3.1. Governing equations

The plane strain model (Khristianovic & Zheltov Reference Khristianovic and Zheltov1955; Geertsma & De Klerk Reference Geertsma and De Klerk1969) assumes that crack deformation and propagation occur in the state of plane strain, as shown in figure 2. The corresponding governing equations can be obtained from the planar model (2.1) by neglecting variation with respect to the vertical coordinate $y$ as

(3.1) \begin{align} & \frac {\partial w}{\partial t}+\frac {\partial q}{\partial x}+\frac {C^{\prime }}{\sqrt {t-t_{0}(x)}}=Q_{0}\delta (x),\quad q=-\frac {w^{3}}{\mu ^{\prime }}\frac {\partial p_{f}}{\partial x},\nonumber\\ & p=p_{f}-\sigma _{0}=-\frac {E^{\prime }}{4\pi }\int _{-\ell }^{\ell }\frac {w+w^{\ell o}}{(x-s)^{2}}\mathrm{d}s,\quad w^{\ell o}=2B^{\prime }C^{\prime }\sqrt {t-t_{0}(x)},\nonumber\\ & w=\frac {K^{\prime }}{E^{\prime }}(\ell -x)^{1/2},\quad x\rightarrow \ell ,\nonumber\\ & Q_{0}t=2\int _{0}^{\ell (t)}w(x,t)\mathrm{d}x+4C^{\prime }\int _{0}^{\ell (t)}\sqrt {t-t_{0}(x)}\mathrm{d}x. \end{align}

Here the net pressure is introduced as $p$ , the fracture half-length is $\ell$ and the global volume balance is also written for future reference, assuming a symmetric fracture.

Figure 2.

Sketch of a plane strain fluid-driven fracture.

3.2. General scaling

To proceed with the analysis and to find self-similar solutions, we employ the following scaling:

(3.2) \begin{align} &x=\ell (t)\xi ,\quad \ell (t)=L(t)\gamma (t),\nonumber \\ &p_{f}(x,t)-\sigma _{0}=P(t)\varPi (\xi ,t),\quad w(x,t)=W(t)\varOmega (\xi ,t), \end{align}

where $L$ , $P$ , $W$ are time-dependent scales that need to be determined. By transforming the derivatives as

(3.3) \begin{equation} \left .\frac {\partial }{\partial x}\right |_{x}=\frac {1}{\ell }\left .\frac {\partial }{\partial \xi }\right |_{\xi },\quad \left .\frac {\partial }{\partial t}\right |_{x}=\left .\frac {\partial }{\partial t}\right |_{\xi }-\xi \frac {\dot {\ell }}{\ell }\frac {\partial }{\partial \xi }, \end{equation}

the governing equations (3.1) become

(3.4) \begin{align} & G_{v}\left [\frac {\dot {W}}{W}\varOmega t+\dot {\varOmega }t-\xi \left (\frac {\dot {\gamma }t}{\gamma }+\frac {\dot {L}t}{L}\right )\frac {\partial \varOmega }{\partial \xi }\right ]-\frac {1}{G_{m}}\frac {1}{\gamma ^{2}}\frac {\partial }{\partial \xi }\left (\varOmega ^{3}\frac {\partial \varPi }{\partial \xi }\right )+\frac {G_{c}}{\sqrt {1-\theta (\xi)}}=0,\nonumber\\ & \varPi =-\frac {G_{e}}{4\pi \gamma }\int _{-1}^{1}\frac {\varOmega (\zeta)}{(\xi -\zeta )^{2}}\mathrm{d}\zeta -\frac {G_{p}}{2\pi \gamma }\int _{-1}^{1}\frac {\sqrt {1-\theta (\zeta)}}{(\xi -\zeta )^{2}}\mathrm{d}\zeta ,\nonumber\\ & \varOmega =G_{k}\gamma ^{1/2}(1-\xi )^{1/2},\quad \xi \rightarrow 1,\nonumber\\ & \frac {1}{2\gamma }=G_{v}\int _{0}^{1}\varOmega \mathrm{d}\xi +2G_{c}\int _{0}^{1}\sqrt {1-\theta (\xi)}\mathrm{d}\xi , \end{align}

while the dimensionless groups are

(3.5) \begin{align} &G_{v}=\frac {WL}{Q_{0}t},\quad G_{m}=\frac {\mu ^{\prime }Q_{0}L}{W^{3}P},\quad G_{c}=\frac {C^{\prime }L}{Q_{0}t^{1/2}},\nonumber \\ &G_{e}=\frac {E^{\prime }W}{PL},\quad G_{p}=\frac {E^{\prime }B^{\prime }C^{\prime }t^{1/2}}{PL},\quad G_{k}=\frac {K^{\prime }L^{1/2}}{E^{\prime }W}. \end{align}

Note that the newly introduced backstress-related dimensionless group is related to other parameters as $G_{p}=B^{\prime }G_{e}{G_{c}}/{G_{v}}$ . The dimensionless group $G_v$ characterizes the fracture volume relative to the injected volume; $G_m$ captures the effect of fluid viscosity; $G_c$ quantifies leak-off volume relative to the injected volume; $G_e$ reflects the relation between the opening and the pressure; $G_p$ measures the influence of backstress; $G_k$ represents fracture toughness.

3.3. Backstress-dominated $B$ -vertex

Backstress inevitably accompanies the pore-pressure diffusion induced by leak-off. Formally, the $B$ -vertex scaling is defined by requiring $G_c$ , $G_e$ and $G_p$ to be of the order of one. These three conditions express the simultaneous leading-order balance between leak-off, elastic deformation and poroelastic backstress, while other terms are assumed to be subdominant. In the present formulation we set $G_c=G_e=G_p=1$ , which establishes the characteristic length, opening and pressure scales. This common definition of the $B$ -scaling is used for the plane-strain, radial and constant-height geometries, so that the resulting $B$ -vertices can be compared within the same physical balance. According to the result in (3.5), the corresponding scales are

(3.6) \begin{equation} L=\frac {Q_{0}t^{1/2}}{C^{\prime }},\quad W=B^{\prime }C^{\prime }t^{1/2},\quad P=\frac {E^{\prime }B^{\prime }C^{\prime 2}}{Q_{0}}. \end{equation}

The remaining dimensionless groups are

(3.7) \begin{equation} G_{v}=B^{\prime },\quad G_{m}=\frac {\mu ^{\prime }Q_{0}^{3}}{B^{\prime 4}E^{\prime }C^{\prime 6}t},\quad G_{k}=\frac {K^{\prime }Q_{0}^{1/2}}{B^{\prime }E^{\prime }C^{\prime 3/2}t^{1/4}}. \end{equation}

The storage group $G_v$ follows directly from the definitions in (3.5). For $t\to \infty$ , we have $G_{m}\rightarrow 0$ and $G_{k}\rightarrow 0$ , and therefore the corresponding terms are negligible in the $B$ -vertex. In this situation, given that the fracture length is proportional to $t^{1/2}$ (see (3.6)), the leak-off history is simplified to $\theta =t_{0}/t= (x/\ell )^{2}=\xi ^{2}$ , and the contribution of the back stress in the elasticity equation in (3.4) can be rewritten as

(3.8) \begin{equation} \varPi _b=-\frac {1}{2\pi \gamma }\int _{-1}^{1}\frac {\sqrt {1-\zeta ^{2}}}{(\xi -\zeta )^{2}}\mathrm{d}\zeta =\frac {1}{2\gamma }. \end{equation}

Since $G_{m}\rightarrow 0$ , the lubrication equation in (3.4) results in $\varPi =\mathrm{const}$ . Given the uniform pressure and constant contribution of the backstress, the elasticity equation in (3.4) can be solved for fracture width, which essentially represents a solution for a Griffith fracture (Sneddon & Elliot Reference Sneddon and Elliot1946). Expansion of the general solution for fracture opening $\varOmega =k\sqrt {1-\xi ^{2}}$ near the tip leads to

(3.9) \begin{equation} \varOmega =\sqrt {2}k(1-\xi )^{1/2},\quad \xi \rightarrow 1. \end{equation}

Comparison of this result with the propagation criterion in (3.4) leads to $k=G_{k}\gamma ^{1/2}/\sqrt {2}$ , and since $G_{k}\rightarrow 0$ , we have $\varOmega \rightarrow 0$ . The fracture length can be calculated from the global volume balance equation in (3.4) as $\gamma =1/\pi$ , while the pressure has only contribution of the backstress

(3.10) \begin{equation} \varPi =\varPi _b=\frac {\pi }{2}. \end{equation}

To summarize the above, the corresponding dimensionless self-similar solution for the backstress-dominated limit is

(3.11) \begin{equation} \varPi _{b0}=\frac {\pi }{2},\quad \varOmega _{b0}=0,\quad \gamma _{b0}=\frac {1}{\pi }, \end{equation}

which we refer to as the $B$ -vertex. The result $\varOmega _{b0}=0$ is a leading-order result in the $B$ -scaling. It indicates that no aperture of order $W_b=B'C't^{1/2}$ is resolved at the $B$ -vertex, rather than that the physical aperture is exactly zero. A finite aperture is recovered when the subdominant toughness or viscous contribution is retained in the transition problem, as is shown later in this section. By using the scaling relations (3.2) and (3.6), the dimensional form of the above solution is

(3.12) \begin{equation} p_{b0}=\frac {\pi }{2}\frac {E^{\prime }B^{\prime }C^{\prime 2}}{Q_{0}},\quad w_{b0}=o(B'C't^{1/2}),\quad \ell _{b0}=\frac {Q_{0}}{\pi C^{\prime }}t^{1/2}. \end{equation}

As can be seen from the result, in this limiting case or regime, the net pressure is balanced at leading order by the poroelastic backstress, so that no aperture is resolved at the $B$ -scaling. In addition, the fracture length is identical to that for the leak-off-dominated regimes that assume no backstress (Adachi & Detournay Reference Adachi and Detournay2008).

3.4. Parametric space

The next goal is to modify the existing parametric space for the considered plane-strain fracture geometry (Dontsov Reference Dontsov2017) and to introduce the effect of poroelastic backstress. The dimensionless variables that define the original diagram are

(3.13) \begin{equation} \tau =\frac {tE^{\prime }C^{\prime 6}}{\mu ^{\prime }Q_{0}^{3}},\quad K_{m}=\left (\frac {K^{\prime 4}}{\mu ^{\prime }E^{\prime 3}Q_{0}}\right )^{1/4}. \end{equation}

In the absence of backstress, there are four limiting regimes: storage-viscosity $M$ ; leak-off-viscosity $\tilde {M}$ ; storage-toughness $K$ ; leak-off-toughness $\tilde {K}$ . The presence of backstress adds an additional backstress-dominated regime or $B$ -vertex, which is defined in the previous section.

In contrast to the approach in Dontsov (Reference Dontsov2017), in which an approximate solution is used to determine boundaries of fracture regimes, we are going to utilize an alternative approach that uses only vertex solutions. While the latter approach is simpler, it is much less accurate and provides only an approximate result. Such an approach was also adopted in Dontsov (Reference Dontsov2019). Nevertheless, this is the only option at this point since all the vertex solutions are known, while the approximate solution for the whole parametric space is currently unavailable. Specifically, to construct the parametric space, the following metric is used:

(3.14) \begin{equation} p_{i0}=\epsilon p_{j0}\left (\frac {\ell }{2},t\right ),\quad \epsilon p_{i0}=p_{j0}\left (\frac {\ell }{2},t\right ), \end{equation}

where subscripts $i$ and $j$ ( $i\neq j$ ) correspond to $M$ , $\tilde {M}$ , $K$ , $\tilde {K}$ and $B$ vertex solutions. The tolerance parameter $\epsilon$ is specified below through the calibration procedure used to construct the approximate parametric space. Given two individual regimes $i$ and $j$ , (3.14) allow us to calculate the boundaries of the transition zone, which are straight lines in the $(\log (\tau ),\log (K_m))$ parametric space. The combination of all such transition zones then defines the location of the vertex solutions. Note that net pressure is used as a metric and it is evaluated in the middle between the injection point and the fracture tip to avoid singularities. The value of a small parameter $\epsilon$ will be determined later. The limiting solutions for the problem have been thoroughly investigated in the past, including the $M$ -vertex (Adachi Reference Adachi2001; Adachi et al. Reference Adachi, Detournay and Savitski2001; Adachi & Detournay Reference Adachi and Detournay2002), the $\tilde {M}$ -vertex (Adachi & Detournay Reference Adachi and Detournay2008), the $K$ -vertex (Adachi Reference Adachi2001; Detournay Reference Detournay2004; Garagash Reference Garagash2006) and the $\tilde {K}$ -vertex (Bunger et al. Reference Bunger, Detournay and Garagash2005). The vertex solutions are also summarized in Hu & Garagash (Reference Hu and Garagash2010) and Dontsov (Reference Dontsov2017). The vertex solutions for fluid pressure that are used in (3.14), including the new $B$ -vertex, are listed in table 1.

Table 1.

Vertex pressure solutions for a plane strain hydraulic fracture.

Table 1. Long description

Panel A: The table lists vertex pressure solutions for a plane strain hydraulic fracture. It includes columns for Vertex, Expression, and Net pressure. Panel B: The Vertex column contains symbols M, M-tilde, K, K-tilde, and B. Panel C: The Expression column shows mathematical expressions for each vertex. Panel D: The Net pressure column provides net pressure values for each vertex.

