Cheng. Parameters Affecting the Proppant Distribution in Multiple Growing Hydraulic Fractures

Rock Mechanics and Rock Engineering
https://doi.org/10.1007/s00603-024-03872-z
ORIGINAL PAPER
Parameters Affecting the Proppant Distribution in Multiple Growing
Hydraulic Fractures
Shaoyi Cheng1,2,3 · Bisheng Wu1,2,3 · Guangjin Wang4 · Zhaowei Chen5 · Yang Zhao1,2,3
· Tianshou Ma6
Received: 30 December 2023 / Accepted: 17 March 2024
© The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2024
Abstract
The uniformity of the proppant distribution among multiple hydraulic fractures is of great importance in the production
period after hydraulic fracturing treatments. To facilitate the understanding of the proppant transport along multiple hydraulic
fractures, we present a numerical model predicting the proppant transport inside multiple growing hydraulic fractures. The
model incorporates an empirical formula for the slurry and proppant transport to a fully coupled in-house hydraulic fracturing simulator, i.e. DeepFrac, based on the dual boundary element method and finite volume method. This model focuses on
the propagation and interaction of multiple 2D plane strain fractures driven by mixture of fluid and proppant and neglects
the perforation friction and proppant inertia. A bifurcation analysis is carried out to identify the dimensionless parameters
that control the competitive growth of multiple hydraulic fractures. A dimensional analysis indicates that the propagation
of multiple fractures and the proppant transport in these fractures are controlled by five dimensionless parameters, i.e. the
dimensionless toughness, dimensionless proppant size, dimensionless time when proppant is injected, dimensionless time
of gravitational settling and injected proppant concentration. The effectiveness of these dimensionless groups is verified first
by numerical simulations. Then a parametric study of each dimensionless parameter is carried out to reveal their roles in the
proppant distribution. The results show a strong dependence of proppant distribution on the stress interaction between fractures,
which highlights the importance of the time when proppant is injected. A wider window period for the proppant entering each
fracture can be obtained when K′ < 0.4. The ratio of two time scales is found to control the settling behavior of proppant.
Severe gravitational settling of proppant can lead to an uneven proppant distribution among multiple hydraulic fractures.
Highlights
• A numerical model predicting the proppant transport in multiple simultaneously growing hydraulic fractures is developed.
• Five dimensionless parameters controlling the proppant transport behavior in multiple hydraulic fractures are deduced.
• The impact of the stress shadow effect, proppant size, proppant injection timing, proppant settling and proppant injection
concentration on the proppant distribution is revealed.
Keywords Multiple hydraulic fractures · Proppant transport · Bifurcation condition · Dimensional analysis
* Bisheng Wu
wu046@mail.tsinghua.edu.cn
3
* Yang Zhao
yangzhao2021@mail.tsinghua.edu.cn
Department of Hydraulic Engineering, Tsinghua University,
Beijing 100084, China
4
1
State Key Laboratory of Hydroscience and Engineering,
Tsinghua University, Beijing 100084, China
Faculty of Land Resources Engineering, Kunming University
of Science and Technology, Kunming, Yunnan, China
5
2
Key Laboratory of Hydrosphere Sciences of the Ministry
of Water Resources, Tsinghua University, Beijing 100084,
China
CNPC Engineering Technology R&D Company Limited,
Beijing, China
6
State Key Laboratory of Oil and Gas Reservoir Geology
and Exploitation, Southwest Petroleum University,
Chengdu 610500, Sichuan, China
Vol.:(0123456789)
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S. Cheng et al.
1 Introduction
The extraction of unconventional resources, such as shale
gas and oil, heavily relies on horizontal drilling and multistage hydraulic fracturing. In hydraulic fracturing process,
multiple hydraulic fractures (HFs) driven by high-pressure
fluid initiate and propagate from a horizontal well. At some
point, proppant is pumped in together with the fracturing
fluid and transports along the advancing fracture, and keeps
the fracture open after shut-in to provide a conductive path
for fluid flow from reservoir to wellbore (Zhang et al. 2017).
Therefore, a uniform proppant distribution in multiple HFs
is crucial to ensure sufficient fracture conductivity and effective well stimulation (Cipolla et al. 2009; Warpinski et al.
2009; Yu et al. 2015). However, field data has frequently
reported the uneven proppant distribution among different
fractures within a fracturing stage, leaving a large portion
of the reservoir unstimulated (Gu and Mohanty 2014; Mao
et al. 2021). The production logs of more than 100 horizontal wells from six basins indicated that only 1/3 of perforation clusters contribute to 2/3 of gas production (Miller
et al. 2011). The stress shadow effect may play an important role in this phenomenon because strong stress interaction between fractures can suppress the initiation of some
perforations, but the uneven proppant distribution is also
considered as a major cause (Daneshy 2011). Therefore, it
is important to investigate the key parameters controlling
the proppant transport behavior and the uniformity of the
proppant distribution.
The propagation of multiple HFs has been extensively
studied for decades. Compared to the single fracture case,
the transient interaction between multiple HFs poses great
difficulty in analyzing the growth behavior of the fracture
system. The compression stresses exerted by neighboring
fractures dynamically change the flow rate of each fracture, making the problem intractable when using analytical
approaches (Zhang et al. 2018). Some theoretical studies
analyze the multiple HFs from an energy perspective and
investigate the energy required to propagate multiple HFs
under different propagation regimes (Bunger 2013; Bunger
et al. 2013). These studies highlight the importance of the
energy dissipation mode in the growth of multiple HFs.
However, the analysis of the input power requires strong
assumptions that neglect the fluid partitioning among HFs
and fracture curving. Indeed, keeping an equal rate of flow
entering all HFs throughout the injection duration is the key
to ensure a simultaneous growth of all fractures (Nikolskiy
and Lecampion 2020), which highlights the importance of
finding the full solution to the problem. A number of contributions have been made that focus on stress interference
between fractures (Wu and Olson 2013; Peirce and Bunger
2015), propagation regimes (Bunger 2013; Dontsov and
Suarez-Rivera 2020), perforation friction (Lecampion and
Desroches 2015; Nikolskiy and Lecampion 2020) and in situ
stresses (Cheng et al. 2022b; Cheng et al. 2023b; Kresse
et al. 2012), and the effect of these parameters on the simultaneous growth of HFs are characterized by dimensionless
groups. Nonetheless, the relationship between these factors
and the fracture pattern remains unclear.
Modeling proppant transport and placement poses an
additional challenge to the analysis of multiple HFs. In
general, proppant transport is simulated using the Eulerian–Lagrangian or Eulerian–Eulerian method. In the Eulerian–Lagrangian method, the proppant transport is solved
by the Lagrangian method, while the fluid flow in the fractures is solved by the Eulerian method (Dontsov and Peirce 2015a). This approach is computationally expensive in
modeling multiple HFs since the movement of each proppant particle needs to be tracked (Wen et al. 2022). In the
Eulerian–Eulerian method, the mixture of proppant and fluid
is solved using a continuum approach, where a constitutive
relation for the proppant is needed to describe the rheology of suspension flow. In hydraulic fracturing problems,
the slurry is typically treated as Newtonian and its effective
viscosity is given by an empirical function (Roostaei et al.
2018). In contrast, Dontsov and Peirce (2014) developed
the governing equations for slurry flow and proppant transport based on an empirical formula proposed by Boyer et al.
(2011). This model accounts for the non-uniform particle
distribution across the fracture channel, and the transition
from Poiseuille’s flow to Darcy’s flow is captured. This
method is computationally efficient and has been widely
used (Dontsov and Peirce 2015b; Shiozawa and McClure
2016; Wang et al. 2018; Luo et al. 2023).
Achieving a uniform proppant distribution is quite challenging because many factors can influence the proppant
transport and settlement in the fracture, including proppant
size (Vincent 2012), fluid rheology (Dontsov and Peirce
2014), rock mechanical properties (Fredd et al. 2000), nearwellbore tortuosity (Qu et al. 2021), and natural fractures
(Gu et al. 2014). Experimental and theoretical studies on
proppant distribution among perforations show that the proppant tends to enter the last perforation cluster, which can be
explained by the inertial effect of proppant at the intersection
between wellbore and perforation (Crespo et al. 2013; Dontsov 2023). It should be noted that the proppant distribution
in multiple fractures differs from that in multiple perforations because the fracture width and fluid flux vary when
the fractures grow, imposing a non-negligible influence on
the proppant transport behavior. Various numerical methods including computational fluid dynamics (Bokane et al.
2013), discrete element method (Yi et al. 2018; Mao et al.
2021) and material point method (Raymond et al. 2015), are
used to simulate the proppant transport through perforations
and fractures. However, many studies mainly focus on the
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Parameters Affecting the Proppant Distribution in Multiple Growing Hydraulic Fractures
proppant transport and slurry flow in the fracture channel,
while fracture propagation and fluid–solid coupling between
slurry and fracture are neglected. In fact, the stress interference between multiple propagating fractures can significantly
impact the proppant transport behavior (Germanovich et al.
