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ABSTRACT Nuclear Magnetic Resonance (NMR) is the only logging technique available to estimate pore-size distributions. However, quantitative interpretation of NMR data can become uncertain in carbonate rocks because of unaccounted diffusive coupling between existing pore scales. The objective of this study is to assess the relative importance of diffusive coupling and temperature using practical examples encountered in the interpretation of NMR data.

The core of our work is based on the analysis of experimental NMR measurements and their comparison with numerical simulation results. Our numerical simulation algorithm consists of Monte-Carlo random walks in three dimensions and was specifically designed to account for two-phase fluid saturations in the presence of a bimodal pore-size distribution. We present numerical simulation results that reproduce MRIL experimental data acquired in carbonate rocks exhibiting bimodal pore-size distributions and two-phase fluid saturations. Simple interpretation models are derived to include NMR coupling effects by way of cross-interactions between the fluids borne within the different types of pores. Such models have been subsequently used to assess the importance of diffusion coupling. Experimental data acquired from rock core samples was also used to assess the influence of temperature on the estimation of movable fluids volumes otherwise determined assuming a constant formation T2cutoff . It is shown that temperature has moderate effect on T2 distribution, T2cutoff, and BVI determination. The diffusive coupling effect is more significant on mapping T2 distribution to pore size distribution than on the determination of BVI. INTRODUCTION Quantifying pore structures of carbonate rocks is known to be much more challenging than in the case of sandstones. The genesis of carbonate rocks is commonly associated with both chemical and biological processes that can cause large variations

† Now with Baker Hughes Inteq, Celle, Germany.

of pore size, shape, and texture, as well as of the degree of dissolution/cementation. Carbonate rocks also exh ibit a large degree of heterogeneity in the form of (a) partially isolated pores and (b) presence of both intra- and inter-connected pores. Petrographic or standard core analysis techniques often face serious technical difficulties to determine whether an existing “vug” is hydraulically isolated or not from the rest of the pore structure. Similarly, it is difficult to isolate and/or to identify in the laboratory whether secondary, inter-connected micropores provide significant hydraulic connectivity to the overall pore structure. The detection and quantification of such structural properties and of their impact on pore connectivity and fluid transport is central to the characterization of carbonate reservoir quality and producibility.

As the only logging technique currently available to characterize pore size distributions, Nuclear Magnetic Resonance (NMR) logging has been used to assess a handful of key petrophysical parameters, such as irreducible bulk volume (BVI), clay-bound water volume (CBW), irreducible bulk movable bulk volume (BVM), effective (φe) and total porosity (φT), and permeability index (k). All of these parameters are related to pore structure. Thus, an understanding of how NMR measurements respond to variations of pore structure is essential to quantify petrophysical formation properties.

Most of the currently available NMR interpretation models and parameters thereof, such as T2cutoff, spectral BVI, CBW, Timur-Coates permeability index, etc., are based on the concept of pore-size distributions. The latter concept is most suitable in cases where a deterministic relationship exists between pore size and throat size. Although these models have been applied successfully in interpreting sandstones, and to some extent in carbonate formations, modifications are sometimes required to accommodate for local geological and petrophysical knowledge. Some modifications are of empirical nature (Quintero et al., 1999; Allen et al., 2001), while others stem from more fundamental approaches (Ramakrishnan et al., 1999; Godefroy et al., 2001).

Analysis of NMR Diffusion Coupling Effects in Two-Phase Carbonate Rocks: Comparison of Measurements with Monte Carlo Simulations

E. Toumelin, C. Torres-Verdín, The University of Texas at Austin S. Chen, and D. M. Fischer†, Baker Atlas

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METHODOLOGY Even though large variations are expected in

the spatial distribution of pores in carbonates, it is important to quantify their effect on petrophysical properties, and on the assessment of how significantly such variations may affect log interpretation. The main thrust of this paper is to shed light to the dependence of NMR measurements performed on carbonate rocks on temperature and diffusive coupling. We carry out this analysis by comparing NMR measurements performed on rock core samples with numerical simulations that take into account diffusive coupling and temperature effects. First, we briefly describe the core samples used in our study from the viewpoint of their pore structure and of their propensity to exhibit diffusive coupling. Subsequently, we describe the NMR experimental protocol used to obtain core measurements at different temperatures and fluid saturations. We show how both the NMR measurements and the petrographic analysis can be combined to synthesize a parametric numerical model of bimodal porosity suitable for the description of the pore complexity exhibited by the core samples. The comparison between numerical simulation results and experimental measurements is intended to explore the role played by diffusive coupling in relation with the influence of temperature and a specific BVI cut-off. Lastly, cumulative porosity curves derived from both NMR experiments and numerical simulations are used to assess the discrepancy of BVI and permeability index values otherwise obtained by making use of a T2cutoff independent from temperature and diffusion coupling. We conclude by discussing the need to correct T2cutoff values in log analysis when temperature and diffusion coupling effects are significant.

