This paper reports on a new model for hydraulic fracture propagation. The mathematical model uses the following basic concept: a fracture propagates in a predetermined precursor plane based on the three-dimensional stress field in the reservoir. Specifically, the fracture propagation relies on the following principles: (a) the injected fluid enters the pore space in a 3-D volumetric manner before any fracturing occurs; (b) fracturing occurs when pore pressure becomes at least larger than the minimum confining stress; (c) after fracture initiation, the fracture propagates in a plane perpendicular to the minimum stress; (d) in the presence of rock permeability variation, the injected fluid can channel along high permeability streaks before fracturing occurs, which can lead to fracture initiation within the permeability channels; and (e) for layered reservoirs, the vertical flow is affected by the layer permeability which should affect the shape of the fracture plane. The fracture extension with time is determined using flow equations both in the fracture and the surrounding matrix. Two approaches are used to determine the fracture width as a function of time and space. The first method (implemented in this paper) assumes that the fracture width has an elliptical shape. The second approach would use fracture face deformation, computed from rock elasticity modeling during fracture propagation, to calculate fracture aperture. The approach has been successfully used to match a fracture created in a laboratory setting.
Several numerical models have been developed during the last sixty years, from –simple analytical models to sophisticated numerical models that incorporate more complex physics. The geometry of –planar the hydraulic fractures is defined by length, width, and height. The exact geometry of the hydraulic fracture can be captured by a numerical model via the interaction of pore pressure and stress conditions in the node surrounding the pore space. Two dimensional (2D) and three dimensional (3D) numerical models imply that the total variables of interests within these models (i.e. width, length, and/or height) are two- and three- dimensional, respectively. There are two types of 3D models commonly used in industry. They are the pseudo-three dimensional models (P3D) and the planar three dimensional models (PL3D). The former ones are extensions of the Perkins, Kerns and Nordgren (PKN) analytical model to simulate hydraulic facture propagation in layered reservoir. In contrast to the PKN model, these models are not restricted by a fixed height as long as the proportionality of the vertical and lateral growth satisfies the critical assumption of local elastic compliance. An example of recent 3D models, which is based on the PKN model was presented by Charoenwongsa et al [1]. Also, Adachi and his co-authors, in their recent study, presented rigorous mathematical formulations to define propagation regimes and the criterion of the height growth that validates such P3D models. This referred study cited those P3D models developed in the 1980s including Settari and Cleary (1982 and 1986), Palmer and Carroll (1982-1983), Palmer and Craig (1984), Advani et al (1990) [2].
