# Mechanical response of cement and shale admixtures under cyclic triaxial loading monitored by in-situ synchrotron micro-computed tomography

Dag W. Breiby, Aldritt Scaria Madathiparambil, Fazel Mirzaei, Kim Robert

Tekseth, Benoît Cordonnier, Nicolaine Agofack, Pierre Cerasi, François

Renard, Basab Chattopadhyay

► To cite this version:

Dag W. Breiby, Aldritt Scaria Madathiparambil, Fazel Mirzaei, Kim Robert Tekseth, Benoît Cordonnier, et al.. Mechanical response of cement and shale admixtures under cyclic triaxial loading monitored by in-situ synchrotron micro-computed tomography. Cement and Concrete Research, 2024, 186, pp.107682. 10.1016/j.cemconres.2024.107682. hal-05099478

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# Mechanical response of cement and shale admixtures under cyclic triaxial loading monitored by in-situ synchrotron micro-computed tomography

Aldritt Scaria Madathiparambil<sup>a</sup>, Fazel Mirzaei<sup>a</sup>, Kim Robert Tekseth<sup>a</sup>, Benoît Cordonnier<sup>b,c</sup>,
Nicolaine Agofack<sup>d</sup>, Pierre Cerasi<sup>d</sup>, François Renard<sup>b,e</sup>, Basab Chattopadhyay<sup>a</sup>,
Dag W. Breiby<sup>a,f,*</sup>

## ARTICLE INFO

### Keywords:

Dynamic computed tomography

Fatigue

Cement composite

Shale

## ABSTRACT

Understanding the mechanical behaviour of natural soils as mixed with cement for stabilization is crucial for civil engineering developments. The response of cement-soil admixtures when subjected to cyclic loads is a largely unexplored topic, despite the importance of understanding fatigue in these ubiquitous construction materials. We present cyclic loading experiments on Portland cement mixed with fragmented shale fragments using triaxial testing, monitored with synchrotron-based µCT. Through digital volume correlation (DVC), the temporal evolution of the displacement, volumetric, and von Mises equivalent strain fields were obtained. We observed in detail the fatigue damage evolution during cyclic loading and found that following high-strain deformation of the much softer shale fragments, the ultimate failure of the samples occurred in the adjacent cement matrix. The failure mechanism under periodic stress and its relevance for accelerated laboratory testing of slow degradation long-term processes are of key importance to technical infrastructure, including subsea CO₂ storage.

## 1. Introduction

Natural clay-rich soils and shales are often unsuitable for civil engineering developments such as buildings, roads, bridges, and dams due to inappropriate geotechnical and engineering properties [1–3]. By introducing cement or other additives into these soils for stabilization, their mechanical properties can be improved, see e.g., [4,5]. Shales are commonly encountered in the construction of highway embankments and other large-scale engineering structures [6,7], and are also encountered as caprocks in abandoned oil reservoirs considered for CO₂ storage [8]. In all these geotechnical applications, stabilization of the shales is commonly achieved by injecting cement. Shales are thin laminar fine-grained rocks composed predominantly of siliciclastic materials and their granulometric composition is a mixture of clays and powder or silt particles [9]. The tailings from shale quarries can be used in their natural state as embankments but are unsuitable as subgrade and base materials unless strengthened [10]. Studies have reported the use

of cement and concrete combined with shale, shale dust, and shale ash to be used as admixtures in stabilization processes, and their mechanical properties have been studied [10,11]. It is important to understand the long-term mechanical response, fatigue and load-bearing ability of the cement-shale admixtures used for different applications.

Generally, underground masses are exposed to systematic multi-level cyclic loading and unloading events over a wide range of timescales, corresponding to sub-seconds for external loads, via annual loads due to seasonal variations, to geological times for naturally occurring large-scale processes like seismicity. The wearing effects of mechanical stress do not necessarily appear immediately and may accumulate over time, substantially complicating the analysis. In a recent study [12], we observed from triaxial testing with monotonic stress increments the importance of small inelastic deformation events to explain the mechanical behaviour of shale samples, emphasizing the need for systematic loading-unloading cycles to highlight the permanent damage. As the topic of cement-rock interface stability is highly important also to fields

<footnote><sup>a</sup> Department of Physics, Norwegian University of Science and Technology (NTNU), Høgskoleringen 5, 7491 Trondheim, Norway</footnote>

<footnote><sup>b</sup> The Njord Centre, Departments of Geosciences and Physics, University of Oslo, Oslo, Norway</footnote>

<footnote><sup>c</sup> ESRF - The European Synchrotron, CS40220, Grenoble 38043, France</footnote>

<footnote><sup>d</sup> SINTEF Industry, Applied Geoscience Department, Pb 4760 Sluppen, 7465 Trondheim, Norway</footnote>

<footnote><sup>e</sup> ISTerre, University Grenoble Alpes, University Savoie Mont Blanc, CNRS, IRD, University Gustave Eiffel, Grenoble, France</footnote>

<footnote><sup>f</sup> Department of Microsystems, University of South-Eastern Norway (USN), 3184 Borre, Norway</footnote>

<footnote>* Corresponding author at: Department of Physics, Norwegian University of Science and Technology (NTNU), Høgskoleringen 5, 7491 Trondheim, Norway.
E-mail address: dag.breiby@ntnu.no (D.W. Breiby).</footnote>
like CO₂ storage, there are many studies looking into the mechanical properties of cement in contact with rocks, see e.g., [13]. Along a different vein, calcareous oil shale is even considered as a raw material for clinker production [14]. Despite its relevance, there is a lack of both experimental triaxial testing and theory development to gain a better understanding of the cyclic loading mechanical behaviour of cement-shale mixtures.

