Structural Improvement of Differential Motion Assembly in In Situ Pressure-Preserved Coring System Using CFD Simulation

: In situ pressure-preserved coring (IPP-Coring) is one of the most efﬁcient methods for identifying the scale of the oil and gas content. However, the differential motion assembly of the IPP-Coring system often undergoes ball and ball seat seal failure and sticking due to surface erosion, and a greater pressure drop may unexpectedly trigger the assembly. This paper addresses these issues by improving the hydraulic structure of an assembly based on a deep understanding of the ﬂow characteristics in the assembly, thus increasing the success rate of the IPP-Coring. Computational ﬂuid dynamics (CFD) was employed to investigate ﬂows in a differential motion assembly. The effects of the diameter and outlet structure of the ball seat on the ﬂuid status, velocity, and pressure distribution were thoroughly analyzed. When the ball seat diameter increased from 30 to 40 mm, the maximum velocity and pressure drop decreased to 0.55 and 0.2 times their original values, respectively. There was a severe vortex area in the differential motion assembly due to the presence of the ball seat, but changing the outlet structure in the ball seat to an arc structure decreased the length of the vortex area and the ﬂuid velocity near the wall to 0.7 and 0.4 times, respectively, compared with those with the original right-angled structure. In addition, the pressure drop decreased to 0.33 times the original value. Thus, the hydraulic structure of the assembly was improved, and a 40 mm diameter ball seat and an arc-shaped ball seat outlet were selected. Particle trajectory and erosion calculation results showed that the improved structure has a lower particle velocity and less impact on the wall, and the average erosion rate is only 0.42 times the value of the original structure. Due to the better erosion resistance and smaller pressure drop, the improved structure shows promise for ﬁeld performance.


Introduction
With the exhaustion of resources at shallow depths in the earth, the development of resources is gradually occurring at greater depths [1][2][3].At present, the depth of coal mining exceeds 1500 m, metal mining has reached 4350 m, and oil and gas resource mining has reached 7500 m [4][5][6][7].The exploitation and utilization of deep resources require accurate knowledge of reservoir scale and properties.Coring is a vital method used in exploration; it can obtain core samples from subsurface formations.By analyzing core samples, reliable information about the rock and fluid properties can be obtained, and then the scale of the reservoir content can be identified [3,8].However, for conventional coring technology, the change in pressure during tripping can expel fluid and gas components from the core sample, resulting in a large error between the reserve assessment values and the real situation of the deep resource [9][10][11][12][13].Field tests conducted by Bjorum et al. have shown that for conventional coring technology, more than 50% of the fluid and gas can be lost from a sample due to fluid expulsion during tripping [14].
To overcome this issue, in situ pressure-preserved coring (IPP-Coring) technology has been introduced for deep exploration [15][16][17][18][19].This technology avoids fluid expulsion by maintaining the core pressure at the bottom hole pressure, thereby making it possible to accurately assess the scale of the reservoir contents [20][21][22][23].At present, scholars around the world have performed much research on in situ pressure-preserved coring technology and developed many types of IPP-Coring equipment.For example, to obtain cores from offshore gas hydrates for research, the pressure core barrel (PCB), advanced piston corer (APC), pressure core sampler (PCS), hydrate autoclave coring equipment system (HY-ACE), and pressure temperature core sampler (PTCS) have been developed by different countries and institutions in the past few years [24][25][26][27][28][29][30].For onshore coring for deep oil and gas exploration, the GWY194-70BB and GW-CP194-80A pressure coring systems are two typical types of IPP-Coring equipment; their pressure preservation capacities can reach 20 and 60 MPa, respectively [31,32].In current IPP-Coring devices, their pressure preservation principle is mainly the closure of a ball valve or flap valve to form a sealed chamber to maintain the in situ pressure of the core sample.The closure of the valves depends on the relative movement between the core (inner) pipe and the outer pipe through a mechanical device such as an overshot assembly or a hydrodynamic-pressure lifting device such as a differential motion assembly.In onshore deep oil and gas exploration, due to the small inner diameters of common oil drill pipes, tripping the mechanical device in the hole has been a problem; usually, it is necessary to use specialized drill pipes, which wastes considerable effort and cost.Therefore, relative movement between the core tube and outer tube often depends on the hydrodynamic-pressure lifting device in onshore deep oil and gas explorations, namely, the differential motion assembly in this study [32].
The working principle of the differential motion assembly used to realize the pressure preservation operation of the IPP-Coring system for onshore oil and gas explorations is shown in Figure 1.In the coring operation, the upper section of the IPP-Coring system is successively connected to the drill collars and drill pipes in the bottom hole assembly.When coring begins, the drilling fluid passes through the inside of the differential motion assembly, and the valve cover in the valve assembly stays open due to the constraint of the inner pipe.At the end of drilling, the rock core enters the in situ pressure-preserved core chamber.At this time, a steel ball is dropped into the hole and dropped onto the ball seat, which changes the drilling fluid path from the interior to the outer part of the differential motion assembly, as shown by the red arrow.Under the action of hydrodynamic pressure, two limit pins are cut off, and the differential motion assembly is lifted to drive the inner pipe up over the valve cover.Then, the valve cover automatically closes under the action of gravity to maintain the in situ pressure.According to this working principle, the differential motion assembly plays a vital role in the process of IPP-Coring during onshore oil and gas exploration, and it directly determines the success rate of IPP-Coring.Unfortunately, in the field service process, due to the complex deep well environment characterized by high pressure and temperature, high flow rates, and solids content of the drilling fluid, the differential motion assembly is often faced with the failure of the seal of the ball and ball seat as well as sticking caused by surface erosion.In addition, in the case of large positive displacement pumps, the differential motion assembly may be lifted at the wrong time (the pin is cut off before the ball is dropped) due to the greater pressure drop, which contributes to the failure of IPP-Coring.Therefore, it is necessary to improve the hydraulic structure of the differential motion assembly to decrease the surface erosion and pressure drop to further increase its reliability and service life.To improve the hydraulic structure, it is necessary to have a deep insight into the flow characteristics of the drilling fluid in the differential motion assembly.Computational fluid dynamics (CFD) can effectively analyze the flow characteristics of downhole tools [33][34][35][36][37][38].However, there has been little research on flow characteristics in differential motion assemblies in the IPP-Coring systems used in deep oil and gas exploration.The effects of the internal size and structure of the differential motion assembly on the flow velocity and pressure distribution are unclear, and it is impossible to target the design of the hydraulic structure to reduce erosion and pressure drop.As a result, the operational reliability and service life of differential motion assemblies have been low, which would result in a low success rate of the IPP-Coring.To increase the operational reliability and service life of differential motion assemblies and increase the success rate of IPP-Coring, this study employed CFD with the standard k-ε turbulence model to study the characteristics of flows with different rates in a differential motion assembly during IPP-Coring.The effects of the diameter of the ball seat and ball seat outlet structure on the fluid status, velocity, and pressure distribution were analyzed.Further, the hydraulic structure of the differential motion assembly was improved, and field experiments demonstrated the erosion resistance of the improved structure.To improve the hydraulic structure, it is necessary to have a deep insight into the flow characteristics of the drilling fluid in the differential motion assembly.Computational fluid dynamics (CFD) can effectively analyze the flow characteristics of downhole tools [33][34][35][36][37][38].However, there has been little research on flow characteristics in differential motion assemblies in the IPP-Coring systems used in deep oil and gas exploration.The effects of the internal size and structure of the differential motion assembly on the flow velocity and pressure distribution are unclear, and it is impossible to target the design of the hydraulic structure to reduce erosion and pressure drop.As a result, the operational reliability and service life of differential motion assemblies have been low, which would result in a low success rate of the IPP-Coring.To increase the operational reliability and service life of differential motion assemblies and increase the success rate of IPP-Coring, this study employed CFD with the standard k-ε turbulence model to study the characteristics of flows with different rates in a differential motion assembly during IPP-Coring.The effects of the diameter of the ball seat and ball seat outlet structure on the fluid status, velocity, and pressure distribution were analyzed.Further, the hydraulic structure of the differential motion assembly was improved, and field experiments demonstrated the erosion resistance of the improved structure.

