Cluster Analysis Assisted Float-Encoded Genetic Algorithm for a More Automated Characterization of Hydrocarbon Reservoirs *

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1 Intelligent Control an Autoation, 03, *, ** oi:0.436/ica.03.***** Publishe Online ** 03 ( Cluster Analysis Assiste Float-Encoe Genetic Algorith for a More Autoate Characterization of Hyrocarbon Reservoirs * Norbert Péter Szabó, Mihály Dobróka,, Réka Kavana 3 Departent of Geophysics, University of Miskolc, Miskolc, Hungary MTA-ME Applie Geoscience Research Group, University of Miskolc, Hungary 3 DACHS GbH., Miskolc, Hungary Eail: norbert.szabo.ph@gail.co, obroka@uni-iskolc.hu, kavreka@gail.co Receive **** 03 Abstract A genetic algorith base joint inversion etho is presente for evaluating hyrocarbon-bearing geological forations. Conventional inversion proceures routinely use in the oil inustry perfor the inversion processing of borehole geophysical ata locally. As having barely ore types of ata than unknowns in a epth, a set of arginally over-eterine inverse probles has to be solve along a borehole, which is a rather noise sensitive proceure. For the reuction of noise effect, the aount of overeterination ust be increase. To fulfill this requireent, we suggest the use of our interval inversion etho, which inverts siultaneously all ata fro a greater epth interval to estiate petrophysical paraeters of reservoirs to the sae interval. A series expansion base iscretization schee ensures uch ore ata against unknowns that significantly reuces the estiation error of oel paraeters. The knowlege of reservoir bounaries is also require for reserve calculation. Well logs contain inforation about layer-thicknesses, but they cannot be extracte by the local inversion approach. We showe earlier that the epth coorinates of layer-bounaries can be eterine within the interval inversion proceure. The weakness of etho is that the output of inversion is highly influence by arbitrary assuptions ae for layer-thicknesses when creating a starting oel (i.e. nuber of layers, search oain of thicknesses). In this stuy, we apply an autoate proceure for the eterination of rock interfaces. We perfor ultiiensional hierarchical cluster analysis on well-logging ata before inversion that separates the easuring points of ifferent layers on a lithological basis. As a result, the vertical istribution of clusters furnishes the coorinates of layer-bounaries, which are then use as initial oel paraeters for the interval inversion proceure. The iprove inversion etho gives a fast, autoatic an objective estiate to layer-bounaries an petrophysical paraeters, which is eonstrate by a hyrocarbon fiel exaple. Keywors: Hierarchical Cluster Analysis; Genetic Algorith; Well-Logging; Interval Inversion. Introuction Borehole geophysical easureents are extensively use in hyrocarbon exploration for collecting high resolution in-situ inforation about subsurface geological forations. Well-logging ata easure by ifferent physical principles are recore along epth in the for of well logs []. The priary ai of the interpretation of observations is the lithological separation of the succession of strata an the estiation of thicknesses an petrophysical properties of forations (such as porosity, water saturation, ineral content, pereability etc.) for the calculation of oil an gas reserves. The ost avance tools of well log analysis are base on geophysical inversion ethos. By assuing a petrophysical oel one can calculate theoretical well logs, which are copare to real ones easure in the borehole. The initial oel is progressively refine in an iteration proceure until a proper fit is achieve between the preictions an observations. The estiate oel obtaine in the last iteration step is accepte as the solution of the inverse proble that represents the ost probable geological structure. The atheatical basis an coputer ipleentation of traitional well-logging inversion ethos, which solve the inverse proble epth by epth separately, can be foun in [], [3], [4]. The positions of layer-bounaries cannot be extracte fro the local ata set by this inversion etho instea they are picke out on the lithological logs anually. A novel inversion e-

