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Mathematics
in Geosciences, September 2001 - June 2002
Talk
Abstracts:
IMA
Minisymposium:
March
13-15, 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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