Penny-shaped hydraulic fracture with fluid lag in impermeable elastic medium

Abstract

This paper revisits the classic problem of a penny-shaped hydraulic fracture propagating in an impermeable elastic medium. The fracture is driven by a viscous fluid injected at a constant rate. Motivated by the need to clarify the nature of the early-time solution under conditions of negligible toughness, the study uncovers the existence of a new similarity solution, denoted as the $S$-vertex, which complements the three already known similarity solutions ($O$-, $M$-, $K$-vertex). The considered problem accepts therefore 4 similarity solutions, which all have a power law dependence on time. Interpreting these solutions as either early-, intermediate-, or late-time asymptotics naturally leads to defining the $OSMK$ phase diagram, where the similarity solutions are represented by the vertices of the rectangular domain. The evolution of a radial fracture can be pictured as following a trajectory in the $OSMK$ space that starts at the $O$-vertex and ends at the $K$-vertex. Each trajectory in the $OSMK$ parametric space corresponds to a particular value of the dimensionless parameter $\eta$, which can be interpreted as the ratio of two independent time scales. If $\eta \ll 1$, i.e, the toughness is very small in a relative sense, the trajectory connect the $O$- and $K$-vertex by following the edges $OS$, $SM$, and $MK$ of the $OSMK$ domain. Under these conditions, the complete fracture evolution is thus characterized by two intermediate time asymptotes, the $S$- and the $M$-vertex. Finally, the paper describes an efficient polynomial-based algorithm to construct the $O$- and $S$-vertex solutions, and its extension to an implicit time-stepping algorithm to track the evolution of the hydraulic fracture in the parametric space.