METHOD FOR MEASURING RHEOLOGICAL PROPERTY OF DRILLING FLUID BY USING CURVED PIPE ON SITE

Abstract

A method for measuring a rheological property of a drilling fluid by using a curved pipe on site includes: step 1: deriving relationship constants between friction coefficients of a drilling fluid through offline checking; step 2: calculating Rei according to fci; step 3: calculating an actual shear stress τwi and a shear rate γi of the drilling fluid in the on-site curved pipe according to the relationship constants and Rei; step 4: establishing a plurality of on-site models according to τwi and γi; step 5: determining an optimal on-site model according to correlations between τwi and predicted shear stresses of the plurality of on-site models; and step 6: performing on-site measurement on the rheological property of the drilling fluid according to the optimal on-site model. The method avoids inaccurate rheological measurement due to different types of drilling fluids and improves the measurement accuracy for different types of drilling fluids.

Description

TECHNICAL FIELD

The present invention relates to the field of oil well construction and, in particular, to a method for measuring a rheological property of a drilling fluid by using a curved pipe on site.

BACKGROUND

The properties of the drilling fluid (mud) play a crucial role in optimizing drilling operations. Among them, the density and rheological properties of the drilling fluid play a significant role in the optimal management of wellbore pressure. Therefore, it is necessary to carry out accurate measurement of the density and rheological properties of the drilling fluid in narrow-window drilling operations, especially in advanced drilling techniques, including managed pressure drilling (MPD) and dual gradient drilling (DGD).

At present, the rheological properties of the drilling fluid are mainly measured by a rotation method and a pipe flow method. In the pipe flow method, the online rheological measurement device uses a straight pipe for measurement. However, due to its large size, it requires significant changes to the site space, resulting in greatly limited on-site applications and poor measurement accuracy.

SUMMARY

A technical problem to be solved by the present invention is to provide a method for measuring a rheological property of a drilling fluid by using a curved pipe on site, so as to improve the accuracy of the rheological measurement of the drilling fluid.

To solve the above technical problem, the present invention adopts the following technical solution: a method for measuring a rheological property of a drilling fluid by using a curved pipe on site, including the following steps:

step 1: deriving relationship constants between friction coefficients of a drilling fluid through offline checking;

step 2: calculating a Reynolds number R_ei of the drilling fluid in an on-site curved pipe according to a friction coefficient f_ci of the drilling fluid in the on-site curved pipe;

step 3: calculating an actual shear stress τw_i of the drilling fluid in the on-site curved pipe according to the relationship constants between the friction coefficients of the drilling fluid and the Reynolds number R_ei of the drilling fluid in the on-site curved pipe, where i denotes a number of times the drilling fluid flows through the on-site curved pipe, which is a positive integer not less than 2;

step 4: establishing a plurality of on-site models according to the actual shear stress τw_i and a shear rate γ of the drilling fluid;

step 5: determining an optimal on-site model according to correlations between the actual shear stress τw_i and predicted shear stresses of the plurality of on-site models; and

step 6: performing on-site measurement on the rheological property of the drilling fluid according to the optimal on-site model.

The working principle and beneficial effects of the present invention are as follows: The present invention derives the relationship constants between the friction coefficients of the drilling fluid through offline checking and can derive the relationship constants for different types of drilling fluids. The present invention avoids inaccurate rheological measurement due to different types of drilling fluids and improves the measurement accuracy for different types of drilling fluids. In addition, the present invention determines the optimal on-site model according to the correlations between the actual shear stress τw_i and the predicted shear stresses of the plurality of on-site models, so as to ensure the accuracy of on-site measurement of the drilling fluid.

The following improvement may be further made by the present invention based on the above technical solution.

Further, step 1 may include:

step 11: calculating a friction coefficient f_ck of the drilling fluid in an offline curved pipe and a friction coefficient f_sk of the drilling fluid in an offline straight pipe, where, k denotes a number of times the drilling fluid flows through an offline pipe, which is a positive integer not less than 2;

step 12: establishing a plurality of prediction models according to an actual friction coefficient ratio y_k, where y_k=f_ck/f_sk;

step 13: determining an optimal prediction model according to correlations between the actual friction coefficient ratio y_k and predicted friction coefficient ratios of the plurality of prediction models; and

step 14: deriving the relationship constants between the friction coefficients of the drilling fluid according to the optimal prediction model.

The beneficial effects of the above further solution are as follows. The present invention measures and calculates the actual friction coefficients of the drilling fluid in offline curved and straight pipes and establishes the plurality of prediction models. The present invention determines the optimal prediction model according to the correlations between the actual friction coefficient ratio y_i and the predicted friction coefficient ratios of the plurality of prediction models. The present invention ensures the accuracy of selecting the prediction model. The present invention uses the relationship constants between the friction coefficients of the selected prediction model for subsequent on-site measurement on the rheological property of the drilling fluid. The present invention analyzes the correlations through the plurality of prediction models to ensure the accuracy of the relationship constants between the friction coefficients of different types of drilling fluids. Therefore, the present invention avoids the inaccuracy caused when a single relationship constant between the friction coefficients is applied to the on-site measurement of the drilling fluid.

The following improvement may be further made by the present invention based on the above technical solution.

Further, in step 11, f_ck may be expressed by formula (1):

$\begin{array}{cc}{f}_{ck}=\frac{d_{\mathrm{tc}1}}{2{\rho }_1*v_{ck}^2}*\frac{\Delta P_{ck}}{\Delta L_{ck}}& \left(1\right)\end{array}$

where, d_tc1 denotes an inner diameter of the offline curved pipe, and has a unit of m;

ρ₁ denotes a density of an offline drilling fluid, and has a unit of kg/m³;

v_ck denotes a flow velocity of the drilling fluid at the k-th time in the offline curved pipe, and has a unit of m/s; and

ΔP_ck/ΔL_ck denotes a measured average pressure difference in the offline curved pipe, and has a unit of kPa/m; and ΔP_ck denotes a total pressure difference in a pipe section with a length of ΔL_ck, and has a unit of kPa;

in step 1, f_sk may be expressed by formula (2):

$\begin{array}{cc}{f}_{\mathrm{sk}}=\frac{d_{\mathrm{ts}1}}{2{\rho }_1*v_{\mathrm{sk}}^2}*\frac{\Delta P_{\mathrm{sk}}}{\Delta L_{\mathrm{sk}}}& \left(2\right)\end{array}$

where, d_ts1 denotes an inner diameter of the offline straight pipe, and has a unit of m;

ρ₁ denotes a density of the drilling fluid, and has a unit of kg/m³;

v_sk denotes a flow velocity of the drilling fluid at the k-th time in the offline straight pipe, and has a unit of m/s; and

ΔP_sk/ΔL_sk denotes a measured average pressure difference in the offline straight pipe, and has a unit of kPa/m; and ΔP_sk denotes a total pressure difference in a pipe section with a length of ΔL_sk, and has a unit of kPa.

The beneficial effects of the above further solution are as follows. The present invention measures the average pressure differences of the straight pipe and the curved pipe and calculates the friction coefficient f_ck of the drilling fluid in the curved pipe and the friction coefficient f_sk thereof in the straight pipe, so as to ensure the calculation accuracy of the friction coefficients.

The following improvement may be further made by the present invention based on the above technical solution.

Further,

there may be at least three prediction models, namely:

a first prediction model:

ŷ _1k =a*D _nk ^b +C  (3)

a second prediction model:

$\begin{array}{cc}{\hat{y}}_{2k}=1+\frac{a*D_{nk}^b}{70+D_{nk}}& \left(4\right)\end{array}$

a third prediction model:

ŷ _3k=1+a*(log₁₀D_nk)^b  (5)

where

ŷ_1k denotes a predicted friction coefficient of the first prediction model;

ŷ_2k denotes a predicted friction coefficient of the second prediction model;

ŷ_3k denotes a predicted friction coefficient of the third prediction model; and

a, b and c denote the relationship constants between the friction coefficients of the drilling fluid, respectively;

where, D_nk denotes a Dean number of the drilling fluid at the k-th time in the offline curved pipe, and is expressed by formula (6):

$\begin{array}{cc}{D}_{nk}=\frac{{\rho }_1*v_{\mathrm{ck}}*d_{\mathrm{tc}1}}{{\mu }_1}*\sqrt{\frac{d_{\mathrm{tc}1}}{D_{c1}}}& \left(6\right)\end{array}$

where

μ₁ denotes a viscosity of an offline drilling fluid, and has a unit of Pa·s; and

v_ck denotes a flow velocity of the drilling fluid at the k-th time in the offline curved pipe, and has a unit of m/s.

The advantages of the above further solution are as follows. The present invention designs a plurality of prediction models involving the Dean number, which ensures the calculation accuracy.

The following improvement may be further made by the present invention based on the above technical solution.

Further,

step 13 may specifically include:

expressing a correlation R₁₁² between the actual friction coefficient ratio y_k and a predicted friction coefficient ratio of the first prediction model by formula (7):

$\begin{array}{cc}{R}_{11}^2=1-\frac{\stackrel{m}{\sum _{k=1}}{\left(y_k-\widehat{y_{1k}}\right)}^2}{\stackrel{m}{\sum _{k=1}}{\left(y_k-\stackrel{\_}{y}\right)}^2}& \left(7\right)\end{array}$

expressing a correlation R₁₂² between the actual friction coefficient ratio y_k and a predicted friction coefficient ratio of the second prediction model by formula (8):

$\begin{array}{cc}{R}_{12}^2=1-\frac{\stackrel{m}{\sum _{k=1}}{\left(y_k-\widehat{y_{2k}}\right)}^2}{\stackrel{m}{\sum _{k=1}}{\left(y_k-\stackrel{\_}{y}\right)}^2}& \left(8\right)\end{array}$

expressing a correlation R₁₃² between the actual friction coefficient ratio y_k and a predicted friction coefficient ratio of the third prediction model by formula (9):

$\begin{array}{cc}{R}_{13}^2=1-\frac{\stackrel{m}{\sum _{k=1}}{\left(y_k-\widehat{y_{3k}}\right)}^2}{\stackrel{m}{\sum _{k=1}}{\left(y_k-\stackrel{\_}{y}\right)}^2}& \left(9\right)\end{array}$

comparing R₁₁², R₁₂² and R₁₃² in terms of magnitude, and selecting a prediction model with a maximum correlation as an optimal prediction model;

where

m denotes a number of samples;

y_k denotes the actual friction coefficient ratio; and

y denotes an average actual friction coefficient ratio.

The beneficial effects of the above further solution are as follows: The present invention selects the final model for offline calibration through the correlations between the actual friction coefficient ratio y_i and the predicted friction coefficient ratios of the prediction models, which ensures the accuracy of selecting the offline model.

The following improvement may be further made by the present invention based on the above technical solution.

