Saturation effects on the joint elastic-dielectric ...

Journal of Applied Geophysics 129 (2016) 36–40

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Saturation effects on the joint elastic–dielectric properties of carbonates Tongcheng Han ⁎, Michael Ben Clennell, Marina Pervukhina, Matthew Josh CSIRO Energy Flagship, 26 Dick Perry Avenue, Kensington, WA 6151, Australia

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Article history: Received 24 March 2015 Received in revised form 23 March 2016 Accepted 29 March 2016 Available online 6 April 2016 Keywords: Cross-property relations Elastic wave velocity Dielectric permittivity Carbonate

a b s t r a c t We used a common microstructural model to investigate the cross-property relations between elastic wave velocities and dielectric permittivity in carbonate rocks. A unified model based on validated self-consistent effective medium theory was used to quantify the effects of porosity and water saturation on both elastic properties (compressional and shear wave velocities) and electromagnetic properties (dielectric permittivity). The results of the forward models are presented as a series of cross-plots covering a wide range of porosities and water saturations and for microstructures that correspond to different predominant aspect ratios. It was found that dielectric permittivity correlated approximately linearly with elastic wave velocity at each saturation stage, with slopes varying gradually from positive at low saturation conditions to negative at higher saturations. The differing sensitivities of the elastic and dielectric rock properties to changes in porosity, pore morphology and water saturation can be used to reduce uncertainty in subsurface fluid saturation estimation when co-located sonic and dielectric surveys are available. The joint approach is useful for cross-validation of rock physics models for analysing pore structure and saturation effects on elastic and dielectric responses. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Quantifying fluid content within rocks is an important goal in reservoir characterization. While dielectric measurement is useful in determining water saturation without knowing its salinity (Chew, 1988; Schmitt et al., 2011), it can be difficult to distinguish between low dielectric constant hydrocarbons (e.g., oil and gas, with dielectric permittivity of 2 and 1, respectively). Similarly, the elastic velocity of reservoir rocks is also affected by fluids in the pores, and elastic methods based on the fluid effect on the elastic moduli and density of a rock are conventionally employed to determine hydrocarbon saturation (Domenico, 1976; Khazanehdari and Sothcott, 2003; Renaud et al., 2009; Lan et al., 2011). Since elastic and dielectric methods measure complementary but independent petrophysical properties of reservoir formations that are related through porosity and fluid properties within the pores, the joint interpretation of co-located elastic and dielectric data could offer a better way to quantify fluid saturation provided the saturation effects on the joint elastic–dielectric properties are well understood. Whereas there are numerous experimental and theoretical investigations of the saturation effects on elastic and dielectric properties of carbonates (e.g., Gregory, 1976; Knight et al., 1998; Baechle et al., 2005; Seleznev et al., 2004, 2006; Vanorio et al., 2008; Markov et al., 2012), few studies of the joint elastic–dielectric properties exist in the open literature. ⁎ Corresponding author. E-mail address: [email protected] (T. Han).

http://dx.doi.org/10.1016/j.jappgeo.2016.03.029 0926-9851/© 2016 Elsevier B.V. All rights reserved.

Carrara et al. (1994) proposed an electro-seismic model to evaluate porosity and degree of fluid saturation in carbonates, and the model was tested by Carrara et al. (1999) by measuring compressional wave velocity and electrical resistivity on carbonates with varying porosity and brine saturation. Because the problem of low frequency electrical conductivity (or reciprocally electrical resistivity) is mathematically equivalent to that of high frequency dielectric permittivity (Carcione et al., 2007), the electro-seismic model given by Carrara et al. (1994) can be adapted to a joint elastic–dielectric model. However, this electroseismic model is essentially a harmonic average equation that is known to perform poorly at high porosities (e.g., Berryman, 1995). Kazatchenko et al. (2004) proposed a method for joint modelling of acoustic velocities and electrical conductivity from unified microstructure (defined as the geometry of the rock-forming materials and how they are arranged in the rock) of rocks based on effective medium approximation of two component media, composed of grains, which constitute a solid frame and pores saturated by a fluid. They used this method to calculate the acoustic velocities and electrical conductivity for carbonate formations with a primary pore system. Although dielectric permittivity can be modelled in place of electrical conductivity, Kazatchenko et al. (2004) did not give explicit joint elastic–electrical (dielectric) correlations. Furthermore, because the model was developed for a 2-phase medium, the effect of fluid saturation on joint elastic–electrical (dielectric) properties could not be modelled and investigated. Pervukhina and Kuwahara (2008) modelled both elastic and electrical properties of equilibrium interfacial energy controlled microstructures

