Velocities in Seismic Application
John M Hooper
Section 1
Log Conditioning for the Purposes of Improved
Well Ties
Velocity Ground Truth from Well Log Data
Sonic logs do not directly measure seismic velocities for a number of reasons. This
section examines one path for deriving seismic velocities using well log data based on the
convolutional model. This model specifies that the seismic trace is the convolution of the
reflection coefficient series with the wavelet where T, W and R are the seismic trace,
wavelet and reflectivity series, respectively as shown by equation 1.
T=W*R
Equation 1
Seismic reflectivity, under the simplest case for normal incident assumptions, is given by
equation 2. Velocities in this equation are derived from sonic well log measurements
which are used with layer density measurement to calculate acoustic impedance as the
product of layer velocity and density.
R(1,2) =
VI (2)ρ(2) − VI (1)ρ(1)
VI (2)ρ(2) + VI (1)ρ(1)
Equation 2
The well log measurements provide ground truth for the seismic velocities. The
following sections will document that the field measurements are not always accurate.
Corrections may need to be made for borehole conditions that contaminate the field
measurements of both velocity and density.
Velocity Measurement
There are two types of sonic tools for the determination of the p-wave velocity. They are
the threshold-based tools and the full-waveform tools. The full-waveform tool also
provides the shear-wave velocity in some configurations in addition to the p-wave
velocity. Under this description, the differences in the tools may only be that the fullwaveform sonic records the waveforms for later processing, while the threshold-based
tools record travel time between different arrivals as detected by specified thresholds for
each receiver.
Figure 1 shows the difference between the motions of the shear and the compressional
waves with downward wave propagation. The particle motion for shear waves is shown
to oscillate perpendicular to the motion of the wavefront. The particle motion for
compressional waves is parallel to the motion of the wavefront.
The creation of the zero-offset synthetic only requires knowledge of the compressional
wave velocity. Creation of the more general amplitude versus offset or AVO synthetic
(discussed in a following chapter) requires knowledge of both the compressional wave
velocity and the shear wave velocity.
1
Figure 1: Shear (left) and compressional (right) waves
The next section presents the determination and correction of the compressional-wave
velocity. A following section presents the determination of the shear-wave velocity.
Standard Sonic Tool
The first sonic tools utilized a single source and a single receiver. The time required for
the compressional wave to travel from the source to the receiver was recorded as the
amplitude at the receiver exceeded an operator specified threshold. The record was
typically on film, but later a tape record was also created. An example of the sonic
waveforms is shown by Figure 2. Because these tools were very sensitive to borehole
enlargement, the velocity measurements were typically very poor.
2
Source Fires
Amplitude
Compressionalwave Arrivals
Shear-wave
Arrivals
Mud-wave Stoneley-wave
Arrivals
Arrivals
Time
Figure 2: Idealized wave-train
R2 Ampl
This tool was replaced with a version which was designed to compensate for the specific
effects caused by tilting of the sonde within the borehole. Initial versions utilized two
sources and two receivers and averaged the travel-times between the receiver arrivals to
achieve compensation. Later versions such as the long spaced sonic achieved the similar
effects by processing. The measurement of delta t is shown by Figure 3 to be the
difference in time to threshold amplitude measured by a pair of receivers.
Threshold
R1 Ampl
Time
Threshold
Δt
Figure 3: Delta-t measurement
3
Cycle-Skipping
R2 Ampl
This measurement of delta t can be in error if the mechanism does not pick the same,
relative portions of the waveform. Figure 4 illustrates the situation of low compressionalwave amplitude. In this cartoon, the picked, first-arrival time at receiver number 2 is in
error by a one cycle. This produces a value of delta t that is too large. This effect is
termed cycle skipping. It is also possible for the picked time at receiver number 1 to also
be picked too late, thus producing a cycle skip that is too small or even negative. This
error is termed “road noise.’ In addition, fluctuation in the amplitude of the either of the
detectors fluctuates the extracted value for delta t, even without cycle-skipping. These
fluctuations are termed threshold errors. A systematic change in the amplitude at either
receive introduces a systematic bias in the determination of delta t.
Threshold
R1 Ampl
Time
Threshold
Δt
Figure 4: Cycle-Skipping
4
IN DUC TION
RESISTIVIT Y
( O H M M)
D EP T H
GA M M A RAY ( A P I)
0
FE E T
200
0.1
SONIC
TRANSI T TIME
(us /f t )
DENSITY (g/cc)
100 1.7
2.7
200
100
Cycle Skipping
Gas
5000
5200
Figure 5: Cycle-Skipping Example
Figure 5 provides an example of cycle skipping of the sonic log. In addition to the
obvious spikes, all of the values in the gas zone are probably the result of cycle-skipping
because the transit time is longer than expected for the sonic log of gas sand. It is
generally essential to edit the well log to remove the influence of the cycle skipping
before the creation of the synthetic to obtain reasonable seismic velocities in subsequent
steps.
Full-waveform sonic tool
The digital record of the full-waveform sonic tool differs from the traditional, thresholdbased sonic tool in the determination of the sonic velocity. The full-waveform tool
provides a surface recording of the measured amplitude as a function of time as recorded
by each of a series of receivers in the tool. Figure 6 provides a schematic configuration of
this type of tool.
5
Detectors
P-wave
Source
Figure 6: Full-waveform sonic
Compressional wav
e
The full-waveform sonic tool is able to overcome the problems of cycle skipping and
threshold error artifacts because the data can be processed to maximize semblance to
effectively pick the same portion of the waveform from all receivers.
Figure 7: Full-waveform sonic traces
Figure 7 shows a cartoon of a series of seven traces (amplitude as a function of time),
obtained from seven borehole detectors. These traces are recorded at the surface of the
6
earth for later analysis. Correlation between successive traces provides trace-to-trace,
relative time delay information. This time delay information, along with knowledge of
the receiver separations, permits the interval velocity to be calculated over the length of
the receiver array. The same method is similarly employed for the determination of the
shear-wave velocities found in later arrivals.
The full-waveform-based determination of the p-wave velocity is more robust against
cycle skipping and threshold error artifacts. Figure 8 shows a comparison of the use of
the two types of sonic processing methods used in the determination of the compressional
wave travel-times. The data for this figure comes from the same sonic tool. The full
waveforms were both recorded for later analysis and also picked by the customary field
hardware. The horizontal axis is the compressional-wave traveltime in millisecond/foot
determined from an analysis of the full-waveform sonic. The vertical axis is the
compressional-wave traveltime in milliseconds/foot determined from the customary,
threshold-based sonic log tool of the very same data. The previous section described the
automatic operation of this sonic tool.
Threshold errors
120
Cycle skips
Threshold Detection
110
100
90
80
70
Road noise
60
50
40
40
50
60
70
80
90
100
110
120
Waveform analysis
Figure 8: Sonic velocity comparison
Ideally, there would be perfect agreement between the two sonic measurements. In that
case, all of the observed values would lie along a straight line. The small deviations from
the straight line are most likely due to the threshold errors in the threshold-based sonic.
Those errors will be less than one-quarter of the sonic wavelength. Most likely, the larger
deviations originate from errors introduced by cycle skipping and road noise.
7
Log Digitizing Errors
Before the mid-1980’s, the results of most sonic logs were recorded on paper in the
recording unit. (In more recent time, the results are recorded on magnetic tape.) In order
to be used to generate synthetic traces, the paper logs are digitized by hand or with
optical character recognition software to produce digital information for the computer
programs. Unfortunately, this transcription process introduces a new potential for errors.
Figure 9 shows just such a digitization error. After the well was partially drilled to a
depth of 14050, it was logged. Then casing was set to that depth. After further drilling,
the well was again logged, with the logging run coming up to the 13900 foot depth. So,
the shallowest portion of the second log run was through the casing.
100
Re-scaled Data
90
80
Microseconds/foot
70
60
50
40
30
20
13500
Original Data
1st log run
Poor Log
2nd log run
Casing
13600
13700
13800
13900
14000
14100
14200
14300
Depth (feet)
Figure 9: Sonic Digitizing Error
Turning to the actual curves, the lower curve is the original, digitized curve. In this case
the digitization missed a scale change on the paper log at the junction of the 1st log run
and the 2nd log run. The upper magenta curve reflects the proper scale. The middle
portion of the log is a region of an unreliable sonic. It was logged in the cased portion of
the borehole. Some commercial digitized databases contain sonic logs that have been
digitized on the wrong scale for approximately 10% of available wells. Note that the
average sonic of 57 microseconds/foot between 13,900’ and 14,050’ is the approximate
velocity of sound in steel. This portion of the sonic is unreliable even after correction for
the log scale change.
8
Because the units are in reciprocal of the velocity, the original digitization would have
implied significantly faster velocities than appropriate. Because the reflection coefficient
is proportional to the differences in the velocities divided by the sum of the velocities
(see Equation 2), the error of the synthetic seismogram would have been significant at the
14,050’ depth.
Log data is out of range for area
Experience in an area often provides the seismic-petrophysicist with an understanding of
the valid range of expected sonic velocities. In mature, hard-rock areas the velocities are
typically high and porosity is relatively low. In the shallow portions of basins with
geologically recent sedimentation, anticipated velocities are typically quite low and
porosity is high.
TRANSI T TIME
Sonic
(us/ft)
2.7
200
100
e Skipping
Cycle
Minimum
Maximum
Figure 10: Expected range of valid sonic
9
The expected log response is developed by looking at sonic logs from multiple wells in a
depth range. This implies that a geophysicist, undertaking a one well modeling project in
an unfamiliar area that does contain other sonic control could overlook portions of logs
with serious problems that should be corrected before modeling. Figure 10 shows the use
of typical thresholds on a sonic log. The previously noted cycle skipping would be
flagged for further analysis based on this bad log test. Also note that there are excursions
of sonic log that are below accepted minimum travel times. These high velocities are
possible indications of “road noise”.
Enlarged borehole and washouts
Ask a geophysicist how they evaluate the quality of the sonic and you typically get a
response such as, “I look at the caliper!” This would be a rather difficult qualitative
method to teach. Also it may be misleading. Just because a caliper log indicates that the
borehole is quite large, doesn’t necessarily mean that the sonic log must be corrupt. The
shallow portions of a well are drilled with large bits. If the borehole is a constant size,
then standard borehole correction algorithms already discussed will generally perform
well on the sonic log. However, when enlarged portions of the borehole are found at the
same depths that the sonic log shows excessive cycle-skipping, then these algorithms are
probably overwhelmed by changing hole geometry. Figure 11 shows schematic washout
enlargement of a borehole.
Figure 11: Large borehole and washouts
Another metric that can be used to flag bad borehole conditions is a derivative of the
caliper log. The borehole rugosity is mathematically the absolute value of the first
derivative of the caliper. It measures how rapidly the size of the wellbore is changing.
An example of the rugosity is shown by Figure 12.
10
Figure 12: Caliper and rugosity
The caliper and two depth shifted caliper curves are shown on the right. The rugosity
curve is shaded red to show rapid borehole enlargement.
Dissimilarity between sonic and estimated sonic
The first step in editing sonic measurements is to recognize when those measurements are
suspect. However zones that require correction, necessitate that an estimated sonic curve
be merged with valid sonic logs adjacent to the suspect intervals. One of the first
published methods for estimating a velocity from other logs was the Faust equation. We
will cover Faust in more detail in the next section. This method of sonic quality control
compares the measured sonic log to an estimated sonic from Faust, Gardner, or another
relationship. If the two curves are too dissimilar, then the sonic is suspected to be
invalid.
11
esi
stiv
So
ity
nic
Re Syn
the
st .
t
Sy
nth ic
etic
p R
V
V
p S
on
ic
Figure 13 shows an example of a synthetic seismogram from a measured sonic log
compared to a synthetic seismogram from the Faust relationship in the same well. In this
example, the synthetic seismograms are quite similar even though the character of the
velocity from the resistivity and sonic logs have some obvious differences.
1.6 sec
3.2 sec
Figure 13: Synthetics from sonic and resistivity curves
Figure 14 shows an example where a Faust sonic is overlain on a measured sonic. There
are some obvious places where the measured sonic shows cycle skipping and “road
noise”, but the character of the match is pretty good over all.
12
4000
2000
6000
Vp
8000
10000
12000
2500
Depth
3000
Cycle Skips
3500
4000
4500
5000
Sonic
Vp from resistivity
Figure 14: Velocity from sonic and Faust from resistivity fair to good match
Figure 15 shows an example where there are brine sands that have very low resistivity
response in comparison to their velocities. This example also suffers from brine flowing
into the wellbore and the background velocity is a poor match as well.
4000
6000
Vp
Example
of poor
6000
8000
10000match
12000
6500
Depth
7000
7500
8000
Brine Sands
8500
Cycle
9000
skip
Sonic
Vp from resistivity
Figure 15: Velocity from sonic and Faust from resistivity poor match
13
The object of this example shown by Figures 14 and 15 is to demonstrate that poor
matches between reconstructed curves and measured sonic curves are not always because
the sonic curve is bad. This is an interpretive process. Care must be exercised to apply
corrections only when there is reason to believe that the final synthetic will be improved
as a result.
Bad sonic log indicator summary
9
9
9
9
9
9
9
Digitized on wrong scale
Logged through casing
Cycle skipping & road noise
Logs are outside the range of expected values.
The caliper indicates an excessive borehole size.
The caliper indicates washouts that create significant rugosity of the borehole
A portion of the sonic looks very dissimilar to a reconstructed or estimated log.
14
Faust, Gardner and statistical methods used
to estimate sonic and other logs.
This section examines the use of empirical relationships to predict missing sonic, density
and shear sonic logs as required for acoustic or elastic modeling. Early petrophysical and
geophysical relationships predate application of personal computers and primarily were
based on data which was hand plotted on various types of graph paper. The most
common types of plots were rectangular or linear grids, semi-log grids and log-log grids.
While not universally true, a common claim was that all data plot as a straight line on
log-log paper. It is interesting that many of the standard empirical relationships such as
“Faust”, “Gardner”, “Kim/Rudman” and even the “Archie” equation are Power fit
equations which can be derived from linear plots on log-log paper.
One other important consideration in dealing with the published equations from the
1950’s is that many of these relationships were determined in what are now mature
geologic basins. Even more significant is that they were established with older,
consolidated rocks and do not particularly work well in very young, unconsolidated
clastic sediments found in shallow depths in Tertiary basins.
A final consideration is that access to personal computers and data mining software
currently allows the derivation of locally determined relationships that may predict
required elastic logs better than a 60 year old equation from rocks with a different age
and history from another continent.
Faust estimation of the sonic
As stated before, one of the first published methods for estimating a velocity from other
logs was the Faust equation. We will cover Faust in more detail in this section. The
Faust equation uses resistivity measurements to estimate velocity based on an empirical
relationship. The resistivity and velocity are partially controlled by formation effective
stress in similar ways. However the response of resistivity to pore fluid content may
require a correction prior to implementation of the published equation. Hydrocarbons
and matrix have near infinite resistivity. Therefore one of the forms of the Archie
equation can be used to correct for variation in saturation.
15
Equation 3 relates the observed reservoir resistivity to an equivalent wet reservoir
resistivity. This may be viewed as “fluid substitution” for resistivity measurements
where Sw is brine saturation, Rt is “true” or measured deep resistivity, and R0 is the
calculated wet reservoir resistivity. Derived from the “Archie equation” we have,
R0 = SW2 Rt
Equation 3
Equation 4 is a general version of the Faust equation that relates the p-wave velocity for
the wet reservoir case to the wet resistivity. The value of C can range between 1500 and
3000. Crossplot regressions from a portion of the well that contains both the resistivity
and the velocity logs provide the appropriate value of C. Depth is in units of feet as
published and R0 is in units of ohm-meters/meter2.
Vp(ft/sec) = C (R0 * Depth)(1/6)
Equation 4
Figure 13 – 15 show prior examples of the Faust equation as used for sonic log QC. The
Faust equation isn’t the only relationship used with resistivity logs. Kim/Rudman
prediction of the sonic is shown by Equation 5.
Dt(usec/ft) = C1 + C2R0e
Equation 5
The Gardner equation is a popular equation, typically used to estimate density from
velocity. However if only density is available, the equation can be used to solve for
velocity as shown by Equation 6. The Gardner is another empirical formula. The typical
default values for C1 and e are 0.23 and 4 respectively when solving for velocity with
density in units of g/cc.
Vp(ft/sec) = (ρ/C1)e
Equation 6
The last empirical relationship that I will present is Castagna’s “Mudrock” relationship in
Figure 7. It was initially used to predict shear velocity from compressional velocity, but
can be used to estimate Vp in very low velocity gas sands where the fluid arrival masks
the compressional arrival.
