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CO2 Saturation, Distribution and Seismic Response in Two-Dimensional Permeability Model Hamid Behzadi, Vladimir Alvarado,*,† and Subhashis Mallick‡ † ‡
Chemical and Petroleum Engineering Department, University of Wyoming, Laramie, Wyoming 82071, United States Geology and Geophysics Department, University of Wyoming, Laramie, Wyoming 82071, United States
bS Supporting Information ABSTRACT: Carbon dioxide capture and storage (CCS) has been actively researched as a strategy to mitigate CO2 emissions into the atmosphere. The three components in CCS are monitoring, verification, and accounting (MVA). Seismic monitoring technologies can meet the requirements of MVA, but they require a quantitative relationships between multiphase saturation distributions and wave propagation elastic properties. One of the main obstacles for quantitative MVA activities arises from the nature of the saturation distribution, typically classified anywhere from homogeneous to patchy. The emerging saturation distribution, in turn, regulates the relationship between compressional velocity and saturation. In this work, we carry out multiphase flow simulations in a 2-D aquifer model with a log-normal absolute permeability distribution and a capillary pressure function parametrized by permeability. The heterogeneity level is tuned by assigning the value of the DykstraParson (DP) coefficient, which sets the variance of the log-normal horizontal permeability distribution in the entire domain. Vertical permeability is a 10th of the horizontal value in each gridcell. We show that despite apparent differences in saturation distribution among different realizations, CO2 trapping and the Vp-Sw Rock Physics relationship are mostly functions of the DP coefficient. When the results are compared with the well accepted limits, GassmannWood (homogeneous) (A Text Book of Sound; G. Bell and Suns LTD: London, 1941) and GassmannHill (patchy) models, the Vp-Sw relationship never reaches the upper bound, that is, patchy model curve, even at the highest heterogeneity level in the model.
’ INTRODUCTION Carbon dioxide capture and storage (CCS) has been proposed as a viable approach to mitigate CO2 emissions from large point sources such as coal fired power and chemical processing plants.1,2 Monitoring, verification, and accounting (MVA) of CO2 are three components in CCS. The primary objectives of MVA is to identify and quantify CO2 migration in geological media and help to identify any leakage of stored CO2. These objectives highlight the importance of multiphase flow and its controlling properties, namely relative permeability and capillary pressure, which govern how phases migrate in subsurface porous media and consequently how rock elastic properties change during storage operations. Hysteresis of multiphase flow functions plays a significant role in CCS. In addition, hysteresis trapping can be enhanced if, for example, CO2 is injected at the bottom of the formation. In this approach, once CO2 is injected at the bottom of an aquifer, it will spontaneously migrate upward, leading to consecutive drainage and imbibition processes. Moreover, as CO2 moves up, it dissolves in water and is stored. These two storage mechanisms are characterized by being fast, typically in the scale of years, as opposed to hundreds of years or more for more permanent storage. Saadatpour3 showed that local capillary trapping may significantly contribute to phase distribution and trapping. A region r 2011 American Chemical Society
with high entry capillary pressure may act as a barrier to drainage by CO2 and redirect the flow laterally. Therefore, this mechanism is associated with heterogeneous systems. The redirection of flow increases the odds of both hydraulic and dissolution trapping. In the first part of this paper, we consider hysteresis and local capillary trapping effects on trapping, and consequently on saturation distribution. This will impact elastic properties, as shown in the second part of this work. Mineral trapping is not included in this work, but it should become important after hundreds of years. The second part of this study focuses on seismic monitoring. Monitoring is an essential tool to design and fine-tune improved oil recovery, optimize CO2 storage and mitigate leakage risks. To accomplish this, it becomes necessary to map saturation and pressure throughout the formation over the time span of interest. Time-lapse seismic has been utilized to map these changes mostly qualitatively.47 A quantitative interpretation of saturation changes may be possible, if the case-specific velocitysaturation relationship for a system is known. In practice, there are two main velocity-saturation (Vp-Sw) models based on two extremes of phase-distribution type: uniform Received: June 12, 2011 Accepted: September 22, 2011 Revised: August 22, 2011 Published: September 22, 2011 9435
