Modeling Competitive Adsorption between Methane and Water on

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Modeling Competitive Adsorption between Methane and Water on Coals Quanshu Zeng,† Zhiming Wang,*,† Brian J. McPherson,‡ and John D. McLennan‡ †

State Key Laboratory of Petroleum Resources and Prospecting, College of Petroleum Engineering, China University of Petroleum, Beijing 102249, China ‡ Energy & Geoscience Institute at the University of Utah, Salt Lake City, Utah 84108, United States ABSTRACT: Natural gas produced from coals, or coalbed methane (CBM), is a significant component of the energy portfolio for many countries. One challenge associated with CBM production is associated water. Specifically, coalbeds in situ contain significant amounts of water, and ideally this water is removed by pumping prior to the primary recovery of CBM to lower pressure and stimulate methane desorption. Such a prior water production can be challenging because desorption depends on the occurrence state of methane and water in situ, e.g., how much of each fluid is adsorbed or otherwise. Accordingly, primary objectives of this analysis include quantifying both the occurrence state of methane and water of different coals for a range of coalbed properties and conditions, and specifically quantifying the impact of coal moisture on methane desorption. Ultimate and proximate analysis and methane adsorption tests were first conducted on several coal samples from different basins. Simplified local density (SLD) theory was then tailored and applied to describe the adsorption characteristics of specific methane/water mixtures for each coal. Then, a fluid mixing rule was introduced to characterize competitive adsorption processes and a minimum potential energy method was applied to distinguish primary and secondary water adsorptions. Analysis of all resulting data included a regression analysis to obtain best fit parameters. Finally, an analytical reserve estimation method for methane and water was developed to quantify the extent of potential original reserves, and results of the method were compared to forecasts by conventional simulators. Combined results of all analyses suggest that both methane and water adsorptions decrease with temperature. While water adsorption decreases monotonically with pressure, methane adsorption first increases then decreases with pressure. Maximum methane adsorption occurs at approximately 10 MPa pressure. Water will compete with methane for adsorption sites, thus degrading the effective methane adsorption capacity of coal. Both methane and water adsorptions by coal can be effectively described by a competitive adsorption model, but the impact of moisture on methane desorption varies with temperature and pressure. With water divided into adsorbed and free states and with methane occurring in adsorbed, free, and dissolved forms, the proposed reserve estimation method predicted approximately the same original reserves as existing simulators predict, and also quantified how reserves change during CBM recovery.

1. INTRODUCTION Since coalbeds in situ usually contain significant amounts of water,1 understanding the occurrence characteristics of water and its impact on methane adsorption/desorption can improve quantitative forecasts of original reserves and well performance.2−6 Although it has been widely recognized that water can reduce methane adsorption on coal significantly, many studies assume that water effectively blocks the coal surface, and that blockage is what limits the accessibility of methane to coal.7−10 However, we contend that both methane and water are adsorbed on coal surfaces, or more specifically, methane and water are competitive with respect to adsorption, and the nature of the competitive adsorption changes with pressure and temperature. In general, methane/water mixture adsorption characteristics in coals are difficult to describe because of uncertainty associated with characteristics of supercritical methane in situ and water’s polar molecule interactions. In this study, ultimate and proximate analysis and methane adsorption tests were first conducted on several fresh coal samples from different basins. Simplified local density (SLD) theory was then tailored and applied to describe the adsorption characteristics of methane/water mixtures on coal, and a fluid mixing rule was introduced to characterize the competitive © XXXX American Chemical Society

adsorption relationship. Additionally, a minimum potential energy method was used to distinguish between primary and secondary water adsorption. Next, five key parameters were regressed with measured isotherm data to obtain their best fit. Last but not least, a reserve estimation method for methane and water was developed to describe original reserves and associated variability. The remainder of this article is organized as follows: Section 2 presents the experimental method and procedures; Section 3 presents the competitive adsorption model between methane and water; Section 4 presents the experimental and fitting results; and Section 5 presents the proposed reserve estimation method.

2. EXPERIMENT 2.1. Specimens. Several coal samples from CBM basins of the United States and China were selected for adsorption capacity assessment. Results of ultimate and proximate analyses with ASTM standards11,12 for each coal are summarized in Table 1. Five coals were evaluated (Table 1): BWBC refers to Received: July 5, 2017 Revised: September 7, 2017 Published: September 11, 2017 A

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Energy & Fuels Table 1. Ultimate and Proximate Analysis of Coals Tested ultimate analysis

proximate analysis

adsorbent

C%

H%

O%

N%

S%

MEMC %

MFC %

MV %

MA %

rank

BWBC coal SJF coal PRW coal O4 coal EYWG3 coal

80.55 59.11 53.28 67.02 84.20

4.86 5.09 6.04 4.23 3.09

3.23 17.37 34.08 21.99 5.68

1.70 1.36 0.74 0.87 1.03

0.92 0.90 0.39 0.66 0.54

1.94 6.10 23.78 11.03 1.80

56.39 40.79 37.48 44.02 78.48

32.93 36.94 33.27 39.72 14.26

8.74 16.17 5.47 5.23 5.46

high-volatile A bituminous lignite lignite lignite low-volatile bituminous

Figure 1. Schematic diagram of the GAI-100 gas adsorption isotherm system.

