4.1 Near-field asymptote

The near-tip asymptote of the fracture opening is governed by linear elastic fracture mechanics (LEFM). In this region, the fracture width evolves with the square root of distance to the tip and the toughness characteristic length scale $\ell _{k}$ (see (4.1)):

which takes the following form in the scaling previously defined:

For the strictly zero toughness case ( $\unicode[STIX]{x1D705}=0$ ), the width evolves as $\unicode[STIX]{x1D709}^{3/2}$ which corresponds to the higher-order term in the linear elastic fracture mechanics asymptotic development (see e.g. Rice Reference Rice1968; Garagash Reference Garagash2009). In the complete problem, the extent of this LEFM region depends on the viscous length scale $L_{\unicode[STIX]{x1D707}_{o}}$ as well as the value of the dimensionless toughness (i.e. the corresponding lag size).

4.2 Far-field asymptotes

In the case where both the fluid lag and toughness are negligible ( $\unicode[STIX]{x1D705}\ll 1,\unicode[STIX]{x1D6EC}\ll 1$ ), the behaviour of the solution is governed by the coupling between lubrication flow and elasticity which yields a different power-law dependence of the opening with distance from the tip. The solution for a power-law rheology (for zero lag and zero toughness) have been obtained in Desroches et al. (Reference Desroches, Detournay, Lenoach, Papanastasiou, Pearson, Thiercelin and Cheng1994). Such a solution corresponds to a far-field asymptote valid at a distance from the tip where the effect of dimensionless toughness and the presence of the lag vanishes (see e.g. Garagash & Detournay (Reference Garagash and Detournay2000), Garagash (Reference Garagash2009), Garagash et al. (Reference Garagash, Detournay and Adachi2011) for discussion). For a Carreau rheology, as the average shear rate $V/w$ decreases as one moves away from the fracture tip, we can expect to see two or more viscosity dominated asymptotes. Far away from the fracture tip, i.e. for very low shear rate, one should recover the viscosity asymptote for a Newtonian with the low shear rate viscosity (see figure 1), i.e.

where

This $m_{o}$ -asymptote in the scaling previously defined in § 3 has the following form:

As we move closer to the fracture tip, the fluid will exhibit shear thinning. The fracture opening should thus eventually follow the viscosity asymptote for a power-law fluid. We will refer to this power-law asymptote – first derived by Desroches et al. (Reference Desroches, Detournay, Lenoach, Papanastasiou, Pearson, Thiercelin and Cheng1994) – as $m_{n}$ :

where

We can re-express this asymptote using the expression of the power-law consistency $M$ as function of the Carreau rheological parameters (see (1.1)). Moreover, using the low shear rate scaling defined in the § 3, such $m_{n}$ asymptote $\displaystyle (\unicode[STIX]{x1D705}=\unicode[STIX]{x1D6EC}=0)$ reads:

where

Finally, as we move closer to the fracture tip, for larger average shear rate, it may well be that another Newtonian viscous asymptote linked to the Newtonian behaviour at large shear rate $\unicode[STIX]{x1D707}_{\infty }$ could be observed (see (4.12)). In the low shear rate scaling, this large shear rate asymptote is given by:

or alternatively in dimensionless form in the lag scaling:

The evolution of the complete solution between these different asymptotes as well as the extent of the regions where these asymptotes may be valid will depend on the values of the different dimensionless parameters governing the problem, namely $\unicode[STIX]{x1D705},\unicode[STIX]{x1D6FC},n$ and $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ .

6.1 Effect of dimensionless toughness $\unicode[STIX]{x1D705}$

We first investigate the effect of dimensionless toughness $\unicode[STIX]{x1D705}$ for a given set of rheological dimensionless parameters representative of a Carreau fluid: $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3},n=0.5,$ and $\unicode[STIX]{x1D6FC}=10^{2}$ . The profile of dimensionless fracture opening with distance from the tip for different values of dimensionless lag size/ dimensionless toughness is displayed in figure 4, while the corresponding dimensionless net pressure profiles (in semi-log) and the apparent viscosity $\unicode[STIX]{x1D6E4}$ are displayed in figures 5 and 6. We can directly observe the extent of the lag on the net pressure profiles which depart from the value $\unicode[STIX]{x1D6F1}=-1$ at the fluid front $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D6EC}$ . The relation between the dimensionless toughness and the dimensionless lag size for these simulations are better grasped in figure 7. The dimensionless lag size $\unicode[STIX]{x1D6EC}$ is decreasing function of dimensionless toughness $\unicode[STIX]{x1D705}$ . This is similar to the Newtonian case (see Garagash & Detournay Reference Garagash and Detournay2000) with the difference that for a given value of dimensionless toughness, due to shear thinning, the dimensionless lag is always smaller compared to the Newtonian case (see figure 10 for more simulations with different shear-thinning index $n$ , and figure 8 for different transition shear-stress $\unicode[STIX]{x1D6FC}$ ).

Figure 4.

Dimensionless fracture opening $w/\unicode[STIX]{x1D716}L_{\unicode[STIX]{x1D707}_{o}}$ along the fracture for $\unicode[STIX]{x1D705}=0,0.002,0.008,0.043,0.33,0.74$ (with corresponding lag size $\unicode[STIX]{x1D6EC}=0.049,0.048,0.047,0.041,0.008,8.3\times 10^{-8}$ ) in log–log scale for $\unicode[STIX]{x1D6FC}=10^{2}$ , $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3}$ , and $n=0.5$ . The dashed lines correspond to the asymptotic solution in the toughness dominated regime ( $\unicode[STIX]{x1D705}$ -asymptote), power-law viscosity regime ( $m_{n}$ -asymptote) and low shear rate Newtonian viscosity regime ( $m_{o}$ -asymptote).

The dimensionless opening profiles (figure 4) also evolve in a similar way to that in the Newtonian case with a region dominated by the LEFM toughness $\unicode[STIX]{x1D705}$ -asymptote (4.3) near the fracture tip (for $x\ll L_{\unicode[STIX]{x1D707}_{o}}$ ) and a region in the far field (for $x\gg L_{\unicode[STIX]{x1D707}_{o}}$ ) dominated by the low shear rate viscosity asymptote $m_{o}$ (4.6). However, here due to the shear-thinning nature of the Carreau rheology, an intermediate region following the power-law dominated asymptote $m_{n}$ can be observed for intermediate distances between the LEFM near-tip region and the far-field low shear rate viscosity region. It is worth noting the particularity of the zero toughness case ( $\unicode[STIX]{x1D705}=0$ ) for which the dimensionless lag is maximum and the opening in the near-tip LEFM region evolves as $\unicode[STIX]{x1D709}^{3/2}$ (see figure 4).

