7 -A207 540 - O LINEAR ATTENUI t ollf NEcw~ ism IN sft ?- T ft ER IlT - /11 STRAIN BASED ON SALMO..(U) MISSION RESEARCH CORP SANTA

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Nonlinear Attenuation Mechanism in Saltat Moderate Strain Based on Salmon Data

G. D. McCartorW. R. Wortman

Mission Research Corporation736 State StreetP.O. Drawer 719Santa Barbara, CA 93102

December 1988 r -

AY 5Scientific Report No. 3 A U

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Nonlinear Attention Mechanism in Salt at Moderate Strain Based onSalmon Data

12. PERSONAL AUTHOR(S)G. D. McCartor; W. R. Wortman

13a. TYPE OF REPORT 13b. TIME COVERED 14. Date of Report (Year, Month, Day) 15. PAGE COUNTScientific #3 From To- 1988 December 44

16. SUPPLEMENTARY NOTATION

17. COSATI CODES 18. SUBJECT TERMS (continue on reverse If necessary and identify by block numbe1FIELD GROUP SUB-GROUP Salmon

Nonlinear AttenuationPartial Shear Failure

19. ABSTRACT (Continue on reverse if necessary and Identify by block number)' O.in order to describe the seismic pulse or source function from UGTs outside the region of non 1neliir attenuation,data from the Salmon event (5.3 IT in salt) have been examined to serve as the basis for a description of a

mild nonlinear attenuation mechanism. It is found that a precursor in the Salmon pulses can be at.joIuted to a

partial shear failure of the medium which operates above a compresional strain threshold of about ID.? Whenthis loss mechanism is included along with a linear Q of about 10, the Salmon pulses in the moderate strain

regime are nearly reproduced in both amplitude and shape. Using this result the pulse can be propagated out

* to a range for which no further shear failure occurs and it can serve as a linear source function_.. , , ,

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TABLE OF CONTENTS

Section Page

1 INTRODUCTION ............................ 1

2 MODERATE STRAIN ATTENUATION DATA IN SALT. 3

3 MECHANISM FOR SALMON ELASTIC PRECURSOR . 6

4 COM M ENTS ............................... 17

REFERENCES .............................. 19

Appendix

A WILKINS FINITE DIFFERENCE EQUATIONS .......... A-1

REFERENCES .............................. A-5

B INCLUSION OF DAY-MINSTER Q INFINITE DIFFERENCES ....................... B-i

REFERENCE ...... ............................... B-4

.ftef |

LIST OF ILLUSTRATIONS

Figure Page

1 Salmon radial velocity pulses seen at 166, 225, 276, 318, 402 and 660meters range (time zero on plot corresponds to initial signal arrival at166 meters which was at 0.036 seconds after detonation) .......... 7

2 Calculated pulses at Salmon ranges for pure plastic medium ..... 8

3 Stress-strain curve for partial shear failure ................... 11

4 Calculated pulses at Salmon ranges for shear failure ............ 12

5 Cowboy Q estimates as a function of peak strain ............... 13

6 Calculated pulses at Salmon ranges for shear failure and Q=10 . . . 15

7 Calculated pulses at Salmon ranges for smoothed shear failure andQ=1O. ......... ................................... 16

B-1 Absorption band Q and propagation speed C for three values of thePad6 index m for Qo = 20, ,r = 1.59 x 10 - 3 and r2 = 1.59 x 10-1 . . B-3

iv

SECTION 1

INTRODUCTION

In order to characterize a distant seismic pulse according to the energywhich generated it, a source function' is needed which can be used to initiate theseismic signal at sufficient range from the event that the subsequent propagation canbe described in linear terms according to the properties of the earth. Thus an ob-served seismic signal implies a source function which may be compared with knownfunctions for discrimination and yield estimation. Therefore, the properties of suchsource functions must be known to use this technique. Near-field data from variousevents have been taken. The smallest strains observed, due to practical free fieldinstrument placement selection, are typically no less than 10 5 and are often muchgreater. If the pulse at this extreme range undergoes no further significant nonlinearmodification, its characterization can supply the needed source function. However,if any additional nonlinear changes are important, a useful source function cannotbe determined. Consequently, it is important to characterize any possible nonlin-ear attenuation of moderate strain pulses, preferably through methods which allowgeneralization to all placement media of interest.

There is a long history of development of numerical methods for calculationof strongly nonlinear behavior in the near field of UGTs.2'3 The gross nonlinear ef-fects of vaporization, crushing, cracking and plastic behavior induced by strains muchgreater than 10- 3 must be taken into account to obtain an understanding of near fielddata taken at NTS. Complex equations of state or constitutive relations includingeffective stress and porosity are needed for a description.4 These methods generallyapply for strains greater than 10-3; at lesser strains linear behavior is assumed. How-ever, as indicated in the following section, it seems clear that some residual nonlineareffects continue out to strains less than 10- . Ideally one would like to have a phys-ically based model of any nonlinear effects in order to use existing near field data todefine a more distant linear source function or at least to determine what effect, if any,the nonlinearities over the moderate strain range have on the linear source function.

