Effective elastic properties of randomly fractured soils: 3D numerical experiments

This paper is concerned with numerical tests of several rock physical relationships. The focus is on effective velocities and scattering attenuation in 3D fractured media. We apply the so-called rotated staggered finite-difference grid (RSG) technique for numerical experiments. Using this modified grid, it is possible to simulate the propagation of elastic waves in a 3D medium containing cracks, pores or free surfaces without applying explicit boundary conditions and without averaging the elastic moduli. We simulate the propagation of plane waves through a set of randomly cracked 3D media. In these numerical experiments we vary the number and the distribution of cracks. The synthetic results are compared with several (most popular) theories predicting the effective elastic properties of fractured materials. We find that, for randomly distributed and randomly orientated non-intersecting thin penny-shaped dry cracks, the numerical simulations of Pand S-wave velocities are in good agreement with the predictions of the self-consistent approximation. We observe similar results for fluid-filled cracks. The standard Gassmann equation cannot be applied to our 3D fractured media, although we have very low porosity in our models. This is explained by the absence of a connected porosity. There is only a slight difference in effective velocities between the cases of intersecting and non-intersecting cracks. This can be clearly demonstrated up to a crack density that is close to the connectivity percolation threshold. For crack densities beyond this threshold, we observe that the differential effective-medium (DEM) theory gives the best fit with numerical results for intersecting cracks. Additionally, it is shown that the scattering attenuation coefficient (of the mean field) predicted by the classical Hudson approach is in excellent agreement with our numerical results. I N T R O D U C T I O N The derivation and validation of accurate relationships between pore structure and elastic properties of porous rocks is an ongoing problem in geophysics, material science and solid mechanics. Understanding the interactions between rock, pore space and fluids, and how they control rock properties is crucial to a better understanding of acoustic and seismic data. A range of different effective-medium theories (see Mavko, Mukerji and Dvorkin 1998 and references therein) give ex∗E-mail: [email protected] pressions for the overall properties of fractured media if the wavelength is large compared with the size of inclusions. There is a general agreement among these theories for a dilute concentration of inclusions. However, there are considerable differences for higher concentrations. Therefore, it is necessary to validate the different analytical predictions with experimental (e.g. Carvalho and Labuz 1996; Hudson, Pointer and Liu 2001) or numerical data. With this in mind, Saenger and Shapiro (2002) presented efficient and accurate finite-difference (FD) computer simulations of wave propagation and effective elastic properties in 2D fractured media. The present C © 2004 European Association of Geoscientists & Engineers 183 gpr407 GPR-xml.cls Apr l 14, 2004 12:4 184 E.H. Saenger, O.S. Krüger and S.A. Shapiro paper is an extension of this work to 3D fractured media. Spring network techniques (e.g. Garboczi and Day 1995; Garboczi and Berryman 2001; Ursenbach 2001) provide an alternative numerical method for studying the elastic moduli of porous media. All these methods are currently restricted to isotropic materials where Poisson’s ratio cannot be chosen arbitrarily. Attenuation effects also cannot be described with these methods, because they treat the static case only. Finite-difference methods discretize the wave equation on a grid. They replace spatial derivatives by FD operators using neighbouring points. This discretization can cause instability problems on a staggered grid (Virieux 1986) when the medium contains high-contrast discontinuities (strong heterogeneities). These difficulties can be avoided by using the rotated staggered grid (RSG) technique (Saenger, Gold and Shapiro 2000). Since the FD approach is based on the wave equation without physical approximations, the method accounts not only for direct waves, primary reflected waves and multiply reflected waves, but also for surface waves, head waves, converted reflected waves, and diffracted waves observed in ray-theoretical shadow zones (Kelly et al. 1976). Additionally, it accounts for the proper relative amplitudes. Consequently, we use this numerical method for our considerations of 3D fractured materials. This paper consists of two main parts. In the first part, we review several theoretical predictions of effective elastic properties of fractured media. In the second part, we validate the predictions numerically. We explain our simulation set-up with a detailed estimation of sources of errors, and discuss the numerical results.