Abstract

A three-dimensional fully-coupled poroelastic displacement discontinuity method is developed and used to analyze the temporal variation of opening and slip of a natural fracture in a reservoir in response to the sudden application of fluid pressure on the fracture surfaces. Numerical results show that a hydraulic fracture opens in an increasing manner with time as the rock moves towards a drained state under the applied stress. The applied pore pressure induces a time-dependent closure caused by the rock dilation. On the other hand, poroelastic analysis of a natural fracture subjected to shear shows that the fracture slip decreases with the time in response to a pore pressure-induced increase in the normal stresses on the joint.

1. THEORY OF POROELASTICITY

One of the main features of the deformation of fluidsaturated porous rock is its transient nature which is related to the presence of a fluid diffusion process, and is described by the linear theory of poroelasticity [1]. Since the pioneering work of Biot, the theory of poroelasticity has been reformulated by a number of investigators, such as Rice and Cleary [2]. The coupled constitutive equations of poroelastic material under isothermal conditions are:

(Equation in full paper)

where eij and sij are respectively the strain and stress of the solid matrix, p and ? are the pore pressure and pore volume, respectively, dijd is the Kronecker delta. The material constants are the shear modulus, G, Biot coefficient a, the drained and undrained Poisson’s ratios v and vu, and Skempton’s pore pressure coefficient B . These equations describe the fundamental aspects of the deformation of fluid-saturated porous rock namely; the variation of the volumetric response of the rock to pore pressure changes, and variation of pore pressure due to the application of a mean stress. The three-dimensional field equations for the poroelastic rock deformation can be presented as a Navier equation with a coupling term, and a diffusion equation:

(Equation in full paper)

where ui is the solid displacement in the i direction, e is the volumetric strain, and the other notations are the same as those defined previously.

2. DISPLACEMENT DISCONTINUITY METHOD

The displacement discontinuity (DD) method is an indirect boundary element method which is based on the fundamental solutions of a point DD in an infinite elastic or poroelastic medium. This technique has been used extensively in mining and hydraulic fracturing [3-5]. It is a boundary method and has the advantage of reducing the dimensions of the problem by one. The formulation of DD can be based on the solution of a constant line or square DD in an infinite elastic medium [3].

Alternatively, a point displacement DD can be integrated over an area (square, triangle or quadrilateral) to form elements which form the building block of the DD method. The stresses and displacements caused by a three dimensional point DD in a poroelastic medium is given in [6,7].

A fracture in a poroelastic medium can be viewed as a surface across which the solid displacements.

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