Numerical simulation of multicomponent gas diffusion and flow in coals for enhanced coalbed methane recovery
Introduction
Carbon dioxide sequestration in coalbeds for enhanced coalbed methane (ECBM) recovery has recently attracted attention due to increasing concerns about greenhouse gases and the emerging commercial significance of coalbed methane production all over the world. While considerable efforts have been made in modeling the gas sorption kinetics for CBM/ECBM production, significant limitations exist in current models. One predominant deficiency is that most of the currently available CBM/ECBM models are based on a single-component unipore diffusion formulation. These models are not suitable to simulate the simultaneous counter flow of into, and out of, the coal matrix. A further complication is the wide range in the scale of the pore spaces typically occurring within coal.
The pore structure of coal is highly heterogeneous, and it depends on the coal type and rank (Laxminarayana and Crosdale, 1999, Unsworth et al., 1989). Moreover, gas diffusion in coals is significantly influenced by coal type, microstructure and secondary mineralization (Crosdale et al., 1998, Gamson et al., 1993, Gamson et al., 1996). According to the studies of Gamson et al. (1996), gas transport in coal seams, in terms of coal type and rank, can be classified into two types: a two-step transport process (i.e., viscous flow followed by diffusion) and a multi-step transport process. The multi-step process consists of gas diffusion in micropores ; gas diffusion through partly blocked microfractures, which may consist of mesopores (2–50 nm) and macropores ; gas flow through open, un-mineralized microfractures wide); and then gas movement through main fractures (0.1–2 mm wide). It is believed that CBM/ECBM production is largely controlled by the multi-step transport process. Therefore, accurate description of multicomponent gas diffusion in coals with multi-modal pore structure is one of the critical issues for the correct simulation of -ECBM recovery.
Generally gas diffusion in coals is controlled by the surface sorption process and the diffusion of gas through the coal matrix. These two processes are usually lumped together in numerical models. So far the work found in open literature primarily focuses on the adsorption capacity and pays less attention to the gas diffusion dynamics, in particular, the counter-diffusion of different gases when gas mixture including more than one gas species is involved. Counter-diffusion is defined herein as the diffusion out of micropores of one gas while another gas diffuses into the same micropores.
Three types of models are currently available to simulate gas diffusion in coals: equilibrium models, non-equilibrium models and bidisperse diffusion models. A comprehensive summary of equilibrium and non-equilibrium models with a pseudo-steady state approach and an unsteady state approach for methane diffusion in coals has been presented by King and Ertekin, 1989a, King and Ertekin, 1989b. All of the models reviewed basically treated the gas diffusion process as an instantaneous or one-step diffusion. The pseudo-steady state approach (Warren and Root, 1963) and unsteady state approach (King and Ertekin, 1988, King and Ertekin, 1989a, King and Ertekin, 1989b) are now used in most commercial CBM simulators to simulate coal gas sorption kinetics. The former represents the matrix response to changes in gas partial pressure in a lumped parameter fashion, neglecting the true spatial variation of gas concentration. This approach is only accurate once the concentration gradients in the reservoir have stabilized, after an extended period, and is less suitable for early stages of production (Kolesar et al., 1990a, Kolesar et al., 1990b). The unsteady-state approach accounts for the effects of the gas concentration gradient, which seems a more rigorous approach, but at the cost of significantly more numerical burden.
More recently, researchers have realized that gas diffusion in coals with multi-scale pores can be described better by a bidisperse diffusion model (Clarkson and Bustin, 1999b, Cui et al., 2004; Shi and Durucan, 2003a, Shi and Durucan, 2003b), which assumes two-step gas diffusion in a coal matrix: surface diffusion in the microporous system and pore diffusion in the meso/macropore system. However, it is still difficult to apply such a model in CBM/ECBM simulations due to the extremely complicated pore structure in coals and fluid flow mechanisms associated with ECBM recovery. The key issues for bidisperse diffusion modeling are how to obtain a reasonable representation of the multi-scale pore structure for numerical models; the concentration-dependent diffusivities; and the counter-diffusion processes that must occur during ECBM production.
Fick's law of diffusion is a widely used method to describe the coalbed gas diffusion process because of its simplicity. However its validity is severely restricted and likely to be misleading in many practical situations where the diffusion coefficients of the fluid species depend on composition. The Maxwell–Stefan (MS) diffusion formulation applies to multi-component fluid diffusion and deals rigorously with the interactions between multicomponent gas molecules. It is therefore more appropriate than Fick's law for describing multicomponent gas diffusion dynamics and consequently should be applied in the CBM/ECBM models.
This paper presents an alternative bidisperse diffusion model to address the issues discussed above. The model adopts a bidisperse pore structure in the coal matrix based on the approach proposed by Ruckenstein et al. (1971). The concentration-dependent micropore diffusivity is described with a polynomial and estimated by comparing the model prediction with experimental data; the diffusion dynamics of mixed gases are simulated using the MS diffusion formulation. The model is used to simulate the combined gas diffusion and viscous flow in the bulk coal and validated against previously published experimental data.
Section snippets
Mathematical model
The following assumptions are used to model the gas diffusion and flow in the coal matrix:
- (I)
The coal matrix is treated as a cylindrical cell surrounded by main fractures. It contains particles with uniform radius, between which are open microfractures (Fig. 1a).
- (II)
The particles have bimodal pore structure, containing uniform radius microporous micro-particles with the space between micro-particles making up the meso/macroporosity (Fig. 1b).
- (III)
Gas flow through open microfractures is assumed to be
Numerical technique
A loose coupling algorithm is adopted in this work for numerically solving the highly coupled model equations, as illustrated in Fig. 2. Such a numerical technique has been used in simulation of coupled processes with high accuracy (Minkoff et al., 2003). Our case is a coupled system dealing with the different processes: gas sorption equilibrium; gas diffusion and flow; and matrix shrinkage and swelling. Among these processes, while the equations for gas sorption equilibrium and matrix
Sensitivity tests
The model is sensitive to some “unknown” parameters, such as tortuosity, kinetic properties relating to pore scales, and component interactions for binary or multiple component system. For a pure component, the kinetic property can be described by a ratio of micropore diffusion to macropore diffusion , as defined earlier. For a binary-component (e.g. in this case) system the ratio of micropore diffusivity of to that of is further introduced as a model parameter to account for
Conclusions
A simulation model for prediction of mixed gases counter-diffusion and flow dynamics in coals is developed in this paper. The model accounts for three-scale porosities and multi-component gas diffusion and flow in coals. Gas diffusion in coal matrix is treated as bidisperse pore diffusion by using MS analysis. A modified ARI model is incorporated into the model to describe the coal shrinkage and swelling induced by gas desorption and adsorption.
The sensitivity of the model to parameters
Notation
matrix of inverted MS effective macropore diffusivities, matrix of inverted MS effective micropore diffusivities, differential swelling coefficients of component i, dimensionless matrix shrinkage compressibility due to methane desorption, 1/Pa mole concentration of gas component i in open microfractures, mole concentration of gas component i in macropores, initial gas mole concentration of component i in macropores, total concentration,
Acknowledgments
This work was supported by the Australian Research Council (ARC), Australian Postgraduate Award (APA) and CSIRO Ph.D. scholarship.
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