AD-A250 355lll'lfr 11 r I I I : I ill!

AN APFpROXDUlIC ANALYTICAL MNDUL OF SHOCK WAVES FROK01ME21=003ND NUCLA EXPLOSIONS

F. K. LambB. W. CallenJ. D. Sullivan

University of Illinois at Urbana-ChampaignDepartment of Physics1110 West Green Street DUrbana, Illinois 61801 D I

6LECTIMAY 2 9

December 1990

Scientific Report No. 1

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An Approximate Analvtlial Model of Shock Waves from Contract

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6 AUTHOR(S) PR 8A10F. K. Lamb TA DAB. 1U. Callen WI] A LJ. D. Sullivan_____________

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13. ABSTRACT (Max'rnurn 200 words)

We dis1cuISS an approximate analytical model for the hydrodynamic evolution of thehokfront produced 1w an explosion in a homogeneous medium. The model assumes a

particular relation between the energy of the explosion, the density of the medium

into whlicih thle shock wave is expanding, and the particle speed immediately' behind

th, !ilock front. TIhe assumed relation is exact at early times, when thle Shock wave

is,- strong and self-similar. Comparison with numerical simulations shows that the

relat ion remlains approximately valid even at later times, when the Shock wave is

neitiler strong nor s-elf-similar. Tilhe model allows one to investigate bow thle

teVi)lui 1ofl thtie shock wave is i nfluienced by the properties of the ambient medium.

11he ,;io(-k front radius. vs. t ime Curves predicted by the model agree well wi th

nnekrical ;imul~lat ion-s of t.xplo'.;ions- in quartz and wet tuff and with data from four

11111(0rgrollnd nuc10lear tfes Is conduct. d in granite, basalt, and wet tuf f t-:lon thle

ot I icia I vie I ds are, assu~med . Fits of the model to data from thle hvdrodvnuami c phase

of th -'(, t ... -t iJve viilds- that aire within 87 of the official yields.

14, SUBJECT TERMS 15 NUMBER OF PAGES

iilr'sh41old JI Vs B/lIT1 rt t , lirodvinam it methiods, '-hock waves, 74

1111(1 rl~I~rotmi 111111 .1r xi]sI 16, PRICE CODE

17. SECURITY CLASSIFICATION 18 SECURITY CLASSIFICATION 19 SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT

OF REPORT OF THIS PAGE OF ABSTRACT

l'lll~~ ~~ Uis if Id iic asftid 'c i e d SARS r~'l tardlard 1 0-r 298 (P~v 2 89)

Contents

1. Introduction I

2. Model 4Assumptions 4Predicted Radius vs. Time 6

3. Comparisons with Analytical Models and Numerical Simulations 10Expression for f 10Assessment of Particle-Speed Predictions 12Assessment of Radius vs. Time Predictions 13

4. Comparisons with Field Data 16Radius vs. Time Curves 17Yield Estimation 20

5. Summary and Conclusions 25

Appendix: Comparison with Heusinkveld's Model 27

Acknowledge me nts 30

References 32

Figure Captions and Tables 36

Acoession For

NTIS GRA&IDTIC TABUnImn.unced 5 ]

By

tR1 4LributlolfAvailabl]ty Codes

-f Ava die a d/or

1. INTRODUCTION

Shock wave methods have long been used to estimate the yields of nuclear explosions,both in the atmosphere (see, for example, Sedov [1946]; Taylor [1950b]) and underground(sc,, for exani)le, Johnson,, Higgins, and Violet [1959]; Nuckolls [1959]). All such methods

are based on the fact that the strength of the shock wave produced by an explosion

increases with the yield, all other things being equal. As a result, the peak pressure,peak density, and shock speed at a given radius all increase monotonically with the yield.

Hence, by comparing measurements of these quantities with the values predicted by a

mnodel of the evolution of the shock wave in the relevant ambient medium, the explosive

yield can be estimated. Shock wave methods for determining the yields of underground

nuclear explosions are of increasing interest as one means of monitoring limitations on

ulderground nuclear testing. These methods were first introduced as a treaty-monitoring

tool in the original Protocol of the Peaceful Nuclear Explosions Treaty of 1976 [U. S. Arms

Control and Disarmament Agency, 1990a]. Hydrodynamic methods were explored fiurther

in a joint U.S.-U.S.S.R. verification experiment [U. S. Department of State, 1988] and havenow been incorporated in new protocols to the Threshold Test Ban and Peaceful Nuclear

Explosions Treaties [U. S. Arms Control and Disarmament Agency, 1990b].Most shock wave algorithms for estimating the yields of underground nuclear

explosions have focused on the so-called hydrodynamic phase (see Lamb [1988]), because

the evolution of the shock wave during this phase is relatively simple. The energy released

by a nuclear explosion initiall.y emerges from the nuclear device as nuclear radiation, fission

fragments, and thermal electromagnetic radiation (see Glasst one and Dolan [1977], pp. 12-

25 and 61-63). At the very earliest times, energy is carried outward by the expandingweapon debris and radiation. As this debris and radiation interact with the surrounding

medium, a strong sho -k wave forms and begins to expand. The evolution of the explosion

during this phase can be followed using the equations of hydrodynamics and radiation

transport. However, within -- 10-100 its, depending on the yield and the composition

and distribution of matter surrounding the nuclear charge, the outward flow of energy viaradiation becomes unimportant and the explosion can be described using the equations

of hydrodynamics alone. At this point the explosion enters the (purely) hydrodynamic

p)has(. The radial stress produced by the shock wave at the beginning of this phase greatlyexceeds the critical stress at which the surrounding rock becomes plastic, so that to a

good approximation the shocked medimn can be treated as a fluid. As the shock wave

ex)alds, it weakens. Eventually. the strength of the rock can no longer be neglected, the

flid approximation fails, and the livdrodynainic )hae ends. Yield estimation methods

that ise nmeasuromeits made dhring the hydrodynamic phase are called hydrodynamic

rru th od.,.

All hydrodynamic methods require a model of the evolution of the shock wave. Models

in recent or current use range in sophistication from an empirical power-law formula that

supposes the evolution is completely independent of the medium (Bass and Larsen [1977];

see also Heusinkveld [1982]; Lamb [1988]) to multi-dimensional numerical simulations

based on detailed equations of state (for recent examples of one-dimensional simulations,

see Moss [1988]; King et al. [1989]; lMoran and Goldwire [1990]). When detailed equation of

state data are available, state-of-the-art numerical simulations are expected to be highly

accurate, at least for spherically-symmetric, tamped explosions in homogeneous media.

Nevertheless, a simple analytical model of the shock wave produced by such explosions

that allows one to determine how the evolution depends on the Hugoniot and the yield is

useful for several reasons. First, detailed equations of state are available only for a few

geologic media. Second, large codes can be run for only a limited number of cases. Third

and most importantly, an analytical model is more convenient than numerical simulations

for analyzing how the evolution is affected by the properties of the ambient medium.

This is the first of several papers in which we investigate the evolution of the shock

wave produced by a spherically-symmetric explosion in a homogeneous medium during the

hydrodynamic phase. Such a shock wave is necessarily spherically symmetric. Here we

investigate a simple analytical model. In this model, the compression of the medium at

the shock front is treated exactly, using the Rankine-Hugoniot jump conditions and the

Hugoniot of the ambient medium. The rarefaction of the shocked fluid that occurs as the

shock front advances is treated approximately, via an ansatz relating the specific kinetic

energy of the fluid just behind the shock front to the mean specific energy within the

shocked volume. This model was proposed by Lamb [1987], who showed that it is exact

for strong. self-similar shock waves. Lamb [1987] also made a preliminary comparison of

the shock front radius vs. time curves predicted by the model with data from several

underground nuclear explosions and numerical simulations. The model was proposed

independently by Moss [1988], who compared its pre(lictions with particle speed data

from underground nuclear explosions and nume'ric l simulations. Their results showedthat the model provides a useful approximate description of the shock wave evolution

throughout the hydrodynamic phase. The model is similar in spirit to one proposed earlier

by Heusinkveld [1979, 1982], but is more satisfactory theoretically and appears to provide

a more accurate description of underground nuclear explosions, as shown in an appendix.

In § 2 we first discuss the assumptions on which the model is based, including the

an.,atz relating the specific kinetic energy of the fluid just behind the shock front to the

mean specific energy ; ithin the shocked volume. Next, we combine the ansatz with the

Hugoniot of the ambient medium expressed as a relation between the shock speed D and

the post-shock particle speed u to obtain a first-order ordinary differential equation that

describes the motion of the shock front. We show that, solutions of this equation of motion

2

can be expressed in terms of simple analytical functions when the D vs. ul relation is

piecewise-linear. Since an arbitrary D vs. u, relation can be represented to any desired

accuracy by an appropriate piecewise-linear relation, the radius vs. time predictions of the

model for an arbitrary Htugoniot can always be expressed as a sum of simple analytical

functions. Alternatively, the equation of motion can be integrated numerically to find the

model predictions for any prescribeld tiigoniot. In practice, the latter approach is often

more convenient. The model also givs the shock speed, post-shock density, post-shock

particle speed, and post-shock pr'ssure as functions of the shock front radius or the elapsed

time, the yield of the explosion, and the Hugoniot of the ambient medium.

In § 3 we assess the accuracy of the model. We first show that the an.,atz is exact

for a shock wave that is strong and self-similar. We then compare this ansatz with results

from numerical simulations, and find that it is also remarkably accurate for spherica! shock

waves that are neither strong nor self-similar. Finally, we compare the radius vs. time and

particle velocity vs. radius curves predicted by the model with the corresponding curves

obtained from nmnerical simulations of underground nuclear explosions. We conclude that

the model with point-source boundary conditions provides a remarkably good description

of the spherically-symmetric shock waves produced by such explosions.

In § 4 we show that the radius vs. time curves given by the analytical model of § 2

provide an excellent description of the field data from four underground nuclear tests

conducted by the United States, despite the fact that these tests are not point explosions

and that the ambient media may be nonuniform. In fact, the model sometimes describes

the data accurately even well beyond the hydrodynamic phase of the explosion. When the

model and the Hugoniots of § 3 and § 4 are used to estimate yields using data from the

hydrodynamic phase of these four nuclear explosions, the resulting estimates are within

8% of the official yields. For comparison, when the numerical simulations described in

j 3 are fitted to the same data, the resulting yield estimates are within 9% of the official

yields. Our lack of knowledge of the geometry of these tests, of the way in which the data

was gathered, and, in the case of one explosion, of the medium in which the explosion

occuirred, make it difficult to assess whether the relatively small differences between the

various yield estimates are due to errors in the radius vs. time data, departures from

splerical synnet ry dime to asphericity of the source and/or inhomogeneity of the ambient

znediiun. uncertixIties in the yield standard, or inadequacies of the models. The U. S.

Do.partment of State [1936ab] has claimed that hydrodynamic methods are accurate to

Wxithill 15% (at lde 9,5'4 c',l filiv level) of radiochernical yield estimates for tests with

vield,' greater than 50 kt in tlhe geolgi c ii, lia found at the Nevada Test, Site (see also

1'. S COm i,-.';S, OftiiC of TCchmio)l,,y Assm,.nt 119881; Lamb [19881). Thus, the analytical

mw iIlel ,,f § 2 appears to 1,e comp;etitive with other models for purposes of yield estimat ion.

A pclimtinary accoumt )f ti, work lms been iven by Callen et a]. [1990b).

3

2. MODEL

In this section, we first present the fundamental assumptions of the model and derivethe resulting equation of motion for the shock front. We then solve this equation of motion

and discuss the scalings allowed by the shock-front radius vs. time curve predicted by the

model.

Assump tions

The model assumes that the shock wave is purely hydrodynamic, i.e., that transport

of energy via radiation is negligible and that the stress produced by the shock wave is muchlarger than the critical stress at which the medium becomes plastic. The model assumesfurther that the medium ill which tile shock wave is propagating is homogeneous, and thattile shock wave is spherically symmetric at the time the model first applies. The shockwave therefore remains spherically symmetric. As the shock wave expands and weakens,

the strength of the ambient medium eventually becomes important. At this point the

model is no longer applicable.

