Model order reduction in porous media flow simulation using quadratic bilinear formulation

In this work, the saturation equation in the two phase flow model is transformed into quadratic bilinear form by introducing auxiliary variables. This new formulation is the first step for applying training-free reduced order modeling. There is no approximation involved in this transformation and it only increases the size of the problem linearly, before applying any model order reduction technique. Although this might seem counterintuitive to increase the size of the system, it can improve the basis selection and consequently the reduced system. Also, this formulation allows certain properties of the original model to be preserved in the reduced-order model, such as stability and passivity. A projection-based model order reduction on this form of system will yield a reduced-order model without the need to use anymore approximation such as discrete empirical interpolation method for evaluating nonlinear terms. Although in this work only the proper orthogonal decomposition is applied, this formulation is the first step for training free model order reduction after addressing the existing challenges. One of the main challenges is the dependency of the saturation equation on flux which is changing at different time steps, and thus makes the saturation equation time varying. The numerical results of applying the proposed formulation with POD model order reduction on a two-phase immiscible flow show a substantial reduction in the computational complexity.