Controlling effects of differential swelling index on evolution of coal permeability

Chunzhong Jing, Zhnfng Zho, Xiwi Zhng, Jishn Liu, Drk Elsworth,Gungli Cui

a Key Laboratory of Ministry of Education on Safe Mining of Deep Metal Mines, Northeastern University, Shenyang,110819, China

b Beijing Key Laboratory for Precise Mining of Intergrown Energy and Resources, China University of Mining and Technology, Beijing,100083, China

c Oil & Gas Technology Research Institute, Changqing Oilfield Company, Xi’an, 710018, China

d Department of Chemical Engineering, School of Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia

e Department of Energy and Mineral Engineering, G3 Centre and Energy Institute, The Pennsylvania State University, University Park, PA16802, USA

Abstract Coal permeability measurements are normally conducted under the assumption that gas pressure in the matrix is equalized with that in fracture and that gas sorption-induced swelling/shrinking strain is uniformly distributed within the coal.However,the validity of this assumption has long been questioned and differential strain between the fracture strain and the bulk strain has long been considered as the primary reason for the inconsistency between experimental data and poroelasticity solutions. Although efforts have been made to incorporate the impact into coal permeability models,the fundamental nature of those efforts to split the matrix strain between fracture and coal bulk remains questionable. In this study, a new concept of differential swelling index (DSI) was derived to theoretically define the relation among sorption-induced strains of the coal bulk, fracture, and coal matrix at the equilibrium state. DSI was a function of the equilibrium pressure and its magnitudes were regulated by the Langmuir constants of both the matrix and the coal bulk. Furthermore, a spectrum of DSI-based coal permeability models was developed to explicitly consider the effect of differential strains.These models were verified with the experimental data under the conditions of uniaxial strain, constant confining pressure, and constant effective stress.

2020 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords:Coal permeability Differential swelling behavior Gas adsorption Equilibrium pressure

1. Introduction

As an “unconventional”gas resource,coalbed methane(CBM)is a natural product of coalification process (Moore, 2012). CBM has long been considered as a source of disasters for underground coal mining,but now is recognized as a valuable energy resource(Flores,1998). Therefore, reasonable disposal of CBM released from underground coal mining can not only reduce gas explosion hazard but also recover this clean energy. Depending on reservoir conditions, CBM can be extracted using reservoir-pressure depletion method or enhanced CBM (ECBM) recovery methods. For ECBM,nitrogen (N2) and/or carbon dioxide (CO2) are injected into coal seam to desorb and displace CBM(Puri and Yee,1990;Gunter et al.,1997).

One of the key reservoir parameters for CBM production and CO2sequestration in coal seams is coal permeability.Gas is mainly stored in a form of adsorption in coal seams,and both CBM recovery and CO2injection trigger a series of desorption or adsorption associated processes in the coal seams (Chen et al., 2012; Perera et al., 2013). During CBM production, the fracture pressure drawdown induces gas desorption from the coal matrix, and the desorbed gas flows into the fracture network. During desorption, the coal matrix shrinks. As a direct consequence of this matrix shrinkage, the fractures may dilate, and fracture permeability increases correspondingly (e.g. Harpalani and Schraufnagel,1990a).Conversely, CO2geological sequestration and CO2-ECBM may reverse these processes.

Significant experimental efforts have been made to investigate the characteristics of coal permeability evolution.Coal permeability experiments can be generally divided into stress-controlled(constant confining pressure and constant effective stress) and displacement-controlled ones (uniaxial strain condition and constant volume condition) (Shi et al., 2018). Under following four boundary conditions, permeability shows different evolution characteristics:

(1) Laboratory-measured coal permeability in terms of adsorbing gasses such as CH4and CO2can change from reduction to enhancement instantaneously under constant confining pressure condition (Robertson, 2005; Pini et al., 2009;Siriwardane et al., 2009; Wang et al., 2011, 2019; Vishal et al., 2013).

(2) Most experimental data (Connell et al., 2010a; Lin and Kovscek, 2014; Meng et al., 2015; Chen et al., 2019; Wei et al., 2019a) exhibit significant changes under constant effective stress condition: the permeability declines initially and then remains stable as the gas pressure increases.

