Transport in Porous Media (2020) 135:287–308
https://doi.org/10.1007/s11242-020-01474-1

Petrophysical Correlations for the Permeability of Coquinas(Carbonate Rocks)
M. C. O. Lima1 · E. M. Pontedeiro1,3 · M. Ramirez2 · A. Boyd1 · M. Th. van Genuchten3,5 ·L. Borghi4 · P. Couto1 · A. Raoof3
Received: 10 March 2020 / Accepted: 2 September 2020 / Published online: 4 October 2020
© The Author(s) 2020

Abstract
The pore structure of many carbonate formations is known to be very complex and heterogeneous. Heterogeneity is manifested by the presence of different types, sizes, and shapesof pores resulting from sedimentation and diagenetic actions. These complexities greatly
increase uncertainties in estimated rock hydraulic properties in that different permeability
values may occur for samples having similar porosities. In order to understand the effects ofpore structure and heterogeneity, petrophysical analyses were performed on coquina samples from the Morro do Chaves Formation (Barremian, Sergipe-Alagoas Basin), which isan analogue of Brazilian Pre-salt oil reservoirs of Itapema Formation in the Santos Basin.
Routine core analyses, and NMR and MICP measurements were carried out to obtain pore
body and pore throat distributions. Obtained T2 relaxation times were converted to poresize radii by matching the NMR and MICP curves. Pore-scale imaging and pore networkmodelling were performed using microCT scans and the PoreFlow software, respectively.
Calculated permeabilities using PoreFlow showed excellent agreement with the routine
laboratory measurements. Samples having pore bodies with a higher coordination number showed much larger permeabilities at similar porosities. This study includes a statistical analysis of various features that caused the observed differences in permeability of thecoquinas, including the role of connectivity of the entire porous system. Limitations andchallenges of the various techniques, and the imaging and pore-scale flow simulations, arediscussed.
Keywords Carbonate rocks · Nuclear magnetic resonance · Image processing · PoreFlow ·Permeability* A. Raoofa.raoof@uu.nl
1

Department of Civil Engineering, COPPE, Federal University of Rio de Janeiro, Rio de Janeiro,
Brazil2

Department of Mechanical Engineering, COPPE, Federal University of Rio de Janeiro,
Rio de Janeiro, Brazil

3

Department of Earth Sciences, Utrecht University, Utrecht, The Netherlands

4

Department of Geology, LAGESED, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

5

Center for Environmental Studies, CEA, São Paulo State University, Rio Claro, São Paulo, Brazil

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1 Introduction
Carbonate rocks contain approximately 50% of the world’s hydrocarbon reserves (Teles
et al. 2016). Because of their wide distribution and importance, carbonate rocks are now
increasingly being studied at a range of scales and from various perspectives, especiallysince the discovery of Brazilian Pre-salt reservoirs in the Santos Basin, an Eastern Brazilian Continental Margin basin. Petrophysical analyses of these rocks indicate extreme
heterogeneity in their lithology. Sedimentation and diagenetic action much affected theirpore and particle size distributions and original shape (Sun et al. 2016), ultimately reflectedin the permeability and percolation properties of these rocks (Lima et al. 2019). Understanding the petrophysical properties of carbonate rocks is essential for better assessmentand exploration of hydrocarbon reservoirs (Sun et al. 2016). Like other physical properties of porous materials, permeability is a function of the complex microstructure of therock involved (Mostaghimi et al. 2013). Although permeability is often assumed to be a
relatively simple function of porosity (Kaufmann 2009), in reality this property is also afunction of grain size, pore geometry, and the pore size distribution, including pore connectivity and tortuosity (Zhou et al. 2019).
Nuclear magnetic resonance (NMR) and other imaging techniques are very much facilitating rapid and non-destructive petrophysical characterizations of carbonate rocks andother porous media (Cnudde and Boone 2013; Fleury and Romero-Sarmiento 2016; Liet al. 2017). With NMR, the pore space interpretation is evaluated indirectly through theexcitation of the nuclear spins, mainly 1H contained in the saturating fluid molecules. In
addition to porosity, the technique is able to provide a pseudo-pore size distribution and,
consequently, provides an attractive means for estimating the permeability (Chi and Heidari 2016). Typically, the pore size distribution of a rock sample drives the distribution oftransverse relaxation times (referred to as T2), determined by mathematical inversion ofrelaxation measurements (Buttler et al. 1981). At the limit of the fast diffusion regime, therelaxation time, T2, of a fluid saturating a porous medium is given by( )V= 𝜌2 T2(1)
S
pore

where 𝜌2 represents the surface relaxativity. This equation shows that relaxation times of
fluids inside pores are inversely proportional to the pore surface to volume ratio, definedas S/V in Eq. (1), with the scaling of this relationship being the surface relaxativity, ρ2.
Relaxation at pore surfaces is known to be much faster than that of the bulk fluid phasebecause of interactions between the solid and fluid molecules in the immediate vicinityof surfaces (Chang et al. 2004). Since 𝜌2 represents the strength of the relaxation at poresurfaces, it depends on properties specific to the solid surface and the fluid involved, fromwhich a relationship between relaxation time and pore size can be established (e.g. Wong
1999). The surface relaxativity represents the decay speed of a polarized spin when it interacts with the pore wall. This parameter hence depends on the pore surface, mineralogy, andpresence of clay, as well as the saturating fluid involved, thus imposing great uncertainties
in terms of its measurement (Souza et al. 2013).
The pore system of a rock can be represented by a pore network consisting of porebodies (corresponding to the largest voids) and pore throats (connections between thepore bodies) that are responsible for flow (Dong and Blunt 2009; Raoof et al. 2013;
among others). A useful analysis complementary to NMR is the characterization of
the distribution of pore apertures (e.g. their radii) using mercury intrusion capillary

