John R. Fanchi - Principles of Applied Reservoir Simulation, Second Edition
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102
Principles
of
Applied
Reservoir Simulation
The
N.E. Nash Unit
has a
gradual
dip
from
north
to
south.
The
Misener
sandstone reservoir
is
bounded above
by the
Woodford shale,
on the
flanks
by
the
Sylvan shale,
and
below
by the
Viola limestone.
The
Viola
limestone does
allow
some
aquifer
support
for the
Misener sandstone.
One of the
primary tasks
of the
study
was to map
the
N.E.
Nash
Unit.
Two
sets
of
maps were prepared: conventional hand-drawn maps,
and a set of
maps
based
on a
geostatistical
analysis
of
the
field.
The
hand-drawn maps correspond
to
the
deterministic approach
in
which
a
single realization
is
used, while
the
geostatistical maps correspond
to a
stochastic image
of the
reservoir.
A
geostatistical analysis
was
performed using
42
well control points
to
calculate structural tops, gross thickness, net-to-gross ratio,
and
porosity.
A
cross-plot
between
porosity
and
core
permeability
yielded
a
relationship
for
calculating permeability
from
porosity. From this data, directional
semi-
variograms
(Table
11
-2)
were prepared
to
describe
the
spatial continuity
of
each
parameter.
The
semi-variograms
represent parameter changes
as
functions
of
distance
and
direction.
For a
detailed technical discussion
of
geostatistics,
see
a
text
such
as
Isaaks
and
Srivastava
[1989].
Hebert,
et
al.
[1993]
have published
some geostatistical software that
is
compatible with BOAST
II.
Table 11-2
Semi-Variogram
Model
Goal:
Model
spatial
correlation
of
data
with
semi-variance
y(h)
Semi-Variance
Value
of
spatially distributed
property
at
point
x
it
for
example,
(j),
Spatial vector
or
"lag" distance between data point
at
x
t
+ h and
data
point
at
x
f
.
"Lag"
A is a
vector with length
and
direction.
N(h)
Number
of
data pairs approximately separated
by
vector
h.
When
two
sets
of
maps were compared,
the
hand-drawn maps were
found
to
be
more homogenous than
the
geostatistical maps.
The
geostatistical maps
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103
exhibited
the
large-scale
trends shown
in the
hand-drawn maps,
but
contained
more
local variability. This
was not
surprising, since additional heterogeneity
is
expected
to
arise
as a
result
of
geostatistical mapping.
The
choice
of
final
maps
was
based
on
management priorities:
minimize
the
risk of
drilling
a dry
hole
on the flanks of the field, and
complete
the
study
before
water breakthrough occurred
in the
remaining
oil
producers.
The
geostatistical model
satisfied
both
of
these criteria.
The
main
flow
path
in the
reservoir
was
narrower
in the
geostatistically generated maps than
in the
hand-
drawn
maps,
and the
geostatistical realization
could
be
modified
in
a
day or
two.
Once
a set of
maps
was
chosen,
the
history match
process
could begin.
Tracer
information
in the
form
of
salinity
changes
was
useful
in
helping
identify
sources
of
injection water
as the
water
was
produced. This
was
valuable
in
defining
flow
channels that could
not
otherwise
be
inferred.
In
some areas,
transmissibility
and
porosity changes were needed
to
match water
cut and
reservoir pressure.
The
geostatistical realization used
in the
N.E. Nash study
was
just
a
single
realization.
It was
selected because
it
satisfied constraints imposed
by
previous
volumetric
and
material
balance
studies.
If
these
constraints were
not
available
or
were less reliable, which would
be the
case early
in the
life
of a field, a
geostatistical
study would require
the
use
of
multiple
realizations
to
characterize
the
reservoir.
This raises
the
question
of how
many realizations
are
necessary.
Value
—*-Avg
Figure 11-5. Running average.
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104
Principles
of
Applied
Reservoir Simulation
Figure
11-5
shows
a
random sampling
from a
discrete
probability-
distribution.
A
running average
is
also plotted.
