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GEOPHYSICS, VOL. 63, NO. 3 (MAY-JUNE 1998); P. 948956, 10 FIGS., 2 TABLES.

Framework for AVO gradient and intercept interpretation

John P. Castagna, Herbert W. Swan, and Douglas J. Foster

ABSTRACT

Amplitude variation with offset (AVO) interpretationmay be facilitated by crossplotting the AVO intercept(A) and gradient (B). Under a variety of reasonable

petrophysical assumptions, brine-saturated sandstonesand shales follow a well-defined background trend inthe AB plane. Generally, A and B are negatively cor-related for background rocks, but they may be posi-tively correlated at very high VP/VS ratios, such as mayoccur in very soft shallow sediments. Thus, even fullybrine-saturated shallow events with large reflection co-efficients may exhibit large increases in AVO.

Deviations from the background trend may be indica-tive of hydrocarbons or lithologies with anomalous elas-tic properties. However, in contrast to the common as-sumptions that gas-sand amplitude increases with offset,or that the reflection coefficient becomes more negativewith increasing offset, gas sands may exhibit a variety

of AVO behaviors. A classification of gas sands basedon location in the AB plane, rather than on normal-incidence reflection coefficient, is proposed. According

to this classification, bright-spot gas sands fall in quad-rant III and have negative AVO intercept and gradient.These sands exhibit the amplitude increase versus off-set which has commonly been used as a gas indicator.

High-impedance gas sands fall in quadrant IV and havepositive AVO intercept and negative gradient. Conse-quently, thesesands initially exhibit decreasingAVO andmay reverse polarity. These behaviors have been previ-ously reported and are addressed adequately by existingclassification schemes. However, quadrant II gas sandshave negative intercept and positive gradient. Certainclassical bright spots fall in quadrant II and exhibitde-creasing AVO. Examples show that this may occur whenthe gas-sand shear-wave velocity is lower than that of theoverlying formation. Common AVO analysis methodssuch as partial stacks and product (A B) indicators arecomplicated by this nonuniform gas-sand behavior andrequire prior knowledge of the expected gas-sand AVO

response. However, Smith and Gidlows (1987) fluid fac-tor, and related indicators, willtheoreticallywork forgassands in any quadrant of the AB plane.

INTRODUCTION

Amplitude variation with offset (AVO) analysis for nonbri-ght-spot reservoirs is, in certain instances, facilitated by cross-plotting extracted seismicparameters (Smith and Gidlow, 1987;Hilterman, 1987). As pointed out by Smith and Gidlow (1987)and others, the seismically extracted AVO intercept (A) andgradient (B) terms, in the absence of hydrocarbon-bearingstrata, often form a well-defined background trend whencrossplotted. The deviation from this background trend can be

a hydrocarbon indicator.Interpretation of such A versus B crossplots is complicated

by (1) a lack of intuitive feel on the part of the interpreterfor the physical significance of the B term, (2) the confound-ing effects of the seismic wavelet, (3) scatter caused by poor

Manuscript received by the Editor August 26, 1996; revised manuscript received October 10, 1997.University of Oklahoma, SEC-810, School of Geology and Geophysics, 100 E. Boyd St., Norman, OK 73019-0628. E-mail: [email protected] Exploration and Production Technology, a division of Atlantic Richfield Co., 700 G Street, Rm 1340, Anchorage, AK 99501. E-mail:[email protected] ARCO Exploration and Production Technology, currently Mobil Exploration and Production Technology Center, P.O. Box 650232,Dallas, TX 75265.c 1998 Society of Exploration Geophysicists. All rights reserved.

seismic data quality, and (4) nonpetrophysical influences onthe extracted seismic parameters A and B. The purpose of thispaper is to address item (1) by providing simple formulas andrules of thumb that aid AVO crossplot interpretation.

One approach to facilitate interpretation is to keep the Aterm (whichis taken to be the normal-incidence P-wave reflec-tion coefficient,RP , and consequently is readily interpretable inphysical terms) for one axis of the crossplot, but to recombineA and B to provide a second axis more directly related to rock

properties. Hilterman (1987) and Verm and Hilterman (1995)use the change in Poissons ratio across the interface () forthe second axis, whereas Castagna (1991) and (1993) uses theshear-wave reflection coefficient (RS). Both these approachesuse the Wiggins et al. (1983) approximation, and assume that

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AVO Interpretation 949

the compressional to shear-wave velocity ratio (VP/VS) is 2,or at least known (see Spratt et al. 1993). The penalty for thisparameterization, of course, is that the quantities to be inter-preted can no longer be readily related to the observed CDPgather by physical inspection.

