Experimental study and theoretical interpretation of saturation effect on ultrasonic velocity in tight sandstones under different pressure conditions

Experimental study and theoretical interpretation of saturation effect on ultrasonic velocity in... Summary Understanding the influence of lithology, porosity, permeability, pore structure, fluid content and fluid distribution on the elastic wave properties of porous rocks is of great significance for seismic exploration. However, unlike conventional sandstones, the petrophysical characteristics of tight sandstones are more complex and less understood. To address this problem, we measured ultrasonic velocity in partially saturated tight sandstones under different effective pressures. A new model is proposed, combining the Mavko–Jizba–Gurevich relations and the White model. The proposed model can satisfactorily simulate and explain the saturation dependence and pressure dependence of velocity in tight sandstones. Under low effective pressure, the relationship of P-wave velocity to saturation is pre-dominantly attributed to local (pore scale) fluid flow and inhomogeneous pore-fluid distribution (large scale). At higher effective pressure, local fluid flow gradually decreases, and P-wave velocity gradually shifts from uniform saturation towards patchy saturation. We also find that shear modulus is more sensitive to saturation at low effective pressures. The new model includes wetting ratio, an adjustable parameter that is closely related to the relationship between shear modulus and saturation. High-pressure behavior, Acoustic properties, Wave propagation 1 INTRODUCTION A porous rock is described as partially saturated if the pore space contains two or more different fluids. The propagation of body waves (P- and S-waves) in partially saturated rock is complicated, which poses great challenges for seismic oil and gas exploration (Gist 1994; Muller et al.2010; Si et al.2016). Therefore, theoretical and experimental studies of elastic wave propagation in partially saturated rock have always been a focus of petrophysical research. Exploring the relationship between seismic response and fluid saturation can provide the basis for amplitude variation with offset analysis, fluid prediction and time-lapse seismic techniques. The relationship between velocity and saturation is mainly affected by lithology, the properties of each fluid component, the method of fluid saturation (imbibition and drainage), the fluid distribution pattern, frequency, pressure and temperature (Muller et al.2010; Ding et al.2014; Khalid & Ahmed 2016; Ding et al.2017). Gregory (1976) measured the ultrasonic velocity of partially saturated sedimentary rocks. The results showed that the petrophysical properties of low-porosity rocks are more sensitive than those of high-porosity rocks to fluid saturation (see also Kahraman 2007; Si et al.2016). Some experiments (Endres & Knight 1991; King et al.2000; Lebedev et al.2009) found that imbibition and drainage processes can cause saturation hysteresis, which means that velocities may not be consistent at the same saturation level. Murphy (1984) measured the velocities of tight sandstones in seismic, sonic, and ultrasonic frequency bands, and found significant deviation between the P-wave velocity predicted by the Biot–Gassmann (Gassmann 1951; Biot 1956a,b) theory and that measured in the sonic and ultrasonic frequency bands. However, most of the experiments described above employed atmospheric or constant effective pressure measurements. Due to the development of micro-cracks and the complex petrophysical characteristics of tight sandstones, changes in effective pressure affect the distribution pattern of pore fluid, velocity dispersion, and the velocity–saturation relationship. However, research on this aspect remains limited (Smith et al.2009; Mo et al.2015). In addition, due to the limited measurement accuracy and the insensitivity of S-wave velocity to fluid in conventional sandstones, research into the relationship between the S-wave velocity and fluid saturation of tight sandstones is often neglected. To quantitatively explain the relationship between velocity and saturation, many empirical formulas (Brie et al.1995; Karakul & Ulusay 2013) and theoretical models (Norris 1993; Carcione et al.2003; Toms et al.2006) have been presented. When wave frequencies are sufficiently low, the effective modulus of pore fluid in partially saturated rocks is calculated using Wood's law (Johnson 2001). Therefore, Gassmann's theory is termed the Gassmann–Wood model when applied to partially saturated rock. This model has been well demonstrated in studies by Murphy (1984) and Mavko & Mukerji (1998). At high frequencies, the propagation of waves can induce a local (pore scale) pore pressure gradient, under which the pore fluid is moving between pores of different scales, causing dispersion and attenuation of elastic waves. In such scenarios, the Gassmann–Wood model alone has often proven ineffective (Toms et al.2006; Muller et al.2010). Gist (1994) explained the relationship between ultrasonic velocity and saturation in a wide range of rocks by combining the gas pocket model (White 1975) and the local flow theory. Mavko & Nolen-Hoeksema (1994) proposed another model based on the Mavko–Jizba (MJ) relations (Mavko & Jizba 1991). They considered that ultrasonic velocities were controlled by saturation heterogeneity at two scales: (1) differences between large stiff pores and thin soft pores, and (2) differences between saturated patches and partially saturated patches at a scale much larger than any pore. The MJ model is based on the idea that the pore space has binary structure (contains stiff and compliant pores). This model adequately describes the pressure dependence of rock velocity (Chapman et al.2002). The expressions of the MJ model are satisfied for typical liquids and most rocks, but might not be valid for gas saturated rocks (Gurevich et al.2009). Gurevich et al. (2009, 2010) derived a more general MJ model, which we term the Mavko–Jizba–Gurevich (MJG) model. This improvement eliminates the restriction of pore fluid modulus, possibly enabling the MJG model to be used for partially saturated rocks. Therefore, our aim is to extend the MJG model to explore the pressure dependence of velocity in partially saturated tight sandstones. In this paper, the theory of elastic wave propagation in partially saturated rocks is briefly reviewed, and a new combined model is proposed based on the MJG model and the White model. Further, we describe the experimental process in detail. The validity of the new model is verified using experimental data, and the dependence of the elastic properties of tight sandstones on saturation and effective pressure is explained and summarized. 2 THEORY 2.1 Theoretical background for partially saturated porous media A considerable amount of research (Mavko & Mukerji 1998; King et al.2000; Si et al.2016) has used Gassmann's theory (Appendix A in the Supporting Information) to estimate wave velocities in partially saturated rocks. These studies show that for partially saturated rock, the pore pressure induced by each fluid is different due to the differing in bulk modulus of each pore fluid component. With uniform mixing of the pore fluid components, the mixed fluids can be homogeneously distributed in pore space (called uniform saturation). In another case, the fluid is inhomogeneously distributed in the pores of the rock. Some regions (patches), on a scale containing many pores, are partially saturated, while other regions containing many pores are entirely saturated, which is termed patchy saturation (Mavko & Nolen-Hoeksema 1994). White (1975) proposed a model that can describe the variation of velocity and attenuation with frequency, and rock porosity, permeability, and saturation. The White model can describe both uniform and patchy saturation patterns, and also their transition trends, by adjusting the gas pocket radius (r). Dutta & Ode (1979) developed a theoretical derivation of a more rigorous elastic wave velocity and attenuation of the White model according to Biot's theory. This modified White model is described in Appendix B in the Supporting Information. We use Travis Peak tight gas sandstone (Mavko & Jizba 1991) to simulate different saturation patterns and the effect of the gas pocket radius (parameter r) on the saturation dependence of the P-wave velocity. Table 1 gives the main parameters used in the calculation. As shown in Fig. 1, when the radius r is <0.01 mm, the P-wave velocity trend predicted by the White model is similar to that for uniform saturation (lower bound). The turning point of the trend gradually moves to low saturation as the radius of the gas pocket is increased. When r is sufficiently large, the White model's predictions are consistent with the values for patchy saturation (upper bound). In addition, the White model is similar to the Gassmann–Wood model in predicting the S-wave velocity of partially saturated rocks, and the two models predict that the shear modulus is independent of saturation. Figure 1. View largeDownload slide Simulation of P-wave velocity as a function of saturation in a sample of Travis Peak tight gas sandstone, using the White model. Uniform (solid blue) and patchy (solid red) saturation patterns obtained by the Gassmann model. Figure 1. View largeDownload slide Simulation of P-wave velocity as a function of saturation in a sample of Travis Peak tight gas sandstone, using the White model. Uniform (solid blue) and patchy (solid red) saturation patterns obtained by the Gassmann model. Table 1. Material properties of Travis Peak tight gas sandstone (TGS) and fluid (Mavko & Jizba 1991; Mavko et al.2009). Material  Bulk modulus (GPa)  Shear modulus (GPa)  Density (g cm−3)  Viscosity (η/Pa s)  Travis Peak TGS  38  35  2.63  –  Gas  0.0022  0  0.0012  0.00011  Water  2.25  0  1  0.001  Material  Bulk modulus (GPa)  Shear modulus (GPa)  Density (g cm−3)  Viscosity (η/Pa s)  Travis Peak TGS  38  35  2.63  –  Gas  0.0022  0  0.0012  0.00011  Water  2.25  0  1  0.001  View Large 2.2 Modelling squirt flow dispersion At ultrasonic frequencies, elastic waves in saturated rocks display velocity dispersion and attenuation under the action mechanism of wave-induced fluid flow. Dispersion is the variation of velocity with frequency (Muller et al.2010). Dispersion can be caused by wave induced fluid flow, which can take place at the macroscale, mesoscale or microscale. Many studies (Gregory 1976; Gist 1994; Mavko & Jizba 1994; Bhuiyan & Holt 2016) have reported that the dispersion of tight sandstones is mainly caused by mesoscopic and/or microscopic fluid flow. Therefore, to quantitatively describe ultrasonic velocity as a function of saturation and effective pressure in tight sandstones, we combine the MJG model describing the microscopic fluid flow, and the White model describing the mesoscopic fluid patches into a new model that we call the MJGW model. We first estimate the microscopic fluid flow effects in partially saturated rock based on the MJG model. The elastic modulus of the partially saturated frame with water in the compliant pores only is defined as an ‘unrelaxed wetted frame’. Compared with the original MJ model, the MJG model is not limited to liquid saturation, and can therefore be used to simulate partial saturation. The