Vp(m/sec) = 1.16 * Vs + 1360
Equation 7
16
Sonic log reconstruction summary
9 These popular relationships are empirical, without theoretical basis
9 They work better in deeper, older, more mature basins in contrast to young,
shallow, Tertiary sediments
9 They should be tested against measured sonic values when ever possible prior to
use.
Density log editing
The value of the reflection coefficient depends on both the velocity and the density. The
well logs also supply an independent measure of the density. Before considering how a
density log should be edited, a review of the how the density log is obtained would be
useful. Then we’ll consider some of the borehole issues that impact the measurements
and QC options unique to the density.
Density measurements
Figure 16 provides a cartoon of the density tool while recording. The tool measures
gamma rays that back-scatter from a gamma ray source. For accurate measurements, the
tool must be tightly pressed against the borehole wall. The gamma rays predominately
interact with electrons. Because the electron density is proportional to the bulk density,
the amount of gamma ray back-scattering is closely proportional to the bulk rock density.
Gamma-ray
Source
Figure 16: Density tool in recording mode.
17
Density measurement issues and inaccuracies
Washouts
The accuracy of the determination of bulk rock density may be decreased because of the
following phenomena. The tool must be in close proximity to the borehole wall.
Therefore it is more sensitive than the sonic to washouts shown by Figure11.
Clay Hydration
Some clays absorb fluids from the drilling mud, causing their physical properties and
chemistry to change. In particular, if the shale contains smectites, they soak up water
from low salinity, water based drilling mud. With clay hydration, the shales typically lose
their physical integrity and slough off the borehole walls.
Stress relaxation
Figure 17 cartoons stress relaxation caused by the release of the confining stresses due to
the presence of the wellbore. This may cause fracturing adjacent to the well bore.
Because the logging tools measure the rock properties closely adjacent to the well bore,
those measurements may not be representative of the formation. This generally has a
much greater impact on the sonic than the density log.
Figure 17: Cracks near borehole from stress relaxation
Mudcake and barite invasion of the formation
Figure 18 illustrates the influence of changes in the borehole diameter on the
determination of the density. The lower log is a direct, caliper measurement of the
18
borehole diameter. The drill bit size was six inches in diameter. The upper curve is the
correction term applied to the density measurements. (Atlas/Western/Baker terms this a
Zcor log, a correction to the Zden log, where Zden is simply the density log.
Schlumberger uses the delta rho terminology.) The count rates from the two different
gamma-ray detectors (shown in Figure 41) provides the correction term. In the ideal case,
Δρ should be zero. While some random error is acceptable, values of Δρ greater than .1
(or less than -.1) indicate an inaccurate density determination. Δρ is not equal to zero if
only one of the detectors is in contact with the side of the borehole. Thus, Δρ reacts to
changes in borehole diameter.
Poor Log
Because of mud cake, the hole diameter may occasionally be less than the bit diameter.
Notice the correlation of the non-zero values of the Δρ measurement and the washout
zones. The density will be in error for those depths that have large values of Δρ and/or
large changes in the caliper with depth.
Good Log
Poor Log
0.2
0.4
1
0.5
0
-0.5
Poor Log
Delta
Rho
-1
0
0.6
0.8
1
Rugosity
Figure 18: Δρ and Rugosity provide evidence of density quality
Most wells are drilled with drilling mud to provide lubrication, pressure equilibrium and
to bring the cuttings to the surface. In permeable rock, the more liquid portions of the
mud filters into the rock, leaving the solids behind as a mudcake coating on the surface of
the hole. For this reason, density tools employ two detectors as shown in Figure 19.
19
The counts recorded by each detector are used to correct for the density log for presence
of mudcake. This correction term is always applied in the field before the density log is
plotted or digitally recorded.
Mudcake
Figure 19: Δρ correction necessitated to correct for mudcake
Density reconstruction (or estimation)
The density log can be easily calculated from a theoretical basis if the volumes and
densities of the individual rock constituents are known. However there are frequent
situations where either the mineralogy of the rock or the volumetric analysis of the
constituents is unknown. A frequent empirical relationship used to estimate the density
from velocity is the Gardner equation. It was given in Equation 6 as a means of
estimating Vp from density. It can also be used when velocity is known and density
unknown as shown by Equation 8 where density is in g/cc and velocity in ft/sec. The
published coefficients for Gardner are 0.23 for C1 and 0.25 for e.
ρ = C1 * VP e
Equation 8
Although the Gardner equation is commonly used, it is a poor choice in many areas.
Figure 20 shows a plot of density on the X axis and sonic on the Y axis. Common
petrophysical relationships are shown on this plot including the Time-Average equation,
The Raymer-Hunt-Gardner relationship and Critical porosity for sandstone. These
relationships are typically used to determine porosity from velocity or sonic travel time.
This plot also shows the limiting isostress and isostrain mixing bounds for quartz and
brine. These are the Reuss and Voigt relationships, respectively. Elastic materials can
not theoretically fall outside these mixing bounds. The Gardner relationship is also
plotted on Figure 20. Note that the Gardner relationship violates the Reuss theoretical
mixing relationship for densities less than about 2 g/cc.
20
Sonic-Density Cross Plot
200
Sonic (usec/ft)
180
160
140
120
100
80
60
40
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
Density (g/cc)
Voigt limit
Reuss limit
Time Average
RHG
Critical Porosity
Gardner
Figure 20: ρ (density) versus Dt (sonic) with isostress and isostrain
n limits
This plot suggests that the Gardner relationship is poor for low density or high porosity
sediments. Note however that Gardner appears to project through the matrix point for
sandstone.
The lightest colored of the two upper curves in Figure 21 shows the result of a theoretical
calculation of density as a function of depth for a single well. The darker of the upper
curves is the result of the density log in the same well. For the most part, the two curves
track each other. However, there are notable exceptions at depth for which the density log
provided a significantly lower density than indicated by the theoretical calculation. As
shown by the lower curves, this departure occurs at the depths of increased borehole
problems as flagged by the pair of curves at the bottom of the plot. These flags for
rugosity and Δρ are zero set to unity for questionable conditions and zero elsewhere.
Density QC
3
Density & QC
2.5
2
1.5
1
0.5
0
11400
11450
11500
11550
11600
11650
11700
Depth
Density
D Rho
Rugosity
Density Est
Figure 21: ρ (density) reconstruction from sum of constituents
21
Bad density log indicator summary
9
9
9
9
9
9
9
Digitized on wrong scale
Logged through casing
Logs are outside the range of expected values.
The Δρ curve indicates the density is probably beyond the range of correction
The caliper indicates an excessive borehole size.
The caliper indicates washouts that create significant rugosity of the borehole
A portion of the density looks very dissimilar to a reconstructed or estimated log.
Shear Sonic measurement, QC and
reconstruction
ave
Tube
w
wave
Shea
r
Compressional wav
e
If the only acoustic synthetic seismograms are desired, shear sonic is an unnecessary
concern. However for offset or AVO synthetics, shear velocity is a requisite input. Shear
waves are body waves that travel slower than compressional waves. This precludes the
use of first arrival, threshold detection methods to measure their velocities. Early shear
velocity measurements were determined from processing recorded waveforms on existing
sonic tools. These monopole sourced measurements were only useful where shear
velocity was faster than the mud arrival. Figure 22 shows a cartoon of delay time from
adjacent receivers on a full waveform sonic record.
Figure 22: Determining shear traveltimes from sonic waveform records
Figure 23 shows a waveform record from one (near trace) receiver of a long spaced sonic
tool in a “hard rock” basin.
22
Time
Shear wave arrival
Compressional wave arrival
Depth
Figure 23: P wave and S wave arrivals on sonic waveform records
Figure 24 shows another portion of the borehole record where the shear velocity is
sufficiently slow to be overwhelmed by the borehole fluid arrival. Because a monopole
shear is a refracted body wave, no shear measurement is possible when it is slower than
the fluid arrival.
Fluid line
Inadequate shear signal
Compressional wave arrival Depth
Figure 24: P wave, S wave and fluid arrivals on sonic waveform records
One of the incentives for development of Equation 7, the Castagna et al. “mudrock”
equation was to be able to estimate shear velocity when shear logs were unavailable.
Because of the problems in detecting Vs when shear velocities were slower than borehole
mud arrivals, this was a ubiquitous problem prior to the development of the dipole sonic
tool. The dipole tool generates a flexural wave that is capable of being processed to yield
a shear sonic log in slow shear velocity environments where the shear arrival is later than
the mud arrival. Figure 25 shows how modern dipole tools accentuate shear arrivals and
minimize fluid responses.
23
Shear Arrival
Semblance
“Velocities”
Near Traces
Fluid line
Depth
Figure 25: S wave and fluid arrivals, dipole waveforms and semblance plot
Modern dipole sonic logs provide very good shear velocity measurements when these
tools are run with appropriate parameters and properly processed.
Shear Sonic QC
The quality control for the shear sonic is similar to that for the compressional sonic.
However there is one additional tool available. Poisson’s Ratio can be calculated from
Vp and Vs. For elastic materials under small strain, (PR) Poission’s Ratio must fall
between 0.5 and 0. If PR is outside these limits, either Vp, Vs or both measurements are
invalid. Equation 9 provides the formula to calculate dynamic Poisson’s Ratio.
PR = (0.5 - (VS / VP)2) / (1- (VS / VP)2)
Equation 9
Bad shear sonic log indicator summary
9
9
9
9
Mislabeling of waveform mode which looks like cycle skipping
Logs are outside the range of expected values.
A portion of the shear sonic looks very unlike a reconstructed or estimated log.
Poisson’s Ratio calculated from Vp and Vs is outside valid range
24
Well Ties (time to depth and velocity)
Review: It was necessary to show the issues with measured log data prior to a discussion
of using those data to tie the depths from logs to the two-way travel time measurements
which are the native recording of seismic data. Sonic logs, density logs and shear sonic
logs may not be representative of subsurface rock properties and may require corrections.
Additionally, the seismic data may be noisy at the well locations. Noise resident in the
seismic data as well as reflectivity errors caused by log quality problems both degrade the
synthetic seismogram to surface seismic data tie.
Examining the process of tying well and seismic data reveals several issues:
1) There may be considerable uncertainty in the initial time to depth (TD) trial,
requiring significant adjustments to the starting TD model.
2) The processing intent of the phase and polarity of the seismic wavelet may or may
not be known, but even if known the intent may not have been achieved. In other
words, the seismic wavelet, W cannot be reliably assumed.
3) Well log velocities frequently require adjustment in the form of ‘stretch &
squeeze’ prior to obtaining the best final tie.
4) The well may be deviated requiring that the synthetic be tied to portions of several
seismic traces.
Well tie example with wavelet extraction
An example of the well tie process using commercial, Hampson-Russell software
follows. Obviously this isn’t the only choice in synthetic seismogram software, but it is
readily available. This example used well ties in three wells to determine the seismic
wavelet, the time – depth relationship and is contrasted to nearby checkshot TD
relationships in other wells.
Figure 26 shows the original sonic and density curves in black, edited curves in blue and
the sonic after stretch and squeeze in red. A blue trace of the resulting synthetic
seismogram is shown on the right panel. Note the magnitude of corrections applied in
editing the sonic and density logs.
25
Figure 26: Synthetic seismogram overlay with Dt & ρ curves and edits
Also you might note that the middle portion of the density log is missing and had to be
estimated from other curves. Figure 27 shows an enlargement of the synthetic – seismic
tie after editing and wavelet determination.
Figure 27: Synthetic seismogram overlay in red on black seismic traces
26
The synthetic tie in Figure 27 is quite good. There is little doubt that most events can be
accurately tied to the appropriate depths, lithology, fluid content and stratigraphy in this
well. The real test is if these results are repeatable. Figure 28 shows a second well of the
program.
Figure 28: Synthetic seismogram overlay with Dt & ρ curves and edits
Note the short interval of measured sonic in this well. Also note that it is much slower
than the estimated and final sonic curves. This is another example where the sonic was
digitized on the wrong scale. Review of the field records documented the correct log
scale before subsequent editing. Figure 29 shows the detailed synthetic – seismic tie.
Figure 29: Synthetic seismogram overlay in red on black seismic traces
27
What is the extracted wavelet and how similar would it be to the wavelet extracted if you
didn’t correct the sonic and density logs prior to well tie and wavelet extraction? Figure
30 shows the cross correlation between the synthetic seismogram from the edited logs
and the seismic data. The correlation is about 0.74, or a pretty decent tie.
Statistical measure
Of synthetic tie using
edited acoustic logs
Figure 30: Synthetic seismogram (blue), black seismic traces and cross correlation
Figure 31 shows the level of improvement above in comparison with the well tie from the
raw log data.
Statistical measure
Of synthetic tie using
raw acoustic logs
Figure 31: Synthetic seismogram from raw logs (blue), black seismic traces and
cross correlation
The maximum correlation is 0.16 between the synthetic seismogram and the seismic data.
Of interest is that this point of correlation is based on an erroneous time – depth tie as
well. The correct time – depth tie is at zero lag.
28
Edited Dt, Rhob
Raw Dt, Rhob
Figure 32: Extracted Wavelet comparison for edited and raw logs.
Figure 32 provides another indication of the value of correcting the log data before
performing well ties and wavelet extractions. The wavelet on the right is from the
uncorrected, raw input sonic and density log data. The wavelet on the left is extracted
from corrected log data. The synthetic match from Figure 30 and the extracted wavelet
on the left side of Figure 32 show how much additional fidelity the output velocity from
the synthetic to seismic tie is generated with this step.
Steps to consider in producing a quality well tie
Some steps will be software dependent, but there are some basic issues that must be
considered regardless of the platform used for generating synthetic seismograms. This is
at least a partial list that should be considered.
29
Collocation
Although an experienced interpreter in an area may have good judgment concerning the
correlation between specific reflectors and formation tops from well, an interpreter in a
new basin must go through this interpretive step of collocating reflections and well log
tops. There is an implicit assumption that all data used to map geological units are in fact
from those units. It is uncommon, but also not unheard of to determine that seismic
reflections being interpreted for sequence stratigraphy are not sufficiently close to the
formations being mapped that they satisfy the conditions of collocation.
Sonic logs almost never are recorded from the bottom of the well to the surface.
Typically, the shallowest portion of the well isn’t logged at all, much less with a sonic.
Additionally, sometimes the seismic datum is uncertain. That means that when making a
synthetic tie with seismic data, there may be considerable initial uncertainty about the
best initial time to depth estimate. Historically, some software may require that the
interpreter provide a minimum of one time – depth pair even when no checkshot is
available in the well. Other software uses the shallowest measured sonic value, assumes
a constant velocity for all points shallower and provides a gross estimate for the TD
function for the entire well. Both of these programs require knowledgeable user input
early in the procedure.
Figure 33 shows a typical initial poor tie using the Hampson-Russell software. The red
curves are edited ρ and Dt. The blue traces are the synthetic seismogram, and the red
traces are replicates of the composite seismic trace nearest the wellbore. The crosscorrelation on the right shows a region of better tie about 150 milliseconds above the
initial trial.
Figure 33: Initial well tie. Blue synthetic, red is composite seismic trace.
30
Figure 34 shows the shifted synthetic. This was only a bulk shift and is equivalent to
changing the shallowest velocity layer above the top of the sonic log. Note that although
the envelope of the cross-correlation suggests this is the best tie within a 200 millisecond
window, the synthetic appears to be close to 180 degrees out of phase with the seismic
data.
Figure 34: Shifted well tie. Blue synthetic, red is composite seismic trace.
Phase estimation
Simple bulk shift in this case provided reasonable ties and phase estimates with 500
millisecond windows. However if significant stretch / squeeze must be applied to correct
the low frequency portion of the borehole measurement to match the seismic data, then it
will be necessary to either apply a subset of available check-shots or look at small
correlation windows between the synthetic and seismic data to remove an initial coarse
correction. Figure 35 shows three windows where phase estimates between seismic data
and a zero phase synthetic have been calculated. The statistics from multiple windows
and multiple wells provide a way to verify if the phase of the seismic data is relatively
constant. The average phase determined for these three windows is 156 degrees +/- a6
degrees.
31
1200 to
1700
2100 to
2600
3000 to
3500
172o
156o
140o
Figure 35: Windowed phase estimates from one well and 3D seismic data
The next step is to rotate the seismic data (opposite sign of phase tie) by 156 degrees
prior to performing detailed stretch and squeeze shown in Figure 36.
Figure 36: Peak to peak stretch & squeeze after phase rotation.
32
If detailed peak to peak & trough to trough stretching forces a well tie prior to phase
determination, there is a strong possibility that the resulting phase determination will be
near zero degrees of phase. Conversely if peaks on the synthetic are tied to troughs on
the seismic data then the phase may be forced to near 180o. There is an additional test of
reasonableness, however. The correction should be examined before it is applied to the
sonic data and time-depth relationship for the well as shown by Figure 37.
