A Different Perspective on the Forchheimer and Ergun Equations

SPE Journal ◽  
2016 ◽  
Vol 21 (05) ◽  
pp. 1501-1507 ◽  
Author(s):  
Deming Mao ◽  
John M. Karanikas ◽  
Phillip S. Fair ◽  
Ionut D. Prodan ◽  
George K. Wong
Keyword(s):  
Shale Gas ◽  
Reynolds Numbers ◽  
High Rate ◽  
Hydraulic Fractures ◽  
Focused Attention ◽  
Inertia Effect ◽  
Forchheimer Equation ◽  
Wide Range ◽  
Converging Flow ◽  
Fixed Beds

Summary The need to better estimate the performance of hydraulically fractured wells in shale-gas formations has focused attention on the description of the non-Darcy flow in the region of the hydraulic fracture near the borehole where the converging flow may be increasingly dominated by the inertia effect. The analogy between high-rate flow in proppant-filled hydraulic fractures—best described by the Forchheimer equation—and the flow of reactants in fixed beds or granular beds in catalysis—widely described by the Ergun equation—may offer insights on how to estimate the value of the coefficient β across the complete range of velocities (or, equivalently, Reynolds numbers). In this paper, we show that, although the value of β varies over a wide range depending on the porosity (ϕ) and permeability (K) of the porous medium, the value of the product βK is confined in the range of 10−5 to 10−4 m, and a correlation is obtained with narrower uncertainty compared with published ones.

Vertical Growth of Hydraulic Fractures in Layered Formations

2021 ◽  
Author(s):  
Vibhas J. Pandey ◽  
Vamegh Rasouli
Keyword(s):  
Pressure Distribution ◽  
Solid Mechanics ◽  
High Rate ◽  
Design Parameters ◽  
Hydraulic Fractures ◽  
Vertical Wells ◽  
Wide Range ◽  
Net Pressure ◽  
Fracture Height ◽  
Injection Rates

AbstractFracture growth in layered formations with depth-dependent properties has been a topic of interest amongst researchers because of its critical influence on well performance. This paper revisits some of the existing height-growth models and discusses the evaluation process of a new and modified model developed after incorporating additional constraints.The net-pressure is the primary driver behind fracture propagation and the pressure distribution in the fracture plays an important role in vertical propagation, as it supplies the necessary energy for fracture advancement in the presence of opposing forces. The workflow adopted for this study included developing a preliminary model that solves a system of non-linear equations iteratively to arrive at fracture height versus net pressure mapping. The theoretical results were then compared to those available in the literature. The solution set was then extended to a 100-layer model after incorporating additional constraints using superposition techniques.The predicted outcomes were finally compared to the fracture height observations made in the field on several treatments.A reasonable agreement between model-predicted and observed height was observed when a comparison between the two was made, for most cases.The majority of these treatments were pumped in vertical wells, at low injection rates of up to 8.0 bbl/min (0.021 m3/s) where net pressures were intentionally restricted to 250 psi (1.72 MPa) in order to prevent fracture rotation to the horizontal plane.The leak-off was minimal given the low permeability formations. In some cases, however, the pumping parameters and fluid imparted pressure distribution appeared to dominate. Overall, it was apparent that for a slowly advancing fracture front, which is the case in low injection rate treatments, the fracture height could be predicted with reasonable accuracy. This condition could also be met in high rate treatments pumped down multiple perforation clusters such as in horizontal wells, though fracture-height measurement may not be as straightforward as in vertical wells.The model developed under the current study is suitable for vertical wells where fracture treatments are pumped at low injection rates. The solid-mechanics solution that is presented here is independent of pumping parameters and can be readily implemented to assist in selection of critical design parameters prior to the job, with a wide range of applicability worldwide.

Turbulent three-dimensional dielectric electrohydrodynamic convection between two plates

2012 ◽  
Vol 696 ◽  
pp. 228-262 ◽  
Author(s):  
A. Kourmatzis ◽  
J. S. Shrimpton
Keyword(s):  
Spatial Data ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Energy Budgets ◽  
Turbulent Regime ◽  
Scalar Flux ◽  
Wide Range ◽  
Scalar Variance

