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1.
Boundary conditions are required to close the mathematical formulation of unstable density‐dependent flow systems. Proper implementation of boundary conditions, for both flow and transport equations, in numerical simulation are critical. In this paper, numerical simulations using the FEFLOW model are employed to study the influence of the different boundary conditions for unstable density‐dependent flow systems. A similar set up to the Elder problem is studied. It is well known that the numerical simulation results of the standard Elder problem are strongly dependent on spatial discretization. This work shows that for the cases where a solute mass flux boundary condition is employed instead of a specified concentration boundary condition at the solute source, the numerical simulation results do not vary between different convective solution modes (i.e., plume configurations) due to the spatial discretization. Also, the influence of various boundary condition types for nonsource boundaries was studied. It is shown that in addition to other factors such as spatial and temporal discretization, the forms of the solute transport equation such as divergent and convective forms as well as the type of boundary condition employed in the nonsource boundary conditions influence the convective solution mode in coarser meshes. On basis of the numerical experiments performed here, higher sensitivities regarding the numerical solution stability are observed for the Adams‐Bashford/Backward Trapezoidal time integration approach in comparison to the Euler‐Backward/Euler‐Forward time marching approach. The results of this study emphasize the significant consequences of boundary condition choice in the numerical modeling of unstable density‐dependent flow.  相似文献   

2.
The fractional advection–dispersion equation (FADE) known as its non-local dispersion, has been proven to be a promising tool to simulate anomalous solute transport in groundwater. We present an unconditionally stable finite element (FEM) approach to solve the one-dimensional FADE based on the Caputo definition of the fractional derivative with considering its singularity at the boundaries. The stability and accuracy of the FEM solution is verified against the analytical solution, and the sensitivity of the FEM solution to the fractional order α and the skewness parameter β is analyzed. We find that the proposed numerical approach converge to the numerical solution of the advection–dispersion equation (ADE) as the fractional order α equals 2. The problem caused by using the first- or third-kind boundary with an integral-order derivative at the inlet is remedied by using the third-kind boundary with a fractional-order derivative there. The problems for concentration estimation at boundaries caused by the singularity of the fractional derivative can be solved by using the concept of transition probability conservation. The FEM solution of this study has smaller numerical dispersion than that of the FD solution by Meerschaert and Tadjeran (J Comput Appl Math 2004). For a given α, the spatial distribution of concentration exhibits a symmetric non-Fickian behavior when β = 0. The spatial distribution of concentration shows a Fickian behavior on the left-hand side of the spatial domain and a notable non-Fickian behavior on the right-hand side of the spatial domain when β = 1, whereas when β = −1 the spatial distribution of concentration is the opposite of that of β = 1. Finally, the numerical approach is applied to simulate the atrazine transport in a saturated soil column and the results indicat that the FEM solution of the FADE could better simulate the atrazine transport process than that of the ADE, especially at the tail of the breakthrough curves.  相似文献   

3.
ABSTRACT

Forward–backward solute dispersion with an intermediate point source in one-dimensional semi-infinite homogeneous porous media is studied in this paper. Solute transport under sorption conditions, first-order decay and zero-order production terms are included. The first type of boundary condition is taken as a constant point source at an intermediate point from where forward and backward solute dispersion is examined. The Laplace transform method is adopted to solve the governing equation analytically. All the analytical results are obtained in graphical form to investigate the forward–backward solute transport in porous media for various hydrological input data. The graphical nature of the analytical solution is compared with numerical data taken from existing literature and similar results are obtained. Also, numerical solution of the governing equation is obtained by the Crank-Nicolson finite difference scheme and validated with the analytical solution, which demonstrates good agreement between them. Accuracy of the solution is also observed by using RMSE.  相似文献   

4.
Vedat Batu 《Ground water》2010,48(4):560-568
Using a steady-state mass conservative solute transport analytical solution that is based on the third-type (or flux-type or Cauchy) source condition, a method is developed to estimate the degradation parameters of solutes in groundwater. Then, the inadequacy of the methods based on the first-type source-based analytical solute transport solution is presented both theoretically and through an example. It is shown that the third-type source analytical solution exactly satisfies the mass balance constraint at the inlet location. It is also shown that the first-type source (or constant source concentration or Dirichlet) solution fails to satisfy the mass balance constraint at the inlet location and the degree of the failure depends on the value of the degradation as well as the flow and solute transport parameters. The error in the first-type source solution is determined with dimensionless parameters by comparing its results with the third-type source solution. Methods for estimating the degradation parameter values that are based on the first-type steady-state solute transport solution may significantly overestimate the degradation parameter values depending on the values of flow and solute transport parameters. It is recommended that the third-type source solution be used in estimating degradation parameters using measured concentrations instead of the first-type source solution.  相似文献   

