Well Modeling in the Multiscale Finite Volume Method for Subsurface Flow Simulation

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.

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“…The proposed approach to handle concentrated sources is fundamentally different from the one proposed in [35], as is shown in Section 3.2.…”

Section: Introductionmentioning

confidence: 93%

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

Juanes

Dub

2008

Comput Geosci

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“…Applications of these techniques to real reservoir problems presuppose the ability of dealing with complex wells that might exhibit complex geometries and penetrate several blocks of the numerical grid. Recently, such a model has been proposed for the MSFV method [10]. In this paper, an alternative approach is presented.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, an alternative approach is presented. Like in [10], additional well basis functions are used to account for the new degrees of freedom. Differently, however, these well functions have the same support as the original basis functions, which is computationally advantageous and allows to take advantage of the formalism recently developed to include gravity and capillary pressure effects [8].…”

Section: Introductionmentioning

confidence: 99%

Well modeling in the multi‐scale finite‐volume context

Jenny

Lunati

2007

Proc Appl Math and Mech

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An extension of the multi-scale finite-volume (MSFV) method is devised, which allows to simulate flow and transport in reservoirs with complex well configurations. The new framework fits nicely into the data structure of the original MSFV method and it has the important property that no large patches covering the whole well are required. For each well, an additional degree of freedom is introduced. While the treatment of pressure-constraint wells is trivial, since the well-bore reference pressure is explicitly specified, additional equations have to be solved to obtain the unknown pressure values of rate-constraint wells. Numerical simulations of a difficult test case with multiple complex wells demonstrate the ability of the new framework to capture the interference between the various wells and the reservoir accurately.

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“…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)). …”

Section: Introductionmentioning

confidence: 99%

Monotone Multiscale Finite Volume Method for Flow in Heterogeneous Porous Media

Wang¹,

Hajibeygi²

2014

Proceedings

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SUMMARYThe MultiScale Finite-Volume (MSFV) method is known to produce non-monotone solutions. The causes of the non-monotone solutions are identified and connected to the local flux across the boundaries of primal coarse cells induced by the basis functions. We propose a monotone MSFV (m-MSFV) method based on a local stencil-fix that guarantees monotonicity of the coarse-scale operator, and thus the resulting approximate fine-scale solution. Detection of non-physical transmissibility coefficients that lead to non-monotone solutions is achieved using local information only and is performed algebraically. For these `critical' primal coarse-grid interfaces, a monotone local flux approximation, specifically, a TwoPoint Flux Approximation (TPFA), is employed. Alternatively, a local linear boundary condition is used for the basis functions to reduce the degree of non-monotonicity. The local nature of the two strategies allows for ensuring monotonicity in local sub-regions, where the non-physical transmissibility occurs. For practical applications, an adaptive approach based on normalized positive off-diagonal coarse-scale transmissibility coefficients is developed. Based on the histogram of these normalized coefficients, one can remove the large peaks by applying the proposed modifications only for a small fraction of the primal coarse grids. Though the m-MSFV approach can guarantee monotonicity of the solutions to any desired level, numerical results illustrate that employing the m-MSFV modifications only for a small fraction of the domain can significantly reduce the non-monotonicity of the conservative MSFV solutions.

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Well Modeling in the Multiscale Finite Volume Method for Subsurface Flow Simulation

Cited by 70 publications

References 15 publications

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

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

Well modeling in the multi‐scale finite‐volume context

Monotone Multiscale Finite Volume Method for Flow in Heterogeneous Porous Media

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