Figure 5.

Dimensionless apparent viscosity $\unicode[STIX]{x1D6E4}$ along the fracture starting from the fluid front $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D6EC}$ for $\unicode[STIX]{x1D705}=0$ ( $\unicode[STIX]{x1D6EC}=0.049$ ) and $\unicode[STIX]{x1D705}=0.74$ ( $\unicode[STIX]{x1D6EC}=8.3\times 10^{-8}$ ) in log–log scale for $\unicode[STIX]{x1D6FC}=10^{2}$ , $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3}$ , and $n=0.5$ .

Figure 6.

Dimensionless fluid pressure $p/\unicode[STIX]{x1D70E}_{o}$ along the fracture for $\unicode[STIX]{x1D705}=0,0.002,0.008,0.043,0.33,0.74$ in semi-log scale for $\unicode[STIX]{x1D6FC}=10^{2}$ , $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3}$ , and $n=0.5$ . The dashed lines correspond to the asymptotic solution of pressure in the power-law viscosity regime ( $m_{n}$ -asymptote) and in the Newtonian viscosity regime ( $m_{o}$ -asymptote).

The same evolution can be seen in the net pressure profiles (figure 6). The different viscosity regimes as function of distance from the tip are also visible in figure 5 where the dimensionless apparent viscosity $\unicode[STIX]{x1D6E4}$ is plotted along the fracture for the two cases $\unicode[STIX]{x1D705}=0$ ( $\unicode[STIX]{x1D6EC}=0.049$ ) and $\unicode[STIX]{x1D705}=0.74$ ( $\unicode[STIX]{x1D6EC}=8.3\times 10^{-8}$ ). The largest shear rate and therefore the lowest tangent apparent viscosity is always located at the fluid front. For the largest lag case, a smaller shear-thinning region can be observed (see figure 5). On the other hand, for a very small lag ( $\unicode[STIX]{x1D705}=0.74$ ), the fluid strongly shear thin, although the value of the tangent viscosity at the fluid front ( $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D6EC}$ ) remains larger than the large shear rate value $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ . It is important to note that the large shear rate asymptote is absent for all the simulations performed with this given set of rheological parameters ( $n=0.5$ , $\unicode[STIX]{x1D6FC}=10^{2}$ , and $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3}$ ).

Figure 7.

Extent of the regions where the numerical solution can be approximated by the different asymptotes ( $\unicode[STIX]{x1D705}$ -asymptote,  $m_{n}$ -asymptote and $m_{o}$ -asymptote) with accuracy of $10^{-2}$ as function of dimensionless toughness with $\unicode[STIX]{x1D6FC}=10^{2},\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3},n=0.5$ . The position of the fluid lag is also displayed for comparisons.

Figure 8.

Dimensionless lag length $\unicode[STIX]{x1D706}/L_{\unicode[STIX]{x1D707}_{o}}$ versus the dimensionless toughness $\unicode[STIX]{x1D705}$ for a Carreau rheology for different values of $\unicode[STIX]{x1D6FC}$ ( $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3},n=0.5$ ). The corresponding results for a power-law rheology are also displayed for $\unicode[STIX]{x1D6FC}=10^{5}$ and $\unicode[STIX]{x1D6FC}=10^{3}$ for comparison.

The relative extent of these different asymptotic regions depends on the values of the dimensionless problem parameters. In order to picture the variation of the size of these asymptotic regions as function of dimensionless toughness $\unicode[STIX]{x1D705}$ , we determine from our numerical results the spatial locations ( $\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D705}}$ , $\unicode[STIX]{x1D709}_{n}$ and $\unicode[STIX]{x1D709}_{o}$ ) where our numerical results for the fracture opening are within 1 % of these different asymptotes. The results are displayed in figure 7 for that same set of parameters ( $\unicode[STIX]{x1D6FC}=10^{2}$ , $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3}$ , and $n=0.5$ ) where we see that the extent of the power-law and low shear rate viscosity regions do not significantly change with dimensionless toughness while the LEFM near-tip region shrinks when the dimensionless toughness decreases (and the lag size increases) as expected. We also see that the fluid lag region is always localized in the toughness asymptotic region (for large $\unicode[STIX]{x1D705}$ ) – or in the transition between the toughness and power-law region (small $\unicode[STIX]{x1D705}$ ).

6.2 Effect of the dimensionless transition shear stress $\unicode[STIX]{x1D6FC}$

The value of the dimensionless transition shear stress ratio $\unicode[STIX]{x1D6FC}$ reflects the extent of the low shear rate plateau: $\unicode[STIX]{x1D6FC}$ is inversely proportional to the critical shear rate $\dot{\unicode[STIX]{x1D6FE}}_{c}$ (see (3.8)). For large $\unicode[STIX]{x1D6FC}$ , the fluid shear thins faster and as a result the dimensionless lag is getting smaller for a similar value of dimensionless toughness. This can be clearly seen in figure 8 where the $\unicode[STIX]{x1D6EC}-\unicode[STIX]{x1D705}$ evolution is plotted for different values of $\unicode[STIX]{x1D6FC}$ for $n=0.5$ and $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3}$ .

More interestingly, the structure of the solution also changes for larger value of $\unicode[STIX]{x1D6FC}$ . We present in figure 9 the evolution of the dimensionless fracture opening and apparent viscosity for a large dimensionless transition shear stress ratio $\unicode[STIX]{x1D6FC}=10^{7}$ while the shear-thinning index $n$ and extent $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ are kept the same as in the previous section: $n=0.5$ and $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3}$ . For this case, the dimensionless opening profiles (for different $\unicode[STIX]{x1D705}$ ) involve an additional viscosity asymptote corresponding to the large shear rate $m_{\infty }$ viscosity asymptote located in between the toughness and shear-thinning asymptotic regions. Even though the toughness region increases with $\unicode[STIX]{x1D705}$ , the validity of the $m_{\infty }$ asymptote roughly starts at the same distance from the fracture tip $\unicode[STIX]{x1D709}\simeq 0.01$ for all values of $\unicode[STIX]{x1D705}$ . This can also be clearly seen in the apparent viscosity profiles depicted on figure 9(b): the apparent viscosity close to the fluid front reaches the large shear rate value at about $\unicode[STIX]{x1D709}\approx 0.1$ .

