Brian RussellBrian Russell ##, Larry Lines , Larry Lines ##, , Dan Hampson*, and Todor Todorov*. Dan Hampson*, and Todor Todorov*.

##CREWES Consortium, University of CalgaryCREWES Consortium, University of Calgary

**Hampson-Russell SoftwareHampson-Russell Software Ltd. Ltd.

Combining geostatistics and Combining geostatistics and multiattribute transforms – multiattribute transforms – A channel sand case studyA channel sand case study

IntroductionIntroduction

• In this talk, we will look at a new approach to integrating well log and seismic data, which involves post-stack inversion, geostatistics, and multiattribute transforms.

• This method will be applied to data slices extracted from multiple 3D volumes.

• We will illustrate this approach using the Blackfoot dataset.

The Blackfoot surveyThe Blackfoot survey

This map shows the location of the Blackfoot survey area, with the portion used in this study outlined in red. The objective, a Glauconitic channel within the Lower Cretaceous Mannville formation, is shown running north-south on the map. The survey was recorded in October, 1995 for PanCanadian Petroleum.

Alberta

Calgary

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Channel

Base map from the Blackfoot surveyBase map from the Blackfoot survey

This map shows the 12 wells in the seismic survey area, and Xline 18. Note that we have rotated the map from the previous display.

Xline 18

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Correlating the logs with the seismic dataCorrelating the logs with the seismic data

This figure shows the correlation of well 14-09 with the seismic data, where the synthetic trace is in blue the and the seismic trace is in red. The sonic and porosity logs are on the right. The top and base of sand are also shown.

Seismic line from volumeSeismic line from volume

This figure shows Xline 18 from the seismic volume, showing correlated sonic logs from two intersecting wells, and the picked channel top.

Line from inverted seismic volumeLine from inverted seismic volume

This figure shows Xline 18 from the inverted volume. The color key indicates impedance.

Acoustic impedance sliceAcoustic impedance slice

This map shows the arithmetic average of the acoustic impedance over a 10 ms window below the channel top event. Notice the channel in green (low impedance) on the left of the map.

Channel

Wells showing average porosityWells showing average porosity

This map shows the average porosity over the zone of interest at each well. Notice the high porosity (purple) values in the center left.

HighPorosity

Initial crossplotInitial crossplot

This is the crossplot between the well porosities and acoustic impedance values. The red line is the regression fit, and the correlation is –0.65.

Regression applied to inversion sliceRegression applied to inversion slice

This map shows the application of a the regression line from the previous slide to the inversion slice. Notice that the wells do not tie.

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Map-based geostatisticsMap-based geostatistics

• Map-based geostatistics involves producing three types of maps:

– Optimal maps (Best Linear Unbiased Estimates) from sparse well data (kriging).

– Maps that incorporate both sparse well data and a secondary seismic attribute (cokriging and kriging with external drift, or KED).

– Conditional simulations of a range of equally probable maps.

• In this talk, we will focus on the kriged and cokriged maps.

• Both of these maps are based on the variogram.

VariogramsVariograms

(a) The figure above shows the variogram from the 12 wells on the map, using a spherical variogram.

(b) This figure shows the seismic variogram, again with a spherical fit. By using the Markov-Bayes linear assumption, we can scale this variogram for both the kriged and cokriged maps.

Kriged resultKriged result

This map shows the result of applying kriging to the 12 wells within the 3D seismic survey, using the scaled seismic-to-seismic variogram.

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Estimating the errorEstimating the error

• To see the error associated with kriging, we usually display the error variance. But this is simply the “theoretical” error, and will go to zero as the variance of the input values goes to zero.

• A better measure of the error is the cross-validation error, which is found by successively leaving out well values and comparing their correct values to the predicted value.

• We will use the standard deviation of the cross-validation error as our measure of success.

• The next two slides show these different errors for the kriging example.

Kriging error varianceKriging error variance

The error variance map. As expected the error is small at the wells and gets larger away from the wells.

Kriging cross-validation errorKriging cross-validation error

The cross-validation error map, displaying the absolute error at each well in % porosity. The standard deviation is 3.25%.

Cokriged resultCokriged result

This map shows the collocated cokriging result, using impedance as the secondary variable. Note the “imprint” of the kriged map. The standard deviation of the validation error is 2.91%, better than for kriging.

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• In the multiattribute transform using multilinear regression, we compute M+1 weights such that the log value L(x,y) at a particular map value is a weighted sum of M attributes Ai :

• The solution to this problem can be found by using a standard least-squares technique.

• A key problem is deciding which attributes to use.

• Another important consideration is which of the attributes are statistically significant.

• The next slide shows this approach pictorially.

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The multiattribute transformThe multiattribute transform

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valuelogL

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The multiattribute map transformThe multiattribute map transform

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Attribute map 1

Attribute map 2

Attribute map M

This figure shows the multiattribute map transform approach in schematic form. We need to compute the weights wi which, when multiplied by the attribute values, will produce the log value.

