MODELING OF GAS SORPTION AND DIFFUSION BEHAVIOR AND IMPLICATIONS ON COALBED METHANE PRODUCTION

The Pennsylvania State University

The Graduate School

MODELING OF GAS SORPTION AND DIFFUSION BEHAVIOR AND IMPLICATIONS ON COALBED METHANE PRODUCTION

A Dissertation in

Energy and Mineral Engineering by

Yun Yang

© 2020 Yun Yang

Submitted in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

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The dissertation of Yun Yang was reviewed and approved by the following:

Shimin Liu

Associate Professor of Energy and Mineral Engineering Dissertation Advisor

Chair of Committee

Derek Elsworth

Professor of Energy and Mineral Engineering

Sekhar Bhattacharyya

Associate Professor of Energy and Mineral Engineering Chair of Mining Engineering Program

Chris Marone

Professor of Geosciences

Mort Webester

Professor of Energy Engineering

iii ABSTRACT

Exploration of coalbed methane (CBM) in North America started from the 1970s as the oil crisis shifted the interest to potential natural gas resources in coalbeds. Unlike conventional natural gas reservoirs, coal acts as both source and reservoir for hydrocarbon, where 90-98% of gas in the coal seam is adsorbed at its internal surface of coal matrices. Previous studies have demonstrated that pore structure is a key factor determining gas storage and transport behaviors of CBM reservoirs. This study established an analytical relationship between pore structure and gas sorption and diffusion characteristics of coal. My holistic study can be broadly divided into two parts, including theoretical modeling (Chapter 2) and experimental study (Chapter 3). Theoretical models have been proposed to quantify gas storage capacity and diffusion coefficient of coal by directly using pore structure parameters as physical inputs. The proposed models are calibrated and validated by laboratory data, and the results are presented in Chapter 4. The theoretical analysis and experimental work conducted in these three Chapters are further coupled into gas production simulator to define the unique production profile for mature CBM wells in San Juan basin (Chapter 5). The knowledge of pore structure alteration and its influence in gas-solid interactions of coal is employed to examine the applicability of a waterless fracturing technique, cryogenic fracturing in CBM reservoirs (Chapter 6).

A pore structure-gas sorption model has been proposed in Chapter 2. This model is validated against experimental data measured by sorption apparatus depicted in Chapter

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findings of my thesis on the relationship between pore structure and gas sorption behavior. Gas adsorption volume has long been recognized as an important parameter for CBM formation assessment as it determines the overall gas production potential of CBM reservoirs. As the standard industry practice, Langmuir volume (VL) is used to describe the

upper limit of gas adsorption capacity. Another important parameter, Langmuir pressure (PL), is typically overlooked because it does not directly relate to the resource estimation.

However, PL defines the slope of the adsorption isotherm and the ability of a well to attain

the critical desorption pressure in a significant reservoir volume, which is critical for planning the initial water depletion rate for a given CBM well. Qualitatively, both VL and PL are related to the fractal pore structure of coal, but the intrinsic relationships among

fractal pore structure, VL, and PL are not well studied and quantified due to the complex

pore structure of coal. In this thesis, a series of experiments were conducted to measure the fractal dimensions of various coals and their relationship to methane adsorption capacities. The thermodynamic model of the gas adsorption on heterogonous surfaces was revisited, and the theoretical models that correlate the fractal dimensions with the Langmuir constants were proposed. Applying the fractal theory, adsorption capacity (𝑉𝐿) is

proportional to a power function of specific surface area and fractal dimension, and the slope of the regression line contains information on the molecular size of the adsorbed gas. We also found that 𝑃𝐿 is linearly correlated with sorption capacity, which is defined as a

power function of total adsorption capacity (𝑉_𝐿) and a heterogeneity factor (ν). This implies that PL is not independent of VL, instead, a positive correlation between 𝑉_𝐿 and 𝑃_𝐿 has been

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inversely related to coal rank (Kim, 1977; Pashin, 2010), and Langmuir pressure is positively related to coal rank. It was also found that 𝑃_𝐿 is negatively correlated with adsorption capacity and fractal dimension. A complex surface corresponds to a more energetic system, which results in an increase in the number of available adsorption sites and adsorption potential, which raises the value of 𝑉𝐿 and reduces the value of 𝑃𝐿.

A pore structure-gas diffusion model is developed in Chapter 2. This model is validated against experimental data measured by sorption apparatus depicted in Chapter

3, and the validation results are presented in Chapter 4. Here presents an abstract of the

findings of the research on the relationship between pore structure and gas diffusion behavior. Diffusion coefficient is one of the key parameters determining the coalbed methane (CBM) reservoir economic viability for exploitation. Diffusion coefficient of coal matrix controls the long-term late production performance for CBM wells as it determines the gas transport effectiveness from matrix to fracture/cleat system. Pore structure directly relates to the gas adsorption and diffusion behaviors, where micropore provides the most abundant adsorption sites and meso- and macro-pore serve as gas diffusive pathway for gas transport. Gas diffusion in coal matrix is usually affected by both Knudsen diffusion and bulk diffusion. A theoretical pore-structure-based model was proposed to estimate the pressure-dependent diffusion coefficient for fractal porous coals. The proposed model dynamically integrates Knudsen and bulk diffusion influxes to define the overall gas transport process. Uniquely, the tortuosity factor derived from the fractal pore model allowed to quantitatively take the pore morphological complexity to define the diffusion for different coals. Both experimental and modeled results suggested that Knudsen

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diffusion dominated 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 factors directly depended on the Knudsen number. The proposed model was validated against experimental data, where the developed automated computer program based on the Unipore model can automatically and time-effectively estimate the diffusion coefficients with regressing to the pressure-time experimental data. This theoretical model is the first-of-its-kind to link the realistic complex pore structure into diffusion coefficient based on the fractal theory. The experimental results and proposed model can be coupled into the commercially available simulator to predict the long-term CBM well production profiles.

Chapter 5 presents a field case study to model long-term production behavior for

mature CBM wells. CBM wells in the fairway of the San Juan basin are in the mature stage of pressure depletion, experiencing very low reservoir pressure. These mature wells that have been successfully producing for more than 20 years exhibit long-term hyperbolic decline behavior with elongated production tails. Permeability growth during primary production is a well-known characteristic of fairway reservoirs and was historically interpreted to be the dominant factor causing the production tail. Several experimental works observed that the diffusion coefficient of the San Juan coal sample also varied with pressure. However, the pressure-dependent nature of gas diffusion in the coal matrix was neglected in most simulation works of CBM production. This may not significantly mis-predict the early and medium stage of production behavior when permeability is still the

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primary controlling parameter of gas flow. Prediction errors are elevated considerably for these late-stage fairway wells when diffusion mass flux takes the predominant role of the overall flowability. A novel approach to implicitly incorporate the evolution of gas diffusion during pressure depletion in the flow modeling of fairway reservoirs was proposed in this Chapter, where the derived diffusion-based matrix permeability model converts gas diffusivity into Darcy's form of matrix permeability. This modeling of matrix flow enables the direct use of lab measurements of diffusivity as input to the reservoir simulator. The calculated diffusion-based permeability also increases with pressure decrease. The matrix and cleat permeability growths are then coupled into the numerical simulator to history-match the field production of multiple CBM wells in the fairway region. The established numerical model provides satisfactory matches to field data and accurately predicts the elongated production tail in the late decline stage. Sensitivity analyses were conducted to examine the significance of accurate modeling of gas diffusion flow in CBM production throughout the life span of the fairway wells. The results show that the assumption on constant matrix flowability leads to substantial errors in the prediction of both peak gas production rate and long-term declining behavior. Accurate modeling of gas diffusive in the matrix is essential in production projection for the mature fairway CBM wells. The integration of gas diffusivity growth into production simulation improves the prediction of gas production rates and the estimation of ultimate recovery for the late-stage fairway reservoirs.