The opening in the shear-thinning region actually does not coincide exactly with the $m_{n}$ power-law asymptote (red dashed line in figure 9 a). This is the case for all dimensionless toughness values. This difference is rooted in the behaviour of the Carreau viscosity at large shear rate. If we zoom in the region of large $\dot{\unicode[STIX]{x1D6FE}}>1000$ in figure 1, one can observe that the transition from the power-law behaviour to the Newtonian high shear rate behaviour occurs over a large range of shear rate for a Carreau rheology. This large transition region can be clearly seen in the apparent viscosity profiles (figure 9 b), where it departs from the large shear rate value at $\unicode[STIX]{x1D709}\approx 0.1$ and reaches the power-law branch at approximately $\unicode[STIX]{x1D709}\approx 100$ (or more). As a result the opening profile beyond the large shear rate viscosity asymptotic region are shifted from the $m_{n}$ power-law asymptote although they exhibit a similar power exponent. This difference is negligible for sufficiently small dimensionless shear stress $\unicode[STIX]{x1D6FC}$ as observed in the previous section (see e.g. figure 4), where the large shear rate viscosity asymptote was not visible. The difference between the Carreau opening profile and the power-law asymptote in the shear-thinning region becomes more significant for large $\unicode[STIX]{x1D6FC}$ . It is also worthwhile to point out that for the simulations for $\unicode[STIX]{x1D6FC}=10^{7}$ reported in figure 9(a), the extent of our computational domain ( $\unicode[STIX]{x1D709}_{max}=10^{4}$ ) was not sufficient to observe the low shear rate asymptotic region.

It is also worth noting that the value of dimensionless toughness (and thus the extent of the lag region) has an effect limited to the near-tip region, such that the transition between the different far-field viscosity asymptotes are not influenced by $\unicode[STIX]{x1D705}$ .

Figure 9.

(a) Dimensionless fracture opening $w/\unicode[STIX]{x1D716}L_{\unicode[STIX]{x1D707}_{o}}$ along the fracture for $\unicode[STIX]{x1D705}=0,0.013,0.045,0.085,0.11$ (with corresponding lag size $\unicode[STIX]{x1D6EC}=0.0004,0.00025,5.8\times 10^{-5},2.5\times 10^{-6},8.3\times 10^{-8}$ ) in log–log scale for $\unicode[STIX]{x1D6FC}=10^{7}$ , $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3}$ and $n=0.5$ (the dashed lines correspond to the asymptotic solution of opening in the toughness dominated regime ( $\unicode[STIX]{x1D705}$ -asymptote), high shear rate Newtonian viscosity regime ( $m_{\infty }$ -asymptote) and the power-law viscosity regime ( $m_{n}$ -asymptote)). (b) Corresponding dimensionless apparent viscosity $\unicode[STIX]{x1D6E4}$ starting from the fluid front $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D6EC}$ for $\unicode[STIX]{x1D705}=0,0.11$ .

6.3 Effect of the fluid shear-thinning index $n$

The shear-thinning index $n$ dictates how fast the Carreau fluid transition from the low to high shear rate viscosity. For the same value of dimensionless toughness $\unicode[STIX]{x1D705}$ , same shear-thinning extent $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ and transition shear stress $\unicode[STIX]{x1D6FC}$ , a smaller shear-thinning index results in a significantly smaller dimensionless fluid lag as can be seen in figure 10. This result is expected as the extent of the fluid lag is governed mostly by the value of the apparent viscosity near the fluid front. The apparent viscosity actually gets toward the large shear rate limit $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ near the fluid front for small $n$ (steeper shear-thinning branch).

Figure 10.

Dimensionless lag length $\unicode[STIX]{x1D706}/L_{\unicode[STIX]{x1D707}_{o}}$ versus the dimensionless toughness $\unicode[STIX]{x1D705}$ for different values of $n$ , ( $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3},\unicode[STIX]{x1D6FC}=10^{2}$ ).

6.4 Effect of the viscosity ratio $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$

The overall magnitude of the shear-thinning behaviour (between low and high shear rate) is governed by the ratio $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ . We have so far set this ratio to $10^{-3}$ , a value corresponding to a relatively large shear-thinning magnitude. We now vary this ratio as well as the dimensionless toughness while keeping $\unicode[STIX]{x1D6FC}=10^{3}$ and $n=0.5$ . The lag size–dimensionless toughness relation for different values of $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ can be seen in figure 11. As expected, the fluid lag is smaller for smaller $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ (larger shear-thinning magnitude). For $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=1$ , we obviously recover the Newtonian case.

Figure 11.

Dimensionless lag length $\unicode[STIX]{x1D706}/L_{\unicode[STIX]{x1D707}_{o}}$ versus the dimensionless toughness $\unicode[STIX]{x1D705}$ for a Carreau rheology with respect to $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ ( $\unicode[STIX]{x1D6FC}=10^{3},n=0.5$ ). The results for a power-law rheology are also displayed for $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3}$ .

6.5 Comparison between the Carreau, power-law and Ellis models

The numerical solver previously discussed can be easily adapted to solve the same problem for a different fluid rheology. Here, we investigate the differences obtained if one use either a power-law or an Ellis model to model the fluid shear-thinning behaviour instead of the more complete Carreau rheology. We refer to figure 1 for the difference between these models. The only difference in the solution between the different rheologies lies in the lubrication relation (2.6). As a result, the scalings for the power-law and Ellis models are slightly different than for Carreau. The details of the lubrication equations and the corresponding scalings are described in appendices B and C respectively for these two rheologies. The solution for the power-law rheology can actually be re-expressed in the Carreau scaling (see appendix B) using the expression of the power-law consistency parameter as a function of the Carreau rheological parameters (see (1.1)). The solution for the Ellis model is expressed with the same low shear rate viscosity scaling than for the Carreau rheology but the dimensionless parameter $\unicode[STIX]{x1D6FC}_{e}$ related to the transition from the low shear rate plateau to the shear-thinning branch is defined differently for the Ellis model (see appendix C, equation (C 3)).

