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TECHNICAL PAPERS

# Plane-Strain Propagation of a Fluid-Driven Fracture: Small Toughness Solution

[+] Author and Article Information
Dmitry I. Garagash1

Department of Civil and Environmental Engineering, Clarkson University, Potsdam, NY 13699-5710garagash@clarkson.edu

Emmanuel Detournay

Department of Civil Engineering, University of Minnesota, Minneapolis, MN 55455

1

To whom correspondence should be addressed.

J. Appl. Mech 72(6), 916-928 (Apr 10, 2005) (13 pages) doi:10.1115/1.2047596 History: Received September 30, 2004; Revised April 10, 2005

## Abstract

The paper considers the problem of a plane-strain fluid-driven fracture propagating in an impermeable elastic solid, under condition of small (relative) solid toughness or high (relative) fracturing fluid viscosity. This condition typically applies in hydraulic fracturing treatments used to stimulate hydrocarbons-bearing rock layers, and in the transport of magma in the lithosphere. We show that for small values of a dimensionless toughness $K$, the solution outside of the immediate vicinity of the fracture tips is given to $O(1)$ by the zero-toughness solution, which, if extended to the tips, is characterized by an opening varying as the $(2∕3)$ power of the distance from the tip. This near tip behavior of the zero-toughness solution is incompatible with the Linear Elastic Fracture Mechanics (LEFM) tip asymptote characterized by an opening varying as the $(1∕2)$ power of the distance from the tip, for any nonzero toughness. This gives rise to a LEFM boundary layer at the fracture tips where the influence of material toughness is localized. We establish the boundary layer solution and the condition of matching of the latter with the outer zero-toughness solution over a lengthscale intermediate to the boundary layer thickness and the fracture length. This matching condition, expressed as a smallness condition on $K$, and the corresponding structure of the overall solution ensures that the fracture propagates in the viscosity-dominated regime, i.e., that the solution away from the tip is approximately independent of toughness. The solution involving the next order correction in $K$ to the outer zero-toughness solution yields the range of problem parameters corresponding to the viscosity-dominated regime.

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## Figures

Figure 1

Sketch of a plane-strain fluid-driven fracture

Figure 2

Zero- (after Adachi and Detournay (16)) and first-order terms in the outer solution expansion 13 for the normalized opening Ω¯, (a), and net-pressure Π, (b). Solution for Ω¯1 and Π1 is shown for n=1 (dotted lines) and n=10 (solid lines) terms in series 21.

Figure 3

Semi-infinite fluid driven fracture

Figure 4

LEFM tip boundary layer solution: Variation of dimensionless (a) opening Ω̂ and (b) net-pressure Π̂ with the distance from the tip ξ̂. Asymptotic expansions of the solution in the near-field (near-tip) and the far-field are shown by long dash lines.

Figure 5

Matching of the inner and outer solutions; structure of the composite solution

Figure 6

Small toughness outer solution for (a) opening Ω and (b) net-pressure Π for various values of K={0,0.75,1,1.25}. The finite-toughness solution for K=1 and K=1.25 is shown by open circles (after Adachi (20)). The ascending parts of the pressure curves correspond to the continuation of the outer solution into the near tip region where the outer solution is no longer valid and the solution has to be given by the LEFM boundary layer solution.

Figure 7

Comparison between the small toughness outer solution and uniformly valid composite solution for (a) opening Ω and (b) net-pressure Π for K=1. The composite solution is shown by dashed line, the rest of the curves as in Fig. 6.

Figure 8

Variation of dimensionless fracture half-length γ with toughness K. Zero-toughness, (23,16), and zero-viscosity, (21-22), solutions are shown by dashed lines. Small toughness and large toughness asymptotic solutions are shown by solid lines. The finite toughness solution (after Adachi (20)) is shown by open circles.

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