Isentropic Decompression of Fluids from Crustal and Mantle Pressures

VOL. 84, NO. B4 JOURNAL OF GEOPHYSICAL RESEARCH APRIL 10, 1979

IsentropicDecompression of Fluids From Crustal and Mantle Pressures

SUSAN WERNER KIEFFER 1 AND JOAN MARIE DELANY

Departmentof Earth andSpace Sciences, University of California,Los Angeles,California 90024

Possiblethermodynamic histories of H:O andCO: decompressingisentropically from crustal and upper mantlepressures are examinedusing graphs of entropyversus density with contoursof constantpressure and massfraction. These graphs are particularlyuseful for problemsin dynamicprocesses because, in additionto providingthermodynamic information about entropy, density, pressure, phase, and--if more than onephase is present--massfraction vapor, the graphsallow visualizationof the soundspeed, the parameterwhich controls the rateof propagationof disturbancesin manyfluid dynamics and geophysical problems.The soundspeed is representedby the verticalgradient of the isobarson the entropy-density graphsand is thus easilyenvisioned across phase changes or as a functionof pressure,mass fraction vapor, or density,as the 'topography'represented by the isobariccontours. This representationis especiallyuseful for illustratingthe low soundspeeds of two-phaseliquid-gas systems, e.g., the speedof a few metersper secondcharacteristic of water-steammixtures at l-bar pressureas contrasted to 1500m s-• in liquidwater or 450 m s-• in steam.Two simpleinequalities are derivedfrom conservationof energy, mass,and momentumto clarify conditionsunder which flow processesin single-component,single- phasesystems are 'approximatelyisentropic.' Application of theseinequalities to specificproblems of gas dynamics,ascent of magmaand shockdecompression of gases,liquids, and solidssuggests that under manyrealistic conditions, rapid magma emplacement and shock decompression of gasesand liquids may be consideredto be approximatelyisentropic processes. However, shockdecompression of solidsis probablynot an isentropicprocess because viscous dissipation could contribute significantly to entropy productionand to volumechanges not accountedfor in an isentropicequation of state.Entropy-density graphsof H:O along representativecrustal geotherms show that isentropicascent of H:O from crustal depthscauses partial vaporizationof a liquid phaseas pressures decrease toward the surfacepressure of 1 bar, whereasisentropic ascent from greaterdepths (pressures greater than 20-50 kbar, dependingon the geothermchosen) causes partial condensationof a vapor phase near the surface.A comparisonof the thermodynamichistory of H:O and CO: originatingat depthscharacteristic of kimberliteor carbonatite sourceregions shows that H:O goesthrough a condensationphase change near the surface(supercritical fluid • vapor • vapor + liquid) but CO,. expandsentirely into the vapor-alone field. Becausethe condensationphase change in theH:o systemmight significantly alter fluid flow properties, H:O andCO: cannotbe consideredto behavein a qualitativelysimilar manner in the dynamicsof eruptionof kimberlite or carbonatite.The entropy-densitygraphs are also usedto examinethe isentropicrelease of H:O from shockHugoniot states, under the commonly used assumption that thermodynamicequilibrium is maintained.Upon releasefrom pressures below 275 kbar, partial vaporization of a liquidphase occurs: upon releasefrom higherpressures, partial condensationof a vapor phaseoccurs.

1. INTRODUCTION propagatedin a material. Thus, for example,if we ask about The behavior of materialsunder rapid decompression,ei- the rate at which a disturbance such as a vent opening or ther from crustal pressures,as in the case of volcanism, or pressurerelease is propagated into a magma chamber, the from evenhigher pressures, as in the caseof meteoriteimpact, sound speedof the magma must be known before the answer is stronglydetermined by the soundspeed of the decompress- can be determined. The 'dynamic' variable, the sound speed, ing medium (for example,see McGetchin and Ullrich [1973], dependson the 'static' state variables(pressure P, densityp, Sanford et al. [1975], Kieffer [1975, 1977a]for discussionsof and entropy $) through the identity the effectof soundspeed during volcanicor geysereruptions; seeZeldovich and Razier [1966, chapter 1], for a discussionof C:=( CO•-•-p )s (1) decompressionfrom shockstates). In many casesthe decom- pressingmaterials are multiphase, single- or multiple-com- This can be demonstrated as follows (after Zeldovich and ponent mixtures,such as water-steamor magma-gasmixtures. Razier [1966] or many fluid dynamicstexts): The soundspeeds of multiphasemixtures are generallysuch Consider the density and pressureto be written as their complex functions of many variables (the proportions of undisturbedvalues po and Po, respectively,plus small changes phasespresent, pressure, temperature, bubble size,latent heat, Ap and Ap, accompanyingthe fluid motion. For a uniform and degreeof equilibrium[Kieffer, 1977b])that simplerepre- fluid and, for simplicity, for a one-dimensionalcase, the Eu- sentationof the soundspeed as a functionof thesevariables is lerian equation of continuity gives difficult.This paper introducesa simplegraphical representa- COAp COu tion of soundspeeds in multiphasesystems and demonstrates cot--po cox (2) the thermodynamicand fluid dynamic conclusionswhich can be obtained from the representation. and the equation of motion is The speedof soundis an importantfluid dynamicsparame- ter becauseit is the velocity at which small disturbancesare pocot- cox=- s cox (3) •Now at U.S. Geological Survey, Flagstaff, Arizona 86001. where t is time, x is the spatial coordinate, u is the particle Copyright ¸ 1979 by the American GeophysicalUnion. velocity(assumed to be small), and S is entropy. In all formu- Paper number 8B1064. 1611 0148-0227/79/008B- 1064501.00 1612 KIEFFERAND DELANY:ISENTROPIC DECOMPRESSION OF FLUIDS

