Herschel-Bulkley Fluid Research Articles

Roll Waves in Mudflow Modeled as Herschel–Bulkley Fluids

Roll Waves in Mudflow Modeled as Herschel–Bulkley Fluids

Self-similar solution of the first Stokes problem for non-Newtonian fluids with power-law viscosity

The first Stokes problem is considered for flows of non-Newtonian fluids with effective viscosity varying in accordance with the power law. Self-similar solutions are constructed for the fluids, whose mechanical behavior is described by the Ostwald – de Waele and Herschel–Bulkley rheological models with the corresponding restrictions on the nonlinearity degree exponent n. It is shown that for the Ostwald – de Waele fluid self-similar solutions exist only for 0 n 1, which corresponds to the pseudoplastic behavior. At the same time, self-similar solutions for the Herschel–Bulkley fluid can be obtained only at n 1, when this fluid exhibits the properties of dilatancy.

Recognizing the Most Accurate Herschel-Bulkley Fluid Pressure Estimation Method for Annulus: An Update to API RP 13D, 7th Edition

Summary The Herschel-Bulkley (HB) model is widely regarded as highly accurate for modeling non-Newtonian fluid flow, applicable across diverse fields such as chemical, mechanical, and petroleum engineering. The American Petroleum Institute and other references recognize the HB model as the most accurate and recommend the industry use it for drilling hydraulics. Pressure estimation models are beneficial for applying the HB model in field settings, as numerical solutions may not be feasible due to the computational demands and time constraints, especially challenging for high-frequency real-time pressure loss estimation (at least 1 Hz). Past literature has presented four estimation methods for the HB fluid, consisting of Zamora and Lord (1974), Merlo et al. (1995), Gjerstad and Time (2015), and the generalized method. However, the recommended practice API RP 13D (2023), “Rheology and Hydraulics of Oilwell Drilling Fluids,” has considered the most traditional Zamora and Lord (1974) as their chosen estimation method for industry use while neglecting to identify the most accurate model. Therefore, to update API RP 13D (2023), this work addresses this shortcoming by investigating the performance of different methods in estimating pressure losses while providing user-friendly, straightforward procedures applicable for field use. To recognize the most accurate estimation method for industry use, different methods’ pressure losses were estimated and compared with the numerical solution as well as experimental data to find the errors. The results showed that the Merlo et al. (1995) method provides the most accurate estimates, with the least average absolute percent error (APE) of maximum 4% underestimating the numerical solution, followed by Zamora and Lord’s error overestimating the numerical solution. The errors may slightly vary depending on the input parameters used, including diameters, mud properties, and circulation rate. The effect of the circulation rate on the accuracy of different methods was also investigated; except for the generalized method, the accuracy of the other methods increases with increasing the circulation rate. This work is a very innovative practical work for finding annular pressure losses accurately without using complex numerical solutions.

Derivation and numerical resolution of 2D shallow water equations for multi-regime flows of Herschel–Bulkley fluids

This paper presents mathematical modelling and simulation of thin free-surface flows of viscoplastic fluids with a Herschel–Bulkley rheology over complex topographies with basal perturbations. Using the asymptotic expansion method, depth-averaged models (lubrication and shallow water type models) are derived for 3D (three-dimensional) multi-regime flows on non-flat inclined topographies with varying basal slipperiness. Starting from the Navier–Stokes equations, two flow regimes corresponding to different balances between shear and pressure forces are presented. Flow models corresponding to these regimes are calculated as perturbations of the zeroth-order solutions. The classical reference models in the literature are recovered by considering their respective cases on a flat-inclined surface. In the second regime case, a pressure term is non-negligible. Mathematically, it leads to a corrective term to the classical regime equations. Flow solutions of the two regimes are compared; the difference appears in particular in the vicinity of sharp changes of slopes. Nonetheless, both regime models are compared with experiments and are found to be in good agreement. Furthermore, numerical examples are shown to illustrate the robustness of the present shallow water models to simulate viscoplastic flows in 3D and over an inclined topography with local perturbations in basal elevation and basal slipperiness. The derived models are adequate for direct (engineering and geophysical) applications to real-world flow problems presenting Herschel–Bulkley rheology like lava and mud flows.

