CHAPTER TWENTY-ONE: SEISMIC PETROPHYSICS 1
Editing and Modeling Logs

Table of Contents
Introduction
21.00: Introduction To This Chapter
21.01: Seismic Petrophysics and Seismic Modeling
21.02: Seismic Petrophysics and Well Log Modeling
21.03: Logs Used for Seismic Petrophysics

Log Editing Models
21.04: Log Editing Concepts
21.05: Seismic Check Shots
21.06: Editing Sonic Logs With SRS and VSP Data
21.07: Modeling Sonic and Density Logs With Trend Data
21.08: Modeling Sonic and Density Logs From Resistivity Data
21.09: Modeling Sonic and Density Logs From Neutron and Gamma Ray Data
21.10: Modeling Sonic and Density Logs With Regression

Lithology and Fluid Replacement Models
21.11: Modeling Sonic and Density From Log Response Equation
21.12: Modeling the Sonic Log in Vuggy Porosity
21.13: Modeling the Sonic Log Response From Biot-Gassmann Equations

Using the Log Models
21.14: Integrating the Sonic Log
21.15: Acoustic Impedance and Reflection Coefficients
21.16: Quicklook Log Analysis Calculations for Geophysicists
1. Nomenclature for Log Analysis Models
2. Nomenclature for Seismic Models
21.17: Elastic Properties of Rocks
21.18: In Conclusion
21.19: Exercises for Chapter Twenty-One
21.20: Bibliography for Chapter Twenty-One

TABLE 21.01: Elastic Properties of Rocks

Continue to Chapter Twenty-Two

Publication History: This Chapter formed part of Chapter Ten of Volume Two of The Log Analysis Handbook, a self published series of course notes covering geological and geophysical aspects of log analysis. First published in 1978, revised 1985, 1993. Completely revised and re-organized for this electronic edition Sep 2002. Section 21.11 was also published as "Determination of Seismic Response Using Edited Well Log Data" by E. R. Crain and J. D. Boyd, CSEG, October 1979. ** Best Paper Award, CSEG, 1979** Sections 21.00 thru 21.03 rewritten Jun 2003 and published as "The New Role of Petrophysics in Seismic Interpretation", CSEG Recorder, Sept 2003.

CHAPTER TWENTY-ONE: SEISMIC PETROPHYSICS 1
Editing and Modeling Logs

21.00 Introduction To This Chapter
The role of petrophysics in seismic interpretation has taken a major leap forward in the past ten years, resulting from important advances in seismic data processing techniques, particularly seismic inversion, attribute analysis, and amplitude versus offset methods that showed we could estimate reservoir properties from such data. Coupled with the recent advances in dipole shear sonic logging, new vistas in seismic interpretation, dubbed seismic petrophysics, have opened.

Geophysical well logs suffer from many borehole and environmental problems that need to be repaired before being used for calibrating seismic models or seismic interpretations. A primary aim of the geophysicist/petrophysicist is to create a synthetic seismic trace from EDITED log data that accurately represents the seismic response of the subsurface. This is accomplished by editing, repairing, or reconstructing the log data. Using unedited logs for seismic purposes is a waste of time and money and, in the worst case, can lead to very expensive exploration and development mistakes.

If the synthetic seismic trace is a good representation of the real seismic response, then the edited logs can be used effectively as aids to interpretation of the advanced seismic products.

Consequently, the role of the petrophysicist has also evolved; she must now be competent in log reconstruction as well as conventional log analysis, and must understand the petrophysical needs and limitations of the inversion, attribute, or AVO results. Unfortunately, logs are not perfect measures of in-situ rock properties and seismic data is severely band-limited compared to log data, so there are many compromises to be made. A significant change in mindset is also needed, as most of the log repairs (with the exception of fluid replacement) take place in the non-reservoir intervals - intervals that are not usually of interest to petrophysicists.

Geophysicists engaged in seismic interpretation seldom use logs to their full advantage. This sad state is caused, of course, by the fact that most geophysicists are not experts in log analysis. They rely heavily on others to edit the logs and do the analysis for them. But, many petrophysicists and log analysts have no ides what geophysicists need from logs, or even how to obtain the desired results. That's a particularly vicious "Catch-22".

Education, practical solutions, appropriate software, and practice are the keys to success. In order for geophysicists and petrophysicists to communicate well, each must know something of the other's specialty. Chapters Twenty through Twenty-Five provide theory, practical methods, and case histories to accomplish this goal.

Chapter Twenty describes in detail how sonic and density logs are recorded and how the elastic properties of rocks are derived from these measurements. This Chapter (Chapter Twenty-One) describes practical log editing and repair procedures. Chapter Twenty-Two contains Case Histories showing results of these methods. Chapters Twenty-Three through Twenty-Five deal with how these transformed logs and synthetic seismograms are used to calibrate inverted seismic sections, seismic attribute interpretations, and amplitude versus offset studies..

