Early-Time Shut-In for Plane-Strain Hydraulic Fractures

The quasi-static balance of momentum relates the fracture opening w(x, t) and the net pressure \(p_{{f}}(x,t)-\sigma _o\) (the difference between the fluid pressure and the far field confining stress) via a boundary integral equation (see for example Hills et al. 1996). In view of the problem symmetry, it can be written on one wing of the fracture:where \(\ell\) is the half-length of the plane-strain fracture.

The propagation mechanism is modulated by a constant fracture toughness following the linear elastic fracture mechanics by assuming a negligible non-linear process zone. The stress intensity factor \(K_\textrm{I}\) must equal the fracture toughness \(K_\textrm{Ic}\) at any given time during the propagation and can only stay below the toughness when the fracture is at arrest. As a consequence, the following condition must be satisfied at all times.\(K_\textrm{I}=K_\textrm{Ic}\) can be alternatively expressed in the form of the classical square-root asymptote near the fracture tip (Rice 1968).

Assuming zero compressibility, the fluid mass conservation in the elastic deformable fracture reduces towhere \(\ell _{{f}}(t)\) denotes the current fluid front position, and q(x, t) the local fluid flux inside the fracture. The fluid is injected at the fracture center under a constant injection rate \(Q_o\) (in \(\textrm{m}^2/\textrm{s}\) under plane-strain conditions) until a shut-in occurs at \(t=t_s\). The flux entering one wing of the fracture thus writes:which can be translated into the form of the global fluid volume balance by integrating the continuity Eq. (7) for the fluid:with

Under the assumption of a Newtonian fluid and a laminar flow inside the fracture, the fluid flux can be approximated by Poiseuille’s law:

The fluid front lags behind the fracture front with its velocity \(\dot{\ell }_{{f}}\) equal to the mean fluid velocity q/w at \(x=\ell _{{f}}\) (Stefan condition):The fluid is vaporized inside the fluid lag with a cavitation pressure \(p_{cav}\) much smaller than the liquid pressure \(p_{{f}}\) in the fluid-filled part and the in-situ confining stress \(\sigma _o\).

The fracture opening and fluid flux are zero at the fracture tip.

One models the fluid lag nucleation starting from a negligibly small fracture. The initial conditions correspond to the linear elastic solutions of a uniformly pressurized static flaw with a fluid pressure slightly larger than the in-situ stress \(\sigma _o\) (\((p_{{f}}-\sigma _o)/\sigma _o \ll 1\)).

When the fracture toughness dominates the propagation at the time of shut-in (\(\mathcal {K}_m \gg 1\)), the fluid pressure is uniform inside the fracture with the stress intensity factor equal to the fracture toughness. Any further fracture extension will lead to a drop in the stress intensity factor and an immediate arrest. The fracture behaviour (the pulse toughness-dominated solution or the \(K^{[V]}\)-vertex solution) thus corresponds to the solution of a linear elastic fracture under uniform far-field load. The fracture half-length \(\ell _{{a}}\), the fracture opening \(w_{{a}}\) and the net pressure at arrest write as follows.In absence of a fluid lag, the fracture propagates without arrest when the fracture toughness is zero \(K_{\textrm{Ic}}=0\). The solution, denoted as the pulse viscosity dominated solution or the \(M^{[V]}\)-vertex solution, is self-similar and has been obtained numerically using the Gauss–Chebyshev quadrature and Barycentric interpolation techniques in Liu and Lu (2023). One recalls here the expression for the fracture half-length.The \(M^{[V]}\)-vertex solution implies that the fracture half-length evolves with time in a way that \(\ell \sim t^{1/6}\). Note that such a time evolution of the fracture length may also appear during the transition to the final fracture arrest for \(K_\textrm{Ic}> 0\) as shown in Fig. 2.

