Check Shot Correction

Check Shot Correction Theory  This theory is used used in the Log Check Shot Correction function. 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 measurements start start at the subsurface, subsurface, usually just just below the bottom of the well well casing so there there is no steel casing separating the logging tool f rom the rock. Note that the loose upper surface and the water-bearing rock near the surface must be cased before any logging tools can be run, so there will never be sonic data for the very top of a well.  The depth of a sonic log is measured measured as the depth of the tool below the kelly kelly bushings on on the drilling floor (kelly bushings are the metal parts that grab and rotate the kelly, and as the kelly holds the drill pipe with the drill bit at its end, the "K.B.'s" therefore therefore rotate the drill bit).  The transit times times are made over a set tool distance distance for each depth sample, sample, and the interval velocity is derived over that distance (see Velocity Definitions). Definitions). From these velocities, a time-depth curve can be calculated. The time refers to the time a vertically traveling signal takes to reach that depth.  The resulting resulting integrated integrated time-depth time-depth curve will usually usually require require correction correction to a seismic datum, which could well be the surface surface itself, but is usually the the horizon just below the loose loose overburden. overburden. Then this log 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 (V  (V   ) 1 of the first layer to the surface (i.e. the section that was cased and never logged).  That velocity is usually usually assumed assumed equal to the first measured measured velocity. velocity. If the well is deviated (not drilled straight down), then measured depths must also be corrected to the true vertical depth. Fortunately, this information is always available for deviated wells.  The sonic log will not perfectly perfectly match the seismic seismic data because: because: a. The seismic and log datums are usually different. b. The first layer layer velocity velocity is unknown. c. Errors in calculating calculating time-depths time-depths form form logs accumulate. d. The interpreted seismic data may be mis-positioned mis-posit ioned if migrated improperly. improperly. e. The seismic data may have time stretch caused by frequency-dependent absorption absorpti on and f. Surface seismic data are subject to greater dispersion dispersio n and absorption absorpti on than the sonic data  Therefore,  Therefore, check shots shots are used used to improve the depth-time depth-time conversion. conversion. They are also needed needed to Extraction  method. SeeRoy correct sonic logs for the Roy White Wavelet Extraction method. SeeRoy White Theory. Theory. Note that vertical seismic profiles (VSP's) are treated as seismic data, not checkpoint data, in HR software.

The Correction correction  adapts the sonic log velocities and/or the log time-depth curve to match  The check shot correction adapts the time-depth relationship obtained from surface seismic data. From a raw sonic log V  , since V = z/t  , we can derive a time-depth curve  z  t  as:  as:  z 

(1) 1

Alternatively, we can input  z  t   directly. Matching the time-depth curve  z  t  with independently acquired check shot data (t  1, z  1), (t  2, z  2),  z  t , which we have to compensate with the check shot N), we usually see discrepancies with  z  correction.

¼ (tN  ,

We calibrate the time-depth curve  z  t , slicing it into pieces and forcing it to go through the check shot points. We could then obtain a corrected sonic as the derivative of the corrected time-depth curve, 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 the check shot data. 2. The time-depth curve (and optionally the sonic log) are "check shot corrected" using the drift curve. 3. We only change the time-depth curve where there is sonic log data. If there is a gap in the sonic log 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 (t  1, z  1), (t  2, z  2), ¼ (t  t  at a "few" isolated check shot depths, but we want to compute N, z  N) and the time-depth curve  z  interpolated drifts di (i = 1, 2, ¼ M) along the whole time-depth curve t(z) which has as many samples as the sonic log itself. Our problem is as follows: Given: da =

(t , z ) = check shot times t  measured at depth  z for check shot number a a 

a 

a 

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

da =t   ) a  –t(z  a 

Wanted:

for each check shota = 1, 2,¼ N 

(2)

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

di = d(z   )  z  i = Drift( i; {z  a, a =

a 

da})

1, 2,¼ N 

i = 1, 2,¼ M 

(3)

whereM>>N

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

da}

obtained

Note: 1. As time always increases, the check shot data and the time-depth curves are monotonically increasing functions (as depth increases, time must increase), but the drift curve, representing an error, 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 are assumed to be 2-way times. HR provides 3 ways to calculate the function Drift(  z  ;  {z  a, da}) in equation (3): Drift Description Honors Piecewise linear interpolation between data Data points Linear  z  ( ;  {z  a, da}) points z  ( a, da) and  z  ( a+1, da+1) a = 1, 2, ¼ A bent line of straight segments, with the N-1 data points being the segment ends. Polynomial(n) (  z  ;  {z  a,

da})

Least squares fit of an n-th degree

None. May no 2

Resembles a gentle curve

polynomial through all data points  z  ( a, a = 1, 2,¼ N

da)

as well.

