Reactive Solute Transport Research Articles

Theory for reactive solute transport through clay membrane barriers

The theoretical development for one-dimensional, coupled migration of solutes with different ionic mobilities through clay soils that behave as ion-restrictive membranes, referred to as clay membrane barriers (CMBs), is presented. The transport formulation is based on principles of irreversible thermodynamics and accounts explicitly for coupling effects of hyperfiltration (ultrafiltration) and chemico-osmotic counter-advection associated with clay membrane behavior in the absence of electrical current. Since, by definition, no solute can enter a “perfect” or “ideal” membrane, the concept of an implicit coupling effect, such that the effective salt-diffusion coefficient, D s* approaches zero as the chemico-osmotic efficiency coefficient, ω approaches unity is introduced. The theoretical development also illustrates that, even in the absence of membrane behavior, traditional advective–dispersive transport theory based on a constant value of D s* for the solutes may not be appropriate for simulating transient transport in reactive (ion exchanging) systems. This potential limitation is illustrated through simulations for solute mass flux involving the migration of a binary salt solution (KCl) through a clay barrier with exchange sites saturated with a single exchangeable cation (e.g., Na +) that enters the pore solution upon ion exchange with the salt cation (K +).

Lithium sorption to Yucca Mountain tuffs

The Li ion has been used as a reactive tracer in field tests performed in the saturated and unsaturated-zone in volcanic tuffs at Yucca Mountain, Nevada. Lithium sorbs weakly by cation exchange and permits field-scale testing of laboratory-based predictions of reactive-solute transport. A series of laboratory studies show that Li sorption is nonlinear and varies with lithology in the different horizons of the tuff. In particular, both Li sorption and Li-specific cation-exchange capacity vary as functions of tuff mineralogy, and can be predicted given quantitative X-ray diffraction data. These results indicate that Li sorption is dominated by clay and zeolite minerals, and that sorption coefficients can be predicted given mineralogic analysis results.

Cesium and Niobium transport through poorly cemented sandstone from Krasnoyarsk-26 (Russian Federation): From Batch to Transport experiments

ABSTRACTWe have studied the sorption of Cs and Nb onto poorly cemented sandstone (drillcore from – 250 m depth) from the radioactive waste disposal site at Krasnoyarsk-26 (Russian Federation). Results were evaluated in terms of KD-values and Langmuir isotherms. These values have been used to model the transport of Cs and Nb through a sandstone column using the RETRASO code (REactive TRAnsport of Solutes), which is able to simulate simultaneously groundwater flow, heat transport and multicomponent reactive solute. Good agreement between experimental data and modeling has been achieved for both elements. Cs transport has been modeled using rapid sorption equilibrium, while for Nb a kinetically controlled transport has been considered. The results obtained in this work have been used to predict the 1D transport of Cs and Nb taking into account some parameters of the hydrogeology of Krasnoyarsk-26.

Stochastic Analysis of Reactive Solute Transport in Heterogeneous, Fractured Porous Media: A Dual-Permeability Approach

A nonlocal, first-order, Eulerian stochastic theory is developed for reactive chemical transport in a heterogeneous, fractured porous medium. A dual-permeability model is adopted to describe the flow and transport in the medium, where the solute convection and dispersion in the matrix are considered. The chemical is under linear nonequilibrium sorption and first-order degradation. The hydraulic conductivities, sorption coefficients, degradation rates in both fracture and matrix regions, and interregional mass transfer coefficient are all assumed to be random variables. The resultant theory for mean concentrations in both regions is nonlocal in space and time. Under spatial Fourier and temporal Laplace transforms, the mean concentrations are explicitly expressed. The transformed results are then numerically inverted to the real space via Fast Fourier Transform method. The theory developed in this study generalizes the stochastic studies for a reactive chemical transport in a one-domain flow field (Hu et al., 1997a) and in a mobile/immobile flow field (Huang and Hu, 2001). In comparison with one-domain transport, the dual-permeability model predicts a larger second moment in the longitudinal direction, but smaller one in the transverse direction. In addition, various simplification assumptions have been made based on the general solution. The validity of these assumptions has been tested via the spatial moments of the mean concentration in both fracture and matrix regions.

