A fractal model of thermal–hydrological–mechanical interaction on coal seam

https://doi.org/10.1016/j.ijthermalsci.2021.107048Get rights and content

Abstract

Although the microstructure of matrix has a significant impact on coal thermal–hydrological–mechanical interaction, this effect has not been included in the permeability analysis of the coal bed methane (CBM) extraction. Previous studies have typically investigated the relationship between coal porosity and permeability through classical cubic permeability model, neglecting the contribution of the coal seam microstructure to permeability. In this paper, we proposed a new fractal model by defining the permeability of the coal as a function of temperature and effective stress, and characterized the permeability by two microstructural parameters of coal with a thermal variable: (1) fractal dimension of the fracture; (2) maximum fracture length; (3) coal seam temperature. And this fractal model is applied to couple the gas flow, thermal conduction and deformation of the coal. The results show that the fractal permeability model is more effective in characterizing the thermal conduction and seepage processes in coal seam than the classical cubic permeability model. Compared with the cubic-law permeability model, the permeability changes about 19.62% with the different fractal dimension, and about 95.01% with the different maximum fracture length. As the fractal dimension increases from 1.25894 to 1.25926, the gas pressure decreases by 185,117.5 Pa. Furthermore, permeability decreases with the increase of coal seam temperature, and different coal parameters have various contributions to the structural parameters. However, the classical cubic permeability model cannot capture these conclusions.

Introduction

As an essential part of the world's energy sources, more and more scholars are focusing on the issue of coal bed methane (CBM) migration. Coal seams have a highly complex fracture-pore network, which is the main conduit for CBM transport [1,2]. Many factors, such as gas adsorption-desorption and matrix deformation, have a significant effect on the fracture-pore structure. And the evolution of the complex microstructure has a significant contribution to coal seam permeability [[3], [4], [5]]. Quantitative analysis of the effects of coal seam microstructure on CBM extraction during the coupling of coal thermal conduction and gas flow is of major theoretical and practical importance [[6], [7], [8]].

For the study of the coal seam microstructure, methods such as digital core, equivalent models of cubic-law permeability and fractal theory are typically used. Using FIB-SEM, Shaina obtained the micron-scale digital cores and compared them with scanning electron microscope features, and the porosity, organic content and pore connectivity were analyzed [9]. Combining the conclusive features of SEM and based on the concept of the pore network model, Zhang proposed a method for calculating permeability by combining the pore network model with percolation theory [10]. Ju proposed a fractal reconstruction method based on digital cores in order to obtain the distribution of microfluidic fields and their variation with structural parameters [11]. However, the models are usually at the micron scale with long calculation times, and make it difficult for macroscopic reservoir simulations [12]. Yu proposed the fractal theory applicable to rock networks, and developed the fractal permeability model for pore networks [13]. Miao extended the fractal theory to fracture networks and developed a fractal model for dual-porosity media [[14], [15], [16]]. The studies of Au et al. [17], Guarracino [18], and Ye et al. [19] also show that the microscopic network structure makes a significant contribution to coal seepage. However, previously published microstructure-permeability models typically neglect the effects of multiple factors such as gas adsorption-desorption, thermal conduction, and the matrix deformation [[20], [21], [22]].

The current study of gas behaviour incorporating adsorption-induced deformation and thermal effects is usually based on a coupled model of Darcy's law, the thermal conduction equations and the matrix deformation. Temperature contributes significantly to the deformation of the matrix and the migration of gases during the extraction process [[23], [24], [25]]. Noorishad [26] evaluated the coupled thermal-hydraulic mechanical characteristics of rock and investigated the effects of thermal stresses on permeability through the deformation change of the fractures. In addition to matrix deformation, coal seam temperature also makes a significant contribution to the adsorption-desorption effects. At higher temperatures the Langmuir constant decreases resulting in a lower initial slope of the isotherm [27]. Sorption capacity decreases with an increase temperature, indicating that the geometry and number of potential sorption sites also changes with temperature. This explains why coal sorption isotherms change shape with increasing temperature; in general, the Langmuir volume decreases and the Langmuir pressure increases [28]. Zhu [29] proposed a general model to describe the evolution of coal porosity under the combined influence of gas pressure, thermally induced solid deformation, thermally induced gas adsorption change, and gas-desorption-induced solid deformation. Qu [30] developed a new thermal model with gas flow and matrix deformation and suggested that the increase in temperature leads to the larger cleat aperture and the higher coal permeability. However, models for permeability analysis are usually based on classical cubic-law permeability equation which neglects the effects of coal microstructure.

