Missing log data interpolation and semiautomatic seismic well ties usingdata matching techniques
Sean Bader1, Xinming Wu1, and Sergey Fomel1
Abstract
Relating well-log data, measured in depth, to seismic data, measured in time, typically requires estimatingwell-log impedance and a time-to-depth relationship using available sonic and density logs. When sonic anddensity logs are not available, it is challenging to incorporate wells into integrated reservoir studies becausethe wells cannot be tied to seismic. We have developed a workflow to estimate missing well-log information,automatically tie wells to seismic data, and generate a global well-log property volume using data matchingtechniques. We first used the local similarity scan to align all logs to constant geologic time and interpolatemissing well-log information. Local similarity is then used to tie available wells with seismic data. Finally,log data from each well are interpolated along local seismic structures to generate global log property volumes.We use blind well tests to verify the accuracy of well-log interpolation and seismic well ties. Applying our work-flow to a 3D seismic data set with 26 wells achieves consistent and verifiably accurate results.
careful integration of multiple data sets in an attemptto understand the distribution of subsurface rock prop-erties. An important step of integrating multiple datasets is the seismic well ties, in which well logs are usedto calibrate the seismic data, which has lower verticalresolution than the well logs. The calibration typicallyinvolves estimating a reflectivity series and time-to-depth relationship (TDR) using the available sonic anddensity logs (White and Simm, 2003). In plays wheresonic and density logs are not acquired in every well,estimating missing logs is an essential step for integrat-ing well-log and seismic data sets.
A simple linear interpolation of missing log data be-tween wells enables estimation of a reflectivity seriesand TDR; however, it does not account for variationsin lithology or structure. Several methods have beenproposed to estimate missing logs that could be usedfor a more accurate seismic well tie. Gardner’s equation(Gardner et al., 1974) has been shown to provide a rea-sonable relationship between sonic and density for alarge number of brine-saturated rock types. In addition,the Faust method (Faust, 1953) and Smith method(Smith, 2007) provide empirical relationships betweenresistivity and sonic logs. Saggaf and Nebrija (2003)note a high interdependence of different log types andapply regularized back-propagation neural networks to
estimate missing portions of sonic logs. Each methodassumes that specific well logs are collected in everywell to carry out the estimation.
An alternative approach is to assume that rock prop-erties do not vary significantly in lateral space, whichallows using density and sonic logs from a nearby wellto estimate a TDR. Because this assumption does nottake into consideration structural or stratigraphic var-iations in lithology, applying a TDR generated at onewell to a nearby well may result in a mistie with the seis-mic data. To account for these variations, the well mustbe correlated to a common geologic time. Wheeler andHale (2014) and Wu et al. (2018) use dynamic timewarping (DTW) (Berndt and Clifford, 1994; Hale, 2013)to correlate multiple well logs. Shi et al. (2017a) use thelocal similarity scan (LSIM) (Fomel, 2007a) to optimallysort and flatten multiple well logs. Once the well logsare flattened or aligned in geologic time, which is analo-gous to a stratigraphic correlation, missing well-log sec-tions can be estimated from the available data byhorizontal interpolation. Bader et al. (2018) flatten welllogs from depth to relative geologic time domain andinterpolate a missing sonic log using several availablesonic logs and several empirical relationships assumingfluid variations have a negligible effect on the well logs.
With a complete well-log suite, including those sonicand density logs estimated by interpolation, we are ableto further tie the wells to seismic data. The manual seis-
1The University of Texas at Austin, Bureau of Economic Geology, John A. and Katherine G. Jackson School of Geosciences, University Station,Box X, Austin, Texas 78713-8924, USA. E-mail: [email protected]; [email protected]; [email protected].
Manuscript received by the Editor 9 February 2018; revised manuscript received 13 October 2018; published ahead of production 07 January2019.This paper appears in Interpretation, Vol. 7, No. 2 (May 2019); p. 1–15, 20 FIGS., 2 TABLES.
Interpretation / May 2019 1Interpretation / May 2019 1
mic well tie involves matching common reflectorsbetween the modeled synthetic and seismic data bystretching and squeezing the synthetic until a desiredcorrelation between the data sets is achieved (Whiteand Simm, 2003). To reduce interpreter bias and im-prove consistency between multiple seismic well ties,several automatic methods have been proposed. Muñozand Hale (2012) use DTW to automatically align realand synthetic seismograms; this approach is extendedto automatically and simultaneously tie multiple wellsto seismic by estimating a synthetic image to tie withthe seismic image ensuring lateral consistency of thewell ties (Muñoz and Hale, 2015). Furthermore, Wu andCaumon (2017) show that laterally consistent seismicwell ties can be achieved by using DTW to correlate syn-thetic and seismic data that are “flattened” to relativegeologic time. An alternative approach to carry outthe seismic well tie is LSIM; Herrera et al. (2014) com-pare DTW with LSIM, showing that both methods cansuccessfully compute a seismic well tie. Their studyshows that using DTW can achieve a higher correlationbetween synthetic and seismic data compared withLSIM; however, the resulting TDR using DTW showsan undesirable oscillatory behavior due to stretchingand squeezing.
