Finite element modeling of seismic attenuation due to fluid flow in partially saturated rocks Beatriz Quintal*, ETH Zurich, Holger Steeb, Ruhr-University Bochum, Marcel Frehner, University of Vienna, and Stefan M. Schmalholz, ETH Zurich Summary The finite element method is used to solve Biot’s quasi- static equations of consolidation. We perform 1D and 2D numerical creep tests of partially saturated porous rocks to calculate the frequency-dependent seismic attenuation and phase velocity from the modeled stress-strain relations. The resulting attenuation and velocity dispersion are due to fluid flow induced by pressure differences between mesoscopic-scale regions of the rock fully saturated with different fluids (White’s model). Comparisons of our numerical results with analytical solutions show accuracy for a wide range of frequencies. The algorithm is applied to a 1D partially saturated rock with a random distribution of saturation. We show that the numerical results for the random distribution can be approximated with a volume average of analytical solutions for periodic media. Introduction Attenuation of seismic waves in partially saturated rocks is of great interest because it has been observed that gas and oil reservoirs often exhibit high attenuation (e.g., Dasgupta and Clark, 1998; Rapoport et al., 2004), especially at low frequencies (Chapman et al., 2006). Data, from both laboratory and field, and theoretical work show that attenuation can be related to an increase in reflectivity in the low-frequency range (Korneev et al., 2004; Quintal et al., 2009). Goloshubin et al. (2006) showed three examples of field data in which oil-rich reservoirs exhibit increased reflectivity at low seismic frequencies (around 10 Hz). At low seismic frequencies, wave-induced fluid flow on the mesoscopic scale is presumably the major cause of wave attenuation and velocity dispersion in partially saturated porous rocks (e.g., Norris, 1993; Johnson, 2001; Pride and Berryman, 2003a, b). The mesoscopic scale is the scale much larger than the pore size, but much smaller than the wavelength. White (1975) and White et al. (1975) were the first to introduce the wave-induced fluid flow mechanism for a 3D model of a water-saturated medium with spherical gas-saturated inclusions and a 1D layered model. In White’s model, a partially saturated rock is represented by a poroelastic solid with regions fully saturated by one fluid and regions fully saturated by another fluid. Wave-induced fluid flow is caused by pore pressure differences between the two regions. Dutta and Odé (1979a, b) showed that wave-induced fluid flow can be modeled using Biot’s equations (Biot, 1962) for wave propagation in poroelastic media with spatially varying petrophysical parameters. Several theoretical studies, based on White’s model and Biot’s theory (Biot, 1962), provide various closed-form analytical solutions for seismic attenuation in porous saturated media with periodic mesoscopic-scale heterogeneities of particular geometries, such as layered media or media with spherical inclusions (e.g., Johnson, 2001; Pride and Berryman, 2003a, b). There are also closed-form analytical solutions for randomly layered media (e.g., Gurevich and Lopatnikov, 1995), however, they are restricted to infinite media and to particular autocorrelation functions. Müller and Gurevich (2005) showed that significant differences in the magnitude and frequency dependence of attenuation are caused by only the use of different autocorrelation functions. Additionally, in the low-frequency limit, 1/Q (Q is the quality factor, 1/Q is a measure of attenuation) scales differently in infinite randomly layered media, compared to periodically layered media or finite randomly layered media. For infinite random media, 1/Q is proportional to the square root of frequency, while for periodic and finite random media it is proportional to frequency (Müller and Rothert, 2006). Thus, stable and accurate numerical solutions for seismic attenuation in porous saturated media with mesoscopic- scale heterogeneities are required, for example, for: (i) heterogeneities with complicated geometries, (ii) finite random media with arbitrary distribution pattern, or (iii) media containing more than two heterogeneities, such as partial saturations with more than two fluids. Calculating seismic attenuation due to wave-induced fluid flow with numerical algorithms for wave propagation in poroelastic media (e.g., Zhu and McMechan, 1991) is computationally inefficient because wave propagation, fluid flow and fluid pressure diffusion occur on different time scales. An efficient method is a quasi-static creep test, suggested by Masson and Pride (2007), in which they solved Biot’s equations (Biot, 1962) for wave propagation in poroelastic media with the finite difference method. In this study, we performed similar quasi-static creep tests for calculating seismic attenuation as suggested by Masson and Pride (2007), however, we solved a simpler mathematical problem, Biot’s equations of consolidation (Biot, 1941), in which inertia forces do not play a significant role and therefore are excluded. Attenuation due to wave-induced fluid flow is controlled by fluid pressure diffusion. For calculating the amount of attenuation, it is sufficient to model only the pressure diffusion. We used the finite element method to solve Biot’s equations of consolidation in the u-p formulation (Zienkiewicz and Shiomi, 1984). We show that our numerical scheme is powerful and accurate in calculating attenuation and velocity dispersion due to the 2564 SEG Denver 2010 Annual Meeting © 2010 SEG