SPE-160870-PA (Correlations Between NMR-Relaxation Response and Relative Permeability From Tomographic Reservoir-Rock Images)

Correlations Between NMR-RelaxationResponse and Relative Permeability From

Tomographic Reservoir-Rock ImagesTariq M. AlGhamdi, SPE, Saudi Aramco, C.H. Arns, SPE, University of New South Wales, and

R.Y. Eyvazzadeh, SPE, Saudi Aramco

Summary

Nuclear magnetic resonance (NMR) is typically used in the petro-leum industry to characterize pore size and to identify fluids infully and partially saturated reservoir samples. Although theNMR-relaxation response can be used to estimate the permeabil-ity of the rock, it may also provide information about the fluid dis-tribution for multiphase systems that could lead to the estimationof the effective permeability of fluids at partial saturations and thederivation of relative permeability to assess hydrocarbon recov-ery. By use of a random-walk method, we simulate the NMRresponse as a function of saturation on tomographic images ofBentheimer and Berea sandstone as well as Ferroan dolomite sam-ples. Fluid distributions are simulated for fully water-wet condi-tions by use of a morphological capillary-drainage transform,allowing the calculations of the saturations directly on the imagescorresponding to capillary pressure. The magnetic susceptibilityof minerals and fluids is used to calculate the internal magneticfields from the material distributions of solids and fluids quanti-fied by X-ray-diffraction (XRD) analysis. We show that the loga-rithmic mean of the NMR T2 distribution is a robust measure ofpermeability, and it results in strong correlations between NMRresponse and the relative permeability of both fluids. The ob-served relative permeability from NMR in our work is in excellentagreement with image-based relative permeability calculations byuse of the lattice Boltzmann method (LBM). We compare ourNMR results for the wetting phase to published experimentalresults on Bentheimer and Berea sandstone samples, and weobserve excellent agreement. By use of NMR numerical calcula-tions, we demonstrate that internal gradients aid the establishmentof relative permeability correlations for the nonwetting phase.

Introduction

NMR is increasingly used to estimate pore-size distribution andto identify fluids (Kleinberg 1996; Strange et al. 1996; Akkurtet al. 1998; Sun and Dunn 2004). One of the main advantages ofNMR is its ability to provide an estimate of permeability (Curwenand Molaro 1995; Dunn et al. 1999; Hidajat et al. 2002). Anessential tool in reservoir description is the estimation of perme-ability from NMR responses by empirical formulae. The NMRT2 relaxation time response for the Carr-Purcell-Meiboom-Gill(CPMG) sequence is a function of parameters such as saturating-fluid bulk relaxation time T2b, surface relaxivity q, surface area topore volume (PV), diffusion coefficient D, the gyromagnetic ratioof the proton c, the local magnetic field G, and the CPMG echo-spacing time tE (Bloembergen et al. 1948). For a single pore andfast diffusion, the NMR T2 relaxation is described as the follow-ing (Straley et al. 1994; Akkurt 1995):

1

T2

¼ 1

T2b

� �þ q

S

V

� �þ D

12ðcGtEÞ2 ð1Þ

In the weak-coupling regime (also known as the fast-diffusionlimit), the process of NMR relaxation results in a magnetizationdecay as a function of the relaxation times of the individual pores.The magnetization decay resulting from spin-spin relaxation canthen be expressed as

MðtÞ ¼XðaiÞM0exp

�t

T2

� �ð2Þ

Here, M(t) is the magnetization decay as function of elapsedtime, ai accounts for the fraction of pore size, and M0 is the equi-librium magnetization.

Correlations between NMR responses and permeability werefirst established by Seevers (1966) by use of T1 relaxation times,both for laboratory measurements and in the borehole. NMR T2

correlations with permeability have been introduced by Kenyonet al. (1988) and superseded spin-lattice relaxation measurementsbecause transverse relaxation responses gave better results forsandstones and can be measured much faster. A wide range ofrelationships between NMR spin-lattice relaxation responses andphysical properties, including permeability, is given by Banavarand Schwartz (1987), Sen et al. (1990), Dunn et al. (1999), andArns et al. (2005). Kenyon’s NMR permeability model by use oftransverse relaxation time T2 became much recognized (Straleyet al. 1994; Westphal et al. 2005), even for carbonates, as a resultof the advantage of faster acquisition to obtain relaxationmeasurements.

