Towards a better understanding of nuclear magnetic resonance (NMR) measurements on well logs and rock cores - - final report

Object:

To extend and continue the finished project called 'Improved Characterization of Rock Cores Obtained by a Combined Use of Multivariate Analysis Techniques and Magnetic Resonance (NMR)’ and to explore new ideas connected to NMR. One major goal is to extract useful information from NMR logging data. The signal-to-noise ratio in such data is poor and makes data handling and algorithm design a challenge.

Another major goal is to extract information on rock permeability from NMR measurements. Permeability has so far been hard to predict. A possible strategy is to vary the inter-echo spacing. This may shed light on pore geometry. Another possible strategy is to combine different types of NMR experiments, such as T₁/T₂ measurements and PFG-NMR. If new algorithms and data handeling routines turn out to be successful, then user friendly implementations should result.

Status:

This project was a continuation of the project 'Improved characterization of Rock Cores obtained by a combined use of Multivariate analysis techniques and Magnetic Resonance (NMR)', which was lasted from 2005-10-14 to 2007-10-13. The fist task undertaken in the combined four year period was the implmentation of Kaufmann's algorithm for fitting a sum of first order exponentials to measured data. This unstable and tricky numerical problem is central to the analysis of NMR data. Then, Kaufmann's algorithm was used as the starting point for a Newton-Raphson algorithm which fits second order exponentials. The gradient and the Hessian matrix are computed using analytical expressions. The resulting MATLAB implementation is called ANAHESS.

The number of components in the second order fit is determined using the Bayesian information criterion. During the testing and implementation of ANAHESS, extensive comparisons were done with a widely used inverse Laplace transform computer program which uses a Tikhonov regularization algorithm. We used the earth mover's distance metric in order to compare the results from computations on data having added artificial noise. Our tests show that ANAHESS compares fovourably, producing simpler models with as good or better solution stability in the presence of noise. Parts of this work were presented at international conferences [1,2,3].

This work has resulted in the article 'ANAHESS, a new second order sum of exponentials fit algorithm, compared to the Tikhonov regularization approach, with NMR applications'. The article is currently a draft, very soon ready for submission. Related to this was the contribution to 'Separation of two dimensional diffusion and relaxation time distributions from oil/fat and moisture in food', a presentation by Dr. Geir Sørland at the 8th International Conference on the Applications of Magnetic Resonance in Food Science, Nottingham, UK, july 19, 2006, and a subsequent publication [4].

Another major constituent of this project has been the use of multivariate methods in order to model various properties of rock cores. This was done on various data sets containing sand stone cores and calcite cores, and wettability, porosity and rock type are modelled well using such multivariate models. Permeability proved, as expected, to be more difficult. Studies did however show a relationship between signal sensitivity to varying interecho spacing and permeability.

Kaufmann's algorithm was also applied to oil well NMR logging data. However, the signal-to-noise ratio in the presently available data was so poor that good conclusions were hard to discern. Also, an inverse Laplace transfor algorith based on stretched exponentials was designed and implemented. The resulting implementaton proved so unstable and sensitive to noise that further extensions of this work were abandoned.

At the end of this project, much of the MATLAB computer code has been annotated, cleaned up and made available to all the involved parties. This project hass been completed with the close collaboration with Dr. Geir Sørland, Anvendt Teknologi AS.

Publications: