16. Acoustic Waves in Rocks

8/12/2019 16. Acoustic Waves in Rocks

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16COUSTIC WAWSIN ROCKS

INTRODUCTIONThe previous chapter notes the use of the measured interval transit time forthe determination of porosity. It also suggests that, in addition to porosity,other rock properties such as lithology and environmental effects such aspressure might be important factors determining the acoustic velocity. Whatare the important factors which affect the acoustic velocity of rocks? Howcan these factors be related to elastic constants which are representative of therock? These are some of the questions to be addressed in this chapter.As background, we begin with a description of the experimentalprocedure used in investigating acoustic rock properties. This is followed bya review of some of the laboratory data accumulated by a wide variety ofresearchers who have studied the acoustic properties of rocks. This workshows that the velocity of acoustic propagation is a rather more complexphenomenon than would be implied by the simple Wyllie time-averageequation. As a consequence of the laboratory observations, some models ofrocks which attempt to describe actual rock acoustic properties have beendeveloped. The approach used in several models is described. Returning tologging applications, the final portion of the chapter is an introduction tosome aspects of acoustic waves in boreholes and to the propagation ofacoustic energy at the interface between materials with different properties.

359


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Acoustic Waves in Rocks 361

Figure 16-2. Details of a differential pressure cell for the simulation of overburdenand pore pressure. Adapted from Wyllie?sandstone samples can be seen in Fig. 16-3. The porosity is plotted as afunction of the measured travel time and the straight line is the time-averagefit to the data. The matrix and fluid travel times used in the fit are given inthe figure. These measurements were made with no confining pressure, andthe scatter may represent other factors which are important, such as texture.The effect of matrix is clearly seen in Fig. 1 6 4 , in which porosity is plottedversus travel time for quartz and calcite core samples. The matrix velocity ofcalcite is somewhat greater than that of quartz, as indicated in the figure. Asummary of some of the fluid and maaix velocities for both shear andcompressional waves is shown in Table 16-1.

To study the effect of total confining pressure on velocities,measurements are made with the pore fluid at atmospheric pressure, and theconfining pressure (the pressure in the fluid surrounding the rubber jacket ofFig. 16-2) is varied. The result of this type of measurement in samples of


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362 Well Logging for Earth Scientists

A

>.v

a 20-15-

.-

10-

I

5-0

35[ \ Average ftisec r n i s )Matr ix 19500 (5944)30 Fluid 5000 (1524)

0 0

251 *.%D

Figure 16-3. Measurements of the effect of porosity on compressional transit timefor sandstone samples. From Wyllie.4

30-25--t20-a.>:- 15-

0a.10-

5-

\ Calcite Wsec ( m / s )Matrix 22500 (6858)Brine 5235 (1596)

o \

\QuartzMatrix 19200 (5852)Brine 5235 1596)

ol I I I I I \ I \ J100 90 80 70 60 50 40

A t cseclft

Figure 16-4. Effect of rock composition on the relationship between porosity andtransit time. From Timur.


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1 3 1

2 12-

>-

10

olomite

21 I

I I I I I I I I IWater saturated Sandstone

AP=Po.PfT c c c AP=O verbu rdenressure- -+--- -0- A P = Z psi x 10)

+- - -4 - P=l ps i x l o3-- -+- ---o--- P=O

I I I I

I6 = 6

Limestone


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Acoustic Waves in Rocks 365types of pore fluids: water, kerosene, and air. However, the magnitude of thevelocity change as a function of the differential pressure on the sampledepends upon the fluid. What is even more striking, in Fig. 16-7, is thereversed behavior of the compressional and shear velocities.The effect on the compressional and shear velocities of porous rocksunder confining pressures has also been studied. Representative data isshown in Fig. 16-8. The first conclusion that can be drawn from this datasummary is that the effect of water saturation will be largest for thecompressional (or p) waves. Note that the shear wave behavior hardlychanges under the two extreme conditions considered. This is because liquidsdo not support a shear wave. However, especially at low porosities, thecompressional velocity is quite sensitive to water saturation. The differencein behavior for shear and compressional waves leads naturally to a techniquefor distinguishing gas-bearing formations. The ratio of the two velocities canbe used as a gas indication.