Bingham and herschel-bulkley fluids flow regimes in rough-walled rock fractures

Flow of typical non-Newtonian fluids such as cement grouts can experience different regimes as the Reynolds number (Re) changes, understanding of which is important for design and operation of rock grouting in various rock engineering applications. Here, flow regimes of representative non-Newtonian fluids, i.e., Bingham and Herschel–Bulkley (H–B) fluids, are numerically investigated with experimental validations. Three tensile rock fracture surfaces originated from a fine-grained sandstone, a medium-grained sandstone and a medium-grained granite samples are used to create rough-walled fracture models with variable aperture structures. Flow of groundwater, Bingham and H–B fluids through these fractures is numerically simulated respectively, by solving the full mass and momentum conservation equations with the Re ranging from 0.01 to 1000. The regimes for these fluids flowing through the fractures are characterized. Laboratory flow tests are conducted in a cylindrical granite fracture sample to verify the characterized flow regimes. The results reveal important differences of flow regimes between Newtonian and non-Newtonian fluids. Specifically, the transmissivity for water flow is constant when Re is relatively small until the Re reaches certain critical values; the transmissivity for Bingham and H–B fluids flow increases with increasing Re until asymptotically reaches certain peak values, followed by a descending stage when Re is relatively large. The critical (water) and peak (Bingham and H–B fluids) values are affected by surface roughness, that is, a rougher surface results in smaller critical and peak values as well as greater discrepancies compared to the analytical solutions based on the smoothed parallel plates model. These results show that a peak or optimum transmissivity is achievable in a specific range of Re (Re = 10–100 for the fractures studied). This new finding can potentially help optimize the injection pressure or flow rate in rock grouting practices.

Scraping of a thin layer of viscoplastic fluid

We analyze the scraping of a thin layer of viscoplastic fluid residing on a horizontal surface by a translating rigid scraper. This motion generates a mound of fluid upstream of the scraper and a residual layer behind it, both of which are modelled using a shallow layer theory for a Bingham or Herschel-Bulkley fluid. The flows ahead of and behind the scraper are coupled by the motion in the gap under the scraper, which is driven both by the translation of the scraper and by the induced pressure gradient due to the difference in flow thickness upstream and downstream. When the gap between the scraper and the underlying surface is sufficiently small, we find that a steady state emerges after a relatively long transient and that en route to this state, the unsteady dynamics exhibit a variety of regimes that are self-similar to leading order. We construct these solutions explicitly and derive key scalings for the temporal development of the flowing viscoplastic layer, as well as identifying the timescales at which there are transitions between the regimes. These predictions are confirmed by comparison with results from the numerical integration of the full system. Finally, we report results from preliminary laboratory experiments, which are compared with predictions from the shallow-layer theory, obtaining reasonable agreement once a slip boundary condition is included in the model, as motivated by experimental observations. Published by the American Physical Society 2024

Optimisation of Synchronous Grouting Mix Ratio for Shield Tunnels

During shield construction in underground spaces, synchronous grouting slurry is poured between the surrounding rock and tunnel lining to ensure stability. For synchronous grouting slurries, few studies have investigated the relationship between the rheological parameters and physical properties, grout-segregation mechanism, and anti-segregation performance. Therefore, we explored the relationships between the slurry rheological parameters, segregation rate, and bleeding rate. Cement, sand, fly ash, and bentonite were used to prepare the slurry, and the effects of different polycarboxylate water-reducing agents and dispersible latex powder dosages were studied. The rheological parameters of 16 groups of uniformly designed slurries were tested, and the data were fit using the Herschel–Bulkley model. The optimal mix ratio lowered the slurry segregation rate, and its rheological behaviour was consistent with the Herschel–Bulkley fluid characteristics. High-yield-shear-stress synchronous grouting slurries with high and low viscosity coefficients were less likely to bleed and segregate, respectively. The optimised slurry fluidity, 3 h bleeding rate, 24 h bleeding rate, segregation rate, coagulation time, and 28 days compressive strength were 257.5 mm, 0.71%, 0.36%, 3.1%, 6.7 h, and 2.61 MPa, respectively, which meet the requirements of a synchronous grouting slurry of shield tunnels for sufficiently preventing soil disturbance and deformation in areas surrounding underground construction sites.

