Adaptive enriched Galerkin methods for miscible displacement problems with entropy residual stabilization

Abstract

We present a novel approach to the simulation of miscible displacement by employing adaptive enriched Galerkin finite element methods (EG) coupled with entropy residual stabilization for transport. In particular, numerical simulations of viscous fingering instabilities in heterogeneous porous media and Hele-Shaw cells are illustrated. EG is formulated by enriching the conforming continuous Galerkin finite element method (CG) with piecewise constant functions. The method provides locally and globally conservative fluxes, which is crucial for coupled flow and transport problems. Moreover, EG has fewer degrees of freedom in comparison with discontinuous Galerkin (DG) and an efficient flow solver has been derived which allows for higher order schemes. Dynamic adaptive mesh refinement is applied in order to save computational cost for large-scale three dimensional applications. In addition, entropy residual based stabilization for high order EG transport systems prevents any spurious oscillations. Numerical tests are presented to show the capabilities of EG applied to flow and transport.

Analytical and variational numerical methods for unstable miscible displacement flows in porous media

“…Particularly, discontinuous Galerkin methods have some important advantages over continuous methods when hanging nodes are present in the adapted grid. For a more recent contribution on these topics, including an overview of the current state of the art in mesh adaptation, we refer the reader to the recent adaptive simulations on miscible displacement flow in [174].…”

Enriched Galerkin methods for two-phase flow in porous media with capillary pressure

“…Here we apply the discontinuous Galerkin (DG) IIPG (incomplete interior penalty Galerkin) method for the flow problem to satisfy the discrete sum compatibility condition [21,55,70]. Mathematical stability and error convergence of EG for a single phase system is discussed in [51,52,55] The EG finite element space approximation of the wetting phase pressure p w (x,t) is denoted by P w (x,t) ∈ V EG h,l (T h ) and we let P k w := P w (x,t k ) for time discretization, 0 ≤ k ≤ N. We set an initial condition for the pressure as P 0 w := Π h p w (·, 0). Let p k+1 in , p k+1 out , u k+1 N and f k+1 are approximations of p in (·,t k+1 ), p out (·,t k+1 ), u N (·,t k+1 ) and f (·,t k+1 ) on Γ D , Γ N and Ω, respectively at time t k+1 .…”

“…The bilinear form of EG coupled with an entropy residual stabilization is employed for modeling the transport system (19) with high order approximations [55]. Here, again we apply DG IIPG method although other interior penalty methods can be utilized.…”

“…with ε < 1 as chosen in [13,36,55]. Finally, the local entropy viscosity for each step is defined as…”

“…However, EG has substantially fewer degrees of freedom in comparison with DG and a fast effective high order solver for pressure whose cost is roughly that of CG [51]. EG has been successfully employed to realistic multiscale and multi-physics applications [55,53,54]. An additional advantage of EG is that only those subdomains that require local conservation need be enriched with a treatment of high order non-matching grids.…”