Multiscale Finite Volume Method Research Articles

ON NUMERICAL UPSCALING FOR FLOWS IN HETEROGENEOUS POROUS MEDIA

AbstractA numerical upscaling approach, NU, for solving multiscale elliptic problems is discussed. The main components of this NU are: i) local solve of aux-iliary problems in grid blocks and formal upscaling of the obtained results to build a coarse scale equation; ii) global solve of the upscaled coarse scale equation; and iii) reconstruction of a fine scale solution by solving local block problems on a dual coarse grid. By its structure NU is similar to other methods for solving multiscale elliptic problems, such as the multiscale finite element method, the multiscale mixed finite element method, the numerical subgrid upscaling method, heterogeneous mul-tiscale method, and the multiscale finite volume method. The difference with those methods is in the way the coarse scale equation is build and solved, and in the way the fine scale solution is reconstructed. Essential components of the presented here NU approach are the formal homogenization in the coarse blocks and the usage of so called multipoint flux approximation method, MPFA. Unlike the usual usage of MPFA as a discretization method for single scale elliptic problems with tensor discontinuous coefficients, we consider its usage as a part of a numerical upscaling approach. An aim of this paper is to compare the performance of NU with the one of MsFEM for ceratin multiscale problems. In particular, it is shown that the resonance effect, which limits the application of the Multiscale FEM, does not appear, or it is significantly relaxed, when the presented here numerical upscaling approach is applied

Analysis Of Upwind Method For Piecewise Deterministic Markov Processes

AbstractA numerical upscaling approach, NU, for solving multiscale elliptic problems is discussed. The main components of this NU are: i) local solve of aux-iliary problems in grid blocks and formal upscaling of the obtained results to build a coarse scale equation; ii) global solve of the upscaled coarse scale equation; and iii) reconstruction of a fine scale solution by solving local block problems on a dual coarse grid. By its structure NU is similar to other methods for solving multiscale elliptic problems, such as the multiscale finite element method, the multiscale mixed finite element method, the numerical subgrid upscaling method, heterogeneous mul-tiscale method, and the multiscale finite volume method. The difference with those methods is in the way the coarse scale equation is build and solved, and in the way the fine scale solution is reconstructed. Essential components of the presented here NU approach are the formal homogenization in the coarse blocks and the usage of so called multipoint flux approximation method, MPFA. Unlike the usual usage of MPFA as a discretization method for single scale elliptic problems with tensor discontinuous coefficients, we consider its usage as a part of a numerical upscaling approach. An aim of this paper is to compare the performance of NU with the one of MsFEM for ceratin multiscale problems. In particular, it is shown that the resonance effect, which limits the application of the Multiscale FEM, does not appear, or it is significantly relaxed, when the presented here numerical upscaling approach is applied.

Well modeling in the multi‐scale finite‐volume context

AbstractAn 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. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Adaptive fully implicit multi-scale finite-volume method for multi-phase flow and transport in heterogeneous porous media

We describe a sequential fully implicit (SFI) multi-scale finite volume (MSFV) algorithm for nonlinear multi-phase flow and transport in heterogeneous porous media. The method extends the recently developed multiscale approach, which is based on an IMPES (IMplicit Pressure, Explicit Saturation) scheme [P. Jenny, S.H. Lee, H.A. Tchelepi, Adaptive multiscale finite volume method for multi-phase flow and transport, Multiscale, Model. Simul. 3 (2005) 50–64]. That previous method was tested extensively and with a series of difficult test cases, where it was clearly demonstrated that the multiscale results are in excellent agreement with reference fine-scale solutions and that the computational efficiency of the MSFV algorithm is much higher than that of standard reservoir simulators. However, the level of detail and range of property variability included in reservoir characterization models continues to grow. For such models, the explicit treatment of the transport problem (i.e. saturation equations) in the IMPES-based multiscale method imposes severe restrictions on the time step size, and that can become the major computational bottleneck. Here we show how this problem is resolved with our sequential fully implicit (SFI) MSFV algorithm. Simulations of large (million cells) and highly heterogeneous problems show that the results obtained with the implicit multi-scale method are in excellent agreement with reference fine-scale solutions. Moreover, we demonstrate the robustness of the coupling scheme for nonlinear flow and transport, and we show that the MSFV algorithm offers great gains in computational efficiency compared to standard reservoir simulation methods.

Multiscale finite-volume method for compressible multiphase flow in porous media

The Multiscale Finite-Volume (MSFV) method has been recently developed and tested for multiphase-flow problems with simplified physics (i.e. incompressible flow without gravity and capillary effects) and proved robust, accurate and efficient. However, applications to practical problems necessitate extensions that enable the method to deal with more complex processes. In this paper we present a modified version of the MSFV algorithm that provides a suitable and natural framework to include additional physics. The algorithm consists of four main steps: computation of the local basis functions, which are used to extract the coarse-scale effective transmissibilities; solution of the coarse-scale pressure equation; reconstruction of conservative fine-scale fluxes; and solution of the transport equations. Within this framework, we develop a MSFV method for compressible multiphase flow. The basic idea is to compute the basis functions as in the case of incompressible flow such that they remain independent of the pressure. The effects of compressibility are taken into account in the solution of the coarse-scale pressure equation and, if necessary, in the reconstruction of the fine-scale fluxes. We consider three models with an increasing level of complexity in the flux reconstruction and test them for highly compressible flows (tracer transport in gas flow, imbibition and drainage of partially saturated reservoirs, depletion of gas–water reservoirs, and flooding of oil–gas reservoirs). We demonstrate that the MSFV method provides accurate solutions for compressible multiphase flow problems. Whereas slightly compressible flows can be treated with a very simple model, a more sophisticate flux reconstruction is needed to obtain accurate fine-scale saturation fields in highly compressible flows.

