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Modeling velocity in carbonates using a dual porosity DEM model

Gregor T. Baechle*, Arnout Colpaert**, Gregor P. Eberli* and Ralf J. Weger*

*Comparative Sedimentology Laboratory, University of Miami, USA**Statoil Research Center, Trondheim, Norway

Summary

The differential effective medium theory is used to modelthe velocity of carbonates with two predefined end-member

pore types and under dry and water saturated conditions.The dual porosity DEM takes into account input parametersderived from digital image analysis of thin sections. In

particular the respective amount of microporosity andmacroporosity and the aspect ratio of the macropores areincorporated. A conceptual aspect ratio of 0.1 for

micropores and a measured aspect ratio of 0.5 formacropores is used as input parameters for the differentialeffective medium (DEM) model. The model predicts that

the compliant micropores have a strong influence on thesonic velocity of porous carbonates because increasingconcentrations of micropores reduce the rock stiffness. Themodel values are compared to high frequency (1MHz)

laboratory velocity measurements. These velocity predictions with the dual porosity DEM model showsignificant better velocity prediction than empirical models,e.g. the Wyllie times average equation. We obtain a root-

mean-square-error of 392 m/s when comparing predictedwith measured velocity values. Our results also show that adifferential effective medium model that uses measuredinput parameters from quantitative digital image analysis

improves estimates of acoustic properties of carbonates.

Introduction

Elastic moduli are affected directly by three influencing

factors: rock framework, pore fluid and pore space. Indirectfactors, such as, changes in temperature and pressure havethe potential to modify the effect of the direct factors on

elastic moduli. Carbonate rocks, in contrast to sandstone,display complex pore structures with an astonishing rangeof pore sizes and pore shapes. Although the pore shape is

the most significant rock property, affecting the elastic property of the rock (Wang, 2001), it can not be easilyquantified. In comparison to the pore shape, the pore size isrelatively easily to measure and quantify.

A positive correlation between pore size and velocity wasfirst been observed by Hamilton et al. (1956). Anselmettiand Eberli (1993) observed a relationship between pore

type and velocity, where rock samples containing moldicand intraparticle porosity have a higher velocity than

samples containing micro-moldic porosity andmicroporosity.Effective medium models have the potential to capture theeffect of those pore geometries on acoustic properties.

Many experimental and numerical studies use the aspect

ratios as pore type indicators. The aspect ratio is either

assigned (Goldberg and Gurevich, 1998, Xu and White,1995 and Markov et al., 2005), derived from velocity-

pressure measurements (Sun and Goldberg, 1997), derivedfrom joint inversion of acoustic and resistivitymeasurements (Kazatchenko et al., 2004) or estimated

using neural networks (Yan et al., 2002). Only a fewstudies used aspect ratios determined from thin sections and

they proved not to capture the observed velocity variations(Colpaert et al., 2007 and Rossebø et al., 2005). Effectivemedium models with various concentrations of pores andassigned pore shapes or pore types have been used to

characterize the acoustic properties of carbonate rocks(Brie, 2001 and Kumar at al., 2005).In this study we emphasize the importance of separating theeffects of micropores and macropores when evaluating

acoustic properties of carbonates (Baechle et al., 2004).Consequently, we divide the pore space into stiffmacropores characterized by high aspect ratio pores andcompliant micropores characterized by low aspect ratio

pores, using a similar DEM approach as Xu and White(1995) and Markov et al. (2005). Using digital imageanalysis we measured the aspect ratio of the macropores,and the amounts of micropores and macropores. In the

modeling, these input parameters are used to solve for the best-fit aspect ratio of the micropores to predict velocity.The model results are compared to the laboratorymeasurements to estimate the validity of the model.

Data Set and Method

Our study is based on mining the CSL database for

carbonate samples with a wide range of pore and rock typesand ages from ten to hundreds of million years. For allthese samples, digital images, p- and s-wave velocity,

porosity, mineralogy (calcite or dolomite) and grain densityis known.The dolostones are dominantly re-crystallized and havevuggy and inter-crystalline pore types. The limestones

consist of of floatstone and frame/boundstones, grainstonesand packstones, with a wide range of dominant pore types:interparticle, intraparticle, moldic, vuggy and

microporosity.Digital image analysis is performed on thin sections that

are produced from an end piece of each core plug sample .To visually separate the rock from the pore space, therock’s pore space is saturated with blue epoxy. A full thinsection digital image at a resolution of 6 to 7 microns per

