Incorporating Posterior‐Informed Approximation Errors Into a Hierarchical Framework to Facilitate Out‐of‐the‐Box MCMC Sampling for Geothermal Inverse Problems and Uncertainty Quantification

Abstract

We consider geothermal inverse problems and uncertainty quantification from a Bayesian perspective. Our main goal is to make standard, "out-of-the-box" Markov chain Monte Carlo (MCMC) sampling more feasible for complex simulation models by using suitable approximations. To do this, we first show how to pose both the inverse and prediction problems in a hierarchical Bayesian framework. We then show how to incorporate so-called posterior-informed model approximation error into this hierarchical framework, using a modified form of the Bayesian approximation error approach. This enables the use of a "coarse," approximate model in place of a finer, more expensive model, while accounting for the additional uncertainty and potential bias that this can introduce. Our method requires only simple probability modeling, a relatively small number of fine model simulations and only modifies the target posterior-any standard MCMC sampling algorithm can be used to sample the new posterior. These corrections can also be used in methods that are not based on MCMC sampling. We show that our approach can achieve significant computational speedups on two geothermal test problems. We also demonstrate the dangers of naively using coarse, approximate models in place of finer models, without accounting for the induced approximation errors. The naive approach tends to give overly confident and biased posteriors while incorporating Bayesian approximation error into our hierarchical framework corrects for this while maintaining computational efficiency and ease of use. Key Points: • We consider geothermal inverse problems and uncertainty quantification from a Bayesian perspective • We present a simple method for incorporating posterior-informed approximation errors into a hierarchical Bayesian framework • Our method makes standard out-of-the-box MCMC sampling feasible for more complex models while correcting for bias and overconfidence Supporting Information: • Supporting Information S1 , M. J. (2020). Incorporating posterior-informed approximation errors into a hierarchical framework to facilitate out-of-the-box MCMC sampling for geothermal inverse problems and uncertainty quantification. Water Resources Research, 56, e2018WR024240. https://

Bayesian Calibration of a Natural State Geothermal Reservoir Model, Krafla, North Iceland

“…Compared to other approaches that construct approximate posterior distributions using reduced‐order models in the context of natural state geothermal reservoir models, such as delayed acceptance schemes that calculate the model reduction error dynamically (Cui et al., 2011; Cui, Fox, Nicholls, & O’Sullivan, 2019; Cui, Fox, & O’Sullivan, 2019), the approach of Maclaren et al. (2020) is relatively easy to implement, requiring a relatively small number of fine model simulations to construct the posterior‐informed model approximation error. Despite a huge reduction in the dimensionality of the coarse model compared to the fine model (from 17,193 to 2,551 grid blocks), the naïve posterior is strongly informative (Figure 13), suggesting that the physics of the fine model are adequately captured in the coarse model and that the approximation error calculated using the naïve posterior yields relevant error estimates.…”

“…A numerical method that accurately models flow of multi-phase, variably miscible fluids, with a range of applicability extending to >375°C 3. A hierarchical Bayesian approach incorporating a posterior-informed approximation error model Here, we briefly describe the numerical method and hierarchical Bayesian approach, which have been published previously (Maclaren et al, 2020;Pruess et al, 2012), and focus on description of the model setup.…”

“…Here, we briefly describe the numerical method and hierarchical Bayesian approach, which have been published previously (Maclaren et al., 2020; Pruess et al., 2012), and focus on description of the model setup.…”

“…Cui et al (2011) had each cell correspond to a different rock type, and found up to one order of magnitude standard deviation in spatially restricted areas. Maclaren et al (2020) found even higher uncertainties in permeability for certain rock types, ranging up to 3 or 4 orders of magnitude, but used a model with fewer rock types and fewer observational constraints from wells.…”

“…We adapt the Bayesian hierarchical approach of Maclaren et al. (2020) to perform the model calibration and quantify the uncertainty in the inferred permeability structure. This study highlights the potential for Bayesian approaches to enable better consideration of geologic uncertainty during natural state model calibration.…”