Granular Matter (2017) 19:32 DOI 10.1007/s10035-017-0718-5
ORIGINAL PAPER
Numerical simulation of transport and placement of multi-sized proppants in a hydraulic fracture in vertical wells Guodong Zhang1,2 · Marte Gutierrez2 · Mingzhong Li1
Received: 24 August 2016 © Springer-Verlag Berlin Heidelberg 2017
Abstract The placement of proppant in hydraulic fractures, which is governed by slurry flow, proppant transport and settlement, can significantly affect the conductivity of the fracture, and this will subsequently affect the productivity of the hydraulically fractured wells. To investigate proppant transport mechanisms and placement profiles in a hydraulic fracture, computational fluid dynamics coupled with discrete element method is used to model slurry flow and micro-mechanical interactions of proppant particles in a fracture. The effect of the solid phase is introduced in terms of volumetric porosity and particle interaction forces, and a contact model is used to simulate the particle-to-particle and the particle-to-wall interactions. The particles are modelled as real spheres, and for a two-dimensional computational model, an out-of-plane length of the maximum particle diameter is assumed. The proppants are injected into the fracture through seven perforations to mimic the slurry flow in a fracture in vertical wells. When the proppants are injected into a fracture driven by thin fracturing fluids, they quickly settle out of the fluid and accumulate on the fracture bottom, forming a proppant dune. A three-layer flow pattern forms in the fracture consisting of: (1) a stationary proppant bed at the bottom, (2) a proppant-fluid mixture layer above it,
B B
Guodong Zhang
[email protected] Marte Gutierrez
[email protected] Mingzhong Li
[email protected]
1
College of Petroleum Engineering, China University of Petroleum (East China), Qingdao 266580, China
2
Department of Civil and Environment Engineering, Colorado School of Mines, 308 Coolbaugh Hall, 1012 14th St., Golden, CO 80401, USA
and (3) a clean fluid layer at the top. During the early stages, the motion of the proppants is governed by suspension, settlement and fluidization, and the injected proppants mainly settle and accumulate near the wellbore. When the equilibrium height of the dune is reached, the dune changes to a proppant bank. Then the later injected proppants are transported to overshoot the bank and dragged deeper into the fracture. A new proppant transport mechanism, vortex, is observed, which governs the proppant motion after they leave the proppant bank. Keywords Particle-fluid flow · Multi-sized proppants · CFD–DEM · Micro-mechanics · Particulate modeling
List of symbols Cd d di dmax Er max fb f¯drag Fc h He i Kn Ks m n p Re p
Turbulent drag coefficient Particle diameter (m) Diameter of particle i (m) Maximum particle diameter (m) Maximum relative error (%) Particle body force per unit volume (N) Drag force applied to each individual discrete particle (N) Contact force (N) Fluid cell size (m) Proppant bank equilibrium height (m) Particle index Normal stiffness at the contact (N/s) Shear stiffness at the contact (N/m) Effective system mass (kg) Particle number Fluid pressure (Pa) Particle Reynolds number
123
32
Page 2 of 15
t U v vin j vs vtail Vf Vn Vs W
Time (s) u¯ = average velocity of all particles in a given fluid element (m/s) Average relative velocity between particles and fluid (m/s) Fluid velocity (m/s) Fluid injection rate (m/s) Sweep rate (m/s) Tail-in rate (m/s) Particle falling velocity (m/s) Relative normal velocity at the contact (m/s) Relative shear velocity at the contact (m/s) Perforation height (m)
Greek symbols β γn γs δn δs ε εp θ θe θinj μ ρf ρp
A coefficient Critical normal damping ratio Critical shear damping ratio Overlap defined to be the relative contact displacement in the normal direction (m) Overlap defined to be the relative contact displacement in the shear direction (m) Porosity Particle injection volumetric concentration Settling angle (◦ ) Erosion angle (◦ ) Injection angle (◦ ) Fluid dynamic viscosity (Pa s) Fluid density (kg/m3 ) Particle density (kg/m3 )
1 Introduction As one of the most efficient completion technologies, hydraulic fracturing is widely used to economically develop oil and gas reservoirs in the petroleum industry, especially for unconventional resources (shale gas, tight gas and tight oil, etc.). During the hydraulic fracturing treatment, high pressurized liquid is injected, and huge fracture systems, which provide high conductive channels for oil and gas flow from reservoirs to the wellbore, are formed. In order to keep the facture open, the proppant-laden fluid is injected after the fracture is created [1]. The injected proppants can prevent the fracture from closing and maintain significant conductivity for the fracture after shut-down. After the pressure in the fracture is reduced, the region of the fracture un-propped by proppants will close, and the conductivity of the whole fracture will significantly decrease. Therefore, it is of significant importance to identify where the injected proppants go. However, since proppants were first used in hydraulic
