Uncertainty in Geothermal Numerical Models: a Contribution from Geostatistics and Bayesian Methods

Abstract

Numerical models of geothermal reservoirs are a valuable tool to understand the processes that control subsurface flow and to help engineers and scientists to manage these resources. However, there are a number of uncertainties that modellers have to face to generate reliable predictions. The problem of geological uncertainty was addressed by geostatistical simulation. Multiple-point statistics (MPS) algorithms were introduced. The main feature of a MPS algorithm is that it relies on a training image and through the use of multiple training images spatial uncertainties in subsurface flow problems are addressed. Methods for global uncertainty and sensitivity analysis (UA/SA) are mainly based on the Monte Carlo approach. In these analyses the simulator will be run many times with different combinations of parameters and an ensemble of outputs will be gathered from which sample statistics can be calculated. The main limitation of this approach is that a very large number of samples from the simulator are necessary to obtain reliable estimates for uncertainty analyses, which is not feasible for expensive models. As an alternative to Monte Carlo methods the Bayesian emulation methodology was presented. The idea behind this method is that a statistical approximation to an output can be generated from a small number of runs of the simulator. This approximation may then be used to produce fast estimates of a model output for different combinations of parameters. This methodology was applied to a series of problems in UA/SA with a considerably saving of the number of simulator runs. Additionally, the problem of global calibration was addressed with the use of surrogate models and adaptive sampling and efficient solutions were obtained for nonlinear, multimodal problems, where standard, derivative-based methods may achieve convergence to low-quality solutions. The use of surrogate models was tested in low and high dimensional problems and the results show in general a good agreement between the estimated outputs and the available observations. An important feature of surrogate model calibration is that it allows the inclusion of categorical variables into the analysis, which was shown to be an efficient strategy to include geological uncertainties into a calibration problem.