Triaxial testing is an established methodology for characterizing the mechanical properties including structural evolution of static and dynamic microstructures [8,15]. Even though different imaging techniques like scanning electron microscopy (SEM), also combined with focused ion beam-milling (FIB-SEM), have been extensively utilized to obtain high-resolution images, also of shales [16], these techniques are destructive, and generally cannot be combined with triaxial tests. Another efficient experimental technique, digital image correlation (DIC, in 2D) or digital volume correlation (DVC, in 3D) based on optical imaging, is frequently employed for investigating the progressive failure of rocks under different loading conditions by optically measuring minute changes of the rock surface [15,17]. This approach has an obvious advantage of easy availability and relatively low instrumental cost. DIC has also been employed to study the progressive failure of rocks under cyclic loading [18].

As a non-destructive microscopy technique, X-ray microcomputed tomography (µCT) has been increasingly applied for the investigation, visualization, and quantification of geological samples in three dimensions. With the µCT technique, radiographic X-ray projections are obtained from different angles, and computer algorithms are used to obtain a greyscale 3D tomogram showing the interior of the sample, with a resolution typically in the micron range [19,20]. Each voxel in a tomogram represents the effective X-ray attenuation coefficient, which is a function of density and atomic number of the material [19]. The reconstructed images can then be further processed to extract properties like porosity and grain size distribution.

In recent years, X-ray transparent triaxial cells have been developed to carry out mechanical testing to investigate the mechanical behaviour of geomaterials while continuously monitoring the sample using X-ray imaging techniques [21–23]. Using µCT in conjunction with a triaxial loading apparatus like HADES [21], it is currently possible to investigate the pore network, micro-textures, fracture evolution and failure of porous samples under various loading conditions, including compression, tension, shearing and fracture [24]. With new technological advances, time-resolved X-ray microtomography (“4D-CT”) has opened the opportunity of examining dynamic processes such as mechanical response and fluid flow in porous geomaterials [22,25–27]. We note that X-ray µCT has helped in investigating the stress- and time-dependent deformation of many samples, ranging from polymers [28] to rocks [26], including cement behaviour under CO₂ injection [22]. Neutron computed tomography is also increasingly used for dynamic studies in rocks [29], also for cement [30]. Recently, other µCT techniques based on diffraction contrast [31] and coherent imaging have been used to study rocks and other porous materials [32,33] and cement by high-resolution X-ray ptychography [34]. DVC based on time-resolved X-ray imaging has been demonstrated to provide insights into the development of strain fields within rock samples [22,26,35–38].

For cement and concrete, there is a comprehensive literature on X-ray microscopy studies, see for example Chung et al for an overview [39]. A further selection includes fracture studies in 2D [40] and 3D [41], and erosion studies of cement pastes [42]. The CT studies by Hong et al on rubber-cement composites [43] and by Praneeth et al on biochar additions to cement mortar [44] are specifically mentioned here as examples of a cement composite comprising a soft material. Wellbore cement for carbon storage is another branch of high interest, with selected references including [22,25,45].

In this article, we present a study of the mechanical properties of consolidated shale – cement mixed samples exposed to cyclic triaxial loading, as monitored using in situ 4D-CT and analysed with DVC. We

present data clearly showing the importance of inelasticity and fatigue to understand the mechanical response of the samples. As we shall discuss based on the results, the continuous imaging of the sample facilitates an improved understanding of the deformation mechanisms in composite materials containing both a brittle (cement) and ductile (Draupne shale) component phase.

## 2. Materials and methods

### 2.1. Sample preparation

The samples used in this study were prepared from Portland cement (G type, manufactured by Norcem AS) and Draupne shale obtained from the North Sea. The Draupne shale formation is a caprock varying in thickness from a few meters up to >150 m and is considered a suitable location for permanent storage of the greenhouse gas CO₂ [8]. Draupne shale is a low-permeable, homogeneous, and anisotropic black shale with a porosity of about 12% [8].