As a starting point, to evaluate the accuracy of the new approach and to determine the value of $\epsilon$ , we first recalculate the original $M\tilde {M}\tilde {K}K$ parametric space. As was mentioned previously, the parametric space is effectively defined by the boundaries of the transition regimes. They are characterized by corresponding transition parameters. These are defined as (see e.g. Dontsov (Reference Dontsov2017))

(3.15) \begin{align} \tau _{m\tilde {m}}=\tau _{\tilde {m}b}=\tau _{mb}=\tau , \quad \tau _{k\tilde {k}}'=\tau _{\tilde {k}b}'=\tau _{kb}'=\tau K_{m}^{-4},\quad K_{m,mk}=K_{m,\tilde m\tilde k}=K_m. \end{align}

Specifically, there are two boundaries, the first indicating the beginning of the transition and the second one representing the end of the transition. Superscripts ‘ $(1)$ ’ and ‘ $(2)$ ’ are used to denote the aforementioned two boundaries.

Table 2 summarizes results for the transition regions that are obtained by substituting the vertex solutions from table 1 into (3.14). The parameter $\epsilon$ is set to 0.210 to make the newly calculated transitions as similar as possible to the target values that are calculated in Dontsov (Reference Dontsov2017). Specifically, this parameter is defined as

(3.16) \begin{equation} \epsilon =\arg \min _{\epsilon }\sum _{i}|\log a_{i}(\epsilon )-\log b_{i}|, \end{equation}

where $a_{i} (\epsilon )$ represents each of the transition boundaries and $b_{i}$ corresponds to the target values. As can be seen from the results in table 2, the new approach that uses exclusively vertex solutions is able to adequately estimate boundaries for all of the transitions.

Table 2.

Boundaries of the transition between vertex solutions $M$ , $\tilde {M}$ , $\tilde {K}$ , $K$ for $\epsilon =0.210$ for plane strain hydraulic fracture geometry.

Table 2. Long description

The table has four columns and four transition groups, each containing two boundary rows. The columns are labeled Transition edge, Scaled expression, Target, and Result. The rows are labeled with different transition edges: MK, M–M-tilde, K–K-tilde, and M-tilde–K-tilde. Each row contains values for the scaled expression, target, and result. For example, the first row under MK shows the scaled expression as epsilon times Pi sub m0(0.5) over Pi sub k0 equals K sub m,mk superscript (1) to the power of 4/3, with a target value of 0.70 and a result value of 0.58. The table provides a detailed comparison of calculated transition boundaries against target values for different transition edges in plane strain hydraulic fracture geometry.

By further applying this approach to the transitions featuring the $B$ -vertex, we obtain the results that are summarized in table 3. Note that there are two critical values of $B'$ , namely

(3.17) \begin{equation} B'_{k}=1.93\times 10^{-3},\quad B'_{m}=2.04\times 10^{-3}, \end{equation}

that determine if the corresponding leak-off-dominated regimes exist. If $B'\gt B'_k$ , then there is no leak-off-toughness regime, while if $B'\gt B'_m$ , then there is no leak-off-viscosity regime. The critical values of $B'$ are obtained using $\tau _{\tilde {k}b}^{\prime (1 )}=\tau _{k\tilde {k}}^{\prime (2 )}$ and $\tau _{\tilde {m}b}^{ (1 )}=\tau _{m\tilde {m}}^{ (2 )}$ , respectively.

Table 3.

Boundaries of the transition between $M$ , $\tilde {M}$ , $\tilde {K}$ , $K$ vertices and the $B$ -vertex for $\epsilon =0.210$ for plane strain hydraulic fracture geometry.

Table 3. Long description

Panel A: The table lists transition edges, their definitions, and bounds. Transition edges include K-tilde B, M-tilde B, K B, and M B. Definitions are given in terms of tau and B-prime. Bounds are provided for each transition edge with specific values and inequalities. Panel B: The definitions and bounds are detailed with mathematical expressions and inequalities, specifying the conditions under which each transition occurs.

Figure 3.

Parametric space for a plane-strain hydraulic fracture with backstress: (a) $B^{\prime }\lt B'_{k}\,,B^{\prime }\lt B'_{m}$ , with leak-off regimes; (b) $B^{\prime }\gt B'_{k}\,,B^{\prime }\gt B'_{m}$ , without leak-off regimes. See (3.17) for the definition of $B'_{k}$ and $B'_{m}$ . The red lines delineate the toughness-storage ( $K$ ) vertex, while the blue lines delineate the viscosity-storage ( $M$ ) vertex. The right-hand region, bounded by the brown solid and dashed lines, corresponds to the backstress-dominated ( $B$ ) regime. For small $B^\prime$ , the region enclosed by the magenta boundaries defines the toughness-leak-off ( $\tilde K$ ) vertex, whereas the region enclosed by the green boundaries defines the viscosity-leak-off ( $\tilde M$ ) vertex. For large $B^\prime$ , both the $\tilde M$ and $\tilde K$ vertices disappear. The numbered markers correspond to the cases discussed in § 6.

Figure 3 shows the parametric space that is calculated assuming a small value of $B'$ (leak-off-dominated regimes present) and large $B'$ (no leak-off-dominated regimes). The lines represent boundaries of the transitions between the regimes, as summarized in tables 2 and 3. As can be seen from the result, the increasing values of $B'$ shift the $B$ -vertex towards the no leak-off regimes and once $B'$ exceeds a critical value, leak-off-dominated regimes cease to exist. This means that for such conditions, the effect of backstress is crucial and cannot be neglected.

Specifically, the qualitative behaviour at large times of fracture propagation depends primarily on the magnitude of $B'$ , characterized by two independent critical values $B'_{k}$ and $B'_{m}$ , which differ by only approximately 5 % and are therefore effectively indistinguishable at the level of approximation of the present regime-map construction. They are associated with the transitions from the $\tilde {K}$ - and $\tilde {M}$ -regimes, respectively, as follows:

1. (i) $B'=0$ ;

   both transition boundaries tend to infinity, indicating that without the poroelastic coupling the fracture never enters the backstress-dominated regime, i.e.

   (3.18) \begin{align} K\rightarrow \tilde {K},\quad M\rightarrow \tilde {M}; \end{align}

2. (ii) small $B'$ ;

   when $B'\lt B'_{k}$ and $B'\lt B'_{m}$ , the $\tilde {K}B$ and $\tilde {M}B$ boundaries shift to the right of the original $\tilde {K}$ and $\tilde {M}$ regimes, so that the fracture successively evolves through one of the two paths

   (3.19) \begin{align} K\rightarrow \tilde {K}\rightarrow B,\quad M\rightarrow \tilde {M}\rightarrow B; \end{align}

3. (iii) large $B'$ ;

   when $B'$ exceeds $B'_{k}$ and $B'_{m}$ , both $\tilde {K}$ - and $\tilde {M}$ -vertices vanish, and the fracture behaviour is entirely governed by the $B$ -regime at large times, i.e. the time evolutions are

   (3.20) \begin{align} K\rightarrow B,\quad M\rightarrow B. \end{align}

It should be noted here that the constructed parametric space is approximate, whereby the location of the boundaries may have some error. Also, the actual solution never coincides with the vertex solution. Instead, it asymptotically approaches it. As a result, different levels of tolerance lead to different quantitative results. However, since the result is presented on a logarithmic scale, such deviations lead to relatively small differences. Finally, the constructed parametric space can be used to determine the dominant mechanism that governs hydraulic fracture propagation for a given set of parameters. The addition of the $B$ regime specifically allows us to estimate whether poroelastic backstress is important or not for the considered case.

3.5. Transition solutions

To further investigate the problem, we focus on leak-off dominated cases, when the poroelastic effect can be significant. According to the physical meaning of dimensionless groups, there are three evolution parameters: $B^{\prime }$ , $G_m$ and $G_k$ . The limiting case of $B^\prime \rightarrow 0$ is discussed, corresponding to the case when the storage term in the global volume balance can be neglected. This is consistent with observations made in the previous section, showing that the leak-off-dominated regimes disappear when $B^\prime =O(1)$ . For the toughness-dominated case, we have $G_m\rightarrow 0$ . The parameter $G_k$ controls the transition from the $\tilde {K}$ -vertex to the $B$ -vertex. For the viscosity-dominated case, we have $G_k\rightarrow 0$ . The parameter $G_m$ controls the transition from the $\tilde {M}$ -vertex to the $B$ -vertex.

For the toughness-dominated case, we analyse the $\tilde K B$ transition. With $G_m\rightarrow 0$ , the governing equations in (3.4) are simplified to

(3.21) \begin{equation} \begin{aligned} &\varPi _{\tilde {k}}+\varPi _{\tilde {k}}^{b}=\mathrm{const},\quad \varPi _{\tilde {k}}=-\frac {1}{4\pi \gamma _{\tilde {k}b}}\int _{-1}^{1}\frac {\varOmega _{\tilde {k}b}}{(\xi -\zeta )^{2}}\mathrm{d}\zeta ,\quad \varPi _{\tilde {k}}^{b}=-\frac {1}{2\pi \gamma _{\tilde {k}b}}\int _{-1}^{1}\frac {\sqrt {1-\theta (\zeta ,t)}}{(\xi -\zeta )^{2}}\mathrm{d}\zeta ,\\ &\varOmega _{\tilde {k}b}=G_k \gamma _{\tilde {k}b}^{1/2}(1-\xi )^{1/2},\quad \xi \rightarrow 1,\quad \frac {1}{2\gamma _{\tilde {k}b}}= B^{\prime }\int _{0}^{1}\varOmega _{\tilde {k}b} \mathrm{d}\xi + 2\int _{0}^{1}\sqrt {1-\theta (\xi ,t)}\mathrm{d}\xi . \end{aligned} \end{equation}

The elasticity equation and constant pressure condition in the above lead to the following general expression for fracture width:

(3.22) \begin{equation} \varOmega _{\tilde {k}b}=k\sqrt {1-\xi ^{2}}-2\sqrt {1-\theta (\xi ,t)}. \end{equation}

Further, when $B^\prime \to 0$ , the storage term in the global volume balance becomes negligible. As a result, the dimensionless half-length $\gamma =\ell /L$ is constant and the fracture half-length scales as $\ell \sim t^{1/2}$ . The leak-off history simplifies to $\theta =t_0/t=(x/\ell )^2=\xi ^2$ , and the integral term is time-independent. Given this result and accounting for the propagation condition, we obtain the solution

(3.23) \begin{equation} \varPi _{\tilde {k}b} = \frac {\sqrt {\pi }}{4\sqrt {2}}G_k+\frac {\pi }{2},\quad \varOmega _{\tilde {k}b} = \frac {G_k}{\sqrt {2\pi }}\sqrt {1-\xi ^2},\quad \gamma _{\tilde {k}b} = \frac {1}{\pi }, \end{equation}

whose dimensional form is

(3.24) \begin{align} p_{\tilde {k}b}=\frac {\sqrt {\pi }}{4\sqrt {2}}\frac {K^{\prime }C^{\prime 1/2}}{Q_0^{1/2}t^{1/4}} + \frac {\pi }{2}\frac {E^{\prime }B^{\prime }C^{\prime 2}}{Q_{0}},\quad\!\! w_{\tilde {k}b}=\frac {1}{\sqrt {2\pi }} \frac {K^{\prime }Q_0^{1/2}t^{1/4}}{E^{\prime } C^{\prime 1/2}} \sqrt {1-\xi ^2},\quad\!\! \ell _{\tilde {k}b}=\frac {Q_{0}}{\pi C^{\prime }}t^{1/2}. \end{align}

The solution converges to the classical toughness-dominated $\tilde {K}$ -vertex solution when $G_k\gg 1$ , while it converges to the backstress-dominated $B$ -vertex solution when $G_k\rightarrow 0$ . This form of the solution can be used to quickly estimate the contribution of backstress for the high-toughness, high-leak-off scenario.

It is important to note that the problem of a toughness-dominated plane strain hydraulic fracture with leak-off and backstress has been investigated in Liu et al. (Reference Liu, Zhang and Detournay2026). The general case of two-dimensional diffusion around the fracture is considered. In the limit of one-dimensional diffusion and Carter’s leak-off, the previously obtained result coincides with (3.24), as shown in Appendix B.

