Automatic Determination of Lithology From Well Logs Pierre Delfiner, SPE, Etudes et Productions Schlumberger Olivier Peyret, SPE, Schlumberger Overseas S.A., Brunei Oberto Serra, Schlumberger Technical Services Summary. A procedure combining modern wireline measurements with a lithofacies data base has been developed to produce an automatic lithologic description of the formations crossed by a well. The database lithofacies are defined from petrographic knowledge and translated in terms of log responses. The assignment of depth levels to a lithofacies is done with the data base and with a discriminant function (Bayesian decision rule). External knowledge can be taken into account by use of artificial intelligence methods'. A confidence factor is produced for each result. . Logs currently in the data base are the density, neutron, sonic transit time, gamma ray, photoelectric cross section, and concentrations of thorium, potassium, and uranium. Major lithofacies groups represented in the data base include sandstones, limestones, dolomites, shales, coals, and evaporites. These are subdivided by introducing cement and special minerals and by considering porosity ranges. The construction of the data base is a critical step. It is largely empirical and requires careful calibration against intervals with well-known lithologies (e.g., from cores). The data base can be tuned to local conditions. The procedure has been tested in several environments and compared with cores and mud log descriptions. A detailed lithologic column can be produced at the wellsite and used in decision making. The results can also serve as input for further geologic studies of facies and sequences or for quantitative evaluation of formations. Introduction Subsurface lithology is traditionally determined from core or cutting analysis. Cores are generally not continuous and consequently do not provide a complete description of formations crossed by a' well. It can be difficult to restore the components and the thickness of the lithologic column from cuttings. This is because of mud swirling, influence of caving (fall-down of wall fragments), loss of some constituents (salts and silts), or even the total loss of circulation. As a result, the lithology column based on those data is not sufficiently accurate and precise for quantitative use. Well logs give a practically continuous survey of the formations crossed by a well. They allow measurement of apparent thickness and of real thickness if dipmeter data are taken into account. Burke et al. 1 and Clavier and Rust 2 have shown that well log responses can give a good idea of the lithology. With the increase of physical parameters recorded by modern logging tools-e.g., photoelectric cross section, natural or induced gamma ray spectrometry (GRS), and dielectric constant-it becomes more obvious that their combination can give a good lithologic description of the formations. This evidence was the basis for the concept of "electrofacies" (Serra and Abbott 3 ) defined as "the set oflog responses which characterizes a bed and permits it to be distinguished from other beds" (Serra 4 ). Applied to openhole logs, this electrofacies is an equivalent of the lithofacies that, according to Moore,5 is the "total sum of the lithological characteristics (including both physical and biological Copyright 1987 Society of Petroleum Engineers SPE Formation Evaluation, September 1987 characters) of a rock." It is not always obvious, however, to translate this electrofacies in terms of geologically meaningful rocks. A procedure combining modern wireline measurements with a lithofacies data base created from logs (strictly speaking, an electrolithofacies data base) has proved to be effective in this translation. The lithofacies are previously defined from petrographic knowledge and translated in terms oflog responses. This forms a data base of rocks (sandstone, limestone, etc.) as opposed to minerals (quartz, calcite, etc.). Levels of log data are assigned to a given lithofacies by use of the data base and a discriminant function. The objective in this approach is to define the lithology by use of a set of log responses applied at the wellsite and at the computing center. Approach The basic idea is to represent a set of n log readings at a given depth level as a point in n-dimensional space, the log space. An electrofacies is then materialized as a cluster in this space-i.e., an area with a relatively high concentration of points. The points are close because the log responses are similar. Our task is to attach to each level of the well the cluster, or group, to which it belongs. Statistical theory provides two types of approach. 