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Effect of hydraulic fracture closure on the void ratio of proppant particles in coalbed methane reservoir
Journal Pre-proof Effect of hydraulic fracture closure on the void ratio of proppant particles in coalbed methane reservoir Shengyong Hu, Shuwen Guan, Guorui Feng, Dandan Han, Yunbo Chen PII:
S1875-5100(19)30363-4
DOI:
https://doi.org/10.1016/j.jngse.2019.103111
Reference:
JNGSE 103111
To appear in:
Journal of Natural Gas Science and Engineering
Received Date: 16 July 2019 Revised Date:
14 December 2019
Accepted Date: 14 December 2019
Please cite this article as: Hu, S., Guan, S., Feng, G., Han, D., Chen, Y., Effect of hydraulic fracture closure on the void ratio of proppant particles in coalbed methane reservoir, Journal of Natural Gas Science & Engineering, https://doi.org/10.1016/j.jngse.2019.103111. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
The author Shengyong Hu is mainly responsible for the overall layout design of this article and responding to review comments. The author Shuwen Guan is mainly responsible for writing the manuscript paper and responding to review comments. The author Guorui Feng is responsible for providing financial support for the project and revising the original manuscript. Authors Dandan Han and Yunbo Chen participated in drawing the picture of this paper.
Dear Editor: Please find the manuscript entitled: “Effect of hydraulic fracture closure on the void ratio of proppant particles in coalbed methane reservoir”, by Feng et al. for your consideration in “Journal of Natural Gas Science and Engineering”. The work described was original research that has not been published previously. All the authors listed have approved the manuscript that is enclosed. Fracture closures have direct effects on the void ratio distributions of proppant particles filled in hydraulic fractures and then influence the fluid pathways of coalbed methane (CBM) within those fractures. In this paper, a numerical simulation method was used to investigate the void ratio distributions of proppant-filled fracture under different compression amounts of 0%, 5%, 10%, 15% and 20%. The results showed that the void ratio distributions had exhibited an inverted “U” shape, with a trend of decreasing first, followed by remaining constant, and then increasing from the bottom to the top of the fracture, which could be divided into three zones: bottom loose zone (BLZ), middle compaction zone (MCZ) and top loose zone (TLZ). At the same compression, the small coordination numbers in the BLZ and TLZ had accounted for a larger proportion, and the average coordination numbers and void ratios in the two zones were 0.8-0.9 and 1.3 times than those in the MCZ, respectively. In addition, the proportion of the large coordination numbers in the different layers had increased, and the layered average coordination numbers had increased linearly, which had resulted in the layered void ratios decreasing linearly with the increases in the compression amounts. Yours faithfully Corresponding author: Guorui Feng
E-mail: [email protected] Tel:+86-351-6010177 Fax:+86-351-6010177
1
Effect of hydraulic fracture closure on the void ratio of proppant particles in coalbed
2
methane reservoir
3
Shengyong Hua, c*, Shuwen Guana, Guorui Feng b, c*, Dandan Hana, Yunbo Chena a.
4 5
College of Safety and Emergency Management Engineering, Taiyuan University of Technology, Taiyuan
030024, China
6
b.
College of Mining Engineering, Taiyuan University of Technology, Taiyuan 030024, China
7
c.
Green Mining Engineering Technology Research Center of Shanxi Province, Taiyuan 030024, China
8
Abstract
9
Fracture closures have direct effects on the void ratio distributions of proppant
10
particles filled in hydraulic fractures and then influence the fluid pathways of coalbed
11
methane (CBM) within those fractures. In this paper, a numerical simulation method was
12
used to investigate the void ratio distributions of proppant-filled fracture under different
13
compression amounts of 0%, 5%, 10%, 15% and 20%. The results showed that the void
14
ratio distributions had exhibited an inverted “U” shape, with a trend of decreasing first,
15
followed by remaining constant, and then increasing from the bottom to the top of the
16
fracture, which could be divided into three zones: bottom loose zone (BLZ), middle
17
compaction zone (MCZ) and top loose zone (TLZ). At the same compression, the small
18
coordination numbers in the BLZ and TLZ had accounted for a larger proportion, and the
19
average coordination numbers and void ratios in the two zones were 0.8-0.9 and 1.3 times
20
than those in the MCZ, respectively. In addition, the proportion of the large coordination
21
numbers in the different layers had increased, and the layered average coordination numbers
22
had increased linearly, which had resulted in the layered void ratios decreasing linearly with
23
the increases in the compression amounts.
24
KEY WORDS: coalbed methane, hydraulic fracture, proppant particle, void ratio 1
_________________ *
Corresponding author. E-mail address: [email protected] (S.Y. Hu); [email protected] (G.R. Feng).
1
1. Introduction
2
Coalbed methane (CBM), which is widely distributed around the world, is an enormous
3
resource (Wang et al. 2018, Kong et al. 2017). There are many countries with rich CBM, such
4
as the United States, Russia, Canada, China and Australia (Huang et al. 2019, Boger et al.
5
2014). However, it has been found that most coal reservoirs generally have low permeability
6
(Huang et al. 2019, Li et al. 2018, Kang et al. 2018). In recent years, hydraulic fracturing
7
technologies which utilize surface-well methods have become effective ways to improve the
8
permeability of coal reservoirs, and subsequently enhance the production levels for CBM
9
(Zhang 2014, Wang et al. 2018, Liu et al. 2019). During hydraulic fracturing treatments, the
10
proppant particles are usually injected to already formed open fractures (Suri et al. 2019).
11
Once pumping is ceased, the pressure inside the fractures decreases, which narrows the
12
fracture width under closure stress (Wang and Elsworth, 2018). The proppants keep the
13
fractures open and improve the fracture conductivity (Wang and Sharma 2018), as shown in
14
Figure 1. The void ratio distributions of proppant particles in proppant-filled fractures under
15
different compression amounts can directly affect the flow paths of the fluids (including
16
methane and water) through the fractures, thereby significantly affecting the variation of
17
permeability or conductivity of those fractures. In addition, it has been assumed that they will
18
significantly affect the production of CBM. Therefore, in order to improve the CBM
19
production results, it has become necessary to study the effect of hydraulic fracture closure on
20
the void ratio of proppant particles in a proppant-filled fracture.
21
Recently, major efforts have been undertaken to study the permeability, conductivity
22
variations of proppant-filled fractures and hydraulic behavior of particles. For example, Li et 2
1
al. (2017) investigated the permeability changes of fractures supported by placed proppants
2
under various compression strengths by establishing a new constitutive model. Also, Guo et al.
3
(2017) studied the permeability variations of proppant-filled fractures under different
4
compression states using a CT scanning method. In another related study, Santos et al. (2018)
5
developed a coupled numerical model for the evaluations of fracture conductivity variations
6
under different compression statuses. Tan et al. (2018) investigated the permeability changes
7
of a proppant-filled fractures under various load conditions using a microscopic X-ray
8
computed tomography method. In the studies conducted by Hou et al. (2017), an experimental
9
model was established to investigate the conductivity variations of fractures with
10
discontinuous proppant distributions under different loading conditions. Also, Tan et al. (2017)
11
studied the permeability changes of natural fractures supported with proppants at different
12
compression stages by establishing an analytical model. A Discrete Element Method was
13
adopted by Shamsi et al. (2017) to investigate the conductivity variations of fractures filled
14
with proppants at various compression states. Cao et al. (2016) proposed new prediction
15
models to study the conductivity variations of fractures under different closure conditions in
16
which a digital image processing method was utilized. The permeability variations of
17
fractures filled with proppant packs at various closure conditions were examined by Neto et al.