Chapter 6 investigates the applicability of cryogenic fracturing in exploiting CBM

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3. Cryogenic fracturing using liquid nitrogen is a waterless and environmentally-friendly

formation stimulation method to effectively create a complex fracture network and dilatated nano- and micro- pores within coal matrix that greatly enhances gas transport in coal matrix as well as cleats. However, the development of cryogenic fracturing is still at its infancy. Before large-scale field implementation, a comprehensive understanding of the fracture and pore alteration will be essential and required. For pore-scale investigation, this chapter focuses on the induced pore structural alterations due to cryogenic treatment and their effects on gas sorption and diffusion behaviors. The changes in the pore structure of coal induced by cyclic nitrogen injections were studied by physical adsorption at low temperatures. A micromechanical model was proposed to simulate the microscopic process and predict the degree of deterioration due to low temperature treatments. As a common characteristic of modeled results and experimental results, the total volume of mesopore and macropore increased with cryogenic treatment, but the growth rate of pore volume became much smaller as freezing-thawing were repeated. Pores in different sizes experienced different degrees of deterioration. In the range of micropores, no significant alterations of pore volume occurred with the repetition of freezing and thawing. In the range of mesopores, pore volume increased with the repetition of freezing and thawing. In the range of macropores, pore volume increased after the first cycle of freezing and thawing but decreased after three cycles of freezing and thawing. Because of pore structural alterations, cryogenic treatment enhanced gas transport process as the diffusion coefficients of the freeze-thawed coal samples were increased by 18.76% and 30.18% in the adsorption and desorption process. For the studied Illinois coal sample, repetitive applications of

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cryogenic treatment reduced macropore volume and increase mesopore volume. For the tested sample, the diffusion coefficient of the coal sample that underwent three cycles of freezing-thawing was lower than that of the coal sample that underwent a single cycle of freezing and thawing. The outcome of this study provides a scientific justification for post-cryogenic fracturing effect on diffusion improvement and gas production enhancement, especially for high “sorption time” CBM reservoirs.

For fracture-scale investigation, Chapter 6 develops a non-destructive geophysical technique using seismic measurements to probe fluid flow through coal and ascertain the effectiveness of cryogenic fracturing. A theoretical model was established to determine fracture stiffness of coal inverted from wave velocities, which serves as the nexus that correlates hydraulic with seismic properties of fractures. In response to thermal shock and frost forces, visible cracks were observed on coal surfaces that deteriorated the mechanical properties of the coal bulk. As a result, the wave velocity of the frozen coal specimens exhibited a general decreasing trend with freezing time under both dry and saturated conditions. For the gas-filled specimen, both normal and shear fracture stiffness monotonically decreased with freezing time as more cracks were created to the coal bulk. For the water-filled specimen, the formation of ice provoked by cryogenic treatment leads to the grouting of the coal bulk. Accordingly, fracture stiffness of the wet coal initially increased with freezing time and then decreased for longer freezing time. Coalbed with higher water saturation is preferred in the application of cryogenic fracturing because fluid-filled cracks can endure larger cryogenic forces before complete failures, and the contained water aggravates breaking coal as ice pressure builds up from volumetric expansion of

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water-ice phase transition and adds additional splitting forces on the pre-existing or induced fractures/cleats. This study also confirms that the stiffness ratio is sensitive to fluid content. The measured stiffness ratio varied between 0.7 and 0.9 for the dry coal, and it was less than 0.3 for the saturated coal. The outcome of this study provides a basis for a realistic estimation of stiffness ratio for coal for future discrete fracture network modeling.

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TABLE OF CONTENT

LIST OF FIGURES ... xiv

LIST OF TABLES ... xx ACKNOWLEDGEMENTS ... xxii Chapter 1 INTRODUCTION ... 1 1.1 Background ... 1 1.2 Problem Statement ... 3 1.3 Organization of Thesis ... 7

Chapter 2 THEORETICAL MODEL ... 9

2.1 Gas Sorption Modeling in CBM ... 9

2.1.1 Literature Review ... 9

2.1.2 Fractal Analysis ... 12

2.1.3 Pore Structure-Gas Sorption Model ... 13

2.2 Gas Diffusion Modeling in CBM ... 22

2.2.1 Literature Review ... 22

2.2.2 Diffusion Model (Unipore Model) ... 28

2.2.3 Pore Structure-Gas Diffusion Model ... 33

2.3 Summary ... 41

Chapter 3 EXPERIMENTAL WORK ... 45

3.1 Coal sample procurement and preparation ... 45

3.2 Low-Pressure Sorption Experiments ... 47

3.3 High-Pressure Sorption Experiment ... 48

3.3.1 Void Volume ... 49

3.3.2 Ad/Desorption Isotherms ... 51

3.3.3 Diffusion Coefficient ... 53

3.4 Summary ... 54

Chapter 4 RESULTS AND DISCUSSION ... 56

4.1 Coal Rank and Characteristics ... 56

4.2 Pore Structure Information ... 57

4.2.1 Morphological Characteristics ... 57

4.2.2 Pore size distribution (PSD) ... 58

4.2.3 Fractal Dimension ... 60

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4.4 Pressure-Dependent Diffusion Coefficient ... 67

4.5 Validation of Pore Structure-Gas Sorption Model ... 70

4.6 Validation of Pore Structure-Gas Diffusion Model ... 78

4.7 Summary ... 87

Chapter 5 FIELD APPLICATION TO CBM WELLS IN SAN JUAN BASIN ... 90

5.1 Overview of CBM Production ... 90

5.2 Reservoir Simulation in CBM ... 92

5.2.1 Numerical Models in CMG-GEM ... 92

5.2.2 Effect of Dynamic Diffusion Coefficient on CBM Production ... 94

5.3 Modeling of Diffusion-Based Matrix Permeability ... 97

5.4 Formation Evaluation ... 101

5.5 Field Validation (Mature Fairway Wells) ... 103

5.5.1 Location of Studied Wells ... 105

5.5.2 Evaluation of Reservoir Properties ... 107

5.5.3 Reservoir Model in CMG-GEM ... 114

5.5.4 Field Data Validation ... 116

5.5.5 Sensitivity Analysis ... 121

5.6 Summary ... 127

Chapter 6 PIONEERING APPLICATION TO CRYOGENIC FRACTURING ... 129

6.1 Introduction ... 129

6.2 Mechanism of Cryogenic Fracturing ... 130

6.3 Research Background ... 132

6.3.1 Cleat-Scale ... 132

6.3.2 Pore-Scale ... 133

6.4 Experimental and Analytical Study on Pore Structural Evolution ... 134

6.4.1 Coal Information ... 136

6.4.2 Experimental Procedures ... 137

6.4.3 Micromechanical Analysis ... 142

6.5 Freeze-thawing Damage to Nanoporous Network of Coal Matrix ... 146

6.5.1 Gas Kinetics ... 146

6.5.2 Pore Structure Characteristics ... 155

6.5.3 Application of Micromechanical Model ... 169

6.6 Experimental and Analytical Study on Fracture Structural Evolution ... 174

6.6.1 Background of Ultrasonic Testing ... 174

6.6.2 Coal Specimen Procurement ... 176

6.6.3 Experimental Procedures ... 177

6.6.4 Seismic Theory of Wave Propagation Through Cracked Media ... 179

6.7 Freeze-thawing Damage to Cleat System of Coal ... 193

6.7.1 Surface Cracks ... 194

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6.7.3 Fracture Stiffness ... 201

