Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 13, No. 3, 2024 

 

265 

Real‐time	Optimization	Method	of	Drilling	Fluid	
Rheological	Model	Based	on	Pipe	Viscosity	
Measurement	
Hai Yang, Peng Zhang 

Southwest Petroleum University, Chengdu, Sichuan 610500, China 

 

Abstract: The rheology of drilling fluid is crucial for bottom hole pressure control, hydraulic fracturing, and wellbore stability. 
Currently, manual methods like rotational viscometers are used on drilling sites, which have long measurement cycles and are 
prone to human interference. Existing online measurement devices with pipe viscometers are affected by fluid pulsation from 
diaphragm pumps and motor vibrations, making it hard to accurately determine and monitor rheological parameters in real-time. 
To address these issues, this paper proposes a real-time optimization method for the rheological model of drilling fluid based on 
a pipe viscometer. First, the rheological model of drilling fluid and the calculation method for piep viscometer parameters were 
analyzed, and a pulsation interference suppression model for the pressure difference in the rheological measurement pipe based 
on empirical mode decomposition was established. Then, preprocessing and regression analysis of the drilling fluid rheological 
model were carried out, and a real-time optimization method based on univariate linear regression and the golden section search 
algorithm was constructed. Finally, an experimental platform for online monitoring of drilling fluid rheology using a single 
measuring pipe was set up. Experimental results show that the proposed method effectively suppresses pulsation interference on 
the pressure difference in the measuring pipe. The shear stress-shear rate rheological model curve of the drilling fluid better 
matches the manual rotational viscometer parameters compared to directly measured parameters. The real-time optimization 
accuracy of the drilling fluid rheological model exceeds 95%, significantly improving the real-time measurement accuracy of 
drilling fluid rheological parameters. 

Keywords: Drilling fluid, Rheological parameters, Real-time measurement, Pipeline viscometer, Rheological model real-time 
optimization. 

 

1. Introduction 
Drilling fluid is the blood of drilling and plays an important 

role in the drilling process. The drilling fluid not only has the 
effect of carrying cuttings to complete the wellbore 
purification and stabilizing the formation without collapse, 
but also can control the bottom hole pressure balance, avoid 
causing well kick and blowout, and prevent accidents such as 
sticking. [1],[2]. 

At present, the measurement of drilling fluid rheological 
properties is still based on traditional manual measurement 
methods [3]. The commonly used drilling fluid performance 
measuring instruments include rotary viscometer, funnel 
viscometer and vibrating viscometer. These instruments 
generally use off-line, manual or semi-automatic methods to 
achieve measurement. The test process is complex and takes 
a long time. Sometimes it is necessary to manually calculate 
the measurement results. If an accident occurs in the wellbore, 
it cannot be known at the first time and the best treatment time 
is missed. Real-time online measurement of drilling fluid 
performance has become a research hotspot. 

The real-time measurement technology of drilling fluid 
rheology mainly includes the following three methods: (1) 
automatic rotary viscometer, (2) tuning fork viscometer, (3) 
pipeline viscometer. Broussard et al. [4] developed an real-
time rotating Couette viscometer and conducted experiments 
in water-based and oil-based mud wells. It demonstrates the 
possibility of automatic drilling fluid measurement using an 
real-time rotating Couette viscometer. However, the 
viscometer of this scheme is easily blocked by gel particles 
and solid particles. Miller et al. [5]introduced a system for 

continuous measurement and recording of mud density and 
viscosity using instruments based on tuning fork technology. 
When a significant change in density or viscosity occurs, a 
manual calibration instrument assembly using a standard 
Malvern funnel viscometer is required. Since the real-time 
rotating Couette viscometer is easily clogged, instruments 
based on tuning fork technology require manual calibration. 
Sercan Gul [6] and Knut Taugbøl [7] used a pipeline 
viscometer to test the rheological properties of drilling fluid. 
In the actual measurement, the use of high-power diaphragm 
pump to extract drilling fluid will cause serious pulsation of 
drilling fluid in the pipeline, and will also cause vibration of 
the pipeline, which will seriously affect the measurement 
accuracy. 

In order to optimize the rheological model of drilling fluid, 
this paper proposes a real-time optimization method for 
measuring the rheological properties of drilling fluid by single 
pipe. By controlling the drilling fluid to flow through the 
measuring pipe at different flow rates, the pressure difference 
at different flow rates can be obtained, which can significantly 
improve the measurement accuracy of drilling fluid 
performance and the optimization accuracy of drilling fluid 
rheological model. In view of the influence of the diaphragm 
pump on the measurement, Xu Ming et al. [8] analyzed the 
suppression mechanism of the fluid pulsation of the hydraulic 
accumulator on the basis of the accumulator modeling. The 
simulation model of the servo vibration generator was 
established by Amesim, and the good suppression effect of 
the accumulator on the fluid pulsation was verified by 
simulation. J. Berrebi et al. [9] modeled a flow pulsation 
model containing harmonics for the sampling error caused by 



 

266 

improper processing of the flow time average value, and 
finally reduced the sampling error through correct average 
processing; tianyi Zhou et al. [10]analyzed the dynamic 
characteristics of the differential pressure signal of gas-liquid 
two-phase flow in a 4mm diameter pipe, designed a 
differential pressure acquisition system, and used EMD and 
SVM to complete the flow pattern recognition. In order to 
accurately identify the stable pressure difference data, 
considering the regular pulsation of the diaphragm pump and 
the stable vibration characteristics of the motor, a feature 
recognition method of pressure difference parameters based 
on empirical model decomposition is proposed to ensure the 
accurate identification of effective pressure difference data 
under stable flow conditions. 

