Check Shot Correction Theory

This theory is used in the Log Check Shot Correction function.

Why Use Check Shots

Sonic (velocity) well log tools measure discrete transit times of the rock adjacent to the well bore.The measurements start at the subsurface, usually just below the bottom of the well casing so thereis no steel casing separating the logging tool from the rock. Note that the loose upper surface andthe water-bearing rock near the surface must be cased before any logging tools can be run, so therewill never be sonic data for the very top of a well.

The depth of a sonic log is measured as the depth of the tool below the kelly bushings on the drillingfloor (kelly bushings are the metal parts that grab and rotate the kelly, and as the kelly holds the drillpipe with the drill bit at its end, the "K.B.'s" therefore rotate the drill bit).

The transit times are made over a set tool distance for each depth sample, and the interval velocityis derived over that distance (see Velocity Definitions). From these velocities, a time-depth curve canbe calculated. The time refers to the time a vertically traveling signal takes to reach that depth.

The resulting integrated time-depth curve will usually require correction to a seismic datum, whichcould well be the surface itself, but is usually the horizon just below the loose overburden.  Then thislog is used to create a depth-time table, through:

 

 

As the time to a layer depends on all the velocities above that layer, it includes the unmeasured"first" velocity (V1) of the first layer to the surface (i.e. the section that was cased and never logged).That velocity is usually assumed equal to the first measured velocity.

If the well is deviated (not drilled straight down), then measured depths must also be corrected tothe true vertical depth. Fortunately, this information is always available for deviated wells.

The sonic log will not perfectly match the seismic data because:   a.  The seismic and log datums are usually different.   b.  The first layer velocity is unknown.   c.  Errors in calculating time-depths form logs accumulate.   d.  The interpreted seismic data may be mis-positioned if migrated improperly.   e.  The seismic data may have time stretch caused by frequency-dependent absorption and short-period multiples.   f.  Surface seismic data are subject to greater dispersion and absorption than the sonic data recorded in the well.

Therefore, check shots are used to improve the depth-time conversion. They are also needed tocorrect sonic logs for the Roy White Wavelet Extraction method. See Roy White Theory.

Note that vertical seismic profiles (VSP's) are treated as seismic data, not checkpoint data, in HRsoftware.

The Correction

The check shot correction adapts the sonic log velocities and/or the log time-depth curve to matchthe time-depth relationship obtained from surface seismic data.

From a raw sonic log Vz, since V = z/t, we can derive a time-depth curve tz as:

   (1)

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Alternatively, we can input tz directly.

Matching the time-depth curve tz with independently acquired check shot data (t1, z1), (t2, z2), ¼ (tN,zN), we usually see discrepancies with tz, which we have to compensate with the check shotcorrection.

We calibrate the time-depth curve tz, slicing it into pieces and forcing it to go through the checkshot points.  We could then obtain a corrected sonic as the derivative of the corrected time-depthcurve, but we will apply a more direct correction.

The check shot correction is done in 2 steps:

1. A drift curve is interpolated to measure the discrepancy between the time-depth curve and thecheck shot data.

2. The time-depth curve (and optionally the sonic log) are "check shot corrected" using the driftcurve.

3. We only change the time-depth curve where there is sonic log data. If there is a gap in the soniclog curve, any check shot data in that gap will not be used.

Drift Curve

We can only measure the discrepancies da (a = 1, 2, ¼N) between check shot data (t1, z1), (t2, z2), ¼(tN, zN) and the time-depth curve tz at a "few" isolated check shot depths, but we want to computeinterpolated drifts di (i = 1, 2, ¼ M) along the whole time-depth curve t(z) which has as manysamples as the sonic log itself.

Our problem is as follows:

Given:             (ta, za) = check shot times ta measured at depth za for check shot number a

da = measured time of check shot #a – (time of time-depth curve at depth za)

da = ta –t(za)   for each check shot a = 1, 2, ¼ N       (2)

 

Wanted:   interpolated drift samples di at all depths zi of the time-depth curve

di = d(zi) = Drift(zi; {za, da})      i = 1, 2, ¼ M   (3)

a = 1, 2, ¼ N              where M>>N

The function Drift is a function of depth z and should honor all calibration points {za, da} obtainedfrom check shot data.

