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GEOPHYSICS, VOL. 71, NO. 6 �NOVEMBER-DECEMBER 2006�; P. F165–F171, 7 FIGS., 2 TABLES.10.1190/1.2356909

eismic reflections of gas hydrate from perturbational forward modeling

ngrid Cordon1, Jack Dvorkin1, and Gary Mavko1

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ABSTRACT

We perturb the elastic properties and attenuation in theArctic Mallik methane-hydrate reservoir to produce a set ofplausible seismic signatures away from the existing well.These perturbations are driven by the changes we impose onporosity, clay content, hydrate saturation, and geometry. Thekey is a data-guided, theoretical, rock-physics model that weadopt to link velocity and attenuation to porosity, mineralogy,and amount of hydrate. We find that the seismic amplitude isvery sensitive to the hydrate saturation in the host sand and itsporosity as well as the porosity of the overburden shale. How-ever, changes to the amount of clay in the sand only weaklyalter the amplitude. Attenuation, which may be substantial,must be taken into account during hydrate reservoir charac-terization because it lowers the amplitude to an extent thatmay affect the hydrate-volume prediction. The spatial struc-ture of the reservoir affects the seismic reflection: A thinly-layered reservoir produces a noticeably different amplitudethan a massive reservoir with the same hydrate volume.

INTRODUCTION

An estimated 7�105 Tcf of methane is trapped in gas hydratesround the world �Kvenvolden, 1993�. This makes hydrates a poten-ial future energy source and has prompted a recent increase in hy-rate research worldwide. Gas hydrate is an icelike crystalline solidormed from water molecules that encase and trap gas molecules.ydrate develops and is stable at high pressure and low temperature

onditions in the sediments of ocean basins and arctic regions. Aommon seismic manifestation of offshore methane hydrate is a bot-om-simulating reflector �BSR� caused by the impedance contrastetween sediment containing solid gas hydrate and underlying sedi-ent containing free gas. A BSR is often located at the base of the

as-hydrate stability zone, below which the temperature is too highor gas hydrate to exist. A BSR is parallel to the water-bottom reflec-

Manuscript received by the Editor March 2, 2005; revised manuscript rece1Stanford University, Stanford Rock Physics Laboratory, Geophysics D

vorkin@standford.edu; mavko@stanford.edu.2006 Society of Exploration Geophysicists.All rights reserved.

F165

or and has the opposite polarity. Hydrates have also been found on-hore beneath permafrost, e.g., in the Mackenzie River delta in Can-da.Areflector associated with the base of the hydrate-stability zonenshore is not well pronounced �Collett and Dallimore, 2002�. Rath-r, the seismic response shows high- and low-amplitude anomaliesinked to alternating high-impedance, hydrate-bearing sand andow-impedance sediment without gas hydrate. Although the gas-hy-rate reservoir can often be detected and delineated from its higheismic amplitude, rigorous rock-physics treatment can lead to accu-ate hydrate quantification.

Synthetic seismic modeling is a powerful tool for quantifying aonventional reservoir in terms of lithology and porosity as well asore fluid and pressure. Usually, an existing well is selected from anrea geologically similar to the prospect under examination. Thenorosity, lithology, and pore fluid in the well are perturbed accordingo an anticipated scenario away from the well to create a plausiblearth model. The corresponding changes in the elastic properties arealculated, and synthetic traces are generated and compared to realeismic data. The underlying supposition is that if the real and syn-hetic seismic responses are analogous, the properties and conditionsn the subsurface creating these responses are similar as well. Sys-ematic perturbations to the model help create a record of seismicignatures of lithology, porosity, and fluid away from well control,hus setting realistic expectations for reserve detection and quantifi-ation as well as helping to optimize the site-specific selection ofeismic attributes. The same methodology is applicable to a meth-ne-hydrate reservoir where the main variables are hydrate concen-ration and distribution as well as the background elastic properties.

