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Journal of Petroleum Science and Engineering

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Rock physics modelling of elastic properties of organic shale consideringkerogen stress and pore pressure distribution

Xuehui Hana, Junguang Niea, Junxin Guob,∗, Long Yangc, Denghui Xua

a School of Geosciences, China University of Petroleum (East China), Qingdao, 266580, ChinabDepartment of Earth and Space Sciences, Southern University of Science and Technology, Shenzhen, 518055, Chinac Xinjiang Laboratory of Petroleum Reserve in Conglomerate, Kelamay, Xinjiang, 834000, China

A R T I C L E I N F O

Keywords:Organic shaleKerogen stressPore pressureElastic properties

A B S T R A C T

Rock physics model for the elastic properties of organic shale is important for shale oil/gas exploration andproduction. Up until now, little work has been done to investigate the influence of the distributions of the stressin kerogen (kerogen stress) and the pore pressure on the elastic properties of organic shale. In this work, we usedthe Kuster- Toksöz (KT) model and Gassmann equation to study such effects. The Gassmann equation was ap-plied for the homogeneous kerogen stress or pore pressure case whereas KT model was suitable for the in-homogeneous case. Four cases with different combinations of kerogen stress and pore pressure distributions, forthe elastic properties of organic shale, were analysed and the corresponding models were given. Based on thesemodels, a numerical example was studied. The results showed that the distributions of kerogen stress and porepressure significantly affect the elastic properties of organic shale. The elastic moduli under the inhomogeneouskerogen stress or pore pressure distributions are larger than the homogeneous case, whose magnitude dependson porosity and kerogen content. Furthermore, the joint effects of kerogen stress and pore pressure distributionsare similar to those of kerogen stress due to the much smaller effects of pore pressure. Hence, it is essential toconsider the effects of the distributions of kerogen stress and pore pressure when building the rock physics modelfor the elastic properties of organic shale. This work revealed the importance of kerogen stress and pore pressuredistributions on the elastic properties of organic shale and hence is helpful for the shale oil/gas exploration andproduction.

1. Introduction

Organic shale forms an important type of unconventional reservoirsin the world. Nowadays, it has contributed a large amount to the pro-duction of oil/gas worldwide, especially in the USA. The successfuldevelopment of shale oil/gas has attracted more and more attention onthe study of the exploration and development methods of organic shalereservoirs. Since advanced geophysical methods have been widely usedto investigate formation velocity (Wang et al., 2015a, 2017) and frac-tures (Wang et al., 2015b), the seismic methods and sonic loggingmeasurements are often used to detect and characterize the physicalproperties of the organic shale, such as its porosity, kerogen maturity,and oi/gas saturation, among many others (e.g., Mondol, 2014; Yanget al., 2015; Vernik, 2016; Wang et al., 2016; Zhao et al., 2016; Li et al.,2017). The feasibility of obtaining these physical properties of organicshale from seismic or sonic logging measurement data is based on thefact that these properties have great influence on the elastic moduli of

organic shale (e.g., Yenugu and Vernik, 2015; Vernik, 2016). Hence, toaccurately interpret the seismic or sonic measurement data, the re-lationship between the physical and elastic properties of organic shaleneeds to be established via the rock physics models. Different withconventional sandstone, the organic shale contains kerogen and isusually composed of complex minerals (Shaw and Weaver, 1965; Wanget al., 2009). This makes the elastic properties of organic shale quitedifferent from the sandstone. In addition, organic shale usually exhibitsthe properties of transversely isotropy due to the alignment of kerogenand minerals, which further complicates its elastic properties (e.g.,Hornby et al., 1994; Hornby, 1998; Lonardelli et al., 2007). Hence, rockphysics models considering the characteristics of organic shale need tobe built to link its physical properties to the corresponding elasticproperties.

Up until now, various rock physics models have been proposed topredict the elastic properties of organic shale. Hornby et al. (1994)predicted the elastic properties of organic shale through combining the

https://doi.org/10.1016/j.petrol.2018.11.063Received 19 April 2018; Received in revised form 23 November 2018; Accepted 25 November 2018

∗ Corresponding author.E-mail address: guojx@sustc.edu.cn (J. Guo).

Journal of Petroleum Science and Engineering 174 (2019) 891–902

Available online 04 December 20180920-4105/ © 2018 Elsevier B.V. All rights reserved.