Study on the resistance coefficient of hot dry rock cuttings in Herschel-Bulkley fluid: experiments and modeling

Most hot dry rock geothermal wells are small angle directional wells, and rock cuttings easily accumulate at the bottom of the borehole to form a cuttings bed, causing accidents such as drill sticking, reducing the rate of penetration, and drilling tool breakage. Accurately calculating the resistance coefficient and settling velocity of hot dry rock cuttings can improve cuttings transportation efficiency, design and optimize drilling hydraulic parameters, and is crucial to solving borehole cleaning problems. Through visual experiments, this paper obtained experimental data on the settlement of 167 groups of spherical pellets, 153 groups of granite cuttings, and 174 groups of carbonate cuttings in the Herschel-Bulkley fluid. First, a prediction model for the resistance coefficient of spherical pellets consistent with Herschel-Bulkley fluid was established. Based on this, form factor-Roundness is introduced as the starting point, and two prediction models for the resistance coefficients of granite cuttings and carbonate cuttings in the Herschel-Bulkley fluid were established. The average relative errors between the resistance coefficient model predictions and experimental measurements are 9.61% for granite cuttings and 6.59% for carbonate cuttings. The average relative errors between the predicted and measured values of settlement velocity are 7.27% for granite cuttings and 6.21% for carbonate cuttings, respectively, which verifies the accuracy and reliability of the prediction model. The research results can provide a theoretical basis and engineering application guidance for optimizing drilling fluid rheology and circulation displacement in engineering.

Torsional parallel plate flow of Herschel–Bulkley fluids with wall slip

The effect of wall slip on the apparent flow curves of viscoplastic materials obtained using torsional parallel plate rheometers is analyzed by considering Herschel–Bulkley fluids and assuming that slip occurs above the slip yield stress τc, taken to be lower than the yield stress, τ0. When the rim shear stress τR is below τc, the exerted torque is not sufficient to rotate the disk. When τc<τR≤τ0 the material is still unyielded but exhibits wall slip and rotates as a solid at half the angular velocity of the rotating disk. Finally, when τR>τ0, the material exhibits slip everywhere and yields only in the annulus r0≤r≤R, where r0 is the critical radius at which the shear stress is equal to the yield stress and R is the radius of the disks. In the general case, the slip velocity, which varies with the radial distance, can be calculated numerically and then all quantities of interest, such as the true shear rate, and the two branches of the apparent flow curve can be computed by means of closed form expressions. Analytical solutions have also been obtained for certain values of the power-law exponent. In order to illustrate the effect of wall slip on the apparent flow curve and on the torque, results have been obtained for different gap sizes between the disks choosing the values of the rheological and slip parameters to be similar to reported values for certain colloidal suspensions. The computed apparent flow curves reproduce the patterns observed in the experiments.

Numerical investigation of turbulent flow of Herschel–Bulkley fluids in a concentric annulus with inner cylinder rotation

In the present work, under-resolved direct numerical simulation (UDNS) is used to study the turbulent flow of Herschel–Bulkley fluids in a concentric annular region with the rotation effect of the inner cylinder. The current numerical method is verified against the first- and second-order statistics of the velocity field with the large-eddy simulation (LES) data available in the literature for the Reynolds number of 8,900. The influence of the flow behavior index (n= 0.65, 0.70, and 0.75), the Bingham number (Bn= 0.10, 0.25, and 0.40), and the Rotation number (N= 0, 0.15 and 0.30) on the flow characteristics are explored. The instantaneous flow quantities, including contours of the axial velocity and viscosity and vortical structures, and mean flow features, such as the first- and second-order turbulence statistics, mean viscosity profiles, pressure gradient, and skin friction coefficients, are investigated. The results show that weaker Reynolds stress tensor components are generated as the n value is reduced and the Bingham number increases. Moreover, raising the rotation rate increases the magnitudes of turbulent statistics and makes the velocity fluctuations more asymmetrical.

