castagna_etal_1985_gpy000571[1].pdf

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ophysics VOL. 50, NO. 4 (APRIL 1985); P. 571-581, 25 FIGS., 2 TABLES.

compressional-wave and shear-wave velocitieselastic silicate rocks

*, M. L. Batzle*, and R. L. Eastwood*

ABSTRACTNe w velocity data in addition to literature data

derived from sonic log, seismic, and labora torymeasurements are analyzed for elastic silicate rocks.These data demonstrate simple systematic relationshipsbetween compressional and shear wave velocities. Forwater-saturated elastic silicate rocks, shear wave veloci-ty is approxim ately linearly related to compressionalwave velocity and the compressional-to-shear velocityratio decreases with increasing compressional velocity.Labora tory data for dry sandstones indicate a nearlyconstant compressional-to-shear velocity ratio withrigidity approxima tely equal to bulk modu lus. Idealmodels for regu lar packings of spheres and crac kedsolids exhibit behavior similar to the observed water-saturated and dry trends. For dry rigidity equal to drybulk m odulus, Gassm anns eq uations predict velocitiesin close agreemen t with da ta from the water-saturatedrock.

INTRODUCTIONThe ratio of compressional to shear wave velocity (VP/V,) or

and other rock-forming minerals isreflection seismology and formation evaluation.

paper, we investigate the VV , ratio in binary andmixtures of quartz, clays, and fluids.

The classic paper by Pickett (1 963 ) popularized the use of the

Figure 1, reproduced from Picketts paper, shows thes. Nations (1974) Eastwood and C astagna (1983)

et al. (198 4) indicated tha t VP/V, for binary m ix-

Figure 2 shows compressional and shear wave velocities forin the literature (e.g., Birch, 196 6;

1982 ). These velocities are calculated from single

crystal data and represent isotropic agg regates of grains. Alsoplotted is an extrapolation of Tosa yas (19 82) empirical relationfor VP and V, in shaly rocks to 100 percent clay and zeroporosity. The pos ition of this clay point depends upon theparticular clay mineral, and it is plotted only to indicateroughly the neighborhood in which velocities of clay mineralsare to be expected. Interpretation of sedimentary rock veloci-ties should be done in the context of these mineral velocities.

We establish general VP/V, relationships for elastic silicaterocks by comparing in-situ and laboratory data with theoreti-cal model data . Available velocity information is examined fordata from water-saturated mudrocks and sandstones. Weexamine laboratory data from dry sandstone and compare withsimple sphere pack and cracked media theoretical model data.Data from water-saturated rocks are similarly investigated. Theresults of the relationships established between VP and V, arethen applied to calculations of rock dynamic moduli. Finally,the general VP-< trends versus depth are estimated for GulfCoast elastics.

EXPERIMENTAL TECHNIQUESWe have combined a variety of in-situ and laboratory

measurements for elastic silicate rocks that includes our dataand data extracted from the literature. In-situ compressionaland shear wave velocities were obtained by a number of sonicand seism ic methods which are described in detail in the litera-ture cited. Due to the development of shear wave logging, it isnow possible to obtain shear wave velocities routinely over awide range of borehole/lithologic conditions with few samplingproblems and in quantities which were never before available(Siegfried and Castagna, 1982).Both our laboratory data and laboratory data we assembledfrom the literature were obtained using the pulse transmissiontechnique. Compressional and shear wave velocities are deter-mined for a sample by the transit time of ultrasonic pulses(approximately 200 kHz to 2 MHz). The jacketed sample isplaced in either a pressure vessel or loa d fram e so that stressescan be applied. Tem perature and pore fluid pressure can also becontrolled. Details of the techniques are found in Gregory(1977) or Simmons (1965).

received by the Editor March 12, 1984; revised manuscript received October 12, 1984.Oil and Gas Company, Exploration and Production Research, P. 0 Box 2819, Dallas, TX1985 Society of Exploration Geophysicists. All rights reserved.571

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FIG. 1. Laboratory measurements on limestones? dolomites,and sandstones from Pickett (1963). VP= compress ronal veloci-ty, v, = shear velocity.