modified White model is then used to describe the saturation of the remaining pore space. To overcome the limitations of the original MJ model for fluid bulk modulus, Gurevich et al. (2009,2010) derive a more general MJ model according to the effective pressure relaxation approach (Murphy et al.1986) and the Sayers–Kachanov discontinuity formalism (Sayers & Kachanov 1995), which we term the MJG relations. Detailed derivation refers to Gurevich et al. (2009, 2010), and the final result is as follows:   $$\frac{1}{{{K_{{\rm{mf}}}}({P,\omega })}} = \frac{1}{{{K_h}}} + \frac{1}{{\frac{1}{{\frac{1}{{{K_{{\rm{dry}}}}\left( P \right)}} - \frac{1}{{{K_h}}}}} + \frac{{3i\omega \eta }}{{8{\phi _c}\left( P \right){\alpha ^2}}}}}$$ (1)  $$\frac{1}{{{\mu _{{\rm{mf}}}}\left( {P,\omega } \right)}} = \frac{1}{{{\mu _{{\rm{dry}}}}\left( P \right)}} - \frac{4}{{15}}\left( {\frac{1}{{{K_{{\rm{dry}}}}\left( P \right)}} - \frac{1}{{{K_{{\rm{mf}}}}\left( {P,\omega } \right)}}} \right),$$ (2)where Kmf(P, ω) is bulk modulus of the modified frame with frequency dependence and pressure dependence, which can represent a relaxed or unrelaxed state. μmf(P, ω) is the corresponding shear modulus of the modified frame. Kh is the bulk modulus of dry rock under high effective pressure under which it is generally assumed that the compliant pores of the rock have been completely closed. Kdry(P) and μdry(P) are the pressure-dependent bulk and shear modulus of the dry rock, respectively. η is the viscosity of the fluid, ϕc(P) is compliant porosity, α represents the aspect ratio of compliant pores. The characteristic frequency fc of the local flow dispersion can usually be written as   $${f_c} = \alpha _m^3\frac{K}{\eta },$$ (3)where K represents the bulk modulus of rock and αm represents the mean aspect ratio of pores. For stiff pores (αm≈1), if the rock is saturated with water, fc is much higher than 1 MHz. For compliant pores (αm ≪ 1), if the rock is saturated with water, fc may be less than 1 MHz. Therefore, the main mechanism of pore-scale dispersion is wave induced fluid flow between compliant and stiff pores. According to eqs (1) and (2), frequency- and pressure-dependent, unrelaxed wetted (partially saturated) frame bulk modulus Kps(P, ω) and shear modulus μps(P, ω) can be written as follows:   $$\frac{1}{{{K_{{\rm{ps}}}}\left( {P,\omega } \right)}} = \frac{1}{{{{\left( {{K_{{\rm{dry}}}}} \right)}_{{P_{{\rm{ps}}}}}}}} + \frac{1}{{\frac{1}{{\frac{1}{{{K_{{\rm{dry}}}}\left( P \right)}} - \frac{1}{{{{\left( {{K_{{\rm{dry}}}}} \right)}_{{P_{{\rm{ps}}}}}}}}}} + \frac{{3i\omega \eta {S_c}}}{{8{\phi _c}\left( P \right){\alpha ^2}}}}}$$ (4)  $$\frac{1}{{{\mu _{{\rm{ps}}}}\left( {P,\omega } \right)}}= \frac{1}{{{\mu _{{\rm{dry}}}}\left( P \right)}} - \frac{4}{{15}}\left( {\frac{1}{{{K_{{\rm{dry}}}}\left( P \right)}} - \frac{1}{{{K_{{\rm{ps}}}}\left( {P,\omega } \right)}}} \right)$$ (5)where $${( {{K_{{\rm{dry}}}}} )_{{P_{{\rm{ps}}}}}}$$represents the measured dry bulk modulus at the equivalent closing effective pressure, Pps (this parameter can be determined by reference to Mavko & Nolen-Hoeksema (1994)), and μdry(P) is shear modulus of the dry rock. Eq. (4) is compared with eq. (1), and a new variable Sc is added to represent the saturation of the liquid in compliant pores. According to eqs (1) and (3), we know that the viscous resistance of the liquid in the compliant pores is the main cause of the unrelaxed state of the rock framework. Therefore, the parameter Sc can be used to describe the unrelaxed frame modulus of partially saturated rock. The relationship between Scand Sw (the water saturation of rock) can be written as   $${S_c} = \rm{WR} \times {S_w}\frac{\phi }{{{\phi _c}}},$$ (6)where ϕ and ϕc are the total porosity and compliant porosity of the rock, respectively (Mavko & Nolen-Hoeksema 1994). Most liquid entering the rock will fill the stiff pores, and a small proportion enters the compliant pores, which is defined as the wetting ratio (WR). As seen in eq. (6), WR is related to the saturation of rock, and determines the saturation of the compliant pores. In particular, when Sc = 0, there is no liquid in the compliant pores, and the right-hand side of eq. (4) reduces to the dry modulus Kdry(P), as expected. When Sc = 1, the modulus of the unrelaxed wetted frame reaches its maximum. 2.3 Modelling saturation dependence To investigate how the MJGW model describes the effect of water saturation on ultrasonic velocities, we simulated Travis Peak tight gas sandstone (Mavko & Jizba 1991). According to Gurevich et al. (2010), the α (aspect ratio of compliant pores) of this sandstone is 0.01. The compliant porosity is estimated by the method described by Shapiro (2003) and Pervukhina et al. (2010). The porosity of this tight sandstone is 8.0 per cent, assuming the permeability is 0.09 mD, and the other parameters are shown in Table 1. For the MJGW model, the two most important parameters are the wetting ratio and the gas pocket radius. Taking Fig. 2(a) as an example, the wetting ratio is 0.44 per cent, and when the gas pocket radius increases from 0.01 mm to 0.3 mm, the saturation pattern of the P-wave velocity changes from uniform saturation to patchy saturation. Comparing Figs 2(a)–(c) indicates that when the gas pocket radius is the same, the rock shows local fluid flow at lower saturation levels as the wetting ratio increases from 0.44 per cent to 3 per cent. Therefore, the wetting ratio determines the effect of local fluid flow on P-wave velocity at different saturation levels, and the gas pocket radius determines the saturation pattern. Figure 2. View largeDownload slide Simulation of P-wave velocity as a function of saturation in a sample of Travis Peak tight gas sandstone, using the MJGW and Gassmann models. Wetting ratios (a) 0.44 per cent, (b) 0.8 per cent, (c) 3 per cent. Figure 2. View largeDownload slide Simulation of P-wave velocity as a function of saturation in a sample of Travis Peak tight gas sandstone, using the MJGW and Gassmann models. Wetting ratios (a) 0.44 per cent, (b) 0.8 per cent, (c) 3 per cent. As seen from Fig. 3(a), the shear modulus simulated by the MJGW model is dependent on saturation, and the wetting ratio controls the shear modulus versus saturation relationships. The White model considers that the gas pocket radius has no effect on the shear modulus, and thus, the dependence of the shear modulus on saturation is only caused by the local fluid flow. According to eq. (4), the compliant pores contain fluid that causes stiffening of the framework, and the shear modulus also produces a dispersion in the relationship of eq. (5), so that the dependency of shear modulus on saturation is closely related to the wetting ratio. Similarly, the MJGW model can also describe the dependence of S-wave velocity on saturation. When the wetting ratio is low, the S-wave velocity first decreases, and then increases, with increase in saturation; conversely, when the wetting ratio is high, the S-wave velocity first increases, and then decreases, with increase in saturation (Fig. 3b). Figs 2 and 3 show that the P- and S-wave velocities of the fully saturated sandstone predicted by the Gassmann model are lower than those from the MJGW model because the former does not take into account local fluid flow. Figure 3. View largeDownload slide Simulation of (a) shear modulus and (b) S-wave velocity as a function of saturation in a sample of Travis Peak tight gas sandstone, using the MJGW and Gassmann models. Figure 3. View largeDownload slide Simulation of (a) shear modulus and (b) S-wave velocity as a function of saturation in a sample of Travis Peak tight gas sandstone, using the MJGW and Gassmann models. 3 EXPERIMENTAL STUDY 3.1 Experimental procedures We selected three samples from a tight gas reservoir in northwest China, and cut them into standard core columns of diameter approximately 1 inch and length of approximately 2 inches. We carefully polished the ends of the sample to reduce errors in the length and velocity measurement. The samples were then vacuum-oven-dried at 70 °C for more than 4 d. A large number of studies (King et al.2000; Mavko et al.2009) showed that a small amount of moisture may cause significant changes in the rock matrix. We therefore placed the dried samples in a room under constant environmental conditions (humidity 45–55 per cent, temperature 23–25 °C) for several hours, thereby obtaining dry samples with insignificant moisture content. The porosity and permeability (Klinkenberg-corrected) of the samples were measured by a porosity and permeability testing system (BenchLab 7000EX). The precision of porosity measurement is within ± 0.03 porosity units, and the error of permeability is approximately 8 per cent (American Petroleum Institute (API) 1998). X-ray diffraction (XRD) showed that the mineral compositions of the three samples were similar, with quartz and feldspar, accounting for about 89 per cent. The samples had permeability <1 mD and porosity <10 per cent (see Table 2), which reflects the low-porosity/low-permeability petrophysical characteristics of the tight sandstones. The selected samples were from the same reservoir, and analyses of cast thin sections and SEM images showed they had similar pore structures. As seen in Fig. 4(a), the pore structure of sample S3 is complex with poor connectivity; the pore type is mainly intergranular dissolved pores, and includes a small number of intragranular pores. Fig. 4(b) shows the existence of microcracks and compliant pores in sample S3, and the pore aspect ratios of these cracks and pores are very small. The presence of these pores may cause changes in rock velocity, and if these pores contain fluids, the velocity changes may be more complex (Smith et al.2009). Figure 4. View largeDownload slide (a) Thin section image of sample S3; (b) SEM image of sample S3. Figure 4. View largeDownload slide (a) Thin section image of sample S3; (b) SEM image of sample S3. Table 2. Measured parameters of the core samples (porosity and permeability were measured at 2 MPa effective pressure). Sample  Porosity (per cent)  Permeability (mD)  Density (g cm−3)  Clay (per cent)  Quartz (per cent)  Feldspar (per cent)  Calcite (per cent)  Muscovite (per cent)  S1  6.48  0.023  2.50  5.7  58.8  30.5  1.8  3.2  S2  6.71  0.069  2.47  5.5  68.6  21.4  2.1  2.4  S3  7.22  0.131  2.41  4.7  65.2  24.3  3.0  2.8  Sample  Porosity (per cent)  Permeability (mD)  Density (g cm−3)  Clay (per cent)  Quartz (per cent)  Feldspar (per cent)  Calcite (per cent)  Muscovite (per cent)  S1  6.48  0.023  2.50  5.7  58.8  30.5  1.8  3.2  S2  6.71  0.069  2.47  5.5  68.6  21.4  2.1  2.4  S3  7.22  0.131  2.41  4.7  65.2  24.3  3.0  2.8  View Large The P- and S-wave velocities of the three tight sandstones were measured at different saturation conditions, from dry to fully saturated with fluid, at the temperature of approximately 20 °C. For each saturation, the ultrasonic velocities at eight different confining pressures (2, 5, 10, 20, 30, 40, 50, and 60 MPa) were measured. In the experimental measurements, we consider that pore pressure in partially saturated tight sandstones can be neglected, so that the effective pressure is equal to the confining pressure applied to the sample (Mayr & Burkhardt 2006; Mikhaltsevitch et al.2016). The velocity was measured three times directly after the effective pressure reached the set