Prediction of permeability from NMR response: surface relaxivity heterogeneity

Transcription

1 Prediction of permeability from NMR response: surface relaxivity heterogeneity C. H. Arns,*, A. P. Sheppard, M. Saadatfar, and M. A. Knackstedt,2 Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Australian National University, Canberra, Australia 2 School of Petroleum Engineering, University of New South Wales, Sydney, Australia * Corresponding Author: christoph.arns@anu.edu.au Copyright 2006, held jointly by the Society of Petrophysicists and Well Log Analysts (SPWLA) and the submitting authors. This paper was prepared for presentation at the SPWLA 47 th Annual Logging Symposium held in Veracruz, Mexico, June 4-7, ABSTRACT NMR responses are commonly used in reservoir characterization to estimate e-size information, formation permeability, as well as fluid tent and type. Difficulties arise in the interpretation of NMR response as an estimator of permeability due to internal gradients, diffusion coupling, surface-relaxivity heterogeneity, and a possible breakdown of correlations between e and striction sizes. Here we sider several scenarios of surface relaxivity heterogeneity for a set of sandstones and a set of carbonate rock in a numerical NMR study based on Xray micro-ct data. We have previously demonstrated the ability to image, visualize, and characterize sedimentary rock in three dimensions (3D) at the e/grain scale via X-ray microcomputed tomography. We also numerically tested the influence of structure and diffusion-coupling on NMRpermeability correlations. Here we sider surface relaxivity heterogeneities due to e partitioning, mineralogy, e size, and saturation history. We partition the e space and solid phase into regions of es and grains. These partitions could reflect different mineralogy for weakly coupled e systems, or differences in mineralogy for the grains. Further, we use a morphological drainage simulation technique to partition the e space in terms of invasion radius or throat size, reflecting surface relaxivity heterogeneity due to the saturation history of immiscible fluids, which could cause e.g. pressure dependent adsorption on surfaces and/or changes in wettability. Finally, we use the cept of covering radius to assign a surface relaxivity due to e size. For each sample, four sandstones and three carbonates, we sider distributions of surface relaxivity based on above partitions, keeping the mean surface relaxivity stant, simulate the magnetisation decay, and derive a e size distribution through an inverse Laplace transform assuming stant surface relaxivity. Further, we test the effect of these heterogeneities on NMR-permeability correlations based on the log-mean of the relaxation time distributions for two frequently used empirical NMR-permeability cross-correlations. At the scales probed here, surface relaxivity heterogeneity changes the prefactor in the equations for sandstones only minimally, while the prefactor is changes orders of magnitudes for carbonates. The influence of surface relaxivity heterogeneity on the quality of the fit for individual samples is small. INTRODUCTION The estimation of permeability through cross-correlations from other physical measurements on rocks is a classical task in petrophysics and has a long history in well logging. Of the measurements available, the NMR relaxation is the one, which typically correlates best to permeability (e.g. (Sen et al., 990)). One reason is that the estimation of permeability requires length scale information, which the NMR relaxation response provides, since the relaxation process is typically trolled by the surface to volume ratio of the e space (Wayne and Cotts, 966; Brownstein and Tarr, 979; Kenyon et al., 986; Kenyon et al., 988; Kenyon, 992; Song et al., 2000). Under the assumption of stant surface relaxivity, weak coupling between es, and fast diffusion within es, the magnetisation decays uniformly within each e and the decay can be written as N [ M(t) = M 0 (t) a p exp t ], () T 2p p= where M 0 is the initial magnetization, p is a e label, a i is the fractional e volume, t is time, and the transverse relaxation time T 2 of the es is given by = + ρ S p. (2) T 2p T 2b V p

