J. Phys. Chem. 1983, 87, 4180-4185
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Natural Gas Hydrate Deposits: A Review of in Situ Properties C. F. Pearson, P. M. Halleck,' P.
L. McOulre, R. Hermes, end M. Mathews
Los Alamos National Laboratory, Geophysics Group, M S J977, Los Alamos, New Mexico 87545 (Received:August 23, 1982; I n Final Form: March 23, 1982)
Hydrates of natural gas exist in nature in the Arctic regions and underneath the sea floor. If the trapped gases can be released, they are a huge potential energy resource with worldwide reservoir estimates ranging as high as lo7 trillion cu f t (TCF). The Los Alamos hydrate project has concentrated on three activities. First, we have evaluated techniques to produce gas from hydrate deposits to determine critical reservoir and production variables. Second, we have predicted physical properties of hydrate-containing sediments both for their effects on production models and to allow us to develop geophysical exploration and reservoir characterization techniques. Third, we are measuring properties of synthetic hydrate cores in the laboratory. Exploration techniques can help assess the size of potential hydrate deposits and determine which production techniques are appropriate for particular deposits. Unfortunately, so little is known about the physical properties of hydrate deposits that it is difficult to develop geophysical techniques to locate or characterize them. However, because of the strong similarity between hydrates and ice, empirical relationship between ice composition and seismic velocity, electrical resistivity, density, and heat capacity that have been established for frozen rocks may be used to estimate the physical properties of hydrate deposits. The resistivities of laboratory permafrost samples probably follow a variation of Archie's equation, pf/pt = C-TSw'-n, where pt and pf are the thawed and frozen resistivities of the sample, Tis temperature, S, is the unfrozen water content, and n and C are empirical constants. Using multiple linear regression techniques, we calculate C and n for a variety of lithologic types. The compressional wave velocities of partially frozen sediments (Vp) are related to the velocity of the matrix (V,) and to the liquid (V,) and solids (V,) phases present in the pores by the well-known three-phase rule: l / V p = [$(S,)/V,] + [+(lS,)/ V,/ + [(l- 4)/ Val, where is the porosity. Clearly, both the resistivities and seismic velocities are functions of the unfrozen water content; however, resistivities are more sensitive to changes in S,, varying by as much as three orders of magnitude, which may allow the use of electrical resistivity measurements to estimate the amount of hydrate in place. We estimated the unfrozen water content, assuming that the dissolved salt in the pore water is concentrated as a brine phase as the hydrates form. Using this technique, we estimated the brine content as a function of depth, assuming several temperature gradients and pore water salinities. We find that hydrate-bearing zones are characterized by high seismic velocities and electrical resistivities compared to unfrozen sediments or permafrost zones. Surprisingly, hydrates may also have higher resistivities than permafrost deposits, due to lower amounts of unfrozen water contained by the hydrate deposits. The presence of hydrates also tends to lower the heat capacities and densities of sediments, but these effects are comparatively small and not useful for exploration.
+
Introduction Hydrates of methane and other constituents of natural gas, which have been known for years as a laboratory curiosity and an occasional problem in gas transmission limes, have only recently been discovered in nature.' Gas hydrate reserves are restricted to permafrost regions in the Arctic on land and to the continental slopes and rises offshore. While naturally occurring methane hydrate deposits are new to science, they are fairly widespread in nature' with some estimates of the worldwide reservoir ranging as high as 1.5 X 10l8 m3,2enough to stimulate interest in hydrates as a possible energy source. Unfortunately, very little is known about the physical properties of natural gas hydrate deposits in nature, making their detection by remote geophysical surveys very difficult. In this paper we review experimental sonic and resistivity measurements on hydrates, hydrate-bearing sediment, and permafrost. We conclude that hydrate layers are characterized by anomolously high sonic velocities and resistivities, both of which are functions of the amount of liquid water associated with the hydrates in the rock matrix. Using an analogy between hydrate-bearing sediments and permafrost, we propose simple quantitative relationships between liquid water content and the elec(1) Chersky, N.; Makogon, Y. Oil Gas Inuest. 1970,10,82. (2) Barraclough, B. L. Los Alamos National Laboratory Report LA8368-MS, June, 1980. 0022-365418312087-4180$01.50/0
trical resistivities and sonic velocities of the deposits. Finally, we present electrical and sonic well-logging data that substantiates the basic conclusions of this paper.
