Evaluation of Matrix Swelling Behavior in Shale Induced by Methane

Article Cite This: Energy Fuels 2019, 33, 4986−5000

pubs.acs.org/EF

Evaluation of Matrix Swelling Behavior in Shale Induced by Methane Sorption under Triaxial Stress and Strain Conditions Yu Pang,†,∥ Shengnan Chen,*,† Mohamed Y. Soliman,‡ Stephen M. Morse,§ and Xiaofei Hu∥ †

Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, T2N 1N4, Canada Department of Petroleum Engineering, University of Houston, Houston, Texas 77024, United States § Department of Civil and Environmental Engineering, Michigan Technological University, Houghton, Michigan 49931, United States ∥ State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Chengdu University of Technology, Chengdu, 610059, PR China

Downloaded via BUFFALO STATE on August 1, 2019 at 07:28:52 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

‡

ABSTRACT: Gas sorption can lead to the volumetric swelling of the shale matrix and reduction of the effective pore volume, which further impacts the gas transportation in micro- and nanopores in shale. At present, it is very challenging to directly measure the pore volume shrinkage (i.e., pore volumetric strain) in the laboratory. In this study, an innovative method is proposed to quantify the pore volumetric strain resulting from the sorption-induced matrix swelling in shale. More specifically, Gibbs methane sorption capacities of the Barnett and Eagle Ford shale core samples were determined via volumetric and gravimetric methods, respectively. Meanwhile, the bulk volume swelling of the shale core sample was also measured during the gas sorption process. Correlations between the sorption-induced bulk and pore volumetric strains were developed to calculate the pore volume shrinkage ascribed to the gas sorption, which was further validated with the measured gas sorption capacities. It is found that the pore volume shrinkage is 3.7−4.8 times greater than the bulk volume swelling during the gas sorption process for the Barnett shale, while such a ratio reaches as high as 59.8−67.1 times for the Eagle Ford shale. In addition, the sorptioninduced bulk and pore volumetric strains follow a power law relationship for both Barnett and Eagle Ford shales, which yields a nearly identical absolute gas sorption isotherm with the ones determined using volumetric and gravimetric methods. The results of this study give insight into the feasibility of characterizing the sorption-induced pore volume shrinkage in shales and illustrate the benefits of applying both gravimetric and volumetric methods to evaluate the gas sorption behaviors.

1. INTRODUCTION Gas sorption capacity is an important parameter for the organicrich shale gas reservoirs, which affects the gas storage, matrix swelling, and gas flow mechanisms in the nanoporous shale.1−4 The adsorbed and absorbed gases (sorbed gas) account for a large proportion of the original gas-in-place in the shale gas formations.3,5−7 During the gas sorption process, the sorbed gas leads to the swelling of the shale matrix and thus reduces the effective pore volume for the free gas.8−11 In addition, the pore volume change and the gas sorption capacity are also sensitive to the effective stress.12,13 Therefore, understanding and quantifying the gas sorption capacity under the reservoir in situ stress and temperature conditions are essential to reliably estimate the gasin-place, predict the gas production rates, and characterize the pore structure. In general, there are two types of methods to measure the gas sorption amount in the laboratory, i.e., the volumetric method and the gravimetric method.14−16 For both methods, the Gibbs gas sorption amount is determined by executing the experimental procedure twice with a nonsorptive gas (e.g., helium) and a sorptive gas (e.g., methane), respectively. Parameters such as the pore volume, mass, and volume of the testing core sample are obtained from the measurement with the nonsorptive gas primarily. Then, they are used to calculate the Gibbs gas sorption amount via the measurement with the sorptive gas. Among these two methods, the volumetric method has been widely used in previous studies to measure the gas © 2019 American Chemical Society

sorption capacities and characterize the rock deformation and matrix permeability of the shale and coal samples.1,10,17−21 The volumetric method does not directly measure the gas sorption amount but calculates the sorption amount by the real gas law. Such a method expands an amount of the sorptive gas by transferring the gas from a pressurized reference cell into an evacuated sorption cell under the isothermal condition. The bulk volumetric strain, rock permeability, and gas sorption amount can be determined at the same time by placing the intact core sample under the triaxial stress condition during the test. Compared with the volumetric method, the gravimetric method exposes a core sample to a pure sorptive gas under constant temperature and directly records the changes in the balance reading from the magnetic suspension sorption system.4,16,22,23 Similar to the volumetric method, the gravimetric method also has been accepted and utilized to carry out many light hydrocarbons and carbon dioxide sorption tests in the literature.23−26 Recently, gas sorption models, density functional theory, and molecular simulation methods have been widely used to describe the gas sorption capacity in shales and coals.3,21,27−33 The traditional methods apply various gas sorption models to depict the gas sorption behavior in shale or coal under the Received: March 8, 2019 Revised: May 21, 2019 Published: May 22, 2019 4986

DOI: 10.1021/acs.energyfuels.9b00708 Energy Fuels 2019, 33, 4986−5000

Article

Energy & Fuels Table 1. Porosity, Total Organic Carbon, Vitrinite Reflectance, and Mineralogy for the Studied Shale Core Samples mineral composition in weight percentage (wt %) core sample

helium porosity (%)

TOC (wt %)

VrO (%)

illite

kaolinite

chlorite

illite/smectite

quartz

feldspar

calcite

pyrite

Barnett Eagle Ford

7.17 9.74

12.87 5.24

0.45 0.67

18 0

3 8

3 0

9 0

54 17

6 0

0 74

7 1

sorbed gas and the absolute gas sorption amount can be calculated once the sorption-induced pore volume shrinkage is determined. In this study, bulk and pore volumetric strains measured with the nonsorptive and sorptive gases were utilized to describe the shale swelling behaviors resulting from the geomechanical and gas-sorptive effects. Empirical models were first proposed to describe the relationship between the pore volumetric strain and bulk volumetric strain due to gas sorption. Such a relationship enables us to investigate how the sorption-induced bulk swelling at the core-scale transmits to the sorption-induced pore shrinkage at the micro- and nanopore scale. The proposed models were further applied to evaluate the pore volume change caused by the gas sorption and then predict the absolute gas sorption amount in shales. Furthermore, the predictions were validated with the calculated absolute gas sorption amount using the modified Dubinin−Astakhov (D-A) equation. Conclusions and findings from this work clearly reveal the difference between the sorption-induced bulk and pore volumetric strains of the shales and advance the understanding of the pore volume change attributed to the geomechanical and gas-sorptive effects.

reservoir conditions through matching the sorption isotherm to the classification of International Union of Pure and Applied Chemistry (IUPAC).18,28,34−36 Nevertheless, the gas sorption models are applicable to describe the absolute gas sorption amount rather than the Gibbs gas sorption amount determined from the laboratory measurements. In general, the absolute gas sorption amount is calculated through dividing the measured Gibbs gas sorption amount by a conversion term (1 − ρgas/ ρsorp).15 For measuring the Gibbs gas sorption amount in the laboratory, the pore volume occupied by the sorbed gas is not considered because the density of the sorbed gas cannot be measured or calculated directly. Hence, the pore volume for the free gas is not corrected to the effective pore volume at each pressure equilibrium. Consequently, the major challenge of quantifying the absolute gas sorption amount based on the Gibbs gas sorption amount is to determine the density of the sorbed gas. Typically, the density of the sorbed gas is obtained by performing regression analysis on the Gibbs gas sorption amount with a typical gas sorption model (e.g., Langmuir, D-R, D-A, and BET). In addition, density functional theory and molecular simulation can evaluate the Gibbs gas sorption capacity on the basis of the density profile and the trajectories and velocities of molecules, respectively.34 Then, the absolute gas sorption capacity can be calculated accordingly. Alternatively, predicting the pore volume occupied by the sorbed gas might be another option to obtain the absolute gas sorption capacity.22,27 Furthermore, the swelling behavior of the adsorbent induced by the gas sorption is closely dependent on the absolute gas sorption amount. The swelling behavior related to the bulk volume of the shale and coal samples has been intensively investigated in the previous studies.1,2,11,13,37−39 The bulk volumetric strain measured by the nonsorptive gas reveals the geomechanical (poroelasticity) effect on the rock deformation, whereas such a strain measured by the sorptive gas represents both the geomechanical and gas-sorptive effects on the rock deformation. The sorption-induced bulk volumetric strain could be differentiated from the poroelasticity-induced bulk volumetric strain.40 Additionally, measuring the pore volumetric strain by the nonsorptive gas (e.g., helium) is accepted as the method to determine the pore compressibility at the stresscontrolled conditions.41 However, such strain cannot be determined by the sorptive gas because the pore volume change resulting from the sorption-induced matrix swelling is not wellrecognized. In some studies, the sorption-induced pore volumetric strain was assumed to be equivalent to the sorption-induced bulk volumetric strain in the permeability models of coal.42,43 Nevertheless, the result of permeability measurement under the triaxial stress and strain condition with the coal sample indicated that the sorption-induced pore volumetric strain might be proportional to the sorption-induced bulk volumetric strain.13 Therefore, finding out a proper link between the sorption-induced pore and bulk volumetric strains is critical to understand the effect of sorption-induced matrix swelling on the pore volume change in shale. The density of

