Received: 29 December 2023

Accepted: 22 April 2024

DOI: 10.1002/vzj2.20348
Vadose Zone Journal

ORIGINAL ARTICLE
S p e c i a l S e c t i o n : Tr i b u t e t o R i e n v a n G e n u c h t e n , R e c i p i e n t o f t h e 2 0 2 3 Wo l f P r i z e f o r A g r i c u l t u r eModeling sub-resolution porosity of a heterogeneous carbonate
rock sample
William Godoy1
Elizabeth M. Pontedeiro1,2
Rafael A. B. R. Barros3
Enno T. de
Vries2,4
Amir Raoof2
Martinus Th. van Genuchten2,5
Paulo Couto1
1 Department of Civil Engineering, Federal

University of Rio de Janeiro, Rio de Janeiro,
Brazil2 Department of Earth Sciences, Utrecht

University, Utrecht, The Netherlands
3 Nanotechnology Engineering Program,Federal University of Rio de Janeiro, Rio deJaneiro, Brazil
4 Hydrogeology Department, KWR Water

Research Institute, Nieuwegein, The
Netherlands
5 Department of Nuclear Engineering,Federal University of Rio de Janeiro, Rio deJaneiro, Brazil
Correspondence
William Godoy, Department of Civil
Engineering, Federal University of Rio deJaneiro, UFRJ, Ilha do Fundão, RJ, Brazil.
Email: wmgodoy97@gmail.com
Assigned to Associate Editor Jirka Simunek.

Abstract
Accurately estimating the petrophysical properties of heterogeneous carbonate rocks
across various scales poses significant challenges, particularly within the context ofwater and hydrocarbon reservoir studies. Digital rock analysis techniques, such as
X-ray computed microtomography and synchrotron-light-based imaging, are increasingly employed to study the complex pore structure of carbonate rocks. However,
several technical limitations remain, notably the need to balance the volume ofinterest with the maximum achievable resolution, which is influenced by geometricproperties of the source–detector distance in each apparatus. Typically, higher resolutions necessitate smaller sample volumes, leading to a portion of the pore structure(the sub-resolution or unresolved porosity), that remain undetected. In this study,
X-ray microtomography is used to infer the fluid flow properties of a carbonate
rock sample having a substantial fraction of porosity below the imaging resolution. The existence of unresolved porosity is verified by comparisons with nuclear
magnetic resonance (NMR) data. We introduce a methodology for modeling the subresolution pore structure within images by accounting for unresolved pore bodies
and pore throats derived from a predetermined distribution of pore throat radii. The
process identifies preferential pathways between visible pores using the shortest distance and establishes connections between these pores by allocating pore bodies andthroats along these paths, while ensuring compatibility with the NMR measurements.
Single-phase flow simulations are conducted on the full volume of a selected heterogeneous rock sample by using the developed pore network model. Results are thencompared with petrophysical data obtained from laboratory measurements.

1
Abbreviations: μCT, X-ray computed microtomography; MDF, maximum
distance factor; NCC, number of closest connections; NCZ, number ofclosest zones; NMR, nuclear magnetic resonance; PNM, pore networkmodel; PSD, pore size distribution; REV, representative elementary volume.

INTRODUCTION

Reliable estimates of the petrophysical properties of heterogeneous carbonate rocks, crucial for water and hydrocarbon reservoir studies as well as CO2 sequestration analyses, present significant challenges across various scalesThis is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided
the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.
© 2024 The Author(s). Vadose Zone Journal published by Wiley Periodicals LLC on behalf of Soil Science Society of America.
Vadose Zone J. 2024;e20348.
https://doi.org/10.1002/vzj2.20348

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GODOY ET AL.