(b) (o) The equation (and similar equationsfor Ap and u) have two i families of solutions: 10o Ap = Ap(x- ct) Ap = AP(x- ct) u = u(x- ct)(Sa) Ap = Ap(x + ct) Ap = AP(x + ct) u = u(x + ct)($b) wherec is the positiveroot, c - + (c"P/&p)s •/•'. These solutions ;RITICAL describedisturbances which propagatein the positiveand POINT negativex directionswith the velocityc, e.g., in the first family of solutionsabove, Ap, Ap, or u are constantif x = ct + const, and the disturbancehp or AP thereforepropagates in the +x direction. Therefore c is the speedat which disturbancesare propagatedthrough materials. This brief reviewis a reminderthat, althoughthe derivative (&P/&p)s is normally thought of in terms of the compres- sibility,K = p(&P/&p)s, a staticparameter, it is importantin the form c•' - (&P/&p)s in dynamicproblems as well. In cases 75 5O where the sound speedhas been determinedfrom pressure- 125 densitydata at constantentropy and comparedwith the sound 100 speeddetermined from dynamic ultrasonicdata, the agree- ment between the static and dynamic methods is good to within about 10%(the comparisonshown by Kieffer [1977b, Figure 2] is typical). Differencesmay be due to experimental uncertainties,dissipation in a dynamic experiment,or dis- persio_n(frequency dependence). bor lsentropicprocesses arenot only 0f geologic interest inthe propagation of small-amplitude disturbancesas discussed (v) above,but alsoin a numberof problemsinvolving large pressure, density,or temperaturedisturbances. In theseproblems, processeswhich can be demonstratedto be adiabatic(dQ = 0, wheredQ is the heat transferred)are frequentlyassumed to be 10-3 isentropic (dS - 0, where dS is the entropy change). Two examplesof processesassumed to be adiabaticand isentropic are ascent of magma [Rumble, 1976] and releasefrom shock compressionduring meteoriteimpacts and laboratoryshock experiments[Rice and Walsh, 1957; McQueen and Marsh, 1960;Ahrens and Rosenberg,1968]. Although adiabaticproc- esses,being irreversible,are not necessarilyisentropic (see discussionbelow), the assumptionof constant entropy so greatly simplifiesthe mathematicaland thermodynamictreat- 10-4 I i , , , I1[I I ment of many problems,particularly those of geologicsystems t 5 10 for which material propertiesare complex,that it is usefulto apply the isentropicassumption whenever reasonably justified. Entropy,Joule cj K Thus in section2 of this paper we considera criterionaccord- FiB. 1. Entropy-densityrelations for H•.O.Entropy is relativeto a ing to which processesmay be considered'approximately triple-pointentropy, $ (triple point) = 0. Data are from K•nan •! a/.'s isentropic.' [1969]steam tables. The sin81e-phasefield is shownas 1•)and the two- We describein section3 entropy-densitygraphs which we phase(liquid + vapor) field as 2•). Contoursoœ constant pressure (isobars)are shownin 25-bar increments.In the two-phasefield, have found offer a simplevisual representation of a numberof contoursof constantmass fraction vapor (isopleths)are shown. thermodynamicvariables involved in isentropicprocesses. In Dashedlines (a) and (b) are discussedin the text;(a) intersectsthe section 4 we demonstratehow the graphs can be used to two-phaseresion at l-bar pressure. examine the behavior of volatiles (1) ascendingin volcanic systemsoriginating at differentdepths within the earth, and (2) decompressingfrom a shockHugoniot state.Entropy-density graphs for the two systemsconsidered, H•.O and CO•., are lations it is assumedthat the particle motionsin the distur- presentedin Figures I and 2, respectively. banceare isentropic,so that a small changein pressureis relatedto a smallchange in densityby hp = (&p/&O)shp. 2. ISENTROPIC PROCESSES Denote (&P/&0)s as c: without, for the moment,any con- notation of sound speed.A simplewave equationcan be We propose below criteria which might be usedto check obtainedfrom (2) and(3) by differentiating(2) with respectto if single-component,single-phase flows are 'approximately time,and (3) with respectto x andeliminating &•'u/&t &x. The isentropic.'In the following discussionit is assumedthat resultingequation has the form of a classicalwave equation: the usual results of thermodynamicscan be applied to materialmotions. This can generallybe done if the instanta- a•- Ap a•- Ap (4) neouslocal thermodynamicstate is consideredand if the rates of changeof the parametersdescribing the localstate are not KIEFFER AND DELANY: ISENTROPIC DECOMPRESSIONOF FLUIDS 1613 too large. In particular, it is assumedthat the characteristic (a) times of the problem (to) are larger than relaxation times (r) for material changes(such as phase changes,nucleation and growth processes,etc.), so that flows are 'slow' in terms of departure from thermodynamicequilibrium. (The word slow takes on a wide range of values, dependingon whether the relaxation times are those associated with molecular trans- lations, rotations,'vibrations, or dissociationsin gas flow (r = (s) 10-•ø-10-5 s) or those of nucleation and growth of metamor- RIPL•POINT phic mineralsin viscoussolids (r perhapsas large as 10TM s)!) It t00 is also assumed that the substances considered are fluidlike in i that they cannot support shear stressesand that they are 5oo describedby linear stressand heat conductionrelations. Similar criteria for isentropicflow have been given by Bat- chelor [1967, pp. 164-171] and Thompson[1972, chapter 3]. The criteria follow from consideration of the total material derivative of a generalizedequation-of-state of the form P = eo, s):

Dt - s•- + o Dt (6a) or

DP_c2DDt -•-p + (PP)-• oDS Dt (6b) In theseexpressions, D/Dt is the total derivative,P is pressure, p is density,S is entropy, c is the adiabatic soundspeed, and t is time. For an isentropicprocess, S is constant and therefore the equation-of-stateis a function of densityonly, P = P(p). t0-:' This would be true in the equationsabove if c•' Dp >> •- o-•- (7) The partialderivatives on the right-handside can be expressed in termsof the fluid flow parametersthrough use of the continuity, momentum,and energyequations. As thereare subtle assumptionsabout the relativemagnitudes of forces,circum- i0-3 stancesof the particularflow underconsideration, and proper- I 5 1o ties of the fluid in the followingsection, the readeris referred to Thompson[1972, pp. 138-144]for details.For example,the Entropy,doule (]-1 K-t equationstake accountof the gravitationalfield, but it is not Fig. 2. Entropy-densityrelations for CO•_.Entropy is relative to importantif 10<< CoX'/g,where 10 is a characteristiclength, Co is S(0 K) = 0. Data are from Newitt et al. [ 1956].The single-phasefield is shown as 1½; the two-phaseregions (solid + vapor)and (liquid + the soundspeed, and g is the accelerationof gravity. It is vapor) as 2½; and the three-phaseregion boundedby the triple-point assumednegligible in the followingdiscussion. The resulting wedgeas 3½.Note the small area of the (solid + vapor) field which lies approximationfor flow to be consideredapproximately isen- below the lower triple point, at higher entropiesand lower densities tropic is than the (liquid + vapor) field. Contours of constantpressure (isobars) are shownin 10-bar incrementsat pressuresgreater than 10 bars. In the two-phase field, contours of constant mass fraction vapor IV.ul>> 1(•v) [I'+kV•'T] (8) (isopleths) are shown.Dashed line (a) intersectsthe two-phaseregion at 1-bar pressure. where u is the flow velocity, C• is the specificheat, v is the volume,T is the temperature,F is the dissipationfactor (which representsthe part of the work goinginto deformation),k is U = uluo P = lo•' rluuo •' • = T/To the thermalconductivity, V' u is the divergenceof the velocity (•u/•x + •u/•y + •u/•z in Cartesiancoordinates), and V•'T Re = pouolo/#= uolo/v Pr = #Ce/k = is the Laplacian of the temperature(P•'T/Px •' + P•'T/Py•' + •:T/•z •'in Cartesiancoordinates). The physicalinterpretation where the length10 chosen is a characteristiclength over which of this equationis that the overallrate of volumechange (on a characteristicvelocity change u0, and a characteristictemper- the left-handside of the equation)must be large comparedto ature changeTo, are obtained(in a time to), i.e., spatialdiffer- the rate of volume changedue to viscousheating (first term on entiationof a quantityis assumedto changeits magnitudeby the right) and thermal conduction(second term on the right) 10-•. We assumea Newtonian fluid and denote by # the ordi- for an isentropicprocess. The equationis most easilyconsid- nary shear viscosityand v the kinematic viscosity;p0 is the ered for generalsituations by forming nondimensionalvari- referencedensity and K is the thermal diffusivity.Re is the ables as follows: classicalReynolds number and Pr is the Prandtl number. The 1614 KIEFFERAND DELANY:ISENTROPIC DECOMPRESSION OF FLUIDS

