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5.1
Sonic / Velocity Log Models
VELOCITY
TIME
VA, t
VELOCITY
DEPTH
Velocity Models
from Well Logs
Sonic / Velocity Log Models
How might we derive velocity models from sonic logs?
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5.2
Sonic / Velocity Log Models
We derive velocity models from logs:
Interval Velocities from: -
Integrating sonic logs
Blocking logs in depth and in time
Instantaneous Velocities from: -
Fitting functions
Sonic / Velocity Log ModelsIntegration
example of log
Although the sonic and
velocity logs represent
instantaneous velocity - and
well come to that later - it is
relatively easy to compute
the interval velocity directly
from these logs when they
have integrator tick marks.
- Count the integrator tickmarks and divide the
integrated time into the
interval thickness.
Data Courtesy of Amoco
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5.3
Sonic / Velocity Log Models
There are some simple techniques that can be
applied to sonic or velocity logs in order to derive
interval velocities when integration ticks are illegible,
or are not there. These methods are suitable for use
with paper copies of logs.
Sonic / Velocity Log Models
The first step in the procedure is to
divide the log up into blocks.
Both sonic and velocity logs can be
blocked digitally, a Walsh filter is
used in suitable software.
Each block should average the log
over the chosen interval.
After Marsden, Leading Edge, August 1992.
Blocking
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5.4
Sonic / Velocity Log ModelsBlocking
Data courtesy of ARCO British Ltd.
Manually when blocking
logs and segments
exhibit gradients, the
gradients may be picked.
The intersections of the
gradients define the
block boundaries and the
mid point of the slope
within the block defines
the velocity.
Sonic / Velocity Log Models
Interval velocity
Most often we will want to find
the average interval velocity over
some interval from the blocked
log.
We require to know the velocities
of the blocks and the
proportional thicknesses.
The proportional thickness will
depend on whether the log has a
linear time or depth scale.
Blocking
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5.5
Sonic / Velocity Log Models
Time AverageVELOCITY
DEPTH
VIj, zj, tj
VI, z
When the log is in depth:
VI = z / t and t = tj = (zj/VIj)
VI = z / (zj/VIj)
z / VI = (zj/VIj)
Putting rj for the ratio zj / z then:
1 / VI = (rj/VIj)
we average slowness,
preserving travel time, hence
it is time averaged.
Blocking
Sonic / Velocity Log Models
Lets apply this equation to a layer
made up of two lithologies
deposited alternately:
1 / VA = r1 / V1 + r2 / V2
Geologists will recognize this as
Wylies time-average equation.
It is used when the seismic
wavelength is less than 5 x the bed
cycle length.
Wylies Time-Average Equation
VELOCITY
DEPTH
V1 V2
PR
OPORTION
r1 PROPORTIONr2
Blocking
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5.6
Sonic / Velocity Log Models
Depth AverageVELOCITY
TIME
VIj, tj, zj
VI, t
When the log is in time:
VI = z /t and z = VIj tj VI = (VIjtj) / t
Putting rj for the ratio tj / t then:
VI = rjVIjwe average velocity, this
preserves thicknesses,hence thickness or depth
averaged.
Blocking
Sonic / Velocity Log Models
Lets apply this equation to a
layer made up of two lithologies
deposited alternately:
VI = r1V1 + r2V2
It is used when the seismic
wavelength is less than 5 x the
bed cycle length.
VELOCITY
TIME
V1 V2
PR
OPORTION
r1 PROPORTIONr2
Blocking
Depth Average
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Well Logs
1. Use a straight edge to block the log. What interval velocities do you find for the two
formations?
2. Use the integration marks to calculate interval velocities for the two formations.
Exercise 5.1
4090 sec/ft
100ft
DatacourtesyofARCOB
ritishLtd. F
ormationB
FormationA
5.7
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5.8
Sonic / Velocity Log Models
Formation velocities are not often constant, they usually vary
with depth.
The variation can be described analytically with an equation.
Functions
Sonic / Velocity Log Models
The velocity log is a log of the instantaneous velocity of
the formation in the direction of the borehole and we saw
earlier there are three well known formulae for Vi:
The popular V0,K Vi = V0 + Kz
(Slotnicks equation)
Evjens function Vi = V0 (1 + Kz)nalso known as a modified Faust function.
Fausts formula Vi = Kz1/n
Instantaneous Velocity
Functions
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5.9
Sonic / Velocity Log Models
Instantaneous Velocity
Functions
Velocity logs approximated from VSP time-depth data are
usually noisy compared to using the digital velocity log or a sub-
sampled version of the log. This is because the times are not
measured to a sufficient degree of accuracy and also because
the values are not repeatable to sufficient accuracy. When
possible it is better to use the log rather than the approximation.