Characteristics of a Particle’s Incipient Motion from a Rough Wall in Shear Flow of Herschel–Bulkley Fluid

Characteristics of a Particle’s Incipient Motion from a Rough Wall in Shear Flow of Herschel–Bulkley Fluid

Rheology-based wall function approach for wall-bounded turbulent flows of Herschel–Bulkley fluids

Modeling fully developed turbulent flow for Herschel–Bulkley (HB) fluids in pipes is a long-standing challenge. Existing semi-empirical, theoretical, and numerical methods are either inconsistent with experimental data or are validated for low Reynolds numbers. This study focuses on validating a novel approach using rheology-based wall functions within Reynolds-averaged Navier–Stokes solvers. Simulations of wall shear stress and velocity profiles were conducted across a wide range of Reynolds numbers using a single-phase HB fluid, with measurements taken both upstream and downstream of a 90° pipe bend. Two turbulence closure models, the k–ε model and the Reynolds stress model, were employed with the wall function implemented as a specified shear boundary condition. Results demonstrate significant improvements over the Newtonian-based models, such as standard wall function by Launder–Spalding or with available semi-empirical models, achieving strong statistical correlations and minimal deviation (from the experimental findings) at high Reynolds numbers. The study also examines the utility of the wall viscosity Reynolds number and assesses the reliability of semi-empirical models for HB fluids. These findings offer valuable insights for enhancing modeling accuracy in complex fluid flow scenarios, with potential applications spanning industries like mining, chemical processing, petroleum transportation, and sanitation systems, providing practical alternatives to costly experimental procedures in pipe systems.

A case study on the evolution of rigid zones within the permanent two-dimensional Herschel-Bulkley fluid flow

The formation and development of undeformed regions during the flow of viscoplastic fluids can significantly affect fluid flow behaviour. They can impair the efficiency and effectiveness of industrial operations that use viscoplastic fluids. This paper uses numerical analysis to examine the emergence of rigid zones during non-stationary Herschel-Bulkley fluid flow across a square plate. The impact of pressure and yield stress on the behaviour of rigid zones over time is explored, and the development of the rigid zone area is shown. This work produced mathematical correlations that describe the relationship between the area of rigid zones and both the time of flow and the yield stress, as well as the area of rigid zones and the stagnation time.

Research on measurement method of drilling fluid rheological parameters based on helical pipe

Monitoring drilling fluid rheological parameters can provide a basis for assessing downhole complexities and analyzing well control risks. Among various methods for measuring drilling fluid rheological parameters, the pipe flow method allows for real-time online measurements. However, it faces limitations due to the long length of pipes and challenges in miniaturization. Based on a survey of domestic and international research, this study conducted numerical simulations to analyze the differences in flow pressure drop of drilling fluid between helical pipes and straight pipes. It clarified the impact of helical pipe structural parameters, drilling fluid process parameters, and rheological parameters on drilling fluid flow pressure drop. Utilizing the constitutive equations of Bingham, power-law, and Herschel-Bulkley fluids, a regression model was established to convert the flow pressure drop of drilling fluid between helical pipes and straight pipes. The reliability of the numerical simulation results was verified using a self-developed drilling fluid flow pressure measurement device, which yielded an average relative error of only 2.13%. These research findings provide a theoretical basis for the development of drilling fluid rheological parameter monitoring devices based on helical pipes and the corresponding software.

