Incremental correlation of multiple well logs following geologicallyoptimal neighbors

Xinming Wu1, Yunzhi Shi1, Sergey Fomel1, and Fangyu Li2

Abstract

Well-log correlation is a crucial step to construct cross sections in estimating structures between wells andbuilding subsurface models. Manually correlating multiple logs can be highly subjective and labor intensive.We have developed a weighted incremental correlation method to efficiently correlate multiple well logs fol-lowing a geologically optimal path. In this method, we first automatically compute an optimal path that startswith longer logs and follows geologically continuous structures. Then, we use the dynamic warping technique tosequentially correlate the logs following the path. To avoid potential error propagation with the path, we modifythe dynamic warping algorithm to use all the previously correlated logs as references to correlate the current login the path. During the sequential correlations, we compute the geologic distances between the current log andall of the reference logs. Such distances are proportional to Euclidean distances, but they increase dramaticallyacross discontinuous structures such as faults and unconformities that separate the current log from the refer-ence logs. We also compute correlation confidences to provide quantitative quality control of the correlationresults. We use the geologic distances and correlation confidences to weight the references in correlating thecurrent log. By using this weighted incremental correlation method, each log is optimally correlated with all thelogs that are geologically closer and are ordered with higher priorities in the path. Hundreds of well logs fromthe Teapot Dome survey demonstrate the efficiency and robustness of the method.

IntroductionWell-log correlation aims to identify corresponding

depth samples among well logs. Each set of such cor-responding samples belongs to the same geologic layerwith similar rock properties (Wheeler and Hale, 2014).Manually identifying multiple sets of correspondingsamples among many well logs is highly labor intensiveand subjective. Therefore, computer-based automaticmethods have been proposed to correlate single pairsof logs and multiple logs.

The earliest automatic methods (e.g., Rudman andLankston, 1973; Mann and Dowell, 1978) use the cross-correlation algorithm to compute shifts that correlatea single pair of logs. The crosscorrelation algorithm,however, is accurate only to estimate slowly varyingshifts. Therefore, some authors (e.g., Smith and Water-man, 1980; Waterman and Raymond, 1987) propose touse a dynamic waveform matching method to betterestimate rapidly varying shifts. Such a dynamic warp-ing method is further improved by incorporatinghuman interpretations (e.g., Lineman et al., 1987;Wu and Nyland, 1987; Lallier et al., 2012, 2016) andseismic constraints (Julio et al., 2012) into well-logcorrelations.

The crosscorrelation and dynamic warping methodscan be directly applied for correlating multiple well logsby choosing a reference and then independently align-ing all the other logs to the reference (Le Nir et al.,1998). This independent pair-wise correlation method,however, is not able to compute consistent and globallyoptimal correlations of multiple logs. Fang et al. (1992)compute three pairwise correlations among a cycle ofthree logs and repeatedly perform such a three-passcycle processing to obtain multiple-log correlations.Wheeler and Hale (2014) first compute shifts for allpossible pair-wise correlations among multiple logswithout choosing a correlation path, and then theycompute the least-squares fit of the shifts to obtainconsistent correlations of the logs. Lallier et al. (2016)sequentially compute pair-wise correlations following apath, and then they iteratively reduce inconsistent cor-relations until a stable and minimal cost correlation isachieved.

In this paper, we present a highly efficient method, asshown in Figure 1, to compute robust correlations ofmultiple well logs. First, we discuss how to computea geologically optimal path that starts with longer logs,follows shorter distances, and avoids faults and uncon-

1University of Texas at Austin, Austin, Texas, USA. E-mail: xinming.wu@beg.utexas.edu; yzshi08@utexas.edu; sergey.fomel@beg.utexas.edu.2University of Oklahoma, Norman, Oklahoma, USA. E-mail: clintfangyu@gmail.com.Manuscript received by the Editor 22 January 2018; revised manuscript received 12 May 2018; published ahead of production 06 June 2018;

published online 17 July 2018. This paper appears in Interpretation, Vol. 6, No. 3 (August 2018); p. T713–T722, 7 FIGS.http://dx.doi.org/10.1190/INT-2018-0020.1. © 2018 Society of Exploration Geophysicists and American Association of Petroleum Geologists. All rights reserved.

