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3.2 Optimization with brine cycling We now consider cases with varying amounts of brine cycling. These optimizations include additional parameters that define the timing, duration and pumping fraction of the injection events, as detailed in Section 2.1. The total pore volume (PV) of brine injected over the life of the project is specified in each case. We consider PVs of 0.01, 0.03, 0.09 and 0.27 as fractions of the core region pore volume. These values correspond to 0.0036, 0.0108, 0.0324, and 0.0972 pore volumes of the storage region (the entire 25 × 25 × 8 block region shown in Figure 2.6), or equivalently, to around 0.14, 0.43, 1.3 and 3.9 times the volume of injected CO2 (at typical aquifer conditions). Three runs are performed for each PV using the same initial guesses for CO2 well placement and control as in Section 3.1. The initial guess for the brine cycling parameters, which is the same in each run, is s0 = [50, 100, 200] years, l0 = [10, 10, 10] years, and p0 = 1/(N K) = [1/12,..., 1/12]. This means that the initial guess entails three cycling events at 50, 100 and 200 years, each lasting for ten years, with equal pumping volume fractions for each well and cycling event.

Results for the HJDS optimizations are presented in Figure 3.8 and Table 3.1.

The figure displays the progress of the optimization for the best run for each value of PV, while the table presents the final optimization results for all three runs. In the table, the value in bold represents the best optimum achieved. It is evident that, for a given value of PV, the optima for the three runs differ slightly.

Table 3.1: Objective function values for the initial guess and optimized solution for each HJDS run.

The best performing run for each PV is shown in bold

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Figure 3.8 clearly shows that increasing the PV of brine cycled decreases the optimized mobile CO2 fraction.

It is also evident that the optimizations are quite


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effective for these systems; for example, for the case of 0.03 PV brine cycling, the initial objective function is 0.252 while the optimized objective function is 0.125. We additionally observe that significantly more function evaluations are required for the cases with brine cycling than for the case without brine cycling. This is consistent with the fact that the search space is of a higher dimension for systems with brine cycling (41 versus 24).

We now present more detailed results for the 0.03 PV case. Figures 3.9 and 3.10 show the optimization results for well placement, CO2 injection, and brine cycling.

The well locations are similar to those in Run 1 for the case with no brine cycling (Figure 3.3(b)), though they are not identical. In addition, we again see from Figure 3.10(a) that the well that injects the most CO2 (Well 4 in this case) lies in the east portion of the aquifer.

The optimization results for the brine cycling events (Figure 3.10(b)) are also of interest. There is a relatively small amount of brine cycled around the time injection stops, and then two events toward the upper bound of 500 years. By having such a long time between cycling events, the optimizer allows CO2 to accumulate at the top of


the aquifer before the final brine injection. This would be expected to enhance, if not maximize, the impact of both dissolution and hysteresis in trapping the accumulated CO2. Also of note is that the late-time cycling events in the optimized solution last for only one year, which is the minimum duration allowed.

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Figure 3.9: CO2 injector positions and vertically-averaged fraction of mobile CO2 at 1000 years for the best result (Run 3) for the case with 0.

03 PV brine cycling. The inscribed box is 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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0.4 0.4 0.2 0.2

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Figure 3.11: CO2 trapping mechanisms with time for 0.

03 PV brine cycling Figures 3.11(a) and 3.11(b) show the evolution of CO2 trapping for the initial guess and optimized (Run 3) solutions for 0.03 PV brine cycling. The reduction in mobile CO2 fraction is accounted for by an increase of 0.05 in dissolved CO2 fraction and an increase of 0.08 in immobile CO2 fraction. The immediate reduction in mobile CO2 during brine cycling events is clearly visible in both figures. For example, in Figure 3.11(b), we see sharp reductions in mobile CO2 at around 25 years and again at around 500 years.

The effect of brine cycling is also evident in Figures 3.12(a) and 3.12(b), which show isosurfaces of the mobile CO2 before and after the late-time brine cycling events in the optimized solution. The ‘holes’ in the middle of three of the plumes represent areas where CO2 has either dissolved into the injected brine or been immobilized via residual trapping. It is thus evident that the optimizer ‘steers’ the plumes (by optimally placing wells and adjusting rates) such that brine injection will significantly reduce mobile CO2.

All of the optimization results for the best runs at various PV values (presented in Table 3.1) can be combined into a single plot. This plot, shown in Figure 3.13 (the dashed curve simply connects the points), represents the bi-objective optimization Pareto front for the competing objective functions, mobile CO2 fraction and cycling PV. These objectives are competing because, although we would like to minimize both, it is clear that by increasing cycling PV we act to decrease mobile CO2 fraction.


(a) Before the two late-time brine cycling (b) Ten years after the final brine cycling events (486 years) event (511 years) Figure 3.12: Isosurfaces for mobile CO2 saturation of 0.1 in the optimized solution for 0.03 PV brine cycling (Run 3) The Pareto front provides the minimum mobile CO2 fraction that can be achieved for a given PV of brine cycling. It can also be read to provide the minimum PV of brine that must be injected to achieve a given fraction of mobile CO2. We note finally that, because mobile CO2 fraction can be related to risk and PV of brine cycling to cost, this plot presents the risk-cost tradeoffs associated with this CO2 storage operation.


