# Introducing a Rheology Model for Non-Newtonian Drilling Fluids

Authors: Masoud Rashidi; Ahmad Sedaghat; Biltayib Misbah; Mohammad Sabati; Koshy Vaidyan; Ali Mostafaeipour; Seyyed Shahabaddin Hosseini Dehshiri; Khalid Almutairi; Alibek Issakhov
Journal: Geofluids
 (2021)

---

## Abstract

An API standard drilling fluid was investigated from laminar to turbulent flow conditions using an in-house-built viscometer at speeds from 200 to 1600 RPM. A power-based method was applied to obtain the apparent viscosity and the shear stress of the water-based drilling mud (WBM) in the annulus of the viscometer. Then, a MATLAB optimization program was developed to obtain model parameters for five rheology models integrated in a generalized Herschel-Bulkley-Extended (HBE) model and two widely used 4-parameter models in drilling industry. It is found that the Bingham, Cross, and HBE rheology models have precisely matched the WBM measurements in the viscometer. A generalized Reynolds number was applied to determine the Darcy friction factor although the PL (power law model) and the HB (Herschel-Bulkley model) exhibited a nonrealistic negative shift from the laminar friction factor.


---

## Body

## 1. Introduction

Energy is one of the most important needs of human life and plays an important role in the development of countries. Energy consumption is the amount of energy consumed in a year. It includes all energy harnessed from every energy source applied towards operation across all industrial and technical sectors, in every country [1]. World energy consumption has consequences for the socio-economic-political sphere [2, 3]. Today, with the dramatic growth of population and the development of industries, the demand for energy in the world has increased [4, 5]. Many countries are facing energy supply problems to meet the demand of the industrial, agricultural, and residential sectors [6]. The growing global demand for energy in many different countries [7] and the declining production of oil from existing oil sources have renewed engineers’ interest in discovering new opportunities in deep and unconventional deep-water hydrocarbon reservoirs [8]. Extraction of crude oil from reservoirs is becoming more challenging due to various factors including increasing level of asphaltene, needs to drill deeper wells in search of new reservoirs, offshore explorations, and horizontal wellbore drilling. In tribology studies, the main attention is given to friction between two or more surfaces which is affected by asperity, abrasiveness, and shape of surfaces which are mainly dependent on materials of surfaces [9], whilst in rheology studies, behavior of fluids is sought. In oil and gas wells, drilling fluid exhibits shear thinning behavior which is known as thixotropy [10].The drilling process is successful when the drilling fluid used in this process is effective [11]. The drilling method for oil and gas wells is drilling of a telescopic hole from the surface to the reservoir which can be kilometers away from the surface. Drilling is accomplished with the use of a drilling bit connected to a long string of drill pipe. By applying weight and rotation on the bit, the bit crushes the rock into small fragments, the cuttings [11, 12]. Drilling fluid is circulated from the surface, through the drill pipe to the bit face; it lifts the generated cuttings and brings them to the surface, where separation equipment removes the cuttings from the drilling fluid, which is circulated, with the help of powerful pumps, back to the wellbore [11, 13]. Drilling fluids perform additional functions [11, 13–15]; mainly, they control subsurface pressures, stabilize the exposed rock, prevent contamination of subsurface formation hydrocarbon fluids, provide buoyancy, and cool and lubricate the bit. Such fluids must be engineered so that they can perform efficiently in harsh environments, and it must be ensured that they do not damage the formations which are being drilled.In oil and gas wellbore drilling industry, drilling fluids are highly complex and are prepared using certain additives to provide desirable tribology and rheology properties for specific drilling methods and reservoir conditions. The drilling fluids are generally classified as water-based mud (WBM), inhibitive WBM, oil-based mud (OBM), synthetic-based mud (SBM), foam fluids, gas or air, emulsions, and so on [16, 17]. WBM fluids are preferred drilling fluids because of their environmentally friendly characteristics and lower operation costs, but OBMs are preferred fluids in deeper wellbore drilling and high-pressure-high-temperature (HPHT) drillings. This is partially attributed to the ionic nature of water molecules in WBM with a positive charge at the hydrogen side that attracts a negative charge at the oxygen side of water molecules by van der Waals forces. Hence, WBM is susceptible to higher internal friction as compared to OBM. Therefore, OBM drilling fluids are preferred over WBM fluids due to better lubrication properties.The drilling industry is continuously seeking drilling fluids with enhanced mechanical, physical, chemical, and thermal properties that are also environmentally friendly in all exploration and extraction of oil and gas reservoirs [18]. To increase drilling speed and to overcome difficulties particularly in HPHT deep well drilling including increased drag force and torque, it is required to use “smart drilling liquids” that are usually referred to drilling fluids with additives such as organomolybdenum substances, organic friction reducers, and nanoparticles [19].Fakoya and Ahmed [20] have studied rheological properties of WBM and OBM and developed generalized empirical models for apparent viscosity of OBM and WBM. Kumar et al. [21] suggested a generalized shear rate model for drilling fluids using FANN viscometer measurements. They claimed that the generalized shear rate model can be used for any rotational coaxial-cylinder viscometer with any fluids particularly Herschel-Buckley-type drilling fluids.In well drilling operations, it is important to correctly determine the hydrodynamic behavior of drilling fluids in the well annulus space within “the operating window” to sustain correct pressure to avoid instability and fractural collapse of the well. The hydrodynamic nature of mud between the drill string and well bore is very complex due to the non-Newtonian nature of drilling fluid, laminar or turbulent flow condition, and eccentric rotation of the well string [22]. Classical lab equipment referred as the Taylor-Couette system (TCV) can be a good model for studying the wellbore drilling process. TCV consists of two coaxial rotational cylinders to study viscous flows in the gap of the system. For low rotational speed and low Reynolds number flows, the flow inside the gap is steady and azimuthal and is referred to as circular Couette flow. Couette [23] developed his system to be used as a viscometer. Later in 1923, Taylor [24] reported flow instability in TCV which opened the new science of hydrodynamic instability and proved the concept of no-slip condition on a wall. In general, TCV uses the inner cylinder as the rotating one and the outer one as the stationary one. Taylor developed linear stability theory by studying TCV with the inner cylinder being rotating and the outer cylinder being stationary. Taylor showed that the gap fluid flows become unstable if the rotational speed of the inner cylinder exceeds a particular limit.TCV is one of the widely used geometries in turbulence modelling. The main advantages of TCV are that all power input of the system through an external electric motor is known and the energy loss on wall surfaces can be related to friction torque and rotational speed of TCV [25]. Many different flow regimes were observed in TCV including the Taylor vortex, modulated wavy vortex, wavy vortex, turbulent vortex, and random-pattern turbulent flows [26]. Many top scientists have studied Taylor-Couette flow regimes from laminar flow and its transition to instabilities and then to turbulence. Chandrasekhar [27] and Drazin and Reid [28] presented comprehensive theoretical solutions to many Taylor-Couette flow-type problems.In the present work, a smaller Taylor-Couette viscometer (TCV) similar to [29] is designed and manufactured completely in Kuwait, with 100 ml volume for fluids with circular speeds of 0-1600 rpm, to investigate experimentally skin friction of laminar and turbulent flows. Experimental measurements are conducted at different rotational speeds of the Taylor-Couette system. The WBM was tested at different rotational speeds of TCV. The experimental results in laminar and turbulent flow regimes of TCV are reported here.


## 2. Research Methodology

### 2.1. High-Speed Viscometer

The Taylor-Couette viscometer is a confined cavity between two cylinders with the same axis which are also closed from the top and the bottom by two disks as shown in Figure1. Rotation of one or both cylinders creates a rotational confined viscous flow inside the gap of cylinders which are also influenced by the top and the bottom solid disk boundary conditions to make it a completely 3D complex flow system. Fluid flows in the TCV have been deeply investigated for a variety of flow regimes and flow patterns by many researchers.Figure 1
Sketch of the Taylor-Couette viscometer.Many complex flow phenomena have been observed in the Taylor-Couette system which is widely explored for studying linear and nonlinear hydrodynamic stability [30] and also for developing turbulence models [25]. Andereck et al. [26] reported a number of complex flow regimes for flows between two counterrotating cylinders in TCV such as basic Couette flow and appearance of Ekman cells, development of primary instabilities, wavy-vortex flow, low Reynolds modulated wavy-vortex flow, interpenetrating laminar spirals, wavy interpenetrating spirals, intermittency, transition region, spiral turbulence, turbulent flow, and hysteresis. For flow between corotating cylinders in TCV, they reported basic flow, primary instability, twisted vortices, wavy vortex boundary flows, and high azimuthal-wave number flows.The most complete flow model for viscous flows is Navier-Stokes equations discussed next for TCV.


### 2.2. Taylor-Couette Flow Equations

The Navier-Stokes equations in three dimensionsr,θ,z for incompressible viscous flows with constant dynamic viscosity (μ) and constant density (ρ) may be expressed as follows [31]:
(1)1r∂∂rrur+1r∂∂θruθ+∂uz∂z=0,(2)ρDurDt−uθ2r=−∂p∂r+Fr+μ∇2ur−urr2−2r2∂uθ∂θ,(3)ρDuθDt+uθurr=−1r∂p∂θ+Fθ+μ∇2uθ−uθr2−2r2∂ur∂θ,(4)ρDuzDt=−∂p∂z+Fz+μ∇2uz.In equations (1)–(4), ur,uθ,uz and Fr,Fθ,Fz are velocity and body force components in r,θ,z directions, respectively. p is the static or thermodynamic pressure, and the operators D/Dt and ∇2 are material derivative and Laplace operators defined by [31]
(5)DDt=∂∂t+ur∂∂r+uθr∂∂θ+uz∂∂z,∇2=∂2∂r2+1r∂∂r+1r2∂2∂θ2+∂2∂z2.


### 2.3. Stress Tensor in Newtonian Flows

For Newtonian incompressible flows, a stress tensor for an infinitesimal volume of fluid is obtained as follows [31]:
(6)σrr=−p+2μ∂ur∂r,(7)σθθ=−p+2μ1r∂uθ∂θ+urr,(8)σzz=−p+2μ∂uz∂z,(9)σrθ=μ1r∂ur∂θ+∂uθ∂r−uθr,(10)σrz=μ∂ur∂z+∂uz∂r,(11)σθz=μ1r∂uz∂θ+∂uθ∂z.


