Gedeon Dagan

Solute transport in heterogeneous porous formations

Solute transport in porous formations is governed by the large-scale heterogeneity of hydraulic conductivity. The two typical lengthscales are the local one (of the order of metres) and the regional one (of the order of kilometres). The formation is modelled as a random fixed structure, to reflect the uncertainty of the space distribution of conductivity, which has a lognormal probability distribution function. A first-order perturbation approximation, valid for small log-conductivity variance, is used in order to derive closed-form expressions of the Eulerian velocity covariances for uniform average flow. The concentration expectation value is determined by using a similar approximation, and it satisfies a diffusion equation with time-dependent apparent dispersion coefficients. The longitudinal coefficients tend to constant values in both two- and three-dimensional flows only after the solute body has travelled a few tens of conductivity integral scales. This may be an exceedingly large distance in many applications for which the transient stage prevails. Comparison of theoretical results with recent field experimental data is quite satisfactory. The variance of the space-averaged concentration over a volume V may be quite large unless the lengthscale of the initial solute body or of V is large compared with the conductivity integral scale. This condition is bound to be obeyed for transport at the local scale, in which case the concentration may be assumed to satisfy the ergodic hypothesis. This is not generally the case at the regional scale, and the solute concentration is subjected to large uncertainty. The usefulness of the prediction of the concentration expectation value is then quite limited and the dispersion coefficients become meaningless. In the second part of the study, the influence of knowledge of the conductivity and head at a set of points upon transport is examined. The statistical moments of the velocity and concentration fields are computed for a subensemble of formations and for conditional probability distribution functions of conductivity and head, with measured values kept fixed at the set of measurement points. For conditional statistics the velocity is not stationary, and its mean and variance vary throughout the space, even if its unconditional mean and variance are constant. The main aim of the analysis is to examine the reduction of concentration coefficient of variation, i.e. of its uncertainty, by conditioning. It is shown that measurements of transmissivity on a grid of points can be effective in reducing concentration variance, provided that the distance between the points is smaller than two conductivity integral scales. Head conditioning has a lesser effect upon variance reduction.

A comparison of seven geostatistically based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow

This paper describes the first major attempt to compare seven different inverse approaches for identifying aquifer transmissivity. The ultimate objective was to determine which of several geostatistical inverse techniques is better suited for making probabilistic forecasts of the potential transport of solutes in an aquifer where spatial variability and uncertainty in hydrogeologic properties are significant. Seven geostatistical methods (fast Fourier transform (FF), fractal simulation (FS), linearized cokriging (LC), linearized semianalytical )LS), maximum likelihood (ML), pilot point (PP), and sequential self-calibration (SS)) were compared on four synthetic data sets. Each data set had specific features meeting (or not) classical assumptions about stationarity, amenability to a geostatistical description, etc. The comparison of the outcome of the methods is based on the prediction of travel times and travel paths taken by conservative solutes migrating in the aquifer for a distance of 5 km. Four of the methods, LS, ML, PP, and SS, were identified as being approximately equivalent for the specific problems considered. The magnitude of the variance of the transmissivity fields, which went as high as 10 times the generally accepted range for linearized approaches, was not a problem for the linearized methods when applied to stationary fields; that is, their inverse solutions and travel time predictions were as accurate as those of the nonlinear methods. Nonstationarity of the “true” transmissivity field, or the presence of “anomalies” such as high-permeability fracture zones was, however, more of a problem for the linearized methods. The importance of the proper selection of the semivariogram of the log10 (T) field (or the ability of the method to optimize this variogram iteratively) was found to have a significant impact on the accuracy and precision of the travel time predictions. Use of additional transient information from pumping tests did not result in major changes in the outcome. While the methods differ in their underlying theory, and the codes developed to implement the theories were limited to varying degrees, the most important factor for achieving a successful solution was the time and experience devoted by the user of the method.

Transport of kinetically sorbing solute by steady random velocity in heterogeneous porous formations

A Lagrangian framework is used for analysing reactive solute transport by a steady random velocity field, which is associated with flow through a heterogeneous porous formation. The reaction considered is kinetically controlled sorption–desorption. Transport is quantified by the expected values of spatial and temporal moments that are derived as functions of the non-reactive moments and a distribution function which characterizes sorption kinetics. Thus the results of this study generalize the previously obtained results for transport of non-reactive solutes in heterogeneous formations (Dagan 1984; Dagan et al. 1992). The results are illustrated for first-order linear sorption reactions. The general effect of sorption is to retard the solute movement. For short time, the transport process coincides with a non-reactive case, whereas for large time sorption is in equilibrium and solute is simply retarded by a factor R = 1+ K d , where K d is the partitioning coefficient. Within these limits, the interaction between the heterogeniety and kinetics yields characteristic nonlinearities in the first three spatial moments. Asymmetry in the spatial solute distribution is a typical kinetic effect. Critical parameters that control sorptive transport asymptotically are the ratio e r between a typical reaction length and the longitudinal effective (non-reactive) dispersivity, and K d . The asymptotic effective dispersivity for equilibrium conditions is derived as a function of parameters e r and K d . A qualitative agreement with field data is illustrated for the zero- and first-order spatial moments.