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Technical papers

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formities. In computing such an optimal path, we con-struct a pseudovelocity map by taking into account aseismic structural discontinuity map and a smoothedlog-length map. From the pseudovelocity map, we solvefor an eikonal equation to compute a traveltime map,which is further used to sort the logs in a geologicallyreasonable order. Then, we discuss an incrementalcorrelation method to use an increasing number of pre-viously correlated logs as references to correlate a sub-sequent log in the correlation path. In correlating thecurrent log to the multiple references, geologicallycloser references are weighted more than the furtherones that are separated from the current log by faultsand unconformities. We also discuss how to computecorrelation confidences to quantitatively evaluate thecorrelation results. Such confidences are also used toweight the references (previously correlated logs) forcorrelating the current log. Finally, we apply a similarincremental correlation method to simultaneously com-pute consistent correlations of multiple types of logs.

Geologically optimal pathSequentially correlating multiple well logs is effi-

cient, but it often requires an optimal path to obtainreliable results. An optimal path should start with rela-tively long logs and follow geologically continuousstructures.

Figure 2 shows the method that we use to computesuch an optimal path for the 165 density logs (the yel-low circles in Figure 2a) of the Teapot Dome survey.In this method, we first count the number of samplesin each log to compute a normalized log-length map

(Figure 2b), which is further smoothed by applying aGaussian smoothing filter with a smoothing half-widthsigma ¼ 10 (samples). In this smoothed and normalizedlog-length map lðxÞ(Figure 2c), the logs with relativelylonger length are denoted by higher values (white col-ors). The log length will be an important considerationin properly ordering the logs for a sequential correlationmethod. We expect the relatively longer logs (with geo-logically more complete rock properties) to be orderedat the beginning in a sequential correlation method be-cause these beginning logs will serve as references forcorrelating the subsequent logs. However, log length isnot the only consideration of choosing a proper corre-lation order of the logs. For example, in Figure 2b, thelongest log is denoted by the red arrow, but we may notwant to choose it as the reference at the very beginningbecause it is located at the edge of the survey and is faraway from all the other logs. The Gaussian smoothingapplied to the log length map helps to take into consid-eration the length of the nearby logs and avoid choosingthe outlier longest log as the reference to start with.After the Gaussian smoothing, the smoothed longestlogs are now located in the middle of the survey, as de-noted by the red circle in Figure 2c.

A proper correlation path should also avoid passingacross structural discontinuities. We use a correspond-ing 3D seismic image to compute the attributes, suchas coherence (Marfurt et al., 1999), fault likelihood(Hale, 2013b; Wu and Hale, 2016), and unconformitylikelihood (Wu and Hale, 2015), that can highlight struc-tural discontinuities including faults and unconform-ities. We stack the 3D attribute volume verticallyover time or depth to obtain a 2D spatial map of thestructural discontinuities. As an example, we computea 3D fault-likelihood image from the 3D Teapot Domeseismic image and we vertically stack the likelihood im-age to obtain a map dðxÞ of fault zone show in Figure 2d.Such a discontinuity map is also normalized suchthat 0 ≤ dðxÞ ≤ 1.

With these two maps, the log length (lðxÞ) and seis-mic structural discontinuities (dðxÞ), we further definea pseudovelocity map by vðxÞ ¼ lðxÞ½1 − dðxÞ� (Fig-ure 2e), where relatively higher values denote areaswith longer logs and more continuous structures. Tocompute a path following longer logs and more continu-ous structures, we place a wave source (the red circle inFigure 2e) at the position with the highest velocity andwe propagate the wave in this velocity map to computea traveltime map by solving the following eikonal equa-tion:

∇tðxÞ · ∇tðxÞ ¼ 1

v2ðxÞ : (1)

We compute the traveltime map tðxÞ (Figure 2f) bysolving a finite-difference approximation to the aboveeikonal equation using an iterative algorithm similar tothat discussed by Jeong et al. (2007) and Hale (2009).In this traveltime map (Figure 2f), the black curves

Figure 1. The workflow of the proposed methods to first or-der well logs in a geologically reasonable path and then se-quentially correlate the logs using a weighted incrementalcorrelation method.

