Geological Survey of Denmark and Greenland Bulletin 33, 2015, 13-16 13 Down-hole permeability prediction – a chemometric wire-line log feasibility study from a North Sea chalk well Kim H. Esbensen, Niels H. Schovsbo and Lars Kristensen Permeability in chalk depends primarily on porosity but also on other factors such as clay and quartz content, and can theoretically be described by the Kozeny equation using empirically determined constants (Mortensen et al. 1998; Røgen & Fabricius 2002). Recent attempts to predict per- meability from wire-line logs have shown that compressional velocity within operative chalk units, defi ned by specifi c surface and hydraulic properties established from stratigra- phy and core plugs, can provide excellent well permeability predictions (Alam et al. 2011). High-quality predictions de- pend on a solid knowledge of a multitude of parameters of the relevant ‘operative rock types’. Th e more detailed this a priori knowledge is, the better predictions can be achieved. But this approach may, or may not, be fast enough for well- site operations or when core data are lacking. In this study, we illustrate a situation for direct permeability prediction if only well-site, wire-line logs are available. Th is pilot study is based on multivariate descriptor rela- tionships, specifi cally aimed at direct permeability predic- tion, using all immediately available wire-line characteristics and/or core (plug) information in a top-down mode with sequential exclusion of non-correlated, irrelevant variables. We show prediction-model results based on [log] data only and on [log + plug] data. Other relevant descriptors could be included in an augmented X-matrix, such as quantitative core and facies descriptions while still retaining the fast well- site perspective. However, such data were not included in this feasibility study. Material and methods Core and log data are from the M-1X well in the Danish part of the North Sea; core data were collected in the mid- 1990s during a multi-disciplinary reservoir study (Dons et al. 1995). Th e M-1X well intersects the Danian Ekofi sk For- mation and the Maastrichtian Tor Formation (Kristensen et al. 1995). Core analysis included determinations of con- ventional He-porosity and air permeability, whole-rock Ca, Mg, Fe, Mn and Sr concentrations, δ13C and δ18O isotope ratios, per cent carbonate and per cent non-carbonate. Be- fore data analysis all concentrations were corrected to rep- resent weight pr. volume. M-1X was drilled in 1971 on the Dan Field structure (Fig. 1), and encountered a c. 200 m thick hydrocarbon-bearing zone in the chalk. Petrophysical evaluation shows the top reservoir is at 1800 m; a gas cap was encountered down to 1880 m and the oil–water contact was found at 2036 m. A 192 m long core was collected from the hydrocarbon-bearing zone with a core recovery of c. 75%. Wire-line logs included gamma ray (GR), sonic, formation density, spontaneous potential (SP), calliper, induction log (deep resistivity), lateral log (deep resistivity), micro-lateral log (shallow resistivity; MLL) and short normal resistivity (medium resistivity). Core data depth and well-log readings were adjusted and aligned applying an estimated common depth shift of 3 m. Log readings were sampled for each plug depth to ensure a common plug-log training data set. Two chemometric techniques were used, Principal Com- ponent Analysis (PCA) and Partial Least Squares (PLS) regression. PCA transforms a matrix of measured data (N samples, P variables), X, into sets of projection sub-spaces de- lineated by Principal Components (each a linear combination of all P variables), which display variance-maximised interre- © 2015 GEUS. Geological Survey of Denmark and Greenland Bulletin 33, 13–16. Open access: www.geus.dk/publications/bull 4°E Gas field in chalk Oil field in chalk Inversion zone Fault zone Studied well 55°30´N Dan Field M-1X 10 km UK Germany Norway Denmark The Nether- lands 500 km Fig. 1. Location of well M-1X in the Dan Field in the Danish part of the North Sea. 1414 lationships between samples and variables respectively (Mar- tens & Næs 1989; Höskuldsson 1996; Esbensen 2010). PCA score plots display groupings, or clusters, between samples based on compositional similarities, as described by the vari- able correlations (shown in accompanying loading plots), and also quantify the proportion (%) of total data-set variance that can be modelled by each component, see Fig. 2. All data analy- ses in this work are based on auto-scaled data [X-X(avr)/std]. PLS regression replaces the classical multiple linear regression and allows direct correlations to be modelled between y and the multivariate X data, among other compensating for debilitating co-linearity between x- variables, (Martens & Næs 1989; Höskuldsson 1996; Es- bensen 2010). PLS regression models are used extensively in science, technology and industry for prediction pur- poses where the critical success factor is proper validation (Esbensen & Geladi 2010). Both PCA and PLS result in informative score plots, loading plots (PLS: loading- weights) and prediction validation plots, which are the prime vehicles for detailed interpretation of complex data relationships. PLS components are based on [X,y] covariance optimisation, but the scientifi c interpretation of the derived scores and loading-weights plots follows procedures which are identical to the PCA. Validation was based on a test set prepared before modelling: As the M-1X data set is limited, it was sorted with respect to the RHOB Mg NonC Fe DT SP GR Mn MLL Sr IL Por Perm SN LL Ca Carb Ekofisk Fm Tor Fm Hod Fm δ18O δ13C PCA 1 (38%) –8 –4–6 –2 0 2 4 PC A 2 (3 1% ) 6 4 2 0 –2 –4 –6 –8 PCA 1 (38%) –0.4 –0.2 0.0 0.2 0.4 0.4 0.2 0.0 –0.2 –0.4 PC A 2 (3 1% ) A B Fig. 2. Principal component analysis. A: Loading and B: Score relations for the full training data set (Ekofisk, Tor and Hod Formations). The plot models 69% of the total data variance, the proportions are shown along each component axis (38 + 31%). A: abbreviations see Fig. 3. –6 –4 –2 0 2 4 6 8 P L S 2 (3 5 % , 2 % ) Reference air permeability (mD) 0 2 4 6 8 P re d ic te d a ir p e rm e ab il it y ( m D ) PLS 1 PLS 2 PLS 4 PLS 6 % e x p la in e d y -v ar ia n c e PLS1 (39%, 86%) PLS 5 PLS 7PLS 3 DT GR IL LL MLL RHOB SN SP Carb NonC Ca Mg Fe Mn Sr Por Perm PLS1 (30%, 86%) –0.4 –0.2 0.0 0.2 0.4 0.6 P L S 2 ( 3 5 % , 2 % ) δ18O δ13C y = 0.44 + 0.88x r2 = 0.83 C D BA 4 2 0 –2 –4 –6 –8 0.6 0.4 0.2 0.0 –0.2 –0.4 –0.6 100 80 60 40 20 0 8 6 4 2 0 Fig. 3. PLS regression model [log + plug] variable set; full training set with Ekofisk, Tor and Hod Formations. A: PLS X-space score plot (t1-t2). B: Corresponding loading-weights plot (w1-w2). C: Modelled y-variance. D: Prediction versus reference plot. Two outliers were deleted from the original data set. Proportions of total data vari- ance modelled shown along each PLS component [X%, y%]. GR: gamma ray. DT: compressive wave interval travel time. R HOB: formation density. IL: induction log. LL: lateral log. MLL: micro lateral log. SN: short normal resistivity. SP: spontaneous potential. Por: He-porosity. Perm: air permeability. Ca: calcium. Mg: magnesium. Fe: iron. Mn: manganese. Sr: strontium. carb: carbonate volume content [calculated]. NonC: non-carbonate [calculated as 100% – carbonate volume %]. For data analysis, concentrations were transformed to weight per volume rock values. Legend see Fig. 2 15 full permeability range before being randomly split into two independent data sets, i.e. the training versus the test set, securing a realistic prediction performance valida- tion (Esbensen 2010; Esbensen & Geladi 2010). Results Th ere is a marked and fundamental diff erence in rock prop- erties between the Ekofi sk Formation and the Tor and Hod Formations (Fig. 2). Th e Ekofi sk Formation shows a high concentration of non-carbonate, Fe and Mn and high GR and MLL levels. Th ese characteristics are well-known from the North Sea region, which forces a cautious approach to data set defi nition. Th e developed permeability model may, or may not, apply to both the Tor and the Hod Formations and the Ekofi sk Formation. Th is will depend on whether the relationships between the X data from the three formations are similar with respect to correlation to permeability. A two-component PLS model on the full (log + plug) vari- able set predicts permeability with satisfactory validation re- sults as seen in the prediction versus reference plot in Fig. 3 (slope 0.88; r2 = 0.83), suggesting that the PLS model leads to better permeability estimates than normally achieved from conventional poro-perm plots. Conventional statistics per- taining to a fi tted linear regression model between predicted (y) versus reference (x) values are used to express the degree of prediction strength: slope and regression coeffi cient, r2. For both these modelling indices the criterion is to be as close to 1.00 as possible. Such validation statistics must be based on proper validation (Esbensen & Geladi 2010). Th e permeabil- ity model is primarily carried by positively correlated Por, LL, IL, SN and negatively correlated RHOB and GR, but several other log and composition variables also have minor, but sig- nifi cant infl uence. From the loading-weights plot it is diffi cult to resolve any fully irrelevant variables; PLS models benefi t from using a full X-variable complement; variable selection is not needed in this case. Variable relationships are interpreted in the more appropriate PLS loading-weight plots; a technical detail not to be elaborated on here, as interpretation follows the same principles (Martens & Næs 1989; Esbensen 2010). Figure 4 shows permeability prediction only based on log data (Ekofi sk Formation excluded), simulating a situation in which there are only well-site, wire-line logs available for the fastest possible permeability prediction. Th e validation results for this model (slope 0.77; r2 = 0.75) are