Bayesian Uncertainty Quantification for Channelized Reservoirs via Reduced Dimensional Parameterization
Abstract
:1. Introduction
2. Geological Models, Parameterization, and the Bayesian Inverse Problem
2.1. Fine and Coarse Models
2.2. The Bayesian Inverse Problem
2.3. Parameterization of Permeability Field
2.3.1. Parameterization of Interfaces
2.3.2. Parameterization within Facies
3. Posterior Error Introduced by Truncation
4. MCMC Sampling from the Truncated Posterior
5. Numerical Results
5.1. Convergence Estimation
5.1.1. Single Facies
5.1.2. Channelized Reservoirs
5.2. Matching Permeability with Reduced Parameters
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Acknowledgments
Conflicts of Interest
Abbreviations
K-L | Karhunen–Loève |
MCMC | Markov chain Monte Carlo |
MsFEM | Multiscale finite element method |
M-H | Metropolis Hastings |
PDE | Partial differential equation |
Velocity of phase j | |
k | Permeability |
Relative permeability to phase j | |
S | Water saturation |
p | Pressure |
f | Fractional flux of water |
Total mobility | |
Fractional flow | |
Outflow boundary | |
Normal velocity field | |
Observed fractional flow data | |
Fractional flow obtained by running the forward model to permeability k | |
Random error | |
Error variance | |
w | Pseudo-velocity field |
Pseudo-time | |
The interface | |
Eigen value from the K-L expansion of permeability | |
Eigen vector from the K-L expansion of permeability | |
K-L coefficient | |
Coefficient for the velocity field | |
Spatial basis for the velocity field | |
Y | Log permeability |
Posterior | |
Prior | |
Flowmap | |
T | Time to flight |
Appendix A
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M | ||
---|---|---|
5 | 1.111681 | 0.081809 |
10 | 0.750662 | 0.106264 |
15 | 0.517555 | 0.063635 |
20 | 0.337901 | 0.030207 |
25 | 0.189272 | 0.017931 |
30 | 0.071924 | 0.011225 |
M | ||
---|---|---|
5 | 1.176697 | 0.308118 |
10 | 0.820661 | 0.191601 |
15 | 0.566938 | 0.119590 |
20 | 0.378454 | 0.059173 |
25 | 0.248267 | 0.033023 |
30 | 0.123347 | 0.014965 |
5 | 5 | 5 | 0.526235 | 0.786077 | 0.853727 | 0.109464 |
10 | 5 | 5 | 0.367011 | 0.786077 | 0.853727 | 0.116172 |
10 | 10 | 10 | 0.367011 | 0.530798 | 0.477141 | 0.051925 |
15 | 10 | 10 | 0.253542 | 0.530798 | 0.477141 | 0.093109 |
15 | 15 | 10 | 0.253542 | 0.365967 | 0.477141 | 0.053869 |
20 | 15 | 15 | 0.169250 | 0.365967 | 0.210844 | 0.047356 |
20 | 20 | 15 | 0.169250 | 0.238932 | 0.210844 | 0.019996 |
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Mondal, A.; Wei, J. Bayesian Uncertainty Quantification for Channelized Reservoirs via Reduced Dimensional Parameterization. Mathematics 2021, 9, 1067. https://doi.org/10.3390/math9091067
Mondal A, Wei J. Bayesian Uncertainty Quantification for Channelized Reservoirs via Reduced Dimensional Parameterization. Mathematics. 2021; 9(9):1067. https://doi.org/10.3390/math9091067
Chicago/Turabian StyleMondal, Anirban, and Jia Wei. 2021. "Bayesian Uncertainty Quantification for Channelized Reservoirs via Reduced Dimensional Parameterization" Mathematics 9, no. 9: 1067. https://doi.org/10.3390/math9091067