Improved Initialization of Non-Linear Solvers in Numerical Simulation of Flow in Porous Media with a Deep Learning Approach

Abstract

Abstract Partial Differential Equations (PDEs) have a wide list of applications in modeling complex processes including flow in porous materials. Solution of these equations that are mostly highly non-linear is generally possible using numerical algorithms that are carried out by iterative approaches like Newton's method, where the calculations to find the solution at a new time step are started using an initial guess of the unknown variables. The computational efficiency of the calculations is highly dependent on the closeness of these initial guesses to the exact values. As a routine, solvers pick the solutions at the previous timestep as the kickoff point for Newton's method. Improvement of this starting point at each time step can reduce the time-to-solution of the solver. This study focuses on using a Deep Learning (DL) algorithm for optimization of a PDE solver and improvement of the computational efficiency of simulation of flow in porous media by providing more efficient initial guesses for the unknown variables. In this work, a 1D gravity-capillary driven two-phase flow problem with a fully implicit Newton's solver was hired as the base numerical model. A fully connected neural network (NN) was initialized and added to the numerical solver, at the point before starting each timestep. The data from the most important features and the target properties were collected from a series of simulation cases and the DL model was trained with the Adam optimizer. After training and testing, the default initialization approach (i.e., solution at previous timestep) was replaced by the hybrid DL-based approach that provides an initial guess for cells with high saturation gradients. Saturation gradients at the previous timestep, location, and mobility ratio of both phases are selected as the input features. The results showed that applying the developed algorithm to numerical simulation problems reduced the run-time in the range of 15-35% in different cases, while the required processing time of the DL model was only around 1-3% of the whole simulation. The model performed acceptably when the effective parameters are like porosity, permeability and capillary pressure deviated from the range of the training data in order of 100%. The model performance declined when this deviation increased. This hybrid initialization approach showed the possibility of applying DL methodologies for the improvement of the numerical simulation processes.