Reactive transport modeling of processes controlling the distribution and natural attenuation of phenolic compounds in a deep sandstone aquifer

Reactive solute transport modeling was utilized to evaluate the potential for natural attenuation of a contaminant plume containing phenolic compounds at a chemical producer in the West Midlands, UK. The reactive transport simulations consider microbially mediated biodegradation of the phenolic compounds (phenols, cresols, and xylenols) by multiple electron acceptors. Inorganic reactions including hydrolysis, aqueous complexation, dissolution of primary minerals, formation of secondary mineral phases, and ion exchange are considered. One-dimensional (1D) and three-dimensional (3D) simulations were conducted. Mass balance calculations indicate that biodegradation in the saturated zone has degraded approximately 1–5% of the organic contaminant plume over a time period of 47 years. Simulations indicate that denitrification is the most significant degradation process, accounting for approximately 50% of the organic contaminant removal, followed by sulfate reduction and fermentation reactions, each contributing 15–20%. Aerobic respiration accounts for less than 10% of the observed contaminant removal in the saturated zone. Although concentrations of Fe(III) and Mn(IV) mineral phases are high in the aquifer sediment, reductive dissolution is limited, producing only 5% of the observed mass loss. Mass balance calculations suggest that no more than 20–25% of the observed total inorganic carbon (TIC) was generated from biodegradation reactions in the saturated zone. Simulations indicate that aerobic biodegradation in the unsaturated zone, before the contaminant entered the aquifer, may have produced the majority of the TIC observed in the plume. Because long-term degradation is limited to processes within the saturated zone, use of observed TIC concentrations to predict the future natural attenuation may overestimate contaminant degradation by a factor of 4–5.

Evaluating the sensitivity of a subsurface multicomponent reactive transport model with respect to transport and reaction parameters

The input variables for a numerical model of reactive solute transport in groundwater include both transport parameters, such as hydraulic conductivity and infiltration, and reaction parameters that describe the important chemical and biological processes in the system. These parameters are subject to uncertainty due to measurement error and due to the spatial variability of properties in the subsurface environment. This paper compares the relative effects of uncertainty in the transport and reaction parameters on the results of a solute transport model. This question is addressed by comparing the magnitudes of the local sensitivity coefficients for transport and reaction parameters. General sensitivity equations are presented for transport parameters, reaction parameters, and the initial (background) concentrations in the problem domain. Parameter sensitivity coefficients are then calculated for an example problem in which uranium(VI) hydrolysis species are transported through a two-dimensional domain with a spatially variable pattern of surface complexation sites. In this example, the reaction model includes equilibrium speciation reactions and mass transfer-limited non-electrostatic surface complexation reactions. The set of parameters to which the model is most sensitive includes the initial concentration of one of the surface sites, the formation constant ( K f) of one of the surface complexes and the hydraulic conductivity within the reactive zone. For this example problem, the sensitivity analysis demonstrates that transport and reaction parameters are equally important in terms of how their variability affects the model results.

Biodegradation during contaminant transport in porous media: 3. Apparent condition-dependency of growth-related coefficients

The biodegradation of organic contaminants in the subsurface has become a major focus of attention, in part, due to the tremendous interest in applying in situ biodegradation and natural attenuation approaches for site remediation. The biodegradation and transport of contaminants is influenced by a combination of microbial and physicochemical properties and processes. The purpose of this paper is to investigate the impact of hydrodynamic residence time, substrate concentration, and growth-related factors on the simulation of contaminant biodegradation and transport, with a specific focus on potentially condition-dependent growth coefficients. Two sets of data from miscible-displacement experiments, performed with different residence times and initial solute concentrations, were simulated using a transport model that includes biodegradation described by the Monod nonlinear equations and which incorporates microbial growth and oxygen limitation. Two variations of the model were used, one wherein metabolic lag and cell transport are explicitly accounted for, and one wherein they are not. The magnitude of the maximum specific growth rates obtained from calibration of the column-experiment results using the simpler model exhibits dependency on pore-water velocity and initial substrate concentration ( C 0) for most cases. Specifically, the magnitude of μ m generally increases with increasing pore-water velocity for a specific C 0, and increases with decreasing C 0 for a specific pore-water velocity. Conversely, use of the model wherein observed lag and cell elution are explicitly accounted for produces growth coefficients that are similar, both to each other and to the batch-measured value. These results illustrate the potential condition-dependency of calibrated coefficients obtained from the use of models that do not account explicitly for all pertinent processes influencing transport of reactive solutes.