0.0 0.5 1.0 1.5 2.0 2 5 3.0 35 4.0 45 50vs (KM i SEC)FIG. 2. Compressional and shear velocities for some minerals.

lm ,o V S K M I SECGREEN RIVER SH*LE; POOIO. GREGORY. AN0 GRAY (1988,

. AXIAL LO*owG 10,cca PSI Yp= 4.39. s,=s,=2.42. HYoRosTATlC LO*DING. 10,mPsI p=4_4.s,= 2.wJ.vs, = 2.64TOSAYA HWOCLAY POINT VP = 3.4, Y* = 1.60 COTTON ALLEY SILTSTONE0 PIERRE SHALE

X ARM DATA

FIG. 3. Ultrasonic laboratory measurements for various mud-rocks.

OBSERVATIONS IN MUDROCKSWe define mudrock as elastic silicate rock composed pri-

marily of clay- or silt-sized particles (Blatt et al., 19 72). Lithifiedmuds are composed primarily of quartz and clay minerals.Owing to the difficulty associated with handling of mostmudrocks, laboratory measurements on these rocks are notcommonly found in the literature. Measurements that do existare generally biased toward highly lithified samples.

Figure 3 is a V,-versus-V, plot of laboratory measurem entsfor a variety of water-saturated mudro cks. For reference, linesare drawn from the clay-point velocities extrapolated fromTosa yas data (VP = 34 km/s, C; = i.6 kmjsj to caicite andquartz points. The data are scattered about the quartz-clay line,suggesting that I$ and V, are principally controlled by mineral-ogy.In-situ sonic and field seismic measu rements in mu drocks(Figure 4) form a well-defined line given by

VP= 1.16v, + 1.36, (1 )where the velocities are in km/s. In view of the highly variablecomposition and texture of mudrocks, the uniform distributionof these data is surprising. We believe this linear trend isexplained in part by the location of the clay point near a linejoining the quartz point with the velocity of water. We hypoth-esize that, as the porosity of a p ure clay increases, compres-sional and shear velocities decrease in a nearly linear fashion asthe water point is approache d. Similarly, as quartz is added topure clay, velocities increase in a nearly linear fashion as thequartz point is approached. These bounds generally agree withthose inferred from the empirical relations of Tosaya (1982); theexception is for behavior at very high porosities. The net resultis that quartz-clay-water ternary mixtures are spread along an

l.O-.

00 I I I I I I I0.0 0.5 1.0 1.5 2.0 2.5 3.0 35vs , KMlSECn KOERPERICH, 1979, SILTSTONE, SONIC LOG (15 KHZ)0 EASTWOOD AND CASTAGNA. 1983. W O LFCAM P SHALE. SONIC LOG (10 KHz)A OIL SHALE. SONIC LOG. (25 KHz,

+ LlNGLE AND JONES. 1977. DEYONIAN SHALE, SONIC LOG+ HAMILTON, 1979, PIERRE SHALE0 HAMILTON. ,979, GRAYSON SHALE0 HPIMILTON. 1979. JAPANESE SHALE. LASH, 1980. GULF COAST SEDIMENTS, VERTICAL SEISMIC PROFILEA SHALE, SONIC LOG, INVERTED STONELEY WAVE VELOCITIES. (1 KHZ,

-... HAMILTON. ,979. MUDSrONES--- ESENIRO, 1981. GULF co*.9 SEDIMENTS. SURF*CE WAVE INVERSION

@ TOSAY*S CLAY POINT (EXTR*POLATION FROM L*BOR*ToRY OAT*,

FIG. 4. Compressional and shear wave velocities for mudrocksfrom in-situ sonic and field seismic measurem ents.



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P-wave and S-wave Velocities in Clastic Rocks 573elongate triangular region loosely defined by clay-water,quartz-clay, and quartz-water lines.

Figure 4 and equation (1) indicate that V,iK for mudro cks ishighly variab le, ranging from less than 1.8 n quartz-rich rocksto over 5 in loose, water-saturated sediments. This is a directresult of the nonzero intercept at the water point.

Compressional and shear wave velocities obtained by soniclogging in geopressured argillaceous rocks of the Frio forma-tion that exhibit clay volume s in excess of 30 percent, as deter-mined by a neutron-density crossplot, are plotted in Figure 5.Most of the data are consistent with equation (1)and Figure 4.Figure 6 is a plot of sonic log data in shaly intervals reported byKithas (1976 ). As with Figure 5, these data are well described byequation (1).

