Mathematical Problems in Engineering

Volume 2015, Article ID 582963, 5 pages

http://dx.doi.org/10.1155/2015/582963

## Modelling Gas Diffusion from Breaking Coal Samples with the Discrete Element Method

^{1}Department of Mathematics and Information Technology, Hanshan Normal University, Chaozhou, Guangdong 521041, China^{2}School of Engineering and Technology, Central Queensland University, North Rockhampton, QLD 4702, Australia

Received 13 November 2014; Accepted 19 January 2015

Academic Editor: Sebastian Anita

Copyright © 2015 Dan-Ling Lin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1–4]. Flow through the cleat system (or the fracture system) is believed to be pressure-driven laminar flow and can be described by Darcy’s law. Flow through the matrix is a diffusion process, which can be described by Fick’s diffusion law where the concentration gradient in the matrix is the driving force. From a microscopic point of view, the two kinds of flow patterns start from the desorption of gas which was initially adsorbed in the coal matrix, followed by the diffusion through the matrix into the cleat and then the flow through the cleat system into a production well or a drainage borehole. Hence, the gas production rate is mainly controlled by the gas diffusivity in the matrix and gas permeability in the cleat system.

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 [5–10]. These important mechanisms should be implemented in a coal and gas outburst model which will permit certain parameters to be varied so that their effects on the outburst can be quantitatively studied. However most of the current models do not model solid fracture and fragmentation explicitly. Free flow of fluid and two-way interactions between the solid and fluid are also missing in most of the existing outburst models. To overcome these difficulties, we have developed a new outburst model which couples two well-developed numerical approaches: the discrete element method (DEM) and the lattice Boltzmann method (LBM) [11, 12]. The DEM is used to model the deformation and fracture of solid, while LBM simulates fluid flow, including free flow and Darcy flow. These two methods are coupled in a two-way process: the solid part provides a moving boundary condition and transfers momentum to the fluid, and the fluid exerts a dragging force to the solid. The new model includes the most recognized factors of outbursts, including deformation, fracture and fragmentation of solids, free flow of fluid, Darcy flow, diffusion, desorption of gas, and two-way coupling of solid and fluid. The preliminary results with small scale simulations suggest that the new model has potentials to numerically investigate the underlying mechanism and interaction of contributing factors of outbursts.

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.

#### 2. Implementation of Fluid Diffusion in DEM

The DEM is a widely used numerical tool to model the behaviour of rock and granular materials [13]. In DEM simulations, the specimen to be modelled is represented as an assembly of indivisible particles interacting with their nearest neighbours. By applying boundary conditions and solving Newton’s laws for each particle, the complex behaviour of the material may be simulated. The Esys-Particle, an open source DEM code, is used in this study. Detailed information about the model and the code can be found in the literature [14].

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 for each particle . For two contacted particles and , the fluid exchange at each time step is determined by Fick’s first law of diffusion: where is the diffusion coefficient of the link at particle scale. For the regular packing in 2D case, we can derive theoretically the relation between and the macroscopic diffusion coefficient [15]:

#### 3. Numerical Verifications of (2) with 1D Diffusion Simulation

The method described in the book of Crank [16] is adopted in this study to determine the macroscopic diffusion coefficient. In this approach, we assume a constant diffusion coefficient and consider the case of one-dimensional diffusion in a medium bounded by two parallel planes, for example, the planes at and . For diffusion through a plane sheet or membrane of thickness with diffusion coefficient , the diffusion equation is

If the boundary and initial conditions are the analytical solution of this problem is [16]

Let be the total amount of gas desorbed by the sheet at time and the corresponding amount theoretically after infinite time; we have

The value of for which , written as , is given by [16]

The macroscopic diffusion coefficient is found to be

Numerical tests have been carried out to calculate the macroscopic diffusion coefficients based on (8). In these tests, we use the regular packing which consists of 2006 particles with the same size of 1 unit and 5824 bonds (Figure 1). The width of the sheet in vertical direction is . We keep the initial concentration and the boundary condition (top), (bottom). Desorption can only occur on the top and bottom boundaries but is not allowed from the left and right boundaries. Therefore this is exactly a one-dimensional problem described in [16].