SHEAR-WAVE VELOCITY ESTIMATION IN POROUS ROCKS.pdf

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Geophysical Prospecting 40, 195-209,1992

S H E A R - W A V E V E L O C I T Y E S T I M A T I O N I NP O R O U S R OC K S: T H E O R E T IC A L F O R M U L A T I O N ,P R E L IM I N A R Y V E R I F IC A T I O N A N D

A P P L I C A T I O N S 'M . L . G R E E N B E R G ' and J . P. C A S T A G N A 3

A B S T R A C TGREENBERG,.L . and CASTAGNA,.P. 1992. Shear-wave velocity estimation in porous rocks :theoretical formulation, preliminary verification and applications. Geophysical Prospecting 40,

Shear-wave velocity logs are useful for various seismic interpretation applications,including bright spot analyses, amplitude-versus-offset analyses and multicomponent seismicinterpretations. Measured shear-wave velocity logs are, however, often unavailable.

We developed a general method to predict shear-wave velocity in porous rocks. If reli-able compressional-wave velocity, lithology, porosity and water saturation data are available,the precision and accuracy of shear-wave velocity prediction are 9% and 3%, respectively.The success of our method depends on: (1) robust relationships between compressional- andshear-wave velocities for water-saturated, pure, porous lithologies; (2) nearly linear mixinglaws for solid rock constituents; (3) first-order applicability of the Biot-Gassmann theory toreal rocks.

We verified these concepts with laboratory measurements and full waveform sonic logs.Shear-wave velocities estimated by our method can improve formation evaluation. Ourmethod has been successfully tested with data from several locations.

195-209.

I N T R O D U C T I O NExplanations for porous rock response to elastic waves require knowledge of shear-wave velocity, yet data are often unavailable. Elastic wave theories (e.g. Kuster andToksoz 1974; OConnell and Budiansky 1974) that predict velocities invoke numer-ous simplifications which cannot account for the diversity of sedimentary rocks.A combination of empirical relations and robust aspects of theoretical modelare required if a generalized method for predicting shear-wave velocity is to besuccessful.

While many empirical studies have focused on predicting velocities from porosityReceived April 1991, revision accepted September 1991.ARC0 Oil and Gas Company, Houston, TX, U.S.A.* Atlantic Richfield Corp., Plano, TX, U.S.A.

195


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196 M . L . G R E E N B E R G A N D J . P. C A S T A G N Aand rock composition, these have necessarily had limited success, because velocityalso depends on effective stress, porous rock structure (pore shape distribution) anddegree of lithification. An alternative approach to shear-wave velocity predictionexists, because these factors affect compressional- and shear-wave velocity in asimilar way and because compressional-wave velocity data are widely available. Wehave investigated the use of measured compressional-wave velocity, with porosityand lithology data, to predict shear-wave velocity.Relationships between compressional- and shear-wave velocities are well-knownfor brine-saturated, pure lithologies (Pickett 1963). These relations are readilyapplied to mixed lithologies because, as predicted from Hashin and Shtrikman(1963) bounds, the solid rock constituents combine almost linearly.A general method for shear-wave velocity prediction must account for rockswhich are not brine saturated. Biots (1956) theoretical work on elastic-wave propa-gation in porous media, once coupled to measurable elastic parameters by Geertsmaand Smit (1961), showed that Gassmanns (1951) equations are generally applicableto statistically isotropic, porous rocks in the limit of zero-frequency wave propaga-tion. When applied to real rocks, Gassmanns equations yield a first-order predic-tion for the dependence of elastic velocities on the properties of pore-filling fluids(e.g. Castagna, Batzle and Eastwood 1985).We developed a method for predicting shear-wave velocity in porous, sedimen-tary rocks which couples empirical relations between shear- and compressional-wave velocities with Gassmanns equations. Mixed lithologies and fluids areaccounted for. Laboratory measurements are used for method verification. We findthat the mean predicted shear-wave velocity precision (2 standard deviations) is 9%and the accuracy is better than 3%. We provide preliminary verification of ourtechnique using well log data. Shear-wave velocity errors in well log applications arefound to be comparable to laboratory results.

