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The Impact of Sub-Resolution Porosity of X-ray
Microtomography Images on the Permeability
Cyprien Soulaine, Filip Gjetvaj, Charlotte Garing, Sophie Roman, Anna
Russian, Philippe Gouze, Hamdi Tchelepi
To cite this version:
Cyprien Soulaine, Filip Gjetvaj, Charlotte Garing, Sophie Roman, Anna Russian, et al.. The Impact of
Sub-Resolution Porosity of X-ray Microtomography Images on the Permeability. Transport in Porous
Media, Springer Verlag, 2016, 113 (1), pp.227 - 243. �10.1007/s11242-016-0690-2�. �hal-01692755�
Transp Porous Med (2016) 113:227–243
DOI 10.1007/s11242-016-0690-2
The Impact of Sub-Resolution Porosity of X-ray
Microtomography Images on the Permeability
Cyprien Soulaine
1
· Filip Gjetvaj
2
· Charlotte Garing
1
· Sophie Roman
1
·
Anna Russian
2
· Philippe Gouze
2
· Hamdi A. Tchelepi
1
Received: 26 November 2015 / Accepted: 4 April 2016 / Published online: 20 April 2016
© Springer Science+Business Media Dordrecht 2016
Abstract There is growing interest in using advanced imaging techniques to describe the
complex pore-space of natural rocks at resolutions that allow for quantitative assessment of the
flow and transport behaviors in these complex media. Here, we focus on representations of the
complex pore-space obtained from X-ray microtomography and the subsequent use of such
‘pore-scale’ representations to characterize the overall porosity and permeability of the rock
sample. Specifically, we analyze the impact of sub-resolution porosity on the macroscopic
(Darcy scale) flow properties of the rock. The pore structure of a rock sample is obtained using
high-resolution X-ray microtomography (3.16
3
µm
3
/voxel). Image analysis of the Berea
sandstone sample indicates that about 2 % of the connected porosity lies below the resolution
of the instrument. We employ a Darcy–Brinkman approach, in which a Darcy model is used
for the sub-resolution porosity, and the Stokes equation is used to describe the flow in the
fully resolved pore-space. We compare the Darcy–Brinkman numerical simulations with core
flooding experiments, and we show that proper interpretation of the sub-resolution porosity
can be essential in characterizing the overall permeability of natural porous media.
Keywords Sub-resolution porosity · Microporosity · Pore-scale simulation ·
Darcy–Brinkman formulation · Berea sandstone · Microtomography
1 Introduction
Improvements in imaging technologies and step changes in computing power are making it
possible to image natural porous media with high resolution. Digitized images of the complex
pore-space in three dimensions (3D) are often used to analyze the flow dynamics and compute
B
Cyprien Soulaine
csoulain@stanford.edu
1
Energy Resources Engineering, Stanford University, 367 Panama St., Stanford, CA 94305, USA
2
Géosciences Montpellier, UMR 5243 CNRS/INSU, Université de Montpellier 2, 34095 Montpellier,
France
123
228 C. Soulaine et al.
flow-related quantities, such as permeability, relative permeability, and capillary–pressure
relations. The process of imaging natural rocks and modeling the flow dynamics using the
high-resolution representations of the pore-space is often referred to as digital rock physics.
The application areas include reservoir engineering, subsurface hydrology, and subsurface
CO
2
sequestration. Specifically, X-ray microtomography (micro-CT) permits 3D imaging of
the pore structure of rock samples that are a few cubic millimeters in size at a resolution a
micron (e.g., images made up of 2000
3
voxels with a voxel size of one cubic micrometer;
Wildenschild and Sheppard 2013; Blunt et al. 2013). On the flow simulation side, the compu-
tational fluid dynamics (CFD) community has invested significant efforts to develop flexible
and computationally efficient software platforms for high-resolution numerical simulation of
complex flows. Combined with modern high-performance computing (HPC) architectures
and solution algorithms, CFD flow simulators can employ models with complex 3D grids on
the order of a billion cells (Trebotich and Graves 2015). Numerical solutions of the Stokes
equations using realistic representations of the 3D pore geometry, which can be obtained from
micro-CT images, are used to obtain the (single-phase) velocity v field in the pore-space.
