Longitudinal Macrodispersivity Research Articles

Stochastic Inversion of a Tomographic Pumping Test: Identifying Conductivity Horizontal Correlation and Longitudinal Macrodispersivity

AbstractTwo important properties for modeling flow and transport in heterogeneous aquifers are the horizontal integral scale (Ih) of hydraulic logconductivity (Y = ln K) and the longitudinal macrodispersivity (αL). Estimating the former generally requires K measurements in many wells, which are typically not available. In this work, we present a method for estimating Ih and αL from hydraulic tomography (HT) pumping tests. The approach is based on stochastic inversion, assuming head and K are stationary random space functions. Head variance and mean head gradient are calculated from measured head data at large distances from the pumping well and for large times from the start of pumping. These are then used in a theoretical formula for head variance in uniform mean flow to obtain Ih and subsequently αL, requiring only previous knowledge of the vertical integral scale (Iv) and the logconductivity variance . The method is applied to data from an HT pumping test at the Boise Hydrogeophysical Research site and results for Ih and αL are in the range previously reported in the literature. Particularly promising results are found for αL, which is seen to be robust, with relatively little variation for different values of Iv and .

Evidence Based Estimation of Macrodispersivity for Groundwater Transport Applications.

The scope of this work is to discuss the proper choice of macrodispersion coefficients in modeling contaminant transport through the advection dispersion equation (ADE). It is common to model solute concentrations in transport by groundwater with the aid of the ADE. Spreading is quantified by macrodispersivity coefficients, which are much larger than the laboratory observed pore-scale dispersivities. In the frame of stochastic theory, longitudinal macrodispersivity is related to the hydraulic conductivity spatial variability via its statistical moments (mean, variance, integral scales), which are generally determined by geostatistical analysis of field measurements. In many cases, especially for preliminary assessment of contaminant spreading, these data are not available and ad hoc values are adopted by practitioners. The present study aims at recommending dispersivity values based on a thorough analysis of tens of field experiments. Aquifers are classified as of weak, medium, and high heterogeneity and for each class a range of macrodispersivity values is recommended. Much less data are available for the transverse macrodispersivities, which are significantly smaller than the longitudinal one. Nevertheless, a few realistic values based on field data, are recommended for applications. Transport models using macrodispersivities can predict mean concentrations, different from the local ones. They can be used for estimation of robust measures, like plumes spatial moments, longitudinal mass distribution and breakthrough curves at control planes.

Seeing macro-dispersivity from hydraulic conductivity field with convolutional neural network

Building a quantitative relation between the spatial heterogeneity of the hydraulic conductivity fields and the macroscale behavior of solute transport is fundamental for groundwater environment problem. In this work, the deep learning technique is explored to build the functional mapping between the hydraulic conductivity field and the longitudinal macro-dispersivity. We examine the capability of the deep neural network in estimating macro-dispersivities of conductivity fields with different variances. The universality of the trained deep neural network is investigated. Comparisons of the neural network results and the reference values (macro-dispersivities from transport simulation) suggest the promising potential of deep learning technique in porous media with moderate heterogeneity. For a given size of training datasets, the deep neural network produces better macro-dispersivity estimation for the conductivity field with smaller variance. The trained neural network by conductivity fields with larger variance has stronger universality for macro-dispersivity estimation. This study demonstrates that deep neural network can be an effective alternative for estimating macroscale behavior of solute transport by directly interpreting hydraulic conductivity fields.

Effects of local transverse dispersion on macro-scale coefficients of oxygen-limited biodegradation in a stratified formation

The correct characterization of macro-scale contaminant transport and transformation rates is an important issue for modeling reactive transport in heterogeneous aquifers. While previous studies have investigated field-scale heterogeneity of transport and biochemical properties, the effects of local transverse dispersion on macro-scale transport and transformation rates have not been well understood. In this paper, the process of oxygen-limited biodegradation in a stratified aquifer is analysed by spectral perturbation approach, and longitudinal macrodispersivity, effective biodegradation rate, effective retardation factor and effective velocity are derived for the coupled transport equations of a system consisting of a contaminant and an oxidizing agent (oxygen). The effects of local transverse dispersion on these macro-scale coefficients are studied. It is shown that local transverse dispersion can smooth the heterogeneity in biodegradation and sorption processes and enlarge effective biodegradation rate and retardation factor. The local transverse dispersion can also limit the effects of heterogeneity in biodegradation process on longitudinal macrodispersivities and effective velocities for the contaminant and dissolved oxygen. But the effects of heterogeneity in sorption process on the contaminant macrodispersivity is likely to be magnified by local transverse dispersion.

