Prediction Method for Wellbore Abrasion and Actual Diameter in Close-Spacing Multi-Cluster Temporary Plugging Fracturing

Abstract

The present invention provides a method for predicting borehole abrasion and actual diameter in close-cut multi-cluster temporary plugging fracturing. Data such as the top and bottom of a single cluster, the number of perforations within the cluster, and the cluster spacing are obtained, and a two-phase flow model of single-cluster multi-perforation in a DPM horizontal well is established to calculate the erosion amount under a certain sand-fixing amount in the borehole. The diameter and density parameters of the temporary plugging ball are obtained, a DEM model is established, the parameters of the DDPM model are initialized, the volume fraction is calculated, and momentum is exchanged through SCALARS to achieve two-way solid-liquid coupling calculation. The calculated data is substituted into the formula to obtain the ratio range of the actual borehole diameter to the temporary plugging ball diameter. According to the diameter of the temporary plugging ball used on-site, the actual borehole diameter and abrasion rate during construction can be calculated, effectively improving the setting rate of the temporary plugging ball, increasing the complexity of the fracture, enlarging the effective transformation volume, and enhancing the production.

Description

Technical Field

[0001] The present invention belongs to the field of enhanced oil and gas production, and specifically relates to a method for predicting borehole abrasion and actual diameter in closely spaced multi-cluster temporary plugging fracturing. Background Art

[0002] Unconventional resources such as shale oil and gas, tight gas, etc. have special reservoir spaces and formation processes, and are characterized by low porosity and low permeability. Most shale gas wells need to undergo reservoir stimulation technologies such as hydraulic fracturing to obtain commercial exploitation value. Nowadays, the horizontal well staged fracturing technology has been widely applied in the industrial exploitation of shale gas. Along with the increasing maturity of key technologies, the growth rate of production increase has slowed down. Meanwhile, in the North American region, by shortening the cluster spacing and increasing the proppant dosage, the process technology has achieved a substantial increase in the single-well production capacity. According to statistics, the shale gas production in the United States reached 5.264×108 m3 in 2017, which is 8.8 times that of 10 years ago. Currently, in work areas such as Fuling and Changning, the closely spaced multi-cluster fracturing technology is mainly adopted, supplemented by the temporary plugging and diversion technology in refracturing, which can effectively improve the opening degree of perforation holes, increase the fracturing stages without changing the number of segments, and increase the complexity of the fracture network. It is a key auxiliary technology.

[0003] The common approaches in the temporary plugging and diversion process are temporary plugging agents and temporary plugging balls, and mechanical temporary plugging is more widely used. The size of the perforation hole plays a crucial role in whether the temporary plugging ball can be successfully seated. Due to the abrasion of fracturing fluid and proppant, etc., the size of the downhole hole changes. Therefore, exploring the abrasion situation of the borehole and predicting the actual diameter of the downhole borehole are of great significance for fracturing operations. Summary of the Invention

[0004] Aiming at the deficiencies of the prior art, the present invention provides a method for predicting borehole abrasion and actual diameter in closely spaced multi-cluster temporary plugging fracturing, which can predict the borehole abrasion situation, calculate the actual diameter of the borehole under construction conditions, and improve the seating rate of the temporary plugging ball.

[0005] To achieve the above object, the present invention provides a method for predicting borehole abrasion and actual diameter in closely spaced multi-cluster temporary plugging fracturing, including the steps of:

[0006] Based on the DPM horizontal well single-cluster multi-perforation two-phase flow model, establish a borehole flow field and velocity field, and calculate the borehole erosion amount;

[0007] Based on the temporary plugging ball force model, according to the DEM-FLUENT solid-liquid two-way coupling mathematical model, simulate the movement of the temporary plugging ball in the horizontal wellbore, and combine the simulation results of the movement of balls with different diameters in the horizontal wellbore established by the DPM horizontal well single-cluster multi-perforation two-phase flow model to obtain the calculation method for the ratio of the temporary plugging ball diameter to the perforation hole diameter, calculate the actual diameter of the borehole, and deduce the borehole abrasion rate;

[0008] Among them, the ratio K of the actual diameter of the perforation hole to the diameter of the temporary plugging ball C is as follows:

[0009]

[0010] In the formula: F 1 is the reaction force when the ball contacts point A, ΔP is the pressure difference for the ball to seal the perforation, d m is the diameter of the ball, d is the perforation diameter.

