In this paper, geological prior information is incorporated in the classification of reservoir lithologies after the adoption of Markov random fields (MRFs). The prediction of hidden lithologies is based on measured observations, such as seismic inversion results, which are associated with the latent categorical variables, based on the assumption of Gaussian distributions. Compared with other statistical methods, such as the Gaussian mixture model or k-Means, which do not take spatial relationships into account, the hidden MRFs approach can connect the same or similar lithologies horizontally while ensuring a geologically reasonable vertical ordering. It is, therefore, able to exclude randomly appearing lithologies caused by errors in the inversion. The prior information consists of a Gibbs distribution function and transition probability matrices. The Gibbs distribution connects the same or similar lithologies internally, which does not need a geological definition from the outside. The transition matrices provide preferential transitions between different lithologies, and an estimation of them implicitly depends on the depositional environments and juxtaposition rules between different lithologies. Analog cross sections from the subsurface or outcrop studies can contribute to the construction of these matrices by a simple counting procedure.

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Hidden Markov Models (HMMs) have been applied to predict reservoir lithologies using seismic inversion results as inputs. This approach takes into account the conditional probabilities between different lithologies, i.e. the vertical transitions in sedimentary sequences. These properties are used as prior geological information. In order to relate the seismic inversion results to the true well-log data, HMMs need to be trained based on the Expectation-Maximization theory. Application of the resulting model on a synthetic example from the Book Cliffs (Utah, USA) showed that most lithologies are classified correctly, even for some thin layers. A comparison with point-wise methods in which data samples are treated independently from each other, such as k-means and fuzzy logic classifiers, leads to the conclusion that the spatial correlation in HMMs allows better lithological predictions because the prior information accounts for the geological depositional processes. A real case study with data from the Vienna Basin (Austria) is performed, in which lithologies in a 3D cube are obtained based on properties from seismic inversions, via trained HMMs. While the vertical sequences are shown to be reasonably well predicted, the horizontal continuities are not. This indicates that the future research should focus on the lateral geological relationships.

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The coupled Markov chain model can be used to simulate reservoir lithologies between wells, by conditioning them on the observed data in the cored wells. However, with this method, only the state at the same depth as the current cell is going to be used for conditioning, which may be a problem if the geological layers are dipping. This will cause the simulated lithological layers to be broken or to become discontinuous across the reservoir. In order to address this problem, an actively conditioned process is proposed here, in which a tolerance angle is predefined. The states contained in the region constrained by the tolerance angle will be employed for conditioning in the horizontal chain first, after which a coupling concept with the vertical chain is implemented. In order to use the same horizontal transition matrix for different future states, the tolerance angle has to be small. This allows the method to work in reservoirs without complex structures caused by depositional processes or tectonic deformations. Directional artefacts in the modeling process are avoided through a careful choice of the simulation path. The tolerance angle and dipping direction of the strata can be obtained from a correlation between wells, or from seismic data, which are available in most hydrocarbon reservoirs, either by interpretation or by inversion that can also assist the construction of a horizontal probability matrix.

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In this study, geological prior information is incorporated in the classification of reservoir lithologies using the Markov Random Field (MRF) technique. The prediction of hidden lithologies in seismic data is based on measured observations such as seismic inversion results, which are associated with the latent categorical variables derived from the distribution of Gaussian assumptions. The Hidden Markov Random Field (HMRF) approach can connect similar lithologies laterally (horizontally) while ensure a geologically reasonable stratigraphic (vertical) ordering. It is, therefore, able to exclude randomly appearing lithologies caused by errors in the inversion. In HMRF, the prior information consists of a Gibbs distribution function and transition probability matrices. The Gibbs distribution connects similar lithologies and does not need a geological definition derived from non-case-related information. The transition matrices provide preferential transitions between different lithologies and an estimation of these matrices implicitly depends on the depositional environments and juxtaposition rules between different lithologies.@en

We study a reconstruction algorithm for the general inverse scattering problem based on the estimate of not only medium properties, as in more conventional approaches, but also wavefields propagating inside the computational domain. This extended set of unknowns is justified as a way to prevent local minimum stagnation, which is a common issue for standard methods. At each iteration of the algorithm, (i) the model parameters are obtained by solution of a convex problem, formulated from a special bilinear relationship of the data with respect to properties and wavefields (where the wavefield is kept fixed), and (ii) a better estimate of the wavefield is calculated, based on the previously reconstructed properties. The resulting scheme is computationally convenient since step (i) can greatly benefit from parallelization and the wavefield update (ii) requires modeling only in the known background model, which can be sped up considerably by factorization-based direct methods. The inversion method is successfully tested on synthetic elastic datasets.