Liquid Phase Model
During running, the drilling fluid flows through the differential motion assembly.The continuity equations and momentum equations are as follows [39]: where ρ is the liquid phase density, → u represents the instantaneous velocity vector, P is pressure, µ is the molecular viscosity, I is the unit tensor, ρ → g is the gravitational body force, and S M is the added momentum caused by the discrete phase.
The standard k-ε model is very popular in industrial flow and heat transfer simulations because of its robustness and economy, and it has shown reasonable accuracy for fully developed turbulent flow conditions [40].For the differential motion assembly, during the IPP-Coring process for deep oil and gas exploration, the drilling fluid, which is characterized by a high velocity (pump displacement can reach 20~40 L/s), high pressure, and high solid particle content, circulates through it.Due to the high velocity and the resulting high Reynolds number, the flow in the differential motion assembly is usually turbulent.Thus, the standard k-ε model, which has reasonable accuracy for a wide range of turbulent flows and a fast convergence rate [40], was employed in this study to gain an insight into the flow characteristics in the differential motion assembly.The equations are as follows [39,41,42]: where µ T is the turbulent eddy viscosity; k and ε are the turbulent kinetic energy and the turbulent dissipation rate, respectively; u i denotes the velocity component in the i-direction; x i and x j are the coordinates in space; C ε1 , C ε2 , and C µ are model constants; S k and S ε represent the source terms defined by the discrete phase caused by the particle damping and turbulence eddies; σ k and σ ε denote the turbulent Prandtl numbers; and P k is the kinetic energy of the turbulence caused by the mean velocity gradients.In the standard k-ε model, C µ = 0.99, C ε1 = 1.44,C ε2 = 1.92, σ k = 1.0, and σ ε = 1.3 [40,43].After the velocity and pressure fields in the fluid domain were solved by the momentum equation, the k-ε model was used to obtain the turbulent kinetic energy and its dissipation rate.Then, the results of the k-ε model were incorporated into the momentum equation to account for the effect of turbulence and the flow characteristics were obtained.

Particle Motion Model
During IPP-Coring, the motion trajectories of the particles in the drilling fluid can be obtained by solving the motion equation in the Lagrangian domain, and the motion governing equation of the particles can result from Newton's second law [42,44]: where m p denotes the particle mass, → v p is the particle velocity, and

and
→ F vm are the drag force, gravity force, buoyancy, pressure gradient force, and added mass force, respectively.Those forces can be expressed as [41,44]: Appl.Sci.2023, 13, 4108 → → where C D is the drag coefficient, R e r is the relative Reynolds numbers, ρ p is the density of the particle, and d p is the diameter of the particle [41].

Erosion Model
Due to the impingement of the particles in the drilling fluids on the surface of the differential motion assembly, erosion is inevitable.In this study, the E/CRC model was used to calculate the erosion rate, and the equations are as follows [45,46]: F(α) = 5.40α − 10.11α 2 + 10.93α 3 − 6.33α 4 + 1.42α 5 (14) where E is the erosion rate, C is a coefficient of the erosion model, F s is a coefficient of the particle shape, BH is the Brinell hardness of the material, v is the impact velocity of the particles, and α is the incidence angle of the particles colliding against the wall.In this study, the round sands were set as the erosion particles, and 42CrMo steel was the material used for the differential motion assembly.Therefore, C = 15.59 × 10 −7 , F s = 0.2, BH = 240, and n = 1.73.After the impact velocity and angle of the particles were calculated by the above turbulence and particle motion model, the erosion rate can be quantified using Equations ( 13) and (14) in CFD.

Computational Domain
In this paper, the CFD module in COMSOL Multiphysics was used to conduct the CFD simulation of the flow channel of the differential motion assembly.According to the structure of the differential motion assembly, a 3D computational domain was built.Because of the symmetry of the fluid domain of the motion assembly and considering the cost of computational time, 1/4 of the fluid domain of the differential motion assembly was selected as the computational domain.Figure 2 shows the mesh and boundary conditions of the flow domain of the differential motion assembly.Hexahedral grids with high accuracy were built using sweep algorithms.The density of the drilling fluid was set as 1000 kg/m 3 .To study the effect of the fluid viscosity, fluid viscosities of 1000, 100, 10, and 1 mPa•s were employed.The mass flow rate of the particles was set to 0.5 kg/s, and the density and the diameter of the sand particles were set as 2850 kg/m 3 and 6 × 10 −5 m, respectively.According to the real working conditions of the differential motion assembly, the boundary conditions were set as follows: Inlet: The boundary condition of the inlet was set as "fully developed flow", and flow rates of 0.01, 0.02, 0.03, and 0.04 m 3 /s were set at the inlet to study the flow characteristics.
Wall: The inner surface of the differential motion assembly was set as the wall, no slip was selected for the wall, and the wall roughness parameter and the roughness height were set as 0.26 and 3.2 µm, respectively.In addition, the wall boundaries were set as "bounce" for the particles.
Outlet: The boundary condition of the outlet was set as the pressure condition, and according to the pump pressure in the field, the value of the pressure at the outlet was set as 30 MPa.In addition, the wall condition of the outlet for the particles was set as "Freeze".
Wall: The inner surface of the differential motion assembly was set as the wall, no slip was selected for the wall, and the wall roughness parameter and the roughness height were set as 0.26 and 3.2 μm, respectively.In addition, the wall boundaries were set as "bounce" for the particles.
Outlet: The boundary condition of the outlet was set as the pressure condition, and according to the pump pressure in the field, the value of the pressure at the outlet was set as 30 MPa.In addition, the wall condition of the outlet for the particles was set as "Freeze".The velocity distributions along the radial direction at a line 150 mm away from the inlet under a flow rate of 0.04 m 3 /s and fluid viscosity of 100 mPa•s were used to analyze the grid independency.The results of the grid independence study are shown in Figure 3; it shows that when the number of grids is more than 108,134, the solution is grid independent.