6, One approach is to determine the clusters from the data in each well. This "clustering" has the advantage ofletting the data "speak for themselves" and reveals their subtle differences. However, the geologic interpretation of the clusters must be ~epeated each time. A procedure known as FAcloLoa™ has been developed to analyze electrofacies by clustering. 3,7 303 RHOB~ , · : ·· .. . . . : :..: ., . .. LITHIC SANDSTONE :. ..............• LIMESTONE 25·35 PU FELDSPAR N ... ";".00 QUARTZ . • ....,.\> ' .. """"".""'''''.''''''''.,.''~''''''''''''''''''''''''''''''''''.'''' . , , .. , , . 15.0 25.0 35.0 ijS.Q· NPHI a ROCK FRAGMENTS PEF ~ , ACCESSORY MINERALS HEAVY MINERALS (pyrite, hematite) HEAVY RADIOACTIVE MINERALS (z:ircon, monaz:lte, mica) LIGNITE FRAGMENTS CEMENT ~ CALCITE ,. :LITHIC SANDSTONE is c.:t-.:::"oo----:r::-------".-:t,5-:-,.0-----::!25i:"":,.0--~:r.:---:::-:iij •. o NPHI b DOLOMITE ANHYDRITE HALITE CLAY Fig.1-Projections of a lithic sandstone and a 25- to 35-p.u. limestone electrofacies on (a) density/neutron crossplot and (b) P e/denslW crossplot. The second approach, used here, is "classification" (also called "discriminant analysis"). The grolJ.ps are specified in a lithofacies data base and' then each depth level is assigned to the correct group by use of an appropriate discriminant function. The procedure has the advantage of being fully automatic because the interpretation work is done only once; however, acorrect definition of the 4ata base is critical to obtain good results. There are various classification methods that differ mainly by the criterion, or distance measure, used for assigning a sample to a group., The distance measure can be uniform for all rocks or can vary with the rock. Busch et al. 8 have proposed a method in which the discriminant functions are calibrated on core data for each field, but the same distance measure is used for all rocks. In our approach, the distance measure varies with the rock, reflecting the internal dispersion of log readings for each rock. Each rock is represented in the data base as a volume in the n-dimensional space of logs. A volume.is used rather than a point to allow for variations resulting from both geology and data acquisition. By definition, a rock has mineral' proportions that vary within certain ranges. For example, a quartzose sandstone is composed of 90 to 100% quartz, and 0 to 10% other material (feldspar, mica, clay, or calcite). Rocks also vary in porosity, texture' and fluid content. Data acquisition introduces variations of its own, independent of lithology, by way of pad contact, borehole rugosity, invasion profile, mud effect, and shoulder effects. In the case of nuclear tools, there are also statistical fluctuations in count rates. 304 Fig. 2-Classification of sandstones· (adapted from Pettijohn 10,11). The size of e~ch volume reflects the amplitude of all these variations. For example, gypsum is a lithofacies with a well-fixed composition and is represented by a small volume in the data base. By contrast, the volume representing shaly sands is quite large. In practice, this volume is divided into subcategories, keeping in mind the tradeoff between robustness and resolution: a large electrofacies volume "absorbs" more variations due to acquisition statistics but may also hide real lithologic differences. Each electrofaciesvolume is usually represented as an ellipsoid in n-dimensional space by reference to the multivariate Gaussian distribution. The model is simple and yet flexible enough to capture the main features of an electrofacies-range on eacn log and directions of main elongation. Fig. 1 shows the representation of two electrofacies, a lithic sandstone (quartz, feldspar, rock fragments, and chlorite) and a 25- to 35-p.u. limestone, in the three-dimensional (3D) space defined by the density, neutron, and photoelectric cross-sectional (P e) lithodensity logs. From the neutron/density crossplot alone, it is not possible to distinguish between the two rocks; however, they are well separated by the P e' Generally, the more dimensions, the better the discrimination. Construction of the Lithofacies Data Base A data base of about 150 rocks was designed to cover the most·abundant sedimentary rocks. Some igneous rocks have been included because they are encountered in sedimentary series as basements or as intrusions of lava flow (weathered granite reservoirs would be counted as SPE Formation Evaluation, September 1987 OPEN HOLE LOGS CORE DESCRIPTION LITHOLOGICAL t-L--T-H-0-rF.