lower, but still acceptable for direct on-site permeability screening based on Fig. 4. PLS regression model (logs only). A: PLS X-space loading weights plot (t1-t2). B: Prediction versus reference plot. Proportions of total data variance modelled shown along each PLS compo- nent [X%, y%]. Legend see Fig. 2, abbreviations see Fig. 3. –0.6 –0.4 –0.2 0.0 0.2 0.4 0.6 PLS 1 (71%, 84%) IL 0.4 0.2 0.0 –0.2 –0.4 –0.6 –0.8 6 4 2 0 P re d ic te d a ir p e rm e ab il it y ( m D ) P L S 2 ( 1 3 % , 4 % ) 0 2 4 6 8 Reference air permeability (mD) y = 0.61 + 0.77x A B RHOB DT SP MLL LL Perm SN GR r2 = 0.75 Fig. 5. Reservoir properties versus depth. A: Wire-line density log and core porosity measurements. B: Predicted air permeability based on the model presented in Fig. 4, compared with reference permeability (core measure- ments). The reservoir is gas filled from 1800 to 1880 m and oil filled down to a depth of 2036 m. The permeability model does not apply to the Eko- fisk Formation (red rectangle). 1800 1850 1900 1950 2000 2050 Tor Hod Ekofisk A B 2.4 2.0 0 5 10 4020 D ep th (m ), m d Core porosity (%) Permeability predictor (X=logs, Tor and Hod), mD Density (g/cm3) Core measurement 1616 contemporaneous log data alone. Th e results in Figs 3 and 4 indicate that the Tor Formation can be modelled equally well with, or without, the Hod Formation. Figure 5 shows stratigraphic permeability results for the all-logs prediction model (Fig. 4), plotted together with measured core porosity (%) and density. An all-logs predic- tion model is fully able to characterise the Hod and Tor Formations, but not the Ekofi sk Formation. For the latter, additional core information is necessary (Fig. 3). Discussion Th e compositional diff erence between the Ekofi sk Forma- tion and the Tor Formation has also previously been stud- ied by multivariate data analysis (Kunzendorf & Sørensen 1989), pointing to a relationship between reservoir quality and geochemistry. Røgen & Fabricius (2002) showed that these compositional and textural relations are also refl ected in specifi c surface area diff erences between the formations, and thus in permeability and porosity diff erences. Our analysis shows that high permeability is closely re- lated to high porosity, and to high resistivity (Fig. 3; LL, IL, SN), whereas low permeability is related to high density and high GR, high non-carbonate content and thus to impure chalk with high concentrations of Mn, Fe and Mg. Røgen & Fabricius (2002) also showed that quantitative mineral data can help to explain permeability values better. Our analysis also shows that permeability predictions from wire-line logs alone strongly depend on the sonic and resistivity logs (Fig. 4; DT, IL, LL, SN and SP). Th ese fi nd- ings complement those of Alam et al. (2011) in which perme- ability was also predicted but based on the sonic log alone (DT). Our analysis further shows that it is pos sible to model permeability more comprehensively by including the full set of readily available wire-line logs. Conclusions Th e present study confi rms that multiple parameters control permeability levels. Both log data and core data can be used advantageously in direct PLS prediction; there are real ben- efi ts in including the full set of available well-site parameters. Prediction of permeability from models based on log infor- mation alone is useful for screening purposes, whereas per- meability prediction from models based on both log data and core data are, not surprisingly, signifi cantly better. Which approach to use depends on the context in which permeabil- ity prediction is used, especially on the time available for se- curing the additional core information from the laboratory. Th is study shows that direct well-site permeability predic- tion is feasible. Improvements can be made by adding stand- ard He-porosity data and other easily measured conventional laboratory core parameters. Th e feasibility study was based on a 192 m long chalk interval in a single well only. Th e data- base can be extended to include more of the comprehensive core data available from the Danish North Sea. Based on an augmented data set, it is in principle an easy task to refi ne this pilot study to investigate the more general limits of the feasibility demonstrated. A parallel study based on a similar approach using log data and logs + core data also proved successful for prediction of ‘functional rock types’ for other lithologies than chalk, i.e. Alum Shale (Schovsbo et al. 2015). 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Schovsbo, N.H., Esbensen, K.H., Nielsen, A.T., Derbez, E., Gaucher, E.C., Poirier-Coutansais, X., Riou, A., Tallone, P. & Milton-Taylor, D. 2015: Rock types in the Scandinavian Alum Shale resource play: defi ni- tions and predictions. 77th EAGE Conference & Exhibition, Madrid, 1–4 June, 2015. Abstract. Authors’ address Geological Survey of Denmark and Greenland, Øster Voldgade 10, DK-1350 Copenhagen K, Denmark. E-mail: ke@geus.dk