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In Equation 5.5, σSF and σOA are calculated separately for each mj, from observation data dobs (mj ), whereas σBHP is treated as a constant, similar to the θ parameters used in Chapter 4. The reason for these different treatments is that the magnitude of σOA, in particular, depends strongly on the true model. For example, a model with a low-permeability leak will have much smaller pressure differences in the overlying aquifer than a model with a high-permeability leak. Specifically, σOA is calculated as

–  –  –

that is, we take the standard deviation of the BHP data dobs,BHP (mj ) between realw,t izations in the prior, and average this value over each injection well w and time step t. Our assumption here is that these σOA, σSF and σBHP values are indicative of the differences between models that are close to the true model, or equivalently, that the contributions to the misfit function from each data type will be comparable when these scalings are used. Analysis of the misfit function from several history matching examples verified that this is indeed the case. These scalings will be modified


by the data weight parameters wSF, wOA and wBHP, which will be determined via optimization.

The value for σBHP was calculated to be 100 psi. The value of σSF ranges from about 50 to 100 psi, depending on the model. This is a relatively small degree of variability, and a single standard deviation value may be appropriate for these data.

Conversely, the standard deviations for data in the overlying aquifer (σOA ) display a much broader range, from 0.2 psi to 70 psi, depending on the model. Figure 5.7 presents the σOA value as a function of the kz,l value of the leak. These points were generated for each of the 2000 single-leak cases simulated in the previous section. It is evident that σOA varies substantially and displays a clear relationship with kz,l. If we were to set σOA to a constant for all cases (analogous to the treatment used in Chapter 4), the pressure data in the overlying aquifer would be largely ignored in the history matching algorithm for models with small leaks. This issue did not arise in Chapter 4 because the data variability was fairly consistent from model to model.

Figure 5.7 also suggests that noise may become significant when attempting to detect small leakages (e.

g., when kz,l ≤ 0.1). Although the typical accuracy of pressure

5.3. HISTORY MATCHING LEAKAGE sensors that would likely be used in carbon storage monitoring applications is about ±0.1 psi (Sun et al., 2013), other sources of noise such as precipitation (which may affect overburden pressure) may cause pressure fluctuations of ±1 psi. In coastal storage projects, ocean tides may cause significant pressure variation, though these effects and other factors affecting overall aquifer pressure changes could be quantified.

Future work should incorporate some level of data noise, along the lines of the singleleak case analyzed by Sun & Nicot (2012), for more complicated leakage cases such as the multi-leak scenarios addressed here.

As mentioned previously, the data weight parameters wSF, wOA and wBHP are selected to optimize the expected performance of history matching, which is quantified ˆ in terms of Λ. Mathematically, we wish to solve the minimization problem

–  –  –

The calculation of m∗ is fast because no additional simulations are required. The j ˆ using this method is typically larger than what we could achieve using a value of Λ ˆ formal history matching algorithm to minimize S. In this work, we take Ns = 2000 and use the set of realizations that were previously generated for the single-leak and multi-leak cases considered in Section 5.2.

Since there are only one or two independent optimization variables in Equation 5.8 (assuming, without loss of generality, that wBHP = 1 − wSF − wOA in the datarich scenario and wBHP = 1 − wOA in the data-scarce scenario), we can perform an


–  –  –

Figures 5.8–5.

11 provide important insight regarding the appropriate data weight parameters for the examples in this chapter. In each figure, the optimum weights are labeled with a red circle, although it is apparent that these optima generally lie in ˆ relatively large regions with similar values of E{Λ}. This suggests that our history matching results are not likely to be overly sensitive to the precise values used for the data weight parameters. The surface plots also provide information about how we expect our history matching procedure to perform without certain types of data.

The three corners of the triangle correspond to using only one type of data, while the optimum value along an edge corresponds to using two types of data. The key values from the plots are provided below in Table 5.1.