In order to determine this, we are attempting to develop nonlinear consti-tutive relations for the mildly nonlinear attenuation in the moderate strain regime(scaled ranges of 102 m/kT / 3 to 104 m/kT1/ 3 or strains from 10- 3 to 10 - 6 ) for explo-

sively generated seismic pulses. The relations are to be used to determine the characterof seismic pulses in the linear low-strain regime (beyond 104 m/kT 1/ 3) which may then

1

serve as source functions for regional or teleseismic propagation. A linear source func-tion can be found by numerical propagation of an experimentally well known initialpulse through the moderate strain range to account for mild nonlinear attenuation.

Experimental data provide significant restrictions on candidate constitutiverelations. Attenuation in a salt medium from Salmon, 5s - 0 Cowboy," - 13 Livermoresmall scale explosions,' 4 Rockwell damped oscillation experiments' and New Eng-land Research ultrasonic pulse attenuation,16 with one exception, are fairly consistentinternally. The first three provide a detailed indication of the change of amplitudeand shape of explosively driven pulses. Proposed constitutive relations must repro-duce these data. The data from the wide range of yields in salt indicate that yield' /3

scaling applies with a remarkable precision; if all times and distances are scaled byyield" 3 , the scaled amplitudes and shapes of pulses from all experiments are nearlythe same."7 Thus the initial pulse on entering the moderate strain regime must scale atsome small scaled range and the subsequent attenuation must result from constitutiverelations which have no time or space scales which are fixed by the medium. Thisprovides a significant reduction in the domain of allowable nonlinear behavior. Forexample, the relations may be a function of the strain but the may not depend on thestrain rate.

In order to investigate nonlinear constitutive relations, a standard numericaltime stepping method is used. The technique which we have used is that of takingthe observed Salmon initial velocity pulse at small range (166 meters) as a sourceand comparing the resulting pulses as they are propagated through material subjectto candidate constitutive relations. The results are compared with observed signalsat larger ranges. For any constitutive relation the effective Q associated with theattenuation may be determined but it must be emphasized that nonlinear attenuationcannot be properly described by a Q function; still Q may sometimes be useful forcomparison with past work. The fundamental comparison of the data with calculationsis not in terms of the Q but in terms of reproduction of waveform including bothamplitude and shape.

2

SECTION 2

MODERATE STRAIN ATTENUATION DATA IN SALT

The most comprehensive attenuation data, over the range of moderatestrains, exists in the medium of salt. For explosive sources the laboratory resultsof Larson 14 covers strains from 10-' to 10 - 3 , the Salmon field test5 covers strains for10- 3 to 10 - 4 while the Cowboy field test" series covers strains from 10 - 4 to 10 - 5 .

The Rockwell' 5 laboratory decaying oscillations complete the salt data by rangingfrom 10- 5 to 10-8. The NER' 6 laboratory ultrasonic pulse propagation experimentsgo from less than 10- 6 to more than 10- . The essential results of these experimentswill now be reviewed.

Larson's laboratory data were taken from the effects of a series of smallchemical explosives in blocks of pressed salt. Velocity data were taken at severalgauges such that the totality of the examples from all shuts covered peak velocityto compressional velocities, which is approximately the peak strain, of 10- 1 to 10 - 3 .

Using a triplet of records from a single shot, it was estimated that over peak strainsof 1.4 x 10 - 3 , 7.0 x 10 - 4 and 4.6 x 10 - 4 , the Q changes from 12.5 to 24.9. The cornerfrequency for this small scale experiment was about 5 x 10' Hertz.

The Salmon data were generated by a 5.3 kT nuclear explosion in salt) Thework by McCartor and Wortman 7 as well as McLaughlin and Gupta8 shows that thehigh quality data clearly indicate a high level of attenuation. It is estimated that forpeak strains from 4 x 10 - 3 to 3 x 10 - 4 , the effective Q is to order of 10 at a cornerfrequency of about 6 Hertz. This Q appears nearly constant,7 perhaps increasingmildly with range,8 over the order of magnitude strain range available. However, inview of the fact that small strain data from other experiments, including the decoupledevent Sterling8 which used the Salmon cavity, indicate a much larger Q, it seems likelythat there are residual nonlinearities at the extreme range of the Salmon data so thatQ must increase at larger ranges (and so smaller strains).