2 N. P. SZABÓ ET AL. tho that inverts a ata set of a greater epth interval to estiate the variation of petrophysical paraeters an thicknesses along a borehole was evelope by the Departent of Geophysics, University of Miskolc. The inversion ethoology nae interval inversion has been applie to solve several hyrocarbon exploration probles [5], [6], [7], [8], [9]. Multivariate statistical ethos are proven to be powerful tools in foration evaluation [0]. Regression, factor an cluster analysis are wiely use to fin correlations between petrophysical variables, to reuce proble iensionality or explore non-easurable backgroun variables as well as to separate observe ata into istinct groups of ifferent lithological characters. The principles of clustering techniques are etaile in [], to which a variety of petrophysical applications can be foun in the literature, e.g. in [], [3], [4], [5]. In this stuy, we apply the hierarchical for of cluster analysis to separate nearly hoogeneous bes of shaly-san sequences on a lithological basis. This technique is able to specify the coorinates of layer bounaries in an autoatic proceure, which are then use as input for solving a global optiization-base interval inversion proceure. The workflow enables a fast an autoatic estiation of layer-bounaries an petrophysical paraeters to provie the log analysts an reservoir engineers with accurate an reliable inforation for planning the exploitation of hyrocarbon fiels.. Theory of Interpretation Metho.. Well-Logging an Inversion Several books eal with the basics of well-logging surveying an interpretation ethos, aong the any was written chiefly for petroleu geologists [6], [7]. The classification of well logs can be ae by paraeter sensitivities [8], which in this particular case infor how the ata are influence by the petrophysical paraeters, separately. Accoringly, three ain groups of easureents can be istinguishe, i.e. lithology, porosity an saturation sensitive logs. In hyrocarbon exploration relatively high cost easureents are ae to increase the probability of fining an econoically valuable oil/gas fiel. Norally, a fiel ata set consists of natural gaa-ray intensity (GR), spectral gaa-ray intensity such as potassiu (K), thoriu (TH), uraniu (U), an spontaneous potential (SP) ata (lithology logs). Porosity logs coprise foration ensity (DEN), neutron porosity (NPHI) an acoustic traveltie (AT) ata. Resistivity tools easure the foration resistivity with ifferent penetration epth fro the borehole wall. In this stuy, true resistivity (RT) log is applie that represents the correcte resistivity of the part of foration not invae by the rilling u (saturation log). The observe ata are inverte to erive petrophysical properties of forations such as effective porosity, water saturation at ifferent penetration epth, ineral volues, shale content an pereability that cannot be easure irectly, but necessary for the calculation of hyrocarbon reserves. Since the easure ata can be relate to petrophysical paraeters by probe response equations [5], thus the latter can be estiate by an iterative inversion proceure. The principles of geophysical inversion an relate issues are etaile in [9]. Inversion ethos are preferre as they use all suitable logs instea of one or two logs to reuce the estiation error of petrophysical paraeters. To further increase the perforance of inverse oeling as uch goo quality prior inforation as possible shoul be given an proper optiization strategy ust be pose when they are applie... Local Inverse Moeling Let be the colun vector of oel paraeters such as porosity (POR), water saturation in the flushe zone invae by u filtrate (SX0), water saturation of unisturbe foration (SW), shale volue (VSH), quartz content (VSD) in a given epth. Well-logging ata types easure at the sae easuring point are also represente in a colun vector (). If we consier the ata set liste in Section., an inverse proble with 5 unknowns an 9 ata has to be solve in each epth. In this case, the overeterination (ata-to-unknowns) ratio is.8. Theoretical (calculate) well-logging ata are connecte to the petrophysical oel nonlinearly as (c) =g(), where g represents