Geophysical Journal International

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1 Geophysical Journal International Geophys. J. Int. (2016) 205, GJI Marine geosciences and applied geophysics doi: /gji/ggw019 Linking the pressure dependency of elastic and electrical properties of porous rocks by a dual porosity model Tongcheng Han, 1 Boris Gurevich, 1,2 Marina Pervukhina, 1 Michael Ben Clennell 1 and Junfang Zhang 1 1 CSIRO Energy Flagship, Perth, Australia. tongcheng.han@csiro.au 2 Department of Exploration Geophysics, Curtin University, Perth, Australia Accepted 2016 January 13. Received 2016 January 13; in original form 2015 October 13 SUMMARY Knowledge about the pressure dependency of elastic and electrical properties is important for a variety of geophysical applications. We present a technique to invert for the stiff and compliant porosity from velocity measurements made as a function of differential pressure on saturated sandstones. A dual porosity concept is used for dry rock compressibility and a squirt model is employed for the pressure and frequency dependent elastic properties of the rocks when saturated. The total porosity obtained from inversion shows satisfactory agreement with experimental results. The electrical cementation factor was determined using the inverted porosity in combination with measured electrical conductivity. It was found that cementation factor increased exponentially with increasing differential pressure during isostatic loading. Elastic compressibility, electrical cementation factor and electrical conductivity of the saturated rocks correlate linearly with compliant porosity, and electrical cementation factor and electrical conductivity exhibit linear correlations with elastic compressibility of the saturated rocks under loading. The results show that the dual porosity concept is sufficient to explain the pressure dependency of elastic, electrical and joint elastic-electrical properties of saturated porous sandstones. Key words: Electrical properties; Microstructures; Acoustic properties. 1 INTRODUCTION Understanding the pressure dependency of elastic and electrical properties is of great importance for a range of geophysical applications from overpressure prediction in sedimentary rocks, seismic and electrical monitoring of hydrocarbon production and CO 2 leakage from storage reservoirs to constraining rock properties in the deep Earth s mantle. Recent development in marine controlledsource electromagnetic (CSEM) techniques has enabled measurements of electrical resistivity offshore, providing independent and complementary information to seismic data for improved geophysical inversion of pore fluid type and saturation. The success of the joint quantitative interpretation of colocated seismic and CSEM survey data, however, will depend on the knowledge of the interrelationships between elastic and electrical properties or reservoir rocks (e.g. Hoversten et al. 2006; Harris et al. 2009), among which the pressure dependency of joint elastic-electrical properties is one of the key relationships. The dependence of elastic and electrical properties on pressure (e.g. Eberhart-Phillips et al. 1989; Mahmood et al. 1991; Zimmerman 1991; Jing et al. 1992; Khaksar et al. 1999) is usually explained by closure or dilation of pores and cracks with a broad distribution of pore stiffnesses, identified with different aspect ratios (Cheng & Toksöz 1979; Zimmerman 1991; Meglis et al. 1996; Tod 2002; David & Zimmerman 2012). According to elasticity theory, the more rounded pores within a solid matrix are found to be the least compressible, whereas compliance increases progressively when pores become more elongated and flattened. While an approach accounting for a continuous distribution of pore shapes remains valid, it has been recognized that the pore space of many rocks can be plausibly simplified to a dual distribution (Bernabe 1988, 1991; Shapiro 2003; Kaselow & Shapiro 2004; Shapiro & Kaselow 2005): stiff pores, which form most of the pore space, and compliant (or soft) pores, which mostly account for the pressure dependency of the elastic and electrical properties (e.g. Dvorkin et al. 1995; Chapman et al. 2002). The amount of stiff porosity of a rock can be estimated by fitting a linear trend to the data in the high-pressure range (where compliant porosity is assumed closed) of total porosity experimentally measured as a function of pressure and the compliant porosity is then obtained as the difference between total and the stiff porosity (Mavko & Jizba 1991). Alternatively, the stiff and compliant porosity can be inverted from compressional and shear wave velocity measurements on dry samples using the workflow described by Pervukhina et al. (2010) and de Paula et al. (2012). However, 378 C Crown copyright 2016.

