Mihály Dobróka

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.

New petrophysical model describing the pressure dependence of seismic velocity

Seismic data are increasingly applied to predict the characteristics of reservoirs, as their quality improves. Since the change of pressure is a major component in exploitation of reservoirs, a thorough understanding of the influence of pressure on seismic velocity is also important. In this study we introduce the first results of the developed petrophysical model which describes the pressure dependence of seismic velocity. The model is based on the idea that microcracks in rocks open and close under the change of pressure. Laboratory measurements are presented on several sandstone samples. Longitudinal wave velocities were measured at various incremental pressures increased from 0 to 20 MPa. During the measurements, the pulse transmission technique was used and the parameters of the model were determined by using a linearized inversion method. The inversion results proved that the proposed petrophysical model well applies in practice.

On the use of Steiner’s weights in inversion-based Fourier transformation: robustification of a previously published algorithm

In our previous paper (Dobróka et al. Acta Geod Geophys Hung 47(2):185–196, 2012) we proposed a new robust algorithm for the inversion-based Fourier transformation. It was presented that the Fourier transform and its variants responds very sensitively to any little measurement noise affected an input data set. The continuous Fourier spectra are assumed as a series expansion with the scaled Hermite functions. The expansion coefficients are determined by solving an over-determined inverse problem. Here, we use the new Steiner’s weights (previously called the weights of most frequent values or abbreviated as MFV), where the scale parameter can be determined in an internal iteration process. This method results a very efficient robust inversion method in which we calculate the Steiner weights from iteration to iteration into an IRLS procedure. The new method using the Steiner’s weights is also numerically tested by using synthetic data.

Fourier transformation as inverse problem — An improved algorithm

This paper presents a new algorithm for the inversion-based 1D Fourier transformation. The continuous Fourier spectra are assumed as a series expansion with the scaled Hermite functions as square-integrable set of basis functions. The expansion coefficients are determined by solving an over-determined inverse problem. In order to define a quick and easy-to-use formula in calculating the Jacobi matrix of the problem a special feature of the Hermite functions are used. It is well-known, that the basic Hermite functions are eigenfunctions of the Fourier transformation. This feature is generalized by extending its validity for the scaled Hermite functions. Using the eigenvalues, given by this generalization, a very simple formula can be derived for the Jacobi matrix of the problem resulting in a quick and more accurate inversion-based Fourier transform algorithm. The new procedure is numerically tested by using synthetic data.

In-mine geoelectric investigations for detecting tectonic disturbances in coal seam structures

The methods of in-mine seam-sounding and transillumination (geoelectric tomography) for the detection of tectonic disturbances of coal seams were developed by the Department of Geophysics of the University of Miskolc in the 1970–80’s with the effective support of the former “Borsod” Coal Mines Ltd.The paper gives an overview about the theory of seam-sounding and a special geoelectric tomographic inversion, and introduces the in-mine geoelectric seam-sounding and transillumination measurement systems using vertical electrode dipoles. In the second part the paper, the results of an in-mine geoelectric measurement are presented, which was carried out in order to detect tectonic disturbances of the Miocene aged coal seams situated in Slovakia. As results of the geophysical investigation, the authors forecasted the tectonic features in the coal seam. The company confirmed the results by independent information about seam disturbances and tectonic features arising from the excavation of the investigated area.

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.

On the Reduced Noise Sensitivity of a New Fourier Transformation Algorithm

In this study, a new inversion method is presented for performing one-dimensional Fourier transform, which shows highly robust behavior against noises. As the Fourier transformation is linear, the data noise is also transformed to the frequency domain making the operation noise sensitive especially in case of non-Gaussian noise distribution. In the field of inverse problem theory it is well known that there are numerous procedures for noise rejection, so if the Fourier transformation is formulated as an inverse problem these tools can be used to reduce the noise sensitivity. It was demonstrated in many case studies that the method of most frequent value provides useful weights to increase the noise rejection capability of geophysical inversion methods. Following the basis of the latter method the Fourier transform is formulated as an iteratively reweighted least squares problem using Steiner’s weights. Series expansion was applied to the discretization of the continuous functions of the complex spectrum. It is shown that the Jacobian matrix of the inverse problem can be calculated as the inverse Fourier transform of the basis functions used in the series expansion. To avoid the calculation of the complex integral a set of basis functions being eigenfunctions of the inverse Fourier transform is produced. This procedure leads to the modified Hermite functions and results in quick and robust inversion-based Fourier transformation method. The numerical tests of the procedure show that the noise sensitivity can be reduced around an order of magnitude compared to the traditional discrete Fourier transform.

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.