Modelling the impact of physical and chemical heterogeneity on solute leaching in pyritic overburden mine spoils

Spatial variability of physical and chemical properties may affect acidification and acid mine drainage in sulfide-bearing overburden mine spoils. This study compares heterogeneous with vertically layered and horizontally averaged spatial distributions of chemical components in a generic 2D-vertical spoil cross-section with heterogeneous distributions of water and air contents and of water flux densities. The initial spatial distributions of the fraction of sulfur and the pre-oxidized zones follow autocorrelated random functions. The initial background chemistry distribution is correlated to the degree of pre-oxidation using five classes of differing initial chemical systems. Processes considered include variably saturated water flow, oxygen diffusion, shrinking-core kinetics of pyrite oxidation, multicomponent reactive solute transport, and geochemical equilibrium reactions between aqueous and mineral components. Spatial distributions were generated using the SGSIM-GSLIB geostatistical simulation algorithm and estimated parameters. Results show differences in vertical concentration profiles of chemical components and in integrated breakthrough curves at 30 m depth between the heterogeneous and the layered case for both—high and low—sulfur content scenarios. Differences are relatively large in the beginning and with respect to the distribution of solid phase components. A heterogeneous sulfide mineral distribution results in vertical spreading of the oxidation front while precipitation and dissolution of the secondary minerals affects the acid mine drainage, by locally retarding and releasing the solutes.

Modeling Biogeochemical Processes in Leachate-Contaminated Soils: A Review

During subsurface transport, reactive solutes are subject to a variety of hydrological, physical and biochemical processes. The major hydrological and physical processes include advection, diffusion and hydrodynamic dispersion, and key biochemical processes are aqueous complexation, precipitation/dissolution, adsorption/desorption, microbial reactions, and redox transformations. The addition of strongly reduced landfill leachate to an aquifer may lead to the development of different redox environments depending on factors such as the redox capacities and reactivities of the reduced and oxidised compounds in the leachate and the aquifer. The prevailing redox environment is key to understanding the fate of pollutants in the aquifer. The local hydrogeologic conditions such as hydraulic conductivity, ion exchange capacity, and buffering capacity of the soil are also important in assessing the potential for groundwater pollution. Attenuating processes such as bacterial growth and metal precipitation, which alter soil characteristics, must be considered to correctly assess environmental impact. A multicomponent reactive solute transport model coupled to kinetic biodegradation and precipitation/dissolution model, and geochemical equilibrium model can be used to assess the impact of contaminants leaking from landfills on groundwater quality. The fluid flow model can also be coupled to the transport model to simulate the clogging of soils using a relationship between permeability and change in soil porosity. This paper discusses the different biogeochemical processes occurring in leachate-contaminated soils and the modeling of the transport and fate of organic and inorganic contaminants under such conditions.

Methods and Results of Numerical Modeling of Reactive Solute Transport in River Flow

The equation of advection–diffusion transport of a reactive solute in a stream is solved by different methods. Procedures for evaluating the parameters of a four-node finite-difference approximation used to numerically integrate this equation are proposed. Numerical experiments based on the developed numerical procedures, software, and models of solute transport allowed, in particular, the evaluation of the sensitivity of pollutant concentration estimates to variations in the parameters of interaction processes and hydrodynamic dispersion for a lowland river.

Modeling Transport of Protons and Calcium Ions in an Alginate Gel Bead System: The Effects of Physical Nonequilibrium and Nonlinear Competitive Sorption

Solute transport in heterogeneous soil systems is determined by a combination of physical and chemical processes. Heterogeneity can cause physical nonequilibrium, which results in asymmetrical breakthrough curves with early initial breakthrough and tailing. In the case of reactive transport, the shape of the breakthrough curve is also effected by the sorption isotherm and by multicomponent effects. The combined effects of physical nonequilibrium and nonlinear multicomponent chemistry are the subject of this study. We show how an experimental system composed of alginate gel beads, together with two-region transport modeling, can be used to study transport phenomena in reactive heterogeneous systems. Proton transport through the gel column with ion exchange of protons and calcium ions was measured and simulated. The results show the combined effects of nonlinear sorption and of competition, which add to the effects of physical nonequilibrium. The results emphasize the importance of a thorough understanding of both physical and chemical processes for a reliable prediction of reactive solute transport.