In this study, a new gas multi-field coupling seepage model is established, which can consider the influence of the above two factors simultaneously, that is, multi-field coupling effects and micro-pore structure. Then the evolution of gas pressure, temperature and matrix deformation in coal seam at different mining times is studied, and the evolution mechanism of the coal microstructure with different parameters such as seam temperature, time and thermal parameters is also analyzed.

Gas-bearing coal seam is the typical porous media, this is interpreted in Fig. 1 [31]. Based on fractal theory, we proposed a thermal conduction seepage model for coal seam, which included the microstructural contributions and the effects of adsorption-desorption. Additionally, the following assumptions were made to simplify the calculation [32,33]:

  • (1)

    Coal deformation is linear elasticity;

  • (2)

    Gas saturated in pores;

  • (3)

    Gas viscosity is constant;

  • (4)

    The effects of heterogeneity and anisotropy are not considered.

The change of porosity will occur when there is a temperature change caused by methane production and other factors in the coal seam, which is equivalent to affecting the shrinkage of coal seam matrix. Therefore, the effect of the coal matrix can be described in terms of thermal expansion-contraction, so as to approximate the adsorption-induced strain. In this regard, the constitutive relation of non-isothermal gas desorption coal seam can be expressed by Ref. [21]:σij=2Gεij+2Gν12νεkkδijαpδijKαTTδijKεsδijwhere G is shear modulus, ν is Poisson's ratio, δij is Kronecker delta, and p is the pressure in fractures. K is the bulk modulus withK=2G(1+ν)/3 and(12ν)=E/3(12ν). αT is thermal expansion coefficient. α is Biot's coefficient with α=1K/Ks. εs=αsgVsg is the matrix shrinkage strain caused by the desorption of gas in coal, where Vsg is the content of the absorbed gas.

Based on the above assumptions, displacement and strain have the following relationship [34]:εij=12(ui,j+uj,i)where εij is the component of the total strain. ui,j and uj,i are the components of the displacement.

The equilibrium equation is given by Ref. [34]:σij,j+fi=0where fi is the component of force and σij,j is the component of stress tensor.

Substituting Eqs. (2), (3) into Eq. (1), we can obtain:Gui,kk+G12μuk,kiαp,iKαTT,iKεs,i+fi=0

Eq. (4) is the governing equation for coal deformation considering the combined effect of coal seam gas pressure, thermal effects and adsorption resolution resulting in strain on the matrix.

The number of fractures whose lengths are in the infinitesimal range l to l+dl can be expressed as [13]:dN(l)=DflmaxDfl(Df+1)dlwhere Df is the fractal dimension for fractures, N is the number of fissures in a coal unit, and lmax is the maximum length of fractures. The fractal dimension for length distribution can be expressed as [13]:Df=2lnφlnlminlmaxwhere lmin is the minimum length of fractures and φ is the porosity of coal seam fracture. In general, lmin<<lmax, and in two dimensions, 0<Df<2. The total flow rate through all the fractures can be expressed as [15]:Q=β312μDf(1cos2γsin2θ)4DfΔpL0lmax4where β=a/l, γ is the azimuth angle of a fracture, θ is the dip angle of a fracture, and L0 is the side length of a unit cell of fracture system. According to Darcy's law, Q=kAfμΔpL0, where k is the permeability of the coal seam, and Af is the cross-sectional area of the characteristic unit of the fracture network.

Therefore. the permeability equation can be expressed as [14]:k=β312AfDf(1cos2γsin2θ)4Dflmax4

Eq. (8) is the permeability model in coal seam.

The methane content of the gas containing the absorbing and free phases, defined as:m=ρφg+ρgaρcVsgwhere the density of gas is ρg, ρga refers to gas density under standard conditions, and the coal density is ρc. Vsg is the gas absorption volume.

Liang [35] defined the gas absorption volume Vsg as:Vsg=VLpp+PLexp[c21+c1p(Tar+TTt)]where the VL is Langmuir volume constant at Tt, PL is Langmuir pressure constant at Tt. Tt is the reference temperature for desorption/adsorption, Tar is the absolute reference temperature in the stress-free state, (Tar+T) is the temperature of the coal seam. c1 is the pressure coefficient and c2 is the temperature coefficient.