A cement slurry was prepared according to the American Petroleum Institute (API) standards with a cement-to-water ratio of 0.44. The Draupne shale was crushed using a manual morter, and the pieces selected according to their sizes by mechanical sieving. Samples were prepared using two size categories of Draupne shale: sample A1 with 1–2 mm diameter shale fragments and A2 with 2–3 mm diameter shale fragments. The shale was mixed with Portland G cement. The slurry was poured into a rubber tube, making efforts to avoid trapping of air bubbles in the samples, and subsequently the tube ends were sealed. The rubber tubes used had an inner diameter of 5 mm. The specimens were left to cure for 5 days at room temperature and atmospheric pressure. The cured cement-shale samples were subsequently cut using a sharp blade into lengths of 10 mm, and the sample ends polished at right angles using finely grained sandpaper (grit 220) to facilitate the mechanical testing. The third sample B was prepared without cement, thus with only Draupne shale fragments of diameter 1–3 mm, which were compacted inside the rubber sleeve used in the experiment. Brine solution (3.7 wt%) prepared using table salt (NaCl) was used as a pore fluid for all the experiments. The samples were saturated with the pore fluid after installation in the sample cell, to be tested under realistic geotechnical conditions.

### 2.2. Micro computed tomography and HADES triaxial rig

A series of 4D-CT measurements under in situ stresses and ambient temperature were carried out to monitor the evolution of the deformation in the samples under cyclic loading at the ID19 microtomography beamline at the European Synchrotron Radiation Facility (ESRF). The cylindrical core samples (5 mm diameter, 10 mm height) were triaxially deformed under in situ conditions using the X-ray transparent HADES rig [21]. This apparatus has a low X-ray attenuation for photon energies >60 keV and can apply stresses both axially (up to 200 MPa) and radially (up to 100 MPa) at temperatures up to 200 °C, while the pore fluid pressure can be varied from 0.1 to 100 MPa. The measurements were carried out using a polychromatic “white” X-ray beam with a representative photon energy of 70 keV, collimated to a rectangular cross-section covering the whole sample. During each CT scan, the sample stage with the entire triaxial cell was rotated from 0° to 180° while the applied axial and confining stresses were maintained constant. Each scan lasted 80 s during which a series of 2500 radiographic transmission exposures of the three-dimensional sample were recorded. The voxel size of the reconstructed tomograms was 6.5 µm, with a spatial resolution roughly two to three times the voxel size.

The pore fluid pressure was controlled by two hydraulic pumps in the HADES apparatus; one to control the pore fluid inlet, and the other for the outlet, establishing a constant flow rate during the experiment.

Two independent additional pumps were used to control the confining pressure and the axial load. A Viton™ fluoropolymer
elastomer encased the rock sample, and the confining pressure was applied onto this jacket using silicone oil. All the cyclic loading tests were performed under a constant confining pressure of 20 MPa and a pore pressure of 10 MPa, and at room temperature. These testing conditions were chosen based on our previous studies of Draupne shales [12] and are representative for conditions ~1 km below the seafloor, assuming an average seafloor density of 2000 kg m⁻³. The axial stress was changed stepwise and held constant while the CT projection images were acquired. During the experiment, the projection images were monitored to ensure that the jacket was always in tight contact with the sample.

Each sample was placed axially between two stainless steel pistons and the interface contacts were not lubricated. The lower piston was fixed, and the samples, one at a time, were loaded axially by the movable upper piston. For sample A1, the axial stress loading amplitude was 10 MPa with 10 cycles for the three different base-levels of 25, 40 and at 50 MPa. For samples A2 and B, 3 cycles of 7 loops with 10 MPa amplitude were carried out, based at 25, 45 and 65 MPa. The loading rate was 1 MPa/min. While the friction in the setup, notably the O-rings, could contribute to the hysteresis, this effect is from previous experience low [21,36]. A linear variable differential transformer (LVDT) incorporated into the HADES rig was used to measure the axial shortening, and thus

the global (averaged) axial strain $\epsilon_1(t)$ of the sample. The time-resolved axial strain $\epsilon_1(t)$ was also obtained from the 3D X-ray images by measuring the shortening of the samples between the two pistons. The lateral strain $\epsilon_3(t)$ was estimated by measuring the change in diameter across the middle part of the sample.

## 2.3. Digital volume correlation analysis

The open-source software Tomowarp2 [38] was used for DVC analysis, yielding the incremental displacement fields $\mathbf{u}(\mathbf{r}, t)$ between consecutive pairs of tomograms. The DVC algorithm divides the 3D volumes into sub-volumes, and there is considerable uncertainty related to the optimal choice of search parameters, and how they influence the returned displacement values [38]. Here, we chose to work with a node spacing of 20 voxels (130 µm) and a search window of size 10 voxels which have proven reasonable in our previous experiments [35]. These parameters were kept constant for all the analyses. For the DVC calculations, we used a region of interest (ROI) fully contained within the cement/shale samples. From the 3D displacement field $\mathbf{u}(\mathbf{r}, t)$ the second-rank three-dimensional infinitesimal strain tensor field $\epsilon(\mathbf{r}, t)$ was calculated. From the infinitesimal strain tensor, the volumetric and von Mises equivalent strain were calculated.