Part of the energy released in any nuclear explosion escapes without contributing to

tile energy of the shock wave (see Glasstone and Dolan [1977], pp. 12-13). Thus, the yieldmeasured by hydrodynamic methods is less than the total energy released in the explosion.

Here we are concerned exclusively with the hydrodynamic phase of the explosion, and hence

the yield It to which we refer is the so-called hydrodynamic yield, namely, the energy thatcontribiutes to the formation and evolution of the shock wave. The model assumes thatI' is constant in time. This is expected to be an excellent approximation during the

hydrodynamic phase.

The Rankine-Hugoniot jump conditions express conservation of mass, momentum, and

energy across the shock front (see, for example, Zel'dovich and Raizer [1967, Chapter I]).The model is based on approximate forms of the jump conditions, which are nevertheless

extiiiiely accurate itnder the cJIiiltions of inteivst. The model neglects the pressure P0 of

the unshocked ambient medium in comparison with the pressure pl of the fluid just behind

the shock front. Since p, is > I GPa for the times and shock radii of interest, whereas p0 is

-20 Pa, neglecting p0 is an excellent approximation. The model also neglects the specific

internal energy E0 of the unshocked medium in comparison with the specific internal energyil of the fllid just behind the shock front. This approximation is also highly accurate,

since c is greater than ,0 for post-shock particle speeds u greater than about 1'50 m/s,

and ul is >1 km/s for the times and shock front radii of interest.

With these approximatioiis, the Rankine-Hugoniot equations, written in the frame in

which the imshocked material is at rest, become

p1(D - u) poD, (1)

4

poDu = P, (2)

and

PO P1 2 1 = 1 E (3)

where D = dRIdi is the speed of the shock front, and P0 and P, are the densities just

ahead of and just behind the front. Equation (3) shows that the energy pi(1/po - l/pi)

acquired by a unit mass of the nledimn as a result of shock compression is divided equally

between kinetic energy of bulk motion and the increase in the specific internal energy. The

shock speed D is related to the post-shock particle speed u by the Hugoniot

D - D(ul), (4)

which depends on the inedium.Without loss of generality, the specific kinetic energy of the fluid just behind the

shock front can be related to the mean specific energy within the shocked volume via the

expression2t f 43Wa ° , 5

where f is a dimensionless factor that generally depends on the equation of state of the

ambient medium and the radius of the shock front. A key assumption of the model is that

f is independent of the shock front radius R for all hock fr9,!t radii of interest. We assessthe validity of this ansatz in the next section, where we show that it is exact when the

shock wave is strong and is approximately valid throughout the hydrodynamic phase of

the explosion.The model treats the compression of the ambient medium at the shock front exactly,

since the jump conditions and the Hugoniot are correctly incorporated. On the other hand,

the rarefaction that occurs as a shocked fluid element is left behind by the advancing shock

front, is treated only indirectly, and approximately, via the parameter f. The value ofthis parameter depends on the density, velocity, and specific internal energy distributions

within the shocked volume, distributions that would be determined in a full hydrodynamic

calculation of the strmctime and evolution of the shock wave. In order to carry out such

a calculation, knowledge of the equation of state off the Hugoniot (i.e., along the releaseadiabat) is required. This requirement is sidestepped in the model by assuming that f is

independent of R. The parameter f is then the only free paranieter in the model.The be,,st value of f to uqe for explosiOns ill a given rock can be determined by fitting

the po st-shock particle-spee-d relation (5) (or the relations for the shock speed, shock front

raldius, and post -shock prcssure that follow from it) to data from numerical simulations or1a ta froim actiial ulnderuiiild exi,!(soizs in that rock. Once f is determined, the model

provides a description of the properties and evolution of the shock wave produced by an

explosion of any yield in the same medium.

Predicted Radius vs. Time

With the assumption that f is independent of R, the right side of equation (4) becomes

a known function of R and hence equation (4) becomes a first-order ordinary differential

equation for R. This equation can be integrated directly to determine the radius of the

shock front as a function of time. Solutions of the shock front equation of motion can

be expressed in terms of simple analytical functions when tile shock speed is a linear or

piecewise-linear function of the post-shock particle speed, as we now show.

Linear Hugoniois.-Experimental studies of shock waves in solids (see, for example,

ZeJ'dovich and Raizer [19671, Chapter XI) have shown that for many materials, the

relation between the speed D of a shock front and the particle speed ul just behind it

is approximately linear for large ul, that is

D(ui) ; A + Bul , (6)

for some constants A and B. In general, the D(ul) relation deviates from this high-speed

relation as the post-shock particle speed falls. If we assume for the moment that D(ul)

can be adequately represented by a single linear relation of the forn (6) over the full range

of u i that is of interest, we can obtain an interesting and useful analytical solution for the

motion of tile shock front.

First, for convenience we introduce the dimensionless variables

x = R/L and 7-t/T, (7)

where

3fIV B2 and T= . (8)L 4irpoA 2 A

The characteristic length L and the characteristic time T depend on the medium through

the constants Po, A, B, and f, and scale as the cbile root of the yield IV. Making use of

relation (5) and the characteristic length L, the equation (6) becomes

D= ,4 1 +. (9)(it R

This equation shows that the length L is the radius that separates the strong shock regime,

where D -( R -- /2, from the low-pressure plastic wave regime, where D ; const. In non-

dimensional form, equation (9) is

+ (10)

6

The general solution of equation (10) is

r- 7o = h(x) - h(xo), (11)

where 7o = tolT and x) = Ro/L. Here R 0 is the radius of the shock front at to, the time

at which the evolution of the shock wave is first described by the model. The function

h( x) in equation (11) is given by

x (+ 2V'+lI) _2 [7r+tl- (2V x-l)]h(x) =x+-ln(I±-j-+ 1 ) -1 n +tai -' (2V/ ) (12)

For a point explosion, x0 = 0 at T0 = 0. For such explosions, the function x(xo, 7 0 , 7)

defined implicitly by equation (11) becomes, at small radii (x < 1),

x(T) (5/2)2/572/5, (13)

which is the well-known temporal behavior of a strong, self-similar shock wave produced

by a point explosion [Sedov, 1959]. At large radii (x > 1), this function becomes

x(T) const. + 7, (14)

which describes a constant-speed plastic wave (this is sometimes referred to a.s a bulk

wave). Equation (11) thus provides an interpolation between the strong shock wave and

the low-pressure plastic wave regimes.

Within the assumptions of the model, an explosion is completely defined by its yield

IV and the ambient mediun, which in turn is completely defined by the quantities Pa,

A, B, and f. The shock front radius vs. time curve for an explosion of any yield in any

medium can be generated from the function x(xo, 70 , r) by using the relation

Ro to t(15)R(t) = L x ( R ° T ) .(5

For a point explosion, this simplifies to

R(t) = L x(t/T). (16)

The radius vs. tmine c,:rve (15) satisfies a scaling involving the yield WV and the

properties A, B, anl p. of the aml)ient medium. In particular, relation (15) implies

that if the radius vs. tmine curve for (,xplosion i is known, then the radius vs. time curve

f,)r a secolnd 'xpl)osi,,n j can be gcnerated, provide(I that p0, A, B, f, and 1V are known

"()I- )t h explosionis a11d the initial ratiii and times -0 ,. to,, Roj, and To) satisfy

RN, = IL L,,)Ro, and to) = (T,/T,)Too. (17)

Under these conditions, the radius vs. time curve RI(t) for ezplosion j is given in terms

of the curve R.(t) for explosion i by the similarity transformation

R,(t) = (Li/L,) R,(Tjt/T,). (18)

The required scaling (17) is satisfied trivially if both explosions are point explosions. Thesimilarity transformation (18) can be used to shed light on the physical origin of the so-

called "insensitive interval" tnd to develop optimal weighting schemes for radius vs. timedata (Lamb, et a]. [1991]; for preliminary accounts, see Lamb et a]. [19891 or Callen et

al. [1990a]).

A special case of equation (18) that we use in the next sections is the case of explosionsin identical ambient media. According to equation (18), the radius vs. time curves of two

such explosions satisfy

Rj(t) (W 3/w,) (19)

provided that

noj = (WI/W,) / 3 Ro, and toj = (T,4 j/IW)l/3 To,. (20)

In other words, the radius vs. time curves scale with the cube-root of the yield if the initialradii and times scale with the cube-root of the yield. This result illustrates the moregeneral point that cube-root scaling does not follow from the hydrodynamic equations

and the jump conditions alone; in addition, the relevant properties of the hydrodynamic

source must scale [Lanb et al., 19911)]. The required scaling of the source is again satisfiedtrivially if both explosions are point explosions. This is consistent with the known validity

of cube-root scaling during the hydrodynamic phase for point explosions in uniform media

(King et al. [1989]; Lamb et al. [19911)]).So far, we have discussed the predictions of the model for the post-shock particle speed

u as a function of R (eq. [5]), shock speed D as a function of R (eq. [9]), and shock front,

radius R as a function of tine (eq. [15). The model also predicts the evolution of other

qiantities of interest, including the IMass density, spe)ccific internal elirgy, aild pressure

i1nI11edliatelv behind the shock front. Expressions for these quantities can be obtaineld

from the junip conditions (1), (2). and (3) by subst i tit ing expressions (5) and (9) for ul

and D.

The predift,,d post-shock mass density is

X 3' / 2 4-1 "P1 = (z1112 4- 1 - B-1 ) Po (21)

8

where x = RIL is the dimensionless shock front radius. For x < 1, p1 ; [B/(B - 1)]po,

which is the limiting value for a strong shock wave. For large radii, Pi approaches Po, as

it must. The predicted post-shock specific internal energy is

A' 1(22)

while the predicted post-shock pressure p, isp0.4 ( 1 + 1 (3

Bu x31 xii'3- B + 7 (23)

For small radii (x < 1), pi po(A 2/B)x - , whereas for large radii, Pi (poA/B)x - a 2 .

Arbitrary Hugoniots.- Although for many materials the Hugoniot at high particle

speeds (or equivalently, at high pressures) is well-described by a single linear relation of

the form (6), the Hugoniot at lower particle speeds usually deviates from the high-speed

relation. If the linear relation that is valid at high particle speeds could be extrapolated

to small un, the constant A would correspond to the low-pressure plastic wave speed co.

However, such an extrapolation usually is not valid. In granite, for example, A is about

3 kin/s,. whereas Co is about 4 km/s.

Even if the H-ugoniot is not linear over the range of ul of irterest, it can still be

represented to any desired ,ccuracy by a sequence of piecewise-linear segments. In this

cast,, equation (10) still describes the motion of the shock front within each segment of

the H1ugoniot, but at each break in D( t ) new Hugoi.iot parameters A and B must be

introdluced. While it is possible to write the radius vs. time curve for a piecewise-linear

Hiigoniot with an arbit rary number of segments as a sum of standard functions, in practice

it is more convenient to treat this case by integrating the shock front equation of motion (9)

numerically.

In integrating equation (9), we handled the transitions between different linear

segments of the Hugoniot as follows. The transitions occur at a sequence of fixed points

in u I, which, for a given yield, are related to a sequence of radii by equation (5). After

each time step, we computed the new value of the particle speed from equation (5) and

compared it with the particle speed ua at the junction of the (i - I)st segment of the

Hugoniot and the ith segment. Wlien the newly computed value of ul dropped below uin the next integration step we replaced the constants .4,1 and Bi-, that described the

previouis segment of th,, Hugmniot with the constants A, and Bi that described the current

segmnent. The transition points etween the diffe-rent linear segments of the Hugoniot are

not readily apparent in the resulting radius vs. time curve, because steps occur only in the

.CcoTId derivative of tHie shock front radius with respect to time; both R( t) and its first

-hrivative are cort ilt 101 is.

