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ock physics models for fluid and stress dependency in reservoir rocks are essential for quantification and interpretation of 4D seismic signatures during reservoir depletion and injection. However, our ability to predict the sensitivity to pressure from first principles is poor. The current state-of-the-art requires that we calibrate the pressure dependence of velocity with core measurements. A major challenge is the fact that consolidated rocks often break up during coring, and hence the stress sensitivity is likely to be overpredicted in the laboratory relative to the in-situ conditions (Furre et al., 2009). For unconsolidated sands, acquisition of core samples is not very feasible due to the friable nature of the sediments. One physical model that has been applied to predict pressure sensitivity in unconsolidated granular media is the Hertz-Mindlin contact theory. Several authors (Vernik and Hamman, 2009, among others) have suggested empirical models with fitting parameters that correlate with microcrack intensity, soft porosity, and aspect ratio of the rock, and feasibility studies can be undertaken based on assumptions about these parameters. These models may not be easy to use for poorly to moderately consolidated sandstones with contact cement, where crack parameters and aspect ratios are difficult to quantify. In this study, we suggest an approach to predict stress sensitivity in cemented sandstones using nonuniform contact theory combined with modified Hashin-Shtrikman elastic bounds. We assume that the cemented rock will consist of a binary mixture of cemented and uncemented grain contacts, or “patchy cementation” (Figure 1). Assuming that the cemented “stiff” grain contacts are stress-insensitive and the unconsolidated “loose” grain contacts are stress-sensitive according to Hertzian contact theory, this hybrid model will allow us to predict the pressure sensitivity in cemented sandstones. Note that in this paper “static” refers to reservoir parameters that do not change during 4D time-lapse (e.g., mineralogy, porosity, and cement volume), whereas “dynamic” refers to reservoir parameters that normally change in 4D studies (e.g., saturation and pressure). Static rock physics modeling If we wish to predict the seismic velocities of a rock, knowing only the porosity, mineralogic composition, and the elastic moduli of the mineral constituents, we can at best predict the upper and lower bounds of the seismic velocities. However, if we know the geometric details of how the mineral grains and pores are arranged relative to each other, we can predict more exact seismic velocities. There are several models that account for the microstructure and texture of rocks, and these in principle allow us to go the other way: to predict the type of rock and microstructure from seismic velocities. The rock physics 90
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Figure 1. Thin section showing example of “patchy” cement. Some grain contacts of a reservoir sandstone are clearly contact cemented, whereas others are loose and uncemented. Pressure sensitivity in such reservoirs will be due to the “loose” grain contacts.
Figure 2. Rock physics diagnostics implies the use of rock physics models to predict rock texture from elastic properties versus porosity (Avseth et al., 2005).
diagnostics technique was introduced by Dvorkin and Nur (1996) as a means to infer rock microstructure from velocityporosity relations. This diagnostic is conducted by adjusting an effective-medium theoretical model curve to a trend in the data, assuming that the microstructure of the sediment matches that used in the model. Avseth et al. (2005) describe three heuristic rock physics models that have been used to diagnose the rock texture of medium- to high-porosity sandstones: (a) the friable sand model; (b) the contact cement model; and (c) the constant cement model (Figure 2). These models are made by first de-
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fining the elastic properties of the “end members.” At zero porosity, the rock must have the properties of mineral. At the high-porosity limit, the elastic properties are determined by elastic contact theory. We interpolate between these two end members by using either upper or lower Hashin-Shtrikman bounds. The upper bound explains the theoretical stiffest way to mix load-bearing grains and pore-filling material, while the lower bound explains the theoretical softest way to mix these. Hence, we have found that the upper bound is a good representation of contact cement, while the lower bound accurately describes the effect of sorting. It is found that rocks with very little contact cement (a few percent) are not well described by the Hashin-Shtrikman upper bound, because there is a large stiffening effect during the very initial porosity reduction as cement fills in the microcracks between the contacts. Then, it is not realistic to interpolate between the high-porosity and zero-porosity end members. We therefore include a high-porosity contact cement model (i.e., the Dvorkin-Nur model) that takes into account the initial cementation effect. These models have been used to quantify depositional sorting and diagenetic cement volume in reservoir sandstone intervals (Avseth et al., 2009). A major shortcoming with the Dvorkin-Nur contact-cement model is that is does not include pressure sensitivity. It is assumed that the cemented grain contacts immediately lose pressure sensitivity as the cementation process initiates. From in-situ observations, we know that cemented reservoirs can have significant pressure sensitivity. This could either be related to fractures not captured by the microstructural scale model, or by a patchy cementation where some grain contacts are cemented and others are loose. In this study, we consider the latter