Dario Grana

Probabilistic petrophysical-properties estimation integrating statistical rock physics with seismic inversion

A joint estimation of petrophysical properties is proposed that combines statistical rock physics and Bayesian seismic inversion. Because elastic attributes are correlated with petrophysical variables (effective porosity, clay content, and water saturation) and this physical link is associated with uncertainties, the petrophysical-properties estimation from seismic data can be seen as a Bayesian inversion problem. The purpose of this work was to develop a strategy for estimating the probability distributions of petrophysical parameters and litho-fluid classes from seismics. Estimation of reservoir properties and the associated uncertainty was performed in three steps: linearized seismic inversion to estimate the probabilities of elastic parameters, probabilistic upscaling to include the scale-changes effect, and petrophysical inversion to estimate the probabilities of petrophysical variables andlitho-fluid classes. Rock-physics equations provide the linkbetween reservoir properties and velocities, and lin...

Bayesian Gaussian Mixture Linear Inversion for Geophysical Inverse Problems

A Bayesian linear inversion methodology based on Gaussian mixture models and its application to geophysical inverse problems are presented in this paper. The proposed inverse method is based on a Bayesian approach under the assumptions of a Gaussian mixture random field for the prior model and a Gaussian linear likelihood function. The model for the latent discrete variable is defined to be a stationary first-order Markov chain. In this approach, a recursive exact solution to an approximation of the posterior distribution of the inverse problem is proposed. A Markov chain Monte Carlo algorithm can be used to efficiently simulate realizations from the correct posterior model. Two inversion studies based on real well log data are presented, and the main results are the posterior distributions of the reservoir properties of interest, the corresponding predictions and prediction intervals, and a set of conditional realizations. The first application is a seismic inversion study for the prediction of lithological facies, P- and S-impedance, where an improvement of 30% in the root-mean-square error of the predictions compared to the traditional Gaussian inversion is obtained. The second application is a rock physics inversion study for the prediction of lithological facies, porosity, and clay volume, where predictions slightly improve compared to the Gaussian inversion approach.

Bayesian inversion of time‐lapse seismic data for the estimation of static reservoir properties and dynamic property changes

Seismic conditioning of static reservoir model properties such as porosity and lithology has traditionally been faced as a solution of an inverse problem. Dynamic reservoir model properties have been constrained by time-lapse seismic data. Here, we propose a methodology to jointly estimate rock properties (such as porosity) and dynamic property changes (such as pressure and saturation changes) from time-lapse seismic data. The methodology is based on a full Bayesian approach to seismic inversion and can be divided into two steps. First we estimate the conditional probability of elastic properties and their relative changes; then we estimate the posterior probability of rock properties and dynamic property changes. We apply the proposed methodology to a synthetic reservoir study where we have created a synthetic seismic survey for a real dynamic reservoir model including pre-production and production scenarios. The final result is a set of point-wise probability distributions that allow us to predict the most probable reservoir models at each time step and to evaluate the associated uncertainty. Finally we also show an application to real field data from the Norwegian Sea, where we estimate changes in gas saturation and pressure from time-lapse seismic amplitude differences. The inverted results show the hydrocarbon displacement at the times of two repeated seismic surveys.

Petroelastic and geomechanical classification of lithologic facies in the Marcellus Shale

Log-facies classification at the well location allows determination of the number of facies, the facies definition, and the correlation between facies and rock properties along the well profile. In unconventional reservoirs, because of the necessity for hydraulic fracturing in shale gas and shale oil reservoirs, facies classification should account for petroelastic and geomechanical properties. We developed a facies classification methodology based on the expectation-maximization algorithm, a statistical method that allows finding the most likely facies classification and the associated probability distribution, given the set of geophysical measurements in the borehole. We applied the proposed workflow to a complete set of well logs from the Marcellus shale and developed the corresponding facies classification from log properties measured and computed in three different domains: petrophysics, rock physics, and geomechanics. In thne preliminary well-log and rock-physics analysis, we identify three main lithofacies: limestone, shale, and sandstone. The application of the classification method provided the vertical sequence of the three lithofacies and their pointwise probability of occurrence. A sensitivity analysis was finally evaluated to investigate the impact of the number of input variables on the classification and the effects of cementation and kerogen.

The link between seismic inversion, rock physics, and geostatistical simulations in seismic reservoir characterization studies

Reservoir models are typically based on geostatistical simulations of facies and their physical properties, such as porosity, permeability, and fluid. Seismic data are often used to guide these simulations. However, seismic response directly depends only on the elastic properties of the subsurface, which may be or may not be linked to the physical parameters used in reservoir simulation.

