Hydrodynamic analysis method for Bingham fluid in eccentric annulus of shaft

Abstract

The invention discloses a hydrodynamic analysis method for Bingham fluid in a shaft eccentric annulus, and relates to the technical field of oil and gas wells. The method comprises the following steps: constructing an extended reach well model according to design parameters of an extended reach well, enabling Bingham fluid to flow in an eccentric annulus of the extended reach well model, determining a non-shear-plug position of the Bingham fluid, obtaining shear stress and velocity distribution of the Bingham fluid in the eccentric annulus, determining dimensionless volume flow of the Bingham fluid in the eccentric annulus, and determining a non-shear-plug position of the Bingham fluid in the eccentric annulus. And based on the friction resistance pressure drop gradient of the Bingham fluid in the concentric annulus in the extended reach well model, numerical simulation software is utilized to calculate the friction resistance pressure drop gradient ratio of the Bingham fluid, the friction resistance pressure drop gradient of the Bingham fluid in the eccentric annulus is determined, and hydrodynamic analysis of the Bingham fluid in the shaft eccentric annulus is completed. According to the method, eccentric annulus dynamics and hydraulics numerical analysis are combined, the Bingham fluid friction resistance pressure drop gradient ratio is accurately obtained, and a foundation is laid for guiding drilling operation of an extended reach well.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of oil and gas wells, and particularly relates to a method for fluid dynamics analysis of Bingham fluid in an eccentric annulus of a wellbore. BACKGROUND

[0002] In the process of extended reach well drilling operation, accurately predicting the fluid dynamics and hydraulic characteristics of drilling fluid in the annular space of the wellbore under the yield Bingham mode is the theoretical basis for accurately evaluating the real-time equivalent circulating density (ECD for short), and is crucial for the determination and effective management of wellbore pressure. Theoretical and experimental studies have shown that the pressure loss and fluid velocity distribution in the annular space of the wellbore are significantly affected by the fluid rheological properties, wellbore eccentricity and annular geometry.

[0003] The research on the eccentric annulus flow of non-Newtonian fluid has evolved from analytical exploration to multi-method cooperation. Early research mainly focused on Newtonian fluid, using the bipolar coordinate system analytical method, but it is difficult to extend to non-Newtonian fluid due to its complexity. After 1980, with the simplified model becoming the mainstream method, some scholars proposed using the slit model to equivalent the eccentric annulus to a variable height slit, and solving the power-law fluid flow by integrating the motion equation; another part of the scholars proposed the concentric annulus superposition method, which discretizes the eccentric annulus into infinite concentric ring elements, and obtains the analytical solution of power-law and Bingham fluid by combining geometric equivalence and rheological equation integration.

[0004] With the improvement of computing power, since 2000, the research on the eccentric annulus flow of non-Newtonian fluid has gradually adopted the analysis method combining numerical simulation and experiment. However, the existing theoretical model and analysis method mostly focus on the analysis of the dimensionless shear stress, shear rate and volume flow of the eccentric annulus relative to the concentric annulus, and a systematic method for fluid dynamics analysis of Bingham fluid in the eccentric annulus of the wellbore has not been established, especially the analysis method of the friction pressure drop gradient in the eccentric annulus. Therefore, it is urgent to propose a method for fluid dynamics analysis of Bingham fluid in the eccentric annulus of the wellbore, which can provide a basis for quantitatively obtaining the drilling hydraulic parameters of the extended reach well and guiding the drilling operation of the extended reach well. SUMMARY

[0005] The application aims to solve the problems of the prior art, and proposes a method for fluid dynamics analysis of Bingham fluid in the eccentric annulus of the wellbore, which combines the numerical analysis method of eccentric annulus dynamics and hydraulics, realizes the accurate acquisition of the friction pressure drop gradient of Bingham fluid in the eccentric annulus, and can quantitatively obtain the distribution of shear stress, shear rate and flow rate of Bingham fluid in the eccentric annulus, thereby providing a theoretical basis and technical support for guiding the drilling operation of the extended reach well.

[0006] The application specifically adopts the following technical scheme: A method for hydrodynamic analysis of a Bingham fluid in an eccentric annulus of a wellbore, comprising the following steps: Step 1, constructing a large displacement well model with an eccentric annulus inside according to design parameters of the large displacement well; Step 2, setting the drilling fluid in the large displacement well model as the Bingham fluid, and determining the shear-free plug position of the Bingham fluid in the large displacement well model; Step 3, obtaining the shear stress and velocity distribution of the Bingham fluid in the eccentric annulus; Step 4, determining the dimensionless volume flow rate of the Bingham fluid in the eccentric annulus; Step 5, based on the frictional pressure drop gradient of the Bingham fluid in the concentric annulus of the large displacement well model, using numerical simulation software to calculate the frictional pressure drop gradient ratio of the Bingham fluid, determine the frictional pressure drop gradient of the Bingham fluid in the eccentric annulus, and complete the hydrodynamic analysis of the Bingham fluid in the eccentric annulus of the wellbore.

[0007] Preferably, in step 1, a large displacement well model is constructed according to the design parameters of the large displacement well; The large displacement well model includes a drill string model and a wellbore model, the drill string model is eccentrically arranged in the wellbore model, and the annular space between the outer wall of the drill string model and the inner wall of the wellbore model is the eccentric annulus; A polar coordinate system is constructed in the large displacement well model, with the center position of the drill string model as the origin of the polar coordinate system , the positive direction of the polar axis in the polar coordinate system points to the widest direction of the eccentric annulus, and the polar angle corresponding to the positive direction of the polar axis is 0°, and the polar angle increases in the counterclockwise direction; The vertical distance between the center of the wellbore model and the center of the drill string model is the eccentric distance , the eccentricity of the drill string model is determined according to the eccentric distance : ; In the formula, is the outer diameter of the drill string model; is the inner diameter of the wellbore model; The polar coordinate equation of the wellbore model is: ; In the formula, is the polar radius, which represents the distance between the origin of the polar coordinate system and the inner wall of the wellbore model; The annular gap between the drill string model and the wellbore model is: ;​​ Where, is the annular gap, which is used to represent the distance between the outer wall of the drill string model and the inner wall of the wellbore model on the polar angle azimuth line.