Flow Characteristics in the Differential Motion Assembly
The viscous effect is significant, and it is of great concern to the dynamic response of structures.Thus, in this study, the flow characteristics in the differential motion assembly with different fluid viscosities were studied first.The velocity streamlines in the The velocity distributions along the radial direction at a line 150 mm away from the inlet under a flow rate of 0.04 m 3 /s and fluid viscosity of 100 mPa•s were used to analyze the grid independency.The results of the grid independence study are shown in Figure 3; it shows that when the number of grids is more than 108,134, the solution is grid independent.
according to the pump pressure in the field, the value of the pressure at the outlet wa as 30 MPa.In addition, the wall condition of the outlet for the particles was set as "Free

Flow Characteristics in the Differential Motion Assembly
The viscous effect is significant, and it is of great concern to the dynamic respons structures.Thus, in this study, the flow characteristics in the differential motion assem with different fluid viscosities were studied first.The velocity streamlines in

Flow Characteristics in the Differential Motion Assembly
The viscous effect is significant, and it is of great concern to the dynamic response of structures.Thus, in this study, the flow characteristics in the differential motion assembly with different fluid viscosities were studied first.The velocity streamlines in the differential motion assembly with different fluid viscosities are shown in Figure 4.It can be seen that a vortex area appeared in the differential motion assembly, and that the size of the vortex area increased as the viscosity of the fluid decreased from 1000 mPa•s to 1 mPa•s.To quantify the effect of the fluid viscosity on the flow velocity, we extracted the velocity at a monitoring line 220 mm away from the flow inlet, as shown in Figure 5.As seen in Figure 5, the maximum velocity in the monitoring line increased with the increase in the fluid viscosity, and the maximum in the monitoring line was 68 m/s for a fluid viscosity of 1000 mPa•s.These results demonstrate that increased fluid viscosity can increase the local velocity of the flow and decrease the size of the vortex area.
of the vortex area increased as the viscosity of the fluid decreased from 1000 mPa•s to 1 mPa•s.To quantify the effect of the fluid viscosity on the flow velocity, we extracted the velocity at a monitoring line 220 mm away from the flow inlet, as shown in Figure 5.As seen in Figure 5, the maximum velocity in the monitoring line increased with the increase in the fluid viscosity, and the maximum in the monitoring line was 68 m/s for a fluid viscosity of 1000 mPa•s.These results demonstrate that increased fluid viscosity can increase the local velocity of the flow and decrease the size of the vortex area.For the real working condition of the differential motion assembly, the fluid viscosity is usually about 100 mPa•s.Thus, we set the fluid viscosity as 100 mPa•s to study the effect of the flow rate and the inner structure.The flow characteristics of drilling fluid in the differential motion assembly, such as flow velocity and pressure distribution, directly affect the operational reliability and life of the assembly.Figure 6 shows the velocity streamlines in the differential motion assembly for different inlet flow rates.As the inlet flow rate increased, the velocity of the fluid in the differential motion assembly increased remarkably.The maximum velocity of 65 m/s appeared at the inlet of the ball seat for an inlet flow rate of 0.04 m 3 /s.Although the diameter of the flow domain increased abruptly from 30 to 90 mm after the drilling fluid flowed through the ball seat, the drilling fluid near the axis still flowed a distance of about 200 mm at a relatively large velocity (e.g., at approximately 64 m/s when the inlet flow rate was 0.04 m 3 /s) before the velocity of the fluid gradually of the vortex area increased as the viscosity of the fluid decreased from 1000 mPa•s to 1 mPa•s.To quantify the effect of the fluid viscosity on the flow velocity, we extracted the velocity at a monitoring line 220 mm away from the flow inlet, as shown in Figure 5.As seen in Figure 5, the maximum velocity in the monitoring line increased with the increase in the fluid viscosity, and the maximum in the monitoring line was 68 m/s for a fluid viscosity of 1000 mPa•s.These results demonstrate that increased fluid viscosity can increase the local velocity of the flow and decrease the size of the vortex area.For the real working condition of the differential motion assembly, the fluid viscosity is usually about 100 mPa•s.Thus, we set the fluid viscosity as 100 mPa•s to study the effect of the flow rate and the inner structure.The flow characteristics of drilling fluid in the differential motion assembly, such as flow velocity and pressure distribution, directly affect the operational reliability and life of the assembly.Figure 6 shows the velocity streamlines in the differential motion assembly for different inlet flow rates.As the inlet flow rate increased, the velocity of the fluid in the differential motion assembly increased remarkably.The maximum velocity of 65 m/s appeared at the inlet of the ball seat for an inlet flow rate of 0.04 m 3 /s.Although the diameter of the flow domain increased abruptly from 30 to 90 mm after the drilling fluid flowed through the ball seat, the drilling fluid near the axis still flowed a distance of about 200 mm at a relatively large velocity (e.g., at approximately 64 m/s when the inlet flow rate was 0.04 m 3 /s) before the velocity of the fluid gradually For the real working condition of the differential motion assembly, the fluid viscosity is usually about 100 mPa•s.Thus, we set the fluid viscosity as 100 mPa•s to study the effect of the flow rate and the inner structure.The flow characteristics of drilling fluid in the differential motion assembly, such as flow velocity and pressure distribution, directly affect the operational reliability and life of the assembly.Figure 6 shows the velocity streamlines in the differential motion assembly for different inlet flow rates.As the inlet flow rate increased, the velocity of the fluid in the differential motion assembly increased remarkably.The maximum velocity of 65 m/s appeared at the inlet of the ball seat for an inlet flow rate of 0.04 m 3 /s.Although the diameter of the flow domain increased abruptly from 30 to 90 mm after the drilling fluid flowed through the ball seat, the drilling fluid near the axis still flowed a distance of about 200 mm at a relatively large velocity (e.g., at approximately 64 m/s when the inlet flow rate was 0.04 m 3 /s) before the velocity of the fluid gradually decreased.However, due to the abrupt increase in the diameter of the flow domain at the outlet of the ball seat, a severe vortex was present near the wall of the differential motion assembly.The size of the vortex area showed almost no change for different inlet flow rates, which may be attributed to the same inner hydraulic structure.A vortex is a harmful flow characteristic that causes drilling fluid, which carries particles such as barite, to continuously impact and severely erode the inner wall of the differential motion assembly.Unfortunately, the inner wall with a diameter of 90 mm is a motion surface, and after the ball is dropped to trigger the differential motion assembly, this wall moves upward under hydrodynamic pressure to realize in situ pressure preservation.The erosion of this wall causes sticking when the differential motion assembly moves upward and leads to the failure of the IPP-Coring system.