value, and another three times 10 min later (Mayr & Burkhardt 2006). The experiment showed that the system (rock and fluid) was able to reach a steady state after 10 min. The average of all six results measured at a given effective pressure is used as the final measured value. Ultrasonic velocities were measured by the BenchLab 7000EX acoustic measurement system, and the centre frequency of the transducers was 0.5 MHz. The system has very good velocity measurement performance. The P- and S-wave waveforms of sample S3 are given in Figs 5(a) and (b), respectively, where the saturation of the sample is 24 per cent. The shapes of the waveforms are similar at different effective pressures. As the effective pressure increases, the amplitude of the wave gradually increases and the first arrival time gradually decreases. These high-quality waveforms ensure the accuracy of the first arrival pick-up. We pick up the first arrival of the shear wave according to the method discussed by Yi & Ning (2016). According to the method of Hornby (1998), we estimate that the errors for P- and S-wave velocities are approximately 1 per cent, and the errors for bulk modulus and shear modulus are less than 2 per cent. Based on the uncertainty analysis procedure described by Adam et al. (2006), the standard deviations are calculated and given in the form of error bars in the figures. For most figures, the length of the error bars is smaller than the size of the symbols so that they cannot be seen, which indicates small errors. Figure 5. View largeDownload slide Ultrasonic measurement waveform of sample S3: (a) P-wave signals received at different effective pressures; (b) S-wave signals received at different effective pressures. The red arrow represents the first arrival position. Figure 5. View largeDownload slide Ultrasonic measurement waveform of sample S3: (a) P-wave signals received at different effective pressures; (b) S-wave signals received at different effective pressures. The red arrow represents the first arrival position. To obtain different saturations, the sample was first placed in a container filled with distilled water. By this self-absorption method, the saturation of the three tight sandstones can reach 50–70 per cent. The sample was weighed at given time intervals to obtain different saturations. To achieve higher saturation, the samples were placed in a vacuum vessel containing distilled water and evacuated for 2 d, thereby achieving water saturation of up to 80 per cent. We then used the vacuum saturation equipment to pressurize the samples for more than one week at 30 MPa. In this way the samples can approach total fluid saturation, but because the pore structure of the samples is complex and some pores are almost disconnected, there is no guarantee that the samples can reach 100 per cent fluid saturation (Murphy 1984; Verwer et al.2010). To allow the water to have sufficient time to be distributed in the sample, after the desired saturation was established, the sample was wrapped in aluminium foil and placed under vacuum for 24 hr prior to conducting velocity measurements (Mayr & Burkhardt 2006). 3.2 Experimental results We compared the measurements, and found that the three tight sandstones showed similar velocity variation from dry to fully saturated with water. Fig. 6 shows the variation trend of the Vp (P-wave velocity) of the three samples as a function of saturation and effective pressure. At low effective pressure, the Vp of the samples increases or decreases slowly with increase in water saturation, then starts to increase rapidly when a certain saturation range (50–70 per cent) is reached. At high effective pressure, the Vp shows an approximately linear trend with the increase in saturation. In addition to the change of the trends, the difference between the saturated and dry P-wave velocities, ▵Vp, also changes: with increasing effective pressure, ▵Vp gradually decreased in all three samples, but is always positive. Figure 6. View largeDownload slide Vp of samples measured at different pressure and saturation: (a) S1; (b) S2; (c) S3. Figure 6. View largeDownload slide Vp of samples measured at different pressure and saturation: (a) S1; (b) S2; (c) S3. Fig. 7 shows the Vs (S-wave velocity) of three samples as functions of saturation and effective pressure. At low effective pressure, Vs hardly changes with increasing saturation, but increases gradually as saturation exceeds a certain range. At high effective pressure, Vs exhibits a decreasing trend. The ▵Vs (saturated and dry S-wave velocity difference) of the three samples showed similar changes: As the effective pressure increases, ▵Vs gradually changes from positive to negative, which is explained in detail in the following section. Since the samples were from the same reservoir with similar physical properties and pore characteristics, we selected sample S3 as being representative for further analysis below. Figure 7. View largeDownload slide Vs of samples measured at different pressure and saturation: (a) S1; (b) S2; (c) S3. Figure 7. View largeDownload slide Vs of samples measured at different pressure and saturation: (a) S1; (b) S2; (c) S3. 4 THEORETICAL INTERPRETATION To explain the experimental results, we used the MJGW model to predict the trend of the velocities of sample S3 at saturation for 2, 30, and 60 MPa representing low-, medium-, and high-pressure conditions, respectively. The prediction process is as follows. First, the compliant porosity is estimated by the method described in Appendix  C. Then, the velocities in rock (measured dry and fully saturated with fluid) are used as input parameters, and the aspect ratio of the compliant pores α = 0.01 is obtained using eq. (1). The results (see Fig. 3) show that the trend of Vs with saturation is only related to the wetting ratio and is independent of the gas pocket radius. Therefore, the Vs with saturation is described by adjusting the wetting ratio. Finally, after determining the wetting ratio, the variation of Vp with saturation can be predicted simply by adjusting the gas pocket radius. The parameters involved in the calculation are shown in Tables 1–3. Table 3. The main parameters used in the MJGW model and permeability measurements of sample S3. For 2 MPa effective pressure, the wetting ratios differ at low- and high-saturation ranges. Pressure (MPa)  Permeability (mD)  Gas pocket radius (mm)  Saturation (per cent)  Wetting ratio (per cent)  2  0.131  0.03  0–75  1.5        75–100  5  30  0.043  0.04  0–55  1.5  60  0.012  0.2  0–100  0  Pressure (MPa)  Permeability (mD)  Gas pocket radius (mm)  Saturation (per cent)  Wetting ratio (per cent)  2  0.131  0.03  0–75  1.5        75–100  5  30  0.043  0.04  0–55  1.5  60  0.012  0.2  0–100  0  View Large 4.1 Effect of saturation on P-wave velocity under different effective pressures At 2 MPa (Fig. 8a), for saturation > 40 per cent, the saturation dependence of Vp shows a large deviation from that predicted by the Gassmann–Wood (solid green line) and White (dotted purple line) models. However, Vp predicted by the MJGW model (dashed red line) shows good agreement with the measured values. The results of the White model show patchy saturation effect at saturation > 70 per cent. Therefore, by comparing the results of the White and MJGW models, we attribute the slow increase of Vp within the saturation range 40–70 per cent mainly to local fluid flow. When saturation exceeds 70 per cent, the rapid increase of Vp is mainly attributed to the combined action of patchy saturation effect and local fluid flow. Similarly, at 30 MPa (Fig. 8b) with saturation of 30–60 per cent, Vp is mainly affected by local flow, and when saturation exceeds 60 per cent, Vp is influenced by both these mechanisms. Figure 8. View largeDownload slide Experimental measurements of P-wave velocity versus values predicted by various models for sample S3 at effective pressures of (a) 2 MPa, (b) 30 MPa and (c) 60 MPa (the MJGW and White models’ predictions are consistent). Figure 8. View largeDownload slide Experimental measurements of P-wave velocity versus values predicted by various models for sample S3 at effective pressures of (a) 2 MPa, (b) 30 MPa and (c) 60 MPa (the MJGW and White models’ predictions are consistent). At 60 MPa (Fig. 8c), the compliant pores in sample S3 are completely closed, and the sample is no longer affected by local fluid flow. In this case, the MJGW predictions match those of the White model. As shown in Fig. 8(c), due to the heterogeneous distribution of the mixed fluid in the pore space, the Vp trend is close to the patchy saturation pattern. In addition, Fig. 8 also shows that with increasing effective pressure the Vp trend gradually changes from uniform to patchy saturation pattern. This change in saturation pattern can be described by the diffusion length L (Mavko & Mukerji 1998), and the critical diffusion length Lc is expressed as follows:   $${L_c} = \sqrt {{{k{K_{{\rm{fl}}}}} / {\eta f}}} ,$$ (7)where k is the permeability of the rock, Kfl is the bulk modulus of the fluid with the highest viscosity in the pore fluid, η is the characteristic viscosity of the fluid, and f is the frequency. When L < Lc, the pore fluid pressure has sufficient time to reach equilibrium through the fluid flow; in this case, the saturation pattern is uniform. When L > Lc, the pore fluid pressure cannot reach equilibrium in a short time, and the fluid shows patchy saturation pattern. The diffusion length L can be expressed by the gas pocket radius r and the water saturation Sw (Gist 1994):   $$L = r\left( {{{\left( {1 - {S_w}} \right)}^{ - 1/3}} - 1} \right).$$ (8) As can be seen from eq. (8), L gradually increases with increasing water saturation. At 2 MPa, Lc≈0.021 mm can be calculated from the parameters in Tables 2 and 3, and when water saturation exceeds 70 per cent, L is greater than 0.021 mm, at which point the Vp trend changes from uniform to patchy saturation pattern. The permeability of the sample gradually declines as effective pressure increases, resulting in a decrease in Lc. The decrease of Lc indicates that the diffusion length L can reach the critical diffusion length Lc at low saturation. Therefore, for 60 MPa effective pressure, the trend of Vp changes from uniform saturation to patchy saturation when the saturation exceeds 20 per cent. Comparing the measured values and theoretical predictions confirms that our proposed MJGW model can describe the dependence of the tight sandstone Vpon saturation and effective pressure conditions. Under low effective pressure, the relationship between Vp and saturation is mainly affected by the local (pore scale) fluid flow and the inhomogeneous distribution of pore fluid (large scale). As the effective pressure increases, the local fluid flow gradually decreases, and the Vp trend gradually shifts away from uniform saturation and approaches patchy saturation. 