OBSERVATIONS IN SANDSTONES

The trend of Picketts (196 3) laboratory data for clean water-saturated sand stones (Figure 7) coincides precisely with equa-tion (1) established for mudrocks. The correspondence of V,/V,for sandstones and m udrocks is not entirely e xpected. Figure 8is a plot of sonic log V, and v, data in sandstones exhibiting lessthan 20 percent neutron-density clay volume in the Frio forma-tion. Except for some anomaloud, ~1 ow !/,!Ys ratios~ ndicativeof a tight gas sandstone (verified by conventional log analysis),the data again fall along the water-saturated line established formudrocks. Also falling along this line are sonic velocities for anorthoquartzite reported by Eastwood and Castagna (1983).In-situ m easurem ents for shallow marine s ands compiled byHam ilton (197 9) fall above the line. Sonic log velocities report-ed by Backus et al. (1979) and Leslie and M ons (1982) in cleanporous brine sand s tend to fall slightly below the line (Figure 9).

Figure 10 is a compilation of our laboratory data for water-saturated sandstones with d ata from the literature (Do menico,1976: Gregory, 1976; King, 1966; Tosaya, 1982; Johnston,1978; Murphy, 1982; Simmons, 1965 : Hamilton 1971). To firstorder, the data are consistent with equation (1); however, theyare significantly biased toward higher V, for a given VP. Thelocation of some sandstone data on the mudrock water-saturated line, although other data fall below this line, is pre-sum ably related to the sandstone texture and/or clay content.

Following the lead of Tosaya (1982) we used multiple linearregression to determine the dependence of sonic waveform-derived compressional and shear wave velocity on porosity andclay content for the Frio form ation. W e applied conven tionallog analysis to determine porosity and volume of clay (V,,) romgam ma ray, neu tron, and density logs. The resulting relation-ships for the Frio formation are

V,, (km/s) = 5.81 - 9.42+ - 2.21 V,, (2a)an d

V, (km/s) = 3.89 - 7.07+ - 2.04V,, (2b)The correlation coefficient r for both of these relationships is.96. The equations of Tosaya (1982 ) for laboratory data a re

V, , (km/s) = 5.8 - 8.64 - 2.4V,, , wan d

V, (km/s) = 3.7 - 6.34 - 2.1 V,, (24The sim ilarity between our equations and Tosa yas is evident.

FIG. 5. Compressional and shear wave velocities for geo-pressured shaly rocks of the T rio form ation from sonic logs.

FIG. 6. Sonic data from Kithas (1 976) mostly for shales.

FIG. 7. Trend of sandstone compressional and shear wave ve-locities from Pickett (1963).

From Tosa yas equations the sonic properties of zero poros-ity clay are: P-wave transit time = 89.6 us/ft, S-wave transittime = 190.5 us/ft, and VP/L; = 2.125. The correspondingvalues for the Frio formation clay are: P-wave transittime = 84.7 us,/ft, S-wave transit t ime = 165.1 ps/ft, andv,.:v, = 1.9 5.



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(4

IO-

0.0 I I I I I00

I I0.5 1.0 1.5 2.0 2.5 3.0 3.5

VS KM I SEC

W

9. Sonic log velocities for clean water-saturated sand-(a) Backus et al. (1979),(b) Leslie and Mons (1982).

0.0 0.5 10 1.5 20 2.5 3.0 3.5 4.0v,. KM I SE CFIG. 10. Com pilation of laboratory data for water-saturatedsandstones, including ARC0 and literature data.

. EASTwoclD a CASTAGNA (lsss,. 0RTHooARRITE. SONIC LOG. 10 KHZ0 FM0 FORMATION SANDSTONES, SONIC LOG (IS KHz,

--- MAlLTON (?WS,, SANDS

FIG. 8. Sonic log velocities in sandstones.

Some algebraic manipulation yields equations which ex-plicitly reveal the dependence of V,/V, on porosity and volumeof clay. From equations (2a) and (2b) we get

c;;v, = 1.33 + .63/(3X9 - 7.074)for clean sand and

(3a)

VP/V, = 1.08 + 1.61/(3X9 - 2.04v,,) (3b)for zero-porosity sand/clay m ixtures. These equations revealthat increas ing porosity o r clay content increas es VP/V, and tha tthe velocity ratio is more sensitive to porosity changes.

DRY SANDSTONES:LABORATORY DATA AND IDEAL MODELS

According to Gregory (1 977) Poissons ratio is abou t 0.1(corresponding to V,,/V, o 1.5) for most dry rocks and un-consolidated sands, and it is independent of pressure. Figure 11is a crossplot of V,,/V, versus VP for d ry and water-saturatedBerea sandstone. Note that the water-saturated points are rea-sonably close to the relationship defined by equation (I),whereas the dry points a re nearly constan t at a VP/V, of about1.5. Figure 12 is a compilation of laboratory compressiona l andshear wave velocities for dry sandstones. The field data ofWhite (196 5) for loose sands also are included. The data fit aline having a constant VP/K ratio of 1.5.