outlet of the ball seat, a severe vortex was present near the wall of the differential motion assembly.The size of the vortex area showed almost no change for different inlet flow rates, which may be attributed to the same inner hydraulic structure.A vortex is a harmful flow characteristic that causes drilling fluid, which carries particles such as barite, to continuously impact and severely erode the inner wall of the differential motion assembly.Unfortunately, the inner wall with a diameter of 90 mm is a motion surface, and after the ball is dropped to trigger the differential motion assembly, this wall moves upward under hydrodynamic pressure to realize in situ pressure preservation.The erosion of this wall causes sticking when the differential motion assembly moves upward and leads to the failure of the IPP-Coring system.To further study the drilling fluid velocity distributions at the vortex area, we extracted the velocity at a monitoring line 220 mm away from the flow inlet, as shown in the inset illustration of Figure 7.The velocity distributions at the monitoring line with different inlet flow rates are shown in Figure 7. Along the radial direction, the flow area could be divided into the central jet area and vortex area according to the flow characteristics of the drilling fluid.In the central jet area, the velocity was stable at a larger value in the area with a radius less than 7 mm, such as 64 m/s for the inlet flow rate of 0.04 m 3 /s.As the inlet flow rate decreased from 0.04 to 0.01 m 3 /s, this velocity value gradually decreased to 16 m/s.With increasing radius, the velocity started to decrease and then gradually entered the vortex area where the velocity gradually decreased to nearly 0 m/s with increasing radius and then increased slowly.This trend conformed to the typical characteristics of the vortex.Near the wall with a diameter of 90 mm, the velocity of the vortex was relatively slow, less than 7.3 m/s.As the inlet flow rates decreased from 0.04 to 0.01 m 3 /s, the velocity near the wall decreased from 7.3 to 1.9 m/s.At low velocity and small impact angles, the particles in the drilling fluid could still remove material from the wall with a diameter of 90 mm by plowing, which resulted in erosion [47].As the flow rate increased from 0.01 to 0.04 m 3 /s, although the size of the vortex area did not increase, the vortex velocity near the wall increased by 3.8-fold, which may lead to more severe surface erosion.To further study the drilling fluid velocity distributions at the vortex area, we extracted the velocity at a monitoring line 220 mm away from the flow inlet, as shown in the inset illustration of Figure 7.The velocity distributions at the monitoring line with different inlet flow rates are shown in Figure 7. Along the radial direction, the flow area could be divided into the central jet area and vortex area according to the flow characteristics of the drilling fluid.In the central jet area, the velocity was stable at a larger value in the area with a radius less than 7 mm, such as 64 m/s for the inlet flow rate of 0.04 m 3 /s.As the inlet flow rate decreased from 0.04 to 0.01 m 3 /s, this velocity value gradually decreased to 16 m/s.With increasing radius, the velocity started to decrease and then gradually entered the vortex area where the velocity gradually decreased to nearly 0 m/s with increasing radius and then increased slowly.This trend conformed to the typical characteristics of the vortex.Near the wall with a diameter of 90 mm, the velocity of the vortex was relatively slow, less than 7.3 m/s.As the inlet flow rates decreased from 0.04 to 0.01 m 3 /s, the velocity near the wall decreased from 7.3 to 1.9 m/s.At low velocity and small impact angles, the particles in the drilling fluid could still remove material from the wall with a diameter of 90 mm by plowing, which resulted in erosion [47].As the flow rate increased from 0.01 to 0.04 m 3 /s, although the size of the vortex area did not increase, the vortex velocity near the wall increased by 3.8-fold, which may lead to more severe surface erosion.Pressure contours in the differential motion assembly with different inlet flow rates are shown in Figure 8.There was an obvious pressure drop when the drilling fluid flowed through the differential motion assembly.In this study, the outlet pressure was set to 30 MPa during the calculation.With an increase in the inlet flow rate from 0.01 to 0.04 m 3 /s, the inlet pressure increased from 30.1 to 31.4 MPa, and the pressure drop ∆P between the inlet and outlet of the differential motion assembly increased by 14-fold.Due to the greater pressure drop of 1.4 MPa when the inlet flow rate was 0.04 m 3 /s, under the long-term impact of the drilling fluid, there is a risk of cutting the pins before the ball is dropped, which may trigger the differential motion assembly at the wrong time and result in the failure of the IPP-Coring process.Therefore, when a deep oil and gas IPP-Coring operation is conducted under large pump displacement, it is necessary to replace the pins with those of greater shear strength or diameter.In addition, the lowest pressure of 28.3 MPa (inlet flow rate 0.04 m 3 /s) occurred at the inlet of the ball seat of the differential motion assembly where the flow velocity was extremely fast, up to 65 m/s.Therefore, severe erosion may occur in this zone due to the complex flow characteristics.In general, there was a severe vortex area in the differential motion assembly due to the presence of the ball seat and, with the increase in inlet flow rate, the velocity of the drilling fluid and the pressure drop between the inlet and outlet increased remarkably.These conditions may increase the erosion rate on the inner surface of the differential motion assembly and the risk of cutting the pin at the wrong time, which would damage the differential motion assembly and lead to the failure of the IPP-Coring process.Pressure contours in the differential motion assembly with different inlet flow rates are shown in Figure 8.There was an obvious pressure drop when the drilling fluid flowed through the differential motion assembly.In this study, the outlet pressure was set to 30 MPa during the calculation.With an increase in the inlet flow rate from 0.01 to 0.04 m 3 /s, the inlet pressure increased from 30.1 to 31.4 MPa, and the pressure drop ΔP between the inlet and outlet of the differential motion assembly increased by 14-fold.Due to the greater pressure drop of 1.4 MPa when the inlet flow rate was 0.04 m 3 /s, under the long-term impact of the drilling fluid, there is a risk of cutting the pins before the ball is dropped, which may trigger the differential motion assembly at the wrong time and result in the failure of the IPP-Coring process.Therefore, when a deep oil and gas IPP-Coring operation is conducted under large pump displacement, it is necessary to replace the pins with those of greater shear strength or diameter.In addition, the lowest pressure of 28.3 MPa (inlet flow rate 0.04 m 3 /s) occurred at the inlet of the ball seat of the differential motion assembly where the flow velocity was extremely fast, up to 65 m/s.Therefore, severe erosion may occur in this zone due to the complex flow characteristics.In general, there was a severe vortex area in the differential motion assembly due to the presence of the ball seat and, with the increase in inlet flow rate, the velocity of the drilling fluid and the pressure drop between the inlet and outlet increased remarkably.These conditions may increase the erosion rate on the inner surface of the differential motion assembly and the risk of cutting the pin at the wrong time, which would damage the differential motion assembly and lead to the failure of the IPP-Coring process.

