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

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

Journal of Petroleum Science and Engineering 174 (2019) 891–902 Contents lists available at ScienceDirect Journal of Petroleum Science and Engineeri...

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

Contents lists available at ScienceDirect

Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol

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

T

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

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

A R T I C LE I N FO

A B S T R A C T

Keywords: Organic shale Kerogen stress Pore pressure Elastic properties

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

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



organic shale (e.g., Yenugu and Vernik, 2015; Vernik, 2016). Hence, to accurately interpret the seismic or sonic measurement data, the relationship between the physical and elastic properties of organic shale needs to be established via the rock physics models. Different with conventional sandstone, the organic shale contains kerogen and is usually composed of complex minerals (Shaw and Weaver, 1965; Wang et al., 2009). This makes the elastic properties of organic shale quite different from the sandstone. In addition, organic shale usually exhibits the properties of transversely isotropy due to the alignment of kerogen and minerals, which further complicates its elastic properties (e.g., Hornby et al., 1994; Hornby, 1998; Lonardelli et al., 2007). Hence, rock physics models considering the characteristics of organic shale need to be built to link its physical properties to the corresponding elastic properties. Up until now, various rock physics models have been proposed to predict the elastic properties of organic shale. Hornby et al. (1994) predicted the elastic properties of organic shale through combining the

Corresponding author. E-mail address: [email protected] (J. Guo).

https://doi.org/10.1016/j.petrol.2018.11.063 Received 19 April 2018; Received in revised form 23 November 2018; Accepted 25 November 2018 Available online 04 December 2018 0920-4105/ © 2018 Elsevier B.V. All rights reserved.

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homogeneous stress and fluid pressure in kerogen and pores respectively (HSHP); b) inhomogeneous kerogen stress and homogeneous pore pressure (ISHP); c) homogeneous kerogen stress and inhomogeneous pore pressure (HSIP); d) Inhomogeneous stress and fluid pressure in kerogen and pores, respectively (ISIP). The corresponding rock physics models for the elastic properties of organic shale for these four cases are proposed. Hence, the effects of the distributions of the kerogen stress and pore pressure on the elastic properties of organic shale can be studied by comparing these four cases. To focus on analysing the stress and fluid pressure distribution effects, we ignore the anisotropic properties of organic shale in this paper and assume the shale to be isotropic. The models taking into account the effects of anisotropy can be developed in the future based on the isotropic models given in this paper.

self-consistent approximation (SCA) and the differential effective medium theory (DEM). Vernik and Landis (1996) analysed the influence of kerogen shape and distributions on the effective elasticity of organic shale using a modified Backus averaging method. Lucier et al. (2011) modelled the elastic moduli of organic shale through the Hashin-Shtrikman lower bounds and investigated the effects of kerogen and gas saturation based on it. Zhu et al. (2012) treated the kerogen as the inclusions and then the Gassmann equations for solid substitution are used to study the relations between kerogen content and the acoustic velocities of organic shale. Guo et al. (2013) studied the brittleness of the Barnett Shale by using the Backus averaging and the isotropic SCA. Li et al. (2015) developed a rock physics model to investigate the influence of kerogen content and kerogen porosity on the elastic properties of organic shale and the corresponding AVO responses of shale layers based on the Kuster and Toksoz (KT) theory and the SCA method. Zhao et al. (2016) proposed the rock physics scheme to predict the variation of the elastic behaviours of organic shale with the kerogen maturity degree. The prediction results are in good agreement with the ultrasonic and log measurement data. Chen et al. (2016) considered the effects of the multi-inclusion and the interfacial transition zone on the elastic moduli of organic shale and proposed the corresponding multiscale model to quantify these effects. Due to the complex structures and distributions (both in positions and orientations) of pores and kerogen in the organic shale, the distributions of stress in kerogen (kerogen stress) and pore pressure are usually complicated (e.g., Vermylen and Zoback, 2011; Roshan and Aghighi, 2012). However, to date, little work has been done to investigate their influence on the elastic properties of organic shale. The different distributions of kerogen stress and pore pressure may have great effects on the elastic moduli of organic shale, and therefore, the seismic and sonic logging responses of organic shale reservoirs. Hence, it is essential to develop the physics models to study these effects. In this paper, we consider four cases of stress and fluid pressure distributions in the kerogen and the pores of organic shale (Fig. 1): a)

