Jo Eidsvik & Henning Omre, Department of Mathematical Sciences, NTNU
Geophysical reservoir characterization is critical for understanding properties of the subsurface. This is important for instance in relation to petroleum companies’ workflows. We describe statistics research conducted at NTNU on Bayesian inversion for improved geophysical reservoir characterization. Supported by the Norwegian Research Council and petroleum companies, the Uncertainty in Reservoir Evaluation (https://wiki.math.ntnu.no/ure , URE) and Geophysics and Applied Mathematics for Exploration and Safe production (https://wiki.math.ntnu.no/games/start , GAMES) projects have developed novel inversion methods.
The vision in the projects has been to
- provide creative mathematically based solutions to recognized challenges in reservoir evaluation,
- develop methodologies for analysis of spatial and spatio-temporal phenomena.
Hence, while it has been important to refine mathematical representations to include more sophisticated features and data, and to develop fast scalable methods for Bayesian inference in spatial applications, it has also been essential to apply methods on relevant real-world datasets. University research labeled as applied has limited value if it cannot be applied by users. Also, the history of statistics shows that work on new data types and bottle-neck problems in industry often feed back into novel statistical methodological development.
Background
Seismic waves and rock physics relations were established a long time ago. Along with regularization terms and data processing, they have been useful in geophysical data inversion. But as one seeks improved oil recovery and aims to produce smaller volumes of resources near existing infrastructure, the modeling gets more sophisticated, and solutions are required at a finer scale. Reliable predictions and uncertainty quantification are then key to decision support systems. The topic of probabilistic inversion has hence grown substantially in recent years.
Early approaches to probabilistic inversion for reservoir characterization were not very successful because they intermixed interpretations and data in several steps of the inversion, leading to underestimation of uncertainty and biased predictions. In many ways, Buland et al. (2003) re-defined seismic inversion in a probabilistic setting: This Bayesian view separates
a) the likelihood model linking reservoir properties to the data,
b) the prior model containing geoscience belief and spatial correlations.
This approach leads to a modular setup which is easily explainable and reproducible. The paper also introduced a linearized likelihood expression where a posterior solution is conjugate with a Gaussian prior for fast probabilistic inversion. Going beyond Gaussian solutions, Eidsvik et al. (2004) and Larsen et al. (2006) developed methods for probabilistic inversion to discrete reservoir variables of interest (geological facies or fluid saturation).
We will next illustrate some of the recent contributions made by PhD students in the URE and GAMES projects. These students have contributed with new statistical model and methodological development for challenging data from the industry. Most graduates have gone to energy companies, continuing to use skills developed during their PhD.
Lithology and Fluid prediction
Among the many reservoir properties, the fluids (oil, gas, water) and lithologies (rock types) characteristics are of course particularly important because they directly influence decisions about reservoir development. Rimstad et al. (2012) predicted fluids and facies distributions at the Alvheim field in the North Sea. They built a Bayesian hierarchical model which enables an understanding of the geological burial history as well as predictions of lithologies and fluids in different zones of the reservoir. Figure 1 shows the graphical model used in the paper.
Figure 1: Graphical model of parameters, reservoir variables, well observations and seismic data. A Bayesian hierarchical model is constructed based on established physical relations and data from similar reservoir situations. (Fig. from Rimstad et al. (2012).)
Their analysis confirmed the belief that parts of the Eastern reservoir segments were filled with oil-sands rather than gas-sands (see predictions in Figure 2). This fluid difference entails a substantial economic benefit. Going beyond earlier results, their model facilitated a principled approach for uncertainty quantification related to these resources.
Figure 2: Facies prediction of gas (red) and oil (green) based on six well logs and seismic amplitude versus offset data at the Alvheim field in the North Sea. Transparent colors reflect that the facies classes are most likely shale or water-saturated sands. (Fig. from Rimstad et al. (2012).)
Their analysis confirmed the belief that parts of the Eastern reservoir segments were filled with oil-sands rather than gas-sands (see predictions in Figure 2). This fluid difference entails a substantial economic benefit. Going beyond earlier results, their model facilitated a principled approach for uncertainty quantification related to these resources.
Figure 2: Facies prediction of gas (red) and oil (green) based on six well logs and seismic amplitude versus offset data at the Alvheim field in the North Sea. Transparent colors reflect that the facies classes are most likely shale or water-saturated sands. (Fig. from Rimstad et al. (2012).)
Fjeldstad et al. (2021) used related methods building on hierarchical models to conduct Bayesian inversion results for North Sea reservoirs. Their approach relied on a discrete latent variable which induces mixture components in the reservoir properties, which can give more reliable inversion results leading to improved decision support systems.
Ensemble-based geophysical inversion
Rather than building fully specified parametric models for Bayesian inversion solutions, there has lately been a drive towards ensemble-based approaches forming an approximate posterior solution to the inverse problem. Gineste et al. (2020) developed such an ensemble-based approach for seismic waveform inversion, where assumptions of linearity in the likelihood models are no longer needed. Instead, it relies on prior realizations and the generation of synthetic geophysical datasets to estimate model-data correlations. And then these trained correlations are formed to assimilate data in the prior realizations. Figure 3 shows the inversion result from a dataset provided by BP in the UK.
Figure 3: Left) Seismic waveform data in a depth profile (offset [km] on the first axis and seismic traveltime [s] on the second axis). Middle) Prior ensemble of pressure-wave velocity with a BP well log in black and prior mean (light blue) and spread (dark blue). Right) Posterior ensemble of the pressure-wave velocity showing that the well log is reproduces and the uncertainty is clearly reduced. (Fig. from Gineste et al., 2020.)
Spremic et al. (2024) similarly used ensemble-based inversion for predicting oil and gas saturations in a North Sea reservoir study. They relied on local conditioning to seismic AVO observations and assimilation of well log data. Predictions enabled probabilistic statements that can be valuable infill drilling programs and increased oil and gas recovery from the producing field.
References
Buland, A., & Omre, H. (2003). Bayesian linearized AVO inversion. Geophysics, 68(1), 185-198. https://doi.org/10.1190/1.1543206
Eidsvik, J., Avseth, P., Omre, H., Mukerji, T., & Mavko, G. (2004). Stochastic reservoir characterization using prestack seismic data. Geophysics, 69(4), 978-993. https://doi.org/10.1190/1.1778241
Gineste, M., Eidsvik, J., & Zheng, Y. (2020). Ensemble-based seismic inversion for a stratified medium. Geophysics, 85(1), R29-R39. https://doi.org/10.1190/geo2019-0017.1
Fjeldstad, T., Avseth, P., & Omre, H. (2021). A one-step Bayesian inversion framework for 3D reservoir characterization based on a Gaussian mixture model—A Norwegian Sea demonstration. Geophysics, 86(2), R221-R236. https://doi.org/10.1190/geo2020-0094.1
Larsen, A. L., Ulvmoen, M., Omre, H., & Buland, A. (2006). Bayesian lithology/fluid prediction and simulation on the basis of a Markov-chain prior model. Geophysics, 71(5), R69-R78. https://doi.org/10.1190/1.2245469
Rimstad, K., Avseth, P., & Omre, H. (2012). Hierarchical Bayesian lithology/fluid prediction: A North Sea case study. Geophysics, 77(2), B69-B85. https://doi.org/10.1190/geo2011-0202.1
Spremić, M., Eidsvik, J., & Avseth, P. (2024). Bayesian rock-physics inversion using a localized ensemble-based approach—With an application to the Alvheim field. Geophysics, 89(2), R95-R108. https://doi.org/10.1190/geo2022-0764.1