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** The experimental setup with stress and strain history. a) Applied stress history, and b) the corresponding axial displacement (LVDT) readings for samples A1, A2 and B. c) Principal sketch of a triaxial rig for X-ray CT measurements. The sample is contained within a rubber sleeve upon which the confining pressure $p_{conf}$ is applied using silicone oil. In the experiments, the externally controlled pore pressure was kept the same at both ends of the sample, $p_{pore in} = p_{pore out}$. The white arrows indicate the applied axial stress $\sigma_1$.</figcaption></figure>
Texture-based segmentation and calculation of the strain fields were done using MATLAB™ (Mathworks Inc.) and the visualizations were made using the open-source software ParaView (www.paraview.org). The strain field patterns were segmented based on their texture, using the stdfilt.m function which performs standard deviation filtering. The resulting texture images were thresholded and processed to create binary masks, effectively segmenting the original images into regions of shale or cement based on their texture characteristics. ROIs for visualizations were selected using the Matlab function imellipse.m, separating the cement-shale specimen from other structures like the rubber sleeve. The segmented regions were visualized using imshow3D.m. Vector plots were made using quiver.m.

# 3. Results and discussion

## 3.1. Macroscopic stress–strain evolution

An axial stress-loading pattern was programmed for each sample, and the samples A1, A2 and B were exposed to cyclically varying stress, as shown in Fig. 1 (see also SI, Figs. S1–S3). The applied stress program consisted of repeated axial stress cycles (loading and unloading) at pre-

defined stress base levels. The corresponding time resolved LVDT readings (~strain) are also shown, and this response, representing a global average for the different samples, will be discussed before a detailed presentation of the CT data is given. The axial displacement (LVDT) measurements for the two cement/shale samples A1 and A2 exhibited similar response patterns. The abrupt increase of deformation early in the experiments corresponds to the samples settling inside the test apparatus in response to the increasing stress conditions.

The axial stress, axial strain and lateral strain of the cement-shale samples subjected to cyclic loading are shown in Fig. 2. During cycle I, all samples exhibited yielding as seen in the stress-strain responses (see also SI, Fig. S1), but the imposed stress variations were too small to give discernible strain oscillations. In cycle II, a more pronounced increase in the axial deformation (LVDT readings) with time can be noted, clearly demonstrating fatigue of the samples. During cycle III significant weakening of the samples A1 and A2 was observed with a noticeable increase in the axial displacement. The materials exhibited plasticity by their continued increase in permanent deformation with increasing number of loading cycles. For sample B, consisting of lightly compacted shale fragments, the displacement limit for the LVDT was rapidly reached and the experiment thus had to be terminated already after the

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Stress-strain plots for the three samples. The axial (ε₁) and lateral strain (ε₃) are plotted versus the axial stress σ₁. a,d) Sample A1. b,e) Sample A2. c,f) Sample B. Shown in (d), (e) and (f) is the axial strain (ε₁) vs stress plot for a zoomed-in range of ε₁. The markers indicate the experimental measurements, while the solid lines are guides to the eye.</figcaption></figure>
4th loop of cycle II (Fig. 1). Compacting axial strains (shortening of the sample) are expressed as positive values ($ε_1 > 0$), and dilation strains are expressed as negative values in this study. From the test results, it is evident from Fig. 3 (for sample A2) that the axial strain hysteresis loops became denser during the cyclic loading of the samples, being a clear signature of strain hardening.

For samples A1 and A2 (Fig. 2) at lower stress levels (cycles I and II), the hysteresis loops were tighter, and the evolution of the stress-strain loops signify compaction-induced hardening of the sample. At the

higher stress levels (III), the loops were sparser, and the stress-strain loops changed from loose to compact towards the end of the cycle. This behaviour could be attributed to irreversible local deformations, including opening and closing of microcracks, which might have accelerated the fatigue damage of the samples. Such microcracks were not observed directly because they would have an aperture smaller than the spatial resolution of the images. We note that the origin of creep in cement has been proposed to be a consequence of nanoparticle sliding [46]. Such nanoscale processes are beyond the spatial resolution that could be accessed in the experiments reported here.

After each cycle, the samples had undergone additional deformation, i.e., the residual strain had increased. Furthermore, the samples were compacted and became stiffer after each cycle. This mechanical behaviour resembles that reported for cement mortar under triaxial deformations [47]. This effect becomes more evident when considering the enlarged view of the evolution of the axial strain with increasing axial stress (Fig. 2). From Fig. 2d–f), we observe that a higher and permanent deformation occurred in the samples at the beginning of each cycle. Later in cycles I and II, A1 and A2 exhibited more pronounced irreversible deformation. In cycle III, there was no overlap between the subsequent loops, contrary to what was observed in the other cycles, which suggests that the deformation happening in the sample was permanent and irreversible.

For sample B (Fig. 2), which consisted of loosely compacted Draupne shale fragments, the initial packing height was about 9.2 mm. After loading into the HADES sample cell, the sample got markedly more tightly packed, with the measured height being 7.6 mm at the start of the experiment. The same stress protocol as for A2 was applied.

