Calculating structural and geometrical parameters by laboratory experiments and X-Ray microtomography : a comparative study applied to a limestone sample

Introduction Conclusions References Tables Figures


Introduction
Characterizing the pore rock :::: 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 CO 2 in case of geological sequestration for example. Porosity is one of the most important petrophysical parameters as it indicates : a :::: key ::::::::::::: petrophysical ::::::::::: parameters ::::::::: indicating : the total volume of oil, gas or water that can be contained in a reservoir. It is essential to differentiate total porosity and ::: the ::::: total :::::::: porosity ::::: from ::: the : open connected or effective porosity : , : which is the real accessible porosity ::::::: fraction ::: of :::::::: porosity :::::::::: accessible : by any fluid. Connectivity as well as permeability and tortuosity indicate the extraction or injection facility ::::: allow :: 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 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 (Archie, 1942;Fick, 1855). 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 (Gillham et al., 1984), contaminant remediation (e.g., Becker and Shapiro, 2003), CO 2 geological storage (Hatiboglu and Babadagli, 2010), and oil and gas recovery (Cui et al., 2004(Cui et al., , 2009). Diffusion of hazardous gas or liquid chemicals in construction materials has also become a concern for public health and security (Nestle et al., 2001a, b). Therefore, knowledge of diffusion processes and rates in rocks is important to a better understanding of these problems ::::: 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 realized ::::::::: 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.
3 Currently, a lot of petrophysical, geometrical, transport and mechanical parameters are computed using 3-D X-ray microtomography (XMT) images (Alshibli and Reed, 2012). 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 (Remeysen and Swennen, 2008). 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 (Ketcham and Carlson, 2001). 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 (Arns et al., 2001), diffusivity (Schwartz et al., 1994;Knackstedt et al., 2006;Promentilla et al., 2009;Gouze and Luquot, 2011), permeability (Martys et al., 1999;Arns et al., 2004Arns et al., , 2005a, pore size distribution (Dunsmuir et al., 1991;Prodanovic et al., 2010;Garing et al., 2014;Luquot et al., 2014a) and linear elasticity (Roberts and Garboczi, 2000;Arns et al., 2002;Knackstedt et al., 2006) can be calculated on large three-dimensional digitized grids (over 10 9 voxels). More details on this technique, acquisition and data computing step can be found in Taina et al. (2008); Cnudde and Boone (2013) and Wildenschild and Sheppard (2013). In geosciences, the internal structure of a great diversity of geological samples has been examined by radiographic imaging mainly in the last 50 years (Calvert and Veevers, 1962;Hamblin, 1962;Baker and Friedman, 1969;Herm, 1973;Sturmer, 1973;Bjerreskov, 1978;Monna et al., 1997;Louis et al., 2007;Schmidt et al., 2007).
Yet, to our knowledge, no study has been conducted to explores ::::::: explore : the potential links that can be made between computed variables :::::::: variables :::::::::: computed : from XMT images and those that are traditionally measured in the laboratory for a limestone core rock sample, before and after a dissolution process. Only one similar study was done by Lamande et al. (2013) 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.

4
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, surface-to-volume ratio, permeability, pore size diameter distribution, effective diffusion coefficient, and tortuosity. These parameters are those needed for numerical modeling to evaluate oil and gas deposit volume and extraction flow rate for example. Measuring the pore structure of the ::::::::::: Quantifying ::: the ::::: pore :::::::: network :::::::::::::: characteristics ::: of :: a : same sample before and after a dissolution experiment allows to apply our methodology to two different samples ::::: 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.

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 Luquot et al. (2014b) and Roetting et al. (2015). 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 Reatto et al. (2008) using 6 speed increments up to 4500 rpm. The maximum suction that can be applied to the sample at 4500 rpm is 213 kPa.
(1), we can convert the measured θ(P ) into an equivalent θ(r p ) curve. This curve is actually a cumulative poresize distribution; the water content θ(r p ) indicates the combined volume of all pores with opening radius less than r p . The measurements were performed twice in the two directions to evaluate the sample anisotropy. Six days were necessary to acquire two times ::::: twice the retention curves in both directions and dry and saturate again 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 Boving and Grathwohl (2001) and Luquot et al. (2014a).
Through-diffusion experiments were performed to determine the effective diffusion coefficient and tortuosity/constrictivity ratio before and after the dissolution experiment (Li and Gregory, 1974). The same methodology as the one developed by Luquot et al. (2014b) 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 mol L −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 (NaN 3 ) 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 Luquot et al. (2014b). The aqueous diffusion coefficient for 7 iodide (D aq = 1.86 × 10 −9 m 2 s −1 ) (Robinson and Stokes, 1959) was used to calculate the effective diffusion coefficient D eff . The effective diffusion coefficient was calculated using the equation (van Brakel and Heertjes, 1974;Crank, 1975): 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 C 0 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 van Brakel and Heertjes (1974): 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: Through-diffusion experiment are time consuming because of the slow diffusion rate of a liquid tracer, even for porous limestone and sandstone samples (Boving and Grathwohl, 2001). One to two weeks are needed for each measurement.