1997; Fisher et al. 2004; Siddhamshetty et al. 2019). The
stress interaction between fractures can dynamically change
the fracture width, tortuosity and flow rate which are important aspects in proppant transport. Nevertheless, the influence
of the stress shadow effect on proppant transport has not been
well studied. While certain studies have addressed the proppant distribution under different fracture spacing (Siddhamshetty et al. 2019; El Sgher et al. 2023), proppant injection
time (Hu et al. 2018), particle properties (Zhou et al. 2017),
the controlling parameters of the process remain unclear, and
a comprehensive understanding is lacking.
In this work, we focus on the growth of multiple HFs
and the effect of stress shadow on the proppant transport
behavior. First, a dimensionless analysis is carried out and
five dimensionless parameters are obtained, among which
one parameter is related to the fracture growth behavior,
three parameters are related to the proppant transport, and
one parameter is related to the time when the proppant is
injected. A bifurcation analysis is performed to reveal the
existence of a scaling relationship between the dimensionless numbers and the fracture pattern. The bifurcation condition that separates the fracture growth in small and large
time scales is derived by incorporating the fluid net pressure
and the interacting stress from neighboring fractures (Cheng
et al. 2023a). The numerical model combines a fully coupled
hydraulic fracturing simulator (Cheng et al. 2022a) and a
proppant transport model developed by Dontsov and Peirce
(2014). After presenting several verifications of the scaling
arguments, we explore the parameters controlling the proppant distribution among multiple HFs.
of fluid injection with a constant rate of Q0, and the proppant
flux into each fracture is allowed to change dynamically due
to the fluid partitioning among HFs. To make the problem
tractable, some assumptions are made as follows:
• The rock matrix is considered to be an isotropic and lin-
ear elastic solid, and its deformation is under plane strain
condition. The in situ stresses are uniform and isotropic.
• The fluid injected to the fractures is treated as incompressible and Newtonian, and the fluid flow inside the
fracture is considered to be laminar. The fluid leak-off
from fracture into the rock matrix is neglected.
• The fracture propagation is described under the framework of linear elastic fracture mechanics, and the fracture
front coincides with the fluid front.
• The slurry flow in the wellbore and proppant turning
from wellbore to a perforation are neglected. The pressure drop caused by perforation friction is also neglected.
All proppant particles are spherical with the same radius.
2.1 Solid Deformation
In the framework of linear elastic fracture mechanics, the
tractions and the displacements on fracture surfaces can be
represented by a pair of boundary integral equations (Hong
and Chen 1988; Portela et al. 1992)
cij (x� )ui (x� ) +
∫Γ
Tij (x� , x)uj (x)dΓ =
∫Γ
Uij (x� , x)tj (x)dΓ
1
𝜎 (x� ) + Sijk (x� , x)uk (x)dΓ = Dijk (x� , x)tk (x)dΓ
∫Γ
∫Γ
2 ij
(1)
2 Problem Formulation
The problem geometry for multiple equidistant HFs simultaneously propagating in the rock medium is shown in Fig. 1.
In this model, we account for the mechanical deformation
of the rock, fracture propagation, fluid flow in the fracture
channel, fluid partitioning between different propagating
fractures, and proppant transport among these fractures.
Initially, there are N evenly spaced parallel fractures with
a spacing of D. The length and the aperture of the fractures
are denoted by l and w, respectively. After the fracture initiation, the fractures are allowed to curve when the stress
field changes, and the fluid flux of each HF can vary due
to the stress interaction between fractures. The proppant is
injected at a constant concentration ϕ0 after a certain period
Fig. 1 Schematic of multiple HFs equally spaced at distance D
injected at constant pumping rate Q0 and proppant concentration ϕ0
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S. Cheng et al.
where x′ and x are the coordinates of the source point and
field point on the boundary, respectively. Tij and Uij are the
Kelvin fundamental solution for traction and displacement,
respectively, while Sijk and Dijk are linear combinations of
derivatives of the Kelvin solution. ui and σi denote the displacement and stress components, respectively. cij = δij/2 for
a smooth boundary at point x′, and δij is the Dirac function.
Note that the kernels Tij, Uij, Sijk and Dijk exhibit different
order of singularities when the field point approaches the
source point, the integrals in Eq. (1) represent Cauchy and
Hadamard principal value integrals for the strong and hypersingular case (Portela et al. 1992).
If the deformation of non-fracture boundaries, such as
wellbore or outer boundaries of the model, is neglected,
Eq. (1) can be further simplified. Since the fracture width
is generally much smaller than the fracture length, the fluid
flow in the fracture width direction is often negligible when
modeling fluid-driven fractures (Zhang et al. 2011), and the
magnitude of the fluid pressure on both fracture surfaces can
be considered equal. Using the properties of the Kelvin solutions (Portela et al. 1992), Eq. (1) can be reduced to one
integral equation
tj (x� ) = −nj (x� )
∫ Γ+
Sijk (x� , x)Δuk (x)dΓ
(2)
where nj is the unit outward normal to the boundary, Γ+ represents one of the fracture surfaces, and Δuj is the discontinuous displacement component across the fracture surfaces.
2.2 Slurry Flow and Proppant Transport
The solution of slurry flow and proppant transport is based
on an empirical model for a Newtonian fluid and spherical
particles (Boyer et al. 2011; Dontsov and Peirce 2014). The
mass balance equations for the slurry and proppant are written as follows
𝜕w
+ ∇ ⋅ qs = 0
𝜕t
(3)
)
𝜕 𝜙w
+ ∇ ⋅ qp = 0
(4)
where w is the fracture opening, 𝜙 = 𝜙∕𝜙m is the normalized
particle volumetric concentration, here ϕ denotes the average concentration across the fracture width and ϕm = 0.585
is the maximum allowed concentration (Boyer et al. 2011).
qs and qp are the flux vectors for slurry and proppant, respectively, and are defined as follows
)
(
w3 ̂ s
w
Q 𝜙,
∇p
q =−
12𝜇
a
s
(6)
where μ is the clear fluid viscosity, p is the fluid pressure,
g is the gravitational acceleration, a is the particle radius,
and Δρ = ρp − ρf are the density difference between the proppant and fluid. Note that the proppant can transport through
curving fractures, the proppant flux caused by gravity is projected to the fracture surface when the fracture deflects from
the vertical direction. Therefore, the unit outward normal
of the fracture surface nx is added to Eq. (6). A blocking
function, B, is introduced to capture the bridging effect of
proppant particles, and is given by
B
(
w
a
)
=
(
) (
){
)]}
[ (
w
1
w
w
1 + cos 𝜋 M + 1 −
H
−M H M+1−
2
2a
2a
2a
(
)
w
+H
−M−1
2a
(7)
where H is the Heaviside step function, M denotes “several”
particle diameters, which means that the particle bridging
occurs at a location where the fracture width is smaller than
M times the diameter of a particle, and M = 3 is taken in this
paper, as is commonly used in the simulation of proppant
transport (Dontsov and Peirce 2015b; Luo et al. 2023).
̂ s, Q
̂ p, and G
̂ p, are used
Three dimensionless functions, i.e. Q
in Eqs. (5) and (6) to characterize the interaction between the
proppant and slurry flow
(
)
2
̂ s 𝜙, w = Qs (𝜙) + a 𝜙D
Q
a
w2
(
)
2
p
w Q (𝜙)
̂ p 𝜙, w =
Q
(8)
a
2
w Qs (𝜙) + a2 𝜙D
)
(
2 s
p
̂ p 𝜙, w = Gp (𝜙) − w G (𝜙)Q (𝜙)
G
a
w2 Qs (𝜙) + a2 𝜙D
(
)𝛼
where D = 8 1 − 𝜙m ∕3𝜙m describes the permeability of
the packed particles, and 𝛼 = 4.1 (Dontsov and Peirce 2014).
Qd, Qp, Gd, and Gp are univariate functions of 𝜙 , which can
be found in the work by Dontsov and Peirce (2015b).
2.3 Fracture Propagation Criteria
(
𝜕t
)
)
(
(
2
̂ p 𝜙, w Δ𝜌nx
̂ p 𝜙, w qs − a w BgG
qp = BQ
a
12𝜇
a
(5)
As fracture curving is allowed in the present model, the maximum tensile hoop stress criteria is adopted to calculate the
propagation direction under mixed-mode loading (Erdogan
and Sih 1963), which is given by
KI sin 𝜃 + KII (3 cos 𝜃 − 1) = 0
]
[
𝜃 3
𝜃
cos KI cos2 − KII sin 𝜃 = KIc
2
2 2
(9)
where θ is the propagation angle, KI and KII are the stress
intensity factors (SIFs) of Mode I and II, respectively. KIc
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Parameters Affecting the Proppant Distribution in Multiple Growing Hydraulic Fractures
denotes the Mode I fracture toughness of rock. The tip
asymptote for the propagating fracture is given by
√
32 1 − 𝜈 2
KIc (l − x)1∕2 , l − x ≪ l
w=
𝜋 E
(10)
E� = E∕(1 − 𝜈 2 )
where E is the Young’s modulus, v is the Poisson’s ratio and
l is the fracture length. Equation (9) is used in numerical
simulation, while Eq. (10) is used in scaling analysis.
