Comparative verification of discrete and smeared numerical approaches for the simulation of hydraulic fracturing

Abstract

The numerical treatment of propagating fractures as embedded discontinuities is a challenging task for which an analyst has to select a suitable numerical method from a range of options. Since their inception in the mid-80s, smeared approaches for fracture simulation such as non-local damage, gradient damage or more lately phase-field modelling have steadily gained popularity. One of the appeals of a smeared implicit fracture representation, the ability to handle complex topologies with unknown crack paths in relatively coarse meshes as well as multiple-crack interaction and multiphysics, is a fundamental requirement for the numerical simulation of hydraulic fracturing in complex situations which is technically more difficult to achieve with many other methods. However, in hydraulic fracturing simulations, not only the prediction of the fracture path but also the computation of fracture width and propagation pressure (frac pressure) is crucial for reliable and meaningful applications of the simulation tool; how to determine some of these quantities in smeared representations is not immediately obvious. In this study, two of the most popular smeared approaches of recent, namely non-local damage and phase-field models, and an approach in which the solution space is locally enriched to capture a strong discontinuity combined with a cohesive-zone model are verified against fundamental hydraulic fracture propagation problems in the toughness-dominated regime. The individual theoretical foundations of each approach are discussed and differences in the treatment of physical and numerical properties of the methods when applied to the same physical problems are highlighted through examples.

Appendices

Appendix

A Rescaling of the phase-field energy functional (non-dimensionalization)

The scaling of Eq. (23) to achieve a non-dimensional format proceeds as follows. Let $\psi _0$, $x_0$, $u_0$, and $\tau _0 =p_0$ be scaling factors, and

$$\begin{aligned} \psi&= \psi _0\tilde{\psi } \end{aligned}$$

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$$\begin{aligned} {\mathbf {x}}&= x_0\tilde{{\mathbf {x}}} \end{aligned}$$

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$$\begin{aligned} {\mathbf {u}}&= u_0\tilde{{\mathbf {u}}} \end{aligned}$$

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$$\begin{aligned} p&= p_0\tilde{p} \end{aligned}$$

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$$\begin{aligned} \bar{{\mathbf {t}}}&= \tau _0\tilde{{\mathbf {t}}} \quad \mathrm {and} \quad \varrho {\mathbf {b}} = \tau _0\tilde{\varrho {\mathbf {b}}} \end{aligned}$$

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where $\tilde{()}$ is a dimensionless variable. Thus, Eq. (23) becomes

$$\begin{aligned} \begin{aligned} E(\mathbf {u},d,p)&= \psi _0 u_0^{2} x_0^{n-2} \int _{\varOmega } d^{2} \tilde{\psi }^{+}(\tilde{\mathbf {u}})+\tilde{\psi }^{-}(\tilde{\mathbf {u}})\, \mathrm {d}\varOmega -\tau _0 x_0^{n-1} u_0 \int _{\partial _{\mathrm {N}} \varOmega } \tilde{\mathbf {t}}\cdot \tilde{\mathbf {u}} \, \mathrm {d}\varGamma -\\&\quad - \tau _0 x_0^{n-1} u_0 \int _\varOmega \tilde{\varrho {\mathbf {b}}}\cdot \tilde{\mathbf {u}} \, \mathrm {d}\varOmega + x_0^{n-1} \frac{G_\mathrm {c}}{4c_w} \int _\varOmega \left( \frac{w(d)}{\ell } + \ell \left| \nabla d \right| ^{2} \right) \, \mathrm {d}\varOmega \\&\quad + p_0 x_0^{n-1} u_0 \int _\varOmega \tilde{p} \tilde{\mathbf {u}} \cdot \nabla d \, \mathrm {d}\varOmega \\ \end{aligned} \end{aligned}$$

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Dividing both sides by $\psi _0 u_0^{2} x_0^{n-2}$,

$$\begin{aligned} \begin{aligned} \tilde{E}(\tilde{\mathbf {u}},d,\tilde{p})&= \int _{\varOmega } d^{2} \tilde{\psi }^{+}(\tilde{\mathbf {u}})+\tilde{\psi }^{-}(\tilde{\mathbf {u}}) \, \mathrm {d}\varOmega -\frac{\tau _0 x_0}{u_0 \psi _0} \int _{\partial _{\mathrm {N}} \varOmega } \tilde{\mathbf {t}}\cdot \tilde{\mathbf {u}} \, \mathrm {d}\varGamma -\\&\quad - \frac{\tau _0 x_0}{u_0 \psi _0} \int _\varOmega \tilde{\varrho {\mathbf {b}}}\cdot \tilde{\mathbf {u}} \, \mathrm {d}\varOmega + \frac{G_\mathrm {c} x_0}{4c_w \psi _0 u_0^{2}} \int _\varOmega \left( \frac{w(d)}{\ell } + \ell \left| \nabla d \right| ^{2} \right) \, \mathrm {d}\varOmega \\&\quad + \frac{p_0 x_0}{u_0 \psi _0} \int _\varOmega \tilde{p} \tilde{\mathbf {u}} \cdot \nabla d \, \mathrm {d}\varOmega \\ \end{aligned} \end{aligned}$$

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and setting

$$\begin{aligned} x_0= & {} \frac{\psi _0 u_0^{2}}{G_\mathrm {c}} \end{aligned}$$

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$$\begin{aligned} \tau _0= & {} p_0= \frac{\psi _0 u_0}{x_0} \end{aligned}$$

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yields the numerically favourable formulation

$$\begin{aligned} \begin{aligned} \tilde{E}(\tilde{\mathbf {u}},d,\tilde{p})&= \int _{\varOmega } d^{2} \tilde{\psi }^{+}(\tilde{\mathbf {u}})+\tilde{\psi }^{-}(\tilde{\mathbf {u}})\, \mathrm {d}\varOmega - \int _{\partial _{\mathrm {N}} \varOmega } \tilde{\mathbf {t}}\cdot \tilde{\mathbf {u}} \, \mathrm {d}\varGamma - \\&\quad - \int _\varOmega \tilde{\mathbf {t}}\cdot \tilde{\mathbf {u}} \, \mathrm {d}\varOmega + \frac{1}{4c_w} \int _\varOmega \left( \frac{w(d)}{\ell } + \ell \left| \nabla d \right| ^{2} \right) \, \mathrm {d}\varOmega + \int _\varOmega \tilde{p} \tilde{\mathbf {u}} \cdot \nabla d \, \mathrm {d}\varOmega \\ \end{aligned} \end{aligned}$$

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