Modelling of inherent anisotropy in sedimentary rocks

doi:10.1016/S0020-7683(01)00110-X

International Journal of Solids and Structures

Volume 39, Issue 3, February 2002, Pages 637-648

https://doi.org/10.1016/S0020-7683(01)00110-X Get rights and content

Abstract

This paper presents a constitutive relation for modelling the inelastic response of sedimentary rocks. The inherent anisotropy of this class of materials is described by employing a second-order microstructure tensor, whose eigenvectors define the principal material triad. Higher-order dyadic products of this tensor are incorporated in the distribution function, which specifies the directional dependence of strength parameters. The mathematical formulation is applied to model the mechanical characteristics of Tournemire shale. Several triaxial tests are simulated, at various initial confining pressures, for samples tested at different orientation relative to the loading direction. The results are compared with the available experimental data.

Introduction

Among sedimentary rocks, the most widespread are shale, siltstone and claystone. These rocks, which are formed by deposits of clay and silt sediment, exhibit strong inherent anisotropy, manifesting itself in a directional dependence of deformation characteristics. The anisotropy is strongly related to the microstructure, in particular the existence of bedding planes which mark the limits of strata and can be easily identified by a visual examination. The study of the mechanical behaviour of sedimentary rocks, especially shale and mudstone, is of particular interest to the oil exploration industry as well as to civil and mining engineering. These materials are often unavoidable in foundations of a broad range of civil structures, in underground excavations as well as tunnelling.

Over the last few decades, an extensive research effort has been devoted to study the mechanical behaviour of anisotropic rocks. Comprehensive references on this topic can be found in a number of review papers (see e.g., Amadei (1983), Kwasniewski (1993) and Ramamurthy (1993)). One significant direction of research has been that involving the experimental component. In general, several experimental studies were carried out (cf. Donath (1961), McLamore and Gray (1967), Hoek (1968), Atwell and Sandford (1974), Lerau et al. (1981) and Hoek (1983)) and the main focus was the directional dependence of rock strength. The results generally indicate that the maximum axial compressive strength is associated with configurations in which the bedding planes are either parallel or perpendicular to the loading direction. At the same time, the minimum strength is typically associated with failure along the weakness plane, which corresponds to sample orientations within the range 30–60°.

In parallel with experimental studies, extensive research has been carried out on formulation of appropriate general failure criteria. An extensive review on this topic, examining different approaches, is provided in the article by Duveau et al. (1998). In general, relatively little work has been done on the description of progressive failure in this class of materials, which stems primarily from difficulties associated with the formulation of the problem. The rigorous approach, based on general representation theorems for tensorial functions (Boehler and Sawczuk, 1970, Boehler and Sawczuk, 1977), is very complex and has never been applied to any practical problem.

The objective of this paper is to propose an approach which retains the mathematical rigour and, at the same time, is pragmatic, i.e. simple enough to be applied to solve some practical engineering problems. The formulation incorporates a scalar anisotropy parameter which is expressed in terms of mixed invariants of the stress and structure-orientation tensors. Such a parameter has recently been introduced by Pietruszczak and Mroz (2001) in the context of specification of the conditions at failure. The main focus of this work is on constructing a complete plasticity framework for describing the deformation process in anisotropic sedimentary rocks. In the next section, the formulation of the problem is outlined, followed by a discussion on the identification of material functions involved. The formulation is applied to study the behaviour of Tournemire shale. In particular, several triaxial tests are simulated at different initial confining pressures. The emphasis is on modelling of the dependence of deformation characteristics on the orientation of the sample relative to the direction of loading.

Section snippets

A constitutive model for anisotropic rocks

Consider first the specification of the conditions at failure. Following the framework developed in Pietruszczak and Mroz (2001), assume that the failure criterion can be expressed in a simplified form $F=F(σ, a)=F(tr σ, tr σ^{2}, tr σ^{3},η)=0$ In the above equation, $a$ is a microstructure tensor and η is a scalar anisotropy parameter which represents the projection of this tensor on a suitably defined loading direction $l$ , i.e. $η=a_{ij} l_{i} l_{j}$ In order to define the loading vector $l$ , consider the principal triad $e^{(i)}$ ,

On identification of material functions/parameters

The formulation outlined in the previous section has been applied to model the mechanical characteristics of a shale taken from the Tournemire site in the Massif Central, France. A comprehensive experimental program has been carried out at Laboratoire de Mécanique de Lille, and the results have been reported by Niandou (1994) and Niandou et al. (1997). The primary minerals for this shale are kaolinite (27.5%), quartz (19%), illite (16.5%) and calcite (15%) and the porosity is in the range of

Numerical Simulations

In order to illustrate the performance of the model, a number of triaxial tests, as reported by Niandou (1994) and Niandou et al. (1997), have been simulated. The tests, carried out on Tournemire shale, involved different initial confining pressures as well as different orientations of the samples relative to the loading direction. The results of numerical simulations are provided in Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7.

Fig. 3 shows the mechanical characteristics for vertical samples, α=0,

Final remarks

In this work, a mathematical formulation has been put forth for the description of deformation characteristics of anisotropic sedimentary rocks. The primary focus was to propose an approach which is rigorous, but at the same time is simple enough to be implemented for the solution of practical engineering problems. The framework has been illustrated by numerical examples simulating a series of triaxial tests carried out on Tournemire shale. It appears that the basic trends in mechanical

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Modelling of inherent anisotropy in sedimentary rocks

Abstract

Introduction

Section snippets

A constitutive model for anisotropic rocks

On identification of material functions/parameters

Numerical Simulations

Final remarks

Int. J. Rock Mech. Min. Sci.

Int. J. Solids Struct.

Int. J. Solids Struct.

J. Mech. Phys. Solids

Rock Anisotropy and the Theory of Stress Measurements

Intrinsic shear strength of a brittle anisotropic rock – I: experimental and mechanical interpretation

Int. J. Rock Mech. Min. Sci. Geomech. Abstr.

Equilibre limite des sols anisotropes

J. de Mécanique

On yielding of oriented solids

Acta Mechanica

Experimental study of shear failure in anisotropic rocks

Geol. Soc. Am. Bull.

Assessment of some failure criteria for strongly anisotropic materials

Mech. Cohesive Frict. Mater.