Partially Saturated Porous Media: A Dynamic Boundary Element Formulation
Martin Schanz
Institute of Applied Mechanics
Graz University of Technology
Abstract
A lot of applications, especially, in geomechanics require the computation of waves in porous media [5], e.g., earthquake waves in soil. Soil is a partial saturated poroelastic material which sometimes can be modelled by a saturated theory, however, sometimes a partial saturated theory is necessary. Having waves in semi-infinite domains in mind a boundary element formulation for such materials seems to be preferable. But, this works only if a linear theory for partial saturated poroelasticity is used.
The partial saturated continuum consists of an elastic solid skeletton and a wetting and non- wetting interstitial fluid. A theory for such a three phase material can be derived based on the mixture theory [1]. Transforming the time dependent set of partial differential equations to Laplace domain allows to reduce the unknowns to the physical necessary set of solid displace- ments and both pore pressures (see [2]).
The fundamental solutions of such an elliptic coupled set of partial differential equations can be found using the method of Hörmander. Their singular behavior is determined by developing the exponential functions in these solutions in a power series. This shows that the fundamental solutions are at most weakly singular and the singular behavior is similar to elastostatics and acoustics [3].
The integral equations can be deduced based on the weighted residual technique. Obviously, for the final integral equations not only the weak singular fundamental solutions are required as well their co-normal derivatives and respective traction representations are needed. The latter ones cause strong singular integrals. These can be regularized by partial integration [4]. The final weak singular integral equation is discretized with standard polynomial shape functions in the spatial variable. Applying the convolution quadrature for time discretisation yield, finally, a time stepping procedure for dynamic processes in partial saturated poroelastic media.
The validation of this method is done with the help of a 1-d semi-analytical solution for a column [2]. The sensitivity on the spatial as well as on the temporal discretisation is presented. Finally, waves in a poroelastic half space are studied as well as the mode of action of an open trench for vibration isolation.
References
[1] Lewis, R. W.; Schrefler, B. A.: The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media. John Wiley and Sons, Chichester, 1998.
[2] Li, P.; Schanz, M.: Wave Propagation in a One Dimensional Partially Saturated Poroelastic Column. Geophys. J. Int., 184(3), 1341-1353, 2011.
[3] Li, P.; Schanz, M.: Time Domain Boundary Element Formulation for Partially Saturated Poroelas- ticity. Eng. Anal. Bound. Elem., 37(11), 1483-1498, 2013.
[4] Messner, M.; Schanz, M.: A Regularized Collocation Boundary Element Method for Linear Poroe- lasticity. Comput. Mech., 47(6), 669-680, 2011.
[5] Schanz, M.: Wave Propagation in Viscoelastic and Poroelastic Continua: A Boundary Element Approach, Vol. 2, Lecture Notes in Applied Mechanics. Springer-Verlag, Berlin, Heidelberg, New York, 2001.