## Abstract

Offshore gravimetric monitoring has been introduced as a complement to seismic monitoring of fields with moving fluids. The Sleipner field in the North Sea is a fully operational carbon capture and storage facility, where $CO2$ is injected for storage. Gravimetric measurements are one of the geophysical monitoring methods applied, and the data have been used to estimate the in situ density and dissolution of the $CO2$. We defined a Bayesian inversion of gravimetric data, and we used this to analyze gravimetric data at Sleipner field. In our approach, we included spatial uncertainty in the model and performed a Bayesian analysis of the in situ $CO2$ density and dissolution. We also analyzed the impact of mass changes due to gas production from the Ty Formation. Our estimates were comparable with published results.

## Introduction

The Sleipner field in the North Sea is a fully operational carbon capture and storage facility. The $CO2$ is injected in a saline aquifer — the Utsira Sand, and the migration of the gas is monitored by time-lapse seismic surveys, time-lapse seabed gravimetric measurements, electromagnetic surveys, and seabed mappings. The key aims of the monitoring are to ensure safe storage of $CO2$ within the storage reservoir, to keep track of the distribution and migration of the gas through the reservoir and possibly into adjacent strata, and for early detection of leakage toward the seabed (Chadwick et al., 2006). Interpretation of time-lapse seismic surveys gives a good image of the distribution and migration of the $CO2$ plume, including the depth, whereas gravimetric measurements provide complementary constraints on the in situ density of the injected $CO2$.

For the Sleipner field, the first gravimetric measurement in August 2002 serves as the baseline for time-lapse gravimetric surveys. As $CO2$ is injected into the storage reservoir, it fills the pore space, displaces the brine, and reduces the total density of the rock. The density decrease has an effect on the local strength of gravity. Monitoring these changes in the gravity strength can give constraints on the in situ density of $CO2$. For the Sleipner field, there is a significant uncertainty in the storage reservoir condition regarding temperature and pressure for all stages in the process, that is, before, during, and after injection. At the predicted temperature and pressure conditions, the $CO2$ is close to the critical point of phase transition. This means that small changes in reservoir temperature may cause large changes in the $CO2$ density. The spatial variability, which has not been incorporated in the analysis previously, might give important contributions to the uncertainty.

Previous work by Nooner et al. (2007) uses time-lapse gravity measurements together with forward gravity modeling based on seismic data and reservoir simulation models to constrain the in situ density of $CO2$ within the Utsira Formation. They provide a best-fit average in situ $CO2$ density of approximately $530\xb165\u2009\u2009kg/m3$ (95% confidence interval). Later work by Alnes et al. (2008) has reprocessed the original gravity data, resulting in a new best-fit average density of $CO2$ of $760\u2009\u2009kg/m3$, with an upper bound of $770\u2009\u2009kg/m3$ and a lower bound of $640\u2009\u2009kg/m3$ at 95% confidence. Both of these works base their analysis on time-lapse data from 2002 to 2005. Alnes et al. (2011) include gravimetric measurements from 2009 and provide an updated best-fit average density of $720\xb180\u2009\u2009kg/m3$, assuming there is no dissolution of $CO2$ in the formation water. They also combine the gravimetric data with improved temperature measurements to provide an upper bound on the dissolution of $CO2$ in formation brine.

Bayesian inversion of geophysical data contributes to the analysis of geophysical data by mapping the uncertainty in data into uncertainty about earth parameters (Tarantola and Valette, 1982; Duijndam, 1988a, 1988b; Scales and Tenorio, 2001; Malinverno and Briggs, 2004; Bosch et al., 2007). Several types of geophysical data with different sets of models have been studied within this context (Gouveia and Scales, 1998; Buland et al., 2003; Gunning and Glinsky, 2004; Hoversten et al., 2006; Buland et al., 2011). Rock-physics models combined with geophysical data can be used to obtain a quantitative prediction of properties in the subsurface reservoirs, e.g., Mukerji et al. (2001). It is common to use a two-step approach by first performing a Bayesian inversion of the geophysical data and next interpret the model in light of the uncertainties in the inversion (Doyen, 2007; Buland et al., 2008). Hauge and Kolbjørnsen (2014) present the Bayesian inversion of gravimetric data and show quantitatively the information content held by the data in terms of which features that are resolved.

In this paper, we use the two-step approach to geophysical inversion; that is, we define a Bayesian inversion of gravimetric data and show how this can be used for analyzing properties derived from gravimetric data. In particular, we do a Bayesian analysis of gravimetric data from the Sleipner project. We show how a geostatistical framework makes it possible to account for the spatial uncertainty and include multiple sources of information. We also discuss how the Bayesian approach can be used to analyze the contributions due to gas production at the Ty Formation. Our estimates for in situ $CO2$ density and dissolution at the Sleipner injection site are in good agreement with estimates provided in the existing literature.