2. SHALE CORE SAMPLES In this study, two Barnett shale core samples (plugs) were drilled and extracted from the long outcrop. One of the core samples was employed to measure the methane sorption amount via the volumetric method and meanwhile detect the sorption-induced bulk volumetric strain. Another one was crushed to measure the methane sorption capacity via the gravimetric method and determine the properties and mineral compositions of the shale core sample. Similarly, two Eagle Ford shale core samples (plugs) were prepared in the same manner to conduct the mentioned measurements. In addition, the porosity of the shale core sample was measured using the helium expansion method, which will be presented in section 3. The total organic carbon (TOC) content and vitrinite reflectance (VRo) were determined from the Rock-Eval pyrolysis. Furthermore, the sample mineralogy of the shale core sample and the clay minerals was detected by the X-ray diffraction (XRD) analysis. The description of the rock properties and mineral compositions for the Barnett and Eagle Ford shale core samples is listed in Table 1. 3. EXPERIMENTAL MEASUREMENTS 3.1. Methane Sorption Capacity Measured by the Volumetric Method. As discussed previously, the pore volume occupied by the sorption-induced matrix swelling of shale cannot be measured in the laboratory, which means that the effective pore volume for the free gas cannot be obtained directly. Therefore, the laboratory measurements can only provide the Gibbs (excess) gas sorption capacity. Nevertheless, the previous studies implied that the gas sorption amount and the bulk volumetric strain have a positively correlated relationship1,2 and the sorption-induced pore volumetric strain may be proportional to the bulk volumetric strain.13 Thereby, the volumetric method coupled with the stress and strain measurements may be able to detect the sorption-induced pore volume shrinkage on the basis of the geomechanical and gas-sorptive characteristics, which leads to an 4987


Article

Energy & Fuels

Figure 1. Schematic of the testing device for gas sorption (volumetric method) and bulk strain measurements. accurate determination of the absolute gas sorption capacity of shale. For this purpose, an experimental device was built to simultaneously measure the bulk volumetric strain and the Gibbs methane sorption capacity of the intact shale core sample (plug) under the triaxial stress and strain conditions. The schematic of the device is presented in Figure 1. To achieve the isothermal condition, the temperatures in the reference cell and the triaxial core holder (surrounded by the dashed line) are held to be equivalent (±2 °F) using two electric rubber heating belts. During the experiment, the shale core sample (plug) is settled in the triaxial core holder. In this study, the prepared 1.5 × 3-in. shale core samples from the Barnett and Eagle Ford shale gas reservoirs were used to conduct the measurements. The weights of the Barnett and Eagle Ford shale core samples are 200.18 and 205.20 g, respectively. The confining stresses in the axial and radial directions are provided by a frame load and a Quizix pump, respectively. Moreover, the initially pressurized testing gas using an Isco pump is stored in the reference cell. The pressure at the reference cell and the upstream of the triaxial core holder and the pressure at the downstream of the triaxial core holder are recorded using the high-accuracy pressure transducer #1 and #2, respectively. The accuracy of the two pressure transducers is ±0.08% BSL (linearity, hysteresis, and repeatability combined). In addition, the bulk volumetric strains of the testing core sample during the gas sorption measurements are acquired by the strain gauge system. In this study, the LabVIEW data acquisition system was applied to record the measured bulk volumetric strains and calibrate the attached strain gauges. Theoretically, the space available in the triaxial core holder for the gas storage is identical to the effective pore volume of the shale core sample under the reservoir condition with stresses in both axial and radial directions. Herein, in light of the gas expansion method, helium as a nonsorptive gas is first used to measure the bulk and pore volumetric strains. Subsequently, a sorptive gas, methane, is used to measure the bulk volumetric strain. To mimic the reservoir conditions and also avoid the gas leakage, the axial stress is set at 6000 psi (41.37 MPa) to represent the overburden stress, and the radial stress is set at 5000 psi (34.47 MPa) to represent the horizontal stress. Additionally, the temperature is set at the reservoir temperature of 150 °F (65.5 °C) to maintain the

isothermal condition during the measurements. The detail of the experimental measurement procedures is presented in Appendix A. Applying the real gas law and mass conservation, the gas sorption amount can be determined as follows. The initial number of moles for the helium in the reference cell and the connector volume (nr) is expressed as

PV 1 r + c = z1nr RT

(1)

where z1 is the gas compressibility factor in the reference cell, Vr+c is the volume of the reference cell plus the connector volume (Vr+c = Vr + Vc1). The Vr is the volume of the reference cell and Vc1 is the volume of the connector from valve #2 to valve #3. Similarly, the initial number of moles for the helium in the triaxial core holder and the connector volume (nt) is calculated by P2Vpi + c = z 2ntRT

(2)

where z2 is the gas compressibility factor in the triaxial core holder, Vpi+c is the initial volume in the triaxial core holder (Vpi) plus the connector volume (Vc2) at P2 (Vpi+c= Vpi+ Vc2). The Vpi is the effective volume in the triaxial core holder, which is equal to the pore volume of the shale core sample at P2 measured by helium. The Vc2 is the volume of the connector from valve #3 to valve #4. Furthermore, after reaching the equilibrium, the total number of moles for the helium in the system can be expressed as P3(Vr + c + V p′ + c) = z 3(nr + nt)RT

(3)

where z3 is the gas compressibility factor in the triaxial core holder at equilibrium, V′p+c is the pore volume in the triaxial core holder plus the connector volume at the equilibrium pressure (V′p+c = V′p+ Vc2). Combining eqs 1−3, the volume of triaxial core holder plus the connector volume can be given as

V p′ + c =

P2Vpi + cz 3 PV 1 r + cz 3 + − Vr + c z1P3 z 2P3

(4)

The V′p+c in eq 4 is assumed to be identical to the Vpi+c at a low equilibrium pressure (P3), which indicates that the pore volume of the shale core sample is assumed to be constant at the low equilibrium pressure. In other words, the pore volume will not change from P2 to P3 4988


Article

Energy & Fuels

Figure 2. Experimental setup to measure the gas sorption using the gravimetric method. suspension sorption system shown in Figure 2.45 This facility enables one to continuously record the balance reading during the gas sorption process, which offers a way to calculate the gas sorption amount without using the real gas law and thus greatly increases the accuracy of measurement. The resolution of the device is 0.01 mg, and its reproducibility is ±0.02 mg. The relative error is less than 0.002% of the measured value. The detailed calculation of gravimetric method can be found in Appendix B. In this study, the nonsorptive gas (helium) and the sorptive gas (methane) were used as the testing gases to perform the gas sorption experiments for the two studied shale core samples, respectively. Setting the testing temperature at 150 °F (65.5 °C), same as the temperature in the volumetric method mentioned previously, the Gibbs sorption isotherms of methane on the Barnett and Eagle Ford shale core samples were determined.

when these two pressures are very low. Thus, the Vpi+c at P2 equals to zero is determined using eq 5.