a set of probe response functions. Substituting the initial guess of oel paraeters to the epirically erive response equations one can calculate a local ata set in the easuring point. The solution of the inverse proble is foun at the inial isfit between the observe an preicte ata [9]. The Eucliean nor of the eviation between easure an calculate ata vectors is applie as an objective function for the optiization E k N () k σ k (c) k in, () where σ k is the variance of the k-th well log epening on the probe type an borehole conitions (N is the nuber of applie probes). To optiize Equation (), the Weighte Least Squares etho (WLSQ) is use, where the actual oel is graually refine by = (0) +, where (0) is the initial oel an is the oel correction vector. By introucing the iagonal weighting atrix W kk =/ k (k=,, N) incluing ata variances,

3 N. P. SZABÓ ET AL. 3 the vector of oel corrections in a given iteration step can be estiate by δ G T WG G T W δ, () where G enotes the Jacobi's atrix containing partial erivatives of ata with respect to oel paraeters an is the ifference between the easure an calculate ata vector (T is the sybol of atrix transpose). The quality check of inversion results is base on the following connection between the uncertainties of observe ata an oel paraeters [9] T cov McovM, (3) where cov an cov enote the ata an oel covariance atrices, respectively, an M is the generalize inverse atrix of the actual inversion etho. The latter can be expresse with the proper cobination of atrix G. If we know ata variances fro cov, Equation (3) gives the estiation errors of oel paraeters at the en of the inversion proceure. They are obtaine fro the ain iagonal of oel covariance atrix as,i=sqrt(iag(cov ii )). For easuring the istance between the observe an calculate ata the RMS error is norally use..3. Genetic Algorith-base Interval Inversion Linearize optiization ethos work properly an quickly only when an initial oel sufficiently close to the solution is available. However, in cases when poor prior inforation or rather noisy ata is provie the WLSQ proceure can be easily trappe in a local iniu of Equation (). To avoi localities a global optiization etho such as Genetic Algorith (GA) is use that searches the absolute extree of the objective function. GA belongs to the class of evolutionary algoriths that solves optiization probles using the analogy of natural selection of living populations [0]. Nowaays the ost preferre variant is the Float-Encoe GA that iproves a oel population represente by oel paraeters fro the oain of real nubers in an iteration proceure []. In the population each iniviual has a fitness value representing its survival capability. During the genetic process the fittest iniviuals reprouce ore successfully in the subsequent generations than those who have relatively low fitness. To achieve the best solution, the fitness function is axiize by using genetic operations in a rano optiu-seeking proceure. Fro the point of view of well-logging inverse proble a petrophysical oel has large fitness when the isfit is relatively sall between the observe an calculate ata. In well-logging inversion norally oel take part in the selection process. To reach the absolute axiu of fitness function a proper cobination of genetic operators such as selection, crossover, utation an reprouction is use (GA search in Figure ). Accoring to our experience after soe tens of thousans generations (iteration steps) the fittest iniviual of the last generation can be accepte as the optial petrophysical oel. More etails of the GA-base inversion proceure can be foun in [5], [9]. Local inversion ethos (Chapter..) process barely ore ata than unknowns, where the accuracy of solution highly epens on the noise level of ata an the initial guess of the petrophysical oel. To increase the overeterination of the inverse proble, we efine a set of probe response functions as (z)=g((z)), which is vali in a greater epth interval. In the response functions the ata an oel paraeters are varying with epth. To iscretize the epth variations of petrophysical unknowns we suggest a series expansion technique i q Q i (z) B Ψ (z) (4) where i enotes the i-th