Abstract
The static bulk modulus of unconsolidated sands is essential for predicting the in situ effective pressure to reduce drilling risks in deepwater reservoirs; however, the dynamic bulk modulus is often more broadly available from seismic or well logging data. Therefore, it is tempting to investigate the relationship between static and dynamic bulk moduli. We perform a series of ultrasonic velocity measurements on 21 deepwater reservoir sands from the Gulf of Mexico (GoM) to study the static and dynamic bulk moduli simultaneously. Both room-dry and brine-saturated ultrasonic velocities are measured under hydrostatic stress conditions to derive the dynamic bulk moduli. Under brine-saturated conditions, if the pore pressure is kept constant, the pore volume change with the confining pressure can be monitored accurately by a digital pump, which is subsequently used to estimate the static bulk modulus. The experimental results suggest that both the static and dynamic bulk moduli decrease upon pressure unloading. The pressure-dependent bulk moduli are modeled using the Hertz-Mindlin contact theory at the critical porosity and combined with the modified Hashin-Shtrikman lower bound for other porosities. The results suggest that the theoretical estimates can serve as the lower bound for the dynamic bulk modulus and the upper bound for the static bulk modulus. Under room-dry conditions, the static-to-dynamic modulus ratio decreases from a value approaching 0.8 to approximately 0.25 with decreasing differential pressure. Moreover, the effects of brine saturation on the relationship between the static and dynamic moduli are investigated using Gassmann’s equation. The brine saturation substantially reduces the difference between the static and dynamic bulk moduli, making the static-to-dynamic modulus ratio approach unity, which may be relevant to the in situ reservoir rock properties.
1. Introduction
Deepwater sand reservoirs in the Gulf of Mexico (GoM), which comprise turbidites associated with early hydrocarbon charging, are major areas of focus for U.S. oil and gas companies [1, 2]. Deepwater turbidites near the seafloor consisting of unconsolidated loose sands are typically overpressured [3]. Drilling such reservoirs may cause long-term shallow water flow, sand production, formation compaction due to depletion, and wellbore instability or casing damage [4–6]. To reduce potential drilling hazards, it is of great significance to investigate the elastic and mechanical properties of deepwater unconsolidated sands.
Bulk modulus is one of the mechanical properties used in numerous geoengineering issues [7–10]. The bulk modulus of rocks could be determined by either the stress-strain relation (i.e., static measurement) or the propagating elastic wave velocities (i.e., dynamic measurement) [11]. The static bulk modulus characterizes the mechanical stability of subsurface reservoirs under most in situ loading conditions because rocks deform statically underground [12, 13]. However, the static moduli of subsurface rocks cannot be directly measured in situ. Therefore, the relationship between static and dynamic moduli is vital because the dynamic moduli can be broadly available from seismic or well logging data [14].
The relationship between static and dynamic bulk moduli has been extensively investigated in consolidated sandstones through numerous laboratory-based measurements [15–23]. Cheng and Johnston [24] concluded that the static-to-dynamic bulk modulus ratio for dry Navajo and Berea sandstones varies from approximately 0.5 at atmospheric pressure to approximately unity at 200 MPa. Jizba and Nur [25] found that the static bulk modulus is almost equal to the dynamic modulus at high-stress levels and becomes approximately 50% smaller than the dynamic bulk modulus at low-stress levels after measuring 43 dry tight gas sandstones. Wang et al. [11] investigated the effects of pressure magnitude and history on the static and dynamic bulk moduli of three outcrop sandstones. They observed that the discrepancy between static and dynamic moduli tends to diminish during the initial pressure unloading. These experimental results suggest that the static bulk modulus of sandstones is almost always lower than the dynamic modulus, especially at low confining pressures when the microcracks and soft pores are open [26, 27]. The microstructural alterations affect the static properties with strain amplitudes larger than 10-4 more than the dynamic properties with strain amplitudes in the range of 10-9 to 10-6 [28]. However, most of these investigations focus on consolidated sandstones under room-dry conditions. Seldom research mentions the effect of fluid saturation on the pressure-dependent rock mechanical properties, given that fluid saturation might be more common for in situ rocks.
