SESolid EarthSESolid Earth1869-9529Copernicus PublicationsGöttingen, Germany10.5194/se-7-441-2016Calculating structural and geometrical parameters by laboratory
measurements and X-ray microtomography: a comparative study applied
to a limestone sample before and after a dissolution experimentLuquotLindalinda.luquot@idaea.csic.eshttps://orcid.org/0000-0002-4389-3019HebertVanessaRodriguezOlivierInstitute of Environmental Assessment and Water Research (IDAEA),
Hydrogeology Group (UPC-CSIC), c/ Jordi Girona 18, 08034 Barcelona, SpainVoxaya SAS, Cap Omega, Rond-Point Benjamin Franklin, CS 39521,
34960 Montpellier, FranceLinda Luquot (linda.luquot@idaea.csic.es)24March20167244145630October201526November20154February20169March2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://se.copernicus.org/articles/7/441/2016/se-7-441-2016.htmlThe full text article is available as a PDF file from https://se.copernicus.org/articles/7/441/2016/se-7-441-2016.pdf
The aim of this study is to compare the structural, geometrical and
transport parameters of a limestone rock sample determined by X-ray
microtomography (XMT) images and laboratory experiments. Total and
effective porosity, pore-size distribution, tortuosity, and effective
diffusion coefficient have been estimated. Sensitivity analyses of
the segmentation parameters
have been performed. The limestone rock sample studied here has
been characterized using both approaches before and after a reactive
percolation experiment. Strong dissolution process occurred during
the percolation, promoting a wormhole formation. This strong
heterogeneity formed after the percolation step allows us to apply our
methodology to two different samples and enhance the use of
experimental techniques or XMT images depending on the rock
heterogeneity. We established that for most of the parameters
calculated here, the values obtained by computing XMT images are in
agreement with the classical laboratory measurements. We
demonstrated that the computational porosity is more informative
than the laboratory measurement. We observed that pore-size distributions
obtained by XMT images and laboratory experiments are slightly
different but complementary. Regarding the effective diffusion
coefficient, we concluded that both approaches are valuable and give
similar results. Nevertheless, we concluded that computing XMT
images to determine transport, geometrical, and petrophysical
parameters provide similar results to those measured at the
laboratory but with much shorter durations.
Introduction
Characterizing the rock pore structure such as the porosity, the total
pore–rock interface and the connectivity is essential to evaluate oil or gas
production or volume of stored CO2 in case of geological
sequestration, for example. Porosity is a key petrophysical parameter,
indicating the total volume of oil, gas, or water that can be contained in
a reservoir. It is essential to differentiate the total porosity from the
open connected or effective porosity, which is the fraction of porosity
accessible by any fluid. Connectivity as well as permeability and tortuosity
allow us to quantify the ability to extract or inject fluids. The higher the
connectivity and permeability are and the lower the tortuosity is, the higher
the extraction or injection flow rate will be. In consolidated rock aquifers,
the groundwater flow occurs through discrete openings, i.e. fractures, and
only to a small extent in the pore-network of the rock matrix. The migration
of solutes and solvents in fractured rock aquifers is therefore determined by
the relative contribution of advective flow compared to matrix diffusion
transverse to the flow direction, resulting in the retardation of the solute. The
advection transport is controlled by the permeability of the reservoir
(mainly controlled by fracture or preferential flow path as wormhole or karst
conduit), whereas the diffusion through the matrix depends of the effective
diffusion coefficient, which is linked to tortuosity, effective porosity and
cementation factor . Diffusion in rock, of either
gas or liquid phase, could be the rate-limiting or dominant process in many
scenarios, such as geologic disposal of radioactive waste
, contaminant remediation
e.g.,, CO2 geological storage
, and oil and gas recovery
. Diffusion of hazardous gas or liquid
chemicals in construction materials has also become a concern for public
health and security . Therefore, knowledge of diffusion processes and
rates in rocks is important to better
understand these issues.
To evaluate all these parameters, different laboratory petrophysical
techniques are usually used. Some of them are time consuming and
destructive methods, such as mercury pore-size measurement or BET
(Brunauer–Emmett–Teller) surface analysis. Moreover, these different
techniques are performed at different sample scales and may require
prior preparation, such as thin section for SEM (Scanning Electron
Microscope) analysis, for example. Yet, most of the laboratories
techniques used and discussed in this article are fully validated.
Currently, a lot of petrophysical, geometrical, transport and mechanical
parameters are computed using 3-D X-ray microtomography (XMT) images
. High resolution 3-D XMT is a powerful technique
used to image and characterize the internal structure and geometry of natural
and/or artificial objects. It is a non-destructive method that does not need
prior sample treatment, such as impregnation, thinning or polishing
. In this technique, contiguous sequential
images are compiled to create 3-D representations that may be digitally
processed to obtain relevant quantitative geometric and/or morphological
parameters . 3-D data provides access to some
very important geometric and topological characteristics such as size, shape,
orientation distribution of individual features and that of their local
neighbourhoods, connectivity between features and network, composition, etc.
Computational techniques have progressed to the point where material
properties such as conductivity , diffusivity
(; ;
; ), permeability
(; ; ;
), pore-size distribution
(; ;
; ) and linear elasticity
(; ;
) can be calculated on large three-dimensional
digitized grids (over 109 voxels). More details on this technique,
acquisition and data computing step can be found in
and
. In geosciences, the internal structure
of a great diversity of geological samples has been examined by radiographic
imaging mainly in the last 50 years
(; ;
; ;
; ;
; ;
).