oel paraeter, B q is the q-th expansion coefficient, Ψ q is the q-th basis function, Q i is the requisite nuber of coefficients escribing the i-th unknown. Basis functions in Equation (4) are known an arbitrarily chosen. For instance, in hoogeneous layers a cobination of heavisie functions (u) is avantageous to use as basis function. Fro one han, using the basis function q(z)=u(z-z q- )-u(z-zq), each petrophysical paraeter in the q-th layer (where Ψ q (z)=) can be escribe by one series expansion coefficient. On the other han, we can introuce Z q-, Z q upper an lower epth coorinates of the q-th layer as unknown in the inverse proble [5]. For escribing inhoogeneous intervals polynoial basis functions can be use [7]. The oel paraeter vector to be eterine by inversion is =[B,Z] T, where B an Z vectors contain all series expansion coefficients given in Equation (4) an layer-bounary coorinates (or thicknesses) in the processe interval, respectively. (i) q Figure. Workflow of the GA-base inversion proceure. q

4 4 N. P. SZABÓ ET AL. The fitness function of the GA process is inversely connecte to the objective function of the well-logging inverse proble. We teste two types of fitness functions. The first one follows the iea of traitional inversion ethos represente by the objective function in Equation () P N c F ( ax, (5) ) p k where P is the total nuber of easuring points. The iniization of the weighte least squares criterion in Equation (5) leas to optial solution as well-logging ata have ifferent agnitues an easureent units. The only weakness of weighting by variances is that we have to know stanar eviations of all ata types in each epth. In fact, the variances of ata in ost of the cases are not known (just fro literature given for probe types), because norally we easure only once in a epth point. Our experience shows that an optial solution can be given by noralizing the iniviual ata ifferences by the easure ata P N c F () ax. (6) p k To copare the perforance of the F an F base interval inversion proceures a four-layere petrophysical oel was efine (Table ). We calculate synthetic ata by the exact oel paraeters (POR, SX0, SW, VSH, VSD). To synthetic well logs 5% Gaussian istribute noise was ae to prouce quasi easure ata substituting real easureents. The input of the inversion proceure was a noisy well-logging ata set incluing 400 ata (0 processe length, 0. sapling interval, 7 types of well logs). Series expansion was evelope for escribing a layer-wise hoogeneous oel. There were 0 unknowns against ata, thus the overeterination ratio (70) was alost 40 ties higher than that of local inversion (Section..). In Equation (5) the stanar eviations of ata were calculate epirically for each ata types layer by layer. For the characterization of isfit we use the relative ata an oel istances. The first easures the ifference between the easure an calculate ata, the latter quantifies the eviation between the estiate an (exactly) known oel D D () () PN LM P L N p k M l i t li e li e li 00(%), 00(%), where (e) an (t) enote the estiate an target oel, respectively (L is the nuber of layers, M is the nuber of oel paraeters). The outputs of inversion progra runs in Table show highly accurate estiation results. We let the search run till 5,000 generations in both cases (Figure ). The inversion proceures were stable an the ata istance of the solutions base on Equation (7) was aroun 5.% (Figure a). The oel istance for fitness function F was 0.85%, while that of F was 0.6% (Figure b). It was conclue that the application of both fitness functions le to the sae (global) optiu. In case of F a faster convergence towars the optiu was foun after iteration 000, an a little higher retrieval accuracy was inicate as the relative iproveent of oel istance was 37% copare to the case of F. c (7) Table. GA interval inversion results for synthetic case. Layer Petrophysical paraeters as inversion unknowns POR SX0 SW VSH VSD * 0.0 ** 0.0 *** * Maxiization of fitness function weighte by ata variances (F ). ** Maxiization of fitness function weighte by observe ata (F ). *** Exactly known (target) oel paraeter Figure. Convergence of interval inversion proceures.