Figure 37: TD function, drift curve and correction to input sonic from detailed tie
If the correction is quite large and oscillates between stretching the sonic and shrinking it,
and the drift curve swings from plus to minus and back, then the tie is probably forced to
match beyond what is acceptable. Some of the tie points should be deleted until the TD
relationship, sonic match and correction are all reasonable.
When the individual well ties are satisfactory, a multiwell wavelet should be extracted
and compared to those from single wells. Figure 38 is a multiwell wavelet extracted from
a tie over 1.8 seconds of seismic data. Because of the quality of the wavelet, the length
of the window and the quality of the cross correlation from this example, the probability
of proper phase determination and time-depth relationships for each well are quite high.
33
Figure 38: Composite extracted wavelet on left with phase & amplitude spectrum
on right.
Time – Depth relationships from synthetic to
seismic ties.
One of the main objects of tying a synthetic seismogram to surface seismic data is to
derive a quality time – depth relationship. The following slides document how the TD
functions change and improve through the process of editing well logs and stepping
through the process of tying several wells. Figure 39 shows integrated raw sonic logs for
three wells.
T D c u rv e s , 3 w e lls ra w s o n ic c u rv e s
4500
4000
3500
2WTT
3000
2500
2000
r
1500
1000
500
0
0
5000
10000
15000
20000
25000
De p t h
W e ll 3
W e ll 2
W e ll 1
Figure 39: TD curves from raw integrated sonic logs. Shallowest value projected to
surface.
34
TD curves, 3 wells edited sonic curves
4000
3500
3000
2WTT
2500
2000
1500
1000
500
0
0
5000
10000
15000
20000
25000
Depth
Well 3
Well 2
Well 1
Figure 40: TD curves from edited sonic logs. Shallowest value projected to surface.
Figure 40 shows that simply editing the input raw curves improve the relationship
between the three wells. The green and red curves are from wells with traditional BHC
sonic logs and threshold detection algorithms. The blue curve is from a well that was
logged with a dipole sonic and semblance processing.
TD curves, 3 wells after editing & bulk shift
4000
3500
3000
2WTT
2500
2000
1500
1000
500
0
0
5000
10000
15000
20000
25000
Depth
Well 3
Well 2
Well 1
Figure 41: TD curves from edited, bulk shifted sonic logs, changing the shallowest
velocity between log top and surface.
Figure 41 demonstrates that application of a bulk shift of the synthetic additionally
improves the TD relationship obtained by the edited logs.
35
TD curves, 3 wells after editing, bulk shift & stretch
4000
3500
3000
2WTT
2500
2000
1500
1000
500
0
0
5000
10000
15000
20000
25000
Depth
Well 3
Well 2
Well 1
Figure 42: TD curves from edited, bulk shifted sonic logs, following stretch and
squeeze independently between wells.
Figure 42 shows the final improvement of the TD function through most of the synthetic
ties following detailed correlation of the synthetic seismogram to the seismic data
through minor adjustments by stretch and squeeze. Note that in this case the three wells
were situated along structural strike with similar overburden and therefore actually have
similar velocity profiles. If large lateral velocity changes occur over a 3D seismic survey,
resulting TD curves would not be expected to overlay to this degree.
TD curves, 3 wells editing & stretch compared to area checkshots
4000
3500
3000
2WTT
2500
2000
1500
1000
500
0
0
5000
10000
15000
20000
25000
Depth
Well 3
Well 2
Well 1
Well 4 checkshot
Well 5 checkshot
Figure 43: TD curves from sonic logs and 2 area checkshots.
36
Figure 43 provides the addition of two checkshot surveys to the plot shown in Figure 42.
The checkshot survey for Well 5 is not on strike with the wells containing the sonic logs
and the checkshot in Well 4. This demonstrates that wells in a program may not have the
exact same TD function. It also shows that a checkshot survey is not necessarily superior
to a well with an accurately tied synthetic seismogram. However it is not uncommon for
interpreters to “share” TD relationships between wells and / or use a somewhat distant
checkshot survey to tie wells and seismic data. Figures 44 and 45 provide examples of
how unreliable this can be.
T D c u rv e s , 2 a re a c h e c k s h o ts
2400
2300
2200
2100
2WTT
2000
1900
1800
> 180 mils time
difference between
checkshots
1700
1600
1500
1400
10000
10500
11000
11500
12000
12500
13000
13500
14000
14500
15000
De p t h
W e ll 4 c h e c k s h o t
W e ll 5 c h e c k s h o t
Figure 44: Time vs Depth curves from two area checkshots. 180 ms difference
T D c u r v e s , 2 a r e a c h e c k s h o ts
2400
2300
2200
2100
2WTT
2000
1900
1800
1700
1600
~ 1350’ depth difference
between checkshots
1500
1400
10000
10500
11000
11500
12000
12500
13000
13500
14000
14500
15000
De p th
W e ll 4 c h e c k s h o t
W e ll 5 c h e c k s h o t
Figure 45: Time vs Depth curves from two area checkshots. 1350’ or 410 meters
difference in depth shown at 1.9 seconds Two-way-travel-time.
37
Structural consistency
The purpose of tying wells to seismic data is usually to be able to produce an integrated
interpretation that honors all well and seismic data. If possible, all wells within a seismic
survey should be tied with quality synthetic seismograms. After the initial ties are made,
the formation tops picked from correlated wells should be posted and key horizons
picked. If one or more wells do not appear to consistently honor the resulting integrated
interpretation, the interpreter should ask the following questions:
o Is the problem poor signal to noise on the seismic data?
o Does structural complexity permit the wells to tie as projected, but leaves
an initially questionable picked horizon?
o Does stratigraphic complexity such as onlapping or top lapping events
result in interference between adjacent seismic reflections that create more
than one possible way to pick the data?
o Are there significant errors with one or more of the well ties?
Figure 46: Quality well tie in deeper section
Figure 46 shows a well tie for the Viola formation. The first well is located closest to the
A’ end of the arbitrary seismic line through the 3D survey. The arrows show points of
the six quality well ties through the 3D survey shown on Figure 47. The horizon is
picked across this relatively simple structure and tested by subsequent well ties. The
process is repeated on other horizons and well tops as shown by Figure 48.
38
A
No reliable tie
A’
Viola
A’
A
No reliable tie
A horizon
Figure 47: Points of quality well ties track honor interpreted seismic
A’
A’
A
Figure 48: TD’s place tops at each well, tracking events consistently
39
Consistent extracted wavelet phase coupled with TD relationships that provide a
reasonable integrated interpretation and a feasible solution. Less than this optimum
requires additional investigation.
Well Ties via Synthetic Seismograms
9 A trial estimated TD will be required to determine an initial rough tie
9 The phase of the seismic data must be determined before detailed correlations are
possible.
9 Detailed stretch and squeeze corrections should not subsample the resident
wavelet.
9 The reasonableness of the resulting correction of the sonic log velocities should
be verified.
9 The consistency of the final TD relationship and the drift curve should be
checked.
9 A final wavelet should be extracted to use for future modeling.
9 Well ties with multiple wells should tie the seismic data in such a way that makes
in terms geologic structure.
40
Well Ties depend on seismic data quality
This section further examines the path for deriving seismic velocities using well log and
seismic data based on the convolutional model. This model, as stated before specifies
that the seismic trace is the convolution of the reflection coefficient series with the
wavelet where T, W and R are the seismic trace, wavelet and reflectivity series,
respectively as shown by equation 1.
T=W*R
Equation 3
Note that “T” in Equation 1 is the seismic trace. Just as reflectivity from the well logs is
assumed to be error free, also the seismic trace is assumed to be free of noise or at least
that the noise in the seismic trace is small relative to the signal. If seismic noise is a
problem, it should be addressed as early in an interpretation project as possible.
Hunton
Target
Figure 49: Shot records from project – AGG applied
The data in this project was relatively short offset and modest fold. Prestack filtering was
applied during processing to improve signal to noise, improve velocity and statics
picking and improve the quality of the migration. Figure 49 shows an example of a shot
record. Some primaries were visible on the raw shot records. This is a good indication
that data will be usable after processing. However for an improved well tie, improved
processing is indicated.
41
Figure 50: Raw gathers pre-migration – AGG applied
xxxxxx
Figure 51: Filtered gathers pre-migration – AGG applied
Figures 50 and 51 contrasts raw versus gather filtering prior to migration. Figure 52 on
the next page shows the difference volume.
42
Figure 52: Differenced gathers, before & after filtering – AGG applied
Figure 53: Stack of raw gathers – AGG applied
Figure 53 is a stack of the raw gathers which were displayed on Figure 50. Figure 54 on
the next page is a stack of the conditioned gathers, displayed on Figure 51.
43
Figure 54: Stack of filtered gathers – AGG applied
Figure 55: Stack of differenced gathers – AGG applied
Figure 55 is a stack of the differenced gathers which were displayed on Figure 52. This
is the noise that was removed from the stack, prior to migration.
44
Figure 56: PSTM of filtered gathers – AGG applied
Figure 57: Stack of Filtered PSTM of filtered gathers – AGG applied
Figure 56 is a stack of the prestack time migrated gathers which were previously filtered
prestack and subsequently had residual statics and velocity analysis applied.
45
Figure 57 shows a stack of filtered PSTM gathers. Additional improvement is subtle.
Figure 58: Well tie to unfiltered (and unmigrated) stack
Figure 59: Well tie to filtered - migrated stack
Figure 58 shows the well tie with the raw stack. Figure 59 shows the same well tie with
two passes of prestack filtering and PSTM. The visual improvement of the synthetic to
seismic tie is obvious. The cross correlation has improved from 0.672 to 0.774 with
prestack filtering. While the improvement in correlation is statistically significant, the
quality of the visual correlation appears to be past the threshold of acceptability. Note
that the tie to the unfiltered data utilized the velocity function produced by the final tie
with the prestack conditioned PSTM data.
46
Seismic data improvement
9 Noise can exist both in the well data and in the seismic data prior to attempts to
perform synthetic to seismic well ties.
9 Improvements to either or both input data sources will improve the final velocity
derived from the procedure.
9 New prestack filtering algorithms can also improve the stack response sufficiently
to improve the quality of well ties.
47
Section 2
Checkshots and VSPs for well ties
Well shooting, Check-shots and VSPs
A vertical seismic profile (VSP) is always a check-shot, but a check-shot may not be a
VSP. A checkshot usually records little more than down going energy. The VSP records
both down going and reflected up-going energy.
Check-Shots
Check-shots surveys are used to provide one-way travel time between a source and a
down-hole detector. Typically the source must be offset from the well by at least a
minimum distance. The correction for this offset and the depth of a shot hole (if any) is
given by equation 1 and diagramed by Figure 1.
VAVG = (Z – d) / (TSG * Cos(θ))
Equation 1
shot
s
x
d
Z
θ
geophone
G
VAVG = (Z – d) / (TSG * Cos(θ))
Figure 1: Schematic of a check-shot acquisition
1
Vertical Seismic Profiles
Surface seismic reflection data are typically acquired, processed and displayed in twoway-travel-time. Well logs, geologic models, and well plans are usually created and
displayed in depth. An accurate velocity log is needed to create a time-depth relationship
to link these two ways of recording and displaying subsurface data. A properly processed
VSP provides this velocity relationship and usually produces the most accurate way to
relate time to depth.
VSPs - The best
links between
surface seismic
data (time) and
subsurface
formation
properties (depth).
Figure 2: Artist’s conception of graduate students collecting a VSP
The point of this cartoon in Figure 2 is that a VSP is “real” seismic data that uses seismic
sources and receivers as opposed to synthetic traces calculated from sonic and density
logs. A synthetic seismogram previously discussed produced from corrected sonic and
density logs can produce acceptable time-depth ties, but under many circumstances the
VSP is the most accurate and reliable source for time-depth ties.
Figure 3 shows a typical vibrator truck that can be used for acquiring land surface seismic
data or as a VSP source. A VSP is acquired with a surface source and a down hole
receiver that may be sequentially located over a range of depths in the borehole. Sources
for VSPs are similar to ones used to acquire surface seismic data, so their characteristics
(frequency content) are similar, which means that they produce data which can readily be
matched to the surface seismic data.
2
Baseplate
Figure 3: VSP Vibrator Source for land acquisition
The VSP receivers must be firmly clamped to the borehole wall to record a reliable wave
field at seismic frequencies, ranging from a few hertz to perhaps over 200 Hz. Figure 4
shows the sonde that is attached to a wireline cable and the locking arm used to attach the
sonde firmly to the borehole wall.
Locking arm
Geophone
module
Figure 4: VSP Sonde with geophone module and locking arm
3
The inset shows the geophone module with three mutually perpendicular receivers.
Currently technology is available to use up to 80 separate 3-component sondes on a
single wireline with sonde separations ranging from 20-50ft. With special cables, 100 or
more 3-component, wall-locking stations can be deployed via tubing.
Figure 5 shows a schematic illustration of a downhole receiver used to record wavefields
within a borehole. The downhole receivers are located within a sonde that typically
contains three mutually orthogonal geophones. For the purposes of this discussion of
VSPs, we will consider use of only the vertical or “Z” component of the tool. Use of all
the components for multi-component analysis of wavefields is a growing field, but
beyond the scope of this initial introduction. The sonde must be clamped to the borehole
wall through the use of a locking arm to make good contact for recording the passing
wavefields. Some contractors offer the use of multiple sondes to increase acquisition
efficiency. The source is initiated several times at each clamped, downhole position so
that summing can be used to improve the signal-to-noise ratio of each depth level
recorded.
Vertical
Horizontal 1
Horizontal 2
Figure 5: Multi-component Geophone Module
4
Figure 6: Rig-site - The VSP Headquarters
Similar to the checkshot survey, Figure 6 reminds us that the VSP is acquired during the
time when a well is being drilled, usually after standard wireline logs have been acquired
at various casing points. Sometimes VSPs are acquired at intermediate casing points to
be use to “predict ahead of the bit” or are acquired prior to setting casing. Thus, like
logs, there is only a brief time window during the process of drilling a well that this type
of experimental data can be acquired. Wells can be re-entered to record VSPs, but this
usually involves shutting in production and pulling strings of casing, which is a very
expensive process.
VSP Example
An AGC’d plot of a field VSP record contains the information that allows separation of
up-going and down-going wavefields based on moveout with respect to depth. Figure 7
displays the output of a VSP illustrating these characteristic dipping events and its
surface seismic like character. Time is on the vertical axis and depth is on the horizontal
axis of this plot. Depth increases to the right, but time increases from top to bottom.
5
Time
Depth
Figure 7: Display of VSP after AGC
The geometry of acquisition allows the wavefields to be separated as shown in Figure 8
where the raypaths are drawn with large horizontal exaggeration for clarity.
PR+FBT=2WT
Figure 8: Display of VSP Raypaths
First, we will look at raypaths from having a source on the surface and a receiver in a
borehole. The VSP cases we will discuss in this course will only address a situation
where the source offset is small and the raypaths are nearly vertical, hence the name
6
Vertical Seismic Profile. On the right side of the plot are the collection of raypaths which
are called downwaves. Downwaves are defined as waves approaching the receiver from
above. These include both direct arrivals from the source as well as multiples. The
downwaves will include all multiples which had their last reflection above the receiver.
On the left are shown the collection of raypaths known as upwaves. These arrivals
include both primary reflections and multiples which have last reflected from below the
receiver. One special characteristic of the VSP geometry is that the downwave first
arrival time is the exact time needed to complete a two-way travel time of a primary
reflection event. This observation will be used later to help with processing and is
denoted on the slide by the equation PR+FBT=2WT.
Geophone Levels (Depth)
Time
De pth
Depth
Source Offset
Figure 9: Display of VSP Raypaths & Reflections
Figure 9 shows both the ray paths and the data generated from direct transmissions and
reflections as they would be recorded from several depths in the borehole. There are a
number of coherent events shown including first arrivals, primary reflections, multiples,
and tube waves. Beginning just above the deepest blue layer, imagine a ray traveling to
the deepest geophone position labeled 1. It would be recorded just before the reflection
from the blue layer. As the geophone is moved up the borehole to position 2, the direct
arrivals appear sooner in time while the reflections from the blue layer arrive
progressively later in time. As the geophone continues to be positioned higher in the
borehole, at level 4 the reflection from both the blue and green layers are recorded.
Typically there are several shots collected for each receiver position. These are edited
and summed to create a single trace per depth level. Figure 10 shows an example. This
plot has been scaled for plotting as it would have been recorded, but without AGC.
7
500ft
100 ft level spacing
Depth
Figure 10: Field recorded VSP – No AGC
Figure 10 also shows several shallow levels shot with increased distance between levels
to produce check shots to provide check shot quality velocity control. Only the levels
shot at the uniform 100ft intervals will be used for the VSP since the wider spaced levels
will be aliased for upcoming reflections.