T

self-consistent approximation (SCA) and the differential effectivemedium theory (DEM). Vernik and Landis (1996) analysed the influ-ence of kerogen shape and distributions on the effective elasticity oforganic shale using a modified Backus averaging method. Lucier et al.(2011) modelled the elastic moduli of organic shale through the Ha-shin-Shtrikman lower bounds and investigated the effects of kerogenand gas saturation based on it. Zhu et al. (2012) treated the kerogen asthe inclusions and then the Gassmann equations for solid substitutionare used to study the relations between kerogen content and theacoustic velocities of organic shale. Guo et al. (2013) studied the brit-tleness of the Barnett Shale by using the Backus averaging and theisotropic SCA. Li et al. (2015) developed a rock physics model to in-vestigate the influence of kerogen content and kerogen porosity on theelastic properties of organic shale and the corresponding AVO responsesof shale layers based on the Kuster and Toksoz (KT) theory and the SCAmethod. Zhao et al. (2016) proposed the rock physics scheme to predictthe variation of the elastic behaviours of organic shale with the kerogenmaturity degree. The prediction results are in good agreement with theultrasonic and log measurement data. Chen et al. (2016) considered theeffects of the multi-inclusion and the interfacial transition zone on theelastic moduli of organic shale and proposed the corresponding multi-scale model to quantify these effects.

Due to the complex structures and distributions (both in positionsand orientations) of pores and kerogen in the organic shale, the dis-tributions of stress in kerogen (kerogen stress) and pore pressure areusually complicated (e.g., Vermylen and Zoback, 2011; Roshan andAghighi, 2012). However, to date, little work has been done to in-vestigate their influence on the elastic properties of organic shale. Thedifferent distributions of kerogen stress and pore pressure may havegreat effects on the elastic moduli of organic shale, and therefore, theseismic and sonic logging responses of organic shale reservoirs. Hence,it is essential to develop the physics models to study these effects. Inthis paper, we consider four cases of stress and fluid pressure dis-tributions in the kerogen and the pores of organic shale (Fig. 1): a)

homogeneous stress and fluid pressure in kerogen and pores respec-tively (HSHP); b) inhomogeneous kerogen stress and homogeneouspore pressure (ISHP); c) homogeneous kerogen stress and in-homogeneous pore pressure (HSIP); d) Inhomogeneous stress and fluidpressure in kerogen and pores, respectively (ISIP). The correspondingrock physics models for the elastic properties of organic shale for thesefour cases are proposed. Hence, the effects of the distributions of thekerogen stress and pore pressure on the elastic properties of organicshale can be studied by comparing these four cases. To focus on ana-lysing the stress and fluid pressure distribution effects, we ignore theanisotropic properties of organic shale in this paper and assume theshale to be isotropic. The models taking into account the effects ofanisotropy can be developed in the future based on the isotropic modelsgiven in this paper.

2. Theory

To predict the elastic properties of the organic shale for the fourcases of stress and fluid pressure distributions shown in Fig. 1, K-Tmodel (Kuster and Toksöz, 1974) and Gassmann equations (Gassmann,1951; Ciz and Shapiro, 2007) will be used. In the KT model, it assumesthat the inclusions are isolated from each other. This means that thestress or fluid pressure can't be equilibrated throughout the inclusionspace. Hence, this model can predict the elastic properties of the rockswith the inhomogeneous distribution of stress or fluid pressure in theinclusion space. Conversely, the Gassmann equations for fluid/solidsubstitution (Gassmann, 1951; Ciz and Shapiro, 2007) assume that theinclusions are interconnected with each other, and therefore, the stressor the fluid pressure in the inclusions can be equilibrated throughoutthe inclusion space. Hence, the Gassmann equations can be used toestimate the elastic properties of rocks with connected inclusion space,in which the stress or the fluid pressure is homogeneous. In the fol-lowing, we will combine these two theories to predict the elasticproperties of the organic shale for the four cases of stress and fluid

Fig. 1. Four different cases for the distributions ofkerogen stress and pore pressure. (a) homogeneouskerogen stress and pore pressure. (b) inhomogeneouskerogen stress and homogeneous pore pressure. (c)homogeneous kerogen stress and inhomogeneouspore pressure. (d) inhomogeneous kerogen stress andpore pressure. Note that the ellipse with dashed andsolid boundaries represent the kerogen and thepores, respectively. The homogeneous and in-homogeneous stress and fluid pressure distributionsare denoted by blue and red colours, respectively.

