Determination of pore size distribution in sedimentary rocks by proton nuclear magnetic resonance

Determination of pore size distribution in sedimentary rocks by proton nuclear magnetic resonance* James J. Howardt Department of Geology and Geophysics, Yale University, New Haven, CT 06511, USA

and William E. Kenyon Schlumberger-Doll Research, Ridgefield, CT 06877, USA

Received25 October 1990; revised21 October 1991; accepted26 October 1991 The distribution of pore sizes in sedimentary rocks can be determined from laboratory measurements of proton nuclear magnetic resonance (NMR) longitudinal relaxation (T1) in water-saturated rocks. The details of the longitudinal relaxation curve are converted into a T1 distribution curve by approximating the data with a sum of single exponential decays. The non-linear least-squares minimization procedure includes a regularization term that reduces excursions in the computed density function, thus producing smooth and robust results. The T1 distribution curve is scaled to pore size distributions by a phenomenological parameter that describes the strength of the proton interaction between the fluid and the6pore wall. This parameter is relatively constant in sandstones, but is an order of magnitude less with greater variability in carbonates. There is excellent qualitative and quantitative agreement between pore size distributions determined by NMR and independently determined information on pore sizes for a variety of sedimentary rocks. The laboratory derived 7-1 distributions are used to estimate the irreducible water saturation in sandstones by identifying the fraction of pores less than 4/~m in diameter. Keywords: pore size distribution; nuclear magnetic resonance; sedimentary rocks

Introduction For petrophysicists interested in the evaluation of formations, proton nuclear magnetic resonance (NMR) measurements on fluid-saturated rocks allows the estimation of rock properties that depend on pore size (Timur, 1969; Loren and Robinson, 1970; Kenyon et al., 1988). Most of the early studies on NMR measurements emphasized direct correlations between measured NMR relaxation and the macroscopic fluid flow properties of rocks, namely permeability, irreducible water saturation and movable fluids. These correlations are successful as a result of the connection between proton relaxation and pore size (Gallegos and Smith, 1988; Davies et al., 1990). This paper illustrates that meaningful pore size distributions can be obtained from an analysis of proton relaxation measurements, which allow the estimation of flow properties. The connection between proton NMR and pore size results from the strong relaxing effect of the grain surface on proton magnetization in water. This effect is illustrated by comparing the proton longitudinal relaxation behaviour of water in Berea sandstone, which has pore sizes of about 50-100 p.m, with the

*Presented at the meeting 'Geophysical Properties of Sedimentary Rocks', London, UK, 15-16 May 1990 tPresent address: Phillips Petroleum Research, 116 Geoscience Building, Bartlesville, OK 74004, USA

same water in a test-tube, with a 'pore size' equal to 20 000/~m (Figure 1). Water in the test-tube has a Tl value of 3.9 s and relaxes as a single exponential. Water in Berea sandstone has a T1 value of about 0.21 s and relaxes more gradually than a single exponential. The shorter value of T1 for the Berea sandstone corresponds to a smaller pore size. The data in Figure 1 is presented as linear signal strength rather than the conventional log measurement space as curve fitting is performed in linear terms (Kenyon et al., 1988). Curve fit errors at low magnetization values are exaggerated by the use of the log magnetization axis and are not a true reflection of measurement errors dominated by the random noise process which is uncorrelated with the signal. The details of the shape of the longitudinal relaxation curve for water in Berea sandstone can be converted into a curve of pore size distribution. The conversion requires an approximation of the measured relaxation curve by a sum of single exponential decays to obtain a curve called a T1 distribution curve. The NMR T~ distribution is then rescaled to pore size by the application of a surface dependent parameter. This paper illustrates the range of T~ distributions observed for a variety of sedimentary rocks. Independent measurements of pore size are compared with the pore size obtained from the NMR measurements. Applications of the pore size information to fluid flow properties important to the evaluation of formations are also illustrated.