5 N. P. SZABÓ ET AL. 5 After efining a proper fitness function, the upper an lower bounaries of oel paraeters an the control paraeters of genetic operators ust be specifie (specifying search oain in Figure ). The optial values of series expansion coefficients B are estiate by the GA-base interval inversion etho, which are substitute into Equation (4) to prouce the vertical istributions of petrophysical paraeters. The eterination of layer-thicknesses cannot be accoplishe with local inversion ethos. They have to be a priori known before starting a local inversion proceure. In interval inversion only the lower an upper liits of thicknesses are require that eans a non-autoatic processing step. To autoate this phase, we use cluster analysis for proucing the estiates of layer-thicknesses that can be treate as constant or unknown in the interval inversion proceure..4. Hierarchical Cluster Analysis Clustering ethos can be effectively use for the grouping of well-logging ata in such a way that the N iensional objects specifie by ata sets easure fro given epths are ore siilar than others observe fro ifferent epths. Fro the point of view of our etho, it is of great iportance that objects connecte to the sae cluster efine approxiately the sae lithologic character, while other clusters represent issiilar ones. Aggloerative clustering ethos buil a hierarchy fro the observe objects by progressively erging clusters. At the beginning, we have as any clusters as iniviual eleents. In the first step, the closest points are couple together to for a new cluster. In each following step, the istances between objects are re-calculate an the proceure is continue until all eleents are groupe into one cluster. In this stuy, we use the atrix of Eucliean istances as a easure of issiilarity between the pairs of observe objects. During the proceure the istances between the eleents of the sae group are iniize while they are axiize between the clusters at the sae tie. During the reconnection of clusters we follow the War's linkage criterion that iniizes the eviances of (x i -c), where x i is the i-th object an c is the centroi (average of eleents) of the given cluster []. The result of cluster analysis is a enrogra that shows the steps of clustering as it provies the hierarchy of clusters an the connections between the at ifferent istances. We use cluster analysis as a preliinary ata processing step before inverse oeling (Figure ). The input of clustering is the coplete ata set originate fro the logge interval. By fining the siilarities between raw ata the objects are groupe into clusters. The log of clusters correlates well to the lithology variation along a borehole. The change in the group nuber of clusters appearing on the log gives the positions of layer bounaries, which can be rea autoatically by coputer processing. The estiate layer-bounary coorinates as iportant a priori inforation for constructing the starting oel serve as input for the interval inversion proceure. 3. Hyrocarbon Fiel Application We teste the inversion etho on a borehole geophysical ata set easure fro a Hungarian hyrocarbon well (Well No. ). We use GR, K, U, TH, DEN, NPHI, AT, RT logs as input for the interpretation proceure. In the processe interval a seientary coplex ae up of four unconsoliate shaly-san bes is foun. The separation between DEN an NPHI logs confirs the presence of gas in the porous an pereable forations. Accoring to the workflow in Figure, at first a siultaneous cluster analysis of the 8 logs was perfore. We specifie three lithological categories: shale, shaly san an san. At this stage of interpretation, this resolution is enough for fining the layer-bounaries, because the relative volues of rock-foring san an shale can be estiate later in the inversion processing phase. For easuring the istance between the observe objects we use a stanarize Eucliean istance, where each atu in the su of squares is inversely weighte by the saple variance of that ata type. We assigne leaf noe nubers for each object in the original ata set, where soe leaf noes correspon to ultiple objects (Figure 3b). We create the hierarchical cluster tree of 30 objects by using the War's linkage algorith (Figure 3a). Figure 3. Result of cluster analysis in Well No..

