Calculating the macrodispersion coefficient of the ensemble averaged solute transport equation in the discrete domain

Recognizing the spatial heterogeneity of hydraulic parameters, many researchers have studied the solute transport by both groundwater and channel flow in a stochastic framework. One of the methodologies used to up‐scale the stochastic solute transport equation, from a point‐location scale to a grid scale, is the cumulant expansion method combined with the calculus for the time‐ordered exponential and the calculus for the Lie operator. When the point‐location scale transport equation is scaled up to the grid scale, using the cumulant expansion method, a new dispersion coefficient emerges in the dispersive term of the solute transport equation in addition to the molecular dispersion coefficient. This velocity driven dispersion is called ‘macrodispersion’. The macrodispersion coefficient is the integral function of the time‐ordered covariance of the random velocity field. The integral is calculated over a Lagrangian trajectory of the flow. The Lagrangian trajectory depends on the following: (i) the spatial origin of the particle; (ii) the time when the macrodispersion is calculated; and (iii) the mean velocity field along the trajectory itself. The Lagrangian trajectory is a recursive function of time because the location of the particle along the trajectory at a particular time depends on the location of the particle at the previous time. This recursive functional form of the Lagrangian trajectory makes the calculation of the macrodispersion coefficient difficult. Especially for the unsteady, spatially non‐stationary, non‐uniform flow field, the macrodispersion coefficient is a highly complex expression and, so far, calculated using numerical methods in the discrete domains. Here, an analytical method was introduced to calculate the macrodispersion coefficient in the discrete domain for the unsteady and steady, spatially non‐stationary flow cases accurately and efficiently. This study can fill the gap between the theory of the ensemble averaged solute transport model and its numerical implementations. Copyright © 2011 John Wiley & Sons, Ltd.     

An exact solution of solute transport by one-dimensional random velocity fields

The problem of one-dimensional transport of passive solute by a random steady velocity field is investigated. This problem is representative of solute movement in porous media, for example, in vertical flow through a horizontally stratified formation of variable porosity with a constant flux at the soil surface. Relating moments of particle travel time and displacement, exact expressions for the advection and dispersion coefficients in the Focker-Planck equation are compared with the perturbation results for large distances. The first- and second-order approximations for the dispersion coefficient are robust for a lognormal velocity field. The mean Lagrangian velocity is the harmonic mean of the Eulerian velocity for large distances. This is an artifact of one-dimensional flow where the continuity equation provides for a divergence free fluid flux, rather than a divergence free fluid velocity.     

An exact solution of solute transport by one-dimensional random velocity fields

The problem of one-dimensional transport of passive solute by a random steady velocity field is investigated. This problem is representative of solute movement in porous media, for example, in vertical flow through a horizontally stratified formation of variable porosity with a constant flux at the soil surface. Relating moments of particle travel time and displacement, exact expressions for the advection and dispersion coefficients in the Focker-Planck equation are compared with the perturbation results for large distances. The first- and second-order approximations for the dispersion coefficient are robust for a lognormal velocity field. The mean Lagrangian velocity is the harmonic mean of the Eulerian velocity for large distances. This is an artifact of one-dimensional flow where the continuity equation provides for a divergence free fluid flux, rather than a divergence free fluid velocity.     

On macrodispersion in uniform – radial divergent flow through weakly heterogeneous aquifers

The transport of instantaneously injected conservative solute through a well in the formation of random conductivity is analyzed. The solute is advected by the recharging well flow with the uniform background gradient. The longitudinal trajectory variance is derived for the central mean streamline. It is shown that the solute is spread as in a radial flow at small travel distances and as in a uniform flow far from the well. Closed form expressions of the longitudinal trajectory variance and macrodispersivity are derived for the case of small scale heterogeneity. It is shown that the macrodispersivity is bounded between the asymptotic macrodispersivities pertinent to the well and uniform flows.     

Solute Migration from the Aquifer Matrix into a Solution Conduit and the Reverse

A solution conduit has a permeable wall allowing for water exchange and solute transfer between the conduit and its surrounding aquifer matrix. In this paper, we use Laplace Transform to solve a one‐dimensional equation constructed using the Euler approach to describe advective transport of solute in a conduit, a production‐value problem. Both nonuniform cross‐section of the conduit and nonuniform seepage at the conduit wall are considered in the solution. Physical analysis using the Lagrangian approach and a lumping method is performed to verify the solution. Two‐way transfer between conduit water and matrix water is also investigated by using the solution for the production‐value problem as a first‐order approximation. The approximate solution agrees well with the exact solution if dimensionless travel time in the conduit is an order of magnitude smaller than unity. Our analytical solution is based on the assumption that the spatial and/or temporal heterogeneity in the wall solute flux is the dominant factor in the spreading of spring‐breakthrough curves, and conduit dispersion is only a secondary mechanism. Such an approach can lead to the better understanding of water exchange and solute transfer between conduits and aquifer matrix. Highlights:

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Incorporating Super‐Diffusion due to Sub‐Grid Heterogeneity to Capture Non‐Fickian Transport

Numerical transport models based on the advection‐dispersion equation (ADE) are built on the assumption that sub‐grid cell transport is Fickian such that dispersive spreading around the average velocity is symmetric and without significant tailing on the front edge of a solute plume. However, anomalous diffusion in the form of super‐diffusion due to preferential pathways in an aquifer has been observed in field data, challenging the assumption of Fickian dispersion at the local scale. This study develops a fully Lagrangian method to simulate sub‐grid super‐diffusion in a multidimensional regional‐scale transport model by using a recent mathematical model allowing super‐diffusion along the flow direction given by the regional model. Here, the time randomizing procedure known as subordination is applied to flow field output from MODFLOW simulations. Numerical tests check the applicability of the novel method in mapping regional‐scale super‐diffusive transport conditioned on local properties of multidimensional heterogeneous media.     

A numerical method of moments for solute transport in physically and chemically nonstationary formations: linear equilibrium sorption with random K<Subscript>d</Subscript>

A Lagrangian perturbation method is applied to develop a method of moments for solute flux through a three-dimensional nonstationary flow field. The flow nonstationarity stems from medium nonstationarity and internal and external boundaries of the study domain. The solute flux is described as a space-time process where time refers to the solute flux breakthrough through a control plane (CP) at some distance downstream of the solute source and space refers to the transverse displacement distribution at the CP. The analytically derived moment equations for solute transport in a nonstationarity flow field are too complicated to solve analytically, a numerical finite difference method is implemented to obtain the solutions. This approach combines the stochastic model with the flexibility of the numerical method to boundary and initial conditions. The developed method is applied to study the effects of heterogeneity and nonstationarity of the hydraulic conductivity and chemical sorption coefficient on solute transport. The study results indicate all these factors will significantly influence the mean and variance of solute flux.     

Bridging the gap between numerical solutions of travel time distributions and analytical storage selection functions

Recent advancements in analytical solutions to quantify water and solute travel time distributions (TTDs) and the related StorAge Selection (SAS) functions synthesize catchment complexity into a simplified, lumped representation. Although these analytical approaches are efficient in application, they require rarely available long‐term and high‐frequency hydrochemical data for parameter estimation. Alternatively, integrated hydrologic models coupled to Lagrangian particle‐tracking approaches can directly simulate age under different catchment geometries and complexity, but at a greater computational expense. Here, we bridge the two approaches, using a physically based model to explore the uncertainty in the estimation of the SAS function shape. In particular, we study the influence of subsurface heterogeneity, interactions between distinct flow domains (i.e., the vadose zone and saturated groundwater), diversity of flow pathways, and recharge rate on the shape of TTDs and the SAS functions. We use an integrated hydrology model, ParFlow, linked with a particle‐tracking model, SLIM, to compute transient residence times (or ages) at every cell in the domain, facilitating a direct characterization of the SAS function. Steady‐state results reveal that the SAS function shape shows a wide range of variation with respect to the variability in the structure of subsurface heterogeneity. Ensembles of spatially correlated realizations of hydraulic conductivity indicate that the SAS functions in the saturated groundwater have an overall weak tendency toward sampling younger ages, whereas the vadose zone gives a strong preference for older ages. We further show that the influence of recharge rate on the TTD is tightly dependent on the variability of subsurface hydraulic conductivity.     

Travel time and trajectory moments of conservative solutes in three dimensional heterogeneous porous media under mean uniform flow

We present expressions satisfied by the first statistical moments (mean and variance–covariance) of travel time and trajectory of conservative solute particles advected in a three-dimensional heterogeneous aquifer under uniform in the mean flow conditions. Closure of the model is obtained by means of a consistent second-order expansion in σ_Y (standard deviation of the log hydraulic conductivity) of (statistical) moments of quantities of interest. As such, the results obtained are nominally limited to mildly non-uniform fields, with σ_Y < 1. Resulting mean and variance of particles travel time and trajectory are functions of first and second moments and cross-moments of trajectory and velocity components. Our solution is applicable to infinite domains and is free of distributional assumptions. As an important application of the methodology we obtain closed-form expressions for the unconditional mean and variance of travel time and particle trajectory for isotropic log-conductivity domain characterized by an exponential variogram. This allows us to recover the non linear behavior of mean travel time versus distance, in agreement with numerical results published in the literature, as well as a non-linear effect in the mean trajectory. The analysis of trajectory variance allows recovering some known results regarding transverse macro-dispersion, evidencing some limitations typical of perturbation theory.     