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pressure (MICP). MICP techniques were designed initially to estimate the permeabilityof relatively small and irregular samples (Purcell 1949), as opposed to measurements by
gas injection, which require more regular formats to be accurate. MICP measurements
are usually taken by injecting the non-wetting fluid under pressure, with a series ofpressure increments until the sample is completely filled. Pore bodies with their larger
throats are invaded then typically first at the lower pressures, followed by connecting
smaller pore throats at the higher pressures (Brown 2015). By hypothesizing the poresystem as a bundle of capillary tubes with different diameters, each capillary pressure
increment, Pc, can be related to the pore throat radius, Rt, through the Washburn equation (Washburn 1921):

Pc = −

2𝜎 cos 𝜃
Rt(2)

where σ is the interfacial tension of mercury with air (dynes/cm2 or mN/m2) and θ is thecontact angle of mercury with air (°).
An important application of NMR and MICP techniques is the integration of theirresults to determine the surface relaxativity, a parameter that is essential for calibration of
the relaxation time curves T2 to obtain the pore size distribution (Wong 1999; Fleury andRomero-Sarmiento 2016). Approximating the pore network as a bundle of cylindrical capillary tubes of radius r, the pore structure can be correlated with T2 by means of the surface
relaxativity ρ2 according to:( )S
2𝜋rl 2== → rcylinder = 2𝜌2 T2(3)
V cylinder 𝜋r2 lrwhere l is the length of the capillary. The value 2 in the above equation is a pore shapeparameter needed to convert the ratio (S/V) in Eq. (1) to pore throat radii, which can bechanged to 3 for spherical pores. Thus, once a correlation between the NMR and MICP
curves is obtained, ρ2 can be adjusted until the two curves are consistent (Souza et al.
2013). Combining the NMR and MICP measurements will not yield good results if no correlation exists between the pore size distribution curves and the corresponding pore throats(Mesquita et al. 2016).
Basic and special petrophysical studies are crucial for accurate reservoir descriptions and related computer simulations. These techniques allow measurements of verysmall (nano- to micron-scale) pores, leading to results that are independent of their shape(Oliveira et al. 2019). To obtain a better understanding, in addition to calculating porosity,
pore connections need to be observed. Beginning in the 1990s, 3D image analysis has beenused extensively in the petroleum industry to more quickly and reliably obtain rock sample
information (Dernaika et al. 2018; Jin et al. 2018). X-ray microtomography images makeit now possible to visualize in more detail the internal part of a sample, thus corroborating
qualitative and quantitative evaluations of the physical properties and the flow dynamics ofa sample (Cnudde and Boone 2013; Ravlo et al. 2016). This approach addresses deficiencies that larger-scale laboratory measurements often have. However, the extraction of information from 3D images still involves inherent ambiguities such as properly distinguishing
between pore bodies and pore throats. To address some of these issues, algorithms havebeen developed to improve the extraction of the networks, sometimes referred to as “networking” or “skeletonization” (e.g. Dong and Blunt 2009). The combined use of laboratory
experiments and 3D image analyses should ensure a better understanding of the structureof the rock pore network.

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Another integration of 3D imaging techniques and NMR results to estimate the permeability of carbonate rock samples was suggested by Chi and Heidari (2016). They developed a new approach to estimate directional permeability by incorporating a connectivity
factor into a conventional NMR-based permeability model. The anisotropic permeability
used for calibration and subsequent tests were obtained using lattice Boltzmann modelling (LBM) of microtomographic images. Still, as pointed out by Mostaghimi et al. (2013)
and others, LBM methods are computationally very demanding, which restricts their use insome cases.
Recently, pore network models (PNMs) have become popular to improve the description of multi-phase systems. Their use has improved our understanding of macro-scale (orcontinuum-scale) fluid flow and contaminant transport processes by highlighting the basic
pore-scale basis of the various processes involved (Raoof et al. 2013; de Vries et al. 2017).
Pore-scale models are not necessarily a substitute for experimental methods, but rather
serve as a complement to better understand the experimental data and their uncertainties,
and allowing one to replicate experimental conditions (Raoof et al. 2012; Gundogar et al.
2016).
This study focuses on the effects of pore structure, pore size distribution, and connectivity on the relationship between porosity and the permeability. For this purpose, we characterized porosity and permeability data of samples using basic petrophysics. NMR and
MICP experiments were performed to obtain the pore and throat size distribution curves,which helped to obtain a better description of the pore systems involved. Microtomographic images were analysed in attempts to better understand and describe the connectivity of the entire system.