2. Theory To predict the elastic properties of the organic shale for the four cases of stress and fluid pressure distributions shown in Fig. 1, K-T model (Kuster and Toksöz, 1974) and Gassmann equations (Gassmann, 1951; Ciz and Shapiro, 2007) will be used. In the KT model, it assumes that the inclusions are isolated from each other. This means that the stress or fluid pressure can't be equilibrated throughout the inclusion space. Hence, this model can predict the elastic properties of the rocks with the inhomogeneous distribution of stress or fluid pressure in the inclusion space. Conversely, the Gassmann equations for fluid/solid substitution (Gassmann, 1951; Ciz and Shapiro, 2007) assume that the inclusions are interconnected with each other, and therefore, the stress or the fluid pressure in the inclusions can be equilibrated throughout the inclusion space. Hence, the Gassmann equations can be used to estimate the elastic properties of rocks with connected inclusion space, in which the stress or the fluid pressure is homogeneous. In the following, we will combine these two theories to predict the elastic properties of the organic shale for the four cases of stress and fluid Fig. 1. Four different cases for the distributions of kerogen stress and pore pressure. (a) homogeneous kerogen stress and pore pressure. (b) inhomogeneous kerogen stress and homogeneous pore pressure. (c) homogeneous kerogen stress and inhomogeneous pore pressure. (d) inhomogeneous kerogen stress and pore pressure. Note that the ellipse with dashed and solid boundaries represent the kerogen and the pores, respectively. The homogeneous and inhomogeneous stress and fluid pressure distributions are denoted by blue and red colours, respectively.

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

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

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

−1

−1 2 (K d−11 − K gr ) ⎞ ⎛ Ks = ⎜K d−11 − −1 −1 −1 ⎟ ϕkg (Kkg − K gr ) + (K d−11 − K gr ) ⎠ ⎝

(1)

−1

−1 2 (Gd−11 − Ggr ) ⎞ ⎛ Gs = ⎜Gd−11 − −1 −1 −1 ⎟ ϕkg (Gkg − Ggr ) + (Gd−11 − Ggr ) ⎠ ⎝

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

(2)

where K d1 and Gd1 are the bulk and shear moduli of the effective solid phase before filling the kerogen into the inclusion space, respectively; Kgr and Ggr are the effective bulk and shear moduli of the minerals composing the organic shale, respectively; Kkg and Gkg are the bulk and shear moduli of the kerogen, respectively; ϕkg is the relative volume of the kerogen in the solid phase, which can be calculated as follows:

If the inclusion space is interconnected and the frequency of the acoustic wave is low enough, the kerogen stress and the pore pressure will have enough time to equilibrate. Hence, the kerogen stress and the pore pressure will be homogeneous. Under this condition, Gassmann equations can be used to predict the elastic properties of the organic shale. First, the Gassmann equation for solid substitution is applied to 893

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

′ ϕkg

ϕkg =

1 − ϕf

Q1 =

(3)

′ is the relative volume of the kerogen with reIn equation (3), ϕkg spect to the whole organic shale; ϕf is the porosity of the organic shale filled with fluid. Since K d1 and Gd1 are the bulk and shear moduli for the solid phase of organic shale before filling the inclusion space with the kerogen, these dry elastic properties of the solid phase can be obtained by using the KT model (Kuster and Toksöz, 1974; Mavko et al., 2009) as follows:

K d1 =

Gd1 =

βgr = Ggr

(4)

(5)

where:

ζgr =

P1 =

Ggr (9K gr + 8Ggr ) 6(K gr + 2Ggr )

3K gr + Ggr 3K gr + 4Ggr

(6)

Ksat = K d2 +

(1 − K d2/ Ks )2 ϕf Kf

K gr παkg βgr

(9)