For the viscosity-dominated case, we analyse the $\tilde {M}B$ transition. With $G_k\rightarrow 0$ , the governing equations in (3.4) are simplified to

(3.25) \begin{align} &-\frac {1}{G_m}\frac {1}{\gamma _{\tilde {m}b}^{2}}\frac {\partial }{\partial \xi }\left (\varOmega _{\tilde {m}b}^{3}\frac {\partial \varPi _{\tilde {m}b}}{\partial \xi }\right ) + \frac {1}{\sqrt {1-\theta (\xi ,t)}}=0, \quad \varPi _{\tilde {m}b}=\varPi _{\tilde {m}}+\varPi _{\tilde {m}}^b, \nonumber\\ &\varPi _{\tilde {m}}=-\frac {1}{4\pi \gamma _{\tilde {m}b}}\int _{-1}^{1}\frac {\varOmega _{\tilde {m}b}}{(\xi -\zeta )^{2}}\mathrm{d}\zeta ,\quad \varPi _{\tilde {m}}^b=-\frac {1}{2\pi \gamma _{\tilde {m}b}}\int _{-1}^{1}\frac {\sqrt {1-\theta (\zeta ,t)}}{(\xi -\zeta )^{2}}\mathrm{d}\zeta ,\nonumber\\ &\varOmega _{\tilde {m}b}=0,\quad \xi \rightarrow 1,\quad \frac {1}{2\gamma _{\tilde {m}b}} = B^{\prime }\int _{0}^{1}\varOmega _{\tilde {m}b} \mathrm{d}\xi + 2\int _{0}^{1}\sqrt {1-\theta (\xi ,t)}\mathrm{d}\xi . \end{align}

Further, when $B^\prime \rightarrow 0$ , the storage term in the global volume balance can be neglected. Similarly to the $\tilde {K}B$ transition, it can be shown that $\ell \sim t^{1/2}$ and $\theta =\xi ^{2}$ in this limit, and that the integral terms with $\theta$ appearing in the elasticity and global volume balance equations are both constant. Therefore, the backstress contributes only to the pressure solution, while the length and opening solutions are identical to those for the leak-off-viscosity dominated $\tilde {M}$ -vertex solution. The $\tilde {M}$ -vertex solution for pressure $\varPi _{\tilde {m}0}$ , opening $\varOmega _{\tilde {m}0}$ and fracture length $\gamma _{\tilde {m}0}$ is converted to the $B$ -scaling, which leads to the following solution for the transition from the $\tilde {M}$ -vertex to the $B$ -vertex:

(3.26) \begin{equation} \varPi _{\tilde {m}b} = G_m^{1/4}\varPi _{\tilde {m}0} + \frac {\pi }{2},\quad \varOmega _{\tilde {m}b} = G_m^{1/4}\varOmega _{\tilde {m}0}, \quad \gamma _{\tilde {m}b} = \gamma _{\tilde {m}0} = \frac {1}{\pi }, \end{equation}

where the subscript $\tilde {m}0$ denotes the classical viscosity-leak-off-dominated $\tilde {M}$ -vertex solution (Adachi & Detournay Reference Adachi and Detournay2002), while the subscript $\tilde {m}b$ denotes the transition solution from the $\tilde {M}$ -vertex to the $B$ -vertex under the $B$ -scaling. The dimensional form of the above solution is

(3.27) \begin{align} p_{\tilde {m}b} = \left ( \frac {\mu ^{\prime }E^{\prime 3}C^{\prime 2}}{Q_0 t} \right )^{1/4}\!\! \varPi _{\tilde {m}0} + \frac {\pi }{2} \frac {E^{\prime }B^{\prime }C^{\prime 2}}{Q_{0}},\quad\!\! w_{\tilde {m}b} = \left ( \frac {\mu ^{\prime }Q_0^3 t}{E^{\prime }C^{\prime 2}} \right )^{1/4} \varOmega _{\tilde {m}0} ,\quad\!\! \ell _{\tilde {m}b} = \frac {Q_{0}}{\pi C^{\prime }}t^{1/2}. \end{align}

The above result converges to the $B$ -vertex solution (3.11) when $G_m\rightarrow 0$ , while when $G_m\gg 1$ , the $\tilde {M}$ solution is recovered.

It is also possible to calculate the correction to the storage-toughness $K$ -vertex solution for small leak-off and backstress, as indicated in Appendix C.

Figure 4.

Sketch of a radial fluid-driven fracture.

4.1. Governing equations

The radial, or penny-shaped, model assumes that the fracture propagates within a single plane and grows symmetrically from the injection point, as illustrated in figure 4. Mathematically, the governing equations (2.1) can be simplified by introducing the radial coordinate $r=\sqrt {x^2+y^2}$ and fracture radius $R$ as

(4.1) \begin{align} & \frac {\partial w}{\partial t}-\frac {1}{\mu ^{\prime }}\frac {1}{r}\frac {\partial }{\partial r}\left (rw^{3}\frac {\partial p}{\partial r}\right )+\frac {C^{\prime }}{\sqrt {t-t_{0}(r)}}=Q_{0}\delta (r),\nonumber\\ & p(r,t)=p_{f}-\sigma _{0}=-\frac {E^{\prime }}{2\pi R}\int _{0}^{R}M\left (\frac {r}{R},\frac {r^{\prime }}{R}\right )\frac {\partial (w+w^{\ell o})}{\partial r^{\prime }}\mathrm{d}r^{\prime }, \quad w^{\ell o}(r,t) =2B^{\prime }C^{\prime }\sqrt {t-t_{0}(r)},\nonumber\\ & \int _{0}^{R}(w(r,t)+2C^{\prime }\sqrt {t-t_{0}(r)})r\mathrm{d}r=\frac {Q_{0}t}{2\pi },\nonumber\\ & w\rightarrow \frac {K^{\prime }}{E^{\prime }}(R-r)^{1/2},\quad r\rightarrow R. \end{align}

Here $\delta (r)$ represents a unit point source at the origin in the fracture plane. The kernel for the elasticity equation is

(4.2) \begin{equation} M(\rho ,s)=\begin{cases} \frac {1}{\rho }K\left (\frac {s^{2}}{\rho ^{2}}\right )+\frac {\rho }{s^{2}-\rho ^{2}}E\left (\frac {s^{2}}{\rho ^{2}}\right ), & \rho \gt s,\\ \frac {s}{s^{2}-\rho ^{2}}E\left (\frac {\rho ^{2}}{s^{2}}\right ), & \rho \lt s. \end{cases} \end{equation}

The functions $K (\boldsymbol{\cdot })$ and $E (\boldsymbol{\cdot })$ denote the complete elliptic integrals of the first and the second kind, respectively. These governing equations closely follow those for a radial hydraulic fracture without the backstress (Madyarova Reference Madyarova2003; Kovalyshen Reference Kovalyshen2010; Dontsov Reference Dontsov2016a ).

4.2. General scaling

By following the procedures used in previous research (Madyarova Reference Madyarova2003; Kovalyshen Reference Kovalyshen2010), we introduce the following scaling:

(4.3) \begin{align} &r=R(t)\rho ,\quad R(t)=L(t)\gamma (t),\nonumber \\ &w(r,t)=W(t)\varOmega (\rho ,t),\quad p(r,t)=P(t)\varPi (\rho ,t), \end{align}

where $\rho$ is the dimensionless radial coordinate ( $0\leq \rho \leq 1$ ). By changing the derivatives as

(4.4) \begin{equation} \left .\frac {\partial }{\partial r}\right |_{r}=\frac {1}{L\gamma }\left .\frac {\partial }{\partial \rho }\right |_{\rho },\quad \left .\frac {\partial }{\partial t}\right |_{r}=\left .\frac {\partial }{\partial t}\right |_{\rho }-\rho \frac {\dot {R}}{R}\left .\frac {\partial }{\partial \rho }\right |_{\rho }, \end{equation}

the governing equations (4.1) become

(4.5) \begin{align} & G_{v}\left [\left (\frac {\dot {W}t}{W}\varOmega +\dot {\varOmega }t\right )-\rho \left (\frac {\dot {L}t}{L}+\frac {\dot {\gamma }t}{\gamma }\right )\frac {\partial \varOmega }{\partial \rho }\right ]-\frac {1}{G_{m}}\frac {1}{\gamma ^{2}\rho }\frac {\partial }{\partial \rho }\left (\rho \varOmega ^{3}\frac {\partial \varPi }{\partial \rho }\right )+\frac {G_{c}}{\sqrt {1-\theta (\rho ,t)}}\nonumber\\&\quad =\frac {1}{\gamma ^2}\delta (\rho ),\nonumber\\ & \varPi =-\frac {G_{e}}{2\pi \gamma }\int _{0}^{1}M(\rho ,\rho ^{\prime })\frac {\partial \varOmega }{\partial \rho ^{\prime }}\mathrm{d}\rho ^{\prime }-\frac {G_{p}}{\pi \gamma }\int _{0}^{1}M(\rho ,\rho ^{\prime })\frac {\partial }{\partial \rho ^{\prime }}\sqrt {1-\theta }\mathrm{d}\rho ^{\prime },\nonumber\\ & G_{v}\int _{0}^{1}\varOmega \rho \mathrm{d}\rho +2G_{c}\int _{0}^{1}\sqrt {1-\theta }\rho \mathrm{d}\rho =\frac {1}{2\pi \gamma ^{2}},\nonumber\\ & \varOmega =G_{k}\gamma ^{1/2}(1-\rho )^{1/2},\quad \rho \rightarrow 1. \end{align}

The dimensionless groups that appear in the above equations are

(4.6) \begin{align} &G_{v}=\frac {WL^{2}}{Q_{0}t},\quad G_{m}=\frac {\mu ^{\prime }Q_{0}}{W^{3}P},\quad G_{c}=\frac {C^{\prime }L^{2}}{Q_{0}t^{1/2}},\nonumber\\ &G_{e}=\frac {E^{\prime }W}{PL},\quad G_{p}=B^{\prime }\frac {G_{c}}{G_{v}}G_{e},\quad G_{k}=\frac {K^{\prime }L^{1/2}}{E^{\prime }W}. \end{align}

Similarly to the plane strain geometry, the dimensionless group $G_v$ characterizes the fracture volume relative to the injected volume; $G_m$ captures the effect of fluid viscosity; $G_c$ quantifies leak-off volume relative to the injected volume; $G_e$ reflects the relation between the opening and the pressure; $G_p$ measures the influence of backstress; $G_k$ represents fracture toughness.

4.3. Backstress-dominated $B$ -vertex

We aim to obtain a self-similar solution in the backstress-dominated regime. The backstress scaling can be defined by setting $G_{c}=G_{e}=G_{p}=1$ (similar to the plane strain case), with $G_{v}=B^{\prime }$ , which results in the following scales:

(4.7) \begin{equation} L=\left (\frac {Q_{0}t^{1/2}}{C^{\prime }}\right )^{1/2},\quad P=\frac {B^{\prime }E^{\prime }C^{\prime 3/2}t^{1/4}}{Q_{0}^{1/2}},\quad W=B^{\prime }C^{\prime }t^{1/2}, \end{equation}

and the two time-dependent dimensionless parameters,

(4.8) \begin{equation} G_{m}=\frac {\mu ^{\prime }Q_{0}^{3/2}}{B^{\prime 4}E^{\prime }C^{\prime 9/2}t^{7/4}},\quad G_{k}=\frac {K^{\prime }Q_{0}^{1/4}}{B^{\prime }E^{\prime }C^{\prime 5/4}t^{3/8}}. \end{equation}

Further, the backstress-dominated limiting solution occurs when $t\to \infty$ , and therefore $G_{m}\rightarrow 0$ and $G_{k}\rightarrow 0$ . This implies that the length $\gamma _{b0}$ is independent of time and $R=L\gamma \sim t^{1/4}$ . Consequently, the arrival time is related to the coordinate as $r\sim t_{0}^{1/4}$ , and the leak-off history is simplified to $\theta =t_{0}/t= (r/R )^{4}=\rho ^{4}$ . Since $G_{m}\rightarrow 0$ , the lubrication equation in (4.5) leads to the constant pressure solution, while the elasticity equation in (4.5) can be written as

(4.9) \begin{equation} \varPi _{b0}=-\frac {1}{2\pi \gamma _{b0}}\left [\int _{0}^{1}M(\rho ,\rho ^{\prime })\frac {\partial }{\partial \rho ^{\prime }}\big(\varOmega _{b0}+2\sqrt {1-\rho ^{\prime 4}}\big)\mathrm{d}\rho ^{\prime }\right ]=\mathrm{const}. \end{equation}

The latter equation can be inverted to obtain

(4.10) \begin{equation} \varOmega _{b0} = k \sqrt {1-\rho ^{2}}-2\sqrt {1-\rho ^{4}},\quad \varOmega _{b0}\rightarrow \big(\sqrt {2}k - 4\big)s^{1/2},\quad s\rightarrow 0, \end{equation}

where the second equation represents expansion near the fracture tip $s=1-\rho \rightarrow 0$ , and the constant $k$ is to be determined from the propagation condition. The elasticity equation also provides the value of the constant pressure as $\varPi _{b0} = \pi k / (8 \gamma _{b0})$ . Comparing the asymptote near the tip with the propagation condition in (4.5) gives

(4.11) \begin{equation} \sqrt {2}k - 4 = G_k \gamma _{b0}^{1/2}. \end{equation}

Finally, substituting the opening (4.10) into the global volume balance equation in (4.5) and using the fact that $G_k=0$ in (4.11), allows us to find the full limiting solution as

(4.12) \begin{align}\varPi _{b0} & =\frac {2^{-3/2}\pi }{\gamma _{b0}},\nonumber\\ \varOmega _{b0} & =2^{3/2}\sqrt {1-\rho ^{2}}-2\sqrt {1-\rho ^{4}},\nonumber\\ \gamma _{b0} & =\left [\frac {2^{5/2}\pi B^{\prime }}{3}+\frac {\pi ^{2}}{2}(1-B^{\prime })\right ]^{-1/2}. \end{align}

The scaling relations (4.3) and (4.7) further allow us to write the dimensional form of the above solution as

(4.13) \begin{align}p_{b0} & =2^{-3/2}\pi \left [\frac {2^{5/2}\pi B^{\prime }}{3}+\frac {\pi ^{2}}{2}(1-B^{\prime })\right ]^{1/2}\frac {B^{\prime }E^{\prime }C^{\prime 3/2}t^{1/4}}{Q_{0}^{1/2}},\nonumber\\ w_{b0} & =2^{3/2}B^{\prime }C^{\prime }t^{1/2}\sqrt {1-\rho ^{2}}-2B^{\prime }C^{\prime }t^{1/2}\sqrt {1-\rho ^{4}},\nonumber\\ R_{b0} & =\left [\frac {2^{5/2}\pi B^{\prime }}{3}+\frac {\pi ^{2}}{2}(1-B^{\prime })\right ]^{-1/2}\left (\frac {Q_{0}t^{1/2}}{C^{\prime }}\right )^{1/2}. \end{align}

As can be seen from the result, time dependence of the radius is similar to the leak-off dominated solutions. However, this solution features constant pressure and zero toughness. In the absence of backstress, a radially symmetric solution with uniform pressure corresponds to the toughness-controlled limit. If the toughness is set to zero, this limit gives a zero-aperture solution. In the present case, however, a non-zero aperture is obtained because the poroelastic backstress produces a spatially non-uniform compressive stress on the fracture faces.

