SIAM J. NUMER. ANAL. c  2007 Society for Industrial and Applied Mathematics Vol. 45, No. 1, pp. 389–420 FINITE ELEMENT METHODS FOR THE SIMULATION OF WAVES IN COMPOSITE SATURATED POROVISCOELASTIC MEDIA ∗ JUAN E. SANTOS † AND DONGWOO SHEEN ‡ Abstract. This work presents and analyzes a collection of finite element procedures for the simulation of wave propagation in a porous medium composed of two weakly coupled solids saturated by a single-phase fluid. The equations of motion, formulated in the space-frequency domain, include dissipation due to viscous interaction between the fluid and solid phases with a correction factor in the high-frequency range and intrinsic anelasticity of the solids modeled using linear viscoelasticity. This formulation leads to the solution of a Helmholtz-type boundary value problem for each temporal frequency. For the spatial discretization, nonconforming finite element spaces are employed for the solid phases, while for the fluid phase the vector part of the Raviart–Thomas–Nedelec mixed finite element space is used. Optimal a priori error estimates for global standard and hybridized Galerkin finite element procedures are derived. An iterative nonoverlapping domain decomposition procedure is also presented and convergence results are derived. Numerical experiments showing the application of the numerical procedures to simulate wave propagation in partially frozen porous media are presented. Key words. poroviscoelasticity, finite element method, error estimate, domain decomposition AMS subject classifications. 65C20, 65N30, 65N55, 86-08 DOI. 10.1137/050629069 1. Introduction. Wave propagation in composite porous materials has appli- cations in many branches of science and technology, such as seismic methods in the presence of shaley sandstones [8], frozen or partially frozen sandstones [29, 10, 11], gas-hydrates in ocean-bottom sediments [12], and evaluation of the freezing condi- tions of foods by ultrasonic techniques [26]. A recent review of the theory of wave propagation in fluid-saturated porous media can be found in [7]. A theory to describe wave propagation in frozen porous media was first presented by Leclaire, Cohen-Tenoudji, and Aguirre Puente [24]. This model, valid for uniform porosity, predicts the existence of three compressional and two shear waves; the veri- fication that additional (slow) waves can be observed in laboratory experiments was published by Leclaire, Cohen-Tenoudji, and Aguirre Puente [25]. Later, Carcione and Tinivella [12] generalized this theory to include the interaction between the solid and ice particles and grain cementation with decreasing temperature, used as a parame- ter to determine the bulk water content. Also, Carcione, Gurevich, and Cavallini [8] applied this theory to study the acoustic properties of shaley sandstones, assuming that sand and clay are nonwelded and form a continuous and interpenetrating porous composite skeleton. Both frozen porous media and shaley sandstones are examples of porous materials where the two solid phases are weakly coupled or nonwelded, i.e., both solids form a continuous and interacting composite structure, interchanging mechan- ∗ Received by the editors April 12, 2005; accepted for publication (in revised form) May 26, 2006; published electronically February 15, 2007. http://www.siam.org/journals/sinum/45-1/62906.html † CONICET, Departamento de Geof´ ısica Aplicada, Facultad de Ciencias Astron´omicas y Geof´ ısicas, UNLP, Paseo del Bosque S/N, La Plata, 1900, Argentina, and Purdue University, West Lafayette, IN 47907 (santos@fcaglp.unlp.edu.ar). ‡ Department of Mathematics, Seoul National University, Seoul 151–747, Korea (sheen@snu.ac.kr). The research of this author was supported in part by Korea Research Foundation grant KRF-2004- C00007 and Korea Science and Engineering Foundation grant KOSEF R14-2003-019-01000-0. 389