PORE STRUCTURE OF THE SAMPLES The three samples considered in this paper are compacted, micritized limestones retrieved from the same well but exhibiting different facies. Backscatter electron (BSE) imaging was used to analyze the pore structure of these samples and to measure pore-size dis tributions and absolute porosities φBSE for pores larger than 2 microns. Scattering electron microscope (SEM) was also used to provide high-resolution spatial images of the pore-matrix structure. A complementary laboratory core-plug measurement was performed to assess total porosity φT, which in turn allowed

the quantification of micro-porosity otherwise unaccounted by the BSE analysis, i.e.,

BSETmicro φφφ −= . Pore geometry analysis helped to identify

several pore types within the samples, namely: § Type 1: Large primary inter-connected pores

with average diameters larger than 60 µm. Pores exhibit variable cementation, thereby providing partial hydraulic coupling with the surrounding micro-porosity region.

§ Type 2: Large secondary dissolution pores, with average diameters within the range of 20-60 µm. These pores are largely hydraulically isolated and hence remain uncoupled to other existing pore types.

§ Type 3: These are pores developed between circumgranular cement crystals ranging in diameter from 5 to 15 µm. Due to their proximity with smaller micro-porosity regions, they can be regarded as coupled with the latter.

§ Type 4: Pores developed from residual matrix bearing irreducible water saturation. These pores constitute the micro-porosity regions of the smallest pore size associated with φmicro . Porosities and average radii exhibited by pore

types 1 through 3 were quantified using the pore-size distribution yielded by the BSE analysis. In terms of pore-size distributions, the main features of the samples are as follows: § Sample A mostly exhibits a combination of

small-size (types 3 and 4) hydraulically coupled porosities amounting to 13.6% of the bulk volume, with 10% vuggy (type 2), isolated macro -porosity. An example of this pore structure is provided by the SEM images shown in Figs. 1a and 1 b, where one can easily recognize abundant micro-porosity as well as a few isolated vugs. A low water permeability of 4 mD was measured in this sample.

§ Sample B mainly consists of 8.1% primary macro-porosity (type 1) partially hydraulically coupled with 5.1% micro-porosity (type 4), in addition to 5% isolated secondary macro-porosity (type 2). The SEM images of Figs. 2a and 2b clearly show the dual porosity exhibited by this limestone sample. The same figures show a remarkable structural organization of matrix grains into packs intrinsically micro-porous. The permeability of sample A to water was measured at 70 mD.

§ Sample C exhibits the highest porosity among all the samples reported in this paper. The major part of the porosity is associated with primary macro-porosity regions of type 1 (17%

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porosity), while the remaining 4% porosity is evenly shared by the other three pore types. Figs. 3a and 3b are SEM images of the macro-porous structure exhibited by sample C, also evidencing very good cementation along the macro-pore surface. The highly connected macro-pore structure justifies a high measured water permeability equal to 980 mD.

As illustrated in Fig. 4, the BSE-based pore-size distribution allowed us to identify pore types 1 through 3 as pore modes. The SEM images were also used to define the micro-pore size (mode 4) in the range of 0.2 to 0.4 µm. NMR EXPERIMENTS Nuclear magnetic resonance measurements of the three core samples were performed with a MARAN ULTRA system operating at a frequency of 2.3 Mhz. The samples were first measured with 100% water saturation (Sw = 1) in the temperature range between 4 and 65 ºC. Subsequently, the same samples were desaturated to their irreducible water saturation (Swir) state by means of centrifuging. All NMR measurements for the irreducible water saturated samples were acquired at regulated room temperature (22 ºC).