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pressure distributions.It should be noted here that as we focus on studying the elastic wave

propagation in the organic shale, the elastic properties investigated inthis paper are dynamic elastic moduli, which means the strains inducedby the stresses are usually smaller than 10−6 (Mikhaltsevitch et al.,2016). For such small strains, the organic shale should exhibit theproperties of linear elasticity.

2.1. Case a: homogeneous stress and fluid pressure in kerogen and poresrespectively (HSHP)

If the inclusion space is interconnected and the frequency of theacoustic wave is low enough, the kerogen stress and the pore pressurewill have enough time to equilibrate. Hence, the kerogen stress and thepore pressure will be homogeneous. Under this condition, Gassmannequations can be used to predict the elastic properties of the organicshale. First, the Gassmann equation for solid substitution is applied to

calculate the bulk and shear moduli, Ks and Gs, of the effective solidphase as follows (Ciz and Shapiro, 2007):

= ⎛

⎝⎜ −

−− + −

⎞

⎠⎟

−− −

− − − −

−

K KK K

ϕ K K K K( )

( ) ( )s dd gr

kg kg gr d gr11 1

1 1 2

1 111 1

1

(1)

= ⎛

⎝⎜ −

−− + −

⎞

⎠⎟

−− −

− − − −

−

G GG G

ϕ G G G G( )

( ) ( )s dd gr

kg kg gr d gr11 1

1 1 2

1 111 1

1

(2)

where Kd1 and Gd1 are the bulk and shear moduli of the effective solidphase before filling the kerogen into the inclusion space, respectively;Kgr and Ggr are the effective bulk and shear moduli of the mineralscomposing the organic shale, respectively; Kkg and Gkg are the bulk andshear moduli of the kerogen, respectively; ϕkg is the relative volume ofthe kerogen in the solid phase, which can be calculated as follows:

Fig. 2. The procedure of calculating the elastic properties of organic shale for Case a (homogeneous kerogen stress and pore pressure).

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=′

−ϕ

ϕ

ϕ1kgkg

f (3)

In equation (3), ′ϕkg is the relative volume of the kerogen with re-spect to the whole organic shale; ϕf is the porosity of the organic shalefilled with fluid.

Since Kd1 and Gd1 are the bulk and shear moduli for the solid phaseof organic shale before filling the inclusion space with the kerogen,these dry elastic properties of the solid phase can be obtained by usingthe KT model (Kuster and Toksöz, 1974; Mavko et al., 2009) as follows:

=+ −

+ +K

K K G ϕ K P G

K G ϕ K P

( 4/3 ) 4/34/3d

gr gr gr kg gr gr

gr gr kg gr1

1

1 (4)

=+ −

+ +G

G G ζ ϕ G Q ζ

G ζ ϕ G Q

( )d

gr gr gr kg gr gr

gr gr kg gr1

1

1 (5)

where:

=+

+ζ

G K GK G(9 8 )

6( 2 )grgr gr gr

gr gr (6)

=PK

πα βgr

kg gr1

(7)

= ⎛

⎝⎜ +

++ ⎞

⎠⎟Q

Gπα G β

Gπα β

15

18

( 2 )4/3gr

kg gr gr

gr

kg gr1

(8)

with αkg being the aspect ratio of the kerogen and βgr having the fol-lowing form:

=+

+β G

K GK G

33 4gr gr

gr gr

gr gr (9)

After obtaining the elastic moduli of the effective solid phase, thebulk and shear moduli of the dry organic shale, Kd2 and Gd2, can beobtained in a similar way as Kd1 and Gd1 using the KT model throughequations (4) and (5). However, it should be noted that Kgr and Ggr needto be replaced by Ks and Gs, respectively. Furthermore, the fraction andaspect ratio of the kerogen, ϕkg and αkg, also need to be replaced by thecorresponding values of the pores ϕf and αf , respectively.

Finally, due to the homogeneous fluid pressure distribution in thepores, the bulk and shear moduli, Ksat and Gsat, of the organic shale canbe obtained by saturating the dry pores with the fluid using theGassmann equation for fluid substitution (Gassmann, 1951):

= +−

+ −−K K K K(1 / )

sat dd s

ϕ

K

ϕ

KKK

22

2

1f

f

f

sd

s

22 (10)

=G Gsat d2 (11)

Fig. 3. The procedure of calculating the elastic properties of organic shale for Case b (inhomogeneous kerogen stress and homogeneous pore pressure).

