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We present analytical tip region solutions for fracture width and pressure when a power law fluid drives a plane strain fracture in an impermeable linear elastic solid. Our main result is an intermediate asymptotic solution in which the tip region stress is dominated by a singularity which is particular to the hydraulic fracturing problem. Moreover this singularity is weaker than the inverse square root singularity of linear elastic fracture mechanics. We also show how the solution for a semi-infinite crack may be exploited to obtain a useful approximation for the finite case.

[1] A numerical model has been developed for fluid-driven opening mode fracture growth in a naturally fractured formation. The rock formation contains discrete deformable fractures, which are initially closed but conductive because of their preexisting apertures. Fluid flow that develops along fractures depends on fracture geometry defined by preexisting aperture distribution, offsets along a fracture path, and intersections of two or more fractures. The model couples fluid flow, elastic deformation, and frictional sliding to obtain the solution, which depends on the competition between fractures for permeability enhancement. The fractures can be opened by fluid pressure that exceeds the normal stress acting on them and by interactions with intersecting closed fractures experiencing Coulombtype frictional slip. The Newtonian fluid is assumed to flow through the conductive fractures according to a lubrication equation that relates the cube of an equivalent hydraulic aperture to fracture conductivity. The rock material is assumed to be impermeable and elastic. This paper provides the governing equations for the multiple fracture systems and the solution methods used. Flow distribution and fracture growth in conductive fracture sets are simulated for a range of geometric arrangements and hydraulic properties. Numerical results show that elastic interaction between fracture branches plays a controlling role in fluid migration, although initial apertures can give rise to a preferential fluid flow direction during the early stage. In the presence of offsets, fracture segments subject to strong compression are difficult to open hydraulically, and their resulting smaller permeability can increase overall upstream fracture pressure and opening. The patterns of fluid flow become more complicated if fractures intersect each other. A portion of injected fluid is lost into closed empty fractures that cut across the main hydraulic fracture, and this delays the pressure increases required for fracture growth past the crosscutting fracture. The nonlinear fluid loss rate depends on the geometric complexities of the fracture sets and on the fluid viscosity. Sometimes fracture growth can be accelerated by the fast fluid transport along an intersected, relatively conductive natural fracture.Citation: Zhang, X., R. G. Jeffrey, and M. Thiercelin (2009), Mechanics of fluid-driven fracture growth in naturally fractured reservoirs with simple network geometries,

Summary This paper presents an evaluation of two fundamentally different stress models: an elastic model, which is based on linear transverse isotropic elasticity, and a failure model which is based on the concept that rocks are in an equilibrium state of shear failure. The models are first evaluated using physical parameters measured on core, pore pressure and in-situ stress data from the Gas Research Institute's pressure and in-situ stress data from the Gas Research Institute's Staged Field experiments in East Texas, It is shown that the elastic model and the failure model provide satisfactory predictions for most of the lithologies encountered. However, predictions for most of the lithologies encountered. However, the failure model is more accurate for predicting stress in soft shales. An example of stress predictions based on log derived elasticity parameters is also shown which gives stress estimations comparable to core based predictions. Introduction The influence of rock lithology on the state of stress has been the subject of an increasing number of studies during the last decades. These studies were initially applied to the determination of fracture pressure gradient to prevent fracturing while drilling then to containment of hydraulic fractures. More recently, the prediction and control of sanding and wellbore instability has increased the need for stress estimation. Initially, it was the lack of accurate stress measurements in rocks which led to the estimation of fracture gradient using a stress model. Various models were proposed but the elastic uniaxial strain model became quickly commonly used in the petroleum industry, certainly due to its simplicity. This model predicts the minimum stress from a knowledge of the overburden, pore pressure and Poisson's ratio. The Poisson's ratio used in the early predictions was either Poisson's ratio used in the early predictions was either constant or determined from previous fracture gradient measurements. In 1973, Anderson et al proposed to use log measurements to derive a better estimation of Poisson's ratio. They established a relationship between Poisson's ratio and formation shaliness as estimated using sonic compressional wave velocity and density. This relationship showed that the fracture gradient may be dependent on the lithology, if the assumptions of the elastic uniaxial strain model are valid. At the end of the 70's hydraulic fracturing dominated the subject of stress estimation. Analytical studies showed that fractures are easily contained by a stress contrast of about 3.5 MPa (500 psi) between the reservoir and the adjacent layers. Such a stress contrast, which was lithology dependent, was measured by Brechtel et al and Warpinski et al. Rosepiler used the elastic uniaxial strain model to predict stress containment and evaluated the success of the predict stress containment and evaluated the success of the predictions using acid fracturing data. Rosepiler found a predictions using acid fracturing data. Rosepiler found a good agreement between prediction and observation. More conclusive field validations could not be made without accurate stress, pore pressure and rock property measurements. A lack of physical basis was also the main criticism of stress models based on elasticity.