Analytical Solution for Wave Propagation in Stratified Poroelastic Medium. Part I: the 2D Case

We are interested in the modeling of wave propagation in poroelastic media. We consider the biphasic Biot’s model in an infinite bilayered medium, with a plane interface. We adopt the Cagniard-De Hoop’s technique. This report is devoted to the calculation of analytical solutions in two dimensions. The solutions we present here have been used to validate numerical codes. Key-words: Biot’s model, poroelastic waves, analytical solution, Cagniard-De Hoop’s technique. ∗ EPI Magique-3D, Centre de Recherche Inria Bordeaux Sud-Ouest † Laboratoire de Mathématiques et de leurs Applications, CNRS UMR-5142, Université de Pau et des Pays de l’Adour – Bâtiment IPRA, avenue de l’Université – BP 1155-64013 PAU CEDEX in ria -0 03 05 39 5, v er si on 1 24 J ul 2 00 8 Solution analytique pour la propagation d’ondes en milieu poroélastique stratifié. Partie I : en dimension 2 Résumé : Nous nous intéressons à la modélisation de la propagation d’ondes dans les milieux infinis bicouches poroélastiques. Nous considérons ici le modèle bi-phasique de Biot. Cette partie est consacrée au calcul de la solution analytique en dimension deux à l’aide de la technique de Cagniard-De Hoop. Les solutions que nous présentons ici ont été utilisés pour la validation de codes numériques. Mots-clés : Modèle de Biot, ondes poroélastiques, solution analytique, technique de Cagniard de Hoop. in ria -0 03 05 39 5, v er si on 1 24 J ul 2 00 8 Analytical solution 3 Introduction Many seismic materials cannot only be considered as solid materials. They are often porous media, i.e. media made of a solid fully saturated with a fluid: there are solid media perforated by a multitude of small holes (called pores) filled with a fluid. It is in particular often the case of the oil reservoirs. It is clear that the analysis of results by seismic methods of the exploration of such media must take to account the fact that a wave being propagated in such a medium meets a succession of phases solid and fluid: we speak about poroelastic media, and the more commonly used model is the Biot’s model [1, 2, 3]. When the wavelength is large in comparison with the size of the pores, rather than regarding such a medium as an heterogeneous medium, it is legitimate to use, at least locally, the theory of homogeneization [4, 12]. This leads to the Biot’s model [1, 2, 3] which involves as unknown not only the displacement field in the solid but also the displacement field in the fluid. The principal characteristic of this model is that in addition to the classical P and S waves in a solid one observes a P “slow” wave, which we could also call a “fluid” wave: the denomination “slow wave” refers to the fact that in practical applications, it is slower (and probably much slower) than the other two waves. The computation of analytical solutions for wave propagation in poroelastic media is of high importance for the validation of numerical computational codes or for a better understanding of the reflexion/transmission properties of the media. Cagniard-de Hoop method [5, 7] is a useful tool to obtain such solutions and permits to compute each type of waves (P wave, S wave, head wave...) independently. Although it was originally dedicated to the solution to elastodynamic wave propagation, it can be applied to any transient wave propagation problem in stratified medium. However, as far as we know, few works have been dedicated to the application of this method to poroelastic medium. In [11] the analytical solution to poroelastic wave propagation in an homogeneous 2D medium is provided. In order to validate computational codes of wave propagation in poroelastic media, we have implemented the codes Gar6more 2D [9] and Gar6more 3D [10] which provide the complete solution (reflected and transmitted waves) of the propagation of wave in stratified 2D or 3D media composed of acoustic/acoustic, acoustic/elastic, acoustic/poroelastic or poroelastic/poroelastic layers. The codes are freely downloadable at http://www.spice-rtn.org/library/software/Gar6more2D and http://www.spice-rtn.org/library/software/Gar6more3D. We will focus in this paper on the 2D poroelastic case, the 2D acoustic/poroelastic case is detailed in [8] and the three dimensional cases will be the object of forthcoming papers. The outline of the paper is as follows: we first present the model problem we want to solve and derive the Green problem from it (section 1). Then we present the analytical solution to the wave propagation problem in a two-layered 2D poroelastic (section 2). Finally we show how the analytical solution can be used to validate a numerical code (section 3). RR n° 6591 in ria -0 03 05 39 5, v er si on 1 24 J ul 2 00 8 4 Diaz & Ezziani 1 The model problem We consider an infinite two dimensional medium (Ω = R) composed of two homogeneous poroelastic layers Ω+ = R×] − ∞, 0] and Ω = R × [0,+∞[ separated by an horizontal interface Γ (see Fig. 1). We first describe the equations in the two layers (§1.1) and the transmission conditions on the interface Γ (§1.2), then we present the Green problem from which we compute the analytical solution (§1.3). Ω+ Ω First Layer