After obtaining the elastic moduli of the effective solid phase, the bulk and shear moduli of the dry organic shale, K d2 and Gd2 , can be obtained in a similar way as K d1 and Gd1 using the KT model through equations (4) and (5). However, it should be noted that Kgr and Ggr need to be replaced by Ks and Gs, respectively. Furthermore, the fraction and aspect ratio of the kerogen, ϕkg and αkg , also need to be replaced by the corresponding values of the pores ϕf and αf , respectively. Finally, due to the homogeneous fluid pressure distribution in the pores, the bulk and shear moduli, Ksat and Gsat, of the organic shale can be obtained by saturating the dry pores with the fluid using the Gassmann equation for fluid substitution (Gassmann, 1951):

Ggr (Ggr + ζgr ) − ϕkg Ggr Q1 ζgr Ggr + ζgr + ϕkg Ggr Q1

(8)

with αkg being the aspect ratio of the kerogen and βgr having the following form:

K gr (K gr + 4/3Ggr ) − 4/3ϕkg K gr P1 Ggr K gr + 4/3Ggr + ϕkg K gr P1

8Ggr 4/3Ggr ⎞ 1⎛ 1+ + 5⎜ παkg (Ggr + 2βgr ) παkg βgr ⎟ ⎠ ⎝

(7)

Gsat = Gd2 894

+

1 − ϕf Ks



K d2 Ks2

(10) (11)

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

where Kf is the fluid bulk modulus. The whole procedure of obtaining the elastic properties of the organic shale for Case a is shown in Fig. 2.

P2 =

Q2 = 2.2. Case b: inhomogeneous kerogen stress and homogeneous pore pressure (ISHP)

Gs =

(15)

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

(12)

Ggr (Ggr + ζgr ) + ϕkg (Gkg − Ggr ) Q2 ζgr Ggr + ζgr − ϕkg (Gkg − Ggr ) Q2

8Ggr Kkg + 2/3(Gkg + Ggr ) ⎞ 1⎛ 1+ +2 5⎜ 4Gkg + πα (Ggr + 2βgr ) Kkg + 4/3Gkg + παβgr ⎟ ⎠ ⎝

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

K gr (K gr + 4/3Ggr ) + 4/3ϕkg (Kkg − K gr ) P2 Ggr K gr + 4/3Ggr − ϕkg (Kkg − K gr ) P2

(14)

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

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

Ks =

K gr + 4/3Gkg Kkg + 4/3Gkg + παβgr

(13)

where: 895

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Then, the kerogen is filled into the inclusion space and its effects on the elastic properties of the solid phase can be estimated by using the Gassmann 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 inhomogeneous fluid pressure can be obtained using the KT model as follows:

3.2. Results

Ks (Ks + 4/3Gs ) + 4/3ϕf (Kf − Ks ) P3 Gs

Ksat =

Gsat =

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

Ks + 4/3Gs − ϕf (Kf − Ks ) P3

3.2.1. Effects of stress distributions in kerogen on the elastic properties of organic shale Fig. 6 shows the elastic properties of the organic shale for Case a (HSHP) and Case b (ISHP) under different porosities and kerogen contents. It can be found that the bulk and shear moduli, as well as Young's moduli, all decrease with the porosity and kerogen content. This is due to the fact that both the elastic moduli of the kerogen and the fluid are much smaller than those of the minerals. Hence, the increase of the kerogen content or the porosity will result in the decrease of the elastic moduli of the organic shale. The influence of the distributions of the kerogen stress on the elastic properties of organic shale can be analysed by comparing the elastic properties of the organic shale for Case a and b. It can be seen that the organic shale with homogeneous distribution of kerogen stress has obviously lower elastic moduli than the inhomogeneous case, especially for the bulk moduli. The reason is that, in the case with homogeneous stress distribution, the stress increase is averaged over the whole volume of the kerogen inclusion space, whereas it cannot be released in the inhomogeneous case. Hence, the organic shale is under the relaxed state for the homogeneous stress distribution case, but unrelaxed state for the inhomogeneous case. The elastic moduli of the organic shale are thus lower under the homogeneous stress distribution than under the inhomogeneous stress distribution. The effects of stress distributions in kerogen can be quantified by the relative change of the elastic moduli between Case a and b, as shown in Fig. 6d. The relative change is calculated by first obtaining the difference between the elastic moduli for Case a and b, then dividing it by the elastic moduli for Case a. It can be found that both the relative changes of the shear and Young's moduli increase slightly with the porosity, whereas that for the bulk moduli shows a reverse trend with the porosity. Furthermore, it can be observed that the relative changes of the shear and Young's moduli are close to each other, while that for the bulk moduli is much higher. This indicates that the influence of the stress distributions in the kerogen on the bulk modulus is much higher than that on the shear and Young's moduli. The shear and Young's moduli are affected by the stress distributions in a similar manner, whereas the bulk moduli are influenced differently. When the kerogen content increase, we can observe that the relative changes of the elastic moduli increase almost linearly with the kerogen content. This implies that the influence of the stress distributions depends linearly on the kerogen content. It can also be seen that the effects of the stress distributions are more obviously influenced by the kerogen content than the porosity.