0264-8172/92/020139-07 © 1992 B u t t e r w o r t h - H e i n e m a n n Ltd

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where M(t) is the proton magnetization recovery time t, Mo is the initial magnetization and P is the relaxivity of the surface in units of length/time. Most sedimentary rocks fall within this 'fast diffusion limit' where diffusion is rapid enough to supply the walls with sufficient protons. The strength of the surface relaxation mechanisms is the rate-limiting step and controls the overall relaxation rate. The result is uniform decay of the nuclear spins within the pore. Decay at the surface is fast because of the hindered rotation of water molecules at the surface and additional dipole interactions of the protons with unpaired electron spins in paramagnetic ions located on the pore wall. If only a monolayer of water was present, those protons would show very short magnetic relaxation times. However, for the present example of rapid diffusion of protons between the bulk and surface volumes of fluid, a single relaxation time Tim is measured. l/Tim = 1/Tjb + (StV)klTl~m

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where Tlb is the relaxation time constant of bulk water, S / V is the pore surface area to volume ratio and ~. is the

Principles of proton NMR relaxation in rocks Measurements of proton relaxation by NMR are unique among petrophysical techniques in that both volume and rate information are obtained. The amplitude of the NMR signal is proportional to the density of proton spins and thus provides a porosity measurement, comparable with that provided by neutron logging tools. Of interest here is the more detailed information carried in the rate of decay of the proton magnetization and its connection with pore size. The NMR experiment measures the rate at which nuclear spin states return to equilibrium. In an applied magnetic field, proton magnetization is defined by a non-equilibrium distribution of nuclear spin states. The removal of the applied field results in the relaxation of the proton nuclear spins to their equilibrium distribution. Longitudinal relaxation behaviour is often parameterized by a single TI value, even when it is not best described by a single exponential decay. The relaxation is often referred to as decay, although in typical experiments such as the inversion recovery used here, the final state is magnetized and it is actually the build-up rate which is observed. For protons in water saturating a porous medium, the measured longitudinal relaxation curve is affected by bulk and surface decay processes. The surface effect often dominates, as shown by the comparison between bulk water and water in Berea sandstone. The surface dominated magnetization decay of protons depends on the strength of the surface decay, 9, a characteristic length, a, and on the self-diffusion coefficient of the water, D. When diffusion is fast enough, in simple geometries when paiD < < 1 (Brownstein and Tarr, 1979; Banavar and Schwartz, 1989), the magnetization decays as a single exponential with a time constant M(t) = Mo exp(-t/Tl) 140

(1)

M a r i n e and P e t r o l e u m G e o l o g y , 1992, V o l 9, A p r i l

thickness of the surface monolayer that by itself relaxes with time constant TL~,r,- For most sedimentary rocks, Tlb is significantly longer than Tim, so that the desired surface relaxation time is roughly the same as the measured time. If each pore decays as a single exponential and the pores decay independently of each other, then an assembly of pores in a sedimentary rock will decay as a sum of exponentials

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If a measured longitudinal relaxation curve can be expressed as a sum of single exponential relaxation terms, that sum should represent the pore size distribution when properly rescaled. In the past, standard practice was to fit the measured longitudinal relaxation curve with N taken to be two or three. This was based more on computational concerns than on any insight into sedimentary rocks, as most sedimentary rocks clearly do not possess only two or three different pore sizes. There has been interest in expressing the measured longitudinal relaxation curve as a continuous distribution of exponentials (Schmidt et al., 1986; Gallegos and Smith, 1988; Kroeker and Henkelman, 1986; Halperin et al., 1989; Davies and Packer, 1990). Numerically, the problem is to perform a robust calculation when the number of terms is large; 35-50 single exponential terms are typically used to provide a meaningful pore size distribution. Generally, the minimization of squared-fit error with such a large number of terms is considered an ill-posed problem, i.e. any solution is not unique and small variations in the data result in dramatically different answers (Whittal and MacKay, 1989; Kenyon et al., 1989). The

Pore size distribution measurements with 1H-NMR: J. ,I. Howard and W. E. Kenyon numerical instability of the least-squares solution is transverse plane, where they precess and thus can be avoided by adding a regularizing term to the fit error measured. Thirty five recovery times between 70/xs and minimizing the sum. The regularization term has and 20 s were selected to cover the range of observed the effect of minimizing excursions in the computed proton relaxation rates. The resulting inversion density function, thus producing a smooth result. recovery curve was converted to a T~ population curve The regularization term forces neighbouring points in with 35 values of T1 using a non-linear optimization the density function to have similar intensities and procedure and a constant regularization term. rejects solutions with a large number of peaks. The In addition to the T1 distribution curves, a mean T1 minimization also can be arranged to allow only value was obtained by fitting a stretched exponential positive values in the density function. The new method model uses user-specified decay times covering the range of interest, in contrast to the earlier two or three (5) M(t) = Mo exp(-t/Tx) ~' exponential representations in which the /'1 values were chosen as part of the fit procedure. Thus, the new method essentially allows the data to determine the to the longitudinal relaxation curve (Kenyon et al., appropriate form of model to be fitted (Kenyon et al., 1988). The TI,~ term obtained was used as a mean /'1 1989). value and is essentially the time for longitudinal The T~ distribution curves that are generated by this magnetization to relax within 1/e of its equilibrium data inversion process closely represent the variability value. As the measured TI~ value was considerably less and breadth of pore size distributions observed in than Tlb, tenths of a second compared with 3.9 s for sedimentary rocks. Comparison of these TI distribution water at 10 MHz, the mean T~ value was not corrected curves with pore size information obtained from for bulk contributions. independent image analysis and mercury porosimetry Independent pore size information was obtained measurements provides insights into the magnitude and from petrographic image analysis and from mercury variability of the surface relaxation parameter. porosimetry. In petrographic image analysis, pore

