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Limits of 2D NMR InterpretationTechniques to Quantify Pore Size,

Wettability, and Fluid Type: A NumericalSensitivity Study

Emmanuel Toumelin, SPE, and Carlos Torres-Verdín, SPE, U. of Texas at Austin, and Boqin Sun and Keh-Jim Dunn,Chevron Energy Technology Co.

SummaryTwo-dimensional (2D) NMR techniques have been proposed asefficient methods to infer a variety of petrophysical parameters,including mixed fluid saturation, in-situ oil viscosity, wettability,and pore structure. However, no study has been presented to quan-tify the petrophysical limitations of such methods. We address thisproblem by introducing a pore-scale framework to accuratelysimulate suites of NMR measurements acquired in complex rock/fluid models. The general pore-scale framework considered in thispaper is based on NMR random walks for multiphase fluid diffu-sion and relaxations, combined with Kovscek’s pore-scale modelfor two-phase fluid saturation and wettability alteration. We usestandard 2D NMR methods to interpret synthetic data sets fordiverse petrophysical configurations, including two-phase satura-tions with different oil grades, mixed wettability, or carbonatepore heterogeneity.

Results from our study indicate that for both water-wet andmixed-wet rocks, T2 (transverse relaxation)/D (diffusion) maps arereliable for fluid typing without the need for independently deter-mined cutoffs. However, significant uncertainty exists in the esti-mation of fluid type, wettability, and pore structure with 2D NMRmethods in cases of mixed-wettability states. Only light oilwettability can be reliably detected with 2D NMR interpretationmethods. Diffusion coupling in carbonate rocks introducesadditional problems that cannot be circumvented with current 2DNMR techniques.

IntroductionWettability state and oil viscosity can play a significant role in theNMR response of saturated rocks. This property of NMR mea-surements has been discussed in recent papers (Freedman et al.2003) for particular examples of rock systems. However, to date,no systematic study has been published of the reliability and ac-curacy of NMR methods to assess fluid viscosity and wettability,including cases of mixed wettability. This paper quantifies thesensitivity of 2D relaxation/diffusion NMR techniques to mixedwettability and fluid viscosity in generic rock models.

Given that measurements are often made on rock samples withuncertain petrophysical properties and therefore uncertain corre-sponding measurement contributions, the work described in thispaper is based on the numerical simulation of pore-scale systems.We introduce a general numerical model that simultaneously in-cludes immiscible fluid viscosities, water or mixed wettability,variable fluid saturations and history, and disordered complexity ofrock structure. Geometrical fluid distributions at the pore scalewere considered a function of pore size, saturation history, andwettability following Kovscek et al.’s model of mixed-oil-wetrocks (1993). We simulated suites of NMR measurements with

random walkers within these pore-scale geometries, and subse-quently inverted them into relaxation/diffusion NMR maps. Theobjective of this paper is to assess the accuracy of 2D NMR in-terpretation techniques to detect fluid and wettability types, and toquantify pore-size distributions.

The first section of the paper summarizes the principles andlimitations of current NMR petrophysical interpretation. We thensummarize our pore-scale modeling procedure, its assumptions,and limitations. Subsequent sections analyze simulation resultsobtained for drainage and imbibition involving water-wettabilityand mixed-oil-wettability with partial saturations of water and dif-ferent hydrocarbon types in a generic clay-free rock model. Next,we consider the case of coupled carbonate rocks with emphasis onthe assessment of wettability and microporosity.

Model and Limits of NMRPetrophysical InterpretationAnalytical Approximations of NMR Decay and RelaxationTimes. Conventional interpretation of NMR measurements per-formed on saturated rocks is based on the assumptions that (1) thefluid protons within a pore of given size relax independently fromprotons residing in other pores, and (2) only one fluid is present ineach pore. The measured NMR temporal decay, M, is then ex-pressed as a multiexponential decay function of the form

M�t� = �s,f

As,f exp�−t�T2s,f�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

where t is time, s is the index of a pore size encountered in the porespace, f is the index of the saturating fluids, As,f is the partialporosity associated with pores of size s filled with fluid f, and T2

s,f

is the apparent transverse relaxation time associated with the samepores. If fluids exhibit substantial contrasts of hydrogen index, thepartial porosities in Eq. 1 are weighted by their respective fluidhydrogen-index values.

We define T2Bf as the intrinsic (or bulk) relaxation time of fluid

f, and �f as the surface relaxivity at the interface between rock andfluid f which dissipates energy depending on the rock/fluid pair.The quantity (S/V)s defines the ratio of relaxing surface over vol-ume of a pore of size s. Finally, DB

f defines the bulk diffusivity offluid f and Df its effective diffusivity in the presence of a tortuousporous medium. For a standard CPMG radio-frequency pulsingsequence in the fast-diffusion limit (i.e., where �f (V/S)s<D B

f ), theapparent transverse relaxation time is given by the harmonic av-erage (Fukushima and Roeder 1981)

1

T2s,f

=1

T2Bf

+ �f�S

V�s

+�� G TE�2D f

12. . . . . . . . . . . . . . . . . . . . . . (2)

In this equation, � is the proton gyromagnetic ratio, G is theaverage magnitude of the background magnetic field gradient overthe spatial zone of investigation, and TE the interecho time or theinterval between two radio-frequency pulses.

