prediction of permeability from nmr response: surface ...people.physics.anu.edu.au/~tjs110/2005-2008...

SPWLA 47th Annual Logging Symposium, June 4-7, 2006

Prediction of permeability from NMR response: surfacerelaxivity heterogeneity

C. H. Arns1,*, A. P. Sheppard1, M. Saadatfar1, and M. A. Knackstedt1,2

1Department of Applied Mathematics, Research School of Physical Sciences and Engineering,Australian National University, Canberra, Australia

2School of Petroleum Engineering, University of New South Wales, Sydney, Australia* Corresponding Author: [email protected]

Copyright 2006, held jointly by the Society of Petrophysicists and Well Log An-alysts (SPWLA) and the submitting authors.

This paper was prepared for presentation at the SPWLA 47th Annual LoggingSymposium held in Veracruz, Mexico, June 4-7, 2006.

ABSTRACT

NMR responses are commonly used in reservoir char-acterization to estimate pore-size information, formationpermeability, as well as fluid content and type. Diffi-culties arise in the interpretation of NMR response asan estimator of permeability due to internal gradients,diffusion coupling, surface-relaxivity heterogeneity, anda possible breakdown of correlations between pore andconstriction sizes. Here we consider several scenarios ofsurface relaxivity heterogeneity for a set of sandstonesand a set of carbonate rock in a numerical NMR studybased on Xray micro-CT data.

We have previously demonstrated the ability to image,visualize, and characterize sedimentary rock in three di-mensions (3D) at the pore/grain scale via X-ray micro-computed tomography. We also numerically tested theinfluence of structure and diffusion-coupling on NMR-permeability correlations. Here we consider surface re-laxivity heterogeneities due to pore partitioning, miner-alogy, pore size, and saturation history.

We partition the pore space and solid phase into regionsof pores and grains. These partitions could reflect differ-ent mineralogy for weakly coupled pore systems, or dif-ferences in mineralogy for the grains. Further, we use amorphological drainage simulation technique to partitionthe pore space in terms of invasion radius or throat size,reflecting surface relaxivity heterogeneity due to the sat-uration history of immiscible fluids, which could causee.g. pressure dependent adsorption on surfaces and/orchanges in wettability. Finally, we use the concept ofcovering radius to assign a surface relaxivity due to poresize.

For each sample, four sandstones and three carbonates,we consider distributions of surface relaxivity based on

above partitions, keeping the mean surface relaxivity con-stant, simulate the magnetisation decay, and derive a poresize distribution through an inverse Laplace transform as-suming constant surface relaxivity.

Further, we test the effect of these heterogeneities onNMR-permeability correlations based on the log-meanof the relaxation time distributions for two frequentlyused empirical NMR-permeability cross-correlations. Atthe scales probed here, surface relaxivity heterogeneitychanges the prefactor in the equations for sandstones onlyminimally, while the prefactor is changes orders of mag-nitudes for carbonates. The influence of surface relaxiv-ity heterogeneity on the quality of the fit for individualsamples is small.

INTRODUCTION

The estimation of permeability through cross-correlationsfrom other physical measurements on rocks is a classicaltask in petrophysics and has a long history in well log-ging. Of the measurements available, the NMR relax-ation is the one, which typically correlates best to per-meability (e.g. (Sen et al., 1990)). One reason is thatthe estimation of permeability requires length scale in-formation, which the NMR relaxation response provides,since the relaxation process is typically controlled by thesurface to volume ratio of the pore space (Wayne andCotts, 1966; Brownstein and Tarr, 1979; Kenyon et al.,1986; Kenyon et al., 1988; Kenyon, 1992; Song et al.,2000). Under the assumption of constant surface relax-ivity, weak coupling between pores, and fast diffusionwithin pores, the magnetisation decays uniformly withineach pore and the decay can be written as

M(t) = M0(t)

N∑

p=1

ap exp

[

− t

T2p

]

, (1)

where M0 is the initial magnetization, p is a pore label, ai

is the fractional pore volume, t is time, and the transverserelaxation time T2 of the pores is given by

1

T2p=

1

T2b+ ρ

Sp

Vp. (2)

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Here T2b is the bulk relaxation time, Sp/Vp is the surface-to-pore-volume ratio of the pore space, and ρ the surfacerelaxation strength. The multi-exponential distributioncorresponds to a partition of the pore space into N groupsbased on the Sp/Vp values of the pores. This is the clas-sical picture of the NMR relaxation response used e.g. instandard NMR logging tools to derive a relaxation timeor pore size distribution. Analog expressions exist forderiving length scales using higher diffusion eigenmodes(Song et al., 2000; Song, 2003).