Sonic / Velocity Log Models
We have Vi data
and we want a
function in Vi so -
The natural
temptation is to
fit the function to
the velocity data,right?
Slotnick
Functions
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5.10
Sonic / Velocity Log Models
The curve obtained from the
velocity log is not a particularly
good representation of the
time depth data.
Functions
Slotnick
Sonic / Velocity Log Models
We get a better function for
depth conversion if we use
the time-depth values
directly to obtain estimates
of the parameters using
optimisation.
Functions
Slotnick
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5.11
Sonic / Velocity Log Models
Comparison of the
two results of
finding a linear
function of V0 to
represent the data.
One function
represents the
time-depth data
best whilst thesecond represents
the velocity curve
better.
Functions
Slotnick
Sonic / Velocity Log Models
Fausts power
function derived
from the estimated
instantaneous
velocity values.
This function
clearly does not
represent the data
very well in either
in the shallow or
deep section.
Functions
Faust
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5.12
Sonic / Velocity Log Models
In the time-depth domain it is
a worse fit than the linear
increase of instantaneous
velocity with depth.
Once again a better fit could
be achieved working directly
from the time depth values.
Functions
Faust
Sonic / Velocity Log Models
Deriving the parameters in the
time-depth domain by
optimisation gives a much
better result!
Functions
Faust
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5.13
Sonic / Velocity Log Models
Comparison of the
two results of
finding a Faust
function to
represent the data.
Here the time-
depth derived
function fits the
velocity curvebetter in the
shallow section but
is worse at depth.
Functions
Faust
Sonic / Velocity Log Models
Evjens function,
derived from the
velocity data, is
much better fit than
the previous
functions.
Functions
Evjen
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5.14
Sonic / Velocity Log Models
Although the velocity log
was well matched by this
function the time depth
curve isnt.
Note the very small value
for K and the very large
value for n.
Functions
Evjen
Sonic / Velocity Log Models
Optimising the fit to the time
depth data produces a much
better curve to use for depth
conversion.
Note the large changes in K
and n.
When the values of K and nare extreme and change in
this way it is preferable to use
a two parameter function.
Functions
Evjen
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5.15
Sonic / Velocity Log Models
The fit to the
velocity curve
is also a good
fit with the
curve
ignoring the
unreasonable
high velocity
spikes.
Functions
Evjen
Sonic / Velocity Log Models
The functions obtained in the time-depth domain by deriving
parameters so that the depth conversion error is minimised
produce the most accurate representations of the data for
depth conversion purposes.
Functions
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Well LogsExercise 5.2
This is a digital exercise and uses SOLVER to optimise the fitted function to
the data in the time-depth domain
.
The exercise is to plot the velocity log, fit Fausts function to obtain initial values for
V0and n. You will then see that the data does not honour the time-depth data
so you will then optimise the function in the time-depth domain.
Open the Ex 5.2.xls spreadsheet. Note that we are given a starting depth of 1 ft
instead of 0 ft. This is to facilitate the fitting of Fausts function to the data set
(a power function of depth at zero depth cannot be evaluated).
1. In cell D5 type V log and in cell D6 type ft/sec
2. In cells D7 and D8 enter 4850 for the water velocity.
3. In cell D9 enter the formula =1000000/C9 to compute the Instantaneous
velocity from the given sonic log value and then fill the column.
5.16
4. Create a chart of the instantaneous velocity, column D, (on the y axis) against
depth, z, (on the x axis). Format to taste.
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Well Logs
5.17
Exercise 5.2 continued
5. Select the data series in the chart, right click, and fit a trendline, select the power
function and from the options tab select display equation on the chart. This
function is a Faust function with the power term being equal to 1/n in Fausts
equation.
6. Type Vo = in cell C2, 1/n = in cell C3 and n = in cell C4. Right justify theseentries.
7. In cell D2 enter the value of Vo from the equation displayed on the chart
(3068.3), and in cell D3 the value of 1/n from the displayed equation (0.1252).
Calculate the value of n in cell D4 from the value in D3.
8. Type the headings z estimate in cell E5 and z error in cell F5.
9. Now compute the z values in column E using the Faust equation for a single layer
depth conversion (z = [(n-1)V0t / n]n / (n-1)). Use the values of V0 and n from cells
D2 and D4.