Besides providing optimal weight parameters, Table 5.1 reveals a number of interesting properties that may affect the efficacy of history matching leakage scenarios.

First, the expected leakage saturation error between true models and their history matched counterparts is larger in the multi-leak case than in the single-leak case (for example, the expected error is 8.19 in the single-leak data-rich scenario, and 20.10 in the multi-leak data-rich scenario). This is probably due, at least in part, to the fact that more CO2 is released in the multi-leak case.


–  –  –

Second, we see that the most important data appear to be the pressure data in the ˆ overlying aquifer. There is only a small reduction in E{Λ} when we use only these data in all cases. For example, using only the overlying aquifer sensor data increases the expected error from 8.19 to 8.36 in single-leak cases and from 20.10 to 20.22 in multi-leak cases. This is useful information because it is less expensive and less risky to put monitoring wells in the overlying aquifer than in the storage formation itself.

The downside, however, is that monitoring wells in the overlying aquifer cannot be used to determine the location of the CO2 plume within the storage formation, as was considered in Chapter 4. We note finally that using only BHP data appears to provide less accurate matches when compared with using sensor data from the overlying aquifer. This suggests that history matching to BHP data alone may not provide accurate leakage assessments, and motivates the need for monitoring wells in carbon storage projects.

In the next section, we will present results using formal history matching in singleleak and multi-leak cases, for both the data-rich and data-scarce scenarios.


5.3.4 Results for single-leak cases We select five single-leak ‘true’ realizations from the prior, which provide the observed data dobs (mj ) in Equation 5.2. These realizations are denoted as mj for j ∈{SLC1,...,SLC5}, where SLC stands for single-leak case. PSO is applied to determine the n = 3L + ℓ1 + ℓ2 = 63 optimization variables that minimize the misfit ˆ function S. The best PSO solution is denoted m∗,rich in the data-rich scenario or j ∗,scarce in the data-scarce scenario, where j ∈{SLC1,...,SLC5}.

mj Figure 5.12 shows the progression of three PSO runs for SLC3 in the data-rich scenario. In all three cases, the normalized misfit function (by normalized, we mean divided by the expected misfit of the prior model) is reduced by almost three orders of magnitude. Similar behavior was observed in most of our PSO runs. The leakage locations determined in all three of these PSO runs were within one or two grid blocks of the true leak location, and the leakage fraction was also very similar from run to run. Since all three runs gave solutions that were close to the true model after 200 PSO iterations, we use only a single PSO run with 200 iterations for the other single-leak history matches.

Table 5.2 provides key data for the five true realizations and their history matched solutions.

Leakage locations, in terms of grid block indices (il, jl ), leakage block permeabilities kz,l, the fraction of CO2 that has leaked over 500 years FCO2, and the ˆ saturation error in the overlying aquifer Λ, are provided. The superscripts ‘true,’ ‘rich’ and ‘scarce’ indicate the true model and history matched models. The true realizations were selected to represent leakage cases with different amounts of leaked CO2. True model SLC1 has little leakage (less than 1% of injected CO2 ), while true model SLC5 has a large amount of leakage (23% of injected CO2 ). When the true ˆ model has more leakage, the value of Λ will generally be higher.

Table 5.2 shows that the history matched leakage location for single-leak cases is generally close to the true location, particularly for the data-rich scenario.

It also shows that the leakage fraction is matched fairly accurately in most cases, with the data-rich scenario providing better results in four of the five cases. We note that the history matched value of kz,l often has a poorer match when kz,l is high (e.g., the history matched kz,l for SLC3 is only 31% of the true value for the data-rich scenario).

This may be due to a lack of sensitivity to leakage permeabilities for kz,l greater than


–  –  –

Figure 5.12: Progress of the minimization (misfit corresponding to the best PSO particle is plotted) for three history matching runs for SLC3 in the data-rich scenario.