After obtaining stable pressure difference data, the 
corresponding shear stress and shear rate of the fluid can be 
calculated. The core problem of drilling fluid rheology is to 
study the relationship between shear stress and shear rate of 
various drilling fluid systems [11],[12] . At present, the least 
square method is mostly used to determine the rheological 
parameters of drilling fluid. The least square method has an 
important application when the drilling fluid is a linear 
rheological model. Its modeling is easy and the calculation 
process is relatively simple [13],[14]. However, when the 
drilling fluid is a nonlinear rheological model ( power law 
model, Carson model, Herschel-Barclay model, etc. ), the 
linear least squares method cannot meet the characteristics of 
the minimum variance of the calculation results and is no 
longer applicable. Many scholars have proposed a variety of 
methods for determining rheological parameters. Ojewumi et 
al. [15] ( 2018 ) used the error variance statistical parameters 
of five rheological models to propose a rheological model 
evaluation strategy of nonlinear least squares estimation 
algorithm, and realized the optimization of fluid rheological 
models. For the five rheological models commonly used in 
drilling fluids, Gucuyener [16] ( 2022 ) used the linear least 
squares method to parameterize the three-parameter nonlinear 
model, and realized the rapid solution of rheological 
parameters. However, for the nonlinear fluid rheological 
model, the versatility and calculation accuracy of the model 
still need to be improved. For a given drilling fluid system 
and formulation, real-time measurement should be performed 
to select the most suitable rheological model to describe its 
actual rheology. 

In view of the above problems, this paper proposes a fault-
tolerant optimization method of drilling fluid rheological 
model based on real-time monitoring device of drilling fluid 
performance. By measuring the pressure difference and flow 
rate at both ends of the pipeline, by controlling the flow rate 
through the measuring pipeline, different pressure difference 
values can be obtained. By solving different pressure 

difference and flow rate parameters, the shear stress-shear rate 
curve of the drilling fluid is calculated, and the rheological 
model of the measured drilling fluid is judged by using the 
linear regression-golden section search algorithm, and the 
final rheological model of the drilling fluid is obtained. 

2. Rheological Model and Principle of 
Drilling Fluid 

2.1. Common rheological models of drilling 
fluid 

The rheological models of drilling fluid are: Bingham 
model, power law model, Carson model and power law model 
with yield value (also known as Herschel-Barr-Clay model). 
The relationship between shear stress and shear rate obtained 
by the measurement and calculation of drilling fluid can 
accurately describe the rheological properties of drilling fluid. 
[17],[18],[19].  

At present, the rheological models commonly used in the 
field are mainly divided into two-parameter rheological 
models: Bingham model, power-law model and Carson model. 
Three-parameter rheological model: Herschel-Bulkley (H-B) 
model. 

The constitutive equation of drilling fluid can be written as 
𝜏 𝑓 𝜏, 𝛾 .  

The rheological model of drilling fluid commonly used in 
the drilling site and its corresponding rheological parameter 
calculation formula are shown in table 1. 

2.2. Rheological measurement principle of pipe 
viscometer 

The calculation of the rheological properties of the pipeline 
viscometer is based on the pressure loss transformation of the 
fluid in the pipeline viscometer. The rheological properties of 
the fluid are divided into two types according to different 
measurement principles: constant speed type and constant 
pressure type. The former mainly measures the pressure 
difference, and the latter mainly measures the flow rate. 
Usually in practical engineering, most of the high-pressure 
tubular rheological instruments used are constant-speed type. 
The principle of using a constant-speed tubular rheometer to 
measure fluid performance parameters is based on the Hagen-
Poiseuille theorem [20]. Pressure sensors are installed at both 
ends of the flow pipe. When the fluid flows through the flow 
pipe at a certain flow rate, the pressure at different positions 
in the pipe will change. Therefore, the viscosity of the fluid 
can be calculated according to the parameters such as the 
length and diameter of the flow pipe. The measurement 
principle is shown in Figure 1: 

 
Table 1. Common drilling fluid rheological model and its corresponding rheological parameter calculation formula 

Name of rheological model Characteristics of rheological model 
Constitutive equation of 

rheological model 

Bingham model 
The Bingham rheological model is one of the simplest rheological models, which is 
suitable for describing Newtonian fluids, that is, the shear stress is proportional to 

the shear rate. At any shear rate, the viscosity of the fluid remains unchanged. 
0   

 

Power Low model 
The power-law rheological model is suitable for non-Newtonian fluids, and there is 
a power-law relationship between shear stress and shear rate. With the increase of 

shear rate, the viscosity of the fluid will change. 

nK 
 

Casson’s model 
The Carson rheological model combines the characteristics of Newtonian fluid and 
non-Newtonian fluid, and is suitable for describing complex non-Newtonian fluid, 

such as fluid containing particles or multiphase fluid. 

1/2 1/2 (1/2) 1/2 c    
 

Herschel-Bulkely model 
The HB rheological model is a more flexible rheological model, which has wider 

applicability and can describe more complex fluid behavior. The model combines a 
variety of rheological properties, including shear stress, time dependence, etc. 

0
nK   

 



 

267 

 
In Table 1, 𝜏——shear stress, 𝑃𝑎; 𝜏 —— yield value, 𝑃𝑎; 

𝛾——shear rate, 𝑠 ; 𝜂——Bingham plastic viscosity, 𝑃𝑎. 𝑠; 
𝜂 —— Carson plastic viscosity, 𝑃𝑎. 𝑠 ; 𝐾——consistency 
coefficient, 𝑃𝑎. 𝑠, 𝑛——flow index. 