Bayesian seismic inversion based on rock-physics prior modeling for the joint estimation of acoustic impedance, porosity and lithofacies

We propose a Bayesian approach for seismic inversion to estimate acoustic impedance, porosity and lithofacies within the reservoir conditioned to post-stack seismic and well data. The link between elastic and petrophysical properties is given by a joint prior distribution for the logarithm of impedance and porosity, based on a rock-physics model. The well conditioning is performed through a background model obtained by well log interpolation. Two different approaches are presented: in the first approach, the prior is defined by a single Gaussian distribution, whereas in the second approach it is defined by a Gaussian mixture to represent the well data multimodal distribution and link the Gaussian components to different geological lithofacies. The forward model is based on a linearized convolutional model. For the single Gaussian case, we obtain an analytical expression for the posterior distribution, resulting in a fast algorithm to compute the solution of the inverse problem, i.e. the posterior distribution of acoustic impedance and porosity as well as the facies probability given the observed data. For the Gaussian mixture prior, it is not possible to obtain the distributions analytically, hence we propose a Gibbs algorithm to perform the posterior sampling and obtain several reservoir model realizations, allowing an uncertainty analysis of the estimated properties and lithofacies. Both methodologies are applied to a real seismic dataset with three wells to obtain 3D models of acoustic impedance, porosity and lithofacies. The methodologies are validated through a blind well test and compared to a standard Bayesian inversion approach. Using the probability of the reservoir lithofacies, we also compute a 3D isosurface probability model of the main oil reservoir in the studied field.

Petro-Elastic Log-Facies Classification Using the Expectation–Maximization Algorithm and Hidden Markov Models

Log-facies classification methods aim to estimate a profile of facies at the well location based on the values of rock properties measured or computed in well-log analysis. Statistical methods generally provide the most likely classification of lithological facies along the borehole by maximizing a function that describes the likelihood of a set of rock samples belonging to a certain facies. However, most of the available methods classify each sample in the well log independently and do not account for the vertical distribution of the facies profile. In this work, a classification method based on hidden Markov models is proposed, a stochastic method that accounts for the probability of transitions from one facies to another one. Differently from other available methods where the model parameters are assessed using nearby fields or analogs, the unknown parameters are estimated using a statistical algorithm called the Expectation–Maximization algorithm. The method is applied to two different datasets: a clastic reservoir in the North Sea where four litho-fluid facies are identified and an unconventional reservoir in North America where four lithological facies are defined. The results of the applications show the added value of the introduction of a vertical continuity model in the facies classification and the ability of the proposed method of inferring model parameters such as facies transition probabilities and facies posterior distributions. The application also includes a sensitivity analysis and a comparison to other statistical methods.

Sequential Simulations of Mixed Discrete-Continuous Properties: Sequential Gaussian Mixture Simulation

We present here a method for generating realizations of the posterior probability density function of a Gaussian Mixture linear inverse problem in the combined discrete-continuous case. This task is achieved by extending the sequential simulations method to the mixed discrete-continuous problem. The sequential approach allows us to generate a Gaussian Mixture random field that honors the covariance functions of the continuous property and the available observed data. The traditional inverse theory results, well known for the Gaussian case, are first summarized for Gaussian Mixture models: in particular the analytical expression for means, covariance matrices, and weights of the conditional probability density function are derived. However, the computation of the weights of the conditional distribution requires the evaluation of the probability density function values of a multivariate Gaussian distribution, at each conditioning point. As an alternative solution of the Bayesian inverse Gaussian Mixture problem, we then introduce the sequential approach to inverse problems and extend it to the Gaussian Mixture case. The Sequential Gaussian Mixture Simulation (SGMixSim) approach is presented as a particular case of the linear inverse Gaussian Mixture problem, where the linear operator is the identity. Similar to the Gaussian case, in Sequential Gaussian Mixture Simulation the means and the covariance matrices of the conditional distribution at a given point correspond to the kriging estimate, component by component, of the mixture. Furthermore, Sequential Gaussian Mixture Simulation can be conditioned by secondary information to account for non-stationarity. Examples of applications with synthetic and real data, are presented in the reservoir modeling domain where realizations of facies distribution and reservoir properties, such as porosity or net-to-gross, are obtained using Sequential Gaussian Mixture Simulation approach. In these examples, reservoir properties are assumed to be distributed as a Gaussian Mixture model. In particular, reservoir properties are Gaussian within each facies, and the weights of the mixture are identified with the point-wise probability of the facies.

Factor analysis prediction of effective stress from measurable rock attributes and calibration data

We show how to apply the statistical factor analysis method to effective stress prediction from seismic attributes. Two examples are discussed, one based on a laboratory dataset and relevant to the unloading mechanisms of abnormal pore pressure generation, and the other based on well data and a seismic section, relevant to the compaction disequilibrium mechanism. Arguably, this method of seismic interpretation is superior to the existing deterministic and statistical approaches as it reduces the multitude of available seismic attributes to a small number of factors most relevant to the effective stress and,in addition, provides a mathematically systematic way of arriving at most probable effective stress values as well as the confidence range.

Changing Scale And Domain of a Petrophysical And Elastic Properties Based Log-facies Classification

Summary The aim of this work is to present a methodology for logfacies classification integrating petrophysical properties derived from formation evaluation analysis and elastic properties computed through a rock physics model. Furthermore a change of scale and domain of a log-facies classification is here presented in order to solve the problem related to reconciling seismic and log scales. The methodology consists of 3 steps: 1) rock physics model calibration and scenarios simulation; 2) log-facies classification; 3) histogram upscaling of log facies. The methodology was applied to real log data from a well in the West Africa deep offshore. The results show the improvement of the classification obtained by integrating both elastic and petrophysical data and the coherency of the upscaling of the original log-facies from a fine scale in depth domain to a coarse seismic scale in time.