A CPMG sequence was used to obtain echo trains of up to 5000 echoes with time spacing, TE, of 1 ms, and in the absence of an applied magnetic field gradient. Measurements were repeated a sufficiently large number of times to ensure signal-to-noise ratios greater than 100. We made use of a discrete, multi-exponential relaxation model with a sufficiently wide T2 bin range to invert the echo trains into transverse relaxation (T2) distributions. The inversion technique employs a curvature-smoothing regularization (Chen et al., 1999) to stabilize the solutions in the presence of noise. Figure 5 shows the T2 distribution obtained from NMR measurements performed at Sw = 1 and Swir for each of the three core samples. NMR SIMULATION MODEL The algorithm used to simulate numerically the NMR response of carbonate rocks replicates conditional Monte Carlo random-walk procedure. It is based on the algorithm reported by Ramakrishnan et al. (1999), further generalized to allow the inclusion of a microscopic description of the diffusivity effect under a constant magnetic field gradient.

Model geometry The simulation model was constructed with a three-dimensional (3D) bimodal pack of spheres intended to replicate the pore-structure geometry of carbonate formations, in which grains are arranged into packs. As in the example of Fig. 2, the simulation model accounts for a micro-porosity region immediately surrounding the grains, while different types of macro -porosity (characterized by different pore sizes and degrees of cementation, depending on diagenesis) exist between the packs. The geometrical construction of the 3-D model is based on the following assumptions:

- Bimodal pore-size distribution, - Periodic geometry, - Isotropic pores, - Spherical, compacted micro-grains, - Spherical, compacted grain packs. At the micro- (macro-) scale, each grain (grain

pack) is constructed with 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 on the macro-pore space can be included into the model to represent partial saturations of immiscible non-wetting phase, while the micro-porosity can only be filled with wetting fluid. Figure 6 shows a 3D period of this 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 cell. Random-walk algorithm At each iteration of the Monte Carlo simulation procedure, a fictitious proton is randomly chosen in the space formed by the available pore volume. The same proton is thereafter displaced to a new location yielded by a random-walk algorithm that operates according 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 a First-Passage-Time technique (Zheng and Chiew, 1989) is implemented to displace the proton to its new location.

The First-Passage-Time strategy is based on the analytical solution of the unbounded diffusivity equation. This yields a stochastic relationship between the mean square displacement, R2, during each step, and its duration, ∆t, with a maximum of probability value close to the near the free space bulk diffusivity, i.e.,

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tR

Do ∆><

=6

2

. (1)

In the above relationship, R is limited to the largest displacement allowed by the surrounding fluid boundaries, while ∆t is constrained by the time sampling period TE corresponding to the echo time of the NMR tool. Since no magnetic field gradient is enforced in the simulations, the signal generated by fluid protons is simply proportional to the population of magnetized protons sampled with period TE. The First-Passage-Time method is particularly suited for the simulation of proton displacements in large pore spaces as it allows the simulation of large displacements in a few steps. However, close to surfaces of material discontinuity, the First-Passage-Time strategy does not converge and instead a shift to a conventional random walk of fixed step ε is necessary.

If contact is made with a surface of material discontinuity, then the proton magnetization decays with a probability equal to

oDp

ρε= , (2)

where ρ is the surface relaxivity of the interface (Bergman et al., 1995). If no magnetization decay occurs, the proton rebounds specularly at the surface of the interface. Once a large-enough number of proton trajectories are simulated with the Monte Carlo algorithm, convergence to the average time solution is insured and the total magnetization signal can be inverted into a T2 distribution (generally a few hundreds to a thousand proton trajectories are needed for convergence depending on the modeled pore-size distribution). Considering the total relaxation of the pore fluid

( ) 1222 /1/1 −+= BS TTT , (3)

where T2S is the surface relaxation and T2B is the bulk relaxation, we remark that the dual random-walk algorithm introduced by Ramakrishnan et al. (1999) only models the effect of T2S. Because T2B is solely a function of fluid properties, the algorithm reported in this paper can take into account bulk relaxation effects by multiplying the simulated magnetization decay times the TE-sampled temporal decay exp(–t/T2B) before performing the T2 inversion. An in-depth description of this algorithm including a number of

benchmark examples can be found in Toumelin, 2002.