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denote traveltime contours or wave-fronts, which generally follow the highvelocity values in Figure 2e.

From this computed traveltime map,we can extract traveltimes at all thewell-log positions; smaller times shouldcorrespond to longer log lengths, morecontinuous structures, and shorter dis-tances from the source position. There-fore, we can simply sort all the logs(Figure 3a) by their corresponding trav-eltimes to order the logs in a geologi-cally optimal path that starts withlonger logs, follows shorter distances,and avoids discontinuous structures,such as faults and unconformities. Withthis method, the randomly ordered logsin Figure 3a are ordered in Figure 3b,where the longer ones have been re-arranged with higher priorities on leftand will be correlated earlier. We canalso observe that the density measure-ments are laterally more continuous inthe optimally ordered logs (Figure 3b)compared with those in the randomlyordered ones (Figure 3a). Figure 4ashows the same density logs but arepoorly ordered, where the short logsare ordered at the beginning, whereasthe relatively long logs are ordered atthe end. In the next section, we will ap-ply the same correlation method to theoptimally (Figure 3b) and poorly (Fig-ure 4a) ordered logs for comparison.

Although the logs are ordered in ageologically optimal path, it can stillbe challenging to obtain accurate corre-lations of the logs using a sequential cor-relation method. The missing data andmeasurement errors within the logsmay generate correlation errors, whichcan propagate and accumulate withthe path. In addition, the logs orderedclose to each other in this geologicallyoptimal path can be spatially far awayfrom each other. In the next section,we propose a weighted incremental cor-relation method to deal with these po-tential problems.

Weighted incremental correlationTo avoid potential error propagation

with the correlation path, we propose touse all the previously correlated logs asreferences to correlate the current log,so that each log is optimally correlatedto all the logs with higher priorities inthe path. Because the number of refer-ence logs increases with the correlation

Figure 2. (a) In total, 165 Teapot Dome logs are displayed on a time slice of thecorresponding 3D seismic volume. A map of the normalized length of all the logs(b) before and (c) after Gaussian smoothing. (d) A map of a fault zone is com-puted from the seismic fault likelihood. Maps of the (c) smoothed well-log lengthand (d) a fault zone are used to define (e) a pseudovelocity map, (f) where a wavesource is placed at the position with the highest velocity to compute a traveltimemap. A geologically optimal correlation path of the logs is found by sorting trav-eltimes at all the log positions.

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path, we call this method incremental correlation. In ad-dition, we compute geologic distances between the cur-rent log and all the previously correlated logs and weuse the distances to weight the references in correlatingthe current log. Such distances are proportional to theEuclidean distances between the current log and the

reference logs, but they increase dramatically acrossfaults or unconformities.

Weighted incremental dynamic warpingIn this method of computing correlation shifts uk½i�

for each kth log f k½i�, we use all the previously corre-

Figure 3. (a) Randomly ordered density logs. (b) Geologically optimally ordered density logs. (c) The optimally ordered densitylogs are correlated by using the weighted incremental correlation method, and the corresponding correlation confidences areshown at the bottom.

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lated logs as references and solve the following con-strained minimization:

argminuk½i�

XN

i¼1

Xk−1

j¼1

wj;kjf j½iþ uj½i�� − f k½iþ uk½i��jp;

subject to umin ≤ uk½i� ≤ umaxandjuk½i� − uk½i − 1�j ≤ ε;