Regarding the initial and boundary conditions, initially all
fractures are dry. The total injection rate Q0 is assumed to be
constant, and the influx of each fracture is allowed to vary
dynamically. Here we also neglect the pressure drop along the
horizontal well and the perforation friction. Thus, the wellbore
pressure at the entry of each fracture is equal:
p(1)
(t) = p(2)
(t) = ⋯ = p(N)
(t),
f ,entry
f ,entry
f ,entry
N
∑
qientry (t) = Q0
i=1
(11)
where the superscript denotes the fracture number. It is
assumed that the proppant distribution along the wellbore is
uniform when the proppant is injected. Therefore, the proppant flux at the fracture entry is given by
p,i
qentry = 𝜙0 qs,i
entry ,
(12)
i = 1, 2, … , N
The slurry flux and the proppant flux are all zeros at the
fracture tip, namely
qs |x=l = 0,
qp |x=l = 0
(13)
3 Scaling
In this section, we demonstrate the nondimensionalized
governing equations for the coupled processes of slurry
fluid flow, proppant transport, solid deformation and fracture propagation. The solution to this problem aims to
determine the fracture width, w, fluid pressure, p, fracture
length, l, fluid flux, q, proppant migration distance, l p,
and proppant concentration, 𝜙 , under the influence of the
fracture spacing, D, injection rate, Q0, proppant particle
radius, a, concentration of proppant injected, 𝜙0 , proppant
injection time, tp, and three material parameters, E', K',
and μ', which are given by (Detournay 2004)
E
,
E =
1 − 𝜈2
�
( )1∕2
2
KIc ,
K =4
𝜋
�
�
𝜇 = 12𝜇
(14)
Equation (14) is introduced to keep equations uncluttered by numerical constants, so we refer to E', K', and
μ' as the elastic modulus, fracture toughness and fluid
viscosity. Note that the Poisson’s ratio is lumped into E'.
To reduce the computational effort and facilitate a deeper
understanding of the proppant transport in multiple HFs,
the following transformation is used
𝜉=
x
,
L∗
𝛾=
l
,
L∗
Ω=
w
,
W∗
Π=
p
,
P∗
𝜓=
q
,
Q∗
𝜏=
t
T∗
(15)
where ξ, γ, Ω, Π, ψ, and τ are the dimensionless coordinates,
fracture half-length, fracture width, fluid pressure, fluid flux
and time, respectively. L*, W*, P*, Q* and T* are their corresponding characteristic values. After substituting Eq. (15) to
Eqs. (2) –(13), the governing equations, boundary and initial
conditions become.
• Elasticity equation
Πj (𝜉 � ) = −nj (𝜉 � )Pe
∫ Γ+
(
)
Sijk 𝜉 � , 𝜉, Ps Ωk (𝜉)dΓ(𝜉)
(16)
• Mass continuity equations for slurry and proppant
Ẇ ∗
̇ ∗ + Pf ∇𝜉 ⋅ 𝜓 s = 0
ΩT ∗ + ΩT
W∗
(17)
Ẇ ∗
̇
̇ ∗ 𝜙 + Pf ∇𝜉 ⋅ 𝜓 p = 0
𝜙T ∗ Ω + ∗ ΩT ∗ 𝜙 + ΩT
W
(18)
where ∇𝜉 = L∗ ∇ is the dimensionless gradient operator.
• Fluxes for slurry and proppant
)
(
Ω
s
3̂s
∇𝜉 Π
𝜓 = −Pm Ω Q 𝜙,
(19)
Pa
)
)
(
(
̂ p 𝜙, Ω nx
̂ p 𝜙, Ω 𝜓 d − Pg BG
𝜓 p = BQ
Pa
Pa
(20)
• Fracture propagation criterion
Ω = Pk (𝛾 − 𝜉)1∕2 ,
𝜉→𝛾
(21)
• Boundary condition at the fracture inlet
N
∑
𝜓 i |𝜉=0 = Pq
(22)
𝜓 p |𝜉=0 = P𝜙 𝜓 s |𝜉=0
(23)
i=1
• Boundary condition at the fracture tip
𝜓 s |𝜉=𝛾 = 0,
𝜓 p |𝜉=𝛾 = 0
(24)
Ten dimensionless groups emerging from the dimensionless equations are obtained as follows
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S. Cheng et al.
Pe =
E� W ∗
,
P∗ L∗
D
Ps = ∗ ,
L
Q∗ T ∗
,
W ∗ L∗
Pf =
Q
Pq = ∗0 ,
Q
W ∗3 P∗
K � L∗1∕2
a
,
P
=
, Pa = ∗
k
𝜇� Q∗ L∗
E� W ∗
W
tp
a2 gW ∗ Δ𝜌
Pg =
,
P
=
, P𝜙 = 𝜙0
t
𝜇� Q∗
T∗
Pm =
where tp denotes the time when the proppant injection starts.
The five characteristic quantities can be obtained by imposing five constraints on Eq. (25). Note that the dimensionless
unknowns γ, Ω, Π, ψ should be in the same order, dimensionless groups Pe, Pf , Pq are set to 1. We choose the fracture spacing D as the fracture length scale, thus Ps = 1. The
last constraint depends on the propagation regimes (Detournay 2016). The viscosity scaling is defined by imposing
Pm = 1, and the corresponding characteristic quantities are
given by
)1∕4
� D6
𝜇
∗
,
Lm
= D, Q∗m = Q0 , Tm∗ =
E� Q30
( �
)1∕4
( �
� )1∕4
𝜇 Q0 D2
𝜇 Q0 E 3
∗
Wm∗ =
,
P
=
m
E�
D2
(
(26)
where the subscript m denotes the viscosity scaling. The rest
of the dimensionless groups in Eq. (25) become
Pk,m = K =
Pg,m =
a2 gΔ𝜌D1∕2
(𝜇� 3 Q30 E� )1∕4
,
(
a4 E�
�
𝜇 Q0 D2
( 4 � 3 )1∕4
tp E Q0
Pt,m =
,
𝜇� D6
K�
,
�
(𝜇 Q0 E� 3 )1∕4
)1∕4
Pa,m =
P𝜙,m = 𝜙0
(27)
where K refers to the dimensionless toughness. The toughness scaling is defined by imposing Pk = 1, and the characteristic quantities are given by
Lk∗ = D, Q∗k = Q0 , Tk∗ =
K ′ D3∕2
K ′ D1∕2 ∗
K′
, Wk∗ =
, Pk = 1∕2
E ′ Q0
E′
D
(28)
where the subscript k denotes the toughness scaling. The
remaining dimensionless groups become
�
𝜇� Q0 E 3
aE�
, Pa,k = � 1∕2
�4
K
KD
E� Q0 tp
a2 gΔ𝜌K � D1∕2
Pg,k =
, Pt,k = � 3∕2 ,
𝜇� Q0 E�
KD
Pm,k = M =
(29)
P𝜙,k = 𝜙0
where M refers to the dimensionless viscosity. These
two scalings are related to the dimensionless viscosity or
toughness:
Tk∗
=
∗
Tm
(25)
P∗k
Pg,k
Wk∗
=
=
=
∗
∗
Pm
Wm
Pg,m
(
Pa,k
Pa,m
)−1
(
=
Pt,k
Pt,m
)−1
= K = M−1∕4
(30)
Therefore, the transition between the two scalings can
be understood in terms of either M or K . For the purpose of simplicity, the viscosity scaling in the subsequent
analysis is used unless specified elsewhere. The dimensionless numbers in Eq. (27) account for different aspects
that control the proppant migration in multiple HFs. The
dimensionless toughness K indicates the energy dissipation mode during the propagation of HFs. Pa is the ratio
between particle radius and characteristic fracture width.
Pa characterises the effect of gravitational settling, and 𝜙0
denotes the injected proppant concentration.
The dimensionless number that describes the proppant
gravitational settling is defined by comparing the velocity of the proppant settling and fluid flow as shown in
Eq. (20). However, it is not a straightforward indicator
in determining when the gravitational settling dominates
the proppant transport, which may hinge on the intrinsic
time scale that relates to the gravitational settling. The
characteristic velocity of the gravitational settling v∗g can
be written as
p
v∗g =
qg
W∗
∼
a2 gΔ𝜌
𝜇�
(31)
Here two time scales are defined. The first one refers to
the time when the migration distance of proppants reaches
the characteristic fracture length for the gravitational settling, i.e.
