SPWLA 46 th Annual Logging Symposium, June 26-29, 2005 NMR PETROPHYSICAL PREDICTIONS ON DIGITIZED CORE IMAGES C. H. Arns 1,* , A.P. Sheppard 1 , R. M. Sok 1 , and M.A. Knackstedt 1,2 1 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: [email protected]Copyright 2005, 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 46 th Annual Logging Symposium held in New Orleans, Louisiana, United States, June 26-29, 2005. ABSTRACT NMR is a popular logging technique used to estimate pore size information, formation permeability, wettabil- ity and irreducible water saturation. Quantitative inter- pretation of NMR data is based on a set of fundamen- tal assumptions (e.g., pore isolation and fast diffusion). These assumptions establish the quantitative link between NMR response and petrophysical predictions. While there is a need to test these assumptions directly, to date no quantitative study on reservoir core material has been un- dertaken. The ability to digitally image reservoir rock in 3D, calculate petrophysical properties directly from the images coupled with a comprehensive simulation tool to numerically generate a range of NMR response data may help to address this need. In this paper we image a large set of reservoir cores in- cluding sandstones and carbonates at the pore scale using high resolution micro-CT. A set of petrophysical proper- ties are measured directly on the cores including surface- to-volume, permeability and pore size distribution. The permeabilities of the cores range from 10 mD to several Darcies. Realistic multiphase fluid distributions are de- rived by simulation of drainage. We then simulate the NMR responses on the same core images using a com- prehensive NMR simulator. The internal magnetic field is derived numerically from applied magnetic fields and susceptibility distributions and the phase evolution of the magnetic spins calculated with a random walk method. NMR responses currently include inversion recovery (T1) and CPMG (T2), and the longitudinal and transversal signals are monitored simultaneously. The interpretation of the signals acquired is done by standard 1D Laplace inversion to calculate the pore size distribution from T2 responses, and with a 2D inverse Laplace transform for fluid typing. In a preliminary study we compare predictions of petro- physical properties from the interpretation of the NMR response to direct calculations on the images. The foun- dational assumption of pore isolation is directly tested by partitioning of the pore space. This allows one to calcu- late the coupling constants and magnetisation exchange between pores, or between macro- and micro-porous re- gions. Further, the sensitivity of the responses to vari- ations in relaxivities or the presence of magnetic impu- rities is studied. Fluid typing is performed on a shaly sandstone sample. INTRODUCTION NMR techniques are usually employed in the petroleum industry to either predict permeability or for fluid typing. The former application uses the surface relaxation mech- anism or internal fields to derive a length scale (Brown- stein and Tarr, 1979; Kenyon et al., 1986; Kenyon et al., 1988; Kenyon, 1992; Song et al., 2000), which can then be used in permeability correlations (Banavar and Schwartz, 1987; Sen et al., 1990). The method relies on the application of the fast diffusion limit, in which the magnetisation of an isolated pore decays over time as a single exponential: (Wayne and Cotts, 1966; Brownstein and Tarr, 1979) M (t)= M 0 (t) exp - t T 2 , (1) where M 0 is the initial magnetization and the transverse relaxation time T 2 is given by 1 T 2 = 1 T 2b + ρ S p V p , (2) with S p /V p the surface-to-pore-volume ratio of the pore space, T 2b the bulk relaxation time of the fluid that fills the pore space and ρ the surface relaxation strength. For small pores or large ρ the bulk relaxation contribution is considered negligible and 1 T 2 = 1 T 2s = ρ S p V p . (3) 1 MMM
Transcript
SPWLA 46th Annual Logging Symposium, June 26-29, 2005
C. H. Arns1,*, A.P. Sheppard1, R. M. Sok1, 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 2005, 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 46th Annual LoggingSymposium held in New Orleans, Louisiana, United States, June 26-29, 2005.
ABSTRACT
NMR is a popular logging technique used to estimatepore size information, formation permeability, wettabil-ity and irreducible water saturation. Quantitative inter-pretation of NMR data is based on a set of fundamen-tal assumptions (e.g., pore isolation and fast diffusion).These assumptions establish the quantitative link betweenNMR response and petrophysical predictions. While thereis a need to test these assumptions directly, to date noquantitative study on reservoir core material has been un-dertaken. The ability to digitally image reservoir rock in3D, calculate petrophysical properties directly from theimages coupled with a comprehensive simulation tool tonumerically generate a range of NMR response data mayhelp to address this need.
In this paper we image a large set of reservoir cores in-cluding sandstones and carbonates at the pore scale usinghigh resolution micro-CT. A set of petrophysical proper-ties are measured directly on the cores including surface-to-volume, permeability and pore size distribution. Thepermeabilities of the cores range from 10 mD to severalDarcies. Realistic multiphase fluid distributions are de-rived by simulation of drainage. We then simulate theNMR responses on the same core images using a com-prehensive NMR simulator. The internal magnetic fieldis derived numerically from applied magnetic fields andsusceptibility distributions and the phase evolution of themagnetic spins calculated with a random walk method.NMR responses currently include inversion recovery (T1)and CPMG (T2), and the longitudinal and transversalsignals are monitored simultaneously. The interpretationof the signals acquired is done by standard 1D Laplaceinversion to calculate the pore size distribution from T2responses, and with a 2D inverse Laplace transform forfluid typing.
In a preliminary study we compare predictions of petro-physical properties from the interpretation of the NMRresponse to direct calculations on the images. The foun-dational assumption of pore isolation is directly tested bypartitioning of the pore space. This allows one to calcu-late the coupling constants and magnetisation exchangebetween pores, or between macro- and micro-porous re-gions. Further, the sensitivity of the responses to vari-ations in relaxivities or the presence of magnetic impu-rities is studied. Fluid typing is performed on a shalysandstone sample.
INTRODUCTION
NMR techniques are usually employed in the petroleumindustry to either predict permeability or for fluid typing.The former application uses the surface relaxation mech-anism or internal fields to derive a length scale (Brown-stein and Tarr, 1979; Kenyon et al., 1986; Kenyon etal., 1988; Kenyon, 1992; Song et al., 2000), which canthen be used in permeability correlations (Banavar andSchwartz, 1987; Sen et al., 1990). The method relies onthe application of the fast diffusion limit, in which themagnetisation of an isolated pore decays over time as asingle exponential: (Wayne and Cotts, 1966; Brownsteinand Tarr, 1979)
M(t) = M0(t) exp
[
− t
T2
]
, (1)
where M0 is the initial magnetization and the transverserelaxation time T2 is given by
1
T2=
1
T2b+ ρ
Sp
Vp, (2)
with Sp/Vp the surface-to-pore-volume ratio of the porespace, T2b the bulk relaxation time of the fluid that fillsthe pore space and ρ the surface relaxation strength. Forsmall pores or large ρ the bulk relaxation contribution isconsidered negligible and
1
T2=
1
T2s= ρ
Sp
Vp. (3)
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Real rocks contain a network of pores of different sizesconnected by pore throats which restrict diffusion. If in-terpore diffusion is considered negligible, each pore canbe considered distinct and the magnetization within in-dependent pores assumed to decay independently. Thedecay can then be described as
M(t) = M0(t)N
∑
i=1
ai exp
[
− t
T2i
]
, (4)
where ai is the volume fraction of a pore i decayingwith relaxation time T2i. The multi-exponential distribu-tion corresponds to a partition of the pore space into Ngroups based on the Sp/Vp values of the pores. Analogexpressions exist for deriving length scales using higherdiffusion eigenmodes (Song et al., 2000; Song, 2003).Given a pore, grain and fluid partitioning, the fast diffu-sion (Eqn. 3-4) limit can be adapted to varying surfacerelaxivity and different fluids by accounting for the num-ber of active surfaces N in each pore
1
T2=
1
NVp
N∑
i=1
ρiSi . (5)
In the case of partially saturated pores one would haveto treat each fluid partition of a pore separately. How-ever, the information needed to calculate this fast diffu-sion limit is prohibitive and generally not available.