6.8 Summary ... 214

Chapter 7 CONCLUSIONS ... 219

7.1 Overview of Completed Tasks ... 219

7.2 Summary and Conclusions ... 220

APPENDIX A: USER INTERFACE IN MATLAB FOR THE ESTIMATION OF DIFFUSION COEFFICIENT ... 231

APPENDIX B: MATLAB PROGRAM TO DERIVE FRACTURE DENSITY ... 238

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LIST OF FIGURES

Figure 1-1: Illustration of multi-scale and multi-mechanism gas flow in a CBM reservoir. CBM production data. Source: DringInfo.inc. ... 3 Figure 1-2: Workflow of the theoretical and experimental study. ... 8 Figure 2-1: Graphical illustration of fractal dimension (Df): (a) For a smooth

surface, Df = 2; (b) For irregular surfaces, 2 < Df < 3. ... 13

Figure 2-2: Conceptual model of collisonal frequency at smooth and rough surfaces. ... 16 Figure 2-3: Diffusion regimes in coal matrix can be categorized as Knudesn

diffusion, viscous diffusion and bulk diffusion, controlled by Knudsen number. ... 24 Figure 2-4: User interface of unipore model based effective diffusion coefficient

estimation in MATLAB GUI... 31 Figure 2-5: Flowchart of the automated computer program for effective diffusion

coefficient estimation in MATLAB GUI ... 32 Figure 2-6: Fractal pore model. ... 35 Figure 2-7: Diagnostic plot of reciprocal diffusion coefficient (𝐷𝑝 − 1) vs. 𝑃 to

determine the dominant diffusion regime. Plot (b) is updated from plot (a) by considering the weighing factor of individual diffusion mechanisms and Knudsen diffusion coefficient for porous media. ... 41 Figure 3-1: Location and geologic information of Luling, Sijaizhuang and Xiuwu

coalmine. The Luling coal mine is located in the outburst-prone zone as separated by the F32 fault. ... 46

Figure 3-2: (a) Experimental adsorption setup (reference cell and sample cell); (b) Data acquisition system; (c) Schematic diagram of an experimental adsorption setup. ... 49 Figure 4-1: N2 adsorption-desorption results from four coal samples from Northeast

China. ... 58 Figure 4-2: The pores size distribution of the selected coal samples calculated from

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Figure 4-3: Fractal analysis of N2 desorption isotherms. ... 62 Figure 4-4: Results of methane adsorption tests and the corresponding Langmuir

isotherm curves. ... 65 Figure 4-5: Sample experimental results of CH4 sorption rate at 0.55 MPa for

Xiuwu-21 and Luling-10. ... 68 Figure 4-6: Variation of the experimentally measured methane diffusion

coefficients with pressure. ... 70 Figure 4-7: Relationships of fractal dimension (D1, D2) and Langmuir’s parameters

(VL, PL). ... 72

Figure 4-8: Relationship of Langmuir pressure (PL) to sorption capacity (X1=VLν). .. 76

Figure 4-9: Relationships of Langmuir’s volume (VL) to monolayer coverage

estimated by gas molecules with unit diameter (X2=σDf/2). ... 76

Figure 4-10: Relationship of Langmuir pressure (PL) to sorption capacity evaluated

from monolayer coverage (X3 = (SσDf/2 + B)ν). ... 77

Figure 4-11: Variation of bulk diffusion coefficient (DB) and Knudsen diffusion

coefficient (DK,pm) at different pressure stages for Sijiazhuang-15. ... 80

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*)). ... 81 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. ... 83 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). ... 85 Figure 4-15: Comparison between experimental and theoretical calculated

diffusion coefficients of the studied four coal samples at same ambient pressure. ... 85 Figure 5-1: (a) Structure contour map of San Juan Fruitland Formation. (b)

Application of Arp's decline curve analysis to gas production profile of San Juan wells. The deviation is tied to the elongated production tail. ... 92

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Figure 5-2: Modelling of gas transport in the coal matrix. ... 98 Figure 5-3: Workflow of simulating CBM production performance coupled with

pressure-dependent matrix and cleat permeability curves. ... 104 Figure 5-4: Blue dots correspond to the production wells investigated in this work.

The yellow circle marked offset wells with well-logging information available. ... 105 Figure 5-5: The production profile of the studied fairway well with the exponential

decline curve extrapolation for the long-term forecast. ... 106 Figure 5-6: Example of using a gamma-ray log and bulk density log to identify coal

layers and determine the net thickness of the pay zone for reservoir evaluation. The well-logging information is accessed from the DrillingInfo database (DrillingInfo, 2020) . ... 108 Figure 5-7: Fairway coalbed pressure-dependent permeability multiplier curve:

Po=1542 psi ... 113 Figure 5-8: Evolution of diffusion coefficient and corresponding equivalent matrix

permeability with pressure for San Juan coal. Data on the diffusion coefficient is provided by Wang and Liu (2016). ... 114 Figure 5-9: Rectangular numerical CBM model with a vertical production well

located in the center of the reservoir ... 116 Figure 5-10: Relative permeability curves for cleats used to history-match field

production data. ... 119 Figure 5-11: Matrix permeability growth during pressure depletion employed in the

matching process. ... 119 Figure 5-12: History-matching of the field gas production data of two fairway

wells: (a) Well A and (b)Well B (shown in Figure 5-4) by the numerical simulation constructed in CMG. ... 121 Figure 5-13: Effect of cleat and matrix permeability growth on gas production. The

solid grey lines correspond to comparison simulation runs with constant matrix/cleat permeability evaluated at initial condition. The grey dashed lines correspond to comparison simulations runs with constant matrix/cleat permeability estimated at average reservoir pressure of the first 4000 days. ... 125 Figure 6-1: Mechanism of cryogenic fracturing. Damage mechanism A derives

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contraction applied by sharp heat shock. Damage mechanism C is stimulated by the frost-heaving pressure. ... 132 Figure 6-2: The experimental system. (a) is a freeze-thawing system, where the

coal sample is first water saturated in the glassware beaker and then subject to cyclic liquid nitrogen injection. In between the successive injections, the sample is thawed at room temperature. The freeze-thawed coal samples and the raw sample are sent to the subsequent measurements ((b) and (c)). (b) is the experimental setup for measuring the gas sorption kinetics. This part of the experiment is to evaluate the change in gas sorption and diffusion behavior of coal after cryogenic treatment. (c) is the low-pressure adsorption system for the determination of surface area and porosimetry of pore structure of the coal sample. This step is to evaluate the pore-scale damage caused by the cryogenic treatment to the coal sample. ... 140 Figure 6-3: The process diagram of freeze-thawing treatment: (a) freezing

operation; (b) thawing operation. ... 141 Figure 6-4: Hori and Morihiro’s model of fractured micropore (Hori and Morihiro,

1998). The nanopore system of coal is modeled as a micro cracked solid. The pair of concentrated forces normally acting on the crack center represents the crack opening forces produced by the freezing action of pore water. ... 143 Figure 6-5: Results of methane ad/desorption tests and the corresponding Langmuir

isotherm curves for raw, 1F-T, and 3F-T coal. ... 149 Figure 6-6: The role of PL acting on the adsorption and desorption process. ... 150

Figure 6-7: Measured CH4 diffusion coefficients for raw coal, 1F-T coal, and 3F-T coal at different pressure stages. ... 151 Figure 6-8: Effect of surface heterogeneity on surface diffusion. (a) and (c) describe

surface diffusion along a rough surface. (b) describes surface diffusion along a flat surface. Less energy is required to initiate surface diffusion along a flat surface than a rough surface. ... 154 Figure 6-9: The role of multilayer of adsorption on surface diffusion. In desorption,

the already built-up multiple layers of adsorbed molecules smoothened the rough pore surface. Greater surface diffusion happens in the desorption process than the adsorption process. ... 154 Figure 6-10: Low-pressure N2 adsorption isotherm at 77K for the raw, 1F-T, and

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Figure 6-11: The adsorption isotherm of nitrogen at 77K on raw coal sample fitted by the BET equation and GAB equation. The solid curves are theoretical, and the points are experimental. The grey area Aad is the area under the fitted

adsorption isothermal curve by the GAB equation. ... 160 Figure 6-12: The desorption isotherm of nitrogen at 77K on raw coal sample fitted

by the GAB equation (n*=0) and the modifed GAB equation (n*=1, 2). The grey region is the area under the desorption isothermal curve fitted by the quadratic GAB equation. ... 163 Figure 6-13: PSD calculated from N2 sorption isotherm using the BJH model for

the raw, 1F-T, and 3F-T coal samples. ... 165 Figure 6-14: CO2 sorption isotherms at 273 K of the raw, 1F-T, and 3F-T coal

samples. ... 166 Figure 6-15: Micropore size distribution curves of CO2 adsorption for the raw,

