Tip region of a hydraulic fracture driven by a laminar-to-turbulent fluid flow

2016 ◽  
Vol 797 ◽  
Author(s):  
E. V. Dontsov
Keyword(s):  
Fluid Flow ◽  
Turbulent Flow ◽  
Numerical Solution ◽  
Hydraulic Fracture ◽  
Asymptotic Solutions ◽  
Turbulent Regime ◽  
Zone Size ◽  
Turbulent Fluid Flow ◽  
Fully Developed Turbulent Flow ◽  
Steady Propagation

The focus of this study is to analyse the tip region of a hydraulic fracture, for which a fluid flow inside the crack transitions from the laminar to the turbulent regime away from the tip. To tackle the problem, a phenomenological formula for flow in pipes has been adapted to describe flow in a fracture through the concept of a hydraulic diameter. The selected model is able to capture laminar, turbulent and transition regimes of the flow. The near-tip region of a hydraulic fracture is analysed by focusing on steady propagation of a semi-infinite hydraulic fracture with leak-off, for which the aforementioned phenomenological formula for the fluid flow is utilized. First, the distance from the tip within which a laminar solution applies is estimated. Then, expressions for asymptotic solutions that are associated with fully developed turbulent flow inside the semi-infinite hydraulic fracture are derived. Finally, the laminar zone size and the asymptotic solutions are compared with the numerical solution, where the latter captures all regimes of the fluid flow.

scholarly journals An examination of turbulent flow with an ultramicroscope

1932 ◽  
Vol 135 (828) ◽  
pp. 656-677 ◽  
Keyword(s):  
Fluid Flow ◽  
Turbulent Flow ◽  
Flow Pattern ◽  
Wave Length ◽  
Turbulent Fluid Flow ◽  
Turbulent Fluid ◽  
Dark Background ◽  
Internal Motions ◽  
Ordinary Light

1. Hitherto, the majority of researches into the character of turbulent fluid flow have been concerned with the motions of relatively large molar masses of fluid, and the methods used to obtain visual impressions of the flow pattern have usually involved the introduction into the fluid of particles of extraneous matter, such as aluminium particles, oil drops, etc. It is questionable whether such methods are permissible for the examination of microturbulence, especially near the boundary of the fluid where the scale of the turbulence is small, since if the particles introduced are comparable in size with the molar masses, their internal motions may not be faithfully represented. In a study of this kind of motion it is very desirable therefore to avoid any such interference with the flow, and the ultramicroscope offered a possible means of doing this provided the difficulties in applying the instrument could be surmounted. 2. The principle of the ultramicroscope depends on the fact that minute particles usually present in most fluids, but invisible in ordinary light even under the most powerful microscope, become visible when intensely illuminated provided they are seen against a dark background. Particles whose shapes are not discernible, because they are smaller than the wave-length of light, then become visible as bright points of light.

Pressure Drop for Fully Developed Turbulent Flow in Circular and Noncircular Ducts

2012 ◽  
Vol 134 (6) ◽  
Author(s):  
Zhipeng Duan ◽  
M. M. Yovanovich ◽  
Y. S. Muzychka
Keyword(s):  
Fluid Flow ◽  
Pressure Drop ◽  
Turbulent Flow ◽  
Square Root ◽  
Sectional Area ◽  
Cross Sectional ◽  
Physical Behavior ◽  
Frictional Pressure Drop ◽  
Fully Developed Turbulent Flow ◽  
The Cross

The objective of this paper is to furnish the engineer with a simple and convenient means of estimating frictional pressure drop in ducts and the original physical behavior can be clearly reflected. Fully developed turbulent flow frictional pressure drop in noncircular ducts is examined. Simple models are proposed to predict the frictional pressure drop in smooth and rough noncircular channels. Through the selection of a novel characteristic length scale, the square root of the cross-sectional area, the effect of duct shape has been minimized. The proposed models have an accuracy of 6% for most common duct shapes of engineering practice and can be used to predict pressure drop of fully developed turbulent flow in noncircular ducts. It is found that the hydraulic diameter is not the appropriate length scale to use in defining the Reynolds number to ensure similarity between the circular and noncircular ducts. By using the Reynolds number based on the square root of the cross-sectional area, it is demonstrated that the circular tube relations may be applied to noncircular ducts eliminating large errors in estimation of pressure drop. The square root of the cross-sectional area is an appropriate characteristic dimension applicable to most duct geometries. The dimensionless mean wall shear stress is a desirable dimensionless parameter to describe fluid flow physical behavior so that fluid flow problems can be solved in the simple and direct manner. The dimensionless mean wall shear stress is presented graphically and appears more general and reasonable to reflect the fluid flow physical behavior than the traditional Moody diagram.

scholarly journals Turbulent flow effects in hydraulic fracture propagation in permeable rock

