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1.
The inherent heterogeneity of geological media often results in anomalous dispersion for solute transport through them, and how to model it has been an interest over the past few decades. One promising approach that has been increasingly used to simulate the anomalous transport in surface and subsurface water is the fractional advection–dispersion equation (FADE), derived as a special case of the more general continuous time random walk or the stochastic continuum model. In FADE, the dispersion is not local and the solutes have appreciable probability to move long distances, and thus reach the boundary faster than predicted by the classical advection–dispersion equation (ADE). How to deal with different boundaries associated with FADE and their consequent impact is an issue that has not been thoroughly explored. In this paper we address this by taking one-dimensional solute movement in soil columns as an example. We show that the commonly used FADE with its fractional derivatives defined by the Riemann–Liouville definition is problematic and could result in unphysical results for solute transport in bounded domains; a modified method with the fractional dispersive flux defined by the Caputo derivatives is presented to overcome this problem. A finite volume approach is given to numerically solve the modified FADE and its associated boundaries. With the numerical model, we analyse the inlet-boundary treatment in displacement experiments in soil columns, and find that, as in ADE, treating the inlet as a prescribed concentration boundary gives rise to mass-balance errors and such errors could be more significant in FADE because of its non-local dispersion. We also discuss a less-documented but important issue in hydrology: how to treat the upstream boundary in analysing the lateral movement of tracer in an aquifer when the tracer is injected as a pulse. It is shown that the use of an infinite domain, as commonly assumed in literature, leads to unphysical backward dispersion, which has a significant impact on data interpretation. To avoid this, the upstream boundary should be flux-prescribed and located at the upstream edge of the injecting point. We apply the model to simulate the movement of Cl− in a tracer experiment conducted in a saturated hillslope, and analyse in details the significance of upstream-boundary treatments in parameter estimation.  相似文献   

2.
The results of a series of high-resolution numerical experiments are used to test and compare three nonlinear models for high-concentration-gradient dispersion. Gravity stable miscible displacement is considered. The first model, introduced by Hassanizadeh, is a modification of Fick’s law which involves a second-order term in the dispersive flux equation and an additional dispersion parameter β. The numerical experiments confirm the dependency of β on the flow rate. In addition, a dependency on travelled distance is observed. The model can successfully be applied to nearly homogeneous media (σ2 = 0.1), but additional fitting is required for more heterogeneous media.The second and third models are based on homogenization of the local scale equations describing density-dependent transport. Egorov considers media that are heterogeneous on the Darcy scale, whereas Demidov starts at the pore-scale level. Both approaches result in a macroscopic balance equation in which the dispersion coefficient is a function of the dimensionless density gradient. In addition, an expression for the concentration variance is derived. For small σ2, Egorov’s model predictions are in satisfactory agreement with the numerical experiments without the introduction of any new parameters. Demidov’s model involves an additional fitting parameter, but can be applied to more heterogeneous media as well.  相似文献   

3.
In this paper we present a reliable algorithm, the homotopy perturbation method, to construct numerical solutions of the space–time fractional advection–dispersion equation in the form of a rapidly convergent series with easily computable components. Fractional advection–dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium. The fractional derivatives are described in the Caputo sense. Some examples are given. Numerical results show that the homotopy perturbation method is easy to implement and accurate when applied to space–time fractional advection–dispersion equations.  相似文献   

4.
River water quality models usually apply the Fischer equation to determine the longitudinal dispersion coefficient (Dx) in solving the advection–dispersion equation (ADE). Recently, more accurate formulas have been introduced to determine Dx in rivers, which could strongly affect the accuracy of the ADE results. A numerical modelling-based approach is presented to evaluate the performance of various Dx formulas using the ADE. This approach consists of a finite difference approximation of the ADE, a MATLAB code and a MS Excel interface; it was tested against the analytical ADE solution and demonstrated using eight well-known Dx formulas and tracer study data for the Chattahoochee River (USA), the Severn (UK) and the Athabasca (Canada). The results show that Dx has an important effect on tracer concentrations simulated with the ADE. Comparison between the simulated and measured concentrations confirms the appropriate performance of Zeng and Huai’s formula for Dx estimation. Use of the newly proposed equations for Dx estimation could enhance the accuracy of solving the ADE.  相似文献   