Vpi + c =

z 2z 3P1 − z1z 2P3 Vr + c z1z 2P3 − z1z 3P2

(5)

In eqs 4 and 5, all the gas compressibility factors (z-factor) are calculated by the Peng−Robinson EOS.44 The volume of the reference cell (Vr) and the volume of the two connectors (Vc1 and Vc2) are measured in advance as known variables with value 80 cm3, 1.472 cm3, and 3.06 cm3, respectively. Analogously, the initial number of moles for methane in the reference cell and the connector volume (n′r) is expressed as PV 1 r + c = z1nr′RT

(6)

Also, the initial number of moles for methane as the free gas in the triaxial core holder and the connector volume (n′t) is calculated by P2V pi′ + c = z 2nt′RT

4. CALCULATION METHOD AND WORKFLOW OF SHALE SWELLING The swelling of the shale matrix is attributed to the in situ or effective stress change and the gas sorption behavior. The results of the bulk and pore volumetric strains obtained during the helium (nonsorptive) and methane (sorptive) expansion measurements are presented in this section, which demonstrates the effect of the geomechanical effect and the combination of the geomechanics and gas-sorption on the shale rock deformation, respectively. 4.1. Swelling Characterized via the Measured Strains by Helium (He). The bulk volumetric strain increment (dεb) and the pore volumetric strain increment (dεp) are detected as the result of the changes in the geomechanical stress when the nonsorptive helium is used as the testing gas. The dεb and dεp represent the expansion of the bulk and pore volumes in shale due to the increase of the pore pressure. In the experiment, the geomechanical effect is the confining pressure provided by the hydraulic oil and the pore pressure offered by the helium (Figure 1). Therefore, the dεb and dεp are normally expressed in terms of the rock compressibility as follows.41

(7)

where V′p+c is the initial volume in triaxial core holder (V′pi) plus the connector volume (Vc2) at P2 of methane (Vpi+c= V′pi+ Vc2). After reaching the pressure equilibrium, the total number of moles for methane in the system can be expressed as P3(Vr + c + V p′′+ c) = z 3(nr′ + nt′ − nsorp)RT

(8)

V p′′+ c = V pi′ + c − V psorp

(9)

V p′′ = V p′′+ c − Vc2

(10)

The V′′ is the effective pore volume of shale core sample at equilibrium pressure (P3) measured by methane. Herein, the V′pi+c is equal to Vpi+c when the P2 equals to zero. It is impossible to solve the amount of sorbed gas (nsorp) because the pore volume occupied by the sorbed gas (Vsorp p ) cannot be found. Therefore, the conventional volumetric methods utilize the V′p+c measured by helium in eq 4 to replace the V′′p+c in eq 8. In other words, it ignores the pore volume occupied by the sorbed gas (Vsorp p ). Therefore, the experimentally measured amount of sorbed gas (nsorp) is the Gibbs sorption amount. However, the bulk volumetric strain can be measured using the strain gauge system. If the pore volumetric strain can be predicted based on the bulk volumetric strain, the pore volume occupied by the sorbed gas can be obtained, and thus the effective pore volume for free adsorptive gas (V′′p) can be calculated. 3.2. Methane Sorption Capacity Measured by the Gravimetric Method. The sorption isotherms of methane on Barnett and Eagle Ford shale core samples were measured using the magnetic

dεbHe =

dεpHe = 4989

−dVb = C bcdPc − C bpdPp V bi −dVp V pi

(11)

= Cpc dPc − Cpp dPp (12) DOI: 10.1021/acs.energyfuels.9b00708 Energy Fuels 2019, 33, 4986−5000

Article

Energy & Fuels

which the Cpp can be calculated following eq 16. In summary, for experimental measurements by helium, the dVb is determined according to the measured bulk volumetric strain and the dVp is computed using the real gas law. 4.2. Swelling Characterized via the Measured Strains by Methane (Me). In terms of the sorptive methane, the total bulk volume increment (dVbMe) and the total pore volume increment (dVpMe) are not only controlled by the geomechanical effect but also by the gas-sorptive effect. The dVbMe and dVpMe are presented in eqs 18 and 19 as

where dVb and dVp are the bulk and pore volume increments, respectively. Cbc and Cpc are the bulk and pore compressibilities with various confining pressure and constant pore pressure, respectively, and C bp and Cpp are the bulk and pore compressibilities with various pore pressure and constant confining pressure, respectively. Vib and Vip are the initial bulk and pore volumes at the reference condition. In this study, the reference condition is the shale core sample under axial pressure at 6000 psi (41.37 MPa), radial pressure at 5000 psi (34.47 MPa), pore pressure at 0 psi (0 MPa), and temperature at 150 °F (65 °C). On the basis of the experimental condition mentioned previously, eqs 11 and 12 can be simplified because of the constant confining pressure (Pc). dεbHe

= −C bp,He dPp

(13)

dεpHe

= −Cpp,He dPp

(14)

C bp,He

Cpp,He =

1 ijjj ∂Vp yzzz jj z i j ∂Pp zz V p,He k { Pc

(18)

dVpMe = dVpHe + dV ps

(19)

s

where dVb refers to the sorption-induced bulk volume swelling and dVsp refers to the sorption-induced pore volume shrinkage. Accordingly, the total bulk strain increment (dεMe b ) and the total pore strain increment (dεMe p ) can be written as follows.

where the definition of bulk compressibility (Cbp,He) and pore compressibility (Cpp,He) are given as 1 ij ∂V yz = i jjjj b zzzz V b,He jk ∂Pp z{ Pc

dVbMe = dVbHe + dV bs

(15)

dεbMe = dεbHe + dεbs

(20)

dεpMe = dεpHe + dεps

(21)

dεsb

dεsp

where and are the bulk and pore volumetric strain increments induced by methane sorption, respectively. According to eqs 11 and 20, the dεbads can be determined by subtracting the dεHe from the dεMe under the identical pressure and b b temperature condition. However, the dεMe p cannot be measured directly, and thus the dεsp cannot be calculated. Moreover, similar to eqs 13 and 15, the dεMe b and Cbp,Me can be expressed as

(16)

where Vb is the bulk volume, Vib,He is the initial bulk volume determined by helium, Vp is the pore volume, Vib,He is the initial pore volume determined by helium, Pc is the confining pressure, and Pp is the pore pressure. To measure the swelling or shrinkage of the bulk volume with respect to the effective stress, strain gauges are usually used to record the bulk volumetric strains during the test. In this study, two rosette strain gauges were attached on the surface of shale core samples to determine the strains in the axial and radial directions as shown in Figure 3. The average strain in the axial

dεbMe = −C bp,MedPp C bp,Me =

i

(22)

y

1 jjj ∂Vb zzz j z i V b,Me jjk ∂Pp zz{

(23)

Pc

dεMe p

However, the definitions of and Cpp,Me may be different from eqs 14 and 16, respectively. The reason might be that the matrix swelling of shale leads to the reduction of pore volume, and the reduction of pore volume due to the gas sorption is larger than the increase of pore volume due to the effective stress change. In this study, the dεMe p and Cpp,Me are defined by eqs 24 and 25, respectively. dεpMe = Cpp,Me dPp Cpp,Me = −

Figure 3. Two rosette strain gauges attached to the testing core sample.