Tittmann" has studied the attenuation of flexural and torsional harmonicoscillations of salt samples. It has been found that the attenuation, expressed interms of Q' as a function of strain amplitude, tends to be nearly constant for strainsfrom 10- 8 to 10- 6 , then increase for greater strains. The attenuation is a decreasingfunction of confining pressure with an average Q of approximately 200 at 10 8 strainat a frequency of 400 Hertz for confinement consistent with the explosions.

F-. I

3

The regime covered by the COWBOY experiments overlaps that for Salmon,by beginning at strains of about 5 x 10- 3 and extending out to 10- '. The COWBOYexperiments consisted of a series of shots over a range of yields, both coupled anduncoupled, such that the individual events generally had only a few instruments inoperation. Several authors have combined these data by invoking the experimentallycompelling evidence for simple scaling, which suggests that the velocity pulse datafrom various yields can be equated by scaling all times and distances by the cube rootof the yield. This is particularly dramatic when peak velocity is plotted against scaledrange giving a consistent curve over about ten orders of magnitude in yield. Larsonhas pointed out that the full velocity pulses, as well as their peak values, tend to bepreserved through scaling.

Trulio' ° has analyzed the scaling-combined Cowboy data to estimate Q asdictated by decrease in peak displacement with range by using one decade at a time inthe frequency domain. He has found that attenuation decreases as range increases ina fashion inconsistent with linearity expressed as dispersive harmonic potential waves.If the coupled Cowboy data are fit assuming a Q independent of range (a possibilitydue only to scatter in the data), Q must be a function of frequency ranging fromabout 5 at low frequencies (1 Hz at Salmon scale) to nearly 100 at high frequencies(32 Hz at Salmon scale). However, simple scaling is inconsistent with this fit sincescaling requires linearity coupled with a Q independent of frequency. If the decayof the reduced displacement potential is fit by the form exp(-wr/2cQ), it is seenthat simple scaling can result so long as, for constant c, Q is a function of a variablewhich is unchanged by scaling - such as wr. Trulio has shown that the a range ofdata from Salmon, through Cowboy and Cowboy Trails, do have the property that aneffective Q can be expressed as a function of wr for all the experiments. Obviously thisindicates that at fixed frequency, the attenuation expressed in terms of Q, is dependentupon range. Since it is assumed that the medium is approximately homogeneous,the implication is clear that the attenuation must be amplitude dependent and sononlinear.

Minster and Day" have used the scaled peak velocity data for COWBOYto determine if these data require and amplitude (nonlinear) or frequency dependentQ for consistency. They determined that a Q which consists of a small constant(consistent with small strain data from Tittmann) plus a term proportional to thepeak strain provides a good fit to the data. The observed attenuation effects do notfirmly indicate the need for a frequency dependent Q but indicate that an amplitudedependent Q provides a more convincing fit than a constant Q. Based on the smallstrain Q, which is several hundred as seen in the Rockwell experiments, it is concludedthat there must be nonlinear attenuation in the COWBOY strain regime.

4

Coyner 16 of New England Research has carried out a series of laboratoryexperiments for which compressional and shear ultrasonic pulses consisting of abouttwo cycles at 100-200 kliz were propagated through a sample. Attenuations werecalculated using a spectral ratio technique. Variation of the attenuation with peakstrain amplitude and confining pressure were determined. Experiments were carriedout with several materials including Sierra White Granite, Berea Sandstone and DomeSalt. For the dome salt it was found that over a strain range of 5 x 10- 7 to 3 x 10 - 5

and for a confining load range of 0.1 to 1 MPa, the P-wave attenuation is nearlyconstant and can be described by a Q of about 20. There is no particular evidence ofnonlinearity in these data alone.

In summary, the salt data appear to support the hypothesis that explosivelygenerated pulses encounter nonlinear attenuation for strains much greater than 10- 6.No detailed knowledge of the attenuation mechanism currently exists but there doesappear to be a consistency in than the explosively generated pulses closely obey simplescaling which suggests that the mechanism must be rate independent. The effectiveQ for this process increases with increasing frequency and increases with decreasingstrain. In the next section we shall attempt to exploit a feature of the Salmon near-field pulses which suggests a physical mechanism which can account for the attenuationdata.

5

SECTION 3

MECHANISM FOR SALMON ELASTIC PRECURSOR

The Salmon data have a feature which may be useful in understanding somenonlinear effects of attenuation. Each of the pulses experimentally observed at sixsensors at ranges from 166 meters to 660 meters exhibit a discontinuity in the slopeupon the initial steep rise in pulse velocity, as seen in Figure 1. This appears as atoe-like behavior in the leading edge of the velocity profile which has been describedby Perret5 as an "elastic precursor" to the main pulse. The absolute amplitude ofthe toe remains approximately constant with range, at a particle velocity of about0.5 m/s, while its amplitude relative to the peak increases with range. This precursoramplitude corresponds to a compressional strain level of E - 10-4.The leading edgeof the pulses (i.e., that disturbance earliest in time) propagates at a speed of about4.7 km/sec while the pulse peaks, always after the toe, propagate at a speed of about3.7 km/sec. The elastic compressional speed of mild disturbances in this salt mediumfound from independent measurements is typically about 4.6 km/sec. This indicatesthat the precursor signal seen in the Salmon data is due to elastic behavior while thesubsequent pulse suffers a lower propagation speed due to some relaxation or plasticbehavior. If propagation were purely plastic the sequence of pulses seen at the Salmonsensor sites would be as shown in Figure 2.

