Thus, in this study, a coupled proppant transportation equation is used to simulate the proppant transport process in transverse fractures of horizontal wells. Field data are used to verify the accuracy of this model. Several numerical simulations are carried out under different conditions to study the proppant transport process and determine the critical factors of proppant distribution. These studies can provide a better understanding of the proppant transport process in transverse fractures of horizontal wells, which is helpful to the proppant schedule design.
Due to uncertainty in the hydraulic fracture, the proppant design remains a great challenge in the oil and gas industry. Some authors have previously used numerical and experimental methods in this research area to better understand the proppant movement in hydraulic fractures. Sahai et al., investigated the proppant transport in complex hydraulic fractures, and they found the proppant size, density, and pump rate all had an impact on proppant transport [ 5 ]. Wang et al., analyzed the influences of injection time, injection rate, fracturing fluid viscosity, and proppant combination type on the migration and sedimentation law of a proppant in single and branch fractures. They demonstrated that when the fracture morphology is single, the viscosity of the fracturing fluid is recommended to be between 30 and 60 mPa&#;s. When the fracture morphology is complex, the recommended value is between 40 and 50 mPa&#;s [ 6 ]. Wang et al., focused on modeling and examining proppant movement with respect to the diversion of energy. The experimental results indicate that proppant breakage, embedding, and particle migration are harmful to fracture conductivity. With the increase in closure pressure to 50 MPa, large embedding of the proppant occurs, and damage to the conductivity increases from 12.7% to 85.6% [ 7 ]. Suri et al., studied the effect of fracture roughness on proppant transport in hydraulic fractures using the Joint Roughness Coefficient and a three-dimensional multiphase modeling approach. They believed that the interproppant and proppant wall interactions become dominant, which adds turbulence to the flow [ 8 ]. Merzoug et al., discussed proppant placement efficiency considering the hydraulic fracture and natural fracture interaction. They revealed that the effect of the pre-existing fracture friction angle and the angle of approach, as well as the differential horizontal stress on hydraulic fracture and natural fracture interaction mechanisms and proppant transport and placement [ 9 ]. Zheng et al., introduced a CFD (Computational Fluid Dynamics) -DEM (Discrete Element Method) technique to investigate the effects of different roughness characterization parameters on the efficiency of proppant transport using supercritical CO. They suggested higher pump power is required for efficient proppant transport using supercritical CO; otherwise, sand plugs may occur [ 10 ]. Although various experimental and numerical research studies have been performed on proppant transport at vertical or horizontal wellbores, the proppant transport process at transverse fractures has rarely been reported in the literature. In particular, the flow pattern of the fracturing fluid and the proppant delivery law in transverse fractures in horizontal wells differ significantly from that of conventional hydraulic fracturing [ 11 15 ]. In recent years, some authors have tended to consider more complex fracture conditions, such as fracture surface roughness. Through the reconstruction of rock surfaces using 3D techniques, the comparison between the proppant transport behavior for the smooth and rough fractures can be discussed. In addition, under this condition, the fracturing fluid velocity and viscosity and particle density and size can be investigated. In previous studies, numerical simulation research was mainly performed to understand the proppant transport mechanism in hydraulic fractures. Moreover, the CFD-DEM method has been commonly used in the field of proppant transport, and its validated effectiveness has been recognized. However, when the fracture geometry is complex, such as in secondary branch fracture development, the computational load for the proppant transport is heavy and cannot be applied for the field application. Considering multiphase flow simulation and the coupling between proppant transport and crack propagation, numerical modeling is more complex [ 16 21 ]. The radial flow zones around the wellbore of horizontal wells can be observed, while the two-dimensional flow of the fracturing fluid should be considered. Furthermore, proppant transportation in horizontal wells is rarely studied [ 22 24 ]. To offset the calculation efficiency, the analytical modeling of proppant transportation is necessary.