A second approach, which avoids this difficulty, is to inter-pret the A versus B crossplot directly with the help of refer-ence points such as the fluid angle of Foster et al. (1993) andthe fluid line of Foster et al. (1997). With some reasonablepetrophysical assumptions, we will show that equations for thebackground trendor fluid line canbe derived, and that theresulting relationships are simple and easy to apply. Combinedwith rules of Koefoed (1955), these equations provide a simpleframework for the interpretation of AVO crossplots.

ANGULAR REFLECTION COEFFICIENTS

Let us consider two semi-infinite isotropic homogeneouselastic half-spaces in contact at a plane interface. For an in-cident plane wave, the reflection coefficient variation with an-gle of incidence is given by the well-known Knott-Zoeppritzequations. These equations are unwieldy and defy direct phys-

ical insight. Fortunately, for small variations in layer parame-ters and angles of incidence commonly encountered in seismicreflection applications, these equations can be accurately ap-proximated (e.g., Bortfeld, 1961; Richards and Frasier, 1976;Aki and Richards, 1980; Shuey, 1985). Following Swan (1993),we express the Aki and Richards (1980) form of the Richardsand Frasier (1976) approximation in terms of the angular re-flection coefficients A, B, and C:

R() A + B sin2() + Csin2()tan2() (1)

where R is thereflection coefficient as a function of theaverageangle (taken as a further approximation to be the angle ofincidence) with

A =1

2

VP

VP +

; (2)

B =1

2

VP

VP 2

VS

VP

22VS

VS+

; (3)

C =1

2

VP

VP , (4)

where VP is the change in compressional velocity across theinterface (VP2 VP1), VP is the average compressional ve-locity across the interface [(VP2 + VP1)/2], is the change indensity across the interface (2 1), is the average density

across the interface [(2 + 1)/2], VS is the change in shearvelocity across the interface (VS2 VS1), and VS is the av-erage shear velocity across the interface [(VS2 + VS1)/2], withVP1, VS1, 1 and VP2, VS2, 2 being themediumproperties in thefirst (overlying) and second (underlying) media, respectively.Note that A is the linearized form of the normal-incidencecompressional-wave reflection coefficient (RP ).

RELATIONSHIPS BETWEEN A AND B

The parameters VP , VS, and are often highly correlated,with deviations attributable to hydrocarbons or unusual

lithologies. These correlations imply relationships between theangular reflection coefficients A and B.

Gardners equation

We assume that the background consists of brine-saturated sandstones and shales and, to first order, that densityis a constant factor times the velocity raised to some arbitrary

power g. Gardner et al. (1974) showed that g is about equal to1/4 for most sedimentary rocks. For small changes in velocity,we have

g

VP

VP . (5)

This relationship can be violated strongly at lithologic bound-aries, however. The significance of such deviations is discussedbelow.

VS versus VP relationships

Within a sequence of fully brine-saturated rocks of similarage and composition, we can often assume, to a first order, a

linear relationship betweenV

P andV

S without lithologic dif-ferentiation. For example, Castagna et al. (1985) showed thatbrine-saturated sandstones and shales often roughly follow agiven trend, referred to as the mudrock line. (Because vari-ance about this trend depends on local geology, we cannotoffera global variance number; however, deviations on the orderof 5% are common. Such deviations can significantly effectAVO analysis.) Rather than restrict ourselves to the particularcoefficients of themudrock line, let us assume simply that, foraparticulardepth windowat a given locality, there is some linearrelationship between VP and VS:

VP = mVS + c, (6)

where m and c are empirical coefficients.

General intercept versus gradient equation

Combining equations (2) through (6) yields a general equa-tion for background reflections in the AB plane:

B =A

1 + g

1 4

VS

VP

2

m+ g

VS

VP

. (7)

This general relationship allows for arbitrary perturbations ofthe Gardner and VP versus VS trend curves. We now considerfour special cases for the background trend.

The background trend: Constant VP/VS and constant density

IfVP/VS is constant, c must vanish, and

m = VP/VS = VP /VS = VP/VS. (8)

Setting = 0 (for constant density) in equation (2) while sub-stituting g = 0 and equation (8) into equation (7) yields

A = 12VP

VP (9)

and

B =

1 8

VS

VP

2A. (10)


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950 Castagna et al.