In absence of a fluid lag, the fracture with a finite non-zero toughness continues to grow after shut-in, with the elastic energy stored prior to the shut-in partially balanced by the creation of new fracture surfaces. Such post-shut-in fracture growth stops when the fracture length reaches the \(K^{[V]}\)-solution at \(\ell =\ell _{{a}}\). Approximating the fracture half-length \(\ell _s\) at shut-in with the self-similar solutions (Garagash and Detournay 2005; Garagash 2006a) (by assuming that the plane-strain fracture is driven under a constant injection rate before shut-in), one defines \(1-\ell _s/\ell _{{a}}\) as the potential fracture extension after the stop of fluid injection.where \(\gamma _m\) is the self-similar solution for constant injection rate in the viscosity dominated scaling (Garagash and Detournay 2005; Garagash 2006a) and \(L_m\) the length scale in the viscosity scaling as shown in Table 1. This ratio of the fracture length \(\ell _s/\ell _{{a}}\) is solely a function of the dimensionless toughness \(\mathcal {K}_m\). As illustrated in Fig. 2a, when \(\mathcal {K}_m \gtrapprox 2.596\), the fracture half-length \(\ell _s\) is equal to or larger than 95% of the final arrest fracture half-length \(\ell _{{a}}\), and the fracture front could barely advance after the stop of fluid injection.

One further investigates numerically the post-shut-in growth of a plane-strain hydraulic fracture and its subsequent arrest employing a spectral solver. One displays in Fig. 2b the evolution of the fracture half-length as a function of \(t/t_s\) for different values of \(\mathcal {K}_m\). For small values of \(\mathcal {K}_m\), the hydraulic fracture is in the viscosity dominated regime when the shut-in occurs. One observes a transition from the small toughness solution (with \(\ell \sim t^{2/3}\)) (Garagash 2006a) to the pulse viscosity dominated solution (28) (with \(\ell \sim t^{1/6}\)) (Liu and Lu 2023). Such a transition is much shortened for larger values of \(\mathcal {K}_m\), where more energy dissipation is owed to the creation of fracture surfaces. For \(\mathcal {K}_m \gtrapprox 4.3\) (Garagash 2006a), the fracture growth is dominated by fracture toughness at the time of shut-in, leading to an immediate arrest. A diagram describing such post-shut-in propagation of a plane-strain hydraulic fracture with zero fluid lag is illustrated in Fig. 1 together with that of a continuous fluid injection.

In this section, one focuses on the post-shut-in growth accounting for the presence of a non-negligible fluid lag. The fluid lag can be significant for small and intermediate values of dimensionless toughness especially when the shut-in occurs at the early time of hydraulic fracture growth. To better describe the fracture behaviour after the shut-in, one defines the volume efficiency coefficient as the ratio between the total volume of the created fracture V and the injected fluid volume \(V_{{f}}\).One also defines the length efficiency coefficient \(\ell /\ell _{{a}}\) as the ratio between the fracture half-length \(\ell\) and the fracture half-length at arrest \(\ell _{{a}}\) under the pulse injection.

The post-shut-in fracture behaviour is a function of both the dimensionless toughness and dimensionless shut-in time. One shows in Fig. 3 the volume and length efficiency parameters upon shut-in as a function of these two dimensionless parameters. Note that these results only describe the fracture growth at the shut-in time, neither before nor after.

The volume efficiency is always greater than one owing to a non-zero volume of the fluid cavitation. A more significant effect of the viscous fluid flow or a larger dimensionless toughness \(\mathcal {K}_m\) tends to lead to a more efficient injection with the fracture volume larger than the volume of injected fluid. The volume efficiency decreases towards one as the shut-in occurs later, and the fluid front tends to catch up with the fracture front until the fracture is fully filled with fluid.

The fracture length efficiency also decreases as the shut-in occurs later (Fig. 3). When the effect of viscous fluid flow is more dominant during the fracture growth (smaller dimensionless toughness \(\mathcal {K}_m\) values), the decrease in length efficiency becomes more sensitive to the increase of the shut-in time \(t_s/t_{om}\). When the fluid lag vanishes at \(t_s/t_{om} \approx 1\), \(\ell _s/\ell _{{a}}\) converges towards zero-lag solutions, and one obtains the same conclusion as shown in Fig. 2a: a smaller \(\mathcal {K}_m\) leads to a larger difference in fracture extension between upon shut-in and at arrest.

