Validation of Pore Structure-Gas Diffusion Model

As the diffusion process controls the gas influx from matrix towards the cleat/fracture system, it dominates the long-term well performance of CBM after the fracture storage is depleted (Wang and Liu, 2016). The estimation of diffusion coefficient based on pore structure is critical to determine the production potential of a given coal formation. Apparently, diffusion process is slower for coal pore in a smaller size or having a more complex structure. As mentioned above, the diffusive gas influx is controlled by combined Knudsen and bulk diffusions. The theoretical values of the diffusivity under

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these two diffusion modes was calculated based Eq. (2-37) and Eq. (2-39) and the results are listed in Table 4-6. It should be noted that the expression of 𝐷_𝐵 given in Eq. (2-37) is derived for open space and independent of the solid structure. For porous media, a multiplication of porosity is added to the expression of 𝐷_𝐵 that considers volume not occupied by the solid matrix (Maxwell, 1881; Rayleigh, 1892; Weissberg, 1963).

Table 4-6: Theoretically calculated bulk diffusion coefficient (DB) and Knudsen diffusion coefficent of porous media (DK,pm).

The overall diffusion coefficient (𝐷_𝑝) was then defined as a weighted sum of Knudsen diffusion and bulk diffusion given in Eq. (2-41). To estimate the weighing factor (𝑤_𝐾) of each mechanism, it is critical to determine the critical Knudsen number (𝐾𝑛^∗) and for 𝐾𝑛 > 𝐾𝑛^∗, a pure Knudsen diffusion can be assumed. Examination of the manner in which 𝐷_𝑝 varies with pressure using the diagnostic plot (Figure 2-7(b)) is intuitively helpful to identify the pressure interval for pure Knudsen flow. One challenging aspect of applying the diagnostic plot is the uncertainty about the sensitivity of 𝐷_{𝐾,𝑝𝑚} to the change in pressure. If 𝐷_{𝐾,𝑝𝑚} is not very sensitive to pressure, a small variation in pressure will not have an apparent change of 𝐷_𝑝 at low pressure stages and under pure Knudsen diffusion.

Then a relative flat line can be found in a plot of 𝐷_𝑝⁻¹ vs. P at low pressure. It corresponds

Pressure [MPa] 0.55 1.38 2.48 4.14 6.07 8.07

Theoretical Diffusion Coefficient [× 10¹⁰⁡𝑚²/𝑠]

DB DK,pm DB DK,pm DB DK,pm DB DK,pm DB DK,pm DB DK,pm

Xiuwu-21 104.77 67.60 42.27 54.94 23.88 48.22 14.69 43.15 10.42 39.90 8.20 37.77 Luling-9 41.87 19.22 16.89 12.26 9.54 9.24 5.87 7.26 4.16 6.13 3.28 5.44 Luling-10 38.47 21.54 15.52 13.73 8.77 10.35 5.39 8.13 3.83 6.86 3.01 6.10 Sijiazhuang-15 262.48 51.02 105.89 30.29 59.82 21.81 36.79 16.50 26.11 13.55 20.56 11.81

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to a pressure interval of pure Knudsen flow, and the contribution from bulk diffusion is ignored as the intermolecular collision strongly correlated with pressure. Figure 4-11 shows the change in 𝐷_𝐵 and 𝐷_{𝐾,𝑝𝑚} with pressure for Sijiazhuang-15 sample. Figure 4-12 demonstrates the application of using diagnostic plot to identify diffusion mechanism.

Figure 4-11: Variation of bulk diffusion coefficient (DB) and Knudsen diffusion coefficient (DK,pm) at different pressure stages for Sijiazhuang-15.

0 2 4 6 8

0 5 10 15 20 25

30 D_B

D_K,pm

Diffusion Coefficient (m2/s)

Pressure (MPa)

× 10⁻⁹

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Figure 4-12: Diagnostic plot of reciprocal diffusion coefficient (DP-1) vs. P to specify pressure interval of pure Knudsen flow (P < P*) and critical Knudsen number (Kn*= Kn (P*)).

In Figure 4-11, bulk diffusion was subject to much greater variation than Knudsen diffusion over the pressure range of interest. Consequently, a relatively flat line was found at low pressure interval (𝑃 𝑃^∗) in the diagnostic plot (Figure 4-12) for a pure Knudsen diffusion. Effective diffusion coefficient (𝐷_𝑝⁻¹) is then equivalent to 𝐷_{𝐾,𝑝𝑚} and weighing factor (𝑤_𝐾) equals to one. The critical Knudsen number (𝐾𝑛^∗) is determined at the inflection point, where 𝑃 = 𝑃^∗. As pressure increases, pore wall effect diminishes as mean free path of gas molecules shortens and bulk diffusion becomes important. Then at about 2.5 MPa, 𝐷_𝑝⁻¹ was subject to a greater variation in terms of pressure variation since 𝐷_𝐵 is directly proportional to mean free path and inversely proportional to the pressure. The dividing pressure between pure Knudsen diffusion and combined diffusion for tested coal