21.01 Seismic Petrophysics and Seismic Modeling
Seismic petrophysics is a term used to describe the conversion of seismic data into meaningful petrophysical or reservoir description information, such as porosity, lithology, or fluid content of the reservoir. Until recently, this work was qualitative in nature, but as seismic acquisition and processing have advanced, the results are becoming more quantitative. Calibrating this work to well log - ground truth - can convert the seismic attributes into useful reservoir exploration and development tools. Since there are an infinity of possible inversions, it is pretty important to find the one that most closely matched the final edited logs or the computed results from those logs.

A seismic petrophysics study aimed at quantifying porosity is shown in Figure 21.01.

FIGURE 21.01: Seismic petrophysics study for porosity

This example used a geo-statistical package to distribute the dense "fuzzy" seismic attribute data between the sparse, "accurate" well log data. The logs, or log analysis results, in turn are calibrated to core, well test, and production data before being used to control seismic interpretation. The use of geostatistics to map seismic attributes onto well logs is a relatively new phenomenon

21.02 Seismic Petrophysics and Well Log Modeling
Unfortunately, it takes a fair amount of effort to compare seismic results to log data. The logs will usually require some kind of editing or modeling or both. Comparison of seismic results to log data may indicate that further processing of the seismic is needed, and the calibration cycle is repeated, often several iterations are needed. In other cases, it is the logs that need further editing.

Log modeling or editing is required because logs don’t see the same rock and fluid mixtures that the seismic signal sees. Drilling fluid invasion removes gas or oil near the wellbore, replacing it with water and altering the sonic and density log response from the reservoir's undisturbed values. Compensating for invasion is called "fluid replacement". Fluid replacement calculations are also used in "what-if" scenarios to see what a gas filled reservoir might look like on seismic. Such models are usually run post-mortem, after a lovely seismic bright spot was drilled to find an equally lovely porous water zone. Maybe the models should be run BEFORE drilling?

The author and John Boyd presented a practical solution for fluid replacement in 1979, based on the log response equation and a "pseudo-travel time" for typical gases. Since then, at least a dozen, more rigorous but less friendly, solutions have been published: Castagna, Greenberg and Castagna, Aki and Richards, Batzie and Wang, Toksoz et al among others. Most are based on extensions of early work (late 1950's) by Biot, Gassmann, and later, Domenico. The final tally on fluid replacement calculations for gas effect on the sonic log is not in, especially in shallow, unconsolidated, or underpressured reservoirs.

Fluid replacement calculations for the density log are straight forward, with no pitfalls if the gas or oil PVT properties are known. How well do you know the reservoir engineer down the hall?

Mechanical or chemical rock alteration due to drilling usually reduces sonic velocity and density in the environment measured by the logging tool. This effect is somewhat subtle but pervasive or it can be catastrophic as in hole breakouts. It can be repaired by using information from other log curves (in the case of bad density data), or checkshot or VSP data to calibrate the sonic log. But many common sense rules for using checkshots are ignored because the software doesn't think like a human petrophysicist.

Acoustic frequency differences have to be accounted for, especially when shear velocity is measured. High frequency shear velocity (lab measurements and sometimes sonic log data) is faster than low frequency (seismic) data. Anderson's 1984 paper provides useful information but is weak on specific recommendations.

Poor log response due to bad hole condition or faulty logs may be an even more serious problem, as in Figure 21.02 at left. Check-shots, offset well data, other logs, and common sense are used to correct for this.

FIGURE 21.02: Rough sonic log corrected where it needs it

The log should be edited only where it needs it using common sense rules grounded in local and regional trends. Few practitioners have hip pockets full of sonic and density trend data applicable to their current projects.

Again, at least a dozen authors have provided more or less practical solutions, such as Ausburn, Faust, Smith, Fischer and Good, Crain and Boyd, Patchett.

Calibration methods come in three flavours: good, bad, and really ugly. Block shifting a log is really ugly. Rescaling and delta-T minimum methods are better but still ugly. Discreet editing where the log needs it, or more sophisticated curve fitting techniques based on other logs, are pretty good approaches. The ugly methods are fast and mostly useless, as most of the false reflectivity is still there. The good methods take more effort, but you get what you pay for.

In other cases, no appropriate logs exist, so sonic and density data have to be created by transforming some other available log. Most of the methods used to repair bad hole effects will also generate complete sonic or density logs. In the worst case, a set of geological tops, lithology descriptions, and an offset well log will suffice, especially if only the density log is missing.

