Gas flow regimes judgement in nanoporous media by digital core analysis

Wenhui Song; Hua Liu; Weihong Wang; Jianlin Zhao; Hai Sun; Dongying Wang; Yang Li; Jun Yao

doi:10.1515/phys-2018-0062

Article Open Access

Gas flow regimes judgement in nanoporous media by digital core analysis

Wenhui Song , Hua Liu , Weihong Wang , Jianlin Zhao , Hai Sun , Dongying Wang , Yang Li and Jun Yao

Published/Copyright: August 13, 2018

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From the journal Open Physics Volume 16 Issue 1

Abstract

A method to judge shale gas flow regimes based on digital core analysis is proposed in this work. Firstly, three-dimensional shale digital cores in an anonymous shale formation in the Sichuan Basin are reconstructed by a Markov Chain Monte Carlo (MCMC) algorithm based on two-dimensional Scanning Electron Microscope (SEM) images. Then a voxel-based method is proposed to calculate the characteristic length of the three-dimensional shale digital core. The Knudsen number for three-dimensional shale digital cores is calculated by the ratio of the molecular mean free path to the characteristic length and is used to judge the flow regimes under different reservoir conditions. The results indicate that shale gas flow regimes are mainly located at the slip flow and transition flow region. Furthermore, adsorption has no obvious influence on the free gas flow regimes. Because adsorption only exists in organic pores, three-dimensional inorganic pores and organic pores in the Haynesville shale formation are reconstructed by a MCMC algorithm based on two-dimensional SEM images. The characteristic lengths of the three-dimensional inorganic pores and three-dimensional organic pores are both calculated and gas flow regimes in organic pores and inorganic pores are judged.

Keywords: shale gas; digital core; flow regimes; characteristic length; nanoporous media

PACS: 51.20.+d; 91.60.Np

1 Introduction

With the rapid decline in conventional reserves, unconventional resources such as tight gas and shale gas reservoirs play an increasingly important role in the North American energy industry and have gradually become a key component of the world energy supply [1, 2, 3]. In shale gas reservoirs, gas is stored in organic pores and inorganic pores with nm-scale pore size and extremely low matrix intrinsic permeability [4, 5, 6, 7]. Zou et al. [8] reported that pore sizes in shale gas reservoirs range from 1 to 200 nm and that shale matrix intrinsic permeability ranges from 10⁻⁹ to 10⁻³ × 10⁻¹⁵ m² [9, 10]. Gas flows in the forms of free gas and adsorbed gas in an organic pore system, while gas flows in the form of free gas in an inorganic pore system [11, 12, 13]. The size of shale nanopores approach the molecular mean free path. Therefore, the continuity assumption becomes invalid [14, 15, 16]. The Knudsen number is defined as the ratio of the molecular mean free path to the pore radius [17, 18]. At different Knudsen numbers, the corresponding gas flow regime in Figure 1 differs.

Figure 1

Transition of flow regimes based on the Knudsen number

Lin et al. [19] reviewed recent advances on understanding gas flow processes in unconventional porous rocks. Javadpour [20] described gas flows in the forms of Knudsen diffusion and slip flow regime and proposed a model for gas flow in a nanopore duct. Darabi and Javadpour et al. [21] incorporated Knudsen diffusion and surface roughness into the gas flow model by Maxwell theory. However, the Javadpour model [20] includes one empirical coefficient known as the tangential momentum accommodation coefficient (TMAC) and accurate prediction of TMAC for different shale samples is not available. Beskok and Karniadakis [22] developed a unified Hagen–Poiseuille-type model valid in all flow regimes and this model has been adopted by Civan et al. [18, 23, 24] to consider the intrinsic permeability, porosity, and tortuosity of porous media. Freeman et al. [25] claimed that shale gas flows in the transitional flow regime. Landry et al. [26] deemed that gas flow in nanopores at reservoir pressure – temperature conditions falls within the slip flow and early transition flow regime. Wu et al. [27] considered shale gas flows in the forms of continuum flow, slip flow and transition flow. Sun et al. [28] concluded that Knudsen diffusion can be ignored when pressure is larger than 1 MPa. Although there are different viewpoints on shale gas flow in transition flow regime, most of the current studies suggest that shale gas flows in the slip flow regime at typical reservoir conditions [3, 29, 30].