10. Next compute the z error values as z z-estimate (column A column E).
11. Enter the text error = in cell E4.
12. In cell F4 compute the RMS error of the values in column E
=sqrt(sumsq(F7:F86)/count(F7:F86))
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Well Logs
5.18
Exercise 5.2 continued
13. Create a chart of the observed time-depth values (put time on the x axis and
show depth increasing downward). Next copy the z estimate values and add
them to the chart, using the paste special function from the edit menu. You
may format the data series and legend labels. The estimated depth series
does not agree with the observed data.
So that we can compare the results that are obtained after optimisation with those
from the function fitted to the log we need to duplicate the function
parameters and the depth computation
14. In cell G2 enter the value of V0 displayed in D2, and in cell G3 the value of
n displayed in cell D4.
15. Type the headings z estimate in cell G5 and z error in cell H5.
16. Now compute the z values in column G using the Faust equation for a single
layer depth conversion. Use the values of V0 and n from cells G2 and G3.
17. Next compute the z error values as z z-estimate (column A column G).
18. In cell G4 type the label error = and in cell H4 compute the RMS error forthe column G values =sqrt(sumsq(G7:G86)/count(G7:G86))
19. Now use solver to minimise the error value in cell G4 by changing the
values in cells G2 and G3.
20. Next copy the updated z estimate values in column G and add them to the
time-depth chart, using the paste special function from the edit menu.
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5.19
Sonic / Velocity Log Models
When we fit an analytic function to well control to
build a macrovelocity model then the model derived
at the well should be applicable elsewhere.
The predicted velocity has to be geologically
reasonable at depths other than those for which it
was derived.
We need to be able to extrapolate with the function.
Extrapolation
Functions
Sonic / Velocity Log Models
ExtrapolationThe constant velocity
is unsuitable for
extrapolation, the
linear function little
better.
Evjens function will
predict velocities which
are geologically
reasonable at all
depths.
VELOCITY
x 1000 ft/s
DEPTH
After Marsden et al, Leading Edge, 1995
x1000ft
Functions
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5.20
Sonic / Velocity Log Models
Functions derived by fitting relatively short segments of the
velocity log do not predict interval travel times all that well, i.e.
they tend to give large depth conversion errors (or misties). The
longer the segments of log used to derive the parameters the
less uncertainty there is in the parameters and the better the
function for depth conversion.
Large range,
small uncertainty.
Small range,
large uncertainty.
Functions
Sonic / Velocity Log Models
Combining Logs
Three velocity logs
overlie one another so
closely that they can
all use the same
analytical function.
VELOCITY
DEPTH
After Marsden et al, Leading Edge, 1995
Functions
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5.21
Sonic / Velocity Log Models
Three velocity logs,
although recorded at
different depths,
clearly belong to the
same trend so one
analytical function will
serve the area.
VELOCITY
DEPTH
Combining Logs
After Marsden et al, Leading Edge, 1995
Functions
Sonic / Velocity Log Models
Seismically derived
velocities were added
to the three velocity
logs seen on the
previous slide,
extending the range
and adding even
more stability to thefunction.
Adding Seismic Velocities
VELOCITY
DEPTH
After Marsden et al, Leading Edge, 1995
Functions
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5.22
Sonic / Velocity Log Models
These two logs have
similar slopes but their
V0s are different. We
cannot use a single
analytical function.
It would be appropriate
to map the variation in
V0 given sufficient well
control.
VELOCITY
D
EPTH
Combining Logs
After Marsden et al, Leading Edge, 1995
Functions
Sonic / Velocity Log Models
These three logs exhibit
different slopes and
different intercepts.
It is necessary to map the
variation in both
parameters in the area
from whence they came.
Or to try and use seismic
velocities
VELOCITY
DEPTH
Combining Logs
After Marsden et al, Leading Edge, 1995
Functions
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5.23
Sonic / Velocity Log Models
Combining logs can reduce the uncertainty in the
constants of the fitted function.
The different wells must have the same geological
conditions;
same lithologies
same diagenetic history
same pore pressure
same tectonic history.
Sonic / Velocity Log Models
In this example there
was clear evidence
from the seismic data
that tectonic inversion
was the only real
geological difference.
The slope K is almostconstant.
The V0 value varies with
tectonic inversion.
Tectonic Inversion
VELOCITY
DEPTH
After Marsden et al, Leading Edge, 1995
Geology
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5.24
Sonic / Velocity Log Models
DiagenesisFor a single lithology:
Velocity logs shows little
variation in velocity
shallow in the section. At
depth the log varies with
porosity. This is a
reflection of diagenesis.
Diagenesis influences K.
