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Dynamic characterization of shale systems by dispersion andattenuation of P- and S-waves considering their mineral compositionand rock maturity

Oscar C. Valdiviezo-Mijangos n, Rubén Nicolás-Lopez

Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas 152, Delegación Gustavo A. Madero, 07730 México, DF, Mexico

a r t i c l e i n f o

Article history:Received 19 November 2013

Accepted 31 July 2014Available online 12 August 2014

Keywords:

unconventional reservoirs

shale characterization

effective properties

wave dispersion

wave attenuation

self-consistent method

a b s t r a c t

The dynamical elastic effective properties of gas–oil shale systems are found by using the self-consistentmethod and from them, dispersion and attenuation of P - and S -waves are calculated for a wide range of

frequencies including seismic, sonic and ultrasonic bands. The mathematical model has the virtue of

considering solid and uid inclusions embedded in a matrix solid taking into account the volume

fraction of each one. The mineral composition estimated from geological and petrophysical data is fed

into the theoretical model in order to calculate the effective mechanical properties. The solid frame may

be composed of clay, quartz or carbonate dominated lithotype; and the complement volume is occupied

by pore uids (water, heavy oil or dry gas), kerogen or solid inclusions. From that, typical patterns are

established by the dispersion and attenuation of elastic waves in shale systems considering their

mineralogy and maturity. In the acoustical branch, the results of the modeling have already been

validated with laboratory data. Quartz and carbonate dominated lithologies exhibit very similar elastic

responses and clay dominated lithotype shows a reverse reaction. These results are very useful tools to

analyze and interpret the seismic response of target zones in oil and gas shale formations. Also, they aim

to discriminate: drained, water-saturated, immature, mature or post-mature shale systems.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

An investment project on unconventional reservoirs becomes

successful when the target shale formation produces hydrocar-

bons at commercial rates (KPMG, 2011; Bustin, 2012; Vawter,

2013). Therefore, it is essential to estimate accurately the quality

and eld distribution of mature and post-mature source rocks

from seismic data. Many efforts are focused on developing metho-

dologies for associating uid saturation and mineralogy data

within the analyses of rock seismic responses at full frequency

band. Thus, well-known geophysics analyses supported by the-

ories of wave propagation are used in the exploration and

development stages. However, to develop analytical solutionsand build up standardized laboratory techniques the complexity

increases when it is required to quantify in situ volume percent of

gas–oil shale constituents from the bulk elastic properties or in the

opposite case.

On the one hand, several geoscientists have frequently stressed

the importance of estimating key parameters as the total organic

content, brittleness, maturity and mineralogy of source rocks in

order to properly characterize unconventional reservoirs (Rickman

et al., 2008; Passey et al., 2010; Glorioso and Rattia, 2012; Sun

et al., 2013). On the other hand, there are experimental and

theoretical modeling techniques such as the self-consistent

method, extensively applied on heterogeneous media character-

ization integrating the analysis of P - and S -waves dispersion and

attenuation (Sabina and Willis, 1988). Furthermore, these sorts of

methods have been extended to thermoelastic analyses of hetero-

geneous composites (Valdiviezo-Mijangos et al., 2009).

In this work, the self-consistent method has been extended

using laboratory data observed in measurements of frequency-

dependent P - and S -wave velocities on sedimentary rock samples

(Batzle et al., 2006). Furthermore, this method is used to reconcilethe elastic properties observed at seismic (30–50 Hz), sonic

( 8–24 kHz) and ultrasonic ( 1 MHz) frequencies by a scaling

factor similar to that used in Sams et al. (1997). As a result of the

nonexistence of direct eld measurements of seismic velocity from

source-rock mineralogy, and inconsistent experimental datasets;

herein is followed by a coherent indirect process: (a) forming

mineral composition of the lithotypes from ternary diagrams

based on clay, quartz and carbonate of source rocks, Fig. 1. (b)

Applying a self-consistent scheme briey described in Eqs. (1)–(13).

