The pressure distribution around a growing crack

Gordeyev, Yu.N.; Entov, V. M.

doi:10.1016/s0021-8928(98)80005-1

Cited by 12 publications

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“…Assuming that l(t) = at 1/2 , Gordeyev and Entov [29] derived a similarity solution to the two-dimensional pressure diffusion equation, which yields a leak-off velocity of the same form as (1.8). In this case the fracture is growing sufficiently rapidly for the leak-off process to be one-dimensional, a situation that is likely to persist for power laws in which the fracture evolves more rapidly:…”

Section: Problem Formulation and Dimensional Resultsmentioning

confidence: 99%

An Asymptotic Framework for Finite Hydraulic Fractures Including Leak‐Off

Mitchell¹,

Kuske²,

Peirce³

2007

SIAM J. Appl. Math.

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Abstract. The dynamics of hydraulic fracture, described by a system of nonlinear integrodifferential equations, is studied through the development and application of a multiparameter singular perturbation analysis. We present a new single expansion framework which describes the interaction between several physical processes, namely viscosity, toughness, and leak-off. The problem has nonlocal and nonlinear effects which give a complex solution structure involving transitions on small scales near the tip of the fracture. Detailed solutions obtained in the crack tip region vary with the dominant physical processes. The parameters quantifying these processes can be identified from critical scaling relationships, which are then used to construct a smooth solution for the fracture depending on all three processes. Our work focuses on plane strain hydraulic fractures on long time scales, and this methodology shows promise for related models with additional time scales, fluid lag, or different geometries, such as radial (penny-shaped) fractures and the classical Perkins-Kern-Nordgren (PKN) model.

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Section: Problem Formulation and Dimensional Resultsmentioning

confidence: 99%

An Asymptotic Framework for Finite Hydraulic Fractures Including Leak‐Off

Mitchell¹,

Kuske²,

Peirce³

2007

SIAM J. Appl. Math.

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“…Following Gordeyev and Entov (1997), consider a circular planar hydraulic fracture of radius a [L] situated within an infinite porous medium characterized by a storage coefficient, S [M −1 LT 2 ] and permeabilities, k z [L 2 ] and k r [L 2 ] in the z and r direction, respectively, where r [L] and z [L] are distances parallel and normal to the fracture plane, respectively. The origin of the fracture is situated at r = 0 and z = 0 (see Fig.…”

Section: The Leak-off Modelmentioning

confidence: 99%

“…The hydraulic pressure is further defined by the boundary value problem (see Fig. 1b) (Gordeyev and Entov 1997)…”

Section: The Leak-off Modelmentioning

confidence: 99%

“…The issue is that explicitly simulating the three-dimensional flow field around the fracture generally involves the application of a numerical model. Gordeyev and Entov (1997) present an analytical method based on a similarity solution to the three-dimensional leak-off problem. However, unfortunately, their similarity transforms require that the fracture pressure is constant and the fracture radius grows as a square root of time.…”

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confidence: 99%

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Hydraulic Fracture Propagation with 3-D Leak-off

