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GEOPHYSICS, VOL. 74, NO. 1 �JANUARY-FEBRUARY 2009�; P. E1–E15, 17 FIGS., 1 TABLE.10.1190/1.3033212

ediment with porous grains: Rock-physics modelnd application to marine carbonate and opal

ranklin Ruiz1 and Jack Dvorkin1

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ABSTRACT

We offer an effective-medium model for estimating theelastic properties of high-porosity marine calcareous sedi-ment and diatomite. This model treats sediment as a pack ofporous elastic grains. The effective elastic moduli of the po-rous grains are calculated using the differential effective-me-dium �DEM� model, whereby the intragranular ellipsoidal in-clusions have a fixed aspect ratio and are filled with seawater.Then the elastic moduli of a pack of these spherical grains arecalculated using a modified �scaled to the critical porosity�upper Hashin-Shtrikman bound above the critical porosityand modified lower �carbonates� and upper �opal� Hashin-Shtrikman bounds below the critical porosity. The best matchbetween the model-predicted compressional- and shear-wave velocities and Ocean Drilling Program �ODP� datafrom three wells is achieved when the aspect ratio of intra-granular pores is 0.5. This model assigns finite, nonzero val-ues to the shear modulus of high-porosity marine sediment,unlike the suspension model commonly used in such deposi-tional settings. The approach also allows one to obtain a satis-factory match with laboratory diatomite velocity data.

INTRODUCTION

Many empirical and theoretical rock-physics relations and mod-ls deal with siliciclastic sediment composed of solid grains or car-onates with inclusions. However, large areas on the earth are cov-red with deposits of microscopic, hollow, calcareous or siliceousossil skeletons. This study concentrates on velocity-porosity-min-ralogy relations for such sediment textures.

Calcareous sediments cover about 68% of the area in the Atlanticcean, 36% in the Pacific Ocean, and 54% in the Indian Ocean. The

otal coverage is about 48% of the world’s seafloor �Sverdrup et al.,942�. In most cases, calcium carbonate is transferred to the seafloor

Manuscript received by the Editor 24August 2007; revised manuscript rec1Stanford University, Stanford, California, U.S.A. E-mail: fjruiz@stanford2009 Society of Exploration Geophysicists.All rights reserved.

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y biological activities. Organisms use dissolved calcium carbonateo construct their skeletons. The remains of the microorganisms set-le to the seafloor and form a bed of calcareous sediment �Mohamed-lhassan and Shang, 2003�. In deep water, shallow buried calcareousarine sediment is composed largely of minute skeletons �porous

rains�. When this sediment is deposited, its porosity may be as highs 0.7–0.8 �Fabricius, 2003�. Burial and the resulting compaction re-uce porosity to approximately 0.5–0.6 within the first few hundredeters below the seafloor. The elastic properties of this overburden

nd their relation to porosity, mineralogy, and stress are importantor properly imaging targets located beneath this calcareous sedi-ent.Another widely distributed deposit with porous grains is diato-ite, which can be part of the overburden �e.g., in the North Sea� or

ydrocarbon reservoirs �Monterey Formation, California, U.S.A.�.iatomite is composed of the fossilized skeletal remains of micro-

copic plants called diatoms. Diatoms are made of siliceous skeletonnd are found in almost every aquatic environment. Because theirell wall is composed of hydrated silica �SiO2 .nH2O�, they are wellreserved in the sediments �Mohan et al., 2006�.

Diatoms have the unique ability to absorb water-soluble silicaresent in their natural environment to form a rigid, highly porouskeletal framework of amorphous silica. Atomic force microscopyAFM� analysis of live diatoms reveals the nanostructure of thealve silica to be composed of a conglomerate of packed silicapheres �Crawford et al., 2001; Losic et al., 2007�. Hamm et al.2003� perform real and virtual loading tests on diatom cells, usingalibrated glass microneedles and finite-element analysis. Theyhow that the frustules are remarkably strong by virtue of their archi-ecture and the material properties of the diatom silica.

Diatomite is structurally close to calcareous sediment becauseoth have a biogenic origin and thus are composed of the skeletalarts of organisms. Calcareous and diatomite materials have inter-ranular and intragranular porosity. Diatoms precipitate silica fromeawater as amorphous opal �opal-A�. After deposition, silicarogresses from opal-A toward quartz, the stable phase, through anntermediate phase, opal-CT. Each transition occurs through disso-

July 2008; published online 10 December [email protected].

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ution and reprecipitation �Chaika, 1998; Chaika and Dvorkin,000�.

Empirical relations combined with the theoretical Gassmann1951� approach between porosity, mineralogy, and P- and S-waveelocity �VP and VS, respectively� have been developed for shallowuried marine calcareous sediment by workers such as Hamilton1971, 1976�, Hamilton et al. �1982�, and Richardson and Briggs1993�. A theoretical model by Wood �1955� assumes that shallowediment is a suspension of solid particles in water; the model esti-ates the bulk modulus of this suspension as the harmonic average

f the solid and fluid components.However, Hamilton �1971�, Hamilton et al. �1982�, and Wilkens

t al. �1992� show that shallow buried marine deposits do transmithear waves. This implies that contacts exist between the grains,hich means the suspension model is inadequate for such sedi-ents. Hamilton �1971� points out the suspension model is invalid

or marine sediments, which have some rigidity but still can be usedo obtain a maximum estimate of VS. Wilkens et al. �1992� observehat Wood’s estimation of the bulk modulus of the most porous sedi-

ents is fairly close to the dynamic bulk modulus.To address this situation and to estimate the shear modulus and,

ventually, VS, Hamilton �1971� and Wilkens et al. �1992� assumehat the effective bulk modulus K is given by the suspension modelnd that the compressional modulus M is calculated from the mea-ured VP and bulk density ��b� as �bVP

2. Then the shear modulus G is3/4��M �K� and Vs � �G/�b.

A slightly different approach is proposed in Hamilton �1971� andn Hamilton et al. �1982�, where the decimal logarithm of the dry-rame bulk modulus Kdry relates to porosity � as 1.87355–3.41062�,here the modulus is in gigapascals and porosity is in fractions ofnity. Then the saturated-rock bulk modulus K is computed fromdry using Gassmann’s �1951� fluid substitution. Finally, the mea-

ured VP is used to calculate M, and G is obtained from M and K ashown above.