Tg∗ =
𝜇� D
L∗
= 2
∗
vg
a gΔ𝜌
(32)
and the other one is used to describe the time when the fracture width is big enough for proppant transport. The characteristic velocity of the fracture width v∗w is defined by
v∗w ∼
Q
W∗
= 0
∗
T
D
(33)
and the time scale when the fracture width reaches the particle size is given by
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Parameters Affecting the Proppant Distribution in Multiple Growing Hydraulic Fractures
Tw∗ =
aD
a
=
v∗w
Q0
(34)
The relationship between the two time scales as given
in Eqs. (32) and (34) defines the significance of the gravitational settling. If Tg∗ ≫ Tw∗ , it will take a long time for the
gravitational settling to kick in, and the proppant concentration distribution will be hardly influenced by gravity. If
Tg∗ ∼ Tw∗ , the effect of gravitational settling will become
significant soon after the proppant is injected. If Tg∗ ≪ Tw∗ ,
the proppant transport will be dominated by gravitational
settling. Therefore, a new dimensionless number can be
defined to measure the visibility of gravitational settling by
comparing the two time scales
Tg∗
𝜇� Q
G= ∗ = 3 0
Tw
a gΔ𝜌
(35)
This dimensionless number contains only parameters
related to fluid and proppant. and does not vary with scaling, including the scaling for multiple HFs in this paper and
the scaling for a single HF as proposed by Detournay (2004).
Given the simplicity and universality of Eq. (35), it is used
to describe the effect of gravitational settling instead of Pg.
In summary, the propagation of multiple HFs and the
proppant transport inside them are found to be controlled by
five dimensionless parameters: dimensionless toughness, K,
dimensionless particle radius, A , dimensionless time when
proppant injection begins, T , dimensionless time for gravitational settling, G , and injection concentration 𝜙0 , which
are given by
=
K′
′ 3 1∕4
,
=
(𝜇 ′ Q0 E )
( 4 ′ 3 )1∕4
t p E Q0
=
,
𝜇 ′ D6
(
a4 E ′
)1∕4
𝜇 ′ Q0
,
𝜇 ′ Q0 D2
=
a3 gΔ𝜌
,

𝜙0
(36)
4 Bifurcation Analysis
The stability analysis of multiple growing fractures has been
widely used in studies concerning thermal shock problems
(Nemat-Nasser et al. 1978; Bazant and Tabbara 1992; Bahr
et al. 2010; Hofmann et al. 2011). The simultaneous growth
of multiple HFs shares many similarities with the thermal
shock cracks, only that the force that drives the fracture
propagation differs. In this section, we derive the bifurcation condition that separates the growth of HFs at small- and
large-time limits using the analogy between thermal cracks
and HFs.
Two limiting cases exist for the growth of multiple HFs,
as shown in Fig. 2. At the beginning, the effect of interaction is small, and all fractures tend to grow simultaneously
with evenly distributed fluid flux. When the hydraulic fracturing has proceeded long enough, the stress shadow effect
becomes so significant that the inner fracture is completely
arrested, and the fluid is redistributed to the outer fractures.
Here we consider a rigid transition between these two conditions. Before the transition, all fractures grow simultaneously with no stress shadow effect. Since the fracture growth
is quasi-static, the SIF of all fractures equals the rock fracture toughness, KIc, i.e.
KIout = KIin = KIc
K̇ out = K̇ in = 0
(37)
I
I
where KIout and KIin are the mode I SIFs of the outer and
inner fractures, respectively. The overdot denotes the derivative with respect to time. Assume that the transition occurs
over a small time increment dt, where the inner fracture
stops immediately and the other keep growing, which can
be expressed by
(
)
KIout lout + dlout , lin + dlin , t + dt = KIc
(
)
KIin lout + dlout , lin + dlin , t + dt < KIc
(38)
dlout > 0, dlin = 0
where lout and lin are the half-length of the outer and inner
fractures, respectively. dlout and dlin are the corresponding
length increment. The first-order expansion of Eq. (38) is
given by
𝜕KIout ||
(
)
KIout lout + dlout , lin + dlin , t + dt ≈ KIout ||t +
| dl
𝜕lout ||t out
𝜕KIout ||
𝜕KIout ||
+
| dt = KIc
| dlin +
𝜕lin ||t
𝜕t ||t
𝜕KIin ||
(
)
|
KIin lout + dlout , lin + dlin , t + dt ≈ KIin | +
| dl
|t 𝜕lout | out
|t
𝜕KIin ||
𝜕KIin ||
+
| dl +
| dt < KIc
𝜕lin ||t in
𝜕t ||t
(39)
Substituting Eqs. (37) and (38) to Eq. (39), then subtracting
the two equations in Eq. (39) and forcing it to equality yields
𝜕KIout
𝜕lout
−
𝜕KIin
𝜕lout
=0
(40)
which defines a bifurcation point between the small- and
large-time limit. It should be noted that the above derivation does not include the fluid pressure and stress interaction
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S. Cheng et al.
Fig. 2 Schematic of a smalltime limit and b large-time
limit. In the small-time limit, all
fractures grow simultaneously
and the fluid is equally distributed. In the large-time limit, the
inner fracture stops propagation
and the fluid is equally distributed to the outer fractures
effect, thus the bifurcation condition is similar to that defined
in thermal shock problems (Bahr et al. 2010). After using the
integral form of the SIFs in Eq. (40) (Rice 1968), we obtain
KIout = 2P∗
If =
∫0
1
√
√
)
lout ( out
D out in in
I − 𝜆 IΩ
Is KI = 2P∗
𝜋 f
𝜋b
Π
1 − 𝜉2
d𝜉, IΩ =
∫0
1
Ωd𝜉, Is =
√
)
lin ( in
D in out
I I
I −
𝜋 f
𝜋a Ω s
3𝜋 2 45𝜋 4
𝜒 −
𝜒
4
16
(41)
where λ ∈ (0,1) is a constant that accounts for the asymmetric loading of the interaction stress for the outer fractures.
If, IΩ and Is are integrals related to the fluid pressure, fracture width and interaction stress, respectively, and Is can be
written explicitly using the dimensionless half-length of the
inner fracture χ = l/D. Substituting Eqs. (41) to (40), we have
(
)
If
45 45
3𝜆
− 𝜆 𝜒3 +
𝜒−
=0
4
16
4
IΩ
(42)
which is a cubic equation in terms of χ and its discriminant
is given by
(
Δ = −27
) 2
(
)
45 45 2 If
27 45 45
𝜆3 < 0
− 𝜆
−
𝜆
−
4
16
16 4
16
I2
(43)
Ω
indicating that Eq. (42) has a single real root. The SIFs in
Eq. (40) belong to the prior-bifurcation period that near the
bifurcation point, the profiles of the dimensionless fracture
width and pressure in If, IΩ correspond to the single HF case.
When neglecting the fluid lag and fluid leak-off, the selfsimilar evolution of a plane strain HF is controlled by the
dimensionless toughness (Detournay 2004), which is given
by
[
K� =
�
K4
(
)
E� 3 𝜇� Q0 ∕N
]1∕4
= N 1∕4 K
(44)
where N is the number of fractures. As the transition
between the simultaneous and preferential growth is rigid,
χ is supposed to be fixed after the bifurcation point. Since
only one real root exists in Eq. (42), we can find that there is
a one-to-one relationship between K′ and χ. According to the
scaling analysis, the role of proppant transport is characterized by a group of dimensionless numbers Q = {A, T, G, 𝜙0 }.
Therefore, the solution to Eq. (42) can be written as
(
)
𝜒 = f K� , Q
(45)
It should be noted that the bifurcation analysis incorporates the number of fractures N into the dimensionless
toughness, which is not included in the dimensionless
parameters derived by nondimensionalizing the governing
equations. This is because N only appears in the integral
domain of elasticity equation and the flux conservation condition in Eq. (22), where N cannot be explicitly extracted to
the dimensionless groups. The bifurcation analysis reveals
the relationship between the dimensionless numbers and the
fracture pattern. The fracture spacing is lumped to the characteristic quantities and dimensionless numbers instead of an
additional parameter that needs extra analysis, which makes
it easier and more efficient for further analysis.
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Parameters Affecting the Proppant Distribution in Multiple Growing Hydraulic Fractures
5 Numerical Scheme
The solution to the problem consists of two parts, one for
the fracture propagation driven by slurry flow and the other
deals with the migration of proppant among these HFs. The
general framework of the numerical scheme follows the one
proposed by Dontsov and Peirce (2015b), which splits the
algorithm into two steps. First, the slurry flow and fracture
propagation are solved for a given proppant concentration at
the previous time step. Then the proppant distribution at the
current time step is updated. The flow chart of the numerical
scheme is shown in Fig. 3.