Time
Depth
Figure 11: Field recorded VSP – with AGC
To see both the downwaves and upwaves, a gain function or AGC can be applied to the
VSP portion of the data shown on Figure 11. Recall that reflection coefficients average
8
around .04 or less so that events tracked away from the first arrival time are lower in
amplitude by a factor of 25 or more.
Figure 12: VSP Downwave – with AGC
By subtracting the first arrival times from each level (after adding 30ms) we can see the
aligned downwaves. Of course the steeply dipping events are now the reflected upwaves.
Depth
Figure 13: VSP First Arrival Window – with AGC
9
Amplitude (arb)
If we take a closer look at the first arrivals on Figure 13, we can see the short period, near
surface reverberations and longer period multiples. We can also see how the upwaves
interfere with these multiples (and vice versa). Recalling that the traces have been gained
and scaled for displays, if we calculate the amplitude or energy (amplitude squared) in
the 80ms time gate between the red arrows for the raw data, we can get an idea of the
losses of the downward propagating seismic wavefield.
Depth (ft)
Figure 14: VSP Amplitude decay with depth
Figure 14 shows two measures of the loss amplitude of the propagating wavefield using
the mean absolute and RMS amplitude within the 80ms time gate on the previous Figure
13. The data points that show increased scatter near a casing point in the well bore at
~6700.
Now returning to the downgoing wavefield, we want evaluate applying a deconvolution
operator to also remove multiples. From the downwave display on Figure 15 below, we
can observe where the multiples are strongest. The operator length indicated is 350ms,
although this is somewhat arbitrary. It is a matter of judgment to determine the operator
length. Also note that the reflected upwaves are “noise” on the downwaves and should
be removed to determine the true downwave wavefield. This can be done by using a
median filter or other multi-trace operations.
10
Decon
operator
Figure 15: VSP Downwave with AGC
Figure 16: VSP Downwave – post Deconvolution
Using the first 350ms window as the primary downgoing wavelet for designing an
inverse (decon) operator to “zero phase” the downwaves, we get the data as shown above
in Figure 16. Notice that the inverse operator can be used for every trace to capture the
slow variation of the frequency content with depth, however, this can results in noise and
upwaves affecting the inverse operator adversely. A better approach is to use a running
5-trace mean or median filter to average out the upcoming energy and noise in the
estimated downgoing wavefield before calculating and applying the inverse. Also notice
that the longer operator allows the removal of the longer period multiples up to the
operator length. Some longer period multiples are beginning to reappear just below
11
350ms. Also note the very strong first arrivals compared to the upcoming events and the
residual downgoing events.
Depth
xxxxxxxxxxxxxxxxxxxx
Figure 17: VSP Upwave – AGC – Decon - Mix
Figure 17 displays the VSP after using a median, or f-k, or other type of multi-channel
filter to remove the residual downgoing energy, muting off the residual direct arrival,
since it was so strong, and then shifting to twice the first arrival time to align events
horizontally at their two-way time. The first shift gets back to VSP time and the second
to two-way time. This two-way time should now correspond to the seismic two-way
travel time.
12
Inside
Corridor
Upwaves
Figure 18: VSP Upwave – Corridor and Inside Stacks
The zero-phased, gained upwaves can be summed to produce a single trace that
represents the reflection signal at the well location. This single trace is often reproduced
several times for display to create a more seismic-like display. To optimize the
information in this stacked or summed trace, different approaches can be used for the
summing. One approach is to keep a sliding time gate after the direct arrival and mute
later times. This results in what is sometimes called a corridor stack, which has the
advantage of maintaining the highest S/N ratio signal that is also closest to zero phase
and to eliminating any multiples with a period longer than the corridor length. It also
helps eliminate reflection signal distortion when there is geologic dip or slight deviation
to the borehole. The other stack is produced from summing the shallowest recorded
levels, say 5 traces. This trace will sometimes match the seismic better because it more
nearly duplicates the geometry of the surface experiment, but it is often contaminated
with multiples and other noise. The corridor and inside stacks have been shows adjacent
to the upwave display on Figure 18 above.
13
Seismic (-135deg)
or
h
de
rid
r
nt nsi
o
y
I
C
S
Sonic Log
Vel
Time (sec)
Depth (ft)
Figure 19: VSP Montage – Seismic, VSP & Synthetic seismogram
Figure 19 is a specialty VSP Montage display designed to show several important
features of the VSP. Starting of the left is a short segment of the surface seismic data
near the borehole (5 inlines and 5 Xlines). Next comes the VSP Corridor stack, the
synthetic made from the sonic and density logs, the VSP Inside stack, and finally the VSP
upwaves. Note that all of these panels are zero phase by construction. The aqua colored
line shows the corridor stack window. Above the VSP upwaves is a plot of the sonic log.
Recall that the scale along the top for the VSP is depth, hence the sonic log is displayed
for these same depths. The red horizontal line indicates a strong reflection event that can
be found on all the display panels. When this line is extrapolated to first arrival time and
projected vertically, we can see the portion of the sonic log and depth that corresponds to
this time, hence a very precise time-depth time can be established that is accurately tied
to the seismic data.
Seismic data phase from VSP
The VSP can be processed to zero phase because the downgoing wavelet is recorded at
the first break. The following is a procedure that can be used to determine the phase of
the surface seismic data from a collocated VSP.
Returning to the seismic and VSP corridor stack, the Figure 20 shows the 5 ILs and 5XLs
or 25 traces from the original seismic data. The VSP corridor stack has been inserted in
the middle of the display and the seismic trace nearest the well repeated on each side of
the well. Recall that the VSP corridor stack has been converted to zero phase since the
downgoing wavefield was used to de-phase it and define the polarity. The seismic data,
14
on the other hand, is shown at the phase resulting from the original field acquisition and
processing. Note there appears to be a slight time shift between the stronger events. The
red arrows indicate the time gate for which a transfer function will be computed between
the corridor stack trace and seismic traces nearest the well.
Figure 20: VSP Corridor Stack and Surface Seismic Data
When a transfer function or match filter is run for the time gate indicated between the
VSP corridor stack and the seismic trace nearest the well, Figure 21 below shows the
resulting phase is about -135 degrees.
Transfer Function
or Inverse Wavelet
Amplitude
180
0
Phase
-180
Figure 21: TFUN VSP Corridor Stack & Surface Seismic Data
15
Recall the convolutional formula shown here as equation 2. R is reflectivity, W is the
wavelet and T is the seismic trace.
T=W*R
Equation 2
Equations 4 and 5 show the Fourier equivalent definition for convolution, in other words,
that convolution is equivalent to multiplying the amplitude spectra and summing the
phase spectra of the traces. Now in the case of the transfer function if Tc is the zero
phase VSP corridor stack, and if Ta is the seismic trace with phase W, then the inverse
operator must have phase = to the negative of the seismic phase W.
TC = TA * TW
Equation 3
Amp(TC) = Amp(TA) X Amp(TW)
Equation 4
Amplitude spectra multiplied:
Phase spectra added (where Phase(TC) = 0 for the VSP :
Phase(TC) = Phase(TA) + Phase(TW)
Equation 5
16
Figure 22: VSP Corridor Stack & Zero Phase Surface Seismic Data
After applying the transfer function to the seismic data, it is now close to zero phase and
matches the VSP corridor stack trace phase very accurately in the analysis time gate
1150-2100ms. The visual match of the seismic and VSP data is now more apparent on
Figure 22. Verifying the phase by checking the transfer function, the phase is
demonstrated to be close to zero on Figure 23 below.
Amplitude
180
0
Phase
-180
Figure 23: TFUN VSP Corridor Stack & 0 Phase Surface Seismic Data
17
Well Ties via Checkshots and VSPs
9 A VSP is the preferable means of tying time to depth because it records depth of
first arrival, signature of source and is similar in frequency content to surface
seismic data.
9 A checkshot provides a crude estimate of velocity over relatively large depth
windows. Older checkshots may be inaccurate for numerous reasons, but provide
no direct methods to verify the accuracy of the derived velocities.
9 The phase of the seismic data must be determined before detailed correlations are
possible and it can be determined by a quality VSP.
9 Additionally a VSP may be used to determine propagation energy losses.
9 Finally, a VSP may be used to determine the origin of some multiples.
This section largely the courtesy of Dr. Robert J. Corbin of Corbin Geophysical, Inc.
18
Section 3
Petrophysics for Quantitative Seismic Modeling
Petrophysics for Quantitative Seismic Modeling
Rock physics analysis prior to synthetic seismogram modeling may be initiated from raw
logs or from the output of a qualified log analyst. Most geoscientists would benefit from
working with a petrophysicist when possible. Likewise, few classic log analysts have
much experience in the esoteric methods of Gassmann fluid replacement modeling. This
makes the development of natural work teams with complementary expertise quite useful.
First steps in seismic petrophysical modeling.
Most petrophysicists approach the task of analyzing logs in a fairly similar manner. This
is a partial list of steps in the order performed:
1. Load digital data and log header information
2. Review curve availability over known depth ranges
3. Determine “unusual” curve names from mnemonic lists, curve units and
descriptions in log headers.
4. Quality control of curve data (similar to sonic and density editing steps).
5. Determining if all curves are “on depth” with each other and depth shifting as
necessary
6. Perform environmental corrections to curves if necessary.
7. Determine lithology
8. Determine porosity
9. Determine formation water resistivity
10. Determine saturation
11. Determine movable hydrocarbons
12. Determine hydrocarbon type (gas or oil) if present.
13. Determine bulk volume hydrocarbons, net reservoir feet and net pay.
The first two steps are software dependent.
Because step three can be an impossible task if the petrophysicist must start with
randomly renamed log curves, most prefer to start with, or at least have access to field
records or at least those from the initial service provider (log acquisition company).
We have previously reviewed the methods for QC for the sonic and density curves. The
QC of GR, resistivity curves, neutron and Pe curves to name a few is beyond the scope of
this course.
Depth shifting of log curves relative to each other is software specific. It also requires
knowledge of how different measurements correlate to each other and which curves are
collected on different log runs into the borehole.
1
Environmental corrections are tool specific. For example correcting the GR curve for
borehole effects requires a caliper; the mud weight in the borehole at the time of logging;
knowledge of mud type such as KCl mud; and knowledge if the GR tool was run in a
centralized or decentralized state. This is worthwhile knowledge for a log analyst to have
available and should always be digitized or loaded into a LAS header when possible.
It is not unusual for rock physicists to load the log data and jump immediately to step
number seven. In purely clastic sediments, lithology determination is typically a
determination of shale or clay volume. The remaining volume of clastic sediments is
usually considered to be sandstone.
There are a number of log based formulas used to calculate shale volume. The GR is one
of the most popular methods, but there are areas where the use of the GR is precluded by
abundant radioactive (non-shale) minerals such as K-spar. The following formula in
Equation 1 is the basis of a VSHGR (gamma ray based shale volume) calculation where
GRCN, GRSH and GRLOG are the gamma ray value for clean (non-shaly) reservoir, shale
and the log measurement, respectively.
VSHGR = (GRLOG – GRCN) / (GRSH – GRCN)
Equation 1
Equation 1 is a linear volumetric equation. It implies that the volume of shale ranges
linearly with changing GR measurement from the clean end member to the shaly end
member. We should note here that there are a plethora of nonlinear VSHGR equations
that some petrophysicists utilize. However there is limited theoretical basis for using any
nonlinear VSHGR formula. A test for nonlinear shale volume from GR will be presented
after neutron – density shale volume is considered. Figure 1 shows a GR curve,
environmentally corrected GR and VSHGR from linear volume.
Figure 1: VSHGR from environmentally corrected GR log
2
The SP or spontaneous potential curve can be used to estimate shale volume under some
circumstances. The SP only develops when there is at least some permeability in the
reservoir (non-shale) intervals. Additionally, there is a hydrocarbon effect that
suppresses the SP curve in the presence of any shale and the SP becomes very insensitive
when formation resistivity is close to mud filtrate resistivity. For the stated reasons, the
SP is frequently the curve of last choice for shale volume calculations. Equation 2 is the
basis of a VSHSP (SP based shale volume) calculation where SPCN, SPSH and SPLOG are
the gamma ray value for clean (non-shaly) reservoir, shale and the log measurement,
respectively. Figure 2 shows VSHSP from the input SP curve.
VSHSP = 1 - (SPLOG – SPSH) / (SPCN – SPSH)
Equation 2
Figure 2: VSHSP from a baseline treatment of the SP log
The neutron and density logs can be used together to calculate shale volume. There are a
couple of extra steps that precede the actual shale volume calculation. First the neutron
must be corrected from limestone to sandstone matrix if it was recorded in the more
typical format. This is typically one of the environmental corrections available in most
professional petrophysical software packages. Second density porosity on a sandstone
3
matrix must be calculated as well. ΦD, ρMA, ρLOG, and ρf are density porosity, matrix
density, measured bulk density from the log, and fluid density respectively as found in
equation 3.
ΦD = (ρMA – ρLOG) / (ρMA – ρf)
Equation 3
The neutron – density shale volume is a very good quantitative measure of shale volume
if both input curves are valid and if the reservoir is not a gas sand. Gas has the opposite
effect that shale does on the neutron and density curves and results in an underestimation
of shale volume. Equation 4 is the classis form of the VSHND (neutron – density) based
shale volume calculation where ΦN, ΦD, ΦN SH, and ΦD SH are the neutron porosity,
density porosity, neutron porosity of shale, and density porosity of shale, respectively.
VSHND = (ΦN – ΦD) / (ΦN SH – ΦDSH)
Equation 4
Figure 3: VSHND based on neutron and density porosity logs
Figure 3 shows ΦN and ΦD in the right track. The greater ΦN is than ΦD, the more shale
is indicated. The track on the left is the NMD, or neutron minus density. This is the
numerator of Equation 4. Petrophysicists use the NMD combined log to parameterize
shale volume calculations. Figure 4 shows NMD crossplotted against the GR curve to
4
determine if a “nonlinear” GR relationship should be used to compute shale volume.
Because the scatter-plot is linear, no such computation is indicated.
Figure 4: NMD on X axis with GR on Y axis linear trend indicated
An inspection of Equation 4 shows that the only difference between the numerator
(NMD) and the denominator is that the denominator is NMD in shale. This means that
once NMD has been calculated, then VSHND can be calculated by selecting an
appropriate NMD value for shale in the zone of interest rather than selecting ΦN SH and
ΦD SH independently. Thus Equation 5 is
VSHND = NMD / NMD SH
Equation 5
It is important to note the implications of the VSHND calculation. The ΦN and ΦD curves
are expected to be equal when no shale is present. Therefore if a limestone matrix is used
to calculate porosity, then NMD will be negative in clean quartz sandstone. This will
result in minor shaliness calculating as clean (after truncating volumes between 0 and 1).
Alternately, if a quartz matrix is used to calculate porosity in a limestone, then clean
limestone will appear to contain some shale volume. It is important to know the basic
mineralogy prior to interpretation.
Figure 5 displays a crossplot of NMD against SP. Note the lack of decent linearity
between these curves in contrast to the similar crossplot against GR on Figure 4.
5
Figure 5: NMD on X axis with SP on Y axis. No linear trend indicated
Figure 6: NMD on X axis; Conductivity on Y axis. Scattered linear trend indicated
Figure 6 shows some degree of linear fit between NMD and conductivity. However there
is only a 0.65 correlation in contrast to the 0.83 between NMD and GR. Conductivity is
impacted by the presence of hydrocarbons and changes in the contrast between shale
6
resistivity and brine resistivity. Conductivity was selected for shale volume rather than
resistivity because the laminated shaly sand saturation model uses a linear relationship
between conductivity and shale volume. The conductivity relationship is provided by
Equation 6 where VSHC, CondLOG, CondCN, and CondSH are shale volume from
conductivity, measured conductivity, clean formation conductivity, and shale
conductivity respectively.
VSHC = (CondLOG – CondCN) / (CondSH – CondCN)
Equation 6
There is a more common shale volume equation in the literature than the one provided for
linear conductivity in Equation 6 above. Equation 7 below provides that for VSHREST,
where VSHREST, RSH, RT, RCN, and b are the shale volume from resistivity, resistivity of
shale, true formation resistivity, clean formation resistivity, and the exponent, “b”,
respectively. When b is 1, this equation provides the same results as equation 6, provided
that all the resistivities are equal to 1000 / conductivity for logs and parameters.
Otherwise, b generally varies between 1 and 2.
VSHREST = (RSH / RT) * (RCN – RT) / (RCN – RSH)(1/b)
Equation 7
Figure 7 below shows shale volume calculated continuously by the four different
methods. With exceptions, the smallest of a select calculation of shale volume is used as
the final indicator of shale. This is because for SP, resistivity / conductivity, and poorly
calibrated GR and neutron – density, generally errors tend to over predict shale volume.