123
G. Zhang et al.
fracturing, the question of proppant placement in the fracture has remained, for the most part, unanswered [2]. For conventional fracturing fluids, their ability to transport the proppants is strong due to high viscosity, and the injected proppants almost remain suspended after shut-down. On the contrary, when the proppants are driven by thin fracturing fluids such as slick-water, which is commonly used in developing unconventional resources, the proppant settlement becomes prominent, so the proppant transport and placement will be totally different from those in high viscous fracturing fluids. It is estimated that more than 80% of the fracturing fluids used in hydraulic fracturing treatment in the United States is slick-water [3]. Therefore, to identify the transport mechanisms and placement of the proppants driven by such thin fracturing fluids can help to improve fracturing design, and this will eventually significantly promote the development of unconventional resources. Slot experiments have been used for decades to investigate proppant transport mechanisms and placement profiles in a fracture, and this is helpful to improve the understanding of the macro-mechanics of the proppants driven by thin fracturing fluids [4–6]. Kern et al. [7] first conducted an experiment for sand-water slurry transport in a slot. The STIM-LAB consortium has been collecting data on proppant transport in slots for more than 20 years [4]. The previous experimental results show that when the proppants are injected into a slot, they will quickly settle out of the fluid and accumulate at the slot bottom, forming a proppant dune as shown in Fig. 1. During the early injection stages, the injected proppants mainly accumulate near the wellbore. As the proppants are continuously injected, the height of the dune increases and finally comes up to an equilibrium height, so the proppant dune changes to a proppant bank. Until then, proppants injected later can be transported to overshoot the proppant bank and dragged deeper into the fracture. During the development stages of the proppant dune, a three-layer flow pattern in the slot is observed consisting of: (1) a stationary proppant bed at the slot bottom, (2) a proppant-fluid mixture layer like a “traction carpet” above the proppant bed, and 3) finally a clean fluid layer at the top of the slot [4,5,8]. For different development stages of the proppant dune, the proppant transport mechanisms are different. In the early development stages, proppant settlement and fluid drag gov-
Fig. 1 Schematic of the proppant transport in a hydraulic fracture [4]
Numerical simulation of transport and placement of multi-sized proppants in a hydraulic…
Page 3 of 15
32
Fig. 2 Three particle transport styles above the sand dune: surface creep, saltation and suspension [9] Fig. 3 Schematic of the proppant transport in the fracture after the proppant bank is formed [10]
ern the proppant motion. When the three-layer flow pattern forms, fluidization becomes the main transport mechanism. The fluidization caused by fluid lift, which plays a central role in particle suspension in channel flows, is proposed as an efficient particle transport mechanism [4]. Fluidization of the proppant particles occurs when fluid turbulence “lifts” the proppants off the stationary bed. The following three primary transport models of the proppants in the fluidization layer were proposed by Mack et al. [9] as shown in Fig. 2. 1. Surface creep: When the proppant-laded fluid flows across the top of the stationary proppant bed at a flow rate higher than the critical starting velocity, proppant particles will roll or slide along the surface of the stationary proppant bed. 2. Saltation: As the flow rate increases, parts of the proppant particles are lifted off the stationary proppant bed and travel downstream before falling back and being resuspended. 3. Suspension: When the flow rate exceeds the critical suspension velocity, some proppant particles are suspended and transported by fluids. Additionally, during the later development stages after the proppant bank is formed, the formed proppant bank hinders the flow stream, so the injected proppants are vertically lifted to overshoot the proppant bank and dragged deeper into the fracture [8]. A more prominent fluidization layer forms near the wellbore as shown in Fig. 3 [10]. Once the proppant bank forms, the proppant transport in the slot is primarily governed by fluidization and sedimentation. Moreover, when the injected proppants leave the proppant bank at a high flow rate, a high turbulence regime at the top of the proppant bank creates a vortex, which makes the proppant particles travel in a circular motion back toward the bank [11]. Therefore, due to the viscosity difference, the flow field of the low viscosity fluid is different from that of high viscosity fluid. This causes
the transport micromechanics of the injected proppants to be change in different stages of proppant transport. Numerical simulation is an important method to investigate the hydrodynamics of the particle-fluid slurry flow. Although the transport and distribution of the proppants in the fracture has been modeled for decades using different numerical models, most of the models are 2-D and different assumptions were proposed. In the earlier models, the effective-fluid was presented to approximate the particle-fluid slurry depending on the particle volumetric concentration [12–14]. The slurry flow was modeled by Eskin [15,16] considering fracture shear and fluid leak-off, but the gravitational settling, fluid-particle and particle-particle interactions were neglected, so uniform particle distribution along the fracture was observed. The particle-fluid interaction was taken into account by Tsai et al. [17]. Considering the particle settlement and particle-fluid interaction, Dontsov and Peirce [18–20], and Shiozawa and McClure [21] simulated the proppant transport and distribution in the fracture. To the best of our knowledge, the only work, taking the particle-particle interaction into account, was carried out by Blyton et al. [22], but they only focused on the proppant distribution, and the proppant transport mechanisms and flow patterns have not been investigated. Ultimately, the development of the simulation on the proppant transport in the fracture is slow, and the interaction between particles and particles is almost neglected in the earlier studies. However, for the flow of dense particle-liquid slurry, the velocity difference between particles and fluids is large, and collisions between particles and particles become more frequent. Therefore, the effect of particle-particle interaction on the flow field cannot be neglected, and this will subsequently affect the proppant transport and placement in the fracture. Coupled computational fluid dynamics and discrete element modeling (CFD–DEM) are promising method that trace the trajectories of individual particles, and inves-
123
32
Page 4 of 15
G. Zhang et al.
tigate the micromechanics of particles in the slurry flow [23–25]. In this paper, the micromechanics of the proppants in the fracture in vertical wells were studied, and the transport mechanisms and placement of the proppants were investigated further using CFD–DEM. The effects of the fluid viscosity, injection rate, uneven perforation and open-hole completion were analyzed, and two processes of sweep and tail-in during the hydraulic fracturing treatments were modeled.
2 CFD–DEM coupling in PFC2D When the proppant-fracturing fluid slurry flows in a fracture, the proppant particles and the fluid will mutually interact. Additionally, the interactions of the proppant-proppant and proppant-wall are also prominent, and this will eventually affect the proppant transport and placement in the fracture. In this section, a numerical solution to the problem of particleliquid mixture flow between two parallel plates is described to mimic the slurry flow in the fracture. A commercially available Particle Flow Code, PFC2D , implemented with coupled CFD–DEM approach is used in this study. Fluid flow is modeled in PFC2D based on modified Navier–Stokes equations [26], while a contact model is used to simulate particleparticle and particle-wall interactions. The contact force combining normal and shear components is shown in Eq. (1). The first term in both of the parentheses is dominant by elastic deformation of the interactional particles, while the second term refers to the damping effect. The slip behavior of the interactional particles is checked by calculating the maximum allowable shear contact force. F c = K n δn −2γn m K n |Vn | + K s δs − 2γs m K s |Vs | (1) where F c is the contact force (in N); K n is the normal stiffness at the contact (in N/m); K s is the shear stiffness at the contact (in N/m); δn (in m) is the overlap defined to be the relative contact displacement in the normal direction; δs (in m) is the overlap defined to be the relative contact displacement in the shear direction; γn is the critical normal damping ratio; γs is the critical shear damping ratio; m is the effective system mass (in kg); Vn is the relative normal velocity at the contact (in m/s); and Vs is the relative shear velocity at the contact (in m/s). The modified Navier–Stokes equations are used to model fluid flow in PFC2D [26]. The effect of solid phase is introduced in terms of volumetric porosity and coupling force. The momentum equation and continuity equation for an incompressible fluid in a porous medium are described as follows,