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 3.** Stress-strain curves for cycles I, II and III for sample A2. Note the different intervals on the ordinate axis. The markers indicate the experimental measurements, while the solid lines are guides to the eye.</figcaption></figure>

## 3.2. Sample deformation observed with µCT

Analyzing the µCT data for the 3 samples allowed us to follow the deformation mechanisms including fatigue and fracture. With the limited time available at the synchrotron, it was not possible to measure additional samples. To summarize, the samples A1 and A2 exhibited similar strain response to the applied stress program, however, the A1 sample did not fracture before the full program had been applied. Conversely, sample B was too soft to sustain the test program and reached the mechanical limit of the apparatus already in the middle of the second cycle. The data from sample A2 was good enough to be analysed by DVC using Tomowarp2, and also exhibited regions without shale fragments that proved quite helpful for interpreting the data. Based on these considerations, it was decided to focus the attention in this article to A1 and A2, with a more detailed discussion of the latter.

To better appreciate the stress-strain dynamics for sample A2, “zoomed-in” stress-strain plots for each of the three stress cycles are shown in Fig. 3. During the first stress loop of cycle I, the inelastic strain increased by >15 %, clearly showing that there was a significant irreversible plastic deformation, which is accompanied with energy dissipation. During the subsequent stress loops in cycle I, the strain response was increasingly reversible, suggesting strain hardening and indicating that there was no further energy dissipation by irreversible deformations. This behaviour repeated itself in cycles II and III, however, there was clear fatigue in the sample as the accumulated strain increased markedly with the higher stress level. To summarize, it is seen in Figs. 2 and 3 that the cyclic loading and unloading curves of each loop did not coincide, exhibiting clear hysteresis. The first stress loop of each cycle gave a significant irreversible compaction of the sample, and a large residual deformation was observed between the start and the end of each cycle, demonstrating that permanent damage had been done to the sample. The observed behaviour is consistent with sample fatigue.

The evolution of the irreversible axial strain with the number of cycles for the three samples is plotted in Fig. S2. The evolution of the irreversible axial strain at each axial stress levels is similar. The irreversible axial strain increased with increasing cycles. Similar observations have been reported for sandstones and cement mortar samples
[47,48]. A clear “loose – tight – loose” behaviour can be observed in the stress-strain curves; while the irreversible axial strains in cycle I and II tended to decrease towards the end of these cycles, for cycle III, a steady increase was observed towards sample failure. In other words, in cycles I and II the sample underwent a partially reversible deformation, while in cycle III the sample experienced permanent deformation.

Following the analysis by Peng et al. [49], the irreversible strain is the difference between the strains at the beginning and end of a stress loop. In our case, the number of sampling points for each loop was 4. The irreversible axial strain $\epsilon_{ir,n}^a$ was calculated as

$$ \epsilon_{ir,n}^a = \epsilon_{nf} - \epsilon_{1,i}, \qquad (1) $$

where $\epsilon_{nf}$ and $\epsilon_{1,i}$ refer to strains after the nth loop and at the start of loop 1, respectively. The irreversible strain $\epsilon_{ir,n}^a$ is marked explicitly for the 1st loop in cycle I for sample A2 in Fig. 3.

The relationship between irreversible axial strain and unloading strain for samples A1 and A2 under cyclic loading is shown in Fig. 4. A linear relationship was noted for the irreversible axial strain with unloading strain, observed from the linear regression analysis using the relation

$$ \epsilon_{ir}^a = \alpha \epsilon_u^a + \beta. \qquad (2) $$

where $\epsilon_{ir}^a$ is the irreversible axial strain and $\epsilon_u^a$ is the axial strain after unloading, $\alpha$ and $\beta$ are fitting parameters, as listed in Table 1.

The reconstructed 3D µCT volumes of A1 and A2 allow appreciating the packing of the shale fragments within the cement matrix and to observe their structural evolution during the experiments (Fig. 5). Sample A1 had shale fragments of diameter ~1–2 mm distributed throughout the cement matrix. For A2 the shale particles were mainly present at the top and bottom parts of the sample. Minor cracks were initially observed in both samples A1 and A2 from the beginning of the experiments, as indicated in Fig. 5. These cracks were later observed to decrease in size for A1 and to completely close for A2. Additionally, a failure fracture formed specifically in sample A2, indicated by the green arrow in Fig. 5. Different orientations of sample A2 post-fracture are shown in Fig. S6.

### 3.3. Detailed discussion of the displacement field

After investigating the macroscopic responses of the samples for the different steps, the damage initiation and evolution until failure was studied. DVC analysis was performed to obtain a detailed understanding of the deformation process and fatigue evolution. Successive pairs of tomograms were analysed to visualize the displacement fields and subsequently the spatial distribution the compaction and the von Mises equivalent strain within the sample. For DVC analysis, the RoI was chosen to be fully contained within the rock sample to avoid edge effects

from the jacket, such as the rubber sleeve, or surface irregularities on the sample. Note that for each of the presented images, the average displacement field within a homogeneous 3D region not containing shale fragments has been subtracted, thus effectively removing trivial (minor) sample translations and shifts. The following discussion focuses on the three cycles of sample A2, cf. Figs. 6 and 7, where we have chosen to give particular attention to two loops in each of these cycles: loop 3 is representative of the sample after having stabilized from the stress increment (cf. Fig. 3), while loop 7 represents the end of the cycle.

