Identification of nonlinear conservation laws for multiphase flow based on Bayesian inversion - Evje, Steinar; Skadsem, Hans Joakim; Nævdal, Geir (Nonlinear dynamics, 2023)
Conservation laws of the generic form c t + f ( c ) x = 0 play a central role in the mathematical description of various engineering related processes. Identification of an unknown flux function f ( c ) from observation data in space and time is challenging due to the fact that the solution c ( x , t ) develops discontinuities in finite time. We explore a Bayesian type of method based on representing the unknown flux f ( c ) as a Gaussian random process (parameter vector) combined with an iterative ensemble Kalman filter (EnKF) approach to learn the unknown, nonlinear flux function. As a testing ground, we consider displacement of two fluids in a vertical domain where the nonlinear dynamics is a result of a competition between gravity and viscous forces. This process is described by a multidimensional Navier–Stokes model. Subject to appropriate scaling and simplification constraints, a 1D nonlinear scalar conservation law c t + f ( c ) x = 0 can be derived with an explicit expression for f ( c ) for the volume fraction c ( x , t ). We consider small (noisy) observation data sets in terms of time series extracted at a few fixed positions in space. The proposed identification method is explored for a range of different displacement conditions ranging from pure concave to highly non-convex f ( c ). No a priori information about the sought flux function is given except a sound choice of smoothness for the a priori flux ensemble. It is demonstrated that the method possesses a strong ability to identify the unknown flux function. The role played by the choice of initial data c 0 ( x ) as well various types of observation data is highlighted.