Simulations of hydrofracking in rock mass at mesoscale using fully coupled DEM/CFD approach
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Abstract
The paper deals with twodimensional (2D) numerical modelling of hydrofracking (hydraulic fracturing) in rocks at the mesoscale. A numerical model was developed to characterize the properties of fluiddriven fractures in rocks by combining the discrete element method (DEM) with computational fluid dynamics (CFD). The mechanical behaviour of the rock matrix was simulated with DEM and the behaviour of the fracturing fluid flow in newly developed and preexisting fractures with CFD. The changes in the void geometry in the rock matrix were taken into account. The initial 2D hydrofracking simulation tests were carried out for a rock segment under biaxial compression with one injection slot in order to validate the numerical model. The qualitative effect of several parameters on the propagation of a hydraulic fracture was studied: initial porosity of the rock matrix, dynamic viscosity of the fracking fluid, rock strength and preexisting fracture. The characteristic features of a fractured rock mass due to a highpressure injection of fluid were realistically modelled by the proposed coupled approach.
Keywords
CFD DEM Fluid flow Fracture Hydrofracking Mesoscale Rock1 Introduction
 (a)
explanation of the complex mechanism of the initiation and propagation of fractures in rocks due to the activity of the high fluid pressure (higher than the tensile strength of rocks) during hydrofracking (by taking the existing prediscontinuities and voids into account) and
 (b)
description of this mechanism at the mesoscale by applying an advanced mathematical model, based on the threedimensional (3D) discrete element method (DEM) fully coupled with fluid flow (the socalled a coupled discretecontinuum methoddiscrete and continuous domains coexist in one system).
Since hydraulic fracturing strongly depends on the heterogeneous mesostructure of rocks, the discrete element method (DEM) is a suitable numerical tool for investigating a nonuniform formation process of complex fracture patterns at the mesoscopic level. The fracturing fluid system is assumed to be continuous and is described by locally volumeaveraged Navier–Stokes equations to be solved by computational fluid dynamics (CFD). In general, the numerical coupled calculations will be carried out for a twophase (fluid and gas) turbulent flow of incompressible viscous fluid by taking the mass, momentum and heat transport into account in existing and newly developing fractures in rocks. In the first research step, we developed a simplified dynamic hydromechanical mesoscale model of hydrofracking, based on DEM and CFD to characterize the properties of fluiddriven fractures in rocks. DEM was used to capture the mechanical behaviour of the rock mass that was represented by a set of representative discrete spherical elements interacting through elasticbrittle bonds that can break to form fractures. It was coupled with fluid mechanics via the computational fluid dynamics (CFD) to describe laminar fracturing fluid flow in voids and channels between discrete elements using a novel socalled Virtual Pore Network (VPN) approach.
The aim of the current paper is to present simulation results on checking the capability of the fully coupled simplified approach DEM/CFD to describe the process of hydrofracking in rocks at the mesolevel. The approach was formulated for laminar incompressible viscous onephase fluid flow in isothermal conditions. A series of simple twodimensional (2D) hydraulic fracturing simulations were performed on a small rock segment subjected to a pressurized fracturing fluid under biaxial compression conditions. This small rock segment had a very simplified particulate mesostructure and contained solely one injection slot. The qualitative effect of the following parameters on the hydraulic fracture geometry, fracturing fluid velocity and pressure was studied in 2D simulations: initial microporosity of the rock matrix, rock matrix strength, dynamic viscosity of the fracking fluid and presence of a preexisting macrocrack. The innovative elements of our approach with respect to other existing DEM/CFD models in the literature are the following: (1) coexistence of two domains (a discrete and continuous one) in one physical system (the sum of domain geometries creates a complete physical system), (2) precise determination of the geometry and topology change of voids and fractures during hydrofracking, (3) remeshing method of voids and fractures, (4) transformation schema of computation results from the old grid (before remeshing) to the new grid (after remeshing) and (5) detailed tracking of the fluid volume fraction in voids and fractures (rock voids can be fully or partially filled with the fluid). In our approach, every single pore is discretized by a number of triangles. Thus, the pressure field in a single pore is spatially variable while in existing DEM/CFD models, the pressure field in a single pore is constant. The flow path is also reproduced in a single pore in contrast to existing DEM/CFD models. In addition, the huge pressure gradients in a single pore are captured while in existing DEM/CFD models the pressure gradient in a single pore is equal to zero. Thus, the resulting forces from the fluid are more realistic on our approach (magnitude and direction). The hydraulic fracture propagation can be investigated in our approach in rocks that are initially partially filled with the fluid (what is more realistic). In addition, the particles floating (e.g. proppant particles) in fractures prefilled with the fluid may precisely be traced.
