共查询到20条相似文献，搜索用时 15 毫秒
1.
We extend the particle-tracking method to simulate general multi-rate mass transfer (MRMT) equations. Previous methods for single-rate equations used two-state Markov chains and found that the time a particle spends in the mobile state between waiting time epochs is random and exponentially distributed. Using Bochner’s subordination technique for Markov processes, we find that the random mobile times are still exponential for the stochastic process that corresponds to the MRMT equations. The random times in the immobile phase have a distribution that is directly related to the memory function of the MRMT equation. This connection allows us to interpret the MRMT memory function as the rate at which particles of a certain age, measured by residence time in the immobile zone, exit to become mobile once again. Because the exact distributions of mobile and immobile times are known from the MRMT equations, they can be simulated very simply and efficiently using random walks. 相似文献
2.
This study formulates and analyzes continuous time random walk (CTRW) models in radial flow geometries for the quantification of non-local solute transport induced by heterogeneous flow distributions and by mobile–immobile mass transfer processes. To this end we derive a general CTRW framework in radial coordinates starting from the random walk equations for radial particle positions and times. The particle density, or solute concentration is governed by a non-local radial advection–dispersion equation (ADE). Unlike in CTRWs for uniform flow scenarios, particle transition times here depend on the radial particle position, which renders the CTRW non-stationary. As a consequence, the memory kernel characterizing the non-local ADE, is radially dependent. Based on this general formulation, we derive radial CTRW implementations that (i) emulate non-local radial transport due to heterogeneous advection, (ii) model multirate mass transfer (MRMT) between mobile and immobile continua, and (iii) quantify both heterogeneous advection in a mobile region and mass transfer between mobile and immobile regions. The expected solute breakthrough behavior is studied using numerical random walk particle tracking simulations. This behavior is analyzed by explicit analytical expressions for the asymptotic solute breakthrough curves. We observe clear power-law tails of the solute breakthrough for broad (power-law) distributions of particle transit times (heterogeneous advection) and particle trapping times (MRMT model). The combined model displays two distinct time regimes. An intermediate regime, in which the solute breakthrough is dominated by the particle transit times in the mobile zones, and a late time regime that is governed by the distribution of particle trapping times in immobile zones. These radial CTRW formulations allow for the identification of heterogeneous advection and mobile-immobile processes as drivers of anomalous transport, under conditions relevant for field tracer tests. 相似文献
3.
Diffusive mass exchange of non‐reactive substances in dual‐porosity porous systems – column experiments under saturated conditions 
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下载免费PDF全文 Diffusive mass exchange into immobile water regions within heterogeneous porous aquifers influences the fate of solutes. The percentage of immobile water is often unidentified in natural aquifers though. Hence, the mathematical prediction of solute transport in such heterogeneous aquifers remains challenging. The objective of this study was to find a simple analytical model approach that allows quantifying properties of mobile and immobile water regions and the portion of immobile water in a porous system. Therefore, the Single Fissure Dispersion Model (SFDM), which takes into account diffusive mass exchange between mobile and immobile water zones, was applied to model transport in well‐defined saturated dual‐porosity column experiments. Direct and indirect model validation was performed by running experiments at different flow velocities and using conservative tracer with different molecular diffusion coefficients. In another column setup, immobile water regions were randomly distributed to test the model applicability and to determine the portion of immobile water. In all setups, the tracer concentration curves showed differences in normalized maximum peak concentration, tailing and mass recovery according to their diffusion coefficients. These findings were more pronounced at lower flow rates (larger flow times) indicating the dependency of diffusive mass exchange into immobile water regions on tracers' molecular diffusion coefficients. The SFDM simulated all data with high model efficiency. Successful model validation supported the physical meaning of fitted model parameters. This study showed that the SFDM, developed for fissured aquifers, is applicable in porous media and can be used to determine porosity and volume of regions with immobile water. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
4.
