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1.
Three-dimensional analytical solutions for solute transport in saturated, homogeneous porous media are developed. The models account for three-dimensional dispersion in a uniform flow field, first-order decay of aqueous phase and sorbed solutes with different decay rates, and nonequilibrium solute sorption onto the solid matrix of the porous formation. The governing solute transport equations are solved analytically by employing Laplace, Fourier and finite Fourier cosine transform techniques. Porous media with either semi-infinite or finite thickness are considered. Furthermore, continuous as well as periodic source loadings from either a point or an elliptic source geometry are examined. The effect of aquifer boundary conditions as well as the source geometry on solute transport in subsurface porous formations is investigated.  相似文献   

2.
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   

3.
Many studies indicate that small‐scale heterogeneity and/or mobile–immobile mass exchange produce transient non‐Fickian plume behavior that is not well captured by the use of the standard, deterministic advection‐dispersion equation (ADE). An extended ADE modeling framework is presented here that is based on continuous time random walk theory. It can be used to characterize non‐Fickian transport coupled with simultaneous sequential first‐order reactions (e.g., biodegradation or radioactive decay) for multiple degrading contaminants such as chlorinated solvents, royal demolition explosive, pesticides, and radionuclides. To demonstrate this modeling framework, new transient analytical solutions are derived and are inverted in Laplace space. Closed‐form, steady‐state, multi‐species analytical solutions are also derived for non‐Fickian transport in highly heterogeneous aquifers with linear sorption–desorption and matrix diffusion for use in spreadsheets. The solutions are general enough to allow different degradation rates for the mobile and immobile zones. The transient solutions for multi‐species transport are applied to examine the effects of source remediation on the natural attenuation of downgradient plumes of both parent and degradation products in highly heterogeneous aquifers. Results for representative settings show that the use of the standard, deterministic ADE can over‐estimate cleanup rates and under‐predict the cleanup timeframe in comparison to the extended ADE analytical model. The modeling framework and calculations introduced here are also applied for a 30 year groundwater cleanup program at a site in Palm Bay, Florida. The simulated plume concentrations using the extended ADE exhibited agreement with observed long concentration tails of trichloroethene, cis 1,2 DCE, and VC that remained above cleanup goals.  相似文献   

4.
The Laplace Transform Galerkin (LTG) method and the Arnoldi modal reduction method (AMRM) have been implemented in finite element schemes designed to solve mass transport problems in porous media by Sudicky [Sudicky, E.A., Water Resour. Res., 25(8) (1989) 1833-46] and Woodbury et al [Woodbury, A.D., Dunbar, W.S., & Nour-Omid, B., Water Resour. Res., 26(10) (1990) 2579-90]. In this work, a comparative analysis of the two methods is performed with attention focused on efficiency and accuracy. The analysis is performed over one- and two-dimensional domains composed of homogeneous and heterogeneous material properties. The results obtained using homogeneous material properties indicate that for a given mesh design the LTG method maintains a higher degree of accuracy than does the AMRM. However, in terms of efficiency, the Arnoldi attains a pre-defined level of accuracy faster than does the LTG method. It is also shown that for problems involving homogeneous material properties the solution obtained using the LTG method on a coarse mesh is comparable in terms of solution time and accuracy to that obtained using the AMRM on a fine mesh. Comparisons similar to those performed using homogeneous material properties are also performed for the case where the hydraulic conductivity field is heterogeneous. For this case, the level of accuracy achieved by the AMRM and the LTG method are similar. However, as with the analysis involving homogeneous material properties, the AMRM is found to be more efficient than the LTG method. It is also shown that for heterogeneous material properties, use of the LTG method under high grid Peclet conditions can be potentially problematic.  相似文献   

5.
《Advances in water resources》2007,30(6-7):1668-1679
Under steady-state conditions, the degradation of contaminant plumes introduced continuously into an aquifer is controlled by transverse dispersion when the other reacting compound is provided from ambient groundwater. Given that the reaction is instantaneous and longitudinal dispersion can be neglected, the length of the plume is inversely proportional to the transverse dispersion coefficient. In typical scenarios of natural attenuation, however, the considered reaction is biotic and kinetic. The standard model of bioreactive transport relies on double-Monod kinetics and pseudo first-order biomass decay. Under these conditions, a fraction of the injected mass flux remains beyond the length of the plume determined for the instantaneous reaction. We present an analytical framework to derive the steady-state concentration distributions of the dissolved compounds and the biomass from the concentration distribution of a conservative compound, assuming double-Monod kinetics and two different models describing biomass decay. The first biomass-decay model assumes a constant first-order decay coefficient, while the second assumes that the decay coefficient depends upon the electron-acceptor concentration. We apply the method to the case of a line-injection in two-dimensional uniform flow. In general, the bioreactive concentration distributions are similar to the distributions computed for an instantaneous reaction. The similarity is greater when the biomass decay coefficient is assumed to depend on the electron-acceptor concentration rather than being constant.  相似文献   

