Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 4. Species transport fundamentals

This work is the fourth in a series of papers on the thermodynamically constrained averaging theory (TCAT) approach for modeling flow and transport phenomena in multiscale porous medium systems. The general TCAT framework and the mathematical foundation presented in previous works are built upon by formulating macroscale models for conservation of mass, momentum, and energy, and the balance of entropy for a species in a phase volume, interface, and common curve. In addition, classical irreversible thermodynamic relations for species in entities are averaged from the microscale to the macroscale. Finally, we comment on alternative approaches that can be used to connect species and entity conservation equations to a constrained system entropy inequality, which is a key component of the TCAT approach. The formulations detailed in this work can be built upon to develop models for species transport and reactions in a variety of multiphase systems.     

Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 2. Foundation

This paper is the second in a series that details the thermodynamically constrained averaging theory (TCAT) approach for modeling flow and transport phenomena in porous medium systems. In this work, we provide the mathematical foundation upon which the theory is based. Elements of this foundation include definitions of mathematical properties of the systems of concern, previously available theorems needed to formulate models, and several theorems and corollaries, introduced and proven here. These tools are of use in producing complete, closed-form TCAT models for single- and multiple-fluid-phase porous medium systems. Future work in this series will rely and build upon the foundation laid in this work to detail the development of sets of closed models.     

Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 3. Single-fluid-phase flow

This work is the third in a series of papers on the thermodynamically constrained averaging theory (TCAT) approach to modeling flow and transport phenomena in multiscale porous medium systems. Building upon the general TCAT framework and the mathematical foundation presented in previous works in this series, we demonstrate the TCAT approach for the case of single-fluid-phase flow. The formulated model is based upon conservation equations for mass, momentum, and energy and a general entropy inequality constraint, which is developed to guide model closure. A specific example of a closed model is derived under limiting assumptions using a linearization approach and these results are compared and contrasted with the traditional single-phase-flow model. Potential extensions to this work are discussed. Specific advancements in this work beyond previous averaging theory approaches to single-phase flow include use of macroscale thermodynamics that is averaged from the microscale, the use of derived equilibrium conditions to guide a flux–force pair approach to simplification, use of a general Lagrange multiplier approach to connect conservation equation constraints to the entropy inequality, and a focus on producing complete, closed models that are solvable.     

Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 1. Motivation and overview

We give several examples of weaknesses in classical, empirically derived models of transport phenomena in porous medium systems. We also place recent attempts to develop improved multiscale porous medium models using averaging theory in context and note deficiencies in these approaches. These deficiencies are found to arise in part from the manner in which thermodynamics is introduced into a constrained entropy inequality, which is used to guide the formation of closed models. Because of this, we briefly examine several established thermodynamic approaches and outline a framework to develop macroscale models that retain consistency with microscale physics and thermodynamics. This framework will be detailed and applied in future papers in this series.     

Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 7. Single-phase megascale flow models

This work is the seventh in a series that introduces and employs the thermodynamically constrained averaging theory (TCAT) for modeling flow and transport in multiscale porous medium systems. This paper expands the previous analyses in the series by developing models at a scale where spatial variations within the system are not considered. Thus the time variation of variables averaged over the entire system is modeled in relation to fluxes at the boundary of the system. This implementation of TCAT makes use of conservation equations for mass, momentum, and energy as well as an entropy balance. Additionally, classical irreversible thermodynamics is assumed to hold at the microscale and is averaged to the megascale, or system scale. The fact that the local equilibrium assumption does not apply at the megascale points to the importance of obtaining closure relations that account for the large-scale manifestation of small-scale variations. Example applications built on this foundation are suggested to stimulate future work.     

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.     

Stochastic modeling of reactive transport in wetlands

This study describes the development of a general model for reaction in and performance of spatially heterogeneous bioreactors such as treatment wetlands. The modeled domain possesses local-scale velocities, reaction rates and transverse dispersion coefficients that are functions of an underlying heterogeneity variate representing one or more controlling biophysical attributes, for example, reactive surface area (submerged plant) density. Reaction rate coefficients are treated as related to local velocities in an inverse square fashion via their mutual dependence upon the variate. The study focuses on the solution for the steady-state case with constant inlet concentration. Results compare well with exact solutions developed for laterally-bounded systems in which the heterogeneity is represented explicitly. Employing the bicontinuum analogue of a second-order model, an expression for an effective longitudinal dispersion coefficient as a function of travel distance is developed using the method of moments. The result provides insights into the behavior of concentration as transverse mixing drives the system asymptotically toward Fickian longitudinal dispersion. The model may represent an improvement over other approaches for characterizing treatment wetland performance because it accounts for evolving shear flow dispersion, and because parameters are few in number, physically based, and invariant with mean velocity.     