-~-C-IE-S--_f COLUMN SUB MAJOR FROM LOGS definition definition _J.ftJL_ 46 p.u. RHOS -15 150 :::I::t':~J~~':t:.:~ 1"~'f.. I""'""'O"t. ,eckstone, .... WICke "onl wnh common '"hr'titln"ula. CALCAREOUS 0-10 PU PURE ANHYDRITE CALCAREOUS 0·10 PU TIGHT DOLOMITIC CALCAR.EOUS 0-10 PU ANHYDRITIC DOLO EVAP DOLO LIME DOLO PURE ANHYDRITE EVAP LIME Chicklnwi,••n,'Ydrita TIGHT OOLOMITIC LI'htohw••' .... tom.d w '..""r.ye.le.r . CALCAREOUS 0·10 PU . ....llIoun•••O,. ••c....ton••. ,.,fllnn. tn' ,'I'nt,onl .. n.... 'hin,I,.'ll'y'OIO""1 IH••• "I'•. chickln :;~~i:::::~:: In· .•n. ,,'" ••"..".r., 'om'd~ ,II',., •••• I'11 ~r lum •• EVAP ANHYDRITIC LJME PURE ANHYDRITE EVAP MEDIUM POROUS10·25P ionll.lckt"onl C"telltn wi,. PURE- ANHYDRITE ANHYDRITIC ,rain,.tnetnlll,.I..... ,.c.....n•. wi.hocc•• • .,It. LIME DOLO il" LIME TIGHT 0-10 PU I ,"h.. · L'ltttOliv••, • .,.Iome'- PURE ANHYDRITE ANHYDRITIC EVAP DOLO DOLOMITIC 25-35 PU LIME I'URE ANHYDRITE EVAP ANHYDRITIC DOLOMITIC 10-25 PU DOLO LIME PURE ANHYDRITE EVAP ANHYDRITIC DOLO TIGHT 0·10 I'U LIME. =.T:~r:~:~~r.:c:;, ....1'Inde.ton, In' "C"'lonewllh,"h... • dri'l nMMI•••nd I• .,., :~itC.hiCII.nWI,• • n"". -r---::::_ DIrk ".,,..,,•. w• .,., In' ,.,a"lllamlne,•• hm.m......en••• ,....m.reu.w.c.... l'I"""',ICIl.'I . Wlth,.""''''',,11 "'f.'tcar,." Fig. 3-Comparison of lithofacies description from logs and cores in a carbonate and evaporite series. arkoses). Metamorphic rocks have not been introduced because of lack of control and lack of logs recorded in such formations. The logs currently introduced in the data base are natural gamma ray; thorium, uranium, and potassium contents as computed from natural GRS; density; neutron porosity (or apparent hydrogen index); sonic transit time; and P e' Other possible additions are sonic waveforms, following the work of Hoard, 9 and results from Gamma Ray Spectrometry Tool (GST™) measurements, which provide the concentrations in elements (iron, calcium, silicon, sulfur, oxygen, carbon, hydrogen, chloride, etc.). Three approaches can be considered for the construction of the data base. Theoretical Approach. This is based on petrographic classifications found in the literature and on tool response SPE Formation Evaluation, September 1987 equations. For example, detrital rocks have been classified by Pettijohn 10 into quartzose sandstones, arkoses, graywackes, and shales, as illustrated in Fig. 2. For each type, the ranges of the mineralogical and elemental compositions result from analyses of typical rock samples reported by Pettijohn et ai. 11 These compositions can be converted to well log responses by use of tool response equations and the well logging parameters of minerals established by Edmundson and Raymer. 12 The influence of porosity, of accessory miner'.lls, and of data acquisition statistics is also considered. Interpretive Approach. This method is based on interpretation of crossplots validated with core or cutting data. Ellipsoids are defined from their two-dimensional projections on crossplots, which are ellipses. Each ellipse is specified by the minimum and maximum on each log and 305 GLOBAL RESULTS OPEN HOLE LOGS c( ~ IX a.. LITHOLOGIC COLUMN LITHOFACIES SUB MAJOR definition definitio I(AOLINITIC ;i ..~_·-I+-----t=''-''1-oI-:::1-~r--f''''''-'''';;~~~~~'''''';''~-ICOMPACTEDKAOLINITI KAOLINITIC WITH HEAYY MINERALS KAOLINITIC POROUS RADIOACTIYE UNCOMPACTED KAOLINITIC PEAT UNCOMPACTED KAOLINITIC LO.OSE HEAVY SILTY SAND SHAL COAL SHAL CLEAN 30-45 PU SAND CLEAN. 