2021 ◽  
Author(s):  
Evgenii Kanin ◽  
Dmitry Garagash ◽  
Andrei Osiptsov
Keyword(s):  
Fluid Flow ◽  
General Solution ◽  
Turbulent Flow ◽  
Flow Regime ◽  
Hydraulic Fracture ◽  
Fracture Propagation ◽  
Linear Elastic ◽  
Permeable Rock ◽  
Hydraulic Fracture Propagation ◽  
Laminar Model

This chapter considers a model for a radial hydraulic fracture propagation in a permeable, linear elastic rock formation driven by a point source fluid injection. The linear elastic fracture mechanics theory controls the quasi-static propagation. The hydraulic fracturing fluid is slickwater -- pure water solution with polymeric additives which allow reducing the fluid flow friction in the wellbore and fracture in reservoir field applications. We focus on the possible transformation of the fluid flow regime inside the fracture channel from laminar to turbulent with distance from the fracture front. We assume that the turbulent friction of slickwater is described by the maximum drag reduction asymptote, while Carter's law governs the leak-off into the permeable rock. The solution is obtained numerically using the algorithm based on the Gauss-Chebyshev quadrature and Barycentric Lagrange interpolation techniques. We compute solution examples for typical field cases and demonstrate a significant impact of the turbulent flow regime during the initial few minutes of propagation, namely, shorter radius and wider maximum aperture than the laminar model provides. Moreover, we observe higher fluid pressure values at the wellbore within tens of minutes of the start of the injection. This leads to a larger hydraulic pumping power requirement than the laminar model predicts. We also find that the fluid leak-off into the permeable rock enhances the turbulent flow effect in the fracture when compared to the impermeable rock case. In order to analyze the parametric dependence of the general solution, we convert the governing equations into the dimensionless form. We perform an extensive exploration of the normalized solution in space of two non-dimensional parameters, leak-off and characteristic Reynolds numbers, and normalized time. Specifically, we determine the applicability domains of the limiting propagation regimes to frame the general solution, investigate the alterations of the crack characteristics depending on the governing parameters, and identify zones where the turbulent flow is important.

Drag Calculations for Vehicles in Very Long Tubes From Turbulent Flow Theory

1981 ◽  
Vol 103 (2) ◽  
pp. 361-366 ◽  
Author(s):  
I. Sud ◽  
J. B. Chaddock
Keyword(s):  
Fluid Flow ◽  
Turbulent Flow ◽  
Numerical Solution ◽  
High Speed ◽  
Limited Sample ◽  
Fully Developed Flow ◽  
Similarity Hypothesis ◽  
Annular Gap ◽  
Flow Techniques ◽  
Good Agreement

Fundamental turbulent flow techniques are used to examine the annular fluid flow occurring in an idealized model of a high speed ground transportation system. The model consists of a smooth cylindrical train moving in an infinitely long tunnel, creating an annular gap with one wall in motion. The developing, and the fully developed, flow regions are separately analyzed by using fundamental relations of turbulent flow and Von Karman’s similarity hypothesis. The relevant equations are developed and numerical solution procedures presented. Limited sample calculations show good agreement with existing empirical and experimental results.

scholarly journals Slickwater hydraulic fracture propagation: near-tip and radial geometry solutions

2019 ◽  
Vol 880 ◽  
pp. 514-550 ◽  
Author(s):  
Brice Lecampion ◽  
Haseeb Zia
Keyword(s):  
Hydraulic Fracturing ◽  
Turbulent Flow ◽  
Hydraulic Fracture ◽  
Early Time ◽  
High Rate ◽  
Hydraulic Fractures ◽  
Late Time ◽  
Turbulent Regime ◽  
High Molecular Weight Polymer ◽  
Molecular Weight Polymer

We quantify the importance of turbulent flow on the propagation of hydraulic fractures (HF) accounting for the addition of friction reducing agents to the fracturing fluid (slickwater fluid). The addition in small quantities of a high molecular weight polymer to water is sufficient to drastically reduce friction of turbulent flow. The maximum drag reduction (MDR) asymptote is always reached during industrial-like injections. The energy required for pumping is thus drastically reduced, allowing for high volume high rate hydraulic fracturing operations at a reasonable cost. We investigate the propagation of a hydraulic fracture propagating in an elastic impermeable homogeneous solid under a constant (and possibly very high) injection rate accounting for laminar and turbulent flow conditions with or without the addition of friction reducers. We solve the near-tip HF problem and estimate the extent of the laminar boundary layer near the fracture tip as a function of a tip Reynolds number for slickwater. We obtain different propagation scalings and transition time scales. This allows us to easily quantify the growth of a radial HF from the early-time turbulent regime(s) to the late-time laminar regimes. Depending on the material and injection parameters, some propagation regimes may actually be bypassed. We derive both accurate and approximate solutions for the growth of radial HF in the different limiting flow regimes (turbulent smooth, rough, MDR) for the zero fracture toughness limit (corresponding to the early stage of propagation of a radial HF). We also investigate numerically the transition(s) between the early-time MDR regime to the late-time laminar regimes (viscosity and toughness) for slickwater fluid. Our results indicate that the effect of turbulent flow on high rate slickwater HF propagation is limited and matters only at early times (at most during the first minutes for industrial hydraulic fracturing operations).