Computation of the pressure and velocity fields for single-phase flow at the pore-scale is used
to obtain the permeability tensor, K, of a representative elementary volume (REV) of porous
structures (Spanne et al. 1994; Arns et al. 2005; Kainourgiakis et al. 2005; Zhan et al. 2009;
Khan et al. 2011; Thovert and Adler 2011; Blunt et al. 2013; Mostaghimi et al. 2013; Andrä
et al. 2013a, b; Guibert et al. 2015b).
Many natural rocks, such as carbonates, have pore-spaces with complex heterogeneities
that span a very wide range of scales. A fraction of the pore-network may have features
that are smaller than the voxel size. This is the case even with the most advanced micro-CT
devices (e.g., synchrotron-based micro-CT with a voxel size below 500 nm; Arns et al. 2005;
Garing et al. 2014; Hebert et al. 2015). For natural rocks that exhibit multimodal pore-size
distributions, the ‘segmentation’ of the raw images is a crucial step, since different methods
can lead to very different pore-space geometries. Multithresholding segmentation algorithms
improve the quality of differentiation. Nevertheless, one ends up with three classes: voxels
occupied completely by solid minerals, voxels that represent void space, and voxels that are
below the instrument resolution. This last category, which may be composed of solid material,
void space, or both, is referred to as micropores. Ignoring such microporosity can lead to
a significant underestimation of the overall pore-space connectivity, as illustrated in Fig. 1.
The red and blue regions in the figure correspond to zones that are unambiguously identified
as void and solid phases, respectively; the purple and green areas denote microporous zones.
If during the segmentation process the purple region is identified as a solid phase, then the
flow resistance in the highlighted throat will be overestimated. This interpretation would
lead to less flow through this particular area and more flow through the adjacent pore-throats.
If such complex features are present across the 3D space, the erroneous distribution of the
preferential flow paths can lead to significant errors in characterizing the flow capacity of
the rock. It is also possible for microporous material to act as connecting macropores, as
illustrated by the green region in Fig. 1. It follows that ‘bad’ segmentation of the raw image
can misrepresent the degree of connectivity. That, in turn, is likely to lead to significant
errors in the computed local velocity field, v, and produce significant errors in assessing the
overall permeability of the sample. Such complexities are not restricted to carbonates. For
example, it has been demonstrated that Berea sandstones may contain between 2 and 10 %
of microporosity (Churcher et al. 1991; Tanino and Blunt 2012).
For example, non-Fickian dispersion in natural rocks has been attributed—in part—to
diffusion in the microporous matrix (see Gouze et al. 2008, and reference therein). Most of
the works published so far assume that diffusion is the primary transport mechanism in the
123
The Impact of Sub-Resolution Porosity of X-ray… 229
Fig. 1 Schematic of a micro-CT rock sample image. The red and blue regions are identified as solid minerals
and void spaces, respectively. The green and purple a reas denote microporous regions
microporosity (Haggerty and Gorelick 1995; Carrera et al. 1998; Gouze et al. 2008; Shabro
2011; Gjetvaj et al. 2015). The validity of this assumption is questionable, especially if the
microporous region is located in a percolating pathway, as illustrated by the green area in
Fig. 1. In such a case, convection in the microporous region may play an important role and
should be included into the modeling of tracer transport. To capture the multiscale nature
of the pore-space, the microporous regions can be represented using parallel bonds in a
pore-network model (PNM; Ioannidis and Chatzis 2000; Bekri et al. 2005; Youssef et al.
2008; Bauer et al. 2011; Jiang et al. 2013; Mehmani and Prodanovi´c 2014; Bultreys et al.
2015; Prodanovi´c et al. 2015 ). With such a technique, often referred to as a dual PNM, the
macroporosity is treated using a standard PNM reconstructed from the micro-CT images,
and the microporous bonds are considered as continuum porous medium. The dual PNM
representation is easily extended to the flow of multiple fluid phases (e.g., oil and water).