Laboratory percolation experiment
We performed the flow-through experiment using the apparatus presented in Luquot et al. (2014b). 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 8 permeability using Darcy's Law. We injected an acidic solution through sample Bvl at constant flow rate Q = 16 mL h −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 g L −1 of acetic acid with 2.184 g L −1 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 cm 3 , or some 90 pore volumes of sample Bvl (initial pore volume = 1.62 cm 3 ).
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: : . The volume of dissolved calcite (∆V calcite (t)) is calculated as: where υ is the calcite molar volume (3.7 × 10 −5 m 3 mol −1 ), and ∆C Ca is the difference between the outlet and inlet calcium concentration. Therefore, we can calculate the samplescale porosity change during the percolation experiment using the following equation: where V is the total sample volume and φ 0 is the initial sample porosity. The error (∆C Ca ) in the change of calcium concentration was estimated using the Gaussian error propagation 9 method (Barrante, 1974). 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. 2.1.2.

Images acquisition
We acquired the 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 pxls. 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 µm 3 . 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 around ::::: 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 Bvl be and Bvl af .

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
One ::::::::::::: Segmentation :: is :::: one : of the most important task :::: step in image analysisis segmentation. It consists in gathering voxels belonging :::: that :::::::: belongs to a same object and assigning then ::::: them a single common value. Voxels identified to the matrix constitute the solid phase and ::::: which : have the highest intensity and appear in the brightest gray levels. Pores with size larger than the voxel size (here, 7.42 µm) are entirely captured by the camera and appear in darkest gray levels in the image, forming a phase called ::::::: referred :: to ::: as : void phase or resolved porous phase. We call ::::: name : subresolved porous space the area appearing in intermediate gray levels and formed by matrix (calcite in the present case) and pores of size smaller than the voxel size. Note that this phase is sometimes denoted by the ambiguous term microporous phase :::::::::::: "microporous ::::::: phase" whereas resolved porous phase is often called macroporous phase :::::::::::: "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.
In terms of numerical core analysis, computing the porosity requires to determine the relative fraction of the void phase volume and to estimate the pore volume in the subresolved porous space. An image ::: first : segmentation algorithm based on a region growing method was used to isolate the void phase. An additional image segmentation was then realized :::::::::: conducted to isolate the subresolved porous space and compute its volume fraction in the 11 sample voxel. We can define the sample subresolved porosity Φ S as: where ξ x and Φ x respectively denote the volume fraction and the porosity of the subresolved porous space. The total porosity is then given by: where Φ R 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 Φ x . We first compute the mean grayscale values corresponding to solid and void phases, respectively denoted by G m and G v . Following (Mangane et al., 2013) and (Luquot et al., 2014a), the gray level value of a voxel belonging to the subresolved porous phase is linearly related to Φ x as follows:

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. This enables to assess the robustness of the segmentation parameters determined by the user.

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 (Mecke, 2000;Vogel et al., 2010) . Here, we focused on the surface-to-volume ratio sometimes referred as specific surface. Statistical measurements such as chord length distribution functions (Torquato, 2002;Luquot et al., 2014a) 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 (Sen, 2004). Consider a large number N of (virtual) diffusing particles, initially uniformly distributed in the void phase, and randomly moving following a Brownian motionThe : , :::: the diffusion coefficient D(t) characterizes their ability to disperse in the void phase, probing its structure. Denote ::: We ::::::: denote : by x i (t) the position of the ith particle at time t and σ(t) 2 their mean square displacement, that is: Then according to Einstein (1956): If d 0 denotes the diffusion coefficient in an unbounded domain and τ the tortuosity, then when t → ∞, D(t) tends to an asymptotic value D ≈ d 0 /τ .

Skeleton and properties
The skeleton of a three-dimensional object is a one-dimensional reduction, centered 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 Siddiqi and Pizer (2008). 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 Lee et al. (1994) was used for this work.