2 Pressure dependent physical properties 379 the variation of porosity with pressure is not routinely measured in the ultrasonic or electrical experiments. Furthermore, simultaneous elastic velocity and electrical conductivity (or reciprocally electrical resistivity) measurements can only be made on saturated rocks and not dry rocks. This prevents the simple application of the dual porosity concept to the interpretation of the pressure dependency of joint elastic-electrical properties of reservoir rocks. In this paper, we present a technique for inversion of stiff and compliant porosity from velocity measured as a function of differential pressure (difference between confining and pore pressure) on saturated rocks, based on existing models. The relationships between the obtained compliant porosity and the elastic (compressibility, inverse of bulk moduli), electrical (electrical conductivity and cementation factor) and the joint elastic-electrical properties of saturated clean reservoir sandstones published by Han et al. (2011a,b) are then analysed. The results offer new insights into the pressure dependency of the joint elastic-electrical properties of reservoir sandstones. 2 INVERSION OF POROSITY FROM VELOCITY OF SATURATED ROCKS 2.1 Dual porosity model and dry rock compressibility Shapiro (2003) explains the pressure dependency of the elastic properties of a dry rock in terms of a dual distribution of porosity, namely, stiff porosity φ s, which decreases linearly with increasing pressure, and compliant (or soft) porosity φ c, which decreases exponentially with increasing pressure. Shapiro (2003) shows that if the total porosity φ of a rock consists of stiff and compliant porosities of so defined, φ = φ s + φ c, (1) then the pressure dependency of the dry rock compressibility C dr (C dr = 1/K dr,wherek dr is the dry rock bulk modulus) can be expressed as C dr (P) = C drs (1 + θ s φ s + θ c φ c ), (2) where, C drs is the stiff limit compressibility of the dry porous rock at high enough confining stress with all compliant porosity closed, θ s and θ c >> θ s are the pressure sensitivity coefficients for stiff and compliant pores, respectively, and φ s = φ s φ s0 is the deviation of the stiff porosity from its zero-pressure value φ s0. Shapiro (2003) further shows that the linear variation of stiff porosity and exponential change of compliant porosity with pressure P can be written as φ s = φ s0 + φ s = φ s0 P ( C drs C gr ), (3) and φ c = φ c0 exp ( θ c C drs P), (4) where φ c0 is the compliant porosity at zero pressure, and C gr = 1/K gr is the compressibility of the solid grain. In many cases, θ s φ s << θ c φ c so that eq. (2) reduces to C dr (P) = C drs (1 + θ c φ c ). (5) The linear dependence of dry rock compressibility on compliant porosity (eq. 5) and exponential decay of compliant porosity with pressure (eq. 4) has been tested experimentally and verified for sandstones by Pervukhina et al.(2010). 2.2 Squirt model for saturated rock compressibility Squirt flow, a phenomenon of wave-induced pressure relaxation and fluid flow between pores of different shapes and/or orientations (e.g. Mavko & Nur 1975, 1979; Jones 1986), is a major cause of elastic wave dispersion and attenuation in fluid-saturated sandstones. When an elastic wave propagates through a fluid-saturated medium, relatively compliant pores are deformed to a greater degree than the relatively stiff pores, giving rise to local pressure gradients within the fluid phase, and resulting in fluid flow and corresponding viscous fluid flow losses. At very low frequencies, there is sufficient time for the pressure gradient to equilibrate and the pore fluid can easily flow in and out of every pore, so that the drained frame moduli are essentially the same as the dry rock moduli. However, at higher frequencies, viscous effects isolate the thinnest pores with respect to fluid flow, so that the frame is only partially drained and is stiffer than at low frequencies (Mavko & Jizba 1991; Gurevich et al. 2009; de Paula et al. 2012). Based on the dual porosity concept of Murphy et al. (1986), Gurevich et al. (2010) derived an expression for the frequency dependent compressibility of the modified frame, a hypothetical frame in which the stiff pores are assumed to be dry while the compliant pores are saturated with a liquid, as 1 C mf (P,ω) = C drs +, (6) 1 C dr (P) C drs + 3iωη 8φ c(p)α 2 where ω (ω = 2πf,wherefis the frequency in Hz) is the angular frequency of the elastic wave, η is the viscosity of the liquid saturating the rock and α is the aspect ratio of the compliant pores. The squirt model (eq. 6) shows that the elastic compressibility of a rock is dependent on the elastic wave frequency, the fluid properties and the geometry of the pores. With an increase in the elastic wave frequency or fluid viscosity, there is less time for the local pressure gradient to equilibrate, so that more fluid is trapped in the compliant pores, leading to a decrease in the compressibility (an increase in the bulk modulus) of the frame. The same is true for compliant pores with decreasing aspect ratio. The squirt model is consistent with Gassmann equation (Gassmann 1951) and Mavko Jizba equations (Mavko & Jizba 1991) at low and high frequencies, respectively, and with the piezosensitivity model of Shapiro (2003). 2.3 Porosity estimation The pressure dependent elastic compressibility of the rock fully saturated with a liquid C f can be obtained by substituting eqs. (1), (3), (4) and (5) and the modified frame compressibility eq. (6) into Gassmann equation (Gassmann 1951) C sat (P) [ φs0 P ( ) C drs C gr + φc0 exp ( θ c C drs P) ]( ) C f C gr = C gr +. [φ 1 + s0 P(C drs C gr)+φ c0 exp( θ cc drs P)](C f C gr) C drs iωη C gr + C drs [1+θcφ c0 exp( θcc drs P)] C drs 8φ c0 exp( θcc drs P)α 2 (7) In eq. (7), the saturated rock compressibility C sat can be calculated using the measured velocities as a function of differential pressure and the bulk density (d 0 ) of the saturated sample determined at zero-pressure C sat = 1/[(Vp 2 (4/3)V s 2)d 0], C gr is estimated from the mineral analysis of the rock, aspect ratio α of the compliant pores cannot be directly measured but is found to be 0.01 to give a good fit of a range of lithology (Gurevich et al. 2010), and coefficients C drs, θ c, φ c0 and φ s0 are fitting parameters which can be

