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.

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=-2σcos⁡αP, where σ is the surface tension (72.3±0.5 mNm-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=1.86×10-9 m2s-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.

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=16 mLh-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 =1.62 cm3).

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 (3.7×10-5 m3mol-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 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.

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.

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 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.

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).

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.

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.

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 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.

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.

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 .

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. .

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<34.99 µm (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.

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<10.23 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 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.

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 .

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≈8 h, which is surely associated with the breakthrough of the main dissolution path (wormhole).

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.

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. .

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.

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 ).

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).

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.