ENCYCLOPEDIA OF EARTH SCIENCES
PHYSICAL PROPE~
Because most of the earth is not accessible for di-rect observation, it is important to understand asmuch as possible about the physical properties ofrocks. In many geological settings or studies (e.g.,
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those concerning oil exploration, crustal structure,mantle convection), the only information we havecomes from indirect observations of gravity andmagnetic anomalies and/or seismic velocities. Inorder to make an assessment of the composition orstructure of the subsurface for inaccessible re-gions, we need to know as much as possible aboutthe physical properties of the rocks that composethe subsurface. The rock properties most oftenstudied include the density, elastic constants (bulkmodulus, shear modulus, Poisson's ratio), thermalconductivity, coefficient of thermal expansion,magnetic susceptibility, electrical conductivity,electrical resistivity, and viscosity. A detailed dis-cussion of all these properties is beyond the scopeof this entry, so we will focus on the elastic con-stants, which can be directly related to the seismicvelocity, density, coefficient of thermal expansion,and viscosity.
Elastic Constant and Seismic Velocities
An isotropic solid is one in which the response ofthe solid to an imposed stress is independent of theorientation of the solid. Only cubic minerals aretruly isotropic. Because of the nature of the crys-tal structure of minerals, almost all minerals areslightly anisotropic, with as much as a 10 percentdifference in their elastic constant depending onthe orientation. However, an aggregate of crystalsin various random orientations is often assumed tobe isotropic. We begin our discussion of elastic con-stants with the isotropic case.
For an isotropic, homogeneous solid there is alinear relation between stress and strain, under thelimit of infinitesimal strains. The limit of infinitesi-mal strains is a reasonable approximation for thedeformation caused by seismic waves as they travelthrough the earth. For an isotropic system, thenumber of independent elastic constants reducesto two. We can express the relationship betweenstress (0-) and strain (8) as
(1)
where oij is equal to 1 if i =1=j and to zero if i =1=j,and».. and J.Lare the two independent Lame con-stants. The elastic properties of an isotropic mate-rial can be described by elastic moduli: the shearmodulus, J.L,and the bulk modulus or incompressi-bility, K, defined as
K =3».. + '2J.L
3('2)
We can measure the bulk modulus with the follow-ing experiment. Given a uniform cube of an iso-tropic material with a volume of V, at a startingpressure P, we increase the pressure to a value ofP + AP and measure the change in volume AV.The bulk modulus is given by
AP dPK = -V.
AV = d In V(3)
Another useful parameter is Poisson's ratio, v. Wecan measure Poisson's ratio with the following sim-ple experiment. We take a long, cylindrical rodand subject it to a uniform stress along its axis ofsymmetry. Poisson's ratio is defined as minus theratio of the strain normal to the stress axis to thestrain along the stress axis (i.e., ratio of thinning toelongation or thickening to contraction):
v= -- (4)
Poisson's ratio is useful in earth science because itcan be expressed as a function of the ratio vp/vs ofthe velocities of pressure (vp) and shear (vs) seismicwaves. We can write the seismic velocities in termsof the elastic moduli or the Lame parameters:
(5)
(6)
Hence,
(7)
In the crust, we often find vp = v'3 v., which corre-sponds to v = 0.25. In this case, »..= J.L,so there isonly one independent elastic modulus. Such a ma-terial is called a Cauchy solid.
Since the mid-1960s, the seismic velocity of alarger number of rocks and minerals has beenmeasured. These have provided a basic under-standing of many factors that influence seismic ve-
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locity and attenuation of rocks believed to be abun-dant constituents of the continental lithosphere.
The effect of confining pressure on velocities.has been reported in a number of investigations.An example of data for a typical crystalline rock(Twin Sisters Dunite, an olivine-rich igneous rock)is shown in Figure 1. The characteristic shape ofthe curve of velocity versus pressure is attributedto the closure of microcracks. As can be seen fromFigure 1, much of the closure takes place over thefirst 100 mega pascals (MPa). Velocities measuredin crystalline rocks at pressures up to 3,000 MPademonstrate that changes in velocity with pressuredo not approach those of single crystals until theconfining pressure is above 1,000 MPa. Even atthese high pressures, solid contact between themineral components is probably only approximatebecause some porosity has originated from anistro-pic thermal contraction of the minerals.
Fewer data are available on the influence oftemperature of rock velocities. It has been wellknown that the application of temperature to arock at atmospheric pressure results in the creationof cracks that often permanently damage the rock.Thus, reliable measurements of the temperaturederivatives of velocity must be obtained at confin-ing pressures high enough to prevent crack forma-tion. In general, pressures of 200 MPa are suffi-cient for temperature measurements to 300°C.
A wide variety of techniques have been em-ployed to measure the influence of temperatUre onrock velocities. An example of data showing theinfluence of temperature on velocities is shown in
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8.6
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... .. ..u51
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8.0 .
