Norbert Péter Szabó

Extending the Application of a Shale Volume Estimation Formula Derived from Factor Analysis of Wireline Logging Data

A multivariate statistical procedure is developed for the estimation of shale volume in clastic sedimentary formations. The method offers an alternative for extracting the shale content from borehole geophysical measurements. Factor analysis of various well-logging data types generates a new well log that correlates with the shale content of shaly-sandy rocks. The mathematical relationship between shale volume and factor scores is represented by a nonlinear equation, which seems to be applicable for data sets originating from different sedimentary basins. A comparative study is made between three different data sets originating from Hungary and the United States of America in order to check the validity of the proposed empirical formula. Shale volumes predicted from factor analysis are confirmed by estimates from independent deterministic and inverse modeling. Petrophysical information derived by factor analysis of logs recorded in deep wells can be used for a more accurate and reliable estimation of effective porosity and absolute permeability of reservoir rocks, for decreasing the estimation error of inversion estimates and for reducing the ambiguity in the solution of the well-logging inverse problem.

Interval inversion of borehole data for petrophysical characterization of complex reservoirs

Wireline logging surveys are routinely used for the reconnaissance and quantitative characterization of multi-mineral hydrocarbon structures. The interpretation of well-logging data, however, is quite a challenging task, because the conventionally used local inversion procedure becomes either an underdetermined or a slightly overdetermined problem that may result in poor parameter estimation. In order to determine the petrophysical model composed of several parameters, such as specific volumes of matrix components, water saturation, primary and secondary porosity and numerous zone-parameters, in a more reliable way a new inversion methodology is required. We suggest a joint inversion technique for the estimation of model parameters of multi-mineral rocks that inverts data acquired from a larger depth interval (hydrocarbon zone). The inverse problem is formulated assuming homogeneous intervals within the zone to get a highly overdetermined inversion procedure. The interval inversion method has been applied to shaly sandy hydrocarbon reservoirs, in this study, that is used for the estimation of petrophysical parameters of complex reservoirs. Numerical results with synthetic and field data demonstrate the feasibility of the inversion method in investigating carbonate and metamorphic structures.

Factor analysis of borehole logs for evaluating formation shaliness: a hydrogeophysical application for groundwater studies

