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The models used here contain relatively few grid blocks because large numbers of simulations must be performed for optimization and data assimilation, and these computations would be prohibitive with fine-scale models. Ideally, the coarse simulation models would be constructed via a detailed upscaling of underlying high-resolution models (though such an approach is not applied here). The upscaled model would then contain properly computed upscaled permeabilities (or transmissibilities) as well as coarse-scale relative permeability and capillary pressure functions. See Chen & Durlofsky (2006) for a discussion of upscaled models for two-phase flow simulations.

The optimization determines the locations of four horizontal CO2 injection wells, each of maximum length 1304 m (the length of 3 grid blocks). The maximum amount of CO2 that a well can inject in any time interval is 40% of the total CO2 injected (this maximum acts to limit the pressure increase in the vicinity of the well). Four horizontal water injectors are also used along with producers in boundary blocks in cases with water cycling events, as described previously (see Figure 2.1).

Figure 2.5: 39 × 39 × 8 areal grid with 25 × 25 × 8 storage aquifer shown as white square in the center The aquifer model is heterogeneous with porosity φ generated using sequential Gaussian simulation (Remy et al.

, 2009). Porosity is normally distributed with mean ¯ φ = 0.2 and standard deviation σφ = 0.025. The porosity distribution is truncated on the interval [0.15, 0.25]. A spherical variogram model is used, with correlation lengths of cx = 8∆x (in Chapter 3) and cx = 24∆x (in Chapters 4 and 5) in the x-direction, cy = 8∆y in the y-direction, and cz = ∆z in the z-direction, where ∆x,

2.3. AQUIFER MODEL Figure 2.6: 25 × 25 × 8 storage aquifer showing log-permeability field. Injection wells can be placed within the inscribed 15 × 15 × 8 ‘core’ region ∆y, ∆z designate grid-block dimensions. This defines the prior realizations in cases with geological uncertainty. When injection wells and sensor wells are drilled, the porosity values from a particular realization, taken to be the actual (true) model, are included as hard data.

Permeability (kx ) is related to porosity using the relation ¯ φ−φ kx = exp a + b, (2.12) σφ where a = 4.2 and b = 1.5 in the model in Chapter 3, resulting in a log-normal permeability distribution for kx with a median kx of 67 md and minimum and maximum values of 3 md and 1300 md. Examples in Chapters 4 and 5 use a slightly more heterogeneous model, with a = 4.5 and b = 2 in Equation 2.12. This results in a log-normal permeability distribution between about 2 md and 5000 md with a median of 90 md. In all examples we specify ky = kx and kz = 0.1kx.

Relative permeability curves, shown in Figure 2.7, were derived by applying the Brooks-Corey parameters found by Krevor et al. (2012) for a Berea sandstone. The connate water saturation is Swc = 0.11 with corresponding maximum CO2 relative permeability of krg (Swc ) = 0.95. The Brooks-Corey exponents are Nw = 6 and Ng = 5. Note that krw is very low for Sw ≤ 0.5 (i.e., Sg ≥ 0.5). Hysteresis is represented using Killough’s method (Killough, 1976) with a Lands trapping coefficient


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Figure 2.7: Relative permeability curves (Krevor et al.

, 2012) for water (drainage only) and CO2 (drainage and imbibition) of 1. Capillary pressure effects are not included in our models. Limited investigation showed that the impact of capillary pressure was small if the capillary pressure curves were taken to be the same in all grid blocks. However, it has been shown in Saadatpoor et al. (2010) and Krause et al. (2009) that heterogeneity in capillary pressure can be an important effect in CO2 trapping, and this should be considered in future work (though we note that capillary heterogeneity can significantly increase simulation time, and may also require much finer grid resolution).

All simulations are performed using the CO2STORE option in the ECLIPSE simulator. The equation of state calculations within the simulator match experimental data from Spycher & Pruess (2005). When CO2 dissolves in the aqueous phase, the increased density leads to gravity-driven transport, though this effect may be underestimated due to the large grid blocks used in our models. The initial salinity of the in-situ water is 10,000 ppm. Salt precipitation is not modeled, though water mobility is affected by salt concentration. As we mentioned previously, we do not include mineralization in this study. Our simulations also do not resolve detailed density-driven instabilities, as computed by, e.g., Riaz et al. (2006). Incorporating such effects in

2.3. AQUIFER MODEL practical engineering models could be accomplished in theory by computing appropriate coarse-scale flux functions from detailed fine-scale results, but robust procedures for this upscaling are not yet available.

The simulations in this work are all run fully implicitly, with time steps of about 50 days during the injection period and up to 3650 days, or 10 years, after injection is completed (fluid movement slows dramatically during this time so we can use large time steps). More time steps are required for cases with brine cycling, since this entails additional injection. We typically use approximately 350 time steps for 1000-year simulations without brine cycling and 450 time steps for 1000-year simulations with brine cycling. Run times for a single 1000-year simulation are around 3-5 minutes, and up to 7 minutes for cases with brine cycling. In some optimization examples, we run 500-year or 100-year simulations. These typically reduce the run time of a single simulation by about 30 seconds to one minute. Some data assimilation examples require only 10-year, 30-year or 100-year simulations, and none of these include brine cycling. Run times for those examples are typically around 1 minute, 2 minutes and 3 minutes, respectively.


Optimization results with knowngeology

In this chapter we present extensive optimization results for optimization examples where the geology is known. The objective function in most cases is the fraction of mobile CO2 in the formation after 1000 years, though other objective functions and shorter time frames are also considered. Results for cases without and with brine cycling are provided. Hooke-Jeeves Direct Search (HJDS) is applied unless otherwise indicated (when this work was performed, the requisite simulations were run serially because we did not have access to multiple software licenses). We also investigate the sensitivity of optimization results to grid refinement.