1F-T, and 3F-T coal samples. ... 167 Figure 6-16: Proportional variation of pore sizes for different F-T cycles. ... 169 Figure 6-17: Various estimates of pore volume expansion (Upper case and Lower

case) due to cyclic liquid nitrogen injections, according to the micromechanical model (solid line). The grey area is the range of estiamtes specified by the two extreme cases. The computed results are compared with the measured pore volume expansion determined from experimental data listed in Table 6-4 (scatter).Vpi is the intial pore volume or the pore volume of the raw coal sample. Vpf is the pore volume after freezing and thawing corresponding to the pore

volume of 1F-T sample and 3F-T sample. ... 173 Figure 6-18: An intact coal specimen (M-2) before freezing. ... 177 Figure 6-19: Experimental equipment and procedure. ... 179 Figure 6-20: The fracture model: random distribution of elliptical cracks in an

otherwise in-contact region. ... 180 Figure 6-21: The workflow of seismic interpretations of fracture stiffness for coal

specimens subject to cryogenic treatments. ... 194 Figure 6-22: Evolution of surface cracks in a complete freezing-thawing cycle for

(a) dry coal specimen (b) wet coal specimen. Major cracks are marked with

red lines in the images of surface cracks taken at room temperature i.e., pre-existing surface cracks and surface cracks after completely thawing. ... 196

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Figure 6-23: Recorded waveforms of compressional waves at different freezing times for (a) #1 dry coal specimen and (b) #2 saturated coal specimen ... 198 Figure 6-24: Variation of seismic velocity with freezing time for (a) dry coal

specimen, (b) wet coal specimen. ... 200 Figure 6-25: Under dry condition (M-1), the variation of normal and tangential

fracture stiffness and tangential/normal stiffness ratio with freezing time. ... 204 Figure 6-26: Under wet condition (M-2), variation of normal and tangential fracture

stiffness and tangential/normal stiffness ratio with freezing time. ... 209 Figure 6-27: Effect of the presence of water and ice on fracture stiffness. A

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LIST OF TABLES

Table 2-1: Sorption kinetic experiments of methane performed in the various literature HVB and LVB are high and low volatile bituminous coals. Sub is sub-bituminous coals. Diffusion coefficient is derived from unipore model. ... 27 Table 3-1: Proximate analyses and vitrinite reflectance of the coal samples used in

this study. ... 46 Table 3-2: Void volume for each sample estimated with multiple injections of

Helium. ... 51 Table 4-1: Mean pore diameter, specific surface area, and pore volume of the coal

samples analyzed during this study. ... 59 Table 4-2: Fractal dimensions of the four coal samples. ... 62 Table 4-3: The fractal dimension, mean free path and tortuosity factor based on the

fractal pore model and estimated at the specified pressure stage (i.e., 0.55, 1.38, 2.48, 4.14, 6.07, and 8.07 MPa). ... 63 Table 4-4: Langmuir parameters for methane adsorption isotherms. ... 66 Table 4-5: Parameters used in the analysis of pore characteristics and its effect on

CH4 adsorption on coal samples. ... 74

Table 4-6: Theoretically calculated bulk diffusion coefficient (DB) and Knudsen

diffusion coefficent of porous media (DK,pm). ... 79

Table 5-1: Investigated logs for coalbed methane formation evaluation ... 102 Table 5-2: Coal characteristics interpreted from well-logging information in four

offset wells. ... 109 Table 5-3: Input parameters for Liu and Harpalani model on the permeability

growth ... 113 Table 5-4: Coal seam properties used to history-match field data of two fairway

wells ... 118 Table 6-1: Langmuir’s parameters for raw, 1F-T, and 3F-T coal. The bracket

indicates the percentage increase in PL of 1F-T and 3F-T coal with respect to PL of raw coal. An increase in PL is preferred in gas production as it promotes

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Table 6-2: Measured diffusion coefficients of raw coal, 1F-T coal and 3F-T coal (Draw, D1F-T, D3F-T) in the adsorption process and desorption process and the

corresponding increase in the diffusion coefficient due to freeze-thawing cycles (ΔD1F-T, ΔD3F-T). ... 152

Table 6-3: BET surface area, parameters of GAB adsorption model and quadratic GAB desorption model of nitrogen experimental sorption data with their corresponding correlation coefficients (R2_{), the areas under the best adsorption}

and desorption fitting curves (Aad, Ade), and the respective hysteresis index of

raw coal, 1F-T coal, and 3F-T coal samples. ... 157 Table 6-4: Peak pore diameter, mean pore diameter, total pore volume with its

distribution in different pore sizes after the different number of freeze-thawing cycles. ... 168 Table 6-5: Coal properties used in the proposed deterioration analysis ... 171 Table 6-6: Physical properties of two coal specimens used in this study. ... 177 Table 6-7: Crack density (𝑁 ) and average half-length ( 𝑎 ), aperture ( 𝑏 ) and

ellipticity (𝑒) of cracks determined from the automated computer program. ... 202 Table 6-8: Thermophysical parameters used in modeling heat transfer in the

freezing immersion test. The heat capacity (Cp) and heat conductivity (𝑘𝑐) of

the saturated coal specimen (M-2) were measured at room temperature of 25 ℃ following the laser flash method (ASTM E1461-01) . ... 208

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ACKNOWLEDGEMENTS

I would like to express my gratitude to my primary supervisor, Dr. Shimin Liu, who guided me throughout this entire Ph.D. study for three and half years. His patience, enthusiasm, and immense knowledge make me passionate about my research and my Ph.D. life an enjoyable journey. I could not have a better advisor and mentor.

I would also like to thank my doctoral committee members, Dr. Derek Elsworth, Dr. Sekhar Bhattacharyya, and Dr. Chris Marone, who have provided their valuable suggestions and insights on this research and taught me a great deal about scientific research. I also wish to acknowledge the help provided by Dr. Luis Ayala and Dr. Hamid Emami as my master advisor. Their advice and assistance taught me the way to conduct professional research.

I am also grateful for my colleagues Ang Liu, Guijie Sang, Qiming Huang, Long Fan, Xiaowei Hou, who were good colleagues and provided me kind help in the laboratory work. A special thank also goes to my best friends in the U.S. and China, Yuzhe Cai and Peiwen Yang, for their support and time spending with me during my graduate study.

I would also like to thank my parents in China, Chunhe Yang and Jun Yang. They always listened to my words and helped me get through all the hard times I encountered during my life in the U.S. Thanks for their unconditional love. I also want to thank my boyfriend, Haoming Ma as a perfect companion of my life.

Chapter 1

1.1 Background

Exploration of coalbed methane (CBM) in North America started with the early activities conducted by US Bureau of Mines experiments in Alabama and Pennsylvania. Then it came to prominence in the 1980s as the oil crisis shifted the interest to potential natural gas resources in coalbeds. CBM classified by energy industry is an unconventional resource and an important natural gas source. According to Energy Information Administration (EIA), the proven coalbed methane reserves of the U.S. was 11.8 trillion cubic feet (TCF) in 2017. The CBM production in 2017 was 0.98 TCF that accounted for 3.0% of total natural gas production in the U.S. (EIA, 2018). CBM is considered as an environmentally friendly fuel because its combustion emits no ash, no toxins, and less greenhouse gas emission compared to oil, coal, or even wood (Al-Jubori et al., 2009). The extraction of CBM from coal seam also prevents underground coal-mine gas outbursts and benefits safe mining operations. For these advantages, CBM is expected to be an essential sector in the future energy portfolio.