Some models are made by "cut and paste", for example thickening or thinning a reef or pinching-out a sand bar to see what happens to the seismic signature. Splicing realistic data from one well to another in a geologically sensible manner can create any number of plausible models. The more models you create, the more likely you will find one that matches your seismic.

Smoothing and filtering may also be performed on raw or edited logs to extract only those frequencies that are likely to be recorded in real seismic data. Cut and paste, and filtering, are fairly obvious operations and are not dealt with further here.

A competent petrophysicist working closely with the geophysicist can provide the needed expertise to solve these problems and generate useful log data. When integrated with the geologist and reservoir engineering members of the team, very credible interpretations will result.

21.03 Logs Used to Aid Seismic Petrophysics
The two logs most used by geophysicists are the sonic (also called acoustic) log) and the density log, because these two rock properties determine the acoustic impedance and hence the reflection coefficients of the rock layers. A synthetic seismogram can be calculated from these data.

Raw logs should NEVER be used for this purpose - editing and modeling are nearly always required.

Most other log curves are useful to the geophysicist. For example, the neutron, density, photoelectric effect, and spectral gamma ray (both natural and induced) can be used to determine lithology quite accurately. This knowledge assists seismic modeling and inversion or attribute interpretation.

Even the lowly gamma ray log plotted on a two-way time scale on a seismic section can be an invaluable aid to horizon picking and interpretation, since it is one of the best shale indicators available.

Computed log analysis results, such as shale volume, porosity, lithology, and hydrocarbon fill are very informative when displayed on a seismic section, as shown in the illustration in Figure 21.03 at the right. Notice the strong reflections caused by even thin gas zones (pink colour on the log analysis).

FIGURE 21.03: Log analysis results showing hydrocarbon fill (pink) plotted on two-way time scale with VSP data.

These properties are all derived from appropriate log analysis techniques. They are generally called log analysis results, petrophysical properties, or computer processed interpretations (CPI). They often provide the "ground truth" for calibrating attribute or inversion interpretation.

Modern sonic logs, called full wave, array, or dipole sonic tools, record the complete sonic waveform instead of just the travel time of the first arrival. This allows us to process each wavetrain to determine shear wave and Stoneley wave travel time (and hence velocity) as well as the more usual compressional wave travel time.

Thus shear wave synthetics can be constructed to calibrate shear wave seismic sections. Lithology analysis and direct hydrocarbon detection are sometimes possible from a comparison of compressional and shear velocities. These can be verified by the compressional and shear synthetic seismograms. A transform of shear and compressional data, either from logs or seismic, into Poisson's Ratio helps distinguish between hydrocarbon and lithology variations.

21.04 Log Editing Concepts
There are two major facets of log editing:
1. recognize bad data,
2. substitute better data.

Sounds easy! But most of us underestimate the severity of the problem. Bad data can be easily recognized in cases of obvious noise, such as cycle skips on the sonic log or hole washouts, as in Figure 21.04. It may be difficult in subtle hole condition changes, different lithologies, borehole weathering, and undetected or unrecorded log calibration problems.

FIGURE 21.04: Sonic log before and after edit

If logs were perfect, editing would not be required. However, logs can suffer from a number of problems, such as:
1. misidentification of curves or scales
2. miscalibration
3. electronic failure
4. human failure
5. noise
6. depth discrepancies
7. poor borehole conditions
8. improper tool choice for the hole conditions
9. environmental effects such as temperature, mud salinity, mud type, mud weight
10. bed boundary and bed thickness effects
11. deviated boreholes

Good judgment, interpretation, and background data from offset wells are needed in order to substitute better data.

If a log cannot be repaired, note this fact and consider your task complete - don't use the data if it isn't any good.

Logical use of other log curves in the well, or in offset wells, plus regional trend data prepared in advance by the analyst, will be the basis for most edits.

On older sonic logs, the worst cases are caused by cycle skipping, which results from a large rough borehole, a poor logging tool, a sleepy logging engineer, gas in the borehole, or gas in the formation. On uncompensated logs, spikes caused by hole size changes must be removed. On modern array or full wave sonic logs, missing data due to low amplitude signals must be interpolated.

Rock alteration due to drilling affects both the sonic and density logs. An example is given in Figure 21.05. If regional trends for sonic and density data are known for each major lithology (shale, sand, carbonates), these can be used to draw a more reasonable log.

FIGURE 21.05: Sonic and density edited for rock alteration

On density logs, the worst cases are caused by large or rough borehole, which often occurs in shale sections, in stress relieved carbonates, and in gas bearing formations. An example of a reconstructed density log, corrected for bad hole and rock alteration is shown in Figure 21.06.

