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1.
《Advances in water resources》1998,22(3):257-272
This work presents a rigorous numerical validation of analytical stochastic models of steady state unsaturated flow in heterogeneous porous media. It also provides a crucial link between stochastic theory based on simplifying assumptions and empirical field and simulation evidence of variably saturated flow in actual or realistic hypothetical heterogeneous porous media. Statistical properties of unsaturated hydraulic conductivity, soil water tension, and soil water flux in heterogeneous soils are investigated through high resolution Monte Carlo simulations of a wide range of steady state flow problems in a quasi-unbounded domain. In agreement with assumptions in analytical stochastic models of unsaturated flow, hydraulic conductivity and soil water tension are found to be lognormally and normally distributed, respectively. In contrast, simulations indicate that in moderate to strong variable conductivity fields, longitudinal flux is highly skewed. Transverse flux distributions are leptokurtic. the moments of the probability distributions obtained from Monte Carlo simulations are compared to modified first-order analytical models. Under moderate to strong heterogeneous soil flux conditions (σ2y≥1), analytical solutions overestimate variability in soil water tension by up to 40% as soil heterogeneity increases, and underestimate variability of both flux components by up to a factor 5. Theoretically predicted model (cross-)covariance agree well with the numerical sample (cross-)covarianaces. Statistical moments are shown to be consistent with observed physical characteristics of unsaturated flow in heterogeneous soils.©1998 Elsevier Science Limited. All rights reserved 相似文献
2.
《Advances in water resources》2005,28(5):429-439
We present expressions satisfied by the first statistical moments (mean and variance–covariance) of travel time and trajectory of conservative solute particles advected in a three-dimensional heterogeneous aquifer under uniform in the mean flow conditions. Closure of the model is obtained by means of a consistent second-order expansion in σY (standard deviation of the log hydraulic conductivity) of (statistical) moments of quantities of interest. As such, the results obtained are nominally limited to mildly non-uniform fields, with σY < 1. Resulting mean and variance of particles travel time and trajectory are functions of first and second moments and cross-moments of trajectory and velocity components. Our solution is applicable to infinite domains and is free of distributional assumptions. As an important application of the methodology we obtain closed-form expressions for the unconditional mean and variance of travel time and particle trajectory for isotropic log-conductivity domain characterized by an exponential variogram. This allows us to recover the non linear behavior of mean travel time versus distance, in agreement with numerical results published in the literature, as well as a non-linear effect in the mean trajectory. The analysis of trajectory variance allows recovering some known results regarding transverse macro-dispersion, evidencing some limitations typical of perturbation theory. 相似文献
3.
The hydraulic conductivity of heterogeneous porous media depends on the distribution function and the geometry of local conductivities at the smaller scale. There are various approaches to estimate the effective conductivity Keff at the larger scale based on information about the small scale heterogeneity. A critical geometric property in this ‘upscaling’ procedure is the spatial connectivity of the small-scale conductivities. We present an approach based on the Euler-number to quantify the topological properties of heterogeneous conductivity fields, and we derive two key parameters which are used to estimate Keff. The required coefficients for the upscaling formula are obtained by regression based on numerical simulations of various heterogeneous fields. They are found to be generally valid for various different isotropic structures. The effective unsaturated conductivity function Keff (ψm) could be predicted satisfactorily. We compare our approach with an alternative based on percolation theory and critical path analysis which yield the same type of topological parameters. An advantage of using the Euler-number in comparison to percolation theory is the fact that it can be obtained from local measurements without the need to analyze the entire structure. We found that for the heterogeneous field used in this study both methods are equivalent. 相似文献
4.
Christina Schornberg Christian Schmidt Edda Kalbus Jan H. Fleckenstein 《Advances in water resources》2010
Analytical solutions to the one-dimensional heat transport equation for steady-state conditions can provide simple means to quantify groundwater surface water exchange. The errors in exchange flux calculations that are introduced when the underlying assumptions of homogeneous sediments and constant temperature boundary conditions are violated were systematically evaluated in a simulation study. Temperatures in heterogeneous sediments were simulated using a numerical model. Heterogeneity in the sediments was represented by discrete, binary geologic units. High contrasts between the hydraulic conductivities (K) of the geologic units were found to lead to large errors, while the influence of the structural arrangement of the units was smaller. The effects of transient temperature boundary conditions were investigated using an analytical equation. Errors introduced by transient boundary conditions were small for Darcy-velocities > 0.1 m d− 1 in the period near maximum and minimum annual surface water temperatures. For smaller fluxes, however, errors can be large. Assuming steady-state conditions and vertical flow in homogeneous sediments is acceptable at certain times of the year and for medium to high exchange fluxes, but pronounced geologic heterogeneity can lead to large errors. 相似文献
5.