4.2 Effect of fluid saturation on S-wave velocity under different effective pressures In Fig. 9, the Vs predicted by the Gassmann model (solid green line) at 2 and 30 MPa deviates considerably from the measured values. However, the MJGW model's predictions (dashed red line) at three effective pressures are in good agreement with the measured values. According to the Gassmann model, the shear modulus will remain constant during saturation. Assuming that this conclusion is established, then according to eq. (A.7) in the Supporting Information, it can be concluded that Vs should gradually reduce as saturation increases. As shown in Fig. 9(c), Vs gradually decreased with increasing saturation at 60 MPa pressure. However, at low effective pressure, we found that the sample showed increasing Vs trend with increase in saturation (Figs 7 and 9a–b). As seen from eq. (A.7) in the Supporting Information, the density of partially saturated rocks can be measured accurately; therefore, it is inferred that the increasing trend of Vs in tight sandstones is likely related to the change of shear modulus. Figure 9. View largeDownload slide Experimental measurements of S-wave velocity versus values predicted by different models for sample S3: (a) 2 MPa; (b) 30 MPa; (c) 60 MPa. Figure 9. View largeDownload slide Experimental measurements of S-wave velocity versus values predicted by different models for sample S3: (a) 2 MPa; (b) 30 MPa; (c) 60 MPa. Khazanehdari & Sothcott (2003) define the increasing/decreasing shear modulus of rock saturated with fluid as shear strengthening/weakening respectively. Some more recent studies show that the fluid types and their interactions with the matrix, effective pressure, frequency, dispersion and pore structure are the most important factors affecting the shear modulus strengthening/weakening (e.g. Adam et al.2006; Lebedev et al.2014; Bhuiyan & Holt 2016; Li et al.2017). However, very few studies have examined the mechanism of shear modulus changes in partially saturated tight sandstones. To explore this problem, we used the MJGW model to predict the shear modulus of partially saturated sample S3 at different effective pressures. As shown in Fig. 10, the MJGW model (dashed red line) can satisfactorily describe the sample's variation in shear modulus with saturation at these three effective pressures. According to eqs (1) and (3), the wetting ratio determines the fluid content in the compliant pores, which in turn influences the unrelaxed effect of the fluid in the compliant pores. Therefore, the wetting ratio also determines the change of shear modulus. Figure 10. View largeDownload slide Experimental measurements of shear modulus versus values predicted by different models for sample S3: (a) 2 MPa; (b) 30 MPa; (c) 60 MPa. Figure 10. View largeDownload slide Experimental measurements of shear modulus versus values predicted by different models for sample S3: (a) 2 MPa; (b) 30 MPa; (c) 60 MPa. As seen from Table 3, the average wetting ratio is obtained by the best fit of the predictions to the measured values. At 2 MPa pressure, the average wetting ratio of sample S3 within the saturation range of 0–75 per cent is 1.5 per cent. The low wetting ratio indicates that only a comparatively small proportion of fluid enters the soft pores; consequently, the change in shear modulus is not obvious at saturation <40 per cent (Fig. 10a). With increasing saturation, there is gradual increase in fluid accumulated within the soft pores, and local fluid flow becomes stronger, such that the shear modulus increases as saturation increases. When saturation exceeds 75 per cent, the average wetting ratio is increased to 5 per cent, and thus, the slope of the change in shear modulus is slightly increased. The change of wetting ratio within different saturation ranges is related to the saturation process of the sample. At low saturation, the sample is self-absorbent to the corresponding saturation without applying effective pressure. Therefore, the proportion of the fluid entering the soft pores is very small. The high saturation levels were obtained by placing the sample in a vacuum-pressurized saturated apparatus. Under high effective pressure, the water is forced to enter the sample, thereby achieving a higher wetting ratio compared with low-saturation conditions. At effective pressure of 30 MPa, when saturation exceeds 55 per cent, the wetting ratio is zero, indicating that the compliant pores have been fully filled with fluid. In this case, the unrelaxed effect has reached its maximum, and the shear modulus does not change as saturation continues to increase (Fig. 10b). The compliant pores in the sample are completely closed at 60 MPa; the MJGW and Gassmann models agree that the fluid content has no effect on shear modulus; and the predicted results are in agreement with the measured values (Fig. 10c). Our experiment showed shear strengthening in the saturated sample, whereas the Gassmann model considers that shear modulus will remain constant during saturation, thereby underestimating the S-wave velocity of the sample. This is the main cause of the deviation between Gassmann predictions and the measured values in Figs 9(a) and (b). The MJGW model can describe the change in shear modulus with saturation, which confirms that local fluid flow is the main mechanism leading to shear strengthening in the tight sandstones. Some studies have found that rocks with low permeability and low fluidity, especially carbonate and tight sandstones, can cause dispersion even at seismic frequencies (Adam et al.2006; Batzle et al.2006; Lebedev et al.2014). Therefore, it is necessary to study the relationship between shear modulus and fluid saturation in tight sandstones at low frequency. This relationship has important implications for fluid identification and other seismic techniques. 5 CONCLUSIONS We propose a new combined model based on the MJG relations and the White model. The new model not only includes microscale and mesoscale fluid flows, but also considers the fluid distribution pattern. Comparison with experimental data demonstrates that the proposed model can describe and explain how ultrasonic velocity in tight sandstones is dependent on saturation and effective pressure conditions. Under low effective pressure, the saturation dependence of P-wave velocity is mainly affected by local (pore scale) fluid flow and the inhomogeneous distribution of pore fluid (large scale). At higher effective pressure, the local flow gradually decreases, and P-wave velocity gradually trends from uniform toward patchy saturation. An interesting finding is that shear modulus is also saturation- and pressure-dependent, and is more sensitive to fluid saturation at lower effective pressure. An adjustable parameter, wetting ratio, in the new model is closely related to the change in shear modulus. This finding suggests that S-wave velocity and shear modulus, which are considered insensitive to the fluid in the conventional reservoir, are likely to play important roles in fluid identification of tight sandstones reservoirs. Mechanisms of pore-scale and mesoscopic fluid-flow are usually considered to be important at ultrasonic frequencies but may also play a role at seismic and logging frequencies. We have only studied the ultrasonic range, and thus, it is necessary to further investigate these fluid flow mechanisms at lower frequencies. Acknowledgements We appreciate the editors and referees for their thoughtful comments. We are very grateful to Prof. Boris Gurevich from Curtin University for his valuable comments on this study. We also thank Prof. Tongcheng Han from China University of Petroleum for the helpful suggestions. We acknowledge financial sponsorship from the National Science and Technology Major Project (2016ZX05007-006) and the National Natural Science Foundation (41474112). REFERENCES Adam L., Batzle M., Brevik I., 2006. Gassmann's fluid substitution and shear modulus variability in carbonates at laboratory seismic and ultrasonic frequencies, Geophysics , 71( 6), F173– F183. https://doi.org/10.1190/1.2358494 Google Scholar CrossRef Search ADS   American Petroleum Institute (API), 1998. Recommended Practices for Core Analysis, API Recommended Practice 40 , 2nd edn, American Petroleum Institute (API). Batzle M.L., Han D.H., Hofmann R., 2006. Fluid mobility and frequency-dependent seismic velocity—direct measurements, Geophysics , 71( 1), N1– N9. https://doi.org/10.1190/1.2159053 Google Scholar CrossRef Search ADS   Bhuiyan M.H., Holt R.M., 2016. Variation of shear and compressional wave modulus upon saturation for pure pre-compacted sands, Geophys. J. 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Prospect. , 64( 4), 799– 809. https://doi.org/10.1111/1365-2478.12404 Google Scholar CrossRef Search ADS   Mo S.Y., He S.L., Lei G., Gai S.H., Liu Z.K., 2015. Effect of the drawdown pressure on the relative permeability in tight gas: A theoretical and experimental study, J. Nat. Gas Sci. Eng. , 24, 264– 271. https://doi.org/10.1016/j.jngse.2015.03.034 Google Scholar CrossRef Search ADS   Muller T.M., Gurevich B., Lebedev M., 2010. Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks — A review, Geophysics , 75( 5), 75A147– 75A164. https://doi.org/10.1190/1.3463417 Google Scholar CrossRef Search ADS   Murphy W.F., 1984. Acoustic measures of partial gas saturation in tight sandstones, J. geophys. Res. , 89( B13), 11 549–11 559. https://doi.org/10.1029/JB089iB13p11549 Google Scholar CrossRef Search ADS   Murphy W.F., Winkler K.W., Kleinberg R.L., 1986. 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Elastic piezosensitivity of porous and fractured rocks, Geophysics , 68( 2), 482– 486. https://doi.org/10.1190/1.1567215 Google Scholar CrossRef Search ADS   Si W.P., Di B.R., Wei J.X., Li Q., 2016. Experimental study of water saturation effect on acoustic velocity of sandstones, J. Nat. Gas Sci. Eng. , 33, 37– 43. https://doi.org/10.1016/j.jngse.2016.05.002 Google Scholar CrossRef Search ADS   Smith T.M., Sayers C.M., Sondergeld C.H., 2009. Rock properties in low-porosity/low-permeability sandstones, Leading Edge , 28( 1), 48– 59. https://doi.org/10.1190/1.3064146 Google Scholar CrossRef Search ADS   Toms J., Müller T.M., Ciz R., Gurevich B., 2006. Comparative review of theoretical models for elastic wave attenuation and dispersion in partially saturated rocks, Soil Dyn. Earthq. Eng. , 26( 6–7), 548– 565. https://doi.org/10.1016/j.soildyn.2006.01.008 Google Scholar CrossRef Search ADS   Verwer K., Eberli G., Baechle G., Weger R., 2010. Effect of carbonate pore structure on dynamic shear moduli, Geophysics , 75( 1), E1– E8. https://doi.org/10.1190/1.3280225 Google Scholar CrossRef Search ADS   Walsh J.B., 1965. The effect of cracks on the compressibility of rock, J. geophys. Res. , 70( 2), 381– 389. https://doi.org/10.1029/JZ070i002p00381 Google Scholar CrossRef Search ADS   White J.E., 1975. Computed seismic speeds and attenuation in rocks with partial gas saturation, Geophysics , 40( 2), 224– 232. https://doi.org/10.1190/1.1440520 Google Scholar CrossRef Search ADS   Yi D., Ning L., 2016. Dependencies of shear wave velocity and shear modulus of soil on saturation, J. Eng. Mech. , 142( 11), 1– 8. SUPPORTING INFORMATION Supplementary data are available at GJI online. Supplementary Material'+ÜGJI-S-17-0615.docx Appendices A and B are available in the Supporting Information of the publisher as online-only. Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the paper. APPENDIX C When the MJGW model is used, it is necessary to first estimate the compliant porosity, which cannot be directly measured. Two estimation methods are most commonly used. One is similar to that described by Shapiro (2003) and Pervukhina et al. (2010) to estimate the compliant porosity based on the pressure dependence of the dry sample elastic modulus. The other is to measure total porosity at different pressures. At high pressure, it is assumed that all the compliant pores have been completely closed. The red line in Fig. C1(a) shows the stiff porosity content obtained by fitting; the compliant porosity can be obtained by subtracting the stiff porosity from the total porosity. Compared with the previous method, this method requires more accurate porosity measurement, and the estimated compliant porosity is more accurate (Walsh 1965; Mavko & Jizba 1991; Pervukhina et al.2010). The porosity of sample S3 under different pressures was measured by a BenchLab 7000EX porosity measurement system. Fig. C1(b) shows the estimated compliant porosity of the sample. The proportion of compliant porosity to total porosity is very small, and the compliant pores are close to complete closure at 50 MPa. Figure C1. View largeDownload slide (a) Total porosity measured under different pressures, and stiff porosity obtained by fitting. (b) Estimated compliant porosity. Figure C1. View largeDownload slide (a) Total porosity measured under different pressures, and stiff porosity obtained by fitting. (b) Estimated compliant porosity. © The Author(s) 2017. Published by Oxford University Press on behalf of The Royal Astronomical Society. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Geophysical Journal International Oxford University Press