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Figure 3.13: Pareto front for the bi-objective optimization involving mobile CO2 fraction and PV of brine cycling



3.3 Optimization with different objective functions In the results presented thus far, the objective function J was the fraction of mobile CO2 after 1000 years. Other optimization time frames and objective functions may also be of interest. We have therefore additionally tested the optimization procedure with several other objective functions: mobile CO2 fraction at 100 years, time-averaged mobile CO2 fraction over the 1000-year project, and the time-average of the total mobility of CO2 in the top layer of the aquifer over 500 years, defined in Equation 2.2 (this objective function will be used extensively in Chapter 4). We denote the fractions of mobile CO2 at 1000 years, 100 years and the time-average as J1000, J100 and Jave. Similarly, the time-average of the total mobility of CO2 in the top layer over 500 years is denoted Jλ. The optimized solutions for the various cases are designated u∗, u∗, u∗ and uλ. Optimizations aimed at minimizing J100, Jave 1000 100 ave and Jλ were performed for a case with no brine cycling (Run 1) and for a case with

0.03 PV brine cycling (Run 3). For the case where we seek to minimize J100 with

0.03 PV brine cycling, some of the constraints and initial guesses for optimization variables were changed (e.g., cycling now must start by 95 years, maximum duration of cycling is 25 years). In each case, the initial guess is denoted u0. The results for these optimizations are presented in Table 3.2.

The table shows results for the objective functions J1000, J100, Jave and Jλ using the four sets of optimized control parameters (u∗, u∗, u∗ and u∗ ). For the values 1000 100 ave λ shown in bold, u∗ and the objective function correspond. Comparing (bold) values for the first three diagonal entries for the case with no brine cycling, we see that the minimum mobile CO2 fraction at 100 years (0.330) is larger than the minimum fraction at 1000 years (0.220) and the time-averaged result (0.264). It is also evident that the benefit from optimization is comparable in all three cases, specifically a reduction in mobile CO2 from the base case (u0 ) of about 0.1. We additionally note that, even when the u∗ determined for one objective function is applied for a different time frame, optimization still provides benefit over the base case. For example, if we perform the optimization to minimize J1000 but then assess J100, the mobile CO2 fraction at 100 years, designated J100 (u∗ ), is 0.344, which is only slightly higher than the mobile CO2 fraction in the optimization problem that seeks to minimize J100 (J100 (u∗ ) = 0.330).


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As mentioned previously, the objective Jλ quantifies the time-average total mobility in the top layer of the aquifer. Since mobility is proportional to relative permeability, and relative permeability increases more than linearly with Sg, the optimal solution u∗ will favor configurations that reduce the overall CO2 in the top layer while λ also avoiding regions of high CO2 saturation in the top layer. Consequently, optimal solutions for Jλ tend to provide improvement to the other objectives, though the converse is not necessarily true. This is because optimal solutions for other objectives may result in regions of high CO2 saturation in the top layer, leading to a value of Jλ that may exceed the initial guess. This is evident for the case without brine cycling.

For the case with brine cycling, both Jλ (u∗ ) and Jλ (u∗ ) are very close to the initial guess value.

Table 3.2: Optimization results using different objective functions.

Values in bold are for cases where u∗ and the objective function correspond

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3.4 Sensitivity to grid refinement Because the optimizations considered here require hundreds to thousands of function evaluations, it is important that the simulation models run efficiently. In our optimizations, we therefore used a relatively coarse model to achieve reasonable run times. It is, however, useful to assess the impact of grid resolution on the optimized solutions obtained by HJDS. To address this issue, we compare solutions obtained during the optimization (using the 39 × 39 × 8 grid-block model for the case without brine cycling) to a refined model containing 64 × 64 × 16 blocks. The finer model was obtained by refining the central region of the original model by a factor of two in each direction.

A cross plot comparing mobile CO2 fraction on the coarse (39 × 39 × 8) model and mobile CO2 fraction on the fine model is presented in Figure 3.14. The runs on the coarse grid are simply solutions at different iterations of the optimization for Run 1. The fine-grid runs are simulations of the same well configurations (placed appropriately in the 64 × 64 × 16 model) and injection schedules. As the optimization progresses, solutions move down the diagonal – from upper right to lower left. It is evident that there is reasonable correspondence in mobile CO2 fraction between the models, though there is an offset of 0.01 or more for most points. We note that refining by a factor of two in each direction represents a relatively modest level of grid refinement. These results nonetheless suggest that the original model is able to provide useful optimization results.

We can proceed a step further by viewing the 39 × 39 × 8 grid-block model as, essentially, a surrogate for the ‘true’ 64 × 64 × 16 model. There are many ways in which surrogate models can be used for optimization. One approach is to first perform the optimization using the surrogate model (either for some number of iterations or until convergence) and to then switch to the ‘true’ 64 × 64 × 16 model and perform additional optimization iterations. The idea with this treatment is that the bulk of the optimization can be accomplished using the much more efficient coarse model, though the final optimum is determined using the ‘true’ model.

This procedure was in fact tested and the results are presented in Figure 3.15.

Following optimization with the original model (solid curve), we switch to the ‘true’


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Figure 3.14: Cross plot of fine-scale and coarse-scale mobile CO2 fractions for various points during the optimization (case with no brine cycling, Run 1) model (dashed curve).

There is an immediate jump in the objective function, followed by a gradual decrease with iteration. The optima achieved by the coarse and fine models are close, 0.22 and 0.23 respectively. It is also worth noting that only two of the rate variables changed during the fine-scale optimization (the well positions did not change), which again indicates the efficacy of optimization on the coarse model.

This procedure could be continued with increasingly refined models if necessary. Additionally, finer-scale permeability and porosity descriptions, if available, could be introduced as the grid is refined.


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0.3 0.28 0.26 0.24

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