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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.

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., QC 20230613

This paper employs stochastic analysis to investigate the combined effect of temporal and spatial variability on the temporal variance of baseflow in large catchments. The study makes use of the well-known aggregated reservoir model, representing the catchment as a network of parallel linear reservoirs. Each reservoir models a sub-catchment as an independent unit whose discharge temporal variation is characterized by a response time. By treating the rainfall-generated recharge and the sub-catchment response times as random variables, the statistical temporal moments of total baseflow are quantified. Comparisons are made between the temporal variance of baseflow in the aggregated reservoir model and that of a single homogeneous reservoir to define an upscaled response time. The analysis of the statistical moments of the random baseflow reveals that the number of reservoirs N has a weak impact on baseflow variance, with ergodic conditions achieved even with a small number of reservoirs. The study highlights that the ratio between the recharge correlation time and the geometric mean of the sub-catchment response times plays a critical role in baseflow damping and the upscaled response. The results indicate that the dynamics of baseflow generation depend not only on the catchment hydro-geological structure but also on the variability of the input signal. This research underscores the importance of understanding the combined influences of hydro-geological factors and recharge input variability for baseflow prediction under uncertainty. The present study should be regarded as a first step, setting the theoretical framework for future research toward incorporating field data.

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.

Six conceptually different transport models were applied to the MADE-1 field tracer experiment as a first major attempt for model comparison. The objective was to show that complex mass distributions in heterogeneous aquifers can be predicted without calibration of transport parameters - solely making use of structural and flow data. The models differ in their conceptualization of the heterogeneous aquifer structure, computational complexity, and use of conductivity data obtained from various observation methods (Direct Push Injection Logging - DPIL, Grain Size Analysis, Pumping Tests and Flowmeter). They share the same underlying physical transport process of advection by the velocity field solely. Predictive capability is assessed by comparing results to observed longitudinal mass distributions of the MADE-1 experiment. The decreasing mass recovery of the observed plume is attributed to sampling and no physical process like mass transfer is invoked by the models. Measures like peak location and strength are used in comparing the modeled and measured plume mass distribution. Comparison of models reveals that the predictions of the solute plume agree reasonably well with observations if the models are underlain by a few parameters of close values: mean velocity, a parameter reflecting log-conductivity variability and a horizontal length scale related to conductivity spatial correlation. The robustness of the results implies that conservative transport models with appropriate conductivity upscaling strategies of various observation data provide reasonable predictions of plumes longitudinal mass distribution as long as key features are taken into account.

Transverse dispersion, or tracer spreading orthogonal to the mean flow direction, which is relevant e.g, for quantifying bio‐degradation of contaminant plumes or mixing of reactive solutes, has been studied in the literature less than the longitudinal one. Inferring transverse dispersion coefficients from field experiments is a difficult and error‐prone task, requiring a spatial resolution of solute plumes which is not easily achievable in applications. In absence of field data, it is a questionable common practice to set transverse dispersivities as a fraction of the longitudinal one, with the ratio 1/10 being the most prevalent. We collected estimates of field‐scale transverse dispersivities from existing publications and explored possible scale relationships as guidance criteria for applications. Our investigation showed that a large number of estimates available in the literature are of low reliability and should be discarded from further analysis. The remaining reliable estimates are formation‐specific, span three orders of magnitude and do not show any clear scale‐dependence on the plume traveled distance. The ratios with the longitudinal dispersivity are also site specific and vary widely. The reliability of transverse dispersivities depends significantly on the type of field experiment and method of data analysis. In applications where transverse dispersion plays a significant role, inference of transverse dispersivities should be part of site characterization with the transverse dispersivity estimated as an independent parameter rather than related heuristically to longitudinal dispersivity.

The driving mechanism of tracer transport in aquifers is groundwater flow which is controlled by the heterogeneity of hydraulic properties. We show how hydrodynamics and mass transfer are coupled in a general analytical manner to derive a physically-based (or process-based) residence time distribution for a given integral scale of the hydraulic conductivity; the result can be applied for a broad class of linear mass transfer processes. The derived tracer residence time distribution is a transfer function with parameters to be inferred from combined field and laboratory measurements. It is scalable relative to the correlation length and applicable for an arbitrary statistical distribution of the hydraulic conductivity. Based on the derived residence time distribution, the coefficient of variation and skewness of residence time are illustrated assuming a log-normal hydraulic conductivity field and first-order mass transfer. We show that for a low Damkohler number the coefficient of variation is more strongly influenced by mass transfer than by heterogeneity, whereas skewness is more strongly influenced by heterogeneity., QC 20161118