ADAPTATION OF THE SIMULATION MODEL TO REPLICATE THE NMR RESPONSE OF CORE SAMPLES The purpose of the simulation work is to ascertain the influence of pore coupling on the NMR response of the available core samples. Once a quantification is made of the proportion of actively coupled pores, the main task of the simulation work consists of determining the best bimodal porosity model that honors the NMR measurements. On the hydraulic isolation of vuggy porosity Within each sample, the irreducible water saturations Swir measured by NMR is larger than the fractional micro -porosity φmicro / φT . As a consequence, it is assumed that the entire water-wet micro-porosity is filled with connate water, and that the complementary amount of irreducible water (Swir φT – φmicro) resides in other porosity regions. However, the pore analysis defines the vuggy pores (type 2) as isolated and sealed by cementation. In other words, under this assumption, no fluid can leave the vuggy macro-pores without fracturing the rock.

Analysis of the NMR experimental results described in Fig. 5 shows that, for each sample, the long-T2 peak corresponding to the water-filled vugs at Sw = 1 disappears at Swir. This phenomenon implies that the water filling the vuggy porosity was effectively removed from this region during the centrifugation process. Such an observation, in turn, disproves the initial assumption of hydraulically isolated vugs. In consequence, NMR measurements performed on the three available cores provide strong evidence that vugs are not totally isolated hydraulically from the rest of the porosity regions.

Determination of the model parameters By describing vugs as part of a coupled pore system, their NMR response can be explained by the superposition of coupled modes. We assume that the total NMR response of the porous medium is dictated by the mutual responses of the hydraulically connected porosity modes coupled two-by-two. Such an assumption is of course amenable to our numerical simulation algorithm. For instance, the macro-porosity region (mode 1 or 2) is expected to couple with the pores of type 3

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(peripheral porosity located between the cement crystals) more than with that of type 4 (intrinsic micro -porosity). On the contrary, the two macro-porosity modes corresponding to primary and secondary macro-porous regions are exp ected to exhibit no mutual interaction.

In order to make consistent use of our 3D simulation model, we choose the average radii and porosities of the main pore modes as input parameters. The pore information derived from modes 1 trough 3 is adapted from the pore-size distribution analysis. However, the average radius of type 4 micro-pores has to be determined for each sample so that it honors the spatial range observed on the corresponding SEM images and at the same time allows a consistent reproduction of the NMR measurement. Furthermore, as demonstrated by Ramakrishnan et al. (1999), the influence of the rock surface relaxivity ρ (defined as the inverse of the product of T2S and the surface-to-volume ratio S/V of the pore) is crucial in the NMR response of hydraulically coupled pores. In sandstones, the almost linear relationship between T2S and S/V directly yields an estimate of ρ. By contrast, the presence of diffusive coupling in carbonates prevents a simple determination for ρ. The choice of NMR parameters in our simulation models hence represents an iterative process with two degrees of freedom, namely, the average micro -porosity radii r4, and the surface relaxivities ρ. A practical way to solve for the required parameters is to make use of the NMR responses of the desaturated samples. We remark that all the simulations assumed that the saturating brine exhibited the same properties observed in the laboratory measurements: temperature of 22°C, and T2B = 2300 ms.

Sample A The NMR measurements performed on the first sample yielded a high irreducible water saturation of 0.524. This value is larger than the fractional micro -porosity (0.424), thereby implying that, at Swir, the detected NMR signal is produced by water both within the micro-porosity (mode 4) and within the next smaller pores of mode 3. Therefore, by making use of the average pore radius of Mode 3, r3 = 6 µm, the selection of an equivalent model consisted in finding a satisfactory combination of (r4 , ρ) couplets such that the bimodal porosity simulation constructed with the remaining parameters (φ4 = 10%, r4; φ3 = 3.8%, r3 = 6 µm; ρ) matched the NMR measurements shown in Fig. 5 at Swir. Simulation

models were tested for r4 = 0.2, 0.4 and 0.5 µm, and for ρ = 1, 2, 3, and 5 µm/s, which correspond to normal values for carbonates. The irreducible water saturation was matched by defining a gas blob radius within the macro-porosity region of the model representing pore mode 3. Given the low gas hydrogen index, it was assumed that the pore volume associated with gas created no magnetization signal. A best fit was obtained for r4 = 0.5 µm and ρ = 2 µm/s, and by using a gas blob of radius equal to 6 µm.