(3) An ingenious experimental work was conducted by Mitra et al. (2012). In their experiment, coal sample was held under uniaxial strain condition.The coal sample was taken from San Juan basin. It was not permitted to shrink laterally as a result of gas desorption by adjusting the confining pressure.The results showed that coal permeability increases continuously with decrease in gas pressure.These observations are consistent with the field observations made in different parts of the San Juan basin(Gierhart et al., 2007).

(4) A permeability test for coal was conducted under constant volume condition to investigate the coalecleat interactions using helium (Wang et al., 2017). The results showed that permeability increases as the pore pressure increases.

Various coal permeability models have been proposed to explain the variability of coal permeability under laboratory or field conditions. Based on applicable boundary conditions, major coal permeability models were classified into two groups (Liu et al., 2011a): permeability models under conditions of uniaxial strain and that under conditions of variable stress.In these models(Gray,1987;Palmer and Mansoori,1996;Shi and Durucan,2003b),coal permeability is defined as a function of the coal bulk strain.It is also assumed that the coal bulk strain caused by gas adsorption is equal to that of the fracture(Cui et al.,2007;Zhang et al.,2008).However, this assumption may not be valid for coal (Zang et al.,2015).

This discrepancy between assumption and reality has long been recognized.In the study(Liu et al.,2010a),the concept of“internal swelling stress”was introduced to account for the impact of matrix swelling/shrinkage on the fracture aperture changes.In this study,the matrix strain was divided into two parts: one for the fracture component and the other for the bulk strain.Based on this concept,permeability models under conditions of uniaxial strain and constant external stress were developed.

In other studies (e.g. Connell et al., 2010a; Lu and Connell,2010), a constant ratio of the adsorption-induced fracture strain increment to the sorption-induced coal bulk strain increment was introduced. By rearranging the Shi-Durucan model, they found that the ratio was significantly larger than (approximately 50 times) the sorption-induced coal bulk strain. Liu et al. (2010b)proposed the concept of elastic modulus reduction ratio to allocate matrix swelling strain to the fracture and the matrix. By assuming that only part of total swelling strain contributes to fracture aperture change and the remaining to coal bulk deformation, a permeability model under variable stress conditions was developed (Chen et al., 2012).

To explain the phenomenon of coal permeability switch from reduction to rebound under constant confining pressure condition,the concepts of critical pressure and critical time were introduced to define the switch from local swelling to macro-swelling due to gas diffusion from fractures to matrices (Liu et al., 2011b). Before the critical time,coal permeability was determined by the internal volumetric transformations (local swelling) between matrix swelling and fracture void compaction; after the critical time, coal permeability was determined by the external boundaries (macroswelling). Similarly, the concept of critical swelling area was introduced to define the relation between swelling transition and coal permeability evolution under variable stress conditions (Qu et al., 2014). Subsequently, numerous coal permeability models were proposed(e.g.Guo et al.,2014;Wang et al.,2014;Peng et al.,2014b;Lu et al.,2016;Liu et al.,2017a).In these models,a constant strain splitting factor was used to define the ratio of the facture sorption strain to the matrix sorption strain.In a recent study(Peng et al., 2017a), the heterogeneous distribution of internal swelling was considered by extending the concept of strain splitting factor to strain splitting function. All strain splitting-based permeability models are summarized in Table 1.For sake of simplicity,only strain splitting-based parameters are presented in the table. More detailed information can be found in the original works.

As mentioned above, the impact of the differential strain between the fracture strain and the bulk strain is a challenging issue.To address this issue,a concept of differential swelling index (DSI)was introduced in this study to define the relation among sorptioninduced deformations of the coal bulk, fracture, and coal matrix.Furthermore, a spectrum of DSI-based coal permeability models was developed to explicitly consider the effect of differential strains.

