Quantification of Multi-Phase Fluid Saturations in Complex Pore Geometries From Simulations of Nuclear Magnetic Resonance Measurements

We develop a numerical algorithm to simulate nuclear magnetic resonance (NMR) measurements in the presence of constant magnetic field gradients. The algorithm is based on Monte Carlo conditional random walks in restricted and unrestricted space. Simulations can be performed of threedimensional (3D) porous media that include both arbitrary bimodal pore distributions and multi-phase fluid saturations. The ability to account for the presence of a constant external magnetic field gradient allows us to replicate actual well logging conditions that include the effect of CMPG pulse sequences at a microscopic level. This is accomplished by simulating pulse acquisition techniques that include multiple inter-echo times (TE) similar to those currently used by the well-logging industry. Benchmark examples are presented to validate the accuracy and internal consistency of our algorithm against previously published results for the case of a null magnetic field gradient. Validation examples are also presented against actual NMR measurements performed on core samples of carbonate rock formations. Interpretation work is focused to the petrophysical assessment of both partial oil/water saturations and pore structures exhibiting hydraulic coupling. Simulation examples are designed to quantify whether the inclusion of diffusion under a magnetic field gradient can improve the interpretation of multi-phase fluid saturations when hydraulic coupling is significant. The simulation algorithm sheds light to new NMR data acquisition strategies that could be used to improve the detection and quantification of (a) fluid types, (b) complex fluid saturations, and (c) complex pore geometries. Introduction Presence of hydraulic (or diffusion) pore coupling in carbonate rocks (mainly grainstones) challenges conventional NMR interpretation techniques. Although pore coupling phenomena are commonplace in the majority of rock pore systems, they become relevant to assess NMR measurements when the following three conditions are met: (a) microporosity regions are present within the grains, (b) micro-pores are hydraulically well connected (not cemented) to outer macro-porous regions exhibiting low surface-to-volume ratios, and (c) rock surface relaxivity is low enough to prevent decay of proton magnetization within the macro-porosity before protons can enter the micro-porous regions. In addition, fluid diffusivity must be sufficiently large in order for hydraulic coupling to be significant within the time scale of NMR measurements. This is usually the case only for water and light hydrocarbons. As a result, fluid magnetization will be exchanged between microand macro-pore regions, and no obvious relationship will exist between NMR transverse relaxation (T2) distribution and pore-size distribution. Figure 1 illustrates conditions (a) and (b) described above with an example of scanning electron microscope (SEM) images of a carbonate rock exhibiting hydraulic coupling. Varying inter-echo times (TE) is a common NMR data acquisition technique used for in-situ reservoir fluid identification. However, in the case of NMR measurements performed in carbonate rocks, there exist no published reports dealing with the impact of diffusion coupling on hydrocarbon typing and quantification using multiple-TE logging techniques. The objectives of this paper are twofold: (a) to develop a simulation algorithm capable of reproducing NMR measurements in complex pore geometries under a variety of experimental conditions and, in particular, under the influence of a constant magnetic field gradient, and (b) to provide simulation examples that will help assess the validity of fluid phase discrimination using multi-TE measurements in porous media exhibiting hydraulic coupling. In the past, numerical models of NMR decay have been proposed based on periodic bimodal packs of spheres that accounted for surface relaxation effects. We have reproduced and extended Ramakrishnan et al.’s Monte Carlo algorithm to account for the effect of an external constant magnetic field SPE 77399 Quantification of Multi-Phase Fluid Saturations in Complex Pore Geometries From Simulations of Nuclear Magnetic Resonance Measurements E. Toumelin, SPE, C. Torres-Verdín, SPE, The University of Texas at Austin, and S. Chen, SPE, Baker Atlas 2 E. TOUMELIN, C. TORRES-VERDIN, AND S. CHEN SPE 77399 gradient of the type enforced by modern NMR tools, and in the presence of a non-wetting phase. The first part of this paper introduces the Monte Carlo simulation algorithm applicable to a bimodal pack of spheres in the presence of a constant magnetic field gradient. A subsequent section describes examples of numerical simulation that illustrate the versatility of the algorithm. Finally, we derive and interpret simulation examples intended to address specific issues of fluid discrimination using multiple inter-echo times in the presence of diffusion coupling. We address four specific cases of NMR T2 distributions (two unimodal and two bimodal) by modeling several possible combinations of pore structure, hydraulic coupling, and fluid distribution. These case studies were designed such that different pore configurations created identical NMR signal at low values of TE, but did exhibit differences at high values of TE. Model for the Simulation of NMR Decay The algorithm developed to simulate numerically NMR magnetization decay in carbonate rocks makes use of conditional Monte Carlo random walks. It is based on the algorithm presented by Ramakrishnan et al., further generalized to include a microscopic description of diffusivity effects in the presence of a constant magnetic field gradient. The porous medium can either include micro-and macroporous regions (grainstone model) to form a bimodal pore distribution, or can exhibit a single pore size with solid grains (wackestone model). Model geometry. The simulation model was constructed with a three-dimensional (3D) bimodal pack of spheres that replicates the complex pore-structure geometry encountered in some carbonate formations, namely micro-porous grainstones, where the solid micro-grains are arranged into packs. As in the example shown in Fig. 1, the model adopted for the simulations accounts for a micro-porosity region immediately surrounding the grains, while different types of macroporosities (characterized by different sizes and degrees of cementation, depending on diagenesis) exist between the packs. The geometrical construction of the 3D model of a bimodal pack of spheres is based on the following assumptions: Bimodal pore-size distribution, Periodic geometry, Isotropic pores, Spherical, compacted micro-grains, and Spherical, compacted grain packs. At the micro(respectively, macro-) scale, each grain (respectively, grain pack) of the model is defined as a sphere inscribed into a concentric cubic cell. If the sphere does not completely fill its cubic unit, then the complementary volume is considered filled with fluids. Furthermore, fixed blobs centered in the macro-pore space can be included in the model to represent partial saturations of immiscible non-wetting oil phase, while the micro-porosity can only be filled with irreducible water. Figure 2 shows a 3D period of the assumed bimodal pore structure. A large variety of pore sizes can be modeled by adjusting the values of the grain (or grain pack) radii, and the sizes of the corresponding cubic cells. Unimodal pore-size distributions can be constructed with the same basic model by allowing no micro-porosity. The NMR response of uncoupled bimodal pore structures, i.e., porous media where no hydraulic coupling occurs, can be simulated by superimposing the response of each isolated pore mode. Dual random-walk strategy. At each iteration of the NMR numerical simulation algorithm, a fictitious proton is randomly chosen in the available pore space. This proton is thereafter displaced to a new location yielded by a continuous random walk operating according to two possible strategies. Depending on the proximity of the proton to the surface of material discontinuity (pore wall or fluid phase interface), either a conventional random-walk strategy, or Zheng and Chiew’s First-Passage-Time technique can be implemented to displace the proton to its new location. The First-Passage-Time strategy was developed to allow macroscopic jumps in the bulk pore space and hence to speed up the otherwise prohibitively long random-walk process. It is based on the analytical solution of the unbounded diffusivity equation, which yields a stochastic relationship between the mean square displacement, < R 2 >, during a given step, and its duration, ∆t, with a maximum of probability close to the free space bulk diffusivity, i.e.,