The modelling of the fluiddriven fracture propagation in rocks comprises the coupling of different physical mechanisms including deformation of the solid skeleton induced by the fluid pressure on fracture surfaces, flow of the pore fluid along new fractures and through the region of surrounding existing fractures and pronounced temperature changes. Hydraulic fracturing results in complex fracture patterns composed of preexisting and new ones that greatly influence the process at the global scale [40]. Due to difficulties in performing experimental works on the fracture network propagation in situ and at the laboratory scale, the numerical modelling becomes an essential tool in analyses of hydraulic fracturing [61].

multiphase fluid flow in both macrovoids, microvoids and prediscontinuities of rocks (faults, veins, bedding planes) that may be partially filled with both a fluid and gasphase. The rock pressure is close to the hydrostatic pressure at the borehole depth,

nonisothermal conditions, wherein both the temperature difference between the wellbore fluid and rocks and rapid size changes of fractures cause pronounced thermal stresses,

high velocity of a hydraulic fracture propagation process that contributes to extremely large and rapid topology changes of voids and fractures.
There are two main approaches for modelling the propagation of hydraulically driven complex fracture patterns: (1) continuumbased models and (2) discontinuous mesoscale models at the grain level. The continuumbased models [5, 21, 41, 57] use continuum formulations and coarsegrid meshing for the fluid part. The fluid flow and solid–fluid interactions are defined at the mesoscale using empirical relations (e.g. Darcy’s law). There is no direct coupling at the local scale: forces acting on the individual particles are defined as a function of a mesoscale averaged fluid velocity obtained from permeability–porositybased estimates. The solution of the continuum problem provides a fluid velocity and momentum exchange between two phases at each node of the mesh. The individual particle behaviour is not accurately reproduced, and this fact limits their application to problems such as strain localization, microcracking, local heterogeneities in porosity and internal erosion by transport of fines that are all inherently heterogeneous on the microscale. The solutions can be obtained with classical numerical methods such as finite differences, finite elements and finite volumes. A great advantage of the continuumbased models is an affordable CPU cost. However, they need a series of phenomenological assumptions that rely on a number of empirical relationships. In consequence, they require a new calibration procedure for each new type of rock mesostructure. The continuumbased mesoscale models are obviously unable to fully describe mesoscale coupled thermal, hydraulic and mechanical effects [5]. In addition, continuum models usually assume a homogeneous and an isotropic rock structure that is not realistic [9] and thus, the models do not capture interaction between the hydraulic fracturing and discrete fracture network. It is of major importance to take into account a heterogeneous mesostructure of rocks and a realistic pattern of preexisting discontinuities that affect the shape and range of hydraulic fracture propagation. Summing up, the main drawback of currently used continuum hydraulicmechanical models [5, 9, 10, 15, 18, 21, 41, 56, 57, 61, 64] is the lack of the detailed treatment of the geometry at the mesostructural level. In addition, the rock system is initially fully prefilled with the fluid and only onephase fluid flow is taken into account. Simple approaches are used to track fluid flow in voids and fractures.