Figure 7: Plots of VSH from GR, SP, ND & Cond (left to right)
7
The red boxes shown on Figure 7 highlight depths where SP for example either shows
much more or much less shale than the other 3 methods.
Porosity Calculations
The density porosity equation has already been provided by Equation 3. Repeating it
below for completeness, the input variables are ΦD, ρMA, ρLOG, and ρf are density
porosity, matrix density, measured bulk density from the log, and fluid density
respectively. Obviously, the QC steps provided under the earlier sonic and density
editing section for the purposes of generating well ties should be addressed prior to this
calculation.
ΦD = (ρMA – ρLOG) / (ρMA – ρf)
Equation 3
The neutron log was one of the first porosity logs invented. Early neutron logs were
recorded on a somewhat arbitrary API scale. These curves are related to porosity by a
LOG10 relationship. The modern neutron porosity curve is generally pre-calculated to a
limestone matrix. If quartz or dolomite matrix is more representative then software or
“chart book” corrections for mineralogy differences are required. In some dominantly
clastic sections, however a quartz matrix will be preferred and the logs will be recorded
with that bias. Note that the first step in using a neutron log is to determine what selected
matrix was used before it was written to a digital file.
The sonic tool was developed at about the same time as the old neutron logs. Both of the
old tool were related to porosity in either empirical or heuristic methods. The modern
neutron currently provides a quality porosity measurement. The sonic is still
problematic. Early log analysts employed corrections attributed to lack of compaction to
the sonic porosity when it failed to match neutron – density or core values. Equation 8 is
a form of the heuristic Wyllie time average relationship. ΦS, ΔTMA, ΔTLOG, and ΔTf are
sonic porosity, matrix travel time, measured sonic travel time from the log, and fluid
travel time respectively.
ΦS = (ΔTLOG – ΔTMA) / (ΔTf – ΔTMA)
Equation 8
The Raymer-Hunt-Gardner (RHG) relationship for the sonic, or more specifically for
velocity is shown by Equation 9 where V, VMA, Vf, and ΦS are log velocity, matrix
velocity, fluid velocity and sonic porosity, respectively.
8
V = (1 - ΦS)2 * VMA + ΦS * Vf
Equation 9
Solving Equation 9 for porosity in terms of sonic travel time, results in the following two
equations where ΦS, ΔTMA, ΔTLOG, and ΔTf are sonic porosity, matrix travel time,
measured sonic travel time from the log, and fluid travel time respectively.
ΦS = – α – (α 2 + (ΔTMA / ΔTLOG – 1)1/2
Equation 10
α is a variable supplied by equation 11 and calculated from ΔTMA, and ΔTf
α = ΔTMA / (2 * ΔTf) – 1
Equation 11
The RHG relationship was determined empirically to better fit field measured porosity
values. It doesn’t require compaction correction terms in soft rock like the Wyllie time
average equation does.
Many petrophysicists would prefer to use density porosity, then neutron porosity and
finally sonic porosity if only one porosity tool is available. Unfortunately, the density
probably has the greatest tendency to exhibit borehole problems because it requires a
very small standoff from the borehole wall for valid measurements.
TERMS
The term “Porosity” means fractional volume of void space in a rock to a geologist. The
petrophysical literature is littered with quasi-related terms. Two terms that are generally
discussed and calculated nearly simultaneously as a requirement for shaly sand analysis
are “Total Porosity” and “Effective Porosity”.
The SPWLA glossary defines “Total Porosity” as all void space in a rock and matrix
whether effective or noneffective. Total porosity includes that porosity in isolated pores
and adsorbed water on grain or particle surfaces. From other sources, total porosity is
sometimes assumed to be the free and bound water, plus hydrocarbon volume in the rock.
Bound water is the adsorbed water above.
9
The SPWLA glossary defines “Effective Porosity” as interconnected pore volume
occupied by free fluids. The glossary also states that “porosity usually means effective
porosity”.
A review of the porosity equations relative to shaly sand analysis provides additional
clues about actual versus nomenclatural usage. The relationship between total and
effective porosity is given by Equation 12. In this equation regardless of the porosity tool
used for initial measurements, ΦT, Φeff, ΦSH and VSH are total porosity, effective
porosity, shale porosity, and shale volume.
Φeff = ΦT – ΦSH * VSH
Equation 12
In practice, the porosity calculated directly from log measurement is always assumed to
be ΦT. Shale porosity is assumed to be the total porosity measurement obtained from
pure shale. The implication is that the input for Equation 12 is the apparent porosity
measured in shale and the apparent porosity measured in reservoir rock are actually
apparent porosity measurements. Also, in practice nothing precludes isolated pores such
as vugs in carbonate from inclusion in the effective porosity volume.
Figure 8: “Total” and “Effective” porosity from Neutron - Density
10
Figure 8 shows effective porosity calculated from apparent neutron – density sandstone
porosity. Note the very large apparent porosity in the shales, but very low effective
porosity. In terms of conventional reservoirs, it would be unreasonable to expect to
replace significant volumes of that apparent porosity with hydrocarbons. Also, the total
porosity curve is apparent porosity based on a quartz matrix. Although some claims have
been made that “shale matrix” is the same as quartz, that claim is unverified.
Saturation Calculations
The Archie equation has been used for decades to provide saturation calculations based
on well logs. SW, RW, RT, Φ, a, m, and n are water saturation, resistivity of formation
water, true formation resistivity, porosity, the “tortuosity” coefficient, cementation factor
and saturation exponent, respectively.
SWn = (a * RW) / (Φm * RT)
Equation 13
The Equation 13, describing Archie’s empirical relationship was developed in the
laboratory using measurements on core samples. There is no theoretical support for this
relationship, however some petrophysicists believe the “a” term should always be unity
because that would represent conditions where resistivity was measured in formation
water, RW. However, the use of a variable “a” term allows a better empirical fit to actual
rock samples. Special core, “laboratory” analysis is used to determine a, m and rarely n
from good quality core samples. RW may be calculated from produced waters,
determined from a log-log porosity versus resistivity (Pickett) plot, or picked from a
calculated RWA curve.
Note that “Archie” is only applicable for clean formations. When significant shale is
present, then one of the many shaly sand saturations equations must be used. The “Dual
Water” saturation model is shown by Equation 14. Vsh, a, m, n, Rw, Rt, Phit, and PHIsh
are as described above. Rsh is the resistivity of representative shale.
Sw = (-(((Vsh * PHIsh) / PHIt) * (1 / (Rsh / (a /
PHIsh^m)) – 1 / Rw)) + ((((Vsh * PHIsh) / PHIt) * (1 / (Rsh / (a
/ PHIsh^m)) – 1 / Rw))^2 – 4 * (1 / Rw) * (-1 / (Rt / (a /
PHIt^m))))^(1/n))/(2 / Rw)
Equation 14
11
Complex mineralogy
Clean and shaly formations may be handled with either the Archie procedure or a shaly
sand method. However, some settings have combinations of matrix minerals or
interbedded carbonates and clastic rocks. Understanding these rocks requires additional
techniques. Multimineral petrophysics originated as a solution of simultaneous
equations. It was motivated by a desire for a better porosity solution than available from
monomineral end-members. VL, VQ, and VP are the volumes for limestone, quartz and
porosity, respectively, using bulk density and neutron porosity input curves. Because you
have a summation equation such that all constituents must add to 1.0 a fully determined
set of equation allows solution for porosity, plus one mineral for each input log curve.
One key assumption is that the log curves sum volumetrically. Also you must be able to
parameterize the mineral matrix response for each mineral, for each log. Then the
simultaneous equations are solved for the mineral constituents.
ρ = ρL * VL + ρQ * VQ + ρP * VP
Equations
ΦN = ΦL * VL + ΦQ * VQ + ΦP * VP
15a – 15c
1.0 =
VL
+
VQ
+
VP
Figure 9 on the next page shows the response of two different, fully determined
Multimineral solutions. They are similar, but not identical. The differences result from
unique methods handling calculated negative mineral volumes. Just because you have a
fully determined solution doesn’t require that the mineral volumes be physically possible.
12
Fully determined A
GR, Dt, N, D, Pe
Fully determined B
GR, Dt, N, D, Pe
Figure 9: Comparison between two different Multimineral algorithms.
Key to this particular analysis is the porosity in the limestone at the bottom of the well.
Although the two methods are somewhat different in terms of mineralogy, there is no
calculated porosity in the target horizon.
13
Petrophysics for Seismic Modeling summary
9 Shale volume, porosity and saturation must be determined to understand the rock
sufficiently to begin modeling.
9 Multimineral analysis may be required when working with complex stratigraphy.
9 Calculations of mineralogy and porosity are generally based on equations of state
and thus well behaved.
9 The popular saturation relationships are empirical, without theoretical basis, but
have been used none the less with reasonable success for decades.
14
Lithology – Impact on velocity and density
Density is well behaved with linear volume mixing relationships. Velocity depends not
only on what the mineral constituents are, but how they are distributed and connected
through the media. Some models provide useful insight.
Density
The Multimineral simultaneous solver solutions for Equation 15a in the last section
provided the formula for mixing any known volumes of any known densities. 15a is
restated again below where ρ is the bulk density, ρL is lime matrix density, ρQ is quartz
matrix density, and ρP is the density of the pore filling fluid. AlsoVL is volume
limestone, VQ is volume quartz, and VP is pore volume.
ρ = ρL * VL + ρQ * VQ + ρP * VP
Equation 15a
The Multimineral solution for a mixture of quartz and either limestone of dolomite is
shown by the graphs in Figure 10. As expected, the lines connecting the end member
lithologies are linear.
Density mixing (no porosity)
2.9
Dol
2.85
Density g/cc
2.8
2.75
2.7
LS
2.65
Qtz
2.6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frac LS
LS Dol
LS Qtz
Figure 10: Impact of change in end member mineralogy on bulk density.
15
Compressional Velocity
Calculation of velocity is a function of both the constituents and the distribution of those
constituents throughout the media. The following plot on Figure 11 utilizes the VoigtReuss-Hill mixing average. The Voigt and Reuss relationships are the isostrain and
isostress limiting bounds for modulus mixing. Dynamic moduli are related to velocity
and density. The Hill average is simply the midpoint between the theoretical limits that
can exist in an elastic body. Again, Limestone is mixed with quartz and dolomite to
present an expectation for velocity change caused by a change in mineralogy.
Vp Hill mixing average (no porosity)
7200
Dol
7000
Vp (m/sec)
6800
6600
6400
LS
6200
6000
Qtz
5800
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frac LS
LS Dol
LS Qtz
Figure 11: Impact of change of end member mineralogy on compressional velocity.
Note that neither the line connecting quartz and limestone, or the line connecting
limestone and dolomite are perfectly linear. However the later is much more linear than
the former.
16
Porosity – Impact on velocity and density
Density is well behaved with linear volume mixing relationships. Velocity depends on
the volumes of the mineral constituents and porosity, but also is impacted by how the
volume is distributed and connected throughout the media. There are a number of
porosity models for velocity.
Density
The density changes linearly as porosity and mineral volumes are changed by incremental
volumes. Again, the Multimineral simultaneous solver solutions for Equation 15a in the
prior section provided the formula for mixing any known volumes of any known
densities. Figure 12 below graphically demonstrates the impact of porosity compared to
changes in mineralogy on rock density.
Density mixing with porosity
3
Dol
2.8
LS
Qtz
2.6
Density g/cc
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Porosity
LS & porostiy
Dolomite & porosity
Quartz & porosity
Figure 12: Impact of change in porosity with 3 minerals on bulk density.
Compressional Velocity
Calculation of velocity is a function of both the constituents and the distribution of those
constituents throughout the media. The following plot on Figure 13 utilizes the VoigtReuss-Hill mixing average.
17
Velocity from Hill average mineral mixing with porosity
8000
Dol
7000
LS
Qtz
6000
Vp (m/sec)
5000
4000
3000
2000
1000
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Porosity
LS & porostiy
Dolomite & porosity
Quartz & porosity
Figure 13: Impact of porosity change on compressional velocity VR Hill mixing.
This figure graphically demonstrates the impact of porosity compared to changes in
mineralogy on the compressional velocity of the rock. The change in porosity is largest
at small porosity values and minimal through the range of very high soft sediment
porosities. Figure 14 below shows the same variation of porosity with mineralogy based
on the popular heuristic, Wyllie time average relationship.
Velocity from Wyllie time average
8000
7000
LS
Qtz
Dol
Vp (m/sec)
6000
5000
4000
3000
2000
1000
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Porosity
LS & porostiy
Dolomite & porosity
Quartz & porosity
Figure 14: Impact of porosity change on compressional velocity Wyllie
18
Finally, for hypothetical velocity modeling relationships, the Raymer-Hunt-Gardner
relationship is shown in Figure 15 below.
Velocity from RHG relationship
8000
Dol
7000
LS
Qtz
6000
Vp (m/sec)
5000
4000
3000
2000
1000
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Porosity
LS & porostiy
Dolomite & porosity
Quartz & porosity
Figure 15: Impact of porosity change on compressional velocity via RHG
Porosity and mineralogy impact on
compressional velocity and density.
9 Density is well behaved with volumetric mixing of all known constituents
regardless of whether they are mineral or porosity (or fluid).
9 Velocity is more complex. There are also a number of relationships, but the
theoretical relationships only provide upper and lower limits on velocity as a
function of mineralogy and porosity mixing relationships.
9 The well known sonic to porosity transforms are all empirical.
19
Velocity modeling with starting porosity
When the impact of porosity change on velocity is modeled starting with insitu porosity
and velocity, it is preferable for the method used to honor insitu measurements and be
reversible. Using the empirical methods above, but insitu porosity will typically result a
modeled “insitu” velocity that doesn’t match actual insitu measured velocity. This is an
undesired artifact of the typical velocity – porosity transforms. After conditioning the
insitu logs and seismic data to optimize a synthetic seismogram to surface seismic tie,
having another mental adjustment to apply in testing porosity variations is not
appreciated. However there is a work-around.
Restating Equations 8 and 9 for the Wyllie time average and RHG equations where:
For the Wyllie time average relationship. ΦS, ΔTMA, ΔTLOG, and ΔTf are sonic porosity,
matrix travel time, measured sonic travel time from the log, and fluid travel time
respectively.
ΦS = (ΔTLOG – ΔTMA) / (ΔTf – ΔTMA)
Equation 8
The Raymer-Hunt-Gardner (RHG) relationship for the sonic, or more specifically for
velocity is shown by Equation 9 where V, VMA, Vf, and ΦS are log velocity, matrix
velocity, fluid velocity and sonic porosity, respectively.
V = (1 - ΦS)2 * VMA + ΦS * Vf
Equation 9
Equation 8 can be solved for ΔTMA resulting in Equation 16. The other variables remain
as defined for Equation 8.
ΔTMA = (ΔTLOG – ΔTf * ΦS) / (1 – ΦS)
Equation 16
Equation 9 can also be solved for ΔTMA resulting in Equation 17. Again, the other
variables remain as defined for Equation 9.
VMA = (V – ΦS * Vf) / (1 - ΦS)2
Equation 17
20
We will address a parameter called “dry frame bulk modulus” when we address
Gassmann fluid replacement modeling, also known as “Fluid Substitution”. In this
porosity modeling method for either the Wyllie time average or the RHG method, the
impact of the dry frame is unaccounted in a specific formalism. However the uncertainty
in both matrix and dry frame are incorporated into the apparent matrix parameters which
are calculated on a sample by sample basis over the reservoir zones that are modeled.
The first step of this method is to calculate ΔTMA for Wyllie and / or VMA for RHG.
These matrix parameters are held constant as is the fluid parameter. Then the model
porosity is substituted in for equation 9 and velocity is calculated. Equation 8 can be
solved for ΔTLOG, and restated as Equation 18.
ΔTLOG = ΔTMA * (1 – ΦS) + ΔTf * ΦS
Equation 18
Because the calculated matrix, fluid and porosity will remain constant at insitu
conditions, the velocity can be back calculated to derive the insitu velocity as well.
The uncertainty of the porosity modeling in this way can be tested at other depths in the
same well where porosity and velocity are also known. The matrix parameter is
determined for all sampled reservoir sections. The insitu porosity from another depth is
used to model velocity with this method which can then be compared to the insitu
velocity. An example for Wyllie and RHG is shown by Figure 16 on the next page.
