# Phenomenological Over-Parameterization of the Triple-Fitting-Parameter Diffusion Models in Evaluation of Gas Diffusion in Coal

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2})-enhanced CBM production have flourished in many countries [1]. The development of these projects requires detailed and reliable information on gas sorption and flow [2]. Gas sorption determines methane (CH

_{4}) reserves [3,4] and CO

_{2}sorption capacity in coal [2,5]. Since gas sorption is normally assumed instantaneous [6,7,8], CBM production and CO

_{2}injection rates are mainly determined by gas flow processes. Gas flow in coal is often considered as a two-stage process: laminar flow in cleats and diffusion in coal matrices [9,10,11]. This paper focuses on gas diffusion and does not include the laminar flow in cleats.

## 2. Summary of the TFP Models

#### 2.1. Ruckenstein Bidisperse Model

^{−3}and Equation (2) reduces to [13,22,26]:

^{2}and Equation (2) reduces to [26]:

^{−3}< $\alpha $ < 10

^{2}, both macropore sorption and micropore sorption are important and Equation (2) should be used.

#### 2.2. Fickian Diffusion-Relaxation Model

#### 2.3. Double Exponential Model

## 3. Experimental Measurements of Fractional Sorption Curves

#### 3.1. Sample

#### 3.2. Experimental Setup

#### 3.3. Measurement Procedures

- (1)
- Implement the sample into the sample cell.
- (2)
- Evacuate the air in the system and shut all valves.
- (3)
- Opening V1.
- (4)
- Open V2 slowly and inject CH
_{4}gas from a gas cylinder into the reference cell. - (5)
- When the gas pressure in the reference cell increased to about 1.0 MPa, close V2 and open V3 rapidly.
- (6)
- Record gas pressure change in the sample cell with a time interval of one second.
- (7)
- When the gas pressure in the sample cell kept invariant in two hours, terminate recording the gas pressure and close V3.
- (8)
- Repeat (3)–(6) two times.

#### 3.4. Measured Fractional Adsorption Curves

## 4. Modeling the Measured Fractional Adsorption Curves with the TFP Models

#### 4.1. Modeling Configuration

^{th}iteration step and ${\left(\frac{{M}_{t}}{{M}_{\infty}}\right)}_{n-1}$ is the computed fractional sorption at the (n − 1)th iteration step.

^{−3}. However, many studies misused this model by expanding the upper bound of $\alpha $ to unity. In order to evaluate this misusage, modeling with the RB model was conducted twice, one with the assumption of 0 < $\alpha $ < 10

^{−3}and the other with the assumption of 0 < $\alpha $ < 1.0.

#### 4.2. Modeling Results

^{−3}decreases with increasing gas pressure. The DE-fast rate parameter and the RB-macropore diffusivity with the assumption of 0 < $\alpha $ < 1.0 are also negatively proportional to gas pressure. However, the RB-micropore diffusivity with the assumption of 0 < $\alpha $ < 1.0, the FDR-diffusivity and relaxation rate, and the DE-slow rate parameter show non-monotonic dependence on gas pressure.

^{−3}, the RB model with the assumption of 0 < $\alpha $ < 1.0, the FDR model, and the DE model, respectively. The RB model with the assumption of 0 < $\alpha $ < 10

^{−3}had only one quasi-optimal fit at 0.81 MPa while the other three models had multiple quasi-optimal fits at each equilibrium pressure.

^{−5}s

^{−1}to 2.02 × 10

^{−5}s

^{−1}when θ

_{a}increased from 0.10 to 0.90.

## 5. Discussion

#### 5.1. Phenomenological Over-Parameterization of the TFP Models

^{−3}have only one quasi-optimal fit when fitting to the measured fractional adsorption curve at 0.81 MPa, the fluctuation ratios are quite high. This high fluctuation ratio indicates that the quasi-optimal fit at 0.81 MPa is anomalous and can be omitted artificially. However, the fluctuation ratios of the other three TFP models are moderate and all lower than 10. The quasi-optimal fits of these models cannot be omitted artificially. Generally, the RB model with the assumption of 0 < $\alpha $ < 1.0 and FDR model produced greater fluctuation ratios than the DE model did. The RB model with the assumption of 0 < $\alpha $ < 1.0 produced higher fluctuation ratios than the RB model with the assumption of 0 < $\alpha $ < 10

^{−3}when fitting to the fractional adsorption curves at 0.35 MPa and 0.64 MPa. Therefore, expanding the upper bound of $\alpha $ in curve fitting makes the RB model phenomenologically over-parameterized.