The calculation of groundwater reserves in shaly sand aquifers requires a reliable estimation of effective porosity and permeability; the amount of shaliness as a related quantity can be determined from well log analysis. The conventionally used linear model, connecting the natural gamma-ray index to shale content, often gives only a rough estimate of shale volume. A non-linear model is suggested, which is derived from the factor analysis of well-logging data. An earlier study of hydrocarbon wells revealed an empirical relationship between the factor scores and shale volume, independent of the well site. Borehole logs from three groundwater wells drilled in the northeastern Great Hungarian Plain are analyzed to derive depth logs of factor variables, which are then correlated with shale volumes given from the method of Larionov. Shale volume logs derived by the statistical procedure are in close agreement with those derived from Larionov’s formula, which confirms the validity of the non-linear approximation. The statistical results are in good accordance with laboratory measurements made on core samples. Whereas conventional methods normally use a single well log as input, factor analysis processes all available logs to provide groundwater exploration with reliable estimations of shale volume.RésuméLe calcul de réserves d’eau souterraine dans des aquifères sablo-argileux nécessite une estimation fiable de la porosité efficace et de la perméabilité; la quantité d’argilosité comme paramètre relatif peut être déterminée à partir de l’analyse des logs de forage. Le modèle linéaire communément utilisé, associant l’index d’émission gamma à la teneur en argiles donne souvent une estimation grossière du volume d’argiles. Un modèle non linéaire est suggéré, qui est dérivé de l’analyse factorielle des données de logs de forages. Une étude antérieure de forages pour hydrocarbures avait révélé une relation empirique entre les scores de l’analyse factorielle et le volume d’argiles, indépendamment de l’emplacement du forage. Les logs des trois forages d’eau exécutés au Nord Est de la Grande Plaine Hongroise sont analysés pour déduire les variables des logs pour l’analyse factorielle, qui sont ensuite corrélées avec le volume d’argiles donné par la méthode de Larionov. Les logs de volume d’argiles dérivés de la procédure statistique sont en accord étroit avec ceux dérivés de la formule de Larionov, ce qui confirme la validité de l’approximation non linéaire. Les résultats statistiques sont en bon accord avec les mesures de laboratoire faites sur des carottes. Alors que les méthodes conventionnelles utilisent normalement le log unique d’un forage comme entrée, l’analyse factorielle considère tous les logs disponibles pour fournir une exploration des aquifères avec une estimation fiable du volume des argiles.ResumenEl cálculo de las reservas de agua subterránea en acuíferos de arenas arcillosas requiere una estimación confiable de la porosidad eficaz y de la permeabilidad; la magnitud de la arcillosidad como una cantidad relacionada puede ser determinada a partir de análisis de registros de pozos. El modelo lineal convencionalmente usado, que conecta el índice de rayos gamma natural con el contenido de arcilla, a menudo da sólo una estimación grosera del volumen de arcilla. Se sugiere un modelo no lineal, desarrollado del análisis factorial a partir de datos de registros de pozos. Un estudio anterior de pozos hidrocarburíferos reveló una relación empírica entre los factores determinantes y el volumen de arcilla, independiente de la localización del pozo. Se analizaron los registros de de tres pozos de agua subterránea perforados en el noreste de la planicie Great Hungarian para deducir los registros en profundidad de los factores determinantes, los cuales fueron luego correlacionados con los volúmenes de arcilla dados a partir del método de Larionov. Los registros del volumen de arcilla deducidos por el procedimiento estadístico están en cercana coincidencia con aquellos derivados de la fórmula de Larionov’s, lo cual confirma la validez de la aproximación no lineal. Los resultados estadísticos están en un buen acuerdo con las mediciones de laboratorio hechas sobre muestras testigos. Mientras que los métodos convencionales normalmente usan un solo registro de pozo como entrada, el análisis factorial procesa todos los registros disponibles para proveer a la exploración de agua subterránea con estimaciones confiables del volumen de arcilla.KivonatAz agyagos homok tárolók édesvíz-készletének számítása az effektív porozitás és a permeábilitás megbízható becslését igényli; a karotázs szelvények kiértékeléséből meghatározható az agyagtartalom, mint ehhez kapcsolódó mennyiség. A hagyományosan alkalmazott lineáris modell, mely a természetes gamma indexhez köti az agyagtartalmat, gyakran csak durva becslés ad. A cikkben egy nemlineáris modellt javaslunk, amely mélyfúrási geofizikai szelvények faktoranalíziséből származik. Egy korábbi, szénhidrogén-kutató fúrásban történő alkalmazás terület-független tapasztalati összefüggést mutatott a faktor értékek és az agyagtartalom között. Jelen cikkben az Alföld észak-keleti részén fúrt három vízkutató fúrás szelvényeit dolgozzuk fel, abból származtatjuk a faktorok mélységszelvényeit, majd azokat a Larionov módszerrel kapott agyagtartalom szelvényekkel hasonlítjuk össze. A statisztikai módszerrel kapott agyagtartalmak jó egyezést mutatnak a Larionov egyenlettel számítottakkal, amely megerősíti a nemlineáris közelítés érvényességet. A statisztikai eredmények jó összhangban állnak magmintákon végzett laboratóriumi mérések eredményeivel. Míg a hagyományos módszerek általában egyetlen fúrólyuk szelvényt használnak fel bemeneti adatként, addig a faktoranalízis az összes rendelkezésre álló szelvényt feldolgozza, mellyel a vízkutatás számára megbízható agyagtartalom becslés valósítható meg.ResumoO cálculo das reservas de água subterrânea em aquíferos de areia argilosa requere uma estimação fiável da porosidade efetiva e da permeabilidade; e do teor de argilas, como uma quantidade relativa pode ser determinada a partir da análise de testemunhos de sondagem. O modelo linear convencional usado, correlacionando o índice de raio-gama natural com o conteúdo de argila, dá muitas vezes apenas uma estimativa grosseira do volume de argilas. É sugerido um modelo não linear, derivado do fator de análise dos dados dos testemunhos de sondagem. Um estudo anterior de furos de hidrocarbonetos revelou uma relação empírica entre os resultados fatoriais e o volume de argilas, independentemente dos locais dos furos. São analisados os testemunhos de sondagem de três furos de água subterrânea perfurados no nordeste da Grande Planície Húngara, a fim de obter perfis de profundidade das variáveis fatoriais, os quais são depois correlacionados com os volumes de argila dados pelo método de Larionov. O volume de argilas obtido por procedimento estatístico está de acordo com os obtidos a partir da fórmula de Larionov, a qual confirma a validade da aproximação não linear. Os resultados estatísticos encontram-se de acordo com as medições de laboratório feitas em amostras de testemunhos. Enquanto os métodos convencionais usam normalmente um furo simples como entrada, a análise fatorial processa todos os testemunhos disponíveis, a fim de providenciar a exploração de águas subterrâneas com estimações mais fiáveis sobre o volume de argilas.