As compared with usual continuum mechanics methodologies in most of existing numerical studies, discontinuous mesoscale models at the grain level [such as the discrete element method (DEM)] are more realistic since they allow for a direct simulation of the mesostructure and are very useful for studies of local physical phenomena at the mesolevel such as the mechanism of the initiation, growth and formation of cracks [37, 40, 54]. DEM allows for a better understanding of local mesostructural phenomena that evidently affect a macroscopic rock behaviour. The strength and deformational features of rock masses are strongly affected by persistence, spacing, orientation and mechanical properties of geological structures. It is thus essential to accurately describe the fundamental behaviour of preexisting structures when studying the stability of a rock mass. The first DEM model for rocks was formulated by Potyondy and Cundall [43]. Rock fracturing was captured by the rupture of bonds whose strength was characterized by a constant maximum acceptable force in tension and a cohesivefrictional maximum acceptable force in shear. Different types of bonds were proposed to simulate discontinuities [31, 35, 45]. The most universal contact formulation was proposed by Scholtès and Donze [45], based on the identification and reorientation of each discrete element interaction that crossed the plane representing a discontinuous surface.
Various methods were developed to model fluid flow in pores and fractures at the grain level when using DEM. The commonly used approach to describe fluid flow and predict interaction mechanisms between flowing fluid and particles is the pore network modelling. It assumes that fluid flows through channels connecting pores that accumulate pressure. A simplified laminar viscous Poiseuille flow is assumed in channels [32, 47, 60] and no flow or Stokes flow is taken in pores [4, 40, 55]. The Poiseuille flow in channels describes laminar Newtonian fluid flow between two plates (in general nonparallel) and is based on Reynold’s equations of a classical lubrication theory. The Stokes flow takes place in pores when advective inertial forces are small as compared with viscous forces (the Reynolds number less than 1). The pore network model is built through a weighted Delaunay triangulation over the discrete element packing. Preexisting fractures and hydraulically driven fractures are treated as a series of channels connected facetoface or by cell grids, depending on fluid flow model. The finite volume method is applied to solve the governing equations of motion. In this approach, the edges of triangles connect the gravity centres of discrete elements. The geometry of a fluid domain (voids) is thus not accurately reproduced. The triangles do not reproduce a geometry but volumes of voids. A single triangle covers partially particles and a void. The void volumes are the difference between the areas of triangles and areas of particles in triangles and next multiplied by 1 m (in the zdirection). One pore is always discretized by one triangle. This approach is also called the porescale finite volume (PFV) method [4, 55]. The model may describe either incompressible or compressible fluid. Due to its simplicity, PFV overcomes the problem of the high computational cost of mesoscale models without introducing all phenomenological assumptions of continuumbased methods [23]. The models meet the following simplifying assumptions [4, 32, 40, 47, 55, 60]: (a) the singlephase fluid flow (full saturation) in voids and fractures; the only phase is a fluid (the gasphase is neglected), (b) the incompressible viscous laminar fluid flow (turbulent flow conditions are neglected) and (c) isothermal conditions in rock and flowing fluid (thermal stresses are not taken into account).
There exist also several combined solutions using DEM connected to the Smooth Particle Hydromechanics (SPH) approach to describe the fluid behaviour [2, 14, 34, 44, 53]. Other coupled approaches which were successfully applied to fluidparticle system simulations is DEM combined with the lattice Boltzmann method [3, 33] that comes from the kinetic theory of gases. However, this approach requires huge computation time and is rarely employed in modelling of hydrofracking. Recently several coupled DEM/CFD simulation results were described in different hydromechanical engineering problems [12, 30, 46, 58, 59, 62, 63].
The paper is arranged as follows. After Introduction (Sect. 1) and a brief overview of existing continuous and discontinuous approaches for simulating hydrofracking in rocks, the discrete element method is summarized in Sect. 2 and the fluid flow model in Sect. 3. The coupling of DEM/CFD is discussed in Sect. 4 and the model calibration in Sect. 5. Section 6 reports on numerical study results of hydrofracking in a rock specimen. Finally, some concluding remarks are offered in Sect. 7.
2 DEM for rocks
The advantage of the mesoscale modelling (in particular under 3D conditions) is that it directly simulates mesostructure and thus may be used to comprehensively study the mechanism of the initiation, growth and formation of localized zones and cracks that greatly affect the macroscopic behaviour of frictionalcohesive materials. It may also be used for studying different local phenomena at the aggregate level (e.g. force chains and vortexstructures) to predict the early fracture process [24, 36]. The disadvantages are the huge computational cost.