with AJP-100 pore volume calibrator. Helium with purity of 99.999% was used for pore volume calibration while methane with purity of 99.99% was used as the adsorbate. 2.3. Testing Process. (1) Place 80−100 g prepared ground sample inside the adsorption cell, connect it to the adsorption system, immerse it in the oil bath, and heat to experimental temperature. (2) Check air tightness of the system with helium, the leakage testing pressure should excess the maximum operation pressure. The tightness meets experiment requirement only if the pressure change is less 6 × 10−3 MPa after 12 h. (3) Vacuum the system for 30 min, inject a certain amount of helium into the reference cell with AJP-100 pore volume calibrator, record the pressure after the system reaches equilibrium, and calculate the volume of reference cell with eq 1, or

Blue Creek coal seam, Black Warrior basin, SJF refers to Fruitland coal seam, San Juan basin, PRW refers to Wyodak coal seam, Powder River basin, O4 refers to 4# coal seam, Ordos basin, and EYWG3 refers to 3# coal seam, Eastern Yunnan Western Guizhou basin. After recovery, the samples were wrapped in plastic bubble wrap and transported to the laboratory. Since particle size has little impact on methane adsorption,13,14 the samples were ground into 40−50 mesh to accelerate adsorption equilibrium. Since this study focuses on the impact of moisture on adsorption, equilibrium water content preprocessing was conducted for each coal sample, consulting the ASTM D1412 standard.15 2.2. Apparatus. Adsorption capacity tests were conducted using the Gas Adsorption Isotherm System GAI-100,16 as shown in Figure 1. The system comprises three standalone adsorption cells, with each cell capable of testing up to 80−100 g ground sample at one time. The system includes a digitally controlled oil bath for temperature control, with a maximum operating temperature of 177 °C and an accuracy of ±0.1 °C. The system uses a gas booster and reference cells to adjust the adsorption pressure, and its maximum operating pressure is 69 MPa with an accuracy of ±0.03 MPa. All of the transducers (Figure 1) were connected to a data acquisition and control system, to allow continuous monitoring and collection of pressure and temperature data. System feedback controls boundary conditions if real-time adjustment required. Prior to adsorption tests, the pore volume of each adsorption cell was standardized

cal pHe Vcal cal Z He RTcal

=

ref pHe Vref ref Z He RT

=

equ pHe (Vref + Vp) equ Z He RT

(1)

pcal He

where, is the pressure of helium inside calibrator, Pa; Vcal is the volume of calibrator, m3; Zcal He is the gas compressibility factor of helium inside calibrator, dimensionless; R is the gas constant, 8.314 J/(K·mol); Tcal is the temperature inside calibrator, K; pref He is the pressure of helium inside reference cell, Pa; Vref is the volume of reference cell, m3; Zref He is the gas compressibility factor of helium inside reference cell, B

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Energy & Fuels dimensionless; T is the temperature, K; pequ He is the pressure of helium inside equilibrium cell, Pa; and Zequ He is the gas compressibility factor of helium inside equilibrium cell, dimensionless. (4) Open the valve between adsorption and reference cells slowly, record the pressure after the system reaches equilibrium, and calculate the volume of adsorption cell with eq 1. (5) Vacuum the system for 30 min, inject methane into the reference cell, record the pressure after the system reaches equilibrium, and calculate the injection amount with eq 2, or inj (1) NCH 4

=

equilibrium, dimensionless; Nfree CH4(k) is the free methane amount after the kth equilibrium, mol; pequ CH4(k) is the pressure of methane th after the k equilibrium, Pa; Zequ CH4(k) is the gas compressibility factor of methane after the kth equilibrium, dimensionless; and th Nads CH4(k) is the adsorbed methane amount after the k equilibrium, mol. (8) Divide the adsorption amount with adsorbent mass, plot the adsorption isotherm, or ads nCH = 4

ref pCH (1)Vref 4

ref ZCH (1)RT 4

4

(3)

inj free ads NCH (1) = NCH (1) − NCH (1) 4 4 4

(4)

where, is the free methane amount after the first equilibrium, mol; pequ CH4(1) is the pressure of methane after the first equilibrium, Pa; Zequ CH4(1) is the gas compressibility factor of methane after the first equilibrium, dimensionless; and Nads CH4(1) is the adsorbed methane amount after the first equilibrium, mol. (7) For subsequent increments denoted by k below, close the valve, inject more methane into the reference cell to increase pressure, after the kth injection, the injection, free, and adsorbed amounts can be calculated with eqs 5 to 7, or Nfree CH4(1)

inj NCH (k ) 4

=

free NCH (k ) = 4

ref pCH (k)Vref 4

ref ZCH (k)RT 4

−

equ pCH (k − 1)Vref 4

equ ZCH (k − 1)RT 4

(5)

equ pCH (k)(Vref + Vvoid) 4

equ ZCH (k)RT 4

μ(z) = μ bulk = μff (z) + μfs (z)

(6)

inj free (i) − NCH (k ) ∑ NCH 4

i=1

4

(9)

where, z is the distance between adsorbate molecule and adsorbent surface, m; μ(z) is the chemical potential at z position, J/mol; μbulk is the chemical potential of bulk phase, J/mol; μff(z) is the chemical potential caused by the adsorbate− adsorbate interaction at z position, J/mol; and μfs(z) is the chemical potential caused by the adsorbate−adsorbent interaction at z position, J/mol. Since slit shape is assumed for coal pores while adsorbate molecules located between the parallel slit walls, adsorbate molecules have interactions with both walls (Figure 2), or

k ads NCH (k ) = 4

(8)