Mathias

Reeuwijk

2009

Transp Porous Med

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Hydraulic fracture models typically couple a fracture elasticity model with a geological reservoir model to forecast the rate of fluid leak-off from the propagating fracture. The most commonly used leak-off model is that originally specified by Carter, which involves the assumption that the fracture is embedded within an infinite homogenous porous medium where flow only occurs perpendicular to the fracture plane. The objectives of this paper are:(1) to show that assuming one-dimensional leak-off can lead to erroneous conclusions, (2) to present a robust numerical methodology for simulating three-dimensional leak-off from propagating hydraulic fractures, and (3) to present and compare a new analytical method based on assuming three-dimensional flow of an incompressible fluid through an incompressible porous formation from a circular planar fracture. Provided the fluid and formation compressibility can be ignored within the reservoir flow model, the three-dimensional leakoff from a circular planar fracture can be written in closed-form as a function, which depends linearly on fracture pressure and radial extent. This simple expression for leak-off can be easily coupled to a range of circular fracture elasticity models. As a comparison example, the Carter model, our new function and a three-dimensional numerical model of the full problem are coupled to the PK-radial fracture model. Comparison with the numerical model shows that our new function overestimates fracture growth during intermediate times but accurately predicts both the early and late-time asymptotic behavior. In contrast, the Carter model fails to replicate both the early and late-time asymptotic behavior. Our new function additionally improves on the Carter model by not requiring the evaluation of convolution integrals and allowing easy evaluation of both the spatial leakage flux distribution across the fracture face and the three-dimensional pressure distribution within the porous formation. List of symbolsa Radius of hydraulic fracture [L] a D = a/a f Dimensionless radius of hydraulic fracture [-] a f = {[E/(1 − ν 2 )](Qt f ) 2 /G} 1/5 Characteristic fracture radius [L] E Young's modulus [ML −1 T −2 ] G = K 2 Ic /E Specific surface energy of the rock [MT −1 ] k = (k r k z ) 1/2 Mean permeability [L 2 ] k r Permeability in r -direction [L 2 ] k z Permeability in z-direction [L 2 ] K Bulk modulus [ML −1 T −2 ] K Ic Critical mode I stress intensity factor [ML −1/2 T −2 ] M Biot modulus [ML −1 T −2 ] P Hydraulic pressure [ML −1 T −2 ] P d Hydraulic pressure within the fracture [ML −1 T −2 ] P dD = P d /P f Dimensionless hydraulic pressure within the fracture [-]Radial distance parallel to the fracture plane [L] r D = r/a f Dimensionless radial distance parallel to the fracture planeQt f Dimensionless volume of fracture [-] z Distance perpendicular to the fracture plane [L] z = (k r /k z ) 1/2 z Re-scaled distance perpendicular to the fracture plane [L] z D = z/a f Dimensionless distance perpendicular to the fracture plane [-] α Biot effective stress coef...

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“…Although, several investigations looked at the propagation of a fracture driven by a low viscosity fluid, when fluid diffusion is fully two-dimensional [11,12,13], the study of injection into a stationary, pre-existing fracture has not yet received due attention. One of the foci of this study is to identify solutions corresponding to the limiting cases of the small and large injection time (1-D and fully-developed 2-D diffusion, respectively), and the solution in the intermediate regime corresponding to the evolution between the two limiting cases.…”

Section: Introductionmentioning

confidence: 99%

Pressurization of a PKN Fracture in a Permeable Rock During Injection of a Low Viscosity Fluid

Sarvaramini¹,

Garagash²

2013

Effective and Sustainable Hydraulic Fracturing

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The aim of the present work is to investigate injection of a low-viscosity fluid into a pre-existing fracture within a linear elastic, permeable rock, as may occur in waterflooding and supercritical CO2 injection. In conventional hydraulic fracturing, high viscosity and cake building properties of injected fluid limit diffusion to a 1-D boundary layer incasing the crack. In the case of injection of low viscosity fluid into a fracture, diffusion will take place over wider range of scales, from 1-D to 2-D, thus, necessitating a new approach. In addition, the dissipation of energy associated with fracturing of the rock dominates the energy expended to flow a low viscosity fluid into the crack channel. As a result, the rock fracture toughness is an important parameter in evaluating the propagation driven by a low-viscosity fluid. We consider a pre-existing, un-propped, stationary Perkins, Kern and Nordgren's (PKN) fracture into which a low viscosity fluid is injected under a constant flow rate. The fundamental solution to the auxiliary problem of a step pressure increase in a fracture [1] is used to formulate and solve the convolution integral equation governing the transient crack pressurization under the assumption of negligible viscous dissipation. The propagation criterion for a PKN crack [2] is then used to evaluate the onset of propagation. The obtained solution for transient pressurization of a stationary crack provides initial conditions to the fracture propagation problem.

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The pressure distribution around a growing crack

Cited by 12 publications

References 1 publication

An Asymptotic Framework for Finite Hydraulic Fractures Including Leak‐Off

An Asymptotic Framework for Finite Hydraulic Fractures Including Leak‐Off

Hydraulic Fracture Propagation with 3-D Leak-off

Pressurization of a PKN Fracture in a Permeable Rock During Injection of a Low Viscosity Fluid

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