Class Note for EECS 833 at KU
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Date Created: 02/06/15
INTRODUCTION TO WELL LOGS And BAYES THEOREM EECS 833 27 February 2006 Geoff Bohling Assistant Scientist Kansas Geological Survey geoffkgskuedu 8642093 Overheads and resources available at http peoplekuedugbohlingEECS833 Critical Petroleum Reservoir Properties Petroleum geologists and engineers are interested in the properties of the rocks constituting oil and gas reservoirs In particular they are interested in characterizing the distribution of the following properties in the subsurface Porosity The volumetric proportion of void space in a given volume of rock includes a range of pore sizes from 0012 mm 102000 microns in sedimentary rocks Pore size is generally a function of grain size larger grains larger pores and occurs in a variety of ways including spaces between grains andor crystals as molds of dissolved grains or as vugs which are large pores resulting from dissolution of a portion of the rock The pore space is lled with uids including water oil and gas Porosity is expressed as a percentage or a decimal ratio Permeability A property characterizing the ease with which uids ow through the rock dependent on both porosity and pore space geometry Permeability is generally expressed in millidarcies md Relative Saturations The proportion of pore space occupied by water oil and gas are also important since these in uence the overall volume of oil or gas in the reservoir and the relative permeability of the formation to each uid Oil cannot ow through a water lled pore Facies Facies is a rather vague term that in general means rock type It is vague because rocks can be categorized in different ways depending on the goals of the analysis Petroleum geologists are generally interested in categorizing rocks according to their ability to store and transmit uids that is their porosity and permeability For example sandstones generally have relatively high porosity and permeability while shales have high porosity but low permeability The application that Marty Dubois will describe involves carbonate rocks limestones and dolomites which can display a wide range of porosities and permeabilities depending on how they were formed and their history subsequent to original deposition The facies distribution is a first order control on the distribution of porosity and permeability in a reservoir Geophysical W ireline Well Logs Many wells drilled for petroleum exploration are logged meaning that a tool or tools containing various sensors is lowered into the borehole and then drawn back up while the sensors measure various properties of the surrounding rock The measured properties generally include electrical acoustic and nuclear properties of the surrounding medium the combined properties of both the rock matriX and the uids in the pore space The resulting records of measured properties versus depth are variously referred to as wireline logs because the tools are lowered on a wireline well logs geophysical logs or just plain logs These logs give indirect information regarding the distribution of the critical reservoir properties discussed above and log analysts spend most of their time trying to infer reservoir properties from logs Some of the more common logs include Gamma Ray Measures natural radioactivity of the surrounding rock The most abundant radioactive isotopes those of Potassium Thorium and Uranium occur in higher concentrations in the clay minerals that constitute shales so the gamma ray log is particularly useful for distinguishing shales from other rocks Density and Neutron Porosity Two different sensors providing estimates of the porosity the first based on the bulk density of the surrounding rock and the second based on neutronabsorbing capability of the medium These two porosity estimates tend to have complementary biases so they are often combined to get a more accurate estimate Electrical Resistivity Because of the large contrast in electrical resistivity between most rockforming minerals highly resistive and any contained uids more conductive variations in bulk electrical resistivity are dictated primarily by variations in porosity and in the relative saturations of the uids in the pore space Given a porosity estimate from one or both of the density logs described above the resistivity log is generally used to estimate the relative saturations of water oil and gas Photoelectric factor Measures the photoelectric absorption capacity of the surrounding medium sensitive to the mineralogical composition of the rocks and thus good for discriminating lithology rock type sensu stricto In somewhat idealized form a set of logs Versus depth in the borehole in feet 7 the numbers in the center track might look like the following This is actually a synthetic example With logs computed from a perfectly known sequence of rock types drawn in