Yet, to our knowledge, no study has been conducted to explore the
potential links that can be made between variables computed from XMT
images and those traditionally measured in the laboratory for
a limestone core rock sample, before and after a dissolution
process. Only one similar study was done by
on soil material using a medical scanner with
a large pixel size (0.6 mm). Nevertheless, op. cited authors
focused their study to only a few parameters.
Here, we use high resolution X-ray microtomography images to
characterize the structural and geometrical parameters of a limestone
core rock sample percolated by an acidic solution. We computed and
experimentally measured, total and effective porosity, pore-size diameter
distribution, effective diffusion coefficient, and tortuosity. These
parameters are those needed for numerical modelling to evaluate oil and
gas deposit volume and extraction flow rate, for example. Quantifying
the pore network characteristics of a same sample before and after a dissolution
experiment allows to apply our methodology to two different pore networks
and enhance the use of experimental techniques or XMT images depending
on the rock heterogeneity. The focus of the present study is to
articulate the potential of variables estimated using XMT images and
how these estimates compare with, and complement, traditional
laboratory-based measurements.
Materials and methods
In this section, we describe the different laboratory measurements and XMT
images computations to evaluate the various petrophysical, geometrical
and hydrodynamic parameters. The list of these parameters and the corresponding methods are summarized in Table .
Summary of the measured parameters and the corresponding methodologies. (seg.: segmentation, Psd: pore-size distribution, Msd:
mean square diplacement, TPP: total porous phase, RPP: resolved porous phase, CRPP: connected resolved porous phase).
The rock sample used in this study is an oolitic limestone almost
composed of calcite (CaCO3) and is named Bvl in the paper.
This limestone is commonly referred to as Beauval rock and is coming from Beaunotte in Dordogne
region in France. It is characterized by a beige colour with some
shells. The mean connected porosity is usually comprised between 9
and 13 %, according to general information provided by quarry mining companies.
The core sample diameter and length are respectively 2.5
and 2 cm.
Laboratory petrophysical characterization
In order to characterize the different geometrical and structural
properties of the rock sample before and after the percolation
experiments, we used different classical laboratory experiments. These
different techniques have been performed following a home-methodology
to avoid too many drying and wetting sample steps.
First of all, to evaluate the effective (connected) porosity, we used
the triple weighing method. We first measured the dry sample weight
after leaving the sample during 48 h in an oven at
40 ∘C. We then saturated the sample, starting by a sample
vacuum step. Afterwards, we left the calcite equilibrated water
penetrate into the pore structure and we weighed the saturated and
submerged sample weights. We weighed four times the sample during the
dry, saturated and submerged steps. This classical method is very time consuming
and requires 4 entire days.
Then, we took advantage of the saturated sample state to evaluate the
pore-size distribution by measuring the retention curve of the sample using
a centrifuge and applying various rotation rates as previously done by
and . The technique consists
in applying a high gravity field to an initially saturated sample and
measuring the drained volume of water. For this purpose, we used
a Rotina® 420R centrifuge following the
methodology of using six speed increments up to
4500 rpm. The maximum suction that can be applied to the sample at 4500 rpm
is 213 kPa. This technique allows us to measure the effective
capillary size distribution and the retention curve θ(P), where
θ is the volumetric water content and P is the capillary pressure
(minus suction). By capillarity theory (Young-Laplace equation with
cylindrical approximation), the minimum radius of a pore that drains at P
is given by:
rp=-2σcosαP,
where σ is the surface tension (72.3±0.5mNm-1,
) and α is the contact angle (40±8∘, ). Using Eq. (),
we can convert the measured θ(P) into an equivalent θ(rp) curve. This curve is actually a cumulative pore-size
distribution; the water content θ(rp) indicates the
combined volume of all pores with opening radius less than rp.
The measurements were performed twice in the two directions to evaluate the
sample anisotropy. Six days were necessary to acquire the retention curves in
both directions two times, dry, and re-saturate the sample for the second
measurement.
After the centrifuge step, we dried the sample again in an oven during
48 h at 40∘ and measured again the dry sample
weight. We then repeated the triple weighing method to evaluate the
initial porosity and took advantage of the saturation state of the
sample to perform through-diffusion experiment. Classically, the
effective diffusion coefficient as well as the tortuosity factor is
measured by liquid phase conservative tracer test (usually iodine) as
presented by and
.
Through-diffusion experiments were performed to determine the
effective diffusion coefficient and tortuosity/constrictivity ratio
before and after the dissolution experiment
. The same methodology as the one developed by
has been used here. The diffusion cell
apparatus consisted of two acryl-glass cells of equal size and
volume. The reservoir cell contained a 0.02 molL-1 of
potassium iodide tracer solution, whereas the sink cell did not contain
any tracer at the beginning. We used Beauval rock equilibrated water
in both reservoirs and we added several milligrams of sodium azide
(NaN3) to prevent biofilm formation. The mounted rock sample was
sandwiched between the sink and reservoir cell. During the
experiment, iodide ions diffuse from the reservoir cell into the sink
cell through sample Bvl. An iodide-specific electrode from
Cole–Parmer Instrument CO was used to measure the iodide
concentration in the sink cell. More details about the procedure can
be found in . The aqueous diffusion
coefficient for iodide (Daq=1.86×10-9m2s-1) was used
to calculate the effective diffusion coefficient Deff. The
effective diffusion coefficient was calculated using the equation
:
Deff=βlC0,
where β is the slope of solute mass vs. time, which is
obtained from linear regression of data in the steady-state range, l
is the rock sample thickness and C0 is the concentration in the
reservoir cell. Then, it is possible to determine the tortuosity
coefficient using the definition of the effective diffusion
coefficient in a porous water-saturated media proposed by
:
Deff=Daqϕδτ2,
where τ is tortuosity and δ is constrictivity. These
coefficients are sometimes gathered together into an empirical
exponent m, denoted cementation factor, as in the following
equation:
Deff=Daqϕm
Through-diffusion experiment are time consuming because of the slow
diffusion rate of a liquid tracer, even for porous limestone and
sandstone samples . One to two weeks are
needed for each measurement.