Banavar and Schwartz (1987) were the first to measure NMR-relaxation responses at partial saturations but suggested thatindustry should perform NMR measurements in mixtures of oiland water to match the reality of hydrocarbon reservoirs, whichfrequently are partially saturated. Their partial-saturation meas-urements depicted the behavior of the ratios of spin-lattice relaxa-tion times at multiple saturations to the NMR response at 100%saturations. Straley et al. (1991) were among the first to addresspartial saturations directly. By use of a 10-MHz NMR instrument,they performed drainage-centrifuge experiments on clay-rich low-permeability sandstone samples by air and kerosene so that onlythe water phase contributes to the NMR spin-lattice relaxation.They noted during water drainage that the long peak of T1 disap-pears as saturation decreases, and the short T1 remains the same.These observations led to the derivation of free-fluid index byNMR in agreement-centrifuge experiments.

More recently, NMR responses of partially saturated rockshave been used to give information about fluid flow (Chen et al.1993; Tessier and Packer 1998; Toumelin et al. 2003). Chen et al.(1993) conducted drainage experiments on Bentheimer sandstonewith nitrogen to displace water from the sample and to measureNMR spin-lattice relaxation (T1) at different saturations. Theyintroduced a power-law model to describe the decrease in T1

relaxation time with decreasing water saturations. Although notdirectly deriving a relative permeability curve for the wettingphase, they suggested that this is feasible.

Chen et al. (1994) reported the wetting-phase relative perme-ability from water/nitrogen experiments at a magnetic-fieldstrength of 2 Tesla and room-temperature conditions on Ben-theimer and Berea sandstone in addition to a limestone carbonatereservoir sample. From the experiments, profiles of spin-lattice

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Copyright VC 2013 Society of Petroleum Engineers

This paper (SPE 160870) was accepted for presentation at the 2012 SPE Saudi ArabiaSection Technical Symposium and Exhibition, AlKhobar, Saudi Arabia, 8–11 April 2012, andrevised for publication. Original manuscript received for review 31 March 2012. Revisedmanuscript received for review 25 March 2013. Paper peer approved 24 July 2013.

REE160870 DOI: 10.2118/160870-PA Date: 28-August-13 Stage: Page: 1 Total Pages: 9

ID: jaganm Time: 18:14 I Path: S:/3B2/REE#/Vol00000/130029/APPFile/SA-REE#130029

2013 SPE Reservoir Evaluation & Engineering 1

relaxation time at multiple saturations were generated. The rela-tionship by Katz and Thompson relating T1 to morphologicallength scales (Katz and Thompson 1987) was used in combinationwith Archie’s law (Archie 1942) to derive the relative permeabil-ity for the water phase from NMR T1 measurements.

Further progress in the assessment of dynamic flow throughreservoir cores was made with NMR-saturation imaging (Rothwelland Vinegar 1985; Chen et al. 1992; Kulkarni et al. 1998). Kul-karni et al. used NMR imaging to map the saturation profiles of oiland water for a limestone carbonate sample. This led to an estimateof two-phase-flow functions such as relative permeability and cap-illary pressure. The experiment showed the possibility of derivingrelative permeability by means of NMR measurements, but theexperiment was limited to water saturations greater than 40%.

Xue (2004) used a 50-MHz NMR-imaging setup to determinefluid saturation, and he used the model of Chen et al. (1994) toderive the wetting-phase relative permeability on a sandstonesample. Spin-lattice NMR T1 measurements in full and partial sat-urations were gathered by displacing water by nitrogen from thesample and resolving T1 spectra locally. They concluded that theiranalysis is not validated and might be subject to flow-stabilityconditions governing relative permeability. Ioannidis et al. (2006)repeated the NMR experiments of Chen et al. (1994) and Xue(2004) on glass beads for transverse relaxation and predicted rela-tive permeability successfully with high field at a 500-MHz fre-quency. They modified the Chen et al. (1994) model by replacingspin-lattice time T1 with transverse relation time T2 in the perme-ability correlations.

The interpretation of NMR-relaxation responses is compli-cated by the nature of the Laplace inversion needed. Simulationsof NMR responses assist these interpretations and can provideadditional insights. The most accurate NMR-response simulationsare carried out on high-resolution computed-tomography (CT)images, mainly with random-walk techniques (Kim et al. 2000;Valckenborg et al. 2002; Arns et al. 2011). The work presented byArns et al. (2011), Toumelin et al. (2003), and Talabi (2008) high-lighted advancement in NMR numerical modeling for porousmedia on high-resolution tomographic images and 3D pore-net-work models at multiple saturations. Earlier simulation methodsdid not account for internal gradients, which would lead to errone-ous results if iron minerals are present (LaTorraca and Dunn1995; Keating and Knight 2010). Newer simulation techniquesinclude internal-gradient effects by explicitly calculating the in-ternal-field distributions (e.g., Valckenborg et al. 2002, Arns et al.2011, and Chen et al. 2011).

