Computer Simulation of the Seismic Wave Propagation in Poroelastic Medium

Abstract

This article presents an algorithm for the numerical solution of an initial-boundary value problem for a symmetric t-hyperbolic system of partial differential equations. This problem is based on continual filtration model, which describes the propagation of seismic waves in a poroelastic medium saturated with a fluid characterized by such physical parameters as the propagation velocities of longitudinal P- (fast and slow) and transverse S-waves, the density of the medium materials, and porosity. The system of linearized equations of saturated porous media is formulated in terms of physical variables of the velocity–stress tensor of the porous matrix and the velocity–pressure of the saturating fluid in the absence of energy dissipation. The solution is implemented numerically using an explicit finite difference upwind scheme built on a staggered grid to avoid the appearance of oscillations in the solution functions. The program code implementing parallel computing is developed in the high-performance Julia programming language. The possibility of using the approach is demonstrated by the example of solving the problem of propagation of seismic waves from a source located in the formation. Computational experiments based on real data from oil reservoirs have been implemented, and dynamic visualization of solutions consistent with the first waves arrival times has been obtained.

A Combined Approach for Predicting the Distribution of Harmful Substances in the Atmosphere Based on Parameter Estimation and Machine Learning Algorithms

“…ε is the ε-approximation of the right-hand side of Equation (1) at time stamp t = t n defined as in [63].…”

Simulation of Wave Propagation Using Finite Differences in Oil Exploration

“…where C includes all the constants of Equation ( 5), and F ij is the numerical solution obtained from the numerical implementation of Equation (10), keeping only the dependence on the z coordinate (14). Making L 2 = error, by plotting Log(error) vs. Log(∆x), changing the number of points in x, one has a straight line of slope 1.8, in agreement with the expected second order, as shown in Figure 4.…”

“…For the numerical solution to be stable, the following stability criterion for the standard second-order finite difference scheme for the wave equation is established, as reported in [9,10], and is given by ( 15):…”

“…At each time step, the wavefield propagated at the internal nodes according to Equation (10), and then propagated along the lateral and bottom boundaries employing the difference relations given in [10]. In this work, the relation max (∆x, ∆z) ≤ λ/4 was used, where λ represents the wavelength [11].…”

“…Sabino [9] employed a GPU-based cluster environment for the development of scalable solvers for a 3D wave-propagation problem with finite difference methods. Bliyeva [10] presented an algorithm for the numerical solution of an initial-boundary value problem for a symmetric t-hyperbolic system of partial differential equations, which describes the propagation of seismic waves in a poroelastic medium saturated with a fluid. Chao [11] developed a fourthorder finite difference scheme for the numerical modeling of elastic waves in the frequency domain, which can provide a stable numerical solution with fewer grid points per wavelength.…”