Study on the influence of non-newtonian fluid characteristics on the hydraulic performance and internal flow field of multiphase pump

As a pivotal prospective clean energy source, the secure and efficient extraction of natural gas hydrate has emerged as a forefront concern within the global oil drilling industry in recent years. In this study, a self-developed multiphase pump adapted to "gas-liquid-solid" three-phase mixing is used to extract natural gas hydrate slurries which have non-Newtonian characteristics. Prior research on pumps has primarily focused on water, introducing a significant disparity in handling slurries containing hydrates. This paper investigates the hydraulic performance and flow field variations of a multiphase pump using three different concentrations of Herschel-Bulkley non-Newtonian fluids that exhibit rheological properties similar to natural gas hydrate slurries. Additionally, a viscous Newtonian fluid with seawater as the liquid phase medium is studied for comparison. The results show that the shear-thinning properties of the non-Newtonian fluid improve the efficiency of the pump. However, excessive concentration negatively affects the hydraulic properties. The flow field characteristics such as apparent viscosity and turbulent kinetic energy are closely related to the pump structure and fluid properties. Compared with viscous Newtonian fluid, the non-Newtonian fluid's also have a greater effect on the distribution of gas-solid phases.

Analyzing Local Shear Rate Distribution in a Dual Coaxial Mixing Bioreactor Handling Herschel–Bulkley Biopolymer Solutions through Computational Fluid Dynamics

For the aeration of highly viscous non-Newtonian fluids, prior studies have demonstrated the improved efficacy of dual coaxial mixing bioreactors fitted with two central impellers and a close clearance anchor. Evaluating the effectiveness of these bioreactors involves considering various mixing characteristics, with a specific emphasis on shear rate distribution. The study of shear rate distribution is critical due to its significant impact on the mixing performance, gas dispersion, and homogeneity in aerated mixing systems comprising shear-thinning fluids. Although yield-pseudoplastic fluids are commonly employed in various industries, there is a research gap when it comes to evaluating shear rate distribution in aerated mixing bioreactors that utilize this fluid type. This study aims to investigate shear rate distribution in an aerated double coaxial bioreactor that handles a 1 wt% xanthan gum solution, known as a Herschel–Bulkley fluid. To achieve this goal, we employed an experimentally validated computational fluid dynamics (CFD) model to assess the effect of different mixing configurations, including down-pumping and co-rotating (Down-Co), up-pumping and co-rotating (Up-Co), down-pumping and counter-rotating (Down-Counter), and up-pumping and counter-rotating (Up-Counter) modes, on the shear rate distribution within the coaxial mixing bioreactor. Our findings revealed that the Up-Co system led to a more uniform local shear distribution and improved mixing performance.

A Review of the Settling Law of Drill Cuttings in Drilling Fluids

During the drilling process, cuttings settle under the action of gravity, which easily results in the formation of a cuttings bed, which then results in wellbore cleaning problems. The settling law of cuttings in drilling fluid is essentially a problem of solid–liquid two-phase settling. This study analyzes and summarizes the effects of the wall effect, the rheology of the fluid, particle shape irregularity, and particle concentration on the settling rate of particles and clarifies the problems faced by current research on the settling rate of particles and the development direction. Studies have shown that walls exert additional blocking effects on particles, thus reducing their settling velocity. The shear thinning effect of non-Newtonian fluids such as power-law fluids and Herschel–Bulkley fluids will reduce the viscosity of the liquid, thus increasing the settling velocity of the particles. Compared with spherical particles, irregular particles will obtain higher resistance in the fluid, leading to a decline in the particle settling velocity. The mutual interference between particles will result in an increase in the drag force on the particles and a decline in the settling velocity. However, when the particles are aggregated, the settling velocity will increase. This study can provide theoretical guidance for predicting the migration law of cuttings during the drilling of horizontal wells, and it has important significance for enriching the theory of solid–liquid two-phase flow.