In this work, we use digital images of a set of samples (Ben-theimer and Berea sandstone and Ferroan dolomite) and calculatethe petrophysical properties of porosity, permeability, and relativepermeability directly on the tomographic images. NMR-relaxationresponses are simulated at low field (2 MHz) as a function of satu-ration and internal gradients are accounted for. First, NMR-relax-ation responses for both fluid phases at fully saturated states ofeach phase (100% water and 100% oil) are simulated to calculate

the absolute permeability of both phases. Then, the NMR re-sponse of each phase at different saturations is simulated. Afterthe extraction of spin-spin relaxation-time behavior with satura-tion, we adopt and modify the Schlumberger-Doll research (SDR)NMR-permeability equation to accommodate the relation ofNMR T2 relaxation response as a function of saturation. Excellentresults are observed matching NMR relative permeability toimage-based (LBM) relative permeability, particularly for thenonwetting phase. Finally, we compare our wetting-phase relativepermeability results for Bentheimer and Berea sandstone withpublished results.

Methodology

Sample Characterization. In this work, we consider three sam-ples: Berea and Bentheimer sandstone and a Ferroan dolomitefrom the Middle East. XRD results on these samples revealed thatBerea is composed of 86% quartz and a mixture of different clayminerals (6% kaolinite, 3.5% ankerite, 2.9% illite, and 1% feld-spar). The Bentheimer sandstone exhibits 2 to 3% of kaoliniteclay mineral. The Ferroan dolomite contains 90% dolomite andapproximately 10% ankerite; this type of dolomite is calledFerroan because of the existence of iron-bearing minerals. Allthree samples display a relatively homogeneous microstructurewith high porosity and permeability. In Fig. 1, the samples wereimaged at high resolution with X-ray-CT at a field of view of20483 voxels with an accessible resolution of approximately 3 lm(Sakellariou et al. 2007).

Image Processing and Analysis. The sample processing of high-resolution tomographic images is a critical step (Sheppard et al.2004) for the accurate definition of PV and solid volumes, respec-tively. Raw tomograms exhibit a wide range of different noisetypes and image artifacts, and a set of sophisticated filters is usedto accurately quantify the solid, pore, and intermediate phases.The final process is the segmentation of the sample into two orthree phases by defining the volumes of pore, solid, and clay ormicroporosity from the intermediate phase. Sample dimensionsand characteristics are given in Table 1. The reported porositycorresponds to the porosity derived from the tomographic imagesafter segmentation. For voxels containing a mixture of materials(e.g., in clay regions or for microporous voxels), a straight vol-ume-weighted arithmetic-average susceptibility is calculated. It isassumed that the clay regions in the samples presented here con-tain 50% clay minerals and 50% water. Experimental cation-exchange capacity (Ma and Eggleton 1999) of the different claytypes would assist in the determination of the actual volume ofwater that can be absorbed in the clay porosity.

Numerical Simulation

Image-Based Fluid Saturations. The fluid distributions in thesamples are numerically simulated directly on the voxelated

(a) (b) (c)

Fig. 1—Segmented tomographic images of (a) Bentheimer sandstone, (b) Berea sandstone, and (c) Ferroan dolomite.



2 2013 SPE Reservoir Evaluation & Engineering

tomographic images by means of a capillary-drainage transform(Hilpert and Miller 2001) with open-boundary conditions fromall sides, mimicking the standard mercury-intrusion experiment.A particular saturation corresponds to a certain capillary radius.Fig. 2 shows different saturation profiles simulated on the tomo-

graphic images for the three reservoir samples used in this work.The saturation maps are calculated on the full image. For the fol-lowing simulations, a central region of interest of order 10003

voxel of the saturation map is chosen as simulation domain (seeTable 1) to avoid capillary end effects and at the same time pro-vide a sufficient representative elementary volume for transportcalculations. The simulated-fluid distributions are approximatingwater-wet steady-state fluid-flow conditions (Hussain 2011).

