Research on the erosion law of shale gas casing perforation hole during fracturing

During the fracturing process of shale gas wells, the high-speed injection of sand-carrying fluid throttles at the perforation casing hole, resulting in erosion at the hole, increasing the geometric parameters of the orifice and even producing cracks, affecting the fracturing construction effect and safety. Therefore, the flow field, the movement trajectory of particles and the erosion law of the perforated holes under different displacement and pump pressure were analyzed by using numerical simulation software and Euler-Lagrangian particle tracking theory, and the simulation analysis results were verified by the full-size physical erosion test of the perforated orifice of the casing. The analysis results show that with the increase of displacement, the erosion rate of the eyelet continues to increase, and the erosion rate of the eyelet in the direction of the liquid inlet is significantly greater than that of the bottom of the casing, and the degree gradually decreases in all directions. Increasing the fracturing construction displacement not only increases the erosion rate and diffusion area at the borehole, but also increases the erosion rate of the pipe wall near the borehole, which will accelerate the thinning of the wall thickness of the casing and affect the service safety of the perforated section casing.


Introduction
In recent years, multi cluster fracturing in long intervals of horizontal wells is an effective means for the development of unconventional oil and gas reservoirs [1] .With the continuous increase of proved reserves of Low-permeability Oilfields, the state and enterprises have made more and more efforts to develop them.Large displacement and high sand ratio hydraulic fracturing technology is an important means for the development of low-permeability and ultra-low-permeability reservoirs [2] .In the process of hydraulic fracturing, the sand carrying fluid will produce a throttling effect at the perforation hole, causing severe erosion at the hole, increasing the geometric parameters of the hole, and even producing cracks at the connection, affecting the construction safety and effect [3,4] .Scholars at home and abroad have done a lot of research on erosion by using numerical simulation analysis software and fitting formula of test data [5] .Hole erosion has a significant effect on the distribution of fluid and proppant during hydraulic fracturing.However, the direct data of hole size before and after fracturing can only be obtained through ground experiments.Lilly and tymons [6,7] used a downhole video technology to directly measure the geometry of hole erosion on site, which can explain all variables affecting the erosion rate.So as to improve the effectiveness of fracturing operation.S robinson [8] used the acoustic system to evaluate the hole erosion in the perforated section of a horizontal well.After quantifying the collected data, the hole diameter, hole erosion range, hole erosion offset and other information are obtained.The author uses the numerical simulation software to simulate the erosion of the sand carrying fluid on the perforations, and analyzes the erosion law of the flow field near the perforations under different displacement, in order to provide a scientific basis for the optimization of the field perforation and fracturing parameters [9] .

Establishment of Casing Erosion Model 2.1 A subsection sample
The flow of sand-carrying fluids in pipelines is a typical two-phase flow of liquid and solid.Due to the low concentration of particles in the fluid, the interactions between particles and the influence of particle volume fraction on the continuous phase are not considered.The discrete phase model (DPM) is used to simulate the fluid flow in the pipeline in a Lagrangian coordinate system.The flow field is complex at the perforation site, and there are phenomena such as backflow and rotation inside the pipeline.The Realizable κ-ε model is derived from the Standard κ-ε model by modification and improvement.To ensure the accuracy of the calculation, the Realizable κ-ε turbulence model is adopted to study the erosion at the perforation site.The transport equations for turbulent kinetic energy κ and turbulent dissipation ε are as follows: In the formula, C = max(0.43,η/(η+5)), C = 1.9; μt = ρCμk2/ε, is the turbulent viscosity coefficient, Cμ is the model constant; t is time, s; ρ is the density of the continuous phase, kg/m3; κ is the turbulent kinetic energy, J; ε is the turbulent dissipation rate, J/s; xi, xj are displacements, m; μi is the velocity coordinate component, m/s; Gκ is the turbulent kinetic energy κ induced by the average velocity gradient, J; μ is the fluid dynamic viscosity coefficient; σκ, σε are the Prandtl numbers corresponding to the turbulent kinetic energy κ and the turbulent dissipation rate ε, σκ = 1.0, σε = 1.2.
The erosion rate formula used in the DPM model is as follows: In the equation, C(dp) is the particle diameter function with a constant value of 1.8e-9; α is the impingement angle of the particle on the wall in radians; f(α) is the function of impingement angle; υ is the relative velocity between the particle and the wall in m/s; and b(υ) is the function of relative velocity with a constant value of 2.6.