Impact of fractional probability distributions on statistics of hydraulic conductivity,dynamics of groundwater flow and solute transport at a low-permeability site

The fractional Brownian motion (fBm) and fractional Lévy motion (fLm) can easily describe the geometry and the statistical structure of hydraulic conductivity (K) for real-world. However, the fBm and fLm models have not been systematically evaluated when building the K field for a low-permeability site. In this study, both the fBm and fLm are used to simulate the low-K field at NingCheGu (NCG), Tianjin, China. Groundwater flow and solute transport are then computed using MODFLOW and MT3DMS, respectively, and the influence of the fBm/fLm models for K on groundwater flow and solute transport is discussed. Results show that the fLm fits better the statistics of the low-K medium than fBm, and the random logarithmic K (LnK) field generated by fLm is more stable because the resultant LnK field captures more of the measured properties at the field site than that generated by fBm. In contrast, the LnK generated by fBm is more likely to form both high-K channels and low-K barriers. The fBm therefore predicts more extreme behaviours in flow and transport, including the preferential flow, low-concentration blocks and solute retention. The overall groundwater renewal period and solute travel time for the fLm simulation are slightly shorter than those for fBm. The impacts of the fLm and fBm models on the statistics of the resultant LnK fields and the dynamics of groundwater flow and solute transport revealed by this study shed light on the selection and evaluation of the fractional probability distribution models in capturing the K fields for low-K media.     

Perspective on theories of non-Fickian transport in heterogeneous media

Subsurface fluid flow and solute transport take place in a multiscale heterogeneous environment. Neither these phenomena nor their host environment can be observed or described with certainty at all scales and locations of relevance. The resulting ambiguity has led to alternative conceptualizations of flow and transport and multiple ways of addressing their scale and space–time dependencies. We focus our attention on four approaches that give rise to nonlocal representations of advective and dispersive transport of nonreactive tracers in randomly heterogeneous porous or fractured continua. We compare these approaches theoretically on the basis of their underlying premises and the mathematical forms of the corresponding nonlocal advective–dispersive terms. One of the four approaches describes transport at some reference support scale by a classical (Fickian) advection–dispersion equation (ADE) in which velocity is a spatially (and possibly temporally) correlated random field. The randomness of the velocity, which is given by Darcy’s law, stems from random fluctuations in hydraulic conductivity (and advective porosity though this is often disregarded). Averaging the stochastic ADE over an ensemble of velocity fields results in a space–time-nonlocal representation of mean advective–dispersive flux, an approach we designate as stnADE. A closely related space–time-nonlocal representation of ensemble mean transport is obtained upon averaging the motion of solute particles through a random velocity field within a Lagrangian framework, an approach we designate stnL. The concept of continuous time random walk (CTRW) yields a representation of advective–dispersive flux that is nonlocal in time but local in space. Closely related to the latter are forms of ADE entailing fractional derivatives (fADE) which leads to representations of advective–dispersive flux that are nonlocal in space but local in time; nonlocality in time arises in the context of multirate mass transfer models, which we exclude from consideration in this paper. We describe briefly each of these four nonlocal approaches and offer a perspective on their differences, commonalities, and relative merits as analytical and predictive tools.     

Modified Mixed Lagrangian–Eulerian Method Based on Numerical Framework of MT3DMS on Cauchy Boundary

MT3DMS, a modular three‐dimensional multispecies transport model, has long been a popular model in the groundwater field for simulating solute transport in the saturated zone. However, the method of characteristics (MOC), modified MOC (MMOC), and hybrid MOC (HMOC) included in MT3DMS did not treat Cauchy boundary conditions in a straightforward or rigorous manner, from a mathematical point of view. The MOC, MMOC, and HMOC regard the Cauchy boundary as a source condition. For the source, MOC, MMOC, and HMOC calculate the Lagrangian concentration by setting it equal to the cell concentration at an old time level. However, the above calculation is an approximate method because it does not involve backward tracking in MMOC and HMOC or allow performing forward tracking at the source cell in MOC. To circumvent this problem, a new scheme is proposed that avoids direct calculation of the Lagrangian concentration on the Cauchy boundary. The proposed method combines the numerical formulations of two different schemes, the finite element method (FEM) and the Eulerian–Lagrangian method (ELM), into one global matrix equation. This study demonstrates the limitation of all MT3DMS schemes, including MOC, MMOC, HMOC, and a third‐order total‐variation‐diminishing (TVD) scheme under Cauchy boundary conditions. By contrast, the proposed method always shows good agreement with the exact solution, regardless of the flow conditions. Finally, the successful application of the proposed method sheds light on the possible flexibility and capability of the MT3DMS to deal with the mass transport problems of all flow regimes.     