Permeability and Relative Permeability

Absolute and relative permeability are calculated with the LBM(Gunstensen and Rothman 1991; Shan and Chen 1993), which iswell-suited to deal with multiphase flow in complex geometries.By use of the static-fluid distributions with no-slip boundary con-ditions, permeability for each phase is calculated at different satu-rations by means of a single-phase lattice Boltzmann calculation.

TABLE 1—SUMMARY OF PETROPHYSICAL PROPERTIES OF

THE SAMPLES

Sample

Size

(voxels)

Resolution

(lm) Porosity

Permeability

(md)

Bentheimer 800 2.9 0.23 2,777

Berea 1080 2.83 0.19 523

Ferroan dolomite 1020 2.18 0.18 642

(d) (e) (f)

(g) (h) (i)

(a) (b) (c)

Fig. 2—Simulated drainage and imbibition by use of capillary-drainage mechanism. (a through c) Bentheimer sandstone—(a)100%, (b) 50%, and (c) 25% Sw; (d through f) Berea sandstone—(d) 100%, (e) 80%, and (f) 44% Sw; (g through i) Ferroan dolomitecarbonate—(g) 100%, (h) 75%, and (i) 25% Sw; (white is the invading nonwetting phase; black is the defending wetting phase; grayis the solid phase; and red is the clay region).




This allows relative permeability for each phase to be calculatedaccording to

Krw ¼Kw;eff

Kabs

ð3Þ

Kro ¼Ko;eff

Kabs

: ð4Þ

This direct image-based calculation of relative permeability byuse of numerically derived fluid distributions was validated experi-mentally on Bentheimer sandstone (Hussain 2011). In his steady-state experiment, capillary end effects were minimized by controlledhigh injection rates so no further effects on the residual saturationwould occur. The absolute-permeability values for the samples aregiven in Table 1, and they agree well with experimental data forBentheimer and Berea available in the literature.

NMR Simulation

Surface Relaxation. The spin-relaxation time of a saturated po-rous system is numerically calculated with a lattice random-walkmethod (Mendelson 1990; Bergman et al. 1995). Initially, thewalkers are placed randomly in the 3D pore space. At each time-step i, the walkers are moved from their initial position to a neigh-boring site and the clock of the walker is advanced by si¼ e2/(6D0), where D0 is the bulk-diffusion constant of the relevantfluid, reflecting Brownian dynamics, and e is a small fraction ofthe voxel size. We treat each random walk as the movement of aspin packet with initial strength Mw (t¼ 0). At each timestep (i) oflength (si), the strength of the walker is reduced by the survivalprobability Si, with the strength of the walker at time t¼Risi

given by

MwðtÞ ¼ Mwð0ÞY

i

Si ð5Þ

Here Si¼ SbSs, where Sb¼ exp (–ti/Tb) for bulk relaxation andSs¼ 1–� for surface relaxation. For steps within the same fluid,Ss¼ 1. The killing probability � is related to the surface relaxivityq (q here is fixed) by means of

A� ¼ qeD0

� �þ O

qeD

� �2

ð6Þ

Here, A is a correction factor of order 1 (we take A¼ 3/2),accounting for the details of the random-walk implementation(Mendelson 1990; Bergman et al. 1995). This leads to

SS ¼ 1� 6qsi

eAð7Þ

Dephasing Because of Internal Gradients. To capture thedephasing of random walkers caused by internal fields, we modelthe phase accumulation of the random walkers for the CPMGsequence explicitly. For this, one needs to derive the local mag-netic-field strength. For the cases considered here, the internalfield, which results from susceptibility contrast between mineralsand fluids, is accurately described by a dipole approximationbecause the magnetic susceptibilities of all components are smallcompared with unity. The internal magnetic field is numericallycalculated on the tomographic image by assigning an effectivemagnetic susceptibility to each voxel and convoluting the dipole

field around the susceptibility field (Arns et al. 2011). We calcu-late that the dipole field [when the distance from the dipole center(r) is larger than the radius of the dipole (a)] is given as

~Bdipole ¼l0

4p3ð~m~r Þ~r � ~mr2

r5

� �ð8Þ

For the dipole field inside the sphere, when r is less than ad,

~Bdipole ¼2

3l0 ~m ð9Þ

Here, l0 is the magnetic permeability of the vacuum and m isthe dipole magnetic field for a unit volume of the lattice spacing(resolution of tomographic image). When we apply an externalmagnetic field (H) to the sample, the susceptibility field will resultin an induced internal magnetic-field response (Binternal). We usethis to model the internal gradient distribution on high-resolutiontomographic images at different fluid saturations for a givenapplied static magnetic field that is oriented in the z-direction.The phase evolution of a spin with reference to the Lamor fre-quency at the starting position x0¼ c Bz(0) is given by