The radius vs. timt' ciiirve predicted by the model for an arbitrary Hugoniot satisfies the

,'ute-rott scaling n 'hat ian (19. providd' t hat the initial conditions satisfy equation (20).

3. COMPARISONS WITH ANALYTICAL MODELS

AND NUMERICAL SIMULATIONSIn this section we assess the accuracy of the model. We first derive a general expression

for the dimensionless factor f, and show that the constancy of f is exact for a point

explosion in a homogeneous medium when the shock wave is strong. 1 We then explore the

validity of relation (5) with f constant when the shock is no longer strong, by comparing

predictions of the model with numerical simulations of underground nuclear explosions in

quartz and wet tuff.

Expression for f

In order to evaluate the accuracy of the an.satz that f is constant, we make use of

the assumption that the hydrodynamic energy of the matter interior to the shock front isconserved, that is

R(t)I- 4r p(r,t) [1 u2 (r,t) + e(r,t)] r 2 dr = const. (24)

To turn equation (24) into a relationship between ul and W, we first introduce the time-

dependent dimensionless radius = r/R(t). Then, the distributions p(r,t), u(r,t), and

E(r,t) inside the shocked volume may be rewritten, without loss of generality, as

p(r,t) =g( ,t)pl(t), u(r,t)= w(.,t)u(t), and e(r,t) = e( ,t)Ei(t), (25)

where pi(t), ul(t), and EI(t) are the mass density, particle speed, and specific internal

energy just behind the shock front (where = 1). It will he convenient to express thepost-shock mass density pi in terms of the pre-shock density P0 via the dimensionless

factor

K(t) = Pi/P0. (26)

1 A .!,ong shock wave is one in which the speed of the shock front is much larger

than the speed of sound in the iuidisturbed rock, the pressure behind the bhock front is

predominantly thermal, and the ratio of the density immediately behind the shock front tGthe density ahead of the front is close to its limiting value. Such shock waves have special

properties. In particular, the shock wave produced by a point explosion is self-similar while

it remains strong (see Zel dovich and Raizer [1967], Chapters I and XII). Tile condition

that a shock wave be strong is not the same as the condition that the shock produce a

radial stress greater than the critical stress at which the rock becomes plastic. The latter is

the hydrodynamic condition, which is usually satisfied for some time after the shock wave

is no longer strong (see § 4).

10

Using equations (25) and (26), equation (24) can be rewritten as

-,,(t)i4 I VR p K (t),t) [_Iui 2 (t) IV( ',t) + E I(t) e( , t)] ' d47R O 1 (27)

[( 2(t) K(t) J (t) [w2( ,t) + C( ,t)] 2 d(

where in the last line we have used equation (3). Comparison of equation (27) with the

'nsatz (5) gives a useful expression for the dimensionless factor f, namely,

I =3 (t) 9( ,t) [W 2( ,t) + C( ,t)] 2dA . (28)f (t) -- f

Equation (28) is merely a re-expression of equation (24) and therefore is completely general.It shows that f(t) depends on the density, velocity, and specific internal energy distributions

within the shocked volume at time t. We now investigate the value of f(t) and its variation

with time.

Strong shock interval.--Consider for simplicity a point explosion during the interval

when the shock wave is strong. As noted above, during this interval the ratio of the

density Pi behind the shock front to the density P0 ahead of the shock front approaches a

limiting value (see Zel'dovich and Raizer [1967], p. 708). Thus, K is independent of timeand independent of TV in this interval. Moreover, during the strong shock interval the

shock wave produced by a point explosion is self-similar. Therefore, the profiles g, w, and

c are also independent of time and independent of W. Thus, f is independent of time andindependent of W in the strong shock interval.

For a medium that is adequately described by a Mie-Griineisen equation of state with a

constant Griineisen coefficient, the value of f in the strong shock interval can be calculatedby comparison with the solution for a self-similar shock wave produced by a strong point

explosion [Sedov, 1946, 1959; Taylor, 1950a] as follows.

The Mie-Griineisen equation of state assumes that the total pressure p is the sum of

two parts: a thermal pressure PT, which depends on the temperature and density, and a

cold pressure p, which (epends only on the density, that is,

, =pTp, T) + 1P,(P) = pFr- + p(p), (29)

where CW is the thermnal conponent of the internal energy and P is the Griineisen coefficient

(see. for example, Zel'dovich and Raizer [19671, p. 697). The thermal pressure PT increaseswith the strength of the shock, whereas the cold pressure p, is bounded, since p approaches

a limiting value. Thus, in the strong shock interval the cold pressure term in equation (29)

can Ie neglected (see Z,,'dovich and Raizer [1967], pp. 708 709). If in addition the

11

Griineisen coefficient is constant, this equation of state has the form considered by Sedov

and Taylor in their solution.

The dependence of f on r in the strong shock interval can be calculated from

equation (28) using Sedov's solution for the functions K, g(e), w( ), and e( ) (see, forexample, Landau and Lifshitz [1987], pp. 403-406, for explicit expressions for K, g, u', and

e). The result is shown in Figure 1. When the shock wave is no longer strong, or when

it never was strong, a value of f different from that given by Figure 1 may give a more

accurate description of the shock wave evolution.

Actual nuclear tests are not point explosions but are generated by aspherical sources

of finite size. In part to give the shock wave time to become more spherically symmetric,

radius vs. time measurements are usually made at scaled radii -- 2-5 m/kti/ 3 for tests with

yields ---150 kt (at larger radii, the hydrodynamic approximation is no longer valid). At

these radii, the strong-shock expression for f shown in Figure 1 is no longer accurate. As

we now show, f - 0.53 appears to give a relatively accurate description of the evolution

of ohock waves in granite and wet tuff during the interval in radius where measurements

are usually made.

Ass eisment of Particle-Speed Predictions

The behavior of f when the shock wave is not strong can be investigated by comparing

the predictions of the ansatz (5) with shock wave data from actual and simulated nuclear

explosions.Lamb [19871 showed that the radius vs. time curves predicted by equations (4) and (5)

agree fairly well with radius vs. time data from a numerical simulation of a nuclear explosion

in wet tuff by the P-15 (CORRTEX) Group at Los Alamos National Laboratory and with

field data from the Piledriver and Cannikin nuclear tests, which were conducted in granite

and basalt, respectively. A more detailed comparison of the radius vs. time curves predicted

by the model with data from numerical simunn1tions is presented at the end of this section.

The predictions of the model are compared with field data from underground nuclear tests

in § 4.

A more direct test of the ansatz (5) can be made by comparing the post-shock particle

sp(ed it predicts with post-shock particle speed data from nuclear tests and numericalsintlations. Perret and Ba.ss [1975] have summnarize(d a large collection of particle speed

data obtained from un(tergroul(! nuclear explosions. A\lMo.ss [19881 has shown that these

data agree fairly well with the scaling u ox R - 31

1 predicted by relation (5), for particle

speeds > 1 km/s. These data appear roughly consistent with this scaling even for particle

speeds as low as -10'- km/s. Moss [1988] also com)ared the an.4atz (5) with post-shock

particle speeds from his numerical simulations of 125 kt nuclear ex)losions in quartz and

wet tuff. He found that for particle speeds between 1 and 30 km/s, both the radius and the

12

(I('Isity dlependenice of his grm iite anid vwi t iff (dita are accurately dlescribed1 by relation (5)wIt I f =70,53.

To assess the ail7atz (5~) furthecr, we comipare it withI post -shock p~article speed data

obtained from simiulatijons of 100 k t nutclear explosions in quartz and wet t uff. These

simulations were perfo rmed by IeI( Los Alamos C'OIRRTEX group using the radiation

hvdrocode do-Mcribed by Cox vt al. [19661. 1In order to compare equation (5) withI the

'Siilat tionls, we have 1i;1(1 to 1 ci )iist ruclt thle po st -shock particle speeds using appropriate

11 ig ilot s and the radlil is vs. t irne ci irves obt ailied fromn the simulations. The radii is

vs. t ine curves were kindly iovd toi us by D. Ei hers (private comnmunicat ion, 19,S7).

The reconstruct ion iPtOCeSS can (list ort thle particle sp)eed curve if thlt Huigoniot used

inI thle reconstruct ion (differs from thle ingi niot uised1 iII the simnulat ion. Throughou t this

pper, when modlelinig shock waves iII quartz we uise the Hugoniot data compiled by Kingof ;11. [1 9S9] fromx several souirces [Cluig anjd Simmnons, 1969; Al 'tshtulor et al., 19717:

XV-tckorle, 1962; Mc Qtucc et al., 1977; lagan, 1984]. These data are shown inI Figure 2.

Ail expanidedl view of t hc~ low-ul sectin iof the (lit ais shown inI Figure 3. W~hen comparing

with the quiartz siinulatti ins of the Los Alamos CORRTEX groilp, we use a piecewise-linear

represent ation)1 of thle dit a comnxpiled by1 King et a)., using their interp~olation at low post-

shock particle speeds ( Ind~icatedl by thle (lash-(otted line in Figure 3). In modeling shockwvaves InI wet t iff, we uise thle piecewise- inea~r Hlugoniot given byKn a.[99,wIc

is shownvi in Figure 4. The light solid curves in Figuries 3 and 4 show where the post-shock

Jprsuri' calculatedl froni thle juni) cond~it io n (2) is 15 C Pa. For the reasons discussed in4, wve adopt t huspns; as mnarking thle end~ of the hydrodynamnic phase. W\e believe

these 11ungoiniots are very close to thle Huigoijiots uised in the niumerical simulations, b~ut we

cannot rulle out the possibility of some dist i rtinn.

Flire 5 shows that relation (5) with f =0.53 providles an excellent description of the

post -shock rpart ice speed data from thle simulated explosion in quartz, for particle speeds

from -30 down to -0.G ki/s. Figuire 6 shows that relation (5) with f =0.53 also provides

an excellent dlescript ion of thle pos4t -shock particle speed dat a from the simulated exIplosiOfl

inI ect tutff. for particle, spec us froi n - 40 (Imi wi to -1 kim /s.

011 tine bai)sP o)f these, connlijIns.011 we( Conclude that relation (5) withi f =0.53I' r Idj( ls a Ko iideci i of l~wie claitii a betivw' thle yv ld thle miass (density of thle

amet ii i aduditui.i thle raditins o f thle shocak ftro nt, anid t he pi st -shock particle speed (hiring

tlte hivi odvulainic piluuse of tilie ex ilos'().Ii n. tcl idinig t inies wvien thec shock wave is rio longer

strolig.

A i A. ?lcltof Rad u.i ius. Timf' Pridu honru

III o)r(ler to asesfiurt her t li ace iii;nc of thle imo del , we compare the radliuis vs. time

ci ir-ves that it pre hict, witlin thle corrrpmiinnitg ci rves p~redictedl by inumerical simuil at ions

13

of underground nuclear explosions in quartz and wet tuff. We set f equal to 0.53 and use

point-source boundary conditions when solving equation (10) here and throughout this

paper.

Quartz.---As described above, thc Los Alamos CORRTEX Group (D. Eliers et al.) has

simulated a 100 kt nuclear explosion in quartz. We compIared the present model with this

simulation, using both a linear description of the quartz Hugoniot and the More complete

piecewise-li near description dliscussed above. These Hugoxiiots are indicated respectively

by the dashed and dash-dotted lines in Figures 2 and 3. Thle mass density used in the

mnodel was the same as that used in the simulation, nanely 2650 kg/i 3 .

Figure 7 comp~ares the radii predicted by the model withi the radii predlictedl by the

slimulation. Tile top panel shows these radii as functions of tinie, wherea-s the bottom

panel displays the relative difference

(Rdta(QO rde~) (30)

between these radl. The dashed curve is the value of that results, from using the linear

description of the Hiigoniot in the analytical mnodel, whereas the (lashi-(otted curve is the

result given by using the piecewise-hlear Hugoiot. When the linear approximation to the

Hugoniot is used, the absolute value of 6 is less than 3% before 0.7 ins but rises to '-12%) by

~-5 mns. As expected from the behavior of the actual Hugoniot. the radii predicted by the

linear approximation are systeniati1cally too large at late tunies. When the more accurate

piecewi se- linear Hugoniot is uised, b is niever mnore than 1.8%.

