Problems In Heterogeneous Media Research Articles

Solving inhomogeneous inverse problems by topological derivative methods

We introduce new iterative schemes to reconstruct scatterers buried in a medium and their physical properties. The inverse scattering problem is reformulated as a constrained optimization problem involving transmission boundary value problems in heterogeneous media. Our first step consists in developing a reconstruction scheme assuming that the properties of the objects are known. In a second step, we combine iterations to reconstruct the objects with iterations to recover the material parameters. This hybrid method provides reasonable guesses of the parameter values and the number of scatterers, their location and size. Our schemes to reconstruct objects knowing their nature rely on an extended notion of topological derivative. Explicit expressions for the topological derivatives of the corresponding shape functionals are computed in general exterior domains. Small objects, shapes with cavities and poorly illuminated obstacles are easily recovered. To improve the predictions of the parameters in the successive guesses of the domains we use a gradient method.

Application of the boundary element method to three-dimensional potential problems in heterogeneous media

The present work discusses a solution procedure for heterogeneous media three-dimensional potential problems, involving nonlinear boundary conditions. The problem is represented mathematically by the Laplace equation and the adopted numerical technique is the boundary element method (BEM), here using velocity correcting fields to simulate the conductivity variation of the domain. The integral equation is discretized using surface elements for the boundary integrals and cells, for the domain integrals. The adopted strategy subdivides the discretized equations in two systems: the principal one involves the calculation of the potential in all boundary nodes and the secondary which determines the correcting field of the directional derivatives of the potential in all points. Comparisons with other numerical and analytical solutions are presented for some examples.

Modelling of highly-heterogeneous media using a flux-vector-based Green element method

One of the modern techniques for solving nonlinear problems encountered in flow in porous media is the Green element method (GEM). It combines the high accuracy of the boundary element method with the efficiency and versatility of the finite element method. The high accuracy of the GEM comes from the direct representation of the normal fluxes as unknowns. However, in the classical GEM procedure the difficulties imposed by a large number of normal fluxes at each internal node are typically overcome by approximating them in terms of the primary variable, and this can lead to a diminution of the overall accuracy, particularly in applications to heterogeneous media. To maintain the high accuracy, another approach was proposed, namely the ‘flux-vector-based GEM’, or ‘ q -based’ GEM. According to this approach, only two and three unknown flux components are required for a node in two- and three-dimensional domains, respectively. An important advantage of this approach is that the flux vector at each node is determined directly. We present a comparison between the results obtained using the classical GEM and those obtained using the ‘ q -based’ GEM for problems in heterogeneous media with permeability changes of several orders of magnitude. In some simple examples of such situations, the classical GEM fails to produce physically sensible results, whilst our novel development of the ‘ q -based’ GEM is in general agreement with both the boundary element method and the control volume method. However, in order to acquire results to the same degree of accuracy, these latter two approaches are less efficient than the ‘ q -based’ GEM. This new approach is thus suitable for overcoming the difficulties present when modelling flow in heterogeneous media with rapid and high-order changes in material parameters. Within geological problems, we can therefore apply it to flow through layered sequences, across partially-sealing faults and around wells. The ‘ q -based’ GEM approach presented here is developed and applied for rectangular grids, and we provide details of the extension to triangular finite-element-type meshes in two dimensions.

A Novel Multigrid Based Preconditioner For Heterogeneous Helmholtz Problems

An iterative solution method, in the form of a preconditioner for a Krylov subspace method, is presented for the Helmholtz equation. The preconditioner is based on a Helmholtz-type differential operator with a complex term. A multigrid iteration is used for approximately inverting the preconditioner. The choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with Fourier analysis. Multigrid analysis results are verified by numerical experiments. High wavenumber Helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner.

Centered and staggered Fourier derivatives and Hilbert transforms

When ringing derivatives are used in seismic modeling, migration, and other wave propagation problems in heterogeneous media, they interact with boundaries of blocks or layers of different physical properties, generating noncausal and nonphysical waves. These spurious waves can be as strong as reflected or converted waves and other legitimate physical events, leading to errors and misinterpretations. Therefore, compact (i.e., short and nonringing) derivatives should be used.