(16)

Gs (Gs + ζs ) − ϕf Gs Q3 ζs Gs + ζs + ϕf Gs Q3

(17)

where:

ζs =

Gs (9Ks + 8Gs ) 6(Ks + 2Gs )

(18)

P3 =

Ks Kf + παβs

(19)

Q3 =

Kf + 2/3(Gf + Gs ) ⎞ 1⎛ 8Gs +2 ⎜1 + ⎟ 5⎝ παf (Gs + 2βs ) Kf + παf βs ⎠

(20)

with βs having the following form:

βs = Gs

3Ks + Gs 3Ks + 4Gs

(21)

The whole procedure of obtaining the elastic properties of organic shale 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 interconnected or the frequency of the acoustic wave is high enough, the kerogen stress and the pore pressure are both inhomogeneous. In this case, the effects of kerogen and fluid on the elastic properties of organic shale can both be estimated by using the KT model. First, the elastic properties of the effective solid phase can be calculated using equations (12) and (13), which is same as Case b. Then, the elastic moduli of the saturated organic shale can be obtained using the same equations as Case c, as shown in equations (16) and (17). The whole procedure of obtaining the elastic properties of the organic shale for Case d is shown in Fig. 5. 3. Numerical example 3.1. Parameters To show the effects of the distributions of the kerogen stress and pore pressure on the elastic properties of organic shale, we study the saturated shale samples containing mature kerogen. Following Zhao et al. (2016), the mineral composition of the samples is as follows: 20% clay, 20% carbonate, 30% quartz, and 30% feldspar. The elastic moduli of these minerals are shown in Table 1. The effective elastic moduli of the mineral mixture can be obtained by using the Voigt-Reuss-Hill averaging approach (Mavko et al., 2009). The bulk and shear moduli of mature kerogen are 5 GPa and 3.5 GPa, respectively (Yan and Han, 2013; Zhao et al., 2016). The other parameters of the organic shale samples are assumed as follows: the bulk modulus of the saturating fluid (brine) is 2.18 GPa; the aspect ratios of the kerogen inclusion and pores 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 the fractions of the kerogen in the samples (kc) are 0.05 and 0.1, respectively.

3.2.2. Effects of fluid pressure distributions in pores on the elastic properties of organic shale The effects of fluid pressure distributions in pores on the elastic properties of organic shale can be studied by comparing the results for Case a (HSHP) and Case c (HSIP), as shown in Fig. 7. We can also observe the decrease of the elastic moduli with the kerogen content and the porosities due to the lower moduli of the kerogen and the fluid compared to the minerals. We can see that the effects of the fluid pressure distributions on the elastic properties of organic shale are much smaller than those of stress distributions. Compared to the shear and Young's moduli, the effects of the fluid pressure distributions on the bulk moduli are slightly larger. This implies that the fluid pressure distributions have different influence on the bulk moduli and the shear moduli (or Young's moduli). 896

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

pressure distribution on the elastic properties of organic shale, we also show the relative change of the elastic moduli between Case a and Case d in Fig. 8d. It can be seen that the joint effects of the stress and fluid pressure distributions on the bulk moduli are larger than those on the shear and Young's moduli. This is similar to the individual effects of the stress or fluid pressure distributions. The previous analysis shows that the 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 are found to increase slightly with the porosity. In terms of the effects of the kerogen, we can obviously find that the joint effects of the stress and fluid pressure increase with the kerogen content, which are similar to the individual effects of the stress and fluid pressure. In summary, we can see that the effects of stress distributions in kerogen and pore pressure have obvious influence on the elastic properties of organic shale. In particular, the effects of the stress distributions are much larger than those of fluid pressure distributions. Hence, it is important to consider the effects of stress and fluid pressure distributions in kerogen and pores when calculating the elastic moduli of the organic shale.