[0008] Preferably, in step 2, the drilling fluid in the extended reach well model is set as a Bingham fluid, and the fluid parameters of the Bingham fluid are set according to the fluid properties of the Bingham fluid; The Bingham fluid flows in the eccentric annulus within the extended reach well model to form a shear-free plug flow region. The plug flow region is located in the middle region of the eccentric annulus. The width of the plug flow region is independent of the annular gap. The shear-free plug position of the Bingham fluid in the eccentric annulus is determined according to the width of the plug flow region. The calculation formula for the plug flow area width is: ; in, ; ; Where, is the width of the plug flow region; is the upper boundary of the plug flow region; is the lower boundary of the plug flow region; is the natural logarithm; is the dynamic shear force of Bingham fluid; is the eccentric annulus friction pressure drop gradient of Bingham fluid.

[0009] Preferably, in step 3, the flow domain of the Bingham fluid at any polar angle in the eccentric annulus is divided into three sub-regions, namely, the inner sub-region, the plug flow sub-region and the outer sub-region, wherein the flow domain range of the inner sub-region is , the drainage area of ​​the plug flow sub-basin is , the drainage area of ​​the outer subdomain is ,in, is the radial coordinate of the Bingham fluid in the eccentric annulus, which is used to indicate the radial position of the Bingham fluid in the eccentric annulus; In the inner subdomain, the flow rate of the Bingham fluid is With the polar diameter The outer diameter of the drill string model increases with the increase of The flow rate of Bingham fluid is 0; in the plug flow subdomain, the flow rate of the Bingham fluid is Constant, , is the maximum flow rate of the Bingham fluid; in the outer subdomain, the flow rate of the Bingham fluid Polar diameter As the diameter increases, it decreases. The flow rate of Bingham fluid is 0.

[0010] Preferably, the shear stress and flow rate of the Bingham fluid in the inner subdomain, plug subdomain and outer subdomain are different; The momentum conservation equation in the inner subdomain is: ; In the formula, is the shear stress of the Bingham fluid; The rheological model of the Bingham fluid in the inner subdomain is: ; In the formula, is the plastic viscosity of the Bingham fluid; The shear rate equation of the Bingham fluid in the inner subdomain is obtained by combining the momentum conservation equation in the inner subdomain with the rheological model of the Bingham fluid: ; The flow rate equation of the Bingham fluid in the inner subdomain is obtained by integrating the boundary conditions of the inner subdomain: ; In the plug subdomain, the shear rate equation of the Bingham fluid in the plug subdomain is determined by: at the lower boundary of the plug subdomain, and at the upper boundary of the plug subdomain; ; The momentum conservation equation in the outer subdomain is: ; The rheological model of the Bingham fluid in the outer subdomain is: ; The shear rate equation of the Bingham fluid in the outer subdomain is obtained by combining the momentum conservation equation in the outer subdomain with the rheological model of the Bingham fluid: ; The flow rate equation of the Bingham fluid in the outer subdomain is obtained by separating variables and integrating the boundary conditions of the outer subdomain: ; According to the flow rate equations of the Bingham fluid in the inner subdomain, plug subdomain and outer subdomain, the velocity distribution of the Bingham fluid in the eccentric annulus is determined.

[0011] Preferably, in step 4, based on the symmetry of the eccentric annulus, the eccentric annulus frictional pressure drop gradient when the Bingham fluid flows in the eccentric annulus is determined as: ; wherein, ; ; wherein, is the eccentric annulus friction pressure drop gradient of the Bingham fluid; is the dimensionless discharge of the Bingham fluid in the eccentric annulus; is the volume flow rate of the Bingham fluid in the eccentric annulus; the dimensionless discharge of the Bingham fluid in the eccentric annulus is simplified by using numerical integration method the integral is calculated in the integral interval [0, ], and the following is obtained: ; an intermediate quantity is set to simplify the expression of the dimensionless discharge of the Bingham fluid in the eccentric annulus, and the expression of the dimensionless discharge of the Bingham fluid in the eccentric annulus is simplified as: ; wherein, ; wherein, is the intermediate quantity; the composite Simpson rule is used for numerical integration, the integral interval is divided into multiple sub-intervals according to the preset angle, and then the calculation formula of the intermediate quantity is obtained according to the Simpson formula expansion as follows: ; wherein, ; wherein, is the sub-interval serial number, is the total number of sub-intervals; is the weight coefficient of the i-th sub-interval; is the integral function; the integral function values at each angle are weighted and summed according to the weight coefficient, multiplied by the coefficient , and then substituted into the volume flow rate calculation formula of the Bingham fluid in the eccentric annulus to calculate the dimensionless discharge of the Bingham fluid in the eccentric annulus.

[0012] Preferably, in step 5, when the eccentricity of the drill string model in the large displacement well model is 0, the center of the drill string model coincides with the center of the wellbore model, and the annular space between the outer wall of the drill string model and the inner wall of the wellbore model changes into a concentric annulus; when the Bingham fluid flows in the concentric annulus, the concentric annulus friction pressure drop gradient of the Bingham fluid is expressed as:​ ; wherein, is the concentric annulus frictional pressure drop gradient of the Bingham fluid; is the yield stress of the Bingham fluid; is the average flow velocity in the annulus; Combining the dimensionless discharge of the Bingham fluid in the concentric annulus, the simplified expression of the concentric annulus frictional pressure drop gradient is: ; wherein, ; ; ; wherein, is the dimensionless discharge of the Bingham fluid in the concentric annulus; is the inner boundary radial coordinate of the plug flow region in the concentric annulus; is the right boundary radial coordinate of the plug flow region in the concentric annulus; The ratio of the eccentric annulus frictional pressure drop gradient of the Bingham fluid to the concentric annulus frictional pressure drop gradient of the Bingham fluid is calculated under the same discharge, and the Bingham fluid frictional pressure drop gradient ratio is determined as: ; wherein, is the Bingham fluid frictional pressure drop gradient ratio; Combining the simplified expression of the dimensionless discharge of the Bingham fluid in the eccentric annulus and the simplified expression of the concentric annulus frictional pressure drop gradient, we get: ; Substituting the calculation formula of the dimensionless discharge of the Bingham fluid in the eccentric annulus , we get the implicit function of the Bingham fluid frictional pressure drop gradient ratio as: ; Since the width of the plug flow region in the middle and the upper boundary are variables, and the width of the plug flow region in the middle and the upper boundary are functions of the eccentric annulus frictional pressure drop gradient , we get the width of the plug flow region in the middle and the upper boundary about the Bingham fluid frictional pressure drop gradient ratio : ; ; ​The implicit function of the Bingham fluid friction pressure drop gradient ratio is solved iteratively to construct a Bingham fluid eccentric annulus friction pressure drop gradient ratio calculation equation, and the following is obtained: ; In the formula, is a calculation function of the eccentric annulus friction pressure drop gradient ratio; The Bingham fluid friction pressure drop gradient ratio is the root of the Bingham fluid eccentric annulus friction pressure drop gradient ratio calculation equation, and the following is obtained by using the Newton-Raphson method through Taylor expansion linear approximation: ; In the formula, is the number of iterations; is the derivative of the eccentric annulus friction pressure drop gradient ratio in the th iteration calculation; The derivative of the eccentric annulus friction pressure drop gradient calculation function is solved by using a numerical differentiation method, and the following is obtained: ; In the formula, is the difference between the Bingham fluid friction pressure drop gradient ratios before and after updating.