2 SPWLA 47th Annual Logging Symposium, June 4-7, 2006 Here T2b is the bulk relaxation time, Sp /Vp is the surfaceto-e-volume ratio of the e space, and ρ the surface relaxation strength. The multi-exponential distribution corresponds to a partition of the e space into N groups based on the Sp /Vp values of the es. This is the classical picture of the NMR relaxation response used e.g. in standard NMR logging tools to derive a relaxation time or e size distribution. Analog expressions exist for deriving length scales using higher diffusion eigenmodes (Song et al., 2000; Song, 2003). Apart from weak coupling between es and background microosity, which we discussed in a previous paper (Arns et al., 2005b), above interpretation technique assumes a stant surface relaxivity. In this study, we sider the impact of surface relaxivity heterogeneity on NMR-permeability correlations by numerically deriving NMR responses and transt properties under very trolled ditions on realistic microstructures, using images acquired by X-ray µct techniques (Sakellariou et al., 2004b) and established algorithms to calculate transt properties on those images (Arns et al., 200; Arns et al., 2002; Arns et al., 2004b; Knackstedt et al., 2004; Arns et al., 2005a; Arns et al., 2005c). The paper is organised as follows: in the next section we introduce the images used in this study and the partitioning techniques needed to assign surface relaxivity distributions to the surface area of the rocks. The following section details the NMR response simulation with a focus on the different surface relaxivity distributions employed. The third section reviews two common NMR-permeability crosscorrelations, before we present results in section four and draw clusions in a final section. [d] [e] [f] [g] [h] Figure : Slices through the Xray density maps of the sandstone [a-d] and carbonate [e-g] samples comprising this study. Castlegate sandstone (FS), same as, 2cm apart (FS2), unsolidated sand (US), [d] clayey Regolith (CRS), [e] outcrop limestone (OC), [fg] vuggy carbonates (VC and VC2).; [h] three-phase segmentation of [d]. IMAGE PROCESSING Samples - In this study, we sider the impact of surface relaxivity heterogeneity on NMR-permeability correlations for four sandstones and three carbonates. All samples have been imaged on the ANU high-resolution X-ray µct facility (Sakellariou et al., 2004b; Sakellariou et al., 2004a). A slice of the Xray density map of each sample is shown in Fig. and basic features of the samples are summarized in Table. Two of the sandstones are fluvial deposits (Castlegate sandstone, FS and FS2), taken from top and middle sections of the same plug, the third is an unsolidated sand (US), and the fourth is a clayey Regolith sample (CRS). Of the carbonates, one is an outcrop limestone (OL) originating from South Australia, while the others are vuggy carbonates (VC and VC2) of West Texan and Middle Eastern origin with various degrees of internectivity. Table : Characteristics of the sandstone (top four) and carbonate samples (bottom three) of this study. is the voxel size, and t the total resolved image osity. Sample FS FS2 US CRS OC VC VC2 Segmentation - Partitioning of the Xray-density data into two (solid, void) or three phases (solid, clay, void) is per- 2 Description fluvial fluvial unsolidated clayey Regolith outcrop limestone vuggy vuggy [µm] t

3 formed in a multi-stage procedure using a modified version of the verging active tour method outlined in (Sheppard et al., 2004), which we used before (Arns et al., 2005b). An example of a resulting segmentation is given in Fig. 2b, and for a three-phase segmentation of sample CRS in Fig. h. In this study we chose to interpret the clayey region at intermediate Xray-density of sample CRS as solid phase. The same applies to the microous regions of the carbonate samples VC and VC2. Topological e partitioning - To assign surface relaxivity based on the topological cept of es, we need to partition the e space into simple geometric cells (e bodies), separated by narrow strictions taken to be volumeless (throats). An account of this technique has been given elsewhere (Arns et al., 2005b). It basically involves the derivation of the medial axis of the nected e space, followed by a topology serving breakdown of the medial axis into separate bodies using distance information, and a final step of merging e centers at small separation. An example of a resulting e partitioning is given in Fig. 2c. This partitioning can reflect different micro-environments, which are weakly coupled through narrow strictions. Grain partitioning - Grain partitioning has only been carried out on the sandstones. While it is essentially the inverse problem of e partitioning, it is significantly simpler. We used a method based on identifying watersheds of the Euclidean distance map of the grain space (Saadatfar et al., 2005). An example of a resulting e partitioning is given in Fig. 2d. Partitioning based on e size - To partition the e space based on the geometric cept of e size, we use the maximal inscribed radius partitioning. This partitioning technique is based on a classical mathematical description of the morphology in terms of basic geometrical quantities (Serra, 982). More complete and generic descriptions of the basic cepts and techniques are given elsewhere (Hilpert and Miller, 200; Thovert et al., 200; Arns et al., 2005c). In a succession of morphological operations each voxel gets assigned the radius of the