In recent years, a series of breakthroughs have been made in the exploration and development of unconventional oil and gas worldwide, and the production of unconventional oil and gas is growing rapidly and becoming increasingly prominent in the global energy supply. The application of the hydraulic fracturing technique on horizontal wells has been successfully applied in the unlocking of unconventional reservoirs, and the final hydraulic fracturing effectiveness has a close relationship with the proppant distributions [ 1 4 ]. Due to the fact that horizontal wells are usually drilled along the direction of the minimum horizontal principal stress, multiple transverse fractures can be generally created. The connection between the transverse fracture and horizontal wellbore can result in special flow conditions, thus affecting the proppant flow into fractures. Thus, understanding the proppant transport along the transverse fractures of horizontal wells is necessary.
2. Model Establishment
Horizontal wells are generally oriented along the direction of the minimum horizontal ground stress; thus, horizontal well fracturing usually forms transverse fractures perpendicular to the axis of the wellbore. Regardless of the fracture geometry model adopted and the fracture morphology, due to the unique internal boundary conditions, there exists a region of a radial flow of fracturing fluid near the wellbore, which is a distinctive feature of fracturing fluid flow within the transverse fracture of horizontal well fracturing. This fracturing fluid flow characteristic can have an impact on the fracture morphology and proppant delivery, and, at the same time, due to the presence of the radial flow field, it may increase the complexity of fractures in the near-wellbore zone and generate additional friction; thus, there is a high risk of sand plugging when sand-carrying fluids are flowing in it. The hydraulic fracturing model usually consists of three basic control equations, which are the mass conservation equation of the injected fluid, the rock fracture mechanics equation that relates the fracture width to the fluid pressure distribution inside the fracture, and the fluid flow equation that describes the pressure and fluid flow inside the fracture. These are the basic equations for describing and controlling the hydraulic fracturing process, and the dynamics of the fracture can be obtained by the coupling of the three equations. The dynamic fracture expansion process can be obtained by solving the coupled solution of the above three equations. In addition, in order to describe the loss of fracturing fluid into the formation during the fracturing process, the transportation of proppant, and the distribution of the temperature field in the fracture and the formation, the hydraulic fracturing model should also include the equation for the loss of fracturing fluid filtration, the equation for the transportation of proppant, and the equation for calculating the temperature field, etc. The model should also include the equations of fracturing fluid filtration, proppant transportation, and temperature field calculation, as shown in Figure 1
2.1. Assumptions
To accurately characterize the proppant transport along the transverse fracture of a horizontal well, the modeling of proppant transport should follow several assumptions.
(1) The formation rocks are linear elastomers, and fracture cracking and expansion satisfy linear elastic fracture mechanics;
(2) A single shot hole cluster can be simplified as an annular cut in the wellbore wall, and the hydraulic fracture initiates radially at this annular cut;
(3) The fracture height is well controlled by the spacer, and the fracture extends only within the reservoir;
(4) The fracturing fluid is an incompressible Newtonian fluid, which flows in a two-dimensional laminar flow in the fracture;
(5) The flow of fracturing fluid in the fracture is between two parallel walls, and the influence of gravity on the flow of fracturing fluid is not considered;
(6) The rate of loss of fracturing fluid at a point in the fracture depends on the time that the point is exposed to the fracturing fluid and satisfies the Carter loss equation, but the loss of fracturing fluid from the formation does not affect the fluid pressure distribution in the fracture;
(7) The velocity gradient of the fracturing fluid in the fracture length and fracture height directions is negligible compared to the velocity gradient of the fracturing fluid across the width of the fracture;
(8) The effect of temperature field changes within the fracture on the rheology of the fracturing fluid is not considered.
The basic axis system can be illustrated, as shown in Figure 2 , and the related boundary condition can be set. The fracture height is equal to the pay zone thickness due to stress confinement between the pay zone and the boundary zone; thus, two-dimensional flow in the x-axis and y-axis can be assumed.
2.2. Continuity Equations
d
x,
d
y, and
d
z, respectively.