Results
The temporal evolution of the increase in calcium concentration, ∆C Ca , during the percolation experiment as well as the inlet and outlet fluid pH are presented in Fig. 1. Dissolution reaction occurred during the percolation experiment. Indeed, the ∆C Ca is always positive (∆C Ca > 0) indicating Ca release in the outlet fluid. Moreover, the outlet pH ( Fig. 1) is higher than the inlet one which corroborates proton consumption and thus calcite dissolution , as described as follow: + + :::: (Eq. :::::: (R1)). Dissolution reaction may induce porosity increase and other geometrical, structural and hydrodynamical parameter changes. Evaluating and characterizing these changes are primordial :::::::: essential : to develop predictive models of reactive-transport processes such as those occurring for example during CO 2 geological storage, fracking processes, oil and gas exploitation, acid mine drainage or seawater intrusion among others.
3.1.2 Pore size distributions ::::::::::: 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. 2.1.2. We measured the retention curve twice. We first placed the sample in order to apply the centrifuge force :: by :::::::: draining ::: the ::::::: sample ::::: both in the flow direction of the percolation experiment and then we placed the sample in the opposite direction ::: and ::::::::::: counterflow ::::::::: direction :: in ::::: order : to evaluate the pore-size :::: pore ::::: size anisotropy. If the pore-size distribution was heterogeneous, then we should get different retention curves due to the gradient of the 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 Luquot et al. (2014b). Initially, the pore-size distribution is homogeneous; the initial retention curves are similar in both sample orientations (Fig. 2). 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. 2, 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. 3.
We can note that after the dissolutionexperiment :::: 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. 2) that the second huge difference in ::::: major :::::::::: difference :::::::: between : the retention curves , between before and after the dissolution experiment, :::::::::: dissolution : appears for pore radii 10.23 < r p < 34.99 µm (suction between 5.11 et 17.47 kPa). After the dissolution experiment, less pores of such radii are presented ::::::: present : through the sample, indicating that most of the dissolution occurred in these pores. Moreover, some differences are observed ::: 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. We can observe that the differences are not so large ::::: They ::: are ::::::::: however :::::: minor :::::: when : compared with other experiments with strong dissolution localization (Luquot et al., 2014a). Most of the difference :::::::::::: discrepancy is visible for intermediate and smallest pores (0.48 < r p < 10.23 µm). 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 smaller ::::: lower when the sample is placed opposite to the flow direction, implying that some dissolution occurred even ::::::::: 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 slightly higher than before the experiments, implying a small decrease in pore size at the outlet. This mechanism of pore clogging is marginal here.  Table 2. The slope of the iodide increment after :::: After : the dissolution experimentincreases, indicating an increase of , : the diffusion coefficient . The latter ::