20·30 PU CLEAN 30·45 PU POROUS RAOIOACTIYE LOOSE HEAYY SILTY SILTY IlRAYWACKE 35'45 PU UNCOMPACTED KAOLINITIC SHAL SAND SHAL KAOLINITIC L1GNITIC DR BITUMIN. COAL PEAT L1GNITIC OR BITUMIN. SHAL PEAT COAL Fig. 4-Lithofacies description from logs in a sand,shale, and coal series. a correlation coefficient. The lithofacies are defined manually by a geologist who interprets the crossplots in mineralogical and petrographic terms. Simple statistical techniques, such as histograms and Z-plots, help to determine ellipsis boundaries. Experimental control is obtained by analysis of all possible log data crossplots in wells where the composition is known either from core or cutting analyses or from regional geologic knowledge. Empirical Approach. This approach is based on fitting a log model on cores. If lithologies are picked from cores, it is possible to build a log model by analyzing the distribution of log responses in front of the conis; however, this requires that cores. and complete logging suites for all rocks be entered into the data base. This is usually not possible except in field studies. Alternatively, a clustering analysis (e.g., by FACIOLOG) may be used to define the electrofacies . 306 The procedures actually used to construct the lithofacies data base is a mixture of the three approaches, making use of cores whenever available, but also of tool response equations and of crossplots to cover the other cases. Thus a general data base was builtto integrate data from different wells representing various types of lithologies. The data base includes 30 sandstones, 25 shales, 30 limestones, 25 dolomites, 25 evaporites, 3 coals, 10 igneous rocks, and 4 miscellaneous rocks. . To account for the influence of porosity on log responses, porous rocks have been subdivided into several porosity ranges .. Porosity is assumed to be filled with water. The effect of gas or light hydrocarbon on the logs has to be detected and corrected beforehand. Sandstones have been defined following Pettijohn's 10 classification. Cement (calcareous, dolomitic, or halitic) and accessory minerals have also been introduced-e.g., heavy minerals (pyrite and hematite), heavy radioactive SPE Formation Evaluation, September 1987 SH 5S 5S ~~ SH SS,;SH 7ZS0 55 SS,;SH 7250 SH COAL SH COAL "'Ot~ II. co.o4 6 ~~ . 10 CJ»(. J ~""', ........~, 7300 CI1"" . ~iii!lilrr-r---------------1 SH '"!!!---- 7300 59 "1 Go el' - ~", la, "fo.' ):) 50..1... I ~1I..&.'" - "'/~·· . o -.Q.,l».\>; 2l,"~"J ..... Q.W.JI'"t+; eua.QQ. • .) ... "- 5S SH S5,;5H .+. 11DC~ ,~~~~ c':S~ flO . . J.f'c...<~ 7350 - - - - t-ti;IOO~S.:T'-al:;:.:-,~.-(:'t~ ... ::---~,\'1:-)""':' ' '::-:'f'o(:-,-occ.-.-ftA""':'""':'-;.-ss--~r-II-' -~ .:~-- -..: fI\d.&a.( \I. cas4 6r1i-. ~ GO"", .I\:>, ......a 0.... , ...""\. , f'i~ , _~ - "'to· c\.... COAL 7350 v- - - SH ,cit ~ .......", ~ M", - . ~ ~) c:l..- 1 , . -nL "', elf'- .... ,'(J II\", '"1~. ""'. J N" , dMA,I. -\ll,\-, Fig. 5-Comparison of lithofacies description from logs with a mud-log description from cuttings. minerals (zircon, monazite, and micas), and lignite fragments. For shale lithofacies, four grades of compaction have been considered (tight, compacted, average, and uncompacted). When natural GRS data are available, the clay mineral type is deduced from the thorium and potassium values and their ratios. The introduction of induced GRS measurements will provide even finer clay typing. Special shale lithofacies have been included .(lignitic or bituminous shales, and shales with heavy minerals, reflecting the influence of limonite, siderite, or pyrite). Carbonates have been subdivided into limestones (less than 25 % dolomite), dolomitic limestones (25 to 50 % dolomite), calcareous dolomites (50 to 75 % dolomite), and dolomites (more than 75 % dolomite). Other lithofacies have also been introduced to account for the presence of silica (especially chert), anhydrite, heavy minerals, clay (argillaceous), or radioactive minerals (apatite). Rarer carbonates, such as siderite and ankerite, have also been introduced for completeness. Evaporites are well known and e"sy to recognize from log responses, especially if they are present in beds thick enough to make the influence of adjacent beds negligible. Pure evaporites have very well-defined log responses and in principle could be represented as points in the electroSPE Formation Evaluation, September 1987 facies data base. They have been represented as volumes, however, to account for minor changes in composition and statistical variations of the log responses related to borehole conditions or acquisition statistics. Organic rocks have been differentiated in