Laboratory percolation experiment
We performed the flow-through experiment using the apparatus presented
in . The setup allows to inject at constant
flow rate up to four different solutions in parallel through four
distinct core samples. Various pressure sensors and a differential
pressure sensor enable to monitor the inlet and outlet pressures and
then calculate the sample permeability using Darcy's law. We injected
an acidic solution through sample Bvl at constant flow rate Q=16mLh-1 during 9 h at room temperature and
pressure. The injected solution was an acetic acid at pH =3.5 and
buffered at 500 mM. We prepared the injected solution by mixing
28.43 gL-1 of acetic acid with 2.184 of
sodium acetate. During the dissolution experiment, we continuously
recorded the inlet and outlet pH and the pressure drop between the
inlet and outlet of the sample to calculate the sample
permeability. The total injected fluid was 146 cm3, or some
90 pore volumes of sample Bvl (initial pore volume =1.62cm3).
Outlet water was sampled periodically, acidified to prevent mineral
precipitation, and analyzed for concentrations of Ca using inductively
coupled plasma-atomic emission spectrophotometry (ICP-AES, IDAEA,
Spain). Reaction progress and porosity changes are calculated from the
difference between injected and percolated waters, knowing that
calcite is the only mineral composing the sample. Calcite dissolution is described as follows:
CaCO3+H+⇌Ca2++HCO3-..
The volume of
dissolved calcite (ΔVcalcite(t)) is calculated as follows:
ΔVcalcite(t)=υQ∫t′=0t′=tΔCCa(t′)dt′,
where υ is the calcite molar volume (3.7×10-5m3mol-1), and ΔCCa is the
difference between the outlet and inlet calcium
concentration. Therefore, we can calculate the sample-scale porosity
change during the percolation experiment using the following equation:
ϕ(t)=ϕ0+ΔVcalcite(t)V,
where V is the total sample volume and ϕ0 is the initial
sample porosity. The error ϵ(ΔCCa) in the
change of calcium concentration was estimated using the Gaussian error
propagation method . The calculated error is
propagated to the porosity estimation. After the percolation
experiment, we characterized the core rock sample using the same
methodology as before, described in Sect. .
X-ray microtomography imagesImages acquisition
X-ray microtomography images were acquired on the ID19 beamline at ESRF
(European Synchrotron Radiation Facility), Grenoble (France). The acquisition
was done in white beam configuration, using a ROI of
2048×1690 pixels. The sample was placed at 1.7 m and different filters
were used in this configuration (2.8 mm of Al and 0.35 mm of
W) to achieve an energy of 71.1 keV with a gap of 57. The voxel size
was 7.42 µm3. We acquired 4998 radiographies in 360∘,
41 references and 20 dark images to reduce the noise during the 3-D
reconstruction. The acquisition time for each radiography was 0.25 s
which induces a total acquisition time for the entire sample (two scan steps)
of about 1 h (taking the motor movements into account). Two 3-D
images were acquired: one for the sample before percolation experiment and
one after, respectively named Bvlbe and
Bvlaf.
Image processing and parameter extraction
Analysis of the XMT images allows us to quantify the volume and morphology of the pore structure identified
during the segmentation process. Using the 3-D pore
representation, one can estimate its total and connected porosity
and geometrical properties, such as its surface area and pore-size
distribution. The processed images and results were mostly computed
with Voxaya's software. The same methodological framework was
applied to both images.
Filtering and region of interest extraction
The very first step in the image processing workflow consists in
isolating the region of interest: a cylindrical mask is applied on the
image in order to extract its relevant part. A median filter is then
used to remove noise while preserving edges of the structures.
Segmentation and porosity calculation
Segmentation is one of the most important step in image analysis. It
consists in gathering voxels that belongs to a same object and assigning
them a single common value (see Fig. which illustrates the segmentation step). Voxels identified to the matrix
constitute the solid phase which have the highest intensity and appear
in the brightest grey levels. Pores measured to be larger than the voxel
size (here 7.42 µm) are entirely captured by the camera
and appear in darkest grey levels in the image, forming a phase referred to as void phase
or resolved porous phase. We name sub-resolved porous space
the area appearing in intermediate grey levels and formed by matrix
(calcite in the present case) and pores measured to be smaller than the voxel
size. Note that this phase is sometimes denoted by the ambiguous term
“microporous phase” whereas resolved porous phase is often called
“macroporous phase”. Presence of pores smaller that the voxel size can
be confirmed for example by microscopic observations on thin sections
or a priori knowledge of the rock.
Histogram of grey scale values of Bvl sample images before the
dissolution experiment. The threshold value n is indicated in the graph. On
the top, a 2-D slice of Bvl before the dissolution experiment illustrates the
pore initial pore structure on the left and the binary image next to the
threshold step on the right.