The model was then verified for the remaining porosity modes. Figure 7 shows the cross-coupling NMR responses corresponding to modes 2, 3 and 4, where each coupled mode pair is weighted by its total porosity. If the pore system were uncoupled, its NMR response would be obtained from the superposition of the T2S values of its main pore modes. In Fig.7, the individual T2S response of each mode is described in a way that allows qualitative assessment of the coupling effect between each mode pair. It can be seen that mode 3 only slightly influences the remaining pore modes by causing no appreciable time shift in the T2 distribution. In consequence, the effective response of the whole pore system can be explained as controlled mainly by the coupling between the micro-porosity (mode 4) and the vuggy porosity (mode 2). A reasonable approximation of the sample NMR response can then be obtained by considering their coupled response at Sw = 1, weighted by the total porosity. The T2 distribution of the corresponding equivalent simulation model positively compares to the measurements (Fig. 8), although the micro-porosity peak is rendered sharper by the numerical simulations. This exercise validates the Swir-based parameters and the assumption of full hydraulic coupling between existing pores.

Sample B A similar method was used to construct a model that could replicate the NMR response of the second core sample. The latter exhibits an irreducible water saturation Swir = 0.277, which falls remarkably close to its fractional micro-porosity φmicro / φT = 0.263. Without much error, the micro-pore size r4 of the model was calibrated to 0.45 µm so that the individual response of mode 4 matched the NMR measurement performed at Swir for a surface relaxation of 2 µm/s. In this sample, however, three additional coupled pore modes coexist and hence entail some simplifying assumptions. It was assumed that the diffusion

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process within such system was controlled by the hydraulic coupling of all pore modes with the one nested closest to the cement crystal, and also by the direct coupling of both types of macro-porosities with micro-porosity. Figure 9 describes the simulation results yielded by these cross-pore coupling modes. The simulations were performed by taking into account the respective porosities for each mode pair, along with their individual T2S response when regarded as uncoupled. As can be seen in this figure, the overall magnetization associated with micro-porosity regions remains close to the uncoupled response of mode 4. However, the magnetizations borne by the macro-porosities shift to lower relaxation times, spread over the whole T2S range of the macro-porosities, and slightly fill the gap with mode 4.

The coupling process taking place over the entire pore structure is expected to reinforce two T2 peaks centered at 70 and 2000 ms, before accounting for bulk relaxivity, with a valley of T2 relaxations between the two peaks. As an illustration, Fig. 10 provides a comparison of the experimental NMR measurements with the sum of the coupled simu lations of Fig. 9 (obtained by assuming a bulk relaxation of 2300 ms). Clearly, Fig. 10 shows that the superposition of the coupled NMR responses is a sufficiently good approximation to explain the relative importance of the T2 modes in relation to their respective porosities. However, the bridge between both peaks cannot be explained by this first step of cross-coupling analysis. A next step of refinement of the simulation process would consist of iterating the coupling effects by considering equivalent bimodal models corresponding to the simulated coupled T2 distribution. Such an exercise, however, is beyond the scope of this paper. Sample C We applied a similar interpretation/simulation approach to analyze the pore-size distribution associated with the third core sample characterized in Fig. 4. However, the sizable dominance of macro-pores and the high concentration of cement crystals along the periphery of the macro-porosity region (Fig. 3b) prevented a satisfactory numerical representation by way of our hydraulically coupled model. Nevertheless, as pointed out below, the same sample provided us with an interesting technical challenge to explain the influence of temperature on NMR laboratory measurements.

The ensuing section of this paper are devoted to the understanding and assessment of temperature effects on the NMR response of coupled carbonate rocks. TEMPERATURE EFFECT ANALYSIS In order to test over NMR response to temperature variations, we analyze the measurements which were made at temperatures ranging from 4 to 65°C. Figure 11 displays the corresponding T2 distributions and shows that temperature has a different impact on the 3 samples in spite of the fact they are filled with the same fluid. This temperature effect is relatively significant on sample A, moderate on sample C, and negligible on sample B. More samples were measured that were not reported in this paper, and exhibited a wide temperature sensitivity of T2 spectra from sample to sample. No universal conclusions could be formulated; however, the samples exhibiting temperature dependence demonstrated an increase of T2 with temperature.

We assume that the behavior of the total transverse relaxation with temperature can be attributed to the following factors: (a) The bulk relaxation, proportional to

temperature and inversely proportional to the fluid viscosity.