largest sphere which lies within the e space and covers the voxel (covering radius transform, CRT). An example of a resulting inscribed radius map is given in Fig. 2e. This partitioning can mimic the effects of internal gradients (stronger in smaller es), or of fluid distribution history, where it reflects the fluid distribution at variable equilibrium pressure. Invasion radius partitioning - The nected e space is partitioned by assigning to each voxel the radius of the largest sphere, which can penetrate from the outer boundary of the sample to cover the voxel. This simulation technique mirrors the boundary ditions of standard mercury intrusion experiments, e.g. a fixed capillary pressure is associated with a e entry radius (capillary drainage transform, CDT). The center of the invading sphere is allowed to move such that the sphere does not overlap the solid (Hilpert and Miller, 200). An example of a resulting invasion radius map is given in Fig. 2f. This partitioning reflects the history of the fluid distribution at breakthrough pressure. NMR RESPONSE SIMULATION Surface relaxation simulation - The spin relaxation of a saturated ous system is simulated by using a lattice random walk method (Mendelson, 990; Bergman et al., 995). Initially the walkers are placed randomly in the 3D e space. At each time step the walkers are moved from their initial position to a neighboring site and the clock of the walker advanced by t = ɛ 2 /(6D 0 ), where ɛ is the lattice spacing and D 0 the bulk diffusion stant of the fluid, reflecting Brownian dynamics. The lattice is made periodic by mirroring the structure in all directions. An attempt to go to a site of another phase will kill the walker with probability ν/6, 0 ν (Mendelson, 990). The killing probability ν is related to the surface relaxivity ρ via Aν = ρɛ D 0 + O ( (ρɛ D ) 2 ), (3) where A is a correction factor of order (here, we take A = 3/2) accounting for the details of the random walk implementation (Bergman et al., 995). Surface relaxivity distributions - Here we adapt Eqn. (2) to varying surface relaxivity by accounting for N active surface patches of surface area S p and surface relaxivity ρ p within each well-nected region of volume V p = + T 2p T 2b NV p N i= ρ pi S pi = T 2b + T 2Sp. (4) Eqn. can then be applied again with the more general definition of the surface relaxation time T 2Sp of a nected region p. We are now in a position to assign a distribution of surface relaxivities, while keeping the mean surface relaxation time T 2Sp stant. In order to explore the influence of surface relaxivity heterogeneity at different length scales on permeability predictions and e size distributions, we assign spatially varying surface relaxivities ρ( x) to the e-solid interface using five different methods:. Constant surface relaxivity: ρ( x) = ρ = st. We use the index to indicate this. 3

4 [d] [e] [f] Figure 2: Illustration of the image segmentation and partitioning steps for a clean sandstone. Shown are slices of a central subsection of a solidated sand dataset (400 2 voxel, voxel size 4.93 µm). All morphological calculations were carried out on a much larger subset (n x n y n z in Table 2), e.g. boundary effects are minimal. Xray density, segmented image, topological e micro-environment partitioning, [d] grain partitioning, [e] e size or irreducible saturation partitioning, [f] invasion radius partitioning. 4

5 2. Using the topological e partitioning (Fig. 2c, index ), assign each e with equal probability either the surface relaxivity ρ = 5 3 ρ or ρ 2 = 3 ρ, i.e. ρ /ρ 2 = 5, where ρ is the average surface relaxivity. 3. Using the grain partitioning (Fig. 2d, index ), assign each grain with equal probability either the surface relaxivity ρ = 5 3 ρ or ρ 2 = 3 ρ. 4. Using the geometric e size partitioning or covering radius transform (Fig. 2e, index ), assign each surface voxel a surface relaxivity based on an inverse relationship: ρ(r) /r, where r denotes the inscribed radius of the nearest e voxel. The inverse relationship can reflect stronger internal gradients in fined spaces. In terms of fluid saturation history it could reflect time intervals, in which the large curvature of the non-wetting phase allowed deposition of relaxive substances through the wetting phase. 