(117d) Discrete Approximation Model for the Optimal Planning of CO2 Injection into Deep Saline Aquifers
AIChE Annual Meeting
2024
2024 AIChE Annual Meeting
Computing and Systems Technology Division
10C: Planning and Operation of Energy Systems
Monday, October 28, 2024 - 1:33pm to 1:54pm
The storage of CO2 into deep saline aquifers mostly occurs at natural formation pressures. Reservoirs are generally represented as open systems from which the brine can flow laterally and make room for the injected CO2 that is trapped in the porous media (Nordbotten et al., 2005). Although pressure build-up is not as relevant as for closed systems (e.g., depleted oil and gas reservoirs), pressure signals during injection can propagate far beyond the CO2 migration front (âplumeâ), on the scale of tens to even hundreds of kilometers (Birkholzer et al., 2015). Given the relatively large depth of these reservoirs (more than 1500 m, yielding pressures over 15 MPa and temperatures over 50ºC) the CO2 is stored in supercritical state, with a density around 600 kg/m3 (Zou and Durlofsky, 2023). In turn, typical brine salinities are over 10,000 ppm. Despite carbon sequestration into deep aquifers being a permanent and environmentally friendly disposition of CO2, continuous measurement, monitoring, verification, and reporting during sequestration must be carefully recorded by companies (Air Products, 2024).
The optimization problem addressed in this work can be defined as follows. Given: (a) a multi-layer saline aquifer with given geological characterization, (b) a set of vertical wells to inject CO2 at predetermined layers of the aquifer, (c) an available amount of CO2 that can be distributed and injected into the wells over time, (d) a long-term, multiperiod planning horizon. The goal is then to determine the injection rates and the corresponding pressures at each single well over time, in order to maximize the total amount of CO2 ultimately sequestered by the end of the time horizon (i.e., maximize pore space occupation with CO2). The model accounts for two main constraints: (i) not exceeding maximum bottom hole pressures (injectivity), and (ii) not exceeding the borders of the control volume with the CO2 plume in the long term (containment).
The prediction of reservoir pressure gradients and CO2 migration is proposed with simplified models to solve the optimization problem by means of mathematical programming tools rather than using sampling or heuristic strategies as has been reported in the literature (Cameron and Durlofsky, 2012; Sun and Dusrlofsky, 2019). More specifically, we seek to overcome limitations of previous contributions in the field based on metaheuristics like particle swarm optimization and differential evolution (Zou and Durlofsky, 2023), which require numerous simulations and do not guarantee optimality after hours of computation. However, the development of a proper prediction model, of reasonable dimensions for optimization purposes, is challenging. To build the model we rely on a discrete space-time representation, much coarser than typical simulation models, including material balances and Darcyâs law equations to track the CO2 front over time. Buoyancy effects are also modeled with detail due the significant difference of densities between water and supercritical CO2 (Celia et al., 2015). Finally, dynamic pressure propagation curves are also predicted from Darcyâs law applied to multiphase flows, evaluating changes along the horizontal (radial) dimension of the system.
In the current version of the model, we present a simplified grid-based framework deployed in 2D that promises to scale reasonably to real instances and 3D configurations. The problem can be solved with commercial solvers and guarantee an optimality gap relative to relaxations, as opposed to heuristics/sampling based algorithms. We have solved two case studies involving two injection wells each. In the first case, the reservoir is divided into 24 grid blocks over 4 layers, and CO2 injection is planned for the first 4 years. However, CO2 will continue migrating after well shut-down, and containment condition is imposed for a total of 10 years. The resulting QCP model comprises 4,574 continuous variables, 1760 bilinear constraints and a total of 9,841 equations. After 196 sec of CPU time using Gurobi solver 11.0, the optimality gap is below 0.5% yielding a solution that takes advantage of 18.5% of the porous media.
A second experiment with a finer discretization is also solved with our approach, comprising a total of 84 grid blocks, 7 layers and 20 annual periods along the time horizon. In this larger instance, the maximum utilization factor of the reservoir reaches 13.1%. The optimization model has 16,220 continuous variables, 5,960 bilinear constraints and a total of 34,080 equations. As a consequence of its larger size, the solver can close the optimality gap below 1% only after 3,075 seconds. Nontrivial strategies guiding the distribution of CO2 among the wells over time are the most interesting results from the experiments.
As for conclusions, the proposed mathematical programming model can help operators solve a very challenging problem faced by CCS industries: how to optimally and safely manage injection rates at different wells, efficiently sequestering CO2 into brine reservoirs. We have devised a mathematical framework seeking to maximize the lifetime of CO2 injection, while satisfying pressure, CO2 plume extension and saturation constraints. Next steps seek to obtain more accurate surrogate models for pressure gradients, as well as analyzing other trapping mechanisms through simulation. Besides, extensions to 3D grids and sensitivity to uncertain parameters is also a path that is worth exploring.
References
Air Products and Chemicals, Inc. âLouisiana Clean Energy Complexâ https://www.airproducts.com/energy-transition/louisiana-clean-energy-complex (Apr. 2024)
Birkholzer JT, Oldenburg CM, Zhou Q. (2015) CO2 migration and pressure evolution in deep saline aquifers. International Journal of Greenhouse Gas Control, 40: 203-220 https://doi.org/10.1016/j.ijggc.2015.03.022
Cameron DA, Durlofsky LJ (2012) Optimization of well placement, CO2 injection rates, and brine cycling for geological carbon sequestration. International Journal of Greenhouse Gas Control, 10: 100-112 https://doi.org/ 10.1016/j.ijggc.2012.06.003
Celia MA, Bachu S, Nordbotten JM, Bandilla KW. (2015) Status of CO2 storage in deep saline aquifers with emphasis on modeling approaches and practical simulations. Water Resources Research, 51: 6846-6892 https://doi.org/ 10.1002/2015WR017609
Nordbotten JM, Celia MA, Bachu S. (2005) Injection and Storage of CO2 in Deep Saline Aquifers: Analytical Solution for CO2 Plume Evolution During Injection. Transport in Porous Media, 58: 339-360 https://doi.org/10.1007/s11242-004-0670-9
Sun W, Durlofsky LJ (2019) Data-space approaches for uncertainty quantification of CO2 plume location in geological carbon storage. Advances in Water Resources, 123: 234-255 https://doi.org/10.1016/ j.advwatres.2018.10.028
Zou A, Durlofsky LJ (2023) Integrated Framework for Constrained Optimization of Horizontal/Deviated Well Placement and Control for Geological CO2 Storage. SPE J., 28: 2462â2481 https://doi.org/10.2118/212228-PA