Coalbed incorporate unique gas transport and storage mechanism that differs from conventional reservoirs. Coal acts as both source and reservoir for the gas, where 90-98% of methane is adsorbed in a liquid-like dense phase at the internal surface of coal matrix by

2

physical adsorption with the remaining small amount of gas compressed in open void spaces in the natural fracture network by pressure mechanism (Gray, 1987; Harpalani and Chen, 1997a; Levine, 1996). The sorbed gas content of coal depends on mineral content, total organic content, coal rank, moisture content, petrology, gas composition as well as reservoir conditions (Busch and Gensterblum, 2011; Yee et al., 1993). Migration of methane in a CBM reservoir starts from desorption from the internal coal surface followed by the diffusion in coal matrix, which is subject to the diffusion coefficient and gas concentration gradient. After diffusing through the matrix, the gas reaches the naturally occurring fractures (cleats) and evolves to Darcy flow controlled by the permeability of coal and pressure gradient (Figure 1-1). The rate of viscous Darcian flow through the cleat network depends on the distribution of cleat presented in coalbed. The rate of gas diffusion depends on the pore properties of the coal matrix. Production of gas from a CBM reservoir is intuitively affected by both diffusion coefficient and permeability of coal (King, 1985; Kumar, 2007). At the late stage of a CBM production well (i.e. mature wells), coal permeability might not be the bottle-neck for the overall gas production as commonly believed, and instead, diffusion process dominates overall well production performance since the matrix to cleat influx is limited (Wang and Liu, 2016).

3

Figure 1-1: Illustration of multi-scale and multi-mechanism gas flow in a CBM reservoir. CBM production data. Source: DringInfo.inc.

1.2 Problem Statement

Coal is a complex polymeric material with a convoluted pore structure (Clarkson and Bustin, 1999a). Coal exhibits a broad pore size distribution ranging from micropores (< 2 nm) to mesopores (2-50 nm) and macropores (>50 nm) according to the International Union of Pure and Applied Chemistry (IUPAC) classification (Schüth et al., 2002). As

0 5 10 15 20 25 30 0 50 100 150 Prod uctio n r ate (M cf/d ay) time (yrs) Desorption from internal pore surface Diffusion in coal matrix

Butt cleat Face cleat Darcy’s flow Log (nm) >3 2 1 0 Dominated by

Darcy’s flow Dominated by Diffusion + Desorption

Short-term Long-term

Well information: Pennsylvanian Formation Central Appalachian Basin Total producing life: 28 yrs

4

micropores provide the greatest internal surface area, the proportion of microporosity is a dominant factor of gas storage in coal. The distribution of mesopores and macropores provide free gas storage and transport pathway for gas molecules that dominates gas diffusion rate in coal. Pore structure has an immerse effect on gas storage and transport behavior in coal matrix (Smith and Williams, 1984).

Extensive research have been performed on understanding the effect of pore structure on gas sorption and diffusion behavior of coal. Pore structure of coal is known to be complex in occurrence that does not converge to a traditional Euclidean geometry. The application of fractal theory provides an intuitive description of heterogeneous structure of coal (Pfeifer and Avnir, 1983). Coal with a convoluted pore structure typically have high adsorption energy, a great number of adsorption sties as well as elevated gas storage capacity. On the other hand, coal with a homogenous structure is favorable for gas desorption and diffusion. Fractal analysis serves as a powerful tool of characterizing the complexity of pore structure of coal. The effect of fractal dimension on gas adsorption capacity has been studied in several works (Cai et al., 2013; Li et al., 2015; Liu and Nie, 2016; Wang et al., 2018a; Wang et al., 2016; Yao et al., 2008). However, their works were limited to qualitative analysis derived from experimental measurements. A quantitative modeling of gas sorption capacities by using pore structure information as direct inputs is still lacking in the literature. For CBM production, diffusion coefficient is another important parameter as it directly related to the matrix permeability and is a required input in most reservoir simulators such as CMG-GEM, ARI-COMET, IHS-FASTCBM. However, as coal exhibits ultralow matrix permeability, direct permeability measurements

5

on coal matrix is subject to great uncertainties. As an alternative, diffusion coefficient measured by particle method varies with pressure, but no unified trend persists (Charrière et al., 2010; Mavor et al., 1990a; Nandi and Walker, 1975; Pillalamarry et al., 2011; Wang and Liu, 2016). Theoretical understanding on the change of diffusion coefficient of coal during pressure depletion is still obscure in the previous studies.

A mechanistic based understanding on the correlation between pore structure and gas transport mechanism of coal is highly desireable to be established. This is because pore structural parameters including pore size, pore shape and pore volume is closely related to coal rank and coal composition (e.g., fixed carbon, moisture, mineral constituent, vitrinite, inertinite and others) that control gas diffusion characteristics of coal. A dual porosity model (Warren and Root, 1963) that depicts coal as large fractures (secondary-porosity system) and much smaller pores (primary-porosity system) is commonly applied to describe the physical structure of coal for gas transport simplification, which is widely adopted in commercial CBM simulators such as CMG-GEM, IHS-FASTCBM. Diffusion coefficient or sorption time is a required input in all these numerical simulations. Therefore, it is critical to couple gas diffusion into CBM simulation that requires a comprehensive understanding on the pressure-dependent diffusion behavior. Nevertheless, the application of dual-porosity model to simulate CBM production always treats the high-storage matrix as a source feeding gas to cleats with a constant diffusion coefficient, which violates its pressure-dependent nature. As discussed, the traditional modeling approach may not significantly mis-predict the early and medium stage of production behavior since the permeability is still the dominant controlling parameter. However, the prediction error will

6

be substantially elevated for mature CBM wells which diffusion mass flux dominates total gas production. It is crucial to accurately model gas diffusion in coal matrix and properly weigh the contribution of diffusional flux from matrix to cleats and Darcian flux through cleats to the overall gas production.

Even with the improved understanding of gas sorption and diffusion on coal, the CBM development is still challenging due to the low permeability, high fracture density, high formation compressibility. CBM reservoir stimulation is commonly required for the coal formations. The conventional hydraulic fracturing can effectively increase the stimulated reservoir volume (SRV) through fracture generation; however, it has no influence on the diffusion enhancement for low diffusion coals. Therefore, the exotic formation stimulation should be pursued and investigated for simultaneously increasing SRV as well as the micropore dilation for the diffusion enhancement. Cryogenic fracturing is one of candidates for this purpose and its effectiveness should be investigated for future application.

The objective of this Dissertation was to predict gas storage and transport properties of coalbed based on pore structure information. The study aimed at an improved understanding on the change of gas diffusion coefficient or matrix permeability of coal during CBM production that is critical for accurate analysis of production data and forecasting of well performance.

7

1.3 Organization of Thesis

The present study can be separated into four tasks: theoretical models, experimental work, field application, and fundamental research on cryogenic fracturing. Figure 1-2 outlines the workflow of the theoretical (Chapter 2) and experimental studies (Chapter

3). Two sets of theoretical models were developed for both gas sorption and diffusion

characteristics and their relationship with pore structure of coal (Chapter 2). Correspondingly, sorption experiments were conducted at high-pressure for obtaining sorption isotherms and diffusion coefficient, and at low-pressure for characterizing nanoporous network of coal (Chapter 3). Then theoretical models were validated against laboratory data (Chapter 4). The theoretical and analytical methodology developed in

Chapter 2 and Chapter 3 on the quantification of gas diffusion in coal matrix was applied

to history-match field production for mature CBM wells in San Juan Basin (Chapter 5).

Chapter 6 presents another application of theoretical and analytical methodology

developed in Chapter 2 and Chapter 3, which is the development of cryogenic fracturing in CBM exploration. This research is conducted at multi-scale ranging from micropores to large apertures of coal utilizing the experimental setup depicted in Chapter 3 and the theoretical analysis in Chapter 2 to evaluate the effectiveness of this waterless fracturing technique on the enhancement of gas production. Chapter 7 presents the conclusion based on the results of experimental and analytical work.