In order to compare the solution obtained with these different rheological models, we set the rheological parameters of the different models to the ones reproducing the rheology of an HPG fluid (see figure 1, and table 1 for the corresponding rheological parameters of the different models). It is worth re-emphasizing that the large shear rate/low viscosity limit is absent from both the power-law and Ellis models. The power-law rheology also overestimates the viscosity for low shear rate. The deviation of the responses obtained with these two models can be grasped by comparing them with the solution obtained with the Carreau rheology.

We perform simulations for two different values of transition shear stress and dimensionless toughness: $\unicode[STIX]{x1D6FC}=7\times 10^{3}$ , $\unicode[STIX]{x1D705}=0.01$ , and $\unicode[STIX]{x1D6FC}=10^{6}$ , $\unicode[STIX]{x1D705}=0.002$ respectively. In order to set the dimensionless shear stress, we set the ratio $\unicode[STIX]{x1D70E}_{o}^{2}/E^{\prime }$ and thus using the rheological parameters reproducing the HPG fluid, obtain the corresponding transition shear stress for the Ellis model: $\unicode[STIX]{x1D6FC}_{e}=4975$ and $7.2\times 10^{5}$ (for $\unicode[STIX]{x1D6FC}=7\times 10^{3}$ , and $10^{6}$ respectively). The results of the power-law model are also dependent on both $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ and $\unicode[STIX]{x1D6FC}$ via the relation between the consistency index and the Carreau parameters. The dependence of the power-law results on $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ is weak however as can be seen from the contribution of $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ to the correspondence between the power-law – Carreau scaling $(1+\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o})^{1/n}\simeq 1$ (see (B 5) in appendix B).

A similar structure for the fracture opening profile is observed for the Ellis and Carreau in figure 12 for the case $\unicode[STIX]{x1D6FC}=7\times 10^{3}$ , $\unicode[STIX]{x1D705}=0.01$ . The toughness region is larger for the Carreau model compared to the other two models as well as the lag size (Carreau $\unicode[STIX]{x1D6EC}=8.7\times 10^{-4}$ , Ellis $\unicode[STIX]{x1D6EC}=3\times 10^{-4}$ , and power law $\unicode[STIX]{x1D6EC}=1.9\times 10^{-4}$ ). As one moves away from the tip ( $\unicode[STIX]{x1D709}>10^{-2}$ ), the Ellis model follows closely the Carreau solution all the way to the far-field zero shear rate viscosity asymptote $m_{o}$ (slope in $2/3$ ). Similarly, the power-law model also follows closely the Carreau solution in the intermediate power-law region but starts to deviate in the far field ( $\unicode[STIX]{x1D709}>10^{5}$ ) where the low shear rate plateau (absent from the power-law model) starts to dominate. For such a value of dimensionless transition shear stress ( $\unicode[STIX]{x1D6FC}=7\times 10^{3}$ ), the tangent apparent viscosity of the Carreau model did not reach the large shear rate Newtonian plateau near the fluid front, and thus the $m_{\infty }$ asymptote is not visible in the corresponding Carreau opening profiles.

On the contrary, for $\unicode[STIX]{x1D6FC}=10^{6}$ and $\unicode[STIX]{x1D705}=0.002$ , the large shear rate asymptote is reached near the fluid front for the Carreau model and the $m_{\infty }$ asymptote is clearly visible in the opening profile (figure 12 b). As a result, both the power-law and Ellis models – which do not capture the large shear rate viscosity – are significantly off from the Carreau rheology for that case. This difference observed in the transition between the near-tip toughness asymptote and the far-field shear-thinning regions increases with increasing $\unicode[STIX]{x1D6FC}$ . The values of the corresponding dimensionless lag sizes are respectively Carreau $\unicode[STIX]{x1D6EC}=3.3\times 10^{-4}$ , Ellis $\unicode[STIX]{x1D6EC}=8\times 10^{-8}$ and power law $\unicode[STIX]{x1D6EC}=1.2\times 10^{-8}$ . The power-law and Ellis models actually always underestimate the value of the fluid lag for any value of dimensionless shear stress and toughness. This is due to their over-estimation of the shear-thinning behaviour over the complete range of shear rates.

Figure 12.

Dimensionless fracture opening $w/\unicode[STIX]{x1D716}L_{\unicode[STIX]{x1D707}_{o}}$ along the fracture in log–log scale for Carreau (with $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3}$ ), Ellis, power-law rheology: (a) $\unicode[STIX]{x1D6FC}=7\times 10^{3}$ ( $\unicode[STIX]{x1D6FC}_{e}=4975$ , Ellis model), $n=0.46$ and $\unicode[STIX]{x1D705}=0.01$ , (b) $\unicode[STIX]{x1D6FC}=10^{6}$ ( $\unicode[STIX]{x1D6FC}_{e}=7.2\times 10^{5},$ Ellis model), $n=0.46$ and $\unicode[STIX]{x1D705}=0.002$ .

7.1 Scaling

The limiting case of large dimensionless toughness $\unicode[STIX]{x1D705}\gg 1$ corresponds to a vanishing lag. As can be observed from (3.5), large confining stress $\unicode[STIX]{x1D70E}_{o}$ implies large dimensionless toughness. A situation that will necessarily occur at great depth in the sub-surface $(\unicode[STIX]{x1D70E}_{o}>1~\text{MPa})$ . The limit of zero lag is therefore particularly important in hydraulic fracturing practice. Moreover, as observed previously, the effect of the fluid lag is localized near the tip region and does not readily influence the transition between the different viscosities power-law asymptotic regions (see figure 7 for example). We will therefore analyse in more depth the influence of the different governing parameters on the extent of these different asymptotic regions on the zero-lag case.