5.
For transient, high frequency, and pulse like excitation of structures in the near field of strong earthquakes, the classical design approach based on relative response spectrum and mode superposition may not be conservative. For such excitations, it is more natural to use wave propagation methods. In this paper (Part I), we review several two-dimensional wave propagation models of buildings and show results for theoretical dispersion curves computed for these models. We also estimate the parameters of these models that would correspond to a seven-story reinforced concrete building in Van Nuys, California. Ambient vibration tests data for this building imply vertical shear wave velocity βz=112 m/s and anisotropy factor βx/βz=0.55 for NS vibrations, and βz=88 m/s and βx/βz=1 for EW vibrations. The velocity of shear waves propagating through the slabs is estimated to be about 2000 m/s. In the companion paper (Part II), we estimate phase velocities of vertically and horizontally propagating waves between seven pairs of recording points in the building using recorded response to four earthquakes.  相似文献   

6.
三角网格有限元法能够准确模拟复杂构造和复杂介质条件下的地震波场,数值频散和稳定性条件是地震波数值模拟中参数选择的主要依据.基于均匀的线性三角网格单元,根据结构刚度矩阵的组装原理以及平面波理论,推导了集中质量矩阵下两种网格结构的声波频散函数以及稳定性条件,并对数值频散特性以及稳定性进行了详细研究:三角网格单元中波动的数值频散除了受到空间采样间隔、单元网格纵横比和波传播方向等常规因素的影响外,还受到网格布局的影响,过锐或过钝的三角单元会对波动数值频散产生不良的影响,不同类型的单元网格、单元纵横比对应着不同的稳定性条件,正三角单元中的波动具有较好的数值频散特性,其数值各向异性(频散随波传播方向的变化)效应最弱,稳定性条件也较为宽松.最后通过数值模拟直观地验证了以上分析结果,为有限元正演三角网格的剖分和参数的设置提供一定的理论依据.  相似文献   

7.
We consider 3D steady flow of fresh water over a salt water body in a confined aquifer of constant thickness D, with application to a pumping well in a coastal aquifer. With neglect of mixing, a sharp interface separates the two fluid bodies and an existing analytical solution, based on the Dupuit assumption, is adopted. The aim is to solve for the mixing between the fresh and salt waters for αT/D  1 (αT transverse dispersivity), as field studies indicate that αT = O(10−3 − 10−2 m). The mixing zone around the interface is narrow and solutions by existing codes experience numerical difficulties. The problem is solved by the boundary layer (BL) approximation, extending a method, applied previously to two-dimensional flows. The BL equations of variable-density flow are solved by using the Von Karman integral method, to determine the BL thickness and the rate of entrainment of salt water along the interface. Application to the pumping well problem yields the salinity of the pumped water, as function of the parameters of the problem (well discharge, seaward discharge, well distance from the coast and density difference).  相似文献   

8.
Pressure–volume–temperature relations have been measured to 32 GPa and 2073 K for natural magnesite (Mg0.975Fe0.015Mn0.006Ca0.004CO3) using synchrotron X-ray diffraction with a multianvil apparatus at the SPring-8 facility. A least-squares fit of the room-temperature compression data to a third-order Birch–Murnaghan equation of state (EOS) yielded K0 = 97.1 ± 0.5 GPa and K′ = 5.44 ± 0.07, with fixed V0 = 279.55 ± 0.02 Å3. Further analysis of the high-temperature compression data yielded the temperature derivative of the bulk modulus (∂KT/∂T)P = −0.013 ± 0.001 GPa/K and zero-pressure thermal expansion α = a0 + a1T with a0 = 4.03 (7) × 10−5 K−1 and a1 = 0.49 (10) × 10−8 K−2. The Anderson–Grüneisen parameter is estimated to be δT = 3.3. The analysis of axial compressibility and thermal expansivity indicates that the c-axis is over three times more compressible (KTc = 47 ± 1 GPa) than the a-axis (KTc = 157 ± 1 GPa), whereas the thermal expansion of the c-axis (a0 = 6.8 (2) × 10−5 K−1 and a1 = 2.2 (4) × 10−8 K−2) is greater than that of the a-axis (a0 = 2.7 (4) × 10−5 K−1 and a1 = −0.2 (2) × 10−8 K−2). The present thermal EOS enables us to accurately calculate the density of magnesite to the deep mantle conditions. Decarbonation of a subducting oceanic crust containing 2 wt.% magnesite would result in a 0.6% density reduction at 30 GPa and 1273 K. Using the new EOS parameters we performed thermodynamic calculations for magnesite decarbonation reactions at pressures to 20 GPa. We also estimated stability of magnesite-bearing assemblages in the lower mantle.  相似文献   