3 380 T. Han et al. used to compute the variation of stiff and compliant porosity with differential pressure. In this study, we use the non-linear Levenberg Marquardt algorithm (Levenberg 1944; Marquardt 1963) for approximation of all exponential dependencies. We fit experimentally measured compressibility of saturated rocks expressed by the real part of eq. (7) to get the best-fitting values of C drs, θ c, φ c0 and φ s0 and then calculate stiff, compliant and total porosity as a function of differential pressure. 2.4 Test on laboratory data examples To illustrate the applicability of eq. (7), we use it to predict the porosity from the velocity measurement of saturated rocks and compare the inverted porosity with laboratory measurement. Since stiffness of pores is a relative measure, there is freedom to separate stiff and compliant porosity from the total porosity measurement (e.g. Mavko & Jizba 1991; Pervukhina et al. 2010). Therefore, a single comparison of stiff or compliant porosity between inversion and measurement will always show a higher degree of success. To avoid this and to make the comparison more convincing, we compare the total porosity instead of stiff or compliant porosity. It is believed that a satisfactory prediction of the measured total porosity is a better indication of the applicability of the proposed inversion technique. We first consider the Navajo sandstone sample (Coyner 1984). The ultrasonic compressional ( 1 MHz) and shear wave velocity was measured on a sample fully saturated with Benzene at 10 MPa pore pressure (K f = 1.21 GPa, d = 0.88 g cm 3,andη = 0.6 cp), and the variation of porosity with differential pressure was determined using the strain measurement. The grain bulk modulus is taken to be K gr = 39 GPa, and the compliant pore aspect ratio it set to be α = By best-fitting the measured wet rock compressibility as shown in Fig. 1(a), the parameters C drs, θ c and φ c0 (φ s0 = φ 0 φ c0,whereφ 0 is the zero pressure total porosity of the rock that is usual measured in the laboratory) are determined to compute the variation of total porosity with pressure, a comparison of which with measured total porosity is given in Fig. 1(b). The inverted total porosity is within ±0.3 per cent of the measured data. We also consider a water-saturated sample of Berea sandstone (Han et al. 1986). The sample was fully saturated with water at pore pressure of 1 MPa (K f = 2.25 GPa, d = 1.0 g cm 3 and η = 1.0 cp) with velocity measured at ultrasonic frequency ( 1 MHz) and the variation of porosity with pressure monitored with a pore-pressure intensifier while the pore pressure was kept constant. By using grain bulk modulus K gr = 39 GPa (Gurevich et al. 2010) and compliant pore aspect ratio α = 0.01, the measured rock compressibility is fitted with eq. (7) to obtain parameters C drs, θ c, φ c0 and φ s0,which are further employed to calculate the variation of porosity. The comparison of the best-fitted wet rock compressibility and inverted total porosity with measurement are illustrated in Figs 2(a) and (b), respectively. The inverted total porosity is within ±0.7 per cent of measured data. The total porosity inverted from the velocity measured on the two saturated sandstones shows satisfactory agreement with the measurement, with accuracy better than the error-bar of porosity determined as a function of differential pressure, which is generally agreed to be around ±1 per cent (M. Lebedev and J. Dautriat, personal communication). It is also shown from these two examples and the samples presented by Gurevich et al.(2010) that a compliant pore aspect ratio α = 0.01 can be representative to all of these samples. Therefore, unless otherwise stated, a compliant pore aspect Figure 1. Comparison of (a) elastic compressibility and (b) porosity between measurement (circles) and model prediction (lines) for the saturated Navajo sandstone (Coyner 1984) as a function of differential pressure. The inverted total porosity is within ±0.3 per cent of measured data. ratio fixed at this constant value will be employed for the estimation of porosity for the rest of the work. 3 LINKING THE PRESSURE DEPENDENCY OF ELASTIC AND ELECTRICAL PROPERTIES Having outlined the dual porosity and squirt models for the calculation of elastic compressibility of saturated rocks, and validated the inversion technique for the variation of porosity with differential pressure on laboratory data, we proceed to apply the workflow to compute the porosity of the clean sandstones published by Han et al. (2011a,b), the relationships between the obtained compliant porosity and the elastic compressibility, electrical conductivity and the joint elastic-electrical properties of the sandstones are then analysed. The sample dataset consists of 8 clean sandstones with volumetric clay content less than 1 per cent and benchtop porosity (φ 0 ) determined using a helium porosimeter ranging from to per cent, as given in Table 1. The samples were fully saturated with 35 g l 1 NaCl brine (K f = 2.25 GPa, d = g cm 3 and η = 1.0 cp) with conductivity of 4.69 S m 1, and the ultrasonic compressional and shear wave velocity (with accuracy of ±0.3 per cent) and electrical conductivity (at 2 Hz frequency with accuracy of