Vadose Zone Journal(Sun et al., 2017). Digital rock analysis techniques are now
used extensively to decode the complex pore structuresinherent of carbonate rocks. They typically involve X-ray
computed microtomography (μCT) (Blunt, 2017; Bultreys
et al., 2015; Godoy et al., 2019; Suhrer et al., 2020), focusedion beam scanning electron microscopy (Devarapalli et al.,2017; Dvorkin et al., 2011), and/or synchrotron-light based
imaging (Ferreira et al., 2019; Pak et al., 2023). These studies
aim to navigate the trade-off between the volume of interest(VOI) pertaining to the volumetric dimensions of a rock sample and the resolution, which is tied to the smallest visible
structures within the rock images (Carrillo et al., 2022).
A persistent hurdle in these analyses is accurate estimationof sub-resolution porosity, also known as unresolved porosity(Archilla et al., 2016; Blunt, 2017; Lucas et al., 2020). This
fraction of porosity, not detectable due to resolution limitations, plays a pivotal role in fluid flow studies of carbonate
rocks. Carbonate rocks often exhibit multi-modal pore size
distributions (PSDs), involving a wide range of pore throat
sizes from nanometers to macropores spanning several centimeters in diameter (Drexler et al., 2022; Silveira et al., 2022).
For typical core samples with diameters around 5 cm, the μCT
resolution is confined to a few micrometers (Ferreira et al.,2023; Lima et al., 2022). Achieving higher resolutions necessitates either focusing on a smaller sample area (thus reducing
the VOI) or physically segmenting the sample for multiple
scanning processes (Drexler et al., 2022; Lima et al., 2022).
Two widespread modeling approaches are utilized todeduce the petrophysical properties from μCT imaging data.
One approach involves direct fluid flow simulations within
the segmented images by employing either lattice Boltzmann
methods (McClure et al., 2021; Vogel et al., 2005) or recreating an equivalent pore system through finite element meshtechniques (Saxena et al., 2017). A second strategy introduces
an intermediary step to simplify the pore geometry and topology, thereby facilitating the generation of an approximate
network of pore bodies and pore throats for subsequent use ina pore network model (PNM) (Bultreys et al., 2016; Gostick
et al., 2016; Raoof et al., 2013). PNM approaches offer theadvantage of lower computational demands for conducting
single- and multi-phase fluid flow simulations compared toother numerical methods, thus enabling the study of larger
sample volumes (Blunt, 2017; Godoy et al., 2019) and circumventing the need to calculate petrophysical parameters notbased on a representative elementary volume (REV) of thesample.
Despite the efficiency of PNMs for larger volumes, a balance must be reached between rapid computation and theaccuracy of results since the entire pore system cannot be
represented once a specific resolution threshold is chosen.
To enhance the prediction of petrophysical parameters, anadditional modeling step can be integrated during the pore
network generation phase. Alves et al. (2021), for this pur-Core Ideas∙ Using digital rock analysis, we generated andreconnected sub-resolution pores of a carbonate
rock sample.
∙ A workflow is presented to allow the reconnectionof detectable and sub-resolution pores.
∙ We compared simulated absolute permeabilities ofdifferent pore networks with laboratory measurements.
∙ Digital rocks generated with a pore network model
can serve as a complementary tool for studying
fluid flow patterns.

pose, introduced a method to link μCT attenuation values with
sub-resolution porosity. Suhrer et al. (2020) similarly exploredthe relationship between resolved and unresolved porosity
in studies with digitized plug scale samples having varying resolutions. In these studies, petrophysical contributedby the unresolved porosity regions were based on data fromthe resolved porosity before conducting lattice Boltzmann or
finite element simulations. Bultreys et al. (2016) employed“micro-links” to reconnect sub-resolution areas with theresolved porosity, adding pore throats from the unresolved
porosity regions to clusters within the defined macropore
region. Jiang et al. (2013) adopted a different tactic by creating a stochastic network at the microscale that connects to
the resolved porosity region through pore throats.
This study seeks to address sub-resolution porosity byintroducing a novel approach inspired by the work of deVries et al. (2017) and Bultreys et al. (2016). The approach ispredicated on the formation of pore bodies and pore throats
according to a predetermined PSD. Our proposal involves
reconnecting sub-resolution pores with those across the entire
plug volume, rather than limiting the scope to only a portionof the sample.

2
2.1

MATERIALS AND METHODS
Sample preparation

The methodology outlined in this study was applied to a carbonate rock sample sourced from the Sergipe-Alagoas Basin,
located in northeastern Brazil. More specifically, the sample
was obtained from the Morro do Chaves Formation in São
Miguel dos Campos, being the geographical location of previous research conducted by Godoy et al. (2019). The selected
sample, designated as 1–4, was extracted from a geologic
layer known as Bed 2B (Figure 1), as identified in the work ofCorbett and Borghi (2013).

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Vadose Zone Journal

3 of 14F I G U R E 1 (a) Image of the Morro do Chaves Formation in the Mina São Sebastião quarry in the city of São Miguel dos Campos, Brazil, (b)identification of layer (Bed 2B) from which samples 1–4 was taken (highlighted by a red point), and (c) a vertical view of the extracted sample(source: Modified from Godoy [2019]).
T A B L E 1 Parameters related to the acquisition and reconstruction of data of plug 1–4 as studied using the Bruker Skyscan 1173 X-ray
computed microtomography (μCT) equipment.

Pixel size (µm)

Sourcevoltage (kV)

9.97

130

61

18.53

130

61

24.95

130

61

4

Current (µA)

Ring artifact
correction factor

Smoothing factor

Beam hardening
correction (%)

5

4

15

0.5/0.5135

20

0.5/0.5320

0.5/0.5

The sample underwent a series of preparatory steps toensure its suitability for further analysis. Initially, the top andbottom surfaces of the plug were flattened to create uniformity. The sample was subsequently subjected to a cleaning
process for a period of 21 days using methanol and toluene.
Following cleaning, the sample was dried in an oven at 60˚C
for a duration of 24 h. These preparatory procedures were conducted in accordance with methodologies previously adopted
by Hoerlle et al. (2018). The final dimensions of the plug were
measured to be approximately 35 mm in diameter and 40 mm
in height.