18
(2015), using a simple mathematical model. Khanna et al. (2012) proposed a simplified
19
theoretical approach to study the conductivity variations of narrow fractures filled with sparse
20
monolayers of proppant particles under various loading conditions. In another related study,
21
Bolintineanu et al. (2017) examined the conductivity variations of fractures under the
22
different closure degrees by adopting both discrete element simulation and finite element flow 3
1
simulation methods. Kamalgharibi et al. (2016) and Nikkhah et al. (2015; 2015)
2
experimentally investigated the effects of various parameters on the stability of spherical
3
nanoparticles dispersed in different base fluids. Sarafraz et al. (2014) studied the convective
4
boiling characteristics of dilute dispersions of nanoparticles in a base fluid under various
5
operating conditions; Sarafraz and Peyghambarzadeh (2012) experimentally investigated the
6
effects of different volumetric concentrations of nanoparticles on nucleate boiling heat
7
transfer. Also, Salari et al. (2015; 2016) conducted an experimental study to investigate the
8
effects of different diameters and mass concentration in different nano-fluids on the variation
9
of heat transfer coefficient. And Arya et al. (2017) investigated the impacts of various mass
10
concentration of carbon nanotubes and filling ratio of working fluid on thermal performance
11
of heat pipe by using an experiment method. However, most researchers had mainly focused
12
on the entire permeability or conductivity variation of the proppant-filled fracture under
13
various closure degrees, while the effects of different compression amounts on the layered
14
and overall void ratio distributions of the proppant-filled fractures as the basis of studying the
15
fluid migrations within fractures and entire permeability and conductivity variation,
16
respectively, had been ignored by most scholars.
17
In this study, a Lagrange framework and a discrete element method were developed to
18
investigate the void ratio distributions of proppant particles filled in a fracture at various
19
compression amounts of 0%, 5%, 10%, 15% and 20%, respectively. Also, the effects of the
20
different compression amounts on the overall and layered void ratio distributions of the
21
proppant particles in a proppant-filled fracture were extracted, which were analyzed using the
22
coordination number. 4
1
2. METHOD
2
2.1 Mathematical model
3
Figure 2 shows the mechanical conditions between particle i and particle j. The contact
4
force (FCij) between the proppant particle i and the proppant particle j was written as
5
follows: = +
6 7 8
(1)
Then, the normal contact force (Fnij) between the proppant particles i and j was as follows: = − − ∙
9
(2)
10
In which k represents the normal elastic coefficient; is the normal deformation
11
of the proppant particles; indicates the normal damping coefficient; represents the
12
relative speed between i and j, m/s; and denotes the unit normal vector.
13
The tangential contact force (Ftij) between i and j was given by the following: = − −
14 15
(3)
The slip speed ( ) of the proppant particles could then be written as follows: = − ∙ + + ×
16
(4)
17
In which k indicates the tangential elastic coefficient; is the tangential
18
deformation of the proppant particles; represents the tangential damping coefficient; r
19
denotes the position of the center of gravity of the proppant particle; and and indicate
20
the rotational angular velocities of i and j, rad/s, respectively.
21 22
Therefore, when > ，the tangent forces (Ftij) between i and j can be written as follows: 5
= −
1
2
Where μ represents the sliding friction coefficient; the unit vector
3
Then, the line speed (U) of the proppant particle was as follows:
6 7
10 11 12
=
. "
(6)
Where $ denotes the initial velocity of the proppant particle, m/s; % indicates the proppant particle mass acceleration, m/s2; and ∆t is the time-step, s. Then, the angular velocity (ω) of the proppant particle could be written as follows: ω = $ + % ∆'
8 9
= $ + %∆'
4 5
(5)
(7)
Where $ denotes the initial angular velocity of the proppant particle, rad/s; and % represents the angular acceleration of the proppant particle, )*- , . + It was then determined that the joint force (F(t)) of the proppant particle at t time was as follows: .'/ + 01 = m%.'/ + α0.'/
13
(8)
14
Where m indicates the mass of the proppant particle, kg; g denotes the gravity
15
acceleration, m/s2; %.'/ represents the acceleration of the proppant particle at t time; .'/
16
is the velocity of the proppant particle at t time; and 3 indicates the energy dissipation..
17 18
The combined moment (T (t)) of the proppant represents the energy dissipation particle at t time was as follows:
19
4.'/ = 5ω%.'/ + 35ω.'/
20
In which I indicates the moment of inertia of the proppant particle, kg·m2; ω%.'/
21
represents the angular acceleration of the proppant particle at t time, )*- , ; and ω(t) is the +
22
angular velocity of the proppant particle at t time, )*-+. 6
(9)
1 2
∆
Therefore, at '7 = '$ + , , the line speed (.'7 /) and the angular velocities (ω.'7 /) of the proppant particles were as follows:
3
.'7 / = 8'$ −
∆ 9+ ,
%.'$ /∆'
(10)
4
ω.'7 / = ω 8'$ −
∆ 9+ ,
ω%.'$ /∆'
(11)
5 6
∆ ,
It was determined that at ', = '7 + ，the displacements (.', /) of the proppant particles were as follows:
7
.t , / = .'$ / + %.'7 /∆'
(12)
8
ω.t , / = ω.'$ / + ω%.'7 /∆'
(13)
9
Where %.'$ / and %.'7 / represent the acceleration of proppant particles at t $ and
10
t7 time, respectively, m/s2; and ω%.'$ / and ω%.'7 / are the angular accelerations of
11
proppant particles at t $ and t7 time, respectively, )*- , . +
12
2.2 Physical model
13
In the present study, the proppant particles were spherical particles. The quartz sand
14
proppant particles with a typical and common diameter (D) range of 0.425-0.850 mm were
15
selected for the simulation process (Fu et al. 2016; Zhang et al. 2019; Peters et al. 2017).
16
The mechanical parameters of the aforementioned proppant particles are shown in Table 1.
17
The model dimensions of 177.8 mm length, 38.1 mm width and 5.26mm thickness (in
18
Figure 3) were established according to the dimensions of the API (American Petroleum
19
Institute) conductivity cell (Santos et al. 2010; Hou et al. 2017; Liang et al. 2018). In
20
addition, the proppant particles filled in the fracture at various compression amounts (ε) (the
21
proportion of the proppants’ reductions in height to the original proppants’ heights) of 0%,
22
5%, 10%, 15%, and 20%, respectively, were simulated. Figure 4 shows the process of 7
1
model establishment. According to the Figure 4, first, proppant particles were generated in
2
the fracture, particles’ positions and velocities information was initialized, the load was
3
applied on the fracture and a reasonable time step was set. Then, the Discrete Element
4
Method was used to calculate the forces of particles (including the gravities, contact forces
5
between particles, forces between particles and the fracture). Next, using the Newton’s
6
second law obtained the accelerations, velocities and positions of particles, and the
7
computing time was updated. After that, the particles’ states under different compression
8
amounts were obtained by time step iteration. Finally, once the particles achieved the
9
designed final compression, the computation completed.
