# DEM multi-scale insights on the stress, energy and crack propagation in proppant-fractured rock collisions

Sarath C R Nallala, Huan He, Kostas Senetakis

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Sarath C R Nallala, Huan He, Kostas Senetakis. DEM multi-scale insights on the stress, energy and crack propagation in proppant-fractured rock collisions. Computers and Geotechnics, 2025, 183, 10.1016/j.compgeo.2025.107234. hal-05105646

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Submitted on 10 Jun 2025

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Research Paper

# DEM multi-scale insights on the stress, energy and crack propagation in proppant-fractured rock collisions

Sarath C.R. Nallala<sup>a ID</sup>, Huan He<sup>b</sup>, Kostas Senetakis<sup>c,*</sup>

<figure><img src="image_2.png" /></figure>

## ARTICLE INFO

### Keywords:

Proppant

Fractured rock

Collision

Energy transfer

DEM

## ABSTRACT

The collision behavior of an impactor (proppant simulant) colliding on weak clastic rock with pre-existing microcracks is examined in this work at multi-scales. Using discrete element-based analysis, the linear parallel bond contact model is used to simulate the rock base, while the Hertz-Mindlin contact model is used to capture the non-linear force-displacement relationship between grain-block contacts. A systematic parametric study is conducted to assess the impact of microcrack size, number, angle, and position on the damage potential of the impactor colliding the base rock. The results indicate that some of these parameters significantly influence the damage potential while others have minimal effects. Multiscale assessment is conducted by comparing stress and energy distribution results at the macroscale, penetration and contact formation information at the mesoscale and studying the discrete fracture network at the microscale. The findings of this study provide important insights into the collision behavior of proppants with clastic rocks, particularly during hydraulic fracturing operations and can potentially aid in the modeling of these systems at larger scales.

## 1. Introduction

With the depletion of conventional oil and gas resources, the exploration of hydrocarbons from unconventional deposits has become increasingly important. Unconventional energy sources such as shale gas, geothermal energy, coal gas, and methane hydrate are typically stored in impermeable layers, making extraction and use challenging for the petroleum industry. To increase reservoir output and improve extraction, a hydraulic fracturing technology capable of maintaining a consistent hydrocarbon flow must be chosen (Davies et al., 2012; Bandara et al., 2020; Zhu et al., 2024). This technology uses proppants, such as quartz sand grains or ceramic/synthetic grains, along with high-pressure fluids to stimulate reservoir formations and create fractures, which releases the enclosed hydrocarbons and maintains the required hydraulic conductivity in the fractures (Maslowski and Labus, 2021). Fig. 1 shows a schematic of an unconventional horizontal and vertical type reservoir, with the zoomed-in view of the clastic rock fractures highlighting the proppants entering the fractured spaces and becoming embedded.

Advanced hydrocarbon exploration technologies require vast knowledge of the reservoir, such as the type of rock formations, deposit

pressure, temperature, and horizontal or vertical type of well (Morales 2012; Maslowski and Labus, 2021; Wang et al., 2023; Deng et al., 2024; Li et al., 2024). Additionally, reservoir rock characteristics like strength, stiffness, and hydraulic conductivity must be considered when designing a rock fracturing exercise, including the type and size of proppants, type and viscosity of fluid, and the rate of pressure application to create cracks/fractures (as shown in Fig. 1). Use of waterless (i.e., gas based) hydraulic fracturing technologies (e.g., Wang et al., 2016; Zhang et al., 2018; Yan et al., 2020) at greater depth of exploration is also being investigated owing to the requirement of high velocities/pressure of fluid-proppant mixture to cause damage to the clastic rock surface. Irrespective of the type of fracturing technology used, the consistency of the reservoir output is another important challenge which depends on unforeseen factors occuring at meso and microscale where fracture fluid-proppant mixture interacts with rock surfaces (Bandara et al. 2018, Bandara et al., 2020). Therefore, to improve the predictability of the reservoir behavior, multiscale assessment (i.e., at macro, meso and microscale) of the proppant-rock, fluid-rock, and proppant-fluid-rock interaction is essential to draw useful inferences.

Recent studies have focused on assessing the influence of mechanical stresses generated in the proppant pack on crushing and embedment

<footnote><sup>a</sup> Institut de Physique du Globe de Paris, Paris, France</footnote>

<footnote><sup>b</sup> Southeast University, Nanjing, China</footnote>

<footnote><sup>c</sup> Department of Architecture and Civil Engineering, City University of Hong Kong, Hong Kong, China</footnote>

<footnote>* Corresponding author at: Department of Architecture and Civil Engineering, City University of Hong Kong, Hong Kong, China.
E-mail address: ksenetak@cityu.edu.hk (K. Senetakis).</footnote>
into rock, and how this impacts the hydraulic conductivity of the fractured rock. Other pertinent issues related to proppant damage and embedment result in generation of fines which can migrate thereby further decreasing the fracture conductivity (Palish et al., 2007; De Silva and Ranjith., 2019; Mittal et al., 2018; Ahamed et al., 2019; Bandara et al., 2020; Aslannezhad et al., 2021; Maslowski and Labus, 2021). Overall, these studies show that the proppant damage may cause significant reduction in the fracture pack hydraulic conductivity. This proppant damage and embedment into the rock is the result of excessive compressive stress development in the fracture (few hundred of Mega pascals) or due to the high velocity impact of the proppants with rock surface (illustrated in Fig. 1). While some research has been conducted on the former (Maslowski and Labus, 2021; Bandara et al., 2021), very little is known about the latter (Kasyap et al., 2022; Reddy et al., 2023). To address this gap, the present study aims at obtaining new insights by numerically simulating the mechanism of proppant impacting a fractured rock base.