Once each well is tied to the seismic data, the highspatial coverage of seismic can be used to understandlateral variations in log properties. Several methodshave been proposed to interpolate log data along localseismic structures. Assuming that available log data areproperly tied to seismic and conforms to seismic imagefeatures, Hale (2010) uses image guided blended neigh-bor interpolation (Hale, 2009) for seismic guided well-log interpolation. Alternatively, Karimi et al. (2017)show that predictive painting (Fomel, 2010) can be usedto interpolate log data along seismic structures to gen-erate accurate starting models for poststack inversion.
Fomel (2016) presents a fast interpolation algorithm forinterpolating scattered data to a regularly sampled grid.Interpolation along seismic structure using well-logdata generates log property volumes that conform towell-log and seismic data sets. Wu (2017) proposes tocompute such a structurally conformable model in theflattened space, in which the seismic and well-log dataare unfaulted and unfolded.
In this paper, we address limitations brought aboutby missing well-log data as well as challenges associ-ated with achieving consistent seismic well ties andpropose a workflow that integrates the data matchingtechniques, LSIM and predictive painting, to estimatemissing logs, tie synthetic seismograms to seismic, andfinally, interpolate all available well-log data along seis-mic structures. We use cross validation with a blind welltest to test the consistency of seismic well ties. We ap-ply our method to tie 26 wells and the 3D Teapot Domeseismic data set.
Teapot Dome data setWe test the proposed workflow by using the Teapot
Dome seismic and well-log data set that were madeavailable by the U.S. Department of Energy andRMOTC. Approximately 1300 wells have drilled into thestructure targeting nine reservoirs between 300 and5500 ft measured depth (Harbert, 2012). The well-logdata set contains 900 wells, and we select a subset of26 wells to test our methods. Also available is a 3D seis-mic data set (188 crosslines, 345 inlines, sampled at0.002 s) that is acquired over the selected wells. Figure 1is a time slice through the 3D seismic volume at 0.72 sillustrating the extent of the 3D seismic data and showsthe location of each well. From the time slice, we observethe dome structure and several faults that bisect thestructure. These structures are obvious in the crosslineshown in Figure 2 and will result in different depth tops
Figure 1. Time slice through seismic data at 0.72 s. The starsindicate the location of each well. The purple well is used asthe reference well for missing log data interpolation.
Figure 2. Crossline 126 from available 3D seismic data. Thedome structure and several faults that bisect the structure canbe observed.
2 Interpretation / May 2019
and thicknesses among rock units between wells. Anoverview of the data set is presented by Harbert (2012).
As discussed later in this paper, not all wells containsonic and density logs that are required to relate well-log data (in depth) to seismic data (in time), makingintegration of well-log and seismic data sets challeng-ing. In addition, as the number of wells increases, itbecomes more challenging to ensure consistency andaccuracy between multiple well ties. Our workflow ad-dresses these limitations and provides a method forvalidating the results.
Missing log data estimation and seismic well tiesWe propose to estimate missing well-log data by first
aligning all well logs to a common geologic time. Align-ing well logs is analogous to stratigraphic correlationand allows us to interpolate missing sonic and densitylogs using the available logs. After missing well logs areinterpolated, we remove the alignment shifts, thus shift-ing available and interpolated logs back to the well’soriginal domain. The predicted velocity and densitywell logs provide the minimum required logs to forwardmodel a synthetic seismogram assuming no changes influids between wells. In the next step, synthetic seismo-grams are modeled using the complete well-log suites ateach well location and then are semiautomaticallymatched with a nearby real seismic trace.
Data alignment using local similarityMatching data sets involves aligning similar wave-
forms between two data sets. Whether aligning twologs from different wells or aligning a modeled syn-thetic seismogram with a seismic trace, we focus onmatching a response that corresponds to similar lithol-ogies between the two data sets or a common relativegeologic time. In comparing two data sets, our purposeis to estimate the warping function Sk required to alignone data set hk to a reference data set rk:
rkðtÞ ≈ hkðSkðtÞÞ: (1)
We can represent the warping func-tion SkðtÞ as follows:
SkðtÞ ¼ tþ gkðtÞ; (2)
where t denotes the original indepen-dent axis and gkðtÞ is the shift requiredto match the data sets as defined inequation 1.
We estimate the warping shifts gkðtÞby using the LSIM method based on thecorrelation coefficient, which can beused to quantify the quality of the matchbetween data sets (Hampson-Russell,1999). The LSIM method begins withthe observation that the correlation co-efficient only provides one number to
describe the match; however, we are interested inunderstanding the local changes in the data sets’ simi-larity. Therefore, the LSIM method computes local sim-ilarity ct, which is a function of time t. The square of ccan be split into a product of two factors (Fomel,2007a):
c2t ¼ rt � ht; (3)
where rt and ht are the regularized least-squared inver-ses (Appendix A). This problem is posed as a regular-ized inversion where regularization operator is definedusing shaping regularization and designed to enforcesmoothness (Fomel, 2007b). To visualize LSIM, the in-version is calculated for a series to shifts. The results ofthis calculation are accumulated and displayed on a“similarity scan” as shown in the following synthetic ex-ample. From the similarity scan, we automatically pick
a) b)
Figure 3. (a) Reflectivity series in time convolved with (b) a30 Hz Ricker wavelet to model a seismogram.