T. Han et al. / Journal of Applied Geophysics 129 (2016) 36–40

that might be typical for rock in the high temperature environment of the lower crust and upper mantle. The rock was assumed to be fully saturated with melt or brine. They applied the model to the data of collocated magnetotelluric and seismic tomography experiments. Based on a comprehensive study of the pressure and petrophysical control on the joint elastic–electrical properties of reservoir sandstones (Han et al., 2011a, 2011b), Han et al. (2011c) developed a joint elastic– electrical effective medium model for sandstones with pore-filling clay minerals, which can be mathematically adjusted to model the saturation effects on the joint elastic–dielectric properties of carbonates. Carbonate rocks without clay are expected to have simpler electrical responses than shaly sandstones. However, assumptions in the model for sandstones might be different from those for carbonates (e.g., critical porosity of 0.5 used for sandstones in the combined self-consistent approximation and differential effective medium model may not be valid for the carbonate case), and therefore the proposed model needs to be tested before it is applied to carbonates. This paper studies theoretically the saturation effects on the joint elastic–dielectric properties of carbonates for the understanding of elastic and electromagnetic wave propagation phenomena as well as quantifying hydrocarbon content in partially saturated carbonates. Based on validated self-consistent effective medium models for elastic velocity and dielectric permittivity, we show for the first time the crossproperty relations between elastic velocity and dielectric permittivity (the joint elastic–dielectric properties) of carbonates with a unified microstructure (the microstructure described by the elastic and dielectric models are consistent) and the effects of porosity and water saturation on the joint elastic–dielectric properties. The results show the potential for estimating in situ carbonate porosity and hydrocarbon saturation using joint velocity–permittivity crossplots from co-located sonic and dielectric surveys.

2. Methodology Effective medium models suitable for simultaneous simulation of the effect of saturation on both elastic velocity and dielectric permittivity include averages, complex refraction-index method (CRIM), selfconsistent (SC) models and differential effective medium (DEM) models among others (Carcione et al., 2007). The averages and the CRIM do not specify the geometric details of how the inclusions are arranged relative to each other and therefore can only predict the upper and lower bounds of the effective properties (Mavko et al., 2009). DEM models (Berryman, 1995) can be extended to give good estimations of a 3-phase elastic and electrical medium (e.g., Han et al., 2011c); however they require to designate one phase as the connected background medium into which the other phases are embedded and different results will be given depending on which constituent is chosen as the background. It has been shown (e.g., Han et al., 2011c) that the host background in the DEM model should be solid grains to give a good estimate of measured elastic velocity but should be the conductive phase to predict electrical resistivity. The choice of different connected host background in the elastic and electrical (dielectric) simulation implies the microstructure in each case is different and therefore is inconsistent. In the SC models (Berryman, 1995), on the other hand, all the constituents are equivalent and connected without one single component playing the role of host matrix for the others distributed as isolated inclusions. Therefore, SC models with the same microstructure are suitable for modelling of both elastic and dielectric effective properties of composite materials.