123
ρf
∂ε v + ρ f v · ∇ (ε v ) = −ε∇ p + μ∇ 2 (ε v ) + fb ∂t
∂ε + ∇ (ε v) = 0 ∂t
(2) (3)
where ρ f is the fluid density (in kg/m3 ); ε is the porosity; t is the time (in s); v is the fluid velocity (in m/s); p is the fluid pressure (in Pa); μ is the fluid dynamic viscosity (in Pa s); and fb is the particle body force per unit volume (in N). The interaction between particles and fluid is modeled taking into account both the average particle force on fluid and the average fluid force on the particles for each computational fluid cell. The drag force applied by particles to fluid on each computational fluid cell is defined as follows. fb = β U
(4)
where β is a coefficient defined below, and U is the average relative velocity between particles and fluid (in m/s), defined as follows: U = u¯ − v
(5)
where u¯ is the average velocity of all particles in a given fluid element (in m/s). The coefficient β depends on the porosity of fluid element. For low porosity (ε < 0.8), Ergun’s correlation [27] is used as shown in Eq. (6a), which is derived by observing pressure drop for fluid flowing through porous materials. On the righthand side of the correlation, the first term in the parentheses is dominant at low Reynolds numbers and low porosity, while the second term begins to dominant when turbulent effects occur. For high porosity (ε ≥ 0.8), the expression is derived from the corrected nonlinear drag force exerted on a spherical particle by fluid [28]. β=
(1 − ε) U¯ ε < 0.8 150 − ε) μ + 1.75ρ d (1 f d 2 ε2
4 U¯ ρ f (1 − ε) β = Cd ε ≥ 0.8 3 dε1.7
(6a) (6b)
where d is the particle diameter (in m), and Cd is the drag coefficient [29] defined in terms of the particle Reynolds number Re p as follows: Cd =
⎧ ⎨ 24 1+0.15 Re0.687 p Re p
⎩ 0.44 U¯ ρ f εd¯ Re p = μ
Re p < 1000 Re p > 1000
(7)
(8)
The body force term fb has a unit of force per unit volume. The force applied to each particle is proportional to
Numerical simulation of transport and placement of multi-sized proppants in a hydraulic…
vinj
(9)
A SIMPLE fluid coupling scheme [30] for incompressible viscous flow is applied to fluid cells within a fixed rectangular geometry. Particle motion is calculated using the law of motion and superposition of all the relevant forces. Between the law of motion and force-displacement law in the mechanical (DEM) calculation, the pressure and velocity field of the fluid is calculated by the CFD code, and this process is repeated in each time step for the whole simulation duration. A critical time-step, which is related to the minimum eigenperiod of the total system, is estimated at the start of each cycle to guarantee the computed solution stable tcrit =
m/K
(10)
where tcrit is the critical time-step (in s); m is the mass (in kg) and K is the contact stiffness (in N/m). The particles in PFC2D are real spheres, and the fluid cells are assumed to have an out-of-plane width of the maximum particle diameter in the model [31]. Therefore, the 2-D geometry for the particles and flow field can be predictive for the pseudo 3-D case in PFC2D , and the PFC2D can accurately represent 2-D dynamics of particle collisions. During hydraulic fracturing treatments, the fracture aperture is small, so the widthwise transport of particle-fluid slurry is commonly ignored in previous studies. For narrow fracture, the widthwise movement of injected particles is also restricted, so the particle dynamics in such a fracture can be approximately treated as 2-D. Since the PFC2D can represent 2-D particle collision, it is appropriate to use the 2-D code to model proppant transport in a fracture during hydraulic fracturing. However, since the 2-D particle interaction cannot accurately represent real particle 3D dynamics, the multiple collision effect on particle dynamics is somewhat underestimated in the simulation. Actually, the development on the simulation of proppant transport is slow, and most of the earlier models are 2D. In addition, the interactions between particles have not been widely considered in previous proppant transport simulations. The simplifications can significantly cause more unrealistic particle dynamic behavior and subsequently affect the prediction of proppant placement in a fracture. Therefore, although the 2-D dynamics in PFC2D cannot accurately represent realistic particle interaction, the work presented in this paper is envisioned to improve understanding of the micromechanics of proppant particles in fractures by considering micromechanical interactions between particles.
Non-slip boundary
32
5 cm
f¯drag
f¯b 1 = π d3 6 (1 − ε)
Non-slip boundary
Constant pressure
the volume of each particle. The drag force applied to each individual discrete particle is as follows.
Page 5 of 15
17 cm
Fig. 4 Diagram of the fracture in vertical wells
A rectangular channel with a length of 17 cm and a height of 5 cm was used to represent the fracture in this work as shown in Fig. 4. Seven perforations were set at the left boundary, and a velocity boundary condition was applied on fluid cells of the perforations. The fluid, which flows across the domain from the left boundary to the right boundary, was injected through the perforations in a parallel method to mimic the injection process of proppant-fracturing fluid slurry in a fracture in vertical wells. Non-slip fluid boundary condition was applied at the contacts with the upper and lower limits, and a constant pressure boundary condition was applied on the right boundary. Multi-sized particles with diameter range of 0.5 mm ≤ d ≤ 0.8 mm, which obeys the normal distribution, were used to represent the actual proppants. The parameters for the proppant, fluids and model used are summarized in Table 1. Before proppant injection, the fluid was only injected at an injection rate of Vinj and adequate simulation cycles were conducted to reach steady flow field in the calculation domain. Then particles, which were under the same injection rate as fluid, were generated at the perforations. The particle diameter was set as random value in the range of 0.5 mm ≤ d ≤ 0.8 mm, which obeys the normal distribution. Therefore, the volumetric rate of the injection slurry is Vin j W d max , where W is the height of the perforation and dmax is the maximum particle diameter, which indicates the out-of-plane width of the 2-D fracture. Thevolumetric n 1 3 rate of the injected particle can be calculated by i=1 6 π di , so the injection volumetric concentration can be calculated as follows εp =
n 1 i=1
6
π di3 /Vinj W dmax
(11)
where ε p is the injection volumetric concentration. For high-concentration proppant transport, the particleparticle interactions, and the particle-fracture wall interaction will increase, which will affect simulation convergence. In PFC2D flow simulation, the fluid cell size should be set at least two times larger than particle diameter, so all of the particles in a fluid cell are exerted with the same fluid drag force. This can allow PFC2D to deal with a higher particle concentration problem. In addition, the high-concentration issue has
123
32
Page 6 of 15
Table 1 Properties of particles, fluid and model
G. Zhang et al.
Property
Symbol
Unit
Value
Particle diameter
d
m
5 × 10−4 − 8 × 10−4
Particle density
ρp
kg/m3
2600
Fluid density
ρf
kg/m3
1000
Fluid dynamic viscosity
μ
Pa s
5 × 10−3
Particle contact normal stiffness
Kn
N/m
103
Particle contact shear stiffness
Ks
N/m
103
Critical normal damping ratio
γn
–
0.65
Critical shear damping ratio
γs
–
0.65
Fluid injection rate
vinj
m/s
0.15
Particle injection volumetric concentration
εp
–
0.0873
been discussed in a previous paper [25], and the results show that the PFC2D can deal with a high particle concentration problem up to 39%, which is far larger than proppant concentration used in hydraulic fracturing treatment. In this paper, a small particle injection concentration of 8.73% was used, and the highest particle concentration occurs on the bottom of a fracture, where the settled particles accumulate to form a particle dune, wherein the particle concentration can reach a particle piling concentration of 52%.
