Herschel-Bulkley Fluid Research Articles

Experimental footprints of a water-rich depletion layer in the Herschel–Bulkley pipe flow of solidifying polyelectrolytes

A fundamental understanding of the transition from fluid-like to gel-like behavior is critical for a range of applications including personal care, pharmaceuticals, food products, batteries, painting, biomaterials, and concrete. The pipe flow behavior of a Herschel–Bulkley fluid is examined by a combination of rheology, ultrasound imaging velocimetry, and pressure measurements together with modeling. The system is a solution of 0.50 wt. % polyelectrolytes of sulfated polysaccharides in water that solidifies on cooling. Fluids with different ionic strengths were pumped at various rates from a reservoir at 80 °C into a pipe submerged in a bath maintained at 20 °C. The fluid velocity, pressure drop ΔP, and temperature were monitored. The same quantities were extracted by solving continuity, energy, and momentum equations. Moreover, the modeling results demonstrate that the local pressure gradient along the pipe dPdx|x is related to the local yield stress near the pipe wall τywall|x, which explains the variations of dPdx|x along the pipe. Experimental results show much lower values for ΔP compared to those from modeling. This discrepancy is exacerbated at higher ionic strengths and smaller flow rates, where fluid shows a higher degree of solidification. The tabulated experimental ΔP data against the solidification onset length Lonset (where the fluid is cool enough to solidify) along with the ultrasound imaging velocimetry associate these discrepancies between experiments and models to a depletion layer of ∼1 μm, reflecting the lubrication effects caused by the water layer at the wall.

Study on the Influence of Tripping Operation on Annular Transient Surge-Swab Pressure of Herschel–Bulkley Fluid

Abstract The accurate prediction of transient surge-swab pressure, during the tripping operation under a narrow pressure window, is an important premise to ensure the safety of tripping operation and drilling. Different rheological modes of drilling fluid will have a great influence on the surge-swab pressure in wellbore, while the Herschel–Bulkley mode can simulate the rheological characteristics of actual drilling fluid in a better way. Therefore, the influence of drilling fluid flow friction under different rheological modes was mainly considered according to the transient flow model. Meanwhile, the influence of pumpage on transient surge-swab pressure during tripping operation was also considered with Herschel–Bulkley fluid-based tripping operation bottomhole transient surge-swab pressure calculation model established and model verification carried out. The following research results were obtained. First, compared with the Herschel–Bulkley mode, the power-law mode and Bingham mode will overestimate the flow friction of drilling fluid. Second, the surge-swab pressure peak values of different mud systems are water-based mud > polymer mud > oil-based mud. Third, the peak value of bottomhole transient surge-swab pressure increases with the increase of drilling fluid yield value, consistency coefficient, and flow behavior index, and the influence law is similar. Fourth, the tripping velocity and pumpage will directly affect the peak value and variation rule of bottomhole transient surge-swab pressure. Herschel–Bulkley fluid-based tripping operation bottomhole transient surge-swab pressure study helps control the surge-swab pressure during tripping operations and it is of great significance for lean, safe, and efficient drilling in narrow pressure window formations.

ON THE BLOCKING FORCE OF STEADY-STATE FLOW OF HERSCHEL-BULKLEY FLUID

The paper is devoted to the study of the blocking force of Herschel-Bulkley fluid in the case of steady-state flow. To this aim, we consider a mathematical model which describes the steady-state flow of a Herschel-Bulkley fluid in a bounded domain. We give the mathematical formulation of the blockage phenomenon and we establish the existence of blocking force. We also focus on behaviour of the flow with respect to the blocking force.

