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3.5 Summary In this chapter we applied procedures for optimizing the geological sequestration of CO2 when the geology is known. Examples involving only CO2 injection, as well as cases that also included brine cycling, in which brine was periodically injected at the top of the model and produced at the bottom, were considered. Optimization variables included CO2 injection well locations and injection rate schedules as well as the timing and injection/production volumes of brine (for cases with brine cycling).

A Hooke-Jeeves Direct Search algorithm was used for all of the results presented.

Most of the optimizations sought to minimize the fraction of mobile CO2 in the system at the end of the 1000-year lifespan of the CO2 storage operation. Results using several other objective functions were also provided. Clear decreases in mobile CO2 over the course of the optimizations were observed. It was also evident that brine injection has a strong impact on mobile CO2. For example, in the absence of brine injection, the optimized fraction of mobile CO2 was 0.220. This was reduced to 0.125 and 0.072 with 0.03 and 0.09 PV (respectively) of brine injection. In fact,


a plot of minimum mobile CO2 fraction versus PV of brine injected represents the Pareto front for a bi-objective optimization involving these two variables as objectives.

The impact of grid resolution on the optimization results was also considered, and a strategy that involves optimizing using coarse and fine models (in sequence) was described and applied.

In general, we observed optimal solutions to display several interesting behaviors.

First, optimal injection well locations tended to be toward the boundaries of the allowable region, with the wells placed relatively far apart, presumably to provide a large contact area between CO2 and brine to increase dissolution. Second, optimal injection rates were observed to follow a pulsing behavior, which may act to increase dissolution and residual trapping by maximizing the interaction between brine and CO2. Finally, when brine cycling was applied for the examples in this chapter, optimized brine cycling events tended to occur late in the simulation. This allows the injected brine to affect as large a volume of CO2 as possible. The optimal timing of these events is, however, dependent on the objective function considered (mobile CO2 fraction at 1000 years in this case). When the objective function involves timeaveraging, as it will in Chapter 4, optimal brine cycling events occur earlier.

Chapter 4

Closed-loop aquifer management

In this chapter, we introduce a ‘closed-loop’ optimization framework for the management of aquifer storage operations in cases where the geological model is uncertain. We provide a description of the major components of the closed-loop framework including a-priori sensor placement optimization and data assimilation using the Karhunen-Lo`ve (K-L) parameterization. Extensive results for a variety of cases e are presented.

The optimizations in this chapter are an extension of those presented in Chapter 3, the key differences being in the objective function, the optimizer and the treatment of geological uncertainty (which is not addressed in Chapter 3). The aquifer models used for optimization and data assimilation in this chapter are similar to the model used in the optimization examples in Chapter 3, though there are some differences in the variogram model as well as in the relationship between porosity and permeability.

4.1 General framework Because of the long duration and large spatial scales involved in carbon storage, in addition to geological and other uncertainties, practical storage operations cannot be expected to perform in full accordance with initial model predictions. Extensive monitoring, involving measurements of different types and resolution, will however be conducted over the life of the operation. This data can be used in various ways – in real time, to optimize the injection operation, and over longer time frames, to


monitor the CO2 plume location or to detect existing or possible leaks. To achieve this outcome, we propose a ‘closed-loop’ optimization framework for storage aquifer management. As discussed in Chapter 1, closed-loop approaches have been developed for oil field operations (Sarma et al., 2006; Jansen et al., 2005), and many of the existing components can be adapted for use in carbon storage operations.

The general framework, illustrated schematically in Figure 4.1 (this figure was also shown in Chapter 1), entails the solution of a sequence of optimization problems. The specific goals of the computations are to determine CO2 injection well locations and time-varying rates to minimize the risk of leakage, and to perform data assimilation to provide models that can accurately predict long term CO2 plume movement. Our measure of risk involves the time-averaged mobility of CO2 that is trapped directly beneath the cap rock (by mobility, we mean the ‘gas’ or supercritical-phase relative permeability multiplied by density and divided by viscosity). Other risk measures could also be used, such as the fraction of mobile CO2 in the formation (as was used in Chapter 3), but we view CO2 mobility in the layer below the cap rock (which is the uppermost layer in our models) as perhaps the most direct measure of the risk of leakage.

Because the geological model is uncertain, optimizations based on an initial model, or even on a set of prior realizations, will be suboptimal for the actual subsurface system. However, because data are collected as the storage operation progresses, a data assimilation or history matching procedure can be applied. This enables us to update the geological model (or set of models) and to then ‘re-optimize’ well settings for the remainder of the operation using the new, and presumably more accurate, geological description. This data assimilation component can also be framed as an optimization problem.

Our methodology proceeds as follows. We assume that a prior model of the aquifer geology is available. In our example this is a Gaussian model for porosity and permeability, with specified (three-dimensional) spatial correlation structure. Using a set of geostatistical realizations sampled from the prior distribution, we determine the optimal well locations, and initial estimates for optimal time-varying CO2 injection rates for each well. We also present a procedure for determining the optimal locations for some number of sensors (monitoring wells). Optimality in this context entails


Figure 4.1: Flow chart for closed-loop aquifer management

placing sensors such that the expected error in the prediction of the CO2 plume location (over a 500-year time period) is minimized. Data assimilation is accomplished using a K-L parameterization of the geological model, as described in Sarma et al.

(2006). The overall set of procedures is applied to several synthetic aquifer models.

