Numerical Methods and Bayesian Inference for Nuclear Magnetic Resonance (NMR)

Abstract

Within the petrophysical community, there has always been a strong interest in producing accurate estimates of how much oil is contained in a particular reservoir and how easy it is to extract it. To do that, few logging samples are first taken from the selected reservoir. The estimates for the full reservoir are then based on the analysis of the logging samples. How easy it is to extract the oil is related on the internal porosity structure of the rock: the larger the pores, the easier it is to extract the oil.

In order to recover this porosity structure, petrophysicists make use of Nuclear Magnetic Resonance (NMR) decay measurements. With the help of a physical model, petrophysicists have always applied Tikhonov regularization to solve this problem. Within this thesis we explore some new advancements in the Tikhonov regularization algorithms. By proposing Krylov subspace methods alternatives to the currently employed algorithm, we shown how the computational limit on the pixel resolution of the solution can be finally surpassed. However, although results are of direct interpretation, this technique still lacks information about the uncertainty of the solution identified.

To surpass this issue, within this thesis we reformulate this problem in the Bayesian framework as well. By introducing a new custom MCMC sampler for the NMR-like problem, we provided a full posterior exploration and uncertainty quantification analysis of the retrieved solution. This result has been further extended to the related quantities of interest. We enriched the robustness to this result with a careful analysis of the convergence by employing the uniform and ergodicity theory setting. Finally, we shown how this Bayesian structure con be further employed to provide a proper statistical comparison between the physical assumptions underlying the choice of the solution, i.e., continuous vs atomic.

In Section 2.1 we propose, and numerically explore, a new 1D Tikhonov algorithm. In Section 2.2, starting from the 1D algorithm, we introduce, and numerically explore, two new algorithms for the 2D problem setting. A comparison with the currently employed VSH algorithm is then provided in the numerical simulations paragraphs.

From the Bayesian point of view, in sections 3.2.1-3.2.7 we introduce step by step a new MCMC algorithm designed to deal with the various complexities of the 1D problem. In Section 3.3 we show some results on the uniform and geometric ergodicity of the algorithm. Within these results, we show that if a technical condition is satisfied, then convergence of the algorithm can be achieved with a geometric rate. In Section 3.4 we highlight the performances of this algorithm on real rock dataset. Finally, a comparison between continuous and atomic solution assumptions is provided in Section 3.4.4.

This work has been presented at the Numerical Analysis seminar organised by the Department of Mathematics of the University of Bath (2022), and in a poster at the PhD students connection day organised by Schlumberger at its site in Cambridge (2019). Further planning includes to present this work to the reference conference on the topic, the Magnetic Resonance in Porous Media conference (Tromso 2024). Two separate publications, one for the Tikhonov part and one for the Bayesian one, are also planned although the appropriate journal for the topic has not been identified yet.

Date of Award	21 Feb 2024
Original language	English
Awarding Institution	University of Bath
Supervisor	Tony Shardlow (Supervisor) & Silvia Gazzola (Supervisor)