5. Using the invasion radius partitioning or capillary drainage transform (Fig. 2f, index ), assign each surface voxel a surface relaxivity based on an inverse relationship: ρ(r) /r, where r notes the invasion radius of the nearest e voxel. The same argument as above applies. All distributions are renormalised, such that ρ( x) = ρ, after the surface relaxivity assignment stage of the algorithm. Here the average runs over all e-solid surfaces of the sample. In summary, the different distributions of surface relaxivity are assigned such that the mean surface relaxivity is the same for all distributions for a given sample. It should be mentioned, that the geometrically based partitions show much more variability of surface relaxivity compared to the topologically based e and grain partitions by definition. The expected effect would be a better mixing in terms of surface relaxivities by diffusion. Note, that modes of small surface relaxivity over large areas - which can appear in large es using e.g. the e size surface relaxivity partitioning - are damped, since we always sider bulk relaxation using the bulk relaxation rate of water (T 2b = 2.876s). Inverse Laplace transform - The relaxation time distribution is derived by fitting a sum of exponentials to the magnetisation decay M(t) using a bounded least square solver (Stark and Parker, 995) combined with Tikhonov regularisation (Lawson and Hansen, 974). The L-curve method (Hansen, 992) is used to choose the optimal regularisation parameter. PERMEABILITY CROSS-CORRELATIONS Permeability calculation - Permeability is calculated using the mesoscopic lattice-boltzmann method (LB) (Martys and Chen, 996; Qian and Zhou, 998). It can be shown that the macroscopic dynamics of the solution of a discretized Boltzmann equation match the Navier-Stokes equation. Due to its simplicity in form and adaptability to complex flow geometries one of the most successful applications of the LB method has been to flow in ous media (Chen et al., 992; Frisch et al., 986; Rothman, 988; Ferreol and Rothman, 995; Martys and Chen, 996). In this study we applied a pressure gradient by a body force (Ferreol and Rothman, 995), used closed boundary ditions perpendicular to the flow and mirrored boundaries parallel to the pressure gradient, resulting in a system size of L L 2L. Permeability was measured over the L L L original image of the simulated system. Formation factor - The ductivity calculation is based on a solution of the Laplace equation with charge servation boundary ditions and has been detailed before (Arns et al., 200). We assign to the matrix phase of the sandstone a ductivity σ m = 0 and to the (fluidfilled) e phase a normalized ductivity σ fl =. A potential gradient is applied in each coordinate direction, and the system relaxed using a jugate gradient technique to evaluate the field. The formation factor given by F = σ fl /σ eff, is used. Permeability correlations - Permeability correlations are usually based on the logarithmic mean of the relaxation time [ = exp i a i log(t 2i ) i a i ], (5) which is assumed to be related to an average V p /S p or e size. Commonly used NMR response/permeability correlations include the osity as in (Banavar and Schwartz, 987; Kenyon et al., 988) k = a b T c 2lm, (6) with classical factors a =, b = 4, c = 2, or the Formation factor F as in k = af b T c 2lm, (7) with standard factors b =, c = 2. The use of c = 2 in Eqns. 6-7 implies a unit of a as surface relaxivity squared. In our fits of the permeability correlations we scale the value of a by /(6ρ) 2 to make the prefactor dimensionless; this implies that any difference in the prefactor arises only from structural influences. The length scale associated with T 2 (the e size derived from the 5

6 NMR signal) is given by d = 6 ρ. We have previously shown that one can obtain useful estimates of petrophysical properties from simulations at the scale of a few mm 3 (Arns et al., 2004a; Arns et al., 2005a; Arns et al., 2005b). Each sample is divided into subregions and for each subregion of all samples the permeability and NMR surface relaxation response is calculated. This means that one obtains 00 individual samples per image (see Table 2) and therefore a relationship between k,, for each rock. In all fits the mean residual error 2 s 2 (log0 (k calc ) log = 0 (k emp )). (8) n 2 is minimised and the correlation coefficient (kemp k emp )(k calc k calc ) R = [ (kemp k emp ) 2 (k calc k calc ) 2] /2 calculated. (9) Table 2: Analysed sections of the samples. n x n y n z notes the size (in voxel) of the sections for calculating all partitions and morphological analysis. n is the voxel length of the cubic subsets used for the derivation of cross-correlations of physical properties over osity and N their number. For permeability and ductivity there are 3N results (along the x-,y-, and z-axis of the tomogram). Sample ɛ[µm] n x n y n z n N FS FS US CRS OC VC VC RESULTS Surface relaxivity heterogeneity in sandstones - In this section we analyse the effect of surface relaxivity heterogeneity on the permeability-nmr cross-correlations for a set of four sandstones. In all simulations the average surface relaxivity was kept stant at ρ = 6 µm/s. We apply all five surface relaxivity distributions introduced above. In particular, we compare stant surface relaxivity results to scenarios for e partitioning, grain mineralogy, e size, and capillary pressure history. Fig. 3 shows the influence of the heterogeneity in ρ on the log mean relaxation times. [d] Figure 3: Log mean (surface) relaxation times for four sands and five different surface relaxivity distributions. FS, FS2, US, [d] CRS. 6