AbstractThe fundamental mechanisms responsible for the creation of electrohydrodynamically driven roll structures in free electroconvection between two plates are analysed with reference to traditional Rayleigh–Bénard convection (RBC). Previously available knowledge limited to two dimensions is extended to three-dimensions, and a wide range of electric Reynolds numbers is analysed, extending into a fully inherently three-dimensional turbulent regime. Results reveal that structures appearing in three-dimensional electrohydrodynamics (EHD) are similar to those observed for RBC, and while two-dimensional EHD results bear some similarities with the three-dimensional results there are distinct differences. Analysis of two-point correlations and integral length scales show that full three-dimensional electroconvection is more chaotic than in two dimensions and this is also noted by qualitatively observing the roll structures that arise for both low (${\mathit{Re}}_{E} = 1$) and high electric Reynolds numbers (up to ${\mathit{Re}}_{E} = 120$). Furthermore, calculations of mean profiles and second-order moments along with energy budgets and spectra have examined the validity of neglecting the fluctuating electric field ${ E}_{i}^{\ensuremath{\prime} } $ in the Reynolds-averaged EHD equations and provide insight into the generation and transport mechanisms of turbulent EHD. Spectral and spatial data clearly indicate how fluctuating energy is transferred from electrical to hydrodynamic forms, on moving through the domain away from the charging electrode. It is shown that ${ E}_{i}^{\ensuremath{\prime} } $ is not negligible close to the walls and terms acting as sources and sinks in the turbulent kinetic energy, turbulent scalar flux and turbulent scalar variance equations are examined. Profiles of hydrodynamic terms in the budgets resemble those in the literature for RBC; however there are terms specific to EHD that are significant, indicating that the transfer of energy in EHD is also attributed to further electrodynamic terms and a strong coupling exists between the charge flux and variance, due to the ionic drift term.

scholarly journals Focusing of Particles in a Microchannel with Laser Engraved Groove Arrays

Biosensors ◽  
2021 ◽  
Vol 11 (8) ◽  
pp. 263
Author(s):  
Tianlong Zhang ◽  
Yigang Shen ◽  
Ryota Kiya ◽  
Dian Anggraini ◽  
Tao Tang ◽  
...  
Keyword(s):  
Open Space ◽  
Lateral Displacement ◽  
Reynolds Numbers ◽  
Secondary Flows ◽  
Particle Sedimentation ◽  
Nanoparticle Separation ◽  
Wide Range ◽  
Fs Laser ◽  
Channel Bottom ◽  
Myoblast Cells

Continuous microfluidic focusing of particles, both synthetic and biological, is significant for a wide range of applications in industry, biology and biomedicine. In this study, we demonstrate the focusing of particles in a microchannel embedded with glass grooves engraved by femtosecond pulse (fs) laser. Results showed that the laser-engraved microstructures were capable of directing polystyrene particles and mouse myoblast cells (C2C12) towards the center of the microchannel at low Reynolds numbers (Re < 1). Numerical simulation revealed that localized side-to-center secondary flows induced by grooves at the channel bottom play an essential role in particle lateral displacement. Additionally, the focusing performance proved to be dependent on the angle of grooves and the middle open space between the grooves based on both experiments and simulation. Particle sedimentation rate was found to critically influence the focusing of particles of different sizes. Taking advantage of the size-dependent particle lateral displacement, selective focusing of micrometer particles was demonstrated. This study systematically investigated continuous particle focusing in a groove-embedded microchannel. We expect that this device will be used for further applications, such as cell sensing and nanoparticle separation in biological and biomedical areas.

Sustaining Research-Based Practices in Reading

1999 ◽  
Vol 20 (5) ◽  
pp. 263-287 ◽  
Author(s):  
Janette K. Klingner ◽  
Sharon Vaughn ◽  
Marie Tejero Hughes ◽  
Maria Elena Arguelles
Keyword(s):  
Instructional Practices ◽  
High Rate ◽  
Strategic Reading ◽  
School Year ◽  
Wide Range ◽  
Sustained Use ◽  
Development Experience ◽  
Collaborative Strategic Reading ◽  
Research Based Practices ◽  
Over Time

This study examined the extent to which the reading instructional practices learned by a cohort of teachers who participated in an intensive, yearlong professional development experience during the 1994-1995 school year have been sustained and modified over time. Teachers learned three multileveled practices—partner reading, collaborative strategic reading, and making words—that promote gains in reading for students from a wide range of achievement levels. Teachers were observed and interviewed 3 years later to determine the extent to which they continued to implement the practices, the ways in which they modified them, and factors that influenced their sustained use of the practices. With the exception of one teacher, all the teachers sustained one or more of the three practices at a high rate.