2.2

Imaging data analysis

For the pore structure analyses, we imaged the sample using
the Bruker Skyscan 1173 equipment. The imaging process
was carried out at three distinct resolutions to facilitate
the identification of pore throat connections that may exist
within sub-resolution regions. The multi-resolution approachallowed for a comprehensive examination of the sample’s pore
network. Following the acquisition of μCT images, a filtering
step was implemented using the non-local means filter method
described by Buades et al. (2005). Table 1 shows the primaryparameters associated with the imaging data acquisition.
Measurements of the porosity and absolute permeability
were conducted using the permeameter-porosimeter system
DV-4000 (Weatherford Laboratories). The effective porosity was determined through helium (He) gas injection, while

Al/Cu
filter (mm/mm)

the absolute permeability (Kabs ) was measured using nitrogen(N2 ), with Darcy’s law applied to a steady flow regime. The
helium porosity was found to be 0.147, and the N2 Kabs was
33.19 mD.
Estimates of the effective porosity were also obtained using
nuclear magnetic resonance (NMR) techniques (Blunt et al.,2013; Sullivan et al., 2007), which involved measuring the T1
and T2 relaxation times. This procedure was performed after
the μCT imaging phase, with the sample being saturated in asaline solution. The solution consisted of 50,000 ppm sodium
chloride (NaCl) and had a density of 1.039 g/cm3 at a standard temperature of 25˚C. Full saturation of the samples wasachieved by applying an isostatic pressure of 2000 psi. To confirm the effectiveness of the saturation process, a saturationindex was calculated at the end of the procedure. Table 2 outlines the primary parameters associated with the saturation
process.
Table 3 shows a comparison between the porosity measurements obtained using the DV-4000 gas permeameter andthose derived from the NMR techniques. Ensuring a close
match between these two methods is crucial for accurately
constructing the PSD curve, since discrepancies could leadto misinterpretations of the results. The PSD, based on NMR
techniques and pore radii, was determined following themethodology of Luna et al. (2016) by converting T2 data intoradii under the assumption that the pores are spherical, with asurface area factor (S) set to 3.0.
The PSD and cumulative porosity curves for samples
1–4 are shown in Figure 2. The PSD curve indicates that a

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TA B L E 2

GODOY ET AL.

Vadose Zone Journal

Parameters from the saturation index (SI) calculation.

SampleDried mass (g)Saturated mass (g)Brine density (g/cm3 )Pore volume (cm3 )SI (%)

1–4

82.21

87.43

1.039

5.26

95.51

T A B L E 3 Porosity values obtained with permeameter and nuclear
magnetic resonance (NMR) techniques.

T A B L E 4 Sub-resolution and detectable porosities from thereconstructed digital pore systems of samples 1–4.

SamplePorosity
NMR (%)

Porosity
porosimeter (%)

Relativedifference (%)

Pixel size(µm)

Sub-resolution
porosity (%)

Detectable
porosity (%)

NMR
porosity (%)

1–4

14.26

14.7

3.0

9.97

3.51

10.75

14.26

18.52

4.96

9.30

14.26

24.95

5.49

8.77

14.26

significant portion of the porosity is due to pores having radii<10 μm, although the peak distribution occurs between 40
and 50 μm. The sample had an average porosity typical ofcarbonate rocks (e.g., Mazzullo, 2004), but a relatively low
permeability. The cumulative porosity curve further showsthat approximately 20% of the porosity is contributed bypores with radii up to 10 μm, an important aspect of thispetrophysical property.
The N2 Kabs value of samples 1–4 was determined to be
33.19 mD. This contrasts with other samples from the same
Bed 2B (Godoy et al., 2019), which had far higher absolutepermeability values (in the hundreds of mD) and a radius
NMR peak distribution spanning hundreds of microns. Thissuggests that connectivity within samples 1–4 may have been
influenced predominantly by the smaller pore throats associated with a secondary peak at approximately 0.1 μm in
Figure 2a compared with the main peak between radii of about
1 μm and 500 μm.
The segmentation process for the μCT images utilized a
combination of NMR data and digital petrophysical rockdata, employing various automated segmentation techniques(Kittler & Illingworth, 1986; Otsu, 1979). This approachenabled the selection of an optimal threshold value essential for the reconstruction of a calibrated three-dimensional

Abbreviation: NMR, nuclear magnetic resonance.

model, reflecting the porosity estimates derived from the μCT
images. It is important to note that the spatial resolution of
the structures is influenced not only by the pixel size of thedetector but also by pixel blur, which is a result of the X-ray
tube focal point dimensions. For the segmentation process, weused the Avizo 9.5 (Thermo Scientific) software (FEI), withthreshold selection being a key step. Given that porosity estimation is based on the digital model generated from the μCT
images, the pixel size is a critical factor in this process. Table 4
presents the porosity values obtained from both the NMR data
and the reconstructed μCT images, according to the imaging
resolution.