The zero-lag assumption leads to the disappearance of the pressure boundary condition $p=\unicode[STIX]{x1D70E}_{o}$ and the net pressure becomes singular at the fracture tip (see Garagash & Detournay (Reference Garagash and Detournay2000), Garagash et al. (Reference Garagash, Detournay and Adachi2011) for discussion on the Newtonian case). As a result, another scaling – following Garagash et al. (Reference Garagash, Detournay and Adachi2011) – has to be adopted (see appendix D). We can choose for example to scale the width, net pressure and distance from the tip using the scaling for the transition between the toughness and low shear rate viscosity asymptote. We therefore use a new characteristic opening $w_{m_{o}k}$ , pressure $p_{m_{o}k}$ and length $\ell _{m_{o}k}$ (see table 2) and express the solution as:

where $\bar{\unicode[STIX]{x1D6FA}}$ , $\bar{\unicode[STIX]{x1D6F1}}$ and $\bar{\unicode[STIX]{x1D709}}$ are the dimensionless width, net pressure and spatial coordinates. The length scale $\ell _{m_{o}k}$ characterizes the transition from the toughness dominated regime to the $m_{o}$ viscosity regime. It is defined as the distance $x$ from the tip where the toughness asymptote and the low shear rate viscosity asymptote intersect: $w\sim \ell _{k}^{1/2}x^{1/2}\sim \ell _{m_{o}}^{1/3}x^{2/3}$ (see § 4 for the expression of the different asymptotes). Similarly $w_{m_{o}k}$ and $p_{m_{o}k}$ are the corresponding characteristic pressure and width where these two asymptotes intersect.

Table 2.

Characteristic scales for the zero-lag case for the three different rheology. The low shear rate viscosity Newtonian scaling is used for both the Ellis and Carreau model.

In such a zero-lag scaling, the solution of the problem depends now on three remaining dimensionless parameters: the fluid index $n$ , dimensionless transition shear stress and the extent of the shear-thinning magnitude $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ . The dimensionless transition shear stress is now defined slightly differently due to the different definition of the characteristic length scale and pressure for this zero-lag case. We obtain (see appendix D for more details):

The value of this dimensionless transition shear stress varies significantly depending on both fluid and rock properties. It may actually span an interval as wide as $\bar{\unicode[STIX]{x1D6FC}}\in [10,10^{7}]$ , taking values for shear-thinning fluid (see table 1) and typical ranges of rock properties.

Table 3.

Transition length scales between the different asymptotic regions expressed as a function of the asymptotes characteristic scales (4.1) and directly as function of the problem parameters.

The general solution follows the same structure as in the non-zero-lag case. In the region near the fracture tip, the fracture toughness is the dominating dissipative process and govern the fracture width. As we move away from the tip, the dissipation in the fluid takes over as the dominant mechanism. Depending on the dimensionless transition shear stress $\bar{\unicode[STIX]{x1D6FC}}$ , fluid index $n$ and extent of the shear thinning $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ , three different viscosity asymptotic regions can be observed: high shear rate viscosity asymptote $m_{\infty }$ for large values of the wall shear stress when the low shear rate Newtonian viscosity plateau $\dot{\unicode[STIX]{x1D6FE}}_{c}$ is small enough (large $\bar{\unicode[STIX]{x1D6FC}}$ ), power-law viscosity asymptote $m_{n}$ in an intermediate shear-thinning region and the low shear rate viscosity asymptote $m_{o}$ for small values of shear rate/large distances from the tip.

The extent, occurrence and transition between these asymptotic regions can be estimated based on the different limiting asymptotes (see § 4), by defining transition length scales where two given asymptotes intersect. For example, one can define $\ell _{m_{\infty }k}$ as the transition length scale between the $k$ -asymptote to the $m_{\infty }$ -asymptote as the distance $x$ from the tip where the two asymptotes are comparable: $w\sim \ell _{k}^{1/2}x^{1/2}\sim \ell _{m_{\infty }}^{1/3}x^{2/3}$ such that we obtain $x=\ell _{m_{\infty }k}=\ell _{k}^{3}/\ell _{m_{\infty }}^{2}$ . We can also define a transition length scale between the toughness $k$ and power-law $m_{n}$ asymptotes, the distance $x$ where $w\sim \ell _{k}^{1/2}x^{1/2}\sim \ell _{m_{n}}^{n/(2+n)}x^{2/(n+2)}$ , i.e. $x=\ell _{m_{n}k}=(\ell _{k}^{2+n}/\ell _{m_{n}}^{2n})^{1/(2-n)}$ . Similarly, we can define $\ell _{m_{\infty }m_{n}}$ and $\ell _{m_{o}m_{n}}$ as the boundaries of the high shear rate power-law viscosity and low shear rate viscosity power-law asymptotic regions. The expressions of these different transition length scales can be found in table 3 as function of the problem parameters as well as function of the length scales $\ell _{k}$ , $\ell _{m_{\infty }}$ , $\ell _{m_{n}}$ and $\ell _{m_{o}}$ .

Figure 13.

Occurrence of the different viscous asymptotes as a function of $\bar{\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ for $n=0.46$ . Three distinct regions separate the parametric space: region A defined as $\bar{\unicode[STIX]{x1D6FC}}<\bar{\unicode[STIX]{x1D6FC}}_{l_{n}}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o},n)$ where the solution transitions from $k$ directly to $m_{o}$ , region B defined as $\bar{\unicode[STIX]{x1D6FC}}_{l_{n}}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o},n)<\bar{\unicode[STIX]{x1D6FC}}<\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o},n)$ where three asymptotes $k$ , $m_{n}$ and $m_{o}$ can be observed and region C defined for $\bar{\unicode[STIX]{x1D6FC}}>\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o},n)$ where up to four asymptotes $k$ , $m_{\infty }$ , $m_{n}$ and $m_{o}$ may be observed. The dot-dashed orange lines refer to the coordinates of the numerical simulations performed in figure 14.

