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.

This work has been done in the framework of the EU FP7 PEOPLE project under contract number PIAP-GA-2009-251475-HYDROFRAC.

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:

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

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

$$ u(x) = u_{0} \psi_{j} (t)h(x). $$

(63)

where functions $\psi_{j} (t)$ and h(x) are specified below. On substitution of (63) into (26) one finds:

$$ \begin{aligned} q_{l} (t,x) = & \gamma u_{0} \left[ {\frac{1}{\beta }} \right.\left( {xh^{3} (x)h^{{\prime }} (x)} \right. \\ & \left. {\left. { + 3\left( {h^{3} (x)h^{{\prime }} (x)} \right)^{{\prime }} } \right) - h(x)} \right]\psi_{j}^{\alpha } (t), \\ \end{aligned} $$

$$ w_{0} (t) = u_{0} \psi_{j} (t), $$

(64)

where two sets of the benchmark solutions can be considered. For the first one we choose ψ ₁(t) = e ^γt and β = 2/3, α = 1, while for the second,ψ ₂(t) = (a + t)^γ and β = 2γ/(3γ + 1), α = (3γ − 1)/γ.Corresponding crack lengths are defined in (25) and (30), respectively. Finally, the injection flux rate is computed from the boundary condition (16)₁:

$$ q_{0} (t) = - \frac{{u_{0}^{4} \psi_{j}^{4} (t)}}{L(t)}h^{3} (0)h^{{\prime }} (0), $$

(65)

while the initial condition reads W(x = u ₀ ψ _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 $q_{l} (t,x) = O((1 - x)^{1/3} ,x \to 1$

The presented benchmarks allow one to test numerical schemes in various fracture propagation regimes (accelerating/decelerating cracks) by choosing proper values of the parameter γ [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).