(c) Utilizing the denition of dimensionless frequency ð Ω ¼

2π af =V p; 0o Ω o2:5Þ to conciliate the seismic and ultrasonic

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/petrol

Journal of Petroleum Science and Engineering

http://dx.doi.org/10.1016/j.petrol.2014.07.041

0920-4105/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author. Tel.: þ 52 55 9175 7041.

E-mail address: [email protected] (O.C. Valdiviezo-Mijangos).

Journal of Petroleum Science and Engineering 122 (2014) 420–427

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frequency responses (Kinra and Anand, 1982). (d) The results

modeling were validated with seismic velocities measured onsedimentary rock samples (Batzle et al., 2006). The experimental

data reported were measured 5 Hz–2 kHz utilizing stress–strain

approach. Unfortunately, only a data point was calculated using

standard pulse-transmission technique at 0.8 MHz. It is hoped that

the missed data of measured wave velocities would adjust to a

characteristic radius according to Kinra and Anand (1982) and

Sabina and Willis (1988). Even though it is known to measure

wave velocity throughout dispersive bandwidth is very complex.

Also, in Fig. 1, the idea is to depict clearly the dependence between

inclusion radius and frequency. Finally, this process was applied to

three widespread shale plays: clay, quartz and carbonate domi-

nated lithologies to describe the seismic response patterns,

maturity and dynamic elastic properties.

2. Self-consistent equations

The self-consistent equations used in this work were developed

by Sabina and Willis (1988). They generalized the simple static

equation of Budiansky (1965) and Hill (1965) to dynamic problems

using the Galerkin approximation. A few equations are shown

below to understand the complete idea of the method; more

details are presented in Sabina and Willis (1988). Although, shales

could exhibit different geometry shapes dened by aspect ratio

(ar); i.e., arr0.05 denotes layer, 0.05oarr0.5 to lenses and

0.5oarr1 is for patches (Sone and Zoback, 2013), herein as an

approximation to model shale, the patches are chosen with aspect

ratio equal to 1. The self-consistent equations for an isotropic

matrix containing isotropic spherical inclusion are as follow:

C 0 ¼ C n þ 1 þ ∑n

r ¼ 1

α r hr ðkÞhr ð kÞðC r C n þ 1Þ½I þ S ðr Þ

x 1 ð1Þ

ρ0 ¼ ρn þ 1 þ ∑n

r ¼ 1

α r hr ðkÞhr ð kÞð ρr ρn þ 1Þ½I þ M ðr Þ

t 1 ð2Þ

where C n þ 1 is the tensor of elastic moduli and ρn þ 1 is the density

of the matrix where different types of inclusions are embedded, an

inclusion of type r having tensor of elastic moduli C r and density ρr , r ¼ 1; 2;…; n. Each inclusion of type r is of the same size and

shape and α n þ 1 is dened as follows:

∑n þ 1

r ¼ 1

α r ¼ 1: ð3Þ

Both, matrix and inclusion are considered as isotropic, with C r characterized by bulk modulus κ r and shear modulus μr . The

corresponding symbolic notation used by Hill (1965) is

C r ¼ ð3κ r ; 2 μr Þ

This is very useful because the product of isotropic two

tensors A and B dened by A ¼ ð3κ A; 2 μ AÞ and B ¼ ð3κ B; 2 μBÞ is

AB ¼ ðð 3κ AÞð3κ BÞ; ð2 μ AÞð2 μBÞÞ. In Appendix A, this fact will be

explained in detail.

The term S x; M t and I of Eqs. (1) and (2) are computed from the

Green functions for an isotropic homogeneous media, Eqs. (A6),

(A10), and (A13). Therefore, their denitions are

S x ¼ ε p

3κ 0 þ4 μ0;

1

5

2εV p

3κ 0 þ4 μ0þεV p

μ0 ð4Þ

Fig. 1. Dispersion and attenuation at different inclusion radiuses used as scaling factor for full-frequency band. (a) Compressional wave velocity, V p . (b) Shear wave velocity,

V s . (c) Compressional wave attenuation, Imðk p aÞ. (d) shear wave attenuation, ImðksaÞ. (Experimental data were digitalized from Batzle et al. (2006).)