It is widely recognised that the simple picture of mag-netisation decay is not always applicable to reservoir rocks.The length scale estimate can be affected by surface re-laxivity heterogeneities(Øren et al., 2002), internal fields(Hurlimann, 1998; Sen and Axelrod, 1999; Appel et al.,2001; Dunn, 2001; Dunn, 2002), by being outside thefast diffusion limit (Zhang and Hirasaki, 2003), or by be-ing outside the weak coupling regime (Cohen and Mendel-son, 1982; McCall et al., 1991; McCall et al., 1993;Zielinski et al., 2002; Toumelin et al., 2003). The corre-lation to permeability also implies a correlation betweenNMR response and pore size; this correlation has re-cently been questioned (Lonnes et al., 2003). Most stud-ies are either experimental or based on simulations in rel-atively simple geometries.
The second major area of application of NMR in reser-voir rocks is fluid typing (see e.g. (Looyestijn, 1996;Hurlimann et al., 2003)). This application has been mademuch more reliable by using 2D NMR techniques. How-ever limitations exist in the Laplace inversion techniquerequired (Sun and Dunn, 2004).
In recent years a new methodology has evolved com-bining X-ray µCT imaging techniques with numericalcalculations of petrophysical properties (Auzerais et al.,1996; Knackstedt et al., 2004; Arns et al., 2004a; Arns,
2004; Arns et al., 2005). It has advantages in beingable to study petrophysical parameters under very con-trolled conditions in realistic heterogeneous structures.In this paper we numerically predict NMR responses anda range of petrophysical properties on a suite of 3D im-ages of reservoir rock derived via micro-CT (Arns et al.,2005). This allows us to directly address assumptionsused in NMR interpretation and to test common correla-tions between response and permeability and fluid-typing.The paper is organised as follows: firstly, the experimen-tal setup and datasets underlying this study are brieflyintroduced. Secondly, the petrophysical calculations aredetailed. The result section then presents a preliminaryanalysis of NMR data on realistic microstructures includ-ing studies of;
• length scales used in permeability - relaxation timecorrelations, showing that prefactors in these cor-relations can vary by an order of magnitude de-pending on the sample structure,
• the sensitivity of the length scale prediction to het-erogeneity in surface relaxivities, which for the sam-ple discussed results in a factor of two differencein predicted permeability,
• the sensitivity of the length scale prediction to dif-fusion coupling with clay regions, affecting per-meability by up to 45% for the considered case,
• partial saturation effects, showing a non-linear re-lationship of water saturation to log mean relax-ation time,
• a detailed analysis of pore-pore coupling using apore partitioning and the concept of topologicaldistance, showing that all rocks considered hereare at least in the weak coupling regime. The over-all effect on predicted permeability is small forclastics, but more problematic for carbonates.
• fluid typing on a shaly sandstone sample at differ-ent partial saturations (water/oil).
MATERIALS AND METHODS
This section describes the samples used in this study, theacquisition of images and the computational techniquesused to calculate their physical properties.
Image acquisition
A set of 20 sandstones, including five cores from an off-shore gas well, four cores from an offshore oil well, fourpoorly consolidated cores from a prospective oil reser-voir, a thinly bedded sandstone, a shaly sandstone, fourFontainebleau sandstone samples, and one Berea sand-stone sample were considered as well as two carbonates,
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a vuggy carbonate and an outcrop limestone. All havebeen imaged via high-resolution X-ray µCT (Sakellar-iou et al., 2004b; Sakellariou et al., 2004a). Details ofthe experimental methodology and most of the numer-ical procedures has been described in previous papers(Arns et al., 2001; Arns et al., 2002; Arns et al., 2004b;Knackstedt et al., 2004; Arns et al., 2005) as has thepermeability analysis for the Fontainebleau (Arns et al.,2004b) and the oil and gas reservoir samples (Arns etal., 2004a). Most images are recorded at a resolution∼5µm with a field of view of ∼8 mm. Here we giveinformation for those samples, which are considered formost extensive analysis in the following sections. Ta-ble 1 gives the dimensions of these samples and Fig. 1slices through the raw tomograms. Sample A is a cleansandstone (Fontainebleau), sample B a shaly sandstone,sample C a vuggy multi-scale carbonate, and sample Danother highly porous carbonate (limestone) from an out-crop in South Australia showing large well resolved fea-tures.
Image segmentation and partitioning
Two-phase segmentation: To carry out quantitative anal-ysis on tomographic images one has to phase separate thepore space, and possibly various mineral phases. Thisis not trivial as the X-ray density does not show a cleardistinction between phases and simple thresholding willfail. Here, phase separation into two phases was achievedusing a multi-stage procedure (Sheppard et al., 2004).
Three-phase segmentation: Partitioning of a dataset intothree phases (void, clay and solid) is performed on the X-ray density data using a modified version of the converg-ing active contours method outlined in (Sheppard et al.,2004). First we identify voxels that can easily be classi-fied into one of the three phases. Any voxel whose gradi-ent value is above a threshold tg is considered to be nearan interface and therefore is classified as “undecided” bythe initial thresholding. The other voxels are classifiedas follows: if the voxel intensity is less than t0 then it isconsidered void, if it is between t1 and t2, then it is clas-sified clay, while voxels with intensity greater than t3 areset to solid. The remaining voxels are undecided. Thevoxels classified as void, clay or solid are used as seedregions for the converging active contours method. Anexample of a three-phase partitioning is given in Fig. 2.
Pore partitioning: For the purposes of calculating thecoupling constants measured using NMR, we need topartition the pore space into simple geometric cells (porebodies), separated by narrow constrictions (throats). It ispossible to do this by performing a watershed transfor-mation on the Euclidean distance data, but this approachleads to a partitioning, whit topological properties thatcannot be controlled.
