geophysics today (a survey of the field as the journal celebrates its 75th anniversary) || 3....

Chapter 3

Borehole Geophysics and Rock Properties

chapter03.indd 33chapter03.indd 33 9/11/10 11:55 AM9/11/10 11:55 AM

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35

Rock-physics diagnostics of depositional texture, diagenetic alterations,and reservoir heterogeneity in high-porosity siliciclastic sedimentsand rocks — A review of selected models and suggested work flows

Per Avseth1, Tapan Mukerji2, Gary Mavko3, and Jack Dvorkin3

ABSTRACT

Rock physics has evolved to become a key tool of reservoirgeophysics and an integral part of quantitative seismic inter-pretation. Rock-physics models adapted to site-specific dep-osition and compaction help extrapolate rock propertiesaway from existing wells and, by so doing, facilitate early ex-ploration and appraisal. Many rock-physics models are avail-able, each having benefits and limitations. During early ex-ploration or in frontier areas, direct use of empirical site-spe-cific models may not help because such models have beencreated for areas with possibly different geologic settings. Atthe same time, more advanced physics-based models can betoo uncertain because of poor constraints on the input param-eters without well or laboratory data to adjust these parame-ters. A hybrid modeling approach has been applied to silici-clastic unconsolidated to moderately consolidated sedi-ments. Specifically in sandstones, a physical-contact theory�such as the Hertz-Mindlin model� combined with theoreticalelastic bounds �such as the Hashin-Shtrikman bounds� mim-ics the elastic signatures of porosity reduction associatedwith depositional sorting and diagenesis, including mechani-cal and chemical compaction. For soft shales, the seismicproperties are quantified as a function of pore shape and oc-currence of cracklike porosity with low aspect ratios. A workflow for upscaling interbedded sands and shales usingBackus averaging follows the hybrid modeling of individualhomogenous sand and shale layers. Different models can beincluded in site-specific rock-physics templates and used forquantitative interpretation of lithology, porosity, and porefluids from well-log and seismic data.

INTRODUCTION

Rock physics provides a link between geologic reservoir parame-ters �e.g., porosity, clay content, sorting, lithology, saturation� andseismic properties �e.g., acoustic impedance, P-wave/S-wave veloc-ity ratio VP /VS, bulk density, and elastic moduli�. Rock-physicsmodels can be used to interpret observed sonic and seismic veloci-ties in terms of reservoir parameters or to extrapolate beyond theavailable data range to examine certain what-if scenarios, such asplausible fluid or lithology variations. Along this line, rock physicscan be used to forecast seismic response to assumed reservoir andoverburden properties and conditions.

Rock-physics models also help infer �diagnose� rock texture ofsandstones or shales if we know porosity and elastic-wave velocity.Such diagnostics assume that, e.g., if velocity-porosity data fall on atheoretical cemented-rock trend, the rock is cemented. This seem-ingly circular logic helps us better understand rock properties be-yond elasticity. For example, if rock is cemented, one may expecthigher strength than in uncemented rock of the same porosity andmineralogy. It is also likely that the permeability �at the same porosi-ty� of cemented rock is higher than that of uncemented rock. This ef-fect has a simple physical explanation: Loose pore-filling material�or noncontact cement� increases the specific surface area and thusdecreases permeability, as opposed to pore-filling material occurringas contact cement �Bosl et al., 1998; Dvorkin and Brevik, 1999�.

Local geologic trends can help constrain rock-physics models.Such trends can be split into two types: compactional and deposi-tional. If we can predict the expected change in seismic response as afunction of depositional environment or burial depth, we will in-crease our ability to locate hydrocarbons, especially where little orno well-log information is available. Understanding the geologicconstraints in an area of exploration reduces the range of expectedvariability in rock properties and hence reduces the uncertainties inseismic reservoir prediction.

Manuscript received by the Editor 11 January 2010; revised manuscript received 11 May 2010; published online 14 September 2010.1Odin Petroleum, Bergen, Norway. E-mail: [email protected] University, Energy Resource Engineering Department, Stanford, California, U.S.A. E-mail: [email protected] University, Department of Geophysics, Stanford Rock Physics Laboratory, Stanford, California, U.S.A. E-mail: [email protected]; dvorkin@

stanford.edu.© 2010 Society of Exploration Geophysicists.All rights reserved.

GEOPHYSICS, VOL. 75, NO. 5 �SEPTEMBER-OCTOBER 2010�; P. 75A31–75A47, 20 FIGS.10.1190/1.3483770


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36 GEOPHYSICS Today

Several workers �e.g., Vernik and Nur, 1992; Dvorkin and Nur,1996; Anselmetti and Eberli, 1997; Florez, 2005� recognize that theslope of velocity-porosity �or impedance-porosity� trends in sand-stones is highly variable and depends largely on the geologic processthat controls porosity �Figure 1�. Relatively steep velocity-porositytrends for sandstones are representative of porosity variations con-trolled by diagenesis, i.e., porosity reduction from pressure solution,compaction, and cementation. Hence, we often see steep velocity-porosity trends when examining data spanning a great range ofdepths or ages. The classical empirical trends of Wyllie et al. �1956�,Raymer et al. �1980�, Han �1986�, and Raiga-Clemenceau et al.�1988� show versions of the steep, diagenetically controlled veloci-ty-porosity trend. On the other hand, porosity change resulting fromvariations in sorting and clay content tend to yield much flatter ve-locity-porosity trends, meaning that porosity controlled by sedimen-tation is generally expected to yield flatter trends, which we some-times refer to as depositional trends. Data sets from narrow depthranges or individual reservoirs often �though not always� show thisbehavior �Avseth et al., 2005�.

A range of different models can be used in rock-physics analysis�Dræge et al., 2006a; Mavko et al., 2009�. Every model has certainadvantages and limitations. We follow Box and Draper �1987� in be-lieving that “All models are wrong, but some are useful.” Most rock-physics models relevant to the scope of this paper are aimed at de-scribing relations between measurable seismic parameters and rock/fluid properties. Although our intent is not to review all models ex-haustively, many fall within three general classes: theoretical, em-pirical, and heuristic. We provide a condensed discussion ofdifferent modeling approaches as well as a more detailed analysis ofour hybrid approach, where we combine contact theory �granularmedia� or pore-shape-constrained models with heuristic bounds topredict sedimentary microstructure and geologic trends from elasticproperties.

In particular, diagenetic trends, which connect newly depositedsediment on the Reuss elastic bound with the mineral point at zeroporosity, often can be described accurately using the upper Hashin-Shtrikman �HS� bound �Hashin and Shtrikman, 1963�. In fact, wesometimes refer to it as a modified upper HS bound because we use it

to describe a mixture of the newly deposited sediment at critical po-rosity with additional mineral instead of describing a mixture ofmineral and pore fluid. A slight improvement over the modified up-per HS bound as a diagenetic trend for sands can be obtained bysteepening the high-porosity end. An effective way to do this is touse Dvorkin’s model �Dvorkin and Nur, 1996� for cementing graincontacts. The contact-cement model captures the rapid increase inelastic stiffness of a sand with little change in porosity as the first bitsof cement are added. The depositional or sorting trends can be de-scribed with a series of modified HS lower bounds. By combining acontact-cement model with such sorting trends, we can create linesof constant depth but variable texture, sorting, and/or clay content�Avseth et al. �2000� also refer to these as constant-cement lines�. Fi-nally, we demonstrate how we can use these models to quantify therock texture of sandstones based on North Sea well data.

To interpret the observed seismic contrast, we also need to knowthe rock properties of shales. These require more complex rock-physics models to capture cracklike pore shapes, intrinsic mi-croporosity, and diagenetic mineral transitions. An upscaling ap-proach is discussed to model the effective rock-physics properties ofsands interbedded with shales.

ROCK-PHYSICS MODELS — AN OVERVIEW

Theoretical models

Theoretical models are primarily continuum-mechanics approxi-mations of the elastic, viscoelastic, or poroelastic properties ofrocks. Among the most famous are the poroelastic models of Biot�1956�, who was among the first to formulate the coupled mechani-cal behavior of a porous rock embedded with a linear viscous fluid.The Biot equations reduce to the famous Gassmann �1951� relationsat zero frequency; hence, we often refer to Biot-Gassmann fluid sub-stitution. Elastic models tend to be bounds, inclusion models, dis-placement-discontinuty models, contact models, computationalmodels, and transformations.

Bounds such as the Voigt-Reuss �VR� or HS are the silent heroesof rock models. They are extremely robust and relatively free of ide-alizations and approximations, other than representing the rock as anelastic composite. Originally, bounds were treated only as describ-ing the limits of elastic behavior; some geophysicists even consid-ered them of limited usefulness. However, they have turned out to bevaluable mixing laws that allow accurate interpolation of sorting andcementing trends and are the rigorously correct equations to describesuspensions and fluid mixtures.