The displacement fields for sample A2 at different stages of the cyclic loading test show details of the mechanical response. During the four steps of cycle I loop 3, the axial stress was low and the main observation from Fig. 6a) is that the displacement field exhibits a certain degree of localization to the regions containing shale fragments (compare with Fig. 5b, see also Fig. 7). A tendency can already be observed that the displacement field had the highest magnitude when approaching the maximum stress upon loading (step ii; 30–35 MPa) and minimum stress upon unloading (step iv; 30–25 MPa).

An apparently random distribution of displacements across the sample was observed at the start of loop 3. As the axial stress was increased from 25 to 35 MPa, the displacement field near the movable piston at the top of the sample was oriented in the direction of the applied stress (i.e., down). A disordered displacement field is noted in the regions containing shale fragments. After the axial stress was unloaded from 35 to 30 MPa, the displacement field redistributed to become randomly oriented. When the axial stress was further reduced back to 25 MPa, the vector field near the top piston reoriented and pointed upward, i.e., in the opposite direction compared to the preceding loading steps, demonstrating that the sample relaxed and confirming that the response was, at least partially, elastic.

At the beginning of cycle II, the displacement field was disordered when the axial stress was increased from 45 to 50 MPa (Fig. 6b). On further increasing the axial stress from 50 to 55 MPa, the displacement field aligned and increased in magnitude. When unloading the axial stress from 55 to 50 MPa, we still observed the similar spatial pattern of the displacement fields as before, but the magnitude of the displacements had decreased. On further decreasing the axial stress from 50 to 45 MPa, the direction of the displacement vector field reversed, implying that the sample relaxed. On investigating the direction of the displacement vectors in the shale region with the fracture, the top region of the shale fragment displaced in the -z direction while the lower part moved in the +z direction, thereby leading to the closure of the fracture in the shale. For loop 7 of cycle II, qualitatively similar dynamics was seen, but the magnitude of the displacement vectors in the loading part of the loop (50 to 55 MPa) was markedly lower than the corresponding part in loop 3. At the end of loop 7, the sample exhibited larger displacements compared to the end of loop 3. All these observations are consistent with plastic-elastic partially irreversible damages.

<figure><img src="image_8.png" /><figcaption>Fig. 4. Irreversible axial strain vs. unloading strain. a) Sample A1, and b) sample A2. The green lines are linear fits to the data. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
<table><caption>**Table 1**

Results of fit to the data in Fig. 4. R² is a measure of the goodness of fit.</caption><thead><tr><th rowspan="2">Sample</th><th colspan="3">Cycle I</th><th colspan="3">Cycle II</th><th colspan="3">Cycle III</th></tr><tr><th>α</th><th>β</th><th>R<sup>2</sup></th><th>α</th><th>β</th><th>R<sup>2</sup></th><th>α</th><th>β</th><th>R<sup>2</sup></th></tr></thead><tbody><tr><td>A1</td><td>0.770</td><td>-0.010</td><td>0.995</td><td>0.790</td><td>-0.010</td><td>0.991</td><td>0.832</td><td>-0.011</td><td>0.998</td></tr><tr><td>A2</td><td>0.47</td><td>0.002</td><td>0.973</td><td>0.723</td><td>0.001</td><td>0.989</td><td>0.843</td><td>0.003</td><td>0.998</td></tr></tbody></table>

<figure><img src="image_9.png" /><figcaption>**Fig. 5.** Perspective view (x,z) of segmented shale fragments in a cylinder-shaped RoI within samples a) A1 and b) A2. The shale fragments are represented in red, and the cement in grey. The black arrows point to pre-existing fractures present in the samples prior to the experiment. The green arrow in (b-iv) points to the final fracture at the end of the experiment. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

For cycle III, at the beginning of loop 3 (65 to 70 MPa), the displacement field vectors were pointing in the -z direction at the top, consistent with sample compression (Fig. 6c and Fig. 7c). Note the central region of the sample where the density and the magnitude of the displacement vectors were low, signifying the presence of the comparably stiff cement phase. As the axial stress was increased from 70 to 75 MPa, the magnitude of the displacement field reached a maximum. It is interesting to note that in the central cement-rich region, where the displacements were negligible at the lower stress level, the displacements had now become more pronounced, indicating that at this stress level also the cement phase was being slightly deformed. Upon decreasing the axial stress from 75 to 70 MPa, a similar pattern of the

displacement fields was observed: the magnitude of the vector field was reduced, but, crucially, its direction was maintained, demonstrating that at this stage, the integrity of the sample was completely lost. At the end of the loop (step iv; 70–65 MPa), we see that the displacement fields were distributed similarly as in the previous step iii, yet with a lower magnitude. Similar observations are made for cycle III loop 7 but with a lower magnitude as compared to cycle III loop 3. These displacement field patterns also demonstrate that the sample was further fatigued in cycle III and that the damage done to the sample was irreversible.
<figure><img src="image_10.png" /><figcaption>**Fig. 6.** Evolution of displacement fields in the (x,z) plane, obtained by DVC at different loading stages for sample A2. a) Cycle I, b) cycle II, and c) cycle III. Each cycle is represented by two loops (3 and 7), each with 4 steps of loading and unloading. The displacement vectors are multiplied by a factor of 500 for better visualization.</figcaption></figure>