To gauge the implications for real systems, one considers typical values relevant to the laboratory- and field-scale hydraulic fractures and report in Table 2 the corresponding characteristic scales and dimensionless numbers for different types of injection.

Laboratory hydraulic fracturing injection is often performed in samples with limited dimensions under limited confinement. Assuming relatively low confinement applied on the rock samples, one considers here the injection of a highly viscous and a barely viscous fluid [with experimental conditions similar to those in Liu et al. (2020) and Liu and Lecampion (2021, 2023)] leading respectively to viscous flow/fluid lag dominated growth regime (Lab injection A) and fracture toughness dominated growth regime (Lab injection B). When the fracture toughness dominates the fracture propagation (Lab injection B), the fracture is uniformly pressurized and will immediately arrest upon shut-in. Otherwise, the fracture tends to present a temporary arrest then followed by a restart of the propagation. Note that the restart of the fracture propagation may occur quite late after the shut-in. Direct observation of such temporary arrest has rarely been reported in the laboratory (Liu 2021; Liu and Lecampion 2023) since it necessitates long-period monitoring of the fracture behaviour/dimensions after the fluid shut-in. Apart from the temporary arrest, an immediate arrest of the fracture front may also likely occur in lag/viscosity dominated experiments like Lab injection A: for an early shut-in \(t_s/t_{om} \ll 1\), the fracture front may stop growing immediately at the stop of injection, and the fluid front gradually evolves towards the fracture front until it catches up with the fracture tip or reaches the sample boundary.

Most in-situ industrial injections are performed at a depth from 1.5 km down to 4 km, corresponding to a minimum confining stress of around tens of MPa. Under such confinement, one evaluates the characteristic scales for two injection scenarios: a micro-HF test with the injection of a less-viscous type of slick water, and a well stimulation operation with the injection of a more-viscous type of slick water. As shown in Table 2, in both cases, the timescale \(t_{om}\) is small and indicates the presence of a negligible fluid lag. The temporary arrest behaviour, which necessitates the presence of a fluid lag, is possibly excluded in typical industrial injections. For the micro-HF test, the injection is often performed at a small injection rate, and the fracture growth is characterized by a dimensionless toughness \(\mathcal {K}_m\) of less than two. As a result, the fracture either continues to propagate after the shut-in or stops growing immediately when the fracture toughness dominates the propagation at the stop of injection (which happens under the conditions of a smaller injection rate, less viscous fluid, and rocks with larger fracture toughness). Different from the micro-HF test, the well stimulation is characterized by a larger injection rate and a longer injection period. The viscosity likely governs the fracture growth upon shut-in, and the fracture may directly continue propagating afterwards. In terms of the injection efficiency (30), the fluid lag vanishes fast in both scenarios, and the fracture volume most likely equals to or becomes smaller than the volume of injected fluid due to a probable leak-off. The injection efficiency is thus probably smaller than one for most industrial injections.

