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Investigation of intermingled fractal model for Organic-rich shale Caoxiong Li, Mian Lin, Lili Ji, Wenbin Jiang, and Gaohui Cao Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.7b00834 • Publication Date (Web): 09 Aug 2017 Downloaded from http://pubs.acs.org on August 10, 2017
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Investigation of intermingled fractal model for Organic-rich shale Caoxiong Li1,2, Mian Lin1,2*,Lili Ji1, Wenbin Jiang1, Gaohui Cao1,2 1 Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China 2 School of Engineering Science, University of Chinese Academy of Sciences, Beijing 100049, China *Corresponding author:
[email protected] ABSTRACT Organic-rich shales have abundant micro-nanopores and slits. Most micro-nanopores exist in organic matter and pyrites. Pore size distribution significantly influences shale gas flow. In micro-nanopores of organic matter and pyrites, pore size distribution has a self-imitating or fractal property. This property is helpful in upscaling calculation. In this work, intermingled fractal model for Organic-rich shale is investigated. First, on the basis of scanning electron microscope (SEM) images of shale samples in Longmaxi shale formation, fractal units are built in representative elementary surfaces of organic matter and pyrites. Second, Intermingled fractal models (IFM) are then built considering the organic matter fractal pore units, the pyrite fractal pore units, and slits. Influences of fractal parameters on apparent gas permeability of IFM are investigated. Third, an upscaled shale model is successfully built by mirror image method. This upscaling method is verified by comparing the apparent gas permeability between upscaled models and experimental results. The upscaling method based on fractal theory can significantly shorten the calculation time and provide a promising way for apparent gas permeability calculation in the upscaled model. Keywords Intermingled Fractal Model (IFM); Fractal; SEM; Organic-rich shale; Micro scale flow
1、 、INRTODUCTION A shale play, where most hydrocarbon is stored in tight matrix blocks, contains assemblage rock, reservoir, and caprock. This reservoir has ultra-low permeability and porosity with abundant micro-nanopores, especially organic pores1. Many problems occurs in the developing process2. Especially, gas flow mechanism of shale is an complex problem. Investigating the flow mechanism of shale is important for gas production, hydraulic fracturing, and production forecasting of shale. Shale has relatively strong heterogeneity with complex pore space and various pore types. Pore space is not easy to present precisely and effectively in a permeability calculation model. Many studies have adopted the rebuilding method for pore spaces, which can be generally classified in two ways. One is building a theoretical pore model based on statistical data, and the other is refactoring pore space based on core scanning data. Since the early twentieth century, theoretical pore space has developed from the parallel tube bundle model to the network model. Fatt3 first combined parallel tube bundle based on actual pore size distribution with the network model and studied the relationship between pore space and apparent permeability. Fishcher et al.4 introduced capillary pressure curve in the pore network model and studied the ACS Paragon Plus Environment
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relationship between confine pressure and relative permeability. Bryant et al.5,6 introduced stress deformation into the pore network model to study the effect of confine pressure on permeability. Song et al.7 investigated non-Darcy shale permeability under different stress conditions, taking both organic and inorganic matter into consideration. Jiang et al.8 studied gas transport in pore space based on high-precision pore network extraction algorithm. Meng et al.9,10 studied the influence of auto-removal mechanism on shale permeability. Yuan et al.11,12 examined nanofluid adsorption and liquid damage on permeability in nano pore space. Peng et al. 13,14
used thin-section image analysis to express pores to estimate permeability and
investigated multiphase fluid flow in shale . Shabro et al.15 established an apparent gas permeability model of shale, including diffusion, slip, and adsorption–desorption effect in kerogen. These factors significantly affect gas flow rate. Upscaling is essential. In recent years, computer-scanning-based methods have considerably improved with the development of computer science. Traditional core experiment data and nanoscale computer simulation data must be combined. Numerous researchers have studied upscaling methods from the microscopic to the mesoscopic or macroscopic scale16. On the basis of the network model, Ahmadi et al.17 used volume-average upscaling method to investigate the diffusion coefficient of non-wetting phase in porous media. Frykman et al.18 studied upscaling method based on statistical information of geology. Gauiter et al.19 combined Bayesian theory with upscaling method to study physical parameters of rocks. Pore space in cores have fractal properties in a specific scale20-25,so does shale26, thereby suggesting that such space can be rebuilt as fractal models on the basis of their self-imitating properties. Fractal models provide an efficient way to express pore size distribution to calculate permeability. Many permeability models have been developed based on fractal theory27-29. Yu et al.30 first developed a fractal in-plane permeability model for various fabrics. Xu and Yu31 developed a permeability-prediction model for homogeneous porous media by introducing cross-section parameters. Cai et al.32,33 established an effective permeability model based on spontaneous imbibition by using the self-imitating properties of porous media. Zheng and Yu investigated gas flow and diffusivity in fractal porous media and derived related analytical models34,35. Shou et al.36 developed a difference–fractal model to predict the permeability of fibrous porous media. Cai37 introduced low velocity non-Darcy term in low-permeability porous medium. Zhang38 introduced the Knudsen diffusion term in the gas permeation equation of the fractal model. Pia and Sanna39 first built intermingled fractal units (IFU), which made pores easier to express. In following years, they developed the IFU model, which makes it easier to use in the calculation of conduction properties, such as permeability and imbibition40-42. The innovation of IFU model greatly simplifies the expression of pore size distribution.
