GEOPHYSICS, VOL. 58, NO. 12 (DECEMBER 1993); P. 1726-1738, 13 FIGS.

Tomographic reconstruction of near-borehole slownessusing refracted borehole sonic arrivals

Brian E. Hornby*

ABSTRACT

Two-dimensional (2-D) reconstructions of the near­borehole slowness field are computed using arrivaltimes of refracted borehole sonic arrivals. First-arrivaltraveltimes, derived from both computer simulationsand field data from full-waveform sonic tools, wereinverted for the near-borehole formation slownessboth axially along the borehole and radially away fromthe borehole. The inversion is nonlinear; the solutionis obtained by means of a series of linear inversionsfollowed by provisional ray tracings. Each iterationinvolves the application of a tomographic reconstruc­tion algorithm similar to those used in seismic cross­well tomography or medical imaging applications.

The technique was demonstrated using ray-theoreticexamples to simulate radial variations in slowness. Inaddition, full-waveforms were generated using two­and-a-half-dimensional (2.5-D) FDM computer mod-

INTRODUCTION

The primary function of borehole sonic measurements isthe estimation of compressional-wave formation slownessusing traveltimes acquired via a first motion detection pro­cess. For a basic, two receiver tool, formation slowness isestimated by subtracting the arrival times between tworeceivers and dividing by the inter-receiver spacing. Moresource positions and receivers may be used to attempt tocompensate for such effects as sonde tilt, borehole wash­outs, bed boundary effects, and inaccuracies in individualtraveltime picks.

First, let us consider the case of a simple, two receiversonic instrument. Figure la shows first-arrival raypaths forthe case of a two receiver tool in a borehole surrounded bya homogeneous formation, and Figure lb shows estimatedfirst-arrival raypaths for the case of a damaged zone extend­ing some distance from the borehole. Darker shading indi-

els. The finite-difference method (FDM) computermodels were used to test the validity of the ray­theoretic approximation used in the inversion schemeand to simulate the full-waveform sonic tool responsefor both radial and axial changes in formation proper­ties.

Field data examples highlighted radial changes information slowness caused by two separate mecha­nisms: water take up by swelling shales and themechanical breakdown of the near-borehole rock re­sulting from stress relief caused by the drilling pro­cess. Finally, refracted sonic arrivals from near-bore­hole bed boundaries were identified in a horizontalwell setting. Using refractions arriving beyond theheadwave, a 2-D map of formation slowness wascomputed in the reservoir away from the borehole.Interpretation of the slowness map resulted in anestimation of the stand-off of the horizontal boreholefrom the reservoir boundary.

cates slower formation velocity. Provided that the offsetfrom the source to the receivers is sufficiently long such thatthe first-arrivals probe the undamaged formation, the slow­ness estimate for both cases will be nearly identical and willrepresent the undamaged formation slowness. However,note that the actual value for the arrival times will be greaterfor the damaged formation case and will depend on theactual path the signal takes away from the borehole. It isapparent that simply subtracting traveltimes to arrive at aslowness value actually discards valuable information re­garding the near-borehole formation properties.

In this paper, I demonstrate a technique to use boreholesonic traveltimes to reconstruct a two-dimensional (2-D)map offormation slowness near the borehole. The inversionis nonlinear; the solution is obtained by means of a series oflinear inversions followed by provisional ray tracings. Eachiteration involves the application of a tomographic recon­struction algorithm similar to those used in seismic cross-

Manuscript received by the Editor January 16, 1992; revised manuscript received April 15, 1993.*Schlumberger Cambridge Research, High Cross, Madingley Road, Cambridge CB3 OEL, England.© 1993 Society of Exploration Geophysicists. All rights reserved.

1726

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Reconstruction of Near-borehole Slowness 1727

(1)

b)

well tomography or medical imaging applications. The dis­tance into the formation probed by the technique depends onthe source-to-receiver offset and on the near-borehole slow­ness profile.

Potential applications for this technique include:

I) Depth of investigation for the sonic tool: Determineslowness and distance of penetration away from theborehole. Valuable tie in with AIT' [Array InductionImager, (Hunka et ai., 1990; Howard, 1992)] measure­ments; produce logs of both resistivity and slowness asa function of distance from the borehole.

2) Wellbore stability: Indication of formation damagecaused by stress relief because of the drilling process.

3) Altered zone imaging; quantification of swelling clayeffects.

4) Correction of shallow reading devices: Pad devices, forexample density measurements, have shallow depths ofinvestigations and will read the altered formation prop­erties. If one has a reconstruction of formation slow­ness away from the borehole, then one may be able touse that to correct the pad measurement to the undam­aged formation properties. This may be especiallyimportant for generation of synthetic seismograms.

5) Determination of amplitude/coupling effects and trav­eltime delays (statics) for cross-well surveys and single­well reflection imaging.

6) Cased hole: Imaging of caves caused by sanding duringthe production process; determination of cement thick­ness and location, formation to casing distance, changesin formation properties caused by the cementing process.

7) Invaded zone: Change in saturating fluid propertiesaway from the borehole. Borehole fluid invaded zone togas or oil saturated formation.