[a] [b]
[c] [d]
Figure 1: Slices through the raw tomograms (2048 ×2048 voxel): [a] Fontainebleau sandstone (segmented,4803), [b] shaly sandstone, [c] vuggy carbonate, [d] fea-ture rich limestone.
Table 1: Basic description of the tomographic images interms of sample type, resolution [µm], segmented andanalysed image subsection, and pore volume fraction.Sandstone B is a shaly sandstone with an unresolvedclay/silt fraction of 4.83%.
Sample Res. Size [voxel] φ
Sandstone A 5.68 480 × 480 × 480 17.7Sandstone B 5.60 944 × 1208 × 1480 8.93Carbonate C 1.08 960 × 960 × 1920 15.6Limestone D 3.02 900 × 900 × 1800 50.7
[a] [b]
Figure 2: Illustration of the three-phase segmentation ona slice of the shaly sandstone sample (A). [a] raw tomo-graphic data, [b] phase separated into 3 phases; black(grain), white (pore) and grey (clay).
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Figure 3: Illustration of the pore partitioning on a 2003
subsection of the pore network of sample B.
Our approach is significantly more involved. We com-mence with Euclidean distance data, and perform distance-ordered homotopic thinning to determine the voxels thatcomprise the skeleton, or medial axis, of the pore space.The medial axis defines a network, in which nodes, clus-ters of highly connected voxels, are connected throughlinks composed of 2-connected voxels. We analyse eachlink in the network, calculating its length to width ra-tio, and determining the number and magnitude of theconstrictions along its length. If the link is short and rep-resents a minor constriction then the nodes at each endof the link may be merged. On the other hand, if thelink consists of a number of distinct constrictions, addi-tional nodes may be inserted along its length. Mergingcan only occur if it doesn’t alter the topology of the net-work; if two nodes are already connected to each otherby a single link, then merging them will collapse a ringof the network. Node merging and link division is of crit-ical importance in identifying representative pore bodies.
Having determined the topology of the network of porebodies, we now partition the pore space, i.e. determine towhich pore body each voxel belongs. This is achieved byfirst performing a Voronoi tessellation of the pore space,using the link voxel strings as seeds. This associateseach voxel in the pore space with the link that it’s clos-est to, and is done using Dijkstra-Sethian fast marching(Sethian, 1999). Each link region is then divided into twopieces, with the minimum constriction in the link sepa-rating the pieces. We do this by breaking the medial axisvoxels in the link at the minimum constriction, then us-
Figure 4: Illustration of the invasion of a non-wettingfluid into the water saturated shaly sandstone sample A.Colours: the solid grain phase is black, the clay regiondark grey, the defending fluid light grey, and the invadingfluid white.
ing a Voronoi tesselation starting from the two medialaxis pieces to partition the other voxels in the link. Anillustration of the pore partitioning resulting is given inFig. 3.
Drainage simulation
Multi-phase fluid distributions are generated by drainagesimulations directly on the voxelated image. A fixed cap-illary pressure is associated with a pore entry radius. In-vading the defending fluid by a sphere of that radius fromall boundaries mirrors standard mercury intrusion bound-ary conditions. The center of the invading sphere is al-lowed to move such that the sphere does not overlap thesolid(Hilpert and Miller, 2001). An example of a result-ing fluid distribution is given in Fig. 4.
Formation factor
The conductivity calculation is based on a solution ofthe Laplace equation with charge conservation bound-ary conditions and has been detailed before (Arns et al.,2001). We assign to the matrix phase of the sandstone aconductivity σm = 0 and to the (fluid-filled) pore phasea normalized conductivity σfl = 1. A potential gradientis applied in each coordinate direction, and the systemrelaxed using a conjugate gradient technique to evaluatethe field. The formation factor given by F = σfl/σeff, isused.
Permeability calculation
Permeability is calculated using the mesoscopic lattice-Boltzmann method (LB) (Martys and Chen, 1996; Qianand Zhou, 1998). It can be shown, that the macroscopicdynamics of the solution of a discretized Boltzmann equa-tion match the Navier-Stokes equation. Due to its sim-plicity in form and adaptability to complex flow geome-tries one of the most successful applications of the LB
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SPWLA 46th Annual Logging Symposium, June 26-29, 2005
method has been to flow in porous media (Chen et al.,1992; Frisch et al., 1986; Rothman, 1988; Ferreol andRothman, 1995; Martys and Chen, 1996). In this studywe applied a pressure gradient by a body force (Ferreoland Rothman, 1995), used closed boundary conditionsperpendicular to the flow and mirrored boundaries par-allel to the pressure gradient, resulting in a system sizeof L × L × 2L. Permeability was measured over theL × L × L original image of the simulated system.
NMR Response Simulation
Surface relaxation: The spin relaxation of a saturatedporous system is simulated by using a lattice randomwalk method (Mendelson, 1990; Bergman et al., 1995;Valckenborg et al., 2002). Initially the walkers are placedrandomly in the 3D pore space. At each time step thewalkers are moved from their initial position to a neigh-boring site and the clock of the walker advanced by ∆t =ε2/(6D0), where ε is the lattice spacing and D0 the bulkdiffusion constant of the fluid, reflecting Brownian dy-namics. The lattice is made periodic by mirroring thestructure in all directions. An attempt to go to a site ofanother phase will kill the walker with probability ν/6,0 ≤ ν ≤ 1 (Mendelson, 1990). The killing probability νis related to the surface relaxivity ρ via
Aν =ρε
D0+ O
(
(ρε
D
)2)
, (6)
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). The walkers ini-tially start with zero phase and accumulate phase accord-ing to their path. Since in a real experiment one woulddemodulate the signal with the Lamor frequency of thestatic applied field, the phase accumulation per step is
∆φ = γ∆t(B(~r) − B0), (7)
where γ is the gyromagnetic ratio. It is assumed, thatany spin-lattice relaxation also destroys phase coherency,thus T2 ≤ T1.
Dephasing and code validation: The decay of the spin-echo amplitude in porous media is governed by a set oflength scales, in particular the structural length lS =V/S, the dephasing length lg = (D0/(γg))1/3 (with gbeing the field gradient strength), and the diffusion lengthlD =
√D0τ . At short times in the free diffusion regime,
where lD is the shortest length and using the pulse se-quence π
2 − te − π − te − signal, the spin-echo decay isgiven by
M(2τ, g) = M0 exp[1
12γ2g2D0(2τ)3]. (8)
At longer diffusion times, in the motional averaging regime,where lS is the shortest length scale, one has (Robertson,
Figure 5: Comparison of analytical results and simula-tions for the decay of the spin-echo intensity under [a]free diffusion and [b-d] restricted diffusion inside a n-dimensional sphere for various field gradients [b] line,[c] circle, and [d] sphere.
