Transport in chemically and mechanically heterogeneous porous media: V. Two-equation model for solute transport with adsorption

In this paper we develop the two-equation model for solute transport and adsorption in a two-region model of a mechanically and chemically heterogeneous porous medium. The closure problem is derived and the coefficients in both the one- and two-equation models are determined on the basis of the Darcy-scale parameters. Numerical experiments are carried out for a stratified system at the aquifer scale, and the results are compared with the one-equation model presented in Part IV and the two-equation model developed in this paper. Good agreement between the two-equation model and the numerical experiments is obtained. In addition, the two-equation model is used, in conjunction with a moment analysis, to derive a one-equation, non-equilibrium model that is valid in the asymptotic regime. Numerical results are used to identify the asymptotic regime for the one-equation, non-equilibrium model.     

Transport in chemically and mechanically heterogeneous porous media. II: Comparison with numerical experiments for slightly compressible single-phase flow

In this paper we compare the theory developed in Part I (Adv. Water Resour., 19 (1996) 24–47) with numerical experiments for transient, one-dimensional (in the large-scale averaged sense) flow in nodular and stratified systems. The comparison is limited to systems that are isotropic at the Darcy scale, and that are relatively simple when compared to the complex nature of most geological systems. Good agreement between theory and experiment is obtained, and this suggests that the simplifications made in the development of the three closure problems are acceptable. The closure problems have been used to calculate the permeability tensors and the exchange coefficient for a wide range of conditions of practical importance.     

Transport in chemically and mechanically heterogeneous porous media. I: Theoretical development of region-averaged equations for slightly compressible single-phase flow

Transport processes in heterogeneous porous media are often treated in terms of one-equation models. Such treatment assumes that the velocity, pressure, temperature, and concentration can be represented in terms of a single large-scale averaged quantity in regions having significantly different mechanical, thermal, and chemical properties. In this paper we explore the process of single-phase flow in a two-region model of heterogeneous porous media. The region-averaged equations are developed for the case of a slightly compressible flow which is an accurate representation for a certain class of liquid-phase flows. The analysis leads to a pair of transport equations for the region averaged pressures that are coupled through a classic exchange term, in addition to being coupled by a diffusive cross effect. The domain of validity of the theory has been identified in terms of a series of length and timescale constraints.In Part II the theory is tested, in the absence of adjustable parameters, by comparison with numerical experiments for transient, slightly compressible flow in both stratified and nodular models of heterogeneous porous media. Good agreement between theory and experiment is obtained for nodular and stratified systems, and effective transport coefficients for a wide range of conditions are presented on the basis of solutions of the three closure problems that appear in the theory. Part III of this paper deals with the principle of large-scale mechanical equilibrium and the region-averaged form of Darcy's law. This form is necessary for the development and solution of the region-averaged solute transport equations that are presented in Part IV. Finally, in Part V we present results for the dispersion tensors and the exchange coefficient associated with the two-region model of solute transport with adsorption.     

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.     

A meshless method to simulate solute transport in heterogeneous porous media

We derive a meshless numerical method based on smoothed particle hydrodynamics (SPH) for the simulation of conservative solute transport in heterogeneous geological formations. We demonstrate that the new proposed scheme is stable, accurate, and conserves global mass. We evaluate the performance of the proposed method versus other popular numerical methods for the simulation of one- and two-dimensional dispersion and two-dimensional advective–dispersive solute transport in heterogeneous porous media under different Pèclet numbers. The results of those benchmarks demonstrate that the proposed scheme has important advantages over other standard methods because of its natural ability to control numerical dispersion and other numerical artifacts. More importantly, while the numerical dispersion affecting traditional numerical methods creates artificial mixing and dilution, the new scheme provides numerical solutions that are “physically correct”, greatly reducing these artifacts.     

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.     

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.     

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.     

A multidimensional streamline-based method to simulate reactive solute transport in heterogeneous porous media

We present a new streamline-based numerical method for simulating reactive solute transport in porous media. The key innovation of the method is that both longitudinal and transverse dispersion are incorporated accurately without numerical dispersion. Dispersion is approximated in a flow-oriented grid using a combination of a one-dimensional finite difference scheme and a meshless approximation. In contrast to previous hybrid alternatives to incorporate dispersion in streamline-based simulations, the proposed scheme does not require a grid and, hence, it does not introduce numerical dispersion. In addition, the proposed scheme eliminates numerical oscillations and negative concentration values even when the dispersion tensor includes the off-diagonal coefficients and the flow field is non-uniform. We demonstrate that for a set of two- and three-dimensional benchmark problems, the new proposed streamline-based formulation compares favorably to two state of the art finite volume and hybrid Eulerian–Lagrangian solvers.     

