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The permeability of fractured coal relies on the geometrical parameters of the fracture network, such as orientation, extension, aperture and density of fractures [19]. Mechanical loading leads to stress redistributions inside the sample, which is responsible for microcrack evolution and eventually permeability changes. According to experiments, dilatancy has significant impact on permeability, simply a single fracture and small shear displacement can cause a drastic increase in permeability [20].

Both models are set-up in parallel by duplicating the microstructure obtained from CT. Size of particles (diameter) corresponds to size of zones (edge length) in the continuum-based model. PFC3D is used to simulate the microcrack/microfracture evolution. FLAC3D is used to simulate the gas flow. At predefined time step intervals, crack data are transferred from PFC3D to FLAC3D. Corresponding zones in FLAC3D are assigned with hydraulic parameters based on the crack data and flow-only numerical simulations are performed to obtain the permeability evolution (one-way coupling).

Crack data, including type, position, radius, normal direction and aperture, are exported from the particle-based model via ASCII files. Each crack covers a circular region with thickness in the continuum model. By calculating the distances from nearby zone centers to the crack center, each crack can be represented by one or several zone elements. New fluid-mechanical parameters are assigned to these zones. Figure 8 illustrates in principle how grouped zones represent fractures with different orientations.

An own-developed script is used to translate the crack characteristics into zone properties. As an example, Figure 9 shows four types of cracks. The model contains 240,000 small zones divided into six element types (two intact matrix types for coal and inclusions and four crack types: matrix shear (fracsm) and matrix tensile (fractm) cracks as well as inclusion shear (fracsi) and inclusion tensile (fracti) cracks). The cracks generated between matrix and inclusion are also included in the inclusion cracks.

The crucial point of the simulation is the coupling between crack/fracture propagation and permeability evolution. Theoretical relationships are developed for a single crack characterized by type, size and aperture, and corresponding hydraulic attributes such as porosity and permeability based on previous works [23,24]. The permeability k of the entire sample model at a given loading state is a combination of the permeability of the initial sample (k0) and the changes caused by cracks/fractures (Δk). The cracks are classified as shear cracks and tensile cracks. These two types have different effects on the hydraulic properties.

For a given crack/fracture aperture, flow rate is proportional to pressure difference [25]. Based on the cubic law, under laminar flow conditions fluid flow rate through a narrow channel can be described by the following equation:

Shearing of non-planar cracks/fractures is related to dilation, which results in an aperture increase. A given increment in shear displacement (Δδ) leads to a positive change in aperture (ΔW), according to previous research [26]. This change can be calculated based on the tangent of the dilation angle (ψ):

The particles are in hexagonal close-packed structure. Four initial adjacent particles (P1, P2, P3 and P4) form a regular tetrahedron structure as shown in Figure 10. The mechanism of crack opening calculation is illustrated separately for tensile and shear cracks. The stress-induced breakage of bonds leads to a rearrangement of the particles. The tensile normal displacements between two particle layers follow the relationship shown in Equation (5) and illustrated in Figure 11.

where DT is the new distance from P4 to plane (P1, P2, P3); Di is the initial distance from P4 to plane (P1, P2, P3); WT is the tensile crack width; Dap is the particle surface aperture caused by tensile cracking; rp is the radius of the particles.

where shear displacements Δδ1 and Δδ2 are calculated considering different shear orientations; DS1 and DS2 are distances from C4 to plane (C1, C2, C3) after shearing; WS1 and WS2 are the shear crack induced widths. The dominant shear dilation angle ψ2 is chosen to calculate the crack width according to Equation (7). It is assumed that there is only one crack in each zone. Overall permeability (kz) of one zone is the sum of two parts, as given by the following equation:

The coalescence of cracked zones contributes to local and overall gas permeability. The permeability is achieved for tensile and shear cracks by substituting WT and WS into Equation (9), respectively. The crack width has a major influence on permeability (see first component in Equation (9)). The initial zone permeability has a minor influence, but the value of the second term cannot be neglected when considering the corresponding closed crack.

According to the bond-deformability criterion, a default crack width Was of 0.05 mm is set to newly generated cracks in FLAC3D. Theoretical volumetric strain εVT of the model is accumulated. On the other hand, the volumetric strain εVP is calculated by the PFC3D model. A scaling coefficient cbl is applied to individual crack widths by comparing εVT and εVP. As shown in Equation (10), the coefficient cbl is assigned to determine the permeability kcl of new cracked zones.

