Abstract
In numerical simulation of hydraulic fracture propagation, tangent component of the fluid velocity generally considered to be neglected near the crack front. Then Reynolds transport theorem yields that the limit of the particle velocity coincides with the vector of the front propagation speed. We use this fact in combination with the Poiseuille-type equation, which implies that the particle velocity is always collinear to pressure gradient. We show that this specific feature of the hydraulic fracture problem may serve to simplify tracing the front propagation. The latter may be traced without explicit evaluation of the normal to the front, which is needed in conventional applications of the theory of propagating interfaces. Numerical experiments confirm that, despite huge errors in pressure and even greater errors in its gradient, the propagation speed, statistically averaged over a distance of a mesh size, is found quite accurate. We conclude that suggested method may simplify numerical simulation of hydraulic fractures driven by Newtonian and non-Newtonian fluids.
Highlights
Numerical modeling is of significance for hydraulic fracture (HF) design [13]
This leads to the necessity of the normal to the fracture front evaluation to find the velocity vector entering the discretized continuity equation
The numerical results show that the method avoiding explicit evaluation of the normal may be used in practical calculations
Summary
The mathematical formulation of the problem, used for modeling, includes the continuity equation and the movement equation of the Poiseuille type [14, 15] Combined, they contain second spatial derivatives of the pressure. Errors in the pressure gradient, which has non-integrable singularity, are naturally even greater; and they further aggravate when evaluating the second derivatives of the pressure For this reason, conventional schemes of HF modeling avoid evaluation of the pressure in tip elements. This leads to the necessity of the normal to the fracture front evaluation to find the velocity vector entering the discretized continuity equation In its turn, this leads to the need for using conventional methods of the theory of propagating interfaces [2, 16].
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More From: International Journal of Engineering & Technology
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