Apart from weak coupling between pores and backgroundmicroporosity, which we discussed in a previous paper(Arns et al., 2005b), above interpretation technique as-sumes a constant surface relaxivity. In this study, we con-sider the impact of surface relaxivity heterogeneity onNMR-permeability correlations by numerically derivingNMR responses and transport properties under very con-trolled conditions on realistic microstructures, using im-ages acquired by X-ray µCT techniques (Sakellariou etal., 2004b) and established algorithms to calculate trans-port properties on those images (Arns et al., 2001; Arnset al., 2002; Arns et al., 2004b; Knackstedt et al., 2004;Arns et al., 2005a; Arns et al., 2005c). The paper is or-ganised as follows: in the next section we introduce theimages used in this study and the partitioning techniquesneeded to assign surface relaxivity distributions to thesurface area of the rocks. The following section detailsthe NMR response simulation with a focus on the differ-ent surface relaxivity distributions employed. The thirdsection reviews two common NMR-permeability cross-correlations, before we present results in section four anddraw conclusions in a final section.

IMAGE PROCESSING

Samples - In this study, we consider the impact of sur-face relaxivity heterogeneity on NMR-permeability cor-relations for four sandstones and three carbonates. Allsamples have been imaged on the ANU high-resolutionX-ray µCT facility (Sakellariou et al., 2004b; Sakellariouet al., 2004a). A slice of the Xray density map of eachsample is shown in Fig. 1 and basic features of the sam-ples are summarized in Table 1. Two of the sandstonesare fluvial deposits (Castlegate sandstone, FS1 and FS2),taken from top and middle sections of the same plug, thethird is an unconsolidated sand (US), and the fourth is aclayey Regolith sample (CRS). Of the carbonates, one isan outcrop limestone (OL) originating from South Aus-tralia, while the others are vuggy carbonates (VC1 andVC2) of West Texan and Middle Eastern origin with var-ious degrees of interconnectivity.

Segmentation - Partitioning of the Xray-density data intotwo (solid, void) or three phases (solid, clay, void) is per-

[a] [b]

[c] [d]

[e] [f]

[g] [h]

Figure 1: Slices through the Xray density maps of thesandstone [a-d] and carbonate [e-g] samples comprisingthis study. [a] Castlegate sandstone (FS1), [b] same as[a], 2cm apart (FS2), [c] unconsolidated sand (US), [d]clayey Regolith (CRS), [e] outcrop limestone (OC), [f-g] vuggy carbonates (VC1 and VC2).; [h] three-phasesegmentation of [d].

Table 1: Characteristics of the sandstone (top four) andcarbonate samples (bottom three) of this study. ε is thevoxel size, and φt the total resolved image porosity.

Sample Description ε [µm] φt

FS1 fluvial 5.60 .258FS2 fluvial 4.93 .216US unconsolidated 6.72 .292CRS clayey Regolith 9.41 .098

OC outcrop limestone 3.02 .507VC1 vuggy 2.60 .187VC2 vuggy 8.51 .084

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formed in a multi-stage procedure using a modified ver-sion of the converging active contour method outlinedin (Sheppard et al., 2004), which we used before (Arnset al., 2005b). An example of a resulting segmentationis given in Fig. 2b, and for a three-phase segmentationof sample CRS in Fig. 1h. In this study we chose tointerpret the clayey region at intermediate Xray-densityof sample CRS as solid phase. The same applies to themicroporous regions of the carbonate samples VC1 andVC2.

Topological pore partitioning - To assign surface relax-ivity based on the topological concept of pores, we needto partition the pore space into simple geometric cells(pore bodies), separated by narrow constrictions taken tobe volumeless (throats). An account of this techniquehas been given elsewhere (Arns et al., 2005b). It ba-sically involves the derivation of the medial axis of theconnected pore space, followed by a topology conserv-ing breakdown of the medial axis into separate bodiesusing distance information, and a final step of mergingpore centers at small separation. An example of a result-ing pore partitioning is given in Fig. 2c. This partition-ing can reflect different micro-environments, which areweakly coupled through narrow constrictions.

