Sparsity-based compressive reservoir characterization and modeling by applying ILS-DLA sparse approximation with LARS on DisPat-generated MPS models using seismic, well log, and reservoir data

In the earth sciences, there is only one single true reality for a property of any dimension whereas many realization models of the reality might exist. In other words, a set of interpreted multiplicities of an unknown property can be found but only one unique fact exists and the task is to return from the multiplicities to the uniqueness of the reality. Such an objective is mathematically provided by sparse approximation methods. The term "approximation" indicate the sufficiency of an interpretation that is close enough to the true mode, i.e. reality. In geosciences, the multiplicities are provided by multiple-point statistical methods. Realistic modeling of the earth interior demands for more sophisticated geostatistical methods based on true available images, i.e. the training images. Among available MPS methods, the DisPat algorithm is a distance-based MPS method which generate appealing realizations for stationary and nonstationary training images by classifying the patterns based on distance functions using kernel methods. Advances in nonstationary image modeling is an advantage of the DisPat method. Realizations generated by the MPS methods form the training set for the sparse approximation. Sparse approximation is consisted of two steps, i.e. sparse coding and dictionary update, which are alternately used to optimize the trained dictionary. Model selection algorithms like LARS are used for sparse coding. LARS optimizes the regression model sequentially by choosing a proper number of variables and adding the best variable to the active set in each iteration. Out of numerous training dictionary methods given in the literature, the ILS-DLA is a variant of the MOD algorithm where the latter is inspired by the GLA and the whole trained dictionary is sequentially updated by alternating between sparse coding and dictionary training steps. The ILS-DLA is different from the MOD for addressing the internal structure of the dictionary by considering overlapping or non-overlapping blocks and modifying the MOD algorithm according to the internal structure of the trained dictionary. The ILS-DLA is faster than the MOD in the sense that it inverts for smaller blocks constructing the trained dictionary rather than inverting for the entire block. The subject of this paper is an integration study between sparse approximations from image processing and compressed sensing, multiple-point statistics from the field of geostatisitcs, and the geophysical methods and reservoir engineering from the branch of petroleum science. This paper specifically emphasizes the utilization of image processing in solving reservoir complexities and enhancing reservoir models.