Validity of permeability prediction from NMR measurements

MÉTHODOLOGIE / METHODOLOGY

C. R. Acad. Sci. Paris, Chimie / Chemistry 4 (2001) 869–872 © 2001 Académie des sciences / Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés S1387160901013433/FLA

Validity of permeability prediction from NMR measurements Marc Fleury*, Françoise Deflandre, Sophie Godefroy Institut français du pétrole, 1 et 4, av. de Bois-Préau, 92852 Rueil-Malmaison, France Received 14 May 2001; accepted 10 October 2001

Abstract – Low field nuclear relaxation measurements applied to porous media can provide a wide variety of information. One important use of NMR measurements in the petroleum industry is the estimation of in-situ permeability as a function of depth. Such information is not available from any other tool and is critical for oil recovery predictions. A large number of empirical relationships have been published without clear explanation of their physical origin. We present some understanding and illustration of the link between NMR relaxation measurements and permeability, which is useful to select the appropriate law as a function of the geological context. © 2001 Académie des sciences / Éditions scientifiques et médicales Elsevier SAS permeability / NMR / relaxation / porous media / oil recovery / logging

Résumé – Les mesures de relaxation bas champ procurent une information variée sur la structure des milieux poreux ainsi que sur la nature des fluides en place. Dans le domaine pétrolier, une utilisation cruciale de ces mesures concerne l’estimation de la perméabilité in situ dans les puits. Des lois empiriques sont généralement utilisées, mais leur origine physique est mal comprise. Nous proposons une explication de l’origine des corrélations mesurées ainsi qu’une stratégie générale pour sélectionner le type de loi en fonction du contexte géologique. © 2001 Académie des sciences / Éditions scientifiques et médicales Elsevier SAS perméabilité / RMN / relaxation / milieux poreux / récupération du pétrole / diagraphies

1. Introduction NMR transverse relaxation time (T2) measurements are now routinely performed in situ to characterize oil fields. This is performed at low magnetic fields (corresponding to proton Larmor frequency between 500 kHz and 4 MHz). The in-situ measurements of T2 allow the continuous determination of porosity, pore size distribution, saturation, viscosity and permeability. While the physical phenomenon underlying the determination of the first four petrophysical parameters mentioned above are well established, the permeability determination is more controversial and is based on empirical relationships, more or less successful. Permeability is of primary importance for oil recovery predictions and reservoir characterization, and there

is no in-situ direct and continuous measurement of permeability available. The relaxation time T2, measured by a CPMG [1] sequence, is function of three terms according to (for a single pore, equation (1)): 1 = 1 +q S + D cGT 2 共 2V E兲 T2 T2B 12

(1)

where T2B is the bulk relaxation time of the fluid saturating the porous media, q2 the surface relaxivity, S and V the surface and volume of the pore respectively, D the molecular diffusion, γ the gyromagnetic ratio of the proton, G the effective magnetic field gradient accross the pore and TE the inter-echo time of the CPMG sequence. When the saturating fluid viscosity is low (e.g. water, light oil), the pore size small enough (typically < 100 µm), and/or the surface relaxivity not too high, the relaxation time T2 is dominated

* Correspondence and reprints. E-mail address: [email protected] (M. Fleury).

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M. Fleury et al. / C. R. Acad. Sci. Paris, Chimie / Chemistry 4 (2001) 869–872 Table. Some permeability correlation laws found in the literature.