The first part shown in Fig. 3 solves the fluid–solid
coupling problem between the fracture deformation and
slurry flow. The dual integral equations describing the
solid deformation in Eq. (1) are solved using the dual
boundary element method (DBEM) (Hong and Chen
1988; Portela et al. 1992). The key to numerically solve
the boundary integral equations lies in the accurate evaluation of the singular integrals of the kernel functions.
Thus, different numerical integration schemes are used
when dealing with integrals with different kinds of singularities (Portela et al. 1992). Since the discretization is
only required on the boundaries when using DBEM, we
use the quadratic one-dimensional element to discretize
the fracture surfaces. The slurry flow is solved using the
finite volume method (FVM), and the mesh used is nearly
the same as that for the DBEM, except that the constant
element is used. Namely, the fluid pressure is considered
uniform within one element. The system of equations is
solved in a fully coupled manner, and fracture length,
discontinuous displacements, fluid pressure and flux are
solved simultaneously. Since the matrix of the fluid flow
equation contains the fracture width, an iterative solution
scheme is employed. The iteration starts with an initial
guess consisting of the asymptotic solution near the crack
tip (Detournay 2016), thus it takes only 3–4 iterations to
reach convergence. After solving the coupled equation,
the stress intensity factors are calculated to check if any
fracture satisfies the propagation criteria. If so, the mesh
is updated by adding one boundary element to the tip, and
then the results from the previous mesh are mapped to the
new mesh. Details of the numerical implementation can be
found in our previous work (Cheng et al. 2022a).
The second part solves the proppant concentration
distribution based on the results of the first part, and an
explicit finite difference scheme is employed to solve
Eq. (4). Since the first part uses an implicit scheme to
solve the fluid–solid coupling problem, the time step can
be relatively large and it may be not compatible with the
Courant–Friedrichs–Lewy (CFL) condition (Dontsov and
Peirce 2015b). Therefore, we need to check whether the
time step fits the CFL condition in every loop. If the time
increment does not satisfy the CFL condition, the time
increment will be divided into several small steps and the
proppant concentration will be updated step by step. In
addition, extra care on the stability of the algorithm should
be taken because the abrupt change of the particle velocity when the particles become packed tends to result in
a divergent result. This problem can be solved using an
upwind scheme to calculate the difference of the proppant flux between elements. More details of the numerical
scheme can be found in Dontsov and Peirce (2015b).
6 Numerical Results
In this section, the effect of the dimensionless parameters
on proppant transport is studied. We focus our numerical
investigation on the case of three fractures as a demonstration. First, the effectiveness of the derived scaling arguments
is verified. Then, the role of each dimensionless parameter
is explored.
6.1 Effectiveness of the Scaling Arguments
Fig. 3 Flow chart of the numerical scheme
To verify the effectiveness of the scaling arguments derived
in Sect. 3, a series of numerical simulations are performed
to check whether the evolution of different physical quantities only depends on the proposed dimensionless numbers.
Firstly, we examine the effectiveness of the dimensionless
toughness, K′ , dimensionless proppant radius, A , dimensionless proppant injection time, T , and injected proppant
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S. Cheng et al.
concentration 𝜙0 . Three simulation cases with different
parameter combinations are designed, while the dimensionless numbers are kept the same: K� = 1.0 , A = 0.05,
T = 1.21, G → ∞, 𝜙0 = 0.2 . Then the effectiveness of the
parameter controlling gravitational settling, G , is checked
by case 4–5. The parameter settings of the two cases differ while the dimensionless numbers are kept the same:
Table 1 Simulation parameters in verifying the dimensionless numbers
Parameters
Case 1
Case 2
Case 3
Case 4
Case 5
E/GPa
ν
μ/(Pa s)
Q0/(10–4 ­m2/s)
KIc/(MPa ­m1/2)
D/m
a/(10–4 m)
Δρ/(103 kg/m3)
tp/s
24.0
0.2
0.1
2.00
1.0
10.0
0.50
0.0
60.0
0.2
30.0
0.2
0.2
0.512
1.0
15.0
0.49
0.0
344.5
0.2
15.0
0.2
0.3
2.73
1.0
20.0
1.13
0.0
198.9
0.2
40.0
0.2
0.005
3.088
1.1
12.0
1.65
5.125
641.1
0.1
30.0
0.2
0.003
8.333
1.0
15.0
2.24
3.683
402.5
0.1
𝜙0
K� = 0.76 , A = 0.328, T = 32.8, G = 102.0 , 𝜙0 = 0.1. The
simulation parameters of cases 1–5 are listed in Table 1.
Figure 4 shows the evolution of different quantities
of cases 1–3, where the effect of gravitational settling is
neglected and thus the proppant distribution in the fracture
is symmetric. The original data between cases are quite different due to the differences in the simulation parameters,
as shown in Fig. 4a–d. When normalizing the data with the
characteristic quantities in Eq. (26), the original data curves
collapse into two curves, one of which represents the outer
fractures denoted by #1 and #3, and the other represents the
inner fracture denoted by #2. The fluid entering the inner
fracture decreases due to the stress shadow effect and gradually the fracture stops growing, causing the proppant to stop
moving in the inner fracture, as shown in Fig. 4g. There
is only one curve for the normalized injection pressure as
shown in Fig. 4h, because the pressure drop along the wellbore and the perforation friction are neglected as indicated
in the boundary condition in Eq. (11).
Next, we verify the dimensionless parameter G controlling
the gravitational settling through case 4–5. Figure 5 shows
the proppant distribution in the outer fracture when 𝜏 = 55.4.
After normalizing the coordinates with the characteristic
Fig. 4 a–d The evolution of fluid flux, fracture width at the wellbore, proppant migration distance, and injection pressure when K� = 1.0,
A = 0.05, T = 1.21, G → ∞, 𝜙0 = 0.2. e–h The corresponding dimensionless data normalized by Eq. (26)
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Parameters Affecting the Proppant Distribution in Multiple Growing Hydraulic Fractures
length, the proppant concentration profiles of the two cases
collapse into a single curve. It can be noticed that the effect
of gravitational settling on proppant distribution has become
significant. The proppant moves faster under gravity in the
lower side of the fracture, causing the proppant to migrates
farther along the negative axis than along the positive axis as
shown in Fig. 5. The data collapse, observed across a wide
range of parameter settings, confirms the effectiveness of the
dimensionless numbers. This forms the basis for analyzing
the proppant transport behavior in terms of dimensionless
groups instead of individual physical parameters.
6.2 Effect of Dimensionless Toughness K′
and Dimensionless Proppant Size A
A uniform proppant distribution cannot be achieved if the
proppant fails to enter the fracture. Due to the stress shadow
effect, the width growth of the inner fracture can be suppressed, causing the proppant unable to enter and transport
in the fracture. Between the small- and large-time limits, the
evolution of the width of the inner fracture is not monotonic
as shown in Fig. 4f. Therefore, it is crucial to identify the
appropriate timing for the proppant injection to ensure the
proppant transport in the inner fracture.
The minimum fracture width for the proppant transport
can be obtained by examining the blocking function in the
proppant transport model, as given by Eq. (7). If the ratio
between the fracture width at the injection point and particle
radius w/a < 2 M, then the width is too small for the proppant to enter the fracture. After substituting the characteristic
quantities into w and a, we obtain the dimensionless form
of the criterion estimating whether the proppant enters the
inner fracture
Ωin
0
A
≥ 2M = 6
(46)
where the superscript “in” denotes the inner fracture, and the
subscript 0 denotes the fracture width at the injection point.
As indicated in the bifurcation analysis, the propagation of
multiple HFs before the proppant injection is controlled by
the dimensionless toughness. Therefore, the evolution of Ωin
0
is controlled by the dimensionless toughness K′ and dimensionless time τ.
Figure 6 shows the temporal change of Ωin
in the viscos0
ity-dominated (K′ < 1) and toughness-dominated (K′ > 4)
cases. Since the scalings for the two limiting regimes are
different, the evolution of Ωin
in the two regimes is depicted
0
separately. When K′ approaches 0, which indicates that
most of the energy is dissipated in viscous flow, Ωin
tends
0
to increase monotonously, as shown in the area below the
contour line Ωin
= 0.8 in Fig. 6a. As K′ becomes larger,
0
tends to increase at the beginning and then decrease as
Ωin
0
shown in the area between the contour line Ωin
= 0.5. This
0
phenomenon indicates that a window period exists for the
proppant to enter the inner fracture, and the timing for the
proppant injection should be carefully designed. If the proppant injection is too late and misses the window period, the
proppant can never enter the inner fracture before it closes.
It is worth mentioning that when K′ > 0.4 , the window
period for the proppant entering the inner fracture become
stable. For example, when K′ > 0.4, the window period for
Fig. 5 Proppant concentration
distribution along the outer fracture a before normalization and
b after normalization. The time
for case 4 and case 5 in (a) is
1083.5 s and 680.2 s, respectively, and the corresponding
dimensionless time is 𝜏 = 55.4
in (b)
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S. Cheng et al.