−

−

V bi − Vb dV = − ib i Vb Vb

1 ijjj ∂Vp yzzz jj z i j ∂Pp zz Vp,Me k { Pc

(25)

Vp is the pore volume for the free methane, Vib,Me bulk volume determined by methane, and Vip,Me

where is the initial is the initial pore volume determined by methane. Theoretically, the Vib,Me is equal to the Vib,He, and the Vip,Me is equal to the Vip,He at the reference condition (i.e., Pp = 0 MPa) because no gas sorption takes place. 4.3. Proposed Models for Sorption-Induced Bulk and Pore Volumetric Strains. In the study, several models were proposed to determine the sorption-induced pore volumetric strain (εsp) and thus calculate the corresponding effective pore volume (V′′p) for the free gas. Since the sorption-induced bulk volumetric strain (εsb) can be determined directly via the experiments, we developed the following models (Table 2)

direction and the average strain in the radial direction are applied to calculate the bulk volumetric strain by46 εb = εa + 2 × εr =

(24)

(17)

where εb is the bulk volumetric strain, εa is the average strain in the axial direction, and εr is the average strain in the radial direction. Please refer to the following standards47,48 for details on the strain calculation and measurement. Consequently, the Cbp can be calculated following eqs 15 and 17. For the nonsorptive gas (i.e., helium), the pore volume at each equilibrium pressure can be calculated by the eq 4, from 4990


Article

Energy & Fuels Table 2. Function Models for the Correlation between εsp and εsb models

correlation between dVps and dVbs

function

linear

− εps

quadratic

− εps = a·(εbs)2 + b·εbs + c

exponential

− εps = a exp(b·εbs)

power law

− εps = a(εbs)b

Langmuir-type

− εps = a·

=

a·εbs

dV ps = a·ϕi ·(− dV bs)

+b

dV ps = a·ϕi ·

(− dV bs)2 + b·ϕi ·(− dV bs) Vbi

ij − dV bs yz zz·V dV ps = a expjjjjb· zz pi k Vbi {

ij − dV bs yz zz ·V dV ps = a·jjjj zz pi k Vbi { P d V sp = a·ϕi · ·(− dV bs) P+b b

P ·εbs P+b

Figure 4. Flowchart of calculation workflow to determine the correlation between εsp and εsb.

which express sorption-induced pore volumetric strain as a function of the sorption-induced bulk volumetric strain. In Table 2, the a and b are fitting parameters and ϕi is the initial porosity at the reference condition. The negative sign

denotes the reverse direction of sorption-induced bulk volumetric strain compared with the direction of sorptioninduced pore volumetric strain. Herein, the sorption-induced swelling of bulk volume (tensile strain) is presented by a 4991


Article

Energy & Fuels

Figure 5. Pore volume versus pore pressure by helium for Barnett and Eagle Ford core samples.

measurements via the volumetric and gravimetric methods, respectively. Furthermore, the average absolute percent error (% AAD) is used to evaluate the quality of the regression analysis. The definition of average absolute percent error is given as

negative value and the sorption-induced shrinkage of the pore volume (compressive strain) is presented by a positive value. After simplification, the correlation between dVsp and dVsb for each of the models is demonstrated as well. For the linear function, the parameter b is set to be zero, which represents that the dVsp will not take place if the dVsb is not detected. 4.4. Langmuir and Modified Dubinin-Astakhov (D-A) Equations. In general, the Langmuir equation, which is the most acknowledged method to describe the measured Gibbs sorption amount, is expressed as3 ij ρgas yzz j zz· P na = nojjj1 − jj ρsorp zzz P + PL k {

%AAD =

1 N

N

∑ i=1

ncalc − nref nref

× 100 (28)

where N is the number of the data point. ncalc is the value calculated by models and nref is the reference value. For the % AAD1, the ranges of the four fitting parameters, i.e., no, ρsorp, D, and n, in the modified D−A equation are set as 0−1 mmol/g, 200−423 kg/m3, 0 to 0.5, and 1.5−3 respectively according to the previous studies.3,28 The main steps in the workflow are concluded as

(26)

where na denotes the Gibbs gas sorption amount per mass adsorbent, no indicates the maximum sorption amount, PL is the Langmuir pressure, ρsorp and ρgas are the densities of sorbed gas and free gas. In addition, the modified D−A equation also enables to depict the Gibbs sorption amount by3 ÅÄÅ i ÑÉn o l o ij ÅÅ j ρsorp yzÑÑÑ | ρgas yzz o o o j z jj Å o o zzÑÑÑ o zz expm −D·ÅÅlnjj na = nojj1 − } j z z Å Ñ o o j z jj Å Ñ o o ρsorp zz ρ j z Å Ñ o o gas Å Ñ o k { {ÖÑ o (27) ÇÅ k n ~

(1) Measure the bulk volume swelling and the Gibbs methane sorption capacities of the intact Barnett and Eagle Ford shale core samples simultaneously under the triaxial stress and strain conditions (volumetric method). (2) Measure the Gibbs methane sorption capacities of the Barnett and Eagle Ford shale core samples using the magnetic suspension sorption system (gravimetric method) to check the accuracy of the measured Gibbs sorption through the volumetric method. (3) Propose linear, quadratic, exponential, power law, and Langmuir-type function models to represent the possible correlations between the sorption-induced pore and bulk volumetric strains. Calculate the pore volume occupied by the sorption-induced matrix swelling and then determine the absolute methane sorption amount on the basis of the developed function models. (4) Fit the measured Gibbs methane sorption amount via the gravimetric method with the modified D−A equation using the nonlinear regression technique and calculate the absolute methane sorption amount. (5) Adjust the fitting parameters (regression analysis) in both the modified D−A equation and the proposed function models to guarantee that the volume of sorbed methane and the absolute methane sorption amount derived from the gravimetric method are identical to those derived from the volumetric method.

where D refers to the parameter related to the affinity of the sorbent for the gas, and n implies the parameter related to surface heterogeneity. Using the density of sorbed gas to replace the pressure in the original D−A equation, the modified D−A is more flexible to be utilized to evaluate the measured Gibbs sorption amount. In general, n > 2 signifies the molecular sieve carbons or the carbonaceous solids with highly homogeneous micropores, but n < 2 represents the adsorbent with heterogeneous micropores.49,50 In this study, the modified D− A equation is selected to evaluate the methane sorption isotherm measured using the magnetic suspension sorption system. The curve fitting results will be presented in section 5. 4.5. Calculation Workflow. A workflow for determining the proper relationship between the sorption-induced pore and bulk volumetric strains in shale is presented in Figure 4. The nGibbs‑v, nGibbs‑g, nAbs‑v, and nAbs‑g denote the Gibbs or absolute gas sorption amount determined via the volumetric (v) and sorp gravimetric (g) methods, respectively. Vp‑v and Vsorp p‑g are the pore volumes occupied by the sorbed gas calculated based on the 4992


Article

Energy & Fuels

Figure 6. Bulk volume versus pore pressure by helium for Barnett and Eagle Ford core samples.

Figure 7. Bulk volume versus pore pressure by methane for Barnett and Eagle Ford core samples.

as the initial pore volumes in eq 21. As a consequence, the Cpp,He,Bar and Cpp,He,EF are determined as 1.51 × 10−3 1/MPa and 4.73 × 10−4 1/MPa, respectively. Furthermore, the changes in the bulk volume and sample weight normalized bulk volume versus the pore pressure for the Barnett and Eagle Ford shale core samples are exhibited in Figure 6. Similarly, the trend lines in red with the high R2 values indicate that the bulk volume is linearly correlated with the pore pressure at constant confining pressure. In addition, the bulk volumes of the Barnett and Eagle Ford shale core samples under the reference condition is found to be 83.448 cm3 and 84.7593 cm3, respectively. They are used as the initial bulk volumes in eq 20. Accordingly, the Cbp,He,Bar and Cbp,He,EF are equal to 1.25 × 10−4 1/MPa and 5.06 × 10−5 1/MPa, respectively. Finally, the pore strain increment (dεpHe) and the bulk strain increment (dεbHe) can be calculated for the Barnett and Eagle Ford shale core samples based on the Cpp,He,Bar, Cpp,He,EF, Cbp,He,Bar, and Cbp,He,EF. Results of Pore and Bulk Volumes Measured by Methane (Me). The changes in the pore volume with the pore pressure cannot be calculated directly using the real gas law because the sorbed methane that causes the matrix swelling no longer

(6) Conduct regression analysis on the total % AAD using the genetic algorithm to determine the most appropriate relationship between the sorption-induced bulk and pore volumetric strains.