M E T H O D O R E S T I M A T I N GH E A R - W A V E V E L O C I T YDevelopment of coupled equations

Empirical relations between body shear-wave velocity pi and compressional-wave velocity a, in brine-filled rocks of pure (monomineralic) lithology have beenadequately represented by polynomials (e.g. Castagna, Batzle and Kan 1992). Shear-wave velocity & in a homogeneous composite (multimineralic) brine-filled rock, canbe approximated by averaging the harmonic and arithmetic means of the constitu-ent pure porous-lithology shear-wave velocities. This averaging is analogous toobtaining a Voigt-Reuss-Hill average for the elastic moduli. For a homogeneouscomposite with compressional-wave velocity a,, the porosity can be partitionedamong L constituents such that a1- a2- . - aL - ac. This partitioning approx-imation improves as the porosity increases or the constituent grain velocities con-verge. These observations specify an approximate relation between pc and ac forbrine-filled rocks given by


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S H E A R -W A V E V E L O C I T Y E S T I M A T I O N 197where a,, p, are the compressional- and shear-wave velocities, respectively, of acomposite rock, L is the number of pure (monomineralic) porous constituents, X i isthe dry lithology volume fraction of lithological constituent i, a i j are the empiricalcoefficients and 0 I N ihe order of polynomial i.Gassmann (195 1) obtained equations for a homogeneous isotropic rock relatinga, and & to the crystalline frame and the pore-fluid properties. The effects ofvarying the fluid saturation on a, and pc can be estimated (e.g. Domenico 1976) bycombining Gassmann's equations, the mass balance for a composite rock density,and Wood's (1957) equation; the latter is applied to estimate the homogeneouslymixed fluid incompressibility. By combining these equations with (1) and compositegrain incompressibility, an estimation of & for rocks of mixed lithology at arbitrarybrine saturation can be obtained.For homogeneous, isotropic, linearly elastic media, the body-wave velocities aregiven by

a; = ( K c + [I4/31Pc)/Pc (24and

D; = PC I PC (2b)where K , is the incompressibility, p, is the shear modulus and p c is the massdensity. Gassmann's equations can be expressed as extensions of (2), relating K , and,U to the frame, grain and pore-filling fluid properties (e.g. Domenico 1976). ThusK, = KD + (r+ 4 ) 2 / ( r K > 1+ 4 K i ' ) (34and

P c { K 4, sw>= P C { K 4, 11 = P C { K 41 ? (3b)where r = (1- 4 - K $ K ), and 4 is the volume fraction of pore space (porosity),X is the vector of dry constituent mineral volume fractions, { X i } ,S , is the volumefraction of pore space filled with brine, K,, is the incompressibility of homoge-neously mixed mineral grains, K , = K,{X, 4 } is the incompressibility of compositesolid rock frame (skeleton) and K,, is the incompressibility of homogeneously mixedpore-filling fluids.The frame moduli KD and p c are implicitly dependent on rock texture anddegree of lithification and explicitly dependent on lithology and porosity. ,For an arbitrary homogeneous mixture of pore-filling fluids and minerals, themass balance, Wood's equation and the Voigt-Reuss-Hill estimate for mixed grainincompressibility, are respectively,

gc.


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198 M . L . G R E E N B E RG A N D J . P. C A S T A G N Awhere p , and K , are the brine mass density and incompressibility, respectively,pnwand K , , are the non-wetting fluid mass density and incompressibility, respectively,and pi and K i are the dry constituent i mass density and incompressibility, respec-tively.Solution of coupled equations

In order to estimate the shear-wave velocity pc for a rock with brine saturationS , , 2)-(4) must be combined to give a brine-saturated compressional-wave velocityestimate so that (1) can be applied. Equation (1) estimates the brine-saturated shear-wave velocity, which is then related to the shear-wave velocity at s, using (3b) andratios obtained from (2b) and (4a). Measurements of the compressional-wave veloc-ity tcC (typically at S,), lithology, porosity and brine saturation are required, as arethe specification of the constituent properties { a i j , , ,K , ,pnw K , , ,pi, Ki}.