Additionally, unlike consolidated sandstones, unconsolidated sands are frequently portrayed with high porosity and high permeability [3]. The elastic properties of unconsolidated sands have been extensively studied from a dynamic perspective [29–33] by using the traditional ultrasonic velocity measurement system. However, unconsolidated sands are usually with low frame stiffness and loose cementation. It might be problematic in measuring the stress-strain relations once strain gauges or Linear Variable Differential Transformer (LVDT) are used. Static measurements based on strain gauges have a high probability of being affected by the strain gauge quality, the attachment quality, and the nonparallel end caps [21]. Zimmer [12] applied corrections to the axial strain for the strain gauge hysteresis and the end effects when measuring lab-repacked loose sand samples. Muqtadir et al. [34] reported that the radial LVDT gauges in their triaxial testing system failed to track a large displacement when measuring repacked loose sands from Saudi Arabia. Additional corrections are required to obtain reasonable strain data. The above factors may introduce significant uncertainties in calculating the static bulk modulus using the stress-strain relations.
In this study, we use an ultrasonic velocity measurement system to simultaneously measure the static and dynamic bulk moduli of GoM deepwater reservoir unconsolidated sand samples. We attempt to investigate the roles of pressure and fluid saturation in the relationship between the static and dynamic bulk moduli. The static bulk modulus is estimated by the pore volume change between two pressure steps at a constant pore pressure condition, while the dynamic bulk modulus is calculated from the ultrasonic velocity measurements. This method provides a reliable way to analyze the static and dynamic bulk moduli of unconsolidated sands and avoids the uncertainties caused by the strain gauge method. First, we briefly introduce the samples, experimental setup, and experimental procedures. Second, we introduce the methodologies used to calculate the static and dynamic bulk moduli. Third, we discuss the effects of differential pressure and brine saturation on the relationship between the static and dynamic bulk moduli with the aid of the Hertz-Mindlin contact theory [35, 36] at the critical porosity, the modified Hashin-Shtrikman lower bound [37], and Gassmann’s equation [38].
2. Materials and Methods
2.1. Sample Characterizations
The deepwater reservoir sand cores are obtained from two wells (wells A and B) at a water depth of ~1220 m in the GoM. The cores are drilled from two reservoir formations at depths of ~3660 m (well A) and ~5400 m (well B) [33]. The prepared core samples are machined to right cylinders with a diameter of ~3.8 cm. Subsequently, the core cylinders are placed in an electric oven for drying at 60°C.
According to Boyle-Marriott’s law, dried samples’ porosity and grain density are measured using a helium porosimeter at a pressure of 0.7 MPa [21]. The dry bulk density is calculated using the sample weight and volume. Figure 1 shows the dry bulk and grain densities as a function of the porosity. According to porosity, deepwater sands are classified into two categories. Seven samples from well A have higher porosities ranging from 0.3–0.35. Fourteen samples from well B have lower porosities of 0.24–0.3. The dry bulk density linearly increases with decreasing porosity, indicating a process of compaction or cementation. The grain density is approximately 2.65 g/cm3. Figure 2 shows the thin section images for #1 from well A and #14 from well B. Two samples are relatively clean and are composed of fine-grained quartz with almost no cementation. Sample #1, which has a higher porosity, has better initial sorting than sample #14.
2.2. Experimental Setups and Procedures
All samples are measured under both room-dry and brine-saturated conditions. The samples are saturated with the connate formation brine, which has a salinity of 100000 ppm and a bulk density of 1.07 g/cm3. The experiments are carried out using a custom-built ultrasonic velocity measurement apparatus [39], as shown in Figure 3. The sample, jacketed with a Tygon sleeve, is held between a pair of steel end caps. Two digital pumps independently control the confining and pore pressures. The confining pressure is applied by injecting silicone oil into the pressure vessel. In contrast, the pore pressure is applied by controlling the pore fluid volume within the sample. Under in situ conditions, both reservoir sands are overpressured with the differential pressure (i.e., the overburden pressure minus the pore pressure) around 13.8 MPa (2000 psi) for the well A sands and 27.6 MPa (4000 psi) for the well B sands [33]. Whatever the result from room-dry or brine-saturated measurements, the maximum differential pressure is applied to the in situ differential pressure and then decreases to 10.3, 6.9, 3.4, 1.4, and 0.7 MPa for well A sands and 20.7, 13.8, 10.3, 6.9, 3.4, 2.1, and 0.7 MPa for well B sands. During the brine-saturated measurement, the pore pressure is kept constant at 6.9 MPa.