^{−3}predicted a negative gas pressure dependence of diffusivity in this paper. This dependence can be interpretable and is normally attributed to coal swelling and decreasing molecular diffusion rate with increasing gas pressure [30]. Therefore, the anomalously non-monotonic dependences discussed above may be due to the misuse of the RB model out of its boundary condition by expanding the upper bound of $\alpha $ to unity.

_{a}< 0.95). This indicates that most CH

_{4}adsorption occurred in macropores for the samples used in this study. However, low-temperature nitrogen adsorption experiments showed that the samples used for CH

_{4}adsorption were dominant in pores lower than 10 nm (Figure 6). No visible macropores were found in these samples. The literature have observed that gas adsorption mainly occurs in micropores rather than in macropores. Radliński et al. [33] scattered CO

_{2}adsorption in coal with small angle X-ray and the results show that smaller pores are filled prior to larger pores. Therefore, micropore adsorption should be dominant and high macropore adsorption fractions are anomalous.

#### 5.2. Implications for Future Work

## 6. Conclusions

- (1)
- The measured adsorption rate was negatively proportional to gas pressure. This correlation can be attributed to coal swelling and decreasing molecular diffusion rate with increasing gas pressure.
- (2)
- The TFP models had better performance in curve fitting than the UP model did. The gas pressure dependence of diffusivity/rate parameter was dependent on the chosen model. The UP-diffusivity and the RB-diffusivity with the assumption of 0 < $\alpha $ < 10
^{−3}decreased with increasing gas pressure. The RB-macropore diffusivity with the assumption of 0 < $\alpha $ < 1.0 and the DE-fast rate parameter were also negatively proportional to gas pressure. However, the RB-micropore diffusivity with the assumption of 0 < $\alpha $ < 1.0, the FDR-diffusivity and relaxation rate, and the DE-slow rate parameter showed non-monotonic dependence on gas pressure. - (3)
- In addition to optimal fits, the TFP model produced multiple quasi-optimal fits that were highly close to the corresponding optimal fit in fit curves. This issue was defined as phenomenological over-parameterization in this paper. Phenomenological over-parameterization makes optimal fit non-unique and all optimal and quasi-optimal fits re potentially optimal. Phenomenological over-parameterization also makes the modeling results of the TFP models uninterpretable in physics.
- (4)
- Expanding the upper bound of $\alpha $ made the RB model phenomenologically over-parameterized in curve fitting. This misusage induced the literature-reported anomalous non-monotonic dependence of diffusivity on gas and coal properties such as gas pressure, moisture content, and particle size. Although the correct usage of RB model with the assumption of 0 < $\alpha $ < 10
^{−3}is absent of phenomenological over-parameterization, anomalously high macropore adsorption fraction was regressed when fitting to the fractional adsorption curves measured in this study. - (5)
- Although the TFP models have better performance in curve fitting than the UP model does, their modeling results are seemingly uninterpretable in physics. Their excellent curve fitting performance may be due to the usage of three fitting parameters. Clarkson and Bustin [14] developed a numerical bidisperse diffusion model that has only two fitting parameters. This also has excellent performance in curve fitting and the results can be physically interpretable. Therefore, this numerical bidisperse diffusion model may be a good alternative to the TFP models in future studies.