Perret suggests an elastic-plastic material behavior might account for thedata in perhaps one of two ways. First, the precursor could develop at large strain,where an elastic limit is exceeded from radii much smaller than instrumented forSalmon, and continue to propagate in front of a following plastic wave. Second, itmay be that the precursor develops in the moderate strain region if dome salt has anelasto-plastic nature at such strains. In either case, the modulus of salt must be afunction of the strain - that is, the medium is nonlinear.

If the precursor develops at large strain there must be an elastic limit beyondwhich plastic behavior provides a lower modulus. When such a medium is dynamicallyloaded beyond the elastic limit, a leading pulse at the elastic limiting stress is generatedfollowed by a larger amplitude but slower plastic wave. If the elastic-plastic transitionis not sharply defined, the resulting pulse could consist of a gently rising leadingelastic front which smoothly merges with the main plastic pulse much as seen inthe Salmon data. Perret points out that if some energy from the plastic componentis fed to the elastic portion during propagation, the amplitude of the elastic piece

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could remain nearly constant, but this is quite speculative. What is known is thata variety of laboratory cxperiments show that strong impulsive sources can produceelastic precursors in a variety of media. For example, work of Ahrens and Duvall18

with planar pulses in quartz generated by explosives exhibits an apparent elastic stresslimit of about 70 kbar corresponding to strains of about 10-1. This produces a leadingedge, described as the elastic shock, which propagates at a speed in excess of that ofthe deformational portion of the pulse which follows. The elastic shock propagateswith an equivalent modulus which is greater that the static modulus at this high stress.It is speculated that the elastic wave is supported by a higher than equilibrium shearstress. After the elastic component has passed, the shear stress is apparently reducedto the static value by a plastic or fracture process. This experiment as well as others('which show an elastic precursor consistent provide elastic limits of many or tens ofkilobars in contrast with the Salmon data which gives a precursor amplitude of about5 bars. Consequently this mechanism does not seem a likely means of accounting forthe Salmon data which give a nearly constant and small precursor amplitude whichseems to begin near the 166 meter sensor range rather than be well developed by thistime.

The second possibility indicated by Perret is that of having the precursordevelop locally in the observation region based on moderate strain plastic behavior.While there is no accepted dynamical equation of state for dome salt, Perret pointsout that dome salt is known to be highly plastic - under static conditions, it is nearlyhydrostatic. Thus plastic deformation of salt at moderate stresses is apparently nor-mal. In order to account for the precursor data in Salmon, the equation of state wouldhave to provide linear behavior up to a threshold (a threshold of about 5 bars, muchless that the 70 kbars found for the elastic shock discussed above) and through somedeformation of the material, relax the modulus abruptly (on the Salmon time scale)to a value which provides a compressional propagation speed about 20% less than forinfinitesimal strains in undisturbed material. Having a threshold at the observed andnearly constant precursor amplitude avoids having to account for the constancy ofthe precursor by arguments of convenience about feeding energy back from the plasticportion of the wave.

As an equation of state model which can be consistent with the Salmon data,consider a medium for which the shear modulus permanently (that is, does not recoveruntil the pulse is past) decreases upon having a critical strain threshold exceeded; thecompressional modulus before and after exceeding the strain threshold reflects thecompressional speeds at the beginning and peak of the Salmon pulses, respectively.This will be referred to as a shear failure model. Depending upon the relation betweenthe compressional and shear moduli, complete shear failure may occur, meaning that

9

the elastic shear modulus, ju, goes to zero. For the example used in this discussion, thecompressional speed decreases to 80% of its original value. We have taken the Lain6constants A and pi to have a ratio of 2. Thus a decrease of the compressional speed,((A + 21L)/p)1/2, where p is the density, of 20% corresponds to reducing A to about38% of its elastic value when A is held fixed. Note that the reduction of modulus at afixed strain is consistent with scaling since the strain is a unitless quantity and thereis no rate dependence. The scaling restriction does require that the relaxation timeof the modulus change be short compared with any representative time scale of thedata; we take the transition to be instantaneous. Figure 3 indicates the stress-straincurve which results from this model for the modulus. In order to determine the effecton the pulse propagation we used the observed Salmon velocity at 166 meters as thesource. The calculations were carried out using a standard finite difference methodswhose details are provided in Appendix A.