To this end, our goal is to show how to forecast the onshore seis-ic signatures of gas hydrate, depending upon the amount of hy-

rate in the pore space, thickness and distribution of gas-hydrate lay-rs, as well as the porosity and lithology of a gas-hydrate reservoir.he approach is rock-physics-driven, perturbational, synthetic seis-ic modeling.Three elastic interval properties are needed in synthetic seismic

eneration: the P- and S-wave velocities and bulk density. Inelasticttenuation may affect the amplitude as well. The presence of gas hy-rate has been shown to have a noticeable effect on attenuation

ruary 16, 2006; published online October 20, 2006.nt, 397 Panama Mall, Mitchell #319, Stanford, California 94305. E-mail:

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Guerin and Goldberg, 2002� and thus must be accounted for in syn-hetic seismic generation.

Amodel that links the velocity and attenuation to basic rock prop-rties and gas-hydrate concentration is a key component of perturba-ional synthetic seismic modeling. Here we adopt a data-driven, the-retical, rock-physics model that links the elastic moduli and seis-ic attenuation to mineralogy, porosity, and amount of methane hy-

rate in the pore space.

able 1. Gas hydrate model input parameters.

Bulk modulus�GPa�

Shear modulus�GPa�

Density�g/cm3�

lay 21 7 2.58

uartz 36.6 45 2.65

ydrate 7.7 3.2 0.91

ater 2.2335 0 1.008

ethane 0.11 0 0.23

igure 1. Measured open-hole, well-log curves from Mallik 2L-38.a� Gamma ray. �b� Water saturation. �c� Porosity. �d� P-wave veloci-y �modeled P-wave velocity in red�. �e� S-wave velocity �modeled-wave velocity in red�. �f� Poisson’s ratio.

This study focuses on a theoretical model based on open-hole datarom the Mallik 2L-38 well on Richard’s Island in the Mackenzieiver Delta, Canada �Figure 1�. By varying the reservoir properties,e produce a catalogue of expected seismic responses from plausi-le geologic scenarios in proximity to our data source. In addition toxploring the effect of elastic properties on the seismic responses,e also investigate the effect of seismic attenuation in the presencef gas hydrates.

DATA SET

The data used were collected as part of the JAPEX/JNOC/GSCJapan Petroleum Exploration Company/Japan National Oil Com-any/Geological Survey of Canada� gas-hydrate research programo explore hydrate accumulation in the Arctic through the investiga-ion of core samples, well logs, and vertical seismic profiling at the

allik 2L-38 well �Dallimore et al., 1999�. The well was drilled in998 to a depth of 1150 m. The base of the permafrost is at 613 m,nd, based on borehole temperature measurements, the base of theydrate-stability zone is predicted at 1100 m depth. The well is lo-ated on an anticline where hydrate accumulation, about 100 mhick and reaching concentrations above 80% of the pore space, restst the crest of the structure. At Mallik 2L-38, hydrate fills the porepaces of deltaic, Tertiary, unconsolidated, highly porous sands andravels that are interbedded with nonhydrate-bearing, silty sedimentUchida et al., 2001; Collett and Dallimore, 2002�. At Mallik 2L-38,ydrate occupies clean, blocky sand and fining-upward sequences,nd the base of each hydrate interval is marked by a sharp decline inelocity. A layer of free gas below the base of the hydrate-stabilityone is approximately 1.5 m thick �Collett and Dallimore, 2002�.

The typical log response identifying the presence of gas hydrate isn increase in both P- and S-wave velocities �Figure 1� and in electri-al resistivity, which are used to obtain the hydrate saturation. Theeason for these characteristic responses is that as the hydrate fills theores, the volume available to the formation water is reduced and theolid hydrate becomes part of the rock frame, thus increasing thelastic moduli of the host rock sediment.

