scholarly journals An approximate solution for a penny-shaped hydraulic fracture that accounts for fracture toughness, fluid viscosity and leak-off

2016 ◽  
Vol 3 (12) ◽  
pp. 160737 ◽  
Author(s):  
E. V. Dontsov
Keyword(s):  
Fracture Toughness ◽  
Approximate Solution ◽  
Hydraulic Fracture ◽  
Fluid Viscosity ◽  
Solution Map ◽  
Fracture Width ◽  
Volume Balance ◽  
Wide Range ◽  
Design Calculations ◽  
Fluid Leak

This paper develops a closed-form approximate solution for a penny-shaped hydraulic fracture whose behaviour is determined by an interplay of three competing physical processes that are associated with fluid viscosity, fracture toughness and fluid leak-off. The primary assumption that permits one to construct the solution is that the fracture behaviour is mainly determined by the three-process multiscale tip asymptotics and the global fluid volume balance. First, the developed approximation is compared with the existing solutions for all limiting regimes of propagation. Then, a solution map, which indicates applicability regions of the limiting solutions, is constructed. It is also shown that the constructed approximation accurately captures the scaling that is associated with the transition from any one limiting solution to another. The developed approximation is tested against a reference numerical solution, showing that accuracy of the fracture width and radius predictions lie within a fraction of a per cent for a wide range of parameters. As a result, the constructed approximation provides a rapid solution for a penny-shaped hydraulic fracture, which can be used for quick fracture design calculations or as a reference solution to evaluate accuracy of various hydraulic fracture simulators.

Propagation regimes of buoyancy-driven hydraulic fractures with solidification

2016 ◽  
Vol 797 ◽  
pp. 1-28 ◽  
Author(s):  
E. V. Dontsov
Keyword(s):  
Hydraulic Fracture ◽  
Transfer Problem ◽  
Solidification Rate ◽  
Hydraulic Fractures ◽  
Porous Rock ◽  
Linear Elastic ◽  
Volume Balance ◽  
Elastic Fracture ◽  
Steady Propagation ◽  
Fluid Leak

This study investigates the propagation of a semi-infinite buoyancy-driven hydraulic fracture in situations when the fluid is able to solidify along the crack walls. Such problems occur when hot magma ascends from a chamber due to buoyancy forces and solidifies by interacting with colder rock. In the model, the solidification rate is calculated assuming a one-dimensional heat transfer problem, in which case it becomes mathematically equivalent to Carter’s leak-off model, which is commonly used to describe the fluid leak-off from a hydraulic fracture into a porous rock formation. In order to construct a mathematical model for a buoyancy-driven hydraulic fracture with solidification, the aforementioned thermal problem is combined with (i) linear plane-strain elasticity to ensure equilibrium of the rock surrounding the fracture, (ii) linear elastic fracture mechanics to determine the fracture propagation, (iii) lubrication theory to capture the viscous fluid flow inside the crack and to account for the effect of buoyancy, and (iv) volume balance of the magma. To address the problem, the governing equations are first rewritten in terms of one integral equation with a non-singular kernel, which significantly simplifies the analysis and the procedure for obtaining a numerical solution. The latter solution is shown to obey a multiscale behaviour near the fracture tip that is fully resolved by the numerical scheme. In order to understand the structure of the solution and to quantify the regimes of propagation (and the associated transitions), a thorough analysis of the problem has been performed. Finally, the developments are applied to investigate the non-steady propagation of a buoyancy-driven fracture that is fed by a constant flux.

Experimental Study of Hydraulic Fracturing in an Impermeable Material

1983 ◽  
Vol 105 (2) ◽  
pp. 116-124 ◽  
Author(s):  
M. B. Rubin
Keyword(s):  
Experimental Data ◽  
Hydraulic Fracturing ◽  
Hydraulic Fracture ◽  
Critical Parameters ◽  
Laboratory Model ◽  
Finite Width ◽  
Fracture Width ◽  
Fracture Length ◽  
Numerical Predictions ◽  
Fluid Leak

A laboratory experiment was conducted to study hydraulic fracturing in an impermeable material (PMMA). Quantitative experimental data were obtained to compare with numerical predictions for a simple hydraulic fracture treatment that is not complicated by the effects of fluid leak-off and proppant transport. The borehole pressure, the pressure in the fracture at three locations, the fracture width at one location, and the fracture length were measured as functions of time during propagation of a vertically contained hydraulic fracture. The experimental data are compared with the predictions of simple solutions and the results indicate that when the finite width of the laboratory model is included in the analysis, the comparison between theory and experiment is quite good. The results also indicate that the assumption of a uniform pressure distribution in the fracture is adequate to accurately predict the critical parameters (fracture width and length) even when the fracturing fluid is very viscous.