Ensemble and effective dispersion in three-dimensional isotropic fractal media

We determine the time-dependent behavior of the dispersion coefficient for transport in formations with isotropic log-conductivity fields showing fractal behavior.We consider two different dispersion coefficients for point-like injection: (1) the ensemble dispersion coefficients, defined as half the rate of change of the second central moments of the ensemble-averaged concentration distribution and (2) the effective dispersion, which is half the rate of change of the expected second central moments. Our results show, that the two longitudinal macrodispersion coefficients steadily grow with time and remain different at all times in a fully fractal regime, indicating that no Fickian transport regime is ever reached. The resulting effective longitudinal transport model is consequently a fractional advection–dispersion equation. In the semi-fractal regime, a Gaussian transport regime is reached eventually. However, compared to the case of a classic non-fractal regime, the transient non-Gaussian regime lasts much longer. In the transverse direction, the two dispersion coefficients approach the same large-time limit also in fractal media highlighting the fundamental difference between longitudinal and transverse dispersion.

Effects of Local Transverse Dispersion on Macro-scale Coefficients of Decaying Solute Transport in a Stratified Formation

Estimating the values of dispersion and biochemical reaction rates of heterogeneous aquifers is critical to predicting the temporal evolution and fate of reactive solutes. While previous studies have investigated field-scale heterogeneity of transport and biochemical properties of porous media, effects of local dispersion have not been well understood. In this paper, longitudinal macro-dispersivity, effective decay rate, and effective solute velocity are derived for a stratified aquifer, and the effects of local dispersion, especially the local transverse dispersion, are studied. It is shown that the inclusion of local transverse dispersion leads to enlarged effective decay rate, and that ignoring it may significantly underestimate the effective rate. The Damkohler (Da) number and the coefficient of variation (CV) of decay rate have slight influence to macro-coefficients under very small Pe number (with large local transverse dispersion). However, Da number has growing effect on the asymptotic effective decay rate with the decrease in Pe number, and results in constant asymptotic values regardless of Da number under the condition with very large Pe number. Larger CV of decay rate leads to smaller effective decay rate and effective velocity, and longitudinal macro-coefficient. The longitudinal macro-dispersivity is found to depend on the correlation between the hydraulic conductivity and the decay rate if the local longitudinal dispersion is spatially variable.

The Boundaries Influence in Trending Hydraulic Conductivity Fields

We analyze the impact of the boundary conditions on flow and transport solutions for the case of a two-dimensional heterogeneous medium with a linear trend in the mean log-conductivity. The influence of the boundaries is analyzed for different values of the domain size and aspect ratio, and different directions of the trend respect to the mean flow direction. The analysis involving a steady, spatially nonstationary velocity field, is developed by using the stochastic finite element method. Results of this study show that the impact of the trend in finite domains may disagree with that obtained in unbounded domains and it is more complex. The trend alters the result obtained for the medium with stationary conductivity, leading to a larger or smaller value of the macrodispersion coefficients, depending on the size and aspect ratio of the domain, and this effect may vary with the travel distance from the solute source. Moreover in limited domains here analyzed also for particles starting at distance of several integral scales from the boundaries and traveling in the middle of the domain, the behavior of macrodispersion coefficients is affected by the boundary conditions and the influence of the trend on longitudinal and transverse macrodispersion coefficients may vary with the imposed constraints.

