Ernesto Della Rossa

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...

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.

Ensemble-Based Multiobjective Optimization of On/Off Control Devices Under Geological Uncertainty

We consider robust ensemble-based (EnOpt) multiobjective production optimization of on/off inflow-control devices (ICDs) for a sector model inspired by a real-field case. The use of on/off valves as optimization variables leads to a discrete control problem. We propose a reparameterization of such discrete controls in terms of switching times (i.e., we optimize the time at which a particular valve is either open or closed). This transforms the discrete control problem into a continuous control problem that can be efficiently handled with the EnOpt method. In addition, this leads to a significant reduction in the number of controls that is expected to be beneficial for gradient quality when using approximate gradients. We consider an ensemble of sector models where the uncertainty is described by different permeability, porosity, net/gross ratios, and initial water-saturation fields. The controls are the ICD settings over time in the three horizontal injection wells, with approximately 15 ICDs per well. Different optimized strategies resulting from different initial strategies were compared. We achieved a mean 4.2% increase in expected net present value (NPV) at a 10% discount rate compared with a traditional pressure-maintenance strategy. Next, we performed a sequential biobjective optimization and achieved an increase of 9.2% in the secondary objective (25% discounted NPV to emphasize shortterm production gains) for a minimal decrease of 1% in the primary objective (0% discounted NPV to emphasize long-term recovery gains), as averaged over the 100 geological realizations. The work flow was repeated for alternative numbers of ICDs, showing that having fewer control options lowers the expected value for this particular case. The results demonstrate that ensemble-based optimization work flows are able to produce improved robust recovery strategies for realistic field-sector models against acceptable computational cost.

Improved initial ensemble generation coupled with ensemble square root filters and inflation to estimate uncertainty

The performance of the Ensemble Kalman Filter method (EnKF) depends on the sample size compared to the dimension of the parameters space. In real applications insufficient sampling may result in spurious correlations which reduce the accuracy of the filter with a strong underestimation of the uncertainty. Covariance localization and inflation are common solutions to these problems. The Ensemble Square Root Filters (ESRF) is also better to estimate uncertainty with respect to the EnKF. In this work we propose a method that limits the consequences of sampling errors by means of a convenient generation of the initial ensemble. This regeneration is based on a Stationary Orthogonal-Base Representation (SOBR) obtained via a singular value decomposition of a stationary covariance matrix estimated from the ensemble. The technique is tested on a 2D single phase reservoir and compared with the other common techniques. The evaluation is based on a reference solution obtained with a very large ensemble (one million members) which remove the spurious correlations. The example gives evidence that the SOBR technique is a valid alternative to reduce the effect of sampling error. In addition, when the SOBR method is applied in combination with the ESRF and inflation, it gives the best performance in terms of uncertainty estimation and oil production forecast.

Petrophysical properties estimation in a crosswell study integrated with statistical rock physics

SUMMARY In this paper we propose a methodology for petrophysical properties estimation from crosswell seismic data, based on a probabilistic approach. The methodology is divided into two steps: crosswell seismic inversion and probabilistic petrophysical properties estimation by means of statistical rock physics model integration. We apply the proposed workflow on a crosswell seismic section in an oil clastic reservoir in North Africa, and the results show the possibility of identifying porosity and volcanic rock fraction with the associated uncertainty.

A Functional Data Analysis Approach to Surrogate Modeling in Reservoir and Geomechanics Uncertainty Quantification

Uncertainty quantification for geomechanical and reservoir predictions is in general a computationally intensive problem, especially if a direct Monte Carlo approach with large numbers of full-physics simulations is used. A common solution to this problem, well-known for the fluid flow simulations, is the adoption of surrogate modeling approximating the physical behavior with respect to variations in uncertain parameters. The objective of this work is the quantification of such uncertainty both within geomechanical predictions and fluid-flow predictions using a specific surrogate modeling technique, which is based on a functional approach. The methodology realizes an approximation of full-physics simulated outputs that are varying in time and space when uncertainty parameters are changed, particularly important for the prediction of uncertainty in vertical displacement resulting from geomechanical modeling. The developed methodology has been applied both to a subsidence uncertainty quantification example and to a real reservoir forecast risk assessment. The surrogate quality obtained with these applications confirms that the proposed method makes it possible to perform reliable time–space varying dependent risk assessment with a low computational cost, provided the uncertainty space is low-dimensional.

Integrated Probabilistic Seismic Reservoir Characterization Workflow

The application of a multi-step process of reservoir characterization conditioned with seismic-derived attributes and associated uncertainty is presented. The workflow consists of: a log-facies classification integrating petrophysical properties derived from formation evaluation analysis and elastic properties computed through a rock physics model; a bayesian linearized seismic inversion; a probabilistic estimation of petrophysical properties; and a seismic facies classification. This methodology introduces some improvements with respect to traditional workflows: log-facies are discriminated and classified in a petro-elastic space, both in depth and time domain, handling scale changes for the elastic logs and the discrete log-facies; the rock properties distribution is described by a Gaussian Mixture Model, rather than a Gaussian Model; the conditional probabilities of elastic properties are estimated at coarse scale taking into account the uncertainty associated to the scale change. This conditional probability is combined with the probability of elastic properties from the Bayesian inversion to obtain the posterior probability of petrophysical properties. Then, litho-fluid classes are identified based on petrophysical properties probabilities and on log-facies classification. Since log-facies are coherent both in the geological and geophysical domain, probability volumes of petrophysical properties and reservoir log-facies are easily integrated in the hierarchical reservoir modelling workflow.