10
3. RESULTS AND DISCUSSION
11
3.1 Void ratios of the different layers
12
The model illustrated in Figure 3 was equally re-divided into nine thin layers along the
13
Z direction under different compression amounts. In the current study, the number of
14
particles directly connected to the aimed particle was referred to as the coordination number
15
of the aimed particle (German 2014). Also, the coordination numbers of proppant particles
16
were used to analyze the void ratio distributions of the proppant particles in a
17
proppant-filled fracture under different compression amounts.
18
Figure 5 shows the layered coordination number frequency distributions of the
19
proppant particles. As can be seen in the figure, as the coordination numbers of the proppant
20
particles increased, the coordination number distribution frequency had also basically
21
increased. This was followed by a decreasing trend. It was found that the small coordination
22
numbers had accounted for a large proportion of Layer 1 and Layer 9. Meanwhile, the large 8
1
coordination numbers had accounted for a large proportion on the remaining layers. The
2
average coordination numbers of the proppant particles in Layer 1 and Layer 9 were
3
observed to be relatively small, which had resulted in the void ratios of the proppant
4
particles being higher in those two layers.
5
Figure 6 shows the three-dimensional coordination number distributions of the
6
proppant particles in the proppant-filled fracture. It can be seen in the figure that when the
7
compression amount was 0%, the proppant particles were in a natural accumulation state.
8
Moreover, the overall coordination number distributions of the proppant particles were
9
observed to be basically the same from the bottom to the top of the proppant-filled fracture.
10
It was found that as the compression amounts increased, the coordination numbers of the
11
proppant particles had first increased, and then decreased from the bottom to the top of the
12
proppant-filled fracture. The blue particles shown in Figures. 6(b) to 6(e) can be seen to be
13
distributed in the bottom-most and top-most areas, which indicates that the coordination
14
numbers of the proppant particles in the bottom-most and top-most areas of the fracture had
15
remained smaller. This had caused the void ratios of the proppant particles in the
16
bottom-most and top-most areas of the fracture to be higher.
17
Figure 7 illustrates the void ratio distributions of the proppant particles of the different
18
layers in the proppant-filled fracture. Figure 8 shows this study’s comparison of void ratios
19
and average coordination numbers of the proppant particles. As detailed in Figures 7 and 8,
20
the void ratios of the proppant particles had first decreased, then remained constant, and
21
then increased from the bottom to the top of the proppant-filled fracture. Also, the void ratio
22
distribution within the proppant-filled fracture had exhibited an inverted “U” shape. In this 9
1
study, the proppant particles in the fracture were divided into three zones as follows: bottom
2
loose zone (BLZ), middle compaction zone (MCZ) and top loose zone (TLZ). Meanwhile,
3
the average coordination number distributions of the proppant particles had presented an
4
exactly opposite trend as follows: first increasing, remaining constant, and then decreasing.
5
It was observed that the average coordination numbers of the proppant particles in Layer 1
6
and Layer 9 were the lowest, while the void ratios of the proppant particles were found to be
7
the highest in those layers. This was followed by the proppant particles in Layer 2 and
8
Layer 8, which has exhibited higher average coordination numbers and smaller void ratios.
9
It was determined that this was due to the increases in the average coordination numbers of
10
proppant particles causing decreases in the void ratios of the proppant particles. In particular,
11
the small coordination numbers of the proppant particles in Layer 1 and Layer 9 had
12
accounted for a large proportion, and the average coordination numbers of the proppant
13
particles were smaller. This had the effects of a lower total number of proppant particles
14
contacting each other and fewer connections between the proppant particles. As a result, the
15
channels formed by the gaps between the proppant particles which provided paths for fluid
16
flow were wider, and the void ratios of proppant particles in those areas were higher. In
17
contrast, the large coordination numbers of the proppant particles in the remaining layers
18
had accounted for a large proportion, and the average coordination numbers of proppant
19
particles were larger. Furthermore, the total number of proppant particles which had come in
20
contact with each other had increased and there were more connections formed between the
21
proppant particles. This had caused the channels formed by the gaps between the proppant
22
particles, in order to provide paths for fluid flow, to become narrower, and the void ratios of 10
1
the proppant particles in those areas were noticeably smaller.
2
3.2 Overall void ratios
3
Figure 9 shows the coordination number frequency distributions of the proppant
4
particles under different compression amounts. It can be seen in the figure that the main
5
distribution ranges of the coordination numbers had moved in the direction of the large
6
coordination numbers with the increases in the compression amounts. Also, the proportion
7
of the large coordination numbers had increased, which then caused the average
8
coordination numbers of the proppant particles to also increase. Subsequently, the void
9
ratios of the proppant particles became smaller.
10
Figure 9 details the manners in which the connections between the proppant particles
11
had recombined when the compression amounts had changed. Specifically, the connections
12
between some of the proppant particles were interrupted, and some of the proppant particles
13
had also made new connections. By taking Figure 9(a) as an example, it can be seen that
14
when the coordination number of the proppant particles was 0, the distribution frequencies
15
were 0.07 and 0 at the compression amounts of 0% and 5%, respectively. Therefore, it was
16
assumed that the coordination number 0 of the proppant particles with the distribution
17
frequency of 0.07 had indicated that new connections had been established and coordination
18
numbers of 1 or more had been reached. These results had indicated that proppant particles
19
had recombined, and some of the proppant particles which had originally shown no contacts
20
were now in contact with other proppant particles. Also, it was precisely those newly
21
established connections which had caused the originally open channels between the
22
proppant particles to become closed. It was observed that when the compression amounts 11
1
were increased to 10% and 15%, the distribution frequencies of the coordination numbers
2
10, 11, and 12 were 0.201, 0, 0 and 0, 0, 0.016, respectively. It was found that with the
3
distribution frequency of 0.021 of the coordination number 10, it could be assumed that
4
there were proppant particles of 0.016 adding connections to the distribution frequency of
5
the coordination number 12. Therefore, the remaining distribution frequency 0.005 was not
6
included in the coordination numbers 10, 11, and 12, and must then be returned to the
7
coordination numbers below 10. When the compression amount was increased to 20%, it
8
was found that the distribution frequency of the coordination number 12 had been reduced
9
to 0. Therefore, the distribution frequency must be returned to the coordination numbers
10
below 12. It was determined that this had indicated that the connections between some of
11
the proppant particles were interrupted and the proppant particles had become disconnected.
12
Also, it was found that those exact lost connections had caused the originally closed
13
channels between the proppant particles to be open wider. It was determined that although
14
the number of newly established connections for the proppant particles was much larger
15
than the number of lost connections with the increases in the compression amounts, the
16
proportion of the large coordination numbers had increased. The layered average
17
coordination numbers of the proppant particles had also increased, which had caused the
18
layered void ratios to become decreased.
19
Figure 10 details the relationship between the average coordination numbers of the
20
proppant particles and the compression amounts. Generally speaking, the layered average
21
coordination numbers were linearly increased with the increasing compression amounts.
22
This was due to the fact that when the compression amounts had increased, although the 12
1
connections between some proppant particles were interrupted, the majority of the proppant
2
particles had formed new connections. The proportion of the large coordination numbers
3
had become increased, and the average coordination numbers of the proppant particles had
4
also increased. It was determined that the cause was that the void ratios of the proppant
5
particles had become smaller, and it had become more difficult for the fluid to flow through
6
the channels of the proppant-filled fracture.