nondimensionalform of the condition that a flow processbe Liquidsand solids. At temperaturesbelow 3000 K, Ta for approximatelyisentropic is then liquidsand solidsmay be expectedto be of order0.1 or less,so (10a) and(10b) becomeslightly less restrictive than for gases: IV.ul>> CToP + 1v:f ) (9) •e1 cvro uø•'I<< 10 (12a) (Here v0is the specificvolume, l/p0.) In general,I.ul << , (U/Uo)+ (X/1o)and, specifically,V.U = 0 for incompressible and flow. However,if the flow is compressible,then V.U • 1. We assume from here on that we are dealing with fluid flow 1 1 situationsin which compressibilityis important. If the dimen- << lo (12b) sionlessvariables have been properly chosen, then V•'• and•' are approximatelyunity and the inequalitybecomes Generalization beyond these equations is not warranted be- causeprocesses of geologicinterest can involve Reynoldsnum- bers ranging from 10-'7 to llY (at least) and Prandtl numbers 1 >> (10a) ranging from 1 to 10•'ø. The samplesof fluid flow consideredin this paper are (1) ascentof fluids or magma in volcanicerup- tions and (2) decompressionof fluids from the shock state. Therefore only four dimensionlessparameters determine Ascentof magma: Viscositiesof magmasvary from 10•' to whetheror not the flow is approximatelyisentropic: (1) the 104P for basalticor andesiticmagmas to 105-10• P for granitic ratio of inertialto viscousforces given by the Reynoldsnum- magmas.Basaltic magmasfrequently exhibit high Reynolds ber; (2) the dimensionlessvolume changegiven by T/vo' number flow when they are observedon the surface,e.g., in (•v/•T)l, = Ta, where a is the thermal expansioncoeffi- Hawaiian lava rivers where u0 can be 10a cm s-' and 10about cient;(3) the ratio of kineticto internalenergy given by 10a cm, the Reynolds number will be greater than unity for Ci,T; and(4) the ratio of viscousto thermaldiffusivity given by viscositiesless than 106,i.e., for most basalticmagmas. Shaw the Prandtl number. If it is assumed that the two terms on the [1965, p. 138] has postulated that emplacementof granitic right-handside do not cancelat all points in the flow field, magmas may also occur at Reynolds numbers appreciably (10a) reducesto the two subsidiaryconditions greater than unity, as high as 10a. Therefore most magma 1 uo2 emplacementand flow conditions(to the extentthat they can <<1 (10b) be idealizedas single-component,single-phase systems) may be approximatelyisentropic, the most notableexception being (the viscousinequality) and emplacementof granitic magmasat low Reynoldsnumbers. The ascentof volatiles with or through a magma must be 1 1 consideredseparately. If the volatilescomprise a small volume Kr << (0c) fraction of the magma, and if they are dispersedin small (the thermal inequality) vesiclesof a separateliquid or gasphase, heat transferbetween the volatile phaseand the silicatephase might make the ascent Special Cases of the volatile phase more nearly isothermalthan adiabatic. Gases. In general, for gases,Ta • 1,.so (10b) and (10c) Even thoughthe total entropy of the systemmight be constant, reduce further to the entropy of the individual phasesmight vary. Thus when the volatile fraction is small, the real ascentbehavior may lie 1 uo2 between isothermal and adiabatic. << 1 (11a) Re C•,To For large volatile contents, as in eruptions of kimberlites