## Method

We present here the Bayesian inversion of gravimetric data using a geostatistical methodology, and we show how to include the contributions from an additional source of mass changes.

### Bayesian inversion of gravimetric data

In the Bayesian approach, it is required that a priori knowledge about the field under study is quantified. In the field of reservoir characterization, there is substantial knowledge about modeling of rock-physics properties at the geomodel scale (Dubrule, 2003; Eidsvik et al., 2004; Doyen, 2007). This is, therefore, a natural scale to build the a priori information into the model. This model is, however, too fine to make efficient computations; thus, we perform an intermediate step of upscaling to provide the model with uncertainty on a computational/coarse scale grid. To assess the error in the upscaling, we generated 100 models from the a priori distribution and computed the gravimetric response using the fine-scale and the upscaled approaches. The increase in the uncertainty added in the error standard deviation when including this component is less than 1%. The coarse-scale grid is thus more than sufficient for resolving the information content in the gravimetric data. By introducing the geomodel as a reference, we are able to include rock physical and geostatistical knowledge such that the variability at the coarse-scale model is physically based.

### Including additional reservoirs in inversion model

We will perform the Bayesian analysis of data from the Sleipner injection project. These data are influenced by gas production from the Ty Formation, which is located laterally close to the Utsira injection site. To incorporate this in our analysis, we need to introduce an additional reservoir with density changes.

### Bayesian analysis of inversion results

## Bayesian analysis of the Sleipner gravimetric data

The gravimetric data analyzed here are from the Sleipner injection site. We discuss the 2002–2005 and 2002–2009 changes. The data have been processed to remove the effects of the underlying Ty Formation, which is a gas-producing field deep below the Utsira formation and located laterally very close to the injection site (Alnes et al., 2008, 2011).

### Defining statistical models

We consider analysis with and without accounting for the uncertainty in the Ty Formation. In addition, we do the analysis with an open a priori distribution and an a priori distribution constrained by pushdown from seismic data. The pushdown is the change in time interpretation of a seismic reflector below the reservoir when comparing two surveys. Substitution of $CO2$ into the rock reduces the seismic velocity and causes the reflector to appear deeper on the monitor survey. An overview of the inversion cases is given in Table 1.

We carry out gravimetric inversion with the Sleipner data on a storage reservoir extending 2500 m in the east direction and 5500 m in the north direction and with thickness of 200 m. The top of the reservoir is placed 820 m below mean sea level. The geomodel grid is resolved into $100\xd7220\xd7100$ cells with cells of size $25\xd725\xd72\u2009\u2009m3$. The computational grid that we perform the inversion on is resolved into $10\xd722\xd74$ blocks with size $250\xd7250\xd750\u2009\u2009m3$. The lateral geometry of the data and inversion region is given in Figure 1. When building the a priori model, information from alternative sources might be used to provide a rationale for the assumptions.

According to Alnes et al. (2011), temperature considerations suggest that the average $CO2$ density in the Utsira region is $0.675\xb10.02\u2009\u2009g/ccm$. Using the center of this interval and if we in addition neglect dissolution and use the injected mass in Table 2, we can provide a rough estimate for the mass change using equation 14. In the basic a priori distribution, we divide this equally in all grid cells in the inversion region. For the a priori distribution constrained by seismic, we divide the mass evenly along a vertical profile. Because the pushdown gives a strong indication of the presence of $CO2$, we reduce the uncertainty in density outside the pushdown region. Moreover, because the magnitude of the pushdown is an indication of the amount of $CO2$ in a profile of the $CO2$, we distribute the mass changes proportional to the pushdown change, i.e., the pushdown from 2001 to 2006 and 2001 to 2008, respectively. Note that because we allow full variability around this mean, the a posteriori density change will not be proportional to the pushdown. Figure 2 shows a smoothed version of the pushdown for the years 2001, 2006, and 2008, respectively.

The Ty Formation is located approximately 2300 m below mean sea level. The lateral region with the disturbance is approximately described by a square stretching from the southwestern point with Universal Transverse Mercator (UTM) coordinates (434, 6466) km and 3.5 km to the east and 8.5 km to the north; see Nooner et al. (2007). The thickness of the zone is very small compared to the resolution in gravimetric data; thus, we model only the spatial distribution of lateral changes. The ranges are set to be 500 m in both lateral directions. The gravimetric data analyzed in this section were processed to remove the effect of the Ty Formation (Alnes et al., 2008, 2011). Therefore, we consider the bias in equation 10 to be removed, but the uncertainty remains; thus, only the covariance is required. Moreover, Alnes et al. (2008) estimate that the total mass has an uncertainty in the order of 5 MT. To give room for spatial deviations from the mean scale, the uncertainty is set such that the total mass change has a standard deviation of 5 MT. In a cell of the upscaled grid of $500\xd7500\u2009\u2009m$, the uncertainty is 0.26 MT, which corresponds to $1050\u2009\u2009kg/m3$.