## Author Contributions

## Funding

## Acknowledgement

## Conflicts of Interest

## Nomenclature

${C}_{a}$ | Sorbate concentration in macropores mol·m^{−3} |

${C}_{i}$ | Sorbate concentration in micropores, mol·m^{−3} |

${C}_{sa}$ | Sorbate concentration on macropore surfaces, mol·m^{−2} |

${C}_{si}$ | Sorbate concentration on micropore surfaces, mol·m^{−2} |

${D}_{a}$ | Macropore diffusion coefficient, m^{2}·s^{−1} |

${D}_{a}^{\prime}$ | Equivalent macropore diffusivity, s^{−1} |

${D}_{F}^{\prime}$ | Equivalent Fickian diffusivity in the FDR model, s^{−1} |

${D}_{i}$ | Micropore diffusion coefficient, m^{2}·s^{−1} |

${D}_{i}^{\prime}$ | Equivalent micropore diffusivity, s^{−1} |

${D}_{u}^{\prime}$ | Equivalent unipore diffusivity, s^{−1} |

${H}_{a}$ | Henry constant, m^{3}·m^{−2} |

${k}_{\mathrm{f}}$ | Fast sorption rate parameter in the DE model, s^{−1} |

${k}_{R}$ | Relaxation rate parameter in the FDR model, s^{−1} |

${k}_{s}$ | Slow sorption rate parameter in the DE model, s^{−1} |

${M}_{0}$ | Initial sorption amount, mol |

${M}_{t}$ | Sorption amount at time t, mol |

${M}_{\infty}$ | Sorption amount at infinite time, mol |

${P}_{0}$ | Initial gas pressure, MPa |

${P}_{t}$ | Gas pressure at time $t$, MPa |

${P}_{\infty}$ | Final gas pressure, MPa |

${Q}_{res}$ | Residual (unoccupied) sorption capacity in the DE model, mol |

$R$ | Universal gas constant, J·mol^{−1}·K^{−1} |

${R}_{a}$ | Macropore radius, m |

${R}_{i}$ | Microsphere radius, m |

${S}_{a}$ | Macropore surface area, m^{2} |

$T$ | Temperature, K |

$V$ | Void volume of the experimental system, m^{3} |

${z}_{0}$ | Compressibility factors at ${P}_{0}$, dimensionless |

${z}_{t}$ | Compressibility factors at ${P}_{t}$, dimensionless |

${z}_{\infty}$ | Compressibility factors at ${P}_{\infty}$, dimensionless |

$\alpha $ | Ratio of equivalent micropore diffusivity to equivalent macropore diffusivity, fraction |

${\theta}_{a}$ | Macropore sorption fraction in the RB model, fraction |

${\theta}_{\mathrm{f}}$ | Sorption fraction of the fast sorption stage in the DE model, fraction |

${\theta}_{F}$ | Sorption fraction due to Fickian diffusion in the FDR model, fraction |