coal, lignite, and peat according to the small changes observed on log values. Some igneous rocks have been introduced into the data base. Their mineralogical composition is reflected mainly by density and sonic transit time, as well as thorium, potassium, and uranium content. These rocks have a high resistivity when they are pure and are not weathered or fractured. Statistical Analysis A statistical procedure, known as a Bayesian decision rule, uses the data base to classify depth levels. The principle is to attach a probability distribution of log values to each lithofacies and then to identify from which level the set of log readings most likely originated. The possibility that a depth level does not belong to any of the specified lithofacies is also considered. Within each lithofacies, the vector X of log responses, suitably transformed, is assumed to have an n-variate Gaussian distribution whose density is denoted by P(XIF i ). This corresponds with the ellipsoids of the data 307 BIOTURBATED? EXAMPLE OF COMBINATION OF LITHO GEODIP CLUSTER FOR LITHOLOGY FOR TEXTURE FOR STRATIGRAPHY DRAPE LAMINATED LEGEND MASSIVE _______I~ Fig. 6-Combination of lithofacies description from logs with base that are interpreted as the volumes containing 95 % of the points. These distributions allow calculation of the probability of the log readings by use of the lithofacies. What is needed, however, is the probability of the lithofacies P(F j IX) given the log readings, called the posterior probability ofF j • It can be obtained from Bayes' formula: pjP(XIF j) EjpjP(X\Fj ) , where Pi =the prior probabilities of the lithofacies before any logging data were collected. They 'represent the information that may have been available on the lithology. For example, if there are only carbonates in the well, P j may be set to zero for all sandstones. When no information is available, all lithofacies are given an equal chance and the p;' s are all equal. Values other than these may be derived from external geologic knowledge of the area of interest. The lithofacies selected as the answer at a given depth level is simply that which maximizes the posterior probability: The computed posterior probability provides an index of confidence in the result. A point falling outside of all 95 % ellipsoids is classified as "unidentified." The procedure described (which can be shown to minimize the risk of misclassification) is applied successively to all levels of the studied interval in combination with some special logics reviewed here. Examples Fig. 3 shows an example of the lithologic column obtained in carbonates. The interval is typical of the Upper Jurassic 308 GEODIP and CLUSTER CI.USTER I results. in the Middle East. There are two levels of description: one indicates the major lithology (e.g., limestone, dolomite, or evaporite), and the other qualifies it (e.g., anhydritic dolomite). A posterior probability curve is shown for the major and secondary descriptions indicating, on a scale of 0 to 1, the confidence in each of the results. For example, all anhydrite'beds are detected witb a confidence of 1, except the fourth one from the bottom where the confidence is only 0.7, because the bed is not pure anhydrite as evidenced by the density, which is too low for pure anhydrite. A core description is shown on the side for comparison and indicates that the interval is partially dolomitized with nodules of anhydrite. Thus, the lithology has been correctly identified by the logs used in this example (density, neutron, gamma ray, and sonic). A more complete description from the cores requires texture- and structure-sensitive tools, such as the dipmeter. Fig. 4 shows results from a complex shale/sand lithology, Coal (more precisely, peat) is detected and clay minerals are identified as kaolinite by use of the thorium and potassium logs; this has been confirmed by core analysis. Fig. 5 shows the match between a lithologic description from logs and a wellsite litholog established from drill cuttings. The agreement is quite good in most of the well, with a correct identification of shales, sandstones, and coal beds. In the upper section, between 7,250 and 7,255 ft [2210 and 2211 m], however, the mud log indicates coal, whereas log-derived lithology indicates sandstone. Conversely, coal beds appear on the logs