Multiscale finite-volume formulation for multiphase flow in porous media: black oil formulation of compressible, three-phase flow with gravity

Abstract

Most practical reservoir simulation studies are performed using the so-called black oil model, in which the phase behavior is represented using solubilities and formation volume factors. We extend the multiscale finite-volume (MSFV) method to deal with nonlinear immiscible three-phase compressible flow in the presence of gravity and capillary forces (i.e., black oil model). Consistent with the MSFV framework, flow and transport are treated separately and differently using a sequential implicit algorithm. A multiscale operator splitting strategy is used to solve the overall mass balance (i.e., the pressure equation). The black-oil pressure equation, which is nonlinear and parabolic, is decomposed into three parts. The first is a homogeneous elliptic equation, for which the original MSFV method is used to compute the dual basis functions and the coarse-scale transmissibilities. The second equation accounts for gravity and capillary effects; the third equation accounts for mass accumulation and sources/ sinks (wells). With the basis functions of the elliptic part, the coarse-scale operator can be assembled. The gravity/capillary pressure part is made up of an elliptic part and a correction term, which is computed using solutions of gravity-driven local problems. A particular solution represents accumulation and wells. The reconstructed fine-scale pressure is used to compute the finescale phase fluxes, which are then used to solve the nonlinear saturation equations. For this purpose, a Schwarz iterative scheme is used on the primal coarse grid. The framework is demonstrated using challenging black-oil examples of nonlinear compressible multiphase flow in strongly heterogeneous formations.

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

“…Inspired in the multiscale finite element method, a multiscale finite volume method was proposed in [24]. This method also preserves mass conservation at the coarse and fine scales (see also [25,29]). …”

The multiscale restriction smoothed basis method for fractured porous media (F-MsRSB)

“…The latter property makes them unique compared with alternative advanced solvers, such as multigrid methods [27]. Recent developments of the MSFV method include extensions to compressible and compositional non-linear displacements [28,29], unstructured grids [14,30], and fully-implicit simulations [31]. While these important developments, combined, cast a promising framework for next-generation simulators, they have been focused mainly on addressing challenges due to complex fluid physics, highly heterogeneous rock properties, and complex computational mesh geometries.…”

Monotone Multiscale Finite Volume Method for Flow in Heterogeneous Porous Media

“…MSFV, which was first proposed for heterogeneous elliptic pressure equations by Jenny et al (2003), can be viewed as a locally conservative extension of the MultiScale Finite Element (MSFE) method by Hou and Wu (1997). Recent developments of the MSFV method allow for compositional effects and complex wells, making it a promising approach for the next-generation of reservoir flow simulators (Jenny et al (2006); Lee et al (2008); Zhou et al (2011); Zhou and Tchelepi (2008); Hajibeygi and Jenny (2009) ;Hajibeygi and Tchelepi (2014); Wolfsteiner et al (2006); Jenny and Lunati (2009); Lee et al (2009) ;Hajibeygi et al (2012)). …”