Impact of the anisotropy of fracture toughness on the propagation of planar 3D hydraulic fracture

Abstract

Sedimentary rocks often exhibit a transverse isotropy due to fine scale layering. We investigate the effect of the anisotropy of fracture toughness on the propagation of a planar 3D hydraulic fracture perpendicular to the isotropy plane: a configuration commonly encountered in sedimentary basins. We extend a fully implicit level set scheme for the simulation of hydraulic fracture growth to the case of anisotropic fracture toughness. We derive an analytical solution for the propagation of an elliptical hydraulic fracture in the toughness dominated regime—a shape which results from a particular form of toughness anisotropy. The developed numerical solver closely matches this solution as well as classical benchmarks for hydraulic fracture growth with isotropic toughness. We then quantify numerically the transition between the viscosity dominated propagation regime at early time—where the fracture grows radially—to the toughness dominated regime at large time where the fracture reaches an elliptical shape in the case of an elliptical anisotropy. The time scale at which the fracture starts to deviate from the radial shape and gets more elongated in the direction of lower toughness is in accordance with the viscosity to toughness transition time-scale for a radial fracture defined with the largest value of fracture toughness. Similarly, the toughness dominated regime is fully reached along the whole fracture front when the time gets significantly larger than the same transition time-scale defined with the lowest value of toughness. Using different toughness anisotropy functions, we also illustrate how the details of the complete variation of fracture toughness with propagation direction governs the final hydraulic fracture shape at large time. Our results highlight toughness anisotropy as a possible hydraulic fracture height containment mechanism as well as the need for its careful characterization beyond measurements in the sole material axes (divider and arrester) directions.

Appendices

A The elliptical hydraulic fracture—toughness dominated propagation solution

We derive here from the solution of a mode I uniformly pressurized elliptical fracture, the propagation solution for the growth of such fracture induced by injection of an inviscid fluid (zero viscosity). This corresponds to the toughness dominated regime of hydraulic fracture propagation. We consider an elliptical fracture propagating in the plane $(e_1,\,e_3)$ (thus with normal $e_2$, as per the configuration of Fig. 1.

The fracture width of an elliptical pure mode I fracture under a uniform pressure p perpendicular to the fracture surface is given by Green and Sneddon (1950):

$$\begin{aligned} w =\frac{4b\,p}{E^\prime \, E(k)}\left( 1-\frac{x_1^{2}}{a^{2}}-\frac{x_3^{2}}{b^{2}}\right) ^{\frac{1}{2}}, \end{aligned}$$

(25)

where $E^\prime$ is the plain strain elastic modulus, E(k) is the complete elliptical integral of the second kind, $k=\sqrt{1-b^{2}/a^{2}}$ is the eccentricity of the ellipse and $a,\;b$ are the major and minor axis lengths (in direction $e_1$ and $e_3$) respectively. Let us consider a point $(x_1,x_3)$ inside the elliptical fracture with $(x_1^f,x_3^f)$ as its closest point projection on the fracture front (see Fig. 17). Let $\rho$ be the distance between these two points and $\alpha$ be the angle inscribed by the line joining the two points on the major axis ($e_1$-axis in this case). Note that from elementary geometry, we know that this line is perpendicular to the tangent on the front at the point $(x_1^f,x_3^f)$, hence giving the propagation direction of the fracture. By using the ellipse property, we have

$$\begin{aligned} \tan \alpha =\frac{a^{2}}{b^{2}} \frac{x_3}{x_2} , \end{aligned}$$

(26)

we can thus write

$$\begin{aligned}&1-\left( \frac{x_1^{f}}{a}\right) ^2 -\left( \frac{x_3^f}{b}\right) ^2\nonumber \\&\quad =1-\frac{(x_1-\rho \cos \alpha )^{2}}{a^{2}}-\frac{(x_3-\rho \sin \alpha )^{2}}{b^{2}}\nonumber \\&\quad =\frac{2\rho x_1 \cos \alpha }{a^{2}}+\frac{2\rho x_3 \sin \alpha }{b^{2}}+O(\rho ^{2})\nonumber \\&\quad =\frac{2\rho x_1^{2}b^{2}}{a^{2}\sqrt{b^{4}x_1^{2}+a^{4}x_3^{2}}}+\frac{2\rho x_3^{2}a^{2}}{b^{2}\sqrt{b^{4}x_1^{2}+a^{4}x_3^{2}}}+O(\rho ^{2}).\nonumber \\ \end{aligned}$$

(27)

Fig. 17

The point $(x_1,x_3)$ inside the fracture and its closest point projection $(x_1^f,x_3^f)$ on the elliptical fracture front

Introducing $\beta$ such that

$$\begin{aligned} \begin{aligned} \left\{ \begin{matrix}x_1=a\cos \beta \\ x_3=b\sin \beta \end{matrix}\right. \end{aligned}, \end{aligned}$$

(28)

we obtain

$$\begin{aligned} \begin{aligned}&1-\left( \frac{x_1^{f}}{a}\right) ^2 -\left( \frac{x_3^f}{b}\right) ^2\\&\quad =2\rho \left( \frac{b^{2}\cos ^{2}\beta +a^{2}\sin ^{2}\beta }{a^{2}b^{2}}\right) ^{\frac{1}{2}}+O(\rho ^{2})\\&\quad =2\rho \left( \frac{\cos ^{2}\beta }{a^{2}}+\frac{\sin ^{2}\beta }{b^{2}}\right) ^{\frac{1}{2}}+O(\rho ^{2}). \end{aligned} \end{aligned}$$