To better understand the effect of backstress, it should be noted that the corresponding term for the radial fracture is

(4.14) \begin{equation} \varSigma _{b0}=-\frac { 1 }{\pi \gamma _{b0}}I(\rho ),\quad I(\rho )=\int _{0}^{1}M(\rho ,\rho ^{\prime })\frac {\partial }{\partial \rho ^{\prime }}\sqrt {1-\rho ^{\prime 4}}\mathrm{d}\rho ^{\prime }, \end{equation}

in which the integral can be evaluated numerically. As can be seen from the result shown in figure 5, the magnitude of the backstress is non-uniform and is higher near the fracture tip. This is different from the corresponding solution for the plane-strain fracture, for which the backstress is uniform. The higher stress magnitude compresses the fracture near the tips and allows a zero-toughness, constant-pressure solution. A conceptually similar situation occurs for pseudo-3-D fracture with constant stress barriers (Adachi et al. Reference Adachi, Detournay and Peirce2010; Dontsov & Peirce Reference Dontsov and Peirce2015), for which the stress barriers compress the fracture tips and there is a solution with zero toughness and constant pressure.

Figure 5.

Numerical value of the integral $I(\rho )$ that quantifies backstress for a radial fracture.

4.4. Parametric space

By following a similar process to that for the plane-strain fracture geometry, the next goal is to construct the parametric space for a single radial hydraulic fracture with backstress. Such a parametric space was constructed for the case of zero backstress in Dontsov (Reference Dontsov2016a ), and it was quantified by the following two dimensionless variables:

(4.15) \begin{equation} \tau =\frac {t K^{\prime 9}}{\left (\mu ^{\prime 5}E^{\prime 13}Q_{0}^{3}\right )^{1/2}},\quad \phi =\frac {\mu ^{\prime 3}E^{\prime 11}C^{\prime 4}Q_{0}}{K^{\prime 14}}. \end{equation}

To construct the updated parametric space, we employ the same approach as was used for the plane-strain fracture. Namely, the parametric space is constructed based on the limiting vertex solutions. In particular, the following metric is used:

(4.16) \begin{equation} p_{i0}=\epsilon p_{j0}\left (\frac {R}{2},t\right ),\quad \epsilon p_{i0}=p_{j0}\left (\frac {R}{2},t\right ), \end{equation}

where subscripts $i$ and $j$ ( $i\neq j$ ) correspond to $M$ , $\tilde {M}$ , $K$ , $\tilde {K}$ and $B$ vertex solutions. Similarly to the plane-strain case, the tolerance parameter $\epsilon$ is specified below through the calibration procedure used to construct the approximate parametric space. Equations (4.16) quantify boundaries of the transition zones (straight lines in the $(\log (\tau ),\log (\phi ))$ parametric space) and define the location of the vertex solutions. As for the plane strain geometry, net pressure evaluated in the middle between the injection point and the fracture tip is used as a metric to avoid singularities.

The limiting solutions for the radial hydraulic fracture geometry have been thoroughly investigated, including the $M$ -vertex (Savitski & Detournay Reference Savitski and Detournay2001, Reference Savitski and Detournay2002; Peirce & Detournay Reference Peirce and Detournay2008), the $\tilde {M}$ -vertex (Peirce & Detournay Reference Peirce and Detournay2008), the $K$ -vertex (Savitski & Detournay Reference Savitski and Detournay2002; Bunger et al. Reference Bunger, Detournay and Garagash2005) and the $\tilde {K}$ -vertex (Bunger et al. Reference Bunger, Detournay and Garagash2005). The vertex solutions are also investigated in Madyarova (Reference Madyarova2003) and Dontsov (Reference Dontsov2016a ). Table 4 summarizes the vertex solutions for pressure, including the new $B$ -vertex. Note that for the $B$ -vertex, we use the approximation $\gamma _{b0}\simeq 0.43$ , since the result is not very sensitive to the value of $B^{\prime }$ . This corresponds to $B^{\prime }=0.5$ .

Table 4.

Vertex pressure solutions for a radial hydraulic fracture.

Table 4. Long description

Panel A: The table lists vertex pressure solutions for a radial hydraulic fracture. It includes columns for Vertex, Expression, and Net pressure. Each row provides a specific vertex, its corresponding mathematical expression, and the net pressure value. Panel B: The Expression column contains mathematical formulas with variables and constants. Panel C: The Net pressure column lists numerical values and expressions for net pressure corresponding to each vertex.

By following the same procedure as for the plane strain geometry, we first recalculate the original $M\tilde {M}\tilde {K}K$ parametric space to determine the optimal value of $\epsilon$ . Since the parametric space is effectively defined by the boundaries of the transition regimes that are characterized by corresponding transition parameters, these need to be defined. To this end, by following the results in Dontsov (Reference Dontsov2016a ) and adding the corresponding transitions to the $B$ vertex, we have

(4.17) \begin{align} \tau ^{\prime }_{m\tilde {m}}=\tau ^{\prime }_{\tilde {m}b}=\tau ^{\prime }_{mb}=\tau \phi ^{9/14}, \quad \tau _{k\tilde {k}}'=\tau _{\tilde {k}b}'=\tau _{kb}'=\tau \phi ^{5/6}, \quad \tau _{mk}=\tau , \quad \tau ^{\prime }_{\tilde {m}\tilde {k}}=\tau \phi ^{-1/2}. \end{align}

As for the plane strain geometry, there are two boundaries: the first indicating the beginning of the transition, and the second representing the end of the transition. Superscripts ‘ $(1)$ ’ and ‘ $(2)$ ’ are used to denote the aforementioned two boundaries.

Table 5 summarizes the results for the transition regions that are obtained by substituting the vertex solutions from table 4 into (4.16). The parameter $\epsilon$ that is calculated according to

(4.18) \begin{equation} \epsilon =\arg \min _{\epsilon }\sum _{i}\left |\log a_{i}(\epsilon )-\log b_{i}\right |, \end{equation}

is determined as 0.403. This makes the newly calculated transitions as similar as possible to the target values calculated in Dontsov (Reference Dontsov2016a ). As for the plane strain case, $a_{i} (\epsilon )$ represents each of the transition boundaries and $b_{i}$ corresponds to the target values.

Table 5.

Boundaries of the transition between vertex solutions $M$ , $\tilde {M}$ , $\tilde {K}$ , $K$ for $\epsilon =0.403$ for radial hydraulic fracture geometry.

Table 5. Long description

The table has four columns: Transition edge, Scaled expression, Target, and Result. It contains data for different transition edges labeled as MK, M−M-tilde, K−K-tilde, and M-tilde−K-tilde. Each row provides a scaled expression, its target value, and the resulting value. For example, for the transition edge MK, the scaled expression is (ε * Πm0(0.5) / Πk0)^(15/2) with a target value of 4.54 x 10^−2 and a result of 2.66 x 10^−1. The table systematically compares these values across different transition edges and scaled expressions.

The same approach is applied to the transitions including the $B$ -vertex. Table 6 summarizes the results. There are two critical values of $B'$ , namely

(4.19) \begin{equation} B'_{k}=3.40\times 10^{-3},\quad B'_{m}=2.16\times 10^{-2}, \end{equation}

that determine if the corresponding leak-off-dominated regime exists. If $B'\gt B'_k$ , then there is no leak-off-toughness regime, while if $B'\gt B'_m$ , then there is no leak-off-viscosity regime. The latter critical values of $B'$ are obtained using $\tau _{\tilde {k}b}^{\prime (1 )}=\tau _{k\tilde {k}}^{\prime (2 )}$ and $\tau _{\tilde {m}b}^{\prime (1 )}=\tau _{m\tilde {m}}^{\prime (2 )}$ , respectively.

Table 6.

Boundaries of the transition between $M$ , $\tilde {M}$ , $\tilde {K}$ , $K$ vertices and the $B$ -vertex for $\epsilon =0.403$ for radial hydraulic fracture geometry.

Table 6. Long description

A table with three columns: Transition edge, Definition, and Bounds. The table contains four transition groups, each with two boundary values. Each row provides the definition and bounds for each transition edge. The transition groups are between K-tilde and B, between M-tilde and B, between K and B, and between M and B. Each transition edge has specific definitions and bounds associated with it, presented in mathematical notation.

The corresponding radial parametric spaces for small and large $B'$ are illustrated in figure 6. The qualitative behaviour depends primarily on the magnitude of $B'$ in relation to critical values $B'_{k}$ and $B'_{m}$ :

1. (i) $B'=0$ ;

   both transition boundaries tend to infinity, indicating that without poroelastic coupling, the fracture never enters the backstress-dominated regime at large times, i.e.

   (4.20) \begin{align} M\rightarrow \mathrm{transition}\rightarrow \tilde {K}, \end{align}
   where the transition can occur via $\tilde M$ or $K$ vertices, or can bypass both of them;

2. (ii) small $B'$ ;

   when $B'\lt B'_{k}$ and $B'\lt B'_{m}$ , the $\tilde {K}B$ and $\tilde {M}B$ boundaries shift to the right of the original $\tilde {K}$ and $\tilde {M}$ regimes, and the fracture evolves through

   (4.21) \begin{align} M\rightarrow \mathrm{transition}\rightarrow B, \end{align}
   where, depending on the parameters, the transition can pass through $\tilde M$ , $\tilde K$ , or $K$ and $\tilde K$ vertices;

3. (iii) large $B'$ ;

   when $B'$ exceeds $B'_{k}$ and $B'_{m}$ , both $\tilde {K}$ - and $\tilde {M}$ -vertices vanish, and the fracture behaviour is entirely governed by the $B$ -regime at large times, namely

   (4.22) \begin{align} M\rightarrow \mathrm{transition}\rightarrow B, \end{align}
   where the transition can occur via the $K$ vertex, or the solution can go directly from $M$ to $B$ vertex.

From a practical perspective, since the critical values $B'_{k}$ and $B'_{m}$ are relatively small, the fracture at large times generally evolves directly towards the $B$ -regime.

Figure 6.

Parametric space for a radial hydraulic fracture with backstress: (a) $B^{\prime }\lt B'_{k}\,,B^{\prime }\lt B'_{m}$ , with leak-off regimes; (b) $B^{\prime }\gt B'_{k}\,,B^{\prime }\gt B'_{m}$ , without leak-off regimes. The values of $B'_{k}$ and $B'_{m}$ are defined in (4.19). The red lines delineate the toughness-storage ( $K$ ) vertex, while the blue lines delineate the viscosity-storage ( $M$ ) vertex. The upper right-hand region, bounded by the brown solid and dashed lines, corresponds to the backstress-dominated ( $B$ ) regime. For small $B^\prime$ , the region enclosed by the magenta boundaries defines the toughness-leak-off ( $\tilde K$ ) vertex, whereas the region enclosed by the green boundaries defines the viscosity-leak-off ( $\tilde M$ ) vertex. For large $B^\prime$ , both the $\tilde M$ and $\tilde K$ vertices disappear. The numbered markers correspond to the cases discussed in § 6.

Figure 6. Long description

Panel A: A line graph shows the parametric space for a radial hydraulic fracture with backstress and leak-off regimes. The x-axis is labeled ‘log τ’ and the y-axis is labeled ‘log φ’. The graph includes various colored lines and regions representing different regimes: toughness-storage (red lines), viscosity-storage (blue lines), backstress-dominated (brown lines), toughness-leak-off (magenta boundaries), and viscosity-leak-off (green boundaries). Markers labeled 1(a) and 1(b) indicate specific cases. Panel B: Another line graph shows the parametric space without leak-off regimes. The x-axis is labeled ‘log τ’ and the y-axis is labeled ‘log φ’. Similar to Panel A, it includes colored lines and regions representing different regimes: toughness-storage (red lines), viscosity-storage (blue lines), and backstress-dominated (brown lines). Markers labeled 2(a) and 2(b) indicate specific cases.

4.5. Toughness-leak-off solution with backstress

It is possible to obtain an approximate transition solution between the $\tilde {K}$ - and $B$ -vertex. Following the $B$ -scaling for the radial fracture, we keep $G_k$ as the transition parameter, while $G_{m}\rightarrow 0$ since toughness domination is assumed. The parameter $G_k$ is related to the dimensionless evolution parameters by $G_k = B^{\prime -1} \tau ^{-3/8} \phi ^{-5/16} = (B^{\prime 8/3}\tau _{\tilde {k}b}^\prime )^{-3/8}$ . Also note that we need to set $B^{\prime }\rightarrow 0$ . As indicated on the parametric space in figure 6, the $\tilde K B$ transition occurs only for small values of $B^\prime$ , which is similar to what we had for the plane strain case. The goal is to solve (4.5) under these assumptions.

Figure 7.