Application of a maritime CFD code to a benchmark problem for non-Newtonian fluids: the flow around a sphere

The ship’s resistance and manoeuvrability in shallow waters can be adversely influenced by the presence of fluid mud layers on the seabed of ports and waterways. Fluid mud exhibits a complex non-Newtonian rheology that is often described using the Herschel–Bulkley model. The latter has been recently implemented in a maritime finite-volume CFD code to study the manoeuvrability of ships in the presence of muddy seabeds. In this paper, we explore the accuracy and robustness of the CFD code in simulating the flow of Herschel–Bulkley fluids, including power-law, Bingham and Newtonian fluids as particular cases. As a stepping stone towards the final maritime applications, the study is carried out on a classic benchmark problem in non-Newtonian fluid mechanics: the laminar flow around a sphere. The aim is to test the performance of the non-Newtonian solver before applying it to the more complex scenarios. Present results could also be used as reference data for future testing. Flow simulations are carried out at low Reynolds numbers in order to compare our results with an extensive collection of data from the literature. Results agree both qualitatively and quantitatively with literature. Difficulties in the convergence of the iterative solver emerged when simulating Bingham and Herschel–Bulkley flows. A simple change in the interpolation of the apparent viscosity has mitigated such difficulties. The results of this work, combined with our previous code verification exercises, suggest that the non-Newtonian solver works as intended and it can be thus employed on more complex applications.

Effect of Catheter and Stenosis on Solute Diffusion in Non-Newtonian Blood Flow through a Catheterized Stenosed Artery

The presence of stenosis at the wall of the artery lead to further cardiovascular diseases such as heart attack, stroke and many more. Treatment of a stenosed artery includes the insertion of a catheter through the artery which affects the blood flow and solute dispersion. This present study focuses on the effect of catheter radius and stenosis height on the blood flow and solute dispersion behavior. The problem is modelled using the Herschel-Bulkley fluid to represent the blood rheology, with catheter and stenosis as the boundary conditions. Analytical solutions in integral form are obtained by solving the momentum equation and Herschel-Bulkley constitutive equation. The integrals are numerically evaluated using the Simpson’s 3/8 rule and Regula-Falsi method to obtain the blood velocity. The obtained velocity is employed into the unsteady convective-diffusion equation and solved using the generalized dispersion model (GDM) to analyse the behaviour of solute diffusion. The influence of catheter radius and stenosis height on the diffusion coefficient and mean concentration of solute are observed. Results show that the diffusion coefficient decreases as the catheter radius and stenosis height increases. A decrease in diffusion coefficient simultaneously increases the solute mean concentration.

Exact Analysis of Unsteady Convective Diffusion in Herschel-Bulkley Fluid Flow- Application to Catheterised Stenosed Artery

One of the major causes of cardiovascular disease is atherosclerosis or stenosis. This study is designed to improve the current body of knowledge regarding the condition by inserting a long thin tube called a catheter to widen the narrow part in the artery. The study reviewed the effects of catheter radius, yield stress, and power law index on the velocity distribution, and transport coefficients of solute. A mathematical model is deployed to investigate the dispersion of solute in the flow of a Herschel-Bulkley (H-B) fluid in an annulus, whereas the dispersion process is studied using the generalised dispersion model (GDM) by solving the convective diffusion equation. Resultan tly, the velocity reduces following an increase in the yield stress, catheter size, and power law index. Meanwhile, the dispersion coefficient exhibits a same behaviour as the aforementioned parameters ascend considerably. The dispersion coefficient alterations occurred rapidly for small values of time and became significantly constant following an increase in the time values. Conclusively, this study can be useful in dispersion of a drug to the affected artery where an abnormal plaque was formed.

Laminar–turbulent intermittency in pipe flow for an Herschel–Bulkley fluid: Radial receptivity to finite-amplitude perturbations

We investigate the laminar-to-turbulent transition for non-Newtonian Herschel–Bulkley fluids that exhibit either a shear-thinning or shear-thickening behavior. The reduced-order model developed in this study also includes the effect of yield-stress for the fluid. Within our model framework, we investigate how the Newtonian dynamics change when significant non-Newtonian effects are considered either via the flow index n or the yield-stress τ0 or both. We find that an increase in τ0 as well as a decrease in n lead to a delayed transition if a perturbation of the given turbulent intensity is injected at various radial locations. As the radial position of the injection for the perturbation is varied in this study, our reduced-order model allows for the investigation of the flow receptivity to the finite-amplitude perturbations and to their radial position of inception. We observe that, for a given mean flow profile, the same perturbation becomes more prone to induce turbulence the closer it approaches the wall because of its initial amplitude being relatively higher with respect to the local mean flow. An opposite trend is found when the perturbation amplitude is rescaled on the local mean flow.