6.
The influence of source zone concentration reduction on solute plume detachment and recession times in fractured rock was investigated using new semianalytical solutions to transient solute transport in the presence of advection, dispersion, sorption, matrix diffusion, and first-order decay. Novel aspects of these solutions are: (1) the source zone concentration behavior is simulated using a constant concentration with the option for either an instantaneous reduction to zero concentration or an exponentially decaying source zone concentration initiated at some time (t*) after the source is introduced, and (2) different biodegradation rates in the fracture and rock matrix. These solutions were applied for sandstone bedrock and revealed that biodegradation in the matrix, not the fracture, may be the most significant attenuation mechanism and therefore may dictate remediation time scales. Also, instantaneous and complete source concentration reduction in aged plumes may not be beneficial with respect to plume response because back-diffusion can sustain plume migration for long periods of time. Moderate source zone concentration reduction has a similar impact on the rate of advance of the leading edge of the plume as aggressive concentration reduction. If the source zone concentration reduction half-life is less than the plume decay half-life, then volatile organic compound (VOC) mass sequestered in the rock matrix will ultimately dictate plume persistence and not the presence of the source zone.  相似文献   

7.
Two analytical solution methods are presented for regional steady-state groundwater flow in a two-dimensional stratified aquifer cross section where the water table is approximated by the topographic surface. For the first solution, the surficial aquifer is represented as a set of dipping parallel layers with different, but piecewise constant, anisotropic hydraulic conductivities, where the anisotropy is aligned with the dip of the layered formation. The model may be viewed as a generalization of the solutions developed by [Tóth JA. A theoretical analysis of groundwater flows in small drainage basins. J Geophys Res 1963;68(16):4795–812; Freeze R, Witherspoon P. Theoretical analysis of regional groundwater flow 1) analytical and numerical solution to the mathematical model, water resources research. Water Resour Res 1966;2(4):641–56; Selim HM. Water flow through multilayered stratified hillside. Water Resour Res 1975;11:949–57] to an multi-layer aquifer with general anisotropy, layer orientation, and a topographic surface that may intersect multiple layers. The second solution presumes curved (syncline) layer stratification with layer-dependent anisotropy aligned with the polar coordinate system. Both solutions are exact everywhere in the domain except at the topographic surface, where a Dirichlet condition is met in a least-squared sense at a set of control points; the governing equation and no-flow/continuity conditions are met exactly. The solutions are derived and demonstrated on multiple test cases. The error incurred at the location where the layer boundaries intersect the surface is assessed.  相似文献   

8.
The Laplace domain solutions have been obtained for three-dimensional groundwater flow to a well in confined and unconfined wedge-shaped aquifers. The solutions take into account partial penetration effects, instantaneous drainage or delayed yield, vertical anisotropy and the water table boundary condition. As a basis, the Laplace domain solutions for drawdown created by a point source in uniform, anisotropic confined and unconfined wedge-shaped aquifers are first derived. Then, by the principle of superposition the point source solutions are extended to the cases of partially and fully penetrating wells. Unlike the previous solution for the confined aquifer that contains improper integrals arising from the Hankel transform [Yeh HD, Chang YC. New analytical solutions for groundwater flow in wedge-shaped aquifers with various topographic boundary conditions. Adv Water Resour 2006;26:471–80], numerical evaluation of our solution is relatively easy using well known numerical Laplace inversion methods. The effects of wedge angle, pumping well location and observation point location on drawdown and the effects of partial penetration, screen location and delay index on the wedge boundary hydraulic gradient in unconfined aquifers have also been investigated. The results are presented in the form of dimensionless drawdown-time and boundary gradient-time type curves. The curves are useful for parameter identification, calculation of stream depletion rates and the assessment of water budgets in river basins.  相似文献   

9.
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   

10.
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11.
A previous study [Water Resour Res 39 (3) (2003) doi:10.1029/2002WR001338] questioned the validity of the traditional advection–dispersion equation for describing gas flow in porous media. In an original mathematical derivation presented in Part 1 [Adv Water Resour, this issue] we have demonstrated the theoretical existence of two novel physical phenomena which govern the macroscopic transport of gases in porous media. In this work we utilize laboratory experiments and numerical modeling in order to ascertain the importance of these novel theoretical terms. Numerical modeling results indicate that the newly derived sorptive velocity, arising from closure level coupling effects, does not contribute noticeably to the overall flux, under the conditions explored in this work. We demonstrate that the newly discovered “slip coupling” phenomenon in the mass conservation equation plays an important role in describing the physics of gas flow through porous solids for flow regimes of both environmental and industrial interest.  相似文献   