Further,

in step 2, f_ci may be expressed by formula (10):

$\begin{array}{cc}{f}_{\mathrm{ci}}=\frac{d_{\mathrm{tc}2}}{2{\rho }_2*v_{\mathrm{ci}}^2}*\frac{\Delta P_{\mathrm{ci}}}{\Delta L_{\mathrm{ci}}}& \left(10\right)\end{array}$

where, d_tc2 denotes an inner diameter of the on-site curved pipe, and has a unit of m;

ρ₂ denotes a density of an on-site drilling fluid, and has a unit of kg/m³;

v_ci denotes a flow velocity of the drilling fluid at the i-th time in the on-site curved pipe, and has a unit of m/s; and

ΔP_ci/ΔL_ci denotes a measured average pressure difference in the on-site curved pipe, and has a unit of kPa/m; and ΔP_ci denotes a total pressure difference in a pipe section with a length of ΔL_ci, and has a unit of kPa.

The beneficial effects of the above further solution are as follows. The present invention can accurately calculate the friction coefficient f_ci of the drilling fluid in the on-site curved pipe.

The following improvement may be further made by the present invention based on the above technical solution.

Further,

in step 2, the Reynolds number R_ei of the drilling fluid in the on-site curved pipe may be calculated as follows:

when an offline model is the first prediction model, the Reynolds number R_ei of the drilling fluid in the on-site curved pipe satisfies formula (12):

$\begin{array}{cc}{f}_{\mathrm{ci}}=\frac{16}{R_{\mathrm{ei}}}\left(a*{\left(R_{\mathrm{ei}}*\sqrt{\frac{d_{\mathrm{tc}2}}{D_{c2}}}\right)}^b+c\right)& \left(12\right)\end{array}$

when the offline model is the second prediction model, the Reynolds number R_ei of the drilling fluid in the on-site curved pipe satisfies formula (13):

$\begin{array}{cc}{f}_{\mathrm{ci}}=\frac{16}{R_{\mathrm{ei}}}*\left[1+\frac{a*{\left(R_{\mathrm{ei}}*\sqrt{\frac{d_{\mathrm{tc}2}}{D_{c2}}}\right)}^b}{70+\left(R_{\mathrm{ei}}*\sqrt{\frac{d_{\mathrm{tc}2}}{D_{c2}}}\right)}\right],& \left(13\right)\end{array}$

and

when the offline model is the third prediction model, the Reynolds number R_ei of the drilling fluid in the on-site curved pipe satisfies formula (14):

$\begin{array}{cc}{f}_{\mathrm{ci}}=\frac{16}{R_{\mathrm{ei}}}*\left[1+a*{\left({\log }_{10}\left(R_{\mathrm{ei}}*\sqrt{\frac{d_{\mathrm{tc}2}}{D_{c2}}}\right)\right)}^b\right].& \left(14\right)\end{array}$

The beneficial effects of the above further solution are as follows. The present invention calculates the Reynolds number R_ei of the drilling fluid in the on-site curved pipe and adopts different calculation methods according to different offline models, thereby ensuring the accuracy of the calculation of the Reynolds number R_ei of the drilling fluid in the on-site curved pipe.

The following improvement may be further made by the present invention based on the above technical solution.

Further,

in step 3, the actual shear stress τw_i of the drilling fluid in the on-site curved pipe may be expressed by formula (15):

$\begin{array}{cc}{\tau }_{wi}=\frac{8{\rho }_2*v_{ci}^2}{R_{ei}}& \left(15\right)\end{array}$

where

v_ci denotes the flow velocity of the drilling fluid at the i-th time in the on-site curved pipe, and has a unit of m/s; and

ρ₂ denotes the density of the on-site drilling fluid, and has a unit of kg/m³;

the shear rate γ_i of the drilling fluid is expressed by formula (16):

$\begin{array}{cc}{\gamma }_i=\frac{8*v_{\mathrm{ci}}}{d_{\mathrm{tc}2}}*\frac{3*N_i+1}{4*N_i}& \left(16\right)\end{array}$

where, N is expressed by formula (17):

$\begin{array}{cc}{N}_i=\frac{d\left(\ln {\tau }_{w_i}\right)}{d\left(\ln \frac{8*v_{\mathrm{ci}}}{d_{\mathrm{tc}2}}\right)}& \left(17\right)\end{array}$

The beneficial effects of the above further solution are as follows. The present invention ensures the calculation accuracy.

The following improvement may be further made by the present invention based on the above technical solution.

Further,

there may be at least three on-site models, namely:

a first on-site model:

τŵ_1i=YP−PV*γ_i  (18)

a second on-site model:

τŵ_2i=K*γ_iⁿ  (19)

a third on-site model:

τŵ_2i=τ₀+K*γ_iⁿ  (20)

where

YP denotes a yield strength of the on-site drilling fluid, and has a unit of Pa;

PV denotes a plastic viscosity of the on-site drilling fluid, and has a unit of Pa·s;

n denotes a fluidity index of the on-site drilling fluid, and is dimensionless;

K denotes a consistency coefficient of the on-site drilling fluid, and has a unit of Pa·s{circumflex over ( )}n; and

τ₀ denotes a dynamic shear stress of the on-site drilling fluid, and has a unit of Pa.

The beneficial effects of the above further solution are as follows. The present invention selects the rheological parameters through three different on-site models to ensure the optimal on-site models available for different drilling fluids.

The following improvement may be further made by the present invention based on the above technical solution.

Further,

step 5 may specifically include:

expressing a correlation R₂₁² between the actual shear stress τw_i and a predicted shear stress of the first on-site model by formula (21):

$\begin{array}{cc}{R}_{21}^2=1-\frac{\stackrel{m}{\sum _{i=1}}{\left(\tau w_i-\tau {\hat{w}}_{1i}\right)}^2}{\stackrel{m}{\sum _{i=1}}{\left(\tau w_i-\stackrel{\_}{\tau w}\right)}^2}& \left(21\right)\end{array}$

expressing a correlation R₂₂² between the actual shear stress τw_i and a predicted shear stress of the second on-site model by formula (22):

$\begin{array}{cc}{R}_{22}^2=1-\frac{\stackrel{m}{\sum _{i=1}}{\left(\tau w_i-\tau {\hat{w}}_{2i}\right)}^2}{\stackrel{m}{\sum _{i=1}}{\left(\tau w_i-\stackrel{\_}{\tau w}\right)}^2}& \left(22\right)\end{array}$

expressing a correlation R₂₃² between the actual shear stress τw_i and a predicted shear stress of the third on-site model by formula (23):

$\begin{array}{cc}{R}_{23}^2=1-\frac{\stackrel{m}{\sum _{i=1}}{\left(\tau w_i-\tau {\hat{w}}_{3i}\right)}^2}{\stackrel{m}{\sum _{i=1}}{\left(\tau w_i-\stackrel{\_}{\tau w}\right)}^2}& \left(23\right)\end{array}$

comparing R₂₁², R₂₂² and R₂₃² in terms of magnitude, and selecting an on-site model with a maximum correlation as a final model;

where

m denotes a number of samples;

τw_i denotes the actual shear stress; and

τw denotes an average actual shear stress.

The beneficial effects of the above further solution are as follows: The present invention selects the final model for on-site measurement through the correlations between the actual shear stress τw_i and the predicted friction coefficient ratios of the on-site models, ensuring the accuracy of selecting the on-site model.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flowchart of an on-site measurement method according to a first embodiment of the present invention; and

FIG. 2 is a flowchart of an offline checking method according to the first embodiment of the present invention.

DETAILED DESCRIPTION OF THE EMBODIMENTS

Principles and features of the present invention are described below with reference to the drawings. The described embodiments are only used to explain the present invention, rather than to limit the scope of the present invention.

FIG. 1 is a flowchart of an on-site measurement method according to a first embodiment of the present invention.

A method for measuring a rheological property of a drilling fluid by using a curved pipe on site includes the following steps:

Step 1: Derive relationship constants between friction coefficients of a drilling fluid through offline checking.

Step 2: Calculate a Reynolds number R_ei of the drilling fluid in an on-site curved pipe according to a friction coefficient f_ci of the drilling fluid in the curved pipe.

Step 3: Calculate an actual shear stress τw_i of the drilling fluid in the on-site curved pipe according to the relationship constants between the friction coefficients of the drilling fluid and the Reynolds number R_ei of the drilling fluid in the on-site curved pipe, where i denotes a number of times the drilling fluid flows through the on-site curved pipe, which is a positive integer not less than 2.

Step 4: Establish a plurality of on-site models according to the actual shear stress τw_i and a shear rate γ of the drilling fluid.

Step 5: Determine an optimal on-site model according to correlations between the actual shear stress τw_i and predicted shear stresses of the plurality of on-site models.

Step 6: Perform on-site measurement on the rheological property of the drilling fluid according to the optimal on-site model.

The working principle and beneficial effects of the embodiment of the present invention are as follows. The present invention derives the relationship constants between the friction coefficients of the drilling fluid through offline checking and can derive the relationship constants for different types of drilling fluids. The present invention avoids inaccurate rheological measurement due to different types of drilling fluids and improves the measurement accuracy for different types of drilling fluids. In addition, the present invention determines the optimal on-site model according to the correlations between the actual shear stress τw_i and the predicted shear stresses of the plurality of on-site models, so as to ensure the accuracy of on-site measurement of the drilling fluid.

In this embodiment, when the on-site measurement on the rheological property of the drilling fluid is carried out by the optimal on-site model, if the friction of the curved pipe changes, Step 7 is performed. That is, according to the friction of the curved pipe, the entire method starting from Step 1 is repeated outside a fixed operation time. The fixed operation time refers to a normal operation time of the on-site measurement equipment.

FIG. 2 is a flowchart of an offline checking method according to the embodiment of the present invention.

Step 1 includes:

Step 11: Calculate a friction coefficient f_ck of the drilling fluid in an offline curved pipe and a friction coefficient f_sk of the drilling fluid in an offline straight pipe, where, k denotes a number of times the drilling fluid flows through an offline pipe, which is a positive integer not less than 2.

Step 12: Establish a plurality of prediction models according to an actual friction coefficient ratio y_k, where, y_k=f_ck/f_sk.

Step 13: Determine an optimal prediction model according to correlations between the actual friction coefficient ratio y_k and predicted friction coefficient ratios of the plurality of prediction models.

Step 14: Derive the relationship constants between the friction coefficients of the drilling fluid according to the optimal prediction model.

The present invention measures and calculates the actual friction coefficients of the drilling fluid in offline curved and straight pipes and establishes the plurality of prediction models. The present invention determines the optimal prediction model according to the correlations between the actual friction coefficient ratio y_i and the predicted friction coefficient ratios of the plurality of prediction models. The present invention ensures the accuracy of selecting the prediction model. The present invention uses the relationship constants between the friction coefficients of the selected prediction model for subsequent on-site measurement on the rheological property of the drilling fluid. The present invention analyzes the correlations through the plurality of prediction models to ensure the accuracy of the relationship constants between the friction coefficients of different types of drilling fluids. Therefore, the present invention avoids the inaccuracy caused when a single relationship constant between the friction coefficients is applied to the on-site measurement of the drilling fluid.