MODEL FOR ELASTIC PROPERTIES

A theoretical model appropriate for predicting seismic signaturest the Mallik location is the Dvorkin et al. �2003� gas-hydrate modelsee Appendix A�, which assumes that sediment is unconsolidatednd the hydrate that fills the pore space acts as a component of theoad-bearing mineral frame. The hydrate generated in the pore spacef the host rock �a� reduces the pore volume available to water andb� alters the elastic properties and density of the newly formed solidrame.

We model the mineral frame to be composed of quartz and clayith zero clay content in the sand and 15% in the shale. The water

aturation is determined from the resistivity curve using equations inollett �1998�. The resulting hydrate saturation of the pore space isetween 80% and 90%, slightly exceeding the values reported in Xund Chopra �2003�.

Other model inputs are listed in Table 1, where the pore-waterulk modulus and density are determined from the Batzle and Wang1992� formulas as a function of temperature �15°C�, pore pressure10 MPa�, and salinity �8000 ppm, according to Cranston, 1999�.he elastic moduli and density of the solid hydrate are from Helger-d �2001�. The mineral properties are from Mavko et al. �1998�. Ad-

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itional parameters used in the model �such as the critical porositynd coordination number� are given inAppendix A.

The model curves for P- and S-wave velocities and the resultingoisson’s ratio are superimposed on the data in Figure 1. The mod-led values are close to the measured values except for the short in-erval at the base of the well. Most importantly, the model honors thealient features of onshore sediment containing gas hydrate and ac-urately quantifies the observed increase in the elastic-wave velocityrom the presence of gas hydrate.

To emphasize the effect of gas hydrate on elastic properties andurther assess the validity of the model, we crossplot the P- and-wave velocities versus the porosity of the mineral frame in Figure. The data points are color coded by gas-hydrate saturation. Theodel curves overlying data in this figure are calculated in the poros-

ty range between 0% and 40%, each for fixed gas-hydrate saturationnd clay content. These curves are in good agreement with the data.

The model thus established enables us to perturb the crucial inputsporosity, lithology, and gas-hydrate content� to forward model theeismic signatures of an onshore hydrate reservoir for various geo-ogic scenarios away from well control.

ELASTIC FORWARD MODELING

We use this rock-physics model to identify methane-hydrate-res-rvoirAVO responses caused by changes in the reservoir and in non-eservoir parameters, and we produce templates that can be used toescribe the formation away from the well. The first step is to set theanges for porosity, clay content, and hydrate saturation in the sedi-ent �Table 2� to represent situations not encountered within the ex-

sting data. Next, we calculate the corresponding elastic propertiesrom the rock-physics model and then plot them in the impedance-oisson’s ratio plane. Finally, two points are selected in this plane,ne for the overburden and the other for the reser-oir, and then the AVO response at this interfaces calculated and plotted �Figure 3�.

In this figure, the polygons in the impedance-oisson’s ratio plane represent shale, wet sand,as sand, and sand containing gas hydrate, withorosity, clay content, and hydrate saturationarying within the selected ranges. The shaleolygon is colored light blue, and the wet and gasand polygons are colored dark blue. The hydrateand polygon is color coded by gas hydrate satu-ation. This display shows that in sands, thehange in the elastic properties increases with in-reasing hydrate saturation although the increas-ng clay content �up to 20%� has a small effect onhe elastic properties.

We first model the effect of gas-hydrate satura-ion on the AVO response �Figure 3a� by fixinghe elastic properties of the overburden shale andomputing the reflections at the interface betweenhe shale, wet sand, and sand with high gas-hydrate saturation. Thewo computed AVO signatures are drastically different from eachther. While the shale–wet-sand interface produces a weak Class IVO response, the shale–gas-hydrate interface produces a strong re-ponse.