2. Concept of differential swelling index (DSI)

As shown in Introduction, coal permeability models can be classified into four categories. (1) The gas sorption-induced strain results in fracture strain only(Palmer and Mansoori,1996;Shi and Durucan,2003a);(2)The gas sorption-induced strain results in coal bulk strain only (Gray, 1987; Zhang et al., 2008); (3) The gas sorption-induced coal bulk strain equals the fracture strain(Zimmerman et al.,1986; Cui and Bustin, 2005); and (4) The ratio between two of the gas adsorption-induced three strain components (matrix strain, fracture strain and bulk strain) is a constant(Lu and Connell, 2010). The assumption that gas sorption-induced strain results in fracture strain only is not consistent with some experimental observations and significantly overestimates the effects of matrix swelling on permeability changes(Robertson,2005).The assumption that adsorption only causes coal bulk strain, by contrast, would significantly underestimate the effect of adsorption. The third hypothesis that the adsorption-induced coal bulk strain is the same as the fracture strain may also underestimate the fracture strain(Connell et al.,2010b).The strain splitting approach(Zang et al., 2015; Peng et al., 2017a) is generally empirical. In following section, a theoretical approach is developed.

2.1. DSI model

Coal is a typical porous medium that consists of both matrices and fractures. In this study, the butt/face cleat system, fractures,bedding plies, and joints are all called the fracture system.Microscopic experimental techniques can be used to investigate sorption-induced strains in coal(Mao et al.,2015). It is found that with adsorption of gases, the matrix adsorption deformationmakes the internal fracture compressed and the coal bulk swells(Karacan, 2003; Pone et al., 2009; Dawson et al., 2012). Three sorption-induced strains corresponding to three strain components were defined in this study:representing the sorption/desorption-induced coal bulk strain, matrix strain, and fracture strain, respectively. The adsorption-induced strain is the ratio of adsorption-caused volume fluctuations against original counterpart, which can be measured by a strain gage; the sorption-induced matrix strain,can be measured for a small sample containing only coal matrix and no cleats; however,is more difficult to measure experimentally (Connell et al., 2010a).To study the microstructure of coal,micro-computed tomography(micro-CT)was applied(Ramandi et al.,2016),and it showed that coal is formed with a complex matrix-fracture system (Fig.1a). A representative elementary volume (REV) of the coal structure is depicted in Fig.1b, in which the value of initial coal bulk volumeVb0can be expressed by the sum of initial fracture volumeVf0and initial matrix volumeVm0, i.e.Vb0Vf0þVm0. Coal adsorption characteristics can be generally measured under unconstrained conditions (Peng et al., 2017a). With gas adsorption, coal bulk swells and fracture is compressed.For the initial equilibrium stage as illustrated in Fig.1c at the initial equilibrium pressurep0,the volumes of coal bulk, matrix and fracture can be expressed asrespectively,wheredenote the adsorption-induced strains of coal bulk, coal fracture and coal matrix under pressurep0, respectively. Similarly, as the pressure increases, the volume for the current equilibrium stage with equilibrium pressurepas shown in Fig.1d can then be written asrespectively.

Table 1Summary of strain splitting-based permeability models.

Fig.1. Illustration of adsorption-induced differential swelling behavior:(a)Micro-CT image of coal structure(Ramandi et al.,2016);(b)representative elementary volume(REV);(c)Initial equilibrium stage (initial equilibrium pressure p0); and (d) Current equilibrium stage (equilibrium pressure p).

Bycomparing the current stage and initial stage,we can obtain the volumetric balance between coal matrix,coal fracture and coal bulk:

Eq. (2) explicitly defines the relation among three strain components.Experimentaldatashowthatthemeasuredmatrixandcoalbulk sorption strainscan befittedinto Langmuirtype curves(Harpalani and Schraufnagel,1990b;Mitra et al.,2012;Wei et al.,2019b):

Although the fracture system controls the evolution of permeability, it is almost impossible to measure the adsorption strain of the fracture in experiment. The previous work on this issue has indicated that the value of pore strain was around 50 times the bulk strain for the coal under hydrostatic conditions (Connell et al.,2010b; Chen et al., 2012). As discussed above, the differential swelling/shrinking behavior defines the relation among the sorption-induced coal fracture,coal matrix and coal bulk strains.To further define the relation betweenthe differential swelling/shrinking index (DSI),f, was defined asSubstituting the definition of coal swelling strain Eq.(3)into Eq.(2)gives

In Eq. (4), DSI is a function of initial porosity, adsorption properties of coal matrix and coal bulk, and equilibrium pressure. All these parameters can be measured directly in laboratory. If the Langmuir pressure constants of coal bulk and coal matrix are assumed to be equal,i.e.PLbPLm,the pressure-related terms on the right side of Eq. (4) disappear, then this index is degenerated into a constant ratio:

For this special case,the difference between Langmuir strain of coal matrix and coal bulk defines the differential swelling behavior.If the Langmuir strain constants are fixed, the ratios remain unchanged. This is consistent with the current assumptions (Lu and Connell, 2010; Chen et al., 2011). For the homogeneous case, all properties of the coal matrix and coal bulk are the same everywhere,i.e.PLmPLband εLmεLb.In this ideal case,DSI is equal to one, which is consistent with the theory of poroelasticity (Cui and Bustin, 2005; Zhang et al., 2008).

2.2. DSI-based permeability model

In this section,DSI was included in the coal permeability model.It is commonly assumed that the fracture system controls the gas flow in coal while the flow in the coal matrix can be neglected(Purl et al., 1991). Therefore, the evolution of permeability is mainly controlled by the behavior of fracture system (Sparks et al.,1995).Based on volumetric balance, the volumetric deformation of the coal bulk dVband the fracture dVfconsists of two parts(Gray,1987;Connell et al.,2010a),i.e.one caused by effective stressand the other by gas adsorptionAccordingly,for the strain increment,one can obtain thatBased on this,the volumetric strain of coal bulkdVb=Vband the volumetric strain of fracturesdVf=Vfcan be expressed as follows(Detournay and Cheng,1993; Zhang et al., 2008):

Solving Eqs. (8) and (9), we obtain

Substituting Eqs. (6) and (7) into Eq. (10) yields

As the bulk modulusKbis commonly several orders of magnitude larger than the fracture volume modulusKf(Peng et al.,2017a), by assuming thatthe fracture compressibility can be defined as(Lu et al., 2016):

Integrating Eq. (12) with time gives

By substituting the definition Eq. (4) of DSI into Eq. (13) yields

为了保证语料标注的质量,两位标注者同时对语料库进行标注,目的是进行标注结果的一致性检测.本工作采用Passoneau[15]提出的语料库指代标注可靠性计算方法,并根据Krippendorff[16]的alpha系数来表示两位标注者标注结果之间的一致性.该方法通过一个距离度量来表示指代链之间的相似度,然后通过alpha系数计算指代链之间的相似度距离来表示不同标注者标注结果之间的一致性.

The relationship between permeability and fracture porosity can be described by the cubic law(Chilingar,1964;Chen et al.,2013;Cui et al., 2020) as

wherekis the permeability, andk0denotes the initial value of permeability.

Solving Eqs.(14)and(15),general form of the new permeability model can be expressed as

If equal adsorption-induced fracture and bulk strains are assumed (e.g. Cui and Bustin, 2005; Zhang et al., 2008), the adsorption-induced fracture and bulk strains cancel each other out.Assuming that the material is homogeneous, and the DSI is a constant (f1), then the permeability model is degenerated as

Eq. (17) is the permeability model derived from the classical theory of poroelasticity.

2.3. Permeability models under different boundary conditions

The general permeability model defined by Eq. (16) can be applied to a range of boundary conditions from constant confining pressure to constant volume, as illustrated in Fig. 2.

2.3.1. Constant confining pressure condition

Under the constant confining pressure condition as shown in Fig.2a,the change of the confining pressureis zero.In this circumstance, the permeability model can be expressed as

By substituting DSI (Eq. (4)) into Eq. (18), the permeability model in the case of constant confining pressure in an expansion form is

For the case of constant effective stress condition as shown in Fig. 2b, the difference between confining stress and pore pressure remains constant and the increment of effective stress is equal to zero (Cui et al., 2018):

Substituting Eq. (20) into Eq. (16), the permeability model for the case of constant effective stress becomes

By substituting Eq. (4) into Eq. (21), the permeability model in an expansion form is

2.3.3. Uniaxial strain condition

For the case of uniaxial strain as shown in Fig. 2c, the vertical external stress remains constantthe horizontal effective stress increments can be calculated byand non-zero strain occurs in the vertical/axial direction and zero strain in the horizontal direction.The coal bulk volumetric strain increment can be represented asεbIf the coal bulk strain induced by gas adsorption is isotropicthe coal bulk strain increment in three directions can be expressed as

where v is the Poisson’s ratio,andEis the elastic modulus.