Effect of the Diameter of the Ball Seat on the Flow Characteristics
According to the analysis of the flow characteristics of the drilling fluid in the differential motion assembly, the flow had a higher velocity at the inlet of the ball seat, a severe vortex was present at the outlet of the ball seat near the wall, and the pressure drop reached 1.4 MPa when the inlet flow rate was 0.04 m 3 /s.These characteristics will reduce the reliability and service life of the differential motion assembly.To improve these characteristics and further increase the success rate of IPP-Coring operations in deep oil and gas exploration, it is necessary to improve the local structure of the differential motion assembly.Section 3.1 shows that the change in the flow status was largest at the ball seat.Therefore, we studied the effect of the diameter of the ball seat on the flow characteristics for ball seat diameters of 30, 35, and 40 mm. Figure 9 shows the effect of the diameter of the ball seat on the flow velocity.With the increase in the ball seat diameter, the velocity of the drilling fluid in the differential motion assembly decreased remarkably.The maximum velocity in the differential motion assembly with a 40 mm ball seat diameter decreased to 36 m/s, which was only 0.55 times that of the original assembly with a 30 mm ball seat diameter.In addition, the position of the maximum velocity changed; it was no longer at the inlet of the ball seat.This meant that the erosion at the inlet of the ball seat could be Appl.Sci.2023, 13, 4108 10 of 18 improved more after the diameter of the ball seat was increased to 40 mm.In addition, in Figure 9, the axial length of the vortex area decreased to 0.85 times that of the original differential motion assembly with a 30 mm ball seat diameter when the ball seat diameter increased to 40 mm.To further study the velocity distributions in the vortex area, we extracted the velocity at the monitoring line for ball seats with different diameters, as shown in Figure 10.As the ball seat diameter increased from 30 to 40 mm, the velocity at the axis decreased from 64 to 35 m/s, and the velocity in the central jet area markedly decreased.Moreover, the distance of the vortex center (the place the velocity was near 0 m/s) with the axis increased from 29 to 33 mm.This meant that the radial length of the vortex area also decreased.Therefore, it can be concluded that the increase in the diameter of the ball seat decreased the size of the vortex area.Figure 10 also shows that when the diameter of the ball seat increased from 30 to 40 mm, the velocity near the wall in the vortex area decreased to 0.67 times that of the model with the original ball seat diameter.The decreases in the size of the vortex area and the vortex velocity near the wall would reduce erosion by drilling fluid and the risk of sticking when the differential motion assembly was triggered.fore, we studied the effect of the diameter of the ball seat on the flow characteristics for ball seat diameters of 30, 35, and 40 mm. Figure 9 shows the effect of the diameter of the ball seat on the flow velocity.With the increase in the ball seat diameter, the velocity of the drilling fluid in the differential motion assembly decreased remarkably.The maximum velocity in the differential motion assembly with a 40 mm ball seat diameter decreased to 36 m/s, which was only 0.55 times that of the original assembly with a 30 mm ball seat diameter.In addition, the position of the maximum velocity changed; it was no longer at the inlet of the ball seat.This meant that the erosion at the inlet of the ball seat could be improved more after the diameter of the ball seat was increased to 40 mm.In addition, in Figure 9, the axial length of the vortex area decreased to 0.85 times that of the original differential motion assembly with a 30 mm ball seat diameter when the ball seat diameter increased to 40 mm.To further study the velocity distributions in the vortex area, we extracted the velocity at the monitoring line for ball seats with different diameters, as shown in Figure 10.As the ball seat diameter increased from 30 to 40 mm, the velocity at the axis decreased from 64 to 35 m/s, and the velocity in the central jet area markedly decreased.Moreover, the distance of the vortex center (the place the velocity was near 0 m/s) with the axis increased from 29 to 33 mm.This meant that the radial length of the vortex area also decreased.Therefore, it can be concluded that the increase in the diameter of the ball seat decreased the size of the vortex area.Figure 10 also shows that when the diameter of the ball seat increased from 30 to 40 mm, the velocity near the wall in the vortex area decreased to 0.67 times that of the model with the original ball seat diameter.The decreases in the size of the vortex area and the vortex velocity near the wall would reduce erosion by drilling fluid and the risk of sticking when the differential motion assembly was triggered.In addition to velocity changes, pressure changes also affect the service performance of the differential motion assembly.The larger pressure drop not only increases the load on the drilling pump and further results in unnecessary costs but can also increase the risk of cutting the pin before the ball is dropped, which would unexpectedly trigger the differential motion assembly and cause the IPP-Coring process to fail.Therefore, we studied the effect of the diameter of the ball seat on the pressure distribution.As shown in Figure 11, as the diameter of the ball seat increased from 30 to 40 mm, the pressure at the inlet of the differential motion assembly decreased from 31.4 to 30.3 MPa, and the pressure drop ΔP of the differential motion assembly with a 40 mm ball seat diameter was only 0.2 times that of the assembly with a 30 mm ball seat diameter.In summary, when the ball seat In addition to velocity changes, pressure changes also affect the service performance of the differential motion assembly.The larger pressure drop not only increases the load on the drilling pump and further results in unnecessary costs but can also increase the risk of cutting the pin before the ball is dropped, which would unexpectedly trigger the differential motion assembly and cause the IPP-Coring process to fail.Therefore, we studied the effect of the diameter of the ball seat on the pressure distribution.As shown in Figure 11, as the diameter of the ball seat increased from 30 to 40 mm, the pressure at the inlet of the differential motion assembly decreased from 31.4 to 30.3 MPa, and the pressure drop ∆P of the differential motion assembly with a 40 mm ball seat diameter was only 0.2 times that of the assembly with a 30 mm ball seat diameter.In summary, when the ball seat diameter increased from 30 to 40 mm, the maximum velocity decreased to 0.55 times that of the original differential motion assembly with a ball seat diameter of 30 mm.The axial length of the vortex area and the vortex velocity near the wall decreased to 0.85 times and 0.67 times that for the original ball seat diameter, respectively.In addition, the pressure drop of the differential motion assembly with a ball seat diameter of 40 mm was only 0.2 times that of the original assembly.Thus, increasing the diameter of the ball seat could effectively reduce the erosion of the drilling fluid on the inner surface of the differential motion assembly and the risk of cutting the pin before the ball is dropped, thereby improving the reliability and service life of the differential motion assembly.Notably, the larger diameter of the ball seat in the differential motion assembly means that a larger drop ball should be used to trigger the differential motion assembly when a core enters the IPP-Coring system.However, due to the constraint of the inner diameter of the drilling tool, the diameter of the drop ball should not be too large; otherwise, it will be difficult to smoothly drop the ball onto the ball seat.As shown in Figure 2, the diameter of the inlet of the differential motion assembly was only 50 mm; therefore, a ball seat with a diameter of 40 mm may be the most appropriate.

Effect of the Outlet Structure of the Ball Seat on the Flow Characteristics
Increasing the diameter of the ball seat improved the flow status of the drilling fluid.However, due to the abrupt change of the diameter in the ball seat outlet, a severe vortex and turbulence occurred within the differential motion assembly.To reduce these behaviors, it is necessary to improve the outlet structure of the ball seat.
To study the effect of the outlet structure of the ball seat on the flow characteristics, and considering machining and structural strength factors, three outlet structures of the ball seat-arc, cone, and original right-angled structure-were designed, as shown in Figure 12.Their effect on the velocity is shown in Figure 13.The internal velocity distribution of the differential motion assembly of the cone outlet at the ball seat was similar to that of the original right-angled outlet.Their maximum velocities in the differential motion assembly were both 36 m/s at the axis behind the ball seat outlet.Notably, the arc ball seat outlet structure greatly decreased the fluid velocity in the differential motion assembly, and the maximum velocity was only 34 m/s at the ball seat inlet.In addition, the axial length of the vortex area in the differential motion assembly with a cone outlet structure at the ball seat was 211 mm, which was only 9 mm less than that for the original right-angled outlet.However, the axial vortex length in the differential motion assembly with the arc outlet structure decreased to 0.7 times that with the original right-angled outlet.