Specifically, in step 11, f_ck is expressed by formula (1):

$\begin{array}{cc}{f}_{ck}=\frac{d_{\mathrm{tc}1}}{2{\rho }_1*v_{ck}^2}*\frac{\Delta P_{ck}}{\Delta L_{ck}}& \left(1\right)\end{array}$

where, d_tc1 denotes an inner diameter of the offline curved pipe, and has a unit of m;

ρ₁ denotes a density of an offline drilling fluid, and has a unit of kg/m³;

v_ck denotes a flow velocity of the drilling fluid at the k-th time in the offline curved pipe, and has a unit of m/s; and

ΔP_ck/ΔL_ck denotes a measured average pressure difference in the offline curved pipe, and has a unit of kPa/m; and ΔP_ck denotes a total pressure difference in a pipe section with a length of ΔL_ck, and has a unit of kPa;

in step 1, f_sk is expressed by formula (2):

$\begin{array}{cc}{f}_{ck}=\frac{d_{\mathrm{ts}1}}{2{\rho }_1*v_{\mathrm{sk}}^2}*\frac{\Delta P_{\mathrm{sk}}}{\Delta L_{\mathrm{sk}}}& \left(2\right)\end{array}$

where, d_ts1 denotes an inner diameter of the offline straight pipe, and has a unit of m;

ρ₁ denotes a density of the drilling fluid, and has a unit of kg/m³;

v_sk denotes a flow velocity of the drilling fluid at the k-th time in the offline straight pipe, and has a unit of m/s; and

ΔP_sk/ΔL_sk denotes a measured average pressure difference in the offline straight pipe, and has a unit of kPa/m; and ΔP_sk denotes a total pressure difference in a pipe section with a length of ΔL_sk, and has a unit of kPa.

where,

$\begin{array}{cc}{d}_{\mathrm{tc}1}=\sqrt{\frac{4*V_1}{\pi *\mathrm{le}{n}_1}}& \left(24\right)\end{array}$

where, V₁ denotes a total volume of the offline curved pipe, and has a unit of m³; and

len₁ denotes an length of the offline curved pipe, and has a unit of m.

In this embodiment, there are at least three prediction models, namely:

a first prediction model:

ŷ _1k =a*D _nk ^b +c  (3)

a second prediction model:

$\begin{array}{cc}{\hat{y}}_{2k}=1+\frac{a*D_{nk}^b}{70+D_{nk}}& \left(4\right)\end{array}$

a third prediction model:

ŷ _3k=1+a*(log₁₀D_nk)^b  (5)

where

ŷ_1k denotes a predicted friction coefficient of the first prediction model;

ŷ_2k denotes a predicted friction coefficient of the second prediction model;

ŷ_3k denotes a predicted friction coefficient of the third prediction model; and

a, b and c denote the relationship constants between the friction coefficients of the drilling fluid, respectively;

where, D_nk denotes a Dean number of the drilling fluid at the k-th time in the offline curved pipe, and is expressed by formula (6):

$\begin{array}{cc}{D}_{nk}=\frac{{\rho }_1*v_{\mathrm{ck}}*d_{\mathrm{tc}1}}{{\mu }_1}*\sqrt{\frac{d_{\mathrm{tc}1}}{D_{c1}}}& \left(6\right)\end{array}$

where

μ₁ denotes a viscosity of an offline drilling fluid, and has a unit of Pa·s; and

v_ck denotes a flow velocity of the drilling fluid at the k-th time in the offline curved pipe, and has a unit of m/s.

Step 13 specifically includes:

Express a correlation R₁₁² between the actual friction coefficient ratio yk and a predicted friction coefficient ratio of the first prediction model by formula (7):

$\begin{array}{cc}{R}_{11}^2=1-\frac{\stackrel{m}{\sum _{k=1}}{\left(y_k-\widehat{y_{1k}}\right)}^2}{\stackrel{m}{\sum _{k=1}}{\left(y_k-\stackrel{\_}{y}\right)}^2}& \left(7\right)\end{array}$

express a correlation R122 between the actual friction coefficient ratio y_k and a predicted friction coefficient ratio of the second prediction model by formula (8):

$\begin{array}{cc}{R}_{12}^2=1-\frac{\stackrel{m}{\sum _{k=1}}{\left(y_k-\widehat{y_{2k}}\right)}^2}{\stackrel{m}{\sum _{k=1}}{\left(y_k-\stackrel{\_}{y}\right)}^2}& \left(8\right)\end{array}$

express a correlation R132 between the actual friction coefficient ratio y_k and a predicted friction coefficient ratio of the third prediction model by formula (9):

$\begin{array}{cc}{R}_{13}^2=1-\frac{\stackrel{m}{\sum _{k=1}}{\left(y_k-\widehat{y_{3k}}\right)}^2}{\stackrel{m}{\sum _{k=1}}{\left(y_k-\stackrel{\_}{y}\right)}^2}& \left(9\right)\end{array}$

compare R₁₁², R₁₂² and R₁₃² in terms of magnitude, and select a prediction model with a maximum correlation as an optimal prediction model;

where

m denotes a number of samples;

y_k denotes the actual friction coefficient ratio; and

y denotes an average actual friction coefficient ratio.

In this embodiment, in step 2, f_ci is expressed by formula (10):

$\begin{array}{cc}{f}_{\mathrm{ci}}=\frac{d_{tc2}}{2{\rho }_2*v_{\mathrm{ci}}^2}*\frac{\Delta P_{\mathrm{ci}}}{\Delta L_{\mathrm{ci}}}& \left(10\right)\end{array}$

where, d_tc2 denotes an inner diameter of the on-site curved pipe, and has a unit of m;

ρ₂ denotes a density of an on-site drilling fluid, and has a unit of kg/m³;

v_ci denotes a flow velocity of the drilling fluid at the i-th time in the on-site curved pipe, and has a unit of m/s; and

ΔP_ci/ΔL_ci denotes a measured average pressure difference in the on-site curved pipe, and has a unit of kPa/m; and ΔP_ci denotes a total pressure difference in a pipe section with a length of ΔL_ci, and has a unit of kPa; and

d_tc2 is expressed by formula (11):

$\begin{array}{cc}{d}_{tc2}=\sqrt{\frac{4*V_2}{\pi *{\mathrm{len}}_2}}& \left(11\right)\end{array}$

where, V₂ denotes a total volume of the on-site curved pipe, and has a unit of m³; and

len₂ denotes a length of the on-site curved pipe, and has a unit of m;

Specifically, in step 2, the Reynolds number R_ei of the drilling fluid in the on-site curved pipe is calculated as follows:

when an offline model is the first prediction model, the Reynolds number R_ei of the drilling fluid in the on-site curved pipe satisfies formula (12):

$\begin{array}{cc}{f}_{\mathrm{ci}}=\frac{16}{R_{\mathrm{ei}}}\left(a*{\left(R_{\mathrm{ei}}*\sqrt{\frac{d_{\mathrm{tc}2}}{D_{c2}}}\right)}^b+c\right)& \left(12\right)\end{array}$

when the offline model is the second prediction model, the Reynolds number R_ei of the drilling fluid in the on-site curved pipe satisfies formula (13):

$\begin{array}{cc}{f}_{\mathrm{ci}}=\frac{16}{R_{\mathrm{ei}}}*\left[1+\frac{{a^{*}\left(R_{\mathrm{ei}}*\sqrt{\frac{d_{\mathrm{tc}2}}{D_{c2}}}\right)}^b}{70+\left(R_{\mathrm{ei}}*\sqrt{\frac{d_{\mathrm{tc}2}}{D_{c2}}}\right)}\right]& \left(13\right)\end{array}$

when the offline model is the third prediction model, the Reynolds number R_ei of the drilling fluid in the on-site curved pipe satisfies formula (14):

$\begin{array}{cc}{f}_{\mathrm{ci}}=\frac{16}{R_{\mathrm{ei}}}*\left[1+{a^{*}\left({\log }_{10}\left(R_{\mathrm{ei}}*\sqrt{\frac{d_{\mathrm{tc}2}}{D_{c2}}}\right)\right)}^b\right].& \left(14\right)\end{array}$

Specifically, in step 3, the actual shear stress τw_i of the drilling fluid in the on-site curved pipe is expressed by formula (15):

$\begin{array}{cc}{\tau }_{wi}=\frac{8{\rho }_2\star v_{ci}^2}{R_{ei}}& \left(15\right)\end{array}$

where

v_ci denotes the flow velocity of the drilling fluid at the i-th time in the on-site curved pipe, and has a unit of m/s; and

ρ₂ denotes the density of the on-site drilling fluid, and has a unit of kg/m³;

the shear rate γ_i of the drilling fluid is expressed by formula (16):

$\begin{array}{cc}{\gamma }_i=\frac{8^{*}{v}_{\mathrm{ci}}}{d_{tc2}}*\frac{3^{*}{N}_i+1}{4^{*}{N}_i}& \left(16\right)\end{array}$

where, N is expressed by formula (17):

$\begin{array}{cc}{N}_i=\frac{d\left(\ln {\tau }_{w_i}\right)}{d\left(\ln \frac{8^{*}{v}_{\mathrm{ci}}}{d_{tc2}}\right)}& \left(17\right)\end{array}$

In this embodiment, there are at least three on-site model, namely:

a first on-site model:

{circumflex over (τ)}w_1i=YP+PV*γ_i  (18)

a second on-site model:

{circumflex over (τ)}w_2i=K*γ_iⁿ  (19)

a third on-site model:

{circumflex over (τ)}w_2i=τ₀K*γ_iⁿ  (20)

where

YP denotes a yield strength of the on-site drilling fluid, and has a unit of Pa;

PV denotes a plastic viscosity of the on-site drilling fluid, and has a unit of Pa·s;

n denotes a fluidity index of the on-site drilling fluid, and is dimensionless;

K denotes a consistency coefficient of the on-site drilling fluid, and has a unit of Pa·s{circumflex over ( )}n; and

τ₀ denotes a dynamic shear stress of the on-site drilling fluid, and has a unit of Pa.

