Efficient pseudo-spectral solvers for the PKN model of hydrofracturing

Abstract

In the paper, a novel algorithm employing pseudo-spectral approach is developed for the PKN model of hydrofracturing. The respective solvers compute both the solution and its temporal derivative. In comparison with conventional solvers, they demonstrate significant cost effectiveness in terms of balance between the accuracy of computations and densities of the temporal and spatial meshes. Various fluid flow regimes are considered.

Appendix: Numerical benchmarks

Let us define a set of benchmark solutions useful for testing different numerical solvers. Consider a class of positive functions $C_+(0,1)$ described in the following manner:

$$\begin{aligned}&C_+(0,1)=\left\{ h\in C^2(0,1)\cap C[0,1],\right. \\&\left. \lim _{x\rightarrow 1-}(1-x)^{\!-\!1/3}h(x)\!=\!1,\quad h(x)> 0,\quad x\in [0,1)\right\} . \end{aligned}$$

By taking an arbitrary $h\in C_+(0,1)$, one can build a benchmark solution for the normalized formulation of the problem as:

$$\begin{aligned} u(x)=u_0\psi _j(t)h(x). \end{aligned}$$

(63)

where functions $\psi _j(t)$ and $h(x)$ are specified below. On substitution of (63) into (26) one finds:

$$\begin{aligned}&q_l(t,x) = \gamma u_0\left[ \frac{1}{\beta }\left( x h^3(x)h^{\prime }(x)\right. \right. \nonumber \\&\qquad \quad \qquad \left. \left. +3\left( h^3( x)h^{\prime }(x)\right) ^{\prime }\right) - h( x)\right] \psi _j^\alpha (t),\nonumber \\&w_0(t) = u_0\psi _j(t), \end{aligned}$$

(64)

where two sets of the benchmark solutions can be considered. For the first one we choose $\psi _1(t)=e^{\gamma t}$ and $\beta =2/3$, $\alpha =1$, while for the second, $\psi _2(t)=(a+t)^\gamma$ and $\beta =2\gamma /(3\gamma +1)$, $\alpha =(3\gamma -1)/\gamma$. Corresponding crack lengths are defined in (25) and (30), respectively. Finally, the injection flux rate is computed from the boundary condition (16)$_1$:

$$\begin{aligned} q_0(t)=-\frac{u_0^4\psi ^4_j(t)}{L(t)}h^3(0)h^{\prime }(0), \end{aligned}$$

(65)

while the initial condition reads $W(x)=u_0\psi _j(0)h(x)$.

Note that when taking the function $h(x)$ from the class $C_+(0,1)$ in the following representation:

$$\begin{aligned}&h(x)=(1-x)^{1/3}(1+s(x)), \\&s\in C^2[0,1], \quad s( x)>-1,\quad x\in [0,1) \end{aligned}$$

$q_l$ automatically satisfies the condition $q_l(t,x)=O((1-x)^{1/3}),x\rightarrow 1.$

The presented benchmarks allow one to test numerical schemes in various fracture propagation regimes (accelerating/decelerating cracks) by choosing proper values of the parameter $\gamma$ [see Mishuris et al. (2012)]. Additionally, if one reduces the requirements for the smoothness of $s(x)$ near $x=1$, assuming that $h\in C^2[0,1)\cap H^\alpha (0,1)$ the benchmark can serve to model singular leak-off regimes [compare with (61)].

Note that the zero leak-off case cannot be described by the aforementioned group of benchmarks. However, an analytical benchmark for this regime, represented in terms of a rapidly converging series, has been developed in Linkov (2011c).