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Environmental Science & Technology and patchy. In GassmannWood8 model (uniform), the size of patches is smaller than the fluid diffusion length. In contrast, if the size of patches is larger than the fluid diffusion length, then there is no pressure communication between gas pockets during one wave cycle. In this latter case, the elastic wet rock properties can be estimated using GassmannHill equation.9 These two bounds are also named low and high frequency limits. However, in reality, the system-specific velocity-saturation relationship is in the intermediate frequency domain. There are very few velocity-saturation models available in the open literature. White’s model10 is limited to regular distribution with simple geometry, whereas Muller11 provides the velocitysaturation relationship for partially saturated rock. Muller assumes that the distribution of fluid patches form a two-phase, self-affine mono fractal,11 not considering geology and multiphase flow. Muller then introduces this fractal saturation map into Toms’ model12 to compute elastic rock properties and consequently Vp-Sw relationship as a function of the Hurst exponent, ν, which controls the persistence or antipersistence of the random fractal. In this model, each cell is saturated with either a brine or a gas, but not both. In our study, saturation map is the response of multiphase flow functions to cells properties of the static model. In summary, we first build a geological model with randomly uncorrelated permeability distribution and carefully allocate rockfluid properties, that is, relative permeability and capillary pressure. An equation of state is used to predict thermodynamic fluid properties. The saturation and elastic rock properties are simulated on fine-scale models and then upscaled to a coarser one using Backus’s method,13 in order to derive velocity-saturation relationships.
’ MATERIALS AND METHODS There are four main CO2 storage mechanisms, namely structural, hysteresis, dissolution, and mineral trapping. However, local heterogeneity may result in significant contrast in static and dynamic properties, as to produce preferential paths as well as local barriers to flow in heterogeneous systems, a process coined local capillary trapping.3 This trapping mechanism is different from hysteresis trapping in terms of scale and magnitude. The simulations are carried out in a compositional mode, which more realistically represents solubility of CO2 in brine. It also allows us to track a more realistic thermodynamical behavior, and hence it captures density changes accurately. Structural Model. Local capillary trapping effects on buoyancy-driven migration is a central motivation of this work. The model consists of a 2D aquifer with a source of CO2 at the bottom. In addition, the system boundaries, namely top, bottom and sides represent no-flow conditions, which should enhance countercurrent flow. The model has three main sections: source at the bottom where CO2 is initially located, main reservoir body in the middle, at very fine resolution of 1 m-X 0.3 m-Z cells and a sink at top, where CO2 migrates. Fine-scale models are of interest to capture the fine details of multiphase flow, trapping and velocity-saturation (Vp-Sw) relationship. Grid size influences trapping and Vp-Sw relationship. For instance, if grid size is coarse, then buoyancy may overcome capillary entry pressure. Petrophysical Model. Permeability and the corresponding capillary pressure are assigned on per grid-cell basis, while porosity and relative permeability are constant throughout the domain. The distribution of horizontal permeability is randomly