4 Pressure dependent physical properties 381 Figure 3. Variation of the measured (circles) and modelled (lines) elastic compressibility with differential pressure for all the saturated sandstone samples. Figure 2. Comparison of (a) elastic compressibility and (b) porosity between measurement (circles) and model prediction (lines) for the saturated Berea sandstone (Han et al. 1986) as a function of differential pressure. The inverted total porosity is within ±0.7 per cent of measured data. ±2 per cent) were measured simultaneously at differential pressures of 60, 40, 26, 20, 15 and 8 MPa (the pore fluid pressure was kept at 5 MPa) during the unloading cycle using the ultrasonic pulse-echo system (McCann & Sothcott 1992) and electrical circumferential system (Han et al. 2015), respectively. Detailed descriptions of the sample preparation, experimental procedure and experimental results can be obtained from Han et al. (2011b). A grain bulk modulus K gr = 37 GPa is assumed for all the samples in the calculation. The variation of the physical properties (i.e. elastic compressibility, electrical conductivity and cementation factor of the saturated rocks) with differential pressure for the samples are given in the Appendix. 3.1 Elastic properties The measured variation of elastic compressibility of the saturated rocks (C sat ) with differential pressure is shown by circles in Fig. 3. The compressibility is calculated using a constant bulk density of the saturated rock measured at zero-pressure conditions. In fact, the bulk density of a sample will increase slightly with increasing differential pressure, leading to a decreasing computed rock compressibility, especially for a compliant rock. However, Best (1992) demonstrated that the length change of a consolidated sandstone sample from differential pressure range of 0 60 MPa was less than about 0.3 per cent, which gives 0.61 per cent increase in bulk density (and 0.61 per cent decrease in calculated rock compressibility) on assumption of a water-saturated sample with 15 per cent initial porosity and grain density of 2.6 g cm 3. Moreover, Pervukhina et al. (2010) have estimated density perturbations from axial strain measurements assuming that all deformations are isotropic and found that the error of the compressibility calculation caused by using the zero-pressure density is less than 0.4 per cent for all their dry sandstone samples. Since there is no information available about the density at elevated pressures for the samples in our study, it is reasonable to assume an error of 1 per cent in the determined rock compressibility. However, the error will not significantly affect the approximation results. Also shown in Fig. 3 is the best-fitted elastic compressibility of the samples using eq. (7) with fitting parameters φ c0, C drs and θ c tabulated in Table 1. The obtained pressure sensitivity coefficients for compliant pores θ c ranges from to , well within the order of 10 2 or larger, theoretically estimated by Shapiro (2003) Table 1. Laboratory measurement and fitting parameters for all the sandstone samples. Beestone Covered E3 E4 E5 E6 Stoneraise 1VSF φ 0 (per cent) φ c0 (per cent) C drs (1/GPa) θ c C sats (1/GPa) θ satc m s θ mc σ s (S m 1 ) θ σ c

5 382 T. Han et al. compressibility. This can be interpreted in terms of the squirt flow in the rocks: when the pressure is high enough where all the compliant pores are closed, there is no squirt effect and the difference between the saturated and dry rock compressibility (C sats < C drs )is only caused by the liquid in the stiff pores; when the pressure is low when all the compliant pores are open, a proportion (the proportion depends on the frequency of the elastic wave) of these compliant pores for the saturated samples will be filled with a liquid as there is not sufficient time for the liquid to move in or out of the compliant pores at the ultrasonic frequency employed, therefore the compressibility of the saturated rocks is much smaller than that of the dry rocks due to the extra effects of liquid in the compliant pores. This explanation is consistent with the higher pressure dependency of dry rock velocity than saturated rock velocity experimentally observed by many authors (e.g. Johnston 1978; Coyner 1984; Han 1986; Khaksar et al. 1999; Agersborg et al. 2008). Figure 4. A comparison of measured (circles) and linearly fitted (lines) dependence of saturated rock compressibility on compliant porosity of all the samples. Correlation coefficients of the linear fit are given in Table 2. based on various effective medium theories for penny-shaped cracks (e.g. Mavko et al. 2009). The determined initial compliant porosity φ c0 of less than 0.17 per cent is also consistent with the results of Shapiro (2003) and Pervukhina et al.(2010). In the original dual porosity model of Shapiro (2003), the dry rock compressibility is linearly correlated with compliant porosity as pressure varies, as shown in eq. (5). Our results of saturated rock compressibility shown in Fig. 4 demonstrate that the elastic compressibility of saturated rocks also shows a linear correlation with compliant porosity. In a similar way to the dual porosity model for dry rocks, we propose to demonstrate this linear C sat φ c relationship using an expression C sat (P) = C sats (1 + θ satc φ c ), (8) where C sats and θ satc are the stiff-limit of the saturated rock compressibility and the pressure sensitivity coefficients related to the compliant pores for saturated compressibility, respectively. The values of C sats and θ satc are obtained by linear fit of eq. (8) to the measured data and are given in Table 1, with correlation coefficients tabulated in Table 2. The linear dependence of saturated rock compressibility on compliant porosity indicates that the closing of compliant pores with pressure is the main cause of the exponential decrease of the elastic compressibility of the saturated rocks. It is interesting to note that the coefficient θ satc is much smaller than the magnitude of θ c as shown above. While