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Particle scale diffusion is implemented in the discrete element code, Esys-Particle. We focus on the question of how to calibrate the particle scale diffusion coefficient. For the regular 2D packing, theoretical relation between micro- and macrodiffusion coefficients is derived. This relation is then verified in several numerical tests where the macroscopic diffusion coefficient is determined numerically based on the half-time of a desorption scheme. To further test the coupled model, we simulate the diffusion and desorption in the circular sample. The numerical results match the analytical solution very well. An example of gas diffusion and desorption during sample crushing and fragmenting is given at the last. The current approach is the first step towards a realistic and comprehensive modelling of coal and gas outbursts.

The migration of gas (i.e., methane or CO_{2}) in coal plays an important role in the process of coalbed methane recovery or gas drainage prior to coal mining. It is generally accepted that gas transport in coal mainly occurs in two stages: gas flow within the coal matrix and flow in the cleat system which forms the natural fractures in coal [

Meanwhile, gas diffusion and gas flow in the cleat system are also found to be major contributing factors to the start of coal and gas outbursts. An outburst of coal and gas is the sudden release of a large amount of gas in conjunction with the ejection of coal and possibly associated with rocks. Previous studies have suggested that the major contributing factors of outbursts include stress condition, gassiness of coal seams, geological structures, and mechanical and physical properties of coal [

As the first step towards a realistic and comprehensive modelling for coal and gas outbursts, we mainly focus on the coupling of diffusion mechanism with the DEM model in this paper. Since the DEM models require some input parameters at particle scale which are not directly linked to the macroscopic parameters, calibration of the input parameters has to be carefully carried out. In this paper, we first introduce the implementation of diffusion mechanism into the DEM code and then discuss how to determine the particle scale diffusion coefficient so as to reproduce the macroscopic diffusion coefficient. Several numerical tests are carried out to verify the results by comparing with analytical solutions.

The DEM is a widely used numerical tool to model the behaviour of rock and granular materials [

In this study, diffusion is implemented into the DEM code in the following way. Solid particles are treated as porous materials. It is assumed that the voids inside particle are much smaller than the particle sizes. Therefore the porosity is just an average concept for each particle. There is an average and uniform concentration

The method described in the book of Crank [

If the boundary and initial conditions are

Let

The value of

The macroscopic diffusion coefficient is found to be

Numerical tests have been carried out to calculate the macroscopic diffusion coefficients based on (

Snapshots of concentration distribution at three different times (

The fractional loss

The fractional loss

The relation between

To further test the coupled model, we simulate diffusion and desorption in a circular (or cylinder) sample. We assume that the cylinder length is infinite, diffusion coefficient (

The initial and boundary conditions can be expressed as

Similarly, the analytical solution of the fractional loss can be obtained [

Several numerical tests have been carried out to further test the coupled model based on the calculated macroscopic diffusion coefficients in Section

Snapshots of concentration distribution of a cylindrical sample at three different times (

The fractional loss

In coal mines, the coal sample may be crushed to quickly measure the desorbed gas content and residual gas. This process is modelled by moving two rigid loading walls towards each other in vertical direction. In this case neighboring particles are bonded and the bonds can break if the stress condition is reached, explicitly modelling microscopic fracturing events. Figure

Diffusion and desorption when the sample is under crushing. Colours represent fluid concentration in particles, with red for high and blue for low.

Figure

The fluid velocity during the crushing process. Colours represent macroscopic fluid velocity in the LBM grids, with red for high and blue for low.

In this paper, particle scale diffusion was implemented in the open source discrete element code, the Esys-Particle. Diffusion of gas occurs between particles since each particle was regarded as a porous particle. We specifically focused on how to calibrate the particle scale diffusion coefficient. For the regular 2D packing, we derived theoretically the relation between micro- and macrodiffusion coefficients. This relation was verified in several numerical tests where the macroscopic diffusion coefficient can be determined numerically based on the half-time of a desorption scheme. To further test the coupled model, we simulated diffusion and desorption in the circular sample, and the numerical results match the analytical solution very well. We also simulated the process of diffusion and desorption while the sample is loaded and crushed into small pieces. Desorption occurs at the solid surface when the particle is exposed to the fluid. In this case the analytical solution is not available. Since diffusion process can be described by Fick’s diffusion law or diffusion equation at macroscopic scale, this process can easily be modelled using continuum approaches (i.e., finite difference, finite element method). However, once fractures occur, the continuum based methods are not suitable to model the discontinuity caused by large deformation of solids. The discrete element method does not suffer from this limitation. This is the major advantage of the approach presented in this paper.

The authors declare that there is no conflict of interests regarding the publication of this paper.

_{2}sorption modeling for coalbed methane production and CO

_{2}sequestration