5. RESULTS AND DISCUSSION 5.1. Volumetric Method. Results of Pore and Bulk Volumes Measured by Helium (He). The Barnett and Eagle Ford shale core samples were used to conduct the helium expansion tests first at the multiple-step pressures. The pore volume at each equilibrium pressure can be calculated by eq 4. Figure 5 displays the changes of the pore volume and pore volume normalized to sample weight with the pore pressure. The trend lines in red with the high R2 values indicate an agreement of the linear relationship between the pore volume and the pore pressure at constant confining pressure. Thus, the pore compressibilities of both shale core samples (Cpp,He,Bar and Cpp,He,EF) measured by the helium at constant confining pressure should be constants.41,51 Moreover, the intercepts of linear trend lines in Figure 5 indicate that pore volumes of the Barnett and Eagle Ford shale core samples under the reference condition are equal to 5.9793 cm3 and 8.2526 cm3, respectively. They are used 4993


Article

Energy & Fuels

Figure 8. Gibbs gas sorption and curve fitting results of Barnett shale measured via volumetric and gravimetric methods.

Figure 9. Gibbs gas sorption and curve fitting results of Eagle Ford shale measured via volumetric and gravimetric methods.

belongs to the free gas, whereas the changes in bulk volume with the pore pressure can be determined based on the measured bulk volumetric strain. Figure 7 exhibits the changes in the bulk volume and sample weight normalized bulk volume versus the pore pressure. The trend lines in red with the high R2 values indicate a linear relationship between the bulk volume and the pore pressure at constant confining pressure. The Vib,Me is supposed to be identical to the Vib,He because no gas sorption occurs when the pore pressure equals zero. Finally, the bulk compressibilities of the Barnett and Eagle Ford shale core samples measured by methane (Cbp,Me,Bar and Cbp,Me,EF) are equal to 6.04 × 10−4 1/MPa and 7.73 × 10−5 1/MPa, respectively. It is worthwhile to note that the dεbs can be calculated by eq 20 Me s because the dεHe b and dεb can be measured directly. The dεp s and dVp can be estimated using the developed function models and V′′p are finally presented in Table 2, and then the Vsorp p determined. After the V′′p values are obtained, the absolute gas sorption amount is calculated based on eqs 8−10. 5.2. Gravimetric Method. The Gibbs sorption of the methane on the Barnett and Eagle Ford shale core samples was measured via the magnetic suspension sorption system. Figures

8 and 9 demonstrate the Gibbs sorption isotherms determined using the gravimetric method (nGibbs‑g) associated with the Gibbs sorption isotherms determined using the volumetric method (nGibbs‑v). In light of the volumetric method, replacing the V′′p+c measured by methane in eq 8 by the V′p+c measured by helium in eq 4, the Gibbs methane sorption isotherm can be obtained by solving the eqs 6−8. Moreover, the Gibbs sorption isotherms for the two shale core samples are fitted with the Langmuir and modified D−A equations, and the curve fitting results are also displayed in Figures 8 and 9, respectively. The fitting parameters of the Langmuir equation are presented in Table 3, and the fitting parameters of the modified D−A equation are presented in Table 4. It can be observed that the Gibbs sorption amounts measured via the volumetric and gravimetric methods are nearly equivalent. A polynomial Table 3. Fitting Parameters of Langmuir Equation

4994

sample

no (mmol/g)

PL (MPa)

ρsorp (kg/m3)

Barnett Eagle Ford

0.5 0.65

21.27 24.71

413.1 423


Article

Energy & Fuels Table 4. Fitting Parameters and % AADs for the Proposed Function Models sample

no (mmol/g)

ρsorp (kg/m3)

D

Barnett Eagle Ford

0.4436 0.5701

311.4 397.3

0.4685 0.4821

Barnett Eagle Ford

0.4446 0.5609

309.4 400.4

0.4685 0.4644

Barnett Eagle Ford

0.4164 0.5764

335 375.8

0.4185 0.4989

Barnett Eagle Ford

0.4448 0.5713

309.2 354.3

0.469 0.4911

n

% AAD1

Linear 1.5 5.5013 1.5 7.0846 Quadratic 1.5 5.5141 1.53 6.7070 Power Law 1.5 3.0528 1.514 5.7100 Langmuir Type 1.5 5.5382 1.56 5.0325

% AAD2

% AAD3

total % AAD

a

b

3.7598 3.5023

5.5762 6.8785

14.8372 17.4654

59.52 627

0 0

3.6876 3.4150

5.6088 7.3601

14.8105 17.4821

0.0497 20.44

60.41 616.03

5.3178 3.9371

4.9256 7.2449

13.2962 16.8920

31.83 971.97

0.9 1.045

3.6787 4.3717

5.6005 6.1977

14.8174 15.6019

60.51 831.34

0.0096 1.22

a result, the dVsp is 3.69−4.79 times the dVbs for the Barnett shale core sample, and the dVsp is 59.83−67.12 times the dVbs for the Eagle Ford shale core sample as the pressure increases from 0 to 18 MPa as displayed in Figure 10.

function on the order of six is generated to calculate the Gibbs sorption amount at each testing pressure. In this study, the pressures start from 2 to 17 MPa with a 1 MPa interval are selected for the % AAD calculation based on eq 28. The % AADs of the nGibbs‑v fitting in with the nGibbs‑g are equal to 6.673 and 2.803 for Barnett and Eagle Ford shale core samples, respectively. The % AADs are used to examine whether the gas sorption measurements carried out in the first and second steps in the calculation workflow are qualified. It is found from Figures 8 and 9 that both Langmuir and modified D−A equations can properly describe the Gibbs methane sorption isotherm for the Barnett shale core sample. However, neither can depict the Gibbs methane sorption isotherm for the Eagle Ford shale core sample, where the modified D−A equation deviates from the measured Gibbs methane sorption at pressures higher than 15 MPa and the Langmuir equation deviates such data under all the testing pressures. Compared with the Langmuir equation, the modified D−A equation has a broader application to the measured Gibbs sorption in the laboratory and is selected to perform the regression analysis in this study. 5.3. Relationship Between εsp and εsb. Each of the proposed function models to express εsp as a function of εsb (Table 2) are utilized to calculate the effective pore volume (V′′p) and the absolute gas sorption amount following the calculation workflow (Figure 4). The fitting parameters and total % AAD (total % AAD = % AAD1 + % AAD2 + % AAD3) are demonstrated in Table 4. The fitting parameters and % AADs of exponential function are not provided because no solutions can be found following the calculation workflow. Table 4 reveals that the power law function is the most appropriate expression for εsp as a function of εsb for the Barnett shale core sample, while the Langmuir-type function gives a best match for εsp as a function of εsb for the Eagle Ford shale core sample. Nevertheless, the parameter b in the Langmuir-type function that typically represents the Langmuir pressure (PL) is very small for the Eagle Ford shale and even negligible for the Barnett shale. Although the power law function does not provide the best match for the Eagle Ford shale, the total % AAD of the power law is less than those of the linear and quadratic functions. In addition, the power law function is the best match for the Barnett shale with the minimal % AAD compared to the rest functions. However, the Langmuir-type function gives a nearly identical % AAD to the % AADs provided by the linear and quadratic functions for the Barnett shale. Therefore, we suggest using the power law relationship between the εsp and the εsb for both Barnett and Eagle Ford shale core samples in this study. As

Figure 10. dVsb versus dVsp for Barnett and Eagle Ford shale core samples.