Probabilistic sensitivity and modeling of two-dimensional transport in porous media

A reliability approach is used to develop a probabilistic model of two-dimensional non-reactive and reactive contaminant transport in porous media. The reliability approach provides two important quantitative results: an estimate of the probability that contaminant concentration is exceeded at some location and time, and measures of the sensitivity of the probabilistic outcome to likely changes in the uncertain variables. The method requires that each uncertain variable be assigned at least a mean and variance; in this work we also incorporate and investigate the influence of marginal probability distributions. Uncertain variables includex andy components of average groundwater flow velocity,x andy components of dispersivity, diffusion coefficient, distribution coefficient, porosity and bulk density. The objective is to examine the relative importance of each uncertain variable, the marginal distribution assigned to each variable, and possible correlation between the variables. Results utilizing a two-dimensional analytical solution indicate that the probabilistic outcome is generally very sensitive to likely changes in the uncertain flow velocity. Uncertainty associated with dispersivity and diffusion coefficient is often not a significant issue with respect to the probabilistic analysis; therefore, dispersivity and diffusion coefficient can often be treated for practical analysis as deterministic constants. The probabilistic outcome is sensitive to the uncertainty of the reaction terms for early times in the flow event. At later times, when source contaminants are released at constant rate throughout the study period, the probabilistic outcome may not be sensitive to changes in the reaction terms. These results, although limited at present by assumptions and conceptual restrictions inherent to the closed-form analytical solution, provide insight into the critical issues to consider in a probabilistic analysis of contaminant transport. Such information concerning the most important uncertain parameters can be used to guide field and laboratory investigations.     

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.     

Analysis of biodegradation in a 3-D heterogeneous porous medium using nonlinear stochastic finite element method

Probabilistic analysis by Monte Carlo Simulation method (MCSM) is a computationally prohibitive task for a reactive solute transport involving coupled PDEs with nonlinear source/sink terms in 3-D heterogeneous porous media. The perturbation based stochastic finite element method (SFEM) is an attractive alternative method to MCSM as it is computationally efficient and accurate. In the present study SFEM is developed for solving nonlinear reactive solute transport problem in a 3-D heterogeneous medium. Here the solution of the biodegradation problem involving a single solute by a single class of microorganisms coupled with dynamic microbial growth is attempted using this method. The SFEM here produces a second-order accurate solution for the mean and a first-order accurate solution for the standard deviation of concentrations. In this study both the physical parameters (hydraulic conductivity, porosity, dispersivity and diffusion coefficient) and the biological parameters (maximum substrate utilization rate and the coefficient of cell decay) are considered as spatially varying random fields. A comparison between the MCSM and SFEM for the mean and standard deviation of concentration is made for 1-D and 3-D problem. The effects of heterogeneity on the degradation of substrate and growth of biomass concentrations for a range of variances of input parameters are discussed for both 1-D and 3-D problems.     

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.     

Approximation of interfacial properties in multiphase porous medium systems

We investigate the estimation of interfacial areas, curvatures, and common curve lengths in multiphase porous medium systems. Algorithms are developed to obtain estimates of these quantities based upon a variety of potential data sources and estimation approaches. The accuracy of the derived approximations are evaluated as a function of the data type and resolution of the data. The methods advanced improve upon standard approaches now in use and show excellent accuracy at resolutions on the order of five lattice points per minimum radius of curvature of the object being resolved. Finally, we suggest a promising class of extensions that could lead to further improvements in the accuracy of such methods.     

Discontinuous Galerkin methods for simulating bioreactive transport of viruses in porous media

Primal discontinuous Galerkin (DG) methods are formulated to solve the transport equations for modeling migration and survival of viruses with kinetic and equilibrium adsorption in porous media. An entropy analysis is conducted to show that DG schemes are numerically stable and that the free energy of a DG approximation decreases with time in a manner similar to the exact solution. Combining results for free and attached virus concentrations, we establish optimal a priori error estimates for the coupled partial and ordinary differential equations of virus transport. Numerical results suggest that DG can treat bioreactive transport of viruses over a wide range of modeling parameters, including both advection- and dispersion-dominated problems. In addition, it is shown that DG can sharply capture local phenomena of virus transport with dynamic mesh adaptation.     

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.     

Reactive transport in stratified flow fields with idealized heterogeneity

A two-dimensional equation governing the steady state spatial concentration distribution of a reactive constituent within a heterogeneous advective–dispersive flow field is solved analytically. The solution which is developed for the case of a single point source can be generalized to represent analogous situations with any number of separate point sources. A limiting case of special interest has a line source of constant concentration spanning the domain’s upstream boundary. The work has relevance for improving understanding of reactive transport within various kinds of advection-dominated natural or engineered environments including rivers and streams, and bioreactors such as treatment wetlands. Simulations are used to examine quantitatively the impact that transverse dispersion (deviations from purely stochastic-convective flow) can have on mean concentration decline in the direction of flow. Results support the contention that transverse mixing serves to enhance the overall rate of reaction in such systems.     

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.     

An integrated approach for modeling solute transport in streams and canals with applications

A new modeling approach for solute transport in streams and canals was developed to simulate solute dissolution, transport, and decay with continuously migrating sources. The new approach can efficiently handle complicated solute source feeding schemes and initial conditions. Incorporating the finite volume method (FVM) and the ULTIMATE QUICKEST numerical scheme, the new approach is capable of predicting fate and transport of solute that is added to small streams or canals, typically in a continuous fashion. The approach was tested successfully using a hypothetical case, and then applied to an actual field experiment, where linear anionic polyacrylamide (LA-PAM) was applied to an earthen canal. The field experiment was simulated first as a fixed boundary problem using measured concentration data as the boundary condition to test model parameters and sensitivities. The approach was then applied to a moving boundary problem, which included subsequent LA-PAM dissolution, settling to the canal bottom and transport with the flowing canal water. Simulation results showed that the modeling approach developed in this study performed satisfactorily and can be used to simulate a variety of transport problems in streams and canals.