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where Kf is the fluid bulk modulus.The whole procedure of obtaining the elastic properties of the or-

ganic shale for Case a is shown in Fig. 2.

2.2. Case b: inhomogeneous kerogen stress and homogeneous pore pressure(ISHP)

If the kerogen inclusion space in the organic shale is not inter-connected but the pore space is interconnected, then it is possible thatthe stress inside the kerogen is not homogeneous but the pore pressureis homogeneous given that the frequency of the acoustic wave is lowenough. In this case, the effects of kerogen on the elastic properties oforganic shale can be taken into account by using the KT model, whereasthe influence of fluid saturation in the pores can be considered using theGassmann equation. Hence, the bulk and shear moduli of the effectivesolid phase, Ks and Gs, can be calculated as follows (Kuster and Toksöz,1974; Mavko et al., 2009):

=+ + −

+ − −K

K K G ϕ K K P G

K G ϕ K K P

( 4/3 ) 4/3 ( )4/3 ( )s

gr gr gr kg kg gr gr

gr gr kg kg gr

2

2 (12)

=+ + −

+ − −G

G G ζ ϕ G G Q ζ

G ζ ϕ G G Q

( ) ( )( )s

gr gr gr kg kg gr gr

gr gr kg kg gr

2

2 (13)

where:

=+

+ +P

K GK G παβ

4/34/3

gr kg

kg kg gr2

(14)

= ⎛

⎝⎜ +

+ ++

+ ++ +

⎞

⎠⎟Q

GG πα G β

K G GK G παβ

15

18

4 ( 2 )2

2/3( )4/3

gr

kg gr gr

kg kg gr

kg kg gr2

(15)

Same as Case a, after obtaining Ks and Gs, the elastic properties ofthe dry organic shale, Kd2 and Gd2, can be calculated through the KTmodel using equations (4) and (5), with Kgr, Ggr, αkg , and ϕkg replaced byKs, Gs, αf , and ϕf , respectively. Then, the elastic properties of the sa-turated organic shale can be computed using the Gassmann equation forfluid substitution (Gassamann, 1951), as shown in equations (10) and(11). The whole procedure of obtaining the elastic properties of theorganic shale for Case b is shown in Fig. 3.

2.3. Case c: homogeneous kerogen stress and inhomogeneous pore pressure(HSIP)

When the inclusion space for kerogen is interconnected and thefrequency of the acoustic wave is low enough, the kerogen stress will behomogeneous. However, if the pore space is not interconnected, thefluid pressure will not be equilibrated and thus will be inhomogeneous.Under this condition, the elastic properties of the effective solid phasecan be calculated using the same method as Case a. We can first cal-culate the elastic properties of the solid phase without filling thekerogen into the inclusion space, as shown in equations (4) and (5).

Fig. 4. The procedure of calculating the elastic properties of organic shale for Case c (homogeneous kerogen stress and inhomogeneous pore pressure).

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Then, the kerogen is filled into the inclusion space and its effects on theelastic properties of the solid phase can be estimated by using theGassmann equation for solid substitution, as shown in equations (1) and(2). After we obtain the elastic properties of the effective solid phase,the elastic properties of the saturated organic shale with in-homogeneous fluid pressure can be obtained using the KT model asfollows:

=+ + −

+ − −K

K K G ϕ K K P G

K G ϕ K K P

( 4/3 ) 4/3 ( )4/3 ( )sat

s s s f f s s

s s f f s

3

3 (16)

=+ −

+ +G

G G ζ ϕ G Q ζ

G ζ ϕ G Q

( )sat

s s s f s s

s s f s

3

3 (17)

where:

=+

+ζ G K G

K G(9 8 )

6( 2 )ss s s

s s (18)

=+

P KK παβ

s

f s3

(19)

⎜ ⎟= ⎛⎝

++

++ +

+⎞⎠

Q Gπα G β

K G GK πα β

15

1 8( 2 )

22/3( )s

f s s

f f s

f f s3

(20)

with βs having the following form:

= ++

β G K GK G

33 4s s

s s

s s (21)

The whole procedure of obtaining the elastic properties of organicshale for Case c is shown in Fig. 4.