The estimated ultimate recovery (EUR) of gas reservoirs is highly dependent on the pore structure characteristics [31, 32, 33, 34, 35, 36]. Characteristics of pore systems in Sichuan Basin shale formation in China have been studied in literature [37, 38, 39, 40]. However, the current analysis and results are mostly based on two-dimensional SEM images and the three dimensional pore structure is not fully understood. Though three dimensional organic matter and inorganic matter can be observed by Focused Ion Beam-Scanning Electron Microscopes (FIB-SEM) [41, 42, 43, 44], three dimensional organic pores and inorganic pores cannot be directly imaged and divided due to the limited resolution. On the other hand, the MCMC method is widely used for image processing research [45]. The MCMC method was applied to generate pore space of real heterogeneous porous media [46] and was also used to analyze shale pore structures [47] in our previous study.

Most of the current studies on gas flow in shale needs a predefined gas flow regime based on the Knudsen number to select the corresponding gas flow model. The Knudsen number for three-dimensional porous media is generally defined by the mean free path divided by the characteristic length of porous media. Previous work established a characteristic length analytical solution based on the two dimensional conceptual porous media [48, 49]. However, the characteristic length has not been calculated in the realistic subsurface porous media. With the recent advances in imaging techniques, detailed pore structure can be captured based on digital images, for example SEM and FIB-SEM. In this study we propose an algorithm to accurately calculate the characteristic length based on a digital image and Knudsen number for three-dimensional porous media to determine gas flow regimes and flow models. Three-dimensional shale digital cores in an anonymous shale formation in Sichuan Basin are first reconstructed by the MCMC algorithm based on two-dimensional SEM images. Then a voxel-based method is proposed to calculate the characteristic length of the shale digital core. The Knudsen number for the three-dimensional shale digital core is calculated by the ratio of the mean free path to the characteristic length and is used to judge the flow regimes under different reservoir conditions. Because adsorption only exists in organic pores, three-dimensional inorganic pores and three-dimensional organic pores are reconstructed respectively by the MCMC algorithm based on two-dimensional SEM images. Characteristic lengths of three-dimensional inorganic pores and three-dimensional organic pores are both calculated and free gas flow regimes in organic pores and inorganic pores are judged. Shale gas permeability at different reservoir conditions is also analyzed.

2 MCMC digital core reconstruction method

A Markov chain can be used for describing systems that follow a chain of linked events. Subsequent events depend only on the current state of the system. Suppose X(t) is a known random process describing the state of a process at time t₀. If the state of random process X(t) in the future time t(t > t₀) only depends on the state at time t₀ and is uncorrelated with the state before time t₀, this random process X(t) is called a Markov process. During a Markov process, the relationship between “PAST” and “FUTURE” can be described as “FUTURE” linked to the “PAST” through “NOW”, and if “NOW” is confirmed, “FUTURE” is irrelevant to the “PAST”. Suppose a system is composed of n voxels (i = 1,…,n), and X = X₁,…,X_n indicates the state of the voxels. Therefore, X_i means that site i has a state x_i. Specifically, for a certain voxel s, Λ_s represents all of the other points except s. The neighbor of voxel s, N_s must exist and can be described as:

p(χs|χ(Λ−s))≈p(χs|χ(Ns))(1)

Assume V_LMN = {(l,m,n):0<l≤ L,0<m≤ M,0<n≤ N} represents L rows, M columns, and N layers of a rectangular grid filled with cube voxels. (i,j,k) represents the intersected voxel of row i, column j and layer k, and its associated state is expressed as X_ijk. V_ijk represents a rectangular parallelepiped array of voxels, and its associated state is expressed as vector X(V_ijk). Any (i,j,k)∈ V_LMN has the joint probability function:

p(x(Vijk))=∏l=2i∏m=2j∏n=2kp(xlmn|x(l−1)mn,xl(m−1)n,xlm(n−1))(2)

The conditional probability of each voxel for its random Markov field is:

p(xijk|{xlmn(l,m,n)≠(i,j,k)})=p(xijk|{xlmn(l,m,n)∈N(ijk)}(3)