As an initial effort, the elastic threshold was taken at a compressional strainof 10-4; the resulting pulse sequence at the ranges to observation stations for Salmonis as shown in Figure 4. Note that the character of the calculated precursor is muchlike that seen experimentally, in Figure 1, in that the leading feature is drawn out, thetransition to the main pulse takes place at a constant amplitude and the peak nowmoves at a significantly lower speed. Still the amplitude of the main peak does notdecrease as quickly as the data indicate.

When the modulus decreases, the elastic energy in the pulse also decreasesin a manner approximately proportional to the square of the compressional wavespeed. Since the modulus reduction is permanent, this energy is lost to the pulse andgoes into heating the medium. For the parameter used, over a full cycle for whichmost of the pulse exceeds the critical strain, approximately one-third of the originalelastic energy will be lost. This corresponds to an effective Q of about 13 for peakstrains well in excess of 10 -' (for small strains less than this threshold, there willbe no loss). This value of Q is far less than that expected for very small strainsbut it is still more than the 5 to 10 seen for Salmon attenuation. The addition of amoderate level of linear attenuation consistent with that seen for small to moderatestrains in other experiments will improve the agreement (for example, the NER datasuggest Q ;. 20). More importantly, the use of a partial shear failure mechanism willautomatically terminate once the pulse weakens so that the peak strain falls belowthe critical strain threshold value. This will produce a sharply changing effective Q ina manner suggested by the Cowboy data as given in Figure 5. Thus this first attemptto reproduce the Salmon data is encouraging but there remain differences which maybe accounted for by refining the details of the attenuation mechanism.

10

FAILURE POINT

Figure 3. Stress-strain curve for partial shear failure.

11

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PEAK STRAINFigure 5. Cowboy Q estimates as a function of peak strain. The

horizontal bars represent the two record strain values whilethe vertical bars indicate the spread of Q over a decade offrequency. The fit of Minster and Day is provided. (From Ref.13). 13

The attenuation from partial shear failure does not produce an amplitudefor the pulse at 660 meters which is as small as that seen experimentally. More atten-uation can be added to attempt to match better the data by employing a linear Q ofsufficient value. A method of inclusion of a linear absorption band Q in time steppingfinite difference methods has been demonstrated by use of Pad6 approximants. Ourapplication of this method is outlined in Appendix B. This formalism was employedusing a target Q of 10 with a range of half amplitude frequencies of 1 to 100 Hertz (Qrises above 20 beyond these values). The sequence of pulses which result using boththe partial shear failure and a linear Q of 10, starting with the Salmon pulse at 166meters, is shown in Figure 6. The amplitudes for the main peaks now are in substan-tial agreement with the data and the length and amplitude of the precursor are alsoreproduced fairly well. Still there remains a very abrupt transition from precursor tomain pulse which is clearly sharper than the experimental data.

One further refinement has been applied to the partial shear failure modelin order to avoid the abrupt transition between precursor and pulse. Since there iscertain to be a range of material properties even in the relatively homogeneous saltdome, it is reasonable to require a range of thresholds for the shear failure. The modelhas been altered to produce a variation of failure threshold values of compressionalstrain over a range of 30% with a constant probability about the 10- 4 value. Each cellin the finite difference calculation is given its own threshold which is randomly drawnon this basis. The set of pulses at the Salmon instrument ranges then calculated isgiven in Figure 7. The result is a smoother transition from precursor to main pulse ina manner which is quite similar to the actual Salmon data shown in Figure 1. Whileit is probably possible to achieve a detailed fit to the data by further such refinements,this is not a very meaningful thing to do since the mechanisms are not understood tothe required level of detail. The important point is that it is possible to reproducethe data to a substantial degree using only a few physically based parameters.

14

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SECTION 4

COMMENTS

This work indicates that the elastic precursor or leading toe seen in Salmonnear-field, moderate strain, velocity data is reproduced rather well with the hypothesisof partial shear failure which is activated for the duration of the pulse when thecompressional strain exceeds 10'. This also gives an attenuation mechanism whichaccounts for much of the energy loss seen in the decay of pulses from Salmon withrange. However, the overall attenuation produced is not quite adequate to account forthat seen in the data. The addition of a linear absorption band attetuation, whichis active over much of the significant frequency range appropriate to Salmon andwhich has a Q of 10, then provides a propagation model which nearly reproduces thesignals at ranges beyond 166 meters when the observed signal at this range is usedas the source. Furthermore, this threshold mechanism provides a transition to moremodest attenuation at small strains which is required to be consistent with Cowboydata. While there is no assurance that this mechanism applies in other than the saltmedium, Perret points out that "elastic precursors" of a similar character have beenseen in UGT pulses in both alluvium and dolomite.