${\varphi}_{a}$ | Macroporosity, fraction |

${\varphi}_{i}$ | Microporosity, fraction |

## References

- Moore, T. Coalbed methane: A review. Int. J. Coal Geol.
**2012**, 101, 36–81. [Google Scholar] [CrossRef] - Busch, A.; Gensterblum, Y. CBM and CO
_{2}-ECBM related sorption processes in coal: A review. Int. J. Coal Geol.**2011**, 87, 49–71. [Google Scholar] [CrossRef] - Bustin, R.; Clarkson, C. Geological controls on coalbed methane reservoir capacity and gas content. Int. J. Coal Geol.
**1998**, 38, 3–26. [Google Scholar] [CrossRef] - Bodden, W.R.; Ehrlich, R. Permeability of coals and characteristics of desorption tests: Implications for coalbed methane production. Int. J. Coal Geol.
**1998**, 35, 333–347. [Google Scholar] [CrossRef] - Li, D.Y.; Liu, Q.F.; Weniger, P.; Gensterblum, Y.; Busch, A.; Krooss, B. High-pressure sorption isotherms and sorption kinetics of CH
_{4}and CO_{2}on coals. Fuel**2010**, 89, 569–580. [Google Scholar] [CrossRef] - Crosdale, P.; Beamish, B.; Valix, M. Coalbed methane sorption related to coal composition. Int. J. Coal Geol.
**1998**, 35, 147–158. [Google Scholar] [CrossRef] - Smith, D.M.; Williams, F.L. Diffusion models for gas production from coal: Determination of diffusion parameters. Fuel
**1984**, 63, 256–261. [Google Scholar] [CrossRef] - Bhowmik, S.; Dutta, P. Adsorption rate characteristics of methane and CO
_{2}in coal samples from Raniganj and Jharia coalfields of India. Int. J. Coal Geol.**2013**, 113, 50–59. [Google Scholar] [CrossRef] - Pan, Z.J.; Connell, L. Modelling permeability for coal reservoirs: A review of analytical models and testing data. Int. J. Coal Geol.
**2012**, 92, 1–44. [Google Scholar] [CrossRef] - Wang, K.; Zang, J.; Wang, G.D.; Zhou, A.T. Anisotropic permeability evolution of coal with effective stress variation and gas sorption: Model development and analysis. Int. J. Coal Geol.
**2014**, 130, 53–65. [Google Scholar] [CrossRef] - Zang, J.; Wang, K. A numerical model for simulating single-phase gas flow in anisotropic coal. J. Nat. Gas Sci. Eng.
**2016**, 28, 153–172. [Google Scholar] [CrossRef] - Zang, J.; Wang, K. Gas sorption-induced coal swelling kinetics and its effects on coal permeability evolution: Model development and analysis. Fuel
**2017**, 189, 164–177. [Google Scholar] [CrossRef] - Wang, K.; Zang, J.; Feng, Y.; Wu, Y. Effects of moisture on diffusion kinetics in Chinese coals during methane desorption. J. Nat. Gas Sci. Eng.
**2014**, 21, 1005–1014. [Google Scholar] [CrossRef] - Clarkson, C.; Bustin, R. The effect of pore structure and gas pressure upon the transport properties of coal: A laboratory and modeling study. 2. Adsorption rate modeling. Fuel
**1999**, 78, 1345–1362. [Google Scholar] [CrossRef] - Nandi, S.P.; Walker, J.; Philip, L. Activated diffusion of methane from coals at elevated pressures. Fuel
**1975**, 54, 81–86. [Google Scholar] [CrossRef] - Pillalamarry, M.; Harpalani, S.; Liu, S. Gas diffusion behavior of coal and its impact on production from coalbed methane reservoirs. Int. J. Coal Geol.
**2011**, 86, 342–348. [Google Scholar] [CrossRef] - Staib, G.; Sakurovs, R.; Gray, E. Dispersive diffusion of gases in coals. Part II: An assessment of previously proposed physical mechanisms of diffusion in coal. Fuel
**2015**, 143, 620–629. [Google Scholar] [CrossRef] - Staib, G.; Sakurovs, R.; Gray, E. Dispersive diffusion of gases in coals. Part I: Model development. Fuel
**2015**, 143, 612–619. [Google Scholar] [CrossRef] - Charrière, D.; Pokryszka, Z.; Behra, P. Effect of pressure and temperature on diffusion of CO
_{2}and CH_{4}into coal from the Lorraine basin (France). Int. J. Coal Geol.**2010**, 81, 373–380. [Google Scholar] [CrossRef] - Švábová, M.; Weishauptová, Z.; Přibyl, O. The effect of moisture on the sorption process of CO
_{2}on coal. Fuel**2012**, 92, 187–196. [Google Scholar] [CrossRef] - Zhang, J. Experimental Study and Modeling for CO
_{2}Diffusion in Coals with Different Particle Sizes: Based on Gas Absorption (Imbibition) and Pore Structure. Energy Fuels**2016**, 30, 531–543. [Google Scholar] [CrossRef] - Pan, Z.J.; Connell, L.; Camilleri, M.; Connelly, L. Effects of matrix moisture on gas diffusion and flow in coal. Fuel
**2010**, 89, 3207–3217. [Google Scholar] [CrossRef] - Zheng, G.Q.; Pan, Z.J.; Tang, S.H.; Ling, B.C.; Lv, D.W.; Connell, L. Laboratory and modeling study on gas diffusion with pore structures in different-rank Chinese coals. Energy Explor. Exploit.
**2013**, 31, 859–877. [Google Scholar] [CrossRef] - Guo, J.; Kang, T.; Kang, J.; Zhao, G.; Huang, Z. Effect of the lump size on methane desorption from anthracite. J. Nat. Gas Sci. Eng.
**2014**, 20, 337–346. [Google Scholar] [CrossRef] - Crank, J. The Mathematics of Diffusion; Clarendon Press: Gloucestershire, UK, 1975. [Google Scholar]
- Ruckenstein, E.; Vaidyanathan, A.; Youngquist, G. Sorption by solids with bidisperse pore structures. Chem. Eng. Sci.
**1971**, 26, 1305–1318. [Google Scholar] [CrossRef] - Berens, A.; Hopfenberg, H. Diffusion and relaxation in glassy polymer powders: 2. Separation of diffusion and relaxation parameters. Polymer
**1978**, 19, 489–496. [Google Scholar] [CrossRef] - Busch, A.; Gensterblum, Y.; Krooss, B.; Littke, R. Methane and carbon dioxide adsorption–diffusion experiments on coal: Upscaling and modeling. Int. J. Coal Geol.
**2004**, 60, 151–168. [Google Scholar] [CrossRef] - Karacan, C.O. An effective method for resolving spatial distribution of adsorption kinetics in heterogeneous porous media: Application for carbon dioxide sequestration in coal. Chem. Eng. Sci.
**2003**, 58, 4681–4693. [Google Scholar] [CrossRef] - Cui, X.J.; Bustin, R.; Dipple, G. Selective transport of CO
_{2}, CH_{4}, and N_{2}in coals: Insights from modeling of experimental gas adsorption data. Fuel**2004**, 83, 293–303. [Google Scholar] [CrossRef] - Staib, G.; Sakurovs, R.; Gray, E. A pressure and concentration dependence of CO
_{2}diffusion in two Australian bituminous coals. Int. J. Coal Geol.**2013**, 116–117, 106–116. [Google Scholar] [CrossRef] - OriginLab Corp. OriginPro 9.0 User Guide. 2015. Available online: http://www.originlab.com/doc/User-Guide (accessed on 20 March 2019).
- Radliński, A.P.; Busbridge, T.L.; Gray, E.M.A.; Blach, T.P.; Cookson, D.J. Small angle X-ray scattering mapping and kinetics study of sub-critical CO
_{2}sorption by two Australian coals. Int. J. Coal Geol.**2009**, 77, 80–89. [Google Scholar]

