Sonic and Velocity Log Models

7/28/2019 Sonic and Velocity Log Models

1/30

5.1

Sonic / Velocity Log Models

VELOCITY

TIME

VA, t

VELOCITY

DEPTH

Velocity Models

from Well Logs


How might we derive velocity models from sonic logs?


2/30

5.2


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


3/30

5.3


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.


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


4/30

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.


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


5/30

5.5


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


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


6/30

5.6


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


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


7/30

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


8/30

5.8


Formation velocities are not often constant, they usually vary

with depth.

The variation can be described analytically with an equation.

Functions


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


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.


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


10/30

5.10


The curve obtained from the

velocity log is not a particularly

good representation of the

time depth data.

Functions

Slotnick


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


11/30

5.11


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


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


12/30

5.12


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


Deriving the parameters in the

time-depth domain by

optimisation gives a much

better result!

Functions

Faust


13/30

5.13


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


Evjens function,

derived from the

velocity data, is

much better fit than

the previous

functions.

Functions

Evjen


14/30

5.14


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


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


15/30

5.15


The fit to the

velocity curve

is also a good

fit with the

curve

ignoring the

unreasonable

high velocity

spikes.

Functions

Evjen


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


16/30

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.


17/30

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))


18/30

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.


19/30

5.19


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


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


20/30

5.20


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


Combining Logs

Three velocity logs

overlie one another so

closely that they can

all use the same

analytical function.

VELOCITY

DEPTH


Functions


21/30

5.21


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


Functions


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


Functions


22/30

5.22


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


Functions


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


Functions


23/30

5.23


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.


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


Geology


24/30

5.24


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.


Geology


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


25/30

5.25


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.


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


26/30

5.26


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


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.


Overpressure

Lithology

gamma sonic

4 6 4 6

Geology


27/30

5.27


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


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.


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


28/30

5.28


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


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


29/30

5.29


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


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


30/30

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

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Sonic and Velocity Log Models