In addition to accounting for much of the Salmon data, the shear failuremechanism also, when applied to different yield events in salt, will produce simplescaling as observed over a wide range of explosive events.

The fact that the reduction in compressional wave speed is attributed toshear failure, rather than alteration of some other material property, is largely amatter of consistency with past thinking on modes of material behavior; there is nodirect experimental link to shear properties. The general agreement with data whichresults here could just as well have been produced by any method which reduces thecompressional modulus in the required amount.

The fact that the NER experiments give an apparently linear Q of about 20for small strains of 10-' while Sterling data at comparable strains have much largerQ is a matter of some concern. However it known that the attenuation of a materialcan be a strong function of the liquid content. Spencer2 0 made a series of attenuationmeasurements in several types of rock in dry and water saturated states. He found thatdry rocks tend to show only modest attenuation while the water saturated sampleshad minimum Q's of from 4 in sandstone to about 40 in limestone. He found that theattenuation for saturated rocks is quite frequency dependent with peaks at 10 to 1000

17

lIz depending on the material, lie further found that the attenuation spectrum isconsistent with dispersion in the modulus so that the mechanism is apparently linear.This sort of result indicates that much care must be taken in comparing experimentalresults taken for different levels of water saturation and at different frequencies evenwhen strain levels are quite low.

There is a second area where different experiments appear to yield differentresults. The work of Tittmann on decaying oscillations of salt and other bars giveapparent nonlinear behavior but they also show vary mild attenuation in the linear

regime. As observed by Coyner, the attenuation from these multiple cycle experimentsseem to be consistently much less that from propagation (single pulse) experiments atcomparable strains and frequencies or from hysteresis loop experiments. For nonlinearattenuation, thete does not need to be any particular relation between the effective Qfrom pulse and oscillatory experiments - the detailed nature of the mechanism mustbe known to relate these. For the example of partial shear failure, multiple cycleexperiments will produce very small attenuation estimates since all the failure willtake place by the time a single cycle has been completed - after that only the intrinsicattenuation will be active.

It should be emphasized that the use of a Q description for attenuationwhich is clearly nonlinear is probably not useful, except as a crude estimate of thedegree of amplitude reduction. In fact, the use of Q in nonlinear cases can provide very

deceptive results since the frequency and range dependence are generally mixed. Itwould be best to describe the attenuation mechanism directly in terms of an equationof state. This will allow the application to different experiments. Put another way, theresult of a nonlinear attenuation for a propagation experiment can always be used tofind an effective Q by, say, use of spectral ratios but the same mechanism will producea different effective Q if the character of the initial waveform is changed. The effective

Q is not a robust quantity for nonlinear attenuation.

We expect to use the finite difference propagation code, which was developedas a tool for our current work on Salmon, as a testbed for development of nonlinearconstitutive relations. This avoids the use of Q and concentrates directly on the

physical mechanisms which produce attenuation. Ideally one would like to express aconstitutive relation in terms of parameters known to be important, such as porosity,

effective stress, crack density, crack growth, etc., in order to allow a model which canbe generalized to different media.

18

REFERENCES

1. Examples of such considerations are: Mueller, R.A. and J.R. Murphy, SeismicCharacteristics of Underground Nuclear Detonations, Bull. Seis. Soc. Am., 61,1975, 1971; and Murphy, J.R., Seismic Coupling and Magnitude/Yield Relationsfor Underground Nuclear Detonations in Salt, Granite, Tuff/Rhyolite and ShaleEmplacement Media, CSC-TR-77-0004, Computer Sciences Corporation, 1977.

2. Cherry, J.T., N. Rimer, W.O. Wray, Seismic Coupling from a Nuclear Explosion:The Dependence of the Reduced Displacement Potential on the Nonlinear Behav-ior of the Near Source Rock Environment, SSS-R-76-2742, Systems, Science andSoftware, Sept 1975.

3. Riney, T.D., et a]., Constitutive Models and Computer Techniques for GroundMotion Predictions, DNA 3180F, Systems, Science and Software, March 1973.

4. Cherry, J.T., N. Rimer, W.O. Wray, Verification of the Effective Stress and AirVoid Porosity Constitutive Models, VSC-TR-83-1, S-CUBED, August 1982.

5. Perret, W.R., Free-Field Particle Motion from A Nuclear Explosion in Salt, PartI, Project Dribble, Salmon Event, VUF-3012, Sandia, 1967.

6. Rogers, L.A., Free-Field Motion Near a Nuclear Explosion in Salt: ProjectSalmon, J. Geophys. Res. 71, 3425 (1966).

7. McCartor, G., and W. Wortman, Experimental and Analytic Characterizationof Nonlinear Seismic Attenuation, MRC-R-900, Mission Research Corp., March1985.