**Figure 3.**Optimal and quasi-optimal fit curves when fitting the RB model with the assumption of 0 < $\alpha $ <1.0 to the measured fractional adsorption data at 0.81 MPa: (

**a**) θ

_{a}= 0.10; (

**b**) θ

_{a}= 0.20; (

**c**) θ

_{a}= 0.30; (

**d**) θ

_{a}= 0.40; (

**e**) θ

_{a}= 0.50; (

**f**) θ

_{a}= 0.60; (

**g**) θ

_{a}= 0.70; (

**h**) θ

_{a}= 0.80; (

**i**) θ

_{a}= 0.90; and (

**j**) θ

_{a}= 0.25 (optimal fit). Note that the red background color highlights the optimal fit.

**Figure 4.**Regular residual curves of the optimal and quasi-optimal fits when fitting the RB model 0 < $\alpha $ < 1.0 to the measured fractional adsorption data at 0.81 MPa.

**Figure 5.**Gas pressure dependence of the diffusivity regressed by the ERB model: (

**a**) macropore diffusivity; (

**b**) micropore diffusivity. Note that the black solid line indicates a monotonically negative gas pressure dependence, the red solid line indicates a monotonically positive gas pressure dependence, and the green solid line indicates a non-monotonic gas pressure dependence.

**Figure 6.**Experimental results of low-temperature nitrogen adsorption in the coal samples used for fractional adsorption measurements: (

**a**) adsorption-desorption curves; (

**b**) pore size distribution. The low-temperature nitrogen adsorption data were measured by using a JW-BK112 pore analyzer (Beijing JWGB SCI & Tech Co., Ltd., China, 2016). The used samples were 0.18–0.25 mm. The pore size distribution data were computed with the BJH theory.