1966; Neuman, 1974; Sukstanskii and Yablonskiy, 2002)
M(2τ, g) = M0 exp[−c1r4sgr
2dg(1 − c2r
2sgr
−2dg )] , (9)
where rsg is the ratio lS/lg , rdg is the ratio lD/lg , andc1 and c2 are geometrical constants. For spheres c1 =108/175, c2 = 747/20, for circles c1 = 7/36 and c2 =297/14, and for the planar case c1 = 1/720 and c2 =51/28. The numerical code is tested by comparison tothese analytical results in Fig. 5, using the diffusion con-stant of water (D0 = 2500 µm2/s), and d = 2 µm,where d is the diameter of the n-dimensional sphere. Theagreement is excellent (10000 random walkers were usedfor the simulation).
Analysis of diffusion coupling: The delineation of theindividual pores in some rocks via pore partitioning al-lows tracking of the interpore movements of the randomwalkers within the pore space and enables us to calculatethe number of crossings between adjacent pores at anyone time step. To minimise the sensitivity to resolutionand step size of the random walk, we record a numberof parameters. First we log the number and direction awalker crossed a pore-pore “interface”. We also have asecond counter, in which we only record interface cross-ings, which do not reverse the last interface crossing. Athird counter records the starting pore and end pore of arandom walk. For this application we start a fixed num-ber of random walkers at every pore voxel, reflecting thehydrogen index of the fluid present. A fourth counterrecords the mean first passage time across the pore-poreinterfaces.
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Inverse Laplace transform: The relaxation time distri-bution is obtained by fitting a multi-exponential decay toM(t) or M(t1, t2). Both the one- and the two-dimensionalinversion routines use a bounded least squares solver (Starkand Parker, 1995) combined with Tikhonov regularisa-tion (Lawson and Hansen, 1974). For the inversion ofone-dimensional decay curves the L-curve method is used(Hansen, 1992) to find the optimal regularisation param-eter. The 2D inversion routine follows closely the imple-mentation of (Venkataramanan et al., 2002) with kernelcompression to reduce memory requirements and com-puting time.
Permeability correlation
Permeability correlations are usually based on the loga-rithmic mean T2lm of the relaxation time
T2lm = exp
[
−∑
i ai log(T2i)∑
ai
]
, (10)
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, (11)
with classical factors a = 1, b = 4, c = 2, or the Forma-tion factor F as in
k = aF bT c2lm, (12)
with standard factors b = −1, c = 2.
The use of c = 2 in Eqns. 11-12 implies a unit of aas surface relaxivity squared. In our fits of the perme-ability correlations we scale the value of a by 1/(6ρ)2
to make the prefactor dimensionless; this implies thatany difference in the prefactor arises only from struc-tural influences. The length scale associated with T2
(the pore size derived from the NMR signal is given bydT2lm = 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.,2005). Each sample is divided into subregions and foreach subregion of all samples the permeability and NMRsurface relaxation response is calculated. This meansthat one obtains 100 individual samples per image andtherefore a relationship between k, φ, T2lm for each rock.In all fits the mean residual error
s2 =
∑
(log10(kcalc) − log10(kemp))
n − 2
2
. (13)
is minimised and the correlation coefficient
R =
∑
(kemp − kemp)(kcalc − kcalc)[∑
(kemp − kemp)2(kcalc − kcalc)2]1/2
(14)
calculated.
RESULTS
NMR Permeability Correlations
Structural influences: In these simulations simple sur-face relaxation was assumed with constant ρ = 16µm/sfor all sandstone samples and no bulk relaxation. Forthe carbonate samples C and D we used ρ = 5µm/s.We illustrate data obtained for two sets of samples inFig. 6 – the gas reservoir samples and the poorly consoli-dated core. The gas reservoir cores were part of the studyfrom (Arns et al., 2004a) which exhibited a broad rangeof permeability (varying over three orders of magnitude)across a small range of φ (15% < φ < 25%). While scat-ter in the data is observed the correlation is quite good.The poorly consolidated sands all have Darcy range per-meability and the match is excellent across all values. InTable 2 we summarise the NMR relaxation response cor-relations with permeability, based on the log-mean of therelaxation time distribution. We immediately note thatthe use of (Eqn. 12) dramatically increases the correla-tion coefficient compared to the use of Eqn. 11. Thisshows the important role a measure of F can make tomore accurate permeability estimation. The prefactors inEqns. 11 and 12 are quite consistent for samples fromthe same reservoir (particularly the poorly consolidatedcores B2-B5). The variation in the prefactor is quitesmall for all sandstone samples for both correlations. Theprefactor of Eqn. 12 is similar for the two carbonates. Incontrast the best prefactor for Eqn. 11 is almost an or-der of magnitude smaller than observed for the sandstonesamples.
Variability of ρ: We choose sample B to test the sen-sitivity of the permeability correlations to varying ρ bymaking use of the three-phase partitioning of the sampleinto grain phase, clay regions, and pore space. Surfacerelaxivities are set separately for the grain and clay re-gion surfaces (resolution is 5.6µm) with ratios 1:1, 1:4,and 1:10 in such a way, that the mean surface relaxiv-ity stays constant. Three sets of simulations are run with〈ρ〉 ∈ {4, 7.9, 16}µm/s. No bulk relaxation term is in-cluded in this simulation, so even random walkers start-ing far away from a surface will relax eventually on thesurface. Accordingly, T2lm is caused by surface relax-ation only and can be comparably long. The results aresummarised in Table 3 and Fig. 7.
As can be seen in Table 3, the derived length scale forpermeability correlations increases both with higher meansurface relaxivity and with higher heterogeneity of thesurface relaxivity at constant mean surface relaxivity. Areason for this could be due to the heterogeneous claydistribution (Fig. 2). The clay is compartmentalized andthe regions of different surface relaxivity do not commu-nicate with each other. This may prevent averaging as
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SPWLA 46th Annual Logging Symposium, June 26-29, 2005
Table 2: NMR response correlations to permeability. Thesandstone samples gas 1-4, oil 1-5 and B2-B5 come fromthree different fields. Added are the standard benchmarkrocks Berea and Fontainebleau (Fb, 4 samples), as wellas the four samples featured in Fig. 1 and Table 1. Sam-ples C and D are carbonates. The prefactor a, mean resid-ual error s2 and correlation coefficient R have indices.“1” refers to Eqn. 11 and “2” refers to Eqn. 12.
Table 3: Sensitivity analysis of T2lm upon varying sur-face relaxivity ρ of the clay (ρc) and grain surfaces (ρg)with constant mean 〈ρ〉 for a shaly sandstone sample.The macroporous (void) fraction is .0894 and the frac-tion of the clay region .0483. The specific void:grain sur-face area is Sg = 5.64 mm−1 and the specific void:clay-region surface area Sc = 1.33 mm−1 (at image resolu-tion).