3. MODELING 3.1. Simplified Local Density Theory. Rangarajan18 first proposed simplified local density (SLD) theory by combining mean-field approximation theory and density functional theory, believing that adsorption is the resultant effect of adsorbate− adsorbate and adsorbate−adsorbent molecule interactions. Several assumptions were made for SLD theory,18−21 including (1) The chemical potential of any point above adsorbent surface is equal to the bulk phase chemical potential. (2) The chemical potential of any point above adsorbent surface is the sum of adsorbate−adsorbate and adsorbate− adsorbent interactions. (3) The adsorbate−adsorbent interaction of any point above adsorbent surface is independent of the temperature and the number of molecules. (4) Coal pores are considered as perfect slits with temperature and pressure distributed uniformly. (5) All adsorbate and adsorbent molecules are spherical, except for the adsorbate molecules touching slit walls. Therefore, the chemical potential of any point above adsorbent surface should be the same, or

equ pCH (1)(Vref + Vp) equ ZCH (1)RT 4

mads

where, nads CH4 is the methane adsorption amount per unit mass adsorbent, mol/kg; Nads CH4 is the methane adsorption amount, mol; and mads is the adsorbent mass, kg. (9) Change system temperature, repeat from Step (3). (10) Change the specimen, repeat from Step (1). According to the testing processes described here, the compressibility factor of helium and methane should be determined prior to pore volume calibration and adsorption amount calculation. Both compressibility factors used were from the National Institute of Standards and Technology (NIST) database.17

(2)

where, Ninj CH4(1) is the methane amount injected into reference cell at the first time, mol; pref CH4(1) is the pressure of methane inside reference cell after the first injection, Pa; and Zref CH4(1) is the gas compressibility factor of methane inside reference cell after the first injection, dimensionless. (6) Open the valve between adsorption and reference cells slowly, record the pressure after the system reaches equilibrium, and calculate the amount of free and adsorbed methane with eqs 3 and 4, respectively, or free (1) = NCH 4

ads NCH 4

(7)

Ninj CH4(k) th

where, is the methane amount injected into reference cell at the k time, mol; pref CH4(k) is the pressure of methane inside reference cell after the kth injection, Pa; Zref CH4(k) is the gas compressibility factor of methane inside reference cell after the kth injection, dimensionless; pequ CH4(k − 1) is the pressure of methane after the (k − 1)th equilibrium, Pa; Zequ CH4(k − 1) is the gas compressibility factor of methane after the (k − 1)th

μfs (z) = NA[Ψfs(z) + Ψfs(Ls − z)] C

(10)

DOI: 10.1021/acs.energyfuels.7b01931 Energy Fuels XXXX, XXX, XXX−XXX

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The chemistry potentials of bulk and adsorption phases can be described in fugacity forms, or ⎛f ⎞ μ bulk = μ0 + RT ln⎜⎜ bulk ⎟⎟ ⎝ f0 ⎠

(15)

⎡ f (z ) ⎤ μff (z) = μ0 + RT ln⎢ ads ⎥ ⎢⎣ f0 ⎥⎦

(16)

where, interactions among adsorbate molecules can be described by a fluid equation of state. The Patel−Teja− Valderrama equation of state23 can describe the thermodynamic properties of methane and water with a reasonable accuracy, and thus was selected to calculate density and fugacity of the bulk phase and fugacity of the adsorption phase. The physical properties of methane and water involved in this study are summarized in Table 2 and can be expressed by p aρ 1 = − RTρ 1 − bρ Rθ[1 + (b + c)ρ − bcρ2 ] (17)

Figure 2. Slit pore characterization in SLD theory.

where, where, interactions between adsorbent and adsorbate molecules can be effectively represented by the Lennard-Jones potential function.22 Specifically, the interaction between adsorbate molecules and carbon planes after the fourth layer are ignored, or

a = (0.66121 − 0.761057Zc)

[1 + (0.46283 + 3.58230ωZc + 8.19417ω 2Zc2)

10 ⎧ ⎪1⎛d ⎞ ⎜ fs ⎟ Ψfs(z) = 4πρatoms d 2fsεfs⎨ ⎪ ⎩ 5 ⎝ z′ ⎠

(1 −

⎡ ⎤4 ⎫ ⎪ dfs 1 − ∑⎢ ⎥⎬ 2 i = 1 ⎣ z′ + (i − 1)dss ⎦ ⎪ ⎭ 4

εff × εss

(12)

d + dss dfs = ff 2

2

T /Tc )]

(18)

b = (0.02207 + 0.20868Zc)

RTc pc

(19)

c = (0.57765 − 1.87080Zc)

RTc pc

(20)

(11)

where, εfs =

R2Tc2 pc

where, μ0 is the reference chemical potential, J/mol; f bulk is the fugacity of bulk phase, Pa; f 0 is the reference fugacity, Pa; fads(z) is the fugacity of adsorption phase at z position, Pa; p is the pressure, Pa; ρ is the density, mol/m3; a is the attraction term, J·m3·mol−2; b is the repulsion term, m3/mol; c is the polarity term, m3/mol; Zc is the critical compression factor, dimensionless; Tc is the critical temperature, K; pc is the critical pressure, Pa; and ω is the eccentric factor, dimensionless. To calculate the fugacity of the bulk phase and adsorption phase, the PTV EOS can be rearranged as