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A method is presented to link prior geological information, from wells and interpretation, to full waveform inversion at reservoir scale. The method converts the layer-based prior information to grid-based property probability density distributions that are highly non-Gaussian and is based on Bayes' Rule. The likelihood function for the unconstrained inversion is based on the Hessian of the inversion kernel and the minimum residual energy in the objective function. Good results have been obtained from a synthetic case study based on a very realistic outcrop model (Book Cliffs,Utah). Also results from a real data case study will be shown. @en

Hidden Markov Model has been applied to predict the reservoir lithologies by using seismic inversion results as inputs. This method can take the conditional probability between different states or lithologies into account which is the vertical correlation in geology. In order to consider the misfit between the inversion results and the true well-logging data, the model needs to be trained. The application on a field example is quite successful in which most of lithologies have been predicted correctly even for some thin layers. However, this method is only 1D which means that the lateral continuity has not been considered yet. @en

In the context of carbon capture and storage (CCS), quantitative estimation of injected CO2 is of vital importance to verify if the process occurs without any leakage. From a geophysical perspective this is challenging as a CO2 plume has a severe imprint on seismic data. While this makes delineation of the plume rather straightforward, for quantitative interpretation a technique is required that takes complex wave propagation, including multiple scattering and mode conversions into account. In this contribution to the Special Topic, a wave-equation based AVO technique is discussed and successfully demonstrated on a seismic dataset from the Sleipner site. The technique solves the exact wave-equation which means that all complex wave propagation effects mentioned above are properly modelled. The scheme naturally inverts for compressibility and shear compliance and these parameters are more closely related to saturation, porosity and lithology than the conventional impedances. From compressibility the CO2 saturation can be found and the total amount of injected CO2 is calculated by integration over the plume. The result obtained was found to be in good agreement with the known value at the time when the data was acquired. @en

A previous geological and petrophysical model of the fluvio-deltaic Book Cliffs outcrops contained eight lithotypes, within each of which a number of lithologies were grouped. While this model was an adequate representation of the overall depositional architecture, for reservoir-geological purposes the potential reservoir and non-reservoir lithologies needed to be separated. Here, a new and more detailed geological model is presented in which more differentiation is put on the potential reservoir lithologies. This new model contains 12 lithologies with layers down to 1 m in thickness. Assuming a burial depth of 3 km and an average clay content, representative rock physical properties are assigned to lithologies based on published data. After the model thickness has been stretched by a factor of 4 in order to represent a more realistic reservoir, a full-waveform forward seismic response is modelled. These data are used as inputs into an iterative elastic wave-equation-based inversion scheme, with the goal to retrieve the rock properties and layer geometries. The results of this conceptual study show that sandstone units in the shoreface and distributary channels, which are potential reservoirs, are successfully identified. The recovery of medium parameters has a high resolution because the non-linear relationship between rock properties and the seismic data has been exploited. @en

Inversion results from seismic data of a synthetic example based on the Cretaceous fluvio-deltaic Book Cliffs outcrops in Utah (USA) have been used to extract the reservoir parameters. The input data sets are compressibility and shear compliance which are from the full elastic wave-equation based inversion method. A fuzzy logic inference algorithm has been applied in which the lithology templates are based on well-logging data. The membership functions of the lithologies are constructed firstly. Then inversion results are used to predict the reservoir lithology. It is suggested that this classification method performs well because most of the time the same or similar lithologies have been predicted. However, this approach heavily depends on the input inversion results and therefore the full elastic wave-equation based inversion has been chosen.@en