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However, by ordering the polling directions as described in Section 2.2.3, this optimization required us to access the computational cluster only 101 times. Thus this approach provides an improvement in computational efficiency of more than a factor of three for this case.

Normalized data misfit −1 −2

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Figure 4.16: Progression of the GPS minimization of data misfit Figure 4.

17 shows typical history match results for (a) pressure over the first 100 years for a particular sensor and (b) CO2 breakthrough times for all eight blocks associated with a single sensor well. In both figures, the history matched solution is seen to be nearly identical to the true solution. Expected results from a-priori models do, however, display some error, as would be expected. The somewhat abrupt shifts in the pressure profiles in Figure 4.17(a) correspond to changes in the CO2 injection


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Figure 4.17: Comparison of history matched solution with the true model (TR1) and with expected values from the prior model The results above demonstrate the ability of the data assimilation algorithm to provide a geological model that matches observed data.

The real purpose of history matching is, however, to generate models that can provide reliable predictions of future performance. Our specific interest here is in predicting the time-averaged CO2 mobility in the top layer, so our measure of the efficacy of the history matched model will be the error in this quantity.

Table 4.3 compares the error in top layer CO2 mobility (with error computed as in Equation 4.

9 for T = 500 years) between the true and history matched models.

Results are presented for the five geological models TR1 - TR5. The second column in the table displays the error for the prior model, which is essentially the difference (in the sense of Equation 4.9) between simulated top layer mobility results for the particular geological realization and results averaged over the 10 prior realizations.

The latter correspond to expected results given prior information. History matching clearly acts to reduce this error. For the monitoring scenario with four wells and 30 years of data, the average error is reduced by 29%, while for monitoring with eight wells and 100 years of data, it is reduced by 46%. The increased accuracy observed through use of more wells and a longer monitoring period is consistent with

4.5. DATA ASSIMILATION RESULTS Table 4.3: Errors in prior and history matched models for five geological realizations

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Errors in the top layer mobility for the history matched model (generated using eight sensor wells and 100 years of data) and for the prior model are displayed in Figure 4.18. These results are for geological model TR1. For this model, history matching reduces the mobility error from 2021 to 1125 (a 44% reduction). The decrease in error is evident in the figure, particularly to the left of W2 and W3 and around W4. We found that the performance of the history matching was not particularly sensitive to small changes in the sensor locations and data weight parameters.

For example, when the optimal sensor locations and data weight parameters for TR1 were used in history matches for TR2 to TR5, the reduction in average error (relative to that for the prior model) for the eight-well, 100-year cases was 43%. This is close to the 46% value we obtained when locations and data weight parameters were optimized for each true geology individually.

The results in this section demonstrate that, by collecting data at monitoring and injection wells and then applying data assimilation procedures, it is possible to generate history matched models that provide improved predictions of key quantities such as top layer CO2 mobility. The basic procedures presented here could be extended to incorporate additional data types (e.g., 4D seismic, other sensor data) or to enable the representation of different kinds of geological models, such as those characterized by multipoint spatial statistics. Multiple history matched models could also be generated, using a framework based on sampling from the posterior distribution


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Figure 4.18: Error in time-averaged mobility in top layer for history matched and prior models, for true model TR1 (along the lines of Equation 4.

5) or other approaches such as ensemble Kalman filtering, to provide a measure of prediction uncertainty. Following extensions along these lines, history matched models could be used in practical settings for the real-time optimization of CO2 injection and for the prediction of long-term CO2 migration.

4.6 Summary In this chapter, we developed and applied a set of computational procedures for the closed-loop management and optimization of geological carbon storage operations.

The optimization procedure described in Chapter 3 was extended to treat cases with uncertain geology. A data assimilation (history matching) procedure, which uses injection well bottomhole pressures and measurements at sensors (monitoring wells) as observed data, was also developed. The geological model was represented using a Karhunen-Lo`ve (K-L) expansion, which enables the use of relatively few optimization e parameters in the data assimilation. The resulting history matched model can be used for the real-time optimization of CO2 injection and for the prediction of longtime plume migration. A technique for optimizing the locations of the sensors was also presented. Particle Swarm Optimization (PSO) was used to jointly optimize

4.6. SUMMARY well locations and CO2 injection rates, while Generalized Pattern Search (GPS) was applied for the minimization required for data assimilation.

Results for a range of problems demonstrated the capabilities of our optimization procedures. The objective function considered was the time-averaged total CO2 mobility in the top layer of the model, which was intended to represent a measure of potential environmental or economic damage due to leakage through the cap rock.

Optimization results for deterministic models demonstrated substantial reduction, around 39% on average, in this quantity relative to default cases. The use of brine cycling, in which brine is produced at the bottom of the aquifer and reinjected at the top, was shown to provide even greater reductions, particularly when large volumes of brine were cycled. This is consistent with our findings in Chapter 3.

Less advantage was gained from optimization when the geological model was uncertain. In addition, the maximum benefit from closed-loop operation (with a-priori optimal well locations) was shown to be only 7% in this case. This limited benefit occurred because the well locations were determined using prior models, when very little geological information was available. As a result, well locations could be significantly suboptimal for particular realizations, and optimizing the injection strategy had relatively little impact on top layer CO2 mobility. Data assimilation results demonstrated that pressure and CO2 breakthrough time measurements at sensors, combined with BHP data from the injection wells, can significantly improve the accuracy of the geological model. Specifically, using eight monitoring wells and 100 years of data, we found that the average error in top layer mobility was reduced by 46% relative to the average error in results from the prior model.