This type of research studies is more prominent in other engineering disciplines related to study of granular/particle and solid/block interactions. For example, studies on the impact behavior of impactor-block systems have been conducted using grain scale experiments (Luo et al., 2021; Luo and Senetakis, 2022). This involves dropping an impactor, such as a spherical or sub-rounded grain, from a height onto a solid base and measuring the rebound height and angle of the impactor particle. Through this process, the coefficient of restitution (COR) of the impactor-block system is measured whose value is an indicator of the elastic and plastic nature of the colliding particles. Such experiments have been carried out on natural granular material such as quartz sand, clastic and granitic (crystalline) rocks and artificial materials such as chrome steel balls, glass beads, rubber blocks, concrete blocks and ceramic particles (Banks et al., 2005; Crüger et al., 2016; Sandeep et al., 2020; Luo et al., 2021, 2022a, 2022b; Luo and Senetakis, 2022; He and Senetakis, 2020). From the analysis of impact experiments, surprising results have been observed when different combinations of materials are used (e.g., glass beads or chrome steel balls falling on rubber base or granite rock base), or by varying the size of the impactor or due to the presence of a viscous fluid at the impactor-block interface. However, these experiments are limited to cases of low-velocity impact and cannot be used to study the high-velocity impact of proppants on clastic rocks, as seen in unconventional reservoirs. To solve this problem, the present

study employes numerical simulations to analyze the impact process of the proppant-rock system, providing multiscale insights into the mechanisms that take place when a proppant impacts a clastic rock at high velocities. Conceptually, the model development to perform the present study builds upon the previous numerical simulations by Luo et al. (2022a, 2022b), Kasyap et al. (2022) and Reddy et al. (2023) taking a step further in quantifying the influence of fracturing within the rock mass as explained in the successive paragraph. These previous studies have developed a numerical platform to analyze at multi-scale the collision mechanisms and energy transfer in a particulate medium with calibration macroscopic parameters based on grain-scale experiments (i. e., particle colliding a block), predominantly relying on COR experimental measurements.

In recent years, numerical studies have also been conducted to examine the impact problem using both the Finite Element Method (FEM) and the Discrete Element Method (DEM). Researchers have used FEM to simulate the stress evolution and energy dissipation characteristics of impactor-block systems considering different engineering materials (Rivera et al., 2019; Zhou et al., 2022). On the other hand, the discrete element method has been used to simulate particle impacting a granular or cemented base rock/block such as clastic rocks or weak sandstones. For example, Shen et al. (2017, Shen et al. 2019) used DEM to simulate more complex processes such as impact induced rock fragmentation or the interaction between a rock block and soil buffer, which phenomena are technically difficult to analyze through experimental observation and measurement. Similarly, other researchers focused on understanding the more basic mechanisms (i.e., force network and fracture network formation, stress transfer, energy evolution) that are influenced by size, shape and velocity of the impactor hitting the base block (Bourrier et al., 2008; Aryaei et al., 2010; Awasthi et al., 2015; Shen et al., 2017; Jiang et al., 2018; Ye et al., 2021). Furthermore, recently, few attempts were made to assess the influence of pre-existing fractures/joints or impact induced fracture in the rocks on their elastic and plastic behavior of the respective impactor-block systems (Aziznedjad et al., 2018; Kasyap et al., 2022). The results of these studies have presented a new perspective on the impact problem, emphasizing the need to assess the impact mechanism at multiple scales (macro, meso, and micro). In this context, Reddy et al. (2023) conducted a discrete element analysis that simulated proppants (quartz sand) as impactors colliding with a block representing clastic rock. This

<figure><img src="image_3.png" /><figcaption>Fig. 1. Schematic of the proppant-rock interaction during hydraulic fracturing treatment in an Unconventional shale gas reservoir</figcaption></figure>
fundamental study aimed to understand how the size, velocity, and ki-
netic energy of the impactor influence the damage potential within an
impactor-block system, analyzing the problem at multi-scales para-
metrically. However, there are still unexplored areas related to the
collision behavior of impactor-block systems, for example the role of
microcracks characteristics when the collision involves fractured rock;
the type and position of the crack, its angle (orientation) and length, or
the presence of multi-cracks, are some of the areas that need to be
further addressed.

The insights gained from previous research have provided both the
motivation and the conceptual foundation necessary for the present
study, which aims to simulate the impact problem between proppants
and fractured rock analogues to siltstone or sandstone while considering
the complexities of microcracks of varying thickness, length, and angle.
This study’s primary objectives include conducting a comprehensive
multiscale assessment. At the macroscale, the stress and energy distri-
bution results are compared to understand the overall impact response;
at the mesoscale, the data on penetration and contact formation is
evaluated to assess how the surfaces evolves during the interaction of
impactor with the fractured rock. At the microscale, the discrete fracture
network is examined to gain insights into microcrack behavior during
the impact event. The findings are expected to have direct applications
in geoscience and petroleum engineering, providing valuable insights
into proppant behavior in fractured environments, while also offering
indirect applications in other engineering fields involving impact
mechanisms. By addressing these objectives, the study seeks to enhance
the understanding of impact interactions in fractured rock systems, ul-
timately contributing to the development of improved engineering
practices in related fields.

## 2. Methodology

In this study, the whole impact process of proppants interacting with
a fractured rock during unconventional reservoir exploration was
assessed using a simple yet pertinent model. The model is composed of
an impactor and block system, simulated through the Discrete Element
Method. For this purpose, a clastic rock variant was simulated using the

linear parallel bond contact model (Potyondy and Cundall, 2004; Ning et al., 2016) and interaction between the proppant and rock base was defined using Hertz-Mindlin contact model (Mindlin, 1949; Mindlin and Deresiewicz, 1953). The significance of the chosen linear parallel bond contact model for the rock is that it provides the required mathematical formulation to bond particles/grains using geometrical contact proper- ties such as particle radius, bond length, bond area/radius and me- chanical contact properties including density, linear and normal and tangential stiffness, compression and tensile bond strength, friction, respectively which are the essential features of clastic rock. On the other hand, the Hertz-Mindlin model which defines the interaction between proppant and rock simulant is prominently used to define the non-linear force-displacement response between two non-contacting bodies (i.e., particle-particle or particle-block or block-block systems) such as in the present analysis.

The DEM model of the rock base simulated in the present study is a cuboid as shown in Fig. 2. The rock base constitutes of a cloud of 478,239 spherical bonded particles with particle radius ranging from 0.3 to 0.6 and solid density of 2,650 kg/m³. Similarly, the impactor (prop- pant simulant) of radius 12.5 mm, which is less than quarter of the rock base width and 27.78 times the average ball radius of the bonded par- ticles in the rock base is chosen for the present study. It is to be noted that the above geometrical dimensions are chosen such that the multi- scale mechanisms observed from the present study shall truly represent the impact behavior of the proppant-rock system. For example, the size of proppant is typically 0.3 mm (correspond to mesh 40/70) which is about 30 times average size of slit particles in sandstone type rocks (Bandara et al., 2021; Kaysap et al., 2022). Furthermore, the dimensions of the rock base were chosen to avoid any boundary effects on the resulting response from the impact of the proppant. The properties of the DEM model simulated in the present study are detailed in Tables 1-2 and the failure envelope of the clastic rock variant considered in this study are presented in Fig. 2. Numerical Unconfined Compressive Strength (UCS) test was performed on the chosen clastic rock variant, and the resulting axial stress (σa) versus axial strain (εa) curve, along with the crack number, are presented in Fig. 3. From Fig. 3, it can be seen that the UCS value of the sample was 2.95 MPa, corresponding to an axial strain