A solute flux approach to transport in heterogeneous formations: 1. The general framework

It is common to represent solute tranport in heterogeneous formations in terms of the resident concentration C(x, t), regarded as a random space function. The present study investigates the alternative representation by q, the solute mass flux at a point of a control plane normal to the mean flow. This representation is appropriate for many field applications in which the variable of interest is the mass of solute discharged through a control surface. A general framework to compute the statistical moments of q and of the associated total solute discharge Q and mass M is established. With x the direction of the mean flow, a solute particle is crossing the control plane at y = η, z = ζ and at the travel (arrival) time τ. The associated expected solute flux value is proportional to the joint probability density function (pdf) g1 of η, ζ and τ, whereas the variance of q is shown to depend on the joint pdf g2 of the same variables for two particles. In turn, the statistical moments of η, ζ and τ depend on those of the velocity components through a system of stochastic ordinary differential equations. For a steady velocity field and neglecting the effect of pore-scale dispersion, a major simplification of the problem results in the independence of the random variables η, ζ and τ. As a consequence, the pdf of η and ζ can be derived independently of τ. A few approximate approaches to derive the statistical moments of η, ζ and τ are outlined. These methods will be explored in paper 2 in order to effectively derive the variances of the total solute discharge and mass, while paper 3 will deal with the nonlinear effect of the velocity variance upon the moments of η, ζ and τ

Subsurface flow and transport : a stochastic approach

Part I. Introduction: 1. Stochastic modeling of flow and transport: the broad perspective Gedeod Dagan Part II. Subsurface Characterization and Parameter Estimation: 2. Characterization of geological heterogeneity Mary P. Anderson 3. Application of geostatistics in subsurface hydrology Javier Samper 4. Formulations and computational issues of the inversion of random fields Jesus Carrera, Agustin Medina, Carl Axness and Tony Zimmerman Part III. Flow Modeling and Aquifer Management: 5. Groundwater in heterogeneous formations Peter Kitanidis 6. Aspects of numerical methods in multiphase flows Richard E. Ewing 7. Incorporating uncertainty into aquifer management models Steve Gorelick Part IV. Transport in Heterogeneous Aquifers: 8. Transport of inert solutes by groundwater: recent developments and current issues Yoram Rubin 9. Transport of reactive solutes Vladimir Cvetkovic 10. Nonlocal reactive transport with physical and chemical heterogeneity: linear nonequilibrium sorption with random rate coefficients Bill X. Hu, Fei-Wen Deng and John Cushman 11. Perspectives on field scale application of stochastic subsurface hydrology Lynn W. Gelhar Part V. Fractured Rocks and Unsaturated Soils: 12. Component characterization: an approach to fracture hydrology Jane C. S. Long, Christine Doughty, Akhil Datta-Gupta, Kevin Hestir and Don Vasco 13. Stochastic analysis of solute transport in partially saturated heterogeneous soils David Russo 14. Field scale modeling of immiscible organic chemical spills Jack Parker Part VI. A View to the Future: 15. Stochastic approach to subsurface flow and transport: a view to the future Shlomo Neuman.

Dispersion of a passive solute in non-ergodic transport by steady velocity fields in heterogeneous formations