7
Figure 11 shows the relationship between the layered void ratios of the proppant
8
particles and the compression amounts. As can be seen in the figure, the layered void ratios
9
of the proppant particles had tended to decrease linearly with the increases in the
10
compression amounts. This was due to the fact that the proppant particle system was in a
11
natural accumulation state when the compression amount was 0%. Then, as the compression
12
amounts increased, due to the mutual squeezing effects between the proppant particles, and
13
the proppant particle system had become denser. The number of newly established
14
connections for the proppant particles was much larger than the number of lost connections,
15
and the average coordination numbers of the proppant particles had increased. As a result,
16
only a few channels were open. Meanwhile, the majority of the fluid channels formed by
17
the gaps between the proppant particles in order to provide pathways for the methane and
18
gas fluid had become narrow or even closed. This had resulted in the void ratios of the
19
proppant particles becoming smaller, and the flow of fluids had become more difficult
20
within the channels of the proppant-filled fracture.
21
Figure 12 details this study’s comparison of the average coordination numbers of the
22
proppant particles in the three examined zones. Figure 13 shows this study’s comparison of 13
1
the void ratios of the proppant particles within the three zones under the different
2
compression amounts. It can be seen in the figures that the average coordination numbers of
3
the proppant particles had decreased while the void ratios of the proppant particles had
4
increased with the increases in the compression amounts. The coordination numbers of the
5
proppant particles in the MCZ were observed to be the highest. Meanwhile, the void ratios
6
of proppant particles in the MCZ were found to be the lowest at the same compression
7
amounts. This was determined to be due to the fact that the large coordination numbers of
8
the proppant particles in the MCZ had accounted for a large proportion and the average
9
coordination numbers of the proppant particles were found to be relatively high. There were
10
more connections between the proppant particles and less open channels between proppant
11
particles in that zone, which had caused the void ratios of the proppant particles in the MCZ
12
to be the lowest. As the compression amounts had been increased, the number of contacts
13
between the particles had generally increased. Also, the average coordination numbers had
14
generally increased and the open channels had decreased. This had resulted in decreases in
15
the void ratios. In addition, the gaps between void ratios of the proppant particles in the
16
BLZ, TLZ and MCZ had increased. This was due to the fact that the gaps between the
17
average coordination numbers of the proppant particles in the BLZ, TLZ and MCZ had
18
increased. The proppant particles in the MCZ had become more connected and fewer
19
channels were now open. Finally, the average coordination numbers and void ratios of the
20
proppant particles in the BLZ and TLZ were determined to be 0.8-0.9 and 1.3 times greater
21
than those in the MCZ, respectively.
14
1
3.3 Model verifications
2
Figure 14 shows the comparison of the numerical simulations and the existing
3
experimental results. It can be seen in the figure that the line features of the numerical
4
simulation results and the existing experimental results (Ma et al. 2014) were the same. In
5
other words, the overall void ratio of the particles had linearly decreased with the increases
6
in the compression amounts, and the relationship between the numerical simulation result
7
line and the other three lines of the existing experimental results was almost parallel.
8
Moreover, it was found that the void ratio of the particles had increased with the increases
9
in the ranges of the particle sizes. Therefore, based on the validation conducted above, it is
10
shown that the model generally provides relatively good results, and the model results are
11
considered to have successfully illustrated that the proppant particles shapes, model
12
parameters, and method used in this research study are feasible.
13
4. CONCLUSIONS
14
1. The void ratios of the proppant particles had first decreased, then remained constant,
15
finally increased from the bottom to the top of the proppant-filled fracture. Also, the void
16
ratio distribution of the proppant particles in the fracture had exhibited an inverted “U”
17
shape.
18
2. The proppant particles within the fracture could be divided into three zones: BLZ,
19
MCZ and TLZ. At the same compression, the small coordination numbers in the BLZ and
20
TLZ had accounted for a larger proportion, and the average coordination numbers and void
21
ratios in the two zones were 0.8-0.9 and 1.3 times than those in the MCZ, respectively.
22
3. As the compression amounts increased, the number of newly established 15
1
connections for the proppant particles was found to be much larger than the number of lost
2
connections. Also, the proportion of the large coordination numbers in the different layers
3
increased, and the layered average coordination numbers increased linearly, which resulted
4
in the layered void ratios decreasing linearly.
5
ACKNOWLEDGEMENTS
6
This research was supported by the Joint Funds of the National Natural Science
7
Foundation of China (U1710258, U1710121). This work was also supported by the Program
8
for the Outstanding Innovative Teams of Higher Learning Institutions of Shanxi and the
9
Training Program of First-class Discipline for Young Academic Backbone of Taiyuan
10
University of Technology.
11
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Wang, H. Y., Sharma, M. M. (2018). Modeling of hydraulic fracture closure on proppants with proppant settling. Journal of Petroleum Science and Engineering, 171, 636-645. Wang, J. H., Elsworth D. (2018). Role of proppant distribution on the evolution of hydraulic fracture conductivity. Journal of Petroleum Science and Engineering, 166, 249-262.
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Zhang, H. J., Liu, S. H., Xiao, H. P. (2019). Tribological properties of sliding quartz sand
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particle and shale rock contact under water and guar gum aqueous solution in hydraulic
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fracturing. Tribology International, 129, 416-426.
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Zhang, J. C. (2014). Numerical simulation of hydraulic fracturing coalbed methane reservoir. Fuel, 136, 57-61.
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Zheng, W. B., Tannant, D. D. (2019). Influence of proppant fragmentation on fracture
17
conductivity -Insights from three-dimensional discrete element modeling. Journal of
18
Petroleum Science and Engineering, 177, 1010-1023.
19
21
1
Figure Captions
2
Figure 1. CBM fluid in a proppant-filled fracture during the application of hydraulic
3
fracturing technology.
4
Figure 2. Mechanical conditions between particle i and particle j.
5
Figure 3. Three-dimensional physical compression model of a proppant-filled fracture.
6
Figure 4. Numerical simulation flow chart of establishing the model.
7
Figure 5. Layered coordination number frequency distributions of the proppant particles: (a)
8
ε = 0%; (b) ε = 5%; (c) ε = 10%; (d) ε = 15%; (e) ε = 20%.
9
Figure 6. Three-dimensional coordination number distributions of the proppant particles: (a)
10
ε = 0%; (b) ε = 5%; (c) ε = 10%; (d) ε = 15%; (e) ε = 20%.
11
Figure 7. Void ratio distributions of the proppant particles of the different layers.
12
Figure 8. Comparison results of the void ratios and average coordination numbers of the
13
proppant particles: (a) ε = 0%; (b) ε = 5%; (c) ε = 10%; (d) ε = 15%; (e) ε = 20%.
14
Figure 9. Coordination number frequency distributions under the different compression
15
amounts: (a) Layer 1; (b) Layer 2; (c) Layer 3; (d) Layer 4; (e) Layer 5; (f) Layer 6; (g)
16
Layer 7; (h) Layer 8; (i) Layer 9.
17
Figure 10. Relationship between the average coordination numbers and the compression
18
amounts.
19
Figure 11. Relationship between the void ratios of the proppant particles and the
20
compression amounts.
21
Figure 12. Comparison of the average coordination numbers of the proppant particles in the
22
three zones. 22
1
Figure 13. Comparison of the void ratios of the proppant particles in the three zones.