In addition, Figure 11 exhibits the absolute gas sorption amounts calculated using the power law function and the modified D−A equation, respectively. It is found that the absolute gas sorption amounts calculated by these two methods are nearly identical for the Barnett and Eagle Ford shale core samples. The good matching illustrates that estimating the dVsp based on the power law relationship between the dVsp and dVsb is a feasible and practical method to determine the effective pore volume during the gas sorption measurements via the volumetric method. It is noted that the fitting parameters, i.e., a and b, in the power law function cannot be directly estimated, but they are closely related to the absolute gas sorption amount and the weight percentage of TOC and clay minerals in shale. Figure 12 displays the relationship between the dVsb and the absolute gas sorption capacity of shale (nAbs‑g). The dVsb of the Eagle Ford shale is much less than such value of the Barnett shale, whereas the values of nAbs‑g for the Barnett and Eagle Ford shales are close. As expected, the trend of dVsb versus nAbs‑g is parallel to the trend of dVsb versus dVsp, which implies that the dVsp is proportional to the nAbs‑g. According to Figures 10 and 12, it can be found that, although the lower percentage of TOC and clay minerals in the Eagle Ford shale core sample leads to a much smaller dVsb, the dVsp may be independent of the percentage of TOC and clay minerals in the shale core sample because the dVsp 4995


Article

Energy & Fuels

Figure 11. Absolute gas sorption for Barnett and Eagle Ford shale core samples (v and g represent the volumetric and gravimetric methods).

Figure 14 presents the changes of the effective pore volume and sample weight normalized effective pore volume versus the pore pressure for the Barnett and Eagle Ford shale core samples. The effective pore volume decreases with the increase of the pore pressure, indicating that the more methane sorbed in shale core sample, the more matrix swelling takes place. According to the linear trend lines in Figure 14, the pore compressibilities of the Barnett and Eagle Ford shales measured by methane (Cpp,Me,Bar and Cpp,Me,EF) can be calculated by eq 25 as 2.38 × 10−2 1/MPa and 1.77 × 10−2 1/MPa, respectively. Herein, the bulk and pore compressibilities measured by helium and methane are summarized in Table 5. It is found that the bulk and pore compressibilities measured by helium (Cbp,He and Cpp,He) for the studied shales, which are solely controlled by the geomechanical effect, are in line with such measured values reported in the previous studies under the equivalent effective stress condition.52−54 Furthermore, the bulk and pore compressibilities measured by helium (Cbp,He and Cpp,He) are less than the corresponding ones measured by methane (Cbp,Me and Cpp,Me) for both Barnett and Eagle Ford shales. This implies that the bulk and pore volume increments caused by both the geomechanical and gas-sorptive effects is more distinct than those resulting from the geomechanical effect alone. In addition, the swelling caused by methane sorption has a larger influence on the pore compressibility because Cpp,Me is 1−2 orders larger than Cpp,He, whereas Cbp,Me is on the same order of Cbp,He. Finally, it is noted that the Cbp,He, Cpp,He, Cbp,Me, and Cpp,Me for the Barnett shale core sample with higher TOC (12.87%) and clay (33%) contents is larger than such values for the Eagle Ford shale core sample with lower TOC (5.24%) and clay (8%) contents. Therefore, the shale with higher TOC and clay contents might be more prone to be deformed by the geomechanical and gas-sorptive effects.

Figure 12. dVsb versus absolute gas sorption for Barnett and Eagle Ford shale core samples.

and nAbs‑g for the Eagle Ford shale are roughly identical to those values for the Barnett shale. Therefore, we may predict that the dVsb mainly depends on the TOC and clay minerals contents of the shale core sample and the dVsp is linearly related to the absolute gas sorption capacity of the shale core sample. 5.4. Bulk Swelling and Pore shrinkage of Shale. In this study, the bulk swelling and pore shrinkage of shale due to the methane sorption are investigated from two aspects. First, the sorption-induced bulk volume swelling of the shale core sample is determined by comparing the bulk volumetric strains measured by helium and methane separately. It is reported that the sorption-induced bulk swelling (εsb) is roughly linear to the absolute gas sorption amount in previous studies.1,2 Figure 13 provides the plots of εsb correlating with nAbs‑g for the Barnett and Eagle Ford shale core samples. As predicted, the sorptioninduced bulk swelling is linearly proportional to the absolute gas sorption amount for the Barnett and Eagle Ford shale core samples, which is consistent with the published literature.1,2 Second, the effective pore volume in the methane sorption measurement via the volumetric method can be determined using the power law relationship between the εsp and the εsb.

6. CONCLUSIONS In this paper, a series of experimental and analytical investigations are conducted to study the bulk volume swelling and pore volume shrinkage in shale resulting from the matrix swelling during the methane sorption process. The main conclusions and findings of this study are summarized as follows. 4996


Article

Energy & Fuels

Figure 13. Sorption-induced bulk volumetric strain correlates with absolute gas sorption (negative sign indicates the tensile strain, dimensionless).

Figure 14. Effective pore volume versus pore pressure measured by methane for Barnett and Eagle Ford shale core samples.

Table 5. Summary of Bulk and Pore Compressibilities Measured by Helium and Methane helium

measured gas sorption capacities using both volumetric and gravimetric methods in the laboratory, and results show that almost identical absolute gas sorption isotherms are derived. The determined power law correlation for the Barnett and Eagle Ford shales reflects that the sorption-induced pore volume shrinkage at the micro- and nanopore scale scale cannot be simply assumed to be identical to the sorption-induced bulk volume swelling at the core scale. (3) In this study, the sorption-induced pore volume shrinkage (dVsp) is much larger than the sorption-induced bulk volume swelling (dVsb) for both Barnett and Eagle Ford shale core samples. It is predicted that the dVsb may be dependent on the weight percentage of TOC and clay minerals, and the dVsp is likely to be linearly correlated with the absolute gas sorption capacity of shale. More investigations should be conducted to further verify this prediction. (4) The bulk and pore compressibilities determined using the sorptive methane are larger than such values determined

methane

sample

Cbp,He (MPa−1)

Cpp,He (MPa−1)

Cbp,Me (MPa−1)

Cpp,Me (MPa−1)

Barnett Eagle Ford

1.25 × 10−4 5.06 × 10−5

1.51 × 10−3 4.73 × 10−4

6.04 × 10−4 7.73 × 10−5

2.38 × 10−2 1.77 × 10−2

(1) Methane sorption measurements via the volumetric and gravimetric methods offer an almost equivalent Gibbs gas sorption amount for the Barnett and Eagle Ford shale core samples. The proposed empirical models on the sorptioninduced bulk and pore volumetric strains coupled with the calculation workflow can effectively determine the effective pore volume during the gas sorption measurements. (2) The sorption-induced bulk and pore volumetric strains follows a power law relationship for both Barnett and Eagle ford shales. The relationship was examined with the 4997


Article

Energy & Fuels

■

where Δm denotes the balance reading. mSC and mS indicate the weights of the sample container and testing sample, respectively. mA indicates the absolute gas sorption amount. Moreover, VSC, VS, and VA represent the volumes of the sample container, testing sample, and sorbed gas, respectively. Additionally, ρg is the density of free gas. During the experiments with the helium, mSC, mS, VSC, and VS can be measured by excluding mA and VA because the gas sorption does not take place. Hence, the remaining variables are VA and mA when the measurements are carried out with the methane. Assuming the VA equals to zero, the excess (Gibbs) sorption is derived from eq B-2 as

using the nonsorptive helium. This finding indicates that the changes in bulk and pore volumes caused by both the geomechanical and gas-sorptive effects are more distinct than those resulting from the geomechanical effect alone. In addition, the changes in bulk and pore volumes of the shale core sample might be more remarkable when it contains high TOC and clay contents.