Resistivities There are no reported laboratory resistivity measurements on hydrate-containing sediments; liowever, there is an extensive body of literature on the electrical properties of partly frozen sediments. Because ice and hydrates3 are electrical insulators the resistivities of permafrost and hydrate deposits are largely controlled by the unfrozen brine inclusions which are nearly always found in partially frozen sediments. Archie's law4 p = ap,$-"S,-" is an empirical relationship between water content and the resistivity of water-saturated sediments. Here p is the resistivity of the sediments, p, is the pore water resistivity, is porosity, S, is the fraction of the porosity occupied by liquid water, and a, m,and n are empirically derived parameters. This equation also applies to rocks where the pore spaces are partially filled with ice or hydrates. However, as the amount of liquid water decreases, S, and pw are both reduced, S, because some of the available pore
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(3) Davidson, D. W. 'Clathrate Hydrates in Water, a Comprehensive Treatise"; Franks, F., Ed.; Plenum: New York, 1973; Vol. 2, pp 115-234. (4) Archie, C. E.Trans. AIME 1942,146, 54-62.
0 1983 American Chemical Society
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space is now filled with a solid nonconductor, and pw because the dissolved salts are concentrated in the remaining unfrozen water. If the brine is not very near saturation, the effect of hydrate or ice formation on pw is relatively easy to quantify because an increase in salt concentration causes a linear decrease in p,. Because hydrates and ice exclude all of the dissolved salts as they form, the salt concentration of the brine inclusions is inversely proportional to the volume fraction of liquid water, assuming that the sediments were initially water saturated. In addition, the resistivity of aqueous solutions increases exponentially with decreasing temperatures. Including both the temperature and concentration effects, the resistivity of a partially frozen brine at temperature Tis thus proportional to (C)TSwwhere C is a constant. Substituting this relationship into Archie's equation and dividing by the resistivity a t 0 "C, we find that the ratio of frozen (pf) and thawed ( p t ) resistivities is Pf/Pt = C-T
s l-n W
(1)
In all these calculations we have assumed that the hydrate-containing sediments were fuUy saturated with water. Hydrate deposits in nature, however, are also quite likely to coexist with free natural gas. The effect of natural gas on sonic velocities is quite difficult to quantify but partial gas saturation in Archie's equation can be easily taken into account. If gas exists in the pore spaces of a rock it has the effect of further reducing S,, the percent water saturation in eq 1, because there are now two nonconducting phases in the pores, gas and hydrates. Thus, if free gas is present it will tend to increase resistivities by a factor (1 - SJ" where S, is the percent of the porosity filled with gas.
Estimating the Empirical Constants in Archie's Equation To use this relationship, one must know the empirical constants C and n. Fortunately, Pandit and King? Hoyer et al.! Desai and Moore,' and Dumass have all examined the relationship between formation temperatures, pore water resistivities, and formation resistivities. In all four studies, rock or soil samples, ranging in composition from sandstones and limestones to fine sands and silts, were saturated with dilute brines with known composition. Resistivities of these samples were then measured repeatedly as the temperature was varied from above the freezing point to -21.1 "C, the eutetic of the water NaCl system. Because the samples were saturated with saline water, the pore water did not completely freeze when the temperature was decreased below the freezing point. As ice formed in the pores, the dissolved salts concentrate in the remaining liquid or brine phase. However, ice can only form until the brine concentration reaches a critical value where the freezing point of the brine equals the temperature. Using the well-known freezing point depression curves for the water NaCl system, we estimated the concentration of the brine phase and the amount of unfrozen water present in the sample. We then used this data to test eq 1 and estimate the value of n and C, the empirical constants. Clearly this technique assumes that the salinity measured outside the rock is equal to the in situ salinity. ( 5 ) Pandit, B. I.; King, M. S. Can. J. Earth Sci. 1979, 16, 1566-80. (6) Hoyer, W. A.; Simmons,S. 0.; Spann, M. M.; Watson, A. T. Trans.
SOC.R o c . Well Log Anal. 1976, 1-15. (7)Desai, K. P.; Moore, E. J. Trans. Soc, Prof. Well Log Anal. 1967, 1-27. (8)Dumas,M. C. M.S. Thesis, Colorado School of Mines, Golden, CO, 1962.