Wet tuff. - --The Los Alanios ('Ofl TEX Group ( D. Eilers et, al. ) has also simnulated a

100 kt nuclear explosion in sat urat ed wet tufif. N~e compared thle luesellt model with this

simulation, again using both a linear (lescriptioti If the Wet tuff Huigoiot and the more

compllete Jpece wi se- linear dlescrip~tion of King et al. [1989]. These Hugoiot~s are indlicatedl

respcl~evely by the dashed and solid lines in Figure 4. The imiass denity used in the model

was tie same a-s t hat used in thle slimulmt on. namxmel% 1950 kg/in.

Fiure 8 compares. the radii predlictedl by Iwf axnalyVtical model withI thle radii p~redlicted

by tOe simulation. When the linear approximiation to the L11igoiiliot is ilsedl. the absoluite

value of is always less than 9(/(. Againl, aLS expected froin the 1behiavior of the actunal

Hugoriot, the rad1i predicted by th a hnr apri )r)XI mma u are svstemnat icallv too sumallat late tinies. Wheni the nmore acciuiate plci(-A-('vi,(-inc(ar Hlglgui t is used, the relative(

difference is never more than 6i, ailf.I ss5 than 2(/( afte'r 0.6 ins.

Dt.4riiu.ion.. -These compajrison ()5 f the raditis vs. t ilme ciurves p)red(ict ed by the mlodlel

With1 tlhe ra hilis vs t rue CurIVes pri-cto y 1 1 mhini'rcal sinnulat 11)11 couifirmi the earlier

asse'Su nenlt' .Wh ich Nws h1 -.' 1 on l ii pa riSI( IIII f 1wak particle veh I(i ties. that th li uu el

withI f set equal to 0.53 pi oviles ahn excellent descripi l of sleial sitrcshock

14

waves from underground nuclear explosions in granite and wet tuff, during much of the

hvdrodynantic phase. Therefore, we shall adopt this value when comparing the model with

field d(ta from underground nuclear explosions.

15

4. COMPARISONS WITH FIELD DATA

In this section we use radius vs. time data froin four underground nuclear tests

conducted by the United States to assess the usefulness of the analytical model. The

four data sets we consider are from thle nuclear tests code-nam-ed Piledriver, Cannikin, and

Chlberta, and from a test that we call NTS-X, since its official namne remains classified.

The radius vs. time data from the first three tests were obtained using the SLIFER

technique [Heiusinkveld al Holzer, 1964]. These (data were kindly provided to us by

Ml. Hteusirikveld [1986; 1987, p)rivate coniiiiiicatioii]. The radius vs. time data from the

test we call NTS-X were taken froin Hetusinkveld [1979];, the measureenit technique used

to obtain these (data was not reported. To our knowledge, no radius vs. time measurements.

mnade using the miore recently dlevelopedl CORRTEX technique [Vircliow et aL., 1980] are

publicly available.

Any at tempt to compare Inio(els or simulations of spherically-symmietric explosions Ii

uniformn Media with dat a fromn und(ergroulnd nuclear tests mnust confront at thle outset thle

fact that the shock wave prod~uced( by such a test evolves fr-omi an aspherical source of finite

size lmit( a ineiumii that is at least somiewhiat inogeneous (see Lamb [1988] and Lamb et

a]. [1991c]). In comparing the predlictions of the mlodel of § 2 with (data from nuclear tests,

we adopt thle particular solution that corresponds to a point exp~losion. For this solution.

culbe-root scaling is exact. WVe also assume cube-root scaling is valid when comparing the

results of the nunnerical slinumatioiis with d1ata from nuclear tests. Since these simulations

follow thle shock wave produced by anl initial source of hite size, cube-root scaling is at

best oitly approximately valid for thlese slinulat jolis.

Iii im"iig cube-root scaling, we are tacitly assuminig that thle fiite size of the source,

the aspliericit y of the exiilosii, and~ aN\' lihnhio geiilt ies iii the amxient miediumr have

ai ne1 i'it , fFve . thof i in the sliilatiomis and Mi the actual test, by the time the shock

front has expaii(ledl to thle radii at which thec romiiparison is, madle. Although shock waves

produnced b 1 undergrounld explosmoiis Ili uniforn inedia (10 tend to become more spherical

with im e, thle proplerties of the ,oiirce cani somiet ilnes have a sigmiifi 'ant effect during the

hydrdn amic lase [Moraii aind Guhlwirc. 19S9; Land) ct ;11.. 19911)]. Unfortunately, we

are mnable to assess (direct ly t he v-alldity of our assumiptionis, bieca use we lack detailed

knowledge of the sourices uised Ii thle wiuinerical simuulat ions, the conditions under which

the r1(1(1 'ir tests were cond uct ed . and1 t he wayN iM which thli field dlat a was collected1.

Wec also lack (let ailed kinowledge 4f how t he official yields were dlet ermnined for these

four evets iii ng the official v jelds to assess hv"d rodl I i) rnic et hods, we are Implicitly

assi 11111 h1g that they are'f a"Cliltate anld iii1leperidlelt of hiydrodviuiaic methods. However.

the )V( icedui re, by which offic i ;l vie l Is art, det eriiiiiie I is ktioii m to be complex. anld is not

pu1 i lv available. It Is p X'ssil d i on1 ' 1 c CaVseS t hat thle official yields, Imiax actually be less

accuirte than the hivdrmlvr uain ic N ich I '% iinae.NI e ver, thle official yield det erniinat iou

procediure usually makes use of information derived from hydrodynamnic methods, as well

as radiochemnical and other methods. If so, the official yield obviously is not independent

of the hydrodynamnic yield. Fuirthermore, in some cases the material properties used to

ob~talin hiydrodlyninic vieldl estimates may have been adljusted to give better agreement

with estimates ob~tainedl using other meth1od(s. Thle coin parisons in this section show that

(le51 )ite the complexity of llndergrouII~ nuclear explosions, b)oth the analytical model and

lie nmnerical sinodlat 11)11 accurately dlescribe) the shock waves produced by the riuclear

tests coilsidere(l here, when the off ic .al vieldls are used.

A solution of the analyt ical miodel is determiined by specifying the Hugoniot, the

value of the paramieter f, and time yield. The Hugoniot canl in principle be determined

fro lIi ab oratoryV measu remients miade onl sampijles taken from the emplacement and sat ellite

holes. Unfortunmately. if such inasurernts were mnade for the four events analyzed here.

they are not publicly available. Therefore, we used generic Himgoniot data characteristic of

he Ambient inedimixi of each explosion. For the reasons dliscussedl in the preceding section,

we ulsed f =0.33 thlroughiout the present analysis.

We first asqsess the accuracy of the analytical model in predicting the radius of the

shock front by comp~arin~g the radius vs. tine curves it gives with radius vs. time data from

lhe four nuclear tests cit( above. We then investigate thle usefulness of the model for

yield est imation by fit ting, it to) ri-hs vs. t ine dlat a fromx these tests, treatinig the yield as

Radit4 vs. Time Citriei

InI comparing the radius vs. tuime p~redictions with field data, we generally used

eitber the subset of the available dlata that fell within the hydrodynamic interval defined

])elow, or, where statedl, certain larger (data sets. However, for NTS-X, we followed the

recommendation of Heusinkifeld [1979] and( omitted the first nine data points from our

analysis. For Chiberta, the first seven points were inconsistent with each other and with

the remaining points, and hence these seven points were also omfittedl from our analysis.

Wenow (iscus5 the analysis of each event in turn.

Pledritucr. The Pilcedriiver event was an exp~losion condulcted in granite at thme Nevada

Test Site onl 2 .luie 1966 and had an announmced yield of 62 kt [U. S. Department of

Erergy, 19S71. 111 tiiodllig tis explosion, we considered the simple linear and piecewise-

linecar appr uxilimat ion)" to Ohe quartz Hiigoiiiot showni resp~ect ively by the dashed and solid

limie, in Figu res 2 and 3. We assui cd t hat thle granite surrounding the nuclear dlevice had

aI density equmiil to the s'tanldard den-sity of quartz. nainiel 2650 kg/in3 , and that the yield

of thle exp~losionm was 62 0t. Wec thlen Hit egratedl thle (different ial equmat ion (10) as describcd

ii 2.

17

Figure 9 compares the predictions of the analytical model with thedaafo

Piledriver. The left panel shows the radius as a function of time, whereas the right panel

displays the relative difference

6 ( ,.ta(O Rodel(t)) (31)

between the predicted and mneasuredl radii, to allow a miore detailed assessment of the

accuracy of the model. In b)oth panels, the dashed curve is the result given by the simple

linear applroximationi to the Hugoniot, whereas the solid curve is thle result given by the

piecewise -linear description of the full Hugoniot.

As expected, the radii given by the simple lineiar and the liecewise-line(ar Haigoniots are

very similar at early times, but deviate significanitly from one, another at later times. When

the full Hugomiot is used, the relative (lifferelice h ewe the measured andl predicted radli

is never miore than 7"A and is lc,;s thani 4A after 0.6 ins. Whleni the simple linear Hugonliot

1 is ed for all particle Speeds, thle absolute v alue of h is less than 7X before 0.6 ms but rises

to -'1 (/( after 1.2 ins. The radii predicted by the simiple linear Haigoniot, are systemnatically

too large after 0.6 nis because this approximation gives shock speeds that are systematically

too high when the particle speed is low (see Fig. 3). For reference, the peak pressure drops

to 15 GPa at about 2.8 ins. As discussed below, we adlopted this pressure as marking the

en(I of the hydrodynamnic phase.

Cannikin. -The Cannikin event was an explosion conducted in basalt at Amchitka

Island. Al.'.ska, on 6 November 1971. The official yield of this event remains, classified; the

U. S. Department of Energy [1 9S71 has said only thlit it wats less than 5 megatons. The

data frorii Cannikmn that were given to us hiad beeni scaled by ividling both ile radius andthe timue mnceasurements by the cube-root of the officMil yield in kilotons. If culbe-root scaling

were exaict, this would make the radlius vs. time curve aippjwr idlent ical to the curve that

would result from dletonat ion of ii 1 kt (levice iii the samiie miediuim. As noted above, cube-

roo t scaliuig may not always be accurate for tiiidlrgr( mnd inucleair explosions. However.

sinice the analyt ical miodel we are exlor(ing exhibli ts (,xat cube-root scaling, comparisons

of t his niodel with scaled and iliscaled dat a will givec the samec resuflt. IVe, therefore treated(

lhe ditti from Cannikin as thbough it had been proIi iced by at 1 kt explosion.

To construct a Hugoniot for Canviklmn. We aa1-,a the daiti aon Vatcaville basalt obtainedI

by Jorwes o al. [19681 and Ahlrenms and Gregsoi)j 119641. Ttiese (mt a andl thle 1iecewise-linifa

and slinple lineatr Hu goiot s thait we conistruict ed fr ()iii them are- shown in Figure 10. We

asslaiiled thle rock surrounding the explosion had a dlensity of 2860 kg/in3 , equal to the

dlensity of the samiples ineasured- lV .Jonles et al.

Figure 11 compares the radii prIedict ed by tile ail iclmne del with tilie raldii measured

duiring Can ntkin. Again, the left panel shows, the ra lii is as at funct ion of time, wherews thle

18

right panel displays the relative difference between the predicted and measured radii. When

the piecewise-linear approximation to the full Hugoniot is used, the relative difference

between the radii is always less than 3'/. When the simple linear Hugoniot is used for

all particle speeds, the magnitude of 6 is less than 5% before 0.22 ms, but increases after

this time, reaching 14cc at 0.6 ins, near the end of the data set. As in Piledriver, the radiipredicted by th, simple linear Hug inot are systematically too large after 0.1 ins because

this approximation gives shock speeds that are systematically too high when the particle

speed is low. For reference, the peak pressure falls to 15 GPa at about 0.7 scaled mis.