 

 
Figure 1. Pipe drilling fluid rheological model measurement schematic diagram 

 
The shear stress of the pipe wall can be obtained by the 

pressure difference between the two ends of the measuring 
pipe measured by the differential pressure sensor : 

 

 4
pD

w l                  (1) 
 
In above equations, D is the inner diameter of the pipe, p is 

the pressure difference between the two ends of the measuring 
pipe, l is the length of the measuring pipe. 

the wall shear rate 𝛾  inside the pipe can be expressed as : 
 

 
 
 

8

w

ln

w ln

1 8
3

4

v
Dd

d

v

D   



          (2) 

 
Let: 

 
 
 

w

8

ln

ln v
D

d

d
N                  (3) 

 
There are : 
 

 3 1 8
w 4

N v
N D                  (4) 

 
where 𝛾 is the shear rate at the pipe wall, 𝑠  ; 𝑣 is the 

velocity of the fluid through the measuring pipe, 𝑚/𝑠 ; N is 
a dimensionless parameter. According to the measured value, 
N can be obtained in the velocity gradient curve of 𝑙𝑛 𝜏
𝑙𝑛 .  

After the N value is determined, the rheological parameter 
values of each rheological model can be obtained. 

3. Parameter Calculation Model of 
Drilling Fluid Single Pipe 
Differential Pressure Signal Based 
on Empirical Mode Decomposition 

Pipeline measurement method to measure drilling fluid 
rheological model, the accuracy of measurement depends on 
the drawing accuracy of the flow curve, and the premise of 
accurately drawing the flow curve is to obtain accurate and 
stable pressure difference and flow rate values. When 
measuring the physical quantities of pressure difference and 
flow rate, because the equipment will produce diaphragm 

pump pulsation and motor vibration, the pressure difference 
signal measured by the pressure difference sensor will be 
affected, resulting in complex pulsation interference. It is 
impossible to obtain the accurate pressure difference signal of 
the drilling fluid in the constant current state, which seriously 
affects the calculation accuracy of the rheological parameters 
of the drilling fluid, so that the optimal drilling fluid 
rheological model cannot be optimized. 

Empirical Mode Decomposition ( EMD ) is a signal 
processing technology. EMD is proposed by Norden E. 
Huang in 1998, which is used to decompose nonlinear and 
non-stationary signals into several Intrinsic Mode Functions 
( IMFs ) for signal analysis and processing. The pulsating 
interference signal generated by mechanical action is a 
nonlinear and non-stationary signal, so the empirical mode 
decomposition is used to process the signal. 

 There are some limitations in the use of EMD : ( 1 ) The 
number of local extreme points and zero-crossing points must 
be equal or at most one in the whole time range of the 
function ; ( 2 ) At any time point, the envelope of the local 
maximum ( upper envelope ) and the envelope of the local 
minimum ( lower envelope ) must be zero on average. The 
process of EMD decomposition of pressure difference signal 
is as follows : 

(1) For a given differential pressure curve ( )tu  , find the 

maximum point and the minimum point ;  
(2) the maximum point and the minimum point are 

connected by different curves, and the maximum point 

envelope max( )tf   and the minimum point envelope min( )tf  

are given respectively.  

(3) The average curve ( )ta   = max( ) min( )( ) / 2t tf f  is 

obtained by averaging the two envelope curves.  
(4) Subtracting the average curve, we get a new function 

1 ( ) ( )( ) t tr t u a  , 1( )r t  is an IMF component.  

(5) Replacing the original ( )tu   with the newly obtained 

curve 1( )r t , and then repeating steps 1-4 to obtain different 

( )ir t  ( 2,3,4, )i   , until ( )ir t  can not meet the limiting 

condition cut-off, so there are : 
 



 

268 

 ( )
1

n

t i n
i

u IMF r


               (5) 

 
𝐼𝑀𝐹 , 𝐼𝑀𝐹 , ⋯ 𝐼𝑀𝐹  is the intrinsic mode function of the 

differential pressure signal, and 𝑟   is the residual, which 
represents the oscillation trend of the signal. 

The original pressure difference signal is decomposed by 
EMD to obtain multiple intrinsic mode functions, and each 
stationary signal is related to different factors. In order to find 
the IMF component of the pressure difference stationary 
signal related to the fluid pulsation, he kurtosis sensitive to 
the fluctuation is selected as the characteristic parameter of 
the steady flow pressure difference signal. By conducting 
kurtosis analysis, we identify the intrinsic mode function 
(IMF) component that encompasses the pressure difference 
signal under stable flow conditions, and then, through 
spectrum analysis, it is determined that the kurtosis K of the 
pulsation frequency is a numerical statistic, which reflects the 
distribution characteristics of the vibration signal and is a 
normalized fourth-order central moment. This statistic is very 
sensitive to sharp changes in variables, such as impact 
vibration. The larger the value, the stronger the pulsation. The 
expression is: 

 

 
4

4

[ ( ) ] ( )x t x r x dx
K





+∞

∞              (6) 

 
In above equations, 𝑥 is the instantaneous amplitude, �̅� 

is the mean amplitude, 𝑟  is the probability density, and 𝜎 
is the standard deviation.  

For discretely sampled differential pressure fluctuation 
data, the formula is 

 

 4

1

1
( )

N
i

i i

x x
K

N 


               (7) 

 
In above equations, 𝑥  is the signal value, �̅� is the signal 

mean, N is the sampling length, and 𝜎   is the standard 
deviation. 