I Mark of Schlumberger

a)

Distance from borehole ~

8) Horizontal well logging. Locate bed boundaries awayfrom the borehole and reconstruct the slowness fieldbetween the horizontal borehole and the nearby bedboundaries. Possible results range from a simple esti­mation of the stand-off of the horizontal borehole to thereservoir top to imaging slowness changes caused bythe presence of a gas cap.

METHOD

The goal is to develop a formulation that will give forma­tion slowness in the rock around and away from the boreholefrom the sonic traveltimes. For the damaged zone case Iassume radial symmetry; formation properties are assumedto be invariant with azimuth in the very near borehole region(less than several feet or so from the borehole). Let x =(x r' X z) denote radial and vertical coordinates within therock volume and let /lex) be the unknown slowness function.Suppose we have N receivers in the sonic tool, and the dataset consists of L source positions with all receivers recordedat each source firing. The total number of signal paths in thedata set is K = LN. Traveltimes recorded at the receiversare denoted by t k, k = I, ... , K. Using a high-frequencyapproximation, each tk may be written as

tk = ITk(lI) u(x) ds ,

where ds is arc length and Tk(u) denotes the curve, con­necting source, and receiver, which yields the minimumtraveltime. The relation between t k and /lex) is nonlinearbecause the raypath Tk(u) depends upon the desired result,the slowness function /lex).

Inversion involves a sequence of ray tracing and linearinversions. First, the system is linearized by the use of an

Damaged formation

~_~ First arrival ray paths

o Receiver

Source

Distance from borehole ~

FlG. 1. (a) First-arrival raypaths for the case of a homogeneous formation and (b) first-arrival raypaths for thecase of a damaged zone extending some distance from the borehole. Darker shading indicates slowerformation velocity.

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1728 Hornby

initial slowness model ii. For sonic logging, the initialslowness model may be estimated from computations of fluidslowness, borehole diameter, and virgin formation slowness(estimated as the difference of traveltimes at the largestsource-receiver offset pair). Traveltimes produced by theinitial model are given as

(7)

where Mj is the number of raypaths actually passing throughthe jth cell. An iteration n is defined to have occurred onlyafter all K rays have been processed. An averaged correc­tion factor is then computed and used to perturb the slow­ness matrix. For sonic logging this form of SIRT corre­sponds to perturbing the matrix after corrections arecomputed for all possible source-receiver combinations in agiven data set.

COMPUTER SIMULATIONS

J1a kj J1t kn + I n (6)Uj = Uj + 2 •

I j J1 akj

ART requires that the slowness matrix be updated after eachraypath is traced. An extension to ART is SIRT [simulta­neous iterative reconstruction technique (Dines and Lytle,1979)]. This technique requires that the slowness matrix beupdated only after all K raypaths are traced. An averagedcorrection factor is then applied to the slowness matrix.Although convergence is slower than "pure" ART, SIRT is,in general, more robust. The SIRT equivalent of equation (6)IS

each arrival. Subsequent iterations will define different ray­paths as the slowness function is perturbed, until the slow­ness function and raypaths approach the true slownessfunction and raypath for each source-receiver pair.

Correction factors J1up may be estimated by algebraicreconstruction techniques (ART) (Herman et al., 1971). Astandard ART algorithm is

(3)

(2)

k=I,2, ... ,K,J

t;: = 2: J1akjiijj = I

tk "" ( ii(x) dsJTk(il)

To construct a system of linear equations that can be solvednumerically, a discrete approximation is created for theslowness function u(x). A grid is created that representsdistances radially away from the borehole and verticallyalong the borehole axis. The size of each cell was chosen tobe 15.24 em in length (along the borehole) and 5.08 em inwidth. The length of the cell reflected the inter-receiverspacing, and the width was chosen to give the highestresolution in the radial direction. The direction of signalpropagation is mostly along the borehole; therefore, thehighest resolution result should be in the radial direction, ororthogonal to the direction of the longest signal paths. Ifound that changing the size of the bins did not overly affectthe result; smaller bin sizes could be used that createdessentially the same image.

An initial guess for the slowness function ii(x) is assignedto each grid cell. Using a shooting method, rays are thentraced though the slowness matrix and first-arrival travel­times are computed. The estimated traveltime t k is repre­sented as

Note that the initial raypath is defined by the unperturbedslowness function ul. Because the problem is nonlinear, wemust solve for both the slowness function and raypath of

Here, J1up is a correction to the cell slowness Uj such thatthe residual error J1tk is reduced or equal to zero forsubsequent iterations. This correction is then applied to eachcell through which the kth ray passes.

where J1a kj is the length of the ray k which penetrates cell j,J is the total number of cells intersected by the ray k, n is theiteration number, and iil = iij is an initial estimate for the(average) slowness of cell j. For each iteration n, theresidual error M r is the difference between observed andcomputed traveltimes

Ray-theoretical model

A ray-theoretical model was used to compute a syntheticdata set used to demonstrate the technique. The modelassumes a damaged zone extending away from the borehole.The slowness profile ranges from 472 p.,s/m (144 p.,s/ft) at theborehole wall to 394 p.,s/m (120 p.,s/ft) at a distance of .46 m(1.5 ft) from the borehole wall. Figure 2a shows a plot of themodel along with first-arrival raypaths for one source posi­tion in the borehole. There are 24 receivers and source­receiver offsets ranging from 0.91 m (3 ft) to 4.1 m (13.5 ft).