2 Geological Context
Our studies were carried out using carbonate rock samples (coquinas) from the SergipeAlagoas Basin, located in Northeastern Brazil, which at present constitutes a passive margin basin evolved from rifting processes that caused the opening of the South Atlantic
Ocean in the Cretaceous. These processes have been widely studied since they provide themost complete sedimentary succession of the Eastern Continental Margin of Brazil (Feijó
and Vieira 1991; Souza et al. 2013). The Morro do Chaves Formation, belonging to thisbasin, is defined as a coquina (carbonate) succession of the local Jiquiá stage (late Barremian to early Aptian) interbedded with siliciclastic rocks, varying between 50 and 350 m
in total thickness (Coqueiro Seco Formation and Rio Pitanga Formation) (Campos Neto
et al. 2007). Studies of this formation are often made using outcrops (Tavares et al. 2015;
Borghi and Corbett 2013) and cores (Carvalho et al. 2000; Rigueti et al. 2020) that areclose analogues of off-shore oil-bearing Pre-salt reservoirs of the Santos (Fm. Itapema;
Carlotto et al. 2017) and Campos (Fm. Coqueiros, Carvalho et al. 2000) basins in Brazil.
The term coquina was defined by Schäfer (1972) as a carbonate rock composed of skeletal parts, whose location and composition are governed by the laws of sedimentology
rather than biological laws. Coquinas from the Morro do Chaves formation are composedof bivalve shells and ostracodes that originally lived in shallow and oxygenated waters(Azambuja et al. 1998). The shells were subjected to rework and transportation and were
deposited on beaches and platforms through storm-induced currents (Carvalho et al. 2000;
Teixeira 2012).

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Tavares et al. (2015) conducted a facies study at the Atol Quarry, located in São
Miguel dos Campos, Alagoas (Fig. 1), in order to understand the composition of the rocks
involved. Their study showed a predominance of fragmented shells, without micrite, whichwere deposited in underwater bars, above the base level of good weather wave action. Highshell fragmentation and the absence of micrite would indicate constant rework in a moderate to high energy environment. Whole shell facies with micrite presence were deposited
between the base level of good weather and storm wave action. Below the base level ofstorm wave action, clay deposits occurred under suboxide to anoxic conditions.
Diagenetic actions were the main modifying agents of carbonate rocks. Coquinas of theMorro do Chaves Formation were altered through these processes involving neomorphism,
physical and chemical compaction, and micritization (Tavares et al. 2015). These processes
impacted directly the pore systems of the rocks, leading to the preservation of primaryporosity, the creation of secondary porosity, and pore obliteration. As a consequence, avery heterogeneous porous framework was generated, reflected especially in their permeability (Corbett et al. 2016).

3 Materials and Methods
For our study, we used six plugs (Table 1) taken from core UFRJ 02-SMC-02 drilled in
the Atol Quarry (Fig. 2). The cylindrical coquina samples (plugs), having dimensions of
2.5 cm (diameter) versus 5 cm (height), consisted of bioclasts resulting from shell fragmentation of transported lacustrine bivalves and subsequent impact by several stages ofdiagenesis. Due to these changes, the samples are very complex by containing unusually
heterogeneous pore systems.
After plugging, the samples were cleaned to obtain a completely emptied (dried) porous
system. For this, we employed the Soxhlet method (McPhee et al. 2015), which is widelyused in the oil industry. Possible rock fluids in the coquinas, such as hydrocarbons andsalts, were removed using toluene and methanol, respectively. After cleaning, the samples were subjected to oven drying at 60 °C and 40% humidity for a total of 24 h. These

Fig. 1  Location of the Atol Quarry location at São Miguel dos Campos, Alagoas, Brazil

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Calcirudite

Calcirudite

Calcirudite

Calcirudite

Calcirudite
Calcirudite

87.35

88.00

122.45136.85180.00187.95

Trace amounts of siliciclastic sand and ostracodes
Predominance of entire shells

1% ostracodes and trace amounts of gastropod. Predominance of entire shells
12% siliciclastic sand

5% siliciclastic sand. Predominance of entire shells

30% siliciclastic sand. Predominance of fragmented
shellsClassification Matrix

SampleCore depth(m)Table 1  Geological descriptions of plugs from the Morro do Chaves Formation

Neomorfism, physical and chemical compaction, dissolution of siliciclastic grains, kaolinite precipitates insecondary pores
Neomorfism, physical and chemical compaction, dissolution seams, styloliths
Neomorfism, physical and chemical compaction, kaolinite precipitates in secondary pores
Neomorfism, physical compaction
Neomorfism, physical compaction, dissolution

Neomorfism, physical compaction

Diagenesis

Vugs, moldic
Vugs, intercrystalline

Vugs, moldic

Intraparticle, intercrystalline

Interparticle, moldic

Interparticle

Pore type

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Fig. 2  Schematic of sample
87.35 (right) taken from core
02-SMC-02 (left). Sample 87.35
contained 30% siliciclastic sandtogether with predominance offragmented shellsconditions were used to not degrade possible clays contained within the sample. After drying, the samples were cooled in a desiccator for one hour and then submitted to basic petrophysics experiments.
Routine core analysis is a non-destructive technique used to determine the porosity andpermeability as based on Boyle’s law and Darcy’s law, respectively. Results obtained witha combined gas permeameter–porosimeter (AP-608, from Coretest Systems Inc., Reno,
NV) provided a reference for all other petrophysical measurements (McPhee et al. 2015).
For the NMR measurements, we saturated the samples in a saline solution consisting of
30,000 ppm KCl brine with a density of 1.0167 g/cm3 and subjected to a hydrostatic pressure of 138 bar for 24 h (Fig. 3). Even though the coquinas contained very little if any clay,
we used brine in part to avoid or limit possible shrink/swell processes (e.g. Ewy 2014).
Saturation was confirmed by calculating saturation indexes, whose values were all above
95% indicating satisfactory saturation. The samples were afterwards immersed in a fluid
container.
After saturation, the samples were wrapped in Teflon to minimize pore emptying during the NMR measurements. Using a Geospec 2 MHz device (Oxford Instruments) anda Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence (Carr and Purcell 1954), which
consists of a 90° radiofrequency pulse followed by a 180° pulse train lagged at π/2 of thefirst pulse, echo intensity measurements were taken with a signal-to-noise ratio above 100.
Application of inverse Laplace transforms on the data produced time distribution curves of
the transverse relaxation (T2), which were calculated using LithoMetrix™ software fromGreen Instruments Technology (GreenImaging Techn., Frederiction, NB, Canada). The
main parameters of the NMR acquisitions are listed in Table 2.
Because of their destructive nature during the MICP measurements, we removed therock fragments from the plug surface. The fragments, having a volume of approximately
2 cm3, were subjected to a vacuum of 50 to 900 µm Hg inside a penetrometer filled withmercury. Pressure increments were applied, up to a final pressure of 60,000 psi, to obtain
accessible pore volumes at different pressures. The data allowed us to estimate pore throat
radius distribution curves, and incremental and cumulative porosity amplitudes. The MICP
experiments were performed using Micrometrics Auto Pore IV equipment.

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Fig. 3  Details of the saturation
process of the samples and theirstorage in brine for subsequent
NMR measurements

Table 2  Parameters applied to
the NMR experiments

Parameter

Value

Total echos

16,250

Echo time
Number of averages
Waiting time between averages
90° pulse width
180° pulse width
Temperature

200 µs64
10 s
12.04 µs
24.24 µs
25 °C

The samples were subjected to X-ray microtomography imaging in order to understandthe pore framework of the coquinas. Imaging was performed in three sequential steps:
acquisition of high-resolution (16-bit) x-ray images using Skyscan model 1173 microtomography, image reconstruction using N­ Recon® software (Skyscan/Bruker, v.1.6.9.4) asdetailed by Godoy et al. (2019), and image processing using Avizo ­Fire® 9.5 software(ThermoFisher Scientific, Massachusetts, US). For the acquisition, we used a 1-mm aluminium filter to reduce noise in the images, while the equipment was calibrated with avoltage of 80 kV and 100 µA. The plugs were rotated 360° with an angular pitch of 0.5°.
For the treatment of images, a non-local means filter was used to minimize artefacts andnoise caused during acquisition (Lin et al. 2016). One of the steps consisted of segmenting
the images, which allows the pores to be identified, and separating them from the sample’s
mineral framework using the threshold tool (Otsuki et al. 2006; Lore 2018). Threshold values were determined by comparing pixel size with the NMR curves, with ­T2 values firstbeing converted to pore radii (Eq. 3). After this, using the auto-skeleton tool (Lore 2018),
the segmented pore space was transformed into tubes (made up of sequentially smaller

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spheres) and spheres, equivalent to the pore throats and pore bodies, respectively. The output data were used subsequently in the PoreFlow software (Raoof et al. 2013) to assess thedigital permeability and to obtain a more complete understanding of the entire pore systemand fluid flow within the pore system.

4 Results and Discussion
Results of the core analyses, such as of porosity, permeability, and grain density, as well as
the NMR and ADI results for porosity, are presented in Table 3. As can be seen, the samples had porosity values ranging from 8.7 to 16.7% and permeabilities from 8.7 to 245.1
mD. The grain density data reflect a predominance of calcite in the composition of thesamples (2.71 g/cm3 in general). The NMR porosity results furthermore show a good correlation with the routine porosity measurements, which suggest that the results were satisfactory and the samples were well saturated. MICP results for the low-permeability samples presented values close to routine porosities, but were lower for the high-permeability
samples. This finding can be explained by the pore size in each sample group: generally,
the low-permeability samples contained mostly only relatively small pores in the samples
used for the measurements, whereas the higher permeability samples had larger pores notpresent in the fragments used for the porosity.
Figure 4 shows a plot of porosity versus permeability using the routine data. The plotshows data from all samples according to their petrophysical properties, highlighted by twogroups: one with a relatively low permeability and the other with a high permeability. Two
low-permeability samples had relatively low porosities, and two high-permeability samples had higher porosities, as expected and discussed by Corbett et al. (2016) who noted
substantial changes in permeability with only slight changes in the porosity. However, twosamples (122.45 and 187.95) had nearly the same porosity but showed a large difference intheir permeability (to be discussed later).
Figure 5 shows the pore size distributions as a function of the relaxation time using
five size partitions as employed by Silva et al. (2015). The curves show considerable heterogeneity among the samples, consistent with the variability in the permeability versus
porosity relationships shown in Fig. 4. The pore partitions of Silva et al. (2015), consistedof micropores (­T2 values up to 1 ms), a transition from micropores and mesopores (from1 to 10 ms), mesopores (10 to 100 ms), a transition from mesopores to macropores (100to 1000 ms), and macropores (above 1000 ms). As indicated by Gonçalves et al. (2017), it
is not surprising that most partitions actually contributed to the flow process in our study

