Abstract

We present a new continuum damage mechanics constitutive model for brittle geomaterials based on irreversible thermodynamics with internal state variables. Damage is incorporated by a vector-valued internal state variable that can be interpreted as the local void density in a cross-section with normal n within a representative material volume element. The thermodynamic basis, free energy function, damage evolution equation, and damage-induced inelastic compliance are derived. It is shown that damage causes strain-softening and positive dilatancy, as well as decrease of the material stiffness. Comparisons of the model predictions with well-known experimental data on rocks and concrete under uniaxial and triaxial compression stresses are given. This model could be valuable in appropriate Petroleum Geomechanics cases.

Introduction

Reliable and realistic constitutive laws for geomaterials such as shales, poorly consolidated sandstones and North Sea Chalks for use in analytical and computational procedures is a key aspect of petroleum geomechanics. The importance of constitutive laws has increased with the development of more sophisticated finite element analysis and other numerical techniques for the solution of complex problems. When realistic constitutive laws are incorporated into mathematical models, they can be used to address practical engineering problems such as borehole stability, hydraulic fracture, coupling of transport processes and shearing of reservoirs and overburden strata. However, the range of constitutive models that can be used within mathematical models is limited both by theoretical aspects and by a deficiency of detailed behavioral data on which to base the models. This may well be the weakest link in the simulation of petroleum geomechanics problems at the present time. Hence, the development of general, basic and realistic constitutive models to predict deformations and stresses remains an important task.

During loading to failure, dense geomaterials such as the shales, sandstones and limestones encountered in drilling and reservoir mechanics usually exhibit strain-softening, large positive dilatancy, a gradual decrease of stiffness with increased loading, an increase of peak strength with increasing confining pressure, and a gradual brittle-ductile transition at high confining pressure. Less commonly, at the stresses generated at depth by drilling and production practices, contractile behavior and liquefaction can take place. These loading responses can not be satisfactorily described by classical plasticity theory, which is nevertheless generally successful in modeling ductile behavior.

Although numerous investigations have been conducted on constitutive relationships for geomaterials, progress is slow because of the complexity of the material responses to loads, and the great variability in parameters. Continuum Damage Mechanics (CDM) provides a viable constitutive framework for efficient modeling of geomaterials. CDM is characterized by the introduction of a special internal state variable, called a damage variable, which represents in an average sense irreversible microstructural rearrangements. Damage variable is regarded as a continuous measure of the material degradation in the constitutive equations (Krajcinovic, 1984; Kachanov, 1986; Bazant, 1986; and Lemaitre, 1992). CDM also can be incorporated into mathematical models in a formal manner so that the basic thermodynamics laws are always satisfied, something that many plasticity approaches fail to achieve.

In this paper, a novel continuum damage mechanics model for geomaterials based on irreversible thermodynamics with internal state variables is developed. The model incorporates damage in the material response function as a set of internal vector variables representing irreversible microstructural rearrangement and regarded as a continuous measure of the material degradation. Several comparisons of available experimental testing data with numerical predictions are provided to validate the theory.

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