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Research Papers

Modeling Turbulent Hydraulic Fracture Near a Free Surface

[+] Author and Article Information
Victor C. Tsai1

 Seismological Laboratory, California Institute of Technology, Pasadena, CA 91125 tsai@caltech.edu

James R. Rice

 School of Engineering and Applied Sciences, and Department of Earth and Planetary Sciences, Harvard University, Cambridge, MA 02138 rice@seas.harvard.edu

1

Corresponding author.

J. Appl. Mech 79(3), 031003 (Apr 05, 2012) (5 pages) doi:10.1115/1.4005879 History: Received September 19, 2011; Revised January 11, 2012; Posted February 06, 2012; Published April 04, 2012; Online April 05, 2012

Motivated by observations of the subglacial drainage of water, we consider a hydraulic fracture problem in which the crack grows parallel to a free surface, subject to fully turbulent fluid flow. Using a hybrid Chebyshev/series-minimization numerical approach, we solve for the pressure profile, crack opening displacement, and crack growth rate for a crack that begins relatively short but eventually becomes long compared with the distance to the free surface. We plot nondimensionalized results for a variety of different times, corresponding with different fracture lengths, and find substantial differences when free-surface effects are important.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic for turbulent hydraulic fracture

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Figure 2

Leading order pressure term p0 corresponding with a0w0=a0[(1-x̂)/2] 6/7 for different values of L/H (0.02 = black, 1 = blue, 2 = gray, and 5 = red). (Colors may be viewed in the online version of the paper.)

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Figure 3

Crack openings wk , from k = 1 to 6, for different values of L/H. Odd wk are in solid lines and units are on the left axis. Even wk are in dashed lines and units are on the right axis. Blue, green and red colors correspond to 1,3,5 for odd and 2,4,6 for even, respectively. (Color may be viewed in the online line version of the paper.)

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Figure 4

Scaled pressure p̂(x̂) plotted for different values of L/H (colors are as in Fig. 2). (Colors may be viewed in the online version of the paper.)

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Figure 5

Scaled opening ŵ(x̂) plotted for different values of L/H (colors as in Fig. 4, with additional orange curve at L/H = 3.3). (Colors may be viewed in the online version of the paper.)

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Figure 6

Scaled average opening ŵavg versus L/H (colors as in Fig. 5). Solid line is the polynomial fit discussed in the text. (Colors may be viewed in the online version of the paper.)

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Figure 7

Scaled crack-tip velocity φ≡Utip/US versus L/H (colors as in Fig. 5). Solid line is the polynomial fit discussed in the text. (Colors may be viewed in the online version of the paper.)

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