4.2 A-priori placement and control optimization The initial step in our framework involves the a-priori optimization of CO2 injection well placement and injection rate control variables. We view this as an ‘a-priori optimization’ because at this stage we do not have any flow data, such as injection well pressures or CO2 breakthrough times at monitoring wells. Geological models that honor such flow data (which are constructed using the data assimilation procedure described below) are referred to as posterior or history matched models.

In the a-priori optimization, we account for geological uncertainty by optimizing over multiple geological realizations. These realizations are generated using the Stanford Geostatistical Modeling Software or SGeMS (Remy et al., 2009). Using SGeMS we are able to generate Gaussian permeability fields, in which the spatial structure is defined by the two-point correlation function (using a variogram model), or permeability fields characterized by multipoint statistics. The latter requires that we


provide a training image, which defines the spatial patterns to be represented in the realizations. In this work we consider Gaussian permeability fields, though many of our treatments are also applicable for more general systems. As appropriate, we will indicate the components of our framework that require modification for these more general systems. In the case of variogram based models, the key quantities required for model construction are the variogram parameters (and variogram type) and any hard data such as porosity or permeability in well blocks, which are honored in all realizations.

Various quantities can be chosen as the objective function to be minimized in the optimizations. In Chapter 3, we minimized the fraction of mobile CO2 in the formation after a long post-injection equilibration period (100 years to 1000 years). This acted to minimize the amount of CO2 that was structurally trapped, and facilitated increased dissolution trapping and residual trapping (recall that mineralization is not included in any of our simulations). In the optimizations here, we instead minimize a measure of the total mobility of the free CO2 that lies at the top of the formation, just below the cap rock. We believe this to be a better indicator of the potential damage that could occur if the cap rock was in some way compromised, although it is a slightly less intuitive quantity.

All simulations here are again performed using the CO2STORE option in the ECLIPSE simulator (Schlumberger, 2010). We minimize expected total mobility, where the expectation is computed by averaging over multiple a-priori geological models, as described below. We use Particle Swarm Optimization (PSO) for these optimizations, which was described in Chapter 2. This method searches globally, in contrast to the local HJDS optimization method used in the examples in Chapter 3.

Unlike HJDS, PSO parallelizes naturally, and we make extensive use of available parallel resources in the computations reported in this chapter. PSO does not require derivative information; it views the simulator simply as a ‘black box.’ In the a-priori optimizations, we determine CO2 injection well locations and time-varying injection rates for each well. We also present a-priori optimizations that include brine-cycling events.


–  –  –

where u contains the optimization variables we seek to determine, m defines the geological model parameters (e.g., porosity and permeability in every grid block), and Jλ is the objective function to be minimized. Note that we use the notation Jλ to differentiate between this objective and that used for most of the optimizations in Chapter 3. As discussed in Chapter 2, the optimization variables in u define the placement and injection rates for N CO2 injection wells over M time periods, in addition to the timing and rates for K brine cycling events (if such events are included). The various bound and inequality constraints on u define a convex feasible region denoted Ω ⊂ Rn, where n is the number of linearly independent optimization variables. Recall that n = 3N + M (N − 1) for cases without brine cycling and n = 3N + M (N − 1) + 2K + N K − 1 for cases with brine cycling. Refer to Section 2.1 for a definition of the optimization variables u and the constraints that form Ω.

The objective function Jλ is defined as the time-average of the total CO2 mobility in the top layer of the model. The total CO2 mobility in the top layer is computed by ˆ summing the CO2 mobility (λi ) in all grid blocks i in the top layer. In the examples considered, the grid blocks in which CO2 can reside are all of the same dimension, so no additional area or volume weighting is required in Equation 4.1. The CO2 mobility

in any grid block is computed as:

–  –  –

through the cap rock. Thus, by minimizing Jλ, we reduce the risk associated with ˆ the storage project. In this work, we will use units of lbmol/[RB·cP] for λ (these are the units reported by the flow simulator). This can be expressed in SI units using the conversion 1 lbmol/[RB·cP] = 2.85×106 mol/[m3 ·Pa·s].

Due to the nature of the relative permeability curves for CO2 -brine systems (see Figure 2.7), CO2 mobility increases more than linearly with CO2 (gas) saturation.

Thus, in addition to increasing CO2 trapping by mechanisms other than structural trapping, and delaying the appearance of free gas at the top of the model, the optimization should also result in more uniform CO2 saturation distributions in the top layer. Physically, the minimization of Jλ entails finding the well locations and time-varying injection rates that take maximum advantage of the interplay between geological heterogeneity, CO2 transport, and trapping mechanisms. The density and viscosity terms in Equation 4.2 depend primarily on pressure and do not vary as much as the saturation-dependent relative permeability term. Consequently, density and viscosity variations with pressure have a comparatively small impact on the optimizations.

In the case of uncertain geology, we express the optimization problem in terms of

the minimization of the expected value of Jλ over multiple realizations:

min E{Jλ (u, m) | m ∈ m1,..., mR }, (4.3) u∈Ω⊂Rn where Jλ (u, m) is defined in Equation 4.1 and m1,..., mR designate the geological parameters that define R equiprobable realizations (in this work we use R = 10).

Thus, a single evaluation of the objective function for cases with uncertain geology will involve R simulations.

As described in Section 2.1, the injection wells are prescribed to be horizontal, in the bottom layer of the model, and oriented along a coordinate direction. Since wells are prescribed to start and end in the centers of grid blocks, the well placement optimization variables are rounded to integer values within PSO.

We can obtain a Pareto front by minimizing Jλ (or E{Jλ }) for various specified brine-cycling pore volumes. This curve again defines the optimal trade-off between a measure of risk (Jλ or E{Jλ }) and a measure of cost (brine-cycling pore volume).

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