In terms of numerical core analysis, computing porosity requires
to determine the relative fraction of the void phase volume and to
estimate the pore volume in the sub-resolved porous space. A first
segmentation algorithm based on a region growing method was used to
isolate the void phase. An additional image segmentation was then
conducted to isolate the sub-resolved porous space and compute its
volume fraction in the sample voxel. We can define the sample
sub-resolved porosity ΦS as follows:
ΦS=ξxΦx,
where ξx and Φx respectively denote the volume
fraction and the porosity of the sub-resolved porous space. The total
porosity is then given by the following:
ΦT=ΦR+ΦS,
where ΦR is the sample resolved porosity
(that is, the volume fraction of the void phase). Assuming that the
sample is chemically homogeneous, it is possible to estimate
Φx. We first compute the mean greyscale values corresponding to
solid and void phases, respectively denoted by Gm and
Gv. Following and
, the grey level value Gx of a voxel belonging to
the sub-resolved porous phase is linearly related to Φx as
follows:
Φx=(Gm-Gx)/(Gm-Gv).
Evaluating errors in porosity calculation
Estimating the uncertainty occurring in the porosity calculation can
be achieved from an XMT image by performing, for example, several
image segmentation with different, but close, input parameters and
then calculating the associated porosities. For instance, if an image
is segmented using a basic thresholding technique, then one can perform
extra segmentations by varying threshold values by one or two units and
computing the associated porosities. This enables to assess
the robustness of the segmentation parameters determined by the user.
Connected components of the pore space
A clustering algorithm derived from
enables to assess the connectivity of the pore space by identifying
neighbouring voxels that are connected to one another and assigning
a distinct label to each connected component. In this study, the
largest connected components of the pore space were extracted for both
resolved and sub-resolved porosity.
Geometric parameters
Many geometric parameters can be computed from the segmented image of
the pore space, namely the interface surface area, its global
curvature, and the Euler characteristics .
Here, we focused on the pore-size distribution and the surface-to-volume ratio
sometimes referred as “specific surface”.
The imaged pore space is used to quantify pore network characteristics such
as pore size. The pore-size distribution is evaluated from XMT images using
a Voxaya module. According to , the pore-size
distribution function gives the probability that a random point in the pore
phase lies at a distance x and x+dx from the nearest point on
the pore–solid interface. This is achieved by computing the Euclidean
distance from each voxel of the pore space to the interface, using a distance
transform algorithm based on . Equivalently, one can
consider this distance to be the radius of the largest sphere centred at this
voxel and inscribed in the pore space. Yet, each sphere that is fully
included in a larger one have no significant contribution to the pore space
geometry and can thus be removed. In other words, this method returns the
count of inscribed spheres that are maximal in the sense of inclusion.
Statistical measurements such as chord-length distribution functions
were also calculated. The chord-length
distribution function is linked to a probability density of random
chords corresponding to a virtual mean pore diameter depending on each
x, y, and z direction. It thus provides information on the
sample anisotropy.
Diffusion coefficient
Diffusion experiments can be simulated on the void space image following the
methodology described in . Consider a large number N of
(virtual) diffusing particles, initially uniformly distributed in the void
phase, and randomly moving following a Brownian motion, the diffusion
coefficient D(t) characterizes their ability to disperse in the void phase,
probing its structure. We denote by xi(t) the position of the ith
particle at time t and σ(t)2 their mean square displacement, that
is:
σ(t)=1N∑i=1N(xi(t)-xi(0))2,
Then according to :
σ(t)2=6D(t)t.
If d0 denotes the diffusion coefficient in an unbounded domain and τ
the tortuosity, then when t→∞, D(t) tends to an
asymptotic value D≈d0/τ.
Skeleton and properties
The skeleton of a three-dimensional object is a one-dimensional
reduction, centred inside this object, preserving its geometrical and
topological features. It provides a simplified representation of
a shape: the skeleton of a cylinder, for instance, consists of its
axis of rotational symmetry. Formal definitions can be found in
.
The skeleton is particularly known to be a tool of great interest to
investigate large objects with complex geometry, such as large
microtomography images of porous media. An implementation of the
classical thinning algorithm described in was used
for this work.
Results
The temporal evolution of the increase in calcium concentration, ΔCCa, during the percolation experiment as well as the inlet and
outlet fluid pH are presented in Fig. . Dissolution reaction
occurred during the percolation experiment. Indeed, the ΔCCa is always positive (ΔCCa>0) indicating Ca
release in the outlet fluid. Moreover, the outlet pH (Fig. ) is
higher than the inlet one which corroborates proton consumption and thus
calcite dissolution (Reaction ). Dissolution reaction may induce
porosity increase and other geometrical, structural and hydrodynamical
parameter changes. Evaluating and characterizing these changes are essential
for developing predictive models of reactive-transport processes such as
those occurring during CO2 geological storage, fracking processes,
oil and gas exploitation, acid mine drainage, or seawater intrusion among
others.
Inlet (black) and outlet (red) pH variation during the
percolation experiment as well as variation in calcium
concentration(blue).
Initial porosity measurements on sample Bvlbe
were performed by the triple weighing method (TW) and give
us an initial porosity ϕi(TW) of 16.06±0.44 %
(4 measurements), which is slightly higher than the one provided by
the quarries miner companies. This porosity is the connected porosity
which only takes into account the open pores connected to one of the
sample surface. After the dissolution experiment, the same methodology
was applied four times and we measured a final porosity
ϕf(TW)=20.26±0.73 %.