between 7,232 and 7,242 ft [2204 and 2207 m], while the mud log indicates sandstone. This difference occurs because the coal beds are positioned at drilling breaks on the mud log; the high SPE Formation Evaluation, September 1987 penetration rate of the drill bit in sandstone has led to ~r­ roneous location of the coal at 7,250 ft [2210 m], whIle it actually came from above. Therefore, the disagreement is not really about lithology but about depth, and the answer from logs is more reliable because logs have better depth control. Special Logics A straightforward level-by-level application of the Bay~s ' decision rule is not sufficient to solve some of the practI. cal problems that may arise (see Serra et al. 13). In principle, logs processed for lithology determinati9n have been corrected for borehole effects (temperature, rugosity, etc.). But if the effect is too serious (caving, baryte or potassium salt in the mud, or gas), the affected logs either will define wrong electrofacies or will result in unidentified levels. In such cases, the affected logs are skipped for the determination of the lithofacies and a flag appears to indicate that the answer is not as reliable as in other zones. Light hydrocarbons, and especially gas, may.have an overwhelming effect on the responses of the densIty, neutron, and sonic logs, and may completely overshadow lithologic differences. Thus, light hydrocarbons are detected beforehand and their effect on density and neutron is corrected. Correction for the sonic log is more difficult, so this log is simply skipped in gas-bearing intervals. To avoid holes in the data base, which result in many unidentified levels, the ellipsoids have been defined with partial overlap. This may create instability when points are at the boundary between two lithofacies. To overcome this problem, a vertical-continuity logic has been introduced. Because mineralogical or chemical composition is not the only parameter used in the classif~cation, tw.o .rocks with similar compositions but very dIfferent ongIns or textures may be represented approximately by the same volume in the n-dimensional space. For example, some graywackes and volcanic agglomerates both have th~ c?mposition of a trachy-andesite. In such a case, the dIstInCtion can be made only by introducing external knowledge. The allocation program is guided by specifying the main lithologic type in a given interval to disallow some lithofacies. This information may be derived from the analysis of other logs, such as spontaneous potential, caliper, and resistivity. Alternatively, local geologic knowledge bases holding formation tops and general lithologic composition may be used. For example, "7,900 ft [2408 m] Top Arab-D" indicates the presence of carbonates .below this depth. Geologic knowledge bases can be bUllt and consulted with artificial intelligence methods. In thin beds, log responses are influenced by surrounding beds and the deflection is often not enough to allow correct identification, although it may allow detection. Thus, a minimum bed thickness can be specified with the capability of retaining thinner beds of some very characteristic lithology (e.g., coal, shale, or even sand beds). Applications The first application is, of course, the display of a lithologic column, which can help the geologist at the wellsite to cross-check and depth-match mud logs; to select tops and bottoms of zones of interest, locations of sidewall core samples, and fluid sampling or pressure measSPE Formation Evaluation. September 1987 urement depths for the Repeat Formation Tester (RFT™); to define intervals for a drillstem test; and to provide input to a wellsite formation evaluation "quick look." Applications at the office include well-to-well correlation (in conjunction with other information); mapping of isoliths-i.e., lithofacies thicknesses, after conversion to. true vertical thickness; correlation with seismic data, especially vertical seismic profiles; and automa~ic ~ho~ce of the mineralogical model needed for a quantItatIve Interpretation, and hence automatic selection of the parameters. . The log-derived lithologic column becomes an espeCially powerful geologic tool when combi~ed. with. hi~h­ resolution dipmeter results. An example IS gIven In FIg. 6, which shows a composite display of the resistivity curves