Limits of the Relaxation Model. The model described previouslyassumes that the protons relax within one pore independently fromthe surrounding pores. When the bulk and diffusion terms in the

Copyright © 2006 Society of Petroleum Engineers

This paper (SPE 90539) was first presented at the 2004 SPE Annual Technical Conferenceand Exhibition, Houston, 26–29 September, and revised for publication. Original manuscriptreceived for review 5 September 2004. Revised manuscript received 17 March 2006. Paperpeer approved 27 March 2006.

354 September 2006 SPE Journal

right side of Eq. 2 are negligible compared to the surface term�f (S/V )s, T2

s,f is considered proportional to the pore size (V/S )s.Consequently, the T2 spectrum obtained from 1D Laplace inver-sion of Eq. 1 is assimilated to the rock’s pore-size distribution.This assimilation becomes invalid if the protons diffuse throughpores of markedly different sizes before their NMR signal decays.In such cases, the NMR response of protons becomes representa-tive of the average size of the pores that are connected within thesample. Known as diffusion coupling, this phenomenon was quan-tified for the first time by Ramakrishnan et al. (1999) with modelsof micritized carbonate rocks that exhibited a combination of lowsurface relaxivity and high pore-size contrast between intergranu-lar and intragranular porosity regions.

Irreducible water saturation is usually quantified by integratingthe area under the T2 spectrum for relaxation times smaller thangiven T2 cutoffs. Hydraulic permeability can also be estimatedfrom either the calculated irreducible water saturation or the T2

spectrum itself (Kenyon 1997). However, in the presence of dif-fusion coupling, the default T2 cutoffs are inadequate to separatethe contributions of irreducible, capillary-bound, and free-watercomponents; substantial errors in the estimation of irreduciblesaturation and permeability may ensue (Toumelin et al. 2003a).Pore-scale geometries associated with multiphase saturations andmixed wettabilities also affect the prediction quality of such asurface relaxation model. For instance, in an oil-wet (OW) con-figuration, the model considered by Eqs. 1 and 2 assumes that theentire pore surface is OW and that water protons within thosepores will not be affected by surface relaxation. As described in thefollowing sections, it is now commonly understood that OW poresremain water-wet (WW) in the least accessible pendular rings,thereby giving rise to mixed-oil-wet (MOW) conditions. Conse-quently, when water invades OW pores, this water does not sys-tematically relax in bulk mode because it can also be affected bysurface relaxation when diffusing across pendular rings. The cor-responding NMR response will be markedly different from thatpredicted by Eqs. 1 and 2.

Restricted Diffusion. For the case of unbounded diffusion, Df isequal to the fluid self-diffusivity DB

f . In porous media and in thepresence of other immiscible fluid phases, Df decreases with timebecause fluid may be isolated into snapped-off blobs or emulsions,or because pore throats and other immiscible fluids present in the

pores restrict free displacement (Latour et al. 1993). The asymp-totic value of Df obtained after very long diffusion times (in theorder of 106 ms for water) can reach very low values, down to asmall percentage of bulk diffusion for OW cases. The duration ofNMR experiments, however, does not usually exceed 103 ms. Inthis time frame, Df is not expected to decrease by a factor largerthan 2 or 3, whereupon the impact of restricted diffusion on waterNMR decay is relatively small in saturated rocks. Light hydrocar-bons, however, exhibit values of self-diffusivity that are higherthan that of water. As a consequence, the impact of restricteddiffusion on NMR measurements within the 103 ms duration ofsignal acquisition can be significant.

2D NMR. Measurement methods have been developed for im-proved fluid typing based on special acquisition sequences and 1Drelaxation inversion techniques (Chen et al. 2000; Sun and Dunn2004; Hürlimann et al. 2002). For general applications, however,the most promising techniques are based on 2D diffusion/relaxation inversion of suites of NMR magnetization decays ac-quired with CPMG-like sequences and different echo times (Hür-limann and Venkataramanan 2002; Sun and Dunn 2002). By vary-ing TE, variable emphasis is put on the third term (the diffusionterm) of the right side of Eq. 2. Proper knowledge of the value ofG and 2D Laplace inversion isolates the experimentally controlleddiffusion term from the other two intrinsic bulk and surface relax-ation terms. When the fluids saturating the pore space exhibit lowT2 contrasts but high D contrasts, this technique discriminates fluidsignals while 1D NMR inversion does not. The main theoreticalrestriction of the method lies in the accuracy with which G isknown to properly quantify diffusion effects (Hürlimann and Ven-kataramanan 2002). The present work shows that the geometrydescribed by multiphase fluid saturations at the pore scale alsoinfluences the accuracy of T2/D map interpretation.

Simulation MethodologyRock/Fluid Model. The rock model is based on 3D disorderedpacks of submillimeter spherical grains (Fig. 1a) that were alteredto represent the impact of geological events. Porosity reductionthrough compaction and cementation was achieved by increasingthe grain radii without displacing the grain centers. Delaunay tes-sellation was then performed to partition the bulk pack volume intotetrahedral cells joining every four nearest grains (Fig. 1b) (Glad-kikh and Bryant 2003). For each Delaunay cell, the coordinates ofan effective pore center were calculated as the center of fluidopenings left between the grains within each tetrahedron (Fig. 2).The aperture of each pore was defined as the largest distancebetween the pore center and the four surrounding grains (apices ofthe tetrahedron). The simulations used realistic saturation valuesfor openhole well-logging measurements, and included the pres-ence of oil-based-mud (OBM) or water-based-mud (WBM) fil-trate invasion.