21
Model verification
130
120
110
100
Measured Dt
Final TAvg
Final RHG
90
Linear (Final TAvg)
Linear (Final RHG)
80
y = 1.1952x - 15.787
R2 = 0.9005
70
y = 1.0266x - 1.8409
R2 = 0.9184
60
50
40
40
50
60
70
80
90
100
110
120
130
Modeled Dt
Figure 16: Impact of porosity change on compressional velocity via RHG
The Wyllie porosity model is presented by the dark blue diamonds. The magenta squares
are the RHG model. The measured sonic is on the Y axis and the Models are scaled on
the X axis. Note that both methods have calculated R-squared values over 0.9 However
the Wyllie model has a poorer fit to a 1:1 line than does the RHG model. The scatter of
horizontal points at a single modeled Dt show the variation in repeatability using
different matrix values at depths other than where they were initially derived.
Velocity modeling from Porosity changes
9 It is possible to model velocity as a function of porosity in such a way that it
honors insitu porosity and velocity and is a reversible process.
9 It is possible to model velocity as a function of porosity is such a way as to
explain over 90% of the variation in measured velocity in some cases.
22
Fluid Replacement Modeling - Velocity &
Density
The Gassmann equation is a low frequency theoretical model of p-wave velocity behavior
(essentially seismic frequency ranges). It may be used to convert any given saturation to
a newly desired saturation. Equation 19 shows the dependency of the p-wave velocity on
the rock physics parameters, bulk modulus and shear modulus.
Vp =
4 (Shear modulus)
3
Density
Bulk modulus +
Equation 19
Equation 20 and Figure 17 review the definition of the bulk modulus. Bulk modulus can
be related to change in volume related to change in applied hydrostatic pressure.
Bulk Modulus =
ΔP
ΔVolume/Vol ume
Equation 20
P+ΔP
P+ΔP
A
P+ΔP
ΔP
+
P
P+ΔP
Figure 17: Bulk modulus determined via static experiment.
Figure 18 provides a schematic cartoon of the fluid substitution process. Fluid
substitution characterizes the total rock in components of matrix material, the frame
(architecture) and the fluid itself. Then the properties of a second fluid replace the insitu
conditions and the velocity can be computed from the model of the moduli of the new
rock containing the second fluid. The frame is calculated by knowledge of the matrix and
fluid moduli, the elastic properties of the total rock (VP, VS and ρ) and the porosity.
23
Rock
Fluid
Matrix
Frame
Fluid
Rock
Figure 18: Fluid Substitution process
In other words, if we know the velocity of the rock, the bulk modulus of the initial fluid
and the bulk modulus of the matrix, then we can determine the bulk modulus of the dry
rock frame. Knowing the bulk modulus of the new fluid, old matrix and frame, we can
determine the bulk modulus of the new rock. The Gassmann equation relates rock
velocity to these component bulk moduli. Details of Gassmann are shown by Equation
21 where; K = Bulk Modulus, μ = Shear Modulus, ρ = Density, and Φ = Porosity.
Subscripts f = Pore Fluid, d = Dry Rock, and ma = Matrix.
⎡
⎤
(1 − K d /K ma )2
4
VP2 = ⎢K d + μd +
⎥ /ρ
(
)
3
−
−
+
1
Φ
K
/K
/K
Φ/K
⎥
d
ma
ma
f ⎦
⎣⎢
Equation 21
The Gassmann Equation 21 originates from the Equation 19 for the p-wave velocity,
which is restated below in Equation 22 with variable terms similar to those just used
above.
4 ⎤
⎡
VP2 = ⎢K + μ⎥ /ρ
3 ⎦
⎣
Equation 22
Gassmann can also be recast in terms of bulk modulus as shown by Equation 23
K = Kd +
(1 − K d /K ma )2
(1 − Φ − K d /K ma )/K ma + Φ/K f
Equation 23
24
There is another version of Gassmann that eliminates the absolute requirement to
determine dry frame modulus by providing the following relationship shown in Equation
24. This solution relates the original conditions to the final conditions in terms of bulk
modulus. We will discuss the benefits of examining dry frame later.
Kfl2
Ksat2
Ksat1
Kfl1
_
_ __________
______
_ ________
_ ______
Kma – Ksat2 Φ(Kma – Kfl2)
Kma - Ksat1 Φ(Kma - Kfl1)
Equation 24
The dry rock bulks modulus is almost never measured directly. This requires a rock
sample and measurement in the laboratory. Even then, rock properties change with very
little moisture, resulting in uncertainty concerning the accuracy of the laboratory
measurement anyway. Dry frame is the bulk modulus for the rock when the pore fluid
exerts no pressure, when the pore fluid is allowed to escape the rock when the rock is
compressed. The dry rock modulus is typically determined indirectly when all of the
other Gassmann parameters are known.
A couple of historical observations: First, the Gassmann equation was not used for fluid
substitution until the mid-70’s, even though it had been derived in the mid-50’s. Second,
there is a more sophisticated fluid substitution equation, the Gassmann-Biot equation.
This more sophisticated equation accounts for the actual flow of the fluid through the
pores and is believed by some to be more accurate at higher frequencies. This claim isn’t
universally accepted because “Biot flow” doesn’t account for all dispersion observed
between laboratory measurements and low frequency seismic measurements.
Additionally Gassmann-Biot is dependent upon the knowledge of the permeability and
fluid viscosity. If the permeability is not used as an input, then the relationship is
mathematically equivalent to the low frequency Gassmann equation, even if the user
refers to the fluid substitution as a “Gassmann-Biot” fluid substitution.
A change in the saturation does not alter the shear modulus. This was an assumption
when derived by Gassmann, but has been subsequently shown to be a derivative of
Gassmann by Berryman et al. Shear modulus remains constant under small strains,
because the shear modulus is a rock property that has no volume change.
Fluid properties
Care must be taken in the estimation of the bulk modulus of the fluid. The fluid’s bulk
modulus (and, hence, its velocity) are calculated by Batzle et al from formation
temperature and pressure, plus specific gravity of gas (G), brine salinity, the API of the
hydrocarbons and the gas/oil ratio (GOR).
The following graph illustrates the influence of the temperature dependence on free gas
velocity. As a measure in the change in the fluid compressibility, Figure 19 shows the
25
change in the value of the p-wave velocity with the formation temperature for three
different values of the specific gravity of gas; defined as G. (Note that the specific gravity
of air is defined as 1.)
4500
Gas Vp (ft/sec)
8000 psi, (81.1% variation)
Gas specific gravity
1.2 G
0.8 G
0.6 G
2000
50
Temperature (degrees F)
450
Figure 19: VP as a function of temperature and gas gravity
Figure 19 shows an 81% variation in the velocity of gas as a function of temperature and
gas gravity that a single well might encounter from surface to total drilled depth.
Equation 22 is shown again below to demonstrate the impact of density on the final rock
velocity. Remember from the petrophysical discussion that density mixes volumetrically
with all other constituents. Figure 20 shows the change in gas density as a function of
changing pore pressure. The density shown below is the total rock density, but it is
somewhat modified by the gas saturation. Note that as density decreases with increased
gas saturation, that this tends to increase velocity. However in almost all cases the
decrease on the fluid modulus more than compensates for the density effect.
4 ⎤
⎡
VP2 = ⎢K + μ⎥ /ρ
3 ⎦
⎣
Equation 22
26
Gas Density (g/cc)
.5
1.2 G
0.8 G
200 Degrees F (335% variation)
0
2000
Pore Pressure (psi)
0.7 G
12000
Figure 20: Gas density as a function of pore pressure and gas gravity
Change in temperature drives a change in gas density shown in Figure 20. Note that gas
density may change by 335% in a single well in the study area. Rock density and thus
velocity will change as a result.
Sonic measurements to Seismic velocities
Frequency dispersion has been noted on numerous occasions by many authors (Castagna
& Hooper, 2000). Typically the higher frequency sonic measurements are higher
velocity than expected when performing fluid substitution from a hydrocarbon
(particularly gas) case to predict elastic properties of a 100% brine filled reservoir.
Figure 21, modified after Murphy’s ultrasonic and resonate bar measurements in the
laboratory present the problem.
27
Frequency Dispersion
ic
Son
6
cies
n
e
u
freq
5
Lo
gF 4
req 3
ue
nc
y
S
2
(H
z)
2.25
c
ismi
e
s
ce
urfa
1
.3 100
80
%
60
40
ra
Satu
Gas
2.0
0
Vp (km/sec)
lab
nic
o
s
a
Ultr
nts
me
e
r
asu
me
20
tion
Figure 21: Velocity, Gas saturation and measurement frequency
Note that sonic frequencies are on the cusp of significant velocity change between the
high frequency laboratory measurements and low frequency seismic measurements.
Older 25Khz BHC sonic tools tend to exhibit this problem to a greater degree than do
modern low frequency 1 to 2Khz dipole sonic logs.
Rock physicists have attempted to handle this dispersion with Biot flow, squirt flow, and
patchy saturation models. Brie used a power mixing law to mimic field observations and
Castagna et al used a Voigt – Reuss mixing exponent to handle patchy segregation.
The following method uses a Castagna or Brie treatment to fluid substitute from insitu
measurements with a sonic to a 100% brine saturated state. Figure 22 shows this first
step in the fluid substitution. For this example, the gas saturation is 20% and the velocity
is measured at dipole sonic frequency. The Castagna & Brie methods use the low
frequency Gassmann equation (Equation 21) with one change. The Kf, fluid modulus is
calculated by a semi-empirical mixing relationships between gas and brine. This
increases the fluid modulus which makes the model rock stiffer. If measured shear sonic
data is available, then the p-wave velocity may be approximated through the use of the
shear-wave log data and empirical vp – vs relationships. The exponents for the fluid
modulus is varied until the Vp/Vs relationship is optimized to match measured shear
velocity.
28
Sonic insitu to brine
ic
Son
6
ies
c
n
ue
q
e
fr
2.25
5
Lo
gF 4
req 3
ue
nc
y
ace
f
r
u
S
2
(H
z)
2.0
mi c
s
i
e
s
60
40
0
Vp (km/sec)
Dipole Sonic
Frequency
Observed sonic
measurement
20
tion
ra
Satu
s
a
%G
1
80
.3 100
After Murphy
Figure 22: Velocity, Gas saturation and measurement frequency at insitu
conditions
The velocity is now located at position 2, with 100% brine saturation at dipole sonic
frequency shown by Figure 23.
Partial Gas -> Brine Sonic
Observed sonic
measurement
ic
Son
6
cies
n
e
u 1
freq
c
ismi
e
s
ace
f
r
u
S
20
5
Lo
gF 4
req 3
ue
nc
y
2.25
2
2.0
0
Vp (km/sec)
2
40
n
ratio
u
t
a
.3 100
as S
%G
Figure 23: Velocity at brine filled conditions and at measurement frequency.
(H
z)
1
80
60
29
Sonic to Seismic (Brine)
2
ic
Son
6
ies
c
n
ue 1
q
e
fr
Em
pir
ic
3
5
Lo
gF 4
req 3
ue
nc
y
ace
f
r
u
S
2
(H
z)
1
80
.3 100
al
%
2.25
2.0
m ic
seis
40
0
Vp (km/sec)
Observed sonic
measurement
20
n
ratio
u
t
a
S
Gas
60
Figure 24: Velocity at brine filled conditions sonic to seismic frequency.
Batzle observed with sonic to low frequency laboratory measurements that 100% brine
saturated rocks didn’t appear to exhibit any velocity dispersion as a function of frequency
changes. Therefore the result at position 2 can be considered to be at seismic frequency
(position 3) as well. This is diagrammed on Figure 24.
Brine-> fractional Gas (Seismic)
Em
pir
ic
ic
Son
6
ies
c
n
l
ue
q
e
ca
i
r
t
f
e
r
eo
h
T
2.25
mic
s
i
e
ce s
urfa
2.0
5
Lo
gF 4
req 3
ue
nc
y
S
2
(H
z)
al
1
.3 100
80
%
40
0
Vp (km/sec)
Observed sonic
measurement
20
n
ratio
u
t
a
S
Gas
60
Figure 25: Velocity modeled at partial gas saturation and seismic frequency
30
Figure 25 shows that after corrections for frequency, Gassmann can be used to model any
fluid type or saturation conditions. These output velocities are typically used to make
decisions about seismic amplitude responses of surface seismic data.
Gassmann Fluid Substitution
Vs =
Shear Modulus
Density
Force
θ
VS2 = μ /ρ
Independent of saturation
1) Calculate shear modulus from sonic & density
2) Fluid substitute density
3) Hold shear modulus constant
4) Calculate shear velocity
Figure 26: Fluid substitution for measured shear velocity
Figure 26 shows four steps in fluid substitution of the shear velocity. Shear modulus is
required to remain constant for valid Gassmann relationships. That means that the shear
velocity changes with the square root of the change in density. Density changes linearly
with the change in the fluid density. This means that in fluid substitution from brine to
gas, rock density decreases modestly so shear velocity INCREASES slightly. At the
same time Vp decreases. This means that the shear contribution to change in Vp/Vs ratio
or Poisson’s Ratio is to magnify the impact of the Vp change slightly. Figure 27 shows
an example of multiple fluid substitutions for Vp, Vs and density in a sand shale
sequence with initial 100% brine saturation.
Vs
Vp
ρ
Gas
Brine
Clay
Sand
Porosity
Figure 27: Fluid substitution for Vp, Vs and density
31
Vp
ρ
Vs
-0.3
-0.2
Sand
R0
0
-0.1
_
brine
0.1
oil
0.2
gas
0.3
measured
Figure 28: Synthetic response for AVO “A” or zero offset, following
fluid substitution
Vp
ρ
Vs
-0.3
-0.2
Sand
R30
0
-0.1
_
brine
0.1
oil
0.2
gas
0.3
measured
Figure 29: Synthetic response for 30o angle stack, following fluid substitution
Figures 28 and 29 show synthetic reflection responses for zero offset and 30o angle stack
respectively for brine, oil and gas compared with the measured responses. Gas has much
greater amplitude separation than oil does in comparison with the brine filled response
based on this modeling.
32
Composite Matrix for Gassmann Fluid Substitution
Laboratory measurements have been published for a number of minerals. Required input
parameters are either Vp, Vs and density of the mineral or the calculated bulk modulus.
Vp is related to bulk modules, shear modulus and density as previously stated in Equation
22.
4 ⎤
⎡
VP2 = ⎢K + μ⎥ /ρ
3 ⎦
⎣
Equation 22
Again, as when velocity to porosity transforms were considered, recognition must be
made concerning the fact that the modulus of the mineral content cannot be explicitly
calculated. Only the limiting moduli can be determined. Equation 25 presents the Voigt,
isostrain mixing average. This is the least compressible theoretical mixing bound for “i”
different minerals with volume fractions “f”. The Reuss, isostress mixing bound is the
most compressible mixing bound shown by Equation 26. The Hill average is simply the
arithmetic average of the Voigt and Reuss bounds.
KV = Σ fiKi
Equation 25
KR = 1 / Σ fi / Ki
Equation 26
KVRH = (KV + KR) / 2
Equation 27
The Hill average is assumed to be “close enough” to the true mineral modulus that errors
in fluid substitution from using it are generally believed to be small. When individual
contributing mineral moduli are relatively similar, this is probably true.
However, there is another mixing relationship called the Hashin-Shtrikman bounds which
calculates a narrower range of differential effective media for the mineral matrix. This
media has a representative physical basis which is the volume of one mineral comprises a
central sphere and the second forms a spherical shell surrounding the first. Similar
spheres are modeled to fill all the solid (matrix) volume. The stiffest combination or
upper bound is determined to be when the stiffer of the two mineral is in the spherical
shell and the most compliant bound is when the stiffest mineral is in the central sphere.
This mixing relationship computes upper and lower bounds from cross-coupled bulk
modulus of the minerals with their shear modulus.
33
Berryman extended the original work from simply a combination of two minerals to
include as many minerals as are included in the rock, with known mineral volumes and
moduli.
Fluid Replacement Modeling - Velocity &
Density Summary
9 The process is fairly involved and requires quality petrophysical input, including
porosity, mineral volumes, fluid properties and saturations as well as
measurements of Vp, Vs and density at known initial conditions.
9 Frequency dispersion is not uncommon resulting in high frequency sonic
measurements having higher velocities than expected by the low frequency
Gassmann equation.
9 Multimineral analysis may be required unless mineralogy is known to be
relatively simple.
34
Section 4
Depth Conversion
Depth Conversion
As stated in a prior section, surface seismic reflection data are typically acquired,
processed and displayed in two-way-travel-time (2WTT). Well logs, geologic models,
and well plans are usually created and displayed in depth. An accurate velocity log is
needed to create a time-depth relationship to link these two ways of recording and
displaying subsurface data. A properly processed VSP which can be tied to the surface
seismic data usually produces the most accurate way to relate time to depth, closely
followed by quality synthetic seismogram ties to the surface seismic data. However, an
interpreter frequently must convert to depth without either of these preferred methods.
The following section addresses some methods that are utilized.