2.4. Case d: inhomogeneous stress and fluid pressure in kerogen and pores,respectively (ISIP)

If both the kerogen inclusion space and the pore space are not in-terconnected or the frequency of the acoustic wave is high enough, thekerogen stress and the pore pressure are both inhomogeneous. In thiscase, the effects of kerogen and fluid on the elastic properties of organicshale can both be estimated by using the KT model. First, the elasticproperties of the effective solid phase can be calculated using equations(12) and (13), which is same as Case b. Then, the elastic moduli of thesaturated organic shale can be obtained using the same equations asCase c, as shown in equations (16) and (17). The whole procedure ofobtaining the elastic properties of the organic shale for Case d is shownin Fig. 5.

3. Numerical example

3.1. Parameters

To show the effects of the distributions of the kerogen stress andpore pressure on the elastic properties of organic shale, we study thesaturated shale samples containing mature kerogen. Following Zhaoet al. (2016), the mineral composition of the samples is as follows: 20%clay, 20% carbonate, 30% quartz, and 30% feldspar. The elastic moduliof these minerals are shown in Table 1. The effective elastic moduli ofthe mineral mixture can be obtained by using the Voigt-Reuss-Hillaveraging approach (Mavko et al., 2009). The bulk and shear moduli ofmature kerogen are 5 GPa and 3.5 GPa, respectively (Yan and Han,2013; Zhao et al., 2016). The other parameters of the organic shalesamples are assumed as follows: the bulk modulus of the saturatingfluid (brine) is 2.18 GPa; the aspect ratios of the kerogen inclusion andpores are 0.1 and 0.25, respectively (Sayar and Torres-Verdín, 2016;Zhao et al., 2016); the sample porosity ranges from 0 to 0.1; and thefractions of the kerogen in the samples (kc) are 0.05 and 0.1, respec-tively.

Using these parameters and the theory presented above, we cancalculate the bulk and shear moduli, as well as the Young's moduli[9KG/(3K+G)], of the organic shale under different stress and fluidpressure distributions. Their influence on the elastic properties of theorganic shale can thus be analysed.

3.2. Results

3.2.1. Effects of stress distributions in kerogen on the elastic properties oforganic shale

Fig. 6 shows the elastic properties of the organic shale for Case a(HSHP) and Case b (ISHP) under different porosities and kerogencontents. It can be found that the bulk and shear moduli, as well asYoung's moduli, all decrease with the porosity and kerogen content.This is due to the fact that both the elastic moduli of the kerogen andthe fluid are much smaller than those of the minerals. Hence, the in-crease of the kerogen content or the porosity will result in the decreaseof the elastic moduli of the organic shale. The influence of the dis-tributions of the kerogen stress on the elastic properties of organic shalecan be analysed by comparing the elastic properties of the organic shalefor Case a and b. It can be seen that the organic shale with homo-geneous distribution of kerogen stress has obviously lower elasticmoduli than the inhomogeneous case, especially for the bulk moduli.The reason is that, in the case with homogeneous stress distribution, thestress increase is averaged over the whole volume of the kerogen in-clusion space, whereas it cannot be released in the inhomogeneouscase. Hence, the organic shale is under the relaxed state for thehomogeneous stress distribution case, but unrelaxed state for the in-homogeneous case. The elastic moduli of the organic shale are thuslower under the homogeneous stress distribution than under the in-homogeneous stress distribution.

The effects of stress distributions in kerogen can be quantified by therelative change of the elastic moduli between Case a and b, as shown inFig. 6d. The relative change is calculated by first obtaining the differ-ence between the elastic moduli for Case a and b, then dividing it by theelastic moduli for Case a. It can be found that both the relative changesof the shear and Young's moduli increase slightly with the porosity,whereas that for the bulk moduli shows a reverse trend with the por-osity. Furthermore, it can be observed that the relative changes of theshear and Young's moduli are close to each other, while that for thebulk moduli is much higher. This indicates that the influence of thestress distributions in the kerogen on the bulk modulus is much higherthan that on the shear and Young's moduli. The shear and Young'smoduli are affected by the stress distributions in a similar manner,whereas the bulk moduli are influenced differently. When the kerogencontent increase, we can observe that the relative changes of the elasticmoduli increase almost linearly with the kerogen content. This impliesthat the influence of the stress distributions depends linearly on thekerogen content. It can also be seen that the effects of the stress dis-tributions are more obviously influenced by the kerogen content thanthe porosity.