FIGURE 21.06: Sonic and density editing based on lithology and trend analysis

It is sometimes difficult to discriminate coal and salt beds from rough hole effects (they often go together), so recourse must be made to other logs or sample descriptions. Needless to say, no two analysts will do exactly the same job of editing. An example of salt interbedded in carbonates and evaporites is shown in Figure 21.07. Although, the logs show great activity and the caliper shows a large hole, the log readings are valid and consistent with the lithology descriptions. No edits are needed.

FIGURE 21.07: Salt beds look initially like bad density log - neutron and GR give clues

Contrast this example with Figure 21.08, in which the density log is badly affected by large and rugged hole conditions. An edit is definitely needed here. Although the sonic log is a bit noisy, it really doesn't need any editing.

FIGURE 21.08: Genuine bad hole condition affecting density - sonic and caliper are clues

Even resistivity logs may need edits. Figure 21.09 shows a noisy induction log, run in a salt mud by mistake, compared to one from a nearby well in fresh mud. Since resistivity logs are used to edit sonic logs, it pays to be sure that they are valid before using them for this purpose.

FIGURE 21.09: Induction log affected by salt mud (left). fresh mud case (right - don't use a bad log as a guide to editing another bad log

When in doubt, we feel that the more severe editing should be done first, and adjustments towards leniency be made after the first few response computations have been reviewed. Integrated time discrepancies are the most obvious clues to over edited or under edited data, and usually the offending zone can be identified readily, when compared to seismic section character, check shot data, or VSP data.

It is not unethical to edit, correct, repair, or otherwise modify a log, if corrections are needed and made properly. Some people are horrified by the concept of modifying logs arbitrarily, preferring to believe either the service company can never be wrong or that bad data should not be used. This attitude results in interpretation errors or wasted data.

The watchword in editing is CAUTION ! Try to edit the garbage, but leave in all legitimate anomalies.

21.05 Seismic Check Shots
The seismic reference survey (SRS), often called a seismic check shot survey, is designed as a calibration mechanism for reflection seismic data. In such a survey, seismic velocities are measured in the borehole by recording the time required for a seismic pulse generated by a surface energy source to reach a geophone anchored at various levels in the borehole. Conventional surveys use a single geophone enclosed in a pressure housing.

Older check shot (seismic reference survey) data should be used with extreme care. Experience has shown that time breaks and first break times are often difficult to pick and adjusting the log to such data is sometimes worthless. This problem is complicated further in deviated holes. Modern vertical seismic profiles and multi-geophone borehole seismic strings suffer from fewer problems than checkshot surveys because they use digital timing circuits and digital data recording.

Recent advances have made it possible to use a series of geophones spaced equally along a cable. More flexibility in geophone placement and closer spacing between recordings is achieved with this approach. On early versions, recording was analog so only first breaks were picked to obtain travel time and hence velocity to a depth.

Currently, vertical seismic profiles are made, which record the full seismic trace received downhole at each detector. Automatic first break detection provides the time-velocity-depth data, and a properly processed display of traces is a relatively noise free seismic section near the wellbore.

The recorded travel times are used to calibrate the sonic log, which then becomes the basic seismic calibration reference. A time versus depth plot is produced from these data (Figure 21.10). The calibrated sonic and the density logs (Figures 21.11 and 21.12) are used to construct a synthetic seismogram, which allows identification of reflecting horizons by reference to the seismic response at the wellbore.

FIGURE 21.10: Seismic Reference Survey (Checkshots) and computed results

FIGURE 21.11: Sonic calibrated to SRS checkshots and reconstructed density log

FIGURE 21.12: Time to Depth conversion from SRS checkshots

The tool lowered into the borehole consists of:
- velocity sensitive geophones
- amplifier circuits
- hydraulic anchoring system

At the surface, there will be:
- air guns
- air compressor
- reference hydrophone
- extra surface hydrophones if required
- high speed recorder (self developing film)
- control panel (amplifiers, filters)
- digital tape recorder

The anchored geophone permits releasing cable tension, thus eliminating transmission of much of the surface generated noise. This allows the use of an air gun as a power source thereby obviating explosives and all the attendant safety hazards and logistical complications.

The entire well shooting operation can be carried out by the same crew that performs the logging operation thus simplifying personnel movements. Surveys can be run in open or cased (single string) hole.

The geometry of an SRS survey is shown in Figure 21.11. The calculations take raw arrival times (slant path) and convert them to vertical (straight ray) paths.