The end caps contain piezoelectric transducers to transmit and receive both P- and S-wave signals. The central frequency for P- and S-wave transducers is 0.2 MHz. Ultrasonic velocities are calculated using the sample length and the arrival time picked from an individual waveform. The errors in calculating velocities, caused by sample length changes and ambiguities in determining the first-arrivals, are less than 2% and 4% for P- and S-waves, respectively. Figure 4 shows the P- and S-wave velocities ( and ) as a function of the differential pressure () for two samples from well A (#4) and well B (#16) under both room-dry and brine-saturated conditions. Both P- and S-wave signals of dry samples are difficult to detect at low differential pressures, although the brine saturation improves the detection of P-wave signals, as shown in Figure 4. In addition to the velocities, the porosity at each differential pressure step is simultaneously determined by the pore volume change [21]. Porosity measurements are possible only when the pore space is filled with brine at a constant pore pressure [30]. The porosity at the lowest differential pressure (0.7 MPa) is assumed to be equal to that measured by the helium porosimeter at 0.7 MPa (the error can be negligible considering the small pressure difference). With the assumption that the grain volume remains constant [27], the porosities during pressure unloading are calculated based on the porosity at 0.7 MPa and the pore volume changes. The pore volume change at each pressure step is derived from the amount of brine absorbed into the pore space from the digital pump with decreasing confining pressure. A waiting time of 30–60 min is required for the pressure equilibrium at each pressure step. Given the high resolution of the digital pump, minor errors may be induced at different pressure conditions. Figure 5 shows the porosity evolution with decreasing differential pressure for all samples from wells A and B.
3. Determination of Dynamic and Static Bulk Modulus
3.1. Determination of Dynamic Bulk Modulus
As shown in Figure 4, both the room-dry and brine-saturated velocities demonstrate a power-law relationship with the differential pressure . The coefficient is the P- or S-wave velocity at the unit differential pressure, while characterizes the pressure dependence of ultrasonic velocities. In the Hertzian grain contact model [36], equals 6, which depicts a case with perfect grain contacts. For room-dry deepwater sand samples, the value is approximately 6 or less (Figure 6) and is systematically larger than that for laboratory-repacked sand samples (about 4 for repacked Ottawa sands in Domenico [30]). Additionally, the replacement of gas with brine significantly increases the P-wave velocity but slightly decreases the S-wave velocity (Figure 4), which might be attributed to the increased frame stiffness and bulk density with the brine saturation [33]. The brine saturation substantially reduces the rate of increase with differential pressure for the P-wave velocity but slightly affects the S-wave velocity (Figure 6). The values for the brine-saturated P-wave velocities could reach more than 15. Moreover, the brine saturation induced a larger reduction in the pressure dependence of P-wave velocities for samples from well A (Figures 4 and 6), which might be attributed to the better initial sorting of samples with larger porosities, as shown in Figure 2 [40].
Figure 7 shows the dynamic bulk modulus plotted against the differential pressure for two samples from well A (#4) and well B (#16) under room-dry and brine-saturated conditions. A power-law relationship can approximate the pressure dependence of both the room-dry and brine-saturated moduli: . The coefficient is the dynamic bulk modulus when the differential pressure is 1 MPa. The value denotes the pressure dependence of the dynamic bulk modulus. The higher the value, the lower the increase rate of the dynamic bulk modulus with differential pressure. Figure 8 shows the values for all measured sand samples under both room-dry and brine-saturated conditions. Under room-dry conditions, the value is distributed between 1.8 and 5.4. The brine saturation substantially stiffens the rock (Figure 7) and increases the value to more than 10 (Figure 8).
3.2. Determination of Static Bulk Modulus
Figure 9 shows the dry static bulk modulus () plotted against the differential pressure () for two samples from wells A (#4) and B (#16). For #4, with a porosity of 0.324, decreases almost linearly with decreasing differential pressure. For #16, with a porosity of 0.284, the decreasing trend of follows two slopes separated at a differential pressure of approximately 20 MPa. Over the applied pressure range, decreases by 95.4% and 92.4% for #4 and #16, respectively, which are larger than the decrements of in Figure 7 (37.4% and 47.8% for #4 and #16, respectively).
Additionally, the static bulk modulus at brine-saturated conditions () could be predicted using Gassmann’s equation (Equation (2)). The calculation inputs are , , , and . and are the bulk moduli of the intrinsic mineral material and the pore fluid, respectively. Figure 9 shows as a function of the differential pressure for #4 and #16. In general, the fluid saturation induces a significant increase in the static bulk modulus. However, the increase in at different pressures tends to be constant.