Synthetic model
Tim
e (s
)
Shifted synthetic model Matched synthetic model
0.4
0.2
0
Tim
e (s
)0.
40.
20
Tim
e (s
)0.
40.
20
a) b) c)
Figure 4. (a) Modeled seismogram, (b) modeled seismogram shifted by 40 ms,and (c) realigned seismogram using the shifts estimated from the LSIM.
Interpretation / May 2019 3
the series of shifts along the entire length of the refer-ence data set that optimally aligns the two data sets(Fomel and Jin, 2009).
To illustrate the alignment of two data sets using lo-cal similarity, we use two examples based on the simplemodel shown in Figure 3. In our first example, we applya 40 ms shift to the modeled synthetic seismogramand use LSIM to estimate the shifts to realign the shiftedsynthetic model with the original synthetic model(Figure 4). Estimation of the shifts for the first exampleis visualized in an LSIM shown in Figure 5. In our sec-ond example, we add 15% random noise to the reflec-tivity model and convolve the noisy reflectivity with a30 Hz Ricker wavelet to create a noisy reference trace.LSIM is used to estimate the shifts to realign the shifted
synthetic model with the noisy reference trace inFigure 6. Estimation of the shifts for the second exam-ple is visualized in an LSIM shown in Figure 7. From oursynthetic examples, we observe that shifts can be accu-rately estimated to align a modeled seismogram with anoise-free and noisy reference seismograms. We useshifts estimated from local similarity to align multiplewell logs and perform seismic well ties.
Missing log data estimationBuilding on our previous work (Bader et al., 2018),
we estimate a complete sonic log using all other soniclogs in the data set and compare it against the actualsonic log from the well. These results are also comparedagainst a conventional approach for estimating missingsonic logs. We then extend the approach to honor truewell-log values for well logs that have incomplete or par-tial well logs.
There are several potential sources of informationthat can be used to constrain the estimation of missinglog data: (1) the same well-log type at other well loca-tions, (2) other well logs within the same well, and(3) the seismic data. We focus on using other well logsof the same type in our estimation of a missing log. Ingeneral, we include information from all other wells inour estimation:
26664W1
W2
..
.
WN
37775~l ≈
26664W1
bl1W2
bl2...
WNblN
37775; (4)
where our estimated log ~l is a weighted function of welllogs from different wells denoted by the subscript k. Ifwe simplify the prediction to one unknown log and oneknown log, equation 4 simplifies to the following linearrelationship:
WkðzÞ~lðzÞ ≈WkðzÞblkðSkðzÞÞ; (5)
where WkðzÞ weights the specific valueused to estimate the missing log value~lðzÞ from the available well logblkðSkðzÞÞ. To estimate a missing log ateach depth sample, we must first removethe structural and stratigraphic variationsbetween the well logs by correlating thewell logs to common geologic time usingthe function SkðzÞ, based on the shifts es-timated from LSIM. The correlation isdone by selecting a well-log type that isavailable in all wells, for example, thegamma ray log. We then select one refer-ence gamma ray log and estimate thefunction SkðzÞ that aligns all remaininggamma ray logs to the reference. Thefunction SkðzÞ is applied to the remaining
Relative time shift (s)
Shift scan – Initial
Tim
e (s
)
Figure 5. Similarity scan and picked optimal shifts (the whitecurve). Warm colors represent high similarity, whereas coolcolors represent low similarity.
Noisey synthetic model Shifted synthetic model Matched synthetic modela) b) c)
Tim
e (s
)0.
40.
20
Tim
e (s
)0.
40.
20
Tim
e (s
)0.
40.
20
Figure 6. (a) Noisy reference trace, (b) modeled seismogram shifted by 40 ms,and (c) realigned seismogram using the shifts estimated from the LSIM (right).
4 Interpretation / May 2019
well logs to align all well logs (density and velocity) toconstant geologic time.
We design the weightWkðzÞ, in equation 5, as a prod-uct of two factors: the distance between the unknownand available well logs and the caliper value at thatdepth, which measures the size of the borehole at eachdepth. We assume that the borehole is drilled to be aspecific diameter and deviations, measured by the cal-iper, from this anticipated borehole size likely indicatesan inaccurate log measurement. Although many envi-ronmental factors may impact the well-log data, for sim-plicity we weight the log values in our inversion basedon caliper information. Thus, WkðzÞ can be expressedas
WkðzÞ ¼ ϕðjx − xkjÞ � CkðSkðzÞÞ; (6)
where ϕðjx − xkjÞ is a radial basis function, xk is the welllocation, x is the well with a missing log, and Ck isinversely proportional to the deviation between the ex-pected and actual caliper value at each depth.
There are several different radial basis functions(Powell, 1985). We chose to implement the inverse mul-tiquadratic radial function
which gives a larger weight to a well closer to the un-known well as compared with a well farther away.