In the absence of three-dimensional information, three mutually perpendicular independent two-dimensional images can be used to reconstruct a three-dimensional chain. The core slice data from three mutually perpendicular planes xy, yz and xz are extracted and the corresponding binary images are obtained. The double voxel method is applied to simultaneously produce two new voxels, i.e. voxel (i, j, k) and (i, j+1, k), during the modeling process. Therefore, the reconstruction steps for the three-dimensional digital core are as follows. The porosity of the horizontal core slices is used as the conditional probability and the voxels on the first line of the first layer are simulated along the y-direction. The second voxel of the first line is simulated using the 2-neighborhood model, and after that, the 3-neighborhood model is used at the beginning of the simulation of the third voxel. Its conditional probability is calculated by the two-dimensional core slice on the xy plane (Figure 3(b)). The first layer is simulated along the x direction line by line. For double elements (i,j) and (i,j+1), 3 and 4 neighborhoods are used when simulating the edge voxels, and 5 and 6 neighborhoods are used when simulating the internal voxels. The conditional probability is derived by the two-dimensional core slice on the xy plane (Figure 3(c)). Each layer of voxels is simulated along the Z-direction and the three-dimensional model is reconstructed (Figure 3(d)).

Figure 2

Sketch map of a heterogeneous porous media simulation based on three perpendicular thin sections

Figure 3

The three dimensional digital core reconstruction process based on the MCMC method (white elements with lines are the “NOW” simulating voxel; white elements with dotted lines are the “FUTURE” simulating voxel; dark elements are the voxels already produced, while light elements are the neighbors of the voxel currently being simulated, and they both belong to “PAST” voxels)

3 Reconstruction of Sichuan Basin shale digital core

Three-dimensional shale digital cores in Figure 5 and Figure 6 are reconstructed based on binary SEM images (Figure 4) using the above mentioned MCMC algorithm. The blue colour represents pore phase and the red color represents matrix phase. In three-dimensional shale digital cores, pore phase is represented by number 0 and matrix is represented by number 1. Detailed parameters of shale digital cores are given in Table 1.

Figure 4

Binary SEM images of shale samples (white represents matrix while black represents pore phases)

Figure 5

Three dimensional shale digital cores reconstructed by MCMC

Figure 6

Pore phases of three dimensional shale digital cores

Table 1

Parameters of shale digital cores

Core number	Geometrical size	Voxel size	Resolution	Porosity
1	1096nm×1096nm×1096nm	100×100×100	10.96nm	0.23
2	448nm×448nm×448nm	100×100×100	4.48nm	0.15
3	282nm×282nm×282nm	100×100×100	2.82nm	0.12
4	282nm×282nm×282nm	100×100×100	2.82nm	0.16
5	448nm×448nm×448nm	100×100×100	4.48nm	0.07

4 Voxel based characteristic length calculation method

For single tubes, the analysis on gas flow regimes has been well studied [20, 50]. The Knudsen number for a single tube and a three-dimensional porous media can be written respectively as:

Kn=λr(4)

Kn=λlc(5)

λ=kBT2πdm2Pg(6)

l_c is referred to as the characteristic length of porous media and can be calculated based on the chord length [48]. The chord lengths are those discontinuous lines distributed in pore phase when a line is used to cut porous media. The porous media characteristic length indicates the degree of gas diffusion in porous media and its physical meaning is the free path when gas transports within the porous media. The porous media characteristic length has been used to analyze gas transport ability in loose packings of spheres [51, 52] and can be calculated by [48, 49]:

lc=∫0∞zp(z)dz(7)

Chord lengths are the distributions of lengths between intersections of lines with the interface (Figure 7). A chord is then a special line segment with its end points on the interface and all other points in one of the two phases. Here chord lengths in pore phase are calculated. p(z) is the chord length distribution function in pore phases. p(z)dz is the probability of finding a chord of length between z and z+dz in pore phase.

Figure 7

Schematic of chord-length measurements for a cross section of porous media. The chords are defined by intersection of lines with two-phase interface, closed curve area refers to rock matrix [32]

Previous studies have not applied Equation (5) to real rock samples. Here we propose a method to calculate the characteristic length based on digital image. The chord length in the binary digital image can be represented as the length of continuous number 0 in the pore phase. A regional example in two dimensions is shown in Figure 8, the voxel-based chord length is represented by the number of continuous 0 on red line. In three-dimensional pore phase, the first chord length is measured in the bottom slice on the XY plane and then the measuring window moves up in the Z direction layer by layer. Finally, chord length for three-dimensional pore phase can be obtained and Equation (8) is used to calculate the characteristic length of the three-dimensional pore phase. A validation of the proposed voxel based characteristic length calculation method can be found in the next section.

lc_3D=∫0∞z_volp(z_vol)dz_vol×γ(8)

Figure 8

Illustration of chord length measurements in XY cross section

Gas adsorption is generally assumed to be a monolayer adsorption [53] and can be characterized with the Langmuir isotherm. Considering the influence of an adsorption layer, first adsorbed gas coverage on the pore wall is calculated by Equation (9) and the effective characteristic length of the three-dimensional pore phase is given in Equation (10).