In the vicinity of the wellbore, there exists a region of a radial flow of fracturing fluid, whose flow direction can be decomposed into two components, horizontal and vertical. Thus, in the following model derivation process, the two-dimensional continuity equations and differential equations of motion of the fracturing fluid in the x&#;y planes are established and combined with the boundary conditions and the initial conditions to determine the distribution of the flow field inside the fracture. Figure 3 illustrates the control body unit of a hydraulic fracture. Any one control body unit inside the fracture is selected as the study object, and its length, height, and width arex,y, andz, respectively.
&#;
ρ
s
v
x
w
&#;
x
d
x
d
y
d
t
+
&#;
ρ
s
v
y
w
&#;
y
d
x
d
y
d
t
+
2
ρ
f
v
l
d
x
d
y
d
t
=
&#;
&#;
ρ
s
w
&#;
t
d
x
d
y
d
t
(1)
According to the law of the conservation of mass, the reduction in the mass of the control body at time dt must be equal to the difference between the outgoing and incoming masses:
&#;
ρ
s
(
x
,
y
,
t
)
w
(
x
,
y
,
t
)
&#;
t
+
&#;
&#;
ρ
s
(
x
,
y
,
t
)
q
(
x
,
y
,
t
)
+
2
ρ
f
v
l
(
x
,
y
,
t
)
=
0
(2)
The continuity equation for the sand-carrying fluid is:
In this formula,
ρ
s
is the density of the sand-carrying liquid,
kg
/
m
3
;
ρ
f
is the density of the pure fracturing fluid,
kg
/
m
3
;
v
x
is the flow rate of fracturing fluid along the
x
-axis,
m
/
s
;
v
y
is the flow rate of fracturing fluid along the
y
-axis,
m
/
s
;
v
l
is the filtration velocity of fracturing fluid along the
z
-axis to the formation,
m
/
s
;
q
x
is the flow rate per unit fracture height in the
x
-direction,
m
3
/
s
;
q
y
is the flow rate per unit fracture length in the y-direction,
m
; and
w
is the crack width,
m
.
ρ
s
=
c
ρ
p
+
(
1
&#;
c
)
ρ
f
(3)
The density of the sand-carrying liquid can be calculated by the following formula:
In this formula,
ρ
p
is the density of proppant,
kg
/
m
3
; and
c
is the proppant volume concentration, decimal.
&#;
w
(
x
,
y
,
t
)
&#;
t
+
&#;
&#;
q
(
x
,
y
,
t
)
=
&#;
2
v
l
(
x
,
y
,
t
)
(4)
The continuity equation of pure fracturing fluid is:
v
l
=
C
t
t
&#;
τ
(5)
The filtration velocity of the carrier fluid can be calculated by the Cater filtration equation:
In the equation,
C
t
is the overall filtrate coefficient for the liquid,
m
/
s
;
t
is the hydraulic fracturing duration,
s
; and
τ
is the time at which filtrate begins at a certain point in the fracture,
s
.
1
C
t
=
1
C
1
+
1
C
2
+
1
C
3
(6)
The overall filtrate coefficient is calculated as follows:
&#;
&#;
Ω
&#;
0
t
2
v
l
d
x
d
y
d
t
&#;
&#;
Ω
&#;
0
t
&#;
w
&#;
t
d
x
d
y
d
t
+
&#;
0
t
Q
i
d
t
=
0
(7)
During the hydraulic fracturing process, a portion of the injected fracturing fluid from the surface is used to expand the volume of the fractures, while another portion is lost or filtrates into the formation. Therefore, the total mass conservation equation for the fracturing fluid is as follows:
In the
Qi
equation, represents the injection rate of the fracturing fluid at the wellbore perforation site,
m
3
/
s
.