[0011] The prediction method steps of perforation hole abrasion and actual diameter are as follows:

[0012] ① Calculate the perforation hole erosion amount according to the known sand volume;

[0013] ② Calculate the ratio Kc of the actual diameter of the perforation hole to the diameter of the temporary plugging ball;

[0014] ③ During the actual fracturing ball-sealing perforation operation, detect the temporary plugging balls that can effectively seal the perforation holes;

[0015] ④ Calculate the actual diameter of the perforation hole according to the diameter of the effective temporary plugging ball;

[0016] ⑤ Calculate the perforation hole abrasion rate according to the actual diameter of the perforation hole.

[0017] Among them, the DPM horizontal well single-cluster multi-perforation two-phase flow model is an incompressible fluid and steady-state solid-liquid two-phase flow model, including the continuity equation and the momentum conservation equation. The fluid turbulence equation uses the standard k-ε model; in the particle force, the drag force, the Reynolds number, and the added mass force are considered. The solid erosion effect is considered for the particles and the horizontal wellbore and perforation holes. The Ahlert calculation model is used for the calculation.

[0018] ER = AF s v n1 f(α)

[0019] In the formula: ER is the erosion rate, mm / h; A is the correlation coefficient; n 1 is the particle impact velocity exponent; Fs is the roundness coefficient of the particle, indicating the sharpness of the particle surface. When the particle is an irregular sharp corner, Fs = 1.0; when the particle is an ellipsoidal shape, Fs = 0.5; when the particle is a spherical shape, Fs = 0.2.

[0020] Among them, a method combining the fluid dynamics software (FLUENT) and the discrete element method (DEM) is adopted. FLUENT is used to calculate the flow process of the liquid in the horizontal wellbore, and the fluid flow field is established. DEM is used to obtain parameters such as the position, volume, and velocity of the particles, initialize the particle information in DDPM in FLUENT at the current step, activate the DDPM model in the Euler model, calculate the volume fraction, and exchange momentum through the built-in SCALARS to achieve the solid-liquid two-way coupling calculation. The momentum exchange equation between the particles and the fluid in the calculation model is as follows:

[0021]

[0022] In the formula: F is the momentum transfer amount from the continuous phase to the discrete phase, (kg·m) / s; μ is the viscosity of the fluid, (N·S) / m2; ρp is the particle density kg·m 3 ; d p is the particle diameter, m; R e is the relative Reynolds number, dimensionless; u p and u are the velocity of the particle and the fluid velocity respectively, m / s; C D is the drag coefficient, dimensionless; m p * is the mass flow rate of the particles, kg / s; △t is the time step, s; F other represents other forces per unit mass, N / kg;

[0023] The mass exchange equation from the discrete phase to the continuous phase is as follows:

[0024]

[0025] In the formula: M is the mass transfer amount from the discrete phase to the continuous phase, kg / s 2 ; △m p and m .p，0 represent the change in the mass flow rate of the particles and the initial mass flow rate of particle injection respectively, kg / s; m p,0 is the initial mass of the particle, kg;

[0026] In the DDPM model and the DEM model, the motion trajectory of the discrete phase is predicted and calculated by balancing various forces acting on the particles in the Lagrangian reference frame. Using Newton's second law, the equation controlling the particle motion is as follows.

[0027]

[0028]

[0029] Where: m is the mass of the particle 2, kg; x is the displacement of the particle moving from position 1 to position 2, s; t is the time taken to move from position 1 to position 2, s; V represents the vector velocity of the particle, m / s; F drag 、F virtual _mass, F gravitation and F other respectively represent the vectors of the drag force, virtual mass force, gravitational force and other acting forces on the particle, N.

