Michal Wrobel

Efficient pseudo-spectral solvers for the PKN model of hydrofracturing

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.

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 (Doklady Phys 56(8):436–438, 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.

Energy release rate in hydraulic fracture: Can we neglect an impact of the hydraulically induced shear stress?

Abstract A novel hydraulic fracture (HF) formulation is introduced which accounts for the hydraulically induced shear stress at the crack faces. It utilizes a general form of the elasticity operator alongside a revised fracture propagation condition based on the critical value of the energy release rate. It is shown that the revised formulation describes the underlying physics of HF in a more accurate way and is in agreement with the asymptotic behaviour of the linear elastic fracture mechanics. A number of numerical simulations by means of the universal HF algorithm previously developed in [Wrobel M., Mishuris G. (2015) Hydraulic fracture revisited: Particle velocity based simulation. International Journal of Engineering Science , 94: 23–58] are performed in order to: (i) compare the modified HF formulation with its classic counterpart and (ii) investigate the peculiarities of the former. Computational advantages of the revised HF model are demonstrated. Asymptotic estimations of the main solution elements are provided for the cases of small and large toughness. The modified formulation opens new ways to analyse the physical phenomenon of HF and also improves the reliability and efficiency of its numerical simulations.

Fluid velocity based simulation of hydraulic fracture: a penny shaped model—part I: the numerical algorithm

In the first part of this paper, a universal fluid velocity based algorithm for simulating hydraulic fracture with leak-off, previously demonstrated for the PKN and KGD models, is extended to obtain solutions for a penny-shaped crack. The numerical scheme is capable of dealing with both the viscosity and toughness dominated regimes, with the fracture being driven by a power-law fluid. The computational approach utilizes two dependent variables; the fracture aperture and the reduced fluid velocity. The latter allows for the application of a local condition of the Stefan type (the speed equation) to trace the fracture front. The obtained numerical solutions are carefully tested using various methods, and are shown to achieve a high level of accuracy.

Numerical method for solving joint thermo-diffusive problems in an infinite combined domain with thin resistant interphase.

This work deals with a class of Boundary Value Problems describing joint thermo-diffussive fields in an infinite combined domain, which consists of two subdomains, matched by a thin intermediate layer. The main problem is reduced to an equivalent one given in the bounded subdomain, with non-local boundary condition on the transmission surface. Such a condition incorporates all the information about the infinite subdomain and the intermediate layer. The equivalent problem is solved by means of Finite Element Method in frames of Matlab package. As it is not possible to introduce the non-local boundary conditions along a part of the boundary directly into FEM code, a dedicated iterative subroutine is constructed. Effectiveness of the method has been checked on selected benchmarks. Accuracy and convergence of the procedure have been addressed in the analysis.

Numerical Modelling of Mass Transfer Problems in Combined Domains with Non-Linear Imperfect Transmission Conditions

This work deals with a nonlinear mass transfer problem in an infinite combined domain, consisting of half-space matched to a bounded part via a thin intermediate layer. The latter exhibits high contrast material properties, whereas its thickness is assumed to be negligible in comparison with the dimensions of the bounded subdomain. The corresponding problem is reduced to an auxiliary one, defined only in the bounded region with a non-local boundary condition on the transmission surface, which is solved numerically by means of FEM. To introduce the boundary condition, a special iterative subroutine based on the classic Robin type boundary condition is constructed. The accuracy of the procedure and the range of its applicability are investigated for an analytical benchmark. Numerical results for an axisymmetrical stationary mass transfer problem are presented.

Fluid velocity based simulation of hydraulic fracture—a penny shaped model. Part II: new, accurate semi-analytical benchmarks for an impermeable solid

In the first part of this paper a universal fluid velocity based algorithm for simulating hydraulic fracture with leak-off was created for a penny-shaped crack. The power-law rheological model of fluid was assumed and the final scheme was capable of tackling both the viscosity and toughness dominated regimes of crack propagation. The obtained solutions were shown to achieve a high level of accuracy. In this paper simple, accurate, semi-analytical approximations of the solution are provided for the zero leak-off case, for a wide range of values of the material toughness and parameters defining the fluid rheology. A comparison with other results available in the literature is undertaken.

Coupled FEM-BEM Approach for Axisymetrical Heat Transfer Problems

This work deals with a stationary axisymmetrical heat transfer problem in a combined domain. This domain consists of half-space joined with a bounded cylinder. An important feature of the problem is the possible flux singularity along the edge points of the transmission surface. Domain decomposition is used to separate the subdomains. The solution for an auxiliary mixed boundary value problem in the half space is found analytically by means of Hankel integral transform. This allows us to reduce the main problem in the infinite domain to another problem defined in the bounded subdomain. In turn, the new problem contains a nonlocal boundary conditions along the transmission surface. These conditions incorporate all basic information about the infinite sub-domain (material properties, internal sources etc.). The problem is solved then by means of the Finite Element Method. In fact it might be considered as a coupled FEM-BEM approach. We use standard MATLAB PDE toolbox for the FEM analysis. As it is not possible for this package to introduce directly a non-classical boundary condition, we construct an appropriate iterative procedure and show the fast convergence of the main problem solution. The possible solution singularity is taken into account and the corresponding intensity coefficient of the heat flux is computed with a high accuracy. Numerical examples dealing with heat transfer between closed reservoir (filled with some substance) and the infinite foundation are discussed.