Piecewise heterogeneous media that the earth presents are composed of large-scale boundary structures and small-scale volume heterogeneities. Wave propagation in such piecewise heterogeneous media can be accurately superposed through the generalized Lippmann–Schwinger integral equation (GLSIE). Two different Born series modeling schemes are formulated for the boundary–volume integral equation with 2-D antiplane motion (SH waves). Both schemes decompose the resulting boundary–volume integral equation matrix into two parts: the self-interaction operator handled with a fully implicit manner, and the extrapolation operator approximated by a Born series. The first scheme associates the self-interaction operator with each boundary itself and the volume itself, and interprets the extrapolation operator as the cross-interaction between each boundary and other boundaries/volume scatterers in a subregion. The second scheme relates the self-interaction operator to each boundary itself and its cross-interaction with the volume scatterers on both sides, and expresses the extrapolation operator as both the direct and indirect (through the volume scatterers) cross-interactions between different boundaries in a subregion. By eliminating the displacement field from the volume scatterers, the second scheme reduces the dimension of the resulting boundary-volume integral equation matrix, leading to a faster convergence than the first scheme. Both the numerical schemes are validated by dimensionless frequency responses to a heterogeneous alluvial valley with the velocity perturbed randomly in the range of ca 5–20 %. The schemes are applied to wave propagation simulation in a heterogeneous multilayered model by calculating synthetic seismograms. Numerical experiments, compared with the full-waveform numerical solution, indicate that the Born series modeling schemes significantly improve computational efficiency, especially for high frequencies. 相似文献
6.
Marco Bianchi Hui-Hai Liu Jens T. Birkholzer 《Stochastic Environmental Research and Risk Assessment (SERRA)》2013,27(5):1081-1091
Laboratory experiments in rock samples collected from clay-rich formations indicate that the effective molecular diffusion coefficient (D) is a heterogeneous and anisotropic property. Since laboratory measurements of D are representative of a very small volume, upscaling is necessary in order to incorporate these data in large-scale numerical models of diffusive transport. In this work we address the problem of the estimating the equivalent diffusion coefficient (D eq ), in terms of total diffusive flux, in a three-dimensional domain characterized by a heterogeneous and anisotropic spatial distribution of D. D eq was estimated from the results of steady-state diffusive transport simulations through several realizations of the D field. The ensemble averages of D eq from fields with different degrees of heterogeneity and anisotropy were then compared with estimates from analytical upscaling expressions based on stochastic as well as power-averaging approaches. These expressions are largely based on similar expressions developed for calculating the effective hydraulic conductivity in heterogeneous and anisotropic domains. Comparisons showed that stochastic expressions provide accurate estimates of D eq only for fields characterized by low heterogeneity. Within the range of heterogeneity and anisotropy considered, our results showed that a power-averaging expression is very accurate in predicting D eq especially when the parameter p i is estimated through fitting of the numerical results. Nonetheless, the relationship between this parameter and the anisotropy ratio is linear. 相似文献
7.
8.
In adapting the prestack migration technique used in seismic imaging to the inversion of ground‐penetrating radar (GPR) from time‐ to depth‐sections, we show that the theoretical integral formulation of the inversion can be applied to electromagnetic problems, albeit with three assumptions. The first two assumptions concern the electromagnetic characteristics of the medium, primarily that the medium must be perfectly resistive and non‐dispersive, and the third concerns the antennae radiation pattern, which is taken to be 2D. The application of this adaptation of the inversion method is confirmed by migrating actual GPR measurements acquired on the test site of the Laboratoire Central des Ponts et Chaussées. The results show good agreement with the geometry of the structures in the medium and confirm that the possible departure from the assumption of a purely resistive medium has no visible effect on the information concerning the geometry of scattering and reflecting structures. The field experiments also show that prestack migration processing is sufficiently robust with regard to the assumption of a non‐dispersive medium. The assumption of a 2D antennae radiation pattern, however, produces artefacts that could be significant for laterally heterogeneous media. Nevertheless, where the medium is not highly laterally heterogeneous, the migration gives a clear image of the scattering potential due to the geometry of structural contrasts in the medium; the scatterers are well focused from diffraction hyperbolae and well localized. Spatial geometry has limited dimensional accuracy and positions are located with a maximum error equal to the minimum wavelength of the signal bandpass. Objects smaller than one wavelength can nevertheless be detected and well focused if their dielectric contrasts are sufficiently high, as in the case of iron or water in gneiss gravels. Furthermore, the suitability of multi‐offset protocols to estimate the electromagnetic propagating velocity and to decrease the non‐coherent noise level of measurements is confirmed. Our velocity estimation is based on the semblance calculation of multi‐offset migrated images, and we confirmed the relevance of this quantification method using numerical data. The signal‐to‐noise ratio is improved by summing multi‐offset results after the addition of random noise on measurements. Thus the adaptation of prestack migration to multi‐offset radar measurements significantly improves the resolution of the scattering potential of the medium. Limitations associated with the methods used here suggest that 3D algorithms should be applied to strongly laterally heterogeneous media and further studies concerning the waveform inversion are necessary to obtain information about the electric nature of the medium. 相似文献
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10.