8

Figure 1-2: Workflow of the theoretical and experimental study.

Validation of Theory2

Understanding gas production mechanism

regarding to pore structure of coal

Theory

Experiment

Pore structure-Gas

kinetic Model Gas Kinetic Pore Structure

Theory 1 Theory 2 _{Experiment (CH}High P Sorption

4) Low P Sorption Experiment Adsorption Capacity Adsorption Rate Transport Rate Heterogeneity Pore structure-Sorption Model Pore structure-Diffusion Model Validation of Theory1

9

Chapter 2

THEORETICAL MODEL

2.1 Gas Sorption Modeling in CBM

Modeling of gas adsorption behavior is critical for resource assessment as well as production forecasting of coal reservoirs. As coal incorporates a nanoporous network, sorption characteristics including adsorption capacity and adsorption pressure are closely related to pore structure attributes. However, the mechanism of how these microscale characteristics of coal affect gas adsorption behavior is still poorly understood. This section develops a pore structure-gas sorption model that can predict gas sorption isotherms based on pore structure information. This model provides a direct evaluation method to link the micro-pore structure with the sorption behavior of coal.

2.1.1 Literature Review

Extensive research (Budaeva and Zoltoev, 2010; Cai et al., 2013; Li et al., 2015; Wang et al., 2018a; Wang et al., 2016) have been performed on the fundamental relationship between methane adsorption and pore structure in coals, where a dual porosity model describes the complex structure of coal (Warren and Root, 1963). Typically, macro- (>50 nm) and mesopores (2-50 nm) most likely provide transport pathways, and micropores (< 2 nm) give the greatest internal surface area and hence gas storage capacity (Ceglarska-Stefańska and Zarębska, 2002; George and Barakat, 2001; Harpalani and Chen, 1997; Laubach et al., 1998). Coal pores, distributed in a three-dimensional (3D) space, are

10

hard to model accurately using traditional Euclidean geometric methods and do not converge to Euclidean geometry (Mandelbrot, 1983; Wang et al., 2016). The concept of fractal geometry raised by Mandelbrot (1983) proves to be a powerful analytical tool that provides an intuitive description of the pore structure of coal by characterizing the pore size distribution over a range of pore sizes with a single number (i.e., fractal dimension, 𝐷𝑓). Different values of 𝐷𝑓 were found to be between 2 and 3 for different sized pores,

which is frequently applied to quantify the heterogeneity of pore surface and volume for coals (Pfeifer and Avnir, 1983). A value of fractal dimension close to 2 corresponds to a more homogenous pore structure. Otherwise, the pore structure becomes more complex as 𝐷_𝑓 approaches 3. Among different methods of quantifying fractal dimension, low-pressure N2 adsorption/desorption is the most time- and cost-effective technique, where fractal

Brunauer-Emmett-Teller (BET) model and fractal Frenkel–Halsey–Hill (FHH) models have been effectively applied to evaluate irregularity of pore structure (Avnir and Jaroniec, 1989; Brunauer et al., 1938a; Cai et al., 2011). In the fractal analysis, two distinct values of fractal dimensions (𝐷₁ and 𝐷₂) can be derived from low- and high-pressure intervals of N₂ sorption data. The two fractal dimensions reflect different aspects of pore structure heterogeneity interpreted as the pore surface (𝐷1) and the pore structure fractal dimension

(𝐷2) (Pyun and Rhee, 2004). Higher value of 𝐷1 characterizes more irregular surfaces

giving more adsorption sites. Higher value of 𝐷2 corresponds to higher heterogeneity of

the pore structure and higher liquid/gas surface tension that diminishes methane adsorption capacity (Yao et al., 2008). It has been shown that sorption mechanisms may change at different pressure stages that causes the fractal dimension of pore surface (𝐷₁) differs from

11

fractal of pore volume (𝐷2) (Li et al., 2015). Clearly, fractal dimensions are closely tied to

adsorption behavior of the coal.

The sorption isotherm is commonly used to describe gas sorption capacity. Different adsorption models are developed to mathematically model the gas sorption isotherms of coals, including Langmuir, BET, Barrett-Joyner-Halenda (BJH), density functional theory (DFT) model, etc. (Zhang and Liu, 2017). Among all these models, the Langmuir model is the most straightforward and widely accepted model. Langmuir’s constants, 𝑃𝐿 and 𝑉𝐿,

define the shape of sorption isotherm, where 𝑉_𝐿 describes the ultimate gas storage capacity and 𝑃_𝐿 changes the slope of the sorption isotherm. Some works (Cai et al., 2013; Li et al., 2015; Liu and Nie, 2016; Wang et al., 2018a; Wang et al., 2016; Yao et al., 2008) have attempted to correlate fractal dimension with Langmuir’s parameters, but only based on experimental results with limited theoretical analysis. Among these reported studies, the empirical correlations were not universally consistent for different sets of coal samples. Specifically, Yao et al., (Yao et al., 2008) found significant binomial correlations between 𝑉_𝐿 and fractal dimensions (𝐷₁ and 𝐷₂). Liu and Nie (Liu and Nie, 2016) claimed 𝑉_𝐿 increased linearly with fractal dimensions, but Li et al., (Li et al., 2015) observed that 𝑉𝐿

was affected negatively by 𝐷₂ and correlated positively with 𝐷₁. Some qualitative interpretations were made on these relationships as a high value of 𝐷₁ means irregular surfaces of coals, which provides abundant adsorption sites for gas molecules resulting in high adsorption capacity but the physical mechanism of 𝐷2 acting on 𝑉𝐿 was not well

analyzed. Besides, 𝑃𝐿 was observed to be strongly related to 𝐷2 in Liu and Nie (Liu and

12

inconsistent empirical correlations imply that the mechanism of fractal dimensions acting on gas sorption behavior is still not clearly understood.

2.1.2 Fractal Analysis

The fractal dimension (𝐷_𝑓) of surfaces characterizes surface irregularity, and it has a value between 2 and 3 (Pfeifer and Avnir, 1983). A rougher surface incorporates a value of 𝐷𝑓 approaching 3 as illustrated in Figure 2-1. For coal, the fractal surface is analyzed

using a fractal BET model and a fractal FHH model (Avnir and Jaroniec, 1989; Brunauer et al., 1938a; Cai et al., 2011).

In this current study, the FHH model was used to determine surface fractal dimension from 𝑁₂ sorption isotherm data. The fractal dimension is determined by

ln (V V0

) = 𝐴 ln (ln (P0

𝑃)) + 𝐸 ( 2-1 )

where 𝑉/𝑉0 is the relative adsorption at the equilibrium pressure 𝑃, 𝑉0is a monolayer

adsorption volume, 𝑃₀ is gas saturation pressure, 𝐸 is the y-intercept in the log-log plot , and 𝐴 is the power-law exponent used to determine the fractal dimension of the coal surface (𝐷_𝑓) (Qi et al., 2002). Two distinct formulas were proposed to correlate 𝐴 to 𝐷_𝑓 by (Liu and Nie, 2016):

𝐷_𝑓 = 𝐴 + 3 ( 2-2 )

and

13

Eq. (2-2) was used to determine 𝐷 from the slope 𝐴 as Eq. (2-3) would consistently yield an unreasonably high value of fractal dimension (Yao et al., 2008). Typically, two linear parts were observed in the log-log plot of ln⁡(𝑉

𝑉0) vs. ln (ln ( P0

P)) corresponding to

high- and low-pressure adsorption. The fractal dimension (𝐷 ) of the coal surface is obtained from the slope of the straight line (𝐴).

Figure 2-1: Graphical illustration of fractal dimension (Df): (a) For a smooth surface, Df =

2; (b) For irregular surfaces, 2 < Df < 3.