Equation (10) is equivalent to equation (14) in Foster et al.(1997).

The background trend: Constant VP/VS and Gardner density

Instead of assuming constant density, we now honorGardnersrelation(5). Settingg = 1/4and m = VP /VS (con-

stant VP/VS) yields

A = 58VP

VP (11)

and

B =4

5

1 9

VS

VP

2A. (12)

Equation (12) indicates that lines of constant VP /VSshould pass through the origin. Furthermore, the slope ofthe predicted background trend depends only on the back-ground VP /VS ratio. Figure 1 shows that as VP /VS in-creases, the slope of the background trend becomes more pos-itive (the trend rotates counterclockwise for A plotted alongthe x-axis).

It is useful to note that for the Wiggins approximation(VP /VS = 2.0), from equations (3) and (4) we have B = Airrespective of density and velocity relationships. Thus, scat-ter introduced by violation of the Gardner equation should besmall when the background VP /VS is about 2.

Since

RP RS 1

2

VP

VP

VS

VS

, (13)

where RS is the normal-incidence shear-wave reflection co-efficient, it is evident that RP RS is always about zero forconstant VP /VS. Furthermore, for the particular case whenV

P/V

S =

2.0, we have

R P RS (A + B)/2 = 0. (14)

Also note in equation (12) that B = 0 when VP /VS = 3.

FIG. 1. Zero-offset reflection coefficient (A) versus the AVOgradient (B) assuming constant VP/VS and Gardners rela-tion. The background trend rotates counterclockwise as VP/VSincreases.

The background trend: Linear VP versus VS

Although constant VP /VS may be a useful assumptionfor limited depth ranges, a more general assumption is a linearrelationship between VP and VS. From equation (6), we havedirectly

VP = mVS + c; (15)

VP

VS= m =

VP

VS

VP c

VP

; (16)

VS

VS=

VP

VP

VP

VP c

, (17)

and

RP RS 1

2

VP

VP

c

VP c

. (18)

From equation (18), we can view RP RS as the P-wave ve-locity contrast scaled by a slowly varying background term.RP RS will be magnified as the background velocity ap-

proaches the constant c from above and diminished at highbackground velocities. Similarly, the pseudo-Poissons ratioreflectivity, as defined by Smith and Gidlow (1987) and popu-larized by Vermand Hilterman(1995)amongothers is given by

2(RP RS), (19)

where is the change in Poissons ratio across the interfaceand is the average Poissons ratio.

Substituting g = 1/4 and

VS

VP =

1

m

1

c

VP

(20)

into equation (7) yields

B =4

5

1

1

(mVP )2(VP c)(9VP c)

A, (21)

with A given by equation (11).According to equation (21):

1) The background trend passes through the origin (A =B = 0).

2) As shown in Figure 2, as VP decreases [and VP /VSincreases according to equation (15)], the backgroundtrendslope becomes morepositive (rotates counterclock-wise). This figure assumes the shale mudrock trend slopem of 1.16 andan interceptc of 1.36 km/sgivenin Castagnaet al. (1985). Note that the background trend slopechanges dramatically for velocities lower than 2.5 km/s.One might expect greater difficulty in establishing a well-defined background trend at these lower velocities.

3) These conclusions are similar to those for a constantVP /VS ratio. However, for a given VP /VS, theslopes are not necessarily equivalent for the two cases.Keep in mind, that for a linear VP VS relationship, VP/VSvaries across the interface rather than remaining con-stant. This results in a different background trend at agiven VP /VS.


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4) Equation (21) reduces to equation (12) when c is zero.

Assuming constant density (g = 0), rather than Gardnersrelation, gives

B =

1

8

m

VS

VP

A =

1 8

VS

VP

2VP

VP c

A,

(22)

which is equivalent to equation (15) of Foster et al. (1997).

Background trend comparison

Figure 3 shows the calculated slope of the background trend(B/A) as a function ofVP /VS for

1) Constant density and constant VP/VS from equation (10)2) Gardner density and constant VP/VS from equation (12)

FIG. 2. Zero-offset reflection coefficient (A) versus theAVO gradient (B) assuming a linear VP versus VS trend(m = 1.16, c = 1.36 km/s) and Gardners relation. The back-ground trend rotates counterclockwise as VP decreases.

