Abstract: For a liquid sample with unrestricted diffusion in a constant magnetic field gradient g, the increase R in R 2 =1/T 2 for CPMG measurements is 1/3(τγg)2 D, where γ is magnetogyric ratio, τ is the half the echo spacing TE, and D is the diffusion constant. For measurements on samples of porous media with pore fluids and without externally applied gradients there may still be significant pore-scale local inhomogeneous fields due to susceptibility differences, whose contributions to R 2 depend on τ. Here, diffusion is not unrestricted nor is the field gradient constant. One class of approaches to this problem is to use an “effective gradient” or some kind of average gradient. Then, R 2 is often plotted against τ 2, with the effective gradient determined from the slope of some of the early points. In many cases, a replot of R 2 against τ instead of τ 2 shows a substantial straight-line interval, often including the earliest available points. In earlier work [G.C. Borgia, R.J.S. Brown, P. Fantazzini, Phys. Rev. E 51 (1995) 2104; R.J.S. Brown, P. Fantazzini, Phys. Rev. B 47 (1993) 14823] these features were noted, and attention was called to the fact that very large changes in field and gradient are likely for a small part of the pore fluid over distances very much smaller than pore dimensions. A truncated Cauchy–Lorentz (C–L) distribution of local fields in the pore space was used to explain observations, giving reduced effects of diffusion because of the averaging properties of the C–L distribution, the truncation being at approximately ±1/2χB 0, where χ is the susceptibility difference. It was also noted that, when there is a narrow range of pore size a, over a range of about 40 of the parameter ξ =1/3χνa 2/D, where ν is the frequency, R 2 does not depend much on pore size a nor on diffusion constant D. Examples are shown where plots of R 2 vs τ show better linear fits to the data for small τ values than do plots vs τ 2. The present work shows that, if both grain-scale and sample-scale gradients are present for samples with narrow ranges of T 2, it may be possible to identify the separate effects with the linear and quadratic coefficients in a second-order polynomial fit to the early data points. Of course, many porous media have wide pore size and T 2 distributions and hence wide ranges of ξ. For some of these wide distributions we have plotted R 2 vs τ for signal percentiles, normalized to total signal for shortest τ, again showing initially linear τ-dependence even when available data do not cover the longest and/or shortest T 2 values for allτ values. For the examples presented, both the intercepts and the initial slopes of the plots of R 2 vs τ increase systematically with signal percentile, starting at smallest R 2. [Copyright &y& Elsevier]