An inert solute is convected by a steady random velocity field, which is associated with flow through a heterogeneous porous formation. The log conductivity and the velocity are stationary random space functions. The log conductivity Y is assumed to be normal, with an isotropic two-point correlation of variance σY2 and of finite integral scale I. The solute cloud is of a finite input zone of lengthscale l. The transport is characterized with the aid of the spatial moments of the solute body. The effective dispersion coefficient is defined as half of the rate of change with time of the second spatial moment with respect to the centroid. Under the ergodic hypothesis, which is bound to be satisfied for l/I [Gt ] 1, the centroid moves with the mean velocity U and the longitudinal dispersion coefficient [dscr ]L tends to its constant, Fickian, limit. Under a Lagrangian first-order analysis in σY2 it has been found that [dscr ]L = σY2 UI.This study addresses the computation of the effective longitudinal dispersion coefficient for a finite input zone, for which ergodic conditions may not be satisfied. In this case the centroid trajectory and the second spatial moments are random variables. In line with a previous work (Dagan 1990) the effective dispersion coefficient DL is defined as half the rate of change of the expected value of the second spatial moment for large transport time. The aim of the study is to derive DL and its dependence upon l/I and in particular to determine the conditions under which it tends to the ergodic limit [dscr ]L. The computation is carried out separately for a thin body aligned with the mean flow and one transverse to it. In the first case it is found that DL is equal to zero, i.e. the streamlined body does not disperse in the mean. This result is explained by the correlation between the trajectories of the leading and trailing edges, respectively, once the latter reaches the position of the first. The relatively modest increase of the mean second spatial moment is effectively computed. In the case of a thin body initially transverse to the mean flow, DL may reach the ergodic limit [dscr ]L for a ratio l/I of the order 102. For smaller values, DL is found to be bounded from above, and its maximum depends on l but not on I. The uncertainty caused by the randomness of the velocity field is manifested in the trajectory of the centroid rather than in the effective dispersion.

A solute flux approach to transport in heterogeneous formations: 2. Uncertainty analysis

Uncertainty in the mass flux for advection dominated solute movement in heterogeneous porous media is investigated using the Lagrangian framework developed in paper 1 by Dagan et al. (this issue). Expressions for the covariance of the mass flux and cumulative mass flux are derived as functions of the injection volume and sampling area size relative to the scale of heterogeneity. The result is illustrated for solute advection in three types of heterogeneous porous media: stratified formations, two- and three-dimensional porous media; small perturbation approximation is used for the two- and three-dimensional cases. Variances of the mass flux and cumulative mass flux are evaluated as functions of the injection volume (area) scale versus log-hydraulic conductivity integral scale. The greatest decrease in coefficient of variation (CV) of the mass flux is for the source scale 1–5 times the hydraulic conductivity integral scale; further increase in the source size decreases CV comparatively less. The variance of the cumulative mass flux (or total discharge) indicates that for the source size of 20 hydraulic conductivity integral scales, the transport conditions are almost ergodic. The present results also indicate that the cumulative mass flux is a relatively robust quantity for describing field-scale solute transport.

Concentration fluctuations in aquifer transport: a rigorous first-order solution and applications

Flow and transport take place in a formation of spatially variable conductivity K(x). The latter is modeled as a lognormal stationary random space function. With Y=lnK, the structure is characterized by the mean 〈Y〉, the variance σY2, the horizontal and vertical integral scales Ih and Iv. The fluid velocity field V(x), driven by a constant mean head gradient, has a constant mean U and a stationary two-point covariance. Transport of a conservative solute takes place by advection and by pore-scale dispersion (PSD), that is assumed to be characterized by the constant longitudinal and transverse dispersivities αdL and αdT. The local solute concentration C(x, t), a random function of space and time, is characterized by its statistical moments. While the mean concentration 〈C〉 was investigated extensively in the past, the aim here is to determine the variance σC2, a measure of concentration fluctuations. This is achieved in a Lagrangean framework, continuous limit of the particle-tracking procedure, by adopting a few approximations. The present study is a continuation of a previous one (Dagan, G., Fiori, A., 1997. The influence of pore-scale dispersion on concentration statistical moments in transport through heterogeneous aquifers. Water Resour. Res., 33, 1595–1606) and extends it as follows: (i) it is shown that the indepence of the advective component of a solute particle trajectory from the trajectory component associated with PSD, is a rigorous first-order approximation in σY2. This independence, that was conjectured in the work of Dagan and Fiori (Dagan, G., Fiori, A., 1997. The influence of pore-scale dispersion on concentration statistical moments in transport through heterogeneous aquifers. Water Resour. Res., 33, 1595–1606), simplifies considerably the solution; (ii) the covariance of two-particle trajectories, needed in order to evaluate σC2, is rederived, correcting for an error in the previous work. The general results are applied to determining CVC=σC/〈C〉 at the center of a small solute body, of initial size much smaller than Ih=Iv, as function of σY2, t′=tU/I and Pe=UI/DdT=I/αdT. Though PSD reduces considerably CVC as compared with advective transport (Pe=∞), its value is still quite large for time intervals of interest in applications. This finding is in agreement with the analysis of field data by Fitts (Fitts, C.R., 1996. Uncertainty in deterministic groundwater transport models due to the assumption of macrodispersive mixing: evidence from the Cape Cod (Massachussets, USA) and Borden (Ontario, Canada) tracer tests. J. Contam. Hydrol., 23, 69–84).