Mass balance calculation from the dissolution experiments were
performed using Eqs. () and () and the final
porosity was evaluated using the initial porosity ϕf(chem)
using the initial porosity ϕi(TW) and the ΔCCa
presented in Fig. . We obtained a final porosity
ϕf(chem)=19.37±0.56 % (propagating the error
of the TW method on the initial porosity value and the one from the
chemical analysis). The slightly higher increase in porosity,
measured by the TW method, can be explained by the connection of initially non-
connected pores to the new connected porosity resulting from the dissolution
process whereas the porosity calculated by the mass
balance is affected by the dissolution only. The different porosity
measurements and calculations are summarized in Table .
Determined porosities [%] (final one by mass balance calculation (chem),
before and after by triple weighing technique (TW), and before and after as well as
total (tot.), effective (open), resolved (res.) and sub-resolved (subres.) by XMT images
(XMT)), effective diffusion coefficients Deff [m2s-1] (from
through laboratory experiment (I-) and XMT images (XMT)), permeability k [m2]
and total fluid–rock interface (from BET measurement and XMT images (Minko)) for samples Bvlbe and Bvlaf.
ϕ(TW)ϕ(XMT)ϕ(chem)Deff(I-)Deff(XMT)kS(BET)S(Minko)%% %m2s-1m2s-1m2m2g-1m2g-1res.subres.tot.openBvlbe16.0610.426.7117.1315.94–1.43×10-123.75×10-124.01×10-140.34890.0018Bvlaf20.2615.627.1722.8021.7519.372.96×10-112.26×10-111.16×10-12–0.0021Pore-size distribution
We obtained the pore-size distribution for sample Bvl before and after
the dissolution experiment from the retention curve (RET) as explained in
Sect. . We measured the retention curve by draining
the sample both in the flow and counterflow direction in order to evaluate
the pore-size anisotropy. If the pore-size distribution was heterogeneous, then we
should get different retention curves due to the gradient of
capillary pressure inside the sample. Specifically, large pores at the
inlet (but not at the outlet) will desaturate at small suctions
applied to the inlet (i.e., for small rpm when the inlet is placed
outside in the centrifuge). Reversely, if the sample is rotated,
those pores will only desaturate when suction is large enough to drain
any of the outlet pores. The net result is that the measured curve
will exhibit directional dependence as previously observed by
.
Initially, the pore-size distribution is homogeneous; the initial
retention curves are similar in both sample orientations
(Fig. ). After the dissolution experiment, due to calcite
dissolution and porosity increase, the retention curves vary from the
initial one. Moreover, we can observe in Fig. , that
after the dissolution experiment, some heterogeneity appeared along
the sample inducing different shapes for the retention curves acquired
in both directions. The corresponding pore-size distributions are
presented in Fig. .
Retention curves before and after the dissolution experiment
for sample Bvl.
The results show that after dissolution, the amount of pores
of radii larger than 102.27 µm increases drastically. The
increase in largest pore size is similar whatever the sample direction
indicating that these pores are well connected together and broke
through the sample. We can also observe (Fig. ) that the
second major difference between the retention curves before and
after dissolution appears for pore radii 10.23<rp<34.99µm (suction between 5.11 and
17.47 kPa). After the dissolution experiment, fewer pores of
such radii are present through the sample, indicating that most of
the dissolution occurred in these pores.
Pore volume content for different pore-size diameters before
and after the dissolution experiment. The data have been extracted
from the Psd and C-l numerical measurements and RET laboratory
acquisition. For the latter, the distributions after dissolution
were obtained both by draining in the flow direction and in the
opposite direction.
Iodine concentration measured in the sink cell during
diffusion experiment through sample Bvl before and after the
dissolution experiment.
The results also display some differences in the pore-size distribution after
the dissolution experiment depending on the sample orientation. These
differences highlight some heterogeneous dissolution inducing different pore
diameter changes along the sample. They are however minor when compared with
other experiments with strong dissolution localization
. Most discrepancy is visible for intermediate and
smallest pores (0.48<rp<10.23µm). Moderately large
pores (radius around 10 µm, suction at 17.47 kPa) are
better connected to the inlet than to the outlet (that is, they drain better
when the sample is placed opposite to the flow direction, i.e., dragging
toward the inlet, than otherwise). The proportion of the smallest pores is
consistently lower when the sample is placed opposite to the flow direction,
implying that dissolution also occurred in these pores at the inlet. However,
the water contents obtained with the sample placed in the flow direction for
high suctions (small pore sizes) were higher than before the experiments,
implying a small decrease in pore size at the outlet.
Diffusion coefficient
Figure displays the results of the two iodide diffusion
experiments performed before and after the dissolution experiment on
sample Bvl. The curves show the time evolution of iodide at the sink
reservoir, which is proportional to the cumulative mass of iodide that
has diffused through the sample Bvl until time t. Only the
steady-state phase is reported here, when the iodide concentration in
the sink cell increases linearly with time. Linear regression of the
steady-state portion yields the effective diffusion coefficient
(Eq. ). Experimental results are summarized in
Table . After the dissolution experiment, the diffusion coefficient is increased
by 1 order of magnitude, as suggested by the noticeable increase in the slope of iodide
increment after dissolution. This increase is linked to
a decrease of the tortuosity coefficient τ from 14.43 (highly
tortuous pore skeleton, see Fig. ) to 3.57. These values
of effective diffusion coefficients and tortuosity, as well as their evolution
with dissolution, are coherent with other
previous laboratory measurement done on limestone samples
.