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uncorrelated according to a log-normal Gaussian distribution. The DykstraParson coefficient (DP) parametrizes the variance of the log-normal distribution conveniently. A value of DP = 0 means that the variance is zero. The larger the value of DP, the larger the variance and therefore the system is more heterogeneous. Vertical permeability is assigned to be 10 times smaller than the grid-cell horizontal permeability. The base case is generated with DykstraParson coefficient 0.7 (DP = 1 e(σK)) and a permeability mean of 200 md. Leveret J-function is utilized to parametrize capillary pressure as a function of permeability. sffiffiffi Pc k ð1Þ JðSw Þ ¼ σ cos θ ϕ where θ is the contact angle, and k and ϕ are the absolute permeability and porosity, respectively. Chequet14 showed that contact angle variation of 4050° and 1525° for quartz and mica respectively, can occur as pressure changes. Chequet15 and Chalbaud16 also showed that pressure affects IFT. However, wettability and interfacial tension (IFT) are assumed constant. Hysteresis is limited to the relative permeability model, similar to Killough’s model.17 A maximum gas residual saturation of 0.286 is used in all simulations. Capillary pressure and relative permeability are presented in Figure S1, Supporting Information (SI). Compressional-Wave Formulation. Compressional wave velocity in the matrix is a function of bulk and shear moduli, and density of the medium. Compressional or longitudinal wave or P-wave, Vp, can be estimated in partially saturated rock through Gassmann’s equation, eq 2,18 where Ksat, μ, and Fsat denote the saturated-bulk, shear moduli and density of the saturated rock. Based on this equation, bulk modulus is function of saturations of phases present in the pores, while shear modulus is not, therefore, μsat = μ (μ = 8.4 GPa). vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u uKsat 4 μ t 3 ð2Þ Vp ¼ Fsat Gassmann equation relates saturated bulk modulus to fluid-bulk, dry-bulk, matrix and porosity denoted by Kf, Kd (Kd = 8.3 GPa) Km (Km = 38 GPa) and ϕ, as follows: Kd 2 1 Km Ksat ¼ Kd þ ð3Þ ϕ 1 ϕ Kd þ 2 Kf Km Km We assume ideal mixing, therefore, mixing properties equal pure phase properties weighted by saturation. GassmannWood,8 eq 4, and GassmannHill,19 eq 5, represent the lower and upper bounds of mixing, i.e. uniform and patchy.9 The uniform case describes a mixing level where all pores have the same fraction of phases while patchy corresponds to situation where some pores are saturated with CO2 and the others saturated by water (no hydraulic connectivity between gas pockets in a single wave cycle). The mixing is somewhere between these two bounds, which might be estimated by Brie’s model.20
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1 Sw 1 Sw ¼ þ Kf Kw KCO2
ð4Þ
Kf ¼ Sw Kw þ ð1 Sw Þ KCO2
ð5Þ
Kf ¼ ðSw Þe Kw þ ð1 Sw Þe KCO2
ð6Þ
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where Sw, Kw (Kw = 3 GPa) and KCO2 (KCO2 = 0.046 GPa) are water saturation, and water and CO2 moduli. Brie’s formula, eq 6, becomes Hill’s model, eq 5, when e = 1 and approaches Wood’s model, eq 4, when e is high. The saturated-bulk density, Fsat, can be estimated through eqs 7 and 8. In this study, the density of each phase is calculated using an equation of state, which provides better estimates of density. Fsat ¼ Fd þ ϕ Ff
ð7Þ
Ff ¼ Sw Fw þ ð1 Sw Þ FCO2
ð8Þ
where Ff, Fd, Fw, and FCO2 are fluid, dry-bulk, water and CO2 density. Compressional Wave Velocity Upscaling. In the numerical simulation, properties are homogeneous at grid scale. Therefore, GassmannWood equation can be used to compute velocity as function of saturation, bulk and shear moduli. If the simulation is a seismic scale then velocity-saturation follows GassmannWood equation. Therefore, use of the finer scale and its consequent upscaling is necessary to calculate Vp-Sw relation. Here, the simulation resolution is 1 m-X 0.3 m-Z and Backus13 method has been utilized to upscale data to the seismic scale. Bakcus13 technique involves averaging the elastic moduli and bulk density of thin layers/grids into average properties similar to coarser thick layer/grid. This method has been successfully used for upscaling well logs to seismic wavelength.21 Sequential Backus averaging assumes a normal incidence ray path. That is, the stack of thin layers is horizontal and wavefront is normal to bedding, which is the case in this particular simulation. However, upscaling should be executed carefully because it may not preserve exact Vp-Sw relationship as Mukerji and Mavko22 showed. Backus’s method can produce variability of Vp-Sw relationship with a good estimate if Backus number B, eq 9, is less than a third.23 Liner and Fei23 results show that the simulated wave-field is similar for the original model and the averaged one if B is less than a third (scattered limit). Under this limit, scattered field and transmitted field will be preserved intact. B¼
f L0 minðVs Þ
ð9Þ
where f, L0 , and min(Vs) are the seismic frequency, averaging length and minimum shear velocity after upscaling. Here, we assumed the seismic frequency to be 35 Hz.