Effect of the Outlet Structure of the Ball Seat on the Flow Characteristics
Increasing the diameter of the ball seat improved the flow status of the drilling fluid.However, due to the abrupt change of the diameter in the ball seat outlet, a severe vortex and turbulence occurred within the differential motion assembly.To reduce these behav- sembly were both 36 m/s at the axis behind the ball seat outlet.Notably, the arc ball seat outlet structure greatly decreased the fluid velocity in the differential motion assembly, and the maximum velocity was only 34 m/s at the ball seat inlet.In addition, the axial length of the vortex area in the differential motion assembly with a cone outlet structure at the ball seat was 211 mm, which was only 9 mm less than that for the original rightangled outlet.However, the axial vortex length in the differential motion assembly with the arc outlet structure decreased to 0.7 times that with the original right-angled outlet.To quantitatively study the effect of the outlet structure of the ball seat on the velocity distribution in the differential motion assembly, we set two velocity monitoring lines (shown in black and white) along the axial and radial directions, respectively.The velocity distributions at the two monitoring lines for different outlet structures at the ball seat are shown in Figure 14.In Figure 14a, according to the structural characteristics of the differential motion assembly, the computational domain of the differential motion assembly was divided into the inlet, ball seat, and outlet sections.The change in the outlet structure at the ball seat had almost no effect on the axial velocity distribution of the inlet section.However, when the drilling fluid entered the ball seat and outlet sections, the velocity of the arc structure was significantly less than that of the cone structure and the original right-angled structure.Notably, in the ball seat section of the arc structure, the drilling fluid first rose to 33 m/s and then started to decrease at the arc of the ball seat section.However, for the cone and original right-angled structure, the fluid continued to accelerate in the ball seat section.After entering the outlet section, the fluid continued to advance at a velocity greater than 35 m/s for approximately 130 mm before the velocity began to gradually decrease.In addition, as shown in Figure 14b, the maximum velocity of the central jet zone of the arc structure was only 28 m/s, which was 0.8 times that of the cone and the right-angled structure, and the vortex velocity near the wall of the arc structure was only 0.4 times that of the cone and right-angled structures.Therefore, it can be concluded that the arc structure in the ball seat outlet effectively decreased the fluid velocity in the differential motion assembly and improved the severe vortex and turbulence behaviors in the differential motion assembly.Furthermore, the erosion of the inner surface of the differential motion assembly by the drilling fluid could be improved.To quantitatively study the effect of the outlet structure of the ball seat on the velocity distribution in the differential motion assembly, we set two velocity monitoring lines (shown in black and white) along the axial and radial directions, respectively.The velocity distributions at the two monitoring lines for different outlet structures at the ball seat are shown in Figure 14.In Figure 14a, according to the structural characteristics of the differential motion assembly, the computational domain of the differential motion assembly was divided into the inlet, ball seat, and outlet sections.The change in the outlet structure at the ball seat had almost no effect on the axial velocity distribution of the inlet section.However, when the drilling fluid entered the ball seat and outlet sections, the velocity of the arc structure was significantly less than that of the cone structure and the original right-angled structure.Notably, in the ball seat section of the arc structure, the drilling fluid first rose to 33 m/s and then started to decrease at the arc of the ball seat section.However, for the cone and original right-angled structure, the fluid continued to accelerate in the ball seat section.After entering the outlet section, the fluid continued to advance at a velocity greater than 35 m/s for approximately 130 mm before the velocity began to gradually decrease.In addition, as shown in Figure 14b, the maximum velocity of the central jet zone of the arc structure was only 28 m/s, which was 0.8 times that of the cone and the right-angled structure, and the vortex velocity near the wall of the arc structure was only 0.4 times that of the cone and right-angled structures.Therefore, it can be concluded that the arc structure in the ball seat outlet effectively decreased the fluid velocity in the differential motion assembly and improved the severe vortex and turbulence behaviors in the differential motion assembly.Furthermore, the erosion of the inner surface of the differential motion assembly by the drilling fluid could be improved.
gradually decrease.In addition, as shown in Figure 14b, the maximum velocity of the tral jet zone of the arc structure was only 28 m/s, which 0.8 times that of the cone the right-angled structure, and the vortex velocity near the wall of the arc structure only 0.4 times that of the cone and right-angled structures.Therefore, it can be conclu that the arc structure in the ball seat outlet effectively decreased the fluid velocity in differential motion assembly and improved the severe vortex and turbulence behavio the differential motion assembly.Furthermore, the erosion of the inner surface of the ferential motion assembly by the drilling fluid could be improved.Figure 15 shows the effect of the outlet structure of the ball seat on the pressure distribution.There was a low-pressure zone in the ball seat section of the differential motion assembly with the arc ball seat outlet structure because the flow status changed at the arc.In addition, the pressure drops at the inlet and outlet of the differential motion assembly with the arc ball seat outlet structure were only 0.1 MPa, which was only 0.33 times that of the cone outlet and the right-angled outlet.In summary, the arc outlet ball seat not only greatly reduced the velocity of the drilling fluid and the severe vortex and turbulence behaviors in the differential motion assembly but also decreased the pressure drops in the differential motion assembly.Therefore, this outlet structure of the ball seat can reduce erosion by the drilling fluid and the risk of cutting the pin before the ball is dropped, which would improve the reliability and service life of the differential motion assembly as well as the success rate of the IPP-Coring procedure in deep oil and gas exploration.Figure 15 shows the effect of the outlet structure of the ball seat on the pressure distribution.There was a low-pressure zone in the ball seat section of the differential motion assembly with the arc ball seat outlet structure because the flow status changed at the arc.In addition, the pressure drops at the inlet and outlet of the differential motion assembly with the arc ball seat outlet structure were only 0.1 MPa, which was only 0.33 times that of the cone outlet and the right-angled outlet.In summary, the arc outlet ball seat not only greatly reduced the velocity of the drilling fluid and the severe vortex and turbulence behaviors in the differential motion assembly but also decreased the pressure drops in the differential motion assembly.Therefore, this outlet structure of the ball seat can reduce erosion by the drilling fluid and the risk of cutting the pin before the ball is dropped, which would improve the reliability and service life of the differential motion assembly as well as the success rate of the IPP-Coring procedure in deep oil and gas exploration.