Attribute slicesAttribute slices

• We have already seen one of the slices that will be used in our multiattribute transform: the impedance slice.

• The next three slides will show the other slices used. Each attribute (except trace length) was derived from the seismic volume by taking an RMS average over a 10 msec window below the picked top of sand.

• The following attribute slices were extracted:– Seismic amplitude– Amplitude envelope– Instantaneous phase– Instantaneous frequency– Integrated seismic trace– Trace length – the total length of the trace over the window

Seismic amplitude slicesSeismic amplitude slices

The map in (a) shows the RMS average of the seismic amplitude over a 10 ms window below the channel top event, whereas the map in (b) shows the RMS average of the amplitude envelope over the same window. Notice that the two slices are very close in appearance.

(a) Seismic amplitude slice. (b) Amplitude envelope slice.

Instantaneous phase and frequency slicesInstantaneous phase and frequency slices

The map in (a) shows the RMS average of the instantaneous phase over a 10 ms window below the channel top event, whereas the map in (b) shows the RMS average of the instantaneous frequency over the same window.

(a) Instantaneous phase slice. (b) Instantaneous frequency slice.

Integrated trace and trace length slicesIntegrated trace and trace length slices

The map in (a) shows the RMS average of the integrated trace over a 10 ms window below the channel top event, whereas the map in (b) shows the total trace length over the same window.

(a) Integrated trace slice. (b) Total trace length slice.

Computational issuesComputational issues

• We are now ready to compute the multilinear regression map, which will be a linear combination of the previous maps.

• We will also include a non-linear option by computing transforms (inverse, square root, etc) of the data.

• First, we will look at the correlation coefficients between the wells and each of the attribute slices.

• We then compute the best combination of attributes using a technique called step-wise regression.

• Finally, we decide which attributes are significant using a validation technique in which the target well is left out in jackknife fashion.

Correlation coefficients for all the slicesCorrelation coefficients for all the slices

This table shows the correlation coefficients between the well porosity values and all of the attribute slices, sorted by decreasing correlation coefficient.

Validation error plotValidation error plot

This is the validation error for the 5 attributes used in the multiattribute process. The red line leaves out the target well and shows that only the first 3 attributes should be used.

Validation error and weightsValidation error and weights

The table at the top shows the numerical values from the previous slide and also shows that the best non-linear fit is between the square root of porosity and the inverse attributes. The bottom table shows the weights.

Only use the first three attributes based on validation error.

Weights

Multilinear regression resultMultilinear regression result

This map is the result of applying the multilinear regression weights, shown in the previous slide, to the attributes. Note that the result is in pseudo-porosity.

Channel

New crossplotNew crossplot

This is the new crossplot between the well porosity and the pseudo-porosity from the multiattribute transform. Note that the correlation coefficient has gone up to 0.81.

Combining multilinear regression Combining multilinear regression with geostatisticswith geostatistics

• We will now combine the multilinear regression result with the well values using geostatistics.

• That is, the multiattribute transform will replace the inversion slice as the secondary variable.

• The first step is to re-compute the seismic to seismic variogram.

• We will then compute the cokriging result.• Finally, we will perform a statistical analysis of the

results.

New seismic variogramNew seismic variogram

This is the seismic variogram used in the final cokriging process. Note that this is an exponential fit, rather than the spherical fit used in the earlier variograms.

Cokriging with the multiattribute transformCokriging with the multiattribute transform

This map shows the result of applying the collocated cokriging process using the multiattribute transform as the secondary variable. The standard deviation of the validation error is now 2.3%.

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Map reviewMap review

Note the increase in geological information as we move from (a) the kriged map with wells alone to (b) cokriging with inversion, and finally to (c) cokriging with the multiattribute transform. However, all three maps match the wells.

(b) Cokriging with impedance, Std. Dev.=2.91 %

(c) Cokriging with multiple attributes,Std. Dev.= 2.33 %

(a) Kriging, Std. Dev.= 3.25 %

ConclusionsConclusions

• In this paper, we have combined geostatistics with multiattribute map analysis.

• Traditional cokriging uses a single secondary attribute. When we used impedance as this attribute, we saw a strong “imprint” from the wells.

• Using a multiattribute transform, we were able to get a better fit between the wells and the porosity map.

• After a second pass of cokriging, the final map was more realistic from an exploration point of view.

• Statistically, the standard deviation of the cross-validation error was smallest when we used cokriging with the multiattribute transform.

Future WorkFuture Work

• In this paper, we concentrated on the multi-linear transform. In future work, use of the Probabilistic Neural Network will be explored.

• We will also incorporate the converted wave volume from the Blackfoot dataset.

AcknowledgementsAcknowledgements

• We wish to thank our colleagues at both

CREWES and at Hampson-Russell Software

for their input to this study.

• We also wish to thank the sponsors of the

CREWES consortium.