9.
We investigate the spatiotemporal nonlocality underlying fractional-derivative models as a possible explanation for regional-scale anomalous dispersion with heavy tails. Properties of four fractional-order advection–dispersion equation (fADE) models were analyzed and compared systematically, including the space fADEs with either maximally positive or negative skewness, the time fADE with a temporal fractional-derivative 0<γ<10<γ<1, and the extension of the time fADE with 1<γ<21<γ<2. Space fADEs describe the dependence of local concentration change on a wide range of spatial zones (i.e., the space nonlocality), while time fADEs describe dynamic mass exchange between mobile and multiple immobile phases and therefore record the temporal history of concentration “loading” (i.e., the time-nonlocality). We then applied the fADEs as models of anomalous dispersion to four extensively-studied, regional-scale, natural systems, including a hillslope composed of fractured soils, a river with simultaneous active flow zones and various dead-zones, a relatively homogeneous glaciofluvial aquifer dominated by stratified sand and gravel, and a highly heterogeneous alluvial aquifer containing both preferential flowpaths and abundant aquitards. We find that the anomalous dispersion observed at each site might not be characterized reasonably or sufficiently by previous studies. In particular, the use of the space fADE with less than maximally positive skewness implies a spatial dependence on downstream concentrations that may not be physically realistic for solute transport in watershed catchments and rivers (where the influence of dead-zones on solute transport can be described by a temporal, not spatial, fractional model). Field-scale transport studies show that large ranges of solute displacement can be described by a space nonlocal, fractional-derivative model, and long waiting times can be described efficiently by a time-nonlocal, fractional model. The unknown quantitative relationship between the nonlocal parameters and the heterogeneity, and the similarity in concentration profiles that are solutions to the different nonlocal transport models, all demonstrate the importance of distinguishing the representative nonlocality (time and/or space) for any given regional-scale anomalous dispersion process.  相似文献   

10.
The Rouse formula and its variants have been widely used to calculate the steady-state vertical concentration distribution for suspended sediment in steady sediment-laden flows, where the diffusive flux is assumed to be Fickian. Turbulent flow, however, exhibits fractal properties, leading to non-Fickian diffusive flux for sediment particles. To characterize non-Fickian dynamics of suspended sediment, the current study proposes a Hausdorff fractal derivative based advection-dispersion equation(H...  相似文献   

11.
It has been known for many years that dispersivity increases with solute travel distance in a subsurface environment. The increase of dispersivity with solute travel distance results from the significant variation of hydraulic properties of heterogeneous media and was identified in the literature as scale-dependent dispersion. This study presents an analytical solution for describing two-dimensional non-axisymmetrical solute transport in a radially convergent flow tracer test with scale-dependent dispersion. The power series technique coupling with the Laplace and finite Fourier cosine transform has been applied to yield the analytical solution to the two-dimensional, scale-dependent advection–dispersion equation in cylindrical coordinates with variable-dependent coefficients. Comparison between the breakthrough curves of the power series solution and the numerical solutions shows excellent agreement at different observation points and for various ranges of scale-related transport parameters of interest. The developed power series solution facilitates fast prediction of the breakthrough curves at any observation point.  相似文献   