Table 3  Estimated sample hydraulic parameter values using core analyses, NMR, and MICP
ρgrains (g/cm3)

ϕroutine (%)

87.35

2.70

10.2

5.25

9.4

9.3

88.00
122.45136.85180.00187.95

2.71
2.72
2.71
2.71
2.70

8.7
12.3
16.0
16.7
12.2

5.21
5.65
245.0
245.1
152.4

9.0
12.1
15.1
16.3
12.4

8.3
12.3
10
13.2
8

Sample core
depth (m)Kroutine (mD)ϕNMR (%)

ϕMICP (%)

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Fig. 4  Plot of permeability versus porosity for six core samples. The plot shows a non-unique relationshipbetween permeability and porosity. Samples 187.95 and 122.45 had similar porosity but different permeabilities, while samples 88, 87.35, and 122.45 had similar permeabilities but different porosities

Fig. 5  Plot of the relaxation time for different samples using a five-partition approximation. Curves aremostly non-overlapping, indicating that the samples are heterogeneous. This is consistent with variations in
the permeability and porosity values of the samples shown in Fig. 4

since carbonate rocks are known to have pore sizes that generally vary over three ordersof magnitude. Based on this information, we note that samples 87.35 and 88, which havea lower permeability, show a bimodal distribution, with most of the pores being in themesoporous region. On the other hand, sample 122.45 had a very well-defined bimodal
distribution, but with a predominance of macropores. Of the high-permeability samples,
sample 187.95 had a remarkable unimodal distribution at or near the macroporosity region.
This demonstrates that even having the same porosity as sample 122.45, its permeability islinked more to the pore throats, which are responsible for flow through the porous space.
Samples 136.85 and 187.95 did have very similar distributions, demonstrating their petrophysical similarity.
The MICP measurements produced curves of the capillary pressure as a function of
mercury saturation, from which pore size distributions could be extracted (Moctezuma

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297

et al. 2004). Figure 6 shows that some variability is apparent between the measured capillary–saturation curves. The curves show an increase in capillary pressure consistent withmercury entry into pores having increasingly smaller diameters. Samples 87.35, 88 and122.45 show that 30% of the pores were already accessed with pressures up to 10 psi, thus
characterizing the larger pore throat apertures. About 70% of the pores in these samples
were intruded at a pressure of 100 psi, while the smallest pores making up 15% of the total
were intruded with higher pressures. Knowing that these samples have lower porosities andpermeabilities, one can understand that their pore space is interconnected by means of narrow throats.
For samples with a higher porosity and permeability, different capillary pressure curves
were obtained. For example, sample 136.85 required slightly higher pressures throughout
the mercury intrusion test. This indicates that even with smaller pores, the sample had
good connectivity among the pores, thus contributing to flow in the sample. Sample 187.95
showed that around 60% of its pores were intruded with pressures up to 20 psi, which suggests that these pores had relatively large throats. To access the remaining pore space, aprogressive increase was required, which caused a difference in the slope of the curve,reflected by a reduction in the pore throats. Sample 180 differed in this respect from theothers: 70% of its pores were accessed with pressures up to 20 psi, characterizing poreswith large throats. The remaining pores were accessed using higher pressures, characterized by the slope of the curve. The low-permeability samples, for the most part, hence had
similarities in terms of pore throat size due to the slight increase in capillary pressures,with a considerable increase occurring in only a small pore fraction. High-permeability
levels presented differences in the intrusion pressures, demonstrating that although thesamples had similar permeabilities, they contained a different pore network structure.
Transformations of the T2 distributions to pore size were carried out by overlapping the
NMR curve over the MICP curve and shifting it so that the two peaks of two curves overlap (Mesquita et al. 2016; Zhang and Hebing 2014). Once visually verified that both distributions (i.e. of mercury saturation and T2) correlated well, we adjusted ρ2 in Eq. (3) until
the distributions had the same maximum, as can be seen in Fig. 7. Sometimes, it is necessary to move the peaks away to obtain the same end points for the two curves (as seen forsamples 122,45 and 187,95). While this is an approximation, the approach has been foundFig. 6  Plot of the MICP measurements used to calculate capillary pressure curves as a function of mercury
saturation