Using time‐domain reflectometry to characterize shallow solute transport in an oak woodland hillslope in northern California,USA

The natural heterogeneity of water and solute movement in hillslope soils makes it difficult to accurately characterize the transport of surface‐applied pollutants without first gathering spatially distributed hydrological data. This study examined the application of time‐domain reflectometry (TDR) to measure solute transport in hillslopes. Three different plot designs were used to examine the transport of a conservative tracer in the first 50 cm of a moderately sloping soil. In the first plot, which was designed to examine spatial variability in vertical transport in a 1·2 m² plot, a single probe per meter was found to adequately characterize vertical solute travel times. In addition, a dye and excavation study in this plot revealed lateral preferential flow in small macropores and a transport pattern where solute is focused vertically into preferential flow pathways. The bypass flow delivers solute deeper in the soil, where lateral flow occurs. The second plot, designed to capture both vertical and lateral flow, provided additional evidence confirming the flow patterns identified in the excavation of the first plot. The third plot was designed to examine lateral flow and once again preferential flow of the tracer was observed. In one instance rapid solute transport in this plot was estimated to occur in as little as 3% of the available pore space. Finally, it was demonstrated that the soil anisotropy, although partially responsible for lateral subsurface transport, may also homogenize the transport response across the hillslope by decreasing vertical solute spreading. Copyright © 2002 John Wiley & Sons, Ltd.     

Modeling Transport in Transient Ground-Water Flow: An Unacknowledged Approximation

Abstract. During unsteady or transient ground-water flow, the fluid mass per unit volume of aquifer changes as the potentiometric head changes, and solute transport is affected by this change in fluid storage. Three widely applied numerical models of two-dimensional transport partially account for the effects of transient flow by removing terms corresponding to the fluid continuity equation from the transport equation, resulting in a simpler governing equation. However, fluid-storage terms remaining in the transport equation that change during transient flow are, in certain cases, held constant in time in these models. For the case of increasing heads, this approximation, which is unacknowledged in these models'documentation, leads to transport velocities that are too high, and increased concentration at fluid and solute sources. If heads are dropping in time, computed transport velocities are too low. Using parameters that somewhat exaggerate the effects of this approximation, an example numerical simulation indicates solute travel time error of about 14 percent but only minor errors due to incorrect dilution volume. For horizontal flow and transport models that assume fluid density is constant, the product of porosity and aquifer thickness changes in time: initial porosity times initial thickness plus the change in head times the storage coefficient. This formula reduces to the saturated thickness in unconfined aquifers if porosity is assumed to be constant and equal to specific yield. The computational cost of this more accurate representation is insignificant and is easily incorporated in numerical models of solute transport.     

Modeling Variably Saturated Subsurface Solute Transport with MODFLOW‐UZF and MT3DMS

The MT3DMS groundwater solute transport model was modified to simulate solute transport in the unsaturated zone by incorporating the unsaturated‐zone flow (UZF1) package developed for MODFLOW. The modified MT3DMS code uses a volume‐averaged approach in which Lagrangian‐based UZF1 fluid fluxes and storage changes are mapped onto a fixed grid. Referred to as UZF‐MT3DMS, the linked model was tested against published benchmarks solved analytically as well as against other published codes, most frequently the U.S. Geological Survey's Variably‐Saturated Two‐Dimensional Flow and Transport Model. Results from a suite of test cases demonstrate that the modified code accurately simulates solute advection, dispersion, and reaction in the unsaturated zone. Two‐ and three‐dimensional simulations also were investigated to ensure unsaturated‐saturated zone interaction was simulated correctly. Because the UZF1 solution is analytical, large‐scale flow and transport investigations can be performed free from the computational and data burdens required by numerical solutions to Richards' equation. Results demonstrate that significant simulation runtime savings can be achieved with UZF‐MT3DMS, an important development when hundreds or thousands of model runs are required during parameter estimation and uncertainty analysis. Three‐dimensional variably saturated flow and transport simulations revealed UZF‐MT3DMS to have runtimes that are less than one tenth of the time required by models that rely on Richards' equation. Given its accuracy and efficiency, and the wide‐spread use of both MODFLOW and MT3DMS, the added capability of unsaturated‐zone transport in this familiar modeling framework stands to benefit a broad user‐ship.     

