Models of three-dimensional (3D) fracture propagation are being developed to study the effect of variations of stress and rock properties on fracture height and bottomhole pressure (BHP). Initially a blanket sand bounded by zones of higher minimum in-situ stress is considered, with stresses symmetrical about both the pay-zone axis and the wellbore. An elliptical fracture perimeter is assumed. Fluid flows are one-dimensional (1D) Newtonian in the direction of the pay zone. Two models, FL1 and FL2, are developed. In FL1, a discontinuous stress variation is approximated by a y2 variation in the vertical coordinate, and the fracture criterion, Ki = Kc, is satisfied at both major and minor axes. The net pressure at the tip, Lf, of the long axis required by the boundary condition Ki = Kc does not seem crucial in determining fracture height or BHP (compare with one group of published models that assumes p = 0 at Lf). Model FL2 properly represents the discontinuous stresses, and satisfies Ki = Kc at the wellbore but not at the tip of the long axis. A parametric study is made, with both models, of the comparative effects of stress contrast, Kc, pay-zone height, h, and Young's modulus, E, on fracture height and BHP. Results indicate that Kc does not have as much effect as either E or, at least for large stress contrasts. Model FL2 suggests the possibility of a rapid growth in fracture height as is reduced. Such modeling may be able to give an upper or "safe" limit on the pumping parameters ( and ) to ensure good containment. When the stress contrast is high, 700 psi [4826 kPa], an analytic derivation of BHP appears to be a good approximation for the parameters we use, if everywhere the fracture height is assumed equal to the pay zone height. Although leakoff is neglected here, subsequent modeling results show that, for leak off coefficients 0.001 ft- min [3.9 × 10 -5 m.s ], the results herein are a good approximation to the case when leak off is included.
In their essence, models of hydraulic fracture propagation involve elasticity theory and fluid mechanics. The first is concerned with the fracture opening or width, w(p), as a function of net pressure on the fracture faces, while the second is concerned with the pressure drop, p(w), caused by the flow of viscous fluids in the fracture. Simultaneous solution of these equations includes a boundary condition that often takes the form Ki = Kc, where Ki is the stress-intensity factor at a point on the fracture tip, and Kc is the fracture toughness. The final solution is very complex in 3D, when a vertical fracture can expand vertically as well as horizontally along the pay zone. Thus, the first solutions were essentially two-dimensional (2D), and they assumed that the fracture height, hf, was fixed at the pay zone height, h. The 2D solutions were clustered in two groups as summarized by Nordgren, Perkins, and Geertsma and Haafkens. The first grouping, based on a model by Christianovich and Zheltov, assumed that the sides of an elongated, vertical fracture were parallel (i.e., free slippage between the pay and bounding zones, or no vertical stiffness). Other papers in this grouping included Geertsma and de Klerk, Daneshy and Settari.