Simulation of Hydraulic Heterogeneity and Upscaling Permeability and Dispersivity in Sandy-Clay Formations

Since 1963, radioactive waste has been injected in deep artesian aquifers of Cretaceous terrigenous deposits in western Siberia. It is well known that geologic heterogeneity strongly affects contaminant transport in unconsolidated formations. To predict the long-term migration of this radioactive waste the effective hydraulic and macrodispersion parameters are estimated by using a three-dimensional high-resolution hydraulic heterogeneity model. The heterogeneous model of the injection area is developed by applying transition probability geostatistics. This model is used to simulate local steady-state groundwater flow and advective transport, leading to numerical estimates of effective hydraulic conductivity and macrodispersion parameters. Mean seepage velocities and effective longitudinal macrodispersion are calculated from observed breakthrough curves for a conservative tracer. Results show that mean horizontal lengths exceed vertical lengths by a factor of more than 30. As a result, vertical effective hydraulic conductivity is two orders of magnitude less than the horizontal effective conductivity. Observed breakthrough curves exhibit long tails and appear to be non-Fickian. Estimated effective longitudinal macrodispersivity in the vertical direction is one order of magnitude less than that in the horizontal direction. Under a Fickian framework, this implies that dispersion modeling for regional transport simulations requires an anisotropic-media dispersion model.

How accurate is predictive modeling of groundwater transport? A case study of advection, macrodispersion, and diffusive mass transfer at the Forsmark site (Sweden)

[1] We study predictive modeling of groundwater transport that accounts for three mechanisms: mean advection, macrodispersion, and mass transfer. A general methodology is presented and applied to a series of nonsorbing tracer tests along multiple pathways on scales ranging from ca. 70 to 300 m, in a highly heterogeneous aquifer at Forsmark (Sweden). The mean water residence time cannot be predicted well using a simple water balance model. Longitudinal macrodispersivity λL (L) and a mass transfer parameter group ψ(1/T) are extrapolated from the control tracer experiments, to yield accurate predictions of tracer discharge, once the mean water residence time is constrained. A relatively simple modeling framework based on Fickian macrodispersion and diffusion seems to be adequate for reproducing the tracer discharge in this complex and highly heterogeneous porous media.

Numerical assessment of 3D macrodispersion in heterogeneous porous media

Hydrodynamic dispersion is a key controlling factor of solute transport in heterogeneous porous media. It critically depends on dimensionality. The asymptotic macrodispersion, transverse to the mean velocity direction, vanishes only in 2D and not in 3D. Using the classical Gaussian correlated permeability fields with a lognormal distribution of variance σY 2 , the longitudinal and transverse dispersivities are determined numerically as a function of heterogeneity and dimensionality. We show that the transverse macrodispersion steeply increases with σY 2 underlying the essential role of flow lines braiding, a mechanism specific to 3D systems. The transverse macrodispersion remains however at least two orders of magnitude smaller than the longitudinal macrodispersion, which increases even more steeply with σY 2 . At moderate to high levels of heterogeneity, the transverse macrodispersion also converges much faster to its asymptotic regime than do the longitudinal macrodispersion. Braiding cannot be thus taken as the sole mechanism responsible for the high longitudinal macrodispersions. It could be either supplemented or superseded by stronger velocity correlations in 3D than in 2D. This assumption is supported by the much larger longitudinal macrodispersions obtained in 3D than in 2D, up to a factor of 7 for σY 2 = 7.56.

Gravity-driven fingers in fractures: Experimental study and dispersion analysis by moment method for a point-source injection

In this study, we investigate the behavior of a dense contaminant injected from a point-source in a fracture. Our experimental model is a transparent Hele-Shaw cell, 0.5mm of aperture. A saline solution is injected locally representing the point-source pollution. A Laser Induced Fluorescence (LIF) method provides concentration measurement of the pollution plume. Two propagation patterns have been observed: one and two-finger plumes. If the upper part of the plume is stable over time regardless of the second configuration, the moment when the plume separates into two fingers is highly dependent on both injection flow-rate and contaminant concentration. To further investigate the dispersion process inside the fracture, experimental results are interpreted by the spatial and time moment methods. Resulting dispersivities and plume propagation mean velocity are compared to theoretical values derived from a modified Taylor-Aris dispersion tensor. The longitudinal macro-dispersion obtained suggests an asymptotical behavior of the plume spread regardless of the studied configurations. Experimental local dispersivities derived from time and space moments proved to be close at large times to theoretical values predicted by the density-dependent dispersion tensor (Oltéan et al., 2004). Based on those observations the mechanism behind the separation of the plume into two fingers is believed to be significantly impacted by the pre-asymptotic behavior of the dispersion tensor.

