Géza Seriani

Acoustic and electromagnetic properties of soils saturated with salt water and NAPL

Abstract Electrical, seismic, and electromagnetic methods can be used for noninvasive determination of subsurface physical and chemical properties. In particular, we consider the evaluation of water salinity and the detection of surface contaminants. Most of the relevant properties are represented by electric conductivity, P-wave velocity, and dielectric permittivity. Hence, it is important to obtain relationships between these measurable physical quantities and soil composition, saturation, and frequency. Conductivity in the geoelectric frequency range is obtained with Prides model for a porous rock. (The model considers salinity and permeability.) Whites model of patchy saturation is used to calculate the P-wave velocity and attenuation. Four cases are considered: light nonaqueous phase liquid (LNAPL) pockets in water, dense nonaqueous phase liquid (DNAPL) pockets in water, LNAPL pockets in air, and DNAPL pockets in air. The size of the pockets (or pools), with respect to the signal wavelength, is modeled by the theory. The electromagnetic properties in the GPR frequency range are obtained by using the Hanai–Bruggeman equation for two solids (sand and clay grains) and two fluids (LNAPL or DNAPL in water or air). The Hanai–Bruggeman exponent (1/3 for spherical particles) is used as a fitting parameter and evaluated for a sand/clay mixture saturated with water. Prides model predicts increasing conductivity for increasing salinity and decreasing permeability. The best-fit exponent of the Hanai–Bruggeman equation for a sand/clay mixture saturated with water is 0.61, indicating that the shape of the grains has a significant influence on the electromagnetic properties. At radar frequencies, it is possible to distinguish between a water-saturated medium and a NAPL-saturated medium, but LNAPL- and DNAPL-saturated media have very similar electromagnetic properties. The type of contaminant can be better distinguished from the acoustic properties. P-wave velocity increases with frequency, and has dissimilar behaviour for wet and dry soils.

LOW AND HIGH ORDER FINITE ELEMENT METHOD: EXPERIENCE IN SEISMIC MODELING

The finite element method (FEM) is a numerical technique well suited to solving problems of elastic wave propagation in complex geometries and heterogeneous media. The main advantages are that very irregular grids can be used, free surface boundary conditions can be easily taken into account, a good reconstruction is possible of irregular surface topography, and complex geometries, such as curved, dipping and rough interfaces, intrusions, cusps, and holes can be defined. The main drawbacks of the classical approach are the need for a large amount of memory, low computational efficiency, and the possible appearance of spurious effects. In this paper we describe some experience in improving the computational efficiency of a finite element code based on a global approach, and used for seismic modeling in geophysical oil exploration. Results from the use of different methods and models run on a mini-superworkstation APOLLO DN10000 are reported and compared. With Chebyshev spectral elements, great accuracy can be reached with almost no numerical artifacts. Static condensation of the spectral elements internal nodes dramatically reduces memory requirements and CPU time. Time integration performed with the classical implicit Newmark scheme is very accurate but not very efficient. Due to the high sparsity of the matrices, the use of compressed storage is shown to greatly reduce not only memory requirements but also computing time. The operation which most affects the performance is the matrix-by-vector product; an effective programming of this subroutine for the storage technique used is decisive. The conjugate gradient method preconditioned by incomplete Cholesky factorization provides, in general, a good compromise between efficiency and memory requirements. Spectral elements greatly increase its efficiency, since the number of iterations is reduced. The most efficient and accurate method is a hybrid iterative-direct solution of the linear system arising from the static condensation of high order elements. The size of 2D models that can be handled in a reasonable time on this kind of computer is nowadays hardly sufficient, and significant 3D modeling is completely unfeasible. However the introduction of new FEM algorithms coupled with the use of new computer architectures is encouraging for the future.

Optimal blended spectral-element operators for acoustic wave modeling

Spectral-element methods, based on high-order polynomials, are among the most commonly used techniques for computing accurate simulations of wave propagation phenomena in complex media. However, to retain computational efficiency, very high order polynomials cannot be used and errors such as numerical dispersion and numerical anisotropy cannot be totally avoided. In the present work, we devise an approach for reducing such errors by considering modified discrete wave operators. We analyze consistent and lumped operators together with blended operators (weighted averages of consistent and lumped operators). Furthermore, using the operator-blending approach and a novel dispersion analysis method, we develop optimal spectral-element operators that have increased numerical accuracy, without resorting to very high order operators. The new operators are faster and computationally more efficient than consistent operators. Our approach is based on the tensor product decomposition of the element matrices into 1D f...