The emergence of stochastic subsurface hydrology stemmed from the realization that the random spatial variability of aquifer properties has a profound impact on solute transport. The last four decades witnessed a tremendous expansion of the discipline, many fundamental processes and principal mechanisms being identified. However, the research findings have not impacted significantly the application in practice, for several reasons which are discussed. The paper discusses the current status of stochastic subsurface hydrology, the relevance of the scientific results for applications and it also provides a perspective to a few possible future directions. In particular, we discuss how the transfer of knowledge can be facilitated by identifying clear goals for characterization and modeling application, relying on recent recent advances in research in these areas., QC 20170210

We analyze the probability distribution of the hazard attenuation factor for a noncarcinogenic reactive compound captured by a well in heterogeneous porous formations. The hazard attenuation factor is defined as the ratio between the hazard index HI at a detection well and at the source. Heterogeneity of the aquifer is represented through the multi-indicator model (a collection of blocks of independent permeability) while flow and transport are solved by the means of the self-consistent approach that is able to deal with any degree of heterogeneity. Due to formation heterogeneity, HI is a random variable and similar for hazard attenuation index. The latter can be fully characterized by its cumulative distribution function (CDF), which in turn can be related to the statistics of the travel time of solute particles, from the source to the detection well. The approach is applied to the case of a solute which undergoes decay and a well with a screen much smaller than the correlation scale of hydraulic conductivity. The results show that the probability of exceeding a given acceptable threshold of the hazard index is significantly affected by the level of heterogeneity comparable to the one observed for the MADE site, and the distance between the source and the well., QC 20161124

QC 20160816

The emergence of stochastic subsurface hydrology stemmed from the realization that the random spatial variability of aquifer properties has a profound impact on solute transport. The last four decades witnessed a tremendous expansion of the discipline, many fundamental processes and principal mechanisms being identified. However, the research findings have not impacted significantly the application in practice, for several reasons which are discussed. The paper discusses the current status of stochastic subsurface hydrology, the relevance of the scientific results for applications and it also provides a perspective to a few possible future directions. In particular, we discuss how the transfer of knowledge can be facilitated by identifying clear goals for characterization and modeling application, relying on recent recent advances in research in these areas.

No abstract is available for this article.

The article presents a few recent developments advanced by the authors in a few key areas of stochastic modeling of solute transport in heterogeneous aquifers. First, a brief review of the Lagrangian approach to modeling plumes longitudinal mass distribution and temporal (the breakthrough curve) mass arrival, is presented. Subsequently, transport in highly heterogeneous aquifers is analyzed by using a recently developed predictive model. It relates the non-Gaussian BTC to the permeability univariate pdf and integral scale, with application to the MADE field observations. Next, the approach is extended to transport of reactive solute, combinnig the effects of the random velocity field and multirate mass transfer on the BTC, with application to mass attenuation. The following topic is modeling of the local concentration field as affected by mixing and dilution due to pore scale dispersion. The results are applied to the analysis of concentration measurements at the Cape Cod field experiment. The last section incorporates the results of the preceding ones in health risk assessment by analyzing the impact of concentration prediction on risk uncertainty. It is illustrated by assessing the effect of identification of macrodispersivity from field characterization and transport modeling, upon the probability of health risk., QC 20151117

Spreading of conservative solutes in groundwater due to aquifer heterogeneity is quantified by the macrodispersivity, which was found to be scale dependent. It increases with travel distance, stabilizing eventually at a constant value. However, the question of its asymptotic behavior at very large scale is still a matter of debate. It was surmised in the literature that macrodispersivity scales up following a unique scaling law. Attempts to define such a law were made by fitting a regression line in the log-log representation of an ensemble of macrodispersivities from multiple experiments. The functional relationships differ among the authors, based on the choice of data. Our study revisits the data basis, used for inferring unique scaling, through a detailed analysis of literature marcodispersivities. In addition, values were collected from the most recent tracer tests reported in the literature. We specified a system of criteria for reliability and reevaluated the reliability of the reported values. The final collection of reliable estimates of macrodispersivity does not support a unique scaling law relationship. On the contrary, our results indicate, that the field data can be explained as a collection of macrodispersivities of aquifers with varying degree of heterogeneity where each exhibits its own constant asymptotic value. Our investigation concludes that transport, and particularly the macrodispersivity, is formation-specific, and that modeling of transport cannot be relegated to a unique scaling law. Instead, transport requires characterization of aquifer properties, e.g., spatial distribution of hydraulic conductivity, and the use of adequate models., QC 20151118