Hydraulic conductivity explored by factor analysis of borehole geophysical data

A multivariate statistical method is presented for providing hydrogeological information on groundwater formations. Factor analysis is applied to borehole logs in Hungary and the USA to estimate the vertical distribution of hydraulic conductivity of rocks intersected by the borehole. Earlier studies showed a strong correlation between a statistical variable extracted by factor analysis and shale volume in primary porosity rocks. Hydraulic conductivity as a related quantity can be derived directly by factor analysis. In the first step, electric and nuclear logs are transformed into factor logs, which are then correlated to hydraulic properties of aquifers. It is shown that a factor explaining the major part of variance of the measured variables is inversely proportional to hydraulic conductivity. By revealing the regression relation between the above quantities, an estimate for hydraulic conductivity can be given along the entire length of the borehole. Synthetic modeling experiments and field cases demonstrate the feasibility of the method, which can be applied both in primary and secondary porosity aquifers. The results of factor analysis show consistence with those of the Kozeny-Carman method and hydraulic aquifer tests. The application of the statistical analysis of well logs together with independent ground geophysical and hydrogeological methods serves a more efficient exploration of groundwater resources.RésuméUne méthode statistique multivariée est présentée pour fournir de l’information hydrodynamique sur des formations hydrogéologiques. L’analyse factorielle est appliquée à des diagraphies de forage de Hongrie et des USA pour estimer la distribution verticale de la conductivité hydraulique des roches recoupées par le forage. Des études antérieures ont montré une forte corrélation entre la variable statistique extraite par analyse factorielle et le volume d’argile présent dans la porosité primaire des roches. La conductivité hydraulique en tant que quantité relié à ce volume d’argile peut être directement déduite par analyse factorielle. En première étape, les diagraphies électriques et nucléaires sont transformées en logs factoriels, qui sont ensuite corrélés aux propriétés hydrodynamiques des aquifères. Il est montré que le facteur expliquant la majeure partie de la variance des variables mesurées est inversement proportionnel à la conductivité hydraulique. En révélant la relation de régression entre les paramètres définis ci-dessus, une évaluation de la conductivité hydraulique peut être donnée sur toute la longueur du forage. Des expériences de modélisation synthétique et des cas de terrain démontrent la faisabilité de la méthode qui peut être appliquée aussi bien aux aquifères à porosité de type primaire que secondaire. Les résultats de l’analyse factorielle sont cohérents avec ceux de la méthode de Kozeny-Carman et des essais hydrauliques d’aquifère. La combinaison de l’analyse statistique de diagraphies de forages avec des méthodes géophysiques et hydrogéologiques de terrain permet une plus efficace exploration des ressources en eau souterraine.ResumenSe presenta un método estadístico multivariado para proporcionar información hidrogeológica en formaciones de agua subterráneas. El análisis factorial se aplica a registros de perforaciones en Hungría y los EEUU para estimar la distribución hidráulica vertical de la conductividad hidráulica de rocas intersectadas por la perforación. Estudios anteriores mostraron una fuerte correlación entre una variable estadística extraída por análisis factorial y el volumen de arcilla en rocas de porosidad primaria. La conductividad hidráulica como una cantidad relacionada se puede derivar directamente por el análisis factorial. En el primer paso, los registros eléctricos y nucleares se transforman en registros factoriales, que luego se correlacionan con las propiedades hidráulicas de los acuíferos. Se muestra que un factor que explica la mayor parte de la varianza de las variables medidas, es inversamente proporcional a la conductividad hidráulica. Al ponerse de manifiesto la relación