<figure><img src="image_4.png" /><figcaption>Fig. 2. Discrete element model of the proppant-rock system along with the failure envelope of the weak clastic rock variant considered in this study.</figcaption></figure>
<table><caption>**Table 1**

Summary of properties/parameters for the rock base.</caption><thead><tr><td>No</td><td>Property/Parameter</td><td>Weak clastic rock</td></tr></thead><tbody><tr><td>1</td><td>Contact model</td><td>Linear parallel bond model</td></tr><tr><td>2</td><td>Ball radius (mm)</td><td>0.3–0.6</td></tr><tr><td>3</td><td>Density (kg/m³)</td><td>2650</td></tr><tr><td>4.</td><td>Normal bond stiffness (k<sub>pb</sub><sup>0</sup>) (Pa/m)</td><td>5 × 10<sup>11</sup></td></tr><tr><td>5.</td><td>Tangential bond stiffness (k<sub>pb</sub><sup>1</sup>) (Pa/m)</td><td>2.5 × 10<sup>11</sup></td></tr><tr><td>6.</td><td>Friction angle (θ)</td><td>32°</td></tr><tr><td>7.</td><td>Cohesion (c) (Pa)</td><td>5 × 10<sup>6</sup></td></tr><tr><td>8.</td><td>Tensile strength (σ<sub>t</sub>) (Pa)</td><td>0.5 × 10<sup>6</sup></td></tr><tr><td>9.</td><td>Normal contact stiffness (k<sub>n</sub>) (N/m)</td><td>5 × 10<sup>5</sup></td></tr><tr><td>10.</td><td>Tangential contact stiffness (k<sub>t</sub>) (N/m)</td><td>2.5 × 10<sup>5</sup></td></tr><tr><td>11.</td><td>Interparticle friction (μ)</td><td>0.25</td></tr><tr><td>12.</td><td>Local damping coefficient (η)</td><td>0.5</td></tr></tbody></table>

<table><caption>**Table 2**

Summary of properties/parameters considered during the interaction of the impactor (proppant) with the block (rock base).</caption><thead><tr><td>No</td><td>Property/Parameter</td><td>Input value/description</td></tr></thead><tbody><tr><td>1.</td><td>Contact model</td><td>Hertz-Mindlin</td></tr><tr><td>2.</td><td>Impactor radius (mm)</td><td>12.5</td></tr><tr><td>3.</td><td>Density (kg/m³)</td><td>2650</td></tr><tr><td>4.</td><td>Shear modulus (G) (GPa)</td><td>28</td></tr><tr><td>5.</td><td>Poisson's ratio (ν)</td><td>0.2</td></tr><tr><td>6.</td><td>Interparticle friction (μ)</td><td>0.1</td></tr><tr><td>7.</td><td>Normal contact damping coefficient (ζ<sub>n</sub>)</td><td>0.1</td></tr><tr><td>8.</td><td>Impactor velocity (V<sub>g</sub>) (m/sec)</td><td>15</td></tr></tbody></table>

of 0.27 %. Based on the UCS value, the rock variant may be classified as a weak rock (Geo-guide 3, 2017). The contact parameters of the numerically simulated block, as presented in Table 1, were selected to represent weak clastic rock, consistent with the numerical study conducted by Reddy et al. (2023). The impact behavior of the rock is typically quantified using the coefficient of restitution (COR), which is determined based on the rebound height. Experimental studies by Sandeep et al. (2020) and Luo et al. (2022b) reveal that COR values range from 0.7 to 0.9 for strong rock/block surfaces and from 0.3 to 0.6 for medium-strength to weak rock surfaces. These findings validate the extensive parameter study conducted by Reddy et al. (2023) on the influence of

block stiffness and strength on COR.

As mentioned earlier, the primary focus of this study is to evaluate the impact behavior of a proppant on a fractured rock base or a rock containing pre-existing microcracks. To achieve this, a pre-existing microcrack or fissure was introduced into the rock base at a specific depth, with varying crack or fissure angles considered. Conceptually, this approach reflects the natural geological processes in clastic rocks, such as siltstone or sandstone, where cracks or fractures form, and some silt grains are displaced or removed due to geological activities. Similar methodologies for simulating cracks in rocks have been employed in various experimental and numerical studies (e.g., Xu and Li, 2017; Li et al., 2021; Liu et al., 2022). As loading is applied, the particles in the clastic rock simulation effectively demonstrate how cracks propagate along the edges of the fractured portion. Additionally, the silt granules that constitute the clastic rock are observed to move into the crack, altering the roughness between the two crack interfaces. This movement also influences the micro- and mesoscale quantities analyzed in this study, providing insights into the dynamic behavior of fractured rock systems under impact conditions.

Fig. 4 shows the schematic of the designated single and dual fracture/cracks in the rock base. The notations *d* and *w* represent the height and width of the clastic rock simulant (considering 2D sectional view) and *l*<sub>*f*</sub> represent the length, *d*<sub>*f*</sub> represents the position of the microcrack from the top of rock base where as *α*<sub>*f*</sub> is the fissure/crack angle of the induced fissure in the rock sample. It is to be noted that the length and width of the crack are taken as equal (which can only be observed in 3D view). In the present analysis, for the case of single fracture/crack (Fig. 4 (a)), a fixed value of *l*<sub>*f*</sub> and *d*<sub>*f*</sub> are chosen as 25 mm (which is equal to the impactor size) and thickness of the crack (*t*<sub>*f*</sub>) is considered as 1 mm which is the general value of thickness for a microcrack. Furthermore, the effect of fracture/crack angle on the impact behavior is investigated by considering *α*<sub>*f*</sub> = 0°, 30°, 45°, 60°, and 90°, respectively, two separate analyses were conducted to investigate the influence of length of microcrack and thickness of the microcrack on the impact behavior. This is achieved by increasing the length of crack (*l*<sub>*f*</sub>) to 37.5 mm (which is equal to 1.5 times the impactor size and 0.5 times the rock base width) and by considering the thickness of microcrack as 2 mm. On the other hand, second set of analysis were conducted focusing on the influence of dual cracks on the impact response of the proppant-rock system.

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Unconfined compressive strength (UCS) of the clastic rock variants of the DEM simulated rock sample</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Schematic illustration of the microcrack in the clastic rock geometry considered in the present study a) Single crack of thickness, $t_f$ b) Dual cracks of thickness, $t_f$</figcaption></figure>

From Fig. 4(b), it can be observed that two micro cracks are considered, one at a depth of 25 mm (equal to impactor size) and other at a depth of 50 mm (two times the impactor size) from the top of the rock base. The length, width and thickness of the crack are same as the previous case (similar to the case of single crack) whereas two values of $\alpha_f$ i.e., 0° and 30° were considered to investigate the effect of dual cracks on the resulting impact response of the proppant-rock system. It is to be noted that the impactor velocity ($V_s$) of 15 m/sec was used to introduce the required kinetic energy to cause plastic deformation on the rock base. The comparison of the impact response of the above cases is made with the impact response of proppant-rock system with no cracks (i.e., intact rock) to obtain useful insights. Overall, ten numerical analyses were conducted (as shown in Table 3) to study the impact behavior of proppant-fractured rock system and the results are presented in the subsequent sections.

