Improved Interpretation of Nuclear Magnetic Resonance T1 and T2 Distributions for Permeability Prediction: Simulation of Diffusion Coupling for a Fractal Cluster of Pores
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NMR relaxometry is a powerful tool for inferring porosity and permeability data. In practice, measured magnetization decay curves are inverted for relaxation time distributions. Subsequently, one presumes a linear relationship between the pore radius distribution and the T1 and T2 distribution, for longitudinal and transverse magnetization, respectively. The fundamental equations used are based on a pore model, in which pores are assumed to be isolated from each other with respect to the NMR process and have smooth walls. The present study is based on a geometrical pore space model with connected pores and structured pore walls. The physical processes of surface relaxation, irreversible dephasing of magnetic spins and diffusive proton exchange between pores, are described by a system of differential equations. The solution yields a set of exponential functions representing the relaxation time distribution. We describe the difference between the distributions obtained for diffusion coupling and for isolated pores. With diffusion coupling on, the spectral width of the T1 distribution is strongly reduced, which indicates that the influence of large and small radii according to the T1-pore radius relationship is mixed to some extent. For a fractal pore space structure, where large pores are surrounded by adjacent minor pores, the T1 distribution does not resolve these substructures. Nevertheless, permeability values calculated from the logarithmic mean relaxation time T1,LM are quite the same for diffusion coupling and for isolated pores. The T2 distribution for diffusion coupling is little constricted and gives a better resolution of the pore wall structures than the corresponding T1 distribution. The permeability values from T2 distributions agree with the values from longitudinal magnetization, provided that we use a corrected relaxation time T2,corr, accounting for the dependence of the surface relaxivity ρ2 on pore radius. The study shows that radius distributions calculated from a T1 and from a T2 distribution differ from one another and both present an altered image of the true pore radius distribution. In practice, this has no serious influence on estimating permeability of medium- to high-permeability sandstones with the currently applied methods. The presented methodology of calculating the NMR response of pore space models with diffusion coupling may facilitate understanding porosity-permeability relationships of different rock types such as carbonate rocks with micro-porosity.
KeywordsNMR relaxation diffusion pore radius distribution permeability fractals
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