# Permeability estimation from NMR logs in a limestone geothermal aquifer

Marc Fleury, Maxime Catinat, Benjamin Brigaud, Miklos Antics, Pierre Ungemach

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Marc Fleury, Maxime Catinat, Benjamin Brigaud, Miklos Antics, Pierre Ungemach. Permeability estimation from NMR logs in a limestone geothermal aquifer. European Geothermal Congress 2022, European Geothermal Energy Council, Oct 2022, Berlin, Germany. **hal-05294985**

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# Permeability estimation from NMR logs in a limestone geothermal aquifer

Marc Fleury*<fnref n="1" />, Maxime Catinat<fnref n="2" />,<fnref n="3" />, Benjamin Brigaud<fnref n="2" />, Miklos Antics<fnref n="3" />, Pierre Ungemach<fnref n="3" />

**Keywords:** Carbonate, Paris Basin, NMR, Permeability, Facies, Dogger.

## ABSTRACT

We performed a laboratory study in relation to 2 doublets located in Bobigny (Paris basin). In addition to conventional logs, NMR data were acquired in 2 wells, while a third deviated well was cored over a 35 m interval in the target layer (Dogger formation). Petrographical and petrophysical analysis were performed on about 70 samples: thin sections, water permeability and formation factor, and laboratory NMR $T_2$ distribution. Hence, we could establish a classification using mainly 4 facies and understand the influence of microporosity and micritization in the flow properties. The range of measured permeability was large [0.05 – 1000 mD] and we could evaluate different relationships between permeability and a combination of porosity and $T_2$ distribution. Since $T_2$ distributions can provide the fraction of microporosity in the total porosity, there exist several possibilities to generate a representative unique macropore size which should be included into the permeability relationship. For one facies, permeability was clearly independent of porosity over a porosity range of [0.12-0.22], illustrating the well-known difficulty of predicting permeability in carbonates. We present the general approach for establishing NMR $T_2$ permeability relationships and discuss their significance.

## 1. INTRODUCTION: PERMEABILITY PREDICTION IN CARBONATE SYSTEMS

Permeability is a key parameter for geothermal performance but its determination from well logging data is notoriously uncertain in carbonate systems (Lucia F.J., 2007; Minh C. et al., 1997). Nuclear Magnetic Resonance (NMR) logging data are now routinely acquired in the petroleum industry, in addition to more conventional Gamma-ray, density neutron or resistivity logs but this is not a common practice when drilling geothermal wells. The benefit over cost ratio should however be seriously considered and we provide here some arguments based on an example in the Paris basin at the Bobigny site. Fundamentally, the so-called NMR $T_2$ relaxation time distribution measured by the logging instrument is a proxy for the pore size distribution and it is therefore a

key information for evaluating permeability, still indirect but much more relevant than porosity alone. Contrary to sandstones, in carbonates the porosity-permeability relationship may not exist depending both on lithology and diagenetic effects. Hence standard NMR-porosity-$T_2$ relationship should be carefully applied and/or calibrated with core samples. When this is done with enough understanding of the depositional environment and the geological context, it can be applied safely to other wells in which NMR logs are available.

## 2. BACKGROUND

Relaxation time $T_1$ or $T_2$ obtained from NMR measurements at low magnetic field have long been recognized as permeability indicators in combination with porosity (Babadagli & Al-Salmi, 2004). Since such information can be acquired by logging tools in open hole conditions after drilling a well, it provides a significant improvement of permeability prediction from well logging data. The main reason is that the relaxation time $T_1$ or $T_2$ is a proxy of the pore size according to (Godefroy et al., 2001):

$$ \frac{1}{T_2} = \rho_2 \frac{S}{V} + \frac{1}{T_{2B}} \quad [1] $$

where $T_2$ is the relaxation time measured in a single pore of volume V and surface S, $T_{2B}$ is the relaxation time of the bulk liquid saturating the pore ($T_{2B} \approx 3000$ ms for water at 30°C), and $\rho_2$ is the surface relaxivity, a property of solid-liquid NMR interactions. Equation [1] has been written using $T_2$ but it is also valid using $T_1$, however not routinely measured in practice. The ratio V/S has a dimension of length and characterizes the pore body which is different from the pore throat governing permeability. Natural porous media often comprises a distribution of pore sizes, therefore generating a distribution of relaxation time $T_2$. The information on pore size can be used for evaluating permeability using the following type of relationship:

$$ k = C \rho^{\alpha} T_{R}^{b} \quad [2] $$

where $\Phi$ is porosity, $T_R$ is a relaxation time calculated from the measured distribution and expressing one representative pore size governing permeability; there exist a large number of other formulas (Babadagli &