The rock physics of diatomites is addressed by Chaika �1998�,ho performed laboratory velocity measurements on samples from

he Monterey Formation �California�. However, at that time, no the-retical model was available to explain the observed relations be-ween velocity and porosity �Chaika and Dvorkin, 2000�.

Dvorkin and Prasad �1999� and Prasad and Dvorkin �2001� haventroduced a theoretical model for siliciclastic high-porosity marineediment. This model extends the soft-sand model of Dvorkin andur �1996� into the high-porosity range between the critical porosity

a) b) c)

igure 1. �a� Schematic representation of a rock with porous grains. �urves for PGSO and PGST �as labeled�. These models are the same ietween the total critical porosity � tc and one. �c� The total porosityorosity according to equation 1. Each line is computed for a fixed ing, starting with zero �the lowest diagonal line� and ending at one �t

ine� in increments of 0.2.


nd one. It connects the two end points, one given by the soft-sandodel at the critical porosity and the other for pure pore fluid at po-

osity one, by using the modified upper Hashin-Shtrikman boundHashin and Shtrikman, 1963�. In this model, the critical porosity issed as an intermediate elastic end point so the sediment frame canave nonzero elastic moduli above the critical porosity.

The principal modification to the model introduced here is to treatigh-porosity sediment as a pack of porous elastic grains that repre-ent minute calcareous or siliceous skeletons. The effective elasticoduli of the grains are calculated using the differential effective-edium �DEM, Appendix A� model, where the ellipsoidal inclu-

ions have a fixed aspect ratio and are filled with seawater. The elas-ic moduli of a pack of these grains are calculated using a modifiedscaled to the intergranular critical porosity� upper Hashin-Shtrik-an bound above the critical porosity and a modified lower Hashin-htrikman bound below the critical porosity. We apply this newodel to three Ocean Drilling Program �ODP� well data sets and

how that it matches the data, especially VS, better than previousodels.This principle of theoretically replacing the actual mineral with a

orous material can be applied, as appropriate, to various mineralo-ies and used with any of the existing rock-physics models, includ-ng traditional relations by Wyllie et al. �1956�, Raymer et al. �1980�,nd Krief et al. �1990�. We use the approach to mimic Chaika’s1998� diatomite data by including porous solids in the modified up-er Hashin-Shtrikman bound �Gal et al., 1998�. The main result ofur work is to introduce the porous-grain concept into existing rock-hysics models. By using well-log and laboratory data, we show thathis approach helps match relevant data where other relations fail.

PACK OF POROUS GRAINS

To model carbonate sediment, we propose a porous-grain–soft-and �PGSO� model, which treats the sediment as a pack of porouslastic grains �Figure 1a�. In this model, the intergranular porosity � i

s defined as

� i �� t � �g

1 � �g, �1�

here �g is the internal porosity of the grains, defined as the ratio ofhe intragranular pore volume to the total grain volume, and � t is the

total porosity. When � i � 0, � t � �g; when �g

� 0, � t � � i �Figure 1c�. The value � t can becalculated from the bulk density �b as

� t ��s � �b

�s � �w, �2�

where �s is the density of the mineral and �w is thedensity of water. In clean, calcareous sediment,the neutron porosity also provides an accurate es-timate for � t.

The effective bulk and shear moduli of thegrain material, Kg and Gg, are calculated using theDEM method �Norris, 1985�, where the ellipsoi-dal inclusions of the volumetric concentration �g

�the intragranular porosity� have a fixed aspect ra-tio and are filled with seawater �Appendix A�.

city-porosityorosity rangeintergranularular porosityer horizontal

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Elastic properties of porous-grain sediment E3

Once the elastic properties of a grain are determined, we assumehat the granular sediment is a pack of such grains. The critical po-osity �c of such packs is about 0.40 �Nur et al., 1998�. At �c, the to-al porosity is � tc � �g � �c�1 � �g�. To calculate the elastic mod-li of the pack, we examine two porosity domains: one where � i

�c and the other where � i ��c.In the former domain, the elastic model connects two end points in

he elastic-modulus-porosity plane: the effective moduli at � i � 0the moduli of the porous grain� and the moduli of a dense randomack of identical elastic porous spheres �with water-filled inclu-ions� at � i � �c. To interpolate between these two end points, wese the lower Hashin-Shtrikman bound, rescaled from porosity of–1 to a range of 0–�c �as in the soft-sand model of Dvorkin and Nur1996��. Specifically, the effective bulk and shear moduli of the dryranular frame, Kdry and Gdry, comprised of water-saturated porousrains are

Kdry �� i

�c

KHM �4

3GHM

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1 �� i

�c

Kg �4

3GHM

��1

�4

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�c

GHM � Z1�

1 �� i

�c

Gg � Z1�

�1

� Z1,

Z1 �GHM

6�9KHM � 8GHM

KHM � 2GHM� , �3�

here KHM and GHM are the Hertz-Mindlin �Mindlin, 1949� modulit the critical porosity.

For a dense random pack of identical elastic spheres with theertz-Mindlin contacts, KHM and GHM are �e.g., Mavko et al., 1998�

KHM ��3 C2�1 � � i�2Gg2

18�2�1 � �g�2 P ,

GHM �5 � 4�g

5�2 � �g��3 3C2�1 � � i�2Gg

2

2�2�1 � �g�2 P ,

�g �1

2

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2

3

Kg

Gg�

1

3

, �4�

here � g is Poisson’s ratio of the porous grains, P is the differentialressure acting upon the pack, and C is the average number of con-acts that each grain has with its neighboring grains �the coordinationumber�.