The Darcy law can be expressed as:q˙g=kμpwhere q˙g is the derivative of qg, which is the Darcy velocity.

The equation for gas flow in a coal seam can be expressed by the law of conservation of mass as follows:mt+(ρgqg)=Qswhere t is the overall time, and ρg is the gas density.

Combining Eqs. (9), (10), (11), (12) and the ideal gas law, we can obtain:RMgmt=φtp(Tar+T)+ptφ(Tar+T)Ttφp(Tar+T)2+paρcVLY1Y2Ta×[(c2c1Y3(1+c2p)2+1(p+pL)2)ptc11+c1pTt]where Y1=pL(p+pL), Y2=exp(c21+c2p(Tar+TTt)), Y3=(Tar+TTt).

Cui proposed the porosity model as follows [36]:dφ=1K(αφ)(dσ+dp)where the mean stress σ=σkk/3, and σkk=σ11+σ22+σ33. Thus:φ=α(αφ0)exp{1K[(σσ0)+(pp0)]}where the initial stress of the matrix is σ0 with the pressure p0 and the porosity φ0.

According to the assumptions at the beginning of this section, the matrix strain induced by the gas adsorption-desorption can be expressed as [29]:εS=αsgVsgwhere αsg is the coefficient of sorption-induced volumetric strain, and the gas absorption volume Vsg can be obtained by Eq. (10). And the volume strain can be obtained based on Eq. (1):εv=1K(σ_+αp)+αTT+εs

Substituting Eq. (10) into Eq. (16), we can obtain:εst=εLY1Y2×[(c2c1Y3(1+c2p)2+1(p+pL)2)ptc11+c1pTt]where εL=αsgVL. Substituting Eq. (15) into Eq. (17) and φ differentiates t, considering (εv+p/KsεSαTT)1, we can get:φt=(αφ)(εvt+1KsptεstαTTt)

Substituting Eqs. (18), (19) into Eq. (13), we can obtain:RMgQs=1(Tar+T)[φ+(αφ)pKs]pt[φp(Tar+T)2+(αφ)pαTTar+T]Tt1μ(pkTar+T)p+Y1Y2(paρcVLTa(αφ)pαsgVLTar+T)×[(c2c1Y3(1+c2p)2+1(p+pL)2)ptc11+c1pTt]+p(αφ)Tar+Tεvt

Eq. (20) is the governing equation for gas migration of coal seam. And the fractal permeability k can be obtained by Eq. (8) in section 2.2.

Neglect the thermal-filtration effect, then the two components, heat conduction and fluid heat convection, together make up the total heat flux [29]. That is:q˙T=λMT+ρgCgq˙g(T+Tar)where q˙T is the thermal flux, ρg is the mass density of the gas, and Cg is the gas specific heat constant. λM is the heat conduction coefficient:λM=(1φ)λs+λgφwhere λs, λg are the thermal conductivities of rock and gas respectively. Therefore, the gradient of heat flux is:q˙T=λMTλM2T+ρgCgq˙g(T+Tar)+Cg(T+Tar)(ρgq˙g)=λM2T(1φ)λsTλgφT+ρgCgq˙g(T+Tar)+Cg(T+Tar)(ρgq˙g)

The energy conservation equation of heat flow and the relationship of unit heat capacity are as follows [29,[37], [38], [39]]:[(ρC)M(T+Tar)]t+(T+Tar)Kgαgq˙g+(T+Tar)KαTεvt=q˙T(ρC)M=φ(ρgCg)+(1φ)(ρsCs)where (ρC)M is the specific heat capacity of coal, ρs is the density of coal, Cg and Cs are the heat constants of gas and solid. Kg is the bulk modulus of gas, αg=1/T is the thermal expansion coefficient.

Simultaneous Eqs. (23), (24), (25), we can obtain:[(ρC)M(T+Tar)]t=(ρC)M(T+Tar)t+(T+Tar)(ρC)Mt=(ρC)M(T+Tar)t+(T+Tar)Csρs(1φ)t+(T+Tar)Cgρgφt

Substituting Eqs. (11), (19) and the ideal gas law into Eq. (26), we can obtain:(ρC)MTt(Tar+T)[Kgαg(kup)KαTεvt]=λM2T+ρgapTaCgPa(Tar+T)kupT+(λs+λg)(αφ)×{(εv+pKsαTT)εLY1Y2p[(c1c2pY3(1+c1p)2+Y1)pc21+c1pT]}T

The coal solid has a high heat capacity and therefore, the thermal process is quite slow. The gas-process develops relatively quickly in fractures, while the progress of solid is significantly slower. Compared with these timescales, the propagation rate of elastic stress can be regarded as spontaneous. The multi-field coupling model consists of Eqs. (4), (8), (19), (20), (27).

