Steady crack growth in elastic–plastic fluid-saturated porous media

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Abstract

An asymptotic solution is obtained for stress and pore pressure fields near the tip of a crack steadily propagating in an elastic–plastic fluid-saturated porous material displaying linear isotropic hardening. Quasi-static crack growth is considered under plane strain and Mode I loading conditions. In particular, the effective stress is assumed to obey the Drucker–Prager yield condition with associative or non-associative flow-rule and linear isotropic hardening is adopted. Both permeable and impermeable crack faces are considered. As for the problem of crack propagation in poroelastic media, the behavior is asymptotically drained at the crack-tip. Plastic dilatancy is observed to have a strong effect on the distribution and intensity of pore water pressure and to increase its flux towards the crack-tip.

Introduction

Parts of earth's shallow crust infiltrated with ground water or oil have a mechanical response complicated by elastic–plastic deformation coupled with diffusion of pore fluid. Hydraulic fracturing is often used in these media to enhance gas or oil production. In addition to the interest in this technique, time-dependent geophysical phenomena such as propagation of aseismic slips and following deformations have focussed research on fracture mechanics in viscous or inviscid poroelastic materials (Rice and Cleary, 1976, Rice and Simons, 1976, Cleary, 1978, Rudnicki, 1985, Rudnicki, 1991, Atkinson and Craster, 1991, Rudnicki and Koutsibelas, 1991, Desroches et al., 1994). However, neglecting plastic behavior of the solid skeleton represents only a first approximation. For instance, Johnson and Cleary, 1991, van den Hoek et al., 1993, Papanastasiou, 1997 suggest that linear elastic crack propagation models may underestimate the down-hole pressure measured during field operations for hydraulic fracturing. As a matter of fact, it will be shown in this paper — with reference to steady crack propagation — that plastic dilatancy strongly affects the pore pressure distribution at the crack-tip. There are few contributions that account for the elastic-plastic behavior of fluid-saturated porous media when analyzing crack propagation. Plastic dilatancy effects at the crack-tip were analyzed by van den Hoek et al. (1993) and Mohr-Coulomb elastoplasticity was considered during crack propagation by Papanastasiou and Thiercelin, 1993, Papanastasiou, 1997, using a combined finite difference/finite element technique.

In this article, an asymptotic solution is obtained for steady-state crack propagation under plane strain, mode I conditions. Propagation occurs in a porous, fluid-saturated material characterized by an elastoplastic skeleton obeying a Drucker–Prager yield criterion with volumetric non-associative flow law and isotropic hardening. The technique used to solved this asymptotic problem was initiated by Amazigo and Hutchinson (1977) and developed in various directions, but always for a single phase solid, by Achenbach et al., 1981, Zhang et al., 1983, Ponte Castaneda, 1987, Ostlund and Gudmundson, 1988, Bose and Ponte Castaneda, 1992, Bigoni and Radi, 1993, Bigoni and Radi, 1996, Radi and Bigoni, 1993, Radin and Bigoni, 1994, Radi and Bigoni, 1996, Herrmann and Potthast, 1995. Here, the mechanical behavior of the elastic-plastic medium is described through coupled constitutive equations developed, in the framework of the theory of mixtures, by Loret and Harireche (1991), where the Darcy's law is used to model the fluid diffusion process. Since the ductility of these materials is small compared with metals, as shown by the experimental data reported by Khan et al., 1991, Khan et al., 1992 for Berea sandstone, it is reasonable to consider the small deformation incremental theory. Moreover, both problems of permeable and impermeable crack faces are considered.

Like for poroelasticity, the behavior is asymptotically drained at the crack-tip. The results demonstrate that the explicit coupling between plastic dilatancy and fluid compressibility yields a peculiar behavior of the pore pressure near the crack-tip. Dilatancy increases the flux of water towards the crack-tip, but also the water pressure since in this asymptotic analysis the latter is not controlled at finite distance from the crack-tip. Moreover, the flux of fluid together with plasticity effects may dissipate the amount of supplied energy, leading to a reduction of the energy available to fracture the material.

Section snippets

Constitutive equations

We refer to the elastic–plastic model for fluid-saturated porous media proposed by Loret and Harireche (1991), that is summarized here. Within the context of small deformation incremental theory, both the strain rate ε̇ of the solid skeleton and the rate of fluid mass content per unit reference volume ζ̇ are decomposed into an elastic part and a plastic part, denoted by the respective indices ( )e and ( )p, namelyε̇=ε̇e+ε̇p,ζ̇=ζ̇e+ζ̇p.

The elastic strain rates ε̇e and ζ̇e are related to the

Crack propagation problem

The problem of a plane crack propagating at constant velocity c along a rectilinear path in an infinite medium is now considered. The mechanical behavior of the material is described by the incremental elastic–plastic constitutive model presented in Section 2. This framework allows for possible sectors of elastic unloading which may develop in the proximity of the crack-tip, during crack propagation. A cylindrical coordinate system (O, er, eϑ, e3) moving with the crack-tip towards the ϑ=0

Results

As mentioned previously, the singularity s, the stress, velocity fields and amplitudes of plastic and elastic sectors at the crack-tip in the drained material are available from Bigoni and Radi, 1993, Radi and Bigoni, 1993: in this analysis, these sectors are defined by rays emanating from the crack-tip. For convenience, we report in Fig. 1 the variation with μ of the stress singularity and the elastic unloading and plastic reloading angles, and in Fig. 2 the stress and velocity angular

Conclusions

Steady-state crack propagation has been analyzed in plane strain, Mode I conditions in an elastic-plastic fluid-saturated porous material. The performed analysis is restricted to the leading order terms of the crack-tip fields, neglecting inertial effects. Under this approximation, it is shown that the asymptotic pore pressure and flux fields can be uncoupled from the stress and velocity singular fields. In other words, to solve the problem it is possible, first to determine all asymptotic

Acknowledgements

Financial support of M.U.R.S.T. ex 60%–1998 ‘Propagazione dinamica delta frattura in materiali porosi saturi’ (E.R.), M.U.R.S.T. Cofin–1998 ‘Structural integrity assessment of large dams (D.B.) and GdR Géomécanique des Roches Profondes (B.L.) are gratefully acknowledged.

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