In the � i ��c domain, we also consider two end points: one at � i

�c and the other at � i � 1. At the former, the effective elasticoduli Kdry and Gdry are given by equation 4 as KHM and GHM; at the

atter, they are zero. In between these end points, we use �Dvorkinnd Prasad, 1999�


Kdry ��1 � � i

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1 � �c

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� i � �c

1 � �c

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�1

� Z1. �5�

ollowing Gassmann’s �1951� fluid substitution equations, we as-ume the shear modulus of the fully water-saturated sediment is thatf the dry frame �Gsat � Gdry� and its bulk modulus is

Ksat � Kg

� iKdry � �1 � � i�KfKdry

Kg� Kf

�1 � � i�Kf � � iKg � KfKdry

Kg

, �6�

here Kf is the bulk modulus of seawater.Finally, the elastic P- and S-wave velocities are

VP ��Ksat �

4

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�b, VS ��Gsat

�b. �7�

A counterpart to PGSO is the porous-grain–stiff-sand �PGST�odel. The only difference between the two is for � i ��c �Figure

b�. In this porosity range, the same two end points, one at zero po-osity and the other at the critical porosity, are connected by the mod-fied upper Hashin-Shtrikman bound �Gal et al., 1998�. As a result,e obtain

Kdry �� i

�c

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�

1 �� i

�c

Kg �4

3Gg�

�1

�4

3Gg,

Gdry � � � i

�c

GHM � Z2�

1 �� i

�c

Gg � Z2�

�1

� Z2,

Z2 �Gg

6�9Kg � 8Gg

Kg � 2Gg� . �8�

n the above models, the total porosity always should be larger orqual to the intragranular porosity. That is why the velocity-porosityurves in Figure 1b are within the �g �� t �1 interval.

The same approach, whereby we treat the solid phase as made oforous grains, is, in principle, applicable to any of the existing rock-hysics models. An example of this approach is the Wood–porous-rain �WPG� model, a modification of the Wood-Hamilton �Hamil-on, 1971� method to obtain a maximum estimate of shear velocityWilkens et al., 1992�, presented inAppendix B.

To apply the staged upscaling scheme explained above, where werst applied DEM to upscale porous grains and then the modifiedashin-Shtrikman bounds to account for pores between the grains,e assumed that the intragranular ellipsoidal micropores were much

maller than those pores among grains.


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nfluence of aspect ratio and intragranular porosity onlastic properties

Figure 2 illustrates the effects of varying aspect ratio �AR� and �g

n the P- and S-wave impedance �IP and IS, respectively� of pure cal-ite material, as predicted by PGSO for dry and water-saturated sedi-ent. The solid-phase bulk and shear moduli and density used in this

xample are Ks � 76.8 GPa, Gs�32 GPa, and �s � 2.71 g/cm3, re-pectively. The fluid’s bulk modulus and density are 2.391 GPa and.034 g/cm3, respectively. The differential pressure is 1.61 MPa.

igure 2. Sensitivity of the P- and S-wave impedance to intragranularonditions, using the PGSO model. In this example, the intragranularrows in the third-column frames indicate the direction of increasingerential pressure, critical porosity, and coordination number are 1.61


Clearly, if �g � 0, there is no effect of the aspect ratio of the intra-ranular inclusions on the impedance. However, as we increase �g,he effect of aspect ratio becomes more pronounced; for �g 0.3 andR � 0.01, both IP and IS in the dry sediment are essentially zero.his is not so for IP in water-saturated sediment. Here, the effect of

he pore fluid on the P-wave propagation appears to be especiallyignificant in low-aspect-ratio, very compliant intragranular pores.

Figure 3 displays similar plots for PGST and the same modelingarameters. The effects of �g and aspect ratio on the impedance are

ratio and porosity at dry �upper two rows� and wet �lower two rows�ity varies from 0.0 to 0.5, from left to right, in increments of 0.1. Theratio. The aspect ratios used are 0.01, 0.03, 0.1, 0.2, and 1.0. The dif-0.38, and 9, respectively.

aspectr porosaspectMPa,


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ssentially the same as for PGSO. The difference between the twoodels is manifested in the abrupt change in the impedance-porosity

ehavior at � tc. Other input parameters, such as differential pressurend pore-fluid compressibility, may also influence IP and IS signifi-antly �Dvorkin and Prasad, 1999�. In water-saturated sediment, theatter exhibits larger sensitivity to pressure than the former. Also,arge fluctuations in the salinity of the seawater, such as those thatan occur because of proximity to a salt dome, may affect stronglyhe compressibility of seawater and, as a result, the velocity and im-edance in the soft sediment.

Figure 4 illustrates the effects of varying aspect ratio and �g on the- and S-wave velocities of pure calcite material, as predicted by

igure 3. Same as Figure 2 but using the PGST model.


GSO and PGST for water-saturated sediment. Different curves cor-espond to different values of �g. In this example, �g varies from 0.0o 0.5 in increments of 0.1.

POROUS-GRAIN-MODEL SCENARIOS

We consider a saturated porous grain as a linear elastic solid withllipsoidal inclusions filled with compressible fluid. The PGSO andGST models can be applied to approximate three different porous-rain aggregate scenarios, depending on the effective fluid connec-ivity of the intragranular porosity in the grain. Each scenario is andealized representation of the rock pore-space morphology but isess idealized for existing synthetic materials. Depending on the


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ock’s nature at a microscopic scale, one of the following differentescriptions might be more appropriate than the others. Three sce-arios are possible:

� The intragranular inclusions are isolated and the intragranularpores are not connected with the intergranular pores �Figure5a�.

� The intragranular inclusions are connected and the intragranu-lar pores are not connected with the intergranular pores �Figure5b�.

� The intragranular inclusions are connected and the intragranu-lar pores are connected with intergranular pores �Figure 5c�.

We imagine realizing each of these scenarios by physically creat-ng or eliminating the pore-to-pore connections or by changing theeriod of the externally applied stresses to be faster or slower than

igure 4. Sensitivity of the P- and S-wave velocities to intragranularitions, using PGSO �upper two rows� and PGST �lower two rows� mle, the intragranular porosity varies from 0.0 to 0.5 in increments of 0ate the direction of increasing intragranular porosity �g. The differeal porosity, intragranular aspect ratio, and coordination number are 1nd 9, respectively.

b)a) c)

igure 5. Three porous-grain scenarios. �a� The intragranular inclusihe intragranular inclusions are connected. �c� The intragranular incd and the intragranular porosity is connected with the intergranular p


he pore-to-pore diffusion times. When we speak of high-frequencyersus low-frequency response, the implication is that the connec-ivity is controlled effectively by the diffusion time versus frequencyMavko and Jizba, 1991; Mukerji and Mavko, 1994; Dvorkin et al.,995; Mukerji et al., 1995�.