We examine the prediction capability of two approximate models (Multi-Rate Mass Transfer (MRMT) and Continuous Time Random Walk (CTRW)) of non-Fickian transport, by comparison with accurate 2-D and 3-D numerical simulations. Both nonlocal in time approaches circumvent the need to solve the flow and transport equations by using proxy models to advection, providing the breakthrough curves (BTC) at control planes at any x, depending on a vector of five unknown parameters. Although underlain by different mechanisms, the two models have an identical structure in the Laplace Transform domain and have the Markovian property of independent transitions. We show that also the numerical BTCs enjoy the Markovian property. Following the procedure recommended in the literature, along a practitioner perspective, we first calibrate the parameters values by a best fit with the numerical BTC at a control plane at x(1), close to the injection plane, and subsequently use it for prediction at further control planes for a few values of sigma(2)(gamma) <= 8. Due to a similar structure and Markovian property, the two methods perform equally well in matching the numerical BTC. The identified parameters are generally not unique, making their identification somewhat arbitrary. The inverse Gaussian model and the recently developed Multi-Indicator Model (MIM), which does not require any fitting as it relates the BTC to the permeability structure, are also discussed. The application of the proxy models for prediction requires carrying out transport field tests of large plumes for a long duration., QC 20160216

Detailed information on the spatial structure of hydraulic conductivity (K) is important for understanding and predicting groundwater flow and transport. Direct-push injection logging (DPIL) is a promising technology for rapid measurement of K in unconsolidated formations. This technology was used to gain information on the highly heterogeneous aquifer at the Lauswiesen test site in Germany. Using a large body of DPIL and direct-push slug testing measurements, we characterize the structure of K on scales not previously possible. Two new applications of DPIL are put forward: (1) use of raw DPIL measurements of relative conductivity K to characterize the spatial distribution of K and (2) transformation of K measurements to K values based on their statistical moments. The DPIL results are compatible to those obtained using more conventional methodologies. The main achievement of the methodology is the possibility to delineate deterministic aquifer subunits as well as the identification of the statistical parameters of the log conductivity for each subunit. In particular, the horizontal integral scale I, a parameter affecting solute transport, is difficult and costly to identify using other approaches. Nevertheless, further studies are needed to clarify questions on low K measurements and the nature of the relationship between K and K.

We analyse the mass arrival (breakthrough curve) at control planes at × of a plume of conservative solute injected at time t = 0 in the plane × = 0. The formation is of random three-dimensional stationary and isotropic conductivity K, characterized by the univariate normal distribution f(Y), Y = lnK, and the integral scale I. The flow is uniform in the mean, of velocity U, and longitudinal transport is quantified by f(z,x), the probability density function (pdf) of travel time r at x. We characterize transport by an equivalent longitudinal macrodispersivity αL(x), which is proportional to the variance of the travel time. If αL is constant, transport is coined as Fickian, while it is anomalous if αL increases indefinitely with x. If f(z,x) is normal (for × I), transport is coined as Gaussian and the mean concentration satisfies an ADE with constant coefficients. For the subordinate structural model transport is anomalous, in spite of the closeness of the conductivity distribution to the lognormal one. To further analyse anomalous behaviour, a relationship is established between the shape of f(K) for K→0 and the behaviour of αL, arriving at criteria for normal or anomalous transport. The model is used in order to compare results with the recent ones presented in the literature, which are based on the Continuous Time Random Walk (CTRW) approach. It is found that a class of anomalous transport cases proposed by CTRW methodology cannot be supported by a conductivity structure of finite integral scale., QC 20141103

A contaminant plume of mass Mo is inserted at time t = 0 at an injection plane at × = 0 in an aquifer of spatially variable conductivity K. The log-conductivity Y = InK is modelled as stationary and isotropic, of univariate distribution f(Y), and of finite integral scale I. The flow of water is uniform in the mean (natural gradient) and the plume is of large transverse extent relative to the integral scale. Advective transport and longitudinal spread are quantified by the solute mass arrival ("breakthrough curve", BTC) M(t,x) at a control plane at × > I. For a large plume (ergodic conditions) the relative mass flux μ(t,x) = (l/Mo)M/t is approximately equal to the probability density function of travel times of solute particles f(τx) and the latter is used to analyse transport. f(τx) is derived by adopting a structural model of the aquifer that contains spherical or cubic inclusions of uniform size and of independent Y that fill the space. Such a structure can represent any formation of given f(Y) and I. The flow and transport solutions are obtained by a simple semianalytical model and by accurate numerical simulations. The travel time distribution at few control planes is determined for a log-normal f(K) first. Under the assumption of weak heterogeneity, i.e. for small variance σy 2 and for x»I, the travel time distribution is symmetrical and Gaussian. Subsequently, by using the semi-analytical model and numerical simulations we derive f(τx) for a highly heterogeneous formation of σ y 2 = 2. The main finding is f(τx) is highly skewed due to the presence of a thin, but long tail, for large travel times. The tail is of significance to applications that deal with aquifer pollution and remediation. The tail is related to the large residence time of solute particles in blocks of low conductivity. A simple relationship is established between the tail of f(Y) for low K and that f(τx) for large τ. To further examine the impact of the log-conductivity distribution on, QC 20141031