O.C. Valdiviezo-Mijangos, R. Nicolás-Lopez / Journal of Petroleum Science and Engineering 122 (2014) 420 –427 421

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ðM t Þij ¼ 1

3 ρ0δ ijð3εV p 2εV s Þ ð5Þ

I ¼16π 2a3

3k2γ ðεγ 1Þ ð6Þ

where a is the inclusion radius, kγ ¼ ω=γ and γ ¼ V p or V s is

required. Eqs. (4) and (5) are dened in Sabina and Willis (1988)

in Eqs. (3.5). The εγ corresponds to Eq. (A7) and it is expressed as

εγ ¼ 3ð1 ikγ aÞ

ðkγ aÞ3

½ sin ðkγ aÞ kγ a cos ðkγ aÞ eikγ a ð7Þ

where

V p ¼ κ 0 þ4 μ0=3

ρ0

1=2ð8Þ

and

V s ¼ μ0

ρ0

1=2: ð9Þ

Hence the self-consistent equations with n ¼1, i.e., for just oneinclusion are

κ 0 ¼ κ 2 þ α 1h1ðkÞh1ð kÞðκ 1 κ 2Þ

1þ3ðκ 1 κ 0ÞεV p =ð3κ 0 þ4κ 0Þ ð10Þ

μ0 ¼ μ2 þ α 1h1ðkÞh1ð kÞð μ1 μ2Þ

1 þ2ð μ1 μ0Þ½2 μ0εV p þ ð3κ 0 þ4 μ0ÞεV s =½5 μ0ð3κ 0 þ4 μ0Þ

ð11Þ

ρ0 ¼ ρ2 þ α 1h1 kð Þh1ð kÞð μ1 μ2Þ

1þð ρ1 ρ0Þð3εV p 2εV s Þ=ð3 ρ0Þð12Þ

where

h1ðkÞ ¼3ð sin ka ka cos ka Þ

ðkaÞ3 ð13Þ

Eq. (13) corresponds to Eq. (3.17) in Sabina and Willis (1988) for

a spherical inclusion.

It should be noted that for more than one inclusion, n41,

Eqs. (1) and (2) are used with the properties of the other inclu-

sions to get the whole self-consistent equations. Eqs. (10)–(12)

must be solved simultaneously using a numerical method, for

example the xed point. With the effective μ0; κ 0 and ρ0, it is

possible to calculate the compressional and shear waves (Eqs. (8)

and (9)) as well as their corresponding attenuations. The measure

of attenuation is obtained by Imðk paÞ and ImðksaÞ (Auld, 1973;

Sabina and Willis, 1988). More details on how to solve the non-linear equation systems take into account several inclusions are in

Valdiviezo-Mijangos (2002). In the following section, there are

numerical results about different combinations of shale composi-

tion and their maturity with uid inclusions.

Fig. 2. Ternary diagram of the mineral composition for global shale plays. It is based on clay, quartz and carbonate to point out argillaceous (north), siliceous (southwest) and

calcareous (southeast) source rocks, respectively. Brown solid line corresponds to the experimental data adapted from Rickman et al. (2008) and Passey et al. (2010). (For

interpretation of the references to color in this gure legend, the reader is referred to the web version of this article.)


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3. Numerical results and discussion

Notice that the total porosity and saturation are commonly

measured by bulk volume fractions (v/v) and the mineral concen-

tration is always stated by mass fractions (wt%); both types of

petrophysics data are related to the density of pure minerals. In

this work, the modeling of dynamical elastic properties (Figs. 3–5)

is carried out using bulk volumetric fraction of the shale consti-

tuents based on conventional log interpretation. For just oneinclusion the volumetric fraction α is easily related to the dis-

criminated layer thickness by length-wave resolution Lr and

inclusion radius as Lr ¼ að4π =3α Þ1=3 (Kinra and Anand, 1982),

which is used as a scaling factor throughout full-frequency band.