3 Results and discussion 3.1 Transport mechanisms Due to the inefficiency of low viscosity fracturing fluids to transport proppants, the proppant micromechanics in such thin fluids are totally different from those in high viscosity fracturing fluids. In this section, proppant transport mechanisms driven by thin fluids in a fracture in vertical wells are investigated. Figure 5 shows particle transport and placement in the fracture in vertical wells. The particle size is distinguished by different colors, and red refers to the largest particles while the smallest particles are marked by blue. Three distinct developmental stages before reaching the final proppant placement profile are obtained, and different mechanisms of proppant transport are observed as described below. 1. Stage 1: proppant dune formation—When the proppants are injected into the fracture, they quickly settle out of the fluid and accumulate on the fracture bottom. An injection angle (θinj = 26◦ ) of the injected proppants forms near the wellbore, which denotes the angle between the horizontal direction and the proppant transport direction governed by the proppant settlement and fluid drag. As the accumulation of the injected proppants on the fracture bottom, a proppant dune forms as shown in Fig. 5a.
123
Since the channel size is small, the transport time of the injected proppants before settling on the fracture bottom is short. Therefore, it is clear that the injected large and small proppants cannot separate from each other when they are transported in the fracture, and they are almost uniformly mixed. Because high flow rate causes a far transport distance of the injected proppants, the dune is flat with a wide spread. A three-layer flow pattern is observed: a stationary proppant bed at the bottom, a proppant-fluid mixture layer above it, and a clean fluid layer at the top. The proppant-fluid mixture layer is composed by a fluidization layer and a suspension layer. The fluidization layer is above the stationary proppant bed, and the proppants roll, slide and jump on the stationary bed forming a “traction carpet”. The suspension layer is at the top of the proppant-fluid mixture layer, and the proppants are suspended in the fluid. Therefore, the transport mechanisms during this stage are suspension, fluidization and settlement. 2. Stage 2: proppant dune development—As the proppants are continuously injected, the height of the proppant dune increases, and the dune gradually hinders the flow stream. Then the transport directions of the injected proppants are changed, and the proppants injected through the lower perforations are vertically lifted to overshoot the dune. During this stage the injected proppants mainly accumulate near the wellbore, so a slanted surface forms with a settling angle [11] of θ = 11◦ as shown in Fig. 5b. The shape of the top slanted surface is affected by the erosion caused by the transported proppants and flowing fluid. Due to the hindering of the proppant dune on the flow stream, the flow field at the back side of the dune greatly weakens. During this stage, the injected proppants cannot be transported to the farthest side of the dune, so a more slanted surface forms with a large settling angle of θ = 33◦ . For the proppants injected through the lower perforations, they are lifted upward to overshoot the dune, so
Numerical simulation of transport and placement of multi-sized proppants in a hydraulic…
θinj=26
Page 7 of 15
32
θ=11 θ=33
(a)
(b)
4.36 cm
Small particles
θe=75
(c)
(d)
Fig. 5 Particle transport and placement in the fracture in vertical wells. The particles, with a diameter spread of 0.5 mm ≤ d ≤ 0.8 mm that obeys normal distribution, are randomly injected into the fracture. The
largest particles are marked by red while blue indicates the smallest particles. a t = 2.5 s, b t = 5 s, c t = 7.5 s, d t = 10 s (color figure online)
the fluidization in this location governs the proppant motion. 3. Stage 3: proppant bank formation-As the dune height increases, an equilibrium height (He = 4.36 cm) is obtained and then, the proppant dune changes to a proppant bank. During this stage, a critical flow rate is obtained, and the proppant amount of settlement and liftoff on the bank comes to balance, so the height of the bank becomes stable. The injected proppants are lifted upward to overshoot the bank to settle on the back side of the bank, so the bank only develops lengthwise. A two-layer flow pattern is observed as shown in Fig. 5c consisting of a stationary proppant bed at the bottom of the fracture, and a proppant-fluid mixture layer above the proppant bed. At the top of the mixture layer, the proppants in the suspension layer remain suspended before settling on the proppant bank. In comparison, when the proppants in the fluidization layer leave the bank, a vertical vortex governs the proppant motion, and it drags the proppants in a circular motion back toward the bank. As the proppant bank develops lengthwise approaching the fracture tip, the flow fields near the back side of the bank greatly change because of the restraint of the fracture tip. Then a stronger vortex governs the proppant transport, and the injected proppants are largely captured by the fracture tip as shown in Fig. 5d. Additionally, because the small particles can be dragged for a longer distance, a small region of small particles forms at the innermost side of the bank. Due to the large flow rate in the fracture, the erosion caused by the flowing fluid is severe, and a large particle-free region forms near the wellbore. An erosion angle, which is defined as the angle between the bank front surface and the horizontal plane
as illustrated by θe in Fig. 5d, is used to describe the erosion degree, and a large erosion angle of θe = 75◦ is observed. 3.2 Effect of fluid viscosity The ability of fracturing fluids to transport the proppants is carefully focused on by petroleum engineers, and one of important parameters to evaluate this ability is fluid viscosity. According to Stokes Law, particle settling velocity is inversely proportional to fluid viscosity, causing a particle slowly settle out of a high viscous fluid. In contrast, the drag force is proportional to fluid viscosity, so the transport distance of a particle driven by a viscous fluid is long. In general, both of the effects cause a particle to be dragged a longer distance by a higher viscosity fluid. Therefore, the profile of the proppant placement can be significantly affected by the viscosity of the fracturing fluids. Figure 6 shows the proppant placement in the fracture driven by a higher viscosity fracturing fluid. It is clear that the injected proppants are transported a longer distance, forming a wide dune. The injection angle is divided into two angles during the original injection stage as shown in Fig. 6a. At the top of the mixture layer, the injection angle is the same to that of the lower viscosity fluid. At the bottom of the mixture layer, the high viscosity fluid causes the fluidization to be stronger, and the injected proppants are dragged a longer distance, so a small injection angle (θinj = 12◦ ) forms. Because of the wide dune, it will take a longer time to develop the dune and form the final proppant bank. After the proppant bank is formed, the proppant-free region behind the bank is small, and no vortex forms at the back of the bank. When the injected proppants leave the bank, settlement and fluid drag govern their movement. In