4. Comparison between Dynamic and Static Bulk Moduli
Figure 10 exhibits the relationships between the dynamic and static bulk moduli for all deepwater sands under both room-dry and brine-saturated conditions. The wet dynamic bulk modulus () is (a) calculated from the measured ultrasonic velocities and (b) predicted from Gassmann’s equation. The wet static bulk modulus () is predicted from using Gassmann’s equation for Figures 10(a) and 10(b). The color bar is scaled by the porosity magnitude. The symbol size is scaled by the differential pressure magnitude, with a larger size denoting a larger differential pressure. From both dynamic and static perspectives, the brine-saturated bulk modulus is systematically greater than the room-dry modulus. The dynamic bulk modulus () was greater than the static one (), except for a few samples under relatively high-pressure conditions.
Figure 10(a) shows that the decreasing porosity affects more than at a specific pressure. The difference between the dynamic and static bulk moduli increases with decreasing porosity. Compared with samples from well A, the samples from well B deviate more from the one-to-one correlation line. For a specific sample, the decreasing differential pressure decreases more than . Under room-dry conditions, the ratio of to could decrease from values approaching unity to values as low as 0.2 with the decreasing pressure. In contrast, the ratio between and is less affected by the differential pressure (ranging from ~1.1 to ~0.7) under brine-saturated conditions.
Moreover, comparing Figures 10(a) and 10(b), the difference between the static and dynamic bulk modulus is less pronounced when is predicted using Gassmann’s equation (Figure 10(b)). This might be attributed to the significant dispersion induced when measuring ultrasonic velocities in fully brine-saturated sands [42–45]. In addition, the pressure dependence of ultrasonic velocities potentially causes by the core damage or cracks during coring [48] might contribute to the data scatter in Figure 10(a).
5. Discussion
5.1. Theoretical Modeling
Figure 11 illustrates the modified Hashin-Shtrikman lower bounds for the bulk modulus at six different pressures. The bounds have two endmembers. The left endmember at zero porosity has the bulk modulus of pure quartz. The pressure-dependent bulk moduli given by the Hertz-Mindlin theory at critical porosity () are incorporated into the right endmember. The dashed lines indicate the case when the pure quartz grains are subjected to room-dry conditions. It is noticeable that the bulk modulus increases with increasing differential pressure over the entire porosity range, consistent with that typically observed in sandstones.
Moreover, the low-frequency, brine-saturated bulk moduli at various porosities and pressures are calculated from the dry data using Gassmann’s equation (Equation (2)), as shown by the solid lines in Figure 11. In the calculation, the brine bulk modulus () is 2.8 GPa. After replacing the gas with brine, the pressure dependence over the entire pressure range becomes narrower than under room-dry conditions.
Figure 12 compares the dynamic and static bulk moduli for unconsolidated sands, accompanying the bounds predicted using the Hertz-Mindlin-Hashin-Shtrikman (HMHS) model (Figure 11) at the differential pressures of 3.4 MPa, 6.9 MPa, 10.3 MPa, 13.8 MPa, 20.7 MPa, and 27.6 MPa. The wet dynamic bulk modulus is directly calculated from the measured wet ultrasonic velocities () or predicted from the measured room-dry velocities using Gassmann’s equation (). The wet static bulk modulus is predicted from the room-dry static bulk modulus using Gassmann’s equation (). It is impossible to precisely illustrate the relationship between the dynamic and static bulk moduli using the HMHS model. However, to some extent, the pressure-dependent modified Hashin-Shtrikman lower bound can serve as the lower bound for the dynamic bulk modulus and the upper bound for the static bulk modulus (Figure 12).
From a dynamic perspective, the theoretical bounds provide better predictions for low-modulus dry samples (samples from well A) when the differential pressure is below 13.8 MPa. After Gassmann’s fluid substitution (, solid blue circles), the relative errors for theoretical estimations are reduced compared to the dry case (, open black circles). The error improvement might be attributed to the pore fluid playing a role in reducing the dry modulus variations, as shown in Figures 7 and 8 [32]. Additionally, in contrast to , exhibits more scattering above the theoretical bounds. This might be caused by the strong dispersion during ultrasonic measurements [41, 45]. From a static perspective, the relative errors between the measured data and theoretical results gradually decrease with increasing differential pressure. The larger errors at relatively low pressures might be caused by stress-induced cracks upon pressure unloading [11, 28]. In addition, the brine saturation significantly reduces the relative errors of the theoretical estimates, which is more apparent at differential pressures of 3.4 MPa, 6.9 MPa, and 10.3 MPa.
