Reduced-order Models for Nonlinear Optimization
M. Rousset and L. Durlofsky
Production of unconventional oil resources, such as oil sands in the province of Alberta, Canada, may contribute significantly to the future energy supply of the planet. It is indeed estimated that several trillions of oil barrels remain under the ground along the Athabasca River only. This extremely viscous oil however, cannot be recovered using traditional production techniques. One approach that has been developed consists in using in-situ electrical heaters in order to upgrade the oil sands directly within the subsurface. Such a technique shall however be carefully designed to remain economically viable and environmentally safe. It appears that building numerical models and applying powerful optimization procedures may be necessary to achieve these goals.
Numerical simulation of the oil sands in-situ upgrading process is however a difficult challenge. Indeed many physical and chemical phenomena must be represented in the model. In addition to the traditional transport of fluids through porous media and to the change of properties with varying pressure, in-situ upgrading (IUP) involves transport of heat and several chemical reactions. The changes versus pressure and temperature are very large. It is also necessary to account for a large number of components in the fluid. These and other complications lead to a large number of variables in the model, together with important nonlinearities. For this reason, the computation of such models is extremely CPU intensive. And although the potential benefits can be important, performing optimization with such models can lead to extremely long calculation times. A more efficient simulation capability is therefore very much needed.
Figure 1. Synthetic 2D reservoir used for thermal simulation.
Motivated by this need, this work proposes to investigate the use of reduced-order models for nonlinear thermal applications. As a first step, the proposed approach is to eliminate most of the complexity of IUP simulation. A single-phase system is used where viscosity and compressibility depend strongly on pressure and temperature, in order to reproduce some of the nonlinearities of the initial problem. A reduced order model is then created using the trajectory piecewise linearization procedure. This relatively new reduced-order modeling technique is based on a piecewise linearization combined with a proper orthogonal decomposition. It has previously been applied successfully on two-phase oil-water problems leading to runtime speed-up factors of two or three orders of magnitude.
The trajectory piecewise linearization (TPWL) procedure requires running a number of full-order training simulations (typically 1 to 4 runs) and saving some data during these runs (states and Jacobian of the system). This data is then used to compute the solution of any new set of well controls in a much faster way. Stanford's General Purpose Research Simulator (GPRS) is used to run the full-order simulations.
Figure 2. Performance of a TPWL reduced-order model.
Using the bi-dimensional model pictured in Figure 1 with three producer wells (black) and four in-situ heaters (red), the performance of a TPWL reduced model using four training runs is shown in Figure 2.
A comparison with the solution of the full order model clearly shows that oil production rates were reproduced accurately by the reduced-order model. The runtime speed-up factor achieved here is 33. Although significant, the full acceleration potential is certainly not reached with this reservoir model. Indeed, larger speed-up factors are expected with larger and more complex reservoir models.