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Fig. 7  Fitting of the NMR and MICP curves

to provide accurate permeability values for modelling applications (Mesquita et al. 2016;
Rios et al. 2016; Souza et al. 2013). Yet, because of the complexity of the samples, a better
option is to simply perform a visual fitting in which the micropores from MICP are combined with the NMR results, and so on. Samples 87.35, 88, 122.45, 180 and 187.95 generated very reasonable fits for NMR and MICP curves, which visually appear quite similar.
The distributions suggest that the pore throats merged slowly with the main channels in
the permeable part of the system (Basan et al. 1997). In contrast, sample 136.85 showed
important differences between the two curves. The NMR curve exhibited a triple porosity
distribution dominated very much by the larger pores. By contrast the triple porosity MICP

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curve is predominated mostly by mesoporosity. Knowing that MICP is sensitive mostly
only to the throats and NMR to the entire pore space, the sample seems typical of a situation where the larger pores are surrounded by smaller pores, with MICP assigning volumes
first to the smaller pores since the larger pores are invaded by mercury only after invasionof the smaller pores. The complexity of sample 136.85 is a common characteristic of carbonate rocks. Sample 122.45 is similarly interesting in that the NMR curve shows a higher
percentage of pores in the micropore and mesopore regions as compared to the throats in
the MICP curve. We conclude that the pores of this sample must have relatively better connections, relative to the other samples. As noted earlier, the NMR curve of sample 136.85
had most of its pores within the meso-/macro-region, while the MICP curve showed a triple porosity system with a predominance of mesopores. Figure 7 further contains important information about the surface relaxativity, ρ2. The range of values of ρ2 we obtained(between 6.5 and 15 µm/s) is very typical of carbonate rocks.
For the pore radius distributions, we used thresholds based on the percentage of visible
pores at the given resolution, as developed by Hoerlle et al. (2018). After segmentation, theconnected pores were transformed into cylindrical pore throats and spherical pore bodies
using the skeletonization tool, resulting in a series of nodal points, segments, and nodes
that served as input to calculate porosity and permeability of the sample using PoreFlow
software (Raoof et al. 2013). Considering routine core analysis as a calibration technique(McPhee et al. 2015), PoreFlow results showed a good agreement with the measurements(provides shown in Table 4). While the porosities obtained with PoreFlow were smaller,
the calculated and measured permeabilities were similar for most samples. This is further
shown in Fig. 8, which indicates excellent agreement between the PoreFlow predictions
and the routine measurements, well within a factor of 3 generally considered acceptablefor permeability estimations using NMR (e.g. Kenyon et al. 1988). We hence conclude thatthe pore system observed at this resolution was responsible for flow within the samples assimulated.
Table 3 showed that samples 122.45 and 187.95 had the same porosity (12%) but different permeability values (5.65 and 152.4 mD, respectively). To better understand thisdifference, PoreFlow was used to simulate fluid flow within the pore topologies of these
two samples. All image components were analysed for pore connectivities using Paraview (Ayachit 2015), with results shown in Fig. 9. The images display the connected pores
responsible for the permeability of the samples. Figure 9a–b shows the pore clusters ofsample 122.45, with each cluster having a different colour to indicate a set of connected
pores. In total, 1748 pore clusters were recognized. Figure 9b shows the largest pore cluster

Table 4  Comparison of porosities and permeabilities obtained with the routine core analyses and calculatedusing PoreFlow
Samples core depth(m)

ϕroutine (%)ϕPoreFlow (%)

Kroutine (mD)

KPoreFlow (mD)

87.35

10.2

8.1

4.57

4.58

88.00
122.45136.85180.00187.95

8.7
12.3
16.7
16.7
12.2

6.1
4.2
13.2
10.8
6.8

5.21
5.65
245
245.1
152.4

4.4
3.7
249.2
236.7
138.9

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Fig. 8  Plot comparing the routine permeability measurements with the PoreFlow predictions

responsible for most of the fluid flow through the entire sample. Figure 9c–d shows similar results for sample 187.95, with Fig. 9c exhibiting as much as 724 distinct (i.e. separate from each other) pore clusters making up the pore network of the rock. The different
makeup of the two main percolating clusters in Fig. 9 is a reflection of the different volume of connecting pores of the two samples and their impacts on the permeability: sample
122.45 had a smaller percolating cluster with a lower volume of pores and consequently alower permeability, while sample 187.95 had a much higher volume of connecting porespertaining to the main cluster, leading to a much higher permeability. The large fractionof non-percolating clusters in sample 122.45 (Fig. 9a) must have been responsible for thedeviation between the NMR and MICP curves for this samples as shown in Fig. 7. While
NMR captures the entire pore system, the MICP measurements are more likely affectedby the presence of complex non-percolating pore clusters since this method is based on
the flow of mercury into the sample. Similar correlations for other images were found to
be useful for explaining why certain samples with similar total porosity provided different permeability values. As explained well by Lucia (1999), diagenesis typically reduces
porosity, redistributes the pore space, and alters the prevailing permeability and capillary
characteristics. The resulting rock alterations may well have caused most or all of the permeability modifications and pore complexities of our samples.
The connectivity data of each pore body were also analysed in terms of the overall pore
network of samples 122.45 and 187.95. The coordination number of the pores (also called
pore connectivity) represents the average number of pore bodies which are connected withadjacent pores (Sahimi et al. 2012; Raoof and Hassanizadeh 2010). This parameter is afundamental feature of pore networks and has a noticeable impact on calculations of thehydraulic conductance of porous rocks (Chen et al. 2003; Vasilyev et al. 2012). Figure 10
provides a comparison of the coordination numbers of the same two samples (i.e. of 122.45
and 187.95) obtained with the ­Avizo® software. Sample 187.95 has much higher counting
bins, especially when looking at three coordination numbers, than coquina 187.95. Thisimplies a much better pore continuity and hence a higher permeabilities of sample 187.95
at the same porosity.
Another way to explain the permeability differences is through a pore body-throat
analysis. Using the skeleton of the microCT images, one can quantify the volume and13

