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Talk abstract
IMA Minisymposium:
Numerical Methods in the Geosciences
March 13-15, 2002

Mathematics in Geosciences, September 2001 - June 2002

Material from Talks

Clint Dawson (TICAM, University of Texas at Austin)  clint@brahma.ticam.utexas.edu

The local discontinuous Galerkin method for flow and transport problems

Flow and transport problems are at the heart of most geoscience applications. These problems are characterized by rough coefficients, advection dominance, point sources and sinks, etc. Numerical schemes which preserve mass conservation and provide stable solutions in the presence of high gradients are desirable. In this talk, we will discuss a method recently proposed for handling these problems called the local discontinuous Galerkin method (LDG). This method is a type of classical mixed method, whereby one solves for the solution and its gradient or flux. This method is locally conservative, allows for local high order approximation, has built-in stability mechanisms such as upwinding, and allows for non-conforming mesh. Variants of the scheme will be discussed and applications to flow in porous media and shallow water will be presented.

Rick H. Dean (TICAM, University of Texas at Austin)  rhd@ticam.utexas.edu

Mixed impem/implicit techniques for compositional modeling of flow in porous media

Compositional simulations of porous media typically involve 5 to 20 components per grid block for complex EOR processes. Because of this, implicit techniques require a large amount of memory and each time step can take a significant amount of CPU time. Impem (IMplicit Pressure Explicit Mass) techniques require less memory and use less CPU time per time step. But impem techniques often require many more time steps for the same simulation. Mixed impem/implicit techniques attempt to capture the benefits of both techniques while avoiding the drawbacks that are inherent in each technique. Mixed impem/implicit techniques are discussed for simulations that contain multiple grids where some grids use a time discretization that is implicit and other grids use a time discretization that is impem. In addition, linear solution techniques are discussed for mixed impem/implicit problems. Examples of mixed impem/implicit simulations will be presented.

Jim Douglas, Jr. (Department of Mathematics, Purdue University)  douglas@math.purdue.edu

A Locally Conservative Eulerian-Lagrangian Finite Element Method and Applications

Joint work with Felipe Pereira, and Carlos Roman.

A locally conservative Eulerian-Lagrangian method (LCELM) is described for the numerical solution of two-phase, immiscible displacement in a 3-D porous medium. This method, introduced by Douglas, Pereira, and Yeh for two-dimensional problems, is directly applicable to scalar nonlinear transport problems. We first describe the method and then discuss some issues related to its implementation. The result of a numerical experiment will be presented.

Louis J. Durlofsky (Department of Petroleum Engineering, Stanford University)  lou@pangea.Stanford.EDU

Upscaling and Gridding of Geologically Complex Systems for Reservoir Flow Simulation
Slides:    html    pdf    powerpoint

Geostatistical reservoir descriptions are generally much too detailed for direct use in reservoir flow simulations. Upscaling procedures are required to coarsen these detailed reservoir descriptions to scales more suitable for flow calculations. In this talk, several upscaling procedures, appropriate for both moderate and high degrees of upscaling, will be described and applied. A technique for the improved calculation of coarse scale equivalent permeability tensors, which entails the use of a border region of fine grid cells around the target coarse block, will be described. This technique is then combined with a flow-based grid generation procedure that is able to provide flexible, structured grids for geometrically complex, generally anisotropic systems. The permeability upscaling and gridding techniques will each be shown to lead to improvements in the accuracy of coarse scale reservoir descriptions relative to reference fine grid results. When used in combination, the overall methodology provides significantly enhanced coarse models.

To achieve higher levels of upscaling, subgrid models of transport are required. A methodology for representing subgrid effects in the saturation (water transport) equation, based on the use of volume averaging and the approximate modeling of higher moments, will be presented. The technique couples local fine scale fluctuations with global coarse scale information to provide a subgrid model that is driven by large scale flow behavior. For a series of model problems, it will be shown that this technique provides much more accurate results than coarsened models that do not contain a subgrid treatment.

James M. Hyman (Los Alamos National Laboratory)  mac@t7.lanl.gov

Subgrid scale modeling of flow through heterogeneous porous media

We are deriving numerical methods to model the fine scale structure of heterogeneous porous media on a coarser scale. The multigrid approach provides an automatic approach to transform a fine grid description of a subsurface formation into a coarse grid averaged model. The up-scaled diffusion coefficients capture the average flow rates and pressure of the original formation. The upscaled equations are then solved with a mimetic finite-difference algorithm derived specifically for diffusion problems in strongly heterogeneous and non-isotropic media. These difference approximations are especially effective on problems with rough coefficients or highly nonuniform grids. This is joint work with Misha Shashkov and Stanley Steinberg.