Grain partitioning - Grain partitioning has only been car-ried out on the sandstones. While it is essentially theinverse problem of pore partitioning, it is significantlysimpler. We used a method based on identifying water-sheds of the Euclidean distance map of the grain space(Saadatfar et al., 2005). An example of a resulting porepartitioning is given in Fig. 2d.

Partitioning based on pore size - To partition the porespace based on the geometric concept of pore size, weuse the maximal inscribed radius partitioning. This par-titioning technique is based on a classical mathemati-cal description of the morphology in terms of basic ge-ometrical quantities (Serra, 1982). More complete andgeneric descriptions of the basic concepts and techniquesare given elsewhere (Hilpert and Miller, 2001; Thovert etal., 2001; Arns et al., 2005c). In a succession of morpho-logical operations each voxel gets assigned the radius ofthe largest sphere which lies within the pore space andcovers the voxel (covering radius transform, CRT). Anexample of a resulting inscribed radius map is given inFig. 2e. This partitioning can mimic the effects of inter-nal gradients (stronger in smaller pores), or of fluid dis-tribution history, where it reflects the fluid distribution atvariable equilibrium pressure.

Invasion radius partitioning - The connected pore spaceis partitioned by assigning to each voxel the radius ofthe largest sphere, which can penetrate from the outer

boundary of the sample to cover the voxel. This simula-tion technique mirrors the boundary conditions of stan-dard mercury intrusion experiments, e.g. a fixed capil-lary pressure is associated with a pore entry radius (cap-illary drainage transform, CDT). The center of the invad-ing sphere is allowed to move such that the sphere doesnot overlap the solid (Hilpert and Miller, 2001). An ex-ample of a resulting invasion radius map is given in Fig.2f. This partitioning reflects the history of the fluid dis-tribution at breakthrough pressure.

NMR RESPONSE SIMULATION

Surface relaxation simulation - The spin relaxation of asaturated porous system is simulated by using a latticerandom walk method (Mendelson, 1990; Bergman et al.,1995). Initially the walkers are placed randomly in the3D pore space. At each time step the walkers are movedfrom their initial position to a neighboring site and theclock of the walker advanced by ∆t = ε2/(6D0), whereε is the lattice spacing and D0 the bulk diffusion constantof the fluid, reflecting Brownian dynamics. The lattice ismade periodic by mirroring the structure in all directions.An attempt to go to a site of another phase will kill thewalker with probability ν/6, 0 ≤ ν ≤ 1 (Mendelson,1990). The killing probability ν is related to the surfacerelaxivity ρ via

Aν =ρε

D0

+ O

(

(ρε

D

)2)

, (3)

where A is a correction factor of order 1 (here, we takeA = 3/2) accounting for the details of the random walkimplementation (Bergman et al., 1995).

Surface relaxivity distributions - Here we adapt Eqn. (2)to varying surface relaxivity by accounting for N activesurface patches of surface area Sp and surface relaxivityρp within each well-connected region of volume Vp

1

T2p=

1

T2b+

1

NVp

N∑

i=1

ρpiSpi =1

T2b+

1

T2Sp. (4)

Eqn. 1 can then be applied again with the more generaldefinition of the surface relaxation time T2Sp of a con-nected region p. We are now in a position to assign a dis-tribution of surface relaxivities, while keeping the meansurface relaxation time 〈T2Sp〉 constant.

In order to explore the influence of surface relaxivity het-erogeneity at different length scales on permeability pre-dictions and pore size distributions, we assign spatiallyvarying surface relaxivities ρ(x̄) to the pore-solid inter-face using five different methods:

1. Constant surface relaxivity: ρ(x̄) = ρ = const.We use the index con to indicate this.

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[a] [b]

[c] [d]

[e] [f]

Figure 2: Illustration of the image segmentation and partitioning steps for a clean sandstone. Shown are slices of acentral subsection of a 20483 consolidated sand dataset (4002 voxel, voxel size 4.93µm). All morphological calcula-tions were carried out on a much larger subset (nx × ny × nz in Table 2), e.g. boundary effects are minimal. [a] Xraydensity, [b] segmented image, [c] topological pore micro-environment partitioning, [d] grain partitioning, [e] pore sizeor irreducible saturation partitioning, [f] invasion radius partitioning.

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2. Using the topological pore partitioning (Fig. 2c,index por), assign each pore with equal probabilityeither the surface relaxivity ρ1 = 5

3ρ or ρ2 = 1

3ρ,

i.e. ρ1/ρ2 = 5, where ρ is the average surfacerelaxivity.