CUNITETWIN SISTERS RANGE
WASHINGTON
7.6a 300 600 900 1200 1500 1800 2100 2400 2700 3000
PRESSURE MPa
Figure 1. Laboratory measurement of seismic Pvelocity as a function of pressure for TwinSisters Dunite (olivine-rich igneous rock).
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Figure 2. Laboratory measurement of seismicvelocity as a function of temperature forserpentinite (metamorphic basalt or gabbro)from the Mid-Atlantic Ridge.
400
Figure 2. Increasing temperature decreases ve-locities, whereas increasing pressure increasesvelocities. Thus, in a homogeneous crustal re-gion, velocity gradients depend primarily on thegeothermal gradient. The change of velocitywith depth is given by
dV(av
)dP
(av
)dT
& = ap T dz + aT p & (8)
where z is depth, T is temperature, and P is pres-sure. For regions with normal geothermal gradi-ents (25-40°C/km), the change in compressionalvelocity with depth dV p/dz is close to zero. How-ever, in the high heat-flow regions, crustal velocityreversals are expected if compositional changeswith depth are minimal.
. Amplitudes of seismic waves decrease with in-creasing distance from their source. This propertyis called seismic wave attenuation. Seismic wave at-tenuation has great potential as a tool to yield abetter understanding of the anelastic properties,and hence the physical state, of rocks in the earth'sinterior.
.
The three parameters most often reported asthe attenuation are the seismic quality factor Q,also referred to as the specific attenuation Q-l, theattenuation coefficient, a, and the logarithmic de-crement, S. These are related for low-loss materials(Q> 10) by
(9)
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where V is the phase velocity and f is the fre-quency. In both the field and laboratory, difficul-ties arise in separating the intrinsic dissipation ofthe rock, that is, processes by which seismic energyis converted into heat, from geometric spreading,transmission losses, scattering, and other factors.Nevertheless, the utilization of laboratory attenua-tion measurements to tie seismic data to the anelas-tic properties of rocks is promising, and the refine-ment of laboratory techniques and the theoryconcerning the mechanisms involved has yieldedand will continue to supply valuable insights intothe structure and composition of the continentalcrust and upper mantle.
All investigations have found that Q increaseswith increasing confining pressure. Laboratorymeasurements show a sharp increase in Q at lowpressures, which then levels off at high pressures, aresponse similar to that observed for velocities.The form of the Q versus P curve is generally at-tributed, therefore, to the closure of microcracks.As with velocity measurements, few researchershave studied attenuation as a function of tempera-ture for rocks of the lithosphere. At temperaturesbelow the boiling point of the rock's volatiles, Qappears to be temperature-independent, andabove this Q increases, indicating outgassing ofpore fluids and/or thermal cracking. At the onsetof partial melting, Q decreases.
Seismic velocities are the most often used prop-erties of rocks. Seismologists have produced one-dimensional (radial) models of seismic velocityfrom the surface to the center of the earth (Figure3). While seismic velocities generally increase grad-ually as a function of pressure (depth), an abruptjump in seismic velocity over a small pressure(depth) range usually indicates a change in chemi-cal composition or a change in the solid phase ofthe material. For example, the jump in vp and Vsatapproximately 400-km depth corresponds with thepressure and temperatures of the olivine [(Mg,FehSi04] to wadslayite phase change observed inthe laboratory. Seismic velocity models place animportant constraint on the composition of the in-terior of the earth. Because the seismic velocities ofminerals such as olivine, pyroxene, garnet, andperovskite behave differently as a function ofdepth, one can attempt to match the seismicvelocity curve with simple models of mantlecomposition. At 2,900-km depth, there is anabrupt change in both density and seismicvelocity. This marks the chemical boundary
NORTH ARM-BAY OF ISLANDSVp(km/sec)
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Figure 3. Seismic velocities as a function ofdepth from the Preliminary Reference EarthModel (PREM).
between the silicate mantle and the liquid ironouter core.
In the 1990s, seismic velocities have played animportant role in understanding the structure andcomposition of the earth's crust. An excellent ex-ample is offered by combining oceanic crustal seis-mic studies with laboratory measurements of thevelocity structure of ophiolites. Ophiolites, on landexposures of oceanic crust and upper mantle, con-tain a stratified sequence of rocks which, from topto bottom, consists of marine sediments, pillow ba-salts, dikes, gabbros, and peridotites. The labora-tory-determined velocity structure of ophiolites(see Plate 32) matches very well with field measure-ments of velocities in oceanic basins. The initialrapid increase of velocity extending to depths of 3km originates from a decrease in porosity withdepth. At depths between 4 and 7 km, velocitiesare fairly constant. This region contains mainlygabbro and metagabbroic rocks. The rapid in-crease in velocity encountered at 7 km is similar tofield observations of velocity changes at the Mo-horoviCic discontinuity. The ultramafic sections of
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ophiolites show 6-8 percent anisotropy originatingfrom preferred orientation of olivine and pyrox-ene. This anisotropy correlates well with uppermantle seismic anisotropy measured with marineseismiC surveys.