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Fig. 9  Distribution of pore clusters in two rock samples: a) all pore clusters in sample 122.45 and b) its
main pore cluster responsible for flow through the sample and its permeability; c) all pore clusters of sample 187.95 and d) its main pore cluster. This figure explains why sample 187.95 with a similar porosity as
sample 122.45 has a much higher permeability (as indicated in Fig. 4)

size of these elements, which in their totality should be a reflectance of the complexity of the pore space of the rocks. Figure 11 exhibits the pore throat and pore body
distributions of samples 122.45 and 187.95. The plots show that sample 187.95 hadlarger element volumes, which is consistent with its higher permeability compared tosample 122.45. Regarding the pore sizes obtained from the images, the microCT values were higher than those determined by NMR using the MICP match. A comparisonof these two techniques assumes equivalence between the pore body and pore throat
sizes, which may not be appropriate for heterogeneous carbonate samples. Still, using

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Fig. 10  Comparison of the distribution of coordination numbers for samples 122.45 and 187.95

Fig. 11  Comparison of the pore bodies and pore throats of samples 122.45 and 187.95

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this conversion, the permeability as determined from the images was satisfactory, thus
confirming applicability of the technique.
We noticed that the pore throats found in the samples using microCT images had values
higher than those determined using MICP. As mentioned before, MICP results are based on
the Washburn equation (Washburn 1921), which approximates the porous system as a bundle of cylindrical tubes. When analysing carbonate rocks (especially coquinas), one must
consider heterogeneity in the pore systems in terms of different pore shapes. The images inFig. 12 show difference between the pore systems of two samples. Although the images arein two dimensions (XY slices from microCT), it is possible to recognize that the intraparticle pores of sample 122.45 are different from the larger vugs of sample 187.95. To better
understand these pores, our next step was to scan the samples with higher resolution (3 µm)
in order to observe the resolution impact on unresolved pore throats.
Finally, we note that surface relaxativity, ρ2, may be a complicating factor for this study.
Since ρ2 does not have an exact form of calculation but only provides approximations, thisparameter generally is subject to much uncertainty, also in our study. Another factor that
influences the NMR results is the bulk effect on the larger pores. Above 2.5 s (the saturating fluid relaxation time of the samples), the spins no longer completely collide with thepore walls of the pore network inside the sample, which may cause the measured relaxation
time to be less reliable (Kenyon 1997). This causes uncertainty in the size of especially thelarger pores.

5 Concluding Remarks
This study focused on the effects of pore structure, pore size distribution, and connectivity
to better understand why significant permeability variations occur in carbonate rock samples having similar porosity values. Routine core analyses, NMR and MICP studies were
performed on coquinas samples from the Morro do Chaves Formation, which is considereda close analogue of Brazilian pre-salt oil reservoirs. Integration of the various methods
produced estimates of the surface relaxativity needed to transform relaxation times into