[0030] Among them, the gravitational force of the temporary plugging ball is expressed as:

[0031]

[0032] Where: d m is the ball diameter, ρ m is the ball density, and g is the gravitational acceleration;

[0033] The buoyancy force received is:

[0034]

[0035] Where: ρ w is the density of the fracturing fluid;

[0036] The fluid carrying force received by the temporary plugging ball is:

[0037]

[0038] R e = ρd m |u - u m | / μ

[0039] Where: u is the fluid flow velocity, u m is the migration velocity of the temporary plugging ball, μ is the fluid viscosity, R e is the Reynolds number, C d is the drag coefficient. When R e > 1000, C d = 0.44. When R e ≤ 1000, (1 + 0.15R e 0.0687 ) / (R e / 24).

[0040] Different from the prior art, the present invention provides a method for predicting borehole abrasion and actual diameter in close-spaced multi-cluster temporary plugging fracturing. By establishing two-dimensional and three-dimensional models for numerical analysis of the two-phase flow of fracturing fluid and proppant sand grains, the two-phase flow field of the horizontal wellbore and the borehole can be obtained, and the movement trajectories of sand grains in the horizontal wellbore and perforations can be simulated. The erosion amount under a given sand volume can be calculated through the erosion model; based on the numerical model of the DEM-FLUENT two-way coupling model, combined with the force model of the temporary plugging ball, the temporary plugging critical condition of the temporary plugging ball in the horizontal well is proposed. A method for calculating the borehole abrasion rate and the actual diameter is obtained, which is of great significance for improving the setting rate of the temporary plugging ball. It helps to improve the fracture complexity, increase the effective stimulated volume, and enhance the production in areas using the close-spaced multi-cluster temporary plugging fracturing technology, especially in the areas of normal-pressure shale gas. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 It is a schematic diagram of 3D geometric modeling and numerical model of a horizontal well and single-cluster multi-perforations in the method for predicting borehole abrasion and actual diameter in close-spaced multi-cluster temporary plugging fracturing provided by the present invention;

[0042] Figure 2 It is a schematic diagram of the velocity change curve during the process of the temporary plugging ball setting in the borehole in the method for predicting borehole abrasion and actual diameter in close-spaced multi-cluster temporary plugging fracturing provided by the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0043] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0044] As Figure 1 shown; the present invention provides a method for predicting borehole abrasion and actual diameter in close-spaced multi-cluster temporary plugging fracturing, including the steps:

[0045] Based on the DPM two-phase flow model of single-cluster multi-perforations in a horizontal well, establish the borehole flow field and velocity field, and calculate the borehole erosion amount;

[0046] Based on the force model of the temporary plugging ball, according to the DEM-FLUENT solid-liquid two-way coupling mathematical model, simulate the movement of the temporary plugging ball in the horizontal wellbore, and combine the simulation results of the movement of balls with different diameters in the horizontal wellbore established by the DPM two-phase flow model of single-cluster multi-perforations in a horizontal well to obtain the calculation method for the ratio of the diameter of the temporary plugging ball to the diameter of the perforation borehole, calculate the actual diameter of the borehole, and deduce the borehole abrasion rate;

[0047] wherein, the ratio K of the actual diameter of the borehole to the diameter of the temporary plugging ball Cis:

[0048]

[0049] In the formula: F 1 is the reaction force of the ball in contact with point A, ΔP is the pressure difference for the ball to set and seal the perforation, d m is the diameter of the ball, d is the perforation diameter.

[0050] The prediction method steps of the wellbore abrasion and the actual diameter are as follows:

[0051] ① Calculate the wellbore erosion amount according to the known sand volume;

[0052] ② Calculate the ratio Kc of the actual wellbore diameter to the temporary plugging ball diameter;

[0053] ③ During the actual fracturing ball setting and wellbore sealing construction, detect the temporary plugging balls that can effectively seal the wellbore;

[0054] ④ Calculate the actual wellbore diameter according to the effective temporary plugging ball diameter;

[0055] ⑤ Calculate the wellbore abrasion rate according to the actual wellbore diameter.