Velocities in single crystals of the common rock-forming minerals vary significantly with propaga-tion direction. In general, for a given propagationdirection in anisotropic media such as single crys-tals, there are three waves, one compressional andtwo shear. Their vibration directions form an or-thogonal set, which usually are not parallel or per-pendicular to the propagation direction. Thepropagation of waves is related to the single-crystalelastic constants through the Christoffel equation,which gives the three velocities for each directionas roots of a cubic equation. Details of wave propa-gation are related to the crystal symmetry. Mostmetamorphic rocks and some cumulate igneousrocks have preferred mineral orientations that areusually related to cleavage, foliation, or banding. Itfollows that many rocks are seismically anisotropicin a manner similar to single crystals. Compres-sional wave velocities vary with propagation direc-tion and two shear waves travel in a given directionthrough the rock with different velocities. This lat-ter property of anisotropic rocks, termed shearwave splitting, has recently been observed in sev-eral crustal and upper mantle regions.
Density and the Coefficient ofThermal Expansion
The density of rock is important for understand-ing gravity anomalies. Density depends on thechemical composition of the rock, the mineralstructure, and the void space between minerals.Gravity anomalies are the result of mass anomaliesat the surface of the earth (for example, changes inelevation) as well as mass anomalies within theearth (for example, mineral deposits or sedimen-tary basins). The relationship between a massanomaly, dm, and the resulting gravity anomaly,dg, is
dg = G dmlr2 = Gdp V/rZ (10)
where V is the volume of the mass anomaly and dpis its density difference from the surroundingrocks, G is the universal gravitational constant(6.67 X 10-11 m3/kg S2), and r is the distance be-
tween the mass anomaly and the point where grav-ity is being measured. If we knew the distributionof mass anomalies within the earth exactly, wecould integrate equation (10) over the whole earthand calculate the observed gravity everywhere. Inroutine gravity surveys, it is not uncommon tomeasure anomalies as small as 0.00001 percent ofthe average earth gravity to detect subsurfacestructure. From equation (10) it is clear that thereis a trade-off between the size of the body (V) andthe density difference (dp). Thus, the better weknow the density of the subsurface material, themore accurately we can estimate the size of thebody of interest.
The influence of temperature on density is alsoimportant in under~tanding crustal and mantle dy-namics. For all earth materials, the density of arock or mineral increases with increasing pressureand the density decreases with increasing tempera-ture. The relationship between density and tem-perature is measured by the coefficient of thermalexpansion, a, which is defined as
peT) = po( I - aT) (11)
The coefficient of thermal expansion for themineral olivine is approximately 3 x 1O-5/0C.Thus, a temperature change of 1,000° changes thedensity of the material by 3 percent. This densitydifference provides the force that drives mantleconvection and, ultimately, plate tectonics. Densityis also affected by pressure (mainly at mantledepths), changes in phase, and changes in compo-sition. Thus, convection in the mantle is more com-plicated than a simple, uniform fluid as in mosttank experiments of convection in the laboratory.
Viscosity
While earthquakes provide evidence that the earthbehaves elastically on short timescales, on thetimescaIes of postglacial rebound or mantle con-vection, the earth behaves like a viscous fluid. Theviscosity of the mantle is one of the most importantproperties for understanding mantle flow, but it isalso one of the most poorly constrained properties.
Laboratory measurements of deformation indi-cate that the viscosity of upper mantle mineralssuch as olivine is a strong function of temperatUre,pressure, grain size, and stress (Karato and \Vu,1993). For a temperature increase of 100 K, the
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viscosity decreases by an order of magnitude atconstant stress. An increase of deviatoric stress by afactor of two decreases the viscosity by an order ofmagnitude. Other factors, such as partial pressureof oxygen and water content, may also have impor-tant effects, but are less well studied.
Because of the large difference between timeand space scales in the laboratory and the mantle,estimates of mantle viscosity based on modelinglarge-scale geophysical observations (e.g., postgla-cial rebound) play an important role in our under-standing of mantle viscosity structure; however,viscosity models deduced from these observationsare not unique and require additional assump-tions. For example, in modeling postglacial re-bound, the thickness and extent of the ice sheetthrough time cannot be determined directly. Theuncertainty in the ice sheet model adds to the com-plexity of the problem. In addition, the theoreticalmodels are often greatly simplified to keep themmathematically tractable.
From these studies, several classes of viscositymodels appear, one with essentially a uniform vis-cosity throughout the mantle and one with a lowviscosity channel beneath the lithosphere and ahigher viscosity in the lower mantle. At present, itseems that the observational constraints are notstrong enough to exclude one of these models.Perhaps more worrysome, however, is that evenless is known about the effect of lateral viscosityvariations on surface observables.
Bibliography
ANDERSON,D. L. Theory oj the Earth. Boston 1989.KARATO,S. 1., and P. Wu. "Rheology of the Upper Man-
tle: A Synthesis." Science 2670 (1993 .
SCOTT KINGNIKOLAS CHRISTENSEN
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