(2)

where i ¼ 1;2; : : : ; N represents the sample index in awell logs; f j½iþ uj½i��; j ¼ 1; : : : ; k − 1 represent all thepreviously correlated logs; and uj½i� are the previouslycomputed correlation shifts for the jth log. The positiveexponential number p is set to be one in all examples inthis paper. The shift ranges umin and umax are defined tobe umin ¼ −250 (samples) and umax ¼ 250 (samples),which is large enough to correlate the logs in the TeapotDome survey. The value ε is the shift strain, which de-fines squeezing (uk½i� − uk½i − 1� ≤ 0) and stretching(uk½i� − uk½i − 1� ≥ 0) (Hale, 2013a) of a well log incorrelation. In the Teapot Dome example, there areno significant angular unconformities (missing layers)

apparent in themeasuredwell logs; therefore, we choosea small strain ε ¼ 0.1 (−0.1 ≤ uk½i� − uk½i − 1� ≤ 0.1) toavoid unreasonably large stretching and squeezing andto preserve the length of the logs during the correlation.

The term wj;k denotes a weighting map that dependson the jth and kth logs. We define such a map bycombining three factors: (1) geologic distance betweenthe two logs, (2) missing data in both logs, and (3) cor-relation confidence (equation 3) of the jth log. We setwj;k ¼ 0 when the correlation confidence of a referencelog is smaller than some threshold or either the log sam-ple f j½iþ uj½i�� or f k½iþ uk½i�� is missing. This is helpfulto avoid using poorly correlated reference logs andmissing data for the correlation.

For other reference logs with high correlation confi-dences, we define the weights wj;k with geologic distan-ces that are defined as traveltimes tj;k from the currentlog (f k½i�) position to the locations of all the previouslycorrelated logs (f j½iþ uj½i��; j ¼ 1; : : : ; k − 1). Then,we compute a new traveltime map by placing a wavesource at the kth log position and solving the sameeikonal equation (equation 1) with the same pseudove-

Figure 4. (a) Density logs are poorly ordered with short logs at the beginning, whereas the relatively longer ones at the end.(b) The poorly ordered density logs are correlated by using the weighted incremental correlation method.

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locity map (Figure 2e). From the new traveltimemap, we can simply extract the traveltimes tj;k at thereference-log positions and define such traveltimes asgeologic distances between the current log f k½i� andthe reference logs f j½iþ uj½i��; j ¼ 1; : : : ; k − 1. Theextracted traveltimes or geologic distances will be pro-portional to the Euclidean distances and will dramati-cally increase across faults and unconformities wherethe pseudo velocities are set to be significantly low(nearly zeros). In correlating the current log to thereference logs, we set larger weightswj;k correspondingto smaller times tj;k so that the geologically closer refer-ences are weighted more than those geologically fur-ther ones that may be spatially far away from thecurrent log or geologically separated from the currentlog by faults and unconformities.

To solve the above weighted and constrained mini-mization, we use a dynamic programming algorithm

with three steps of computing alignment errors,forward accumulation, and backward tracking, asdiscussed by Hale (2013a). In the first step, the align-ment error map e½i; uk� is a 2D array with respect tothe sample index (i ¼ 0;1; : : : ; N) and shift lag(umin ≤ uk ≤ umax). Each error in this map is computedas the summation of p norms of the differences betweenthe current log and all the previously correlated refer-ence logs as defined in equation 2. To find the globallyoptimal correlation shifts that minimize the objectivefunction in equation 2 is equivalent to finding an optimalpath that passes though globally minimum alignment er-rors in the map e½i; uk�. However, it is often difficult toextract such an optimal path directly from the align-ment error map. The dynamic programming methodperforms a second step of forward accumulating thealignment errors to obtain an accumulated error mapfor the minimum path picking. In this second step,

Figure 5. (a) The geologically optimally ordered gamma-ray logs are (b) correlated by using the weighted incremental correlationmethod and the corresponding correlation confidences are shown at the bottom of (b).

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the strain constraint (juk½i� − uk½i − 1�j ≤ ε) is incorpo-rated into the dynamic programming method by con-trolling the accumulation of errors. In the forwardaccumulated error map, the globally minimum path isoften more obvious than in the original error mapand can be picked by back tracking the minimum accu-mulated errors. The same strain constraint is also incor-porated in this back-tracking process as in the secondstep of forward accumulation.