2
Figure 14. Comparison of the numerical simulations and the existing experimental results
3
(Ma et al. 2014).
4
Table Caption
5
Table 1. Mechanical parameters of the proppant particle.
23
Parameters
Values
Reference used
Particle density
2,650 kg/m3
Zheng and Tannant 2019
Gravity acceleration
9.81 m/s2
Li et al. 2018
Poisson’s ratio
0.17
Reinicke et al. 2010
Numerical simulation method was used to study compression of proppants.
Void ratio distribution of the proppants exhibited an inverted “U” shape.
Bottom and top void ratios of the proppants were higher.
Bottom and top average coordination numbers of the proppants were smaller.
Declaration of Interest Statement： We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.
S1875-5100(19)30363-4
DOI:
https://doi.org/10.1016/j.jngse.2019.103111
Reference:
JNGSE 103111
To appear in:
Journal of Natural Gas Science and Engineering
Received Date: 16 July 2019 Revised Date:
14 December 2019
Accepted Date: 14 December 2019
Please cite this article as: Hu, S., Guan, S., Feng, G., Han, D., Chen, Y., Effect of hydraulic fracture closure on the void ratio of proppant particles in coalbed methane reservoir, Journal of Natural Gas Science & Engineering, https://doi.org/10.1016/j.jngse.2019.103111. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
The author Shengyong Hu is mainly responsible for the overall layout design of this article and responding to review comments. The author Shuwen Guan is mainly responsible for writing the manuscript paper and responding to review comments. The author Guorui Feng is responsible for providing financial support for the project and revising the original manuscript. Authors Dandan Han and Yunbo Chen participated in drawing the picture of this paper.
Dear Editor: Please find the manuscript entitled: “Effect of hydraulic fracture closure on the void ratio of proppant particles in coalbed methane reservoir”, by Feng et al. for your consideration in “Journal of Natural Gas Science and Engineering”. The work described was original research that has not been published previously. All the authors listed have approved the manuscript that is enclosed. Fracture closures have direct effects on the void ratio distributions of proppant particles filled in hydraulic fractures and then influence the fluid pathways of coalbed methane (CBM) within those fractures. In this paper, a numerical simulation method was used to investigate the void ratio distributions of proppant-filled fracture under different compression amounts of 0%, 5%, 10%, 15% and 20%. The results showed that the void ratio distributions had exhibited an inverted “U” shape, with a trend of decreasing first, followed by remaining constant, and then increasing from the bottom to the top of the fracture, which could be divided into three zones: bottom loose zone (BLZ), middle compaction zone (MCZ) and top loose zone (TLZ). At the same compression, the small coordination numbers in the BLZ and TLZ had accounted for a larger proportion, and the average coordination numbers and void ratios in the two zones were 0.8-0.9 and 1.3 times than those in the MCZ, respectively. In addition, the proportion of the large coordination numbers in the different layers had increased, and the layered average coordination numbers had increased linearly, which had resulted in the layered void ratios decreasing linearly with the increases in the compression amounts. Yours faithfully Corresponding author: Guorui Feng
E-mail: [email protected] Tel:+86-351-6010177 Fax:+86-351-6010177
1
Effect of hydraulic fracture closure on the void ratio of proppant particles in coalbed
2
methane reservoir
3
Shengyong Hua, c*, Shuwen Guana, Guorui Feng b, c*, Dandan Hana, Yunbo Chena a.
4 5
College of Safety and Emergency Management Engineering, Taiyuan University of Technology, Taiyuan
030024, China
6
b.
College of Mining Engineering, Taiyuan University of Technology, Taiyuan 030024, China
7
c.
Green Mining Engineering Technology Research Center of Shanxi Province, Taiyuan 030024, China
8
Abstract
9
Fracture closures have direct effects on the void ratio distributions of proppant
10
particles filled in hydraulic fractures and then influence the fluid pathways of coalbed
11
methane (CBM) within those fractures. In this paper, a numerical simulation method was
12
used to investigate the void ratio distributions of proppant-filled fracture under different
13
compression amounts of 0%, 5%, 10%, 15% and 20%. The results showed that the void
14
ratio distributions had exhibited an inverted “U” shape, with a trend of decreasing first,
15
followed by remaining constant, and then increasing from the bottom to the top of the
16
fracture, which could be divided into three zones: bottom loose zone (BLZ), middle
17
compaction zone (MCZ) and top loose zone (TLZ). At the same compression, the small
18
coordination numbers in the BLZ and TLZ had accounted for a larger proportion, and the
19
average coordination numbers and void ratios in the two zones were 0.8-0.9 and 1.3 times
20
than those in the MCZ, respectively. In addition, the proportion of the large coordination
21
numbers in the different layers had increased, and the layered average coordination numbers
22
had increased linearly, which had resulted in the layered void ratios decreasing linearly with
23
the increases in the compression amounts.
24
KEY WORDS: coalbed methane, hydraulic fracture, proppant particle, void ratio 1
_________________ *
Corresponding author. E-mail address: [email protected] (S.Y. Hu); [email protected] (G.R. Feng).
1
1. Introduction
2
Coalbed methane (CBM), which is widely distributed around the world, is an enormous
3
resource (Wang et al. 2018, Kong et al. 2017). There are many countries with rich CBM, such
4
as the United States, Russia, Canada, China and Australia (Huang et al. 2019, Boger et al.
5
2014). However, it has been found that most coal reservoirs generally have low permeability
6
(Huang et al. 2019, Li et al. 2018, Kang et al. 2018). In recent years, hydraulic fracturing
7
technologies which utilize surface-well methods have become effective ways to improve the
8
permeability of coal reservoirs, and subsequently enhance the production levels for CBM
9
(Zhang 2014, Wang et al. 2018, Liu et al. 2019). During hydraulic fracturing treatments, the
10
proppant particles are usually injected to already formed open fractures (Suri et al. 2019).
11
Once pumping is ceased, the pressure inside the fractures decreases, which narrows the
12
fracture width under closure stress (Wang and Elsworth, 2018). The proppants keep the
13
fractures open and improve the fracture conductivity (Wang and Sharma 2018), as shown in
14
Figure 1. The void ratio distributions of proppant particles in proppant-filled fractures under
15
different compression amounts can directly affect the flow paths of the fluids (including
16
methane and water) through the fractures, thereby significantly affecting the variation of
17
permeability or conductivity of those fractures. In addition, it has been assumed that they will
18
significantly affect the production of CBM. Therefore, in order to improve the CBM
19
production results, it has become necessary to study the effect of hydraulic fracture closure on
20
the void ratio of proppant particles in a proppant-filled fracture.
21
Recently, major efforts have been undertaken to study the permeability, conductivity
22
variations of proppant-filled fractures and hydraulic behavior of particles. For example, Li et 2
1
al. (2017) investigated the permeability changes of fractures supported by placed proppants
2
under various compression strengths by establishing a new constitutive model. Also, Guo et al.