123
32
Page 8 of 15
G. Zhang et al.
θinj=26 θinj=12
(a)
(b)
4.18 cm
(c)
(d)
Fig. 6 Proppant placement in the fracture driven by a higher viscosity fracturing fluid (μ = 10−2 Pa s). The particles, with a diameter spread of 0.5 mm ≤ d ≤ 0.8 mm that obeys normal distribution, are randomly
injected into the fracture. The largest particles are marked by red while blue indicates the smallest particles. a t = 2.8 s, b t = 5.6 s, c t = 8.4 s, d t = 11.2 s (color figure online)
addition, because the ability of high viscosity fluid to transport the proppant is stronger, a smaller equilibrium height (He = 4.18 cm) is developed. Ultimately, the transport and placement of the proppants is very sensitive to fluid viscosity, and little viscosity change can cause proppant placement to be significantly different. Sahai et al. [8] have evaluated the proppant transport and placement in a complex fracture system by flowing proppantslickwater slurry through a complex slot configuration in a low-pressure condition. Figure 7 shows the experimental results in a vertical slot system with one “T” intersection. For the proppants being transported in the primary slot, it is clear that during the early stages, a proppant dune forms on the fracture bottom with a wide spread as shown in Fig. 7a. As the proppants are continuously injected into the fracture, the height of the dune increases, and the dune gradually hinders the flow stream. Therefore, during these stages as shown in Fig. 7b, c, the injected proppants mainly settle near the wellbore. However, due to the effect of the secondary fracture, the flow field in the primary slot behind the secondary fracture is significantly weakened. This subsequently reduces the drag force of the fluid on the proppant particles, so the injected proppant particles are more inclined to settle and accumulate at the front of the bank. Therefore, a more obvious particle peak forms at the front of the bank during the experiment of Sahai et al. [8] compared with the simulation result. When the equilibrium height of the dune is reached, the proppant dune changes to a proppant bank, then the later injected proppants are lifted to overshoot the bank and transported deeper into the fracture. After the final profile of the proppant bank is developed, a large proppant-free region is also observed as shown in Fig. 7f. As can be observed, the proppant transport and placement during different stages in the experiment are
identical to the simulation results in this work, so the coupled CFD–DEM approach is effective in modeling the micromechanics of proppant flow and transport in a fracture in vertical wells.
123
3.3 Effect of injection rate Since it is the fluid drag that governs the proppant horizontal transport, the injected proppants can be dragged a longer distance, forming a wider dune when they are injected at a high rate. Figure 8 shows the proppant placement in a fracture at a high injection rate (vin j = 0.25 m/s). Due to the high injection rate, the horizontal transport ability of the proppants strengths, causing a small injection angle of θinj = 18◦ . As the height of the proppant dune increases, although the dune hinders the flow stream, the injected proppants still can be dragged a long distance by the high velocity fluid. Therefore, during the middle development stages of the proppant dune, the injected proppants settle and accumulate along the whole dune, and the height of the dune at different locations almost develops at the same time. It is clear from Fig. 8d that due to the erosion caused by the transported proppants and flowing fluid, a large proppantfree region forms near the wellbore. Because the part of the fracture that is un-propped by the proppants will close after shut-down, the productivity of the fractured wells will be significantly reduced. This phenomenon should be taken into consideration in developing injection strategies in fracturing design. High injection rate can transport the proppants deeper into the fracture, but a large proppant-free region also can form near the wellbore, which will reduce the fracturing efficiency.
Numerical simulation of transport and placement of multi-sized proppants in a hydraulic…
Page 9 of 15
32
Fig. 7 Experimental results on proppant transport and placement in the fracture [8]
θinj=18
(a)
(b) Particle-free region
(c)
(d)
Fig. 8 Proppant placement in the fracture at a high injection rate (vinj = 0.25 m/s). The particles, with a diameter spread of 0.5 mm ≤ d ≤ 0.8 mm that obeys normal distribution, are randomly injected into
the fracture. The largest particles are marked by red while blue indicates the smallest particles. a t = 1.6 s, b t = 3.2 s, c t = 4.8 s, d t = 6.4 s (color figure online)
3.4 Effect of uneven perforation
which also can induce a vertically asymmetrical fracture. Thus, the perforations may non-uniformly distribute along the fracture during some fracturing treatments. When the proppant-fracturing fluid slurry is injected into the fracture through non-uniform perforations, the fluid field will be different from that of uniform injection, causing the proppant transport and placement to be different. In order to study the effect of non-uniform injection on the proppant transport and
Perforation is one of the most common completion methods, which can significantly affect the productivity of wells [32–34]. For vertically heterogeneous reservoirs, the fracture will asymmetrically propagate upward and downward, causing the fracture to be vertically asymmetrical. In addition, an interlayer can hinder the propagation of the fracture,
123
32
Page 10 of 15
G. Zhang et al.
(a)
(b)
4.47 cm
(c)
(d)
Fig. 9 Proppant placement in the fracture injected through lower perforations. The particles, with a diameter spread of 0.5 mm ≤ d ≤ 0.8 mm that obeys normal distribution, are randomly injected into the
fracture. The largest particles are marked by red while blue indicates the smallest particles. a t = 3.3 s, b t = 6.6 s, c t = 9.9 s, d t = 13.2 s (color figure online)
placement, two proppant injection scenarios from the lower and upper perforations are simulated.
2. Upper-perforation injection—Similarly, the proppantfracturing fluid slurry is injected into the fracture through five upper perforations to mimic excessive downward propagation of the fracture. Figure 10 shows the proppant placement in the fracture injected through the five upper perforations. It is clear that although the proppants are injected into the fracture though the upper perforations with a high suspension height, because of small injection volumetric flow rate, the injected proppant still quickly settle out of the fluid and accumulate on the fracture bottom forming a short dune. Because the injection position moves upward, the flow field near the bottom of the fracture weakens, and the drag effect of the flowing fluid on the proppant decreases. During the early injection stage, the height of the formed proppant dune is small, and the dune has little effect on the flow field. As the injection continues, the height of the dune increases, and the dune gradually hinders the flow stream causing the injected proppants to be vertically lifted. Clearly, during the dune developmental stage, the proppant placement is almost the same to that of uniform injection. It is observed from Fig. 10d that due to lack of the erosion caused by flowing fluid at the lower position of the fracture, the proppant-free region near the wellbore is filled, and there only remains a small open region. Thus, the injection scenario of the proppants through upper perforations can improve the proppant placement near the wellbore.