Figure 2b represents the starting model for the inversion.The background "unaltered" formation slowness of394 p.,s/m (120 p.,s/ft) is used to fill in the initial image grid; noassumptions regarding radial changes in the near-boreholeslowness are made for the starting model. Figure 2c is theresult of the SIRT inversion after 50 iterations; the imagingprocess demonstrates an accurate reconstruction of themodel shown in Figure 2a.

(5)

(4)J

J1tJ: = tk - t;: = 2: J1a kjJ1uj.j~l

Fig. 2. (a) Model used to simulate a damaged zone extending away from the borehole. The slowness profile ranges from472 p.,s/m (144 p.,s/ft) at the borehole wall to 394 p.,s/m (120 p.,s/ft) at a distance of 46 m (1.5 ft) from the borehole wall. There are24 receivers and source-receiver offsets ranging from 0.91 m(3 ft) to 4.1 m (13.5 ft). (b) Starting model for the inversion. Thebackground formation slowness is used to fill in the initial image grid; no assumptions regarding the near-borehole slowness aretaken. (c) Result of the inversion for 50 iterations; the imagingprocess demonstrates a reasonable reconstruction of the modelshown in Figure 2a.

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Reconstruction of Near-borehole Slowness 1729

a)4 4

490g gQl Ql

480 (5 3 (5 3.L:. .L:.

~ 470 ~ ~0 0E .c .cen 460 Cl Cl

:J C C';; 450 0 2 0 2tii tiim 440 Ql QlC U U

~ 430c c1ll 11l

Ci5 4201i5

is is410

400

390 0 00 .5 1.0 0 .5 1.0

Distance from borehole (m) Distance from borehole (m)

b) 4 4

490g gQl Ql

480 (5 3 (5 3.L:. .L:.

~ 470 ~ ~E

0 0.c .cen 460 Cl Cl:J C C';; 450 0 2 0 2tii tiim 440 Ql QlC U U

~ 430c c1ll 1llCi5 420 is is

410

400

390 0 00 .5 1.0 0 .5 1.0

Distance from borehole (m) Distance from borehole (m)

c)4 4

490g gQl Ql

480 (5 3 (5 3.L:. .L:.

~ 470 ~ ~0 0

E .c .cen 460 Cl Cl:J C C';; 450 0 2 0 2tii tiim 440 ~ ~c~ 430

c c

1ll 1llCi5 420 is is

410

400

390 0 00 .5 1.0 0 .5 1.0

Distance from borehole (m) Distance from borehole (m)

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1730 Hornby

FINITE·DIFFERENCE METHOD COMPUTER SIMULATIONS

c)5

I!I

• II4 • II

iii• II

E ..3 II

IICD •I/) •is II • I!I

2 II•• I!III••I!I

00.00 0.01 0.02

Travel time difference (ms)

Two data sets are analysed. The first data set, taken in analtered shale/damaged zone sandstone environment, usesmultiple logging runs with different spacer sections between

from FDM derived waveforms. The excellent agreementbetween the two methods indicates that a high-frequency,ray-theoretical assumption is reasonable for this application.

Figure 4a is a plot of the model used to produce a morecomplex FDM simulation. This model approximates a situ­ation where a soft, altered shale is overlain by a hardlimestone bed. This is a "worst case" scenario in terms offormation changes; velocities change from very fast to veryslow, creating a huge contrast in impedance between thetwo layers. Both the effects of the sharp bed boundary andthe response in a fast, unaltered, formation can be studiedusing this simulation. Figure 4b is the result of the inver­sion process using SIRT and 50 iterations. Note that thereconstructed slowness for both the altered zone sectionand the hard limestone section closely matches the inputmodel for the FDM simulation. The agreement in both thealtered shale and in the unaltered limestone is encouraging.Also on Figure 4b note that the sharp transition caused bythe altered shale to limestone interface is handled very well;the transition zone is smooth and no large deviations areobserved.

EXPERIMENTAL FIELD DATA

FIG. 3. (a) Synthetic waveforms, computed using Randall'scode, and using the same formation model as shown inFigure 2a. First-arrival traveltime picks are indicated on theplot. (b) A comparison of the FDM traveltime picks with theray-theoretical traveltimes. (c) Traveltime differences. Lessthan .02 ms difference is seen between the ray-theoretictraveltimes and traveltimes picked from FDM derived wave­forms. The excellent agreement between the two methodsindicates that a high-frequency, ray-theoretical assumptionis reasonable for this application.