[0056] Specifically, when implementing the prediction method of the present invention, establish two-dimensional and three-dimensional models of the horizontal wellbore and related perforation clusters, and encrypt the boundary layer network of the inner wall of the horizontal wellbore and the perforation inner wall. The input data required for modeling includes perforation distribution, cluster top, cluster bottom, cluster length, number of perforations in the cluster, cluster spacing, perforation diameter and length;

[0057] Based on the Ahlert erosion model and DPM solid-liquid two-phase flow, calculate the perimeter, area, erosion area radius, erosion rate of the casing at the perforation, and estimate the wear amount at the wellbore according to the on-site fracturing time;

[0058] Modeling with the DEM model, the input data includes the temporary plugging ball diameter, ball density, fracturing fluid density, viscosity and displacement, and obtain parameters such as the position, volume, and velocity of the temporary plugging ball;

[0059] Use the temporary plugging ball parameters obtained by the DEM to initialize the temporary plugging ball information in the DDPM in FLUENT, activate the DDPM model, and calculate the volume fraction; the DEM-FLUENT calculation model uses a double-precision solver, and the turbulent model selects the standard k-ε turbulent model. Since the ball will collide with the wall surface, select to check the wall enhancement function, the algorithm is SIMPLE, and the momentum equation, volume fraction equation, turbulent kinetic energy equation, and turbulent dissipation rate equation all use the first-order upwind method;

[0060] Simulate the movement trajectory of the temporary plugging ball, calculate the pressure change before and after the temporary plugging ball setting in the perforation cluster section of the horizontal well, and obtain ΔP;

[0061] Input data of different temporary plugging ball diameters and ball densities in EDEM, export the migration trajectory line of the temporary plugging ball in the horizontal wellbore and the total velocity of the temporary plugging ball, plot the velocity change curve of the temporary plugging ball during the setting of the perforation hole, and calculate the reaction force F when the temporary plugging ball is set 1 ; as Figure 2 shown

[0062] Substitute the calculated ΔP and F 1 into the following formula to calculate K C range;

[0063] During on-site construction, record the diameter, density and number of temporary plugging balls used during the fracturing process, and judge whether the temporary plugging ball successfully sets the perforation hole through the construction curve;

[0064] Substitute the diameter of the effectively set ball into the following formula to calculate the actual diameter of the perforation, and then the perforation abrasion rate under the action of the proppant can be obtained;

[0065] Based on the DPM solid-liquid two-phase flow model calculation and Albert's abrasion model, establish the flow field in the horizontal wellbore and the perforation hole, calculate the erosion amount under a certain amount of sand fixation, and based on the force model of the temporary plugging ball and the DEM-FLUENT solid-liquid two-way coupling mathematical model, calculate the effective temporary plugging critical conditions of the temporary plugging ball in the horizontal well, obtain the calculation method of the ratio of the temporary plugging ball diameter to the perforation hole diameter, and calculate the actual diameter of the perforation hole and the abrasion rate.

[0066] In the above solution, the DPM solid-liquid two-phase flow model is an incompressible fluid and steady-state solid-liquid two-phase flow model. This model includes the continuity equation and the momentum conservation equation, and the standard k-ε model is used for the fluid turbulence equation.

[0067] In the above solution, the abrasion model adopts Albert model,

[0068] ER = AF s v n1 f(α)

[0069] In the formula: ER is the erosion rate, mm / h; A is the correlation coefficient; n 1 is the particle impact velocity exponent; Fs is the roundness coefficient of the particle, indicating the sharpness of the particle surface. When the particle is an irregular sharp corner, Fs = 1.0. When the particle is an ellipsoidal shape, Fs = 0.5. When the particle is a spherical shape, Fs = 0.2.

[0070] In the above solution, the force model of the temporary plugging ball is:

[0071] The gravity of the temporary plugging ball can be expressed as:

[0072]

[0073] Where: d m is the sphere diameter, ρ m is the sphere density, and g is the acceleration due to gravity.

[0074] The buoyant force received is:

[0075]

[0076] Where: ρ w is the density of the fracturing fluid.

[0077] The fluid carrying force on the temporary plugging ball is:

[0078]

[0079] R e = ρd m |u - u m | / μ

[0080] Where: u is the fluid velocity, u m is the migration velocity of the temporary plugging ball, μ is the fluid viscosity, R e is the Reynolds number, C d is the drag coefficient. When R e > 1000, C d = 0.44. When R e ≤ 1000, (1 + 0.15R e 0.0687 ) / (R e / 24). In the above scheme, the coupled model of DEM - DDPM initializes the particle information in DDPM in FLUENT at the current step by using DEM to obtain parameters such as the position, volume, and velocity of the particles, activates the DDPM model in the Eulerian model, calculates the volume fraction, and exchanges momentum through the built - in SCALARS to achieve the solid - liquid two - way coupling calculation.