We determine the relevance of Multi-Rate Mass Tansfer (MRMT) models (Haggerty and Gorelick, 1995) to general diffusive porosity structures. To this end, we introduce Structured INteracting Continua (SINC) models as the combination of a finite number of diffusion-dominated interconnected immobile zones exchanging with an advection-dominated mobile domain. It directly extends Multiple INteracting Continua framework (Pruess and Narasimhan, 1985) by introducing a structure in the immobile domain, coming for example from the dead-ends of fracture clusters or poorly-connected dissolution patterns. We demonstrate that, whatever their structure, SINC models can be made equivalent in terms of concentration in the mobile zone to a unique MRMT model. We develop effective shape-free numerical methods to identify its few dominant rates, that comply with any distribution of rates and porosities. We show that differences in terms of macrodispersion are not larger than 50% for approximate MRMT models with only one rate (double porosity models), and drop down to less than 0.1% for five rates MRMT models. Low-dimensional MRMT models accurately approach transport in structured diffusive porosities at intermediate and long times and only miss early responses. 相似文献
5.
The late tail of the breakthrough curve (BTC) of a conservative tracer in a regional-scale alluvial system is explored using Monte Carlo simulations. The ensemble numerical BTC, for an instantaneous point source injected into the mobile domain, has a heavy late tail transforming from power law to exponential due to a maximum thickness of clayey material. Haggerty et al.'s (2000) multiple-rate mass transfer (MRMT) method is used to predict the numerical late-time BTCs for solutes in the mobile phase. We use a simple analysis of the thicknesses of fine-grained units noted in boring logs to construct the memory function that describes the slow decline of concentrations at very late time. The good fit between the predictions and the numerical results indicates that the late-time BTC can be approximated by a summation of a small number of exponential functions, and its shape depends primarily on the thicknesses and the associated volume fractions of immobile water in "blocks" of fine-grained material. The prediction of the late-time BTC using the MRMT method relies on an estimate of the average advective residence time, t(ad). The predictions are not sensitive to estimation errors in t(ad), which can be approximated by L/v , where v is the arithmetic mean ground water velocity and L is the transport distance. This is the first example of deriving an analytical MRMT model from measured hydrofacies properties to predict the late-time BTC. The parsimonious model directly and quantitatively relates the observable subsurface heterogeneity to nonlocal transport parameters. 相似文献
6.
《Advances in water resources》2005,28(6):643-660
This work presents a highly efficient numerical scheme for solving immiscible, advection-dominated two-phase flow in heterogeneous porous media. The pressure equation is decoupled from the saturation equation using an IMPES approach, while the advective terms are decoupled from the capillary diffusive terms in the saturation equation through sequential operator splitting. The parabolic and hyperbolic equations are approximated in time by implicit and explicit schemes, respectively. Damped Newton linearization is applied to the implicit non-linear diffusive step. Mixed hybrid finite elements are applied to the global pressure equation and to the regularized capillary diffusion term. For both linear systems arising from the approximation procedure, an AMG preconditioned conjugate gradient solver is used. A finite volume scheme with slope limiter is applied to the advective step. Numerical comparison with standard preconditioners demonstrates the reliability of the proposed AMG-preconditioner. Benchmark examples illustrate the robustness of the method. 相似文献
7.
More theoretical analysis is needed to investigate why a dual‐domain model often works better than the classical advection‐dispersion (AD) model in reproducing observed breakthrough curves for relatively homogeneous porous media, which do not contain distinct dual domains. Pore‐scale numerical experiments presented here reveal that hydrodynamics create preferential flow paths that occupy a small part of the domain but where most of the flow takes place. This creates a flow‐dependent configuration, where the total domain consists of a mobile and an immobile domain. Mass transfer limitations may result in nonequilibrium, or significant differences in concentration, between the apparent mobile and immobile zones. When the advection timescale is smaller than the diffusion timescale, the dual‐domain mass transfer (DDMT) model better captures the tailing in the breakthrough curve. Moreover, the model parameters (mobile porosity, mean solute velocity, dispersivity, and mass transfer coefficient) demonstrate nonlinear dependency on mean fluid velocity. The studied case also shows that when the Peclet number, Pe, is large enough, the mobile porosity approaches a constant, and the mass transfer coefficient can be approximated as proportional to mean fluid velocity. Based on detailed analysis at the pore scale, this paper provides a physical explanation why these model parameters vary in certain ways with Pe. In addition, to improve prediction in practical applications, we recommend conducting experiments for parameterization of the DDMT model at a velocity close to that of the relevant field sites, or over a range of velocities that may allow a better parameterization. 相似文献
8.