Inclusion models usually approximate the rock as an elastic solidcontaining cavities �inclusions�, representing the pore space. Be-cause the inclusion cavities are more compliant than solid mineral,they have the effect of reducing the overall elastic stiffness of therock in an isotropic or anisotropic way. Most inclusion models as-sume that the pore cavities are ellipsoidal or penny shaped �Eshelby,1957; Walsh, 1965; Kuster and Toksöz, 1974; O’Connell and Budi-ansky, 1974, 1977; Cheng, 1978, 1993; Hudson, 1980, 1981, 1990;Crampin, 1984; Johansen et al., 2002�. Berryman �1980� generalizesthe self-consistent description so that the pores and grains are con-sidered to be ellipsoidal inclusions in the composite. Inclusion mod-els have contributed tremendous insights as elastic analogs of rockbehavior. However, their limitation to idealized �and unrealistic�pore geometries makes comparing the models to actual pore micro-geometry difficult.

6000

5000

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ave

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city

(m/s

)

0 20 40 60 80 100

Reuss bound

Voigt avg.

Processes thatgive sediment strength:compaction, stress,diagenesis

Newly depositedclean sand

Suspensions

SuspensionsSand-clay mixt.SandClay-free sandstoneClay-bearing sandstone

Porosity (%)

Figure 1. P-wave velocity versus porosity for a variety of water-satu-rated sediments, compared with the Voigt-Reuss bounds. Data fromYin �1992�, Han �1986�, and Hamilton �1956�.


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Chapter 3: Anisotropy 37

Displacement discontinuity or excess compliance models havebeen used to model the long-wavelength elastic behavior of rockswith aligned fractures �Schoenberg and Douma, 1988; Sayers andKachanov, 1995; Schoenberg and Sayers, 1995�. These do not as-sume elliptical inclusions but approximate the fractures as infinitesi-mally thin planes of displacement discontinuity, parameterized bytheir normal and shear compliances, representing the extra compli-ance of the fractures relative to the background rock. Use of thesemodels requires estimating the fracture compliances �or stiffnesses�empirically or with model-based methods such as penny-shapedcracks.

Contact models approximate the rock as a collection of separategrains whose elastic properties are determined by the deformabilityand stiffness of their grain-to-grain contacts. Most of these �Digby,1981; Walton, 1987; Norris and Johnson, 1997; Makse et al., 1999;Bachrach and Avseth, 2008� are based on the Hertz-Mindlin �HM;Mindlin, 1949� solution for the elastic behavior of two elasticspheres in contact. The key parameters determining the stiffness ofthe rock are the elastic moduli of the spherical grains and the area ofgrain contact, which results from the deformability of the grains un-der pressure. Dvorkin and Nur �1996� describe the effect of addingsmall amounts of mineral cement at the contacts of spherical grains.As with inclusion models, spherical contact models are useful elasticanalogs of soft sediments, but they suffer from their extremely ideal-ized geometries. They are not easy to extend to realistic grain shapesor distributions of grain size. Furthermore, the most rigorous part ofthe contact models is the formal description of a single grain-to-grain contact. To extrapolate this to a random packing of spheres re-quires sweeping assumptions about the number of contacts per grainand the distribution of contact forces throughout the composite.

Computational models are a fairly recent phenomenon �e.g.,Bakke and Øren, 1997; Keehm, 2003; Knackstedt et al., 2009�. Inthese, the actual grain-pore microgeometry is determined by carefulthin section or computed-tomography �CT� scan imaging. This ge-ometry is represented by a grid, and the elastic, poroelastic, or vis-coelastic behavior is computed using finite-element, finite-differ-ence, or discrete-element modeling. Clear advantages of these mod-els are freedom from geometric idealizations and the ability to elasti-cally quantify features observed in thin sections or 3D real rock im-ages.

Transformations include models such as the Gassmann �1951� re-lations for fluid substitution, which are relatively free of geometricassumptions. The Gassmann relations take the measured VP and VS

at one fluid state and predict the VP and VS at another fluid state. Ber-ryman and Milton �1991� present a geometry-independent schemeto predict fluid substitution in a composite of two porous media hav-ing separate mineral and dry-frame moduli. Mavko et al. �1995� de-rive a geometry-independent transformation to take hydrostatic ve-locity versus pressure data and predict stress-induced anisotropy.Mavko and Jizba �1991� present a transformation of measured dryvelocity versus pressure to predict velocity versus frequency in flu-id-saturated rocks.

Empirical models

Empirical models do not require much explanation. Generally, theapproach is to assume some function form and then to determine co-efficients by calibrating a regression to data. Some of the best-known models are Han’s �1986� regressions for velocity-porosity-

clay behavior in sandstones, the Greenberg-Castagna �1992� VP /VS

relations, and the Gardner et al. �1974� VP-density relations.Empirical relations are sometimes disguised as theoretical. For

example, the popular model of Xu and White �1995� for VS predic-tion in shaly sands is based on the Kuster-Toksöz ellipsoidal inclu-sion formulation. One unknown aspect ratio is assigned to representthe compliant clay pore space, and a second unknown aspect ratio isassigned to represent the stiffer sand pore space. These aspect ratiosare determined empirically by calibrating to training data. In otherwords, this is an empirical model in which the function form of theregression is taken from an elastic model. It is useful to rememberthat all empirical relations involve this two-step process of a model-ing step to determine the functional form, followed by a calibrationstep to determine the empirical coefficients.

Heuristic models

Heuristic models are what we might call pseudotheoretical.Aheu-ristic model uses intuitive, though nonrigorous, means to argue whyvarious parameters should be related in a certain way. The best-known heuristic rock-physics model is the Wyllie time average, re-lating velocity to porosity: 1 /V�� /Vfluid� �1�� /Vmineral. Atface value, it looks as if there might be some physics involved. How-ever, the time-average equation is equivalent to a straight-ray, zero-wavelength approximation, which makes no sense when modelingwavelengths that are very long relative to grains and pores. The Wyl-lie equation is sometimes a useful heuristic description of clean, con-solidated, water-saturated rocks, but it is certainly not a theoreticallyjustifiable one.

Other very useful heuristic models are the use of the HS upper andlower bounds to describe cementing and sorting trends. Certainly theHS curves are rigorous bounds for mixtures of different phases.However, we use the bounds as interpolators to connect the mineralmoduli at zero porosity, with moduli of well-sorted end members atcritical porosity. We give heuristic arguments justifying why an up-per-bound equation might be expected to describe cementing, whichis the stiffest way to add mineral to a sand, and why a lower-boundequation might be expected to describe sorting. However, we are un-able to derive these bounds from first principles.

Bound-filling models

The bound-filling models that follow provide simple equationsfor families of curves spanning the range between upper and lowerbounds on elastic moduli and can be used to describe the stiffness ofrocks that fall in the range between upper and lower bounds. Themodified HS and modified Voigt averages are useful depth-trendlines for sand and chalk sediments. The bounding-average method�BAM� provides a heuristic fluid-substitution strategy that seems towork best at high frequency �Marion, 1990�. The isoframe model al-lows one to estimate the moduli of rocks composed of a consolidatedgrain framework with inclusions of nonload-bearing grain suspen-sions. These models are based on isotropic linear elasticity.All of thebound-filling models discussed here contain at least some heuristicelements, such as the interpretation of the modified upper bounds.

The VR and HS bounds yield precise limits on the maximum andminimum possible values for the effective bulk and shear moduli ofan isotropic, linear elastic composite. Specifically, for a mixture oftwo components the HS bounds can be written as


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38 GEOPHYSICS Today

KHS��K1�f2

�K2�K1��1� f1�K1�4

3�m��1 , �1�

�HS��1

�f2

��2��1��1� f1��1��m

6�9Km�8�m

Km�2�m��1 ,

�2�

where subscripts 1 and 2 refer to properties of the two componentshaving bulk moduli K1 and K2, shear moduli �1 and �2, and volumefractions f1 and f2. Most commonly, these bounds are applied to de-scribe mixtures of mineral and pore fluid, as illustrated in Figure 2.

Equations 1 and 2 yield the upper bound when Km and �m are themaximum bulk and shear moduli of the individual constituents andthe lower bound when Km and �m are the minimum bulk and shearmoduli of the constituents. The maximum �minimum� shear modu-lus might come from a different constituent than the maximum �min-imum� bulk modulus. For example, this would be the case for a mix-ture of calcite �K�71; ��30 GPa� and quartz �K�37; ��45 GPa�.

The modified HS bounds also use equations 1 and 2. However,with modified bounds, the constituent end members are selected dif-ferently, such as a mineral mixed with a fluid-solid suspension �Fig-ure 2� or a stiffly packed sediment mixed with a fluid-solid suspen-sion. The critical-porosity model �Nur et al., 1991, 1995� identifies acritical porosity �c that separates load-bearing sediments at porosi-ties � � �c and suspensions at porosities � � �c. Modified HS �orVR� equations can be constructed to describe mixtures of mineraland the unconsolidated fluid-solid suspension at critical porosity.Different values of �c produce a family of curves between the upperand lower bounds. The modified upper HS curve has been observedempirically to be a useful trend line describing, for example, how theelastic moduli of clean sandstones evolve from deposition throughcompaction and cementation �Gal et al., 1998�. The modified upperHS curve, constructed as such, is not a rigorous bound on the elasticproperties of clean sand, although sandstone moduli are almost al-ways observed to lie on or below it.