FIG. 3. B/A versus VP /VS for various background trendassumptions: constant VP/VS and constant density, constantVP/VS and Gardner density, mudrock VP versus VS and con-stant density, mudrock VP versus VS and Gardner density. Inall cases, B/A increases with increasing background VP /VS.

3) Constant density and linear VP versus VS computed forthe mudrock trend (m = 1.16 and c = 1.36 km/s), usingequation (7)

4) Gardner density (g = 1/4) and linear (mudrock) VP ver-sus VS from equation (7).

For the small elastic contrasts implicit in the Aki and Richards(1980) approximations, a linear relationship exists between the

background B and A. [Foster et al. (1997) show how this lin-ear relationship is violated for large elastic contrasts, but onlyif VP /VS = 2.] B/A increases with increasing backgroundVP /VS for all cases. It is important to note that the densityrelationship is less important than the VP versus VS relation-ship. The constant VP/VS and linear VP versus VS assumptionsgive distinctly different background trends. For the constantVP/VS cases, B/A becomes positive at VP/VS ratios of about 3and higher. Thus, a large negative A can result in a large nega-tive B leading to an amplitude magnitude increase with offsetfor nonpay background rocks. This could occur in very uncon-solidated sands and shales where VP and VS are not as wellcorrelated as in well consolidated rocks. See the comparison ofHamiltons (1979) data to the mudrock trend in Castagna et al.

(1985). For the linear VP versus VS relationship, B/A becomespositive only at very high VP /VS ratios approaching 10.

DEVIATIONS FROM PETROPHYSICAL RELATIONSHIPS

Imperfect adherence to the assumed background petro-physical relationships introduces scatter about the backgroundtrendonan A versusB crossplot.A key issuefor AVO interpre-tation is the magnitude of this variation relative to deviationscaused by hydrocarbons. Smith and Gidlow (1987) provide anexample where violation of the Gardner density assumptionis a second-order effect on the observed seismic data. In ourexperience, for binary sandstone/shale intervals, scatter dueto nonpetrophysical factors (i.e., anything that causes seismicamplitudes to differ from reflection coefficients) is often larger

than scatter introduced by imperfect adherence to Gardnerdensity and linear VP versus VS assumptions. As noted above,scatter due to violation of the density assumption is minimalat VP /VS ratios close to 2. For a given P-wave velocity,salt exhibits unusually low density (Gardner et al., 1974) and,therefore, may deviate significantlyfrom the background trend.Petrophysical trend curves for various lithologies are given inCastagna et al. (1993).

Of course, the most interesting deviation from backgroundpetrophysical relationships results from replacement of brinein the pore space by hydrocarbons, particularly gas, for whichthe effects are most pronounced. Assuming no associatedgeological or chemical effects, the mechanical replacement ofgas for brine using Gassmanns equations reduces VP /VS

and causes VP and to become more negative. Thus, byequations (2) and (3), for a top-of-sand reflection, partial gassaturation causes both A and B to become more negative thanfor the corresponding fully brine-saturated sand. This is illus-trated in Figure 4, which shows calculated AVO intercepts andgradientsfor pairs of shale/gas sand andshale/brinesand reflec-tors. Theshale over brine-sand reflection coefficientsvary fromstrongly positive to strongly negative,but were selected to havethe same average P-wave velocity. Intercepts and gradientswere then calculated using Gardners relation andthe mudrocktrend,and were found to lie along the straight linepredicted by


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952 Castagna et al.

equation (15). Gassmanns equations were then used to sub-stitute gas for brine in the sands. The reflection coefficientsfor the corresponding gas sands also fall along a straight lineto the lower left of the background trend. Each tie line movestoward more negative A and B from brine sand to gas sand.

Figure 5 shows brine sandgas sandtie linesfor 25 setsof pri-marily in situ sonic-log measurements in brine sands, gas sands,and shales reported in Castagna and Smith (1994). As noted in

FIG. 4. Brine sandgas sand tie lines for shale over brine-sandreflections having an average P-wave velocity of 3 km/s andwhich conform to Gardner and mudrock petrophysical trendcurves (g = 0.25, m = 1.16, c = 1.36 km/s) and Gassmannsequations.