Asymptotic dispersion for two‐dimensional highly heterogeneous permeability fields under temporally fluctuating flow

Temporal fluctuations of water flux have been investigated as a mechanism that strongly enhances transverse dispersion in heterogeneous media. Unfortunately, most results have been obtained by linear stochastic theories on permeability fields of limited variability. Worse, results are inconsistent regarding the impact of fluctuations on longitudinal dispersion, which motivates our work to find the effect of temporal velocity fluctuations on macrodispersion. We perform numerical Monte Carlo simulations for highly variable permeability fields of up to 800 correlation lengths. We find that fluctuations longitudinal to the main flow direction hardly modify macrodispersion because they do not alter the flow lines. Fluctuations transverse to the main flow direction not only increase transverse dispersion, which is well known, but also reduce the longitudinal macrodispersion in a significant and consistent way, which contradicts previous findings. The reduction of the longitudinal dispersion is comparable to the increase of transverse dispersion. Most surprisingly, for high heterogeneity, temporal fluctuations cause total (longitudinal plus transverse) macrodispersion to drop with respect to the steady state one. Enhancement of the transverse macrodispersion comes from both the increase of the transverse velocity variability and Lagrangian correlation. Reduction of the longitudinal macrodispersion results from the reduction of the Lagrangian correlation of the longitudinal velocity. That is, temporal fluctuations reduce longitudinal spreading both by breaking the fastest velocity paths on the plume front and by letting solute bypass the low‐permeability zones that tend to block or trap the solute in steady state flow conditions.

Influence of temporal fluctuations and spatial heterogeneity on pollution transport in porous media

The combined influence of temporal fluctuations and spatial heterogeneity on non-reactive solute transport mechanisms in porous media can be understood by performing simulations of steady and unsteady flow and transport in heterogeneous media. The study focuses on issues such as the degree of heterogeneity, correlation length, separation of the combined effects of temporal and spatial variations, and ergodicity conditions under unsteady flow conditions. It is shown that the effect of temporal variations on solute transport is masked by the strong effect of spatial heterogeneity. There is no obvious difference in plume shape between steady and unsteady flow conditions; the first and the second spatial moments of the plume of the unsteady-state flow condition fluctuate around the steady-state flow condition with the same period of oscillations as the input signal at small storage coefficient (S ≤ 0.001). At a relatively high standard deviation in hydraulic conductivity and a small storage coefficient, the unsteady flow condition sharpens the temporal variations in macrodispersion coefficients. The magnitude of the longitudinal macrodispersion coefficient under unsteady flow condition is almost doubled at the maximum values. However, the transverse macrodispersion coefficient fluctuates around zero. The Kubo number and Peclet number ranges are 1.2–64 and 10–250, respectively.

Effect of permeability variations on solute transport in highly heterogeneous porous media

The effect of aquifer heterogeneity on flow and solute transport in two-dimensional isotropic porous media was analyzed using the Monte Carlo method. The two-dimensional logarithmic permeability (ln K) was assumed to be a non-stationary random field with its increments being a truncated fractional Lévy motion (fLm). The permeability fields were generated using the modified successive random additions (SRA) algorithm code SRA3DC [ http://www.iamg.org/CGEditor/index.htm]. The velocity and concentration fields were computed respectively for two-dimensional flow and transport with a pulse input using the finite difference codes of MODFLOW 2000 and MT3DMS. Two fLm control parameters, namely the width parameter ( C) and the Lévy index ( α), were varied systematically to examine their effect on the resulting permeability, flow velocity and concentration fields. We also computed the first- and second-spatial moments, the dilution index, as well as the breakthrough curves at different control planes with the corresponding concentration fields. In addition, the derived breakthrough curves were fitted using the continuous time random walk (CTRW) and the traditional advection-dispersion equation (ADE). Results indicated that larger C and smaller α both led to more heterogeneous permeability and velocity fields. The Lévy-stable distribution of increments in ln K resulted in a Lévy-stable distribution of increments in logarithm of the velocity (ln v). Both larger C and smaller α created sharper leading edges and wider tailing edges of solute plumes. Furthermore, a relatively larger amount of solute still remained in the domain after a relatively longer time transport for smaller α values. The dilution indices were smaller than unity and increased as C increased and α decreased. The solute plume and its second-spatial moments increased as C increased and α decreased, while the first-spatial moments of the solute plume were independent of C and α values. The longitudinal macrodispersivity was scale-dependent and increased as a power law function of time. Increasing C and decreasing α both resulted in an increase in longitudinal macrodispersivity. The transport in such highly heterogeneous media was slightly non-Gaussian with its derived breakthrough curves being slightly better fitted by the CTRW than the ADE, especially in the early arrivals and late-time tails.