Transition solution between the $\tilde {K}$ -vertex and the $B$ -vertex for a radial hydraulic fracture: (a) normalized opening profiles for different values of $G_k$ ; (b) variation of the pressure versus $G_k$ . The dashed lines indicate the $\tilde K$ and $B$ vertex solutions. The vertical dash–dotted lines in (b) mark the corresponding values of transition boundaries indicated in the legend. The parameter $G_k$ is related to the dimensionless evolution parameters as $G_k = B^{\prime -1} \tau ^{-3/8} \phi ^{-5/16} = (B^{\prime 8/3}\tau _{\tilde {k}b}^\prime )^{-3/8}$ .

Based on (4.10) and (4.11) that utilize the constant pressure assumption or $G_{m}\rightarrow 0$ , if we keep $G_k$ as the transition parameter, the opening solution can be written as

(4.23) \begin{equation} \varOmega _{\tilde {k}b}=\left (\frac {G_k \gamma _{\tilde {k}b}^{1/2}}{\sqrt {2}} + 2\sqrt {2} \right ) \sqrt {1-\rho ^{2}} - 2\sqrt {1-\rho ^{4}}, \end{equation}

with the corresponding pressure solution $\varPi _{\tilde {k}b} = \pi (G_k \gamma _{\tilde {k}b}^{1/2}/\sqrt {2} + 2\sqrt {2}) / (8 \gamma _{\tilde {k}b})$ . Since $G_v=B^{\prime }\rightarrow 0$ , the storage term can be neglected in the global volume balance, yielding $R\sim t^{1/4}$ , which is similar to both the $\tilde K$ - and $B$ -vertex. Substituting (4.23) into the global volume balance equation in (4.5) gives the fracture length $\gamma _{\tilde {k}b}=\sqrt {2}/\pi$ . Finally, the pressure solution is calculated by substituting the opening and fracture length into the elasticity equation in (4.5), and the full transition solution can be written as

(4.24) \begin{align}\varPi _{\tilde {k}b} & =\frac {2^{-7/2} \pi }{\gamma _{\tilde {k}b}}\left (G_k \gamma _{\tilde {k}b}^{1/2} + 4 \right ),\nonumber\\ \varOmega _{\tilde {k}b} & = 2^{-1/2} \left (G_k \gamma _{\tilde {k}b}^{1/2} + 4 \right ) \sqrt {1-\rho ^{2}} - 2\sqrt {1-\rho ^{4}},\nonumber\\ \gamma _{\tilde {k}b} & = \dfrac {\sqrt {2}}{\pi }. \end{align}

This solution is identical to the $\tilde {K}$ -vertex solution (Bunger et al. Reference Bunger, Detournay and Garagash2005; Dontsov Reference Dontsov2016a ) when $G_k\gg 1$ . At the same time, when $G_k\ll 1$ , the transition solution degenerates to the $B$ -vertex solution (assuming $B^\prime \rightarrow 0$ ).

Figure 7 illustrates the transition solutions for different values of $G_k$ . Note that the $\tilde {K}$ -vertex solution written in the $B$ -scaling is

(4.25) \begin{equation} \varPi _{\tilde {k}0} =\frac {2^{-7/2} \pi }{\gamma _{\tilde {k}b}}G_k \gamma _{\tilde {k}b}^{1/2},\quad \varOmega _{\tilde {k}0} = 2^{-1/2} G_k \gamma _{\tilde {k}b}^{1/2} \sqrt {1-\rho ^{2}}. \end{equation}

As shown in figure 7(a), the opening profile coincides with the $\tilde {K}$ -vertex solution when $G_k \gg 1$ and progressively approaches the $B$ -regime profile (grey dashed curves) with decreasing $G_k$ . It is also interesting to observe nearly zero toughness behaviour near the fracture tip for small values of $G_k$ . Figure 7(b) further illustrates that the pressure gradually transitions from the $\tilde {K}$ -vertex solution to the $B$ -regime solution as $G_k$ decreases. The boundaries from table 6 are indicated as vertical dash–dotted lines.

The corresponding dimensional version of the solution is

(4.26) \begin{align} &p_{\tilde {k}b}(\rho ,t) = \frac {2^{-3/2}\pi }{\gamma _{\tilde {k}b}} \frac {B^{\prime }E^{\prime }C^{\prime 3/2}t^{1/4}}{\ Q_{0}^{1/2}} + \frac {2^{-7/2}\pi }{\gamma _{\tilde {k}b}^{1/2}} K^{\prime }\left (\frac {C^{\prime }}{Q_{0}t^{1/2}}\right )^{1/4},\nonumber\\ &w_{\tilde {k}b}(\rho ,t) =2^{-1/2}\left \{ 4B^{\prime }C^{\prime }t^{1/2}+\frac {K^{\prime } \gamma _{\tilde {k}b}^{1/2}}{E^{\prime }}\left (\frac {Q_{0}t^{1/2}}{C^{\prime }}\right )^{1/4}\right \} \sqrt {1-\rho ^{2}}-2B^{\prime }C^{\prime }t^{1/2}\sqrt {1-\rho ^{4}},\nonumber\\ &R_{\tilde {k}b}(t) =\gamma _{\tilde {k}b}\left (\frac {Q_{0}t^{1/2}}{C^{\prime }}\right )^{1/2},\quad \gamma _{\tilde {k}b} = \dfrac {\sqrt {2}}{\pi }. \end{align}

As can be seen from the result, the solution consists of the $\tilde {K}$ -vertex solution and features additional terms with $B^\prime$ that represent the effect of backstress.

5.1. Governing equations

The classical constant height fracture model, or PKN model (Perkins & Kern Reference Perkins and Kern1961; Nordgren Reference Nordgren1972), assumes that the fracture height is constant and is equal to $H$ , while the fracture length is $\ell \gg H$ , so that the fracture is relatively long. This leads to the so-called local elasticity assumption, whereby fracture width in each vertical fracture cross-section is determined by fluid pressure in this cross-section, which is constant within the cross-section. Consequently, in the absence of backstress, this leads to an elliptically shaped fracture opening within each cross-section, while the magnitude of the opening depends on the value of fluid pressure. However, once the backstress is included, the cross-sectional solution is no longer elliptical since the backstress adds a non-uniform loading. The schematic diagram of the modified model is shown in figure 8.

Figure 8.

Sketch of a constant height fluid-driven fracture with backstress.

By following the results in Appendix D, the governing equations are written in terms of the effective opening and fluid flow that are defined as

(5.1) \begin{equation} \bar {w}=\frac {1}{H}\int _{-h/2}^{h/2}w(x,y,t)\mathrm{d}y,\quad \bar {q}_{x}=-\frac {1}{12H\mu }\frac {\partial p}{\partial x}\int _{-h/2}^{h/2}w^{3}\mathrm{d}y. \end{equation}

Here $\mu$ is fluid viscosity, and the local height is defined by

(5.2) \begin{equation} h=\sqrt {1-\beta ^{2}}H,\quad \beta (x,t)=\frac {\sigma _{b}(x,0,t)}{p(x,t)}=\frac {2E^{\prime }B^{\prime }C^{\prime }\sqrt {t-t_{0}(x)}}{\pi Hp(x,t)}, \end{equation}

where $E'$ and $C'$ are defined in (2.3), and $t_{0} (x )$ denotes the time at which the fracture front reached the point $x$ . The dimensionless parameter $\beta \in [0,1]$ signifies the ratio between the magnitude of the backstress and the net pressure, with $\beta =1$ representing the closed limiting case.

The opening and pressure are related by the local elasticity equation

(5.3) \begin{equation} \bar {w}=(1-\beta)^{2}\frac {\pi Hp}{2E^{\prime }}, \end{equation}

which, for the limiting case of $\beta =1$ , becomes

(5.4) \begin{equation} \bar {w}=0,\quad p=\frac {2E^{\prime }B^{\prime }C^{\prime }\sqrt {t-t_{0}(x)}}{\pi H}. \end{equation}

The latter result stems from (5.2) and represents the case of a practically closed fracture (or the fracture with negligible opening), in which fluid net pressure balances the backstress.

The combination of (5.2)–(5.4) allows us to write the relation between the $\beta$ coefficient and the average fracture width as

(5.5) \begin{align} \bar {w}\beta &=(1-\beta)^{2}B^{\prime }C^{\prime }\sqrt {t-t_{0}(x)},\quad \beta \lt 1,\nonumber\\ \bar {w}&=0,\quad \beta =1. \end{align}

The first relation in (5.5) may be used for the whole range $0\leq \beta \leq 1$ , since it also gives $\bar {w}=0$ when $\beta =1$ . It is therefore important not to divide this relation by $1-\beta$ , as doing so would remove the regular $\beta =1$ limiting case.

The lubrication equation written in terms of the effective width is

(5.6) \begin{equation} \frac {\partial \bar {w}}{\partial t}+\frac {\partial \bar {q}_{x}}{\partial x}+\frac {C^{\prime }}{\sqrt {t-t_{0}(x)}}=\frac {Q_{0}}{H}\delta (x),\quad \bar {q}_{x}=-\frac {f_{q}(\beta )}{1-\beta ^{2}}\frac {\bar {w}^{3}}{\pi ^{2}\mu }\frac {\partial p}{\partial x}. \end{equation}

As can be seen from the result, it differs from the classical equation by a multiplier that depends on $\beta$ , while $f_q(\beta )=O(1)$ is defined in Appendix D. This multiplier accounts for the reduction of fracture width and the associated hydraulic conductivity caused by backstress.

Following Sarvaramini & Garagash (Reference Sarvaramini and Garagash2015), the propagation condition is

(5.7) \begin{equation} \bar {w}(\ell ,t)=\frac {\sqrt {\pi H}K_{Ic}}{E^{\prime }}. \end{equation}

This condition is independent of $\beta$ , since $t=t_0$ and hence $\beta =0$ at the lateral fracture tip.

Finally, the global volume balance can be obtained by integrating (5.6) as follows:

(5.8) \begin{equation} \int _{0}^{\ell (t)}\left (\bar {w}(x,t)+2C^{\prime }\sqrt {t-t_{0}}\right )\mathrm{d}x=\frac {Q_{0}t}{2H}. \end{equation}

These governing equations represent a model for a constant height hydraulic fracture with poroelastic backstress. The primary additional quantity is the poroelastic parameter $\beta$ , which measures the ratio between the backstress and the net pressure. Its spatial variation reflects the exposure time of the fracture surface to leak-off: $\beta =0$ at the fracture tip and increases towards the inlet, where the cumulative leak-off effect is stronger. The poroelastic effects can compress the fracture and reduce the effective height near the inlet, as schematically shown in figure 8. Overall, the elasticity and lubrication equations are affected by backstress, while the propagation condition is independent of $\beta$ . When $\beta =0$ and the backstress is negligible, these equations reduce to the classical constant-height fracture model based on local elasticity.

5.2. General scaling

By following the approach of Adachi (Reference Adachi2001), we start addressing the problem by using the following scaling:

(5.9) \begin{equation} x=\xi \ell (t),\quad \ell (t)=L(t)\gamma (t),\quad \bar {w}(x,t)=W(t)\varOmega (\xi ,t), \end{equation}

where $\xi \in [0,1]$ is the dimensionless coordinate. By calculating the derivatives as

(5.10) \begin{equation} \left .\frac {\partial }{\partial x}\right |_{x}=\frac {1}{\ell }\left .\frac {\partial }{\partial \xi }\right |_{\xi },\quad \left .\frac {\partial }{\partial t}\right |_{x}=\left .\frac {\partial }{\partial t}\right |_{\xi }-\xi \frac {\dot {\ell }}{\ell }\frac {\partial }{\partial \xi }, \end{equation}

the scaled governing equations and the boundary conditions for the constant height fracture problem (5.5)–(5.8) become

(5.11) \begin{align} & G_{v}\left [\frac {\dot {W}}{W}\varOmega t+\dot {\varOmega }t-\xi \left (\frac {\dot {\gamma }t}{\gamma }+\frac {\dot {L}t}{L}\right )\frac {\partial \varOmega }{\partial \xi }\right ]-\frac {2}{\pi ^{3}G_{m}}\frac {1}{\gamma ^{2}}\frac {\partial }{\partial \xi }\left [\frac {f_{q}(\beta )}{1-\beta ^{2}}\varOmega ^{3}\frac {\partial }{\partial \xi }\left (\frac {\varOmega }{(1-\beta)^{2}}\right )\right ] \nonumber\\ &\quad +\frac {G_{c}}{\sqrt {1-\theta (\xi ,t)}} =\frac {1}{\gamma }\delta (\xi),\nonumber\\ &\varOmega \beta =(1-\beta)^{2}G_p\sqrt {1-\theta (\xi ,t)},\quad \varOmega (1)=\sqrt {\pi }G_{k},\nonumber\\ &G_{v}\int _{0}^{1}\varOmega (\xi ,t)\mathrm{d}\xi +2G_{c}\int _{0}^{1}\sqrt {1-\theta (\xi ,t)}\mathrm{d}\xi =\frac {1}{2\gamma }, \end{align}

where the history function is defined as $\theta (\xi ,t )=t_{0} (x )/t$ , the poroelastic parameter $\beta =\beta (\xi ,t )$ varies spatially and also depends on time, while the dimensionless groups are

(5.12) \begin{align} &G_{v}=\frac {WLH}{Q_{0}t},\quad G_{m}=\frac {\mu Q_{0}L}{E^{\prime }W^{4}},\quad G_{c}=\frac {C^{\prime }LH}{Q_{0}t^{1/2}},\nonumber \\ &G_p=\frac {B^{\prime }C^{\prime }t^{1/2}}{W}=B^{\prime }\frac {G_{c}}{G_{v}},\quad G_k=\frac {K_{Ic}H^{1/2}}{E^{\prime }W}. \end{align}

The dimensionless group $G_v$ characterizes the fracture volume relative to the injected volume; $G_m$ captures the effect of fluid viscosity; $G_c$ quantifies leak-off volume relative to the injected volume; $G_p$ measures the influence of backstress; $G_k$ represents fracture toughness. The dimensionless formulation allows us to reduce the number of problem parameters and to focus on specific limiting solutions by carefully choosing the values of such parameters.