Experimental study and theoretical interpretation of saturation effect on ultrasonic velocity in tight sandstones under different pressure conditions

, Volume 212 (3) – Mar 1, 2018
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Abstract

Summary Understanding the influence of lithology, porosity, permeability, pore structure, fluid content and fluid distribution on the elastic wave properties of porous rocks is of great significance for seismic exploration. However, unlike conventional sandstones, the petrophysical characteristics of tight sandstones are more complex and less understood. To address this problem, we measured ultrasonic velocity in partially saturated tight sandstones under different effective pressures. A new model is proposed, combining the Mavko–Jizba–Gurevich relations and the White model. The proposed model can satisfactorily simulate and explain the saturation dependence and pressure dependence of velocity in tight sandstones. Under low effective pressure, the relationship of P-wave velocity to saturation is pre-dominantly attributed to local (pore scale) fluid flow and inhomogeneous pore-fluid distribution (large scale). At higher effective pressure, local fluid flow gradually decreases, and P-wave velocity gradually shifts from uniform saturation towards patchy saturation. We also find that shear modulus is more sensitive to saturation at low effective pressures. The new model includes wetting ratio, an adjustable parameter that is closely related to the relationship between shear modulus and saturation. High-pressure behavior, Acoustic properties, Wave propagation 1 INTRODUCTION A porous rock is described as partially saturated if the pore space contains two or more different fluids. The propagation of body waves (P- and S-waves) in partially saturated rock is complicated, which poses great challenges for seismic oil and gas exploration (Gist 1994; Muller et al.2010; Si et al.2016). Therefore, theoretical and experimental studies of elastic wave propagation in partially saturated rock have always been a focus of petrophysical research. Exploring the relationship between seismic response and fluid saturation can provide the basis for amplitude variation with offset analysis, fluid prediction and time-lapse seismic techniques. The relationship between velocity and saturation is mainly affected by lithology, the properties of each fluid component, the method of fluid saturation (imbibition and drainage), the fluid distribution pattern, frequency, pressure and temperature (Muller et al.2010; Ding et al.2014; Khalid & Ahmed 2016; Ding et al.2017). Gregory (1976) measured the ultrasonic velocity of partially saturated sedimentary rocks. The results showed that the petrophysical properties of low-porosity rocks are more sensitive than those of high-porosity rocks to fluid saturation (see also Kahraman 2007; Si et al.2016). Some experiments (Endres & Knight 1991; King et al.2000; Lebedev et al.2009) found that imbibition and drainage processes can cause saturation hysteresis, which means that velocities may not be consistent at the same saturation level. Murphy (1984) measured the velocities of tight sandstones in seismic, sonic, and ultrasonic frequency bands, and found significant deviation between the P-wave velocity predicted by the Biot–Gassmann (Gassmann 1951; Biot 1956a,b) theory and that measured in the sonic and ultrasonic frequency bands. However, most of the experiments described above employed atmospheric or constant effective pressure measurements. Due to the development of micro-cracks and the complex petrophysical characteristics of tight sandstones, changes in effective pressure affect the distribution pattern of pore fluid, velocity dispersion, and the velocity–saturation relationship. However, research on this aspect remains limited (Smith et al.2009; Mo et al.2015). In addition, due to the limited measurement accuracy and the insensitivity of S-wave velocity to fluid in conventional sandstones, research into the relationship between the S-wave velocity and fluid saturation of tight sandstones is often neglected. To quantitatively explain the relationship between velocity and saturation, many empirical formulas (Brie et al.1995; Karakul & Ulusay 2013) and theoretical models (Norris 1993; Carcione et al.2003; Toms et al.2006) have been presented. When wave frequencies are sufficiently low, the effective modulus of pore fluid in partially saturated rocks is calculated using Wood's law (Johnson 2001). Therefore, Gassmann's theory is termed the Gassmann–Wood model when applied to partially saturated rock. This model has been well demonstrated in studies by Murphy (1984) and Mavko & Mukerji (1998). At high frequencies, the propagation of waves can induce a local (pore scale) pore pressure gradient, under which the pore fluid is moving between pores of different scales, causing dispersion and attenuation of elastic waves. In such scenarios, the Gassmann–Wood model alone has often proven ineffective (Toms et al.2006; Muller et al.2010). Gist (1994) explained the relationship between ultrasonic velocity and saturation in a wide range of rocks by combining the gas pocket model (White 1975) and the local flow theory. Mavko & Nolen-Hoeksema (1994) proposed another model based on the Mavko–Jizba (MJ) relations (Mavko & Jizba 1991). They considered that ultrasonic velocities were controlled by saturation heterogeneity at two scales: (1) differences between large stiff pores and thin soft pores, and (2) differences between saturated patches and partially saturated patches at a scale much larger than any pore. The MJ model is based on the idea that the pore space has binary structure (contains stiff and compliant pores). This model adequately describes the pressure dependence of rock velocity (Chapman et al.2002). The expressions of the MJ model are satisfied for typical liquids and most rocks, but might not be valid for gas saturated rocks (Gurevich et al.2009). Gurevich et al. (2009, 2010) derived a more general MJ model, which we term the Mavko–Jizba–Gurevich (MJG) model. This improvement eliminates the restriction of pore fluid modulus, possibly enabling the MJG model to be used for partially saturated rocks. Therefore, our aim is to extend the MJG model to explore the pressure dependence of velocity in partially saturated tight sandstones. In this paper, the theory of elastic wave propagation in partially saturated rocks is briefly reviewed, and a new combined model is proposed based on the MJG model and the White model. Further, we describe the experimental process in detail. The validity of the new model is verified using experimental data, and the dependence of the elastic properties of tight sandstones on saturation and effective pressure is explained and summarized. 2 THEORY 2.1 Theoretical background for partially saturated porous media A considerable amount of research (Mavko & Mukerji 1998; King et al.2000; Si et al.2016) has used Gassmann's theory (Appendix A in the Supporting Information) to estimate wave velocities in partially saturated rocks. These studies show that for partially saturated rock, the pore pressure induced by each fluid is different due to the differing in bulk modulus of each pore fluid component. With uniform mixing of the pore fluid components, the mixed fluids can be homogeneously distributed in pore space (called uniform saturation). In another case, the fluid is inhomogeneously distributed in the pores of the rock. Some regions (patches), on a scale containing many pores, are partially saturated, while other regions containing many pores are entirely saturated, which is termed patchy saturation (Mavko & Nolen-Hoeksema 1994). White (1975) proposed a model that can describe the variation of velocity and attenuation with frequency, and rock porosity, permeability, and saturation. The White model can describe both uniform and patchy saturation patterns, and also their transition trends, by adjusting the gas pocket radius (r). Dutta & Ode (1979) developed a theoretical derivation of a more rigorous elastic wave velocity and attenuation of the White model according to Biot's theory. This modified White model is described in Appendix B in the Supporting Information. We use Travis Peak tight gas sandstone (Mavko & Jizba 1991) to simulate different saturation patterns and the effect of the gas pocket radius (parameter r) on the saturation dependence of the P-wave velocity. Table 1 gives the main parameters used in the calculation. As shown in Fig. 1, when the radius r is <0.01 mm, the P-wave velocity trend predicted by the White model is similar to that for uniform saturation (lower bound). The turning point of the trend gradually moves to low saturation as the radius of the gas pocket is increased. When r is sufficiently large, the White model's predictions are consistent with the values for patchy saturation (upper bound). In addition, the White model is similar to the Gassmann–Wood model in predicting the S-wave velocity of partially saturated rocks, and the two models predict that the shear modulus is independent of saturation. Figure 1. View largeDownload slide Simulation of P-wave velocity as a function of saturation in a sample of Travis Peak tight gas sandstone, using the White model. Uniform (solid blue) and patchy (solid red) saturation patterns obtained by the Gassmann model. Figure 1. View largeDownload slide Simulation of P-wave velocity as a function of saturation in a sample of Travis Peak tight gas sandstone, using the White model. Uniform (solid blue) and patchy (solid red) saturation patterns obtained by the Gassmann model. Table 1. Material properties of Travis Peak tight gas sandstone (TGS) and fluid (Mavko & Jizba 1991; Mavko et al.2009). Material  Bulk modulus (GPa)  Shear modulus (GPa)  Density (g cm−3)  Viscosity (η/Pa s)  Travis Peak TGS  38  35  2.63  –  Gas  0.0022  0  0.0012  0.00011  Water  2.25  0  1  0.001  Material  Bulk modulus (GPa)  Shear modulus (GPa)  Density (g cm−3)  Viscosity (η/Pa s)  Travis