Contrast in capillary pressure of heterogeneous permeable media can have a significant effect on the flow path in two-phase immiscible flow. Very little work has appeared on the subject of capillary heterogeneity despite the fact that in certain cases it may be as important as permeability heterogeneity. The discontinuity in saturation as a result of capillary continuity, and in some cases capillary discontinuity may arise from contrast in capillary pressure functions in heterogeneous permeable media leading to complications in numerical modeling. There are also other challenges for accurate numerical modeling due to distorted unstructured grids because of the grid orientation and numerical dispersion effects. Limited attempts have been made in the literature to assess the accuracy of fluid flow modeling in heterogeneous permeable media with capillarity heterogeneity. The basic mixed finite element (MFE) framework is a superior method for accurate flux calculation in heterogeneous media in comparison to the conventional finite difference and finite volume approaches. However, a deficiency in the MFE from the direct use of fractional flow formulation has been recognized lately in application to flow in permeable media with capillary heterogeneity. In this work, we propose a new consistent formulation in 3D in which the total velocity is expressed in terms of the wetting-phase potential gradient and the capillary potential gradient. In our formulation, the coefficient of the wetting potential gradient is in terms of the total mobility which is smoother than the wetting mobility. We combine the MFE and discontinuous Galerkin (DG) methods to solve the pressure equation and the saturation equation, respectively. Our numerical model is verified with 1D analytical solutions in homogeneous and heterogeneous media. We also present 2D examples to demonstrate the significance of capillary heterogeneity in flow, and a 3D example to demonstrate the negligible effect of distorted meshes on the numerical solution in our proposed algorithm. 相似文献
9.
《Advances in water resources》2005,28(1):33-42
A numerical solution that is significantly more general than other semi-analytical solutions is presented for governing equations describing advective–dispersive transport with multirate mass transfer between mobile and immobile domains. The new solution approach is general in the sense that it does not impose any restrictive assumption on the spatial or temporal variability of advective and dispersive processes in the mobile domain. A single integro-differential equation (IDE) is developed for the concentration in the mobile domain by separating the concentration in the immobile domain from the set of two partial differential equations. The solution to the IDE requires the evaluation of a temporal integral of the concentration in the mobile domain, which is a function of the Laplace transform of the distribution of the mass transfer rate coefficient. The Laplace transform is not limited to flow fields with known constant velocities. The solutions for one- and two-dimensional examples obtained using the new approach agree with those obtained by existing semi-analytical and numerical approaches. 相似文献
10.
Gas exsolution and flow during supersaturated water injection in porous media: I. Pore network modeling 总被引：1，自引：0，他引：1
Weishu ZhaoMarios A. Ioannidis 《Advances in water resources》2011,34(1):2-14
Degassing and in situ development of a mobile gas saturation take place when an aqueous phase saturated with gas at a pressure higher than the subsurface pressure is injected in water-saturated porous media. In the first part of this work, a pore network model is used to study the key physical aspects of this novel and hitherto unexplored way of introducing a gas phase in the subsurface. Following heterogeneous nucleation, growth of gas phase clusters driven by convective diffusion of solute from the bulk aqueous phase, is shown to result in a ramified pattern of gas-occupied pores, which is controlled by capillary and buoyancy forces. The interplay between mass transfer and immiscible displacement processes, namely gas cluster coalescence, mobilization under the action of buoyancy forces and fragmentation resulting from capillary instabilities, is seen to favour the propagation of a stable gas saturation front. Pore network model predictions of the macroscopic mass transfer rate coefficient are in fair agreement with a recently published empirical correlation. 相似文献
11.