We developed an iterative method since an analytical solution is intractable. Aniteration parametrization permitting a robust choice of starting model (seed) isrequired, as multiple solutions (mostly non-physical) exist. Since typical in situ varia-tion of compressional-wave velocity with brine saturation is less than 20%, ascheme to optimize the difference between compressional-wave velocities at S , and100% brine saturation ( S , = 1)can be initiated.

Using subscript S to denote properties at brine saturation S, and subscript 1 todenote properties at 100%brine saturation, we can write

@lC= 1 + J)@SC, (5 )where 6 is a slack variable with a probable optimal value in the range O.M.2.

Combining (1x4)yields a second estimate of cl lc (sic below), which can becompared with ( 5 ) to establish an iteration convergence criterion. a ic can beobtained in four steps (Fig. l), provided (4a, b, c) are pre-evaluated at measured S ,and S , = 1.Step 1. Use (5) and apply (1) followed by (2b) at S , = 1:

Step 2. Apply (2a) and (3b) at S , using p c from step 1 :K sc = Ps c d c - 4/31Pc.

Step 3. Solve (3a) for K D and apply result at S, using Ksc from step 2:KD = KgC(AKSC - )/(AKgC - + [KSdKgCI) ,

whereA = +Kg>' + (1 - +)K,&.

Step 4. Apply (2a) and (3a) at S , = 1 using p c from step 1 and KD from step 3:

When the measurements and the required shear-wave velocity are for 100% brine-saturated rock, 6 = 0 and iterations are unnecessary. In this case, steps 1, 2 and 3


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SHEAR-WAVE VELOCITY E STIM ATIO N 199

Solve at SW = 1:

Step 2:

Step 3:

Step 4:

I equation 3a for K l c It no

Solve at measuredSW:.quation 2a for ale'estimate error = ~ I C ' - ic-

terate -----++FIG.1. Procedure for estimating shear-wave velocity, given measurements of compressional-wave velocity, lithology, porosity and fluid saturations. Constituent properties and a, - pctrend coefficients are required a priori (e.g. Tables 1 and 2 for grain properties; Batzle andWang (1992) for fluid properties). Pre-evaluate equations 4 or { p l c , p,,, K I f c , K,, , , K, ,} .Nomenclature defined in text.

still may be followed to estimate the frame moduli, although (1) is sufficient toestimate plc.

M O D E LV E R I F I C A T I O NWe verified our model using laboratory and well log measurements of sedimentaryrocks of quartz, calcite, dolomite and clay minerals. Clay is defined here as the drymineral fraction of shale and is assumed to have the properties of illite.


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200 M . L. G R E E N B E R G A N D J . P . C A S T A G N ATABLE . Representative regression coefficients forshear-wave velocity (&[km/s]) versus com pressional-wave velocity (ac, [km/s]) in pure porous lithologies:pc = ai2a; + a,, ac + aio Castagna et al . 1992).

Lithology ai2 ai1 aioSandstone 0 0.80416 -0.85588Limestone -0.05508 1.01677 - .03049Dolomite 0 0.58321 -0.07175Shale 0 0.76969 -0.86735

Our method for shear-wave velocity estimation requires a priori knowledge ofthe polynomial coefficients for pure porous lithology pc versus M~ trends, and of theintrinsic mineral and fluid properties. Castagna et al. (1992) reported coefficients ofultrasonic pc versus tlC polynomial regressions for sandstone, limestone, dolomiteand shales (Table 1). M ineral mass density, compressional-wave velocity and shear-wave velocity were given by Ca rm ichae l(l989) for qua rtz an d calcite, And erson andLieberman (1966) for dolom ite, and Eastw ood and Castagn a (1986) for illite.Mineral incompressiblities are determined from mineral velocities, using (2) (Table2). For most sedimentary basins, the dependence of intrinsic mineral properties onpressure an d tem perature is negligible.