The figure
shows
that
the
running
average does
not
stabilize,
or
approach
a
constant value, until
at
least
20
trials
have been
completed.
This
is a
large number
of
realizations
if
history matching
is
needed
for
each realization. Indeed,
it
would
be
an
unacceptably
large number
of
realizations,
in
most cases, because
of the
time
it
takes
to
perform
a
history
match,
Multiple realizations
can
also
confuse
people
who
are
not
closely involved
with
the
modeling process because they
do not
have
a
single picture
of the
reservoir.
On the
other hand,
the use of
multiple realizations makes
it
possible
to
quantify
the
uncertainty
associated
with
our
limited knowledge
of
properties
distributed
spatially throughout
the
field.
Table
11-3
summarizes
the
advantages
and
concerns associated with geostatistics. There
is no
established procedure
for
selecting
one or
more realizations
for
history matching
from a set of
geostatistically derived realizations.
One
procedure
is
described
by
Rossini,
et
al.
[1994],
An
application
of
reservoir geostatistics
in the
context
of a
multidisciplinary
study
is
presented
by
Wang,
et al.
[1998].
Table 11-3
Geostatistics
Advantages
+
Realism
4
Quantifies uncertainty
Concerns
+
Cost
and
confusion
of
multiple realizations
+
History matching still necessary
to
account
for
model
discontinuities
such
as
channeling
4
History matching complicated
by
factors such
as
probabilistically
generated
heterogeneity
Exercises
Exercise
11.1
(A) Run
EXAM1.DAT
and
record
the final
time,
final
pressure
and
initial
oil
volume
(B)
Multiply
the
volume
of the
reservoir
in
EXAM
1
.DAT
by
0.5,10
and
100.
This
can be
done
by
altering
the
gridblock
size (see Chapter
24.1.1).
Make
a
table showing
the final
time,
final
pressure,
and
initial
oil
volume
for
each case.
(C) How
does
the
change
in
volume
affect
the
pressure
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105
performance
of the
model
as a
function
of
time?
Exercise
11.2
Repeat Exercise
11.1,
but
make
the
volume changes
by
modifying
the
grid dimensions using
the
modification option presented
in
Chapter
24.1,2.
Exercise
11.3
Roll
a
pair
of
dice
50
times
and
record
the
results. Calculate
a
running average
by
calculating
a new
average
after
each trial (roll
of the
dice),
Plot
the
running average
for
each trial.
How
many trials
are
necessary
before
the
average stabilizes, that
is, the
average approaches
a
constant value?
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Chapter
12
Reservoir
Structure
The
physical size
and
shape
of the
reservoir
may be
inferred
from
several
methods that serve
as
sources
of
information
for
defining
the
large-scale
structure
of
the
reservoir. These
information
sources
are
briefly
reviewed below.
12.1
Giga
Scale
Seismic measurements discussed
in the
literature
by
authors such
as
Ausburn,
et
al.
[
1978],
McQuillin,
et
al.
[
1984],
Sheriff
[
1989]
and
Dorn
[
1998]
provide much
of the
Giga
Scale
information that
can be
directly used
to
characterize
a
reservoir. Historically, seismic analyses have been
of
interest
primarily
as a
means
of
establishing
the
structural size
of the
reservoir. People
did
not
believe that seismic data could resolve
sufficient
detail
to
provide
information
beyond overall reservoir structure.
But
that view
has
changed with
the
emergence
of
4-D
seismic monitoring
and
reservoir geophysics
[for
example,
see
Richardson,
1989;Ruijtenberg,etal.,
1990;
Anderson,
1995;He,etal.,
1996;
Johnston,
1997;;
Fanchi,
et al.
1999].
It is
therefore worthwhile
to
introduce
some basic geophysical concepts
within
the
context
of
the
reservoir management
function.
Seismic waves
are
vibrations that propagate
from
a
source, such
as an
explosion,
through
the
earth until they encounter
a
reflecting surface
and are
reflected
into
a
detector, such
as a
geophone. Figure
12-1
shows
a
seismic trace.
Each
trace represents
the
signal received
by a
detector. Changes
to the
amplitude
106
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107
of
seismic waves occur
at
reflectors.