’ RESULTS The evolution of the CO2 saturation is depicted in Figure 1. Comparison between Figure 1b and Figure 1d shows that local capillary trapping does not keep CO2 at high saturation for a long period of time, though it stabilizes it at a still significant values. The CO2 saturation becomes essentially stationary after 40 years, but the simulation time was extended to 140 years for completeness. Saturation distribution changes from realization to realization as fluid flows (Figure 3). The rest of this section is organized as follow: first heterogeneous and homogeneous system are compared, and then accessibility ratio and trapping saturation among different realizations are compared. The Vp-Sw relationship as a function of time, specific realization and heterogeneity level are presented in the last part. Accessibility ratio is defined here as the ratio of CO2 accessed porous area to the total area, which is equivalent to sweep efficiency in the oil recovery context.
Figure 1. CO2 upward migration snapshots for DP = 0.7 at different times, simulation start time (a), 7 years (b), 40 years (c), and 140 years (d).
Homogenous vs Heterogenous. Heterogeneity increases CO2 upward migration time 1.5 to 3 fold. The maximum average gas saturation in the body of the homogeneous system happens after 8 years, which is much sooner than that of heterogeneous systems, e.g. 26, 28, and 33 years for realizations 1, 2, and 3. Figure 2 shows how the saturation for three different realizations (DP = 0.7) increases at early time and then stabilizes at the end of simulation. These results are consistent with Flett’s observations.24 He constructed different geostatistical models honoring the same distribution. In the models, porosity and permeability are subject to change. Flett concluded that heterogeneity can serve as an additional containment mechanism, although he did not consider capillary pressure heterogeneity. Heterogeneity results in more fragmented migration and lower accessibility ratio. The modeled accessibility ratio and trapped saturation are 100% and 9% in the homogeneous model, while they are 65% and 20% in the heterogeneous model, respectively. The accessibility ratio and gas trapping for different DP value are shown in Table 1. As expected, the average hydraulic trapping increases as DP increases, while the accessibility ratio reduces. The multiplication of these two parameters tells us that the total hydraulic trapping is higher in the heterogeneous system, therefore, the higher the heterogeneity, the more hydraulic trapped gas there will be the system.
total hydraulic trapping ¼ average hydraulic trapping accessibility ratio The effect of capillary pressure heterogeneity is often overlooked in the literature. Figure S2, SI, shows the contribution of capillary pressure heterogeneity on total hydraulic trapping versus time as well as accessibility ratio . It is observed that capillary 9437
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Figure 2. Average gas saturation as a function of time for three realizations at DP = 0.7, and a homogeneous mode. Gas saturation increases at early time and stabilizes at the end.
Table 1. Accessibility Ratio and Trapped Gas Saturation for Different Heterogeneity Level after 140 Years DP 0
accessibility ratio (%) 100
average trapped gas saturation (%) 9.0
0.3
97.8
9.8
0.5
80.9
12.3
0.7
63.6
19.5
0.9
42.2
30.3
pressure heterogeneity may increase the total hydraulic trapping up to 21%. Trapping saturation and Accessibility ratio. Saturation maps vary from realization to realization as shown in Figure 3. However, trapping saturation and accessibility ratio are the same. To show this, DP values of 0.3 0.5, 0.7, and 0.9 are used to generate a wide range of heterogeneity in the model. Trapping saturation and accessibility ratio are plotted as functions of the DP in Figure 4. As the error bar shows, trapping and accessibility do not vary much from realization to realization at a given value of DP. The standard deviation (error bar size) of the accessibility ratio is relatively high for DP = 0.9, perhaps because of the small system size. Vp-Sw Relationship. Elastic rock properties at each fine scale cell are computed based on Wood model and then upscaled using Backus method (B = (1/3), upscaled grids are 56 times larger in Z direction). Finally, Vp and Sw are plotted for each individual coarse grid. Figure 5 shows the Vp-Sw trend for realizations 1, 2, and 3 after 5 years when DP = 0.7. Although various saturation
Figure 3. Gas saturation distribution after 26 years for three different realizations, DP = 0.7.
distribution maps can be seen for these realizations (Figure 3), they follow similar Vp-Sw trend. Additional realizations results were consistent with this observed response. Therefore, as trapping and accessibility ratio, velocity-saturation relationship is similar among different realizations for the same DP value. The Vp-Sw relationship becomes more GassmannWood’s equation-like as time progresses, mainly because the saturation distribution becomes more uniform with time (Figure S3, SI). The velocity vs saturation is plotted for years 2013, 2015, 2020, 2040, 2060, and 2150 (simulations start in year 2010). Results for 9438
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Figure 4. Trapped gas saturation and accessibility ratio vs time for various levels of heterogeneity; these values are calculated for different realizations and their standard deviations are plotted as error bars.
