Lagrangian formulation for penny-shaped and Perkins-Kern geometry models

Basic theories for vertical penny-shaped and Perkins-Kern (PK) geometry models have been developed with a Lagrangian formulation combined with a virtual-work analysis. The Lagrangian formulation yields a pair of nonlinear equations in Rf or Lf and bf, the fracture radius or length and half-width. By introduction of a virtual-work analysis, a simple equation is obtained that can be solved numerically. This equation is written in a form that can be used to determine fracture geometry when the fluid-loss coefficient of the fracturing fluid is known. Also, this equation, coupled with a material-balance equation after shut-in, can be used to analyze pressure-decline data after shut-in to determine the effective fluid-loss coefficient and fracture geometry. Geometry prediction of the new theory agrees well with other methods for the same models.