A Bayesian Optimization Approach to the Simultaneous Extraction of Intrinsic Physical Parameters fromT1 andT2 Relaxation Responses

Abstract

Summary 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.

A Bayesian optimization approach to the extraction of intrinsic physical parameters from T2 relaxation responses

“…As adapted from DT-ISW introduced in [18], CO-OPT is composed of five components: (a) observables, (b) NMR forward solver, (c) cost-function, (d) Bayesian optimization, and (e) solution analysis. Fig.…”

“…The advantage of Bayesian optimization has been leveraged to recover physical, geometrical, or structural parameters where expensive cost functions are involved [15][16][17], and in particular, complex multi-physics, multi-parameter, multi-modal problem arising in NMR relaxometry [11]. Furthermore, Bayesian optimization has been integrated with transfer learning for multi-objective optimization, and in particular, for by simultaneous optimization of 𝑇 and 𝑇 distributions [18]. In both studies, a multi-modal search strategy, comprising a multi-start L-BFGS-B optimizer searching for local optimum solutions, and a global optimizer social-learning particle swarm optimizer (SL-PSO) for global optimum solutions, are applied to recover all (major) local optimal solutions, i.e., potentially identifying multiple physically valid solution sets.…”

“…In this study, we adapt the dual-task inverse solution workflow (DT-ISW) developed for simultaneous fitting of 𝑇 and 𝑇 distributions in [18] to the context of simultaneous optimization of 𝑇 distributions associated with various values of temperature, referred to as concurrent optimization (CO-OPT). We demonstrate the performance of CO-OPT on a Bentheimer sandstone sample by the identification of five intrinsic parameters controlling the NMR relaxation response: one shared quantity, i.e., the tortuosity in clay regions, together with four unique quantities, i.e., transverse surface relaxivity of quartz and effective transverse relaxation time in clay regions, at two values of temperature 𝜏 and 𝜏 .…”

Bayesian Optimization With Transfer Learning: A Study on Spatial Variability of Rock Properties Using NMR Relaxometry

“…Finally, decrease the λ from a higher value with strong smoothing effect but low spectral resolution to a lower value with weak smoothing but high spectra resolution until the misfit between T 2 distributions inverted from high and low SNR reach a certain threshold. The procedure was proposed for inversion of Bentheimer sandstone T 2 magnetisation decays at SNR of [50,400] (Li et al., 2021), and was then applied to inversion of both T 1 and T 2 data at SNR = 100 (Li et al., 2022a).…”

“…Raw and processed data for the complete optimization process are available as supporting information as Figures – and Tables –. The numerical methods for the calculation of the NMR responses and the respective algorithms for the more basic aspects of the inverse solution workflow are introduced in detail in references (Li et al., 2021, 2022a). The NMR responses were calculated with an in‐house software, namely the NMR package of morphy, version 1.9.40.…”