A new finite integral transform pair for hyperbolic conduction problems in heterogeneous media

The study of heat conduction in situations involving very short times cannot be set about with the help of Fourier's law. This model which cannot be justified in case of local non-equilibrium, leads, in this case, to results which do not agree with the observations. When the continuity hypothesis is posed, Fourier's law may nevertheless be adjusted. Among the different proposed models the Cattaneo–Vernotte model is one of the most credible. It is used in this work. The resulting hyperbolic problem in a non-homogeneous medium is posed in its primitive form of coupled system of partial differential equations where the unknowns are the temperature and the flux density, i.e. a four-component vector field. Thanks to a dot product which is suited to these fields, the existence of two orthogonal families of complex vector-eigenfunctions is shown. A finite-integral transform technique which is based on this dot product is then applied to the primitive system. It produces an infinite set of uncoupled ordinary differential equations: a separated expansion form of the solution vector field is thus obtained. For multifilmed media (1-D cases), it is shown that the notion of transfer matrix which is familiar in pure diffusion context generalizes naturally.

Boundary Element Methods for Flow Problems in Heterogeneous Media.

The boundary element method (BEM) owes its elegance, with regard to the computational accuracy and efficiency, to the existence of the fundamental solution for the governing equation. For heterogeneity problems, there arises difficulty in finding the corresponding fundamental solution; and consequently, the conventional BEM can not be applied to such problems. Among some attempts found in the literature to overcome this shortcoming and to broaden applicability, two kinds of unconventional BEM, the accelerated perturbation BEM and the complex variable BEM, are reviewed.The accelerated perturbation BEM is developed for solving continuous heterogeneity problems. The governing equation is decomposed into various order perturbation equations, for which the fundamental solution can be found. These equations are solved by the BEM, and the complete solution to the original problem is recovered by summing the perturbation solutions. The Padé approximant is applied to improve the convergence of a perturbation series.The complex variable BEM is developed for solving discrete heterogeneity (such as fractures) problems. The difficulty, in modeling singular flow behavior near the tips of thin objects, is dealt by virtue of a conformal mapping technique. The use of complex variables enables us to bridge the gap between the singular solution in a mapped plane and the non-singular solution in a physical plane.To exemplify the practical usefulness of these models, application problems are considered, including heterogeneity detection through type curves, well-to-well displacement predictions, and horizontal-sink productivity estimations. Sound mathematical foundations ensure the high degree of computational accuracy even for heterogeneity problems.

Generalized Boundary Integral Equation for Transient Heat Conduction in Heterogeneous Media

Using the generalized boundary integral method developed by the authors for steady heat conduction in heterogeneous media as a point of departure, a generalized dual reciprocity boundary element method (BEM) is presented for the solution of transient heat conduction problems in heterogeneous media. In the process, new interpolating radial basis functions are dee ned. This method retains the boundary-only discretization feature of the BEM. Two- and three-dimensional numerical examples provide validation of the proposed method. Excellent agreement is found between analytical and BEM-computed results.

The Rosseland approximation for radiative transfer problems in heterogeneous media

We address the problem of the validity of the Rosseland approximation in a binary mixture when the opacities of the two materials exhibit a high contrast ratio. We derive a new formulation for the set of radiative equations. Some numerical simulations on effective opacities are presented.

A parallel implementation of the boundary element method for heat conduction analysis in heterogeneous media

A direct boundary element formulation is implemented on a parallel computer using MPI (message passing interface). This implementation solves potential problems, specifically heat conduction problems in heterogeneous media. Single-zone problems, where at least one boundary condition is known at every node on the boundary, produce dense, asymmetric coefficient matrices which are relatively easy to distribute and solve in parallel. Multi-zone problems, where no boundary conditions are specified at nodes on the heterogeneity interfaces, produce coefficient matrices which contain some unpopulated blocks. The parallel data distribution of the coefficient matrix for the multi-zone case is performed so that the populated blocks are evenly distributed across the processors to balance the computational load. Several application problems are solved to find the effect of various input parameters on the effective thermal conductivity. The numerical results agree well with experimental data and analytical models over the volume fractions and thermal conductivity ratios considered.