pixel consists of several digital images that are stitched

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Modeling the effects of pore structure on velocity

together. The full color image is segmented into a blackand white binary image, with the white features

representing the pore space (Weger, 2006). The imaged pore space is macroporosity, representing pores, which are

larger than 30 microns. Subtracting the amount of thismacro-porosity from the total plug porosity determinesmicroporosity (Anselmetti and Eberli, 1997). The shapesof the visible pores are measured as the mean of the aspect

ratio of the entire pore population, describing the ratio between minimum and maximum axis diameter of the best-fit ellipse around a selected pore.Ultrasonic compressional wave and shear wave velocity

measurements are performed on carbonate plugs at afrequency of ~ 1 MHz. The samples were measured underdry conditions with ambient atmospheric pore pressure andunder water saturated conditions, at 2 MPa pore pressure,

the differential hydrostatic pressures was kept at 10 MPa.All data presented are taken from the loading cycle.

Differential effective medium (DEM) theory

We model the elastic properties of rocks with different poretypes using the differential effective medium model forhigh porosity rocks. A detailed review of the DEM theory

is given by Zimmerman (1991).In our model we separate total porosity into microporosityand macroporosity. We assign aspect ratios with an initial

guess of spherical-stiff macropores and compliantmicropores. The p-wave and s-wave velocity is thencalculated iteratively solving the DEM differentialequations (Berryman, 1992), by adding small portions of

pores with assigned aspect ratios into the host material (non porous limestone or dolostone), until total porosity is

attained:

[ ] )(P)K K ()(K d

d)1( )2(**

2

* φ−=φφ

φ−

(1)

[ ] )(Q)()(d

d)1(

)2(**

2

*φµ−µ=φµ

φφ−

(2) where,

K 1, µ1 is the bulk and shear moduli of the initial hostmaterial, which is a function of the rock’s mineralogy, in

our case dolomite or calcite, and K 2, µ2 is the bulk andshear moduli of the inclusions, which is defined by thefluid saturation of the pores.

The geometrical coefficients P and Q depend on the aspectratio of the elliptical pores (Berryman, 1980). One of the

key aspects of our approach is that macropores aresignificantly stiffer than micropores. The macropores can

be mathematically modeled by average high aspect ratio pores and the micropores with average low aspect ratio

pores. The pore volumes associated to macropores and

micropores is equal to the total porosity. The informationabout the pore volume fractions of the two pore size

populations is derived from quantitative digital imageanalysis.

In addition to these pore size proportion measurements, theaverage aspect ratio of the macropores has been measuredto characterize the pore shape. In addition to thesemeasurements, we know the fluid content and the

mineralogy of our samples. Thus, the only two independentvariables in our model are porosity and the aspect ratio ofmicropores.We first begin the selection of aspect ratios for the two pore

populations: compliant micropores with small aspect ratio(< 0.15) and near-spherical macropores with high aspectratios of 0.5. Next we minimize the root-mean-square-errorof theoretically predicted and experimentally determined

compressional and shear wave velocities in a best-fit procedure by adjusting the micropore aspect ratio.We are able to predict the compressional and shear wavevelocities if the following parameters are known: mineral

moduli, fluid bulk modulus, grain density, fluid density,image macroporosity and microporosity and their aspectratios. The aspect ratio of the macropores and microporesare given by our empirical best-fit procedure. The amount

of macroporosity and microporosity is calculated usingdigital thin section images.

Results

Measured velocity

The p-wave velocity of the 250 samples measured underwater saturated condition range from 6.5 to 2 km/s (Fig.1a). The porosity covers a wide range, from zero to 55%

porosity (Fig. 1a). The limestone samples cover a similarwide range of velocity and porosity than the dolostones. Alarge scatter of velocity exists at a given porosity, e.g. thevelocity spans from ~ 2.5 km/s to ~ 5.2 km/s at porosity of30%. Similarly, the porosity varies at given velocity, e.g.the porosity ranges from 20 to over 50% at a velocity of 3.5

km/s.The 160 samples measured under dry conditions shows arange of porosity from 2 to 50%. The velocity ranges from6.5 to 2.5 km/s. Velocity – porosity data above 35%

porosity is less abundant than in the dataset measured underwater-saturated conditions. With one exception, thedolostones show velocities above 3.5 km/s, lacking the

“slow” p-wave velocity.Vp/Vs shows increasing scatter with increasing porosity,

under dry and water saturated conditions. The bulk of thesamples show Vp/Vs ratio between 1.75 and 2.2 underwater-saturated conditions and Vp/Vs ratio between 1.6 and2 under dry conditions.