5.2. Relationship between Static and Dynamic Bulk Modulus
Figure 10 shows that the static bulk modulus is almost lower than the dynamic bulk modulus for unconsolidated sands, regardless of the room-dry or brine-saturated state. However, the magnitude of the static–dynamic difference is highly associated with the differential pressure, saturation, frequency, and sample porosity [11, 18, 21, 23, 28, 53]. To quantify the effects of the above factors on the relationship between static and dynamic bulk modulus, we derive the fitted static-to-dynamic bulk modulus ratios. The fitted static-to-dynamic bulk modulus ratios are derived by linearly fitting the data points with the same symbol but different colors in Figure 10. It should be noted that the intercept is constrained to zero in each fitting process. Figure 13 exhibits the fitted static-to-dynamic bulk modulus ratio as a function of the differential pressure under both room-dry and brine-saturated conditions.
In Figure 13, under room-dry conditions, the fitted static-to-dynamic ratio is approximately 0.27 at a differential pressure of 3.4 MPa. The ratio approaches 0.8 at a maximum pressure of 27.6 MPa. Cheng and Johnston [24] measured the static and dynamic bulk moduli for various dry rocks. They concluded that the static to dynamic modulus ratio could range from ~0.5 at atmospheric pressure to ~1.0 at 200 MPa. The pressure dependence of the static–dynamic discrepancy might be ascribed to the openings of cracks or thin pores upon pressure unloading, which behave differently at different strain amplitudes during the dynamic and static tests [11, 28, 54, 55].
Brine saturation essentially increases both static and dynamic bulk moduli (Figure 12), resulting in an enlarged static-to-dynamic ratio ranging from 0.83 to 0.9. The pore fluid, filling the pore space, would stiffen the rock framework and substantially reduce the effect of opening cracks or pores on the static–dynamic modulus discrepancy. However, there exists a frequency mismatch in the static–dynamic modulus correlation as the wet dynamic bulk modulus is computed from the measured ultrasonic velocities, whereas the wet static bulk modulus is derived from Gassmann’s equation at a low-frequency limit. Subsequently, the correlation is reexecuted when the wet dynamic and static bulk moduli are predicted from the Gassmann fluid substitution, which suggests that the static-to-dynamic ratio is improved to 0.91-0.95.
The relationship between static and dynamic bulk moduli is essential for modeling the formation deformation and predicting the in situ effective stress [4, 5, 7, 8]. The traditional ultrasonic velocity measurement system provides a reliable way to evaluate the static and bulk moduli of unconsolidated sands simultaneously. The dynamic-static bulk modulus relationship in this study is consistent with that for consolidated sandstones in Yan et al. [21] using the same method. Compared to room-dry rock properties, the brine-saturated properties might better represent in situ properties for deepwater loose sand reservoirs. The application of Gassmann’s equation further unifies both the laboratory dynamic and static measurements at the same frequency, making the static and dynamic property correlation more meaningful. When both dynamic and static bulk moduli have experienced Gassmann fluid substitution from laboratory dry rock data, the static and dynamic modulus differences become less pronounced. This might suggest that the dynamic elastic properties can well represent the static elastic properties for the unconsolidated reservoir sands.
6. Conclusions
The traditional ultrasonic velocity measurement system provides a reliable way to evaluate the static and bulk moduli of unconsolidated sands simultaneously. The experimental results indicate that the static bulk modulus is almost always lower than the dynamically derived modulus under any pressure condition. After modeling with the Hertz-Mindlin theory and the modified Hashin-Shtrikman lower bound, the theoretical estimates can serve as the lower bound for the dynamic bulk modulus and the upper bound for the static bulk modulus. However, the difference between the static and dynamic bulk moduli is pressure-sensitive. The fitted static-to-dynamic modulus ratio for all samples decreases from about 0.8 to about 0.25 upon pressure unloading at room-dried conditions.
The fluid saturation effects can also be analyzed with the aid of Gassmann’s equation. Brine saturation substantially increases both static and dynamic bulk moduli. The difference between the static and dynamic bulk moduli is reduced accordingly. After replacing the gas with brine, the static-to-dynamic bulk modulus ratio is improved to 0.91–0.95 over the entire pressure range. Given that the in situ rocks are fluid-saturated, the difference between the static and dynamic bulk moduli for in situ reservoir sands may be insignificant.
Data Availability
The EXCEL data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Fundamental Research Funds for the Central Universities and the Fluids/DHI consortium at the University of Houston.