Fig. 6 Evolution of the dimensionless width, Ωin
, at the entry
0
of the inner fracture in the a
viscosity-dominated and b
toughness-dominated cases
Ωin
≥ 0.5 is τ = 0.6–2.0. When K′ < 0.4, the inner fracture
0
remains open for a longer period, which widens the window period for proppant entry. In the toughness-dominated
is consistent as K′ varies. The
regime, the evolution of Ωin
0
width only increases for a very short time scale, and soon
begins to decrease for the rest of the time period. It can be
noted that the magnitude of Ωin
in the toughness-dominated
0
case is ten times smaller than the viscosity-dominated case.
This result indicates that the proppant transport in the multiple HFs in the viscosity-dominated regime is much easier
than in the toughness-dominated regime.
After determining the evolution of Ωin
, the role of dimen0
sionless proppant radius A can be investigated combining
the proppant entering criterion in Eq. (46). Figure 7 shows
the envelope diagram when A = 0.135, 0.1, 0.08 and the corresponding proppant concentration profile. It can be found
from Fig. 7a, e, i that a larger proppant size corresponds
to a smaller region allowing for the proppant entry. When
A = 0.135, the proppant can enter the inner fracture only
if K′ is small enough, while the time window for proppant
injection is wide since the stress shadow effect is weak
and the fracture width is increasing. When A = 0.08, the
region is not restricted to small K′. However, only a small
time interval meets the proppant entering criterion when
K′ = 0.4–1.2, as shown in Fig. 7i. This phenomenon indicates
that a narrow time window exists for the proppant injection.
Once the proppant injection time misses the time interval,
the width of the inner fracture will be too small for the proppant to enter and transport.
In order to investigate the proppant distribution when the
proppant injection time is within and beyond the window,
three proppant injection times, i.e. τp1, τp2, and τp3 as marked
in Fig. 7a, e, i, are chosen, and the proppant distribution of
the inner (#2) and outer (#1 and #3) fractures is shown in
the second to fourth columns of Fig. 7, respectively. Considering that the time window differs when K′ varies, the
selection of the proppant injection time is also different. For
clarity, the dimensionless parameters used in simulating the
proppant distribution are listed in Table 2. Since the proppant distribution is symmetric, only half of the concentration profile is shown in Fig. 7, where ξ = 0 corresponds to
the injection point. When A = 0.135, τp1, τp2, and τp3 are all
inside the region. Since the simultaneous growth of multiple
HFs is facilitated with small K′, the proppant distribution is
uniform among the inner and outer fractures, as shown in
Fig. 7b–d. As the proppant is injected earlier in Fig. 7b, the
proppant migrates farther than that in Fig. 7c, d. Affected by
stress interaction, the width of the inner fracture is slightly
smaller than that of the outer fracture. Thus, when the proppant approaches the fracture tip, the proppant travels faster in
the outer fracture than in the inner one, as shown in Fig. 7b.
When the proppant size is larger, we set τp1 and τp2 inside the
region, while τp3 is outside the region, as shown in Fig. 7e,
i. Since K′ is increased, the width of the inner fracture will
increase first and then decrease gradually, and the proppant
transport in the inner fracture is restricted to a small distance
away from the injection point as shown in Fig. 7f, g, j, k. It
should be noted that the proppant concentration spikes in
the inner fracture when it is injected at time τp1, as shown in
Fig. 7f, j. This indicates that the fluid is being squeezed out
and the proppant is left in the fracture, which will eventually lead to screen-out. When the proppant is injected in at
time τp3 which is outside the time window, the proppant will
never have the chance to enter the inner fracture, as shown
in Fig. 7h, l.
In the toughness-dominated case, the proppant transport
in the inner fracture becomes more difficult. Considering
that the evolution of Ωin
is nearly identical in the toughness0
dominated case, we choose K� = 5.0 and A = 0.01 as an
example, and the proppant injection starts simultaneously
with the fluid, as shown in Fig. 8a. Note that the scaling
should be changed to the toughness scaling, and the proppant distribution when τ = 26.4 is given in Fig. 8b. Since
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Parameters Affecting the Proppant Distribution in Multiple Growing Hydraulic Fractures
Fig. 7 Proppant distribution in the viscosity-dominated case. The
= 6A is
contour lines of three dimensionless radius that satisfy Ωin
0
shown in (a), (e), and (i). Three cases of different injection times,
τp1, τp2, and τp3, are chosen as a demonstration as shown in the con-
tour, and the corresponding proppant distribution profiles are shown
in the second to the fourth column, respectively. The time capturing
the concentration profile is τ = 13 for (b)–(d), τ = 7.1 for (f)–(h), and
τ = 4.1 for (j)–(l)
Table 2 Dimensionless
parameters used in Fig. 7
Cases
K′
A
τp1
τp2
τp3
G
𝜙0
Figure 7b–d
Figure 7f–h
Figure 7j–l
0.18
0.28
0.50
0.135
0.10
0.08
4.0
1.5
0.8
6.5
3.6
1.7
9.0
6.0
2.7
∞
∞
∞
0.2
0.2
0.2
the proppant is injected at the very beginning, it reaches the
near-tip region of the outer fracture and begins to aggregate
at the transport front. However, the inner fracture propagates
only for a short distance and closes due to the strong interference exerted by neighboring fractures, causing the proppant
to stop transporting soon after entering the fracture.
The proppant transport behavior in the viscosity- and
toughness-dominated cases highlights the role of the stress
shadow effect in the uniformity of proppant distribution.
On the one hand, achieving a uniform proppant distribution among fractures requires a simultaneous growth
of multiple HFs, which is closely related to the stress
interaction between HFs. On the other hand, a strong
stress shadow effect can suppress the width of the inner
fractures, directly inhibiting the proppant transport and
causing the proppant to move at different velocities in the
inner and outer fractures. The results also show the regions
for the proppant entering the inner fracture when using
proppant of different sizes, which reflects the joint effect
of K′ , A , and 𝜏p.
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S. Cheng et al.
Fig. 8 Proppant distribution in
the toughness-dominated case.
The dimensionless parameters
in the simulation of the dashed
line in a are K� = 5.0, A = 0.01,
𝜏p = 0.0, G → ∞, 𝜙0 = 0.2.
b The proppant concentration
profile along the inner and outer
fractures when τ = 26.4
6.3 Effect of Gravitational Settling and Injected
Concentration
The effect of gravitational settling and injection concentration on the proppant transport is investigated in terms of
two dimensionless numbers, i.e. G and 𝜙0 . Here we take
K� = 0.53, A = 0.2 , T = 9.0 , and set G = 10 , 100, 500 and
𝜙0 = 0.3, 0.5, 0.7 to examine the corresponding proppant
distribution.
The proppant distribution in the HFs under different
parameter settings are shown in Fig. 9. It should be noted
that G measures the time scale for the presence of gravitational settling, i.e. a small G indicates that the gravitational
settling will soon dominate the proppant transport. When
G = 10 , the effect of gravitational settling is so strong that
the proppant only moves downward along the fracture and no
proppant exists inside the upper half of the fracture as shown
in Fig. 9a–c. This is a limiting case where the proppant
transport velocity caused by gravity is much larger than that
caused by fluid flow. Since all proppant moves downward, it
quickly aggregates in the lower half of the fracture, leading
to the tip screen-out. The proppant jamming in the lower
half of the fracture reduces the permeability of the fracture
channel, forcing the fluid flow to change from Poiseuille’s
flow to Darcy’s flow and inhibiting the fluid pressure from
reaching the fracture tip. The proppant pack increases the
difficulty in fracture propagation, thus the lower half of the
fracture gradually stops growing, while the upper half of the
fracture continues to propagate. This results in an undesired
situation, where the propped portion of the fracture is short
and limited while a large fraction of the fracture is left unpropped. When G = 100, the effect of gravitational settling
is weakened. The proppant begins to transport in both lower
and upper half of the fracture. However, the proppant moves
further in the lower half of the fracture, and the tip screenout forms in the transport front as shown in Fig. 9d–f. When
G = 500 , the effect of gravitational settling is weaker, and
the proppant distribution in both wings of the fracture is
nearly symmetric, as shown in Fig. 9 g–i.
It is worth mentioning that no proppant is transporting
in the inner fracture in all simulation cases shown in Fig. 9.