2.3. Fluid Flow Equation
v
x
&#;
v
x
&#;
x
+
v
y
&#;
v
x
&#;
y
+
v
z
&#;
v
x
&#;
z
=
&#;
1
ρ
&#;
p
&#;
x
+
μ
ρ
&#;
2
v
x
&#;
x
2
+
&#;
2
v
x
&#;
y
2
+
&#;
2
v
x
&#;
z
2
v
x
&#;
v
y
&#;
x
+
v
y
&#;
v
y
&#;
y
+
v
z
&#;
v
y
&#;
z
=
&#;
1
ρ
&#;
p
&#;
y
+
μ
ρ
&#;
2
v
y
&#;
x
2
+
&#;
2
v
y
&#;
y
2
+
&#;
2
v
y
&#;
z
2
v
x
&#;
v
z
&#;
x
+
v
y
&#;
v
z
&#;
y
+
v
z
&#;
v
z
&#;
z
=
&#;
1
ρ
&#;
p
&#;
z
+
μ
ρ
&#;
2
v
z
&#;
x
2
+
&#;
2
v
z
&#;
y
2
+
&#;
2
v
z
&#;
z
2
(8)
Navier&#;Stokes equations are basic equations describing the flow of viscous fluids. Assuming that the fracture width is very small compared to the scales of the fracture length and height and that the fracture width changes uniformly and smoothly, the three-dimensional Navier&#;Stokes equations can be simplified to two-dimensional equations, which, in turn, lead to the so-called lubrication equations or the so-called cubic law. The Navier&#;Stokes equations for three-dimensional flow are as follows:
v
z
=
0
. This results in the simplification of the aforementioned three-dimensional Navier&#;Stokes equations into a two-dimensional equation:
v
x
&#;
v
x
&#;
x
+
v
y
&#;
v
x
&#;
y
=
&#;
1
ρ
&#;
p
&#;
x
+
μ
ρ
&#;
2
v
x
&#;
x
2
+
&#;
2
v
x
&#;
y
2
+
&#;
2
v
x
&#;
z
2
v
x
&#;
v
y
&#;
x
+
v
y
&#;
v
y
&#;
y
=
&#;
1
ρ
&#;
p
&#;
y
+
μ
ρ
&#;
2
v
y
&#;
x
2
+
&#;
2
v
y
&#;
y
2
+
&#;
2
v
y
&#;
z
2
(9)
Disregarding the flow of fracturing fluid along the width direction of the fracture, we have:. This results in the simplification of the aforementioned three-dimensional Navier&#;Stokes equations into a two-dimensional equation:
v
x
z
=
1
2
μ
z
2
&#;
w
2
4
&#;
p
&#;
x
v
y
z
=
1
2
μ
z
2
&#;
w
2
4
&#;
p
&#;
y
(10)
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By integrating Equation (9) with respect to z twice and substituting the boundary conditions from the equation, we obtain:
z
-direction. Integrating them separately with respect to
z
from
&#;
w
2
to
w
2
,we have the following description:
q
x
=
&#;
w
3
12
μ
&#;
p
&#;
x
q
y
=
&#;
w
3
12
μ
&#;
p
&#;
y
(11)
Equation (11) represents the expressions of the velocity distributions of u and w in the-direction. Integrating them separately with respect tofromto,we have the following description:
&#;
p
(
x
,
y
,
t
)
=
&#;
12
μ
w
3
(
x
.
