Experimental investigation on the characteristics of gas diffusion in shale gas reservoir using porosity and permeability of nanopore scale

Journal of Petroleum Science and Engineering 133 (2015) 226–237

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Experimental investigation on the characteristics of gas diffusion in shale gas reservoir using porosity and permeability of nanopore scale Changjae Kim, Hochang Jang, Jeonghwan Lee n Department of Energy and Resources Engineering, College of Engineering, Chonnam National University, 77 Yongbong-ro, Buk-gu, Gwangju 500-757, Republic of Korea

art ic l e i nf o

a b s t r a c t

Article history: Received 2 September 2014 Received in revised form 20 March 2015 Accepted 10 June 2015 Available online 19 June 2015

The aim of this study is to investigate the diffusion characteristics of nanoscale gas flow in a shale gas reservoir. An experimental apparatus was designed and set up to measure porosity and permeability at the nanopore scale. The measured properties have been used to determine a diffusion coefficient by classification standard of gas flow regime. To investigate the impact of pressure and pore size, the analysis of diffusion flow was conducted using Knudsen and Fick diffusion coefficients. From the results, it was revealed that Knudsen diffusion coefficient gradually increased with the growth of pressure and pore radius and Fick diffusion coefficient was dependent on the gas molecular diameter and temperature. It was also found that Knudsen diffusion coefficient was equal to Fick diffusion coefficient if pore radius is too small. The study implied that the characteristics of gas diffusion should be implemented by using the diffusion coefficient theory based on gas flow regime in shale gas reservoirs. & 2015 Elsevier B.V. All rights reserved.

Keywords: Shale gas Nanopore Flow regime Knudsen diffusion Fick diffusion

1. Introduction Shale gas boom in the United States has opened a new frontier for the world energy market. Shale gas is natural gas trapped within shale formations underground. Gas production from shale formation is one of the main fields focused on the global natural gas development industry (Rutqvist et al., 2013; Zhao et al., 2013). In general, conventional analysis models have used to measure the fine grained, organic rich shale properties. However, the methods yield erroneous data in characterizing the pore system of shale formation. Cui et al. (2009) described that the conventional methods are not practical because it is time consuming and the equipment performance is too inappropriate to measure small pressure changes or flow rates in tight rocks such as shale or coal. In order to measure the shale matrix permeability, Luffel et al. (1993) developed GRI method using crushed samples. Based on the study, crushing samples nearly destroys microfractures in shale so it is effective to measure the shale permeability in nanopores. Luffel et al. (1993) validated the test results of GRI method by comparing with simulation model and it presented that the pressure transient corresponded to simulation results practically. Cui et al. (2009) introduced the method to obtain permeability and diffusivity from crushed shales by considering adsorption. They examined analytically the relative error of n

Corresponding author. Fax: þ82 62 530 1729. E-mail address: [email protected] (J. Lee).

http://dx.doi.org/10.1016/j.petrol.2015.06.008 0920-4105/& 2015 Elsevier B.V. All rights reserved.

method on pressure difference and it was found that pressure change in the experiment causes a significant error for the property measurement. Tinni et al. (2012) examined which experimental factors affect the crushed samples permeability. In the experiment, particle size, pore pressure, gas species, and initial pressure condition were changed to analyze the property variation of crushed samples. From the results, Tinni et al. (2012) proposed that the pore pressure and particle size have a significant effect for the permeability measurement. Darcy equation, used to investigate gas flow in rocks with micropores, cannot be used for shale matrix due to nanopores in shale (Cui et al., 2009; Swami et al., 2012; Javadpour et al., 2007). Roy et al. (2003) designed 2-D model to investigate gas flow in nanopores. The model was based on Knudsen diffusion and Navier–Stoke equation. Roy et al. (2003) also conducted the validation by comparing the model results with that of membrane experiment and it has an error within 5%. Based on the previous study, Javadpour et al. (2007) established a mathematical model which illustrates shale gas diffusion. The model was designed by using Gaussian distribution function. Javadpour et al. (2007) also excludes the viscous effect to consider gas diffusion in shale matrix. The diffusion model was validated by gas desorption data and it presented that the model corresponded with the experiment data practically. After that, Freeman et al. (2011) conducted the analysis of gas flow using DGM (dusty-gas model). The DGM was used to investigate the effects of Knudsen diffusion on gas composition in shale gas reservoir systems. From the literature survey, it was revealed that the previous studies did not consider the type