[0013] Preferably, in step 5, the Bingham fluid friction pressure drop gradient ratio is calculated by using numerical simulation software, which specifically includes the following steps: Step 5.1, input calculation parameters; The calculation parameters include the outer diameter of the drill string model , the inner diameter of the wellbore model , the eccentricity of the drill string model , the volume flow rate of the Bingham fluid in the eccentric annulus , the dynamic shear force , and the plastic viscosity ; Step 5.2, calculate the concentric annulus friction pressure drop gradient of the Bingham fluid ; Step 5.3, calculate the inner boundary radial coordinate and the right boundary radial coordinate of the plug flow region in the concentric annulus; Step 5.4, according to the outer diameter of the drill string model , the inner diameter of the wellbore model , and the inner boundary radial coordinate and the right boundary radial coordinate of the plug flow region in the concentric annulus, calculate the dimensionless discharge of the Bingham fluid in the concentric annulus ; Step 5.5, set the initial value of the friction pressure drop gradient ratio and the convergence threshold ; Step 5.6, iteratively update the calculation of the friction pressure drop gradient ratio ; Calculate the width of the plug flow region , and according to the polar coordinate equation of the wellbore model and the upper boundary of the plug flow region , obtain the polar radius of the plug flow region in each preset direction , and the upper boundary of the plug flow region , determine the integral function in each preset direction , and according to the integral function in each preset direction, perform numerical integration to determine the intermediate quantity , calculate the calculation value and derivative value of the eccentric annulus friction pressure drop gradient ratio of the Bingham fluid, and according to the calculation equation of the eccentric annulus friction pressure drop gradient ratio of the Bingham fluid, obtain the calculation value of the eccentric annulus friction pressure drop gradient ratio of the Bingham fluid, update the friction pressure drop gradient ratio of the Bingham fluid; Step 5.7, obtain the updated friction pressure drop gradient ratio of the Bingham fluid , calculate the difference between the friction pressure drop gradient ratio of the Bingham fluid before and after updating ; Step 5.8, perform convergence check, compare the difference between the friction pressure drop gradient ratio of the Bingham fluid before and after updating with the preset convergence threshold , if the difference between the friction pressure drop gradient ratio of the Bingham fluid before and after updating exceeds the preset convergence threshold , set the updated friction pressure drop gradient ratio of the Bingham fluid as the initial value of the friction pressure drop gradient ratio of the Bingham fluid, return to step 5.6 and continue to perform iterative updating, otherwise, end the iteration; Step 5.9, output the friction pressure drop gradient ratio of the Bingham fluid.

[0014] The beneficial technical effects brought by the present application are: The present application provides a fluid dynamics analysis method for Bingham fluid in eccentric annulus of wellbore, constructs a large displacement well model based on the design parameters of the large displacement well, determines the shear-free plug position, shear stress and velocity distribution of the Bingham fluid in the eccentric annulus by using the large displacement well model, obtains the dimensionless volume flow rate of the Bingham fluid in the eccentric annulus, and accurately calculates the friction pressure drop gradient ratio of the Bingham fluid based on the friction pressure drop gradient of the Bingham fluid in the concentric annulus of the large displacement well model by using numerical simulation software, determines the friction pressure drop gradient of the Bingham fluid in the eccentric annulus, provides technical support for accurately obtaining the distribution of shear stress, shear rate and flow rate of the Bingham fluid in the eccentric annulus, and lays a foundation for fluid dynamics analysis of the Bingham fluid in the eccentric annulus of the wellbore.

[0015] The present invention proposes a fluid dynamics analysis method for Bingham fluid in the eccentric annulus of the wellbore, which makes up for the current lack of a systematic method for analyzing the flow properties of Bingham fluid in the eccentric annulus. The method of the present invention is based on the superposition thinking of variable-diameter concentric annuli, combines the eccentric annulus dynamics analysis with hydraulic numerical simulation, and realizes the quantitative understanding of the hydraulic engineering parameters of extended reach well drilling, providing a basis for guiding the drilling operations of extended reach wells. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 The present invention is a flow chart of a fluid dynamics analysis method for Bingham fluid in an eccentric annulus of a wellbore.

[0017] Figure 2 Schematic diagram of the extended reach well model.

[0018] Figure 3 Annular gap Polar angle 's change curve.

[0019] Figure 4 This is the calculation flow chart of the friction pressure drop gradient ratio of Bingham fluid.

[0020] In the figure, is the origin of the polar coordinate system, is the polar angle, is the outer diameter of the drill string model, is the inner diameter of the wellbore model, is the annular gap, is the eccentricity, is the upper boundary of the plug flow region, is the lower boundary of the plug flow region, is the width of the plug flow region. DETAILED DESCRIPTION

[0021] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0022] Example 1 The present invention proposes a fluid dynamics analysis method for Bingham fluid in an eccentric annulus of a wellbore, such as Figure 1 As shown, the specific steps include: Step 1: Based on the design parameters of the extended reach well, a model of an extended reach well with an eccentric annulus is constructed.

[0023] In step 1, an extended reach well model is constructed according to the design parameters of the extended reach well. The extended reach well model includes a drill string model and a wellbore model. Figure 2 As shown, the drill string model is eccentrically arranged in the wellbore model, and the annular space between the outer wall of the drill string model and the inner wall of the wellbore model is the eccentric annulus.

[0024] A polar coordinate system is constructed in the extended reach well model, with the center of the drill string model as the origin of the polar coordinate system. In the polar coordinate system, the positive direction of the polar axis points to the widest direction of the eccentric annulus, and the polar angle corresponding to the positive direction of the polar axis is is 0°, polar angle Increases in a counterclockwise direction.

[0025] The vertical distance between the center of the wellbore model and the center of the drill string model is the eccentricity , according to the eccentricity Determine the eccentricity of the drill string model for: ; Where, is the outer diameter of the drill string model; is the inner diameter of the wellbore model.

[0026] The polar coordinate equation of the wellbore model is: ; Where, is the polar diameter, used to represent the origin of the polar coordinate system The distance from the inner wall of the wellbore model.