This pore-scale modeling study in saturated porous media shows that compound-specific effects are important not only at steady-state and for the lateral displacement of solutes with different diffusivities but also for transient transport and solute breakthrough. We performed flow and transport simulations in two-dimensional pore-scale domains with different arrangement of the solid grains leading to distinct characteristics of flow variability and connectivity, representing mildly and highly heterogeneous porous media, respectively. The results obtained for a range of average velocities representative of groundwater flow (0.1–10 m/day), show significant effects of aqueous diffusion on solute breakthrough curves. However, the magnitude of such effects can be masked by the flux-averaging approach used to measure solute breakthrough and can hinder the correct interpretation of the true dilution of different solutes. We propose, as a metric of mixing, a transient flux-related dilution index that allows quantifying the evolution of solute dilution at a given position along the main flow direction. For the different solute transport scenarios we obtained dilution breakthrough curves that complement and add important information to traditional solute breakthrough curves. Such dilution breakthrough curves allow capturing the compound-specific mixing of the different solutes and provide useful insights on the interplay between advective and diffusive processes, mass transfer limitations, and incomplete mixing in the heterogeneous pore-scale domains. The quantification of dilution for conservative solutes is in good agreement with the outcomes of mixing-controlled reactive transport simulations, in which the mass and concentration breakthrough curves of the product of an instantaneous transformation of two initially segregated reactants were used as measures of reactive mixing. 相似文献
12.
Matrix diffusion can attenuate the rate of plume migration in fractured bedrock relative to the rate of ground water flow for both conservative and nonconservative solutes of interest. In a system of parallel, equally spaced constant aperture fractures subject to steady-state ground water flow and an infinite source width, the degree of plume attenuation increases with time and travel distance, eventually reaching an asymptotic level. The asymptotic degree of plume attenuation in the absence of degradation can be predicted by a plume attenuation factor, beta, which is readily estimated as R' (phi(m)/phi(f)), where R' is the retardation factor in the matrix, phi(m) is the matrix porosity, and phi(f) is the fracture porosity. This dual-porosity relationship can also be thought of as the ratio of primary to secondary porosity. Beta represents the rate of ground water flow in fractures relative to the rate of plume advance. For the conditions examined in this study, beta increases with greater matrix porosity, greater matrix fraction organic carbon, larger fracture spacing, and smaller fracture aperture. These concepts are illustrated using a case study where dense nonaqueous phase liquid in fractured sandstone produced a dissolved-phase trichloroethylene (TCE) plume approximately 300 m in length. Transport parameters such as matrix porosity, fracture porosity, hydraulic gradient, and the matrix retardation factor were characterized at the site through field investigations. In the fractured sandstone bedrock examined in this study, the asymptotic plume attenuation factors (beta values) for conservative and nonconservative solutes (i.e., chloride and TCE) were predicted to be approximately 800 and 12,210, respectively. Quantitative analyses demonstrate that a porous media (single-porosity) solute transport model is not appropriate for simulating contaminant transport in fractured sandstone where matrix diffusion occurs. Rather, simulations need to be conducted with either a discrete fracture model that explicitly incorporates matrix diffusion, or a dual-continuum model that accounts for mass transfer between mobile and immobile zones. Simulations also demonstrate that back diffusion from the matrix to fractures will likely be the time-limiting factor in reaching ground water cleanup goals in some fractured bedrock environments. 相似文献
13.