1. Lower-perforation injection-In this scenario, the proppant-fracturing fluid slurry is injected through five lower perforations, and the perforation density in this injection section is the same to that of uniform injection. At the top of the fracture, no perforation is set to mimic excessive upward propagation of the fracture. Figure 9 shows the proppant placement in the fracture injected through the five lower perforations. Because the suspension height of the injected proppants is small, the injected proppants quickly settle out of the fluid and accumulate on the fracture bottom. The horizontal transport distance is short, and the formed proppant dune is close to the wellbore with a short spread. As the proppants are continuously injected, the dune height increases, and the dune gradually hinders the flow stream. Subsequently, the injected proppants are vertically lifted and settle near the wellbore, so the proppant dune quickly increases and finally changes to a proppant bank. It is clear from Fig. 9b that the proppant dune is steep due to the accumulation of the early injected proppants. Only when the equilibrium of the bank is reached, can the later injected proppants be transported deeper into the fracture. Since the injection volumetric flow rate is small, the ability of the fluid to drag the injected proppants decreases, so a high proppant bank forms with an equilibrium height of He = 4.47 cm. Ultimately, when the proppant-fracturing fluid slurry is injected into the fracture through the lower perforations, the formed proppant dune is closer to the wellbore and develops faster during the early stage. However, this injection scenario has little to do with the final profile of the proppant bank.
123
3.5 Effect of open-hole completion For oil and gas development, it is important to obtain the most contact between the wellbore and the reservoir to ensure the maximum drainage. Due to the efficiencies and cost savings,
Numerical simulation of transport and placement of multi-sized proppants in a hydraulic…
Fig. 10 Proppant placement in the fracture injected through upper perforations. The particles, with a diameter spread of 0.5 mm ≤ d ≤ 0.8 mm that obeys normal distribution, are randomly injected into the
Page 11 of 15
32
fracture. The largest particles are marked by red while blue indicates the smallest particles. a t = 3.3 s, b t = 6.6 s, c t = 9.9 s, d t = 13.2 s (color figure online)
θinj=22
(a)
(b) θinj=8
(c)
(d)
Fig. 11 Proppant placement in the fracture in vertical wells with openhole completion. The particles, with a diameter spread of 0.5 mm ≤ d ≤ 0.8 mm that obeys normal distribution, are randomly injected into
the fracture. The largest particles are marked by red while blue indicates the smallest particles. a t = 0.64 s, b t = 1.28 s, c t = 1.92 s, d t = 2.56 s (color figure online)
open-hole completions are becoming more acceptable as the primary completion of choice [35,36]. When wells with open-hole completion are fractured, the wellbore is totally open to the fracture, and the proppant-fracturing fluid slurry is injected into the fracture through the opening wellbore. Figure 11 shows proppant placement in the fracture in vertical wells with open-hole completion. In this simulation, the proppants are randomly injected into the fracture from the whole left boundary. The high volumetric flow rate causes a high flow rate of the fluid in the fracture, which drags the proppants to a long distance with a small injection angle of θinj = 22◦ . When the injected proppants are transported to the fracture tip, they are captured and accumulate at the lower position near the fracture tip. The captured proppants near the fracture
tip block the lower position of the fracture, so the flow stream turns upward. As more and more proppants are captured near the fracture tip, the blocking becomes more severe. Due to the upward turn of the flow stream, the transport of the injected proppants gradually overshadows its settlement, so a smaller injection angle of θinj = 8◦ is observed as shown in Fig. 11c. It is clear that when the fracture tip is blocked by the proppants, the flow field becomes approximately horizontal, and almost all of the later injected proppants are transported to the fracture tip. This causes more severe blocking, soon the fracture tip is totally blocked and the fracture tip screen-out takes place, prohibiting the injected proppants and fluid from further transport [19,20]. When the elapse time comes up to 2.56 s, the flow channel is totally blocked by the captured
123
32
Page 12 of 15
G. Zhang et al.
proppants, and the simulation is forced to stop. As can be seen from Fig. 11d, the fracture tip screen-out can cause a big proppant-free region in the fracture, and this will significantly decrease the productivity of the fractured wells after the fracture closing. 3.6 Effect of sweep In order to transport the injected proppants driven by thin fracturing fluids deeper into the fracture, the sweep stage is commonly employed between successive proppant-laden stages during the hydraulic fracturing treatments [8]. During the sweep stage, the fracturing fluid is alone pumped in order to drag the earlier injected proppants deeper into the fracture. In this section, the effect of the sweep on the proppant placement is investigated, and two cases are simulated at the sweep velocity of vs = 0.15 m/s and vs = 0.3 m/s, respectively. In order to mimic the sweep process implemented in the previous experiment [8], when the swept proppants reach the right boundary, they are deleted to mimic the process of proppant discharge from the experimental slot. Figure 12 shows the proppant placement in the fracture during the sweep stage. When the clean fluid is injected, it exerts a drag force on the proppants that early accumulate above on the bank, and
θe=40
θe=65
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 12 Proppant placement in the fracture during the sweep stage. The particles, with a diameter spread of 0.5 mm ≤ d ≤ 0.8 mm that obeys normal distribution, are randomly injected into the fracture. The largest particles are marked by red while blue indicates the smallest particles. a t = 1.25 s, vs = 0.15 m/s, b t = 1.25 s, vs = 0.3 m/s,
123