[0027] The annular gap between the drill string model and the wellbore model for: ; Where, The annular gap is used to represent the distance between the outer wall of the drill string model and the inner wall of the wellbore model on the polar azimuth line. In this embodiment, the annular gap is Polar angle The change curve of Figure 3 shown.

[0028] Step 2: Set the drilling fluid in the extended reach well model as Bingham fluid, and determine the shear plug position of the Bingham fluid in the extended reach well model.

[0029] In this embodiment, the drilling fluid in the extended reach well model is set as Bingham fluid, and the fluid parameters of the Bingham fluid are set according to the fluid properties of the Bingham fluid.

[0030] The Bingham fluid flows in the eccentric annulus in the extended reach well model to form a plug flow region without shear. In this region, the flow velocity of the Bingham fluid is the same at all locations and the shear rate is 0. The upper boundary of the plug flow region is , the lower boundary is , width is , The plug flow region is located in the middle area of ​​the eccentric annulus. The width of the plug flow region is independent of the annular gap. The shear-free plug position of the Bingham fluid in the eccentric annulus is determined according to the width of the plug flow region.

[0031] The calculation formula for the plug flow area width is: ; in, ; ; Where, is the width of the plug flow region, in cm; is the upper boundary of the plug flow region, specifically the distance between the top of the plug flow region and the origin The distance between the horizontal planes; is the lower boundary of the plug flow region, specifically the distance between the bottom of the plug flow region and the origin The distance between the horizontal planes; is the natural logarithm; is the dynamic shear force of Bingham fluid, unit is ; is the eccentric annular friction pressure drop gradient of Bingham fluid, in units of .

[0032] When the eccentricity of the drill string model When it is 0, that is, the center of the drill string model coincides with the center of the wellbore model, and the annular space between the outer wall of the drill string model and the inner wall of the wellbore model changes to a concentric annulus. The calculation formula for the shear-free plug characteristic parameter of the plug flow region in the concentric annulus is the same as the calculation formula for the shear-free plug characteristic parameter of the plug flow region in the eccentric annulus.

[0033] Step 3: Obtain the shear stress and velocity distribution of the Bingham fluid in the eccentric annulus.

[0034] In this embodiment, the flow domain of the Bingham fluid at any polar angle in the eccentric annulus is divided into three sub-regions, namely the inner sub-region, the plug flow sub-region and the outer sub-region. The flow domain range of the inner sub-region is , the drainage area of ​​the plug flow sub-basin is , the drainage area of ​​the outer subdomain is ,in, is the radial coordinate of the Bingham fluid in the eccentric annulus, which is used to indicate the radial position of the Bingham fluid in the eccentric annulus.

[0035] In the inner subdomain, the flow rate of the Bingham fluid is With the polar diameter The outer diameter of the drill string model increases with the increase of The flow rate of Bingham fluid is 0; in the plug flow subdomain, the flow velocity of the Bingham fluid is constant, , is the maximum flow velocity of the Bingham fluid; in the outer subdomain, the flow velocity of the Bingham fluid decreases with the increase of the polar radius, and the flow velocity of the Bingham fluid at the polar radius is 0. The shear stress and the flow velocity of the Bingham fluid in the inner subdomain, the plug flow subdomain and the outer subdomain are different, that is, the shear rate and the shear stress of the Bingham fluid in the inner subdomain, the plug flow subdomain and the outer subdomain present different characteristics.

[0036] The momentum conservation equation in the inner subdomain is:

[0037] ; ; In the formula, is the shear stress of the Bingham fluid.

[0038] The rheological model of the Bingham fluid in the inner subdomain is: ; In the formula, is the plastic viscosity of the Bingham fluid, and the unit is .

[0039] The shear rate equation of the Bingham fluid in the inner subdomain is obtained by combining the momentum conservation equation in the inner subdomain with the rheological model of the Bingham fluid: ; The flow velocity equation of the Bingham fluid in the inner subdomain is obtained by integrating the boundary conditions of the inner subdomain: .

[0040] In the plug flow subdomain, the plug flow subdomain lower boundary is , the plug flow subdomain upper boundary is , that is, in the plug flow subdomain, , , , , so as to determine the shear rate equation of the Bingham fluid in the plug flow subdomain: ; The momentum conservation equation in the outer subdomain is: .

[0041] The rheological model of the Bingham fluid in the outer subdomain is: . ​

[0042] The momentum conservation equation in the outer subdomain is combined with the Bingham fluid rheological model to obtain the shear rate equation of the Bingham fluid in the outer subdomain as follows: .

[0043] The boundary conditions of the outer subdomain are combined to separate variables and integrate to obtain the flow velocity equation of the Bingham fluid in the outer subdomain as follows: .

[0044] According to the flow velocity equations of the Bingham fluid in the inner subdomain, the plug flow subdomain and the outer subdomain, the velocity distribution of the Bingham fluid in the eccentric annulus is determined.

[0045] Step 4, determining the dimensionless volume flow rate of the Bingham fluid in the eccentric annulus.

[0046] In step 4, based on the symmetry of the eccentric annulus, the eccentric annulus frictional pressure drop gradient of the Bingham fluid when the Bingham fluid flows in the eccentric annulus is determined by imitating the dimensionless volume flow rate expression of the Bingham fluid when the Bingham fluid flows in the concentric annulus as follows: ; wherein, ; ; In the formula, is the eccentric annulus frictional pressure drop gradient of the Bingham fluid; is the dimensionless displacement of the Bingham fluid in the eccentric annulus; is the volume flow rate of the Bingham fluid in the eccentric annulus.

[0047] Since the integral interval of the dimensionless volume flow rate expression of the Bingham fluid in the eccentric annulus is [0, ], the functions and in the integral function depend on the polar angle , contain multiple nonlinear terms and power function terms, and it is difficult to obtain a rigorous analytical solution, so the numerical integral method is used to perform integral calculation on the dimensionless volume flow rate of the Bingham fluid in the eccentric annulus.

[0048] The numerical integral method is used to perform integral calculation on the dimensionless displacement of the Bingham fluid in the eccentric annulus to obtain: .

[0049] The expression of the dimensionless displacement of the Bingham fluid in the eccentric annulus is simplified by setting an intermediate variable, and the dimensionless displacement expression of the Bingham fluid in the eccentric annulus is simplified as follows: ; wherein, ; In the formula, is an intermediate quantity.

[0050] That is, the intermediate quantity is decomposed , , and are four parts in total.