3
(2017) studied the permeability variations of proppant-filled fractures under different
4
compression states using a CT scanning method. In another related study, Santos et al. (2018)
5
developed a coupled numerical model for the evaluations of fracture conductivity variations
6
under different compression statuses. Tan et al. (2018) investigated the permeability changes
7
of a proppant-filled fractures under various load conditions using a microscopic X-ray
8
computed tomography method. In the studies conducted by Hou et al. (2017), an experimental
9
model was established to investigate the conductivity variations of fractures with
10
discontinuous proppant distributions under different loading conditions. Also, Tan et al. (2017)
11
studied the permeability changes of natural fractures supported with proppants at different
12
compression stages by establishing an analytical model. A Discrete Element Method was
13
adopted by Shamsi et al. (2017) to investigate the conductivity variations of fractures filled
14
with proppants at various compression states. Cao et al. (2016) proposed new prediction
15
models to study the conductivity variations of fractures under different closure conditions in
16
which a digital image processing method was utilized. The permeability variations of
17
fractures filled with proppant packs at various closure conditions were examined by Neto et al.
18
(2015), using a simple mathematical model. Khanna et al. (2012) proposed a simplified
19
theoretical approach to study the conductivity variations of narrow fractures filled with sparse
20
monolayers of proppant particles under various loading conditions. In another related study,
21
Bolintineanu et al. (2017) examined the conductivity variations of fractures under the
22
different closure degrees by adopting both discrete element simulation and finite element flow 3
1
simulation methods. Kamalgharibi et al. (2016) and Nikkhah et al. (2015; 2015)
2
experimentally investigated the effects of various parameters on the stability of spherical
3
nanoparticles dispersed in different base fluids. Sarafraz et al. (2014) studied the convective
4
boiling characteristics of dilute dispersions of nanoparticles in a base fluid under various
5
operating conditions; Sarafraz and Peyghambarzadeh (2012) experimentally investigated the
6
effects of different volumetric concentrations of nanoparticles on nucleate boiling heat
7
transfer. Also, Salari et al. (2015; 2016) conducted an experimental study to investigate the
8
effects of different diameters and mass concentration in different nano-fluids on the variation
9
of heat transfer coefficient. And Arya et al. (2017) investigated the impacts of various mass
10
concentration of carbon nanotubes and filling ratio of working fluid on thermal performance
11
of heat pipe by using an experiment method. However, most researchers had mainly focused
12
on the entire permeability or conductivity variation of the proppant-filled fracture under
13
various closure degrees, while the effects of different compression amounts on the layered
14
and overall void ratio distributions of the proppant-filled fractures as the basis of studying the
15
fluid migrations within fractures and entire permeability and conductivity variation,
16
respectively, had been ignored by most scholars.
17
In this study, a Lagrange framework and a discrete element method were developed to
18
investigate the void ratio distributions of proppant particles filled in a fracture at various
19
compression amounts of 0%, 5%, 10%, 15% and 20%, respectively. Also, the effects of the
20
different compression amounts on the overall and layered void ratio distributions of the
21
proppant particles in a proppant-filled fracture were extracted, which were analyzed using the
22
coordination number. 4
1
2. METHOD
2
2.1 Mathematical model
3
Figure 2 shows the mechanical conditions between particle i and particle j. The contact
4
force (FCij) between the proppant particle i and the proppant particle j was written as
5
follows: = +
6 7 8
(1)
Then, the normal contact force (Fnij) between the proppant particles i and j was as follows: = − − ∙
9
(2)
10
In which k represents the normal elastic coefficient; is the normal deformation
11
of the proppant particles; indicates the normal damping coefficient; represents the
12
relative speed between i and j, m/s; and denotes the unit normal vector.
13
The tangential contact force (Ftij) between i and j was given by the following: = − −
14 15
(3)
The slip speed ( ) of the proppant particles could then be written as follows: = − ∙ + + ×
16
(4)
17
In which k indicates the tangential elastic coefficient; is the tangential
18
deformation of the proppant particles; represents the tangential damping coefficient; r
19
denotes the position of the center of gravity of the proppant particle; and and indicate
20
the rotational angular velocities of i and j, rad/s, respectively.
21 22
Therefore, when > ，the tangent forces (Ftij) between i and j can be written as follows: 5
= −
1
2
Where μ represents the sliding friction coefficient; the unit vector
3
Then, the line speed (U) of the proppant particle was as follows:
6 7
10 11 12
=
. "
(6)
Where $ denotes the initial velocity of the proppant particle, m/s; % indicates the proppant particle mass acceleration, m/s2; and ∆t is the time-step, s. Then, the angular velocity (ω) of the proppant particle could be written as follows: ω = $ + % ∆'
8 9
= $ + %∆'
4 5
(5)
(7)
Where $ denotes the initial angular velocity of the proppant particle, rad/s; and % represents the angular acceleration of the proppant particle, )*- , . + It was then determined that the joint force (F(t)) of the proppant particle at t time was as follows: .'/ + 01 = m%.'/ + α0.'/
13
(8)
14
Where m indicates the mass of the proppant particle, kg; g denotes the gravity
15
acceleration, m/s2; %.'/ represents the acceleration of the proppant particle at t time; .'/
16
is the velocity of the proppant particle at t time; and 3 indicates the energy dissipation..
17 18
The combined moment (T (t)) of the proppant represents the energy dissipation particle at t time was as follows:
19
4.'/ = 5ω%.'/ + 35ω.'/
20
In which I indicates the moment of inertia of the proppant particle, kg·m2; ω%.'/
21
represents the angular acceleration of the proppant particle at t time, )*- , ; and ω(t) is the +
22
angular velocity of the proppant particle at t time, )*-+. 6
(9)
1 2
∆
Therefore, at '7 = '$ + , , the line speed (.'7 /) and the angular velocities (ω.'7 /) of the proppant particles were as follows:
3
.'7 / = 8'$ −
∆ 9+ ,
%.'$ /∆'
(10)
4
ω.'7 / = ω 8'$ −
∆ 9+ ,
ω%.'$ /∆'
(11)
5 6
∆ ,
It was determined that at ', = '7 + ，the displacements (.', /) of the proppant particles were as follows:
7
.t , / = .'$ / + %.'7 /∆'
(12)
8
ω.t , / = ω.'$ / + ω%.'7 /∆'
(13)
9
Where %.'$ / and %.'7 / represent the acceleration of proppant particles at t $ and
10
t7 time, respectively, m/s2; and ω%.'$ / and ω%.'7 / are the angular accelerations of
11
proppant particles at t $ and t7 time, respectively, )*- , . +
12
2.2 Physical model
13
In the present study, the proppant particles were spherical particles. The quartz sand
14
proppant particles with a typical and common diameter (D) range of 0.425-0.850 mm were
15
selected for the simulation process (Fu et al. 2016; Zhang et al. 2019; Peters et al. 2017).
16
The mechanical parameters of the aforementioned proppant particles are shown in Table 1.
17
The model dimensions of 177.8 mm length, 38.1 mm width and 5.26mm thickness (in
18
Figure 3) were established according to the dimensions of the API (American Petroleum
19
Institute) conductivity cell (Santos et al. 2010; Hou et al. 2017; Liang et al. 2018). In
20
addition, the proppant particles filled in the fracture at various compression amounts (ε) (the
21
proportion of the proppants’ reductions in height to the original proppants’ heights) of 0%,
22
5%, 10%, 15%, and 20%, respectively, were simulated. Figure 4 shows the process of 7
1
model establishment. According to the Figure 4, first, proppant particles were generated in
2
the fracture, particles’ positions and velocities information was initialized, the load was
3
applied on the fracture and a reasonable time step was set. Then, the Discrete Element
4
Method was used to calculate the forces of particles (including the gravities, contact forces
5
between particles, forces between particles and the fracture). Next, using the Newton’s
6
second law obtained the accelerations, velocities and positions of particles, and the
7
computing time was updated. After that, the particles’ states under different compression
8
amounts were obtained by time step iteration. Finally, once the particles achieved the
9
designed final compression, the computation completed.