Specifically, step 5 includes:

express a correlation R₂₁² between the actual shear stress τw_i and a predicted shear stress of the first on-site model by formula (21):

$\begin{array}{cc}{R}_{21}^2=1-\frac{\sum _{i=1}^m{\left(\tau w_i-\tau {\hat{w}}_{1i}\right)}^2}{\sum _{i=1}^m{\left(\tau w_i-\stackrel{\_}{\mathrm{\tau w}}\right)}^2}& \left(21\right)\end{array}$

express a correlation R₂₂² between the actual shear stress τw_i and a predicted shear stress of the second on-site model by formula (22):

$\begin{array}{cc}{R}_{22}^2=1-\frac{\sum _{i=1}^m{\left(\tau w_i-\tau {\hat{w}}_{2i}\right)}^2}{\sum _{i=1}^m{\left(\tau w_i-\stackrel{\_}{\mathrm{\tau w}}\right)}^2}& \left(22\right)\end{array}$

express a correlation R₂₃² between the actual shear stress τw_i and a predicted shear stress of the third on-site model by formula (23):

$\begin{array}{cc}{R}_{23}^2=1-\frac{\sum _{i=1}^m{\left(\tau w_i-\tau {\hat{w}}_{3i}\right)}^2}{\sum _{i=1}^m{\left(\tau w_i-\stackrel{\_}{\mathrm{\tau w}}\right)}^2}& \left(23\right)\end{array}$

compare R₂₁², R₂₂² and R₂₃² in terms of magnitude, and selecting an on-site model with a maximum correlation as a final model;

where

m denotes a number of samples;

τw_i denotes the actual shear stress; and

τw denotes an average actual shear stress.

The application of the first embodiment of offline checking of the present invention is described below.

In this embodiment, the offline curved pipe is a helical pipe, which has a total volume V₁=1.04 l and a length len₁=5.57476 m; the offline straight pipe has an inner diameter d_ts1=0.01056 m; and a first offline drilling fluid has a density ρ₁=1,003 kg/m³.

The inner diameter of the offline curved pipe is calculated as: d_tc1=0.01051 m:

$\begin{array}{cc}{d}_{\mathrm{tc}1}=\sqrt{\frac{4^{*}{V}_1}{{\pi }^{*}{\mathrm{len}}_1}}& \left(4\right)\end{array}$

In this embodiment, the first offline drilling fluid flows through the pipe for k=24 times.

A flow velocity v_sk of the first offline drilling fluid at the k-th time in a straight pipe and a flow velocity v_ck thereof at the k-th time in the curved pipe are shown in the table below.

An average pressure difference ΔP_ck/ΔL_ck of the offline curved pipe and an average pressure difference ΔP_sk/ΔL_sk of the offline straight pipe are also shown in the table below.

The first offline drilling fluid flows through for 24 times, and its friction coefficient f_ck in the offline curved pipe and friction coefficient f_sk in the offline straight pipe are calculated by formulas (1) and (2) and are shown in the table below.

$\begin{array}{cc}{f}_{ck}=\frac{d_{\mathrm{tc}1}}{2{\rho }_1\star v_{ck}^2}*\frac{\Delta P_{ck}}{\Delta L_{ck}}& \left(1\right)\end{array}$

$\begin{array}{cc}{f}_{\mathrm{sk}}=\frac{d_{\mathrm{ts}1}}{2{\rho }_1\star v_{\mathrm{sk}}^2}*\frac{\Delta P_{\mathrm{sk}}}{\Delta L_{\mathrm{sk}}}& \left(2\right)\end{array}$

The corresponding actual friction coefficient ratios y_k are shown in the table below, where y_k=f_ck/f_sk.

In this embodiment, an average viscosity of the first offline drilling fluid is μ₁=0.00711 Pa·s.

The Dean numbers D_nk of the first offline drilling fluid flowing through the curved pipe for 24 times are calculated according to formula (6) and are shown in the table below.

$\begin{array}{cc}{D}_{nk}=\frac{{\rho }_1\star v_{ck}\star d_{\mathrm{tc}1}}{{\mu }_1}*\sqrt{\frac{d_{\mathrm{tc}1}}{D_{c1}}}& \left(6\right)\end{array}$

Number	ΔPck/	ΔPsk/
of times	ΔLck	ΔLsk	vsk	vck	fsk	fck	yk	Dnk
1	0.576	0.590	0.169	0.170	0.106	0.106	1.001	29.557
2	0.651	0.727	0.235	0.237	0.062	0.068	1.091	50.588
3	0.674	0.777	0.264	0.266	0.051	0.058	1.127	61.461
4	0.691	0.830	0.274	0.276	0.049	0.057	1.175	64.702
5	0.718	0.873	0.304	0.307	0.041	0.049	1.187	76.722
6	0.748	0.927	0.327	0.330	0.037	0.045	1.211	85.333
7	0.777	0.979	0.348	0.351	0.034	0.042	1.231	92.785
8	0.807	1.023	0.374	0.377	0.030	0.038	1.240	103.122
9	0.827	1.064	0.397	0.401	0.028	0.035	1.257	113.686
10	0.909	1.169	0.440	0.444	0.025	0.031	1.257	126.935
11	0.936	1.242	0.482	0.486	0.021	0.028	1.297	147.922
12	0.973	1.295	0.503	0.508	0.020	0.026	1.301	155.213
13	0.999	1.350	0.543	0.548	0.018	0.024	1.321	176.185
14	1.031	1.408	0.571	0.576	0.017	0.022	1.334	188.652
15	1.071	1.468	0.591	0.597	0.016	0.022	1.340	194.715
16	1.111	1.525	0.616	0.622	0.015	0.021	1.341	203.773
17	1.111	1.589	0.631	0.637	0.015	0.021	1.398	213.799
18	1.142	1.627	0.665	0.671	0.014	0.019	1.393	230.859
19	1.178	1.694	0.696	0.703	0.013	0.018	1.405	245.566
20	1.196	1.785	0.723	0.729	0.012	0.018	1.459	260.502
21	1.243	1.845	0.769	0.776	0.011	0.016	1.450	283.448
22	1.262	1.904	0.778	0.785	0.011	0.016	1.475	286.099
23	1.294	1.977	0.809	0.817	0.010	0.016	1.494	301.892
24	1.360	2.089	0.879	0.887	0.009	0.014	1.502	338.859

Three prediction models are fitted by the actual friction coefficient ratios yk and the Dean numbers D_nk as follows:

a first prediction model:

ŷ _1k =a*D _nk ^b +c  (3)

where, a=0.035966, b=0.5, and c=0.855298.

a second prediction model:

$\begin{array}{cc}{\hat{y}}_{2k}=1+\frac{a^{*}{D}_{nk}^b}{70+D_{nk}}& \left(4\right)\end{array}$

where, a=0.052896, and b=1.421332.

a third prediction model:

ŷ _3k=1+a*(log₁₀D_nk)^b  (5)

where, a=0.016495, and b=3.709336.

The predicted friction coefficients of the third prediction models are shown in the table below.

	Number
	of times	ŷ1k	ŷ2k	ŷ3k
	1	1.051	1.065	1.069
	2	1.111	1.116	1.119
	3	1.137	1.140	1.143
	4	1.145	1.147	1.149
	5	1.170	1.172	1.173
	6	1.188	1.189	1.189
	7	1.202	1.203	1.203
	8	1.221	1.222	1.221
	9	1.239	1.241	1.239
	10	1.261	1.262	1.260
	11	1.293	1.295	1.292
	12	1.303	1.305	1.303
	13	1.333	1.335	1.332
	14	1.349	1.351	1.348
	15	1.357	1.359	1.356
	16	1.369	1.370	1.368
	17	1.381	1.382	1.380
	18	1.402	1.402	1.401
	19	1.419	1.418	1.418
	20	1.436	1.434	1.435
	21	1.461	1.458	1.460
	22	1.464	1.461	1.463
	23	1.480	1.476	1.479
	24	1.517	1.510	1.516

A correlation R₁₁² between the actual friction coefficient ratio y_k and a predicted friction coefficient ratio of the first prediction model, a correlation R₁₂² between the actual friction coefficient ratio y_k and a predicted friction coefficient ratio of the second prediction model and a correlation R₁₃² between the actual friction coefficient ratio y_k and a predicted friction coefficient ratio of the third prediction model are calculated according to formulas (7), (8) and (9), respectively.

Through calculation, R₁₁²=0.976421, R₁₂²=0.972209 and R₁₃²=0.971412.

To sum up, in this embodiment, for the first offline drilling fluid, the first prediction model is the optimal prediction model. According to the optimal prediction model, the relationship constants between the friction coefficients are a=0.035966, b=0.5 and c=0.855298, which are used for the on-site measurement on the rheological property of the drilling fluid.

The on-site measurement on the rheological property of the drilling fluid in the curved pipe is described below.

In a first embodiment of the on-site measurement, the density of a first on-site drilling fluid is ρ₂=1,003 kg/m³.

The on-site curved pipe is a helical pipe, which has a total volume V₂=1.04 l and a length len₂=5.57476 m.

An inner diameter of the on-site curved pipe is calculated as d_tc2=0.01051 m:

$\begin{array}{cc}{d}_{\mathrm{tc}2}=\sqrt{\frac{4^{*}{V}_2}{{\pi }^{*}{\mathrm{len}}_2}}& \left(11\right)\end{array}$

In this embodiment, a first offline drilling fluid flows through the on-site pipe for i=24 times.

A flow velocity v_ci of the first on-site drilling fluid at the i-th time in the curved pipe and an average pressure difference thereof in the curved pipe ΔP_ci/ΔL_ci are shown in the table below.

The measured parameters of the on-site drilling fluid in the curved pipe are shown in the table below. The flow velocity of the drilling fluid is increased in the curved pipe in an ascending order to keep a laminar flow state of the drilling fluid.

	Rate
Number	of flow	Temperature
of times	(lpm)	(° C.)	ΔPci/ΔLci	ρ2	vci
1	0.8882	32.5	0.590	1003.9	0.1705
2	1.2356	32.5	0.727	1002.7	0.2372
3	1.3858	32.5	0.777	1002.5	0.2660
4	1.4388	32.5	0.830	1003.4	0.2762
5	1.5973	32.5	0.873	1004.1	0.3066
6	1.7200	32.5	0.927	1003.3	0.3302
7	1.8269	32.5	0.979	1004.2	0.3507
8	1.9638	32.5	1.023	1002.7	0.3770
9	2.0876	32.5	1.064	1002.9	0.4007
10	2.3133	32.5	1.169	1002.6	0.4441
11	2.5336	32.5	1.242	1002.8	0.4864
12	2.6463	32.5	1.295	1002.7	0.5080
13	2.8562	32.5	1.350	1002.8	0.5483
14	3.0025	32.5	1.408	1003.2	0.5764
15	3.1087	32.5	1.468	1002.9	0.5968
16	3.2385	32.5	1.525	1003.5	0.6217
17	3.3183	32.5	1.589	1002.9	0.6370
18	3.4960	32.5	1.627	1002.6	0.6711
19	3.6600	32.5	1.694	1003.7	0.7026
20	3.7989	32.5	1.785	1003.3	0.7293
21	4.0419	32.5	1.845	1002.6	0.7759
22	4.0901	32.5	1.904	1002.9	0.7852
23	4.2535	32.5	1.977	1003.3	0.8165
24	4.6207	32.5	2.089	1003.0	0.8870

The first on-site drilling fluid flows through the curved pipe for 24 times, and its friction coefficient f_ci in the curved pipe is calculated by formula (11), which is shown in the table below.