3.2.2. Effects of fluid pressure distributions in pores on the elastic propertiesof organic shale

The effects of fluid pressure distributions in pores on the elasticproperties of organic shale can be studied by comparing the results forCase a (HSHP) and Case c (HSIP), as shown in Fig. 7. We can also ob-serve the decrease of the elastic moduli with the kerogen content andthe porosities due to the lower moduli of the kerogen and the fluidcompared to the minerals. We can see that the effects of the fluidpressure distributions on the elastic properties of organic shale aremuch smaller than those of stress distributions. Compared to the shearand Young's moduli, the effects of the fluid pressure distributions on thebulk moduli are slightly larger. This implies that the fluid pressuredistributions have different influence on the bulk moduli and the shearmoduli (or Young's moduli).

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Fig. 7d shows that the relative change of the elastic moduli of theorganic shale due to the effects of fluid pressure distributions in pores.It can be seen that the relative changes of elastic moduli increase lin-early with the porosity. When the porosity equals to zero, the relativechange of the elastic moduli is also zero as there is no fluid in the or-ganic shale under this condition. Furthermore, the relative change ofthe elastic moduli also increases with the kerogen content. This is dueto the fact that the increase of the kerogen content will decrease theelastic moduli of the dry shale frame, which will then result in the in-crease of the effects of fluid pressure distributions on the elastic moduliof organic shale. This effect is found to be more obvious on the bulkmoduli than the shear and Young's moduli. Similar as Fig. 7a–c, we canalso obviously observe here that the effects of the fluid pressure dis-tributions on the bulk moduli are larger than those on the shear andYoung's moduli. However, these effects are all small compared to thoseof the stress distributions.

3.2.3. Joint effects of stress and fluid pressure distributions in kerogen andpores on the elastic moduli of organic shale

To show the joint effects of the distributions of the kerogen stressand pore pressure on the elastic properties of organic shale, the resultsfor Case a (HSHP) and Case d (ISHP) are given in Fig. 8. Similarly, wecan observe the decrease of the elastic moduli with the porosity and thekerogen content due to the smaller moduli of fluid and kerogen than theminerals in the organic shale. The bulk and shear moduli under theunrelaxed state (Case d) are obviously larger than those under the re-laxed state (Case a). To quantify the joint effects of the stress and fluid

pressure distribution on the elastic properties of organic shale, we alsoshow the relative change of the elastic moduli between Case a and Cased in Fig. 8d. It can be seen that the joint effects of the stress and fluidpressure distributions on the bulk moduli are larger than those on theshear and Young's moduli. This is similar to the individual effects of thestress or fluid pressure distributions. The previous analysis shows thatthe effects of the stress distributions decrease slightly with the porosity,whereas those of fluid pressure distributions increase with the porosity.Here, due to the competing of these two effects, the joint effects arefound to increase slightly with the porosity. In terms of the effects of thekerogen, we can obviously find that the joint effects of the stress andfluid pressure increase with the kerogen content, which are similar tothe individual effects of the stress and fluid pressure.

In summary, we can see that the effects of stress distributions inkerogen and pore pressure have obvious influence on the elastic prop-erties of organic shale. In particular, the effects of the stress distribu-tions are much larger than those of fluid pressure distributions. Hence,it is important to consider the effects of stress and fluid pressure dis-tributions in kerogen and pores when calculating the elastic moduli ofthe organic shale.

4. Discussion

4.1. Comparison with other model

The elastic properties of rocks with isolated pore-filling materialsare calculated using KT model, as shown above. An alternative ap-proach is using the SK model (Sayers and Kachanov, 1991), which canbe written as follows:

=+

KZ

1eff

K N1

s (22)

=+ +

GZ Z1

eff

G N T1 4

1525s (23)

where Keff and Geff are effective rock bulk and shear moduli,

Fig. 5. The procedure of calculating the elastic properties of organic shale for Case d (inhomogeneous kerogen stress and pore pressure).

Table 1Elastic properties of the minerals in the organic shale (Mavko et al., 2009).

Mineral Bulk modulus (GPa) Shear modulus (GPa)

Clay 25 9Carbonate 76.8 32Quartz 37 44Feldspar 37.5 15

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respectively; Ks and Gs are the bulk and shear moduli of the solid phase,respectively; ZN and ZT are the normal and shear compliances inducedby the compliant pores (cracks), which can be expressed as follows:

=− + +

ZL γ γ K G παG

e43 [1 ( 4/3 )/( )]N

s s s fi fi s (24)

=− +

ZG γ G παG

e163 [3 2 4 /( )]T

s s fi s (25)

with Ls (=Ks+4/3Gs) being the P-wave modulus of the solid phase; Kfi

and Gfi being the bulk and shear moduli of the pore-filling material,respectively; α being the pore aspect ratio; e being the fracture density;γs being expressed as follows:

=γ GLs

s

s (26)

Note that the SK model can only be used for rocks with compliantpores, which means the pore aspect ratio should be small. Hence, forthe small pore aspect ratio case, we can replace the KT model with theSK model in the above schemes for the cases with different pore pres-sure and kerogen stress distributions. Using the same elastic moduli asthe numerical example for the matrix, kerogen, and pore fluid, but

different aspect ratios for the kerogen inclusion and pores (0.05 and 0.1respectively), the comparison between these two approaches is shownin Fig. 9.