The Voigt-Reuss-Hill average is an estimate of elastic modulusdefined to lie exactly halfway between the Voigt upper and Reusslower bounds �Figure 3�. A similar estimate can be constructed to liehalfway between the upper and lower HS bounds �Figure 3�. Thesetwo estimates have little practical value except when the constituentend members are elastically similar, as with a mixture of mineralswithout pore space. In this case, an average of upper and lowerbounds yields a useful estimate of the average mineral moduli.

BAM uses the position of porosity/modulus data between thebounds as an indication of rock stiffness. In Figure 4, the HS upperand lower bounds are displayed for mixtures of mineral and water.The data point A lies a distance d above the lower bound; D is thespacing between the bounds at the same porosity. In BAM, it is as-sumed that the ratio d /D remains constant if the pore fluid in the rockis changed, without changing the pore geometry or the stiffness ofthe dry frame. Though not theoretically justified, BAM sometimesgives a reasonable estimate of high-frequency fluid-substitution be-havior.

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Porosity

Hashin-Shtrikmanlower bound

Hashin-Shtrikman average

Voigt

Voigt-Reuss-Hill

Hashin-Shtrikmanupper bound

Figure 3. Hashin-Shtrikman and Voigt-Reuss bounds for bulk modu-lus in a quartz-water system. The Voigt-Reuss-Hill curve is an aver-age of the Voigt upper and Reuss lower bounds. The Hashin-Shtrik-man average curve is an average of the Hashin-Shtrikman upper andlower bounds.

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Porosity



D

A

d

Figure 4. The bounding average method. The position of a data pointA, described as d /D relative to bounds, is assumed to be a measure ofthe pore stiffness.

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Modified upperHashin-Shtrikmanbound

Critical porosity

Figure 2. Hashin-Shtrikman and modified Hashin-Shtrikmanbounds for bulk modulus in a quartz-water system.


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Fabricius �2003� has introduced the isoframe model to describebehavior between the modified upper and lower HS bounds. In thismodel, the modified upper HS curve is assumed to describe the trendof sediments that become progressively more compacted and ce-mented as they trend away from the lower Reuss average �Figure 5�— an empirical result. It is assumed that these rocks contain onlygrains that are load bearing. Sediments that fall below the modifiedupper bound are assumed to contain inclusions of grain-fluid sus-pensions, in which the grains are not load bearing.Afamily of curvescan be generated �Figure 5�, computed as an upper HS mix of frame,taken from the modified upper bound, and suspension, taken fromthe lower Reuss bound. IF is the volume fraction of load-bearingframe, and 1 — IF is the fraction of suspension. The isoframe modu-lus at each porosity is computed from the frame and suspensionmoduli at the same porosity. The calculation is done separately forbulk and shear moduli, from which the P-wave modulus can be cal-culated.

The bulk modulus K of an elastic porous medium can be ex-pressed as

1

K�

1

Kmin�

�

K̃�

, �3�

where Kmin is the mineral bulk modulus and K̃� is the saturated porespace stiffness, given by

K̃� �K� �KminKfluid

Kmin�Kfluid�4�

�Mavko et al., 2009�, where Kfluid is the bulk modulus of the pore flu-id and K� is the dry rock pore-space stiffness defined in terms of thepore volume v and confining stress � c:

1

K�

�1

v

�v�� c

. �5�

Figure 6 shows a plot of bulk modulus versus porosity with con-tours of constant K�. A large K� indicates a stiff pore space, and asmall K� indicates a soft pore space. The value K� �0 correspondsto a suspension.

All of the bound-filling models provided here contain at leastsome heuristic elements, such as the interpretation of the modifiedupper bounds. The different families of curves are parameterized bydifferent quantities with different physical interpretation, e.g., criti-cal porosity �c, the fraction of load-bearing frame IF, or the pore-space stiffness K�. Though somewhat heuristic, these models pro-vide a simple and practical way to model a wide range of velocity-porosity trends.

Our hybrid approach

We have found a hybrid combination of theoretical, empirical,and heuristic models very useful to describe the rock-physics prop-erties of high-porosity clastic sediments. In this sense, we find our-selves thinking more as engineers than physicists — the models thatwork best in practice, for prediction and interpretation purposes,may not be the models founded on the most advanced physical theo-ries.

It started with Han’s �1986� empirical discovery that the relation-ship between velocity and porosity in shaly sands could be well de-scribed by a set of parallel contours of constant clay. Amos Nur�1992� notes that each of these contours have high- and low-porosityintercepts with a clear physical interpretation: in the limit of zero po-rosity, any model should rigorously take on the properties of puremineral; and in the limit of high porosity �the critical porosity�, wehave a suspension that is rigorously modeled with a lower-boundequation. Eventually, Han’s contours have been replaced by modi-fied upper bounds, partly because they fit the data better over a largerange of porosities and partly because we could defend them heuris-tically. We have come to understand that these modified upperbounds describe the diagenetic or cementation trend for sedimentaryrocks.

A modified lower bound is an excellent description of the veloci-ty-porosity sorting trend. Again, this is more of an empirical obser-vation, aesthetically symmetric to the modified upper bound but notrigorously defendable. Dvorkin and Nur �1996� introduces the fria-ble-sand model, which uses a theoretical elastic contact model to de-scribe clean, well-sorted sands combined with a modified lowerbound to interpolate these to lower-porosity, poorly sorted sands.

In summary, we have found the rock-physics modeling approachpresented in this paper to be useful in high-porosity sand-shale envi-ronments. We avoid overmodeling with too much theory that de-100

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0

Porosity

P-w

ave

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ulus

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a)

0 0.1 0.2 0.3 0.4 0.5 0.6

Modified upperHashin-Shtrikman

Criticalporosity

Reuss

IF = 1

0.8

0.60.4

0.2

Figure 5. The isoframe model. The modified upper Hashin-Shtrik-man curve is assumed to describe a strong frame of grains in goodcontact. The Reuss average curve describes a suspension of grains influid. Each isoframe curve is a Hashin-Shtrikman mix of a fractionIF of frame with �1� IF� of suspension.


Reuss

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00 0.2 0.4 0.6 0.8 1

Porosity (volume fraction)

K/K

min

K /K = 0.1phi min

0.4

0.3

0.2

Figure 6. Normalized bulk modulus versus pressure, showing con-tours of constant K�.


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40 GEOPHYSICS Today

pends on model parameters which follow from mathematical conge-niality rather than from geologic processes. We have also found itdisadvantageous to become attached to meticulously derived theo-retical models that can never approach the complexity of nature. Atthe same time, we like to honor physical principles because theymake the models universal. As time goes on, it almost seems that wethrow away more equations and replace them with clever uses of var-ious bounds. Another important driver in our approach is the desireto discover, understand, and quantify elastic properties as a functionof geologic processes. Although not proving this heuristic approachis correct, the supporting argument for the approach is that the effec-tive medium path chosen from geologic concepts does not violatethe physics of the mixture. An important principle states that if amodel falls within elastic bounds, it is realizable �Norris, 1985�.

BRIEF “LIFE STORY” OFA CLASTIC SEDIMENT

Elastic bounds provide a framework for understanding the acous-tic properties of sediments. Figure 1 shows P-wave velocity versusporosity for a variety of water-saturated sediments, ranging fromocean-bottom suspensions to consolidated sandstones. The Voigtand Reuss bounds, computed for mixtures of quartz and water, areshown for comparison. �Strictly speaking, the bounds describe theallowable range for elastic moduli. When the corresponding P- andS-wave velocities are derived from these moduli, it is common to re-fer to them as the upper and lower bounds on velocity.�

Before deposition, sediments exist as particles suspended in water�or air�. As such, their acoustic properties must fall on the Reuss av-erage of mineral mixed with fluids. When the sediments are first de-posited on the water bottom, we expect their properties to still lie on�or near� the Reuss average as long as they are weak and unconsoli-dated. Their porosity position along the Reuss average is determinedby the geometry of the particle packing. Clean, well-sorted sandswill be deposited with porosities near 40%. Poorly sorted sands willbe deposited along the Reuss average at lower porosities. Chalkswill be deposited at high initial porosities, 55%–65%. We some-times call this porosity of the newly deposited sediment the criticalporosity �Nur, 1992�. Upon burial, the various processes that give thesediment stiffness and strength — effective stress, compaction, andcementing — must move the sediments off of the Reuss bound. Withincreasing diagenesis, the rock properties fall along steep trajecto-ries that extend upward from the Reuss bound at critical porosity andtoward the mineral end point at zero porosity. We see below thatthese diagenetic trends can be described once again using thebounds.

ROCK-PHYSICS MODELING OF SEDIMENTARYMICROSTRUCTURE IN HIGH-POROSITY

SANDSTONES

If we want to predict the seismic velocities of a rock, knowingonly the porosity, mineralogic composition, and the elastic moduli ofthe mineral constituents, we can at best predict the upper and lowerbounds of seismic velocities. However, if we know the geometric de-tails of how the mineral grains and pores are arranged relative toeach other, we can predict more exact seismic properties. Severalmodels account for the microstructure and texture of rocks, andthese in principle allow us to go the other way: to predict the type ofrock and microstructure from seismic velocities.