Seevers [3] (60 samples)

2 k = C 共 1 − Swirr 兲 T 1corr U

Kenyon [4] (67 samples)

k=CT

Kozeny–Carman [6]

k = C 12 U 3 S k = C 共 1 − Swirr 兲2 12 U 3 S k = C 21 U 4.4 S wirr 1.69 共 1 − SwA 兲 k=C U PcA 1 − Swirr 2 4 k=C U Swirr

Rapoport and Leas [7] Timur [8] (155 samples) Swanson [9] Coates [10] (150 samples)

冋

2 2ml

冉

U

4

冊

册

C: constant Φ: porosity T1corr: long. relax. time corrected for bulk contribution (see equation (3))

T2ml = exp

兺i Ai

冥

, average

relaxation time in log. scale Swirr: irreducible water saturation S: specific surface area per unit bulk volume SwA: saturation at the capillary pressure PcA

by surface interactions (q2 S/V) and the measurement is sensitive to the radius of the pore body. Natural porous media usually exhibit a distribution of pore size and therefore, the measured relaxation times are distributed over several decades spanning from 1 ms (water in clay) up to 1 s (vugs). In this context, the relaxation time distribution reflects the distribution of pore size (or more rigorously the distribution of V/S) if there is no exchange between pores of different sizes. Experimental data indicate surprisingly a relationship between T2 and permeability k in a wide variety of porous media. However, k is governed by pore connectivity and pore throats rather than pore body [2]. The relationship clearly relies on a correlation between pore bodies and pore throats, which is only theoretically valid in a limited number of cases. We briefly review some of the existing correlation laws used in petroleum sciences. Then, we highlight some theoretical aspects to explain the physical origin of the correlation between pore body and pore throat. Finally, we discuss the relationships found between T2 relaxation times and permeability in two different systems.

2. Correlation laws A large number of correlation laws can be found in the literature. The relationships mostly used are listed in the table, using a petrophysical terminology. We can distinguish two types of correlations: – correlations involving directly T1 or T2 relaxation measurements [3, 4] (Kenyon’s relation has also been supported later by Straley et al. [5] and should be used in a porous medium fully saturated with water); – correlations involving petrophysical quantities [6–10] such as Swirr (the irreducible water trapped in a porous medium after a capillary dominated displacement process), or the specific surface area S; they are used in NMR interpretation when the porous media are saturated with two fluids (water and oil).

870

冤

兺i Ai log 共 T2i 兲

All relationships indicated in the table are supported by laboratory data and permeability is often estimated within a factor of two, sometimes better [3], over several decades in the range of interest for applications. It is striking to see the variation of the exponent in the porosity dependence, spanning from 1 up to 4.4. Note that porosity can be easily measured by NMR. In general, the use of NMR measurements, either for determining the dominant pore size (T2ml) or the irreducible saturation does improve k–U relationships, which are often very weak. We discuss below the first type of correlation.

3. Physical origin We focus here on the porosity and pore size dependence of the permeability, directly linked to NMR measurements. For grain packs and cemented sandstones, Chauveteau et al. [11] demonstrated the following relationship: k = C dg

2

冉

U − U1 U0 − U1

冊

4

(2)

where dg is the grain size (before cementation), U0 the porosity of the initial packing (∼0.4) and U1 the residual porosity at maximum cementation (∼0.02 or 0.03). Equation (2) can be demonstrated using both Navier–Stokes and Darcy’s laws, and are supported by experimental data on model grain packs and model sandstones (Fontainebleau). In particular, the U1 offset in the porosity dependence is clearly demonstrated experimentally using a large suite of Fontainebleau samples, with similar initial grain size. The power 4th in the porosity dependence is directly linked to the pressure drop at the throats in the porous medium, as predicted by Navier–Stokes equation in a cylinder. From the initial grain pack before the consolidation of the sandstone, the permeability will be reduced because of the reduction of the throat size, which also affect porosity (it is assumed that the deposit on the

M. Fleury et al. / C. R. Acad. Sci. Paris, Chimie / Chemistry 4 (2001) 869–872

grains is uniform). Because the grain size is directly related to the pore size, equation (2) gives a strong theoretical support to a relationship between NMR measurements and permeability. Equation (2) clearly predicts k ∝ T22 and is very close to Kenyon’s relationship (table). Note also that the U1 offset in equation (2) is only sensitive when a wide range of porosity is considered. It will be neglected in the correlations presented below.