2.5

2.4

2.0

1.81.2Time(ms)

1.0 1.5Time (ms)

0.6

0.5

-- Raytheory--- FDM

- ..., - -~ ~;:,:~~~~vvvv

~A~fif;f!/:'A/A' j:,A '-

_ c, l."'vv_ ~)!NJ v- ~';!!1~ v_ Y/, v

- '1'~LV",,;(\1'1 v

f)J} -yv.., .vyv

~~v~IV I I

4

o-1-----..-""T"'"---'"'...---.--..----r-....---,-........--l0.0

5....----------------,

0.0

a)

24

Finite-difference method (FDM) computer simulationswere performed using a process developed by Randall et al.(1991). Randall's method can simulate the response of asonic logging tool in a fluid-filled borehole and can handleboth radially and axially varying formation properties. Herenumerical simulations serve two purposes: (1) to verify thatthe ray-theoretical assumption is valid and (2)to examine theresponse of the inversion process where formation proper­ties change as a function of depth (bed boundary effect).

Figure 3a is a plot of synthetic waveforms, computedusing Randall's code, for the model shown in Figure 2a.First-arrival traveltime picks are indicated on the plot.Figure 3b is a comparison of the FDM traveltime picks withthe ray-theoretical traveltimes and Figure 3c is a plot of thetraveltime differences. Less than .02 ms difference is seenbetween the ray-theoretic traveltimes and traveltimes picked

b)

....a>.0E:::Jc:....a>>

'(jjoa>cr:

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Reconstruction of Near-borehole Slowness 1731

the source and the first receiver to crea te a large effectivereceiver array. The second data set is take n in a horizontalwell penetrating an oil-bearing shale and bounded by car­bonate layers. In the horizontal well example I use therefracted borehole sonic arrivals to reconstruct a tomo­graphic image of slowness between the wellbore and thecarbonate-shale boundary.

EXAMPLE 1: SANDSTONE AND CLAY ALTERATION

Full waveform sonic data were acquired in a well in Texas .A research prototype digital sonic tool was used to acquirethe data set. The tool has a uniform array of 12 receive rswith 15.24 cm (6 inch) between successive receivers, and thetran smitter to the first receiver spacing can be varied by

means of spacers . The source is fired at equal spacings of15 .24 em (6 inch) and the full waveforms taken at eachreceiver are recorded on magnetic tape. By combiningmultiple logging passes with different spacers a large aper­ture array can be created. For this experiment a total of sixlogging runs were performed creating a large array of 52waveforms with transmitter/recei ver spacings ranging from0.9 m (3 ft) to 8.5 m (28.5 ft). Record ed data for a singlesource position are plotted in Figure 5. Signal amplitude isplotted for each receiver as a function of arrival time.Compress ional-wave arrivals and a large amplitud e , disper­sive borehole arrival are noted on the plot. The largeamplitude , dispersive borehole arrival is a guided modegenerated by compressio nal-wave arrivals that intersect theborehole wall at angles greater than the critical angle (Chang

2o 1Time (ms)

4

_16§.Ql<5s:!!!.8 12Clc:o(ijQlo

~ 8is

o .5 1.0Distance from borehole (m)

500 Ql<5s:

440Ql....

E 0.0

~ 380Clc:0

III (ij

13 320 Qlc: 0:; c:0 260 ~en is

200

140

a)

4

_16§.Ql

<5.r::.Ql....0 12.0Clc:0(ijQl0c: 8III-IIIis

b)

_16§.

500 Ql<5.r::.Ql

440 ....E 0 12.0

~ 380Clc:0

III (ijIII

320 QlQl 0c: c::;~0 260en is

200

140

o .5 1.0Distance from borehole (m)

o 1Time (ms)

2

FIG. 4. (a) Model used to approximate a soft, altered shale overlain by a hard limestone bed. Velocities change from very fastto very slow, creating a huge contrast in impedance between the two layers. (b) Result of the inversion process using SIRT and50 iterati ons. The recon struction of both the altered zone section and the hard limestone section closely matche s the inputmodel for the FDM simulation.

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FIG. 5. Field data example. Combined data set of 52 receiv­ers created using six logging runs of the experimental sonictool with different spacers between the source and thereceiver array. Transmitter/receiver offsets range from 0.9 mto 8.5 m.

Hornby

and Everhart, 1983; Hornby and Chang, 1985; Hornby andMurphy, 1987). This mode is attenuated by conversion toshear at the borehole wall and is normally not detectable.However, the attenuation is very sensitive to the shearmodulus of the surrounding formation and the arrival willhave significant energy as the Poisson's ratio approaches0.5.

This well is believed to have substantial near-boreholealteration. Evidence of the existence of an altered zone wasdemonstrated by Hornby and Chang (1985). Using the large52 receiver array, both a ray-tracing analysis, based onslowness changes as a function of offset, and a frequency­domain analysis identified signals related to the presence ofdamaged zones of increased slowness near the borehole.Indications of near-borehole alteration in both shale andsandstone sections were obtained. The next step is toanalyse these data to attempt to invert for the formationslowness as a function of distance from the borehole.

543nme(ms)

2

\..l~

~ Compressional ~::=:::: :=::

Headwaves

SDispersive -~

Arrival

1o

52

1732

2.(ms)

TraveltimeSlowness Image

350 450 550Slowness (us/m)

o .4 .8 1.2 .5Distance from borehole (m)

1725~~

1675

2.