scholarly journals Modeling Turbulent Hydraulic Fracture Near a Free Surface

2012 ◽  
Vol 79 (3) ◽  
Author(s):  
Victor C. Tsai ◽  
James R. Rice
Keyword(s):  
Growth Rate ◽  
Fluid Flow ◽  
Free Surface ◽  
Hydraulic Fracture ◽  
Crack Opening ◽  
Pressure Profile ◽  
Numerical Approach ◽  
Chebyshev Series ◽  
Opening Displacement ◽  
Turbulent Fluid Flow

Motivated by observations of the subglacial drainage of water, we consider a hydraulic fracture problem in which the crack grows parallel to a free surface, subject to fully turbulent fluid flow. Using a hybrid Chebyshev/series-minimization numerical approach, we solve for the pressure profile, crack opening displacement, and crack growth rate for a crack that begins relatively short but eventually becomes long compared with the distance to the free surface. We plot nondimensionalized results for a variety of different times, corresponding with different fracture lengths, and find substantial differences when free-surface effects are important.

Resistance to the Flow of Fluids Through Simple and Complex Porous Media Whose Matrices Are Composed of Randomly Packed Spheres

1987 ◽  
Vol 109 (3) ◽  
pp. 268-273 ◽  
Author(s):  
R. M. Fand ◽  
B. Y. K. Kim ◽  
A. C. C. Lam ◽  
R. T. Phan
Keyword(s):  
Porous Media ◽  
Turbulent Flow ◽  
Turbulent Regime ◽  
Complex Media ◽  
Simple Method ◽  
Forchheimer’S Equation ◽  
The Matrix ◽  
Fully Developed Turbulent Flow ◽  
Uniform Diameter ◽  
Different Diameters

Experimental data relating to the flow of fluids through simple and complex porous media whose matrices are composed of randomly packed spheres have been obtained. In this context the term “simple” refers to porous media whose matrices are composed of spheres of uniform diameter, while “complex” refers to matrices composed of spheres having different diameters. It was found that Darcy’s law is valid for simple media within a range of the Reynolds number, Re, whose upper bound is 2.3. The upper bounds of Darcy flow for complex media were found to be consistent with this value. It is shown that the resistance to flow in the Darcy regime can be characterized by taking the Kozeny-Carman constant equal to 5.34 if the characteristic dimension is taken equal to the weighted harmonic mean diameter of the spheres that comprise the matrix. Forchheimer’s equation was found to be valid for simple media within the range 5 ≤ Re ≤ 80. The corresponding bounds for complex media were found to be consistent with this range. It is shown that the resistance to flow in the Forchheimer regime for both simple and complex media can be characterized by adopting the following values of the Ergun constants: A = 182 and B = 1.92. Finally, it is shown that fully developed turbulent flow exists when Re > 120 and that the resistance to flow in the turbulent regime can be calculated using Forchheimer’s equation by adopting the following values of the Ergun constants: A′ = 225 and B′ = 1.61. A simple method for characterizing the behavior of porous media in the transition regions between Darcy and Forchheimer and between Forchheimer and turbulent flow is presented.

Extending the Meshless Local Petrov–Galerkin Method to Solve Stabilized Turbulent Fluid Flow Problems

2018 ◽  
Vol 16 (01) ◽  
pp. 1850086 ◽  
Author(s):  
Nasrin Sheikhi ◽  
Mohammad Najafi ◽  
Vali Enjilela
Keyword(s):  
Fluid Flow ◽  
Turbulent Flow ◽  
Galerkin Method ◽  
Standard Test ◽  
Test Cases ◽  
Turbulent Fluid Flow ◽  
Turbulent Fluid ◽  
Flow Problems ◽  
Laminar And Turbulent Flow ◽  
Flow Through

The aim of this paper is to extend the meshless local Petrov–Galerkin method to solve stabilized turbulent fluid flow problems. For the unsteady incompressible turbulent fluid flow problems, the Spalart–Allmaras model is used to stabilize the governing equations, and the meshless local Petrov–Galerkin method is extended based on the vorticity-stream function to solve the turbulent flow problems. In this study, the moving least squares scheme interpolates the field variables. The proposed method solves three standard test cases of the turbulent flow over a flat plate, turbulent flow through a channel, and turbulent flow over a backward-facing step for evaluation of the method’s capability, accuracy, and validity purposes. Based on the comparison of the three test cases results with those of the experimental and conventional numerical works available in the literature, the proposed method shows to be accurate and quite implemental. The new extended method in this study together with the previously published works of the authors (on extending the meshless local Petrov–Galerkin method to solve laminar flow problems) now, for the first time, empower the meshless method to solve both laminar and turbulent flow problems.

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