and and carbonatites,or perhaps as in phreatic explosions,a dis- persedsolid and/or liquid phasemay be thermallydecoupled 1 1 from the gas phase. In this case, the dispersedphase may <melting curve reducesto become independentof the viscosity: have not, to our knowledge,been measured. It is the addition of contoursof the intensivethermody- namicvariables, pressure and massfraction, onto thesegraphs •eeI •rr I I = u-•'oK<< 10 (13) whichmakes them usefulfor solutionof problemsrelating to For K "• 10 -a cm 2 s -• and characteristic rarefaction velocities of flow processes.The lineswhich cut acrossboth figures, gener- I km s-•, this inequality holdsfor all characteristicdistances l0 ally from upper left to lower right, are isobars.The lineswhich greater than 10-9cm, and hence entropy production and vol- cut across,generally toward the upper right but with strong ume changesdue to thermal conductionare negligible(as is curvaturestoward the saturationcurve, are isoplethsof mass commonly recognizedin the assumptionof the adiabatic na- fractionvapor. Isoplethscannot be plottedthrough the wedge- ture of shock flow). shapedregion boundedby the triple point for CO• becauseof It is not equally obvious that the viscousinequality (12a) the lack of thermodynamicdata on the liquid-solidequilib- can be satisfied.A major problem in considerationof this is rium curve. our lack of knowledgeof effectiveviscosity of solidsat the high The graphsare usefulfor representationof isentropicproc- strain rates characteristicof shockprocesses, but a few limits essesand associatedchanges of stateand thermodynamicvari- can be placed on viscositieswhich would result in satisfaction ables.Isentropic processes are representedby verticallines on of the inequality (12a). For u0 • I km s-1 and Uo2/CpTo• 1, thesegraphs and isentropicdecompression by paths from top (12a) simply requiresthat v << 11Y10(10 in centimeters).If 10is to bottom. Phasechanges and densitychanges with pressure as large as I cm (which it probably is not), then v < 11YP, i.e., increasesor decreasesduring isentropicprocesses are easily the viscositymust be lessthan that of, say, andesiticmagma. seen.At any given pressure(read from the isobars) it is pos- Since10 is probably much smallerthan 1 cm, a correspondingly sibleto read the total entropy,density, and, in the two-phase lower viscosityis required. It is not apparent that the com- region, mass fraction of the phases.Furthermore, the sound pressedsolid undergoingdecompression can attain such low speedwhich by definition is the directionalderivative along viscositiesby any known mechanism(e.g., formation of theto- verticallines of constantentropy, c = (PP/Pp)s •/•, is simply morphic glass, glass lamellae, dislocation motion). On the proportional to the vertical gradient of the isobaric contours. other hand, it has not been demonstrated here that deviations These graphs are constructedon a logarithmic scalebecause from isentropeticitywould causemeasurable changes in shock the great changeof density possiblefor the vapor phasere- equation-of-stateparameters, such as free surfaceparticle ve- quires a logarithmicaxis and the definition of sound speed locities or releasevolumes. In summary, we recommendthat requiresthat both axesthen be logarithmicin orderto preserve the assumptionof isentropicflow in decompressingsolids be a direct proportionality: critically examined to determine if viscousdissipation is con- tributing significantly to entropy production and to volume changesnot accountedfor in an isentropicequation of state. 21ogc=1og(C?••P) (14) We reemphasizehere that this discussionhas been restricted For isobars,c• P = const.Therefore if the spacing(c•p) between to single-component,single-phase systems and that generaliza- isobarson an isentropeis large, c is small, and vice-versa. tion of (6)-(13) is requiredfor applicationto multicomponent By viewing these graphs as 'topo-maps,' the reader can or multiphase systems. obtain a visualimpression of the magnitudeof the soundspeed changeswhich occur acrossphase changes and as a function of 3. DESCRIPTION OF THE GRAPHS pressure,density, or massfraction. Consider two specificex- Considerfirst the entropy-densitygraphs for H•O shownin amples shown on the water-steamgraph (Figure 1): the isen- Figure 1. The saturation curve is shown as a heavy line sepa- trope (a) at 7.4 J g-• K -• and the isentrope(b) at 3 J g-• K -•. rating the two-phase region in the lower left (labeled as 2½) The first of these(a) is locatedentirely within the vapor field at from the liquid-alone and the vapor-aloneregions on the right pressuresabove I bar. The spacingsof the isobaric contours and top (labeled as 1½).At entropiesless than the critical point representtypical vapor sound speedscharacteristic of steam, entropy, the single phaseis liquid and the heavy curve repre- about 450 m s-• at l-bar pressure.At higher pressuresthe sents the boundary between liquid and (liquid + vapor) re- contours are more closely spaced (i.e., the topography is gions. At entropiesgreater than the critical point entropy, the 'steeper'), indicating that the sound speedincreases with in- single phase is vapor and the heavy curve is the boundary creasingpressure. The secondisentrope, (b), passesthrough between vapor and (vapor + liquid) regions. two regions.It is in the liquid field abovethe saturationcurve, On the entropy-densitygraph of CO• (Figure 2) the satura- and the high sound speed of liquid water (1500 m s-•) is tion curve is also shown as a heavy line separatingthe two- representedby the 'cliff' shown by the isobaric contours.The phaseand one-phaseregions. At entropiesless than the critical isentrope lies in the two-phase liquid-vapor field below the point entropy, liquid is in equilibrium with (liquid + vapor) saturationcurve; in this region,the typicallylow soundspeeds acrossthe saturation curve;at entropiesgreater than the criti- of two-phasemixtures [Kieffer, 1977b] are representedas 'flat- calpoint entropy, vapor is in equilibriumwith (vapor + liq- lands' by the isobariccontours. For example,in this region, at uid) acrossthe curve. CO• differs from H•O in that it has a 1-bar pressureand for massfractions lessthan l0 -a, the sound triple point at pressuresof geological interest: at 5.1 bars, speedis a few tens of meters per secondor less;it is 113 m s-• -56.6øC. In entropy-densityspace the triple point appearsas at massfraction 0. I. The soundspeeds represented graphically a wedge-shapedregion cutting up toward the left in Figure 2; it in the two-phaseregion are 'equilibrium sound speeds'in the is open at the top end becauseof the entropy and density terminologyused by Kieffer [1977b, p. 2901]. They are sound discontinuitiesbetween the liquid and solid phasesat the triple speedsunder conditionswhere heat and masstransfer between point. At the far upper left the solid-vaporsaturation curve of the phasesoccur rapidly enoughthat thermodynamicequilib- 'dry ice' is shown. The solid-liquid curve cannot be shown rium betweenthe liquid and vapor phasesis maintained; ef- becauseonly a portion of the melting curve of CO2 has been fectsof surfacetension on soundspeed are not includedin this measured [Bridgman, 1914] and becauseentropies and vol- representati6n. 1616 KIEFFER AND DELANY: ISENTROPIC DECOMPRESSIONOF FLUIDS

