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A Bayesian Approach for In-Situ Stress Prediction and Uncertainty Quantification for Subsurface Engineering

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Abstract

Many subsurface engineering applications require accurate knowledge of the in-situ state of stress for their safe design and operation. Existing methods to meet this need primarily include field measurements for estimating one or more of the principal stresses from a borehole, or optimization methods for constructing a 3D geomechanical model in terms of geophysical measurements. These methods, however, often contain considerable uncertainty in estimating the state of stress. In this paper, we build on a Bayesian approach to quantify uncertainty in stress estimations for subsurface engineering applications. This approach can provide an estimate of the 3D distribution of stress throughout the volume of interest and provide an estimate of the uncertainty arising from the stress measurement, the rheology parameters, and a paucity of measurements. The value of this approach is demonstrated using stress measurements from the In Salah carbon storage site, which was one of the world’s first industrial carbon capture and storage projects. This demonstration shows the application of this Bayesian approach for estimating the initial state of stress for In Salah and quantifying the uncertainty in the estimated stress. Also, an assessment of a maximum injection pressure to prevent geomechanical risks from CO2 injection pressures is provided in terms of the probability distribution of the minimum principal stress quantified by the approach. With the In Salah case study, this paper demonstrates that using the Bayesian approach can provide additional insights for site explorations and/or project operations to make informed-site decisions for subsurface engineering applications.

Highlights

  • This study proposes a Bayesian approach to quantify uncertainty in estimates of initial 3D stress distributions arising from different sources.

  • The approach provides the joint probability of the two horizontal principal stresses, instead of only the mean stress state of each.

  • Adding regional geologic information and stress-related informative priors can reduce uncertainty in estimates of stress and modeling parameters.

  • The approach helps make more reliable geomechanical decisions for subsurface engineering applications compared to a deterministic method.

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Fig. 1

adapted from White et al. (2014)]

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Abbreviations

\({\varvec{x}}\) :

List of uncertain parameters

\(P_{p}\) :

Effective pore pressure

\(\sigma_{h}\) :

Total minimum principal stress

\(\rho_{r}\) :

Rock density

\(\sigma_{H}\) :

Total maximum principal stress

\({{\varvec{\updelta}}}\) :

Kronecker delta

\(\sigma_{v}\) :

Total vertical stress

\(\alpha\) :

Biot’s coefficient

\(\varepsilon_{h}\) :

Minimum horizontal strain

\({\mathbf{C}}\) :

Stiffness tensor

\(\varepsilon_{H}\) :

Maximum horizontal strain

E :

Young’s modulus

\({\varvec{\sigma}}\) :

Total stress tensor

ν :

Poisson’s ratio

\({\varvec{\varepsilon}}\) :

Strain tensor

\({\varvec{D}}\) :

Given information for \({\varvec{x}}\)

\(L_{l}\) :

Lower bound

\(\theta\) :

Mean

\(L_{u}\) :

Upper bound

\(\xi\) :

Standard deviation

\(\mu\) :

Friction coefficient

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Acknowledgements

Funding for this research was provided by the National Risk Assessment Partnership (NRAP) in the US DOE Office of Fossil Energy under DOE contract number DE-AC05-76RL01830. PNNL is operated by Battelle for the US DOE under Contract DE-AC06-76RLO1830. This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

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Authors and Affiliations

  1. School of Civil Engineering, Chongqing University, Chongqing, China

    Ting Bao

  2. Key Laboratory of New Technology for Construction of Cities in Mountain Area, Chongqing University, Chongqing, China

    Ting Bao

  3. Pacific Northwest National Laboratory, 902 Battelle Blvd, Richland, WA, 99354, USA

    Ting Bao & Jeff Burghardt

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  1. Ting Bao
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Correspondence to Ting Bao.

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Ting Bao: Research work done while at Pacific Northwest National Laboratory; currently at Chongqing University.

Appendix 1

Appendix 1

Figure supplement for 3D Bayesian modeling analysis.

See Fig. 15

Fig. 15
figure 15

a 3D model mesh. The model dimension is 20 km × 10 km × 620 m. The x and y axes are the direction of \(\sigma_{H}\) and \(\sigma_{h}\), respectively. b Rescaled model in the x and y axes (rescale factor = 0.01) to show the anticlinal structure considered here. c 3D initial stress state for \(\sigma_{h}\) in terms of the median of posterior \(\sigma_{h}\). Note that the horizontal dimension is too large to visualize the anticlinal structure of the reservoir

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Bao, T., Burghardt, J. A Bayesian Approach for In-Situ Stress Prediction and Uncertainty Quantification for Subsurface Engineering. Rock Mech Rock Eng 55, 4531–4548 (2022). https://doi.org/10.1007/s00603-022-02857-0

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