7 In all cases the added heterogeneity increases the log mean relaxation time by a significant amount. This is expected, since a part of the diffusing spin population will need to travel a larger distance to reach a more relaxive surface, and the diffusion length scale l D t, where t is time. The type of heterogeneity affects differently for the different samples. For all four sandstones, the micro-environment scenario based on the topological e partitioning shows a strong effect, extending the mean relaxation times by more than 0% (Fig. 3). Further, samples FS and FS2 are both solidated sands from the same core plug, imaged at very similar resolution, but as shown in Fig. 3a,b show differences in the log mean surface relaxivity for the grain partitioning scenario. This is likely due to e size and shape, since each e is surrounded by a number of grain surfaces, and they will be closer together for the more compacted and anisotropic region of the core, represented by sample FS2, which also has lower osity. This in turn allows diffusional averaging to take place more effectively. The strong effect of mineralogy exhibited by sample CRS is an effect caused by the presence of clay. Although clay is counted as solid, it has not been partitioned as grains, and therefore been assigned a stant surface relaxivity. This causes spins escaping from a low relaxivity surface, to frequently find only a surface of average relaxivity (the clay fraction is 30%, compared with 0% osity of the sample). This prevents diffusional averaging and the relaxation time increases. Potentially, if information about the underlying stant surface relaxivity is available, the increase in log mean relaxation time, or even the change in the relaxation time spectrum itself, could be used to deduce information about the heterogeneity, e.g. its characteristic length scale and possible mechanisms. Proceeding to NMR-permeability correlations, we ret the correlation coefficients and quality of fits in Table 3 and Figs. 4 and 5 for the two different correlations given in Eqns. 6 and 7 respectively. It can immediately be seen that the cross-correlations are improved by adding a tortuosity parameter according to Eqn. 7. Surface relaxivity heterogeneity has only a small effect on the correlation coefficients R i of the two empirical equations. Also, the scatter of the cross-plot represented by s i is largely unaffected by heterogeneity. However, the prefactors for the permeability-nmr correlations change by up to 25% due to surface relaxivity heterogeneity effects for the solidated sands, and by a larger factor for the unsolidated sand (CRS). This is of the same order as structural effects for Eqn. 6, while structural effects are dominant in Eqn. 7. The added heterogeneity in ρ, as compared to the ρ = st scenario, always leads to a decrease in the prefactor a of the cross-correlations. Table 3: Dependence of the correlations between NMR response and permeability on surface relaxivity heterogeneity for sandstones. Considered are stant surface relaxivity (index ), and distributions based on partitions by es (), grains (), covering radius (), and capillary drainage (). The mean surface relaxivity is ρ = 6 µm/s and the bulk relaxation time T 2b = 2.876s. The prefactor a, mean residual error s 2 and correlation coefficient R have indices. refers to Eqn. 6 and 2 refers to Eqn. 7. a Sample 36ρ 0s 2 00a 2 R 2 36ρ 2 0s 2 2 R 2 FS FS FS FS FS FS FS FS FS FS US US US US US CRS CRS CRS CRS CRS Thus, use of the prefactor for stant ρ will lead to over-predictions of permeability. Since the scatter of the data and the correlation coefficients are robust, the NMRpermeability correlations stay predictive in the presence of surface relaxivity heterogeneity as sidered here, but the change of the prefactor in the correlation indicates that surface relaxivity heterogeneity gives a different apparent ρ for the sample. Surface relaxivity heterogeneity in carbonates - In this section we analyse the effect of surface relaxivity heterogeneity on the permeability-nmr cross-correlations for a set of three carbonates. In all simulations the average surface relaxivity was kept stant at ρ =.5 µm/s. We apply four of the five surface relaxivity distributions introduced above (stant, e topology, e size, and invasion history) - heterogeneity caused by grain mineralogy was not sidered, since the notion of grains in carbonates is not straightforward. We show the influence of the heterogeneity in ρ on the log mean surface 7