We gain some insight into the behavior of dry sandstones byconsidering variou s regular packings of spheres. The reader isreferred to White (1965) and Murphy (1982) for a detaileddiscussion. The dry comp ressional (V,) to shear (Vf) velocityratio, as a function of Poissons ratio (v) of solid spheres, is

v,D,!v,D [(2 - v)/(l - v)] 2 (4a)for a simple cubic packing (SC) of spheres;

V;IVp = [4(3 - v)/(6 ~ 5~)]~ (4b)for a hexagonal close packing (H CP) of spheres; and

v;/v,; = J5 (4c)for a face-centered cubic (FCC) packing. For HCP and FCCpackings propagation directions are (I, 0, 0).



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P-wave and S-wave Velocities in Clastic Rocks 575These- rtiarions are valid~only at elevated confining pres%res

when the packings have sufficiently high bulk and shear modulito propagate elastic waves. The velocity ratio cancels the pres-sure dependence: howe ver, this pressure restriction s hould bekept in mind.

From equations (4a) through (4 ~) we see that dry V,/V, isconstant or dependent on ly upo n Po issons ratio of the spheres(v). For quartz spheres (v Y O .l), dry V,,/V, is virtually the samefor these packings (FCC = 1.41, HCP = 1.43, SC = 1.45).

Dry VV, ratios based on these regular packings of spheres(Figure 13) indicate that the elastic properties of the grains areof secondary importance. Th e max imum range of dry V< isfor SC, and varies only from $ to 4 when Poissons ratio ofthe spheres varies from 0 to 0.5. For all practical purposes, dryVP/V, for packings of common rock-forming minerals is from1.4 to 1.5.

For real rocks, of course, a number of complicating factorsmust be considered. One of these is the presence of micro-fractures or pores of low aspect ratio. One means of studyingthe effects of microfractures is to make velocity measurementson samples before and after cracking by heat cycling. Figure 14shows the results of heat cycling on dry sandstones obtained byAktan and Farouq Ali (1975 ). The effect of the addition ofmicrofractures is to reduce both compressional and shear wavevelocities in a direction parallel to the dry line, maintaining anearly constant VP/V,.

Further insight into the effect of microfractures is providedby modeling the elastic moduli of solids with various poreaspect ratio sp ectra utilizing the formulation of Toksijz et al.(1976). This theory assumes randomly oriented and distributednoninteracting elliptical pores and, therefore, may not be validfor highly porous or shaly rocks.

As an example, we compu ted the V,ll( relationship for theinverted Boise sandstone pore spectrum (Cheng and Toksiiz,1976). This was done by m aintaining the same relative con-centrations amon g the aspect ratio distribution but uniformlyincreasing the entire distribution to increase the total porosity.Our intent was to show that aspect ratio distributions thoughtto be characteristic of actual rocks produce behavior in closeagreement with observed trends. We do not imply that thesedistributions are quantitatively applicable to real rocks. Except

FIG. 11. Ultrasonic measu rements of V IV, for a dry and water-saturated Berea sandstone sample. I%e various points wereobtained at different effective pressures.

FIG. 12. Laboratory measurements for V, and V for dry sand-stones. Note that o ne sandstone with calcite cement plots wellabove the line.

FIG. 1 3. Calculate d V/V, for regular packings of spheres versusPoissons ratio (v) of the sphere material. SC = simple cubicpacking, HCP = hexagonal close packing, FCC = facecentered cubic packing.

FIG. 14. The results of Aktan and Farouq Ali (1975) for severadry sands tones before and after heat cycling. Data are plottedfor measurements at high and low confining pressures.



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15. Calculate d I$ and V,,from the formulation of Cheng

16. Ultrasonic laboratory data for various sandstones

ve pressure conditions and joined by tie lines.

ay be exceeded, the computed line for dry Boisein excellent agreement with the observed dry sand-

ine (Figure 15). It is interesting to n ote that an identicalted line is obtained when all cracks with p ore aspect

than 0.1 are closed. This is consistent with the resultsand Farouq Ali (1975) shown in Figure 14. The

summarize, we examined two extreme cases for dry rocksdifferent m odels: (1) regular packings of spheres and (2)

lids. Both models predict a nearly cons tant VP/V, forstone, which is consistent with e xperimental observa-

SATURATED SANDSTONES:LABORATORY DATA AND IDEAL MODELS

16 shows VP/V, relationships for dry a nd w ater-ndstones. Lines join measu rements made on single

d water-satura ted samples under the same effective pres-

sure conditions. Qualitatively, the data from water-satura tedsandstones are consistent with eq uation (1) but tend towardhigher V, for a given VP.