Fig. 1—Illustration of (a) slightly overgrown Finney pack sampleand (b) corresponding Delaunay cells shade-coded by order ofaperture. All dimensions are given in µm.

Fig. 2—Graphical examples of pore center locations (diamonds)in different tetrahedral geometries. The dashed grid lines arespaced at 20-µm intervals.

355September 2006 SPE Journal

WW conditions are assumed to exist in reservoirs where cap-illary forces are not sufficient to break the native water films at therock surface (Hirasaki 1991), or where no chemical wettabilityalteration occurred. On the other hand, MOW conditions developin pores where oil is present and where the thin water films at therock surface ruptured (Kovscek et al. 1993; Yang et al. 2002). Inthese pores, the oleic phase comes into contact with the OW grainsurface, whereas water remains the wetting phase in the leastaccessible pore regions. In the case of fast (at the field scale) waterinvasion of the MOW pore, as can be assumed for near-wellborefiltrate invasion, water percolates through the bulk of the pore space.This transport mechanism leaves the wetting-oil phase shaped as athermodynamically stable thin lens between the different regionswhich are water filled and pinned on the OW rock surface.

In this paper, the spatial distribution of fluid geometries at thepore scale is based on the same postulates as Kovscek’s model,whereby the distribution of apertures of Delaunay cells controlsthe spatial distribution of fluids. These fluid distributions weredetermined by applying cutoffs to the aperture distribution of eachgrain pack. Fig. 3 describes the corresponding cutoffs and illus-trates the theoretical spatial distribution of the two immisciblefluid phases according to the position of each pore aperture in thehistogram. The histograms shown in Fig. 3 are typical of thegranular packs used in this method. It is also possible to assignpore fluid geometries according to more accurate models of fluiddisplacement at the pore scale (Gladkikh and Bryant 2003; Ørenand Bakke 2003; Toumelin and Torres-Verdín 2005). For an ini-tially WW formation where oil migrated, pore configuration A ofFig. 3 occurs first. If wettability is not altered, then we considerthat the smallest oil-filled pores are occupied by WBM to formconfiguration B. If wettability is altered before mud-filtrate inva-sion, then the thin water films existing between oil and grain break,and configuration C ensues. Once WBM invades this MOW rock,it preferentially invades the largest oil-filled pores and yields thepore configuration D. Subsequent invasion by an oil phase rein-states configurations A or C.

Fig. 4 illustrates the practical 3D distribution of fluids in eachDelaunay cell. We describe the nonwetting phase as an immobilespherical blob centered on the effective pore center (pore type 2).The associated blob diameter depends on the saturation history ofthe rock; it is set large enough to reach the faces of tetrahedral cellsfor cases of high oil saturation, and small enough to prevent oilpercolation across the pore for cases of low oil saturation. Thecomplementary pore volume forms the wetting fluid. We neglectedthe signal contribution from thin films of water in WW rocksbecause of the high diffusion restriction and enhanced surface

Fig. 3—Sketch of 2-phase fluid distributions in the pores as a function of invasion history, wettability, and aperture distribution ofthe pores within the rock model. Vertical bars identify aperture cutoffs between different types of pore-fluid geometry. In thecorresponding star-shaped pore sketches, gray represents solid grains, white represents water (native brine or WBM), and blackrepresents oil (native oil or OBM). Circled numbers for each pore sketch refer to one of the four models considered in therandom-walk simulations (see Fig. 4).

Fig. 4—3D representation of fluid distribution in a Delaunay tet-rahedron as implemented in the random-walk model. Red: oleicphase. Blue: aqueous phase. OW: locally oil-wet surface; WW:locally water-wet surface. The circled numbers identify thepore-scale configurations described in Fig. 3.


relaxation within the nanometer-thin (Hirasaki 1991) films. In thiscase, oil blobs were formally put into contact with the rock surfacewith zero surface relaxivity.

Simulation Guidelines. The NMR response of the porous rockwas simulated with 3D unmeshed diffusive random walks withinthe rock/fluid structures defined earlier. We used an approximatesolution of Bloch-Torrey’s equations adapted from Toumelin et al.(2003b) along diffusion trajectories. The Appendix summarizesthe random-walk procedure and discusses its numerical accuracy.Table 1 describes the parameters used in the simulations for avariety of petrophysical configurations, including different fluidsaturations and wettability conditions for two rock models. Thevalues of surface relaxivity, �, shown in Table 1 were selected tobe representative of sandstones and carbonates (10 to 30 �m/s and1 to 7 �m/s, respectively, for WW rocks), noting that rock/oilsurface relaxivity is 1 to 3 times smaller than rock/water relaxivity(Kenyon 1997). We neglected the surface relaxivity at the oil/water interface.

