Geoelectrical and hydro-chemical monitoring of karst formation at the laboratory scale

. Ensuring sustainable strategies to manage water resources in karst reservoirs requires a better understanding of the mechanisms responsible for conduits formation (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) dissolution (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) features in the rock mass and the development of detection methods for these hydrological and geochemical processes. In this study, we monitored the electrical conductivity of two limestone core samples during controlled dissolution experiments. We interpret the results with a physics-based model describing the porous medium as effective structural parameters that are tortuosity and constrictivity. We obtain that constrictivity is more affected 5 by calcite dissolution compared to tortuosity (cid:58)(cid:58) the (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) electrical


Samples properties and petrophysical characterization
The two core samples of this study are part of a wider published data set (Leger and Luquot, 2021; ?) :::::::::::::::::::::::::::::::: (Leger and Luquot, 2021; . They were cored into an Oxfordian crinoid limestone from the Euville quarry in Nancy, France. The two samples are named E04 and E05. The cores have a length of 32 and 31 mm, respectively, and a diameter of 18 mm. They are surrounded with epoxy resin and PVC pipe for a total diameter of 25 mm (i.e., one inch).

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In addition, X-Ray MicroTomographic (XRMT) images of the two core samples are collected with a pixel size of 12 µm and are analyzed with a homemade :::::: in-house : software to obtain complementary information about the samples structure ::::::: structure :: of ::: the ::::::: samples. In particular, a pore size distribution is obtained from a probabilistic method displaying the chord lengths.

Experimental setup and monitoring
The experimental protocol followed here is the same as the one used in ? and ? :::::::::::::::: Leger et al. (2022a) ::: and ::::::::::::::::: Leger et al. (2022b) 110 . Under atmospheric pressure and temperature conditions, :: an : acidic solution is injected through the samples. The injected solution is composed of water previously balanced with samples of the same rock type mixed with acetic acid (CH 3 CO 2 H), and sodium acetate (CH 3 CO 2 Na). The resulting acid concentration is about 10 −2 mol L −1 buffered with a pH of 4. Figure 1 displays the homemade experimental device used for the percolation experiments.
At the outlet, fluid samplings of 5 mL are continuously collected during day time and punctually during night time ::::::: nighttime.
pH, brine conductivity : , : and cation concentrations are measured for each fluid sample. From calcium (Ca 2+ ) concentration 140 monitoring, we can determine the mass of dissolved calcite through time, and thus, obtain the time variations of the rock sample porosity ϕ (-). At time t i (s) we have ϕ ti = ϕ ti−1 + ϕ dt , where ϕ dt (-) is the porosity difference over the time laps is the core sample volume and V dt (m 3 ) is the pore volume difference over time laps dt. The latter is calculated as follows To relate the electrical monitoring with the porous medium microstructure, we use the Bundle Of Capillary Tubes (BOCT) approach (e.g., Vinogradov et al., 2021). : It uses a bundle of parallel tortuous and constrictive capillaries as a conceptualization of the pore structure (e.g., Guarracino et al., 2014;Rembert et al., 2020). The conceptual geometry is represented in Fig. 2. The BOCT model considers N capillaries , that do not intersect and run with the same orientation. The capillaries are confined within a cylindrical model of length L (m) and radius R (m). They are permitted to have different radius r (m), but 155 a single length l. Thus, the tortuosity τ = l/L (-) is an effective property. Each tortuous and constrictive capillary presents a varying radius r(x) (m) defined with the following sinusoidal expression, wherer (m) is the average capillary radius, r ′ (m) the amplitude of the radius size fluctuation, and λ (m) is the wavelength. The parameter a (-) is the pore radius fluctuation ratio, also called the constriction factor. It is defined by a = r ′ /2r, thus ranging 160 from 0 to 0.5, corresponding to cylindrical pores (r(x) =r) and periodically closed pores, respectively.
From Jackson's formulation based on capillary radius (Jackson, 2008), the number of capillaries is expressed as with n(r)dr the number of capillaries of radius betweenr andr + dr. As many geologic materials present a skewed pore size distribution (e.g., Bennion and Griffiths, 1966;Dullien, 1992), we use a pseudo-fractal law to describe the distribution of 165 capillaries throughout the model (Jackson, 2008;Vinogradov et al., 2021) where 0 < m j < ∞ is a dimensionless exponent.r min andr max (m) are the extreme average pore sizes. They are estimated from the chord length computation using 3D tomography of the samples before and after the dissolution experiments : . Thus, N is calculated for average pore sizesr comprised between these two extremes. The normalization coefficient D = 1 since 170 the pore size distribution from the chord lengths estimation is normalized and expressed in percent. The minus sign of −dN implies that the number of pores decreases when the average radius increases (Soldi et al., 2017;Thanh et al., 2019;Yu et al., 2003).