Nonideal transport of reactive solutes in heterogeneous porous media: 6. Microscopic and macroscopic approaches for incorporating heterogeneous rate‐limited mass transfer

Two major approaches have been used to incorporate heterogeneous rate‐limited mass transfer into mathematical models for solute transport. One focuses on processes operative at the microscopic scale and associated grain‐scale heterogeneity, while the other stresses the macroscopic variability of the medium and the field‐scale behavior of solute transport. In this paper, we examine the conceptual framework and model formulation of these two approaches in an attempt to evaluate potential commonality. Numerical solvers are developed for both sets of governing equations, and the performance of these two models is tested for two systems, each incorporating one of two types of mass transfer mechanisms. The results show that despite differences in conceptualization and formulation, the models produce comparable behavior for smaller‐scale systems. However, greater deviations are observed at larger scales. This suggests that caution should be exercised when using mathematical modeling for elucidating the specific processes that may be influencing reactive‐solute transport for a given system. We also evaluate the impact of microscopic‐scale mass transfer heterogeneity on field‐scale transport in systems for which hydraulic conductivity is spatially variable. The results show that inclusion of locally heterogeneous mass transfer does not appear to significantly influence the mean transport behavior for systems with field‐scale heterogeneity. However, it does appear to influence low‐concentration tailing. For simulations of reactive transport over extended distances, models with locally heterogeneous mass transfer may “preserve” the nonequilibrium effects associated with rate‐limited mass transfer better than models incorporating locally uniform mass transfer when both pore‐scale and field‐scale heterogeneity coexist.

Colloid-facilitated tracer transport by steady random ground-water flow

We study the transport of reactive solute in a three-phase system (water–solid matrix-colloids) in natural porous media. Semianalytical (integral) solutions are derived for the first time, which can be used for computing expected concentration, mass flux, or discharge for the dissolved as well as for colloid-bounded tracer. The results are based on a few simplifying assumptions: advection-dominated transport, linear mass transfer reactions, and steady-state colloidal concentration. Derived semianalytical expressions capture the main features of colloid-facilitated transport (the reversible-equilibrium and irreversible-kinetic sorption of tracers on colloids), and are applicable for the general class of linear sorption processes on the porous matrix. Derived solutions account for spatial variability of flow and sorption parameters, which is relevant for field-scale applications. We apply the theoretical results to the transport of neptunium and plutonium, using flow and transport data from the alluvial aquifer near Yucca Mountain, Nevada. Based on the zeroth and first temporal moment, dimensionless indicators are proposed for assessing the potential impact of colloid-facilitated tracer transport in aquifers. Generic sensitivity curves show the importance of tracer-colloid kinetic rates. Even very low irreversible rates (which will generally be difficult to determine in the laboratory) may yield observable effects for sufficiently long transport times. The obtained results can be used for assessing the significance of colloid-facilitated tracer transport under field conditions, as well as for setting further constraints on relevant parameters which need to be estimated in the field.

Solute flux approach to transport through spatially nonstationary flow in porous media

A theoretical framework for solute flux through spatially nonstationary flows in porous media is presented. The flow nonstationarity may stem from medium nonstationarity (e.g., the presence of distinct geological layers, zones, or facies), finite domain boundaries, and/or fluid pumping and injecting. This work provides an approach for studying solute transport in multiscale media, where random heterogeneities exist at some small scale while deterministic geological structures and patterns can be prescribed at some larger scale. In such a flow field the solute flux depends on solute travel time and transverse displacement at a fixed control plane. The solute flux statistics (mean and variance) are derived using the Lagrangian framework and are expressed in terms of the probability density functions (PDFs) of the particle travel time and transverse displacement. These PDFs are given with the statistical moments derived based on nonstationary Eulerian velocity moments. The general approach is illustrated with some examples of conservative and reactive solute transport in stationary and nonstationary flow fields. It is found based on these examples that medium nonstationarities (or multiscale structures and heterogeneities) have a strong impact on predicting solute flux across a control plane and on the corresponding prediction uncertainty. In particular, the behavior of solute flux moments strongly depends on the configuration of nonstationary medium features and the source dimension and location. The developed nonstationary approach may result in non‐Gaussian (multiple modal) yet realistic behaviors for solute flux moments in the presence of flow nonstationarities, while these non‐Gaussian behaviors may not be reproduced with a traditional stationary approach.

Comparison of split-operator methods for solving coupled chemical non-equilibrium reaction/groundwater transport models

Numerical models of reactive solute transport in groundwater are often solved using an approximate two-step approach that separates the transport and reaction processes. Even though the transport and reaction steps are split, using a novel formulation it is possible to pose the reaction step in general form as a system of ordinary differential equations (ODEs) such that the original, fully coupled model is solved exactly. The reaction step in the ‘standard’ two-step method is shown to be a special case of this general form. In this paper, these two approaches are compared in terms of accuracy and efficiency. The general and the standard ODEs representing the reaction steps are solved using the public domain ODE solver, LSODI. The ODEs representing the standard reaction step are also solved by the fourth-order Runge–Kutta (RK) method. Both the RK method and LSODI are capable of solving the system of ODEs in the standard two-step method. The RK method is found to be the most efficient even though it requires comparatively smaller time steps to yield accurate solutions. The LSODI solution of the general ODEs representing the reaction step was found to be extremely time consuming without any significant gain in accuracy.