ature-pressuredata [Delanyand Helgeson, 1978, Figures 4 and 5] from 400ø-800øC. The entropy-densitycurves thus obtainedfor H•O along representativecontinental geotherms are shownin Figure 3, (L) __..,•.L•,, whichis an enlargementof the top cycleof Figure1. Consider t0ol (usingFigures 3 and 1) the isentropicascent of H•O originat- _ _ ing at variousdepths along thesegeotherms. H20 originatingat crustaldepths (to the left of the square symbolon eachgeotherm) has, according to all geotherms,an c• øøy•' entropyless than the criticalpoint entropyof 4.4 J g-• K -•. Thusupon isentropic ascent, an'y H•O, which is initiallya supercriticalfluid (SCF), ascendsthrough the supercritical I • • I • I t I •tt• CRITICAL T fieldinto the liquid(L) field and theninto the two-phasefield (L + V), at whichpoint part of the liquid vaporizes.From the isoplethsplotted, it is possibleto read the massfraction of ' •25 /t• 25O H•O transformedinto vapor by the time that the H•O reaches the surfacepressure of 1 bar, simply by readingthe mass • • •1 75 200 fraction at the point where the ascentpath crossesthe 1-bar 5 t0 isobar.For example,20-30% of waterascending from the base Entropy,Joule g-• K -• of the crust(or uppermantle according to onegeotherm) with FiB. 3. Entropy-densityrelations for H•O (enlargementofthc top an entropyof 3 J g-• K -• is transformedto vapor. A similar cycle of Figure 1) showin8four typical continental•cothcrms: •co- successionof phasechanges (SCF • L -• L + V) occursif H•O thcrm A, MocOr•or • Bo•, [1974];•cothcrm B, Boyd• •ixon originatesin the uppermantle at any depthwhich gives an [ 1973];•cothcrm C, Bird [ 1975];and •cothcrm D, Mercier ondConer initial entropyless than the criticalpoint entropyof 4.4 J g-• [1974].Th• baseof the crust(40 kin, 12 kbar) is indicatedby a square K -1. on the •cothcrmsA, C, and D. Dashedsegments on the •cothcrms representextrapolations above 800øC. Densitiesand cntropicscorrc- A differentsuccession of phasechanges is observedif H20 spondin• to the •cothcrms arc calculatedas describedin the text. originateswith an entropy higherthan the criticalpoint entropy, e.g., deeper than 100 km on the Mercier and Carter [1974] geotherm;60 km on the Bird [1975] geotherm,or 150 Figures 1 and 2 show a light dashedvertical line which km on the McGregorand Basu [1974] geotherm.The HaO is passesthrough the saturationcurve at 1-barpressure (labeled (a)): any geologicdecompression which has an entropyless

than the value at this line will result in conversion of the fluid i i i at depthinto a multiphasesystem before it reachesthe surface pressureof 1 bar. In the caseof H20 the multiphasesystem is (liquid + vapor) (the ice stability fields are not considered here); in the caseof CO2 the multiphasesystems encountered first by isentropesof decreasingentropy are: (solid+ vapor), (liquid + vapor), (solid + liquid + vapor), and (solid + GEOTHERMS vapor). 104 4. APPLICATION TO GEOLOGIC PROBLEMS To illustratethe utility of thesegraphs, we examinehere two processesassumed to occurisentropically: (1) ascentof mag- •t0 3 matic fluids(e.g., in eruptionsof volcanoesor diatremes)and (2) shockdecompression of HaO from pressuresto 250 kbar. Ascentof magmaticfluids. We have taken H•O and CO• as 10z representativeend membercompositions for magmaticfluids (H•O for normalvolcanic eruptions and, perhaps,some kim- -zOO- berlite eruptions;CO2 for kimberliteand carbonatiteerup- tions). The effectsof dissolvedgases and saltsare not included at this time for lack of thermodynamicdata. In order to specifythe initial state of H•O in the crust and mantle,we mustspecify entropies and densities of H•O along representativecontinental geotherms. (Each geotherm was cal- 10 200400 G00800 t000 t200 culatedto 1200øC.The maximumentropy obtained was 5.2 J Temper0Iure,C g-• K-•.) Entropiesand densities to 800øC(varying pressures, Fig. 4. Pressure-temperaturerelations for H20. Liquidand vapor dependingon the geotherm used) were'obtained from the fields, labeled(L) and (I/), are separatedby the saturationcurve (heavyline). Contoursof constantentropy (isentropes) are shownas internallyconsistent data of Keenanet al. [ 1969],Helgeson and solidlines, labeled in J g-• K -•. Contoursof constantvolume (iso- Kirkham[1974], and Delanyand Helgeson[1978]. Entropies chores)are shownas dashedlines labeled in cm3 g-•. Contoursof and densitiesat temperaturesbetween 800 ø and 1200øCand at constantmass fraction (isopleths) in the two-phaseregion (shown on pressuresgreater than 10 kbar were extrapolatedas follows: FiguresI and 3)cannot be shownin pressure-temperaturespace becausethe two-phase region is degeneratealong the saturation curve. (1) densities,by a linearextrapolation of volume-temperature-The four geothermsA, B, C, andD are the sameas in Figure3, and pressuredata [Delanyand Helgeson,1978, Figure 6]; (2) en- the dashedsegments are the same extrapolated regions. The Hugoniot tropies,by a quadraticleast-squares fit of entropy-temper- [fromRice and l,Ya/sh, 1957] is the sameas shownin Figure7. KIEFFERAND DELANY.' ISENTROPIC DECOMPRESSION OFFLUIDS 1617

'% '\ 405/ GEOTHERMS HGONIOT\•?

(Y3 .•1•40P•5•6.7. j0ules ½.k_.

n_z , (v)

