Computational Modeling of Waves in Composite Saturated Poroviscoelastic Media

다국어 초록 (Multilingual Abstract)

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 [1], frozen or partially frozen sandstones [12,3,4], gas-hydrates in ocean-bottom sediments [5] and evaluation of the freezing conditions of foods by ultrasonic techniques [10].
A theory to describe wave propagation in frozen porous media was first presented by Leclaire et al. [8]. 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 et al. [9]. Later, Carcione and Tinivella [5] generalized this theory to include the interaction between the solid and ice particles and grain cementation with decreasing temperature. Also, Carcione et al. [1] applied this theory to study the acoustic properties of shaley sandstones, assuming that sand and clay are non-welded and form a continuous and inter-penetrating porous composite skeleton. Both frozen porous media and shaley sandstones are two examples of porous materials where the two solid phases are weakly-coupled or non-welded, i.e, both solids form a continuous and interacting composite structure, interchanging mechanical energy. Similar weakly-coupled formulations have previously been proposed. For instance, McCoy [11] has proposed a mixture theory appropriate for the combination of two acoustic phases.
This work presents a differential and numerical model to describe wave propagation in a heterogeneous poroviscolastic frame consisting of two weakly-coupled solid phases saturated by a single phase fluid. The equations of motion, stated in the space-frequency domain, generalizes that presented in [15] and [2] by the inclusion of solid matrix dissipation using a linear viscoelastic model and frequency dependent mass and viscous coupling coefficients. It also generalizes the models of Leclaire et al. [8] and Carcione et al. [5] for the case of uniform porosity, and consequently is the appropriate model to perform numerical simulation in heterogeneous materials.
The numerical procedures presented employ the nonconforming rectangular element defined in [7] to approximate the displacement vector in the solid phases. The dispersion analysis presented in [16] shows that employing this nonconforming element allows for a reduction in the number of points per wavelength necessary to reach a desired accuracy. On the other hand, the displacement in the fluid phase is approximated by using the vector part of the Raviart-Thomas-Nedelec mixed finite element space of zero order, which is a conforming space [14,13].
The error analysis yields optimal a priori error estimates for the global standard and hybridized Galerkin methods.
Numerical simulation of waves in porous media is computationally expensive due to a large number of degrees of freedom needed to calculate wave fields accurately; the use of a domain decomposition iteration is a convenient approach to overcome this difficulty. Here we define a nonoverlapping domain decomposition iterative scheme and derive convergence results similar to those presented in [6] for solving second-order elliptic problems.
This iterative procedure was used for the simulation of waves in a sample of water saturated partially frozen Berea sandstone [2,5], perturbed by a point source at seismic frequencies. The sample has an interior plane interface defined by a change in ice content in the pores, and the snapshots of the generated wave fields show clearly the events associated with the different types of waves.