scenario where loose grain contacts should still be pressure sensitive. As with the Hertz-Mindlin contact theory for loose granular media, the Dvorkin-Nur contact-cement model is also found to often overpredict shear stiffnesses in cemented sandstones. This could be related to nonuniform grain contacts and tangential slip at loose contacts, associated heterogeneous stress chains, and/or relative roll and torsion not taken into account in the contact theory. A reduced shear factor (Ft) has been introduced to honor this “reduced shear effect” in the contact theory and varies between 0 and 1 representing the boundary conditions between no-friction (Walton smooth contact theory) and no-slip (Walton rough/Hertz-Mindlin contact theory) conditions. For loose sands this parameter can be estimated directly from dry rock Poisson’s ratio (Bachrach and Avseth, 2008). For cemented sandstones, this parameter is a pure fitting parameter, yet it has been found to correlate with degree of cementation. Dynamic rock physics modeling One way to heuristically quantify the pressure sensitivity of a rock is to measure the distance between an upper and lower elastic bound at a given pressure. Marion and Nur (1991) introduced the “bounding average method” as a relative measure of pressure sensitivity of a rock (Figure 3). We suggest
Figure 3. The bounding average method. The position of a data point A, described as d/D relative to bounds, is assumed to be a measure of the pore stiffness (after Marion and Nur, 1991).
using a similar approach to quantify the degree of consolidation, and define a weight function, W, depending on where the sandstone data plot between an upper and lower bound in the elastic moduli versus porosity domain: (1) where Kdry is the dry bulk modulus (modeled or observed) of a cemented sandstone at a given porosity, Ksoft is the pressuresensitive soft (lower bound) bulk modulus at same porosity estimated at a given reference pressure, P0, and Kstiff is the pressure-insensitive stiff (upper bound) bulk modulus at this porosity value (Figure 4). The soft bound is the unconsolidated sand model where the reference effective stress (P0) is set to 20 MPa in this study. This represents the effective stress at around 2 km burial depth, which is the depth we expect initial quartz cementation to initiate in the North Sea (Avseth et al., 2009). Any data point falling on this soft bound should represent unconsolidated sands where all grain contacts are stress-sensitive. The stiff bound is defined by increasing the effective stress in the Hertz-Mindlin model so that it mimics the 10% constant cement model. We find that an effective stress of 600 MPa must be selected in order to get this match. For any practical reason, this stiff bound should therefore be considered what happens when all grain contacts are closed, and there is no stress sensitivity in the sandstone data falling on this bound. This gives us a soft and stiff bound with the same shape, which gives a more stable and realistic weight factor estimation for a given porosity value. A separate weight factor should be estimated for shear modulus, because the elastic bounds for shear modulus will be affected by the reduced tangential shear stiffness mentioned above. We define the soft bound with reduced shear factor as Ft = 0. The reduced shear factor for the stiff bound is set to 0.5. Bachrach and Avseth (2008) demonstrated that this parameter is depth-dependent and likely associated with de-
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Figure 4. Simulated elastic moduli versus porosity from constant cement model with cement fraction 0-10%. Black line is the stress-insensitive stiff bound whereas the gray line is the stress-sensitive soft bound shown for 20 MPa. The colors represent cement volume.
gree of diagenesis. This parameter can be further updated in an iterative scheme to fit a calibration data set (not demonstrated here). The weight functions allow us to estimate pressure sensitivity in cemented sandstones. By combining the Walton smooth pressure-sensitive model for unconsolidated sands with the stiff contact cement model, we obtain a modified contact model for heterogeneous contacts that is pressuresensitive via the fraction of unconsolidated grain contacts: (2) (3) Any sandstone data point could be inverted for the weight factor W allowing us to estimate stress curves for each data point. In this study, we simulate synthetic data for a wide range of porosities and cement volumes using the static rock physics models described above (i.e., the constant cement model). The cement volume varies between 0 and 10%, as we assume that if the cement volume is higher than this there will be no stress sensitivity at the grain contacts. Porosity varies between 0 and 0.4. Hence, we create a synthetic data set where we have all possible combinations of porosity and cement volume within these given ranges (Figure 4). Noise is added in order to make the data set more realistic. This will make the nonlinear regression analysis below more stable. Next, we estimate soft and stiff bounds that enclose the simulated data set in the moduli-porosity domain. These bounds are modeled by combining Hertz-Mindlin and lower bound Hashin-Shtrikman as described above. A linear weight function is then defined between the soft and the stiff bounds (Equation 1), both for bulk modulus and shear modulus versus porosity. This weight factor will define the 92
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Figure 5. Simulated data of bulk (upper) and shear (lower) moduli plotted at various effective pressures (0, 10, and 20 MPa), for varying porosity and cement volumes. The stress-sensitive soft bound plane and the stress-insensitive stiff bound plane are indicated. The estimated weight factors determine the stress sensitivity of the data plotting in between. Colors represent cement volume.