•0 (L) •1• I I II mill •{0 II1• • • •0' •0• •03 •04 V01ume,g 3 FiB. 5. Pressure-volumerelations for H=O.The single-phase•cld is shownas 1• with the liquid(•) and vapor(F) rc•ionslabeled separately; the two-phase (liquid + vapor)•cld is shownas 2•. Contoursof constantentropy (iscntropcs) arcshown as solid lines, labeled in J •-' K-'. Contoursof constantmass fraction (isoplcths) arc shown as dashed lines. The three•cothcrms A, B, andC arcthe same as in Figure3, anddashed segments arc the same extrapolated rc•ions. Ocothcrm D, not shown,overlaps A. The Hu•oniot [from •i• •nJ •/•, 1957]is the sameas shownin Figure?. originallyin the supercriticalstate, but as it decompressesit use of more familiar thermodynamicgraphs, we show the passesinto the vapor phase (see e.g., the isentrope descending same adiabats and geothermsdiscussed above on a variety of at 5.0J g-• K-•). If theentropy is lessthan 7.4 J g-• K -•, asit graphsin Figure4 (P-T), Figure5 (P-V), and Figure6 (T-S is for the limitedtemperature range calculated, then the fluid and P-S). As far as possible,the samethermodynamic infor- passesfrom the vapor phase into the two-phase field before it mation and schematicpaths shown in Figures 1 and 3 are reachesthe surface,i.e., liquid water is obtained by con- indicatedon Figures4, 5, and 6. Details are discussedin the densationof the vaporphase upon ascent. Examination of the figure captions. massfraction isopleths obtained on expansionto 1 bar from As a specificexample of this type of analysis,we consider thesegeotherms indicates that at least30%, and as muchas the ascentof H:O from a kimberlite reservoir.McGetchin and 50%(at the criticalpoint entropy),of the vaporfraction may Ullrich [1973, p. 1844] calculatedthat H:O originatingin a condenseon ascentof mantle fluids.This phenomenonis con- kimberlite reservoirat 100-km depth, 1000øCwould ascend trary to our intuitionabout the behaviorof suchsystems. nearly isothermallyto 10-km depth (calculatedtemperature Changesin acousticproperties of thefluids across the phase 954øC).At thispressure and temperature,H:O hasan entropy changesare dramaticallyshown on Figures1 and 3 by varia- of about 5 J g-• K -• and therefore,upon further ascent, passes tionsin isobarspacings. For example,the supercriticalfluid from supercriticalfluid to vapor to (vapor + liquid) as dis- ascendingwith entropy of 3 J g-• K -• hasthe sound speed of a cussedabove. If nucleationof liquid dropletsoccurs rapidly so 'liquid'(approximately 1500 m s-•), relativelyindependent of that the systemmaintains continuous local thermodynamic pressureuntil the two-phasefield is enteredat about55 bars, equilibrium,there will be a discontinuouschange in the sound asshown by theclose ,spacing of theisobars in theliquid field. speedas the two-phaseregion is entered.(McGetchin and There is a dramaticdiscontinuous decrease in the soundspeed, Ullrich assumesupercooling of the vaporand thus do not deal as shownby the changein spacingof the isobars(to about30 with the possibletwo-phase flow problem.) It is well known m s-•) upon entry into the two-phasefield, and a further that soundspeed discontinuities give rise to perturbationsin continuousdecrease to a few metersper secondas the fluid the fluid flow [Zeldovichand Razier, 1966, chapter 1, sections decompressesto 1-bar pressure. In contrast,although a super- 17, 18, 19, chapter 11, section20]. Althoughcalculations of critical fluid ascendingwith entropyof 5 J g-• K -• has the detailsof the fluid flow propertiesare not within the scopeof soundspeed of a liquid at high pressures,the soundspeed this paper, it seemsreasonable that the fluid flow changes graduallydecreases as the pressuredecreases toward the satu- might be of sufficientmagnitude to significantlyaffect the ration curve.As the fluid entersthe two-phasefield at about natureof the kimberliteeruption and the structureof the maar 190-barpressure, there is a slight(about 10%) discontinuity in complexformed during kimberlite eruption because as much thesound speed. With furtherdecompression to 1-bar pressure as 30-50% of the vapor can condensebefore 1-bar pressureis and condensationof a liquidfrom the vaporphase, the sound reached.In particular,since H:O at 5 J g-• K -• entersthe two- speedof the two-phasemixture continuously decreases to a phasefield at a pressureof about 200 bars, structuresat or final value of about 30 m s-•. A detailed discussion of sound abovedepths where this pressureis reached(•< 1 km) might speedsin the water-steamsystem is givenby Kieffer[1977b]. showfeatures related to the transitionto two-phasefluid flow. Althoughsome of the sameconclusions can be obtainedby Becauseof the lack of thermodynamicdata for CO: at examinationof graphsof other pairsof thermodynamicvari- pressuresabove 3 kbar and temperaturesabove 150øC,it is ables,we believethat the entropy-densitygraphs are uniquein not possibleto estimatethe properties of CO: alonggeotherms their abilityto conveyinformation of interestto bothpetrolo- with sufficientaccuracy to warrant the type of analysisdone gistsand geophysicists because information about the sound for H:O alonggeotherms. Instead, we consideronly a simple speed,the firstderivative of a thermodynamicquantity, is so comparisonof the behaviorof CO: andH:O originatingunder easilyconveyed. To allow the readerto crosscheck the con- similar conditions in a kimberlite reservoir. In order to com- clusionswe have obtainedfor H:O againstthose reached by parethe behavior of CO: with H:O, weextrapolate the entropy 1618 KIEFFER AND DELANY: I SENTROPICDECOMPRESSION OF FLUIDS

t200 I I i i II t061• • • • • • I • t100 i i •11 • GEOTHERMS t000 i IIII 105 i 811 9OO i Ill ,.,"' •i III C•. 8OO •!

A/// CO 700 - i i i 600 - i I i Fil ½.m]•-• t.5 _3 __5'/7 ..10 • t5 _5oo - i 400-

300 - (L) (v) 200 I0 • • t00 I .' (L+V) / I I I I 4/i I •/4/ I I I I I X I I 0o 1 2 3 4 5 6 7 8 9 10 '0 t 2 • 4 5 6 7 8 9 Entropy,Joule cj-t K-t Entropy,Joule g-t K-t Fig. 6. (a) Temperature-entropydiagram and (b) pressure-entropydiagram for H,O. The single-phasefield is shownas 14•with liquid (L) and vapor (V) regionslabeled separately; the two-phase(liquid + vapor) field is shownas 24•. Contours of constantpressure (isobars) are shownin Figure 6a as solid linesand of constantmass fraction (isopleths) in Figures6a and 6b as dashedlines. Isochores,labeled in cm3 g-l, are shownas solid lines in Figure 6b. The four geothermsA, B, C, and D are the sameas thoseshown in Figure 3, and the dashedsegments are the sameextrapolated regions. The Hugoniot[from Rice and Walsh, 1957] is the sameas shown in Figure 7. of CO,, known at 3 kbar, 150øC [Newitt et al., 1956], to possiblestructural variations arising from differingemplace- 1000øCthrough the thermodynamicidentity ment dynamicsin H,O-dominatedversus CO,-dominated systems. T dS = dH - V dP (i5) Isentropicdecompression of CO, with lowerentropies than which, at constantpressure, gives 4.7 J g-i K-i (e.g., originatingat 1000øC,but at higherpres- suresthan consideredabove) will result in phasechanges dS dH Cp before1-bar pressure is reachedas shown in Table 1. A rough T T dT (16a) estimateof the magnitudeof the (- V dP) term in (15) suggests or that the decreasein entropydue to pressureincreases will be about0.2 J g- 1K- 1kbar- 1.Therefore for temperaturesgreater dS = dT (16b) than 1000øC(higher temperatures increase entropy) and pres- suresbetween 3 and 12 kbar (40 km), the entropy of CO, Taking Tt = 150øC(423 K), Tt = 1000øC(1273 K), Cp = const should be greater than 3 J g-i K-1. Hence for CO, fluids • 1.5 J g-1 K-1 [Newitt et al., 1956, p. 113], we obtain arisingfrom crustalpressures, only cases1, 2, 3, and4 in Table S(1000øC, 3 kbar) • 5 J g-i K-1. This entropy is, coinci- I presentplausible thermodynamic paths, although P-T condi- dentally, the same as that of H,.O under similar conditions. tionsmight be contrivedto allowcases 5 and 6. Only addi- Examination of Figure 2 showsthat CO,. with this entropy (in tional high-pressure,high-temperature thermodynamic data fact, with any entropy greaterthan 4.7 J g-i K-i, will ascend for CO, will allow specificconclusions to be drawn,but plot- isentropicallyto the surfaceentirely as a vapor phase.Unlike ting of suchdata on theseentropy-density graphs will allow H,.O, CO,. ascendingfrom 10 km, 1000øC will undergo no quick determinationof possiblephase changes during isen- phasechanges during ascentand thereforeshould not be sub- tropic decompressionof CO, from crustaland mantlepres- jected to discontinuitiesin the sound speed.This conclusion sures. holds for systemswith higher initial entropiesor for systemsin Isentropicrelease from shockcompression. The history of which the entropyincreases upon ascent(up to entropiesof 7.4 H,O undergoingshock decompressioncan be examined in j g-1 K-i for which H,.O will also remain in the vapor phase detailwith the useof entropy-densitygraphs if release'adia- entirely to the surface).H,.O and CO,. systemstherefore should bats'are assumedto be isentropesfollowing vertical paths on not be treated as qualitativelysimilar systems.Field evidence the graph and if continuousequilibrium is assumed.The H u- shouldbe soughtat kimberlite and carbonatitelocales, partic- goniotof waterfrom 40 to 450 kbar is shownon Figure7. The ularly those exposedby erosion to depthsof 1 km, to reflect Hugoniot is centeredat 1 bar, 20øC. KIEFFER AND DELANY: ISENTROPICDECOMPRESSION OF FLUIDS 1619