$\begin{array}{cc}{f}_{\mathrm{ci}}=\frac{d_{\mathrm{tc}2}}{2{\rho }_2\star v_{\mathrm{ci}}^2}*\frac{\Delta P_{\mathrm{ci}}}{\Delta L_{\mathrm{ci}}}& \left(10\right)\end{array}$

The Reynolds number R_ei of the on-site curved pipe is calculated according to the selected optimal offline model. In this embodiment, the optimal prediction model is the first prediction model: ŷ_1k=a*D_nk^b+c, where the relationship constants between the friction coefficients are respectively: a=0.035966, b=0.5, and c=0.855298.

The Reynolds number R_ei of the on-site curved pipe is expressed by formula (12):

$\begin{array}{cc}{f}_{\mathrm{ci}}=\frac{16}{R_{\mathrm{ei}}}\left({a^{*}\left({R_{\mathrm{ei}}}^{*}\sqrt{\frac{d_{\mathrm{tc}2}}{D_{c2}}}\right)}^b+c\right)& \left(12\right)\end{array}$

The Reynolds number R_ei of the on-site curved pipe is shown in the table below.

According to the Reynolds number R_ei of the on-site curved pipe, a shear stress τw_i of the drilling fluid in the on-site curved pipe is calculated, which is expressed by formula (15), and is shown in the table below.

$\begin{array}{cc}{\tau }_{\mathrm{wi}}=\frac{8{{\rho }_2}^{*}{v}_{\mathrm{ci}}^2}{R_{\mathrm{ei}}}& \left(15\right)\end{array}$

In this embodiment, an intermediate parameter N_i is calculated by a binomial fitting method according to formula (17), which is shown in the table below.

$\begin{array}{cc}{N}_i=\frac{d\left(\ln {\tau }_{w_i}\right)}{d\left(\ln \frac{8^{*}{v}_{\mathrm{ci}}}{d_{\mathrm{tc}2}}\right)}& \left(17\right)\end{array}$

A shear rate γ_i of the first drilling fluid in the on-site curved pipe is calculated according to formula (16), which is shown in the table below.

$\begin{array}{cc}{\gamma }_l=\frac{8^{*}{v}_{\mathrm{ci}}}{d_{\mathrm{tc}2}}*\frac{3^{*}{N}_i+1}{4^{*}{N}_i}& \left(16\right)\end{array}$

	Number
	of times	fci	Rci	τwi	Ni	γi
	1	0.11	158.92	1.47	0.47	166.99
	2	0.07	263.11	1.72	0.50	225.78
	3	0.06	316.51	1.79	0.51	250.91
	4	0.06	320.14	1.91	0.51	259.74
	5	0.05	384.57	1.96	0.53	286.06
	6	0.05	425.27	2.06	0.53	306.33
	7	0.04	459.71	2.15	0.54	323.95
	8	0.04	516.12	2.21	0.55	346.44
	9	0.04	569.37	2.26	0.55	366.69
	10	0.03	649.48	2.44	0.56	403.50
	11	0.03	751.29	2.53	0.57	439.27
	12	0.03	793.06	2.61	0.58	457.51
	13	0.02	908.07	2.66	0.58	491.39
	14	0.02	975.29	2.73	0.59	514.94
	15	0.02	1008.00	2.84	0.59	532.01
	16	0.02	1064.98	2.91	0.60	552.83
	17	0.02	1073.93	3.03	0.60	565.62
	18	0.02	1186.55	3.05	0.61	594.03
	19	0.02	1267.02	3.13	0.61	620.20
	20	0.02	1301.98	3.28	0.61	642.33
	21	0.02	1458.08	3.31	0.62	680.96
	22	0.02	1444.81	3.42	0.62	688.62
	23	0.02	1521.06	3.52	0.63	714.54
	24	0.01	1750.53	3.61	0.63	772.62

According to the shear stress τw_i and the shear rate γ_i of the first on-site drilling fluid, at least three on-site models are fitted, which are respectively:

a first on-site model:

{circumflex over (τ)}w_1i=YP+PV*γ_i  (18)

where, PV=0.00354, and YP=0.95267.

a second on-site model:

{circumflex over (τ)}w_2i=K*γ_iⁿ  (19),

where K=0.0622, and n=0.6105.

a third on-site model:

{circumflex over (τ)}w_2i=τ₀+K*γ_iⁿ  (20),

where, n=0.7991, K=0.0151, and τ₀=0.581.

Then the following parameters are respectively calculated:

a correlation R₂₁² between the shear stress τw_i of the first on-site drilling fluid and a predicted shear stress of the first on-site model;

a correlation R₂₂² between the shear stress τw_i of the first on-site drilling fluid and a predicted shear stress of the second on-site model; and

a correlation R₂₃² between the shear stress τw_i of the first on-site drilling fluid and a predicted shear stress of the third on-site model.

These parameters are calculated according to formulas (21), (22) and (23), respectively.

$\begin{array}{cc}{R}_{21}^2=1-\frac{\sum _{i=1}^m{\left(\tau w_i-\widehat{\tau w_{1i}}\right)}^2}{\sum _{i=1}^m{\left(\tau w-\stackrel{\_}{\tau w}\right)}^2}& \left(21\right)\\ R_{22}^2=1-\frac{\sum _{i=1}^m{\left(\tau w_i-\widehat{\tau w_{2i}}\right)}^2}{\sum _{i=1}^m{\left(\tau w-\stackrel{\_}{\tau w}\right)}^2}& \left(22\right)\\ R_{23}^2=1-\frac{\sum _{i=1}^m{\left(\tau w_i-\widehat{\tau w_{3i}}\right)}^2}{\sum _{i=1}^m{\left(\tau w-\stackrel{\_}{\tau w}\right)}^2}& \left(23\right)\end{array}$

Through calculation, R₂₁²=0.9950, R₂₂²=0.9951 and R₂₃²=0.9964. The third on-site model has the maximum correlation and is most in line with the actual situation, so the third on-site model is selected as the final model for calculating other viscosity data.

The on-site measurement results of the third on-site model are compared with those of a 6-speed viscometer (Fann35), which shows that the third on-site model is the most suitable.

Measurement	Measuring
instrument	temperature/° C.	θ600	θ300	θ200	θ100	θ6	θ3	PV	YP
Fann35 (control)	34	7	5	4	3	1	0.5	0.002	1.533
System measurement	34	8.6	5.4	4.3	2.9	1.3	1.2	0.0032	1.1538
device

If the viscosity data calculated by the first on-site model is directly selected without performing on-site model optimization, as shown in the table below, the deviation will increase significantly. The difference percentage of viscosity corresponding to θ6 is 0.9/1=90%, the difference percentage of viscosity corresponding to θ3 is 1.4/0.5=280%, and the difference percentage of YP is 0.5804/1.533=38%. The calculation of the preferred third model of the present invention shows that the difference percentage of viscosity corresponding to θ6 is 0.3/1=30%, the difference percentage of viscosity corresponding to θ3 is 0.7/0.5=140%, and the difference percentage of YP is 0.3792/1.533=25%.

In conclusion, compared with the measurement results of Fann35, the calculation results of the optimal on-site model (third on-site model) determined by the correlations of the actual shear stress τw_i and the predicted shear stresses of the on-site models are the most accurate.

Measurement	Measuring
instrument	temperature/° C.	θ600	θ300	θ200	θ100	θ6	θ3	PV	YP
Fann35 (control)	34	7	5	4	3	1	0.5	0.002	1.533
System measurement	34	9.0	5.4	4.2	3.0	1.9	1.9	0.0035	0.9526
device

A second embodiment is described below.

In the second embodiment of offline checking, the offline curved pipe has a total volume V₁=1.04 and a length len₁=5.57476 m; the offline straight pipe has an inner diameter d_ts1=0.01056 m; and a second offline drilling fluid has a density ρ₁=1,953 kg/m³.

The inner diameter of the offline curved pipe is calculated as: d_tc1=0.01051 m:

$\begin{array}{cc}{d}_{\mathrm{tc}1}=\sqrt{\frac{4^{*}{V}_1}{{\pi }^{*}{\mathrm{len}}_1}}& \left(4\right)\end{array}$

In this embodiment, the second offline drilling fluid flows through the pipe for k=17 times.

A flow velocity v_sk of the second offline drilling fluid at the k-th time in a straight pipe and a flow velocity v_ck thereof at the k-th time in the curved pipe are shown in the table below.

An average pressure difference ΔP_ck/ΔL_ck of the offline curved pipe and an average pressure difference ΔP_sk/ΔL_sk of the offline straight pipe are also shown in the table below.

The second offline drilling fluid flows through for 17 times, and its friction coefficient f_ck in the offline curved pipe and friction coefficient f_sk in the offline straight pipe are calculated by formulas (1) and (2), and are shown in the table below.

$\begin{array}{cc}{f}_{\mathrm{ck}}=\frac{d_{\mathrm{tc}1}}{2{{\rho }_1}^{*}{v}_{\mathrm{ck}}^2}*\frac{\Delta P_{\mathrm{ck}}}{\Delta L}& \left(1\right)\\ f_{\mathrm{sk}}=\frac{d_{\mathrm{ts}1}}{2{{\rho }_1}^{*}{v}_{\mathrm{sk}}^2}*\frac{\Delta P_{\mathrm{sk}}}{\Delta L}& \left(2\right)\end{array}$

The corresponding actual friction coefficient ratios y_k are shown in the table below, where y_k=f_ck/f_sk.

In this embodiment, an average viscosity of the second offline drilling fluid is μ₁=0.02639 Pa·s.

The Dean numbers D_nk of the second offline drilling fluid flowing through the curved pipe for 17 times are calculated according to formula (6), which are shown in the table below.