8. McLaughlin, K.L. and I.N. Gupta, Strain and Frequency Dependent AttenuationEstimates in Salt from Salmon and Sterling Near-Field Data, Teledyne Geotech,to be submitted to Geophysical Research Letters, 1986.

9. Murphy, J.R., A Review of Available Free-Field Seismic Date from UndergroundNuclear Explosives in Salt and Granite, CSC-TR-78-0003, Computer SciencesCorporation, September 1978.

10. Trulio, J., Applied Theory, Inc., private communications, 1986.

11. Murphey, B.F., Particle Motions Near Explosions in Halite, J. Geophys. Res.66, 947 (1961).

12. Minster, J.B. and S.M. Day, Decay of Wave Fields Near an Explosive SourceDue to High-Strain Nonlinear Attenuation, J. Geophys. Res. 91, 2113 (1986).

19

13. Wortman, W. and G. McCartor, Nonlinear Seismic Attenuation from Cowboyand Other Explosive Sources, MRC-R-1107, Mission Research Corp., September1987.

14. Larson, D.B., Inelastic Wave Propagation in Sodium Chloride, Bull. Seism. Soc.Am. 72, 2107 (1982).

15. Tittmann, B.R., Non-Linear Wave Propagation Study, SC5361.3SAR, RockwellInternational Science Center, 1983.

16. Coyner, K.B., Attenuation Measurements on Dry Sierra White Granite, DomeSalt and Berea Sandstone, New England Research, September 1987.

17. Trulio, J., Simple Scaling and Nuclear Monitoring, ATR-78-45-1, Applied The-ory, Inc., 1978.

18. Ahrens, T.J. and G.E. Duvall, Stress Relaxation behind Elastic Shock Waves inRocks, J. Geophys. Res. 71, 4349 (1966).

19. Taylor, J.W. and M.H. Rice, Elastic-Plastic Properties of Iron, J. Appl. Phys.34, 364 (1963).

20. Spencer, J.W., Jr., Stress Relaxations at Low Frequencies in Fluid-SaturatedRocks: Attenuation and Modulus Dispersion, J. Geophys. Res. 86, 1803 (1981).

20

APPENDIX A

WILKINS FINITE DIFFERENCE EQUATIONS

The difference method used to solve for the spherical propagation of a pulseis based on that given by Wilkins.' The essential features of this approach are asfollows:

The equation of motion is

POO E 2Er- E0 (

V - Or + r (A-I)

where the principal stresses are

Er =-(P + q) + s, (A-2)Es= -(P +q)+s (A-3)

in terms of the hydrostatic pressure, P, the artificial viscous pressure, q, and the stressdeviators. The reference density is p , the relative volume is V and U is the particlevelocity.

The continuity equation is

V _ 1 a(r2U) (A 4)V r - (Or

where r is spherical radial position.

The energy equation is

t - V[s 1 i, + 2s 2i] + (P + q)V = 0 (A - 5)

where E is the internal energy and c are strains:

i,= oU/or (A-6)S= U/r (A-7)

Artificial viscous pressure is taken from Viscelli's 2 work as:

A-i

p~a BOU

q=CL---0 c r Ar (A-8)

where CL is a constant (taken as 0.5), a is the sound speed and Ar is the cell spacing.

For the elastic case the equation of state is

s, = 21L(i4 - IV/V) (A-9)1

i2 = 2M(i2 - 1-r/V) (A-10)3

= -(A + 2/A)V/V (A-I1)

where A and k are the Lame elastic constants (A + Ls is the bulk modulus).

The finite difference version equations is taken by division of the materialinto N mass intervals

mj+1 + 3 j = 1,...,N (A- 12)

The equation of motion for the velocity for the jth element at the n th time

A[(+7n -I - (XEr)7_ 1 ] + 2Atn(3 n ) (A - 13)j 2

where= (pn + qn-i) + + (A-14)

(En{)j+ {-(pn + qn-1) + s }j+(A-15)

and

1P + .21- . (r ' ) r (A-16)

pr - )+ I ( e +rr) I - ,+ [E ( 2err- i] (Y )} 7)

A-2

At an outside regional boundary J

while at an inside regional boundary

10 (r- r+i) (A-20)

I. = (r+ -r'+)+ (v )- J~ P0) (A-21)

Given the velocity, the positions are advanced by

r -' rf + Un+ At "+A -22

2 2 (A 22

The equation of continuity is+tn+ (pO) F+( n+_. n+,n2

J+j =\m ,+' U+ kri+i ) - U.1 (r7 )2±-i- Xj (A- 23)

with

#rn +I) 1 2 (A -19)

where

1n{_+ + r+,) ,etc. (A25)

defines the half time values. And, finally

A-3

( f +)2 [U+

xj+ (-12 [(j+I, - (U-+)3] (A - 26)

The strains are defined by

n+ 1 n+ 1

2 r,+ .2 ,

(i2 )n + + - (A-28)

rj+ + r(-

and the stress deviators are advanced by

(s,)n+, .S) , + ,)+IAtn+ I Vn +l - Vn

j+) = (S1)j+i + 2 [- + ( (A-29)2 "+2+2 3 Vn+ 2/J++

+ + 2)n ( + I Atn+ 2 - \ 1 (V - 7 ] (A-30)

Artificial viscous pressure is included by

qj~~~( J+_U+-31)q 1+ = -Laip 1/7+ _.L ~ ;( -+2 j+

where CL ; 0.5 (constant) and a in the compressional wave speed.