2.1. Self-consistent models The SC models for elastic velocity and real relative dielectric permittivity (referred to as SC elastic model and SC dielectric model,

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respectively) of a 3-phase medium are given respectively as (Berryman, 1995) 3 X f i K i −K SC P i ¼ 0 i¼1

3 X f i μ i −μ SC Q i ¼ 0;

ð1Þ

i¼1

and 3 X f i εi −ε SC Ri ¼ 0;

ð2Þ

i¼1

where fi, Ki, μ i and εi are the volume fraction, bulk and shear modulus and dielectric permittivity of each constituent, respectively; KSC⁎, μSC⁎ and εSC⁎ are the effective self-consistent bulk and shear modulus and dielectric permittivity of the composite material; and P⁎i, Q⁎i and R⁎i are the coefficients that take into account the geometric factors of the i-th component in the elastic and dielectric self-consistent effective medium, respectively. The volume fractions of solid grains (fg), brine (fb) and hydrocarbon (fh) are given by fg = 1 − ϕ, fb = Swϕ and fh = (1 − Sw)ϕ, respectively, where ϕ is the porosity and Sw is the water saturation in the porosity. 2.2. Test on laboratory examples To test the applicability of the SC models to simulating the effects of saturation on the elastic and dielectric properties of carbonates, we compare the modelled bulk modulus and dielectric permittivity with published laboratory data on carbonates, as shown in Figs 1 and 2, respectively. Whereas the SC dielectric model predicts the variation of dielectric permittivity as a function of water saturation in carbonate well, the SC elastic model reproduces the bulk modulus of carbonate for water saturation below 0.9, above which the dramatic increase in bulk modulus fails to be predicted. In fact the sharp increase in the bulk modulus approaching full water saturation is caused by the break-up of the interconnected gas phase and the filling of the central volumes of the pores with water (Knight and Nolen-Hoeksema, 1990) indicating a significant change in the geometry of the pore-filling fluids occurs at this saturation stage, which therefore could not be modelled by the same geometric

Fig. 1. Comparison of the SC elastic model with laboratory measurement of the ultrasonic bulk modulus as a function of water saturation in a carbonate. Physical properties are used for calcite, water and air as given in Table 1 with fitting parameters of aspect ratios 1, 0.55 and 0.5 for calcite, water and air, respectively. Laboratory data from Adam and Batzle (2008).

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Fig. 2. Comparison of the SC dielectric model with laboratory measurement of the dielectric permittivity as a function of water saturation in an oil and water saturated carbonate. Physical properties are used for calcite, water and oil as given in Table 1 with fitting parameters of aspect ratios 1, 0.1 and 0.1 for calcite, water and oil, respectively. Laboratory data from Seleznev et al. (2004).

factor as used in the less saturated conditions for the water and air components in the SC elastic model. Having tested the validity of the SC elastic and dielectric models, we will confine our further study to the low water saturation range (specifically Sw ≤ 0.9), which is a good reflection of gas reservoirs. The SC modelled elastic velocity and dielectric permittivity are cross-related by using porosity and saturation as the links to arrive at the joint elastic–dielectric properties of carbonates (Carcione et al., 2007). 3. Joint elastic–dielectric properties The SC computed cross-property relationship between compressional wave velocity (Vp) and dielectric permittivity (εr) and its correlation with porosity (ϕ, 0 to 0.4) and water saturation (Sw, 0 to 0.9) is shown in Fig. 3(a) for water–gas saturated carbonates using physical properties of the constituents listed in Table 1. Both porosity and water saturation have a strong effect on the joint Vp–εr relation. Dielectric permittivity shows an approximately linear correlation with compressional wave velocity at each saturation stage (colour contours in Fig. 3a), with slopes varying gradually from positive at low saturation conditions to negative at higher saturations. While Pwave velocity increases with decreasing porosity at any given water saturation, the variation in the slope is an indication of the dielectric permittivity of the water–gas ‘mixture’ in the pores. When water saturation is low, there is only small amount of high permittivity water in the pores resulting in a lower dielectric permittivity of the water–gas mixture than that of the calcite grains, and as a result both dielectric permittivity and P-wave velocity increase when the amount of calcite with relative higher permittivity and higher velocity increases (which is equivalent to a decrease in porosity, and is better illustrated by the comparison of the low saturation dielectric permittivity at low and high porosity as shown by the solid and dashed black curves in Fig. 4, respectively). With an increase in water saturation, the dielectric permittivity of the water–gas mixture increases and gets closer to those of the calcite grains, and therefore the effective permittivity of the carbonate increases less with decreasing porosity than at lower water saturation. At water saturation of about 0.3 (the cross point of the black permittivity curves in Fig. 4), the permittivity of the water–gas mixture reaches that of the calcite grains, and the permittivity of the carbonate will not change with varying porosity. With further increase in water saturation, the permittivity of the pore-filling water–gas mixture exceeds that of the grains leading to a reducing permittivity with increasing velocity (decreasing porosity, see the comparison of the high