To characterize theAVO response caused by changes in propertiesf the overburden �Figure 3b�, we fix the properties of the sand con-aining gas hydrate and vary the shale properties from very soft shaleith high clay content and high porosity to stiffer shale with low clay

Figure 2. �a� Pcolor coded bythe top of the�top� to 20% �and zero hydrtent, respectiv

ontent and low porosity. The results show that the response istrongly affected by the background earth properties.

Finally, we model the AVO signatures at the interfaces betweenas-hydrate sand overlying gas sand and wet sand overlying theame gas sand �Figure 3c�. The AVO response at the interface be-ween gas-hydrate and gas sand belongs to Class IV, while wet sandbove gas sand produces a Class IIAVO response.

MODELING INELASTIC PROPERTIES(ATTENUATION)

One might expect that waves attenuate less in a stiffer and more incompliant medium, i.e., seismic energy losses through sediment

tiffened by the presence of gas hydrate should be reduced as com-ared to wet sand and nonreservoir sediment. However, the oppositeffect has been observed at many methane-hydrate locations, e.g.,he Outer Blake Ridge �Guerin et al., 1999; Wood et al., 2000� and

allik �Sakai, 1999; Guerin and Goldberg, 2002; Pratt et al., 2003�.uerin and Goldberg �2002� empirically relate the P-wave inverseuality factor �Qp

−1� calculated from Mallik 2L-38 log data to the gas-ydrate saturation �Sgh� as

Qp−1 = 0.029 + 0.12Sgh. �1�

his attenuation observed in methane hydrate appears to be intrinsicather than attributable to scattering of seismic energy by the earth’seterogeneity.

able 2. Parameters used for AVO modeling.

Totalporosity

�%�

Claycontent

�%�

Hydratesaturation

�%�

hale 20–35 40–100 0

as and wet sand 30–40 0–20 0

ydrate reservoir 30–40 0–20 30–90

b� S-wave velocities versus porosity of the mineral frame crossplotsydrate saturation of the pore space. The first four model curves fromcorrespond to zero clay content and hydrate saturation from 80%� with a 20% decrement. The bold model line is for zero clay contentration. The two curves underneath it are for 10% and 30% clay con-

d zero hydrate saturation.

- and �gas-h

figurebottomate satu

FsPTrhsdisplayed in the middle column.

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F168 Cordon et al.

Dvorkin and Uden �2004� offer a theoreticalmodel that partly explains attenuation observedat methane-hydrate locations. The mechanismused in their model is the macroscopic squirt flow.The most important condition for this flow to de-velop in well-log, crosswell, and seismic frequen-cy ranges is a large elastic contrast in the rock.The existence of gas hydrate in the pores gener-ates such an elastic contrast. Within the Dvorkinand Uden �2004� model, the low-frequency com-pressional modulus of partially saturated rock isestimated by means of Gassmann’s fluid substitu-tion using the rock’s dry-frame modulus and theharmonic average of the moduli of individual flu-id components as the effective bulk modulus ofthe pore-fluid mix. The high-frequency compres-sional modulus of the partially saturated rock isestimated by assuming that fluid distribution is ir-regular. The difference between these two esti-mates gives rise to noticeable attenuation if elas-tic heterogeneity in the rock is substantial.

The P-wave inverse quality factor computedaccording to equation 1 is compared to the theo-retical values presented by Dvorkin and Uden�2004� in Figure 4. The theoretical values areabout 75% of the empirical values. We find thismatch satisfactory because the model correctlycaptures the order of magnitude of attenuationand, most important, shows discernable attenua-tion in sediment containing methane hydratewhile the attenuation is small outside of these in-tervals.

EFFECT OF ATTENUATION ONSYNTHETIC SEISMIC RESPONSE

To investigate the effect of attenuation on seis-mic response, we calculate synthetic seismic trac-es at Mallik 2L-38 using a ray-tracing algorithm.We employ a 1D layered earth model where 20rays are shot, spanning from zero to 48° at the sur-face and equally spaced by 50 m. The reflectionand transmission angles at the interfaces are com-puted using Snell’s law, and then horizontal dis-tance and traveltimes for each ray in each layerare computed via ray tracing. Critical angles andmultiples are not accounted for; hence, only pri-mary reflections are computed. Seismic traces aregenerated by the convolution of a 90-Hz, zero-phase Ricker wavelet sampled at 0.4 ms, and thereflection and transmission coefficients are deter-mined from Zoeppritz equations.