de regresión entre las cantidades anteriores, una estimación para la conductividad hidráulica ser puede dar a lo largo de toda la longitud de la perforación. Experimentos de modelado sintético y casos de campo demuestran la factibilidad del método, que puede ser aplicada tanto en acuíferos de porosidad primaria como de secundaria. Los resultados del análisis factorial muestran la consistencia con aquellos del método Kozeny-Carman y con los ensayos hidráulicos de acuíferos. La aplicación de los análisis estadísticos de registros de pozos junto con métodos hidrogeológicos y geofísicos de superficie independientes sirve a una más eficiente exploración de los recursos de agua subterránea.摘要本文展示了提供地下水地层水文地质信息的多元统计方法。在匈牙利和美国对钻孔记录采用因子分析法估算了钻孔岩层的水力传导率的垂直分布。较早的研究结果显示因子分析法得出的统计变量和原生孔隙岩层中的页岩量强相关。水力传导率作为一个相关的量可直接从因子分析法得出。首先,电和核记录要转换成因子记录,然后与含水层的水力特性进行对比。结果显示,说明被测变量差异主要部分的因子与水力传导率成反比。通过揭示上述两个量之间的回归关系,就可得出整个钻孔的水力传导率估算值。综合模拟实验和现场实例展示了方法的可行性,该方法可以应用于原生和次生孔隙度含水层。因子分析法结果显示,该法与Kozeny-Carman法和水力含水层实验得出的结果一致。井记录统计分析法与单独的大地地球物理法和水文地质法一起应用可大大提高地下水资源勘查的效率。KivonatA tanulmányban egy többváltozós statisztikai módszert mutatunk be, mellyel hidrogeológiai információkat nyerhetünk a felszín alatti víztároló formációkról. Magyarországi és USA-beli karotázs szelvényeken faktor analízist alkalmazunk a fúrással harántolt kőzetek szivárgási tényezője vertikális eloszlásának becslése céljából. Korábbi tanulmányainkban erős korrelációt mutattunk ki egy faktor analízissel előállított statisztikai változó és az agyagtartalom között elsődleges porozitással rendelkező kőzetek esetén. A szivárgási tényező, mint agyagtartalomtól függő mennyiség a faktor analízissel közvetlenül is származtatható. Első lépésként elektromos és nukleáris szelvényeket transzformálunk faktor szelvényekké, melyeket a vízadók hidraulikai tulajdonságaival hozunk kapcsolatba. Bemutatjuk, hogy a mért változók varianciájának legnagyobb részét magyarázó faktor fordítottan arányos a szivárgási tényezővel. Feltárva a fenti mennyiségek közötti regressziós kapcsolatot, becslés adható a szivárgási tényezőre a fúrólyuk teljes hossza mentén. Szintetikus modellezési kísérletek és mezőbeli alkalmazások mutatják a módszer megvalósíthatóságát, mely mind elsődleges, mind másodlagos porozitású vízadó rétegekben alkalmazható. A faktor analízis eredményei egyezőséget mutatnak a Kozeny-Carman módszerrel meghatározott és a vízadó rétegek hidraulikai tesztjeiből származó eredményekkel. A fúrólyukszelvények statisztikai elemzésének a független felszíni geofizikai és hidrogeológiai módszerekkel együtt történő alkalmazása a felszín alatti vízkészletek még hatékonyabb kutatását biztosítja.ResumoÉ apresentado um método estatístico multivariado para fornecer informação hidrogeológica sobre formações aquíferas. É aplicada análise fatorial a perfis de sondagem na Hungria e nos EUA para estimar a distribuição vertical de condutividade hidráulica das rochas intersetadas pela sondagem. Os primeiros estudos mostram uma forte correlação entre uma variável estatística extraída por análise fatorial e o volume de xisto em rochas de porosidade primária. A condutividade hidráulica, como quantidade relativa, pode ser derivada diretamente por análise fatorial. Num primeiro passo, perfis elétricos e nucleares são transformados em perfis fatoriais, os quais são depois correlacionados com as propriedades hidráulicas dos aquíferos. Mostra-se que um fator que explica a maior parte da variância das varáveis medidas é inversamente proporcional à condutividade hidráulica. Ao revelar a relação de regressão entre as quantidades acima descritas, pode ser dada uma estimativa para a condutividade hidráulica ao longo de toda a extensão do poço. Experiências sintéticas de modelação e casos de campo demonstram a viabilidade do método, o qual pode ser aplicado tanto em aquíferos de porosidade primária como secundária. Os resultados da análise fatorial mostram consistência com os dos métodos de Kozeny-Carman e de ensaios de caudal. A aplicação de análise estatística de perfis de sondagem, em conjunto com métodos geofísicos e hidrogeológicos independentes, melhora e torna mais eficiente a exploração dos recursos hídricos subterrâneos.