Given the compressional and shear wave velocities obtainedin the laboratory for dry sa ndstones, we use Gassm anns (1951)equations to compute velocities when these rocks are saturatedwith water. Gass mann s equ ations are:

KD + QK,= KS-----&+Q'

K,(Ks - KD)Q= $(Ks - KF) (5b)an d

hV = PO> (5c)

pw = @PF + (1 - $)I%, (5d)where K, is the bulk modulus of the wet rock, K, is the bulkmodulus of the grains, K, is the bulk modulus of the dry frame,K, is the bulk modulus of the fluid, uw is the shear modulus ofthe wet rock, pD is the shear mod ulus of the dry rock, pw is thedensity of the wet rock, pr is the density of the fluid, ps is thedensity of the grains, and 4 is the porosity. Figure 17 is a plot ofcompu ted (from Gassm anns equ ations) and meas ured satu-rated compressiona l and shear wave velocities. Although theindividual points do not always coincide, the computed andmeasured data follow the same trend.

We a lso can apply Gas sman ns equations to the equationsfor dry regu lar packing arrangem ents of spheres given byMurphy (1982). Figure 18 shows the relationship between VPand V, for water-saturated SC, HCP, and FCC packings. Notethat the packings of equal density (FCC and HC P) yield vir-tually the same curve. The curves obtained for these simplepackings are qualitatively consistent with the low-velocity ex-perimental data of Ham ilton (1971) and Domenico (1976).

For wa ter-saturated rock, the formulation of Tokso z et al.(1976)predicts that the addition of microfractures will changethe V,,-bc elationship (Figure 19). The inverted Boise sandstonepore aspect ratio spectrum yields a line close to equation (1).Closing all cracks with aspect ratios less than 0.1 yields a linewhich lies below equa tion (1 ). Thus , we might expec t data forclean porous sandstones dominated by equant porosity to heslightly below the line defined by equation (l), whereas tightsandstones with high concentrations of elongate pores wouldtend to lie along this line. Verification of this hypothesis is leftas an objective of future research.

A good description of the water-saturated VP-Vs elationshipfor sandstones is provided by the following formulation. Thedry line established with laboratory data (VP/V, z 1.5) meansthat dry bulk modulus (K,) is approximately equal to dryrigidity (uD)

PO z K,. (6)These are exactly equal when

v;:vp = 1.53. (7 )From equation (5~) it follows that

K D=~D=~W~ (8)Thus, the saturated shear velocity can be obtained from the drybulk modulus by



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P-wave and S-wave Velocities in Clastic Rocks 577

wet bulk modulus is given byKw = pw(V; - $Vf). (10)

s equations [(5a) throughallow comp utation of V, given VP , I$, and the grain an d

We computed shear-wave velocities for sandstone core po-

Similarly, in Table 1 we compare the calculated shea r veloci-to the measu red laboratory values. The differences are

than 5 percent, demo nstrating excellent agreementthe theory. Since part of these differences must b e due to

nt quartz, we consider the agreement rem arkable.Gassm anns equations are strictly valid only at

Further corrections can be applied to ac countThe Holt sand sample, which gives the largest1, illustrates the importance of the as-

bulk m odulus is equal to dry rigidity [equa-e d ry VP/V, for this Holt sand sample is 2.17 (Table

far different from the ratio of 1.53 required for equa lity ofand shear moduli. This variation is probably due to

we can still apply Gassm annsvalues. As shown in

2, these predicted value s are virtually identical to theFor clean sandstones at high press ures and with mode rate

formula (Wyllie et al., 1956)4- l/V, - 1/v;1/v; - 1/v; (11)

Vi is the grain compressional velocity and Vi is the fluidVP-Vs elationship predicted by

17. Measured (open symbols) versus computed (solid sym-and rip using the dry data from Figure 12 and Gas-s equ atrons.

FK. 18. Calculated V, and VP or saturated regular packings ofquartz spheres using the calculated dry velocities and Gas-smann s e quations.

FIG. 19. Calculated V, and VP or Boise sandstone based on theformulation of Cheng and Toksoz (1979).