The synthetic moveout-corrected gathers com-puted from the described ray-tracing algorithmare shown in Figure 5 for the earth model with andwithout P-wave attenuation. In this example,equation 1 is used to estimate Qp

−1. It is clear thatattenuation associated with the presence of meth-

sand. �a� Wetdance versuste saturation.�a� “1-1” cor-le and a clean

elastic half-or the curves

quality factorin and Uden

igure 3. Perturbational AVO modeling at an interface between shale andand, gas sand, shale, and hydrate sand are mapped onto the P-wave impeoisson’s ratio plane. The hydrate sand polygons are color coded by hydrahe numbers indicate the interface used inAVO modeling. For example, in

esponds to the interface between shale and wet sand while “2-2” is for shaydrate reservoir. �b� AVO curves calculated at the interface between twopaces selected are numbered accordingly. �c� Gradient versus intercept f

igure 4. �a� Gas-hydrate saturation versus depth. �b� The P-wave inverseersus depth according to Guerin and Goldberg �2002�, in red, and Dvork2004�, in blue.

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ne hydrate noticeably affects the seismic amplitude. Furthermore,ttenuation can possibly serve as a seismic attribute for methane-hy-rate reservoir characterization.

MODEL-DRIVENFORWARD MODELING

Acombination of rock physics and synthetic seismic modeling al-ows us to explore the effects of hydrate amount and spatial distribu-ion on the amplitude.

In the first example, we alter the hydrate saturation in the originalell by reducing it by a factor of two in the upper part of the interval,

he lower part of the interval, and the entire interval. The resultingraces �Figure 6� are affected by two competing factors: the elasticontrast between the background and sand containing hydrate andttenuation that increases with the increasing elastic contrast. Thisxample shows that attenuation has to be taken into account when in-erpreting seismic data for hydrate saturation because the decay ofhe amplitude with traveltime is similar between the original wellata and the case where the amount of hydrate in the lower part of theell is half of the original amount.In the second example we construct two pseudowells with four

elatively thin sand layers with hydrate and two thicker layers. Theorosity of the sand is the same in both wells. The methane hydrateaturation is different. However, the total volume of hydrate in thenterval �the product of sand porosity, hydrate saturation, and sandhickness� is the same. The resulting synthetic traces �Figure 7� indi-ate that the geometry of the reservoir and hydrate distribution in itffect the seismic response even if the hydrate volume is constant.

CONCLUSION

One approach to interpreting seismic data for the physical state ofock is forward modeling. Lithology, porosity, and fluid in the rock,s well as the reservoir geometry, are perturbed, the correspondinglastic properties are calculated, and then synthetic seismic tracesre generated. Then it is assumed that if the seismic response is simi-

igure 7. Synthetic seismic traces at two pseudowells. �a� Well withour sand layers and hydrate saturation 0.8. �b� Well with two thickand layers and hydrate saturation 0.6. The porosity of the sand inoth wells is 0.30. The clay content in the sand is zero. The porosityf the shale in both wells is 0.25 and clay content is 0.20. Frequencys 40 Hz.

igure 5. Synthetic seismic traces at Mallik 2L-38. The vertical axiss the two-way traveltime. �a� Gas-hydrate saturation. �b� Syntheticeismic gather �near-, mid-, and far-offset� without attenuation. �c�ynthetic seismic gather with attenuation calculated according toquation 1.

igure 6. Synthetic seismic traces at Mallik 2L-38 with hydrate satu-ation perturbed. The vertical axis is the two-way traveltime. Eachase is represented by two plots: The hydrate saturation and synthet-c amplitude versus the two-way traveltime. In the amplitude frames,he trace calculated without attenuation is black and that with attenu-tion is red. �a� The original case. �b� Hydrate saturation in the upperart of the well is half of the original. �c� Hydrate saturation in theower part of the well is half of the original. �d� Hydrate saturation inhe entire interval is half of the original. Frequency is 40 Hz.