Permeability change
The changes in the sample permeability with time k(t) induced by the dissolution experiment is reported in
Fig. . Unsurprisingly, permeability increases due to
calcite dissolution, as previously mentioned by other authors
. The
permeability increase rate (dk/dt) changes
drastically at t≈8h, which is surely associated with the breakthrough
of the main dissolution path (wormhole).
From top to bottom: extracted skeleton on sample
Bvlbe, the formed wormholed in sample
Bvlaf (blue) replaced in sample
Bvlbe where the resolved connected porosity appears
in yellow, extracted skeleton on sample Bvlaf. For
the extracted skeleton images, the blue to red scale colours
corresponds to pore-size increase indicated in pixels (up to
60 pxls for sample Bvlbe and 234 pxls for sample
Bvlaf).
XMT analysisPorosity evolution
The total porosity calculated from the XMT images, on sample
Bvlbe (before dissolution) and Bvlaf
(after dissolution) is 17.13 and 22.80 %, respectively. As explained
in the methodology section, several steps have been performed with
similar parameters in order to estimate the possible error of
assessment on this crucial step. We used five different segmentation
results to calculate the resolved porosity before and after the
dissolution experiment. The results are presented in
Table . We can observe that decreasing the thresholding
value for the sample Bvlbe induces a resolved
phase underestimation up to 13 % whereas increasing the threshold
value only causes an overestimation less than 0.5 %. These
calculations indicate that the smallest threshold values were used to
estimate the resolved phase volume avoiding huge
underestimations. After the dissolution experiment, an error (±1
and 2) on the thresholding n value carries out fewer changes on the
resolved phase estimation. Increasing and decreasing by 1 the
threshold n value after the dissolution experiment has no effect on
the resolved phase estimation (error always lower than
0.26 %). Consequently, all the calculations done on the XMT images
were performed using the segmented images obtained
by setting the threshold value to n.
Sensitivity analysis of the threshold value n value on the relative
fraction of the resolved phase volume (RPV [%]) for samples Bvlbe and Bvlaf with the relative error ζ [%].
Time evolution of permeability k during dissolution
experiment of sample Bvl. The corresponding 3-D images of the
resolved porosity for sample Bvlbe and
Bvlaf are reported for the initial and final
percolation times.
Initially, the total porosity calculated on sample Bvlbe
is characterized by 60.82 % of large pores (resolved porosity) and
39.18 % of small pores, lower than the voxel size (sub-resolved porosity).
After the dissolution experiment, the total porosity is mainly characterized
by a high amount of large pores (resolved porosity) which represents
68.51 % of the total porosity (see Table ). The total porosity
increase is thus mainly controlled by the increase of the resolved porosity.
Figure shows the resolved and sub-resolved porosities for
samples Bvlbe and Bvlaf along the sample length (which
corresponds to the flow direction during the dissolution experiment). We can
observe that both resolved and sub-resolved porosity increase along the sample
length. These porosity increases are homogeneous along the sample except for
the first millimetres of the sample where the resolved porosity increase
faster than in the remaining part of the sample. The same phenomenon is
observed for the sub-resolved porosity, where its increase is higher for the
first 2 mm of the sample. Similar trends of porosity increase due to
carbonate dissolution have been monitored by previous authors. Nevertheless,
no conclusion on the dissolution patterns can be proposed as
and linked the homogeneous
porosity increase profile to homogeneous dissolution whereas
and observed wormhole
formation. These versatile conclusions are due to the complex structure and
pore geometry of the different limestone samples used. Visualising 3-D XMT
images, we can conclude that during the Bvl dissolution experiment a wormhole
was formed, which promoted a homogeneous porosity increase along the sample.
One of the advantages of using 3-D XMT images is the ability to
distinguish the total porosity from the effective one, or in other
words from the connected porosity. Performing a connectivity
computation, we can evaluate which part of the total porosity is
actually contributing to the fluid flow. Table indicates the
volumes of the resolved and sub-resolved phases are indicated as well
as the volume of the connected resolved and sub-resolved phases with
the corresponding fraction of the connected part. We can observe that
initially, the sample is mainly connected through the sub-resolved
phase, but after the dissolution experiment the resolved porous phase
becomes more connected and mostly contribute to the fluid
pathway. This increase in connectivity through the resolved porous
phase can be linked to the wormhole formation which represents
71.47 % of the connected resolved porous phase. Various other
small clusters compose the percolating resolved phase but none of
them is larger than 2 % of the total porous volume. The main
connected path after the dissolution experiment is imaged in
Fig. .
Porosity changes along samples Bvlbe and
Bvlaf. Both resolved and sub-resolved porosities
are plotted.
Chord-length function along x, y and z (P(x,y,z)) for
sample Bvl before and after the dissolution experiment.
Pore-size distribution
The pore-size distribution of the resolved porous phase was calculated for
sample Bvlbe and
Bvlaf using two different methodologies. We
calculated the pore-size distribution (Psd) as explained in
Sect. , computing the radius of the largest
inscribed sphere centred at every point of the pore space, provided
it is maximal for inclusion. We also estimated an equivalent pore-size
distribution by performing statistical measurement and calculating the
chord-length distribution functions (C-l). The chord-length
distribution functions for sample Bvlbe and
Bvlaf are plotted in Fig. for the 3
directions (x, y, z). Results from both methodology (Psd and
C-l) are summarized in Fig. where the pore volume
distribution is scaled according to the experimental RET thresholds.