Revisiting the model-prototype similarity issue arising from investigations of submarine debris flow passing a downscaled pipeline in normal gravity

In this study, the model-prototype similarity issue arising from investigations of submarine debris flow passing a downscaled pipeline in normal gravity is revisited. Theoretical analyses of the similarity issue for Power-law and Herschel–Bulkley fluids passing downscaled pipelines are performed based on the dimensionless governing equations. It is observed that, for a Power-law fluid, the similarity criterions of Reynolds number and power index can be readily satisfied in theory and practice. However, for a Herschel–Bulkley fluid, the similarity criterions of Oldroyd number, Reynolds number, and power index should be used in theory, but hardly be satisfied simultaneously in practice. Instead, the similarity criterion of plastic Reynolds number is suggested as an approximate alternative for a Herschel–Bulkley fluid. Based on these similarity criterions, the scaling laws for Power-law and Herschel–Bulkley fluids are derived, respectively. The applicability of scaling laws is demonstrated based on Computational Fluid Dynamics (CFD) simulations. The scaling laws are fully applicable to a Power-law fluid, while they can only reach an approximate similarity of drag coefficient for a Herschel–Bulkley fluid. These updated insights will facilitate a more rational design of a downscaled model to study the debris flow passing a pipeline in normal gravity.

Numerical correlations for Herschel bulkley fluid flow through an axisymmetric sudden expansion

Numerical correlations for Herschel bulkley fluid flow through an axisymmetric sudden expansion

Stepwise mathematical derivation of the Herschel–Bulkley laminar fluid flow equations—in pipes

Stepwise derivation of flow equations of the Herschel–Bulkley (HB) model is not available in the literature. These equations are crucial for mechanical, chemical and petroleum engineering academia and industries where fundamental works on non-Newtonian fluids may be done to reach future models and estimation methods. Therefore, this work focuses on derivation of laminar flow equations and estimation methods of HB fluids through pipes. In this work, first, stepwise derivation of the HB fluid flow parameters consisting of fluid velocity, flow rate, average velocity and relative velocity equations is presented, followed by a straightforward mathematical model for use in numerical solution. Next, stepwise mathematical derivation of the laminar pressure drop equations by Merlo et al. (An innovative model for drilling fluid hydraulics. Paper presented at the SPE Asia Pacific oil and gas conference, Kuala Lumpur, Malaysia, 1995) and Gjerstad and Time (SPE J 20:1–18, 2014) is presented, and finally practical and user-friendly calculation procedures for different estimation methods are presented. The step-by-step derivation procedures presented in this work contribute to effective learning for engineering students and practitioners in addition to providing a clear example derivation guideline for future researchers to reach other more accurate non-Newtonian hydraulics models and estimation methods.

Analysis of Unsteady Solute Dispersion in a Blood Flow of Herschel-Bulkley through a Catheterized Stenosed Artery

Blockage of blood flow due to cholesterol deposits at the arterial wall, known as stenosis, can lead to conditions such as heart attack and stroke. Treatment such as balloon angioplasty involves the catheterization of an artery where a stented catheter is inflated at the stenosis site to open the narrowed artery. The catheterization of the stenosed artery affects the surrounding blood flow and dispersion process. The present study analyses the effect of catheter radius and stenosis height on the blood velocity and solute dispersion behavior. Herschel-Bulkley fluid is used to model the problem with stenosis as the boundary condition. The momentum equation and Herschel-Bulkley constitutive equation are solved analytically into integral forms. Simpson’s 3/8 rule and Regula-Falsi method were used to evaluate the integral numerically to obtain the velocity. The velocity was utilized to solve the unsteady convective-diffusion equation using the generalized dispersion model (GDM) to obtain the dispersion function. This present research can potentially help the medical field and industry in determining the suitable catheter radius for patients, calculating drug dosage and improving stent catheter design. Results show that the velocity decreases as the catheter radius and stenosis height increase. A decrease in velocity simultaneously increases the solute dispersion function.

