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Phase field modeling of multi-cluster hydraulic fracturing in horizontal wellbore with inconsistent direction to minimum principal stress

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IOP Conference Series: Earth and Environmental Science
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Phase field modeling of multi-cluster hydraulic
fracturing in horizontal wellbore with inconsistent
direction to minimum principal stress
To cite this article: Qianli Lu et al 2021 IOP Conf. Ser.: Earth Environ. Sci. 861 032008
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11th Conference of Asian Rock Mechanics Society
IOP Conf. Series: Earth and Environmental Science 861 (2021) 032008
IOP Publishing
doi:10.1088/1755-1315/861/3/032008
1
Phase field modeling of multi-cluster hydraulic fracturing in
horizontal wellbore with inconsistent direction to minimum
principal stress
Qianli Lu1*, Hang Zhang1*, Jianchun Guo1, Zuwen Tao2, Songgen He3, Le He4, Yiyao
Zhang5, Lei Chen6
1State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest
Petroleum University, Chengdu 610500, China
2Downhole Operation Branch, Sinopec Southwest Petroleum Engineering Company limited,
Deyang 618000, China
3Petroleum Engineering Technology Institute, Sinopec Southwest Oil and Gas Field Company,
Deyang 618000, China
4Downhole Operating Company, CNPC Chuanqing Drilling and Exploration Engineering
Company limited, Chengdu 610000, China
5Shunan Gas Mine, PetroChina Southwest Oil and Gas field Company, Luzhou 646000, China
6Department of Mechanical Engineering, University of Michigan-Dearborn, Dearborn, MI
48128, USA
Abstract. Zhongjiang gas field is a typical narrow channel tight sandstone gas reservoir. In
order to obtain enough horizontal section length in high-quality reservoir, horizontal wellbore
azimuth is basically consistent with direction of narrow channel sand body, resulting in a
certain included angle θbetween wellbore azimuth and in-situ minimum principal stress
orientation. In this study, a seepage-stress-damage coupled dynamic fracture propagation
model is established through phase field method to investigate the effect of angle θon
multi-cluster fracture simultaneous propagation based on geological engineering data of JS-X
well. Propagation of hydraulic fracture is automatically tracked through phase field evolution
without additional criteria or grid direction limitation. Simulation results show that: (1)
Fracture tip of each cluster would approach to stress diversion zone created by adjacent
clusters. Simultaneous propagating fractures of each cluster are prone to deflect, interconnect
with each other and form one single fracture eventually. (2) Increasing cluster number or
decreasing cluster spacing in the same fracturing stage would further promote diverting of each
11th Conference of Asian Rock Mechanics Society
IOP Conf. Series: Earth and Environmental Science 861 (2021) 032008
IOP Publishing
doi:10.1088/1755-1315/861/3/032008
2
cluster fracture and accelerates fractures merging into one single fracture. We suggest that
azimuth of horizontal wellbore in tight gas reservoir should be consistent with in-situ minimum
principal stress orientation to reduce the effect of angle θ.
1. Introduction
Zhongjiang gas field is a typical narrow channel tight sandstone gas reservoir. High-quality reservoirs
are generally distributed in the center of channels. Multi-cluster hydraulic fracturing in horizontal
wellbore is an important technical means to improve development efficiency of Zhongjiang gas field.
In order to obtain enough horizontal section length in high-quality reservoirs, horizontal wellbore
azimuth is basically consistent with direction of narrow channel sand body, resulting in a certain
included angle θbetween wellbore azimuth and in-situ minimum principal stress orientation (Figure.
1). The effect of included angle θon hydraulic fracture propagation of each perforation cluster is not
clear.
Phase field method is an elegant continuous fracture simulation method, which is derived from
Griffith fracture theory based on variational principle. Phase field method does not require preliminary
assumptions on shape of fractures and propagation of fracture is automatically tracked through energy
minimization principle without additional criteria or grid direction limitation.
Aiming at simulating hydraulic fracturing process, current framework of most phase field method
research is how to better couple flow of reservoir domain and fracture domain. Zhou et al. [1] deal
with flow transition domain between two domains. Biot theory and porous media theory (TPM) are
main theories for constructing basic equations of reservoir domain. Darcy flow in reservoir domain
and Reynolds flow in fracture domain [2, 3], and Darcy flow in reservoir domain and Stokes flow in
fracture domain [4, 5] are two combinations for constructing fluid flow in different domains. Lee et al.