2.3

Analysis of sub-resolution porosity

NMR data play a pivotal role in identifying the sizes of poreradii, including both pore bodies and pore throats that are
not discernible or reconstructable due to the μCT resolution
limitations. The absence of μCT data for the smaller pore
structures can result in the creation of disconnected segments

F I G U R E 2 (a) Pore size distribution (PSD) curve obtained from the nuclear magnetic resonance (NMR) data, and (b) the cumulative porosity
curve of samples 1–4 (source: Godoy [2019]).

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within the pore networks. Pore network modeling facilitates the reconnection of these pore systems by generating
sub-resolution pore structures.
The development of our PNM, based on visible porosity
and the examination of input and output boundaries, adheresto the workflow outlined by Godoy et al. (2019). It is essential to recognize that different algorithms and workflows
may yield varying outcomes, as demonstrated by Baychev
et al. (2019). Our methodology for generating sub-resolution
porosity and their connection with previously separated pore
networks is based on the work of de Vries et al. (2017) and
Bultreys et al. (2015, 2016), with some modifications in how
the pores are connected.
The reconstruction of porous media from μCT imagederived networks using PNM techniques involves two mainstages. Initially, the pore space is segmented into discrete elements, followed by measurements of the geometric properties
for each element within the pore network, including radius,
volume, length, and shape descriptors. These methodologies
can be broadly categorized into two types: Topology-central
methods, which focus on the separation of pore space basedon the network topology, and morphology-central methods
based on the central morphology of the pore space, as detailed
by Bultreys et al. (2016). Drawing on this classification, deVries et al. (2017) explored the complete pore structure ofan aggregated soil by differentiating between the macropore
and micropore domains. They developed algorithms to linkpores on the aggregate’s outer surface with adjacent macropores while randomly incorporating micropore aggregates
distributed within the medium.
The core concept revolves around integrating pores andthroats of dimensions smaller than those undetectable due
to sample size limitations of the μCT system. This approachallows for the identification of undetectable porosity basedon pixel size of the μCT images. Consequently, fluid flow
through the entire pore system is facilitated by these smaller
pores, which are interconnected with the larger pore systems,
including vugs if present.
The primary challenge in constructing pore networks is toconnect pore bodies without causing overlaps among multiple pore throats. To address this, we initiated connectivity
studies using a guided random generation method within aconfined volume. The method begins by identifying dissociated neighborhoods, or pore networks, which are interspersedwith zones of indeterminate grayscale values in the μCT
images, suggesting potential connectivity. We emphasizethat these selected zones are likely candidates for subresolution pores, based on the grayscale intensity in the μCT
images.
In the reconnection algorithm, pairs of pore bodies aredesignated: one from the neighborhood requiring connection
and another one from the primary network. These pairs arethen linked with the newly introduced sub-resolution pores.
Prior to establishing connections, it is essential to identify

Vadose Zone Journal

5 of 14which networks within the model are disconnected but still are
believed to be part of a larger, interconnected system. These
networks, also referred to as neighborhoods, consist of a series
of interconnected pores that form a self-contained system,
isolated from the main pore network.
The distance between the separate neighborhoods and themain network often corresponds to the size of a pore throat,with the intervening space being an ill-defined region likely
containing micropores that are undetected during imaging or
overlooked during segmentation. To facilitate these connections, an auxiliary algorithm was developed to provide thenecessary information, as illustrated in Figure 3. The algorithm employs a decision tree structure, where each node
represents a pore body and each branch signifies a connection, grouping all pores into a single network. The method
effectively eliminates connections between two or more porebodies at the same hierarchical level, as well as connections
involving a pore body common to the subsequent level.
Figure 3 outlines the algorithm’s procedural steps:
1. An arbitrary pore is chosen as the “origin” or “reference.”
For instance, pore number 1 is selected, derived from thepore system’s reconstruction based on the grayscale of theμCT images used for the PNM.
2. Pores connected to the origin (pore bodies 2 and 3)
are identified, and their indices are recorded to prevent
duplication. These pore bodies are generated in areas notrecognized as pores post-segmentation due to their grayscale, yet are situated in lighter gray zones.
3. Each pore identified in the preceding step (i.e., pore bodies
2 and 3) is designated as the new origin.
4. Pores connected to any of the new origins (pore bodies 4
and 5) are identified and recorded.
5. Steps 3 and 4 are repeated until no new pores are generated, marking the completion of the network connectivity.
At this stage, the newly generated and connected porebodies link to pore bodies within the original main
network (marked by number 6), which might already beconnected to other pore bodies (as indicated by number 7).
The algorithm concludes when no additional lighter grayscale regions are found, since these areas likely contain
sub-resolution pores. The segmentation step defines theselighter gray regions, potentially housing sub-resolution
pores, and the algorithm is confined to these areas.
6. All pores identified previously are classified as part ofa neighborhood since they are interconnected. The process now restarts with a new arbitrary pore body not yet
associated with any known neighborhood.
7. The algorithm concludes when there are no remaining
ungrouped pores. In this scenario, pore bodies marked bynumbers 6 and 7 could serve as new sources for generating
connections with other neighborhoods, given the presenceof regions whose gray scale may define areas likely tocontain sub-resolution pores.