2、 、INTERMINGLED FRACTAL UNITS AND PERMEABILITY CALCULATION 2.1 Introduction of IFU model ACS Paragon Plus Environment
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Pia et al.39 utilized an approach based on the description of the microstructure of voids using the fractal method called the IFU model. This model comprises several fractal unit types based on different iteration parameters. Each fractal type is revised as Sierpinski carpet. The volume (area in the plane) of the IFU model can expressed as.
Vres = n AVA + nBVB + nCVC
(1)
where Vres is the volume/area of the IFU model; nA , nB , and nC is the number of each units; and VA , VB ,
VC is the volume/area of each unit.
F=3 Nsoild=0 Np=3
λ
(a) i=0
(b) i=1
(c) i=2
iteration space
non-iteration space
void space
F=3 λ
Nsoild=1 Np=1
(d) i=0
(e) i=1
(f) i=2
Figure 1. Schematic diagram of iteration process in two typical basic units with different iteration parameters (like Unit A and Unit B) To make the model understandable, assuming that the void space of the IFU model is combined with units A, B, and C, each unit have different iteration parameters. Figs. 1(a) to (c) and (d) to (f) are two iteration examples to illustrate basic structure of units with different iteration parameters. The white area is void space, the black area is the iteration area, and the shaded area is a non-iteration space. The iteration factor F is number of generated square on each side in one iteration process. In each iteration, the sides of a square are divided by three (F=3), thus obtaining nine sub-squares with some squares removed [for example, in Fig. 1(a), three squares are removed]. In next iteration, in the remaining sub-squares, each one is regarded as a new square and divided into nine sub-squares; some sub-squares will be removed. Particularly, if non-iteration space exits, the space do not take part in the iteration [Figs. 1(d) and (e)]. N p represents the number of squares removed in each iteration. Nsoild represents the number of squares that always remained solid in each iteration. λi represents the edge length of void area in i th iteration. i represents iteration ACS Paragon Plus Environment
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number. In Unit A, these parameters can be expressed as FA , N Ap , N Asoild , λ Ai and iA . The rest parameters in unit B and C can be expressed in the same manner. According to fractal theory, the edge length of the void area in iA th iteration λ Ai is
λ Ai = λ A max / F A ( i
A −1)
(2)
With the number of void areas generated in iA th iteration
N Ai = ( FA − N Asoild − N Ap ) i A −1 × N Ap 2
(3)
where N Asoild is the number of squares that always remain solid. The volume/area of void area with edge length λ Ai is V Ai = λ 2Ai N Ai
(4)
The total volume/area of void area in unit A is i
VA = VA1 + VA2 + L + VAk + L + VAi = ∑ VAk
(5)
k =1
where the total volume/area of void space before k th iteration is
VAk _ accumulate = VA1 + VA2 + L + VAk (6)
2.2 Permeability calculation In this section, apparent gas permeability calculation method based on intermingle fractal distribution is discussed. Non-Darcy effect is negligible for gas flow in nano-scale pores. For a single capillary, Javadpour et al.43 provided gas flow equation considering non-Darcy effect, like slip and Knudsen diffusion effect as:
λn µM q square = 3 3 × 10 RTρ avg
8RT πM
0.5
+
λn 2
8πRT 1 + 32 M
0 .5
2µ 2 A ( Pin − Pout ) − 1 pavg λn α µ L
(7)
Where, λn is equivalent diameter, in square tube is the edge; M is molar mass of natural gas kg/kmol; µ is viscosity of natural gas, Pa.s; R is gas constant, J/mol/K; T is temperature, K; ρ avg is average density of natural gas, kg/m3 ; Pin and Pout are inlet pressure and outlet pressures, Pa; pavg is average gas pressure, Pa; α is tangential momentum accommodation coefficient. The viscosity and density of natural gas change greatly with average pressure, the equation of viscosity and density of natural gas heve been tested by Lee et al.44 in laboratory experiment. A is cross-sectional area, in single tube A = λ2n , L is length of tube, the tortuosity can be expressed by fractal dimension in length as Eq.840
L = LD0 T λ1n− DT
(8)