 

Note:

1.   As time always increases, the check shot data and the time-depth curves are monotonicallyincreasing functions (as depth increases, time must increase), but the drift curve, representing anerror, can have both signs and can increase or decrease as well.

2.   Check shot times can be input as either 1-way or 2-way times. For this discussion, they areassumed to be 2-way times.

HR provides 3 ways to calculate the function Drift(z; {za, da}) in equation (3): Drift Description Honors Linear (z; {za, da})

A bent line of straight segments, with thedata points being the segment ends.

Piecewise linear interpolation between datapoints (za, da) and (za+1, da+1)  a = 1, 2, ¼N-1

 

Data points

Polynomial(n) (z; {za, da}) Least squares fit of an n-th degree None. May not reflect geological boundaries

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Resembles a gentle curve

Least squares fit of an n-th degree

polynomial through all data points (za, da) a = 1, 2, ¼ N 

Low degrees (n = 2 or 3) arerecommended. Higher degrees can inducelarge amplitude oscillations.

 

None. May not reflect geological boundaries

as well.

Spline (z; {za, da})

Resembles a sharply curving and weavingline through every point.

Cubic spline through all data points (za, da) a = 1, 2, ¼ N

 

Data points, first derivatives, minimumoverall curvature

The output of each of these 3 functions can be smoothed, as you can enter the length of thesmoothing operator.

Check Shot Correction

The time-depth curve tz and the sonic log must be corrected using the drift curve dz obtained fromequation (3).

Sonic Log Change: Apply Relative Changes

This option changes only the velocities for layers between the first and last check shot depth.

Under this option from the Check Shot Parameters window, the check shot correction is applied onlyalong the log range, i.e. from the first depth sample (which may be well below the surface) to the lastone. The resulting log will integrate to the desired times but will need a bulk time shift.

The resulting curves Vzcorr and tz

corr will be only relatively correct, because the curves will not becorrected for the first drift value of z1 which bears the log errors accumulated from the surface to thefirst depth sample.

An additional correction will be necessary to have an absolutely correct log.  This is described inthe next section.

Time-depth curve:   Each sample is corrected with the corresponding sample of the drift curve:

   (4)

Sonic Log: For the sonic log, a sample of the drift curve d(zi) expresses the cumulative effect of allthe time corrections dtj applied to all previous sonic samples, including the current one.

 

We can then extract :

    (5)

 

and apply it to the i-th sample of the sonic log.  The correction is applied differently to a velocitycurve Vz or a transit-time curve tz.

Velocity curve: We have to convert the time correction dti into a corresponding velocity correctiondVi, i.e., the velocity change which makes the seismic wave travel dti slower or faster through thedepth interval dzi between the depths zi-1 and zi. If the time-depth curve is expressed in 2-way time,we have:

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       (6)

 

 

Transit-time curve: A transit time expressing a time span spent through a thin layer simply needs tobe corrected with the time correction dti over the depth interval dzi. If we express transit time as 1-way time in microseconds, we have:

    (7)

 

 

Sonic Log Change: Apply all changes

This options changes all the velocities in the log so the new log integrates to the exact desired times.

Under this option from the Check Shot Parameters window, the check shot correction is applied fromthe surface to the last logged depth. The velocity above the first log measurement is "ramped" tohandle the bulk time shift and minimize the effect of spurious reflections on the synthetic.

Sonic Log: For the sonic log, this correction occurs in 2 steps:

1.  The option "Apply relative change" is executed using (6) or (7). The corrected velocity curve

 needs a further adjustment.

2.  We now extend the check shot correction from the first logged depth z1 up to the surface.  The

only information we have is  from (5).  We have accumulated dt1 milliseconds ofsuccessive errors, when logging from the surface to a depth of z1 meters.  We have now to distributethis total error into partial errors occurred during successive simulated logging steps from surface toz1 meters.  We can achieve that by providing extra velocity samples back to the surface.