Effects of Mineralogy

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Modeling the effects of pore structure on velocity

A common perception is that dolomites are stiffer thanlimestones. Dolomites have higher velocities at zero

porosity than limestones, but in our dataset samples of bothlithologies can show a wide range of velocities at given

porosity, depending on the pore type. For example, asucrosic dolostone with small intercrystalline poresdisplays a lower velocity than a recrystallizedintercrystalline-vuggy dolostone with similar porosity.

Likewise limestones dominated by micropores are slowwhile limestones with interparticle vuggy and moldic

porosity show higher velocities at similar porosity.Furthermore, the dolostones and limestones do not show a

distinct pattern in the Vp/Vs-porosity space, thus the Vp/Vsappears to be independent of the mineral type.We conclude that at high porosity the texture of the rockseems to override the effect of mineralogy of the frame.

Poisson’s ratioThe Poisson’s ratio can be derived from the Vp2/Vs2 ratiousing following equation, (Mavko et al., 1998):

υ = 0.5((Vp^2 / Vs^2) – 2) / ((Vp^2/Vs^2) -1)

where υ is the Poisson ratio, Vp = p-wave velocity and Vs

= s-wave velocity.

The correlation between Vp2/Vs2 is better under dryconditions (r 2 > 0.92) than under water-saturated conditions

(r 2

> 0.81). Using dry Vp2/Vs

2 ratio we derived average

limestone and dolostone Poisson’s ratios of 0.261 and0.275, respectively. The extracted water saturated Poisson’sratio of limestone and dolostone are 0.304 and 0.307,

respectively. The mineralogy obviously influences theaverage Poisson ratio less than the fluid type in the pores.

Effects of aspect ratio and pore size on velocity

Macropores can physically be described as near-sphericalellipsoidal inclusions in the rock. Stiff pores aremathematically described with a high aspect ratio. Thisdoes not imply that all macropores show a near-spherical

pore shape. In fact, dominant pore types range fromintraparticle, intercrystalline, moldic to vuggy porosity,resulting in a range of complex pore shape geometries.In this study we characterize the macro-pore shape by the

aspect ratio of pore-surrounding ellipsoid. For the entiresample dataset the aspect ratio varies only insignificantly,

between 0.5 and 0.6 with a mean value of ~ 0.55. This

narrow range of pore aspect ratio is in agreement with othercarbonate image analysis study results (Rossebo et al., 2006

and Colpaert et al., 2007). Consequently the change inaspect ratio can not account for the large scatter of velocityat given porosities.

In contrast, the percentage of microporosity influences the

velocity drastically. In the velocity – porosity space,samples with high percentage of microporosity are

generally slower than rocks with similar porosity but moremacroporosity, indicating that microporosity is more

compliant than the stiff macroporosity (Fig. 1a). Inaddition, when macroporosity is subtracted from the total

porosity, the remaining microporosity shows a decentcorrelation to velocity (Fig. 1b). This indicates that much ofthe macroporosity is ineffective for the acoustic behavior atleast at ultrasonic frequencies.

Forward modelingAs described above, we select aspect ratios of two end-member pore types: small soft pores with aspect ratio of<0.01 and near-spherical pores with aspect ratios of 0.5.Then we try to minimize the root mean square error

(RMSE) between measured and predicted p-wave and s-wave velocity. In order to quantify the quality of the best

2000

2500

3000

3500

4000

4500

5000

5500

6000

6500

7000

0 10 20 30 40 50 60

Porosity

V p ( m / s )

>90%

>80%

>70%

>60%

>50%

<50%

2000

2500

3000

3500

4000

4500

5000

5500

6000

6500

7000

0 10 20 30 40 50 60

Microporosity (%)

V p ( m / s )

>90%

>80%

>70%

>60%

>50%

<50%

Figure 1: (a) Plot of porosity versus p-wave velocity.Velocity at given porosity decreases with increasingamount of microporosity (percentage microporosity incolor coding). (b) Plot microporosity versus p-wavevelocity Table 1: This caption is placed outside the

frame and is followed by a page break.

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