Where L0 is straight length between two sides. DT is fractal dimension in length. It can be calculated by Eq.945
DT = 1 +
lnτ ln(L0 / λ )
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(9)
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τ is tortuosity of tube, can be calculated by Eq.1045： τ = 1 + a ln(1 / ε )
(10)
Where, ε is pore fraction, a is a shape related parameter. To simplify Eq.7, original equation can be expressed in polynomial form:
(
)
λ2n ( Pin − Pout ) qsquare = C1λn + C2λn + C3λn µ L A is cross-sectional area, 8πRT C3 = M
0.5
µ
2
A = λ 2n , parameters
C1 =
µM
(11) 8 RT πM
3 × 10 RT ρ avg 3
0.5
， C2 =
1 ， 32
2 43 − 1 . Based on theory of Javadpour et al. , volumetric gas flux is combined by α
16 p avg
three parts : Knudsen diffusion, Darcy flow and increment of flow caused by slip effect and in Eq. 11, C1 represents Knudsen diffusion part, C2 represents Darcy flow part and C3 represents increment part of flow caused by slip effect. This expression makes flow equation more explicit. In basic fractal units, apparent gas permeability model can be expressed as square tube bundle. Gas flow rate can be expressed as Eq.12 (taking unit A for example) i
(
Q A = ∑ N Ai C1λi + C 2 λi + C3λi 3
4
1
3
) ( P µ−LP in
out
)
(12)
Where N Ai is the number of tube in stage i , determined by Eq.3; Plugging Eq.2 to Eq.5 into Eq.12, gas flow rate of single fractal unit (taking unit A for example) can be expressed as Eq.13 i λi 3 Q A = C1 ∑ N Ai 1− DT λi 1
i λ4 + C 2 ∑ N Ai 1−i D λi T 1
i λ3 + C 3 ∑ N Ai 1−iD λi T 1
( Pin − Pout ) µLDT 0
2 1 − (( FA − N Asoild − N Ap ) / FA2+ DT ) i 2 + DT λ A max N C Ap 1 2 1 − ( FA − N Asoild − N Ap ) / FA2+ DT 2 1 − (( FA − N Asoild − N Ap ) / FA3+ DT ) i 3+ DT ( Pin − Pout ) λ A max = + N Ap C 2 2 µLD0 T 1 − ( FA − N Asoild − N Ap ) / FA3+ DT 2 1 − (( FA − N Asoild − N Ap ) / FA2+ DT ) i 2+ DT + N Ap C 3 λ A max 2 1 − ( FA − N Asoild − N Ap ) / FA2+ DT DT DT DT ( Pin − Pout ) = CA1λ2A+max + CA2 λ3A+max + CA3 λ2A+max µLD0 T
[
]
1 − (( FA − N Asoild − N Ap ) / FA2 + DT ) i 2
Where CA1 = N Ap C1
(13)
1 − ( FA − N Asoild − N Ap ) / FA2 + DT 2
1 − (( FA − N Asoild − N Ap ) / FA2 + DT ) i
1 − (( FA − N Asoild − N Ap ) / FA3+ DT ) i 2
， CA2 = N Ap C 2
1 − ( FA − N Asoild − N Ap ) / FA3+ DT 2
，
2
CA3 = N Ap C 3
1 − ( FA − N Asoild − N Ap ) / FA2+ DT
Hagen-Poiseuille item,
2
， CA1 indicates Knudsen diffusion item ， CA2 indicates
CA3 indicates slip item, this expression makes flow equation more explicit. ACS Paragon Plus Environment
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Q B and QC can be expressed in the same manner:
[
]( P µ−L P
[
]( P µ−L P
DT DT DT QB = CB1λ2B+max + CB2 λ3B+max + CB3 λ2B+max
DT DT DT QC = CC1λC2+max + CC 2 λ3C+max + CC3 λC2+max
in
out
in
out
CB1 = N BpC1
)
DT 0
1 − (( FB − N Bsoild − N Bp ) / FB2+ DT )i 2
Where
)
DT 0
1 − ( FB − N Bsoild − N Bp ) / FB2+ DT 2
1 − (( FB − N Bsoild − N Bp ) / FB3+ DT ) i 2
, CB2 = N Bp C 2
1 − (( FB − N Bsoild − N Bp ) / FB2 + DT ) i
1 − ( FB − N Bsoild − N Bp ) / FB3+ DT 2
2
CB3 = N Bp C 3
1 − ( FB − N Bsoild − N Bp ) / FB2+ DT 2
1 − (( FC − N Csoild − N Cp ) / FC3+ DT ) i
2
,
CC1 = NCpC1
2
CC 2 = N Cp C 2
1 − ( FC − N Csoild − N Cp ) / FC3+ DT 2
1 − (( FC − NCsoild − NCp ) / FC2+ DT )i
,
1 − ( FC − NCsoild − NCp ) / FC2+ DT 2
,
1 − (( FC − N Csoild − N Cp ) / FC2+ DT ) i 2
, CC 3 = N Cp C 3
1 − ( FC − N Csoild − N Cp ) / FC2 + DT 2
3 INTERMINGLED FRACTAL MODEL (IFM) 3.1 Basic information of samples In this paper, samples come from Longmaxi Marine Shale Formation of Lower Silurian in the Sichuan Basin. Sample1 and 2 are twin plugs cut from fresh outcrop of Longmaxi Marine Shale Formation of Lower Silurian in the Sichuan Basin. Sample 3 and 4 are fresh core sample from a well drilled at depth of 2357-2406m in Longmaxi Formation of the Sichuan Basin. These shales contain abundant organic matter, especially graptolites46. Rock mineralogy of samples was analyzed by Energy Dispersive Spectroscopy (DES) in an area of 409.7 µm x 409.7 µm by using a Zeiss–Merlin rock mineralogy analysis machine, with resolution 5nm. Table 1 shows that the main minerals of shale sample 1, 2 and 3 are quartz, clay minerals and feldspar. Dolomite is dominant mineral of Sample 4.