(13)

dcc (14) 2 23 and where, NA is the Avogadro constant, 6.02 × 10 mol−1; Ψfs(z) is the potential energy caused by the interaction between adsorbate molecule at z position and the left wall, J; Ψfs(Ls − z) is the potential energy caused by the interaction between adsorbate molecule at z position and the right wall, J; Ls is the slit width, m; ρatoms is the carbon atom density, 3.82 × 1019 m−2; εfs is the potential energy caused by the interaction between adsorbate and adsorbent molecules, J; εff is the potential energy caused by the interaction between adsorbate and adsorbate molecules, J; εss is the potential energy caused by the interaction between adsorbent and adsorbent molecules, J; dfs is the collision diameter between adsorbate and adsorbent molecule, m; dss is the carbon plane spacing, 3.35 × 10−10 m; z′ is the distance between adsorbate molecule and carbon atoms of the first layer, m; and dcc is the diameter of carbon atom, 1.4 × 10−10 m. z′ = z +

Table 2. Physical Properties of Methane and Water fluid

Tc °C

Pc MPa

Zc dimensionless

ωc dimensionless

d nm

εff/kB K

methane water

−82.59 373.94

4.599 22.065

0.288 0.229

0.011 0.344

0.3758 0.2641

148.6 809.1

D

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Energy & Fuels where, the attraction term of adsorbed phase varies with the relative position of adsorbate molecules. The relation between aads(z) and Ls/dff was obtained by integrating the sum of the interactions between an adsorbate molecule and all the adsorbed molecules around it.18 For Ls/dff ≥ 3,

Water adsorption can be divided into primary adsorption and secondary adsorption according to the configuration of its attachments (Figure 3).

For 3 > Ls/dff ≥ 2,

Figure 3. Schematic diagram of methane/water mixture adsorption on carbonaceous adsorbent.

Primary adsorption is defined as that where water molecules will first adhere to the functional groups above the adsorbent surface. Secondary adsorption is defined as that when water molecules attach to one another, forming water molecules cluster. Therefore, hydrogen bonds should be considered as adsorbate−adsorbent interactions for primary water adsorption, but as adsorbate−adsorbate interactions for secondary water adsorption. The Lennard-Jones potential function for primary water adsorption can be modified as

For 2 > Ls/dff ≥ 1.5, ⎞ aads(z) 3⎛ L = ⎜ s − 1⎟ abulk 8 ⎝ dff ⎠

(25)

where abulk is the attraction term for the bulk phase, J·m ·mol−2; bbulk is the repulsion term for the bulk phase, m3/mol; cbulk is the polarity term for the bulk phase, m3/mol; aads(z) is the attraction term for the adsorption phase, J·m3·mol−2; bads is the repulsion term for the adsorption phase, m3/mol; and cads is the polarity term for the adsorption phase, m3/mol. Equilibrium criteria can be eventually obtained by substituting eqs 10, 15, and 16 into eq 9, or 3

⎡ Ψ (z) + Ψfs(Ls − z) ⎤ fads (z) = fbulk exp⎢ − fs ⎥ kBT ⎦ ⎣

10 ⎧ ⎪1⎛d ⎞ ⎜ fs ⎟ Ψfs(z) = 4πρatoms dfs2εfs⎨ ⎪ ⎩ 5 ⎝ z′ ⎠

1 − 2

(26)

As 2

Ls − 3d ff /8

∫3d /8 ff

[ρads (z) − ρbulk ]dz

(28)

where, ΨHB is the hydrogen bonding between water molecules and functional groups above adsorbent surface,9 1.242 × 10−21 J. It is important to note that SLD theory assumes that all the adsorbate molecules are attached to the adsorbent surface, which will lead to an abnormal water adsorption area in most cases. To better understand the water adsorption mechanism, a minimum potential energy principle28 was introduced to distinguish between primary and secondary adsorption, or

where, kB is the Boltzmann constant, 1.38 × 10−23J/K. According to SLD theory, the Gibbs excess adsorption can be obtained by integrating the density difference between adsorption and bulk phases along the slit length. Since slit pore includes two walls (Figure 2), half the surface area is considered in eq 27, or nGibbs =

⎡ ⎤4 ⎫ ⎪ dfs + ΨHB ∑⎢ ⎥⎬ ⎪ i = 1 ⎣ z′ + (i − 1)dss ⎦ ⎭ 4

s s A s = A water1 + A water2

(27)

Gibbs

where, n is the Gibbs excess adsorption amount, mol/kg; As is the surface area of unit mass adsorbent, m2/kg; dff is the diameter of fluid molecular, m; ρads(z) is the density of adsorption phase at z position, mol/m3; and ρbulk is the density of bulk phase, mol/m3. 3.2. Modifications to SLD Theory for Water Adsorption. Since hydrogen bonding due to water polarity is much larger than any dispersion force,24 water adsorption mechanisms can be quite different from that of nonpolar molecules.25−27

(29)