Peak TGS  38  35  2.63  –  Gas  0.0022  0  0.0012  0.00011  Water  2.25  0  1  0.001  View Large 2.2 Modelling squirt flow dispersion At ultrasonic frequencies, elastic waves in saturated rocks display velocity dispersion and attenuation under the action mechanism of wave-induced fluid flow. Dispersion is the variation of velocity with frequency (Muller et al.2010). Dispersion can be caused by wave induced fluid flow, which can take place at the macroscale, mesoscale or microscale. Many studies (Gregory 1976; Gist 1994; Mavko & Jizba 1994; Bhuiyan & Holt 2016) have reported that the dispersion of tight sandstones is mainly caused by mesoscopic and/or microscopic fluid flow. Therefore, to quantitatively describe ultrasonic velocity as a function of saturation and effective pressure in tight sandstones, we combine the MJG model describing the microscopic fluid flow, and the White model describing the mesoscopic fluid patches into a new model that we call the MJGW model. We first estimate the microscopic fluid flow effects in partially saturated rock based on the MJG model. The elastic modulus of the partially saturated frame with water in the compliant pores only is defined as an ‘unrelaxed wetted frame’. Compared with the original MJ model, the MJG model is not limited to liquid saturation, and can therefore be used to simulate partial saturation. The modified White model is then used to describe the saturation of the remaining pore space. To overcome the limitations of the original MJ model for fluid bulk modulus, Gurevich et al. (2009,2010) derive a more general MJ model according to the effective pressure relaxation approach (Murphy et al.1986) and the Sayers–Kachanov discontinuity formalism (Sayers & Kachanov 1995), which we term the MJG relations. Detailed derivation refers to Gurevich et al. (2009, 2010), and the final result is as follows:   $$\frac{1}{{{K_{{\rm{mf}}}}({P,\omega })}} = \frac{1}{{{K_h}}} + \frac{1}{{\frac{1}{{\frac{1}{{{K_{{\rm{dry}}}}\left( P \right)}} - \frac{1}{{{K_h}}}}} + \frac{{3i\omega \eta }}{{8{\phi _c}\left( P \right){\alpha ^2}}}}}$$ (1)  $$\frac{1}{{{\mu _{{\rm{mf}}}}\left( {P,\omega } \right)}} = \frac{1}{{{\mu _{{\rm{dry}}}}\left( P \right)}} - \frac{4}{{15}}\left( {\frac{1}{{{K_{{\rm{dry}}}}\left( P \right)}} - \frac{1}{{{K_{{\rm{mf}}}}\left( {P,\omega } \right)}}} \right),$$ (2)where Kmf(P, ω) is bulk modulus of the modified frame with frequency dependence and pressure dependence, which can represent a relaxed or unrelaxed state. μmf(P, ω) is the corresponding shear modulus of the modified frame. Kh is the bulk modulus of dry rock under high effective pressure under which it is generally assumed that the compliant pores of the rock have been completely closed. Kdry(P) and μdry(P) are the pressure-dependent bulk and shear modulus of the dry rock, respectively. η is the viscosity of the fluid, ϕc(P) is compliant porosity, α represents the aspect ratio of compliant pores. The characteristic frequency fc of the local flow dispersion can usually be written as   $${f_c} = \alpha _m^3\frac{K}{\eta },$$ (3)where K represents the bulk modulus of rock and αm represents the mean aspect ratio of pores. For stiff pores (αm≈1), if the rock is saturated with water, fc is much higher than 1 MHz. For compliant pores (αm ≪ 1), if the rock is saturated with water, fc may be less than 1 MHz. Therefore, the main mechanism of pore-scale dispersion is wave induced fluid flow between compliant and stiff pores. According to eqs (1) and (2), frequency- and pressure-dependent, unrelaxed wetted (partially saturated) frame bulk modulus Kps(P, ω) and shear modulus μps(P, ω) can be written as follows:   $$\frac{1}{{{K_{{\rm{ps}}}}\left( {P,\omega } \right)}} = \frac{1}{{{{\left( {{K_{{\rm{dry}}}}} \right)}_{{P_{{\rm{ps}}}}}}}} + \frac{1}{{\frac{1}{{\frac{1}{{{K_{{\rm{dry}}}}\left( P \right)}} - \frac{1}{{{{\left( {{K_{{\rm{dry}}}}} \right)}_{{P_{{\rm{ps}}}}}}}}}} + \frac{{3i\omega \eta {S_c}}}{{8{\phi _c}\left( P \right){\alpha ^2}}}}}$$ (4)  $$\frac{1}{{{\mu _{{\rm{ps}}}}\left( {P,\omega } \right)}}= \frac{1}{{{\mu _{{\rm{dry}}}}\left( P \right)}} - \frac{4}{{15}}\left( {\frac{1}{{{K_{{\rm{dry}}}}\left( P \right)}} - \frac{1}{{{K_{{\rm{ps}}}}\left( {P,\omega } \right)}}} \right)$$ (5)where $${( {{K_{{\rm{dry}}}}} )_{{P_{{\rm{ps}}}}}}$$represents the measured dry bulk modulus at the equivalent closing effective pressure, Pps (this parameter can be determined by reference to Mavko & Nolen-Hoeksema (1994)), and μdry(P) is shear modulus of the dry rock. Eq. (4) is compared with eq. (1), and a new variable Sc is added to represent the saturation of the liquid in compliant pores. According to eqs (1) and (3), we know that the viscous resistance of the liquid in the compliant pores is the main cause of the unrelaxed state of the rock framework. Therefore, the parameter Sc can be used to describe the unrelaxed frame modulus of partially saturated rock. The relationship between Scand Sw (the water saturation of rock) can be written as   $${S_c} = \rm{WR} \times {S_w}\frac{\phi }{{{\phi _c}}},$$ (6)where ϕ and ϕc are the total porosity and compliant porosity of the rock, respectively (Mavko & Nolen-Hoeksema 1994). Most liquid entering the rock will fill the stiff pores, and a small proportion enters the compliant pores, which is defined as the wetting ratio (WR). As seen in eq. (6), WR is related to the saturation of rock, and determines the saturation of the compliant pores. In particular, when Sc = 0, there is no liquid in the compliant pores, and the right-hand side of eq. (4) reduces to the dry modulus Kdry(P), as expected. When Sc = 1, the modulus of the unrelaxed wetted frame reaches its maximum. 2.3 Modelling saturation dependence To investigate how the MJGW model describes the effect of water saturation on ultrasonic velocities, we simulated Travis Peak tight gas sandstone (Mavko & Jizba 1991). According to Gurevich et al. (2010), the α (aspect ratio of compliant pores) of this sandstone is 0.01. The compliant porosity is estimated by the method described by Shapiro (2003) and Pervukhina et al. (2010). The porosity of this tight sandstone is 8.0 per cent, assuming the permeability is 0.09 mD, and the other parameters are shown in Table 1. For the MJGW model, the two most important parameters are the wetting ratio and the gas pocket radius. Taking Fig. 2(a) as an example, the wetting ratio is 0.44 per cent, and when the gas pocket radius increases from 0.01 mm to 0.3 mm, the saturation pattern of the P-wave velocity changes from uniform saturation to patchy saturation. Comparing Figs 2(a)–(c) indicates that when the gas pocket radius is the same, the rock shows local fluid flow at lower saturation levels as the wetting ratio increases from 0.44 per cent to 3 per cent. Therefore, the wetting ratio determines the effect of local fluid flow on P-wave velocity at different saturation levels, and the gas pocket radius determines the saturation pattern. Figure 2. View largeDownload slide Simulation of P-wave velocity as a function of saturation in a sample of Travis Peak tight gas sandstone, using the MJGW and Gassmann models. Wetting ratios (a) 0.44 per cent, (b) 0.8 per cent, (c) 3 per cent. Figure 2. View largeDownload slide Simulation of P-wave velocity as a function of saturation in a sample of Travis Peak tight gas sandstone, using the MJGW and Gassmann models. Wetting ratios (a) 0.44 per cent, (b) 0.8 per cent, (c) 3 per cent. As seen from Fig. 3(a), the shear modulus simulated by the MJGW model is dependent on saturation, and the wetting ratio controls the shear modulus versus saturation relationships. The White model considers that the gas pocket radius has no effect on the shear modulus, and thus, the dependence of the shear modulus on saturation is only caused by the local fluid flow. According to eq. (4), the compliant pores contain fluid that causes stiffening of the framework, and the shear modulus also produces a dispersion in the relationship of eq. (5), so that the dependency of shear modulus on saturation is closely related to the wetting ratio. Similarly, the MJGW model can also describe the dependence of S-wave velocity on saturation. When the wetting ratio is low, the S-wave velocity first decreases, and then increases, with increase in saturation; conversely, when the wetting ratio is high, the S-wave velocity first increases, and then decreases, with increase in saturation (Fig. 3b). Figs 2 and 3 show that the P- and S-wave velocities of the fully saturated sandstone predicted by the Gassmann model are lower than those from the MJGW model because the former does not take into account local fluid flow. Figure 3. View largeDownload slide Simulation of (a) shear modulus and (b) S-wave velocity as a function of saturation in a sample of Travis Peak tight gas sandstone, using the MJGW and Gassmann models. Figure 3. View largeDownload slide Simulation of (a) shear modulus and (b) S-wave velocity as a function of saturation in a sample of Travis Peak tight gas sandstone, using the MJGW and Gassmann models. 3 EXPERIMENTAL STUDY 3.1 Experimental procedures We selected three samples from a tight gas reservoir in northwest China, and cut them into standard core columns of diameter approximately 1 inch and length of approximately 2 inches. We carefully polished the ends of the sample to reduce errors in the length and velocity measurement. The samples were then vacuum-oven-dried at 70 °C for more than 4 d. A large number of studies (King et al.2000; Mavko et al.2009) showed that a small amount of moisture may cause significant changes in the rock matrix. We therefore placed the dried samples in a room under constant environmental conditions (humidity 45–55 per cent, temperature 23–25 °C) for several hours, thereby obtaining dry samples with insignificant moisture content. The porosity and permeability (Klinkenberg-corrected) of the samples were measured by a porosity and permeability testing system (BenchLab 7000EX). The precision of porosity measurement is within ± 0.03 porosity units, and the error of permeability is approximately 8 per cent (American Petroleum Institute (API) 1998). X-ray diffraction (XRD) showed that the mineral compositions of the three samples were similar, with quartz and feldspar, accounting for about 89 per cent. The samples had permeability <1 mD and porosity <10 per cent (see Table 2), which reflects the low-porosity/low-permeability petrophysical characteristics of the tight sandstones. The selected samples were from the same reservoir, and analyses of cast thin sections and SEM images showed they had similar pore structures. As seen in Fig. 4(a), the pore structure of sample S3 is complex with poor connectivity; the pore type is mainly intergranular dissolved pores, and includes a small number of intragranular pores. Fig. 4(b) shows the existence of microcracks and compliant pores in sample S3, and the pore aspect ratios of these cracks and pores are very small. The presence of these pores may cause changes in rock velocity, and if these pores contain fluids, the velocity changes may be more complex (Smith et al.2009). Figure 4. View largeDownload slide (a) Thin section image of sample S3; (b) SEM image of sample S3. Figure 4. View largeDownload slide (a) Thin section image of sample S3; (b) SEM image of sample S3. Table 2. Measured parameters of the core samples (porosity and permeability