History Matching of Truncated Gaussian Models by Parallel Interacting Markov Chains on a Reduced Dimensional Space

In oil industry and subsurface hydrology, geostatistical models are often used to represent the spatial distribution of different lithofacies in the reservoir. Two main model families exist: multipoint and truncated Gaussian models. We focus here on the latter. In history matching of lithofacies reservoir model, we attempt to find multiple realizations of lithofacies configuration, that are conditional to dynamic data and representative of the model uncertainty space. This problem can be formalized in the Bayesian framework. Given a truncated Gaussian model as a prior and the dynamic data with its associated measurement error, we want to sample from the conditional distribution of the facies given the data. A relevant way to generate conditioned realizations is to use Markov Chains Monte Carlo (MCMC). However, the dimension of the model and the computational cost of each iteration are two important pitfalls for the use of MCMC. In practice, we have to stop the chain far before it has scanned the whole support of the posterior. Further more, as the relationship between the data and the random field is non-linear, the posterior can be multimodal. Hence, the chain may stay stuck in one of the modes. In this work, we first show how to reduce drastically the dimension of the problem by using a truncated Karhunen-Loève expansion of the Gaussian random field underlying the lithofacies realization. Then we show how we can improve the mixing properties of classical single MCMC, without increasing the global computational cost, by the use of parallel interacting Markov chains at different temperatures. Applying the dimension reduction and this innovative sampling method lowers drastically the number of iterations needed to sample efficiently from the posterior. We show the encouraging results obtained when applying the methodology to a synthetic history matching case.