In order to extend the application of the self-consistent method to

estimate dispersion and attenuation of elastic waves through gas–oil

shale systems, herein the laboratory data measured on sedimentary

rock sample is utilized (Batzle et al., 2006; Wang and Sun, 2010). The

typical properties for the Fox Hill sandstone are assumed density

2200 kg/m3, compressional and shear-wave velocities are 3.91 km/s

and 2.21 km/s respectively. The pore volume fraction 25% is fully

saturated with Glycerin density 1260 kg/cm3 and compressional

wave velocity of 1.92 km/s. The numerical results are depicted in

Fig. 1; it shows four sets of graphs against frequency: P -wave (a) and

S -wave (b) velocities and their attenuations calculated as Imðk paÞ

(c) and ImðksaÞ (d) respectively. Each simulation result includes three

radii ða ¼ ΩV p=2π f Þ as a scaling factor through frequencies: 0.143 m

(dashed line), 1.43 m (solid line) and 14.3 m (dotted line) which

increases coherently with the wavelength resolutions similar to

Wang and Sun (2010). These simulations were calculated by self-

consistent equations in order to elucidate the relationship between

scaling factor and frequency. This methodology is identical to

experimental technique developed by Kinra and Anand (1982, p.

373). For example, when the frequency is 1 kHz the scaling factor is

0.143 m and either frequency is 10 Hz, the corresponding scaling

factor is 14.3 m. This is useful to estimate the thickness of deposi-

tional cycles identied by the well-log and stratigraphic succession

analysis. In the dispersive frequency bandwidths, at low frequency,

constant values (V p ¼3.24 km/s, V s ¼1.72 km/s) of the wave velo-

cities are observed, next, the curves response slightly decreases andthen it increases as a result of “resonance” phenomenon and nally

for higher frequencies the velocity moves up toward sandstone wave

velocities. Another important fact is that the frequency associated

with the inection point of the velocity corresponds to the attenua-

tion peak for the P - and S -wave velocities. Moreover, P -wave

attenuation is always greater than S -wave attenuation. Similar results

also were reported in Sabina and Willis (1988). Also, this kind of the

response curve behavior is completely consistent with the results

obtained by a micro-structural dispersion model modied by Wang

and Sun (2010). The complete four sets of curves are only shifted by

the inclusion radiuses ða ¼ ΩV p=2π f Þ for modeling the experimental

data digitalized from Batzle et al. (2006). It outlines the reconciliation

process among wave velocities measured at different frequencies

using seismic, sonic and ultrasonic techniques. Therefore, this sort of

modeling quanties the frequency range for velocity dispersions and

their corresponding attenuation.

After the previous analysis, the results of application of the self-

consistent method considering the mineral composition of the

source rocks, maturity and the type of inclusion embedded

in it. The mineralogy is made out from published ternary dia-

grams of different commercial shale plays, i.e., Barnett, Eagleford,

Fig. 3. (a) Compressional wave velocity, V p. (b) Shear wave velocity, V s . (c) Compressional wave attenuation, Imðk paÞ. (d) Shear wave attenuation, ImðksaÞ. Frequency-

dependent elastic response of carbonate dominated shale systems at constant inclusion radius. Gas –oil shale systems: drained shale, CA(80%) þCL(20%); water saturated

shale, CA(80%) þCL(5%)þQR(5%)þWA(10%); immature source rock, CA(80%) þCL(10%) þKE(10%); two mature source rocks, CA(80%) þCL(10%) þKE(5%)þHO(5%) and CA

(80%)þ CL(10%) þKE(5%)þDG(5%); two post-mature source rocks, CA(80%) þ CL(10%) þHO(10%) and CA(80%) þCL(10%) þDG(10%).


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Woodford, Marcellous, Muskwa, etc. (Rickman et al., 2008; Passey

et al., 2010). The mineral groups used are mainly represented byclay (clay, illite, smectite and kaolinite), quartz (quartz, feldspars

and albite) and carbonate (calcite, dolomite and siderite). Fig. 2

shows three different arrangements of lithotypes; each vertex

represents 100% (1.0) of clay, quartz and carbonate. Dominated

lithotypes content more than 80% (0.8) of particular mineral, and

calcareous, siliceous and argillaceous shale formations are identi-ed between 50% and 80% (0.5–0.8). Mixed mudstones could be

classied in the core of the ternary diagram.