The rock-physics diagnostics technique was introduced by Dvor-kin and Nur �1996� as a means to infer rock microstructure from ve-locity-porosity relations. This diagnostic is conducted by adjustingan effective-medium theoretical model curve to a trend in the data,assuming that the microstructure of the sediment matches that usedin the model. We review three heuristic hybrid models that have beenused to describe the velocity-porosity-pressure behavior of medium— to high-porosity sediments and rocks: �1� the friable-sand model,�2� the contact-cement model, and �3� the constant-cement model.

A very effective approach is to begin by defining the elastic prop-erties of the end members. At zero porosity, the rock must have theproperties of mineral. At the high-porosity limit, the elastic proper-ties are determined by elastic-contact theory. Then, we interpolatebetween these two end members using upper or lower HS bounds.The upper bound explains the theoretical stiffest way to mix load-bearing grains and pore-filling material, and the lower bound ex-plains the theoretical softest way to mix these. Hence, we have foundthat the upper bound is a good representation of contact cement,whereas the lower bound accurately describes the effect of sorting.Rocks with very little contact cement �a few percent� are not well de-scribed by the HS upper bound because there is a large stiffening ef-fect during the very initial porosity reduction as cement fills in themicrocracks between the contacts. Then it is dangerous to interpo-late between the high-porosity and zero-porosity end members. Wetherefore include a high-porosity contact-cement model that takesinto account the initial cementation effect.

The friable-sand model

Dvorkin and Nur �1996� introduce two theoretical models forhigh-porosity sands. The friable-sand model, or the unconsolidatedline, describes how the velocity-porosity relation changes as thesorting deteriorates. The well-sorted end member is represented as awell-sorted packing of similar grains whose elastic properties are de-termined by the elasticity at the grain contacts. The well-sorted endmember typically has a critical porosity around 40%. The friable-sand model represents poorly sorted sands as the well-sorted endmember modified with additional smaller grains deposited in thepore space. These additional grains deteriorate sorting, decrease po-rosity, and only slightly increase rock stiffness �Figure 7�.

The elastic moduli of the dry, well-sorted end member at criticalporosity are modeled as an elastic sphere pack subject to confining

Ela

stic

mod

ulus

0.30 0.35 0.40

Porosity

Figure 7. Schematic of the friable-sand model and correspondingsedimentologic variation. Elastic modulus increases slightly withdeteriorating sorting and associated increasing amount of pore-fill-ing material.


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pressure. These moduli are given by the HM theory as

KHM��n2�1��c�2�2

18�2�1��2 P�1/3

, �6�

�HM�5�4�

5�2��3n2�1��c�2�2

2�2�1��2 P�1/3

, �7�

where KHM and GHM are the dry-rock bulk and shear moduli, respec-tively, at �c �i.e., depositional porosity�; P is the effective pressure�i.e., the difference between the confining pressure and the pore pres-sure�; � and � are the shear modulus and Poisson’s ratio of the solidphase; and n is the coordination number �the average number of con-tacts per grain�.

Poisson’s ratio can be expressed in terms of the bulk K and shear �moduli as follows:

� �3K�2�

2�3K��. �8�

Assuming hydrostatic pressure, effective pressure versus depth isobtained with the following formula:

P�g�0

Z

�b� fl�dz, �9�

where g is the gravity constant and where b and fluid are the bulkdensity and fluid density, respectively, at a given depth z.

The coordination number n depends on porosity, as shown byMurphy �1982�. The relationship between coordination number andporosity can be approximated by the following empirical equation:

n�20�34� �14�2. �10�

Hence, for a porosity of 0.4, n�8.6.The other end point in the friable-sand model is at zero porosity

and has the bulk K and shear � moduli of the mineral. Moduli of thepoorly sorted sands with porosities between zero and �c are interpo-lated between the mineral point and the well-sorted end member us-ing the lower HS bound. One heuristic argument for this is that add-ing small grains passively in the pore space is the softest way to addmineral to the well-sorted sands; the lower-bound equation is alwaysthe softest way to mix two phases. Another argument follows fromFigure 7. Here, we envision the poorly sorted sand as a few largegrains enveloped by soft “shells” of fine-grained sand. This is the re-alization of a lower HS bound.

At porosity �, the concentration of the pure solid phase �added tothe sphere pack to decrease porosity� in the rock is 1�� /�c, andthat of the original sphere-pack phase is � /�c. Then the bulk Kdry

and shear Gdry moduli of the dry friable-sand mixture are

Kdry��

�c

KHM�4

3�HM

�

1��

�c

K�4

3�HM

�1

�4

3�HM,

�11�

�dry� �

�c

�HM�z�

1��

�c

��z

�1

�z, �12�

where

z��HM

6�9KHM�8�HM

KHM�2�HM� . �13�

The saturated elastic moduli Ksat and �sat can be calculated from Gas-smann’s equations. Density is given by

b��fluid� �1��min, �14�

where min is the mineral density, which equals 2.65 g /cm3 forquartz, and fluid is the fluid density, normally varying from1.0 to 1.15 g /cm3 for saline water. For dry rocks, the fluid density iszero.

Some of the largest uncertainties with the friable-sand model areassociated with heterogeneous grain contacts, tangential slip, andhighly variable coordination number. Several workers demonstratehow the high-porosity end member can be calibrated to a given dataset by adjusting a slip factor �e.g., Bachrach and Avseth, 2008; FaustAndersen and Johansen, 2010� or by changing n �e.g., Florez, 2005;Dutta et al., 2010�. Florez �2005� finds that because of the uncertain-ty in n and the limitations of using an idealized packing model, themodified HS lower bound �i.e., the friable-sand model� may over-predict the velocity increase because of deteriorating sorting. Grainpacking, which texturally can look very similar to the effect of sort-ing but is caused by postdepositional mechanical compaction, oftenfollows a depositional sorting trend in a velocity-porosity crossplot.Florez �2005� finds, however, that packing often yields a slightlysteeper slope than the one predicted from the friable-sand model. Heargues that the combination of packing and sorting could explain thegood match between the friable-sand model and data of sands withincreasing pore-filling material often observed in real in situ well-log measurements.

The contact-cement model

During burial, sands are likely to become cemented sandstones.This cement may be diagenetic quartz, calcite, albite, or other miner-als. Cementation has a more rigid stiffening effect because graincontacts are “glued” together. The contact-cement model assumesthat porosity reduces from the initial porosity of a sand pack as a re-sult of the uniform deposition of cement layers on the surface of thegrains �Figure 8�. The contact cement dramatically increases thestiffness of the sand by reinforcing the grain contacts. In particular,the initial cementation effect will cause a large velocity increase withonly a small decrease of porosity.

The mathematical model is based on a rigorous contact-problemsolution by Dvorkin et al. �1994�. In this model, Kdry and Gdry of dryrock are

Kdry�n�1��c�McSn

6, �15�

and

�dry�3Kdry

5�

3n�1��c��cS

20, �16�


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42 GEOPHYSICS Today

where �c is critical porosity; Ks and �s are the bulk and shear moduliof the grain material, respectively; Kc and �c are the bulk and shearmoduli of the cement material, respectively; Mc�Kc�4�c /3 is thecompressional modulus of the cement; and n is the coordinationnumber, defined as average number of contacts per grain. Parame-ters Sn and S are porportional to the normal and tangential stiff-nesses, respectively. Statistical approximations of the rigorous ce-mentation theory solutions �Dvorkin et al., 1994� are given by thefollowing equations �Dvorkin and Nur, 1996�:

Sn�An��n��2�Bn��n�� Cn��n�, �17�

An��n��0.024153�n�1.3646, �18�

Bn��n��0.20405�n�0.89008, �19�

Cn��n��0.00024649�n�1.9864. �20�

S �A�� ,� s��2�B�� ,� s�� C�� ,� s�,

�21�

A�� ,� s��10�2�2.26� s2�2.07� s

�2.3��0.079�s

2�0.1754�s�1.342, �22�

B�� ,� s�� 0.0573� s2�0.0937� s

�0.202��0.0274�s

2�0.0529�s�0.8765, �23�

C�� ,� s��10�4�9.654� s2�4.945� s

�3.1��0.01867�s

2�0.4011�s�1.8186, �24�

�n�2�c�1�� s��1�� c�

��s�1�2� c�, �25�

� ��c

��s, �26�

� � 2

3��c��

1��c

0.5

, �27�

� c�0.5�Kc

�c�

2

3�

�Kc

�c�

1

3� , �28�

� s�0.5�Ks

�s�

2

3�

�Ks

�s�

1

3� . �29�

A detailed explanation of equations 17–29 and their derivation isgiven in Dvorkin et al. �1994�. Saturated elastic moduli are calculat-ed using Gassmann’s equations. Dry and saturated bulk densities arecalculated using equation 14.