FIGURE 21.13: Checkshot geometry

For straight hole:
_____1: Dhg = Dkbg - Ekb - Dhy
_____2: Tv = Ts * COS (ARCTAN (Ho / Dhg))

For deviated hole:
_____3: Hhg = Hg^2 + Ho^2 - 2 * Hg * Ho * COS (AZM)
_____4: Tv = Ts * COS (ARCTAN (Hhg / Dhg))

In either case:
_____5: Dsrd = Dkbg - Ekb
_____6: Vint = 2 * (Dsrd2 - Dsrd1) / (Tv2 - Tv1)

These calculations provide one-way times versus depth and interval velocities which can be compared to those derived from sonic logs or seismic data. Similar results are also obtained from VSP data.

21.06 Editing Sonic Logs With SRS or VSP Data
Seismic times obtained through the integration of a sonic log usually differ from those obtained by means of a seismic pulse (surface surveys or check shots) for many reasons. These range from basic discrepancies between the two approaches to disturbances in sonic readings caused by cycle skipping, detection of mud arrivals in large holes, formation alteration, and invasion.

Considerable effort has been dedicated in recent years to alleviate this second category of problems. More powerful transducers, sophisticated detection schemes, and long spacing sondes have all led to higher quality logs. Nevertheless, sonic logs are not yet completely free of anomalous effects and the basic discrepancies mentioned above remain, particularly invasion, which cannot be cured by tool design.

Seismic checkshot times are used as a reference to calibrate the sonic log through a process called drift curve correction. The drift curve is a log of the difference between integrated sonic log time and check shot seismic time. When integrated sonic log times are higher than seismic times (the usual case), drift is negative.

Drift is made equal to zero at an arbitrary depth, the tie point, often the top of the sonic log when, as it should be, a checkshot is available at that depth. Drifts are plotted at each shot depth. Then a curve is drawn, as segments of straight lines fitting the drift points as well as possible. The junction of two such segments is called a "knee". A knee should not be necessarily located at a checkshot point, but where there is a change of lithology or of sonic character (see Figure 21.14).

FIGURE 21.14: Plot of sonic log drift correction from checkshot survey

Between two consecutive knees, the sonic log is adjusted to get rid of the drift. Two different methods are in use: block shift method and delta-T minimum method. However, only one method should be used on a given interval. After these adjustments, the sonic log should provide the continuous formation transit time in the undisturbed formation.

These methods work well where there is rock alteration near the well bore, vuggy porosity, or long intervals of gas bearing reservoir. If differences were caused by thin gas zones, or a few bad spots or cycle skips on the log, it is unlikely that either method will provide correct answers. This is due to the fact that both methods apply a small correction over fairly large intervals, instead of a large correction where it is needed. Manual editing before applying these methods often works very well.

If there are many skips or large intervals of bad hole condition, check shots will not directly cure the problem. Other methods, such as those described in the next few Sections, will create a better log, which can then be calibrated with checkshots or vertical seismic profiles data.

1. Block Shift Method:
Calculate total drift between two knees
_____1: D = (T2 - T1) * 1000 / 2 - (Sum (DELTi * INCR)) / 1000

Calculate drift to apply to each data point
_____2: C = 1000 * D / (H2 - H1)
_____3: DELTcor = DELT + C

The block shift is used when drift is small and no single anomaly is the cause of the error.

By looking at Figure 21.15, one can see that a block shift will not always be a satisfying correction. On that example, a long spacing sonic agrees with checkshots and may be assumed to be right. The standard sonic agrees with the long spacing over the cleaner interval. A block shift correction will change the sonic over that interval and impose changes which will generate false reflections on a synthetic seismogram.

FIGURE 21.15: Block shifted sonic log - not recommended but widely used

A Delta-T minimum correction can be applied in such cases, when the sonic log reading is too high and the difference of drifts is big. Shale alteration or skipping are then suspected. A threshold, DTMIN, is chosen: all values of sonic travel time smaller than this threshold are assumed to be good and are not corrected. When the sonic reading exceeds DTMIN, the excess of the sonic value over the threshold is corrected by a reduction factor defined for the interval, as in Figure 21.16.

FIGURE 21.16: DELTA-T minimum correction applied to sonic log

2. Delta-T Minimum Correction Method:
Calculate total drift between two knees on drift curve
_____1: D = (T2 - T1) * 1000 / 2 - (Sum (DELTi * INCR)) / 1000

Calculate drift factor to apply to data
_____2: C = 1 + (D / Sum (DELTi * INCR - DTMIN * INCR) / 1000)

Apply correction if data is above DTMIN
_____3: IF DELT > DTMIN
_____4: THEN DELTcor = C * (DELT - DTMIN) + DTMIN
_____5: OTHERWISE DELTcor = DELT

Other logs which are less affected by environmental effects (neutron, deep resistivity) are generally used to determine the zones over which the sonic readings are correct and hence to choose the appropriate value of DTMIN. It may vary with depth.