Tarek H. Abdoun Da Ha Michael J. O’Rourke Michael D. Symans Thomas D. O’Rourke Michael C. Palmer Harry E. Stewart 《Soil Dynamics and Earthquake Engineering》2009
Seismic ground faulting is a severe hazard for continuous buried pipelines. Over the years, researchers have attempted to understand pipe behavior, most frequently via numerical modeling and simulation. However, there has been little, if any, physical modeling and tests to verify the numerical modeling approaches and assumptions. This paper presents results of five pairs of centrifuge tests designed to investigate the influence of various factors on the behavior of buried high-density polyethylene (HDPE) pipelines subjected to strike-slip faulting. Parameters considered are the soil moisture content, fault offset rate, relative burial depth (H/D), and pipe diameter. The centrifuge test results show that pipe behavior, specifically pipe strain, is nominally not affected by the soil moisture content and fault offset rate when the pipe is subjected to strike-slip faulting. On the other hand, the burial depth ratio (H/D) and pipe diameter influence peak pipe strain, and in some cases, the ground soil failure pattern. 相似文献
11.
Accurate modeling of storage of carbon dioxide (CO2) in heterogeneous aquifers requires experiments of the capillary pressure as function of temperature and pressure. We present a method with which static drainage and imbibition capillary pressures can be measured continuously as a function of saturation at various temperature (T) and pressure (P) conditions. The measurements are carried out at (T, P) conditions of practical interest. Static conditions can be assumed as small injection rates are applied. The capillary pressure curves are obtained for the unconsolidated sand–distilled water–CO2 system. The experimental results show a decrease of drainage and imbibition capillary pressure for increasing CO2 pressures and pronounced dissolution rate effects for gaseous CO2. Significant capillary pressure fluctuations and negative values during imbibition are observed at near critical conditions. The measurement procedure is validated by a numerical model that simulates the experiments. 相似文献
12.
In this paper we present a stochastic model reduction method for efficiently solving nonlinear unconfined flow problems in heterogeneous random porous media. The input random fields of flow model are parameterized in a stochastic space for simulation. This often results in high stochastic dimensionality due to small correlation length of the covariance functions of the input fields. To efficiently treat the high-dimensional stochastic problem, we extend a recently proposed hybrid high-dimensional model representation (HDMR) technique to high-dimensional problems with multiple random input fields and integrate it with a sparse grid stochastic collocation method (SGSCM). Hybrid HDMR can decompose the high-dimensional model into a moderate M-dimensional model and a few one-dimensional models. The moderate dimensional model only depends on the most M important random dimensions, which are identified from the full stochastic space by sensitivity analysis. To extend the hybrid HDMR, we consider two different criteria for sensitivity test. Each of the derived low-dimensional stochastic models is solved by the SGSCM. This leads to a set of uncoupled deterministic problems at the collocation points, which can be solved by a deterministic solver. To demonstrate the efficiency and accuracy of the proposed method, a few numerical experiments are carried out for the unconfined flow problems in heterogeneous porous media with different correlation lengths. The results show that a good trade-off between computational complexity and approximation accuracy can be achieved for stochastic unconfined flow problems by selecting a suitable number of the most important dimensions in the M-dimensional model of hybrid HDMR. 相似文献
13.