8 [d] [d] Figure 4: NMR-permeability cross-correlations for sands using Eqn. 6 for five different surface relaxivity distributions. FS, FS2, US, [d] CRS. Figure 5: NMR-permeability cross-correlations for sands using Eqn. 7 for five different surface relaxivity distributions. FS, FS2, US, [d] CRS. 8

9 relaxation times in Fig. 6. As for the sandstones, heterogeneity causes an increase in the log mean relaxation time. Again, the impact of the different heterogeneity types is sample dependent. Compared to the sandstones, the values of for the carbonates are much larger, and would be as large as 25s, if we ignored the effect of bulk relaxation. This is clearly aphysical. Looking at the individual samples we see that of the three carbonates the surface relaxivity heterogeneity effect caused through the e body partitioning is smallest for sample OC, similar for VS, and large for VS2, with some very high values at low osity. This is not too surprising, since the coupling between es is increasing with osity, and sample OC is very well nected with a osity of 50%, while for VS2 there are some vugs, which have been assigned a small surface relaxivity (0.5µ m/s), and which are poorly nected, preventing effective mixture of the surface relaxation modes. While in the sandstones the most imtant surface relaxivity heterogeneity mechanism was reflected by the e micro-environment scenario, here the strongest effect is exhibited by the capillary pressure scenario. This can be understood by the relative trast in diameter of different pathways through microosity and macroosity. The heterogeneity caused by this trast is distributed over whole regions of es, and prevents effective diffusional averaging between the regions of different surface relaxivity. In trast, the e size partitioning does not tain the nectedness information, surface relaxivity is distributed much more evenly, and motional averaging takes place. Table 4 and Figs. 7 and 8 sumarise the results for the NMR-permeability correlations. Again we note, that correlation coefficients improve significantly through the inclusion of a tortuosity measurement (Eqn. 7). As with the sandstones, the scatter of the data within each sample across different surface relaxivity distributions is essentially unchanged. Compared to the sandstones, the scatter in the permeability-nmr prediction is significantly larger. This could be a matter of scale, since for the purpose of deriving the osity-permeability curve, we needed enough subsets, which in turn dictated a maximal size on the subsets selected. They were typically about voxel in size, as compared to voxel for the sandstones (see Table 2). As Table 4 shows, the prefactor in the empirical correlations varies by about 0% due to effects of variable ρ, which is less than for the four sandstones of Table 3. Compared to the change of prefactor of two orders of magnitude for structural differences, this is small. The reason for this effect is diffusion coupling, as discussed above. Consistent with the discussion on diffusion coupling and heterogeneity length scales, the largest effect of varying ρ is shown by the capillary pressure / fluid saturation history scenario Figure 6: Log mean (surface) relaxation times for three carbonates and four different surface relaxivity distributions. OC, VC, VC2. 9

10 Figure 7: NMR-permeability cross-correlations for carbonates using Eqn. 6 for four different surface relaxivity distributions. OC, VC, VC2. Figure 8: NMR-permeability cross-correlations for carbonates using Eqn. 7 for four different surface relaxivity distributions. OC, VC, VC2. 0

11 Table 4: Dependence of the correlations between NMR response and permeability on surface relaxivity heterogeneity for carbonates. Considered are stant surface relaxivity (index ), and distributions based on partitions by es (), covering radius (), and capillary drainage (). The mean surface relaxivity is ρ = 6 µm/s. The prefactor a, mean residual error s 2 and correlation coefficient R have indices. refers to Eqn. 6 and 2 refers to Eqn. 7. a Sample 36ρ s 2 a 2 R 2 36ρ 2 s 2 2 R 2 OC OC OC OC VC VC VC VC VC VC VC VC CONCLUSIONS We presented a sensitivity study about the influence of surface relaxivity heterogeneity on prefactors and sistency of NMR-permeability correlations for a number of sandstone and carbonate samples. It was found that with increasing surface heterogeneity length scale the log mean relaxation time increases as a result of leaving the fast diffusion limit. This leads to a decrease in the prefactors of the permeability cross-correlations sidered. The correlation coefficients and scatter in the permeability cross-correlations of the two predictive equations were not affected by the presence of surface relaxivity heterogeneity. We note, that for most samples the cross-correlation involving the formation factor gives a tilted slope. This is an effect which lends itself to further study using an Xray micro-ct approach. For sandstones, the effect