The basics of potential theory applied to fracture mechanics are summarized, and closed forms are derived. Thus fracture length, fracture height, fracture width and well bore pressure are determined as a function of time, rock and fluid properties, and stress contrast for Newtonian and non-Newtonian fluids. Fluid filtration is introduced into the formulation. The general formulas are checked, and concordance between the numerical model and analytical results is shown to be good. The Perkins model is shown to be a special case of the above-mentioned closed forms when stress contrast tends towards infinity.
As defined by Mendelsohn (1984), the hydraulic fracturing process consists of the pumping of a viscous sand / fluid slurry down a wellbore under pressure causing a crack in the reservoir rock, with narrow width and large areal extent. The major application of hydraulic fracturing is to improve the flow rate of oil and / or gas from low permeability rock. Small fractures are also developed to measure in situ rock stress and to estimate the filtration coefficient. Numerical models have been designed to compute fracture propagation. However, when using numerical models, a parametric study is essentially limited to the influence of one parameter at a time. We thus need closed forms to make an in depth parametric study. On the other hand, accurate determination of in situ stresses and filtration coefficients from mini-frac tests implies the use of inversion models (Charlez, 1989). In such cases the direct problem solver, i.e. the fracture propagation model, does not have to be sophisticated, but it still has to be true on physical grounds. In deriving our closed forms, we took these two criteria into account. Existing models may be clustered in three groups:
Two-dimensional models such as the ones proposed by Geertsma (1969) or Perkins (1961), where fracture height is assumed to be known.
Pseudo three-dimensional models, where fracture height extent is computed, while assuming plane strain in any cross-section plane.
Three-dimensional models, where fracture height extent is computed without assuming any plane strain. Closed forms have been derived for the first group of models, and they have been extensively described in the literature. Closed forms have also been derived for penny-shaped fractures, i.e, fracture propagation in unconfined media. Advani (1986) has developed a Lagrangian formulation of the problem and has derived the previously mentioned closed forms from it. However, to our knowledge there is no analytical formulation in which stress contrast is taken into account. This paper aspires to be a contribution to such a study when fractures propagate in a moderately to highly confined medium, i.e. when fracture length Lf is greater than 1.25 times fracture height h f
The elastic rock formation is isotropic and homogeneous. The fracture contour in the propagation plane is elliptical (flat elliptical crack, Fig 1). The semi-axes are the fracture length and half-height. The shape in any cross-section plane is computed.