[0051] The composite Simpson rule is used for numerical integration. According to a preset angle, the number of segments is set, the integral interval is divided into a plurality of sub-intervals, and then according to the Simpson formula, the calculation formula of the intermediate quantity is obtained as follows: ; Among them, ; In the formula, is a sub-interval number, is a total number of sub-intervals; is a weight coefficient of the th sub-interval; is an integrand function, and the independent variable of the integrand function is the polar angle , that is, the integrand function in and are functions of the polar angle , that is, according to the polar angle , the and in the integrand function are determined.

[0052] In this embodiment, the composite Simpson rule is used for numerical integration. According to a preset angle, the number of segments is set to 3, the integral interval is divided into 6 equal-length sub-intervals, and an engineering-acceptable calculation result can be obtained. According to the Simpson formula, the calculation formula of the intermediate quantity is obtained as follows: ; In the formula, , , , , , and are weight coefficients, and the values of the weight coefficients of each sub-interval are set according to the polar angle . When , the weight coefficient takes a value of 1, and when ​the weight coefficient takes the value 4 when the weight coefficient takes the value 2 when the weight coefficient takes the value 4 when the weight coefficient takes the value 2 when the weight coefficient takes the value when the weight coefficient takes the value .

[0053] According to the integral function value at each angle and the weighted sum according to the weight coefficient, multiply the coefficient , and substitute into the simplified expression of the dimensionless displacement of the Bingham fluid in the eccentric annulus, to calculate the dimensionless displacement of the Bingham fluid in the eccentric annulus .

[0054] Step 5, based on the frictional pressure drop gradient of the Bingham fluid in the concentric annulus in the extended reach well model, using numerical simulation software, calculate the frictional pressure drop gradient ratio of the Bingham fluid, determine the frictional pressure drop gradient of the Bingham fluid in the eccentric annulus, and complete the fluid dynamics analysis of the Bingham fluid in the eccentric annulus of the wellbore.

[0055] Since the frictional pressure drop gradient of the Bingham fluid in the eccentric annulus is a function of the dimensionless displacement , and the width of the plug flow region in the dimensionless displacement and the upper boundary depend on the eccentric annulus frictional pressure drop gradient of the Bingham fluid, that is, the frictional pressure drop gradient expression of the Bingham fluid in the eccentric annulus and the dimensionless displacement expression can form a closed equation set, but it is difficult to determine the explicit analytical expression of the Bingham fluid frictional pressure drop gradient. Therefore, in this embodiment, based on the frictional pressure drop gradient ratio of the Bingham fluid in the concentric annulus in the extended reach well model, the expression of the frictional pressure drop gradient of the Bingham fluid in the eccentric annulus is determined.

[0056] In the extended reach well model, when the eccentricity of the drill string model is 0, at this time the center of the drill string model coincides with the center of the wellbore model, the annular space between the outer wall of the drill string model and the inner wall of the wellbore model changes to a concentric annulus, and when the Bingham fluid flows in the concentric annulus, the concentric annulus frictional pressure drop gradient expression of the Bingham fluid is: ; In the formula, is the concentric annular friction pressure drop gradient of Bingham fluid; is the dynamic shear force of the Bingham fluid; is the average annular velocity in m / s.

[0057] Combined with the dimensionless displacement of Bingham fluid in the concentric annulus, the simplified expression of the friction pressure drop gradient in the concentric annulus is: ; in, ; ; ; Where, is the dimensionless displacement of the Bingham fluid in the concentric annulus; is the radial coordinate of the inner boundary of the plug flow region in the concentric annulus; is the radial coordinate of the right boundary of the plug flow region in the concentric annulus.

[0058] Calculate the ratio of the eccentric annular friction pressure drop gradient to the concentric annular friction pressure drop gradient of the Bingham fluid at the same displacement, and determine the Bingham fluid friction pressure drop gradient ratio as: ; Where, is the friction pressure drop gradient ratio of the Bingham fluid.

[0059] Therefore, the calculation problem of the friction pressure drop gradient of the eccentric annulus of the Bingham fluid is transformed into solving the ratio of the friction pressure drop gradient of the eccentric annulus of the Bingham fluid to the friction pressure drop gradient of the concentric annulus under the same displacement. Combining the simplified expression of the dimensionless displacement of the Bingham fluid in the eccentric annulus and the simplified expression of the friction pressure drop gradient of the concentric annulus, we obtain: .

[0060] Substitute the dimensionless displacement of Bingham fluid in the eccentric annulus into The calculation formula of Bingham fluid friction pressure drop gradient ratio is obtained The implicit function of is: .

[0061] Due to the intermediate Width of the plug flow region and upper boundary are all variables, and the width of the plug flow region is and upper boundary All are eccentric annulus friction pressure drop gradient The width of the plug flow region is obtained as a function of and upper boundary On the friction pressure drop gradient ratio of Bingham fluid The expression is: ; .

[0062] The implicit function of the Bingham fluid friction pressure drop gradient ratio is iteratively solved, the Bingham fluid eccentric annulus friction pressure drop gradient ratio calculation equation is constructed, and the following is obtained: ; In the formula, is the calculation function of the eccentric annulus friction pressure drop gradient ratio.

[0063] The Bingham fluid friction pressure drop gradient ratio is the root of the Bingham fluid eccentric annulus friction pressure drop gradient ratio calculation equation, and the Newton-Raphson method is used to obtain the linear approximation through Taylor expansion: ; In the formula, is the number of iterations; is the derivative of the eccentric annulus friction pressure drop gradient ratio in the th iteration calculation.

[0064] Since the intermediate quantity is a function of the Bingham fluid friction pressure drop gradient ratio , the explicit expression of cannot be obtained, so the numerical differentiation method is used to solve the derivative of the eccentric annulus friction pressure drop gradient calculation function, and the following is obtained: ; In the formula, is the difference between the Bingham fluid friction pressure drop gradient ratios before and after updating.

[0065] In this embodiment, the numerical simulation software is used to calculate the Bingham fluid friction pressure drop gradient ratio , as shown in Figure 4 , which specifically includes the following steps: Step 5.1, input the calculation parameters, including the outer diameter of the drill string model , the inner diameter of the wellbore model , the eccentricity of the drill string model , the volume flow rate of the Bingham fluid in the eccentric annulus , the dynamic shear force , and the plastic viscosity .

[0066] Step 5.2, calculate the concentric annulus friction pressure drop gradient of the Bingham fluid .

[0067] Specifically, the concentric annulus friction pressure drop gradient of the Bingham fluid The calculation formula is: .

[0068] Step 5.3, calculating the inner boundary radial coordinate and the right boundary radial coordinate of the plug flow region in the concentric annulus . .