Horizontal

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samples were all determined to be 2.5 MPa, i.e. 𝑃^∗ = 2.5⁡MPa. For even higher pressure, the effect of pore wall-molecular collisions can be neglected and 𝐷_𝑝⁻¹ was estimated by 𝐷_𝐵⁻¹. As a result, a linear trend was noted at pressure greater than 6 MPa when bulk diffusion dominates the overall diffusion and 𝑤_𝐾 equals to zero. Using Figure 4-12, we would be able to identify the dominant diffusion mechanism at different pressure stages and evaluate the relative contribution of each mechanism or 𝑤_𝐾 as dictated by Eq. (2-42).

𝑤_𝐾 equals to one for pure Knudsen diffusion and zero for pure bulk diffusion. In the transition regime, no theoretical development has been made on the prediction of diffusion coefficient in coal matrix.

For catalysis, Wheeler (1955) proposed an empirical combination of Knudsen and bulk diffusion coefficient to determine the effective diffusion coefficient of combined diffusion as

𝐷_𝑝 = 𝐷_𝐵(1 − e^{−1/𝐾𝑛}⁡)⁡ ( 4-2 ) In Eq. (4-2), 𝐷_𝑝 approaches to 𝐷_𝐵 as 𝐾𝑛 approaches to zero and mean free path is far less than the pore diameter. 𝐷_𝑝 approaches to 𝐷_𝐾 as 𝐾𝑛 approaches infinity since 𝑒^{−1/𝐾𝑛}≈ 1 − 1/𝐾𝑛. Correspondingly, the weighing factor of Knudsen diffusion (𝑤_𝐾) grows towards higher 𝐾𝑛. However, some built-in limitations are also observed for this theoretical formula. First, it fails to consider the change in the effective diffusive path at different pressures as 𝐷_{𝐾,𝑝𝑚} rather than 𝐷_𝐾 should be involved to describe the diffusion rate under Knudsen regime. Besides, it underestimates 𝑤_𝐾 as Eq. (4-2) implicitly states that pure Knudsen diffusion only occurs for flow with infinite value of 𝐾𝑛. In fact, Knudsen

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flow dominates the overall diffusion once 𝐾𝑛^∗ is reached as illustrated in Figure 4-12.

Instead, 𝑤_𝐾 is assumed to have a linear dependence on 𝐾𝑛 in the transition pressure range and for a combined diffusion. This assumption would be further justified by comparing with the experimental data. Figure 4-13 is a plot of 𝑤_𝐾 vs. 𝐾𝑛 applied to quantify the relative contribution of each diffusion mechanism.

Figure 4-13: A plot of wk as a piecewise function of Kn. The horizontal tails at the low and high interval of Kn correspond to pure bulk and Knudsen diffusion, respectively.

Once the 𝑤_𝐾 is given, the overall diffusion coefficient can be theoretically determined by Eq. (2-41). Experimentally measured diffusion coefficients for methane are presented in Figure 4-6. The results were then compared with theoretical values predicted

0.0 0.1 0.2 0.3 0.4 0.5

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by the relationships proposed by Wheeler (1955) and this study as given in Eq. (4-2) and Eq. (2-41), respectively. Figure 4-14 indicates that the theory of 𝑤_𝐾 developed in this study provided better fit to the experimental measured diffusion coefficient than the one proposed by Wheeler (1955). The improvement in the prediction of diffusivity was more obvious towards low pressure and Knudsen diffusion becomes predominant This is because our method allows for the expected changes in the effective diffusion path. Nevertheless, great discrepancy was still found at low pressure stages compared with the experimental diffusion coefficient. The source of error originates from the accuracy in the estimation of pore structural parameters, which is critical in Knudsen diffusion when pore morphology is important. Besides, the scale of measured diffusion coefficient is three order of magnitudes smaller than the predicted one. This is caused by the presence of surface diffusion. Movement of gas molecules along the pore wall surface contributes significantly to the gas transport of adsorbed species in micropores, where gas molecules cannot escape from the potential field of pore surface (Do, 1998; Dutta, 2009). The relative contribution of surface diffusion and diffusion in pore volume is related to the volume ratio of gas in adsorbed phase and free phase (Kärger et al., 2012). The primary purpose of this work is to predict diffusion behavior of coal based on pore structure. Surface diffusion as an activated diffusion is mainly a function of adsorbate properties, rather than adsorbent properties. To eliminate the effect of the variation in surface diffusion, we conducted the analysis under the same ambient pressure. In Figure 4-15, the experimental measured diffusion coefficients are plotted against the theoretical values determined by Eq. (2-41) for the four coal samples at each pressure stages.