TABLE I: Calculation of Empirical Parameters a lithology sandstone limestone unconsolidated material pooled estimate
n 2.1661 1.834 1.715
1.051 1.051 1.051
1.9386
1.057
C
high low salinity salinity
1.761 1.299
0.551
2.264 1.674 0.710
1.305
Unfortunately, interactions between pore water and grain surfaces can cause exchangeable ions contained in the rock matrix to go into solution, forming an ion halo around each grain boundary. However, these extra ions have an important effect on the water resistivities only if the dissolved ion concentration is initially quite low. In addition, surface chemistry effects can cause pore water to freeze at temperatures below those calculated from the freezing point depression curves. However, Patterson and Smithg show that, for soil samples containing silt and sand, surface chemistry effects depress the freezing point by less than 1 "C, although the effect can be considerably greater if significant amounts of clay are present. We estimated the empirical parameters C and n using multivariate linear regression techniques. In order to estimate n and C using linear regression, we inverted eq 1 and took the log of both sides. log ( p t / p f ) = T log C + ( n - 1) log S, (2) Our dependent variable was thus log (pt/pf) and the first two independent variables were T and log S,. Using this simple regression model, we found that n is equal to 2.01 and C is equal to 1.05. In order to improve the fit between the data and the log linear relationship suggested by eq 2, we calculated an intercept in our regression models. Physically this intercept (a in Table I) has the effect of adding a multiplicative constant to the right side of eq 1. To examine the effects of lithology, we expanded this simple model using qualitative or indicator variables. Ihdicator variables can take on one of two values, depending on whether or not the sample is a member of a particular qualitative class. We used two qualitative variables in the expanded model to test the effect of lithology. The first compared consolidated and unconsolidated sediments, and the second compared limestones with sands and sandstones. We also included cross products between the two qualitative variables and log S, because this allowed us to independently estimate n for the three lithologic classes. Values of the empirical parameters calculated from this regression model are listed in Table I. Figure 1illustrates the scatter of the data about the regression model.
Sonic Velocities The most accurate estimates of the sonic velocities of natural gas hydrates thus far are calculated compressional velocities published by WhalleylO and since confirmed experimentally by Whiffen et al." Following that approach, the compressional wave velocity (V,) is related to the adiabatic Young's modulus E and Poission's ratio v by 1-v (3) = E p ( l + v ) ( l - 2v) is the density.12 However, K , = (1 - 2v)/E and
v:
where p
(9) Patterson, D. E.; Smith, M. W. Can. Geotech J. 1981,18, 131-44.
(10) Whalley, E. J. Geophys. Res. 1980, 85, 2539-42.
(11)Whiffen, B. L.;Kiefte, H.; Clouter, M. J. Geophys. Res. Lett.
1982, 9,645-8.
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Pearson et al.
1.0A
Sandstone
-
c
0
0.01
L
0.10
10
Log (Swl
Log ( S W I 10 D
Pooled Data
c
%
-p‘m
0.10
A
v
0.010 01
0.10
1 .o
0.01
,
0
0.10
11
Log (SWI
Log (SWI Flgure 1. P ,/P, plotted vs. S, for the three lithologic divisions considered in the multiple regression analysis. We have compensated for the effect of temperature in these data. The regression lines have slope n - 1 and intercept log a . The scatter of the data about the lines illustrates the fit to the regression model. K , / K T = 1 - cu2TV/KTCp)where a is the volume thermal expansivity, K, and KT are the adiabatic and isothermal compressibilities, T is temperature, V is molar volume, and C, is the heat capacity at constant pressure. Substituting into eq 3, we obtain
TABLE 11: Velocities of Gas Hydrates
guest C3H8 CH,
co
2
H*S C2H6 C,H,
The ratio of the compressional wave velocities of ice ( VPJ and hydrates ( v p h ) is just the ratio of eq 3 evaluated for ice and hydrates, respectively:
Following Whalley, we assume that Poisson’s ratio and the volume thermal expansivity are equal in hydrates and ice. The isothermal compressibility (KT) of hydrates has not been measured; however, we can estimate it using the following formula
(12)Landau, L.D.; Lifschitz, E. M. ‘Theory of Elasticity”; Pergamon: New York, 1963.