[0069] Specifically, the inner boundary radial coordinate of the plug flow region in the concentric annulus is calculated as: The calculation formula is: ; The right boundary radial coordinate of the plug flow region in the concentric annulus is calculated as: The calculation formula is: .

[0070] Step 5.4, calculating the dimensionless discharge of the Bingham fluid in the concentric annulus according to the outer diameter of the drill string model , the inner diameter of the wellbore model , the inner boundary radial coordinate of the plug flow region in the concentric annulus , and the right boundary radial coordinate of the plug flow region in the concentric annulus . .

[0071] Specifically, the dimensionless discharge of the Bingham fluid in the concentric annulus is calculated as: The calculation formula is: .

[0072] Step 5.5, setting the initial value of the friction pressure drop gradient ratio to 1, and setting the convergence threshold to .

[0073] Step 5.6, iteratively updating the calculation of the friction pressure drop gradient ratio , and the specific process is as follows: Calculating the width of the plug flow region , and the calculation formula is: .

[0074] According to the polar coordinate equation of the wellbore model and the upper boundary of the plug flow region , the polar radius and the upper boundary of the plug flow region at each preset orientation are obtained, the integrand at each preset orientation is determined , the intermediate quantity is calculated by numerical integration according to the integrand at each preset orientation .

[0075] Specifically, the polar coordinate equation of the wellbore model is: ; The upper boundary calculation formula of the plug flow region is: ; The integrand calculation formula is: ; The intermediate amount The calculation formula is: .

[0076] According to the intermediate amount Calculate the calculated value and derivative value of the current Bingham fluid eccentric annulus friction pressure drop gradient ratio.

[0077] In the current iterative calculation, according to the calculation equation of the Bingham fluid eccentric annulus friction pressure drop gradient ratio, the calculated value of the Bingham fluid eccentric annulus friction pressure drop gradient ratio is obtained, and the Bingham fluid eccentric annulus friction pressure drop gradient ratio is updated.

[0078] Specifically, the calculation equation for the Bingham fluid eccentric annulus friction pressure drop gradient ratio is: ; The derivative value of the Bingham fluid eccentric annulus friction pressure drop gradient ratio is: .

[0079] Step 5.7, update the Bingham fluid friction pressure drop gradient ratio , and obtain the updated Bingham fluid friction pressure drop gradient ratio , calculate the Bingham fluid friction pressure drop gradient ratio before and after the update The difference.

[0080] Step 5.8, perform convergence check and update the difference of Bingham fluid friction pressure drop gradient ratio before and after and the preset convergence threshold For comparison, if the Bingham fluid friction pressure drop gradient ratio before and after the update is The difference Exceeding the preset convergence threshold , then set the updated value as the initial value of the Bingham fluid friction pressure drop gradient ratio, return to step 5.7, and continue iterative updating. Otherwise, end the iteration.

[0081] Step 5.9: Output the friction pressure drop gradient ratio of the Bingham fluid.

[0082] Example 2 This embodiment takes a certain extended reach well as an example. The outer diameter of the drill string of the extended reach well is 0.0635m, the inner diameter of the wellbore is 0.108m, and the eccentricity of the drill string in the wellbore is The Bingham fluid friction pressure drop gradient ratio is determined by using the fluid dynamics analysis method of the Bingham fluid in the eccentric annulus of the wellbore as described in Example 1. , a fluid dynamics analysis of Bingham fluid in the eccentric annulus of the wellbore was conducted.

[0083] In the extended reach well model constructed in this embodiment, the outer diameter of the drill string model is The inner diameter of the wellbore model is 0.0635m. The Bingham fluid flows in the eccentric annulus between the drill string model and the wellbore model. The eccentricity of the drill string model is 0.02 m. The volume flow rate of the Bingham fluid in the eccentric annulus is 0.03 , dynamic shear force is 10 , plastic viscosity 0.02 .

[0084] The eccentric annulus friction pressure drop gradient of the Bingham fluid in the extended reach well model is calculated as follows: The average annular flow velocity for: ; The concentric annular friction pressure drop gradient of the Bingham fluid for: ; The radial coordinates of the inner boundary of the plug flow region in the concentric annulus are: ; The radial coordinates of the right boundary of the plug flow region in the concentric annulus are: ; The dimensionless displacement of the Bingham fluid in the concentric annulus is: ;

[0085] After 4 iterative calculations, the Bingham fluid friction pressure drop gradient ratio is obtained. The converged solution is 0.7927. The key parameters in each iterative calculation process are shown in Table 1.

[0086] Table 1 Key parameters of iterative calculation process ;

[0087] Determining the eccentric annulus friction pressure drop gradient of Bingham fluid for: .

[0088] calculate The specific process of shear stress in the azimuth is: ; ; ; .

[0089] The scope of the inner subdomain is , the range of the plug flow subdomain is , the scope of the outer subdomain is .

[0090] Furthermore, the shear stress in the inner subdomain is: ; The shear stress in the plug flow subdomain is: Department, ; Department, ; The shear stress in the outer subdomain is: .

[0091] calculate The shear rate in the direction is: The shear rate of the Bingham fluid in the inner subdomain is: ; Right now ; The shear rate of the Bingham fluid in the outer subdomain is: ; Right now .

[0092] calculate The speed in azimuth is as follows: The flow rate of the Bingham fluid in the inner subdomain is: ; Right now ; The flow rate of the Bingham fluid in the outer subdomain is: ; Right now .

[0093] calculate The shear stress, shear rate and flow rate distribution in the azimuthal direction, the specific process is: Selecting different radial positions, the shear stress, shear rate and flow rate distribution data at each position are calculated, as shown in Table 2.

[0094] Table 2 Shear stress, shear rate and flow rate distribution in the wellbore The shear stress, shear rate and flow rate distribution in the azimuthal direction ;

[0095] According to the shear stress, shear rate and flow rate distribution data at each position in the radial direction of the wellbore, the distribution of the shear stress, shear rate and flow rate in the wellbore is obtained, and the hydrodynamic analysis of the Bingham fluid in the eccentric annulus of the wellbore is completed, thereby providing a theoretical basis and technical support for guiding the drilling operation of the extended reach well.

[0096] Of course, the above description is not a limitation on the present application, and the present application is not limited to the above examples, and the changes, modifications, additions or replacements made by the person skilled in the art within the essential scope of the present application shall also belong to the protection scope of the present application.

