Rui (Forest) Jiang, Dept of Energy Resources Engineering
Accelerating Oil-Water Subsurface Flow Simulation Through Reduced-Order Modeling and Advances in Nonlinear Analysis
This dissertation focuses on two of the most promising proper orthogonal decomposition (POD)-based ROM methods, POD-TPWL and POD-DEIM. A separate (non-ROM) technique to accelerate nonlinear convergence for oil-water problems is presented in the appendix.
In Chapter 2 our focus is on characterizing and predicting the approximation errors in the POD-TPWL method (proper orthogonal decomposition -- trajectory piecewise linearization). Previous researchers found that POD-TPWL may display large errors when new (test) simulations are very different from the training runs used to construct the model. The main source of POD-TPWL error is that resulting from linearization of the highly nonlinear transport system. A theoretical assessment of the TPWL linearization error is achieved by solving the TPWL system for the 1D Buckley-Leverett equation, both analytically and numerically. The 1D solution explains the error patterns observed by previous authors and suggests that a particular measure of the difference between the test solution and the training solution, referred to as TPWL distance, strongly correlates with the TPWL error.
Using this empirical correlation, an error prediction framework based on Lasso regression (a machine learning technique) is proposed. We generate the training data for error prediction through a series of POD-TPWL training simulations to enrich the data without incurring extra full-order runs. Features used for error prediction include the TPWL distance and other parameters related to changes in streamline patterns between training and test runs. Numerical results for a 50x50 channelized reservoir model with five wells demonstrate the effectiveness of the error model and the significance of the TPWL distance for predicting error in oil and water production rates.
In Chapter 3 we present a systematic exploration of the POD-DEIM procedure. POD-DEIM is a method based on POD state reduction combined with the discrete empirical interpolation method (DEIM) for approximating the nonlinear residual using its value evaluated at a fraction of the grid cells. We apply Petrov-Galerkin projection for its stability and relax the requirement applied in many previous POD-DEIM implementations that the number of DEIM interpolation points and the number of columns in the DEIM basis must be equal. The resulting method is similar to another ROM method called Gauss-Newton approximate tensor (GNAT). To provide an accurate estimation of the computational cost, the computational complexity of POD-DEIM is analyzed. The theoretical speedup factor is shown to be O(10-100) depending on the solver implementation and problem features.
Extensive numerical tests are presented to explore POD-DEIM behavior. A parametric study is conducted on the SPE~10 model to examine the dependency between POD-DEIM performance and four of the key POD-DEIM parameters. It is found that larger POD-DEIM parameter values lead to lower error, and that POD-DEIM `inherits' the accuracy of the underlying POD-Only approximation (POD without DEIM) when the number of DEIM interpolation points and the number of columns in the DEIM basis are large enough. We also show that the latter parameter should be sufficiently large to avoid rank deficiency in the linear system. Comparisons among POD-TPWL, POD-Only, and POD-DEIM are presented for a channelized model with six wells over ~1500 test cases. The results show that POD-DEIM is more accurate for oil and water production rates than POD-TPWL, especially when the test schedule is significantly different from the training schedule.