⎛ 3d ⎞ ⎛ 3d ⎞ s Ep = 2 × μwater1⎜ ff ⎟ρ⎜ ff ⎟A water1 ⎝ 8 ⎠ ⎝ 8 ⎠ +

Ls − 3d ff /8

∫3d /8 ff

s dz μwater2 (z)ρ(z)A water2

(30)

s

where, Awater1 is the surface area of primary water adsorption, m2/kg; Awater2s is the surface area of secondary water adsorption, m2/kg; Ep is the total potential energy of system, J; E

DOI: 10.1021/acs.energyfuels.7b01931 Energy Fuels XXXX, XXX, XXX−XXX

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μwater1

⎛ ∂nc ⎞ c i̅ ads = ⎜ ads ⎟ ⎝ ∂ni ⎠θ , V , n

3d ff 8

( ) is the chemical potential caused by primary water

adsorption at the left wall, J/mol; and μwater2(z) is the chemical potential caused by secondary water adsorption at z position, J/mol. 3.3. SLD Theory for Methane/Water Mixture Adsorption. A fluid mixing rule20 was implemented in the fluid equation of state to describe adsorbate−adsorbate interactions, including methane−methane, methane−water, and water− water (Figure 3). Similar to pure component adsorption, the fugacity of methane and water in both phases can be described by the PTV EOS, revised as

(38)

j

where, attraction terms are determined by quadratic mixing rules while repulsion and polarity terms are determined by linear mixing rules, or abulk =

∑ ∑ xixjaijbulk i

aads(z) =

∑ ∑ yi (z)yj (z)aijads(z) i

b bulk =

(39)

j

(40)

j

∑ xibibulk

(41)

i

bads =

∑ yi (z)biads

(42)

i

cbulk =

∑ xicibulk

(43)

i

cads =

and

∑ yi (z)ciads

(44)

i

Specifically, a dimensionless binary interaction parameter, CBIP, is introduced to calculate the attraction terms of an asymmetric and polar system. For methane/water mixture,29 the value of this parameter is 0.5044, with aijbulk = aijads(z) =

j

(33)

⎛ ∂nb bulk ⎞ =⎜ ⎟ ⎝ ∂ni ⎠θ , V , n

(34)

⎛ ∂nc ⎞ c i̅ bulk = ⎜ bulk ⎟ ⎝ ∂ni ⎠θ , V , n

(35)

bi̅

bulk

j

j

⎡ ∂naads(z) ⎤ ai̅ ads(z) = ⎢ ⎥ ⎣ ∂ni ⎦θ , V , n

j

bi̅

ads

⎛ ∂nb ⎞ = ⎜ ads ⎟ ⎝ ∂ni ⎠θ , V , n

j

(45)

aiads(z) × ajads(z) (1 − C BIP)

(46)

where, f bulk is the fugacity of component i in the bulk phase, Pa; i (z) is the fugacity of component i in the adsorbed phase at f ads i z position, Pa; xi is the mole fraction of component i in the bulk phase, dimensionless; xj is the mole fraction of component j in the bulk phase, dimensionless; yi(z) is the mole fraction of component i in the adsorbed phase at z position, dimensionless; yj(z) is the mole fraction of component j in the adsorbed is the attraction term phase at z position, dimensionless; abulk i of component i in the bulk phase, J·m3·mol−2; aads i (z) is the attraction term of component i in the adsorbed phase at is the repulsion term of component i z position, J·m3·mol−2; bbulk i is the repulsion term of comin the bulk phase, m3/mol; bads i is the polarity ponent i in the adsorbed phase, m3/mol; cbulk i is the term of component i in the bulk phase, m3/mol; cads i polarity term of component i in the adsorption phase, m3/mol; is the cross coefficient to calculate the attraction term in abulk ij the bulk phase, J·m3·mol−2; and aads ij (z) is the cross coefficient to calculate the attraction term in the adsorption phase, J·m3·mol−2. For adsorbate−adsorbent interactions, both components can be calculated by the Lennard-Jones potential function. However, water adsorption should be divided into primary and secondary water adsorptions, and hydrogen bonding should be

where, ⎛ ∂nabulk ⎞ ai̅ bulk = ⎜ ⎟ ⎝ ∂ni ⎠θ , V , n

aibulk × ajbulk (1 − C BIP)

(36)

(37) F

DOI: 10.1021/acs.energyfuels.7b01931 Energy Fuels XXXX, XXX, XXX−XXX

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Energy & Fuels additionally considered for primary water adsorption, or ⎧ ⎛ fs ⎞10 ⎪ 1 dCH4 fs fs 2 ⎟ ΨCH4(z) = 4πρatoms (dCH4) εCH4 ⎨ ⎜⎜ ⎪ 5 ⎝ z′ ⎟⎠ ⎩ fs 4 ⎡ ⎤4 ⎫ dCH4 ⎪ 1 ⎥⎬ − ∑⎢ 2 i = 1 ⎢⎣ z′ + (i − 1)dss ⎥⎦ ⎪ ⎭

∑ xi = 1

(55)

∑ yi = 1

(56)

Gibbs excess adsorption of both methane and water can be calculated as Gibbs nCH = 4