[6] derive power-law fluid flow in fracture domain. Phase field parameter is often used as a bridge
between reservoir domain and fracture domain. Chukwudozie et al. [7] establish a unified
fracture-porous media flow model, and it is no longer necessary to use phase field parameters as
indicator functions or weight functions to distinguish different calculation domains.
Subsequent scholars have applied phase field method to study specific problems of hydraulic
fracturing recently. Guo et al. [8] quantitatively study branching mechanism of hydraulic fractures in
heterogeneous formation. Mollaali et al. [9] simulate CO2 fracturing under isothermal condition based
on phase field method. Li and Zhou [10] examine multizone hydraulic fracture propagation in porous
elastic medium. Shovkun and Espinoza [11] investigate propagation of hydraulic fractures in reactive
porous media. Yi et al. [12] simulate hydraulic fracture propagation in porous media with natural
fracture. Liu et al. [13] investigate hydraulic fracturing behavior in bedded shale with pre-existing
fractures. Zeng et al. [14] propose a phase field based discrete fracture model to simulate fluid flow in
fractured porous media. Zhuang et al. [15] examine hydraulic fracture propagation in naturally-layered
porous media. Liu et al. [16] investigate influence of natural cavities on hydraulic fracture
propagation.
In this study, a seepage-stress-damage coupled dynamic fracture propagation model is established
based on phase field method. This model is proposed to investigate the effect of included angle θon
11th Conference of Asian Rock Mechanics Society
IOP Conf. Series: Earth and Environmental Science 861 (2021) 032008
IOP Publishing
doi:10.1088/1755-1315/861/3/032008
3
multi-cluster fracture simultaneous propagation based on engineering geological data of JS-X well in
Zhongjiang gas field.
Figure. 1. Plane graph of seismic amplitude energy of channel sand bodies (favourable tight gas
reservoir) and JS-X well trajectory
2. Phase field modeling
2.1 Phase field
Before establish phase field evolution equation, consider a solid with a fracture set
( is number of space dimension) as shown in Figure. 2, which is characterized by
displacement field and strain field .
Phase field models in the mechanics community originate from variational formulation of Griffith
brittle fracture. From Griffith fracture theory, fracture propagation is a dynamic energy conservation
process, which is descripted as followed:
\* MERGEFORMAT (1)
where stored strain energy , fracture surface dissipation energy and external potential energy
are defined as:
\* MERGEFORMAT (2)
In order to prevent compression failure, elastic energy density function is split into tension part
and compression part,
\* MERGEFORMAT (3)
where energy degradation function is simplified as , and represents
diffusive definition of fracture topology ( stands for intact state, while stands for
complete damaged state).
Based on viscous regularization of rate-independent formulation proposed by Miehe et al. [17],
rate-dependent dissipation function is:
\* MERGEFORMAT (4)
where is local driving force, is threshold value of Griffith fracture energy, and is
variational derivative of fracture surface density functional .
11th Conference of Asian Rock Mechanics Society
IOP Conf. Series: Earth and Environmental Science 861 (2021) 032008
IOP Publishing
doi:10.1088/1755-1315/861/3/032008
4
Fracture surface density functional is defined as:
\* MERGEFORMAT (5)
where length parameter scales width of diffuse fracture as shown in Figure. 2.
Insert three parts of energy functional expressions into energy conservation equation, and apply Gauss
theorem to get:
\* MERGEFORMAT (6)
For viscous model, three-field formulation of energy balance equation proposed by Miehe et al. [9] is:
\* MERGEFORMAT (7)
Phase field evolution equation is acquired from (7), which is expressed as:
\* MERGEFORMAT (8)
2.2 Displacement field
Motion equation of tight sandstone micro element is written as:
\* MERGEFORMAT (9)
where is fluid pressure, is body force acting on the element,
is acceleration, and is density of tight sandstone.
2.3 Fluid pressure field
Continuity equation of fracture flow is:
\* MERGEFORMAT (10)
where is fracture volume ratio, is fluid velocity in direction, and is
source sink term of mass.
Fracture volume ratio depends on fracture width and micro element length ,which is
expressed as .
11th Conference of Asian Rock Mechanics Society
IOP Conf. Series: Earth and Environmental Science 861 (2021) 032008
IOP Publishing
doi:10.1088/1755-1315/861/3/032008
5
Motion equation of fracture plate flow is:
\* MERGEFORMAT (11)
where is fluid viscosity, is reservoir depth and is fluid density. When fluid flow direction
is vertical to fracture normal direction, then . Otherwise, .
