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Flow Based Scale-up of
Heterogeneous Porous Media
using Homogenization and
Wavelet Representat ion
Joe Koebbel and Ryan horna as^
Abstract - Scaling up microscopic heterogeneities t o a macroscopic
level useful in reservoir engineering applications is a difficult problem.
Many of the small scale heterogeneous structures such as shale barriers
that occur deltaic rock formations dictate flow on the larger scale. Most
averaging methods are not not able to maintain the information contained in these types of structures at the macroscopic scale. Another
important example is that of a fractured reservoir where depending on
age, fractures can act as either conduits or barriers to flow through the
rock. The goal of the research presented is to use (1) flow based averaging techniques t o construct a more rigorous way of averaging the rock
properties and (2) use wavelet representation in the averaging method
t o preserve the effect of any microscopic structures that strongly influence the flow. Results from the application of this methodology to the
rock formations at the Ferron Sandstone site in southern Utah will be
presented. Transect data obtained by the Utah Geological Survey in
joint work with Mobil Oil, British Petroleum, and a number of other
research groups will be used as a basis of the numerical studies.
SCALING PROBLEMS IN POROUS MEDIA
Computer modeling of fluid flow in porous media is used to predict the
performance of reservoirs in groundwater flow and petroleum engineering applications. In many cases the porous medium is highly heterogeneous. The
heterogeneity occurs at all scales in the problem and very small scale features
of a reservoir can strongly influence flow of fluids in the reservoir. In most
applications, computer models are needed on a scale that is large relative to
many import ant geological features. Exactly resolving the import ant small
'Associate Professor, Department of Math & Stat, Utah State University, Logan, UT
2 ~ n d e r g r a d u a t eStudent, Department of Math & Stat, Utah State University, Logan,
UT
scale features is impossible and normally some method of averaging is used in
the hope that the bulk or averaged properties will still contain the important
information from the small scale heterogeneities.
For example, fluvial deltaic formations, which comprise a large share of
the known hydrocarbon bearing reservoir formations in the world, are created
by patterns of depositional events that leave possibly large deposits of coarse
grain material such as sand, separated by small amounts of finer grain materials, such as clays and silt. The resulting rock formation from these types of
depositional environments is layered with high ~ermea~bility
regions separated
by thin, relatively low permeability layers. The bulk flow in these formations
is strongly influenced by these very small scale features.
Fractures in a formation are also small scale features that can function as
either conduits or barriers. Just after formation a fracture will function as
a conduit allowing fluid to move from one region to another easily. Later as
minerals deposit along the fracture surface the fracture becomes a barrier to
flow. Each of these regimes is important in reservoir modeling. To understand
how a formation has trapped the hydrocarbons it necessary to be able to model
the early flow history through the fracture and to understand how to produce
the hydrocarbon effectively after the fractures have shut off, the behavior of
the fracture as a barrier to flow must be understood. The effects of these
relatively small scale heterogeneities must be included in the computer model.
AVERAGING PROCESSES AND HOMOGENIZATION
The description will focus on a simple flow model, Darcy's equation, for a
porous medium. The simplest partial differential equation that models flow in
a reservoir is
V . K(x)Vh(x) = f ( x )
(1)
The parameters in the problem are the rock permeability (or conductivity)
tensor, K defined by
in one two and three dimensions, respectively, the pressure head (or hydraulic
head) h, the spatial variable x, and a forcing function f which can be used
to represent wells or confining reservoirs. For simplicity assume that the permeability varies on only two scales which will be termed the microscopic and
macroscopic scales. In what follows the only parameter that will be scaled
up will be the permeability, K . For convenience the field is assumed to be
periodic. The ideas here have been extended to aperiodic permeability fields.
-XI
microscopic scale
macroscopic scale
Figure 1: Illustration of the (a) macroscopic scale and (b) a microscopic view
of a cell.
Figure (1) depicts the two length scales in two spatial dimensions. The
pattern of the permeabilities from a distant or macroscopic view is periodic. As
we zoom in on the cells we see that in each cell the structure of the permeability
can be quite general. It is assumed that the cell is small compared to the
overall size of the reservoir. Next define two length scales; the macroscopic
scale represented by the variable x and the microscopic scale represented by the
variable y. The macroscopic scale involves the pressure and velocity variables
over the entire reservoir, X. The microscopic scale is defined by a single cell
in Figure ( I ) , denoted by Y. The relationship between x and y is y = X / E .