C. Kim et al. / Journal of Petroleum Science and Engineering 133 (2015) 226–237

Nomenclature

cg Dk Dfick Dtransition FR f0 J Ja JD K Ka

Kc Kn k kb M Mg p p0 pr0 pe ∇p R

gas compressibility, Pa  1 Knudsen diffusion coefficient, cm2/s Fick diffusion coefficient, cm2/s transition diffusion coefficient, cm2/s gas residual ratio of remaining gas over total gas to be taken up by sample or 1  FU (FU , gas uptake ratio) the y-intercept of the straight line total mass flux, kg/s/m2 mas flux of gas diffusion, kg/s/m2 mas flux of gas flow, kg/s/m2 apparent transport coefficient, m2/s partial derivative of adsorbate density with respect to gas density gas capacity ratio of void volume of cells over total pore volume of sample Knudsen number permeability, m2 Boltzmann constant, 1.38  10–23 J/K sample mass, g gas molar mass, kg/kmol pressure, MPa initial pressure in sample cell, MPa initial pressure in reference cell, MPa equilibrium pressure in reference and sample cells, MPa pressure gradient, MPa gas constant, J/mol/K

of diffusion. The studies have just researched the gas flow using Knudsen diffusion or Fick diffusion model (Javadpour et al., 2007; Shabro et al., 2009; Sigal and Qin, 2008). From the results, Wang et al. (2013) conducted the investigation of gas flow ability in the shale by considering Knudsen and Fick diffusion. Wang et al. (2013) suggested that the determination of flow regimes should be implemented by Kn primarily, and then taken the diffusion coefficient of system from the flow regime. In order to examine gas diffusion in shale matrix it is essential to determine the shale properties at nanopore scale. However, the previous studies used the core experiment such as pressure pulse decay (PPD) method to determine the low permeability in the shale gas reservoirs. It cannot be the representative of nanopores in shale matrix due to the microfracture in core samples. Thus, the permeability obtained from the PPD method is not valid for the characterization of gas diffusion in nanopores of shale matrix (Kamath et al., 1992; Luffel et al., 1993). This study presents a new apparatus to measure the porosity and permeability of shale matrix at nanopore scale. Knudsen and Fick diffusion model are determined by the measured properties and the diffusion coefficient is used to analyze gas diffusion in the shale matrix. Finally, the impact of the pore radius and pressure are investigated to verify the gas flow on reservoir conditions.

Ra r rA rpore s1

T t Vb Vc Vr Vs z z0 zr0 ze λ μ μB αn ρ ρ0 ρb ρc0

ρr0 ϕ

radius of crushed samples, mm radial distance from the center of spherical sample, cm radius of gas molecular, cm pore radius, nm slope of the straight line part of ln(FR ) versus time at late-time temperature, K time, s bulk volume of samples (including pore space), cm3 total volume of open space in reference and sample cells (Vr þ Vs  Vb ), cm3 reference cell volume, cm3 sample cell volume, cm3 gas compressibility factor (¼ 1.0 for ideal gas) gas compressibility factor in sample cell gas compressibility factor in reference cell gas compressibility factor in reference and sample cells gas phase molecular mean free path, nm viscosity, Pa/s viscosity of fluid in the pore of shale, Pa/s the nth roots of the transcendental Eq. (8) gas density, mol /m3 initial gas density in sample cell, mol/m3 sample bulk density, g/cm3 average initial gas density in void volume of reference and sample cells, mol /m3 initial gas density in reference cell, mol/m3 porosity of porous media, fraction

that the method is practical to measure the shale permeability at nanoscale. GRI technique only used helium, a non-adsorbate gas, to measure the permeability. However, methane which is major component of shale gas has the adsorption effect. It means that the measurement by methane results in erroneous permeability and needs to be corrected for adsorption in the measurement. To solve this problem, Cui et al. (2009) revised the technique to correct the impact of gas adsorption on permeability measurements. Fig. 1 presents a schematic sketch of a pycnometer apparatus to apply the corrected GRI technique. This study proposed early and late time method for determining permeability. The early time method can only calculate the permeability by using the initial pressure data. However, the data quality may be poor due to the kinetic gas expansion from the reference cell into the sample cell. The expansion causes the temperature change in the pycnometer system,

2. Theoretical background 2.1. Permeability measurement of crushed sample GRI technique was introduced by Luffel et al. (1993) to measure the permeability of shale using crushed sample. Luffel et al. (1993) described that this method is rapid to run and crushed samples rarely have micro-fractures which affect the permeability. It means

227

Fig. 1. A schematic sketch of a pycnometer apparatus (Cui et al., 2009).

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Table 1 Flow regime based on Knudsen number (Schaaf and Chambre, 1961).