<footnote n="1">¹ IFP Energies Nouvelles, 1-4 Avenue de Bois-Préau, 92852 Rueil-Malmaison, France</footnote>

<footnote n="2">² Université Paris-Saclay, GEOPS, CNRS, 91405 Orsay, France</footnote>

<footnote n="3">³ GEOFLUID, 165 Rue de la belle étoile, 95700 Roissy CDG, France</footnote>
Al-Salmi, 2004) and equation [2] is adequate for a formation saturated with a single liquid (brine in the present case) whereas many others deal with a formation saturated with oil and water for hydrocarbon production applications. Equation [2] has been quite successful in sandstones and for these formations, the relaxation time $T_R$ is chosen as the log mean value $T_{2lm}$ of the distribution (Dunn et al., 2002). The significance of this choice is related to the existence of clays and if the amount of lining clays increases in the intergranular pore space, $T_{2lm}$ will decrease and yield smaller values; as a result smaller permeabilities are calculated even though porosity may not vary much. For carbonates, there could be a similar reasoning dealing with microporosity and by default in the absence of core for calibrating C, a and b in equation [2], their values are set to C=4, a=2, b=4 and $T_R = T_{2lm}$ (with units K in mD, $T_{2lm}$ in ms and porosity as a volume fraction). This relation will be called the SDR (Schlumberger-Doll Research) default relationship (Kenyon et al., 1988), with the coefficient C adapted to carbonates as opposed to sandstones to take into account essentially the lower surface relaxivity in carbonates. The SDR equation has a theoretical basis derived from grain packing theory (Banavar & Schwartz, 1987; Chauveteau G. et al., 1996) and as such it should be considered as a guide only; the exponents a and b cannot be universal, especially in the context of carbonates. For example, for a dataset of carbonates from the Middle East, $T_R$ was chosen as the mode of the distribution $T_{2mode}$ (Figure 1) and $a=0.60, b=2.44$ (Fleury et al., 2001).

<figure><img src="image_2.png" /><figcaption>**Figure 1:** Example of a distribution measured in the laboratory for a sample of the Dogger formation (depth 2383.7m); the log mean value $T_{2LM}$ and the mode of this distribution are indicated by vertical lines. The cut-off $T_{2cm}$ for separating micro and macroporosity is also shown.</figcaption></figure>

Permeability relationships have been revisited recently for different carbonate systems (Aghda et al., 2018; Ge et al., 2017; Mason et al., 2019), trying to improve the prediction of permeability from the knowledge of porosity and a proxy of the pore body size distribution measured by NMR well logging tool. We proceed here with the same objective.

<figure><img src="image_3.png" /><figcaption>**Figure 2:** Measurements of porosity, water permeability Kw, formation factor FF and $T_2$ distribution on a set of 72 plugs from the Bobigny well (GB4). A typical thin section for each facies is shown with the corresponding color code.</figcaption></figure>
## 3. PETROPHYSICAL AND PETROGRAPHICAL MEASUREMENTS

Two sections of 18m-full size cores were available from a geothermal deviated well (GBD4) located in the Paris region (Bobigny, France). The target were the Middle Jurassic limestones (Dogger) at a depth of about 1600 m (TVD). We extracted from these full-size cores 72 plugs of standard size for petrophysical measurements (diameter 4 cm and length about 6 cm). The trim ends were used for petrographical observations on thin sections and thus there is a close correspondence between petrophysical and petrographical data.

After saturating the plugs with brine (20 g/l NaCl), porosity and NMR $T_2$ distributions were measured on a Magritek instrument (Wellington, New Zealand) with a proton resonance frequency of 2 MHz, close to logging instruments. Then, water permeability together with formation factor were measured in a flooding cell equipped with electrodes at both ends to measure resistivity. Formation factor results will not be discussed here.

## 4. RESULTS

### 4.1 Link between relaxation time distribution and facies

$T_2$ distributions are clearly linked to facies identified in thin sections (Figure 2). In the present set of samples, the following facies were identified:

- F2A: Microporous oncoidal and peloidal packstone to rudstone,

- F2B: Non-porous oncoidal packstone to floatstone,

- F3B: Low to moderately cemented ooid grainstone with bioclasts,

- F3C: Highly porous ooid grainstone.