+ CH,
V,, kmis 100% 80% occupancy occupancy 3.92 3.73 3.35 3.47 3.80 3.78
3.98 3.78 3.46 3.56 3.90
where N is the number of bonds between water molecules, r is the mean bond length, and K 2 is the harmonic force constant of the bond. Because all these quantities can be estimated from spectral or X-ray diffraction data, we can use eq 5 to estimate KT. Because the interactions between the guest molecules and the lattice are much weaker than the bonds between the water molecules in the lattice, K~ does not depend significantly on the guest molecule or the occupancy ratio. The heat capacity of ice is about 37.7 J / ( K mol), and this is probably about equal to the heat capacity of empty hydrate lattice. However, the heat capacity of hydrates is strongly influenced by the guest molecule. Because the guest molecule in the case of methane can rotate freely and vibrate against the crystal lattice, it has a rotational heat capacity of (3/2)Rand a vibrational heat capacity of 3R. So, if we neglect any contribution to the heat capacity due to intramolecular vibrations in the guest, the total heat capacity of the hydrate is 37 + (9)/2RLwhere L is the number of moles of guest/mole of water. Table I1 lists
Natural Gas Hydrate Deposits
The Journal of Physical Chemistry, Vol. 87, No. 21, 1983
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6
WPA NaCl
I
.lo
0
< 100
10
20
Temperature IC1 Flgure 2. Geothermal gradient for land hydrate deposits plotted on Kobayashi’s curves.15 The surface temperature is -6 O C . Depth (m)
typical compressional wave velocities for hydrates. Note that V, can vary 19% depending on composition and occupancy ratio. Seismic velocities of propane and methane hydrate have been measured by Pandit and King13and Whiffen et al.ll Pandit and King measured compressional wave velocities of a maximum of 3.2 km/s, or only 84% of ice for propane hydrate, much lower than the Whalley model would predict. The reasons for this discrepancy are not clear; however, the densities of the hydrate samples used by Pandit and King are 750 kg/m3, only 80% of the values calculated by Whalley.l0 This may imply that the samples contained liquid propane inclusions. Wiffen et al. measured the sonic velocity of methane hydrate using Brillouin spectroscopy. They measured a compressional wave velocity of approximately 91% the velocity of ice. This measurement is in agreement with Whalley’s calculations. The compressional velocity of a mixture of ice and unfrozen brine in the interstitial pore spaces of a rock can be estimated by using a three-phase time-averaged equation, first proposed by Timur14 and since tested by several other authors. The compressional velocity (V ) is related to the velocity of ice (VJ, the velocity of the trine inclusions (V,,), and the velocity of the solid matrix (V,) by
where Si is the fraction of ice in the rock. Because of the similarities between the seismic velocities of ice and hydrates, this equation can probably be used to calculate the velocity of a mixture of hydrates and brine in sedimentary rock.
Physical Properties of Gas Hydrate Deposits and Ice in Situ Natural gas hydrate deposits and permafrost consist of porous sedimentary rock whose pore spaces are partially or completely filled with hydrates or ice. The physical properties of these deposits thus depend on the properties of the rock matrix, the hydrates, and the amount and composition of the unfrozen water or gas that also may be ~
Flgure 3. Velocities and resistivities vs. depth for the geothermal gradients shown in Figure 2. Temperature (F) M
J
O
O
W
,wo
W
Wi% NaCl
50 Deg C/km ___ 70 Deg C/km .lo
i
10
20
Temperature IC1 Flgure 4. Geothermal gradients for ocean bottom hydrate deposit plotted on Kobayashi’scurves.15 Deposits are at a depth of 1 km. The surface temperature is 2 O C .
present in the pores. A liquid water phase will nearly always be present because of ions dissolved in the pore waters. We calculate S, for hydrate solutions by plotting the pressures and temperatures associated with points along the geothermal gradient on a family of hydrate stability curves for different NaCl solutions (after Kobayashi et al.ls) and then using the curves to estimate brine concentration at equilibrium with methane hydrates. We used a very similar method to estimate the brine concentrations in permafrost. Once we calculate the brine phase concentrations, we can easily calculate S, by assuming that none of the dissolved ions diffuse out of the partially frozen sediments. Once S, as a function of depth is known, we can calculate the resistivities and sonic velocities using eq 1and 6. The calculation is shown graphically in Figures 2 and 3. Note that hydrate-bearing sediments have a
~~~~~~~~
(13) Pandit, B. I.; King, M. S. “Proceeding of the Fourth Canadian Permafrost Conference”; National Research Council of Canada: 1981. (14) Timur, Geophysics, 1968, 33, 584-95.
(16) Kobayashi, R.; Withrow, H. J.; Williams, G. B.; Katz, D. L. Roc., Natl. Gasoline Assoc. Am. 1961, 27-31.