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ar, the properties and conditions in the subsurface that give rise tohis response are similar as well. Systematically conducted perturba-ional forward modeling helps create a catalog of seismic signaturesf lithology, porosity, and fluid away from existing wells, and by sooing, sets realistic expectations for hydrocarbon detection and opti-izes the selection of seismic attributes in an anticipated deposition-

l setting. Although fluid substitution methods are well tested andommonly used, lithology and porosity substitution is much morehallenging because it requires the use of a site-specific, yet general,ock-physics model.

Here we show how to conduct such perturbational modeling in aethane-hydrate reservoir. A critical component of this workflow isrational theoretical and data-driven rock-physics model. Once

uch a model is established, it can be used to systematically produceynthetic seismic traces by changing porosity, mineralogy, geome-ry, and hydrate saturation within any geologically plausible ranges.he several examples presented here by no means cover all these

anges. They are only samples of a physics-based methodology thatnables methane-hydrate reservoir characterization by careful andrderly forward modeling.

Even these few examples teach us that �a� the geometry of the res-rvoir and hydrate distribution in it affect the seismic response; �b�ttenuation has to be taken into account during hydrate-reservoirharacterization; and, most importantly, �c� the competing effects oflastic contrast, geometry, and attenuation make seismic interpreta-ion nonunique. This nonuniqueness must be acknowledged anduilt into interpretational workflows.

ACKNOWLEDGMENTS

We thank Tim Collett of the U. S. Geological Survey for providinghe data and Ezequiel Gonzalez of Stanford University for technicalelp. Support was provided by the Stanford Rock Physics Labora-ory.

APPENDIX A

ROCK-PHYSICS MODEL FOR SEDIMENTCONTAINING GAS HYDRATE

The basis of our model is the Dvorkin et al. �1999� model, whichelates the elastic moduli of high-porosity, ocean-bottom sedimentso porosity, pore-fluid compressibility, mineralogy, and effectiveressure. We summarize this model below.

At the critical porosity �c ��c = 0.36 − 0.43, according to Nur etl., 1998�, we calculate the effective bulk KHM and shear GHM modulif the dry rock frame using the Hertz-Mindlin contact theory �Mind-in, 1949�. According to this theory, the radius a of the contact circleetween two spherical elastic grains of radius R is

a = R� 3��1 − ��2n�1 − �c�G

P�1/3

, �A-1�

here P is the differential pressure; G and � are shear modulus of theolid phase and its Poisson’s ratio, respectively; and n is the averageumber of contacts per grain in the sphere pack �from six to nine, ac-ording to Dvorkin and Nur, 1996�. The normal and tangential con-act stiffness �S and S , respectively� between these two grains are

N T

SN =4aG

1 − �, ST =

8aG

2 − �. �A-2�

he effective bulk and shear moduli of the dry frame of the grainack are

HM = n1 − �c

12�SN, GHM = n

1 − �c

20��SN +

3

2ST� . �A-3�

The above expression for the tangential stiffness is valid if theres perfect adhesion between the grains. This stiffness may reduce toero if friction between the grains is absent. To account for possiblelippage at the grain interface, we introduce an ad hoc reduction fac-or � for ST, so that now we use ST/� instead of ST in equation A-3.