A Parallel Spectral Element Method for Acoustic Wave Modeling

The finite element method is well reputed for its great flexibility in solving problems with complex geometries and heterogeneous structures, but, in its classical form, it has fairly low accuracy and poor computational efficiency. This makes an adverse impact on a large-scale numerical simulation of acoustic wavefield propagation. It has been shown by the author and his co-workers that the spectral approach, based on the use of high-order orthogonal interpolating functions (the Spectral Element Method), yields high accuracy with almost no numerical artifacts; it also significantly reduces the simulation computing costs. In this paper, the method is presented in conjunction with an iterative solution technique in such a way that it fully exploits the currently available parallel computers. The underlying algorithm is based on the observations that the assembly of mass and stiffness matrices is not really needed, and that the matrix-vector product required for the iterations can be done concurrently via an element-by-element approach. Moreover, by using a tensor-product sum-factorization scheme, computational and storage requirements can be further reduced as neither element nor global matrices are ever formed. The method is tested on the Cray T3D MPP computer, and its parallel efficiency and speed performance are discussed.

GPR modeling study in a contaminated area of Krzywa Air Base (Poland)

Krzywa Air Base is a former Soviet military base which is highly polluted because of soil contamination during the dismantling of fuel tanks after the 1990 political changes in the U.S.S.R. The contaminated sites are indicated as magazines 1 and 2 in Figure 1. The area is located within the pre‐Sudetic block, where Tertiary and Quaternary deposits lie on Mesozoic crystalline basement. The deposits consist of sands of different grain sizes, gravels, clays, and interbedded silts, with a total thickness ranging from 80 to 100 m. In the extreme cases, the pollution is so intense that the spilled fuel can be extracted in the system of drainage wells. In general, the contamination can be correlated with the geologic structure. As a prelude to acquisition of field data, we simulate and evaluate the expected ground‐penetrating radar (GPR) responses for uncontaminated and contaminated layers.

DFT MODAL ANALYSIS OF SPECTRAL ELEMENT METHODS FOR ACOUSTIC WAVE PROPAGATION

Spectral element methods are now widely used for wave propagation simulations. They are appreciated for their high order of accuracy, but are still used on a heuristic basis. In this work we present the numerical dispersion of spectral elements, which allows us to assess their error limits and to devise efficient numerical simulations, particularly for 2D and 3D problems. We propose a novel approach based on a discrete Fourier transform of both the probing plane waves and the discrete wave operators. The underlying dispersion relation is estimated by the Rayleigh quotients of the plane waves with respect to the discrete operator. Together with the Kronecker product properties, this approach delivers numerical dispersion estimates for 1D operators as well as for 2D and 3D operators, and is well suited for spectral element methods, which use nonequidistant collocation points such as Gauss–Lobatto–Chebyshev and Gauss–Lobatto–Legendre points. We illustrate this methodology with dispersion and anisotropy graphs for spectral elements with polynomial degrees from 4 to 12. These graphs confirm the rule of thumb that spectral element methods reach a safe level of accuracy at about four grid points per wavelength.

WAVE PROPAGATION MODELING IN HIGHLY HETEROGENEOUS MEDIA BY A POLY-GRID CHEBYSHEV SPECTRAL ELEMENT METHOD

A wide range of applications requires the modeling of wave propagation phenomena in media with variable physical properties in the domain of interest, while highly accurate algorithms are needed to avoid unphysical effects. Spectral element methods (SEM), based on either a Chebyshev or a Legendre polynomial basis, have excellent properties of accuracy and flexibility in describing complex models, outperforming other techniques. In the standard SEM approach the computational domain is discretized by using very coarse meshes and constant-property elements, but in some cases the accuracy and the computational efficiency may be seriously reduced. For instance, a finely heterogeneous medium requires grid resolution down to the finest scales, leading to an extremely large problem dimension. In such problems the wavelength scale of interest is much larger but cannot be exploited in order to reduce the problem size. A poly-grid Chebyshev spectral element method (PG-CSEM) can overcome this limitation. In order to accurately deal with continuous variation in the properties, or even with small scale fluctuations, temporary auxiliary grids are introduced which avoid the need of using any finer global grid, and at the macroscopic level the wave field propagation is solved maintaining the SEM accuracy and computational efficiency.

DFT modal analysis of spectral element methods for the 2D elastic wave equation

The DFT modal analysis is a dispersion analysis technique that transforms the equations of a numerical scheme to the discrete Fourier transform domain sampled in the mesh nodes. This technique provides a natural matching of exact and approximate modes of propagation. We extend this technique to spectral element methods for the 2D isotropic elastic wave equation, by using a Rayleigh quotient approximation of the eigenvalue problem that characterizes the dispersion relation, taking full advantage of the tensor product representation of the spectral element matrices. Numerical experiments illustrate the dependence of dispersion errors on the grid resolution, polynomial degree, and discretization in time. We consider spectral element methods with Chebyshev and Legendre collocation points.