History Matching Through a Smooth Formulation of Multiple-Point Statistics

We propose a smooth formulation of multiple-point statistics that enables us to solve inverse problems using gradient-based optimization techniques. We introduce a differentiable function that quantifies the mismatch between multiple-point statistics of a training image and of a given model. We show that, by minimizing this function, any continuous image can be gradually transformed into an image that honors the multiple-point statistics of the discrete training image. The solution to an inverse problem is then found by minimizing the sum of two mismatches: the mismatch with data and the mismatch with multiple-point statistics. As a result, in the framework of the Bayesian approach, such a solution belongs to a high posterior region. Themethodology, while applicable to any inverse problem with a training-image-based prior, is especially beneficial for problems which require expensive forward simulations, as, for instance, history matching. We demonstrate the applicability of the method on a two-dimensional history matching problem. Starting from different initial models we obtain an ensemble of solutions fitting the data and prior information defined by the training image. At the end we propose a closed form expression for calculating the prior probabilities using the theory of multinomial distributions, that allows us to rank the history-matched models in accordance with their relative posterior probabilities.