The spatial distribution of facies is a crucial part of any reservoir model, since it is often one of the main sources of variability in flow. Multi-point statistics is one class of methods for geological facies modeling, and it has developed along two main paths: the statistical model approach and the algorithmic approach. Algorithmic methods tend to produce artifacts in the simulations. This is due to lacking patterns in the training image. Statistical models can interpolate between observed patterns to compute the probability of patterns that are not explicitly present in the training image, and hence artifacts are avoided. With the introduction of Markov mesh models, the statistical approach overcame its original time-consumption problems in parameter estimation and simulation. In this paper we precede yet another step, and formulate a multi-grid Markov mesh model. This combines an advantage originally developed for algorithmic methods, with the consistency and flexibility of the statistical methods. The use of multiple grids ensures that training image patterns at different spatial scales can be efficiently captured.
We present the general formulation of the method. A hierarchy of nested grids is defined, and a Markov mesh model is defined for each grid, but such that it takes into account information also from coarser grids. The result is what we denote a multi-grid Markov mesh model. The framework of generalized linear models and systematic grid specification enable fast parameter estimation. The estimation is done once per grid level. During simulation the coarse patterns are first laid out, and by simulating increasingly finer grids we are able to create patterns at different scales.
We present several 3D examples, illustrating that the multi-grid Markov mesh model can be successfully applied for a range of training images. For each considered training image, the simulation results are quantitatively evaluated by comparing the up-scaled permeability distributions of the generated realizations to the permeability tensor of the training image. Also distributions for facies volume fractions are evaluated.