4. Optimization Model of Drilling Fluid 
Rheological Model 

4.1. Regression analysis 
For the assumed rheological model, the rheological 

parameters that maximize the determination coefficient of the 
fitting curve or minimize the standard error are sought. By 
comparing the calculation results of different rheological 
models, the optimal rheological model is obtained. Numerical 
analysis and optimization theory are used to solve this 
extreme value problem. In this paper, a simple linear 
regression-golden section search algorithm is used to solve 
the problem, which is simple in principle, easy to program and 
can fully meet the accuracy requirements. 

Regression analysis is an effective tool to deal with the 
correlation between variables. According to the simple and 
representative case of sample values 
𝑥 , 𝑦 𝑥 , 𝑦 ⋯ 𝑥 , 𝑦  , 𝑦  and 𝑥  have a linear 

relationship 𝑦 𝑎𝑥 𝑏 . The problem of estimating the 
mathematical expectation of y by using the linear function 

𝑎𝑥 𝑏 . It is called a linear regression problem. For each 
value of the sample has 𝑦: 𝑁 𝑎𝑥 𝑏, 𝜎  , the normal 
assumption 𝑦 𝑎𝑥 𝑏 𝜀 , 𝜀: 𝑁 0, 𝜎   is called a linear 

regression model We take 𝑦
∧

𝑎
∧

𝑥 𝑏
∧
  as the estimation of 

the regression model. Gauss-Markov proves that 𝑎
∧
  and 𝑏

∧
 

are unbiased estimators of 𝑎 and 𝑏, and the variance is the 
smallest among all unbiased estimators using linear functions 
of 𝑦. 

Regression analysis is an effective tool to deal with the 
correlation between variables. According to the sample values 
𝑥 , 𝑦 𝑥 , 𝑦 ⋯ 𝑥 , 𝑦  , 𝑦 𝑓 𝑥   is determined. The 

simplest case is that 𝑦 has a linear relationship with 𝑥, that 
is, 𝑦 𝑎𝑥 𝑏 

Through the appropriate mathematical transformation of 
various rheological models, the rheological parameters of 
drilling fluid were calculated by one-dimensional linear 
regression analysis, and the program was compiled. The 
calculation of the rheological parameters of the Bingham and 
power-law models is simple and easy to implement, while the 
calculation of the Carson and H-B rheological models is 
relatively complex, especially the H-B rheological model, 
which requires computer iteration and interpolation. As far as 
the regression equation itself is concerned, it is not necessary 
to know in advance which rheological model is more 
reasonable for the relationship between the two variables. The 
sum can be obtained by the regression equation, and then 
judged according to the quantitative index of the correlation 
coefficient 𝑅  

4.2. Rheological equation preprocessing 
Four rheological equations are used as approximation 

functions ( For nonlinear equations, they are linearized by 
taking logarithms on both sides of the equation at the same 
time, and their fitting curves are made by using the least 
square method, and their correlation coefficient 𝑅   is 
obtained. By comparing the size of the correlation coefficient 
𝑅  , the most suitable rheological equation can be selected. 
The specific calculation steps are as follows :  

Firstly, the four rheological models are linearized : 

(1) Bingham fluid model : 0     , let y   , 

0 0a  , 1a  , x  , then 0 1y a a x    

(2) Power-law fluid model : 
nK  , take the logarithm 

at both ends of the equation, and get :  
 

 ln ln lnK n                (8) 

 

Suppose lny  , 0 lna K , 1a n , lnx   

(3) The Carson fluid model : 
1/2 1/2 (1/2) 1/2

c     , can 

also be written as  
 

 21/2 (1/2) 1/2( c     ）           (9)  

 

Let 
1/2y  , 

1
0

/2
ca  , 

(1/2)
1a  , 

1/2x  . 

(4) H-B rheological model ( Herschel-Bulkely ) fluid 
model : 𝜏 𝜏 𝐾𝛾  , Transforming 𝜏 𝜏 𝐾𝛾  , we 
take the logarithm at both ends of the formula, and get : 

 



 

269 

 0 )ln ln( ln K n                (10) 

 

Let 0 )ln(y   , 0 lna K , 1a n , lnx  . 

And 𝑅  can be obtained from the following formula :  
 

 
 

 

N 2n
i i

i 1
N 2

i i
1

y kx
2

y

1

i

y

R 








 


            (11) 

 
𝜏 - shear stress, Pa ; 𝜏 - yield value, Pa ; 𝛾 - shear rate, 

𝑠   ; 𝜂  - Bingham plastic viscosity, Pa.s ; 𝜂∞  - Carson 
plastic viscosity, Pa.s ; 𝐾 - consistency coefficient, Pa.s ; n
-flow index. 

4.3. Unary linear regression-golden section 
search algorithm 

The principle of the golden section search method is to 
quickly converge to the optimal value by continuously 
narrowing the search interval. The golden section method, 
also known as the 0.618 method, is a one-dimensional search 
method for the univariate function f(x) to search for the 
minimum point in a given interval [ a, b ]. The " golden 
section " refers to dividing a line segment into two segments, 
so that the ratio of the length of the whole segment to the 

length of the longer segment is equal to the ratio of the length 
of the longer segment to the length of the shorter segment, 
that is, 1 : L = L : ( 1-L ), and the solution is : L = 0.618. 