5.
Hydraulic conductivity distribution and plume initial source condition are two important factors affecting solute transport in heterogeneous media. Since hydraulic conductivity can only be measured at limited locations in a field, its spatial distribution in a complex heterogeneous medium is generally uncertain. In many groundwater contamination sites, transport initial conditions are generally unknown, as plume distributions are available only after the contaminations occurred. In this study, a data assimilation method is developed for calibrating a hydraulic conductivity field and improving solute transport prediction with unknown initial solute source condition. Ensemble Kalman filter (EnKF) is used to update the model parameter (i.e., hydraulic conductivity) and state variables (hydraulic head and solute concentration), when data are available. Two-dimensional numerical experiments are designed to assess the performance of the EnKF method on data assimilation for solute transport prediction. The study results indicate that the EnKF method can significantly improve the estimation of the hydraulic conductivity distribution and solute transport prediction by assimilating hydraulic head measurements with a known solute initial condition. When solute source is unknown, solute prediction by assimilating continuous measurements of solute concentration at a few points in the plume well captures the plume evolution downstream of the measurement points.  相似文献   

6.
Modelling adsorptive solute transport in soils needs a number of parameters to describe its reaction kinetics and the values of these parameters are usually determined from batch and displacement experiments. Some experimental results reveal that when describing the adsorption as first-order kinetics, its associated reaction rates are not constants but vary with pore water velocity. Explanation of this varies but an independent verification of each explanation is difficult because simultaneously measuring the spatiotemporal distributions of dissolved and adsorbed solutes in soils is formidable. Pore-scale modelling could play an important role to address this gap and has received increased attention over the past few years. This paper investigated the transport of adsorptive solute in a simple porous medium using pore-scale modelling. Fluid flow through the void space of the medium was assumed to be laminar and in saturated condition, and solute transport consisted of advection and molecular diffusion; the sorption and desorption occurring at the fluid–solid interface were modelled as linear first-order kinetics. Based on the simulated spatiotemporal distribution of dissolved and adsorbed solutes at pore scale, volumetric-average reaction kinetics at macroscopic scale and its associated reactive parameters were measured. Both homogeneous adsorption where the reaction rates at microscopic scale are constant, and heterogeneous adsorption where the reaction rates vary from site to site, were investigated. The results indicate that, in contrast to previously thought, the macroscopic reaction rates directly measured from the pore-scale simulations do not change with pore velocity under both homogeneous and heterogeneous adsorptions. In particular, we found that for the homogeneous adsorption, the macroscopic adsorption remains first-order kinetic and can be described by constant reaction rates, regardless of flow rate; whilst for the heterogeneous adsorption, the macroscopic adsorption kinetics continues not to be affected by flow rate but is no longer first-order kinetics that can be described by constant reaction rates. We discuss how these findings could help explain some contrary literature reports over the dependence of reaction rates on pore water velocity.  相似文献   

7.
Abstract

A numerical technique is presented whereby aquifer hydraulic diffusivities (D) and macrodispersivities (α) are calculated by linear equations rewritten from flow and solute transport differential equations. The approach requires a GIS to calculate spatial and temporal hydraulic head (h) and solute concentration gradients. The model is tested in Portugal, in a semi-confined aquifer periodically monitored for h and chloride/sulphate concentrations. Average D (0.46 m2/s) and α (1975 m) compare favourably with literature results. The relationship between α and scale (L) is also investigated. In this context, two aquifer groups could be identified: the first group is heterogeneous at the “macroscopic” scale (solute travelled distances <1 km), but homogeneous at the “megascopic” scale. The overall scale dependency in this case is given by an equation of logarithmic type. The second group is heterogeneous at the macroscopic and megascopic scales, with a scale dependency of linear type.