Model Classification | Model Name | Authors |
---|---|---|

Single-fitting-parameter model | Unipore model | Crank [25] |

Numerical unipore model | Clarkson and Bustin [14] | |

Linear driving force model | Charrière et al. [19] | |

Double-fitting-parameter model | Stretched exponential model | Staib et al. [17], Staib et al. [18] |

Numerical bidisperse model | Clarkson and Bustin [14] | |

Triple-fitting-parameter model | Ruckenstein bidisperse model | Ruckenstein et al. [26] |

Fickian diffusion-relaxation model | Berens and Hopfenberg [27] | |

Double exponential model | Busch et al. [28] |

True Density (g/cm^{3}, Dry) | Apparent Density (g/cm^{3}, Dry) | Proximate Analysis | |||
---|---|---|---|---|---|

M_{ad} (%) | A_{d} (%) | V_{daf} (%) | Fixed Carbon (%) | ||

1.49 | 1.40 | 0.61 | 11.02 | 7.50 | 80.87 |

First Injection | Second Injection | Third Injection | |||
---|---|---|---|---|---|

Maximum gas pressure (MPa) | Equilibrium gas pressure (MPa) | Maximum gas pressure (MPa) | Equilibrium gas pressure (MPa) | Maximum gas pressure (MPa) | Equilibrium gas pressure (MPa) |

0.60 | 0.35 | 0.80 | 0.64 | 0.89 | 0.81 |

Model | Fit Parameters | Equilibrium Pressure (MPa) | ||
---|---|---|---|---|

0.35 | 0.64 | 0.81 | ||

UP model | ${D}_{u}^{\prime}$ (s^{−1}) | 2.59 × 10^{−5} | 2.24 × 10^{−5} | 1.75 × 10^{−5} |

${R}^{2}$ (%) | 94.81 | 96.01 | 99.15 | |

RB model (0 < $\alpha $ < 10 ^{−3}) | θ_{a} | 0.962 | 0.956 | 0.986 |

${D}_{a}^{\prime}$ (s^{−1}) | 3.18 × 10^{−5} | 2.82 × 10^{−5} | 1.86 × 10^{−5} | |

${D}_{i}^{\prime}$ (s^{−1}) | 3.18 × 10^{−8} | 2.82 × 10^{−8} | 1.86 × 10^{−8} | |

${R}^{2}$ (%) | 97.02 | 98.12 | 99.27 | |

RB model (0 < $\alpha $ < 1.0) | θ_{a} | 0.416 | 0.527 | 0.250 |

${D}_{a}^{\prime}$ (s^{−1}) | 1.06 × 10^{−4} | 5.52 × 10^{−5} | 4.03 × 10^{−5} | |

${D}_{i}^{\prime}$ (s^{−1}) | 1.48 × 10^{−5} | 1.15 × 10^{−5} | 1.42 × 10^{−5} | |

${R}^{2}$ (%) | 99.71 | 99.21 | 99.42 | |

FDR model | θ_{F} | 0.63 | 0.69 | 0.52 |

${D}_{F}^{\prime}$ (s^{−1}) | 6.45 × 10^{−4} | 3.74 × 10^{−4} | 3.75 × 10^{−4} | |

${k}_{R}$ (s^{−1}) | 9.36 × 10^{−4} | 7.35 × 10^{−4} | 8.99 × 10^{−4} | |

${R}^{2}$ (%) | 99.75 | 99.28 | 99.56 | |

DE model | θ_{f} | 0.58 | 0.62 | 0.48 |

${k}_{\mathrm{f}}$ (s^{−1}) | 1.28 × 10^{−2} | 7.58 × 10^{−3} | 6.87 × 10^{−3} | |

${k}_{s}$ (s^{−1}) | 1.03 × 10^{−3} | 8.43 × 10^{−4} | 9.25 × 10^{−4} | |

${R}^{2}$ (%) | 99.53 | 99.46 | 99.84 |

**Table 5.**Quasi-optimal fit parameters of the RB model with the assumption of 0 < $\alpha $ < 10

^{−3}.

Equilibrium Pressure (MPa) | θ_{a} | ${\mathit{D}}_{\mathit{a}}^{\prime}\left({\mathbf{s}}^{-1}\right)$ | ${\mathit{D}}_{\mathit{i}}^{\prime}\left({\mathbf{s}}^{-1}\right)$ | ${\mathit{R}}^{2}(\%)$ |
---|---|---|---|---|

0.01 | 1.08 × 10^{−1} | 1.08 × 10^{−4} | 99.15 |

Equilibrium Pressure (MPa) | θ_{a} | ${\mathit{D}}_{\mathit{a}}^{\prime}\left({\mathbf{s}}^{-1}\right)$ | ${\mathit{D}}_{\mathit{i}}^{\prime}\left({\mathbf{s}}^{-1}\right)$ | ${\mathit{R}}^{2}(\%)$ |
---|---|---|---|---|