Thus, all the radius data from Cannikin lie within the hydrodynamic region.

Chiberta.- The Chiberta explosion was conducted in wet tuff at the Nevada Test Site

on 1975 December 20. The official yield of this test remains classified; the U. S. Department

of Energy [19871 has said only that it was between 20 and 200 kilotons. Using seismic data.,

Dahhizian and Israelso [1977] estimated that th yield of Chiberta was 160 kt. Like the

dita from Cannikir, the radius vs. t ine data from Chibferta available to us were scaled

1) the cube-root, of the official yield. Fo ;r the reason explained above in connection with

Cara ikin, we treated th data from Chibecrta as though it had been produced by a 1 kt

III modeling Chiberta, we used ie linear and piecewise-linear approximations to the

wet tuff Hugoitiot shown respectively by the dashed and solid lines in Figure 4. We assumed

that the rock siurrounding the device iiplacement hadt a density of 1950 kg/m 3

Figure 12 compares the predictions of the analytical model with the data from

Chiberta. Again, the (lashed curve is the result given by the simple linear Hugoniot,

whereas the solid curve is the result given by the piecewise-linear approximation to the full

Hlugoniot. As before, the radii given by the two approximations are very similar at early

times, but deviate significantly from one another at late times. When the piecewise-linear

Hugoniot is used, the relative difference 61 between the measured and predicted radii is

never more than --44 and is < lV between 0.35 and 1.6 ms. When the simple linear

Hugoniot is used for all particle speeds, the absolute value of b is less than 3% before

0.6 Ins, but increases after this time, re'aching 14%X at 1.6 ms, near the end of the data

set. The radii predicted by the simple linciar Ittugoniot are systematically too small after

0.4 Ins because this appr)xiiimation gives shock speeds that are systenatically too low for

()w particle sp'eds (se Fig. 4). For t his event, the peak pressure fa!ls below 15 GPa at

aboit 0.5 scal.d I ins. Thii ,, a large fr;ait ion of the radius m'asurements were made itside

th h itrodvlmimc reiion.

Y TS- X. The event we call NTS-X was an explosion conducted at the Nevada Test

Site. Radius vs. time ,hata frnom this exp losion were rep)rted by Heisinkveld [1979], who

sitated that the oflici;l yield wt ; 542 kt. Heusinkveld surmised that the ambient medium

19

was satuirated wet tuff, the amibient iiiediuiin of inost tests conducted at the Nevada Test

Site.

In modeling NTS-X we assumedl that the explosion (lid occuir in wet tuff. We followed

the same procedure ti:' _d in modeling Chiberta, except that we as9sunied the yield was

54.2 kt. Figure 13 compares the predictions of the analytical model with the data from

NTS-X. As before, the radii given by the simple linear Huigoiot and by the piecewise-linear

approximation to the full Hugoinot a-re very similar at early times, but deviate- significantly

from one another at later times. The relative difference 6 is never more than than 5%.

when the piece wise- linear Huigoniot is uised. When the simple linear Hugoniot is used for

all particle speeds, 6 is less than 53%c before 2 is, Ibiit inicreases after this time. reaching

17VA at 6 is, near the end of the data set. As iii Chlberta. the radii p)redictedl 1)y the simple

linear 1-lugozio(t are systemat icallyv too small after 0. 1 is becauise this approximation gives

shock speeds that are systematically t oo low for low particle speeds (see Fig. 4). For

reference, the peak pressuire falls below. 15 GPa at abouit 2.0 is. Like Chiberta, a large

fraction of thme radius measurements were made outside time hydrodynamic region.

Yltld Eitzmation

Having shown that the analytical miodel of 2 p~rov'ides a relatively accurate description

of the evoluition of the shock waves prodllmcedl by ulndergroumnd nuclear explosions for several

of the geologic niedlia found at U. S. test sites, ve now considler it~s usefulness in yield

estimation. We do this by adjuisting the assumied yield to give the b~est, fit of the model to

radiuis vs. timie data from the four U. S. nuclear tests, (iscuissed prev'ioiisly.

Iii order to determine the best fit of thme analytical model to a given set of radius

vs. timie data, we need a mneasurie of the goodness of the fit. This should be a function

of thle differenice b~et weeni the predicted arid ineasured shock front radlii. weighted in an

appropriate way. Uzifortmiitcly, the radius dat a bat we were furnished came withot

any immformationm on the randomn and sv'steniat ic erm mms. Ini fact, no~ error information is

aval able for anyv of the currently declassi fied radiuis vs. int data, a large fraction of which

is analilyzed here.

The abse, 4nce of error in formai~t oll iii;mdlt nim i~t o(ee~~ propernmeasure of the

g odiess of the fits amid to (let ermimme the m ic# rt aimim ii of t lie yield estimiates. We therefore

ado( pt ed a very sinmple fit tlng proci'dtire that aih mm d uis to (leterimiimie a imest-fit yield and

to C )iiij)aIf'C is to field (lati a muaaie wit h the aiimvt ical model and withL thre numerical

sirl im lii, of lhe Los A lamnios CO RRTEX gr( ipj. WCe asse-ss the accuirary of thle yield

estlili atc' Im eL wit it, h an alytica1 modlel Ivy cmnpaminmg liemi withI tihe estimates obt ainied

b , fittng mnmieri cal simmiI i isto the samie datia, ;in atppismah called i 7:7i1ated explo.41ion

.scalitty, aiim] by co mparinig t bein withi time official v ed.The precise Algorithm used in

20

dletermnining official yields is uniknowni, bunt presumiably makes use of radiochemnical and

seisinic as well a.', ,ioc-k wave measemenent s, when thlese are( available (see Lamib [19881).Pr Lr.Fr mnlct v, we's in ice thtalved are equally likely a pri ori and

that the ineasi~iremnt vrtr Cr fe diow ;I Gaussian dlist ribut lon. Then the iaximiuin of thelikelihood function can be' found hy iiiiiiiiiiizin~ th 1)oe~ egtdsmo h en

Square dlifferenices I etwe enl thle prede icted( and 1(1icasllre( shock front radii (see, for examle,

Mfathw LCWSl anW~1ker [196411, 14 7). 'ircv we had no information onl the errors of the

ini ivial illeasIiiiiiit s * we ssne Ithat fic he masuremients are unbiased and assigned

thu1(11111i t weight if thuy' I'le ieJl s 'cVct joii (crit eiia (see below) or zero weight if they did

nlot . The inlaximi III111 of tlie tike ii boo' fur icr ic Ci is thlen given by tile iiuir m of the mleasure

WlI II the IWSu111 run I ,OV T tile iici1riit-wd l'cit the pairticular yield estimate.

-111 axuli l ii le)((l an I i e niincr ical simiuilat ions dliscussedl in §2 and 3 are valid

"lliv dltIriiig thle Iliv' imi' a ri C)I, , wilieii thle .,trelit of the amlbient mlediumil Can be

lI11' 'uj'eted. Ilowe'vcr. ti I IA il1 ercc of thl > riulgth Ii'f lie r i~iciincioeases gradually as theshock wave weakens. so v I ieis cl (l dc I'ieid peaik pressumre at which the hdrodyari

)h-ir ci eds. Wiickerlc (I 9GJ foundme t lint III quartz, st rengthi effects can be ignored above

lie' cutica] st mess, wl'iicli is ahout 4 GNP. St udies by GradY ct a]. 11974] of quartz at

lpressuires Above 13 (GPa1 eeorst rateci that s 'remigtliI effects are negligible in this pressure

regiline. Basalt lbecomis plastic (it at critical stress of about 4 GPa [Ahrens and] Gregson,

1964]. The criticalI stres.s for saturated wet tuff is estimated to be -- 1 GPa [Holzer, 1965].

In the present work we have adopted the conventiomn that the hydrodynamnic phase endsin allII these mlaterials when the p~eak pre'ssure falls below 15 GPa. This is a conservative

critcrion, lin the sense the hydrodynanice phase miost likely extends to lower peak pressures.

When fitting the analytical inlodel or the simulated exp~losionl in wet tuff of King et

al. [19S9] to field (lIlt;a, we determined the p)oint at which the peak p~ressure fell below

13 C Pa iulig thle ann lv t il liodll %vith thle liecewise-liiiar represent at ions of the full

IIIig' Cii(t 5 of o 3. \Vlq ii fit ti uig the si iiida1"ted explosion InI SiC) 2 of King et al. [1989] to

field I (at a. we tie'termii rice I thle polit at whuich the peak pressulre fell below 13 GNP using

lie ailvt ical rue del -withI thle approxiniate Ilugonot adopted by King et ad. Plots of the

peakI.ssme'1, pledie tec byI thle atmalvt ical iodel are( givenI InI thle appendix.

We are'( lilt e'j ested InI th e accuraecy of the analyt ical miodel when it is iied with

si qd' ie'rFl t~ uiCt, silice we iisc t his ap~proxinition ii a comInpanion study of how

lie eve Uctt (if thie lie ck wave is Wi(i eiid 1 y thle properties of the ambient mediumil

ilid IC lew thle' plocl f CC'r is affect thle ch~iinrcrist ic radius at which the shock wave becomes

a lm prtssu~e' 1 i ~te wae' ii c! ,I). [1991all; for a prclirriiary account, sce Lamb et

21

a]. [1989], Caflen et a. [1990a)). We therefore compare ,he yields obtained by fitting the

analytical model to the field data using simple linear approxinmations to the Hugoniotswith the yields obtained using the full, piecewise-lillear Hugoniots.

Although the analytical model and the numerical simulations we consider are validonly during the hydrodynamic phase, in some cases they may describe the evolution of the

shock wave adequately even beyond the region where tihe pear stress is large compared withthe critical stress of the medium. Knowing how rapidly these models becomnt inaccurate

when used outside the hydrodynamic region is important for assessing whether they can be

used for yield estimation when the shock wave within the hydrodynamic region is disturbed,

either because the yield is low, causing the hydrodynamic region to be close to the device

canister, or because the geometry of the test is complex (see LanL [1980]). In order to

investigate the accuracy of the analytical model when fit to data taken at relatively large

radii, we first estimated yields using only data taken during the hydrodyx,..mic pha-se as

defined above and then using two successively larger sets of data, defined by successively

lower cutoff pressures. The radius at which the peak pressure predicted by the analytical

model falls below a given pressure depends on the assumed yield. Thus, the number of

data points used in evaluating expression (32) varies with the assumed yield.

Rc.,Uts.-- The results obtained by fitting the analytical model and numerical

sinmlations to field data from the hydrodynamic interval are summarized in Tables 1-4.

The first column in eacli taMe shows which model was used: the analytical model or oneof the numerical simulations discusseJl in § 3. The second column shows which Hugoniot

was used: the simple linear approximation to the generic Hugoniot, tile piecewise-linear

representation of the full generic Hugoniot, tile approximate SiO 2 Hugoniot of King et

,l. [19891, or the wet tuff Hugoniot of King et al. [1989]. The next four columns list

results obtained by fitting the models with the specified Hugoniots to field data from the

hvdrodynamic phase. Shown are the yield estiat e IT,,, the number N of data points

Iied in tile estimate, the root -mean-square diffeiciwe in radius

I - ( , (33)

;1(1 t I w (III.IIit Itv N A I, 1it 'l / 3 for ,; ,' fit. The lat quaiitity cai be us'd to compare

the quality of tle fits achieved for the four e Fnt> B" r Pl1cdriver and NTS-X, the yield

est ilmt(5s are given to th l nearest 0.1 kt, whereas f , Canv ikin amd Chiubta, the estimates

aro givii to the nearest 0.005 kt. Tale 5 c(onipares the resiilts ob tainedI by fitting the

ailcidt jcal model to data fro Im the bvdrods alamic iit ,rval with the results obtained byfittimu¢ to data sets that inch'ld data from bevond he lvdrhodvnamic interval.