APPENDIX A The procedures of the experimental measurements via the volumetric method are demonstrated as follows. (1) Settle the testing core sample in the triaxial core holder. Gradually increase the confining pressure in the axial direction via the frame load and simultaneously open valve 5 and gradually increase the confining pressure in the radial direction via the Quizix pump to reach the designed pressure condition while keeping the pressure in the axial direction 500 psi larger than the pressure in the radial direction. (2) Open valves #2, #3, and #4 and close valve #1. Vacuum the whole system for 12 h and keep the vacuum process in the soft vacuum range and check the strain gauge to guarantee the integrity of the core structure. (3) Close valves #2, #3, and #4. Set the electric rubber heating belt system to the testing temperature. (4) Set up the strain gauge data acquisition system to record the bulk volumetric strain. (5) Open valve #1 and transfer the helium from the gas tank into the Isco pump. (6) Close valve #1 and open valve #2. Pressurize the helium in the Isco pump into the reference cell and then close valve #2. Until the helium pressure reaches stability, the highaccuracy pressure transducer #1 records the equilibrium pressure (P1). Meanwhile, the high-accuracy pressure transducer #2 records the downstream pressure (P2). (7) Open valve #3 to transfer the pressurized helium in the reference cell to the triaxial core holder. (8) Until the high-accuracy pressure transducers #1 and #2 record an equivalent pressure (P3), the pressures at the upstream and downstream reach the equilibrium. (9) Close valve #3 and repeat the steps 5−8 by increasing the reference cell pressure to the next level. (10) Open Valve #4 to release the helium and vacuum the helium for 12 h (keep the vacuum process in the soft vacuum range and check the strain gauge to guarantee the core structure integrity). (11) Repeat steps 5−9 with methane. Importantly, when the two high-accuracy pressure transducers record the identical pressure for methane in step 8, it is necessary to wait for at least 4 h to check the pressure does not drop and to make sure the adsorption equilibrium has been reached.

mEX = Δm − mSC − mS + (VSC + VS) ·ρg

where mEX denotes the Gibbs (excess) gas sorption amount. The conversion from the absolute gas sorption to the Gibbs gas sorption is described in eqs B-3 to B-5. The function between the Gibbs gas sorption and the absolute gas sorption is given by Abs nsorp =

Gibbs nsorp

ij jj1 − k

ρgas y

zz {

ρsorp z

(B-3)

where ρgas and ρsorp are the densities of free gas and sorbed gas, Gibbs respectively. nAbs sorp and nsorp are the absolute and Gibbs gas sorption capacities, respectively. The volume of sorbed gas (Vsorp) that represents the pore volume occupied by the sorption-induced matrix swelling is defined as Vsorp =

Abs nsorp

ρsorp

(B-4)

Substituting eq B-3 into eq B-4, the absolute gas sorption capacity can be expressed as

■

Abs Gibbs nsorp = nsorp + Vsorp·ρgas

(B-5)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Yu Pang: 0000-0002-7247-1632 Shengnan Chen: 0000-0002-1704-1007 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors gratefully acknowledge the funding from an Open Fund (PLC20190801) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Chengdu University of Technology), the Canada First Research Excellence Fund entitled “Global Research Initiative in Sustainable Low Carbon Unconventional Resources”, and Discovery Grant from the Natural Sciences and Engineering Research Council.

■

■

APPENDIX B The gas sorption isotherm obtained via the magnetic suspension sorption system (gravimetric method) is calculated based on the following equations. The mass balance of the gas sorption is written as Δm = mSC + mS + mA − (VSC + VS + VA ) ·ρg

(B-2)

REFERENCES

(1) Heller, R.; Zoback, M. Adsorption of methane and carbon dioxide on gas shale and pure mineral samples. Journal of Unconventional Oil and Gas Resources 2014, 8, 14−24. (2) Chen, T.; Feng, X.-T.; Pan, Z. Experimental study of swelling of organic rich shale in methane. Int. J. Coal Geol. 2015, 150, 64−73. (3) Pang, Y.; Soliman, M. Y.; Deng, H.; Emadi, H. Analysis of Effective Porosity and Effective Permeability in Shale-Gas Reservoirs With

(B-1) 4998


Article

Energy & Fuels Consideration of Gas Adsorption and Stress Effects. SPE J. 2017, 22, 1739. (4) Pang, Y.; Soliman, M. Y.; Deng, H.; Xie, X. Experimental and analytical investigation of adsorption effects on shale gas transport in organic nanopores. Fuel 2017, 199, 272−288. (5) Ambrose, R. J.; Hartman, R. C.; Diaz-Campos, M.; Akkutlu, I. Y.; Sondergeld, C. H. Shale gas-in-place calculations part I: new pore-scale considerations. SPE Journal 2012, 17 (01), 219−229. (6) Yu, W.; Sepehrnoori, K.; Patzek, T. W. Modeling gas adsorption in Marcellus shale with Langmuir and bet isotherms. SPE Journal 2016, 21 (02), 589−600. (7) Kang, S. M.; Fathi, E.; Ambrose, R. J.; Akkutlu, I. Y.; Sigal, R. F. Carbon dioxide storage capacity of organic-rich shales. SPE Journal 2011, 16 (04), 842−855. (8) Haghshenas, B.; Clarkson, C. R.; Bergerson, J.; Chen, S.; Pan, Z. In Improvement in Permeability Models for Unconventional Gas Reservoirs and Model Selection Using Statistical Analysis; SPE/CSUR Unconventional Resources Conference, Canada; Society of Petroleum Engineers, 2014. (9) Pang, Y.; Soliman, M. Y.; Sheng, J. Investigating Gas-Adsorption, Stress-Dependence, and Non-Darcy-Flow Effects on Gas Storage and Transfer in Nanopores by Use of Simplified Local Density Model. SPE Reservoir Evaluation & Engineering 2018, 21, 073. (10) Cui, X.; Bustin, A.; Bustin, R. M. Measurements of gas permeability and diffusivity of tight reservoir rocks: different approaches and their applications. Geofluids 2009, 9 (3), 208−223. (11) Ottiger, S.; Pini, R.; Storti, G.; Mazzotti, M. Competitive adsorption equilibria of CO2 and CH4 on a dry coal. Adsorption 2008, 14 (4−5), 539−556. (12) Hol, S.; Peach, C. J.; Spiers, C. J. Applied stress reduces the CO2 sorption capacity of coal. Int. J. Coal Geol. 2011, 85 (1), 128−142. (13) Connell, L. D.; Lu, M.; Pan, Z. An analytical coal permeability model for tri-axial strain and stress conditions. Int. J. Coal Geol. 2010, 84 (2), 103−114. (14) Gasparik, M.; Rexer, T. F.; Aplin, A. C.; Billemont, P.; De Weireld, G.; Gensterblum, Y.; Henry, M.; Krooss, B. M.; Liu, S.; Ma, X.; Sakurovs, R.; Song, Z.; Staib, G.; Thomas, M.; Wang, S.; Zhang, T. First international inter-laboratory comparison of high-pressure CH4, CO2 and C2H6 sorption isotherms on carbonaceous shales. Int. J. Coal Geol. 2014, 132, 131−146. (15) Sudibandriyo, M.; Pan, Z.; Fitzgerald, J. E.; Robinson, R. L.; Gasem, K. A. Adsorption of methane, nitrogen, carbon dioxide, and their binary mixtures on dry activated carbon at 318.2 K and pressures up to 13.6 MPa. Langmuir 2003, 19 (13), 5323−5331. (16) Dreisbach, F.; Lösch, H.; Harting, P. Highest pressure adsorption equilibria data: measurement with magnetic suspension balance and analysis with a new adsorbent/adsorbate-volume. Adsorption 2002, 8 (2), 95−109. (17) Santos, J. M.; Akkutlu, I. Y. Laboratory measurement of sorption isotherm under confining stress with pore-volume effects. SPE Journal 2013, 18 (05), 924−931. (18) Gasparik, M.; Ghanizadeh, A.; Bertier, P.; Gensterblum, Y.; Bouw, S.; Krooss, B. M. High-pressure methane sorption isotherms of black shales from the Netherlands. Energy Fuels 2012, 26 (8), 4995− 5004. (19) Mishra, B.; Dlamini, B. Investigation of swelling and elastic property changes resulting from CO2 injection into cuboid coal specimens. Energy Fuels 2012, 26 (6), 3951−3957. (20) Zeng, Q.; Wang, Z.; McPherson, B. J.; McLennan, J. D. Theoretical approach to model gas adsorption/desorption and the induced coal deformation and permeability change. Energy Fuels 2017, 31 (8), 7982−7994. (21) Qi, R.; Ning, Z.; Wang, Q.; Zeng, Y.; Huang, L.; Zhang, S.; Du, H. Sorption of methane, carbon dioxide, and their mixtures on shales from Sichuan Basin, China. Energy Fuels 2018, 32 (3), 2926−2940. (22) Wu, T.; Zhao, H.; Tesson, S.; Firoozabadi, A. Absolute adsorption of light hydrocarbons and carbon dioxide in shale rock and isolated kerogen. Fuel 2019, 235, 855−867.