These three different rock-fluid scenarios have different elasticehaviors, varying with intragranular and intergranular porosities,spect ratio of the intragranular inclusions, orientation of the inclu-ions with respect to the applied stress field , and the magnitude,requency, and direction of . All three cases are undrained scenari-s in the sense that the total fluid mass in the rock sample is constant.

We can define different degrees of fluid relaxation that depend onhe degree of effective pore connectivity. Scenario 1 is least relaxedecause there is no pore-to-pore equilibration of pore pressures. Sce-ario 2 is slightly more relaxed because intragrain pores are equili-rated with respect to pore pressures but there is no effective fluid

communication between the intragranular andthe intergranular porosities. At low frequency,pressure equilibrium in material �scenario 3� canbe reached; in this relaxed scenario, all pores areat the same pressure. The combination of porelas-ticity and effective media theory allows one to de-rive high- and low-frequency moduli and to pre-dict elastic-wave dispersion �Le Ravalec andGuéguen, 1996; Boutéca and Guéguen, 1999�.

In this study, to determine the effective elasticproperties of these three different porous-grainscenarios, we use two models: DEM �Norris,1985� and the combination DEM-Gassmann. TheDEM model assumes that saturated inclusionsare isolated with respect to flow; thus, it simulatesvery-high-frequency �VHF� saturated-rock be-havior that may be appropriate to ultrasonic fre-quency ��1 MHz�. Here, VHF refers to fluid-re-lated effects; but because the models are all effec-tive medium models, the wavelengths are stillmuch longer than any scale of grains or pores�Budiansky, 1965; Wu, 1966; O’Connell and Bu-diansky, 1974; Berryman, 1980�. At low frequen-cy when there is time for wave-induced pore-pressure increments to flow and equilibrate�Mavko et al., 1998; Boutéca and Guéguen,1999�, it is better to find the effective moduli fordry inclusions and then saturate them with theGassmann low-frequency relations �Mavko et al.,1998�. The combination DEM-Gassmann modelmay be appropriate at well-log frequencies. Anymodel that gives the low-frequency response orequalized pore pressure is called Gassmann con-sistent �Thomsen, 1985�.

The application of a specific model dependsnot only on the frequency we are interested in andthe description of the medium but also on physi-cal conditions. If the applied stress field on theporous grains is not isotropic, the fluid pressureinduced by depends on the shape and orienta-tion of the inclusions �Kachanov et al., 1995;Shafiro and Kachanov, 1997� as well as on the fre-quency of . If we assume that the applied exter-

ty at wet con-In this exam-arrows indi-

ressure, criti-Pa, 0.38, 0.5,

isolated. �b�are connect-

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al stress field is isotropic and that all inclusions have identical as-ect ratios, then P is the same in all intragranular inclusions. Thus,nder these assumptions any fluid experiences the same pressure inll pores. This satisfies the Gassmann assumption of pore-pressurequilibrium.

If the intragranular inclusions have a preferred orientation, theack becomes anisotropic. To describe its effective elastic proper-ies, the anisotropic version of the DEM method should be applied. Ifhe intragranular inclusions are distributed randomly but the appliedxternal stress is not isotropic, then the pressures in individual ellip-ical pores depend on the orientation of the applied stress and theEM method is inconsistent with the Gassmann theory.The accuracy of inclusion-based models is questionable at high

nclusion concentration �Kachanov, 2007; Grechka, 2008�. At lowrequency, Gassmann predicts the change in effective elastic moduliaused by a variation in the bulk modulus of fluid filling a fully inter-onnected pore space at any inclusion concentration because porosi-y connectivity ensures pressure equilibrium in the pore fluid with-ut detailed information about the microstructure. Thus, in contrasto the inclusion models, Gassmann remains rigorously correct at ar-itrary porosity �Mavko et al., 1998; Grechka, 2008�.

For a porous-grain material of the type mentioned in scenario 1earlier�, the fluid inside the pores will remain isolated at ultrasonicrequency and the elastic properties of the porous grains can be esti-ated using the DEM method. This corresponds to a model with iso-

ated pores or the high-frequency range of acoustic waves �Mavkot al., 1998�. If external isotropic pressure is applied in this pack oforous grains, the intragranular pressure Pg generally will differrom the intergranular pressure Pi.

For a scenario 2 porous-grain material, thelastic properties of the grains can be estimatedsing the DEM method for a high-frequency ap-roximation �such as ultrasonic measurements�.he DEM-Gassmann method can be used for the

ow-frequency approximation �well logs� be-ause the fluid inside the pores can communicatend fluid may flow from one pore to another buto bulk flow takes place through the porous grainO’Connell and Budiansky, 1977�. In this porousrain pack, if external pressure is applied, Pg williffer from Pi.

For a porous-grain material of scenario 3, thelastic properties of the grains can be estimatedsing the DEM method for the high-frequencypproximation and the DEM-Gassmann methodor the low-frequency approximation. If externalressure is applied to the pack of grains and we al-ow the system to equilibrate, Pg will equal Pi. Ifigh-frequency external pressure is applied, de-ending on how high the frequency is with re-pect to flow, the material in scenario 3 may allowuid diffusion inside each porous grain or inter-rains, and it may behave elastically as materialsor scenarios 1 and 2.

In all three scenarios, to account for the un-rained elastic property effects of the intergranu-ar porosity, we can use a combination of the

odified Hashin-Shtrikman bound and Gas-mann if we want a low-frequency velocity ap-roximation �well logs� or a combination of the

Figure 6. TopGassmann �localculation isarrows indicaframes but usi


odified Hashin-Shtrikman and DEM if we want a VHF velocitypproximation �ultrasonic measurements�.