6 6 N. P. SZABÓ ET AL. On the enrogra 3 clusters can be iscriinate at istance 5. The 3D crossplots of clustere well-logging ata can be seen in Figures 4-6. The plots give useful inforation about soe site-specific constants for calculating wellbore ata in the straightforwar oeling phase (well-logging ata preiction in Figure ) as values of physical properties of rock constituents (san, shale) ust be specifie in probe response equations. For instance, the neutron porosity of sans (5 %) an shale (6 %) or the natural gaa-ray intensity of san (40 API) an shale (0 API) can be estiate for the given hyrocarbon zone (Figure 5). Figure 6. Clustere bulk ensity, resistivity an natural gaa-ray intensity log ata in Well No.. Figure 4. Clustere spectral gaa-ray intensity log ata in Well No.. Figure 5. Clustere neutron, acoustic an natural gaa-ray intensity log ata in Well No.. The log of the group nubers of clusters is useful to separate three types of forations (san reservoir, shale, shaly-san reservoir). The layer-bounaries are norally picke out at the places of inflection points on the GR curve. These epths are well inicate by abrupt changes in cluster log. In the given exaple, the layer-bounary coorinates were foun at 4, 8.4, 9.6, 6. We set these coorinates fixe, then petrophysical paraeters were eterine by interval inversion. In the interval inversion phase, we applie a series expansion of oel paraeters (POR, SX0, SW, VSH, VSD) for a cobination of hoogeneous an inhoogeneous layers. Until the epth of 9.6 the 3 layers can be treate as hoogeneous ones. Below 0 the statistically locate layers can be couple as they belong to the sae hyrocarbon reservoir. For the first 3 layers unit step functions, for the rest ones fourth-egree power functions were use as basis functions in Equation (4). The stanalone GA proceure is very tie-consuing. For ecreasing the CPU tie of the inversion proceure we ipleente a hybri optiization technique that is base on the successive cobination of global an linearize optiization ethos [8]. We start the proceure with global optiization (GA) that perfors a rano search in the paraeter space avoiing the local axia of F in Equation (6). Then, after soe 00 generations, in the near vicinity of the absolute axiu, we change for a faster linear optiization etho (Figure 7). We use the Dape Least Squares (DLSQ) etho for the iniization of objective function E * = F in the secon phase with applying proper aping factor to avoi big skips fro the optiu [9]. This etho both accele-

7 N. P. SZABÓ ET AL. 7 rates the rate of convergence of the inversion proceure an gives the estiation errors of oel paraeters as the Jacobi's atrix is calculate by Equation (3) at the en of the proceure. The petrophysical paraeters an their estiation errors are liste in Table. In the 4 th layer, the epth coorinates were properly transfore into the range of 0 an for the polynoial approxiation, where C 0 enotes the coefficient of the 0 th power exponent of epth coorinate as inepenent variable. The inversion results are highly accurate accoring to the values of estiation errors. This is because of the high overeterination of the inverse proble. For further increasing the overeterination, paraeter VSD in the 4 th layer was calculate by the physical constraint VSD=-POR-VSH. The easure logs an inversion results are in Figure 8. The log of the group nubers of clusters is represente in track 6, while the inversion estiates are in tracks 7-8, where the relative volue of water, ovable an irreucible hyrocarbon, pore space, shale, san can be analyze visually. In the 4 th layer, gas is accuulate at the top of the reservoir, while the enser water is situate unerneath. The aount of shale is increasing with epth in the lowest foration. 