Table 2 Reclassification of flow regimes (Wang et al., 2013). Knudsen number (Kn ) range Flow regime

Knudsen number (Kn ) range

Flow regime

Kn r 0.001 0.001o Kn o 0.1 0.1o Kn o 10 Kn Z 10

Viscous flow Slip flow Transition flow Knudsen flow

Kn o 0.001 0.001 o Kn o0.01 0.01o Kn o 0.1 0.1o Kn o 10 Kn 410

which affects the pressure variation. Therefore, the permeability obtained from the early-time method should be compromised. That is the reason why the late-time method is recommended. The late time method provides the permeability based on theoretical analyses, which gives the validity by analytical solution. Eq. 1 describes the change of gas density in the spherical shaped crushed samples (Cui et al., 2009); where μ , cg , Ka , r , ϕ , k indicate the gas viscosity, gas compressibility, partial derivative of adsorbate density with respect to gas density, radial distance from the center of spherical sample, porosity and permeability, respectively. Ka is specifically defined by the relationship among the Langmuir isotherm, skeleton density, and molar volume of gas at standard condition. In this study, however, helium gas was used for the permeability measurement. It means that there are no considerations for the adsorption effect in our experiment.

∂ρ k 1 ∂ ⎛⎜ 2 ∂ρ ⎞⎟ = r ∂t μcg[ϕ + (1 − ϕ)Ka] r 2 ∂r ⎝ ∂r ⎠

(1)

The change of gas density is expressed as the gas residual ratio (FR ) which describes the remaining gas over the total gas to be taken up by sample, as shown in Eq. (2); where, Ka is partial derivative of adsorbate density with respect to gas density, ρ is gas density, ρ0 is initial gas density in sample cell and ρc0 is average initial gas density in void volume of reference and sample cells.

Darcy flow Slippage flow Fick diffusion Transition diffusion Knudsen diffusion

Table 3 Diffusion coefficients based on Knudsen number (Wang et al., 2013). Knudsen number (Kn ) range

Diffusion model

Kn o 0.1 0.1 o Kn o10 Kn 4 10

Fick diffusion coefficient Transition diffusion coefficient Knudsen diffusion coefficient

FR = 1 −

(Kc + 1)(ρc 0 − ρ) ρc 0 − ρ0

(2)

In the equation, the total void volume of the reference cell and sample cells must be much larger than the total gas-storage capacity of the sample particles (i.e.Kc ≫ 50). With experimental pressure data, FR can be calculated by using Eqs. ((3)–5); where, p is pressure, z is gas compressibility factor, R is gas constant, T is temperature, ρb is sample bulk density, Vc is total volume of open space in reference and sample cells (Vr þ Vs  Vb ), M is sample mass, ρ0 is initial gas density in sample cell, Vb is bulk volume of samples (including pore space), Vr is reference cell volume, Vs is sample cell volume respectively.

ρ=

p zRT

Fig. 2. Range of flow regimes in shale and tight gas reservoir (Xiao and Wei, 1990).

(3)

C. Kim et al. / Journal of Petroleum Science and Engineering 133 (2015) 226–237

229

Fig. 3. Schematic diagram of experimental apparatus.

Fig. 4. Experiment procedures to measure porosity and permeability.

ρb Vc

Kc =

M[ϕ + (1 − ϕ)Ka]

ρc 0 =

(4)

ρr 0 Vr + ρ0 (Vs − Vb) Vr + Vs − Vb

(5)

The analytical results obtained from Eq. (2) are converted to semi-log scale. And then, the logarithmical value of FR is approximated by Eq. (6); where s1 is the slope of the straight line, α1 and f0 are the first solution of Eq. (8) and the y-intercept of the straight line, respectively.

ln(FR ) = f0 − s1t s1 =

(6)

Kα12 Ra2

tan α =

⎛ 6 K (K + 1 ) ⎞ ⎟⎟ f0 = ln⎜⎜ 2 2 c c ⎝ Kc α1 + 9(Kc + 1) ⎠

(8)

(9)

Then, the permeability of crushed sample can be obtained by substituting the properties of gas and samples, s1 and α1 as shown in Eq. (10). In Eq. (10), Ka is the adsorbate density with respect to the gas density and it can be expressed as Eq. (11). Eq. (12) is then used to determine the porosity from the equilibrium pressure data.

k= (7)

3α 3 + Kcα 2

Ra2[ϕ + (1 − ϕ)Ka]μcgs1 α12

(10)

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Fig. 5. Shale gas basins map (Ziff Energy, 2012).

∂q ∂ρ

Ka =

ϕ=

⎡ ⎣⎢Vr

(

(11) pr 0 zr0

−

pe ze

) − (V − V )( V( − ) s

pe b z e

b

p0 z0

p0 z0

−

pe ze

)⎤⎦⎥ (12)

2.2. Knudsen diffusion The Darcy equation has been used in reservoir researches because of its simplicity. However, the gas flow in nanopores is different from the Darcy flow. Javadpour (2009) presented the gas flow in shale systems based on Knudsen diffusion. Knudsen diffusion is a means of diffusion that appears when the pore radius of matrix is equivalent to or smaller than the mean free path (λ ) of the particles. The Knudsen diffusion is described by Kn which defined as the ratio of molecular mean free path to a pore radius (rpore ) as shown in Eq. 13. λ and rpore can be expressed as Eqs. ((14) and 15) in which Mg is gas molar mass (Freeman et al., 2011).