[0081] The momentum exchange equation between the particles and the fluid in the calculation model is as follows:

[0082]

[0083] Where: F is the momentum transfer amount from the continuous phase to the discrete phase, (kg·m) / s; μ is the fluid viscosity, (N·S) / m 2 ; ρ p is the particle density kg·m 3 ; d p is the particle diameter, m; R e is the relative Reynolds number, dimensionless; u p and u are the particle velocity and the fluid velocity respectively, m / s; CD is the drag coefficient, dimensionless; mp* is the mass flow rate of the particles, kg / s; Δt is the time step, s; Fother Representing other forces per unit mass, N / kg.

[0084] The mass exchange equation from the discrete phase to the continuous phase is as follows:

[0085]

[0086] Where: M is the mass transfer amount from the discrete phase to the continuous phase, kg / s 2； △m p and m .p，0 respectively represent the change in the mass flow rate of the particles and the initial mass flow rate of particle injection, kg / s; m p,0 is the initial mass of the particle, kg.

[0087] In the DDPM model and the DEM model, the motion trajectories of the discrete phase are predicted by balancing various forces acting on the particles in the Lagrangian reference frame. Using Newton's second law, the equations governing the particle motion are as follows.

[0088]

[0089]

[0090] Where: m is the mass of the particle, kg; x is the displacement of the particle moving from position 1 to position 2, s; t is the time taken to move from position 1 to position 2, s; V represents the vector velocity of the particle, m / s; F drag 、F virtual_mass 、F gravitation and F other respectively represent the vectors of the drag force, virtual mass force, gravitational force and other forces acting on the particle, N.

[0091] Different from the prior art, the present invention provides a method for predicting the borehole abrasion and actual diameter in dense multi-cluster temporary plugging fracturing. By establishing two-dimensional and three-dimensional models for numerical analysis of the two-phase flow of fracturing fluid and proppant sand grains, the two-phase flow field of the horizontal wellbore and the borehole can be obtained and the movement trajectories of the sand grains in the horizontal wellbore and the perforation can be simulated. The erosion amount under a given sand volume can be calculated through the erosion model; based on the numerical model of the DEM-FLUENT two-way coupling model, combined with the force-bearing model of the temporary plugging ball, the temporary plugging critical condition of the temporary plugging ball in the horizontal well is proposed. A method for calculating the borehole abrasion rate and the actual diameter is obtained, which is of great significance for improving the setting rate of the temporary plugging ball. It helps to improve the fracture complexity, increase the effective stimulation volume and enhance the production in areas using the dense multi-cluster temporary plugging fracturing technology, especially in the areas of normal-pressure shale gas.

[0092] The foregoing description of the disclosed embodiments enables those skilled in the art to implement or use the present invention. Various modifications to these embodiments will be apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the present invention. Thus, the present invention is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A prediction method for borehole abrasion and actual diameter in closely spaced multi-cluster temporary plugging fracturing, characterized in that, it includes the steps: Based on the DPM horizontal well single-cluster multi-perforation two-phase flow model, establish the borehole flow field and velocity field, and calculate the borehole erosion amount; Based on the temporary plugging ball force model, according to the DEM-FLUENT solid-liquid two-way coupling mathematical model, simulate the movement of the temporary plugging ball in the horizontal wellbore, and combine the simulation results of the movement of balls with different diameters in the horizontal wellbore established by the DPM horizontal well single-cluster multi-perforation two-phase flow model to obtain the calculation method of the ratio of the temporary plugging ball diameter to the perforation borehole diameter, calculate the actual diameter of the borehole, and deduce the borehole abrasion rate; Among them, the ratio K of the actual diameter of the blast hole to the diameter of the temporary plugging ball C is as follows: Where: F 1 is the reaction force of the ball in contact with point A, ΔP is the pressure difference for the ball to set and seal the perforation, d m is the diameter of the ball, d is the perforation diameter; The steps of the prediction method for borehole abrasion and actual diameter are as follows: ① Calculate the borehole erosion amount according to the known sand amount; ② Calculate the ratio Kc of the actual diameter of the borehole to the diameter of the temporary plugging ball; ③ During the actual fracturing ball setting and borehole plugging construction, detect the temporary plugging balls that can effectively set the boreholes; ④ Calculate the actual diameter of the borehole according to the diameter of the effective temporary plugging ball; ⑤ Calculate the borehole abrasion rate according to the actual diameter of the borehole.