12.
In this work, numerical modeling is used to evaluate and interpret a series of detailed and well‐controlled two‐dimensional bench‐scale conservative tracer tank experiments performed to investigate transverse mixing in porous media. The porous medium used consists of a fine matrix and a more permeable lens vertically aligned with the tracer source and the flow direction. A sensitivity analysis shows that the tracer distribution after passing the lens is only slightly sensitive to variations in transverse dispersivity, but strongly sensitive to the contrast of hydraulic conductivities. A unique parameter set could be calibrated to closely fit the experimental observations. On the basis of calibrated and validated model, synthetic experiments with different contrasts in hydraulic conductivity and more complex setups were performed and the efficiency of mixing evaluated. Flux‐related dilution indices derived from these simulations show that the contrasts in hydraulic conductivity between matrix and high‐permeable lenses as well as the spatial configuration of tracer plumes and lenses dominate mixing, rather than the actual pore scale dispersivities. These results indicate that local material distributions, the magnitude of permeability contrasts, and their spatial and scale relation to solute plumes are more important for macro‐scale transverse dispersion than the micro‐scale dispersivities of individual materials. Local material characterization by thorough site investigation hence is of utmost importance for the evaluation of mixing‐influenced or ‐governed problems in groundwater, such as tracer test evaluation or an assessment of contaminant natural attenuation.  相似文献   

13.
The paper provides an introduction to fundamental concepts of mathematical modeling of mass transport in fractured porous heterogeneous rocks. Keeping aside many important factors that can affect mass transport in subsurface, our main concern is the multi-scale character of the rock formation, which is constituted by porous domains dissected by the network of fractures. Taking into account the well-documented fact that porous rocks can be considered as a fractal medium and assuming that sizes of pores vary significantly (i.e. have different characteristic scales), the fractional-order differential equations that model the anomalous diffusive mass transport in such type of domains are derived and justified analytically. Analytical solutions of some particular problems of anomalous diffusion in the fractal media of various geometries are obtained. Extending this approach to more complex situation when diffusion is accompanied by advection, solute transport in a fractured porous medium is modeled by the advection-dispersion equation with fractional time derivative. In the case of confined fractured porous aquifer, accounting for anomalous non-Fickian diffusion in the surrounding rock mass, the adopted approach leads to introduction of an additional fractional time derivative in the equation for solute transport. The closed-form solutions for concentrations in the aquifer and surrounding rocks are obtained for the arbitrary time-dependent source of contamination located in the inlet of the aquifer. Based on these solutions, different regimes of contamination of the aquifers with different physical properties can be readily modeled and analyzed.  相似文献   

14.
Despite the availability of numerical models, interest in analytical solutions of multidimensional advection‐dispersion systems remains high. Such models are commonly used for performing Tier I risk analysis and are embedded in many regulatory frameworks dealing with groundwater contamination. In this work, we develop a closed‐form solution of the three‐dimensional advection‐dispersion equation with exponential source decay, first‐order reaction, and retardation, and present an approach based on some ease of use diagrams to compare it with the integral open form solution and with earlier versions of the closed‐form solution. The comparison approach focuses on the relative differences associated with source decay and the effect of simulation time. The analysis of concentration contours, longitudinal sections, and transverse sections confirms that the closed‐form solutions studied can be used with acceptable approximation in the central area of a plume bound transversely within the source width, both behind and beyond the advective front and for concentration values up to two orders of magnitude less than the initial source concentration. As the proposed closed‐form model can be evaluated without nested numerical computations and with simple mathematical functions, it can be very useful in risk assessment procedures.  相似文献   

15.
In sedimentary rocks attenuation/dispersion is dominated by fluid-rock interactions. Wave-induced fluid flow in the pores causes energy loss through several mechanisms, and as a result attenuation is strongly frequency dependent. However, the fluid motion process governing the frequency dependent attenuation and velocity remains unclear. We propose a new approach to obtain the analytical expressions of pore pressure, relative fluxes distribution and frame displacement within the double-layer porous media based on quasi-static poroelastic theory. The dispersion equation for a P-wave propagating in a porous medium permeated by aligned fractures is given by considering fractures as thin and highly compliant layers. The influence of mesoscopic fluid flow on phase velocity dispersion and attenuation is discussed under the condition of varying fracture weakness. In this model conversion of the compression wave energy into Biot slow wave diffusion at the facture surface can result in apparent attenuation and dispersion within the usual seismic frequency band. The magnitude of velocity dispersion and attenuation of P-wave increases with increasing fracture weakness, and the relaxation peak and maximum attenuation shift towards lower frequency. Because of its periodic structure, the fractured porous media can be considered as a phononic crystal with several pass and stop bands in the high frequency band. Therefore, the velocity and attenuation of the P-wave show an oscillatory behavior with increasing frequency when resonance occurs. The evolutions of the pore pressure and the relative fluxes as a function of frequency are presented, giving more physical insight into the behavior of P-wave velocity dispersion and the attenuation of fractured porous medium due to the wave-induced mesoscopic flow. We show that the specific behavior of attenuation as function of frequency is mainly controlled by the energy dissipated per wave cycle in the background layer.  相似文献   