Then the average effective stress imposed on the coal can be expressed as

Fig. 2. Schematic diagrams of various boundary conditions.

Substituting Eq.(25) into Eq.(16),we can obtain the governing equation for the permeability change under the uniaxial strain condition:

By substituting Eq.(4)into Eq.(26),the permeability model for the case of uniaxial strain in an expansion form is

2.3.4. Constant volume condition

For the case constant volume condition as shown in Fig. 2d,the displacement of coal sample in each direction is zero, i.e. the total volume of coal remains unchanged. Then, the coal bulk strain increment is zeroand sorptioninduced matrix strain contributes to fracture deformation only.Ma et al. (2011) developed a permeability model under the constant volume boundary condition by making a distinction between mechanical compression-induced strain and matrix shrinking strain. In the experiment, the expansion of coal is controlled by adjusting confining pressure to achieve the goal of constant volume boundary condition (Espinoza et al., 2014).Based on above understanding, substituting dεb0 into Eq. (6)yields

Integrating Eq.(29)gives porosity model for the case of constant volume:

Based on the cubic law and by substituting the DSI into the porosity model, we can obtain the governing equation for the permeability change under the constant volume condition:

By substituting Eq.(4)into Eq.(31),the permeability model for the case of constant volume can be defined in an expansion form as

3. Model verifications

In this section, our models are compared with experimental observations under a spectrum of boundary conditions from constant confining pressure to uniaxial strain. Four sets of experimental data under these boundary conditions (Robertson and Christiansen, 2005; Pini et al., 2009; Connell et al., 2010a; Mitra et al., 2012) were used. In addition, we also compared with the performances of other models for the same data sets(Connell et al.,2010a; Guo et al., 2014).

Table 2Properties of coal and fluid for the experiment conducted by Pini et al. (2009).

Table 3Properties of Anderson coal for the experiment conducted by Robertson and Christiansen (2005).

3.1. Constant confining pressure condition

We compared the model results of Eq. (19) with experimental data under constant confining pressure condition (Robertson and Christiansen, 2005; Pini et al., 2009). Basic parameters for the experiments are listed in Tables 2 and 3. The matrix Langmuir adsorption constants of εLmandPLmwere obtained by matching the experimental data. The comparisons of our model against experimental data are illustrated in Figs. 3 and 4.

As shown in Fig.3,our model results matched the permeability data well for the Sulcis coal.In this case,0:0556,and the Langmuir pressure constant for the matrix and coal bulk were equalUnder these conditions,the DSI was a constantThis indicates that the ratio of the adsorption-induced fracture strain increment to the bulk strain increment remains unchanged as the gas pressure increases. We also compared our model results with the results of the model proposed by Connell et al.(2010a).In this particular case,two lines were overlapped because of the same DSI valueAs shown in Fig.4,our new model results also match the permeability data for the Anderson coal. In this case0:041, the DSI value decreases as the gas pressure increases.

3.2. Constant effective stress condition

We compared our model performance of Eq. (22) with the experimental data (Connell et al., 2010a) for the case of constant effective stress,as shown in Fig.5.The mechanical parameters and the adsorption constants, as listed in Table 4, were taken directly from this study(Connell et al.,2010a).The permeability data were extracted from the original figures using a data extractor.Compared with the original data, there might be some imperceptible errors,but it will not affect the overall trend and the verification process.In this casePLm15:9 MPa and εLm0:022, our new model results matched experimental data very well for the coal sample from Southern Sydney basin. The value of the DSI increases as the gas pressure increases.If a constant DSI value is used,with18:7 and26:2, the matches are less satisfactory.

3.3. Uniaxial strain condition

Fig. 3. Comparisons of our model results with experimental data for the Sulcis coal sample (Pini et al., 2009).

We compared our model performance of Eq. (27) with the experimental data(Mitra et al.,2012)for the case of uniaxial strain,as shown in Fig.6.The parameters are listed in Table 5.It should be noted that the initial porosity, elastic modulus and Poisson’s ratio were not provided in their study, thus we used the initial porosity from Young et al. (1991), elastic modulus and Poisson’s ratio from Shi et al. (2005). As shown in Fig. 6, our model results matched well with the experimental data.In this casePLm2:795 MPa and εLm0:0195, the DSI valuefdecreased with the pressure. As shown in Fig. 6, we also compared the performance of our model with others including SD model, modified SD model (Liu et al., 2012), and our constant DSI model.