Figure 5. Vp-Sw relationship for DP = 0.7 in realizations of 1, 2, and 3, after 5 years.
Figure 6. Fitted Brie’s model to different Vp-Sw relationship data sets.
years 2013, 2015, and 2020 are naturally grouped, while those for 2060 and 2150 appear to fall in the different category. The 2040s relationship sits between the two aforementioned groups. It can be noticed, as expected, that the Vp-Sw relationship for DP = 0.3 is very close to GassmannWood model as opposed to DP = 0.9, which is in between these two bounds. The velocity-saturation relationship for high heterogeneity (DP = 0.9) does not follow GassmannHill model, which is consistent with observed field evidence. The experimental evidence shows that the velocitysaturation relationship is close to the lower bound, Gassmann Wood, despite the existence of patches.11 Brie’s model may not be appropriate for prediction of velocitysaturation relationship. The model is fitted to all four DP sets of data and Brie’s exponent is estimated. It appears that the Brie’
model cannot predict well between upper and lower boundary frequency. Muller also recommended to employ Biot’s equation of dynamic poroelasticity rather than Brie’s experimental model.11 Figure 6 shows how Brie’s model fits to different VpSw data sets. Comparison with Other Results. Lebedev et al.25 utilized CT scan to observe patchy saturation and its effect on ultrasonic velocity. He also compared his observation with numerical simulation results of wave propagation in 2D using a finite-difference solver for Biot’s equations of dynamic poroelasticity. Although the patchy saturation distribution derived from CT scan was not used in the 2D numerical simulation, the results captured the overall behavior of the measured velocity-saturation relationship. Muller’s 11showed that the shape of the Vp-Sw relationship relates to the size and shape of patches. In Muller’s study, ν controlled 9439
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Environmental Science & Technology the anisotropic scaling of a random fractal. The greater ν is, the fewer, but bigger the patchy clusters are. Therefore, for the same saturation, the higher ν is, the more the patches are observed (closer to GassmannHill bound). Our results cover both Lebedev’s and Muller’s results. Based on our results, the velocitysaturation relationship can be shown to differ for different heterogeneity levels, which explains Muller’s conclusions. However, for different realizations of similar heterogeneity, this relationship is essentially identical, which explains Lebedev’s conclusion. In this sense, our modeling efforts help to place apparently opposite results in a coherent context. Limitation. For the sake of simulation numerical stability, the dissolution of CO2 in water is not considered, and therefore, the mixture modulus equals CO2 and water moduli weighted by volume fraction. However, the solubility of CO2 in water can be significant as Gilfillan et al.26 and Behzadi27 showed, and can alter the bulk modulus noticeably at different equilibrium stages, as shown in Vanorio et al.’s laboratory study,28 which means that bulk modulus dynamically changes along with encroaching CO2 flooding front, and this alteration is also valid for matrix itself (reaction with rock). Therefore, CO2-brine-rock interaction not only changes porosity, permeability and density, but also baseline properties for Gassmann fluid substitution scheme. GassmannWood’s equation may not hold at very low saturation and low frequency for liquidgas system where there is phase conversion under pressure oscillation characteristics of the acoustic wave. In this limit, LandauLifshitz’s method might be used, instead. Wood’s model may result in optimistic approximation of velocity and consequently underestimation of gas saturation.29,30 Employing LandauLifshitz’s method, thermodynamically equilibrated, as opposed to Wood’s method, frozen, causes even better detectability of gas fingerprint at very low saturation. LandauLifshitz’s method Shifts down the lower boundary and increases the envelop volume between two boundaries. This increase in the envelop volume helps clearer effect of time and DP on Vp-Sw relationships, that is, separates more Vp-Sw relationship of DP = 0.5 from that of DP = 0.7.