SHETRAN: Distributed River Basin Flow and Transport Modeling System

Physically based spatially distributed (PBSD) river basin models have been available for over 10 years. One of their strengths lies in the way the surface and subsurface are represented as coupled parts of a whole, giving ground-water flows that are controlled by such factors as realistic surface saturation and infiltration, and surface conditions that are controlled by realistic groundwater levels, discharges, and so forth. PBSD sed- iment and solute transport models can be integrated into PBSD river basin modeling systems, and the integrated systems are powerful tools for studying the environmental impacts associated with land erosion, pollution, and the effects of changes in land use and climate, and also in studying surface-water and ground-water resources and their management. SHETRAN takes PBSD river basin modeling a step further, in that multifraction sediment transport and multiple, reactive solute transport are handled within a single system, fully coupled to water flow, and the subsurface is modeled as a fully 3D variably saturated heterogeneous medium. SHETRAN therefore has a substantial capability for addressing environmental and water resources problems that span the traditional disciplines of river basin and ground-water modeling.

Characterizing multiple timescales of stream and storage zone interaction that affect solute fate and transport in streams

The fate of contaminants in streams and rivers is affected by exchange and biogeochemical transformation in slowly moving or stagnant flow zones that interact with rapid flow in the main channel. In a typical stream, there are multiple types of slowly moving flow zones in which exchange and transformation occur, such as stagnant or recirculating surface water as well as subsurface hyporheic zones. However, most investigators use transport models with just a single storage zone in their modeling studies, which assumes that the effects of multiple storage zones can be lumped together. Our study addressed the following question: Can a single‐storage zone model reliably characterize the effects of physical retention and biogeochemical reactions in multiple storage zones? We extended an existing stream transport model with a single storage zone to include a second storage zone. With the extended model we generated 500 data sets representing transport of nonreactive and reactive solutes in stream systems that have two different types of storage zones with variable hydrologic conditions. The one storage zone model was tested by optimizing the lumped storage parameters to achieve a best fit for each of the generated data sets. Multiple storage processes were categorized as possessing I, additive; II, competitive; or III, dominant storage zone characteristics. The classification was based on the goodness of fit of generated data sets, the degree of similarity in mean retention time of the two storage zones, and the relative distributions of exchange flux and storage capacity between the two storage zones. For most cases (>90%) the one storage zone model described either the effect of the sum of multiple storage processes (category I) or the dominant storage process (category III). Failure of the one storage zone model occurred mainly for category II, that is, when one of the storage zones had a much longer mean retention time (ts ratio > 5.0) and when the dominance of storage capacity and exchange flux occurred in different storage zones. We also used the one storage zone model to estimate a “single” lumped rate constant representing the net removal of a solute by biogeochemical reactions in multiple storage zones. For most cases the lumped rate constant that was optimized by one storage zone modeling estimated the flux‐weighted rate constant for multiple storage zones. Our results explain how the relative hydrologic properties of multiple storage zones (retention time, storage capacity, exchange flux, and biogeochemical reaction rate constant) affect the reliability of lumped parameters determined by a one storage zone transport model. We conclude that stream transport models with a single storage compartment will in most cases reliably characterize the dominant physical processes of solute retention and biogeochemical reactions in streams with multiple storage zones.

