A BDF2 characteristic-Galerkin isogeometric analysis for the miscible displacement of incompressible fluids in porous media

Abstract

Incompressible-miscible problems arise in many fields of application where the main objective is to describe the change of the pressure and the velocity during displacement. These problems are usually subject to some complicated features related to the dominance of convection. Therefore, the multiphysical scales in these problems represent a challenging endeavor. In this study, we propose a NURBS-based isogeometric analysis (IgA) combined with an $L^2$-projection characteristic Galerkin method to deal with this class of equations. The advection part is treated in a characteristic Galerkin framework where high-order nonuniform rational B-spline functions are used to interpolate the solution. The resulting semi-discrete equation is solved using an efficient backward differentiation time-stepping algorithm. The accuracy of the method is analyzed through several Darcy’s flow problems with analytical solutions on differently shaped computational domains, including a miscible displacement of an incompressible fluid, and a real problem with a viscous fingering in porous media. The numerical results presented in this study demonstrate the potential of the proposed IgA characteristic Galerkin method to allow for large time steps in the computations without deteriorating the accuracy of the obtained solutions, and to accurately maintain the shape of the solution in the presence of complex patterns on complex geometries.