(b) The surface relaxivity ρ, whose temperature influence depends on the magnetic content of the rock mineralogy and fluid types (Godefroy et al., 2001).

(c) The apparent volume-to-surface ratio seen from the NMR measurements vary in case of coupling. Indeed, when temperature, and hence bulk diffusivity, increases, the probability of proton exchange between the different pore modes increases too. In turn, the apparent volume-to-surface ratio of the pore volume departs from its actual value.

We support this last statement with cross-coupled simulations made for a constant surface relaxivity ρ = 2 µm/s, at temperatures of 4 and 65°C, with the model of sample A. The corresponding T2S results are plotted on Fig. 12 and are compared with the one previously obtained for room temperature (22°C). In this figure, we can see that temperature does not affect the coupled responses involving the vuggy macro-pores of type 2. However, between the smaller pores of types 2 and 4, a pattern is obvious: T2S decreases with temperature.

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At each temperature T, and for each sample, the geometric mean of the experimental T2 distribution was recorded as the measured T2 value. Then, in order to specifically target the effect of the abovementioned factors (b) and (c), we subtract the effect of bulk relaxation T2B from this measured T2 value to obtain an equivalent surface relaxation

( ) 1222 )(/1)(/1)( −−= TTTTTT BS . (4)

This corrected relaxation is plotted with respect to temperature using an Arrhenius diagram in Fig. 13. Regardless of any temperature dependence of ρ, we showed that the presence of coupling is likely to cause a slight decrease of 1/T2S when the reciprocal temperature 1/T increases. However, we notice from Fig. 13 that, in the cases of the coupled samples A and B, the relaxation rate, 1/T2S exhibits either no temperature dependence (sample B), or a behavior opposite to the one dictated by diffusion coupling (sample A). This remark sheds light to the possible impact of temperature on the surface relaxivity ρ, as suggested by Godefroy et al. (2001). In the case of samples A and B, if this assumption is verified, the temperature dependence of ρ creates an increase of 1/T2S with temperature. In other words, the effective surface relaxivity itself increases with temperature.

Sample C, on the other hand, exhibits a behavior opposite to samples A and B. In the absence of any hydraulic coupling, this example cannot be explained by the same mechanism. The origin of temperature dependence is still under investigation.

INFLUENCE ON T2-CUTOFF AND BVI The BVI is determined at irreducible water saturation from the cumulative T2 distribution plots. In Fig. 14 we plot the cumulative porosity distributions for the 3 samples at 100% water saturation at the two extreme temp eratures that were measured, and at Swir measured at room temperature. Also plotted are the results derived from simulation without accounting for pore coupling. From the comparison of these curves, we observe that temperature does have some effect on the BVI determination. Most reservoirs have higher temperature ranges than the highest temperature (65ºC) we measured in this study, thus the BVI difference could be larger at most of reservoir temperatures. In the previous section, we have demonstrated that with accounting for coupling effect, the

simulated T2 distribution resembles the measured ones. The simulated results in Fig. 14 depict the situation of the same pore structures but with uncoupled pores. We see that although the cumulative T2 distributions can be very different, the T2cutoff, hence the BVI value derived from it, varies only moderately. CONCLUSIONS

The study presented in this paper showed a method to interpret NMR measurements of carbonate rock samples in terms of diffusion coupling using a bimodal numerical simulation model. A parallel analysis of experimental measurements and numerical simulations was used to formulate assumptions on the origin of the samples’ behavior regarding temperature. Namely, we supported the assumption of a diffusion-based T2 decrease with temperature while we showed the possibility of an opposite effect of surface relaxation. Finally, by way of comparison of cumulative porosity plots of measurements and simulations, we were able to determine that, in the case of the studied samples, T2cutoff has moderate temperature dependence. The effect of diffusive coupling is more significant on the interpretation of the measured T2 distribution into pore size distribution, than on the determination of the right T2cutoff, and thus BVI.

