Elastoplastic deformation of a porous rock and water interaction
Introduction
Porous chalks have been extensively studied in mining engineering and petroleum industry. In particular, ground subsidence due to pore compaction of chalk is of great interest for off-shore reservoir engineering. A series of experimental studies, including microstructure analysis and mechanical behaviour investigation of various rocks, have been performed, for instance (Elliott and Brown, 1985, Halleux et al., 1990, Wong et al., 1992, Brignoli et al., 1994, Andersen et al., 1992, Risnes and Flaageng, 1999, Risnes et al., 2000, Monjoie et al., 1995, Lord et al., 1998, Papamichos et al., 1997, Schroeder et al., 1998, Homand and Shao, 2000, Ch. Schroeder, 2003, C. Schroeder, 2003, Chen and Hu, 2003, De Gennaro et al., 2004). These works have shown that the mechanical behaviour of this material is very complex, and characterized by various features: such as low material cohesion, strong pressure sensitivity, plastic pore collapse, creep deformation, dependency on porosity and mineralogical compositions. Further, the mechanical behaviour of chalk is strongly sensitive to chemical and physical nature of pore fluid. The saturation of chalk by water-like fluid could induce pore collapse and substantial plastic deformation, and also increase the rate of creep deformation. In the context of oil engineering, water is injected to reservoir in order to maintain pore pressure during oil production. Therefore, the water sensitivity of chalk can generate very serious subsidence problem and economic consequence. A number of constitutive models have been developed for the description of mechanical behaviour of porous chalks. Most of them have been based on the adaptation of classic elastoplastic models used in soil mechanics. During the recent years, in the context of compaction study in oil reservoir, a lot of effort has been made on the modelling of water saturation effects and creep deformation in reservoir chalk. Some conceptual models, for example, that by Piau and Maury, 1994, Piau et al., 1998 , have been developed. The basic idea is to introduce an additional fictive stress tensor which is seen as the responsible to the plastic deformation induced by the water saturation. This kind of models provides a simple conceptual tool for the evaluation of reservoir compaction. However, this approach does not propose a general constitutive modelling of chalk behaviour in complex loading conditions. The physical meaning of the fictive stress concept is not clear and the theoretical background of the model remains confuse. Therefore, it is needed to develop more general constitutive models able to describe the various features of porous chalk including creep deformation and water sensitivity.
In the general framework of plastic modelling of rock materials, many constitutive models have been developed with specific yield functions and plastic potentials (we do not give here an exhaustive list of all these models). However, two basic plastic flow mechanisms are usually identified; the first one is related to plastic shearing of rock matrix, and the second one to volumetric consolidation. The physical mechanism of volumetric consolidation is related to the microstructure of material. In granular and powder materials, the volumetric consolidation may be generated by the rearrangement of grains reducing pore space. In porous cohesive materials like chalk, the volumetric compaction is generally resulted by inelastic pore collapse by breaking contact forces between grains. Depending on rock material and loading condition, one or other plastic mechanism may be privileged in constitutive modelling. For the modelling of two plastic flow mechanisms in geological materials, the classic method is to combine two yield surfaces, a cap surface for plastic consolidation and a cone surface for plastic shearing (DiMaggio and Sandler, 1971, Lade, 1977, Desai, 1980, Chen and Saleeb, 1982, Desai and Siriwardane, 1984, Gens and Nova, 1993, Cristescu, 1996, Lade, 1997, Desai, 2001, Peric and Ayari, 2002). This kind of approach has also been adopted in other engineering materials (Fleck et al., 1992, Khan and Huang, 1995, Gu et al., 2001). For example, Gu