We consider heterogeneous media whose properties vary in space and particularly aquifers whose hydraulic conductivity K may change by orders of magnitude in the same formation. Upscaling of conductivity in models of aquifer flow is needed in order to reduce the numerical burden, especially when modeling flow in heterogeneous aquifers of 3D random structure. Also, in many applications the interest is in average values of the dependent variables over scales larger or comparable to the conductivity length scales. Assigning values of the conductivity Kb to averaging domains, or computational blocks, is the topic of a large body of literature, the problem being of wide interest in various branches of physics and engineering. It is clear that upscaling causes loss of information and at best it can render a good approximation of the fine scale solution after averaging it over the blocks.The present article focuses on upscaling approaches dealing with random media. It is not meant to be a review paper, its main scope being to elucidate a few issues of principle and to briefly discuss open questions. We show that upscaling can be usually achieved only approximately, and the result may depend on the particular upscaling scheme adopted. The typically scarce information on the statistical structure of the fine-scale conductivity imposes a strong limitation to the upscaling problem. Also, local upscaling is not possible in nonuniform mean flows, for which the upscaled conductivity tensor is generally nonlocal and it depends on the domain geometry and the boundary conditions. These and other limitations are discussed, as well as other open topics deserving further investigation. 相似文献
14.
Ravi Ravindra 《Pure and Applied Geophysics》1968,70(1):12-17
Summary Several assumptions are usually made in seismology in the treatment of wave propagational problems in heterogeneous elastic media. These assumptions are pointed out and some of them are critically examined.This paper was presented in a modified form at the meetings of the Canadian Association of Physicists in Toronto in June, 1967.The author was formerly known asRavindra N. Gupta 相似文献
15.
Two different approaches to finite-difference modeling of the elastodynamic equations have been used: the heterogeneous and the homogeneous. In the heterogeneous approach, boundary conditions at interfaces are treated implicitly; in the homogeneous, they are explicitly discretized. We present a homogeneous finite-difference scheme for the 2-D P-SV-wave case. This scheme represents a generalization of earlier such schemes, being able to model media with arbitrary non-uniformities, provided only that all interfaces are aligned with the numerical grid. We perform a detailed comparison of the generalized homogeneous scheme with the analogous heterogeneous scheme, and show the two schemes to be identical for media with a spatially constynt Poisson's ratio. For media where Poisson's ratio is spatially varying, the schemes differ by terms first-order in the spatial step size. However, a comparison of the numerical results produced by the two schemes shows that the resulting differences are negligible for a wide range of values of the Poisson's ratio contrast. 相似文献
16.
《Advances in water resources》1996,19(1):29-47
Transport processes in heterogeneous porous media are often treated in terms of one-equation models. Such treatment assumes that the velocity, pressure, temperature, and concentration can be represented in terms of a single large-scale averaged quantity in regions having significantly different mechanical, thermal, and chemical properties. In this paper we explore the process of single-phase flow in a two-region model of heterogeneous porous media. The region-averaged equations are developed for the case of a slightly compressible flow which is an accurate representation for a certain class of liquid-phase flows. The analysis leads to a pair of transport equations for the region averaged pressures that are coupled through a classic exchange term, in addition to being coupled by a diffusive cross effect. The domain of validity of the theory has been identified in terms of a series of length and timescale constraints.In Part II the theory is tested, in the absence of adjustable parameters, by comparison with numerical experiments for transient, slightly compressible flow in both stratified and nodular models of heterogeneous porous media. Good agreement between theory and experiment is obtained for nodular and stratified systems, and effective transport coefficients for a wide range of conditions are presented on the basis of solutions of the three closure problems that appear in the theory. Part III of this paper deals with the principle of large-scale mechanical equilibrium and the region-averaged form of Darcy's law. This form is necessary for the development and solution of the region-averaged solute transport equations that are presented in Part IV. Finally, in Part V we present results for the dispersion tensors and the exchange coefficient associated with the two-region model of solute transport with adsorption. 相似文献
17.
Summary It is shown that the marked influence of the air-flow on the value of the diffusion coefficient as determined by the dynamic method can be fully explained when heterogeneity of the condensation nuclei in the aerosol is assumed. A theory of the dynamic method for heterogeneous aerosols is given on the assumption that the diffusion loss of a mixture of condensation nuclei of different diffusion coefficients is the sum of the diffusion losse of its components. Procedures for resolving heterogeneous aerosols into their components using the dynamic method are developed. The accuracy attainable by one of the methods which requires no assumptions on the number of components in the aerosol under investigation is illustrated by a numerical example.