We can see that the results given by these two approaches areoverall similar. The discrepancies between them primarily occur forrelatively large porosity cases. This is due to the fact that, KT modelrepresents the effects of pores as reduction of rock elastic moduliwhereas SK model considers that as increase of elastic compliances.This will results in the slightly larger elastic moduli predicted by SKmodel, especially in the relatively large porosities (e.g., Schoenberg andDouma, 1988). However, it can be observed that these two approachesgive similar trends for the elastic moduli for the cases with differentpore pressure and kerogen stress distributions. The differences betweendifferent cases are also similar. This indicates that the approach pro-posed in this paper can model the effects of pore pressure and kerogenstress distributions on shale elastic properties well.

4.2. Extension to the anisotropic case with relatively large porosity andkerogen content

In this paper, we use the KT model to calculate the elastic propertiesof the organic shale under the inhomogeneous kerogen stress or pore

Fig. 6. Comparison of the elastic properties of the organic shale for Case a and b under different porosities and kerogen contents (kc).

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pressure. This model generally works well if the porosity or the kerogencontent is small (Kuster and Toksöz, 1974). However, if the porosity orthe kerogen content is relatively large, the accuracy of this model willdecrease due to the interactions between the kerogen or fluid inclusions(Kuster and Toksöz, 1974). Hence, we need to take into account theinteraction effects under this condition. In order to do so, the Differ-ential Effective Medium (DEM) or Self-consistent Approximation (SCA)can be applied (Mavko et al., 2009). The DEM approach is an iterationprocedure, which can be described as follows (e.g., Cleary et al., 1980;Norris, 1985; Zimmerman, 1991). First, the fluid or the kerogen in-clusions are divided into many portions. Then, they are added into theshale frame one portion at a time using the KT model. This procedure isiterated until all the portions of the fluid or kerogen inclusions areadded into the mineral frame and the effective elastic properties of theorganic shale are thus obtained. As the elastic properties of the shaleframe are updated every time after a new portion of fluid or kerogeninclusion is added, this approach can take into account the effects ofinteractions between the kerogen or fluid inclusions. For the SCA ap-proach, all components of the organic shale, including the mineralframe, the kerogen, and the fluid, are treated as the inclusions in theyet-unknown effective background medium (e.g., Berryman, 1980,

1995). Hence, by replacing the mineral frame with the effective back-ground medium and at the same time treating the mineral frame as onecomponent of the inclusions in the KT model, the equations for theelastic properties of the effective background medium can be obtained.The equations can also be solved through the iteration procedure. Here,the effects of inclusion interactions are considered by using the effectivebackground medium instead of the mineral frame of the organic shale.

An important feature of organic shale is transversely isotropy (e.g.,Hornby et al., 1994; Vernik and Milovac, 2011). Our current approachdoesn't consider the effects of transversely isotropy on the elasticproperties of organic shale. However, this can be easily extended byusing the anisotropic DEM or SCA for the case with inhomogeneousstress distribution in the kerogen or pore pressure (Hornby et al., 1994;Zhao et al., 2016), and anisotropic Gassmann equations for the casewith homogeneous distribution (Brown and Korringa, 1975; Ciz andShapiro, 2007).

4.3. Effects of pores in the kerogen on the elastic properties of organic shale

In the organic shale, while the pore volume in the kerogen may onlycontribute a small portion to the total pore volume in the organic shale,

Fig. 7. Comparison of the elastic properties of the organic shale for Case a and c under different porosities and kerogen contents.