The best run is shown in bold

–  –  –

Table 5.2: Leakage information for single-leak true models and their respective history matched solutions for both the data-rich and data-scarce scenarios

–  –  –

Both of these observations are consistent with expectations.

Maps of the saturation of CO2 at the top of the overlying aquifer at 500 years for each single-leak case and the corresponding history matched solutions are shown in Figure 5.13. These plots display the position of the leak, and can be used to evaluate the performance of the history matching procedure visually. These results show that the predictions in the data-rich scenario appear generally better than those from the data-scarce scenario, consistent with the results in Table 5.2.

History matching will now be performed using only (time-varying) injection-well BHPs as the observed data. This enables us to test the efficacy of our leak detection procedure in the absence of monitoring wells. Figure 5.14(a–e) shows the leak locations and maps of the CO2 saturation in the top layer after 500 years for the history matched solutions (using only BHP data) for SLC1–SLC5. Comparing these solutions with the true solutions depicted in Figure 5.13(a,d,g,j,m), it is clear that history matching using BHP data alone is ineffective for detecting and characterizing leaks in the cap rock. This is consistent with the very low (optimized) weights associated with BHP data, and with the larger expected history matching error when using only BHP data, that were observed in Section 5.3.3.


–  –  –

Figure 5.14: Leak locations and CO2 saturation at the top of the overlying aquifer after 500 years for BHP history matched solutions for SLC1 to SLC5

120 CHAPTER 5.


5.3.5 Results for multi-leak cases We again select five ‘true’ geologies from the set of multi-leak prior models to provide the observed data dobs (mj ) in Equation 5.2. These realizations are denoted as mj for j ∈{MLC1,...,MLC5}, where MLC stands for multi-leak case. In the history matching runs, for both the data-rich and data-scarce scenarios, we use 500 PSO iterations rather than the 200 iterations used for single-leak cases. This is necessary to achieve a reduction in the misfit function of more than two orders of magnitude, which was achieved for the single-leak cases using only 200 iterations (see Figure 5.12).

The need for more PSO iterations is presumably due to the increased number of optimization variables in multi-leak cases (we now have n = 3L + ℓ1 + ℓ2 = 84 variables). For each of the multi-leak cases, a single PSO run is performed.

Table 5.3 provides the key data for the true realizations mj and their history ∗,rich ∗,scarce matched counterparts mj and mj for j ∈{MLC1,.

..,MLC5}. We do not report grid block indices (il, jl ) or permeability values kz,l for individual leaks, as we did in Table 5.2. These values can be seen in Figure 5.15, where leakage locations are represented by white squares and the kz,l value (given in md) is noted next to each leak.

Table 5.3: Leakage information for multi-leak true models and their respective history matched solutions for both the data-rich and data-scarce scenarios

–  –  –

The table shows that the true number of leaks is determined correctly by the data-rich solution three out of five times (MLC1, MLC4 and MLC5), though it is never correctly determined by the data-scarce solution. The fraction of leaked CO2

5.3. HISTORY MATCHING LEAKAGE after 500 years (FCO2 ) estimated in the data-rich scenario is never more than 0.021 (or 2.1% of the total injected CO2 ) away from the true value. The FCO2 value predicted in the data-scarce scenario, by contrast, is overestimated by 0.09 (or 9% of the total injected CO2 ) for MLC3.

ˆ Although the Λ values in MLC1 and MLC4 are quite close for the data-rich and ˆ data-scarce scenarios, the mean Λ is again lower in the data-rich scenario than in the data-scarce scenario (32.4 compared to 47.0), as expected. Nearly all of the ˆ ˆ history matched values for Λ are larger than the E{Λ} value of 20.10 (for the datarich scenario) and 23.19 (for the data-scarce scenario) from our analysis of optimal ˆ data weights in Table 5.1. It may be more difficult to compare E{Λ} to the results in ˆ Table 5.3 in this case, however, because E{Λ} includes data from a range of examples with different numbers of leaks and widely varying FCO2.

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