TraveltimeSlowness Image

350 450 550Slowness (us/m)

o .4 .8 1.2 .5 (ms)Distance from borehole (m)

Uthology

1900

1800 - --.,. - ....

g g.ca. s: 1700<ll a.0 <ll

1850 0

FIG. 6. Altered shale example. An image of near-boreholeslowness is presented alongside first-arrival traveltime picksand a lithology reference. Dark red shading indicates theradial extent of the borehole. The dominant clay mineral issmectite, the radial change in slowness is believed to becaused by borehole fluid invasion into the swelling clays.(lithology key: yellow = sandstone; brown = clay; blue =water; black = residual oil; green = moved oil).

FIG. 7. Sandstone section. The predominant mineral in thissection is quartz, core analysis shows no swelling clayabundance. Radial changes in slowness in the unconsoli­dated sandstone sections is believed to be a damaged zoneeffect caused by stress relief resulting from the drillingprocess.

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Reconstruction of Near-borehole Slowness 1733

Slowness imaging

Digital first-motion detection was used to compute first­arrival times using the method of Eyl et al. (1991) (also seeHsu et aI., 1988). A subset of 20 receivers was chosen forthis analysis, obtained by combining data from two passeswith source to first receiver spacings of 0.9 m (3 ft) and 2. I m(7 ft), respectively. Offsets for the derived 20 receiver arrayrange from 0.9 m (3 ft) to 3.8 m (12.5 ft).

Figure 6 shows the results for a shale section and Figure 7shows the results for a sandstone section. Presented are avolumetric analysis, a slowness image computed using thepreviously described method, and first motion detectedtraveltimes. The volumetric analysis procedure uses mea­sured electrical, nuclear, and acoustic surveys to estimate afluid and lithology volumetric analysis for the formationssurrounding the wellbore. Figure 8 shows a plot of the rmsreconstruction error for the shale section. The rms error isdefined as

(8)

where t m is the measured traveltime, t r is the reconstructedtraveltime, and N is the number of source-receiver positionsused in the reconstruction. After 30 iterations the rms errorremains low, below 2 us: A comparison of reconstructed(dashed) traveltimes and first-arrival traveltime picks (solid)for a section of the shale section is presented in Figure 9. Thegood match between the reconstructed and measured trav­eltimes is encouraging.

Example 1: Interpretation

A special sidewall core analysis was performed on 100sidewall core samples. Figures lOa and lOb show a subset ofthose results. Figure lOa corresponds to core samples takenin the interval represented in Figure 6. The core analysisindicates that the dominant clay mineral is smectite withlesser amounts of kaolinite and illite. Smectite is a swellingclay. Water take up and reaction with the drilling mud can

35.--------------------,

30

25

~poQ;

CIJ::;; 15II:

10

5

drastically change the properties of the clay, resulting in asoftening of the shale in the near-borehole region. Thissoftening effect will create a zone of increased slowness nearthe borehole and is consistent with the results shown inFigure 6. We conclude that the radial change in slownessseen in the shale section was probably caused by boreholefluid invasion into the swelling clays (smectite).

Now consider the upper section shown in Figure 7.Figure lOb is a plot of the core analysis for that section. Thepredominant mineral in this section is quartz and little or nosmectite is found in the sandstone sections. The slow zonenear the borehole is not likely to be caused by borehole fluidinvasion into the swelling clays. What other effect might bethe cause of the disturbance? These sandstone formationsare known to be unconsolidated (Murray, 1961). Stress reliefcaused by the drilling process may damage the weak inter­granular bonds and create a damaged zone around theborehole. A weakening of the frame will tend to slow downthe speed of sound and thus create a zone of increasedslowness around the borehole. A possible interpretation forthe radial variations in the near-borehole slowness for thesandstone section is that they are a result of stress reliefcaused by the drilling process.

EFFECT OF A WASHOUT

In the previous examples, the borehole diameter wasmeasured using a caliper log, and it was used both in theinitial model and as a constraint for the inversion procedure.However, one might ask the question-what would theresult be in the presence of a washout and without using a

1910

-~ 1920-a.Q)

o

10 20 30 40 50Iteration number

60 70 800.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Time (ms)FIG. 8. RMS reconstruction error for the field data example.After 30 iterations the rms error remains low, below 2 us.

FIG. 9. Altered shale example. Comparison of reconstructed(dashed) traveltimes and first-arrival traveltime picks (solid).

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1734 Hornby

caliper log as a constraint? If we assume the bit size to be aninitial estimate, then the borehole diameter change repre­sents a very slow layer placed between the bit size and theformation. The question arises-might it be possible, undergood conditions, to image variations in borehole size as wellas radial variations in formation properties? In Figure 11, aconstant value for the borehole diameter is used in theinversion procedure and is fixed at the bit size of 10 inches.The actual borehole diameter is overlain on the slownessimage. The dark red shading, extending beyond the bit size,indicates slowness values of more than 590 p,s/m(180 p,s/ft),consistent with the borehole fluid speed of 623 p,s/m(190 p,s/ft). The overlay of the slowness image and themeasured borehole radius is very interesting, a good matchis seen between the borehole size variations around thewashout at 2010 m and the variations in the extent of thefluid-filled borehole represented by the dark red shading onthe slowness image.