10
3. RESULTS AND DISCUSSION
11
3.1 Void ratios of the different layers
12
The model illustrated in Figure 3 was equally re-divided into nine thin layers along the
13
Z direction under different compression amounts. In the current study, the number of
14
particles directly connected to the aimed particle was referred to as the coordination number
15
of the aimed particle (German 2014). Also, the coordination numbers of proppant particles
16
were used to analyze the void ratio distributions of the proppant particles in a
17
proppant-filled fracture under different compression amounts.
18
Figure 5 shows the layered coordination number frequency distributions of the
19
proppant particles. As can be seen in the figure, as the coordination numbers of the proppant
20
particles increased, the coordination number distribution frequency had also basically
21
increased. This was followed by a decreasing trend. It was found that the small coordination
22
numbers had accounted for a large proportion of Layer 1 and Layer 9. Meanwhile, the large 8
1
coordination numbers had accounted for a large proportion on the remaining layers. The
2
average coordination numbers of the proppant particles in Layer 1 and Layer 9 were
3
observed to be relatively small, which had resulted in the void ratios of the proppant
4
particles being higher in those two layers.
5
Figure 6 shows the three-dimensional coordination number distributions of the
6
proppant particles in the proppant-filled fracture. It can be seen in the figure that when the
7
compression amount was 0%, the proppant particles were in a natural accumulation state.
8
Moreover, the overall coordination number distributions of the proppant particles were
9
observed to be basically the same from the bottom to the top of the proppant-filled fracture.
10
It was found that as the compression amounts increased, the coordination numbers of the
11
proppant particles had first increased, and then decreased from the bottom to the top of the
12
proppant-filled fracture. The blue particles shown in Figures. 6(b) to 6(e) can be seen to be
13
distributed in the bottom-most and top-most areas, which indicates that the coordination
14
numbers of the proppant particles in the bottom-most and top-most areas of the fracture had
15
remained smaller. This had caused the void ratios of the proppant particles in the
16
bottom-most and top-most areas of the fracture to be higher.
17
Figure 7 illustrates the void ratio distributions of the proppant particles of the different
18
layers in the proppant-filled fracture. Figure 8 shows this study’s comparison of void ratios
19
and average coordination numbers of the proppant particles. As detailed in Figures 7 and 8,
20
the void ratios of the proppant particles had first decreased, then remained constant, and
21
then increased from the bottom to the top of the proppant-filled fracture. Also, the void ratio
22
distribution within the proppant-filled fracture had exhibited an inverted “U” shape. In this 9
1
study, the proppant particles in the fracture were divided into three zones as follows: bottom
2
loose zone (BLZ), middle compaction zone (MCZ) and top loose zone (TLZ). Meanwhile,
3
the average coordination number distributions of the proppant particles had presented an
4
exactly opposite trend as follows: first increasing, remaining constant, and then decreasing.
5
It was observed that the average coordination numbers of the proppant particles in Layer 1
6
and Layer 9 were the lowest, while the void ratios of the proppant particles were found to be
7
the highest in those layers. This was followed by the proppant particles in Layer 2 and
8
Layer 8, which has exhibited higher average coordination numbers and smaller void ratios.
9
It was determined that this was due to the increases in the average coordination numbers of
10
proppant particles causing decreases in the void ratios of the proppant particles. In particular,
11
the small coordination numbers of the proppant particles in Layer 1 and Layer 9 had
12
accounted for a large proportion, and the average coordination numbers of the proppant
13
particles were smaller. This had the effects of a lower total number of proppant particles
14
contacting each other and fewer connections between the proppant particles. As a result, the
15
channels formed by the gaps between the proppant particles which provided paths for fluid
16
flow were wider, and the void ratios of proppant particles in those areas were higher. In
17
contrast, the large coordination numbers of the proppant particles in the remaining layers
18
had accounted for a large proportion, and the average coordination numbers of proppant
19
particles were larger. Furthermore, the total number of proppant particles which had come in
20
contact with each other had increased and there were more connections formed between the
21
proppant particles. This had caused the channels formed by the gaps between the proppant
22
particles, in order to provide paths for fluid flow, to become narrower, and the void ratios of 10
1
the proppant particles in those areas were noticeably smaller.
2
3.2 Overall void ratios
3
Figure 9 shows the coordination number frequency distributions of the proppant
4
particles under different compression amounts. It can be seen in the figure that the main
5
distribution ranges of the coordination numbers had moved in the direction of the large
6
coordination numbers with the increases in the compression amounts. Also, the proportion
7
of the large coordination numbers had increased, which then caused the average
8
coordination numbers of the proppant particles to also increase. Subsequently, the void
9
ratios of the proppant particles became smaller.
10
Figure 9 details the manners in which the connections between the proppant particles
11
had recombined when the compression amounts had changed. Specifically, the connections
12
between some of the proppant particles were interrupted, and some of the proppant particles
13
had also made new connections. By taking Figure 9(a) as an example, it can be seen that
14
when the coordination number of the proppant particles was 0, the distribution frequencies
15
were 0.07 and 0 at the compression amounts of 0% and 5%, respectively. Therefore, it was
16
assumed that the coordination number 0 of the proppant particles with the distribution
17
frequency of 0.07 had indicated that new connections had been established and coordination
18
numbers of 1 or more had been reached. These results had indicated that proppant particles
19
had recombined, and some of the proppant particles which had originally shown no contacts
20
were now in contact with other proppant particles. Also, it was precisely those newly
21
established connections which had caused the originally open channels between the
22
proppant particles to become closed. It was observed that when the compression amounts 11
1
were increased to 10% and 15%, the distribution frequencies of the coordination numbers
2
10, 11, and 12 were 0.201, 0, 0 and 0, 0, 0.016, respectively. It was found that with the
3
distribution frequency of 0.021 of the coordination number 10, it could be assumed that
4
there were proppant particles of 0.016 adding connections to the distribution frequency of
5
the coordination number 12. Therefore, the remaining distribution frequency 0.005 was not
6
included in the coordination numbers 10, 11, and 12, and must then be returned to the
7
coordination numbers below 10. When the compression amount was increased to 20%, it
8
was found that the distribution frequency of the coordination number 12 had been reduced
9
to 0. Therefore, the distribution frequency must be returned to the coordination numbers
10
below 12. It was determined that this had indicated that the connections between some of
11
the proppant particles were interrupted and the proppant particles had become disconnected.
12
Also, it was found that those exact lost connections had caused the originally closed
13
channels between the proppant particles to be open wider. It was determined that although
14
the number of newly established connections for the proppant particles was much larger
15
than the number of lost connections with the increases in the compression amounts, the
16
proportion of the large coordination numbers had increased. The layered average
17
coordination numbers of the proppant particles had also increased, which had caused the
18
layered void ratios to become decreased.
19
Figure 10 details the relationship between the average coordination numbers of the
20
proppant particles and the compression amounts. Generally speaking, the layered average
21
coordination numbers were linearly increased with the increasing compression amounts.