### 3. Results and discussion

#### 3.1. Kinetic energy of the impactor

The kinetic energy imparted by the impactor, due to its high vertical velocity component, is the primary cause of damage to the rock base during the impact process. The kinetic energy of the impactor is composed of three translational and three rotational components in the

x, y, and z directions. This kinetic energy is determined by the momentum and angular momentum of the impactor, and is transferred through the rock upon impact, resulting in fracturing of the rock base. However, the fracturing of the rock base is also influenced by the internal defects such as microcracks or fissures in the rock base. Figs. 5 and 6 shows the kinetic energy variation of the impactor with respect to time for the different scenarios of microcracks in the rock base. It is to be noted that the kinetic energy trend presented in Fig. 5 is based on the time duration between the moments of first and final contacts of the impactor with the rock base. In Fig. 5, the variation of the impactor’s kinetic energy with respect to the angle of the microcrack ($\alpha_f$) is shown. The general trend observed is that as the impactor collides the rock base, the kinetic energy of the impactor decreases to a minimum threshold value, thereafter, starts to increase and reaches a limiting (constant) value as the impactor loses its contact with the rock base. The trend showed in Fig. 5 confirms the probable influence of $\alpha_f$ on the kinetic energy variation of the impactor. Specifically, the existence of microcrack (of thickness 1 mm at a depth of 25 mm equal to the size of impactor and of size 25 mm × 25 mm) increased the time of contact nearly 1.5 to 2 time (based on the microcrack attributes) compared to the case with no crack in the rock base (i.e., intact rock). Besides, the rebound kinetic energy of impactor also decreased significantly which is nearly 10 times of the case of rock base without any cracks.

Observing the kinetic energy variation trend with change in microcrack angle, $\alpha_f$ revealed a very complex response. For example, when the $\alpha_f$ value is 0°, the threshold minimum value of kinetic energy is lesser than the ideal case (i.e., intact rock). However, this behavior completely changes when $\alpha_f$ values of 30°, 45°, 60°, and 90° are considered. The slope of kinetic energy variation curve of the impactor is flatter during the rebound stage when an inclined angle is considered (expect when $\alpha_f$ = 90°) showing that the angle of microcrack increases the angular velocity component of the impactor leading to higher kinetic energy at the threshold minimum point. Furthermore, the rebound duration also changes slightly as the $\alpha_f$ value increases from 30° to 90°, respectively, while the final rebound value of kinetic energy is not predominantly influenced by change in the $\alpha_f$ value.

Fig. 6 provides a comparison of kinetic energy trends for the cases where the fracture width, $l_f$ is taken as 37.5 mm, or the thickness of the crack is increased to 2 mm or when dual microcracks with $\alpha_f$ values of 0° and 30° are considered (shown in Fig. 4(b)). The kinetic energy trends

<table><caption>**Table 3**

Summary of impact simulations presented in this study.</caption><thead><tr><th>Simulation ID</th><th>Number of micro cracks/descriptions</th><th>Angle of the micro crack, α<sub>f</sub> (degrees)</th><th>Length of the microcrack, l<sub>f</sub> (mm)</th><th>Thickness of the microcrack, t<sub>f</sub> (mm)</th></tr></thead><tbody><tr><td>I-PR-01</td><td>Intact rock</td><td>—</td><td>—</td><td>—</td></tr><tr><td>I-PR-02</td><td>Single</td><td>0</td><td>25</td><td>1</td></tr><tr><td>I-PR-03</td><td>Single</td><td>30</td><td>25</td><td>1</td></tr><tr><td>I-PR-04</td><td>Single</td><td>45</td><td>25</td><td>1</td></tr><tr><td>I-PR-05</td><td>Single</td><td>60</td><td>25</td><td>1</td></tr><tr><td>I-PR-06</td><td>Single</td><td>90</td><td>25</td><td>1</td></tr><tr><td>I-PR-07</td><td>Single</td><td>0</td><td>37.5</td><td>1</td></tr><tr><td>I-PR-08</td><td>Single</td><td>0</td><td>25</td><td>2</td></tr><tr><td>I-PR-09</td><td>Dual</td><td>0</td><td>25</td><td>1</td></tr><tr><td>I-PR-10</td><td>Dual</td><td>30</td><td>25</td><td>1</td></tr></tbody></table>
<figure><img src="image_7.png" /><figcaption>Fig. 5. Kinetic energy of the impactor variation against time for a rock base with single microcrack considering different values of microcrack angle, $\alpha_f$</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6. Kinetic energy of the impactor variation against time for a rock base with single and dual microcrack considering different microcrack geometries.</figcaption></figure>

for the case with intact rock base and single microcrack in the rock base with $\alpha_f$ values of 0° and 30° are also presented in Fig. 6 to provide a comparison of the respective parameters (size, angle, and number of micro cracks) on the kinetic energy trend of the impactor. The data suggest that increasing the width of crack from 25 to 37.5 mm results in

substantial increase in impact duration (nearly by 50 µsec) and a decrease in the rebound kinetic energy value by about 10 times. On the other hand, when thickness of the microcrack ($t_f$) is increased from 1 mm to 2 mm, the contact time increases significantly from mere 400 µsec to more than 800 µsec and the flattening of the curve with a very
negligible value of kinetic energy (<1e⁻⁴ J) may result in almost negligible rebound kinetic energy, leading to the impactor being embedded into the rock base. It is also noticed that when two microcracks are considered with α_f values of 0° and 30° (as shown in Fig. 4 (b)), the contact time and rebound kinetic energy changed noticeably. This indicates that the impactor's kinetic energy is influenced by alterations in the force/energy transfer mechanism, which is regulated by the microcrack geometrical parameters. These preliminary observations of the kinetic energy trend of the impactor give an understanding of the force/energy transfer mechanism taking place between the impactor and fractured rock base. Further investigation of the effect of fractures/fissures/cracks in the rock on its impact behavior will be presented in subsequent sections.