22'

Not surprisingly, the( best agreemlent lbetween the( official yield an(1 the yield estimated

by fitting tile analyticad l odel to the raditus vs. t ime data is achieved when a piecewise-

linear represenitation ~ of the( full Huigoiiii t is ulsed arid the miodel is fit only to d~.ta from the

lhydrodLynamnic lpliase. Ini this case, tite dliffer-ence b~etween thle official yield an4 1 the yield

obtained by fitt ing thli anialytical miodel i's VA for Piledrilpcr, 8% for Cannikin, and 7%,/ for49

Chihcrhz.. The dlifferenice bet weeii tw hocl qu('( (oted by Hensinkveld [1979J for NTS-X arid

the yield obt ainied by fittinug tile analytical model is 8%X. For comparison, the dlifferences

bet weenl the official or qjuotedl yields 4 these events andl the yields ob~tained by fitting

the( nauiierical slimulations to (data froiii the hVdrodynainic phiase are 2%/(, 1%c, 7%, and

1%4, respectively. Thus, tite yild estimiates obtainied by fitting the analytical model with

pi ecewise- linear repres4'iitationis of the Ilugoniots to (lata from the hydrodynamiic phiase

are nearly as, accurate as thle yield estiaes obtained by fitting the numerical simullations

to thiese same data.

The agreciment bet ween tie( official yield aiid the yield estimated by fitting the

'Anllt ical model with Siml)e linear Mi1g()iiiot s to (dat a from the hydrodynamic Ipliase i

not as close, bult is still remlarkably goodl. For the event.s inl wet tuff, the estimated yields

differ fromi thle ofhc iil or (pIt('(1 ylld I II ()uv 4 llyV.'. S-A cund 5% for Chiberta. This is

not sUrp)rising, slince tie( sirmiple flimiei jy1l xint (i to thle Hulgoniot is nearly identical to

the( fuill, piecewise ljneam repreSCIn at ioll of the( Hugoniot for the particle speeds encountered

(buIiligr the Ilviy-od"lX'l1i i pihase Ii th l . iw... Figure 4). For the sainie reason, the

jeld of tie( O.nI;Ikln event obt aini ed by using the analytical model with the simple linearappro1(xliatl in to tHie basaldt Hugoiiiot differs from the official yield by only 2%. Although

the relative diffcrence AIV/W obtained using this approximiation to the Hugoniot is smaller

than lie relative differencee obtained using the pie-ewise-linear representation of thle full

1hlugoniot , the quality of tihe fit is soiiiewat porr as shown by the size of ARrms. (see

Tal e 2 ). However, for ft(i Picdirver (event, thle difference between the official yield and the

yield obt ained us lig the, Simlple linear lliigoiiiot is9 -40%, much greater than the difference

whenl thle piecewise linea.r Hi leoli( It i.s used. This is not surprising, since the simple linear

iq yroxiuiiatioml to bhe Si() H Iug{ uiiot is iiiaccuriat e for the particle sp~eeds encountered

duIng niuoit, of the, hydrilviiaI ie i i (see Figures 2 and 3).

O4 (lilile 1144Wv t 14 cilthe t tilf v hel .4( etiii ' henl (:Ita from ouitside the

I ivdro \11 hi;1 nc phlv Ia>114, Inchll11I4I A ul(,Ieaiiigfill st idv of til" effect is only possible for

C hihi/, and it( NT7S- X, sinee;all rq ;dl4 tal tie( available dat a fromi Ccanyikin anid Pilcdriller

Ie W ,It thI lI hvd (I Il I'II I Ik, 1 L ; r4j)I I, As shown ill Table 5, the( estlimated yield. of NTS-X1 cessfrom S to 66.-4 an e I11- .sKt whien dIa ta out to peA pressures of 7.5 and 4.6 (1Pa

are1 1CiiHicia eI. Ie, d iffem emiCs I t wcci t Ite lattler yields a ml the quoted yield of 54.2 kt are2T/ aiid :32X, , r-e-tiv4 V. F4 ) IiCi14 .onl t llt o)tllhr hial' , inClu(ing data out to peak

o)'~~(f 7., l) ih .1 G (IP; iiieT4;is, the estliniatvd N-elId 441ilv slightly, from 0.930) to 0.970

23

and 0.995 kt. The differences between the latter yields and the official yield of 1.00 kt are

3% and 0.5%, respectively.

The large difference in the sensitivity of the Chiberta and NTS-X yield estimates to

inclusion of data from outside the hydrodynamic interval is somewhat surprising, since

both events supposedly took place in wet tuff and the data from both events extend to

approximately the same scaled time (,-0.6 ms/kt/'). However, as explained above, wedo not know eithcr the medium or the yield of NTS-X for certain. Furthermore, we have

no knowledge of any special conditions that may have affected the explosion or the shockwave radius measurements. There does appear to be a systematic difference between the

fits of the analytical model to these two events at late times. Without more information,we are unable to determine whether this difference is due to some difference in the events

themselves, to systematic error in one of the sets of radius measurements, to systematic

error ir the Hugoniot we have used, or to inaccuracy of the analytical model when it is

used so far outside the hydrodynamic region.

24

5. SUMMARY AND CONCLUSIONS

We have explored an approximate analytical model of the evolution, during the

hydrodynamic phase, of the shock wave )roduced by a spherically-symmetric explosion in a

holmogeneous medium. The equation of motion for the shock front treats the compression

of material at the front exactly, using the Rankine-Hugoniot jump conditions and theHugoniot of the ambient medium. The rarefaction behind the shock front is treatedonly approximately through a parainete'r f that describes the distribution of the fluid

variables within the shocked volume. A key assumption of the model is that f remains

constant throughout the evolution of tihe shock wave. The model predicts the evolutionof the particle speed, shock speed, mass density, pressure, and specific internal energy

imme(iately behind the shock front, as well as the shock front radius as a function oftime. For a point explosion, the model exhibits cube-root scaling, in accordance with the

conservation laws for spherically symmetric point explosions in uniform media (see King

et al. [19891 and Lamb et al. (1991b]).

We have shown that the parameter f, which relates the specific kinetic energy of thefluid just behind the shock front to the mean specific energy within the shocked volume,

is constant when the shock wave is strong and self-similar. By comparing the relation

involving f with results from munierical simulations of underground nuclear explosions

in quartz and wet tuff, we have shown that it is nl!o remarkAbly constant even whenthe shock wave is no longer strong, for explosions in these media. Furthermore, we findthat the value of f is relatively independent of the ambient medium, and that f = 0.53adequately reproduces the particle-speed curve extracted from the numerical simulations,

in agreement with the previous results of Moss [1988].

The radius vs. time curves predicted by the model for a point explosion are in excellentagreement with the shock front radii nieasured during underground nuclear tests in granite,

wet tuff, and basalt, when the official yields are assumed and f is set equal to 0.53. If the

iuodel is used with a piecewise-linear approximation to the Hugoniot, the largest differences

between the predicted and measured radii range from 3% to 7% for the different events.

Even when the model is used with a simple linear approximation to the Hugoniot, the

shock front radii that it predicts agree extremely well with the measured radii f0o the

events in wet tuff (Ch.iberta and NTS-X), where the differences are less than 3% and 6%,

respectively, during the hydrodynamic p)hase. For the events in basalt (Cannikin) and

granite (Piledriver), the high-pressure approximation works less well, but the differences

iii the pre(icted and ineasuired radii are still less than 14% (during the hydrodynamic phase.

Tie average differences are substantially less in all cases.\We have shown that the model cal also be used to estimate the yields of underground

lii cl,ar explo'siolns, witi good ressults. When the analytical model is used with point-source boundary conditions and a pi(,cewise-linear representation of the Hugoniot, the

25

yields obtained by fitting the radius vs. time data from the hydrodynamic phase of the

explosions are within 8% of the official yields. For comparison, the yields obtained byfitting numerical simulations carried out by the Los Alamos CORRTEX group to thc

same data are within 7% of the official yields. Thus, the yield estimates obtained using the

analytical model are nearly as accurate as the yield estimates obtained using the numerical

simulations.More generally, the U. S. Department of State has claimed that hydrodynamic

methods are accurate to within 15% (at the 95% confidence level) of radiochemical yield

estimates for tests with yields greater than 50 kt in the geologic media in which tests

have been conducted at the Nevada Test Site (U. S. Department of State [1986a,b]; see

also U. S. Congress, Office of Technology Assessment [19881; Lamb [1988]). Thus, the

analytical model appears to be competitive with existing models for estimating the yields

of underground nuclear tests conducted in relatively uniform media.In a companion paper [Lamb et al., 1991a], we use the analytical model studied here

to investigate hydrodynamic yield estimation algorithms more fully, including optimalweighting of radius vs. time data (a preliminary account of this work has been given in

Lamb et al. [1989] and Callen et al. [1990a]). In a subsequent paper [Lamb et a., 1991b],we anialyze the validity of cube-root scaling for spherically-symmetric underground nuclear

explosions, using similarity transformation methods and numerical simulations to explore

the effects of source size and composition.

26

APPENDIX: COMPARISON WITH HEUSINKVELD'S MODEL

In this appendix, we compare the approximate analytical model of § 2 with the

approximate model proposed by Heusinkveld [1979, 1982]. Both models neglect the specific

internal energy and pressure of the ambient medium. Both also predict radius vs. time

curves that exhibit the temporal behavior characteristic of a strong, self-similar shock wave

at early times, then enter a gradual transition period, and finally exhibit the temporal

behavior of a low-pressure plastic wave. Itowever, Heusinkveld's model differs from the

model of § 2 in several important rsi,,e-ts.

First, Heusinkveld assumed that, the internal energy per unit volume just behind the

shock front, nanely C1 = P16, is a constant fraction fu of the total energy per unit volume

within the shock front, that is,3fHV (Al)4rrR 3

In contrast, the model of § 2 asslinies that the ,qpecific kinetic energy of the fluid just

behind the shock front is a constant fraction f of the total specific energy within the shock

front (see eq. [5]); the pccific internal ent:rgy just behind the shock front is equal to the

specific kinetic energy there (see eq. [3]).

Second, Heusinkveld's model satisfies only the momentum jump condition (2), whereas

the model of § 2 satisfies all three jump conditions (1), (2), and (?). In place of the specific

inite~rnal energy junp conditioi (3), Heusi:::veld assumed that the pressure just behind the

shock front is proportional to a costait coefficient F times the energy per unit volume

there, that is,

pI = Fe . (A2)

As noted in § 3, this is the strong shock limit of the Mie-Griineisen equation of state when

the Griineisen F does not depend on the density. It may be an adequate description of the

equation of state of the shocked miediuni, provided that the Griineisen r is independent of

density and the shock wave is strong. However, the shock waves produced by underground

nuclear explosions are r(hlativly weak during much of their hydrodynamic phase (see

Lamb [1988]).

lheusinkvell also assumeid a simple hiMear relation between D and ul of the form (6).

However, the jump co(nditions (1), (2), and (3), the equation of state (A2), and the D

vs. U relation (G) are mnlutuallv iucomnsistent. For example, if one accepts the mass flux

jurPa ) condition (1), lie mo)ment1 j 1u1111 cndlition (2), and the ansatz (A 1), one finds that

Ihe energy junip c(ondition (3) is icn'onsistent with a linear D vs. uI relation. Alternatively,

if one accepts the D vs. ii relation (G), one is led to the Hugoniot (see Zel'dovcli and

Raizer [1967], p. 710)

1,u(') =(A 3)[BI" -- (3 - 1 )J')

2 A

27

which is inconsistent with the jump conditions (1), (2), and (3) and the equation of

state (A2).Heusinkveld's model gives expressions for the shock speed, the radius vs. time curve,

and the post-shock pressure, post-shock particle speed, and post-shock internal energy that

are qualitatively different from the expressions given by the model of § 2. For example, byequating the prtssure given by expression (A2) with the post-shock pressure given by the

momentum jump condition (2) and making use of the ansatz (AI), Heusinkveld obtaineda quadratic equation involving the shock speed. The solution of this equation is

D [I + (1 + ], (A4)

where

g ( rfHO 1 / (A 5)

is a characteristic length, alalogous to the characteristic length L defined in equation (8).