The main shortcoming with the Dvorkin-Nur contact-cementmodel is that is does not include pressure sensitivity. It is assumedthat the cemented grain contacts immediately lose pressure sensitiv-ity as the cementation process begins. From in situ observations, weknow that cemented reservoirs can have significant pressure sensi-tivity �e.g., Duffaut and Landrø, 2007; Avseth et al., 2009a; Vernikand Hamman, 2009�. This could be related to fractures not capturedby the microstructural-scale model or by a patchy cementationwhere some grain contacts are cemented and others are loose.Hence, the loose contacts should still be pressure sensitive. As withthe HM contact theory for loose granular media, the Dvorkin-Nurcontact-cement model often overpredicts shear stiffnesses in ce-mented sandstones. This could be related to a heterogeneous mixtureof grain contacts and tangential slip at loose contacts �Avseth et al.,2009a� or to relative roll and torsion not taken into account in thecontact theory �e.g., Elata and Berrymann, 1996�. Furthermore, asteep slope in velocity-porosity crossplots may indicate elastic de-formation, pressure solution at grain contacts, or grain interpenetra-tion without associated cementation �Florez and Mavko, 2004�.

The constant-cement model

The constant-cement model, introduced by Avseth et al. �2000�,assumes that sands of varied sorting �and therefore varied porosity�all have the same amount of contact cement. Porosity reduction issolely from noncontact pore-filling material �e.g., deteriorating sort-ing�. Mathematically, this model is a combination of the contact-ce-ment model, where porosity reduces from the initial sand-pack po-rosity to porosity �b as a result of contact-cement deposition andfrom the friable-sand model where porosity reduces from �b as a re-sult of the deposition of the solid phase away from the grain contacts�Figure 9�. Considering a given reservoir, this is the most likely sce-nario because the amount of cement is often related to depth, where-as sorting is related to lateral variations in flow energy during sedi-ment deposition. In these cases, we can refer to this as a constant-depth model for clean sands. However, it is possible that cement canhave a local source and therefore cause a considerable lateral varia-tion in velocity.

To use the constant-cement model, we must first adjust the well-sorted end-member porosity �b that corresponds to the point shown

Ela

stic

mod

ulus

0.30 0.35 0.40


Figure 8. Schematic of the contact-cement model and the corre-sponding diagenetic transformation. Elastic modulus increasesmarkedly with increasing amount of contact cement.


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as an open circle in Figure 9. The dry-rock bulk and shear moduli atthis porosity �Kb and �b, respectively� are calculated from the con-tact-cement model. Equations for the dry-rock bulk Kdry and shear�dry moduli at a smaller porosity � are then interpolated with a lowerbound:

Kdry��

�b

Kb�4

3�b

�

1��

�b

K�4

3�b

�1

�4

3�b �30�

and

�dry� �

�b

�b�z�

1��

�b

��z

�1

�z, �31�

where

z��b

6�9Kb�8�b

Kb�2�b� . �32�

The effect of pore fluid can be accounted for by using Gassmann’s�1951� equations. Dry and saturated bulk densities are calculated us-ing equation 14. Regarding input mineral properties, tables of stan-dard properties of minerals are available in many handbooks andcompilations, such as Ellis et al. �1988�, Carmichael �1989�, andMavko et al. �2009�. Note that it is possible to arrive at the constant-cement line by first moving along the friable-sand line and then add-ing contact cement to the rock �dashed line in Figure 9�, which isconsistent with diagenesis following deposition. The pitfalls andlimitations mentioned for the friable-sand and the contact-cementmodels also apply for the constant-cement model.

ROCK-PHYSICS ESTIMATION OF CEMENTVOLUME AND SORTING — A NORTH

SEA DEMONSTRATION

Using the diagnostic models, we can infer the microstructure fromvelocity-porosity data. With good local validation of the models, wecan even quantify the degree of sorting and cement volume fromthese diagnostic crossplots �Avseth et al., 2009b�. Figure 10 showshow this procedure is done for some well-log data from the North

Sea. In this case, the rock-physics diagnostics are performed in theVS-versus-porosity domain to avoid significant pore-fluid effects.The cement volume is estimated by interpolating between the con-stant-cement volume trends, whereas the sorting is defined by theobserved porosity normalized by the high-end-member porosityalong the given constant-cement trend — the porosity connectingthe constant-cement model with the contact-cement model.

Having estimated cement volume and sorting, we can plot the dataas logs and compare them with other petrophysical logs. Figure 11shows the resulting estimate of cement volume and sorting. Figure11b shows cement volume as magnitude and sorting as superim-posed color. For the studied North Sea sandstone interval starting ataround 2 km depth, we observe a clear depth trend in the cement vol-ume. The sorting, however, shows a more erratic pattern without adepth trend, which is expected because sorting is associated withdepositional trends.

It is essential to verify the presence of initial cementation predict-ed from the rock-physics relations with thin-section observations.Figure 12 shows a thin section from the relatively clean sands in thisstudy; at first glance, the sandstone looks unconsolidated, with

Ela

stic

mod

ulus

0.30 0.35 0.40Porosity (volume fraction)

Constantcement

Contactcement

FriableInitialsandpack

Figure 9. Schematic of three effective-medium models for high-po-rosity sands in the elastic-modulus/porosity plane and correspond-ing microstructure characteristics. The elastic modulus may be com-pressional, bulk, or shear.


4500

4000

3500

3000

2500

2000

1500

1000

500

00 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Shale

V(m

/s)

S Sor

ting

Quartz

Sorting = φ/φc

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

b)

4500

4000

3500

3000

2500

2000

1500

1000

500

0

VS

(m/s

)

Cem

entv

olum

e(%

)

0 0.05


0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Shale

Increasing cement volume

Dvorkin-Nurcontact cement

10

9

8

7

6

5

4

3

2

1

0

–1

Constant-cementtrends

a)

Quartz

Figure 10. Shear-wave velocity-log data versus total porosity andsuperimposed diagnostic rock-physics models. Using the models,we can quantify �a� the cement volume and �b� the degree of sorting.Green data points are shale data with high GR values and for practi-cal reasons are given the value �1 in cement volume and zero insorting.


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44 GEOPHYSICS Today

grains loosely arranged and moderately well sort-ed. A closer investigation, however, reveals thepresence of initial quartz overgrowth coveringoriginal grain surfaces indicated by dust rims �ar-rows, Figure 12�. This observation confirms whatwe see in the rock-physics crossplots of the well-log data. It is interesting that the well-log datawith tens-of-centimeter resolution reflects whatwe observe at the microscale. Furthermore,Avseth et al. �2009b� find a very good match be-tween the rock-physics estimated cement volumeand the total cement estimated from thin sections.

ROCK PHYSICS OF SHALES

Until recently, shales have often been regardedas a unique type of lithology among geophysi-cists, and minor attention has been given to thegreat variance in mineralogy, texture, and porosi-ty of shales during seismic data analysis. This ispartly because the rock properties of clay miner-als are difficult to measure in the laboratory butalso because the oil industry has given little prior-ity to acquiring detailed log data and core samplesin shale sequences. Geologists, however, havedocumented the complexity of shales, and there

exists a vast amount of published literature on their geochemistryand sedimentology �e.g., Bjørlykke, 1998; MacQuaker et al., 2007;Peltonen et al., 2008�. With increased focus on cross-disciplinary in-tegration, geophysicists are starting to incorporate this geologicknowledge into modeling and analyzing geophysical data �e.g.,Dræge et al., 2006b; Brevik et al., 2007; Mondol et al., 2007; Mar-cussen et al., 2008�.

As with sands, we can distinguish between depositional and di-agenetic trends in shales. Depositional trends will affect clay miner-alogy; but in particular, the silt content will have great impact on theseismic properties. Avseth et al. �2005� use a simple isotropic lowerReuss bound to model vertical velocities of silty shales. However,shales have a different composition and texture than sandstones.Therefore, the rock-physics models applicable for sandstones do notnecessarily apply for shales. More rigorous anisotropic modeling ofshales has been performed by Hornby et al. �1994�, Johansen et al.�2002�, Ruud et al. �2003�, and Dræge et al. �2006b�, among others.In this paper, we use the shale compaction model �Dræge et al.,2006b; Ruud et al., 2003� to estimate the anisotropic effective prop-erties in mechanically compacted shales. More details about thismodel are included in the next section and in Avseth et al. �2008�.

ROCK-PHYSICS DEPTH TRENDS

Rock-physics depth trends for sands and shales can be used tostudy the expected seismic signatures of sand-shale interfaces as afunction of depth. We use existing empirical porosity-depth trendsfor sands and shales as input to rock-physics models of VP, VS, anddensity. We can, for example, use HM theory to calculate the veloci-ty-depth trends for unconsolidated sands and shales, whereas Dvor-kin and Nur’s �1996� contact-cement model can be used for cement-ed sands. The depth trends allow us to study the ability to discrimi-nate between pore fluids and lithologies at different depths.