Fig. 3. Computed cross-property relationships between (a) compressional wave velocity (Vp), (b) shear wave velocity (Vs) and dielectric permittivity (ε) and their correlation with porosity (ϕ) and water saturation (Sw) for water–gas saturated carbonates. Aspect ratios are 1, 0.5 and 0.5 for calcite, water and gas, respectively.

saturation dielectric permittivity at low and high porosity as shown by the solid and dashed black curves in Fig. 4, respectively). The negative linear correlation between dielectric permittivity and P-wave velocity at higher water saturation resembles the laboratory obtained positive linear relation between logarithmic electrical resistivity and P-wave velocity in water saturated sandstones by Han et al. (2011b). The effect of varying water saturation on the joint Vp–εr properties at a constant rock porosity is illustrated by the grey curves in Fig. 3(a). Dielectric permittivity increases with the increase of water saturation at all porosities as water has the highest permittivity in the constituents (see also the black curves in Fig. 4). By contrast the variation of PTable 1 Physical properties of the constituents used in the modelling. Medium

Bulk modulus K (GPa)

Shear modulus μ (GPa)

Density d (g/cm3)

Dielectric permittivity ε

Calcite Water Oil Air Gas

71 2.29 – 1.42 × 10−4 0.133

30 0 – 0 0

2.71 1 – 1.225 × 10−3 0.336

8 80 2 1 1


Fig. 4. SC modelled variations of P-wave velocity (Vp) and dielectric permittivity (ε) as a function of water saturation (Sw) for a water–gas saturated carbonate with porosity (ϕ) of 0.1 and 0.3, respectively. Physical properties and aspect ratios of the constituents are the same as in Fig. 3.

wave velocity with water saturation is more complicated, which shows a slight decrease with increasing water saturation at lower porosities and an increasing trend with water saturation at higher porosities. The distinct behaviours of velocity with water saturation at different porosity can be explained as follows: when porosity is low, the elastic modulus of the grain matrix dominates and the addition of water (into the gas saturated rock) has negligible effect on the bulk modulus but slightly increases the density leading to a small reduction in the velocity (as can be seen by the solid grey curve in Fig. 4); as the porosity gets higher, the elastic modulus of the calcite skeleton becomes weaker and the introduction of water into the (gas filling) pores notably increases the bulk modulus of the carbonate (as shown in Fig. 1), which overtakes the density increase effect and thus resulting in the increasing velocity (shown by the dashed grey curve in Fig. 4). The modelled variations of P-wave velocity with saturation agree with the experimental observations of Knight and Nolen-Hoeksema (1990) and Gregory (1976) on low and high porosity rocks, respectively. Fig. 3(b) shows the SC simulated joint S-wave velocity-dielectric permittivity properties of carbonates and their relationship with porosity and water saturation. In a similar way to the Vp–εr relationships at each saturation stage, dielectric permittivity also exhibits approximately linear trends with increasing S-wave velocity (colour contours in Fig. 3b). The gradients of dielectric permittivity with shear velocity change from positive at low saturations (below ~0.3) to negative relations at higher saturation stages, indicating that porosity has a similar effect in affecting the propagation of both P- and S-waves in carbonates at a specific water saturation. While Vs–εr and Vp–εr relations are similar in terms of their correlation with porosity at each water saturation, they are differently impacted by the change of water saturation for a constant rock porosity (see grey constant-porosity curves in Fig. 3). Dielectric permittivity increases dramatically and S-wave velocity reduces slightly with increasing water saturation when porosity is fixed, leading to an approximately parallel pattern of the joint Vs–εr relation at varying porosity levels as shown by the grey curves in Fig. 3(b). The decreasing behaviour of S-wave velocity with increasing water saturation is well established in the literature (e.g., Mavko et al., 2009), and is caused by the increase in bulk density as a result of the introduction of zero-shear modulus water which does not contribute to the shear modulus of the carbonate.