the proppants closer to the wellbore are quickly eroded and dragged deeper into the fracture. The erosion angle quickly decreases from the original value of θe = 75◦ , especially for high sweep velocity. A fluidization layer, which governs the main proppant transport during the sweep stage, forms above the bank as shown in Fig. 12a, b. Three primary proppant transport styles are observed in the fluidization layer, namely, surface creep, saltation and suspension. This observation is identical to the experimental results of STIM-LAB consortium [4,5]. It is clear that the erosion caused by the clean fluid increases with the increase in sweep velocity. Compared with the small sweep velocity of vs = 0.15 m/s, when the sweep stage is implemented at a higher velocity (vs = 0.3 m/s), the proppant bank is more severely eroded with a smaller equilibrium bank height (He = 3.27 cm). Additionally, after the sweep stage, two large proppant-free regions form near the wellbore and above the bank. This will cause pinching of the fracture near the wellbore and a vertical wedge, which will significantly reduce the conductivity of the fracture [37]. Figure 13 shows experimental results on proppant bank erosion in a vertical slot during the sweep stage [8]. It is observed that the formed proppant bank in the primary vertical slot is significantly eroded by the injected clean fluid. A
c t = 2.5 s, vs = 0.15 m/s, d t = 2.5 s, vs = 0.3 m/s, e t = 3.75 s, vs = 0.15 m/s, f t = 3.75 s, vs = 0.3 m/s, g t = 5 s, vs = 0.15 m/s, h t = 5 s, vs = 0.3 m/s, i t = 6.25 s, vs = 0.15 m/s, j t = 6.25 s, vs = 0.3 m/s (color figure online)
Numerical simulation of transport and placement of multi-sized proppants in a hydraulic…
Page 13 of 15
32
Fig. 13 Experimental results on the profile variation of the proppant bank during the sweep stage [8]
large proppant-free region forms near the wellbore, and the equilibrium height of the bank decreases. The sweep process of the proppant bank in experiments is identical to the simulation results in this work. 3.7 Effect of tail-in It is clear from Fig. 12 that big proppant-free region near the wellbore form after the sweep stage, and the fracture in these locations will close after shut-down, which will significantly reduce the conductivity of the fracture. Therefore, a proppant-laden stage should be implemented after the sweep stage in order to hold sufficient quantities of the proppants near the wellbore. To maintain sufficient conductivity near the fracture entrance at the end of the fracturing treatment, a tail-in stage is usually implemented including injection rate slowdown and/or proppant diameter ramp ups [38]. In this section, the final proppant placement during the tail-in process is investigated, and two injection strategies with large proppants (d = 0.7 mm) at the tail-in rate of vtail = 0.15 m/s and vtail = 0.15 m/s are simulated. Figure 14 shows the proppant transport and placement in the fracture during the tail-in stage. It is clear that when the proppants are injected into the fracture, they first settle near the wellbore, filling the large proppant-free region. As the proppants being continuously injected, a new equilibrium height of the proppant bank forms. Then the later injected proppants can be transported
deeper into the fracture. Therefore, the injected proppants during the tail-in stage mainly accumulate near the wellbore, and this can significantly improve the proppant placement near the fracture entrance. Additionally, because the conductivity of the fracture propped by large proppants is higher than that propped by small proppants and the productivity of the fractured wells is mainly governed by the conductivity of the fracture entrance near the wellbore, so the tail-in stage with large proppants can significantly improve fracturing efficiency. It is observed from Fig. 14g, h that comparing with high tail-in rate, the proppant placement is better with a larger equilibrium height when the proppants are tailed into the fracture at low tail-in rate.
4 Conclusions Taking the interactions of particle-particle, particle-wall and particle-fluid into account, a coupled CFD–DEM approach was used to model proppant-fracturing fluid slurry flow in the fracture in vertical wells, and the proppant transport mechanisms and placement profiles were investigated. When the proppants are injected into the fracture, their motion is governed by settlement and fluid drag, and three developmental stages of the proppant dune are observed. During the first stage, the injected proppants quickly settle out of the fluid and accumulate on the fracture bottom, forming a proppant dune.
123
32
Page 14 of 15
G. Zhang et al.
(a)
(b)
(c)
(d)
(e)
(f)
4.58 cm
4.36 cm
(h)
(g) Fig. 14 Proppant placement in the fracture during the tail-in stage. The diameter of the tail-in particles, which are marked by red, is d = 0.7 mm. a t = 0.8 s, vtail = 0.15 m/s, b t = 1.5 s, vtail =
0.1 m/s, c t = 1.6 s, vtail = 0.15 m/s, d t = 3 s, vtail = 0.1 m/s, e t = 2.4 s, vtail = 0.15 m/s, f t = 4.5 s, vtail = 0.1 m/s, g t = 3.2 s, vtail = 0.15 m/s, h t = 6 s, vtail = 0.1 m/s (color figure online)
The proppant transport mechanisms in this stage are settlement, suspension and fluidization, and a three-layer flow pattern (stationary proppant bed, mixture layer and clean fluid layer) is observed. As the proppant dune develops, it gradually hinders the flow stream and stage 2 appears. A more prominent fluidization layer forms near the wellbore, and the injected proppants are vertically lifted and mainly accumulate near the fracture entrance. When the equilibrium height of the proppant dune is achieved, the dune changes to a proppant bank. Then the later injected proppants are transported to overshoot the bank and deeper into the fracture and a new proppant transport mechanism of vortex is observed in stage 3. Due to the effects of fluid viscosity, injection rate, uneven perforation and open-hole completion on the flow field in the fracture, the proppant placement under these conditions are distinct, which will cause different fracture conductivity. Two processes of sweep and tail-in were modeled. The sweep process can transport the earlier injected proppant deeper into the fracture causing big proppant-free region near the wellbore, while the tail-in process can significantly improve the
proppant placement near the fracture entrance. The findings of this simulation are consistent with previously experimental results, and they can help to improve the understanding of proppant micromechanics and placement in the hydraulic facture in vertical wells.
123
Acknowledgements This study was supported by the Program for Changjiang Scholars and Innovative Research Team in University (IRT1294) and the China Scholarship Council (award to Guodong Zhang for one year’s study abroad at Colorado School of Mines). Compliance with ethical standards
Conflict of interest The authors declare that no competing interests exist.