0.35 | 0.30 | 1.63 × 10^{−4} | 1.73 × 10^{−5} | 99.45 |

0.40 | 1.12 × 10^{−4} | 1.51 × 10^{−5} | 99.71 | |

0.50 | 8.18 × 10^{−5} | 1.32 × 10^{−5} | 99.61 | |

0.60 | 6.32 × 10^{−5} | 1.12 × 10^{−5} | 99.29 | |

0.70 | 5.07 × 10^{−5} | 9.26 × 10^{−6} | 98.84 | |

0.64 | 0.20 | 1.24 × 10^{−4} | 1.76 × 10^{−5} | 98.25 |

0.30 | 9.14 × 10^{−5} | 1.55 × 10^{−5} | 98.80 | |

0.40 | 7.14 × 10^{−5} | 1.37 × 10^{−5} | 99.10 | |

0.50 | 5.81 × 10^{−5} | 1.20 × 10^{−5} | 99.21 | |

0.60 | 4.85 × 10^{−5} | 1.02×10^{−5} | 99.18 | |

0.70 | 4.11 × 10^{−5} | 8.30 × 10^{−6} | 99.04 | |

0.80 | 3.55 × 10^{−5} | 5.98 × 10^{−6} | 98.80 | |

0.90 | 3.06 × 10^{−5} | 4.18 × 10^{−6} | 98.44 | |

0.81 | 0.10 | 6.53 × 10^{−5} | 1.59 × 10^{−5} | 99.37 |

0.20 | 4.58 × 10^{−5} | 1.47 × 10^{−5} | 99.41 | |

0.30 | 3.63 × 10^{−5} | 1.38 × 10^{−5} | 99.42 | |

0.40 | 3.09 × 10^{−5} | 1.29 × 10^{−5} | 99.40 | |

0.50 | 2.75 × 10^{−5} | 1.21 × 10^{−5} | 99.39 | |

0.60 | 2.51 × 10^{−5} | 1.11 × 10^{−5} | 99.38 | |

0.70 | 2.32 × 10^{−5} | 1.00 × 10^{−5} | 99.36 | |

0.80 | 2.16 × 10^{−5} | 8.56 × 10^{−6} | 99.34 | |

0.90 | 2.02 × 10^{−5} | 6.06 × 10^{−6} | 99.32 |

Equilibrium Pressure (MPa) | θ_{F} | ${\mathit{D}}_{\mathit{F}}^{\prime}\left({\mathbf{s}}^{-1}\right)$ | ${\mathit{k}}_{\mathit{R}}\left({\mathbf{s}}^{-1}\right)$ | ${\mathit{R}}^{2}(\%)$ |
---|---|---|---|---|

