Spatial Heterogeneity Length Scales in Carbonate Rocks - Springer Link

Appl. Magn. Reson. (2007) 32, 221231 DOI 10.1007/s00723-007-0010-7 Printed in The Netherlands

Applied Magnetic Resonance

Spatial Heterogeneity Length Scales in Carbonate Rocks A. E. Pomerantz, E. E. Sigmund, and Y.-Q. Song Schlumberger-Doll Research, Ridgefield, Connecticut, USA Received 2 June 2006; revised 29 August 2006 © Springer-Verlag 2007

Abstract. Spatially resolved distributions of T2 relaxation times in carbonate rocks are measured with slice-selective multiple spin echo magnetic resonance imaging to study the length scales of heterogeneity in these samples. Single-voxel CarrPurcellMeiboomGill decays are fit to double exponential functions, and the results of those fits are combined into a histogram. We describe a novel qualitative method of assessing the importance of different length scales of heterogeneity, involving comparing various aspects of these histograms to the full-core T2 distributions. Using this technique, it is found that almost all individual voxels relax not only with more than one time constant but indeed with a range of relaxation times that approximates the full breadth of relaxation times for the entire core, indicating significant subvoxel heterogeneity. In addition, different voxels are found to exhibit relaxation times that differ by orders of magnitude, indicating significant heterogeneity between the scale of a voxel (1 mm) and that of the entire core (several centimeters). These results reflect the importance of a broad range of length scales of heterogeneity in these carbonate rocks.

1 Introduction The structure and dynamics of carbonate reservoirs are of crucial importance to the petroleum industry, as carbonate reservoirs in the Middle East are believed to contain approximately half of the worlds oil. Modeling carbonate reservoirs is challenging because the nature of the deposition and diagenesis results in heterogeneity on multiple length scales [14]. Magnetic resonance imaging (MRI) provides a unique opportunity to study porous media on many length scales. Macroscopic information, typically on the millimeter to meter length scale, can be directly inferred from the spatial resolution afforded by MRI. Microscopic information can be obtained from spin relaxation, diffusion, and internal field properties. For example, relaxation times can reflect the sizes of pores, which are often on the micrometer length scale [515]. Hence, information from the micrometer to the meter length scale can be obtained, in principle, from magnetic resonance images. Several groups have used MRI to image rock cores typical of those found in sedimentary reservoirs [13, 14, 1627]. They report spatially resolved porosi-

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ties, relaxation times, diffusion coefficients, and chemical shifts to establish spatial distributions of oil and water saturations and wettings. These reports have shown that relaxation decays within individual voxels rarely can be satisfactorily fit to single exponentials; rather, multiexponentials or stretched exponentials were required to model the decays [1719, 21, 25]. The multiexponential relaxation of individual voxels (typically 1 µl or smaller) implies subvoxel heterogeneity. Here we report a systematic study of the importance of different length scales of heterogeneity in a series of carbonate rocks. Measurements of CarrPurcell MeiboomGill (CPMG) decays for full cores are analyzed by Laplace inversion to yield full-core T2 distributions. A slice-selective multiple spin echo MRI sequence is then used to measure similar decays with spatial resolution. An MRI T2 distribution is obtained by fitting the decays of individual voxels to a double exponential and combining the fitted values into a histogram. The full-core T2 distributions and MRI T2 distributions are then compared to investigate the importance of heterogeneity at different length scales. This comparison between spatially resolved and full-core T2 distributions represents a novel means of characterizing heterogeneity. Using this technique, several conclusions are drawn about the importance of different length scales of heterogeneity in a set of carbonate cores. 2 Material and Methods Twenty-three carbonate cores are taken from a Middle Eastern oilfield well over a depth range of 800 feet. Cores of 2 cm diameter and 3.75 cm length are cut and then thoroughly cleaned and saturated with 0.20 W$m NaCl brine. These rocks contain mostly calcite and dolomite and have permeabilities in the range of 0.44.000 millidarcy. Porosities for the full cores fall in the range of 928 p.u., while individual voxel porosities span almost the full range of 0100 p.u. Of the 23 samples measured, four representative samples (A, B, C, and D) are discussed in detail here; petrophysical properties of these samples are presented in Table 1, and thin-section photographs are presented in Fig. 1. Table 1. Petrophysical properties of samples studied. Sample Porosity (p.u.)