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they are important for the shale oil/gas accumulation and hence theproduction of shale oil/gas. Therefore, it is essential to explore the ef-fects of the pores in kerogen on the elastic properties of organic shale.These effects are ignored in the above analysis, which will only inducesmall errors when the pore volume in the kerogen is small compared toboth the volume of the kerogen and the total porosity. However, whenthe fraction of the pores in the kerogen is relatively large, its effects maybe important and cannot be ignored. Under this condition, we can di-vide the total pores into two sub-pores, one in the kerogen and the otherin the mineral frame of the organic shale. For the sub-pores in thekerogen, its primary influence is on the elastic properties of thekerogen, which then affect those of the organic shale. Hence, we needto first calculate the effective elastic properties of the kerogen. As thefluid in these pores are often not distributed uniformly, the fluid pres-sure is usually inhomogeneous in these pores (Zhao et al., 2016).Hence, we can apply the KT model to obtain the effective moduli of thekerogen if the sub-pores only occupy a small portion of the kerogen.When the fraction of the sub-pores is relatively large in the kerogen, theDEM or SCA approach can be applied. After obtaining the effectiveelastic moduli of the kerogen, the elastic properties of the organic shalecan be calculated using the approaches presented before with the othersub-pores in the mineral frame.

4.4. Inferring microstructures of organic shale from its elastic properties

According to the different stress distributions in the kerogen andpore pressure, we proposed the corresponding model for the elasticproperties of organic shale in this paper. Four cases with homogeneousand inhomogeneous stress and fluid pressure distributions are con-sidered, as shown in Fig. 1. As explained before, the stress and fluidpressure distributions can be affected by the connectivity of the kerogeninclusion space and the pore space, respectively. When the kerogeninclusion space or the pore space is interconnected, the kerogen stressor pore pressure will be homogeneous if the frequency of the acousticwave is low enough. On the contrary, if the kerogen inclusion space orthe pore space is not interconnected, the stress and fluid pressure willbe inhomogeneous. Hence, it is possible to infer the microstructures ofthe organic shale from its elastic properties. In order to do so, we canfirst calculate the elastic properties of the organic shale using the fourmodels presented above respectively. Then, we can match the calcu-lated results with the elastic properties obtained from the measuredacoustic wave velocities and densities. The model that predicts themeasured data best indicates the connectivity of the kerogen inclusionspace and the pore space. Hence, the microstructures of the organicshale can be inferred. It should be noted here that, the measurement

Fig. 8. Comparison of the elastic properties of the organic shale for Case a and d under different porosities and kerogen contents.

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frequency of the acoustic wave should be low enough to make sure thatthe inhomogeneous stress or fluid pressure distributions are not causedby the insufficient time for the stress and fluid pressure to be equili-brated.

5. Conclusions

In this work, the influence of the distributions of the kerogen stressand the pore pressure on the elastic properties of organic shale wasstudied. For this purpose, the KT model was applied for the organicshale with inhomogeneous kerogen stress or pore pressure, whereas theGassmann equations for solid/fluid substitution was used when thekerogen stress or the pore pressure is homogeneous. Four cases withdifferent combinations of stress and fluid pressure distributions werethen investigated and the models for obtaining the correspondingelastic properties of the organic shale were given. Based on thesemodels, a numerical example for the organic shale with different dis-tributions of stress and fluid pressure was studied. The results showedthat, the distributions of stress and fluid pressure have obvious influ-ence on the elastic properties of organic shale. The effective moduliunder the inhomogeneous stress or fluid pressure distributions werefound to be larger than under the corresponding homogeneous dis-tributions. These effects vary with the porosity and the kerogen content.Furthermore, the joint effects of kerogen stress and pore pressure dis-tributions are similar to those of kerogen stress due to the much smallereffects of pore pressure. Hence, it is essential to consider the effects ofstress and fluid pressure distributions when building the rock physicsmodel for the elastic properties of organic shale.

The organic shale considered in this paper is isotropic with lowporosity and kerogen content. When the shale has obvious anisotropicproperties (transversely isotropy) and relatively large porosity andkerogen content, the proposed models can be extended using the ani-sotropic DEM or SCA scheme. Furthermore, the effects of the pores inthe kerogen can also be included in our models by dividing the totalpores into two parts, one in the kerogen and the other in the mineralframe of organic shale. One possible application of the proposed modelsis inferring the microstructures of the organic shale from the measuredacoustic wave velocities. This work revealed the importance of stressand fluid pressure distributions on the elastic properties of organicshale and hence is helpful for the shale oil/gas exploration and pro-duction.

Acknowledgements

We would like to thank the financial support from National NaturalScience Foundation of China, Contract No. U1562108, and NationalKey R&D Program of China, Grant No. 2018YFC0310105.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.petrol.2018.11.063.

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