Bentheimer sandstone image data

• https://doi.org/10.1103/PhysRevApplied.15.054003

Abstract — A general and robust Bayesian optimization framework for the extraction of intrinsic physical properties from an integration of pore-scale forward modeling and experimental measurements of macroscopic system responses is developed. The efficiency of the scheme, which utilizes Gaussian process regression, enables the simultaneous extraction of multiple intrinsic physical properties with a minimal number of function evaluations. Here it is applied to nuclear magnetic resonance (NMR) relaxation responses, paving the way for inverse problem approaches to digital rock physics given its general nature. NMR relaxation responses of fluids in porous media may be described by sums of multiexponential decays resulting in a relaxation time distribution. The shape of this distribution is dependent on intrinsic physical system properties, but also effects like diffusion coupling between different relaxation regimes in heterogeneous porous materials. Forward models based on high-resolution images are employed to naturally incorporate structural heterogeneity and diffusive motion without limiting assumptions. Extracting the required multiple intrinsic parameters of the system poses an ill-conditioned multiphysics multiparameter inverse problem where multiple scales are covered by the underlying microstructure. Exploration of the multidimensional search space given an expensive cost function makes an efficient solution strategy mandatory. We propose a workflow to match experimental measurements with simulations via Bayesian optimization, with special attention paid to the multimodal nature of the topography of the objective function using solution space partitioning. A multimodal search strategy using state-of-the-art evolutionary algorithms and gradient-based optimization algorithms guarantees that the multimodal nature is captured. The workflow is demonstrated on T2 relaxation responses of Bentheimer sandstone, extracting three physical parameters simultaneously: the surface relaxivity of quartz grains, the effective transverse relaxation time, and the effective diffusion coefficient in clay regions. Multiple mathematically sound and physically plausible solutions corresponding to global minimum and multiple local minima of the objective function are identified within a limited number of function evaluations. Importantly, the shape of the experimental T2 distribution is recovered almost perfectly, enabling the use of classical interpretation techniques and local analysis of responses based on numerical simulation.

• https://doi.org/10.2118/210563-PA

Abstract — Nuclear magnetic resonance (NMR) relaxation responses in porous media provide a sensitive probe of the microstructure and yet are influenced by a number of factors which are not easily detangled. Low-field T2 transverse relaxation measurements can be carried out quickly and are frequently used as pore size distributions, while adding T1 longitudinal relaxation measurements provides additional insights into surface properties and fluid content. Here we present an inverse solution workflow extracting related intrinsic physical parameters of the system by fitting experiment and numerical simulation(s). An efficient NMR forward solver for the simultaneous calculation of T1 and T2 responses is introduced, which honors existing inequality relationships between T1 and T2 parameters. We propose a Bayesian optimization approach that jointly identifies T1- and T2-related properties satisfying physical constraints by simultaneously fitting T1 and T2 experiments to simulations. This dual-task inverse solution workflow (DT-ISW) identifies the solution by minimizing the sum of the L2 norm of the fitting residuals of both T1 and T2 distributions into a single objective and jointly models the two highly correlated objectives with high efficiency using the vector-valued Gaussian process (GP) kernel for transfer learning. A multimodal search strategy is used to identify nonunique solution sets of the problem. The workflow is demonstrated on Bentheimer sandstone, identifying five intrinsic physical parameters. The performance of the joint DT-ISW (DT-ISW-J) is compared to a sequential DT-ISW (DT-ISW-S) approach as well as an independent single-task ISW (ST-ISW) of the T1 and T2 responses. Both dual-task versions converge more than two times faster than the single-task version. DT-ISW-J equally minimizes the L2 norm of T1 and T2 fitting residuals whereas DT-ISW-S only preferentially minimizes the objective assigned higher importance. A Pareto optimal solution (POS) is provided to allow operators to subjectively balance the preference of T1 and T2 data fits for the slightly conflicting objectives. The ability to extract five intrinsic physical parameters simultaneously provides new techniques for tracking wettability alteration and assessing the influence of clay amount and distribution on petrophysical property estimates.

• https://doi.org/10.1029/2021WR031590

Abstract — Nuclear magnetic resonance measurements of sedimentary rocks are used to extract various transport properties including hydraulic conductivity and water retention curves. These estimates are controlled by intrinsic physical quantities like surface relaxivities and effective relaxation time and restricted self-diffusion coefficient of water in clay. Sampling these properties on a set of core plugs presents a series of inverse problems where some of the extracted parameters are expected to be similar. To leverage such valuable information, we extend a previously developed single-task inverse solution workflow (ISW) to the multi-task case, transferring the knowledge gained from previous optimization tasks. Two multi-task kernels: intrinsic model of coregionalization (ICM) and linear model of coregionalization (LCM) are compared to capture the underlying correlations. We consider three micro-CT images of Bentheimer sandstone from two different blocks imaged at different resolutions, following different segmentation pathways, and demonstrate our approach for the case of low and high task similarity. In both scenarios the multi-task ISW finds lower fitting residuals and uses only one-third to one-half of the function evaluations required by the single-task ISW. The scalability of the multi-task ISW is demonstrated by transferring knowledge of two completed optimization tasks to a third task, which outperforms the single-task ISW, with ICM showing faster convergence. The observed 4% difference for the values identified for samples from the same block and around 28% difference across blocks indicates significant spatial variability in surface relaxivity of the main mineral component, while effective clay parameters show a significantly higher variability.