Structural Improvement of the Differential Motion Assembly of the IPP-Coring System
From the above results, the differential motion assembly with a 40 mm diameter ball seat and the arc structure in the ball seat outlet shows the lowest pressure drop and fluid velocity.To increase the operational reliability and service life of the differential motion assembly and further increase the success rate of IPP-Coring, this structure was chosen as the improved structure of the differential motion assembly.To quantitatively analyze the erosion of the improved structure, we further calculated the particle trajectories and the erosion of the original and improved differential motion assemblies when the fluid viscosity was set as 100 mPa•s and the inlet flow rate was 0.04 m 3 /s.As shown in Figure 16, compared with the original structure, the particles in the improved structure have a lower velocity and less impact on the wall.Therefore, the improved structure shows lower surface erosion, and the average erosion rate is only 0.42 times the value of the original structure.Due to the lower pressure drop and erosion rate, the improved structure has a smaller risk of sticking and triggering the assembly at the wrong time.

Structural Improvement of the Differential Motion Assembly of the IPP-Coring System
From the above results, the differential motion assembly with a 40 mm diameter ball seat and the arc structure in the ball seat outlet shows the lowest pressure drop and fluid velocity.To increase the operational reliability and service life of the differential motion assembly and further increase the success rate of IPP-Coring, this structure was chosen as the improved structure of the differential motion assembly.To quantitatively analyze the erosion of the improved structure, we further calculated the particle trajectories and the erosion of the original and improved differential motion assemblies when the fluid viscosity was set as 100 mPa•s and the inlet flow rate was 0.04 m 3 /s.As shown in Figure 16, compared with the original structure, the particles in the improved structure have a lower velocity and less impact on the wall.Therefore, the improved structure shows lower surface erosion, and the average erosion rate is only 0.42 times the value of the original structure.Due to the lower pressure drop and erosion rate, the improved structure has a smaller risk of sticking and triggering the assembly at the wrong time.

Field Validation Experiments
According to the CFD simulation results obtained above, the differential motion assembly was redesigned and manufactured.To validate the simulation results, field experiments were conducted in an experimental well in Tangshan, Hebei, China.During the experiments, the differential motion assembly with original and improved structure was installed in the IPP-Coring system, respectively, and then it was lowed into the well with a depth of 30 m.After that, the pump was turned on and the coring operation was conducted.The solid phase in the drilling fluid is mainly quartz sand and the field experimental conditions were summarized in Table 1.

Parameter
Value Flow rate (m 3 /s) 0.01 Density of the drilling fluid (kg/m 3 ) 1050 Funnel viscosity of the drilling fluid (s) 60-70 The photographs of the outlet of the ball seat of the original and improved differential motion assemblies after the experiments are shown in Figure 17.It can be seen that there was obvious erosion damage at the outlet of the ball seat of the original differential motion assembly, while the outlet of the ball seat of the improved differential motion assembly was relatively smooth and no obvious surface damage can be observed.In addition, the circulation pump pressure of the system with the improved differential motion assembly decreased from 0.2 MPa to nearly 0 MPa compared with that of the system with the original differential motion assembly.Therefore, it can be validated that the improved

Field Validation Experiments
According to the CFD simulation results obtained above, the differential motion assembly was redesigned and manufactured.To validate the simulation results, field experiments were conducted in an experimental well in Tangshan, Hebei, China.During the experiments, the differential motion assembly with original and improved structure was installed in the IPP-Coring system, respectively, and then it was lowed into the well with a depth of 30 m.After that, the pump was turned on and the coring operation was conducted.The solid phase in the drilling fluid is mainly quartz sand and the field experimental conditions were summarized in Table 1.The photographs of the outlet of the ball seat of the original and improved differential motion assemblies after the experiments are shown in Figure 17.It can be seen that there was obvious erosion damage at the outlet of the ball seat of the original differential motion assembly, while the outlet of the ball seat of the improved differential motion assembly was relatively smooth and no obvious surface damage can be observed.In addition, the circulation pump pressure of the system with the improved differential motion assembly decreased from 0.2 MPa to nearly 0 MPa compared with that of the system with the original differential motion assembly.Therefore, it can be validated that the improved differential motion assembly shows the best erosion resistance and the lower pressure drop; thus, it has better promise for field performance.

Conclusions
In this study, CFD simulations were performed to investigate the flow characteristics in a differential motion assembly during the IPP-Coring process with different flow rates.The effects of the diameter of the ball seat and the structure of the ball seat outlet on the fluid status, velocity, and pressure distribution were thoroughly investigated.Thus, the hydraulic structure of the differential motion assembly was improved.The specific conclusions are presented as follows: 1.There was a severe vortex area in the differential motion assembly due to the presence of the ball seat.Increased fluid viscosity can increase the local velocity of the flow and decrease the size of the vortex area.In addition, as the inlet flow rate increased from 0.01 to 0.04 m 3 /s, the vortex velocity near the wall and the pressure drops between the inlet and outlet of the differential motion assembly increased by 3.8-and 14-fold, respectively.2. With the increase in the ball seat diameter from 30 to 40 mm, the maximum velocity in the differential motion assembly decreased to 0.55 times that of the original assembly.The axial length of the vortex area and the vortex velocity near the wall decreased to 0.85 and 0.67 times the values of the original assembly.In addition, the pressure drop decreased to 0.2 times that in the original assembly with a 30 mm ball seat diameter.3. Compared with the cone and original right-angled structures, the arc structure at the ball seat outlet of the differential motion assembly showed the best hydraulic properties.Its axial vortex length and vortex velocity near the wall in the differential motion assembly decreased, respectively, to 0.7 and 0.4 times the values for the system with the original right-angled structure at the ball seat outlet.The pressure drop was only 0.33 times that of the system with the original right-angled structure at the ball seat outlet.4. Considering the above results, the ball seat with a 40 mm diameter was selected for the differential motion assembly, and the outlet of the ball seat was set as the arc structure.The calculated results of the particle trajectories and the erosion show that the improved structure has lower particle velocity and less impact on the wall, and the average erosion rate decreased to 0.42 times the original structure.Due to the lower pressure drop and erosion rate, the improved differential motion assembly may have better promise for field performance.

Conclusions
In this study, CFD simulations were performed to investigate the flow characteristics in a differential motion assembly during the IPP-Coring process with different flow rates.The effects of the diameter of the ball seat and the structure of the ball seat outlet on the fluid status, velocity, and pressure distribution were thoroughly investigated.Thus, the hydraulic structure of the differential motion assembly was improved.The specific conclusions are presented as follows: 1.
There was a severe vortex area in the differential motion assembly due to the presence of the ball seat.Increased fluid viscosity can increase the local velocity of the flow and decrease the size of the vortex area.In addition, as the inlet flow rate increased from 0.01 to 0.04 m 3 /s, the vortex velocity near the wall and the pressure drops between the inlet and outlet of the differential motion assembly increased by 3.8-and 14-fold, respectively.

2.
With the increase in the ball seat diameter from 30 to 40 mm, the maximum velocity in the differential motion assembly decreased to 0.55 times that of the original assembly.The axial length of the vortex area and the vortex velocity near the wall decreased to 0.85 and 0.67 times the values of the original assembly.In addition, the pressure drop decreased to 0.2 times that in the original assembly with a 30 mm ball seat diameter.