#### 2.3.1. Taylor-Couette System with the Stationary Outer Cylinder

Dou et al. [32] obtained solutions to the steady-state Navier-Stokes equation in TCV for incompressible Newtonian flows. For steady flows, the velocity component in the radial direction is assumed to be zero, i.e., ur=0, and there are no changes in the tangential direction; hence, ∂/∂θ=0. Therefore, momentum equations in radial and circumferential directions (equations (2) and (3)) are reduced (by neglecting body forces) as follows [32]:
(12)ρuθ2r=∂p∂r,(13)∂∂r∂uθ∂r+uθr=0.Integrating equation (13) and considering boundary conditions at walls of inner and outer cylinders as follows:
(14)r=Ri,uθ=RiΩi,r=Ro,uθ=RoΩo,and using χ=Ri/Ro andψ=Ωo/Ωi, in which Ωiand Ωo are the rotational speeds of the inner and outer cylinders, respectively, the solution for velocity in the circumstantial direction is obtained as [32]
(15)uθ=Aθr+Bθr.ParametersAθ and Bθ are provided as [32]
(16)Aθ=Ωiχ2−ψχ2−1,(17)Bθ=ΩiRi21−ψ1−χ2.The shear stress on the infinitesimal element shown in Figure2 may be extended and rewritten for non-Newtonian and turbulent flows using equation (9) as follows [32]:
(18)τ=σrθ=η∂uθ∂r−uθr,where η is the apparent viscosity provided in Table 1 and at the limit of Newtonian laminar flow reduces to dynamic viscosity (μ). Substituting the tangential velocity from equation (15) into equation (18), one may obtain
(19)τ=ηAθ−Bθr2−1rAθr+Bθr=−2ηBθr2.Figure 2
Calibration of the ACK TCS versus an Anton Paar MCR302 rheometer data [33].Table 1
Generalized Reynolds number definitions for non-Newtonian fluids [34].
Non-Newtonian fluidsGeneralized Reynolds numberHerschel-Bulkley-Extended (HBE)η=τ0/γ̇+Kγ̇n−1+η∞RegenHBE=ρDnu¯2−n/τ0/8D/u¯n+K3m+1/4mn8n−1+η∞3m+1/4mD/u¯n−1m=nK8u¯/Dn+η∞8u¯/D/τ0+K8u¯/Dn+η∞8u¯/DHerschel-Bulkley (HB)η∞=0;η=τ0/γ̇+Kγ̇n−1RegenHB=ρDnu¯2−n/τ0/8D/u¯n+K3m+1/4mn8n−1m=nK8u¯/Dn/τ0+K8u¯/DnPower law (PL)τ0=0;η∞=0;η=Kγ̇n−1; m=nRegenPL=ρDnu¯2−n/K3m+1/4mn8n−1BinghamK=0;n=1;η=τ0/γ̇+η∞RegenBingham=ρDu¯/τ0/8D/u¯+η∞3m+1/4mm=η∞8u¯/D/τ0+η∞8u¯/DNewtonianτ0=0;K=0;η=η∞; n=m=1ReNewtonian=ρDu¯/η∞For TCV with the stationary outer cylinder (ψ=0), the wall shear rate γ̇wi at r=Ri is obtained using equations (17) and (19) as follows:
(20)γ̇wi=τwiηwi=2BθRi2=2Ωi1−χ2=2ΩiRo2Ro2−Ri2.Theηwi in equation (20) is the apparent viscosity.In the ACK TCV in steady rotation, the input electric power to the DC motor is divided into two components: (1) mechanical power formed by friction between rotating cylinders and the bearing systemPmech=TmechΩi and (2) fluid power developed by viscous effects of fluid on wet surfaces of the inner cylinder (Pfluids=TfluidsΩi). The mechanical power Pmech can be easily measured by running TCV without any fluid at different speeds. The fluid power is then obtained by subtracting the reading total power from the frictional power, i.e., Pfluids=Ptot−Pmech, at the same speed (Ωi). The total torque applied on the inner cylinder of TCV similarly consists of the mechanical torque from frictions on bearings and the torque from fluid viscosity and turbulence effects. Hence, the torque from fluid is obtained as follows:
(21)TFluids=PFluidsΩi.The shear stress developed from fluid on the wall (τwi) can be obtained by the fluid torque obtained from equation (21) divided by Ri and then divided by the inner cylinder circumstantial area (A=2πRiL), where L is the length of the cylinder, as follows:
(22)τwi=TFluids2πRi2L.Equation (22) is valid for Newtonian, laminar, and steady flows. For non-Newtonian fluids and turbulent flow condition, a calibration factor (τwi‐non‐Newtonian=βτwi) was used in which the calibration factor β is obtained by fitting the TCV experimental data versus a high-precision rheometer. Hence, the apparent viscosity of the fluid on the wall of the inner cylinder of TCV can then be determined using equations (20) and (22) by
(23)ηwi=τwiγ̇wi=TFluids2πRi2L1−χ22Ωi=TFluidsRo2−Ri24πRi2LΩiRo2.In equation (23), ηwi is valid for both Newtonian and non-Newtonian fluids. For turbulent flows, the apparent viscosity consists of laminar and turbulent components ηwi=ηwi‐laminar+ηwi‐turbulent.Based on [10], we can use the same formulations developed above for Newtonian fluids, i.e., shear stress and shear rate in TCV, for non-Newtonian fluids. However, for consistency curves, we should use formulations presented in Table 1. More rheological models may be found in other sources [10, 17].


### 2.4. Flow Characteristics in TCV

The Reynolds number is a widely used factor for distinguishing different flow regimes in internal and external flows from laminar to turbulent conditions. In TCV, the Reynolds number is usually defined based on inner or outer cylinder linear velocities; however, the better definition is used based on the average velocity in the gap of TCV. Hence, for Newtonian incompressible and steady flows in TCV, the Reynolds number (Re) is defined here as follows:(24)Re=u¯θδν.In equation (24), u¯θ (m/s) is the average circular speed in the cavity of TCV and δ=Ro−Ri (m) is the gap distance between two cylinders (the length scale), and ν=η/ρ (m2/s). The generalized Reynolds number is presented in Table 1 for different Newtonian/non-Newtonian fluids. Based on Table 1, the average velocity in the gap should be applied using the velocity profile in the gap (equation (15)) as follows:
(25)u¯θ=1δ∫RiRouθdr=BθδlnRoRi+Aθ2δRo2−Ri2.For fluids in Table1, the average circular speed u¯θ is given by equation (25) and the gap length δ is used instead of D in the Reynolds definitions.TCV may be used to investigate the friction factor of laminar and turbulent flows; therefore, it is more appropriate to use a dimensionless parameter, i.e., the Darcy friction factor, instead of shear stress on the wall. In TCV, the Darcy friction factor is given by [34]
(26)fDarcy=8τwiρu¯θ2.In equation (26), ρ is the fluid density, τwi is the wall shear stress obtained from measured values of fluid torque using equation (22), and u¯θ is the average velocity in the gap of TCV given in equation (25). The Darcy friction factor in equation (26) is valid for any type of fluids from Newtonian to non-Newtonian and any flow regime from laminar to turbulent. In order to assess validity of experimental values of the ACK TCV against the widely accepted theoretically and experimentally validated friction factor in laminar flows, the laminar friction factor is defined and used as follows [34]:
(27)flaminar=64ReGen.In equation (27), the generalized Reynolds number (ReGen) is obtained from Table 1 for various fluid types.


## 2.1. High-Speed Viscometer

The Taylor-Couette viscometer is a confined cavity between two cylinders with the same axis which are also closed from the top and the bottom by two disks as shown in Figure1. Rotation of one or both cylinders creates a rotational confined viscous flow inside the gap of cylinders which are also influenced by the top and the bottom solid disk boundary conditions to make it a completely 3D complex flow system. Fluid flows in the TCV have been deeply investigated for a variety of flow regimes and flow patterns by many researchers.Figure 1
Sketch of the Taylor-Couette viscometer.Many complex flow phenomena have been observed in the Taylor-Couette system which is widely explored for studying linear and nonlinear hydrodynamic stability [30] and also for developing turbulence models [25]. Andereck et al. [26] reported a number of complex flow regimes for flows between two counterrotating cylinders in TCV such as basic Couette flow and appearance of Ekman cells, development of primary instabilities, wavy-vortex flow, low Reynolds modulated wavy-vortex flow, interpenetrating laminar spirals, wavy interpenetrating spirals, intermittency, transition region, spiral turbulence, turbulent flow, and hysteresis. For flow between corotating cylinders in TCV, they reported basic flow, primary instability, twisted vortices, wavy vortex boundary flows, and high azimuthal-wave number flows.The most complete flow model for viscous flows is Navier-Stokes equations discussed next for TCV.


## 2.2. Taylor-Couette Flow Equations

The Navier-Stokes equations in three dimensionsr,θ,z for incompressible viscous flows with constant dynamic viscosity (μ) and constant density (ρ) may be expressed as follows [31]:
(1)1r∂∂rrur+1r∂∂θruθ+∂uz∂z=0,(2)ρDurDt−uθ2r=−∂p∂r+Fr+μ∇2ur−urr2−2r2∂uθ∂θ,(3)ρDuθDt+uθurr=−1r∂p∂θ+Fθ+μ∇2uθ−uθr2−2r2∂ur∂θ,(4)ρDuzDt=−∂p∂z+Fz+μ∇2uz.In equations (1)–(4), ur,uθ,uz and Fr,Fθ,Fz are velocity and body force components in r,θ,z directions, respectively. p is the static or thermodynamic pressure, and the operators D/Dt and ∇2 are material derivative and Laplace operators defined by [31]
(5)DDt=∂∂t+ur∂∂r+uθr∂∂θ+uz∂∂z,∇2=∂2∂r2+1r∂∂r+1r2∂2∂θ2+∂2∂z2.


## 2.3. Stress Tensor in Newtonian Flows

For Newtonian incompressible flows, a stress tensor for an infinitesimal volume of fluid is obtained as follows [31]:
(6)σrr=−p+2μ∂ur∂r,(7)σθθ=−p+2μ1r∂uθ∂θ+urr,(8)σzz=−p+2μ∂uz∂z,(9)σrθ=μ1r∂ur∂θ+∂uθ∂r−uθr,(10)σrz=μ∂ur∂z+∂uz∂r,(11)σθz=μ1r∂uz∂θ+∂uθ∂z.


