing RS, or scaled reflectivity,

RS = Rcosχ, (17)

results in

RS = Acosχ + B sinχ. (18)

The EI equivalent of equation (18) is then

EEI(χ) = αoρo

[(α

αo

)p(β

βo

)q(ρ

ρo

)r], (19)

where

p = (cosχ + sinχ),

q = −8K sinχ, (20)

r = (cosχ − 4K sinχ).

We call this extended elastic impedance, or EEI.Scaled reflectivity has the property that it ranges from a value

of A at χ = 0◦ to a value of B at χ = 90◦. The EEI equiva-lent to χ = 0◦ is of course acoustic impedance and at χ = 90◦

EEI will have a reflectivity corresponding to B. We refer toEEI(χ = 90◦) as gradient impedance or GI.

Alternatively, by defining

AIo = αoρo, (21)

equation (19) can be written as

EEI(χ) = AIo

[(AIAIo

)cos(χ)( GIAIo

)sin(χ)]. (22)

To reiterate, our intent is not to produce a model that repli-cates observed reflectivity beyond 30◦ and up to the criticalangle. We are defining a useful model that can be constructed

FIG. 2. The EEI functions for various χ values for well204/24a-2. Note the inverse correlation between EEI(χ = 90◦)and EEI(χ =−90◦).

FIG. 3. Comparisons between elastic parameters and equiva-lent EEI curves for well 204/24a-2, showing the high degree ofcorrelation. The EEI function is defined as a function of theangle χ , not the reflection angle θ .


66 Whitcombe et al.

for any real linear combination of A and B, effectively extrap-olating the observations along the sin2

θ axis in either directionbeyond the physically observed range. We will show that thisapproach allows us to define a single function that is propor-

FIG. 4. The correlation coefficients between EEI and an Sw anda gamma-ray curve for a range of values of χ .

FIG.5. Comparison of the optimized EEI curves with the targetSw and gamma-ray logs.

FIG. 6. Maps generated using EEI data sets using χ values tuned to optimize the imaging of lithology (χ =−51.3◦) and fluids(χ = 12.4◦), respectively.

tional to several different elastic parameters, depending on thevalue of χ used.

Figure 2 shows an example from the West of Shetlands well204/24a-2, where we display EEI across the spectrum of χvalues (−90◦ to +90◦). Note that the χ =−90◦ log is propor-tional to the inverse of the χ =+90◦ log, as expected fromequation (22).

We can therefore now give expressions for the elastic con-stants directly in terms of EEI by substituting the values forsin2 θ in terms of f as given in equations (8), (11), and (14). Byassuming values for f of 0.8 (Gardner) and 0.25 for K , theseequations can be simplified as follows:

κ ∝ EEI(χ = 12.4◦)≈EI(θ = 28◦),

λ ∝ EEI(χ = 19.8◦)≈EI(θ = 37◦),

µ ∝ EEI(χ = −51.3◦) (no equivalent real value of θ).

In addition to the rock parameters already considered, equa-tion (19) also predicts, for K = 0.25, relationships between EEIand shear impedance (=βρ) and the ratio α/β:

EEI(χ = −45◦) ∝ (βρ)1.414 (23)

and

EEI(χ = 45◦) ∝(α

β

)1.414

. (24)

So, including these with K , λ, µ, and AI, we have defined afunction that can be made to be approximately proportionalto six elastic parameters by adjusting a single variable χ . Theequation allows for expressions that are interpolations betweenany of these parameters, which lets us fine-tune any propertywe require.

EXAMPLES

Several approximations have been introduced to reach thispoint: the replacement of C by a scaled version of A, fixing K asconstant, and the approximations inherent in the Zoeppritz lin-earizations. To test their validity, an example from the 204/24a-2well is used. In these examples f = 0.76 and K = 0.21 wereused. In each case the correlation coefficient between the elas-tic parameter and the EEI curve was ≥0.97. Figure 3 displays(α/β), λ, K shear impedance, and µ against their equivalent


Extended Elastic Impedance 67

calculated EEI logs. Note the high degree of correlation inall cases. The high correlation between (α/β) and EEI is per-haps surprising, given the constant K assumption [i.e., constant(β/α)2] in the EI and EEI formulation.