Citation Pacheco, F.A.L., 2013. Hydraulic diffusivity and macrodispersivity calculations embedded in a geographic information system. Hydrological Sciences Journal, 58 (4), 930–944.  相似文献   

8.
Exact analytical solutions for two-dimensional advection-dispersion equation (ADE) in cylindrical coordinates subject to the third-type inlet boundary condition are presented in this study. The finite Hankel transform technique in combination with the Laplace transform method is adopted to solve the two-dimensional ADE in cylindrical coordinates. Solutions are derived for both continuous input and instantaneous slug input. The developed analytical solutions are compared with the solutions for first-type inlet boundary condition to illustrate the influence of the inlet condition on the two-dimensional solute transport in a porous medium system with a radial geometry. Results show significant discrepancies between the breakthrough curves obtained from analytical solutions for the first-type and third-type inlet boundary conditions for large longitudinal dispersion coefficients. The developed solutions conserve the solute mass and are efficient tools for simultaneous determination of the longitudinal and transverse dispersion coefficients from a laboratory-scale radial column experiment or an in situ infiltration test with a tracer.  相似文献   

9.
Heterogeneous water flow and solute transport in soils are an important phenomenon and difficult to be characterized. The objectives of this study were to investigate the heterogeneity of solute transport related to heterogeneous soil water flow using dye infiltration experiments, and to characterize heterogeneous water flow and solute transport in soils using the information theory. Field experiments of dye infiltration were performed in four plots. Various information measures were applied to characterize information content and complexity of water flow and solute transport in soils. Information contents and complexities of the maximum and apparent infiltration depths, and the mean and standard deviation of concentrations in the vertical direction of the plots were calculated. More heterogeneous processes of soil water flow and transport result in higher information/complexity values. The probability distributions of mean concentration were similar to those of the corresponding apparent infiltration depths for the plots, indicating that heterogeneity of dye concentrations was closely related to that of soil water flow. However, the range of information entropy and complexity of the water flow sequences was much narrower than that of the sequences of the concentrations. The results suggested that the transport processes were more heterogeneous than the water flow processes. Compared with the probability distributions of flow parameters, the information measures appeared to be a more versatile tool to describe flow and transport heterogeneities in soils.  相似文献   

10.
A Eulerian analytical method is developed for nonreactive solute transport in heterogeneous, dual-permeability media where the hydraulic conductivities in fracture and matrix domains are both assumed to be stochastic processes. The analytical solution for the mean concentration is given explicitly in Fourier and Laplace transforms. Instead of using the fast fourier transform method to numerically invert the solution to real space (Hu et al., 2002), we apply the general relationship between spatial moments and concentration (Naff, 1990; Hu et al., 1997) to obtain the analytical solutions for the spatial moments up to the second for a pulse input of the solute. Owing to its accuracy and efficiency, the analytical method can be used to check the semi-analytical and Monte Carlo numerical methods before they are applied to more complicated studies. The analytical method can be also used during screening studies to identify the most significant transport parameters for further analysis. In this study, the analytical results have been compared with those obtained from the semi-analytical method (Hu et al., 2002) and the comparison shows that the semi-analytical method is robust. It is clearly shown from the analytical solution that the three factors, local dispersion, conductivity variation in each domain and velocity convection flow difference in the two domains, play different roles on the solute plume spreading in longitudinal and transverse directions. The calculation results also indicate that when the log-conductivity variance in matrix is 10 times less than its counterpart in fractures, it will hardly influence the solute transport, whether the conductivity field is matrix is treated as a homogeneous or random field.  相似文献   

11.
Time nonlocal transport models such as the time fractional advection‐dispersion equation (t‐fADE) were proposed to capture well‐documented non‐Fickian dynamics for conservative solutes transport in heterogeneous media, with the underlying assumption that the time nonlocality (which means that the current concentration change is affected by previous concentration load) embedded in the physical models can release the effective dispersion coefficient from scale dependency. This assumption, however, has never been systematically examined using real data. This study fills this historical knowledge gap by capturing non‐Fickian transport (likely due to solute retention) documented in the literature (Huang et al. 1995) and observed in our laboratory from small to intermediate spatial scale using the promising, tempered t‐fADE model. Fitting exercises show that the effective dispersion coefficient in the t‐fADE, although differing subtly from the dispersion coefficient in the standard advection‐dispersion equation, increases nonlinearly with the travel distance (varying from 0.5 to 12 m) for both heterogeneous and macroscopically homogeneous sand columns. Further analysis reveals that, while solute retention in relatively immobile zones can be efficiently captured by the time nonlocal parameters in the t‐fADE, the motion‐independent solute movement in the mobile zone is affected by the spatial evolution of local velocities in the host medium, resulting in a scale‐dependent dispersion coefficient. The same result may be found for the other standard time nonlocal transport models that separate solute retention and jumps (i.e., displacement). Therefore, the t‐fADE with a constant dispersion coefficient cannot capture scale‐dependent dispersion in saturated porous media, challenging the application for stochastic hydrogeology methods in quantifying real‐world, preasymptotic transport. Hence improvements on time nonlocal models using, for example, the novel subordination approach are necessary to incorporate the spatial evolution of local velocities without adding cumbersome parameters.  相似文献   