the depth track and With no in uence from uid Variations but it is fairly realistic For more information on log analysis see httpWWWkgs kueduPRS ReadRocksportalhtml httpWWWkgs kueduPRSInfopdfoilgas loghtml General reference on petroleum geology Selley Richard C Elements of Petroleum Geology Second Edition 1998 Academic Press San Diego Development of Bayes Theorem Terminology PA Probability of occurrence of eventA marginal PB Probability of occurrence of event B marginal PAB Probability of simultaneous occurrence of events A and B joint PAB Probability of occurrence of A given that B has occurred conditional PBA Probability of occurrence of B given that A has occurred conditional Relationship of joint probability to conditional and marginal probabilities PABPABPB or PAB PBAPA So PABPB PBAPA Rearranging gives simplest statement of Bayes theorem P A B P B PA Often B represents an underlying model or hypothesis and A represents observable consequences or data so Bayes theorem can be written schematically as Pm0deldata 0c Pdatam0delPm0del This lets us turn a statement about the forward problem Pdalam0del probability of obtaining observed data given certain model into statements about the corresponding inverse problem Pm0deldala probability that certain model gave rise to ob served data as long as we are willing to make some guesses about the probability of occurrence of that model Pm0del prior to taking the data into account Or graphically Bayes theorem lets us turn information about the probability of different effects from each possible Cause gt effects A we 0 into information about the probable cause given the observed effects cause B O O 0 possible observed causes BO effects A O 0 Illustration styled after Sivia 1996 Figure 11 Assume that B represents one of 11 possible mutually exclusive events and that the conditional probability for the occurrence of A given that B has occurred is PAB In this case the total probability for the occurrence of A is PltAgt 21PAlePBi and the conditional probability that event B has occurred given that eventA has been observed to occur is given by P Alligt130 PABiPBZ PABJPBJ PM PB A 2 That is if we assume that event A arises with probability PAB from each of the underlying states 3 i1 n we can use our observation of the occurrence of A to update our a priori assessment of the probability of occurrence of each state PB to an improved aposleriori estimate PBA DiscreteProbability Example DolomiteShale Discrimination Using Gamma Ray Log Threshold Reservoir with dolomite pay zones and shale nonpay zones Gamma ray log Measures natural radioactivity of rock measured in API units Shales Typically high gamma ray l 10 API units due to abundance of radioactive isotopes in clay minerals somewhat lower in this reservoir 80 API units due to high silt content Dolomite Typically low gamma ray lO15 API units but some hot intervals due to uranium Can characterize gamma ray distribution for each lithology based on core samples from wells in eld Dolomite Shale Mean 258 852 Std Dev 186 149 Count 476 295 Cores are sections of rock extracted from the borehole during the drilling process Cores provide the most direct information on many reservoir properties but they are expensive Their utility is often extended by calibrating log responses to core properties in cored wells and then using the derived relationships to predict those properties from logs in uncored wells 10 Gamma ray distributions for dolomite and shale 004 dolomite 0 o o I 002 Probability Density 001 000 I I I I I I I 0 20 40 60 80 100 120 140 160 Gamma Ray API Units Will use these distributions to predict lithology from gamma ray in uncored wells rst using a simple rule if GammaRay gt 60 call the logged interval a shale if GammaRay lt 60 call it a dolomite ll Using Bayes rule we can determine the posterior probability of occurrence of dolomite and shale given that we have actually observed a gamma ray value greater than 60 Let s de ne events amp probabilities as follows A GammaRay gt 60 Bl occurrence of dolomite Bzz occurrence of shale PBl prior probability for dolomite based on overall prevalence E 60 476 of 771 core samples 1332 prior probability for shale based on overall prevalence E 40 295 of 771 core samples PABl probability of GammaRay gt 60 in a dolomite 7 34 of 476 dolomite samples PABz probability of GammaRay gt 60 in a shale 95 280 of 295 shale samples Then the denominator in Bayes theorem the total probability of A is given by PM PA31PBl PA32PBz 007 060 095 040 0422 12 If we measure a gamma ray value greater than 60 at a certain depth in a well then the probability that we are logging a dolomite interval is 2010 PB IA PABIPB1 007 060 1 PM 0422 and the probability that we are logging a shale interval is PABPB 095 040 PUMA Z PA 0422 090 Thus our observation of a high gamma ray value has changed our assessment of the probabilities of occurrence of