CAUTION: Neither of these methods will adequately correct a sonic log in a gas zone. See below for details. A possible exception is very closely spaced checkshots throughout the entire gas interval, such as through the long gas filled tight sands of the Deep Basin of Alberta.

21.07 Editing the Sonic and Density Logs With Trend Data
Sonic and density logs often contain "noise" or spikes caused by tool malfunction or bad hole effects. If the trend of the log is discernable, we merely trim off the spikes and digitize the log, as in Figure 21.17 (sonic log) and Figure 21.18 (density log).

FIGURE 21.17: Editing sonic with trend analysis

FIGURE 21.18: Editing density with trend analysis

NOTE: When the integration tics on a sonic log are being used, and edits are needed, the integration must also be re-done. This is true after ANY editing method has been applied. See example in Figure 21.17.

When longer intervals are noisy, it may not be possible to identify the background log. It is common to refer to offset wells, where some logs may be better quality, and use the general trend of log values as a guide to editing. A method suggested by Brian Ausburn in 1977 recommended the preparation of composite sonic versus depth and density versus depth graphs from a number of wells. By choosing only data that did not suffer from noise, the graph could indicate the value to use during an edit of a noisy log.

Separate graphs for shales, sandstones, and carbonates are made and used based on known or assumed lithology. Since the gamma ray log is not strongly affected by bad hole, it is used to differentiate shales from other rocks. Generalized geological knowledge is used to differentiate sandstones from carbonates. Since porosity in carbonates does not vary linearly with depth like sandstone does, trend lines for carbonates might not be very useful.

If evaporites such as coal, salt, anhydrite, or potash minerals are present, correct values for these minerals are known and can be inserted (see table below).

21.08 Modeling the Sonic and Density Logs From Resistivity
Resistivity is sometimes transformed into an apparent velocity log with a number of different equations:

1. Faust Method
This method is very old, but is useful in shallow rock sequences, especially clastics. You may need to determine new parameters for each major geologic horizon.

_____1: Vc = KR1 * RESS ^ (1/KR2) * DEPTH ^ (1/KR3)

Where:
__Vc = compressional velocity (ft/sec or m/sec)
__KR1 = Faust constant (2000 to 3400 for depths in feet)
__RESS = resistivity from shallow investigation log (ohm-m}
__DEPTH = depth of layer (ft or m)
__KR2 and KR3 = 6.0 or as determined by regression analysis

The Faust transform can be used when the sonic log is missing, and can be calibrated with offset well data, check shots, or vertical seismic profiles. The method does not account for gas effect.

2. Smith Method
This method uses a simple correlation between resistivity and sonic traveltime:

_____1: DELTc = KR4 * (RESS ^ KR5)

Where:
__DELTc = compressional travel time (usec/ft or usec/m)
__KR4 = Smith constant (90 to 100 for depths in feet)
__RESS = resistivity from shallow investigation log (ohm-m}
__KR5 = -0.15 or as determined by regression analysis

The method does not account for gas effect. You may need to determine new parameters for each major geologic horizon.

3. Fischer - Good Method
This method assumes a fairly sophisticated log analysis can be run on the well in question or on a nearby well. This is needed to obtain a list of water resistivity (RWA) versus depth. Since most sonic log problems are in shales due to bad hole or rock alteration, this calculation is usually possible and should be done continuously or at least zone by zone.

Similarly, the apparent RW in shale (RWSH) is needed, based on an estimate of the shale total porosity (BVWSH). This can be computed continuously or zone by zone from one of the following:

If neutron and density logs are both available and correct:

_____1: BVWSH = (PHIDSH + PHINSH) / 2
_____2: PHIt = (PHID + PHIN ) / 2

If density log is missing or bad:
_____1: BVWSH = 0.95 * PHINSH
_____2: PHIt = PHIN

Where the sonic log is behaving properly or from an offset well that is OK:
_____1. BVWSH = (DELTSH - DELTMA) / (DELTW - DELTMA)
_____2. PHIt = (DELT - DELTMA) / (DELTW - DELTMA)

Then, for each shale zone:
_____3: RWSH = (BVWSH ^ M) * RSH / A

And, for each clean zone:
_____4: RWA = (PHIt ^ M) * RESD / A

For all digitized intervals or computation layers:
_____5: Vshg = (GR - GR0) / (GR100 - GR0)
_____6: Vshs = (SP - SP0) / (SP100 - SP0)
_____7: Vsh = Min (Vshg, Vshs)
_____8: RMIX = 1 / (Vsh / RWSH + (1 - Vsh) / RWA)
_____9: DELTc = DELTMA + (DELTW - DELTMA) * (A * RMIX / RESD) ^ (1/M)
_____10: DELTmod = Min (DELT, DELTc)
_____11: DENSc = DENSMA + (DENSW - DENSMA) * (A * RMIX / RESD) ^ (1/M)
_____12: DENSmod = Min (DENS, DENSc)

When the zone is 100% shale, this equation should return a reasonable travel time. If it doesn't match the log where it is believed to be good, then adjust RWSH or Vsh. In clean zones, adjust DELTMA or RWA if needed. When zones are hydrocarbon bearing, RWA and RESD will both be too high, and the result will be close to correct, but may give a DELTmod that is too low (too high a velocity) or a DENSmod that is too high.