Kn =

λ rpore

(13)

λ=

π 1 RT μ 2 p Mg

(14)

rpore = 2.81709

Fig. 6. Crushed shale samples to measure the rock properties. (a) Mancos shale and (b) Eagle Ford Shale.

k ϕ

(15)

Table 1 shows the classification of flow regimes based on Kn. Note that shale and tight gas reservoirs fall in slip and transition flow (Fig. 2). Knudsen diffusion coefficient (Dk ) is obtained from Eq. (16), after the determination of flow regime. Then, the diffusion coefficient of system is used to describe the mass flux of gas

C. Kim et al. / Journal of Petroleum Science and Engineering 133 (2015) 226–237

231

Fig. 7. Transition of FR and FU in Mancos shale. (a) Mancos shale-1, (b) Mancos shale-2 and (c) Mancos shale-3.

diffusion ( JD ) and finally obtain the total mass flux ( J ) by Eqs. ((17) and 18); where ∇p is pressure gradient and Ja is the mass flux of the gas flow. Javadpour (2009) defined the equations as the motion equations for mass transfer of gas in nanopores.

Dk =

JD =

2rpore 8RT 3 πM

MgDk 103RT

be likely to occur in 0.01o Kn o 0.1. The Fick diffusion coefficient (Dfick ) should be use the viscosity of saturated fluid in porous medium (Sigal and Qin, 2008). The coefficient is defined as Eq. (19) in which kb , μB and rA are Boltzmann constant, viscosity of fluid in the pore of shale, and radius of gas molecular respectively.

(16)

Dfick = ∇p

J = Ja + JD

(17) (18)

2.3. Fick diffusion Fick diffusion describes the diffusion which occurred by the concentration change with time. Besides Knudsen diffusion, Fick diffusion also characterizes the flow ability and two parameters are utilized to analyze the influence of diffusion to gas flow at nanoscale (Wang et al., 2013). Wang et al. (2013) classified flow regimes as shown in Table 2 and showed that Fick diffusion would

kbT 6πμB rA

(19)

JD can be also defined as Eq. (20) by substituting Dk into Dfick . Eq. (21) presents the transition diffusion coefficient which consists of Knudsen and Fick diffusion coefficient. The coefficient is defined by using Bosanquit equation. Table 3 shows the standards for the determination of diffusion coefficient based on Kn (Wang et al., 2013). From the table, it is implicit that acquiring the diffusion coefficient is needed to select the diffusion system after determining Kn .

JD =

MDfick 103RT

∇p

(20)

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Fig. 8. Transition of FR and FU in Eagle Ford shale. (a) Eagle Ford shale-1, (b) Eagle Ford shale-2 and (c) Eagle Ford shale-3.

(

−1

)

1 Dtransition = D−fick + Dk−1

(21)

3. Nanoscale gas flow in shale matrix 3.1. Experimental apparatus and procedures The schematic diagram of experimental apparatus is shown in Fig. 3. The experimental apparatus was designed to obtain the porosity and permeability which used to investigate the gas diffusion in shale matrix. The system of apparatus consists of sample cell, reference cell, and data acquisition system. The sample cell is divided into sample cell cap and body. A temperature transmitter is attached to the top of the cap. It is useful to measure the system temperature continuously during gas flow through sample particles. The O-ring with cell body can prevent gas leakage from sample cell. The structure of reference cell is similar to sample cell, except that the cap is connected with fitting and pressure transducer. Additionally, the variable volume kit was designed to

measure the properties for a small amount of samples. Suarez-Rivera et al. (2012) have shown that crushed samples with a diameter range between 0.84 mm and 1.68 mm provide the repeatability of experimental results. Based on the research, the optimum size of fragments was used to measure the porosity and permeability. Fig. 4 presents the experimental procedure by the apparatus. To conduct the experiment, first of all core samples were crushed into small fragments and then put into the sample cell. The next stage is to inject the helium gas into the reference cell. Tinni et al. (2012) compared the results for any different pressure condition and gas species, in order to examine the effect of initial state for the GRI method. The results showed that the permeability variation was not varied above equilibrium pressure of 200 psi. However, the effect of gas species showed that the result using helium still represented the difference around 250 psi. Based on the research, we have conducted the extrapolation for Tinni et al.'s (2012) data to determine the appropriate equilibrium point and it was revealed that the permeability variation was not significant at 350 psi. From the results, to minimize the variation, we set the charging pressure to 600 psi to set the equilibrium pressure above 400 psi. Additionally, the gas viscosity was

C. Kim et al. / Journal of Petroleum Science and Engineering 133 (2015) 226–237

233

Table 5 Porosity and permeability of the shale samples at nanopore scale. Sample

Permeability (m2)