For two-dimensional fracture propagation, governing equation by combining continuity equation and
motion equation can be obtained:
\* MERGEFORMAT (12)
where is fracture stiffness in normal direction, is fluid compression coefficient, and is
injection rate or leak off rate.
2.4 Numerical implementation and model validation
Numerical solution of above three field equations is realized by finite difference method.
Fluid pressure will be applied to micro element when calculated order parameter exceed critical value,
which is set as 0.9 in this paper.
Flow chart of numerical implementation is presented in Figure. 3.
Reliability and accuracy of present model have been verified by previous study [8, 16].
Figure. 2. a solid with a fracture set : (a) sharp description of a fracture and (b) diffuse
description of a fracture based on phase field method
11th Conference of Asian Rock Mechanics Society
IOP Conf. Series: Earth and Environmental Science 861 (2021) 032008
IOP Publishing
doi:10.1088/1755-1315/861/3/032008
6
Figure. 3. Flow chart of numerical implementation
3. Model construction
Based on engineering geological data of JS-X well, this section establishes basic physical model to
study the influence of included angle θon multi-cluster fracture simultaneous propagation.
Multi-cluster horizontal well fracturing physical model is implemented in 2-D on a 300 m × 200 m
rectangle plane as shown in Figure. 4, which includes a stage of horizontal wellbore section. Direction
of in-situ maximum principal stress and minimum principal stress are along long and short
side of rectangle plane respectively. Angle between horizontal wellbore section and in-situ minimum
principal stress is so-called included angle θ. Multiple perforation clusters are evenly distributed
on horizontal wellbore section with the same cluster spacing and direction of perforation clusters is
along in-situ maximum principal stress .
Input parameters of multi-cluster fracture simultaneous propagation simulation according to
engineering geological data of JS-X well in Zhongjiang gas field are represented in Table. 1.
Figure. 4. Multi-cluster horizontal well fracturing physical model considering included angle θ
11th Conference of Asian Rock Mechanics Society
IOP Conf. Series: Earth and Environmental Science 861 (2021) 032008
IOP Publishing
doi:10.1088/1755-1315/861/3/032008
7
Table. 1. Parameters for hydraulic fracturing simulation
Properties
Value
Maximum horizontal stress
72 MPa
Minimum horizontal stress
62 MPa
Porosity of rock matrix
9 %
Permeability of rock matrix
0.2 mD
Young’s modulus of rock matrix
18000 MPa
Poisson’s ratio of rock matrix
0.22
4. Results and discussion
4.1 Effect of included angle θ
Figure. 5 demonstrates multi-cluster fracture simultaneous propagation simulation results considering
included angle θbased on phase field method. Although every fracture initially propagates along
direction of in-situ maximum principal stress orientation, fracture tip of each cluster would approach
to stress diversion zone (Blue Zone) created by adjacent clusters as shown in stress field distribution
pattern (Figure. 5 (a)). Fractures of each cluster are prone to deflect, interconnect with each other and
form one single fracture eventually. The effectiveness of fracture propagation in each cluster is limited,
which is shown in fluid pressure distribution pattern in fracture of each cluster (Figure. 5 (b)).
Figure. 6 demonstrates multi-cluster fracture simultaneous propagation simulation results under
different included angle θ. As included angle θincreases gradually, number of clusters and cluster
spacing remains unchanged, but the distance between clusters in the direction of in-situ minimum
principal stress decreases. Fracture tip of each cluster become closer to stress diversion zone and easier
to deflect and interconnect. Under the same number of clusters and cluster spacing, presence of
included angle θwould greatly inhibit fracture propagation effectiveness.
Figure. 5. Simulation results of multi-cluster fracture simultaneous propagation under included angle
20°: (a) stress field distribution around perforation cluster and (b) fluid pressure distribution in fracture
of each cluster
11th Conference of Asian Rock Mechanics Society
IOP Conf. Series: Earth and Environmental Science 861 (2021) 032008
IOP Publishing
doi:10.1088/1755-1315/861/3/032008
8
Figure. 6. Simulation results of multi-cluster fracture simultaneous propagation under different
included angle
4.2 Effect of cluster number and cluster spacing
Effect of cluster number and cluster spacing on multi-cluster fracture simultaneous propagation
considering included angle θare investigated. As can be seen from Figure. 7, increasing number of
clusters or decreasing cluster spacing in the same fracturing stage of horizontal wellbore section would
further reduce the distance between clusters in the direction of in-situ minimum principal stress, which
further promotes diverting of each cluster fracture and accelerates fractures merging into one single
fracture. Increasing cluster number would also enhance inter-cluster stress interference as more
fractures are created. Increasing cluster spacing cannot prevent fracture deflecting and coalescing
under existing included angle θ, but it would extend duration of this process.