Dry density derived by factor analysis of engineering geophysical sounding measurements

Statistical processing of engineering geophysical sounding data is applied to the determination of dry density as an important geotechnical parameter of shallow formations. Dry density has been measured in laboratory or in the field as point information. In order to get more extensive results, dry density is extracted from high-resolution penetration logs. Dry density is related to measured bulk density as well as porosity and shale volume computed directly from engineering geophysical sounding data. The most critical parameter for the calculation is water saturation, which is estimated by factor analysis of all penetration logs. This approach is based on a strong linear correlation between water saturation and one of the extracted variables (factors). The interpretation method is tested in twelve penetration holes drilled in a Hungarian location. A 2D multi-well application is presented, in which the spatial distribution of dry density between the drill-holes is given. A local relationship between bulk density and dry density is also estimated. The study demonstrates that dry density can be estimated as continuous and in-situ information to support geotechnical operations in soil environments.

On the application of combined geoelectric weighted inversion in environmental exploration

Geophysical surveying methods are of great importance in environmental exploration. Inversion-based data processing methods are applied for the determination of geometrical and physical parameters of the target model. The use of this geoelectric inversion method is advantageous in environmental research where highly reliable information with large spatial resolution is required. The 2D combined geoelectric inversion (CGI) method performs more accurate parameter estimation than conventional 1D single inversion methods by efficiently decreasing the number of unknowns of the inverse problem (single means that data sets of individual vertical electric sounding stations are inverted separately). The quality improvement in parameter space is demonstrated by comparing the traditional 1D inversion procedure with a 2D series expansion-based inversion technique. The CGI method was further developed by weighting individual direct current geoelectric data sets automatically in order to improve inversion results. The new algorithm was named combined geoelectric weighted inversion, which extracts the solution by a special weighted least squares technique. It is shown that the new inversion methodology is applicable to resolve near-surface structures such as rapidly varying layer boundaries, laterally inhomogeneous formations and pinch-outs.

