A locally conservative variational multiscale method for the simulation of porous media flow with multiscale source terms

Abstract

Multiscale phenomena are ubiquitous to flow and transport in porous media. They manifest themselves through at least the following three facets: (1) effective parameters in the governing equations are scale dependent; (2) some features of the flow (especially sharp fronts and boundary layers) cannot be resolved on practical computational grids; and (3) dominant physical processes may be different at different scales. Numerical methods should therefore reflect the multiscale character of the solution. We concentrate on the development of simulation techniques that account for the heterogeneity present in realistic reservoirs, and have the ability to solve for coupled pressure-saturation problems (on coarse grids). We present a variational multiscale mixed finite element method for the solution of Darcy flow in porous media, in which both the permeability field and the source term display a multiscale character. The formulation is based on a multiscale split of the solution into coarse and subgrid scales. This decomposition is invoked in a variational setting that leads to a rigorous definition of a (global) coarse problem and a set of (local) subgrid problems. One of the key issues for the success of the method is the proper definition of the boundary conditions for the localization of the subgrid problems. We identify a weak compatibility condition that allows for subgrid communication across element interfaces, something that turns out to be essential for obtaining high-quality solutions. We also remove the singularities due to concentrated sources from the coarse-scale problem by introducing additional multiscale basis functions, based on a decomposition of fine-scale source terms into coarse and deviatoric components. The method is locally conservative and employs a low-order approximation of pressure and velocity at both scales. We illustrate the performance of the method on several synthetic cases, and conclude that the method is able to capture the global and local flow patterns accurately.

Near-well upscaling for three-phase flows

“…A number of treatments of near-well effects based on multiscale modeling have also been presented. These include the studies by [1,17,19,21,35]. None of these studies was specifically targeted toward three-phase near-well flow problems, though presumably these formulations could be extended in this direction.…”

Accurate Representation of Near-well Effects in Coarse-Scale Models of Primary Oil Production

“…Optimization procedures, using numerical gradients rather than adjoints, were employed by Hui (2005); Hui and Durlofsky (2005), and Mascarenhas and Durlofsky (2000) for the computation of the coarse-scale parameters. Near-well treatments within the context of multiscale finite element and finite volume methods were presented by Aarnes (2004); Jenny and Lunati (2009);Juanes and Dub (2008); Krogstad andDurlofsky (2009), andWolfsteiner et al (2006), though none of these studies specifically targeted the modeling of dissolved gas problems. Our treatment here differs from that used in multiscale finite element and finite volume formulations as we do not need to reconstruct or even carry fine-scale information during the course of the simulations.…”

Bayesian data assimilation for stochastic multiscale models of transport in porous media.

“…Additionally, the variational approaches can often provide a systematic way of projecting the coarse pressure to a fine scale pressure that exhibits important features seen in a full fine scale solution. Several variational multiscale methods exist, including the Variational Multiscale Method (VMM) [109,110], Heterogeneous Multiscale Methods (HMM) [111,112], Subgrid upscaling [113], Multiscale finite element methods [107,114], and the Multiscale finite volume method [115,116]. All of these methods are very similar and on some specific problems can be equivalent.…”