Table 1. Rock mineralogy of samples Minerals
Samples 1 and 2 vt.(%)
Sample 3 vt.(%)
Sample 4 vt.(%)
Quartz Clay minerals Feldspars Calcite Pyrite Biotite Dolomite TOC Else
60.9 8.9 14.3 1.71 1.53 4.04 2.73 3.0 2.89
18 38 35 2 2 0.3 2 0.97 1.73
6 3 10 1 1 0.2 76 2.41 0.39
Dry sample permeability of gas is measured by pulse-decay permeability measurement under 25 C ACS Paragon Plus Environment
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from three different directions. Other test conditions are shown in Table 2.
Table 2. Test conditions of pulse-decay permeability test Flow parameters
numbers
Flow parameters
numbers
Molecular mass N2
28 g/mol
Average temperature
306.24 K
Molecular mass He
4 g/mol
Average pressure
0.55 MPa
Pressure difference ∆ p
1.1 MPa
Gas constant
8.314 J/(k.mol)
The average gas permeability is calculated by the quadratic polynomial method by the following equations.
P1,t -P2,t p1,0 − p2,0
= e −α t
Where P1,t and P2 , t are inlet and outlet pressure at time t, P1, 0 and P2, 0 are inlet and outlet pressure at initial time, parameter
α
can be expressed as:
α= Where k
kA( p1,0 + p2,0 ) 1 1 ( + ) 2µ L V1 V2
is apparent permeability, A is across area, L is length of sample, µ is gas viscosity,
V 1 and V 2 are inlet and outlet volume. Taking sample 1 as an example, original pulse-decay permeability
measurement curve are shown in Fig. 2, apparent permeability have tested under confining pressure 3MPa, 5MPa, 7MPa, 9MPa. Additionally, since SEM images are tested with no confining pressure, quadratic polynomial method is used to calculate apparent permeability under no confining pressure to make sure the condition unchanged. apparent permeability(nD)
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100 80 60 40 y = 0.5445x2 - 11.677x + 91.36 R² = 0.9989
20 0 0
2
4
6
8
10
Confining pressure(MPa)
Figure 2. Quadratic polynomial method As the result shows, average gas permeability of sample 1 is 91.36 nD as calculated by the quadratic polynomial method. Apparent permeability in other samples have derived in the same way, the results have shown in Table 5. Nitrogen is used in this test. Viscosity, density and compressibility factor of gas changes with average temperature and pressure. We use experimental results by Lee et al.44 ACS Paragon Plus Environment
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Figure 3. SEM image of Longmaxi marine shale SEM is used to investigate the nanoscale pore shape and the spatial distribution of Longmaxi samples. The samples have ion polishing before SEM tests. The scanning area is 380 µm x 380µm with a maximum ACS Paragon Plus Environment
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resolution of 2.5 nm. Total spliced images of two different areas are shown in Figs. 3(a) and (b). Figures 3(c) and (d) are magnified images in a specific area, which clearly show inorganic slits, as well as pores in organic matter and pyrites. Specifically, pores in pyrites are filled in overdeveloped pyrites. In organic-rich shales, organic matter [Fig. 3(e)] has the most organic pores, and pyrites [Fig. 3(f)] contain some organic pores. In addition, inorganic slits scatter in SEM images [Fig. 3(c)]. Micro-nanopores have fractal properties at a specific scale. Pores have fractal characteristics in each component. We generate IFUs for pores in organic matter and pyrites; the pore size distribution of models follows actual samples calculated from the SEM image. Intermingled fractal models (IFM) are built by mixing pores in organic matter, as well as pores in pyrites and slits. Slit density is calculated from the statistical data of SEM images. This model combines well with both microscale mineral content distribution and pore size distribution of each mineral in nanoscale. Shale gas flow in IFM can be calculated easily and effectively considering the non-Darcy effect, such as slip and Knudsen diffusion.
3.2 Organic matter pores Organic matter has abundant pores with a vast distribution. The modeling process of organic matter is shown in Fig. 4 and detailed below.
Figure 4. Modeling process of organic matter (1) RES is obtained from statistical regularity. The heterogeneity of organic matter distribution is relatively strong in a small area. To calculate the smallest RES, four different initial points are chosen (four corners of an image in this example). We then expand the target area starting from the source points (1 pixel on the edge) in the original SEM image and calculate the average grayscale in the sub-square. The variation of the average grayscale will diminish gradually as the pixels on the edge of the sub-square increases. By calculation, as the number of pixels on the edge is larger than 400, average grayscale keeps stable. So image with pixels not smaller than 400 can be regarded as RES. We choose 400 pixels on the edge as RES. ACS Paragon Plus Environment
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(2) After exacted, the RES image is segmented to exact pores. In segmentation process, threshold value is the most important parameter. Pore fraction of RES is regarded the same with experiment porosity. By adjusting grave scale of threshold with a certain value, pore fraction of exacted RES is equal with experiment result. Hence, pores in RES have been derived. Pore cumulative distribution curve (blue curve in Fig. 5) is calculated from the chosen two images [Figs. 3(a) and 3(b)]. (3) The basic IFU model is built, the cumulative pore curve (red curve in Fig. 5) is calculated. Fractal parameters are adjusted to narrow the differences between cumulative curves of the basic IFU model and image calculation. Afterwards, certain parameters of the basic IFU model are determined, as shown in Table 3. 0.3
0.35
SEM IFU
0.25
SEM
0.25
Pore fraction
0.3
Pore fraction
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0.2 0.15 0.1
IFU
0.2 0.15 0.1 0.05
0.05
0
0 1
10
100
1
1000
10
100
1000
Diamater of organic pore(nm)
Diamater of organic pore(nm)
(a)
(b)
Figure 5. Comparison of pore cumulative curve between the IFU model and SEM analysis.