### 2.3.1. Taylor-Couette System with the Stationary Outer Cylinder

Dou et al. [32] obtained solutions to the steady-state Navier-Stokes equation in TCV for incompressible Newtonian flows. For steady flows, the velocity component in the radial direction is assumed to be zero, i.e., ur=0, and there are no changes in the tangential direction; hence, ∂/∂θ=0. Therefore, momentum equations in radial and circumferential directions (equations (2) and (3)) are reduced (by neglecting body forces) as follows [32]:
(12)ρuθ2r=∂p∂r,(13)∂∂r∂uθ∂r+uθr=0.Integrating equation (13) and considering boundary conditions at walls of inner and outer cylinders as follows:
(14)r=Ri,uθ=RiΩi,r=Ro,uθ=RoΩo,and using χ=Ri/Ro andψ=Ωo/Ωi, in which Ωiand Ωo are the rotational speeds of the inner and outer cylinders, respectively, the solution for velocity in the circumstantial direction is obtained as [32]
(15)uθ=Aθr+Bθr.ParametersAθ and Bθ are provided as [32]
(16)Aθ=Ωiχ2−ψχ2−1,(17)Bθ=ΩiRi21−ψ1−χ2.The shear stress on the infinitesimal element shown in Figure2 may be extended and rewritten for non-Newtonian and turbulent flows using equation (9) as follows [32]:
(18)τ=σrθ=η∂uθ∂r−uθr,where η is the apparent viscosity provided in Table 1 and at the limit of Newtonian laminar flow reduces to dynamic viscosity (μ). Substituting the tangential velocity from equation (15) into equation (18), one may obtain
(19)τ=ηAθ−Bθr2−1rAθr+Bθr=−2ηBθr2.Figure 2
Calibration of the ACK TCS versus an Anton Paar MCR302 rheometer data [33].Table 1
Generalized Reynolds number definitions for non-Newtonian fluids [34].
Non-Newtonian fluidsGeneralized Reynolds numberHerschel-Bulkley-Extended (HBE)η=τ0/γ̇+Kγ̇n−1+η∞RegenHBE=ρDnu¯2−n/τ0/8D/u¯n+K3m+1/4mn8n−1+η∞3m+1/4mD/u¯n−1m=nK8u¯/Dn+η∞8u¯/D/τ0+K8u¯/Dn+η∞8u¯/DHerschel-Bulkley (HB)η∞=0;η=τ0/γ̇+Kγ̇n−1RegenHB=ρDnu¯2−n/τ0/8D/u¯n+K3m+1/4mn8n−1m=nK8u¯/Dn/τ0+K8u¯/DnPower law (PL)τ0=0;η∞=0;η=Kγ̇n−1; m=nRegenPL=ρDnu¯2−n/K3m+1/4mn8n−1BinghamK=0;n=1;η=τ0/γ̇+η∞RegenBingham=ρDu¯/τ0/8D/u¯+η∞3m+1/4mm=η∞8u¯/D/τ0+η∞8u¯/DNewtonianτ0=0;K=0;η=η∞; n=m=1ReNewtonian=ρDu¯/η∞For TCV with the stationary outer cylinder (ψ=0), the wall shear rate γ̇wi at r=Ri is obtained using equations (17) and (19) as follows:
(20)γ̇wi=τwiηwi=2BθRi2=2Ωi1−χ2=2ΩiRo2Ro2−Ri2.Theηwi in equation (20) is the apparent viscosity.In the ACK TCV in steady rotation, the input electric power to the DC motor is divided into two components: (1) mechanical power formed by friction between rotating cylinders and the bearing systemPmech=TmechΩi and (2) fluid power developed by viscous effects of fluid on wet surfaces of the inner cylinder (Pfluids=TfluidsΩi). The mechanical power Pmech can be easily measured by running TCV without any fluid at different speeds. The fluid power is then obtained by subtracting the reading total power from the frictional power, i.e., Pfluids=Ptot−Pmech, at the same speed (Ωi). The total torque applied on the inner cylinder of TCV similarly consists of the mechanical torque from frictions on bearings and the torque from fluid viscosity and turbulence effects. Hence, the torque from fluid is obtained as follows:
(21)TFluids=PFluidsΩi.The shear stress developed from fluid on the wall (τwi) can be obtained by the fluid torque obtained from equation (21) divided by Ri and then divided by the inner cylinder circumstantial area (A=2πRiL), where L is the length of the cylinder, as follows:
(22)τwi=TFluids2πRi2L.Equation (22) is valid for Newtonian, laminar, and steady flows. For non-Newtonian fluids and turbulent flow condition, a calibration factor (τwi‐non‐Newtonian=βτwi) was used in which the calibration factor β is obtained by fitting the TCV experimental data versus a high-precision rheometer. Hence, the apparent viscosity of the fluid on the wall of the inner cylinder of TCV can then be determined using equations (20) and (22) by
(23)ηwi=τwiγ̇wi=TFluids2πRi2L1−χ22Ωi=TFluidsRo2−Ri24πRi2LΩiRo2.In equation (23), ηwi is valid for both Newtonian and non-Newtonian fluids. For turbulent flows, the apparent viscosity consists of laminar and turbulent components ηwi=ηwi‐laminar+ηwi‐turbulent.Based on [10], we can use the same formulations developed above for Newtonian fluids, i.e., shear stress and shear rate in TCV, for non-Newtonian fluids. However, for consistency curves, we should use formulations presented in Table 1. More rheological models may be found in other sources [10, 17].


## 2.3.1. Taylor-Couette System with the Stationary Outer Cylinder

Dou et al. [32] obtained solutions to the steady-state Navier-Stokes equation in TCV for incompressible Newtonian flows. For steady flows, the velocity component in the radial direction is assumed to be zero, i.e., ur=0, and there are no changes in the tangential direction; hence, ∂/∂θ=0. Therefore, momentum equations in radial and circumferential directions (equations (2) and (3)) are reduced (by neglecting body forces) as follows [32]:
(12)ρuθ2r=∂p∂r,(13)∂∂r∂uθ∂r+uθr=0.Integrating equation (13) and considering boundary conditions at walls of inner and outer cylinders as follows:
(14)r=Ri,uθ=RiΩi,r=Ro,uθ=RoΩo,and using χ=Ri/Ro andψ=Ωo/Ωi, in which Ωiand Ωo are the rotational speeds of the inner and outer cylinders, respectively, the solution for velocity in the circumstantial direction is obtained as [32]
(15)uθ=Aθr+Bθr.ParametersAθ and Bθ are provided as [32]
(16)Aθ=Ωiχ2−ψχ2−1,(17)Bθ=ΩiRi21−ψ1−χ2.The shear stress on the infinitesimal element shown in Figure2 may be extended and rewritten for non-Newtonian and turbulent flows using equation (9) as follows [32]:
(18)τ=σrθ=η∂uθ∂r−uθr,where η is the apparent viscosity provided in Table 1 and at the limit of Newtonian laminar flow reduces to dynamic viscosity (μ). Substituting the tangential velocity from equation (15) into equation (18), one may obtain
(19)τ=ηAθ−Bθr2−1rAθr+Bθr=−2ηBθr2.Figure 2
Calibration of the ACK TCS versus an Anton Paar MCR302 rheometer data [33].Table 1
Generalized Reynolds number definitions for non-Newtonian fluids [34].
Non-Newtonian fluidsGeneralized Reynolds numberHerschel-Bulkley-Extended (HBE)η=τ0/γ̇+Kγ̇n−1+η∞RegenHBE=ρDnu¯2−n/τ0/8D/u¯n+K3m+1/4mn8n−1+η∞3m+1/4mD/u¯n−1m=nK8u¯/Dn+η∞8u¯/D/τ0+K8u¯/Dn+η∞8u¯/DHerschel-Bulkley (HB)η∞=0;η=τ0/γ̇+Kγ̇n−1RegenHB=ρDnu¯2−n/τ0/8D/u¯n+K3m+1/4mn8n−1m=nK8u¯/Dn/τ0+K8u¯/DnPower law (PL)τ0=0;η∞=0;η=Kγ̇n−1; m=nRegenPL=ρDnu¯2−n/K3m+1/4mn8n−1BinghamK=0;n=1;η=τ0/γ̇+η∞RegenBingham=ρDu¯/τ0/8D/u¯+η∞3m+1/4mm=η∞8u¯/D/τ0+η∞8u¯/DNewtonianτ0=0;K=0;η=η∞; n=m=1ReNewtonian=ρDu¯/η∞For TCV with the stationary outer cylinder (ψ=0), the wall shear rate γ̇wi at r=Ri is obtained using equations (17) and (19) as follows:
(20)γ̇wi=τwiηwi=2BθRi2=2Ωi1−χ2=2ΩiRo2Ro2−Ri2.Theηwi in equation (20) is the apparent viscosity.In the ACK TCV in steady rotation, the input electric power to the DC motor is divided into two components: (1) mechanical power formed by friction between rotating cylinders and the bearing systemPmech=TmechΩi and (2) fluid power developed by viscous effects of fluid on wet surfaces of the inner cylinder (Pfluids=TfluidsΩi). The mechanical power Pmech can be easily measured by running TCV without any fluid at different speeds. The fluid power is then obtained by subtracting the reading total power from the frictional power, i.e., Pfluids=Ptot−Pmech, at the same speed (Ωi). The total torque applied on the inner cylinder of TCV similarly consists of the mechanical torque from frictions on bearings and the torque from fluid viscosity and turbulence effects. Hence, the torque from fluid is obtained as follows:
(21)TFluids=PFluidsΩi.The shear stress developed from fluid on the wall (τwi) can be obtained by the fluid torque obtained from equation (21) divided by Ri and then divided by the inner cylinder circumstantial area (A=2πRiL), where L is the length of the cylinder, as follows:
(22)τwi=TFluids2πRi2L.Equation (22) is valid for Newtonian, laminar, and steady flows. For non-Newtonian fluids and turbulent flow condition, a calibration factor (τwi‐non‐Newtonian=βτwi) was used in which the calibration factor β is obtained by fitting the TCV experimental data versus a high-precision rheometer. Hence, the apparent viscosity of the fluid on the wall of the inner cylinder of TCV can then be determined using equations (20) and (22) by
(23)ηwi=τwiγ̇wi=TFluids2πRi2L1−χ22Ωi=TFluidsRo2−Ri24πRi2LΩiRo2.In equation (23), ηwi is valid for both Newtonian and non-Newtonian fluids. For turbulent flows, the apparent viscosity consists of laminar and turbulent components ηwi=ηwi‐laminar+ηwi‐turbulent.Based on [10], we can use the same formulations developed above for Newtonian fluids, i.e., shear stress and shear rate in TCV, for non-Newtonian fluids. However, for consistency curves, we should use formulations presented in Table 1. More rheological models may be found in other sources [10, 17].


## 2.4. Flow Characteristics in TCV

The Reynolds number is a widely used factor for distinguishing different flow regimes in internal and external flows from laminar to turbulent conditions. In TCV, the Reynolds number is usually defined based on inner or outer cylinder linear velocities; however, the better definition is used based on the average velocity in the gap of TCV. Hence, for Newtonian incompressible and steady flows in TCV, the Reynolds number (Re) is defined here as follows:(24)Re=u¯θδν.In equation (24), u¯θ (m/s) is the average circular speed in the cavity of TCV and δ=Ro−Ri (m) is the gap distance between two cylinders (the length scale), and ν=η/ρ (m2/s). The generalized Reynolds number is presented in Table 1 for different Newtonian/non-Newtonian fluids. Based on Table 1, the average velocity in the gap should be applied using the velocity profile in the gap (equation (15)) as follows:
(25)u¯θ=1δ∫RiRouθdr=BθδlnRoRi+Aθ2δRo2−Ri2.For fluids in Table1, the average circular speed u¯θ is given by equation (25) and the gap length δ is used instead of D in the Reynolds definitions.TCV may be used to investigate the friction factor of laminar and turbulent flows; therefore, it is more appropriate to use a dimensionless parameter, i.e., the Darcy friction factor, instead of shear stress on the wall. In TCV, the Darcy friction factor is given by [34]
(26)fDarcy=8τwiρu¯θ2.In equation (26), ρ is the fluid density, τwi is the wall shear stress obtained from measured values of fluid torque using equation (22), and u¯θ is the average velocity in the gap of TCV given in equation (25). The Darcy friction factor in equation (26) is valid for any type of fluids from Newtonian to non-Newtonian and any flow regime from laminar to turbulent. In order to assess validity of experimental values of the ACK TCV against the widely accepted theoretically and experimentally validated friction factor in laminar flows, the laminar friction factor is defined and used as follows [34]:
(27)flaminar=64ReGen.In equation (27), the generalized Reynolds number (ReGen) is obtained from Table 1 for various fluid types.