The above equations describe the fractal coupling model, including the thermal process, adsorption-desorption effect, gas flow, and deformation of coal. The coupling relationship is shown as Fig. 2, and the process of permeability evolution is shown in Fig. 3. The spatio-temporal evolution of the present model (Eq. (4), (8), (19), (20) and (27)) is non-linear and difficult to obtain analytical solutions for. Therefore, we have chosen the COMSOL Multiphysics software based on the finite element method to solve the model proposed in this paper [29,34,40].

In order to verify the accuracy of the multi-field model (Eq. (4), (8), (19), (20) and (27)) based on the fractal permeability of coal seam, we carried out the simulations based on field tests of mining. The size of simulated area is 568 × 568 m2; fixed on the left, right and bottom sides, with 100psia on the upper boundary, as shown in Fig. 4. Most of the parameters were obtained from Mora and Wattenbarger [41], and unreported parameters were substituted from contemporary literature. The parameters for simulation are shown in Table 1.

We have selected the actual gas production used by Mora and Wattenbarger [41] and compared it with the simulation results, as shown in Fig. 5. It can be seen from Fig. 5 that the good agreement was achieved, proving the validity of the model.

The geometry and the boundary is shown in Fig. 6, containing a well with the diameter of 0.1 m set centrally within a 100 m × 100 m×5 m area. The boundaries of the model are constrained by fixed without any gas flow and thermal exchange at the edges. The initial pressure of the wellbore is Pw0 = 0.1 MPa, and the initial temperature is Tw0 = 300K, while for the coal seam, gas pressure Pw = 3 MPa, temperature Tw = 310K. And Table 2 shows other relevant parameters [29].

Under the seepage of the coal seam is stable, the temperature distribution of the coal seam at different times is shown in Fig. 7.

Fig. 7 shows that with the extraction, the temperature of coal body shows a downward trend under the combined interaction of stress and desorption. The minimum temperature appears near the wellbore, and the temperature range caused by wellbore gradually expands with the progress of mining.

The pressure distribution of the coal seam is shown in Fig. 8. It can be seen the pressure distributions show a consistent trend with the prolongation of mining. That is, the regional pressure decreases as it approaches the wellbore, and the minimum pressure appears near the wellbore. At the same time, with the passage of mining, the range of pressure reduction caused by drilling is expanding.

With the extension of mining time, different stress distributions are produced in the test coal seam, as shown in Fig. 9. With the process of mining, the deformation of the model gradually increases. Under the long-term pressure, the coal is regarded as a high-stress area, and the structural deformation is proportional to the compression time. As the measuring point moves away from the wellbore, the deformation increases with the wellbore restricts the deformation of the coal.

In this section, compared with the traditional cubic heat transfer seepage model, the advantages of the fractal heat transfer model are analyzed. Besides, we analyzed the effects of fractal parameters, coal structure parameters and gas parameters on the seepage process. The results show that the fractal heat transfer model is more suitable for the seepage process.

The fractal heat transfer model proposed in this study enables better analysis of the coal seam microstructure. We analyzed the impact on three main structural parameters, including:

  • (1)

    fractal dimension for fractures;

  • (2)

    maximum length of fractures;

  • (3)

    coal temperature under different fractal dimension.

The classical cubic heat transfer permeability model is as follows:kk0=(φφ0)3where the porosity of the coal seam changes as a result of the seepage process [34]:φφ0=1+(1+ν)(12ν)φ0E(1ν)(pp0)2(12ν)3(1ν)(εLpp+pLεLp0p0+pL)

When the seepage and heat transfer processes are stable, we calculated the contribution of the fractal dimension to the permeability of the coal seam. Fig. 10 shows the differences between the fractal heat transfer model, considering the microstructure, and the classical model.

It can be seen that as the fractal dimension of the fracture increases, so does the permeability of the coal seam. Simultaneously, the permeability is 21.61% different from the initial when the fractal dimension is 1.85. And when the fractal dimension for fractures is about 1.4. Besides, the initial micro-fractures in the coal seam continue to expand and produce new micro-fractures, as the increase of Df, resulting in a significant increase in coal seam permeability. However, this conclusion cannot be drawn because classical heat transfer model ignores the microstructure of the coal seam.