Figure 6 illustrates the effects of frequency on the pure calcite po-ous-grain pack. We show the PGSO and PGST effective P- and-wave velocities �VP and VS, respectively�, computing the effectiveroperties of the porous grains in two ways: �1� using the DEMPGSO-high frequency� method and �2� using the combined DEM-assmann �PGSO-low frequency� method. We calculated for threeifferent intragranular aspect ratios: 0.05, 0.1, and 0.25. The differ-nce between the low- and high-frequency velocities is significantnly for aspect ratios �0.25. For higher aspect ratios, the differenceetween the elastic velocities determined by DEM and by DEM-assmann does not exceed 2%. Based on this, we can infer that theuid flow in the intragranular pores has a significant influence onlyor small aspect ratios. We observed the same behavior for the PGSTodel.

ODP DATA SETS

We studied deep marine carbonate sediment from three ODP wellites: 1172, 998, and 1007. The porosity of this sediment is between.25 and 0.80. For our study, we selected depth intervals with essen-ially pure calcium carbonate content.

Site 998 �Carey and Sigurdsson, 2000� is located on the Caymanise �the Caribbean� between the Yucatan Basin to the north and theayman Ridge and Cayman Trough to the south at a water depth of

: VP and VS PGSO velocities using DEM �high frequency� and DEM-ency� to calculate the effective properties of the porous grains. Ther three intragranular inclusion aspect ratios: 0.05, 0.1, and 0.25. Thedirection of increasing aspect ratio. Bottom frames: Same as topPGST model. LF � low frequency; HF � high frequency.

framesw frequdone fote the


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E8 Ruiz and Dvorkin

190 m. We concentrated on the depth interval between 210 and70 m below the seafloor �mbsf�, which consists of nannofossilhalks.

Site 1172 �Robert, 2004� is located about 150 km southeast ofasmania at a water depth of 2622 m. The interval under examina-

ion �150–350 mbsf� consists of �a� white and light green-gray nan-ofossil ooze and �b� pale yellow and light gray foraminifer-bearingannofossil chalk with increased foraminifer content and minoromponents of clay and volcanic glass.

Site 1007 �Wright and Kroon, 2000� is located on the toe of theestern Great Bahamas Bank slope in 669 m of water. In this site,

he studied depth interval is divided into four units. Unit I, from–203 mbsf, consists of a succession of nannofossil ooze and vari-ble lithified wackestones that transit to packstones with increasingepth. Unit II, from 203–302 mbsf, is of early Pliocene age and con-ists entirely of bioturbated light gray to pale yellow foraminiferannofossil chalk. Unit III, from 302–363 mbsf, displays variableegrees of lithification of foraminifer wackestone, nannofossilhalk, and nannofossil limestone. Unit IV, from 363–696 mbsf, isade of 5- to 10-cm-thick layers of fine-grained packstone and ex-

ibits alternations between densely cemented and weakly cementedediment �0.1–1.0 m thick� that shows evidence of compactionWright and Kroon, 2000�. Petrographic studies and scanning elec-ron microscopy �SEM� images show different degrees of foramini-er cementation in this well �Kenter et al., 2002�.

a) c)

b) d)

igure 7. P-wave velocity versus total porosity for the ODP data setsa� site 1172, �b� site 998, �c� site 1007, and �d� S-wave velocity sitolor coded by depth �mbsf�. Data from Rafavich �1984� that are essow-porosity rock are displayed for reference �black asterisks�. ThGSO and PGST, with the intragranular aspect ratio 0.5, coordinatioorosity 0.38, and intragranular porosity 0.26; DEM with aspect rationtire sediment by treating it as a solid with inclusions; Raymer et aith Castagna et al. �1993�, the latter to obtain the S-wave velocity;

ombined with Castagna et al. �1993�; and the suspension model, wodeled as a porous solid �WPG�. We use the same parameters as in P


For all of these sites, measurements of VP, �b, neutron-porosity,amma-ray, and carbonate �CaCO3� and organic/inorganic carbonontent are available. Site 1172 has, in addition, VS data. Simpleineralogy �the CaCO3 content above 0.85� and absence of clay de-

ermined our choice of these data sources. Wireline logging was con-ucted with successful runs of the Schlumberger-GHMT sonic tooltring �Robert, 2004�.

atching ODP data with models

For a porous-grain material of scenario 1, the fluid inside the poresemains isolated at ultrasonic frequency and the elastic properties ofhe porous grains can be estimated using the DEM method, whichorresponds to a model with isolated pores or the high-frequency ap-roximation.

In Figure 7, we compare the measured VP and VS with curves pro-uced by the DEM �not applied just to the grains but to the entire sed-ment, treating it all as a solid with inclusions�, Raymer et al. �1980�,

yllie �1956�, and Castagna et al. �1993� models as well as PGSOnd PGST. We assume pure calcite porous grains with isolated inclu-ions and fully brine saturated. The matrix in all models is pure cal-ite, and the pore fluid is brine. The bulk modulus and density of therine in each well were calculated according to Batzle and Wang1992� using salinity of 36,000 ppm and site-specific temperaturend pressure. We assumed the same salinity for calculating the den-

sity and bulk modulus of the intragranular water,intergranular water, and bottom seafloor water.

The total porosity � t in each well was calculat-ed from the bulk density �b by assuming that thedensity of the mineral �s is that of calcite, or2.71 g/cm3. The density of water �w varies slight-ly with depth and location but remains very closeto the value used above, 1.034 g/cm3.

The model curves in Figure 7 were producedfor fixed brine properties, which are 2.391 GPafor the bulk modulus and 1.034 g/cm3 for density,averaged among the data sets. The differentialpressure at sites 1007, 998, and 1172 ranges fromapproximately 1–8 MPa, from approximately1–4.5 MPa, and from approximately0.8–2.3 MPa, respectively. In Figure 7, the differ-ential pressures were kept constant: 1.61 MPa forsite 1172, 4.18 MPa for site 1007, and 2.68 MPafor site 998, the averaged pressure values in eachsite depth interval. The DEM model used to cal-culate the elastic moduli of the porous grains in-corporates a single aspect ratio for all data sets.The best fit to the data is for an aspect ratio of 0.1.The parameters used in PGSO and PGST in theentire interval are intragranular porosity, 0.26;critical porosity, 0.38; coordination number, 9;and aspect ratio of the intragranular inclusions,0.5. Among all of these models, PGSO providesthe best match to the velocity data. AlthoughDEM applied to the entire sediment with AR� 0.1 matches VP, it overestimates VS data forsite 1172.