For theoretical modeling of velocity dispersion and attenuation

in gas–oil shale systems at various frequency bands ( f ) considering

their mineralogy and maturity, the main data required are density

( ρ), compressional (V p) and shear wave (V s) velocities, Table 1, and

volumetric fraction (α ) of pure constituents; i.e. clay (CL), quartz

(QR), carbonate (CA) and pore uids named as water (WA), heavyoil (HO) or dry gas (DG). Obviously, whenever possible, its

laboratory measurements will be preferred from available source

rock specimen.

Moreover, the integrated studies of well, sonic and seismic

data, the properly identication and discrimination of shale

systems must be supported by two requirements: the coherent

basis of mineral classication (Fig. 2) and the maturity level

description of source rocks (Table 2). In this work, mineral content

is indicated by volume fractions of clay, quartz and carbonate; and

the maturity is slightly stated by drained, water-saturated, imma-

ture, mature and post-mature source rocks. This general mechanic

classication may be directly correlated with the maturity level,

sedimentological and lithological characteristics of commercial

shale systems (Passey et al., 2010).

Figs. 3–5 exhibit curve patterns of P - and S -wave velocities and

their attenuation as a function of frequency including the effect of mineralogy and shale maturity. Utilizing them, we can elucidate the

elastic response of shales classied in Table 2. Figs. 3 and 4 represent

carbonate (CA) and quartz (QR) dominated shale formations. They

exhibit very similar bearing on the predicted elastic properties in the

acoustic dispersive frequency bandwidth. However, they have quan-

titatively differences. The previous results from their slightly con-

trasting properties (Table 1) assigned for both solid matrixes

corresponding to 80% bulk volumetric fraction. Their patterns are

normally characterized by low-frequency wave velocities smaller

than high-frequency wave velocities. The lowest and highest calcu-

lated data series correspond to drained shales: CA(80%)þCL(20%), QR

(80%)þCL(20%) and water saturated shales: CA(80%)þCL(5%)þQR

(5%)þWA(10%), QR(80%)þCL(5%) þCA(5%)þWA(10%); respectively.

Also, it is clearly observed that dry gas contained into shale formationdrives to higher P - and S -waves attenuation. The rest of shale systems

(Table 2): immature, mature and post-mature source rocks are

similar in their V p and V p curves. An additional sedimentological

and lithological analyses could be seen in Cen and Hersi (2006).

The shale system with clay dominated bearing (Z80% of Clay)

is drastically different from the previous two systems; it clearly

displays reverse curve patterns of the P - and S -wave velocities and

larger values of wave attenuations, Fig. 5. The set of curves are the

same as Figs. 2 and 3, but the solid skeleton is mainly composed of

clay. It is observed that low-frequency wave velocities are greater

than high-frequency velocities as a result of clay elastic properties

smaller than values of carbonate solid inclusions (Table 2).

For instance, the highest values at seismic wave velocities

(V p ¼1.38 km/s, V s ¼0.77 km/s) are only for solid rock identied

Fig. 4. (a) Compressional wave velocity, V p. (b) Shear wave velocity, V p. (c) Compressional wave attenuation, Imðk paÞ. (d) Shear wave attenuation, ImðksaÞ. Frequency-

dependent elastic response of quartz dominated shale systems at constant inclusion radius. Gas–oil shale systems: drained shale, QR(80%)þ CL(20%); water saturated shale,

QR(80%)þCL(5%)þCA(5%)þWA(10%); immature source rock, QR(80%) þCL(10%) þKE(10%); two mature source rocks, QR(80%) þCL(10%) þKE(5%)þHO(5%) and QR(80%) þ

CL(10%) þKE(5%)þ DG(5%); two post-mature source rocks, QR(80%) þCL(10%) þHO(10%) and QR(80%) þCL(10%) þDG(10%).


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as drained shale: CL(80%) þCA(20%) and the lower values are for

post-mature source rocks containing dry gas: (V p ¼1.14 km/s,

V s ¼0.64 km/s) and heavy oil (V p ¼1.21 km/s, V s ¼0.63 km/s). Also,

throughout dispersive frequency bandwidth, the curve distortion

of mature and post-mature source rock series is derived from the

highly contrasting elastic properties among carbonate inclusion

solid and dry gas or heavy oil uids.