Permeability (millidarcy)

Mineralogy (%) Limestone

Dolomite

A B C

19 19 28

25 150 2900

0 95 2

98 5 89

D

15

0.7

68

27

Lithology

Sucrosoic dolomite Oncoids and foraminifera-rich grainstone Moldic porosity with dolomite recrystallization Dolomite crystals in microporous mudstone matrix

Spatial Heterogeneity Length Scales in Carbonate Rocks

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B

A

250 µm

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250 µm

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250 µm

Fig. 1. Thin-section images of four carbonate cores (A, B, C, D). The thin sections were impregnated with blue epoxy and photographed at 20x magnification. The blue regions are large pores between the grains. Small intragranular pores are not directly visible in such image but show as the dark areas within grains. Note that the area shown for each sample is of similar dimension as the voxel size. For samples A, B, and C, it is clear there exists significant heterogeneity of pore space within 1 mm.

Nuclear magnetic resonance (NMR) and MRI experiments are performed at a 2 T static field (Nalorac Cryogenics) using an Avance console (Bruker Biospin) operating at 85 MHz. NMR experiments employ the CPMG sequence [28, 29], in which the peak echo amplitude is recorded for each of 100 echoes with an echo time of 3.7 ms. These decays are analyzed with the one-dimensional (1-D) version of the fast Laplace inversion (FLI) algorithm using a constant value of the regularization parameter [30]. MRI experiments employ a multislice multiecho (MSME) sequence. This sequence effectively performs a CPMG experiment for each voxel; cubic voxels of 1 mm length are used, except where noted. As in the NMR experiments, 100 echoes are collected at an echo time of 3.7 ms. Imaging gradients are adjusted so that diffusion decay in the applied gradients produces a negligible effect on the relaxation behavior. 2.1 Data Analysis For the NMR data on full cores, CPMG decays are measured and analyzed with Laplace inversion to yield a full-core T2 spectrum [30]. For the MRI data, sim-

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pler techniques are used to analyze decays in individual voxels. Previous MRI investigations of similar rocks with similar resolution found that single-voxel relaxation was often multiexponential [1719, 21, 25]. This behavior is also found here, as illustrated in Fig. 2. Hence, individual voxels are analyzed by fitting the decays to a double exponential function:

M (t ) = j f exp( -t / T2f ) + j s exp( -t / T2s ), M0 where T2f and T2s represent the fast and slow decay time constants, and jf and js are the respective weights. The voxel total porosity, j, is defined as j " jf ! js. The use of this more complicated double-exponential model, as opposed to a simpler single-exponential model, is statistically justified on a voxelwise basis by calculating the Akaike information criterion for both single- and double-exponential fits [31]. If the Akaike criterion indicates that the use of the double exponential is not justified, then only the single-exponential fit is retained. Typically, a double-exponential fit is justified in more than 90% of the voxels. Additionally, decays for individual voxels were required to have at least eight echoes above the noise threshold, which is determined as the signal outside the sample. Images were recorded with sufficient signal-to-noise ratio such that less than 5% of voxels had to be discarded because they had such low values of porosity and/or 60

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Echo Time (ms) Fig. 2. T2 decay data and fits for a typical voxel. The measured decay (circles) is poorly fit with a single exponential (dashed line), and is statistically significantly better fit with a double exponential (solid line). The porosity is displayed in absolute porosity units (p.u.), determined by referencing the measured echo amplitude to that of bulk brine extrapolated to zero echo time. The voxel shown was taken from near the center of the sample labeled C in the other figures.


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T2 that this criterion was not met. Finally, the double-exponential fit is found to be quite stable. Even for voxels with the slowest relaxation, the error in determining log10(T2s) is typically less than 0.05. Hence, the error in determining T2s for individual voxels is typically an order of magnitude smaller than the width of the T2s distribution for the core. This fitting process determines two values of T2 (T2f and T2s) for most voxels, and the T2s obtained for these voxels are then weighted and collected in a histogram. For voxels in which the double exponential fit was not justified, the single T2 was weighted and added to the histogram. We call this histogram the MRI T2 distribution. It is known that experimental relaxation decay data often can be modeled by many different functions, such as a double exponential or a stretched exponential. Numerical Laplace inversion with regularization is often considered a preferred method for obtaining the T2 distribution. For MRI data in individual voxels, the low signal-to-noise ratio would result in extra broadening of a Laplace inversion T2 spectrum [32]. On the other hand, the double-exponential fit represents an attractive way to describe the relaxation in individual voxels because it contains enough fitting parameters to provide a high-quality fit to the data without requiring a more complex fit than could be justified from the signal-to-noise ratio of the experiment. Although a double-exponential fit does not necessarily represent the true T2 distribution of the voxel, it suffices to describe average properties of the underlying T2 distribution, e.g., the average T2 and the width and shape of the distribution, and can do so reliably under the stated experimental conditions. Thus, the double-exponential model is used in this article primarily as a way to describe average properties of the voxels. 3 Results 3.1 T2 Distributions Figure 3 compares T2 distributions from four representative cores, showing the full-core T2 distribution and the MRI T2 distribution for each sample. In samples A, B, and C, the full-core T2 distribution is broad, T2 ranging from greater than 1 s to less than 10 ms; the longest of these times are indicative of small vugs. In sample D, on the other hand, the full-core T2 distribution shows a much narrower peak, ranging from 10 to 40 ms, indicative of a relatively homogeneous pore space compared with the other samples. In all samples, the MRI T2 distribution does not exactly reproduce the fullcore T2 distribution; however, the MRI T2 distribution does reflect some qualitative features of the full-core T2 distribution. For example, the MRI T2 distribution is very narrow for sample D and broad for samples A, B and C, consistent with the full-core T2 distribution. The MRI T2 shows a peak at about 0.5 s for samples A, B, and C, consistent with the slow T2 peak in the full-core T2 distribution. The fast peak in the MRI T2 distribution appears to be an average of