$\begin{array}{cc}{D}_{\mathrm{nk}}=\frac{{{\rho }_1}^{*}{v_{\mathrm{ck}}}^{*}{d}_{\mathrm{tc}1}}{{\mu }_1}*\sqrt{\frac{d_{\mathrm{tc}1}}{D_{c1}}}& \left(6\right)\end{array}$

Number	ΔPck/	ΔPsk/
of times	ΔLck	ΔLsk	vsk	vck	fsk	fck	yk	Dnk
1	6.249	8.759	0.80	0.81	0.03	0.04	1.37	119.74
2	7.020	9.999	0.91	0.91	0.02	0.03	1.39	135.57
3	7.447	10.907	0.97	0.98	0.02	0.03	1.43	145.85
4	8.008	12.182	1.07	1.08	0.02	0.03	1.49	166.80
5	8.420	12.888	1.11	1.12	0.02	0.03	1.50	167.88
6	8.583	13.091	1.13	1.14	0.02	0.03	1.49	172.00
7	9.179	14.505	1.21	1.22	0.02	0.03	1.54	184.56
8	9.603	15.250	1.29	1.30	0.02	0.02	1.55	199.90
9	10.128	16.783	1.37	1.38	0.01	0.02	1.62	215.73
10	10.338	17.224	1.41	1.43	0.01	0.02	1.63	224.42
11	10.880	18.257	1.47	1.48	0.01	0.02	1.64	230.18
12	13.825	24.076	1.82	1.83	0.01	0.02	1.70	277.64
13	13.706	24.131	1.83	1.85	0.01	0.02	1.72	284.93
14	14.606	26.401	1.96	1.98	0.01	0.02	1.77	305.94
15	15.228	27.892	2.07	2.09	0.01	0.02	1.79	326.25
16	16.025	29.395	2.14	2.16	0.01	0.02	1.79	333.56
17	16.940	30.332	2.21	2.23	0.01	0.02	1.75	333.69

Three prediction models are fitted by the actual friction coefficient ratios yk and the Dean numbers D_nk as follows:

a first prediction model:

ŷ _1k =a*D _nk ^b +c  (3)

where, a=0.0576, b=0.5, and c=0.745.

a second prediction model:

$\begin{array}{cc}{\hat{y}}_{2k}=1+\frac{a^{*}{D}_{\mathrm{nk}}^b}{70+D_{\mathrm{nk}}}& \left(4\right)\end{array}$

where, a=0.0644, and b=1.4654.

a third prediction model:

ŷ _3k=1+a*(log₁₀D_nk)^b  (5)

where, a=0.0231, and b=3.8241.

The predicted friction coefficients of the third prediction models are shown in the table below.

	Number
	of times	ŷ1k	ŷ2k	ŷ3k
	1	1.376	1.377	1.379
	2	1.416	1.417	1.418
	3	1.441	1.442	1.443
	4	1.489	1.491	1.490
	5	1.492	1.493	1.492
	6	1.501	1.502	1.501
	7	1.528	1.529	1.528
	8	1.560	1.561	1.560
	9	1.591	1.593	1.591
	10	1.608	1.610	1.608
	11	1.619	1.620	1.619
	12	1.705	1.705	1.704
	13	1.718	1.717	1.717
	14	1.753	1.752	1.752
	15	1.786	1.784	1.785
	16	1.797	1.795	1.796
	17	1.798	1.795	1.797

A correlation R₁₁² between the actual friction coefficient ratio y_k and a predicted friction coefficient ratio of the first prediction model, a correlation R₁₂² between the actual friction coefficient ratio y_k and a predicted friction coefficient ratio of the second prediction model and a correlation R₁₃² between the actual friction coefficient ratio y_k and a predicted friction coefficient ratio of the third prediction model are calculated according to formulas (7), (8) and (9), respectively.

Through calculation, R₁₁²=0.9833, R₁₂²=0.9843 and R₁₃²=0.9828.

To sum up, in this embodiment, for the second offline drilling fluid, the first prediction model is the optimal prediction model. According to the optimal prediction model, the relationship constants between the friction coefficients are a=0.0644 and b=1.4654, which are used for the on-site measurement on the rheological property of the drilling fluid.

The on-site measurement on the rheological property of the drilling fluid in the curved pipe is described below.

In a second embodiment of the on-site measurement, the density of a second on-site drilling fluid is ρ₂=1,300 kg/m³.

The on-site curved pipe has a total volume V₂=1.04 l and a length len₂=5.57476 m.

An inner diameter of the on-site curved pipe is calculated as d_tc2=0.01051 m:

$\begin{array}{cc}{d}_{\mathrm{tc}2}=\sqrt{\frac{4^{*}{\nabla }_2}{{\pi }^{*}{\mathrm{len}}_2}}& \left(11\right)\end{array}$

In this embodiment, a second offline drilling fluid flows through the on-site pipe for i=17 times.

A i-th flow velocity v_ci of the second on-site drilling fluid at the i-th time in the curved pipe and an average pressure difference thereof in the curved pipe ΔP_ci/ΔL_ci are shown in the table below.

The measured parameters of the on-site drilling fluid in the curved pipe are shown in the table below. The flow velocity of the drilling fluid is increased in the curved pipe in an ascending order to keep a laminar flow state of the drilling fluid.

	Rate
Number	of flow	Temperature
of times	(lpm)	(° C.)	ΔPci/ΔLci	ρ2	vci
1	4.212	29	8.759	1960.5	0.809
2	4.761	29	9.999	1951.0	0.914
3	5.088	29	10.907	1950.2	0.977
4	5.642	29	12.182	1950.2	1.083
5	5.813	29	12.888	1944.5	1.116
6	5.933	29	13.091	1949.2	1.139
7	6.358	29	14.505	1947.9	1.221
8	6.776	29	15.250	1943.1	1.301
9	7.207	29	16.783	1955.0	1.384
10	7.425	29	17.224	1956.0	1.425
11	7.716	29	18.257	1955.1	1.481
12	9.544	29	24.076	1958.6	1.832
13	9.631	29	24.131	1956.9	1.849
14	10.306	29	26.401	1955.3	1.978
15	10.868	29	27.892	1955.2	2.086
16	11.263	29	29.395	1958.4	2.162
17	11.595	29	30.332	1954.3	2.226

The second on-site drilling fluid flows through the curved pipe for 17 times, and its friction coefficient f_ci in the curved pipe is calculated by formula (11), which is shown in the table below.

$\begin{array}{cc}{f}_{\mathrm{ci}}=\frac{d_{\mathrm{tc}2}}{2{{\rho }_2}^{*}{v}_{\mathrm{ci}}^2}*\frac{\Delta P_{\mathrm{ci}}}{\Delta L_{\mathrm{ci}}}& \left(10\right)\end{array}$

The Reynolds number R_ei of the on-site curved pipe is calculated according to the selected optimal offline model. In this embodiment, the optimal prediction model is the second prediction model:

${\hat{y}}_{2k}=1+\frac{a^{*}{D}_{\mathrm{nk}}^b}{70+D_{\mathrm{nk}}},$

where the relationship constants between the friction coefficients are respectively: a=0.0644 and b=1.4654.

The Reynolds number R_ei of the on-site curved pipe is expressed by formula (12):

$\begin{array}{cc}{f}_{\mathrm{ci}}=\frac{16}{R_{\mathrm{ei}}}\left({a^{*}\left({R_{\mathrm{ei}}}^{*}\sqrt{\frac{d_{\mathrm{tc}2}}{D_{c2}}}\right)}^b+c\right)& \left(12\right)\end{array}$

The Reynolds number R_ei of the on-site curved pipe is shown in the table below.

According to the Reynolds number R_ei of the on-site curved pipe, a shear stress τw_i of the drilling fluid in the on-site curved pipe is calculated, which is expressed by formula (15), and is shown in the table below.

$\begin{array}{cc}{\tau }_{\mathrm{wi}}=\frac{8{{\rho }_2}^{*}{v}_{\mathrm{ci}}^2}{R_{\mathrm{ei}}}& \left(15\right)\end{array}$

An intermediate parameter N_i is calculated according to formula (17), which is shown in the table below.

$\begin{array}{cc}{N}_i=\frac{d\left(\ln {\tau }_{w_i}\right)}{d\left(\ln \frac{8^{*}{v}_{\mathrm{ci}}}{d_{\mathrm{tc}2}}\right)}& \left(17\right)\end{array}$

A shear rate γ_i of the second drilling fluid in the on-site curved pipe is calculated according to formula (16), which is shown in the table below.

$\begin{array}{cc}{\gamma }_i=\frac{8^{*}{v}_{\mathrm{ci}}}{d_{tc2}}*\frac{3^{*}{N}_i+1}{4^{*}{N}_i}& \left(16\right)\end{array}$

Number
of times	fci	Rei	τwi	Ni	γi
1	0.0359	614.0792	16.6964	0.9448	624.1797
2	0.0322	707.0957	18.4418	0.9517	704.3091
3	0.0308	750.4231	19.8311	0.9554	751.8029
4	0.0280	852.8917	21.4602	0.9612	832.4728
5	0.0280	853.1906	22.7020	0.9628	857.2334
6	0.0272	885.4715	22.8450	0.9640	874.7194
7	0.0263	928.1204	25.0111	0.9678	936.3869
8	0.0244	1026.7406	25.6190	0.9714	997.0713
9	0.0236	1074.6661	27.8581	0.9748	1059.5306
10	0.0228	1125.6006	28.2435	0.9765	1091.0656
11	0.0224	1153.9089	29.7386	0.9786	1133.1933
12	0.0193	1418.3999	37.0815	0.9905	1397.4120
13	0.0190	1448.0218	36.9546	0.9910	1409.9395
14	0.0181	1540.8543	39.7384	0.9948	1507.3717
15	0.0172	1654.1138	41.1553	0.9978	1588.2620
16	0.0169	1702.5492	43.0205	0.9998	1645.2677
17	0.0165	1761.7294	43.9646	1.0014	1692.9891

According to the shear stress τw_i and the shear rate γ_i of the second on-site drilling fluid, at least three on-site models are fitted, which are respectively:

a first on-site model:

{circumflex over (τ)}w_1i=YP+PV*γ_i  (18)

where, PV=0.026, and YP=0.241.

a second on-site model:

{circumflex over (τ)}w_2i=K*γ_iⁿ  (19),

where K=0.0278, and n=0.9917.

a third on-site model:

{circumflex over (τ)}w_2i=τ₀+K*γ_iⁿ  (20),

Where, n=0.9991, K=0.0262, and τ₀=0.2146.

Then the following parameters are respectively calculated:

a correlation R₂₁² between the shear stress τw_i of the second on-site drilling fluid and a predicted shear stress of the first on-site model;

a correlation R₂₂² between the shear stress τw_i of the second on-site drilling fluid and a predicted shear stress of the second on-site model; and

a correlation R₂₃² between the shear stress τw_i of the second on-site drilling fluid and a predicted shear stress of the third on-site model.

These parameters are calculated according to formulas (21), (22) and (23), respectively.

$\begin{array}{cc}{R}_{21}^2=1-\frac{\sum _{i=1}^m{\left({\mathrm{\tau w}}_i-\tau {\hat{w}}_{1i}\right)}^2}{\sum _{i=1}^m{\left({\mathrm{\tau w}}_i-\stackrel{\_}{\mathrm{\tau w}}\right)}^2}& \left(21\right)\end{array}$ $\begin{array}{cc}{R}_{22}^2=1-\frac{\sum _{i=1}^m{\left({\mathrm{\tau w}}_i-\tau {\hat{w}}_{2i}\right)}^2}{\sum _{i=1}^m{\left({\mathrm{\tau w}}_i-\stackrel{\_}{\mathrm{\tau w}}\right)}^2}& \left(22\right)\end{array}$ $\begin{array}{cc}{R}_{23}^2=1-\frac{\sum _{i=1}^m{\left({\mathrm{\tau w}}_i-\tau {\hat{w}}_{3i}\right)}^2}{\sum _{i=1}^m{\left({\mathrm{\tau w}}_i-\stackrel{\_}{\mathrm{\tau w}}\right)}^2}& \left(23\right)\end{array}$

Through calculation, R₂₁²=0.998873, R₂₂²=0.996445 and R₂₃²=0.998873. The first and third on-site models have the maximum correlation and are most in line with the actual situation, so the first and third on-site models are selected as the final models for calculating other viscosity data.