10,000

8000

dVI.Z 6000a>-

4000

zooa

\ \\

I I20 40P o r o s i t y Q Percent

Figure 16-8. A summary graph of the effect of the extremes of water saturation oncompressional and shear wave velocities as a function of porosity.The shear wave velocity increases slightly with the density decreasethat results from replacing water with gas. Th e compressionalvelocity decreases when gas replaces water because of the change inthe rock bulk modulus. From Timur.'


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366 Well Logging for Earth ScientistsA qualitative explanation of this effect can be obtained from the basic

parameters which govern velocity. The shear velocity is given byc.ta fixed porosity, if water is exchanged for low density gas, the shear moduluswill not change, but the density will decrease, thus producing the inversionseen in the figure. In the case of the compressional wave, one expression forvelocity is:4 7 he same argument holds for p,which does notchange. However, the argument for density must be overcompensated by achange in the bulk modulus. At a fixed porosity, the replacement of liquid,which has a very large bulk modulus, by gas, which is very compressible,will certainly reduce the contribution of the pore fluid to the overall bulkmodulus of the formation. According to the data, this reduction willdominate the velocity, regardless of the porosity. Just how thecompressibility of the fluid influences the overall rock compressibility is thekind of question which is answered by rock models, to be discussed in thenext section.

The data just considered treat the two extremes of either full water or gassaturation. What about those cases in between? What is the effect of partialsaturation? That there is little sensitivity of the velocity ratio to saturationhas been confirmed by both laboratory and model calculations, which areshown in Fig. 16-9. These curves indicate a large change in thecompressional velocity as soon as the gas saturation attains lo%, and littleafterwards As expected, there is no sensitivity indicated for the shearvelocity.

The final data to be considered here, which deal with the effect of textureon velocity, are shown in Fig. 16-10. In this data, the p-wave and s-wavevelocity in Troy granite are shown as a function of the differential pressureon the sample. For the s-wave, there is little or no change in velocity as afunction of the differential pressure and no noticeable difference between thery and water-saturated sample. However, for the p-wave the result is ratherdramatic. For the ry rock, the velocity increases by about 50% for a modest

differential pressure. This has been attributed to tiny microfissures in theTroy granite, which tend to close under pressure. From this example, it isclear that the texture of the rock must be included in any comprehensive rockmodel.

ACOUSTIC MODELS OF ROCKSClassical wave theory predicts none of the effects discussed above. Thevelocities of the compressional and shear waves are given by simplecombinations of the material moduli and bulk density. Realistic rock modelswill have to provide a means of extracting appropriate elastic constants forfluid-saturated rocks: a method of predicting the average elastic properties


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c.-UIc0

Computed10100

0 0.2 0.4 0.6 0.8 1.0Water Saturat ion

Figure 16-9. The effect of partial water saturation on the compressional and shearvelocity. The change in bulk modulus occurs dramatically with aslight introduction of gas. From Timur.'6.5 I I I@--,a- 13 4- -

vP

----a Sat 'dD r y -

6.5 I

----a Sat 'd -I5.0 iD r y4.5 c4.0I

Troy Grani te2.0-5L - J0 0.2 0.4 0.6 0.8 1.0Differential Pressure K bFigure 16 10. Indication that the texture of rock has a large impact oncompressional velocity. From Timur.'


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368 Well Logging for Earth Scientistsfrom a knowledge of the elastic properties of the constituents and somedetails of the rock fabric.Our goal in using rock models is to be able, from some set of reasonableparameters, to predict the behavior of acoustic velocities under a variety ofconditions, such as confining pressure, differential pressure, and pore fluidproperties. One of the first attempts at such a model was made by Gassman.His model for porous rocks consisted of a porous skeleton filled with fluidand was based on the assumption that the properties of the fluid and skeletonmaterials are known. The major simplification incorporated is that therelative motion between the fluid and skeleton during acoustic wavepropagation is negligible.The basic definitions used in the model are shown in Fig. 16-11.Following the convention of Gassman, the symbol for bulk modulus used hereis k. We start with a knowledge of the material properties of the skeletonmaterial, in particular its density ps and bulk modulus h. The knownproperties of the evacuated skeleton are its porosity , its density p its bulkmodulus E, its shear modulus ji, and its plane wave modulus i.he planewave modulus is taken to be a = + ZJI and is of interest since thecompressional velocity can be obtained from it easily see Problem I). Thefluid properties are denoted by its density pf and bulk modulus kf. The objectof the model is to obtain the average properties of the saturated rock itsdensity p, bulk modulus k, shear modulus p, and plane wave modulus M.