y
,
t
)
q
(
x
,
y
,
t
)
(12)
Below expresses Equation (12) in vector form:
q
x
and
q
y
in Equation (13) by the fracture width yields
w
, the expressions of the average velocities of the fracturing fluid along the
x
- and
y
-directions across the entire fracture width are as follows:
v
f
x
=
&#;
w
2
12
μ
&#;
p
&#;
x
v
f
y
=
&#;
w
2
12
μ
&#;
p
&#;
y
(13)
Dividingandin Equation (13) by the fracture width yields, the expressions of the average velocities of the fracturing fluid along the- and-directions across the entire fracture width are as follows:
&#;
&#;
w
3
(
x
,
y
,
t
)
12
μ
&#;
p
(
x
,
y
,
t
)
&#;
2
v
l
(
x
,
y
,
t
)
=
&#;
w
(
x
,
y
,
t
)
&#;
t
(14)
The fracturing fluid flow can be described as:
2.4. Fracture Width Equation
w
x
,
y
,
t
=
&#;
16
1
&#;
ν
2
E
&#;
z
l
F
τ
+
y
G
τ
τ
2
-
y
2
(15)
England and Green proposed a relationship between normal stress acting on the fracture surfaces due to any arbitrary distribution within the fracture and the corresponding induced fracture width under plane strain conditions:
w
x
,
y
,
t
=
1
&#;
ν
G
h
2
&#;
4
y
2
1
2
p
(
x
,
y
,
t
)
&#;
σ
h
(16)
In the PKN model, for the fracture tip perpendicular to the fracture length, the shape of the fracture is elliptical, and its fracture width equation is as follows:
w
=
1
&#;
ν
G
h
2
&#;
4
y
2
1
2
&#;
j
=
1
n
p
x
,
y
n
&#;
σ
h
(17)
In Equation (17), the fluid pressure within the fracture changes only with respect to the fracture length and remains constant at the fracture&#;height interface. However, for hydraulic fracturing fluid flow in horizontally oriented fractures in the vicinity of the wellbore, there is a radial flow region due to the presence of the fracturing fluid. Therefore, the fluid flow becomes two-dimensional in the x&#;y plane, with pressure gradients in both the x- and y-directions. In this case, to calculate the fracture width using the equation above at any position along the fracture length, we take the average pressure in the fracture&#;height direction at that location to compute the fracture width, as follows:
In the equation,
w
represents the fracture width, m;
h
is the fracture height, m;
ν
is the dimensionless Poisson&#;s ratio;
G
is the shear modulus,
MPa
;
n
is the number of units in the fracture&#;height direction; and
σ
h
is the minimum horizontal in situ stress,
MPa
.
2.5. Proppant Continuity Equation
&#;
(
c
ρ
p
v
p
x
w
)
&#;
x
+
&#;
(
c
ρ
p
v
p
y
w
)
&#;
y
+
&#;
(
c
ρ
p
w
)
&#;
t
=
0
(18)
Using the principle of mass conservation, which states that the net mass inflow of proppant into a control volume is equal to the change in mass of proppant within that control volume, the proppant continuity equation is as follows:
In this equation,
v
p
x
is the horizontal velocity of the proppant,
m
/
s
; and
v
p
y
is the vertical velocity of the proppant.
2.6. Proppant Velocity Equation
Currently, conventional hydraulic fracturing models, when calculating proppant transport, only consider the settling motion of proppant particles within the fracturing fluid and assume that the horizontal transport velocity of proppant is the same as the horizontal flow velocity of the fracturing fluid. In reality, as proppant particles move within the fracture, the fluid velocity within the fracture varies in a parabolic distribution across the fracture width. At the center of the fracture, the shear forces are minimal, and the fracturing fluid velocity is at its maximum, as shown in Figure 4 Proppant particles tend to move toward areas of lower shear forces, i.e., the central region of the fracture. As a result, the horizontal transport velocity of the proppant may be greater than the average fluid velocity. However, during the actual fracturing process, the movement of the proppant within the fracture is also influenced by the fracture surface and proppant concentration, which may lead to the proppant&#;s horizontal transport velocity often being less than the average fluid velocity.
v
p
=
v
p
x
&#;
i
+
v
p
y
&#;
j
(19)
v
p
x
=
v
f
x
&#;
f
h
v
p
y
=
v
f
y
+
v
s
_
c
(20)
Therefore, in the actual hydraulic fracturing process, there exists both vertical velocity slip and horizontal velocity slip between the proppant and the fracturing fluid. In the second chapter, we provided expressions for the proppant settling velocity and horizontal transport velocity. Thus, the proppant&#;s transport velocity can be expressed as:where
v
f
x
is the fracturing fluid velocity in the x-direction,
m
/
s
;
v
f
y
is the fracturing fluid velocity in the y-direction,
m
/
s
;
f
h
is the dimensionless proppant horizontal transport velocity correction factor; and
v
s
_
c
is the corrected proppant settling velocity,
m
/
s
.
