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Research Papers

# Breakdown of a Pressurized Fingerlike Crack in a Permeable Solid

[+] Author and Article Information
Erfan Sarvaramini

Department of Civil and Resource Engineering,
Dalhousie University,
1360 Barrington Street,
Halifax, NS B3H 4R2, Canada
e-mail: erfan.sarvaramini@dal.ca

Dmitry I. Garagash

Department of Civil and Resource Engineering,
Dalhousie University,
1360 Barrington Street,
Halifax, NS B3H 4R2, Canada
e-mail: garagash@dal.ca

Existence of 1D leak-off regime relies on separation of timescales, $t1≪t2,$ or $A≪1$ (Eq. (38)).

Note that the 2D scales (Eq. (34)) are the same for the two crack geometries.

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received February 27, 2015; final manuscript received March 19, 2015; published online April 30, 2015. Editor: Yonggang Huang.

J. Appl. Mech 82(6), 061006 (Jun 01, 2015) (10 pages) Paper No: JAM-15-1113; doi: 10.1115/1.4030172 History: Received February 27, 2015; Revised March 19, 2015; Online April 30, 2015

## Abstract

This paper is concerned with the analysis of a low-viscosity fluid injection into a pre-existing, fingerlike crack within a linear elastic, permeable rock, and of the conditions leading to the onset of the fracture propagation (i.e., the breakdown). The problem is of interest in reservoir waterflooding, supercritical CO2 injection for geological storage, and other subsurface fluid injection applications. Fluid injection into a stationary crack leads to its elastic dilation and pressurization, buffered by the fluid leak-off into the surrounding rock. The solution of the problem, therefore, requires coupling of the crack deformation and the full-space pore-fluid pressure diffusion in the permeable rock. Contrary to the case of propagating hydraulic fractures, when significant part of the energy input is dissipated in the viscous fluid flow in the fracture, we find that the viscous fluid pressure drop inside a stationary fracture can be often neglected (we establish the conditions when one can do so). This, in turn, allows for a semi-analytical solution of the problem using the Green's function method, and, furthermore, for the full analytical treatment of the small/large injection time asymptotics. We apply the transient pressurization solution to predict the onset of the propagation based on the criteria derived from the energy considerations for a fingerlike crack.

###### FIGURES IN THIS ARTICLE
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## Figures

Fig. 1

Injection into a pre-existing fingerlike crack with length 2ℓ and height h (ℓ>>h)

Fig. 2

Injection at a constant rate into a crack in (a) impermeable rock (K = 0) and (b) permeable rock (K = 0.1): evolution of the normalized net pressure Π = p¯/p* with normalized time τ = t/t* at selected positions along the crack ξ = x/ℓ = 0 (inlet),0.1,0.5,1 (tip) in the case of zero neutral crack opening (W0 = 0). The characteristic pressure and time scales are p* = (μQ0E'3ℓ/h4)1/4 and t* = (μℓ5h4/E'Q03)1/4, respectively. Marked points correspond to the onset of approximate pressure uniformity in the crack.

Fig. 3

Comparison of the numerical solution for the normalized leak-off rate in the auxiliary problem of an instantaneous step pressure increase Δp with (a) the small time (Eq. (24)) and (b) the large-time (Eq. (26)) asymptotes. The time and leak-off rate scales are t* = ℓ2/4α and g¯* = 4ΔpSα/ℓ, respectively.

Fig. 4

Evolution of the cumulative leak-off volume in the auxiliary problem (step pressure increase along the crack). The numerical solution is contrasted to small (28) and large-time (29) asymptotes. The time and volume scales are t* = ℓ2/4α and V* = ΔphSℓ2, respectively.

Fig. 5

Evolution of the normalized net pressure p¯/p1 during the transient pressurization of a crack for various values of the crack aspect ratio parameter A = h/(ℓSE'). The 1D (Eq. (38)) and 2D (Eq. (40)) leak-off asymptotic solutions are shown by dashed lines, respectively. The dotted line shows the early time storage-dominated solution (Eq. (39)a). The 1D pressure and time scales are p1 = Q0E't1/h2ℓ and t1 = (1/SE')2(h2/4α), respectively.

Fig. 6

Dependence of (a) uniformity net pressure p¯uni/p* and (b) uniformity time tuni/t* on the nondimensional neutral hydraulic opening W0 = w0(E'/μQ0ℓ)1/4 during the transient pressurization of a mechanically open fracture in impermeable rock. The characteristic pressure and time scales are p*= (μQ0E'3ℓ/h4)1/4 and t* = (μℓ5h4/E'Q03)1/4, respectively.

Fig. 7

Dependence of (a) uniformity net pressure p¯uni/p* and (b) uniformity time tuni/t* on the nondimensional permeability parameter K = k(ℓ1/3E'/Q0μ)3/4 during the transient pressurization of a mechanically open fracture with the aspect ratio parameter A = 0.01 and negligible neutral hydraulic opening W0 = 0

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