the stiff limit of the saturated rock compressibility C sats is slightly lower than the stiff limit of the dry rock compressibility C drs, the decreasing θ satc will indicate that the linear dependence of saturated rock compressibility on compliant porosity is smaller than that of the dry rock 3.2 Electrical properties The most widely employed model for interpreting the electrical conductivity of clean (clay free) rocks is given by Archie (1942) σ = σ f φ m, (9) where σ is the conductivity of the rock fully saturated with water having conductivity of σ f, φ is the total porosity of the rock, and m is the so-called cementation factor (or exponent or coefficient) serving as a description of geometric factor of the rock. The empirically derived Archie s equation (eq. 9) has a theoretical justification through effective medium models (Bruggeman 1935; Hanai 1960, 1961; Bussian 1983). When a saturated porous rock is subjected to an increasing differential pressure, the closure of the compliant pores will lead to a decreasing amount of conductive pore fluid, giving rise to a reduction in the rock conductivity (Fig. 5). The variation of differential pressure will also have an impact on the cementation factor, which was found to either increase (Fatt 1957; Jing et al. 1990) or decrease (Redmond 1962) with increasing pressure. By carefully monitoring conductivity and the change of porosity with pressure on samples covering a large range of porosity, Hausenblas (1995) demonstrated that the cementation factor increased with pressure for a sandstone sample of relatively low porosity and reduced with pressure for sandstones of relatively high porosity. Hausenblas (1995) then speculated that the porosity of samples showing independent cementation factor on pressure fell in the range between 15 and 20 per cent. To study the dependence of the cementation factor on the differential pressure for our samples, we calculate the variation of the cementation factor as a function of differential pressure based on Table 2. Correlation coefficients (R 2 ) of the linear relationships between compliant porosity and elastic compressibility, electrical cementation factor and electrical conductivity with pressure as outlined in eqs (8), (11) and (12), respectively and of the linear relationships between elastic compressibility and electrical cementation factor and electrical conductivity with pressure as given in eqs (13) and (14), respectively for all the sandstone samples. R 2 Beestone Covered E3 E4 E5 E6 Stoneraise 1VSF Eq. (8) Eq. (11) Eq. (12) Eq. (13) Eq. (14)

6 Pressure dependent physical properties 383 Figure 5. Variation of electrical conductivity with differential pressure for all the sandstone samples. Figure 7. A comparison of measured (circles) and linearly fitted (lines) dependence of cementation factor on compliant porosity of all the samples. Correlation coefficients of the linear fit are given in Table 2. Figure 6. Variation of determined cementation factor with differential pressure for all the sandstone samples. the measured conductivity and inverted total porosity from the measured elastic compressibility of saturated rocks, using an equation m = log σ log σ f. (10) log φ The results of the computed variation of cementation factor with differential pressure are given in Fig. 6, which shows an exponential increase of the cementation factor with increasing differential pressure. While increasing differential pressure will always reduce both porosity and conductivity of a rock, the increase or decrease of the cementation factor will depend on which of the porosity and conductivity reduction dominates. For samples with relatively high porosity, the conductivity of the rock is dominated by the large amount of stiff pores and reduces by a small amount as the compliant porosity decreases, and the cementation factor is required to decrease to compensate for the overwhelming reduction in porosity. On the other hand for samples with relatively low porosity, the interconnecting compliant pores dominates the rock conductivity, and the decreasing compliant porosity as a result of the increasing differential pressure will reduce the connective pathways for the electrical currents in addition to the reduction of conductive pore fluid. This leads to a much more reduced conductivity, which results in an increase in the cementation factor. The increasing cementa- Figure 8. A comparison of measured (circles) and linearly fitted (lines) dependence of rock conductivity on compliant porosity of all the samples. Correlation coefficients of the linear fit are given in Table 2. tion factor of the eight sandstones in this study indicates that they are all in the low porosity category and this is consistent with the observations of Hausenblas (1995). The exponential increasing cementation factor with differential pressure suggests that the cementation factor will decrease with increasing compliant porosity as differential pressure gets lower. The results in Fig. 7 show that the decreasing cementation factor with compliant porosity exhibits a linear trend, which can be expressed as m(p) = m s (1 + θ mc φ c ), (11) where m s is the stiff limit cementation factor and θ mc is the pressure sensitivity coefficients related to compliant pores for the cementation factor. The values of m s and θ mc by linear fit of eq. (11) to the measured data are tabulated in Table 1 (see Table 2 correlation coefficients of the linear fit). The decreasing trend of cementation factor with compliant porosity is denoted by the negative value of θ mc. Fig. 8 shows the dependence of rock conductivity on the compliant porosity of all the samples. Again, the correlation can be