The search process of the golden section method : the 
golden section method requires the position of the insertion 
points 𝑥  and 𝑥  to be symmetrical with respect to the two 
ends of the interval [ a b ], that is, the point is taken in a given 
interval [ a b ]. 𝑥1 𝑏 𝐿 𝑏 𝑎 , if 𝑓 𝑥1 𝑓 𝑥2 ，let 
𝑎 𝑥1. If 𝑓 𝑥1 𝑓 𝑥2 , let 𝑏 𝑥2, restart the search. 

 

 
1

2

0.382( )

0.618( )

x a b a

x a b a

  
   

 (12) 

 
Since the Herschel-Barclay fluid is a three-parameter 

model, its characteristics lead to nonlinearity, and linear 
regression analysis alone cannot solve the problem. In this 
paper, the golden section search method is used to optimize. 
Assuming that 𝜏 is known, n and K values can be obtained 
according to regression analysis. This will return to the linear 
problem. According to the 𝜏  value obtained from the 
rheological equation, the golden section search method is 
used to search 𝜏  , and the process is repeated with the 
assumed accuracy until the local minimum value is reached, 
and the final 𝜏  value is obtained. The specific search process 
is shown in the program flow chart, as shown in Figures 2 : 

 

 
Figure 2. Golden section optimization algorithm flow chart 

 

5. Construction of The Experiment 
Platform 

In this paper, an on-line monitoring device for drilling fluid 
rheology was built. The overall frame of the measuring device 
is built of stainless steel rectangular pipeline. The measuring 
pipe is a stainless steel circular pipe of ø16.0 mm × 1.6 m and 
the wall thickness is 2 mm. The pressure difference sensor is 
installed at both ends of the measuring pipe. The outlet end of 
the electric diaphragm pump is connected with the mass 
flowmeter by a stainless steel circular pipe, and the flange 
connection is used between the pipelines to facilitate the 
disassembly and assembly of the pipelines and the sectional 

cleaning. The inlet end of the electric diaphragm pump is 
connected with the drilling fluid tank and the measuring outlet 
end by hose. The viscosity of drilling fluid is greatly affected 
by temperature. In order to eliminate this interference, the 
whole measuring device is now placed in the box. When the 
temperature fluctuation is large, the necessary insulation 
equipment is added to ensure the accuracy of the device 
measurement. The pressure difference parameters, 
temperature, drilling fluid density, mass flow rate, volume 
flow rate and other parameters collected by the equipment are 
transmitted to the industrial computer. On the industrial 
computer, the EMD method of the differential pressure signal 
mentioned in this paper is used to obtain the real differential 



 

270 

pressure signal under the constant current state. Combined 
with the rheological solution model, the calculation of the 
shear stress and shear rate of the drilling fluid is completed. 
Finally, Based on the obtained shear stress and shear rate data, 
the rheological model of the drilling fluid was optimized by 

employing the model selection approach described in this 
paper, resulting in the most accurate rheological model for the 
drilling fluid. The field test diagram of the device is shown 
in Fig.3.  

 

 
Figure 3. Device physical diagram 

 

6. Experimental Result Analysis 
The validation experiment for the proposed model in this 

paper was conducted using an real-time monitoring device for 
the rheological properties of drilling fluid. Firstly, the drilling 
fluid in the mud tank is extracted by an electric diaphragm 
pump, and the drilling fluid flows through the mass flowmeter 
and a single measuring pipe in turn. Real-time data on the 
drilling fluid's density, temperature, flow rate, and the 
pressure difference across the measurement pipe were 
collected. The experiment utilized a variable frequency drive 
to adjust the frequency of the diaphragm pump, with 8 
different frequency settings ranging from low to high. Let 
each frequency last the same time to control the stability of 
the flow rate in the pipeline so as to obtain the pressure 
difference in the steady state of the flow rate in real time. 
Among them, the original data collected by the pressure 
difference sensor after performing a complete pressure 
difference sensor is shown in Figure 4. 

 

 
Figure 4. The original pressure difference of the measuring 

pipe collected by the pressure difference sensor 
 

It can be seen from Fig.4 that the original pressure 
difference parameters of the measuring pipe have complex 

fluctuations and interference. At the same time, in each stable 
flow stage artificially set by the diaphragm pump, the 
measured pressure difference parameters still have large 
fluctuations, and are greatly affected by the pulsation effect 
of the diaphragm pump, pipeline vibration, and measurement 
noise. It has been unable to smoothly find the pressure 
difference parameters in the stable flow stage. Therefore, it is 
impossible to judge the constant current state of the 
diaphragm pump by using the pressure difference parameters 
shown in Fig.4, and it is impossible to obtain the actual stable 
pressure difference parameters of the measuring pipe, which 
leads to the inability to solve the rheological parameters of the 
drilling fluid. 

The above original pressure difference parameters of the 
measuring pipe are brought into the empirical mode 
decomposition model proposed in this paper, and the noise 
interference signals under different modes in the original 
pressure difference signal can be obtained. The empirical 
mode decomposition of the pressure difference parameters of 
the long measuring pipe is shown in Figure 5. 

 

 
Figure 5. Measuring pipe pressure difference parameter 

empirical mode decomposition waveform 
 

It can be seen from Fig.5 that the pressure difference data 



 

271 

are interfered by the errors of multiple different frequency 
bands, which makes it impossible to accurately identify the 
pressure difference data under the actual constant flow rate, 
and affects the calculation of drilling fluid rheological 
parameters. Therefore, the pressure difference parameters 
after empirical mode decomposition are shown in Figure 6. 