Checkshots
Checkshots may seem like the next obvious method to convert 2WTT to depth. There are
some considerations. First, a well must have previously been drilled to acquire a
checkshot. Also, similar to acquiring a VSP, a checkshot requires that a tool be lowered
down the wellbore. It is very unusual to acquire one in an old borehole.
Also while VSP and synthetic seismograms are cross-correlated with the surface seismic
data ensuring collocation of equivalent events. A checkshot yields a table of values of
depth, time, average velocity, and interval velocity between levels. This table is
generally corrected to a datum. However if there are several vintages of seismic data or
processing in an area, those seismic data may be processed to another datum. This means
that the velocity function implied by a checkshot survey may require shifting to match
any given version of seismic data.
TD curves, 3 wells editing & stretch compared to area checkshots
4000
3500
3000
2WTT
2500
2000
1500
1000
500
0
0
5000
10000
15000
20000
25000
Depth
Well 3
Well 2
Well 1
Well 4 checkshot
Well 5 checkshot
Figure 1: TD curves from sonic logs and two area checkshots.
1
Historically, before VSP surveys became common and synthetic seismograms were
poorly calculated, the closest checkshot was used for the first attempt at time-depth
corrections. Figure 1 shows dissimilarity between proximal checkshot data and synthetic
seismogram ties. As Figure 2 shows, even nearby checkshots may provide quite different
velocity profiles.
T D c u rv e s , 2 a re a c h e c k s h o ts
2400
2300
2200
2100
2WTT
2000
1900
1800
> 180 mils time
difference between
checkshots
1700
1600
1500
1400
10000
10500
11000
11500
12000
12500
13000
13500
14000
14500
15000
De p t h
W e ll 4 c h e c k s h o t
W e ll 5 c h e c k s h o t
Figure 2: Time vs Depth curves from two area checkshots. 180 ms difference
Of course, the assumption for using a single remote checkshot to tie a well to seismic
data or convert a map to depth is that the local world is comprised of horizontal isotropic
velocity layers as shown schematically by Figure 3.
Depth (m)
1800 m/s
2100 m/s
1900 m/s
6700 m/s
1950 m/s
6950 m/s
Figure 3: Horizontal layer cake model.
Of course the real world may be far more complicated that this ideal model. Also the
checkshot datum may or may not be available to the interpreter to reference the
appropriate starting “layer” of this model, but the availability of a datum should be
2
reviewed as part of the sensitivity of the depth conversion. Figure 4 shows a checkshot
computation form. The checkshot datum and correction should be listed if available.
Figure 4: Checkshot computation form.
0
0.5
1
2WTT sec
1.5
2
2.5
3
3.5
4
1500
2000
2500
3000
3500
Vi m /sec
Vi
Va
Figure 5: Interval and average velocity, typical spreadsheet computation
3
Figure 5 shows the typical plot made in spreadsheet form given time – depth paired data
from a checkshot. Note that the interval velocity (Vi) curve shows a number of sloping
line segments similar to the average velocity (Va) curve. The type of plot shown by
Figure 5 is not representative of the velocity structure implied by the checkshot data. The
computation is correct, but the plot is made with the interval velocity presented at the
bottom of the interval that it represents. Figure 6 below is more representative, where the
interval velocity is plotted over the entire interval from which it is calculated.
0
0.5
1
2WTT sec
1.5
2
2.5
3
3.5
4
1500
2000
2500
3000
3500
Vi m /sec
Vi
Va
Figure 6: Interval and average velocity, accurate representation.
Generally commercial software presents the interval velocity in appropriate squared off
format. When it does not, the interpreter must determine if Vi is plotted at the top, base
or center of the representative interval.
Another method of integrating checkshots in a basin is to “shoot” formation tops and
build a velocity relationship about the scatter around a best fit line through a number of
checkshots at a formation top. The issues with this method are numerous. First the
checkshots must be corrected to the same datum. After the plots have been generated,
QC is difficult to impossible. Second, a scatter plot of points contains no spatial
information. Distant points and proximal points are weighted similarly. Also there is an
implicit assumption that the velocities of the shallower layers are directly related to the
depth of the formation top.
4
Stacking Velocities
There are no wells available in unexplored basins to provide checkshots, VSP or
synthetic seismogram ties to provide quality time to depth relationships. However depth
estimates are still required.
In unexplored basins there are no wells at the time the initial time – depth (TD)
conversions are required. However to determine depths to various possible targets, plan a
well program and casing requirements and to estimate well costs, a TD conversion is
necessary. One of the first methods of determining a TD relationship from surface
seismic data used the Dix equation to calculate interval velocity (Vi) from normal
moveout velocity (VNMO). The subscripts 1 and 2 are for the upper and lower layer
bounding the calculated interval velocity. The parameter t, is the 2WTT to the respective
layers.
Vi2 = (VNMO22 x t2 – VNMO12 x t1 ) / (t2 – t1))0.5
Equation 1
VNMO is calculated from surveyed offset distance, X and the travel times, TX and T0 at
distance X and zero offset, respectively. The difference in travel time is called ΔT.
Equation 2 provides the relationship.
VNMO =
x
Tx2 − To2
=
x
Equation 2
2To ΔTNMO
VNMO is on rare occasion utilized directly for depth conversion. The assumption is that
VNMO is equivalent to average velocity (VAVG). This is specifically valid only in a
constant velocity earth.
Refraction Velocities
The Dix equation is valid for reflected travel times. However, another analysis method
using seismic refraction, permits determination of velocity from first arrivals. When an
impulse is generated in a homogeneous-elastic material, compressional, or longitudinal
waves propagate from the point of origin spherically. The wavefront velocity (VP) of a
compressional wave is related to material properties as shown by Equation 3.
Vp =
4 (Shear modulus)
3
Density
Bulk modulus +
Equation 3
5
According to Huygen’s principle, every point on a wavefront in a homogeneous-elastic
material generates a spherically propagating longitudinal wave of its own accord. A
raypath may be assumed to propagate from a point source to any point perpendicular to
the wavefront, as if it were a lengthening radius of an expanding sphere. Wavefronts
impinging upon a planar boundary between homogeneous-elastic materials of sufficiently
differing properties, initiate hemispherical propagation of longitudinal wavefronts in each
medium, resolvable to various raypaths.
Special consideration is given to the raypath that has the least travel time because it is
most easily measured by the relationship expressed in Snell’s law of boundary
refractions, modified from its original formulation for the electromagnetic spectrum.
Snell’s law of refraction, expressed by Dix and various others, is commonly shown in a
form similar to Equation 4. R and i are angles from the normal to the interface after
refraction and before impingence, respectively. Vn and Vn+1 are the velocities of the
initial medium and the medium into which the waves refract, respectively.
Vn
Sin i
_____
= _____
Sin R
Vn+1
Equation 4
Depth
The end member case occurs when R = 90o and therefore Sin ic = Vn / Vn+1. ic is the
maximum and critical incidence angle resulting in a least time raypath through the Vn+1
layer along the boundary interface. As the wavefront expressed by this raypath is
transmitted along the boundary interface according to Huygen’s principle, hemispherical
wavefronts are generated in the Vn velocity layer. These least time raypaths travel back
through the Vn layer at the same critical angle from the interface normal.
Distance
Figure 7: Generalize time-distance graph and associated seismic model
6
Figure 7 presents a generalized three layer case for the seismic refraction model where all
layers are planar, horizontal and composed of homogeneous-elastic materials. The lower
portion of the figure presents the raypaths and layers. The upper half of the figure
presents a time versus distance plot where the velocities of the three layers are equal to
the reciprocals of the line slopes of the distance versus time plot.
The thickness of the n-1 layer is given by the general Equation 5. The layers from
surface to greater depth are ordered from thickness Z0 through Zn-1. The intercept times
are designated as variable t with the same subscript as the layer of interest. The velocities
(V) also have the same subscripts as the thickness and intercept times.
Zn-1 = [Vn * Tan(Sin-1(Vn-1/Vn))/2] * [tn – 2Zn-2 *
Zn-2/Vn * Tan(Sin-1(Vn-2/Vn)) - . . . - 2Z0/Vn *
Tan(Sin-1(V0/Vn))]
Equation 5
Refracted arrivals may be distinguished from reflected arrivals on a typical shot profile.
Figure 8 presents a finite difference model of a layered earth with refracted, reflected,
mode converted and multiple arrivals.
Direct arrival
Crossover point
Refracted arrival
Figure 8: Finite difference seismic model showing refracted arrival.
7
The direct arrival travels horizontally from the source to the surface detectors. At the
offset of the crossover point, the first refracted arrival reaches the geophones prior to the
direct arrival. At increased offset the refracted arrival from the third layer may become
the first arrival.
Basic seismic refraction does have limitations. Snell’s law provides that first arrivals will
only return to the surface from critically refracted raypaths. Therefore only layers are
detected if they have progressively increasing velocity. Velocity reversals affect the
refracted raypaths and observed arrival times for deeper horizons, but can not be detected
by this method. One other undetectable type of layer is the appropriately called the
“hidden layer”. Even though this layer is faster than shallower layers, it is undetected
because it is some combination of too thin and insufficient velocity contrast to generate a
first arrival. The head wave from a deeper arrival reaches the geophones first. Because
of the possibility of these two undetected layers, the seismic refraction method has
obvious limitations. The calculated velocities will be reasonably accurate, but the layer
thickness will be in error and therefore so will the depths calculated from the summation
of thicknesses.
There is one additional consideration. Refraction methods require relatively long offsets
to record first arrivals at even moderate depths. Therefore in a practical sense it is
generally limited to shallow and moderate depths of investigation. However extensions
of the technique based on the Generalized Reciprocal Method of seismic refraction
interpretation are frequently used to provide low frequency statics solutions for seismic
reflections surveys where applicable.
Seismic horizon to well top methods
If there are a number of wells that are located in the same geographic area as a 3D
seismic survey, but many of the wells cannot be accurately tied with VSP or synthetic
seismogram methods, a relative correlation method is commonly utilized. Starting
correlations may be a subset of available wells that do have synthetic seismograms, VPS
ties, or a checkshot. Based on this initial correlation, a seismic horizon is interpreted to
be close to the 2WTT expected for formation tops observed in wells. There are a number
of paths available with this form of data.
Velocity mapping
While historically more difficult, the common use of computer workstations with
mapping and grid math capabilities, makes this method relatively accessible. A velocity
map is generated. A 2WTT map is also generated. Subsequently a depth map can be
created with grid mathematics by multiplying the two grids and properly handling the
two way travel time correction.
One major assumption is that well density adequately samples the variation or lateral
change in velocity. This could also be stated that the velocity field varies slowly and
smoothly between well penetrations. Figure 9 presents an example of a velocity map
8
matching a 3D seismic survey. This map is created by dividing 2WTT into the formation
depth at well points and gridding the map from the well points.
One mile
Map of well
velocities
Va/2 (ft/sec)
Figure 9: Velocity map contoured from well control
Figure 10 provides the resulting depth map.
One mile
Depth map (ft)
Figure 10: Depth map from 2WTT and velocity maps
One advantage of this method is that it matches the depths observed in the input wells
quite closely but somewhat dependant on grid cell size. The disadvantages include rapid
9
lateral velocity changes in the section above the map horizon or too few wells to provide
control for the velocity map will result in an inaccurate depth map.
Velocity – depth trend
The basis of this method has its roots in compaction trend observations that were
fundamental to early pore pressure prediction techniques. The observation is that in
compaction driven Tertiary basins, there is a linear relationship between interval velocity
and depth. One of the earlier expressions of this relationship is given by Equation 6
where Vi, V0, k, and Z are interval velocity (or instantaneous velocity), surface velocity,
velocity gradient and depth respectively.
Vi = V0 + kZ
Equation 6
However, given seismic horizons will be created in two-way travel time and Equation 6
utilizes velocity and depth to determine a linear gradient and intercept, some additional
manipulation is required before we can convert time to depth. The following assumption
is that we deal with “half-velocity” instead of correcting for the two way travel path of
reflection data and then correcting back later. If we use the definition of velocity plus
Equation 6 above, we can derive Equation 7 below to calculate depth from the linear
coefficients and 2WTT. T is two-way-travel-time (2WTT) and the other parameters are
identical to those described by Equation 6.
Z = (V0T) / (1 – kT)
Equation 7
Equations 6 and 7 have been used with as little as a single checkshot survey to create a
working relationship for an area. In the context of a working data set with 2WTT and
well depths, the method is also applicable. Although this method may produce some
Time – depth trend
There is another relatively common method to predict depth from 2WTT surfaces and a
set of well tops providing TD pairs. The method simply regresses time against depth. It
is common because of relative simplicity of operation in grid math. The slope is
multiplied by the 2WTT and added to the intercept. Frequently the method appears to
obtain quite linear relationships based on standard R-Squared measurements. It is not
unusual, however to regress a negative and obviously non-physical intercept, realizing
that the actual TD relationship must intersect the origin. Predictably, the method
virtually never honors the actual well depths at the wells.
10
Residual error map
This method is a combination and extension of the velocity mapping and prediction trend
methods. It requires sufficient wells to produce a map and probably should not be
attempted unless there are enough wells to consider applying a velocity map method for
stand alone time to depth conversion.
First depths are predicted via with a Velocity – Depth method; Time – Depth method; or
other velocity prediction. These predictions do not tend to honor the depths at all the
wells. It is a depth map, but perhaps not sufficiently accurate to use for spotting well
locations. This map can be combined with the well depths in one of two ways.
CCK method
Collocated cokriging (CCK) became semi-mainstream when GSLIB (Geostatistical
Software Library and User’s Guide) was published in 1992. This method allows one
variable with high spatial continuity (and reduced accuracy) to be used to constrain
another variable with low spatial continuity and high accuracy. A common usage is to
combine 3D seismic attributes with well data. Because the source code was also
published, many geoscience workstations have some version of incorporated in their
mapping packages, but Spartan help in using it. The method attempts to minimize the
covariogram between the two data types. The result is that the detail in the 3D map based
on the seismic time – structure is warped to closely fit the points of control provided by
the wells.
Error isopach method
The second method is functionally similar to CCK and based on the ancient subsurface
method of combining a two structure maps and the isopach between them on a light table
to improve the structure map on the deeper horizon. Because the deeper structure map
frequently has fewer wells penetrations, providing structural elevation data than the
shallow map, the shallow map generally contains more structural detail. In practice with
a light table, every point where an isopach contour crossed a shallow structure map
contour, a control point was posted on the deep map. These points in addition to the well
tops were utilized to contour the deep structure map. The assumption is that the interval
thickness varies more slowly than the structure. This is particularly valid if most folding
and faulting occurred after deposition of the shallow horizon.
This method has the same initial step as the application of CCK. A trend prediction is
made based on seismic time – structure, resulting in a depth map. This method is
typically uses grid math in a geoscience workstation. This map will have errors at most
wells as stated previously. Second, the depth predictions of the grid are exported from
the workstation only at the well locations. These can usually be imported into a
spreadsheet with the formation tops at the wells and differenced. Then these “error
isopach” values are imported into the workstation and gridded. Finally the error is
removed from the 3D seismic estimate of depth, resulting in a map that honors the wells
and the trends within the seismic data that are below the spatial sampling of the well
control.
11
The following four figures show the process. Figure 11 is the 3D time structure map.
Figure 12 is the trend map in depth from the regression between seismic time and well
tops. Figure 13 is a map of the residual error and Figure 14 is the final depth map.
Figure 11: 2WTT map of structure
Figure 12: Initial Depth map; Regression Trend
12
Figure 13: Residual Error of Initial Depth map
Figure 14: Final Depth map
13
Integrating Seismic Stacking Velocities with
Well Control
The location of well data is typically beyond control of a geoscience interpreter.
However when 3D seismic data is available, stacking velocities or imaging velocities are
available for much reduced cost to provide control in portions of the map areas where
there are no wells. There is a contrast between using seismic velocities in a rank wildcat
basin and in mature exploration provinces. The accuracy requirement for final depth
maps in a mature exploration area is typically at the edge of the ability of technology and
data quality to provide. Figure 15 shows a seismic time – structure map.
Figure 15: 2WTT map of structure
Figure 16 shows the velocity field from the seismic processor based on a tight grid of
stacking velocities. No well control is directly involved with this depth map at this stage.
Figure 17 shows the depth map created with the seismic velocity, horizon map and the
2WTT from the seismic horizon. Note the “egg carton” appearance with questionable
highs and lows. Note that some of these map closures have been previously tested by
well penetrations.
14
Figure 16: Processors Vnmo map at horizon
Figure 17: Depth map from Vnmo and 2WTT maps
15
Figure 18 presents the regression of values from the depth map (Figure 17) on the X axis,
predicting the well tops on the Y axis. Approximately 64% of the variation in depth is
explained at the wells using stacking velocities. Additionally, there should be a 1:1 line
on the scatter plot, however the slope is approximately 0.61 with an intercept of -1370.