Porosity (%)

Mancos shale-1 Mancos shale-2 Mancos shale-3 Eagle Ford shale-1 Eagle Ford shale-2 Eagle Ford shale-3

2.98  10  19 4.32  10  19 7.37  10  19 6.15  10  19 6.73  10  19 4.53  10  19

0.5 0.6 1.2 0.8 1.1 0.9

were obtained. Tinni et al. (2012) described that the pressure drop in GRI method is not visible and the measurement is too rapid to obtain the data. In order to improve the accuracy of measurement, the sampling rate was set to 0.001 s. After the equilibrium in ALPMA is established, the measured data can be converted to FR and the straight line part of solution is analyzed at semi-log scale. Finally, the porosity and permeability are determined using the experimental data. 3.2. Measurement of porosity and permeability at nanopore scale

Fig. 9. FR transition of Mancos shale at semi-log scale.

Fig. 10. FR transition of Eagle Ford shale at semi-log scale. Table. 4 The values of slope and y-intercept for shale samples. Sample

Slope

Y-intercept

Mancos shale-1 Mancos shale-2 Mancos shale-3 Eagle Ford shale-1 Eagle Ford shale-2 Eagle Ford shale-3

 0.01618911091  0.01586379412  0.01413027485  0.01764379885  0.01425728969  0.01200030163

 0.05128168003  0.10319788270  0.07754056832  0.00534814074  0.02355885121  0.01602559412

calculated by simple correlation (Petersen, 1970). After equilibrium pressure is set in the cell, the valve between the cells is opened. The gas is expanded into the sample cell and the pressure data

The designed apparatus was used to measure the porosity and permeability of Mancos and Eagle Ford shale which is two of the major U.S. basins as shown in Fig. 5. The Mancos shale formation is a sedimentary layer which is located in western part of the United States. It is known feature that oil and gas are coexisted in the area. The gas is reported that it is buried at the northern part of basin. Also, the Eagle ford sedimentary layer in Maverick Basin, Texas is known for the most active shale gas development area (Energy Information Administration (EIA), 2011). The shale samples were crushed and sieved with the optimum particle sizes as shown in Fig. 6. After the samples were dried in the electric furnace at 373.15 K, the test was performed at dry condition where the samples would be dried at high temperature. After that, the experiment by ALPMA was conducted to acquire the pressure transient and equilibrium pressure data. For the equilibrium pressure, it was substituted to Eq. (11) to determine the porosity of shale in U.S. basins. The permeability was calculated by Eq. (10). During the measurement, the pressure transient data was converted to FR . Figs. 7 and 8 show the FR and FU with time data obtained from experimental study. The experimental data were to be compromised by curve-fitting to get optimum slope, because the raw data has an error due to the precision of pressure transducer; where FU , gas uptake ratio, is equivalent to 1 − FR and FR means the mass fraction of gas in the open space of the apparatus relative to the total gas that can be taken up by the sample. Initial FR could be come close to 1, due to the initial thermal dynamics happening to the dosed gas. After the pressure stabilization is reached, the gas takes up the pore space in sample particles and FR is nearly approached to 0. As shown these figures, stabilization times of FR are different for each sample. After converting pressure transient data to FR , the FR was displaced at semilog scale. Cui et al.'s (2009) experience have shown that FR data 0.01–0.75 are appropriate to determine the straight line part which used to obtain the slope. According to this study, FR data were approximated as a logarithmic function which expressed by Eq. (6) and FR between 0.01 and 0.75 was displaced as shown in Figs. 9 and 10. In these figures, the FR at semi-log is almost linear. From the graphs, the slope and y-intercept which obtained from FR between 0.01 and 0.75 were determined as shown in Table 4. And then, the slope was used to measure the porosity and permeability of shale samples as shown in Table 5. Luffel et al. (1993) and Cui et al. (2009) described that crushed samples represents the permeability in shale matrix which involves only nanopores. Based on the results, it is implicit that the permeability and porosity reflect the characteristics of nanopores in shale matrix. Therefore, the

234

C. Kim et al. / Journal of Petroleum Science and Engineering 133 (2015) 226–237

Fig. 11. The comparison between experimental and analytical FR for Mancos shale. (a) Mancos shale-1, (b) Mancos shale-2 and (c) Mancos shale-3.

measured properties were used to describe gas diffusion in nanopores of shale.

FR = 1 −

(Kc + 1)(ρc 0 − ρ) ρc 0 − ρ0 ∞

2

n= 1

In case of GRI method, the pressure change during the experiment is significantly small. It creates difficulties to acquire the accuracy of experiment. For this reason, the shale properties measured from ALPMA may have an error due to the uncertainty of transient data. In order to validate the experimental results, Cui et al. (2009) analytical solution was used to produce a series of analytical FR data. For the pycnometer, the gas density in the void volume of the appratus is expressed as Eq. (22) (Carslaw and Jaeger, 1947).