Returning to our original linear relationship, equa-tion 4, the estimated log ~l is a function of available welllogs weighted by each well’s distance and caliper log.By solving the least-squares problem in equation 4, wecan predict a new “pseudo well log” at each depth asfollows:
glðzÞ ¼ PNk¼1 W
2kðzÞblkðSkðzÞÞP
Nk¼1 W
2kðzÞ
: (8)
Teapot Dome well-log exampleWe use wells from the Teapot Dome data set to test
the proposed approach. As discussed previously, morethan 1000 wells have been drilled into the anticlinestructure. We select only a limited subset of 26 deepestwells for our examples. Several wells are missing sonicor density logs making it challenging to integrate theavailable log and seismic data. Table 1 summarizesthe parameters of the initial well-log data set.
From equation 1, to align all wells to constantgeologic time, we estimate the warping function SkðzÞ.Because gamma ray logs are available in all wells, weuse them to estimate the warping function. The deepestgamma ray log is selected as the reference log rðzÞ, andthe well containing this log is denoted by the purple starin Figure 1. The gamma ray and sonic log in the refer-ence well are compared with the gamma ray and soniclog in a different well in Figure 8a and 8b, respectively.The shifts are estimated by matching the gamma ray logfrom each well to the reference gamma log as shown inFigure 9a. The alignment shifts are then applied to theremaining well logs in each well to align all well logsto constant geologic time. Results of aligning a soniclog before and after applying the shifts estimated fromaligning the gamma ray logs are shown in Figures 8band 9b, respectively.
This approach results in well logs that are flattenedalong a common geologic time. The log data were col-lected over several years, with different logging tools,and likely different techniques applied to process thedata. To account for this variability, we normalize thesonic and density logs using the big histogram method(Shier and others, 2004). For normalization, we select15 intervals based on available well-log tops and lithol-ogy variations. We estimate the cumulative mean andstandard deviation for all well-log data in each interval.We assume that distribution of well-log data from eachwell, in each interval, should fall within one standarddeviation of the cumulative mean. With the aligned andnormalized sonic logs, we estimate the missing soniclogs, or sections of sonic logs using equation 8.
We perform a blind well test to validate the proposedapproach by estimating a sonic log in a well in whicha real sonic log is available. For comparison, we useavailable density information and the reverse Gardner
Relative time shift (s)
Shift scan – Initial
Tim
e (s
)
Figure 7. Similarity scan and picked optimal shifts (the whitecurve). Warm colors represent high similarity, whereas coolcolors represent low similarity.
Table 1. Original well-log data statistics.
Log type Wells Sonic Density Caliper Gamma
Number 26 15 22 26 26
Mean length (ft) 4192 2646 3093 4074 4074
Interpretation / May 2019 5
equation (Gardner et al., 1974) to estimate the sonic log;this result is cross plotted against the real sonic log forthe entire well in Figure 10a. When estimating the re-verse Gardner equation, we break the well into 15 inter-vals based on the well tops and changes in lithologyfrom the gamma ray log and we recompute the equationthat best fits the data for each interval. The estimatedsonic log using the proposed approach is cross plottedagainst the real sonic log for the entire well inFigure 10b. We observe significant improvement in theproposed approach over a conventional method forestimating a missing sonic log. Results comparing thesonic log estimated using the proposed approachagainst the real sonic log along two 1600 ft intervalsare shown in Figure 11. Based on the results of our blindwell test using field data, the proposed approach pro-vides a reasonable first-order approximation of the un-known well logs and can be implemented to predictmissing well-log data.
From Table 1, we observe that most wells have sonicand density logs; however, the difference between the
mean length of the sonic/density logs as compared withthe gamma log indicates that several of the well logswere not acquired over specific intervals or have a miss-ing section as shown in Figure 9b. For well logs thathave a missing section or are partially complete, we in-clude the available log data in equation 8 to honor theavailable measurements and interpolate the missing logsections. In Figure 12, we interpolate missing log data inwhich there are holes in the original log and use truewell-log measurements when available.
We applied the proposed approach to all 26 wellsfrom our subset of the Teapot Dome data set to gener-ate complete sonic and density logs for each well.Table 2 summarizes the log data set after estimatingmissing or incomplete logs.
By estimating missing sonic and density logs for eachwell, we increase the number of logs and the section ofthe log available to integrate with the available seismicdata. Using the interpolated sonic and density logs, it ispossible to compute a TDR and reflectivity series to tieany given well to seismic data.
Semiautomatic seismic well tiesAlthough the log data provide one
source of information to understand thesubsurface, an additional source of in-formation is the 3D seismic data set.Seismic well ties can be used to cali-brate seismic data, which has verticallylower resolution but higher spatial cov-erage, whereas the well logs can providevertically higher resolution but are mea-sured only at limited locations.
Synthetic seismograms are modeledindependently for each well. White andSimm (2003) argue that modeling syn-thetic seismograms benefits from block-ing or upscaling of the logs. Followingtheir suggestion, we upscale the sonicand density logs to seismic frequencies(Backus, 1962; Marion et al., 1994) andestimate an initial reflectivity seriesr0ðzÞ in depth assuming no multiples, at-tenuation, or dispersion:
where v0 is the initial, upscaled, P-wavevelocity from sonic in m∕s, ρ is the den-sity in gm∕cm3, and Δz is the samplinginterval of the log in depth. To relate re-flectivity in depth to seismic data intime, we must compute a TDR. Thereare several ways to compute a TDR.Using available checkshot surveys orvertical seismic profiles can provide ac-curate measurements of seismic travel-
a) b)
Figure 8. (a) Median filtered gamma log from the reference well (red) andmedian filtered gamma log from a second well (black) before applying the align-ment shifts. (b) Sonic log from the reference well (red) and sonic log from asecond well (black) before applying the alignment shifts.