Figure 6: Illustration of the deviation of the individualpermeability predictions for two sample subsets. The fitsaccording to Eqn. 12 with parameter a2 (see Table 2) for[a] gas reservoir samples and [b] B2-B5. Note the largescatter around the fit for the gas samples (high s2) com-pared with the tight fit for samples B2-B5 (low s2).
required by the fast diffusion limit. The range of dT2lm
corresponds to a factor of about two in the permeabilityprediction.
Many clay regions are microporous. In that case spinscan diffuse into the clay region and potentially relax muchfaster. We make a preliminary attempt to estimate this ef-fect by adding a 10% background porosity and an effec-tive background diffusion coefficient in and into the clayregion of D0/10. The results are given in Fig. 8 and Ta-ble 4. The effect of this background diffusion is gener-ally a drop in the mean relaxation times, in particular forhigh contrast and for lower 〈ρ〉, since the diffusion intothe clay region adds surface area. This scenario makesclear, that in the presence of clay a good understandingof diffusion coupling with the clay region is needed tointerpret the NMR response in terms of a length scalerelevant for permeability correlations. In this particularcase, the change in the predicted length scale dT2lm isabout 5-20%, implying a change of 10-45% in perme-ability. In current work we are extending this analysis toa wider range of clastics with larger clay fractions.
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Figure 7: Sensitivity of [a] T2lm and [b] dT2lm uponvarying surface relaxivity ρ of the clay-region (ρc) andgrain surfaces (ρg) compared to constant mean 〈ρ〉 for ashaly sandstone sample. The legends note mean surfacerelaxivity 〈ρ〉 and the values for the individual surfaces(see also Table 3).
Table 4: Sensitivity analysis of T2lm upon varying sur-face relaxivity ρ of the clay (ρc) and grain surfaces (ρg)with constant mean 〈ρ〉. Same as Table 3, but with 10%background diffusion into the clay region (see text).
Figure 8: Sensitivity of [a] T2lm and [b] dT2lm uponvarying surface relaxivity ρ of the clay-region (ρc) andgrain surfaces (ρg) compared to constant mean 〈ρ〉 fora shaly sandstone sample assuming a 10% backgroundporosity and an effective background diffusion coeffi-cient in and into the clay region of D0/10.
Partial saturation effects: We describe briefly a methodto study the effect of partial saturation. We use the samesample as in the previous section. The analysis is lim-ited to two global saturations (SW = 35, 60%) resultingfrom drainage simulations on the full image (sample B,944×1208×1480 voxel). From that subsection we selecta 2× 3× 3 matrix of 4003 subsamples. A bulk T2b of 3swas used for both oil and water and the diffusivities of thefluids were identical. Where oil was in contact with thewater or clay phase, the surface relaxivity is set to zero.In Fig. 9 we plot the relaxation time distribution of thewater phase for various saturations. The relationship isnot linear, since increased restriction of the water phase
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Figure 9: Sensitivity of T2lm upon partial saturation ef-fects for sample B. A mean surface relaxivity of ρ =〈16〉µm/s and ratios of 1:1 and 1:10 for the relaxivity ofthe grain surfaces and clay-regions were used (see alsoTable 3). φW is the water volume fraction.
by the oil phase ( implying “smaller” pore size) competeswith blockage of active surfaces by the oil. Table 5 liststhe resulting log mean relaxation time of the water phase,and in a separate column the measured mean log relax-ation time, taking the oil phase into account as well. For
Table 5: Sensitivity analysis of T2lm upon varying sur-face relaxivity ρ of the clay (ρc) and grain surfaces (ρg)with constant mean 〈ρ〉. Same as Table 3, but varyingglobal water saturation SW (see text). T2w notes the logmean T2 of the relaxing water phase.
constant surface relaxivity the log mean relaxation timeof the water phase rises above the 100% water saturatedcase, while for a 1:10 contrast between grain and clay re-gion surface relaxivities the relationship is inverted. Thelog mean relaxation time of all diffusing spins, and in-cluding bulk relaxation, trends monotonically with watersaturation. Heterogeneity in surface relaxivities at partialsaturation has a more pronounced effect than in the fullywater saturated case.
Pore-pore coupling
To analyse the NMR response in terms of pore networkcharacteristics, we first derive some intrinsic character-istics of the network. Namely, the number of pores Np,their average coordination number 〈C〉, and for each pairof pores the minimal number of links (throats) needed totraverse from one pore to the other. This latter quantityis called the topological distance dt. The average 〈dt〉of this Np × Np matrix of topological distances is an in-trinsic characteristic of the network and dependent on thesize of the lattice, increasing with network size.
We choose the four rocks shown in Fig. 1 to analyse theeffect of diffusion coupling. Applying the pore partition-ing, we track in the NMR simulations for each walkerthe pore in which it started, the first arrival times to otherpores, and the travelled topological distance d
(ρ)tt , which
we define as the numbers of pores crossed (per walker)excluding crossings which go directly back to the porevisited before. We further count the total number of thesecrossings at a given topological distance and normaliseby the total number of crossings (crossing intensity). Thiscounting avoids noise from multiple crossings at pore-pore interfaces. While 〈dt〉 would increase indefinitelywith increasing lattice size, the same is not true for 〈dtt〉,since relaxation events or finite acquisition time limit theachievable topological distance. In Table 6 we report themean of the travelled topological distance 〈d(ρ)
tt 〉.. We
Table 6: Characteristics of the pore partitioning used toanalyse the pore-pore coupling strength influencing theNMR relaxation response. Reduced sample sizes wereused, in particular a 4803 section of samples A,C,D anda 6003 section of sample B. Np notes the number of poresand 〈C〉 the average coordination number, and 〈dt〉 and〈dtt〉 are topological length scales (unitless). The indexto dtt indicates the surface relaxivity.
consider two different surface relaxivities, ρ = 3µm/s,and ρ = 16µm/s and no bulk relaxation. All isolatedpores are removed and the coupling in the resulting net-work is analysed. Reduced sample sizes were used; a4803 section of samples A,C,D and a 6003 section ofsample B.
As expected the surface relaxivity has a large impact onthe results. For ρ = 3µm/s, a large number of pores aretraversed by the diffusing spins. For the sands > 10 and
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for the carbonates ≈ 100. For ρ = 16µm/s, the averagetravelled pore distance drops to less than four pores forsands, but remains ≈ 30 for the carbonates. This resultleads to a questioning of the foundational assumption ofpore isolation — all rocks considered here are at least inthe weak coupling regime.
