Abstract: |
Inverse modeling is an essential process to integrate geophysical information in reservoir characterization. We propose a Markov chain Monte Carlo (McMC) workflow consistent with geology, well-logs, seismic data and rock-physics information. The workflow uses Direct Sampling (DS), a multiple-point geostatistical method, for generating realizations from the prior distribution and Adaptive Spatial Resampling (ASR) for sampling from the posterior distribution conditioned to the geophysical data. To produce samples from the posterior probability density is a key issue in any inversion problems posed in a Bayesian framework. Sampling is a more general approach than optimization as it can assess important uncertainties and not just the most likely model. Rejection sampling is the only way to represent perfect posterior pdf. However, since it requires a large number of evaluations of forward model, it is inefficient and not suitable for reservoir modeling. Metropolis sampling is able to perform a reasonably equivalent sampling by forming a Markov chain. The ASR algorithm perturbs realizations of a spatially dependent variable while preserving its spatial structure by subset points. The method is used as a transition kernel to produce a Markov chain of geostatistical realizations. These realizations are then used in a forward seismic model to compute the predicted data which are compared to the observed data. Depending on the acceptation/rejection criterion in the Markov process, it is possible to obtain a chain of realizations aimed either at characterizing the posterior distribution with Metropolis sampling or at calibrating a single realization until an optimum is reached. Thus the algorithm can be tuned to work either as an optimizer or as a sampler. The validity and applicability of the proposed method is demonstrated by results for seismic lithofacies inversion on the synthetic Stanford VI data set of and an actual data set in West Africa. |