12.
To more accurately predict the migration behavior of pollutants in porous media, we conduct laboratory scale experiments and model simulation. Aniline (AN) is used in one-dimensional soil column experiments designed under various media and hydrodynamic conditions. The advection-dispersion equation (ADE) and the continuous-time random walk (CTRW) were used to simulate the breakthrough curves (BTCs) of the solute transport. The results show that the media and hydrodynamic conditions are two important factors affecting solute transport and are related to the degree of non-Fickian transport. The simulation results show that CTRW can more effectively describe the non-Fickian phenomenon in the solute transport process than ADE. The sensitive parameter in the CTRW simulation process is , which can reflect the degree of non-Fickian diffusion in the solute transport. Understanding the relationship of with velocity and media particle size is conducive to improving the reactive solute transport model. The results of this study provide a theoretical basis for better prediction of pollutant transport in groundwater.  相似文献   

13.
时间域常Q黏声波方程,由于含分数阶时间导数项,数值求解需要大量内存,计算效率低,不利于地震偏移的实施.通过一系列近似,可将该方程简化为介质频散效应和衰减效应解耦的分数阶拉普拉斯算子黏声波方程,数值求解内存需求少,计算效率高.本文采用交错网格有限差分逼近时间导数,改进的伪谱法计算空间导数,PML吸收边界去除边界反射,对该方程进行数值离散和地震正演模拟,开展地震数据的黏声介质逆时偏移,实现波场逆时延拓过程中同时完成频散校正和衰减补偿.改善深层构造的成像精度,数值结果表明,基于分数阶拉普拉斯算子解耦的黏声介质地震正演模拟与逆时偏移可大幅度提高地震模拟计算效率,偏移剖面明显优于常规声波偏移剖面,极大改善深层构造的成像品质.  相似文献   

14.
Lateral heterogeneities in the mantle can be caused by thermal, chemical and non-isotropic pre-stress effects. Here, we investigate the possibility of using observations of the glacial isostatic adjustment (GIA) process to constrain the thermal contribution to lateral variations in mantle viscosity. In particular, global historic relative sea level, GPS in Laurentide and Fennoscandia, altimetry together with tide-gauge data in the Great Lakes area, and GRACE data in Laurentide are used. The lateral viscosity perturbations are inferred from the seismic tomography model S20A by inserting the scaling factor β to determine the contribution of thermal effects versus compositional heterogeneity and non-isotropic pre-stress effects on lateral heterogeneity in mantle viscosity. When β = 1, lateral velocity variations are caused by thermal effects alone. With β < 1, the contribution of thermal effect decreases, so that for β = 0, there is no lateral viscosity variation and the Earth is laterally homogeneous. These lateral viscosity variations are superposed on four different reference models which differ significantly in the lower mantle viscosity. The Coupled Laplace Finite Element method is used to predict the GIA response on a spherical, self-gravitating, compressible, viscoelastic Earth with self-gravitating oceans, induced by the ICE-4G deglaciation model.Results show that the effect of β on uplift rates and gravity rate-of-change is not simple and involves the trade-off between the contribution of lateral viscosity variations in the transition zone and in the lower mantle. Models with small viscosity contrast in the lower mantle cannot explain the observed uplift rates in Laurentide and Fennoscandia. However, the RF3S20 model with a reference viscosity profile simplified from Peltier's VM2 with the value of β around 0.2–0.4 is found to explain most of the global RSL data, the uplift rates in Laurentide and Fennoscandia and the BIFROST horizontal velocity data. In addition, the changes in GIA signals caused by changes in the value of β are large enough to be detected by the data, although uncertainty in other parameters in the GIA models still exists. This may encourage us to further utilize GIA observations to constrain the thermal effect on mantle lateral heterogeneity as geodetic and satellite gravity measurements are improved.  相似文献   