3

Fluid: p,, k,Skeleton Mater ia l : p,, k ,Evacuated Skeleton: , p , k , p, M

Average Propert ies: p, k, p, MFigure 16-11. Model of a porous rock according to Gassman. It is considered toconsist of a rigid frame and a saturating fluid. The elastic constantsof the frame material and fluid are known. The model predicts theelastic constants of the ensemble.


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Acoustic Waves in Rocks 369The bulk density is given simply by:

P = OPf + (l+)PS,and the shear modulus will be the same as for the skeleton p. However, theaverage bulk modulus is not obvious but can be determined from thefollowing analysis.In order to determine the bulk modulus, we need first to refer to thedefining equation:

APAVk = -V

Consider what the change in volume of our saturated skeleton is when apressure AP is applied, as shown in Fig. 16-11. First, the applied pressurecan be separated into two components:

AP=AF+APf,where @ is the portion of the overburden pressure which is actuallysupported by the skeleton, and APf is the portion of the applied pressurewhich is supported by the fluid. This applied pressure will induce a changein the volume AV, which is given by the sum of two contributions:

AV = AVf + AVs ,which are the fluid compression and the actual skeleton compression.The first of these two is given quite simply by:

which follows directly from the definition of the bulk modulus. The skeletonchange has two components: one the result of the pressure portion actuallyborne by the skeleton and the other the result of the pressure of the fluid.This is given by:

The first term is the shrinkage of the skeleton constituents, and the secondterm is the shrinkageof the entire skeleton framework. Thus the total volumechange is given by the sum of these two:

The other relation that is available is:1 1 --APf =APVV k, k- = -


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370 Well Logging for Earth Scientistswhich follows from the pressure breakdown specified initially and thedefinition of the bulk modulus. After algebraic manipulation, it can be shownthat the plane wave modulus M is given by that expected for the ry skeletonplus an additional t e rn

-kk,where r = -.In order for this model to be of any use, some relationship between the

bulk modulus of the ry kame k, and the rock matrix k,must be established.Initially, Gassman was able to do this for a very special case in which heapproximated a loose sand by a set of cubic packed spheres: By consideringthe grain contact area as a function of applied stress, he could relate thecompressibility to the particle size and grain properties,

At first inspection, Eq. (1) is unsatisfying. One of the observational factswhich it must predict is the dependence of the compressional velocity v, onthe differential pressure. This dependence, however, is hidden in the term- Reformulations of the Gassman theory for cubic packed spheres byk,.White and Pickett show this dependence explicit1y?s8 This formulationcontains five parameters in addition to effective stress for the prediction ofporosity from At.

A more complete model was developed by Biot? The additionalapproach to reality was to allow relative motion between fluid and the rockskeleton. The fluid motion is assumed to obey Darcy's law (which relatesfluid flow to differential pressure, viscosity, and permeability) so thatadditional terms of permeability and fluid viscosity appear in the analysis.The Biot theory predicts a frequency dependence for compressional wavevelocity. However, in the low frequency limit, the theory reduces to that ofGassman. Further models which deal specifically with the rock teXNre havebeen developed by Kuster and Toksoz, for example. They derive averageelastic parameter for a mamx containing spheroidal cavities of differingaspect ratios. This model has been quite successful in fitting the data, similarto that in Fig. 16-10.