When neglecting buoyancy, magma-driven geological structures, such as sills and dikes, can be approximated by a hydraulic fracture driven by a pulse fluid injection in the plane-strain state (Rubin 1993; Bunger and Cruden 2011). The formation of these geological structures is often characterized by a significant cavitation between the magma front and the fracture front due to the large viscosity of magma and little leak-off (Rubin 1993). Previous studies point out that dikes present a wide range of distribution in the emplacement scaling relations and aspect ratios (Olson 2003; Scholz 2010; Olson and Schultz 2011; Rivalta et al. 2015). This has been attributed to the scale-related fracture apparent toughness of geological formations, the interplay between the process zone and roughness-induced fluid flow deviation (from Poiseuille’s law), and the mixed mode fracturing at small scales (Liu et al. 2019; Liu and Lecampion 2021, 2022; Arachchige et al. 2022). In the following, assuming constant fracture energy, one focuses on the influence of the fluid lag on the dike emplacement. As shown in Table 2, one considers two scenarios related to the magma dike formation under relatively low tectonic stresses. One scenario is characterized by a short-period injection of basaltic magma with a large flux (Magma dike A), and the other is characterized by a long-period injection of a low-end viscous magma (similar to that of Kimberlite) with a small flux (Magma dike B) (Rivalta et al. 2015). For both scenarios, the growth of magma dikes at the shut-in is very likely dominated by the viscous fluid flow and the fluid lag can not be neglected at least during the pulse injection stage. In the case of a very short injection with an extremely large flux, the dike may probably present a temporary arrest. Otherwise, it tends to propagate continuously after the shut-in followed by a permanent arrest. Such a temporary arrest may last for quite a long time, and the magma may cool off and solidify before the restart of the propagation. It is thus very likely that the final geometry of the dikes deviates from the square-root emplacement scaling relation \(w(0) \sim \sqrt{\ell }\) and presents a different aspect ratio. This partly explains the wide range of dike emplacement/aspect ratio observed in nature. As a result, fracture toughness estimation based on linear elastic fracture mechanics and dikes’ emplacement may not be precise. Further investigation accounting for the injection/shut-in history and fluid lag is necessary to better decipher the relation between the emplacement scaling and fracture toughness.

In this study, one assumes an impermeable medium and zero fluid leak. When accounting for a permeable medium, the fracture growth may present more complex behaviours due to fluid leak-off. Adachi and Detournay (2008), Hu and Garagash (2010) and Chen et al. (2018) account for the fluid leaking into the surrounding medium in the case of zero fluid shut-in for a plane-strain hydraulic fracture. They approximate the fluid leak-off using Carter’s law (Howard and Fast 1957; Lecampion et al. 2018), which is a 1-D approximation of the fluid diffusion into the medium. The local fluid continuity equation thus writes as follows.where \(t_0\) is the arrival time of the fluid front at the position x and \(C^{\prime }=2 C_L\) is the effective leak-off coefficient with \(C_L\) the fluid loss constant which can be calibrated from hydraulic fracture injection tests. Such fluid leak-off is a time-dependent process and introduces another timescale \(t_{\tilde{m}m}\) in the dimensional analysis (Adachi and Detournay 2008; Hu and Garagash 2010; Chen et al. 2018; Peirce and Detournay 2022a):This timescale characterizes the transition between storage growth regimes (OMK) and leak-off regimes (\(\tilde{O}\tilde{M}\tilde{K}\)), where \(\tilde{O}\tilde{M}\tilde{K}\) correspond to the leak-off regimes characterized respectively by a significant fluid lag and the dominant effects of viscous fluid flow and fracture surface creation. In the absence of fluid shut-in, the fracture growth transitions between the six distinct growth regimes (OMK‐\(\tilde{O}\tilde{M}\tilde{K}\)) depending on the interplay among \(\mathcal {K}_m\), \(t_{om}\) and \(t_{m\tilde{m}}\). Detailed information on these fracture growth transitions can be found in Adachi and Detournay (2008), Hu and Garagash (2010) and Chen et al. (2018).

In the presence of fluid shut-in, the fluid leak-off tends to favour an immediate fracture arrest upon shut-in, and may even result in the disappearance of the temporary arrest and the continuous propagation after the stop of fluid injection. In addition, as the stress intensity factor at the fracture tip drops from the fracture toughness \(K_\textrm{Ic}\) to zero, fracture deflation and recession may occur, as previously discussed in Peirce and Detournay (2022a) assuming zero fluid lag. The post-shut-in fracture behaviour thus results from the interplay among the dimensionless toughness \(\mathcal {K}_m\), the shut-in time \(t_s\), the timescale characterizing the coalescence of the fracture and fluid fronts \(t_{om}\), and the leak-off timescale \(t_{m\tilde{m}}\). Only when the leak-off timescale is significantly greater than the other timescales (\(t_{m \tilde{m}} \gg t_s\), \(t_{m \tilde{m}} \gg t_{om}\)) can the fracture growth be approximated by the zero leak-off solutions reported in this study.