On the loss of symmetry in toughness dominated hydraulic fractures

Abstract

Fracking, or hydraulic fracturing, is a ubiquitous technique for generating fracture networks in rocks for enhanced geothermal systems or hydrocarbon extraction from shales. For decades, models, numerical simulation tools, and practical guidelines have been based on the assumption that this process generates networks of self-similar parallel cracks. Yet, some field and laboratory observations show asymmetric crack growth, and material heterogeneity is routinely attributed for it. Here, we show that simultaneous growth of multiple parallel cracks is impossible and that a single crack typically propagates asymmetrically in toughness dominated hydraulic fracturing, in which viscous dissipation of the fluid is negligible. In other words, loss of symmetry is a fundamental feature of hydraulic fracturing in a toughness dominated regime and not necessary the result of material heterogeneities. Our findings challenge the assumptions of symmetrical growth of hydraulic fractures commonly made in practice, and point to yet another instability other than material heterogeneity.

Appendix

1.1 Propagation of a single hydraulic crack in an infinite domain

Here we discuss the closed form solutions of a pressurized straight crack in 2D and a penny-shape crack in 3D in more details. We start by recalling classical results (Sneddon and Elliott 1946) that provide an upper bound on the critical propagation pressure in two space dimensions (2D) followed by three dimensions (3D).

For 2D, the normal displacement on the crack is given by Sneddon and Elliott (1946); Sneddon and Lowengrub (1969):

$$\begin{aligned} u_y(x,0^\pm ) = \pm \frac{2p}{E'}\sqrt{l_0^2-x^2}, \end{aligned}$$

and the pressurized crack forms an elliptical cavity of volume

$$\begin{aligned} V := \frac{\mathcal {W}_p(l_0)}{p}, \end{aligned}$$

in the deformed configuration. The work of the pressure force is

$$\begin{aligned} \mathcal {W}_p(l_0) = \frac{2\pi p^2}{E'}l_0^2. \end{aligned}$$

Owing to Clapeyron’s theorem, the elastic energy is given by

$$\begin{aligned} \mathcal {E}_p(l_0) =-\frac{\pi p^2 l_0^2}{E'}, \end{aligned}$$

and the elastic energy release rate with respect to a pressure change, assuming propagation along the x-axis, is

$$\begin{aligned} G_p(l_0) := -\frac{1}{2}\frac{\partial \mathcal {E}_p}{\partial l}(l_0) = \frac{\pi p^2l_0}{E'}. \end{aligned}$$

Assuming a quasi-static evolution driven by an increasing injection pressure, stability in the sense of Griffith criterion $G_p \le G_\mathrm {c}$ is satisfied as long as $p \le p_0$ with

$$\begin{aligned} p_0 := \left( \frac{G_\mathrm {c}E'}{\pi l_0}\right) ^{1/2}, \end{aligned}$$

and the volume of the cavity in the deformed configuration is

$$\begin{aligned} V_0 := 2\left( \frac{\pi G_\mathrm {c}l_0^3}{E'}\right) ^{1/2}. \end{aligned}$$

(4)

Note that $p_0$ is a decreasing function of $l_0$ so that once the injection pressure attains the critical value $p_0$, Griffith stability can no longer be attained.

When the driving parameter is the volume of the fracture in the deformed configuration (or injected fluid volume, assuming that the pressure force is achieved by injecting an incompressible fluid), the situation is different. The elastic energy becomes

$$\begin{aligned} \mathcal {E}_V(l_0) := \frac{E'V^2}{4\pi l_0^2}, \end{aligned}$$

and the elastic energy release rate with respect to a volume change is

$$\begin{aligned} G_V(l_0) = -\frac{1}{2}\frac{\mathrm {d} \mathcal {E}_V}{\mathrm {d} l_0}(l_0) = \frac{E'V^2}{4\pi l_0^3}. \end{aligned}$$

Griffith’s stability for a crack of length $l_0$ is satisfied as long as $V \le V_0$ given in 4. When V reaches $V_0$, the crack must grow while satisfying $G(V,l) = G_\mathrm {c}$, from which we derive that

$$\begin{aligned} p(V) = \left( \frac{2E'G_\mathrm {c}}{\pi V}\right) ^{1/3}, \end{aligned}$$

and

$$\begin{aligned} l(V) = \left( \frac{E' V^2}{4\pi G_\mathrm {c}}\right) ^{1/3}, \end{aligned}$$

i.e. recovering the classical scaling law for the pressure drop in a propagating hydraulic crack (Dean and Schmidt 2009).