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GODOY ET AL.

F I G U R E 3 Schematic of the adopted network generation: (a) real connections (pore throats in black) and generated connections (pore throats
in red) in the pore network; (b) first iteration with definition of the first level (pore bodies generated and connected to previously existing networks);(c) second iteration in which redundant pore throats are discarded; and (d) last iteration, given the non-existence of connections with pore 7, anddetermination of the final network (source: Godoy [2019]).

As previously noted, the creation of connections between
pore bodies does not involve pore throats that intersect with
one another. Due to the significant computational demands
required to verify intersections in large samples, we chose to
generate and reconnect the pore systems while allowing forpotential overlaps in the generated pore throats.
Given the resolutions used during the acquisition of μCT
data for the selected samples, several assumptions werenecessary:
1. The diameters of newly generated pores are inherently limited by the pixel size utilized during the μCT phase. Ifthey were larger, they would have been detected during
imaging.
2. The lower limit of the diameters is not easily determinedgeometrically. Nevertheless, the objective is to create asystem that accurately represents the network’s behavior
rather than its precise structure. Therefore, the diametersmust not be so small as to diminish their influence on
subsequent simulations.
3. For instance, in our images with a pixel size of 9.97 μm,
the minimum and maximum pore size limits were established to be 2.0 μm and 9.1 μm, respectively. This range
is considered to follow a uniform log-normal distribution
due to the lack of a known distribution. The adopted limits

correspond to 20% and 98% of the smaller throat diameters
for the given pixel size.
4. In images with a pixel size of approximately 18.5 μm,
the minimum and maximum pore sizes were estimated tobe 2.0 μm and 18.2 μm, respectively. For a pixel size of
24.95 μm, these limits were set at 2.0 μm and 24.5 μm.
5. The lower limits for both pore bodies and pore throats,regardless of the pixel size of the μCT images, were
uniformly set at 2.0 μm.
6. A log-normal distribution was chosen for the size distribution of the pores.

2.4
Single-phase fluid flow numericalsimulations
Numerical simulations to obtain the absolute permeability(Kabs ) were conducted following the methodology outlined
by Godoy et al. (2019), who used a custom code partiallybased on the foundational work of Raoof et al. (2013). The
simulations considered fluid flow along the z-axis, traversing
from the top to the bottom of the sample. The Kabs calculations assumed laminar flow within the porous media, andemployed Poiseuille’s equation to determine the volumetricflow rate within each pore, that is,15391663, 0, Downloaded from https://acsess.onlinelibrary.wiley.com/doi/10.1002/vzj2.20348 by Martinus van Genuchten - CAPES , Wiley Online Library on [30/05/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

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Vadose Zone Journal

𝑄𝑖𝑗 =

𝜋𝑟4𝑖𝑗
8𝜇𝑙𝑖𝑗

(𝑝𝑗 − 𝑝𝑖 )7 of 14(1)

where Qij represents the discharge through the pore throat
between pore bodies i and j, pi , and pj are the pressures at
two adjacent pore bodies, rij is the radius of the pore throat,lij is the length of the throat, and μ is the dynamic viscosity of
the fluid (water).
For incompressible flow, the volume balance for each pore
body i is calculated using:
𝑁𝑖
∑
𝑗=1

𝑄𝑖𝑗 = 0 (𝑖 = 1, 2.3, … , 𝑁𝑖 )(2)

where Ni is the coordination number of pore body i. Given awell-defined sample volume, the average fluid velocity withina pore, v¯, can be determined as per Raoof et al. (2013):𝑣̄ =𝑄tot 𝐿
𝑉𝑓(3)

where Qtot is the total flow rate through the pore network,
ascertainable at either the inlet or outlet as the sum of all flows,
L represents the length of the pore network, and Vf is the total
volume of the liquid phase within the pore network.
The sample’s absolute permeability (Kabs ) is subsequently
calculated using Darcy’s equation:
𝐾abs =

𝜈𝑄tot 𝐿
𝐴Δ𝑃

(4)

where μ is the fluid’s viscosity (water), ΔP is the pressure differential between the inlet and outlet (top and bottom of thesample, along the z-axis), and A is the cross-sectional areaof the porous medium. For this study, we considered a pressure differential of 5.0 × 10−3 Pa between the upper and lower
boundaries of the PNM along the vertical (z) axis.