ρ = ρc 0 −

ρc 0 − ρ0 (Kc + 1)

2

= 6Kc (Kc + 1) ∑ e−Kαn t / R a

3.3. Validation

∞

2

2

+ 6Kc (ρc 0 − ρ0 ) ∑ e−Kαn t / R a n= 1

1 Kc2αn2 + 9(Kc + 1) (22)

Assuming the particle shape is a spherical shape with a uniform radius and the pressure transient is rarely marked, the above equation is defined as the residual gas fraction (FR ) as shown in Eq. (23).

1 Kc2αn2 + 9(Kc + 1)

(23)

If Kc is much larger than 50, the pressure change in the reference and sample cell become constant, then the Eq. (23) is rearranged as Eq. (24).

FR =

6 π

∞

2 2

2

∑ e−π n Kt / Ra n= 1

1 n2

(24)

By using the Eq. (24), the numerical FR was calculated to compare with the experimental results. In the comparative analysis, Ra was already known and K was computed by Eq. (25). And then, the analytical and experimental FR transition was analyzed by using correlation equation as shown in Eq. (26).

K=

k μcg[ϕ + (1 − ϕ)Ka]

(25)

C. Kim et al. / Journal of Petroleum Science and Engineering 133 (2015) 226–237

235

Fig. 12. The comparison between experimental and analytical FR for Eagle Ford shale. (a) Eagle Ford shale-1, (b) Eagle Ford shale-2, (c) and Eagle Ford shale-3.

Table 6 Pore radius, Knudsen number and diffusion coefficient for shale samples. Sample

λ (nm)

rpore (nm)

Kn

Dk (cm2/s)

Mancos shale-1 Mancos shale-2 Mancos shale-3 Eagle Ford shale-1 Eagle Ford shale-2 Eagle Ford shale-3

3.39 3.37 3.39 3.41 3.43 3.40

21.70 23.90 22.07 24.69 22.03 19.98

0.155 0.142 0.153 0.138 0.155 0.170

9.08  10  2 9.98  10  2 9.22  10  2 10.31  10  2 9.19  10  2 8.34  10  2

Table 7 Knudsen number in different pressure. Pressure step (MPa)

0.69 1.03 1.38 1.72 2.07 ⋮ 39.99 40.33 40.68 41.02 41.37

r=

Kn Mancos shale-2

Eagle Ford shale-2

0.5903 0.3935 0.2952 0.2361 0.1968 ⋮ 0.0102 0.0101 0.0100 0.0099 0.0098

0.6404 0.4269 0.3202 0.2562 0.2135 ⋮ 0.0110 0.0109 0.0109 0.0108 0.0107

3.4. Determination of diffusion coefficient The gas flow in the nanopores of shale matrix may be described as gas diffusion. In this study, the diffusion coefficient was calculated to identify the gas flow due to the diffusion. Assuming the gas flow of methane, Knudsen number was calculated by using Eqs. ((13) and 14). The required parameters in the equations were determined from the experiment. In case of the porosity and permeability, the values were calculated based on Cui's method and used to determine Kn. The pressure is the equilibrium pressure after gas expansion. In addition, the viscosity was calculated by Lee et al. (1966) correlation. And then, the diffusion coefficient was produced by Javadpour (2009) model. Rock properties of shale, Knudsen number and diffusion coefficient are shown in Table 6. Based on the results, it was revealed that pore radius of samples ranges from 15 to 25 nm and Kn varies from 0.14 to 0.20. It means that gas flow in shale samples fall in transition flow. The diffusion coefficient depends on the pore sizes, experimental condition and gas properties. The calculation model of diffusion coefficient reflects these considerations and the distribution for shale samples ranges from 7.0  10  2 to 10.0  10  2 cm2/s.

4. Results and discussion

∑ (xi − x )(yi − y ) (xi − x )2 (yi − y )2

represented that the coefficient is larger than 0.9, as shown in Fig. 12. It means that the experimental data is valid for the measurement of shale properties.