Table 3. IFU model input data for reproducing cumulative pore curve of organic pores SEM1 Unit
SEM2
Unit
SEM3
SEM4
Unit
Unit
Unit
Unit
Unit
Unit
Unit
Unit
Unit
A
B
C
A
B
C
A
B
C
Unit C A
B
Df
1.89
1.77
1.46
1.89
1.77
1.63
1.77
1.77
1.46
1.89
1.77
1.63
n unit
1
5
230
1
23
20
1
9
350
1
8
8
n pore(i=1)
1
2
4
1
2
3
2
2
4
1
2
3
λmax(nm)
191
80
26.67
170
70
50
272
93
34
133
92
40
Iteration
4
3
2
4
3
2
4
3
2
3
3
2
λmin(nm)
7.07
8.89
8.89
6.30
7.78
16.67
10
10.33
11.33
14.77
10.22
13.33
Soild forever
0
0
0
0
0
0
0
0
0
0
0
0
total surface(μm2)
4.33
3.35
12.97
3.3 Pyrite pores
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28.4
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Figure. 6 Modeling process of pyrite parts. Pyrites are scattered in the SEM area. First, the distribution density of pyrites (number of pyrites in this area) is observed from statistical data based on the SEM image. The, the pores in one pyrite are represented as IFUs. Finally, pores in the model are scattered in the IFM with the observed distribution density. The modeling process of the pyrite parts is shown in Fig. 6. (1) All the pyrites are exacted in a relatively large scale [similar to that in Fig. 3(a)], and the distribution density of pyrites is calculated from the statistical data of the SEM image, excluding overdeveloped pyrites. N pyrites are assumed in the area. (2) One pyrite with average diameter is selected and magnified (Fig. 6). The image is segmented to exact pores, and the pore cumulative distribution curve (blue curve in Fig. 7) is calculated. (3) The basic IFU model is built, and the pore cumulative curve (red curve in Fig. 6) is calculated. Fractal parameters are adjusted to narrow the differences between cumulative curves of the IFU model and image calculation. Afterwards, certain parameters of the basic IFU model are determined, as shown in Table 4. The result in Fig. 7(a) is calculated from Fig. 3(a) while Fig. 7(b) is from Fig. 3(b). (4) The basic IFU model is scattered in the area of Figs. 3(a) and (b), and the distribution density of pyrites is calculated.
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0.12
0.08
SEM
Pore fraction
IFU
0.06
SEM
0.1
0.07
Pore fraction
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0.05 0.04 0.03 0.02
IFU
0.08 0.06 0.04 0.02
0.01 0
0 10
100
1000
10
100
Diamater of pyrite pore(nm)
1000
Diamater of pyrite pore(nm)
(a)
(b)
Figure 7. Comparison of pore cumulative curve between the IFU model and SEM analysis in pyrite parts.
Table 4. IFU model input data for reproducing cumulative pore curve of pyrite pores SEM1 Unit
Unit
SEM2 Unit
Unit
Unit
SEM3 Unit
Unit
SEM4 Unit
Unit
Unit A A
B
C
A
B
C
Df
1.89
1.63
1.46
1.89
1.77
1.63
n unit
1
4
10
1
2
n pore(i=1)
1
3
4
1
λmax(nm)
300
185
120
Iteration
3
2
λmin(nm)
33.33
Soild forever
0
Unit Unit B
B
C
A
C
1.89
1.77
1.77
1.89
1.77
--
8
1
2
5
1
3
--
2
3
1
2
2
1
2
--
260
190
115
225
152
90
371
175
--
2
3
2
2
3
2
2
3
3
--
61.66
40
28.89
63.33
38.33
25
50.66
30
41.22
19.44
--
0
0
0
0
0
0
0
0
0
0
--
total surface(μm2)
22.75
8.94
8.9
54.9
3.4 Slits Slits or cracks also have influence on gas permeability. Distribution of slits have relatively strong heterogeneity. They should be observed from different scale. Distribution of slits should be obtained from a relatively larger scale, for the heterogeneous distribution. While the aperture of slits should be observed in larger resolution images. We determine the statistical parameters of inorganic slits by calculating the distribution density of inorganic slits in a large area and their average length and aperture in a magnified image. The method of intermingling the organic matter’s IFU, pyrites’ IFU, and inorganic slits will be discussed in the subsequent section.