The on-site measurement results of the first on-site model are compared with those of the 6-speed viscometer, which shows that the first on-site model is the most suitable.

Measurement	Measuring
instrument	temperature/° C.	θ600	θ300	θ200	θ100	θ6	θ3
Fann35 (control)	29	50	27	19	11	1	0.5
System measurement	29	52.42	26.45	17.79	9.13	0.99	0.73
device

The on-site measurement results of the third on-site model are compared with those of the 6-speed viscometer, which shows that the third on-site model is the most suitable.

Measurement	Measuring
instrument	temperature/° C.	θ600	θ300	θ200	θ100	θ6	θ3	n	K	τ0
Fann35 (control)	29	50	27	19	11	1	0.5	0.9015	0.0490	0.26
System measurement	29	52.42	26.44	17.77	9.10	0.94	0.68	0.9991	0.0262	0.21
device

If the viscosity data calculated by the second on-site model is directly selected without performing on-site model optimization, as shown in the table below, the deviation will increase significantly. The difference percentage of viscosity corresponding to θ100 is 2.13/11=19% and the difference percentage of viscosity corresponding to θ6 is 0.46/1=46%. The calculation of the preferred third model of the present invention shows that the difference percentage of viscosity corresponding to θ100 is 1.9/11=17% and the difference percentage of viscosity corresponding to θ6 is 0.06/1=6%. The calculation of the preferred first model of the present invention shows that the difference percentage of viscosity corresponding to θ100 is 1.83/11=17% and the difference percentage of viscosity corresponding to θ6 is 0.01/1=1%.

Measurement	Measuring
instrument	temperature/° C.	θ600	θ300	θ200	θ100	θ6	θ3
Fann35 (control)	29	50	27	19	11	1	0.5
System
measurement	29	52.43	26.37	17.64	8.87	0.54	0.27
device

In conclusion, compared with the measurement results of Fann35, the calculation results of the optimal on-site models (the first and third on-site models) determined by the correlations of the actual shear stress τw_i and the predicted shear stresses of the on-site models are the most accurate.

A third embodiment is described below.

In the third embodiment of offline checking, the offline curved pipe has a total volume V₁=1.04 l and a length len₁=5.57476 m; the offline straight pipe has an inner diameter d_ts1=0.01056 m; and a third offline drilling fluid has a density ρ₁=1,227 kg/m³.

The inner diameter of the offline curved pipe is calculated as: d_tc1=0.01051 m:

$\begin{array}{cc}{d}_{\mathrm{tc}1}=\sqrt{\frac{4^{*}{V}_1}{{\pi }^{*}{\mathrm{len}}_1}}& \left(4\right)\end{array}$

In this embodiment, the third offline drilling fluid flows through the pipe for k=13 times.

A flow velocity v_sk of the third offline drilling fluid at the k-th time in a straight pipe and a flow velocity v_ck thereof at the k-th time in the curved pipe are shown in the table below.

An average pressure difference ΔP_ck/ΔL_ck of the offline curved pipe and an average pressure difference ΔP_sk/ΔL_sk of the offline straight pipe are also shown in the table below.

The third offline drilling fluid flows through the pipe for 13 times, and its friction coefficient f_ck in the curved pipe and friction coefficient f_sk in the straight pipe are calculated by formulas (1) and (2), and are shown in the table below.

$\begin{array}{cc}{f}_{ck}=\frac{d_{\mathrm{tc}1}}{2{\rho }_1\star v_{ck}^2}*\frac{\Delta P_{ck}}{\Delta L_{ck}}& \left(1\right)\end{array}$ $\begin{array}{cc}{f}_{sk}=\frac{d_{\mathrm{ts}1}}{2{\rho }_1\star v_{\mathrm{sk}}^2}*\frac{\Delta P_{\mathrm{sk}}}{\Delta L_{\mathrm{sk}}}& \left(2\right)\end{array}$

The corresponding actual friction coefficient ratios y_k are shown in the table below, where y_k=f_ck/f_sk.

In this embodiment, an average viscosity of the third offline drilling fluid is μ₁=0.01455 Pa·s.

The Dean numbers D_nk of the third offline drilling fluid flowing through the curved pipe for 13 times are calculated according to formula (6), which are shown in the table below.

$\begin{array}{cc}{D}_{nk}=\frac{{\rho }_1\star v_{ck}\star d_{\mathrm{tc}1}}{{\mu }_1}*\sqrt{\frac{d_{\mathrm{tc}1}}{D_{c1}}}& \left(6\right)\end{array}$

Number	ΔPck/	ΔPsk/
of times	ΔLck	ΔLsk	vsk	vck	fsk	fck	yk	Dnk
1	2.634	3.108	0.56	0.56	0.04	0.04	1.15	85.88
2	2.788	3.381	0.61	0.61	0.03	0.04	1.19	96.66
3	2.881	3.555	0.64	0.65	0.03	0.04	1.21	105.05
4	3.050	3.808	0.69	0.70	0.03	0.03	1.22	114.64
5	3.363	4.360	0.79	0.80	0.02	0.03	1.27	134.93
6	3.447	4.622	0.83	0.84	0.02	0.03	1.31	145.22
7	3.555	4.894	0.87	0.88	0.02	0.03	1.35	154.57
8	3.664	5.160	0.91	0.92	0.02	0.03	1.38	166.17
9	3.784	5.453	0.96	0.97	0.02	0.03	1.41	176.43
10	3.909	5.734	1.01	1.02	0.02	0.02	1.43	190.62
11	4.036	6.001	1.05	1.06	0.02	0.02	1.45	198.90
12	4.177	6.287	1.09	1.10	0.02	0.02	1.47	208.59
13	4.215	6.420	1.11	1.12	0.01	0.02	1.49	211.66

Three prediction models are fitted by the actual friction coefficient ratios y_k and the Dean numbers D_nk as follows:

a first prediction model:

ŷ _1k =a*D _nk ^b +c  (3)

where, a=0.0645, b=0.5, and c=0.5424.

a second prediction model:

$\begin{array}{cc}{\hat{y}}_{2k}=1+\frac{a\star D_{nk}^b}{70+D_{nk}}& \left(4\right)\end{array}$

where, a=0.0045, and b=1.9298.

a third prediction model:

ŷ _3k=1+a*(log₁₀D_nk)^b  (5)

where, a=0.0025, and b=6.2539.

The predicted friction coefficients of the third prediction models are shown in the table below.

Number
of times	ŷ1k	ŷ2k	ŷ3k
1	1.1401	1.1555	1.1547
2	1.1765	1.1827	1.1822
3	1.2035	1.2042	1.2040
4	1.2330	1.2292	1.2291
5	1.2916	1.2828	1.2830
6	1.3196	1.3103	1.3106
7	1.3443	1.3354	1.3358
8	1.3738	1.3668	1.3671
9	1.3991	1.3946	1.3948
10	1.4329	1.4332	1.4333
11	1.4520	1.4557	1.4557
12	1.4739	1.4822	1.4819
13	1.4807	1.4905	1.4902

A correlation R₁₁² between the actual friction coefficient ratio y_k and a predicted friction coefficient ratio of the first prediction model, a correlation R₁₂² between the actual friction coefficient ratio y_k and a predicted friction coefficient ratio of the second prediction model and a correlation R₁₃² between the actual friction coefficient ratio y_k and a predicted friction coefficient ratio of the third prediction model are calculated according to formulas (7), (8) and (9), respectively.

Through calculation, R₁₁²=0.9923, R₁₂²=0.9948 and R₁₃²=0.9949.

To sum up, in this embodiment, for the third offline drilling fluid, the third prediction model is the optimal prediction model. According to the optimal prediction model, the relationship constants between the friction coefficients are a=0.0025 and b=6.2539, which are used for the on-site measurement on the rheological property of the drilling fluid.

The on-site measurement on the rheological property of the drilling fluid in the curved pipe is described below.

In the third embodiment of the on-site measurement, the density of a third on-site drilling fluid is ρ₃=1,227 kg/m³.

The on-site curved pipe has a total volume V₂=1.04 l and a length len₂=5.57476 m.

An inner diameter of the on-site curved pipe is calculated as d_tc2=0.01051 m:

$\begin{array}{cc}{d}_{tc2}=\sqrt{\frac{4^{*}{V}_2}{{\pi }^{*}\mathrm{le}{n}_2}}& \left(11\right)\end{array}$

In this embodiment, the third offline drilling fluid flows through the on-site pipe for i=13 times.

A i-th flow velocity v_ci of the second on-site drilling fluid at the i-th time in the curved pipe and an average pressure difference thereof in the curved pipe ΔP_ci/ΔL_ci are shown in the table below.

The measured parameters of the on-site drilling fluid in the curved pipe are shown in the table below. The flow velocity of the drilling fluid is increased in the curved pipe in an ascending order to keep a laminar flow state of the drilling fluid.

	Rate
Number	of flow	Temperature
of times	(lpm)	(° C.)	ΔPci/ΔLci	ρ2	vci
1	2.926	29	3.11	1227.7	0.5617
2	3.195	29	3.38	1227.5	0.6133
3	3.385	29	3.55	1228.0	0.6497
4	3.640	29	3.81	1226.4	0.6988
5	4.146	29	4.36	1227.3	0.7958
6	4.355	29	4.62	1226.6	0.8360
7	4.563	29	4.89	1226.8	0.8759
8	4.802	29	5.16	1227.3	0.9218
9	5.028	29	5.45	1227.5	0.9652
10	5.313	29	5.73	1226.9	1.0199
11	5.512	29	6.00	1227.9	1.0582
12	5.741	29	6.29	1228.7	1.1021
13	5.815	29	6.42	1226.2	1.1163

The second on-site drilling fluid flows through the curved pipe for 13 times, and its friction coefficient f_ci in the curved pipe is calculated by formula (11), which is shown in the table below.

$\begin{array}{cc}{f}_{\mathrm{ci}}=\frac{d_{tc2}}{2{\rho }_2\star v_{\mathrm{ci}}^2}*\frac{\Delta P_{\mathrm{ci}}}{\Delta L_{\mathrm{ci}}}& \left(10\right)\end{array}$

The Reynolds number R_ei of the on-site curved pipe is calculated according to the selected optimal offline model. In this embodiment, the optimal prediction model is the third prediction model: ŷ_3k=1+a*(log₁₀ D_nk)^b, where the relationship constants between the friction coefficients are respectively: a=0.0025 and b=6.2539.