3.
Compared with the cone and original right-angled structures, the arc structure at the ball seat outlet of the differential motion assembly showed the best hydraulic properties.Its axial vortex length and vortex velocity near the wall in the differential motion assembly decreased, respectively, to 0.7 and 0.4 times the values for the system with the original right-angled structure at the ball seat outlet.The pressure drop was only 0.33 times that of the system with the original right-angled structure at the ball seat outlet.

4.
Considering the above results, the ball seat with a 40 mm diameter was selected for the differential motion assembly, and the outlet of the ball seat was set as the arc structure.The calculated results of the particle trajectories and the erosion show that the improved structure has lower particle velocity and less impact on the wall, and the average erosion rate decreased to 0.42 times the original structure.Due to the

Figure 1 .
Figure 1.(a) Schematic diagram of the IPP-Coring system.(b) Schematic diagram of the working principle of the IPP-Coring system and the differential motion assembly.(c) Photographs of the IPP-Coring system and the differential motion assembly.

Figure 1 .
Figure 1.(a) Schematic diagram of the IPP-Coring system.(b) Schematic diagram of the working principle of the IPP-Coring system and the differential motion assembly.(c) Photographs of the IPP-Coring system and the differential motion assembly.

Figure 2 .
Figure 2. Mesh and boundary conditions of the flow domain in the differential motion assembly.

Figure 2 .
Figure 2. Mesh and boundary conditions of the flow domain in the differential motion assembly.

Figure 2 .
Figure 2. Mesh and boundary conditions of the flow domain in the differential motion assemb

Figure 4 .
Figure 4. Velocity streamlines in the differential motion assembly with different fluid viscosities; the flow rate was 0.04 m 3 /s.

Figure 5 .
Figure 5. Velocity distributions at the monitoring line with different fluid viscosities; the flow rate was 0.04 m 3 /s.

Figure 4 .
Figure 4. Velocity streamlines in the differential motion assembly with different fluid viscosities; the flow rate was 0.04 m 3 /s.

Figure 4 .
Figure 4. Velocity streamlines in the differential motion assembly with different fluid viscosities; the flow rate was 0.04 m 3 /s.

Figure 5 .
Figure 5. Velocity distributions at the monitoring line with different fluid viscosities; the flow rate was 0.04 m 3 /s.

Figure 5 .
Figure 5. Velocity distributions at the monitoring line with different fluid viscosities; the flow rate was 0.04 m 3 /s.

Figure 6 .
Figure 6.Velocity streamlines in the differential motion assembly with different inlet flow rates (0.01~0.04 m 3 /s); the fluid viscosity was set as 100 mPa•s.

Figure 6 .
Figure 6.Velocity streamlines in the differential motion assembly with different inlet flow rates (0.01~0.04 m 3 /s); the fluid viscosity was set as 100 mPa•s.

19 Figure 7 .
Figure 7. Velocity distributions at the monitoring line with different inlet flow rates; the fluid viscosity was set as 100 mPa•s.Pressure contours in the differential motion assembly with different inlet flow rates are shown in Figure8.There was an obvious pressure drop when the drilling fluid flowed through the differential motion assembly.In this study, the outlet pressure was set to 30 MPa during the calculation.With an increase in the inlet flow rate from 0.01 to 0.04 m 3 /s,

Figure 7 .
Figure 7. Velocity distributions at the monitoring line with different inlet flow rates; the fluid viscosity was set as 100 mPa•s.

Figure 7 .
Figure 7. Velocity distributions at the monitoring line with different inlet flow rates; the fluid viscosity was set as 100 mPa•s.

Figure 8 .
Figure 8. Pressure contours in the differential motion assembly with different inlet flow rates (0.01~0.04 m 3 /s); the fluid viscosity was set as 100 mPa•s.

Figure 8 .
Figure 8. Pressure contours in the differential motion assembly with different inlet flow rates (0.01~0.04 m 3 /s); the fluid viscosity was set as 100 mPa•s.

Figure 9 .
Figure 9.Effect of the diameter of the ball seat on the flow velocity; the fluid viscosity was set as 100 mPa•s.

Figure 9 . 19 Figure 10 .
Figure 9.Effect of the diameter of the ball seat on the flow velocity; the fluid viscosity was set as 100 mPa•s.Appl.Sci.2023, 13, x FOR PEER REVIEW 11 of 19

Figure 10 .
Figure 10.Velocity distributions at the monitoring line for ball seats with different diameters; the fluid viscosity was set as 100 mPa•s.

19 Figure 11 .
Figure 11.Effect of the ball seat diameter on the pressure distribution; the fluid viscosity was set as 100 mPa•s.

Figure 11 .
Figure 11.Effect of the ball seat diameter on the pressure distribution; the fluid viscosity was set as 100 mPa•s.

Figure 12 .
Figure 12.Differential motion assembly with different ball seat outlet structures: (a) arc, (b) cone, and (c) right angle.

Figure 13 .
Figure 13.Effect of the outlet structure of the ball seat on the flow velocity; the fluid viscosity was set as 100 mPa•s.

Figure 14 .
Figure 14.(a) Velocity distributions at the black velocity monitoring line with different diamete the ball seat; (b) velocity distributions at the white velocity monitoring line with different diam of the ball seat; the fluid viscosity was set as 100 mPa•s.

Figure 14 .
Figure 14.(a) Velocity distributions at the black velocity monitoring line with different diameters of the ball seat; (b) velocity distributions at the white velocity monitoring line with different diameters of the ball seat; the fluid viscosity was set as 100 mPa•s.

Figure 15 .
Figure 15.Effect of the outlet structure of the ball seat on the pressure distribution; the fluid viscosity was set as 100 mPa•s.

Figure 15 .
Figure 15.Effect of the outlet structure of the ball seat on the pressure distribution; the fluid viscosity was set as 100 mPa•s.

Figure 16 .
Figure 16.(a) Particle trajectories of the original and improved differential motion assembly; (b) the erosion contours of the original and improved differential motion assembly; (c) the average erosion rate of the original and improved differential motion assembly; the fluid viscosity was set as 100 mPa•s and the inlet flow rate was 0.04 m 3 /s.

Figure 16 .
Figure 16.(a) Particle trajectories of the original and improved differential motion assembly; (b) the erosion contours of the original and improved differential motion assembly; (c) the average erosion rate of the original and improved differential motion assembly; the fluid viscosity was set as 100 mPa•s and the inlet flow rate was 0.04 m 3 /s.
Appl.Sci.2023, 13, x FOR PEER REVIEW 16 of 19differential motion assembly shows the best erosion resistance and the lower pressure drop; thus, it has better promise for field performance.

Figure 17 .
Figure 17.Photographs showing the outlet of the ball seat of the original and improved differential motion assemblies after the experiments.

Figure 17 .
Figure 17.Photographs showing the outlet of the ball seat of the original and improved differential motion assemblies after the experiments.