## 3. Drilling Process

The drilling process in an oil and gas industry involves using a drilling rig which rotates a drill string with a drill bit attached. Drilling is achieved by using the drill bit that is attached to a long string of the drill pipe. Digging a wellbore may require drilling from the ground surface to reach deep into an oil or gas reservoir. The weight of a drilling rig on the bit in addition to the rotation of the bit cuts soil and rocks into smaller pieces and transports them out through a fluid circulating system. The drilling process is highly dependent on the performance of circulating well drilling fluids. Drilling fluids are usually WBM or OBM which are pumped from a mud tank into the drilling pipe inside the wellbore and back to the mud tank. The drilling fluids then transport mud and rocks through the cavity space between the drilling pipe and the wellbore back to the ground. A separator device at the ground is used to separate shale pieces from the drilling fluid. The circulating mud system uses a powerful pump to circulate the mud into and from the wellbore.


### 3.1. Functions of Drilling Mud

One of the important factors in wellbore drilling performance is drilling fluids. Drilling liquids are specifically designed to maximize wellbore drilling performances, to minimize risks to drilling rig operations and equipment, and to be safe to the environment. From the economic point of view, it is important to highlight that the wellbore drilling process counts between 20 and 40 percent of the total cost. Therefore, the drilling fluids are needed to be continuously enhanced to supply the following functions in wellbore drilling:(1)
Transport mud and rocks by the drilling bit back into the surface(2)
Drilling mud provides suitable subsurface pressure for sustainable shale during drilling operation(3)
Lubricate the drill string and remove heat generated because of the function of the bit and friction through mechanical movement of the drill string against the wellbore and hydraulic systems(4)
Support sustainable and stable formation of wellbore shale [35]OBMs are commonly used in wellbore drilling at HPHT, but OBMs are expensive and not environmentally friendly. WBMs are less expensive and safe to the environment yet are limited to minimal temperatures and pressures and are more susceptible to breakdown, gel formation, and degradation of shale soft materials if exposed to high temperatures [35]. Therefore, it is continuously needed to modify wellbore drilling technology and enhance OBM and WBM properties to improve the life cycle of drilling equipment and to reduce costs. We only considered WBM in this research mainly because of environment-friendly aspects, lower costs, and more stability compared to the OBM.


### 3.2. Drilling Fluid Properties

In order to enhance drilling mud properties for efficient and safe operation of wellbore drilling, four primary properties including density, viscosity, gel strength, and filtration must be improved [35]. Gel strength is simply referred to as the shear stress of drilling mud at rest. The gel strength is important to withhold the shale formation when the drilling operations stop. Filtration is a measure of how much drilling fluids are recovered after separating from shale materials. Viscosity is basically referred to as fluid internal resistance and is obtained by the ratio of shear stress to shear strain. A diagram that shows the relationship between shear stress and shear strain is referred to as the consistency chart. Figure 3 shows the general consistency chart for different fluids. Rheological terminology used for drilling fluid can be seen in Figure 3, which distinguishes Newtonian and non-Newtonian fluids and provides graphical description of the yield point (YP), plastic viscosity (PV), and typical drilling mud behavior.Figure 3
Consistency chart for Newtonian and non-Newtonian fluids (typical drilling muds) [35].In general, lower viscosity or lower fluid friction of mud fluids provides more shale stability and less fluid loss. In addition, lower viscosity (thinner fluids) provides better cleaning of bits and improves drilling performance; however, thicker mud fluids are sometimes required to extract larger gravel out of the wellbore during drilling operations.


## 3.1. Functions of Drilling Mud

One of the important factors in wellbore drilling performance is drilling fluids. Drilling liquids are specifically designed to maximize wellbore drilling performances, to minimize risks to drilling rig operations and equipment, and to be safe to the environment. From the economic point of view, it is important to highlight that the wellbore drilling process counts between 20 and 40 percent of the total cost. Therefore, the drilling fluids are needed to be continuously enhanced to supply the following functions in wellbore drilling:(1)
Transport mud and rocks by the drilling bit back into the surface(2)
Drilling mud provides suitable subsurface pressure for sustainable shale during drilling operation(3)
Lubricate the drill string and remove heat generated because of the function of the bit and friction through mechanical movement of the drill string against the wellbore and hydraulic systems(4)
Support sustainable and stable formation of wellbore shale [35]OBMs are commonly used in wellbore drilling at HPHT, but OBMs are expensive and not environmentally friendly. WBMs are less expensive and safe to the environment yet are limited to minimal temperatures and pressures and are more susceptible to breakdown, gel formation, and degradation of shale soft materials if exposed to high temperatures [35]. Therefore, it is continuously needed to modify wellbore drilling technology and enhance OBM and WBM properties to improve the life cycle of drilling equipment and to reduce costs. We only considered WBM in this research mainly because of environment-friendly aspects, lower costs, and more stability compared to the OBM.


## 3.2. Drilling Fluid Properties

In order to enhance drilling mud properties for efficient and safe operation of wellbore drilling, four primary properties including density, viscosity, gel strength, and filtration must be improved [35]. Gel strength is simply referred to as the shear stress of drilling mud at rest. The gel strength is important to withhold the shale formation when the drilling operations stop. Filtration is a measure of how much drilling fluids are recovered after separating from shale materials. Viscosity is basically referred to as fluid internal resistance and is obtained by the ratio of shear stress to shear strain. A diagram that shows the relationship between shear stress and shear strain is referred to as the consistency chart. Figure 3 shows the general consistency chart for different fluids. Rheological terminology used for drilling fluid can be seen in Figure 3, which distinguishes Newtonian and non-Newtonian fluids and provides graphical description of the yield point (YP), plastic viscosity (PV), and typical drilling mud behavior.Figure 3
Consistency chart for Newtonian and non-Newtonian fluids (typical drilling muds) [35].In general, lower viscosity or lower fluid friction of mud fluids provides more shale stability and less fluid loss. In addition, lower viscosity (thinner fluids) provides better cleaning of bits and improves drilling performance; however, thicker mud fluids are sometimes required to extract larger gravel out of the wellbore during drilling operations.


## 4. Rheology of Drilling Muds

The classical rheological models for drilling fluids shown in Figure3 are extended by Shah et al. [17] to include a wider classification such as non-Newtonian pseudoplastic and yield pseudoplastic fluids. Summary of the models is presented in Table 1.Fakoya and Ahmed [20] developed more general empirical models than those shown in Table 1 for viscosity of OBM drilling fluids by considering temperature, pressure, volume concentration, and other factors. Kumar et al. [21] also provided theoretically a very complex model for estimation of shear rates.


### 4.1. Generalized Reynolds Number

In non-Newtonian fluid flows, the relation between shear stress and shear strain is nonlinear and the apparent viscosity (η=τ/ρ) is no longer constant. Depending on the apparent viscosity, non-Newtonian fluids are classified as follows [10]:
(1)
Pseudoplastic or shear thinning(2)
Dilatant or shear thickening(3)
Viscoplastic or yield pseudoplastic or Bingham plasticFor analyzing non-Newtonian fluids, the Reynolds number, however, has to be redefined depending on different types of non-Newtonian fluids. Many different models were used for various flow types such as Herschel-Bulkley- (HB-), Ostwald-de Waele- (power law (PL)), and Bingham-type flows. Madlener et al. [34] introduced a new definition for the Reynolds number in non-Newtonian media. Table 1 illustrates various Reynolds number definitions for different non-Newtonian flows [34].In Table1, ρ, D, u¯, η, andη∞ are the density, the length scale (diameter for pipes), the average velocity, the variable apparent viscosity as a function of shear rate, and the infinite viscosity, respectively. K and n are consistency parameters, n is also called the global exponential term, m is a local term, τ0 is the yield stress, and γ̇ is the shear rate (strain rate).


### 4.2. Four-Parameter Models

The more common four-parameter models for drilling fluids are Cross and Carreau models described by [17]
(28)CrossModel:η=η∞+ηo−η∞1+Kγ̇n,(29)CarreauModel:η=η∞+ηo−η∞1+Kγ̇2−n/2.These models are also considered here for modelling WBM measurements.


### 4.3. Optimization Technique

As illustrated in Table1 and Section 4.2, the apparent viscosity is expressed by a function of shear rate (γ̇wi) described by rheology models with maximum four unknown constants, e.g., HBE and Cross models, as follows:
(30)ηwi=fγ̇wi,K,n,τ0orηo,η∞.The formulations given in the previous sections are used to obtain 4 unknown constants in equation (30) for WBM studied here. To determine 4 unknowns based on the data collected from the ACK TCV and formulations given in the previous sections, an optimization method is used to fit the best function to measured data. To fit model parameters given in equation (30) with measured data, the optimization toolbox “fminsearch” in MATLAB was used which uses a regression method to find optimization variables by minimizing an objective function. The objective function used here utilizes a root mean square method based on R2. The routine code in MATLAB is presented in Algorithm 1.Algorithm 1: Optimization routine in MATLAB.
% Main codeInput data: Shear Rate (SR) and Apparent Viscosity (AV)% Optimization solverfunction = @(x)HBE(SR,x,AV);x0=[1 1 1 1];bestx = fminsearch(function, x0)end%% Objective functionfunction R2 = HBE(SR,x,AV)% Herschel-Bulkley-Extended (HBE) modelt0=x(1);K=x(2);n=x(3);einf=x(4);Rheology model function (HBE =…);% Calculate errorError = Objective function


### 4.4. Convergence Criteria

The convergence criteria useR2 to measure the accuracy of the predicted and actual data. We used the following relation to be minimized as the objective function in MATLAB for predicted values “Pred” and measured data for the apparent viscosity (AV) as follows [36, 37]:
(31)Objectivefunction=1−R2=∑Pred−AV2∑AV−AV¯2.Equation (31) presents AV¯ as the average viscosity, AV as the measured value, and “Pred” as the predicted model value. The R2 close to one gives the highest accuracy; therefore, Objectivefunction=1−R2 merges to zero using the optimization method shown in Algorithm 1.