2.1.3 Pore Structure-Gas Sorption Model

Langmuir Isotherm on Heterogenous Surfaces

A type I isotherm describes the sorption behavior of microporous solids where monolayer adsorption forms on the external surface of the adsorbent (Gregg et al., 1967). Coal is typically treated as a microporous medium and behaves like a type I isotherm without exhibiting significant hysteresis in pure component sorption. The most widely applied adsorption model for a type I isotherm is the Langmuir isotherm. Numerous studies (Bell and Rakop, 1986b; Clarkson et al., 1997; Mavor et al., 1990a; Ruppel et al., 1974) on methane adsorption on coal have shown that Langmuir isotherm accurately fits over the range of temperatures and pressures applied. The surface of the adsorbent is assumed to

𝐷 = 2

(a)

2 𝐷 3

(b)

14

be energetically homogenous, and only a single layer of adsorbate is considered to form (Langmuir, 1918). In this study, the Langmuir isotherm equation is used to model the coal adsorption isotherm from high-pressure gas sorption data of dry coals. The classic form of this equation is expressed as:

𝑉 = 𝑃

𝑃 + 𝑃𝐿

𝑉_𝐿 ( 2-4 )

where, 𝑉_𝐿 and 𝑃_𝐿 are two regressed parameters to fit experimental adsorption data in the plots of 𝑃/𝑉 vs. 𝑃.

Langmuir constants (𝑉𝐿 and 𝑃𝐿) are important parameters that greatly impact the field

development of coal reservoir. Langmuir volume (𝑉_𝐿) is a direct indicator of the CBM gas storage capacity. Langmuir pressure (𝑃_𝐿) is closely related to the affinity of a gas on the solid surface and the energy stored in the coal formation. 𝑉_𝐿 is proportional to total number of available sites for adsorption, and is further affected by surface complexity, total adsorption volume and coal composition (Cai et al., 2013). The relationship between 𝑉𝐿

and pore structure was analyzed, where specific surface area (SSA) is comprised of the mesopore and micropore SSA estimated using BET and Dubinin-Radushkevich (DR) models, respectively (Clarkson and Bustin, 1999a; Zhao et al., 2016). 𝑃_𝐿 is an important parameter in CBM production. Mavor et al. (1990a) shows that 𝑃_𝐿 along with gas content data helps determine critical desorption pressure. This pressure is an important parameter that affects the pressure decline performance of CBM reservoirs as discussed in Okuszko et al. (2007). However, how pore structure relates to 𝑃_𝐿 is still poorly understood, and no quantitative relationship was reported to link the 𝑃_𝐿⁡with the pore structure.

15

Crickmore and Wojciechowski (1977) implied that for a system with high enough number of types of adsorption sites, the total rate of the adsorption process is approximated as: 𝑅_𝑡 =𝑑𝜃1 𝑑𝑡 = 𝑘̅̅̅𝑃(1 − 𝜃𝑎 1) 𝑤+1_{− 𝑘} 𝑑 ̅̅̅𝜃₁𝑚+1 _{( 2-5 )}

where 𝜃1 is surface coverage, 𝑤 and 𝑚 are the coefficients of variation of the rate

constants of adsorption and desorption, and 𝑘̅̅̅ and 𝑘_𝑎 ̅̅̅ are the adsorption and desorption _𝑑 constants, respectively, which are averaged over the heterogeneous surfaces. Commonly, the spread of these two distributions are similar or are even treated as equivalent (i.e., 𝑤 = 𝑚). Then the expression of total rate can be simplified to the following equation by replacing coefficient w by coefficient m,

𝑅_𝑡 = 𝑑𝜃𝑡

𝑑𝑡 = 𝑘̅̅̅𝜇(1 − 𝜃𝑎 1)

𝑚+1_{− 𝑘} 𝑑

̅̅̅𝜃₁𝑚+1 _{( 2-6 )}

where 𝜇 is the number of moles of molecules striking a smooth surface per unit area per second and can be determined from molecular dynamics as:

𝜇 = 𝑃

(2𝜋𝑀𝑅𝑇)1/2 ( 2-7 )

where P is the pressure of the gas in free phase, M is the molecular weight, R is universal gas constant, T is temperature.

For a rough surface, the number of collisions would be expected because of multi-reflection as illustrated in Figure 2-2. A surface heterogeneity factor (𝜈)(Jaroniec, 1983) is introduced to characterize the roughness of coal surfaces with a value ranging from 0 to 1. A ν of 1 corresponds to a perfect smooth surface. For a first-order of approximation, the

16

striking frequency is assumed to increase exponentially with surface heterogeneity, which is expressed as 𝜇1/𝜈.

Figure 2-2: Conceptual model of collisonal frequency at smooth and rough surfaces. At equilibrium, surface coverage (𝜃1) is determined by

𝜃₁ = (𝑘̅̅̅′𝑎 𝑘_𝑑 ̅̅̅ ) 𝜈 𝑃 1 + (𝑘̅̅̅′𝑎 𝑘_𝑑 ̅̅̅ ) 𝜈 𝑃 ( 2-8 ) where 𝜈 = 1/(𝑚 + 1) and 𝑘̅̅̅𝑎 ′ = 𝑘̅̅̅(2𝜋𝑀𝑅𝑇)𝑎 −1/2𝜈.

Compared with Langmuir’s equation, the expression of Langmuir’s coefficient (𝑎) for a heterogenous surface is (Avnir and Jaroniec, 1989):

𝑎 = 1 𝑃_𝐿 = ( 𝑘̅̅̅′𝑎 𝑘𝑑 ̅̅̅) 𝜈 ( 2-9 ) The value of 𝜈 ranges from 0 to 1. When 𝜈 = 1, Eq. (2-8) reduces to Langmuir’s model equation, which agrees with the assumption made in the development of Langmuir’s equation (Langmuir, 1918). 𝜈 may be determined from surface roughness or fractal dimension (𝐷_𝑓) with the value ranging between 2 and 3 (Avnir and Jaroniec, 1989). High

17

𝜈 (relatively small 𝐷𝑓) values indicate a smooth pore surface and a low 𝜈 value represents

an irregular surface. Based on this interpretation and assuming a linear correspondence, 𝜈 can be made a function of 𝐷_𝑓 as,

𝜈 = 1 − (𝐷𝑓− 2

2 ) ( 2-10 )

Two interpretations of 𝜈 are given as measures of surface complexity and variation of the reaction rate constants. In most cases, the latter one may not be directly identical to the former one. A coefficient (𝐶) may be necessary to describe the dependence of the spread of reaction rate constants on surface roughness. Langmuir’s coefficient is then given by 𝑎 = (𝑘̅̅̅′𝑎 𝑘𝑑 ̅̅̅) 𝐶𝜈 ( 2-11 )

If a two-dimensional potential box is used to describe an adsorption site, then the adsorption rate constant (𝑘̅̅̅′) is proportional to the rate of molecules impinging on the site _𝑎 (Hiemenz and Hiemenz, 1986),

𝑘𝑎

̅̅̅′ = 𝑔𝑁0(2𝜋𝑀𝑅𝑇)−1/2𝐶𝜈 ( 2-12 )

where, 𝑁0 is the total available sites for adsorption evaluated by Langmuir’s volume (𝑉𝐿)

and 𝑔 is the fraction of the molecules that condenses and is held by surface forces.

Desorption rate constant (𝑘̅̅̅) is composed of a frequency factor (𝑓) and a Bolzmann _𝑑 factor, (𝑘_𝐵).

𝑘𝑑

18

where, 𝑓 is the frequency with which the adsorbed molecules vibrate against the adsorbent, and 𝑄 is the activation energy of desorption, which is approximated by adsorption heat.

The ratio of 𝑘̅̅̅′ and 𝑘_𝑎 ̅̅̅ is directly related to the Langmuir coefficient 𝑎 as, _𝑑

𝑎 = (𝑘̅̅̅′𝑎 𝑘_𝑑 ̅̅̅) 𝐶𝜈 = 1 √2𝜋𝑀𝑅𝑇( 𝑔 𝑓𝑉𝐿𝑒 𝑄/𝑘𝐵𝑇₎ 𝐶𝜈 ( 2-14 ) where, 𝑁0 is replaced by 𝑉𝐿.

