Comparison of theory and experiment for solute transport in weakly heterogeneous bimodal porous media

In this work, the influence of non-equilibrium effects on solute transport in a weakly heterogeneous medium is discussed. Three macro-scale models (upscaled via the volume averaging technique) are investigated: (i) the two-equation non-equilibrium model, (ii) the one-equation asymptotic model and (iii) the one-equation local equilibrium model. The relevance of each of these models to the experimental system conditions (duration of the pulse injection, dispersivity values…) is analyzed. The numerical results predicted by these macroscale models are compared directly with the experimental data (breakthrough curves). Our results suggest that the preasymptotic zone (for which a non-Fickian model is required) increases as the solute input pulse time decreases. Beyond this limit, the asymptotic regime is recovered. A comparison with the results issued from the stochastic theory for this regime is performed. Results predicted by both approaches (volume averaging method and stochastic analysis) are found to be consistent.     

Numerical simulations of non-ergodic solute transport in three-dimensional heterogeneous porous media

Numerical simulations of non-ergodic transport of a non-reactive solute plume by steady-state groundwater flow under a uniform mean velocity, , were conducted in a three-dimensional heterogeneous and statistically isotropic aquifer. The hydraulic conductivity, K(x), is modeled as a random field which is assumed to be log-normally distributed with an exponential covariance. Significant efforts are made to reduce the simulation uncertainties. Ensemble averages of the second spatial moments of the plume and the plume centroid variances were simulated with 1600 Monte Carlo (MC) runs for three variances of log K, _Y²=0.09, 0.23, and 0.46, and a square source normal to of three dimensionless lengths. It is showed that 1600 MC runs are needed to obtain stabilized results in mildly heterogeneous aquifers of _Y²0.5 and that large uncertainty may exist in the simulated results if less MC runs are used, especially for the transverse second spatial moments and the plume centroid variance in transverse directions. The simulated longitudinal second spatial moment and the plume centroid variance in longitudinal direction fit well to the first-order theoretical results while the simulated transverse moments are generally larger than the first-order values. The ergodic condition for the second spatial moments is far from reaching in all cases simulated and transport in transverse directions may reach ergodic condition much slower than that in longitudinal direction.     

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.     

Coupling upscaling finite element method for consolidation analysis of heterogeneous saturated porous media

The coupling upscaling finite element method is developed for solving the coupling problems of deformation and consolidation of heterogeneous saturated porous media under external loading conditions. The method couples two kinds of fully developed methodologies together, i.e., the numerical techniques developed for calculating the apparent and effective physical properties of the heterogeneous media and the upscaling techniques developed for simulating the fluid flow and mass transport properties in heterogeneous porous media. Equivalent permeability tensors and equivalent elastic modulus tensors are calculated for every coarse grid block in the coarse-scale model of the heterogeneous saturated porous media. Moreover, an oversampling technique is introduced to improve the calculation accuracy of the equivalent elastic modulus tensors. A numerical integration process is performed over the fine mesh within every coarse grid element to capture the small scale information induced by non-uniform scalar field properties such as density, compressibility, etc. Numerical experiments are carried out to examine the accuracy of the developed method. It shows that the numerical results obtained by the coupling upscaling finite element method on the coarse-scale models fit fairly well with the reference solutions obtained by traditional finite element method on the fine-scale models. Moreover, this method gets more accurate coarse-scale results than the previously developed coupling multiscale finite element method for solving this kind of coupling problems though it cannot recover the fine-scale solutions. At the same time, the method developed reduces dramatically the computing effort in both CPU time and memory for solving the transient problems, and therefore more large and computational-demanding coupling problems can be solved by computers.     

An efficient,high-order probabilistic collocation method on sparse grids for three-dimensional flow and solute transport in randomly heterogeneous porous media

In this study, a probabilistic collocation method (PCM) on sparse grids is used to solve stochastic equations describing flow and transport in three-dimensional, saturated, randomly heterogeneous porous media. The Karhunen–Loève decomposition is used to represent log hydraulic conductivity Y=ln_Ks$Y=\ln K_s$. The hydraulic head h   and average pore-velocity v$\mathbf{v}$ are obtained by solving the continuity equation coupled with Darcy’s law with random hydraulic conductivity field. The concentration is computed by solving a stochastic advection–dispersion equation with stochastic average pore-velocity v$\mathbf{v}$ computed from Darcy’s law. The PCM approach is an extension of the generalized polynomial chaos (gPC) that couples gPC with probabilistic collocation. By using sparse grid points in sample space rather than standard grids based on full tensor products, the PCM approach becomes much more efficient when applied to random processes with a large number of random dimensions. Monte Carlo (MC) simulations have also been conducted to verify accuracy of the PCM approach and to demonstrate that the PCM approach is computationally more efficient than MC simulations. The numerical examples demonstrate that the PCM approach on sparse grids can efficiently simulate solute transport in randomly heterogeneous porous media with large variances.     