4. Conclusions The ever-increasing clai of oil inustry lain to highly reliable petrophysical inforation requires avance ata processing techniques. In the paper, a quick autoate inversion etho was shown for the interpretation of borehole geophysical ata. The cluster analysis assiste joint inversion etho is highly accurate (assure by great aount of overeterination), autoatic (autoate layer bounary an petrophysical oel estiation), robust an aaptive (evolutionary algorith phase) an fast (hybri optiization technique; siultaneous inversion processing of ata fro the logge interval instea of one point). Figure 7. Convergence of GA+DLSQ interval inversion proceure in Well No.. Layer 3 4 Table. GA interval inversion results in Well No.. Petrophysical paraeters as inversion unknowns POR SX0 SW VSH VSD ( 0.00) * ( 0.0) ( 0.00) ( 0.004) 0.7 ( 0.003) ( 0.0) C 0=0.77 C 0=0.679 ( 0.00) ( 0.0) C =0.06 C =0.59 ( 0.006) ( 8) C =-0.00 C = ( 0.000) ( 0.406) C 3= C 3= ( 0.000) ( 0.633) C 4= C 4=.350 ( 0.007) ( 0.346) ( 0.0) 0.3 ( 0.00) ( 0.0) C 0=0.53 ( 0.00) C =-4 ( 0.05) C =4.07 ( 0.) C 3=-6.75 ( 0.0) C 4=3.696 ( 0.3) 0.58 ( 0.003) 0.07 ( 0.00) 0.34 ( 0.003) C 0=0.069 ( 0.00) C =-0.54 ( ) C =.860 ( 0.49) C 3=-.85 ( 0.37) C 4=0.7 ( 0.3) * Estiation error of the oel paraeter in fractional unit. 0.3 ( 0.003) 0.66 ( 0.00) 0.49 ( 0.003) Deterinistic The inversion etho is not fully autoatic, where the supervison of the log analyst cannot be iscare. To achive a goo an unique solution the prior geological an geophysical inforation ust be built-in by the user properly (chose of site specific constants an response equations in ifferent hyrocarbon zones). Moreover, in case of the global optiization (GA) phase soe experience is neee to set the cobination an control paraeters of genetic operators an to ecie when it is possible to switch over to linear optiization. As a speciality of the interval inversion etho, the basis functions of series expansion can be chosen arbitrarily. The optial set of basis functions to be in use epens on the variation of lithology an pore fluis along a borehole. A trae-off ust be taken between the vertical resolution (nuber of unknowns) an stability (uniqueness) of the inversion proceure as they are inversely proportional. Usually, this is the ost iportant question fro the point of view of constructing an inversion etho. To increase the overeterination of the inverse proble, it is iportant to search for such paraeters that can be fixe uring the inversion proceure. The cluster analysis base interval inversion etho helps to fin lithological siilarities in the ata set, which leas to constrain the inversion process with reliable site constants an layer-thicknesses. This can reuce the uncertainty an abiguity of inversion estiates. In the future, we are planning to fin technical solutions to reach even better spatial resolution of petrophysical properties with preserving stability by using orthogonal polynoial series expansion along the entire logging interval an further evelop the ultiwell applications of the presente inversion etho.

8 8 N. P. SZABÓ ET AL. Figure 8. Processe well logs an result of cluster analysis assiste interval inversion proceure 5. Acknowlegeents The escribe work was carrie out as part of the TÁMOP-4../B-0/ project in the fraework of the New Hungary Developent Plan. The realization of this project is supporte by the European Union, co-finance by the European Social Fun. N. P. Sz. as the leaing researcher of project no. PD thanks to the support of the Hungarian Scientific Research Fun. M. D. as the leaing researcher of project no. K 0944 thanks to the support of the Hungarian Scientific Research Fun. N. P. Sz. thanks to the support of the János Bolyai Research Fellowship of the Hungarian Acaey of Sciences. As a eber of the MTA-ME Applie Geoscience Research Group, M. D. is grateful for the support of the Hungarian Acaey of Sciences. We thank the long-ter co-operation to the Hungarian Oil an Gas Copany. REFERENCES [] D. V. Ellis an J. M. Singer, Well Logging for Earth Scientists, n