12.
Mehl S  Hill MC 《Ground water》2001,39(2):300-307
Five common numerical techniques for solving the advection-dispersion equation (finite difference, predictor corrector, total variation diminishing, method of characteristics, and modified method of characteristics) were tested using simulations of a controlled conservative tracer-test experiment through a heterogeneous, two-dimensional sand tank. The experimental facility was constructed using discrete, randomly distributed, homogeneous blocks of five sand types. This experimental model provides an opportunity to compare the solution techniques: the heterogeneous hydraulic-conductivity distribution of known structure can be accurately represented by a numerical model, and detailed measurements can be compared with simulated concentrations and total flow through the tank. The present work uses this opportunity to investigate how three common types of results--simulated breakthrough curves, sensitivity analysis, and calibrated parameter values--change in this heterogeneous situation given the different methods of simulating solute transport. The breakthrough curves show that simulated peak concentrations, even at very fine grid spacings, varied between the techniques because of different amounts of numerical dispersion. Sensitivity-analysis results revealed: (1) a high correlation between hydraulic conductivity and porosity given the concentration and flow observations used, so that both could not be estimated; and (2) that the breakthrough curve data did not provide enough information to estimate individual values of dispersivity for the five sands. This study demonstrates that the choice of assigned dispersivity and the amount of numerical dispersion present in the solution technique influence estimated hydraulic conductivity values to a surprising degree.  相似文献   

13.
本文基于Kjartansson常Q模型理论,推导了常Q衰减介质中黏声波和黏弹性波的速度-应力方程,并采用基于二项式窗函数的优化交错网格有限差分方法进行了数值模拟,同时引入不分裂的复频移卷积完全匹配层(CPML)吸收边界条件,以消除边界反射.使用基于自适应时间步长记忆方法的中心差分近似时间分数阶导数,与常用的短时记忆方法相比,提高了波动方程的离散化精度和计算效率.通过对比均匀模型下声波的数值解与解析解,验证了算法的精确性,并进一步分析了不同品质因子下地震波的频散及衰减特征.对BP盐丘模型的数值模拟结果可以较好地反映本文数值方法对复杂介质的适应性及频散压制效果.   相似文献   

14.
As is frequently cited, dispersivity increases with solute travel distance in the subsurface. This behaviour has been attributed to the inherent spatial variation of the pore water velocity in geological porous media. Analytically solving the advection–dispersion equation with distance-dependent dispersivity is extremely difficult because the governing equation coefficients are dependent upon the distance variable. This study presents an analytical technique to solve a two-dimensional (2D) advection–dispersion equation with linear distance-dependent longitudinal and transverse dispersivities for describing solute transport in a uniform flow field. The analytical approach is developed by applying the extended power series method coupled with the Laplace and finite Fourier cosine transforms. The developed solution is then compared to the corresponding numerical solution to assess its accuracy and robustness. The results demonstrate that the breakthrough curves at different spatial locations obtained from the power series solution show good agreement with those obtained from the numerical solution. However, owing to the limited numerical operation for large values of the power series functions, the developed analytical solution can only be numerically evaluated when the values of longitudinal dispersivity/distance ratio eL exceed 0·075. Moreover, breakthrough curves obtained from the distance-dependent solution are compared with those from the constant dispersivity solution to investigate the relationship between the transport parameters. Our numerical experiments demonstrate that a previously derived relationship is invalid for large eL values. The analytical power series solution derived in this study is efficient and can be a useful tool for future studies in the field of 2D and distance-dependent dispersive transport. Copyright © 2008 John Wiley & Sons, Ltd.  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.
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   

17.
Analytical solutions for the water flow and solute transport equations in the unsaturated zone are presented. We use the Broadbridge and White nonlinear model to solve the Richards’ equation for vertical flow under a constant infiltration rate. Then we extend the water flow solution and develop an exact parametric solution for the advection-dispersion equation. The method of characteristics is adopted to determine the location of a solute front in the unsaturated zone. The dispersion component is incorporated into the final solution using a singular perturbation method. The formulation of the analytical solutions is simple, and a complete solution is generated without resorting to computationally demanding numerical schemes. Indeed, the simple analytical solutions can be used as tools to verify the accuracy of numerical models of water flow and solute transport. Comparison with a finite-element numerical solution indicates that a good match for the predicted water content is achieved when the mesh grid is one-fourth the capillary length scale of the porous medium. However, when numerically solving the solute transport equation at this level of discretization, numerical dispersion and spatial oscillations were significant.  相似文献   

