Modeling of the effects of wave-induced fluid motion on seismic velocity and attenuation in porous rocks

Abstract

In this thesis, we use the X-ray CT images of Berea sandstones to carry out the numerical study of the effects of wave-induced fluid motion on seismic velocity and attenuation in porous rocks. In numerical modeling, it is possible to control the factors and inputs that are hard to accomplish in laboratory measurements and isolate those of interest that have significant impact on the seismic responses; this can help in understanding the fundamental physics of seismic waves propagating in saturated porous rocks. The ultimate goal of computational rock physics is to supplement the traditional laboratory measurements, that are time consuming and costly, with cheaper numerical experiments that allow the parameter space to be explored more thoroughly. For this purpose, in this thesis we develop a computational method for time domain simulation of wave propagation in poroelastic medium. The medium is composed of an elastic matrix saturated with a Newtonian fluid. The method operates on a digitized representation of the medium where a distinct material phase and properties are specified at each volume cell. The dynamic response to an acoustic excitation is modeled mathematically with a coupled system of equations: elastic wave equation in the solid matrix and linearized Navier-Stokes equation in the fluid. Implementation of the solution is simplified by introducing a common numerical form for both solid and fluid cells and using a rotated-staggered-grid finite-difference scheme which allows stable solutions without explicitly handling the fluid-solid boundary conditions. A stability analysis is incorporated and can be used to select gridding and time step size as a function of material properties. The numerical results are shown to agree with the analytical solution for an idealized porous medium of periodically alternating solid and fluid layers. When applying the linear solver to compute the effective elastic properties of 3D digitized porous rocks, we find discrepancies between the numerical results and the laboratory measurements. The reason for such a problem is the loss of small features, such as cracks and micro-pores, in the digitized matrix of rocks during the imaging process of aggregation. A hybrid approach, combining the numerical computation 3 and the effective media theories, is developed to deduce the lost cracks from a limited number of laboratory measurements. This approach can recover the lost cracks and is capable of predicting the effective elastic properties of the rock matrix. Compared to the traditional inversion schemes based only on the effective media theories, this hybrid scheme has the advantage of utilizing the complex micro-structures of 3D digitized porous rocks that are resolved in the imaging process, and it helps limit the inversion space for crack distribution. In the study of the dynamic and low-frequency responses of saturated porous rocks, we employ the stress-strain calculation in numerical modeling so as to compute the velocities and attenuations of rock samples, the sizes of which are much smaller than the seismic wavelength of interest. For these cases, transmission measurement cannot be used. Realizing the significant contribution of small cracks to the total attenuation, we extend the hybrid approach by incorporating the modified squirt-flow model where a fluid with frequency-dependent bulk modulus is introduced. Therefore, attenuation due to viscous fluid in stiff pores, that are resolved in the imaging process, can be computed numerically. Attenuation due to viscous fluid in compliant pores can be determined by the modified squirt-flow model since we know the crack distribution. In the inversion for crack distribution, besides using the velocities of P- and S-waves measured in laboratory for the dry and water-saturated cases, measured attenuation data of P-waves are also used so as to further constrain the inversion, and to improve the uniqueness of the inversion results. By using such an extended hybrid approach, we are able to predict both the velocities of saturated porous rocks and the attenuations. From numerical study with the linear solver, we can conclude that the linear solver is able to accurately couple elastic solid and viscous fluid and handle high material contrast and the complex micro-structures of 3D digitized porous rocks. The stress-strain calculation is capable of computing the velocities and attenuations of saturated porous rocks the sizes of which are much smaller than the wavelength of interest. The hybrid approach is a practical way to study the seismic properties of saturated porous rocks until high enough resolution digitized data and enough computational resources are available. From the computations, we observe that the small features, such as cracks lost in the imaging process, are critical for accurately predicting velocities and attenuations of saturated porous rocks. Generally, attenuation is more sensitive to the viscosity of the saturating fluid than velocity is, and attenuation due to viscous fluid in compliant pores is greater than that due to viscous fluid in stiff pores.