Both 𝑓 and 𝑔 depend on the affinity of the adsorbate to gas molecules. For many systems, it is expected that these two constants would be equal, resulting in the modified form of Langmuir’s constant.

𝑎 = 1

√2𝜋𝑀𝑅𝑇(𝑉𝐿𝑒

𝑄/𝑘𝐵𝑇₎𝐶𝜈 _{( 2-15 )}

As explained in Crosdale et al. (1998), methane adsorption onto the pore surfaces of coal is dominated by physical adsorption indicated by the reversibility of the equilibrium between free and adsorbed phase, the relatively rapid sorption rate when pressure or temperature are the varied and low heat of adsorption. For a physisorption dominated system, only physical structural heterogeneity is considered, neglecting the effect of surface geochemical properties and functional groups on adsorption energy. As a result, adsorption heat released at a smooth surface is constant for different coal species, denoted as 𝑄𝑠𝑡. In the aspects of physical structural heterogeneity, coal surface with a low value of

𝜈 corresponds to a more heterogeneous structure with a substantial amount of adsorption energy, which may be approximated as proportional to the inverse of heterogeneity factor

19

(1/𝜈). Based on this, 𝑄 is related to the heat of adsorption measured at a perfect smooth surface (𝑄𝑠𝑡) as:

𝑄 = 𝐾𝑄𝑠𝑡

𝐶𝜈 ( 2-16 )

where, 𝐾 is a constant that evaluates how severe 𝑄 changes in response to surface complexity (𝜈), and 𝑄𝑠𝑡 may be approximated as the latent heat of vaporization.

However, an accurate evaluation of the activation energy of adsorption is related to an energy distribution function (𝑓(𝜀) ). As explained by Jaroniec (1983), an explicit solution of 𝑓(𝜀) on microporous media is hard to obtain and for the purpose of a first order approximation, the activation energy of adsorption may be treated as a constant for given gas species and for the temperature at surfaces with similar properties.

Then the Langmuir constant (𝑎) can be expressed as a function of the heterogeneity factor (𝜈), Langmuir’s volume (𝑉_𝐿) , and temperature (𝑇) as:

𝑎 = 1 𝑃_𝐿 = (𝑉𝐿) 𝐶𝜈_{𝐹(𝑇) ( 2-17 )} 𝐹(𝑇) = 1 √2𝜋𝑀𝑅𝑇𝑒 −𝐾𝑄𝑠𝑡/(𝑘𝐵𝑇) _{( 2-18 )}

where 𝐹(𝑇) is a temperature-dependent function and becomes a constant under isothermal condition.

The Langmuir’s volume (𝑉𝐿) is a measure of ultimate adsorption capacity, which is

affected by specific surface area, pore size distribution and fractal dimension (Zhao et al., 2016). Research has been performed (Avnir et al., 1983; Fripiat et al., 1986; Pfeifer and Avnir, 1983) to quantify the sorption capacity of a heterogenous surface, where the number

20

of gas molecules held by the adsorbent has a power-law dependence on surface area, and the exponent describes the irregularity of the surface, i.e., fractal dimension. The adsorption capacity in multilayer adsorption is hard to accurately derive and instead, the power-law relationship is commonly used to correlate the monolayer coverage with the surface area and fractal dimension. This simplification agrees to the assumption made in the development of Langmuir’s isotherm and can be accurately applied in methane adsorption isotherm. In this work, for a two-dimensional surface, a fundamental straight line between log⁡(𝑉_𝐿) and log⁡(𝜎) is used to describe the power-law relationship as:

𝑉_𝐿 = 𝑆(𝜎)𝐷𝑓/2_{⁡ + 𝐵 ( 2-19 )}

where 𝜎 is the specific surface area determined from the monolayer volume of the adsorbed gas by the BET model. 𝑆 and 𝐵 are the slope and intercept in the plot of 𝑉_𝐿 vs. (𝜎)𝐷𝑓/2_.

𝑆 contains all the information of the effect of gas molecular size dependence on adsorption capacity and thus, the fractal dimension is an intensive property (Pfeifer and Avnir, 1983). 𝐵 is a correction factor to consider the variation of gas molecular size among different gas species. It should be noted that in classical fractal theory, the number of adsorbed molecules is related primarily to the surface area of the gas molecules, where the specific surface area of adsorbent measured by the BET model is inversely proportional to the cross sectional area of different molecules (Pfeifer and Avnir, 1983).

To separate the effect of temperature from pore structure on Langmuir pressure (𝑃𝐿),

Eq. (2-17) may be rearranged as:

21

The term ln(𝑉_𝐿𝜈)⁡ is a lump sum of surface roughness and sorption capacity interpreted as a measure of characteristic sorption capacity. For 𝜈 = 1, log 𝑃𝐿 is linearly

related to log 𝑉_𝐿 corresponding to an energetically homogeneous and smooth surface, which agrees with the assumption made in the Langmuir equation. For a complex surface,⁡log⁡(𝑃_𝐿) would change linearly in response to log⁡(𝑉_𝐿𝜈). In the above equation, 𝑃_𝐿 is correlated with sorption capacity and fractal dimension as a representation of surface roughness. The sorption capacity may be approximated by surface area and fractal dimension with Eq. (19). The expression 𝑃𝐿 could be further expanded as:

ln(𝑃_𝐿) = 𝐶 ln((𝑆(𝜎)𝐷𝑓/2_{+ 𝐵)}𝜈_{) + 𝐹(𝑇)⁡ ( 2-21 )}

The pore structure-gas sorption model given in Eqs. (2-19, 2-20, 2-21) were applied to quantitatively investigate the relationship of Langmuir’s constants and pore characteristics. The value of 𝐷_𝑓 and 𝜎 were measured directly through low-pressure N2

adsorption experiments. The Langmuir’s constants were determined by high pressure methane adsorption data. 𝑉𝐿 and⁡𝑃𝐿 are important parameters in CBM production. As

mentioned before, 𝑉_𝐿 indicates the maximum adsorption capacity of coalbed. 𝑃_𝐿 describes the changing slope of the isotherm across a broad range of pressures and addresses gas mobility. 𝑃𝐿 determines the desorption rate, and the higher the PL value is, the easier the

CBM well arrives the critical desorption pressure. Besides, it has been shown that 𝑃_𝐿 is inversely related to coal rank (Pashin, 2010). Typically, a Langmuir isotherm with a larger value of PL maintains slope at higher pressure, which corresponds to a higher initial gas

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2.2 Gas Diffusion Modeling in CBM

This section develops a pore structure-gas diffusion model that correlates gas diffusion coefficient with pore sturctural characteristics of coal. The proposed model provides an intuitive and mechanism-based approach to define the gas diffusion behavior in coal and it can serve as a bridge from pore-scale structure of mass transport for the CBM gas production prediction.

2.2.1 Literature Review

Diffusion is the process that matter (gases, liquids, and solids) tends to migrate and eliminate the spatial difference in composition in such a way to approach a uniform equilibrium state with maximum entropy (Fick, 1855; Philibert, 2005; Sherwood, 1969). The study of diffusion in nanoporous solids came to prominence as such materials have sufficient surface area required for high capacity and activity with extensive application in the petroleum and chemical process industries (Kärger et al., 2012). For transport through the pores with size comparable to diffusing gas molecules, diffusion effects or may even dominate the overall transport rate (Kärger et al., 2010). A comprehensive understanding of the complex diffusional behavior lies the foundation for the technological development of porous materials in adsorption and catalytic processes (Kainourgiakis et al., 2002). As a natural polymer-like porous material, coal behaves like man-made nanoporous materials for its exceptional sorption capacity contributed by nano- to micron-scale pores (Gray, 1987; Harpalani and Chen, 1997; Levine, 1996). Dual porosity model proposed by Warren and Root (1963) well represents the broad size distribution of coal pores, where macro-