In general, the DEM material constants are calibrated with the aid of simple laboratory tests on the material (uniaxial compression, uniaxial tension, shear, biaxial compression). The detailed calibration procedure for frictionalcohesive materials (e.g. concrete) was described in [36, 37] based on real simple standard laboratory tests (uniaxial compression and uniaxial tension) of concrete specimens. The calibration process consisted of running several uniaxial tension and uniaxial compression simulations on a given assembly of discrete elements with the same material constants to reproduce the selected experimental behaviour.
3 Fracturing fluid flow model
3.1 Fluid flow model in channels
In real 3D problems, the fluid flows around the spheres in contact. However, in 2D problems, there is no free space for fluid flow. Therefore, the concept of virtual channels, called S2S, was introduced. The S2S channels connected the centres of gravity of two triangles which were separated by the spheres in contact (Fig. 5b). They were located along a contact line between two neighbouring deformed or nondeformed spheres. Usually, the S2S channels did not coincide with the contact lines between spheres (they intersected one triangle’s edge only). However, if the S2S channel crossed the vertices of triangles, the flow rate was uniformly divided into adjacent edges. The S2S channel existed in the system until the contact between the spheres was lost. After that, free space occurred that was next discretized by the new triangles.
3.2 Fluid flow model in VPs

flow regime was stagnant (the Reynolds number Re ≪ 1) and

fluid was barotropic (\(\rho = \rho (P)\)).
There exist 3 terms affecting the pressure (Eq. 19): the sum of the volumetric fluid flow rates in the channels, the time derivative of the volume and the internal source. The sum of the volumetric fluid flow rates in the channels was the result of fluid flow and was computed in CFD. The time derivative of the volume was due to the counter forces acting on the fluid. It was computed in DEM and next transferred to CFD. The internal source was solely used to define boundary conditions (volumetric flow rate) in boundary pores.
3.3 Discretization and large grid deformations
The discrete and continuous domains were discretized into a triangular grid. The triangular grid of the continuous domain was used to model fluid flow. The algorithm of discretization was based on the Delaunay triangulation. In the first step, the circumference of spheres and voids was discretized. The number of vertices along the circumference was a parameter set by the user. In the second step, the contact lines (Fig. 5b) between the deformed spheres were identified and the additional vertices of triangles were generated. When the distance between some vertices was too small (defined by a critical value), one of those vertices was removed to avoid too small triangles in the final mesh. Finally, the S2S and T2T channels were generated (Figs. 3, 4). The discretization algorithm directly influenced the fluid flow network. The denser was the grid, the more precise fluid flow path was reproduced.
The following material constants are needed for the CFD simulations: reference pressure \(P_{0}\) (Pa), fluid density \(\rho_{0}\) for the reference pressure \(P_{0}\) (kg/m^{3}), fluid bulk modulus K and dynamic fluid viscosity μ (Pa s). The inverse of the bulk modulus gives a fluid compressibility C (1/Pa).
4 Hydromechanical coupling
The extension of the VPN model from 2D into 3D is straightforward. The discrete and continuous domains are discretized using tetrahedrons under 3D conditions (instead of triangles in 2D conditions). The S2S virtual channels are not required (the T2T channels exist only). In 3D, the width of channels is related to the geometry of tetrahedron’s faces (in 2D, the width of S2S and T2T channels is equal to 1 m). The mathematical model of fluid flow is the same under 2D and 3D conditions.