The Effect of Pulsed Blowing on the Boundary Layer of a Highly Loaded Low Pressure Turbine Blade

2013 ◽  
Author(s):  
Marion Mack ◽  
Roland Brachmanski ◽  
Reinhard Niehuis
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Mach Number ◽  
Suction Side ◽  
Reynolds Numbers ◽  
Low Pressure ◽  
Trailing Edge ◽  
Low Pressure Turbine ◽  
Wide Range ◽  
Highly Loaded

The performance of the low pressure turbine (LPT) can vary appreciably, because this component operates under a wide range of Reynolds numbers. At higher Reynolds numbers, mid and aft loaded profiles have the advantage that transition of suction side boundary layer happens further downstream than at front loaded profiles, resulting in lower profile loss. At lower Reynolds numbers, aft loading of the blade can mean that if a suction side separation exists, it may remain open up to the trailing edge. This is especially the case when blade lift is increased via increased pitch to chord ratio. There is a trend in research towards exploring the effect of coupling boundary layer control with highly loaded turbine blades, in order to maximize performance over the full relevant Reynolds number range. In an earlier work, pulsed blowing with fluidic oscillators was shown to be effective in reducing the extent of the separated flow region and to significantly decrease the profile losses caused by separation over a wide range of Reynolds numbers. These experiments were carried out in the High-Speed Cascade Wind Tunnel of the German Federal Armed Forces University Munich, Germany, which allows to capture the effects of pulsed blowing at engine relevant conditions. The assumed control mechanism was the triggering of boundary layer transition by excitation of the Tollmien-Schlichting waves. The current work aims to gain further insight into the effects of pulsed blowing. It investigates the effect of a highly efficient configuration of pulsed blowing at a frequency of 9.5 kHz on the boundary layer at a Reynolds number of 70000 and exit Mach number of 0.6. The boundary layer profiles were measured at five positions between peak Mach number and the trailing edge with hot wire anemometry and pneumatic probes. Experiments were conducted with and without actuation under steady as well as periodically unsteady inflow conditions. The results show the development of the boundary layer and its interaction with incoming wakes. It is shown that pulsed blowing accelerates transition over the separation bubble and drastically reduces the boundary layer thickness.

Predictions of the Effect of Roughness on Heat Transfer From Turbine Airfoils

1998 ◽  
Author(s):  
Anil K. Tolpadi ◽  
Michael E. Crawford
Keyword(s):  
Heat Transfer ◽  
Side Surface ◽  
Suction Side ◽  
Reynolds Numbers ◽  
Surface Finishing ◽  
Transfer Coefficients ◽  
Average Roughness ◽  
Wide Range ◽  
Turbine Airfoils ◽  
Good Agreement

The heat transfer and aerodynamic performance of turbine airfoils are greatly influenced by the gas side surface finish. In order to operate at higher efficiencies and to have reduced cooling requirements, airfoil designs require better surface finishing processes to create smoother surfaces. In this paper, three different cast airfoils were analyzed: the first airfoil was grit blasted and codep coated, the second airfoil was tumbled and aluminide coated, and the third airfoil was polished further. Each of these airfoils had different levels of roughness. The TEXSTAN boundary layer code was used to make predictions of the heat transfer along both the pressure and suction sides of all three airfoils. These predictions have been compared to corresponding heat transfer data reported earlier by Abuaf et al. (1997). The data were obtained over a wide range of Reynolds numbers simulating typical aircraft engine conditions. A three-parameter full-cone based roughness model was implemented in TEXSTAN and used for the predictions. The three parameters were the centerline average roughness, the cone height and the cone-to-cone pitch. The heat transfer coefficient predictions indicated good agreement with the data over most Reynolds numbers and for all airfoils-both pressure and suction sides. The transition location on the pressure side was well predicted for all airfoils; on the suction side, transition was well predicted at the higher Reynolds numbers but was computed to be somewhat early at the lower Reynolds numbers. Also, at lower Reynolds numbers, the heat transfer coefficients were not in very good agreement with the data on the suction side.

CFD Simulation of Fluid Flow Through Porous Media: Application to Decomposed Gases Flow Through Foam Pattern in Lost Foam Casting

Materials ◽  
2003 ◽  
Author(s):  
Sayavur I. Bakhtiyarov ◽  
Ruel A. Overfelt
Keyword(s):  
Cfd Simulation ◽  
Reynolds Numbers ◽  
Phase Flow ◽  
Flow Through Porous Media ◽  
Foam Material ◽  
Element Analysis ◽  
Wide Range ◽  
Flow Through ◽  
Foam Pattern ◽  
Lost Foam

Numerical simulation of decomposed gases through foam pattern was conducted using finite element analysis. A new kinetic model is proposed for gaseos phase flow between molten metal and foam material. The computations were performed for a wide range of Reynolds numbers. The results of the simulations are compared with the experiemental data obtained in this study.