Note that unlike the plane strain and radial geometries, the elasticity related group $G_e$ does not exist in these equations, as the local elasticity equations (5.3) and (5.4) were used to substitute pressure as a function of opening.

5.3. Backstress-dominated $B$ -vertex

In order to obtain the limit of backstress-dominated self-similar solution, we let $G_c=G_{p}=1$ in (5.12), with $G_v=B^\prime$ . Recall that there is no $G_e$ in the formulation, but otherwise the approach is the same as for the other two fracture geometries considered. In this way, we determine the scales

(5.13) \begin{equation} L = \frac {Q_0 t^{1/2}}{C^\prime H},\quad W = B^{\prime }C^\prime t^{1/2}. \end{equation}

For completeness, we also define the scale for net pressure using (5.3) as

(5.14) \begin{equation} P = \frac {E^{\prime }B^{\prime }C^{\prime }t^{1/2}}{H}. \end{equation}

The remaining dimensionless groups or evolution parameters are

(5.15) \begin{equation} G_m = \frac {\mu Q_0^2}{B^{\prime 4} E^\prime C^{\prime 5}Ht^{3/2}},\quad G_k=\frac {K_{Ic}H^{1/2}}{B^{\prime }E^{\prime }C^{\prime }t^{1/2}}. \end{equation}

The $B$ -vertex appears at large times $t\to \infty$ for which both dimensionless groups $G_m$ and $G_k$ vanish. For a self-similar solution, the dimensionless variables are therefore time independent, and since $\ell (t)=L(t)\gamma$ , we obtain $\ell \sim t^{1/2}$ and $\theta =\xi ^{2}$ . For finite scaled flux, taking $G_m\to 0$ in (5.11) gives the first relation below wherever the leading-order opening is positive. Together with $G_k\to 0$ , the limiting equations are

(5.16) \begin{align} &\frac {\varOmega _{b0}}{(1-\beta _{b0})^2} =\mathrm{const},\nonumber\\ &\varOmega _{b0}\beta _{b0}=\left (1-\beta _{b0}\right )^{2} \sqrt {1-\xi ^2},\quad \varOmega _{b0}(1)=0,\nonumber\\ &B^\prime \int _{0}^{1}\varOmega _{b0}(\xi)\mathrm{d}\xi +2\int _{0}^{1}\sqrt {1-\xi ^2}\mathrm{d}\xi =\frac {1}{2\gamma _{b0}}. \end{align}

The first two relations in (5.16) imply that, if positive, the opening increases towards the tip and therefore cannot decrease to zero. This makes it impossible to satisfy the zero-opening condition at the tip. Hence the leading-order opening vanishes throughout the fracture. The second relation then gives $\beta _{b0}=1$ in the fracture interior, while the global volume balance determines the fracture length. Thus, the dimensionless $B$ -vertex solution is

(5.17) \begin{equation} \beta _{b0}=\begin{cases} 1, & 0\leq \xi \lt 1,\\ 0, & \xi =1, \end{cases},\quad \varOmega _{b0}=0,\quad \varPi _{b0}=\frac {2}{\pi }\sqrt {1-\xi ^2},\quad \gamma _{b0}=\frac {1}{\pi }, \end{equation}

where the elasticity relation (5.4) and the scaling relation (5.14) are used to calculate the net pressure. The result $\varOmega _{b0}=0$ should be understood in the same sense as in the plane-strain case, i.e. it is negligible in the given scaling but is not exactly zero. In the fracture interior $0\leq \xi \lt 1$ , $\beta _{b0}=1$ indicates a leading-order balance between net pressure and poroelastic backstress. Since the local elasticity relation contains the factor $(1-\beta )^2$ , no aperture of order $W_b=B'C't^{1/2}$ is resolved at the $B$ -vertex.

Then, the dimensional solution for the $B$ -vertex is obtained from the dimensionless solution (5.17) using (5.13) and (5.14) as

(5.18) \begin{align}\bar {w}_{b0} & =o(B'C't^{1/2}),\nonumber\\ p_{b0} & =\frac {2E^{\prime }B^{\prime }C^{\prime }t^{1/2}}{\pi H}\sqrt {1-\xi ^{2}},\nonumber\\ \ell _{b0} & =\frac {1}{\pi }\frac {Q_{0}t^{1/2}}{C^{\prime }H}. \end{align}

It is also useful to report the corresponding backstress. From the definition (5.2), the backstress for the constant height fracture is

(5.19) \begin{align} \sigma _{b}(x,t)=p(x,t)\,\beta (x,t)=\frac {2E^{\prime }B^{\prime }C^{\prime }\sqrt {t-t_{0}(x)}}{\pi H},\quad \frac {\sigma _{b}}{P}=\varPi \,\beta = \frac {2}{\pi } \sqrt {1-\xi ^{2}}. \end{align}

It is interesting to observe that, in the $B$ -vertex solution (5.18), no aperture of order $B'C't^{1/2}$ is resolved, while the net pressure supports the backstress. This result is qualitatively similar to that for a plane-strain hydraulic fracture (3.12). At the same time, it is different from the radial case (4.13), for which a finite leading-order width profile is obtained at the $B$ -vertex.

5.4. Parametric space

Similarly to the plane strain and radial fracture geometries, the next step is to construct the parametric space for a single constant height fluid-driven fracture with backstress. In the absence of backstress, such a parametric space was constructed in Dontsov (Reference Dontsov2022) and it was quantified by the following two dimensionless variables:

(5.20) \begin{equation} \tau =\frac {2\pi ^{1/2}E^{\prime 4}\mu Q_{0}^{2}t}{H^{7/2}K_{Ic}^{5}}, \quad \phi =\left (\frac {H^{5}K_{Ic}^{6}C^{\prime 4}}{4\pi ^{3}E^{\prime 4}\mu ^{2}Q_{0}^{4}}\right )^{1/4}. \end{equation}

To construct a modified parametric space that accounts for the effect of backstress, we employ the same approach as was used for the plane strain and radial fracture geometries. Specifically, the parametric space is constructed using the limiting vertex solutions using the following relations:

(5.21) \begin{equation} \ell _{i0}^{-1}p_{i0} =\epsilon \ell _{j0}^{-1}p_{j0}\left (\frac {\ell }{2},t\right ),\quad \epsilon \ell _{i0}^{-1}p_{i0}=\ell _{j0}^{-1}p_{j0}\left (\frac {\ell }{2},t\right ). \end{equation}

Similarly to the previously considered fracture geometries, subscripts $i$ and $j$ ( $i\neq j$ ) correspond to $M$ , $\tilde {M}$ , $K$ , $\tilde {K}$ and $B$ vertex solutions. Similarly to the plane-strain case, the tolerance parameter $\epsilon$ is specified below through the calibration procedure used to construct the approximate parametric space. These equations (5.21) quantify boundaries of the transition zones (straight lines in the $(\log (\tau ),\log (\phi ))$ parametric space) and define the location of the vertex solutions. In contrast to the plane strain and radial geometries, a slightly different metric is used, which involves the ratio between the net pressure and fracture length. This is a necessity because the pressure solutions for the $K$ - and $\tilde {K}$ -vertex are identical.

Table 7.

Vertex pressure and length solutions for a constant height fracture.

Table 7. Long description

The table has three columns: Vertex, Expression, and Dimensionless solution. The five vertex groups are K, K-tilde, M, M-tilde and B. Each group gives dimensional net pressure p, fracture half-length ell, their ratio, dimensionless pressure capital Pi, and dimensionless length coefficient gamma. In each group below, p, ell, capital Pi and gamma carry the matching subscript specified for that group. Lowercase pi denotes the mathematical constant. “Prime” denotes a prime attached to a symbol; “tilde” denotes a tilde above a letter. Q subscript zero is the injection rate, and K subscript Ic is fracture toughness.K vertex; matching subscript k0: Pressure p of x and t equals K subscript Ic divided by the square root of H, multiplied by capital Pi. Half-length ell equals E prime times Q subscript zero times t times gamma, divided by the product of K subscript Ic and H raised to three halves. The ratio p divided by ell equals K subscript Ic squared times H, divided by the product of E prime and Q subscript zero, multiplied by t raised to minus one and by capital Pi divided by gamma. Capital Pi equals two divided by the square root of pi. Gamma equals one divided by twice the square root of pi. K-tilde vertex; matching subscript k-tilde zero: Pressure p of x and t equals K subscript Ic divided by the square root of H, multiplied by capital Pi. Half-length ell equals Q subscript zero divided by the product of C prime and H, multiplied by the square root of t and by gamma. The ratio p divided by ell equals K subscript Ic times C prime times the square root of H, divided by Q subscript zero, multiplied by t raised to minus one half and by capital Pi divided by gamma. Capital Pi equals two divided by the square root of pi. Gamma equals one divided by pi. M vertex; matching subscript m0: Pressure p of x and t equals the fifth root of the ratio of mu times E prime to the fourth power times Q subscript zero squared to H to the sixth power. This fifth root is multiplied by t raised to one fifth and by capital Pi of xi. Half-length ell equals the fifth root of the ratio of E prime times Q subscript zero cubed to the product of mu and H to the fourth power. This fifth root is multiplied by t raised to four fifths and by gamma. The ratio p divided by ell equals the fifth root of the ratio of mu squared times E prime cubed to the product of H squared and Q subscript zero. This fifth root is multiplied by t raised to minus three fifths and by capital Pi of xi divided by gamma. Capital Pi of xi equals 1.12 times the cube root of one minus xi, multiplied by the quantity one minus the ratio of one minus xi to 96.Gamma equals 0.38. M-tilde vertex; matching subscript m-tilde zero: Pressure p of x and t equals the fourth root of the ratio of mu times E prime cubed times Q subscript zero squared to the product of C prime and H to the fifth power. This fourth root is multiplied by t raised to one eighth and by capital Pi of xi. Half-length ell equals Q subscript zero divided by the product of C prime and H, multiplied by the square root of t and by gamma. The ratio p divided by ell equals the fourth root of the ratio of mu times E prime cubed times C prime cubed to the product of H and Q subscript zero squared. This fourth root is multiplied by t raised to minus three eighths and by capital Pi of xi divided by gamma. Capital Pi of xi is the product of four factors: the fourth root of the ratio of 32 to pi cubed; the eighth root of eight ninths; the quantity one minus xi raised to three eighths; and one minus the product of the quantity one minus the eighth root of nine eighths and the quantity one minus xi.

The limiting solutions without backstress have been previously investigated (see e.g. Kovalyshen & Detournay (Reference Kovalyshen and Detournay2010) and Dontsov (Reference Dontsov2022)). The vertex solutions for fluid pressure and fracture length, including the new $B$ -vertex, are summarized in table 7.

By following the same procedure as for the previously considered fracture geometries, we first recalculate the original $M\tilde {M}\tilde {K}K$ parametric space to determine the optimal value of $\epsilon$ . Since the parametric space is defined by boundaries of the transition regimes, we need to define these. At this point, we define the following dimensionless transition parameters:

(5.22) \begin{align} \tau _{km}^{\prime }=\tau ,\quad \tau _{k\tilde {k}}^{\prime }=\tau _{\tilde {k}b}^{\prime }=\tau _{kb}^{\prime }=\tau \phi ^{2},\quad \tau _{\tilde {k}\tilde {m}}^{\prime }=\tau \phi ^{-2},\quad \tau _{m\tilde {m}}^{\prime }=\tau _{\tilde {m}b}^{\prime }=\tau _{mb}^{\prime }=\tau \phi ^{10/3}, \end{align}

which include both the original transitions for the case of no backstress, as well as the new transitions associated with the $B$ vertex. Since each transition is associated with the beginning (i.e. departure from one vertex) and the end (i.e. arrival to another vertex), we utilize the superscripts ‘ $(1)$ ’ and ‘ $(2)$ ’ to indicate both boundaries of the transition.

Table 8 summarizes the results for the transition regions that are obtained by substituting the vertex solutions from table 7 into (5.21). The parameter $\epsilon$ is determined by

(5.23) \begin{equation} \epsilon =\arg \min _{\epsilon }\sum _{i}\left |\log a_{i}(\epsilon )-\log b_{i}\right |, \end{equation}

which yields $\epsilon =0.145$ . This makes the newly calculated transitions as similar as possible to the target values that are calculated in Dontsov (Reference Dontsov2022). Similarly to the plane strain and radial cases, $a_{i} (\epsilon )$ represents each of the transition boundaries and $b_{i}$ corresponds to the target values. The agreement in table 8 is less satisfactory than for the plane-strain and radial cases. Since the pressure scales for the $K$ - and $\tilde K$ -vertices are identical, the boundaries are estimated using $p/\ell$ rather than the pressure alone. Moreover, several transition relations involve high powers of the tolerance parameter $\epsilon$ . For instance, the $\tilde K\tilde M$ transition contains the power of eight, so small changes in $\epsilon$ are amplified into orders-of-magnitude changes in the resulting location of the boundary. The constructed parametric space should therefore be interpreted as a qualitative regime map on a logarithmic scale, rather than a precise instrument that identifies transition boundaries.

