Numerical Rock Physics : Effects of parallel crack distributions on effective elastic properties

This paper is concerned with numerical studies of effective elastic properties of cracked solids. We concentrate on two dimensional media containing different patterns of parallel crack distributions. We use the Rotated Staggered Grid (RSG) which allows one to simulate elastic wave propagation very accurately in fractured media. Our aim is to compare the predictions given by several effective medium theories to the effective properties we derive from our numerical experiments. Namely, these are the “Non-interaction approximation (NIA)”, the “Differential scheme (DS)” and an extension of the DS (EDS). According to our results, the DS theory and its extension perform well. Simulations of media containing very few cracks prove that for our setup the effective properties stabilise at low numbers of cracks. Finally, we studied parallel cracks clustered in stacked columns. We found that, as expected, the shielding effects dominate in such patterns. INTRODUCTION Understanding the relations between crack microstructure and effective elastic properties of a solid has always been of great interest. Theoretical predictions of these relations often differ significantly. Exact knowledge of real crack microstructure is usually not available, making a direct comparison between theoretical predictions and experiment data difficult. Numerical simulations allow one to determine the effective properties of solids with known microstructures. They are therefore attractive for testing effective medium theories, if the simulated microstructures correspond to the assumptions of the theories. Here we rely on a finite difference (FD) scheme, the Rotated Staggered Grid, which simulates the propagation of elastic waves very accurately, even for models containing high contrasts in elastic parameters. A detailed description of the procedure is given by Saenger et al. (2000). We present a numerical study of effective wave velocities and of the corresponding elastic moduli of two dimensional fractured media. First, we consider parallel cracks with uncorrelated positions, oriented perpendicular to the direction of shear and compressive wave propagation. The obtained results are compared to the ones of several effective media theories. According to our results, the DS theory performs well. Second, two families of parallel cracks, at an angle of 30◦ to each other, are placed at random in the model area. Waves propagate along the symmetry axes in the two dimensional plane. We find the agreement between our data and the EDS theory to be partly acceptable. Finally, we study models with randomly located clusters of five parallel cracks stacked in columns. Spacings between cracks within the clusters are varied in order to examine the effects of crack interactions in these particular patterns on effective velocities. THEORY This section gives an overview of the effective media theories relevant for our simulations. We consider two dimensional models with randomly distributed parallel cracks. As we work within the plane strain framework, our two dimensional plane can be understood as one of the symmetry planes of a three dimen264 Annual WIT report 2003 sional orthotropic solid. Seismic velocities are linked to the elastic moduli by (see, for example, Mavko et al. (1998), from where notations are taken): c11 = ρgv 2 P (0 ◦), c22 = ρgv 2 P (90 ◦), c44 = ρgv 2 SV (90 ◦) = ρgv 2 SV (0 ◦) (1) where ρg denotes the gravitational density, cpq are elements of the stiffness matrix and vP , vSV are the phase velocities. The stiffness matrix is given by the inverse of the compliance matrix:   < c11 > < c12 > 0 < c12 > < c22 > 0 0 0 < c44 >   =   1−ν − ν(1+ν) E 0 − ν(1+ν) E 1−ν 2 0 0 0 1   −1 . (2) E, ν and G denote the Young’s modulus, the Poisson’s ratio and the shear modulus of the solid, respectively. Brackets <> indicate effective moduli as given by effective media theories described in the following sections. A common parameter to characterise cracked solids is the crack density ρ, which is given by (Bristow, 1960): ρ = 1 A N ∑