Estimation of anisotropy parameters using the P-wave velocities on a cylindrical shale sample Dariush Nadri *1 , Joel Sarout 1 , Andrej Bóna 2 , and David Dewhurst 1 1 CSIRO Earth Science and Resource Engineering, Perth, Australia 2 Department of Exploration Geophysics, Curtin University, Perth, Australia Summary In this paper we present a new approach to the estimation of the Thomsen anisotropy parameters and symmetry axis coordinates from the P-wave traveltime measurements on cylindrical shale samples. Using the tomography-style array of transducers, we measure the ultrasonic P-wave ray velocities to estimate the Thomsen anisotropy parameters for a transversely isotropic shale sample. This approach can be used for core samples cut in any direction with regard to the bedding plane, since we make no assumption about the symmetry axis directions and will estimate it simultaneously with the anisotropy parameters. We use the very fast simulated re-annealing to search for the best possible estimate of the model parameters. The methodology was applied to a synthetic model and an anisotropic shale sample. Introduction Because of flat-lying sedimentation of clay particles, shales often behave as transversely isotropic (TI). Traditional computation of the Thomsen anisotropy parameters in a TI medium relays on a very few measurements of P-waves normal and along the bedding plane, and also one measurement at an oblique angle - often at 45 - to the bedding plane. This also requires measuring the shear wave velocities at normal to and along the bedding plane, which is always hard to do accurately. To avoid complications that may arise from the tilting of the symmetry axis, hence satisfying the VTI or HTI assumptions, majority of shale samples are cored either normal to or along the bedding planes. Within these assumptions, one can simply use the phase or ray velocity equations for a transversely isotropic medium to estimate the anisotropy parameters. Sometimes it is difficult or impossible to obtain cores along these directions such as for deviated wells or for the dipping formations. In laboratory experiments where the acoustic measurements are taken under non-isotropic stress field, the symmetry axis direction may also changes which may violates the transverse isotropy assumption with vertical or horizontal symmetry axis direction. These considerations motivate us to take into account the symmetry axis deviations from normal to the bedding planes, which not only brings more freedom to laboratory acoustic experiments but also allows for tracking the possible change of the symmetry axis during measurements under stress. Here, we use a tomography-style geometry of transducers to measure the ultrasonic ray velocity in a cylindrical sample (Figure 1). We developed a ray-tracing based algorithm to compute the ray velocities, assuming homogeneity of the sample. Using these velocities, we estimate the Thomsen anisotropy parameters and the symmetry axis coordinates using the Very fast simulating re-annealing (VFSR) algorithm by searching for the true solution, and where required, implementing a non-linear conjugate gradient (CG) algorithm to tune the best estimate. We applied the methodology to a synthetic model and a real shale sample. D Methodology To compute the phase velocity we use the parametric solution of the Christoffel equation for a homogenous transverse isotropic medium with vertical symmetry axis given by Ursin and Stovas (2006), ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 4 4 0 0 2 2 2 2 2 2 2 0 0 1/ 1/ 1 1 1 2 4 , , 42 1 1 1 , , (1) v p bp cp b c 2 0 0 , α σ δ γ α α α σ δ γ σγ σ δ γ γ α γ σ γ ε δ β ⎡ ⎤ = − + − − + + − ⎣ ⎦ ⎡ ⎤ =− − = − + + − ⎣ ⎦ − = = − where is the phase velocity, v p is the ray parameter (projection of slowness vector to the bedding plane) and 0 0 , , and , α β ε δ are the Thomsen anisotropy parameters; the anisotropy parameters 0 and 0 α β are the P- and S-waves velocities along the symmetry axis. Because of the relatively small size of the transducers, the measured velocity is the ray velocity (Vestrum, 1994). The magnitude of the ray velocity is related to the phase velocity, ( ) 2 2 2 , (2) V v v θ = +∂ ∂ where is the P-wave ray velocity and V θ is the phase angle. The derivative v θ ∂ ∂ can be expressed as a function of the ray parameter, 2 2 1 , (3) pv v p v v pv p θ − ∂ ∂ ∂ = ∂ + ∂ ∂ where the derivative v p ∂ ∂ can be computed from equation 1. For a shale sample that is not plugged in direction normal to the bedding plane computing the ray velocity for the rays