Exploratory Factor Analysis of Wireline Logs Using a Float-Encoded Genetic Algorithm

In the paper, a novel inversion approach is used for the solution of the problem of factor analysis. The float-encoded genetic algorithm as a global optimization method is implemented to extract factor variables using open-hole logging data. The suggested statistical workflow is used to give a reliable estimate for not only the factors but also the related petrophysical properties in hydrocarbon formations. In the first step, the factor loadings and scores are estimated by Jöreskog’s fast approximate method, which are gradually improved by the genetic algorithm. The forward problem is solved to calculate wireline logs directly from the factor scores. In each generation, the observed and calculated well logs are compared to update the factor population. During the genetic algorithm run, the average fitness of factor populations is maximized to give the best fit between the observed and theoretical data. By using the empirical relation between the first factor and formation shaliness, the shale volume is estimated along the borehole. Permeability as a derived quantity also correlates with the first factor, which allows its determination from an independent source. The estimation results agree well with those of independent deterministic modeling and core measurements. Case studies from Hungary and the USA demonstrate the feasibility of the global optimization based factor analysis, which provides a useful tool for improved reservoir characterization.

Most frequent value based factor analysis of direct-push logging data

An improved iteratively re-weighted factor analysis procedure is presented to interpret engineering geophysical sounding logs in shallow unsaturated sediments. We simultaneously process cone resistance, electric resistivity and nuclear data acquired by direct-push tools to give robust estimates of factor variables and water content in unconsolidated heterogeneous formations. The statistical procedure is based on the iterative re-weighting of the deviations between the measured and calculated data using the Most Frequent Value method famous for its robustness and high statistical efficiency. The iterative approach improves the result of factor analysis for not normally distributed data and extremely noisy measurements. By detecting a strong regression relation between one of the extracted factors and the fractional volume of water, we establish an independent method for water content estimation along the penetration hole. We verify the estimated values of water volume by using a highly overdetermined quality checked interval inversion procedure. The multidimensional extension of the statistical method allows the estimation of water content distribution along both the vertical and horizontal coordinates. Numerical tests using engineering geophysical sounding data measured in a Hungarian loessy-sandy formation demonstrate the feasibility of the Most Frequent Value based factor analysis, which can be efficiently used for a more reliable hydrogeophysical characterization of the unsaturated zone. This article is protected by copyright. All rights reserved

Most Frequent Value Based Factor Analysis of Engineering Geophysical Sounding Logs

A multivariate statistical approach is presented to estimate water saturation in shallow heterogeneous formations. An improved factor analysis algorithm is developed to process engineering geophysical sounding data in a more reliable way. Resistivity and nuclear data acquired by cone penetration tools equipped with geophysical sensors are processed simultaneously to give an estimate to factor logs. The new factor analysis procedure is based on the iterative reweighting of data prediction errors using the highly robust most frequent value method, which improves the accuracy of factor scores in case of non-Gaussian data sets. A strong exponential relationship is detected between water saturation and the first factor log. Tests made on penetration logs measured from a Hungarian well demonstrate the feasibility of the most frequent value based factor analysis approach, which is verified by the results of local inverse modeling.

Evaluation of hydraulic conductivity in shallow groundwater formations: a comparative study of the Csókás’ and Kozeny–Carman model

The Kozeny–Carman equation has achieved widespread use as a standard model for estimating hydraulic conductivity of aquifers. An empirically modified form applicable in shallow formations called Csókás’ formula is discussed, which is based on the relation between the effective grain-size and formation factor of freshwater-bearing unconsolidated sediments. The method gives a continuous estimate of hydraulic conductivity along a borehole by using electric and nuclear logging measurements without the need of grain-size data. In the first step, synthetic well-logging data sets of different noise levels are generated from an exactly known petrophysical model to test the noise sensitivity of the Csókás’ method and to assess the degree of correlation between the results of Csókás’ and Kozeny–Carman model. In the next step, borehole logs acquired from Hungarian sites are processed to make a comparison between the Csókás’ formula and the Kozeny–Carman equation including grain-size data measured on rock samples. The hydraulic conductivity logs derived separately from the Csókás’ and Kozeny–Carman formulae show reliable interpretation results, which are also validated by the Hazen’s formula and statistical factor analysis. The fundamental goal of Professor Csókás’ research was to derive some useful hydraulic parameters solely from well-logging observations. This idea may be of importance today since the input parameters can be determined more accurately by advanced measurement techniques. Hence, the Csókás’ formula may inspire the hydrogeophysicists to make further developments for a more efficient exploration of groundwater resources.