’ DISCUSSION This study sheds light on how heterogeneity affects trapping, storage and velocity-saturation relationship. This is important to quantitatively enhance subsurface CO2 monitoring. This study also shows that limited number of stochastic realizations might be sufficient, at least in terms of producing trapping, storage, VpSw relationship estimates. In this study, capillary pressure was parametrized with respect to permeability. The results show that absolute permeability, and hence capillary pressure heterogeneity significantly affect trapping. Moreover, both static heterogeneity, namely absolute permeability heterogeneity, and dynamic heterogeneity, associated with multiphase flow function distributions, can improve hydraulic trapping, but may reduce accessibility ratio and dissolution trapping. The aforementioned heterogeneity types, that is, static and dynamic, significantly constraint CO2 migration toward and beneath a caprock, consequently reducing the risk of CO2 leakage. This outcome, as shown in the results, denotes the importance of properly incorporating this flow controls, often neglected in modeling exercises. Results for multiple realizations at the value of the Dykstra Parson coefficient clearly show that total hydraulic trapping and accessibility ratio are weak functions of the specific realization of given level of heterogeneity in the random model. Even more interesting in
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the fact that the Rock Physics response of interest for seismic monitoring, for example, the velocity-saturation relation, is mainly a function of the heterogeneity level (DP), a varies insignificantly from realization to realization at the same value of the DykstraParson coefficient. This means that in order to quantify (bookkeep) CO2 in storage sites by means of time-lapse seismic, it is considerably more important to know the nature of the distribution of heterogeneity and not so much the specific details.
’ ASSOCIATED CONTENT
bS
Supporting Information. Relative permeability and capillary pressure, Variation of velocity-saturation through elapsed time and effect of capillary pressure heterogeneity are presented in Figures S1, S2 and S3 respectively. This information is available free of charge via the Internet at http://pubs.acs.org/
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT We acknowledge CMG, Ltd. for providing academic licenses of the software suite. We thank the Enhanced Oil Recovery Institute at the University of Wyoming for financial assistance. Financial support was provided by DOE through grant DEFE0001160. ’ REFERENCES (1) Martinsen, D.; Linssen, J.; Markewitz, P.; V€ogele, S. CCS: A future CO2 mitigation option for Germany? A bottom-up approach. Energy Policy 2007, 35, 2110–2120. (2) Odenberger, O.; Johnsson, F. The role of CCS in the European electricity supply system. Energy Procedia 2009, 1, 4273–4280. (3) Saadatpour, E.; Bryant, S. L.; Sepehrnoori, K. New trapping mechanism in carbon sequestration. Transp. Porous Media 2010, 82, 3–17. (4) M€uller, N.; Ramakrishnan, T.; Boyda, A.; Sakruai, S. Time-lapse carbon dioxide monitoring with pulsed neutron logging. Int. J. Greenhouse Gas Control 2007, 1, 456–472. (5) Chadwick, R. A.; Noy, D.; Arts, R.; Eiken, O. Latest time-lapse seismic data from Sleipner yield new insights into CO2 plume development. Energy Procedia 2009, 1, 2103–2110. (6) Urosevic, M.; Pevzner, R.; Shulakova, V.; Kepic, A.; Caspari, E.; Sharma, S. Seismic monitoring of CO2 injection into a depleted gas reservoir Otway Basin Pilot Project, Australia. Energy Procedia 2011, 4 3550–3557. (7) Preston, C.; Monea, M.; Jazrawi, W.; Brown, K.; Whittaker, S.; White, D.; Law, D.; Chalaturnyk, R.; Rostron, B. IEA GHG Weyburn CO2 monitoring and storage project. Fuel Process. Technol. 2005, 1547–1568. (8) Wood. A Text Book of Sound; G. Bell and Suns LTD: London, 1941. (9) Mavko, G.; Mukerji, T. Bounds on low-frequency seismic velocities in partially saturated rocks. Geophysics 1998, 63, 918–924. (10) White, J. Computed seismic speeds and attenuation in rocks with partial gas saturation. Geophysics 1975, 40, 224–232. (11) Muller, T. M.; Toms-Stewart, J.; Wenzlau, F. Velocity-saturation relation for partially saturated rocks with fractal pore fluid distribution. Geophys. Res. Lett. 2008, 35, L09306. (12) Toms, J.; Muller, T. M.; Curevich, B. Seismic attenuation in porous rocks with random patchy saturation. Geophys. Prospect. 2007, 55, 671–678. 9440
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