θ=Pg/PL1+Pg/PL(9)

lc(Pg)=lc_3D−dmθ(10)

According to Equation (5), Equation (6), Equation (9) and Equation (10), the Knudsen number without adsorption and with adsorption can be given respectively as:

Kn_3D=kBT2πdm2Pg∫0∞z_volp(z_vol)dz_vol×γ(11)

Kn_3D_ads=kBT2πdm2Pg(∫0∞z_volp(z_vol)dz_vol−dmPg/PL1+Pg/PL)×γ(12)

5 Validation of the proposed voxel based characteristic length calculation method

The three-dimensional microscale lattice Boltzmann (LB) model with the D3Q19 discrete velocity model developed in our previous study [54] is adopted to simulate gas flow in a shale digital core. The developed LB model can be applied to simulate gas flow in slip flow and transition flow regimes. The basic evolution equation with the Bhatnagar-GrossKrook (BGK) collision approximation is shown as follows:

fα(rs+eαδt,t+δt)−fα(rs,t)=−1τ(fα−fαeq)+δtFα(13)

Where f_α is the density distribution function of α direction; α = 0, 1, 2, … , 18; r_s is the spatial location of the particles; e_α is the velocity of α direction; t is time; δ_t is time step; τ is the relaxation time; F_α is the force term; fαeq is the local equilibrium distribution function of α direction:

fαeq=wαρ[1+eαucs2+(eαu)22cs4−u22cs2](14)

Where c_s is the lattice sound speed; w_α is the weight factor of α direction. The force term in Equation (13) can be obtained by Hermite expansion. For the D3Q19 model, it can be expressed as [55]:

Fα=wαρ[eαacs2+au:(eαeα−cs2I)cs4](15)

Where a is the acceleration of the force. For gas flow in shale nanopores, the relaxation time should be determined by the Knudsen number. Considering the microscale effect and the effect of the Knudsen layer, the relaxation time can be expressed as [56]:

τe=12+6πKnψ(Kn)N(16)

ψ(Kn) is the modification function and equals 1/(1+2Kn); N is the number of lattices occupied by the characteristic length. As the solid boundaries in shale are rough, the diffuse reflection boundary condition is very appropriate for such boundaries. The discrete format of the diffuse reflection boundary condition is expressed as [57]:

fα=∑ξα′⋅n<0|ξα′⋅n|fα′∑ξα′⋅n>0|ξα′⋅n|fαeq(ρw,μw)fαeq(ρw,μw),(eα−uw)⋅n>0(17)

Where n is the inward unit normal vector; u_w is the wall velocity; the subscript w represents the solid walls; fα′ is the distribution function after streaming. Gas flux Q is calculated in the outlets and the Darcy equation is applied to calculate gas permeability:

k1=QμLAΔP(18)

Detailed parameters are given in Table 2. The studied shale digital core is shown in Figure 10 and pressure drop distribution on the shale digital core is shown in Figure 11. The calculated permeability k₁ is 2.7831×10⁻⁷ μm².

Figure 9

Illustration of chord length measurements in three orthogonal X,Y,Z directions

Figure 10

Shale digital core (blue represents pore phase, red represents matrix phase)

Figure 11

Pressure drop distribution on the shale digital core

Table 2

Parameters of shale digital core for model validation

Properties	Value
Pressure P_g/MPa	30
Temperature/K	358
Voxel size	100×100×100
Porosity	0.14
Tortuosity	2.2
Resolution γ/m	2×10⁻⁹
Pressure gradient/MPa/m	0.1
Pressure drop on the shale digital core/Pa	0.02