et al. (2001) have developed a constitutive model for cold compaction of metal powders. The authors assume the plastic flow of metal powder to be a combination of a plastic shearing mechanism and a consolidation mechanism. For the distortion mechanism, their model employs a pressure sensitive, Mohr–Coulomb type yield criterion and a non-associated flow rule. For the consolidation mechanism the model employs a smooth elliptical yield surface, together with an associated flow rule. On the other hand, in order to avoid singularity point at the intersection between two surfaces, plastic models with a single surface combining cap and cone have alternatively been proposed (Lade and Kim, 1995, Ehlers, 1995, Aubertin et al., 1999, Lewis and Khoei, 2001, Aubertin and Li, 2004, Khoei and Azami, 2005). According to the extensive experimental data, the basic mechanical behaviour of porous chalk can also be described within this framework. However, in the present work, we will give a clear physical interpretation of two plastic mechanisms and propose specific plastic functions taking into account particular features observed in chalk behaviour. In particular, the chalk is seen as a porous material composed of an equivalent solid matrix and a connected porosity. The macroscopic behaviour of chalk depends on the properties of the solid matrix and the porosity. The mechanical properties of the solid matrix are related to three types of contact force, capillary-related liquid contact force, cemented solid contact force and frictional point contact force. Plastic shearing and pore collapse are generated by progressive destruction of these contact forces. For the modelling of plastic shearing, a smooth quadratic yield function will be proposed to better account for the strong pressure sensitivity of chalk. This is an improvement of classic linear Mohr–Coulomb type criteria. For the pore collapse mechanism, we propose an adaptation of a micromechanics based plastic yield criterion; the Gurson’s criterion (Gurson, 1977, Tvergaard, 1990). This criterion can be obtained by using a rigorous homogenization technique for the determination of macroscopic plastic yielding condition of porous materials (Perrin and Leblond, 1990, Perrin and Leblond, 2000, Leblond and Perrin, 1996). The plastic yield condition is explicitly related to the yield stress of the equivalent solid matrix and porosity.
On the modelling of creep deformation in engineering materials, the viscoplastic theory is largely used. Different viscoplastic models, based on either thermodynamic potential or over-stress approach, have been developed; for instance Perzyna, 1966, Cristescu, 1986, Cristescu, 1994, Jin and Cristescu, 1998, Maranini and Yamaguchi, 2001, Haupt and Kersten, 2003, Voyiadjis et al., 2004, Saleeb and Arnold, 2004. Some viscoplastic models have also been proposed for creep deformation of porous chalk (Shao et al., 1995, Collin et al., 2002, De Gennaro et al., 2003). However, although this kind of theory provides a powerful mathematical description of creep deformation, the physical mechanisms of creep in material are not clearly identified. Further, the viscoplastic deformation is described independently with the plastic deformation. Two distinct formulations will be needed respectively for the short term behaviour (elastoplastic model) and for the long term behaviour (viscoplastic model). Recently, the experimental investigations performed by Hellemann et al., 2002a, Hellemann et al., 2002b have shown that the inter-granular pressure solution process due to aqueous solutions may be an essential mechanism of the mechanical degradation in porous chalks, by progressive dissolution of cemented solid surfaces, leading to the reduction of effective contact area. This physical–chemical interaction could be responsible to time dependent deformation of chalk. Based on these analyses, in the present, a physically based unified approach will be proposed. The creep deformation is assumed to be generated by time dependent dissolution of grain contact surface leading to the degradation of elastic and plastic properties. Therefore, the creep deformation and the time independent strains will be described by the same set of formulations.