Summarizing, the P -, S -waves dispersion and their attenuations

were computed including different mineral compositions, type

of pore uids and rock maturity for shale systems (Table 2).

The results condensed at curve patterns aim to analyze and inter-

pret the seismic responses of argillaceous, calcareous and siliceous

shales (Fig. 2). Moreover, it is useful to identify the quality and

distribution of source rocks from seismic and sonic wave

velocities.

4. Concluding remarks

The rock mineral composition represented by clay, quartz and

carbonate as well as the saturating uid rocks is considered in the

dynamic modeling. The analyses of dispersion and attenuation of

elastic waves through wide-frequency bands in shale systems are

carried out using the effective dynamical mechanical properties.

These properties were calculated with the self-consistent method

for patches with aspect radio equal to 1. Useful curve patterns for

P - and S -waves velocity and attenuation are presented to identify

and discriminate argillaceous, siliceous and calcareous shale

systems and uids in the pore. The maturity of the shale sourceformations is included in the analysis of the elastic response

patterns. The obtained patterns will be invaluable to analyze and

interpret the seismic and sonic responses of oil and gas shale

systems; and for the characterization of unconventional reservoirs.

Higher values of P - and S -waves attenuation are estimated in clay

dominated shale and lower values in quartz dominated shale. To

reconcile elastic properties measured at different frequencies in

laboratory, the inclusion radius and dimensionless frequency are

used. All the modeling results are consistent with the volumetric

fraction, discriminated layer thickness, inclusion radius and

length-wave resolution.

Next steps in this research is to consider the transverse

anisotropy and wide range of inclusion shapes like layers, lens

and patches with aspect ratio from 0.05 to 0.99; and to use

Fig. 5. (a) Compressional wave velocity, V p. (b) Shear wave velocity, V s . (c) Compressional wave attenuation, Imðk paÞ. (d) Shear wave attenuation, ImðksaÞ. Frequency-

dependent elastic response of clay dominated shale systems at constant inclusion radius. Gas –oil shale systems: drained shale, CL(80%) þCA(20%); water saturated shale, CL

(80%)þ CA(5%)þQR(5%)þWA(10%); immature source rock, CL(80%) þCA(10%) þKE(10%); two mature source rocks, CL(80%)þCA(10%)þKE(5%)þHO(5%) and CL(80%)þCA

(10%)þ KE(5%)þDG(5%); two post-mature source rocks, CL(80%) þCA(10%) þHO(10%) and CL(80%) þCA(10%) þDG(10%).

Table 1

Density and elastic properties of the pure mineral constituents and pore uidspresent in documented gas–oil shale systems.

Mineral group

and pore uids

Density V p V s Reference

(kg/m3) (km/s) (km/s)

Clay 2650 1.12 0.61 Mavko et al. (2003)

Carbonate 2710 6.64 3.44 Mavko et al. (2003)

Quartz 2640 5.63 3.48 Mavko et al. (2003)

Kerogen 1300 2.25 1.45 Mavko et al. (2003)

Heavy oil 972.5 1.29 _ Batzle and Wang (1992)

Dry gas 51.7 0.505 – Batzle and Wang (1992)

Sea water 1040 1.5 – Batzle and Wang (1992)

The reference temperature is 100 1C and reference pressure is 10 MPa for density,

P - and S -wave velocities.


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available seismic data from shale reservoir to ultimate the model-

ing validation.

Acknowledgments

The authors are thankful to Instituto Mexicano del Petroleo for

giving permission to publish the article.