 

 
Figure 6. The original pressure difference of the measuring 

pipe collected by the pressure difference sensor 
 

It can be seen from Fig.6 that the pressure difference data 
after empirical mode decomposition eliminates a lot of noise 
and fluctuation compared with the original data, and the 
performance is more smooth. At the same time, the eight 
constant flow rate states are set, and the constant phenomenon 
of the pressure difference data can also be directly observed, 
and the simple variance threshold determination method can 
be used directly to judge the set constant current state, and 
read out the pressure difference parameters corresponding to 
each constant current state. After obtaining the stable pressure 
difference parameters, the drilling fluid flow rate under the 
current pressure difference is obtained by the mass flowmeter. 
Combined with the diameter of the experimental device, the 
length of the pipe is measured, and the above shear stress and 
shear rate calculation model is used to obtain the rheological 
parameters. 

In order to illustrate the practicability of the method 
proposed in this paper, 10 groups of drilling fluid experiments 
were carried out. Table 2 lists the typical data samples of 
drilling fluid to configure 10 groups of drilling fluid into the 
drilling fluid rheology real-time measurement device for test. 
For comparison, we have tested the readings of ten drilling 
fluid viscometers, as shown in Table 2. 

 
Table 2. Drilling fluid laboratory test data 

Drilling fluid 
number 

Rotary viscometer readings 
𝜑  𝜑  𝜑  𝜑  𝜑  𝜑  

1 0.6 0.8 2.8 4.4 6.3 9.2 
2 1.5 2 7 12 18 35 
3 19 20 23 23.5 29 33 
4 15 17 30 35 44 61 
5 23 25 49 56 66 74 
6 3 4 14.5 26 37 71 
7 2 4 21 36 50 91.5 
8 3 5 29 48.5 67 119 
9 4 7 35.5 59 79 137 

10 10 14.5 52 83 103 181 
 
There is a fluctuation in the pressure difference, and the 

measured data is preprocessed. Using the above EMD method 
and the pressure difference data from the determination 
condition to the constant current, the corresponding shear 

stress-shear rate is calculated according to the corresponding 
pressure difference and flow rate brought into the above 
model. At the same time, the shear stress-shear rate calculated 
by the reading of the paradigm rotating viscometer is used as 
the standard. The representative 5,7,9 groups of drilling fluids 
were measured and substituted into the shear stress-shear rate 
curve calculated by the model described in this paper. The 
image is shown in Figure 7(a),(b),(c) 

 

 
(a) 

 
(b) 

 
(c) 

Figure 7. Shear stress-shear rate curve of drilling fluid.(a) 
Shear stress-shear rate curves of 5 group of uncalibrated 

drilling fluid before and after pressure difference 
calibration.(b) Shear stress-shear rate curves of 7 group of 

uncalibrated drilling fluid before and after pressure 
difference calibration. (c) Shear stress-shear rate curves of 9 
group of uncalibrated drilling fluid before and after pressure 

difference calibration 
 

It can be seen from the Fig.7 that the shear stress-shear rate 
curve of the uncalibrated drilling fluid, the shear stress-shear 
rate curve of the measuring pipe ( the blue box symbol in the 



 

272 

figure ) and the shear stress-shear rate curve of the standard 
paradigm rotary viscometer ( the red * symbol in the figure ) 
do not completely coincide, and there is a large deviation. 
According to the principle that the drilling fluid flowing 
through the measuring pipe and the drilling fluid measured by 
the rotary viscometer are the same performance drilling fluid, 
the shear stress-shear rate curve of the measuring pipe should 
coincide with the shear stress-shear rate curve of the rotary 
viscometer. The uncalibrated shear stress-shear rate curve of 
the drilling fluid in the Fig.7 has a large deviation from the 
shear stress-shear rate curve of the paradigm rotary 
viscometer. This is due to the pulsation of the diaphragm 
pump and the vibration of the motor during the operation of 
the equipment. Therefore, the differential pressure data 
measured by the differential pressure sensor will be affected, 
resulting in complex pulsation interference. It can be seen 
from the figure that at the same shear rate, the maximum 

difference between the shear stress of the measuring pipe and 
the shear stress of the paradigm rotary viscometer is 19 Pa. 

From the lower part of Fig.7, it can be seen that the shear 
stress-shear rate curve of the measuring pipe after calibration 
can coincide well with the shear stress-shear rate curve of the 
normal rotary viscometer. At the same shear rate, the 
maximum difference between the shear stress of the 
measuring pipe and the shear stress of the paradigm rotary 
viscometer is 3Pa, which can effectively improve the 
optimization accuracy of the drilling fluid rheological model. 

In order to better illustrate the effectiveness of the search 
method based on empirical mode decomposition and 
univariate linear regression-golden section, the shear stress-
shear rate scatter plots measured by the drilling fluid rheology 
real-time detection device of the 10 groups of drilling fluids 
participating in the experiment were brought into the program. 
The results are shown in Fig. 8 (a),(b),(c). 