6 Interpretation / May 2019
times to known depths; however, these surveys arenot available to us, so we must estimate a TDR fromwell sonic logs. We define the initial TDR as a functionof depth at each well,
T0ðzÞ ¼ 2Z
z
zmin
dξ
v0ðξÞ; (10)
where T0 is the initial TDR, zmin is theminimum depth at which sonic informa-tion is available, v0ðzÞ is the initial, up-scaled, P-wave velocity from sonic,and dξ is the depth increment.
The initial TDR relates the initial re-flectivity series r0ðzÞ to time. We inter-polate the resulting reflectivity seriesin time to a regularly sampled grid of0.002 s, which corresponds to the verti-cal sampling of the seismic data:
r0ðtÞ ¼ r0ðT0ðzÞÞ: (11)
We model synthetic seismogramsby convolving r0ðtÞ with a single, zero-phase, wavelet that is representativeof the seismic data’s frequency content.The zero-phase wavelet extracted usingHampson-Russell software is shown inFigure 13.
Using the statistical wavelet in Fig-ure 13 and the initial TDR, we computea synthetic seismogram shown in Fig-ure 14 (green). We then iteratively esti-mate the alignment shifts gk by usingthe LSIM method to match the syn-thetic seismogram (red) to the corre-sponding real seismic trace in Figure 14(black).
In the time domain, the shifts gk;iðtÞat well k are estimated using several iter-ations i of LSIM data matching. Eachiteration estimates a smooth sequenceof shifts to align the synthetic seismo-gram with the seismic trace. Muñozand Hale (2015) and Herrera et al.(2014) observe a relationship betweenthe shifts used to align a synthetic withseismic trace and an updated velocityfunction.
From equation 2, assuming an initialTDR, T0, we arrive at updated estimate
Sk;1ðT0Þ ¼ T0 þ gk;1ðT0Þ (12)
after one iteration of LSIM. We estimatean updated TDR by interpolating ourshifts from time to depth
T1ðzÞ ¼ T0ðzÞ þ gk;1ðT0ðzÞÞ: (13)
Using equation 10, we relate the initial and updatedvelocity log to the initial and updated TDR,
a) b)
Figure 9. (a) Median filtered gamma log from the reference well (red) and analigned gamma log from a second well (black) after applying the estimated align-ment shifts. (b) Sonic log from the reference well (red) and an aligned sonic logfrom a second well (black) after applying the estimated alignment shifts frommatching gamma logs.
a) b)
Figure 10. (a) Real sonic log crossplotted against the sonic log estimated usingthe reverse Gardner equation. (b) Real sonic log crossplotted against the soniclog estimated using the proposed approach.
Interpretation / May 2019 7
dT1ðzÞdz
�dT0ðzÞdz
�−1
¼ v0ðzÞv1ðzÞ
(14)
to solve for the updated velocity log,
v1ðzÞ ¼ v0ðzÞdT0ðzÞdz
�dT1ðzÞdz
�−1: (15)
In our implementation, we use equation 15 to updatethe velocity function after each iteration. Alternatively,we can directly relate the updated velocity log to theinitial velocity log and the estimated shifts. Starting withthe derivative of equation 13,
dT1ðzÞdz
¼�1þ gk;1ðT0ðzÞÞ
dT0
�dT0ðzÞdz
; (16)
we substitute equation 15 and solve for the updatevelocity log as follows:
v1ðzÞ ¼ v0ðzÞ�dgk;1ðT0ðzÞÞ
dT0þ 1
�−1: (17)
We update the velocity function and recomputeequations 9–11 after each iteration of estimating shiftsusing LSIM. We slowly reduce the smoothness enforcedby regularization in LSIM with each iteration to ensurethat stretching and squeezing are not excessive thus re-sulting in an improbable velocity update (White andSimm, 2003).
Prior to performing the seismic well tie, we need tounderstand phase variations and distortions introducedduring the processing and imaging of the seismic data.There are several seismic processing and imaging tech-niques that adjust or correct the seismic data to zerophase. Information on phase adjustments applied tothe data is not available; however, Harbert (2012) inter-prets the deepest continuous reflection to be Precam-brian basement resulting in a positive amplitude (inthis paper, we define a positive amplitude as relatedto a positive reflection coefficient). As provided bythe U.S. Department of Energy and RMOTC, the base-
ment reflection is a negative amplitude.To account for the observed lateraland vertical phase variations, we applylocal skewness correction (Fomel andvan der Baan, 2014) resulting in a zero-phased seismic volume consistent withobservations from Harbert (2012).
Seismic well tie exampleTo demonstrate our approach, we tie
26 wells using the well-log informationsummarized in Table 2 and the phase-ad-justed 3D seismic data. Each seismicwell tie is computed independently bymodeling a synthetic seismogram, esti-mating the shifts using LSIM, and com-puting an updated velocity function.