Data Courtesy of Amoco
Geology
Sonic / Velocity Log Models
Diagenesis
Vp
z
Mechanical rearrangement
Cementation
Different amounts of
cementation and
diagenesis due to
compaction will lead to
different slopes, i.e.,
different values of k. Crystal regrowth
under pressure
Geology
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5.25
Sonic / Velocity Log Models
Rate of Deposition
Geology
Each trend line in the figure is from the
Chalk interval in a different well.
The trend lines are colour coded
according to the major fault block in
which the wells were drilled.
The area was tectonically active during
the deposition of the Chalk and the
Chalk sedimentation accommodatedthe tectonic movements so that the
major influence on the logs trends is
rate of deposition.
Sonic / Velocity Log Models
Vp x 1000 ft/s
d
epthx1000ft
Lithology
ShaleSand
The initial rates of mechanical
rearrangement and
cementation, which sediments
undergo, vary with lithology,
even when all other conditions
are the same.
For sands and shales thisleads to the crossover
phenomenon and the well
known sand line/shale line
concept.
Crossover
Shalelin
e
Sandlin
e
0
5
10
15
142 4 6 8 10 12After Gardener, Gardener & Gregory, 1974, Geophysics.
Geology
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5.26
Sonic / Velocity Log Models
LithologyDifferent lithologies
exhibit different P-wave
velocities as any well
log indicates.
Over any short section
of log it often appears
that K is constant
whilst V0 varies with
lithology.
SANDSHALE
Geology
Sonic / Velocity Log Models
OverpressureOverpressure affects K and V0.
This plot shows sketches of the
gamma ray and sonic logs
through the Tertiary sequence
for two wells from a North Sea
field. (They are displaced so
that the logs are clearly seen.)
An overpressured zone and
Palaeocene sands are the main
influence on the velocity model.
Data Courtesy of Amoco
Overpressure
Lithology
gamma sonic
4 6 4 6
Geology
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5.27
Sonic / Velocity Log Models
OverpressureOverpressure affects K and
V0.
The plot shows the depth and
time to the base of the
Tertiary sequence for all of
the wells from the field. The
main variable is the
overpressured shale which
results in two distinct trends.These trends are seen most
clearly on the next slide.Data Courtesy of Amoco
9400
9450
9500
9550
9600
9650
9700
9750
9800
9850
1.4 1.42 1.44 1.46 1.48
Time
Depth
Overpressure Normal pressure
Linear (Overpressure) Linear (Normal pressure)
Geology
Sonic / Velocity Log Models
OverpressureOverpressure affects
K and V0.
This plot shows depth
and average velocity
for the same wells as
on the previous slide.
The distinction
between the two
groups of data is
even more dramatic
with a reverse slope.
Data Courtesy of Amoco
y = -0.0307x + 6949.8 y = 0.0572x + 6159.9
6600
6620
6640
6660
6680
6700
6720
6740
9400 9500 9600 9700 9800 9900
Depth
AverageVelocity
Overpressure Normal pressureLinear (Overpressure) Linear (Normal pressure)
Geology
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5.28
Sonic / Velocity Log Models
BlockingThe blocking of logs can be used:
to build velocity models from analogue paper records
when resources do not allow digitising,
in conjunction with Backus averaging to find the
effective medium velocity for thinly bedded sequences.
Summary
Sonic / Velocity Log Models
FunctionsWhen working with digital logs:
use the time-depth data rather than velocity values
parameter values are affected by the way in which they
are derived
combine logs to extend the depth or time range over
which the function is derived
We can use linear functions for the velocity model for
when extrapolating over short depth (time) ranges.
We should use a power law function (Faust or Evjen) for large
depth ranges.
Summary
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5.29
Sonic / Velocity Log Models
Lithology
Diagenesis
Rate of Deposition
Pore pressure
Tectonic history
K
9
9
9
V0
9
9
9
9
For Slotnicks equation Vi = V0 + kz the parameters are
influenced by the geology: -
Geological Effects
Summary
Sonic / Velocity Log Models
Using well logs alone:
is a viable approach when the wells adequately sample
the geologic variations / velocity variations
logs can be combined to increase stability of solution
When we have insufficient well control then we have to try and
use seismically derived velocities.
Always be sure to study any available velocity logs in a projectbefore interpreting the seismic data so that you are sure to pick
seismic events that will be required for depth conversion.
Summary
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Well Logs
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
4090140190
Slowness sec/ft
OneW
ayTime,secs
How would you subdivide this log into macrovelocity units?
How would you represent the velocity of each unit?
Exercise 5.3