3.5 Building IFM The foundation process of IFMs based on SEM images is mainly divided into three steps. First, the SEM images are obtained, and the representative elementary surface (RES) is determined. Second, IFUs are built for pores in organic matter and pyrites. Third, IFM is built by IFU and slits. This section discusses the modeling process of organic matter and pyrites.
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IFUs for organic matter and pyrites are presented in section 3.4. In this section, IFM is developed considering pores in organic matter, as well as pores in pyrites and slits. On the basis of this model, the upscaling process is tested by using the mirror image method. The accuracy and reliability of this upscale method is examined by experiment verification. The building process of IFM is shown in Fig. 8. First, we calculated the RES of organic matter, as shown in the top part of Fig. 8. In this scale, the IFU model of organic matter attempts to follow the pore cumulative curve in SEM images of organic matter. Fractal parameters are adjusted to narrow the difference between the curve of the model and the SEM image. Second, we calculated the distribution density and average diameter of pyrites. Overdeveloped pyrites are removed because they have no pores. Within one typical pyrite, pyrite’s IFU model is generated in the same way. By calculating the distribution density and the average length and aperture of slits, we obtain the statistical parameters of slits, as shown in the middle part of Fig. 8. Finally, IFM in RES can be obtained by mixing these three components. Pores in organic matter’s IFU model and pyrites’ are scattered into the IFM randomly with slits. The final pore size distribution also follows the statistical data of the final SEM image. For example, SEM image 1 [Fig. 3(a), with an edge length of 380 µm] has 82 pyrites. The aperture of slits varies from 20 nm to 85 nm of slits, the average aperture is 40 nm, the average length is 5.18 µm, and the average distribution density is 50.3 lines/10,000 µm2. By scattering these components in an area of 380 µm×380 µm, we obtain the IFM, as shown in the lower right corner of Fig. 8. SEM image 2 [Fig. 3 (b)] is calculated in the same way. Permeability can also be derived based on these fractal parameters. The equivalent gas permeability of organic matter, pyrites, and slits can be derived from Eqs. 15 to 17, respectively. Evidently, IFM’s apparent gas permeability is the sum of that of the components, as shown in the bottom part of Fig. 8.
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Figure 8. IFM procedure to calculate permeability
3.6 Apparent gas permeability Then in organic RES(taking organic RES mingled by unit A,B and C for example), gas flow rate can be expressed as：
Q res − org = n A Q A + n B Q B + n C Q C Suggesting the area of organic RES is
(14)
Ares ，volume fraction of TOC is ϕTOC, equivalent gas
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permeability of organic is：
k app _ org = Qres−org
C µL0 ϕ AsumϕTOC + DT + DT + DT = TOC ∑ ni (Ci1λi2max + Ci2 λ3i max + Ci3λi2max ) Ares Asum ( Pin − Pout ) Ares i = A
(15)
Similarly, equivalent gas permeability of pyrite is：
k app _ py = Where, n
py
n py Asum
C
∑ n (Ci i
2+ DT 1 _ py i max__ py
λ
+ DT 2+ DT + Ci2 _ py λ3i max__ py + Ci3 _ py λi max__ py )
(16)
i= A
is the number of pyrite in IFM.
Similarly, equivalent gas permeability of slit is：
k app _ slit = C py where, C1 _ slit =
λ Dn T_ +slit1 n slit 2 C 1 _ slit λ n _ slit + C 2 _ slit λ n _ slit + C 3 _ slit λ n _ slit A sum L D0 T −1
(
µM 3 × 103 RTρ avg
number of slits in final research area.
)
(17)
0.5
0 .5 8RT µ 2 , C 2 _ slit = 1 , C 3 _ slit = 8π RT − 1 , nslit is the 32 πM M 16 p avg α
λ n _ silt is equivalent diameter of slit, C
py
is shape factor, which
can be determined in Veltzke et al.47. If Knudsen diffusion and slip effect are ignored, Ci1 =Ci3 = 0 ， Ci 1 _ py = Ci 3 _ py = 0 ， C 1 _ slit = C 3 _ slit = 0 .Only Darcy item is left. Permeability under the circumstances is traditional Darcy
permeability. Apparent Darcy gas permeability of each component is
kins _ org = kins _ py =
ϕTOC
Asum
kins _ slit = C py
3+ DT 2 i max
i
Ares
n py
C
∑ n Ci λ
(18)
i= A
C
∑ n Ci i
2 _ py
+D λ3i max__ py T
(19)
i= A
DT +1 nslit 2 λ C2 _ slit λn _ slit nD_T slit Asum L0 −1
(
)
(20)
4 FRACTAL PARAMETERS ON PERMEABILITY In the IFM, fractal parameters control the pore size distribution. To examine the relationship between pore size distribution and apparent gas permeability, the influence of fractal parameters on apparent gas permeability is investigated. The primary fractal parameters are Unit number, iteration time i , number of removed block in each iteration N p , and number of blocks that always remain solid in each iteration The permeability is calculated by Eq. (15).
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i=1 i=2 i=3
70 Permeability(nD)
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60 50 40 30 0
1
2
3
4
5
P (MPa) Figure 9. variation of apparent permeability with pressures in different iteration time To make the influence of fractal parameters more understandable, we discussed the influence of iteration time i , number of removed block in each iteration N p and number of blocks that always remain solid in each iteration
N s in unit A of organic matter component. Basic parameters are shown in Table 3.
