Robust Multi-D Transport Schemes with Reduced Grid Orientation Effects

Abstract

In this paper we investigate truly multi-D upwind schemes for simulating adverse mobility ratio displacements in porous media. Due to an underlying physical instability at the simulation scale, numerical results are highly sensitive to discretization errors and hence the orientation of the underlying computational grid. We use modified equations analysis to predict preferred flow angles on structured grids for several popular methods and present a conservative, multi-D framework for designing positive upwind schemes for general velocity fields. After placing the common schemes in this framework, we go on to develop a novel scheme with "minimal" constant transverse (cross-wind) diffusion. Results for miscible gas injection into homogeneous and heterogeneous media demonstrate that truly multi-D schemes, and in particular our new scheme, greatly reduce grid orientation effects and numerical biasing as compared to dimensional upwinding.

Multidimensional upstream weighting for multiphase transport in porous media

“…For compositional problems, equation of state and phase equilibrium calculations tend to be the most computationally costly part of the simulation, making the additional cost of the multi-D stencil calculation more tractable by comparison. This work along with our previous work [25] suggests that multi-D upstream weighting methods could prove very valuable for these problems. Finally, the volume relaxed framework presented in Section 3 makes no assumption of the dimensionality of the domain and therefore is equally applicable to 3-D problems.…”

“…In Kozdon et al [25], they were shown to be equivalent (for constant scalar advection) to SPU, the narrow scheme [33], and Koren's scheme [23], respectively. Other schemes are possible within this framework.…”

“…SMU performed particularly well for all the test problems presented. This is a result specific to the way we have chosen to formulate our governing equations, and for other applications and formulations, different methods may perform more favorably [25]. The methods were extended to more general flux functions which is important for more complex physics as well as different governing equations arising in other applications.…”

“…For porous media problems, this is important because most simulations are performed at scales that result in unstable displacements [32,38] and numerical errors can trigger these instabilities [8]. There have been many efforts to address this so-called grid orientation effect with multi-D schemes [5,10,15,25,36,41], but the methods proposed are either not monotone (positive for scalar, linear advection) for general flow fields or are not fully developed for multiphase flow. In particular, none of the referenced methods have been extended to problems with countercurrent flow due to gravity.…”

“…In Kozdon et al [25], we developed a multi-D framework for scalar advection in a general divergence-free velocity field. Here, a similar method is developed for a wider class of nonlinear problems.…”