A Multiscale Method for Modeling Flow in Stratigraphically Complex Reservoirs

Abstract

A robust and efficient simulation technique is developed based on the extension of the mimetic finite volume method to multiscale hierarchical hexahedral (corner-point) grids via use of the multiscale mixed finite element method. The implementation of the mimetic subgrid discretization method is compact and generic for a large class of grids, and thereby, suitable for discretizations of reservoir models with complex geologic architecture. Flow equations are solved on a coarse grid where basis functions with subgrid resolution accurately account for subscale variations from an underlying fine-scale geomodel. The method relies on the construction of approximate velocity spaces that are adaptive to the local properties of the differential operator. A variant of the method for computing velocity basis functions is developed that utilizes an adaptive local-global algorithm to compute multiscale velocity basis functions by capturing the principal characteristics of global flow. Both local and local-global methods generate subgrid-scale velocity fields that reproduce the impact of fine-scale stratigraphic architecture. By using multiscale basis functions to discretize the flow equations on a coarse grid, one can retain the efficiency of an upscaling method, while at the same time produce detailed and conservative velocity fields on the underlying fine grid. The accuracy and efficacy of the multiscale method is compared to that of fine-scale models and of coarse-scale models with no subgrid treatment for several two-phase flow scenarios. Numerical experiments involving two-phase incompressible flow and transport phenomena are carried out on high-resolution corner-point grids that explicitly represent example stratigraphic architectures found in real-life shallow marine and turbidite reservoirs. The multiscale method is several times faster than the direct solution of the fine-scale problem and yields more accurate solutions than coarse-scale modeling techniques that resort to explicit effective properties. The accuracy of the multiscale simulation method with adaptive local-global velocity basis functions are compared to that of the local velocity basis functions. The multiscale simulation results are consistently more accurate when the local-global method is employed for computing the velocity basis functions.

Validation of the multiscale mixed finite‐element method

“…The early concepts of MsMFE methods for solving Poisson‐type elliptic equations, were introduced in for Cartesian grids, modified to give conservative discretizations in , and later extended to general polyhedral grids in, for example, . The basic idea of the MsMFE method is to construct a special approximation space, consisting of a set of coarse‐scale basis functions with resolution H that satisfy a flow equation locally and hence are consistent with the differential operator at a finer resolution h .…”

“…were introduced in [8] for Cartesian grids, modified to give conservative discretizations in [9], and later extended to general polyhedral grids in, for example, [10,11]. The basic idea of the MsMFE method is to construct a special approximation space, consisting of a set of coarse-scale basis functions with resolution H that satisfy a flow equation locally and hence are consistent with the differential operator at a finer resolution h. The basis functions are usually computed numerically within each element.…”

“…Quite a number of such multiscale methods have been presented in the literature, including dualgrid methods [3][4][5][6], finite-element methods [7], mixed finite-element [8][9][10][11], and finite-volume methods [12][13][14][15][16][17]. Apart from algorithmic differences, all of these methods share the same basic concept of incorporating fine-scale information into the coarse-scale equations via some sort of numerically constructed functions.…”

The Multiscale Finite-Volume Method on Stratigraphic Grids

“…Herein, we start by discussing how to formulate the MsFV method for general unstructured grids and then present a set of coarsening methods that can create the required primal/dual grids for a reasonable class of stratigraphic grids. To this end, we first create a primal coarse grid using techniques developed for the multiscale mixed finite-element method, for which each coarse block can consist of an (almost) arbitrary simply-connected set of polyhedral cells (see e.g., (Aarnes et al 2006(Aarnes et al , 2008Natvig et al 2011;Alpak et al 2012;Lie et al 2012)). Then, the accompanying dual grid is created based on geometrical considerations.…”

Multiscale method for simulating two and three-phase flow in porous media

“…The literature contains a wide range of multiscale methods that are applicable to reservoir simulation, including dual-grid methods (Guerillot and Verdiere 1995;Audigane and Blunt 2004;Arbogast 2002;Arbogast and Bryant 2002), (adaptive) localglobal methods (Chen et al 2003;Chen and Durlofsky 2006), finite-element methods (Hou and Wu 1997), mixed finite-element methods (Chen and Hou 2002;Aarnes 2004;Aarnes and Efendiev 2008;Alpak et al 2011;Pal et al 2012), and finite-volume multiscale methods (Jenny et al 2003;Lee et al 2008;Jenny 2009, 2011;Parramore et al 2012). Although the methods have certain algorithmic differences, they share a common basic concept for incorporating fine-scale into coarse-scale flow equations.…”