Resolved (RPV) and total (TPV) phase volume [mm3] and
connected resolved (connected-RPV) and total (connected-TPV) phase volume
[mm3] with the respective connected resolved (connected-RF) and total
(connected-TF) fraction [%].
Figures and show that
initially, the sample is mainly composed of pores having small and
intermediate diameters. Most of the pores are smaller than
204.5 µm (69.73 %) in diameter and no anisotropy is
observed. After the dissolution experiment, the chord-length
distribution evolved and a certain anisotropy appears. In sample
Bvlaf, the amount of small to intermediate pore
diameter decreased significantly to 11.27 % of pores smaller than
204.5 µm in diameter. Larger pores were formed due to the
dissolution process. A significant amount of pores
presenting diameters comprised between 1 and 3 mm are measured
and pores having a diameter up to 20 mm in the x direction can
be found. It corresponds to the local face dissolution at the sample inlet inducing
large porosity increase (Figs. and ).
Porosity change along samples Bvlbe and
Bvlaf. Both resolved and sub-resolved porosities
are plotted.
The pore-size distribution (Psd) presents similar results than those
obtained by the chord-length function. Some discrepancies are observed
for sample Bvlbe for the largest pore
diameter. With the Psd analysis, the highest pores have a diameter
comprised between 70 and 204.5 µm, whereas with the chord-length function, we calculated initially lower proportion of these
pores and a higher one for the largest pores (diameter higher than
204.5 µm).
Time scale of the different methods used in this study for
both laboratory and XMT images approaches.
Diffusion coefficient
The effective diffusion coefficient Deff(XMT) before and
after dissolution is obtained by computing randomly distributed
particles in the rock pores following the method describe in
Sect. . Figure displays the
results of the two computations performed on
Bvlbe and Bvlaf. The curves
show the mean squared displacement (Msd) for an interval time step
i. For large time steps, the Msd is linearly dependent to the
effective diffusion coefficient . Linear
regression of the steady-state portion yields the effective diffusion
coefficient. The obtained effective diffusion coefficient for Bvlbe
and Bvlaf are summarized in Table . As expected and
previously reported in the literature
, the effective diffusion
coefficient increases after limestone dissolution experiments.
3-D image of the formed wormhole (blue) and initial non-connected
porosity presented in the final wormhole feature (grey).
Comparison and discussion
This section compares the different parameters characterized using, on the one hand,
classical laboratory measurements and on the other hand, XMT
images. We first compared laboratory measurements and computational
analysis duration for each parameter,
in order to evaluate which approach is the most time consuming. The complete
image processing described in Sect. was performed on a workstation equipped with two quad-core Intel Xeon CPU
X5560 @2.80 GHz and 192 GB DDR3 RAM. We can observe in
Fig. that globally, even if the computer used here is
not a high end build, the total analysis time for
Bvlbe and Bvlaf is much
shorter using image processing than the time analysis for performing laboratory
measurements. The total time needed to extract the different
parameters discussed in this article from the XMT images is 23 days,
whereas the time required to determine the same parameters using
laboratory measurements is 60 days. Moreover, some specific
processing (namely skeletonization) were performed using basic,
non-optimized implementations of classical algorithms that can be
found in open source software packages such as ImageJ. Besides, in
most cases, data extracted from XMT images provided more information
than the desired parameters studied in this article. Considering
porosity, only effective porosity can be determinated by
the experimental triple weighing method, whereas both effective and
total porosities can easily be calculated from the XMT images. Total
porosity can be a key parameter when chemical processes such as
dissolution occurred. The final porosity closely depends on the initial
effective porosity, the porosity created by dissolution and part of
the initially closed porosity that the dissolution process made
accessible. Porosity determination with helium pycnometry is a fast and
non-destructive alternative not used in this study.
However, the main drawback of the XMT image analysis is the high dependence
of all parameters on the voxel size. When using different laboratory
techniques to measure the desired parameters, various resolution scales can
be achieved. For example, the total fluid–rock interface determined by BET
measurement has higher resolution than the one determined by XMT images.
Specifically, the grain roughness as well as grains smaller than the XMT
resolution (7.42 µm) cannot be measured and the water–rock
interface area is underestimated a priori.
Nevertheless, for most of the determined parameters, good agreement is
observed between the data computed from XMT images and the data measured
experimentally. As observed in Table , the initial porosity
determined by the triple weighing method is only different from the effective
porosity extracted from XMT images by 0.74 %. After the dissolution
experiment, the estimated porosity is quite similar even if the difference is
more important. The final porosity determined by the mass balance calculation
is the lowest one (ϕf(chem)=19.37±0.56 %). As
explained before, the slight difference between the porosity obtained by
triple weighing and mass balance calculation can be explained by the
connection of initially non-connected pores to the new connected porosity by
the dissolution process. Actually, the porosity calculated by the mass
balance is only affected by dissolution. In this case, if only initial
effective porosity and mass balance are used, the final porosity can be
underestimated. Indeed, as observed in Fig. a
volume of 32 mm3 of non-connected pores is initially present in the final
wormhole feature. The porosity measured by the triple weighing method is
smaller than the one estimated from the XMT images. This difference can be
due to the formation of highly connected pathways (wormholes) percolating the
entire core. This very permeable fluid pathway in the sample after the
dissolution experiment represents an important water leak pathway where some
water can flow out of the sample, leading to an underestimation of the saturated
sample weighing. Regarding porosity measurement and analysis, we can conclude
that porosity calculated from the XMT images is the most reasonable, as we
can distinguish the effective one from the total one. Moreover the time
required to calculate the XMT porosities is much shorter than the one needed
by the triple weighing method.