3

RESULTS AND DISCUSSION

Our analysis of sub-resolution porosity was performed
directly on the μCT images of sample plug 1–4. The images
did not reveal any connections between the largest pore clusters within the system. Figure 4 illustrates the reconstructed
pore network of samples 1–4, with μCT images captured at
pixel sizes of 9.97, 18.53, and 24.95 μm for images (a), (b),and (c), respectively.
Sub-resolution pores were generated using log-normal
distribution given by𝑓 (𝑥 |𝜇, 𝜎) =

1
√

𝑥𝜎 2𝜋

{
exp

−

[ln(𝑥) − 𝜇]
2𝜎 2

}
(5)

F I G U R E 4 Image of plug 1–4, and digital reconstructions of theporosity system based on X-ray computed microtomography (μCT)
images having pixel sizes of (a) 24.95 μm, (b) 18.16 μm, and (c)9.97 μm (source: Godoy [2019]).

where x is the variable representing pore throat radii, μ is themean of the natural logarithm of x, and σ is the standard deviation of the natural logarithm of x. The parameters for the
distribution were chosen as μ = −4.71 and σ = 0.0087, determined through the proposed reconnection algorithm. Figure 5
provides a plot of the generated pore throat radii.
The methodology allows one to control the number of
sub-resolution pore bodies connected by adjusting three key
parameters: the maximum distance factor (MDF), the numberof closest zones (NCZ), and the number of closest connections (NCC). Of these, the parameter NCZ dictates the numberof neighboring clusters to which each cluster can potentiallyconnect, while NCC specifies the minimum number of connections that must exist between each cluster. These twoparameters, in conjunction with the log-normal distribution
characteristics, determine the dimensions of the pore bodies
and throats to be generated, as well as their spatial placement
within the overall network.
The MDF plays a crucial role in defining the maximum
area within which sub-resolution pores can be generated andwhere connections between neighborhoods are possible. Thisparameter, which acts as a multiplier of the arithmetic meanof the pore throat lengths, effectively creates a sphere of influence. The NCZ specifies the number of neighboring regionsthat can be connected within the area defined by MDF, while
the NCC determines the minimum number of connections
that must exist between each neighborhood within the zoneof interest.

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FIGURE 5

Plots of pore throat radii from the selected log-normal distribution to be used for the generation of sub-resolution pores.

The combined effects of the three parameters (MDF, NCZ,
and NCC) significantly influence the estimates of the absolutepermeability (Kabs ). A notable outcome of introducing new
sub-resolution pores is their integration into previously isolated clusters, thus enabling the execution of numerical fluid
flow simulations. This process facilitates connectivity within
the original networks, which previously exhibited Kabs valuesof zero, despite the experimental Kabs being 33.19 mD.
We carried out an initial investigation about the effects of
the MDF, NCZ, and NCC parameters on the PNM results
using μCT images with a pixel size of 18.53 μm. The
input/output boundaries were adjusted to account for up to 2%
along the flow (z) axis, as per the methodology proposed byGodoy et al. (2019). Kabs estimates obtained from the numerical simulation of single-phase flow within the generated
networks are present in Table 5. By varying each parameter individually, we aimed to discern their respective impacts
on the simulation outcomes. Results indicate that NCZ had
the most influence on the Kabs estimates, although the values were closely matched with all parameter combinations.
Nonetheless, all estimated values were approximately one
to two orders of magnitude lower than the experimentallymeasured value.
Given the unpredictability of the impact of the MDF,
the NCZ, and the NCC parameters, they were initially chosen based on fundamental assumptions following preliminary
simulations. These assumptions were as follows:
1. Geometric means were used for determining the dimensions of pore throats connected to the input/output bound-

T A B L E 5 Pore network model (PNM) generated Kabs estimatesusing18.53 μm pixel size X-ray computed microtomography (μCT)
images of samples 1–4 considering 2% of the flow domain for the
input/output boundaries and using geometric averages when generating
the pore throats.
Estimated Kabs

Sample 1–4

Parameters

PNM

MDF

NCZ

NCC

mD

Case 1

8

3

3

0.395

Case 2

8

3

4

0.445

Case 3

8

4

3

0.605

Case 4

9

3

3

0.576

Case 5

9

4

3

0.407

Abbreviations: MDF, maximum distance factor; NCC, number of closest connections; NCZ, number of closest zones.

aries, which accounted for 2% of the network’s dimension
along the z-axis as noted previously.
2. The foundational pore network model was constructedusing μCT images with a pixel size of 18.53 μm. We
anticipated that generating sub-resolution pores, establishing connections between neighborhoods, and conducting
subsequent simulations to calculate the absolute permeability would be significantly more resource-intensive if
the PNM was based on a much smaller pixel size of
9.97 μm.
3. The initial value for MDF was set to eight times the arithmetic mean of the pore throat lengths. This decision wasmade under the premise that lower values would not facilitate sufficient connectivity to yield Kabs estimates that

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Vadose Zone Journal

9 of 14F I G U R E 6 (a) Results of the pressure field based on a pixel size of the X-ray computed microtomography (μCT) derived pore network model(PNM) image of 18.53 μm for Case 3, (b) pore body radii, (c) pore body pressures, showing in lighter colors the connected region, and (d) a zoom of
this region. Pressure scales are measured in Pa, spatial dimensions measured in m (source: Godoy [2019]).