(29)

Note that from the ellipse property (26), the relation between $\beta$ and $\alpha$ is given by

$$\begin{aligned} \tan \alpha =\frac{a}{b}\tan \beta . \end{aligned}$$

(30)

By taking the limit $\rho \rightarrow 0$ and ignoring second order terms, we finally obtain the following asymptote of the fracture width perpendicular to the elliptical fracture front at $(x_f, y_f)$:

$$\begin{aligned} w =\frac{4b\,p}{E^\prime \, E(k)}\sqrt{2\rho }\left( \frac{\cos ^{2}\beta }{a^{2}}+\frac{\sin ^{2}\beta }{b^{2}}\right) ^{\frac{1}{4}} \text { for } \rho \ll 1 \end{aligned}$$

(31)

Comparing the above expression with the mode I Linear Elastic Fracture Mechanics (LEFM) tip asymptote, when $\rho \ll 1$

$$\begin{aligned} \begin{aligned} w =4\sqrt{\frac{2}{\pi }}\frac{K_I}{E^\prime }\sqrt{\rho }, \end{aligned} \end{aligned}$$

(32)

the mode I stress intensity factor $K_I$ can be written as a function of the elliptical parameter $\beta$ as

$$\begin{aligned} \begin{aligned} K_I=\frac{p \sqrt{\pi b}}{E(k)}\left( \sin ^{2}\beta +\frac{b^{2}}{a^{2}}\cos ^{2}\beta \right) ^{1/4} \end{aligned} . \end{aligned}$$

(33)

In order for such an elliptical fracture to keep its aspect ratio $\gamma =a/b$ during its growth, the mode I stress intensity factor must be equal to the fracture toughness at all points along the fracture front: $K_I=K_c$. In other words, for an elliptical fracture to remain elliptical, the material fracture toughness must exhibit a dependence on the propagation direction, i.e. dependent on the angle $\alpha$ between the local propagation direction and the $e_1$-axis. From the expression of the mode I stress intensity factor for a uniformly pressurized elliptical fracture (33), we obtain the following dependence of the material toughness for the fracture to remain elliptical:

$$\begin{aligned} \begin{aligned} K_c=K_{c,3} \left( \sin ^{2}\beta +\frac{1}{\gamma ^{2}}\cos ^{2}\beta \right) ^{1/4}, \end{aligned} \end{aligned}$$

(34)

with $\beta = \arctan ( \gamma \tan \alpha )$. The ellipse aspect ratio $\gamma =a/b$ is directly linked to the square of the ratio of material toughness in direction $e_3$ and $e_1$:

$$\begin{aligned} \gamma = a/b = \left( \frac{K_{c,3}}{K_{c,1}}\right) ^2 \end{aligned}$$

(35)

Assuming such a direction dependence for the fracture toughness ensure that the elliptical fracture will grow with a constant aspect ratio $\gamma$. The condition $K_I=K_c(\alpha )$ all along the front therefore gives the following relation between net pressure and the minor axis length by equalizing Eqs. 33 and 34:

$$\begin{aligned} p(t)=\frac{K_{c,3} E(k)}{\sqrt{\pi b(t)}}. \end{aligned}$$

(36)

By integrating Eq. 25 over the elliptical region, the volume of the fracture is given by

$$\begin{aligned} {\begin{matrix}V_{frac}=\iint _{A}\;w(x_1,x_2) \text { d}x_1\text { d}x_2=\frac{8\,a\,b^{2}\pi p }{3 E^\prime E(k)}\end{matrix}} \end{aligned}$$

(37)

Substituting Eq. (36), we obtain

$$\begin{aligned} V_{frac}(t)=\frac{8 \gamma K_{c,3} \pi ^{\frac{1}{2}}b(t)^{5/2}}{3 E^\prime } \end{aligned}$$

(38)

Finally, assuming an incompressible fluid and a constant injection rate $Q_o$, the evolution of the minor axis b at any given time t is given obtained as

$$\begin{aligned} b(t)=\left( \frac{3Q_o\,t\gamma E^\prime }{8 K_{c,3}\sqrt{\pi }}\right) ^{2/5} \end{aligned}$$

(39)

and $a(t)=\gamma \, b(t)$.

B Numerical verification: viscosity to toughness dominated transition for the isotropic toughness case

To demonstrate that our numerical scheme correctly captures the transition from the viscosity to the toughness dominated regime, we consider a radial fracture propagating in a medium with isotropic toughness ($K_{c,1}=K_{c,3}=K_{c}$)—which thus stay of radial shape. We demonstrate numerically its evolution from viscosity to toughness dominated propagation over approximately eight decades of time, and compare our results to existing analytical solutions. The parameters used in this test case are listed in Table 1. Figure 18 shows the evolution of the fracture radius (top) and fracture width at the injection point (bottom) with the dimensionless time $\tau =t/t_{mk}$. It can be seen that the solution evaluated by ILSA matches very where both the viscosity and toughness dominated analytical solutions.

Table 1 Parameter values used for the test case in Appendix B

Fig. 18

Transition from viscosity dominated to toughness dominated regime for a radial hydraulic fracture: time evolution of the fracture radius (top) and fracture width at the injection point (bottom)