A safe solution is to append a linear velocity ramp uniformly sampled from the surface to thevelocity curve , the depth sampling interval Dz being the smallest depth interval of the velocitycurve.  In other words we have:

           

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Setting k = Madd + 1, we can verify that the ramp ties with the first sample V1 the velocity curve.

 

We extrapolate linearly from velocity to V1 to V0.  But which V0?  The velocity of the first addedsample V0 must be such that the accumulated errors from surface to the first logged sample z1

equals .

Setting V0 = CV1, we have to find C such that:

 

Each depth increment Dzk being constant = Dz and replacing  by the ramp function Ramp(k×Dz), we get:

an equation of degree (Madd -1) for C, which we can solve via a least squares fit algorithm.

Velocity curve: That way we get a complete corrected velocity curve:

  [From (8)]

  [From (6)]

 

which, if integrated, will yield an absolutely correct time-depth curve .

 

Transit-time curve: The corrected transit time curve is the inverse of the corrected velocity curve obtained from (10):

= 1000000 / Ramp (z) if  0 < z < z1 (11)

= 1000 000 /     if  zi > z1

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Time-depth curve: According to (1), the corrected time-depth curve is obtained by integrating the corrected velocitycurve (10)

 

z0 = 0 being the surface.

Sample Problem

Let us use a model inspired by the Ostrander (1984) gas sand model:

Log Data                                             Check Shot Data zi V(zi) ti za ta m m/s TWT ms m TWT ms 1500 3100 967.74 1500 1000.00 2000 2600 1352.36    

      2100 1500.00 2500 3200 1664.86     3000 4100 1908.76    

      3500 2300.00 4000 4400 2363.30    

 

The depths are measured from the surface.  The following figures illustrate the check shot correctionapplied with different options.

  Sonic Log Change Type of interpretation   Apply relative changes Linear   Apply all changes Linear   Apply all changes Spline   Apply all changes Polynomial order

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Figure 1.  Apply relative changes with linear interpolation.

Here the check shot correction applies only on the depth range over which the log was measured.The drift curve has been piecewise linearly interpolated between the check shots and extrapolatedbeyond the last check shot depth. In order to increase the sonic times to match the check shot times,the sonic velocities must be decreased.

The values for this example are shown here:

  zi         ta Vcorr tcorr

m m/s ms ms m/s ms 1500.000 3100.000 967.742 1000.000 3100.000 1000.000 2000.000 2600.000 1352.357 - 2332.710 1428.686 2100.000 - - 1500.000 - - 2500.000 3200.000 1664.857 - 2908.368 1772.521 3000.000 4100.000 1908.760 - 3675.740 2044.575 3500.000 - - 2300.00 - - 4000.000 4400.000 2363.305 - 4143.383 2527.273

 

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Figure 2.  Apply all changes with linear interpolation.

The check shot correction is applied from the surface to the total log depth.  A linear velocity ramp(see equation (8)) is appended to the velocity function already corrected under the option "Applyrelative change".  This enables us to have a corrected time-depth curve extending to zero time atthe surface.

Figure 3.  Apply all changes with spline interpolation.

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Figure 3 also shows the check shot correction applied up to the surface, but using a drift curveinterpolated by a spline function.  This results in a smoother correction.

Figure 4.  Apply all changes with polynomial interpolation of order 1.

This last figure shows that the polynomial fit does not honor the check shot data, but represents abest fit through them.  The resulting correction is less accurate, but still represents a bestcompromise when the drift data have erratic behavior.

Kelly Bushing (KB) Considerations

The depth values can be measured either from surface or from the Kelly Bushing table on the drill rigfloor.

The assumption we use is as follows:   Input Geoview database  Inside Geoview Depth from surface surface surface Depth from KB KB surface

 

In other words, within our software, the check shot correction uses and plots depths from surfaces,and the database stores and exports the depths as they were input.

This is why the check shot plot may have different depths from the one presented by the Show Databutton of the Display Log menu or the Export Well Logs function.

The present version allows only three out of the four possible cases:     Check Shot depths

from:     surface KB Sonic log depths from: surface: Yes No   KB: Yes Yes

All 3 options give identical corrected time-depth curves and velocity curves.

 

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