All examples in this chapter use the geological model depicted in Figure 2.6. This model was produced by Sequential Gaussian Simulation using the Stanford Geostatistical Modeling Software (Remy et al., 2009). As stated previously, porosity values ¯ are normally distributed with mean φ = 0.2, and standard deviation σφ = 0.025.

Permeability is related to porosity via Equation 2.12, with a = 4.2 and b = 1.5. We note that different values for a and b will be used for models in Chapters 4 and 5.

3.1 Optimization without brine cycling We first consider a case with no water cycling. Because HJDS is a local optimizer, it is useful to run the optimization multiple times. Results for three optimization runs,


using different initial guesses for well placement parameters x0 but the same initial guess for injection rate parameters (rw,t = 0.25, or equal fractions at all time), are shown in Figure 3.1. The figure shows the best solution achieved up to the current iteration. It is evident that each optimization run gives a different result, indicating the presence of multiple local optima. The best initial guess has a mobile CO2 fraction of around 0.30, whereas the best optimized solution, Run 1, has a mobile CO2 fraction of 0.22. Thus we see clear improvement from the optimization.

Particle Swarm Optimization (PSO) was also applied for this example and it provided comparable optimal solutions. While PSO required fewer iterations than HJDS (100 compared with around 500), it performed more function evaluations (2000 compared with around 500) because each iteration entailed the simulation of 20 potential solutions (particles). Because PSO, in contrast to HJDS, readily parallelizes, it should be considered if parallel resources (and simulation licenses) are available.

The initial guesses for the well locations for the three runs are depicted in Figure 3.2(a). Each configuration forms a nearly square pattern, with two of the wells extending in the north-south direction and two in the east-west direction. Figure 3.2(b) shows the optimized well locations for each run. Although the solutions are clearly different, many of the wells are positioned in the same general areas, suggesting that certain injection locations lead to reduced mobile CO2. This is presumably related to the detailed structure of the permeability field, although it is difficult to identify direct relationships between particular (local) permeability structures and optimal well positions due to the complicated well-to-well interactions and time-varying injection rates. We do, however, expect optimal well locations and injection schedules to lead to significant horizontal migration of CO2 and to saturation distributions that facilitate residual and dissolution trapping.

We now further investigate the characteristics of the best solution achieved (the optimized result for Run 1). Figures 3.3(a) and 3.3(b) show the well locations along with the vertically-averaged mobile CO2 after 1000 years for the initial guess and optimized solutions. Note that the wells lie in the core aquifer, and thus honor the optimization constraints, and that the injected CO2 has not proceeded to the outer boundary (which defines the edge of the central portion of the aquifer). The CO2 injection fractions for these two solutions at the four injection periods are presented


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0.3 0.28 0.26 0.24 0.22

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in Figures 3.4(a) and 3.4(b). It is evident from Figure 3.4(b) that Wells 2 and 3 inject considerably more CO2 than Wells 1 and 4. This is also reflected in Figure 3.3(b).

Furthermore, we see that Well 1 injects only during the first two periods, while Well 4 is inactive in the first period and injects relatively little during the second period.

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Figure 3.3: Well locations and vertically-averaged fraction of mobile CO2 at 1000 years for Run 1.

The inscribed box denotes the core aquifer, representing the feasible region for CO2 injection wells

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0.2 0.2 0.1 0.1

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Figures 3.5(a) and 3.

5(b) show the evolution of CO2 trapping for the initial guess


and optimal solution in Run 1. The significant reduction in mobile CO2 fraction (from

0.34 to 0.22) is achieved through an increase of 0.07 in immobile (residually trapped) CO2 fraction and an increase of 0.05 in dissolved CO2 fraction. In both cases, there is a negligible fraction of immobile CO2 during the injection phase (first 30 years).

For the base case, the fraction of mobile CO2 decreases until around 300 years, while in the optimized case it continues to decrease throughout the entire time frame.

Both dissolution and residual trapping are enhanced in the optimized case. The enhancement in dissolution trapping likely occurs because the CO2 ‘plume’ is less continuous in the optimized case, which exposes it to a larger volume of brine. Increased residual trapping may result from the fact that there is more trailing-edge area in the optimized case, and it is in these regions that this type of trapping occurs. It is also possible that the CO2 saturation at the trailing edges is higher in the optimized solution (this may be accomplished by the varying injection rates evident in Figure 3.4(b)), which would increase the degree of hysteresis and thus residual trapping. The ‘disconnectedness’ of the mobile CO2 (which facilitates trapping) in the optimized scenario relative to that in the base case is evident in Figure 3.6, where we display mobile CO2 at 30 and 1000 years for the two cases. The ‘isosurfaces’ in the plots depict contours with a mobile CO2 saturation of 0.1 (these plots appear fairly similar if we contour CO2 saturations down to 0.01). It is clear in the figure that the optimization leads to multiple disconnected CO2 regions rather than a single large region.

We note finally that, on average, CO2 migrates upward more slowly in the optimized case, and this also provides for more secure storage. This is evident in Figure 3.7, which displays the total mass fraction of CO2 (in any form) reaching the top layer for the two cases. At the end of the injection phase, the CO2 mass fraction in the top layer is around 0.12 for the initial guess solution, but only 0.03 for the optimized solution. The CO2 mass fraction for the optimized solution remains below that for the base case over the entire period. This is most likely the result of the optimal solution exploiting the detailed heterogeneity structure to increase horizontal migration and increase block saturations during drainage. Both of these effects would result in slowing the upward migration of the CO2 plume, as we observe in the optimal solution.


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0.4 0.4 0.2 0.2

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