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Figure 6. Stressdependent curves for a range of porosities (0.26–0.35), for gas-, oil- and brinesaturated sandstones. Note that the stress sensitivity is larger for brine, than for oil, and we observe almost no stress sensitivity for gassaturated sandstones.
stress sensitivity of the cemented sandstone. Next, we derive effective bulk and shear modulus as a function of effective stress for the cemented sandstones, depending on the estimated weight factors (i.e., consolidation degree) using Equations 2 and 3. Figure 5 shows the simulated data where we are highlighting the porosities ranging from 0.2 to 0.4, for both bulk and shear moduli at different effective pressures (0, 10, and 20 MPa). Here we have indicated the Hertzian stress sensitivity along the plane representing the soft bound, and a “flat” plane with no stress sensitivity along the stiff bound. These plots demonstrate how the simulated data at varying porosity and cement volumes show varying stress sensitivity. Note that the data plotting close to the lower bound show significant pressure sensitivity, whereas the well cemented data plotting close to the stiff bound show no or insignificant stress sensitivity. After fluid substitution to some typical oil and brine properties, we can investigate the stress sensitivity in terms of saturated VP/VS and acoustic impedances. Figure 6 shows stress curves for a selected range of porosities (0.26–0.35), for gas-, oil-, and brine-saturated sandstones where cement volume is 2%. We apply this hybrid staticdynamic model to some real well-log data. Figure
7 shows well-log data from the target zone in two selected wells from the Statfjord and Gullfaks fields, respectively. The Gullfaks reservoir sands in this example are found to be unconsolidated with no or almost no cement. The Statfjord reservoir sandstones are slightly cemented. We include stress curves in the VP/VS versus AI domain for selected porosities and cement volumes, and for different pore fluid types. The results show that when cement volume is approaching zero, the stress sensitivity is increasing drastically when the sands are saturated with oil or brine. The Gullfaks data are plotting close to the curves modeled for 0.5%. The Statfjord reservoir data are plotting between the models where we assume cement volume to be 3–5%. Here we expect much lower stress sensitivity during depletion or injection. This is further documented by Duffaut et al. (TLE, this issue).
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Figure 7. Dynamic rock physics models for Gullfaks and Statfjord reservoir sands. Note that the selected Gullfaks data plots along the more stress sensitive curves than the Statfjord data. This is likely related to the degree of consolidation. (See Duffaut et al. in this issue of TLE for further details.)
Figure 8. Effective pressure versus dry velocities at porosity of 0.3 and cement volume of 0.02. Blue line shows the soft velocities at the given porosity. Black line shows the corresponding stiff velocities. The green points are the simulated data extrapolated from 20 MPa down to 0 MPa for the given combination of porosity and cement volume. The red line is the stress curve predicted by the nonlinear regression.
Nonlinear regression models and dynamic rock physics templates A nonlinear regression is performed on the simulated data set, for porosities ranging from 0.20 to 0.40. Strictly speaking, contact theory is only valid at relatively high porosities (> ~0.20, Jack Dvorkin, personal communication), and pressure sensitivity at lower porosities should be quantified using inclusion models (i.e., aspect ratios; Vernik and Hamman, 2009). Furthermore, the regression easily becomes unstable if we include the whole porosity range, since the shape of the velocity-porosity trends are very different at lower porosities relative to higher porosities. We choose a mathematical formulation similar to the one suggested by Eberhart-Phillips et al. (1989). However, slight modifications were done to obtain satisfactory fit between regression formulas and simulated data. 94
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First we perform regression on the “plane” representing well cemented sandstones, with cement volumes of 0.08–0.1. The resulting dry rock (i.e., rock without fluid) velocities are given by the following equations as a function of porosity, q, and effective pressure, Peff (in MPa):
9SVWLII
3HII (4) (5)
Next, we perform regression on the plane representing unconsolidated sands (i.e., no cement). The simulated data here represent sands with only Walton smooth contacts (i.e., Hertz-Mindlin with reduced shear factor Ft = 0). The resulting dry velocities are given by the following equations as a function of porosity and effective pressure:
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Finally, we define the dry velocities as a function of porosity, effective pressure and cement volume to be a weighted average of the soft and the stiff velocities, with respect to cement volume: (8) , reflecting that the weighting where average will change with effective pressure (in MPa). The same formulation applies to effective shear-wave velocities. Examples of regression formulas for VP and VS versus effective pressure as a function of porosity and cement volume are plotted in Figure 8, together with the exact models from the workflow above. There is overall a nice match between the regression lines and the exact models. This shows that we can use the regression formulas directly to establish dynamic rock physics templates for different types of reservoirs (i.e., varying porosity and cement volume). Figure 9 shows dry and saturated acoustic impedances versus VP/VS derived from the regression formulas above, where we have input all porosities between 0.2 and 0.4, all cement volumes between 0 and 0.1, and all effective pressures between 0 and 40 MPa. Note that saturated VP/VS ratios vary drastically at low cement volumes and low effective pressures. Acoustic impedance changes are correlating strongly with porosity and cement volume for both dry and saturated scenarios. The regression formulas are also tested on real data from Gullfaks and Statfjord fields in Duffaut et al.