One example of a semiautomatic seis-mic well tie is shown in Figure 14. Thesynthetic modeled from the originalsonic log is compared against the syn-thetic modeled from a sonic log updatedby shifts estimated from four iterationsof matching using LSIM and the closesttrace from the phase adjusted seismicdata set. The high-amplitude reflectorsbetween 0.75 and 1.10 s are well-alignedafter four iterations of shifts are esti-mated to update the sonic log.
The initial Backus-averaged sonic,updated sonic, and original sonic logsare shown in Figure 15a. We observethat most adjustments to the sonic logoccur between 3500 and 3900 ft. The ad-justment can also be observed by com-paring the initial and updated TDRs in
Figure 11. We perform a blind well test by estimating a sonic log using the pro-posed approach and comparing the result against the real sonic log. The real soniclog (black) versus the estimated sonic log (magenta) along two 1600 ft intervalsalong the reference log.
8 Interpretation / May 2019
Figure 15b. The differences between synthetic seismo-grams modeled from well-log data and seismic data areusually attributed to either inaccuracies in the seismicphase or seismic migration velocities (White, 1998;Henry, 2000). The bulk shift between the initial and finalTDR is related to the missing shallow velocity section inthe well log. The results in Figure 15 provide an initialqualitative assessment of a seismic well tie to ensurethat the estimated shifts do not result in an improbableupdate to the sonic log. We overlay the modeled andtied synthetic seismogram with the crossline that cutsthrough the well and observe a reasonable tie with theseismic data, even in the presence of a fault (Figure 16).
Validation by interpolation of log data along seismicstructures
To qualitatively assess the result of each seismicwell tie, we interpolate well-log data from wells alongseismic structures. By generating global log property
volumes, we can verify the lateral continuity of anlog property and perform a blind well test to validatea seismic well tie.
Interpolation using predictive paintingTime dip describes how a seismic event changes
from one trace to the next. If available, the local dipscould be used to interpolate log data along seismicstructure and predict an expected log profile in a loca-tion with no well-log data. Similar to Karimi et al.
a) b) c)
Figure 12. (a) The original sonic log (black) versus the estimated sonic log (magenta). (b) The original density log (black) versusthe estimated density log (magenta). The original well-log data are used in the inversion, so in areas where the sonic or density dataare available, the estimated logs match the original data. In the interval between 3500 and 3900 ft, there is a significant deviation inthe (c) caliper log indicating an inaccurate measurement; therefore, the estimated log deviates significantly from the original log.
Table 2. Well-log data statistics after estimating logsfor all wells.
Log type Wells Sonic Density Caliper Gamma
Number 26 26 26 26 26
Mean length (ft) 4192 3991 3639 4074 4074
Interpretation / May 2019 9
(2017), we generate log property volumes by weightingpredictive painting. Predictive painting is defined usingplane-wave destruction filters that measure the localslope of seismic events (Fomel, 2002). The local slopeof seismic events is used to predict one trace from an-other trace and can be used to interpolate a referencewell log through a seismic volume (Appendix B)(Fomel, 2010). The interpolation based on the distancebetween the reference well and any location is the seis-mic data set, as defined in equation 7. The RBF and logproperty volumes generated from data at each welllocation are combined to form a single log propertyvolume using the following interpolant:
Vðx; tÞ ¼P
Nk ϕðjx − xkjÞSkðx; tÞP
Nk ϕðjx − xkjÞ
; (18)
where Sk is the volume created by spreading log datafrom location xk to the entire seismic data set using pre-dictive painting and N is the total number of wells usedin the interpolation.
Computing log property volumesWe use 26 wells to compute the log property distri-
bution throughout the Teapot Dome seismic survey.Complete density and sonic logs are estimated and tiedto the seismic using the proposed approaches men-tioned in the previous sections. Figure 1 is a time slicethrough the 3D seismic volume at 0.72 s and shows thelocation of each well. Figure 17a–17c shows the phaseadjusted seismic data and estimated inline and cross-line dip using plane-wave destruction filters.
The reflection dip is used in the predictive paintingalgorithm and the RBF interpolant from equation 18 togenerate global log property volumes. The inputs to theinterpolated sonic volume are the original sonic loginterpolated to time using a TDR updated from shiftsestimated using four iterations of LSIM matching. Theresults from interpolating sonic logs from 26 well are
shown in Figure 18a. We observe reasonable lateralcontinuity along seismic structures indicating that thereare no significant misties between the well and seismicdata. Similar to the interpolated sonic volume, the inter-polated the density volume shown in Figure 18b hasreasonable lateral continuity along seismic structuresand shows no evidence of a mistie. Qualitative interpre-tation of these results suggests that the estimated wellties are laterally consistent.
Performing a blind well testThe accuracy of the seismic well ties can be addition-
ally checked by removing a well from the interpolationscheme and performing a blind well test. An inconsis-tent seismic well tie results in a misalignment betweenthe predicted and actual log at the well location. Weperform a blind well test using two wells from the 26well data set. Results shown in Figure 19 indicate aclose match between the predicted and actual sonic
Figure 13. Statistical wavelet extracted from the TeapotDome seismic data set.
Well tie example 1
Tim
e (s
)
Figure 14. Synthetic modeled using the initial sonic log(green). Synthetic modeled using the sonic log updated afterfour iterations of matching using LSIM to estimate shifts (red).The closest trace to the well location extracted from the phaseadjusted seismic data (black).