All multiphase saturations were modeled using two immisciblephases: one aqueous (brine or light WBM) and one oleic (nativeoil, OBM, or a homogeneous mixture of both). We selected threerepresentative oil grades: a 1-cp light crude oil assumed to beeasily identifiable with conventional 1D NMR interpretation be-cause its bulk properties were sufficiently different from those ofwater in rocks; a 7-cp medium crude oil assumed to create iden-tification problems with 1D NMR because its bulk relaxation timeswere close to those of water in porous rocks (in the 100-ms range);and a 300-cp heavy crude oil characterized by several componentsthat relax at different rates and that describe the distribution ofbulk relaxation times shown in Fig. 5. At ambient conditions, liveoils are approximately characterized by the correlation

DBf �T2B

f = 5 × 10−6 cm2�s2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)Bulk properties of fluids used in our simulations were selected inconsistency with published experimental results (Hirasaki et al.2002; Lo et al. 2002; Dunn et al. 2002) at ambient conditions.Table 2 summarizes these properties. The random walkers char-

acterizing each fluid were constrained to stochastically honor thesame distributions of bulk relaxation and diffusivity.

We used five different values of echo time (TE�0.3, 1, 3, 9,and 16 ms) for each combination of rock/fluid/wettability/saturation, assuming a homogeneous background magnetic fieldgradient of strength G�16 G/cm. Fig. 6 shows magnetizationdecays obtained for these five values of TE for the case of a 100%water-saturated model. For each value of TE and each rock/fluidgeometry, one set of NMR simulations produces magnetizationdecays for all the water/hydrocarbon mixtures. Each of these simu-lation sets includes one group of random walkers that diffusethroughout the water-filled pore space and one group of walkersthat diffuse throughout the hydrocarbon-filled pore space for eachhydrocarbon type. The contribution of water is therefore identicalfor all water/hydrocarbon mixtures. Total magnetization de-cays calculated for each fluid mixture are an average of the wa-ter and hydrocarbon magnetizations weighted by their respectivefluid saturations.

Inversion and Plotting Conventions. White noise was added toeach simulated magnetization with amplitude equal to 2% of thefirst echo amplitude. Time decays obtained for different values ofTE were then simultaneously processed with a T2/D 2D inversionalgorithm. Inversion results are plotted as bilogarithmic intensitymaps of diffusivity vs. relaxation time. On these crossplots, T2

spectra are described on the vertical axis (with values increasingupward), while the diffusion coefficient is displayed on the hori-zontal axis (with values increasing to the right). Each plot showsa diagonal reference line, which is the fixed linear T2/D oil cor-relation described by Eq. 3. Deviation from this correlation linecan be caused by presence of water, surface relaxation, inhomo-geneous internal gradients, or restricted diffusion. Finally, cumu-lative distributions of T2 and D are plotted on the right and on thetop of the 2D maps. Because these cumulative T2 spectra do not

Fig. 5—Measured distribution of T2 bulk relaxation spectrum forthe heavy crude oil grade used for numerical simulations atambient conditions.

Fig. 6—Examples of magnetization decay simulated in the rockmodel for different values of TE.


contain diffusion information, they are identical to the T2 spectrathat would be obtained from 1D inversion of magnetization decayswith a small TE value or with G�0. On the 2D maps that follow,“+” marks, circles, and squares identify the T2/D response of bulkhydrocarbons, bulk brine, and brine affected by surface relaxationin WW configurations, respectively.

Fluid Content, Typing, and WettabilityAssessment in a Regular CleanSandstone ModelThe first of our two rock models is intended to represent a genericrock with homogeneous and average (16%) porosity. It was con-

structed with a disordered pack of 1,000 identical grains based onFinney’s coordinates (1970), as extensively used in pore-networkmodeling (Gladkikh and Bryant 2003). An initial uniform grainradius of 25 �m was overgrown by 15% to reach 16% porosity.Given these dimensions, all simulations were performed with arandom-walk step of 0.2 �m (see the discussion on numericalaccuracy in the Appendix).

Results Obtained With 1-cp Light Oil Partial Saturation. Fig. 7shows the T2/D maps simulated with partial saturations of waterand 1-cp light oil. As expected, the WW cases (Figs. 7a, 7c, and7e) exhibit two peaks, one for oil and one for water. In all cases,the oil peak is located at its bulk (T2,D) values, whereas the peakrelaxation of water remains constant at 100 ms. For imbibed con-figurations, the 1-cp oil peak exhibits substantial diffusivity dis-persion at the high end of oil saturation (Fig. 7c) and not at the lowend of oil saturation (Fig. 7e). This difference is attributable to thepore-scale distribution of the fluids, which varies with saturation.At low values of oil saturation, oil only fills the largest pores,where no practical restriction is imposed on oil diffusion. By con-trast, at higher values of oil saturation, the oil phase also fills small,unclustered pores, where diffusion becomes restricted.