## 3.2. Assessment of the impact-induced damage in the fractured rock base

When an impactor collides with a rock mass, it imparts kinetic energy into the rock base, which is used to close any pre-existing cracks and cause elastic-plastic deformation in and around the area of impact. This results in the ejection of particles from the area of impact, creating a crater zone and visible cracks around the crater zone area (Lindqvist et al., 1994). If pre-existing cracks are present, they can influence the damage inflicted and crack propagation in the rock. For example, when a proppant simulant collides with the rock, bonds between the particles are broken and cause damage to the rock (as shown in Fig. 7(a)). If a single microcrack is present in the rock as shown in Fig. 7(b) and 7(c), the energy from the impact is dissipated through the movement of

<figure><img src="image_9.png" /><figcaption>Fig. 7. Sectional view of the particle movement before and after impact for the proppant-fractured rock system.</figcaption></figure>
broken particles into this microcrack, based on the angle of the crack. However, if two microcracks are present (as shown in Fig. 7(d)), the impact's influence on the second crack is minimal, as only the broken bonded grains move into the first microcrack, leaving the second microcrack largely untouched. Additionally, the presence of microcracks/fissures in the rock base may also cause other types of fractures such as wing cracks, secondary cracks, and coalescence cracks along the edges of the pre-existing cracks when the proppant simulant collides the rock base (Xu et al., 2017). All these factors cause a change in penetration and contact formation of the impactor (proppant simulant) with the rock base. As an additional contribution, these microcracks can alter the formation of the discrete fracture network, which can significantly affect the force/energy transfer process. The next two subsections attempt to carefully assess these prognostics in order to gain a multiscale perspective of the impact behavior of the proppant-rock system.

### 3.2.1. Proppant contact formation and penetration analysis

During the collision phase, the impactor penetrates the rock mass forming a crater zone (Kaysap et al., 2022). This subsection provides a discussion of the effect of the microcrack geometric parameters on the penetration of the impactor into the rock mass. Figs. 8 and 9 illustrate the penetration profile and contact formation trend for the proppant-fractured rock system considering variations in microcrack geometric parameters and numbers as presented in Table 3.

In Fig. 8, a comparison of the penetration profile as well as the contact formation trend of the proppant-fractured rock system for a single crack case (as shown in Fig. 4(a)) considering different values of $\alpha_f$ i.e., 0°, 30°, 45°, 60°, and 90° is presented. In general, the presence of pre-existing microcracks in the rock mass can have a significant impact on the penetration of the impactor. When a pre-existing crack is present, the impactor is able to penetrate deeper into the rock mass, and the time required to reach maximum penetration is also increased. For example, the maximum penetration depth for the case of impactor (proppant simulant) colliding an intact rock is 1.53 mm whereas the maximum penetration depth is in the range of 2.0 to 2.2 mm when pre-existing cracks are considered. Besides, the time required to reach maximum penetration increased from 140 µsec to around 250 µsec due to the

presence of pre-existing fracture. This is mainly attributed to the movement of the broken bonded particles into the pre-existing fractured space thereby decreasing the resistance to penetration. The penetration depth and contact time during a collision significantly affect the forces experienced and the overall energy transfer between the colliding bodies. A greater penetration depth can lead to increased plastic deformation, while longer contact times can reduce peak impact forces, as the force is distributed over a longer duration, minimizing the risk of damage to the materials involved. As the impact velocity decreases, the contact duration tends to increase, allowing for a more gradual transfer of momentum and energy, as noted by Luo et al. (2022a). Furthermore, the penetration depth and contact time influence the behavior of both the impactor and the block, including how they deform and dissipate energy (Goldsmith, 2001). In scenarios where friction is present, the contact time can also affect the frictional forces at play, further influencing the outcome of the collision (Johnson, 1985).

The number of contacts between the impactor and the rock mass increases due to the presence of the pre-existing crack, allowing more surface area of the impactor to come into contact with the rock base. The angle of the microcrack also plays a role in the penetration depth, with higher angles resulting in increased penetration depth and lower angles resulting in decreased penetration depth. For example, the maximum penetration depth increased from 2.0 mm to 2.16 mm as the $\alpha_f$ value increased from 0° to 60°. However, when $\alpha_f$ value of 90° was chosen, the penetration depth slightly decreased below 2.11 mm revealing that positioning of the crack influences the penetration depth thereby the damage inflicted to the rock base. When the contact formation trend of the impactor was observed for the above stated cases, in general, the number of contacts increased with the presence of the pre-existing crack, however, the angle of microcrack didn't have a significant influence as the contact formation trend shows a zig-zag profile for different values of $\alpha_f$. Specifically, for the case of single crack where $\alpha_f = 90°$, it can be clearly observed that the number of contacts decreased compared to other single crack cases as the impactor touches one of the edges of crack where particles are removed decreasing the contact number.

Fig. 9 shows the penetration depth profile and contact formation trend for the impactor in the rest of the cases presented in Table 3. It can

<figure><img src="image_10.png" /><figcaption>**Fig. 8.** Penetration profile and contact formation trend for the impactor (proppant simulant) colliding a rock base with single microcrack considering different values of crack angle, $\alpha_f$</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Fig. 9.** Penetration profile and contact formation trend for the impactor (proppant simulant) colliding a rock base with single and dual microcrack considering different microcrack geometries.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Crack/fracture propagation characteristics for the impactor (proppant simulant) colliding a rock base with single microcrack considering different values of crack angle, $\alpha_f$</figcaption></figure>
be seen that the increase of the length of the single crack from 25 mm to 37.5 mm resulted in a slight increase in the penetration depth from 2.0 to 2.15 mm, without much change in the corresponding time to reach maximum penetration. However, when the thickness of the crack was increased to 2 mm, the penetration depth increased significantly, by more than 40 % (i.e., from 2.0 to 2.8 mm) compared to the case with a single crack of 1 mm thickness and 25 mm length. Additionally, the contact duration increased to more than 800 µsec, which is significantly higher than all the other cases. The contact formation profile also followed the same trend, with the number of contacts increasing with the increase in the length of the crack or when the thickness of the crack was increased. On the other hand, when two cases of dual microcracks with $\alpha_f$ = 0° and 30° were considered (as shown in Fig. 4(b)), the penetration depth and number of contacts increased (as shown in Fig. 9), but the increase was minimal. This suggests that the cracks near to the rock surface have a predominant influence on the penetration of the impactor into the rock base.

### 3.2.2. Bond breakage analysis of impactor and fractured rock system

Fig. 10 illustrates the damage caused by proppant simulant on the fractured rock base for different cases of single crack i.e., for $\alpha_f$ = 0°, 30°, 45°, 60° and 90° using plots of the discrete fracture network, and fracture number. In the same figure, the evolution of fracture number with impact duration for the case of single crack considering different values of $\alpha_f$ is presented.