3. Using the grain partitioning (Fig. 2d, index grn),assign each grain with equal probability either thesurface relaxivity ρ1 = 5

3ρ or ρ2 = 1

3ρ.

4. Using the geometric pore size partitioning or cov-ering radius transform (Fig. 2e, index crt), assigneach surface voxel a surface relaxivity based onan inverse relationship: ρ(r) ∝ 1/r, where r de-notes the inscribed radius of the nearest pore voxel.The inverse relationship can reflect stronger inter-nal gradients in confined spaces. In terms of fluidsaturation history it could reflect time intervals, inwhich the large curvature of the non-wetting phaseallowed deposition of relaxive substances throughthe wetting phase.

5. Using the invasion radius partitioning or capillarydrainage transform (Fig. 2f, index cdt), assigneach surface voxel a surface relaxivity based onan inverse relationship: ρ(r) ∝ 1/r, where r notesthe invasion radius of the nearest pore voxel. Thesame argument as above applies.

All distributions are renormalised, such that 〈ρ(x̄)〉 = ρ,after the surface relaxivity assignment stage of the algo-rithm. Here the average runs over all pore-solid surfacesof the sample. In summary, the different distributions ofsurface relaxivity are assigned such that the mean sur-face relaxivity is the same for all distributions for a givensample. It should be mentioned, that the geometricallybased partitions show much more variability of surfacerelaxivity compared to the topologically based pore andgrain partitions by definition. The expected effect wouldbe a better mixing in terms of surface relaxivities by dif-fusion. Note, that modes of small surface relaxivity overlarge areas - which can appear in large pores using e.g.the pore size surface relaxivity partitioning - are damped,since we always consider bulk relaxation using the bulkrelaxation rate of water (T2b = 2.876s).

Inverse Laplace transform - The relaxation time distri-bution is derived by fitting a sum of exponentials to themagnetisation decay M(t) using a bounded least squaresolver (Stark and Parker, 1995) combined with Tikhonovregularisation (Lawson and Hansen, 1974). The L-curvemethod (Hansen, 1992) is used to choose the optimal reg-ularisation parameter.

PERMEABILITY CROSS-CORRELATIONS

Permeability calculation - Permeability is calculated us-ing the mesoscopic lattice-Boltzmann method (LB) (Mar-tys and Chen, 1996; Qian and Zhou, 1998). It can beshown that the macroscopic dynamics of the solution of adiscretized Boltzmann equation match the Navier-Stokesequation. Due to its simplicity in form and adaptability tocomplex flow geometries one of the most successful ap-plications of the LB method has been to flow in porousmedia (Chen et al., 1992; Frisch et al., 1986; Rothman,1988; Ferreol and Rothman, 1995; Martys and Chen,1996). In this study we applied a pressure gradient bya body force (Ferreol and Rothman, 1995), used closedboundary conditions perpendicular to the flow and mir-rored boundaries parallel to the pressure gradient, result-ing in a system size of L × L × 2L. Permeability wasmeasured over the L × L × L original image of the sim-ulated system.

Formation factor - The conductivity calculation is basedon a solution of the Laplace equation with charge conser-vation boundary conditions and has been detailed before(Arns et al., 2001). We assign to the matrix phase ofthe sandstone a conductivity σm = 0 and to the (fluid-filled) pore phase a normalized conductivity σfl = 1. Apotential gradient is applied in each coordinate direction,and the system relaxed using a conjugate gradient tech-nique to evaluate the field. The formation factor given byF = σfl/σeff, is used.

Permeability correlations - Permeability correlations areusually based on the logarithmic mean T2lm of the relax-ation time

T2lm = exp

[∑

i ai log(T2i)∑

i ai

]

, (5)

which is assumed to be related to an average Vp/Sp orpore size. Commonly used NMR response/permeabilitycorrelations include the porosity φ as in (Banavar andSchwartz, 1987; Kenyon et al., 1988)

k = aφbT c2lm, (6)

with classical factors a = 1, b = 4, c = 2, or the Forma-tion factor F as in

k = aF bT c2lm, (7)

with standard factors b = −1, c = 2. The use of c =2 in Eqns. 6-7 implies a unit of a as surface relaxivitysquared. In our fits of the permeability correlations wescale the value of a by 1/(6ρ)2 to make the prefactor di-mensionless; this implies that any difference in the pref-actor arises only from structural influences. The lengthscale associated with T2 (the pore size derived from the

5

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NMR signal) is given by dT2lm = 6T2lmρ.