## 4.1. Generalized Reynolds Number

In non-Newtonian fluid flows, the relation between shear stress and shear strain is nonlinear and the apparent viscosity (η=τ/ρ) is no longer constant. Depending on the apparent viscosity, non-Newtonian fluids are classified as follows [10]:
(1)
Pseudoplastic or shear thinning(2)
Dilatant or shear thickening(3)
Viscoplastic or yield pseudoplastic or Bingham plasticFor analyzing non-Newtonian fluids, the Reynolds number, however, has to be redefined depending on different types of non-Newtonian fluids. Many different models were used for various flow types such as Herschel-Bulkley- (HB-), Ostwald-de Waele- (power law (PL)), and Bingham-type flows. Madlener et al. [34] introduced a new definition for the Reynolds number in non-Newtonian media. Table 1 illustrates various Reynolds number definitions for different non-Newtonian flows [34].In Table1, ρ, D, u¯, η, andη∞ are the density, the length scale (diameter for pipes), the average velocity, the variable apparent viscosity as a function of shear rate, and the infinite viscosity, respectively. K and n are consistency parameters, n is also called the global exponential term, m is a local term, τ0 is the yield stress, and γ̇ is the shear rate (strain rate).


## 4.2. Four-Parameter Models

The more common four-parameter models for drilling fluids are Cross and Carreau models described by [17]
(28)CrossModel:η=η∞+ηo−η∞1+Kγ̇n,(29)CarreauModel:η=η∞+ηo−η∞1+Kγ̇2−n/2.These models are also considered here for modelling WBM measurements.


## 4.3. Optimization Technique

As illustrated in Table1 and Section 4.2, the apparent viscosity is expressed by a function of shear rate (γ̇wi) described by rheology models with maximum four unknown constants, e.g., HBE and Cross models, as follows:
(30)ηwi=fγ̇wi,K,n,τ0orηo,η∞.The formulations given in the previous sections are used to obtain 4 unknown constants in equation (30) for WBM studied here. To determine 4 unknowns based on the data collected from the ACK TCV and formulations given in the previous sections, an optimization method is used to fit the best function to measured data. To fit model parameters given in equation (30) with measured data, the optimization toolbox “fminsearch” in MATLAB was used which uses a regression method to find optimization variables by minimizing an objective function. The objective function used here utilizes a root mean square method based on R2. The routine code in MATLAB is presented in Algorithm 1.Algorithm 1: Optimization routine in MATLAB.
% Main codeInput data: Shear Rate (SR) and Apparent Viscosity (AV)% Optimization solverfunction = @(x)HBE(SR,x,AV);x0=[1 1 1 1];bestx = fminsearch(function, x0)end%% Objective functionfunction R2 = HBE(SR,x,AV)% Herschel-Bulkley-Extended (HBE) modelt0=x(1);K=x(2);n=x(3);einf=x(4);Rheology model function (HBE =…);% Calculate errorError = Objective function


## 4.4. Convergence Criteria

The convergence criteria useR2 to measure the accuracy of the predicted and actual data. We used the following relation to be minimized as the objective function in MATLAB for predicted values “Pred” and measured data for the apparent viscosity (AV) as follows [36, 37]:
(31)Objectivefunction=1−R2=∑Pred−AV2∑AV−AV¯2.Equation (31) presents AV¯ as the average viscosity, AV as the measured value, and “Pred” as the predicted model value. The R2 close to one gives the highest accuracy; therefore, Objectivefunction=1−R2 merges to zero using the optimization method shown in Algorithm 1.


## 5. Experimental Setup

### 5.1. TCV Apparatus

The Taylor-Couette viscometer (TCV) consists of two cylinders with the same axis where the outer cylinder is the stator and the inner one is the rotor connected to an electric motor rotating with constant speed. A similar design to the TCV by Dr. Daniel Borrero Echeverry [29] at Willamette University was developed but with smaller size, different drive, and in-house-developed control system. In the ACK TCV shown in Figure 4, the inner cylinder is seated on two sets of the bearing system at the top and bottom of TCV. The outer cylinder is made from acrylic and the inner cylinder from stainless steel. The gap between acrylic and stainless-steel cylinders is filled in with fluids at a hole in the top, and after the experiment, the fluid is discharged from an elastic hose and valve at the bottom. In the TCV design of Willamette University, a stepper motor is used to rotate the inner cylinder. Technical details of the TCV components may be obtained from [29].Figure 4
Taylor-Couette system of ACK for studying rheology of drilling nanofluids with major dimensions: inner radiusRi=3.4cm, outer radius Ro=3.8cm, length L=11.0cm, gap δ=Ro−Ri=0.4cm, and annulus volume of ∀=πRo2−Ri2L=100ml [38, 39].Due to simplicity and safe operation, the Taylor-Couette system is an ideal device for studying various flow patterns and flow instabilities. Moreover, TCV is an outstanding platform to observe various flow patterns. At low speeds, the fluid particles follow circular paths and have no dependency on axial nor azimuthal directions. As rotational speed increases, the flow transition is initiated. At transitional flow condition, instability patterns such as cellular flow type and toroidal vortices appeared. These vortices are called Taylor rolls. By increasing the speed of the inner cylinder, these Taylor rolls become more unstable and create very complex flow patterns, and then, the flow becomes turbulent. Table2 provides the main features and geometry of the ACK TCV.Table 2
Main specification and sizes of the ACK TCV viscometer.
ParametersValuesInner cylinder radius (Ri)34 mmOuter cylinder radius (Ro)38 mmGap width (δ)4 mmWet length of the cylinder (L)110 mmWet annulus volume (∀)100 mlThickness of the outer cylinder3 mmElectric DC motor speed range0 to 1600 RPMMaterial of the inner cylinderStainless steelMaterial of the outer cylinderAcrylicFigure4 shows the manufactured TCV of ACK in Kuwait. We used a DC motor which enables us to run TCV at different speeds from 0 to 1600 rpm and is also easy to measure electric power. The DC motor uses a speed controller which also displays motor speed. The TCV of ACK uses a controller box which uses two Arduino microprocessors, a tachometer, and a rheostat to measure voltage and current of the DC electric motor and angular velocity and to change the speed of the motor, respectively. The DC electric motor rotates at different speeds from 0 to 1600 rpm. We have used the TCV for testing WBM (at room temperature) at speeds from 200 rpm to 1600 rpm with intervals of 200 rpm. The DC electric motor is connected to a rheostat (variable resistance) to control the speed, and it is also connected with two Arduino microprocessors: one is used to measure voltage and the other one to measure the current supplied to the motor. A tachometer is mounted on the side of the device to measure angular velocity of the shaft of the DC motor. Two Arduino microprocessors are programmed to process all sensor measurements and display RPM, rad/s, power (W), and viscosity (cp) on an LCD 16cm×2cm screen. The controller box is produced using 3D prints.At steady-state rotation and turbulent flows, measuring the power and torque exerted by the fluid on the inner cylinder is the best candidate for evaluating fluid resistance in non-Newtonian turbulent flows from a practical point of view.


### 5.2. WBM Preparation

WBMs are prepared using water as the base fluid and several soluble substances such as salts, surfactants, alkalies, polymers, and oils and several insoluble elements such as clay, cuttings, and barite in suspension. Several WBM types have already been used in well drilling industry such as salt, lime, spud, deflocculated substance, gypsum, polymer, potassium, metal hydroxide, and cationic muds [40]. In literature, various improvements are applied to enhance viscosity, yield point, gel, and thermal stability by carefully selecting different components and composition [40].Water-based fluids are prepared in the labs according to the “pilot testing” which is proportioned to small-scale samples based on using 1 g/350 ml of the sample (equivalent to 1 lb/bbl (42 gal)) of the actual mud system. The WBM sample is prepared by adding distilled (deionized) water and the additives as follows [41]:
(1)
Sodium carbonate (soda ash) as a hardener agent (0.2–0.5 g)(2)
Sodium hydroxide (caustic soda) as a pH controller (0.25–0.3 g) to give pH 9-10(3)
Bentonite (10–20 g) as a viscous agent(4)
Starch as a filtration loss controller(5)
Barite as a weighing agentIn this work, we prepared drilling WBM based on the international standard API RP 131 [42]. In the API RP 131 standard, distilled or deionized water is prepared based on ISO 3696:1987 or prepared by passing distilled water through a series of cation and anion exchange resins [42]. Calibration bentonite is obtained according to API 13A. The standard WBM is prepared according to the following steps:
(1)
At a room temperature, 15 g of bentonite is mixed with 350 ml of deionized water in a mud mixer (4% vol bentonite API standard)(2)
At every 5 minutes, mud on the surfaces of the mixer should be dissolved(3)
The above two steps should be repeated for 20-minute durationThe same procedure mentioned above was applied in all our WBM preparations.


## 5.1. TCV Apparatus

The Taylor-Couette viscometer (TCV) consists of two cylinders with the same axis where the outer cylinder is the stator and the inner one is the rotor connected to an electric motor rotating with constant speed. A similar design to the TCV by Dr. Daniel Borrero Echeverry [29] at Willamette University was developed but with smaller size, different drive, and in-house-developed control system. In the ACK TCV shown in Figure 4, the inner cylinder is seated on two sets of the bearing system at the top and bottom of TCV. The outer cylinder is made from acrylic and the inner cylinder from stainless steel. The gap between acrylic and stainless-steel cylinders is filled in with fluids at a hole in the top, and after the experiment, the fluid is discharged from an elastic hose and valve at the bottom. In the TCV design of Willamette University, a stepper motor is used to rotate the inner cylinder. Technical details of the TCV components may be obtained from [29].Figure 4
Taylor-Couette system of ACK for studying rheology of drilling nanofluids with major dimensions: inner radiusRi=3.4cm, outer radius Ro=3.8cm, length L=11.0cm, gap δ=Ro−Ri=0.4cm, and annulus volume of ∀=πRo2−Ri2L=100ml [38, 39].Due to simplicity and safe operation, the Taylor-Couette system is an ideal device for studying various flow patterns and flow instabilities. Moreover, TCV is an outstanding platform to observe various flow patterns. At low speeds, the fluid particles follow circular paths and have no dependency on axial nor azimuthal directions. As rotational speed increases, the flow transition is initiated. At transitional flow condition, instability patterns such as cellular flow type and toroidal vortices appeared. These vortices are called Taylor rolls. By increasing the speed of the inner cylinder, these Taylor rolls become more unstable and create very complex flow patterns, and then, the flow becomes turbulent. Table2 provides the main features and geometry of the ACK TCV.Table 2
Main specification and sizes of the ACK TCV viscometer.
ParametersValuesInner cylinder radius (Ri)34 mmOuter cylinder radius (Ro)38 mmGap width (δ)4 mmWet length of the cylinder (L)110 mmWet annulus volume (∀)100 mlThickness of the outer cylinder3 mmElectric DC motor speed range0 to 1600 RPMMaterial of the inner cylinderStainless steelMaterial of the outer cylinderAcrylicFigure4 shows the manufactured TCV of ACK in Kuwait. We used a DC motor which enables us to run TCV at different speeds from 0 to 1600 rpm and is also easy to measure electric power. The DC motor uses a speed controller which also displays motor speed. The TCV of ACK uses a controller box which uses two Arduino microprocessors, a tachometer, and a rheostat to measure voltage and current of the DC electric motor and angular velocity and to change the speed of the motor, respectively. The DC electric motor rotates at different speeds from 0 to 1600 rpm. We have used the TCV for testing WBM (at room temperature) at speeds from 200 rpm to 1600 rpm with intervals of 200 rpm. The DC electric motor is connected to a rheostat (variable resistance) to control the speed, and it is also connected with two Arduino microprocessors: one is used to measure voltage and the other one to measure the current supplied to the motor. A tachometer is mounted on the side of the device to measure angular velocity of the shaft of the DC motor. Two Arduino microprocessors are programmed to process all sensor measurements and display RPM, rad/s, power (W), and viscosity (cp) on an LCD 16cm×2cm screen. The controller box is produced using 3D prints.At steady-state rotation and turbulent flows, measuring the power and torque exerted by the fluid on the inner cylinder is the best candidate for evaluating fluid resistance in non-Newtonian turbulent flows from a practical point of view.