Expression (A4) is qualitatively different from equation (9), the relationship predicted bythe model of § 2. The radius vs. time curve predicted by Heusinkveld's model can be

obtained by numerically integrating equation (A4). 2

Even though the model of § 2 is self-consistent whereas Heusinkveld's model is

not, both are approximate. Thus, their usefulness is best evaluated by comparing theirpredicti )ns with data from nuclear tests and/or imierical simulations. We show herecomparisons of the predictions of the two models with da a from numerical simulationsfor three reasons. First, the initial conditions of thse siimlations approach that of point

explosions, a simple case that tLe two models each describe. Second, we lack detailed

knowledge of the conditions under which the nuclhar test data were obtained (see § 4).Third, the simulations have reportedly been validat,,d by extensive comparison with data

from undergroumd nuclear tests.In comparing the two models with the resi lti of simulations, we wish to make a

consistent choice of model parameters. We do this by forcing agreement between the two

models at the beginning of the explosion, as follows. At ea'ly tinl(s, the radius vs. tinecurve given by Heusinkveld's model displays the t2/5 dependence characteristic of a strong.

self-similar shock wave, that is,

(75F7fjtBW) i tV'1 (R11() (77juB1 / (AG)

Alltigh Heiisinkveld assnined a simple lhi. ar D vs? uI relation. an arbitrary D vs u

relation can he treated to any desired accuracy lv i-imng a piecewise-linear approximation,as described in § 2.

28

On comparing this curve with the early time curve given by the miodel of § 2, namely,

R~t) 75f B21V 1/5 t/ A7

we se, that if rf 1 1 is set equal to( ff. tilhe two model,; will give identical results at the

lev 'lilnng of tHie explosionI1. Ili tilt t-( )f iii iis that follow, we do this.

Figures AlI and A2 comipare t fet rab it isVs. t imie curves predicted by the two models for

exiilosionis in qulart z arli wet tutff wit I i t lhe ha t a from simulated explosions in the same media

that were descrilbed in § 2. For the explosion inl quartz, we used rfH = 0.325, whereas

for wet tuff we uised F -fil 0.299. For comparison, Heusinkveld obtained rfil = 0.78

fr explosions in alluviumii and wet tuff' wi(l 1.03 for explosions in granite by fitting his

imodel to the particle speed dlata of Pcrret and Bass [1975j at relatively late times; had

wel uised these vadles inl the comparisons, the (discrepancies between Heusinkveld's model

and the simuillations wouild have been iwuch greater. Alt hough the radius vs. time curves

are, integrals of the shock sp~eedls predhictedl by the models and hence tend to smooth out

differences, tlw cuirve predicted by- the analytical inodel of § 2 agrees better with the

simulat ions than does the curve predicted by Heiisinkveld's model.

Addit ionmal and~ imore dlecisiv'e com~parisonms c-al he made between the post-shock

pr-sires and~ pa~rt icle speeds given by thle mnodels. Onl siibst it uting equation ( A4) intolie D) vs. it relation (611. onle finids t hat Ifeulsinkv~eld 's model predicts the post-shock

it B. + 317f11BW 1/2 11 (AS)

At small radii, equation (AS) b~ecomes

Thus.i ,"~ has the samle R-depemdeiwe at simial radii as that given by the ansatz (5) of §~ 2,

01C Ff I, has becii set, equa'l to f B liI'wever. at large radii the post -shock particle speed

predicted by liemisimik veld' I ro dll "cah's withl radiuls accord1 ug to

,, 31J, It'7~ .~ 1? 3 . q (A 1)

Fii.iires, A3 amid A 4 compxiii am I p -. .l ick pa"rticle speeds predicted by the two models-

'with thio'-(, derived, fri iii th .iniiil;Ire-d explosomis Inl quartz anld wet tuff. The R- 1/2

_

lepemleuc~ce pired(ited ) he w ~ 4 2 agreecs mnch beit ter with the particle speed

da a; at laIte tiil,. thai Ioc, th leU ' fpiid(erv predicted by Hieisliikvel ds model. Ill

129

particular, there is no evidence of the break 'n the slope of the u I vs. R curve at R ; g

that is predicted by Heusinkveld's model.

The post-shock pressure predicted by Heusinkveld's model is given by equations (Al)and (A2), and is

P=31 TV (All)

In contrast, the model of § 2 predicts that the post-shock pressure falls off as R- 3 for

R < L, but is proportional to R 3 /2 for R > L (See eq. [23]). Figures A5 and A6 showthat the pressure curves derived from the simulations show such a break at about the right

radius, demonstrating that the model of § 2 is in better agreement with the simulations

than is Heusinkveld's model.

Perret and Bass [1975] show that pressure data from explosions in several geologic

media is well fit by R - 2 .96 out to distances of 8m/W/ 3 , at which point a clear break

occurs. At distances beyond this break, the data are better described by R-' 7 5 . This large

R behavior is more in keeping with the analytical model of § 2 than the R- dependence

at all distances predicted by Heusinkveld's model.

The predictions of the two models differ significantly well before the assumptions ofthe inodel discussed in § 2 become invalid. As discussed in § 4, the hydrodynamic phase

extends at least out to the radius at which the post-shock pressure has fallen to 15 GPa.

Obviously, the ambient pressure of 20 MPa can be neglected throughout the hydrodynamicphase. As noted in f 2, the ambient specific internal energy can be neglected for particle

speeds greater than 0.1 kmi/s; Figures A3 and A4 show that the post-shock particle speed

is actually 1 kii/s or greater throughout the hydrodynamic phase. Figures A1-A6 show

that the differences between the two niodels are already significant at 10 meters, and

incra.se dramatically at larger radii, whereas the post-shock pressure falls to 15 GPa at

25 xitters in quartz and 22 meters in wet tuff. At 23 mieters in quartz, the peak particle

speed predicted by the model we discuss falls right on the curve predicted by the numerical

siniilation, and is 2.5 times larger than the peak particle speed predicted by Heusinkveld's

nodel. which is far below the curve pr((icted by thi simulation.

These comupanisons show that the model (of j 2, which fully incorporates the Rankine-

Hugoni,)t junip conditit)ns and (l,,ws not a-ssimi aly I aicilar equati(mn of state, also agreesbetter with the radius vs. t ime curw,'.- a t the 1,t),t ,Il(,(ck particle speed an(1 pressure data

(herici fromin the simulated explo)sions tlan d.s th,. iiiiel si iggested by Heusinkveld.

A(k,, WCb dq mint.. ,e are ,speidly g rateful to M. leusink-eld for discussions of

slit'k wave propagation in g,'oltgir imitia anid frt rviding SLIFER data from a variety(,f tim letrgro, ntid mlear explosit nII. Wt. w.ish t haink W. Nit ss for detailed discussions of

He ana lytical m(del investisgatd lI henc, and I) D. Eiers aml tHie P-15 CORRTEX group

:30

at Los Alarnos National Laboratory for kindly providing us with copies of the SESAME

equations of state for quartz and wet tuff and for sharing with us the results of their

numerical simulations of nuclear explosions in quartz mid wet tuff. It is a pleasure to thank

T. J. Ahrens, D. D. Eihcrs, R. G. Gcil, R. E. Hill, NV. S. Leith, and G. S. Miller for helpful

discussions of shock wave propagation in geologic media. This research was supported in

part by DARPA through the Geophysics Laboratory under Contract F-19628-88-K-0040.

31

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33

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.) G42 646. 1980.

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Zvl'du;'ich, Ya. B., and Yu. P. Raizer, Physics of Shock Waves and High-Temperattre

Hydrodynaniic Phenomena, Academic Press, New York, 1967 [English translation].

35

Figure Captions

Fig. 1.-Dimensionless energy partition factor f as a function of Grineisen coefficient

r for a strong point explosion in a medium o)eying a Mie-Griineisen equation of state.

Fig. 2.-Hugoniot data for SiO 2 and two of the representations used in calculations

described in the text. The solid line shows the piecewise-linear approximation to the full

Hugoniot while the dashed line shows a simple linear approximation to the high-pressure

portion of the Hugoniot.

Fig. 3.- -Expanded view of SiO 2 Hugoniot data at low pressures and three

representations used in calculations described in the text. The solid line shows the

piecewise-linear approximation to the full Hugoniot while the dashed line shows a simple

linear approximation to the high-pressure portion of the Hugoniot. The latter is clearly

inaccurate at low particle speeds. The dash-dotted segment at low it, is similar to the

approximate Hugoniot used by King et al. [1989] ai.d replaces the corresponding section of

the piecewise-linear Hugoniot when comparisons are made with the numerical simulations

of D. Eilers et al. Also shown is the isobar at 15 GPa, the pressure we have adopted as

marking the end of the hydrodynamic phase.

Fig. 4. Hugoniot data for wet tuff [King et a;., 19891 and two representations used in

calculations described in the text. The solid line shows our piecewise-linear approximation

to the full Hugoniot, while the dashed line shows the simple linear approximation to the

high-pressure portion of the Hugoniot. Also shown is the isobar at 15 GPa, the pressure

we have adopted as marking the end of the hydrodynamic phase.

Fig. 5. - Peak particle speed iI vs. shock front radius R for a 100 kt explosion in SiO 2,

from a iiumerical simulation by D. Eilrs et al. (priwmtt( coimmunication, 1987), compared

with the peak particle speed predicted y)v the analyt ical model (solid line). The analytical

imo h'l describes the data quite well over two decades of particle speed, showing that the

enei y partition an,'atz (eq. [5]) is relatively accurate.

Fig. 6.-- Peak particle speed iiI vs. shock frin radius R for a 100 kt explosion in

w. t tuff, fr if a n,,imerical sinilati )n Iv D. Eilers ,t al. (private C unmnication., 1987).

coinlpared with the peak partile sptd rilictd I v the analyt ical model (solid line).

Aga in. thi ;ianalvtical niodel desc ilIes the dat a quite well over two decades of particle

spe',, I.h, wing that the energy partition an..atz (e'q. [5]) is relatively accurate.

Fin. 7. Comparison of shock fr lt ra(lius vs. t ne curves pre dicted |v the analytical

inodel with radius vs time dat a froim the iinuulerica I simulation of a 100 kt explosion in

36

SiC)2 lby D. Ejiers et al. (private comminication, 1987). Left panel: Predicted radii asfunctions of timie. Right pancd: Relative dlifferenice between radii predicted from the Si0 2

slinnilation and from the analytical iiiodel. Thle (lash-dotted lines show the results when

he piecewise-linear rep~resentat ion of the full Hlugoiiiot (see Figs. 2 and 3) is used in the

analytical miodel; the (Iashiedlinues show the results when the simple linear approximation

to the high-pressure portion of H ie H iw~ont (again see Figs. 2 and 3) is used.

Fig. 8.- -Comnparisori of shock froiit railiis vs. t ilne curves predicted by the analytical

mod)(el with radlius vs. tine diatta fn iu thle in iinerucal simulation of a 100 kt explosion in

wet t uff by D. Eilers et al. (Iri;te' (-()IuII uIi(tiat Ion, 1987). Left panel: Predicted radii

as funuctionis of time. RIght pancl: Relative difference between rad-ii predicted from the

SiC) 2 simulation andl frouu the analyt ical umdel. The (lash-dotted lines show the results

whln'u thle piecewise-linear rep reselt at i( 1 of the full Hugoriiot (see Fig. 4) is used in the

antalytical iodel; thle dashied linies show the results when the slimple linear approximation

to) the high-pressure portion of the litgoiliot (again see Fig. 4) is used.