To investigate the effect of pore coupling on NMR re-sponses, we ran two distinct sets of simulations. Cou-pling “on” is the normal case and walkers can travel freelyin the pore space. In the case of coupling “off” we usethe pore network to restrict diffusing spins to their re-spective pores; we isolate all pores based on the porepartitioning algorithm. Fig. 10 shows the resulting T2s
distributions for the four rocks for two surface relaxivi-ties. For all samples, the change in ρ causes a large shiftin the relaxation time distribution. The log mean relax-ation time (see Table 7) stays nearly constant with eithercoupling on or off. Sharpening of the spectrum occursfor the sandstones only at low surface relaxivity, whilefor the carbonates (C,D) sharpening of the spectrum isapparent also for high surface relaxivities (in particularFig. 10d). There is a small but noticeable change in T2lm
for sample D ( 9%). From this result we can conclude,
Table 7: Log mean relaxation time for samples A-D withand without coupling and for two different surface relax-ivities (ρ = 3µm/s, ρ = 16µm/s). The indices c,n in-dicate coupling or no coupling, and the indices 3,16 thesurface relaxivity.
that pore-pore coupling does not change the length scaleused for permeability correlations if considering constantsurface relaxivity, and therefore the permeability predic-tion is robust under these conditions. Testing this un-der various other conditions will be undertaken in futurestudies.
Since permeability is controlled by restrictions to flowrather than pore size, it is interesting to ask whether dif-fusing spins diffuse far enough during the measurementto experience these restrictions. To gain insight into thisaspect, we plot the crossing intensity of diffusing spinsover the topological distance in Fig. 11, and the first pas-sage time (Kim and Torquato, 1990) over the topologicaldistance in Fig. 12. For sandstone a maximum intensityof crossings is recorded for topological distances of onlyone pore, and for the number of pores crossed (travelled
Figure 10: Relaxation time distributions for samples A-D ([a]-[d]). Pore-pore coupling effects can be seen forρ = 3µm/s, and for the Carbonate samples also for ρ =16µm/s.
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Figure 12: Illustration of diffusion coupling for samplesA-D ([a]-[d]). The effects are more pronounced for thecarbonate samples, in particular D.
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distance) one has about 5 pores (for ρ = 3µm/s). Forthe carbonates this maximum is closer to 10 (30) poresfor ρ = 3(16)µm/s. If we look at the corresponding firstpassage time, we note that the two sandstones again arerelatively similar, both showing a plateau in the first pas-sage time over topological distance. In contrast, sampleD shows almost a monotonic increase in the first passagetime, which would suggest that one is near the tortuos-ity limit (Sen, 2004); it could also be influenced by thehigh coordination number of the network. The multi-scale carbonate (C) shows an irregular behaviour. Thesimulations were run with surface relaxation only. To es-timate the effect bulk relaxation may have on the cross-ing intensity over topological distance, we plot Fig. 11and Fig. 12 next to each other. From the first passagetime distribution one can estimate the crossing intensityat a given topological distance by multiplying with anexponential (exp(−t/T2b)).
Fluid typing
In a previous section of this paper we discussed partialsaturation effects on permeability predictions for sampleB and assigning surface relaxivities only. In this section,we extend our analysis to consideration of 2D NMR cor-relations. Here we report T1-T2 data. We simulate aninversion recovery sequence to acquire the data varyingthe diffusion time td and echo spacing te in the sequence(e.g. Fig. 1c of (Hurlimann and Venkataramanan, 2002)).For both times we consider 150 logarithmically spacedvalues between 1 ms and 10 s. We use susceptibilities χof -1 for water, -10 for oil, 50 for the grain phase, and200 for the shaly region in units of 10−6. Bulk relax-ation times are 3 s for water and 1 s for oil. Further, weapply two different sets of surface relaxivities. The firstset is ρ
(1)wc = ρ
(1)wg = 16µ m/s, where the subscript “c” is
clay, “g” is grain, “w” is water and the superscript indi-cates T1 relaxivities. The second set is ρ
(1)wc = 59µ m/s,
ρ(1)wg = 5.9, a 1:10 contrast in relaxivities. T2 surface re-
laxivities are set by ρ(2)wc = 4ρ
(1)wc and ρ
(2)wg = 2ρ
(1)wg . The
oil phase is assumed to be not in contact with the clay orgrain phase (ρoc = ρog = 0). Further, it is assumed thatthe oil and water phases do not exchange any magnetisa-tion. The diffusion constant of water D0 = 2500µm/s2
is used for both the water and the oil phase. Fig 13 showssimulation data with no gradient applied and all suscep-tibility effects suppressed. Thus surface and bulk relax-ation are independent of the applied fields. The T1-T2
correlation spectrum of the 14% water saturation cases(Fig 13a-b) show one main peak caused by the oil phase,which experiences no surface relaxation, and some wa-ter in larger pores, while the remaining water signal ofsmaller pores is shifted towards smaller T1, T2, since sur-face relaxation is active in both for T1 and T2 relaxation.For the 47% water saturation case (Fig 13c-d), the same
[a] [a’]
[b] [b’]
[c] [c’]
[d] [d’]
Figure 13: T1-T2 correlation on sample B. All suscepti-bility effects are suppressed. [a] SW =14%, set one, [b]SW =14%, set two, [c] SW =47%, set one, [d] SW =47%,set two.
can be observed with the difference, that the water peak,shifted by surface relaxation, is now much stronger. Thiswe already noted in the previous section, i.e. surfacesactive for relaxation become inaccessible due to the dis-tribution of the oil phase. In the case of high water sat-uration the shape of the peak region is different betweenthe two different surface relaxivity contrasts.
Using exactly the same sample subset as in the previ-ous case, and keeping the two different sets of surfacerelaxivities and water saturations, we now apply a staticmagnetic field B0 of 550 Gauss and susceptibilities [SI]of χ = 50×10−6 for the grain phase, χ = 100×10−6 forthe shale region, and χ = −10×10−6 for the fluid phases(Hurlimann, 1998). The internal fields were calculatedaccording to a dipol approximation (Sen and Axelrod,
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[a] [a’]
[b] [b’]
[c] [c’]
[d] [d’]
Figure 14: T1-T2 correlation on sample B including sus-ceptibility effects (B0 = 550G). [a] SW =14%, setone, [b] SW =14%, set two, [c] SW =47%, set one, [d]SW =47%, set two.
1999; Song, 2003). The results are given in Fig. 14. Im-mediately one observes the large shift of the spectrum,particularly of the region with larger T1, to smaller T2
values. This can be explained, since the short T1 part al-ready relaxed, before the T2 part of the acquisition se-quence starts. For longer T1, now both fluids have amechanism to relax close to the surface. This mechanismis stronger for the oil phase, since ∆χ is larger betweenthe oil-grain or oil-clay surfaces.
CONCLUSIONS
In this paper we presented a comprehensive NMR re-sponse simulation tool for Xray-CT images including sim-ulation of T1, T2 relaxation and dephasing with a particletracking method making use of a pore partitioning. Other
partitions, e.g. grain partitions, will be considered in thefuture to allow greater control of varying surface relaxiv-ity. Particular outcomes are listed in the following:
1. Two common permeability correlations to NMRresponses are tested. The correlation which in-cludes the formation factor, and therefore infor-mation about the tortuosity of the sample, workssignificantly better. The range of prefactors foundindicate an order of magnitude variation of perme-ability due to structure across 20 sandstones andtwo carbonates. The NMR response formation fac-tor relationships have very high correlation coeffi-cients.