## 5.2. WBM Preparation

WBMs are prepared using water as the base fluid and several soluble substances such as salts, surfactants, alkalies, polymers, and oils and several insoluble elements such as clay, cuttings, and barite in suspension. Several WBM types have already been used in well drilling industry such as salt, lime, spud, deflocculated substance, gypsum, polymer, potassium, metal hydroxide, and cationic muds [40]. In literature, various improvements are applied to enhance viscosity, yield point, gel, and thermal stability by carefully selecting different components and composition [40].Water-based fluids are prepared in the labs according to the “pilot testing” which is proportioned to small-scale samples based on using 1 g/350 ml of the sample (equivalent to 1 lb/bbl (42 gal)) of the actual mud system. The WBM sample is prepared by adding distilled (deionized) water and the additives as follows [41]:
(1)
Sodium carbonate (soda ash) as a hardener agent (0.2–0.5 g)(2)
Sodium hydroxide (caustic soda) as a pH controller (0.25–0.3 g) to give pH 9-10(3)
Bentonite (10–20 g) as a viscous agent(4)
Starch as a filtration loss controller(5)
Barite as a weighing agentIn this work, we prepared drilling WBM based on the international standard API RP 131 [42]. In the API RP 131 standard, distilled or deionized water is prepared based on ISO 3696:1987 or prepared by passing distilled water through a series of cation and anion exchange resins [42]. Calibration bentonite is obtained according to API 13A. The standard WBM is prepared according to the following steps:
(1)
At a room temperature, 15 g of bentonite is mixed with 350 ml of deionized water in a mud mixer (4% vol bentonite API standard)(2)
At every 5 minutes, mud on the surfaces of the mixer should be dissolved(3)
The above two steps should be repeated for 20-minute durationThe same procedure mentioned above was applied in all our WBM preparations.


## 6. Results of Measurements

### 6.1. Power Consumption

The ACK TCV uses power consumptions to determine rheology. To take into account frictional power loss, the TCV was first calibrated to measure power loss due to mechanical friction. Then, WBM was filled up into the TCV and power consumptions were measured. The TCV reading was conducted by allowing a steady state which is reached as each speed.Unlike stepper motors, the speed controller of the DC motor used cannot provide a fixed preset value for speed. Thus, we used the best fit, to the raw data given in Table3, to obtain power values at precise rotational speed of TCV. Figure 5 compares mechanical power (air), total power (fluid+air), and fluid power for testing TCV without liquid and with WBM.Table 3
Power measurements of TCV for air and WBM.
AirWBMSpeed (rad/s)Power (W)Speed (rad/s)Power (W)24.633.2221.687.8950.355.1045.9714.7170.276.8067.1323.4688.409.1274.7727.6596.9012.44114.7746.44137.6015.20134.5660.00167.5519.30169.2382.07Figure 5
Power models in TCV: mechanical power (air), total power (fluid+air), and fluid power using WBM (4% vol bentonite API standard).As observed in Figure5, the mechanical power (W) of TCV is a linear function of angular speed (rad/s) for air as follows:
(32)Pmech=0.1128Ωi.Figure5 also shows that the base fluid, i.e., WBM, is a second-ordered polynomial function of rotational speed (rad/s):
(33)Pfluid=0.0013Ωi2+0.1553Ωi.R2 accuracy of the fitted functions is very closed to 1, so these are highly accurate. Mechanical power (air) given in equation (30) should be deducted from the total power measurements of WBM to obtain net fluid power using equation (31).


### 6.2. Flow Properties

Flow properties in the ACK TCV including the apparent viscosity, shear stress, shear rate, average velocity, Reynolds number, and Darcy friction factor are obtained from the measurements of the power and the rotational speed and the outlined formulations given in Research Methodology. The results of measurements are listed in Table3 for WBM in the ACK TCV.


### 6.3. Calibration of the ACK TCV

Results of WBM flow in the ACK TCV are presented in Table3 for speeds 0-1600 RPM (200 RPM intervals). First, we calibrated the TCV using high-precision rheometer measurements reported in [33], and the comparison of the viscosity and stress is shown in Figure 2.We investigate the accuracy of these measurements by fitting different rheology models presented in Table1.


#### 6.3.1. Newtonian Rheology Model

In the Newtonian rheology (Table1), it is assumed that τ0=0,K=0,η=η∞, n=m=1, and ReNewtonian=ρDu¯/η∞. Therefore, the model (equation (28)) needs only one parameter, i.e.,η∞. Having applied the optimization program in MATLAB, we compare rheology model parameters with the experimental data (Table 4).Table 4
Flow property measurement in TCV using WBM.
Speed (RPM)Total power (W)Fluid power (W)Fluid torque (N·m)Shear rate (1/s)Shear stress (Pa)Apparent viscosity (cP)Average velocity (m/s)Reynolds numberDarcy friction factor206.997.893.980.1835217.407.8135.920.361.08E+040.4642439.0014.719.890.2151460.989.1519.850.774.85E+040.1210641.0023.4616.280.2426673.1610.3215.331.121.03E+050.0640714.0027.6518.880.2525749.7810.7514.331.251.28E+050.05371096.0046.4434.950.30451150.8912.9611.261.913.02E+050.02751284.9960.0044.440.33021349.3414.0510.412.244.15E+050.02171616.0082.0763.510.37531697.0015.979.412.826.57E+050.0156Using an optimization program in MATLAB, values forη∞=26.5mPa·s and R2=−4.0e−06 are obtained which obviously show the worst fit to measurements. Figure 6 shows the apparent viscosity for which the Newtonian model has obviously failed to capture trends of the apparent viscosity due to non-Newtonian behavior of the WBM. The consistency curve of the Newtonian model has overpredicted measurements as seen in Figure 6.Figure 6
Newtonian model fit to WBM for the stress and the apparent viscosity.


#### 6.3.2. Bingham Rheology Model

For the Bingham model, it is assumed thatK=0,n=1, η=τ0/γ̇+η∞, RegenBingham=ρDu¯/τ0/8D/u¯+η∞3m+1/4m, and m=η∞8u¯/D/τ0+η∞8u¯/D. Therefore, equation (28) needs two parameters, i.e., τ0 and η∞. The generalized optimization MATLAB program was then applied to investigate the suitability of the Bingham model for WBM examined here.The Bingham rheology model has perfectly fitted WBM measurements (see Figure7) with values τ0=10.5Pa, η∞=8.8mPa·s, and R2=1.0.Figure 7
Bingham model fit to WBM withτ0=10.5Pa and η∞=8.8mPa·s.As observed in Figure7, the exact fit is obtained for TCV-measured apparent viscosity and shear stress with the values from the Bingham model.


#### 6.3.3. Power Law (PL) Rheology Model

In the power law (PL) rheology model (Table1), it is assumed that τ0=0,η∞=0, η=Kγ̇n−1, m=n, and RegenPL=ρDnu¯2−n/K3m+1/4mn8n−1. This model needs two parameters, i.e., K and n, to be found by the optimization program.Power law (PL) has nicely fitted WBM measurements withK=2.4826, n=0.2961, and R2=0.9930 (see Figure 8).Figure 8
Power law (PL) model fit to WBM withK=2.4826Pa·sn and n=0.2961.


#### 6.3.4. Herschel-Bulkley (HB) Rheology Model

The Herschel-Bulkley (HB) rheology model is a widely accepted model for drilling fluids. As shown in Table1, the Herschel-Bulkley (HB) model is expressed by η∞=0 and η=τ0/γ̇+Kγ̇n−1 with 3 unknown parameters τ0, K, and n. The generalized Reynolds number is given by RegenHB=ρDnu¯2−n/τ0/8D/u¯n+K3m+1/4mn8n−1 in which m=nK8u¯/Dn/τ0+K8u¯/Dn. The MATLAB optimization program was applied to find the best fit to experimental data using the HB model.The Herschel-Bulkley (HB) rheology model has fitted very well WBM measurements (see Figure9) with τ0=9.6333Pa, K=0.0325Pa·sn, and n=0.8234 with an accuracy of R2=0.9995.Figure 9
Herschel-Bulkley (HB) model for fitting WBM measurements.


#### 6.3.5. Herschel-Bulkley-Extended (HBE) Rheology Model

The Herschel-Bulkley-Extended (HBE) rheology model is the most complete model compared with previous models. The HBE was introduced by Madlener et al. [32, 34] to study non-Newtonian gel fluids. The HBE model is expressed by η=τ0/γ̇+Kγ̇n−1+η∞ which needs four parameters τ0, K, n, and η∞. HBE is seen as the generalized model for all models (see Table 1). The Reynolds number is RegenHBE=ρDnu¯2−n/τ0/8D/u¯n+K3m+1/4mn8n−1+η∞3m+1/4mD/u¯n−1 in which m=nK8u¯/Dn+η∞8u¯/D/τ0+K8u¯/Dn+η∞8u¯/D. In our in-house-developed MATLAB program, the HBE model can be reduced to any of the aforementioned 5 models to apply optimization and to obtain unknown parameters by fitting to experimental data. Here, we examine the HBE model for validation of experimental results for WBM measurements.Bingham and HBE models produced a perfect match with WBM measurements (see Figure10). The 4 parameters of the HBE model are obtained as τ0=10.5Pa, K=3.5720Pa·sn,n=−1.9281, and η∞=8.8mPa·s with 100% accuracy of R2=1.0000.Figure 10
Herschel-Bulkley-Extended (HBE) rheology model fitting with WBM measurements.


#### 6.3.6. Four-Parameter Drilling Models

The Cross model is one of the 4-parameter models widely used for drilling fluids. This model was presented in [17] to study drilling fluids. The Cross model is expressed by η=η∞+ηo−η∞/1+Kγ̇n with 4 unknown parameters ηo, K, n, and η∞. In the developed MATLAB program, the Cross model is fitted for WBM (4% vol bentonite API standard) measurements as illustrated in Figure 11. The 4 parameters of the Cross model are obtained as ηo=1316mPa·s, K=0.1084s,n=1.0337, and η∞=9.0mPa·s with 100% accuracy of R2=1.0000.Figure 11
Cross model fit to WBM.The Carreau model is also one of the 4-parameter models widely used for drilling fluids [17]. The Carreau model is expressed by η=η∞+ηo−η∞/1+Kγ̇2−n/2 with 4 unknown parameters ηo, K, n, and η∞. The Carreau model is fitted to WBM (4% vol bentonite API standard) measurements as illustrated in Figure 12. The 4 parameters of the Carreau model are obtained as ηo=1894.6mPa·s, K=0.1360s, n=−1.0925, and η∞=10.6mPa·s with an accuracy of R2=0.9993. This model is less accurate than the Cross model.Figure 12
Carreau model fit to WBM.