4. T2 measurements We illustrate the possibility of determining permeability by NMR in two systems: a model system of grain packs clearly supporting equation (2), and a suite of reservoir carbonate samples. Figure 1. Permeability-pore size relationship in grain packs. Best fit: k ∝ dp1.99, R2 = 0.988.

4.1. Case 1: grain packs

The model porous medium consists of a series of packing made with non-porous silicon carbide, SiC, grains of different diameters varying in the range 8–150 µm. The grain size distribution for each pack is narrow, yielding a series of porous media with a narrow pore size distribution and a quasi-constant porosity (45 %). To take into account the small fluctuations of porosity, the pore size dp is deduced from the grain size dg, according to equation (3): dp =

共 1 −UU 兲

1/3

dg

(3)

These model porous media, originally prepared to study surface relaxation processes [12], were carefully cleaned to have a quasi-identical surface relaxivity q2. For each grain pack, water permeability was measured using standard procedure [3], and T2 relaxation time was calculated using a mono exponential fit. Due to the large value of some measured relaxation times compared to the bulk value, the T2 measurements were corrected for the bulk contribution according to: T2corr = (1/T2 – 1/T2B)–1

(4)

A linear relationship is observed when plotting T2corr versus grain size [12]. Permeability is also linked to pore size (figure 1), in close agreement with the theoretical prediction (equation (2)). It is also closely linked to the measured relaxation time T2corr, but the best-fit exponent (1.69, figure 2) is far from the expected value of 2. When porosity is not taken into account, the exponent decreases down to 1.50. We conclude that the k –T2 relationship is very sensitive to small fluctuations (or imperfections of the packs) and that the T2 exponent must be adjusted for each case with laboratory data.

Figure 2. Permeability correlation in grain packs with quasiidentical surface relaxivity. Best fit: k estimated ∝ U4 T2corr1.69, R2 = 0.988. The expected relationship is: k ∝ U4 T22.

4.2. Case 2: carbonates

NMR permeability relationships were tested on a series of 61 reservoir carbonate samples. Carbonates are more complex than sandstones in terms of pore structure and often exhibit no k–U relationship. In the present case, large permeability samples are characterized by bimodal pore size distribution (figure 3), while less permeable samples are more unimodal. It appears that permeability is governed by the largest pore population and, therefore, the logarithmic average T2ml as a characteristic time for each sample is not appropriate and decreases the contrast between the samples. When taking instead the position of the peak T2peak in the

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Figure 3. T2 relaxation distribution on four samples from the carbonate suite. The samples were extracted from the same well.

distribution, a relationship can be established (figure 4) with a reasonable correlation coefficient (0.82). Furthermore, porosity has practically no influence on permeability, as seen in figure 3. A best fit using a power law of the form Ua T2b yields a = 0.61 and b = 2.44, confirming a weak dependence with porosity. We conclude that diagenetic processes lead to a decorrelation of porosity and permeability, as in many carbonate systems, but a correlation between pore size and throat size still exist, as shown by the k–T2 relationship.

5. Conclusion Nuclear relaxation measurements can be used to estimate permeability in porous media. It is an indirect estimation relying essentially on a correlation between pore body (determined by NMR) and pore

Figure 4. Permeability correlation for a suite of 61 reservoir carbonate samples. Best fit, k ∝ U0.60 T2peak2.44, R2 = 0.821.

throat (governing permeability). For packs or sandstones made with grains of uniform size, the size of the pore body and pore throat are proportional. For such simple systems, a theoretical relationship between T1 or T2, porosity and permeability can be established and is verified experimentally. It should be used as a guideline to handle more complex porous structure such as those encountered in carbonates. In general, the relationship should be calibrated with laboratory measurements and there is no unique relationship valid. In the best cases (sandstones and unconsolidated media), permeability is determined within a factor of two. In carbonates, the relationship should be established very carefully to take into account specific diagenetic processes.

Acknowledgements. We thank G. Chauveteau for stimulating discussions about permeability–porosity relationships in sandstones.

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