of surface relaxivity heterogeneity was strongest in the e micro-environment scenario based on a topological partitioning of the e space. The heterogeneity caused about a 25% change of the prefactor of the permeability correlations, which is of the same order as the change of the prefactor between samples based on structure. Thus its effect cannot be ignored, if the surface relaxivity distributions chosen here are realistic. For carbonates, the effect of surface relaxivity heterogeneity in the e micro-environment scenario was significant. However, the strongest effect of surface relaxivity heterogeneity was exhibited in the capillary drainage history scenario. Surface relaxivity heterogeneity caused about a 0% change of the prefactor of the permeability correlations, which is two orders of magnitude smaller than the change of the prefactor between samples based on structure. It should be possible to use the change in prefactor to derive information about the length scale of the heterogeneity in a site-specific text. However, sidering the natural heterogeneities of carbonates and the large effect of structural heterogeneity, an application of this technique to carbonates would require careful calibration techniques. We believe that in principal the imtant parameter trolling the effect of surface relaxivity heterogeneity on permeability correlations is its characteristic length scale. Mixed wettability scenarios, where e.g. the surface relaxivity might scale with ρ(r) r rather than ρ(r) /r as in the fluid saturation scenarios sidered here, are expected to give similar results. ACKNOWLEDGEMENTS CHA acknowledges the Australian Government for their supt through the ARC grant scheme (DP055885). The authors also thank the Australian Partnership for Advanced Computing (APAC) for their supt through the expertise program and APAC and the ANU Supercomputing Facility for very generous allocations of computer time. REFERENCES CITED Arns, C. H., Knackstedt, M. A., Pinczewski, W. V., and Lindquist, W. B., 200, Accurate estimation of transt properties from microtomographic images: Geophysical Research Letters, 28, no. 7, Arns, C. H., Knackstedt, M. A., Pinczewski, W. V., and Garboczi, E. G., 2002, Computation of linear elastic properties from microtomographic images: Methodology and agreement between theory and experiment: Geophysics, 67, no. 5, Arns, C. H., Averdunk, H., Bauget, F., Sakellariou, A., Senden, T. J., Sheppard, A. P., Sok, R. M., Pinczewski, W. V., and Knackstedt, M. A., June 2004, Digital core laboratory: Analysis of reservoir core fragments from 3D images: SPWLA, 45th Annual Logging Symposium, paper EEE.

12 Arns, C. H., Knackstedt, M. A., Pinczewski, W. V., and Martys, N., 2004, Virtual permeametry on microtomographic images: Journal of Petroleum Science and Engineering, 45, no. -2, Arns, C. H., Sakellariou, A., Senden, T. J., Sheppard, A. P., Sok, R., Pinczewski, W. V., and Knackstedt, M. A., 2005a, Digital core laboratory: Reservoir core analysis from 3D images: Petrophysics, 46, no. 4, Arns, C. H., Sheppard, A. P., Sok, R. M., and Knackstedt, M. A., June 2005b, NMR petrophysical predictions on digitized core images: SPWLA, 46th Annual Logging Symposium. Arns, C. H., Knackstedt, M. A., and Martys, N., 2005c, Cross-property correlations and permeability estimation in sandstone: Physical Review E, 72, Banavar, J. R., and Schwartz, L. M., 987, Magnetic resonance as a probe of permeability in ous media: Physical Review Letters, 58, Bergman, D. J., Dunn, K.-J., Schwartz, L. M., and Mitra, P. P., 995, Self-diffusion in a periodic ous medium: A comparison of different approaches: Physical Review E, 5, Brownstein, K. R., and Tarr, C. E., 979, Imtance of classical diffusion in NMR studies of water in biological cells: Physical Review A, 9, Chen, H., Chen, S., and Matthaeus, W. H., 992, Recovery of the Navier-Stokes equations using a latticegas Boltzmann method: Phys. Rev. A, 45, no. 8, R5339 R5342. Ferreol, B., and Rothman, D. H., 995, Lattice- Boltzmann simulations of flow through Fontainebleau sandstone: Transt in Porous Media, 20, Frisch, U., Hasslacher, B., and Pomeau, Y., 986, Lattice-gas automata for the Navier-Stokes equation: Physical Review Letters, 56, no. 4, Hansen, P. C., 992, Analysis of discrete ill-posed problems by means of the L-curve: SIAM review, 34, Hilpert, M., and Miller, C. T., 200, Pore-morphology based simulation of drainage in totally wetting ous media: Advances in Water Resources, 24, Kenyon, W. E., Day, P., Straley, C., and Willemsen, J., 986, Compact and sistent representation of rock NMR data from permeability estimation: SPE, Proc. 6st Annual Technical Conference and Exhibition. Kenyon, W. E., Day, P., Straley, C., and Willemsen, J., 988, A three part study of NMR longitudinal relaxation properties of water saturated sandstones: SPE formation evaluation, 