Effect of Inhomogeneous Surface Relaxivity, Pore Geometry and Internal Field Gradient on Nmr Logging: Exact and Perturbative Theories and Numerical Investigations

Nuclear magnetic resonance is widely used as a probe of pore geometry and fluid composition in well logging. One of the critical assumptions often made is that the diffusion of fluid molecules is sufficiently fast to warrant the condition for direct mapping between the surfaceenhanced relaxation rate and the pore geometry. In pores satisfying such a condition, but having a significant spatial variation of surface relaxivity (ρ), one can show that the one-to-one mapping may break down. The degree to which the NMR logging interpretation is affected has not been systematically studied until now. Extra relaxation due to diffusion in internal field gradient is another example where spatially varying relaxation strength may obscure the direct relationship. Better understanding of their interplay may be exploited to our advantage. In this work, we theoretically investigate the interplay between the pore geometry, internal field and the inhomogeneous surface relaxivity. We develop a perturbative framework and compare its results with exact solutions obtained for a class of ρ textures in a pore with simple geometry. Its effect is quantified for a wide range of diffusivity and/or ρ strength. The result allows us to set the bounds for the change in the final slope of the relaxation curve and may serve as a useful guide for logging applications in real rocks with a wide range of pore sizes and fluid diffusivity. We further employ large scale numerical simulations to perform virtual experiments on more complex situations. Internal field and its gradient distributions were obtained and analyzed for up to 1.5cm based on tomograms of carbonate rocks. We find that the texture of ρ based on the internal field gradient induces a small, but observable shift, compared to a random noise generated texture for which no shift is observed. PORE GEOMETRY, δρ AND NMR LOGGING There exist potential pitfalls in the way NMR logs are interpreted (Kleinberg,1996). While it is widely agreed that the method is robust for simple types of porous media, key assumptions for its successful application may become compromised progressively as their geometrical and lithological properties become complex. To be precise, there are three necessary assumptions for the simple mapping between the so-called T2and the pore sizedistributions to work: (1) The pores are practically isolated or periodic so that diffusive coupling(Cohen,1982; de Gennes,1982; Zielinski,2002) among the pores may be neglected. (2) Within each pore, the so-called fast diffusion criterion is satisfied so that the relaxation is controlled by the weak surface relaxation strength, which will be represented as ρ (ρ0, if uniform) from now on, rather than by the diffusive flux. The condition is given in terms of the control parameter as κ ≡ ρ0L/D 1 assuming a cubic pore of volume V = L and diffusivity D of the fluid.(Brownstein,1979) (3) The mapping is based on the assumption, often made without any quantitative justification for a given rock, that ρ is uniform across the pore-grain interface. Extensive investigations were made on the first and the second(McCall,1991; Wilkinson,1991; Bergman,1995; Ryu,2001; Ryu,2009a; Zielinski,2002; Grebenkov,2007) issues, but the third has received relatively scant attention(Ryu,2008; Ryu,2009b; Ryu,2009a; Arns,2006; Valfouskaya,2006). Unless one incorporates all these issues on an equal footing, it becomes difficult to gauge uncertainty in an NMR log interpretation. The aim of this paper is to investigate systematically the nature of their violation and seek quantitative bounds for their experimental signatures, should they occur. This is done by first considering a simple pore with a class of ρ(r) textures which allows exact solutions for nontrivial cases. For a realistic pore geometry, we derive the pore geometry from 3D tomograms(Sheppard,2004) and run random-walk simulations (Ryu,2009b) under a set of controlled spatial profiles of ρ. Our focus is on gaining better understanding of whether and how the intertwined ρ texture and the pore geometry would make the issues acute or negligible. Careful experimental characterization of ρ is an invaluable component toward ultimately 1 ar X iv :0 90 6. 53 27 v1 [ co nd -m at .o th er ] 2 9 Ju n 20 09 SPWLA 50 Annual Logging Symposium, June 21-24, 2009 increasing the utility of NMR logging for challenging environments and such effort is emerging.(Keating,2007) Our theory makes quantitative analyses possible in such investigations as well as guiding log interpretation when such information is unavailable. The evolution of the polarized proton spin density may be analyzed effectively using the eigenmode analysis of the underlying Helmholtz problem. (Brownstein,1979; Ryu,2001; Grebenkov,2007) For continuity, we will employ the notational convention used in our recent paper. (Ryu,2009a) Within the framework, the evolution of the total polarization M(t) is viewed as a superposition of the eigenmodes {φp(r)}, (p = 0, 1, . . .) each with relaxation rate λp. For κ 1, the slowest mode with p = 0 dominates; viewing a porous medium as an ensemble of pores with a distribution P (λ0) of λ0 values, one arrives at the backbone of current NMR log interpretation. Even though validity of the first two assumptions depends critically on the strength of ρ0, its value, however, is often unknown. A popular method to estimate the ρ0 strength for a given porous medium is based on the assumption that the magnetization decays exponentially with its rate λ0 ∼ ρ0S/V . From an NMR probe-derived T2 (∼ 1/λ0) distribution, one may obtain an average < λ0 >, and combined with the < S/V > value (independently measured such as from BET(Kleinberg,1999; Keating,2007)), the estimated ρ0 strength, ρ̃0, may be obtained: ρ̃0 ∼< λ0 > / < S/V >. To illustrate a potential pitfall in this approach, consider Figure 1 which shows how actual λ0 varies as ρ0 (and therefore κ) increases in a simple cubic pore of size V = L. While the linear relationship λ0 ∝ ρ0 holds for κ 1 as indicated by the borken line, the curve bends with an upper bound ≤ λ∞ ≡ D π 2 L2 . This means that the apparent strength of ρ0 as estimated by the above method, will then be upperbounded,