Dep

th(m

)

Shale volume

1200

1400

1600

1800

2000

2200

2400

2600

2800

Cement volume (%)

1200

1400

1600

1800

2000

2200

2400

2600

2800

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Sorting

1200

1400

1600

1800

2000

2200

2400

2600

2800

10

9

8

7

6

5

4

3

2

1

0

–1

a) b) c)

0 0.5 1 0 5 10 0 0.5 1 (%)

Figure 11. Estimated cement volume and sorting versus depth for a North Sea well. Notethe onset of cement starting at around 2000 m depth �corresponding to 70°C� and �b� theincreasing cement volume with depth. In spite of this depth trend, there is variability incement volume that is likely associated with varying shaliness in the sandstones. Sorting,however, shows a more erratic pattern with no depth trend �color in �b��. �a� Shale volumeversus depth; �b� cement volume versus depth �dots� and sorting versus depth �color�; �c�sorting versus depth �dots� and cement volume versus depth �color�.

a)

Quartz cement

b)

Figure 12. Thin sections from Heimdal Formation sands. �a�Aloose-ly packed, poorly consolidated sand. �b�Analysis of a zoomed-in im-age �partially from the red boxed area in �a�� confirms the presence ofquartz overgrowths and contact cement. On detrital quartz grains,we observe dust rims representing the original grain surfaces thathave been covered by quartz cement �arrows�. Feldspar overgrowthand calcite cement also occur, yet quartz cement is dominating.


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Figure 13 is a schematic representation of shale and sand compac-tion curves and a sequence of interbedded turbidite sands and marineshales, typical for the North Sea deep-marine environment of Tertia-ry age. The depositional porosity in shales is normally much higher�60%–80%� than in sands ��40%�, but we expect a shallow cross-over with depth as a result of the mechanical collapse of the shales.The platy clay fabric in the shales is more prone to compaction thanthe assemblage of spherically shaped grains in sands and hence themore rapid mechanical porosity reduction in shales than sands. Dur-ing burial to approximately 2 km depth, sands and shales are ex-posed mainly to mechanical compaction. The marine shales in theTertiary North Sea are very smectite rich, and they have very lowpermeability. In thick, smectite-rich shale masses, it is therefore nor-mal to observe undercompaction and associated overpressure evenat several hundred meters’ burial depth. At about 70°C, however,chemical alteration of smectite will begin, and we expect a mineraltransformation to illite. This is a typical mineral transformation seenin marine shales all over the world �Bjørlykke, 1998�. Bound waterin the smectite layers is released when the temperature reaches thiscritical temperature, resulting in a porosity decrease. Moreover, thepresence of potassium cations �e.g., in feldspar or mica� causesquartz to be produced as a by-product. This quartz can precipitate asmicrocrystalline quartz within the shale matrix �Thyberg et al.,2009�; if connectivity allows, the quartz may precipitate as cementin adjacent sandstones �Peltonen et al., 2008�.

Figure 14 shows well-log data from a North Sea well penetratingsiliciclastic sediments and rocks of Tertiary age �same data investi-gated in Figures 10 and 11�, juxtaposed with rock-physics depthtrends for shales and sandstones. We observe a good match betweenthe calculated velocity-depth trends for different lithologies and thewell-log data. The sandstone rock-physics models as a function ofdepth are modeled by combining HM contact theory for unconsoli-dated sands with the Dvorkin-Nur contact cement model for cement-ed sandstones. The input porosity-depth trends are calibrated withlocal compaction trends according to empirical relations �e.g.,Ramm and Bjørlykke, 1994; Mondol et al., 2007�. The light-bluemodel curves show how the velocities increase drastically for sandsas we go from the unconsolidated regime with only mechanical com-paction to the cemented regime with predominantly chemical com-paction. The onset of quartz cement happens atabout 70°C, corresponding to about 2 km burialdepth.

We apply the shale compaction model �Ruud etal., 2003; Dræge et al., 2006b� to estimate the an-isotropic effective properties in mechanicallycompacted shales. The first seismically importantmineral reaction in shales is commonly the smec-tite-to-illite reaction. The reaction has several im-plications for the shale; the soft smectite is re-placed by stiffer illite that might be distributeddifferently in the rock. The reaction produces wa-ter, the amount of solids is decreased �i.e., illitehas a denser mineral structure than smectite�,quartz is generated as a by-product, and porosityis reduced by chemical compaction. When theshales are moving into the chemical compactionregime, a new set of rock-physics models is ap-plied to estimate the seismic properties. The an-isotropic version of a differential effective medi-um �DEM� model and self-consistent approxima-

tion �SCA� are used to account for the elongated pores and grains inshales �Hornby et al., 1994�. In this study, pores in chemically com-pacted shales are considered to be isolated, but the pores in the me-chanical compaction regime are connected �c.f., Dræge et al.,2006b�. We define a transition zone, where the properties changefrom the mechanical to the chemical regime.

In Figure 15, the initial shale � � 1500 m� is considered to besmectite rich, and the deeper ��2200 m� illite-rich shale is some-what stiffer. The vertical mineral bulk and shear moduli of smectiteare found by calibration to well-log data to be 12.5 and 7.5 GPa, re-spectively. For illite, these elastic moduli are 21 and 7 GPa, respec-tively. The assumption of isotropic mineral moduli for the clay min-erals is probably incorrect, but the assumption may be realistic forassemblages of clay crystals that are randomly scattered and mixedwith larger silt particles of quartz. In the modeling, we assume rela-tively pure shales with 7% quartz. For quartz, the mineral moduli are37 and 44 GPa, respectively. We assume clay density to be the sameas for quartz, 2.65 g /cm3.

There are two counteracting effects on anisotropy. The initial

DepositionPorosity

Pureshale Clean

sandM

echa

nica

lco

mpa

ctio

n

Che

mic

alco

mpa

ctio

nNorth Sea Paleocene

~ 2 kmUnc. sand

QzCem. SS

Qz

Smectite-rich shale

Volcanic tuff

Illite-rich shale

Depth of burial

Smectite + K+ → Illite + SiO2 + H2O

~ 70°C

Figure 13. Schematic of sand and shale compaction. At 70°C, it iscommon to observe a change from mechanical compaction to pre-dominantly chemical compaction in siliciclastic systems. In deepmarine depositional systems, smectite-rich shales experience illiti-zation and release of bound water, causing a porosity reduction andmineralogy change with depth. For quartz-rich sands, initial cemen-tation tends to start at the same depth level. One possible externalsource of cement is in fact derived from the smectite-to-illite transi-tion in embedding shales.

Vsh Vp (m/s) Vs (m/s) Density (kg/m3)

Dep

th(m

)

20003000 4000 1000 2000 30000 0.5 1 2 2.5

0

500

1000

1500

2000

2500

3000

Seafloor

Mechanicalcompaction

Chemicalcompaction

Gas

Oil

Figure 14. Rock-physics depth trends for shales �blue� and sandstones �cyan�, juxtaposedon North Sea well-log data penetrating a Tertiary sequence of siliciclastic sediments androcks. A gas zone is indicated in yellow, and an oil zone is in red. The remaining intervalof the Heimdal Formation is brine filled. The Heimdal is embedded in the Lista Forma-tion shale. The leftmost panel shows the shale volume Vsh.


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46 GEOPHYSICS Today

alignment of grains leads to more aligned pores and increasing an-isotropy. However, decreasing porosity leads to decreasing anisotro-py, culminating in the anisotropic properties of the solid material�i.e., mixture of quartz and clay� at zero porosity. The pores intro-duce higher anisotropy than the solid because pores commonly areweaker orthogonal to the longest axis, whereas the solids in this caseare less dependent on direction of wave propagation. In addition to

VP, VS, and density, we estimate the Thomsen �1986� parameters ofanisotropy, and �, which can be significant for interpreting angle-dependent seismic reflectivity �AVO analysis�. The estimated Th-omsen parameters are within the range of experimental values ofmixed clay �kaolinite and illite� and quartz derived by Voltolini et al.�2009� under a uniaxial effective stress �no lateral strain allowed� of

5 and 50 MPa. They show that pure clay can haveP-wave anisotropy exceeding 40%, but the pres-ence of silty quartz particles within a shale willdrastically reduce the anisotropy.

Wang �2002� finds that � decreases exponen-tially with increasing porosity, whereas has aweak relationship with porosity. The decreasing �with burial depth and decreasing porosity that weestimate from our modeling does not fit withWang’s observations. This is probably a result ofour assumption of isotropic mineral moduli forthe clays �i.e., individual clay crystals have stronganisotropy, yet assemblages of clay crystals caneffectively show lower anisotropy, especiallywhen mixed with quartz� together with the micro-structural representation of chemical compactionthat we include in modeling the illite-rich shales.Chemical compaction from precipitation of mi-crocrystalline quartz in illite-rich shales is report-ed by Thyberg et al. �2009�, and this diageneticprocess may cause stronger vertical bindings be-tween clay particles and hence reduced P-wave

anisotropy with decreasing porosity. It is beyond the scope of this pa-per to verify our anisotropic modeling results with real anisotropymeasurements in these type of shales, but future research should in-deed investigate how different geologic processes in shales affectseismic anisotropy.

ROCK-PHYSICS TEMPLATES (RPTS)

We can combine the depositional and diagenetic trend modelswith Gassmann fluid substitution and make charts or templates ofrock-physics models for predicting lithology and hydrocarbons. Werefer to these locally constrained charts as rock-physics templates�RPTs�, a technology first presented by Ødegaard andAvseth �2003�.Furthermore, we expand on the rock-physics diagnostics as we cre-ate RPTs of seismic parameters — in our case, acoustic impedanceversus VP /VS ratios �Figure 16�. This will allow us to perform rock-physics analysis of well-log data and of seismic data �e.g., elastic in-version results�.