On the Response of a Herschel-Bulkley Fluid Due to a Moving Plate.

In this paper, we study the boundary-layer flow of a Herschel–Bulkley fluid due to a moving plate; this problem has been experimentally investigated by others, where the fluid was assumed to be Carbopol, which has similar properties to cement. The computational fluid dynamics finite volume method from the open-source toolbox/library OpenFOAM is used on structured quad grids to solve the mass and the linear momentum conservation equations using the solver “overInterDyMFoam” customized with non-Newtonian viscosity libraries. The governing equations are solved numerically by using regularization methods in the context of the overset meshing technique. The results indicate that there is a good comparison between the experimental data and the simulations. The boundary layer thicknesses are predicted within the uncertainties of the measurements. The simulations indicate strong sensitivities to the rheological properties of the fluid.

Shear and solid-liquid separation behaviour of anaerobic digested sludge across a broad range of solids concentrations

Due to the non-homogeneous and multiphase nature of anaerobic lagoon constituents, CFD modelling for process optimisation requires continuous functions for shear and solid-liquid separation properties across a large range of solids concentrations. Unfortunately, measurement of existing material properties of anaerobic sludges is limited to only shear or solid-liquid separation, or to a limited solids concentration. In this work, the shear properties of an anaerobic sludge were measured from 0.4 to 12.5 vol%, which corresponds to the solids concentrations seen in lagoons. The sludge showed Newtonian behaviour at 0.4 vol% and Herschel-Bulkley yield stress fluid behaviour for higher concentrations ranging from 0.5 to 12 vol%. We compared multiple approaches to determine relationships between the model fitting parameters of consistency, k, flow index, n, and shear yield stress, τy with solids volume fraction ϕ.The solid-liquid separation properties were measured from sedimentation and filtration experiments to obtain compressibility and permeability properties across all the above-mentioned concentrations, enabling development of hindered velocity sedimentation curves. Comparison to full-scale anaerobic digestate identified that the pilot lagoon sludge had faster sedimentation at a given solids concentration in comparison to the digestate. This is the first study on simultaneous rheological characterisation and solid-liquid separation behaviour of an anaerobic sludge across a wide range of concentrations, thus enabling CFD modelling of the hydrodynamics and performance of anaerobic lagoons.

Enhanced electroosmotic flow, conductance and ion selectivity of a viscoplastic fluid in a hydrophobic cylindrical pore

We have studied the electroosmotic flow (EOF) of a non-Newtonian viscoplastic fluid, modeled as a Herschel-Bulkley (H-B) fluid, through a single hydrophobic nanopore in a uniformly charged solid hydrophobic membrane separating two identical reservoirs. An interfacial slip velocity develops when the viscoplastic fluid is strained over a hydrophobic surface. It is well established in the context of Newtonian fluid that the interfacial slip at the charged surface augments the EOF. For the viscoplastic fluid, the EOF depends on the fluid behavioral index and the yield stress. We have illustrated the impact of the interfacial slip on the EOF, conductance and ion selectivity of the cylindrical nanopore at different yield stresses for both the shear thinning and the shear thickening cases of the H-B fluid. The slip velocity, characterized by the slip length, enhances the average flow and conductance of the pore and this impact is pronounced for the shear thinning case. The unyielded zone, which develops along the central line of the pore, contracts as the slip length is increased. The ion concentration polarization enhances for the shear thinning fluid, however, the slip length creates a marginal increment. The counterion selectivity of the pore is found to be significant for the Newtonian case as compared to the non-Newtonian fluid and the velocity slip enhances the ion selectivity further. We have determined an analytic solution for the EOF of a power-law fluid in a long hydrophobic tube. Our computed solution for the case of a long tube effectively coincides with this analytic solution. An increment in the pore length reduces its conductance but enhances the ion selectivity. The increment of the average EOF for the hydrophobic pore as compared to the no-slip case grows as the pore length is increased.

