The physics that describes the seismic response of an interval of saturated porous rocks with known petrophysical properties is relatively well understood and includes rock physics, petrophysics, and wave propagation models. The main goal of seismic reservoir characterization is to predict the rock and fluid properties given a set of seismic measurements by combining geophysical models and mathematical methods. This modeling challenge is generally formulated as an inverse problem. The most common geophysical inverse problem is the seismic (or elastic) inversion, i.e., the estimation of elastic properties, such as seismic velocities or impedances, from seismic amplitudes and traveltimes. The estimation of petrophysical properties, such as porosity, lithology, and fluid saturations, also can be formulated as an inverse problem and is generally referred to as rock-physics (or petrophysical) inversion. Several deterministic and probabilistic methods can be applied to solve seismic inversion problems. Deterministic algorithms predict a single solution, which is a “best” estimate or the most likely value of the model variables of interest. In probabilistic algorithms, on the other hand, the solution is the probability distribution of the model variables of interest, which can be expressed as a conditional probability density function or a set of model realizations conditioned on the data. The probabilistic approach provides a quantification of the uncertainty of the solution in addition to the most likely model. Our goal is to define the terminology, present an overview of probabilistic seismic and rock-physics inversion methods for the estimation of petrophysical properties, demonstrate the fundamental concepts with illustrative examples, and discuss the recent research developments.

Among the large variety of mathematical and computational methods for estimating reservoir properties such as facies and petrophysical variables from geophysical data, deep machine-learning algorithms have gained significant popularity for their ability to obtain accurate solutions for geophysical inverse problems in which the physical models are partially unknown. Solutions of classification and inversion problems are generally not unique, and uncertainty quantification studies are required to quantify the uncertainty in the model predictions and determine the precision of the results. Probabilistic methods, such as Monte Carlo approaches, provide a reliable approach for capturing the variability of the set of possible models that match the measured data. Here, we focused on the classification of facies from seismic data and benchmarked the performance of three different algorithms: recurrent neural network, Monte Carlo acceptance/rejection sampling, and Markov chain Monte Carlo. We tested and validated these approaches at the well locations by comparing classification predictions to the reference facies profile. The accuracy of the classification results is defined as the mismatch between the predictions and the log facies profile. Our study found that when the training data set of the neural network is large enough and the prior information about the transition probabilities of the facies in the Monte Carlo approach is not informative, machine-learning methods lead to more accurate solutions; however, the uncertainty of the solution might be underestimated. When some prior knowledge of the facies model is available, for example, from nearby wells, Monte Carlo methods provide solutions with similar accuracy to the neural network and allow a more robust quantification of the uncertainty, of the solution.

We have developed a new iterative geostatistical seismic amplitude variation with angle (AVA) inversion algorithm that inverts prestack seismic data, sorted by angle gathers, directly for high-resolution density, P-wave velocity, S-wave velocity, and facies models. This novel iterative geostatistical inverse procedure is based on two key main principles: the use of stochastic sequential simulation and cosimulation as the perturbation technique of the model parameter space and a global optimizer based on a crossover genetic algorithm to converge the simulated earth models toward an objective function, in this case, the mismatch between the recorded and synthetic prestack seismic data. As a geostatistical approach, all the elastic models simulated during the iterative procedure honors the well-log data at its own locations, the marginal prior distributions of P-wave velocity and S-wave velocity, and density estimated from the available well-log data, and the corresponding joint distributions between density versus P-wave velocity and P-wave versus S-wave velocity. We successfully tested and implemented this new algorithm on a synthetic prestack data set that mimicked the main properties of a real reservoir, and on a real seismic data set acquired over a deepwater turbidite oil reservoir. In both cases, the results showed a good convergence between the recorded and synthetic seismic. The synthetic example showed high-resolution inverted petroelastic models that reproduced the true petroelastic models. The inverted petroelastic models retrieved from the real case study found high resolution and do agree with previous seismic reservoir characterization studies.