SPWLA 47 th Annual Logging Symposium, June 4-7, 2006 Prediction of permeability from NMR response: surface relaxivity heterogeneity C. H. Arns 1,* , A. P. Sheppard 1 , M. Saadatfar 1 , and M. A. Knackstedt 1,2 1 Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Australian National University, Canberra, Australia 2 School of Petroleum Engineering, University of New South Wales, Sydney, Australia * Corresponding Author: christoph.arns@anu.edu.au Copyright 2006, held jointly by the Society of Petrophysicists and Well Log An- alysts (SPWLA) and the submitting authors. This paper was prepared for presentation at the SPWLA 47 th Annual Logging Symposium held in Veracruz, Mexico, June 4-7, 2006. ABSTRACT NMR responses are commonly used in reservoir char- acterization to estimate pore-size information, formation permeability, as well as fluid content and type. Diffi- culties arise in the interpretation of NMR response as an estimator of permeability due to internal gradients, diffusion coupling, surface-relaxivity heterogeneity, and a possible breakdown of correlations between pore and constriction sizes. Here we consider several scenarios of surface relaxivity heterogeneity for a set of sandstones and a set of carbonate rock in a numerical NMR study based on Xray micro-CT data. We have previously demonstrated the ability to image, visualize, and characterize sedimentary rock in three di- mensions (3D) at the pore/grain scale via X-ray micro- computed tomography. We also numerically tested the influence of structure and diffusion-coupling on NMR- permeability correlations. Here we consider surface re- laxivity heterogeneities due to pore partitioning, miner- alogy, pore size, and saturation history. We partition the pore space and solid phase into regions of pores and grains. These partitions could reflect differ- ent mineralogy for weakly coupled pore systems, or dif- ferences in mineralogy for the grains. Further, we use a morphological drainage simulation technique to partition the pore space in terms of invasion radius or throat size, reflecting surface relaxivity heterogeneity due to the sat- uration history of immiscible fluids, which could cause e.g. pressure dependent adsorption on surfaces and/or changes in wettability. Finally, we use the concept of covering radius to assign a surface relaxivity due to pore size. For each sample, four sandstones and three carbonates, we consider distributions of surface relaxivity based on above partitions, keeping the mean surface relaxivity con- stant, simulate the magnetisation decay, and derive a pore size distribution through an inverse Laplace transform as- suming constant surface relaxivity. Further, we test the effect of these heterogeneities on NMR-permeability correlations based on the log-mean of the relaxation time distributions for two frequently used empirical NMR-permeability cross-correlations. At the scales probed here, surface relaxivity heterogeneity changes the prefactor in the equations for sandstones only minimally, while the prefactor is changes orders of mag- nitudes for carbonates. The influence of surface relaxiv- ity heterogeneity on the quality of the fit for individual samples is small. INTRODUCTION The estimation of permeability through cross-correlations from other physical measurements on rocks is a classical task in petrophysics and has a long history in well log- ging. Of the measurements available, the NMR relax- ation is the one, which typically correlates best to per- meability (e.g. (Sen et al., 1990)). One reason is that the estimation of permeability requires length scale in- formation, which the NMR relaxation response provides, since the relaxation process is typically controlled by the surface to volume ratio of the pore space (Wayne and Cotts, 1966; Brownstein and Tarr, 1979; Kenyon et al., 1986; Kenyon et al., 1988; Kenyon, 1992; Song et al., 2000). Under the assumption of constant surface relax- ivity, weak coupling between pores, and fast diffusion within pores, the magnetisation decays uniformly within each pore and the decay can be written as M (t)= M 0 (t) N  p=1 a p exp  − t T 2p  , (1) where M 0 is the initial magnetization, p is a pore label, a i is the fractional pore volume, t is time, and the transverse relaxation time T 2 of the pores is given by 1 T 2p = 1 T 2b + ρ S p V p . (2) 1 GG