Calculation of the primary trajectories of seeds and other particles in strong winds

1983 ◽  
Vol 219 (1215) ◽  
pp. 217-217
Keyword(s):  
Equations Of Motion ◽  
Aerodynamic Resistance ◽  
Reynolds Numbers ◽  
Spherical Particles ◽  
Spores And Pollen ◽  
General Terms ◽  
Wide Range ◽  
Strong Winds ◽  
Dispersal Distances ◽  
Enormous Number

The movement of variously dense spherical particles representing a variety of seeds, fruits, spores and pollen, and released from rest into arbitrary winds and a gravitational field is discussed in general terms that account in detail for changes in the quasi-static aerodynamic resistance to motion experienced by such particles during aerial flight. A hybrid analytical-empirical law is established which describes this resistance fairly accurately for particle Reynolds numbers in the range 0—60 000 and that allows for the numerical integration of the equations of motion so as to cover a very wide range of flight conditions. This makes possible the provision of a set of four-parameter universal range tables from which the dispersal distances for an enormous number of practical cases may be estimated. One particular case of particle movement in a region of pseudo-thermal convection is also discussed and this shows how a marked degree of deposition concentration may be induced in some circumstances by such a flow. Botanists and ecologists concerned with seed and particle dispersal in the environment may find the universal range tables of particular interest and use. This is because the tables obviate the need for the integration of the equations of motion when dealing with individual cases and permit an estimation of range purely on the basis of the specified quantities of particle size, density and altitude of release, atmospheric wind speed, density and viscosity, and the acceleration due to gravity.

Experimental Investigation of Subsonic Turbulent Flow Over Single and Double Backward Facing Steps

1962 ◽  
Vol 84 (3) ◽  
pp. 317-325 ◽  
Author(s):  
D. E. Abbott ◽  
S. J. Kline
Keyword(s):  
Experimental Investigation ◽  
Velocity Profile ◽  
Flow Pattern ◽  
Reynolds Numbers ◽  
Area Ratio ◽  
Single Step ◽  
Reattachment Length ◽  
Double Step ◽  
Wide Range ◽  
Geometric Variables

Results are presented for flow patterns over backward facing steps covering a wide range of geometric variables. Velocity profile measurements are given for both single and double steps. The stall region is shown to consist of a complex pattern involving three distinct regions. The double step contains an assymmetry for large expansions, but approaches the single-step configuration with symmetric stall regions for small values of area ratio. No effect on flow pattern or reattachment length is found for a wide range of Reynolds numbers and turbulence intensities, provided the flow is fully turbulent before the step.

Heat Transfer in Rotating Serpentine Coolant Passage With Ribbed Walls at Low Mach Numbers

2015 ◽  
Vol 7 (1) ◽  
Author(s):  
Shang-Feng Yang ◽  
Je-Chin Han ◽  
Salam Azad ◽  
Ching-Pang Lee
Keyword(s):  
Heat Transfer ◽  
Mach Number ◽  
Reynolds Numbers ◽  
Working Fluid ◽  
Transfer Coefficients ◽  
Low Mach Number ◽  
Heat Transfer Coefficients ◽  
Rotation Numbers ◽  
Wide Range ◽  
Mach Numbers

This paper experimentally investigates the effect of rotation on heat transfer in typical turbine blade serpentine coolant passage with ribbed walls at low Mach numbers. To achieve the low Mach number (around 0.01) condition, pressurized Freon R-134a vapor is utilized as the working fluid. The flow in the first passage is radial outward, after the 180 deg tip turn the flow is radial inward to the second passage, and after the 180 deg hub turn the flow is radial outward to the third passage. The effects of rotation on the heat transfer coefficients were investigated at rotation numbers up to 0.6 and Reynolds numbers from 30,000 to 70,000. Heat transfer coefficients were measured using the thermocouples-copper-plate-heater regional average method. Heat transfer results are obtained over a wide range of Reynolds numbers and rotation numbers. An increase in heat transfer rates due to rotation is observed in radially outward passes; a reduction in heat transfer rate is observed in the radially inward pass. Regional heat transfer coefficients are correlated with Reynolds numbers for nonrotation and with rotation numbers for rotating condition, respectively. The results can be useful for understanding real rotor blade coolant passage heat transfer under low Mach number, medium–high Reynolds number, and high rotation number conditions.

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