By analyzing simulation results, we suggest that azimuth of horizontal wellbore in tight sandstone gas
reservoir should be consistent with in-situ minimum principal stress orientation as much as possible to
reduce effect of included angle θ. Besides, increasing cluster spacing and selecting appropriate number
of clusters are beneficial to weaken inter-cluster stress interference effect and slow down fracture
deflecting and interconnecting under existing included angle θ.
11th Conference of Asian Rock Mechanics Society
IOP Conf. Series: Earth and Environmental Science 861 (2021) 032008
IOP Publishing
doi:10.1088/1755-1315/861/3/032008
9
Figure. 7. Simulation results of multi-cluster fracture simultaneous propagation under combinations of
included angle, clusters number and cluster spacing
5. Conclusion
This paper investigates the effect of included angle θbetween wellbore azimuth and in-situ minimum
principal stress orientation on multi-cluster hydraulic fracture simultaneous propagation based on
phase field computational framework. Due to the existence of included angle, fracture tip of each
cluster would be closer to stress diversion zone created by adjacent clusters. Fractures are prone to
deflect, interconnect with each other and form one single fracture eventually. Increasing number of
clusters or decreasing cluster spacing in the same fracturing stage would promote diverting and
coalescing process. Increasing cluster spacing would extend duration of this process.
Simulation results suggest that azimuth of horizontal wellbore in tight sandstone gas reservoir should
be consistent with in-situ minimum principal stress orientation as much as possible. Increasing cluster
spacing and selecting appropriate number of clusters help to slow down fracture deflecting and
interconnecting process under existing included angle.
6. Acknowledgements
This paper is supported by China National Natural Science Funds (No. 51904258), CNPC Innovation
Foundation (2020D-5007-0208) and Sinopec Southwest Oil and Gas Field Company Research Project
(KJ-675-2050).
7. References
[1] Zhou S, Zhuang X, Rabczuk T 2018 A phase-field modeling approach of fracture propagation
in poroelastic media Eng Geol. B240 189-203
[2] Miehe C, Mauthe S 2016 Phase field modeling of fracture in multi-physics problems. Part III.
Crack driving forces in hydro-poro-elasticity and hydraulic fracturing of fluid-saturated porous
media Comput Method Appl M B304 619-655
11th Conference of Asian Rock Mechanics Society
IOP Conf. Series: Earth and Environmental Science 861 (2021) 032008
IOP Publishing
doi:10.1088/1755-1315/861/3/032008
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[3] Miehe C, Mauthe S, Teichtmeister S 2015 Minimization principles for the coupled problem of
Darcy–Biot-type fluid transport in porous media linked to phase field modeling of fracture J
Mech Phys Solids B82 186-217
[4] Mikelić A, Wheeler MF, Wick T 2015 A quasi-static phase-field approach to pressurized
fractures Nonlinearity B28 1371-1399
[5] Mikelić A, Wheeler MF, Wick T 2015 Phase-field modeling of a fluid-driven fracture in a
poroelastic medium Comput Geosci B19 1171-1195
[6] Lee S, Wheeler M.F, Wick T 2017 Iterative coupling of flow, geomechanics and adaptive
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[7] Chukwudozie C, Bourdin B, Yoshioka K 2019 A variational phase-field model for hydraulic
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[8] Guo J, Lu Q, Chen H, Wang Z, Tang X, Chen L 2018 Quantitative phase field modeling of
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[9] Mollaali M, Ziaei-Rad V, Shen Y, Chen L 2019 Numerical modeling of CO2 fracturing by the
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[10] Li K, Zhou S 2019 Numerical investigation of multizone hydraulic fracture propagation in
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[11] Shovkun I, Espinoza DN 2019 Propagation of toughness-dominated fluid-driven fractures in
reactive porous media Int J Rock Mech Min. B118 42-51
[12] Yi L, Li X, Yang Z, Yang C 2020 Phase field modeling of hydraulic fracturing in porous media
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[13] Liu J, Liang X, Xue Y, Fu Y, Yao K, Dou F 2020 Investigation on crack initiation and
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[14] Zeng Q, Liu W, Yao J, Liu J, Dou F 2020 A phase field based discrete fracture model (PFDFM)
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