were measured at 2 MPa effective pressure). Sample  Porosity (per cent)  Permeability (mD)  Density (g cm−3)  Clay (per cent)  Quartz (per cent)  Feldspar (per cent)  Calcite (per cent)  Muscovite (per cent)  S1  6.48  0.023  2.50  5.7  58.8  30.5  1.8  3.2  S2  6.71  0.069  2.47  5.5  68.6  21.4  2.1  2.4  S3  7.22  0.131  2.41  4.7  65.2  24.3  3.0  2.8  Sample  Porosity (per cent)  Permeability (mD)  Density (g cm−3)  Clay (per cent)  Quartz (per cent)  Feldspar (per cent)  Calcite (per cent)  Muscovite (per cent)  S1  6.48  0.023  2.50  5.7  58.8  30.5  1.8  3.2  S2  6.71  0.069  2.47  5.5  68.6  21.4  2.1  2.4  S3  7.22  0.131  2.41  4.7  65.2  24.3  3.0  2.8  View Large The P- and S-wave velocities of the three tight sandstones were measured at different saturation conditions, from dry to fully saturated with fluid, at the temperature of approximately 20 °C. For each saturation, the ultrasonic velocities at eight different confining pressures (2, 5, 10, 20, 30, 40, 50, and 60 MPa) were measured. In the experimental measurements, we consider that pore pressure in partially saturated tight sandstones can be neglected, so that the effective pressure is equal to the confining pressure applied to the sample (Mayr & Burkhardt 2006; Mikhaltsevitch et al.2016). The velocity was measured three times directly after the effective pressure reached the set value, and another three times 10 min later (Mayr & Burkhardt 2006). The experiment showed that the system (rock and fluid) was able to reach a steady state after 10 min. The average of all six results measured at a given effective pressure is used as the final measured value. Ultrasonic velocities were measured by the BenchLab 7000EX acoustic measurement system, and the centre frequency of the transducers was 0.5 MHz. The system has very good velocity measurement performance. The P- and S-wave waveforms of sample S3 are given in Figs 5(a) and (b), respectively, where the saturation of the sample is 24 per cent. The shapes of the waveforms are similar at different effective pressures. As the effective pressure increases, the amplitude of the wave gradually increases and the first arrival time gradually decreases. These high-quality waveforms ensure the accuracy of the first arrival pick-up. We pick up the first arrival of the shear wave according to the method discussed by Yi & Ning (2016). According to the method of Hornby (1998), we estimate that the errors for P- and S-wave velocities are approximately 1 per cent, and the errors for bulk modulus and shear modulus are less than 2 per cent. Based on the uncertainty analysis procedure described by Adam et al. (2006), the standard deviations are calculated and given in the form of error bars in the figures. For most figures, the length of the error bars is smaller than the size of the symbols so that they cannot be seen, which indicates small errors. Figure 5. View largeDownload slide Ultrasonic measurement waveform of sample S3: (a) P-wave signals received at different effective pressures; (b) S-wave signals received at different effective pressures. The red arrow represents the first arrival position. Figure 5. View largeDownload slide Ultrasonic measurement waveform of sample S3: (a) P-wave signals received at different effective pressures; (b) S-wave signals received at different effective pressures. The red arrow represents the first arrival position. To obtain different saturations, the sample was first placed in a container filled with distilled water. By this self-absorption method, the saturation of the three tight sandstones can reach 50–70 per cent. The sample was weighed at given time intervals to obtain different saturations. To achieve higher saturation, the samples were placed in a vacuum vessel containing distilled water and evacuated for 2 d, thereby achieving water saturation of up to 80 per cent. We then used the vacuum saturation equipment to pressurize the samples for more than one week at 30 MPa. In this way the samples can approach total fluid saturation, but because the pore structure of the samples is complex and some pores are almost disconnected, there is no guarantee that the samples can reach 100 per cent fluid saturation (Murphy 1984; Verwer et al.2010). To allow the water to have sufficient time to be distributed in the sample, after the desired saturation was established, the sample was wrapped in aluminium foil and placed under vacuum for 24 hr prior to conducting velocity measurements (Mayr & Burkhardt 2006). 3.2 Experimental results We compared the measurements, and found that the three tight sandstones showed similar velocity variation from dry to fully saturated with water. Fig. 6 shows the variation trend of the Vp (P-wave velocity) of the three samples as a function of saturation and effective pressure. At low effective pressure, the Vp of the samples increases or decreases slowly with increase in water saturation, then starts to increase rapidly when a certain saturation range (50–70 per cent) is reached. At high effective pressure, the Vp shows an approximately linear trend with the increase in saturation. In addition to the change of the trends, the difference between the saturated and dry P-wave velocities, ▵Vp, also changes: with increasing effective pressure, ▵Vp gradually decreased in all three samples, but is always positive. Figure 6. View largeDownload slide Vp of samples measured at different pressure and saturation: (a) S1; (b) S2; (c) S3. Figure 6. View largeDownload slide Vp of samples measured at different pressure and saturation: (a) S1; (b) S2; (c) S3. Fig. 7 shows the Vs (S-wave velocity) of three samples as functions of saturation and effective pressure. At low effective pressure, Vs hardly changes with increasing saturation, but increases gradually as saturation exceeds a certain range. At high effective pressure, Vs exhibits a decreasing trend. The ▵Vs (saturated and dry S-wave velocity difference) of the three samples showed similar changes: As the effective pressure increases, ▵Vs gradually changes from positive to negative, which is explained in detail in the following section. Since the samples were from the same reservoir with similar physical properties and pore characteristics, we selected sample S3 as being representative for further analysis below. Figure 7. View largeDownload slide Vs of samples measured at different pressure and saturation: (a) S1; (b) S2; (c) S3. Figure 7. View largeDownload slide Vs of samples measured at different pressure and saturation: (a) S1; (b) S2; (c) S3. 4 THEORETICAL INTERPRETATION To explain the experimental results, we used the MJGW model to predict the trend of the velocities of sample S3 at saturation for 2, 30, and 60 MPa representing low-, medium-, and high-pressure conditions, respectively. The prediction process is as follows. First, the compliant porosity is estimated by the method described in Appendix  C. Then, the velocities in rock (measured dry and fully saturated with fluid) are used as input parameters, and the aspect ratio of the compliant pores α = 0.01 is obtained using eq. (1). The results (see Fig. 3) show that the trend of Vs with saturation is only related to the wetting ratio and is independent of the gas pocket radius. Therefore, the Vs with saturation is described by adjusting the wetting ratio. Finally, after determining the wetting ratio, the variation of Vp with saturation can be predicted simply by adjusting the gas pocket radius. The parameters involved in the calculation are shown in Tables 1–3. Table 3. The main parameters used in the MJGW model and permeability measurements of sample S3. For 2 MPa effective pressure, the wetting ratios differ at low- and high-saturation ranges. Pressure (MPa)  Permeability (mD)  Gas pocket radius (mm)  Saturation (per cent)  Wetting ratio (per cent)  2  0.131  0.03  0–75  1.5        75–100  5  30  0.043  0.04  0–55  1.5  60  0.012  0.2  0–100  0  Pressure (MPa)  Permeability (mD)  Gas pocket radius (mm)  Saturation (per cent)  Wetting ratio (per cent)  2  0.131  0.03  0–75  1.5        75–100  5  30  0.043  0.04  0–55  1.5  60  0.012  0.2  0–100  0  View Large 4.1 Effect of saturation on P-wave velocity under different effective pressures At 2 MPa (Fig. 8a), for saturation > 40 per cent, the saturation dependence of Vp shows a large deviation from that predicted by the Gassmann–Wood (solid green line) and White (dotted purple line) models. However, Vp predicted by the MJGW model (dashed red line) shows good agreement with the measured values. The results of the White model show patchy saturation effect at saturation > 70 per cent. Therefore, by comparing the results of the White and MJGW models, we attribute the slow increase of Vp within the saturation range 40–70 per cent mainly to local fluid flow. When saturation exceeds 70 per cent, the rapid increase of Vp is mainly attributed to the combined action of patchy saturation effect and local fluid flow. Similarly, at 30 MPa (Fig. 8b) with saturation of 30–60 per cent, Vp is mainly affected by local flow, and when saturation exceeds 60 per cent, Vp is influenced by both these mechanisms. Figure 8. View largeDownload slide Experimental measurements of P-wave velocity versus values predicted by various models for sample S3 at effective pressures of (a) 2 MPa, (b) 30 MPa and (c) 60 MPa (the MJGW and White models’ predictions are consistent). Figure 8. View largeDownload slide Experimental measurements of P-wave velocity versus values predicted by various models for sample S3 at effective pressures of (a) 2 MPa, (b) 30 MPa and (c) 60 MPa (the MJGW and White models’ predictions are consistent). At 60 MPa (Fig. 8c), the compliant pores in sample S3 are completely closed, and the sample is no longer affected by local fluid flow. In this case, the MJGW predictions match those of the White model. As shown in Fig. 8(c), due to the heterogeneous distribution of the mixed fluid in the pore space, the Vp trend is close to the patchy saturation pattern. In addition, Fig. 8 also shows that with increasing effective pressure the Vp trend gradually changes from uniform to patchy saturation pattern. This change in saturation pattern can be described by the diffusion length L (Mavko & Mukerji 1998), and the critical diffusion length Lc is expressed as follows:   $${L_c} = \sqrt {{{k{K_{{\rm{fl}}}}} / {\eta f}}} ,$$ (7)where k is the permeability of the rock, Kfl is the bulk modulus of the fluid with the highest viscosity in the pore fluid, η is the characteristic viscosity of the fluid, and f is the frequency. When L < Lc, the pore fluid pressure has sufficient time to reach equilibrium through the fluid flow; in this case, the saturation pattern is uniform. When L > Lc, the pore fluid pressure cannot reach equilibrium in a short time, and the fluid shows patchy saturation pattern. The diffusion length L can be expressed by the gas pocket radius r and the water saturation Sw (Gist 1994):   $$L = r\left( {{{\left( {1 - {S_w}} \right)}^{ - 1/3}} - 1} \right).