FIG. 20. Comparison of measured VP with calculated V, forsandstones from two w ells as reported by Gregory et al. (198 0).Calculations are based on measured wet VPand porosity usingGassm anns eq uations and dry bulk mo dulus equal to dryrigidity.



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57 8 Castagna et al.Table 1. Holt sand: Comparison of observed water-saturated shear velocities with those calculated using the measured water-saturated c om-pressional velocities and porosities. The calculations were based on Gassmanns equations and the assum ption that KD 2 po.

Rock Reference v, PorosityBerea Johnston (I 978) 3.888 18.4Berea Johnston (1978) 4.335 1 8.4Navajo Johnston (1978) 4.141 16.4Navajo Johnston (1978) 4.584 16.4Gulf Coast sand Gregory (1976) 3.927 21.7Gulf Coast sand Gregory (1976) 3.185 21.7Boise Gregory (1976) 3.402 26.8Boise Gregory (I 976) 3.533 26.8Travis peak Gregory (1976) 4.732 4.45Travis peak Gregory (1976) 4.990 4.45Travis peak Gregory (1976) 4.342 8.02Travis peak Gregory (1976) 5.001 8.02Bandera Gregory (1976) 3.492 17.9Bandera Gregory (1976) 3.809 17.9Ottawa Domenico (1976) 2.072 37.74Sample no. MAR ARC0 data 5.029 1.0Sample no. MAR ARC0 data 5.438 1.0Sample no. MDP ARC0 data 3.377 21.0Sample no. MDP ARC0 data 3.862 21.0Berea ARC0 data 3.642 19.0Berea ARC0 data 3.864 19.0

Predicted Observedv, v,

2.330 2.3022.700 2.5902.520 2.4302.890 2.7102.380 2.3671.730 1.9751.970 1.9602.080 2.0732.860 2.5813.110 3.2842.590 2.6673.180 3.3911.970 2.0322.250 2.240,740 ,8013.200 3.3153.420 3.4961.900 2.0472.320 2.3502.120 1.9922.310 2.267

PercentError1. 24.23. 7:::- 12.4

::10.8-5.3- 2.9-6.2-3.1-7::-3.5-2.2-1.2- 1.36.44. 3

Berea ARC0 data 3.510 19.0 2.000 1.680 19.0Berea ARC0 data 3.740 19.0 2.200 2.130 3.8St. Peter Tosaya (1982) 5.100 6.6 3.250 3.420 -5.0St. Peter Tosaya (1982) 4.880 7.2 3.060 3.060 0.0St. Peter Tosaya (1982) 4.500 4.2 2.610 2.680 -2.6St. Peter Tosaya (1982) 4.400St. Peter Tosaya (1982) 4.400 ::?I 2.630 2.600 1.22.540 2.600 -2.3St. Peter Tosaya (1982) 3.950 18.8 2.380 2.420 - 1.6St. Peter Tosaya (1982) 3.600 19 .6 2.090 2.070 1.0St. Peter Tosaya (1982) 3.170 14.5 1.580 1.560 1.3Holt sand ARC0 data 3.546 16.3 1.990 1.539 29.3

Table 2. Water-saturated sandstones : A rec alculation of velocities forthe Holt Sand (Table I) for which K, # po. N ew water-saturatedvalues are computed using both the VP and V, measured for the dryrock.Holt Sand

Porosity 16.3%Water-satu rated V, (laboratory) 3.546 km/sWater-satu rated V: (laboratory)Water-satu rated K (predicted from porosityand water-saturated V)Percent errorDry V (laboratory)Dry r/f (laboratory)Dry V,/y (laboratory)Water-satu rated V, (predicted from dry data)Percent errorWater-sa turated V, (predicted from dry data)Percent error

1.539 kmjs1.990 km/s29.3%3.466 km/s1.599 km/s2.17 km/s

3.519 km/s- .8%1.540 km/s1%

equations (5), (9), (lo), a nd (11). This time-average line de-scribes the laboratory data from water-sa turated conditionspresented in Figu re 10 extremely well. Reca lling the results ofcrack modeling sh own in Figure 19, one explanation for thevalidity of this empirical formula would be the dominance ofpores of high asp ect ratio.

DISCUSSION OF RESULTS:DYNAM IC ELASTIC MODU LI RELATIONSHIPS

Compressional and shear velocities, along with the density,provide sufficient information to determine the elastic parame -ters of isotropic m edia (Simmons and Brace, 1965). These pa-rameters proved useful in estimating the physical properties ofsoils and characteristics of formations (see, for exam ple, Ri-chart, 1 977). Howev er, the relationships given by equation (1)for mudrocks or G assma nns e quations and equa tion (6) forsandstones fix v, in terms of V. IJenc e, the elastic parame terscan be determined for elastic silicate rocks from conventionalsonic and den sity logs. This explains in pa rt w hy Stein (19 76)was successful in empirically determining the properties ofsands from conv entional logs.