It is important to realize that depending on the dimensionless transition shear stress $\bar{\unicode[STIX]{x1D6FC}}$ , shear-thinning index $n$ and extent $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ , not all of the different asymptotes may be realized in the solution. For example, the high shear rate viscosity regime can only be seen if the solution transitions first from the toughness $k$ asymptote to the $m_{\infty }$ and then $m_{n}$ asymptotes. In other words, if the $m_{n}k$ transition length scale is larger than the $m_{\infty }k$ transition length scale: $\ell _{m_{n}k}>\ell _{m_{\infty }k}$ . This condition $\ell _{m_{n}k}>\ell _{m_{\infty }k}$ can be re-expressed as the boundary of a region in the parametric space $n-\bar{\unicode[STIX]{x1D6FC}}-\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ . For a given $n$ , this region can be seen in figure 13, and expressed in terms of a limiting transition shear stress $\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o},n)$ ( $\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}$ is a decreasing function of $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ ). For $\bar{\unicode[STIX]{x1D6FC}}>\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o},n)$ , the solution exhibits all the possible asymptotes: $k$ , $m_{\infty }$ , $m_{n}$ and $m_{o}$ as one moves away from the fracture tip. This limit can be obtained from the equality $\ell _{m_{n}k}=\ell _{m_{\infty }k}$ and can actually be expressed analytically as function of $n$ and $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ :

Another case worth considering is when $\ell _{m_{n}k}>\ell _{m_{o}m_{n}}$ . This means that the distance at which the solution transition from the power-law to the low shear rate viscosity is smaller than the distance at which the solution transition from the toughness to the power-law region. In other words, because the shear rate decreases as one moves away from the tip, such a situation indicates that no power-law asymptotic region exists. The condition $\ell _{m_{n}k}=\ell _{m_{o}m_{n}}$ therefore defines another boundary of a region in the parametric space where the solution exhibits only two asymptotic regions: the toughness $k$ asymptote near the tip and the low shear rate viscosity $m_{o}$ away from the tip. Similarly, for a given $n$ , this boundary can be recast in terms of a limiting value of the dimensionless transition shear stress $\bar{\unicode[STIX]{x1D6FC}}_{l_{n}}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o},n)$ expressed as function of $n$ and $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ ( $\bar{\unicode[STIX]{x1D6FC}}_{l_{n}}$ increases with $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ ):

The two limits are actually related as

Figure 13 displays these two limits for $n=0.46$ which therefore define three regions in the parametric space. In region A ( $\bar{\unicode[STIX]{x1D6FC}}<\bar{\unicode[STIX]{x1D6FC}}_{l_{n}}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o})$ ), only two asymptotic regions ( $k$ and $m_{o}$ ) are present in the solution. In region B, for $\bar{\unicode[STIX]{x1D6FC}}_{l_{n}}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o})<\bar{\unicode[STIX]{x1D6FC}}<\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o})$ , three asymptotic regions $k$ , $m_{n}$ and $m_{o}$ are present as one moves away from the tip. Finally, $\bar{\unicode[STIX]{x1D6FC}}>\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o})$ defines a third region where all the different asymptotes may be visible: $k$ , $m_{\infty }$ , $m_{n}$ and $m_{o}$ in that order from the tip.

Figure 14.

Numerical bounds and transition length scales (black solid lines) between the different asymptotic regions: (a) variations of $\bar{\unicode[STIX]{x1D6FC}}$ for $n=0.46,$ $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3}$ , (b) variations of $\bar{\unicode[STIX]{x1D6FC}}$ for $n=0.46,$ $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-1}$ , (c) variations of $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ for $n=0.46,$ $\bar{\unicode[STIX]{x1D6FC}}=10^{2}$ and (d)  $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ for $n=0.46,$ $\bar{\unicode[STIX]{x1D6FC}}=10^{5}$ . The regions where the numerical results fall within $1\,\%$ of accuracy of an asymptote are also displayed. For the power-law region, limits where the numerical results are within $1\,\%$ accuracy of the $m_{n}$ asymptote are displayed in red, and in orange when within $5\,\%$ of accuracy.

7.2 Evolution of the different asymptotic regions

In the previous section, based on scaling considerations, we have defined three distinct regions in the parametric space, where two, three or four asymptotic regions may be observed. Here we compare a large series of numerical results (57 simulations in total) with the limits established previously. We plot as function of the scaled distance from the tip $x/\ell _{m_{o}k}$ the limits where the numerical results are within 1 % of the different asymptotes as function of $\bar{\unicode[STIX]{x1D6FC}}$ (for a given $n$ and $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ ) or $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ (for a given $n$ and $\bar{\unicode[STIX]{x1D6FC}}$ ). For comparison, we also plot the analytical asymptotic transitions based on the defined length scales (table 3).

We perform four series of simulations corresponding to two vertical and two horizontal cross-sections of the parametric space $\bar{\unicode[STIX]{x1D6FC}}-\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ (see figure 13). All reported simulations are for $n=0.46$ (HPG fluid index). The value of $n$ modifies the shape of the different boundaries but does not change qualitatively the overall picture. The results of these four series of simulations are summarized in figure 14.

Let us first discuss the cases where we vary $\bar{\unicode[STIX]{x1D6FC}}$ in the range $[10^{-8},10^{8}]$ for two distinct values of the shear-thinning magnitude $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3}$ and $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-1}$ (top plots in figure 14). The overall extent of the different asymptotic regions as a function of tip distance and dimensionless transition shear stress follow the structure presented previously. For small value of $\bar{\unicode[STIX]{x1D6FC}}$ , only the toughness (near the tip) and low shear rate viscosity $m_{o}$ asymptotes are visible. Above a given value of $\bar{\unicode[STIX]{x1D6FC}}$ , the power-law asymptote starts to be visible in our numerical results albeit in a very narrow region (for a relative asymptotic error of $1\%$ ). Finally, for large $\bar{\unicode[STIX]{x1D6FC}}$ for $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-1}$ , the large shear rate viscosity $m_{\infty }$ asymptote can be clearly seen in the numerical results. For these two series of simulations for $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3}$ and $10^{-1}$ , the limiting values $\bar{\unicode[STIX]{x1D6FC}}_{l_{n}}$ under which only the toughness and low shear rate viscosity asymptotes are present are respectively $\bar{\unicode[STIX]{x1D6FC}}_{l_{n}}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-1})=3.03$ , and $\bar{\unicode[STIX]{x1D6FC}}_{l_{n}}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3})=2.55$ . Our numerical results report a smaller value for this lower limit $\bar{\unicode[STIX]{x1D6FC}}_{l_{n}}$ : $\bar{\unicode[STIX]{x1D6FC}}_{l_{n_{num}}}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-1})\simeq 10^{-4}$ and $\bar{\unicode[STIX]{x1D6FC}}_{l_{n_{num}}}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3})\simeq 10^{-2}$ . For the small shear-thinning magnitude case ( $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-1}$ ), the high shear rate viscosity region is visible in figure 14(b) for values of $\bar{\unicode[STIX]{x1D6FC}}$ greater than $\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-1})=2.1\times 10^{4}$ , while the numerical value at which we start to observe the $m_{\infty }$ asymptote is slightly larger $\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty _{num}}}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-1})\simeq 4.6\times 10^{4}$ . Note that for $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3}$ , the high shear rate viscosity region is pushed out of the dimensionless transition shear stress interval investigated here, the analytical expression gives $\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3})=9.1\times 10^{8}$ in that case, and the $m_{\infty }$ asymptote is not visible on our numerical results.