22
This was due to the fact that when the compression amounts had increased, although the 12
1
connections between some proppant particles were interrupted, the majority of the proppant
2
particles had formed new connections. The proportion of the large coordination numbers
3
had become increased, and the average coordination numbers of the proppant particles had
4
also increased. It was determined that the cause was that the void ratios of the proppant
5
particles had become smaller, and it had become more difficult for the fluid to flow through
6
the channels of the proppant-filled fracture.
7
Figure 11 shows the relationship between the layered void ratios of the proppant
8
particles and the compression amounts. As can be seen in the figure, the layered void ratios
9
of the proppant particles had tended to decrease linearly with the increases in the
10
compression amounts. This was due to the fact that the proppant particle system was in a
11
natural accumulation state when the compression amount was 0%. Then, as the compression
12
amounts increased, due to the mutual squeezing effects between the proppant particles, and
13
the proppant particle system had become denser. The number of newly established
14
connections for the proppant particles was much larger than the number of lost connections,
15
and the average coordination numbers of the proppant particles had increased. As a result,
16
only a few channels were open. Meanwhile, the majority of the fluid channels formed by
17
the gaps between the proppant particles in order to provide pathways for the methane and
18
gas fluid had become narrow or even closed. This had resulted in the void ratios of the
19
proppant particles becoming smaller, and the flow of fluids had become more difficult
20
within the channels of the proppant-filled fracture.
21
Figure 12 details this study’s comparison of the average coordination numbers of the
22
proppant particles in the three examined zones. Figure 13 shows this study’s comparison of 13
1
the void ratios of the proppant particles within the three zones under the different
2
compression amounts. It can be seen in the figures that the average coordination numbers of
3
the proppant particles had decreased while the void ratios of the proppant particles had
4
increased with the increases in the compression amounts. The coordination numbers of the
5
proppant particles in the MCZ were observed to be the highest. Meanwhile, the void ratios
6
of proppant particles in the MCZ were found to be the lowest at the same compression
7
amounts. This was determined to be due to the fact that the large coordination numbers of
8
the proppant particles in the MCZ had accounted for a large proportion and the average
9
coordination numbers of the proppant particles were found to be relatively high. There were
10
more connections between the proppant particles and less open channels between proppant
11
particles in that zone, which had caused the void ratios of the proppant particles in the MCZ
12
to be the lowest. As the compression amounts had been increased, the number of contacts
13
between the particles had generally increased. Also, the average coordination numbers had
14
generally increased and the open channels had decreased. This had resulted in decreases in
15
the void ratios. In addition, the gaps between void ratios of the proppant particles in the
16
BLZ, TLZ and MCZ had increased. This was due to the fact that the gaps between the
17
average coordination numbers of the proppant particles in the BLZ, TLZ and MCZ had
18
increased. The proppant particles in the MCZ had become more connected and fewer
19
channels were now open. Finally, the average coordination numbers and void ratios of the
20
proppant particles in the BLZ and TLZ were determined to be 0.8-0.9 and 1.3 times greater
21
than those in the MCZ, respectively.
14
1
3.3 Model verifications
2
Figure 14 shows the comparison of the numerical simulations and the existing
3
experimental results. It can be seen in the figure that the line features of the numerical
4
simulation results and the existing experimental results (Ma et al. 2014) were the same. In
5
other words, the overall void ratio of the particles had linearly decreased with the increases
6
in the compression amounts, and the relationship between the numerical simulation result
7
line and the other three lines of the existing experimental results was almost parallel.
8
Moreover, it was found that the void ratio of the particles had increased with the increases
9
in the ranges of the particle sizes. Therefore, based on the validation conducted above, it is
10
shown that the model generally provides relatively good results, and the model results are
11
considered to have successfully illustrated that the proppant particles shapes, model
12
parameters, and method used in this research study are feasible.
13
4. CONCLUSIONS
14
1. The void ratios of the proppant particles had first decreased, then remained constant,
15
finally increased from the bottom to the top of the proppant-filled fracture. Also, the void
16
ratio distribution of the proppant particles in the fracture had exhibited an inverted “U”
17
shape.
18
2. The proppant particles within the fracture could be divided into three zones: BLZ,
19
MCZ and TLZ. At the same compression, the small coordination numbers in the BLZ and
20
TLZ had accounted for a larger proportion, and the average coordination numbers and void
21
ratios in the two zones were 0.8-0.9 and 1.3 times than those in the MCZ, respectively.
22
3. As the compression amounts increased, the number of newly established 15
1
connections for the proppant particles was found to be much larger than the number of lost
2
connections. Also, the proportion of the large coordination numbers in the different layers
3
increased, and the layered average coordination numbers increased linearly, which resulted
4
in the layered void ratios decreasing linearly.
5
ACKNOWLEDGEMENTS
6
This research was supported by the Joint Funds of the National Natural Science
7
Foundation of China (U1710258, U1710121). This work was also supported by the Program
8
for the Outstanding Innovative Teams of Higher Learning Institutions of Shanxi and the
9
Training Program of First-class Discipline for Young Academic Backbone of Taiyuan
10
University of Technology.
11
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1
Figure Captions
2
Figure 1. CBM fluid in a proppant-filled fracture during the application of hydraulic
3
fracturing technology.
4
Figure 2. Mechanical conditions between particle i and particle j.
5
Figure 3. Three-dimensional physical compression model of a proppant-filled fracture.
6
Figure 4. Numerical simulation flow chart of establishing the model.
7
Figure 5. Layered coordination number frequency distributions of the proppant particles: (a)
8
ε = 0%; (b) ε = 5%; (c) ε = 10%; (d) ε = 15%; (e) ε = 20%.
9
Figure 6. Three-dimensional coordination number distributions of the proppant particles: (a)
10
ε = 0%; (b) ε = 5%; (c) ε = 10%; (d) ε = 15%; (e) ε = 20%.
11
Figure 7. Void ratio distributions of the proppant particles of the different layers.
12
Figure 8. Comparison results of the void ratios and average coordination numbers of the
13
proppant particles: (a) ε = 0%; (b) ε = 5%; (c) ε = 10%; (d) ε = 15%; (e) ε = 20%.
14
Figure 9. Coordination number frequency distributions under the different compression
15
amounts: (a) Layer 1; (b) Layer 2; (c) Layer 3; (d) Layer 4; (e) Layer 5; (f) Layer 6; (g)
16
Layer 7; (h) Layer 8; (i) Layer 9.
17
Figure 10. Relationship between the average coordination numbers and the compression
18
amounts.
19
Figure 11. Relationship between the void ratios of the proppant particles and the
20
compression amounts.
21
Figure 12. Comparison of the average coordination numbers of the proppant particles in the
22
three zones. 22
1
Figure 13. Comparison of the void ratios of the proppant particles in the three zones.
2
Figure 14. Comparison of the numerical simulations and the existing experimental results
3
(Ma et al. 2014).
4
Table Caption
5
Table 1. Mechanical parameters of the proppant particle.
23
Parameters
Values
Reference used
Particle density
2,650 kg/m3
Zheng and Tannant 2019
Gravity acceleration
9.81 m/s2
Li et al. 2018
Poisson’s ratio
0.17
Reinicke et al. 2010
Numerical simulation method was used to study compression of proppants.
Void ratio distribution of the proppants exhibited an inverted “U” shape.
Bottom and top void ratios of the proppants were higher.
Bottom and top average coordination numbers of the proppants were smaller.
Declaration of Interest Statement： We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.