Well-velocity survey

Problems in Exploration Seismology and their Solutions

Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
Chapter	5
Pages	141 - 180
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 5.14a

Figure 5.14a shows data from a well-velocity survey tabulated on a standard calculation form. Calculate the average velocity and interval velocity, and plot graphs of time, average velocity, and interval velocity versus depth using a sea-level datum.

Solution

Figure 5.14a shows only the measured data on a standard form whereas Figure 5.14b shows also calculated values. The successive columns in this form list

1 - Record number

2 - Source (shothole) location

3 - $D_{gm}$, geophone depth with respect to well datum

4 - $D_{s}$, depth of source

5 - $t_{uh}$, uphole time

6 -$t_{c}$, arrival time at reference geophone

7 - ${T}$, arrival time at well geophone plus polarity and quality grades

8 - $D_{gs}$, geophone depth with respect to source elevation; $D_{gs}=D_{gm}-D_{s}-\Delta e=D_{gm}-D_{s}-29$

9 - ${H}$, horizontal distance of source from wellhead

10,11 - tangent, cosine of angle between straight raypath and vertical; $\tan i={\rm {H}}/{\rm {D}}_{gs}$

12 - $T_{g}$, vertical traveltime from source to geophone $=T\cos i$

13 - $\Delta sd$, source to datum elevation difference: $\Delta {\rm {sd}}=D_{s}-D_{e}=D_{s}-78$; a minus sign means that the shot was above datum

14 - time correction for $\Delta {\rm {sd}}$

15 - $T_{gd}$, vertical traveltime from datum to geophone

17 - $D_{gd}$, depth of geophone below datum

18 - $\Delta D_{gd}$, depth difference between successive geophone depths

19 - $\Delta T_{gd}$, time difference between successive geophone arrivals

20 - $V_{i}$ , interval velocity $\Delta D_{gd}/\Delta T_{gd}$

21 - $T_{gs}$, vertical traveltime from source to geophone $=T\cos i$

Figure 5.14a.  Data from a well-velocity survey.

Figure 5.14b.  Velocity calculations for data in Figure 5.14a.

Figure 5.14c.  Results of well velocity survey.

Depths below the datum are positive. The velocity used to correct $\Delta sd$ is $V=D_{gs}/T_{gs}\approx 0.163/0.102\approx 1.60\ {\rm {km/s}}$ (also obtainable from $D_{gd}/T_{gd}$). Note that the column headed $\Delta {\rm {sd/V}}$ is in milliseconds whereas all other times are in seconds. Column #16 headed $T_{gs}/$average is not used.

Figure 5.14c shows average velocity, interval velocity, and time plotted against depth.

Problem 5.14b

How much error in average velocity and interval velocity values would result from (i) time-measurement errors of 1 ms, and (ii) depth-measurement errors of 1 m?

Solution

1. A 1-ms time error produces an error in the average velocity of 0.1% to 1.5% and an error in the interval velocity of 1.5% to 8.3%.
2. A depth error of 1 m produces an error in the average velocity of 0.03% to 0.9% and an error in the interval velocity of 0.5% to 1.5%.

Table 5.14a. Data for least-squares calculation of $V_{o}$ and $a$.

$z\ ({\rm {m)}}$	$V_{i}\ ({\rm {{m/s})}}$	$\Delta z\ ({\rm {m)}}$	$z\ ({\rm {m)}}$	$V_{i}\ ({\rm {m/s)}}$	$\Delta \ z({\rm {m)}}$
236	2200	255	1970	5400	65
441	2400	155	2038	3700	70
551	2600	65	2118	3900	90
661	2500	155	2228	6500	130
820	3100	165	2360	4000	135
973	2700	140	2506	4400	155
1101	3600	115	2643	4300	120
1203	2400	90	2783	4100	160
1350	4800	205	2928	4500	130
1508	4200	110	3043	4800	100
1603	2200	80	3145	5200	105
1730	7600	175	3243	6000	90
1878	5000	120

Problem 5.14c

Determine $V_{o}$ and $a$ for a velocity-function fit to the data in (a) assuming the functional form $V=V_{o}+az$, where $V$ is the interval velocity and $z$ the depth.

Solution

We can find $V_{o}$ and $a$ by (i) plotting the data and measuring the slope $a$ and intercept $V_{o}$ of the best-fit straight line, or (ii) using the least-squares method (see problem 9.33). The former method is difficult because of the large, irregularly spaced jumps in the curve, and therefore we shall use the latter method. We take $z$ as the depth in meters below datum to the center of each interval $\Delta z$ and give each data pair $(V_{i},\;z)$ the weight (see problem 9.33b) $\Delta z$. Using the data in Table 5.14a, we get $V=2400+0.98z$, as shown in Figure 5.14c.

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Table of Contents (book)

Also in this chapter

• Maximum porosity versus depth
• Relation between lithology and seismic velocities
• Porosities, velocities, and densities of rocks
• Velocities in limestone and sandstone
• Dependence of velocity-depth curves on geology
• Effect of burial history on velocity
• Determining lithology from well-velocity surveys
• Reflectivity versus water saturation
• Effect of overpressure
• Effects of weathered layer (LVL) and permafrost
• Horizontal component of head waves
• Stacking velocity versus rms and average velocities
• Quick-look velocity analysis and effects of errors
• Well-velocity survey
• Interval velocities
• Finding velocity
• Effect of timing errors on stacking velocity, depth, and dip
• Estimating lithology from stacking velocity
• Velocity versus depth from sonobuoy data
• Influence of direction on velocity analyses
• Effect of time picks, NMO stretch, and datum choice on stacking velocity

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