Stability of non-linear flow in heterogeneous porous media simulations using higher order and continuity basis functions

Abstract

Using isogeometric analysis, we present a highly optimized FORTRAN code for simulations of three-dimensional non-linear flow problem in heterogeneous media. This includes deriving the weak formulations, discretizing with the forward Euler scheme, and implementing computation of the right-hand side. Some properties of exact solutions of heat transport and non-linear flow in heterogeneous media problems have been derived to validate the numerical solution. Stability analysis of the employed explicit time-stepping scheme has been performed. First, the properties of the discrete form of the heat transfer problem have been systematically investigated, leading to the conjecture concerning the sufficient condition of stability of the scheme, which numerical computations have subsequently confirmed. A similar analysis has been carried out for the non-linear flow in heterogeneous media problem. For this problem, the bound on the time step was conjectured to decrease exponentially, which has been verified experimentally. The simulation’s behavior close to the stability limit has been investigated and found to be considerably more complex than for the heat transport problem. Detailed experimental analysis of the time complexity of the particular parts of the simulator algorithm has been described. In addition, dependence on both mesh size and order of continuity of B-spline basis has been investigated.