One of its main drawback, however, is inherent to PNM, namely, that the extracted pore-
network is an approximation of the real pore structure, and the results of the flow simulations
depend strongly on the transport/invasion rules used for the nodes and the connections.
We employ a strategy that allow us to compute the flow field in the fully resolved pore-space
and also account for the presence of microporous regions. Such regions are defined as material
that may contain pores whose size is smaller than the voxel size, which depends on the imaging
device. Specifically, we describe a hybrid approach that solves the Stokes equations in the
regions identified as pore (void) space and Darcy’s law for flow in the microporous domain.
The Darcy–Brinkman approach has been used effectively to compute accurate flow fields in
bimodal porosity distributions (Knackstedt et al. 2006; Apourvari et al. 2014; Krotkiewski
et al. 2011; Scheibe et al. 2015). Particular attention must be paid to the estimate of the
permeability of the microporous domain, k
micro
. In the absence of additional knowledge
about the geometrical structure of the sub-resolution (sub-voxel) porosity (e.g., from higher
image resolution), k
micro
has to be modeled based on the available image resolution. At
minimum, the permeability should be expressed as a function of the image resolution and
the associated maps of the microporous regions.
123
230 C. Soulaine et al.
Next, we describe the rock sample used in this study, including the imaging method
and the data processing used to determine the different components (i.e., void, solid, and
microporous region). Then, we present a Darcy–Brinkman formulation, including a heuristic
relation for estimating k
micro
from the porosity of the microporous phase obtained from the
X-ray microtomography images. Then, the numerical simulations of eight sub-volumes of the
scanned rock sample are presented, and the impact of the microporosity on the distribution
of the velocity field and the estimation of the overall permeability are discussed.
2 Materials and Methods
This section introduces the X-ray microtomography technique used to scan the Berea sand-
stone sample, the mathematical model developed to obtain the velocity distribution in the
void space, and the setup for the flow simulations.
2.1 X-ray Microtomography
X-ray microtomography is a noninvasive imaging technique that allows for constructing a
3D image of the target object using a set of two-dimensional (2D) radiographs of the X-ray
attenuation properties of the material that makes up the object. Each ‘voxel’ value corresponds
to a measure of linear-absorption coefficients, which for a porous medium depends on the
porosity and the composition of the solid matrix.
2.1.1 Sample
We study a Berea sandstone sample obtained from a larger block with overall porosity and
permeability values of 0.20 and 500 mD, respectively. A mercury intrusion porosimetry test
(AutoPore IV 9500 V1.06 from Micromeritics Instrument Corporation) was conducted on
a sample cored out from a nearby location; the mercury-injection test yields an overall
porosity of 19.41 % and a mean pore-throat diameter of 11.7 µm. The curve of the differential
intrusion, which can be related to the pore-throat size distribution, is shown in Fig. 2. The limit
size separating the ‘macropores’ (diameter above 3.16 µm) from the ‘micropores’ (diameter
below 3.16 µm) is also shown. The figure suggests that the fraction of microporous material
in the sample is quite small.
2.1.2 Data Acquisition
The sample of 6 mm diameter and 6 mm length was imaged at the BM5 beamline of the
European Synchrotron Radiation Facility (ESRF), Grenoble, France. Synchrotrons provide
very high flux of the monochromatic white beam and collimated X-rays resulting in high-
quality, low-noise images with a resolution of a few microns. A total of 3495 projections of
the sample were taken, every 0.051
◦
for angles from 0
◦
to 180
◦
using an exposure time of
0.1 s and an X-ray beam energy of 30 keV. The voxel size was 3.16 µm
3
.
The radiographs were corrected for variations in the X-ray beam intensity and background
noise. Then, the 3D volume of 4667 × 2130 × 2099 voxels was constructed from the radi-
ographs using a single-distance phase-retrieval algorithm (Paganin et al. 2002; Sanchez et al.
2012). Figure 3a presents a numerically computed cross section through the 3D volume. The
black color denotes the ‘macroporosity’ phase (void only), the lighter gray denotes the solid
matrix, and the intermediate gray levels denote the microporous phase (voxels composed of
123