In this discussion we assumed that eq uation (1) holds to firstorder for all elastic silicate rocks, with the understanding thatGassm anns e quations might be used to obtain more preciseresults in clean porous sandstones if necessary. For many rocks,particularly those with high clay content, the addition of watersoftens the frame, thereby re ducing the bulk elastic modu li. Th efollowing empirical relationships are, therefore, not entirelygeneral but are useful for describing the wide variety of datapresented here. As shown in Figure 22, bulk and shear moduliare about equa l for dry sandstones. Adding water causes thebulk modulus to increase. This elTect is most pronounced at



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P-wave and S-wave Velocities in Clastic Rockshigher porosities (lower moduli). Water-saturated bulk modu-lus normalized by density is linearly related to compressionalvelocity (Figure 23):

K-If z 2bV; - b2,PW

b = 1.36 km/s. WIWater-sa turated rigidity norma lized by density is nonlinearlyrelated to wet compressional velocity:

PW- z i (VP - b)2 . (13)PW

fry bulky mod ulus and rigidity norm alized by density aregiven by

(14)Poissons ratio for wate r-saturated rock is approxim ately

linearly related to compressiona l velocity (Figure 24). Poissonsratio of air-saturated rock is constant.

DISCUSSION OF RESULTS:SEISMOLOGY AND FORMATION EVALUATION

In recent years there has been increased use of VP, V,, andVP/V, in seismic exploration for estimation of porosity, lithol-ogy, and saturating fluids in particular statigraphic intervals.The above ana lysis both com plicates and enlightens such inter-pretation. It is clear that clay co ntent increase s the ratio VP/V,,as does porosity. The analyses of Tosaya (1982) and Eastwoodand Castagna (1983) and equations (3a) and (3b) indicate thatVP/V, is less sensitive to variation of clay content than to vari-ation of porosity. However, the range of variation in claycontent m ay be larger. Thus, VP/v, can be grossly dependentupon clay content.

Figure 25 shows VP/V,compu ted as a function of depth in theGulf Coast for noncalcareous shales and clean porous sand-stones that are water-saturated . T he co mpressional velocityand porosity da ta given in Gregory (1977) are used to establishthe variation with d epth. Equa tion (1) is used to predict V, forshales, and Gassm anns equations are used for sandstones. At agiven depth, shale velocity ra tios are on the order of 1 0 percenthigher than sandstone velocity ratios.

FIG. 21. Calculated V, and VPbased on the time-average equa-tion and Gassm anns equa tions.

FIG. 22. The computed relationships between the bulk an dshear moduli (normalized by density) based on the observed vand VP rends.

14.0I -

FIG. 23. The computed relationships between the bulk modulus(normalized by density) and I, based on the observed v, and Vtrends.

FK. 24. The com puted relationships betw een Poissons ratioand VPbased on the observed V, and VP rends.



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580 Castagna et al.5.0

4.0

3.0

2.0

1.0 L0.0 I I I I0 4,000 8,000 12,000 16,000 20,000

DEPTH, FEETFK . 25. V,/V, comp uted as a function of depth for selected GulfCoast shales and water-saturated sands.

These conclusions are somewhat at odds w ith conventionalwisdom that VP/V, equals 1.5 to 1.7 in sandstones and is greaterthan 2 in shales. Clearly, mapp ing net sand from VP/v7 s not a sstraightforward as conventional w isdom would imply.

In elastic silicates, shear wave velocities for elastic moduliestimation or seismic velocity control may be estimated fromequation (1) and/or Gassm anns equations and conventionallogs. Alternatively, our relationships indicate that when VPan dV, are use d together, they can be sensitive indicators of both gassaturation and nonelastic components in the rock. Gas satu-ration w ill move the VP/V, toward the dry line in Figure 16.Nonelastic components can also move the ratio off the mud-rock line, as shown by the Holt sand sample marked as cal-careous in Figure 16.