FIG. 5. The movement of AVO reflection coefficients as gas replaces brine within a sandy layer below shale for 25 sets of shale,brine-sand, and gas-sand velocities and densities reported in Castagna and Smith (1994).

that paper, most of the shear-wave velocities were measureddirectly with dipole or full-waveform sonic logs, or measuredin the laboratory. Although the validity of sonic log readings ingas sands is questionable, the gas sands do tend to plot to thelower left of the brine sands. In addition to experimental error,geological variation such as porosity differences between thebrine sands and corresponding gas sands will cause deviationsfrom predictions made using Gassmanns equations. Note alsothat the background velocity is different for each brine sand,so one should not expect all of the datapoints to fall along asingle background trend. Nevertheless, the overall tendency ofthe gas sands to deviate from the brine sands is clear. This isthe basis for Smith andGidlows (1987) hydrocarbon detectionmethodology using the fluid factor (F), which is a measureof the deviation from the background trend:

F =VP

VP m

VS

VP

VS

VS. (23)

KOEFOEDS RULES AND GAS SAND CLASSIFICATIONS

Koefoed (1955) manually calculated reflection coefficientversus offset using the exact Zoeppritz equations for abouttwenty different models. His observations of the systematics ofreflection coefficient variation are known as Koefoeds rules.Shuey (1985) showed that these rules are readily discernedfrom the Richards and Frasier (1976) approximation. Here, wediscuss two of these rules, as broadened by Shuey (1985), inthe context of an A versus B crossplot (Figure 6). Consider aplane interface between two infinite half-spaces:


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Rule 1.... an increase of Poissons ratio for the underlyingmedium causes an increase of the reflection coefficient at thelarger angles of incidence.

Rule 2.When ... Poissons ratio for the incident mediumis increased, the reflection coefficient at the larger angles ofincidence is thereby decreased.

Although not statedexplicitly by Koefoed(1955), theserulesalso suggest that a decrease in Poissons ratio for the lowermedium causes a decrease in reflection coefficient at largerangles of incidence. This is the case for a classical top of gas-sand reflector as originallydescribed by Ostrander(1984). Oneshould keep in mind that by decrease at larger angles of in-cidence, Koefoed should be interpreted as saying becomemore negative at larger angles of incidence relative to the priorstate. In other words, B is made more negative by adding hy-drocarbons (thereby decreasing Poissons ratio) in the under-lying medium. A is also made more negative by the additionof hydrocarbons. The effect of adding hydrocarbons to a layerotherwise falling on the background trend of Figure 6 is forthe top of the unit to plot below the background trend (more

negative A and B). Conversely, the bottom of a gas sand shouldplot above the background trend (more positive A and B), as-suming the medium immediately underlying the gas sand hasproperties similar to the properties of the medium overlyingthe gas sand. Top of gas-sand amplitude increases with offsetonly for gas sands whose tops plot in the third quadrant of Fig-ure6. Here, A and B arenegative andamplitudebecomes morenegative (and greater in magnitude) with increasing offset.

The class I, II and III curves of Figure 7 show the Rutherfordand Williams (1989) classification for gas sands. Note that thisclassification is based only on the normal-incidence reflectioncoefficient (RP = A). Class I sands are high impedance rela-tive to the overlying shales. Class II sands have low normal-incidence reflectivity (small impedance contrast). Class III

FIG. 6. AVO intercept (A) versus gradient (B) crossplot show-ing four possible quadrants. For a limited time window,brine-saturated sandstones and shales tend to fall along awell-defined background trend. Top of gas-sand reflectionstend to fall below the background trend, whereas bottom ofgas-sand reflections tend to fall above the trend. AugmentedRutherford and Williams (1989) gas sand classes are also indi-cated for reference.

sands are lower impedance than the overlying shales (classicalbright spots), and exhibit increasing reflection magnitude withoffset. The importance of this classification has been in demon-strating that reflection coefficientsneed not increase with offsetfor gas sands as was commonly assumed previously. In par-ticular, for high-impedance gas sands, reflection coefficientsdecrease with increasing offset. Unfortunately, one could con-clude from the first three curves that gas-sand reflection coeffi-cients always become more negative with increasing offset forall three classes. In fact, a fourth class of sand (class IV) existsforwhichthe introductionof gascauses itsreflection coefficientto become more positive with increasing offset, yet decrease inmagnitude with increasing offset.