Malgorzata Peszynska (CSM TICAM, University of Texas at Austin)  mpesz@ticam.utexas.edu  http://www.ticam.utexas.edu/~mpesz

Coupling of models for multiphase flow and transport in porous media with multiple scales

In recent years it has been recognized that the simulation of subsurface flow and transport phenomena must be accompanied by simulation of all the coupled processes that influence, or are influenced by, these phenomena at all relevant scales. However, in spite of the emergence of new computational methodologies and of the dramatic increases in computational power, for many reasons it is still difficult or impractical to formulate and/or implement large comprehensive models, especially, if there is need to account for multiple spatial and temporal scales. As an alternative to comprehensive models, one sees the emergence of multiphysics couplings, in which models can be coupled in a loose (staggered-in-time) fashion, or more tightly, where a solution is obtained by iterating between models.

In this talk first we give two distinct examples of couplings: i) of multiphase flow models connected to a reactive transport model, all defined in the same domain, and ii) the couplings of different multiphase models across interface using mortar spaces. These two types are being merged in an on-going research project. Then we focus on case ii) and discuss the related numerical and computational challenges as well as the applications to upscaling. For the latter we outline the mortar upscaling technique in which no effective parameters need be computed. Finally, we present some preliminary work on mortar adaptivity.

This is joint work with Qin Lu, Manish Parashar, Shuyu Sun, Mary F. Wheeler, and Ivan Yotov.

Mary F. Wheeler (The University of Texas at Austin)  mfw@brahma.ticam.utexas.edu

Locally Conservative Algorithms for Flow and Transport

Joint with Beatrice Riviere.

In the numerical modeling of fluid flow and transport problems, it is necessary for the velocities to be locally conservative on the transport grid. Lack of local mass conservation results in spurious sources and sinks to the transport equation. Local mass conservation can be accomplished through a projection algorithm, but this can be expensive and is generally only first order. It is generally better to use a locally conservative approach from the beginning.

Here we discuss the formulation, analysis and application of several numerical locally conservative algorithms: Discontinuous Galerkin methods, mixed finite element methods, and control volume. We discuss advantages and disadvantages of each of these methods. Numerical results from subsurface and surface flow problems are presented.

Gabriel Wittum (Universitaet Heidelberg, Simulation in Technology)  gabriel.wittum@iwr.uni-heidelberg.de  http://www.iwr.uni-heidelberg.de/~techsim

Multigrid Methods for Porous Media Flow Problems and Applications

Solvers play a crucial role in many simulations determinig the complexity of the whole process. Thus solving often limits the obtainable accuracy and fast solver even are the key to simulate new challenging problems.

In the lecture multigrid methods for the simulation of large porous media flow problems are discussed. Problems like heterogeneity, non-M-matrices etc. are addresseed. Further adaptivity and parallelism is discussed. The software system UG is presented which is based on these strategies. In several application cases the efficiency of the selected approach is shown.

Dongxiao Zhang (Los Alamos National Laboratory)  donzhang@lanl.gov

Nonstationary Stochastic Flow and Transport Theories and their Applications

It has now been well recognized that flow and transport in porous media is strongly influenced by medium spatial variabilities and is subject to uncertainties. Since such a situation cannot be accurately modeled deterministically without considering the uncertainties, it has become quite common to approach the subsurface flow and transport problem stochastically. Many researchers have developed and applied stochastic theories to subsurface flow and transport problems since the late 1970s. This field of research has led to fundamental understandings for flow and transport in random porous media, uncertainty propagation in such media, effective parameters, and scale dependent coefficients. However, its applications to real-world problems have been limited because of a number of simplifying assumptions and treatments such as stationary (statistically homogeneous) medium properties, uniform mean flow, and unbounded domains. This presentation will discuss some recently developed nonstationary stochastic flow and transport theories, which account for nonstationary, multiscale medium features, non-uniform mean flow (including fluid pumping/injecting), and finite domain boundaries. In these theories, general equations governing the statistical moments of flow and transport variables are derived from the original stochastic equations. Owing to the spatial nonstationarities in both the independent and dependent variables, the moment equations must generally be solved by numerical techniques, whose strategies can differ significantly from those in solving the original equations. Some computational examples and results will be shown and future research directions will be discussed.


Material from Talks

Mathematics in Geosciences, September 2001 - June 2002
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