Preconditioning Markov Chain Monte Carlo Simulations Using Coarse-Scale Models

We study the preconditioning of Markov chain Monte Carlo (MCMC) methods using coarse-scale models with applications to subsurface characterization. The purpose of preconditioning is to reduce the fine-scale computational cost and increase the acceptance rate in the MCMC sampling. This goal is achieved by generating Markov chains based on two-stage computations. In the first stage, a new proposal is first tested by the coarse-scale model based on multiscale finite volume methods. The full fine-scale computation will be conducted only if the proposal passes the coarse-scale screening. For more efficient simulations, an approximation of the full fine-scale computation using precomputed multiscale basis functions can also be used. Comparing with the regular MCMC method, the preconditioned MCMC method generates a modified Markov chain by incorporating the coarse-scale information of the problem. The conditions under which the modified Markov chain will converge to the correct posterior distribution are stated in the paper. The validity of these assumptions for our application and the conditions which would guarantee a high acceptance rate are also discussed. We would like to note that coarse-scale models used in the simulations need to be inexpensive but not necessarily very accurate, as our analysis and numerical simulations demonstrate. We present numerical examples for sampling permeability fields using two-point geostatistics. The Karhunen--Loève expansion is used to represent the realizations of the permeability field conditioned to the dynamic data, such as production data, as well as some static data. Our numerical examples show that the acceptance rate can be increased by more than 10 times if MCMC simulations are preconditioned using coarse-scale models.

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

A multiscale method for effective handling of wells (source/sink terms) in the simulation of multiphase flow and transport processes in heterogeneous porous media is developed. The approach extends the multiscale finite volume (MSFV) framework. Our multiscale well model allows for accurate reconstruction of the fine‐scale pressure and velocity fields in the vicinity of wells. Accurate and computationally efficient modeling of complex wells is a prerequisite for field applications, and the ability to model wells within the MSFV framework makes it possible to solve large‐scale heterogeneous problems of practical interest. Our approach consists of removal of the well singularity from the multiscale solution via a local change of variables and the computation of a smoothly varying background field instead. The well effects are computed using a separate basis function, which is superposed on the background solution to yield accurate representation of the flow field. The multiscale well treatment accounts for b...

Adaptive Multiscale Finite-Volume Method for Multiphase Flow and Transport in Porous Media

We present a multiscale finite-volume (MSFV) method for multiphase flow and transport in heterogeneous porous media. The approach extends our recently developed MSFV method for single-phase flow. We use a sequential scheme that deals with flow (i.e., pressure and total velocity) and transport (i.e., saturation) separately and differently. For the flow problem, we employ two different sets of basis functions for the reconstruction of a conservative fine-scale total velocity field. Our basis functions are designed to have local support, and that allows for adaptive computation of the flow field. We use a criterion based on the time change of the total mobility field to decide when and where to recompute our basis functions. We show that at a given time step, only a small fraction of the basis functions needs to be recomputed. Numerical experiments of difficult two-dimensional and three-dimensional test cases demonstrate the accuracy, computational efficiency, and overall scalability of the method.

Solution of shallow water equations using fully adaptive multiscale schemes

The concept of fully adaptive multiscale finite volume methods has been developed to increase spatial resolution and to reduce computational costs of numerical simulations. Here grid adaptation is performed by means of a multiscale analysis based on biorthogonal wavelets. In order to update the solution in time we use a local time stepping strategy that has been recently developed for hyperbolic conservation laws. The adaptive multiresolution scheme is now applied to two-dimensional shallow water equations with source terms. The efficiency of the scheme is demonstrated on several problems with a general geometry, including circular damp breaks, oblique hydraulic jump, supercritical channel flows encountering sudden change in cross-section, and, finally, the bore wave and its interactions. Copyright © 2005 John Wiley & Sons, Ltd.

Direct Simulation of Fluid Transport at Solid Interfaces with a Multiscale Lattice-Boltzmann Finite-Volume Method

Abstract It is shown that the combined use of a mesoscopic lattice Boltzmann solver with finite-volume techniques, both enriched with local-refinement (multiscale) capabilities, permits to describe transport phenomena at fluid-solid interfaces to a degree of detail which may help dispensing with empirical correlations.

Multi-scale finite-volume method for elliptic problems in subsurface flow simulation

In this paper we present a multi-scale finite-volume (MSFV) method to solve elliptic problems with many spatial scales arising from flow in porous media. The method efficiently captures the effects of small scales on a coarse grid, is conservative, and treats tensor permeabilities correctly. The underlying idea is to construct transmissibilities that capture the local properties of the differential operator. This leads to a multi-point discretization scheme for the finite-volume solution algorithm. Transmissibilities for the MSFV have to be constructed only once as a preprocessing step and can be computed locally. Therefore this step is perfectly suited for massively parallel computers. Furthermore, a conservative fine-scale velocity field can be constructed from the coarse-scale pressure solution. Two sets of locally computed basis functions are employed. The first set of basis functions captures the small-scale heterogeneity of the underlying permeability field, and it is computed in order to construct the effective coarse-scale transmissibilities. A second set of basis functions is required to construct a conservative fine-scale velocity field. The accuracy and efficiency of our method is demonstrated by various numerical experiments.