Table 8.

Boundaries of the transition between $M$ , $\tilde {M}$ , $\tilde {K}$ and $K$ vertices for $\epsilon =0.145$ for constant height fracture geometry.

Table 8. Long description

The table has four columns and four transition groups, each containing two boundary rows. The columns are labeled Transition edge, Scaled expression, Target, and Result. The rows are labeled with different transition edges: KM, KK, KM, and MM” with “KM, K–K-tilde, K-tilde–M-tilde, and M–M-tilde. Each row contains values for the scaled expression, target, and result. For example, the first boundary of KM has the scaled expression obtained by multiplying epsilon, capital Pi subscript k0 and gamma subscript m0, dividing by the product of gamma subscript k0 and capital Pi subscript m0 evaluated at 0.5, raising this entire ratio to the power five halves, and then multiplying by twice the square root of pi. This equals tau-prime with subscript km and superscript (1). Both its target and result are 0.11. The table provides detailed numerical values for each transition edge, scaled expression, target, and result.

The same approach is applied to the transitions to the $B$ -vertex. Table 9 summarizes the results. There are two critical values of $B'$ , namely

(5.24) \begin{equation} B'_{k}=3.82\times 10^{-2},\quad B'_{m}=8.70\times 10^{-3}, \end{equation}

that determine if the corresponding leak-off-dominated regimes exist. If $B'\gt B'_k$ , then there is no leak-off-toughness regime, while if $B'\gt B'_m$ , then there is no leak-off-viscosity regime. The latter critical values of $B'$ are obtained using $\tau _{\tilde {k}b}^{\prime (1 )}=\tau _{k\tilde {k}}^{\prime (2 )}$ and $\tau _{\tilde {m}b}^{\prime (1 )}=\tau _{m\tilde {m}}^{\prime (2 )}$ , respectively.

Table 9.

Boundaries of the transition between $M$ , $\tilde {M}$ , $\tilde {K}$ , $K$ vertices and the $B$ -vertex for $\epsilon =0.145$ for constant height fracture geometry.

Table 9. Long description

The table has three columns and four transition groups, each with two boundary values. The columns are labeled Transition edge, Definition, and Bounds. The rows detail the transitions between different vertices and their corresponding definitions and bounds. Row 1: Transition edge, K-tilde B; Definition, τ′k-tilde b = τϕ², B′ < B′k; Bounds, τ′(1)k-tilde b ≃ 2.80 × 10^−2 B′^−2, τ′(2)k-tilde b ≃ 6.34 × 10^1 B′^−2. Row 2: Transition edge, M-tilde B; Definition, τ′m-tilde b = τϕ^10/3, B′ < B′m; Bounds, τ′(1)m-tilde b ≃ 9.08 × 10^−4 B′^−8/3, τ′(2)m-tilde b ≃ 2.69 × 10^1 B′^−8/3. Row 3: Transition edge, K-tilde B; Definition, τ′k-tilde b = τϕ², B′ > B′k-tilde; Bounds, τ′(1)k-tilde b ≃ 1.07 × 10^−1 B′^−1, τ′(2)k-tilde b ≃ 5.07 B′^−1. Row 4: Transition edge, M-tilde B; Definition, τ′m-tilde b = τϕ^10/3, B′ > B′m-tilde; Bounds, τ′(1)m-tilde b ≃ 4.20 × 10^−3 B′^−5/3, τ′(2)m-tilde b ≃ 2.61 B′^−5/3.

Figure 9.

Parametric space for a constant height hydraulic fracture with backstress: (a) $B^{\prime }\lt B'_{k}\,,B^{\prime }\lt B'_{m}$ , with leak-off regimes; (b) $B^{\prime }\gt B'_{k}\,,B^{\prime }\gt B'_{m}$ , without leak-off regimes. The values of $B'_{k}$ and $B'_{m}$ are defined in (5.24). The red lines delineate the toughness-storage ( $K$ ) vertex, while the blue lines delineate the viscosity-storage ( $M$ ) vertex. The upper right-hand region, bounded by the brown solid and dashed lines, corresponds to the backstress-dominated ( $B$ ) regime. For small $B^\prime$ , the region enclosed by the magenta boundaries defines the toughness-leak-off ( $\tilde K$ ) vertex, whereas the region enclosed by the green boundaries defines the viscosity-leak-off ( $\tilde M$ ) vertex. For large $B^\prime$ , both the $\tilde K$ and $\tilde M$ vertices disappear. The numbered markers correspond to the cases discussed in § 6.

The parametric spaces for small and large $B'$ are illustrated in figure 9. When $B'$ exceeds $B'_{k}$ or $B'_{m}$ , the corresponding leak-off regime disappears. For sufficiently large $B'$ , both the $\tilde {K}$ - and $\tilde {M}$ -vertex vanish, and the fracture behaviour is entirely governed by the $B$ -regime at large times. From a practical perspective, since the critical values $B'_{k}$ and $B'_{m}$ are relatively small, the fracture at large times generally evolves directly towards the $B$ -regime.

The qualitative transition behaviour of the solution depends primarily on the magnitude of $B'$ in relation to critical values $B'_{k}$ and $B'_{m}$ :

1. (i) $B'=0$ ;

   both transition boundaries tend to infinity, indicating that without poroelastic coupling, the fracture never enters the backstress-dominated regime at large times, i.e.

   (5.25) \begin{align} K\rightarrow \mathrm{transition}\rightarrow \tilde {M}, \end{align}
   where the transition can occur via $\tilde K$ , $M$ vertices, or can bypass both of them;

2. (ii) small $B'$ ; when

   $B'\lt B'_{k}$ and $B'\lt B'_{m}$ , the $\tilde {K}B$ and $\tilde {M}B$ boundaries shift upward and to the right of the original $\tilde {K}$ and $\tilde {M}$ regimes, and the fracture evolves through

   (5.26) \begin{align} K\rightarrow \mathrm{transition}\rightarrow B, \end{align}
   depending on the parameters, the transition can pass through $\tilde K$ , $\tilde M$ or $M$ followed by $\tilde M$ ;

3. (iii) large $B'$ ; when $B'\gt B'_{k}$ and $B'\gt B'_{m}$ , the qualitative behaviour of the solution is

   (5.27) \begin{align} K\rightarrow \mathrm{transition}\rightarrow B, \end{align}
   where the transition can occur via the $M$ vertex, or the solution can go directly from $K$ to $B$ vertex.

5.5. Transition solutions

The parametric-space analysis reveals an intricate connection between the $B$ -regime and the leak-off-dominated regimes. To elucidate this connection, we derive the transition solutions $\tilde {K}B$ and $\tilde {M}B$ , respectively.

First, we derive the transition solution from $\tilde {K}$ -vertex to $B$ -vertex. For the $\tilde {K}B$ transition, the evolution parameter $G_k$ is related to the dimensionless evolution parameters by $G_k = (\sqrt {\pi } B^\prime \tau ^{1/2} \phi )^{-1}$ . For the toughness-leak-off dominated limiting solution, we set $G_{m}\rightarrow 0$ and assume $G_{v}=B^\prime \ll 1$ . Under these conditions, $\ell =L\gamma \sim t^{1/2}$ , and the leak-off history is simplified to $\theta =t_{0}/t= (x/\ell )^{2}=\xi ^{2}$ . Given these considerations, the simplified governing equations can be obtained from (5.11) as

(5.28) \begin{align} &\frac {\varOmega _{\tilde {k}b}}{ (1-\beta _{\tilde {k}b})^{2}}=\mathrm{const},\,\,\beta _{\tilde {k}b}\lt 1, \quad \varOmega _{\tilde {k}b}(1)=\sqrt {\pi }G_k, \nonumber \\ &2\int _{0}^{1}\sqrt {1-\xi ^{2}}\mathrm{d}\xi =\frac {1}{2\gamma _{\tilde {k}b}}, \quad \varOmega _{\tilde {k}b}\beta _{\tilde {k}b}= (1-\beta _{\tilde {k}b})^{2} \sqrt {1-\xi ^{2}}. \end{align}

These equations can be easily solved and the corresponding dimensionless solution for the $\tilde {K}B$ transition can be summarized as

(5.29) \begin{align} & \varOmega _{\tilde {k}b}=\sqrt {\pi }G_k (1-\beta _{\tilde {k}b})^{2},\quad \gamma _{\tilde {k}b}=\frac {1}{\pi },\nonumber\\ & \varPi _{\tilde {k}b}=\max \left \{ \frac {2}{\sqrt {\pi }} G_k, \frac {2}{\pi }\sqrt {1-\xi ^{2}}\right \} ,\quad 0\leq \xi \leq 1,\nonumber\\ & \beta _{\tilde {k}b}=\min \left \{ 1,\frac {1}{\sqrt {\pi }}G_k^{-1}\sqrt {1-\xi ^{2}}\right \} ,\quad 0\leq \xi \leq 1, \end{align}

where the elasticity relation (5.4) and the scaling relation (5.14) are used to calculate the net pressure.

The corresponding dimensional solution is

(5.30) \begin{align} \bar {w}_{\tilde {k}b} &=\sqrt {\pi }(1-\beta _{\tilde {k}b})^2 \frac {K_{Ic}H^{1/2}}{E^{\prime }},\quad \ell _{\tilde {k}b}=\frac {Q_0t^{1/2}}{\pi C^{\prime }H},\nonumber\\ p_{\tilde {k}b} &=\max \left \{\frac {2K_{Ic}}{\sqrt {\pi H}}, \frac {2E^{\prime }B^{\prime }C^{\prime }t^{1/2}\sqrt {1-\xi ^2}} {\pi H}\right \},\quad 0\leq \xi \leq 1,\nonumber\\ \beta _{\tilde {k}b} &=\min \left \{1, \frac {E^{\prime }B^{\prime }C^{\prime }t^{1/2}\sqrt {1-\xi ^2}} {\sqrt {\pi }K_{Ic}H^{1/2}}\right \},\quad 0\leq \xi \leq 1. \end{align}

The profile of the coefficient $\beta _{\tilde {k}b}$ is shown in figure 10. For large values of the parameter $G_k$ , the profile remains elliptical, while its magnitude increases with decreasing $G_k$ . Once $\beta (\xi )$ reaches $1$ , it is capped and becomes constant. Accordingly, the profile first touches the ‘ceiling’ $\beta =1$ near the inlet and then progressively extends towards the tip as $G_k$ decreases, approaching $1$ at every fixed point $0\leq \xi \lt 1$ as $G_k\to 0$ . This trend indicates that the backstress contribution is the strongest at the inlet and the weakest at the tip. As $\beta$ increases, backstress becomes increasingly dominant, and the net pressure ultimately balances the backstress throughout the whole fracture.

Figure 10.

Profile of the coefficient $\beta$ for the $\tilde {K}B$ solution of a constant-height fracture, with $G_k$ as the evolution parameter.

In order to compare the $\tilde {K}B$ transition solution with the $\tilde {K}$ solution, it is useful to write the latter in the scaled form using the same $B$ scaling. Specifically, the $\tilde K$ solution is recovered from (5.29) in the limit $G_k\to \infty$ , for which $\beta \to 0$ :

(5.31) \begin{equation} \varOmega _{\tilde {k}0}=\sqrt {\pi }G_k,\quad \gamma _{\tilde {k}b}=\frac {1}{\pi },\quad \varPi _{\tilde {k}0}= \frac {2}{\sqrt {\pi }} G_k. \end{equation}

Figure 11 illustrates the solution in terms of spatial variation of the dimensionless fracture opening $\varOmega _{\tilde {k}b}$ and dimensionless pressure $\varPi _{\tilde {k}b}$ for different values of the transition parameter $G_{k}$ . As can be seen from the results, when $G_{k}\gg 1$ the solution approaches the $\tilde K$ solution. At the same time, for $G_{k}\to 0$ , the solution approaches the $B$ solution. It is also interesting to observe that the fracture has the highest opening at the fracture tip, which is counterintuitive. Partially, this is related to using the local elasticity together with toughness propagation condition that implies that there is a finite fracture opening at the tip. As is shown in Dontsov & Peirce (Reference Dontsov and Peirce2016), the usage of non-local elasticity leads to a smooth transition of the fracture opening to zero at the fracture tip, while the local elasticity solution is able to accurately capture the far field behaviour away from the tip. Therefore, the solution for the fracture opening is qualitatively similar to that for a buoyancy-driven hydraulic fracture (Spence & Turcotte Reference Spence and Turcotte1985; Lister Reference Lister1990a , Reference Listerb ; Roper & Lister Reference Roper and Lister2007; Dontsov Reference Dontsov2016b ), for which there is a propagating fracture head with relatively large fracture width, while this head is connected to the injection point by a narrow tail with small aperture. The solutions are qualitatively similar since the backstress is higher near the injection point and lower near the tip, which is qualitatively similar to the external stress behaviour for buoyancy-driven hydraulic fractures.

Figure 11.

Constant-height fracture: profiles of the $\tilde {K}B$ transition solution. (a) Dimensionless opening $\varOmega _{\tilde {k}b}(\xi )$ and (b) dimensionless net pressure $\varPi _{\tilde {k}b}(\xi )$ versus the normalized coordinate $\xi =x/\ell (t)\in [0,1]$ (inlet at $\xi =0$ , tip at $\xi =1$ ). Solid curves correspond to different values of the evolution parameter $G_k$ (see legend). Dashed curves show the limiting vertex solutions: $\varOmega _{\tilde {k}0}$ and $\varPi _{\tilde {k}0}$ for the $\tilde {K}$ vertex ( $G_k \gg 1$ ), and $\varOmega _{b0}$ and $\varPi _{b0}$ for the $B$ regime ( $G_k \to 0$ ).

