A Continuum-Damage-Mechanics Model of Hydraulic Fracturing (includes associated papers 26282, 26303, 27734 and 27738 )
Authors: Valko Peter , Economides M.J. 
Published: December 2007

Summary.
This work introduces a relatively new branch of fracture mechanics, continuum damage mechanics (CDM), as an extension to current hydraulic-fracture-propagation models. The new approach attempts to resolve inconsistencies evident in current model applications. The fluid-flow-constrained tip-propagation boundary condition of the Perkins-Kern-Nordgren (PKN) model is replaced by a new one derived from continuum damage mechanics. This paper demonstrates how "abnormally high treating pressures" can be interpreted by identifying a combined parameter responsible for fracture-propagation retardation. Model results are given in a form suitable for computer representation to allow easy application in treating pressure interpretation and designing fracture treatments.

Introduction
A number of approximating models have described hydraulic-fracture propagation. Common to all are (1) a material-balance equation relating flow across the boundaries while the volume element expands, (2) an elasticity relationship between fracture width and net pressure, (3) a fluid-flow equation relating the flow rate with the pressure gradient in the fracture, and(4) a fracture-tip-propagation criterion. Early 2D models based on ideal fracture geometry and constant height differed mostly with respect to how Sneddon and Eliot's pressured crack solution was applied to obtain an elasticity equation. While these models are elegant closed-form approximations of the fracture propagation and are particularly useful in practical hydraulic fracture design, they use highly simplified or only fluid-flow-related fracture-tip-propagation criteria. Nordgren, who used the continuity equation in a local manner, provided a major improvement. Within these simplified models, the PKN is considered as a more appropriate approximation for long fractures (compared with height), whereas the Khristianovich-Zheltov-Geertsma-deKlerk(KGD) model is suggested to be more appropriate for fractures with significant height compared with length. Much higher fracturing net pressures, irregular pressure profiles, fracture height growth, and out-of-plane fractures forced a departure from simple 2D models. This generated additional-dimension models and prompted the introduction of improved calculation procedures with the fracture toughness concept (Refs. 9 and 10). The wellbore treating pressure is usually the only available direct information related to fracture propagation, and it is natural to demand that the models reproduce it. In spite of the apparently great variety of propagation models, description of certain observed phenomena is surprisingly difficult. Fracture toughness has been used to justify the pressure departure from the pressure profile that the simple 2D models would predict. However, there are two inconsistencies. While the elevated fracturing pressure at small fracture lengths could be interpreted reasonably through the fracture toughness, this pressure departure should decline as the fracture grows. (The stress-intensity factor is proportional to the pressure and the square root of the characteristic dimension; hence, to keep it equal to a specified toughness value, the pressure must decrease with time unless additional concepts, like variable-length fluid lag, are introduced into the model.) In field cases however, contained fractures exhibit continuously increasing fracturing pressures. Because of this inconsistency, treating-pressure profiles often are interpreted in the frame-work of the PKN model, even if more sophisticated models and the necessary rock mechanical data are available. The second inconsistency is that fracture toughness appears to be very much rate-dependent and frequently much larger than laboratory-derived values. In attempts to match field data, a whole range of fracture toughness values was used for the same lithology and the same formation. Some of these discrepancies can be resolved by taking into account the uncertainties in measurements and interpretation of field data. However, reports on abnormally high treating pressures have provided further evidence of the existing inconsistency between observations and models. Medlin and Fitch observed abnormally high treating pressures in massive hydraulic fracturing. Palmer and Veatch defined abnormally high treating pressure as the net fracturing pressure that is multiple of that expected from standard hydraulic fracturing theories. They have chosen the Perkins-Kern model for interpreting the wellbore pressure behavior during a step-rate test whenever a constant height could be assumed. In this framework, the usual pressure increase with time could be well-represented, but the level of the net pressure was underestimated by the model (see Fig. 5 of Ref. 17). If, however, a fracture-toughness-based 3D model is used, the starting value of the pressure curve could be well-reproduced but the time behavior predicted by the model was disappointingly different from the actual one (the computed pressure curve in Fig. 6 of Ref. 17 decreases even with an exponentially increasing injection rate, while the measured values show a steep increase).
JPT
P. 198^