Influence of rheological parameters on cement slurry penetration characteristics of novel oscillating grouting technology

Traditional grouting techniques are primarily based on stable-pressure grouting, with slurry penetration performance limited. Research on the penetration behavior of pulsating slurry is relatively scarce, with even less research focused on the influence of slurry rheological parameters.This paper proposes an efficient oscillating grouting technology (OGT) that can generate pressure pulses and periodically change the direction of slurry motion. Through experiments and numerical simulations, its working mechanism is investigated, and the penetration distance of oscillating grouting and steady pressure grouting is compared under different soil parameters (porosity and particle size). The effect of cement slurry rheological parameters (viscosity of the Newtonian fluid and the K, τ0 and n, of the Herschel-Bulkley fluid) on penetration distance, grouting pressure, pressure pulse amplitude, and oscillating frequency are studied. Finally, the correlation analysis between penetration behavior, oscillating parameters, and rheological parameters is conducted. The results show that the oscillatory switching of the slurry is related to the interaction of vortexes within the tool, and the switching process leads to the generation of pressure pulsations. Under different soil parameters and rheological parameters, the penetration distance of oscillating grouting is larger than that of steady-pressure grouting. Furthermore, due to the pressure recovery process within the oscillatory grouting tool, the oscillating grouting exerts less pressure on the soil, which can prevent soil fracturing. The results also indicate that the oscillatory grouting requires a certain level of flow resistance to achieve optimal effect, but excessive resistance can impede the slurry switching process and reduce the penetration distance. And the optimal values for K, τ0 and n are around 0.015 Pa·sn, 1.5 Pa, and 0.5, respectively. According to the correlation analysis, oscillating frequency is positively correlated with penetration distance. Average pressure and rheological parameters are negatively correlated with penetration distance.

Dynamics of displacement flows of viscoplastic fluids in obstructed eccentric annuli at moderate Reynolds numbers

As a sequel to Mitishita et al. [“Turbulent displacement flows of viscoplastic fluids in obstructed eccentric annuli: Experiments,” Phys. Fluids 34, 053114 (2022)], we present an experimental study of laminar displacement flows in obstructed eccentric annuli. Xanthan gum (XG) solutions (0.35%, 0.50%, or 0.75%) are used to displace a 0.15% viscoplastic Carbopol solution. The eccentricity of the annulus section is set to near 0.5. We study the effect of a solid obstruction in the narrow side of the annulus, similar to that provided by a consolidated residual cuttings bed, and compare the results to unobstructed displacement flows. While we predicted that all displacements would be in the laminar regime, we actually observe mixed regimes where the initial displacement of Carbopol can be transitional or turbulent. With the obstruction on the narrow side of the annulus, we observe the formation of cavities in the Carbopol layer, both upstream and downstream of the obstruction. We believe that the cavities are formed because the obstruction behaves like an abrupt contraction/expansion. This geometric irregularity affects the velocity profiles of the displacing fluid near the obstruction. Once the cavities reach the bottom of the pipe, we observe that the remaining Carbopol layer is more easily eroded. The dynamics of the Carbopol removal also share similarities to cleaning of soil layers in pipes, as described by Palabiyik et al. [“Flow regimes in the emptying of pipes filled with a Herschel–Bulkley fluid,” Chem. Eng. Res. Des. 92, 2201–2212 (2014)].

Numerical study of irreversibility analysis on pulsatile flow of blood through a ω-shaped stenotic artery

The mathematical modeling of blood flow in a stenotic blood vessel is very important for understanding the effects of various geometric and rheological parameters on the normal flow of blood. The present study is concerned with the heat and mass transfer in the blood flow of a non-Newtonian fluid through a ω-shaped stenotic arterial segment. The non-Newtonian behavior of blood is described mathematically by employing the constitutive equation of the Herschel-Bulkley fluid. The wall of the artery is considered to be rigid having ω-shaped constriction in its lumen. The mild stenosis assumption is used to simplify the fully coupled momentum, energy, and concentration equations as well as the constitutive equation of the Herschel-Bulkley fluid. Numerical solutions for fluid velocity, temperature, mass concentration, and entropy generation are evaluated by using the explicit finite difference scheme. The effects of various emerging parameters on the velocity, wall shear stress, flow rate, temperature, concentration, and entropy distributions have been analyzed in detail. A comparison between different fluid models has also been presented. It is found that the temperature in the Herschel-Bulkley fluid is marginally lower than that of the Newtonian, Power law, and Bingham fluids whereas an opposite trend is noticed in the case of mass concentration.