MOW cases exhibit very different behavior depending onwhether drainage or imbibition is involved. Fig. 7b shows theNMR map obtained for drainage at the highest oil saturation. Theeffect of partial surface relaxation between oil and rock unambigu-ously appears as the oil peak displaces across the T2/D oil corre-lation line of Eq. 3. The relaxation time of this oil peak alsodecreases from 1000 ms (in the WW case) to less than 300 ms (inthe MOW case). It is important to note that the T2/D oil correlationline must be accurately determined in an independent fashion;otherwise, the relative shift of the oil peak with respect to thecorrelation line cannot be properly identified. Figs. 7d and 7f as-sume lower oil saturations during imbibition. The response ofwater is clearly influenced by the partial oil wettability, whichincreases the T2 value of the water peak from 100 ms to approxi-mately 500 ms. At 40% oil saturation (Fig. 7d), the oil response isaffected by two factors: (1) partial oil wettability decreases the T2

value of the oil peak from 1000 to 200–500 ms, and (2) restricteddiffusion reduces the oil diffusivity from 5.10−6 cm2/s to 2–3.10−6

cm2/s. As a result, the oil peak continues to intersect the T2/Dcrude oil correlation line; this behavior could be interpreted as thesignature of higher-grade oil, and not necessarily that of a MOWsystem. Simultaneously, the water peak appears halfway betweenthe purely surface-induced relaxation time of the WW effect andthe bulk relaxation that is expected within the OW zones. Thegeometrical complexity of our 3D pore model therefore yieldsresults that cannot be expected from the simplistic representationshown in Fig. 3. When oil saturation further decreases to 20% (Fig.7f), neither water nor oil exhibit OW behavior: the water responseremains similar to that obtained at 40% water saturation, and thelarge pore size of oil-filled pores makes the effect of wettabilityunnoticeable on the corresponding oil response. The water in theisolated WW asperities of the MOW pores forms the water signalaround D�4.10−6 cm2/s and T2�80 ms in Fig. 7f. This low-diffusivity peak gives the only hint toward mixed-wettability. As aresult, Fig. 7f could be easily misinterpreted as the 2D NMR map

Fig. 7—Parametric T2/D maps simulated for partial saturation of1-cp light oil in the clean sandstone model for different valuesof oil saturation (So) and under both WW and MOW assumptions.


of a WW rock with pores 3 to 4 times larger than those populatingthe actual rock model. Because of these seemingly larger pores,usual permeability models in T2

2 would overestimate permeabilityby a factor of 10 to 20. Therefore, when following standard NMRinterpretation guidelines, mixed-wettability could be responsiblefor significant errors in the assessment of wettability, pore size,and NMR permeability.

Results Obtained With 7-cp Medium Oil Partial Saturation.Fig. 8 shows the results of similar simulations performed withsaturation of water and 7-cp medium oil. These results are com-parable to those obtained with 1-cp light oil for WW configura-tions. In the WW cases depicted in Figs. 8a, 8c, and 8e, the oil peakis located at the same (T2,D) coordinates. No substantial diffusivityspread is observed in Fig. 8c, unlike Fig. 7c with 1-cp oil, whichconfirms that restricted diffusion effects observed for oil are lim-ited to the lightest grades only. In MOW fluid configurations, thedrained system considered for high oil saturation in Fig. 8b exhib-its minimal dependency on wettability. In this figure, the watersignal remains unchanged compared to Fig. 8a, whereas the 7-cpoil peak remains located along the T2/D oil correlation line andexhibits a larger diffusivity spectrum. No definitive wettabilityeffect can be assessed solely based on this T2/D representation,unlike the 1-cp case of Fig. 7b. For imbibition cases, however,strong similarities exist between Fig. 7 and Fig. 8. In Figs. 8d and8f, the oil peak shifts toward lower values of T2 and D, and, as aresult, the peak is very similar to the corresponding signature of

heavier oil grade. The water peaks in these figures also extendbetween the bulk and surface-relaxation responses with the sameconsequences as for the case of 1-cp oil saturation. In similarfashion to Fig. 7f, the narrow diffusion spectrum of the oil peak inFig. 8f gives no clue to the MOW nature of the rock. Only alow-diffusivity water signal hints toward restricted diffusion oc-curring in the WW pore corners.

Results Obtained With 300-cp Heavy Oil Partial Saturation.Because of the wide distribution of relaxation times and corre-sponding diffusivities that characterize the heavy oil (Fig. 5), noclear heavy-oil peak appears in the T2/D maps of Fig. 9. Rather,the oil signal is distributed in the form of a delta-shaped region thatpeaks at the maximum (T2,D) values indicated in Table 2 andrepresented with “+” marks in Fig. 9. For the case of drainage(Figs. 9a and 9b), it is not possible to distinguish between WW andMOW configurations because the bulk relaxation times of theheaviest oil components are too low to be affected by surfacerelaxation. For the case of imbibition (Figs. 9c to 9f), the width ofthe heavy-oil signal in (T2,D) space decreases to concentrate at itspeak value. Wettability of heavy oil can only be inferred from thewater signal, which in turn gives rise to the same ambiguitiesassociated with the other oil grades.

Influence of Pore Heterogeneity andDiffusion CouplingOur second rock model describes a 24%-porosity carbonate rockwith large amounts of micritized microporosity and vugs that are

Fig. 8—Same as Fig. 7 for partial saturation of 7-cp medium oil.

Fig. 9—Same as Fig. 7 for partial saturation of heavy oil.


completely coupled. We use this model to test whether the resultsobtained in the previous sections remain valid in coupled carbon-ates, and whether it is possible to detect the presence of coupledmicroporosity using 2D NMR interpretation techniques.