A
seismic reflection occurs
at the
interface
between
two
regions with
different
acoustic impedances.
Model
100ms
—
200 ms —
Seismic response
-
5
10
15
__
.
0
<
i
2
Figure 12-1. Seismic trace
for a
sand wedge
(after
Ruijtenberg,
1990;
reprinted
by
permission
of the
Society
of
Petroleum Engineers).
Acoustic
impedance
is a
fundamental
seismic
parameter.
Acoustic
impedance
is
defined
as
Z
=
p V
where
p is the
bulk density
of the
medium
and
V
is the
compressional velocity
of the
wave
in the
medium. Figure
12-2
3.0
n
28
E
•S2
2.6
o>
2
'
4
C
2.2
0)
Q
2.0
1.8
1.5
2.0 3 4
56789
Velocity
(km/sec)
Figure 12-2. Seismic wave velocity
and
bulk density
of
rock
(after
Telford,
et
al.,
1976; reprinted
by
permission
of
Cambridge University
Press;
after
Gardner,
et
al.,
1974).
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Principles
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Applied
Reservoir Simulation
illustrates
a
correlation between seismic wave velocity
and the
bulk density
of
different
types
of
rock. Further discussion
of
rock properties
and
their relation-
ship
to
seismic variables
can be
found
in the
literature [for example,
Schon
1996],
A
change
in
acoustic
impedance will cause
a
reflection
of the
sound
wave.
The
ability
to
reflect
a
sound wave
by a
change
in
acoustic impedance
is
quantified
in
terms
of the
reflection
coefficient.
The
reflection
coefficient
R
at
the
interface between
two
contiguous layers
is
defined
in
terms
of
acoustic
impedances
as
Z
2
-
Z
n
_
/
£
_
where
subscripts
1 and 2
refer
to the
contiguous layers.
Reflection
coefficient magnitudes
for
typical subsurface interfaces
are
illustrated
in
Table
12-1.
Values
of
reflection
coefficients
at the
sandstone/lime-
stone interface show that reflection coefficient values
can be
relatively
small
In
addition
to
reflection coefficient,
a
transmission coefficient
can be
defined.
The
transmission coefficient
is one
minus
the
reflection coefficient.
Table 12-1
Typical
Reflection
Coefficients
Interface
Sandstone
on
limestone
Limestone
on
sandstone
Ocean bottom
Reflection
Coefficient
0.040
-
0.040
0.11
(soft)
to
0.44 (hard)
Nonzero reflection coefficients occur when
a
wave encounters
a
change
in
acoustic impedance, either because
of a
change
in
compressional velocity
of
the
wave
as it
propagates
from
one
medium
to
another,
or
because
the
bulk
densities
of
the
media
differ.
If
the
change
in
acoustic
impedance
is
large enough,
the
reflection
can be
measured
at the
surface. That
is why gas
tends
to
show
up
as
bright spots
on
seismic data
-
there
is a big
change
in the
density
of
the
fluid.
By
contrast,
the
presence
of
an
oil/water
contact
is
harder
to
observe with seismic
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measurements
because density
differences
between
the oil and
water phases
are
relatively
small
and
result
in
small changes
in
acoustic impedance.
The
seismic trace plots seismic amplitude versus two-way travel time,
or
the
time
it
takes
the
seismic wave
to
propagate
from
the
source
to the
receiver.
One of the
central problems
in
seismic data processing
is to
determine
the
time/depth
conversion.
The
conversion
of
travel time data
to
formation depth
requires that
the
velocity associated with each geologic zone
be
known
or can
be
inferred
as the
wave evolves
with
time. When
the
time/depth conversion
is
applied
to
seismic data,
it can
change
the
relative depths
of
seismic amplitudes
associated with adjacent traces.
Figure
12-3
shows
the
amplitude
and
wavelength
of a
seismic wave
[after
de
Buyl,
et
al.,
1988].
The
sonic
log
response shown
in
Figure 12-3 illustrates
the
relationship between seismic amplitude
and the
sonic log. Sonic logs
are
typically
used
to
calibrate seismic data when seismic data
is
used
in
reservoir
characterization.