By solving such minimizations sequentially for alllogs in the path, we obtain all the correlated logs inFigure 3c. We can observe that all the logs are visuallyaccurately correlated (the same layers in different logsare all horizontally aligned), although a lot of missingsamples and noisy or singular values (e.g., the blue col-ors in Figures 3) are apparent in many of the logs. We donot try to detect these noisy values, andwe eliminate them during the correla-tion. Using weights andmultiple referen-ces in the incremental correlationmethod is helpful to reduce the influ-ence from these noisy values.

In Figure 4b, we apply the same cor-relation method to correlate the samedensity logs, which, however, are notproperly ordered as in Figure 4a. We ob-serve that this correlation method stillworks well to reasonably correlate mostof the density logs, especially theseshallow ones. The misalignments of thelogs, denoted by arrows in Figure 4b,are caused by the poorly chosen correla-tion path, where the reference logs onthe left are short ones containing onlythe shallow rock properties. The mis-aligned logs, however, contain only thedeep properties but miss the shallowproperties and are therefore not able tobe accurately aligned to the referencelogs. This problem was solved by morereasonably order the correlation logsas shown in Figure 3b.

Correlation confidenceTo provide a quantitative quality con-

trol of each correlated log f k½iþ uk½i��,we compute a correlation confidenceck by averaging the zero-lag crosscorre-lations of this log and all the previouslycorrelated logs (f j½iþ uj½i��; j ¼ 1; : : : ;k − 1):

ck ¼1

k − 1

Xk−1

j¼1

hf j; f ki2hf j; f jihf k; f ki

; (3)

where hx½i�; y½i�i denotes the dot prod-uct between two sequences of ½x½i��and y½i�, f j ¼ f j½iþ uj½i�� represents

the previously correlated logs, and f k ¼ f k½iþ uk½i��represents the currently correlated log whose confi-dence is to be computed. The black curve at the bottomof Figure 3c shows the computed correlation confiden-ces for all correlated logs. The confidences of most logsare higher than 0.8, which indicates these logs are well-aligned as shown in the image above the confidencecurve. The two relatively low confidence values (thedashed vertical lines in Figure 3b) highlight the corre-sponding two logs, which are not well-aligned.

Using the same incremental correlation methods, wealso automatically correlate the gamma-ray (Figure 5)and velocity (Figure 6) logs measured in the TeapotDome survey. We observe that most of these logs arealso well-aligned (Figures 5b and 6b) and their corre-sponding correlation confidences are larger than 0.8.

Figure 6. Geologically optimally ordered velocity logs (a) before and (b) aftercorrelation. (b) Horizontal lines in the correlated logs correspond to (c) geologichorizons in the original logs.

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The low-confidence logs are denoted by the verticaldashed lines in Figure 5b. The misalignments and lowcorrelation confidences of these logs are caused by theirlarge measurement errors (17th, 76th, and 102th logs)and missing data (all the other logs highlighted by verti-cal dashed lines in Figure 5b). Although some of theselogs are correlated relatively earlier, these logs are notused as references for the subsequent correlations. Inthese examples, we set the weightswj;k ¼ 0 in equation 2when the correlation confidences are smaller than 0.8 tofilter out these unreliably correlated logs.

After the well-log correlation, all of the logs aremapped to the relative geologic time (RGT) domain,in which the geologic layers in all the logs are horizon-tally aligned. It is convenient to pick horizons across welllogs in this RGT domain and map them back to the origi-nal depth domain to construct lithostratigraphic crosssections. As shown in Figures 3c, 5b, and 6b, the vertical

axis of the correlated logs represents RGT, which meansthat horizontally aligned samples of different logscorrespond to a same geologic time and belong to thesame geologic layer. We can simply draw horizontallines, as in Figure 6b, to pick geologic horizons acrossmultiple logs in this RGT domain. Then, we can simplyadd the computed correlation shifts to the RGT of thesehorizontal lines to obtain the corresponding horizons inthe original depth domain and build a cross section asshown in Figure 6c.