### 3.4. Detailed discussion of the volumetric and von Mises tensor fields

The incremental volumetric strain reveals the distribution of local dilatancy ($\epsilon_V$ < 0) and compaction ($\epsilon_V$ > 0) in the sample. Studying the evolution of the different incremental strain using DVC showed the localization of the higher magnitude of negative volumetric strain (dilations) in the top region of the sample. It might be instructive to remember that as the figures display only a region of the sample, the sample material can protrude laterally outwards during the compression. Fig. 8 shows the evolution of the volumetric strain for cycle III loops 3 and 7 (cf. Fig. S8 for cycles I and II). For cycles I and II, only minor volumetric strain was observed, seen to be localized to the regions containing the shale fragments. For cycle III loops 3 and 7 a different behaviour of the volumetric strain was observed as the axial stress was increased. At the beginning of loops 3 and 7, the axial stress was increased from 65 to 70 MPa, and the presence of localized dilation was

observed with a low magnitude. As the axial stress was increased from 70 to 75 MPa, regions of pronounced dilation became visible, co-localized with shale-cement boundaries within the sample, cf. Fig. 7. With the unloading of axial stress from 75 to 70 MPa, the magnitude of the dilation was reduced but still comparable to the preceding DVC pairs. When the axial stress was further unloaded to 65 MPa, a behaviour resembling the pattern at the beginning of the loop was obtained, still with the compaction and dilation located to the same shale-rich regions.

The von Mises equivalent strain, calculated from the displacement field $u(r,t)$, is used as a proxy to visualize the incremental deviatoric strain in the sample. Fig. 9 shows the evolution of the incremental von Mises equivalent strain in the sample A2 during cycle III, loops 3 and 7 (cf. Fig. S9 for cycles I and II). We observed a progressive development or propagation of deviatoric strain localization within the specimen, again confined to the regions containing the shale fragments. For cycles I and II, a rather homogeneous distribution of the von Mises equivalent
<figure><img src="image_11.png" /><figcaption>**Fig. 7.** Central CT (x,z) cross sections with overlaid displacement field (u_y, u_z). a) At low stress, it can be noted that the displacement magnitude was closely correlated with the location of the shale fragments. b) In cycle II, the strain magnitude remained most pronounced at the shale fragments, however with a new feature being the shear strain across the upper shale-rich part of the sample, with a systematic down-left oriented displacement. c) In cycle III, loop 3, this trend was further developed, with the much larger displacements forming a sharp boundary across a thin cement bridge between two large shale fragments. d) In loop 7 the nature of the displacement field had again changed. The vertically aligned red markers help visualizing the global deformation of the specimen. Note the different scaling factors used for the displacement vectors. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 8.** Evolution of volumetric strain for cycle III, loops 3 and 7, for sample A2. a) The central CT (x,z) slice with the region used for DVC indicated by the red dashed rectangle. b) The volumetric strain I cycle III, loops 3 and 7. The black arrows point to regions of pronounced dilation at cement-shale interfaces in the sample. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

strain across the shale-rich regions was seen (compare Fig. 8a). Conspicuously, there was also an early and sharply defined deformation event taking place during cycle I (Fig. S9), located at the interface between a shale fragment and the cement matrix. We propose that the presence of the higher von Mises strain localized in the region of the shale particles can be attributed to the pseudo-continuous deformation (or “warping”) of these comparatively soft materials during loading and unloading. Similar observations of strain localization along the inclinations have been reported from studies on concrete consisting of aggregates, macro-pores and mortar matrix [48].

It is instructive to note that for a homogeneous sample, the volumetric strain can be obtained as the axial strain plus two times the lateral strain, $\epsilon_V = \epsilon_1 + 2\epsilon_3$. In the present study, the samples were highly inhomogeneous, with comparatively large regions of the softer shale material dispersed irregularly throughout the sample volume.

Consequently, the volumetric strain can only be meaningfully defined either globally for the entire sample as a whole, or locally, as done here with the DVC analysis, see e.g. Fig. 8. As shale fragments are generally known to have highly anisotropic mechanical properties, it would be an interesting continuation of the present study to correlate the anisotropic properties of the shale fragments with the local strain fields from DVC. We note that so-called X-ray tensor tomography can be used to obtain the orientation of mineral grains [51,52]. However, because shale fragments tend to have a pronounced flake-shape, it might even be possible to guess the normal directions to the shale layers directly from the shape of the fragments and to use this information for further analysis.

