Hydraulic Fracturing Model Based on a Three-Dimensional Closed Form: Tests and Analysis of Fracture Geometry and Containment

Abstract

Summary. The numerical model described in this paper was designed to provide a realistic three-dimensional (3D) solution for fracture propagation in minimal computation time. Comparisons with available results show good concordance for a wide range of data. The results of a parametric study are compared with the Nolte analysis of the propagation pressure. Introduction Pseudo-3D and 3D models have recently been developed to calculate the complete geometry of hydraulic fractures. Basic features of these models are given by Mendelsohn and Veatch and Moschovidis. An interesting approach recently proposed by Advani et al. involves a Lagrangian formulation of the problem. This method was pioneered by Blot et al. for two-dimensional (2D) models. We have shown how analytical methods recently developed in fracture mechanics could be used in fracture-propagation models. The method was applied with the simplest hypotheses. The crack was supposed to be of constant height at the wellbore; thus, the in-situ stress was constant and equal to the pay-zone minimal principal stress. For the fluid, constant pressure and no leakoff were assumed. In this paper, the in-situ-stress distribution is introduced, the height of the fracture is computed, the flow of the fluid through the fracture (Fig. 1) along the × and z axes is computed, and the leakoff is taken into account. Modal Description A propagation model can be defined by a set of three coupled equations and -a criterion for propagation. The equation of equilibrium is (1) where b(x, z) = fracture width, p(x, z) = pressure, and a(z) = in-situ stress. The equation for pressure drop is (2) where is the pressure gradient and q is the flow rate. The fluid-balance equation is(3) where qL is the leakoff rate. For the propagation criteria, we use Fa =Fac, where Fa is the stress-intensity factor and Fac is the fracture toughness. Rock and Fracture Description. The elastic rock formation is isotropic and homogeneous. The fracture contour in the propagation plane is elliptical (flat elliptical crack). The semiaxes are the fracture length and half-height. The shape in any plane cross section is not assumed but computed. It is only when the pressure is constant that the cross section is also elliptical in the z direction. In the latter case, the axes in the cross-section plane are the fracture width and the height. The elliptical edge shape of the crack has been chosen after analysis of laboratory tests and numerical computation results. We have conducted laboratory tests with plexiglass blocks, to simulate confined fractures. The successive shapes of the propagation contour can be observed in Figs. 2 and 3. The same results can be seen in tests Thiercelin et al. conducted for confined fractures. Daneshyi also showed experimentally that unconfined fractures extending from a line source have elliptical edge shapes. Stresses and Pressure. A closed form can he obtained for the mechanical equilibrium of a flat elliptical crack, assuming that the loading can be expressed in terms of a polynomial of × and z. In this model, the in-situ stress distribution and the pressure are approximated by quadratic polynomials: (4a) and (4b) where p=pressure polynomial and c=stress polynomial. So far, we have assumed symmetrical loadings with respect to the × axis, although linear terms could be introduced. In fact, most of the software has been written taking into account linear terms for both pressure and stress. For a 2D fracture, Geertsma and de Klerk gave the following closed form for the pressure distribution: where Kp is a proportionality coefficient. We smoothed the quadratic approximation on this solution. The results are shown in Fig. 4. The Geertsma-de Klerk solution is represented with a continuous line and the approximation with a dotted line. The two curves are very similar, and the elastic energy is exactly the same. The in-situ stress polynomial is obtained, smoothing the given distribution on the polynomial expression by the least-squares method. As long as the fracture height is less than the pay-zone thickness minus 10%, the polynomial expression is reduced to the pay-zone minimal principal stress. SPEPE P. 445^