This augmented Rutherford and Williams (1989) classifica-tion for gas sands is also superimposed on Figure 6. Considerclass III gassands falling in quadrant III of Figure 6. They havenegative A and B, and reflection magnitude increases versusoffset. Class IV gas sands, which fall in quadrant II, also havea large negative A but positive B. These are true bright spots,but reflection magnitude decreases with increasing offset. Thisis illustrated in Figure 8 for a low-impedance brine sand whichfalls on the background trend. Although the reflection coeffi-cients are large, the Shuey (1985) two-term approximation isgood to about 30 local angle of incidence. For angles less thanthis, B is positive for both brine and gas sands, and magnitudedecreases with increasing offset in both cases. Since the AVOgradient from a class IV brine sand may be almost identical

FIG. 7. Plane-wave reflection coefficients at the top of eachRutherford and Williams (1989) classification of gas sand.Class IVsands, notdiscussed by Rutherford andWilliams, havea negative normal-incidencereflectioncoefficient, but decreasein amplitude magnitude with offset.


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954 Castagna et al.

to the gradient from a class IV gas sand (see Figures 5 and 8),these gas sands may be difficult to detect by comparing partialoffset stacks.

Table 1 summarizes the possible AVO behavior for thevarious types of gas sand. We find either an AB quadrantidentification or an augmented Rutherford and Williams clas-sification to be more informative than one based only on thenormal-incidence compressional-wave reflection coefficient.

EXAMPLE: LOW-IMPEDANCE GAS SAND OVERLAIN

BY HIGH-VELOCITY UNIT

Class IV gas sands frequently occur when a porous sandis overlain by a high-velocity unit, such as a hard shale (e.g.,siliceous or calcareous), siltstone, tightly cemented sand, or acarbonate. Table 2 gives well log VP , VS, and density for a gassand and an overlying shale and tight unit (courtesy of JeremyGreene, ARCO International Oil and Gas Co.) As shown inFigure 9, when the gas sand is overlain by shale, the AVO in-tercept (A) is large and negative and the AVO gradient (B) isnegative. This falls in quadrant III of Figure 6 and representsa typical well-behaved Rutherford and Williams (1989) class

III gas sand. However, when the overlying unit is a tight unit

FIG. 8. Plane-wave reflection coefficient versus angle of inci-dence for the top of a class IV (quadrant II) gas sand, and thecorresponding brine-sand reflection. The model parametersare: shaleVP = 3.24 km/s, VS = 1.62 km/s, = 2.34 gm/cm

3;brine sandVP = 2.59 km/s, VS = 1.06 km/s, = 2.21 gm/cm

3;gas sandVP = 1.65 km/s, VS = 1.09 km/s, = 2.07gm/cm

3.Thesolid lines are the full Zoeppritz solution. The dashed lines arethe two-term Shuey (1985) approximation.

Table 1. Top gas sand reflection coefficient versus offset behavior for the three Rutherford and Williams (1989) classes IIIIassuming a typical background trend with negative slope. Class IV sands, though not explicitly discussed by Rutherford and

Williams, may be considered to be a subdivision of their class III sands.

Class Relative impedance Quadrant A B Remarks

I Higher than overlying IV + Reflection coefficient (andunit magnitude) decrease with

increasing offsetII About the same as the III or IV Reflection magnitude may in-

overlying unit crease or decrease with offset,and may reverse polarity

III Lower than overlying III Reflection magnitude in-unit creases with offset

IV Lower than overlying II + Reflection magnitude de-unit creases with offset

(Figure 10), the AVO intercept (A) is large and negative, butthe AVO gradient (B) is positive. Thus, although one wouldclassify this reflection as class III based on compressional-wave impedance contrast alone, the reflector falls in quad-rant II of Figure 6, since its amplitude decreases with offset.Furthermore, the same gas sand produces very different AVObehavior depending on its overlying shale. Thus, it is incorrectto classify a reflector based on the properties of the sand alone.

To understand this unusual but highly significant behavior,we refer to the original Richards and Frasier (1976) approxi-mation as given in Aki and Richards (1980):

R() =1

2

1 4VS

2p2

+

1

2cos2

VP

VP

4VS2p2

VS

VS, (24)

FIG. 9. Richards and Frasier (1976) decomposition for aclass III (quadrant III) shale over gas-sand reflection show-

ing the AVO contribution from fractional changes in density, compressional velocity VP , and shear velocity VS.

Table 2. Well log velocities and densities for an East Africangas sand and overlying strata (provided by Jeremy Greene).