(47)

s A CH 4

2

fs 2 Ψ fs1 water(z) = 4πρatoms (d water) εwater

⎧ ⎛ fs ⎞10 ⎪ 1 d water 1 ⎟⎟ − ⎨ ⎜⎜ 2 ⎪ 5 ⎝ z′ ⎠ ⎩

=

Gibbs n water =

⎩ 5 ⎝ z′ ⎠

−

1 2

⎡

i=1

where, AsCH4 and nGibbs water is

(49)

fs εCH = 4

ff εCH × εss 4

(50)

fs εwater =

ff εwater × εss

(51)

fs d water =

4

2

ff dCH + dss 4

2 ff d water + dss 2

(52)

(53)

ΨfsCH4(z)

where, is the potential energy caused by the interaction between methane molecule at z position and the left wall (Figure 2), J; Ψfs1 water(z) is the potential energy caused by the interaction between primary water adsorption and the left wall, J; Ψfs2 water(z) is the interaction potential energy caused by the interaction between secondary water adsorption and the left wall, J ; εfsCH4 is the potential energy caused by interactions between methane and either wall, J; εfswater is the potential energy caused by interactions between water and either wall, J; εffCH4 is the potential energy caused by the interactions among the methane molecules, J; εffwater is the potential energy caused by the interactions among the water molecules, J; dfsCH4 is the collision diameter between methane molecule and carbon atom, m; dffCH4 is the diameter of methane molecule, m; dfsCH4 is the collision diameter between methane molecule and carbon atom, m; flow-solid particles collision diameter of methane molecules and carbon atoms, m; dffCH4 is the diameter of methane molecule, m; dfswater is the collision diameter between water molecule and carbon atom, m; and dffwater is the diameter of water molecule, m. To implement the methane/water mixture adsorption model, each component of the mixture should satisfy the adsorption equilibrium criterion. In addition, both phases should satisfy the mole fraction constraints, or ⎡ Ψ fs(z) + Ψ fs(L − z) ⎤ i s ⎥ f iads (z) = fibulk exp⎢ − i ⎥⎦ ⎢⎣ kBT

⎤ ⎛ 3dff ⎞ ⎛ 3d ⎞ ⎟ywater ⎜ ff ⎟ − ρbulk x water ⎥ ⎢ρads ⎜ ⎝ 8 ⎠ ⎣ ⎝ 8 ⎠ ⎦

s A water2 2

ff Ls − 3/8d water

∫3/8d

ff water

[ρads (z)ywater (z) (58) 2

is the surface area of methane adsorption, m /kg; the water adsorption amount, mol/kg. 3.4. Solution Algorithm. The solution procedure for the methane-water competitive adsorption model is depicted in Figure 4. Input data include pressure, temperature, adsorption parameters, physical properties of water and methane, and bulk phase mole fraction, and these data are used to calculate the densities, fugacity, and chemical potential of both components in the bulk phase. Then, divide the slit pores into n intervals, initialize densities of methane and water in adsorption phase with those in the bulk phase, and the surface area ratio of primary and secondary water adsorptions to 0.01, and calculate chemical potential induced by adsorbate−adsorbent and adsorbate−adsorbate interactions. Next, loop the densities of methane and water in the adsorption phase until both methane and water satisfy a specific adsorption equilibrium criterion. Finally, output the densities of methane and water in adsorption phase, loop the surface area ratio of primary and secondary water adsorptions until the system reaches its minimal potential energy, and calculate the adsorption amounts of methane and water.

where,

fs dCH = 4

[ρads (z)yCH (z) − ρbulk xCH4]dz

− ρbulk x water]dz

4⎫

fs ⎤⎪ d water ⎥⎬ ⎢⎣ z′ + (i − 1)dss ⎥⎦ ⎪ ⎭

∑⎢

ff CH 4

s ⎡ A water1

+

⎧ fs ⎞10 ⎪1⎛d fs 4πρatoms (d water )2 εwater ⎨ ⎜⎜ water ⎟⎟ ⎪ 4

∫3/8d

(57)

⎫ fs ⎡ ⎤4 ⎪ d water ⎢ ⎥ ⎬ + ψHB ∑ ⎢ ⎥⎪ i = 1 ⎣ z′ + (i − 1)dss ⎦ ⎭ 4

(48)

Ψ fs2 water(z)

ff Ls − 3/8dCH 4

4. RESULTS AND AANALYSIS The experimental results of methane and water adsorption on coals with varying temperatures and pressures are shown in Figures 5, 6, 7, 8, and 9. Since adsorption on coals is a reversible and exothermic process, the adsorption rate equals the desorption rate at equilibrium. According to Le Chatelier’s Principle, any change to system conditions will shift the equilibrium to eliminate the change until it reaches a new balance. For example, if temperature increases, system internal energy will increase, causing the adsorbate molecules to desorb faster than adsorb to minimize the temperature variations. Therefore, both methane and water adsorptions decrease with temperature (Figures 5−9). Similarly, if pressure increases, bulk phase concentration will increase, shifting the equilibrium toward adsorption, resulting in adsorption increase. However, given that the adsorption characteristic is represented by Gibbs excess adsorption in this study, methane adsorption will first increase then decrease with pressure, with the inversion pressure about 10 MPa, as shown in Figures 5−9. The surface potential energy will decrease if the adsorption sites have been occupied, thus adsorption capacity is reduced. On the other hand, methane will shift to its supercritical

(54) G

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Figure 4. Solution procedure for competitive adsorption model between methane and water.

state at a pressure higher than its critical value, and in this case, methane has a much larger compressibility in the bulk phase. Since only 1% is assumed for the mole fraction of water in the bulk phase, the partial pressure increment of methane is much more significant than that of water, leading methane molecules

to occupy some adsorption sites of water molecules as pressure increase. In this case, water adsorption will decrease monotonically with pressure. Combining the ultimate and proximate analysis of coals, results suggest that methane adsorption increases with fixed H

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Figure 5. Methane and water adsorption isotherms on BWBC coal with varying temperatures.