We have previously shown that one can obtain useful es-timates of petrophysical properties from simulations atthe scale of a few mm3 (Arns et al., 2004a; Arns et al.,2005a; Arns et al., 2005b). Each sample is divided intosubregions and for each subregion of all samples the per-meability and NMR surface relaxation response is calcu-lated. This means that one obtains 100 individual sam-ples per image (see Table 2) and therefore a relationshipbetween k, φ, T2lm for each rock. In all fits the meanresidual error

s2 =

∑

(log10

(kcalc) − log10

(kemp))

n − 2

2

. (8)

is minimised and the correlation coefficient

R =

∑

(kemp − kemp)(kcalc − kcalc)[∑

(kemp − kemp)2(kcalc − kcalc)2]1/2

(9)

calculated.

Table 2: Analysed sections of the samples. nx ×ny ×nz

notes the size (in voxel) of the sections for calculat-ing all partitions and morphological analysis. n is thevoxel length of the cubic subsets used for the derivationof cross-correlations of physical properties over porosityand N their number. For permeability and conductivitythere are 3N results (along the x-,y-, and z-axis of thetomogram).

Sample ε[µm] nx ny nz n N

FS1 5.6 960 1056 1560 200 140FS2 4.93 928 1074 1600 200 160US 6.72 360 400 960 180 20CRS 9.41 702 702 1872 234 72

OC 3.02 900 900 1800 300 54VC1 2.60 1032 672 1128 224 60VC2 8.51 1140 1140 1140 285 64

RESULTS

Surface relaxivity heterogeneity in sandstones - In thissection we analyse the effect of surface relaxivity hetero-geneity on the permeability-NMR cross-correlations fora set of four sandstones. In all simulations the averagesurface relaxivity was kept constant at 〈ρ〉 = 16µm/s.We apply all five surface relaxivity distributions intro-duced above. In particular, we compare constant sur-face relaxivity results to scenarios for pore partitioning,grain mineralogy, pore size, and capillary pressure his-tory. Fig. 3 shows the influence of the heterogeneity in ρon the log mean relaxation times T2lm.

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Figure 3: Log mean (surface) relaxation times T2lm forfour sands and five different surface relaxivity distribu-tions. [a] FS1, [b] FS2, [c] US, [d] CRS.

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In all cases the added heterogeneity increases the logmean relaxation time by a significant amount. This is ex-pected, since a part of the diffusing spin population willneed to travel a larger distance to reach a more relaxivesurface, and the diffusion length scale lD ∝

√t, where

t is time. The type of heterogeneity affects T2lm differ-ently for the different samples. For all four sandstones,the micro-environment scenario based on the topologi-cal pore partitioning shows a strong effect, extending themean relaxation times by more than 10% (Fig. 3). Fur-ther, samples FS1 and FS2 are both consolidated sandsfrom the same core plug, imaged at very similar reso-lution, but as shown in Fig. 3a,b show differences inthe log mean surface relaxivity for the grain partitioningscenario. This is likely due to pore size and shape, sinceeach pore is surrounded by a number of grain surfaces,and they will be closer together for the more compactedand anisotropic region of the core, represented by sampleFS2, which also has lower porosity. This in turn allowsdiffusional averaging to take place more effectively. Thestrong effect of mineralogy exhibited by sample CRS isan effect caused by the presence of clay. Although clayis counted as solid, it has not been partitioned as grains,and therefore been assigned a constant surface relaxivity.This causes spins escaping from a low relaxivity surface,to frequently find only a surface of average relaxivity (theclay fraction is 30%, compared with 10% porosity of thesample). This prevents diffusional averaging and the re-laxation time increases.

Potentially, if information about the underlying constantsurface relaxivity is available, the increase in log meanrelaxation time, or even the change in the relaxation timespectrum itself, could be used to deduce information aboutthe heterogeneity, e.g. its characteristic length scale andpossible mechanisms.