(26)

Fig. 11 displays the comparison between experimental and analytical FR for Mancos shale. The FR means the mass fraction of gas in the open space of the apparatus relative to the total gas that can be taken up by the sample. Initial FR could be close to 1, due to the initial thermal dynamics occurrence to the dosed gas. After the pressure stabilization is reached, the gas filled up the pore space in the sample particles and FR is approached 0. In case of Mancos shale, the FR transition was matched well, representing the correlation coefficient above 0.9. The analytical procedure was also applied to the transition data of Eagle Ford shale and it

4.1. The effect of pore pressure on the gas diffusion To analyze the impact of pressure to gas diffusion, it was assumed that the pore pressure is 0.69–41.37 MPa. Mancos shale-2 and Eagle Ford shale-2 were selected to conduct the analysis based on Knudsen number. Table 7 shows Kn with different pressure (0.69–41.37 MPa) and the results were plotted as shown in Fig. 11. Fig. 13(a) presents that Kn decreases with the increasing of pore pressure. For Mancos shale-2, initial value of Kn was 0.6145. The decreasing is significant if pore pressure o10 MPa, then close to 0.01. Eagle Ford shale-2 showed that the initial value of Kn is 0.8045. The decreasing is significant if pore pressure o10 MPa

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C. Kim et al. / Journal of Petroleum Science and Engineering 133 (2015) 226–237

Fig. 13. Kn trends of Mancos shale-2 and Eagle Ford shale-2 with different pressure. (a) Linear plot and (b) Log–log plot.

Table 8 Knudsen and Fick diffusion coefficient with different pore radius (5–300 nm).

rpore (nm)

Dk (cm2/s)

5 7.5 10 12.5 15 ⋮ 290 292.5 295 297.5 300

0.020 0.031 0.041 0.052 0.062 ⋮ 1.210 1.221 1.231 1.242 1.252

Dfick (cm2/s)

on the change of Kn , it is found that the change of pore radius is limited due to the negligible matrix compressibility. This consideration restricts the change of λ and it affects Kn . 4.2. The effect of pore radius on the gas diffusion

5.068  10  4

The diffusion coefficient with different pore radius was also determined to investigate the impact of pore radius to the diffusion. Zou et al. (2011) discovered nanopores in the shale gas reservoir based on the SEM and CT reconfiguration technique. Their diameters were ranged from 5 to 300 nm. Sondergeld et al. (2010) presented the pore sizes in shales were varied from 300 to 800 nm by SEM image. Based on the previous study, the pore radius was ranged from 5 to 300 nm. Table 8 shows the Knudsen and Fick diffusion coefficient of shale samples. To analyze the gas diffusion with different pore radius, the results were converted to log-log scale as shown in Fig. 14. Two samples showed that Dk and Dfick are proportional to the increasing of pore radius, while Fick diffusion coefficient remains 5.068  10  4 cm2/s. It is implicit that Fick diffusion coefficient remains constant because the parameter is independent of the pore radius. It depends on the change of gas molecular diameter.

5. Conclusions

Fig. 14. Methane diffusion coefficient in pore radius 5–300 nm.

and it was closed to 0.013. The results were converted to log–log scale as shown in Fig. 13(b). Kn decreases with the increasing of pore pressure and increases with decreasing of pore radius. Based

The gas diffusion in U.S. shale gas reservoirs was studied with an emphasis on nanopore scale. An experimental apparatus was designed and set up to determine porosity and permeability at nanopore scale and the properties were used to determine the diffusion coefficient. From the results, following conclusions have been drawn: (1) The porosity and permeability considering nanopore characteristics in the shale matrix were measured by the developed apparatus. Kn was determined by the measured properties and the gas flow regime was illustrated as transition flow. (2) The Knudsen diffusion coefficients which describe the gas transport process in the shale matrix were determined. The diffusion coefficient ranges from 7.0  10  2 to 10.0  10  2 cm2/s. (3) To investigate the effect of pore radius and pressure, Knudsen and Fick diffusion coefficient were drawn with different pore radius and pressure. The results presented that Knudsen

C. Kim et al. / Journal of Petroleum Science and Engineering 133 (2015) 226–237

diffusion coefficient depends on the two parameters, while Fick diffusion coefficient is dependent on the gas molecular diameter and temperature. (4) Fick diffusion coefficient is less than Knudsen diffusion coefficient for pore size 5–300 nm and the diffusion coefficient could be equal if the pore radius is too small. Thus, the diffusion coefficient of the system in a shale gas reservoir should be determined after selecting the flow regime by Kn .

Acknowledgments This work was supported by the Energy Efficiency & Resources of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea Government Ministry of Trade, Industry & Energy (2011201030001B).