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.4001
,,/’
-1;r.I
I
1945
0
2
-- -
4
6
C
-100
I
-200
5n -300
-400
-50°Clkm --- 70OCikm
I I I 1
I
/
/ -500 1
1957
well.
Resistivity (Pf/Pt)
-E
Depth Iml
Flgure 6, Well-logging data from a deep sea drilling program CDSDP
7O0C/km
-500
Pearson et al.
2
I
3
Velocity (km/sl
Flgure 5. Velocities and resistivities for the geothermal gradients shown in Figure 4.
resistivity higher than both thawed sediments (pf = 10 Q m) and permafrost. A similar calculation for marine deposits is shown in Figures 4 and 5. Sonic velocities also increase when hydrates are present; however, proportionately, the increase is much smaller than is the case with resistivities. Also sonic velocities are not as strongly dependent on the unfrozen water content as the electrical resistivity and thus have a relatively constant high value if hydrates are present. This effect is’potentially important because high sonic velocities are thus a qualitative indicator of the presence of hydrates while electrical resistivities quantitatively indicate the amount of hydrates in place. Well-Logging Data While most of our knowledge of the physical properties of hydrates are the result of laboratory and theoretical studies, a few in situ physical property measurements of known hydrate-containing zones which were made using well-logging techniques. During well-logging measurements, instruments are used to measure various physical properties of the rocks surrounding a wellbore. Because
of the propritary nature of many of these measurements, very little well-logging data have been reported in the literature. The limited amount of well-logging data available tends to qualitatively support our model of the physical properties of hydrate-bearing sediments. Figure 6 shows sonic and resistivity logs from a known hydrate-bearing formation which was encountered by the Deep Sea Drilling Project (DSDP) while drilling a well (DSDP hole no. 570) off the Pacific coast of Guatemala (DSDP Staff 1982). While the entire interval below 1950 m contained interstitial hydrates, the zone from 1952.6 to 1955 m contained a massive layer consisting almost entirely of solid hydrate. Note that the resistivities and sonic velocities of the formation are very high in the massive hydrate-bearing zone, the electrical resistivities increasing by as much as two orders of magnitude. While hydrates were recovered during drilling in the entire region below 1950 m, only the massive hydrate zone shows a large increase in sonic velocity although high resistivity are found over a somewhat larger region. Because both the sonic velocities and the electrical resistivities (Archie’s law) are functions of the porosity, we can use the type of data shown in Figure 6 to develop two independent estimates of the sediment porosity. By comparing these two estimates one can test the validity of the three phase rule and Archie’s equation. Using well-logging data for both land and marine sources and a crossplot analysis technique,16 MathewP demonstrates that hydrate-containing sediments do follow Archie’s equation.
Conclusion Very little published data exist on the physical properties of natural gas hydrates. However, because of the strong similarity between hydrates and ice, empirical relationships originally developed for permafrost are used to estimate the physical properties of hydrate deposits. Four major conclusions are drawn from this study. (1) Hydrate deposits will have much higher resistivities and sonic velocities than similar unfrozen sediments. Hydrates will also tend to reduce the thermal conductivities and densities of sediments, but these effects are comparatively small (usually less than 5%)lSso seismic and electrical methods will be the most useful geophysical (16) Schlumberger Ltd. “Log Interpretation: Principles”; Schlumberger Ltd.: New York, 1972; Vol. 1. (17) Mathews, M. A. Logging Characteristics of Methane Hydrate. 1982 Meeting Society of Profeasional Well Log Analysts, Denver Chapter, Dec 15, 1982. (18) Pearson, Loa Alamos National Laboratory Report LA-9422-MS, June, 1982.