Mathematical modeling of leachate routing in a leachate recirculating landfill

Full-scale application of leachate recirculation has been hampered by an inability to uniformly apply leachate to the waste mass. Much of this problem is due to the varied nature of the disposed waste which promotes preferential routing and short-circuiting of leachate flows. The United States Geological Survey’s Saturated-Unsaturated Flow and Transport (SUTRA) model [Voss C. I. (1984) SUTRA, Saturated–Unsaturated Transport, A Finite-Element Simulation Model for Saturated–Unsaturated, Fluid-Density-Dependent Ground-Water Flow with Energy Transport or Chemically Reactive Single-Species Solute Transport, US Geological Survey, National Center, Reston, VA.] was used to model leachate application to homogeneous anisotropic and heterogeneous waste masses. The homogeneous anisotropic scenario was modeled using vertical hydraulic conductivities of 10 −3 and 10 −4 cm/s while the horizontal hydraulic conductivity was simulated as one order of magnitude higher than these values. The heterogeneous waste mass was simulated by applying statistical relationships to the local hydraulic conductivities. Normal, exponentially increasing, and exponentially decreasing probability density functions (PDFs) were used to model the frequency-hydraulic conductivity relationship. Median hydraulic conductivities of 10 −3 and 10 −4 cm/s with hydraulic conductivity ranges of 10 −1–10 −5 cm/s and 10 −2–10 −6 cm/s, respectively, were simulated for each probability distribution. Leachate was applied 8 h/day via a horizontal trench at a daily rate of 4 m 3/m of trench. Saturation iso-clines are used to illustrate the effect of the waste mass hydraulic parameters on leachate routing. The ability to develop predictions and evaluate a variety of different scenarios without the effort and expense of physical experimentation is a great strength of mathematical models. However, a model’s predictive capabilities must be verified by comparing the model results with field studies. In an attempt to verify model estimations, SUTRA was used to simulate the site recirculation system and operational procedures used at the Delaware Solid Waste Authority Test Cells and the Yolo County Leachate Recirculation Demonstration Project. Cumulative measured and simulated leachate generation were compared. Results from this verification effort indicate that channeled flow is the major leachate movement mechanism that is not well understood at this time.

A Simple Approach to Determine Reactive Solute Transport Using Time Domain Reflectometry

Time domain reflectometry (TDR) possesses potential for determining solute‐transport parameters, such as dispersion coefficients and retardation factors for reactive solutes. We developed a simple method based on peak‐to‐peak measurements of water and solute velocities through the soil using TDR. The method was tested by carrying out unsaturated leaching experiments in the laboratory on two soil columns packed with a South Pacific soil from Maré, which is a ferrasol with variable surface charge. One column was left bare and the other was planted with mustard. Pulses of CaBr2 and Ca(NO3)2 were applied to the surface of either wet or dry soil and then leached by water from a rainfall simulator applied at a steady rate of between 30 and 45 mm h−1 Water and solute transport were monitored by collecting the effluent. Contemporaneous in situ measurements of the water content and electrical conductivity were made using TDR. Transport parameters for the convection–dispersion equation, with a linear adsorption isotherm, were obtained from the flux concentration and the solute resident concentrations measured by TDR. Anion retardations between 1.2 and 1.7, and dispersivities between 1 and 9 mm, were found. Retardations also were calculated using our simple approach based on TDR‐measured water and solute front velocities. These used TDR measurements of soil water content and bulk soil electrical conductivity with time, and were similar to those obtained from the effluent. The agreement suggests TDR could be a valuable in situ technique for obtaining the parameters relating to reactive solute transport through soil.

Reactive solute transport in flow between a recharging and a pumping well in a heterogeneous aquifer

Steady flow between a fully penetrating recharging and pumping well (doublet) takes place in a heterogeneous aquifer. The spatially variable hydraulic conductivity is modeled as a lognormal stationary random function, of anisotropic two‐point covariance. The latter is characterized by the horizontal and vertical integral scales I and Iυ respectively. A tracer, or a reactive solute obeying first‐order kinetics, is injected as a pulse or continuously in the recharging well. Our aim is to determine the flux‐averaged concentration (the breakthrough curve) in the pumping well as a function of time and of the various parameters of the problem, i.e., σY2 (the logconductivity variance), l′ = l/I (l is the distance between wells), and e = Iυ/I (the anisotropy ratio). A simple solution of this difficult problem is achieved by adopting a few simplifying assumptions: (1) the wells are fully penetrating, of length much larger than Iυ and of radius rw much smaller than I, (2) a first‐order solution in σY2 of the flow and transport equations is sought, (3) the anisotropy ratio is small, say e < 0.2, and (4) neglect of the effect of pore‐scale dispersion. After determining the travel time τ mean and variance, the mean flux‐averaged concentration is found by assuming that τ is lognormal. In a homogeneous medium there is a large spreading of the solute signal in the pumping well owing to the variation of the travel time among the streamlines connecting the two wells. The effect of heterogeneity is similar to that of pore‐scale dispersion; that is, it leads to enhanced spreading and in particular to an early breakthrough. The solution has potential applications to aquifer tests and to evaluation of efficiency of remediation schemes and may serve as a benchmark for numerical models.