TABLE 1. lsentropicDecompression Paths for COo.From Crustal or pressuresdoes not appear to exist but shouldbe obtainablein Mantle Pressures to I Bar laboratory experimentswith technologycurrently available, and the experimentswould contribute a great deal to our Entropy RangeS, J g-• K -• PhaseChanges understandingof isentropicprocesses and phasechange kinet- 1. >4.7 SCF• V ics under shock conditions since sufficient thermodynamic 2. 4.7-4.2(TP1) SCF-• V-• (S + V) data existto specifywhat the equilibrium statesshould be. 3. 4.2-3.55 (CP) SCF-• V-•(VonlyatTPl)-•(S + V)or SCF -• V-• (L + V)-• (S + V) 5. CONCLUSIONS 4. 3.55-2.64 (TP2) SCF --, L -• (L + V)--, (S + V) or SCF --, L -• (L only at TP2) -• (S + V) According to the two inequalitiespresented in (10a)and 5. 2.64-1.72 (TP3) SCF -• L -• (L + S) -• (S + V) or SCF (10b), the flow of single-phasemagmatic fluids and the rare- -• L --, (S only at TP3) -• (S + V) faction expansionof low viscosityliquids and gasesmay be 6. <1.72 {SCF • L -, S}* • (S + V) consideredto be approximately isentropic. Entropy-density SCF, supercriticalfluid; S, solid; V, vapor; L, liquid. TP1 refersto graphsprovide a useful way of envisioningchanges in ther- the entropyof vapor at the triple point;TP2, the entropyof liquid at modynamic parameters for isentropic processes.Consid- the triple point; TP3, the entropyof solidat the triple point; and CP, eration of isentropicascent of H:O originatingalong various the entropy of the critical point. geothermssuggests that H:O originatingin the crust(S < 4.4 J *Unknown becausemelting curve data do not exist. g-• K -•) isentropicallydecompresses from a supercriticalfluid to a liquid to a partially vaporizedstate, whereas H:O origi- Hugoniot statesless than 50 kbar have entropiesless than natingin the uppermantle isentropically decompresses from a that required for vaporization of H:O on release to l-bar supercriticalfluid to a vapor to a partially condensedstate, pressure.A vertical releaseisentrope from sucha shockstate with at least 30% and as much as 50% of the vapor condensing. lies entirelywithin the liquid water field at all pressuresabove A similar progressionof phasechanges is predictedfor H:O I bar, as can be seenby examinationof the point where the 1- releasedfrom shockHugoniot states:vaporization on release bar isobar intersectsthe saturation curve on Figures I or 7. from shock states lower than 275 kbar, condensationupon H ugoniot statesabove 50 kbar have entropiessufficient to releasefrom higherpressures. Decompression entirely within allow partial vaporization of H:O on releaseto l-bar pressure. the vaporfield is the mostlikely behaviorof CO: originatingin The isobars and isopleths plotted on the entropy-density kimberlite locales(assuming origin at 100 km, 1000øC, and graphs allow determinationof the approximatepressure at isothermal ascent to 10 km, 1000øC, S = 5 J g-: K-•). A Whichvaporization begins and of the massfraction vaporized complexseries of phasechanges may occur upon isentropic as a function of pressureon each releaseisentrope. For exam- decompressionof CO: from entropiesbelow 4.7 J g-: K -•. ple, upon releasefrom 60 kbar, vaporizationbegins at approxi- H:O and CO: originatingunder similarconditions in kimber- mately 2-barspressure, and •6% of the liquid is vaporizedby lite systemswill not showqualitatively similar thermodynamic decompressionto I bar. Similar behavioroccurs to 275 kbar, or fluid dynamic behavior becauseH:O may partially con- from which decompressionis throughthe critical point (220.9 denseduring ascent,whereas CO: will remain in the vapor- alone field. bars, 4.4 J g-• K -•) and 50% of the liquid is vaporized upon decompression. Upon decompressionfrom pressuresgreater than 275 kbar, a different behavior is encountered.The compressedliquid 3004.0•o4S0 expandsisentropically around the critical point into the vapor '150 phase,and as decompressionbrings the material to pressures lower than the critical pressure,a part of the vapor condenses 405060• to give a liquid fraction. For example,from 300 kbar, liquid toO[- (L) beginscondensing at about 225 bars, and nearly 50% of the vapor condenses.From 500 kbar, liquid beginscondensing at 75 bars and approximately 20% of the vapor condenses.At very high shockpressures (order of a megabar),the unloading "'• O0 - isentropemay remain entirely in the vapor region at all pres-