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Fig. 3. T2 distributions for four carbonate cores (A, B, C, D). The solid line represents the fullcore T2 distribution with data from the full sample analyzed using regularized Laplace inversion, the dotted line represents the T2 distribution, and the dashed line represents the MRI T2 distribution from fitting individual voxels to a double exponential. Each distribution is normalized to unity at its maximum.

the full-core T2 features in the faster region. The widths of the peaks in the MRI distributions are always narrower than in the full-core distribution, potentially reflecting spatial averaging within a voxel in the double-exponential fitting. However, the full-core distributions may be broadened due to the regularization in the inversion algorithm [32]. The inversion-limited resolution of peaks in the T2 spectrum was calculated using the methods of ref. 32 and was found to be comparable to the widths of the measured peaks. This result indicates that width of the peaks in the Laplace inversion spectrum is controlled by the inversion algorithm, so the true peak width may be less than what is shown in the spectra and the true T2 distribution may look more like the MRI T2 distribution. The presence of widely separated peaks (separation exceeding the width of the individual peaks) in the spectra is not influenced by the inversion algorithm, indicating that the good qualitative agreement between MRI and full-core T2 distributions is robust. Figure 3 also shows the spatially resolved distributions of the average T2. The average T2, T2, is defined by 1/T2 " (jf/T2f ! js/T2s)/j using the results


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of the double-exponential fit. For voxels in which the Akaike information criterion determined that a double-exponential fit was not justified, decays were fit to a single exponential and the fitted T2 was taken as T2. As before, the measured values of T2 for each voxel are weighted by the porosity and combined into a histogram to yield the T2 distribution. As shown in Fig. 3, the T2 spectrum exhibits significant width for all samples; however, it does not reproduce the features or the width of both the full-core T2 and MRI T2 distributions. These results indicate the presence of heterogeneity on length scales both above and below that of the voxels (1 mm). Two limiting cases can be considered. At one extreme, if all the heterogeneity were subvoxel, then the entire sample would be a collection of equally heterogeneous voxels. In this limit, the observed T2 distribution would be a delta function. Thus, the variation of T2 over the sample, e.g., the width of the T2 distribution, measures the heterogeneity at length scales above that of the voxel. Consistently, T2 heterogeneity

Position (mm)

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Fig. 4. Spatial maps of T2 for the four carbonate cores (A, B, C, D). The gray scale represents the value of log10(T2/ms). T2 is the average T2 from the double-exponential fits and ranges over two orders of magnitude. The slices shown were taken near the middle of the cores.

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between voxels can be seen in the images presented in Fig. 4, which shows significant spatial heterogeneity at length scales greater than the voxel dimension. At the other extreme, the imaging resolution might be so fine that there is no subvoxel heterogeneity. In this limit, every voxel would exhibit single exponential behavior, and the T2 distribution would be similar to the full-core T2 distribution (assuming the fast-diffusion limit prevails, as is commonly encountered in these samples [5, 9]). Figure 3 demonstrates that the T2 distribution does not match the breadth or the features of the full-core T2 distribution, demonstrating significant heterogeneity below the voxel dimension. This result is examined further in Sect. 3.2. 3.2 Resolution Dependence The importance of subvoxel heterogeneity can be further investigated with MRI experiments at different voxel size: as the imaging resolution improves, the large volume that would comprise one voxel and, correspondingly, give one T2 at a coarse resolution is divided into many voxels yielding a distribution of T2. The MRI experiments were repeated for sample C at four imaging resolutions from 1.53 to 0.53 mm3. MRI T2 distributions for each resolution are shown in Fig. 5a. The T2 distribution is narrowest at the coarsest imaging resolution and broadens for successively finer resolutions, consistent with less averaging for smaller voxels. However, even at the finest imaging resolution, the T2 distribution still does not match the width or contain the features of the full-core T2 spectrum. From the coarsest to the finest resolution, the voxel size decreases by a factor of 27, but the width of the T2 distribution increases by only a factor of 1.4. This slow increase in width implies that it will be hardly possible to achieve sufficient resolution that individual voxels will appear homogeneous.