Appendix A. The kernel functions

Green's function G for the isotropic homogeneous matrix

material takes the form

Gijð xÞ ¼ 1

4πρ0δ ij

eiωj xj=V s

V 2s j xj

1

ω2∂2

∂ xi∂ x j

eiωj xj=V p eiωj xj=V s

j xj

( ) ðA1Þ

where the elastic tensor C 0 is

ðC 0Þijkl ¼ κ 0δ ijδ kl þ μ0ðδ ikδ jl þδ ilδ jk 23 δ ijδ klÞ ðA2Þ

the velocities V p and V s are respectively

V p ¼ κ 0 þð4=3Þ μ0

ρ0 1=2

and V s ¼ μ0

ρ0 1=2

ðA3Þ

The elastic moduli κ 0 and μ0 and density ρ0 may be complex but

are assumed to be such that the square roots in (A3) can both be

chosen with positive real parts and negative imaginary parts so

that, when ω is real and positive, (A1) corresponds to outgoing

waves that decay as j xj increases.

Considering rst the term M t , for a spherical inclusion of

radius a (patches with aspect radio equal to 1) the integral

I ¼

Z j xjoa

dx

Z j x

0joa

dx0eikγ j x x

0j

j x x0j ðA4Þ

where

kγ ¼ ω

γ

ðA5Þ

and γ ¼ V p or V s. The evaluation of (A4) is elementary and gives

I ¼ 16π 2a3

3k2γ ðεγ 1Þ ðA6Þ

where

εγ ¼ 3ð1 ikγ aÞ

ðkγ aÞ3

½ sin ðkγ aÞ kγ a cos ðkγ aÞeikγ a: ðA7Þ

Another term required is

J ij ¼

Z j xjoa

dx

Z j x'joa

dx0 ∂2

∂ xi∂ x j

eikγ j x x0j

j x x0j

ðA8Þ

This is isotropic and so may be evaluated as one-third of its

trace, multiplied by δ ij. Since term in brackets is 4π times the

fundamental solution of a Helmholtz equation, the result is

J ij ¼ 1

3δ ij½4π a

3 þ k2γ I or J ij ¼ δ ij16π 2a3

9 εγ : ðA9Þ

It follows that:

ðM t Þij ¼ 1

3 ρ0δ ijð3εV p 2εV s Þ: ðA10Þ

The term S x can be evaluated similarly. It is an isotropic, fourth-

order tensor with the same symmetries as C 0. With the notation

C 0 ¼ ð3κ 0; 2 μ0Þ ðA11Þ

it follows that:

3κ 0 ¼ 13 ðC 0Þiikk ; 3κ 0 þ10 μ0 ¼ ðC 0Þijij: ðA12Þ

The terms corresponding to 3κ 0, 2 μ0 for S x may thus be

obtained by considering ðS xÞiikk and ðS xÞijij . The details of the

calculation are omitted but the result is

S x ¼εV p

3κ 0 þ4 μ0;

1

5

2εV p3κ 0 þ4 μ0

þεV s μ0 : ðA13Þ

Table 2

Volumetric fractions used for characterization of the elastic response patterns of shale systems. The shale maturity classication and general mechanic description are

included.

No. Shale systems Volumetric fraction Matrix ( I ¼80%)

10 10 5 5

(%) (%) (%) (%)

1 Drained shale: a II II

Solid skeleton, matrix of the shale systems. Any pore uids are not present. b II IIIc III III

2 Water saturated shale: d II III 4

Matrix and whole porous volume is saturated with sea water. e III II 4

f 4 II III

3 Immature source rock: g II 1

Matrix and kerogen prone to generate oil and gas. h III 1

i 1 II III

j II III 1

k III II 1

4 Mature source rock: l II 1 2

m III 1 2Matrix, kerogen and dry gas or heavy oil.

n II 1 3

o III 1 3

5 Post-mature source rock: p II 3

q III 3Matrix and dry gas or heavy oil saturating bulk porosity.r II 2

s III 2

I, II and III are clay, quartzite and carbonate. They are switched among them to characterize the complex mineralogy of different shale matrixes (1 ϕ). 1, 2, 3 and 4 are

kerogen, dry gas, heavy oil and sea water, respectively. [The total porosity (ϕ) could include solid inclusions embedded into the solid matrix (I) and pore uids.]


8/16/2019 1-s2.0-S092041051400240X-main.pdf

8/8

Eqs. (1)–(13) and Eqs. (A1)–(A13) are compiled to describe

the elastic behavior of shale plays by means of self-consistent

equations.

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