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

Bulk modulus (GPa)

Shear modulus (GPa)

Clay Carbonate Quartz Feldspar

25 76.8 37 37.5

9 32 44 15

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

4. Discussion 4.1. Comparison with other model The elastic properties of rocks with isolated pore-filling materials are calculated using KT model, as shown above. An alternative approach is using the SK model (Sayers and Kachanov, 1991), which can be written as follows:

3.2.3. Joint effects of stress and fluid pressure distributions in kerogen and pores on the elastic moduli of organic shale To show the joint effects of the distributions of the kerogen stress and pore pressure on the elastic properties of organic shale, the results for Case a (HSHP) and Case d (ISHP) are given in Fig. 8. Similarly, we can observe the decrease of the elastic moduli with the porosity and the kerogen content due to the smaller moduli of fluid and kerogen than the minerals in the organic shale. The bulk and shear moduli under the unrelaxed state (Case d) are obviously larger than those under the relaxed state (Case a). To quantify the joint effects of the stress and fluid

K eff =

Geff =

1 1 Ks

+ ZN

1 Gs

+

(22)

1 4 Z 15 N

2

+ 5 ZT

(23)

where Keff and Geff are effective rock bulk and shear moduli, 897

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

different aspect ratios for the kerogen inclusion and pores (0.05 and 0.1 respectively), the comparison between these two approaches is shown in Fig. 9. We can see that the results given by these two approaches are overall similar. The discrepancies between them primarily occur for relatively large porosity cases. This is due to the fact that, KT model represents the effects of pores as reduction of rock elastic moduli whereas SK model considers that as increase of elastic compliances. This will results in the slightly larger elastic moduli predicted by SK model, especially in the relatively large porosities (e.g., Schoenberg and Douma, 1988). However, it can be observed that these two approaches give similar trends for the elastic moduli for the cases with different pore pressure and kerogen stress distributions. The differences between different cases are also similar. This indicates that the approach proposed in this paper can model the effects of pore pressure and kerogen stress distributions on shale elastic properties well.

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

ZN =

4 e 3Ls γs [1 − γs + (Kfi + 4/3Gfi )/(παGs )]

(24)

ZT =

16 e 3Gs [3 − 2γs + 4Gfi/(παGs )]

(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:

γs =

Gs Ls

(26)

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

4.2. Extension to the anisotropic case with relatively large porosity and kerogen content In this paper, we use the KT model to calculate the elastic properties of the organic shale under the inhomogeneous kerogen stress or pore 898

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

1995). Hence, by replacing the mineral frame with the effective background medium and at the same time treating the mineral frame as one component of the inclusions in the KT model, the equations for the elastic 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 effective background 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 approach doesn't consider the effects of transversely isotropy on the elastic properties of organic shale. However, this can be easily extended by using the anisotropic DEM or SCA for the case with inhomogeneous stress distribution in the kerogen or pore pressure (Hornby et al., 1994; Zhao et al., 2016), and anisotropic Gassmann equations for the case with homogeneous distribution (Brown and Korringa, 1975; Ciz and Shapiro, 2007).

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

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 only contribute a small portion to the total pore volume in the organic shale, 899

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

4.4. Inferring microstructures of organic shale from its elastic properties

they are important for the shale oil/gas accumulation and hence the production of shale oil/gas. Therefore, it is essential to explore the effects of the pores in kerogen on the elastic properties of organic shale. These effects are ignored in the above analysis, which will only induce small errors when the pore volume in the kerogen is small compared to both the volume of the kerogen and the total porosity. However, when the fraction of the pores in the kerogen is relatively large, its effects may be important and cannot be ignored. Under this condition, we can divide the total pores into two sub-pores, one in the kerogen and the other in the mineral frame of the organic shale. For the sub-pores in the kerogen, its primary influence is on the elastic properties of the kerogen, which then affect those of the organic shale. Hence, we need to first calculate the effective elastic properties of the kerogen. As the fluid in these pores are often not distributed uniformly, the fluid pressure is usually inhomogeneous in these pores (Zhao et al., 2016). Hence, we can apply the KT model to obtain the effective moduli of the kerogen 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, the DEM or SCA approach can be applied. After obtaining the effective elastic moduli of the kerogen, the elastic properties of the organic shale can be calculated using the approaches presented before with the other sub-pores in the mineral frame.