15.
The advection–dispersion equation with spatially variable coefficients does not have an exact analytical solution and is therefore solved numerically. However, solutions obtained with several of the traditional finite difference or finite element techniques typically exhibit spurious oscillation or numerical dispersion when advection is dominant. The mixing cell and semi-analytical solution methods proposed in this study avoid such oscillation or numerical dispersion when advection dominates. Both the mixing cell and semi-analytical solution methods calculate the spatial step size by equating numerical dispersion to physical dispersion. Because of the spatial variability of the coefficients the spatial step size varies in space. When the time step size Δt→0, the mixing cell method reduces to the semi-analytical solution method. The results of application to two cases show that the mixing cell and semi-analytical solution methods are better than a finite difference method used in the study. © 1998 John Wiley & Sons, Ltd.  相似文献   

16.
A new method to estimate the vertical crustal motion from satellite altimetry over land was developed. The method was tested around Hudson Bay, where the observed vertical motion is largely caused by the incomplete glacial isostatic adjustment (GIA) as a result of the Laurentide ice sheet deglaciation since the last glacial maximum (LGM). Decadal (1992–2003) TOPEX/POSEIDON radar altimetry data over land surfaces were used. The results presented here are improved compared to a previous study (Lee, H., Shum, C.K., Kuo, C.Y., Yi, Y., Braun, A., 2008. Application of TOPEX altimetry for solid Earth deformation studies. Terr. Atmos. Ocean. Sci. 19, 37–46. doi:10.3319/TAO.2008.19.1-2.37(SA).) which estimated vertical motion only over relatively flat land surfaces (standard deviation of the height variation <40 cm). In this study, we extended the concept of traditional 1-Hz (one-per-frame) radar altimeter ocean stackfiles to build 10-Hz (10-per-frame) land stackfiles over Hudson Bay land regions, and succeeded in obtaining vertical motion estimates over much rougher surfaces (standard deviation of the height variation <2 m). 90-m C-band Shuttle Radar Topography Mission (SRTM) Digital Elevation Model (DEM) is used as a reference surface to select an optimal waveform retracker, to correct surface gradient errors, and to calculate land surface anomalies. Here, we developed an alternative retracker, called the modified threshold retracker, resulting in decadal vertical motion time series over a 1500 km by 1000 km region covering northern Ontario, northeastern Manitoba, and the Great Lakes region which is at the margin of the former Laurentide ice sheet. The average of the estimated uncertainties for the vertical motion is 2.9 mm/year which is comparable to 2.1 mm/year of recent GPS solutions. The estimated vertical motion is compared with other geodetic observations from GPS, tide gauge/altimetry, GRACE, and several GIA models. The data agree best with the laterally varying 3D GIA model, RF3S20 (β = 0.4) whereas the combination of land altimetry solution with other measurements match best with the models RF3S20 (β = 0.0) or RF3S20 (β = 0.2) in terms of mean and standard deviation of the differences. It is anticipated that this innovative technique could potentially be used to provide additional constraints for GIA model improvement, and be applied to other geodynamics studies.  相似文献   

17.
The Boussinesq equation appears as the zeroth-order term in the shallow water flow expansion of the non-linear equation describing the flow of fluid in an unconfined aquifer. One-dimensional models based on the Boussinesq equation have been used to analyse tide-induced water table fluctuations in coastal aquifers. Previous analytical solutions for a sloping beach are based on the perturbation parameter, N=αcotβ (in which β is the beach slope, α is the amplitude parameter and is the shallow water parameter) and are limited to tan−1(α)βπ/2. In this paper, a new higher-order solution to the non-linear boundary value problem is derived. The results demonstrate the significant influence of the higher-order components and beach slope on the water table fluctuations. The relative difference between the linear solution and the present solution increases as and α increase, and reaches 7% of the linear solution.  相似文献   