dolomite and shale from 60 and 40 based on our prior estimates of overall prevalence to 10 and 90 We can do simple sensitivity analysis with respect to prior probabilities For example if we take prior probability for shale to be 20 meaning prior for dolomite is 80 then get posterior probability of 77 for shale 23 for dolomite if the gamma ray value is greater than 60 API un1ts l3 ContinuousProbability Example DolomiteShale Discrimination Using Gamma Ray Density Functions It is also possible to formulate Bayes theorem using probability density functions in place of the discrete probabilities PAB We could represent the probability density function that a continuous variable X follows in each case as fxBi or more compactly fx Then PBix 12f ltxgtPltBjgt That is if we can characterize the distribution of X for each category B we can use the above equation to compute the probability that event B has occurred given that the observed value of X is x For example based on the observed distribution of gamma ray values for dolomites and shales a gamma ray measurement of 110 API units almost certainly arises from a shale interval because the probability density function for gamma ray in dolomites evaluated at 110 API units f 1 x1 10 is essentially O This form of Bayes theorem lets us develop a continuous mapping from gamma ray value to posterior probability l4 ShaleDolomite Discrimination Using Normal Density Functions Dolomite 1 Shale 2 Mean Y 258 852 Std DeV s 186 149 Count 476 295 f1 x j eXp x Elf21912 xxx expl x Yzsil 52 Normal Approximations for Gamma Ray Distributions 004 Kernel density estimate Normal density estimate 003 a 39 5 dolomite o E 002 5 N 0 2 n 001 39 000 5 30 55 80 105 130 155 Gamma Ray API Units 15 Let q2 PBz represent prior probability for shale prior for dolomite is then PB 1 l qz Let p2x PBzx represent posterior probability for shale posterior for dolomite is then PBlx l p2x So posterior probability for shale given that the observed gamma ray value x is x q2f2x p2 1 q2f1xq2f2x Shale Occurrence Probability Using Normal Densities 11 005 10 09 06 i 39 004 02 prior probability for shale used to compute posterior 39 003 Posterior Probability for Shale solid lines Probability Density dashed lines x 002 039 0 I I t r u normal pdfforshale quot normal pdffor 39 x dolomite quot 02 t a 001 r 01 x 00 i quot I quot i 000 596 0 50 100 150 Gamma Ray API Units 16 Bayes rule allocation Assign observation to class with highest posterior probability For base case prior of 40 for shale 50 posterior probability point occurs at gamma ray of 596 a so Bayes rule allocation leads to basically same results as thresholding at 60 API units But now have means for converting gamma ray to continuous shale probability log 2000 2000 2010 x 2010 2020 E 2020 2030 2030 E 2040 E 2040 a a g g 2050 2050 7 7 D 2000 D 2000 2070 2070 2080 2080 2090 2090 2100 2100 0 30 00 90 120 00 02 04 00 08 10 Gamma Ray API units Posterior Probabilityfor Shale 17 ShaleDolomite Discrimination Using Kernel Density Estimates No need to restrict approach to just normal densities Could use any other form of probability density function for each category including the kernel density estimates shown initially Shale Occurrence Probability Using Kernel Densities 005 10 kernel pdffor E sandstone E 004 A 3 a o 08 I EL 5 g prior probability for shale used 4 c l to com t st 39 39 003 a U I u pu e p0 enor u 5 06 1 a 39l a f a 2 E u g quotI I39 0 02 E n 2 04 a kernel pdf for shale g n y I 0 f g 3 2 3 39 n quot39 quot 001 g 02 I a I I J a h h 00 39i39 quot i 000 Gamma Ray API Units Kernel density estimates are essentially smoothed histograms scaled to represent legitimate probability density functions nonnegative everywhere and integrating to 1 They can be computed using the ksdensity function in Matlab s statistics package 18 Relationship to Discriminant Analysis Could just as easily use multivariate density functions in Bayes theorem For example could be discriminating facies based on a vector of log measurements x rather than a single log If use multivariate normal density functions for each class Bayes rule allocation leads to classical discriminant analysis Assuming covariance matrices all equal for different classes leads to linear discriminant analysis Bayes rule allocation draws linear boundaries between classes in X space Assuming unequal covariance matrices leads to quadratic discriminant analysis Bayes rule allocation draws quadratic boundaries between classes We will talk more about discriminant analysis next time 19 An Excel Workbook implementing the shaledolomite discrimination example is available at huggeoglekueduNgbohling EECSSSS IXle a mw bunmm wmmmunwww 42mm man In Em Em m yr mfu ii iSWGi39h 20 anmAnn 315 V i AV WV mMWn1nhmm rmwwm nm sr w W W w W m y WWW Wm mam 3 mmmnmmwmwz arm 7 r hum MHVMMMMW7U in m a x w WWW W m M 21
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