Despite the existence of detailed but unwieldy acoustic rock models, inpractical logging interpretation the Wyllie time-average equation is widelyused to determine porosity from measurements of At. The justification, ifanything beyond simplicity is required, is due to Geertsma. He showed thatin the limit of low porosity, the Biot theory predicts a simple porositydependence for the inverse compressional velocity:

1- - a + + $ .V,


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Stoneley

Compressiona lIc y

First Motion

ACOUSTIC WAVES IN BOREHOLESThe picture of borehole acoustic logging presented earlier (see Fig. 15-12) isvery simplified. What is implied in this figure is that somethingapproximating the transmitted pulse is also received. However, that this isnot the case can be seen clearly in Fig. 16-12 which shows an actual acousticwaveform recorded with a logging device in a borehole. In this particularexample, three distinct packets of energy are seen. The first is indicated to bethe result of the compressional wave moving through the formation. Thesecond, with a much larger amplitude, is the result of a slower shear wavemoving through the formation. Finally, arriving some 1500 p e c later is awave of large amplitude, known as the Stoneley wave, which is entirely dueto the presence of the borehole, as we. shall see later. Although thetransmitter pulse is brief and fairly uncomplicated, the received signal is quitecomplex, because of the various reflections and interferences which can beproduced as a result of waves traveling in the borehole.

I'I0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Tim e in M ic roseconds

Figure 16-12. A typical acoustic waveform recorded in a borehole. Th ree distinctarrivals are indicated. Courtesy of Schlumberger.In order to understand the two basic arrivals of Fig. 16-12 the wavefronts

produced in a two-dimensional medium of mud and formation have beencomputed for different times after emission in Fig. 16-13. The case takenhere is for a fast, or hard, formation in which the compressional velocity inthe solid is greater than the shear velocity in the solid, which is greater thanthe compressional velocity in the mud, i.e., v, > v, > v d. The first frame, at40 psec after emission, shows a spherical wavefront traveling out from thesource toward the formation. At 70 pec, the pressure wave has hit theborehole and three things have happened: There is a wave reflected back intothe mud, a shear and a compressional wave have been created in theformation, and all of these have begun expanding.


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170

i\

Figure 16-13. Simulation of the wavefronts in a two-dimensional medium consistingof mud and a fast formation. Courtesy of Schlumberger.

Figure 16-14. Much later in the simulation, when the wavefronts have passed thedetector array. Courtesy of Schlumberger.


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Acoustic Waves in Rocks 373If we follow the compressional wave through the rest of the frames, wesee that it goes on expanding in time. At somewhere around 90 psec, theangle it makes with respect to the interface is 90, and from that point on it

travels down the interface with a speed of v,. This is the so-called headwave. It can be thought to create a series of disturbances along the boreholewall which reradiate into the mud as a plane wave, resembling the wave of aboat. The angle of this wake will be related to the ratio of the formationcompressional and mud velocities. Turning our attention to the shear wave,whose expansion is much slower because of its reduced velocity, we s e e thatit does not make a right angle with respect to the interface until somewherebetween 110 and 170 psec, at which point it travels to the right with theshear velocity creating a wave in the borehole mud, just as in thecompressional case.Fig. 16-14 shows the situation after nearly 10 times as much time haselapsed (at 1300 psec). The distance scale has been much enlarged andshows the location of 8 receivers. In this view, the compressional disturbancehas already passed through all 8 receivers, and the next one to arrive will bethe result of the shear wave. It will then be followed by the direct mudarrival, which will be followed shortly by the reflected m ud pulse.

C om p r e s s i o n a l Sh e a r Stoneley

0 5000Time M s e c )

Figure 16-15. Recorded acoustic wavetrains recorded at 13'- 17' from thetransmitter in a fast formation. The compression, shear, and Stoneleyarrivals are clearly separated. The slopes of the lines indicating eachof the three types of first amvals is proportional to their velocity.Courtesy of Schlumberger.


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374 Well Logging for Earth ScientistsAlthough in fast formations the shear signal can often be seen at large

distances from the transmitter, this is not always the case. Returning to datarecorded in a borehole, Fig. 16-15 shows waveforms at five differentreceivers at increasing distances from the transmitter. The shear arrivals arewell-separated from the very low amplitude compressional arrivals, and theseparation improves with increased transmitter-to-receiver distance. However,as shown in Fig. 16-16 for a soft formation, it is clear that there is no sheararrival present. In this example, the mud velocity is greater than theformation shear velocity.