4. Discussion of results Microstructure of constituents strongly impacts both elastic and dielectric properties of rocks that are complex composite materials

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(e.g., Markov et al., 2012; Adam et al., 2013) and thus the relations between elastic and dielectric properties of a rock. The geometry of the inclusions in both SC modelled elastic velocity and dielectric permittivity is modelled in the form of aspect ratio, which has an explicit physical meaning and is also a reasonable approximation for carbonates (e.g., Kazatchenko et al., 2004). The common use of aspect ratio is important in keeping consistency of the geometric details of the constituents in the elastic and dielectric modelling; especially when Archie's cementation coefficient is widely employed in the dielectric (electrical) community for inclusion geometry and though a correlation exists between cementation coefficient and grain aspect ratio (e.g., Archie, 1942; Berg, 2007; Gelius and Wang, 2008; Kennedy and Herrick, 2012). In addition to being consistent in modelling the geometric information (i.e., aspect ratio) of each component, the SC elastic and dielectric models assume symmetry of all constituents indicating all the phases are inter-connected. This resembles the micro-structure of real carbonates where elastic waves propagate mainly through the connected solid phase while electromagnetic waves via the conductive water phase. By simulating consistent inclusion geometry and geometric details of how the inclusions are arranged relative to each other for elastic velocity and dielectric permittivity, the joint elastic–dielectric relations presented in this paper based on SC models are for carbonates with a unified microstructure. The SC models used in this paper assume that each phase, namely, solid grains, water and gas is present in the mixture in the form of idealized ellipsoidal inclusions of a unified aspect ratio. This is not always the case for real carbonate formations which are characterized by a complex structure of pore space that consists of a primary matrix pore system and secondary pores related to vugs and microfractures. Such pore space of carbonate rocks is generally characterized with a broad spectrum of aspect ratios (Kazatchenko et al., 2004). Therefore the joint relations established in this study are valid for carbonates at high effective stresses (defined as confining stress minus pore pressure), at which the effect of low aspect ratio secondary porosity is expected to be smaller (e.g., Shapiro, 2003; Agersborg et al., 2008; Pervukhina et al., 2010). For carbonates at lower pressures with non-negligible secondary porosity, an ‘effective aspect ratio’ can be used. Han et al. (2011c) suggested calculating this effective aspect ratio by averaging aspect ratios of different pore types with weights, which are equal to their volumetric fractions. However further investigation of applicability of this method to different types of carbonates is needed. While both elastic and dielectric properties are frequency dependent (Biot, 1956a, 1956b; Revil, 2013), we simplified our analysis and modelling by restricting ourselves to situations where the inhomogeneities are much smaller than the wavelength and thus, effective medium approach such as SC approximation is applicable. We also restrict our analysis to the situations where energy losses per cycle are small, such that truly propagating waves exist. This applies to electromagnetic waves at high radio frequencies (e.g. hundreds of MHz to GHz). On the other hand for elastic wave velocity, the inter-connected pore network of the SC model limits its applicability to the case of high frequency laboratory ultrasonic experiments with respect to fluid flow (Mavko et al., 2009). To simulate elastic properties at lower seismic frequencies, the same SC scheme might be applied to calculate the elastic properties of dry skeleton and then Gassmann's fluid substitution (Gassmann, 1951) should be used to calculate properties of the rock saturated with a gas–water mixture. However, two possible scenarios should be further taken into account while applying Gassmann's fluid substitution model to partially saturated rocks: inhomogeneous patchy saturation and homogeneous partial saturation. The former corresponds to imbibitions (or injection stage) and is described with Gassmann–Hill relation and the later relates to drainage (or depletion stage) and Gassmann– Wood relation should be used to describe the situation (Mavko et al., 2009; Müller et al., 2010). Experimental measurements of the elastic and dielectric properties show significant hysteresis during the process of imbibition and