Efficient Modeling of Proppant Transport During Three-Dimensional Hydraulic Fracture Propagation

2021 ◽  
Author(s):  
Jiacheng Wang ◽  
Jon Olson
Keyword(s):  
Particle Size ◽  
Hydraulic Fracture ◽  
Three Dimensional ◽  
Fracture Propagation ◽  
Displacement Discontinuity ◽  
Fluid Viscosity ◽  
Fracture Geometry ◽  
Fracture Width ◽  
Proppant Transport ◽  
Hydraulic Fracture Propagation

Abstract We propose an adaptive Eulerian-Lagrangian (E-L) proppant module and couple it with our simplified three-dimensional displacement discontinuity method (S3D DDM) hydraulic fracture model. The integrated model efficiently calculates proppant transport during three-dimensional (3D) hydraulic fracture propagation in multi-layer formations. The results demonstrate that hydraulic fracture height growth mitigates the form of proppant bed, so the proppant placement is more uniform in the hydraulic fracture under a smaller stress contrast. A higher fracturing fluid viscosity improves the suspension of proppant particles and generates a fracture larger in height and width but shorter in length. Lower proppant density and particle size reduce the proppant settling and create more uniform proppant placements, while they do not affect the hydraulic fracture geometry. Moreover, a larger proppant particle size limits the accessibility of the hydraulic fracture to the proppant, so the larger proppant particles do not fill the fracture tip and edge where the fracture width is small.

SAE 1020

Alloy Digest ◽  
1987 ◽  
Vol 36 (2) ◽  
Keyword(s):  
Fracture Toughness ◽  
Corrosion Resistance ◽  
Carbon Steel ◽  
Low Carbon Steel ◽  
Heat Treating ◽  
Low Carbon ◽  
Steel Mills ◽  
Temperature Performance ◽  
Wide Range ◽  
Cold Worked

Abstract SAE 1020 is a low-carbon steel combining good machinability, workability and weldability. It is carburized for use in case-hardened components and it is used for a wide range of applications in the hot-worked, cold-worked, normalized or quenched-and-tempered conditions. Its many uses include bolts, rods, plate applications, machinery components, case-hardened parts, spinning tools and trimming dies. This datasheet provides information on composition, physical properties, hardness, elasticity, and tensile properties as well as fracture toughness. It also includes information on low temperature performance and corrosion resistance as well as heat treating, machining, joining, and surface treatment. Filing Code: CS-113. Producer or source: Carbon steel mills.

ZYMAXX

Alloy Digest ◽  
1991 ◽  
Vol 40 (10) ◽  
Keyword(s):  
Fracture Toughness ◽  
Compressive Strength ◽  
Corrosion Resistance ◽  
Creep Resistance ◽  
Chemical Processing ◽  
Industrial Equipment ◽  
Automotive Applications ◽  
Adverse Conditions ◽  
Chemical Inertness ◽  
Wide Range

Abstract ZYMAXX provides outstanding compressive creep resistance, toughness and chemical inertness at high temperatures and pressures and under adverse conditions. They have a wide range of uses beyond chemical processing, including aerospace and automotive applications, general industrial equipment, home appliances, farm and construction equipment. This datasheet provides information on physical properties, hardness, tensile properties, and compressive strength as well as fracture toughness and creep. It also includes information on corrosion resistance. Filing Code: Cp-18. Producer or source: E. I. Dupont de Nemours & Company Inc..

DURCOMET 100

Alloy Digest ◽  
1993 ◽  
Vol 42 (2) ◽  
Keyword(s):  
Fracture Toughness ◽  
Wear Resistance ◽  
Corrosion Resistance ◽  
Mechanical Strength ◽  
Heat Treating ◽  
Corrosion And Wear ◽  
Annealed Condition ◽  
Excellent Corrosion Resistance ◽  
Wide Range ◽  
Very High

Abstract Durcomet 100 is an improved version of Alloy CD-4 MCu with better corrosion and wear resistance. The alloy is used in the annealed condition and possesses excellent corrosion resistance over a wide range of corrosion environments. Mechanical strength is also very high. This datasheet provides information on composition, physical properties, hardness, and tensile properties as well as fracture toughness. It also includes information on corrosion resistance as well as heat treating and joining. Filing Code: SS-540. Producer or source: Duriron Company Inc.