No Shear

C o m p .

0 1000 2000 3000 4000 51Time in p . e c

DO

Figure 16-16. Acoustic wave trains recorded at 1 intervals in a slow formationwhere the shear arrival is seen to be absent. Courtesy ofSchlumberger.

To understand the lack of shear development, refer to Fig. 1617, whichshows the wavefronts in the two-dimensional simulation of a soft formation atsome time after emission. It is clear that there will be a compressionalarrival. The compressional wavefront is perpendicular to the interface andtravels along it at a speed of vc, which is larger than vmd in this case.However, the shear wave does not make a right angle since v, < v,,+ Thehead wave does not appear, because the condition for constructiveinterference between the disturbance along the borehole wall and itstransmission in the mud does not occur. Thus the wake will not develop.

The last aspect of the borehole sonic waveforms to be considered is theStoneley wave. A good example of this is shown in Fig. 16-18. The fullsonic waveform is shown from a tool in a borehole over a 200 section. Thevariations, due to lithology or porosity changes, are clearly seen in the shear


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Acoustic Wav es in Rocks 375and compressional arrivals. The Stoneley wave is also seen to have somevariation in its arrival time, even though it is a pulse of energy which istraveling only in the borehole. W hat causes the variations in the Stoneleywave velocity?

3000 100 200 300 400 5Z in MM

0

Figure 16-17. Results of the two-dimensional wavefront simulation for a slowformation. The shear velocity is such that a head wave neverdevelops and thus is not observed in the borehole. Courtesy ofSchlumberger.Fig. 16-19 illustrates the basic notions of the Stoneley, or tube wave. Ina liquid-filled tube which has very rigid walls, a low frequency pressure wavewill travel as a nearly plane wave at the compressional velocity of the fluid.This type of phenomenon is responsible for the so-called water hammer. In

this case, however, the sound is produced by the small distortion of the wallsof the pipe carrying the water, which has been suddenly stopped. For aborehole, with semi-rigid confining walls, the speed of the pressuredisturbance is related to the elastic constants of the wall as well as the fluid.White has shown in the low frequency limit that the speed of this tubewave is given by:1/21V t u k = Vm

where vm is the compressional velocity in the mud, v, is the shear velocity inthe formation, and p is the density of the formation? Thus the shear velocityof the form ation v, can be obtained in the absence of a shear arrival from thespeed of the tube wave if the density of the formation is known.


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Shear

400

500

Figure 16-18. A sequence of acoustic waveforms recorded in the borehole withshear and Stoneley arrivals clearly shown. In addition to variations inthe shear velocity, which reflect formation properties, the Stoneleywave is also seen to have velocity variations. Courtesy ofSchlumberger.

Using the elastic constant relationships developed in Chapter 15, the tubewave velocity dependence on the mud bulk modulus B, and the formationshear modulus p can be found from the preceding expression to be:

This illustratesvelocity.

lute =

the role of the borehole wall rigidity in modulating the mud


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Acoustic Waves in Rocks 377Wall

mud

Elast ic Wal l(Borehole)

Vtube < vmudh u b . < v*

Figure 16-19. A simplified representation of the low frequency Stoneley, or tubewave.

References1. Timur, A., Acoustic Logging, in Petroleum Production Handbook,edited by H. Bradley, SPE , Dallas (in press).2. Jordan, J. R., and Campbell, F., Well Logging 11: Resistivity andAcoustic togging, Monograph Series, SPE, Dallas (in press).3. Wyllie, M. R. J., Gregory, A. R., Gardner, L. W., and Gardner, G. H.F., An Experim ental Investigation of F actors Affecting Elastic WaveVelocities in Porous M edia, Geophysics, Vol. 23, 1958, p. 459.4 . Wyllie, M. R. J., Gardner, G. H. F., and Gregory, A. R., SomePhenomena Pertinent t Velocity Logging, JPT, July 1961, pp.629-636.5. Gassman, F., Elasticity of Porous Media, Vierteljahrschr.6. Gassm an, F., Elastic Waves Through a Packing of Spheres,Geophysics, Vol. 16, No. 18, 1951.7. White, J . E., Underground Sound, Elsevier, Am sterdam, 1983.