The
sonic
log
response
in
Figure
12-3
delineates
the top and
base
of a
geologic section.
Sonic
Log
Seismic Wave
Figure 12-3. Seismic wave
and
sonic
log
response.
The
wavelength
of the
seismic wave
is the
velocity
of the
wave divided
by
its frequency.
Alternatively,
the
wavelength
is
the
velocity
in a
given medium
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Principles
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Applied
Reservoir Simulation
times
the
period
of the
wave.
The
frequency
of the
wave
is a
measure
of the
energy
of the
wave
and is
conserved
as the
wave
propagates
from one
medium
to
another.
The
wavelength, however,
can
vary
from one
medium
to
another.
When
waves overlap
- or
superpose
-
they create
a
wavelet,
as
shown
in
Figure
12-4.
The
time duration
associated
with
the
wavelet disturbance
is
denoted
Af.
The
wavelet
has a
velocity
Fin
a
medium,
and the
period
Tis
the
width
of the
wavelet when plotted
as a
trace
on a
time-map
of
seismic data.
The
length
of the
wave
is
equal
to the
velocity
V
times
the
period
T.
Thus,
if the
wavelet
has a
10
millisecond period
and the
velocity
is
5000
feet
per
second
in
a
particular medium, then
the
length
L of
that wavelet
is 50
feet.
Wavelet
V=
velocity
in
medium
T
= A
/
=
period
of
wavelet
Figure
12-4.
Seismic wavelet.
If
seismic
data
has
enough resolving power
to
show
the
reflecting
bound-
aries
of a
geologic
layer,
then
the
amplitudes
of the
seismic waves
may
be
useful
for
further
characterizing
petrophysical
properties
of the
reservoir.
For
example,
suppose
a
reservoir region
is
characterized
by a
porosity
<f>,
permeability
K,
net
thickness
h
net
,
and oil
saturation
S
0
.
Seismic amplitude
may be
correlatable
with
rock quality (for example,
Kh
net
or
§kh
net
)
or oil
productive capacity (for
example,
S
0
<j>
kh
net
).
When
a
correlation does exist between seismic amplitude
and
a
grouping
of
petrophysical parameters,
the
correlation
may
be
used
to
help
guide
the
distribution
of
reservoir properties
in
areas between wells.
Figures
12-5a
and b
show
two
approaches
to
contouring
a set of
values
at
control points.
The
smooth contour lines shown
in
Figure
12-5a
are
preferred
by
mappers
[Tearpock
and
Bischke,
1991
]
unless
the
undulating contour lines
in
Figure 12-5b
are
supported
by
additional
data. Seismic correlations
can be
used
to
justify
the
more heterogeneous contouring style shown
in
Figure 12-5b.
A
growing body
of
literature provides additional discussion
of
this application
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in
the
context
of an
emerging discipline known
as
reservoir geophysics.
For
example,
see de
Buyl,
et
al.
[1988],
Evans [1996],
Blackwelder,
et
al.
f
1996],
Beasley
[1996],
and
Jack
[1998].
Control
Point
Figure 12-5a.
Smooth
contour lines.
Figure
12-5b.
Undulating contour lines.
12.2
Mega Scale
The
Giga Scale helps
define
reservoir architecture,
but is too
coarse
to
provide
the
detail needed
to
design
a
reservoir development plan.
The
Mega
Scale
is the
scale
at
which
we
begin
to
integrate well
log and
well test
data
into
a
working model
of the
reservoir. Table
12-2
illustrates
the
type
of
information
that
can be
obtained
at the
Mega Scale level
from
well
log
data.
The
most
common interpretations
of
each
log
response
are
included
in the
table.
For
example,
a
high gamma
ray
response implies
the
presence
of
shales, while
a low
gamma
ray
response
implies
the
presence
of
clean sands
or
carbonates.
A
combination
of
well logging tools
is
usually needed
to
minimize ambiguity
in
log
interpretation,
as
discussed
by
Brock
[1986].
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