The Civan et al. model [18] in Equation (19) is applied to calculate gas permeability in slip flow regime and transition flow regime. The characteristic length is calculated based on the proposed voxel based method in Equation (8) and the Knudsen number is calculated using Equation (11).

k2=ϕrc28τf(Kn)=ϕlc232τf(Kn)(19)

The flow condition function f(K_n) is given by [22]:

f(Kn)=(1+αKn)(1+4Kn1−βKn)(20)

The parameter α in Equation (20) is a dimensionless rarefaction coefficient and can be written as:

α=12815π2tan−1[4.0Kn0.4](21)

Though the slip coefficient β = −1 is initially considered only to be applicable to slip flow condition, evidence from the DSMC simulations and Boltzmann solutions [58] showed that β = −1 is valid within the full range of flow regimes. The calculated characteristic length in Equation (8) is 11.72nm and the calculated permeability k₂ in Equation (19) is 2.9784×10⁻⁷ μm². k₂ is very close to k₁ which suggests the proposed characteristic length calculation algorithm is correct.

6 Gas transport mechanism judgement and permeability analysis

6.1 Gas flow regimes judgement of Sichuan Basin shale

Firstly characteristic lengths are calculated by Equation (8) based on three-dimensional pore phase of shale digital cores shown in Figure 6. From the calculated results listed in Table 3, the average characteristic length for Sichuan Basin shale is 14.87 nm. Subsequently, the average characteristic length is used in Equation (11) and Equation (12) to study the change in the Knudsen number under different formation pressure and temperature without adsorption effect and with adsorption effect. The relative degree of deviation is defined to study the free gas transport differences in the above two cases:

dev=|Kn_3D−Kn_3D_ads|Kn_3D×100%(22)

Table 3

Characteristic lengths of anonymous shale formation, Sichuan Basin

Core number	Characteristic lengths /nm
1	25.2127
2	11.9197
3	10.5176
4	11.2995
5	15.3914
Average	14.87

The Knudsen number values in Figure 12, Figure 13 are compared with the Knudsen number values in Figure 1. It is can be seen that shale gas flow regimes mainly locate at the slip flow and transition flow region. The value of the relative degree of deviation is larger in high pore pressure than that in low pore pressure. However, the maximum value of the relative degree of deviation is 5%. Therefore, the adsorption effect has no obvious influence on the free gas transport mechanism in the Sichuan Basin shale gas reservoir.

Figure 12

Knudsen number versus temperature and pressure with adsorption effects

Figure 13

Knudsen number versus temperature and pressure without adsorption effects

6.2 Gas flow regimes judgement based on three dimensional organic pores system and inorganic pores system

Based on the aforementioned MCMC method, SEM images (Figure 15) in Haynesville shale [59] are used to reconstruct three-dimensional organic pores and inorganic pores. First we identify organic pores and inorganic pores by the pixel value of red and green colour. Then, the corresponding binary images can be obtained using the maximum class separation distance method proposed by Otsu [60]. Taking a pixel as the basic unit, the pixel value is set to 1 when the pixel is located in the matrix phase and the pixel value is set to 0 when the pixel is located in the pore phase (Figure 16, Figure 17). Finally, the MCMC method is applied to reconstruct three-dimensional organic pores and inorganic pores based on binary images (Figure 18). The voxel sizes of the three-dimensional organic pores and inorganic pores are 800 × 800 × 800.

Figure 14

Relative degree of deviation comparing the above two results

Figure 15

American Haynesville Shale SEM Image

Figure 16

Binary image of inorganic pore in Haynesville shale rocks (white is the rock matrix, and black is the pore space)

Figure 17

Binary image of organic pore in Haynesville shale rocks (white is the rock matrix, and black is the pore space)