For porosity below critical, the bulk Kdry and shear Gdry moduli ofhe dry frame are calculated via the modified lower Hashin-Shtrik-an �H-S� bound �Dvorkin and Nur, 1996�:

Kdry = ��

�c

KHM +4

3GHM

+

1 −�

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GHM + Z+

1 −�

�c

G + Z�

−1

− Z ,

Z =GHM

6�9KHM + 8GHM

KHM + 2GHM� . �A-4�

If the sediment is saturated with pore fluid of bulk modulus Kf,he shear modulus Gsat is the same as that of the dry frame, and theulk modulus Ksat is calculated from Gassmann’s �1951� equation as

Ksat = K�Kdry − �1 + ��KfKdry/K + Kf

�1 − ��Kf + �K − KfKdry/K. �A-5�

Once the elastic moduli are known, the elastic-wave velocitiesre calculated as

Vp = �Ksat +4

3Gsat�

�b, Vs = Gsat

�b, �A-6�

here �b is the bulk density.In a common case of mixed mineralogy, the elastic constants of

he solid phase are calculated from those of the individual mineralonstituents using the Hill �1952� average formula:

K =1

2�i=1

m

fiKi + �i=1

m

fi/Ki�−1� ,

G =1

2�i=1

m

fiGi + �i=1

m

fi/Gi�−1� , �A-7�

here m is the number of mineral constituents, f i is the volumetricraction of the ith constituent in the solid phase, and Ki and Gi are theulk and shear moduli of the ith constituent, respectively. The solidhase density is calculated as

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� = i=1

m

fi�i, �A-8�

here �i is the density of the ith constituent.The differential pressure is calculated as the difference between

he lithostatic and hydrostatic pressure: P = ��b − �w�gD, where �b

s sediment bulk density, �w is water density, g is the accelerationrom gravity, and D is depth below the seafloor.

We account for the presence of gas hydrate by assuming that it isart of the dry frame and acts to reduce porosity and alter the solid-hase elastic properties. The volumetric concentration of gas hy-rate in the pore space is given by Sh = Ch/�, where Ch is the volu-etric concentration of hydrate in the rock.

Hydrate is modeled as a load-bearing component of the frame. Iteduces the original porosity � to �̄ = � − Ch and changes the effec-ive mineral modulus as calculated in equation A-7, where f i shouldow be replaced by

f̄ i =f i�1 − ��

1 − �̄. �A-9�

as hydrate has to be treated as an extra mineral component withraction f̄ h given by

f̄ h =Ch

1 − �̄. �A-10�

The critical porosity used in the application of this model to theallik well is 0.43, the coordination number is 8, and the reduction

actor � is 2.

REFERENCES

atzle, M., and Z. Wang, 1992, Seismic properties of pore fluids: Geophys-ics, 57, 1396–1408.

ollett, T. S., 1998, Well log evaluation of gas hydrate saturations: Transac-tions of the Society of Professional Well Log Analysts, 39th Annual Log-ging Symposium, Paper MM.

ollett, T. S., and S. R. Dallimore, 2002, Integrated well log and reflectionseismic analysis of gas hydrate accumulations on Richards Island in theMackenzie Delta, N.W.T., Canada: CSEG Recorder, 27, no. 10, 28–40.

ranston, R. E., 1999, Pore water geochemistry: JAPEX/JNOC/GSC Mallik2L-38 gas hydrate research well, in S. R. Dallimore, T. Uchida, and T. S.

Collett, eds., Scientific results from JAPEX/JNOC/GSC Mallik 2L-38 gas

hydrate research well, Mackenzie Delta, NWT, Canada: Geological Sur-vey of Canada Bulletin 544, 165–175.

allimore, S. R., T. S. Collett, and T. Uchida, 1999, Overview of science pro-gram, JAPEX/JNOC/GSC Mallik 2L-38 gas hydrate research well, in S.R. Dallimore, T. Uchida, and T. S. Collett, eds., Scientific results fromJAPEX/JNOC/GSC Mallik 2L-38 gas hydrate research well, MackenzieDelta, NWT, Canada: Geological Survey of Canada Bulletin 544 11–17.