To overcome some of this effect, you could substitute the shallow resistivity RESS for RESD and RMF@FT for RWA. You may still need to calibrate the RMF@FT with its own RMFA equation:
_____4A: RMFA = (PHIt ^ M) * RESS / A
_____8A: RMIX = 1 / (Vsh / RWSH + (1 - Vsh) / RMFA)
_____9A: DELTc = DELTMA + (DELTW - DELTMA) * (A * RMIX / RESS) ^ (1/M)
_____10: DELTmod = Min (DELT, DELTc)
_____11A: DENSc = DENSMA + (DENSW - DENSMA) * (A * RMIX / RESS) ^ (1/M)
_____12: DENSmod = Min (DENS, DENSc)

Neither method accounts for the effect of gas, which must be handled separately as in Sections 21.10 and 21.12.

21.09 Modeling Sonic and Density Logs From Neutron and Gamma Ray Logs
One log that is relatively unaffected by noise and bad hole effects is the neutron log. It is a good source of total porosity (PHIt) and can be used in the time average equation to generate a sonic log:
_____1: DELTmod = DELTMA + (DELTW - DELTMA) * PHIN

This can be rewritten in its more usual form as:
_____2: DELTmod = DELTMA * (1 - PHIN) + DELTW * PHIN

Neutron logs can be run through casing and many are available in well files where no sonic or a poor sonic is present. Because neutron and sonic logs respond similarly to shale, no special shale compensation is needed with this method.

The density log is not as strongly affected by shale, so it requires more attention to detail:
_____1: Vshg = (GR - GR0) / (GR100 - GR0)
_____2: Vshs = (SP - SP0) / (SP100 - SP0)
_____3: Vsh = Min (Vshg, Vshs)
_____4: PHIe = PHIN - (Vsh * PHINSH)
_____5: DENSmod = (1 - Vsh - PHIe) * DENSMA + DENSW * PHIe + Vsh * DENSSH

If there is no neutron log to use as a guide, the following will give reasonable results for seismic purposes:
_____1: Vsh = (GR - GR0) / (GR100 -GR0)
_____2: DELTmod = DELTMA * (1 - Vsh - PHIMAX) + DELTW * PHIMAX + DELTSH * Vsh
_____3: DENSmod = DENSMA * (1 - Vsh - PHIMAX) + DENSW * PHIMAX + DENSSH * Vsh

Where:
PHIMAX is determined from offset well data (zoned according to lithology).

PHIN is too low in gas zones, giving DELTmod too low and DENSmod too high. Gas corrections are covered in the Section 21.11 and 21.13.

21.10 Modeling the Sonic and Density Response From Regression
Jay Patchett proposed a sonic editing technique in 1975 for shales, based on the following:
_____1: log (COND) = A0 + A1 * log (DELT - 42) + A2 * log (CEC) + A3 * log (ES)

Where:
__CEC = cation exchange capacity of the shale
__ES = effective stress (psi)

Since CEC is not readily available in most wells, this approach was not terribly practical. However, by recognizing other work that related CEC to gamma ray log response, the equation becomes:

For shale zones:
_____1: log (DELTmod - 40) = KW0 + KW1 * log (RSH) + KW2 * log (GR) + KW3 * log (ES)

A similar equation for density is:
_____2: DENSmod = KX0 + KX1 * GR + KX2 * DEPTH + KX3 * log (RSH)

For sandstones:
_____1: DELTmod = KY0 + KY1 * GR + KY2 * log(ES) + KY3 * PHIrs
_____2: DENSmod = KZ0 + KZ1 * GR + KZ2 * DEPTH + KZ3 * PHIrs

Where:
__PHIrs = porosity from the shallow resistivity log

These models are decidedly not simple and a great deal of calibration is required to make them work. Practitioners should refer to the original paper for details of the method. In addition, a sophisticated multiple linear regression program is required.

21.11 Modeling Sonic and Density From Log Response Equation
All of the modeling techniques described above create logs or portions of logs similar to those in offset wells. None of them are designed to specifically replace the mud filtrate invasion (water) with the gas or oil that was moved away from the wellbore. None allow replacement of one lithology with another, except by cut and paste techniques.