/D ¼ /� /0 ¼XN

j¼1

pjcsi½BzðtjÞ � Bzð0Þ� ; tj ¼Xj

i¼1

si

� � � � � � � � � � � � � � � � � � � ð10Þ

The total-magnetization decay, including dephasing for anindividual spin, is then given by Mw(t)¼Mw(t) � cos(/d) andrecorded for the echo positions (maximal coherence) of theCPMG sequence. An elementwise sum over the magnetizationdecays of the individual walkers finally results in the total-mag-netization decay.

Modeling Parameters. The important material parameters forNMR modeling are surface relaxivity and the susceptibilities ofthe individual components. For surface relaxivities, we use thevalues published by Talabi (2008). We use, as susceptibilities ofthe minerals and fluids, the results of a recent literature survey(Potter et al. 2008). The values for the different parameters arereported in Tables 2 through 4. To derive the volume susceptibil-ity of the minerals, and particularly the intermediate clay phase,we carried out XRD analysis for all three samples. Because mostreservoir fluids are diamagnetic (Potter et al. 2008), we used, asanalogs for the simulation, water and dodecane. We obtained fur-ther fluid properties (see Table 3): hydrogen index (HI), diffusioncoefficients, and bulk-fluid properties from the Halliburton NMRcatalog (Coates et al. 1999).

For our simulations, we define the lattice spacing as being afraction of the resolution of our sample. The resolution is approxi-mately 3 lm, and we typically fine-grained the system by a factorof 10 to achieve a good time resolution of the CPMG pulsesequence. We use between 60,000 and 100,000 walkers per eachsimulation. The NMR T2 pore-size distribution can be obtained byinverting the magnetization-decay profile (Arns et al. 2005).

NMR-Permeability Correlations

NMR is a practical petrophysical tool especially in determiningpermeability and has been widely used (Timur 1969; Sen et al.

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TABLE 2—NMR FLUID-PROPERTIES INPUT PARAMETERS

Parameter Water Clay Oil

HI 0.94 0.47 1.04

D0 (cm2/s) 2.30�10–5 4.10�10–6 8.00�10–6

T2b (seconds) 1 0.1 1

TABLE 3—SURFACE-RELAXIVITY MATRIX FOR MODELING

INTERACTIONS BETWEEN PHASES

q (lm/s) Water Clay Solid Oil

Water 0 10 3 0

Clay 10 10 3 0

Solid 3 3 0 0

Oil 0 0 0 0




1990; Prammer 1994). Expanding the approach to partial satura-tions enables the calculation of effective permeability for the rele-vant fluids. The SDR/NMR-permeability correlation found byKenyon et al. (1988) is described as

KNMR ¼ a � T22lm � /41 ð11Þ

We modify the NMR-permeability equation of SDR to accountfor partially saturated samples by calculating the logarithmicmean T2lm from NMR T2 responses at different saturations. Themodified version of the SDR equation becomes

KNMRðSwÞ ¼ a � T22lmðSwÞ � ð/ � SwÞ4 ð12Þ

Here the porosity in Eq. 11 denotes the fractional porosity of theeffective phases (water/oil), and T2lm is calculated as a function ofsaturation. The final version of the modified SDR equation for rel-ative permeability is then described as

Kr;NMR ¼T2lmðSwÞ

T2lm

� �2

� ðSwÞ4 ð13Þ

T2lm is calculated as the logarithmic mean of the transverse relaxa-tion time T2, and ai presents the volume fraction of pores at differ-ent sizes:

T2lmðSwÞ ¼ exp

XfaiðSwÞ � log½T2iðSwÞ�gX

aiðSwÞð14Þ

Results and Analysis

NMR Response of Samples Saturated With Single Fluid. Wefirst simulate the response on Bentheimer, Berea, and Ferroan-dolomite samples fully saturated with water or oil, respectively.The response of the water phase includes the NMR response of theclay regions, which are considered 50% water-saturated. Conse-quently, the NMR response of water includes a fast-relaxing com-ponent in the clay region and the region around it, which canexchange. The NMR water response is dominated by surface relax-ation, whereas the oil-phase response is mainly a result of bulkrelaxation (Akkurt 2005). Fig. 3 illustrates the NMR responses ofthe saturated rocks. Bentheimer sandstone exhibits only a smallamount of clay and at the same time shows a weaker susceptibilitycontrast. Consequently, the short relaxation-time peak attributed towater-saturated clays is weak. For Berea sandstone, both suscepti-bility contrast and clay fraction are higher, leading to a much morenoticeable clay peak. For Ferroan dolomite, the short relaxation-time peak is caused by a combination of internal gradients and

increased surface relaxivity, making the peak position depend onecho spacing.