Numerical Monte Carlo analysis of the influence of pore‐scale dispersion on macrodispersion in 2‐D heterogeneous porous media

We investigate the influences of pore‐scale dispersion and of larger‐scale permeability heterogeneities on the macrodispersion without the molecular diffusion. Permeability follows a lognormal exponentially correlated distribution characterized by its correlation length λ and its lognormal variance σ2. Macrodispersion is evaluated numerically by using parallel simulations on grids of characteristic size ranging from 200λ to 1600λ. We note αL and αT the pore‐scale longitudinal and transversal dispersivities. For αL/λ < 10−2 and αT/λ < 10−3, the influence of pore‐scale dispersion on the macrodispersion is smaller than 5% of the macrodispersion due only to permeability heterogeneities. Larger dispersivities (αL/λ ≥ 10−2 or αT/λ ≥ 10−3) induce larger effects than those obtained by the semianalytical expression of Salandin and Fiorotto (1998) for σ2 > 1. The effects of local dispersion on the longitudinal macrodispersion remain limited to 25% at most of the macrodispersion due only to permeability heterogeneities. For σ2 > 1, isotropic local dispersion induces a reduction of the longitudinal macrodispersion, whereas anisotropic local dispersion lets it increase. The longitudinal and transverse local dispersions induce opposite effects on the longitudinal macrodispersion, which are respectively an increase and a reduction. The transverse macrodispersion null without local dispersion or molecular diffusion becomes strictly positive with local dispersion. Because of the velocity field heterogeneities, it is amplified by a factor of 2 to 50 from the grid scale to the macro scale. The transverse dispersion is triggered by both longitudinal and transverse local dispersions. A reduction of a factor of 2 of the transverse local dispersion at fixed longitudinal local dispersion yields only a reduction of a factor of 4 at most of the transverse macrodispersion for σ2 ≥ 2.25.

Characteristics and controlling factors of dispersion in bounded heterogeneous porous media

In this study, we present first‐order (in terms of the log conductivity variance) analytical solutions to displacement covariances Xij and macrodispersion coefficients Dij for transport of conservative solutes in two‐dimensional, bounded heterogeneous porous media. These solutions are presented as infinity series and are explicit functions of the statistics of the log hydraulic conductivity. Using numerical examples, the convergence of these solutions in terms of the number of terms in truncated finite summations is first investigated, and the accuracy of these solutions is examined by comparing them with results from Monte Carlo simulations and the semianalytical solutions of Osnes (1998). Through several examples, the general features of time‐dependent Xij and Dij are discussed. Unlike in unbounded domains, the longitudinal macrodispersivity D11 for bounded domains does not approach an asymptotic value at large dimensionless time but instead increases quickly near the downstream constant head boundary. In addition, D11 for bounded domains is always larger than that in unbounded domains, and accordingly, the transverse macrodispersivity D22 in bounded domains is smaller than that in unbounded domains. These differences stem from the lateral no‐flow boundaries in our bounded domain models. We also investigated the effect of domain sizes on Xij and Dij. Our study shows that both Xij and Dij depend not only on the dimensionless size of the domain but also on its aspect ratio. The dependence of Xij and Dij on the particles' initial location has also been investigated at detail. Our results indicate that while this dependence is very small for X11 and D11, the release location has a significant impact on both X22 and D22. Our solutions to both displacement covariances and macrodispersivity are compared against those derived from tracer test data at the Borden site. The comparison shows that our solutions are quite close to observed data, indicating that they may be applicable to predict solute transport at the field scale.