The apparent permeability varies with pressure. Non-Darcy flow effect increases with the decrease in average flow pressure during the percolation process. Apparent permeability is calculated for six different average pore pressures (Fig.9), namely, 0.1, 0.3, 0.55, 1, 2, and 5 MPa, with the average temperature fixed at 306.24 K. The non-Darcy flow effect emerges when the average pressure is less than 1 MPa, especially under 0.3 MPa. Many small pores are generated as the iteration time increases. Initially, the increase of iteration time i significantly affects both Darcy and non-Darcy permeability. Meanwhile, as i increases, this effect diminishes, similar to increases from 3 to 5. The diameter of pores in the fifth iteration is too small and therefore cannot significantly influence the apparent permeability of the model. Non-Darcy effects in gas flow should be considered in the analysis of gas flow in micro-nanopores under low pressure.
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200 Np=1 Np=2 Np=3
180 Permeability(nD)
160 140 120 100 80 60 40 20
0
1
2
3
4
5
P (MPa) Figure 10. variation of apparent permeability with pressures in different removed numbers. Removed number N p is the number of removed blocks in each iteration. N p generally controls the general porosity of the model. Figure 9 shows the variation of apparent permeability with pressures in different iteration numbers. The non-Darcy flow effect increases as the average flow pressure decreases during the percolation process. As the N p increases, the Darcy and non-Darcy permeabilities increase linearly. In the iteration process, as the number of removed pores in each iteration N p increases, the cross section of the model increases linearly, thereby making the permeability of the model increase linearly at the same time.
70 Ns=0 Ns=2 Ns=4
65 60 Permeability(nD)
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55 50 45 40 35 30 0
1
2
3
4
5
P (MPa) Figure 11. variation of apparent permeability with pressures in different solid block number Apparent permeability also varies with solid block number
N s . In contrast to N p , as N s
increases, the pore number of the fractal model decreases because some blocks will never part in the ACS Paragon Plus Environment
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iteration process, thereby decreasing the porosity of the model.
N s should be no less than zero and no
2 greater than F − N p . The variation of the apparent permeability with pressures in different solid block
N s increases, the apparent permeability decrease. While the influence on permeability is not as great as that of N p because N s increases from 0 to 4, the variation of
numbers are shown in Fig. 11. As of
Ns
permeability is significantly smaller than N p .
N s slightly influences the non-Darcy permeability.
Unit number is important for permeability. To make the model understandable, we choose organic matter part. Parameters are from organic matter calculation [Table 3]. Figure 12 shows the influence of Unit numbers. From Fig. 12(a), cumulative distribution of pores can follow SEM result with all three units, lack of any sub-units will deviate cumulative curve. As sub-units increase, more tiny pores will be generated, making permeability increasing, especially under low pressure [Fig. 12(b)]. 200 SEM Unit A+B+C Unit A+B Unit A
0.25 0.20 0.15 0.10
Unit A+B+C Unit A+B Unit A
180 160 Permeability (nD)
0.30
Pore fraction
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140 120 100 80 60
0.05
40
0.00 0
50
100
150
200
20
0
1
Diamater (nm)
(a)
2
3
4
5
P (MPa)
(b)
Figure 12. variation of apparent permeability with pressures in different iteration time
5 UPSCALING The next process is upscaling. The area of the IFM is the same as that in the SEM image, i.e., approximately 0.4 µm×0.4 µm, which is relatively smaller than that of the actual rock sample. Upscaling should be taken into consideration. On the basis of capillary bundle theory, IFM units are spliced with each other. On the edge of the image, some pores may not be complete. To avoid this illogical incompleteness, the mirror image method is used. Vertical neighbor images have been flipped up to down, whereas horizontal neighbor images have been flipped left to right, as shown in Fig. 13.