A Novel Prediction Model of the Drag Coefficient of Shale Cuttings in Herschel–Bulkley Fluid

In the drilling industry, it is of great significance to accurately predict the drag coefficient and settling velocity of drill cuttings falling in the non-Newtonian drilling fluid. However, the irregular shape of drill cuttings and the non-Newtonian rheological properties of drilling fluid (e.g., shear-thinning and yield stress behavior) make it challenging to predict the settling velocity. In this study, the velocity of particle settlement was studied by a visual device and high-speed camera system. Experimental data of the free settlement of 224 irregular drilling cuttings and 105 spherical particles in the Herschel–Bulkley fluid were obtained. A mechanical model dependent on the force balance of settlement particles was adopted to conduct a detailed statistical analysis of the experimental results, and a prediction model of the drag coefficient of spherical particles in the Herschel–Bulkley fluid was established. A two-dimensional shape description parameter is introduced to establish a model for predicting the drag coefficient of irregular-shaped cuttings in a Herschel–Bulkley fluid. The model has high prediction accuracy for the settling velocity of irregular shale cuttings in Herschel–Bulkley fluid. The average relative error is 7.14%, verifying the model’s accuracy.

Modeling lost-circulation in natural fractures using semi-analytical solutions and type-curves

Drilling is a requisite operation for many industries to reach a targeted subsurface zone. During operations, various issues and challenges are encountered, particularly drilling fluid loss. Loss of circulation is a common problem that often causes interruptions to the drilling process and a reduction in efficiency. Such incidents usually occur when the drilled wellbore encounters a high permeable formation such as faults or fractures, leading to total or partial leakage of the drilling fluids.In this work, a semi-analytical solution and mud type-curves (MTC) are proposed to offer a quick and accurate diagnostic model to assess the lost-circulation of Herschel-Bulkley fluids in fractured media. Based on the observed transient pressure and mud-loss trends, the model can estimate the effective fracture conductivity, the time-dependent cumulative mud-loss volume, and the leakage period. The behavior of lost-circulation into fractured formation can be quickly evaluated, at the drilling site, to perform useful diagnostics, such as the rate of fluid leakage, and the associated effective fracture hydraulic properties. Further, derivative-based mud-type-curves (DMTC) are developed to quantify the leakage of drilling fluid flow into fractures. The developed model is applied for non-Newtonian fluids exhibiting yield-power-law (YPL), including shear thickening and thinning, and Bingham plastic fluids. Proposing rigorous dimensionless groups generates the dual type-curves, MTC and DMTC, which offer superior predictivity compared to traditional methods. Both type-curve sets are used in a dual trend matching, which significantly reduces the non-uniqueness issue that is typically encountered in type-curves.Usage of numerical simulations is implemented based on finite elements to verify the accuracy of the proposed solution. Data for lost circulation from several field cases are presented to demonstrate the applicability of the proposed method. The semi-analytical solver, combined with Monte Carlo simulations, is then applied to assess the sensitivity and uncertainty of various fluid and subsurface parameters, including the hydraulic property of the fracture and the probabilistic prediction of the rate of mud leakage into the formation. The proposed approach is based on a robust semi-analytical solution and type-curves to model the flow behavior of Herschel-Bulkley fluids into fractured reservoirs, which can serve as a quick diagnostic tool to evaluate lost-circulation in drilling operations.

Viscoplastic and pseudoplastic flows through fractal fractures

A new mathematical model is developed to examine the coupled effects of yield stress, shear-thinning or shear-thickening property and fractal characteristics of fractures on the flow of Herschel–Bulkley fluid through fractal fractures. The flowrate is integrated in a range of fractures that follow a fractal distribution. Increasing the fractal dimension increases the flowrate for both Herschel–Bulkley fluid and Newtonian fluid, but the increase for the Newtonian fluid flow is more significant. The influence of maximum and minimum trace length ratio on the flowrate ratio of the Herschel–Bulkley fluid over the Newtonian fluid is more significant than the fractal dimension. The scaling exponent has very similar effect as maximum and minimum trace length ratio on the flowrate ratio in the fractal fractures. The flowrate ratio drops very quickly with the increasing modified Bingham number. For both shear-thinning and shear-thickening fluids, increasing the pressure head gradient amplifies the effect of the power-law index.