 

 
(a) 

 
(b) 

 
(c) 

Figure 8. linear regression analysis showed 

0 200 400 600 800 1000 1200

0

20

40

60

80

100

S
he

ar
 s

tr
es

s 
(P

a)

Shear rate(s-1)

Bingham model linear fitting
 1
 2
 3
 4
 5
 6
 7
 8
 9
 10
 1 set of fitting curves
 2 set of fitting curves
 3 set of fitting curves
 4 set of fitting curves
 5 set of fitting curves
 6 set of fitting curves
 7 set of fitting curves
 8 set of fitting curves
 9 set of fitting curves
 10 set of fitting curves

2 3 4 5 6 7
0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Power law model linear fitting

 1
 2
 3
 4
 5
 6
 7
 8
 9
 10
 1 set of fitting curves
 2 set of fitting curves
 3 set of fitting curves
 4 set of fitting curves
 5 set of fitting curves
 6 set of fitting curves
 7 set of fitting curves
 8 set of fitting curves
 9 set of fitting curves
 10 set of fitting curvesSh

ea
r 

st
re

ss
(P

a)

Shear rate(s-1)

0 5 10 15 20 25 30 35
1

2

3

4

5

6

7

8

9

10

Carson model linear fitting
 1
 2
 3
 4
 5
 6
 7
 8
 9
 10
 1 set of fitting curves
 2 set of fitting curves
 3 set of fitting curves
 4 set of fitting curves
 5 set of fitting curves
 6 set of fitting curves
 7 set of fitting curves
 8 set of fitting curves
 9 set of fitting curves
 10 set of fitting curvesS

he
ar

 s
tr

es
s(

Pa
)

Shear rate(s-1)



 

273 

 

The results show that the 10 groups of drilling fluids are 
linearized in Bingham model, power law model , Carson 
model  respectively. 

Because the H-B model cannot be directly linearly fitted. 
Taking drilling fluid 5 as an example, the golden section 
search method quickly converges to the optimal value by 
continuously narrowing the search interval. As shown in 
Fig.9(a), the 𝜏   is searched according to the regression 
analysis and the golden section search method. The golden 
section search process is completed according to the given 
conditions, and the corresponding 𝜏  is obtained. The n and 
K values are obtained by linear regression, and the 
corresponding rheological model of the drilling fluid is 
judged according to 𝑅 . The results in Fig.9 (b) shows that 
the 10 groups of drilling fluids are linearized in H-B model 
respectively. 

 

 
(a) 

 
(b) 

Figure 9. H-B model golden section search process of 
drilling fluid 5 

 

The golden section-linear regression algorithm was used to 
fit the 10 groups of drilling fluids. The 10 groups of drilling 
fluids all regressed the optimal results and obtained the 
corresponding 𝑅  values. Linear regression is performed to 
obtain the correlation coefficient value, which can 
quantitatively determine the most consistent rheological 
model, that is, when the correlation coefficient 𝑅  is closest 
to 1, the theoretical correlation is the largest. From Table 3,4 
and Fig.10, it can be seen that the regression effect of 
Bingham, power law and Carson model in different drilling 
fluid systems is better by converting into linear model, but it 
is only applicable to specific types of drilling fluid. The 
regression effect of the linear regression-golden section 
search algorithm used in the H-B model is better, which can 
better express the rheology of the drilling fluid and has certain 
adaptability. 

 

 
(a) Rotary viscometer optimization results 

 
(b) Rotary viscometer optimization results 

Figure 10. The final calculated 𝑅  histogram 
 

Table 4. Comparison of drilling fluid rheological model optimization results 
Drilling fluid 

number 
Rotary viscometer optimization results Optimization results of drilling fluid on-line monitoring device 

1 Power-law model fluid Power-law model fluid 
2 Bingham model fluid Bingham model fluid 
3 Bingham model fluid Bingham model fluid 
4 Carson model fluid Carson model fluid 
5 Herschel-Bulkely model fluid Herschel-Bulkely model fluid 
6 Bingham model fluid Bingham model fluid 
7 Bingham model fluid Bingham model fluid 
8 Bingham model fluid Bingham model fluid 
9 Carson model fluid Carson model fluid 
10 Carson model fluid Carson model fluid 

2 3 4 5 6 7

0

1

2

3

4

5

H-B model linear fitting
 1
 2
 3
 4
 5
 6
 7
 8
 9
 10
 1 set of fitting curves
 2 set of fitting curves
 3 set of fitting curves
 4 set of fitting curves
 5 set of fitting curves
 6 set of fitting curves
 7 set of fitting curves
 8 set of fitting curves
 9 set of fitting curves
 10 set of fitting curvesS

he
ar

 s
tr

es
s(

P
a)

Shear rate(s-1)

0.
97

96

0.
98

63

0.
95

37

0.
97

17

0.
83

37

0.
99

58

0.
99

49

0.
99

36

0.
98

98

0.
98

74

0.
98

35

0.
86

04

0.
76

88

0.
94

94 0.
99

78

0.
94

54

0.
93

94

0.
97

33

0.
97

81

0.
98

58

0.
98

95

0.
94

71

0.
90

95

0.
98

88

0.
95

24

0.
98

1

0.
98

52

0.
99

24

0.
99

2

0.
99

46

0.
98

35

0.
87

13

0.
77

25

0.
95

37 0.
99

79

0.
94

54 0.
98

94

0.
97

33

0.
97

81

0.
98

63

1 2 3 4 5 6 7 8 9 10
0.0

0.2

0.4

0.6

0.8

1.0

L
in

ea
r 

fi
tt

in
g 

va
ri

an
ce

 o
f 

ro
ta

ry
 v

is
co

m
et

er

Number of drilling fluid groups

 Bingham model variance
 Power law model variance
 Carson model variance
 H-B model variance

