Finite Element Methods for the Simulation of Waves in Composite Saturated Poroviscoelastic Media

Abstract

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 applications 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 conditions 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, and Aguirre Puente [24]. This model, valid for uniform porosity, predicts the existence of three compressional and two shear waves; the verification that additional (slow) waves can be observed in laboratory experiments was published by Leclaire, 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 parameter 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-

Discontinuous Galerkin methods for wave propagation in poroelastic media

“…These change the formal expression of ⌿ , which then has much more complex time dependencies. In the frequency domain, this is not a problem because the convolutional product in equations 10d-10f becomes a normal product ͑Rubino et al, 2007;Santos and Sheen, 2007͒. However, to solve Biot's equations in the time domain with such high-frequency viscodynamic operators, one can apply the concept of memory variables to fit phenomenologically some attenuating laws to that viscous mechanism ͑Carcione, 2001͒. This can be done in a way similar to the introduction of viscoelasticity in the elastic equation system ͑Käser et al, 2007͒.…”

Finite element modeling of SHTE and PSVTM electroseismics

“…Moreover, the domain decomposition technique is devised so as to make straightforward the implementation of the numerical algorithms on parallel computers or distributed systems. A priori optimal error estimates for the discretization of the electroseismic equations using this finite element spaces were derived in Santos (2009), while an a priori optimal error estimates for Biot's equation can be found in Santos and Sheen (2007); and the corresponding dispersion analysis was presented in Zyserman and Santos (2007). On the other hand, finite element procedures to solve Maxwell's equations in 2D and 3D within the frame of magnetotelluric modeling were presented in Santos (1998), Zyserman et al (1999, Douglas et al (2000), Santos and Sheen (2000), and Zyserman and Santos (2000).…”

“…Concerning the computational domain, the presented algorithm can deal with heterogeneous subsurfaces with lateral variations; there is no restriction to homogeneous or horizontally layered models. The computational boundary conditions here used are absorbing boundary conditions, which make the computational border transparent for normally impinging waves (Santos and Sheen, 2007;Santos, 2009). Turning back to the finite element methods, the iterative domain decomposition technique leads to the solution of a large number of relatively small linear systems, easily solvable by direct solvers.…”

Finite element approximation of coupled seismic and electromagnetic waves in fluid‐saturated poroviscoelastic media

“…To show uniqueness for the solution of (4.17), let As each term in the left-hand side of (4.20) is nonnegative, it follows that Next, as in [21], using (4.25) and (4.26) let us take a corner element j of p with two faces contained in p . Without loss of generality, after a transformation we can assume that j = (−1, 1…”

“…A priori optimal error estimates for the discretization of Biot's equations of motion using the finite element spaces described earlier were derived in [21], and the corresponding dispersion analysis was presented in [22]. On the other hand, finite element procedures to solve Maxwell's equations in 2D and 3D within the frame of magnetotelluric modeling were presented in [23][24][25][26][27].…”