Abstract
This paper develops a closed-form approximate solution for a penny-shaped hydraulic fracture whose behaviour is determined by an interplay of three competing physical processes that are associated with fluid viscosity, fracture toughness and fluid leak-off. The primary assumption that permits one to construct the solution is that the fracture behaviour is mainly determined by the three-process multiscale tip asymptotics and the global fluid volume balance. First, the developed approximation is compared with the existing solutions for all limiting regimes of propagation. Then, a solution map, which indicates applicability regions of the limiting solutions, is constructed. It is also shown that the constructed approximation accurately captures the scaling that is associated with the transition from any one limiting solution to another. The developed approximation is tested against a reference numerical solution, showing that accuracy of the fracture width and radius predictions lie within a fraction of a per cent for a wide range of parameters. As a result, the constructed approximation provides a rapid solution for a penny-shaped hydraulic fracture, which can be used for quick fracture design calculations or as a reference solution to evaluate accuracy of various hydraulic fracture simulators.
Highlights
Hydraulic fractures are the fluid-filled cracks that propagate under the influence of a fluid pressure acting along the crack’s surface
Q0δ(r), q where w(r, t) denotes the fracture width, q is the flux in the radial direction, the term that contains C accounts for leak-off via Carter’s model, t0(r) is the time instant at which the fracture front was located at point r, p is the fluid pressure and Q0 is the fluid injection rate
This paper presents a closed-form approximate solution for a penny-shaped hydraulic fracture, whose behaviour is governed by a simultaneous interplay of fluid viscosity, fracture toughness and fluid leakoff into the formation
Summary
Hydraulic fractures are the fluid-filled cracks that propagate under the influence of a fluid pressure acting along the crack’s surface. Even for the simplest geometries, hydraulic fractures are known to obey a complex multiscale behaviour, see e.g. a thorough review paper [27]. This multiscale nature arises both in time, where multiple timescales determine fracture evolution, and space, where the solution may experience variations at different length scales in the tip region. This study develops a closed-form approximate solution that provides results virtually instantly and accurately captures the complex behaviour of a radial hydraulic fracture at all length scales and timescales.
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