Zusammenfassung Es wird gezeigt, dass der bemerkenswerte Einfluss der Luftstromgeschwindigkeit auf die Grösse des mit der dynamischen Methode bestimmten Diffusionskoeffizienten völlig erklärt werden kann, wenn Heterogenität der Kondensationskerne im Aerosol angenommen wird. Eine Theorie der dynamischen Methode für heterogene Aerosole wird unter der Annahme, dass der Diffusionsverlust eines Gemisches von Kondensationskernen mit verschiedenen Diffusionskoeffizienten gleich der Summe der Diffusionsverluste ihrer Komponenten ist, gegeben. Verfahren der Zerlegung heterogener Aerosole in ihre Komponenten mittels der dynamischen Methode werden entwickelt. Die mit einem der Verfahren, welches keine Annahme über die Zahl der in dem zu untersuchenden Aerosol vorhandenen Komponenten benötigt, erreichbare Genauigkeit wird an einem numerischen Beispiel illustriert.相似文献
18.
《Advances in water resources》1999,22(7):681-696
The prediction of contaminant transport in porous media requires the computation of the flow velocity. This work presents a methodology for high-accuracy computation of flow in a heterogeneous isotropic formation, employing a dual-flow formulation and adaptive gridding. The dual equations, describing the hydraulic head and the streamfunction, are numerically solved through finite element approximations. The application of classic finite-element methods requires a rather large number of nodes to represent suitably the flow in high-contrast formations. We present a mesh-adaptive approach that enhances the accuracy of the numerical flow solution for a given computational effort. We rely on an a posteriori error estimator to identify areas where refinements of the finite element mesh are needed or unrefinements are acceptable. We also demonstrate through numerical experiments that the developed methodology efficiently enhances accuracy through successive mesh adaptation. 相似文献
19.
F.A. Radu N. Suciu J. HoffmannA. Vogel O. Kolditz C.-H. ParkS. Attinger 《Advances in water resources》2011,34(1):47-61
This work deals with a comparison of different numerical schemes for the simulation of contaminant transport in heterogeneous porous media. The numerical methods under consideration are Galerkin finite element (GFE), finite volume (FV), and mixed hybrid finite element (MHFE). Concerning the GFE we use linear and quadratic finite elements with and without upwind stabilization. Besides the classical MHFE a new and an upwind scheme are tested. We consider higher order finite volume schemes as well as two time discretization methods: backward Euler (BE) and the second order backward differentiation formula BDF (2). It is well known that numerical (or artificial) diffusion may cause large errors. Moreover, when the Péclet number is large, a numerical code without some stabilising techniques produces oscillating solutions. Upwind schemes increase the stability but show more numerical diffusion. In this paper we quantify the numerical diffusion for the different discretization schemes and its dependency on the Péclet number. We consider an academic example and a realistic simulation of solute transport in heterogeneous aquifer. In the latter case, the stochastic estimates used as reference were obtained with global random walk (GRW) simulations, free of numerical diffusion. The results presented can be used by researchers to test their numerical schemes and stabilization techniques for simulation of contaminant transport in groundwater. 相似文献
20.
Marisol Monterrubio-Velasco F. R. Zúñiga Victor Hugo Márquez-Ramírez Angel Figueroa-Soto 《Journal of Seismology》2017,21(6):1623-1639
The rupture processes of any heterogeneous material constitute a complex physical problem. Earthquake aftershocks show temporal and spatial behaviors which are consequence of the heterogeneous stress distribution and multiple rupturing following the main shock. This process is difficult to model deterministically due to the number of parameters and physical conditions, which are largely unknown. In order to shed light on the minimum requirements for the generation of aftershock clusters, in this study, we perform a simulation of the main features of such a complex process by means of a fiber bundle (FB) type model. The FB model has been widely used to analyze the fracture process in heterogeneous materials. It is a simple but powerful tool that allows modeling the main characteristics of a medium such as the brittle shallow crust of the earth. In this work, we incorporate spatial properties, such as the Coulomb stress change pattern, which help simulate observed characteristics of aftershock sequences. In particular, we introduce a parameter (P) that controls the probability of spatial distribution of initial loads. Also, we use a “conservation” parameter (π), which accounts for the load dissipation of the system, and demonstrate its influence on the simulated spatio-temporal patterns. Based on numerical results, we find that P has to be in the range 0.06 < P < 0.30, whilst π needs to be limited by a very narrow range (0.60 < π < 0.66) in order to reproduce aftershocks pattern characteristics which resemble those of observed sequences. This means that the system requires a small difference in the spatial distribution of initial stress, and a very particular fraction of load transfer in order to generate realistic aftershocks. 相似文献