Pore-scale forces have a significant effect on the macroscopic behaviour of multiphase flow through porous media. This paper studies the effect of these forces using a new volume-of-fluid based finite volume method developed for simulating two-phase flow directly on micro-CT images of porous media. An analytical analysis of the relationship between the pore-scale forces and the Darcy-scale pressure drops is presented. We use this analysis to propose unambiguous definitions of Darcy-scale viscous pressure drops as the rate of energy dissipation per unit flow rate of each phase, and then use them to obtain the relative permeability curves. We show that this definition is consistent with conventional laboratory/field measurements by comparing our predictions with experimental relative permeability. We present single and two-phase flow simulations for primary oil injection followed by water injection on a sandpack and a Berea sandstone. The two-phase flow simulations are presented at different capillary numbers which cover the transition from capillary fingering at low capillary numbers to a more viscous fingering displacement pattern at higher capillary numbers, and the effect of capillary number on the relative permeability curves is investigated. Overall, this paper presents a new finite volume-based methodology for the detailed analysis of two-phase flow directly on micro-CT images of porous media and upscaling of the results to the Darcy scale. 相似文献
14.
《Advances in water resources》1999,23(3):199-216
This paper is concerned with the fast resolution of nonlinear and linear algebraic equations arising from a fully implicit finite volume discretization of two-phase flow in porous media. We employ a Newton-multigrid algorithm on unstructured meshes in two and three space dimensions. The discretized operator is used for the coarse grid systems in the multigrid method. Problems with discontinuous coefficients are avoided by using a newly truncated restriction operator and an outer Krylov-space method. We show an optimal order of convergence for a wide range of two-phase flow problems including heterogeneous media and vanishing capillary pressure in an experimental way. Furthermore, we present a data parallel implementation of the algorithm with speedup results. 相似文献
15.
Eulerian–Lagrangian localized adjoint methods (ELLAMs) provide a general approach to the solution of advection-dominated advection–diffusion equations allowing large time steps while maintaining good accuracy. Moreover, the methods can treat systematically any type of boundary condition and are mass conservative. However, all ELLAMs developed so far suffer from non-physical oscillations and are usually implemented on structured grids. In this paper, we propose a finite volume ELLAM which incorporates a novel correction step rendering the method monotone while maintaining conservation of mass. The method has been implemented on fully unstructured meshes in two space dimensions. Numerical results demonstrate the applicability of the method for problems with highly non-uniform flow fields arising from heterogeneous porous media. 相似文献
16.
Changes in the water table level result in variable water saturation and variable hydrological fluxes at the interface between the unsaturated and saturated zone. This may influence the transport and fate of contaminants in the subsurface. The objective of this study was to examine the impact of a decreasing and an increasing water table on solute transport. We conducted tracer experiments at downward flow conditions in laboratory columns filled with two different uniform porous media under static and transient flow conditions either increasing or decreasing the water table. Tracer breakthrough curves were simulated using a mobile–immobile transport model. The resulting transport parameters were compared to identify dominant transport processes. Changes in the water table level affected dispersivities and mobile water fractions depending on the direction of water table movement and the grain size of the porous media. In fine glass beads, the water flow velocity was similar to the decline rate of the water table, and the mobile water fraction was decreased compared with steady‐state saturated conditions. However, immobile water was negligible. In coarse glass beads, water flow was faster because of fingered flow in the unsaturated part, and the mobile water fraction was smaller than in the fine material. Here, a rising water table led to an even smaller mobile water fraction and increased solute spreading because of diffusive interaction with immobile water. We conclude that changes of the water table need to be considered to correctly simulate transport in the subsurface at the transition of the unsaturated–saturated zone. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
17.
