An integrative approach to robust design and probabilistic risk assessment for CO2 storage in geological formations

Abstract

CO 2 storage in geological formations is currently being discussed intensively as a technology with a high potential for mitigating CO 2 emissions. However, any large-scale application requires a thorough analysis of the potential risks. Current numerical simulation models are too expensive for probabilistic risk analysis or stochastic approaches based on a brute-force approach of repeated simulation. Even single deterministic simulations may require parallel high-performance computing. The multiphase flow processes involved are too non-linear for quasi-linear error propagation and other simplified stochastic tools. As an alternative approach, we propose a massive stochastic model reduction based on the probabilistic collocation method. The model response is projected onto a higher-order orthogonal basis of polynomials to approximate dependence on uncertain parameters (porosity, permeability, etc.) and design parameters (injection rate, depth, etc.). This allows for a non-linear propagation of model uncertainty affecting the predicted risk, ensures fast computation, and provides a powerful tool for combining design variables and uncertain variables into one approach based on an integrative response surface. Thus, the design task of finding optimal injection regimes explicitly includes uncertainty, which leads to robust designs with a minimum failure probability. We validate our proposed stochastic approach by Monte Carlo simulation using a common 3D benchmark problem (Class et al., Comput Geosci 13:451-467, 2009). A reasonable compromise between computational efforts and precision was reached already with second-order polynomials. In our case study, the proposed approach yields a significant computational speed-up by a factor of 100 compared with the Monte Carlo evaluation. We demonstrate that, due to the non-linearity of the flow and transport processes during CO 2 injection, including uncertainty in the analysis leads to a systematic and significant shift of the predicted leakage rates toward higher values compared with deterministic simulations, affecting both risk estimates and the design of injection scenarios.

Uncertainty Quantification in CO2Sequestration Using Surrogate Models from Polynomial Chaos Expansion

“…In the work by Oladyshkin et al, 16 design variables are integrated into the PCE approximations, i.e., each design variable or parameter is assigned with a hypothetical distribution and treated as the uncertainty input (parameters). Consequently, the model output becomes some polynomial function of the model input and the design variables.…”

“…[9][10][11][12][13][14][15][16] These methods mainly fall into two categories: intrusive approaches and nonintrusive ones. Intrusive methods involve substituting the PCE approximations into the governing equations and using a Galerkin technique 17 and a discretization scheme to solve for the coefficients in the expansion.…”

“…12 Non-intrusive methods evaluate the coefficients in the expansion using a small number of model simulations and require no manipulation of underlying partial differential equations. [9][10][11][13][14][15][16] Therefore, the latter approaches are easier to implement and generalize to complex systems. For this reason, we focus on non-intrusive PCE methods.…”

“…16 However, using a hypothesis test, Kopp et al 19 have shown that the raw data used by Oladyshkin et al 16 fail to follow a normal or a lognormal distribution. In addition, using collocation points 16 to solve for the expansion coefficients limits application to low-dimensional PCEs. In related work, 18 PCE models were developed without parametric assumptions but the correlations of uncertain parameters were not explicitly addressed.…”

SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos

“…Oladyshkin and Nowak observed that as every set of random data, as well as a continuous or discrete PDF, can be described using the moments without making any assumptions about the shape or existence of a suitable probability distribution, the moments provided a very general approach to propagate data without requiring the determination of deterministic PDFs. Oladyshkin and Nowak [7] promoted this concept in the geo-sciences by successfully applying it to identify uncertainties in carbon dioxide storage in geological formations [15,16,17] and also for robust design [18]. Moreover, Oladyshkin and Nowak presented a derivation of the optimal orthogonal polynomials from the moments.…”