Rock Model Construction. We synthesized a carbonate rockmodel from core data, core pictures, and NMR measurements of amicritized rock sample exhibiting diffusion coupling [Sample A ofToumelin et al. (2003a)]. This sample includes both dissolutionvugs and abundant micritized intragranular porosity. The imageanalysis reported in Toumelin et al. (2003a) indicates presence ofsecondary-porosity vugs (amounting to 10 of the 24% porosity ofthe sample, with pores sizes in the 15 �m range), primary-porositymacropores (4% porosity, 5 to 15 �m range), and micritized in-tragranular micropores (10% porosity, submicron range). Irreduc-ible water saturation at 60 psi (414 kPa) in air was assumed equalto 52% and mostly filled the micritized porosity. Micritized grainsin the range of 2-�m diameter agglomerate into packs of approxi-mately 50 �m that form microporous grains (Fig. 10). Finally,calcite volumes in the range of 50 to 100 �m are present inthe sample, so that approximately 1⁄3 of the volume can be con-sidered calcite-cemented, while the rest of the rock matrix exhib-its microporosity.

Once more, we used the Finney pack as the skeleton of the rockmodel. The pack was sized and overgrown until porosity, grainsize, and pore size fitted primary intergranular porosity data.Grains were then removed from the pack to match vug size andporosity. Finally, 2⁄3 of the grains of the Finney pack were mademicroporous with a compacted cubic-centered structure (Ra-makrishnan et al. 1999; Toumelin et al. 2003a) to match intra-granular data and resemble the pore structure shown in Fig. 10.The step size for the random walks was reduced from 200 to 20 nmwithin the microporous regions. We also adjusted the homoge-neous rock/water surface relaxivity reported in Table 1 to modifythe T2 distribution simulated at full brine saturation until matchingthe measured T2 spectrum. Fig. 11 shows a good agreement be-tween measurements and simulations at 100% water saturation.This result lends credence to our modeling approach when used tocapture the NMR behavior of the coupled carbonate sample.

Simulation Results in the Coupled Carbonate Model. Earlier,we showed that light-oil saturations are more sensitive to wetta-bility effects than other oil grades. In order to assess the impact ofcoupled pore structure complexity on 2D NMR maps, we com-pared T2/D NMR maps simulated with partial saturations of 1-cpoil in both the coupled carbonate rock model and the clean sand-stone model. Fig. 12 shows the resulting 2D NMR maps at 20and 40% water saturation. Given the high value of irreducible

water saturation associated with the microporous component of therock, values of water saturation of 60% or higher can only beaccounted for as being the result of drainage in the MOW carbon-ate rock model.

From Fig. 12, we first note that no substantial shift of the oilpeak occurs because of wettability change, except in Fig. 12d. Thisbehavior is attributable to the low surface relaxivities in thecarbonate rock model, and consequently the oil signal is mar-ginally affected by surface relaxation. In Fig. 12d, the oil responseis divided between two T2/D peaks corresponding to differentdiffusion regimes in the pore space, including the isolated oilblobs considered at low values of oil saturation. At higher valuesof oil saturation and under the assumption of drainage (Fig. 12f),oil blobs become connected and are responsible for a single T2/Doil peak.

The water response exhibits markedly different behavior. First,all the NMR maps simulated in the coupled carbonate models(Figs. 12c to 12f) exhibit a single water peak. As illustrated in Fig.11, this peak is the product of diffusive coupling and is not rep-resentative of the bimodal pore-size distribution of the rock model.The peak diffusivity of water in Figs. 12c to 12f is lower than thebulk water diffusivity (identified with squares on the figures),which suggests the importance of restricted diffusion within themicroporosity. We expect that a more tortuous microporosity (e.g.,taking the form of disordered micrograins instead of regular cubic-centered packs) would yield lower values of diffusivity for thewater signal. This subtle reduction of water diffusivity indicatespresence of coupled microporosity, but remains marginal. Pres-ence of abundant coupled microporosity also has a noticeableimpact on the MOW results: it connects the isolated WW zonesof MOW mesopores. Consequently, the diffusivity of water in-creases in these isolated zones, and the low-diffusivity signal in-dicated in Fig. 12b becomes attenuated in Fig. 12d. In the latterfigure, a continuous water spectrum characterizes the response ofsaturating water.

In drained configurations, low values of surface relaxation pre-vent measurable differences in the oil and water responses for bothWW and MOW configurations. This behavior indicates that the 2DNMR assessment of low-relaxivity carbonate wettability is im-practical.

Summary and ConclusionsThis paper introduced a simulation methodology that incorporatesdetailed pore-scale geometries of rocks and fluids to quantify thecollective sensitivity of pore structure, fluid saturations, and wet-tability on the NMR response of rocks saturated with two immis-cible fluid phases. We simulated multiple echo-time NMR mag-netization decays based on the assumed geometry and bulk fluid

Fig. 10—High-resolution SEM picture of the carbonate sampleused as the basis of our geometrical model. The dotted curvesdescribe the contour of the microporous grains that form therock matrix in the form of micrite assemblies.

Fig. 11—Comparison of NMR T2 spectra at full brine saturationfor measurements performed on the carbonate sample andsimulations performed on the corresponding rock model.


properties. In turn, the simulated NMR magnetization decays weretransformed into parametric 2D NMR maps. Our simulation meth-odology was generalized to consider generic porosities, rock fa-cies, fluid properties, acquisition sequences, and generic single-gradient background magnetic fields. Despite the fact that varia-tions of these parameters will change the final results, we do notanticipate significant differences in the conclusions stemmingfrom the numerical sensitivity analysis. We also emphasize that thesimulation results described in this paper are limited by the accu-racy of the geometrical model used for MOW fluid distributions.