FLUID AND LITHOLOGY IMPEDANCE

Bulk modulus and Lame’s parameter tend to lie within anarea of EEI space with values of χ from about 10◦ to 30◦,and shear modulus lies within a range of χ from about−30◦ to−90◦. These areas are therefore likely to be good starting pointsto look for optimum fluid and lithology impedance functions,respectively. One way to search for the optimum EEI functionis to maximize the correlation between the EEI curve and atarget curve such as water saturation Sw or gamma for a rangeof χ values.

Figure 4 show the results of correlating a range of EEIcurves with an Sw and gamma-ray curve. All curves have low-frequency trends removed. For these data the highest corre-lation with the Sw curve is at χ = 35◦, higher than the bulkmodulus or Lame’s parameter values. The lithology correlationcoefficient peaks at a positive value for χ = 71◦, with large neg-ative values of χ also having high correlation. Figure 5 showsthe Sw and gamma-ray curves, together with the EEI curveswhich optimally correlate. In practice, noise in the seismic sec-tion, the presence of anisotropy, velocity errors, etc., may meanthat the optimum angle to use with real data may be differentfrom that shown by using these log plots, but the degree ofcorrelation indicates what is possible with good-quality data.

As a second example, in Figure 6 we show maps of averageEEI averaged over a 25-ms time gate measured from top reser-voir for the Forties field in the central North Sea. The images aretuned to shear and bulk modulus. These have been obtained byusing f = 0.8, giving χ angles of−51.3◦ and 12.4◦ (sin2θ valuesof −1.25 and 0.22, respectively). Here we are clearly imagingthe channel systems and, potentially, remaining hydrocarbonswithin these channels.

CONCLUSIONS

We have introduced an extension of the EI concept andhave demonstrated that, by adjusting a single parameter χ ,EEI can provide a good approximation to acoustic impedance,bulk modulus, Lame’s parameter, α/β ratio, shear impedance,and shear modulus and can be optimized as a fluid or lithol-ogy discriminator. Having established a desired value of χ , anequivalent seismic section can be constructed using conven-tional impedance inversion and AVO processing techniques.

ACKNOWLEDGMENTS

We thank Keith Nunn for his comments, which improvedthe submitted manuscript. We also thank the board of BP forpermission to publish this paper.

REFERENCES

Aki, K. I., and Richards, P. G., 1980, Quantitative seismology: W. H.Freeman & Co.

Connolly, P., 1999, Elastic impedance: The Leading Edge, 18, No. 4,438–452.

Dong, W., 1996, A sensitive combination of AVO slope and interceptfor hydrocarbon indication: 58th Conference and Technical Exhibi-tion, Eur. Assn. Geosci. Eng., paper M044.

Gardner, G. H. F., Gardner, L. W., and Gregory, A. R., 1974, Formationvelocity and density—The diagnostic basics for stratigraphic traps:Geophysics, 39, 770–780.

Gassmann, F., 1951, Uber die Elastizitat Poroser Medien: Viertel-jahrschrift der Naturforschenden Gesellschaft in Zurich, 96, 1–23.

Goodway, B., Chen, T., and Downton, J., 1997, Improved AVO fluiddetection and lithology discrimination using Lame petrophysical pa-rameters, “λρ”, “µρ”, and λ/µ fluid stack”, from P and S inver-sions: National convention, Can. Soc. Expl. Geophys., ExpandedAbstracts, 183–186.

Shuey, R. T., 1985, A simplification of the Zoeppritz equations: Geo-physics, 50, 609–614.

Smith, G. C., and Gidlow, P. M., 1987, Weighted stacking for rockproperty estimation and detection of gas: Geophys. Prosp., 39, 915–942.

Whitcombe, D. N., 2002, Elastic impedance normalization: Geophysics,67, 60–62, this issue.


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