10 Interpretation / May 2019
log at both well locations confirming thelateral consistency in the seismicwell ties.
To understand the accuracy of allseismic well ties, we proceed to performblind well tests at each well using theremaining wells as input. Results of theactual versus predicted sonic at all 26wells are cross plotted in Figure 20.
The results shown in Figure 20 indi-cate the predicted sonic matches rea-sonably well with the real sonic at all 26wells giving us confidence that all seis-mic well ties are consistent and the re-sulting TDR for each well accuratelymaps the logs from depth to time.
DiscussionThe proposed approach addresses
several challenges in integrated studies,specifically (1) interpolating missingwell logs at wells that have incompletewell-log suites and (2) providing amethodology for semiautomatically ty-ing wells to seismic and validating theconsistency of the ties. In our example,the proposed approach accuratelycomputed a TDR at all well locationsregardless of the completeness of the in-itial well-log suite. Consequentially, inte-grated studies need not be constrainedto pilot wells in which full log suitesare collected. The proposed approachshould be particularly useful in onshoreplays in which the number of wellsdrilled is much higher compared withthe number of sonic and density logs acquired.
Our method involves interpolation techniques thatassume that rock properties do not vary significantlylaterally. We make several additional assumptionsrelated to the interpolation of missing log data:
1) Gamma logs are matched to estimate the alignmentshifts; therefore, estimated section is limited to sec-tion in each well with available gamma log.
2) All gamma logs are aligned with a single referencegamma log, and the estimated log section is limitedto the stratigraphy found in this reference log. Thisreference well log can be thought of as a type of logthat contains the entire stratigraphic column ob-served in other well logs.
3) We did not perform fluid substitution prior to solv-ing equation 8 for each well. The proposed approachis based on interpolation, and we assume fluid sub-stitution to have a negligible impact on the results.This assumption may present challenges in reser-voirs in which hydrocarbons impact the well-logresponse within the same stratigraphic interval.
Tru
e de
pth
(ft)
Sonic (µs/ft) Time (s)
Tru
e de
pth
(ft)
Figure 15. (a) Estimated sonic log using the proposed approach after Backusaveraging (green). Updated sonic log after four iterations of matching using LSIMto estimate shifts (red). Estimated sonic log after interpolation of missing data(black). (b) TDR from the estimated sonic log (green). Updated TDR after fouriterations of matching using LSIM to estimate shifts (red).
Figure 16. Seismic crossline through the well in Figures 14and 15. We observe a good tie between the modeled syntheticand real seismic data. The sonic and density logs usedto model the synthetic are estimated using the proposedapproach.
Interpretation / May 2019 11
These assumptions may not be valid in geologicallycomplex areas with significant stratigraphic variationssuch as unconformities or channels in which entirestratigraphic units may be absent due to erosion. In ad-dition, the well-log correlation approach may meet sim-ilar challenges as those experienced by conventional,interpreter-driven workflows in which rapid strati-graphic variability (e.g., slope deposits and clinoforms)may not correlate, or they may correlate ambiguously inseveral places, between wells. Although we did not ac-count for changes in the fluid content, the proposedapproach provides a reasonable first-order approxima-tion of the unknown well logs. The predicted velocityand density well logs provide the minimum requiredlogs to forward model a synthetic seismogram andtie the well with real seismic data.
In addition, well-log data interpolation by predictivepainting may result in errors when crossing faults. Weobserve this challenge when comparing the fault inFigure 17a with the log property volumes in Figure 18.The fault in the property volumes is not accounted forduring interpolation. Interpolation schemes that ac-count for discontinuities would further improve results.Recently suggested approaches by Xue et al. (2017) andShi et al. (2017b) address predictive painting acrossfaults.
Our methodology can also be directly impacted byerrors in the seismic data. An incorrect migration veloc-ity will improperly place reflectors, which will result inan incorrect TDR estimated from seismic well ties. In-accuracies in the migration velocity may be a reason forthe need for a velocity log update shown in Figure 15a.
The assumptions we make and errors in migrationcan compound resulting in inaccuracies in our inte-grated study. However, by relating several sources ofinformation (multiple well logs and 3D seismic), we pro-vide an approach and validation technique to minimizethe impact of these challenges and to provide a bettercharacterization of the subsurface.
Figure 17. (a) Phase-adjusted seismic amplitude data, (b) inline dip, and (c) crossline dip estimated using plane-wave destructionfilters.
a)
b)
Figure 18. (a) Interpolated sonic and (b) interpolated den-sity based on logs from 26 wells and the interpolant describedin equation 18. Note that the interpolated log data follow theseismic structure.
12 Interpretation / May 2019
ConclusionWe present a workflow for integrating
available well-log data and 3D seismicdata. As indicated by computational ex-amples in this paper, the proposed work-flow allows us to predict missing orincomplete well-log suites by using LSIMto align all well-log data to common rel-ative geologic time. Therefore, our nextstep of seismic well ties is not limited towells that have complete sonic and den-sity well-log data. The wells, with com-plete or interpolated well-log suites, aretied to a 3D seismic data set using shiftsthat are semiautomatically estimated us-ing LSIM. The shifts are used to updatethe initial TDR and sonic logs providinga qualitative assessment of the seismicwell tie.
