Jørg E. Aarnes

On the Use of a Mixed Multiscale Finite Element Method for GreaterFlexibility and Increased Speed or Improved Accuracy in Reservoir Simulation

In this paper we propose a modified mixed multiscale finite element method for solving elliptic problems with rough coefficients arising in, e.g., porous media flow. The method is based on the construction of special base functions which adapt to the local property of the differential operator. In particular, the method incorporates the effect of small-scale heterogeneous structures in the elliptic coefficients into the base functions and produces a detailed velocity field that can be used to solve phase transport equations at a subgrid scale. The method is mass conservative and accounts for radial flow in the near-well region without resorting to complicated well models or near-well upscaling procedures. As such, the method provides a step toward a more accurate and rigorous treatment of advanced well architectures in reservoir simulation. The accuracy of the method is demonstrated through a series of three-dimensional incompressible two-phase flow simulations.

A Hierarchical Multiscale Method for Two-Phase Flow Based upon Mixed Finite Elements and Nonuniform Coarse Grids

We analyze and further develop a hierarchical multiscale method for the numerical simulation of two-phase flow in highly heterogeneous porous media. The method is based upon a mixed finite-element formulation, where fine-scale features are incorporated into a set of coarse-grid basis functions for the flow velocities. By using the multiscale basis functions, we can retain the efficiency of an upscaling method by solving the pressure equation on a (moderate-sized) coarse grid, while at the same time produce a detailed and conservative velocity field on the underlying fine grid.Earlier work has shown that the multiscale method performs excellently on highly heterogeneous cases using uniform coarse grids. In this paper, we extend the methodology to nonuniform and unstructured coarse grids and discuss various formulations for generating the coarse-grid basis functions. Moreover, we focus on the impact of large-scale features such as barriers or high-permeable channels and discuss potentially problematic flow ...

An Introduction to the Numerics of Flow in Porous Media using Matlab

Even though the art of reservoir simulation has evolved through more than four decades, there is still a substantial research activity that aims toward faster, more robust, and more accurate reservoir simulators. Here we attempt to give the reader an introduction to the mathematics and the numerics behind reservoir simulation. We assume that the reader has a basic mathematical background at the undergraduate level and is acquainted with numerical methods, but no prior knowledge of the mathematics or physics that govern the reservoir flow process is needed. To give the reader an intuitive understanding of the equations that model filtration through porous media, we start with incompressible single-phase flow and move step-by-step to the black-oil model and compressible two-phase flow. For each case, we present a basic numerical scheme in detail, before we discuss a few alternative schemes that reflect trends in state-of-the-art reservoir simulation. Two and three-dimensional test cases are presented and discussed. Finally, for the most basic methods we include simple Matlab codes so that the reader can easily implement and become familiar with the basics of reservoir simulation.

Mixed Multiscale Finite Element Methods for Stochastic Porous Media Flows

In this paper, we propose a stochastic mixed multiscale finite element method. The proposed method solves the stochastic porous media flow equation on the coarse grid using a set of precomputed basis functions. The precomputed basis functions are constructed based on selected realizations of the stochastic permeability field, and furthermore the solution is projected onto the finite-dimensional space spanned by these basis functions. We employ multiscale methods using limited global information since the permeability fields do not have apparent scale separation. The proposed approach does not require any interpolation in stochastic space and can easily be coupled with interpolation-based approaches to predict the solution on the coarse grid. Numerical results are presented for permeability fields with normal and exponential variograms.

Mixed Multiscale Finite Element Methods Using Limited Global Information

In this paper, we present a mixed multiscale finite element method using limited global information. We consider a general case where multiple global information is given such that the solution depends smoothly on these global fields. The global fields typically contain small-scale (local or global) information required for achieving a convergence with respect to the coarse mesh size. We present a rigorous analysis and show that the proposed mixed multiscale finite element methods converge. Some preliminary numerical results are shown. We study a parameter dependent permeability field (a simplified case for general stochastic permeability fields). As for spatial heterogeneities, channelized permeability fields with strong nonlocal effects are considered. Using a few global fields corresponding to realizations of permeability fields, we show that one can achieve high accuracy in numerical simulations.