### Gravimetric inversion

The measurement positions relative to the lateral outline of the reservoir are shown in Figure 1. We will consider two sets of measurements: The first set of data is the change in gravimetric response from 2002 and 2005, and the second set is from 2002 to 2009. The standard deviation for the measurement errors (given as part of the data) $\sigma w$ is approximately $3\u2009\u2009\mu Gal$, and the standard deviation for the scalar level shift is set to $\sigma c=1\u2009\u2009\mu Gal$. Figure 5a and 5b shows the processed data (Alnes et al., 2008, 2011) together with the response from the a priori mean as defined above.

The results of the Utsira inversions are displayed in Figures 6 and 7 for both data sets. The corresponding data fits are displayed in Figure 8a and 8b.

It is interesting to notice that both inversions (see Figures 6 and 7) suggest positive changes slightly off target, but compared to the uncertainties in the standard deviations, these are not significant. For the basic a priori distribution, the northernmost region has been very little influenced by the data and the uncertainty remains almost at the size of the a priori distribution. There is very little visual influence when including the Ty Formation. Figure 9 shows the effect on data caused by including the Ty Formation. Inspecting the inversion of Ty in Figure 10, we find weak indications of a positive contribution in the northern part. The uncertainties in the inversion are, however, large, and the spatial resolution is very poor. This reflects the lacking ability to constrain the spatial model.

### Bayesian analysis of in situ CO_{2} density and dissolution

The target is to analyze the $CO2$ density and dissolution constrained by gravimetric data. Equation 14 relates this to the mass change. Table 3 shows the estimated mass changes and corresponding uncertainties. Note that all a posteriori standard deviations are considerably smaller than the a priori ones; thus, we might conclude that the gravimetric data carry much information about the total mass change. For the unconstrained a priori distributions, the a posteriori uncertainty is still much larger than the estimated effect. Thus, for these cases, we are not able to resolve the mass changes to a significant degree. In general, we see that inclusion of the Ty Formation results in a larger mass reduction; also, the uncertainty in the estimate is increased.

Alnes et al. (2011) report that $k=10.0\u2009\u2009yr$ for the 2002–2009 survey. Assuming a constant rate of injection between 2005 and 2009, we find that $k=7\u2009\u2009yr$ for the 2002–2005 survey. Assuming that the density is in the range $0.4\u22120.8\u2009\u2009g/ccm$ and that the dissolution is in the range 0.0%–10.0%, we can compute the likelihood of the combinations. This is shown in Figure 11 for the seismic constrained cases. We find that 2009 data are better resolved and have the mode of the mean density at higher values than the 2005 data. This reflects the larger uncertainty in the 2005 data. It is also reasonable that the 2005 data could have a lower average density if a larger fraction of the injected $CO2$ has cooled down in 2009. Note that the data do not separate the two effects, that is, density and dissolution. If the focus is on the dissolution and we assume that the average density of $CO2$ from both survey times is independent and has a normal distribution centered in 0.675 and a standard deviation of 0.01, according to computations from Alnes et al. (2011), we can compute the a posteriori distribution for the dissolution by summing out the densities. These a posteriori distributions are shown in Figure 12. The mean value is 0.67% if we neglect the effect of the Ty Formation. Including the Ty Formation in the analysis, the mean increases to 0.72%. In either case, the probability of exceeding 1.76% dissolution is less than 5%.

## Conclusion

We have shown how it is possible to account for spatial variability in gravimetric inversion. For the Sleipner data, the lateral resolution is poor, but when we constrain our model with seismic information, we are able to make quantitative predictions about the $CO2$ density and dissolution. Using the restriction of the spatial average of the density provided by Alnes et al. (2011) and analyzing the 2009 data only, we find an upper limit (1.83%) for dissolution that is rounded down to the number provided by Alnes et al. (2011). Including the data from 2005 as well reduces the upper limit, but the value is still rounded up to the number provided by Alnes et al. (2011). The upper bound we obtain for dissolution is 1.76% per year, whereas the mean value is approximately 0.7% per year. The bound is sensitive to the assumptions related to the average $CO2$ density.

## Acknowledgments

We thank the Research Council of Norway, Statoil, and ExxonMobil for their financial support for the project “Monitoring geological $CO2$ storage: Quantitative $CO2$ prediction with uncertainty from physical modelling and multiple time-lapse data types” under which part of this work has been done. We also thank the Sleipner license partners, Statoil, ExxonMobil, and Total for permission to publish Sleipner field gravimetric data, and finally, we thank A.-K. Furre, H. Alnes, and O. Eiken for useful discussions on this topic.

Biographies and photographs of the authors are not available.