0.35 | 0.50 | 1.14 × 10^{−3} | 1.21 × 10^{−3} | 98.96 |

0.60 | 7.26 × 10^{−4} | 9.93 × 10^{−4} | 99.71 | |

0.70 | 4.84 × 10^{−4} | 7.98 × 10^{−4} | 99.57 | |

0.80 | 3.36 × 10^{−4} | 6.05 × 10^{−4} | 98.90 | |

0.64 | 0.50 | 7.28 × 10^{−4} | 1.09 × 10^{−3} | 98.34 |

0.60 | 5.08 × 10^{−4} | 9.02 × 10^{−4} | 99.08 | |

0.70 | 3.68 × 10^{−4} | 7.26 × 10^{−4} | 99.28 | |

0.80 | 2.74 × 10^{−4} | 5.44 × 10^{−4} | 99.10 | |

0.90 | 2.07 × 10^{−4} | 3.05 × 10^{−4} | 98.65 | |

0.81 | 0.30 | 9.97 × 10^{−4} | 1.21 × 10^{−3} | 98.63 |

0.40 | 6.06 × 10^{−4} | 1.05 × 10^{−3} | 99.36 | |

0.50 | 4.01 × 10^{−4} | 9.20 × 10^{−4} | 99.56 | |

0.60 | 2.85 × 10^{−4} | 8.08 × 10^{−4} | 99.52 | |

0.70 | 2.15 × 10^{−4} | 7.06 × 10^{−4} | 99.43 | |

0.80 | 1.70 × 10^{−4} | 6.00 × 10^{−4} | 99.36 | |

0.90 | 1.39 × 10^{−4} | 4.54 × 10^{−4} | 99.32 |

Equilibrium Pressure (MPa) | θ_{f} | ${\mathit{k}}_{\mathit{f}}\left({\mathbf{s}}^{-1}\right)$ | ${\mathit{k}}_{\mathit{s}}\left({\mathbf{s}}^{-1}\right)$ | ${\mathit{R}}^{2}(\%)$ |
---|---|---|---|---|

0.35 | 0.50 | 1.74 × 10^{−2} | 1.20 × 10^{−3} | 99.09 |

0.60 | 1.18 × 10^{−2} | 9.82 × 10^{−4} | 99.48 | |

0.64 | 0.50 | 1.09 × 10^{−2} | 1.08 × 10^{−3} | 98.80 |

0.60 | 8.00 × 10^{−3} | 8.78 × 10^{−4} | 99.45 | |

0.70 | 6.04 × 10^{−3} | 6.86 × 10^{−4} | 99.12 | |

0.81 | 0.40 | 9.07 × 10^{−3} | 1.04 × 10^{−3} | 99.65 |

0.50 | 6.45 × 10^{−3} | 8.96 × 10^{−4} | 99.83 | |

0.60 | 4.85 × 10^{−3} | 7.58 × 10^{−4} | 99.49 |

Equilibrium Pressure (MPa) | ORB Model | ERB Model | FDR Model | DE Model | ||||
---|---|---|---|---|---|---|---|---|

${\mathit{F}}_{\mathit{R}}\left({\mathit{D}}_{\mathit{a}}^{\prime}\right)$ | ${\mathit{F}}_{\mathit{R}}\left({\mathit{D}}_{\mathit{i}}^{\prime}\right)$ | ${\mathit{F}}_{\mathit{R}}\left({\mathit{D}}_{\mathit{i}}^{\prime}\right)$ | ${\mathit{F}}_{\mathit{R}}\left({\mathit{D}}_{\mathit{i}}^{\prime}\right)$ | ${\mathit{F}}_{\mathit{R}}\left({\mathit{D}}_{\mathit{F}}^{\prime}\right)$ | ${\mathit{F}}_{\mathit{R}}\left({\mathit{k}}_{\mathit{R}}\right)$ | ${\mathit{F}}_{\mathit{R}}\left({\mathit{k}}_{\mathbf{f}}\right)$ | ${\mathit{F}}_{\mathit{R}}\left({\mathit{k}}_{\mathit{s}}\right)$ | |

0.35 | 1.00 | 1.00 | 3.21 | 1.87 | 3.39 | 2.00 | 1.47 | 1.22 |

0.64 | 1.00 | 1.00 | 4.04 | 6.75 | 3.52 | 3.58 | 1.81 | 1.57 |

0.81 | 927.80 | 927.80 | 3.25 | 2.62 | 7.18 | 2.67 | 1.87 | 1.37 |

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## Share and Cite

**MDPI and ACS Style**

Zang, J.; Wang, K.; Liu, A.
Phenomenological Over-Parameterization of the Triple-Fitting-Parameter Diffusion Models in Evaluation of Gas Diffusion in Coal. *Processes* **2019**, *7*, 241.
https://doi.org/10.3390/pr7040241

**AMA Style**

Zang J, Wang K, Liu A.
Phenomenological Over-Parameterization of the Triple-Fitting-Parameter Diffusion Models in Evaluation of Gas Diffusion in Coal. *Processes*. 2019; 7(4):241.
https://doi.org/10.3390/pr7040241

**Chicago/Turabian Style**

Zang, Jie, Kai Wang, and Ang Liu.
2019. "Phenomenological Over-Parameterization of the Triple-Fitting-Parameter Diffusion Models in Evaluation of Gas Diffusion in Coal" *Processes* 7, no. 4: 241.
https://doi.org/10.3390/pr7040241