Fig. 12  2D slices from microCT highlighting pore type

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the pore size distribution. Threshold values of the distributions, based on the percentageof visible pores at a given resolution, were used (maximum porosity visible in the resolution), considering the pore radius distribution. The microCT images were used to obtainthe skeleton, which in turn provided necessary input data (points, segments, and nodes) for
porosity and permeability calculations using the PoreFlow software.
Calculated permeabilities showed excellent agreement with routine laboratory measurements. Differences in the permeability of two coquinas having the same porosity could be
explained using image analysis and statistical correlations, as well as by considering poreconnectivity. Samples having pore bodies with higher coordination numbers showed muchlarger permeabilities at similar porosities. A pore body-throat analysis could also explainsome of the permeability differences. The increase in permeability is related to the presence of larger elements within the connecting pore system. Although the range of pore bodies and pore throats observed on microCT images were higher than determined by NMR
using a MICP match, some aspects may need consideration. For example, when combining
NMR and MICP, an equivalence is assumed between the pore body to pore throat ratio,
which is not necessarily very realistic for carbonate rocks. Pore throats determined using
microCT showed differences with those of the MICP experiments. The manner in whichpressure is converted to pore radius may present for this reason also some uncertainty forthe carbonate rocks. Inaccurate or incomplete definition of some of the NMR parameters,
notably surface relaxativity and bulk effects, may also introduce some uncertainly for especially the larger pores.
Our study in this paper focused on the micro-scale petrophysical properties of relatively
complex samples (coquinas). While macro-scale multi-phase reservoir simulations dependon having petrophysical properties that are reliable at the larger scale, they still must be
as consistent as possible with the micro-scale physical measurements. Required properties
include porosity, absolute and relatively permeabilities (the latter as a function of fluid saturation), contact angles, and various geochemical parameters. They are needed for effective
characterization of carbonate rock formations, including their micro-scale and macro-scale
heterogeneities. These issues are especially important for carbonate oil reservoirs in theBrazilian pre-salt, which are unusually heterogeneous as characterized by a broad spread in
their pore size distributions due to the presence of vugs and even larger pores of different
sizes. Vuggy carbonates are particularly challenging to characterize in terms of relevant bior multi-modal petrophysical properties.
Permeability is one of the key parameters for reservoir assessments and management.
Direct measurements of permeability require laboratory analyses of drilled reservoir cores.
NMR logging provides indirect in situ permeability data, which could be used along withvarious semi-empirical equations to estimate permeability from the measured porosity values. Unfortunately, current permeability models are based on assumptions that often do
not apply to carbonate rocks, such as having a well-connected pore system. Chemical andbiological processes during diagenesis of carbonate rocks typically cause a profound reorganization of the initial pore structure, leading to a broad range of pore types and sizes.
Our rock samples have clearly shown that samples with different porosities can have similar permeability values. This shows that, for such complex samples, Kozeny–Carman-type
relations are not adequate because they assume a unique relation between porosity and permeability. We expect that more complex relations which include topological properties ofthe media should be developed (Parvan et al. 2020).
Another point to consider is how different techniques with their different results cancontribute to a better understanding of complex rocks. Routine core results are important since they provide a calibration for judging other techniques. NMR in principle gives

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exceptionally good results for porosity and the pore size distribution. Still, several aspects
are important in terms of field applications. One aspect relates to measurements that canbe done via a well, during logging, to estimate permeabilities. If the measurements can bemade through the well itself and in the laboratory, this would become a preferred methodin petrophysics. MICP (at first developed to obtain porosities of irregular samples) leads
to porosities and estimates of the permeability from the obtained pore throat distribution
consistent with the imposed boundary conditions. Using microCT images, numerical simulations can be done also to determine porosities and permeabilities. In this case, the resolution will guide all the results to be obtained. The most powerful asset of the latter technique
is its ability to carefully observe the pore system, while the other tools deliver more global
values without direct visualization. We conclude that many available techniques are potentially important to obtain relevant information about a certain rock formation, and hence,that integration of the various techniques and their results will optimize our understanding
about the complex pore system involved.
Finally, we note that pore-scale modelling and upscaling based on its results is key
for reliable characterization of strongly heterogeneous carbonate rock formations, which
typically have representative elementary volumes (REVs) that are far too large for direct
numerical simulations. Correct upscaling must focus on those aspects of the system thatare controlling fluid flow and/or contaminant transport at the relevant scale. Pore networkmodelling (PNM) is one approach to speed up simulations of large porous systems since
flow and transport in very large samples can be simulated, which generally is not feasibleusing direct numerical solutions, especially under common multi-phase flow conditions.
In this study, we attempted to understand and describe the interplay between porosity andabsolute permeability of heterogeneous carbonates at the pore scale and use PoreFlow tosimulate fluid flow through the sample to drive the hydraulic properties. Having accurate
pore structures, PoreFlow can be used to derive more complex constitutive relationships
such as those between saturation and relative permeability as well as capillary pressure(Raoof and Hassanizadeh 2012). A well-integrated methodology is needed to allow pore
network modelling up to REV’s for use at the larger scale. The methodology presented inthis paper, combined with improved analytical correlations between porosity and permeability, may facilitate more realistic representations of the continuum scale.
Acknowledgments This research was carried out in association with the ongoing R&D projects registeredas ANP 19027-2, “Desenvolvimento de infraestrutura para pesquisa e desenvolvimento em recuperação
avançada de óleo – EOR no Brasil” (UFRJ/Shell Brasil/ANP) setting up a advanced EOR Laboratory facility for R&D in Brasil, and ANP 20163-2, “Análise experimental da recuperação de petróleo para as rochas
carbonáticas do pré-sal brasileiro através da injeção alternada de ­CO2 e água”, with samples provided by
Project SACL (Análise geológica sedimentar de sucessões carbonáticas do Cretáceo em uma bacia sedimentar brasileira (ANP n.18993-6), all sponsored by Shell Brasil under the ANP R&D levy as “Compromissode Investimentos com Pesquisa e Desenvolvimento”. This study was financed in part by the Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior- Brasil (CAPES)—Finance Code 001, and carried out withthe support of CNPq, National Council of Scientific and Technological Development—Brazil. We acknowledge Professor Rodrigo Bagueira (UFF-LAR) for helping with saturating the samples and use of the NMR
equipment. We also thank the research teams of LRAP/COPPE/UFRJ and LAGESED/UFRJ.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,
which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as longas you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article
are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to thematerial. If material is not included in the article’s Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directlyfrom the copyright holder. To view a copy of this licence, visit http://creat​iveco​mmons​.org/licen​ses/by/4.0/.

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