7 384 T. Han et al. Figure 9. A comparison of measured (circles) and linearly fitted (lines) correlation between cementation factor and compressibility of all the saturated samples. Correlation coefficients of the linear fit are given in Table 2. Figure 10. A comparison of measured (circles) and linearly fitted (lines) correlation between conductivity and compressibility of all the saturated samples. Correlation coefficients of the linear fit are given in Table 2. approximated by a liner trend σ (P) = σ s (1 + θ σ c φ c ), (12) with σ s and θ σ c given in Table 1 representing the stiff limit and pressure sensitivity coefficients related to compliant pores for the rock conductivity, respectively. The positive linear relationship (correlation coefficient are tabulated in Table 2) between rock conductivity and complaint porosity indicates that while both compliant porosity and conductivity decrease with increasing differential pressure, the amount of the reduction in rock conductivity is linearly related to the decrease in the compliant porosity. 3.3 Joint elastic-electrical properties Since compressibility, cementation factor and conductivity of the saturated rocks are all linearly linked with compliant porosity, it is reasonable to expect a linear correlation between cementation factor and the saturated rock compressibility and between conductivity and compressibility of the saturated rocks. Integrating the linear dependence of saturated rock compressibility on compliant porosity eq. (8) into the correlation between cementation factor and compliant porosity eq. (11), we obtain a linear relationship between cementation factor and compressibility of the saturated rocks, as [ m(p) = m s 1 + θ ( )] mc Csat (P) 1. (13) θ satc C sats The measured and linearly fitted correlation between cementation factor and saturated rock compressibility is given in Fig. 9, which shows that cementation factor decreases linearly (see Table 2 the correlation coefficients for the linear relationship between elastic compressibility and electrical cementation factor of the saturated rocks) with increasing saturated rock compressibility. The correlation between electrical conductivity and elastic compressibility of the saturated rocks can be obtained by combining the linear dependence of conductivity (eq. 12) and saturated rock compressibility (eq. 8) on compliant porosity, given as [ σ (P) = σ s 1 + θ ( )] σ c Csat (P) 1. (14) θ satc C sats The experimentally measured and linearly fitted conductivitycompressibility relationship given in Fig. 10 shows a linear (see Table 2 the correlation coefficients for the linear relationship between elastic compressibility and electrical conductivity of the saturated rocks) increase of electrical conductivity with increasing elastic compressibility of the saturated rocks. 3.4 Relationships among the various coefficients We have demonstrated an exponential pressure dependency of electrical cementation factor and a linear dependence of elastic compressibility, cementation factor and electrical conductivity on compliant porosity of the saturated rocks. In this section we present an empirical analysis of the relationships among the various measured and fitted coefficients given in Table 1, with an aim to gain more insights into the pressure effects on the joint elastic-electrical properties of reservoir sandstones through the dual porosity model. For the linear dependence of saturated rock compressibility, cementation factor and electrical conductivity on compliant porosity obtained in eqs (8), (11) and (12), respectively, we have defined stiff limit compressibility, cementation factor and electrical conductivity of the saturated rocks. A stiff-limit property refers to the property of a hypothetical rock without the compliant porosity (i.e. φ c = 0) and with stiff porosity equal to zero-pressure stiff porosity φ s0. Figs 11(a) and (b) show that the stiff-limit elastic compressibility (C sats ) and electrical conductivity (σ s ) of the saturated rocks increase generally with increasing initial stiff porosity (φ s0 ), indicating that when all compliant porosity vanishes, both elastic compressibility and electrical conductivity are determined by the amount of stiff porosity in the rocks. This, however, is not the case for cementation factor given in Fig. 11(c), which shows an initial reduction and then an increase in the stiff-limit cementation factor (m s ) with zero-pressure stiff porosity, suggesting cementation factor is not uniquely linked with porosity but can be interpreted as a measure of the connectedness of the pore network (e.g. Glover 2009) or the efficiency of available conduction pathways for a given amount of available porosity (Herrick & Kennedy 1994; Clennell 1997). Since both elastic compressibility and electrical conductivity of the stiff limit saturated rocks increase with zero-pressure stiff porosity, an increasing electrical conductivity with elastic compressibility

8 Pressure dependent physical properties 385 Figure 11. Correlations between the various parameters given in Table 1. Colours have the same meaning as in Figs is expected and shown in Fig. 11(d). The approximately linear σ s C sats relationship not only provides information of how the elastic and electrical properties of sandstones are naturally linked with porosity, but also offers an opportunity to estimate one parameter from the other if only one of the parameters is known. The correlation between cementation factor and compressibility of the stiff-limit saturated rocks illustrated in Fig. 11(e) shows much scatter attributing to the additional factors other than porosity that affect cementation factor, i.e. variations in pore space connectedness and tortuosity. The pressure sensitivity coefficients related to compliant pores for elastic compressibility (θ satc ), electrical cementation factor (θ mc ) and electrical conductivity (θ σ c ) of the saturated rocks defined in eqs (8), (11) and (12), respectively determine the sensitivity of a property to the variation of differential pressure. The results in Figs 11(f) (h) show that while θ mc has a lower order of magnitude than θ satc and θ σ c, the absolute values of these sensitivity parameters exhibit broadly positive correlations, implying that the change of cementation factor caused by varying pressure is smaller compared to compressibility and conductivity and the samples with higher