Figure 9. Saturated and dry VP/VS versus AI for porosities of 0.2–0.4, cement volumes of 0–0.1, and effective pressure of 0–40 MPa, using the nonlinear regression formulas extracted in this study.
(6) (7)
Conclusions We have established a heuristic approach to estimate fluid and pressure sensitivity in cemented sandstones. We expand on existing static models of cemented sandstones (Avseth et al., 2005) to account for stress sensitivity using elastic bounds in the porosity-moduli domain, where we define a soft bound to be stress sensitive (Hertz-Mindlin contact theory) and a stiff bound to be insensitive to stress (Dvorkin-Nur contact cement model). Based on the location of a data point (welllog data or inverted seismic data) between these bounds, we are able to quantify expected pressure and fluid sensitivity in elastic and seismic properties (including moduli, velocities, acoustic impedance and VP/VS) of cemented sandstones. We also establish regression formulas that can be used to estimate dynamic rock physics templates for reservoir sandstones where the input parameters are confined to cement volume, porosity, and effective pressure. This approach can be applied to predict the effect of pressure changes for example during 4D monitoring analysis. References Avseth, P., T. Mukerji, and G. Mavko, 2005, Quantitative seismic interpretation; applying rock physics tools to reduce interpretation Risk: Cambridge University Press. Avseth, P., A. Jørstad, A. -J. Wijngaarden, and G. Mavko, 2009, Rock physics estimation of cement volume, sorting, and net-togross in North Sea sandstones: The Leading Edge, 28, 98–108. doi:10.1190/1.3064154 January 2011
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Bachrach, R. and P. Avseth, 2008, Rock physics modeling of unconsolidated sands: Accounting for nonuniform contacts and heterogeneous stress fields in the effective media approximation with applications to hydrocarbon exploration: Geophysics, 73, no. 6, E197–E209, doi:10.1190/1.2985821. Dvorkin, J. and A. Nur, 1996, Elasticity of high-porosity sandstones: Theory for two North Sea data sets: Geophysics, 61, no. 5, 1363– 1370, doi:10.1190/1.1444059. Eberhart-Phillips, D., D. Han, and M. Zoback, 1989, Empirical relationships among seismic velocity, effective pressure, porosity, and clay content in sandstone: Geophysics, 54, no. 1, 82–89, doi:10.1190/1.1442580. Furre, A. –K., M. Andersen, A. S. Moen, and R. K. Tønnesen, 2009, Deriving effects of pressure depletion on elastic framework moduli from sonic logs: Geophysical Prospecting, 57, no. 3, 427–437, doi:10.1111/j.1365-2478.2008.00744.x.
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Marion, D. and A. Nur, 1991, Pore-filling material and its effect on velocity in rocks: Geophysics, 56, no. 2, 225–230, doi:10.1190/1.1443034. Vernik, L., and J. Hamman, 2009, Stress sensitivity of sandstones and 4D applications: The Leading Edge, 28, no. 1, 90–93, doi:10.1190/1.3064152.
Acknowledgments: Thanks to Anders Dræge at Statoil for contributions on the regression analysis. We also acknowledge Kenneth Duffaut, Ola-Petter Munkvold ,Tore Bersås, and Harald Flesche at Statoil, Martin Landrø at NTNU, and Gary Mavko and Jack Dvorkin at Stanford University for valuable input and discussions during this study. Corresponding author:
[email protected]