Naturforsch. Ges. Zurich, Vol 96, No. 1 , 1951.


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378 Well Logging for Earth Scientists8. Pickett, G. R., The Use of Acoustic Logs in the Evaluation ofSandstone Reservoirs, Geophysics, Vol. 25, 1960.9. Biot, M. A., Theory of Propagation of Elastic Waves in Fluid-

Saturated Porous Solids, Journal of the Acoustic Society oAmerica, Vol. 28, 1956.10. Kuster, G., and Toksoz, M. N., Velocity and Attenuation of SeismicWaves in Two-Phase Media, Geophysics Vol. 39, 1974.11. Geertsma, J. Velocity-Log Interpretation: The Effect of Rock BulkCompressibility, SPEJ, Dec. 1961.

1.

2.

3.

4.

ProblemsUsing Table 15-1, find an expression for compressional velocity v, as afunction of the bulk modulus and the shear modulus. The quantityB + -p is known as the plane wave modulus.3Show that the plane wave modulus can be expressed as L + 2p, whereis the Lame constant.Consider two immiscible liquids of bulk modulus kl and k2 which aremixed together with volume fractions V1 and V2. What is the mixinglaw for the bulk modulus of the mixture of two liquids? Write it interms of the volume fractions of the two components.It was seen that the Gassman equation predicts the bulk modulus k of aliquid-filled rock of porosity I to be:

where the subscripts of the bulk moduli are as follows: df refers to thedry frame, ma to the matrix, and f to the fluid.a. Using the mixing law of Problem 2, write the above expressionfor bulk modulus in terms of the state of water saturation.b. Describe how you would obtain a value of f your laboratory

measurements limited you to a judicious choice of compressionalvelocity measurements. An explicit equation for the value ofmay be more than you want to do, but please indicate theprocedure.Given the bulk modulus of a rock sample to be 141.7 GPa and acompressional velocity of 7.8 km/sec, what is an upper limit to be


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Acoustic Waves in Rocks 379expected for the shear velocity? What is the actual shear velocity ifthe density of the rock is 5.00 g/cm3?Fig. 16-7 shows some data for a 25%-porosity sandstone that indicatethe effect of the saturating fluid on compressional and shear velocities.The implied change in apparent formation At for the compressionalarrivals is at a m aximum for the case of no differential pressure.a. Using the Wyllie time-average approach, what value of At (inpsec/ft) would you ascribe to air? The velocities of typicalmaterials in Table 16-1 may be of some use in this calculation.b. If, while logging a sandstone formation, you encountered aformation At corresponding to this minimum velocity and did not

recognize the presence of gas, what porosity would you estimateit to be?The curves for shear waves are inverted on this figure comparedto the compressional. Show, however, that they are in theanticipated order and that the magnitude of the separation is asexpected for no differential pressure.

5 .

c.

Non-Porous Solids V. V.AnhydriteCalciteCement (Cured)DolomiteGraniteGypsumLimestone

SteelWafer-Saturated Porous

Rocks In Situ

20,06020,100'12,00023,00019,70019,00021,00018,900'15,000'20.000

11,400

12,70011,20011,10012,0008,0009SM)

PorosityDolomitesLimestonesSandstones

Shales 7.003-17,000Liquids**

Water (Pure) 4,800Water (100,oO m g i l )of NaCI)Water (200.00 mullof NaC1) 5,500Drilling MudPeaoleum 4,200

r Dry or moist) 1,100Hydrogen 4,250

Gases.'

Methane 1,500Anthmetic Average of Values Along Axes (Wylhe et al., 1956)**At Normal Temperature and Pressure

Table 16-1. Acoustic velocities for various materials. From Timur.'

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16. Acoustic Waves in Rocks