References Carslaw, H.S., Jaeger, J.C., 1947. Conduction of Heat in Solids. Oxford University Press, London, United Kingdom, pp. 198–207. Cui, X., Bustin, A.M.M., Bustin, R.M., 2009. Measurements of gas permeability and diffusivity of tight reservoir rocks: different approaches and their applications. J. Geofluids 9 (3), 208–223. Energy Information Administration (EIA), 2011. Review of Emerging Resources: U.S. Shale Gas and Shale Oil Plays. EIA, Washington D.C., pp. 29–64. Freeman, C.M., Moridis, G.J., Blasingame, T.A., 2011. A numerical study of microscale flow behavior in tight gas and shale gas reservoir systems. J. Transp. Porous Media 90 (1), 253–268. Javadpour, F., Fisher, D., Unsworth, M., 2007. Nanoscale gas flow in shale gas sediments. J. Can. Pet. Technol. 46 (10), 55–61. Javadpour, F., 2009. Nanopores and apparent permeability of gas flow in mudrocks (shales and siltstone). J. Can. Pet. Technol. 48 (8), 16–21. Kamath, J., Boyer, R.E., Nakagawa, F.M., 1992. Characterization of core scale heterogeneities using laboratory pressure transients. SPE Paper 20575. SPE Form. Eval. 7 (3), 219–227. Lee, A.L., Gonzalez, M.H., Eakin, B.E., 1966. The viscosity of natural gases. SPE Paper 1340. J. Pet. Technol. 18 (8), 997–1000. Luffel, D.L., Hopkins, C.W., Holditch, S.A., Shettler, P.D., 1993. Matrix Permeability

237

Measurement of Gas Productive Shales. SPE Paper 26633. SPE Annual Technical Conference and Exhibition, Houston, Texas, 3–6 October. Petersen, H., 1970. The Properties of Helium: Density, Specific Heats, Viscosity, and Thermal Conductivity at Pressure from 1 to 100 Bar and from Room Temperature to about 1800 K. Danish Atomic Energy Commission Research Establishment Risoe, Tranbjerg, Denmanrk, pp. 1–45. Roy, S., Raju, R., Chuang, H.F., Cruden, B.A., Meyyapan, M., 2003. Modeling gas flow through microchannels and nanopores. J. Appl. Phys. 93 (8), 4870–4879. Rutqvist, J., Rinaldi, A.P., Cappa, F., Moridis, G.J., 2013. Modeling of fault reactivation and induced seismicity during hydraulic fracturing of shale-gas reservoirs. J. Pet. Sci. Eng. 107, 31–44. Schaaf, S.A., Chambre, P.A., 1961. Flow of Rarefied Gases. Princeton University Press, Princeton, New Jersey, pp. 1–58. Shabro, V., Javadpour, F., Toress-Verdín, C., 2009. A generalized finite-difference diffusive advective (fdda) model for gas flow in micro and nano porous media. World J. Eng. 6 (3), 7–15. Sigal, R.F., Qin, B.F., 2008. Examination of the importance of self diffusion in the transportaion of gas in shale gas reservoirs. J. Petrophys. 49 (3), 301–305. Sondergeld, C.H., Ambrose, R.J., Rai, C.S., Moncrieff, J., 2010. Micro-structural Studies of Gas Shlaes. SPE Paper 131771. SPE Unconventional Gas Conference, Pittsburgh, Pennsylvania, 23–25 February. Suarez-Rivera, R., Chertov, M., Willberg, D.M., Green, S.J., Keller J., 2012. Understanding Permeability Measurements in Tight Shales Promotes Enhanced Determination of Reservoir Quality. SPE Paper 162816. SPE Canadian Unconventional Resources Conference, Calgary, Alberta, Canada, 30 October–1 November. Swami, V., Clarkson, C.R., Settari, A., 2012. Non-Darcy Flow in Shale Nanopores: Do We Have a Final Answer?. SPE Paper 162665. SPE Canadian Unconventional Resources Conference, Calgary, Alberta, Canada, 30 October–1 November. Tinni, A., Fathi, E., Agarwal, R., Sondergeld, C.H., Akkutlu, I.Y., Rai, C.S., 2012. Shale Permeability Measurements on Plugs and Crushed Samples. SPE Paper 162235. In: SPE Canadian Unconventional Resources Conference, Calgary, Alberta, Canada, 30 October–1 November. Wang, R., Zhang, N., Lui, X., Wu, X., Yan, J., 2013. Characterization of gas flow ability and contribution of diffusion to total mass flux in the shale. J. Appl. Sci. Eng. Technol. 6 (9), 1663–1668. Xiao, J., Wei, J., 1990. Diffusion Mechanism of Hydrocarbon in Zeolites I. Theory. J. Chem. Eng. Sci. 47 (5), 1123–1141. Zhao, Y.L., Zhang, L.H., Zhao, J.Z., Luo, J.X., Zhang, B.N., 2013. “Triple porosity” modeling of transient well test and rate decline analysis for multi-fractured horizontal well in shale gas reservoirs. J. Pet. Sci. Eng. 110, 253–262. Ziff Energy, November 2012. Shale Gas Basins Map. URL 〈http://www.ziffenergy. com/media/chartsofthemonth.aspx〉 (accessed 23.07.14.). Zou, C., Zhu, R., Bai, B., Yang, Z., Wu, S., Su, L., Dong, D., Li, X., 2011. First discovery of nano-pore throat in oil and gas reservoir in china and its scientific value. Acta Pet. Sin. 27 (6), 1857–1864.