Experimental methods NMR measurements were made on water-saturated plugs, 2 cm in diameter and 3.75 cm in length, in a desk-top instrument operating at a proton Larmor frequency of 10 MHz (IBM-Bruker Minispec). These plugs were saturated under a vacuum with a 0.1 M sodium chloride brine and NMR measurements were made in a sealed PTFE holder. The experiments were computer controlled after the system was tuned to resonance and the 90° and 180° pulse lengths were set for each sample. The experiment measured the longitudinal relaxation o r T I curve using a standard inversion recovery procedure (Figure 2). The protons align in the field of the permanent magnet. A pulse from the radio frequency coil then produces a second magnetic field orthogonal to the first, which rotates the magnetic moments of the protons by 180°. The net proton magnetization then relaxes towards its original state for a specified time, after which a second radio frequency pulse rotates the proton spins by 90° into the

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type end-members are obtained from digitized images of pore and non-pore space by unmixing the rough-smooth spectra generated by repeated erosion-dilation operations (Ehrlich et al., 1984). A pore diameter is determined for each pore type end-member, with an overall mean pore diameter determined by -E =

~, pidi i=1

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where Pi is the proportion of each pore type, di is the characteristic diameter associated with each pore type and n is the number of pore types generated for that set of samples. The pores which could not be resolved optically were assigned a nominal diameter of 1/zm and their proportion was determined from the difference between the amount of total optical porosity and the total porosity measured by buoyancy methods. Mercury porosimetry curves were generated using injection pressures ranging from 1 to 60 000 lb/inL Cylindrical samples were used instead of chips to minimize surface effects. The injection pressure P is converted to an effective throat radius r using the Young-Laplace equation r = - 2ycos0/P

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Figure 2 Schematic diagram of laboratory NMR apparatus and the pulse sequence for the inversion recovery experiment

where • is the surface tension for mercury [480 dyn/cm (1 dyn = 10/xN)] and 0 is the contact angle through the wetting fluid (140°). A mercury injection population density curve is calculated to facilitate comparison with NMR and image analysis data. The ordinate of the population density is calculated as l/Vmax (dV/dlog P)

(8)

where V is the intruded volume and Vmax is the intrusion volume at 60 000 lb/in 2.

Marine and Petroleum Geology, 1992, Vol 9, April

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Pore size distribution measurements with ~H-NMR: J. J. Howard and W. E. Kenyon

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Relaxation time constant, msec Figure 3 T~ distribution curves for a suite of microporous cherty sandstones. Short T~ times correspond to small pores, long Ta times to larger pores. The height of dashed box is 0.1 of the total porosity. Multiple curves for each sample represent different plugs taken from a single rock. Variations in curves, e.g. 3N-2D, reflect different amounts of coarse and fine grained laminations in each plug

Results

The relationship between pore size distribution and proton T1 distribution curves is first shown for a set of sandstones containing abundant microporous chert. The T1 distribution curves show bimodal and very broad unimodal distributions (Figure 3). Several curves are shown for each labelled sample, each corresponding to a replicate plug. The repeatability of the conversion from inversion recovery to T~ distribution is shown by the similarity between replicate plugs. A few samples (sample 3N-2D in particular), show significant differences between replicate plugs. These differences reflect differing ratios of coarse grained to fine grained laminations in the different sample plugs. There is a strong qualitative agreement between the shapes of the T~ distributions and the pore size distribution observed visually in thin section. Samples at the top of the figure predominantly contain microporosity within the chert grains, which agrees with the T~ distribution curve. The T] curves indicate that most of the microporosity has very small T~ values of about 10 ms. Samples at the bottom of the figure with longer Tl populations have mainly large intergranular pores (Kenyon et al., 1989). Pore size distributions from image analysis are used as the basis for a quantitative comparison of these cherty sandstones. Image analysis of these samples results in six pore type end-members, plus a seventh to account for pores that are not resolved at the magnification used (Kenyon et al., 1989). To allow a comparison, the NMR T~ distributions are redefined into seven categories centred around the pore diameters corresponding to the pore type end-members 142