The graph shows that the number of bond breakages (fracture number) decrease with an existence of microcrack in the rock base and this trend is also influenced by the angle of the microcrack ($\alpha_f$). For example, the number of fractures reached a value 232,932 when the impactor collides with the intact rock base whereas this value ranges from 123,000 to 143,500 when a pre-existing microcrack (as shown in Fig. 4(a)) is at a closer depth below the top of the rock base is considered. A general observed trend is that the number of fractures (for the rock base with pre-existing crack) decrease with increase in $\alpha_f$ to a certain extent, and thereafter, starts to increase. The fracture number trend shows that the maximum fracture number decreased when $\alpha_f$ increased from 0° to 45°, however, when $\alpha_f$ values of 60° and 90° are considered, the trend gets reversed. It is difficult to predict the exact infliction point as it depends on various factors related to the impactor and block material and energy characteristics. This proves that there is a change in the fracture distribution (i.e., DFN formation) and force transfer mechanism with change in crack angle, $\alpha_f$.

Fig. 10 also shows the discrete fracture network (side view) for representative set of samples to illustrate visually the influence of the microcrack angle on fracture evolution. The discrete fracture network (DFN) formed during the impact process generally constitute a central crater zone (where majority number of fractures occur) along with side cracks, median cracks and radial cracks. These crack features are observed in the DFNs presented in Fig. 10. Visually comparing the DFNs, it can be observed that the crack pattern is clearly influenced by the existence of the microcrack. The existence of microcrack acted as local dampener dissipating the energy by contributing to the movement of broken bonded particles thereby decreasing the number of fractures in the rock base below the microcrack. When the graph plot of the crack number and the DFNs are combinedly observed, it can be noticed the angle of microcrack plays a major role in the local dampening process there by the damage inflicted to the fractured rock base. Another common observation from all the DFNs is that a concentrated cracks group emerges from the edges of the installed microcrack which represent wing cracks or secondary cracks which was also previously reported by previous researcher when analysing models related to fractured rocks, faults among others using experimental or numerical analysis (Xu et al., 2017; Li et al., 2021; Liu et al., 2022).

Similarly, an evaluation was done to determine the impact of other factors related to microcracks, such as size, thickness, and number of

microcracks, on crack progression during the impact process. Fig. 11 presents the fracture progression trend for all the cases listed in Table 3.

From Fig. 11, it can be observed that the increase in length of the microcrack from 25 mm (equal to impactor size) to 37.5 mm (1.5 times the impactor size) caused a minimal decrease (about 10 %) in the number of fractures whereas increasing the thickness of the microcrack from 1 mm to 2 mm drastically decreased the number of fractures by about 42.5 %, respectively. This may be attributed to the huge energy dissipation due to the microcrack (of higher thickness) preventing the fracture propagation in the rock base. The corresponding DFNs for the respective numerical tests (for the above mentioned two cases) also show difference in crack propagation due to the respective parameter change. For example, when the length of the crack is increased from 25 mm to 37.5 mm, the wing and secondary cracks (which are denser in the form of a group) that start from the edges only extend to lesser distance meeting the boundary decreasing the overall number of cracks. On the other hand, when the thickness of the cracks is increased instead overall propagation of the cracks decrease due to local damping of the energy thereby decreasing the overall number of cracks.

Another observation is that when two microcracks are considered (as shown in Fig. 4(a) and listed in Table 3), the crack evolution behavior is surprising, such as, the maximum number of fractures nearly increased 18–20 % when two microcracks separated by 25 mm with $\alpha_f$ values of 0° and 30° are considered compared to single crack case with $\alpha_f$ values of 0° and 30°, respectively. This may be due to the creation of weaker zone in between two microcracks resulting creation of extra stresses causing additional fractures. Specifically, when the curve for the case double microcrack with $\alpha_f$ values of 0° is observed, there is sudden change in curvature. This may be attributed to the triggered failure at weaker zone created in between the two microcracks. In addition, the discrete fracture network (DFN) associated with the two microcracks depicted in Fig. 11 had a significant impact on the fracture pattern and the propagation of the cracks to a larger area of the clastic rock base. Overall, when the all the microcrack cases are compared to the no microcrack case, it can be inferred that existence of a microcrack can decrease the fracture number significantly depending on the geometrical features and numbers of microcracks present in force transfer domain.

## 3.3. Kinetics of proppant-fractured rock system during impact

### 3.3.1. Stress transfer through the fractured rock base

In this subsection, the kinetics of the impact process between the proppant simulant and fractured rock base is analysed by assessing the stress transfer mechanism. When the impactor (proppant simulant) collides the rock base, there is a transfer of forces/stresses. This stress attenuates along the depth and width of the rock base due to the energy dissipation which occurs because of the particle damping or bond breakage and moment (sliding and rotation) of broken bonded particles or due to pre-existing cracks which act as local dampeners (Kasyap et al., 2022). In this study, specific focus is kept on the influence of pre-existing microcracks on the stress transfer occurring in the rock base. For this purpose, a measurement sphere of diameter 10 mm at depth of 37.5 mm (equal to 3 times the impactor radius) from the top of the clastic rock simulant is used to calculate the average normal stress at that level in the rock.

Fig. 12 illustrates the stress variation during the impact for a rock base with no cracks and a rock base with single microcrack with $\alpha_f$ values of 0°, 45° and 90°, respectively. From the figure, the solid line represents the stress in the impactor during the impact process whereas the dashed line represents the average stress at a depth of 37.5 mm (3 times the impactor radius or at three quarters of the rock base) from the top of the clastic rock base simulant. In general, it has been observed that the presence of a microcrack in the rock base decreases the stress in the impactor during the impact due to a decrease in the overall stiffness and strength of the rock base. Another specific observation is that the stress
<figure><img src="image_13.png" /><figcaption>Fig. 11. Crack/fracture propagation characteristics for the impactor (proppant simulant) colliding a rock base with single and dual microcrack considering different microcrack geometries.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 12. Normal Impact stress ($\sigma_i$) transfer trend from the impactor to the rock base with single microcrack considering different values of crack angle, $\alpha_f$</figcaption></figure>
variation profile of the impactor colliding a rock base with microcrack differs in terms of the curve profile. For example, the stress variation of the impactor increased with time to reach a maximum value when the impactor reaches maximum penetration, thereafter, it slowly starts to decrease consistently (with a nearly constant negative slope) to reach a minimum value (almost zero) as impactor rebounds and loses contact with the rock base. However, when the proppant simulant collides a rock base with a microcrack, the trend changes, with the stress variation of the impactor exhibiting a more complex profile. From Fig. 12, it can be seen that when a single microcrack with an angle of $α_f = 0°$ and $45°$ is present in the rock base, there is a distinct bump along the stress decrement path, or the rebound path of the impactor. This bump is more profound when $α_f = 0°$ is considered which was likely caused by the impactor force being orthogonal to the microcrack, resulting in a unique broken particle movement and a sudden resistance. Changing the $α_f$ (e. g., $α_f = 45°$) also affected the bump formed along the stress variation profile. On the other hand, when the rock base has a single microcrack with an angle of $α_f = 90°$, the stress profile is similar to the ideal case with no microcrack, but with a lower impactor stress.