According to the different stress distributions in the kerogen and pore pressure, we proposed the corresponding model for the elastic properties of organic shale in this paper. Four cases with homogeneous and inhomogeneous stress and fluid pressure distributions are considered, as shown in Fig. 1. As explained before, the stress and fluid pressure distributions can be affected by the connectivity of the kerogen inclusion space and the pore space, respectively. When the kerogen inclusion space or the pore space is interconnected, the kerogen stress or pore pressure will be homogeneous if the frequency of the acoustic wave is low enough. On the contrary, if the kerogen inclusion space or the pore space is not interconnected, the stress and fluid pressure will be inhomogeneous. Hence, it is possible to infer the microstructures of the organic shale from its elastic properties. In order to do so, we can first calculate the elastic properties of the organic shale using the four models presented above respectively. Then, we can match the calculated results with the elastic properties obtained from the measured acoustic wave velocities and densities. The model that predicts the measured data best indicates the connectivity of the kerogen inclusion space and the pore space. Hence, the microstructures of the organic shale can be inferred. It should be noted here that, the measurement 900

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Fig. 9. Comparison between the schemes that use KT and SK model (Sayers and Kachanov, 1991) respectively for the cases with different pore pressure and kerogen stress distributions. The solid and dashed lines represent the results that use KT and SK model respectively. The kerogen content equals to 0.05 for all cases.

frequency of the acoustic wave should be low enough to make sure that the inhomogeneous stress or fluid pressure distributions are not caused by the insufficient time for the stress and fluid pressure to be equilibrated. 5. Conclusions In this work, the influence of the distributions of the kerogen stress and the pore pressure on the elastic properties of organic shale was studied. For this purpose, the KT model was applied for the organic shale with inhomogeneous kerogen stress or pore pressure, whereas the Gassmann equations for solid/fluid substitution was used when the kerogen stress or the pore pressure is homogeneous. Four cases with different combinations of stress and fluid pressure distributions were then investigated and the models for obtaining the corresponding elastic properties of the organic shale were given. Based on these models, a numerical example for the organic shale with different distributions of stress and fluid pressure was studied. The results showed that, the distributions of stress and fluid pressure have obvious influence on the elastic properties of organic shale. The effective moduli under the inhomogeneous stress or fluid pressure distributions were found to be larger than under the corresponding homogeneous distributions. These effects vary with the porosity and the kerogen content. Furthermore, the joint effects of kerogen stress and pore pressure distributions are similar to those of kerogen stress due to the much smaller effects of pore pressure. Hence, it is essential to consider the effects of stress and fluid pressure distributions when building the rock physics model for the elastic properties of organic shale. The organic shale considered in this paper is isotropic with low porosity and kerogen content. When the shale has obvious anisotropic properties (transversely isotropy) and relatively large porosity and kerogen content, the proposed models can be extended using the anisotropic DEM or SCA scheme. Furthermore, the effects of the pores in the kerogen can also be included in our models by dividing the total pores into two parts, one in the kerogen and the other in the mineral frame of organic shale. One possible application of the proposed models is inferring the microstructures of the organic shale from the measured acoustic wave velocities. This work revealed the importance of stress and fluid pressure distributions on the elastic properties of organic shale and hence is helpful for the shale oil/gas exploration and production. Acknowledgements We would like to thank the financial support from National Natural Science Foundation of China, Contract No. U1562108, and National Key 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. References Berryman, J.G., 1980. Long-wavelength propagation in composite elastic media. J. Acoust. Soc. Am. 68, 1809–1831. Berryman, J.G., 1995. Mixture theories for rock properties. In: Ahrens, T.J. (Ed.), Rock Physics and Phase Relations: a Handbook of Physical Constants. American Geophysical Union, Washington, DC, pp. 205–228.

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