Defining a cut-off value for microporosity at 120 ms (Figure 1), a value slightly lower than determined in a previous work (Vincent et al., 2011), the non or weakly permeable facies F2A and F2B have components mostly below 120 ms, while the more permeable facies F3B and F3C have components much above 120 ms and with large intensities (i.e. large fraction of macroporosity). The NMR tool can clearly evidence the presence and amount of micro or macroporosity, and this is essential for the prediction of permeability. Indeed, it is the interplay and organisation of the micro and macroporosity that governs permeability, with 2 extreme cases: (i) macropores form a continuous network and micropores are in paralel relative to the flow; in this case, permeability is governed by macropores whatever the contribution of micropores to the total porosity, (ii) micropores are dominant and the flow in macropores is interrupted by microporous zones in series relative to the flow; in this case, permeability is severely decreasing, but not necessarily porosity.

<figure><img src="image_4.png" /><figcaption>**Figure 3:** Test of various relationships for predicting permeability. Green: facies F3C, yellow F3B, red F2B, cian F2C.</figcaption></figure>
It should also be remembered that permeability depends on the power 4th of the pore entry size (pore throat) meaning that a small amount of the macropore network can be responsible for the permeability.

## 4.2 Permeability relationships

Different relationships were tested on the data set described above. The uncertainty of the prediction is expressed as a factor directly calculated from the standard deviation (STD) between the measured and predicted permeabilities; since we consider power laws, a STD value of 3 means that the predicted permeability Kw can statistically be between Kw/3 and Kw*3 with 95% confidence (Table 1).

First, the plot of water permeability Kw vs. porosity (Figure 3A) reveals that for the most permeable facies F3C Kw is nearly constant while porosity varies from 0.22 down to 0.12. This is typical of permeable carbonate systems. It is a facies for which water is flowing through the macropore network independently of the micropores that only affect porosity. For F3B, this effect is less pronounced, and permeability tends to decrease as porosity decreases. The 2 others facies F2A and F2B have very low permeability around 1 mD or less. A simple K-Φⁿ relationship has an uncertainty factor of 4.5 (Table 1). When plotting Kw vs. the mode of the distributions (Figure 3B), it is interesting to see two main groups of points: a first group around 800 ms gathering the most permeable facies for which the amount of macropores is larger, and a second group around 30 ms gathering the less permeable facies for which the amount of micropores is larger; interestingly, there exist some outliers traducing for these samples a nearly equal amount of micro and macropores and so the mode can fall over either at a small or high value. Thus choosing T₂mode as a representative pore size has some relevance but dealing with outliers is difficult.

As expected, equation [2] with the parameters C, a and b adjusted by least square fitting has a lower uncertainty factor of 3.5 (Figure 3C, Table 1). Note that the default SDR relationship has an uncertainty factor of 4.2 (Table 1, graph not shown) and improves only moderately the simple K-Φⁿ relationship. At this stage, it is tempting to add new parameters by subdividing the T₂ distribution into more intervals in addition to micro and macropores to include for example mesopores (Ge et al., 2017). We

suggest here a much simpler approach by calculating the amount of macroporosity from T₂ distribution with a cut-off at 120 ms (Figure 1). With a simple power law Kw vs. macroporosity Φₘ (Figure 3D) we obtain an uncertainty factor of 2.6, the lowest of all relationships (Table 1, 5 points with a macroporosity smaller than 0.02 were not included in the fit).

## 4. CONCLUSION

We tested several relationships for predicting permeability based on a dataset comprising 72 data points of porosity, permeability and NMR T₂ distribution. Thin sections indicate 4 facies, 2 of them being packstone of low permeability. A permeability-porosity relationship does exist but provides a very uncertain prediction. By including the pore size distribution, prediction uncertainties are decreased. The best prediction is obtained using macroporosity instead of total porosity in a power law. The macroporosity is easily obtained from the measured T₂ distribution with a cut-off at 120 ms.

<table><caption>Table 1: Summary of the different relationships tested and uncertainty factor STD.</caption><thead><tr><th>Relationship</th><th>Parameter values</th><th>STD</th></tr></thead><tbody><tr><td>Kw = CΦⁿ</td><td>C=5.81x10⁹<br>n=4.49</td><td>4.49</td></tr><tr><td>Kw = 4ΦⁿT²</td><td></td><td>4.23</td></tr><tr><td>Kw = CΦⁿT²</td><td>C=1859.6<br>a= 6.29<br>b=1.77</td><td>3.46</td></tr><tr><td>Kw = CΦⁿ</td><td>C=2.738x10⁷<br>n=5.18</td><td>2.62</td></tr></tbody></table>

## Acknowledgements

This work is supported by the UPGEO “UPscaling and heat simulations for improving the efficiency of deep GEOthermal energy” program funded by the Agence Nationale de la Recherche [ANR-19-CE05-0032-01].
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