For example, we may want to replace water with gas to see what happens to the seismic signature. Or we could change a dolomite to a limestone, or thicken or thin an existing zone, to see various "what-if" scenarios. A spreadsheet to perform this math is available from the downloads section of this website. Some examples are shown in Chapter Twenty-Two.

The log response equation is the best way to do fluid or lithology replacement, as long as the fluid is reasonably incompressible. Oil and water satisfy this criteria and gas at high pressure also works reasonably well. An alternate approach for gas is given at the end of this Section.

Since conventional log analysis techniques eliminate spurious artifacts on the logs caused by rough borehole, the log analysis results can be fed back to the response equation to create reconstructed logs. This reduces noise on both sonic and density logs that might reduce the quality of synthetic seismograms. Such noise might make the synthetic totally useless or misleading.

This following material was originally published as "Determination of Seismic Response Using Edited Well Log Data" by E. R. Crain and J. D. Boyd, CSEG, October 1979. ** Best Paper Award, CSEG, 1979**

1. Density Log Response
The response of a density log can be described rigorously by a volume weighted summation of the densities of the individual components in the rock. The usual form of this equation is:
_____0: DENS = Sum (DENSi * Vi)

The expansion for well logging situations is:
_____1: DENSmod = PHIe * Sw * DENSW                   (water term)
__________+ PHIe * (1 - Sw) * DENSHY                       (hydrocarbon term)
__________+ Vsh * DENSSH                                       (shale term)
__________+ (1 - Vsh - PHIe) * Sum (Vi * DENSi)        (matrix term)

See below for a discussion of hydrocarbon density.

CAUTION: Synthetics will not tie seismic unless you do this step in all gas zones.

A similar equation can be written using density derived porosity values by replacing each DENS term by its equivalent PHID term:
_____2: PHIDmod = PHIe * Sw * PHIDW                   (water term)
__________+ PHIe * (1 - Sw) * PHIDHY                      (hydrocarbon term)
__________+ Vsh * PHIDSH                                      (shale term)
__________+ (1 - Vsh - PHIe) * Sum (Vi * PHIDi)        (matrix term)

Since many density logs are run on a porosity scale instead of a density scale, this alternate form of the equation may be easier to use in certain cases.

Either equation can be used to calculate what a density log would read given a hypothetical rock/fluid mixture, thus modeling of various formation alternatives is a straight forward mathematical process. It is preferable to guessing or estimating from previous experience.

This equation is rigorous and can be used with real hydrocarbon densities based on the temperature, pressure, and phase relationship of the fluid in question. A chart showing approximate gas density versus depth is shown in Figure 21.17, based on average pressure and temperature data for the western Canadian basin. No correction for vuggy porosity is needed.

FIGURE 21.19: Density of gas at reservoir conditions - default approximation

The straight line on the graph is:
For gas, in Englsih units  (gm/cc and feet),
      1. DENSHYgas = Min (0.8, 0.000038 * DEPTH)

For gas, in Metric Units (Kg/m3 and meters).     
      1a: DENSHYgas = Min (800, 0.125 * DEPTH)

For oil, in Englsih units (gm/cc):
      2. DENSHYoil =  141.5 / (131.5 + API_GR) 

For oil, in Metric  units (Kg/m3):
      2a. DENSHYoil =  141 500 / (131.5 + API_GR) 

Where:
DENSHYgas = density of gas at DEPTH
DENSHYoil = density of oil
DEPTH = depth of reservoir
API_GR = oil gravity

Corrections for the fact that density logs respond to electron density, and not bulk density, can be made, and may be necessary especially in the case of coal or salt beds. The correction factors are supplied in the chart in Figure 21.20. We usually do not make these corrections, because the accuracy needed for computing seismic response does not warrant the effort.

FIGURE 21.20: Correction to density log for Z/A effect of different lithologies

2. Sonic Log Response
An equation similar to that for density can be generated for sound velocity of mixtures. However, it is a summation of travel time weighted by volume and not a summation of velocity components:
_____0: DELT = Sum (DELTi * Vi)

This is called the Wyllie time average equation and is true for many situations where the components are not very compressible, such as water, sandstone, and shale. It does not work too well with gas under low pressure. It is an empirical relationship and is not rigorous. However, the Biot model for sound velocity in mixtures is rigorous, and reduces to Wyllie's equation in most situations (ie: compressibility is very low). The Biot model is discussed later in this Section.

The expansion of this formula for log analysis parallels the density formula:
_____1: DELTmod = PHIe * Sw * DELTW                                 (water term)
__________+ PHIe * (1 - Sw) * DELTHY                                    (hydrocarbon term)
__________+ Vsh * DELTSH                                                    (shale term)
__________+ (1 - Vsh - PHIe) * Sum (Vi * DELTi)                     (matrix term)