Meanwhile, Fig. 11 shows the contribution of maximum fracture length to the permeability. According to Fig. 11, with the other parameters unchanged, the permeability of the coal increases sharply as the maximum length of the fracture increases. When the maximum length is 0.1539, the permeability is 95.01% different from the initial. As the maximum length for fractures increases, the fractures in the coal seam gradually increase, making it easier for gas to flow in the fractures, which results in a significant increase in permeability. However, for the classical cubic model, there is no change in gas seepage with the increase of maximum length for fractures.

As temperature and stress conditions change, the fractal dimension of the cracks also varies significantly. We therefore analyzed the contribution of temperature to the evolution of permeability at different fractal dimensions. As shown in Fig. 12.

From Fig. 12 we can conclude that as the coal seam temperature increases, the permeability at different fractal dimensions all show a decreasing trend, and the trend remains consistent. This is because the higher the temperature, the greater the degree of deformation within the coal body. The original fractures are squeezed shut, which is not conducive to the seepage process.

We calculate the fractal heat transfer model in terms of:

  • (1)

    Both seepage and heat transfer processes reach stable conditions.

  • (2)

    Contribution of coal body temperature, initial temperature of matrix and wellbore, fractal dimension and thermal expansion coefficient to the extraction process at different locations.

  • (3)

    The effect of the fractal dimension on gas pressure.

Based on the previous conclusions, we analyzed the contribution of monitoring location and initial wellbore temperature to the fractal dimension, as shown in Fig. 13. It follows that the fractal dimension tends to decrease at different initial temperatures as the monitoring position moves away from the wellbore, and the rate of decrease is initially sharp and then slow. Besides, the fractal dimension of the coal seam at the same location decreases as the initial wellbore temperature increases. The fractal dimension of fracture reflects the complexity of fractures in coal seams. As a result of drilling and mining, the coal body close to the shaft wall was subjected to a greater degree of stress, resulting in the creation of many new fractures and a significant increase in the fractal dimension of the fractures.

Since the shape of the model is a square with a side length of 100 m, we chose five equally spaced points (at locations (70, 70), (75, 75), (80, 80), (85, 85) and (90, 90)) as representative points for the analysis.

We have analyzed the influence of coal temperature on the fractal dimension at different test points, as shown in Fig. 14.

From Fig. 14 we can conclude that the fractal dimension of the fracture gradually decreases as the temperature of the coal body increases. At the same coal body temperature, the fractal dimension becomes smaller as the location of the monitoring point is gradually moved away from the wellbore.

To further investigate the mechanism of temperature influence on the fractal dimension, we analyzed the contribution of the thermal expansion coefficient αT and the initial temperature of the coal seam T0 to the fractal dimension at different locations, as shown in Fig. 15, Fig. 16.

According to Fig. 15 we can see that as the thermal expansion coefficient increases, the fractal dimension of the coal body increases at different locations. And the closer to the wellbore, the more pronounced the tendency to increase. According to Eq. (4), as the coefficient of thermal expansion increases, the degree of solid deformation increases, more fractures are created and the existing fractures are deformed by extrusion, leading to a significant increase in the fractal dimension. And according to Fig. 16, the fractal dimension has a clear tendency to increase as the initial temperature of the matrix increases. The closer the location to the wellbore, the greater the trend of increase.

The evolution of gas pressure under different fractal dimensions is shown in Fig. 17. It can be seen that the gas pressure of the coal at different locations decreases as the increase of Df. With the increase of fractal dimension, many tiny new fractures are created and the gas from the existing fractures migrates into the new fractures, resulting in a drop in coal seam gas pressure.

Section snippets

Conclusion

In this study, the flow equations for the gas, the deformation equations for the coal, and the thermal conduction equations for seepage are fully coupled by the fractal model. And this coupling quantifies the effect of coal seam microstructural parameters on the thermal conduction, permeability and gas flow evolution of the coal.

Based on the results of our model, the following conclusions can be drawn:

  • The fractal model proposed in this paper can well characterize the quantitative relationships

Declaration of competing interest

The authors declared that they have no conflicts of interest to this work.

Acknowledgements

This work was supported by the Foundation of Key Laboratory of Deep Earth Science and Engineering (Sichuan University) (Grant No. DESE 202103), Postgraduate Research & Practice Innovation Program of Jiangsu Province.

References (41)

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