VDC HOT WORK

Alloy Digest ◽  
1956 ◽  
Vol 5 (4) ◽  
Keyword(s):  
Fracture Toughness ◽  
Die Casting ◽  
Heat Treating ◽  
Steel Company ◽  
Die Steel ◽  
White Metal ◽  
Aisi Type ◽  
Wide Range ◽  
Resistance To Heat

Abstract VDC is an air-hardening 5% chromium die steel recommended for a wide range of hot work applications requiring resistance to heat and abrasion, and is particularly useful for white metal die casting and extrusion applications. It is equivalent to AISI TYPE H13. This datasheet provides information on composition and hardness as well as fracture toughness. It also includes information on heat treating, machining, and joining. Filing Code: TS-45. Producer or source: Latrobe Steel Company.

Buoyancy-driven crack propagation: the limit of large fracture toughness

2007 ◽  
Vol 580 ◽  
pp. 359-380 ◽  
Author(s):  
S. M. ROPER ◽  
J. R. LISTER
Keyword(s):  
Fracture Toughness ◽  
Pressure Gradient ◽  
Viscous Flow ◽  
Numerical Solutions ◽  
Elastic Solid ◽  
Propagation Rate ◽  
Fluid Viscosity ◽  
Viscous Effects ◽  
Vertical Propagation ◽  
Viscous Pressure

We study steady vertical propagation of a crack filled with buoyant viscous fluid through an elastic solid with large effective fracture toughness. For a crack fed by a constant flux Q, a non-dimensional fracture toughness K=Kc/(3μQm3/2)1/4 describes the relative magnitudes of resistance to fracture and resistance to viscous flow, where Kc is the dimensional fracture toughness, μ the fluid viscosity and m the elastic modulus. Even in the limit K ≫ 1, the rate of propagation is determined by viscous effects. In this limit the large fracture toughness requires the fluid behind the crack tip to form a large teardrop-shaped head of length O(K2/3) and width O(K4/3), which is fed by a much narrower tail. In the head, buoyancy is balanced by a hydrostatic pressure gradient with the viscous pressure gradient negligible except at the tip; in the tail, buoyancy is balanced by viscosity with elasticity also playing a role in a region within O(K2/3) of the head. A narrow matching region of length O(K−2/5) and width O(K−4/15), termed the neck, connects the head and the tail. Scalings and asymptotic solutions for the three regions are derived and compared with full numerical solutions for K ≤ 3600 by analysing the integro-differential equation that couples lubrication flow in the crack to the elastic pressure gradient. Time-dependent numerical solutions for buoyancy-driven propagation of a constant-volume crack show a quasi-steady head and neck structure with a propagation rate that decreases like t−2/3 due to the dynamics of viscous flow in the draining tail.

On the Crack Quasi-Static Growth

2019 ◽  
Vol 827 ◽  
pp. 312-317
Author(s):  
Vitalijs Pavelko
Keyword(s):  
Fracture Toughness ◽  
Crack Growth ◽  
Plastic Material ◽  
Small Scale ◽  
Invariant Equation ◽  
Practical Applications ◽  
Wide Range ◽  
Scale Yielding ◽  
Elastic Plastic Material ◽  
Principal Assumption

The theoretical model of quasi-static crack growth in the elastic-plastic material under load variation in a wide range. Small-scale yielding is principal assumption and main restriction of proposed theory. The model of crack growth provides for continues and interrelated both the crack propagation and plastic deformation development. The nonlinear first-order differential equation describes the quasi-static process of crack growth. In dimensionless form this equation invariant in respect to geometrical configuration and material. The critical size of the plastic zone is proposed as the characteristics of material resistance which is directly connected with the fracture toughness, but more convenient in practical applications of invariant equation. The demonstration of solution is performed for the double cantilever beam that widely used as the standard (DCB) sample for measurement of the mode-I interlaminar fracture toughness. he short analysis of some properties of solution of the invariant equation and its application is done.

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