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Marine and Petroleum Geology,

1992,

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9, April

101

10 2

Pore diameter, microns Figure 4 Comparison of pore size distributions from NMR (solid line) and image analysis (broken line) for suite of microporous cherty sandstones

(Figure 4). The similarity of pore size distributions obtained from optical and NMR methods for these samples is fairly good. The redefining of NMR data requires the selection of a value for the surface relaxivity p. In Figure 4, p is 0.001 cm/s. This value agrees with p values obtained by comparing mean T~ with mean pore size. This value of p also is not far from the values of 0.0015 cm/s obtained for glass bead packs (Straley et al., 1987) and 0.00065 cm/s for porous ceramics (Senturia and Robinson, 1970). Tj distributions for a second set of sandstones are more unimodal than those for the set containing microporous chert (Figure ,5). These reservoir sandstones are characterized by better grain sorting and fewer laminations than the cherty sandstones. The predominantly long T1 values of samples AU 4 and AU 17 are reflected in high permeabilities (500 mD) and low mercury porosimetry entry pressures. Image analysis similarly indicates a large average pore diameter of 50/~m. In contrast, sample AU 5, which has a shorter Tj value, is finer-grained, with reduced permeability and an average pore diameter of 20/~m. For all samples of this second set, the T] populations of about 50-85 ms correspond to the presence of clusters of authigenic kaolinite in the larger pores. Relaxation time distributions for several carbonates from a Devonian formation in western Canada can be used to distinguish between different lithofacies and depositional environments (Figure 6). Samples 101 and 105 are shoal margin packstones, with a high intraparticle porosity found in the chambers of large stromatoporoid fragments. Samples 118, 131 and 135 are foreslope wackestones, with larger proportions of fine matrix and less pore space visible in thin section. Sample 009 is a peloid grainstore from a high energy shoal margin with a high interparticle porosity. The

Pore size distribution measurements with ~H-NMR: J. J. Howard and W. E. Kenyon

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intensities of these T~ distributions are normalized to the NMR porosity determined from the intensity of the original inversion recovery curve. The wackestones have less macroporosity and total porosity than the packstones. The most obvious comparison between these carbonate T1 distributions and those for the sandstones is the significantly longer T~ values for the carbonates. Other workers have recognized that proton relaxation in carbonates is less efficient than in sandstones (Timur, u i nlllnnI

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1972; Brown and Gamson, 1960; Senturia and Robinson, 1970). Estimates of the reduced relaxation strength of carbonate pores ranged from one-seventh to one-tenth of that of sandstones. Thus, 7"1 for a pore in a carbonate would be seven to ten times longer than for a sandstone pore of the same size. Several samples from the Middle East also illustrate this reduced relaxation strength of carbonates. The resultant T1 distributions show TI~ values of 300 ms (Figure 7). Mercury porosimetry curves for these samples indicate entry pressures around 100 lb/in 2, which is equivalent to a pore throat diameter of 0.4/zm (Figure 8). The long T1 values in these samples translate into smaller pore sizes than would be expected in sandstones. Interestingly, the small sample to sample variation in pore size seen by NMR and mercury porosimetry conflicts with the hand lens classification of these three samples as grainstone, packstone and wackestone, respectively. Microscopic examination

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Pore size distribution measurements with 1H-NMR: J. J. Howard and W. E. Kenyon 14 reveals that diagenetic alteration has made each sample microporous in spite of their different depositional 12fabric (S. Moshier, personal communication). 5 ,410