NOMENCLATURES Do bulk diffusivity p probability of proton demagnetization at

fluid interface R macroscopic displacement of a step of

First-Passage-Time technique S/V surface-to-volume ratio of pore Swir water saturation in a core sample Swir irreducible water saturation T temperature t time T2 total transversal relaxation T2B bulk transversal relaxation T2S surface transversal relaxation TE echo time of the NMR tool ∆t duration associated with a First-Passage-

Time technique step ε unitary displacement of a step of

conventional random-walk technique applied near the pore wall

ρ surface relaxivity

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ACKNOWLEDGEMENTS We wish to express our gratitude to Baker Atlas for permission to publish these results, and for partially funding this work through internship positions offered to Emmanuel Toumelin and Marion Fischer during the Summer 2001. We also like to thank Drs. Gigi Zhang and Carl Edwards for useful discussions. We acknowledge partial support from the Center of Excellence in Formation Evaluation of The University of Texas at Austin, and from the American Chemical Society under grant ACS PRF#37519-AC9. The Center of Excellence in Formation Evaluation is an industry research consortium jointly sponsored by Baker Atlas, Halliburton, Schlumberger, and Anadarko.

REFERENCES Allen, D.F., Boyd, A., Massey, J., Fordham, E.J.,

Amabeoku, M.O., Kenyon, W.E., Ward, W.B., 2001, The practical application of NMR logging in carbonates: 3 Case Studies: Trans. SPWLA Annual Symposium, Paper K, Houston, Texas.

Bergman, D. J., Dunn, K.-J., Schwartz, L. M., and Mitra, P. P., 1995, Self-diffusion in a periodic porous medium: a comparison of different approaches: Physical Review E, vol. 51, No. 4, pp. 3393-3400.

Chen, S., Fang, S., Georgi, D., Salyer, J., and Shorey, D., 1999, Optimization of NMR Logging Acquisition and Processing: SPE 56766, presented at SPE Annual Tech. Conf. and Exhibit (ATCE), Houston, Texas.

Godefroy, S., Fleury, M., Deflandre, F., and Korb, J.-P., 2001, Temperature effect on NMR surface relaxation: SPE 71700, SPE ATCE, New Orleans, Louisiana.

Kenyon, W. E., 1997, Petrophysical principles and applications of NMR logging: The Log Analyst, March-April issue, pp. 21-43.

Quintero, L., Boyd, A. Gyllensten, A., and El Wazeer, F., 1999, Comparison of Permeability from NMR and Production Analysis in Carbonate Reservoirs: SPE 56798, p. 777, SPE ATCE, Houston, Texas.

Ramakrisnan, T. S., Schwartz, L. M., Fordham, E. J., Kenyon, W. E., and Wilkinson, D. J., 1999, Forward models for nuclear magnetic resonance in carbonate rocks: The Log Analyst, July-August issue, v. 40, No. 4, pp. 260-270.

Toumelin, E., 2002, Monte Carlo simulations of NMR measurements in carbonate rocks under a constant magnetic field gradient: Master’s thesis, The University of Texas at Austin.

Zheng, L. H., and Chiew, Y. C., 1989, Computer simulation of diffusion-controlled reactions in dispersions of spherical sinks: Journal of Chemical Physics, vol. 90, No. 1, pp. 32-327.

ABOUT THE AUTHORS Emmanuel Toumelin is a Research Assistant and a Master’s Candidate in the Department of Petroleum Engineering at The University of Texas at Austin. He has a General Engineering degree from the Ecole Centrale de Lille in France. Carlos Torres-Verdín received a Ph.D. in Engineering Geoscience from the University of California, Berkeley, in 1991. During 1991-1997, he held the position of Research Scientist with Schlumberger-Doll Research. From 1997-1999, he was Reservoir Specialist and Technology Champion with YPF (Buenos Aires, Argentina). Since 1999, he is an Assistant Professor with the Department of Petroleum and Geosystems Engineering of The University of Texas at Austin, where he conducts research in formation evaluation and integrated reservoir characterization. He has served as Guest Editor for Radio Science, and is currently a member of the Editorial Board of the Journal of Electromagnetic Waves and Applications, and an associate editor for Petrophysics (SPWLA). Songhua Chen is a staff scientist and project leader for NMR interpretation development at Baker Atlas Houston Technology Center. Prior to joining Baker Atlas, he was a Research scientist for 5 years with Texas Engineering Experiment Station in College Station, Texas, where he worked in the area of NMR and MRI applications to flow in porous media. Songhua has a B.S. from Nanjing Institute of Technology in China and a Ph.D. from University of Utah, both in Physics. D. Marion Fischer is a scientist with Baker Hughes Inteq at Celle, Germany, since December 2001. Prior to joining Inteq, she spent one and half years at Leibniz Institute of Applied Geosciences in Germany as a postdoc doing NMR laboratory research, and a summer at Houston Research Center of Baker Atlas, U.S.A. Marion received her Diploma (physics) in 1995 and PhD (NMR physics) in 1999, both from the University of Hannover, Germany. Her PhD dissertation topic is on NMR diffusion measurements in solid-state materials.