a

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Population (a. u.)

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0.8 0.6 0.4 0.2 0 0.01

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Fig. 5. MRI T2 distributions at different imaging resolutions compared to the full-core T2 distribution (gray line) for sample C. The imaging resolution is 1.5 mm (solid line), 1.0 mm (dashed line), 0.75 mm (dash-dot line), and 0.5 mm (dotted line). a T2 distribution and b MRI T2 distribution.


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Figure 5b shows the MRI T2 distribution obtained using the double-exponential fit at different resolutions. This distribution again contains narrow peaks at coarse resolution, and those peaks broaden slightly at finer imaging resolutions due to the reduction of spatial averaging. However, the qualitative appearance of the distribution is again unchanged at the different imaging resolutions. For each imaging resolution, approximately 90% of the voxels are fit significantly better with a double exponential than with a single exponential, using the Akaike information criterion. The insensitivity of the double-exponential T2 distribution to the imaging resolution is consistent with the presence of significant heterogeneity both above and below the voxel size. 4 Discussion The MRI T2 distributions (Fig. 3) are the weighted sums of the fast and slow components of the double-exponential fit, as well as of the single-exponential component for voxels in which application of the Akaike information criterion determined that the double-exponential fit was not justified. Figure 6

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Fig. 6. Distributions for the T2f (dashed line) and T2s (dash-dot line) obtained from the double-exponential fit for each of four carbonate cores (A, B, C, D). The dotted line shows the single-exponential T2 distribution for voxels in which a single-exponential fit is adequate; this distribution lies near zero for samples A, B, and C. The full-core T2 distribution is also shown (solid line) for comparison.

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presents these three contributions separately in order to illustrate the voxelto-voxel variation of the individual components. Two observations can be made from these figures. First, the total porosity of voxels that do not support a double-exponential fit is quite small for rock cores A, B, and C; for these cores, at least 95% of voxels were fit significantly better by a double exponential than by a single exponential. For the most homogeneous sample, core D, 65% of voxels were significantly better fit with a double exponential, although the difference between the fast and slow T2 values is much smaller than in the other samples. The second observation is that there is negligible overlap between the fast and slow T2 distributions. Combining these two results, it is demonstrated that almost every voxel relaxes at two different times that are characteristic of the breadth of the relaxation times of the entire core. Certainly, large differences in relaxation times can exist between different voxels, as demonstrated by the width of each component T2 distribution shown in Fig. 6 and by the spatial heterogeneity shown in Fig. 4. However, much of the heterogeneity of relaxation times in the entire core is contained in each individual voxel. Such subvoxel heterogeneity is consistent with the thin-section images of samples A, B and C. Additionally, samples A and B each show a broader distribution of T2s than of T2f, suggesting that there is more spatial variation in the large pores than in the small pores. 5 Conclusion We have used NMR and MRI techniques to obtain distributions of several relaxation properties in carbonate cores. MRI T2 distributions with double-exponential fits qualitatively match the full-core T2 distributions obtained by Laplace inversion of data from the whole sample. We demonstrate that contrasting fullcore T2, T2, and MRI T2 distributions is a useful method for characterization of pore heterogeneity on different length scales. Upon comparison of these distributions, it is evident that almost every rock sample shows significant heterogeneity both below the voxel size and between the size of an individual voxel and the size of the entire sample. In particular, almost every voxel relaxes at two time scales that are characteristic of the breadth of relaxation times of the entire core, emphasizing the importance of heterogeneity at volumes less than 1 µl. Finally, there appears to be more spatial variation in the large pores than in the small pores. Acknowledgments We thank M. Hürlimann, L. Venkataramanan, and P. Tilke for discussions and H. Cho and X. Ren for experimental assistance.


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Authors address: Yi-Qiao Song, Schlumberger-Doll Research, 36 Old Quarry Road, Ridgefield, CT 06877, USA E-mail: [email protected]