Figure 12.

Constant-height fracture: evolution of the inlet values of the $\tilde {K}B$ transition solution as a function of the evolution parameter $G_k$ . Here (a) $\varOmega _{\tilde {k}b}(0)$ and (b) $\varPi _{\tilde {k}b}(0)$ , where $\xi =0$ denotes the inlet. Solid lines show the transition solution, while dashed lines indicate the corresponding limiting vertex solutions: $\varOmega _{\tilde {k}0}(0)$ and $\varPi _{\tilde {k}0}(0)$ for the $\tilde {K}$ vertex ( $G_k \gg 1$ ), and $\varOmega _{b0}(0)$ and $\varPi _{b0}(0)$ for the $B$ regime ( $G_k \to 0$ ). The vertical dash–dotted lines mark the corresponding values of transition boundaries indicated in the legend. For the $\tilde {K}B$ transition, the evolution parameter $G_k$ is related to the dimensionless evolution parameters by $G_k = (\sqrt {\pi } B^\prime \tau ^{1/2} \phi )^{-1} = (\pi B^{\prime 2} \tau _{\tilde {k}b}^\prime )^{-1/2}$ .

Figure 12. Long description

Two line graphs depict the evolution of inlet values of the transition solution as a function of the evolution parameter. Panel A: A line graph shows the evolution of the inlet values of the transition solution for the vertex. The x-axis represents the evolution parameter on a logarithmic scale ranging from 10^−2 to 10^2. The y-axis represents the inlet values of the transition solution ranging from 0 to 200. Solid lines represent the transition solution, while dashed lines indicate the corresponding limiting vertex solutions. Vertical dash-dotted lines mark the corresponding values of transition boundaries. Panel B: A line graph shows the evolution of the inlet values of the transition solution for the regime. The x-axis represents the evolution parameter on a logarithmic scale ranging from 10^−2 to 10^2. The y-axis represents the inlet values of the transition solution on a logarithmic scale ranging from 10^−1 to 10^2. Solid lines represent the transition solution, while dashed lines indicate the corresponding limiting vertex solutions. Vertical dash-dotted lines mark the corresponding values of transition boundaries.

As shown in figure 12, the inlet values of the opening and net pressure along the $\tilde {K}B$ transition evolve differently as the evolution parameter $G_k$ increases. The inlet opening $\varOmega _{\tilde {k}b}(0)$ remains close to the $\tilde {K}$ -vertex value over a wide range of large $G_k$ , and then drops rapidly as $G_k$ decreases towards $O(1)$ , indicating a sharp reduction of the inlet opening along the transition (figure 12 a). In contrast, the inlet net pressure $\varPi _{\tilde {k}b}(0)$ stays nearly constant at small $G_k$ and subsequently increases monotonically for larger values, approaching the corresponding $\tilde {K}$ -regime limit (figure 12 b). In figure 12(b), the inlet net pressure $\varPi _{\tilde {k}b}(0)$ exhibits a sharp corner. The corner occurs at $G_k=1/\sqrt {\pi }$ : the inlet pressure equals $2/\pi$ below this value and follows the $\tilde K$ -vertex relation $2G_k/\sqrt {\pi }$ above it. The boundaries from table 9 are indicated as vertical dash–dotted lines. Together, the two panels show that the $\tilde {K}B$ transition is characterized by a pronounced change of inlet opening and inlet pressure as the evolution parameter increases.

It is also possible to calculate the transition solution from the $\tilde {M}$ -vertex to the $B$ -regime. For the viscosity-leak-off dominated limiting solution, we set $G_{k}\rightarrow 0$ , $G_{v}=B^\prime \ll 1$ . Similar to the $\tilde {K}B$ transition, the leak-off history is simplified to $\theta =t_{0}/t= (x/\ell )^{2}=\xi ^{2}$ . Given these considerations, the simplified governing equations can be obtained from (5.11) as

(5.32) \begin{equation} -\frac {2}{\pi ^{3}G_m}\frac {1}{\gamma _{\tilde {m}b}^{2}}\frac {\mathrm{d}}{\mathrm{d}\xi }\left [\frac {f_{q}(\beta _{\tilde {m}b})}{1-\beta _{\tilde {m}b}^{2}}\varOmega _{\tilde {m}b}^{3}\frac {\mathrm{d}}{\mathrm{d}\xi }\left (\frac {\varOmega _{\tilde {m}b}}{(1-\beta _{\tilde {m}b})^{2}}\right )\right ]+\frac {1}{\sqrt {1-\xi ^{2}}}=\frac {1}{\gamma _{\tilde {m}b}}\delta (\xi), \end{equation}

(5.33) \begin{equation} \varOmega _{\tilde {m}b}(1)=0,\quad 2\int _{0}^{1}\sqrt {1-\xi ^{2}}\mathrm{d}\xi =\frac {1}{2\gamma _{\tilde {m}b}},\quad \varOmega _{\tilde {m}b}\beta _{\tilde {m}b}=(1-\beta _{\tilde {m}b})^{2} \sqrt {1-\xi ^{2}}. \end{equation}

Since $\gamma =1/\pi$ from (5.33), (5.32) can be integrated to obtain

(5.34) \begin{equation} \frac {1}{G_m} \frac {f_{q}(\beta _{\tilde {m}b})}{1-\beta _{\tilde {m}b}^{2}}\varOmega _{\tilde {m}b}^{3}\frac {\mathrm{d}}{\mathrm{d}\xi }\left (\frac {\varOmega _{\tilde {m}b}}{(1-\beta _{\tilde {m}b})^{2}}\right )=\frac {\pi }{2}\arcsin \xi -\frac {\pi ^{2}}{4}. \end{equation}

The above equation is complemented by

(5.35) \begin{align} &\varOmega _{\tilde {m}b}(1)=0,\quad \beta _{\tilde {m}b}(1)=0, \quad \beta _{\tilde {m}b}=1-\frac {\sqrt {(k+2)^{2}-4}-k}{2},\nonumber\\ &k=\frac {\varOmega _{\tilde {m}b}}{\sqrt {1-\xi ^{2}}}, \quad 0\leq \xi \lt 1. \end{align}

The tip conditions $\varOmega _{\tilde {m}b}(1)=0$ and $\beta _{\tilde {m}b}(1)=0$ follow from the propagation condition and from $t=t_0$ at the fracture tip, respectively. The algebraic relation for $\beta _{\tilde {m}b}$ is obtained by solving the quadratic elasticity relation in (5.33). Therefore, since the function $f_{q}$ is known, (5.34) and (5.35) represent an ordinary differential equation that only depends on the dimensionless parameter $G_m$ .

The solution for equations (5.34) and (5.35) is computed using a finite-difference scheme, with $f_{q} (\beta _{\tilde {m}b} )$ evaluated by interpolation. We first present the profiles of the coefficient $\beta _{\tilde {m}b}$ , then show the normalized opening and net-pressure profiles with comparisons with the limiting regimes, and finally report the evolution of the inlet values of the dimensionless opening and net pressure. The inlet solutions are approximated by the values of the first element.

Figure 13.

Constant-height fracture: profiles of the $\tilde {M}B$ transition solution. Backstress coefficient $\beta _{\tilde {m}b}(\xi )$ versus the normalized coordinate $\xi =x/\ell (t)\in [0,1]$ (inlet at $\xi =0$ , tip at $\xi =1$ ). Curves are parameterized by the time-like evolution variable $G_m$ , which decreases as the solution evolves towards the $B$ -regime.

Figure 14.

Constant-height fracture: normalized profiles of the $\tilde {M}B$ transition solution. (a) Normalized opening $\varOmega _{\tilde {m}b}(\xi )/\varOmega _{\tilde {m}b}(0)$ and (b) normalized net pressure $\varPi _{\tilde {m}b}(\xi )/\varPi _{\tilde {m}b}(0)$ versus the normalized coordinate $\xi =x/\ell (t)\in [0,1]$ (inlet at $\xi =0$ , tip at $\xi =1$ ). Solid curves are parameterized by the time-like evolution variable $G_m$ , which decreases as the solution evolves towards the $B$ -regime. Dashed curves indicate the corresponding limiting vertex solutions.

Figure 14. Long description

Panel A: A line graph shows normalized opening profiles of the transition solution. The x-axis represents the normalized coordinate, ranging from 0 to 1, with the inlet at 0 and the tip at 1. The y-axis represents the normalized opening, ranging from 0 to 1. Solid curves are parameterized by the time-like evolution variable, which decreases as the solution evolves towards the B-regime. Dashed curves indicate the corresponding limiting vertex solutions. Panel B: A line graph shows normalized net pressure profiles of the transition solution. The x-axis represents the normalized coordinate, ranging from 0 to 1, with the inlet at 0 and the tip at 1. The y-axis represents the normalized net pressure, ranging from 0 to 1. Solid curves are parameterized by the time-like evolution variable, which decreases as the solution evolves towards the B-regime. Dashed curves indicate the corresponding limiting vertex solutions.

Figure 13 shows the profiles of the backstress coefficient $\beta _{\tilde {m}b}$ for the $\tilde {M}B$ transition solution. For large values of the evolution parameter $G_m$ , the backstress coefficient remains well below one and varies little along the fracture (figure 13). As $G_m$ decreases, $\beta _{\tilde {m}b}$ increases monotonically and approaches one over an increasingly large portion of the fracture, indicating that backstress becomes increasingly dominant relative to the net pressure. These trends indicate a gradual shift from the viscosity-leak-off-dominated regime towards the backstress-dominated response as the evolution parameter decreases.

We proceed to compare the $\tilde {M}B$ transition solution with the $\tilde {M}$ - and $B$ -solutions. When $\beta =0$ , we retrieve the classical viscosity-leak-off-dominated $\tilde M$ -vertex for the constant height fracture (Dontsov Reference Dontsov2022). For the purpose of comparison, we define the $\tilde {M}$ solution as

(5.36) \begin{equation} \varOmega _{\tilde {m}0}=\varOmega _{\tilde {m}b}|_{\beta =0},\quad \varPi _{\tilde {m}0}=\varPi _{\tilde {m}b}|_{\beta =0}. \end{equation}

Figure 14 shows the normalized opening and net-pressure profiles for the $\tilde {M}B$ transition solution. The normalized opening $\varOmega _{\tilde {m}b}(\xi )/\varOmega _{\tilde {m}b}(0)$ exhibits a clear dependence on the transition parameter $G_m$ , particularly over the interior portion of the fracture, indicating that the transition has a non-negligible influence on the opening shape (figure 14 a). As $G_m$ increases, the opening profile moves closer to the $\tilde {M}$ -vertex solution, whereas for smaller $G_m$ it departs more markedly from that limit. In contrast, the normalized net pressure $\varPi _{\tilde {m}b}(\xi )/\varPi _{\tilde {m}b}(0)$ shows a comparatively weaker, though still discernible, dependence on $G_m$ and remains bounded by the two limiting curves shown by dashed lines (figure 14 b). Specifically, the pressure profile approaches the $\tilde {M}$ -vertex solution for $G_m \gg 1$ and the $B$ -regime solution for $G_m \to 0$ . No corresponding $B$ -regime opening profile is shown in figure 14(a), because no $B$ -scaling opening is resolved in that limit.

As shown in figure 15, the inlet values of the opening and net pressure exhibit distinct trends with respect to the evolution parameter $G_m$ . The inlet opening $\varOmega _{\tilde {m}b}(0)$ increases monotonically as $G_m$ increases and departs from the $B$ -vertex limit, eventually approaching the $\tilde {M}$ -vertex at large $G_m$ (figure 15 a). In contrast, the inlet net pressure $\varPi _{\tilde {m}b}(0)$ remains close to its $B$ -vertex value for small $G_m$ , before transitioning to the $\tilde {M}$ -vertex and increasing rapidly as $G_m$ grows (figure 15 b). The boundaries from table 9 are indicated as vertical dash–dotted lines. This figure indicates that both pressure and opening approach the $B$ -vertex for $G_m \to 0$ , and the $\tilde {M}$ -vertex for $G_m\gg 1$ .

Figure 15.

Constant-height fracture: evolution of the inlet values of the $\tilde {M}B$ transition solution as a function of the evolution parameter $G_m$ . Here (a) $ \varOmega _{\tilde {m}b}(0)$ and (b) $ \varPi _{\tilde {m}b}(0)$ , where $\xi =0$ denotes the inlet. Solid curves show the transition solution. Vertical dash–dotted lines mark the corresponding values of transition boundaries indicated in the legend. Dashed lines indicate the limiting vertex solutions: $\varOmega _{\tilde {m}0}(0)$ in (a) and $\varPi _{\tilde {m}0}(0)$ in (b) for the $\tilde {M}$ vertex ( $G_m \gg 1$ ), and $\varOmega _{b0}(0)$ in (a) and $\varPi _{b0}(0)$ in (b) for the $B$ regime ( $G_m \to 0$ ). For the $\tilde {M}B$ transition, the evolution parameter $G_m$ is related to the dimensionless evolution parameters by $ G_m = 1/(2\pi ^3) ( B^{\prime 8/3} \tau \phi ^{10/3})^{-3/2} = 1/(2\pi ^3)(B^{\prime 8/3} \tau ^{\prime }_{\tilde {m}b})^{-3/2}$ .