In fact, this result can be explained by comparing the time
scales related to the gravitational settling and growth of multiple HFs. According to the field data of hydraulic fracturing (Fisher et al. 2004; Bunger et al. 2014), the order of the
parameters is given by
(
(
(
)
)
)
E� ∼ O 1010 Pa, 𝜇 � ∼ O 10−2 ∼ 100 Pa ⋅ s, Q0 ∼ O 10−4 ∼ 10−1 m2 /s
( )
( )
( )
D ∼ O 101 m, g ∼ O 101 m/s2 , Δ𝜌 ∼ O 103 kg/m3
(47)
where O(∙) represents the order of magnitude of a physical
parameter. According to Eq. (32), the time scale Tg∗ can be
rewritten as
(
Tg∗ = D
𝜇� G2
gΔ𝜌Q20
)1∕3
(48)
When G = 10, the time scale for gravitational settling is
Tg∗ ∼ O(2.2 ∼ 1000.0)s, and that for the arrest of inner fracture is Tm∗ ∼ O(0.18 ∼ 100.0)s, which is at least one order
of magnitude smaller than Tg∗. For cases with larger G , the
difference between Tg∗ and Tm∗ will become larger since Tg∗
increases with G2∕3. The relation between the two time scales
indicates that the effect of gravitational settling appears later
than the arrest of the inner fracture. In other words, at the
time when the gravitational setting becomes significant, the
width of the inner fracture has become too small for proppant transport. This embodies the interaction between gravitational settling and stress interference between HFs.
The influence of the injection concentration on the proppant distribution is also shown in Fig. 9. We can find that the
proppant migration distance is nearly identical as 𝜙0 varies,
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Parameters Affecting the Proppant Distribution in Multiple Growing Hydraulic Fractures
Fig. 9 Proppant distribution in multiple HFs when 𝜏 = 31.6. The coordinates have been normalized by D. G is increased from the first to the
third row, while 𝜙0 is increased from the first to the third column
and the tip screen-out behavior is similar to each other, as
shown in Fig. 9a––f. This results from the boundary condition with a constant injection rate applied at the wellbore
and the fluid partitioning among multiple HFs is controlled
by the parameters K′ and G . Since a larger proppant concentration leads to a denser slurry, more energy is needed to
propagate the fracture and maintain a constant injection rate,
leading to a higher injection pressure as shown in Fig. 10b,
c. When the tip screen-out occurs, the propagation of the
lower half of the fracture requires more energy which causes
the pressure to increases, as shown in Fig. 10a, b. Once the
lower half of the fracture stops growing, the injection pressure begins to drop since the propagation of the upper half
of the fracture consumes less energy.
7 Conclusion
In this paper, a numerical model predicting the proppant
transport behavior in multiple simultaneously growing
HFs is developed by combining a fully coupled in-house
hydraulic fracturing simulator, DeepFrac, based on the
DBEM and FVM with an empirical constitutive model
describing the slurry flow and proppant transport. After
a dimensional analysis, a sensitivity study is carried out
to study the proppant transport behavior under different dimensionless parameters. The main conclusions are
drawn as follows:
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
S. Cheng et al.
Fig. 10 Evolution of the injection pressure Π0 when a G = 10, b G = 100, and c G = 500, respectively
(1) The proppant transport in multiple HFs is controlled
by five dimensionless parameters: the dimensionless
toughness, the dimensionless proppant radius, the
dimensionless proppant injection time, the dimensionless time for gravitational settling and the injected
proppant concentration.
(2) A bifurcation condition separating the simultaneous
and preferential growth of multiple HFs is derived,
and the dependence of the fracture pattern on the five
dimensionless parameters is proved.
(3) The stress shadow effect can exert huge impact on the
proppant distribution among HFs. In the viscositydominated regime, the fractures tend to grow simultaneously and produces a larger fracture width, which
benefits a uniform proppant distribution among HFs.
The window period for the proppant entry is wider
when K′ < 0.4 . In the toughness-dominated regime,
the proppant can hardly enter and transport in the inner
fracture because of the significant stress interference
exerted by neighboring fractures.
(4) Severe gravitational settling can lead to uneven proppant distribution and leave a large fraction of the fracture un-propped. The gravitational settling often lags
behind the proppant transport in the inner fracture. A
higher injected proppant concentration can increase the
injection pressure while imposing a limited effect on
the proppant distribution.
We have made a series of simplifications in our analysis. We neglected the mechanical properties of the proppant pack, where the proppant plug can support the fracture once it tends to close. We have also assumed that the
fractures propagate from notches in the wellbore, neglecting
the fracture initiation from the perforations and the fracture
nucleation. These problems highlight the need for a comprehensive three-dimensional model that considers the full
lifecycle of proppant transport in multiple HFs, which is a
challenging problem that requires further research.
Acknowledgements This research is under the support of the Program for International Exchange and Cooperation in Education by the
Ministry of Education of the People's Republic of China (Grant No.
57220500123) and the National Natural Science Foundation of China
(Grant No. 52371279).
Funding This study was funded by the Ministry of Education and the
National Natural Science Foundation of China.
Declarations
Conflict of interest The authors have no competing interests to declare
that are relevant to the content of this article.
References
Bahr H-A, Weiss H-J, Bahr U et al (2010) Scaling behavior of thermal shock crack patterns and tunneling cracks driven by cooling
or drying. J Mech Phys Solids 58:1411–1421. https://​doi.​org/​10.​
1016/j.​jmps.​2010.​05.​005
Bazant ZP, Tabbara MR (1992) Bifurcation and stability of structures
with interacting propagating cracks. Int J Fract 53:273–289.
https://​doi.​org/​10.​1007/​BF000​17341
Bokane A, Jain S, Deshpande Y, Crespo F (2013) Transport and distribution of proppant in multistage fractured horizontal wells: a
CFD simulation approach. OnePetro
Boyer F, Guazzelli É, Pouliquen O (2011) Unifying suspension and
granular rheology. Phys Rev Lett 107:188301. https://​doi.​org/​10.​
1103/​PhysR​evLett.​107.​188301
Bunger AP (2013) Analysis of the power input needed to propagate
multiple hydraulic fractures. Int J Solids Struct 50:1538–1549.
https://​doi.​org/​10.​1016/j.​ijsol​str.​2013.​01.​004
Bunger AP, Menand T, Cruden A et al (2013) Analytical predictions
for a natural spacing within dyke swarms. Earth Planet Sci Lett
375:270–279. https://​doi.​org/​10.​1016/j.​epsl.​2013.​05.​044
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Parameters Affecting the Proppant Distribution in Multiple Growing Hydraulic Fractures
Bunger AP, Jeffrey RG, Zhang X (2014) Constraints on simultaneous
growth of hydraulic fractures from multiple perforation clusters
in horizontal wells. SPE J 19:608–620
Cheng S, Zhang M, Zhang X et al (2022a) Numerical study of hydraulic fracturing near a wellbore using dual boundary element
method. Int J Solids Struct 239–240:111479. https://​doi.​org/​10.​
1016/j.​ijsol​str.​2022.​111479
Cheng S, Zhang M, Chen Z et al (2022b) Numerical study of simultaneous growth of multiple hydraulic fractures from a horizontal
wellbore combining dual boundary element method and finite volume method. Eng Anal Bound Elem 139:278–292. https://d​ oi.o​ rg/​
10.​1016/j.​engan​abound.​2022.​03.​029
Cheng S, Wu B, Han Y et al (2023a) Determination of critical fracturing spacing affecting the simultaneous growth of multiple hydraulic fractures from a horizontal wellbore. In: All days. ARMA,
Atlanta, Georgia, USA, p ARMA-2023–0454
Cheng S, Wu B, Zhang M et al (2023b) Surrogate modeling and global
sensitivity analysis for the simultaneous growth of multiple
hydraulic fractures. Comput Geotech 162:105709. http://​doi.​org/​
10.​1016/j.​compg​eo.​2023.​105709
Cipolla CL, Lolon E, Mayerhofer MJ, Warpinski NR (2009) The effect
of proppant distribution and un-propped fracture conductivity
on well performance in unconventional gas reservoirs. SPE, p
SPE-119368
Crespo F, Aven NK, Cortez J, et al (2013) Proppant distribution in
multistage hydraulic fractured wells: a large-scale inside-casing
investigation. OnePetro
Daneshy A (2011) Uneven distribution of proppants in perf clusters.
World Oil 232:75–76
Detournay E (2004) Propagation regimes of fluid-driven fractures in
impermeable rocks. Int J Geomech 4:35–45
Detournay E (2016) Mechanics of hydraulic fractures. Annu
Rev Fluid Mech 48:311–339. https://​d oi.​o rg/​1 0.​1 146/​a nnur​
ev-​fluid-​010814-​014736
Dontsov EV (2023) A model for proppant dynamics in a perforated
wellbore. Int J Multiph Flow 167:104552. https://​doi.​org/​10.​
1016/j.​ijmul​tipha​seflow.​2023.​104552
Dontsov EV, Peirce AP (2014) Slurry flow, gravitational settling and
a proppant transport model for hydraulic fractures. J Fluid Mech
760:567–590. https://​doi.​org/​10.​1017/​jfm.​2014.​606
Dontsov EV, Peirce AP (2015a) A Lagrangian approach to modelling
proppant transport with tip screen-out in KGD hydraulic fractures. Rock Mech Rock Eng 48:2541–2550. https://​doi.​org/​10.​
1007/​s00603-​015-​0835-6
Dontsov EV, Peirce AP (2015b) Proppant transport in hydraulic fracturing: crack tip screen-out in KGD and P3D models. Int J Solids
Struct 63:206–218. https://​doi.​org/​10.​1016/j.​ijsol​str.​2015.​02.​051
Dontsov EV, Suarez-Rivera R (2020) Propagation of multiple hydraulic fractures in different regimes. Int J Rock Mech Min Sci
128:104270. https://​doi.​org/​10.​1016/j.​ijrmms.​2020.​104270
El Sgher M, Aminian K, Matey-Korley V, Ameri S (2023) Impact of
the stress shadow on the proppant transport and the productivity of the multi-stage fractured Marcellus shale horizontal wells.
