Geological scenario determination using parameterized training images in a Bayesian framework

Advisors

Louis Durlofsky, primary advisor
Tapan Mukerji, advisor
Celine Scheidt, advisor

Abstract

In order to appropriately manage reservoir performance, it is necessary to accurately represent uncertainty in the geological characterization. Geological uncertainty is typically treated by considering multiple realizations generated from a prescribed scenario (often defined in terms of a training image). The geological scenario, however, is also uncertain, though this is often ignored. Instead, one or a few scenarios may simply be selected based on the interpretation of available data and existing knowledge. This approach is likely to result in the selection of an incorrect scenario, which may lead to significant inaccuracies in future predictions. In this work, we devise a more systematic approach that considers a vast prior ensemble of allowable geological scenarios. We then introduce formal optimization to determine the maximum (most probable) a posteriori (MAP) estimate, based on the information provided by observed production data. Predictions are then generated using this MAP scenario. As is common in geomodeling applications, the scenario is defined in terms of a training image (TI), as well as a histogram and a variogram for each rock facies. Construction of the prior ensemble is achieved by introducing continuous parameterizations for the uncertain attributes characterizing the TI, histograms, and variograms. Each possible scenario is then viewed as a statistical model describing the uncertain subsurface. Indeed, many realizations can be generated for each scenario using multipoint statistics (MPS). We then follow a Bayesian approach motivated by formal model selection in statistics, which is based on the posterior probability of the scenarios. Evaluating these probabilities, however, requires computing the marginal likelihoods, and this is achieved by integrating over a large number of realizations for each scenario considered. We consider three existing methods for the estimation of scenario marginal likelihoods. We find that accurate estimates can be obtained with these methods, but they may entail excessive computational cost when measurement errors in the observed data are small, which renders them unsuitable for use in our optimization approach. We then devise a more efficient alternative procedure to rank scenarios with respect to their marginal likelihoods that is not sensitive to the measurement error. This method relies on a particular (small) percentile of the cumulative distribution function (CDF) of mismatches for each scenario. Construction of the CDF requires performing flow simulations for a number of realizations for the scenario considered. We also extend our scenario-ranking procedure to obtain an approximate ranking of scenario posterior probabilities in cases with nonuniform priors. We then introduce a two-step optimization workflow that uses our efficient scenario-ranking procedure for objective-function evaluations. In the first step, a particle swarm optimization (PSO) is performed, in which the objective function is evaluated with a relatively small number (50) of realizations. The PSO solution is then used as an initial guess for a local optimization in the second step. Mesh adaptive direct search (MADS) is used for this step and the required CDFs are evaluated using significantly more realizations (1000) than in the first step. The resulting `optimal' scenario corresponds to the MAP estimate, which can be used to perform flow predictions. When observed data are not very reliable (large measurement error) or are limited, slightly sub-optimal scenarios may still be associated with relatively large posterior probabilities. We propose a criterion to verify that our MAP estimate is an appropriate choice for flow predictions. If this is not the case, we instead consider an ensemble of scenarios, corresponding to those with large posterior probabilities, for flow predictions. Results for two synthetic examples involving oil production via waterflooding are presented. In each example, we perform two runs using `representative' realizations and consistently find that the MAP estimate is an appropriate representation of the `true' scenario. In particular, flow predictions obtained from this scenario are of similar quality to those obtained from the true scenario, and significantly more accurate than predictions from a randomly selected scenario within the prior ensemble. For the first example, we also successfully test the robustness of our procedure to other choices of the `true' subsurface realization. In addition, we consider cases with nonuniform prior probabilities and find that, as expected, results vary with the level of relative measurement error in the observed data. Specifically, for small levels of error, the MAP scenario is similar to that obtained with a uniform prior, while for large levels of error, the most probable a posteriori scenario is quite similar to the most probable a priori scenario. We also demonstrate that our procedure is able to identify a range of possible scenarios in cases where the observed data are more limited. Taken in total, the results show that our overall framework is quite effective at enabling the formal selection of the geological scenario from a vast initial ensemble. As such, it may represent a useful addition to data assimilation workflows.