Fig. 9. -Comparisou of shock front radlius vs. time curves predicted by the analytical

miodel with radius vs. time (data from Piledrizier, a 62 kt explosion in granite. The arrow

iii each panel marks the radius at which the peak pressure drops to 15 GPa, which we have

al tdas the end of the hydrodynmunic phause. Left panel: Predicted andl measured radii

ais functions of time. Right panel: Relative difference between measured and predicted

radii. Thle solid linies show the results when thle piecewise-linear representation of the full

luigomiiot (see Figs. 2 and 3) is used in the analytical model; the dashed lines show the

results when the simple linear appjroximation to the high-pressure portion of the Hugoniot

(a1gain see Figs. 2 and :3) is uisedl. When the lpiecewise-linear Hugoniot is used, the radii

pr1edictedh by the analy~t ical mod)(el dliffer from the mneasuredh radlii by no more than 7% over

the whiole range of the (data. The li(eeise-hiear rep~resentation of the Hugoniot is clearly

superior to the simple linear after ahout 0.5 is.

Fig. 10. luigoniot (hat a for basalt from JIones et a]. [1968] andl Ah~rens and

Greg-son [1964] and~ two re;~ -esemtatioiis usedh in calculations dlescrib~ed inthe text. Theso Id linc 'Shows the( piecew sv liear rceiutscmt at ion of tilie full Hugoniot while thle dashed

linec shows tlhe siruuphe linear approxuiia~t Ion to the high-pressure portion of the Hugoniot.

Fig 1I Co ni10ui(hok front radius vs. t imue curves lpredicte(l by the analytical

rui de with rat Ilii' v ". t i, lar a froi i Cain nikin, anl exp~losion iii basalt with a yield of

Seve-ral Nit, Th11 ieaso iien jem its ll ivr f I OE scalco I to show anu apparent yield of 1 kt (see

text ). Lcft panf 1: Pitohiotod amid nwmieleo ii as fiiictioiis of time. Right panel: Relative

di Iforeiwe lbetwel niea.,l ired a Iri proohict ed radi. The soidi( linics show the results, when the

.37

piecewise-linear representation of the full Hugoniot (see Fig. 10) is used in the analytical

model; the dashed lines show the results when the simple linear approximation to the

high-pressure portion of the Hugoniot (again see Fig. 10) is used. The analytical model

with the piecewise-linear Hugonliot predicts shock front radii that are within 3% of the

measured radii over the full range of the data.

Fig. 12.-Comparison of shock front radius vs. time curves predicted by the analytical

model with radius vs. time data from Chiberta, an explosion in wet tuff with a yield in

the range 20-200 kt. The measurements have been scaled to show an apparent yield of

1 kt (see text). The arrow in each panel marks the radius at which the peak pressure

drops to 15 GPa, which we have adopted as the end of the hydrodynamic phase. Left

panel: Predicted and measured radii as functions of time. Right panel: Relative difference

between measured and predicted radii. The solid lines show the results when the piecewise-

linear representation of the full Hugoniot (see Fig. 4) is used in the analytical model; the

dashed lines show the results when the simple linear approximation to the high-pressure

portion of the Hugoniot (again see Fig. 4) is used. The analytical model with the piecewise-

linear Hugoniot predicts shock front radii that are within 3% of the measured radii over

the full range of the data.

Fig. 13.-Comparison of shock front radius vs. time curves predicted by the analytical

model with radius vs. time data from NTS-X, assuneid to be an explosion in wet tuff with

a yield of 54.2 kt. The measurements have been scaled to show an apparent yield of

1 kt (see text). The arrow in each panel marks the radius at which the peak pressure

drops to 15 GPa, which we have adopted as the end of the hydrodynamic phase. Left

panel: Predicted and measured radii as functions of time. Right panel: Relative difference

between measured and predicted radii. The solid lines show the results when the piecewise-

linear representation of the full Hugouiot (see Fig. 4) is used in the analytical model; the

dashed lines show the results when the simple linear approximation to the high-pressure

portion of the Hugoniot (again see Fig. 4) is used. The analytical model with the piecewise-

linear Huigoniot predicts shock front radii that are within 5% of the measured radii over

the full range of the data.

Fig. Al. Comparison of ti shock fro nt rilii pedicted b3 the analytical inodel

of j 2 (solid line) and the model of ttisinkv,-lit 19S2] (dashed line) with radius data

from a numerical sinmulation of a 100 kt explosion in S1O 2 by D. ELlers et al. (private

communication. 1987). The piecewis,-linear repr,.sentation of the SiO 2 Hugoniot shown

in Figs. 2 and 3 was used in both niodels.

38

Fig. A2.--Coinparison of the shock front radii predicted by the analytical modelof § 2 (solid line) and the model of Husizikveld [1982] (dashed line) with radius datafrom a numerical simulation of a 100 kt explosion in wet tuff by D. Eilers et al. (privatecomnmnication, 1987). The piecewise-linear representation of the wet tuff Hugoniot shown

in Fig. 4 was used in both models.

Fig. A3. --Comparison of the peak particle speed predicted by the analytical modelof § 2 (solid line) and the model of Heisinkveld [1982] (dashed line) with peak particle

speeds from a numerical simulation of a 100 kt explosion in SiO 2 by D. Eilers et al. (privatecommunication, 1987). The piecewise-linear representation of the SiO 2 Hugoniot shown

in Figs. 2 anid 3 was used in the model of Heusinkveld. The peak particle speed predicted

by the analytical model of ,j 2 is independent of the Hugoniot and scales as R- 3 /2.

Fig. A4. Comparison of the pieak particle speed predicted by the analytical model of

§ 2 (solid line) and the mnoel of Hm,imnkv,,ld [1982] (dashed line) with peak particle speedsfr ,i a niunierical siil dati j( ),f a 100 kt exp l)sion in wet tuff by D. Eilers et al. (privatecommunication, 1987). The piecewise linear representation of the wet tuff Hugoniot shownin Fig. 4 was used in tlie model of Heusinkveld. The peak particle speed predi-ted by the

analytical model of j 2 is initdepen(lent of the Hiigoniot.

Fig. A5. Conparisou of the peak pressure predicted by the analytical model of

§ 2 (solid line) and the model of Hetsinkveld [1982] (dashed line) with peak pressuresfrom a numerical simulation of a 100 kt explosion in SiO 2 by D. Eilers et al. (private

communication, 1987). The numerical results are more consistent with the R -1 / 2 variationat large R predicted by the model of § 2 than with the R - variation predicted by the

model of Heusinkveld.

Fig. AG. -- Comparison of the peak pressure predicted by the analytical model of

§ 2 (solid line) and the model of Heusinkveld [1982] (dashed line) with peak pressuresfrom a numerical simulation of a 100 kt explosion in wet tuff by D. Eilers et al. (privatecommunication, 1987). Again, the numerical results are more consistent with the R -3 /2

variation at large R predicted by the model of § 2 than with the R - ' variation predicted

),- the model of Heusinikveld.

39

Tables

1. Yield Estimates for Piledriver

2. Yield Estimates for Cannikin

3. Yield Estimates for NTS-X

4. Yield Estimates for Chiberta5. Effect on Yield Estimates of Including Data from Outside the Hydrodynamic Phase

40

TABLE 1

Yield Estimates for Pledrivera

Modtel Hugoniot VIt (kt) N A-iRrms (ni) ARrms/ 11e'1/ 3

Analytical Imodel Linear Sit), 37.6 22 0.785 0.234

Analytical model Full Si()2 62.5 25 0.312 0.079

Nuinerical simulation King et al. Si0 2 63.4 24 0.349 0.088

a Yield estimates obtained by fitting thil model or simulation to measurements made during

h, hvdrodyna inc phase of the explosion. IV., is the estimat-d yield, N is the number

of data points used in the .ield estimate, and LRrm, is the root-mean square difference

between the measured and pire(lictet shock front radii. The quantity ARrms /1V 1/3est can

be used to compare the quality of th, fits for different explosions. The official yield of

Piledriver was 62 kt [U. S. Department of Energy, 1987].

TABLE 2

Yield estimates for Cannikina

M(otel ttugoniot W,,t (kt) N /Rrm, (In) ARr,s/I t3

Analytical model Linear basalt 0.980 154 0.030 0.031

Analyvtical model Fiill )asalt 0.925 158 0.020 0.021

Nill r'iial siu"n lat ion King et ;Il. Si02 0.990 158 0.037 0.037

' Yield estimiates obtained by fitting lite model or simulation to measurements made duringthe hvdrodyna ini phase of the exlosion. 1It, N ,Rrm, arid ARrm/W 1

/3 have the same

rInarings as in Table 1. The data for Cannikin. have been scaled so that the apparent yield

is I kt (see text).

41

TABLE 3

Yield Estimates for NTS-X'

Model Hugoniot West (kt) N ARrms (m) ARrms/W

Analytical model Linear wet tuff 59.2 30 0.101 0.026

Analytical model Full wet tuff 58.5 34 0.087 0.022

Numerical simulation Full wet tuff 57.9 34 0.084 0.022

' Yield estimates obtained by fitting the miodel or simulation to measurements made during

the hydrodynamic phase of the explosion. TV, N, Rrms, and ARrms/IVV1/ 3 have the

same meanings as in Table 1. The yield of NTS-X is given as 54.2 kt by Heusinkveld119791.

TABLE 4

Yield Estiiates for Chtbertaa

ur r1/3

Mo(lel Hugoniot ,.t (kt ) N N-Rrms (in) ARrrs/l'Ae't

Analytical model Linear wet tuff 0.950 42 0.028 0.028

Analytical model Full wt tuff 0.930 47 0.021 0.021

Numerical simulation Full wet t~iff 0.910 45 0.013 0.014

' Yield estimates obtained by fittin g the model or simnilation to ineasureunenit made during

the hydrodynamic pha.e of the explosion. I,,, N. ARr,.,, and ARrm,/1 / 3 have the

saine meanings as in Table 1. The data for Chtb.rta Aere scaled so that the apparent yield

is 1 kt (see text).

42

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I I I'.0

~J. ~6S

~

U~~~~1~

lr) Cl

(a!) 1 SnIpPI }LIOJJ ~O14~

I I I -4

44

44

44

4 ci4

44

44

44

44

44

44

4

~.~

6H~ 44~

ci

j6

6 6c/c

ci

44

44

44

ci4

44

4

4444 "4-.'

44 0

4

(in) 1 SnIPPI WOJJ )IC)014S

h)

4 I I4

44

44

4 44

44

44

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44 44 4

44

44

444 '4 4

44

4.

*4H-4

4.zS

6

S4

44

4

4. 444

.4. 44

44 4

4N..4

444 4

4

*44 444. 4

4. 44 4

4

Hz -

zCl - -

(in) ~, sniprJ iHOJI rj~oij~

401

'N 30 I

00

10-

0 13 45

Time (ins)Fig. Al1

40

Wet Tuff

30

EV

10

0 123 45

Time (mns)F ig. A'2

100 'I

765 SiO24-

3

-S2-

- 10 -

-~ 765

4

- 3 .4.

3-•

0.14

0"12 4 .5 6 7 8 9 2 -3 4 510

Shock front radius R (m)Fig. A3

59

100 I ' I I I I I

7-64 Wet Tuff3

2-

"- 10 -

7-65-

r. 4-

- 3

2 4 5

104

"'2 5 6 89 3S

60

Shoc frot raius (mFig. A4

104 ' , , '' 105

SiO 2

.44

10 3 10 4 O

1020

101

0 .°

10' 102-ou

10

Shoc frot raius (mFig. A5

614

4 5104 , 105

_ Wet Tuff

-, 10 3 " 10 4

CIL..

02 0310 10

22

S1010

1-• Numerical simulation (100 kt) "i - Analytical model (tis paper) "

... Heusinkveld model

0

8 92 3 4 5 1000

Shock front rcadius R (m)Fig. A6

62

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