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In the paper, we propose a new effective mathematical formulation andresulting universal numerical algorithm capable of tackling various HF modelsin the framework of a unified approach. The presented numerical scheme is notlimited to any particular elasticity model or crack propagation regime. Itsbasic assumptions are: i) proper choice of independent and dependent variables(with the direct utilization of a new one - the reduced particle velocity), ii)tracing the fracture front by use of the speed equation which can be integratedin a closed form and sets an explicit relation between the crack propagationspeed and the coefficients in the asymptotic expansion of the crack opening,iii) proper regularization techniques, iv) improved temporal approximation, v)modular algorithm architecture. The application of the new dependent variable,the reduced particle velocity, instead of the usual fluid flow rate,facilitates the computation of the crack propagation speed from the localrelation based on the speed equation. As a result, the position of the crackfront is accurately determined from an explicit formula derived from the speedequation. The underlying ideas employed in the algorithm are combined togetherproducing a robust and efficient numerical scheme. Its performance isdemonstrated using classical examples of 1D models for hydraulic fracturing:PKN and KGD under various fracture propagation regimes. Solution accuracy isverified against dedicated analytical benchmarks and other solutions availablein the literature. Most of the ideas developed here, can be directly extendedto more general 2D and 3D cases.

Particle velocity based universal algorithm for numerical simulation of hydraulic fractures