Note that the same analysis can be performed in 3D, assuming a penny-shaped crack throughout the evolution with initial radius $R_0$. In this case, the critical pressure and volumes are given by

$$\begin{aligned} p_0 := \left( \frac{\pi G_cE'}{4R_0}\right) ^{1/2}, \end{aligned}$$

and

$$\begin{aligned} V_0 := \frac{8}{3}\left( \frac{\pi G_cR_0^5}{E'}\right) ^{1/2}, \end{aligned}$$

As the critical volume is exceeded, the injection pressure and crack radius are given by

$$\begin{aligned} p(V) = \left( \frac{G_\mathrm {c}^3 E'^2 \pi ^3}{12 V}\right) ^{1/5}, \end{aligned}$$

and

$$\begin{aligned} R(V) = \left( \frac{9 E' V^2}{64 \pi G_\mathrm {c}}\right) ^{1/5}. \end{aligned}$$

Table 1 Parameters used for the simulation of a single fracture in 2D and 3D

Fig. 6

Sketch of of the computational domain geometry. The symmetry axis being a reflection in 2D and a revolution axis in 3D

1.2 Verification simulation

Here we present the verification of our numerical model against the closed form solution of a single hydraulic fracture in 2D and 3D. All computations were performed with the vDef open-source implementation of the variational phase-field approach to fracture (Bourdin 2019) in non-dimensional form².

All the geometric parameters are defined in an invariant geometry (a reflexion axis in 2D and a rotation in 3D) depicted in Fig. 6. Note, however, that we computed the simulations in the full domain. We set up all the geometric and material parameters identically for both problems as summarized in Table 1. To simulate the infinite domain considered in the closed form solutions, we set the edge size of the computational domain to 100 times the initial crack length, and refined the mesh near the expected area of propagation of the crack as shown in Fig. 6.

Fig. 7

Snapshots of phase-field profile for the line crack example at different loadings. Prior to the critical pressure loading (top), after the critical pressure loading (middle) and during the propagation (bottom). The red color represents fully damage material (fracture) and blue undamaged

Fig. 8

A snapshot (view from above) of fracture damage $(\alpha \ge .99)$ for the penny shape crack during the propagation (left). The solid white line indicates the initial crack and black line is the limit of the casing. An off-centered penny shape crack propagation was also observed in a toughness dominated hydraulic fracturing experiment as reported in Bunger et al. (2013) (right)

The initial phase-field function needs to be constructed carefully when performing simulations of crack re-nucleation (Tanné et al. 2018). To obtain a proper phase-field profile, we seeded an initial crack of length strictly less than $l_0$. We then applied a pressure field in the pre-existing crack and monitored its propagation until the length, estimated from the level line $\alpha = 0.8$, reached $l_0$.

Figure 7 shows snapshots of the evolution of the phase-field. The top figure represents the initial damage field obtained by enforcing $\alpha =1$ on a one-element wide strip of length $< l_0$. The center figure shows the damage field associated with a crack of length $l_0$ obtained by pressurizing the initial crack. The lower figure shows the phase-field profile during the propagation phase. Notice the small difference between the phase profile near the crack tips in the first two figures.

Fig. 9

Evolutions of normalized p, V and l for the line fracture (left column figures) and penny shape crack (right column figures). Colored dots refer to numerical results and solid black lines to the closed form solution given in Sect. 1

Figure 8 shows the crack in a 3D computation, by plotting the level surface $\alpha = 0.99$. Whereas the crack remains penny-shaped, it is not symmetrical with respect to the injection source. Such asymmetric growth has also been observed in laboratory experiments (Bunger et al. 2008, 2013).

Figure 9 show the excellent agreement between crack radius and injection pressure from phase-field computations with the closed form solution in 2D and 3D. In both cases as long as the $V \le V_c$ the crack does not grow and for $V>V_c$ the pressure starts to decline as $p \sim V^{-1/3}$ (line fracture) and $p\sim V^{-1/5}$ (penny-shape crack). Note that we have accounted for the “effective” toughness $(G_\mathrm {c})_\mathrm {eff}=G_\mathrm {c}\left( 1+3h/8\ell \, \right)$ induced by the finite discretization size (Bourdin et al. 2008).