In Figures 8–10, we plot the predicted andmeasured VP versus depth in each of the selectedwells. In Figure 11, we plot the predicted and

examination:All plots are

y pure calciteel curves areber 9, criticalapplied to the0� combinedet al. �1956�

he grains arend PGST.

undere 1172.entialle modn numof 0.1l. �198Wylliehere tGSO a


Fv�md

Figure 9. ODP well site 1007. The display is the same as in Figure 5.



measured VS versus depth in site 1172. The inputfor the models used here are �1� the bulk modulusand density of water, which vary with depth ac-cording to the increasing pore pressure �hydro-static� and temperature, and �2� the differentialpressure, which is the integral of the bulk densityminus the density of water with respect to depth.The bulk density data were missing between thesea bottom and the shallowest depth datum. Forpressure estimation, these missing curves wereapproximated by a linear interpolation betweenthe shallowest density data and the density at thesea bottom. The latter was 1.537 g/cm3, as calcu-lated for a calcite sediment with �assumed� 0.70porosity.

The original log curves contained many �ap-parently artificial� spikes. To remove them, wesmoothed the curves by using the arithmetic aver-age for porosity and density and the Backus�1962� average for the velocity. The running win-dow contained 10–15 original depth increments.

In all three wells, PGSO with intragranular in-clusion aspect ratio of 0.5 accurately matched themeasured VP except for the lower-porosity inter-vals in site 1007 �Figure 9�. In these intervals, thesoft-sand model is unsuitable for the rock, whichis apparently more consolidated and harder thanin other parts of the wells. The Miocene section atsite 1007 has been affected by diagenesis; as a re-sult, the sediments appear to be fully lithified be-low � 300 mbsf �Wright and Kroon, 2000�. TheDEM model used with a constant aspect ratio of0.1 also provides a good match to the data, exceptin the high-porosity intervals in sites 1007 and1172 where it is apparently unsuitable. The sus-pension �Wood, 1955� model �not displayed here�strongly underestimates the data, whereas boththe Raymer et al. �1980� and Wyllie et al. �1956�equations �the former not displayed here� overes-timate the velocity. The WPG model, with intra-granular inclusion aspect ratio of 0.5, underesti-mates the VP data in all three wells; however, thereduction decreases at shallow depth where thesediment is more in suspension condition.

The Castagna et al. �1993� VS predictor appliedto the Wyllie et al. �1956� equation overestimatesthe measured VS in the entire interval �Figure 11�.So does the DEM model. The Hamilton-Gass-mann �Hamilton, 1976� model provides a good VS

prediction only for very shallow depths where theporosity exceeds 0.45. The WPG �AR � 0.5�model provides a good VS prediction in the entireinterval. However, PGSO was the only modelthat provided an accurate, consistent match to themeasured VS in site 1172. PGSO used with the in-tragranular aspect ratio, 0.5; coordination num-ber, 9; critical porosity, 0.38; and intragranularporosity, 0.26 accurately matched these data inthe upper interval of the well, above 271 mbsf�Figure 11�.

tent; P-waveorous grainsused in this

e model pre-

igure 8. ODP well site 998. From left to right: total porosity and CaCO3 conelocity as measured and predicted by Wood’s suspension model with pWPG�, Wyllie et al. �1956�, DEM, and PGSO models. The aspect ratiosodeling are listed in the legend. The data are shown as a thin solid line; th

ictions are shown as a thick solid line.


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E10 Ruiz and Dvorkin


To achieve a good match in the lower part ofthe well below 271 mbsf, where the sediment ismore consolidated, we changed these parametersby increasing the coordination number C to 15and simultaneously reducing the intragranularporosity from 0.26 to 0.20, consistent with the on-set of compaction. These changes increase VS butleave VP essentially the same �Figures 10 and 11�.The reason is that �g and C affect the VP/VS ratioand, as a result, � . However, these two ratios aremore sensitive to changes in C than in �g �Figure12�:As C increases, the dry frame becomes stifferand the wet-rock � decreases. As a consequence,different variations of C and �g produce differenteffects on VP and VS, such as simultaneously in-creasing or reducing VP and VS, keeping both ve-locities constant, or keeping VP constant and in-creasing VS, as in our case.

Figure 13 shows a crossplot of VP versus VS asmeasured in site 1172; the Pickett �1963�, DEM,Krief et al. �1990�, Castagna et al. �1993�, andPGSO model predictions are plotted. The firsttwo predictors overestimate VS at any given VP.The Krief et al. �1990� method correctly predictsVS in the high-porosity part of the interval, but theCastagna et al. �1993� method overestimates it.Conversely, the latter provides a satisfactory VS

prediction in the lower-porosity part of the inter-val, but the former underestimates VS. PGSO, ifused with consistently adjusted inputs, correctlypredicts VS in the entire interval.

Finally, in Figure 14, we compare the predictedto measured velocity according to selected mod-els: DEM with aspect ratio of 0.1, Hamilton�1971�, and PGSO. The latter model mimics thedata accurately, but the former two fail.

DIATOMITES

Consider the laboratory measurements byChaika �1998� performed on room-dry diato-mites from three Monterey Formation reservoirsin California. The mineralogy of these samplesincludes opal-A, opal-CT, quartz, clay, analcima,and feldspar as well as organic components �Fig-ure 15�. The elastic moduli and densities of thesecomponents are from Chaika �1998� and Mavkoet al. �1998� and are shown in Table 1.