OnePetro
Erdogan F, Sih G (1963) On the crack extension in plates under plane
loading and transverse shear. J Basic Eng 85:519–525
Fisher MK, Heinze JR, Harris CD, et al (2004) Optimizing horizontal completion techniques in the Barnett shale using Microseismic fracture mapping. In: All days. SPE, Houston, Texas, p
SPE-90051-MS
Fredd C, McConnell S, Boney C, England K (2000) Experimental
study of hydraulic fracture conductivity demonstrates the benefits
of using proppants. SPE, p SPE-60326
Germanovich LN, Astakhov DK, Mayerhofer MJ et al (1997) Hydraulic
fracture with multiple segments I. Observations and model formulation. Int J Rock Mech Min Sci 34:97.e1-97.e19. https://​doi.​
org/​10.​1016/​S1365-​1609(97)​00188-3
Gu M, Kulkarni P, Rafiee M, et al (2014) Understanding the optimum
fracture conductivity for naturally fractured shale and tight reservoirs. SPE, p D021S007R004
Gu M, Mohanty K (2014) Effect of foam quality on effectiveness
of hydraulic fracturing in shales. Int J Rock Mech Min Sci
70:273–285
Hofmann M, Bahr H-A, Weiss H-J et al (2011) Spacing of crack patterns driven by steady-state cooling or drying and influenced by
a solidification boundary. Phys Rev E 83:036104. https://​doi.​org/​
10.​1103/​PhysR​evE.​83.​036104
Hong H-K, Chen J-T (1988) Derivations of integral equations of
elasticity. J Eng Mech 114:1028–1044. https://​doi.​org/​10.​1061/​
(ASCE)​0733-​9399(1988)​114:​6(1028)
Hu X, Wu K, Li G et al (2018) Effect of proppant addition schedule
on the proppant distribution in a straight fracture for slickwater treatment. J Petrol Sci Eng 167:110–119. https://​doi.​org/​10.​
1016/j.​petrol.​2018.​03.​081
Kresse O, Weng X, Wu R, Gu H (2012) Numerical modeling of
hydraulic fractures interaction in complex naturally fractured
formations. OnePetro
Lecampion B, Desroches J (2015) Simultaneous initiation and growth
of multiple radial hydraulic fractures from a horizontal wellbore.
J Mech Phys Solids 82:235–258. https://​doi.​org/​10.​1016/j.​jmps.​
2015.​05.​010
Luo B, Wong GK, Han Y (2023) Modeling of dynamic bridging of
solid particles in multiple propagating fractures. Int J Solids Struct
262–263:112078. https://​doi.​org/​10.​1016/j.​ijsol​str.​2022.​112078
Mao S, Zhang Z, Chun T, Wu K (2021) Field-scale numerical investigation of proppant transport among multicluster hydraulic fractures. SPE J 26:307–323. https://​doi.​org/​10.​2118/​203834-​PA
Miller C, Waters G, Rylander E (2011) Evaluation of production
log data from horizontal wells drilled in organic shales. SPE,
p SPE-144326
Nemat-Nasser S, Keer LM, Parihar KS (1978) Unstable growth of
thermally induced interacting cracks in brittle solids. Int J Solids Struct 14:409–430. https://​doi.​org/​10.​1016/​0020-​7683(78)​
90007-0
Nikolskiy D, Lecampion B (2020) Modeling of simultaneous propagation of multiple blade-like hydraulic fractures from a horizontal well. Rock Mech Rock Eng 53:1701–1718. https://​doi.​org/​
10.​1007/​s00603-​019-​02002-4
Peirce APP, Bunger APP (2015) Interference fracturing: nonuniform
distributions of perforation clusters that promote simultaneous
growth of multiple hydraulic fractures. SPE J 20:384–395.
https://​doi.​org/​10.​2118/​172500-​PA
Portela A, Aliabadi MH, Rooke DP (1992) The dual boundary element method: effective implementation for crack problems. Int
J Numer Meth Engng 33:1269–1287. https://​doi.​org/​10.​1002/​
nme.​16203​30611
Qu H, Tang S, Liu Z et al (2021) Experimental investigation of proppant particles transport in a tortuous fracture. Powder Technol
382:95–106. https://​doi.​org/​10.​1016/j.​powtec.​2020.​12.​060
Raymond S, Aimene Y, Nairn J, Ouenes A (2015) Coupled fluidsolid geomechanical modeling of multiple hydraulic fractures
interacting with natural fractures and the resulting proppant
distribution. OnePetro
Rice JR (1968) Mathematical analysis in the mechanics of fracture.
Fract Adv Treat 2:191–311
Roostaei M, Nouri A, Fattahpour V, Chan D (2018) Numerical simulation of proppant transport in hydraulic fractures. J Petrol Sci
Eng 163:119–138. https://​doi.​org/​10.​1016/j.​petrol.​2017.​11.​044
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
S. Cheng et al.
Shiozawa S, McClure M (2016) Simulation of proppant transport
with gravitational settling and fracture closure in a threedimensional hydraulic fracturing simulator. J Petrol Sci Eng
138:298–314. https://​doi.​org/​10.​1016/j.​petrol.​2016.​01.​002
Siddhamshetty P, Wu K, Kwon JS-I (2019) Modeling and control of
proppant distribution of multistage hydraulic fracturing in horizontal shale wells. Ind Eng Chem Res 58:3159–3169. https://​
doi.​org/​10.​1021/​acs.​iecr.​8b056​54
Vincent M (2012) The next opportunity to improve hydraulic-fracture stimulation. J Petrol Technol 64:118–127
Wang J, Elsworth D, Denison MK (2018) Propagation, proppant
transport and the evolution of transport properties of hydraulic
fractures. J Fluid Mech 855:503–534. https://​doi.​org/​10.​1017/​
jfm.​2018.​670
Warpinski NR, Mayerhofer MJ, Vincent MC et al (2009) Stimulating
unconventional reservoirs: maximizing network growth while
optimizing fracture conductivity. J Can Pet Technol 48:39–51
Wen Z, Zhang L, Tang H et al (2022) A review on numerical simulation of proppant transport: Eulerian–Lagrangian views. J Petrol Sci Eng 217:110902. https://​doi.​org/​10.​1016/j.​petrol.​2022.​
110902
Wu K, Olson JE (2013) Investigation of the impact of fracture spacing
and fluid properties for interfering simultaneously or sequentially
generated hydraulic fractures. SPE Prod Oper 28:427–436. https://​
doi.​org/​10.​2118/​163821-​PA
Yi SS, Wu C-H, Sharma MM (2018) Proppant distribution among multiple perforation clusters in plug-and-perforate stages. SPE Prod
Oper 33:654–665. https://​doi.​org/​10.​2118/​184861-​PA
Yu W, Zhang T, Du S, Sepehrnoori K (2015) Numerical study of the
effect of uneven proppant distribution between multiple fractures
on shale gas well performance. Fuel 142:189–198. https://d​ oi.o​ rg/​
10.​1016/j.​fuel.​2014.​10.​074
Zhang X, Jeffrey RG, Bunger AP, Thiercelin M (2011) Initiation and
growth of a hydraulic fracture from a circular wellbore. Int J Rock
Mech Min Sci 48:984–995. https://d​ oi.o​ rg/1​ 0.1​ 016/j.i​ jrmms.2​ 011.​
06.​005
Zhang X, Wu B, Jeffrey RG, Connell LD, Zhang G (2017) A pseudo3D model for hydraulic fracture growth in a layered rock. Int J
Solids Struct 115:208–223. https://d​ oi.o​ rg/1​ 0.1​ 016/j.i​ jsols​ tr.2​ 017.​
03.​022
Zhang X, Wu B, Connell LD et al (2018) A model for hydraulic
fracture growth across multiple elastic layers. J Petrol Sci Eng
167:918–928. https://​doi.​org/​10.​1016/j.​petrol.​2018.​04.​071
Zhou L, Chen J, Gou Y, Feng W (2017) Numerical investigation of
the time-dependent and the proppant dominated stress shadow
effects in a transverse multiple fracture system and optimization.
Energies 10:83. https://​doi.​org/​10.​3390/​en100​10083
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