Modelling of Multiscale Structures in Flow Simulations for Petroleum Reservoirs

Flow in petroleum reservoirs occurs on a wide variety of physical scales. This poses a continuing challenge to modelling and simulation of reservoirs since fine-scale effects often have a profound impact on flow patterns on larger scales. Resolving all pertinent scales and their interaction is therefore imperative to give reliable qualitative and quantitative simulation results. To overcome the problem of multiple scales it is customary to use some kind of upscaling or homogenisation procedure, in which the reservoir properties are represented by some kind of averaged properties and the flow is solved on a coarse grid. Unfortunately, most upscaling techniques give reliable results only for a limited range of flow scenarios. Increased demands for reservoir simulation studies have therefore led researchers to develop more rigorous multiscale methods that incorporate subscale effects more directly.

An Adaptive Multiscale Method for Simulation of Fluid Flow in Heterogeneous Porous Media

Several multiscale methods for elliptic problems that provide high resolution velocity fields at low computational cost have been applied to porous media flow problems. However, to achieve enhanced accuracy in the flow simulation, the numerical scheme for modeling the transport must account for the fine scale structures in the velocity field. To solve the transport equation on the fine scale with conventional finite volume methods will often be prohibitively computationally expensive for routine simulations. In this paper we propose a more efficient adaptive multiscale method for solving the transport equation. In this method the global flow is computed on a coarse grid scale, while at the same time honoring the fine scale information in the velocity field. The method is tested on both two‐ and three‐dimensional test cases with complex heterogeneous structures. The numerical results demonstrate that the adaptive multiscale method gives nearly the same flow characteristics as simulations where the transpor...

Multiscale Discontinuous Galerkin Methods for Elliptic Problems with Multiple Scales

We introduce a new class of discontinuous Galerkin (DG) methods for solving elliptic problems with multiple scales arising from e.g., composite materials and flows in porous media. The proposed methods may be seen as a generalization of the multiscale finite element (FE) methods. In fact, the proposed DG methods are derived by combining the approximation spaces for the multiscale FE methods and relaxing the continuity constraints at the inter-element interfaces. We demonstrate the performance of the proposed DG methods through numerical comparisons with the multiscale FE methods for elliptic problems in two dimensions.

Multiscale Methods for Subsurface Flow

“There is a growing recognition that the world faces a water crisis that, left unchecked, will derail the progress towards the Millennium Development Goals and hold back human development. Some 1.4 billion people live in river basins in which water use exceeds recharge rates. The symptoms of overuse are disturbingly clear: rivers are drying up, groundwater tables are falling and water-based ecosystems are being rapidly degraded. Put bluntly, the world is running down one of its most precious natural resources and running up an unsustainable ecological debt that will be inherited by future generations.”

A multiscale method for modeling transport in porous media on unstructured corner-point grids

Multiscale solution methods are currently under active investigation for the simulation of subsurface flow in heterogeneous formations. These procedures capture the effects of fine-scale permeability variations through the calculation of specialized coarse-scale basis functions. Most of the multiscale techniques presented to date address subgrid capturing in pressure solution (elliptic/parabolic equations). In this paper we propose a multiscale method for solving transport equations on a coarse grid. In this method the global flow is computed on a coarse grid scale, but information from a fine scale velocity field is used to improve accuracy. The method is applied to incompressible and immiscible two-phase flow on a synthetic geological model with corner-point grid geometry. Although corner-point grids are given on a logically Cartesian format, the resulting grids are unstructured in physical space. The numerical results demonstrate that the multiscale method gives nearly the same flow characteristics as simulations where the transport equation is solved on the scale of an underlying fine grid.