5 Model calibration
5.1 Pure DEM calibration
The real mesostructure of the rock mass was not taken into account. Instead an extremely simple 2D DEM onephase mesoscopic model with spheres was assumed to approximately reproduce the rock mass behaviour at the mesoscale. The specimen included one row of spherical particles in depth. The spheres’ diameter was between 0.7 and 1.3 mm (with the mean diameter equal to d_{50}= 1 mm). An arbitrary microporosity can be achieved in DEM due to that the particles may overlap. The initial microporosity was assumed as p = 1% (corresponding, e.g. to shale rocks). The spheres were put into the box that corresponded to the rock specimen size and shape (with the interparticle friction angle of μ_{c}= 0°) until the desired initial microporosity was obtained. The spheres were allowed to settle until their total kinetic energy became insignificant. Next, all forces between the spheres were deleted due to the particle penetration U and μ_{c} was set on the target value [37]. In order to take the starting configuration into account, the initial overlap was subtracted in each calculation step when determining the normal forces (\(\vec{F}_{\text{n}} = K_{\text{n}} \left( {U_{n}  U_{0} } \right)\vec{N}\), where U_{o} is the initial overlap and U_{n} the overlap in the calculation nsteps). The following material constants were mainly used in all DEM analyses for rock specimens (Sect. 4): E_{c}= 3.36 GPa, υ_{c}= 0.3, C_{c}= 170 MPa and T_{c}= 34 MPa (C_{c}/T_{c}= 5), μ_{c}= 18° and ρ = 2.6 kG/m^{3}. The damping parameter α_{d}= 0.10 did not affect the results [27, 28, 50, 51]. The material constants were calibrated in order to approximately describe laboratory test results for shale rock [17] during quasistatic both uniaxial compression and tension splitting. About 10,000 spheres were used in calculations. For uniaxial compression, the quadratic specimen 100 × 100 mm^{2} was assumed. The bottom and top of specimen were smooth. The prescribed vertical displacement was applied along the top boundary with the constant load velocity of 2 mm/s. For tension splitting, the circular specimen was used with the diameter of D = 100 mm. The vertical displacement was prescribed along the specimen top and bottom boundary by two rigid cylinders to eliminate the effect of boundaries [51] (with the constant load velocity of 2 mm/s).
5.2 CFD calibration
The equilibrium state was reached for the fluid velocity of 0.016 m/s for the case ‘1’ and 0.008 m/s for the case ‘2’. The calculated permeability coefficients of the rock matrix were: \(\kappa = 1.092 \cdot 10^{  17}\) m^{2} for the case ‘1’ (value typical for rocks like fresh limestone and dolomite) and \(\kappa = 2.165 \cdot 10^{  15}\) m^{2} for the case ‘2’ (value typical for sandstone). A realistic onedimensional fluid flow was obtained at the macroscopic level (Fig. 12b). The pressure isolines (Fig. 12c) were almost parallel to the bottom and top wall of the rock specimen (they were qualitatively the same for the cases ‘1’ and ‘2’). The aperture constant β = 0.9 (corresponding to fresh limestone) was chosen for further simulations. Note that the calibration of β has to be always carried out for the specified rock.
5.3 Dependence on time step
Basic material constants assumed for rock matrix and fluid in coupled DEM/CFD calculations of hydrofracking
Symbol  Value  Unit  

Material constants for rock  
Modulus of elasticity of contact  \(E_{\text{C}}\)  3.36  GPa 
Poisson’s ratio of contact  υ _{c}  0.35  – 
Cohesive stress  \(C_{\text{C}}\)  170  MPa 
Tensile normal stress  \(T_{\text{C}}\)  34  MPa 
Interparticle friction angle  μ _{c}  18  ^{o} 
Mass density  ρ  2.6  kG/m^{3} 
Microporosity  p  1  % 
Material constants for fluid  
Dynamic viscosity  μ  4.06·10^{−4}  Pa s 
Bulk modulus/compressibility  K/C  2.5·10^{9}/4.0·10^{10}  Pa/1/Pa 
Reference pressure  P _{0}  0.1  MPa 
