This paper presents a hydraulic fracture model that employs a variable "apparent toughness" dependent on fracture size. An approximate solution of the complete set of governing equations for hydraulic fracture propagation within an elastic permeable medium is formulated using a combination of the Green's function method with numerical techniques. The set of equations couples the hydraulic and solid mechanics sides of the problem including a criterion for equilibrium fracture growth. The criterion is formulated as the Griffith-Irwin type that reflects a growing plastic or damage zone in the vicinity of the crack. An increasing apparent toughness represents the effect of a growing damage zone on fracture propagation resistance. Formation and growth of the damage zone is the material response to stress concentration at the crack front. The introduction of apparent toughness combined with conventional hydraulic fracture equations results in a simulation of the hydraulic fracture processes that agrees well with observations. Examples of laboratory and field test simulations are given and discussed.


This work is concerned with a fluid-driven crack propagating within an elastic permeable medium. The analysis is based on the complete set of hydraulic fracture governing equations (HFGE) that reflects the hydraulic part of the problem, includes the fluid mass (volume) balance and the condition of fluid flow inside the fracture. The solid mechanics part of the problem is presented by singular integral equations for the stress intensity factor and crack opening displacement, as well as the Griffith-Irwin type criterion for equilibrium fracture growth. Solutions of penny-shaped hydraulic fracture using the HFGE with a constant toughness have been constructed and analyzed previously (see, e.g., Abe et al [1], Savitski and Detournay [2]). The assumption of constant toughness leads to a solution that predicts a monotonic decrease in wellbore pressure toward the insitu stress with fracture growth. Though a decreasing wellbore pressure has been reported in some cases (Murdoch [3]), there are numerous laboratory and field observations (Daneshy [4], Shlyapobersky et al [5], Blair et al [6], Johnson et al [7], Roodhart et al [8], Dudley et al [9], Britt et al [10], Wong and Hii [11], Chudnovsky et al [12], Wu et al [13]), which show a different pattern of wellbore pressure with fracture growth under a constant injection rate. Specifically, these show a pattern that contains three noticeably different segments: 1) pressure peak and sharp drop immediately after fracture initiation (transient stage), 2) pressure plateau noticeably higher than the in-situ stress (steady growth stage), and 3) pressure decline after shut-in. The peak pressure is most likely associated with material breakdown and fracture initiation. As the fracture propagates away from the wellbore, the fluid penetrates into the crack crevice and almost entirely fills it. These phenomena reveal themselves as a sharp pressure drop in wellbore pressure vs. time as the fluid lag decreases. The pressure plateau of the second stage suggests that the fluid-filled crack grows under the action of balanced driving and resisting forces. The crack driving force is the elastic energy release rate with fracture growth.

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