Implementation of the transformation field analysis for inelastic composite materials

The transformation field analysis is a general method for solving inelastic deformation and other incremental problems in heterogeneous media with many interacting inhomogeneities. The various unit cell models, or the corrected inelastic self-consistent or Mori-Tanaka fomulations, the so-called Eshelby method, and the Eshelby tensor itself are all seen as special cases of this more general approach. The method easily accommodates any uniform overall loading path, inelastic constitutive equation and micromechanical model. The model geometries are incorporated through the mechanical transformation influence functions or concentration factor tensors which are derived from elastic solutions for the chosen model and phase elastic moduli. Thus, there is no need to solve inelastic boundary value or inclusion problems, indeed such solutions are typically associated with erroneous procedures that violate (62); this was discussed by Dvorak (1992). In comparison with the finite element method in unit cell model solutions, the present method is more efficient for cruder mesches. Moreover, there is no need to implement inelastic constitutive equations into a finite element program. An addition to the examples shown herein, the method can be applied to many other problems, such as those arising in active materials with eigenstrains induced by components made of shape memory alloys or other actuators. Progress has also been made in applications to electroelastic composites, and to problems involving damage development in multiphase solids. Finally, there is no conceptural obstacle to extending the approach beyond the analysis of representative volumes of composite materials, to arbitrarily loaded structures.

Perturbation Boundary Element Method for Heterogeneous Reservoirs: Part 1—Steady-State Flow Problems

Summary A perturbation boundary element method (BEM) for steady-state flow problems in heterogeneous media is developed, and various aspects of the model are tested. The perturbation series obtained by the BEM gives the complete solution for the original governing equation. For highly heterogeneous problems. Padè approximants are effective to accelerate the rate of convergence and to convert a divergent series into a convergent solution. Because the model does not suffer from numerical dispersion and grid orientation effects, it is possible to track streamlines accurately.

Analytical solution of the one‐dimensional time‐dependent transport equation

An analytical model which provides an approximate description of scale‐dependent transport is presented. The model is based on the advection‐dispersion equation but with the dispersion coefficient dependent on the travel time of the solute from a single input source. The time dependence of the dispersion coefficient can assume arbitrary functional forms. The governing equation, which includes a time‐varying dispersion coefficient, linear equilibrium adsorption, and first‐order reaction, is reduced to the heat diffusion equation after a series of transformations. Analytical solutions pertaining to a time‐varying mass injection in an infinite medium with arbitrary initial distribution of concentration and functional dependence of the dispersion coefficient can be easily derived from the model. Since the analytical solution is in an integral form, different temporal variation of the dispersion coefficient over different time intervals can also be incorporated. It is shown that the solution reduces to the well‐known result for the case of constant dispersion and mass injection. In this study, particular solutions for various dispersion functions and mass injection scenarios are presented. These include linear, exponential, and asymptotic variation of the dispersion functions and instantaneous as well as continuous mass injection. The analytical results could model the transport of solute in a hydrogeologic system characterized by a dispersion coefficient that varies as a function of travel time from the input source. It could provide a modeling solution to solute transport problems in heterogeneous media and be used as a suitable model for the inversion problem, especially since more than one fitting parameter is available to fit the field tracer data which exhibit a scale effect.

加速摂動境界要素法による不均質媒体内流動挙動シミュレーション

The boundary element method (BEM) owes its elegance, computational efficiency and accuracy, to the existence of the free-space Green's function for the governing equation. If there is no such solution in a closed form, the BEM cannot be applied, and unfortunately, this is the case for flow problems in heterogeneous media.To overcome this difficulty, the governing equation is decomposed into various order perturbation equations, for which the free-space Green's function can be found. At each level of perturbation, the solution is obtained by the BEM and the summation of various order perturbation solutions gives the complete solution for the original governing equation. To improve the perturbation series, Pade approximants are employed, which accelerate the rate of convergence of a slowly convergent series and convert a divergent series into a convergent series.Two kinds of perturbation boundary element models are introduced: one for transient flow problems, associated with the modified Helmholtz operater in Laplace space and the other for steady-state flow problems, associated with the Laplace operator. The perturbation BEM shows its utility in streamline tracking and well testing problems in heterogeneous media. The sound mathematical foundations ensure the high order of computational accuracy.

Effect of the Sampling Network on Estimates of the Covariance Function of Stationary Fields

AbstractThe covariance function of the stochastic component of various soil properties plays an essential role in the estimation and interpolation of spatial phenomena using kriging techniques and in the solution of transport problems in heterogeneous media using stochastic continuum transport modeling. In this study the effect of the sampling network on the estimates of covariance functions of two‐dimensional, isotropic, second‐order stationary processes, characterized by different underlying correlation scales, was analyzed. Estimates of the covariance function based on a standard systematic sampling network were compared with estimates of the covariance functions based on a modified sampling network. Results of these comparisons showed that the modified sampling network was better suited than the systematic network to resolving the statistical properties of the underlying stochastic process at small separation distances. This may result in an improvement in structure identification. Analyses of the effect of the sampling network used for estimating the covariance function on the results of conditional inference procedures such as kriging showed that the kriging estimates were essentially insensitive to the geometric configuration of the sampling network. Uncertainties about these estimates were reduced when using the modified sampling network, however. In the case of unconditional inferences, such as with stochastic analyses of transport problems in heterogeneous media, predictions of the head variance provided by using estimates of the parameters of the covariance function of the log saturated hydraulic conductivity based on the modified sampling network fit the theoretical head variance better than those based on the systematic sampling network. The effect of the sampling network used for estimating the covariance function on the predicted head variance was significant, particularly when the underlying correlation scale was relatively small, when the porous medium texture became finer, and when the hydraulic conductivity variations were treated as being three‐dimensional.