Density-dependent dispersion in heterogeneous porous media Part I: A numerical study

In this paper, we describe carefully conducted numerical experiments, in which a dense salt solution vertically displaces fresh water in a stable manner. The two-dimensional porous media are weakly heterogeneous at a small scale. The purpose of these simulations, conducted for a range of density differences, is to obtain accurate concentration profiles that can be used to validate nonlinear models for high-concentration-gradient dispersion. In this part we focus on convergence of the computations, in numerical and statistical sense, to ensure that the uncertainty in the results is small enough.Concentration variances are computed, which give estimates of the uncertainty in local concentration values. These local variations decrease with increasing density contrast. For tracer transport, obtained longitudinal dispersivities are in accordance with analytical findings. In the case of high-density contrasts, stabilizing gravity forces counteract the growth of dispersive fingers, decreasing the effective width of the transition zone. For small log-permeability variances, the decrease of the apparent dispersivity that is found is in agreement with laboratory results for homogeneous columns.     

Solute dilution under imbibition and drainage conditions in a heterogeneous structure: Modeling of a sand tank experiment

This study aims at modeling the transport of a conservative tracer in two dimensions, as experimentally observed in a strongly heterogeneous medium under conditions of variable water saturation during drainage and imbibition. Solute transport experiments were conducted in a sand tank containing an artificial packing of three quartz sands of different particle sizes. The packing was characterized by the presence of numerous homogeneous layers (0.5 × 5 × 5 cm) inclined at 45° and randomly distributed in a tank. Six different stationary flow conditions were sequentially established during imbibition and drainage. When a stationary flow regime was reached, several solute pulses were applied at different positions at the upper surface of the sand structure. The transport regime was studied by monitoring the tracer plumes injected as point-like pulses at the surface, as they travelled through the sand bedding.     

Evaluation of an upscaled model for DNAPL dissolution kinetics in heterogeneous aquifers

Estimates of contaminant fluxes from DNAPL sources as a function of time and DNAPL mass reduction are important to assess the long-term sustainability and costs of monitored natural attenuation and to determine the benefits of partial source removal. We investigate the accuracy of the upscaled mass transfer function (MTF) proposed by Parker and Park [Parker JC, Park E. Modeling field-scale dense nonaqueous phase liquid dissolution kinetics in heterogeneous aquifers. WRR 2004;40:W05109] to describe field-scale dissolved phase fluxes from DNAPL sources for a range of scenarios generated using high-resolution 3-D numerical simulations of DNAPL infiltration and long-term dissolved phase transport. The results indicate the upscaled MTF is capable of accurately describing field-scale DNAPL dissolution rates as a function of time. For finger-dominated source regions, an empirical mass depletion exponent in the MTF takes on values greater than one which results in predicted mass flux rates that decrease continuously with diminishing DNAPL mass over time. Lens-dominated regions exhibit depletion exponents less than one, which results in more step-function like mass flux versus time behavior. Mass fluxes from DNAPL sources exhibiting both lens- and finger-dominated subregions were less accurately described by the simple MTF, but were well described by a dual-continuum model of the same form for each subregion. The practicality of calibrating a dual-continuum model will likely depend on the feasibility of obtaining spatially resolved field measurements of contaminant fluxes or concentrations associated with the subregions using multilevel sampling or some other means.     

Density-dependent dispersion in heterogeneous porous media Part II: Comparison with nonlinear models

The results of a series of high-resolution numerical experiments are used to test and compare three nonlinear models for high-concentration-gradient dispersion. Gravity stable miscible displacement is considered. The first model, introduced by Hassanizadeh, is a modification of Fick’s law which involves a second-order term in the dispersive flux equation and an additional dispersion parameter β. The numerical experiments confirm the dependency of β on the flow rate. In addition, a dependency on travelled distance is observed. The model can successfully be applied to nearly homogeneous media (σ² = 0.1), but additional fitting is required for more heterogeneous media.The second and third models are based on homogenization of the local scale equations describing density-dependent transport. Egorov considers media that are heterogeneous on the Darcy scale, whereas Demidov starts at the pore-scale level. Both approaches result in a macroscopic balance equation in which the dispersion coefficient is a function of the dimensionless density gradient. In addition, an expression for the concentration variance is derived. For small σ², Egorov’s model predictions are in satisfactory agreement with the numerical experiments without the introduction of any new parameters. Demidov’s model involves an additional fitting parameter, but can be applied to more heterogeneous media as well.     