2. The prediction method for borehole abrasion and actual diameter in closely spaced multi-cluster temporary plugging fracturing according to claim 1, characterized in that, The DPM horizontal well single-cluster multi-perforation two-phase flow model is an incompressible fluid and steady-state solid-liquid two-phase flow model, including the continuity equation and the momentum conservation equation. The fluid turbulence equation adopts the standard k-ε model; in the particle force, the drag force, the Reynolds number and the added mass force are considered. The solid erosion effect is considered between the particles and the horizontal wellbore and the perforation holes, and the calculation model of Ahlert is used for calculation. ER = AF s v n1 f(α) Where: ER is the erosion rate, in mm / h; A is the correlation coefficient; n 1 is the particle impact velocity exponent; Fs is the roundness coefficient of the particle, indicating the sharpness of the particle surface. When the particle is an irregular sharp angle, Fs = 1.0; when the particle is ellipsoidal, Fs = 0.5; when the particle is spherical, Fs = 0.

2.

3. The prediction method for borehole abrasion and actual diameter in closely spaced multi-cluster temporary plugging fracturing according to claim 1, characterized in that, Adopt the method of combining the fluid dynamics software FLUENT and the discrete element method (DEM). Use FLUENT to calculate the flow process of the liquid in the horizontal wellbore, establish the fluid flow field, use DEM to obtain parameters such as the position, volume, and velocity of the particles, initialize the particle information in DDPM in FLUENT at the current step, activate the DDPM model in the Euler model, calculate the volume fraction, and exchange momentum through the built-in SCALARS to achieve solid-liquid two-way coupling calculation; the momentum exchange equation between the particles and the fluid in the calculation model is as follows: Where: F is the momentum transfer amount from the continuous phase to the discrete phase, (kg·m) / s; μ is the viscosity of the fluid, (N·S) / m2; ρp is the particle density kg·m 3 ; d p is the particle diameter, m; R e is the relative Reynolds number, dimensionless; u p and u are the particle velocity and the fluid velocity respectively, m / s; C D is the drag coefficient, dimensionless; m p * is the mass flow rate of the particles, kg / s; △t is the time step, s; F other Denotes other forces per unit mass, N / kg; The mass exchange equation from the discrete phase to the continuous phase is as follows: Where: M is the mass transfer amount from the discrete phase to the continuous phase, kg / s 2 ; △m p and m .p，0 respectively represent the change in the mass flow rate of the particles and the initial mass flow rate of particle injection, kg / s; m p,0 is the initial mass of the particles, kg; In the DDPM model and the DEM model, the movement trajectory of the discrete phase is predicted and calculated by balancing various forces acting on the particles in the Lagrangian reference system; using Newton's second law, the equation controlling the movement of the particles is as follows: Where: m is the mass of the particle 2, kg; x is the displacement of the particle moving from position 1 to position 2, s; t is the time taken to move from position 1 to position 2, s; V represents the vector velocity of the particle, m / s; F drag and F virtual _mass, F gravitation and F other respectively represent the vectors of the drag force, virtual mass force, gravity and other acting forces on the particle, N.

4. The prediction method for borehole abrasion and actual diameter in closely spaced multi-cluster temporary plugging fracturing according to claim 1, characterized in that, The gravity of the temporary plugging ball is expressed as: where: d m is the sphere diameter, ρ m is the sphere density, and g is the acceleration due to gravity; The buoyancy force received is: Where: ρ w is the density of the fracturing fluid; The fluid carrying force received by the temporary plugging ball is: R e = ρd m |u - u m | / μ where: u is the fluid velocity, u m is the migration velocity of the temporary plugging ball, μ is the fluid viscosity, R e is the Reynolds number, C d is the drag coefficient. When R e > 1000, C d = 0.

44. When R e ≤ 1000, (1 + 0.15R e 0.0687 ) / (R e / 24).

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