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The petrophysicist is interested in whether a given measurement can be used to evaluate amounts of fluids and their producibility. It has previously been shown that NMR measurements can produce good estimates of permeability and irreducible water saturation (Timur, 1969; Loren and Robinson, 1970). It is our contention that it is the relationship between NMR measurements and pore size, illustrated in the previous section, which underlies the ability to estimate flow properties. In this section, we illustrate this contention by examining the relationship between NMR, microporosity and irreducible water saturation. According to the preceding section, microporosity should be readily obtainable from the T1 distribution curve as the fraction of porosity showing T1 values less than some threshold value, i.e. all the pores smaller than a certain size. This is tested in the microporous cherty sandstones by estimating the microporosity as the fraction of mercury intrusion volume at pressures greater than 500 lb/in 2 (0.4 p~m throat diameter). The selection of the cut-off pressure is rather arbitrary; the threshold is chosen as beyond the peak of the differential volume curve, but similar results are obtained when a value of 100 lb/in 2 is used. The microporosity values thus obtained are close to the values obtained by subtracting the optical porosity from buoyancy porosity. An NMR estimate of microporosity is obtained by taking the fraction of porosity which shows T] values of 85 ms. For 9 equal to 0.001 cm/s, 85 ms corresponds to a pore diameter of 4 /xm. The NMR microporosity thus derived usually agrees with the microporosity obtained from porosimetry (Figure 9). Microporosity estimates from NMR are higher than those determined from mercury porosimetry and reflect the fact that the 85 ms cut-off is arbitrary and not based on the best fit for each sample (Straley et al., 1991). In the preceding paragraph, it was assumed that the microporosity can be measured by mercury porosimetry. In principal, mercury porosimetry measures the drainage characteristics of the rock (a non-wetting fluid, mercury, replaces the wetting fluid, air). Thus, the mercury porosimetry curve should

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Figure 10 Comparison of laboratory NMR estimates of free fluid index (FFI) with results of centrifuge desaturation experiments on sandstones from western Canada. Laboratory NMR estimate of FFI also agrees with NML logging result

be a scaled version of the drainage curve obtained by removing water (a wetting fluid) from rock by centrifuging against air (a non-wetting fluid). The scaling factor should be the ratio between the surface tensions of mercury and water (approximately seven). Figure 10 shows the comparison between water removed by centrifuging and the fraction of porosity with T] values greater than 50 ms for a sandstone sequence from western Canada. The cut-off at 50 ms which yields the observed good agreement is rather longer than the 12 ms value used by Timur (1969; 1972), but this may be accounted for by the difference between fitting three exponentials, as Timur did, and the conversion to a T~ distribution as used here. Figure 10 also shows a borehole measured NMR result, in addition to the laboratory measured NMR property described in this paper. In commercial borehole NMR measurements, a quantity called the free fluid index (FFI) is produced which has roughly similar qualities to the laboratory NMR estimate of rnicroporosity. The FFI is the amplitude (calibrated in porosity units) of the signal measured after the tool dead time of approximately 30 ms. The dead time heavily weights the long T~ components of the measured waveform and discriminates against the short T, value components.

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Marine and Petroleum Geology, 1992, Vol 9, April

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Pore size d i s t r i b u t i o n m e a s u r e m e n t s w i t h 1H-NMR: J. J. H o w a r d a n d W.

A comparison of the FFI determined by wireline log and laboratory NMR estimates of the FFI are shown for several samples (Figure 11). The T1 distribution curves for samples that correspond to specific depths on the FFI curve are presented. The samples with a FFI greater than 1 - 2 % have 7"1 distributions skewed towards longer T1 values, whereas the samples with F F I = 0 have a significantly larger proportion of T1 values less than 80 ms. The zone with F F I = 0 has laboratory permeabilities of <0.01 mD and is non-producing. The zone with increased FFI values has measured permeabilities of 10-30 mD and does produce fluids. Conclusions Laboratory NMR longitudinal relaxation curves can be expressed as a sum of single exponentials called a T1 distribution. The number of exponential terms is large enough to provide a useful pore size curve and is robust and repeatable. The T1 distribution curves agree qualitatively with pore size distributions seen in thin sections, with long 7"1 values corresponding to large pores and small T1 values corresponding to small pores. T1 distribution curves are typically unimodal, with the mode corresponding to large pores, with a secondary short T1 value that corresponds to microporosity among clay particles. T1 components range from approximately 1 ms up to 1 s or more, limited by the 7"1 of bulk water, which was 3.8 s in these experiments. The scaling of T1 distributions into pore sizes requires knowledge of the surface relaxation term. Values for p in sandstones are about 0.001 cm/s, whereas carbonates have smaller p values. The laboratory derived T1 distributions can be used to estimate the irreducible water content of a formation, which is an important parameter for reservoir rocks. The connection between NMR measurements and irreducible water saturation is the sensitivity of NMR to pore size. NMR properties can be measured in the borehole which are also sensitive to the pore size distribution and which correlate well with the laboratory measured irreducible water saturation of the rock.

Acknowledgements The authors thank C. Straley for assistance in the laboratory and for discussions. L. McGowan, D. Rossini and P. Dryden are thanked for the variety of petrophysical measurements and geological descriptions they made. R. Ehrlich and E. Etris (University of South Carolina) provided the image analysis results. C. Morriss assisted with the collection of samples and the NML log measurements.

E.

Kenyon

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