would closely align with the experimental value for oursample.
4. Values of 3 were selected for both NCZ and NCC to ensurea sufficient number of connections to enable fluid flow
simulations and obtaining Kabs estimates. This initial setup
hence envisioned three neighborhoods, each with threeconnections.
5. The initial baseline condition for evaluation was established with the reconstructed PNM having MDF, NCZ, andNCC values set to 8, 3, and 3, respectively.
6. Kabs values derived from the PNM could then be benchmarked against the experimental value obtained for thissample, which was 33.19 mD.
Figure 6 illustrates the results of numerical simulations
focusing on the convergence of the pressure field for Case 3,
as detail in Table 6. The radii of the pore bodies were significantly smaller than those derived directly from the PNM using
images with a pixel size of 18.53 μm. Specifically, the directly
measured smallest radii were 9.265 μm, while after the generation and reconnection of sub-resolution pores, the smallest
radii were reduced to 2.830 μm, as shown in Figure 6b.

T A B L E 6 Kabs estimates based on pore network models (PNMs)
generated from 18.53 μm pixel size X-ray computed microtomography(μCT) images for samples 1–4 considering geometric average for porethroats generated for Case 3 using different input/output boundaries.
Kabs numericalsimulation (mD)

Sample 1–4
18.53 µm (PNM)

Input/output
boundaries (%)
2

0.605

Case 3

10

3.85

20

4.37

Table 7 provides similar estimates of Kabs using PNMs
developed now from images with a pixel size of 9.97 μm andincorporating input/output boundaries set at 10% as per the
methodology of Godoy et al. (2019). The pore systems wereconfigured using parameters of values of 9, 4, and 3 for MDF,
NCZ, and NCC, respectively. Kabs values were calculated forvarious averages (harmonic, geometric, and arithmetic) of thepore throat radii. Results show that the PNMs created with ahigher resolution and input/output boundaries defined as 10%of the plug in the flow direction sections aligned more closelywith the experimental value of 33.19 mD, despite all being

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GODOY ET AL.

Vadose Zone Journal

GODOY ET AL.

F I G U R E 7 Results of the pressure field based on a pixel size of the X-ray computed microtomography (μCT) derived pore network model(PNM) image of 9.97 μm for Case 3, (b) pore body radii, (c) pore body pressures, showing in lighter colors the connected region, and (d) a zoom of
this region. Pressure scales are measured in Pa, spatial dimensions measured in m (source: Godoy [2019]).

T A B L E 7 Kabs estimates based on pore network models (PNMs)
generated using a pixel size of 9.97 μm in the X-ray computed
microtomography (μCT) images for samples 1–4, considering 10% of
input/output boundaries and three ways of averaging the pore throat
radii.

Sample 1–4 pore throats

Estimated numerical absolutepermeability (mD)Harmonic average

15.68

Geometric average

31.85

Arithmetic average

51.20

within the same order of magnitude. Notice that the PNM network using geometric means for the pore throats radii yieldeda Kabs value most closely matching the reference value.
Figure 7 presents PNM outcomes derived from μCT images
with a pixel size of 9.97 μm, employing geometric means
for the pore throats, 10% input/output boundaries, and MDF,
NCZ, and NCC values of 9, 4, and 3, respectively. A numerically imposed pressure differential of 5.0 × 10−3 Pa wasagain applied. As previously discussed, the sub-resolution
porosity was estimated from a log-normal distribution of the
sub-resolution pore throats (Equation 5 with μ = 4.71 μm andσ = 0.0087 μm), reflecting a similar distribution pattern for
macro-throats observed in the sample network.
Similarly, as for the PNMs generated with μCT images
having pixel sizes equal to 18.53 μm and 24.95 μm, several
assumptions were made regarding the insertion and reconnection of sub-resolution pores for the pore networks generated
with a lower resolution. For networks obtained with μCT
images having a pixel size of 18.53 μm, numerically estimatedKabs values were consistently one or two orders of magnitude below the experimentally obtained reference value,even after the addition of sub-resolution pores. This discrepancy occurred regardless of the method used to define
the pore throats (arithmetic, geometric, or harmonic), thespecified input/output boundaries (2%, 10%, or 20%), andthe values of MDF, NCZ, and NCC. Conversely, networks
obtained with μCT images using a pixel size of 24.95 μm
did not show any connectivity between the neighborhoods
when using the same parameters for the pore network andsubsequent reconnections.
A significant advantage of our methodology for generating and reconnecting sub-resolution porosity is that thealgorithm seeks connections within topological proximity
between already existing pore bodies and pore throats and

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Vadose Zone Journal

11 of 14F I G U R E 8 Pore network model (PNMs) of samples 1–4 based on 18.53 μm (a) and 9.97 μm (c) pixel size X-ray computed microtomography(μCT) images assuming 10% input/output boundaries, and the pressure fields for those two PNMs (b and d, respectively). Pressure scales aremeasured in Pa (source: Godoy [2019]).