Block-effective macrodispersion for numerical simulations of sorbing solute transport in heterogeneous porous formations

Heterogeneity is prevalent in aquifers and has an enormous impact on contaminant transport in groundwater. Numerical simulations are an effective way to deal with heterogeneity directly by assigning different hydraulic property values to each numerical grid block. Because hydraulic properties vary on different scales, but they cannot be sampled exhaustively and the number of numerical grid blocks is limited by computational considerations, the dispersive effects of unmodeled heterogeneity need to be accounted for. Dispersion tensors can be used to model the dispersion caused by unmodeled heterogeneity. The concept of block-effective macrodispersion tensors for modeling the effects of small-scale variability on solute transport introduced by Rubin et al. [Rubin Y, Sun A, Maxwell R, Bellin A. The concept of block-effective macrodispersivity and a unified approach for grid-scale- and plume-scale-dependent transport. J Fluid Mech 1999;395:161–80] is extended in this paper for use with reactive solutes. The tensors are derived for reactive solutes with spatially variable retardation factors and for solutes experiencing spatially uniform rate-limited sorption. The longitudinal block-effective macrodispersion coefficient is largest for perfect negative correlation between the log-hydraulic conductivity and the retardation factor. Because dispersion tensors, as they are usually implemented in numerical simulations, produce symmetric spreading, the applicability of the concept depends on the portion of the plume asymmetry caused by small-scale variability. The presented results show that the concept is applicable for rate-limited sorption for block sizes of one and two integral scales.     

Modeling non-equilibrium mass transport in biologically reactive porous media

We develop a one-equation non-equilibrium model to describe the Darcy-scale transport of a solute undergoing biodegradation in porous media. Most of the mathematical models that describe the macroscale transport in such systems have been developed intuitively on the basis of simple conceptual schemes. There are two problems with such a heuristic analysis. First, it is unclear how much information these models are able to capture; that is, it is not clear what the model's domain of validity is. Second, there is no obvious connection between the macroscale effective parameters and the microscopic processes and parameters. As an alternative, a number of upscaling techniques have been developed to derive the appropriate macroscale equations that are used to describe mass transport and reactions in multiphase media. These approaches have been adapted to the problem of biodegradation in porous media with biofilms, but most of the work has focused on systems that are restricted to small concentration gradients at the microscale. This assumption, referred to as the local mass equilibrium approximation, generally has constraints that are overly restrictive. In this article, we devise a model that does not require the assumption of local mass equilibrium to be valid. In this approach, one instead requires only that, at sufficiently long times, anomalous behaviors of the third and higher spatial moments can be neglected; this, in turn, implies that the macroscopic model is well represented by a convection–dispersion–reaction type equation. This strategy is very much in the spirit of the developments for Taylor dispersion presented by Aris (1956). On the basis of our numerical results, we carefully describe the domain of validity of the model and show that the time-asymptotic constraint may be adhered to even for systems that are not at local mass equilibrium.     

Numerical investigation of the anisotropic hydraulic conductivity behavior in heterogeneous porous media

The behavior of the mean equivalent hydraulic conductivity normal and parallel to stratification (K₁, and K₂, respectively) is studied here through Monte Carlo simulations of three-dimensional, steady-state flow in statistically anisotropic, bounded, and heterogeneous media. For water flow normal to stratification in strongly heterogeneous porous media (²_Y=3) the value of K₁ is not unique; it ranges from an arithmetic to a geometric, and finally, to a harmonic mean behavior depending on field dimensions, and medium anisotropy. For a fixed anisotropy ratio and variance of Y = ln K, the larger the distance, in the direction perpendicular to stratification, over which water flow takes place, the faster the rate at which, K_H, behavior is approached. However, even for large anisotropy ratios, harmonic mean behavior appears to be a good approximation only for aquifer thickness L₁ that is large enough to allow stratified flow to occur. For small aquifer thickness (L₁/₁<8, where ₁ is the integral scale normal to stratification) the limiting behavior, for large anisotropy ratios, appears to be, instead, that of two-dimensional flow, i.e., water flows primarily parallel to the planes of stratification. When the aquifer thickness is very small compared to the horizontal dimensions (and with relative similar integral scales in the three directions) a behavior resembling arithmetic mean conditions is exhibited, i.e., water flow takes place through heterogeneous, vertical, soil volumes. The geostatistical expressions of Desbarats (1992a) for upscaling hydraulic conductivity values were utilized and closed form empirical relations were developed for the main components of the upscaled hydraulic conductivity tensor.     