Figure 18

(a) Three dimensional inorganic pores (b) Three dimensional organic pores

Based on the statement that gas transports in the forms of free gas and adsorbed gas in organic pores and gas transports only in the form of free gas in inorganic pores, Equation (8) is applied to calculate the characteristic length without adsorption effect in three-dimensional inorganic pores and organic pores. Our calculation results indicate characteristic lengths without adsorption effect in three-dimensional inorganic pores and organic pores are 49.5 nm and 35 nm respectively. Then, the value of the characteristic length without adsorption effect in three-dimensional organic pores of 35 nm is revised by Equation (10) to consider the influence of the adsorption layer. Finally, Equation (11) and Equation (12) are used respectively to calculate the Knudsen number in three-dimensional inorganic pores and organic pores. In Figure 19 and Figure 20, the Knudsen number differs a little in low pore pressure (< 5 MPa). But for shale gas reservoirs, pore pressure usually ranges from 10 MPa to 40 MPa [61] and the Knudsen number is almost the same in three-dimensional organic pores and inorganic pores under this condition. Therefore, free gas flow regimes can be deemed as the same in inorganic pores and organic pores. This may indicate that in macro scale gas flow numerical simulation, inorganic pores and organic pores can be viewed as a continuous system and a dual medium (matrix-micro fracture) mathematical model is suitable to simulate the shale gas production process.

Figure 19

Knudsen number versus temperature and pressure in three dimensional organic pores

Figure 20

Knudsen number versus temperature and pressure in three dimensional inorganic pores

6.3 Shale gas permeability at different reservoir conditions

According to Equation (10), Equation (12) and Equation (19), free gas permeability in consideration of adsorption can be given as:

kfree=ϕlc2(Pg)32τf(Kn_3D_ads)(23)

Surface diffusion takes place within the adsorbed gas to enhance the transport of gas molecules along molecular concentration gradients [62]. Based on Hwang and Kammermeyer’s model [63], combined with methane adsorption experimental data, the surface diffusion coefficient for methane D_s0 when gas coverage is zero can be expressed as:

Ds0=8.29×10−7T0.5exp⁡(−ΔH0.8RT)(24)

The isosteric adsorption heat ΔH is a function of gas coverage. According to Equation (25), the isosteric adsorption heat and gas coverage have a linear relationship and can be given as [64]:

ΔH=ηθ+ΔH(0)(25)

The surface diffusion coefficient in Equation (24) is obtained under a low pressure condition by theory and experiments, and is a function of gas molecular weight, temperature, and gas activation energy, isosteric adsorption heat and is independent of pressure [63]. In order to describe the gas surface diffusion in nanopores of shale gas reservoirs under a high pressure condition, the influence of gas coverage on surface diffusion is considered. Chen et al. [65] used the kinetic method to calculate the surface diffusion coefficient:

Ds=Ds0(1−θ)+κ2θ(2−θ)+{H(1−κ)}(1−κ)κ2θ2(1−θ+κ2θ)2(26)

H(1−κ)=0,κ≥1;1,0≤κ≤1(27)

Where H(1-κ) is Heaviside function. The surface diffusion coefficient at different pressure and temperature conditions is calculated according to Equation (26). Surface diffusion of adsorbed gas molecules can be modelled as the general diffusion process, using the molar flow rate per unit area of the concentration gradient within the adsorbed monolayer as developed in [62]:

Ja=DsdCadx(28)

C_a is calculated assuming Langmuir adsorption and is given by:

Ca=Camaxθ(29)

C_amax can be expressed as [66]:

Camax=Cmaxεks(30)

Combining Equation (28) and Equation (29), molar flow rate in the adsorbed layer is then expressed below:

JA=DsCamaxdθdpπ(lc2−lc2(Pg))dpdx(31)

From Equation (31), volumetric flow rate V_A is:

VA=MρDsCamaxdθdpπ(lc2−lc2(Pg))dpdx(32)

According to Equation (26) and the Darcy law, adsorbed gas permeability can be written as:

ksurface=ϕμMτρDsCamaxdθdp(1−(lc(Pg)lc)2)(33)

Shale gas permeability in Equation (34) can be derived by combining free gas permeability in consideration of adsorption by Equation (23) with surface diffusion of adsorbed gas by Equation (33). Typical shale gas reservoir parameters are given in Table 4. Figure 21 shows that gas permeability increases with the decrease of pressure and the increase of temperature. The gas permeability value ranges from 4.3685 × 10⁻⁷ μm² to 2.2451 × 10⁻⁶ μm² for different reservoir conditions.

kt=ϕlc2(Pg)32τf(Kn_3D_ads)+ϕμMτρDsCamaxdθdp(1−(lc(Pg)lc)2)(34)