suresgreater than 1 bar (entropiesgreater than 7.4 J g-• K-•). _ I i • • • • t t•t• CRITICALPOINT The behavior describedabove is thermodynamicallyre- quired if isentropic conditions are obeyed. The kinetics of (L+V)', nucleationand growth of vapor bubbles(below 275 kbar) and liquid droplets(above 275 kbar) may control the actualcondi- I I I I t t25 250 tions obtained.Zeldovich and Razier [1966,p. 764], in an • 225 {b0r • I • I •/ 50 • • • {75 analysisof the vaporizationof solidsby shockwaves, suggest that the rate of nucleationof vapor bubblesor liquid droplets is so slowthat the isentropesobserved would actually be those Entropy,Joule of superheatedliquids or subcooledvapors. However, there is Fig. ?. Entropy-densityrelations •or H=O (enlargemento[ the top ample evidence from laboratory equation-of-statemeasure- cycle o• Figure 1) showingthe shock Hugoniot ments and petrographicproperties of shockedrocks initially [1957]. Pressuresarc labeled in kilobars. The dashedlinc passes containing water that at least partial vaporization occurs through the critical point. The Rice and Walsh Hugoniot data Figures4, 5, 6, and ? (P, T, p, S, and •) are not internally consistent (summarizedby Kieffer et al. [1976, pp. 73-83]); the degreeto with the D•l•ny • •1•o• [1978] data, which wcrc only indirectly whichequilibrium is obtainedhas not beendetermined. Exper- basedon the Rice and Walsh data throughJ•z•'s [ 1966]cquation-o[- imental evidencefor condensationon releasefrom high shock stat•. 1620 KIEFFER AND DELANY: ISENTROPIC DECOMPRESSION OF FLUIDS

Acknowledgments. This work was accomplishedwhile S.W.K. was magma-gassystems: Geysers, maars and diatremes (extended an Alfred P. Sloan ResearchFellow, and shegratefully acknowledges abstract), in SecondInternational Kimberlite Conference,American the support of the Alfred P. Sloan Foundation. The work was also GeophysicalUnion, Washington,D.C., 1977a. supportedby NASA grant NSG 7052 and NSF grant 77-00789. We Kieffer, S. W., Sound speed in liquid-gas mixtures:Water-air and thank A. Boettcher, F. Busse,G. Schubert, and G. Ernst of UCLA water-steam,J. Geophys.Res., 82, 2895-2904, 1977b. for helpful comments. Kieffer, S. W., P. P. Phakey, and J. M. Christie, Shock processesin porous quartzite:Transmission electron microscope observations REFER!•NCES and theory, Contrib.Mineral. Petrol., 59, 41-93, 1976. MacGregor, I.D., and A. R. Basu, Thermal structureof the litho- Ahrens, T. J., and J. T. Rosenberg,Shock metamorphism:Experi- sphere:A petrologicmodel, Science,185, 1007-1011,1974. mentson quartz and plagioclase,in ShockMetamorphism of Natural McGetchin, T. R., and G. W. Ullrich, Xenoliths in maars and dia- Materials, edited by B. French and N. Short, Mono, Baltimore, tremeswith inferencesfor the moon, Mars, and Venus,J. Geophys. M d., 1968. Res., 78, 1833-1853, 1973. Batchelor, G. K., An Introduction to Fluid Dynamics, 615 pp., Cam- M cQueen,R. G., and S. P. Marsh, Equation of state for nineteen bridge University Press,London, 1967. metallic elementsfrom shock-wavemeasurements to two megabars, Bird, P., Thermal and mechanical evolution of continental con- J. Appl. Phys.,31, 1253-1269, 1960. vergencezones: Zagros and Himalayas, Ph.D. thesis, 423 pp., Mercier, J. C., and N. L. Carter, Pyroxenegeotherms, J. Geophys. Mass. Inst. of Technol., Cambridge, 1975. Res., 80, 3349-3362, 1974. Boyd, F. R., and P. H. Nixon, Structureof the upper mantle beneath Newitt, D. M., M. U. Pal, N. R. Kuloor, and J. A. W. Huggill, Car- Lesotho,Carnegie Inst. WashingtonYearb., 72, 431-449, 1973. bon dioxide, in ThermodynamicFunctions of Gases,vol. l, edited Bradley,J. N., Shock Wavesin •Chemistryand Physics,Methuen, by F. Din, Butterworths, London, 1956. London, 1962. Rice, M. H., and J. M. Walsh, Equation of state of water to 250 Bridgman,P. W., Changeof phaseunder pressure, I, The phase kilobars,J. Chem.Phys., 26, 825-830,1957. diagramof elevensubstances with especial reference to themelting . Rumble,D., The adiabaticgradient and adiabaticcompressibility, curve,Phys. Rev., 3, 153-203,1914. AnnualReport of the Directorof theGeophysical Lab, Carnegie Delany, J. M., and H. C. Helgeson,Calculation of the thermodynamic Inst. WashingtonYearb., 75, 651-655, 1976. consequencesof dehydration in subductingoceanic crust to 100 Sanford, M. T., E. Jones, and T. McGetchin, Hydrodynamicsof KB and >800øC, A mer. J. Sci., 278, 636-686, 1978. caldera-formingeruptions, Geol. Soc. A mer. A bstr. Programs,7, H-elgeson,H. C., and D. H. Kirkham, Theoreticalprediction of the 1257, 1975. thermodynamicbehavior of aqueouselectrolytes at high pressures Shaw, H. R., Comments on viscosity,crystal settlingand convection and temperatures,l, Summaryof the thermodynamic/electrostatic in granitic magmas,Amer. J. Sci., 263, 120-152, 1965. propertiesof the solvent,Amer. J. Sci., 274, 1089-1198, 1974. Thompson, P. A., Compressible-FluidDynamics, 665 pp., McGraw- Jiiza, J., An equation of statefor water and steam,Steam tables in the Hill, New York, 1972. critical region and in the range from 1,000 to 100,000bars, Rozpr. Zeldovich, Ya. B., and Yu. P. Razier, Physicsof Shock Wavesand Cesk. Akad. Ved. Rada Tech. Ved., 76, 3-121, 1966. High-TemperatureHydrodynamic Phenomena, vols. 1 and 2, Aca- Keenan, J. H., F. G. Keyes, P. G. Hill, and J. G. Moore, Steam demic, New York, 1966. Tables, 196 pp., John Wiley, New York, 1969.

Kieffer, S. W., Geysers:.. A hydrodynamicmodel for the earlystages of eruption, Geol. Soc. Amer. Abstr. Programs,7, 1146, 1975. (ReceivedMarch 27, 1978; Kieffer,S. W., Fluid.dynamicsduring eruption of water-steamand acceptedOctober 6, 1978.)