#### 6.3.7. Friction Factor

Darcy friction factor calculations are presented here. For the Newtonian rheology model, the measurement values in the TCV fall in the turbulent region and deviate significantly from the laminar friction factorflaminar=64/Re (see Figure 13). The shift from the laminar friction factor exhibits turbulent flow condition for all speeds.Figure 13
Darcy friction factor for all models versus the laminar model.For the Bingham model, we found that the Darcy friction factorfDarcy=8τwi/ρu¯θ2 presented in [34] is more suitable for comparing TCV flows with the laminar friction factor flaminar=64/Re. As shown in Figure 13, transition from laminar flow (solid line) occurred at ssssBingham=2000 and above, which is the laminar limit in pipe flows. The deviation from the laminar Darcy friction shows arrival in turbulent flow condition. Madlener and Ciezki [43, 44] also concluded that this deviation is an indication of turbulence.Using the power law (PL) for the WBM, it seems that the Darcy friction factor remains with a slight negative shift below the laminar friction factor for the full range of experimental data as shown in Figure13. Madlener et al. [34] reported the negative deviation with no explanation for the reason of that.The HB model in comparison with the Bingham model (highest accuracy) shows that the measured values of the Darcy friction factor have deviated very slightly (negative deviation) from the laminar friction factor (see Figure13).For the HBE model, it is observed from the Darcy friction factor that laminar flow is deviated at the higher Reynolds number of aroundRe=2000 compared with the Bingham model; this is the exact match with a classical upper limit of laminar flows in internal pipe flows. However, the Bingham model may be preferred over HBE due to its simplicity.


## 6.1. Power Consumption

The ACK TCV uses power consumptions to determine rheology. To take into account frictional power loss, the TCV was first calibrated to measure power loss due to mechanical friction. Then, WBM was filled up into the TCV and power consumptions were measured. The TCV reading was conducted by allowing a steady state which is reached as each speed.Unlike stepper motors, the speed controller of the DC motor used cannot provide a fixed preset value for speed. Thus, we used the best fit, to the raw data given in Table3, to obtain power values at precise rotational speed of TCV. Figure 5 compares mechanical power (air), total power (fluid+air), and fluid power for testing TCV without liquid and with WBM.Table 3
Power measurements of TCV for air and WBM.
AirWBMSpeed (rad/s)Power (W)Speed (rad/s)Power (W)24.633.2221.687.8950.355.1045.9714.7170.276.8067.1323.4688.409.1274.7727.6596.9012.44114.7746.44137.6015.20134.5660.00167.5519.30169.2382.07Figure 5
Power models in TCV: mechanical power (air), total power (fluid+air), and fluid power using WBM (4% vol bentonite API standard).As observed in Figure5, the mechanical power (W) of TCV is a linear function of angular speed (rad/s) for air as follows:
(32)Pmech=0.1128Ωi.Figure5 also shows that the base fluid, i.e., WBM, is a second-ordered polynomial function of rotational speed (rad/s):
(33)Pfluid=0.0013Ωi2+0.1553Ωi.R2 accuracy of the fitted functions is very closed to 1, so these are highly accurate. Mechanical power (air) given in equation (30) should be deducted from the total power measurements of WBM to obtain net fluid power using equation (31).


## 6.2. Flow Properties

Flow properties in the ACK TCV including the apparent viscosity, shear stress, shear rate, average velocity, Reynolds number, and Darcy friction factor are obtained from the measurements of the power and the rotational speed and the outlined formulations given in Research Methodology. The results of measurements are listed in Table3 for WBM in the ACK TCV.


## 6.3. Calibration of the ACK TCV

Results of WBM flow in the ACK TCV are presented in Table3 for speeds 0-1600 RPM (200 RPM intervals). First, we calibrated the TCV using high-precision rheometer measurements reported in [33], and the comparison of the viscosity and stress is shown in Figure 2.We investigate the accuracy of these measurements by fitting different rheology models presented in Table1.


### 6.3.1. Newtonian Rheology Model

In the Newtonian rheology (Table1), it is assumed that τ0=0,K=0,η=η∞, n=m=1, and ReNewtonian=ρDu¯/η∞. Therefore, the model (equation (28)) needs only one parameter, i.e.,η∞. Having applied the optimization program in MATLAB, we compare rheology model parameters with the experimental data (Table 4).Table 4
Flow property measurement in TCV using WBM.
Speed (RPM)Total power (W)Fluid power (W)Fluid torque (N·m)Shear rate (1/s)Shear stress (Pa)Apparent viscosity (cP)Average velocity (m/s)Reynolds numberDarcy friction factor206.997.893.980.1835217.407.8135.920.361.08E+040.4642439.0014.719.890.2151460.989.1519.850.774.85E+040.1210641.0023.4616.280.2426673.1610.3215.331.121.03E+050.0640714.0027.6518.880.2525749.7810.7514.331.251.28E+050.05371096.0046.4434.950.30451150.8912.9611.261.913.02E+050.02751284.9960.0044.440.33021349.3414.0510.412.244.15E+050.02171616.0082.0763.510.37531697.0015.979.412.826.57E+050.0156Using an optimization program in MATLAB, values forη∞=26.5mPa·s and R2=−4.0e−06 are obtained which obviously show the worst fit to measurements. Figure 6 shows the apparent viscosity for which the Newtonian model has obviously failed to capture trends of the apparent viscosity due to non-Newtonian behavior of the WBM. The consistency curve of the Newtonian model has overpredicted measurements as seen in Figure 6.Figure 6
Newtonian model fit to WBM for the stress and the apparent viscosity.


### 6.3.2. Bingham Rheology Model

For the Bingham model, it is assumed thatK=0,n=1, η=τ0/γ̇+η∞, RegenBingham=ρDu¯/τ0/8D/u¯+η∞3m+1/4m, and m=η∞8u¯/D/τ0+η∞8u¯/D. Therefore, equation (28) needs two parameters, i.e., τ0 and η∞. The generalized optimization MATLAB program was then applied to investigate the suitability of the Bingham model for WBM examined here.The Bingham rheology model has perfectly fitted WBM measurements (see Figure7) with values τ0=10.5Pa, η∞=8.8mPa·s, and R2=1.0.Figure 7
Bingham model fit to WBM withτ0=10.5Pa and η∞=8.8mPa·s.As observed in Figure7, the exact fit is obtained for TCV-measured apparent viscosity and shear stress with the values from the Bingham model.


### 6.3.3. Power Law (PL) Rheology Model

In the power law (PL) rheology model (Table1), it is assumed that τ0=0,η∞=0, η=Kγ̇n−1, m=n, and RegenPL=ρDnu¯2−n/K3m+1/4mn8n−1. This model needs two parameters, i.e., K and n, to be found by the optimization program.Power law (PL) has nicely fitted WBM measurements withK=2.4826, n=0.2961, and R2=0.9930 (see Figure 8).Figure 8
Power law (PL) model fit to WBM withK=2.4826Pa·sn and n=0.2961.


### 6.3.4. Herschel-Bulkley (HB) Rheology Model

The Herschel-Bulkley (HB) rheology model is a widely accepted model for drilling fluids. As shown in Table1, the Herschel-Bulkley (HB) model is expressed by η∞=0 and η=τ0/γ̇+Kγ̇n−1 with 3 unknown parameters τ0, K, and n. The generalized Reynolds number is given by RegenHB=ρDnu¯2−n/τ0/8D/u¯n+K3m+1/4mn8n−1 in which m=nK8u¯/Dn/τ0+K8u¯/Dn. The MATLAB optimization program was applied to find the best fit to experimental data using the HB model.The Herschel-Bulkley (HB) rheology model has fitted very well WBM measurements (see Figure9) with τ0=9.6333Pa, K=0.0325Pa·sn, and n=0.8234 with an accuracy of R2=0.9995.Figure 9
Herschel-Bulkley (HB) model for fitting WBM measurements.


### 6.3.5. Herschel-Bulkley-Extended (HBE) Rheology Model

The Herschel-Bulkley-Extended (HBE) rheology model is the most complete model compared with previous models. The HBE was introduced by Madlener et al. [32, 34] to study non-Newtonian gel fluids. The HBE model is expressed by η=τ0/γ̇+Kγ̇n−1+η∞ which needs four parameters τ0, K, n, and η∞. HBE is seen as the generalized model for all models (see Table 1). The Reynolds number is RegenHBE=ρDnu¯2−n/τ0/8D/u¯n+K3m+1/4mn8n−1+η∞3m+1/4mD/u¯n−1 in which m=nK8u¯/Dn+η∞8u¯/D/τ0+K8u¯/Dn+η∞8u¯/D. In our in-house-developed MATLAB program, the HBE model can be reduced to any of the aforementioned 5 models to apply optimization and to obtain unknown parameters by fitting to experimental data. Here, we examine the HBE model for validation of experimental results for WBM measurements.Bingham and HBE models produced a perfect match with WBM measurements (see Figure10). The 4 parameters of the HBE model are obtained as τ0=10.5Pa, K=3.5720Pa·sn,n=−1.9281, and η∞=8.8mPa·s with 100% accuracy of R2=1.0000.Figure 10
Herschel-Bulkley-Extended (HBE) rheology model fitting with WBM measurements.


### 6.3.6. Four-Parameter Drilling Models

The Cross model is one of the 4-parameter models widely used for drilling fluids. This model was presented in [17] to study drilling fluids. The Cross model is expressed by η=η∞+ηo−η∞/1+Kγ̇n with 4 unknown parameters ηo, K, n, and η∞. In the developed MATLAB program, the Cross model is fitted for WBM (4% vol bentonite API standard) measurements as illustrated in Figure 11. The 4 parameters of the Cross model are obtained as ηo=1316mPa·s, K=0.1084s,n=1.0337, and η∞=9.0mPa·s with 100% accuracy of R2=1.0000.Figure 11
Cross model fit to WBM.The Carreau model is also one of the 4-parameter models widely used for drilling fluids [17]. The Carreau model is expressed by η=η∞+ηo−η∞/1+Kγ̇2−n/2 with 4 unknown parameters ηo, K, n, and η∞. The Carreau model is fitted to WBM (4% vol bentonite API standard) measurements as illustrated in Figure 12. The 4 parameters of the Carreau model are obtained as ηo=1894.6mPa·s, K=0.1360s, n=−1.0925, and η∞=10.6mPa·s with an accuracy of R2=0.9993. This model is less accurate than the Cross model.Figure 12
Carreau model fit to WBM.