[1] We consider steady groundwater flow of uniform mean in aquifers of random, spatially variable, hydraulic conductivity. Analytical expressions for the statistical moments of mass fluxes of sorbing solutes in presence of pore-scale dispersion are derived, where the reactive solutes undergo first-order sorption kinetics. The developments which lead to the analytical formulation of the solute flux are rigorous in the first-order analysis framework, and results obtained are valid for weakly heterogeneous formations. The methodology is exemplified for a two-dimensional aquifer, assuming that the source is of small transverse extent compared to the heterogeneity length scales. The examples show that pore-scale dispersion has a relatively small effect on the mean point flux, whereas the point flux variance shows much larger sensitivity to pore-scale dispersion. The variance first decreases as the reaction rate departs from the nonreactive limit, but for equilibrium reactions it is of the same order as for nonreactive solutes. The effect of averaging the solute flux over a finite sampling area is also investigated. It is found that for the expected area-averaged flux the mixing effect induced by sampling tends to supersede that caused by pore-scale dispersion. On the contrary, pore-scale dispersion may have a strong effect on the flux variance also when sampling effects are taken into account., QC 20100525

Special approximations are needed in order to analyze the free surface flow in the vicinity of the bow of blunt ships. In the case of thin ships the inner bow flow reduces to a two-dimensional flow in a vertical plane normal to the bow. The situation at the confluence of a blunt bow and the free surface is clarified first. It is shown that there are two possible angles between a free surface and a rigid wall at the stagnation point: (1) if the wall is inclined with respect to the horizontal at an angle larger than 120 deg, the free surface intersects the wall at 120 deg; (2) if the wall is inclined at less than 120 deg, the free surface is horizontal. The latter case is usual for ships.

The two-dimensional free-surface flow generated by a singularity moving with near-critical speed (i.e. with Froude number referred to the water depth near unity) is solved by using the method of matched asymptotic expansions. In the vicinity of the singularity the problem is solved by an infinitesimal wave expansion (inner expansion) while at large distances from the singularity shallow water theory provides the proper solution (outer expansion). A composite expansion provides a uniformly-valid solution for the velocity components, the free-surface profile and the drag force. These two basic approaches of water-wave theory are shown to be solutions of the same flow problem, but valid in different regions. Although the inner expansion satisfies linear equations, the solution depends nonlinearly on the small parameter of the problem (the singularity strength).

The free-surface gravity flow past a cylinder is solved by inner and outer expansions in the potential plane. The linearized solution, based on an expansion in epsilon (the ratio between the radius and the depth of immersion) while F (the depth Froude number) is kept of order one, is subsequently expanded in a small F series. The second-order, nonlinear, terms are computed under this double expansion. The analysis is extended to expansions in the physical plane and to the case of a schematical thin body. The free-wave amplitude and wave resistance asymptotic sequence converges for a thickness Froude number larger than unity. (Author)

The breaking wave in front of blunt bow displacement ships causes most of the wave resistance of such ships. A theoretical analysis of the two-dimensional gravity free-surface flow past a blunt body of semi-infinite length, representing a first approximation of the breaking wave, is presented. The jet model was selected for the representation of the bow momentum loss. The equation of flow are solved by a high draft Froude number expansion. The velocity distribution, the jet thickness and the associated drag are computed by the aid of inner and outer expansions. The results are compared with Baba's (1969) towing-tank tests. (Author)

The gravity flow past a cylinder moving close to a free-surface at high Fr is investigated by the method of matched asymptotic expansions. The inner flow model is that of a nonseparated non-linear gravity-free flow past a doublet. (Author)

The equations of steady free-surface flow with gravity past a ship are expanded in a small Froude number perturbation series. Expressions for the vertical force and the moment (and the sinkage and trim, correspondingly) are derived up to second order. Two examples of flow are solved to first order: a two-dimensional flow past a Rankine half-body in shallow water and the three-dimensional flow past a slender body of revolution. (Author)