Electrical properties
We follow the approach of Rembert et al. (2020), which consists in :: of the expression of the electrical properties at the pore scale 175 before upscaling them considering the bundle of capillaries as an equivalent circuit of parallel conductances. This approach is valid for a negligible surface conductivity and leads to where f (-) is the constrictivity of the porous medium varying from 0 to 1, corresponding to periodically closed capillaries and cylindrical capillaries, respectively. This approach quantifies the effect of pore throats and pore bodies on the medium electrical 180 conductivity.
Neglecting the surface conductivity in Eq. , Eq. (10) leads to the following definition of the formation factor F = τ 2 ϕf .
The generation of dissolved calcium ( Fig. 3d : e) and the increase of porosity (Fig. 3a), permeability (Fig. 3b), and water conductivity difference (Fig. 3e) indicate that calcite dissolution is the main chemical reaction for both experiments. However, calcium difference shows a : slow decrease until the end of the experiment, which can also be seen in the pH curves and the water conductivity difference curves ( Fig. 3c and e), and indicates that the dissolution regime was initially strong and got weaker 215 over time.
Before the percolation time, one can observe that :: the : E05 permeability curve presents variations attributed to complex effects of grain displacement and transitory pore clogging. After percolation time, both E04 and E05 permeability curves 225 present high-frequency noise corresponding to the limit of detection of the sensors of differential pressure. In order to :: To avoid these instabilities, the curves are filtered. After a comparison with the moving average and Butterworth filters, we choose the Savitsky-Golay filter, since it offers a better adjustment. We observe that the filtered curves are well superimposed on the curves of the measurements, except between 10 and 12 h for the dissolution of E05, because of the transient complex effects. The final permeability of E05 is higher than the last value of the temporal series. This discrepancy is also due to the limit of detection of 230 the sensors of differential pressure. However, we did not take this higher value into account when filtering, to be in accordance with the temporal series.
The sample conductivity increases during both experiments (Fig. 3d). At first glance, this increase could be attributed to the increase of the pore water conductivity (Fig. 3e) instead of calcite dissolution. However, one can observe that the normalized formation factor (Fig. 3f) decreases during the experiment. Thus, the sample conductivity and formation factor vari-235 ations are linked to calcite dissolution. It can be observed that for sample conductivity and formation factor, the variations are monotonous, with a rapid initial change followed by a gentler slope until the end of the experiment. The initial slopes are correlated to higher calcium concentrations at the beginning of the experiment (Fig. 3d : e) and are, thus, due to initial strong dissolution regimes. The sample conductivity and formation factor smooth variations show that they are more impacted by porosity increase than by permeability variations (Fig. 3a and b). :::: Over ::: this ::::: time ::::: range, ::: the :::::::::: acquisition :::: rate :: of ::: the ::::::: sample 240 :::::::::: conductivity :: is :::: high :::::::: compared ::: to ::: the :::: time ::::::::: variations. ::: We ::::::: estimate ::: the :::::::: standard :::: error :::::: around : 10 −3 µS m −1 : . Furthermore, the comparison of the results between E04 and E05 shows that with equal water conductivity and close permeability values before the breakthrough, higher porosity induces higher sample conductivity. Compared to the petrophysical measurements, the normalized formation factor curves show good agreement with the measured amplitude difference between the initial and final valuesand : , represented as the final normalized formation factor (Fig. 3f).
3.3 Influence of rock structure on electrical signature: ::: the :::::: major :::: role :: of :::::::::::: constrictivity 285 We have shown in the previous section that the pore size distributions of both samples were successfully describable using pseudo-fractal laws. Thus, we can apply the relationship relating the formation factor and the porosity developed in Eq. (11).
We can see in Fig The constrictivity varies over different ranges of values between E04 and E05 ( Fig. 5c and d). For E05, f is lower by half (i.e., more constrictive) compared to E04, which is in agreement with E04 higher porosity values. In contrast, the evolution 295 with porosity follows roughly the same law for these two samples. This indicates that we are working on two samples of the same rock type with similar pore shapes.
The tortuosity varies little during the experiment for both samples, although it is more pronounced for E05 ( Fig. 5e and f).

