Reduced-order Modeling for Thermal Simulation a Thesis Submitted to the Department of Energy Resources Engineering of Stanford University in Partial Fulfillment of the Requirements for the Degree of Master of Science

The simulation of complex thermal recovery processes such as in-situ upgrading is computationally demanding. Reduced-order modeling techniques allow the representation of high-dimensional computational problems in a reduced mathematical space, where most of the physical behavior can be reproduced. Use of such models can lead to significant reduction in computational demands. This work focuses on the application of the trajectory piecewise linearization (TPWL) procedure to nonlinear thermal problems. The thermal problem considered is a highly idealized representation of the in-situ upgrading process. Thus this work represents a first step in the application of reduced-order modeling for challenging thermal simulation problems. The trajectory piecewise linearization procedure entails running one or more training high-fidelity (full-order) simulations. During these runs, snapshots of the states of the system are recorded at every timestep, along with Jacobian matrices and other derivative information. Proper orthogonal decomposition is then applied to produce a basis matrix for projection into the reduced space. The governing equations of the thermal simulation problem are then linearized around the previously saved states, and projected into the reduced space using the basis matrix. The governing equations solved in this thesis describe the flow of a single component in a single phase coupled to an energy equation. Downhole heaters are modeled by fixing the temperatures of selected grid blocks in the energy equation. The models v that are simulated contain up to 75,000 grid blocks and involve heterogeneous permeability fields. Viscosity is taken to be a strong function of temperature and varies over several orders of magnitude during a simulation. Density is also a strong function of pressure and temperature. Initially, one training run is used to construct the TPWL model. Results are found to be in reasonable agreement with the reference high-fidelity simulations when the heater and bottom hole pressure controls are set close to those used in the training run. When the controls differ significantly from those of the training run, the TPWL accuracy is shown to degrade, sometimes considerably. Accuracy is restored however when two additional training runs are used in the construction of the TPWL model. The same multiple training approach is applied for a more challenging example, which involves more significant nonlinearities. For this case, TPWL results consistently display close agreement with the reference high-fidelity simulations. For the examples considered in this work, the TPWL procedure provides runtime speedups of a factor of 400-500. The overhead requirements for TPWL depend on the number of training runs used. For the examples considered here, TPWL overhead corresponds to the simulation time for 3-10 high-fidelity runs. Thus, it only makes sense to use TPWL if many simulations are to be performed, as would be the case in computational optimization procedures.