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The paper presents novel results, obtained on the basis of the modifiedtheory of hydraulic fractures (HF). The theory underlines significance of thespeed equation. When applied to numerical simulation of HF, the theory revealsthree distinct issues: (i) modeling the central part of a HF; (ii) modeling thenear-front zone; and (iii) tracing changes in the shape of a fracture contour.Modeling the central part leads to a stiff system of ODE in time, what stronglycomplicates its integration. For explicit schemes, it requires small time stepsto meet the CFL condition. For implicit schemes, it requires properpreconditioners. The gains and flaws of the two strategies are discussed. It isnoted that a rough spatial mesh may be used in the central part. Modeling thenear-front zone reveals the vital role of the speed equation for HF modeling byany method. Its asymptotic analysis has resulted in the fundamental concept ofthe universal asymptotic umbrella. For the near-front zone, it is alsoestablished that a notable part of the zone adjacent to the front propagatesvirtually as a simple wave. This implies that the CFL condition of stabilityfor this zone is much less restrictive than for the central part of thefracture. Of essence is also that the wave-like propagation of the near frontzone makes preferable upwind schemes of time stepping. On whole, the analysisimplies that explicit time stepping may be complementing, competitive and evensuperior over implicit integration. Tracing changes in the front shape appearsmerely in 3D problems when the contour changes its form. The problem, beingessentially geometrical, it may be solved separately by various methods,including fast marching, level set and the simplest string/marker methods. In1D cases, it does not arise at all.  Quantitative estimations and numerical examples illustrate the theoreticalconclusions.

Modern theory of hydraulic fracture modeling with using explicit and implicit schemes