Note that that on the microscopic scale as y varies from 0 to 1 the macroscopic
variable x varies from 0 to e.
The goal is to determine a bulk permeability on the region X by scaling
up the information in the small cells. There are some simple cases that can be
discussed without complicated averaging. Consider a model of flow through a
layered porous medium with well defined discontinuities between the different
layers of rock. Assume the permeability in each type of rock is constant. If we
assume that the fluid flow is either parallel or perpendicular to the direction of
flow then the scale up methods that are appropriate are clear. When the flow
is parallel to the discontinuities the arithmetic average should be used for the
permeability and when the flow is perpendicular to the direction of flow the
Permeability = 10
Permeability = 10.0
I
I
Permeability = 1
Figure 2: The symmetric and inverted-L patterns of permeabilities used for
illustrating problems with functional averaging methods. The harmonic, arithmetic and geometric means return diagonal tensors.
harmonic average of the permeabilities should be used. With a little analysis
it is easy to show that if the flow occurs at some other angle of incidence
to the discontinuities the permeability should be between the harmonic and
arithmetic average. The question is what is the appropriate effective value to
used based on the flow? In many cases ad hoc methods are used that combine
harmonic and arithmetic averaging. If the field is log-normal the geometric
average may be appropriate.
Each of these averaging methods is a functional averaging method. That
is, no matter where the signal or data has come from the same averaging technique will be applied. How the data is used in the solution of the flow problem
is not considered. Thus the standard methods of averaging are not conditioned on the underlying physical process. As an illustration of the problem
consider the two permeability patterns shown in Figure (2). A symmetric and
nonsymmetric pattern that looks like an inverted 'L'are shown. Each of these
regions can be divided into 16 equally sized cells; 4 with permeability 1 and
12 with permeability 10. In two dimensions the permeability tensors in this
case would be
where I is the usual 2x2 identity matrix. Suppose that we impose a pressure
gradient so that the fluid will flow from lower left to upper right in the region.
We should expect the fluid to move around the obstacle in both case. the flow
should be symmetric in the first case while asymmetric in the inverted-L case.
If we average the permeability tensor using any functional based method
we will obtain a symmetric tensor which is a multiple of the identity. The
tensors for the arithmetic, harmonic, and geometric averages are K A = 7.75 I
and
= 3.077 I, and I h x 5.623 I, respectively for both the symmetric and
asymmetric patterns. You should note that in all of these cases the principle
flow directions are still lined up along the coordinate axes. Thus using these
averaging methods will produce the same tensor and the same bulk properties.
We need an averaging method that is conditioned to the flow problem we are
trying to solve.
The homogenization procedure explained below will generate tensors
0
6.651
and
K#
=
[
5.574 -0.348
-0.348
6.949
for the symmetric and asymmetric patterns, respectively. For the symmetric
pattern the principle flow directions are lined up along the coordinate axes as
one would expect. In the asymmetric pattern the principle flow directions are
at approximately 76 degrees from the horizontal axis.
In the next section a very brief description of the homogenization process
will be given. The interested reader is referred to Bourgeat, 1984, for theoretical results and to Amaziane, et. al., 1990 for computational aspects of
homogenization. In addition, a code exists for applying homogenization on
rectangular grids Koebbe, 1996. The computer code can be downloaded via
anonymous ftp from
and then uncompressed on a Unix operating system.
The Perturbation Method
The averaging is done over one of the small cells as depicted in Figure (1).
In the work below we will want to use the Darcy velocity in the computations
which is defined by v = -KVh This quantity can be used to reduce the second
order partial differential equation to a system of first order partial differential
equations of the form
The perturbation method will be applied to the system of partial differential
equations defined by Equations (2) and (3).
The parameter E which is related to the size of the microscopic cell is
small; that is 0 < 6 << 1. The perturbation method then assumes that the
variables, h and v, can be expanded in a series in the perturbation parameter t ;
h = ho thl t2hz$ . . and v = vo+tvl t2v2 . The spatial differentiation
where the subscripts denote differentiation
operator V becomes V = V,+
with respect to to the specified variable. The differentiation is decomposed into
two pieces; one at the macroscopic scale, V,, and one at the microscopic scale,
V,. The microscopic derivative is scaled by $ due to the relationship y =
mentioned above.