Governing equation
In two-phase flow in a bend, the different phases mix at a macroscopic scale which is much smaller than the grid scale yet much larger than the molecular scale.All phases occupy the same volume in space, and the volume of each phase in the control volume is represented by the variable volume fraction.Each phase has its own flow parameters, and the phases are coupled through models of interphase energy transfer, momentum transfer, and mass transfer.Each phase has its own mass, momentum, and energy transfer equations.In three-dimensional space coordinates, the governing equations for the mixed medium are as follows: (1) Continuity equation: In the equation, ρ is the mixture density; uj, is the velocity components in different directions; Sm is the total momentum source term, satisfying the momentum conservation with Sμu = 0.
(2) Momentum equation: In In the equation, E represents the total energy of the system, in J/kg.hj represents the enthalpy of component j, in J/kg.T represents the fluid temperature, with a reference temperature of Tref = 298.15K.keff represents the thermal conductivity coefficient, in W/(m•K), and is calculated as keff = k + ke.Jj represents the diffusion flux of component j, while s represents the internal heat source term.

Physical model
The research object is the erosion situation of the perforation holes in the fracturing casing.Based on the actual working conditions, the outer diameter of the casing is 139.7mm, the wall thickness is 12.7mm, and the density is 7.9mm.The diameter of the perforation hole is 11mm.To ensure the fully developed flow of the fluid in the straight pipe section, a section of 1000mm in length is taken.The Ansys Space Claim software is used to establish the flow field domain model, as shown in Figures 1 and 2.

Grid partitioning
A hexahedral structured grid is used when dividing the mesh, and the boundary layer grid is refined at the inlet and orifice to better simulate the flow near the orifice.The results of the grid division for the curved pipe are shown in Figures 3 and 4.  Using slickwater as the continuous phase medium, with a density of 1030 kg/m 3 and a kinematic viscosity of 0.00103 m 2 /s, quartz sand with a particle size of 100 mesh and an apparent density of 2.6 g/cm 3 and a bulk density of 1.45 g/cm 3 is selected as the proppant, with a sand concentration of 260 kg/cm 3 .The erosion of the perforation hole under different displacement and pump pressure conditions during sand-fracturing simulation is simulated, as shown in Table 1.

Assumption
The basic assumptions introduced in the erosion calculation are as follows: the single-phase fluid medium is a slippery water medium; the fluid medium is incompressible and its thermal effects are ignored; the temperature at the wellbore inlet and perforation outlet is constant; and the solid particles are spherical.The boundary conditions are set as follows.

Conclusion
(1) During the casing fracturing process, the erosion at the entry of the perforation hole is the most severe, and the agreement rate between the numerical simulation results and the physical experimental results reaches 85%.
(2) When the sand volume fraction is 15%, with the increase of displacement, the maximum erosion rate on the perforation hole wall continues to rise.When the displacement is 16m3/min, the maximum erosion rate is 1.7 times that of 12m 3 /min.
the equation, p represents the pressure acting on the control volume, Pa； τxx、 τyy, and τzz represent the components of the viscous stress τ in different directions, in Pa; fx, fy, and fz represent the force components of the mass force in different directions, in m/s2.

Fig 1
Fig 1 Fluid Domain Model Diagram.

Fig. 15 .
Fig. 15.The maximum erosion rate on the wall surface of the experimental perforation under different flow rates when the sand content ratio is 15% in the physical experiment.

Fig. 16 .
Fig. 16.Average erosion rate on the wall of physical experimental orifices under different displacement conditions when the sand ratio is 15%.

Table 1 .
Simulation conditions table

Table 2 .
Wall maximum erosion rate and average velocity.