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Fig. 1a: SEM image of sample A at 140x magnification. This picture shows the extensive micritization of the rock matrix and the isolation of scarce macro-pores.

Fig. 2a: SEM image of sample B at 140x magnification. The macro-porosity and the general agglomeration of grains into packs are well defined.

Fig. 3a: SEM image of sample C at 140x magnification. The macroscopic structure looks similar to that of sample B, although the BSE analysis shows the large pores to be more numerous in this sample.

Fig. 1b: SEM image of sample A at 1400x magnification. One can see that the abundant micro-porosity can easily couple with the little macro-porosity present in the rock.

Fig. 2b: SEM image of sample B at 1400x magnification. One notes the presence of cementation crystals at the limit of some micritized grain packs (macro-pore surface).

Fig. 3b: SEM image of sample C at 1400x magnification. The high density of calcite crystals provides strong cementation of the macro-pores.

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Fig. 4: Pore-size modes recognized in samples A, B and C. Plain stems correspond to coupled porosity modes, dash stems, to partially couple d primary porosity, and dot stems, to a priori uncoupled vuggy porosity.

Fig. 5: T2 distributions of NMR measurements performed at ambient temperature on samples A, B, and C. Dashed curves represent the measurements at irreducible water saturation, and solid-line curves, the ones at 100% water saturation.

Fig. 6: Three-dimensional representation of the distribution of fluids and rock matrix at the spatial resolution of both macro- and micro-cells. Gas blobs are centered on the macro-pores. Compaction is simulated by assigning sphere diameters larger than the concentric cube sizes.

Gas

Grain

Water-filled micro-porosity

Water-filled macro-

porosity

Top view

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Fig. 7: Cross-coupled NMR responses of the main modes present in sample A, and comparison with the uncoupled modes. The stems in gray represent the individual T2S response of each mode as if there were no coupling, that is, the ‘original’ location of the T2S peaks before pore coupling

Fig. 9: Cross-coupled NMR responses of the main modes present in sample B and comparison with the uncoupled modes. The stems in gray represent the individual T2S response of each mode as if there were no coupling.

Fig. 8: Comparison of simulations and measurements of sample A at room temperature. Solid-line curves: Sw = 1. Dashed curve: Sw = Swir. Although sharper than the experimental peak at irreducible water saturation, the simulation result yields same T2 peak value and same porosity.

Fig. 10: Comparison of simulations and measurements of sample B at room temperature. The closest representation of this sample’s structure with the numerical model looks almost uncoupled.

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Fig.11: Examples of T2 distributions measured at 4 and 65°C for the three samples A, B and C at Sw = 1. For each sample, the solid-line curve represents the NMR response at 4°C, the dashed curve, at 65°C, and the dotted curve, the reference measurements at irreducible water saturation and regulated 22°C.

Fig.13: Arrhenius diagram showing the experimental influence of temperatures over the NMR response of samples A, B and C.

Fig. 12: Numerical evidence of a temperature influence of the volume-to-surface ratio of coupled pore systems. Simulation results of the model of sample A run for a constant, uniform surface relaxivity ρ = 2 µm/s, at temperatures of 4, 22, and 65°C. Solid-line curves: coupling between modes 3 and 4; dash: between 2 and 3; dots: between 2 and 4. (See also Fig. 7.) One can observe the slight but regular T2 decrease with temperature T of the coupling between the pore modes 2 and 4.

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Fig.14: Cumulative porosity plots of the T2 distributions of samples A, B and C for different temperatures (samples A, B, C) and assuming non-coupling of pores (samples A and B). The dotted curves represent the cumulative T2 porosities at irreducible water saturation. The dashed curves represent the cumulative porosities from the measurements at 65°C, and the solid-line curves, at 4°C. For samples A and B, the thick gray curves stand for the cumulative porosities of the corresponding uncoupled models. The cumulative porosity can be understood as a BVI percentile. The vertical line at 90 ms represents the BVI-T2cutoff usually used in carbonates.