Figure 21

Shale gas permeability versus temperature and pressure

Table 4

Typical shale gas reservoir parameters

Properties	Value
Porosity	0.1
Tortuosity	2.3
Characteristic length l_c /m	1.487×10⁻⁸
Total organic grain volume per total grain volume ϵ_ks	0.01 [66]
Langmuir pressure p_L /MPa	13.789514 [66]
Maximum adsorbed gas concentration C_max /mol/m³	328.7 [66]
Isosteric adsorption heat at zero gas coverage Δ H(0) /J/mol	16000 [67]
Fitting coefficients of isosteric adsorption heat γ /J/mol	−4186 [67]
Ideal gas constant R /J/(mol⋅k)	8.314
Ratio of the rate constant for blockage to the rate constant for forward migration κ, dimensionless	0.5 [67]
Molecular weight M /kg/mol	0.016

7 Conclusions

A method to judge shale gas flow regimes based on digital core analysis is studied. A Voxel-based method is proposed to the calculate characteristic length of a three-dimensional shale digital core. The Knudsen number for MCMC reconstructed three-dimensional shale digital core is calculated by the ratio of the molecular mean free path to the characteristic length and is used to judge the gas flow regime under different reservoir conditions. The results indicate that shale gas flow regimes mainly locate at the slip flow and transition flow region. Furthermore, adsorption has no obvious influence on the free gas flow regimes. Moreover, free gas flow regimes in organic pores and inorganic pores are judged. Our analysis results show that free gas flow regimes can be deemed as the same in two different pore systems and inorganic-organic pores can be viewed as the continuous system in a macro scale gas flow simulation. Gas permeability increases with the decrease of pressure and the increase of temperature.

Acknowledgement

This project was supported by the National Natural Science Foundation of China (No. 51504276, No.51490654), the Fundamental Research Funds for the Central Universities (No.16CX05018A, No. 18CX06008A), the Major Projects of the National Science and Technology (2016ZX05061), Graduate School Innovation Program of China University of Petroleum (YCX2017019).

Conflict of Interest
Conflict of Interests: The authors declare no conflict of interest.

Nomenclature

ϵ_ks: total organic grain volume per total grain volume, dimensionless
κ: ratio of the rate constant for blockage to the rate constant for forward migration
ΔH(0): isosteric adsorption heat at zero gas coverage (J/mol)
η: fitting coefficients of isosteric adsorption heat (J/mol)
γ: resolution(m)
λ: mean free path of molecules(m)
μ: viscosity(Pa⋅s)
Φ: porosity, dimensionless
ρ: gas density (kg/m³)
τ: tortuosity, dimensionless
θ: gas coverage on the pore wall, dimensionless
A: cross section area of shale digital core (m²)
C_amax: maximum adsorbed gas concentration inside the total core sample (mol/m³)
C_a: adsorbed gas concentration (mol/m³)
C_max: maximum adsorbed gas concentration inside the organic matter (mol/m³)
d_m: gas molecular diameter (m) 4 × 10⁻¹⁰
D_s: surface diffusion coefficient (m²/s)
D_s0: surface diffusion coefficient when gas coverage is zero (m²/s)
dev: relative degree of deviation, dimensionless
J_a: molar flow rate per unit area (mol/(m²⋅s))
k_B: Boltzmann constant (J/K),1.3805 × 10⁻²³
K_n: Knudsen number, dimensionless
K_{n_3D}: Knudsen number without adsorption effect
K_{n_3D_ads}: Knudsen number with adsorption effect
k: permeability (μm²)
l_c: characteristic length (m)
l_{c_3D}: characteristic length of three dimensional pore phases (m)
l_c(P_g): characteristic length of three dimensional pore phases with adsorption effect(m)
M_g: gas molecular weight (g/mol)
N(ijk): neighbor of (i,j,k)
n: voxel number
P_g: pore pressure(MPa)
p_L: Langmuir pressure(MPa)
p(z): chord length distribution function in pore phase, dimensionless
Q: gas flux (m³/s)
r: single tube radius (m)
s: certain voxel, dimensionless
t: time(s)
v_g: gas velocity (m/s)
V_ijk: rectangular parallelepiped array of voxels
V_LMN: rectangular grid filled with cube voxels, dimensionless
X(t): Markov process
z_{_vol}: chord length based on voxel unit in three dimensional digital image
z: chord length (m)

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Received: 2017-12-19

Accepted: 2018-06-12

Published Online: 2018-08-13

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/phys-2018-0062

Keywords for this article

shale gas; digital core; flow regimes; characteristic length; nanoporous media

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