The multiple scale differentiation operator and the series expansions are
then substituted into the first order system in Equations (2) and (3). The
result is the system
+ +
:v,
+
+
The perturbation method equates the coefficients multiplying the powers of
the small parameter t on either side of Equations (4) and (5).
The equations for the powers of 6-' and to are:
and
respectively. The coefficients that are multiplied by a power of t greater than
zero are neglected. From this point on the perturbation is routine and the
interested reader can check the references for the details of the analysis.
Computation of the Homogenized Permeability
In the perturbation analysis it is necessary to assume the microscopic pressure variable hl can be written as a combination of the partial derivatives of
the macroscopic pressure variable ho in the following way.
allow the computation of a scaled combination of macroThe multipliers
scopic derivatives and can be thought of as containing the first order fluctuations of the pressure on a microscopic cell, Y.
With this assumption the perturbation analysis results in an averaged first
order system of equations
These averaged or macroscopic equations are of the same from as the original
equations defined for the original heterogeneous medium.
The effective permeability, K#, returned by this perturbation analysis
called the homogenized permeability is defined by
where the notation < - > indicates an integral average over the domain. The
integral average is done for each of the nine components separately. The
problem that we are left to solve is: Find the functions wk(y) that are included
in the form given in Equation (13).
If we substitute the form in Equation (13) into Equation (10) we generate
a system of uncoupled elliptic partial differential equations of the form
where k = 1 , 2 , 3 and e k is the k" unit vector in X3 (or X or X2 in the one
and two dimensional cases, respectively). The solutions of these equations is
done via some standard finite element method. For a piecewise constant permeability a piecewise continuous linear finite element will return exact results
and also the continuity of flux condition is a natural boundary condition for
the Galerkin method.
SPECTRAL SOLUTIONS AND WAVELETS
To illustrate the connection of the homogenization process to wavelet representation consider the one dimensional restriction of Equation (14) to the
unit interval [0, 11. The homogenized permeability is computed using
where the function w ( y ) must satisfy the ordinary differential equation
Physically, K# should be the harmonic average and it is always the case that
the homogenization process agrees with this value.
The problem can be viewed from the point of Sturm-Liouville theory if
we include the boundary conditions. The conditions that are imposed by the
physical problem are (1) continuity of the fluid flux, (2) the average returned
must be the harmonic average, and (3) a 'wavelet' condition which can be
stated as w(0). Solving Equation (15) with the given conditions will produce
a set of orthogonal eigenfunctions (trigonometric functions) that can be used
as a basis for constructing characterizations of the permeability field from the
point of view of Fourier analysis or wavelets. Details of this process will be
presented in the talk and appear in another paper.
REFERENCES
Bourgeat, A., 1984 Homogenization method applied to the behavior of a naturally fissured reservoir, In K.I. Gross, editor, Mathematical Methods in
Energy Research, pages 181-193. SIAM.
Amaziane, B., Bourgeat, A., and Koebbe, J., 1990, Numerical simulation
and homogenization of two-phase flow in heterogeneous porous media,
Hornung, Dogan, and Knaber, editors, Transport in Porous Media 11.
Kluwer Academic Publishers.
Koebbe, J.V., 1996, Homcode: A code for scaling up permeabilities using
homogenization, (to appear as a Utah Geological Survey report).
Bourgeat, A.P., and Hidani A., 1994, Effective model of two-phase flow in a
porous medium of different rock types. Publication de 1'Equipe d'Analyse
Numerique, Lyon-St . Et ienne
Bourgeat , A.P., Kozlov, S.M., and Mikelic, A., April 1993, Effective equations
of two-phase in random media, Publication de 1'Equipe d'Analyse
Numerique, Lyon-St . Etienne
BIOGRAPHICAL SKETCH
Joe Koebbe is an Associate Professor in the Department of Mathematics
and Statistics at Utah State University. He received a PhD in Mathematics
from the University of Wyoming in 1988. Joe works in reservoir simulation
with applications in groundwater flow and petroleum applications.
Ryan Thomas is an undergraduate student at Utah Stat University working
towards a Bachelors degree in mathematics and Masters degree in electrical
engineering. His area of interest is currently in signal processing.