This encouraging comparison indicates that it may bepossible to image hole enlargements behind the casing, forexample caves caused by sanding during the productionprocess, using first-arrival formation traveltimes.

EXAMPLE 2: HORIZONTAL WELL

Data from a horizontal well survey are shown in Figure 12.The well penetrates an oil bearing shale (Bakken shale,Williston Basin) bounded by a carbonate layer. The "shale"is actually a siltstone. Production is through fractures in thereservoir. The reservoir is approximately 9 m thick. Thepurpose of the horizontal well program is to intersect thevertical fractures over a large interval, thus tremendouslyincreasing production over standard drilling methods.

Figure 12a shows a variable density display of the rawwaveform data, for a single receiver, at a constant source­receiver offset of 2.43 m (8 ft). Presented below the wave­form display is a plot of first-arrival times (solid curves) andthe slowness computed using the first-arrival times (solidcurve). The slowness log indicates a jump from carbonate(app. 187 p,s/m) to the shale (app. 280 p,s/m) at about 3839m.Also on Figure 12b is a reference plot of compensatedgamma ray (cgr) and uncompensated gamma ray (sgr). TheBakken shale is characterized by very high gamma rayreadings; note that the gamma ray log indicates a cleartransition to the shale bed at 3825 m. This depth is roughly13.5 m less than indicated by the first-arrival slownessestimation! Refer back to the waveform display. In theinterval from 3825 m to 3839m, the first-arrival time actuallyincreases from 600 to 780 us, For a source-receiver spacingof 2.43 m, one would expect a corresponding increase in theformation slowness of more than 65 uslin. However, little orno change in the slowness, computed using the first-arrivals,is evident. This effect results when the sonic tool hasactually passed into the shale bed at 3825 m. Because of thepresence of the faster layer (carbonate) near the borehole,the arrivals, refracted away from the borehole and throughthe carbonate bed, actually arrive before the arrivals thatpass along the borehole wall through the shale bed. Whenthe faster layer (carbonate) is a critical distance from theborehole, the arrivals will cross and the first-arrival willcome from the shale bed. At 3839 m there is a jump inslowness, indicating that the first arrival now passes throughthe shale bed only. Careful examination of the waveformdisplay confirms this analysis. A clear indication of a cross­over of the two arrivals can be seen at a depth of 3839 m.

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FIG. 10. (a) Special core analysis for the shale section. The dominant mineral is a swelling clay (sm~ctite). The slow region ~ear

the borehole is believed to be caused by water take up by the swelhng clay. (b) Special core analysis for the sandstone section.The predominant mineral in this section is quartz. The slow zone near the borehole (Figure 7) is believed to be caused by stressrelief resulting from the drilling process.

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Reconstruction of Near-borehole Slowness 1735

FIG. 11. Imaging of a borehole enlargement. The dark redshading, extending beyond the bit size, indicates slowne ssvalues of more than 590 uslt« (180 IJ-s/ft), consistent with theborehole fluid speed of 623 IJ-s/m (190 IJ-s/ft). Overlain withthe image is the caliper-measured borehole radius (whitecurve). A good match is seen between the borehole sizevariations around the washout at 2010 m and the recon­structed variations in the extent of the fluid-filled borehole.

Look closer at the waveform display. At 3839 m, thecarbonate arrival is no longer the first-arrival; howe ver ,the arrival can still be observed and can be followed on thewaveform display to a depth of about 3863 m. At that pointthe signal appears to be disturbed , possibly because of anintersection with fractures. Arrival times were picked fromthis signal for the eight recei vers (dotted curves) and slow­ness was estimated using these traveltimes (dotted curve .lower section). Observe that the slowness, computed usingthese secondary signals, is faster than the first-arrival (shale)slowness and is in the range of 164-197 p.s/m (50-60 IJ-s/ft),which is reasonable for the carbonate rock encountered inthis reservoir. This corroborates the identification of thisarrival as one that refracts away from the borehole andthrough the carbonate bed.

Figure 13 is a tomographic inversion of the near-boreholeslowness using arrival times identified as arrivals refractedthrough the carbonate layer. An average slowness value forthe carbonate is used as the initial slowness background . Theshale-carbonate contrast appears on this image as a sharptransition from the slower shale (yellow) to the faster (blue)carbonate layer. Additional information can help us orientthe image and propose an interpretation. Drilling report sindicate that the well actually was passing below the shalebed from 3802 m to 3825 m; the well is deviated up and intothe shale reservoir at a well offset of 3825 m. I propose thatthe well deviates up into the shale reservoir at 3825 m,continues until the stand-off between the wellbore and the

o .2 .4 .6 .5 (ms) 2.Distance from borehole (m)

LINKS TO SURFACE SEISMIC IMAGING

carbonate-shale boundary is approximately 1 m, (fromFigure 13), and then continues nearly parallel with thecarbonate-shale boundary. The result is a clear indication ofthe stand-off of the borehole from the reservoir boundary.and a quantification of the formation slowness in the reser­voir away from the borehole.