J, Phys. Chem. 1983, 87, 4185-4190
techniques when exploring for hydrate deposits. (2) Surprisingly, hydrate deposits that form above the 0 "C isotherm will often have higher resistivities than nearby permafrost deposits. This is because hydrate deposits, which form near the bottom of the permafrost zone, usually contain smaller amounts of unfrozen water than permafrost. This characteristic may be useful in distinguishing ice and hydrates. (3) All of the physical properties of hydrate deposits are functions of the unfrozen water content. This is particularly true of the electrical resistivity that can vary as much as three orders of magnitude. This effect is potentially very useful because it may be possible to calculate the unfrozen water content in hydrate deposits from geophysical measurements. (4) Of the physical properties examined in this report, sonic velocities and resistivities are the most strongly affected by the presence of hydrates. However, unlike resistivities, sonic velocities are not strongly affected by the amount of unfrozen water present. Thus, high sonic velocities provide a qualitative indication of the presence of hydrates whereas resistivities provide a quantitative indication of the amount of hydrate present. Clearly both measurements are potentially valuable in evaluating hy-
4185
drate deposits. Because the resistivities of natural gas hydrate-bearing sediments are strongly affected by the amount of free water present in the deposit, resistivity measurements may be very useful in evaluating hydrate deposits and determining what production methods have the best chance of producing the natural gas. McGuirelgproposes two methods of producing natural gas from hydrate deposits. In the first method, warm water injected into a hydrate deposit dissociates the hydrates and the resulting free gas is recovered from nearby production wells. Unfortunately in order to work, this method requires quite high permeabilities. In a second method (which may produce natural hydrate deposits) reduction of pore pressure causes the hydrates to dissociate. Because the low permeabilities of hydratebearing deposits are caused by hydrates clogging the pore spaces, hydrates with low resistivities, and a high unfrozen water content, may have relatively high permeabilities and thus are candidates for the first production method whereas hydrate deposits with high resistivity may be candidates for the second method. (19) McGuire, P. Los Alamos National Laboratory Report LA-9102MS, Nov, 1981.
Thermal Conductivity of the I c e Polymorphs and the I c e Clathrates M. W. C. Dharma-wardana Division of Physics, National Research Council of Canada, Ottawa K I A OR6, Canada (Received: August 23, f982; I n Final Form: March 17, 1983)
We examine the thermal conductivities of the various forms of ice, and the ice clathrates, noting that previous authors have concluded that these thermal conductivities display no physically meaningful, simple, relationships among them. However, we show that the Debye mean free path formula used within a reasonable scaling model is capable of predicting the observed thermal conductivities of the ices, deuterated ices, and ice clathrates, to good accuracy. The model is least satisfactory for cubic ice. In the case of the clathrates, the large number of atoms per unit cell is sufficient to quench the conductivities to small values. The "anomalous" temperature dependence of the clathrate thermal conductivities can also be explained without ad hoc recourse to special mechanisms or models of glassy solids, when all the factors in the mean free path formula are carefully treated.
I. Introduction Water crystallizes in possibly ten different forms' in which the bonding is essentially identical. The thermal conductivity X(T) of ordinary ice (ice Ih) seems to be well described by Slack's theoretical analysk2 His analysis depends on the abundant experimental information on phonon dispersion and thermal expansion data available for ice Ih. The theoretical approach is of the LeibfriedSchlomann, Julian t ~ p e . ~When ? ~ we consider other types of ice, the available experimental data do not allow a detailed analysis as done by Slack.2 However, given the fact that the different crystal types of ice are variations of the basic wurtzite type lattice which retain the same chemical bonding, viz., hydrogen bonds, one would expect some form of a simple trend to exist among the X(T) of the various polymorphs even if quantitative accuracy is not obtained. As seen from Table I, (1) P. V. Hobbs, "Ice Physics", Clarendon, Oxford, 1974, Chapter 1. (2) G . A. Slack, Phys. Rev. B, 22, 3065 (1980). (3) G. A. Slack, Solid State Phys., 34, 1 (1979). (4) P. G . Klemens, Solid State Phys., 7 , 1 (1958). 0022-3654/83/2087-4185$0 1.5010
the different polymorphs of ice show a variation in the density and in the number of water molecules (n) contained in the unit cell. Some of the ices show long-range proton ordering when the double minima in the hydrogen bonds become appreciably asymmetric. Proton ordering seems to play a special role in ice VI11 which contains two nonbonded but interpenetrating sublattices where the antiferromagnetic proton ordering couples the two sublattices. In all the forms of ice (except ice VIII), the proton subsystem appears to be mostly decoupled from the oxygen lattice, at least as far as lattice vibrations are concerned. Hence the discussion will be concerned with the number of oxygen atoms ( n ) per primitive unit cell. Roufosse and Klemens5 (see also Slack3) have given a theoretical analysis to show that X ( T ) for similar crystals should vary essentially as n-2/3,if thermal conduction by optic phonons could be disregarded. Hence, given the similarity of bond type etc. found in the ice polymorphs, one would expect that X(T) of the ices would follow the n-2/3scaling of Roufosse and Klemens. However, Ross, (5) M. Roufosse and P. G. Klemens, Phys. Rev. B , 7 , 5379 (1968).
0 1983 American Chemical Society