$$ (8) As can be seen from eq. (8), L gradually increases with increasing water saturation. At 2 MPa, Lc≈0.021 mm can be calculated from the parameters in Tables 2 and 3, and when water saturation exceeds 70 per cent, L is greater than 0.021 mm, at which point the Vp trend changes from uniform to patchy saturation pattern. The permeability of the sample gradually declines as effective pressure increases, resulting in a decrease in Lc. The decrease of Lc indicates that the diffusion length L can reach the critical diffusion length Lc at low saturation. Therefore, for 60 MPa effective pressure, the trend of Vp changes from uniform saturation to patchy saturation when the saturation exceeds 20 per cent. Comparing the measured values and theoretical predictions confirms that our proposed MJGW model can describe the dependence of the tight sandstone Vpon saturation and effective pressure conditions. Under low effective pressure, the relationship between Vp and saturation is mainly affected by the local (pore scale) fluid flow and the inhomogeneous distribution of pore fluid (large scale). As the effective pressure increases, the local fluid flow gradually decreases, and the Vp trend gradually shifts away from uniform saturation and approaches patchy saturation. 4.2 Effect of fluid saturation on S-wave velocity under different effective pressures In Fig. 9, the Vs predicted by the Gassmann model (solid green line) at 2 and 30 MPa deviates considerably from the measured values. However, the MJGW model's predictions (dashed red line) at three effective pressures are in good agreement with the measured values. According to the Gassmann model, the shear modulus will remain constant during saturation. Assuming that this conclusion is established, then according to eq. (A.7) in the Supporting Information, it can be concluded that Vs should gradually reduce as saturation increases. As shown in Fig. 9(c), Vs gradually decreased with increasing saturation at 60 MPa pressure. However, at low effective pressure, we found that the sample showed increasing Vs trend with increase in saturation (Figs 7 and 9a–b). As seen from eq. (A.7) in the Supporting Information, the density of partially saturated rocks can be measured accurately; therefore, it is inferred that the increasing trend of Vs in tight sandstones is likely related to the change of shear modulus. Figure 9. View largeDownload slide Experimental measurements of S-wave velocity versus values predicted by different models for sample S3: (a) 2 MPa; (b) 30 MPa; (c) 60 MPa. Figure 9. View largeDownload slide Experimental measurements of S-wave velocity versus values predicted by different models for sample S3: (a) 2 MPa; (b) 30 MPa; (c) 60 MPa. Khazanehdari & Sothcott (2003) define the increasing/decreasing shear modulus of rock saturated with fluid as shear strengthening/weakening respectively. Some more recent studies show that the fluid types and their interactions with the matrix, effective pressure, frequency, dispersion and pore structure are the most important factors affecting the shear modulus strengthening/weakening (e.g. Adam et al.2006; Lebedev et al.2014; Bhuiyan & Holt 2016; Li et al.2017). However, very few studies have examined the mechanism of shear modulus changes in partially saturated tight sandstones. To explore this problem, we used the MJGW model to predict the shear modulus of partially saturated sample S3 at different effective pressures. As shown in Fig. 10, the MJGW model (dashed red line) can satisfactorily describe the sample's variation in shear modulus with saturation at these three effective pressures. According to eqs (1) and (3), the wetting ratio determines the fluid content in the compliant pores, which in turn influences the unrelaxed effect of the fluid in the compliant pores. Therefore, the wetting ratio also determines the change of shear modulus. Figure 10. View largeDownload slide Experimental measurements of shear modulus versus values predicted by different models for sample S3: (a) 2 MPa; (b) 30 MPa; (c) 60 MPa. Figure 10. View largeDownload slide Experimental measurements of shear modulus versus values predicted by different models for sample S3: (a) 2 MPa; (b) 30 MPa; (c) 60 MPa. As seen from Table 3, the average wetting ratio is obtained by the best fit of the predictions to the measured values. At 2 MPa pressure, the average wetting ratio of sample S3 within the saturation range of 0–75 per cent is 1.5 per cent. The low wetting ratio indicates that only a comparatively small proportion of fluid enters the soft pores; consequently, the change in shear modulus is not obvious at saturation <40 per cent (Fig. 10a). With increasing saturation, there is gradual increase in fluid accumulated within the soft pores, and local fluid flow becomes stronger, such that the shear modulus increases as saturation increases. When saturation exceeds 75 per cent, the average wetting ratio is increased to 5 per cent, and thus, the slope of the change in shear modulus is slightly increased. The change of wetting ratio within different saturation ranges is related to the saturation process of the sample. At low saturation, the sample is self-absorbent to the corresponding saturation without applying effective pressure. Therefore, the proportion of the fluid entering the soft pores is very small. The high saturation levels were obtained by placing the sample in a vacuum-pressurized saturated apparatus. Under high effective pressure, the water is forced to enter the sample, thereby achieving a higher wetting ratio compared with low-saturation conditions. At effective pressure of 30 MPa, when saturation exceeds 55 per cent, the wetting ratio is zero, indicating that the compliant pores have been fully filled with fluid. In this case, the unrelaxed effect has reached its maximum, and the shear modulus does not change as saturation continues to increase (Fig. 10b). The compliant pores in the sample are completely closed at 60 MPa; the MJGW and Gassmann models agree that the fluid content has no effect on shear modulus; and the predicted results are in agreement with the measured values (Fig. 10c). Our experiment showed shear strengthening in the saturated sample, whereas the Gassmann model considers that shear modulus will remain constant during saturation, thereby underestimating the S-wave velocity of the sample. This is the main cause of the deviation between Gassmann predictions and the measured values in Figs 9(a) and (b). The MJGW model can describe the change in shear modulus with saturation, which confirms that local fluid flow is the main mechanism leading to shear strengthening in the tight sandstones. Some studies have found that rocks with low permeability and low fluidity, especially carbonate and tight sandstones, can cause dispersion even at seismic frequencies (Adam et al.2006; Batzle et al.2006; Lebedev et al.2014). Therefore, it is necessary to study the relationship between shear modulus and fluid saturation in tight sandstones at low frequency. This relationship has important implications for fluid identification and other seismic techniques. 5 CONCLUSIONS We propose a new combined model based on the MJG relations and the White model. The new model not only includes microscale and mesoscale fluid flows, but also considers the fluid distribution pattern. Comparison with experimental data demonstrates that the proposed model can describe and explain how ultrasonic velocity in tight sandstones is dependent on saturation and effective pressure conditions. Under low effective pressure, the saturation dependence of P-wave velocity is mainly affected by local (pore scale) fluid flow and the inhomogeneous distribution of pore fluid (large scale). At higher effective pressure, the local flow gradually decreases, and P-wave velocity gradually trends from uniform toward patchy saturation. An interesting finding is that shear modulus is also saturation- and pressure-dependent, and is more sensitive to fluid saturation at lower effective pressure. An adjustable parameter, wetting ratio, in the new model is closely related to the change in shear modulus. This finding suggests that S-wave velocity and shear modulus, which are considered insensitive to the fluid in the conventional reservoir, are likely to play important roles in fluid identification of tight sandstones reservoirs. Mechanisms of pore-scale and mesoscopic fluid-flow are usually considered to be important at ultrasonic frequencies but may also play a role at seismic and logging frequencies. We have only studied the ultrasonic range, and thus, it is necessary to further investigate these fluid flow mechanisms at lower frequencies. Acknowledgements We appreciate the editors and referees for their thoughtful comments. We are very grateful to Prof. Boris Gurevich from Curtin University for his valuable comments on this study. We also thank Prof. Tongcheng Han from China University of Petroleum for the helpful suggestions. 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Effect of carbonate pore structure on dynamic shear moduli, Geophysics , 75( 1), E1– E8. https://doi.org/10.1190/1.3280225 Google Scholar CrossRef Search ADS   Walsh J.B., 1965. The effect of cracks on the compressibility of rock, J. geophys. Res. , 70( 2), 381– 389. https://doi.org/10.1029/JZ070i002p00381 Google Scholar CrossRef Search ADS   White J.E., 1975. Computed seismic speeds and attenuation in rocks with partial gas saturation, Geophysics , 40( 2), 224– 232. https://doi.org/10.1190/1.1440520 Google Scholar CrossRef Search ADS   Yi D., Ning L., 2016. Dependencies of shear wave velocity and shear modulus of soil on saturation, J. Eng. Mech. , 142( 11), 1– 8. SUPPORTING INFORMATION Supplementary data are available at GJI online. Supplementary Material'+ÜGJI-S-17-0615.docx Appendices A and B are available in the Supporting Information of the publisher as online-only. Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the paper. APPENDIX C When the MJGW model is used, it is necessary to first estimate the compliant porosity, which cannot be directly measured. Two estimation methods are most commonly used. One is similar to that described by Shapiro (2003) and Pervukhina et al. (2010) to estimate the compliant porosity based on the pressure dependence of the dry sample elastic modulus. The other is to measure total porosity at different pressures. At high pressure, it is assumed that all the compliant pores have been completely closed. The red line in Fig. C1(a) shows the stiff porosity content obtained by fitting; the compliant porosity can be obtained by subtracting the stiff porosity from the total porosity. Compared with the previous method, this method requires more accurate porosity measurement, and the estimated compliant porosity is more accurate (Walsh 1965; Mavko & Jizba 1991; Pervukhina et al.2010). The porosity of sample S3 under different pressures was measured by a BenchLab 7000EX porosity measurement system. Fig. C1(b) shows the estimated compliant porosity of the sample. The proportion of compliant porosity to total porosity is very small, and the compliant pores are close to complete closure at 50 MPa. Figure C1. View largeDownload slide (a) Total porosity measured under different pressures, and stiff porosity obtained by fitting. (b) Estimated compliant porosity. Figure C1. View largeDownload slide (a) Total porosity measured under different pressures, and stiff porosity obtained by fitting. (b) Estimated compliant porosity. © The Author(s) 2017. Published by Oxford University Press on behalf of The Royal Astronomical Society.

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Geophysical Journal InternationalOxford University Press

Published: Mar 1, 2018

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