ABSTRACT: The fundamental solutions for two-dimensional displacement discontinuities and instantaneous and continuous point fluid sources in a homogeneous, isotropic, linear elastic, fluid-saturated porous media are presented. These solutions are used to develop a two-dimensional boundary element program that computes the stress, displacement, pore pressure and fluid flux fields for problems of arbitrary geometry. Several applications are presented.
RÉSUMÉ: On presente les solutions fondamentales en deux dimensions pour discontinuites de deplacement et sources de fluide instantanees et continues dans un milieu homogene, isotrope, elastique lineaire, poreux et sature de fluide. Nous avons developpe un programme aux elements de frontiere qui permet le calcul des contraintes, des deplacements, de la pression interstitielle et de l'ecoulement, de fluide en problèmes de geomatrie arbitraire. Quelques applications sont presentes.
ZUSAMMENFASSUNG: Die fundamental en Lösungen zweidimensionaler Verschiebungsdlskontinuitaten und momentaner und kontinuierlicher Fluessigkeitsquellen in einem homogenen, isotropen, linear elastischen, fluessigkeitsgesattigten, porösen Material werden dargestellt. Mit Hilfe dieser Lösungen wurde ein zweidimensionales Grenzelementprogramm entwickelt, welches Spannungen, Verschiebungen, Porendruecke und Fluβfelder fuer Probleme beliebiger Geometrie berechnet. Mehrere Anwendungen werden gezeigt.
1. INTRODUCTION The important influence that pore pressure has on the deformation behaviour of fluid saturated porous rocks is widely recognized. The pore pressure variation in space and time controls not only the flow of fluid but also the effective stress field in the rock mass. Where the rock mass is jointed, the pore pressure distribution can strongly affect whether slip will occur along preexisting discontinuities. A consistent theory which explicitly accounts for the coupling between the pore fluid and the rock matrix responses was developed by Biot (1941). In this paper we use Biot's theory to develop the two-dimensional fundamental solutions (Green's functions) for the normal (aperture) and shear (ride) displacement discontinuities, as well as instantaneous and continuous point fluid sources in a poroelastic medium. The displacement discontinuity solutions are constructed using ‘point force’ quadrupoles together with an instantaneous point fluid source. These Singular solutions form the basis of the boundary element technique developed in section 3.
2. GREEN'S FUNCTIONS It is assumed that both the material matrix and its Solid components (mineral grains) have a linear elastic behaviour and the fluid flow through the porous Skeleton is laminar. The equations presented here are valid for a fluid-saturated isotropic, homogeneous material under small strain. A tension positive convention is used for the stresses while a positive pore pressure is treated as compressive. Throughout the paper, all subscripts take the values 1, 2, a comma denotes partial differentiation with respect to Xi and the usual summation convention is employed over repeated subscripts.
2.2. Displacement discontinuities The solutions for displacement discontinuities (DDs) can be derived from the point force solutions for a poro-elastic medium (Cleary, 1976, Wiles and Curran, 1982). The normal DO is obtained by applying to the medium two sets of double forces (dipoles). The application of the sets of double forces initiates a diffusion process which causes the magnitude of the DD to change with time. In order to keep the magnitude constant, an instantaneous point fluid sink has to be applied simultaneously with the double forces. To obtain a unidirectional DD one of the two sets of double forces has to be adjusted to prevent lateral movement at the point of application. The shear DD is obtained by applying to the medium two sets of double forces with moment. This creates a state of pure shear, causing no change in volume, and therefore no diffusion process, at the point of application.
3. DISPLACEMENT DISCONTINUITY METHOD The displacement discontinuity method for fluid saturated poroelastic media consists of discretizing the boundary of a body into elements and determining the intensities of the normal and shear DDs and the fluid sources on the elements such that their summed effects reproduce the prescribed boundary conditions.