### 6.3.7. Friction Factor

Darcy friction factor calculations are presented here. For the Newtonian rheology model, the measurement values in the TCV fall in the turbulent region and deviate significantly from the laminar friction factorflaminar=64/Re (see Figure 13). The shift from the laminar friction factor exhibits turbulent flow condition for all speeds.Figure 13
Darcy friction factor for all models versus the laminar model.For the Bingham model, we found that the Darcy friction factorfDarcy=8τwi/ρu¯θ2 presented in [34] is more suitable for comparing TCV flows with the laminar friction factor flaminar=64/Re. As shown in Figure 13, transition from laminar flow (solid line) occurred at ssssBingham=2000 and above, which is the laminar limit in pipe flows. The deviation from the laminar Darcy friction shows arrival in turbulent flow condition. Madlener and Ciezki [43, 44] also concluded that this deviation is an indication of turbulence.Using the power law (PL) for the WBM, it seems that the Darcy friction factor remains with a slight negative shift below the laminar friction factor for the full range of experimental data as shown in Figure13. Madlener et al. [34] reported the negative deviation with no explanation for the reason of that.The HB model in comparison with the Bingham model (highest accuracy) shows that the measured values of the Darcy friction factor have deviated very slightly (negative deviation) from the laminar friction factor (see Figure13).For the HBE model, it is observed from the Darcy friction factor that laminar flow is deviated at the higher Reynolds number of aroundRe=2000 compared with the Bingham model; this is the exact match with a classical upper limit of laminar flows in internal pipe flows. However, the Bingham model may be preferred over HBE due to its simplicity.


## 6.3.1. Newtonian Rheology Model

In the Newtonian rheology (Table1), it is assumed that τ0=0,K=0,η=η∞, n=m=1, and ReNewtonian=ρDu¯/η∞. Therefore, the model (equation (28)) needs only one parameter, i.e.,η∞. Having applied the optimization program in MATLAB, we compare rheology model parameters with the experimental data (Table 4).Table 4
Flow property measurement in TCV using WBM.
Speed (RPM)Total power (W)Fluid power (W)Fluid torque (N·m)Shear rate (1/s)Shear stress (Pa)Apparent viscosity (cP)Average velocity (m/s)Reynolds numberDarcy friction factor206.997.893.980.1835217.407.8135.920.361.08E+040.4642439.0014.719.890.2151460.989.1519.850.774.85E+040.1210641.0023.4616.280.2426673.1610.3215.331.121.03E+050.0640714.0027.6518.880.2525749.7810.7514.331.251.28E+050.05371096.0046.4434.950.30451150.8912.9611.261.913.02E+050.02751284.9960.0044.440.33021349.3414.0510.412.244.15E+050.02171616.0082.0763.510.37531697.0015.979.412.826.57E+050.0156Using an optimization program in MATLAB, values forη∞=26.5mPa·s and R2=−4.0e−06 are obtained which obviously show the worst fit to measurements. Figure 6 shows the apparent viscosity for which the Newtonian model has obviously failed to capture trends of the apparent viscosity due to non-Newtonian behavior of the WBM. The consistency curve of the Newtonian model has overpredicted measurements as seen in Figure 6.Figure 6
Newtonian model fit to WBM for the stress and the apparent viscosity.


## 6.3.2. Bingham Rheology Model

For the Bingham model, it is assumed thatK=0,n=1, η=τ0/γ̇+η∞, RegenBingham=ρDu¯/τ0/8D/u¯+η∞3m+1/4m, and m=η∞8u¯/D/τ0+η∞8u¯/D. Therefore, equation (28) needs two parameters, i.e., τ0 and η∞. The generalized optimization MATLAB program was then applied to investigate the suitability of the Bingham model for WBM examined here.The Bingham rheology model has perfectly fitted WBM measurements (see Figure7) with values τ0=10.5Pa, η∞=8.8mPa·s, and R2=1.0.Figure 7
Bingham model fit to WBM withτ0=10.5Pa and η∞=8.8mPa·s.As observed in Figure7, the exact fit is obtained for TCV-measured apparent viscosity and shear stress with the values from the Bingham model.


## 6.3.3. Power Law (PL) Rheology Model

In the power law (PL) rheology model (Table1), it is assumed that τ0=0,η∞=0, η=Kγ̇n−1, m=n, and RegenPL=ρDnu¯2−n/K3m+1/4mn8n−1. This model needs two parameters, i.e., K and n, to be found by the optimization program.Power law (PL) has nicely fitted WBM measurements withK=2.4826, n=0.2961, and R2=0.9930 (see Figure 8).Figure 8
Power law (PL) model fit to WBM withK=2.4826Pa·sn and n=0.2961.


## 6.3.4. Herschel-Bulkley (HB) Rheology Model

The Herschel-Bulkley (HB) rheology model is a widely accepted model for drilling fluids. As shown in Table1, the Herschel-Bulkley (HB) model is expressed by η∞=0 and η=τ0/γ̇+Kγ̇n−1 with 3 unknown parameters τ0, K, and n. The generalized Reynolds number is given by RegenHB=ρDnu¯2−n/τ0/8D/u¯n+K3m+1/4mn8n−1 in which m=nK8u¯/Dn/τ0+K8u¯/Dn. The MATLAB optimization program was applied to find the best fit to experimental data using the HB model.The Herschel-Bulkley (HB) rheology model has fitted very well WBM measurements (see Figure9) with τ0=9.6333Pa, K=0.0325Pa·sn, and n=0.8234 with an accuracy of R2=0.9995.Figure 9
Herschel-Bulkley (HB) model for fitting WBM measurements.


## 6.3.5. Herschel-Bulkley-Extended (HBE) Rheology Model

The Herschel-Bulkley-Extended (HBE) rheology model is the most complete model compared with previous models. The HBE was introduced by Madlener et al. [32, 34] to study non-Newtonian gel fluids. The HBE model is expressed by η=τ0/γ̇+Kγ̇n−1+η∞ which needs four parameters τ0, K, n, and η∞. HBE is seen as the generalized model for all models (see Table 1). The Reynolds number is RegenHBE=ρDnu¯2−n/τ0/8D/u¯n+K3m+1/4mn8n−1+η∞3m+1/4mD/u¯n−1 in which m=nK8u¯/Dn+η∞8u¯/D/τ0+K8u¯/Dn+η∞8u¯/D. In our in-house-developed MATLAB program, the HBE model can be reduced to any of the aforementioned 5 models to apply optimization and to obtain unknown parameters by fitting to experimental data. Here, we examine the HBE model for validation of experimental results for WBM measurements.Bingham and HBE models produced a perfect match with WBM measurements (see Figure10). The 4 parameters of the HBE model are obtained as τ0=10.5Pa, K=3.5720Pa·sn,n=−1.9281, and η∞=8.8mPa·s with 100% accuracy of R2=1.0000.Figure 10
Herschel-Bulkley-Extended (HBE) rheology model fitting with WBM measurements.


## 6.3.6. Four-Parameter Drilling Models

The Cross model is one of the 4-parameter models widely used for drilling fluids. This model was presented in [17] to study drilling fluids. The Cross model is expressed by η=η∞+ηo−η∞/1+Kγ̇n with 4 unknown parameters ηo, K, n, and η∞. In the developed MATLAB program, the Cross model is fitted for WBM (4% vol bentonite API standard) measurements as illustrated in Figure 11. The 4 parameters of the Cross model are obtained as ηo=1316mPa·s, K=0.1084s,n=1.0337, and η∞=9.0mPa·s with 100% accuracy of R2=1.0000.Figure 11
Cross model fit to WBM.The Carreau model is also one of the 4-parameter models widely used for drilling fluids [17]. The Carreau model is expressed by η=η∞+ηo−η∞/1+Kγ̇2−n/2 with 4 unknown parameters ηo, K, n, and η∞. The Carreau model is fitted to WBM (4% vol bentonite API standard) measurements as illustrated in Figure 12. The 4 parameters of the Carreau model are obtained as ηo=1894.6mPa·s, K=0.1360s, n=−1.0925, and η∞=10.6mPa·s with an accuracy of R2=0.9993. This model is less accurate than the Cross model.Figure 12
Carreau model fit to WBM.


## 6.3.7. Friction Factor

Darcy friction factor calculations are presented here. For the Newtonian rheology model, the measurement values in the TCV fall in the turbulent region and deviate significantly from the laminar friction factorflaminar=64/Re (see Figure 13). The shift from the laminar friction factor exhibits turbulent flow condition for all speeds.Figure 13
Darcy friction factor for all models versus the laminar model.For the Bingham model, we found that the Darcy friction factorfDarcy=8τwi/ρu¯θ2 presented in [34] is more suitable for comparing TCV flows with the laminar friction factor flaminar=64/Re. As shown in Figure 13, transition from laminar flow (solid line) occurred at ssssBingham=2000 and above, which is the laminar limit in pipe flows. The deviation from the laminar Darcy friction shows arrival in turbulent flow condition. Madlener and Ciezki [43, 44] also concluded that this deviation is an indication of turbulence.Using the power law (PL) for the WBM, it seems that the Darcy friction factor remains with a slight negative shift below the laminar friction factor for the full range of experimental data as shown in Figure13. Madlener et al. [34] reported the negative deviation with no explanation for the reason of that.The HB model in comparison with the Bingham model (highest accuracy) shows that the measured values of the Darcy friction factor have deviated very slightly (negative deviation) from the laminar friction factor (see Figure13).For the HBE model, it is observed from the Darcy friction factor that laminar flow is deviated at the higher Reynolds number of aroundRe=2000 compared with the Bingham model; this is the exact match with a classical upper limit of laminar flows in internal pipe flows. However, the Bingham model may be preferred over HBE due to its simplicity.


## 7. Concluding Remarks and Future Works

It is of interest to enhance mud properties particularly at high-speed drilling. In this work, a high-speed viscometer is developed using the classical Taylor-Couette system (TCV) with low volume capacity of 100 ml to operate at speeds from 0 to 1600 RPM. Some standard viscometers merely operate at low speeds below 600 RPM. However, it is inevitable to study rheology of drilling fluids at turbulent flow conditions in a real drilling process. A high-speed TCV power-based system was developed to study rheology of WBM (4% vol bentonite API standard) at laminar to turbulent flow conditions. An optimization program was developed in MATLAB to fit WBM measurements with five classical rheology models extracted from the HBE model and two 4-parameter family models, i.e., Cross and Carreau models. The following conclusions are obtained:(i)
The modelled apparent viscosity and the shear stress are presented and discussed for all models. Bingham, HBE, and Cross models have fitted precisely with the experimental data(ii)
A positive shift from the laminar friction factor was detected at higher Reynolds numbers than 2000 which indicates initiation of turbulent flows. It is also shown that the upper limit of laminar internal pipe flows (ReGen=2000) may be valid for the annulus flow of the TCV(iii)
The optimization methodology developed here can feasibly obtain 4-parameter or more rheology models. This will be useful particularly for rheology model development of nanofluids(iv)
The HB and PL models exhibited the unresolved negative shift from the laminar friction factor to be further investigatedWork currently continues on WBM enhanced by nanoparticles. Nanoparticles are usually very expensive, so the designed ACK TCV with the volume of only 100 ml is a good viscometer for studying nanofluids.



---

*Source: 1344776-2021-08-26.xml*