ABSTRACT
A general method-of-lines numerical approach for modeling gas driven fractures has been developed and tested. The numerical solutions agree very well with known similarity solutions for laminar and turbulent flow. Acceptable accuracy is obtained with a very few numerical grid points and with modest execution times. The model has also been checked against the results of several field tests, in which air was used to drive axisymmetric fractures into permeable tunnel-bed tuff.
INTRODUCTION
Gas driven fracture propagation is a critical consideration in rock blasting (Kutter and Fairhurst, 1971) and well shooting (Warpinski, et al., 1979) technologies and in the containment of underground nuclear tests (Travis and Davis, 1980). The importance of gas penetration into fractures has been previously demonstrated by analytical and numerical studies in which varying degrees of penetration were presumed to occur (0uchterlony, 1982). However, little consideration has been given to the fluid and thermal processes which control this flow into a fracture. The present paper describes a general numerical procedure for modeling gas driven fractures. The method is easy to implement, since it relies on widely available library routines to accomplish many of the numerical tasks. It is very versatile in allowing an arbitrary equation of state, arbitrary friction laws, arbitrary boundary conditions at the inlet, and an arbitrary relationship between the pressure distribution and fracture aperture. Fracture geometry may be either axisymmetric (penny-shaped) or planar (wedge-shaped), as indicated in Figure 1.
Fig. 1. Schematic of planar and axisymmetric geometries which are treated by numerical model. (available in full paper)
A complete explanation of the mathematical model and numerical procedure is given by Nilson and Oriffiths (1983) who initially tested the method under isothermal conditions. Heat transfer from the gas to the surrounding rock was incorporated into the same computational framework by Criffiths, Nilson, and Morrison (1983). The present paper briefly reviews that previous work. The model is extended to include Darcian eepage losses to the surrounding rock, and comparisons are made between numerical calculations and field experiments.
GOVERNING EQUATIONS
The physical model of hydraulic fracturing is well known. Since the displacement field is quasi-steady for tip velocities much less than the wave speed in the rock, the aperture profile w(x,t) depends only on the crack length L(t) and the current pressure distribution P(x,t) in the fluid. For either planar (n = 0) or axisymmetric (n = 1) geometries, the theory of linear elasticity gives (Sheddon and Lowengrub, 1969) (mathematical equation) (available in full paper) where O and V are the shear modulus and Poisson's ratio, respectively; , is the compressive tectonic stress, and O = x/L(t) is the normalized position along the crack. The fracture mechanics of the tip region are accounted for by the following inequality which must be satisfied in order for the fracture to propagate. (Barenblatt, 1962). (mathematical equation) (available in full paper)