9 386 T. Han et al. variation in cementation factor will also show a larger change in elastic compressibility and electrical conductivity. The slopes of the linear dependence of elastic compressibility, cementation factor and electrical conductivity of the saturated rocks on compliant porosity shown in Figs 4, 7 and 8, respectively are determined by the coefficients C sats θ satc, m s θ mc and σ s θ σ c, respectively. The results in Figs 11(i) (k) demonstrate that the slopes of the linear dependence of a property (e.g. elastic compressibility, cementation or electrical conductivity of a saturated rock) on compliant porosity are positively correlated with the pressure sensitivity coefficients related to compliant pores for the same property. While the variation of compliant porosity with pressure covers a small range, the relative change of a physical property with pressure can be large and will dominate the slope. A bigger increase (e.g. cementation factor) or decrease (e.g. compressibility and conductivity) is caused by a higher pressure sensitivity coefficient, as shown above. Finally it is interesting to study the slopes of the linear correlations between cementation factor and compressibility and between conductivity and compressibility of the saturated rocks given in eqs (13) and (14) and shown in Figs 9 and 10, respectively. The m C sat and σ C sat slopes are expressed as dσ dc sat = dm dc sat = ms θmc C satsθ satc and σs θσ c C satsθ satc, respectively. It is shown in Figs 11(l) and (m) that the slopes of the joint electrical-elastic properties are determined by the slopes of the linear dependence of the electrical properties (i.e. electrical cementation factor and electrical conductivity) on compliant porosity. This can be explained by the fact that the absolute variation of saturated rock compressibility with pressure is usually smaller than the variation of electrical properties. The stress sensitivity of electrical conductivity therefore largely determines the slopes of the linear electrical-elastic correlations of the samples. It is already obvious from the above analysis that the slopes of the linear joint elastic-electrical correlations depend on the slopes of the linear dependence of the electrical property on compliant porosity, which in turn are determined by the pressure sensitivity coefficients related to compliant pores for this electrical property. Figs 11(n) and (o) show that the pressure sensitivity coefficients related to compliant pores for cementation factor and electrical conductivity (i.e. θ mc and θ σ c, respectively) are generally negatively correlated with the measured bench-top porosity of the samples (φ 0 ), implying that samples with lower initial porosity will have a higher pressure sensitivity of the electrical properties and a steeper slope for the pressure dependency of the linear joint elastic-electrical properties. This was foreseen by Hausenblas (1995). 4 DISCUSSION We have presented a method to estimate the variation of stiff and compliant porosity with pressure from velocity measurement made as a function of differential pressure on saturated rocks, and explained the pressure dependency of electrical and joint elastic-electrical properties of porous rocks using the obtained compliant porosity. The pressure dependency of dry rock compressibility was computed using the dual porosity model (Shapiro 2003), and Gassmann s (1951) equation was employed to calculate the ultrasonic compressibility of saturated rocks from the frequency dependent modified frame compressibility based on the squirt model (Gurevich et al. 2010). The Gassmann equation was used under the assumption that the visco-inertial relaxation in the porous sandstones is insignificant. For samples where visco-inertial relaxation becomes important, the squirt model can be combined with Biot s (1962) equations of poroelasticity to account for squirt and local and Biot s global flow dissipation mechanisms (e.g. Carcione & Gurevich 2011). However, the use of Biot s model will require an extra unknown parameter, that is tortuosity factor (Clennell 1997), a description of the geometry and spatial orientation of the pores and the connectivity of the pore network, which although can be measured experimentally (Johnson et al. 1982), is not available for the samples in this study. Nevertheless, the good fit of the inverted porosity using eq. (7) with measurement as shown in Figs 1(b) and 2(b) indicates Gassmann s equation gives satisfactory approximation of the saturated rock compressibility at ultrasonic frequency. The aspect ratio for compliant pores was assumed to be 0.01 in this study. This was based on the observation that a compliant pore aspect ratio α = 0.01 gave a satisfactory fit to the variation of both saturated rock compressibility and porosity with pressure for the two laboratory examples shown before and for the samples presented in Gurevich et al. (2010). However, it should be noted that the compliant pore aspect ratio might vary between samples and the employment of a constant value for all the samples may lead to some error. Furthermore, the changing differential pressure might also affect the compliant pore aspect ratio, therefore the constant compliant pore aspect ratio is employed based on the assumption that the increasing differential pressure only reduces the amount of the compliant pores without changing their aspect ratio. Nevertheless, since the compliant pore aspect ratio cannot be directly measured, such an assumption seems necessary and