0.
98

2

0.
98

3

0.
91

4 0.
97

1

0.
88

3

0.
98

2

0.
99

3

0.
99

2

0.
99

4

0.
98

7

0.
93

5

0.
83

3

0.
62

2

0.
91

0.
99

5

0.
9

0.
96

4

0.
97

4

0.
97

8

0.
98

0.
98

3

0.
94

7

0.
83

7

0.
97

7

0.
97

7

0.
96

3

0.
99

1

0.
99

1

0.
99

4

0.
99

6

0.
94

6

0.
87

9

0.
62

6

0.
91

6

0.
99

6

0.
92

6 0.
97

9

0.
98

4

0.
98

5

0.
98

4

1 2 3 4 5 6 7 8 9 10
0.0

0.2

0.4

0.6

0.8

1.0

L
in

ea
r 

fi
tt

in
g 

va
ri

an
ce

 o
f 

dr
il

li
n

g 
fl

ui
d 

on
li

n
e 

m
on

it
or

in
g 

d
ev

ic
e

Number of drilling fluid groups

 Bingham model variance
 Power law model variance
 Carson model variance
 H-B model variance



 

274 

Table 3. Linear regression results of drilling fluid rheological parameters 

Drilling fluid 
number 

Bingham model fluid Power-law model fluid Carson model fluid Herschel-Bulkely model fluid 

𝝉𝟎 𝜼 𝑹𝟐 𝑲 𝒏 𝑹𝟐 𝝉𝒄
𝟏 𝟐 𝜼

𝟏
𝟐 𝑹𝟐 𝝉𝟎 𝑲 𝒏 𝑹𝟐 

1 0.0532 4.674 0.9817 0.5137 0.3296 0.9346 0.1902 1.4441 0.9829 0.0033 0.5138 0.3287 0.9347 
2 0.6408 0.0175 0.9834 0.4994 0.450769 0.8329 0.6669 0.1099 0.947 0.0031 0.4997 -0.7992 0.8331 
3 10.459 0.0074 0.914 0.1012 7.956586 0.6221 3.0467 0.0346 0.8367 0.0044 0.1012 2.0736 0.6221 
4 10.932 0.0188 0.9705 0.2377 5.048545 0.9096 2.7964 0.0808 0.9774 0.0037 0.2378 1.6186 0.9096 
5 19.376 0.0218 0.8832 0.2283 7.921654 0.9952 3.6915 0.0851 0.9774 12.9952 0.7724 -1.9808 0.9959 
6 0.1445 0.04 0.9819 0.5747 0.600856 0.9 0.7011 0.1725 0.9625 0.0047 0.575 -0.5117 0.9001 
7 2.5443 0.0475 0.993 0.6074 0.623504 0.9643 0.9417 0.1866 0.9908 0.0047 0.6077 -0.4746 0.9643 
8 3.0073 0.0603 0.9923 0.6286 0.713481 0.974 1.0162 0.2159 0.9911 0.0034 0.6285 -0.337 0.974 
9 5.409 0.0668 0.9935 0.6013 0.994913 0.9777 1.3197 0.2238 0.9944 0.0048 0.6015 -0.0065 0.9778 
10 12.614 0.0741 0.9872 0.5052 2.346442 0.9801 2.2366 0.2208 0.9958 0.0047 0.5053 0.8521 0.9801 

 
It can be seen from the calculation results that the device 

can measure the rheological properties of drilling fluid in real 
time, automatically select the corresponding rheological 
model of drilling fluid through the algorithm, and 
continuously record the change process of the rheological 
properties of drilling fluid. In this experiment, the method of 
this paper is used to optimize the rheological model of drilling 
fluid. The results are the same as the actual drilling fluid 
rheological model, and the accuracy rate is above 95 %. 

7. Conclusion 
In this paper, aiming at the problems of low measurement 

accuracy and poor stability caused by the influence of fluid 
pulsation, motor vibration and bias error of differential 
pressure measurement in the existing pipeline viscometer 
real-time measurement device, a differential pressure signal 
calibration method based on empirical mode decomposition 
is proposed on the basis of single-pipe differential pressure 
pipeline viscometer. The differential pressure signal 
processing is carried out by using the real-time monitoring 
experimental platform of drilling fluid performance. Firstly, 
the original differential pressure signal of single measuring 
pipe is collected, and the original differential pressure signal 
is decomposed and filtered by the established empirical mode 
decomposition model. The filtered curve can significantly 
capture the state of diaphragm pump under each constant 
current control compared with the original differential 
pressure curve, and can obtain accurate differential pressure 
signal, which provides data support for the subsequent 
calculation of shear stress and shear rate. Then, the 
rheological experiments of 10 groups of drilling fluids with 
different viscosities were completed by using flow rate, 
pressure difference and measuring pipe size parameters. The 
shear stress-shear rate curves were drawn by using the actual 
measured pressure parameters and the calibrated pressure 
values in this paper. The calibrated shear stress-shear rate 
curve can be consistent with the curve of the rotary viscometer, 
and the maximum shear stress difference is reduced from 19 
Pa to 3 Pa. According to the one-dimensional linear 
regression-golden section search algorithm proposed in this 
paper, the above 10 groups of drilling fluids are optimized. 
The optimized results are completely consistent with the 
actual rheological model of drilling fluid, and the automatic 
process of determining the optimal rheological model of 
drilling fluid is completed , with an accuracy rate of more 
than 95 %. The established drilling fluid model optimization 
method is of great significance for monitoring the rheology of 
drilling fluid, optimizing drilling operations, and avoiding 
problems such as wellbore cleaning, sticking, and leakage. 

Acknowledgements 
This work was supported in part by the National Natural 

Science Foundation of China under Grant 52204013, in part 
by the Scientific Research and Technological Development 
Project of CNPC 2023ZZ0603, in part by the National Key 
Research and Development Program of China 
2023YFF0611101, and in part by the Science and Technology 
Cooperation Project of the China National Petroleum 
Corporation (CNPC)-Southwest Petroleum University 
(SWPU) Innovation Alliance under Grant 2020CX040000. 

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