the newly generated sub-resolution pores. For comparison,
Figure 8 presents the results of the pressure fields for the
PNMs obtained with different resolutions. Notice that thegenerated sub-resolution pores facilitated the reconnection ofneighborhoods in the vicinity of the same region as before(as shown in Figure 8b,d). The methodology minimizes thegeneration of new pores in regions initially not consideredcandidates for receiving new pores, particularly considering
the gray scale of the μCT images, which suggests the absenceof such pores.
Although the results are promising, further potential
improvements to our methodology may include:
1. identifying specific regions for the insertion and connection of sub-resolution pore bodies and pore throats. This
process could leverage the grayscale of the μCT images
in selected areas with a high likelihood of containing
sub-resolution porosity;
2. implementing multi-scale imaging techniques to examine certain regions with higher resolution images. Thisapproach would increase confidence in identifying subporosity regions by assigning weights to areas most likely

to contain sub-resolution porosity, as suggested by Ruspini
et al. (2021);
3. correlating the NMR derived pore size distribution withthe distribution curve used during script execution. This
correlation would help in fine-tuning the parameters usedin the PNM model;
4. exploring additional experimental methods to calibrate thepore throat size distribution curves. This could involve
utilizing data from mercury injection capillary pressure(MICP) techniques or acquiring μCT images of specificareas at higher resolutions to achieve more accurate poresystem representations.

4

CONCLUSIONS

Our investigation combined both experimental and numerical analyses of a carbonate rock sample from a Brazilian
basin. Through a combination of direct laboratory measurements and μCT-based imaging techniques, we explored therock matrix composition and digitally reconstructed pore systems. This comprehensive approach shed light on the matrix

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GODOY ET AL.

GODOY ET AL.

Vadose Zone Journaland macropore fluid flow processes within a naturally heterogeneous rock sample. By employing PNM-based numericalsimulations, we were able to estimate the absolute permeability of the entire plug, which aligned well with the measured
REV.
The experimental findings played a crucial role in establishing significant correlations across the entire plug, primarily
to gather accurate information about the pore structure necessary for the modeling phase. Higher resolution μCT images
yielded the most accurate results for permeability, closely
matching the experimental data for the entire plug. To develop
more realistic pore systems for this carbonate rock sample,
we incorporated additional experimental techniques, such as
NMR to confirm prevailing pore distributions and refine thesegmentation process for further PNM studies, particularly
under multiphase flow conditions.
Our sub-resolution porosity investigations demonstrated astrong correlation between the numerical Kabs results andthe laboratory measurement, following the selection of poreconnection parameters. We emphasize that our efforts wereintentionally directed toward establishing a viable digital
workflow for generating and reconstructing sub-resolution
porosity. This strategy was supported by physical data on PSD
obtained using NMR, albeit without conducting further flow
tests to assess potential impacts on pore predictions and associated fluid flow behavior. Future studies will focus on (a)enhancing the methodology for pore generation and reconnection in targeted sample regions; (b) achieving finer control of
the PSDs through the use of NMR and MICP data for calibrating the curves generating pore bodies and throats; and(c) exploring the use of data from two-phase flow simulations for more accurate calibration of the generated networks.
These avenues of research will be pursued in a broader study
dedicated to sub-resolution porosity.
AU T H O R C O N T R I B U T I O N S
William Godoy: Conceptualization; investigation; methodology; software; validation; writing—original draft. Elizabeth
M. Pontedeiro: Investigation; methodology; supervision; validation; writing—review and editing. Rafael A. B. R. Barros:
Conceptualization; methodology; software; writing—review
and editing. Enno T. de Vries: Methodology; software;
writing—review and editing. Amir Raoof: Investigation;
software; supervision; writing—review and editing. Martinus Th. van Genuchten: Conceptualization; investigation;
supervision; writing—review and editing. Paulo Couto:
Investigation; supervision; writing—review and editing.
AC K N OW L E D G M E N T S
The authors would like to thank the Federal University of
Rio de Janeiro (UFRJ) in Brazil and the Utrecht University
in Utrecht, The Netherlands, for supporting this research. We
thank Prof. Ricardo Tadeu from UFRJ (LIN) for providing the

X-ray computed microtomography images with different resolutions. We would also like to thank Prof. Leonardo Borghifrom UFRJ (LAGESED) for the sample selection. The authors
also like to thank William Godoy and UFRJ (COPPE) for thepermission to use the images.
C O N F L I C T O F I N T E R E ST STAT E M E N T
The authors declare no conflicts of interest.
ORCID
William Godoy https://orcid.org/0000-0003-1879-9386
Enno T. de Vries https://orcid.org/0000-0001-9937-1087
Martinus Th. van Genuchten https://orcid.org/0000-00031654-8858
Paulo Couto https://orcid.org/0000-0002-3847-8638
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GODOY ET AL.

How to cite this article: Godoy, W., Pontedeiro, E.
M., Barros, R. A. B. R., de Vries, E. T., Raoof, A., vanGenuchten, M. T., & Couto, P. (2024). Modeling
sub-resolution porosity of a heterogeneous carbonate
rock sample. Vadose Zone Journal, e20348.
https://doi.org/10.1002/vzj2.20348

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