Proceeding to NMR-permeability correlations, we reportthe correlation coefficients and quality of fits in Table 3and Figs. 4 and 5 for the two different correlations givenin Eqns. 6 and 7 respectively. It can immediately beseen that the cross-correlations are improved by addinga tortuosity parameter according to Eqn. 7. Surface re-laxivity heterogeneity has only a small effect on the cor-relation coefficients Ri of the two empirical equations.Also, the scatter of the cross-plot represented by si islargely unaffected by heterogeneity. However, the pref-actors for the permeability-NMR correlations change byup to 25% due to surface relaxivity heterogeneity effectsfor the consolidated sands, and by a larger factor for theunconsolidated sand (CRS). This is of the same order asstructural effects for Eqn. 6, while structural effects aredominant in Eqn. 7. The added heterogeneity in ρ, ascompared to the ρ = const scenario, always leads toa decrease in the prefactor a of the cross-correlations.

Table 3: Dependence of the correlations between NMRresponse and permeability on surface relaxivity hetero-geneity for sandstones. Considered are constant surfacerelaxivity (index con), and distributions based on parti-tions by pores (por), grains (grn), covering radius (crt),and capillary drainage (cdt). The mean surface relax-ivity is 〈ρ〉 = 16µm/s and the bulk relaxation timeT2b = 2.876s. The prefactor a, mean residual error s2

and correlation coefficient R have indices. “1” refers toEqn. 6 and “2” refers to Eqn. 7.

Sample a1

36ρ2 10s2

1R1

100a2

36ρ2 10s2

2R2

FS1con .182 .0628 .689 1.02 .0160 .942FS1por .142 .0658 .681 .797 .0124 .961FS1grn .162 .0653 .686 .908 .0116 .969FS1crt .158 .0663 .686 .885 .0112 .966FS1cdt .170 .0650 .687 .953 .0124 .961

FS2con .228 .156 .620 .887 .0316 .959FS2por .170 .157 .620 .662 .0282 .961FS2grn .197 .155 .623 .768 .0267 .966FS2crt .170 .156 .625 .661 .0255 .966FS2cdt .179 .153 .626 .696 .0258 .970

UScon .202 .0155 .688 1.17 .0037 .939USpor .152 .0181 .655 .887 .0051 .892USgrn .175 .0162 .677 1.02 .0041 .928UScrt .168 .0178 .674 .979 .0041 .914UScdt .178 .0177 .672 1.03 .0041 .912

CRScon .265 1.57 .589 .577 1.28 .806CRSpor .183 1.63 .551 .398 1.33 .780CRSgrn .113 1.64 .547 .245 1.35 .770CRScrt .205 1.56 .600 .447 1.27 .813CRScdt .223 1.57 .613 .485 1.27 .820

Thus, use of the prefactor for constant ρ will lead toover-predictions of permeability. Since the scatter of thedata and the correlation coefficients are robust, the NMR-permeability correlations stay predictive in the presenceof surface relaxivity heterogeneity as considered here,but the change of the prefactor in the correlation indi-cates that surface relaxivity heterogeneity gives a differ-ent apparent ρ for the sample.

Surface relaxivity heterogeneity in carbonates - In thissection we analyse the effect of surface relaxivity hetero-geneity on the permeability-NMR cross-correlations fora set of three carbonates. In all simulations the averagesurface relaxivity was kept constant at 〈ρ〉 = 1.5µm/s.We apply four of the five surface relaxivity distributionsintroduced above (constant, pore topology, pore size, andinvasion history) - heterogeneity caused by grain miner-alogy was not considered, since the notion of grains incarbonates is not straightforward. We show the influ-ence of the heterogeneity in ρ on the log mean surface

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conporgrncrtcdt

[d]

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Figure 4: NMR-permeability cross-correlations forsands using Eqn. 6 for five different surface relaxivitydistributions. [a] FS1, [b] FS2, [c] US, [d] CRS.

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Figure 5: NMR-permeability cross-correlations forsands using Eqn. 7 for five different surface relaxivitydistributions. [a] FS1, [b] FS2, [c] US, [d] CRS.

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relaxation times T2lm in Fig. 6. As for the sandstones,heterogeneity causes an increase in the log mean relax-ation time. Again, the impact of the different heterogene-ity types is sample dependent. Compared to the sand-stones, the values of T2lm for the carbonates are muchlarger, and would be as large as T2lm ≈ 25s, if we ig-nored the effect of bulk relaxation. This is clearly aphys-ical. Looking at the individual samples we see that of thethree carbonates the surface relaxivity heterogeneity ef-fect caused through the pore body partitioning is smallestfor sample OC, similar for VS1, and large for VS2, withsome very high values at low porosity. This is not toosurprising, since the coupling between pores is increas-ing with porosity, and sample OC is very well connectedwith a porosity of 50%, while for VS2 there are somevugs, which have been assigned a small surface relaxiv-ity (0.5µ m/s), and which are poorly connected, prevent-ing effective mixture of the surface relaxation modes.While in the sandstones the most important surface relax-ivity heterogeneity mechanism was reflected by the poremicro-environment scenario, here the strongest effect isexhibited by the capillary pressure scenario. This can beunderstood by the relative contrast in diameter of differ-ent pathways through microporosity and macroporosity.The heterogeneity caused by this contrast is distributedover whole regions of pores, and prevents effective diffu-sional averaging between the regions of different surfacerelaxivity. In contrast, the pore size partitioning does notcontain the connectedness information, surface relaxivityis distributed much more evenly, and motional averagingtakes place.