(23) Belmabkhout, Y.; Frere, M.; De Weireld, G. High-pressure adsorption measurements. A comparative study of the volumetric and gravimetric methods. Meas. Sci. Technol. 2004, 15 (5), 848. (24) Zhou, S.; Xue, H.; Ning, Y.; Guo, W.; Zhang, Q. Experimental study of supercritical methane adsorption in Longmaxi shale: Insights into the density of adsorbed methane. Fuel 2018, 211, 140−148. (25) Shen, Y.; Pang, Y.; Shen, Z.; Tian, Y.; Ge, H. Multiparameter analysis of gas transport phenomena in shale gas reservoirs: Apparent permeability characterization. Sci. Rep. 2018, 8 (1), 2601. (26) Zhao, H.; Wu, T.; Firoozabadi, A. High pressure sorption of various hydrocarbons and carbon dioxide in Kimmeridge Blackstone and isolated kerogen. Fuel 2018, 224, 412−423. (27) Tian, Y.; Yan, C.; Jin, Z. Characterization of methane excess and absolute adsorption in various clay nanopores from molecular simulation. Sci. Rep. 2017, 7 (1), 12040. (28) Clarkson, C. R.; Haghshenas, B. In Modeling of Supercritical Fluid Adsorption on Organic-Rich Shales and Coal; SPE Unconventional Resources Conference, USA, Society of Petroleum Engineers, 2013. (29) Li, Z.; Jin, Z.; Firoozabadi, A. Phase behavior and adsorption of pure substances and mixtures and characterization in nanopore structures by density functional theory. SPE J. 2014, 19 (06), 1096− 1109. (30) Jin, Z.; Firoozabadi, A. Thermodynamic modeling of phase behavior in shale media. SPE Journal 2016, 21 (01), 190−207. (31) Liu, Y.; Li, H. A.; Tian, Y.; Jin, Z.; Deng, H. Determination of the absolute adsorption/desorption isotherms of CH4 and n-C4H10 on shale from a nano-scale perspective. Fuel 2018, 218, 67−77. (32) Wang, S.; Feng, Q.; Javadpour, F.; Hu, Q.; Wu, K. Competitive adsorption of methane and ethane in montmorillonite nanopores of shale at supercritical conditions: A grand canonical Monte Carlo simulation study. Chem. Eng. J. 2019, 355, 76−90. (33) Wang, S.; Feng, Q.; Zha, M.; Javadpour, F.; Hu, Q. Supercritical methane diffusion in shale nanopores: effects of pressure, mineral types, and moisture content. Energy Fuels 2018, 32 (1), 169−180. (34) Lowell, S.; Shields, J. E.; Thomas, M. A.; Thommes, M. Characterization of Porous Solids and Powders: Surface Area, Pore Size and Density; Springer Science & Business Media, 2012; Vol. 16. (35) Singh, H.; Cai, J. A mechanistic model for multi-scale sorption dynamics in shale. Fuel 2018, 234, 996−1014. (36) Sing, K. S. Reporting physisorption data for gas/solid systems with special reference to the determination of surface area and porosity (Recommendations 1984). Pure Appl. Chem. 1985, 57 (4), 603−619. (37) Larsen, J. W. The effects of dissolved CO2 on coal structure and properties. Int. J. Coal Geol. 2004, 57 (1), 63−70. (38) Pini, R.; Ottiger, S.; Burlini, L.; Storti, G.; Mazzotti, M. Role of adsorption and swelling on the dynamics of gas injection in coal. J. Geophys. Res. 2009, 114 (B4). DOI: 10.1029/2008JB005961 (39) Pan, Z.; Connell, L. D. A theoretical model for gas adsorptioninduced coal swelling. Int. J. Coal Geol. 2007, 69 (4), 243−252. (40) Connell, L.; Pan, Z.; Lu, M.; Heryanto, D. D.; Camilleri, M. In Coal permeability and its behaviour with gas desorption, pressure and stress, SPE Asia Pacific Oil and Gas Conference and Exhibition; Society of Petroleum Engineers, 2010. (41) Jaeger, J. C.; Cook, N. G.; Zimmerman, R. Fundamentals of Rock Mechanics; John Wiley & Sons, 2009. (42) Cui, X.; Bustin, R. M. Volumetric strain associated with methane desorption and its impact on coalbed gas production from deep coal seams. AAPG Bull. 2005, 89 (9), 1181−1202. (43) Liu, J.; Chen, Z.; Elsworth, D.; Miao, X.; Mao, X. Evaluation of stress-controlled coal swelling processes. Int. J. Coal Geol. 2010, 83 (4), 446−455. (44) Peng, D.-Y.; Robinson, D. B. A new two-constant equation of state. Ind. Eng. Chem. Fundam. 1976, 15 (1), 59−64. (45) Pang, Y.; Tian, Y.; Soliman, M. Y.; Shen, Y. Experimental measurement and analytical estimation of methane absorption in shale kerogen. Fuel 2019, 240, 192−205. (46) Azeemuddin, M.; Scott, T.; Zaman, M.; Roegiers, J.-C. In StressDependent Biot’s Constant Through Dynamic Measurements On 4999


Article

Energy & Fuels Ekofisk Chalk, DC Rocks 2001, The 38th US Symposium on Rock Mechanics (USRMS); American Rock Mechanics Association, 2001. (47) Soil, A. C. D.-o.; Rock Standard Test Method for Compressive Strength and Elastic Moduli of Intact Rock Core Specimens Under Varying States of Stress and Temperatures; ASTM International, 2010. (48) RP40, Recommended Practices for Core Analysis; American Petroleum Institute, 1998. (49) Puziy, A. Heterogeneity of synthetic active carbons. Langmuir 1995, 11 (2), 543−546. (50) Wu, F.-C.; Wu, P.-H.; Tseng, R.-L.; Juang, R.-S. Description of gas adsorption isotherms on activated carbons with heterogeneous micropores using the Dubinin-Astakhov equation. J. Taiwan Inst. Chem. Eng. 2014, 45 (4), 1757−1763. (51) McKee, C. R.; Bumb, A. C.; Koenig, R. A. Stress-dependent permeability and porosity of coal and other geologic formations. SPE Form. Eval. 1988, 3 (01), 81−91. (52) Mbia, E. N.; Fabricius, I. L.; Krogsbøll, A.; Frykman, P.; Dalhoff, F. Permeability, compressibility and porosity of Jurassic shale from the Norwegian-Danish Basin. Pet. Geosci. 2014, 20 (3), 257−281. (53) Lan, Y.; Moghanloo, R. G.; Davudov, D. Pore compressibility of shale formations. SPE J. 2017, 22 (06), 1778−1789. (54) Davudov, D.; Moghanloo, R. G. Impact of pore compressibility and connectivity loss on shale permeability. Int. J. Coal Geol. 2018, 187, 98−113.

5000


Evaluation of Matrix Swelling Behavior in Shale Induced by Methane

Recommend Documents