from the High-Resolution Dipmeter Tool (HDT™) with correlated events found by the processing . (GEODlp™ program), the lithologic column limited. by the thorium and potassium curves, and the computed dIpS. The drape pattern observed in the upper section of the interval resulting from the dipmeter could suggest several interpretations-e. g., a fluvial channel fill, a bar, or a reef. The limestone lithology, however, rules out a fluvial or deltaic environment and indicates a carbonate buildup in which the upper laminated layers were molded on top of the underlying massive limestone. Further, the formation is dipping to the south, and therefore the top of the reef is to the north. This is where the next well should be (and was successfully) drilled. Conversely, dipmeter information complements the lithologic description. For example, the thick limestone bed in the middle is seen as massive by the dipmeter (little activity and no correlations), whereas· the argillaceous limestones on top are laminated (high activity and many correlations)., A shale interval in the upper part of the well illustrates the case of high activity on dipmeter curves associated with scarce correlations, indicating possible bioturbation. (In coarser clastic rocks, this behavior is typical of conglomerates.) Thus texture, structure, and lith?logy obtained from wireline measurem~nts can be combI~ed to define the complete geologic faCIeS and to determIne the depositional environment. Nomenclature F i = lithofacies i F j = lithofacies j j = dummy summation index Pi = prior probability of F i Pj = prior probability of P,i P = probability of P e = photoelectric cross-section fj.t = sonic transit time X = vector of log response at a given depth level Z = Z-plot crossplot showing values of a third variable P b = bulk density ¢ N = neutron porosity index Acknowledgments We are particularly thankful to I.C. Levert, S. Luthi, C. Dahan, and J. Harry for their contributions to this work. Thanks are also extended to the oil companies for permission to publish the examples. 309 References 1. Burke, J.A., Campbell, R.L. Jr., and Schmidt, A.W.: "The LithoPorosity Cross-Plot, " paper Y presented at the 1969 SPWLA Annual Logging Symposium, Houston, May 25-28. 2. Clavier, C. and Rust, D.H.: "MID-PLOT: A New Lithology Technique," The Log Analyst (1976) 17, 6. 3. Serra, O. and Abbott, H.: "The Contribution of Logging Data to Sedimentology and Stratigraphy," SPEJ (Feb. 1982) 117-31. 4. "Services Techniques Schlumberger," presented at the 1979 Schlumberger Well Evaluation Conference, Algeria. 5. Moore, R.C.: "Meaning of Facies," GSA (1949), memoir 39. 6. Gnanadesikan, R.: Methods for Statistical Analysis ofMultivariate Observations, Wiley & Sons, New York City (1977) 82-120. 7. Wolff, M. and Pelissier-Combescure, J.: "FACIOLOG: Automatic Electrofacies Determination," paper FF presented at the 1982 SPWLA Annual Logging Symposium, Corpus Christi, July 6-9. 8. Busch, J.M., Fortney, W.G., and Berry, L.N.: "Determination of Lithology from Well Logs by Statistical Analysis," paper SPE 14301 presented at the 1985 SPE Annual Technical Conference and Exhibition, Las Vegas, Sept. 22-25. 9. Hoard, R.E.: "Sonic Waveform Logging: A New Way to Obtain Subsurface Geologic Information, " paper XX presented at the 1983 SPWLA Annual Logging Symposium, Calgary, Canada. 310 10. Pettijohn, FJ.: "Classification of Sandstones," J. Geology (1954) 62, 360-65. 11. Pettijohn, F.J., Potter, P.E., and Siever, R.: Sand and Sandstone, Springer-Verlag, New York City (1972). 12. Edmundson, H. and Raymer, L.L. "Radioactive Parameters for COmmon Minerals," paper 0 presented at the 1979 SPWLA Annual Logging Sy'mposium, Tulsa, June 3-6. 13. Serra, 0., Delfiner, P., and Levert, J.C.: "Lithology Determination from Well Logs: Case Studies," paper WW presented at the 1985 SPWLA Annual Logging Symposium, Dallas, June 17-20. 51 Metric Conversion Factors ft x 3.048* E-Ol in. x 2.54* E+OO 'Conversion factor is exact. m em SPEFE Original manuscript received in the Society of Petroleum Engineers office Sept. 16, 1984. Paper accepted for publication Sept. 20, 1985. Revised manuscript received June 19, 1986. Paper (SPE 13290) first presented at the 1984 SPE Annual Technical Conference and Exhibition held in Houston, Sept. 16-19. SPE Formation Evaluation, September 1987