A non-singular integral equation formulation to analyse multiscale behaviour in semi-infinite hydraulic fractures

2015 ◽  
Vol 781 ◽  
Author(s):  
E. V. Dontsov ◽  
A. P. Peirce
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Hydraulic Fracture ◽  
Closed Form Solution ◽  
Form Solution ◽  
Hydraulic Fractures ◽  
Length Scales ◽  
Practical Applications ◽  
Integral Equation Formulation

This study revisits the problem of a steadily propagating semi-infinite hydraulic fracture in which the processes of toughness-related energy release, viscous dissipation and leak-off compete on multiple length scales. This problem typically requires the solution of a system of integro-differential equations with a singular kernel, which is complicated by the need to capture extremely disparate length scales. In this study the governing equations are rewritten in the form of one non-singular integral equation. This reformulation enables the use of standard numerical techniques to capture the complete multiscale behaviour accurately and efficiently. This formulation also makes it possible to approximate the problem by a separable ordinary differential equation, whose closed-form solution captures the multiscale behaviour sufficiently accurately to be used in practical applications. We also consider a similar reformulation of the equations governing the propagation of a buoyancy-driven semi-infinite hydraulic fracture. The resulting numerical solution is able to capture the near-tip multiscale behaviour efficiently and agrees well with published solutions calculated in the large-toughness limit.

A Quantitative Evaluation of Two Classical Approximations Used to Predict the Extent of Vertical Hydraulic Fractures

1983 ◽  
Vol 105 (4) ◽  
pp. 512-527 ◽  
Author(s):  
M. B. Rubin
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Hydraulic Fracture ◽  
Closed Form Solution ◽  
Solution Procedure ◽  
Form Solution ◽  
Critical Parameters ◽  
Hydraulic Fractures ◽  
Fracture Width ◽  
Special Case

An integral equation was developed to predict the critical parameters (fracture width and length) associated with the propagation of a vertical hydraulic fracture and a numerical solution procedure was developed. The effects of the classical approximations of pressure and fracture width were investigated both separately and together. It was found that the effects associated with the pressure approximation were relatively insignificant, whereas those associated with the fracture width approximation were significant, particularly when the formation was only moderately permeable. Finally, an exact closed-form solution of the integral equation was developed for a special case. It was shown that when the formation is only moderately permeable, this solution provides a better approximation of the exact solution than the classical solution of Carter [2].

Approximate Solution Of A Singular Integral Equation With A Multiplicative Cauchy Kernel In The Half-Plane

2008 ◽  
Vol 8 (2) ◽  
pp. 143-154 ◽  
Author(s):  
P. KARCZMAREK
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Half Plane ◽  
Singular Integral ◽  
Trigonometric Polynomials ◽  
Cauchy Kernel

AbstractIn this paper, Jacobi and trigonometric polynomials are used to con-struct the approximate solution of a singular integral equation with multiplicative Cauchy kernel in the half-plane.

A semianalytical and finite-element solution to the unbonded contact between a frictionless layer and an FGM-coated half-plane

2017 ◽  
Vol 24 (2) ◽  
pp. 448-464 ◽  
Author(s):  
Jie Yan ◽  
Changwen Mi ◽  
Zhixin Liu
Keyword(s):  
Integral Equation ◽  
Finite Element ◽  
Singular Integral Equation ◽  
Contact Problem ◽  
Singular Integral ◽  
Material Properties ◽  
Elastic Layer ◽  
Coated Substrate ◽  
Receding Contact ◽  
Contact Size

In this work, we examine the receding contact between a homogeneous elastic layer and a half-plane substrate reinforced by a functionally graded coating. The material properties of the coating are allowed to vary exponentially along its thickness. A distributed traction load applied over a finite segment of the layer surface presses the layer and the coated substrate against each other. It is further assumed that the receding contact between the layer and the coated substrate is frictionless. In the absence of body forces, Fourier integral transforms are used to convert the governing equations and boundary conditions of the plane receding contact problem into a singular integral equation with the contact pressure and contact size as unknowns. Gauss–Chebyshev quadrature is subsequently employed to discretize both the singular integral equation and the force equilibrium condition at the contact interface. An iterative algorithm based on the method of steepest descent has been proposed to numerically solve the system of algebraic equations, which is linear for the contact pressure but nonlinear for the contact size. Extensive case studies are performed with respect to the coating inhomogeneity parameter, geometric parameters, material properties, and the extent of the indentation load. As a result of the indentation, the elastic layer remains in contact with the coated substrate over only a finite interval. Exterior to this region, the layer and the coated substrate lose contact. Nonetheless, the receding contact size is always larger than that of the indentation traction. To validate the theoretical solution, we have also developed a finite-element model to solve the same receding contact problem. Numerical results of finite-element modeling and theoretical development are compared in detail for a number of parametric studies and are found to agree very well with each other.

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