Effects of NaCl and MTGase on printability and gelling properties of extrusion-based 3D printed white croaker (Argyrosomus argentatus) surimi

Surimi has been explored as a promising 3D printed ingredient, but its rheological properties need to be identified for printing accuracy and gelling properties improvement. This study investigated the printability of white croaker surimi mixtures with NaCl and microbial transglutaminase (MTGase) in an extrusion-based 3D printing process and their gel forming characteristics after thermal treatment. The surimi mixture with different levels of NaCl (5–40 g/kg) and MTGase (0–4.0 g/kg) behaved as Herschel-Bulkley fluid. The rheological properties and printing performance screened out the appropriate amount of additive (20 g/kg NaCl and 2.0 g/kg MTGase) suitable for 3D printing. Compared the conventionally prepared surimi gels, 3D printed surimi gels with 70% infilled density had similar or even higher water holding capacity and gel strength, but lower L and whiteness values significantly (p < 0.05) because of the porous structure created by 3D printing. The addition of NaCl and MTGase had significant effect on the 3D printability of surimi, and the mechanical properties of 3D printed surimi gels were comparable to the conventionally prepared ones.

Unsteady solute dispersion in non-Newtonian fluid flow in a tube with wall absorption—Deviation from the Gaussianity

Unsteady solute dispersion in a pulsatile Herschel–Bulkley fluid flow in a tube is reinvestigated to examine the significance of the skewness and kurtosis on the concentration distribution using Aris' method of moments considering Hermite polynomials. This study is also an initiation in the direction of solute dispersion in a pulsatile non-Newtonian flow considering the first five moments. This investigation not only brings in the accuracy in the estimation but also measures the deflection and decrease in the axial mean concentration distribution of a solute in a tube. Significant variations in the skewness and kurtosis coefficients against various values of the flow governing parameters, such as the yield stress τy, the wall absorption parameter β, the power law index a, the Womersley frequency parameter α, and the amplitude of fluctuating pressure component e, are presented graphically along with the variations in the mean concentration distribution of the solute in the tube. For larger values of the Womersley frequency parameter, the occurrence of double frequency period for the convection and dispersion coefficients is noticed, which has significant influence on the skewness and kurtosis coefficients. The results for solute dispersion in Newtonian fluid, Bingham fluid, and power law fluid flows are also reported as special cases of this analysis.

Large-eddy simulation of turbulent pipe flow of Herschel-Bulkley fluids - Assessing subgrid-scale models

This paper investigates the turbulent pipe flow of viscoplastic fluids using large-eddy simulation (LES) and the OpenFOAM package. The Herschel-Bulkley model is employed to describe the viscosity function with shear-thinning and viscoplastic behavior. Many LES studies evaluating different subgrid-scale (SGS) models for Newtonian fluids are available in the literature. However, studies regarding LES and SGS models involving non-Newtonian fluids are scarce, despite the importance of the subject. The present paper focuses on evaluating the performance of the Dynamic Smagorinsky and Wall Adapting Local Eddy-viscosity (WALE) models, and the under-resolved direct numerical simulation (UDNS) with viscoplastic fluids. The first- and second-order turbulence statistics and flow features, such as mean wall shear stress, mean wall viscosity, friction factor, mean shear rate, and mean viscosity profiles are used to assess the different models. Simulations are carried out at two different generalized Reynolds numbers (ReG) of 5000 and 7500. The numerical results computed with LES are compared with direct numerical simulation (DNS) results and experimental data for turbulent pipe flow of a Newtonian fluid to validate the numerical method. LES results for the Herschel-Bulkley fluid show a good agreement for the first- and second-order turbulence statistics compared with DNS data. Moreover, mean flow quantities obtained with LES follow the trend of DNS data.