Table 4 and Figs. 7 and 8 sumarise the results for theNMR-permeability correlations. Again we note, that cor-relation coefficients improve significantly through the in-clusion of a tortuosity measurement (Eqn. 7). As withthe sandstones, the scatter of the data within each sampleacross different surface relaxivity distributions is essen-tially unchanged. Compared to the sandstones, the scat-ter in the permeability-NMR prediction is significantlylarger. This could be a matter of scale, since for thepurpose of deriving the porosity-permeability curve, weneeded enough subsets, which in turn dictated a maximalsize on the subsets selected. They were typically about3003 voxel in size, as compared to 2003 voxel for thesandstones (see Table 2). As Table 4 shows, the prefactorin the empirical correlations varies by about 10% due toeffects of variable ρ, which is less than for the four sand-stones of Table 3. Compared to the change of prefactorof two orders of magnitude for structural differences, thisis small. The reason for this effect is diffusion coupling,as discussed above. Consistent with the discussion ondiffusion coupling and heterogeneity length scales, thelargest effect of varying ρ is shown by the capillary pres-sure / fluid saturation history scenario.

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[b]

0 0.1 0.2 0.3 0.4 0.5φ

1

1.5

2

2.5

T2l

m [s

]

conporcrtcdt

[c]

0.04 0.06 0.08 0.1 0.12φ

1.8

2

2.2

2.4

2.6

2.8

T2l

m [s

]

conporcrtcdt

Figure 6: Log mean (surface) relaxation times T2lm forthree carbonates and four different surface relaxivity dis-tributions. [a] OC, [b] VC1, [c] VC2.

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[a]

103

104

105

κsim [mD]

103

104

105

κ corr [m

D]

conporcrtcdt

[b]

100

101

102

103

104

105

κsim [mD]

100

101

102

103

104

105

κ corr [m

D]

conporcrtcdt

[c]

100

101

102

103

104

κsim [mD]

100

101

102

103

104

κ corr [m

D]

conporcrtcdt

Figure 7: NMR-permeability cross-correlations for car-bonates using Eqn. 6 for four different surface relaxivitydistributions. [a] OC, [b] VC1, [c] VC2.

[a]

103

104

105

κsim [mD]

103

104

105

κ corr [m

D]

conporcrtcdt

[b]

100

101

102

103

104

105

κsim [mD]

100

101

102

103

104

105

κ corr [m

D]

conporcrtcdt

[c]

101

102

103

104

κsim [mD]

101

102

103

104

κ corr [m

D]

conporcrtcdt

Figure 8: NMR-permeability cross-correlations for car-bonates using Eqn. 7 for four different surface relaxivitydistributions. [a] OC, [b] VC1, [c] VC2.

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Table 4: Dependence of the correlations between NMRresponse and permeability on surface relaxivity hetero-geneity for carbonates. Considered are constant surfacerelaxivity (index con), and distributions based on par-titions by pores (por), covering radius (crt), and cap-illary drainage (cdt). The mean surface relaxivity is〈ρ〉 = 16µm/s. The prefactor a, mean residual errors2 and correlation coefficient R have indices. “1” refersto Eqn. 6 and “2” refers to Eqn. 7.