Density at reference pressure  ρ _{0}  977.36  kg/m^{3} 
Channel width  \(h_{ \inf }\)  \(2.5 \cdot 10^{  8}\)  m 
Channel width  \(h_{0}\)  \(3.25 \cdot 10^{  7}\)  m 
Aperture coefficient (Eq. 10)  β  0.9  – 
Reduction facture (Eq. 11)  \(\gamma\)  0.01  – 
Permeability coefficient  \(\kappa\)  \(1.092 \cdot 10^{  17}\)  m^{2} 
Fluid volume fraction  α  0.98  – 
The fluid pressure along the main flow path in the hydraulic fracture is shown in Fig. 13b. The maximum pressure difference of 2.54 MPa (4.2% of the maximum pressure in the main fluid flow path in the hydraulic fracture) was obtained between the time step \(\Delta t_{\text{D}} = 5 \cdot 10^{  8}\) and \(\Delta t_{\text{D}} = 1 \cdot 10^{  9}\). However, a significant difference in the computing time t_{c} was registered (t_{c}= 2 h for \(\Delta t_{\text{D}} = 5 \cdot 10^{  8}\) and t_{c}= 23 h for \(\Delta t_{\text{D}} = 1 \cdot 10^{  9}\)). The computing time was not directly proportional to \(\Delta t_{\text{D}}\) due to the use of the adaptive algorithm in CFD calculations (the shorter the DEM time step \(\Delta t_{\text{D}}\), the less CFD time substeps were needed). The DEM time step of \(1 \cdot 10^{  8}\) s was always assumed in the simulations. The CFD time step was about 2–10 times smaller.
6 Coupled DEM/CFD simulation results
6.1 Initiation and propagation of hydraulic fracture in rock matrix
The hydraulic fracture occurred at the injection slot and propagated in a vertical direction up to the specimen top with some branches in the upper region caused by the specimen heterogeneity (Fig. 15). The final crack width for t = 5.51 ms varied between 0.75 mm and 1.3 mm. Initially, no fluid flow occurred since the fluid needed some time to flood first the microvoids. The clear hydraulic fracture initiated for t = 0.29 ms. The fluid started to damage the rock matrix for t = 2.80 ms, when the pressure was higher than the rock strength and initial confining pressure. The mean fluid velocity was 0.5 m/s (R_{e}= 0–400) (Fig. 16). In the final stage of the hydraulic fracture process (t = 5.51 ms), some large fluid velocity jumps were obtained. The mean fluid velocity was 3.8 m/s; the fluid velocity locally increased up to 11.5 m/s. The maximum and mean Reynolds numbers were 20,300 and 4000 (Fig. 16dC), respectively, i.e. below the limit value of 10^{5}. The fluid zone width was higher than the macrocrack width (up to 5 mm at the maximum in some regions) since the fluid slightly leaked out beyond the hydraulic fracture.
6.2 Effect of initial microporosity of rock matrix
The fluid leakage became also higher with growing initial microporosity. The branching of the macrocrack already happened at the specimen bottom for the high p (p = 10–15%, Fig. 17a, b). The secondary macrocrack was almost horizontal for those cases. The fracture width was about 7 mm for p = 15%.
6.3 Effect of dynamic viscosity of fracking fluid
The results show that the smaller the dynamic viscosity, the faster the hydraulic fracture propagates. The macrocrack length of 20 mm was reached for t = 1.50 ms with μ = 0.4·10^{−4} Pa s and for t = 3.06 ms with μ = 1.0·10^{−4} Pa s (Fig. 18). The smallest width of the hydraulic fracture was 1.5 mm for μ = 0.4·10^{−4} Pa s, and the largest was 2.2 mm for μ = 1.0·10^{−4} Pa s. The strongest fluid leakage was obtained for the smallest viscosity μ = 0.4·10^{−4} Pa s and the weakest one for the highest dynamic viscosity μ = 1.0·10^{−4} Pa s. The highest fluid velocity (23 m/s) took place for μ = 0.4·10^{−4} Pa s and the lowest fluid velocity (17 m/s) for μ = 1.0·10^{−4} Pa s.
6.4 Effect of rock strength
When the rock specimen was the strongest, the hydraulic fracture developed during the longest time and it was the least curved. In addition, no secondary macrocracks were created (that are visible for two weakest specimens). The fluid pressures and velocities of along the main path were similar for all specimens.