Contaminant transport in aquifers with spatially variable hydraulic and sorption properties

We consider migration of contaminants in groundwater and wish to characterize transport globally using spatial and temporal moments. The specific problem addressed in this work is how to simultaneously account for the spatial variability of the hydraulic conductivity, K, and of one or several sorption parameters, P. The Lagrangian framework for reactive transport in aquifers of Cvetkovic and Dagan (1994) and Dagan and Cvetkovic (1996) is extended to incorporate the spatial variability in sorption parameters. For arbitrary sorption reactions, the general result can be used for simplified Monte Carlo simulations, where a three–dimensional advection–sorption problem is reduced to a three–dimensional advection and one–dimensional advection–sorption problem. The first two spatial moments characterize the spatial extent of a contaminant plume and are derived for ergodic transport, for cases of continuous and pulse injection. Expressions for the first three temporal moments which characterize field–scale contaminant discharge, are derived for linear sorption reactions. All the derived expressions for the global transport quantities are given in terms of Lagrangian statistics of the fluid velocity and the sorption parameter(s) random fields. Analytical solutions are provided for a few sorption models which are most frequent in applications: nonlinear equilibrium sorption and linear nonequilibrium sorption. Analytical results are given in terms of Lagrangian statistics of the ‘reaction flow path’, μ, which integrates the sorption parameter along an advection flow path with time as the integration variable. Lagrangian statistics of μ are related to the Eulerian statistics of the hydraulic conductivity, K, and the sorption parameter, P, analytically and using Monte Carlo, particle–tracking simulations. The derived analytical expressions are robust for the considered range of variabilities, when compared to simulation results. For extraction of a contaminant subject to Langmuir sorption, the effect of spatial variability in the sorption capacity on the first two moments of the displacament front is suppressed by the effect of nonlinearity. For linear nonequilibrium sorption, spatial variability in the forward rate coefficient has a more significant influence than in the backward rate, on the first three temporal moments.

The significance of heterogeneity of evolving scales to transport in porous formations

Flow takes place in a heterogeneous formation of spatially variable conductivity, which is modeled as a stationary space random function. To model the variability at the regional scale, the formation is viewed as one of a two-dimensional, horizontal structure. A constant head gradient is applied on the formation boundary such that the flow is uniform in the mean. A plume of inert solute is injected at t = 0 in a volume V0. Under ergodic conditions the plume centroid moves with the constant, mean flow velocity U, and a longitudinal macrodispersion coefficient dL may be defined as half of the time rate of change of the plume second spatial moment with respect to the centroid. For a log-conductivity covariance CY of finite integral scale I, at first order in the variance σY2 and for a travel distance L = Ut ≫ I, dL → σY2UI and transport is coined as Fickian. Ergodicity of the moments is ensured if l ≫ I, where l is the initial plume scale. Some field observations have suggested that heterogeneity may be of evolving scales and that the macrodispersion coefficient may grow with L without reaching a constant limit (anomalous diffusion). To model such a behavior, previous studies have assumed that CY is stationary but of unbounded integral scale with CY ∼ arβ (−1 < β < 0) for large lag r. Under ergodic conditions, it was found that asymptotically dL ∼ aUL1+β, i.e., non-Fickian behavior and anomalous dispersion. The present study claims that an ergodic behavior is not possible for a given finite plume of initial size l, since the basic requirement that l ≫ I cannot be satisfied for CY of unbounded scale. For instance, the centroid does not move any more with U but is random (Figure 1), owing to the large-scale heterogeneity. In such a situation the actual effective dispersion coefficient DL is defined as half the rate of change of the mean second spatial moment with respect to the plume centroid in each realization. This is the accessible entity in a given experiment. We show that in contrast with dL, the behavior of DL is controlled by l and it has the Fickian limit DL ∼ aUl1+β (Figure 3). We also discuss the case in which Y is of stationary increments and is characterized by its variogram γy. Then U and dL can be defined only if γY is truncated (equivalently, an “infrared cutoff” is carried out in the spectrum of Y). However, for a bounded U it is shown that DL depends only on γY. Furthermore, for γY = arβ, DL ∼ aUl2Lβ−1; i.e., dispersion is Fickian for 0 < β < 1, whereas for 1 < β < 2, transport is non-Fickian. Since β < 2, DL cannot grow faster than L = Ut. This is in contrast with a recently proposed model (Neuman, 1990) in which the dispersion coefficient is independent of the plume size and it grows approximately like L1.5.