Depth Conversion
Vnmo
y = 0.6129x - 1369.6
2
R = 0.6364
-3600
Well Depths
-3550
-3500
-3450
-3400
-3350
-3300
-3300
-3350
-3400
-3450
-3500
-3550
-3600
prediction
Vnmo Depth
Linear (Vnmo Depth)
Figure 18: Regression depth map from Vnmo and well tops
Figure 19 is a map showing the results of the error isopach method. While this map
matches the formation tops in the wells quite closely, the noise in the underlying Vnmo
map suggests that depth error is significant. Improvement is required in the velocity
processing before this method can be utilized with this data set.
16
Figure 19: Depth map from Vnmo map and well tops
Depth Conversion
9 Checkshots are useful, but not as reliable as VSP or Synthetic Seismogram well
ties.
9 Refraction velocities are generally restricted to applications at shallow depths
because of acquisition limitations.
9 Seismic horizon to well top relationships match well control, but datum handling
and areal sampling of velocity variation impacts accuracy.
9 Stacking Velocities can be used to estimate the velocity field between wells, but
they require quality processing and integration with available well control
9 Velocity mapping; trend methods and residual error maps provide quality TD
control in mature areas.
17
Section 5
Pore Pressure
Anisotropy
Pore Pressure
Sonic logs were first used in 1965 to quantify pore pressure by Hotmann and Johnson.
Edward Reynolds utilized seismic velocities for the same purpose, using what is known
as the equivalent depth method.
The Hotmann method resulted in a curvilinear relationship between the difference
between a trend line for sonic travel-time decreasing in shale intervals as depth increased
and the observed sonic travel time. The trend was subtracted from the observed value. If
the difference was greater than zero, then increased pore pressure was indicated. The
Hotmann method required utilization of graphical methods, picking pore pressure
gradient of psi / foot from the difference plot. The paper also provided a semi-log
plotting format, however regression treatment of the digital Hotmann data suggested that
an exponential relationship would provide the best fit.
There are a large number of relationships that have been developed to relate velocity,
shale porosity or resistivity to pore pressure. Although the formulations are different,
they have had one assumption in common until the advent of Centroid theory. The
pressures found in porous reservoir are assumed to be the same as the pressure in the
adjacent shale section. Therefore, pressure is measured in permeable formations and
related to shale properties at approximately the same depth.
The Centroid effect
Centroid theory modified the earlier assumption to the extent that the place where the
reservoir pressure and surrounding shale pressures were assumed to be in equilibrium
was as the structural center or centroid of the reservoir. Updip from the centroid, the
reservoir pressure exceeds the prediction based on the shale pressure. Downdip, the
opposite is true and measured or reservoir pressure is less than predicted by adjacent
shale properties.
Compaction Disequilibrium (Undercompaction) vs Unloading Mechanisms
The mechanism for generation of pore pressure could largely be attributed to the inability
of the formation to compact. The theory was that there was inadequate plumbing of
permeable intervals to allow the shales to release their pressures and achieve a normal
pressure gradient. This is a viable mechanism to generate abnormal pore pressures, but it
has been supplemented by the theory of unloading. Unloading is proposed to be related
to fluid expansion, or hydrocarbon generation.
Bower’s formula
Glen Bowers proposed a method to deal with unloading. Hysteresis has been observed in
laboratory experiments that measure velocity as a function of loading and unloading.
Typically, when plotting effective stress against velocity, the slope is steeper when
loading than when unloading. The process is similar to stress hardening of metal. A
Bowers loading and unloading graph is shown by Figure 1.
1
Compaction Curve with Unloading
15000
Vp (ft/sec)
Plastic
ading
typic al unlo
tic
as
l
E
10000
5000
0
5000
10000
15000
Effective Stress (psi)
Figure 1: Bower’s relation of Effective Stress to Velocity
Equation 1 is the formula for Bower’s “Virgin” loading curve relating velocity to
effective stress. V, σ, C, A, and B are velocity, effective stress, and coefficients. In
Bower’s publication, C is shown in to be 5000, velocity is in feet / second and effective
stress or confining stress is in units of psi. In practice, the coefficients are optimized to
best predict shale velocity from adjacent reservoir pressure, given overburden pressure.
V = C + A x σB
Equation 1
Pore pressure is related to effective stress and overburden pressure by Equation 2. OB is
overburden, PP is pore pressure, and σ is effective stress, all in units of psi. The OB
pressure is determined from integrating the overburden density from surface to the depth
of interest.
PP = OB - σ
Equation 2
Via personal communication, Bowers has updated his published formula. This is a
combined method using Bower’s suggestions. It has some advantages. The maximum
velocity is limited. It can be set to mineral matrix if known. The minimum velocity is
similar to the coefficient C in Equation 1. Effective stress is the same in terms of psi and
2
there is a curvature term B. This equation has the form of an exponential with linear
modification.
Equation 3
V = VMA – (VMA – VMIN) x e (-σ / B)
Velocity to Pressure Calibration
Following is an example of multiple wireline pressure measurements related to adjacent
shale velocities. Effective stress is calculated from Equation 2 where pore pressure is
obtained from the wireline MDT pressures and OB is from an integrated density. Figure
2 shows both a loading and an unloading curve. The loading curve asymptotically
approaches VMA at large effective stress. This corrects an issue with the prior
formulation which can project to velocities greater than matrix velocity at high effective
stress. Comparison of Figures 1 and 2 provide insight.
Bower's New Compaction Trend
20000
Vp (ft/sec)
15000
10000
5000
0
0
5000
Data
Vma
Unloading
10000
15000
Effective Stress (psi)
Vmin
Bad Data ?
Best Compaction Fit
Figure 2: Bower’s calibration of Effective Stress to Velocity
3
Pore Pressure Calculation
The basic procedure is as follows: Overburden pressure must be calculated either from
nearby wells or by estimation as presented in the well log QC and editing section.
Effective stress is estimated from calibrated velocity with Equation 4. Equation 4 is
simply Equation 3 solved for effective stress. The coefficients are the same.
σ = - Ln((V – VMA)/(VMIN – VMA)) x B
Equation 4
The unloading case uses the same form of Equations 3 and 4 which is designated for the
loading or virgin compaction curve; however the coefficients VMIN and B will be
different and must be calibrated separately.
Equations 5 and 6 provide the formulation for calculating overburden pressure. Equation
5 provides the incremental pressure for an interval based on the ith thickness of layer Zi
(in feet) and density (g/cc) of the layer. Incremental Overburden Pressure (Pi) is in psi.
Pi = ρ x 0.433 x Zi
Equation 5
Equation 6 is simply the integration or running sum of Pi from the surface to depth. It is
also in units of psi.
OB = ΣPi
Equation 6
Finally, pore pressure is predicted by Equation 2 using input from Equations 6 and 4.
Use of Seismic velocities for Pore Pressure
Considerations: The velocities come from a different source. Seismic sampling is coarse
compared to sonic depth sampling measurements. Density must be estimated from
velocity or inversion. Loading or unloading must be determined from the velocity
profile, other wells in the area or experience. Because the intervals are coarse, they may
contain a mixture of shale and sand. However the relationships used in calibration are
based on pure shale velocities. If the sand shale ratio can be estimated from inversion or
sequence stratigraphic analysis, then superior application of the calibration is possible.
4
Trend analysis and regressions should be based on the midpoint of the coarse interval
defined by the seismic resolution of velocity, not the top or base of the interval.
Note that there is an implied assumption that the pore pressures measured in the wells are
calibrated to the appropriate 2WTT on the seismic data. If this has not been properly
handled, then the velocity relationship to pore pressure calibration will be related to an
incorrect depth. A leading weakness of many pore pressure studies is the failure to
demonstrate that the seismic data have been adequately tied to the calibration wells.
Pore Pressure from Seismic Data
9 A calibration of Effective Stress to velocity must be performed locally or in an
analogous pressure environment.
9 Calibration requires adequate ties of pressures and densities in wells to the
seismic data.
9 Overburden pressure is essential to determine pore pressure. This requires
understanding, measurement or estimation of the density profile.
9 Quality seismic velocities are required as input.
Anisotropy
Anisotropy is one of the major issues impacting the use of surface seismic velocity data
in pore pressure prediction. Some of the difference between the velocities that optimize
well ties with seismic data and depths from VNMO is caused by anisotropy. This is caused
by the fact that horizontal, or bedding parallel velocities are typically faster than vertical,
or normal to bedding velocities. The topic of anisotropy could be extensively developed
over days. This will be an abbreviated version. There is a key relationship based on the
formulations that we have already developed.
An essential parameter, δ (delta) is derived from the work on transverse isotropy by Leon
Thomsen. While Thomsen’s work derived from the compliance tensors has been
described as applicable for “weak” anisotropy, the δ parameter is also described with
Equation 7 and subsequently found applicable for more general application. VPNMO is the
stacking velocity or “short spread moveout velocity”. VPO is the velocity normal to
layering. This is the well velocity or vertical velocity when considering a horizontal
layered case.
VPNMO = VP0 x (1 + δ)
Equation 7
5
Moveout velocity is NOT equal to horizontal velocity as is or has been commonly
assumed. Horizontal velocity or VP90 is related to another Thompson parameter ε.
Thompson assures us that it is very rare in a real rock to find that δ equal to ε epsilon. ε
is provided by Equation 8 where VP90 is the horizontal or at least the bedding parallel
velocity.
VP90 = VP0 x (1 + ε)
Equation 8
Lastly for this discussion is Equation 9 which shows angular dependent velocity, VP(θ).
It introduces the angle from bedding plane normal, θ and the term eta or η’. In practice,
eta is determined by varying the parameter until the far offsets on the gather are as
horizontal as possible. Equation 9 is approximate
VP(θ) = VP0 x (1 + δ(sin2θ + η’sin4θ)
Equation 9
By examining equation 9, the reader will better appreciate that imaging, depth conversion
and pore pressure prediction are impacted by simple transverse isotropy.
Anisotropy
9 The parameter delta relates vertical velocity to moveout velocity.
9 Delta can only be accurately determined when a well has been drilled to measure
vertical velocity or measured in the laboratory.
9 Horizontal velocity is typically faster than vertical velocity, but very rarely related
to delta.
9 Angular velocity can be approximated by a Taylor expansion series for weak
anisotropy.
6
Section 6
Impact of Noise on Velocities
Impact of Noise in Velocity estimation
As shown in the previous section, velocity picking can be subjective when horizons and
well velocities are not provided to the processor. Other issues can also be deleterious to
optimum choices of velocity during the picking stage. One of these obvious issues is
noise. This section discusses random noise, although coherent noise degrades the
processor’s ability to accurately estimate velocity in the section also.
There are several methods of noise filtering which have been available to processors
which work in the prestack domain. These methods transform the data from the time
domain into the frequency or other domain so that the noise is separated from the signal
and can be attacked. Unfortunately, these methods have the undesired side effect of
mixing the signal across the offsets which is deleterious to prestack attributes (AVO,
velocity). FK or velocity filtering is known to move energy across the offset range.
Radon filtering, while less damaging at larger offsets, can cause significant damage to the
prestack data in the presence of some forms of noise. Radon also is less useful at small
offsets where it is most needed for suppression of multiples.
Fortunately, there is a new class of prestack noise tools which operate in the time domain
and some of these have been proven to not mix the signal. As shown in previous
sections, the placement of the velocity picks in a given semblance panel is somewhat
subjective, but depends on identifying high amplitude events which stack together to
form strong semblance events that stand out from the surrounding events. In the presence
of high noise, the response of these events is obscured making identification difficult or
impossible. Figure 1 shows velocity analysis with and without conditioning. Figure 2
shows the comparison between premigration gathers which have been conditioned with
the new method and their counterparts which have the original noise in them, along with
the difference panel which shows the noise removed.
1
Figure 1: Velocity panels showing original gather (top) and conditioned gather
(bottom)
Figure 2: Raw gathers (upper left), conditioned gathers (upper right) and
difference (bottom)
2
Impact of noise in velocity estimation
9 The quality of velocity estimation depends on the level of noise in the data
9 Noisy data impedes the processor’s ability to accurately pick noise from seismic
data
9 Various tools are available to address noise issues in gathers
9 Most are detrimental to prestack attributes because they operate in domains other
than the time domain
9 A new class of filters operates in the time domain and accurately removes noise
from the gathers
Near Surface Statics
Changes in surface topography as well as lateral variation in near surface velocity
produce near static time shifts in the raw seismic traces. These become more apparent
after NMO correction of the data. Early work from the late 1960’s through the early
1980’s developed a surface consistent model of these time static anomalies observed on
specific seismic events.
Δti is the time shift of trace i with shot point location si; receiver or geophone location of
gi; midpoint location Mi and offset Oi. Equation 1 determines the midpoint location.
Equation 2 determines the shot to receiver offset.
Mi = (si + gi) / 2
Equation 1
Oi = (si - gi) / 2
Equation 2
Equation 3 provides the Δti is the time shift of trace i where S(x) and G(x) are shot and
receiver statics corrections at surface location x; and Y(x) and R(x) are a structural term
and residual NMO corrections, respectively.
Δti = S(si) + G( gi) + Y(Mi) + R(Mi) x Oi2
Equation 3
3
The process is described as the solution of a set of under constrained, but over determined
system of equations. Cross correlation finds the most likely maxima between relative
time shifts between similar traces to obtain the time anomalies. These early methods
were limited by signal to noise, initial NMO applied, fold of data, and dominant
frequency of data.
Improvement in statics solutions was obtained when methods allowed some stacking to
improve the base trace. This permitted better solutions in poorer signal to noise data sets.
This process is iterative, resulting in improved statics solutions building a better stacked
trace and subsequent superior model of the static anomaly. Figure 3 presents a scheme
designed to utilize these methods.
Figure 3: Use of cross correlated, stacked traces to improve signal to noise handling
for surface consistent static solutions. (after Ronen and Claerbout)
Considerations: Some seismic processors believe that starting with too many NMO
velocity functions degrades the statics solution. This thesis contends that multiple NMO
corrections will partially adjust the traces in such a way that the statics solution will not
be optimum. They contend that sparse velocities should be used for the initial iteration of
surface consistent statics, followed by better velocity picking and a second round of
statics.
4
Statics Summary
9 The quality of the static correction depends in part on signal to noise in the data
9 Higher fold data improves the determination of a base trace and subsequent static
solutions.
9 Iteration between statics and NMO velocity picking results in a superior product if
allowed to converge.
5
Suggested Workflow for Velocity
Estimation in Processing
Previous sections document many issues in the accurate estimation (picking) of velocities
during processing. These sections indicate how some of the issues can be ameliorated via
close cooperation between the processor and interpreter during the data processing phase
of the project. Accordingly, below is a suggested workflow which helps to optimize the
flow of information between these two disciplines. This is not intended to be a rigid
workflow, as with everything in exploration/production seismology a common-sense
approach is indicated. The prescient geoscientist will use this workflow as a guide and
adjust it to fit the circumstances of the project.
1) initial meeting between the processor and interpreter to review project goals such
as:
a. target depth/time, information expected from processing, i.e. AVO,
velocity, etc.,
b. further products after initial processing, i.e. inversion, pore pressure, etc.
c. Review any available seismic data in the area to determine likely issues
the data will present during the processing.
d. Review available well control and status of logs, i.e. edited acoustic logs,
velocity surveys or VSP’s etc.
2) Initial geometry/trace editing/gain/signal processing steps done by processor
a. initial pass of velocity estimation and statics
b. remove noise using appropriate method
3) Post-stack migration using best velocity available – send to interpreter for
preliminary tie to wells and horizon picking – transfer horizons back to processing
4) second pass of velocities using filtered data, this time incorporating well control
and horizons
a. apply updated velocities to unfiltered data, new pass of residual statics
5) Apply noise removal to updated dataset
6) Choose and migrate target inlines for PSTM velocity estimation, pick PSTM
velocities using horizons and wells to guide picking.
7) Migrate volume using interpolated PSTM velocity function.
8) Use residual velocity program to refine velocity volume from the interpolated
velocity function used in step 7.
9) Provide PSTM gathers and final velocity volume to interpreter for interpretation.
1
Workflow for Velocity Estimation
9 Initial meeting between processor and interpreter provides venue for processor to
absorb goals for the processing in terms of anticipated final use of his work
9 Details of well locations, available edited acoustic logs for velocity control,
velocity surveys and VSP’s, etc should be shared
9 Any existing seismic in the area should be reviewed for expected issues
9 Workflow should incorporate appropriate noise abatement early on to maximize
processor’s ability to pick most accurate velocities.
9 Horizons and wells should be incorporated to ensure picked velocity profile
honors this control.
9 Use of residual velocity program to utilize all seismic velocities available
2