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Figure 4-14: Comparison between experimental and theoretical calculated diffusion coefficient for methane diffusion in Xiuwu-21. Wheeler (1955) is described by Eq. (4-2), and this work is given by Eq. (2-41).

Figure 4-15: Comparison between experimental and theoretical calculated diffusion coefficients of the studied four coal samples at same ambient pressure.

0 2 4 6 8

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The experimental diffusion coefficients were measured at six pressure stages ranging from 0.55 MPa to 8.07 MPa. Therefore, six isobaric lines are presented in Figure 4-15 and each line is composed of 4 points corresponding to the four studied coal samples.

The theoretical diffusion coefficient derived from Eq. (2-41) is a function of pore structural parameters. Overall, it provides good fits to the experimental diffusion coefficients. Due to the presence of surface diffusion, the scale of the theoretical values does not agree with it of the experimental values. But the linear relationships in Figure 4-15 inherently illustrates that pore structure has negligible effect on the transport of gas molecules along the pore surface. Otherwise, the contribution from surface diffusion should vary for different coal samples, and the four points will not stay in the same line.

There is a compelling mechanism that determines the steepness of the linear relationships. Generally, surface diffusion becomes predominant as surface coverage increases and multilayer of adsorption builds up at higher pressure stages. The slope is reduced towards high pressures due to an increase in the contribution from surface diffusion. On the contrary, as the pore surface is smoothed, and the effective diffusive path is shortened with a reduction in the induced tortuosity. This leads to a faster diffusion process with greater mass transport occurring in pore volume, and the lines are expected to be steeper as pressure increases. Under these mechanisms, the lines are steeper at lower pressure stages (𝑃 4⁡MPa) in Figure 4-15. For higher pressures, reverse trend can be found as the lines tend to be horizontal as pressure increases.

87 4.7 Summary

This chapter investigates the validity of theoretical models developed in Chapter 2 using the laboratory measurements from high-pressure and low-pressure sorption experimental setup presented in Chapter 3. This work aims at investigating the effect of pore structure on methane adsorption and diffusion behavior for coal. Major findings of this chapter can be summarized as follows:

• Langmuir isotherm provides adequate fit to experimental measured sorption isotherms of all the bituminous coal samples involved in this study. Based on the FHH method, two fractal dimensions 𝐷₁ and 𝐷₂ referred as pore surface and structure fractal dimension are obtained within low- and high- pressure intervals, which reflects the fractal geometry of adsorption pores (i.e., micropores) and seepage pores (i.e., mesopores and macropores). However, fractal dimensions alone appear not to be strongly correlated to the CH4 adsorption behaviors of coal.

• The pore structure-gas sorption model developed in Chapter 2 well predicts Langmuir constants including gas sorption capacity and gas adsorption pressure based on pore structure information, which is very easy to obtain. Langmuir volume appears to have a linear correspondence with a lump of specific surface area and fractal dimension in a log-log plot. Langmuir pressure is also linearly correlated with a lump of Langmuir volume and fractal dimension in a log-log plot. The correlation is valid for a set of coal with similar rank and composition.

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• The application of the unipore model provides satisfactory accuracy to fit lab-measured sorption kinetics and derive diffusion coefficients of coal at different gas pressures. A computer program in Appendix A is constructed to automatically and time-effectively estimate the diffusion coefficients with regressing to experimental sorption rate data.

• The pore structure-gas diffusion model developed in Chapter 2 is applied to model the pressure-dependent diffusion behavior for fractal coals, where diffusion coefficients are measured from the high-pressure experimental setup constructed in Chapter 3. The proposed model takes the pore structure parameters, including, porosity, pore size distribution and fractal dimension, as inputs and it provides accurate modeling of the variation of diffusion coefficients at different pressures and for different coals.

• Based on fractal pore model, the determined tortuosity factors range from 1.787 to 24.223 for the tested pressure interval between 0.55MPa and 8.07 MPa. The results suggest that the increase in pressure and pore structural heterogeneity resulted in a longer effective diffusion path and a higher value of tortuosity factor affecting the Knudsen diffusion influx in porous media. The pore structural parameters lose their significance in controlling the overall mass transport process as bulk diffusion dominates.

• Both experimental and modeled results suggest that Knudsen diffusion dominate the gas influx at low pressure range (< 2.5 MPa) and bulk diffusion dominated at high pressure range (>6 MPa). For intermediate pressure ranges (2.5 to 6 MPa), combined diffusion should be considered as a weighted sum of Knudsen and bulk diffusion, and the weighing factor directly depends on Knudsen number. The overall diffusion

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coefficient was then evaluated as a weighted sum of Knudsen and bulk diffusion coefficient. At individual pressure stages from 0.55MPa and 8.07 MPa, it provided good fits to the experimentally measured overall diffusion coefficient, which varied from 1.05 × 10⁻¹³ to 9.77⁡ × 10⁻¹²⁡𝑚²/𝑠.

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Chapter 5