The following conclusions stem from our numerical sensitivityanalysis:1. Microstructure and fluid distributions exerted substantial re-

stricted diffusion effects on water and oil signals. Restricteddiffusion of water was particularly marked in microporous re-gions and in isolated WW pore corners; restricted diffusion ofoil was more important for light oil grades in MOW imbibedconditions. The accuracy of NMR interpretation techniques im-proves when they take into account these usually neglect-ed effects.

2. The 2D NMR signature of water within coupled intragranularmicroporosity and within the isolated corners of MOW poresmanifests itself in different ways: the former decreases the dif-fusivity of the entire water signal, whereas the latter extends theT2/D spectrum toward low values of both T2 and diffusivity.

3. Oil typing can be performed unambiguously from T2/D NMRmaps both in WW and drained MOW configurations.

4. In imbibed MOW configurations (i.e., for the case of WBMinvasion in MOW rocks), the 1-cp oil grade was the only oilgrade that was clearly influenced by wettability. The NMR re-sponse of water was an average between bulk response (inOW areas) and restricted-diffusion response (in WW porecorners). Ignoring the wettability origin of the resulting waterpeak causes substantial errors in the assessment of pore sizesand permeability.

5. Oil grades exceeding a few cp of viscosity were not amenable towettability characterization based on the oil NMR response;neither are light-oil grades when surface relaxivity is smallerthan 5 to 10 �m/s for 10-�m pores. In these cases, the waterresponse provided the only evidence of mixed wettability.

6. T2/D NMR techniques are theoretically accurate and reliable toidentify and quantify heavy-oil saturations. Therefore, the suc-cess of heavy-oil NMR applications depends on practical con-siderations such as measurement noise and logging speed.

7. Low values of surface relaxivity present in coupled carbonaterocks prevent the reliable characterization of wettability on thesole basis of oil signal. The possibility of characterizing thewettability of carbonate rocks is constrained either to MOWimbibed cases, where petrophysical interpretation is nontrivialbecause of the multiple pore sizes involved, or to cases wheresurface relaxivity is larger than a few �m/s (i.e., when the effectof diffusion coupling is not significant).

The numerical NMR simulation procedure considered in this paperuses an approximation of Bloch’s equations that is accurate forwell-logging applications using CPMG pulse sequences. Similarnumerical studies can be performed with other NMR formulationsfor various types of pulse sequences, such as stimulated echoes,DDif (Song, 2000), or with CPMG sequences that include multiplewait times (Chen et al. 2000).

Nomenclature

A � partial porosityD � effective diffusivity, cm2/s

DB � bulk diffusivity, cm2/sG � magnetic field gradient strength, Gauss/cmM � magnetizationSo � oil saturationSw � water saturation

t � time, msT2 � transverse relaxation time, ms

T2B � transverse bulk relaxation time, msTE � inter-echo time of the pulse sequence, msV/S � volume-to-surface ratio, �m

� � proton gyromagnetic ratio�2�.4258 rad/(Gauss.s)for hydrogen

�r � norm of a random walker’s 3D displacement duringone step, �m

�t � duration of a random-walk step, �s� � transverse surface relaxivity, �m/s

Acknowledgments

We appreciate stimulating discussions with S. Bryant and M.Sharma concerning the construction of the geometrical poremodel. A note of gratitude goes to Professor Finney for giving usaccess to his data set of monodisperse sphere-pack coordinates.We thank Chevron Corp. for allowing the publication of this paper.Partial funding was provided by the U. of Texas Formation Evalu-ation Consortium, jointly sponsored by Anadarko PetroleumCorp., Baker Atlas, ConocoPhillips, ExxonMobil, Halliburton, theInst. Mexicano del Petróleo, Schlumberger, Shell Intl. E&P,and Total. Additional funding for this project was provi-ded by a research grant from the American ChemicalSoc. (PRF#37519-AC2).

Fig. 12—Comparison of T2/D plots for simulations of 1-cp light-oil saturations in clean sandstone and coupled carbonate rockmodels for several values of fluid saturation and different typesof wettability.


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Appendix—NMR Random-Walk Resolution andAccuracyRandom-Walk Procedure. Random walkers within each saturat-ing fluid are displaced with infinitesimal steps of duration �t andlength �r related through Einstein’s formula,

�r2 = 6 DBf �t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-1)

During this step, the magnetization amplitude M of a walker ismultiplied by exp(−�t/T2), where T2 is adapted from the results ofBergman et al. (1995) and given by the relation

1

T2=

1

T2B+ �

�r

3.84�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-2)

In Eq. A-2, � is equal to 1 when the walker is located within onestep of a relaxing surface of longitudinal relaxivity �, and 0 oth-erwise. The shift of walker spin phase acquired during that step isgiven by

�� = � B �t, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-3)

where � is the proton gyromagnetic ratio and B is the value of thebackground magnetic field exerted by the NMR tool in the rotatingframe centered about the location of the walker. If the walk clockmeets a scheduled RF pulse, then the phase shift accumulated untilthen is reversed. If the clock meets a scheduled acquisition time t,then the recorded signal is equal to the mean of the projections�M�t� cos��0

t �� over the entire population of walkers.