The possibly amb iguous interpretation of lithology an d po-rosity from VP, V,, and Vp/Vs n seismic exploration also appliesto log an alysis. Howev er, because sonic logs are generally apart of usua l logging programs, the interpretation of full wave-form sonic logs should be made in the context of other logginginformation (as we did) to obtain equ ations (2a) and (2b). Otherlogs which are sensitive to clay content and porosity (e.g.,neutron and gamma-ray) are needed to evaluate the details ofhow porosity and clay content separately affect elastic rocks. Ingas-bearing and highly porous zones where the time-averageequation fails, combined VPand V, information may potentiallybe used in porosity determination.

Assuming that the time-average equation works for compres-sional transit times and that dry incom pressibility e quals dryrigidity, Gas smann s eq uations yield a nearly linear relation-ship between shear wave transit time and porosity. This re-lationship is well described by a time-average equation

+= l/v, - 1/v;1.12 s/km - l/V:

where Vi is the grain shear wave velocity. The constant 1.12s/km is used in place of fluid transit time This equation shouldbe superior to the compressional wave time-average equationfor porosity estim ation when there is significant residual gassaturation in the flushed zone.

SUMMARY AND CONCLUSIONS

To first order, we conclude that shear wave velocity is nearlylinearly related to compressional wave velocity for both water-saturated and dry elastic silicate sedimentary rocks. For a givenVP, mud rocks tend toward slightly higher VP/V, than do clea nporous sandstones.

For dry sandstones, VP/V, is nearly co nstant. For wet sand-stones and mud stones, V,lf( decreases with increa sing VP.Water-sa turated sandstone shear wave velocities are consistentwith those obtained from G assma nns equ ations. The wa ter-saturated linear VP-versus-V, rend b egins at VPslightly less thanwater velocity and V, = 0, and it term inates at the compres-sional and shear wave velocities of quartz. The dry sandstonelinear VP-versus-y trend begins at zero velocity and terminatesat quartz velocity. Dry rigidity and bulk modulus are aboutequal.

Theoretical models based on regular packing arrangementsof spheres and on crac ked solids yield I/p-versus-V, trends con-sistent with observed dry and w et trends.

We believe that the observed VP-versus-l/,-relationship forwet elastic silicate rocks results from the coincidental locationof the quartz and clay points. The simple sphere-pack modelsindicate that the low-velocity, high-porosity trends are largelyindependent of the mineral elastic properties. As the porosityapproach es zero, however, the velocities must necessa rily ap-proach the values for the pure mineral. In this case, clay andquartz fall on the extrapolated low-velocity trend.

REFERENCESAktan, T., and Farouq Ali, C. M., lY75, Effect of cyclic and in situheating on the absolute permeabilities, elastic constants, and elec-trical resistivities of rocks: Sot.Petr Eng., 5633.Backus, M. M., Castagna, J. P., and (iregory, A. R., 1979, Sonic logwaveforms from geothermal well. Brazoria Co., Texas: Presented at49th Annual International SEG Meeting, New Orleans.Birch, F., 1966, Compressibility; elastic constants, in Handbook ofphysicai constants. Clark, 5. P., Jr., Id.. Geoi. Sot. Am.. Memoir 97,977174.Blatt, H., Middleton, G. V., and Murray. R. C., 1972, Origin of sedi-mentary rocks: Prentice-Hall, Inc.Cheng, C. H., an d Toksoz, M. N., 1976, Inversion of seismic velocitiesfor the pore aspect ratio spectrum of a rock: J. Geophys. Res., 134,7537-7543Christensen. N. J., 1982, Seismic velocities, in Carmichael, R. S., Ed.,Handbook ofphysical properties of rocks, II: 2-227, CRC Press,.Inc.Domenico, S. N., 1976, Effect of brine-gas mixture on velocity m anunconsolidated sand reservoir: Geophysics, 41.887-894.Eastwood, R. L., and Castagna, J. P.. 1983. Basis for interpretation ofy,/v, ratios in complex lithologies Sot. Prof. Well Log Analysts24th Annual Logging Symp.Ebeniro, J., Wilson, C. R., and Dorman, J., 1983, Propagation ofdispersed compressional and Rayleigh waves on the Texas coastalplain: Geophysics. 48,27-35.Gassmann, F., 1951, Elastic waves through a packing of spheres:Geophysics, 16,673-685.Gregory, A. R., 1977, Fluid saturation effects on dynamic elasticproperties of sedimentary rocks: Geophysics, 41,8955921.--~~ 1977, Aspects of rock physics from laboratory and log data thatare important to seismic interpretation, in Seismic stratigraphy-application to hydrocarbon exploration: Am. Assn. Petr. Geol.,Memoir 26.



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