w
&#;
c
&#;
t
&#;
1
&#;
c
&#;
w
&#;
t
&#;
&#;
&#;
x
1
&#;
c
+
c
ρ
p
ρ
f
1
&#;
h
f
q
x
&#;
&#;
&#;
x
1
&#;
c
q
y
&#;
c
v
s
_
c
ρ
p
ρ
f
=
v
l
(21)
In Equation (20),is the fracturing fluid velocity in the x-direction,is the fracturing fluid velocity in the y-direction,is the dimensionless proppant horizontal transport velocity correction factor; andis the corrected proppant settling velocity,
μ
s
l
u
r
r
y
=
μ
f
1
&#;
c
p
1
&#;
c
p
/
c
m
a
1
&#;
c
m
1
&#;
c
m
(22)
Once the proppant concentration distribution at various points within the fracture is determined, the viscosity of the slurry can be modified using the following equation. The modified slurry viscosity is then used in place of the fracturing fluid viscosity for the calculation of fracture width and flow field distribution in the subsequent time step.
In this equation,
μ
s
l
u
r
r
y
represents the viscosity of the slurry,
mPa
&#;
s
;
c
p
is the viscosity of the fracturing fluid, a dimensionless fraction;
a
1
is the volume concentration of the proppant, a dimensionless fraction;
c
m
is the volume fraction of the proppant when randomly densely packed, a dimensionless fraction; and
a
1
is the first-order viscosity coefficient, a dimensionless fraction. Here, we take
c
m
and
a
1
as 0.64 and 2.5, respectively.
2.7. Initial Conditions and Boundary Conditions
By combining equations, we obtain the system of governing equations for solving proppant transport in a horizontally fractured well. To obtain a unique solution, initial conditions and boundary conditions for these equations must be provided.
(1) Initial conditions
w
x
,
y
,
0
=
0
c
x
,
y
,
0
=
0
(23)
At the initial moment, the fracture width is zero, and the proppant concentration within the fracture is zero, i.e.,:
(2) Boundary conditions
&#;
w
3
12
μ
&#;
p
&#;
n
|
&#;
Ω
p
e
r
f
=
Q
i
c
&#;
Ω
p
e
r
f
,
t
=
c
i
n
j
;
(
t
&#;
t
s
)
(24)
In the wellbore perforation segment, the flow rate of the fracturing fluid is equal to the injection rate. When the proppant-laden slurry injection begins, the proppant concentration at the perforation location is the same as the proppant concentration in the injected slurry:
&#;
w
3
12
μ
&#;
p
&#;
x
|
&#;
Ω
t
i
p
=
0
&#;
w
3
12
μ
&#;
p
&#;
y
|
&#;
Ω
b
o
u
n
d
=
0
&#;
c
&#;
x
|
&#;
Ω
t
i
p
=
0
&#;
c
&#;
y
|
&#;
Ω
b
o
u
n
d
=
0
(25)
At the leading edge of the fracture and the upper and lower boundaries, the flow rate of the fracturing fluid is zero, and the proppant concentration gradient is zero:
Above, by combining the fracture width equation, slurry continuity equation, slurry flow equation, and proppant transport equation with the given initial conditions and boundary conditions, a closed system of governing equations for solving proppant transport is established.
2.8. Numerical Solution
The system of governing equations for hydraulic fracturing fluid flow and proppant transport in horizontally fractured wells, composed of equations, is a system of partial differential equations. These equations are coupled together, making direct analytical solutions difficult; thus, numerical methods are used. By numerically solving the above governing equations, it is possible to obtain the fracture width, fracture internal pressure distribution, fracturing fluid velocity distribution within the fracture, and proppant concentration distribution.
f
h
i
+
1
,
j
q
x
i
+
1
,
j
c
i
+
1
,
j
&#;
f
h
i
,
&#;
1
j
q
x
i
&#;
1
,
j
c
i
&#;
1
,
j
2
Δ
x
+
q
y
i
,
j
+
1
&#;
v
s
_
c
i
,
j
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(26)
This paper primarily employs the finite difference method. By discretizing the governing equations and initial/boundary conditions, corresponding difference equations are constructed. Then, the computational domain is divided into a grid, and these difference equations are iteratively solved on the grid. Ultimately, this approach yields a numerical solution to the problem.
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