3, no. 3, , SPE Kenyon, W. E., 992, Nuclear magnetic resonance as a petrophysical measurement: Nuclear Geophysics, 6, Knackstedt, M. A., Arns, C. H., Limaye, A., Sakellariou, A., Senden, T. J., Sheppard, A. P., Sok, R. M., Pinczewski, W. V., and Bunn, G. F., March 2004, Digital core laboratory: Properties of reservoir core derived from 3D images: SPE, Asia Pacific Conference on Integrated Modelling for Asset Management, SPE Lawson, C. L., and Hansen, R. J., 974, Solving least squares problems: Prentice-Hall. Martys, N. S., and Chen, H., 996, Simulation of multicomponent fluids in complex three-dimensional geometries by the lattice Boltzmann method: Physical Review E, 53, no., Mendelson, K. S., 990, Percolation model of nuclear magnetic relaxation in ous media: Physical Review B, 4, Qian, Y.-H., and Zhou, Y., 998, Complete Galileaninvariant lattice BGK models for the Navier-Stokes equation: Europhysics Letters, 42, no. 4, Rothman, D. H., 988, Cellular-automaton fluids: A model for flow in ous media: Geophysics, 53, no. 4, Saadatfar, M., Turner, M. L., Arns, C. H., Averdunk, H., Senden, T. J., Sheppard, A. P., Sok, R. M., Pinczewski, W. V., Kelly, J., and Knackstedt, M. A., June 2005, Rock fabric and texture from digital core images: SPWLA. Sakellariou, A., Senden, T. J., Sawkins, T. J., Knackstedt, M. A., Turner, M. L., Jones, A. C., Saadatfar, M., Roberts, R. J., Limaye, A., Arns, C. H., Sheppard, A. P., and Sok, R. M., August 2004, An x-ray tomography facility for quantitative prediction of mechanical and transt properties in geological, biological and synthetic systems SPIE, Developments in X-Ray Tomography IV, Sakellariou, A., Sawkins, T.-J., Senden, T.-J., and Limaye, A., 2004b, X-ray tomography for mesoscale physics applications: Physica A, 339, no. -2, Sen, P. N., Straley, C., Kenyon, W. E., and Whittingham, M. S., 990, Surface-to-volume ratio, charge 2

13 density, nuclear magnetic relaxation, and permeability in clay-bearing sandstones: Geophysics, 55, no., Serra, J., 982, Image analysis and mathematical morphology:, volume,2 Academic Press, Amsterdam. Sheppard, A. P., Sok, R. M., and Averdunk, H., 2004, Techniques for image enhancement and segmentation of tomographic images of ous materials: Physica A, 339, no. -2, Song, Y.-Q., Ryu, S., and Sen, P. N., 2000, Determining multiple length scales in rocks: Nature, 406, no. 3, Song, Y.-Q., 2003, Using internal magnetic fields to obtain e size distributions of ous media: Concepts in Magnetic Resonance Part A, 8A, no. 2, Stark, P., and Parker, R., 995, Bounded-variable least-squares: an algorithm and applications: Computational Statistics, 0, no. 2, Thovert, J.-F., Yousefian, F., Spanne, P., Jacquin, C. G., and Adler, P. M., 200, Grain restruction of ous media: Application to a low-osity Fontainebleau sandstone: Physical Review E, 63, Wayne, R. C., and Cotts, R. M., 966, Nuclearmagnetic-resonance study of self-diffusion in a bounded medium: Physical Review, 5, no., structures, and tomographic image processing. M. A. Knackstedt - Mark Knackstedt was awarded a BSc in 985 from Columbia University and a PhD in Chemical Engineering from Rice University in 990. He is a Professor at and Head of the Department of Applied Mathematics at the Australian National University and a visiting Fellow at the School of Petroleum Engineering at the University of NSW. His work has focussed on the characterisation and realistic modelling of disordered materials. His primary interests lie in modelling transt, elastic and multi-phase flow properties and development of 3D tomographic image analysis for complex materials. M. Saadatfar - was awarded a BSc from the Institute of Advanced Studies in Zanjan, Iran and successfully defended his PhD from the ANU in His main research interests lie in the grain partitioning of tomographic images, physics of granular matter and large scale simulation of the elastic properties of ous materials. ABOUT THE AUTHORS C. H. Arns - Christoph Arns was awarded a Diploma in Physics (996) from the University of Technology Aachen and a PhD in Petroleum Engineering from the University of New South Wales in He is a Research Fellow at the Department of Applied Mathematics at the Australian National University. His research interests include the morphological analysis of ous complex media from 3D images and numerical calculation of transt and linear elastic properties with a current focus on NMR responses and dispersive flow. Member: AM- PERE, ANZMAG, D, SPWLA. A. P. Sheppard - Adrian Sheppard received his B.Sc. from the University of Adelaide in 992 and his PhD in 996 from the Australian National University and is currently a Research Fellow in the Department of Applied Mathematics at the Australian National University. His research interests are network modelling of multiphase fluid flow in ous material, topological analysis of complex 3