An indirect assessment on the impact of connectivity of conductivity classes upon longitudinal asymptotic macrodispersivity

Solute transport takes place in heterogeneous porous formations, with the log conductivity, Y = ln K, modeled as a stationary random space function of given univariate normal probability density function (pdf) with mean 〈Y〉, variance σY2, and integral scale IY. For weak heterogeneity, the above mentioned quantities completely define the first‐order approximation of the longitudinal macrodispersivity αL = σY2IY. However, in highly heterogeneous formations, nonlinear effects which depend on the multipoint joint pdf of Y, impact αL. Most of the past numerical simulations assumed a multivariate normal distribution (MVN) of Y values. The main aim of this study is to investigate the impact of deviations from the MVN structure upon αL. This is achieved by using the concept of spatial correlations of different Y classes, the latter being defined as the space domain where Y falls in the generic interval [Y,Y + ΔY]. The latter is characterized by a length scale λ(Y), reflecting the degree of connectivity of the domain (the concept is similar to the indicator variograms). We consider both “symmetrical” and “non‐symmetrical” structures, for which λ(Y′) = λ(−Y′) (similar to the MVN), and λ(Y′) ≠ λ(−Y′), respectively, where Y′ = Y − 〈Y〉. For example, large Y zones may have high spatial correlation, while low Y zones are poorly correlated, or vice versa. The impact of λ(Y) on αL is investigated by adopting a structure model which has been used in the past in order to investigate flow and transport in highly heterogeneous media. It is found that the increased correlation in the low conductive zones with respect to the high ones generally leads to a significant increase in αL, for the same global IY. The finding is explained by the solute retention occurring in low Y zones, which has a larger effect on solute spreading than high Y zones. Conversely, αL decreases when the high conductivity zones are more correlated than the low Y ones. Dispersivity is less affected by the shape of λ(Y) for symmetrical distributions. It is found that the range of validity of the first‐order dispersivity, i.e., αL = IYσY2, narrows down for non‐symmetrical structures.

Heterogeneity in hydraulic conductivity and its role on the macroscale transport of a solute plume: From measurements to a practical application of stochastic flow and transport theory

The spatial variability of hydraulic conductivity in a shallow unconfined aquifer located at North Bay, Ontario, composed of glacial‐lacustrine and glacial‐fluvial sands, is examined in exceptional detail and characterized geostatistically. A total of 1878 permeameter measurements were performed at 0.05 m vertical intervals along cores taken from 20 boreholes along two intersecting transect lines. Simultaneous three‐dimensional (3‐D) fitting of Ln(K) variogram data to an exponential model yielded geostatistical parameters for the estimation of bulk hydraulic conductivity and solute dispersion parameters. The analysis revealed a Ln(K) variance equal to about 2.0 and 3‐D anisotropy of the correlation structure of the heterogeneity (λ1, λ2, and λ3 equal to 17.19, 7.39, and 1.0 m, respectively). Effective values of the hydraulic conductivity tensor and the value of the longitudinal macrodispersivity were calculated using the theoretical expressions of Gelhar and Axness (1983). The magnitude of the longitudinal macrodispersivity is reasonably consistent with the observed degree of longitudinal dispersion of the landfill plume along the principal path of migration. Variably saturated 3‐D flow modeling using the statistically derived effective hydraulic conductivity tensor allowed a reasonably close prediction of the measured water table and the observed heads at various depths in an array of piezometers. Concomitant transport modeling using the calculated longitudinal macrodispersivity reasonably predicted the extent and migration rates of the observed contaminant plume that was monitored using a network of multilevel samplers over a period of about 5 years. It was further demonstrated that the length of the plume is relatively insensitive to the value of the longitudinal macrodispersivity under the conditions of a steady flow in 3‐D and constant source strength. This study demonstrates that the use of statistically derived parameters based on stochastic theories results in reliable large‐scale 3‐D flow and transport models for complex hydrogeological systems. This is in agreement with the conclusions reached by Sudicky (1986) at the site of an elaborate tracer test conducted in the aquifer at the Canadian Forces Base Borden.