References 1. Economides, M.J., Nolte, K.G.: Reservoir Stimulation, 3rd edn. Wiley, Hoboken (2000)
Numerical simulation of transport and placement of multi-sized proppants in a hydraulic… 2. Deshpande, Y.K., Crespo, F., Bokane, A.B., Jain, S.: Computational fluid dynamics (CFD) study and investigation of proppant transport and distribution in multistage fractured horizontal wells. In: SPE Reservoir Characterization and Simulation Conference and Exhibition, Abu Dhabi, UAE, 16–18 Sept 2013 3. Ely, J.W., Fowler, S.L., Tiner, R.L., Aro, D.J., Sicard, G.R., Sigman, T.A.: Slick water fracturing and small proppant. The future of stimulation or a slippery slope? In: SPE Annual Technical Conference and Exhibition, Amsterdam, The Netherlands, 27–29 Oct 2014 4. Patankar, N.A., Joseph, D.D., Wang, J., Barree, R.D., Conway, M., Asadi, M.: Power law correlations for sediment transport in pressure driven channel flows. Int. J. Multiph. Flow 28(28), 1269– 1292 (2002) 5. Wang, J., Joseph, D.D., Patankar, N.A., Conway, M., Barree, R.D.: Bi-power law correlations for sediment transport in pressure driven channel flows. Int. J. Multiph. Flow 29(3), 475–494 (2003) 6. Babcock, R.E., Prokop, C.L., Kehle, R.O.: Distribution of propping agent in vertical fractures. In: Conference Paper Drilling and Production Practice, New York, USA, 1 Jan 1967 7. Kern, L.R., Perkins, T.K., Wyant, R.E.: The mechanics of sand movement in fracturing. J. Pet. Technol. 11, 55–57 (1959) 8. Sahai, R., Miskimins, J.L., Olson, K.E.: Laboratory results of proppant transport in complex fracture systems. In: SPE Hydraulic Fracturing Technology Conference, Woodlands, TX, USA, 4–6 Feb 2014 9. Mack, M., Sun, J., Khadilkar, C.: Quantifying proppant transport in thin fluids: theory and experiments. In: SPE Hydraulic Fracturing Technology Conference, Woodlands, TX, USA, 4–6 Feb 2014 10. Stim-Lab Proppant and Fluid Consortia Notes (1986–2007) 11. Alotaibi, M.A., Miskimins, J.L.: Slickwater proppant transport in complex fractures: new experimental findings and scalable correlation. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, Houston, TX, USA, 28–30 Sept 2016 12. Pearson, J.R.A.: On suspension transport in a fracture: framework for a global model. J. Non-newton. Fluid Mech. 54, 503–513 (1994) 13. Hammond, P.S.: Settling and slumping in a Newtonian slurry, and implications for proppant placement during hydraulic fracturing of gas wells. Chem. Eng. Sci. 50, 3247–3260 (1995) 14. Lecampion, B., Garagash, D.I.: Confined flow of suspensions modelled by a frictional rheology. J. Fluid Mech. 759, 197–235 (2014) 15. Eskin, D., Miller, M.J.: A model of non-Newtonian slurry flow in a fracture. Powder Technol. 182, 313–322 (2008) 16. Eskin, D.: Modeling non-Newtonian slurry flow in a flat channel with permeable walls. Chem. Eng. Sci. 123, 116–124 (2015) 17. Tsai, K., Fonseca, E., Lake, E., Degaleesan, S.: Advanced computational modeling of proppant settling in water fractures for shale gas production. SPE J. 18, 50–56 (2012) 18. Dontsov, E.V., Peirce, A.P.: Slurry flow, gravitational settling and a proppant transport model for hydraulic fractures. J. Fluid Mech. 760, 567–590 (2014) 19. Dontsov, E.V., Peirce, A.P.: Proppant transport in hydraulic fracturing: crack tip screen-out in KGD and P3D models. Int. J. Solids Struct. 63, 206–218 (2015) 20. Dontsov, E.V., Peirce, A.P.: A Lagrangian approach to modelling proppant transport with tip screen-out in KGD hydraulic fractures. Rock Mech. Rock Eng. 48, 2541–2550 (2015)
Page 15 of 15
32
21. Shiozawa, S., McClure, M.: Simulation of proppant transport with gravitational settling and fracture closure in a three-dimensional hydraulic fracturing simulator. J Pet. Sci. Eng. 138, 298–314 (2016) 22. Blyton, C.A., Gala, D.P., Sharma, M.M.: A comprehensive study of proppant transport in a hydraulic fracture. In: SPE Annual Technical Conference and Exhibition, Houston, TX, USA, 28–30 Sept 2015 23. Patankar, N.A., Joseph, D.D.: Modeling and numerical simulation of particulate flows by the Eulerian–Lagrangian approach. Int. J. Multiph. Flow 27, 1659–1684 (2001) 24. Chiesa, M., Mathiesen, V., Melheim, J.A., Halvorsen, B.: Numerical simulation of particulate flow by the Eulerian–Lagrangian and the Eulerian–Eulerian approach with application to a fluidized bed. Comput. Chem. Eng. 29, 291–304 (2005) 25. Tomac, I., Gutierrez, M.: Fluid lubrication effects on particle flow and transport in a channel. Int. J. Multiph. Flow 65, 143–156 (2014) 26. ITASCA: PFC User Manual. Itasca Consulting Group Inc., Mineapolis (2004) 27. Ergun, S.: Fluid flow through packed columns. Chem. Eng. Prog. 48, 89–94 (1952) 28. Wen, C.Y., Yu, Y.H.: Mechanics of fluidization. Chem. Eng. Prog. Symp. 62, 100–111 (1966) 29. Kaushal, D.R., Tomita, Y.: Solids concentration profiles and pressure drop in pipeline flow of multisized particulate slurries. Int. J. Multiph. Flow 28, 1697–1717 (2002) 30. Patankar, S.: Numerical Heat Transfer and Fluid Flow. Hemisphere Pub, New York (1980) 31. ITASCA: PFC2D Version 4.0 Optional Features. Itasca Consulting Group Inc., Mineapolis (2008) 32. Wutherich, K., Walker, K.J.: Designing completions in horizontal shale gas wells: perforation strategies. In: SPE Americas Unconventional Resources Conference, Pittsburgh, PA, USA, 5–7 June 2012 33. Tang, Y.: Optimization of Horizontal Well Completion. The University of Tulsa, Tulsa (2001) 34. Pang, W., Chen, D., Zhang, Z., Jiang, L., Li, C., Zhao, X., Wang, B.: Segmentally variable density perforation optimization model for horizontal wells in heterogeneous reservoirs. Pet. Explor. Dev. 39(2), 230–238 (2012) 35. Yakeley, S., Wood, E., Knebel, M.J.: Contacting the reservoir— benefits of horizontal open-hole completions. In: Conference Paper Offshore Europe, Aberdeen, UK, 1 Jan 2009 36. Rivenbark, M., Appleton, J.: Cemented versus open hole completions: What is best for your well? In: SPE Unconventional Gas Conference and Exhibition, Muscat, Oman, 28 Jan 2013 37. McLennan, J.D., Green, S.J., Bai, M.: Proppant placement during tight gas shale stimulation: literature review and speculation. In: The 42nd US Rock Mechanics Symposium (USRMS), San Francisco, CA, USA, 1 Jan 2008 38. Barasia, A., Pankaj, P.: Tail-in proppant and its importance in channel fracturing technique. In: SPE Bergen One Day Seminar, Bergen, Norway, 2 Apr 2014
123