Gas Diffusion in Coal Powders is a Multi-rate Process

Gas migration in coal is strongly controlled by surface diffusion of adsorbed gas within the coal matrix. Surface diffusion coefficients are obtained by inverse modelling of transient gas desorption data from powdered coals. The diffusion coefficient is frequently considered to be dependent on time and initial pressure. In this article, it is shown that the pressure dependence can be eliminated by performing a joint inversion of both the diffusion coefficient and adsorption isotherm. A study of the log–log slope of desorbed gas production rate against time reveals that diffusion within the individual coal particles is a multi-rate process. The application of a power-law probability density function of diffusion rates enables the determination of a single gas diffusion coefficient that is constant in both time and initial pressure.

Keywords Coal-bed methane · Diffusion coefficient · Gas desorption · Multi-rate 15 1 Introduction 16 There is much interest in measurement of gas diffusion coefficients for coal. Such coeffi-17 cients are required for field-scale coal-bed-methane (CBM) simulators to plan and forecast 18 the performance of CBM production operations. Coal beds generally exhibit an orthogonal 19 set of fractures. Fractures in coal are referred to as cleats. The surrounding blocks of coal 20 are typically referred to as the matrix. Methane gas is adsorbent in coal. Gas adsorption 21 is a pressure dependent process with adsorption increasing with increasing pressure. CBM 22 production involves reducing pressure in the coal bed by fluid production (this can be water 23 and/or gas). The reduced pressure causes gas to desorb and migrate through the cleat system. 24 Due to very small permeability, migration of gas within the coal matrix is dominated  (two diffusion times and a weighting coefficient) to fit the observed data of concern, the 52 physical basis of the conceptual model is weak. The model represents a mixture of particles 53 with two different sizes and/or two different diffusion coefficients. Whilst it is conceivable 54 that there should be a continuum of different particle sizes present in such experiments it is 55 unclear why the distribution should be dominated by two specific sizes in particular.

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More recently, spherical diffusion models with transient diffusion coefficients have been 57 adopted, whereby a spatially uniform diffusion coefficient is defined using a heuristic func-  The fact that a transient diffusion coefficient or a time-fractional derivative is required 75 implies that there is missing physics within the conventional Fickian diffusion model. Previ-76 ous researchers have suggested that the missing physics of concern includes free and Knud-  inated by using a dual-porosity model whereby coal particles are assumed to comprise a 80 micro and macro pore-space. A dual porosity framework will give rise to at least two addi-81 tional fitting parameters as compared to a single porosity diffusion model and will therefore 82 be much better at matching observed experimental data (Zang et al., 2019). Of note is that 83 the gas production data from "particle-method" experiments is presented as desorbed gas 84 volume as a function of time. However, a better diagnostic approach, not typically used in 85 the literature, is to study desorbed gas production rate as a function of time on log-log axes.

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Dual-porosity phenomena will manifest itself as two connected offset straight-lines both 87 with log-log slopes of -0.5, one for the macro-pores and another for the micro-pores. This 88 will similarly be the case for the so-called bidisperse model.  It is also noted that previous research has treated diffusion coefficient to be a function 102 of the initial pressure of the experiment. However, the gas pressure within the packed bed 103 pore-space is assumed to be at atmospheric pressure from the start of the experiment. The 104 fact that different diffusion coefficients are required for different initial pressures points 105 towards potential errors in the gas adsorption isotherm being adopted (which is pressure 106 dependent). A frequently used adsorption isotherm is the Langmuir isotherm, which has two 107 physical parameters. These parameters are generally obtained by calibrating the isotherm to  The objective of this article is to demonstrate that gas production rates from gas desorp-115 tion experiments using ground coal particles can be described using a single static diffusion 116 coefficient that is independent of initial pressure when MRP are accounted for and the asso-  In this article, "particle method" experimental data, previously generated by Dong et al. was maintained at P = P I for 6 hours to ensure gas adsorption equilibrium within the coal 143 particles was achieved. The initial pressures studied were 0.25, 0.5, 1, 2, 3 and 4 MPa.

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The outlet of the reactor vessel was then reduced to atmospheric pressure, P 0 [ML −1 T −2 ], 145 0.101 MPa, by linking the outlet to a closed gas sample bag, exposed externally to atmo-146 spheric conditions. Atmospheric pressure was reached at the vessel outlet after around 5 s 147 and then the vessel was connected to a gas measuring cylinder. The volume of gas entering 148 the measuring cylinder was recorded at different times for a total of 120 minutes.  159 where ρ gI [ML −3 ] is the density of gaseous methane at P = P I .
and 178 Also of relevance is that the desorbed gas production rate, dv d /dt, is found from The v d0 parameter represents the maximum volume of gas per unit mass of coal that can 180 be desorbed from the experiment. The α parameter represents the characteristic diffusion 181 rate of the spherical particle under consideration. and Here we consider the case that f (α) is a truncated power law of the form where k [-] is an exponent and α 0 [T −1 ] and α 1 [T −1 ] represent the minimum and maximum 188 diffusion rates, respectively.

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The multiple rate model applied in our study is based on the existence of a distribution and where Γ(a, x) is the incomplete gamma function (Jameson, 2016).

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Notably, Eq. (11) is problematic because Γ(a, x) is difficult to evaluate for a < 0. How-208 ever, given the recursive relationship (Jameson, 2016) it can be shown that Figs. 1a, c and e show example plots of F against normalised time, α 0 t, for a range of 211 k values and α 1 /α 0 ratios. It can be seen that F equilibrates faster with increasing α 1 and 212 also increasing k. This is because increasing these parameters implies that the PDF for α is 213 increasingly dominated by higher diffusion rates. Also of interest is that, regardless of the green lines for comparison). For π −2 α −1 1 < t < π −2 α −1 0 , the log-log slope of the stochastic 220 power law model is −k (compare with the plots of (α 0 t) −k , shown as black dashed lines). For t > π −2 α −1 0 , the gas production rate quickly drops off as this represents the time at which the 222 system reaches diffusion equilibrium. The π 2 factor is a geometry parameter, characteristic  The above methodology was also applied using the stochastic power law model, Eq.

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(14). However, in this case, FMINSEARCH was used to find optimal values for α 0 and α 1 246 with k [-] being obtained a priori by inspection of the log-log slope of the observed dv d /dt 247 data.

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FMINSEARCH is a local optimisation tool, which is appropriate in this case because 249 the number of free parameters is small and multiple minima in RMSE are not expected.

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FMINSEARCH requires specification of seed values for the unknown model parameters.   . 3).

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An important feature of the simple spherical diffusion model is that the log-log slope 269 of a plot of desorbed gas production rate against time is -0.5. In contrast, the observed data 270 show a log-log slope of around -0.7 (see Figs. 2b and d). Deviations from a -0.5 log-log 271 slope are indicative of multi-rate phenomena (Haggerty et al., 2000). Also of interest is that 272 the observed gas production rates exhibit an exponential cut off at large times, which can 273 be represented by the minimum diffusion rate in our truncated power-law PDF described 274 above.

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An advantage of using the stochastic power law model, described above, is that the log-  Table 1). The same is also true for the desorbed gas production rate (compare Figs. the presence of much smaller particles, which have higher rate coefficients (as high as α 1 ).

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It is also of note that the RMSE is significantly reduced when the stochastic power law 292 model is used instead of the simple spherical diffusion model (see Table 1). Furthermore,  The study relies on an assumption that MRP is due to individual coal-particles being