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drainage, which could be related to changes in geometry of the fluid phases in the pore space of the rock (Knight and Nur, 1987; Knight and Nolen-Hoeksema, 1990). Since the SC models are validated with laboratory measurements on elastic and dielectric behaviours of carbonates during imbibition as shown Figs 1 and 2, respectively, it is expected that the SC modelled joint elastic–dielectric properties are applicable to carbonates at the imbibition stage. However, as the gas– water systems in carbonate reservoirs are stable before production, the hysteresis caused by change of saturation can be negligible and the obtained elastic–dielectric relations can be safely applied. While the joint elastic–dielectric correlations given in this paper can be used as a template for quantification of gas content in gas-bearing carbonates, their application to gas shales (another important type of partially saturated gas reservoirs) might be problematic, not only because the variation of elastic velocity of shales as a function of water saturation needs further investigation but also because the electrical double-layer associated with clay minerals in shales has an excess contribution to the permittivity (e.g., Revil, 2013; Revil et al., 2013) which is out of the scope of the SC dielectric model, as well as gases can be adsorbed on the surface of clay minerals (e.g., Ross and Bustin, 2009; Mengal and Wattenbarger, 2011) in addition to that reside in the free pores, adding further difficulty to quantifying their content. The theoretically derived joint elastic–dielectric relations need to be tested with controlled laboratory data before practical applications. To our knowledge, however, there seems to be a complete lack of these types of data for carbonates, and the next stage of our research is to perform such experiments and determine the more reliable joint elastic– dielectric relations. 5. Concluding remarks We have tested the self-consistent models in modelling the elastic velocity and dielectric permittivity of carbonates as a function of water saturation (up to 0.9 and 1, respectively). Based on the validated SC models, the joint elastic–dielectric properties of carbonates and their strong relationships with porosity and water saturation are analysed. In addition to helping understand the elastic and electromagnetic wave propagations in carbonates, the joint elastic–dielectric relations presented in this paper can serve as a template for blind saturation quantification, one of the goals in reservoir characterization, especially when porosity information is unavailable. Acknowledgements We would like to thank Dr. Tobias Müller for his helpful discussion of the results and CSIRO Energy Flagship for financial support of this work. References Adam, L., Batzle, M., 2008. Elastic properties of carbonates from laboratory measurements at seismic and ultrasonic frequencies. Lead. Edge 27, 1026–1032. Adam, L., van Wijk, K., Otheim, T., Batzle, M., 2013. Changes in elastic wave velocity and rock microstructure due to basalt–CO2–water reactions. J. Geophys. Res. 118, 4039–4047. Agersborg, R., Johansen, T.A., Jakobsen, M., Sothcott, J., Best, A.I., 2008. Effects of fluids and dual-pore systems on pressure-dependent velocities and attenuations in carbonates. Geophysics 73, 35–47. Archie, G.E., 1942. The electrical resistivity log as an aid in determining some reservoir characteristics. J. Pet. Technol. 5, 1–8. Baechle, G., Weger, R., Eberli, G., Massaferro, J., Sun, Y., 2005. Changes of shear moduli in carbonate rocks: implications for Gassmann applicability. Lead. Edge 24, 507–510. Berg, C., 2007. An effective medium algorithm for calculating water saturations at any salinity or frequency. Geophysics 72, E59–E67. Berryman, J., 1995. Mixture theories for rock properties. In: Ahrens, T.J. (Ed.), A Handbook of Physical Constants. American Geophysical Union, pp. 205–228. Biot, M.A., 1956a. Theory of propagation of elastic waves in a fluid saturated porous solid. 1: low-frequency range. J. Acoust. Soc. Am. 28, 168–178.

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