A theoretical study of the estimation of the correlation scale in spatially variable fields: 1. Stationary fields

The correlation scale of the stochastic component of various soil properties plays an essential role in the interpolation of spatial phenomena using kriging techniques and in the solution of transport problems in heterogeneous media using stochastic modeling. In this study the uncertainty of the estimation of the correlation scales of stationary fields was analyzed by generating realizations of a set of two‐dimensional isotropic second‐order stationary functions with given correlation scales. Results from simulations of 100 independent realizations showed that the variance of the estimate of the correlation scale decreases as the number of the sample points n increases and when the size of the correlation scale of the underlying process relative to the field area decreases, whereas the biasedness decreases as n increases arid the underlying correlation scale increases. Results of variogram model fitting and cross‐validation tests involving single realizations showed that both the accuracy of the estimates of the correlation scales and the fitted variogram model consistency increase as both the sample points and the correlation scale of the underlying process increase. Analysis of the effect of the grid size on the estimates of the correlation scale showed that reasonable estimates can be obtained when the distance between the sampling points is smaller than half the range of the underlying process. Estimates of correlation scales based on data along transects may underestimate the correlation scale of the underlying process by a factor of 2 or greater. A comparison between different estimation procedures showed that in general estimates of both the variance Ĉ (0) and the integral scale Ĵ provided by the restricted maximum likelihood procedure are higher than those provided by the maximum likelihood procedure, which, in turn, are higher than those provided by fitting the nonparametric estimates of the variogram using the minimum mean‐square deviation criterion. Results of the objective model validation tests, however, implied that generally all these estimates are accurate and consistent with the data, suggesting that a given realization of a given stationary process may be adequately described by different sets of Ĵ and Ĉ(0). This may not affect the results of kriging estimation which requires the ratio C(0)/J, but may pose problems in stochastic transport modeling which requires the values of Ĵ and Ĉ(0) individually.

Solution of heat-conduction problems in heterogeneous media by the integral relations method

The integral relations method is developed to solve heat-conduction problems in a two-component complex medium and nonstationary filtration for the case of bounded and unbounded domains.

Rapid solution of ray tracing problems in heterogeneous media

abstract The determination of local earthquake hypocenters and orgin times from first-P-arrival times by Geiger's method requires a technique for finding the minimum travel time (and derivatives) between the source and the station. Sophisticated ray tracing techniques have been developed for this purpose for use in complex velocity structures. Unfortunately, the two common techniques, shooting and bending, are generally prohibitively expensive for routine use in data analysis. The bending method is also particularly vulnerable to the problem of local minima in travel time. A method has been developed known as the ray initializer, which can be used to circumvent these problems in many cases. First, the technique can find a reasonable estimate of the minimum-time ray path in a quick and efficient manner. The velocity in a region local to the source and receiver is laterally averaged to yield an approximate layered velocity model. One-dimensional ray tracing techniques are used to find the minimum-time path for this layered structure. The ray path estimate can then be used as the starting path in a bending routine, a procedure resulting in more rapid convergence and the avoidance of local minima. Second, the travel time found by numerical integration along the estimated ray path is an excellent approximation to the actual travel time. Thus, in many cases, the ray initializer can be substituted for a three-dimensional ray tracing routine with a tremendous increase in efficiency and only a small loss in accuracy. It is found that the location of an explosion, derived using the ray initializer, is nearly identical to a complete ray tracing solution, even for a highly complex velocity structure.

An approach to the solution of criticality problems in heterogeneous media by coupling the transport and diffusion theory

This work describes a new technique for the solution of criticality problems. Transport theory, in its integral form, is applied in the multiplying region, whilst diffusion theory is employed in the moderating region. The equations valid for a general geometry are obtained in two-group approximation. A numerical iterative scheme is developed for a multiplying reflected slab. The results obtained for a sample case are also described.