Figure 6. Methane and water adsorption isotherms on SJF coal with varying temperatures.

Figure 7. Methane and water adsorption isotherms on PRW coal with varying temperatures.

Figure 8. Methane and water adsorption isotherms on O4 coal with varying temperatures.

carbon content, while water adsorption increases with equilibrium water content. This means fixed carbon is the

effective component that adsorbs methane while equilibrium moisture does not. On the contrary, water will compete with I

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lignite coals. For the relative dry bituminous coals, their equilibrium moistures are within 2%, and this results in the regressed surface area of methane larger than that of water instead. In addition, since hydrogen bonding between water molecules and hydroxyl groups is larger than that among water molecules, the water molecule will first adhere to hydroxy groups above the adsorbent surface, and then link to one another if coal moisture is large enough. Therefore, the surface area proportion of primary water adsorption relative to secondary water adsorption decrease with equilibrium moisture. With these parameters, methane and water adsorption isotherms on coals with varying temperatures can be calculated, as illustrated by Figures 5−9. The deviations between the predicted and experimental results are all within 1% for methane and within 5% for water. The higher deviation for water is mainly due to two reasons. First, PTV EOS has some defects in describing the thermodynamic properties of water, which may cause the inaccuracy estimation of adsorbate−adsorbate

methane for adsorption sites, thus degrading the adsorption capacity of coal. Prior to calculation of methane and water adsorption isotherms with methane−water competitive adsorption model, five parameters must be determined by fitting experimental results (isotherms). The five parameters include slit length, solid−solid interaction potential energy parameter, surface area of methane adsorption, surface areas of primary and secondary water adsorptions. To reduce the fitting parameters, the former three parameters are obtained by fitting the isotherms of dry coal, with shape adjusted by slit length and the solid−solid interaction potential energy parameter while the magnitude is adjusted by surface area. Given that 1% is assumed for the mole fraction of water in the bulk phase, both the surface areas of primary and secondary water adsorptions can be obtained by fitting the measured isotherms, as summarized in Table 3. Since hydrogen bonds are much larger than dispersion forces, water molecules more easily adsorb on coals than methane molecule, showing up as much more water adsorption on

Figure 9. Methane and water adsorption isotherms on EYWG3 coal with varying temperatures.

Table 3. SLD Fitting Results for Methane and Water Mixture Adsorption adsorbent

temperature °C

Ls nm

εss/kB K

ACH4 m2/g

Awater1 m2/g

Awater2 m2/g

ECH4 %

Ewater %

BWBC coal

35 45 55 65 75 35 45 55 65 75 35 45 55 65 75 35 45 55 65 75 35 45 55 65 75

1.82 1.55 1.48 1.34 1.24 1.69 1.57 1.46 1.36 1.33 1.98 1.71 1.50 1.36 1.26 1.94 1.65 1.45 1.32 1.23 1.35 1.33 1.31 1.29 1.28

40.0 42.2 44.3 47.8 51.6 29.7 33.2 37.0 40.0 44.1 56.5 59.7 59.0 59.0 60.1 43.1 45.8 47.1 48.4 49.7 39.4 43.0 49.5 54.3 57.5

90.4 74.0 60.5 48.9 39.5 74.2 59.1 47.0 38.0 30.3 126.9 104.3 87.3 72.9 60.4 88.5 72.1 59.6 49.3 40.7 264.4 226.9 183.6 144.2 104.6

70.8 58.0 47.4 38.3 30.9 128.7 102.5 81.6 65.9 52.6 330.7 271.8 227.5 190.0 157.4 220.8 179.8 148.7 123.0 101.5 197.4 169.4 137.1 107.7 78.1

6.4 5.2 4.3 3.5 2.8 30.7 24.5 19.5 15.7 12.5 640.2 526.2 440.4 367.8 304.7 105.2 85.7 70.9 58.6 48.4 16.9 14.5 11.8 9.2 6.7

0.21 0.06 0.02 0.22 0.33 0.08 0.02 0.08 0.19 0.14 0.41 0.02 0.15 0.51 0.57 0.17 0.10 0.07 0.08 0.12 0.36 0.47 0.42 0.53 0.33

2.55 3.42 3.31 2.66 2.79 3.37 3.38 3.94 3.02 3.00 2.27 2.77 4.22 3.58 2.80 3.37 2.72 2.97 3.31 3.68 3.02 1.94 3.09 2.80 3.08