18.
When formulated properly, most geophysical transport-type process involving passive scalars or motile particles may be described by the same space–time nonlocal field equation which consists of a classical mass balance coupled with a space–time nonlocal convective/dispersive flux. Specific examples employed here include stretched and compressed Brownian motion, diffusion in slit-nanopores, subdiffusive continuous-time random walks (CTRW), super diffusion in the turbulent atmosphere and dispersion of motile and passive particles in fractal porous media. Stretched and compressed Brownian motion, which may be thought of as Brownian motions run with nonlinear clocks, are defined as the limit processes of a special class of random walks possessing nonstationary increments. The limit process has a mean square displacement that increases as tα+1 where α > −1 is a constant. If α = 0 the process is classical Brownian, if α < 0 we say the process is compressed Brownian while if α > 0 it is stretched. The Fokker–Planck equations for these processes are classical ade’s with dispersion coefficient proportional to tα. The Brownian-type walks have fixed time step, but nonstationary spatial increments that are Gaussian with power law variance. With the CTRW, both the time increment and the spatial increment are random. The subdiffusive Fokker–Planck equation is fractional in time for the CTRW’s considered in this article. The second moments for a Levy spatial trajectory are infinite while the Fokker–Planck equation is an advective–dispersive equation, ade, with constant diffusion coefficient and fractional spatial derivatives. If the Lagrangian velocity is assumed Levy rather than the position, then a similar Fokker–Planck equation is obtained, but the diffusion coefficient is a power law in time. All these Fokker–Planck equations are special cases of the general non-local balance law.  相似文献   

19.
Tidal water table fluctuations in a coastal aquifer are driven by tides on a moving boundary that varies with the beach slope. One-dimensional models based on the Boussinesq equation are often used to analyse tidal signals in coastal aquifers. The moving boundary condition hinders analytical solutions to even the linearised Boussinesq equation. This paper presents a new perturbation approach to the problem that maintains the simplicity of the linearised one-dimensional Boussinesq model. Our method involves transforming the Boussinesq equation to an ADE (advection–diffusion equation) with an oscillating velocity. The perturbation method is applied to the propagation of spring–neap tides (a bichromatic tidal system with the fundamental frequencies ω1andω2) in the aquifer. The results demonstrate analytically, for the first time, that the moving boundary induces interactions between the two primary tidal oscillations, generating a slowly damped water table fluctuation of frequency ω1−ω2, i.e., the spring–neap tidal water table fluctuation. The analytical predictions are found to be consistent with recently published field observations.  相似文献   

20.
The gravimetric parameters of the gravity pole tide are the amplitude factor δ, which is the ratio of gravity variations induced by polar motion for a real Earth to variations computed for a rigid one, and the phase difference κ between the observed and the rigid gravity pole tide. They can be estimated from the records of superconducting gravimeters (SGs). However, they are affected by the loading effect of the ocean pole tide. Recent results from TOPEX/Poseidon (TP) altimeter confirm that the ocean pole tide has a self-consistent equilibrium response. Accordingly, we calculate the gravity loading effects as well as their influence on the gravimetric parameters of gravity pole tide at all the 26 SG stations in the world on the assumption of a self-consistent equilibrium ocean pole tide model. The gravity loading effect is evaluated between 1 January 1997 and 31 December 2006. Numerical results show that the amplitude of the gravity loading effect reaches 10−9 m s−2, which is larger than the accuracy (10−10 m s−2) of a SG. The gravimetric factor δ is 1% larger at all SG stations. Then, the contribution of a self-consistent ocean pole tide to the pole tide gravimetric parameters cannot be ignored as it exceeds the current accuracy of the estimation of the pole tide gravity factors. For the nine stations studied in Ducarme et al. [Ducarme, B., Venedikov, A.P., Arnoso, J., et al., 2006. Global analysis of the GGP superconducting gravimeters network for the estimation of the pole tide gravimetric amplitude factor. J. Geodyn. 41, 334–344.], the mean of the modeled tidal factors δm = 1.1813 agrees very well with the result of a global analysis δCH = 1.1816 ± 0.0047 in that paper. On the other hand, the modeled phase difference κm varies from −0.273° to 0.351°. Comparing to the two main periods of the gravity pole tide, annual period and Chandler period, κm is too small to be considered. Therefore, The computed time difference κL induced by a self-consistent ocean pole tide produces a negligible effect on κm. It confirms the results of Ducarme et al., 2006, where no convincing time difference was found in the SG records.  相似文献   

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