16.
An Eulerian perturbation approach was applied to develop a method of moment for solute transport in a nonstationary, fractured medium. The conceptualized fractured medium is described through a dual-porosity model. Stochastic governing equations for mean concentration and concentration covariance were analytically derived to the first-order accuracy of log-conductivity variance and solved with a numerical method––a finite difference method. The developed method is called a numerical Eulerian method of moment (NEMM). This method was compared with the stationary transport theory [Water Resour. Res. 36(7) (2000) 1665] for predicting mean concentration and its spatial moments. The comparison indicated that the two methods matched very well in predicting first and second spatial moments. NEMM solutions were also compared with Monte Carlo simulations for solute transport in stationary fractured media. The results of the two methods were consistent for calculating small log conductivity variance. The theory was then used to study effects of various parameters and nonstationarity of the medium on flow and transport processes. Results indicated that medium nonstationarity would significantly influence the solute transport process. The nonstationary transport theory relaxes many assumptions adopted in stationary theories and paves the way for applying the NEMM to many environmental projects, especially in analyzing uncertainty of solute transport.  相似文献   

17.
This is Part-II of a two-part article that presents analytical solutions to multi-species reactive transport equations coupled through sorption and sequential first-order reactions. In Part-I, we provide the mathematical derivations and in this article we discuss the computational techniques for implementing these solutions. We adopt these techniques to develop a general computer code and use it to verify the solutions. We also simplify the general solutions for various special-case transport scenarios involving zero initial condition, identical retardation factors and zero advection. In addition to this, we derive specialized solution expressions for zero dispersion and steady-state conditions. Whereever possible, we compare these special-case solutions against previously published analytical solutions to establish the validity of the new solution. Finally, we test the new solution against other published analytical and semi-analytical solutions using a set of example problems.  相似文献   

18.
In this article, the quadrupole method is implemented in order to simulate the effects of heterogeneities on one dimensional advective and diffusive transport of a passive solute in porous media. Theoretical studies of dispersion in heterogeneous stratified media can bring insight into transport artefacts linked to scale effects and apparent dispersion coefficients. The quadrupole method is an efficient method for the calculation of transient response of linear systems. It is based here on the Laplace transform technique. The analytical solutions that can be derived by this method assists understanding of upscaled parameters relevant to heterogeneous porous media.First, the method is developed for an infinite homogeneous porous medium. Then, it is adapted to a stratified medium where the fluid flow is perpendicular to the interfaces. The first heterogeneous medium studied is composed of two semi-infinite layers perpendicular to the flow direction each having different transport properties. The concentration response of the medium to a Dirac injection is evaluated. The case studied emphasises the importance in the choice of the boundary conditions.In the case of a periodic heterogeneous porous medium, the concentration response of the medium is evaluated for different numbers of unit-cells. When the number of unit cells is great enough, depending on the transport properties of each layer in the unit cell, an equivalent homogeneous behaviour is reached. An exact determination of the transport properties (equivalent dispersion coefficient) of the equivalent homogeneous porous medium is given.  相似文献   

19.
We analyze the impact of a linear trend in the mean log-conductivity on the transport of a conservative tracer in a bounded domain. The effects of such a linear trend on solute transport were analyzed in depth for unbounded domains (Rubin and Seong, Water Resour Res 30(11):2901–2911, 1994; Indelman and Rubin, Water Resour Res 31(5):1257–1265, 1995; Water Resour Res 32(5):1257–1265, 1996), whereas studies concerning this special case of medium nonstationarity in finite domains usually focus on head or flow statistics (Guadagnini et al., Stoch Environ Res Risk Assess, 17:394–407, 2003). In this study both ensemble and effective plume moments are provided for an instantaneous release of a solute through a linear source normal to the mean flow direction, by taking into account different sizes of the source. The analysis involving a steady velocity field spatially nonstationary is developed by using the stochastic finite element method. Results show that ensemble moments are affected by increasing trends both parallel and normal to the mean flow direction, but the impact on effective plume moments is very different. A parallel trend does not seem to influence the effective second moments; while a normal trend, although modifies the transverse effective moment only weakly, strongly increases the longitudinal one, especially for large initial sizes of the source. Furthermore, the increase of the particle displacement variance produced by a parallel trend in the finite domain disagrees with the results obtained in an unbounded domain, due to the boundary conditions here considered making both head and velocity moments nonstationary and nonsymmetric.  相似文献   

20.
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