Claims

1. A method of fluid dynamic analysis of a pinger fluid in a wellbore eccentric annulus, characterized by, The method comprises the following steps: Step 1, constructing a large displacement well model with an eccentric annulus according to the design parameters of the large displacement well; Step 2, setting the drilling fluid in the large displacement well model as Bingham fluid, and determining the position of the Bingham fluid without shear in the large displacement well model; Step 3, obtaining the shear stress and velocity distribution of the Bingham fluid in the eccentric annulus; Step 4, determining the dimensionless volume flow rate of the Bingham fluid in the eccentric annulus; Step 5, based on the friction pressure drop gradient of the Bingham fluid in the concentric annulus of the large displacement well model, using numerical simulation software to calculate the friction pressure drop gradient ratio of the Bingham fluid, determine the friction pressure drop gradient of the Bingham fluid in the eccentric annulus, and complete the fluid dynamics analysis of the Bingham fluid in the eccentric annulus of the wellbore.

2. The method for the hydrodynamic analysis of a Bingham fluid in a wellbore eccentric annulus of claim 1, wherein, In the step 1, the large displacement well model is constructed according to the design parameters of the large displacement well; The large displacement well model comprises a drill string model and a wellbore model, the drill string model is eccentrically arranged in the wellbore model, and the annular space between the outer wall of the drill string model and the inner wall of the wellbore model is the eccentric annulus; A polar coordinate system is constructed in the large displacement well model, with the center position of the drill string model as the origin of the polar coordinate system , the positive direction of the polar axis in the polar coordinate system points to the widest direction of the eccentric annulus, the polar angle corresponding to the positive direction of the polar axis is 0°, and the polar angle increases in the counterclockwise direction; The vertical distance between the wellbore model center and the drill string model center is the eccentricity The eccentricity of the drill string model is determined from the eccentricity :​ ； wherein is the outer diameter of the drill string model; is the inner diameter of the wellbore model; The polar coordinate equation of the wellbore model is: ； wherein r is the polar radius, used to indicate the origin of the polar coordinate system distance between the inner walls of the wellbore model an annular gap between the drill string model and the wellbore model is: ； In the formula, is the annular space for indicating the distance between the outer wall of the drill string model and the inner wall of the wellbore model on the polar azimuth line.

3. The method for the hydrodynamic analysis of a Bingham fluid in a wellbore eccentric annulus of claim 2 wherein, In the step 2, the drilling fluid in the large displacement well model is set as Bingham fluid, and the fluid parameters of the Bingham fluid are set according to the fluid properties of the Bingham fluid; The Bingham fluid flows in the eccentric annulus of the large displacement well model to form a shear-free plug flow region, the plug flow region is located in the middle region of the eccentric annulus, the width of the plug flow region is independent of the annular gap, and the position of the Bingham fluid without shear in the eccentric annulus is determined according to the width of the plug flow region; The calculation formula of the plug flow region width is: ； Wherein, ； ； wherein is the width of the plug region; is the upper boundary of the plug region; is the lower boundary of the plug region; is the natural logarithm; is the dynamic shear stress of the Bingham fluid; is the eccentric annular friction pressure drop gradient of the Bingham fluid.

4. The method for the hydrodynamic analysis of a Bingham fluid in a wellbore eccentric annulus of claim 3 wherein, In step 3, the flow field of the Bingham fluid at any polar angle position in the eccentric annulus is divided into three sub-regions, namely an inner sub-region, a plug flow sub-region and an outer sub-region, wherein the flow field range of the inner sub-region is , the flow field range of the plug flow sub-region is , and the flow field range of the outer sub-region is , wherein is the radial coordinate of the Bingham fluid in the eccentric annulus, used to represent the radial position of the Bingham fluid in the eccentric annulus; In the inner subdomain, the flow rate of the Bingham fluid is With the polar diameter The outer diameter of the drill string model increases with the increase of The flow rate of Bingham fluid is 0; in the plug flow subdomain, the flow rate of the Bingham fluid is Constant, , is the maximum flow rate of the Bingham fluid; in the outer subdomain, the flow rate of the Bingham fluid Polar diameter As the diameter increases, it decreases. The flow rate of Bingham fluid is 0.

5. The method for the hydrodynamic analysis of a Bingham fluid in a wellbore eccentric annulus of claim 4 wherein, The shear stress and flow rate of the Bingham fluid in the inner sub-domain, plug flow sub-domain and outer sub-domain are different; The momentum conservation equation in the inner sub-domain is: ； wherein is the shear stress of the Bingham fluid; The rheological model of the Bingham fluid in the inner sub-domain is: ； wherein is the plastic viscosity of the Bingham fluid; The shear rate equation of the Bingham fluid in the inner sub-domain is obtained by combining the momentum conservation equation in the inner sub-domain with the rheological model of the Bingham fluid: ； The flow rate equation of the Bingham fluid in the inner sub-domain is obtained by integrating the boundary conditions of the inner sub-domain: ； In the plug subdomain, at the plug subdomain lower boundary , at the plug subdomain upper boundary , the shear rate equation of the Bingham fluid in the plug subdomain is determined as ； The momentum conservation equation in the outer sub-domain is: ； The rheological model of the Bingham fluid in the outer sub-domain is: ； The shear rate equation of the Bingham fluid in the outer sub-domain is obtained by combining the momentum conservation equation in the outer sub-domain with the rheological model of the Bingham fluid: ； The flow rate equation of the Bingham fluid in the outer sub-domain is obtained by separating variables and integrating the boundary conditions of the outer sub-domain: ； The velocity distribution of the Bingham fluid in the eccentric annulus is determined according to the flow rate equation of the Bingham fluid in the inner sub-domain, plug flow sub-domain and outer sub-domain.

6. The method for the hydrodynamic analysis of a Bingham fluid in a wellbore eccentric annulus of claim 5 wherein, In the step 4, based on the symmetry of the eccentric annulus, the eccentric annulus friction pressure drop gradient when the Bingham fluid flows in the eccentric annulus is determined as: ； Wherein, ； ； wherein is the eccentric annulus friction pressure drop gradient for a Bingham fluid; is the dimensionless displacement of a Bingham fluid in an eccentric annulus; is the volumetric flow rate of a Bingham fluid in an eccentric annulus; The dimensionless displacement of Bingham fluid in eccentric annulus is calculated by numerical integration method In the integral interval [0, ] integral calculation, get: ； The expression of the dimensionless discharge of the Bingham fluid in the eccentric annulus is simplified by setting the intermediate quantity, and the expression of the dimensionless discharge of the Bingham fluid in the eccentric annulus is simplified as: ； Wherein, ； In the formulae, is an intermediate quantity; The numerical integration is performed by using the composite Simpson rule, a preset angle is set to set the number of segments, and the integral interval is divided into multiple sub-intervals According to the Simpson formula, the intermediate quantity The calculation formula is: ； Wherein, ； In the formula, is a sub-interval number, is a total number of sub-intervals; is a weight coefficient of the th sub-interval; is an integrand function; According to the integrand value at each angle and the weighted sum according to the weight coefficient, multiply by the coefficient The volume flow rate of Bingham fluid in eccentric annulus is calculated by inserting it into the calculation formula of volume flow rate of Bingham fluid in eccentric annulus to obtain the dimensionless displacement of Bingham fluid in eccentric annulus .