NMR-Simulation Response at Partial Saturations. NMRresponses for partially saturated samples are derived with the satu-ration methods (capillary drainage) presented before. For waterand oil phases, the simulation parameters are presented in Tables2, 3, and 4. The wetting phase (water) exhibits a nonzero surfacerelaxivity with the solid. However, for the nonwetting fluid, thereare interactions with neither the solid surface nor the wetting fluid.Thus, the nonwetting fluid can relax only by internal-gradienteffects or bulk relaxation. We report the NMR response of bothphases at different saturations in Fig. 4. For the water phase start-ing at a fully water-saturated state, the relaxation time decreasesbecause drainage takes place until the samples are fully drained.This is because of largest pores being drained first, followed bythe next largest. The decrease in relaxation time is governed by

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0.0010

0.05

0.1

Freq

uenc

y

0.15

0.2

0.01 0.1

Fully OilFully Water

T2 (Sec)(a)1 10

0.0010

0.03

0.06

Freq

uenc

y

0.09

0.12

0.01 0.1


T2 (Sec)(b)1 10

0.0010

0.05

0.1

Freq

uenc

y

0.01 0.1


T2 (Sec)(c)1 10

Fig. 3—NMR response of full saturations of oil and water: (a)Bentheimer, (b) Berea, and (c) Ferroan Dolomite.

TABLE 4—VOLUME SUSCEPTIBILITY OF MINERALS AND

FLUIDS USED TO CALCULATE EFFECTIVE SUSCEPTIBILITY

OF THE SAMPLES

Material

Volume Susceptibility

(SI�10–5)

Quartz –1.641

Calcite –1.311

Dolomite –1.37

Kaolinite –1.68

Feldspar –0.1695

Ankerite 36.1

Illite 4.16

Water, H2O –0.9035

Dodecane, C12H26 –1.26

50,000 ppm NaCl brine –0.935

Bentheimer clay region, 50% brine –1.31

Berea clay region, 50% brine 4.71

Ferroan dolomite clay region, 50% clay 2.38




the surface-area/PV ratio on which that water-phase signaldepends.

Diffusional-coupling effects between clay regions andresolved pore space were examined in all three samples. As sam-ples desaturate, the clay regions stay coupled with the free waterbecause diffusion-coupling effects increase. The shift to smallerrelaxation times is explained by the increased surface/volume ra-tio. As a result of this diffusional coupling, larger amplitudes ofshort/transverse relaxation times were visible at lower water satu-rations. The diffusional-coupling effects varied here because bothBerea and Ferroan dolomite exhibited larger peaks than Ben-theimer, and this could be caused by more paramagnetic claysbeing present compared with Bentheimer, which contains diamag-netic kaolinite. These remarks are very crucial in the considera-tion of cutoff analysis defining the bound-fluid index (BFI) and

thus the accuracy of calculating the irreducible water saturation.Considering a cutoff value leads to more-accurate permeabilityestimation in the free-fluid region.

Consider now the NMR-relaxation response of the oil phase.The Bentheimer sandstone illustrated a marginal increase in relax-ation time as oil saturation decreases. In the Berea and the Ferroandolomite samples, an incremental increase and a wider separationin the relaxation time of oil phase are seen as oil saturationdecreases. In addition to bulk-fluid relaxation in the oil phase, in-ternal-field gradients play a significant role in the relaxation of theoil signal. The magnitude of the internal fields on the basis ofrelaxation was observed to be weaker in Bentheimer and strongerin Berea and in Ferroan dolomite. The increase in relaxation timefor Berea and for the Ferroan dolomite with decreasing saturationcan be explained by losing access to smaller pores and crevices,