Eition, Springer, Dorrecht, 007. [] M. Alberty an K. Hashy, Application of ULTRA to log analysis, SPWLA Syposiu Transactions, Paper Z, 984, pp. -7. [3] S. M. Ball, D. M. Chace an W. H. Fertl, The Well Data Syste (WDS): An Avance Foration Evaluation Concept in a Microcoputer Environent, Proc. SPE Eastern Regional Meeting, Pittsburgh, Paper 7034, 987, pp oi:0.8/7034-ms. [4] C. Mayer, GLOBAL, a New Approach to Coputer-Processe Log Interpretation, Proc. 55th SPE Annual Technical Conference an Exhibition, Paper 934, 980, pp. -4. oi:0.8/934-ms. [5] M. Dobróka an N. P. Szabó, Interval Inversion of Well-Logging Data for Autoatic Deterination of Foration Bounaries by Using a Float-Encoe Genetic Algorith, Journal of Petroleu Science an Engineering, Vol , 0, pp oi: 6/j.petrol [6] M. Dobróka, N. P. Szabó an E. Turai, Interval Inversion of Borehole Data for Petrophysical Characterization of Coplex Reservoirs, Acta Geoaetica et Geophysica Hungarica, Vol. 47, No., 0, pp oi: 0.556/AGeo [7] M. Dobróka, P. N. Szabó, E. Cararelli an P. Vass, D Inversion of Borehole Logging Data for Siultaneous Deterination of Rock Interfaces an Petrophysical Paraeters, Acta Geoaetica et Geophysica Hungarica, Vol. 44, No. 4, 009, pp oi:0.556/ageo [8] M. Dobróka an P. N. Szabó, Cobine Global/Linear Inversion of Well-Logging Data in Layer-Wise Hoo-

9 N. P. SZABÓ ET AL. 9 geneous an Inhoogeneous Meia, Acta Geoaetica et Geophysica Hungarica, Vol. 40, No., 005, pp oi:0.556/ageo [9] N. P. Szabó an M. Dobróka, Float-Encoe Genetic Algorith Use for the Inversion Processing of Well-Logging Data, In: A. Michalski, E., Global Optiization: Theory, Developents an Applications, Matheatics Research Developents, Coputational Matheatics an Analysis Series, Nova Science Publishers Inc., Hauppauge NY, 03, pp [0] W. B. Heins, Multivariate Statistical Analysis in Foration Evaluation, SPE California Regional Meeting, Doc. ID 744-MS, 978, pp. -0, oi:0.8/744-ms [] G. N. Lance an W. T. Willias, A general theory of classificatory sorting strategies. Hierarchical systes, The Coputer Journal, Vol. 9, No. 4, 967, pp oi:93/cojnl/ [] J. Kovács, P. Tanos, J. Korponai, I. K. Székely, K. Gonár, K. Gonár-Sőregi an I. G. Hatvani, Analysis of Water Quality Data for Scientists, In: K. Vououris an D. Voutsa, Es., Water Quality Monitoring an Assessent, InTech Open Access Publisher, Rijeka, 0, pp [3] M. Kazierczuk an J. Jarzyna, Iproveent of Lithology an Saturation Deterine fro Well Logging Using Statistical Methos, Acta Geophysica, Vol. 54, No. 4, 006, pp oi:0.478/s y [4] V. Tavakoli an A. Aini, Application of Multivariate Cluster Analysis in Logfacies Deterination an Reservoir Zonation, Case Stuy of Marun Fiel, South of Iran, Journal of Science University of Teheran, Vol. 3, No., 006, pp [5] H. H. Perez, A. Datta-Gupta an S. Mishra, The Role of Electrofacies, Lithofacies, an Hyraulic Flow Units in Pereability Preictions fro Well Logs: A Coparative Analysis Using Classification Trees, SPE Annual Technical Conference an Exhibition, Denver, Paper 8430-MS, 003, pp. -. oi: 0.8/8430-MS [6] G. Asquith an D. Krygowski, Basic Well Log Analysis, n Eition, AAPG, Tulsa, 004. [7] M. H. Rier, The Geological Interpretation of Well Logs, n Eition, Rier-French Consulting Lt., Sutherlan, 00. [8] Á. Gyulai, M. K. Baracza an É. E. Tolnai, The Application of Joint Inversion in Geophysical Exploration, International Journal of Geosciences, 03, Vol. 4, No., pp oi:0.436/ijg [9] W. Menke, Geophysical Data Analysis: Discrete Inverse Theory, Acaeic Press Inc., New York, 984. [0] J. H. Hollan, Aaptation in Natural an Artificial Systes, University of Michigan Press, Ann Arbor, 975. [] Z. Michalewicz, Genetic Algoriths Plus Data Structures Equals Evolution Progras, Springer-Verlag Inc., New York, 99. [] J. H. War, Hierarchical Grouping to Optiize an Objective Function, Journal of the Aerican Statistical Association, Vol. 58, No. 30, 963, pp oi: 80/