We modelled pressure- and temperature-dependent properties of brine, methaneand oil in the subsurface according to Batzle and W ang (1992). In o ur la bora torymeasurements of air-saturated rocks, the effects of extrinsic gas property dependenceon shear-wave velocity estimations were not detectable. For modelling with labor-atory data, we used constant properties for water (mass density = 1.0 g/cm3 andincompressibility = 2.2 G P a ) and nitrogen (mass density = 1.297 x 10 -3 g/cm3 andincompressibility = 1.417 x 10 -4 G Pa ), the latter values are given by Dom enico(1976).

TABLE. Intrinsic elastic properties of representative rock-formingminerals : mass density (p , [g/cm3]), compressional-wave velocity(ag [km/s]), shear-wave velocity (p , [km/s]) and incom press ibility

Mineral p , (g/cm3) ag (km/s) & (km/s) K , (GPa)( K , CGPal).

Quartz 2.649 6.05 4.09 37.88CalciteZ 2.712 6.53 3.36 74.82Dolomite3 2.87 7.05 4.16 76.42Illite4 2.66 4.32 2.54 26.16Estimated from a,, p, and p , using (2). armichael (1989).Anderson and Lieberman (1966). Eastwood and Castagna (1986).


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S H E A R - W A V E V E L O C IT Y E S T I M A T I O N 201VeriJication with laboratory data

Shear-wave velocities were estimated and com pared to labo ratory me asureme ntsfor sets of brine- and air-filled rocks. We incorporated data sets from Rafavich,Kendal and Todd (1984),Han, Nur and M organ (1986)and Tosaya (1982)with ourown. Only those sets with high-quality lithology, porosity, density, compressional-and shear-wave velocity da ta w ere selected. Erro rs in quantitative mineralogies werenot uniform as volumetric da ta was acquired by X-ray diffraction in som e cases andby point count on thin section in others. The mineralogy consisted principally ofquartz, calcite and dolomite. Feldspar, when present, was modelled as clay. Thecombined clay and feldspar content did not exceed 3% of dry mineralogy byvolume. The com bined volumes of q uartz, calcite, dolom ite, clay an d feldspar werenot less than 95% of dry mineralogy; the neglected remainder was principallyanhyd rite for carbo nate sam ples.Laboratory velocity mc zsurements were acquired by ultrasonic methods. Allmeasurements were at ambient temperature. Confining pressures varied from 0 to200 M P a and pore pressures varied from 0 t o 170 M P a. Multiple velocity measure-ments on a rock at different pressures were treated as distinct data sets. For eachsample, mineral densities were adjusted from values in Table 2 (on a mineral frac-tion vo lume-w eighted basis) to satisfy the mass balance (4a).The results of our shear-wave velocity estimation method are compared to mea-sured shear-wave velocities in Figs 2 and 3 for water- and air-filled rocks, respec-tively. Characteristically, large errors in quantitative mineralogy analyses precludethe estimation of the model erro r; however, a com bination of model and da ta errorcan be assessed from regression results. 93%of variance in 336 water-filled sam plesand 89% of variance in 341 air-filled samples are accoun ted for by ou r model, basedo n corre lation coefficients.The upper limit of the modelled shear-wave velocity precision is twice the esti-mated data variance shown in Figs 2 and 3. Fo r w ater-filled rocks, the m odel sh ear-wave velocity variance is ? 83 m/s, whereas for air-filled rocks, f 82 m/s. The meanfractional variance, within the range 2.2-3.8 km/s, is 0.045 for water- and air-filledrocks. The uppe r limit of precision is 9.0%.

We estimate model shear-wave velocity accuracy as the mean error determinedfrom an estimated versus measured shear-wave velocity regression line. There is asmall negative bias in the estimated shear-wave velocities of water- and air-filledrocks since both regression lines lie beneath the diagonal, over the range of mea-sured shear-wave velocities. We estimate mean accuracies of 30 m/s and 62 m/s forwater- and air-filled rocks, respectively, within the range 2.2-3.8 km/s. Mean frac-tional error, within the range 2.2-3.8 km/s, is 0.016 and 0.033 for water- and air-filled rocks, respectively. Th e overall accuracy is no worse th an 3.3%.Our statistical assessment of laboratory data modelling results indicates thatshear-wave velocity estimation is robust. The estimated accuracy (3%) is substan-tially better than the estimated precision (9%), indicating that our method isunbiased. Combining ultrasonic p, versus a, plots with Gassmanns (195 1 ) zero-frequency mo del app ears to yield reasonable results when applied to u ltrasonic da ta.