The impact of local diffusion on longitudinal macrodispersivity and its major effect upon anomalous transport in highly heterogeneous aquifers

Flow and transport are solved for a heterogeneous medium modeled as an ensemble of spherical inclusions of uniform radius R and of conductivities K, drawn from a pdf f ( K ) ( Fig. 1). This can be regarded as a particular discretization scheme, allowing for accurate numerical and semi-analytical solutions, for any given univariate f ( Y ) ( Y = ln K ) and integral scale I Y . The transport is quantified by the longitudinal equivalent macrodispersivity α Leq , for uniform mean flow of velocity U and for a large (ergodic) plume of a conservative solute injected in a vertical plane ( x = 0 ) and moving past a control plane at x ≫ I Y . In the past we have solved transport for advection solely for highly heterogeneous media of σ Y 2 ⩽ 8 . We have found that α Leq increases in a strong nonlinear fashion with σ Y 2 and transport becomes anomalous for the subordinate model. This effect is explained by the large residence time of solute particles in inclusions of low K. In the present work we examine the impact of local diffusion as quantified by the Peclet number Pe = UI Y / D 0 , where D 0 is the coefficient of molecular diffusion. Transport with diffusion is solved by accurate numerical simulations for flow past spheres of low K and for high Pe = O ( 10 2 - 10 4 ) . It was found that finite Pe reduces significantly α Leq as compared to advection, for σ Y 2 ≳ 3 ( Pe = 1000 ) and for σ Y 2 ≳ 1.4 ( Pe = 100 ) , justifying neglection of the effect of diffusion for weak to moderately heterogeneous aquifers (e.g. σ Y 2 ⩽ 1 ) . In contrast, diffusion impacts considerably α Leq for large σ Y 2 due to the removal of solute from low K inclusions. Furthermore, anomalous behavior is eliminated, though α Leq may be still large for Pe ≫ 1 .

Spatial moments analysis of kinetically sorbing solutes in aquifer with bimodal permeability distribution

We analyze the influence of the formation structure upon transport of nonreactive and kinetically sorbing solutes in heterogeneous porous formations. Formation structure is represented through a bimodal distribution of hydraulic conductivity with two scales of spatial variability, which is a simplified, yet realistic, model of spatial variability. Kinetic sorption is modeled using a linear reversible rate expression. We derive new closed‐form solutions of the second ergodic spatial moments of a kinetically sorbing solute. Although these solutions are obtained through a perturbation expansion in term of logconductivity variance, they are not limited to any particular model of spatial variability. We show that the formation structure, as described by the bimodal model, has a significant impact on plume moments of a non‐reactive tracer at moderate contrast between mean hydraulic conductivity of the two hydrofacies, when the two scales of variability have comparable weight. On the contrary, the impact diminishes when bimodality weakens at both low and high contrast. As expected, the large time asymptotic longitudinal macrodispersion coefficient is influenced by the structure only through the variance and integral scale of log‐conductivity, and thus the result for the bimodal model is identical to that of a unimodal Gaussian field sharing the same second‐order statistics. The impact of kinetic sorption depends upon the plume travel time since injection. At early times, sorption kinetics tends to accentuate the difference between the longitudinal macrodispersivity in bimodal and unimodal fields, while at later times the difference is minimal because overall macrodispersivity is dominated by the spreading caused by nonequilibrium sorption. In a bimodal formation transverse macrodispersivity peaks later and declines to zero slower than in a Gaussian unimodal field with the same second‐order statistics. These results demonstrate that even a very simplified facies model, such as the bimodal model relying on a statistical distribution of high‐ and low‐conductivity zones, has important impacts upon global parameters describing the migration of nonreactive and reactive tracers. Consequently, larger effects are expected for quantities such as early arrival times and BTC tailing which are more directly influenced by the spatial patterns of high hydraulic conductivity than global parameters, such as macrodispersivity. Furthermore, these solutions contribute to bridge the gap between stochastic modeling and applications, since bimodal fields can be obtained to honor soft data typically available in applications, for example through conditional simulations using stratigraphic information from well logs.