It is believed that the permeability decreases when stresses are compressive on the fracture. On the other hand, the permeability increases significantly when stresses induce tensile/shear displacements on the cracks/fractures. Compared with flow through macroscopic fractures, the permeability of coal for fluids (gas or liquid) is extremely small.

The pore pressure shows some local variations as documented in exemplary Figure 16 caused by the heterogeneity of the sample, especially the distribution of cracks and their specific hydraulic properties.

In the early loading stage, the permeability remains at a very low value due to two reasons. Firstly, the number of cracks increases slowly, where only 34 new cracks are observed before the axial strain reaches 0.01. Secondly, new microcracks are generated isolated in the sample, and no new flow paths are formed. The existing cracks and fractures remain closed and aperture of opening cracks/fractures become smaller.

When approximately 30% of peak differential stress is reached, the crack number increases drastically, which leads to increasing permeability. Until peak strength (axial strain of about 0.014), the total crack number increases to more than 600. Among them, 22 cracks are detected with noticeable apertures (width larger than 0.05 mm). Significant enhancement of permeability occurs locally, when individual cracks form vertical fractures. Finally, end-to-end flow paths are formed, and flow rate, as well as corresponding permeability, reaches peak values. In the post-failure region, new fractures occur continuously and shear cracks become dominating, but the opening stays restricted by the confinement. As shown in exemplary Figure 17, the overall permeability of the sample C1 remains at a high level after the axial strain reaches 0.025. Please note that we have not expected a close agreement between laboratory tests and numerical simulation results due to the complex inhomogeneity and the restricted resolution of the numerical sample.

A coupling method between PFC3D and FLAC3D is effective in simulating the permeability of damaged coal samples also considering inclusions. A novel approach for defining the hydraulic properties is proposed based on crack behavior and fracture distribution. The properties of both tensile and shear fractures are derived separately based on the mechanical interaction of particles.

Abstract Shale gas exploitation initiated in North America has rapidly extended worldwide. Hydraulic fracturing is an emerging field technique for stimulating the gas reservoir. The study of cracking processes, particularly crack coalescence, is vital for a successful hydraulic fracturing to enhance the gas exploitation. Experimental studies have observed that the size effects of the constituent particles are significant on the cracking behavior of the rock specimens. To further investigate the size effects, the bonded-particle model (BPM), which is based on the discrete element method (DEM), is adopted in the present research. In flaw-containing specimens, by varying the crack resolution (Ψ= a/2R), which is the ratio of half flaw length (a) to particle size (2R), the size effects on cracking behavior under compressive loading are studied. By keeping the flaw length constant, the particle size is varied independently in the BPM analysis. Decreasing the crack resolution increases the first crack initiation stress, but it has no obvious effects on the uniaxial compressive strength. The trajectories of the first cracks and secondary cracks hence generated have a higher resolution and are well-defined in those specimens possessing a higher crack resolution. On the contrary, in lower crack resolution specimens, the macroscopic first cracks appear to be wider and less continuous. These findings from numerical simulation clearly demonstrate particle size effects on cracking behavior. Special attention should be paid to these effects in future numerical study using the bonded particle model. Introduction Shale gas exploitation initiated in North America has rapidly extended worldwide. Hydraulic fracturing is an emerging field technique for stimulating the gas reservoir. The study of cracking processes, particularly crack coalescence, is vital for a successful hydraulic fracturing to enhance the gas exploitation. Different cracking processes are observed in marble and gypsum, which possess different grain sizes (Wong & Einstein, 2009a, 2009b; Wong, 2008). To further investigate such effect, the bonded-particle model (BPM) is adopted in the present research. The BPM, which is one of the DEM-based particle models, has been widely used for rock simulations (Cho, Martin, & Sego, 2007; Hazzard, Young, & Maxwell, 2000; Potyondy & Cundall, 2004) since the particle assembly approach was initially developed by Cundall (1971) and Cundall & Strack (1979). Recently, a time-dependent bond breakage model (Wu, Zhu, & Zhu, 2011), synthetic rock mass approaches (Bahaaddini, Sharrock, & Hebblewhite, 2011; Mas Ivars, Pierce, DeGagné, & Darcel, 2008; Thompson, Mas Ivars, Alassi, & Pradhan, 2011), as well as flat-jointed BPM (Potyondy, 2012) have been developed and used for engineering applications. However, some basic cracking phenomena are not yet fully understood for BPM, such as the effect of particle size on cracking processes. This paper will analyze and discuss this effect. 2b1af7f3a8