The Tip Region of a Fluid-Driven Fracture in an Elastic Medium

Abstract

The focus of this paper is on constructing the solution for a semi-infinite hydraulic crack for arbitrary toughness, which accounts for the presence of a lag of a priori unknown length between the fluid front and the crack tip. First, we formulate the governing equations for a semi-infinite fluid-driven fracture propagating steadily in an impermeable linear elastic medium. Then, since the pressure in the lag zone is known, we suggest a new inversion of the integral equation from elasticity theory to express the opening in terms of the pressure. We then calculate explicitly the contribution to the opening from the loading in the lag zone, and reformulate the problem over the fluid-filled portion of the crack. The asymptotic forms of the solution near and away from the tip are then discussed. It is shown that the solution is not only consistent with the square root singularity of linear elastic fracture mechanics, but that its asymptotic behavior at infinity is actually given by the singular solution of a semi-infinite hydraulic fracture constructed on the assumption that the fluid flows to the tip of the fracture and that the solid has zero toughness. Further, the asymptotic solution for large dimensionless toughness is derived, including the explicit dependence of the solution on the toughness. The intermediate part of the solution (in the region where the solution evolves from the near tip to the far from the tip asymptote) of the problem in the general case is obtained numerically and relevant results are discussed, including the universal relation between the fluid lag and the toughness. [S0021-8936(00)02401-6]

Numerical methods for hydraulic fracture propagation: A review of recent trends

“…Garagash (2009)). Garagash and Detournay (2000) argue that the tip behavior is governed by local plane strain conditions, provided the crack front curvature is not too large. For a hydraulic fracture growing with velocity V , the continuity Eq.…”

“…In this case, where the fluid front C f lags behind the fracture front C, the boundary conditions at the fracture front C f are (for an impermeable medium) p = p cav ≈ 0 and the Stefan condition V i = q i /w Meanwhile, the boundary condition at the fracture front C is w = 0 in complement to the propagation condition K I = K Ic . It has been shown (Garagash and Detournay, 2000) that the fluid and fracture fronts actually coalesce when…”

An Eulerian moving front algorithm with weak‐form tip asymptotics for modeling hydraulically driven fractures

“…In order to minimize the energy expended, fractures in pre-stressed media typically propagate in a plane normal to the least compressive stress. Besides the standard assumptions regarding the applicability of LEFM and lubrication theory, we make a series of simplifications that can readily be justified for the purposes of this contribution: (i) the rock is homogeneous (having uniform values of toughness K I c , Young's modulus E, and Poisson's ratio ), (ii) the fracturing fluid is incompressible and Newtonian (having a viscosity ), (iii) the fracture is always in limit equilibrium, (iv) the rock is impermeable, (v) gravity is neglected in the lubrication equation, and (vi) the fluid front coincides with the crack front, because the lag between the two fronts is negligible under typical high confinement conditions of reservoir stimulation [18,27]. Note that the assumptions regarding the homogeneity of K I c and can be relaxed without any significant changes to the model.…”

“…Indeed, it can be inferred, from the generalized tip asymptote for an HF advancing in a solid with finite toughness [18] and from the thickness c of the asymptotic region, which is of order O(10 −1 ), that the 2 3 asymptote (15) applies if c v −2 . For a penny-shaped fracture, dissipation associated with viscous flow dominates the fracture growth when 1 [26].…”

“…Recent asymptotic analyses [18][19][20][21] of these coupled equations have identified a multi-scale structure in which different physical processes compete. For example, the energy dissipated in driving the viscous fluid through the fracture competes with the energy expended in breaking the rock.…”