Comparison to Analytical Magnetization Decays. Fig. A-1shows that the simulated decays in uncompacted and compactedcubic-centered arrays of grainpacks perfectly match the analyticalexpressions given by Eqs. 1 and 2 with zero gradient. Fig. A-2illustrates the accuracy of our simulations in the presence of amagnetic field gradient in bulk diffusion compared to analyticalexpressions with D=DB.

Effect of Timestep. This issue is not relevant in unbounded dif-fusion, but is crucial in porous media where tight geometricallimits exist. The geometries shown in Fig. 4 include patches ofwetting water in the pore corners or thin lenses of oil, and thereforeare responsible for local diffusion traps if the timestep is too large.The walkers cannot “see” diffusion exits from these traps, whichwill bias the actual diffusivity of the fluid phase. Likewise, in verysmall pores (such as in the intragranular microporosity used for the

Fig. A-1—Comparison between analytical (dots) and simulated(dashes) water NMR decay curves for different configurationsof cubic-centered grain packs, values of apparent pore radius(i.e., apparent S/V), and surface relaxivity. The compaction co-efficient is defined as the ratio between the diameter of thespherical grain and the size of the concentric cube that forms aperiod of the cubic-centered pack.


coupled microporous carbonate model), the timestep must be de-creased accordingly; otherwise, walkers will not diffuse from onemicropore to another. Fig. A-3 describes the numerical errors thatoriginate from inappropriately large timesteps in such complexrock/fluid configurations. In this figure, the left panels show thatthe microporosity-borne water exhibits artificially low diffusion,which disappears when the simulations are performed with a

smaller step size. The middle panels show that water diffusesunrealistically fast in the macropores where the simulations areperformed with a large timestep, as indicated by the 10−4 cm2/sdiffusion peak. Different water peaks coalesce into one peakwhen performing the simulations with a smaller step size. Finally,the panels on the right illustrate the need to adopt a small stepsize in order to honor MOW fluid geometries even in the absenceof microporosity. With a large step size, two water peaks ap-pear, one for the water bulk response in the MOW pores and onefor the water affected by surface relaxation in the pore corners.Once the step size is decreased, the two peaks merge intoone continuous distribution of relaxation times, which suggestshigh connectivity between pore centers and pore corners in ourMOW model.

Emmanuel Toumelin joined Chevron North America E&P as aFormation Evaluation Specialist in early 2006. e-mail:[email protected]. Previously, he was a research assis-tant at the U. of Texas at Austin from 2000 to 2005, where heconducted research on pore-scale modeling and combinedinterpretation of NMR and dielectric rock properties. He holdsan engineering degree from the Ecole Centrale and a PhDdegree in petroleum engineering from the U. of Texas at Austin.Carlos Torres-Verdín held the position of Research Scientistfrom 1991 to 1997 with Schlumberger-Doll Research. From 1997to 1999, he was a reservoir specialist and technology cham-pion with YPF (Buenos Aires, Argentina). Since 1999, he hasbeen with the Dept. of Petroleum and Geosystems Engineer-ing of the U. of Texas at Austin, where he currently holds theposition of Associate Professor and conducts research in for-mation evaluation and integrated reservoir characterization.He has served as Guest Editor for Radio Science, and he iscurrently a member of the Editorial Board of the Journal ofElectromagnetic Waves and Applications. He is an associateeditor for Petrophysics (SPWLA) and SPE Journal. Torres-Verdínis co-recipient of the 2003 and 2004 SPWLA (Petrophysics) BestPaper Awards, and recipient of the 2006 SPWLA TechnicalAchievement Award. He holds a PhD degree in engineeringgeoscience from the U. of California, Berkeley. Boqin Sunhas been a senior research scientist for Chevron EnergyTechnology Co. since 2001. He has been working on var-ious NMR applications in solution, solid, and petrophysics.Previously he was with Schlumberger at its Sugar Land productcenter for 4 years. Before that, he was a post-doctoral associ-ate at the Massachusetts Inst. of Technology. He holds aBS degree from Hanzhou U. and an MS degree from theWuhan Inst. of Physics in 1985, both in physics, and also aPhD degree from the U. of California, Berkeley, in chemistry.Keh-Jim Dunn has been with Chevron Energy Technology Co.since 1981. Before joining Chevron, he worked for GeneralElectric’s research center in Schenectady, New York, for 5years. His main research includes acoustic, electric, dielectric,and neutron transport, and NMR properties of fluid-saturatedporous media. He holds a BS degree in physics from the Natl.Taiwan U. as well as a PhD degree in materials science fromCornell U.

Fig. A-3—Illustration of T2/D NMR map errors caused by step-size inconsistencies. Left-side panels: oil-filled mixed-wet car-bonate model at irreducible water saturation; (top) homoge-neous step-size �r=200 nm; (bottom) �r=200 nm in themacroporosity, 20 nm in the microporosity. Middle panels: WWcarbonate model with 20% oil saturation; (top) homogeneous�r=200 nm; (bottom) dual space-step. Right-side panels: MOWsandstone model (no microporosity), with 60% water; (top) �r=1µm; (bottom) �r=200 nm.

Fig. A-2—Comparison between analytical (dots) and simulated(other curves) decay curves for two oil grades and for differentvalues of the G.TE product (see Eq. 2). Overlapping simulatedcurves show the results of simulations performed with differentvalues of G and TE for equal product values.


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