reasonable and it was found that a moderate variation in the aspect ratio will not affect the main conclusions although the specific values of the fitting parameters given in Table 1 may change. Archie s (1942) equation was used for the calculation of electrical cementation factor with varying differential pressure based on the measured conductivity and inverted porosity with pressure. Archie s equation was designed for clean rocks where the rock forming materials are insulating and the rock conductivity is mainly from the conductivity of the fluid saturating the rocks. For reservoir sandstones that contain significant amount of clay minerals or other types of heavy minerals (e.g. pyrite or glauconite), the electrochemical interactions of the mineral-water system can give rise to an excess surface conductivity (e.g. Rabaute et al. 2003; Revil & Skold 2011; Revil 2013) in addition to the conductivity from the pore fluid, making Archie s equation invalid. This explains the clean sandstones with less than 1 per cent volumetric clay content andfullysaturatedwith35gl 1 NaCl brine selected for study in this work. The determination of cementation factor with varying differential pressure for clay-rich sandstones is the topic of a separate study. The obtained linear correlations between elastic compressibility, electrical cementation factor, electrical conductivity and compliant porosity of the saturated rocks suggest that the pressure dependence of the elastic and electrical properties of porous rocks is mainly controlled by the existence of compliant porosity, and the exponential variation of the elastic and electrical properties with differential pressure is determined by the same exponent C drs θ c that accounts also for the exponential decay of dry rock compressibility with pressure. This is an explicit confirmation of the theoretical derivations of Kaselow & Shapiro (2004) that the argument D of the exponential term of a four-parametric equation X(P) = A + KP B exp( DP) is a universal quantity for elastic moduli and velocities and electrical resistivity of isotropic rocks under hydrostatical loading. The approximately linear relationship between elastic velocity and electrical resistivity as a function of differential pressure was first observed in the work of Han et al. (2011a), who empirically attributed the linear correlations to the closure of low aspect ratio

10 Pressure dependent physical properties 387 pores and micropores being present as cracks either within mineral grains or more probably at grain contacts with increasing stress. This speculation seems to be supported by the work presented in this study by quantitatively linking the pressure dependency of the joint elastic-electrical properties with the complaint porosity of the rocks. The dual porosity concept not only explains the variation of dry rock elastic properties with pressure (e.g. Shapiro 2003; Pervukhina et al. 2010) but also accounts for the stress dependency of elastic, electrical and joint elastic-electrical properties of saturated porous rocks. However, it should be noted that the various linear relations obtained above are for samples under differential pressures (up to 60 MPa) that are not high enough to close all the compliant pores, the relationships might be different when the differential pressure is further increased after all the compliant pores are closed, and this will need future studies. It is also interesting to note that the saturated rock compressibility given in eq. (7) by combining the dual porosity model (Shapiro 2003) for dry rock compressibility and the squirt model (Gurevich et al. 2010) for compressibility of modified rock frame with Gassmann s equation (Gassmann 1951) is complex, indicating the elastic waves will exhibit attenuation. This provides a unique opportunity to study the pressure dependency of elastic attenuation and the corresponding correlations between elastic attenuation and electrical properties of rocks with varying pressure. This approach has the potential to offer a theoretical justification (e.g. Best et al. 2013) of the linear attenuation-resistivity observation of Han et al. (2011a). This will be presented in a separate study. 5 CONCLUSIONS The results and analyses presented above lead to the following conclusions: 1. We have outlined a method to invert for the stiff and compliant porosity from velocity measurement made as a function of differential pressure on saturated rocks. Tests on laboratory examples show that the model gives satisfactory prediction of porosity. The model extends the dual porosity concept from dry rocks to saturated porous rocks. 2. Cementation factor is calculated based on the inverted compliant porosity and the measured electrical conductivity. It shows that cementation factor exhibits an exponential increase with increasing differential pressure for our sandstones with porosity ranging from to per cent. 3. Elastic compressibility, electrical cementation factor and electrical conductivity of saturated rocks are linearly correlated with the compliant porosity of the samples. The slopes of the various linear correlations are determined by the sensitivity of the elastic and electrical properties to the variation of differential pressure. 4. Through the link of compliant porosity, electrical cementation factor and conductivity both show a linear correlation with the elastic compressibility of saturated rocks under loading. Samples with lower zero-pressure porosity tend to have steeper slopes of the pressure dependency of the joint electrical-elastic properties. ACKNOWLEDGEMENTS The authors would like to thank CSIRO Energy Flagship for financial support of this work. We would also like to thank two anonymous reviewers for their valuable comments and suggestions, which significantly improved the manuscript. 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