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3.4 Influence of hydrodynamic properties on rock structure and electrical signature

Impact of the experimental conditions and permeability changes
In this section, we are interested in highlighting :::: focus ::: on : the link between the evolution of hydrodynamic and electrical properties. The permeability is plotted as a function of the formation factor and porosity in Fig. 7. We can first note that the permeability varies over the same ranges of values for both samples, while E04 has a larger porosity (and lower formation 335 factor) compared to E05. As discussed in the literature (e.g., Guarracino et al., 2014;Soldi et al., 2017;Rembert et al., 2020), the pore throats play a primary role on :: in the hydrodynamic behavior of porous media. According to the model used in this study, the studied samples follow the same distribution for constrictivity (Fig. 5), which is consistent with the similarity in permeability. It can be observed that before the breakthrough, the permeability increases little, while the formation factor and the porosity 340 vary strongly. Then, after percolation, the trends are reversed, ::: and : the permeability increases strongly compared to the lower variation of the porosity and the formation factor. This behavior change is very clear for both samples and indicates that monitoring electrical properties allows us to be sensitive to the impact of dissolution on the porous medium long before the sample is percolated.

The characteristic pore size defined through the Johnson length
The comparison of the permeability of E04 and E05 reveals that they are samples with similar initial hydrodynamic properties related to their identical rock type (Table 1). However, their difference of :: in porosity leads to different ranges for the formation factor, and the different imposed hydrodynamic regimes lead to higher final permeability for E05. Moreover, the use of ::::: using 360 a normalized formation factor (Fig. 3) enables to compare :: f) :::::: enables :::::::::: comparing trends but does not allow to quantitatively interpret :::::::::: quantitative ::::::::::: interpretation :: of : the electrical monitoring in term :::: terms : of rock structure. Thus, as recommended by Niu and Zhang (2019), we use the Johnson length as an effective pore size (Fig. 8), that enables to compare :::::::: comparing : the behavior of both samples E04 and E05 regarding other data sets from the literature (Niu and Zhang, 2019;Vialle et al., 2014). The series from Niu and Zhang (2019) comes from calcite dissolution simulation of a digital image of a carbonate mud-365 stone (Wellington Formation, Kansas, USA), that has an initial porosity of 0.13. The fluid transport is assumed as advection dominated (P e > 1 :::::: Pe > 1), and under this transport condition, referring to Pereira Nunes et al. (2016), they impose a transportlimited case, related to wormholing :::::: conduit :::::::: formation, in which the reaction at the solid-liquid interface is limited by the diffusion of reactants to and from solid surfaces (P eDa > 1, where Da = k r /u ::::::::: PeDa > 1, ::::: where :::::::::: Da = k r /u is the so-called Damköhler number, and k r is the local reaction rate).

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The data points from Vialle et al. (2014) correspond to bioclastic limestone (Estaillades limestone, France), composed of more than 99% calcite, and of initial porosity of 0.29. The values for P e ranges :: Pe ::::: range : from 18.5 (for the smaller grains) to 37 (for the larger ones) and Da = 3486 ::::::::: Da = 3486. Meaning that for this data set, the dissolution is also transport controlled and leads to dominant wormhole formation.
Data availability. The data used in this study are available in the Zenodo repository (https://zenodo.org/record/6908522).
Appendix A: Accuracy of the determination of the pseudo-fractal exponent The accuracy of the pseudo-fractal computation for the determination of the pore size distribution is controlled through the error ϵ defined in Eq. (15). Since dissolution does not generate notable variations of the pseudo-fractal exponent m j , we confront the computed error to a range of value m j in Fig. A1. We obtain V-shaped curves for each data set, showing that m j 430 is well-determined for each sample, before and after percolation experiments. Figure A1. The error of the pseudo-fractal law against the exponent mj.