We verify lateral consistency of seis-mic well ties by interpolating log datafrom all available wells along seismicstructures using predictive painting andperforming a blind well test. Our resultsusing well logs and a 3D seismic dataset demonstrate that the proposed ap-proach can consistently and accuratelytie well-log data to seismic.
The approach has some limitations. Inthe field data examples reported in thispaper, we assume that the reference typelog is representative of the entire strati-graphic column found in other wells;this assumption may cause challenges
in plays in which rapid stratigraphic variations such asunconformities, channels, or clinoforms may be present.In addition, we did not account for fluid variability in therocks, which can cause different well-log responses inthe same stratigraphic unit. Although fluid substitutionmay further improve the results, the approach proposedin this paper provides a reasonable, first-order, approxi-mation of the unknown well logs.
Although the uncertainty of the models comparedwith the true earth model is not considered in this pa-per, one can assume that each constraint added to theproposed workflow (fluid substitution and modeling)will cast the model into a range of more accurate re-sults. We show, by using field data examples, that theproposed approach is a feasible workflow for overcom-ing the challenges with the conventional seismic well tieworkflows. Further refinement of the well-log informa-tion should only improve the results.
The purpose of integrating seismic and well-log datais to achieve reservoir characterization that is mostconsistent with all available data and achieves the high-est possible resolution. Our approach includes previ-ously ignored wells in seismic well ties and providesa method for verifying the consistency and accuracyof the results.
a) b)
Figure 19. Predicted (green) and actual (black) sonic logs from two differentwells using a blind well test. The predicted and actual sonic logs match along theentire length of the well log indicating consistency in seismic well ties.
Figure 20. Real sonic log crossplotted against the predictedsonic log from the blind well test for all 26 wells. Each blindwell test used the remaining 25 wells as input.
Interpretation / May 2019 13
AcknowledgmentsWe thank the Rocky Mountain Oilfield Testing
Center for making the Teapot Dome data set available.We thank the sponsors of the Texas Consortium forComputational Seismology (TCCS) for their financialsupport. All computations in this paper were done usingthe Madagascar software package (Fomel et al., 2013).
Data and materials availabilityData associated with this research are available and
can be accessed via the following URL: https://wiki.seg.org/wiki/eapot_dome_3D_survey.
Appendix A
Local similarityIn comparing two data sets, the purpose is to esti-
mate a smoothly varying warping function Sk requiredto align one data set hk to a reference data set rk
rkðtÞ ≈ hkðSkðtÞÞ: (A-1)
We can represent the warping function with the shiftsgkðtÞ as follows:
SkðtÞ ¼ tþ gkðtÞ; (A-2)
where t denotes the original independent axis and gkðtÞare the shifts required to match the data sets as definedin equation A-1. The correlation coefficient can be usedto quantify the quality of the match between data sets(Hampson-Russell, 1999). The LSIM method beginswith the observation that the correlation coefficient(c) only provides one number to describe the data sets;however, we are interested in understanding the localchanges in the data sets’ similarity. Therefore, the LSIMmethod computes local similarity ct, which is a functionof time t. The square of c can be split into a product oftwo factors (Fomel, 2007a):
c2t ¼ rt � ht; (A-3)
where rt and ht are the solutions to the followingregularized least-squares problems, respectively:
minrt
�Xt
ðat − rtbtÞ2 þ R½rt��
minht
�Xt
ðbt − htatÞ2 þ R½ht��: (A-4)
The regularization operator R is implemented usingshaping regularization (Fomel, 2007b) and is designedto enforce smoothness. To estimate the solution, LSIMis calculated for a series of shifts. The results of thiscalculation are accumulated and displayed on a similar-ity scan.
Appendix B
Predictive paintingWe adopt predictive painting to interpolate well-log
data along a seismic structure. Predictive painting is de-fined using plane-wave destruction filters that measurethe local slopes of seismic events (Fomel, 2002). Theplane-wave destruction operator can be written in lin-ear operator notation:
r ¼ Ds; (B-1)
where s is a group of seismic traces from a seismic im-age (s ¼ ½s1s2 : : : sN �T ), r is the destruction residual, andD, the destruction operator, is defined as
D ¼
26666664
I 0 0 : : : 0−P1;2 I 0 : : : 00 −P2;3 I : : : 0
..
. ... ..
. . .. ..
.
0 0 : : : −PN−1;N I
37777775; (B-2)
where I is the identity operator and Pi;j describes theprediction of trace j from trace i by shifting along thelocal slope of the seismic data. Slopes can be estimatedin this way by minimizing the prediction residual oper-ator r using regularized least-squares optimization.The prediction of one trace from another trace (Fomel,2010) can be defined as
sk¼ Pr;ksr; (B-3)
where sk is the unknown trace and sr is the referencetrace. The predictive painting operator is defined as
P1;k ¼ Pk−1;k : : :P2;3P1;2: (B-4)
Predictive painting spreads information along localseismic structures to generate volumes of well-log datafrom a single well-log reference trace providing amethod to predict an expected log profile in a locationwith no well-log data.
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Biographies and photographs of the authors are notavailable.