7. The method for the hydrodynamic analysis of a Bingham fluid in a wellbore eccentric annulus of claim 6 wherein, In step 5, when the eccentricity of the drill string model in the extended reach well model is 0, at this time the center of the drill string model coincides with the center of the wellbore model, and the annular space between the outer wall of the drill string model and the inner wall of the wellbore model changes to a concentric annulus. In step 5, when the eccentricity of the drill string model in the extended reach well model is 0, at this time the center of the drill string model coincides with the center of the wellbore model, and the annular space between the outer wall of the drill string the bingham fluid when flowing in the concentric annulus the expression is ； wherein is the concentric annulus friction pressure drop gradient for a Bingham fluid; is the yield point of the Bingham fluid; is the average annulus flow rate; The simplified expression of the concentric annulus friction pressure drop gradient is combined with the dimensionless discharge of the Bingham fluid in the concentric annulus: ； Wherein, ； ； ； wherein Q* is the dimensionless discharge of the Bingham fluid in the concentric annulus; Ri is the inner boundary radial coordinate of the plug flow region in the concentric annulus; Rr is the right boundary radial coordinate of the plug flow region in the concentric annulus; The ratio between the eccentric annulus friction pressure drop gradient and the concentric annulus friction pressure drop gradient of the Bingham fluid under the same discharge is calculated, and the friction pressure drop gradient ratio of the Bingham fluid is determined as: ； wherein is the ratio of the Bingham fluid frictional pressure drop gradient; Combining the simplified expression of the dimensionless discharge of the eccentric annulus and the simplified expression of the friction pressure drop gradient of the concentric annulus, the following equation is obtained: ； Substitute the dimensionless discharge of Bingham fluid in eccentric annulus into the calculation formula of the dimensionless discharge of Bingham fluid in eccentric annulus The implicit function of the Bingham fluid frictional pressure drop gradient ratio is obtained as follows: ； Because the intermediate quantity The width of the plug region And the upper boundary Are both variables, and the width of the plug region And the upper boundary Are both functions of the eccentric annulus friction pressure drop gradient The width of the plug region And the upper boundary In terms of the ratio of the Bingham fluid friction pressure drop gradient Is ； ； The implicit function of the friction pressure drop gradient ratio of the Bingham fluid is iteratively solved to construct a calculation equation of the friction pressure drop gradient ratio of the eccentric annulus of the Bingham fluid, and the following equation is obtained: ； wherein is a calculated function of the eccentric annulus friction pressure drop gradient ratio; The friction pressure drop gradient ratio of the Bingham fluid is the root of the calculation equation of the friction pressure drop gradient ratio of the eccentric annulus of the Bingham fluid, and the Newton-Raphson method is used to obtain the following equation through Taylor expansion linear approximation: ； wherein is the number of iterations; is the derivative of the eccentric annulus friction pressure drop gradient ratio at the th iteration; The derivative of the calculation function of the friction pressure drop gradient of the eccentric annulus is solved by using the numerical differentiation method, and the following equation is obtained: ； In the formula, is the difference between the ratio of the pre- and post-update Bingham fluid frictional pressure drop gradients.

8. The method for the hydrodynamic analysis of a Bingham fluid in a wellbore eccentric annulus of claim 7 wherein, In step 5, the friction pressure drop gradient ratio of the Bingham fluid is calculated by using a numerical simulation software, and the following steps are included: Step 5.1, input calculation parameters; the outer diameter of the drill string model the inner diameter of the wellbore model the eccentricity of the drill string model the volume flow rate of the Bingham fluid in the eccentric annulus the dynamic shear force and the plastic viscosity ; Step 5.2, calculating the concentric annulus friction pressure drop gradient for Bingham fluids ; Step 5.

3. Calculate the inner boundary radial coordinate of the plug flow region in the concentric annulus and the right boundary radial coordinate ; Step 5.4, outer diameter of the drill string model , inner diameter of the wellbore model , inner boundary radial coordinate of the plug flow region in the concentric annulus , right boundary radial coordinate of the plug flow region in the concentric annulus , dimensionless displacement of the Bingham fluid in the concentric annulus ; Step 5.5, setting the frictional pressure drop gradient ratio of the initial value and the convergence threshold value ; Step 5.6, iteratively update the calculated frictional pressure drop gradient ratio ; Calculating the width of the plug flow region , and then according to the polar coordinate equation of the wellbore model and the upper boundary of the plug flow area Regarding the expression of the Bingham fluid friction pressure drop gradient ratio, obtain the polar diameter at each preset direction and the upper boundary of the plug flow region , determine the integrand at each preset orientation , and then perform numerical integration based on the integrand at each preset position to determine the intermediate quantity , calculate the calculated value and derivative value of the current Bingham fluid eccentric annulus friction pressure drop gradient ratio, then obtain the calculated value of the Bingham fluid eccentric annulus friction pressure drop gradient ratio according to the calculation equation of the Bingham fluid eccentric annulus friction pressure drop gradient ratio, and update the Bingham fluid eccentric annulus friction pressure drop gradient ratio; Step 5.7, obtaining the updated Bingham fluid frictional pressure drop gradient ratio , calculating the difference between the Bingham fluid frictional pressure drop gradient ratio before and after the update ; Step 5.8, performing convergence check, comparing the difference of the ratio of the Bingham fluid friction pressure drop gradient before and after the update with a preset convergence threshold Step 5.8, performing convergence check, comparing the difference of the ratio of the Bingham fluid friction pressure drop gradient before and after the update with a preset convergence threshold Step 5.8, performing convergence check, comparing the difference of the ratio of the Bingham fluid friction pressure drop gradient before and after the update with a preset convergence threshold Step 5.8, performing convergence check, comparing the difference of the ratio of the Bingham fluid friction pressure drop gradient before and after the update with a preset convergence threshold Step 5.8, performing convergence check, comparing the difference of the ratio of the Bingham fluid friction pressure drop gradient before and after the update with a preset convergence threshold Step 5.9, output the friction pressure drop gradient ratio of the Bingham fluid.

Images