6.5 Effect of preexisting fracture
In the case ‘I’, the hydraulic fracture started to propagate towards the preexisting fracture (Fig. 20aA). It reached the preexisting fracture for t = 1.94 ms (Fig. 20bA). The fracking fluid started next to fill in the preexisting fracture (Fig. 20bA). For t = 2.44 ms, the preexisting fracture was totally filled in with the fluid (Fig. 20cA) and the fluid pressure started to grow there (Fig. 20cB). Later the fluid pressure sufficiently increased to damage the rock matrix and the hydraulic fracture slightly moved upwards at both the ends of the preexisting fracture (Fig. 20dA). The moderate fluid leakage from the hydraulic fracture to the rock matrix was obtained during the entire simulation (Fig. 20B). Several single floating particles (separated from the rock mass) appeared in the hydraulic fracture (Fig. 20dA). Thus the VPN model makes it possible to study the effect of proppant particles on the hydraulic fracture opening/closure.
In the case ‘II’, the hydraulic fracture initiated and started to propagate also towards the preexisting fracture (Fig. 21aA). After t = 2.43 ms, it reached the preexisting fracture and the fracking fluid started to flood it (Fig. 21bA). The fracking fluid fully flooded the preexisting fracture for t = 3.26 ms and next the fluid’s pressure began to grow (Fig. 21cA). On the right and left end of the preexisting fracture, the fluid pressure increased enough to damage the rock matrix and extended slightly the preexisting fracture. However, unlike the case ‘I’, the highest fluid pressure’s increase was obtained at the intersection of both fractures. It resulted in the huge damage of the preexisting fracture near this intersection region. The hydraulic aperture of the preexisting fracture increased near the intersection rather than at its ends as in the case ‘I’. After the relatively long time (1.8 ms), the hydraulic fracture damaged the preexisting fracture at the intersection region and resumed the propagation way upwards. In the final stage of the simulation, the hydraulic fracture propagated upwards from the intersection region (Fig. 21dA). Before the preexisting fracture was damaged and totally flooded with the fluid, no floating single particles grains appeared. However, after a significant increase of the fluid pressure, the grains began to separate from the rock matrix and to float. This process was more intensive than in the case ‘I’. The leakage of the fracking fluid in the rock matrix was pronounced during almost the entire simulation (Fig. 21aB–cB). The relative longtime damage of the preexisting fracture (about 2.8 ms) caused a huge fluid’s leak in the rock matrix and consequently the significant pressure loss (Fig. 21dB).
7 Conclusions
This study focused on a hydrofracking (hydraulic fracturing) process in the rock mass at the mesolevel with the use of one injection slot. The novel twoway coupled CFD/DEM approach was used to simulate this complex process by discretizing the geometry of voids in the rock mass during laminar fracturing fluid flow. The model realistically depicted the development of a hydraulic fracture and fracturing fluid velocity and pressure.
The results showed significant effects of the initial porosity of the rock matrix, rock matrix strength, dynamic viscosity of the fracking fluid and presence of a preexisting fracture on the fracture initiation and propagation. The height of the hydraulic fracture and its velocity strongly increased with increasing initial microporosity of the rock matrix and decreasing dynamic viscosity of the fracturing fluid and rock strength, and the lack of a preexisting fracture.
The Virtual Pore Network model enabled to study the effect of floating grains separated from the rock matrix in a hydraulic fracture.
The developed method of tracking fluid fractions in voids allowed for investigating the effect of the fluid’s leakage from the hydraulic fracture into the rock matrix. The fluid’s leakage was in particular pronounced in the case of high rock microporosity and presence of a preexisting macrocrack.
Notes
Acknowledgements
The research works have been carried out within the project: ‘Modelling of hydrofracking in shales’ financed by the National Centre for Research and Development (NCBR) as part of the program BLUE GAS—POLISH SHALE GAS. Contract No. BG1/MWSSSG/13 and within the project ‘Fracture propagation in rocks during hydrofracking experiments and discrete element method coupled with fluid flow and heat transport’ financed by the National Science Centre (NCN) (UMO2018/29/B/ST8/00255).
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