Leakage detection from historymatching

In this chapter, in order to study leakage, we extend our previous geologic model to a three-region model, which includes a storage formation region, a cap rock region and an overlying aquifer region. Many realizations of leakage scenarios are generated and simulated to quantify the fraction of injected CO2 we can expect to leak based on leakage location, leak permeability, and aquifer geology. From these realizations, we select several ‘true’ realizations to test the performance of our history matching algorithm for characterizing leakages in the cap rock using different amounts of monitoring data.

The aquifer geology is represented in the history matching process using the Karhunen-Lo`ve (K-L) parameterization. Leaks in the cap rock are represented using e additional optimization variables. Particle Swarm Optimization (PSO) is applied to minimize the misfit function, which is more complex (due to the leak variables) than the misfit function considered in Chapter 4, where Generalized Pattern Search was applied in the minimization problem.

5.1 Leaky aquifer model In order to model leakage effects, we extend our previous model to include a singlelayer cap rock, which may contain leaks, and a four-layer overlying aquifer, as shown


in Figure 5.1. This figure shows the central portion of the model (25 × 25 × 13 grid blocks, 160 m in thickness, and 10.9 km horizontally), which is contained within a larger 39 × 39 × 13 model (as in Figure 2.5) that extends 230 km horizontally and represents a large-scale regional aquifer. We will henceforth refer to the three regions of the model as the ‘storage formation’ (25 × 25 × 8 lower region), ‘cap rock’ (25 × 25 × 1 middle region) and ‘overlying aquifer’ (25 × 25 × 4 upper region). Four CO2 injection wells are placed and controlled according to the optimal a-priori solution determined in Chapter 4. These are the green wells at the bottom of the model in Figure 5.1. We also consider several vertical pressure monitoring wells, to be used for data assimilation. We will consider a ‘data-rich’ monitoring scenario with nine monitoring wells completed in both the storage formation and overlying aquifer (i.e., both the red and blue wells in Figure 5.1), as well as a ‘data-scarce’ monitoring scenario, with only four monitoring wells that do not penetrate the storage formation (i.e., the red wells in Figure 5.1).

Geologic heterogeneity in the storage formation and overlying aquifer is modeled using the same variogram model as in Chapter 4. An example of the porosity field for one geologic realization is shown in Figure 5.2. Realizations are conditioned to honor, as hard data, the grid block porosities at the four CO2 injection wells and all nine pressure monitoring wells. In our examples hard data in the storage formation are taken from the TR1 model, which was introduced in Chapter 4. Hard data in the overlying aquifer are taken from a randomly chosen unconditioned realization generated using SGeMS (Remy et al., 2009). These particular realizations are depicted in Figure 5.2. The permeability fields in the storage formation and overlying aquifer are calculated from porosity using Equation 2.12 with a = 4.5 and b = 2.

We assume the cap rock to have zero permeability, except at leakage locations, where the vertical permeability of a single block may range from 0.005 to 100 md. The leakage block permeability kz,l is intended to be representative of an upscaled value for an underlying leakage conduit such as a fracture zone (or possibly an abandoned well) within the block. Therefore, the permeability of an actual leakage conduit would likely be significantly higher than the kz,l value we assign to the block. For abandoned wells and fracture zones, permeabilities between 0.1 md and 2000 md have been considered by Nordbotten et al. (2005), Ebigbo et al. (2007), Chabora & Benson

5.1. LEAKY AQUIFER MODEL Figure 5.1: Three-region model depicting the overlying aquifer, cap rock and storage formation. CO2 injection wells are shown in green, and pressure monitoring wells are shown in red or blue. The blue wells are specific to the data-rich monitoring scenario, while the red wells are used in both scenarios (2009) and Sun & Nicot (2012). When these are upscaled to kz,l values for grid blocks of the dimensions considered here, they fall within our (intentionally) large range of upscaled permeabilities. The porosity of all leakage blocks is set to 0.01. Extensive model testing revealed that the porosity of a leakage block has little direct effect on the leakage rate or pressure response at the monitoring wells.

As discussed previously, in order to reduce computational time, our models do not consider capillary pressure heterogeneity, though it has been shown by Krause et al. (2009) that its inclusion may act to restrict flow through low permeability leaks. While this may be an important physical process, it is difficult to include here because our kz,l values represent upscaled permeabilities. For current purposes, we will assume that any capillary pressure effects have been approximately incorporated into the kz,l values (to the extent possible). We note that a method for performing such an upscaling should be considered in future work.

The number of leaks in the cap rock (i.e., grid blocks with nonzero kz,l ) can be anywhere from 0 to L, where L is defined as the number of leakage sites. Each leakage site is characterized by three variables (il, jl, kz,l ), for l ∈ {1,..., L}, defining the grid block indices and the upscaled permeability of the block. When we generate


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Figure 5.2: One geologic realization of the three-region model showing the porosity field in both the overlying aquifer and storage formation.

The cap rock region is shown in black. This particular realization provides hard data at the four CO2 injection well and nine sensor well locations, which are used for conditioning other realizations

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where l may pertain to a particular leak, ⌈...⌉ denotes the ceiling function for conmin version from real to integer values, and η3 = ln(0.005) for single-leak cases or min η3 = −35 for multi-leak cases. With this approach, when η3,l is drawn from a uniform distribution, leakage sites in single-leak cases are always leaks, whereas leakage

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