vorkin, J., and A. Nur, 1996, Elasticity of high-porosity sandstones: Theoryfor two North Sea data sets: Geophysics, 61, 1363–1370.

vorkin, J., A. Nur, R. Uden, and T. Taner, 2003, Rock physics of a gas hy-drate reservoir: The Leading Edge, 22, 842–847.

vorkin, J., M. Prasad, A. Sakai, and D. Lavoie, 1999, Elasticity of marinesediments: Geophysical Research Letters, 26, 1781–1784.

vorkin, J., and R. Uden, 2004, Seismic wave attenuation in a methane hy-drate reservoir: The Leading Edge, 23, 730–732.

assmann, F., 1951, Uber die elastizitat poroser medien: Vierteljahrsschriftder Naturforschenden Gesselschaft, 96, 1–23.

uerin, G., and D. Goldberg, 2002, Sonic waveform attenuation in gas hy-drate-bearing sediments from the Mallik 2L-38 research well, MackenzieDelta, Canada: Journal of Geophysical Research, 107, 1029–1085.

uerin, G., D. Goldberg, and A. Meltzer, 1999, Characterization of in-situelastic properties of gas-hydrate-bearing sediments on the Blake Ridge:Journal of Geophysical Research, 104, 17781–17796.

elgerud, M., 2001, Wave speeds in gas hydrate and sediments containinggas hydrate: A laboratory & modeling study: Ph.D. dissertation, StanfordUniversity.

ill, R., 1952, The elastic behavior of crystalline aggregate: Proceedings ofthe Physical Society, A65, 349–354.

venvolden, K. A., 1993, Aprimer on gas hydrates: U. S. Geological Survey,Professional Paper 1570, 555–561.avko, G., T. Mukerji, and J. Dvorkin, 1998, The rock physics handbook:Tools for seismic analysis in porous media: Cambridge Univ. Press.indlin, R. D., 1949, Compliance of elastic bodies in contact: Transactionsof theAmerican Society of Mechanical Engineers, 71, A-259.

ur, A., G. Mavko, J. Dvorkin, and D. Galmudi, 1998, Akey to relating phys-ical properties to porosity in rocks: The Leading Edge, 17, 357–362.

ratt, R. G., K. Bauer, and M. Weber, 2003, Crosshole waveform tomogra-phy velocity and attenuation images of arctic gas hydrates: 73rd AnnualInternational Meeting, SEG, ExpandedAbstracts, 2255–2258.

akai, A., 1999, Velocity analysis of vertical seismic profiling �VSP� surveyat JAPEX/JNOC/GSC Mallik 2L-38 gas hydrate research well, and relatedproblems for estimating gas hydrate concentration: Geological Society ofCanada Bulletin 544, 323–340.

chida, T., S. R. Dallimore, H. Lu, and T. S. Collett, 2001, Comparison ofnatural gas hydrate occurrence observed in the JAPEX/JNOC/GSC Mallik2L-38 well, Mackenzie Delta, N.W.T. with recently obtained natural gashydrates: The Rock Foundation Convention, Canadian Society of Petro-leum Geologists,Abstract 035-1.ood, W. T., W. S. Holbrook, and H. Hoskins, 2000, In situ measurements ofP-wave attenuation in the methane-hydrate and gas-bearing sediments ofthe Blake Ridge, in C. K. Paull, R. Matsumoto, P. J. Wallace, and W. P. Dil-lon, eds., Proceedings of the Ocean Drilling Program, Scientific Results:National Science Foundation, report 164, 265–271.

u, Y., and S. Chopra, 2003, Rock physics and AVO applications in gas hy-drate exploration: Partners in a New Environment Conference, CanadianSociety of Petroleum Geologists/CSEG Joint Convention, accessed Au-gust 29, 2006, http://www.cseg.ca/conferences/2003/2003abstracts/402

S0131.pdf.