The Reynolds number R_ei of the on-site curved pipe is expressed by formula (12):

$\begin{array}{cc}{f}_{ci}=\frac{16}{R_{ei}}\left(1+a*{\left({\log }_{10}\left(R_{ei}*\sqrt{\frac{d_{tc2}}{D_{c2}}}\right)\right)}^b\right)& \left(12\right)\end{array}$

The Reynolds number R_ei of the on-site curved pipe is shown in the table below.

According to the Reynolds number R_ei of the on-site curved pipe, a shear stress τw_i of the drilling fluid in the on-site curved pipe is calculated, which is expressed by formula (15), and is shown in the table below.

$\begin{array}{cc}{\tau }_{wi}=\frac{8{\rho }_2\star v_{\mathrm{ci}}^2}{R_{ei}}& \left(15\right)\end{array}$

An intermediate parameter N_i is calculated according to formula (17), which is shown in the table below.

$\begin{array}{cc}{N}_i=\frac{d\left(\ln {\tau }_{\mathrm{wi}}\right)}{d\left(\ln \frac{8^{*}{v}_{\mathrm{ci}}}{d_{\mathrm{tc}2}}\right)}& \left(17\right)\end{array}$

A shear rate γ_i of the third drilling fluid in the on-site curved pipe is calculated according to formula (16), which is shown in the table below.

$\begin{array}{cc}{\gamma }_i=\frac{8^{*}{v}_{\mathrm{ci}}}{d_{tc2}}*\frac{3^{*}{N}_i+1}{4^{*}{N}_i}& \left(16\right)\end{array}$

Number
of times	fci	Rei	τwi	γi
1	0.042	438.213	7.072	474.696
2	0.038	491.082	7.521	518.882
3	0.036	534.109	7.764	550.185
4	0.033	589.713	8.124	592.341
5	0.029	699.632	8.888	675.882
6	0.028	739.707	9.271	710.482
7	0.027	779.393	9.661	744.922
8	0.026	838.195	9.953	784.520
9	0.025	885.939	10.327	822.071
10	0.024	970.584	10.518	869.288
11	0.023	1016.091	10.825	902.452
12	0.022	1075.315	11.102	940.432
13	0.022	1080.252	11.316	952.785

According to the shear stress τw_i and the shear rate γ_i of the third on-site drilling fluid, at least three on-site models are fitted, which are respectively:

a first on-site model:

{circumflex over (τ)}w_1i=YP+PV*γ_i  (18)

where, PV=0.0088, and YP=2.98.

a second on-site model:

{circumflex over (τ)}w_2i=K*γ_iⁿ  (19),

where, K=0.6715, and n=0.1126.

a third on-site model:

{circumflex over (τ)}w_2i=τ₀+K*γ_iⁿ  (20),

where, n=0.6716, K=0.1126, and τ₀=0.001.

Then the following parameters are respectively calculated.

a correlation R₂₁² between the shear stress τw_i of the second on-site drilling fluid and a predicted shear stress of the first on-site model;

a correlation R₂₂² between the shear stress τw_i of the second on-site drilling fluid and a predicted shear stress of the second on-site model; and

a correlation R₂₃² between the shear stress τw_i of the second on-site drilling fluid and a predicted shear stress of the third on-site model.

These parameters are calculated according to formulas (21), (22) and (23), respectively.

$\begin{array}{cc}{R}_{21}^2=1-\frac{\sum _{i=1}^m{\left({\mathrm{\tau w}}_i-\tau {\hat{w}}_{1i}\right)}^2}{\sum _{i=1}^m{\left({\mathrm{\tau w}}_i-\stackrel{\_}{\mathrm{\tau w}}\right)}^2}& \left(21\right)\end{array}$ $\begin{array}{cc}{R}_{22}^2=1-\frac{\sum _{i=1}^m{\left({\mathrm{\tau w}}_i-\tau {\hat{w}}_{2i}\right)}^2}{\sum _{i=1}^m{\left({\mathrm{\tau w}}_i-\stackrel{\_}{\mathrm{\tau w}}\right)}^2}& \left(22\right)\end{array}$ $\begin{array}{cc}{R}_{23}^2=1-\frac{\sum _{i=1}^m{\left({\mathrm{\tau w}}_i-\tau {\hat{w}}_{3i}\right)}^2}{\sum _{i=1}^m{\left({\mathrm{\tau w}}_i-\stackrel{\_}{\mathrm{\tau w}}\right)}^2}& \left(23\right)\end{array}$

Through calculation, R₂₁²=0.996314, R₂₂²=0.997775 and R₂₃²=0.997775. The second and third on-site models have the maximum correlation and are most in line with the actual situation, so the second and third on-site models are selected as the final models for calculating other viscosity data.

The on-site measurement results of the second on-site model are compared with those of the 6-speed viscometer, which shows that the third on-site model is the most suitable.

Measurement	Measuring
instrument	temperature/° C.	θ600	θ300	θ200	θ100	θ6	θ3	n	K
Fann35 (control)	29	22	14	10	6.5	1	0.8	0.6521	0.1226
System measurement	29	23.1	14.5	11.1	6.9	1.0	0.7	0.6715	0.1126
device

The on-site measurement results of the third on-site model are compared with those of the 6-speed viscometer, which shows that the third on-site model is the most suitable.

Measurement	Measuring
instrument	temperature/° C.	θ600	θ300	θ200	θ100	θ6	θ3	n	K	τ0
Fann35 (control)	29	22	14	10	6.5	1	0.8	0.6835	0.0950	0.4088
System measurement	29	23.1	14.5	11.1	6.9	1.0	0.7	0.6716	0.1126	0.001
device

If the viscosity data calculated by the first on-site model is directly selected without performing on-site model optimization, as shown in the table below, the deviation will increase significantly. The difference percentage of viscosity corresponding to θ100 is 2.2/6.5=35%, the difference percentage of viscosity corresponding to θ6 is 5/1=500%, and the difference percentage of viscosity corresponding to θ3 is 5.1/0.8=639%. The calculation results of the preferred second and third models of the present invention show that the difference percentage of viscosity corresponding to θ100 is 0.4/6.5=7%, the difference percentage of viscosity corresponding to θ6 is 0.1/5=5% and the difference percentage of viscosity corresponding to θ3 is −0.1/0.8=−17%.

Measurement	Measuring
instrument	temperature/° C.	θ600	θ300	θ200	θ100	θ6	θ3
Fann35	29	22	14	10	6.5	1	0.8
(control)
System	29	23.4	14.6	11.7	8.7	6.0	5.9
measurement
device

In conclusion, compared with the measurement results of Fann35, the calculation results of the optimal on-site models (the second and third on-site models) determined by the correlations of the actual shear stress τw_i and the predicted shear stresses of the on-site models are the most accurate.

The above described are merely preferred embodiments of the present invention, which are not intended to limit the present invention. Any modifications, equivalent replacements and improvements made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for measuring a rheological property of a drilling fluid by using a curved pipe on site, comprising the following steps: step 1: deriving relationship constants between friction coefficients of a drilling fluid through offline checking;step 2: calculating a Reynolds number Rei of the drilling fluid in an on-site curved pipe according to the relationship constants between the friction coefficients of the drilling fluid and a friction coefficient fci of the drilling fluid in the on-site curved pipe, wherein i denotes a number of times the drilling fluid flows through the on-site curved pipe, and i is a positive integer not less than 2;step 3: calculating an actual shear stress τwi of the drilling fluid in the on-site curved pipe according to the Reynolds number Rei of the drilling fluid in the on-site curved pipe, and calculating a shear rate γi of the drilling fluid according to the actual shear stress τwi;step 4: establishing a plurality of on-site models according to the actual shear stress τwi and the shear rate γi of the drilling fluid;step 5: determining an optimal on-site model according to correlations between the actual shear stress τwi and predicted shear stresses of the plurality of on-site models; andstep 6: performing on-site measurement on the rheological property of the drilling fluid according to the optimal on-site model.

2. The method for measuring the rheological property of the drilling fluid by using the curved pipe on site according to claim 1, wherein step 1 comprises: step 11: calculating a friction coefficient fck of the drilling fluid in an offline curved pipe and a friction coefficient fsk of the drilling fluid in an offline straight pipe, wherein k denotes a number of times the drilling fluid flows through an offline pipe, and k is a positive integer not less than 2;step 12: establishing a plurality of prediction models according to an actual friction coefficient ratio yk, wherein, yk=fck/fsk;step 13: determining an optimal prediction model according to correlations between the actual friction coefficient ratio yk and predicted friction coefficient ratios of the plurality of prediction models; andstep 14: deriving the relationship constants between the friction coefficients of the drilling fluid according to the optimal prediction model.

3. The method for measuring the rheological property of the drilling fluid by using the curved pipe on site according to claim 2, wherein in step 11, fck is expressed by formula (1):

4. The method for measuring the rheological property of the drilling fluid by using the curved pipe on site according to claim 2, wherein the plurality of prediction models at least comprise: a first prediction model: ŷ1k=a*Dnkb+c  (3)a second prediction model:

5. The method for measuring the rheological property of the drilling fluid by using the curved pipe on site according to claim 4, wherein step 13 comprises: expressing a correlation R112 between the actual friction coefficient ratio yk and a predicted friction coefficient ratio of the first prediction model by formula (7):

6. The method for measuring the rheological property of the drilling fluid by using the curved pipe on site according to claim 5, wherein in step 2, fci is expressed by formula (10):

7. The method for measuring the rheological property of the drilling fluid by using the curved pipe on site according to claim 6, wherein in step 2, the Reynolds number Rei of the drilling fluid in the on-site curved pipe is calculated as follows: when an offline model is the first prediction model, the Reynolds number Rei of the drilling fluid in the on-site curved pipe satisfies formula (12):

8. The method for measuring the rheological property of the drilling fluid by using the curved pipe on site according to claim 7, wherein in step 3, the actual shear stress τwi of the drilling fluid in the on-site curved pipe is expressed by formula (15):

9. The method for measuring the rheological property of the drilling fluid by using the curved pipe on site according to claim 8, wherein the plurality of on-site models at least comprise: a first on-site model: {circumflex over (τ)}w1i=YP+PV*γi  (18)a second on-site model: {circumflex over (τ)}w2i=K*γin  (19),a third on-site model: {circumflex over (τ)}w2i=τ0+K*γin  (20),whereinYP denotes a yield strength of the on-site drilling fluid, and has a unit of Pa;PV denotes a plastic viscosity of the on-site drilling fluid, and has a unit of Pa·s;n denotes a fluidity index of the on-site drilling fluid, and is dimensionless;K denotes a consistency coefficient of the on-site drilling fluid, and has a unit of Pa·s{circumflex over ( )}n; andτ0 denotes a dynamic shear stress of the on-site drilling fluid, and has a unit of Pa.

10. The method for measuring the rheological property of the drilling fluid by using the curved pipe on site according to claim 9, wherein step 5 comprises: expressing a correlation R212 between the actual shear stress τwi and a predicted shear stress of the first on-site model by formula (21):