0.0010

0.02

0.04

Freq

uenc

y

0.06

0.08

0.01 0.1

100% Sw25% Sw50% Sw

T2 (Sec)(a)1 10 0.1

0

0.05

0.1

Freq

uenc

y

0.15

0.02

1

Fully Oil Saturated

Oil Response - Sw 25%

T2 (Sec)(b)10

0.0010

0.02

0.04

Freq

uenc

y

0.06

0.08

0.01

0.03

0.05

0.07

0.01 0.1

100% Sw80% Sw44% Sw20% Sw

T2 (Sec)(c)1 10 0.1

0

0.05

0.1

Freq

uenc

y

0.15

0.02

1

Fully Oil

Oil Response 50% Sw

T2 (Sec)(d)10

0.0010

0.01

0.02

Freq

uenc

y

0.03

0.04

0.01 0.1

Sw = 100%Sw = 44%Sw = 30%Sw = 10%

T2 (Sec)(e)1 10 0.1

0

0.05

0.1

Freq

uenc

y

0.15

0.02

1

Fully Oil

Oil Response 75% Sw

T2 (Sec)(f)10

Fig. 4—NMR relaxation response at partial saturations of both phases—water (a, c, e) oil (b, d, f)—incorporating internal gradientsfrom susceptibility contrast: (a, b) Bentheimer, (c, d) Berea, and (e, f) Ferroan dolomite.




in which internal gradients are stronger. For both phases, we cal-culated the logarithmic mean of relaxation-time values (Eq. 14) asa function of saturation in all the samples, and we generated theprofiles for each phase (Fig. 5).

Absolute and Relative Permeability From NMR-Relaxation

Measurements. Both phases’ absolute permeabilities are calcu-lated with Eq. 11. To account for the influence of clay on theabsolute permeability of the water phase, we apply cutoff analysison the logarithmic-average mean of relaxation time T2lm in thesame way as one would typically perform for experimental data.The cutoff value used in Bentheimer and Berea sandstone samplesis 33 milliseconds, which is shorter than the cutoff value used forthe Ferroan dolomite sample of 67 milliseconds. Relative perme-ability correlations from numerical NMR-relaxation responses are

calculated from the effective NMR permeability of each fluid atthe different saturations from the NMR response at partial satura-tions from the profiles of the logarithmic mean of relaxation timesof each phase (Fig. 5) with Eq. 12. Relative permeability is calcu-lated by means of Eq. 13 for both water and oil phases. Strongcorrelations are observed in all three samples (Fig. 6) between therelaxation responses of both phases against the relative permeabil-ity calculated directly on the tomographic images of samples bylattice Boltzmann calculation. This suggests that internal gra-dients played a major role in establishing the correlations for bothphases, but powerfully for the nonwetting phase that was modeledwith zero surface relaxation. We validate the simulation results onBerea and Bentheimer sandstone samples with those measuredexperimentally by Chen et al. (1994). Our results are in excellent

0.200

0.5

1T2l

m (

sec) 1.5

2.5

2

0.4 0.6

T2lm (Oil)T2lm (Wat)

Sw(a)0.8 1

0.200

0.5

1

T2l

m (

Sec

)

1.5

2

0.4 0.6

T2lm_OilT2lm_Wat

Sw(b)0.8 1

0.200

0.5

1

T2l

m (

Sec

)

1.5

2

0.4 0.6

T2lm_OilT2lm_Wat

Sw(c)0.8 1

Fig. 5—Logarithmic mean of relaxation-time profiles of eachphase at partial saturations: (a) Bentheimer, (b) Berea, and (c)Ferroan dolomite.

0.200

0.2

0.4

Rel

ativ

e P

erm

eabi

lity

- K

r

0.6

1

0.8

0.4 0.6

Krw_LBMKro_LBMKrw_NMRKro_NMR

Sw(a)0.8 1

0.200

0.2

0.4

Rel

ativ

e P

erm

eabi

lity

- K

r

0.6

1

0.8

0.4 0.6


Sw(b)0.8 1

0.200

0.2

0.4

Rel

ativ

e P

erm

eabi

lity

- K

r

0.6

1

0.8

0.4 0.6


Sw(c)0.8 1

Fig. 6—Relative permeability from NMR response of partiallysaturated samples compared with direct image-based relativepermeability: (a) Bentheimer, (b) Berea, and (c) Ferroandolomite.




agreement with their results and observed relative permeability ofthe wetting phase (Fig. 7). This shows that numerical simulationscan be accurate in modeling fluid-flow dynamics by incorporatingall important elements, including the presence of internal mag-netic fields.