2. Heterogeneity in surface relaxivity is shown to havean effect on permeability predictions of about afactor two for a particular sample. Further, a mod-erate diffusive coupling between the macro-poroussections of the image (resolved) and the clay re-gions (unresolved) leads to another 10-45% changein predicted permeability. Calibration of the back-ground porosity might be facilitated by use of He-and Hg-porosimetry equipment.
3. Partial saturation effects can be masked by hetero-geneity in surface relaxivity. For the sample stud-ied in this paper they affect the log mean relax-ation time in opposite ways. This effect could besevere, if the bulk relaxation time of the oil phasefalls within the surface relaxation time distributionof the water phase.
4. An analysis of pore-pore coupling using a porepartitioning shows that, for all rocks studied in de-tail, diffusion between pores occurs to a significantdegree. For the sandstones considered here, typi-cally five pores are traversed in 2 s with significantremaining signal, while for the carbonates about30 pores are traversed in less than 1 s.
ACKNOWLEDGEMENTS
The authors acknowledge the Australian Government fortheir support through the ARC grant scheme and the Aus-tralian Partnership for Advanced Computing (APAC) fortheir support through the expertise program and APACand the ANU Supercomputing Facility for very gener-ous allocations of computer time. We also thank BHP-Billiton and Woodside Energy who have provided finan-cial support for the facility.
REFERENCES CITED
Appel, M., Freeman, J. J., Gardner, J. S., Hirasaki,G. H., Zhang, Q. G., and Shafer, J. L., 2001, Interpre-tation of restricted diffusion in sandstones with inter-
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nal field gradients: Magnetic Resonance Imaging, 19,535–537.
Arns, C. H., Knackstedt, M. A., Pinczewski, W. V.,and Lindquist, W. B., 2001, Accurate estimation oftransport properties from microtomographic images:Geophys. Res. Lett., 28, no. 17, 3361–3364.
Arns, C. H., Knackstedt, M. A., Pinczewski, W. V.,and Garboczi, E. G., 2002, Computation of lin-ear elastic properties from microtomographic images:Methodology and agreement between theory and ex-periment: Geophysics, 67, no. 5, 1396–1405.
Arns, C. H., Averdunk, H., Bauget, F., Sakellar-iou, 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 corefragments from 3D images:, paper EEE.
Arns, C. H., Knackstedt, M. A., Pinczewski, W. V.,and Martys, N., Virtual permeametry on microtomo-graphic images: J. Petr. Sc. Eng., 45, no. 1-2, 41–46.
Arns, C. H., Sakellariou, A., Senden, T. J., Sheppard,A. P., Sok, R., Pinczewski, W. V., and Knackstedt,M. A., 2005, Digital core laboratory: Reservoir coreanalysis from 3D images: Petrophysics, 46, no. 3, toappear.
Arns, C. H., 2004, A comparison of pore size dis-tributions derived by NMR and Xray-CT techniques:Physica A, 339, no. 1-2, 159–165.
Auzerais, F. M., Dunsmuir, J., Ferreol, B. B., Martys,N., Olson, J., Ramakrishnan, T. S., Rothman, D. H.,and Schwartz, L. M., April 1996, Transport in sand-stone: A study based on three dimensional microtro-mography: Geophysical Research Letters, 23, no. 7,705–708.
Banavar, J. R., and Schwartz, L. M., 1987, Magneticresonance as a probe of permeability in porous media:Phys. Rev. Lett., 58, 1411–1414.
Bergman, D. J., Dunn, K.-J., Schwartz, L. M., andMitra, P. P., 1995, Self-diffusion in a periodic porousmedium: A comparison of different approaches:Phys. Rev. E, 51, 3393–3400.
Brownstein, K. R., and Tarr, C., 1979, Importance ofclassical diffusion in NMR studies of water in biolog-ical cells: Phys. Rev. A, 19, 2446–2453.
Chen, H., Chen, S., and Matthaeus, W. H., 1992, Re-covery of the Navier-Stokes equations using a lattice-gas Boltzmann method: Phys. Rev. A, 45, no. 8,R5339–R5342.
Cohen, M. H., and Mendelson, K. S., 1982, Nuclearmagnetic relaxation and the internal geometry of sed-imentary rocks: J. Appl. Phys., 53, no. 2, 1127–1135.
Dunn, K.-J., 2001, Magnetic susceptibility contrastinduced field gradients in porous media: MagneticResonance Imaging, 19, 439–442.
Dunn, K.-J., 2002, Enhanced transverse relaxation inporous media due to internal field gradients: J. Magn.Res., 156, 171–180.
Ferreol, B., and Rothman, D. H., 1995, Lattice-boltzmann simulations of flow through Fontainebleausandstone: Transport in Porous Media, 20, 3–20.
Frisch, U., Hasslacher, B., and Pomeau, Y., 1986,Lattice-gas automata for the Navier-Stokes equation:Phys. Rev. Lett., 56, no. 14, 1505–1508.
Hansen, P. C., 1992, Analysis of discrete ill-posedproblems by means of the L-curve: SIAM review, 34,561–580.
Hilpert, M., and Miller, C. T., 2001, Pore-morphologybased simulation of drainage in totally wetting porousmedia: Advances in Water Resources, 24, 243–255.
Hurlimann, M. D., and Venkataramanan, L., 2002,Quantitative measurement of two-dimensional distri-bution functions of diffusion and relaxation in grosslyinhomogeneous fields: J. Magn. Res., 157, 31–42.
Hurlimann, M. D., Flaum, M., Venkataramanan, L.,Flaum, C., Freedman, R., and Hirasaki, G. J., 2003,Diffusion-relaxation distribution functions of sedi-mentary rocks in different saturation states: MagneticResonance Imaging, 21, 305–310.
Hurlimann, M. D., 1998, Effective gradients in porousmedia due to susceptibility differences: J. Magn. Res.,131, 232–240.
Kenyon, W. E., Day, P., Straley, C., and Willemsen, J.,1986, Compact and consistent representation of rockNMR data from permeability estimation:, Proc. 61stAnnual Technical Conference and Exhibition.
Kenyon, W. E., Day, P., Straley, C., and Willemsen, J.,1988, A three part study of NMR longitudinal relax-ation properties of water saturated sandstones: SPEformation evaluation, 3, no. 3, 626–636, SPE 15643.
Kenyon, W. E., 1992, Nuclear magnetic resonanceas a petrophysical measurement: Nucl. Geophys., 6,153–171.
Kim, I. C., and Torquato, S., 1990, Effective conduc-tivity of suspensions of hard spheres by brownian mo-tion simulation: J. Appl. Phys., 69, no. 4, 2280–2289.