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(a)
(b) (c) Figure 13. upscaling process by mirror image method
To examine the validation of RES after upscaling, we calculated the variation of the average grayscale with pixels on the edge, as shown in Fig. 14(b). The original points to be enlarged are selected from four corners [Fig. 14(a)]. The squares’ numbers on the edge of the upscaled model should be odd. Therefore, when the research area is larger than 400 pixels on the edge, the average grayscale from different directions tend to be stable, which means that the area of IFM can be regarded as an RES of the upscaled image. 200
···
Average grayscale
200
···
··· 0.4mm
Average grayscale
180
···
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160
direction1 direction2 direction3 direction4
180 160 140 120 100
direction1 direction2 direction3 direction4
80 60 0
140
200
400
600
800
1000
Pixels
120
100
80 0
2cm
10000
20000
30000
40000
50000
Pixels
(a)
(b)
Figure.14 principals of upscale and verification of RES During the upscaling process, the variation of apparent gas permeability should also be carefully considered. In a valid upscaling process, the apparent gas permeability should be stable with the increase of the research area. As described, the apparent gas permeability model is regarded as a capillary bundle with the cross section of IFM, similar to that in Fig. 15(a). Apparent gas permeability literally represents the permeability of the model in unit area. When the restricted research area doubles or triples in size, pore distribution and porosity will remain stable. In the upscaling process, the apparent gas permeability of the model is calculated as shown in Fig. 15(b). The environment parameters are set the same as those of the parameters in Table 2. Figure 15(b) shows that the calculated apparent gas permeability remains stable as the ACS Paragon Plus Environment
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edge length increases from 0.2 mm to 20 mm. In conclusion, the upscaling process is valid. 140 Apparent gas permeability(nD)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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130
SEM1
120
SEM2
110 100 90 80 0
2
4
6
8
10 12 14 16 18 20
Edge length(mm)
(a) (b) Figure 15. Variation of calculated apparent gas permeability in the upscaling process. We conducted experimental verification to examine the validation of this model. Experimental and IFM calculation results are shown in Table 5. Model results are generally close to the experimental data. Meanwhile, some differences also exist. These differences mainly caused by two processes. First is the simulation of pore size distribution. It can be observed in Fig.5 that simulated curve is not exactly coincide with real one. The second is fractal tortuosity calculation process. Although simplification methods based on fractal theory was used, it can significantly shorten the calculation time and provide a promising way for apparent gas permeability calculation. The method in this work also has some room for improvement for the 3D permeability calculation process and we are working on the process. For some marine shale samples, to observe some organic pores below the resolution of SEM image (2.5nm), other observation methods maybe useful, like BET or NLDFT method in nitrogen adsorption-desorption results. A ‘Splice’ method to combine pore size distribution result from NLDFT method and SEM method is innovating in future work. Table 5. Comparison between experimental and IFM permeability data Korg
Kpy
Kslit
KIFM= Korg+ Kpy+ Kslit
Kexp_avg
Error
SEM 1
76.67nD
19.90nD
9.13nD
105.70nD
91.36nD
13.57%
SEM 2
84.97nD
12.78nD
14.13nD
111.88nD
91.36nD
18.34%
SEM 3
137.13nD
2.32nD
1642nD
1784.45nD
2521.7nD
29.52%
SEM 4
6.17nD
1.17nD
1.24nD
8.58nD
6.24nD
37.50%
6 CONCLUSIONS In this paper, the IFM for organic-rich shale is successfully built based on the SEM results of shale samples in Longmaxi Marine Shale Formation of Lower Silurian in the Sichuan Basin. The IFM model contains pores in organic matter, pyrites, and inorganic slits. In the IFM, the influence of fractal parameters on gas permeability is explored. The accuracy and reliability of the upscale process are examined by experiment verification. The apparent gas permeability of the model matches the experimental results. Upscaling is improved by using fractal characteristics. Findings show that obtaining fractal parameters from SEM images is valid. The IFM model takes ACS Paragon Plus Environment
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advantage of the detailed information at the micro-nanoscale and the distribution of the component in a large observation area. The calculation process for fractal parameters is efficient and easy, thus establishing a good foundation for percolation calculation. Non-Darcy effects in gas flow should be taken into account in micro-nanopores under low pressure. Higher iteration times i , removed number N p , and lower solid block number
N s are helpful for increase of permeability. The upscaling process is valid and easy in IFM.
With the use of image flipping method, a small area of IFM based on the SEM image is upscaled to a larger surface. This IFM-based upscaling method can significantly simplify apparent gas permeability calculation for the upscaled model, which may promote the method in the future.
ACKNOWLEDGEMENTS This work is supported by the National Natural Science Foundation of China (Grant No. 41690132), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB10020302), the National Natural Science Foundation of China (Grant No. 41574129) and the National Program on Key Basic Research Project (973 Program, Grant No. 2014CB239004).
4 NOMENCLATURE
Ares = area of organic RES Asum= area of IFM
DT = fractal dimension in length F A = iteration factor of unit A in bacic IFU model
i A = iteration times of unit A in bacic IFU model kapp_ org = equivalent gas permeability of organic matter pores in IFM, m2 kapp _ py = equivalent gas permeability of pyrite pores in IFM, m2 kapp_ slit = equivalent gas permeability of slit, m2 L = length of tube, m
L0 = straight length between two sides, m M = molar mass of natural gas kg/kmol
nA , nB , nC =
number of each units in basic IFU model
N Asoild , N s = number of squares that always remain solid in every iteration in unit A of basic IFU model
NAp = number of squares that remove from every iteration in unit A of basic IFU model n
py
= the number of pyrite in IFM
Pin , Pout = inlet pressure and outlet pressures, Pa pavg = average gas pressure, Pa QA = Gas flow rate in unit A
Qres−org = Gas flow rate of organic matter pores in RES
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R = gas constant, J/mol/K
T = temperature, K
VA , VB , VC = volume(3D)/area(2D) of each unit in basic IFU model VAk _ accumulate= total volume/area of void space before kth iteration
Vres = volume(3D)/area(2D) of the IFU model
α
= tangential momentum accommodation coefficient
λAmax= edge length of the largest void area in unit A of basic IFU model, m λ Ai = edge length of the void area in i A th iteration, m λn = edge of square tube in single tube, m µ = viscosity of natural gas, Pa.s ρavg = average density of natural gas, kg/m3
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