Sample a1

36ρ2 s2

1R1

a2

36ρ2 s2

2R2

OCcon .70 .067 .422 .27 .040 .732OCpor .64 .067 .423 .25 .039 .736OCcrt .67 .067 .419 .26 .040 .733OCcdt .62 .067 .424 .24 .039 .736

VC1con .89 .427 .712 .047 .343 .832VC1por .81 .422 .732 .043 .337 .866VC1crt .84 .427 .709 .045 .343 .824VC1cdt .82 .426 .715 .044 .341 .833

VC2con 66. .36 .445 .333 .129 .874VC2por 63. .35 .449 .316 .129 .874VC2crt 62. .36 .446 .313 .128 .879VC2cdt 60. .36 .448 .301 .126 .880

CONCLUSIONS

We presented a sensitivity study about the influence ofsurface relaxivity heterogeneity on prefactors and con-sistency of NMR-permeability correlations for a num-ber of sandstone and carbonate samples. It was foundthat with increasing surface heterogeneity length scalethe log mean relaxation time increases as a result of leav-ing the fast diffusion limit. This leads to a decrease in theprefactors of the permeability cross-correlations consid-ered. The correlation coefficients and scatter in the per-meability cross-correlations of the two predictive equa-tions were not affected by the presence of surface relax-ivity heterogeneity.

We note, that for most samples the cross-correlation in-volving the formation factor gives a tilted slope. This isan effect which lends itself to further study using an Xraymicro-CT approach.

For sandstones, the effect of surface relaxivity hetero-geneity was strongest in the pore micro-environment sce-nario based on a topological partitioning of the pore space.The heterogeneity caused about a 25% change of theprefactor of the permeability correlations, which is of thesame order as the change of the prefactor between sam-ples based on structure. Thus its effect cannot be ignored,if the surface relaxivity distributions chosen here are re-alistic.

For carbonates, the effect of surface relaxivity hetero-geneity in the pore micro-environment scenario was sig-nificant. However, the strongest effect of surface relaxiv-ity heterogeneity was exhibited in the capillary drainagehistory scenario. Surface relaxivity heterogeneity causedabout a 10% change of the prefactor of the permeabilitycorrelations, which is two orders of magnitude smallerthan the change of the prefactor between samples basedon structure.

It should be possible to use the change in prefactor toderive information about the length scale of the hetero-geneity in a site-specific context. However, consideringthe natural heterogeneities of carbonates and the largeeffect of structural heterogeneity, an application of thistechnique to carbonates would require careful calibrationtechniques.

We believe that in principal the important parameter con-trolling the effect of surface relaxivity heterogeneity onpermeability correlations is its characteristic length scale.Mixed wettability scenarios, where e.g. the surface re-laxivity might scale with ρ(r) ∝ r rather than ρ(r) ∝1/r as in the fluid saturation scenarios considered here,are expected to give similar results.

ACKNOWLEDGEMENTS

CHA acknowledges the Australian Government for theirsupport through the ARC grant scheme (DP0558185).The authors also thank the Australian Partnership for Ad-vanced Computing (APAC) for their support through theexpertise program and APAC and the ANU Supercom-puting Facility for very generous allocations of computertime.

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ABOUT THE AUTHORS

C. H. Arns - Christoph Arns was awarded a Diploma inPhysics (1996) from the University of Technology Aachenand a PhD in Petroleum Engineering from the Univer-sity of New South Wales in 2002. He is a ResearchFellow at the Department of Applied Mathematics at theAustralian National University. His research interests in-clude the morphological analysis of porous complex me-dia from 3D images and numerical calculation of trans-port and linear elastic properties with a current focuson NMR responses and dispersive flow. Member: AM-PERE, ANZMAG, DGG, SPWLA.

A. P. Sheppard - Adrian Sheppard received his B.Sc. fromthe University of Adelaide in 1992 and his PhD in 1996from the Australian National University and is currentlya Research Fellow in the Department of Applied Math-ematics at the Australian National University. His re-search interests are network modelling of multiphase fluidflow in porous material, topological analysis of complex

structures, and tomographic image processing.

M. A. Knackstedt - Mark Knackstedt was awarded a BScin 1985 from Columbia University and a PhD in Chem-ical Engineering from Rice University in 1990. He isa Professor at and Head of the Department of AppliedMathematics at the Australian National University anda visiting Fellow at the School of Petroleum Engineer-ing at the University of NSW. His work has focussed onthe characterisation and realistic modelling of disorderedmaterials. His primary interests lie in modelling trans-port, elastic and multi-phase flow properties and devel-opment of 3D tomographic image analysis for complexmaterials.

M. Saadatfar - was awarded a BSc from the Instituteof Advanced Studies in Zanjan, Iran and successfullydefended his PhD from the ANU in 2006. His mainresearch interests lie in the grain partitioning of tomo-graphic images, physics of granular matter and large scalesimulation of the elastic properties of porous materials.

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