14
SPWLA 46th Annual Logging Symposium, June 26-29, 2005
Knackstedt, M. A., Arns, C. H., Limaye, A., Sakel-lariou, A., Senden, T. J., Sheppard, A. P., Sok, R. M.,Pinczewski, W. V., and Bunn, G. F., March 2004, Dig-ital core laboratory: Properties of reservoir core de-rived from 3D images:, Presented at the 2004 SPEAsia Pacific Conference on Integrated Modelling forAsset Management, 1–14.
Lawson, C. L., and Hansen, R. J., 1974, Solving leastsquares problems: Prentice-Hall.
Lonnes, S., Guzman-Garcia, A., and Holland, R.,June 22-25 2003, NMR petrophysical predictions oncores: NMR petrophysical predictions on cores:, 44thAnnual Logging Symposium.
Looyestijn, W. J., June 1996, Determination of oilsaturation from diffusion NMR logs: Determinationof oil saturation from diffusion NMR logs:, SPWLA37th Annual Logging Symposium, SS1–SS11.
Martys, N. S., and Chen, H., 1996, Simulation of mul-ticomponent fluids in complex three-dimensional ge-ometries by the lattice Boltzmann method: Phys. Rev.E, 53, no. 1, 743–750.
McCall, K. R., Johnson, D. L., and Guyer, R. A.,1991, Magnetization evolution in connected pore sys-tems: Phys. Rev. B, 44, 7344–7355.
McCall, K. R., Guyer, R. A., and Johnson, D. L.,1993, Magnetization evolution in connected pore sys-tems. II. pulsed-field-gradient NMR and pore-spacegeometry: Phys. Rev. B, 48, no. 9, 5997–6006.
Mendelson, K. S., 1990, Percolation model of nuclearmagnetic relaxation in porous media: Phys. Rev. B,41, 562–567.
Neuman, C., 1974, Spin echo of spins diffusing in abounded medium: J. Chem. Phys., 60, no. 11, 4508–4511.
Øren, P. E., Antonsen, F., Rueslatten, H. G., andBakke, S., September 2002, Numerical simulations ofNMR responses for improved interpretations of NMRmeasurements in reservoir rocks:, Presented at theSPE 77th Annual Technical Conference and Exhibi-tion.
Qian, Y.-H., and Zhou, Y., 1998, Complete Galilean-invariant lattice BGK models for the Navier-Stokesequation: Europhys. Lett., 42, no. 4, 359–364.
Robertson, B., 1966, Spin-echo decay of spins diffus-ing in a bounded region: Phys. Rev., 151, no. 1, 273–277.
Rothman, D. H., 1988, Cellular-automaton fluids: Amodel for flow in porous media: Geophysics, 53, no.4, 509–518.
Sakellariou, A., Senden, T. J., Sawkins, T. J., Knack-stedt, 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 tomog-raphy facility for quantitative prediction of mechani-cal and transport properties in geological, biologicaland synthetic systems, Developments in X-Ray To-mography IV, 473–484.
Sakellariou, A., Sawkins, T. J., Senden, T. J., and Li-maye, A., X-ray tomography for mesoscale physicsapplications: Physica A, 339, no. 1-2, 152–158.
Sen, P. N., and Axelrod, S., 1999, Inhomogeneity inlocal magnetic field due to susceptibility contrast: J.Appl. Phys., 89, no. 8, 4548–4554.
Sen, P. N., Straley, C., Kenyon, W. E., and Whitting-ham, M. S., 1990, Surface-to-volume ratio, chargedensity, nuclear magnetic relaxation, and permeabil-ity in clay-bearing sandstones: Geophysics, 55, no. 1,61–69.
Sen, P. N., 2004, Time-dependent diffusion coefficientas a probe of geometry: Concepts in Magnetic Reso-nance Part A, 23A, no. 1, 1–21.
Sethian, J. A., 1999, Level set methods and fastmarching methods: Cambridge University Press.
Sheppard, A. P., Sok, R. M., and Averdunk, H., 2004,Techniques for image enhancement and segmentationof tomographic images of porous materials: PhysicaA, 339, no. 1-2, 145–151.
Song, Y.-Q., Ryu, S., and Sen, P. N., 2000, Determin-ing multiple length scales in rocks: Nature, 406, no.13, 178–181.
Song, Y.-Q., 2003, Using internal magnetic fields toobtain pore size distributions of porous media: Con-cepts in Magnetic Resonance Part A, 18A, no. 2, 97–110.
Stark, P., and Parker, R., 1995, Bounded-variableleast-squares: an algorithm and applications: Com-putational Statistics, 10, no. 2, 129–141.
Sukstanskii, A. L., and Yablonskiy, D. A., 2002, Ef-fects of restricted diffusion on MR signal formation:J. Magn. Res., 157, 92–105.
Sun, B., and Dunn, K.-J., 2004, Methods and limita-tions of NMR data inversion for fluid typing: J. Magn.Res., 169, 118–128.
Toumelin, E., Torres-Verdın, C., Chen, S., and Fis-cher, D. M., 2003, Reconciling NMR measurementsand numerical simulations: Assessment of tempera-ture and diffusive coupling effects on two-phase car-bonate samples: Petrophysics, 44, no. 2, 91–107.
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Valckenborg, R. M. E., Huinink, H. P., v. d. Sande,J. J., and Kopinga, K., 2002, Random-walk simu-lations of NMR dephasing effects due to uniformmagnetic-field gradients in a pore: Phys. Rev. E, 65,21306.
Venkataramanan, L., Song, Y.-Q., and Hurliman,M. D., 2002, Solving Fredholm integrals of the firstkind with tensor product structure in 2 and 2.5 di-mensions: IEEE Trans. on Signal Process., 50, no. 5,1017–1026.
Wayne, R. C., and Cotts, R. M., 1966, Nuclear-magnetic-resonance study of self-diffusion in abounded medium: Phys. Rev., 151, no. 1, 264–272.
Zhang, G. Q., and Hirasaki, G. J., 2003, CPMG relax-ation by diffusion with constant magnetic field gradi-ent in a restricted geometry: numerical simulation andapplication: J. Magn. Res., 163, 81–91.
Zielinski, L. J., Song, Y.-Q., Ryu, S., and Sen, P. N.,2002, Characterization of coupled pore systems fromthe diffusion eigenspectrum: J. Chem. Phys., 117, no.11, 5361–5365.
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, ARMA, DGG.
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 complexstructures, and tomographic image processing.
R. M. Sok: Rob Sok studied chemistry and received hisPhD (1994) at the University of Groningen in the Nether-lands and is currently a Research Fellow in the Depart-ment of Applied Mathematics at the Australian NationalUniversity. His main areas of interest are computationalchemistry and structural analysis of porous materials.
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 is anAssociate Professor at the Department of Applied Math-ematics at the Australian National University and a visit-ing Fellow at the School of Petroleum Engineering at theUniversity of NSW. His work has focussed on the charac-terisation and realistic modelling of disordered materials.His primary interests lie in modelling transport, elasticand multi-phase flow properties and development of 3Dtomographic image analysis for complex materials.
