ABSTRACT:

This paper describes a numerical model to simulate the propagation of a plane-strain (KGD) hydraulic fracture in an elastic, impermeable medium with zero toughness. The fracture is driven by injection of an incompressible fluid with power-law rheology. The numerical model, which is formulated in terms of a moving coordinates system, is based on the displacement discontinuity method and on an explicit finite difference scheme. The accuracy of the algorithm is validated against the available self-similar solution for a Newtonian fluid.

INTRODUCTION

Hydraulic fracturing (HF) is a technique widely used to enhance the flow of oil or natural gas from the reservoir formations towards the extracting wells. The uncertainty about the in situ conditions, the complexity of the mechanisms taking place and the difficulty in obtaining precise measurements of the fracture geometry, make necessary the use of idealized models (e.g., the KGD or "plane-strain" model, the "pennyshaped" or radial model and the PKN model) for studying this process.

Even with these simplified models, the mathematical formulation for the propagation of hydraulic fractures is given by a relatively complicated system of integral and non-linear differential equations. Some analytical solutions of these mathematical models are already available (Spence & Sharp, 1985; Savitski & Detournay, 1999; Garagash, 2000; Savitski, 2000; Adachi, 2001; Adachi et al., 2001). However, these solutions are constructed on the basis of various restrictive assumptions (e.g., constant injection rate; very small or very large material toughness, Newtonian theology for the fracturing fluid, no fluid leakoff, etc.). In order to extend the applicability of these models, it is necessary to release some of these assumptions and consequently, the solution of the governing equations demands the use of numerical techniques.

Commonly, the numerical solution of non-linear problems is restricted to the use of implicit schemes. Explicit schemes are not often applied in this type of problems due to the difficulty in reaching numerical stability, even though the latter are simpler to implement. In this paper, we introduce an explicit finite difference scheme with a moving mesh that can be used to simulate the propagation of a KGD hydraulic fracture. This scheme is shown to be numerically stable, accurate and "flexible", in the sense that additional features (e.g., fluid leak-off, poroelastic effects, etc.) can be easily incorporated into the model (Detournay et al., 1990).

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