18.
Three-dimensional grids representing a heterogeneous, ground water system are generated at 10 different resolutions in support of a site-scale flow and transport modeling effort. These grids represent hydrostratigraphy near Yucca Mountain, Nevada, consisting of 18 stratigraphic units with contrasting fluid flow and transport properties. The grid generation method allows the stratigraphy to be modeled by numerical grids of different resolution so that comparison studies can be performed to test for grid quality and determine the resolution required to resolve geologic structure and physical processes such as fluid flow and solute transport. The process of generating numerical grids with appropriate property distributions from geologic conceptual models is automated, thus making the entire process easy to implement with fewer user-induced errors. The series of grids of various resolutions are used to assess the level at which increasing resolution no longer influences the flow and solute transport results. Grid resolution is found to be a critical issue for ground water flow and solute transport. The resolution required in a particular instance is a function of the feature size of the model, the intrinsic properties of materials, the specific physics of the problem, and boundary conditions. The asymptotic nature of results related to flow and transport indicate that for a hydrologic model of the heterogeneous hydrostratigraphy under Yucca Mountain, a horizontal grid spacing of 600 m and vertical grid spacing of 40 m resolve the hydrostratigraphic model with sufficient precision to accurately model the hypothetical flow and solute transport to within 5% of the value that would be obtained with much higher resolution.  相似文献   

19.
The numerical simulation of long‐term large‐scale (field to regional) variably saturated subsurface flow and transport remains a computational challenge, even with today's computing power. Therefore, it is appropriate to develop and use simplified models that focus on the main processes operating at the pertinent time and space scales, as long as the error introduced by the simpler model is small relative to the uncertainties associated with the spatial and temporal variation of boundary conditions and parameter values. This study investigates the effects of various model simplifications on the prediction of long‐term soil salinity and salt transport in irrigated soils. Average root‐zone salinity and cumulative annual drainage salt load were predicted for a 10‐year period using a one‐dimensional numerical flow and transport model (i.e. UNSATCHEM) that accounts for solute advection, dispersion and diffusion, and complex salt chemistry. The model uses daily values for rainfall, irrigation, and potential evapotranspiration rates. Model simulations consist of benchmark scenarios for different hypothetical cases that include shallow and deep water tables, different leaching fractions and soil gypsum content, and shallow groundwater salinity, with and without soil chemical reactions. These hypothetical benchmark simulations are compared with the results of various model simplifications that considered (i) annual average boundary conditions, (ii) coarser spatial discretization, and (iii) reducing the complexity of the salt‐soil reaction system. Based on the 10‐year simulation results, we conclude that salt transport modelling does not require daily boundary conditions, a fine spatial resolution, or complex salt chemistry. Instead, if the focus is on long‐term salinity, then a simplified modelling approach can be used, using annually averaged boundary conditions, a coarse spatial discretization, and inclusion of soil chemistry that only accounts for cation exchange and gypsum dissolution–precipitation. We also demonstrate that prediction errors due to these model simplifications may be small, when compared with effects of parameter uncertainty on model predictions. The proposed model simplifications lead to larger time steps and reduced computer simulation times by a factor of 1000. Copyright © 2006 John Wiley & Sons, Ltd.  相似文献   

20.
Wang F  Bright J 《Ground water》2004,42(5):760-766
The influence on solute transport of the small-scale spatial variation of aquifer hydraulic conductivity (K) was analyzed by comparing results from fine-grid (2 m by 2 m) simulations of a synthetic heterogeneous aquifer to those from coarse-grid (8 m by 4 m) simulations of an equivalent homogeneous aquifer. Realizations of the K field of the heterogeneous aquifer were generated, using the Monte Carlo approach, from a lognormal distribution with mean log K of 2 (K in m/d) and three levels of log K variance of 0.1, 0.5, and 1.0. Numerical simulation results show that the average standard deviation of point concentrations increased from 1.21 to 5.78 when the value of log K variance was increased from 0.1 to 1.0. The average discrepancy between modeled concentrations (obtained from a coarse-grid deterministic numerical simulation) and the actual mean point concentrations (obtained from fine-grid Monte Carlo numerical simulations) increased from 0.91 to 4.23 with the increase in log K variance. The results from this study illustrate the uncertainty in predictions from contaminant transport models due to their inability to simulate the effects of heterogeneities at scales smaller than the model grid.  相似文献   

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