Chaika �1998� identifies two distinct patternsof porosity reduction in reservoir rocks of theMonterey Formation as they undergo silica di-agenesis from opal-A through opal-CT to quartz.In pattern 1, the porosity reduction appears to re-sult from increased amounts of nonsilica minerals�and quartz, if present� and the grain density in-creases with decreasing porosity. In pattern 2, theamount of opal-CT �and quartz, if present� in-creases, but the fraction of nonsilica minerals isrelatively constant and the grain density decreas-es with decreasing porosity. Because the samples

lcite content;�, DEM, and. The PGSO

actory match

ured and pre-PG, Hamil-

he same as in

igure 10. ODP well site 1172. From left to right: total porosity and XRD-ca-wave velocity as measured and predicted by WPG, Wyllie et al. �1956GSO models. The aspect ratios used in this model are listed in the legendodel, for coordination number 15 and grain porosity 0.20, provides a satisf

o the data below 271 mbsf.

igure 11. ODP well site 1172. From left to right: S-wave velocity as measicted by Wyllie et al. �1956� combined with Castagna et al. �1993�, DEM, Won �1971, 1982�, and PGSO models. The parameters used in this model are tigure 7. PGSO provides a satisfactory match to the data below 271 mbsf.


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rom Cymric contain opal-A, they are the only rocks that could bealled diatomite.

We modeled these data using PGSO and PGST models, assumingiliceous porous grains with isolated inclusions. The model parame-ers, the same for all samples, are intragranular aspect ratio, 0.6; co-rdination number, 10; critical porosity, 0.42; and intragranular po-osity, 0.15.

We observed that PGSO only matches the high-porosity data �notisplayed here�, but PGST �Figure 15� provides a satisfactory matchn the entire porosity range �because above the critical porosity these

odels are the same�. To illustrate the quality of this match, werossplot the PGST-predicted velocity versus theeasured data in Figure 15. In spite of a few out-

ying data points �mostly for VS�, we deem theatch satisfactory and practically usable. This

xample illustrates the utility of the porous-grainpproach with the soft-sand model and with otherxisting models, the stiff-sand model in particu-ar.

DISCUSSION

The porous-grain concept appears to be gener-lly applicable to medium- to high-porosity sedi-ents. Consider Figure 16, which shows VP ver-

us � t for the marine chalk data set used in Nur etl. �1998�. The trend apparent in these data can beatched with a PGSO curve with a constant dif-

erential pressure 4 MPa, C � 9, �c � 0.42, andg � 0.22. Fabricius �2003� uses a similar ma-

ine carbonate data set. Figure 16 implies that theorous-grain model is appropriate for the Fabri-ius �2003� data as well. In his original paper,abricius �2003� explains the velocity behaviorf chalk data with different degrees of burial di-genesis by using an additional �free� parameterisoframe� to fill the space between the lower andpper modified Hashin-Shtrikman bounds withodel curves. Here, we can explain the data usingmodel with physics- and geology-driven pa-

ameters.In the case of opal data sets, Chaika �1998�

oints out that the process of transitioning frompal-A to opal-CT begins with grains in an un-ithified rock of small, hollow �porous� particlesf opal-A where the particle contacts have smallross-sectional areas. Instead of forming over-rowth cements as clastic rocks do, these opalineorous particles dissolve and reprecipitate aspal-CT �Williams et al., 1985�; some new parti-les form and some particles grow. Both of theserocesses result in particle contacts with largerross-sectional areas �equivalent to cementation�ecause the grains and cement in this model areomprised of opal-CT.

To predict the velocities of rocks coming fromhese three fields, Chaika �1998� uses two differ-nt rock-physics models. The first model is thepper Hashin-Shtrikman bound, scaled to theritical porosity to describe the samples from As-

Figure 12. WeThe coordinatvaries from 0.in the top-mid

Figure 13. S- vin mbsf� and pCastagna �199hand frames c


halto and McKittrick. The second model is a combination of theertz-Mindlin theory with the upper Hashin-Shtrikman bound forymric samples. In both models, the total porosity was replacedith intergranular porosity. In our study, we observe that all threeata sets �Asphalto, McKittrick, and Cymric� can be modeled as aack of porous cemented grains and that PGST provides a satisfacto-y match in the entire porosity range.

As with any rock-physics model, the PGSO and PGST modelsave advantages and disadvantages. Among the advantages are �1�he physically and texturally consistent treatment of calcareous sedi-

ent and opalines comprised of porous grains and �2� the consisten-

Poisson’s ratio versus intragranular porosity according to PGSO.ber varies from 4 to 16 in increments of two, and the total porosity

.70. All other PGSO model parameters are kept constant. The arrowe indicates the direction of increasing coordination number C.

-wave velocity. Comparison of site 1172 data �color coded by depthions according to PGSO, DEM, and Pickett �1963� on the left, andKrief et al. �1990�. The two PGSO branches in the left- and right-nd to the upper and lower depth intervals in this well.

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y of model predictions with selected offshore calcareous and on-hore opaline data. The disadvantages are linked intimately to thetructure of the model, which requires input such as intragranularorosity, intragranular inclusion aspect ratio, critical porosity, andoordination number, which are somewhat idealized and not imme-iately available from experimental measurements.

However, this is a common feature of any micromechanical mod-l. One way of resolving the resulting ambiguity is to select inputshat are reasonable, such as the critical porosity varying within the.35–0.45 range and a coordination number varying between 5 and5. We can calibrate these inputs to existing data and then link themo specific geographic locations and depth intervals. Once such a cal-

igure 14. ODP well site 1172. Predicted versus measured P- and-wave velocity, color coded by depth in mbsf. In the upper tworames, we use the DEM �AR � 0.1� and Hamilton-GassmannHamilton, 1971� models; in the lower frame, we use the PGSOodel.
t

bration is accomplished, the model can be used in a predictive modeway from well control.

The disadvantage of PGSO and PGST turns into an advantage ife utilize these models in an exploratory mode. By varying the in-

a)

b)

c)

igure 15. �a� P- and S-wave dry-rock velocity versus porosity aseasured and predicted by PGST. The filled circles are measured P-

nd S-wave velocities on a zero-porosity opal-CT sample. The mea-urements are room dry at 10 MPa confining pressure. �b� PredictedPGST� versus measured velocity. �c� X-ray mineralogical composi-
ion of opaline rock samples. Data from Chaika �1998�.

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uts in reasonable and site-consistent ranges, we can explore thelausible ranges of elastic properties as well as their interrelation.