Supported by the experimental data, we have described in detail the strain accumulation both within and between the cycles. An observation of special interest is the profound interplay between the stiff cement
<figure><img src="image_13.png" /><figcaption>**Fig. 9.** Evolution of von Mises equivalent strain for cycle III loops 3 and 7 in sample A2. The (x,z) plane is shown, see also Fig. 7. Note the virtually absent strain in the cement rich regions.</figcaption></figure>

matrix and the comparatively much softer shale material. The stage is prepared for a full quantitative modelling of the experimental data, which could be the focus of a future study, beyond the scope of the present article.

### 3.5. Discussion of the fracturing process

As described, the experiment was designed to investigate the mechanical properties of shale particles embedded in cement. Both the techniques used for “soil improvement” for civil engineering purposes [4] and the deep underground stabilization or plugging of borewells, in both cases often based on injecting cement, are prone to form heterogeneous samples conceptually similar to those studied in the present study. We have followed the sample response of A2 to cyclic loading and unloading, leading to the failure of the sample. These results are consistent with observations reported for fracturing of micro-concrete under uniaxial and triaxial compression [50]. Specifically, we observe that in the low-strain regime, the shale fragments act as local soft “cushions” taking up the local stress. The imposed small stress excursions resulted in local deformations co-localized with the shale inclusions. Conversely, at larger stress levels, the behaviour changed quite markedly, with subsequent large-scale deformations of the shale-rich regions. While such behaviour appears intuitively reasonable, it has to our knowledge not been previously observed directly.

Regarding the creep, it should be noted that at the levels of stress used in the current study, creep strain should be expected to take place during constant loading. A limitation of our experiment is that the resolution was insufficient to distinguish between different nanoscopic creep mechanisms, and hence one cannot, based on the jacketed-only experiments presented here, pinpoint the contribution of e.g. the delay flux of brine-transport in micropores inside the specimen to the creep strain. Another limitation is the possible existence of nanoscopic fractures that are not resolved by CT, yet are likely to influence the mechanical response of the sample. The duration of each CT scan (~90 s) was too short to result in notable motion blurring by creep for the lower stress levels. On the other hand, towards the end of the experiment (cycle III, loop 7), when the sample was severely weakened, significant axial creep strain of up to 1–2 % was observed.

We note that the size distribution of the shale fragments, also as compared to the total sample diameter, is an important parameter that

should be considered carefully in future experiments. If the shale fragments are of similar dimensions as the sample diameter, the resulting sample tends to be heterogenous or even layered, as seen in Fig. 5. Still, for our purposes in the current article, the heterogeneity proved instrumental in disentangling the response of shale from that of cement.

### 4. Conclusion

In the present work, cyclic triaxial loading compression tests were conducted on Portland G cement-shale composites under in situ conditions relevant for geotechnical applications. X-ray micro computed tomography and deformation fields obtained from digital volume correlation (DVC) were used to investigate the damage and fatigue evolution in the samples. Our results demonstrate that the evolution of the fatigue in the sample and the route towards failure can be described in detail. We calculated and monitored the evolution of the 3D displacement fields, the volumetric strain and the von Mises strain to track the non-homogeneous deformations and fatigue development inside the samples. Our study confirms that the volumetric and the von Mises equivalent strain are highly influenced by the presence of the coarse shale fragments. The methods for investigating fatigue and the analysis techniques presented here reveal the local in-situ mechanics in shale - cement admixture under periodically varying loading. We envision that these methods can be further developed to serve as accelerated aging laboratory testing to understand the long-term stability of cement-shale but also other geological materials for large-scale civil engineering projects such as buildings, roads, and dams, and also CO₂ storage.

#### CRediT authorship contribution statement

**Aldritt Scaria Madathiparambil:** Conceptualization, Data curation, Investigation, Visualization, Writing – original draft. **Fazel Mirzaei:** Methodology, Visualization. **Kim Robert Tekseth:** Investigation, Writing – review & editing. **Benoît Cordonnier:** Investigation, Methodology, Writing – review & editing. **Nicolaine Agofack:** Conceptualization, Methodology, Writing – review & editing. **Pierre Cerasi:** Conceptualization, Funding acquisition, Investigation, Methodology, Supervision, Writing – review & editing. **François Renard:** Conceptualization, Funding acquisition, Methodology, Writing – review & editing. **Basab Chattopadhyay:** Formal analysis, Supervision, Writing – review & editing. **Dag W. Breiby:** Conceptualization, Funding acquisition, Methodology, Project administration, Supervision, Writing – original draft.

#### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

#### Data availability

The data from this study are available from the leading author upon reasonable request.

#### Acknowledgements

We thank Cathrine Ringstad and Ragnhild Skorpa for project management and valuable discussions. We acknowledge the European Synchrotron Radiation Facility (ESRF) for provision of beamtime at beamline ID19. The Norwegian Research Council is gratefully acknowledged for funding through Petromaks2 (#280942 CuttingEdge), NANO2021 (#272248 CompMic), and FRINATEK (#275182 4D-CT).
# Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.cemconres.2024.107682.

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