Initially linear echo-spacing dependence of 1/T2 measurements in many porous media with pore-scale inhomogeneous fields

For a liquid sample with unrestricted diffusion in a constant magnetic field gradient g, the increase R in R2 = 1/T2 for CPMG measurements is 1/3(τγg)2D, 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 R2 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, R2 is often plotted against τ2, with the effective gradient determined from the slope of some of the early points. In many cases, a replot of R2 against τ instead of τ2 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 ±1/2χB0, 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 = 1/3χνa2/D, where ν is the frequency, R2 does not depend much on pore size a nor on diffusion constant D. Examples are shown where plots of R2 vs τ show better linear fits to the data for small τ values than do plots vs τ2. The present work shows that, if both grain-scale and sample-scale gradients are present for samples with narrow ranges of T2, 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 T2 distributions and hence wide ranges of ξ. For some of these wide distributions we have plotted R2 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 T2 values for allτ values. For the examples presented, both the intercepts and the initial slopes of the plots of R2 vs τ increase systematically with signal percentile, starting at smallest R2.