The RPTs are site �basin� specific and honor local geologic fac-tors. Geologic constraints on rock-physics models include lithology,mineralogy, burial depth, diagenesis, pressure, and temperature. Allof these factors must be considered when generating RPTs for a giv-en basin. In particular, it is essential to include only the expectedlithologies for the area under investigation when generating theRPTs. Water depth and burial depth determine the effective pressure,pore pressure, and lithostatic pressure. The pore pressure is impor-tant when calculating fluid properties and determining the effectivestress on the grain contacts of the rock frame carrying the overbur-den.

Pore-fluid sensitivity in reservoir sandstones is highly affected byreservoir heterogeneity and sandstone microstructure, and it is there-fore important to include these geologic factors in the rock-physics

Dep

th(m

)

VP (m/s) VS (m/s) Density (kg/m3) Anisotropy

Delta Epsilon

Smectiterich

Illiterich

1200

1400

1600

1800

2000

2200

2400

2600

28002000 2500 3000 500 1000 1500 2.2 2.4 2.6 0 0.1 0.2

~ 70°C

Smectite + K → Illite + Si + H O2

Figure 15. Modeled rock-physics depth trends of shales, showing the effect of illitizationof marine smectite-rich shales.

Shale

Brine sand

Quartz mineralOil sand

Acoustic impedance

DecreasingVV

PS

Sw

a)

VV

PS

Acoustic impedance

Shale

Oil sand

Initialcementation

Brine SS

Decreasing net to gross

b)

Figure 16. Rock-physics templates �RPTs� can be made from therock physics presented in this paper combined with Gassmann’s the-ory to define regions where expected facies and fluids will plot. Inparticular, the VP /VS ratio is a great fluid discriminator in siliciclasticenvironments. �a� Homogeneous, unconsolidated sands filled withoil are normally well separated from brine-filled sands in an RPT ofVP /VS versus acoustic impedance �AI�. However, the effect of initialcement will reduce the fluid sensitivity of sandstones, and the VP /VS

ratio of cemented brine sandstones can be similar to the VP /VS ratioof unconsolidated sands filled with oil. The effect of N/G �i.e., heter-ogeneity� normally will move data in the opposite direction in aVP /VS-AI crossplot. Hence, �b� oil sands with relatively low net-to-gross can have higher VP /VS than homogeneous brine sands with ini-tial cement.


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analysis. As indicated, initial cement reduces the pressure and fluidsensitivity of sandstones. Figure 16 shows schematically the outlineof a rock-physics template, where calibrated rock-physics modelshave been selected that fit local data observations �well-log data orseismic inversion data� of various lithologies and pore fluids. In acrossplot of acoustic impedance �AI� versus VP /VS, the presence ofdiagenetic quartz cement will move brine-saturated sandstone datato plot in an area of very low VP /VS where we may expect hydrocar-bon-saturated sandstones to plot. On the contrary, reservoir hetero-geneity and decreasing net to gross �N/G� associated with interbed-ded sands and shales tend to move data points in the direction of theshale cluster. The cement effect is a microstructural effect, whereasN/G is a scale effect. When the interbedded shale is relatively softcompared to the sand, the N/G effect will counteract the effect of ce-ment on effective rock stiffness — hence, the opposite directions inthe AI-VP /VS crossplot. Figure 16 demonstrates why it is importantto include and understand these geologic factors when analyzingrock-physics properties and seismic-fluid sensitivity in reservoirsandstones.

Figure 17 shows an RPT including data from two neighboringwells penetrating Paleocene sands in the North Sea. One well pene-trates a thick, turbiditic gas sand with a thin oil leg, whereas the adja-cent well penetrates a turbidite sand filled with oil. It turns out thatthe sandstone quality changes from one well to the other, and thisdrastically distorts the fluid sensitivity to hydrocarbons. The gas-sat-urated top Heimdal sands in well 1 show a small increase in acousticimpedance, but the oil-saturated sands in well 2 show a significantdrop in acoustic impedance. This drastic change in sandstone qualityover a short distance yields a corresponding change in seismic signa-tures �see Avseth et al., 2009b�.

ROCK-PHYSICS MODELING AND UPSCALINGOF SAND-SHALE SEQUENCES

As we move from developing thick, hydrocarbon-filled layers tothinner interbedded and laterally discontinuous sequences, we needto focus on hydrocarbon prediction in heterogeneous reservoirs. One

useful parameter for quantifying the heterogeneity of sands is N/G,the fraction of clean, permeable sand to the complete reservoir in-cluding reservoir sands and intercalating impermeable shales. Wefind N/G to be a useful parameter when we upscale from alternatingthin beds of different lithologies and/or fluid saturations to an effec-tive medium during rock-physics analysis of well-log and seismicdata. It is also a parameter with which geologists are very familiar.Takahashi �2000� formulates one possible methodology to predictsand/shale ratio based on statistical rock-physics simulations of var-ious bedding scenarios. Vernik et al. �2002� predict N/G from P- andS-wave impedance-inversion results. Stovas et al. �2006� use effec-tive medium theory combined with Gassmann theory to predict N/Gand saturation from AVO attributes. Finally, Connolly and Kemper�2007� use an integrated and data-driven approach to predict N/Gfrom turbiditic reservoirs in offshoreAngola.

We suggest a five-step methodology �Figure 18� to model therock-physics properties of interbedded sands and shales with differ-ent pore-fluid saturation scenarios, where the first four steps followthe generation of the homogeneous rock-physics models outlinedabove. Step 1 is to estimate dry bulk and shear moduli Kdry and �dry atcritical porosity using HM contact theory. In step 2, we interpolatebetween the high-porosity end member and the zero-porosity miner-al point, choosing the modified lower or upper HS bound. In step 3,we apply Gassmann theory, perform fluid substitution, and estimateelastic properties of clean sands with uniform, but we vary saturationof brine Sw and hydrocarbons �1�Sw�, for all porosities.

In step 4, we select a characteristic shale to be interbedded withthe clean sand units. We can derive typical shale properties fromwell-log data in an area of study, or we can use rock-physics model-ing. We assume that the shale will be completely impermeable to hy-drocarbons and that the shale layers will only be saturated with brinetrapped during deposition. It is reasonable to assume that the porosi-ty of thin-bedded shales at a given depth will be fairly constant, in

VV

PS

Acoustic impedance

3.0

2.5

2.0

1.54 6 8 10 12

Contact-cement model

Shale model

Brine SS model (2% cement)Gas sand model

I

II

IIIIV

V

3(km/s∗g/cm )

Figure 17. Rock-physics template of VP /VS versus acoustic imped-ance �AI� for target zone �Paleocene� of two North Sea wells. ClusterI is the cap-rock shale in both wells; II comprises the brine sand-stones in both wells; III and IV are reservoir sandstones in well 1filled with oil and gas, respectively; and V is the upper oil zone inwell 2, with VP /VS higher than the brine sandstones and AI lowerthan the gas sandstones in well 1. This is counterintuitive and mustbe explained by difference in sandstone quality �see Figure 16�. Blueis shale, cyan is brine sand, yellow is gas sand, and red is oil sand.

Step 1: Hertz-Mindlin dry sandstone Step 3: Gassmann fluid sub.

Steps 4 and 5: Pick a shaleStep 2: Hashin-Shtrikman interpolation

Kdry

φsh

Kdry

φcφc

Ksat

φc φc

Ksat

K f (P= )dry eff

+ Backus upscaling

Figure 18. Work flow for hybrid rock-physics modeling and Backus-average upscaling of sandstone interbedded with shale. The drysandstone is modeled by combining Hertz-Mindlin contact theoryand Hashin-Shtrikman interpolation �steps 1 and 2�. Gassmann fluidsubstitution is done to estimate the effect of varying gas versus watersaturation in the sand layers �step 3�. We assume a characteristicshale with constant porosity �step 4� to be interbedded with the sandsand use the Backus average �step 5� to estimate effective propertiesfor varying N/G ratios. The dashed red arrows in the last subplot in-dicate that the characteristic shale with constant porosity is interbed-ded with sandstone layers with varying porosity and varying satura-tion.


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48 GEOPHYSICS Today

contrast to the porosity of the thin-bedded sands that are prone tovary with sorting. Hence, we assume a constant characteristic poros-ity for shale �sh.

In step 5, we apply Backus average effective medium theory �e.g.,Gelinsky and Shapiro, 1997; Mavko et al., 2009� to estimate the ef-fective, upscaled anisotropic properties of the interbedded shale-sand sequences. This we do for various N/G values, ranging fromzero to one. The Backus average approximates a stack of alternatingthin layers of two isotropic media as one effective anisotropic medi-um. This medium is characterized by five independent elastic modu-li according to the transverse isotropic elasticity matrix �Mavko etal., 2009�. The various elastic moduli that we need further are foundfrom the velocities, densities, and fractions of the alternating sandand shale layers. Finally, we can derive diagnostic rock-physicsmodels for varying N/G.