Fig. 12 shows the average vertical/normal stresses at the bottom center of the rock base (at a depth of 37.5 mm from the top) using dashed lines to illustrate the effect of microcracks on the rock base. Generally, it is expected that the presence of a microcrack decreases the stress generated at the bottom of the rock, compared to the stress generated in an intact rock (as shown in Fig. 12). Comparing the vertical stress trends for different values of $α_f$, it is evident that the average stress values are lowest when $α_f = 0°$, followed by the single crack case when $α_f = 90°$ and $α_f = 45°$ in third position. This suggests that microcracks act as effective local dampeners when they are perpendicular to the applied force, and that vertical microcracks ($α_f = 90°$) perform better as dampeners than inclined microcracks.

Fig. 13 presents the stress profiles for the rest of the cases of microcracks as mentioned in Table 3. It can be seen that increasing the length of the microcrack from 25 mm to 37.5 mm or the thickness of the

microcrack from 1 mm to 2 mm resulted in a steeper stress drop with a very small bump along the stress decrement path or the rebound path of the impactor.

Additionally, considering two microcracks (as shown in Fig. 4(b)) did not influence the impactor stress, possibly due to one of the microcracks being out of the influencing domain of the impactor force/stress. On the other hand, when looking at the average stresses developed at the bottom center of the rock base, it can be seen that increasing the length or thickness of the microcrack does result in lower stress values (as shown in Fig. 13), though the trend is not consistent. When two microcracks are considered (as seen in Fig. 4(b)), it appears that the stress profile does not vary significantly, indicating that microcracks can only act to absorb energy when they are close to the top of the rock base. This is further supported by the results of Fig. 7, which showed broken particles entering the crack space. This suggests that microcracks can be effective at dampening the energy of an impactor when they are in close proximity to the point of impact.

### 3.3.2. Energy characteristics of proppant-rock system

From an energy perspective, the impact process is a conversion of kinetic energy of the proppant simulant into other forms through energy exchange between proppant simulant and rock base. In the simulations of the proppant-rock system, the proppant is initially given a velocity that imparts kinetic energy, while the rock is stationary and has zero energy. During the impact simulation, the proppant collides with the rock, transferring energy between the two systems. This energy exchange causes the rock to deform and fracture, and the proppant's kinetic energy is dissipated. The important energy components for a bonded particle system (e.g., clastic rock) considered in this study are contact strain energy ($E_s$), bond strain energy ($\bar{E}_s$), kinetic energy ($E_k$), friction energy ($E_f$), and slip energy/damping ($E_d$). Of these components, the first two are internal energy components, meaning they are stored in the rock, while the rest are dissipative energy components. The energy equilibrium equation (Wang et al., 2012) is given as follows:

<figure><img src="image_15.png" /><figcaption>Fig. 13. Normal Impact stress ($σ_i$) transfer trend from the impactor (proppant simulant) to the rock base with single and dual microcrack considering different microcrack geometries.</figcaption></figure>
$$E = E_s + \bar{E}_s + E_k + E_f + E_d \quad (1)$$

In the above equation, the kinetic energy component, $E_k$ can be further divided into imparted kinetic energy of impactor (proppant simulant) ($E_k^i$) and kinetic energy of the block (clastic rock base) ($E_k^b$). This allows for a more precise assessment of each component's contribution to the overall kinetic energy. The mathematical formula for these variables can be found elsewhere in PFC manual (Itasca Consulting Group, 2016). In this study, the energy dynamics of the proppant-rock system is examined, with a particular emphasis on the influence of microcracks in the clastic rock on the energy evolution. The energy evolution of all the energy components is normalized by the imparted initial kinetic energy ($E_k^i$) to assess qualitative energy characteristics during the impact process. For convenience, only important cases which show a major difference in energy distribution trend are presented in this paper.

Fig. 14 shows the energy evolution trend for a representative set of impact simulations, with the subfigures compared to assess the effect of the existence of a single microcrack in the rock base on the energy distribution characteristics during the impact phase. It can be observed

from the subfigures that the applied kinetic energy of the impactor ($E_k^i$) decreases over time to reach a minimum limiting value, and then starts to increase to reach a steady value as the impactor rebounds with the reaction kinetic energy imparted by the rock base onto the proppant simulant. However, this imparted reaction kinetic energy is very small in all the cases due to huge plastic deformation caused due the high velocity impact, weak clastic rock base and the presence of microcracks.

Comparison of Fig. 14(a) (intact rock) with other subfigures reveals that the existence of a microcrack contributes to the rock base kinetic energy, allowing the movement of broken bonded particles into the crack space. The area under rock base kinetic energy is highest when single microcrack with $\alpha_f = 0^\circ$ is considered and decreases as $\alpha_f$ increases from $0^\circ$ to $90^\circ$. On the other hand, the contact strain energy decreases with the introduction of microcrack, which may be due to the energy dissipation caused due to the movement of particles to the microcrack. However, the area under contact strain energy is only slightly affected with the increase of $\alpha_f$ value from $0^\circ$ to $90^\circ$. The position of the peak, however, varies due to the completely different bond breakage patterns (shown in Fig. 10), which represent the changing

<figure><img src="image_16.png" /><figcaption>Fig. 14. Energy distribution in an proppant-rock system with single microcrack comparing the effect of change in microcrack angle, $\alpha_f$</figcaption></figure>
force chain or varied damage inflicted in each case.

Another observation is that the bonded strain energy contribution is negligible in all the cases owing to the weak type of clastic rock simulant considered and the high velocity impact, which resulted in huge force

transfers through the bonded particle in the force network leading to bond breakage rather than energy storage. The other dissipation energy components friction and damping though influenced by the existence of crack, however, the effect of crack angle, $\alpha_f$ on their values is not pre-

<figure><img src="image_17.png" /><figcaption>**Fig. 15.** Energy distribution in an proppant-rock system for a rock base with a) Single microcrack with parameters, $\alpha_f = 0°$, $l_f=25$ mm, $t_f = 2$ mm b) Dual microcracks with parameters, $\alpha_f = 0°$, $l_f=25$ mm, $t_f = 1$ mm, $s_f = 25$ mm.</figcaption></figure>
dictable. For example, comparing Fig. 14(a) and 14(b) shows that the damping contribution increased from 0.42 to 0.58 and friction decreased from 0.43 to 0.36 due to the existence of horizontal microcrack. On the other hand, as the microcrack angle changes, there is some change in the damping and friction values (comparing Fig. 14(b),14(c) and 14(d)), however, the exact reason for this small change is unpredictable owing to the complex change in the force and discrete fracture network caused due to the existence of micro crack along different planes and whose pattern changes with change in microcrack angle.