Stochastic modeling of solute transport in 3-D heterogeneous porous media with random source condition

During probabilistic analysis of flow and transport in porous media, the uncertainty due to spatial heterogeneity of governing parameters are often taken into account. The randomness in the source conditions also play a major role on the stochastic behavior in distribution of the dependent variable. The present paper is focused on studying the effect of both uncertainty in the governing system parameters as well as the input source conditions. Under such circumstances, a method is proposed which combines with stochastic finite element method (SFEM) and is illustrated for probabilistic analysis of concentration distribution in a 3-D heterogeneous porous media under the influence of random source condition. In the first step SFEM used for probabilistic solution due to spatial heterogeneity of governing parameters for a unit source pulse. Further, the results from the unit source pulse case have been used for the analysis of multiple pulse case using the numerical convolution when the source condition is a random process. The source condition is modeled as a discrete release of random amount of masses at fixed intervals of time. The mean and standard deviation of concentration is compared for the deterministic and the stochastic system scenarios as well as for different values of system parameters. The effect of uncertainty of source condition is also demonstrated in terms of mean and standard deviation of concentration at various locations in the domain.     

Transport upscaling using multi-rate mass transfer in three-dimensional highly heterogeneous porous media

A methodology for transport upscaling of three-dimensional highly heterogeneous formations is developed and demonstrated. The overall approach requires a prior hydraulic conductivity upscaling using an interblock-centered full-tensor Laplacian-with-skin method followed by transport upscaling. The coarse scale transport equation includes a multi-rate mass transfer term to compensate for the loss of heterogeneity inherent to all upscaling processes. The upscaling procedures for flow and transport are described in detail and then applied to a three-dimensional highly heterogeneous synthetic example. The proposed approach not only reproduces flow and transport at the coarse scale, but it also reproduces the uncertainty associated with the predictions as measured by the ensemble variability of the breakthrough curves.     

A master equation for reactive solute transport in porous media

The mean value of a density of a cloud of points described by a generalized Liouville equation associated with a convection dispersion equation governing adsorbing solute transport yields a joint concentration probability density. The general technique can be applied for either linear or nonlinear adsorption; here the application is restricted to linear adsorption in one-dimensional transport. The equation generated for the joint concentration probability density is in the general form of a Fokker-Planck equation, but with a suitable coordinate transformation, it is possible to represent it as a diffusion equation with variable coefficients.     

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.     

Analytical solutions for solute transport in saturated porous media with semi-infinite or finite thickness

Three-dimensional analytical solutions for solute transport in saturated, homogeneous porous media are developed. The models account for three-dimensional dispersion in a uniform flow field, first-order decay of aqueous phase and sorbed solutes with different decay rates, and nonequilibrium solute sorption onto the solid matrix of the porous formation. The governing solute transport equations are solved analytically by employing Laplace, Fourier and finite Fourier cosine transform techniques. Porous media with either semi-infinite or finite thickness are considered. Furthermore, continuous as well as periodic source loadings from either a point or an elliptic source geometry are examined. The effect of aquifer boundary conditions as well as the source geometry on solute transport in subsurface porous formations is investigated.     

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.     

Non-Fickian mass transport in fractured porous media

The paper provides an introduction to fundamental concepts of mathematical modeling of mass transport in fractured porous heterogeneous rocks. Keeping aside many important factors that can affect mass transport in subsurface, our main concern is the multi-scale character of the rock formation, which is constituted by porous domains dissected by the network of fractures. Taking into account the well-documented fact that porous rocks can be considered as a fractal medium and assuming that sizes of pores vary significantly (i.e. have different characteristic scales), the fractional-order differential equations that model the anomalous diffusive mass transport in such type of domains are derived and justified analytically. Analytical solutions of some particular problems of anomalous diffusion in the fractal media of various geometries are obtained. Extending this approach to more complex situation when diffusion is accompanied by advection, solute transport in a fractured porous medium is modeled by the advection-dispersion equation with fractional time derivative. In the case of confined fractured porous aquifer, accounting for anomalous non-Fickian diffusion in the surrounding rock mass, the adopted approach leads to introduction of an additional fractional time derivative in the equation for solute transport. The closed-form solutions for concentrations in the aquifer and surrounding rocks are obtained for the arbitrary time-dependent source of contamination located in the inlet of the aquifer. Based on these solutions, different regimes of contamination of the aquifers with different physical properties can be readily modeled and analyzed.