Development and application of Bayesian algorithms for reservoir characterization

The physical properties of the subsurface reservoir, which constitute essential information for optimizing either hydrocarbon exploration and production or CO2 sequestration. Frequently, reservoir characterization uses the changes of the seismic amplitudes with respect to the source-receiver offset (Amplitude versus Offset, AVO) to infer the elastic contrasts at the reflecting interfaces and to detect changes in the lithology and in the rock-saturating fluids. In this thesis, we invert the AVO seismic responses to infer the elastic parameters and, when a calibrated rock-physics model is available, the associated petrophysical parameters. We focus on both target-oriented and interval-oriented AVO inversions of pre-stack seismic data. The former approach only inverts the AVO responses of the target reflection, whereas the interval-oriented method inverts the seismic amplitudes within a time interval including both the target layer and the surrounding geological formations. In all cases, we cast the inversion into a Bayesian framework that allows including prior information (e.g. derived from well logs) inside the inversion framework to mitigate the ill-conditioning of the inverse problem and to stabilize the final solution. We developed four different inversion algorithms all based on numerical Markov chain Monte Carlo (McMC) approaches, that are used to numerically derive accurate uncertainty assessments in case of non-linear forward operators and/or non-Gaussian prior assumptions. The numerical solution of Bayesian inverse problems is computationally expensive and time-consuming, thus we present novel McMC strategies that not only use different strategies to mitigate the ill-conditioning of the AVO inversion but are also devoted to reduce the computational cost of the probabilistic sampling. First, we present target- and interval-oriented McMC elastic inversion suited for non-parametrical and multimodal distributions for the joint estimation of elastic properties (P- and S-wave velocities and density) and litho-fluid facies from pre-stack data. It exploits geostatistical constraints to reduce the ill-conditioning of the elastic AVO inversion, whereas advanced meta-algorithms, as the Parallel Tempering and the Delayed Rejection Scheme, are used to speed up the convergence toward a stable posterior model. Then, differently from the first approach, the second McMC method we discuss assumes an uninformative prior model (i.e. a uniform prior distribution) and uses a transdimensional inversion framework to reduce the ill-conditioning of the inversion procedure. More in detail, the number of model parameters (i.e. the number of layers) is treated as an unknown, and a reversible jump Markov Chain Monte Carlo (rjMcMC) algorithm is used to sample the variable-dimension model space. This inversion scheme provides a parsimonious solution and [Type text] 2 reliably quantifies the uncertainties affecting the estimated parameters. This approach is developed for the estimation of the elastic properties and also petrophysical parameters (i.e. porosity, clay content and water saturation) from the AVO responses, but this transdimensional framework is also employed to implement a data-driven inversion that automatically includes lateral constraints into the target-oriented elastic AVO inversion. It is known that the main drawback of McMC inversion is the considerable number of forward model evaluations needed to attain stable uncertainty estimations, although some strategies can be used to partially reduce this computational burden. To this end, we introduce a numerical Hamiltonian Monte Carlo method for both interval- and target-oriented AVO inversion that exploits the derivative information of the objective function to dramatically speed up the convergence of the sampling toward the stationary regime. Again, geostatistical constraints are used to mitigate the ill-conditioning of the inverse problem. The curse-of-dimensionality issue usually hampers the application of the previously described inversion approaches to simultaneously invert a 2D seismic section or a 3D seismic volume. For example, the previously mentioned interval-oriented McMC inversions must be applied to each seismic gather separately to make the computational cost affordable. For this reason, the last method we present combines a discrete cosine transform (DCT) reparameterization of data and model spaces with a convolutional neural network (CNN) to simultaneously solve the elastic AVO inversion along a 2D section (but the method is also extendible to 3D volumes). On the one hand, the CNN is trained to predict the mapping between the DCT-transformed seismic data and the DCT-transformed 2-D elastic model. On the other hand, the DCT reduces the dimensionality of the input and output of the network, and also acts as a regularization operator in the model space that preserves the lateral and vertical continuity of the elastic properties in the recovered solution. Once trained, the network can estimate the elastic properties along a 2-D section from the observed seismic data in near real-time. In addition, a Monte Carlo simulation framework is used to propagate onto the estimated elastic model the uncertainties related to both noise contamination and network approximation. All the presented approaches have been tested on synthetic data and benchmarked against analytical inversion algorithms, checking also the influence of possible errors in the propagation wavelet and of the presence of noise in the data. In some cases, the implemented inversion strategies have also been applied to actual seismic and well log data pertaining to a 3D land dataset.