Our examples emphasize that existing VS predictors may be accu-ate within certain depth ranges but do not provide correct predic-ions within the entire depth range. However, PGSO provides an ac-urate prediction in the entire depth interval at the expense of vary-ng the model parameters within a reasonable and depositionallyonsistent range.

CONCLUSIONS

Our approach and models appear to apply to sediment with porousrains, such as calcareous and diatomaceous ooze. The parametersf these models — specifically, the intragranular porosity and coor-ination number — can be linked to compaction and diagenesis. Toake such a link predictive, robust, and repeatable, one needs to ex-

lore systematically the applicability of this model to various high-

able 1. Mineralogy of diatomite samples (Chaika, 1998;avko et al., 1998).

ineralBulk modulus

�GPa�Shear modulus

�GPa�Density�g/cm3�

pal-A/CT 14.219 12.580 2.0

uartz 37.88 44.31 2.649

lay 21.83 7.0 2.56

nalcime 55.629 26.231 2.712

yrite 87.91 137.9 5.1

alcite 76.8 32.0 2.71

eldspar 53.36 27.04 2.56

rganic 2.937 2.733 1.3

igure 16. P-wave velocity versus total porosity crossplots for theur et al. �1998� chalk data set. The model curve displayed is PGSO,ith the intragranularAR 0.5, coordination number 9, critical poros-

ty 0.42, and intragranular porosity 0.22.


uality data sets with established geologic records and mineralogies,n analysis beyond the scope of this paper. Our approach, where wereat the solid phase as a porous material, can be used to modify ex-sting rock-physics models. These models can become part of the ar-enal of rock-physics relations used to generate synthetic seismicata as well as in real seismic data interpretation for rock properties.

ACKNOWLEDGMENTS

This work has been supported by the Stanford Rock Physics andorehole Geophysics project and its industry affiliates. The techni-al advice of Gary Mavko, Tapan Mukerji, Amos Nur, and Jonathanayne was highly beneficial.

APPENDIX A

DEM THEORY

DEM theory assumes that a composite material can be construct-d by making infinitesimal changes in an existing composite. If theffective bulk and shear constants of the composite are K*�y� and*�y�, where the volume fraction of the inclusion phase is y, the

quations governing the changes in these constants are �Mavko etl., 1998�

�1 � y�d

dyK*�y� � �K2 � K*�P*2�y�

�1 � y�d

dyG*�y� � �G2 � G*�Q*2�y� , �A-1�

ith initial conditions K*�0� � K1 and G*�0� � G1, where K1 and

1 are the bulk and shear moduli of the initial host material, respec-ively, and K2 and G2 are the bulk and shear moduli of the incremen-ally added inclusions, respectively.

In porous rock, y is the total porosity �. The coefficients P and Qre geometric factors dependent upon the shape of the inclusionMavko et al., 1998�. Here, we use the ellipsoidal inclusions.

The superscript *2 for P and Q indicates factors for the inclu-ions; the superscript * is for the background medium whose bulkodulus is K* and shear modulus is G*. Fluid-saturated cavities are

imulated by setting the inclusion shear modulus to zero.The coefficients P and Q for ellipsoidal inclusions of arbitrary as-

ect ratio are given by

P �1

2Tiijj, Q �

1

2�Tijij �

1

3Tiijj� , �A-2�

here the tensor T relates the uniform far-field strain to the strainithin the ellipsoidal inclusion �Wu, 1966�. In our study, the elasticroperties of the inclusions are set as those of seawater at in situ con-itions, and the matrix properties are for pure calcite.

The DEM model is physically realizable �Norris, 1985� andherefore is always consistent with rigorous effective mediumounds.


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APPENDIX B

APPLICATION OF THE POROUS-GRAINCONCEPT TO WOOD’S MODEL

The Wood �1955� model assumes that in high-porosity unconsol-dated sediment, the P-wave velocity can be estimated as in a suspen-ion of solid particles in the load-bearing fluid:

VP ��KR

�b, �B-1�

here KR is the Reuss average of the bulk moduli of the solid and flu-d phases and where �b is the bulk density of the suspension:

KR�1 � � iKf

�1 � �1 � � i�Kg�1,

�b � � i� f � �1 � � i��g. �B-2�

n the original model, the subscript g refers to the properties of theure nonporous mineral. Here, we modify this approach by assum-ng the grains are porous; therefore, �g��s�1��g� � � f�g and Kg

s determined from DEM. The value � i is the intergranular porosity,ccording to equation 1.

Hamilton �1971� and Wilkens et al. �1992� acknowledge thatuch unconsolidated sediment still supports shear-wave propaga-ion; therefore, its shear modulus is not zero �as assumed by the sus-ension model�. They point out that the suspension model is not val-d for marine sediments, which have some rigidity but still can besed to obtain a maximum estimate of VS. To estimate VS, they calcu-ate the effective shear modulus of sediment as G� �3/4��M �KR�,here the compressional modulus M is calculated from the mea-

ured P-wave velocity VP and bulk density �b as �bVP2. Then, VS

�G/�b. Here, we follow the same ad hoc approach but use porousrains instead of a pure solid. The corresponding model VS curve isisplayed in Figure 11.

An example of calculating VP in a suspension of porous calciterains in seawater for varying intragranular porosity with fixed as-ect ratio as well as for varying aspect ratios with fixed intragranularorosity is displayed in Figure B-1. Our results suggest that Wood’s

a) b)

igure B-1. The P-wave velocity versus porosity in a suspension of pontragranular aspect ratio is fixed at 0.5. The intragranular porosity isthe bold baseline� to 0.4 in increments of 0.1. The velocity increaseragranular porosity. �b� The intragranular porosity is fixed at 0.4. Tect ratio is decreasing from 0.5 �the upper curve� to 0.0 �the curve aine�. The baseline curves in both frames are for suspension withrains.


estimate of the bulk modulus of the most poroussediments is fairly close to the dynamic bulkmodulus, an observation that can be used to esti-mate shear-wave velocity.

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sediment with porous grains: rock-physics model and ......sediment with porous grains: rock-physics...

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