The energy equation is advanced by

+1 n+ n+=()n+, + V. = s).$ (2.)"+ Atn+ (A-32)(E2 j+1 (E2)j+ I+ 2+, 2 j+.t2 2 J+2 2 1+

-

(E,2)7+ : (E2)+ + 2V+.2(2,i ()+.2A (A-33)

-l =E -E + p'+ qn+i [V" 1 V-1 + AE1 + AE21 A-34)

A-4

For the current application, this energy is just a diagnostic and is not fedback through any equation of state.

The hydrostatic pressure for the elastic case is

REFERENCES

1. Wilkins, M.L., "Calculation of Elastic-Plastic Flow," Methods in ComputationalPhysics, Vol. 3, 1964.

2. Viecelli, J.A., "The Linear Q and the Calculation of Decaying Spherical Shocksin Solids," J. Comp. Phys., 12, 187 (1973).

A-5

APPENDIX B

INCLUSION OF DAY-MINSTER Q IN FINITE DIFFERENCES

Day and Minster' have shown how to include an arbitrary linear Q functionin time stepping calculations. For an absorption band an analytic solution is available.An outline of their methods, as easily generalized to the spherical case is given here.

For a single normalized relaxation function, m(t), the stresses and strainsare related by

Er = (A+2 )fm(t-r)E1(r)drT+ 2Af m(t -r)c 2 (T)dr (B-1)

= (2A + 2js) f m(t - r)( 2 (r)dr + Af m(t - r)E,(r)dr (B-2)

Here the strains are

auE -r (B-3)

u

2 - (B-4)r

where u is displacement. Generally one could have two different relaxation functions(e.g., bulk and shear) but we shall not consider this possibility. The A and JL are theusual Lam6 constants which are now the unrelaxed or high frequency moduli of themedium. It stresses can be written in terms of Q corrected strains, e, as

FIr = (A + 2)el + 2Ae 2 (B-5)Ea = (2A + 21i)e 2 + Ael (B-6)

Day and Minster have shown how to express the ej in terms of the Ei

ej = f m(t - r)cidr (B - 7)

using a sequence of m Pad6 approximants to write the integral relation as a differentialrelation. For an absorption band attenuation with relaxation times between r, and r2and with a flat spectrum they show that the integral relation can be replaced by

B-1

m

el(t) = ti(t) - E Zjk(t) (B -- 8)k=i

wheredWkQo I i(t) kzl, m (B - 9)

Here Qo is the target Q in the absorption band, the

1[t= I L - ri) + (qO +il)] (B - 10)

where the tk are the abscissas and Wk are the weights for m-point Gauss-Legendrequadrature. The index m is that of the Padi approximant. As m increases, thesolution converges to the analytic result which in the frequency domain is

2 tnr2 (1 W 2,2\)i/2]1 2i (W(r 2 - 'rl)I0 =1+-tan- W 1 2 (B - 11)

The convergence of the sequence with increasing m is shown in Figure B-1. Forcalculations given in the main text m=5 was used since convergence is near. Use ofsignificantly larger m slows down the calculations of pulse propagation since each madds an additional differential equation which must be carried. In finite differenceform the are advanced for both i by

n+ 2~t n- 1 6 1 ~ n+2' .n.-=j (2- Atl.) (k)l+, Wk -j + ij+1)

-'Atn 2 2+Atni4 xrQo J2 BJ B-12)

and then the

m

i = i E ?k (B - 13)

k=1

These are then used to advance the stresses by

B-2

0v

Cb

4.4

t4 a

cq

z

0

M Go I 1* M

COO/D~ P. bo

B-3

4 pe 2 (B-14)3

= (, + + 262) (B-15)

so

,,,n+l (,, ,+ , )i+j+2= j+ I (B-16)

+ P pj + (B-17)"S2)I P - + (B-18)

are used in place of the procedure applied in Appendix A.

REFERENCE

1. Day, S.M. and J.B. Minster, Numerical Simdation of Attenuated WavefieldsUsing a Pad, Approzimant Method, Geophys. J.R. Astr. Soc. 78, 105 (1984).

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