Similar assessment is conducted on the other two important cases where the thickness of the crack is increased from 1 to 2 mm and when double cracks are considered. Fig. 15(a) shows the energy distribution among different components for the case of single crack of 2 mm thick and with a $\alpha_f$ value of 0°.

Comparing the respective energy components shown in Fig. 15(a) with Fig. 14(a) and 14(b), reveal that the contact strain energy and rock base kinetic energy are considerably influenced by increasing the microcrack thickness. For example, the peak of contact strain energy decreased from 0.276 for intact rock to 0.136 when the thickness of microcrack is 1 mm but this value changed minimal (to about 0.127) with increase in thickness of microcrack to 2 mm, however, the area of contact strain energy decreased with increase in thickness. On the other hand, the kinetic energy of the rock base increased from a negligible value to bell shaped curve with peak 0.09 when a microcrack of thickness 1 mm was introduced. This value of peak further increased to 0.12 with increased area when the thickness of microcrack was increased to 2 mm. The other dissipation energy variables, such as damping and friction, also showed a reasonable variation when the thickness of the microcrack was increased (comparing Fig. 14(b) and 15(a)). This increased thickness had a positive effect on the former and a negative effect on the latter. Furthermore, when comparing the energy evolution of the impactor-block system with a single crack in the rock base (Fig. 14(b)) to the one with double cracks in the rock base (Fig. 15(b)), it is observed that the second microcrack has minimal influence on the energy distribution parameters. These findings are in line with the observations from the previous subsections.

## 4. Conclusions

The purpose of this study is to explore the mechanics and dynamics involved in the collision of an impactor (proppant simulant) with a fractured rock base. Specifically, the study aims to investigate the influence of pre-existing microcracks in the base rock on the damage inflicted by the impactor (proppant simulant). A parametric analysis was conducted to examine the influence of various microcrack characteristics (e.g., size, number, angle, position) on the collision process. To replicate the interaction between sand and bonded silt particles (which form sandstone or shale variants), a scaled model was used with an impactor size 27.78 times larger than the average size of the bonded particles. Each parameter was individually studied to better understand the basic dynamic and fracture mechanisms that may occur in proppant-fractured rock systems, which can be encountered in scenarios such as an exploration of an unconventional reservoir, rockfall problems or any physics problems involving impacts between large grain sizes and bonded rock masses.

From the analysis, it was found that pre-existing microcracks and their angles greatly impact the damage inflicted by an impactor (proppant simulant) on the fractured rock. Our findings demonstrate that pre-existing microcracks significantly influence the force or energy transfer zone, resulting in a greater kinetic energy loss and possible embedment of the impactor into the rock base. Additionally, the angle and thickness of the microcrack play a crucial role in the impactor's kinetic energy transfer, with higher angles and increasing thickness leading to deeper penetration (and contact formation) and greater kinetic energy loss. However, this pattern slightly varies when $\alpha_f = 90°$ where the microcrack is along the line of application of impact force. Furthermore, our

findings suggest that double microcracks have minimal impact, highlighting that pre-existing crack near to the rock surface have a more significant influence on the impactor's penetration. On the other hand, examining the impact of microcracks on the number of bond breakages or fracture number in the rock base caused by an impactor revealed that microcracks act as local dampeners, reducing the number of fractures in the rock base below them. The angle of the microcrack affects fracture number, with a maximum decrease at $\alpha_f$ values of 0° to 45° but a reversal of this trend at $\alpha_f$ values of 60° and 90°. The existence of microcracks also resulted in the emergence of a concentrated group of wing or secondary cracks from the edges of the installed microcracks, which can affect crack propagation during impact. Besides, our analysis showed that thicker and longer microcracks cause fewer fractures due to energy dissipation and the presence of two microcracks may create a weaker zone and lead to additional fractures compared to a single microcrack.

The present numerical results suggest that the presence of a microcrack reduces overall stiffness and strength of the rock base and also causes a noticeable bump along the stress decrement path which is not observed for the case of intact rock during impact. The microcrack angle affects this bump expect when the angle is at 90°, where the stress profile is similar to the ideal case with no microcrack but with a lower impactor stress. On the other hand, comparison of stress at the bottom of the rock base reveal that microcracks act as effective local dampeners when they are perpendicular ($\alpha_f = 0°$) to the applied force, and that vertical microcracks ($\alpha_f = 90°$) perform better as dampeners than inclined microcracks. Increasing the length or thickness of the microcrack results in lower stress values at the bottom center of the rock base, but the trend is not consistent. Two microcracks do not have a significant impact, and microcracks near the top of the rock base perform best, absorbing energy. Broken particles entering the crack space support this inference. Furthermore, the study demonstrated that the presence of a microcrack can decrease the applied kinetic energy of an impactor due to plastic deformation, weak rock base, and microcracks leading to rebound with limited reaction kinetic energy. The existence of a microcrack contributes to the rock base kinetic energy whose value decreases as $\alpha_f$ increases which is almost negligible for an intact rock base. Microcracks decrease contact and bond strain energy and friction and increase damping and kinetic energy of the rock base. Thicker microcracks affect the distribution of energy components, with a significant influence on contact strain energy, rock base kinetic energy and friction but minimal impact on damping. Besides, the introduction of a second microcrack has minimal impact on energy distribution parameters.

Despite providing valuable insights, this study has limitations and opportunities for future research. Future work could focus on simulating impacts on fine-grained rocks like mudrocks, which are computationally challenging due to their complex microstructures. Additionally, incorporating fluid-driven fracture propagation and studying proppant transport in complex fracture geometries using coupled CFD-DEM and FEM-DEM methods could further enhance understanding, particularly for applications in unconventional reservoirs.

### CRediT authorship contribution statement

**Sarath C.R. Nallala:** Writing – review & editing, Writing – original draft, Validation, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Huan He:** Writing – review & editing, Supervision, Software, Investigation, Conceptualization. **Kostas Senetakis:** Writing – review & editing, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization.

### 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.

Acknowledgements

The present study was supported by the SRG-Fd grant, project no. 7005901 (CityU 11210522), the APRC grant, project no. 9610604, and the SIRG grant, project no. 7020027. The finance support of these grants is acknowledged.

Data availability

Data will be made available on request.

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