Poromechanics of an Externally Heated Sphere

Abstract

†Coupled THM processes that describe the mechanical behaviour of fluid saturated porous media are an extension to the classical theory originally proposed by Biot [1]. This extension in now an integral part of geomechanics [2-8], where the concepts are applied in a number of areas including the deep geological disposal of heat emitting high level nuclear fuel waste, thermal stimulation of resource bearing formations and the use of geologic basins for the permanent storage of greenhouse gases converted to supercritical forms. In addition to the consideration of coupled processes of elastic deformations of the porous skeleton, fluid transport in the pore space and heat transfer encountered in a variety of disciplines in the engineering sciences, it is also essential to accurately evaluate the pore fluid pressure responses, where adverse fluid pressures can induce damage in the form of microfracturing of the porous skeletal structure. Investigations dealing with the modelling of THM processes in poro-elastic fluid-saturated media are quite extensive; readers are further directed to recent literature in geomechanics, particularly those conferences devoted to poromechanics and the recent articles on nuclear waste management cited previously, for further information. The present paper includes the study of an idealized problem where the skeleton of the fluid-saturated material exhibits elasto-plastic effects. The treatment of such effects is not generally considered applicable to brittle geomaterials that have a greater susceptibility to damage development in the form of micro-cracks and micro-fissures that can alter the stiffness and fluid transmissivity properties [911]. The consideration of plasticity effects is largely of interest to soft rocks that could display elastoplastic constitutive behaviour in their skeletal responses, particularly in the small strain range [12-16]. Thermo-poroelasto-plasticity (TPEP) of clays has been discussed in connection with engineered clay barriers proposed for high level nuclear waste disposal endeavours [17-19]. Analytical investigations are rare; an example of the consolidation of a poro-elasto-plastic column is given in [20] and the CamClay plasticity model has been applied to the problem of thermal failure in saturated clays [9]. This paper presents a comparison of analytical and computational results for the idealized problem of the external heating of a fluid-saturated sphere with a skeletal response that can be described by elastic or elasto-plastic phenomena. The objectives of the research are to develop a set of benchmark analytical results that can be used to validate computational codes (such as COMSOL™ and ABAQUS™) and their ability to provide inter-code validations. Attention is restricted to porous materials where the pore space is saturated and the skeleton can exhibit mechanical and thermal deformations associated with stresses and temperatures applied to the skeletal phase. We assume that the failure of the porous skeleton can be described by an appropriate elasto-plastic model, characterized by a yield condition, a flow rule and a hardening law. It is also assumed that the thermo-physical properties for the solid phase of the geomaterial are the same, irrespective of whether it exhibits elastic or plastic responses. Also, we assume that the fluid transport properties in the porous medium are the same in both the intact and yielded regions. Governing Equations We denote the total stress tensor in the pore skeleton by (, ) ij i x t σ and the pore fluid pressure by (, ) i px t . The development of the poro-elastic part of the constitutive modelling adopts the procedures described by Selvadurai and Nguyen [5]. The constitutive equation governing the poroelastic response of the fluid-saturated porous medium is given by