Another aspect which should be taken into account in the modelling of chalk behaviour is the strong sensitivity to water saturation. The theoretical framework for the mechanical of saturated porous media was founded by Biot’s theory in studying three-dimensional consolidation of soil (Biot, 1955, Biot, 1973). This framework has been successively extended to include plastic deformation and damage mechanics (Coussy, 1995, de Buhan and Dormieux, 1996, Lydaba and Shao, 2002, Coussy, 2004, Shao et al., 2004). The constitutive models for saturated media have been then extended to partially saturated media, by using the Bishop’s effective stress concept (Bishop and Blight, 1963), and the net stress concept proposed by the so-called Barcelona plastic model (Alonso et al., 1990, Houlsby, 1997). Micromechanical analyses have been performed in order to validate the basic assumptions used in macroscopic models, for instance, Chateau and Dormieux (2002). The key feature of these approaches is to find an appropriate way to take into account the effect of capillary pressure on mechanical behaviour. In this framework, Collin et al. (2002) proposed an elastoplastic model for porous chalks partially saturated by oil and water. The suction was seen as the responsible to water sensitivity of porous chalk. However, due to the microstructure of most porous chalks, the capillary pressure is generally quite small with respect to applied stresses; the suction concept alone cannot fully explain the drastic decrease of mechanical strength in water saturated chalk and in particular the time dependent creep deformation. Water saturation can not only induce a capillary effect but also contribute to the modification of microstructure of chalk. Based on these analyses, in the present work, a new approach is proposed for modelling the effect of water saturation. Two kinds of effects will be taken into account. In short term, water saturation reduces the yield stress and failure strength of equivalent solid matrix due to the decrease of capillary force in the liquid contact. In long term, water saturation enhances the dissolution process of the cemented contact and then emphasizes the time dependent strains. However, the topic of the present paper is limited to the study of water saturation effects on mechanical behaviour of chalk without taking into account effect of pore pressure. Modelling of coupled poromechanical behaviour in partially saturated condition will be investigated in a future work.
A short review of the mechanical behaviour of typical porous chalks is first presented including some new laboratory data. A general and unified constitutive model is then formulated by accounting for the main aspects of chalk behaviour including creep deformation and water saturation effects. The predictive capacity of the proposed model will be examined through numerical simulations of laboratory tests with various loading conditions.
Section snippets
Review of mechanical behaviour of typical porous chalks
The rock chosen in this work is the so-called “Lixhe chalk”, which is representative of highly porous rocks. This chalk is from Upper Campanian age and drilled in the CBR quarry near Liège (Belgium). This material has been studied in a series of previous experimental investigations because its mechanical behaviour is qualitatively close to that of North Sea reservoir chalks (Ch. Schroeder, 2003, C. Schroeder, 2003). It is a very pure chalk, composed of more than 98% of CaCO3, and with less than
Formulation of the constitutive model
According to the previous review on the basic mechanical behaviour of porous chalks, two plastic deformation mechanisms are taken into account; the plastic shear mechanism at low confining pressures and the plastic pore collapse mechanism at high confining pressures. The first mechanism is quite a common mechanism for most cohesive-frictional rock materials. Classical models based on Mohr–Coulomb and Drucker–Prager criteria can be adapted by choosing an appropriate hardening and eventually
Numerical simulations of oil saturated chalk
The proposed model contains 12 parameters to describe basic elastic–plastic behaviour of porous chalk. These parameters correspond to clearly identified deformation mechanisms and can be determined from conventional laboratory tests. The two elastic parameters, E0 and ν0, are obtained from the initial linear part of stress–strain curve during a conventional triaxial compression test. The initial pore collapse stress is easily identified from the transition point from linear to non-linear
Influence of water saturation
According to the previous analysis, the macroscopic mechanical behaviour of porous chalk depends on the mechanical properties of the equivalent solid matrix, which are related to the various contact forces. Among these contact forces, the liquid contact force related to the capillary effect plays an important role in porous chalks. When chalk is saturated with water, the liquid contact force is reduced, leading to the diminution of plastic yield stress of the solid matrix. Thus, the diminution
Creep deformation
As observed from the review of mechanical behaviours of porous chalks, the creep deformation of material is essentially related to the evolution of microstructure, which is characterized by the progressive destruction of various contact forces between grains. According to the experimental investigations performed by Hellemann et al., 2002a, Hellemann et al., 2002b, the inter-granular pressure solution process due to aqueous solutions may be an essential mechanism of the mechanical degradation
Conclusions
Based on the extensive experimental investigations, the elastoplastic deformation of a porous chalk has been studied. Two plastic deformation mechanisms have to be taken into account: plastic pore collapse and plastic shearing. The plastic pore collapse process is an important phenomenon of porous chalks to be taken into account in various applications. A new elastoplastic model has been proposed, taking into account the two plastic mechanisms in relation with the microstructure of porous
Acknowledgement
The present work is partially supported by the Natural Science Foundation of China (NSFC) through the Grant Number 50128908.
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