Conclusions

A random-walk method is used to simulate an NMR response as afunction of saturation on Berea, Bentheimer, and Ferroan dolomitereservoir samples. We perform XRD analysis on all the samples tocalculate magnetic susceptibility by use of the reported literaturevalues of magnetic response to minerals and fluids. The carefulmodeling of the susceptibility of minerals forming the sample,including the clay, is essential because internal gradients from sus-ceptibility contrast between fluid and surrounding solid interfacesenhance the NMR spin-spin relaxation measurements for bothwetting and nonwetting phases. Strong correlations between rela-tive permeability and NMR spin-spin relaxation response areobserved on all three samples. The analysis of the diffusional-cou-pling effects of clay micropores should be considered because itwould distinguish bound-fluid from free-fluid regions. As a result,the use of cutoff analysis to calibrate the logarithmic-mean valueleads to more-accurate permeability estimates that result inenhancements in relative permeability correlations for the wettingphase. Because oil phase does not relax by means of surface relax-ation, internal gradients from susceptibility contrast assisted inestablishing a surface-related/weighted relaxation mechanism.Our results for Bentheimer and Berea sandstone are also in excel-lent agreement with previously published work by Songhua Chenfrom the relation of NMR spin-lattice and saturation for the waterphase.

The work presented here uses a systematic approach thatadopts an NMR experimental procedure. Mineralogy here plays amajor role in the simulation of surface relaxation and mainly thevolume susceptibility of the samples at which the magnitude ofinternal gradients is assessed. The results from this work mightprovide hints for estimating dynamic-flow relative permeabilitymuch faster and with noninvasive methods such as NMR.

This work demonstrates that NMR is capable of accuratelyestimating relative permeability in partially saturated samples.Relative permeability experiments are expensive and time-con-suming. It would be highly desirable to extend this study to more-complex rock, accompanied by experimental measurements.

Future Work and Studies

Further work can be performed to validate the outcomes from thisstudy experimentally by means of both NMR and laboratory ex-perimental relative permeability. Some modeling parameters here,such as clay porosity, can be investigated experimentally to ensureaccuracy. Also, other wettability conditions could be examined toassess the robustness of these relative permeability correlationsfrom NMR compared with experimental analysis.

Acknowledgments

The authors acknowledge the University of New South Wales andAustralian National University for their support through providingthe samples for this work and facilitating the computationsthrough access to the National Computing Infrastructure. TAacknowledges Saudi Aramco for their financial support through aPhD scholarship at the University of New South Wales. CHAacknowledges the Australian Research Council for an AustralianResearch Fellowship (DP0881112) and a Future Fellowship(FT120100216).

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Tariq AlGhamdi is a reservoir engineer working for Saudi Ara-mco in the Reservoir Description and Simulation Department.His responsibilities include the management and petrophysicalevaluation of exploration and gas fields. AlGhamdi is currentlyleading the Unconventional Shale gas team in Saudi Aramco.He holds a BS degree from the University of Tulsa, an MSdegree from Herriot-Watt University, and a PhD degree fromUniversity of New South Wales—all in petroleum engineering.AlGhamdi’s main interests are optimizing petrophysical evalu-ation, and he recently was involved in digital core analysisand numerical simulations of special core analysis and NMR.He has authored or coauthored many papers and journals.

Christoph H. Arns is an associate professor at the School of Pe-troleum Engineering, University of New South Wales and is amember of SPE. His work has focused on the morphologicalanalysis of porous complex media and the numerical calcula-tion of transport and elastic properties as well as NMRresponses. Arns’ primary interest lies in the combination of 3Dtomographic-imaging technology and NMR spectral techni-ques for petrophysical applications. He holds a Diplomadegree in physics from the University of Technology, Aachen,Germany, and a PhD degree in petroleum engineering fromthe University of New South Wales.

Ramsin Eyvazzadeh has been a petrophysicist with Saudi Ara-mco since 2001, working in the Reservoir Description and Simu-lation Department, and is a member of SPE. He holds a BSdegree in petroleum engineering from the University of South-ern California and an MS degree in mechanical engineeringfrom the University of California. Before joining Saudi Aramco,Eyvazzadeh worked with Schlumberger in many countriesacross the world. He has published papers in several differentareas of petrophysics and is currently involved with optimizingsimulation and prediction models through petrophysicalapplications.




Documents

SPE-160870-PA (Correlations Between NMR-Relaxation Response and Relative Permeability From Tomographic Reservoir-Rock Images)