This technique could as well be applied to seismic imagingproblems, such as the determination of the (two-dimension­al) near-surface velocity field from first-arrival times ofdiving/refracted waves.

Velocity structure of the near -surface sediments has beenanalyzed using refracted seismic first-arrival times by vari­ous researchers. Berge and Beskow (1985) inverted seismicfirst-arrival times for a one-dimensional (I-D) estimation ofthe depth-dependent velocity of the seafloor and near-sur­face sediments. Zhu and McMechan (1988) estimated thenear-surface velocity distribution using diving rays. White(\989) and Simmons and Backus (1992) inverted for velocityanomalies and structural interfaces relative to a known(slowly varying) background , and Docherty (1992) solved forweathered layer thickness and lateral velocity using refrac­tion tomography. In all of the se works strong constraints onthe model or substantial a priori information was required ;for estimating velocity anom alies , the slowly varying back­ground model is required. and for estimating the near­surface velocity distribution using diving rays a good initialguess of the laterally homogeneous near-surface velocityprofile was used . In the paper by Docherty (1992), the authorsolves for the thickness of a single layer where the velocit yonly varies laterally, and not as a function of depth . Clearl ythe velocity in the weathered layer will vary as a function ofdepth as well as later ally.

In the present work. a two-dimensional velocity distribu­tion is reconstructed using refracted arrivals, with no a prioriassumptions regarding the near-borehole velocity distribu­tion. The sonic data sets used herein embody a large numberof source positions where each successive source position isequal to the inter-receiver spacing, creating a dense raycoverage of the near-borehole velocity structure. A typicalsonic survey, say for the data acquired in example I,incorporates 20 receiv ers with offsets ranging from 0.9 m(3 ft) to 3.8 m (12.5 ft). Data for 300 m of borehole depthinvolves 2000 source position s resulting in 40 000 source­receiver pairs . Clearl y the inversion process is facilitated bythis large amount of redundant data. If one considers arepresentative seismic center frequency of 40 Hz and arepresentative sonic center frequenc y of 8000 Hz, then thesonic to seismic problem will scale by a factor of 200. Thu san equivalent marine seismic survey will have a source tofirst receiver spacing of 180 m and an interhydrophonespacing of 30 m. For marine surveys, interhydrophonespacings are typically 2 m with typicall y 25 m betweenreceiver groups . With a typical source firing every 25 rn, wecan conclude that a typical seismic survey should have atleast as good ray coverage as the (scaled) borehole sonicsurvey and that tomographic inversion of the near-surfacevelocity field using refracted seismic arrivals is quite reason­able. The depth of penetration of the survey would be limited

TraveltimeSlowness Image

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1736 Hornby

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FIG. 12. Horizontal well example. Variable density display of the raw waveform data, for a single receiver, at a constantsource-receiver offset. Presented below the waveform display is a plot of first-arrival traveItimes (solid curves) and slownesscomputed using the first-arrival traveltime s (solid curve). An apparent refracted arrival from the carbon ate is identified (?) andboth travelt imes and slowness are computed for this arrival (dashed curves).

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Reconstruction of Near-borehole Slowness 1737

by the largest source-receiver spacing of the seismic arrayand the velocity distribution.

CONCLUSIONS

A technique was developed to use traveltime s, computedusing refracted borehole sonic arrival s, to reconstruct for­mation slowness as a function of distance away from theborehole. The technique was illustrated using numericalmodels, computed using both ray-theoretical assumption sand a FDM analysis. Both radial and axial variations inslowness can be handled by the FDM model. This allowedthe investigation of axial variations in formation propert ies(bed boundary effects) coupled with radial variations information properties (damaged zone effects).

The technique was demonstrated using full waveformsonic data acquired in a well that penetrates unconsolidatedsands and swelling shales . Multiple runs of an experimentalfull waveform sonic tool were used to create a large array of52 receivers, of which the first 20 receiver offsets were usedfor the reconstruction. Results for this well showed strongradial changes in slownes s in both an altered shale environ ­ment and for an unconsolidated sandstone sect ion. Watertake up by swelling shales is believed to cause the radialvariations in the shales. Core analysis of samples taken inthis section show a high percentage of swelling clays (smec­tite), corroborating this interpretation. For the damagedsandstone, a preliminary hypothe sis for the cause of thiseffect is that near-borehole damage is caused by the drillingprocess. Sections of the reservoir where the rock strength is

less may be more easily damaged by the drilling proce ss andhence show a larger change in the near-borehole slowness.

Results of this sort , especially if available at the wellsite ,could have impact in drilling and completion, reservoi rtesting (damaged zone/skin), mechanical properties determi­nation , and computation of synthetic seismograms. Notethat for rocks where no radial change s in velocity occur, thetomographic imaging proce ss will not give you any moreinformation on the formation velocities than can be derivedby standard sonic methods. However, confirmation thatthere are indeed no radial variations in velocity is usefulinformation in itself. This enables us to be confident in theassumption that the measured formation velocities and den­sities represent the virgin formation properties.