Porosity evolution
The total sample porosity calculated from the XMT images, on sample Bvl be :::::: (before ::::::::::: dissolution) :::: and :::: Bvl af :::::: (after ::::::::::: dissolution) is 17.13 % , whereas the one calculated on sample Bvl af yields :::: 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 3. We can observe that decreasing the thresholding value for the sample Bvl be in-18 duces a resolved phase underestimation up to 13 % whereas increasing the threshold value only causes an overestimation less than 0.5 %. These calculations indicate that we used 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 less 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 using the thresholding n value :::::: setting :::: the ::::::::: threshold :::::: value :: to :: n.
Initially, the total porosity calculated on sample Bvl be is characterized by 60.82 % of large pores (resolved porosity) and 39.18 % of small pores, lower than the voxel size (subresolved porosity). After the dissolution experiment, the total porosity is mainly characterized by a large :::: high amount of large pores (resolved porosity) which represents 68.51 % of the total porosity (see Table 2 : 3). The total porosity increase is thus mainly controlled by the increase of the resolved porosity. Figure 7 shows the resolved and subresolved porosities for samples Bv lbe and Bv laf along the sample length (which corresponds to the flow direction during the dissolution experiment). We can observe that both resolved and subresolved porosities ::::::: porosity : increase along the sample length. These porosity increases are homogeneous along the sample except for the first millimeters of the sample where the resolved porosity increase faster than in the remaining part of the sample. The same phenomenon is observed for the subresolved porosity, where its increase is higher in ::: 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 Luquot and Gouze (2009) and Menke et al. (2015) linked the homogeneous porosity increase profile to homogeneous dissolution whereas Smith et al. (2013) and Luquot et al. (2014a) 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 Bvl dissolution experiment a wormhole have been formed promoting 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. In Table 4 , ::::::::: indicates the volumes of the resolved and subresolved phases are indicated as well as the volume of the connected resolved and subresolved phases with the corresponding fraction of the connected part. We can observe that initially, the sample is mainly connected through the subresolved 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 composethe :::::::: compose :::: the percolating resolved phase but none of them is higher ::::: larger : than 2 % of the total porous volume. The main connected path after the dissolution experiment is imaged in Fig. 5. 3.2.2 Pore size distributions ::::::::::: distribution Calculation of the :::: The pore size distribution was conducted on : of : the resolved porous phase on :::: was :::::::::: calculated ::: for sample Bvl be and Bvl af using two different methodologies. We calculated the pore size distribution (Psd) as explained in Sect. 2.2.2, computing the radius of the largest inscribed sphere centered at every point of the pore space, provided it is maximal for inclusion. We also estimate ::::::::: 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 Bvl be and Bvl af are plotted in Fig. 8 for the 3 directions (x, y, z). Results from both methodology (Psd and C-l) are summarized in Fig. 3 where the pore volume distribution is scaled according to the experimental RET thresholds.
The pore size distribution (Psd) presents similar results than those obtained by the chord length function. Some discrepancies are observed for sample Bvl be for the largest pore diameter. With the Psd measures :::::::: 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). The effective diffusion coefficient D eff(XMT) before and after dissolution is obtained by computing randomly distributed particles in the rock pores following the method describe in Sect. 2.2.2. Figure 9 displays the results of the two computations performed on Bvl be and Bvl af . 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 (Einstein, 1956). Linear regression of the steady state portion yields the effective diffusion coefficient. The obtained effective diffusion coefficient for Bvl be and Bvl af are summarized in Table 2. As expected and previously reported in the literature (Gouze and Luquot, 2011;Luquot et al., 2014b), the effective diffusion coefficient increases after limestone dissolution experiments.

Comparison and discussion
Here we compare :::: This ::::::: section :::::::::: compares :::: the different parameters characterized using, on the one hand, classical laboratory measurements and on the other hand, XMT images. 21 Before considering these parameters, we ::: We :::: first : compared laboratory measurements and computational analyses durations :::::::: analysis :::::::: duration for each parameter, in order to evaluate which approach is the most time consuming. All :::: The ::::::::: complete image processing described in Sect. 2.2.2 have been 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. 10 that globally, even if the computer used here is not a high end build, the total analysis time for Bvl be and Bvl af is much shorter using images ::::: image : processing than 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 for instance, 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 occured. The final porosity closely depends on the initial effective porosity, the porosity created by dissolution and part of the initial :::::: 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. : Nonetheless :::::::: However, the main drawback of the XMT image analysis is the high dependence :: of ::: all :::::::::: parameter on the voxel sizefor all parameters. 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 a priori underestimated.
Nevertheless, for most of the determined parameters, good agreement is observed between the processed data acquisition and the one :::: data ::::::::: computed ::::: from ::::: XMT :::::::: images :::: and ::: the ::::: ones : measured experimentally. As observed in Table 2, the initial porosity determined by the triple weighing method is much closer to :::: only :::::::: different ::::: from : the effective porosity measured :::::::: extracted : from XMT images . We indeed calculated a :: by : 0.74 %difference. 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 (φ f(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. 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 is :::::::::: represents : an important water leak pathway where some water can flow out the sample, leading to an underestimation of the saturated sample weighing. So, on the porosity parameters :::::::::: Regarding ::::::: porosity :::::::::::::: measurement :::: and :::::::: analysis, we can conclude that the porosity calculated from the XMT images is the more ::::: 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.
We cannot reach the same conclusion :::: 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). In Fig. 3 , we can see :::::: Figure : 3 :::::::: 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 Bvl be , the C-l methods determined a larger :::::: 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 (Nimmo, 2004;Hinai et al., 2014). 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 Knackstedt et al. (2006) and Prodanovic et al. (2010) 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 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 distribution and localize them in the 3-D sample. This fourth method gives the best pore and throat size distribution and their localization (Fig. 5).
The third main parameter measured here :::::::::::::: experimentally :::: and :::::: using ::::: XMT ::::::: images : is the effective diffusion coefficient. By laboratory through diffusion experiments, determination of the diffusion coefficient is time consuming (Fig. 10) 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.