The same conclusion cannot be drawn for the pore-size diameter
distribution. For this parameter, the retention curves acquisition
allows to classify pores with diameters smaller than the XMT images
voxel size (7.42 µm). Figure displays
the pore-size distributions before and after the dissolution
experiment determined by three different methods: measuring the
retention curves of the core sample via laboratory experiments (RET),
calculating the larger sphere inscribed in the pore space (Psd) and
measuring the chord-length distribution function (C-l) on the XMT
images. As already mentioned, we can see that the pore-size
distribution obtained by the retention curves analysis allows to
determine the quantity of pore with radii down to
0.48 µm. Results from Psd and C-l are quite similar. In
sample Bvlbe, the C-l methods determined a higher
amount of very large pores, whereas the Psd methods estimated larger
content of pores with radii comprised between 34.99 and
102.27 µm. After the dissolution experiment, for both
techniques based on the XMT images the fraction of the largest pores
increases significantly. This increase of the amount of large pore is
in agreement with the dissolution process and the formation of
a preferential flow path (wormhole). Comparing with the RET method, we
observed that the RET pore-size distribution underestimates the
largest pores and the smallest pores are certainly overestimated. This
large difference can be attributed to the basic definition of a pore
. Indeed, during retention curves
measurement, the volume of pore estimated for a given suction pressure
correspond to the volume of extracted water through a corresponding
throat. Consequently, this laboratory technique gives a mixed
estimation between pore and throat distribution, underestimating the
largest pores and overestimating the smallest ones. It might be
interesting to simulate drainage and imbition experiments using the XMT
images as previously done by and
in order to compare the laboratory RET
measurements in a further study. To summarize, one should note
that the pore-size distribution obtained by the retention curves indicates the capillary
pressure needed to extract a specific fluid volume, but without any
information about the amount of pores containing this fluid volume and
the respective pore diameters. The two other methods used here to
extract the pore-size distribution using the XMT images allow us to
determine the pore-size diameter at each voxel point with some
anisotropy information (C-l technique). Nevertheless, with these two
methods, we don't have any knowledge about the connectivity and the
accessibility to these pores. Using the extracted skeleton, we can
extract both pore and throat distributions and localize them in the
3-D sample. This fourth method gives the best pore and throat size
distribution and their localization (Fig. ).
The third main parameter measured experimentally and using XMT images is the
effective diffusion coefficient. By the utilization of a laboratory through diffusion experiments,
determination of the diffusion coefficient is time consuming
(Fig. ) and strongly depends on sample length as the diffusion
time increases with the squared length. Using the XMT images, the calculation
of the effective diffusion coefficient is very efficient and performed in
less than 9 min. The results of both methods are presented in
Table . The values obtained by laboratory measurements and
statistical modelling are similar. With both techniques, the effective
diffusion coefficient increased after the dissolution experiment by 1 order
of magnitude. Consequently, the tortuosity coefficient decreased after the
dissolution experiment and both values for Bvlbe and
Bvlaf are similar for laboratory measurements and image
based computations. The statistical estimation using XMT images is a good
option to determine the effective diffusion coefficient as the time needed is
1500 times faster than the utilization of a laboratory through diffusion experiments.
Conclusions
In this paper, we have shown that microtomographic imaging hardware
and computational techniques have progressed to the point where
properties such as effective diffusion coefficient, conductivity and
pore-size distribution can be calculated on large three-dimensional
digitized images of real core rock sample. We demonstrated that for
most of the parameters studied here, the values obtained by computing
XMT images are in agreement with the classical laboratory
measurements. For some parameters, such as the porosity, the
computational one is the more informative, as one can calculate both
total and effective porosity. As discussed here, when dissolution
process occurs, the knowledge of the total porosity can be necessary.
As the definition of pore is highly discussed in the scientific
community, we observed that the pore-size distributions obtained by
XMT images and laboratory experiments are slightly different. We
highlighted advantages and limitations of both approaches: the RET
measurement allows to determine the accessible volume for a given
capillary pressure, whereas Psd and C-l methods extract the maximum
pore volume locally without information of its accessibility.
Concerning the effective diffusion coefficient, we observed that both
approaches are valuable and similar results are
obtained. Nevertheless, the duration of a laboratory through-diffusion
experiment is much longer than the time required by the computational
option (about 1500 times longer).
As a conclusion, computing XMT images to determine transport,
geometrical, and petrophysical parameters provide similar results than
the one measured at the laboratory in only 23 days instead of
60 days for the laboratory option. Moreover, the studied sample presents
both resolved and sub-resolved porosities, which would be the case for any
other type of natural or synthetic porous material, whatever the acquisition
technique. Thus, the framework developed in this work is relevant and can be
easily applied in many contexts.
Furthermore, new developments are expected in a near future favouring
microtomographic imaging at higher resolutions with faster acquisition times
allowing dynamical effects to be imaged,
(). Further
developments using the extracted skeleton will also allow us to extract the
accessible pore volume and the capillary pressure needed to ingress.
Acknowledgements
We would like to acknowledge Arnaud Chabanel from EURL Thomann Hanry
(Carriéres de Vers Est) to provide us the rock sample and
Paul Tafforeau from ESRF for the X-ray microtomography images
acquisition. L. Luquot is funded by the Juan de la Cierva fellowship
(MINECO, Spain).
Edited by: H. Steeb
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