SJF coal

PRW coal

O4 coal

EYWG3 coal

J

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Energy & Fuels Table 4. Coal Seam Properties A c m2

tc m

Φi dimensionless

T °C

ppi MPa

ρc kg/m3

ρwater kg/m3

Swater dimensionless

SCH4 dimensionless

647497.0

9

0.001

45

7.65

1434

990

0.592

0.408

Table 5. Coal Properties MA dimensionless

MEMC dimensionless

pL MPa

VL m3/t

Ls nm

ess/kB K

ACH4 m2/g

Awater1 m2/g

Awater2 m2/g

0.156

0.0672

4.6885

15.2

1.96

16.0

60.6

112.3

29.7

Table 6. Initial Methane and Water in Place GCH4 107sm3

7 3 Gads CH4 10 sm

7 3 Gfree CH4 10 sm

7 3 Gdis CH4 10 sm

Gwater 108kg

8 Gads water 10 kg

8 Gfree water 10 kg

6.1681 6.1233 6.1290 6.1340 6.0315 6.1165 6.1170

6.0995 6.0995

0.0169 0.0169

0.0005

13.0363 9.5426

9.5083

0.0343

GEM ECLIPSE COMET2 SIMED II GCOMP Seidle This Study

molecule interactions. Second, 1% is assumed for the mole fraction of water in the bulk phase, however, this is usually not consistent with the fact.

and ads free Gwater = Gwater + Gwater Gibbs = Actcρc n water +

5. DISCUSSION Volumetric estimation and numerical simulation are two primary methods used to forecast CBM reserves.30 However, current studies focus on coalbed gas, and dissolved gas is often ignored, as suggested by

pActcϕSCH4 ZCH4(p , T )RT (59)

To the authors’ best knowledge, only Seidle proposed a reserve estimation method for coalbed water, as Actcρc MEMC Mrwater

(60)

where, GCH4 is the methane reserve in situ, mol; Gads CH4 is the free adsorbed methane reserve in situ, mol; GCH4 is the free methane reserve in situ, mol; Ac is the area of coal seam, m2; tc is the thickness of coal seam, m; ρc is the density of coal, kg/m3; MA is the ash content, %; MEMC is the equilibrium water content, %; SCH4 is the methane saturation in the cleats, dimensionless; ZCH4 is the gas compression factor of methane in the cleats, dimensionless; and Mrwater is the relative molecular weight of water, kg/mol. To estimate the occurrence state of methane and water in situ, a reserve estimation method for both methane and water is proposed on the basis of methane-water competitive adsorption model, or

6. CONCLUSIONS In this article, a competitive adsorption model is developed to describe the adsorption characteristics of methane/water mixture on coal, wherein the adsorption parameters are obtained by fitting the adsorption isotherms. The following conclusions and recommendations can be obtained on the basis of the study. (1) The adsorption characteristics of methane/water mixture on coals can be described by the proposed competitive adsorption model, and the deviation between predicted

ads free dis GCH4 = GCH + GCH + GCH 4 4 4 Gibbs = Actcρc nCH + 4

pActcϕSCH4 ZCH4(p , T )RT

× ρwater (p , T )sCH4(p , T )

(62)

where, Gwater is the water reserve in situ, mol; is the adsorbed water reserve in situ, mol; Gfree is the free water water reserve in situ, mol; ρwater is the density of water, kg/m3; Gdis CH4 is the dissolved methane reserve in situ, mol; and sCH4 is the solubility of methane in water, dimensionless. Both methane and water adsorption are calculated with a methane-water competitive adsorption model while water density, compression factor, and solubility of methane are drawn from a NIST database.17 The performance of proposed reserve estimation method was validated against a CBM case, and compared to estimates by coalbed simulators31 and by a volumetric method.4 The coal seam properties, coal properties, and original reserves estimated are summarized in Tables 4, 5, and 6, respectively. Additionally, the adsorption parameters were regressed with the methanewater competitive adsorption model. Results suggest that the methane reserve estimation forecasted by this study approximates the forecast by all simulators considered and that by eq 59 above, while the water reserve estimation is consistent with results by eq 60 above. With water divided into adsorbed and free states, and methane divided into adsorbed, free, and dissolved forms, the approach detailed here provides the occurrence states of methane and water in situ and a robust reserve estimation method for coal seam.

4

Gwater =

Mrwater

Gads water

ads free GCH4 = GCH + GCH 4 4 Gibbs = Actcρc nCH (1 − MA − MEMC) + 4

ActcϕSwaterρwater (p , T )

+ ActcϕSwater (61) K

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and experimental results are within 1% for methane and 5% for water. (2) Both methane and water adsorptions decrease with temperature. While water adsorption decreases monotonically with pressure, methane adsorption first increases then decreases with pressure, and the maximum occurs at pressure ∼10 MPa. (3) Fixed carbon is the effective component that adsorbs methane while equilibrium moisture does not; on the contrary, water will compete with methane for adsorption sites, thus degrading the adsorption capacity of coal. (4) The proposed reserve estimation method reveals the occurrence states of methane and water in situ, as well as a robust reserve estimation for a variety of coal seam.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; Phone: +86 1089734958. ORCID

Zhiming Wang: 0000-0002-9301-1942 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Financial support from the National Science and Technology Major Project of China (2016ZX05044005-001), Programme of Introducing Talents of Discipline to Universities (111 Project: B12033), and China Scholarships Council Program (201606440099) are gratefully acknowledged.

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REFERENCES

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DOI: 10.1021/acs.energyfuels.7b01931 Energy Fuels XXXX, XXX, XXX−XXX