Assigment due: March 5, 2012
Two groups:
Group 1:
Develop a MATLAB program that solves the eective conductivity problem
for 2-D problems for a broad range of periodic variability. Method: FourierGalerkin (see the Dykaar and Kitanidis 1992 papers). The met
Selected Web Resources for Fourier Analysis
Web Book by Julius O. Smith
Web Book by Paolo Prandoni and Martin Vetterli
Tables of Transform Pairs by Marc Stoecklin
CN
Chapter 4
CT
Stokes Flow and Darcys Law
In this chapter, we will focus on the linearized ow equations that describe creeping ow.
We examine the fundamental equations of uid mechanics in order to build insights into
the mechanics of non-turbulent viscid
The Virtual Laboratory Approach
In this approach we imitate through mathematical simulation the process of
evaluating eective parameters of a block of a heterogeneous porous medium.
This approach gives us insights into the meaning of eective parameters an
Chapter 0
Taylor Dispersion
We will examine a classical upscaling problem, known as Taylor dispersion.
This problem is interesting and sustantial in itself but it becomes even more
important when we consider that the method that we will use can be general
WATER RESOURCES
RESEARCH,
VOL. 19, NO. 1, PAGES 161-180, FEBRUARY
1983
Three-Dimensional Stochastic Analysis of Macrodispersion
in Aquifers
LYNN W. GELHAR1 AND CARL L. AXNESS
2
New Mexico Institute of Mining and Technology, Socorro, New Mexico 87801
The d
WATER RESOURCES RESEARCH,
VOL. 25, NO. 11, PAGES 2287-2298, NOVEMBER
1989
Numerical Spectral Approach for the Derivation
of Piezometric
THOMAS
VAN
Head
LENT
Covariance
Functions
AND PETER K. KITANIDIS
Department of Civil Engineering, Stanford University,
WATER RESOURCES
RESEARCH,
VOL. 32, NO. 5, PAGES 1197-1207, MAY 1996
Effects of first-order approximations on head and specific
discharge covariancesin high-contrast log conductivity
Thomas
Van Lent
Departmentof Civil and Environmental
Engineering,
SouthDa
function [ran,x] = rfld1d(n,dx,rcovar);
% RFLD1D: Computes a realization of a 1-dimensional stationary
% random field
%
% Call by:
% [ran,x] = rfld1d(n,dx,rcovar)
% Input:
% n - number of nodes at which a value will be computed.
% Must be a power of 2.
%
Chapter 0
Review of Stochastic
Processes
In this chapter we will review some useful denitions and concepts from stochastic processes. When stochastic processes are functions (particularly functions
of spatial coordinates), they are known as random elds. (
Selected Web Resources for Fourier Analysis
Web Book by Julius O. Smith
Web Book by Paolo Prandoni and Martin Vetterli
Tables of Transform Pairs by Marc Stoecklin
January 11, 2012
6:20
World Scienti c Book - 9.75in x 6.5in
Chapter 1
Introduction
In this chapter, we dene the scope of these notes and introduce some key concepts.
1.1
Objectives
Porous media and, more generally, geologic media are complex composite mat
EffectiveConductivity
A series of three papers:
Kitanidis, P. K., "Effective Hydraulic Conductivity for Gradually Varying Flow." Water
Resources Research, 26(6), 1197-1208, 1990.
Dykaar, B. B., and Kitanidis, P. K., "Determination of the Effective Hydraul
The DFT
In our introduction to the numerical spectral approach, we encountered some of
the numerical challenges of computing Fourier coecients and periodic functions
from Fourier coecients. This leads to us the Discrete Fourier Transform (or
DFT).
The DFT
Chapter 0
Fourier Series
Fourier analysis is invaluable in the study of heterogeneity. The simplest application and a good place to start is with periodic media where Fourier-series
analysis applies.
This case is also important in its own right when a REV
Chapter 0
Multi-dimensional Fourier
In most applications, we are interested in variability in two or three dimensions.
Extension of Fourier series and integral to higher dimensions is simple, provided
that we stick with the complex-exponential notation.
0
Review of Stieltjes Integrals
P. K. Kitanidis
February 15, 2012
The Riemann-Stieltjes Integral
Consider an interval [ ] and the partition
= cfw_ = 0 1 =
(1)
The norm (or mesh) of the partition is the length of the longest of these
subintervals, that is,
Chapter 0
Fourier Integral
The Fourier series describes variability in a nite interval, with periodic repetition assumed for outside of that interval, and is convenient to use in numerical
computations because of its discrete nature. The Fourier integral
Review of Gelhar and Axness Approach
P. K. Kitanidis
February 22, 2012
For steady ow with isotropic but nonuniform conductivity = (x) 0:
=0
(1)
which is the same as
ln
2
+
=0
(2)
ln = + , = [ln ] = ln
= + = []
(3)
(4)
Use decomposition:
2
2
+
+
D ecember 29, 2011
13:23
World Scientic Book - 9.75in x 6.5in
Preface
This is a set of class notes for a specialized course on the eects of heterogeneity and
scale on ow and transport in permeable formations. The course is for researchoriented graduate st
function [ran,x,y] = rfld2d(n,dx,rcovar)
% RFLD2D: Generates a realization of a 2-dimensional intrinsic
% random field
% function [ran,x,y] = rfld2d(n,dx,rcovar)
% Inputs:
% n - vector with number of nodes in directions x and y
% dx - vector with incremen