Low degrees (n = 2 or 3) are recommended. Higher degrees can induce large amplitude oscillations. Spline (  z  ;  {z  a,

da})

Resembles a sharply curving and weaving line through every point.

Cubic spline through all data points  z  ( a, a = 1, 2,¼ N

da)

Data points, fi overall curvat

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

Check Shot Correction  The time-depth curve  z  t and the sonic log must be corrected using the drift curve d z  obtained from equation (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 only along the log range, i.e. from the first depth sample (which may be well below the surface) to the last one. The resulting log will integrate to the desired times but will need a bulk time shift. corr corr  The resulting curves V   z   andt   z   will be onlyrelatively correct, because the curves will not be corrected for the first drift value of z  1 which bears the log errors accumulated from the surface to the first depth sample.

An additional correction will be necessary to have an absolutely correct log. the next section. Time-depth curve:

This is described in

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(z i) expresses the cumulative effect of all the time corrections d tj 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 velocity curve Vz or a transit-time curvetz. Velocity curve: We have to convert the time correction dt  i into a corresponding velocity correction dV  i, i.e., the velocity change which makes the seismic wave travel dt  i slower or faster through the depth interval  z   z   z  d i between the depths i-1 and i. If the time-depth curve is expressed in 2-way time, we have: 3

(6)

Transit-time curve: A transit time expressing a time span spent through a thin layer simply needs to d i. If we express transit time as 1be corrected with the time correction dt  i over the depth interval z  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 from the surface to the last logged depth. The velocity above the first log measurement is "ramped" to handle 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 dt  1 milliseconds of  successive errors, when logging from the surface to a depth of z  1 meters. We have now to distribute this total error into partial errors occurred during successive simulated logging steps from surface to  z  1 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 the velocity curve , the depth sampling interval D  z   being the smallest depth interval of the velocity curve. In other words we have:

4

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 V  1 toV  0. But which V  0? The velocity of the first added sample V   z  0 must be such that the accumulated errors from surface to the first logged sample 1 equals

.

Setting V  0 = CV  1, we have to find C such that:

Each depth increment D  z   being constant =  z and D replacing k  (k  ×D  z  ), we get:

by the ramp function Ramp

an equation of degree (M add -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) if zi > z1 = 1000 000 / 5

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

z0 = 0 being the surface.

Sample Problem Let us use a model inspired by the Ostrander (1984) gas sand model: Log Data zi m 1500 2000 2500 3000 4000

V(zi) m/s 3100 2600 3200 4100 4400

ti TWT ms 967.74 1352.36

Check Shot Data za m 1500

ta TW 100

2100

150

3500

230

1664.86 1908.76 2363.30

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

Sonic Log Change Apply relative changes Apply all changes Apply all changes Apply all changes

Type of interpretation Linear Linear Spline 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 extrapolated beyond 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 m 1500.000 2000.000 2100.000 2500.000 3000.000 3500.000 4000.000

m/s 3100.000 2600.000 3200.000 4100.000 4400.000

ms 967.742 1352.357 1664.857 1908.760 2363.305

t  a

corr V 

corr t 

ms 1000.000 1500.000 2300.00 -

m/s 3100.000 2332.710 2908.368 3675.740 4143.383

ms 1000.000 1428.686 1772.521 2044.575 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 "Apply relative change". This enables us to have a corrected time-depth curve extending to zero time at the surface.

Figure 3.

Apply all changes with spline interpolation. 8

Figure 3 also shows the check shot correction applied up to the surface, but using a drift curve interpolated 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 a best fit through them. The resulting correction is less accurate, but still represents a best compromise 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 rig floor.  The assumption we use is as follows: Input Depth from surface Depth from KB

Geoview database surface KB

Inside Geoview surface 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 Data button of the Display Log menu or theExport Well Logs function.  The present version allows only three out of the four possible cases: Check Shot depths from: Sonic log depths from:

surface: KB:

surface Yes Yes

KB No Yes

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

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