Correlating multiple types of logsIn practice, multiple types of logs, corresponding to

different rock properties, are often measured in a samewell. It is worthwhile to simultaneously correlate multi-ple types of logs to obtain consistent correlation shifts.In addition, simultaneous correlation is helpful to com-

Figure 7. Four types of well logs, including (a) velocity, (b) gamma ray, (c) density, and (d) porosity are simultaneously correlatedas shown in (e-h), respectively.

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pute more reliable correlations by incorporating moreconstraints from different types of logs.

To be applied to the simultaneous correlations ofmultiple types of logs, the weighted incremental corre-lation method requires only minor modifications. In thiscase, we use the dynamic programming method to com-pute the same shifts uk½i� for all M types of logs at eachkth well by solving the following minimization:

argminuk½i�

X

i

XM

m¼1

em½i; uk�;

subject to umin ≤ uk½i� ≤ umaxandjuk½i� − uk½i − 1�j ≤ ε;

(4)

where em½i; uk� represents normalized alignment errorsem½i; uk� computed for each mth type of logs:

em½i; uk� ¼Xk−1

j¼1

wm;j;kjf m;j½iþ uj½i�� − f m;k½iþ uk½i��jp:

(5)

In this equation, f m;j denotes previously correlated jth(j ¼ 1;2; : : : ; k − 1) well at the mth log type; uj½i� repre-sents the shifts that are already computed for the jthwell; f m;k is the kth well (at the mth type) to be corre-lated; and uk½i� represents the shifts to be computed forthis kth well. Notice that the shifts uk½i� are independenton the log typem because we expect the shifts to be thesame for all different types of logs at the same well. Theweights wm;j;k are defined similarly to those in equa-tion 2, but they also depend on the log typem. Note thatthe alignment errors (equation 5) are normalized beforesumming them together in equation 4 because the mag-nitude orders of different log types are much differentfrom each other as shown in Figure 7.

Using the correlation shifts computed with thismethod, we simultaneously correlate the four typesof velocity, gamma ray, density, and porosity logs, asshown in Figure 7e–7h, respectively. We can observethat all the well logs are accurately and consistentlyaligned. These 21 wells are still sequentially correlatedfrom well index k ¼ 1 to k ¼ 21. However, the process-ing order (from index k ¼ 1 to k ¼ 21) is not chosen tobe geologically optimal as in the previous examples ofcorrelating a single type of logs.

The processing order shown in Figure 7 is actuallyrandomly chosen for testing the method. We can ob-serve that the all the velocity logs are missing at thefirst eight wells except the first well. In addition, thefour types of logs at the ninth well are all short butare ordered and correlated before many other logs.Therefore, this randomly chosen processing order inFigure 7 is obviously not an optimal order for thesequential correlation. However, the weighted incre-mental correlation method still provides reliable corre-lations as shown in Figure 7e–7h, which indicates that

this proposed method does not highly depend onthe path.

ConclusionWe have discussed methods to compute a geologi-

cally optimal path by solving an eikonal equationdefined with log length and seismic structural disconti-nuities, to sequentially correlate multiple logs by usingan increasing number of references, and to quantita-tively evaluate the correlation results by computing cor-relation confidences.

The geologically optimal path starts with relativelylonger logs, follows shorter Euclidean distances, andavoids geologically discontinuous structures such asfaults and unconformities that are recognized from acorresponding 3D seismic image. To avoid correlationerrors propagating with the path, we use all the previ-ously correlated logs as references to correlate a sub-sequent log. We also compute geologic distances andcorrelation confidences of the reference logs and usethe distances and confidences to weight these referen-ces in correlating the current log. By using weightedand increasing number of references, each log is opti-mally correlated to all the logs that are geologicallycloser and are ordered with higher priorities in the path.This weighted incremental correlation method is highlyefficient and takes less than 1 min to correlate the fourtypes of logs of 21 wells (Figure 7) with an eight-corecomputer.

AcknowledgmentsThis research is supported by the sponsors of the

Texas Consortium for Computation Seismology. Allthe well-log data are provided by the Rocky MountainOilfield Test Center.

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Biographies and photographs of the authors are notavailable.

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