Backward probabilistic model of groundwater contamination in non-uniform and transient flow

Backward location and travel time probabilities, which provide information about the former location of contamination in an aquifer, can be used to identify unknown contamination sources. Backward location probability describes the possible upgradient positions of contamination at a known time in the past, and backward travel time probability describes the time required for contamination to travel from a known upgradient location to an observation point. These probabilities are related to adjoint states of resident concentration, and their governing equation is the adjoint of a forward contaminant transport model. Using adjoint theory to obtain the appropriate governing equation, we extend the backward probability model for conservative solutes to more general non-uniform and transient flow fields. In particular, we address three important extensions, spatially-varying porosity, transient flow and temporally-varying porosity, and internal distributed sources and sinks of solute and water. For the first time we learn that forward and backward location and travel time probabilities are not necessarily equivalent to adjoint states, but are related to them. The extensions are illustrated using a vertically-integrated groundwater model, creating transient flow by a step change in pumping and using areal recharge as an internal distributed source. Both the movement and spread of probabilities are affected. With internal sources of water, there are two interpretations of backward probability, depending on whether or not the source of water is also a source of solute. The results demonstrate how the backward probability model can be applied to other, perhaps more important, non-uniform and transient flow conditions, with time- and space-varying water storage, such as time-varying pumping or unsaturated (or saturated–unsaturated) flow and transport with spatially- and temporally-varying moisture content.     

Numerical method of moments for solute transport in a nonstationary flow field conditioned on hydraulic conductivity and head measurements

The unconditional stochastic studies on groundwater flow and solute transport in a nonstationary conductivity field show that the standard deviations of the hydraulic head and solute flux are very large in comparison with their mean values (Zhang et al. in Water Resour Res 36:2107–2120, 2000; Wu et al. in J Hydrol 275:208–228, 2003; Hu et al. in Adv Water Resour 26:513–531, 2003). In this study, we develop a numerical method of moments conditioning on measurements of hydraulic conductivity and head to reduce the variances of the head and the solute flux. A Lagrangian perturbation method is applied to develop the framework for solute transport in a nonstationary flow field. Since analytically derived moments equations are too complicated to solve analytically, a numerical finite difference method is implemented to obtain the solutions. Instead of using an unconditional conductivity field as an input to calculate groundwater velocity, we combine a geostatistical method and a method of moment for flow to conditionally simulate the distributions of head and velocity based on the measurements of hydraulic conductivity and head at some points. The developed theory is applied in several case studies to investigate the influences of the measurements of hydraulic conductivity and/or the hydraulic head on the variances of the predictive head and the solute flux in nonstationary flow fields. The study results show that the conditional calculation will significantly reduce the head variance. Since the hydraulic head measurement points are treated as the interior boundary (Dirichlet boundary) conditions, conditioning on both the hydraulic conductivity and the head measurements is much better than conditioning only on conductivity measurements for reduction of head variance. However, for solute flux, variance reduction by the conditional study is not so significant.     

Simulation of field‐scale water flow and bromide transport in a cracked clay soil

Single domain models may seriously underestimate leaching of nutrients and pesticides to groundwater in clay soils with shrinkage cracks. Various two‐domain models have been developed, either empirical or physically based, which take into account the effects of cracks on water flow and solute transport. This paper presents a model concept that uses the clay shrinkage characteristics to derive crack volume and crack depth under transient field conditions. The concept has been developed to simulate field average behaviour of a field with cracks, rather than flow and transport at a small plot. Water flow and solute transport are described with basic physics, which allow process and scenario analysis. The model concept is part of the more general agrohydrological model SWAP, and is applied to a field experiment on a cracked clay soil, at which water flow and bromide transport were measured during 572 days. A single domain model was not able to mimic the field‐average water flow and solute transport. Incorporation of the crack concept considerably improved the simulation of water content and bromide leaching to the groundwater. Still deviations existed between the measured and simulated bromide concentration profiles. The model did not reproduce the observed bromide retardation in the top layer and the high bromide dispersion resulting from water infiltration at various soil depths. A sensitivity analysis showed that the amounts of bromide leached were especially sensitive to the saturated hydraulic conductivity of the top layer, the solute transfer from the soil matrix to crack water flow and the mean residence time of rapid drainage. The shrinkage characteristic and the soil hydraulic properties of the clay matrix showed a low sensitivity. Copyright © 2000 John Wiley & Sons, Ltd.