Bayesian uncertainty quantification for flows in heterogeneous porous media using reversible jump Markov chain Monte Carlo methods

In this paper, we study the uncertainty quantification in inverse problems for flows in heterogeneous porous media. Reversible jump Markov chain Monte Carlo algorithms (MCMC) are used for hierarchical modeling of channelized permeability fields. Within each channel, the permeability is assumed to have a log-normal distribution. Uncertainty quantification in history matching is carried out hierarchically by constructing geologic facies boundaries as well as permeability fields within each facies using dynamic data such as production data. The search with Metropolis–Hastings algorithm results in very low acceptance rate, and consequently, the computations are CPU demanding. To speed-up the computations, we use a two-stage MCMC that utilizes upscaled models to screen the proposals. In our numerical results, we assume that the channels intersect the wells and the intersection locations are known. Our results show that the proposed algorithms are capable of capturing the channel boundaries and describe the permeability variations within the channels using dynamic production history at the wells.     

Models of sub-grid variability in numerical simulations of solute transport in heterogeneous porous formations: three-dimensional flow and effect of pore-scale dispersion

Modeling transport of contaminants in the earths subsurface relies on numerical solutions over grids with blocks larger than Darcys scale. The hydraulic conductivity is homogenized over the grid blocks and the plumes spreading is reduced as a consequence of the wiped-out variability. To compensate for this loss Rubin et al. (1999) proposed to augment mixing by block-effective dispersion coefficients, and Rubin et al. (2003) showed, by means of two dimensional simulations, how this concept can be applied in practice. In this paper, we present new solutions of the block-effective dispersion tensor for an axisymmetric exponential covariance model. In addition, we discuss the influence of pore-scale dispersion in both two- and three-dimensional applications.     

Modeling the effects of nonlinear equilibrium sorption on the transport of solute plumes in saturated heterogeneous porous media

Transport of sorbing solutes in 2D steady and heterogeneous flow fields is modeled using a particle tracking random walk technique. The solute is injected as an instantaneous pulse over a finite area. Cases of linear and Freundlich sorption isotherms are considered. Local pore velocity and mechanical dispersion are used to describe the solute transport mechanisms at the local scale. This paper addresses the impact of the degree of heterogeneity and correlation lengths of the log-hydraulic conductivity field as well as negative correlation between the log-hydraulic conductivity field and the log-sorption affinity field on the behavior of the plume of a sorbing chemical. Behavior of the plume is quantified in terms of longitudinal spatial moments: center-of-mass displacement, variance, 95% range, and skewness. The range appears to be a better measure of the spread in the plumes with Freundlich sorption because of plume asymmetry. It has been found that the range varied linearly with the travelled distance, regardless of the sorption isotherm. This linear relationship is important for extrapolation of results to predict behavior beyond simulated times and distances. It was observed that the flow domain heterogeneity slightly enhanced the spreading of nonlinearly sorbing solutes in comparison to that which occurred for the homogeneous flow domain, whereas the spreading enhancement in the case of linear sorption was much more pronounced. In the case of Freundlich sorption, this enhancement led to further deceleration of the solute plume movement as a result of increased retardation coefficients produced by smaller concentrations. It was also observed that, except for plumes with linear sorption, correlation between the hydraulic conductivity and the sorption affinity fields had minimal effect on the spatial moments of solute plumes with nonlinear sorption.     

Eulerian spatial moments for solute transport in three-dimensional heterogeneous,dual-permeability media

A Eulerian analytical method is developed for nonreactive solute transport in heterogeneous, dual-permeability media where the hydraulic conductivities in fracture and matrix domains are both assumed to be stochastic processes. The analytical solution for the mean concentration is given explicitly in Fourier and Laplace transforms. Instead of using the fast fourier transform method to numerically invert the solution to real space (Hu et al., 2002), we apply the general relationship between spatial moments and concentration (Naff, 1990; Hu et al., 1997) to obtain the analytical solutions for the spatial moments up to the second for a pulse input of the solute. Owing to its accuracy and efficiency, the analytical method can be used to check the semi-analytical and Monte Carlo numerical methods before they are applied to more complicated studies. The analytical method can be also used during screening studies to identify the most significant transport parameters for further analysis. In this study, the analytical results have been compared with those obtained from the semi-analytical method (Hu et al., 2002) and the comparison shows that the semi-analytical method is robust. It is clearly shown from the analytical solution that the three factors, local dispersion, conductivity variation in each domain and velocity convection flow difference in the two domains, play different roles on the solute plume spreading in longitudinal and transverse directions. The calculation results also indicate that when the log-conductivity variance in matrix is 10 times less than its counterpart in fractures, it will hardly influence the solute transport, whether the conductivity field is matrix is treated as a homogeneous or random field.