4. Discussion

As shown in Section 3,the DSI plays a key role in comparison of our model results with experimental data.Unlike previous studies,fis a function of gas pressure and regulated by Langmuir constantsPLmandPLb; εLmand εLbfor both matrices and coal bulk,respectively.

In the following,all of the input parameters are taken from Mitra et al. (2012) and are listed in Table 3, and an initial gas pressurep00:5 MPais assumed.The DSI was calculated by use of Eq.(4).Note that the Langmuir pressure constant isPLb4:16 MPa for the experiment data. By changing the matrix Langmuir pressure constantPLm, we can obtain the evolution trends off, as shown in Fig. 7a. The corresponding evolutions of permeability under constant confining pressure, constant effective stress and uniaxial strain conditions are shown in Fig. 7bed, respectively.

Fig. 4. Comparisons of our model results with the experimental data for the Anderson coal sample (Robertson and Christiansen, 2005).

Fig. 5. Comparisons between different model results and the experimental data under constant effective stress condition for Southern Sydney basin coal sample (Connell et al.,2010a).

Table 4Parameters used in matching the permeability for coal core sample from the Southern Sydney basin of Australia.

As shown in Fig. 7a,fdecreases as pressure increases whenPLmPLb,andfremains as a constant whenPLmPLb. As shown in Fig. 7bed,the magnitudes offhave a significant impact on the evolutions of coal permeability under different boundary conditions. For example, coal permeability decreases initially and then recovers as pressure increases whenPLmPLb.These model results illustrate that matrix Langmuir constants control the evolution of coal permeability.

Table 5Parameters for data matching under uniaxial strain condition (Mitra et al., 2012).

As shown in Eq. (4),fis a function of pressure only and independent of boundary conditions. Under the same pressures, we calculated all permeability evolutions for a range of boundary conditions from constant confining stress to constant volume conditions, as shown in Fig. 8. The upper envelope is the conventional poroelasticity solution for the case of constant confining stress, while the lower envelope is the conventional poroelasticity solution for the case of constant volume condition.Solutions of all other cases are within these envelopes.

Fig. 6. Comparisons between different model results and the experimental data under uniaxial strain condition for San Juan basin coal sample (Mitra et al., 2012).

Fig. 7. Permeability evolution with different differential swelling index.

Fig. 8. Permeability model evolution under different boundary conditions.

5. Conclusions

In this study,a concept of DSI is proposed to theoretically define the relation among sorption-induced strains of the coal bulk,fracture and matrix at the equilibrium state.DSI is a function of the equilibrium pressure and its magnitudes are regulated by the Langmuir constants of both the coal matrix and the coal bulk.Furthermore,a spectrum of DSI-based coal permeability models is developed to explicitly consider the effect of differential strains.These models are verified with the experimental data under the conditions of uniaxial strain, constant confining pressure and constant effective stress. Based on the model results and verifications, the following conclusions can be drawn:

(1) When the gas sorption-induced fracture strain is assumed the same as the bulk strain, DSI is equal to unity; when the Langmuir constants for matrix are the same as the ones for coal bulk,DSI is a constant.However,DSI is a function of the equilibrium pressure when the Langmuir constants for matrix are different from the ones for coal bulk.The theoretical development of DSI concept essentially removes the basic assumptions in previous studies.

(2) An equilibrium state is typically assumed when interpreting permeability measurements e representing the assumption that equilibration has been reached and that both sorptioninduced changes in deformation and their impacts on the evolution of permeability have ceased. At the equilibrium state,coal permeability is a function of equilibrium pressure and its magnitudes are regulated by the magnitudes of DSI.

Declaration of Competing Interest

The authors wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Acknowledgments

This work was supported by National Key R&D Program of China(Grant No. 2018YFC0407006), the 111 Project (Grant No. B17009),and the Australian Research Council (Grant No. DP200101293).These supports are gratefully acknowledged.