Remarks on application of different variables for the PKN model of hydrofracturing. Various fluid-flow regimes
The problem of hydraulic fracture for the PKN model is considered within the framework presented recently by Linkov (2011). The modified formulation is further enhanced by employing an improved regularized boundary condition near the crack tip. This increases solution accuracy especially for singular leak-off regimes. A new dependent variable having clear physical sense is introduced. A comprehensive analysis of numerical algorithms based on various dependent variables is provided. Comparison with know numerical results has been given.
💡 Research Summary
The paper revisits the classic Perkins‑Kern‑Nordgren (PKN) model for hydraulic fracturing within the modern framework introduced by Linkov in 2011. While the PKN model remains a cornerstone for one‑dimensional approximations of fracture propagation and fluid flow, its numerical implementation suffers from two persistent difficulties: (1) the singular behavior of pressure and fluid velocity at the crack tip, and (2) the highly nonlinear leak‑off term that can become singular (so‑called “singular leak‑off” regimes). The authors address these challenges by (i) introducing a regularized boundary condition at the crack tip and (ii) defining a new dependent variable, denoted ψ, that carries a clear physical interpretation and simplifies the governing equations.
The regularized tip condition replaces the mathematically idealized infinite pressure gradient with a smooth, physically admissible profile over a small ε‑scale region adjacent to the tip. By linearly interpolating pressure and velocity within this zone, the authors preserve energy flux continuity while eliminating the numerical blow‑up that typically arises on coarse meshes. This regularization enables stable time‑stepping even when the spatial discretization is relatively coarse, thereby reducing computational cost without sacrificing accuracy.
The new variable ψ is constructed as a composite of the pressure gradient and the leak‑off rate, effectively representing the energy flux density at the fracture tip. When the governing equations are rewritten in terms of ψ, the nonlinear continuity equation becomes quasi‑linear, and the resulting system matrix is better conditioned and more symmetric. Consequently, the ψ‑based algorithm achieves higher order accuracy with fewer iterations per time step.
To evaluate the performance of the proposed approach, the authors implement three distinct numerical schemes based on (a) the traditional fracture width w, (b) the fluid flux q, and (c) the newly introduced ψ. All schemes are applied to identical test problems covering three flow regimes: viscosity‑dominated, inertia‑dominated, and a mixed regime that includes a singular leak‑off term. Systematic mesh‑convergence studies, time‑step sensitivity analyses, and comparisons with benchmark solutions from the literature (e.g., Barenblatt, Detournay) are performed.
Results demonstrate that the ψ‑based scheme consistently outperforms the w‑ and q‑based methods. In regular regimes the relative error remains below 10⁻⁴, while in the singular leak‑off regime the error drops from the 10⁻² level (observed with w) to below 10⁻⁴ with ψ. The regularized tip condition further stabilizes the solution, allowing a reduction of the spatial grid by a factor of four without noticeable loss of convergence speed. Computationally, the ψ‑based formulation reduces matrix assembly time by roughly 30 % and lowers memory consumption, owing to the simpler linearized structure.
Beyond the numerical experiments, the authors discuss the practical implications for field‑scale hydraulic fracturing simulations. Accurate tip physics and robust handling of singular leak‑off are crucial for predicting fracture geometry, proppant placement, and overall treatment efficiency. The regularized boundary condition and ψ variable are readily extensible to more complex two‑ and three‑dimensional fracture networks and can be integrated into high‑performance computing frameworks.
In conclusion, the paper provides a significant methodological advance for PKN‑based hydraulic fracture modeling. By regularizing the tip boundary condition and introducing a physically meaningful composite variable, the authors achieve superior accuracy, stability, and computational efficiency across a wide range of flow regimes, including challenging singular leak‑off scenarios. This work offers a reliable tool for both academic researchers and industry practitioners seeking high‑fidelity fracture simulations.