In addition to reconstructing radial changes in formationproperties, the possibility was explored to use refractedsonic arrivals to reconstruct large borehole size changes, forexample those caused by the presence of washouts. In oneexample a good compari son was seen between a caliper logand the extent of the borehole washout as interpreted from areconstructed slowness map. One possible application forimaging of radial changes would be to image caves, locatedbehind the casing string, caused by sanding during production.

Finally, the possibility of using arrivals refracted throughbedding contrasts away from the borehole to recon structslowness variations away from the borehole for high anglehorizontal well drilling. A horizontal well example was usedto illustrate this application . The analysis of arrivals re­fracted through a fast layer overla ying the reservoir was

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FIG. 13. Tomographic reconstruction of near-borehole slowness in a horizontal well setting . The shale-carbonate contrastappears on this image as a sharp transition from the slower shale (yellow) to the faster (blue) carbonate layer.

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1738 Hornby

successfully used to invert for a 2-D map of formationslowness between the borehole and the reservoir boundary.Interpretation of the slowness map resulted in an estimationof the standoff of the horizontal borehole from the reservoirboundary.

This report focused only on one signal, refracted compres­sional (P) wave arrivals and, except for the horizontal wellexample, first motion detection of the compressional wavesignal. Note that shear (S) waves are also refracted awayfrom and back to the borehole. In addition, both P- andS-waves are reflected off near-borehole bedding (Hornby,1989); incorporation of the reflected arrivals into the tomo­graphic imagingprocess is a possibility. A goal is to computerock properties away from the borehole in the form of Vp,

Vs and VplVs images. For horizontal wells, the pay-offsmay include identification of changes in lithology or cemen­tation properties away from the borehole, and identificationof changes in the fluid saturating the rock (water, oil, gas).

ACKNOWLEDGMENTS

Curt Randall provided software used to compute thefinite-difference method results. I would like to thank thereviewers for their very helpful comments and suggestions.

REFERENCES

Berge, A. M. and Beskow, B., 1985, A method to determine thevelocities of the seafloor and near-surface sediments: Geophys.Prosp., 33, 377-399.

Chang, S. K. and Everhart, A. H., 1983, Acoustic waves along afluid-filled borehole with a concentric solid layer: J. Acoust. Soc.Am., 74, Supp. 8.

Docherty, P., 1992, Solving for the thickness and velocity of theweathering layer using 2-D refraction tomography: Geophysics,57, 1307-1318.

Dynes, K. A. and Lytle, R. J., 1979, Computerized geophysicaltomography: Proc. Inst. Elect. and Electron. Eng., 67,1065-1073.

Eyl, K., Kurkjian, A., Lineman, D., Pierce, E., and Steiner, J.,1991, Method and apparatus for determining compressional first­arrival times from waveform threshold crossings provided byapparatus disposed in a sonic well tool, United States Patent No.4,985,873.

Herman, G. T., Lent, A., and Rowland, S., 1971,ART: Mathemat­ics and applications. A report on the mathematical foundationsand on applicability to real data of the algebraic reconstructiontechniques: J. Theor. BioI., 33, 1-32.

Hornby, B. E., 1989, Imaging of near-borehole structure usingfull-waveform sonic data: Geophysics, 54, 747-757.

Hornby, B. E. and Chang, S. K., 1985, A case study of shale andsandstone alteration using a digital sonic tool: Trans SPWLA 26thAnn. Logging Symp.

Hornby, B. E. and Murphy, W. F. III, 1987, VplVs in unconsoli­dated oil sands: Shear from Stoneley: Geophysics, 52, 502-513.

Howard, A. Q. Jr., 1992, A new invasion model for resistivity loginterpretation: The Log Analyst, 33, 96-110.

Hsu, K., Kurkjian, A. L., and Wiggins, R., 1988,Tool deconvolu­tion and borehole compensation of sonic measurements, 58thAnn. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts,117-120.

Hunka, J. F., Barber, T. D., Rosthal, R. A., Minerbo, G. N., Head,E. A., Howard, A. Q., Jr., Hazen, G. A., and Chandler, R. N.,1990, A new resistivity measurement system for deep formationimaging and high-resolution formation evaluation: Proc. Ann.Tech. Conf., v. omega, Formation Evaluation and ReservoirGeology: Soc. Petr. Eng., SPE-20559, 295-307.

Murray, G. E., 1961, Geology of the Atlantic and Gulf coastprovinces of North America, 164.

Randall, C. J., Scheibner, D. J., and Wu, P. T., 1991, Multipoleborehole acoustic waveforms: Synthetic logs with beds and bore­hole washouts: Geophysics, 56, 1757-1769.

Simmons, J. L., Jr., and Backus, M. M., 1992, Linearized tomo­graphic inversion of first-arrival times: Geophysics, 57, 1482­1492.

White, D. J., 1989, Two-dimensional seismic refraction tomogra­phy: Geophys. J., 97, 223-245.

Zhu, X. and McMechan, G. A., 1988, Estimation of near-surfacevelocities by tomography: 58th Ann. Internat. Mtg., Soc. Expl.Geophys., Expanded Abstracts, 1236-1238.

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