We apply this five-step rock-physics methodology to interpretingwell-log data from a North Sea reservoir. We create a crossplot ofacoustic impedance and VP /VS data from the target zone based on thewell-log data in Figure 14. This crossplot is compared to rock-phys-ics models for various N/G made according to the five-step method-ology. The characteristic shale is picked from the cap-rock shaleabove the reservoir. This may be somewhat erroneous because thecap-rock shale is not neccesarily equivalent to the interbedded shalewithin the resevoir. In Figure 19, we can see the various models forN/G of 1.0, 0.9, 0.8, and 0.6. For each of these N/G, we include vary-ing gas saturation within the sand layers. It is interesting to note howthe decrease in N/G will cause a drastic increase of VP /VS, regardlessof porosity, even for high gas saturation in the sands. The acousticimpedances drop drastically with decreasing N/G when the sandshave low porosities but increase slightly when the sands have highporosity. This is of course because of the relative contrast to the inter-calating shale.

The models for varying N/G are valid for scales such that the layerthicknesses are much below the resolution of the elastic wave, andtypically we apply them when the thicknesses are less than about1 /10 of a wavelength �Backus, 1962�.A reservoir sand can plot withN /G�1 in well-log data; the same sand can plot with N /G � 1 inseismic data. Hence, these effective models can be used to determinewhat happens if we go from well-log scale to seismic scale.

The models can also be used to interpret scale effects and N/G val-ues in well-log data if these values happen to be less than one, whichis often the case in turbidite sequences. The reservoir-sand data inthe studied well match nicely with the models for high N/G values�0.8–1.0�, and at first glance it appears that the gas-sand data pointsfall close to the line for homogeneous �N /G�1� sandstone with100% gas. In Figure 20, however, we remove the brine-filled and oil-filled sand data, leaving only the gas-saturated sands with the cap-rock shales. Now we clearly observe the gas-saturated sands span awide range of VP /VS. Some of the gas sands seem to have N/G valuesbetween 0.8 and 0.6, causing VP /VS close to two — a value more typ-ical of brine-saturated sands. This observation is consistent with apatchy saturation behavior. The heterogeneities of the intercalatingshales are causing a geologic control on the saturation pattern. Eventhough the gas saturation is uniform at the scale of the sandstone po-rosity, the propagating sound waves will effectively experiencepatchy saturation when the interbedded sand-shale thicknesses arelarger than the critical diffusion length yet beneath the resolution ofthe sonic or seismic waves.

LIMITATIONS AND CAVEATS

The models for the high-porosity sandstones are based on isotro-pic linear elasticity. All of the bound-filling models provided herecontain some heuristic elements, such as the interpretation of themodified bounds. The grain-contact models are limited by the fol-lowing assumptions: the strains are small; grains are identical, ho-mogeneous, isotropic, elastic spheres; packings are assumed to berandom and statistically isotropic; and the effective elastic constantsare relevant for long-wavelength propagation, where wavelengthsare much longer �more than 10 times� compared to the grain radius.

Furthermore, effective medium elastic constants based on grain-contact models are derived under the mean-field approximation,V

VP

S

Acoustic impedance3(km/s∗g/cm )

3.2

3.0

2.8

2.6

2.4

2.2

2.0

1.8

1.6

1.42 4 6 8 10 12 14

Shale point

S = 100%w

S = 10%w

S = 0%w

N/G = 0.6

N/G = 0.8

N/G = 0.9

N/G = 1.0

Figure 19. Rock-physics template of acoustic impedance versusVP /VS, including models for varying net-to-gross �N/G� and gas sat-uration �1�Sw� created by the five-step procedure. The coloredmodel lines represent the clean-sand models without shale interbed-ding �i.e., N /G�1.0�. The cyan line represents clean sands with100% water saturation Sw. Porosity is 40% at the left end point anddecreases to the right toward the mineral point �which is outside therange of the plot�. The dashed orange line is the corresponding 10%water saturation and 90% gas saturation, whereas the solid orangeline represents 100% gas. These three sandstone lines “rotate” up-ward toward the selected shale point when N/G decreases, indicatedby black lines. For well-log data, the brine sands �cyan� and oil sands�red� fall between N /G�1 and 0.8. Most of the gas sands �yellow�seem to fall on similar porosities, with N/G varying between 1 and0.6.

VV

PS

Acoustic impedance3(km/s∗g/cm )

3.2

3.0

2.8

2.6

2.4

2.2

2.0

1.8

1.6

1.42 4 6 8 10 12 14

Shale point

S = 100%w

S = 10%w

S = 0%w

N/G = 0.6

N/G = 0.8

N/G = 0.9

N/G = 1.0

Figure 20. Studying the gas sands �yellow� in detail, we clearly seethey span a VP /VS of 1.5–2. Most of the gas sands have VP /VS of 1.6–1.7, representative for N/G values of 0.9. Hence, very little shale in-tercalation will cause a significant increase in VP /VS compared to ho-mogeneous, clean sands �N /G�1�.


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which assumes that all grains experience the same mean strain field.This ignores the actual grain-scale heterogeneous strains and stress-es. Mavko et al. �2009� give three caveats about the use of effective-medium models for granular media. First, granular media have prop-erties lying somewhat between solids and liquids and are sometimesconsidered a distinct form of matter. Second, complex behavior aris-es from the ability of grains to move relative to each other, modifytheir packing and coordination numbers, and rotate. Third, observedbehavior, depending on the stress and strain conditions, is some-times approximately nonlinearly elastic, sometimes viscoelastic,and sometimes somewhat fluidic.

Many workers �Goddard, 1990; Makse et al., 1999, 2004; Duffautet al., 2010� show that this complex behavior causes effective-medi-um theory to fail in cohesionless granular assemblies. Closed-formeffective-medium theories tend to predict the incorrect �relative tolaboratory observations� dependence of effective moduli on pres-sure and poor estimates of bulk-to-shear moduli ratio. Shear modulitend to be overpredicted and often require heuristic corrections �e.g.,Bachrach and Avseth, 2008.� Numerical methods, referred to as mo-lecular dynamics or discrete element modeling, which simulate themotions and interactions of thousands of grains, appear to comecloser to predicting observed behavior �e.g., Makse et al., 2004; Gar-cia and Medina, 2006�. Closed-form effective-medium models canbe useful because it is not always practical to run a numerical simula-tion; however, model predictions of the types presented in this sec-tion must be used with care.

In modeling unconsolidated sands, we have assumed normallycompacted �Middleton and Wilcock, 1994� sediments and have esti-mated the effective pressure used in the HM contact theory fromTerzaghi’s principle �equation 9�. This principle implies that com-pression is one dimensional �i.e., considering only the vertical axis�and pore pressures are hydrostatic. This assumption will be violatedand predictions from HM contact theory will be erroneous in areaswith significant stress anisotropy and/or overpressure conditions�e.g., Xu, 2002; Vega, 2003�.

For the sand and shale models presented above, a limitation arisesfor mixed mineralogies. For mixed mineralogies, often an effective-average mineral modulus must be estimated. As one of the inputs,the mineral moduli of clays are variable and not very well known,leading to uncertainties in the model predictions. Nevertheless, wehave found these models useful for high-porosity siliciclastic rocksin many situations. We are confident that future research will lead tobetter models.

CONCLUSIONS

There are many different rock-physics models and modeling ap-proaches for understanding and interpreting the seismic responses ofsubsurface rocks and their pore fluids. We have listed different typesof models, ranging from simple empirical models to more complexphysical and computational models. There are pros and cons associ-ated with each model, yet in the end all models are, strictly speaking,wrong. Still, we have no choice but to find a model, or a combinationof models, useful �i.e., yield reasonable predictions or lay thegroundwork for reliable interpretation� for a given case or scenario.Empirical models are useful because of their simplicity and the lim-ited number of input parameters. However, they can be erroneouswhen applied to areas with a different geologic setting or depth thanthe data on which the models were generated. More complex phys-ics-based models are normally better for extrapolation exercises but

suffer from the fact that many input parameters need to be known inadvance, many of which can become “fudge factors” in the models.Our hybrid modeling approach for sands and sandstones combinesphysical-contact theory with heuristic elastic bounds to predict themicrostructure of these rocks from elastic properties. For shales, werecommend using inclusion-based models to capture the effect ofvarying pore shapes and anisotropy. We have demonstrated usingthese rock-physics models on well-log data from a sequence of Pale-ocene sands and shales in the North Sea.

It is essential to quantify the lithology and rock texture before wecan reliably estimate pore-fluid and pressure effects. Therefore, thelink between geologic processes and rock properties should alwaysbe integrated in the work flow of any seismic-reservoir characteriza-tion study. Finally, one should note the gap between the differentscales, from the pore-scale microstructure to seismic wiggles. Weuse the Backus average as one technique to upscale and estimate ef-fective seismic properties of interbedded sequences. However, clos-ing the gap between different scales remains a challenging problemin rock-physics modeling applied to seismic exploration of hetero-geneous reservoirs.

ACKNOWLEDGMENTS

Thanks to Anders Dræge at Statoil for contributions on buildingthe rock-physics depth trend for shale used in this paper. We also ac-knowledge the sponsors of the Stanford Rock Physics and BoreholeGeophysics Project �SRB� and the Stanford Center for ReservoirForecasting �SCRF�.

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50 GEOPHYSICS Today

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geophysics today (a survey of the field as the journal celebrates its 75th anniversary) || 3....

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