VP VS

Lithology (m/s) (m/s) (gm/cm3)Shale 2900 1330 2.29Tight unit 3250 1780 2.44Gas sand 2540 1620 2.09


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FIG. 10. Richards and Frasier (1976) decomposition for thesame gassand as in Figure 9, butnow overlain with a calcareoustight unit. Its total AVOresponse is now class IV (quadrant II),due to the reversal of its shear velocity component.

where p is the ray parameter (sin / VP ) and is the average of

the angle of incidence and refraction.Equation (24) can be used to approximately decompose

the contributions to the reflection coefficient variation withoffset by changes in VP , VS, and density (). As shown inFigures 9 and 10, the contribution due to the density contrasthas a positive slope in both cases, whereas the contribution dueto the VP contrast has a negative slope. However, the key pa-rameter is the shear-wave velocity contrast. Note that at zerooffset, the shear contrast contributes nothing to the reflectioncoefficient. However, whenVS ispositive(asisthecasefortheshale over gas sand example; Figure 9), the shear contributionbecomes more negative with increasing offset, thereby enhanc-ing the total amplitude increase with offset. On the other hand,when VS is negative (tight unit over gas sand; Figure 10), the

shear contribution becomes more positive with increasing off-set. The net result is a small decrease in the total reflectioncoefficient with increasing offset, thereby resulting in a classIV sand in quadrant II, having a positive B.

Based on equation (24), we can generalize for class III andIV gas sands that an increase in shear-wave velocity across thetop sand interface enhances the amplitude increase with off-set, whereas a decrease in shear-wave velocity diminishes theamplitude increase with offset and may result in an amplitudedecrease with offset. It is important to note that whereas classIV gas sands exhibit unexpected absolute AVO behavior ac-cording to established rules of thumb and are difficult to inter-pret on partial offset stacksor using product (A B) indicators,they do not confound A versus B crossplot based indicators

such as Smith and Gidlows (1987) fluid factor.

CONCLUSIONS AND DISCUSSION

Reasonable petrophysical assumptions for sandstone-shaleintervals result in linear background trends for limited depthranges on AVO intercept (A) versus gradient (B) crossplots.In general, the background trend B/A becomes more posi-tive with increasing background VP /VS. Thus, if too large adepth range is selected for A versus B crossplotting, and back-ground VP /VS varies significantly (as would be caused bycompaction), a variety of background trends may be superim-

posed. This would result in a less well-defined background re-lationship. For very highVP /VS, as would occur in very soft,shallow, brine-saturated sediments, the background trend B/Abecomes positive,and nonhydrocarbon-related reflectionsmayexhibit increasing AVO and show false positive anomalies onA B product displays or improperly calibrated fluid-factorsections.

Deviations from the background trend may be indicative ofhydrocarbons. This is the basis for Smith and Gidlows (1987)fluid factor and related indicators. Inspection of the AB planereveals that gas sands may exhibit variable AVO behavior. Wesuggest that hydrocarbon bearing sands should be classifiedaccording to their location in the AB plane, rather than bytheir normal-incidence reflection coefficient alone.

Class I gas sands have a positive normal-incidence reflec-tion coefficient, lie in quadrant IV, and decrease in amplitudemagnitude with increasing offset faster than the backgroundtrend. Class II gas sands have a small normal-incidence co-efficient (less than 0.02 in magnitude), but achieve a greateramplitude magnitude than the background at sufficiently highoffsets. Polarity reversals are common with this type of reflec-tor, which can lie in either quadrants II, III, or IV. Class IIIgas sands have a strongly negative normal-incidence reflectioncoefficient, which becomes evenmore negative with increasingoffset. These sands lie in quadrant III. Class IV gas sands alsohave a negative normal-incidence reflection coefficient, but liein quadrant II and decrease in amplitude magnitude with off-set. They do so more slowly than background reflections withthe same intercept.

AVO product indicators (such as A times B) will show pos-itive anomalies only for quadrant III gas sands. Interpretationof product indicators andpartial stacksis difficult without priorknowledge of the expected gas-sand quadrant. Alternatively,the fluid factor and related indicators will theoretically workfor any class of gas sand in any quadrant.

ACKNOWLEDGMENTS

Thisresearchwas supported by ARCOExploration and Pro-duction Technology, Mobil Exploration and Production Tech-nology Center, and the Gas Research Institute under con-tract number 5090-212-2050. Special thanks are owed to KenTubman, Jeremy Greene, Tim Fasnacht, and Manik Talwanifor their support.

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