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202 M . L . G R E E N B E R G A N D J . P. C A S T A G N AMEASURED SHEAR-WAVE VELOCITY (km/s)

2.0 2.4 2.8 3.2 3.6 4.0

FIG.2. Com parison of estimated shear-wave velocity with laboratory shear-wavevelocity measurements (DccM,), in km/s, for water-filled rocks. No. samples: 336, least-squaresregression: &, = 0.9965 &M, - 0.0172, variance: f 3.6 m/s, correlation coefficient:+0.9648.Laboratory results were derived from nearly shale-free rocks, due to inherent diffi-culties in laboratory work with shales.

Verijication with well log dataCompressional- and shear-wave velocities are typically measured with welllogging tools at about 10 kHz. In brine-saturated rocks, ultrasonic bc versus a,

trends may be used to estimate pc at well log frequencies because dispersion isnegligible (Eastwood and Castagna 1986). In gas-saturated rocks, however, disper-sion may be significant. A profile of decreasing drilling fluid saturation with radialdistance from the borehole, often observed after drilling in initially gas-saturatedrocks, can further complicate matters, because well logs do not all sample equivalentvolumes of rock.

Empirically, complicating factors in gas-saturated rocks are difficult to separate.When our method for estimating shear-wave velocity was applied to well logs, wefound good agreement with measured shear-wave sonic logs without an explicitdispersion correction. For seismic applications, a dispersion correction (Winkler1986) may be desirable. We preferentially use a fluid saturation estimate derivedfrom logs which sample a rock volume comparable to that sampled by sonic logs.


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S H E A R - W A V E V E L O C IT Y E S T I M A T I O N 203MEASURED SHEAR-WAVE VELOCITY (km/s)

2.0 2.4 2.8 3.2 3.6 4.0

FIG.3. Comparison of estimated shear-wave velocity GOccE)) with laboratory shear-wavevelocity measurements (&)) in km/s, for air-filled rocks. N o . samples : 341, least-squaresregression: = 0.9670&,,, + 0.0447, variance: 86.7 m/s, correlation coefficient:+0.9393.Example 1 : outh Te xa s clasticsA high-quality multichannel sonic waveform-derived shear-wave transit-time log(Aron, Murray and Seeman 1978, paper at 53rd SPE meeting, Houston) from SouthTexas is compared with the shear-wave velocity estimated by an approach compara-ble to the methodology described above (Fig. 4). The shear-wave velocity was ini-tially estimated using lithology determined from a gamma ray log (assuming a linearresponse to shale volume) and the porosity estimated from density log data(assuming constant grain density). Large discrepancies between the estimated andmeasured shear-wave velocities led to a re-examination of the available log datawhich revealed high gamma count rates in relatively shale-free intervals, probablydue to the presence of uranium. The lithology and porosity were subsequently re-estimated by inverting a linear system of equations representing the environmentalresponse of sonic, neutron and density logs. The revised estimates of the shear-wavevelocity agree closely with measurements (Fig. 4).The semblance among array sonic log waveforms at each acquisition depth pro-vides an indication of shear-wave log reliability (Siegfried and Castagna 1982, paperat 23rd meeting of the Society of Professional Well Log Analysts, Corpus Christi).Figure 5 shows the standard deviation of differences between estimated and mea-sured shear-wave transit times, among data with similar (fO.05) semblance, as afunction of semblance for a 0.73 km interval (4800 data points) in the South Texaswell. As the semblance approaches one (i.e. as the transit-time measurement quality


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204 M . L. G R E E N B E R G A N D J . P. C A S T A G N A

POROS

hEYIwn

P-WAVE S-WAVESONICLOGLITHOLOGY ms/krn

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