Figure 15.

Zero-lag case. Dimensionless fracture opening $w/w_{m_{o}k}$ as a function of the dimensionless distance from the tip $x/\ell _{m_{o}k}$ for the three rheologies (see table 2 for the expressions of the characteristic scales $w_{m_{o}k}$ and $\ell _{m_{o}k}$ ). The rheological parameters of the fluid are those of HPG. Influence of $\bar{\unicode[STIX]{x1D6FC}}$ with $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3}$ : (a)  $\bar{\unicode[STIX]{x1D6FC}}=74\times 10^{-8}$ (Ellis $\bar{\unicode[STIX]{x1D6FC}}_{e}=53.4\times 10^{-8}$ ), (b)  $\bar{\unicode[STIX]{x1D6FC}}=74$ (Ellis $\bar{\unicode[STIX]{x1D6FC}}_{e}=53.4$ ; (c)  $\bar{\unicode[STIX]{x1D6FC}}=10^{3}$ (Ellis $\bar{\unicode[STIX]{x1D6FC}}_{e}=2088.7$ ), (d)  $\bar{\unicode[STIX]{x1D6FC}}=46.2\times 10^{3}$ (Ellis $\bar{\unicode[STIX]{x1D6FC}}_{e}=33.4\times 10^{3}$ ) with also the solution for different $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ for the Carreau model. The Ellis model starts to depart from the Carreau in the intermediate region for $\unicode[STIX]{x1D6FC}\geqslant 10^{3}$ even for a large shear-thinning magnitude ( $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3}$ ).

The extent of the toughness region near the fracture tip is roughly constant for small $\bar{\unicode[STIX]{x1D6FC}}$ , then increases with $\bar{\unicode[STIX]{x1D6FC}}$ until reaching a constant value for sufficiently large $\bar{\unicode[STIX]{x1D6FC}}$ (above the limit $\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}$ ) for which the $m_{\infty }$ asymptote becomes visible. For small enough values of $\bar{\unicode[STIX]{x1D6FC}}$ (below $\bar{\unicode[STIX]{x1D6FC}}_{l_{n}}$ ), the distance at which the numerical results can be approximated by the toughness and low shear rate viscosity asymptote are respectively $x\approx 10^{-7}\ell _{m_{o}k}$ and $x\approx 1.3\ell _{m_{o}k}$ which are consistent with the bounds obtained by Garagash et al. (Reference Garagash, Detournay and Adachi2011). Similarly, for the case of large dimensionless transition shear stress $\bar{\unicode[STIX]{x1D6FC}}>\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}$ , the numerical boundaries demonstrate that the $m_{\infty }$ asymptote starts at $x\approx 10^{1}\ell _{m_{o}k}$ from the fracture tip regardless the value of $\bar{\unicode[STIX]{x1D6FC}}$ .

Similarly we have studied the effect of the viscosity ratio in the range $[10^{-3},0.75]$ for two distinct dimensionless transition shear stress $\bar{\unicode[STIX]{x1D6FC}}=10^{2}$ and $10^{5}$ , see figure 14(c,d). In both cases, the extent of the toughness region (scaled by $\ell _{m_{o}k}$ ) is decreasing with larger values of $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ . In fact, the evolution of the toughness region tracks the evolution of $\ell _{m_{n}k}$ and $\ell _{m_{\infty }k}$ as a function of $\bar{\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ for all cases.

For the series of simulation with $\bar{\unicode[STIX]{x1D6FC}}=10^{2}$ , the $m_{o}$ low shear rate viscosity region has the same domain of validity regardless of the value of $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ (figure 14 c). This low shear rate region cannot, however, be observed for the series of simulation with $\bar{\unicode[STIX]{x1D6FC}}=10^{5}$ , where it is estimated to appear for distances away from the tip above $10^{7}\ell _{m_{o}k}$ . A value which is larger than the extent of our computational domain.

The high shear rate viscosity region $m_{\infty }$ can be seen for values of $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ larger than $0.33$ ( $\bar{\unicode[STIX]{x1D6FC}}=10^{2}$ ) and $0.025$ ( $\bar{\unicode[STIX]{x1D6FC}}=10^{5}$ ) respectively from the scaling arguments. These values are in line with the numerical results which gives respectively $(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o})_{num}=0.75$ ( $\bar{\unicode[STIX]{x1D6FC}}=10^{2}$ , where this $m_{\infty }$ region can be seen for the largest value of $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ only), and $(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o})_{num}=0.07$ ( $\bar{\unicode[STIX]{x1D6FC}}=10^{5}$ ).

It is also worthwhile to note that for all the simulations reported in figure 14, the extent of the power-law region where the numerical results obtained are within $1\%$ of the $m_{n}$ asymptote is extremely limited, and significantly smaller than the predictions obtained from scaling arguments.

7.3 Comparison between the Carreau, power-law and Ellis models

In this section, we compare these three models in the zero-lag case and investigate how close the predictions based on Ellis and power-law models are to the complete Carreau model depending on the occurrence of the different asymptotic regions highlighted in figure 13. We use the zero-lag scaling (7.1) in all cases. The expressions of the characteristic opening, pressure and length for the power-law model (see table 2) are different from the Carreau model (while they are similar for the Ellis model). The scaling and dimensionless problem are detailed in appendix D.

Like previously for the lag case, we fix the rheological parameters to the ones reproducing the rheology of an HPG fluid (see table 1 for the different models). We vary the value of the dimensionless transition shear stress $\bar{\unicode[STIX]{x1D6FC}}$ (by varying $E^{\prime 3}V^{2}/K_{Ic}^{4}$ at fix rheological parameters), as well as the shear-thinning amplitude ratio $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ .

We first fix the shear-thinning magnitude $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ at $10^{-3}$ . Figure 15(a–d) exhibits the fracture opening for the three models for four different values of $\bar{\unicode[STIX]{x1D6FC}}$ (small $\bar{\unicode[STIX]{x1D6FC}}$ – figure 15(a), intermediate $\bar{\unicode[STIX]{x1D6FC}}$ – figure 15(b,c) and large $\bar{\unicode[STIX]{x1D6FC}}$ – figure 15 d). As discussed in figure 13, there are two limits that define the behaviour of solution for the Carreau model $(\bar{\unicode[STIX]{x1D6FC}}_{l_{n}},\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }})$ for a given viscosity ratio and power-law index. These values of the dimensionless transition shear stress can be approximated analytically using the transition length scales (see § 7.1). For $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3}$ , the high shear rate viscosity asymptote may become visible for values of $\bar{\unicode[STIX]{x1D6FC}}$ larger than $\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}=9.1\times 10^{8}$ , whereas the power-law region may appear for $\bar{\unicode[STIX]{x1D6FC}}$ larger than $\bar{\unicode[STIX]{x1D6FC}}_{l_{n}}=2.55$ .

For value of $\bar{\unicode[STIX]{x1D6FC}}$ smaller than $\bar{\unicode[STIX]{x1D6FC}}_{l_{n}}$ (see figure 15 a), as expected the Carreau dimensionless opening transition from the toughness $k$ asymptote near the tip to the far-field low shear viscosity asymptote without any visible power-law region. The solution for the Ellis model matches perfectly the solution of the Carreau rheology for that case (figure 15 a). As expected, the prediction of the power-law rheology widely differs from Carreau for this small $\bar{\unicode[STIX]{x1D6FC}}$ case (due to the large effect of the low shear rate plateau). As the value of $\bar{\unicode[STIX]{x1D6FC}}$ exceeds $\bar{\unicode[STIX]{x1D6FC}}_{l_{n}}$ , a power-law region appears just beyond the near-tip toughness region. For $\bar{\unicode[STIX]{x1D6FC}}=74$ (see figure 15 b), the Ellis model still follows closely the complete Carreau rheology – excepting a small deviation in the transition between the power-law and far-field regions (for $\displaystyle x/\ell _{m_{o}k}\in [1-100]$ ). Here again, the power-law model still significantly deviates in the far field due to the effect of the low shear rate viscosity absent from the power-law model. For larger value of $\bar{\unicode[STIX]{x1D6FC}}$ (figure 15 c), the difference between the Ellis and Carreau rheology slightly increases in the intermediate power-law $m_{n}$ region where the effect of the transition from the power-law branch to the high shear rate Newtonian plateau in the Carreau rheology starts to influence the overall solution (see § 6.2 for details) even tough $\bar{\unicode[STIX]{x1D6FC}}$ did not yet reach the bound $\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}$ . However, the Ellis model still match exactly the Carreau solution in the near-tip toughness region and in the far-field low shear rate $m_{o}$ region. The power-law model gets closer to the predictions of the Carreau and the Ellis models except in the far-field low shear rate $m_{o}$ region ( $x/\ell _{m_{o}k}>10^{4}$ ) as expected.

All the results depicted in figure 15(a–c) are for the case of a large shear-thinning amplitude ( $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3}$ ). The effect of its influence can be seen in figure 15(d) for a large value of $\bar{\unicode[STIX]{x1D6FC}}$ ( $46.2\times 10^{3}$ ). For such a large value of $\bar{\unicode[STIX]{x1D6FC}}$ , the power-law region stretches out far away from the tip, actually driving the low shear rate region outside of our numerical grid. Over those length scales, the Ellis and the power-law models are very close to one another. They both differ however from the Carreau rheology outside the near-tip toughness $k$ region. Such a difference increases as the shear-thinning amplitude decreases (i.e. for smaller viscosity ratio $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ ). The power-law region significantly decreases in the Carreau solution with increasing viscosity ratio $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ . The high shear rate viscosity region $m_{\infty }$ dominates the Carreau response in the intermediate field for the case $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-1}$ for $\bar{\unicode[STIX]{x1D6FC}}>\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-1})=2.1\times 10^{4}$ . For both cases $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-2}$ and $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3}$ , the high shear rate viscosity $m_{\infty }$ region is still absent from the Carreau solution ( $\bar{\unicode[STIX]{x1D6FC}}_{l_{n}}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-2})=2.59<\bar{\unicode[STIX]{x1D6FC}}<\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-2})=1.3\times 10^{6}$ and $\bar{\unicode[STIX]{x1D6FC}}<\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}=10^{-3})=9.1\times 10^{8}$ ) but the solution is already significantly affected by the rather long transition from the power-law branch to the high shear rate viscosity intrinsic to the Carreau model.

These comparisons are in line with the structure of the solution with respect to the different limiting asymptotes depicted in the previous section. We notably see that the scaling estimate of $\bar{\unicode[STIX]{x1D6FC}}_{l_{n}}$ and $\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}$ provides the overall trend of the validity of the different models, with the caveat that the transition from the power-law branch to the high shear rate viscosity plateau has a significant effect on the solution even for $\bar{\unicode[STIX]{x1D6FC}}$ lower than $\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}$ . In summary, for values of $\bar{\unicode[STIX]{x1D6FC}}$ sufficiently lower than $\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o},n)$ , the Ellis model reproduces relatively well the results obtained with the Carreau rheology. Deviations start to occur for larger values of the dimensionless shear stress $\bar{\unicode[STIX]{x1D6FC}}$ . The power-law model reproduces poorly the results of the complete Carreau solution in the far-field region in all cases. It may only provide an acceptable approximation up to a distance $x/\ell _{m_{o}k}\approx 10$ from the tip for intermediate values of $\bar{\unicode[STIX]{x1D6FC}}$ in between the limits $\bar{\unicode[STIX]{x1D6FC}}_{l_{n}}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o},n)$ and $\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o},n)$ . This intermediate region between $\bar{\unicode[STIX]{x1D6FC}}_{l_{n}}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o},n)$ and $\bar{\unicode[STIX]{x1D6FC}}_{l_{\infty }}(\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o},n)$ drastically shrinks for increasing $\unicode[STIX]{x1D707}_{\infty }/\unicode[STIX]{x1D707}_{o}$ ratio (i.e. decreasing shear-thinning amplitude).