Angular and Frequency-Dependent Wave Velocity and Attenuation in Fractured Porous Media 总被引：1，自引：0，他引：1
José M. Carcione Boris Gurevich Juan E. Santos Stefano Picotti 《Pure and Applied Geophysics》2013,170(11):1673-1683
Wave-induced fluid flow generates a dominant attenuation mechanism in porous media. It consists of energy loss due to P-wave conversion to Biot (diffusive) modes at mesoscopic-scale inhomogeneities. Fractured poroelastic media show significant attenuation and velocity dispersion due to this mechanism. The theory has first been developed for the symmetry axis of the equivalent transversely isotropic (TI) medium corresponding to a poroelastic medium containing planar fractures. In this work, we consider the theory for all propagation angles by obtaining the five complex and frequency-dependent stiffnesses of the equivalent TI medium as a function of frequency. We assume that the flow direction is perpendicular to the layering plane and is independent of the loading direction. As a consequence, the behaviour of the medium can be described by a single relaxation function. We first consider the limiting case of an open (highly permeable) fracture of negligible thickness. We then compute the associated wave velocities and quality factors as a function of the propagation direction (phase and ray angles) and frequency. The location of the relaxation peak depends on the distance between fractures (the mesoscopic distance), viscosity, permeability and fractures compliances. The flow induced by wave propagation affects the quasi-shear (qS) wave with levels of attenuation similar to those of the quasi-compressional (qP) wave. On the other hand, a general fracture can be modeled as a sequence of poroelastic layers, where one of the layers is very thin. Modeling fractures of different thickness filled with CO2 embedded in a background medium saturated with a stiffer fluid also shows considerable attenuation and velocity dispersion. If the fracture and background frames are the same, the equivalent medium is isotropic, but strong wave anisotropy occurs in the case of a frameless and highly permeable fracture material, for instance a suspension of solid particles in the fluid. 相似文献
18.
A Practical Modeling Framework for Non‐Fickian Transport and Multi‐Species Sequential First‐Order Reaction 下载免费PDF全文
下载免费PDF全文 Daniel K. Burnell Jie Xu Scott K. Hansen Lawrence S. Sims Charles R. Faust 《Ground water》2018,56(4):524-540
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
19.
We present advances in compositional modeling of two-phase multi-component flow through highly complex porous media. Higher-order methods are used to approximate both mass transport and the velocity and pressure fields. We employ the Mixed Hybrid Finite Element (MHFE) method to simultaneously solve, to the same order, the pressure equation and Darcy's law for the velocity. The species balance equation is approximated by the discontinuous Galerkin (DG) approach, combined with a slope limiter. In this work we present an improved DG scheme where phase splitting is analyzed at all element vertices in the two-phase regions, rather than only as element averages. This approximation is higher-order than the commonly employed finite volume method and earlier DG approximations. The method reduces numerical dispersion, allowing for an accurate capture of shock fronts and lower dependence on mesh quality and orientation. Further new features are the extension to unstructured grids and support for arbitrary permeability tensors (allowing for both scalar heterogeneity, and shear anisotropy). The most important advancement in this work is the self-consistent modeling of two-phase multi-component Fickian diffusion. We present several numerical examples to illustrate the powerful features of our combined MHFE–dg method with respect to lower-order calculations, ranging from simple two component fluids to more challenging real problems regarding CO2 injection into a vertical domain saturated with a multi-component petroleum fluid. 相似文献
20.
In modeling solute transport with mobile‐immobile mass transfer (MIMT), it is common to use an advection‐dispersion equation (ADE) with a retardation factor, or retarded ADE. This is commonly referred to as making the local equilibrium assumption (LEA). Assuming local equilibrium, Eulerian textbook treatments derive the retarded ADE, ostensibly exactly. However, other authors have presented rigorous mathematical derivations of the dispersive effect of MIMT, applicable even in the case of arbitrarily fast mass transfer. We resolve the apparent contradiction between these seemingly exact derivations by adopting a Lagrangian point of view. We show that local equilibrium constrains the expected time immobile, whereas the retarded ADE actually embeds a stronger, nonphysical, constraint: that all particles spend the same amount of every time increment immobile. Eulerian derivations of the retarded ADE thus silently commit the gambler's fallacy, leading them to ignore dispersion due to mass transfer that is correctly modeled by other approaches. We then present a particle tracking simulation illustrating how poor an approximation the retarded ADE may be, even when mobile and immobile plumes are continually near local equilibrium. We note that classic “LEA” (actually, retarded ADE validity) criteria test for insignificance of MIMT‐driven dispersion relative to hydrodynamic dispersion, rather than for local equilibrium. 相似文献