Cass T. Miller

Multiphase flow and transport modeling in heterogeneous porous media: challenges and approaches

Abstract We review the current status of modeling multiphase systems, including balance equation formulation, constitutive relations for both pressure-saturation-conductivity and interphase mass transfer, and stochastic and computational issues. We discuss weaknesses and inconsistencies of current approaches based on theoretical, computational, and experimental evidence. Where possible, we suggest new or evolving approaches.

Pore-morphology-based simulation of drainage in totally wetting porous media

We develop and analyze a novel, quasi-static, pore-scale approach for modeling drainage in a porous medium system. The approach uses: (1) a synthetic, non-overlapping packing of a set of spheres, (2) a discrete representation of the sphere packing, and (3) concepts from pore morphology and local pore-scale physics to simulate the drainage process. The grain-size distribution and porosity of two well-characterized porous media were used as input into the drainage simulator, and the simulated results showed good agreement with experimental observations. We further comment on the use of this simulator for determining the size of a representative elementary volume needed to characterize the drainage process.

Modelling the fate of oxidisable organic contaminants in groundwater

Subsurface contamination by organic chemicals is a pervasive environmental problem, susceptible to remediation by natural or enhanced attenuation approaches or more highly engineered methods such as pump-and-treat, amongst others. Such remediation approaches, along with risk assessment or the pressing need to address complex scientific questions, have driven the development of integrated modelling tools that incorporate physical, biological and geochemical processes. We provide a comprehensive modelling framework, including geochemical reactions and interphase mass transfer processes such as sorption/desorption, non-aqueous phase liquid dissolution and mineral precipitatation/dissolution, all of which can be in equilibrium or kinetically controlled. This framework is used to simulate microbially mediated transformation/degradation processes and the attendant microbial population growth and decay. Solution algorithms, particularly the split-operator (SO) approach, are described, along with a brief resume of numerical solution methods. Some of the available numerical models are described, mainly those constructed using available flow, transport and geochemical reaction packages. The general modelling framework is illustrated by pertinent examples, showing the degradation of dissolved organics by microbial activity limited by the availability of nutrients or electron acceptors (i.e., changing redox states), as well as concomitant secondary reactions. Two field-scale modelling examples are discussed, the Vejen landfill (Denmark) and an example where metal contamination is remediated by redox changes wrought by injection of a dissolved organic compound. A summary is provided of current and likely future challenges to modelling of oxidisable organics in the subsurface.

The influence of mass transfer characteristics and porous media heterogeneity on nonaqueous phase dissolution

A two-dimensional multiphase flow and species transport model was developed and applied to the case of nonaqueous phase liquid (NAPL) emplacement and dissolution in both homogeneous and heterogeneous porous media systems. Simulations were performed to observe dissolution rate variations and the degree of NAPL-aqueous phase nonequilibrium as a function of two aqueous phase velocities and five forms of the NAPL-aqueous phase mass transfer formulation. An integrated form of the Damkohler number was introduced to analyze the degree of NAPL-aqueous phase nonequilibrium. Mass removal rates for homogeneous media were insensitive to the form of the NAPL-aqueous phase mass transfer formulation, yielding results similar to a local equilibrium approach for all but one mass transfer formulation. This latter formulation was most sensitive to NAPL saturation and yielded significant nonequilibrium behavior, which was manifested as a decrease in NAPL dissolution rates as the NAPL volume fraction decreased. Variations in mass elution rates between homogeneous and heterogeneous media were observed, with more significant variations found for variances in porous media properties than for horizontal correlation lengths. In heterogeneous media, decreases in dissolution rates were attributed to the existence of relatively immobile regions of NAPL with saturations greater than the residual saturation of the media, so-called NAPL pools. These results illustrate the importance of the statistical characteristics of heterogeneous porous media on NAPL distribution and dissolution processes.

Optimal design for problems involving flow and transport phenomena in saturated subsurface systems

Estimation problems arise routinely in subsurface hydrology for applications that range from water resources management to water quality protection to subsurface restoration. Interest in optimal design of such systems has increased over the last two decades and this area is considered an important and active area of research. In this work, we review the state of the art, assess important challenges that must be resolved to reach a mature level of understanding, and summarize some promising approaches that might help meet some of the challenges. While much has been accomplished to date, we conclude that more work remains before comprehensive, efficient, and robust solution methods exist to solve the most challenging applications in subsurface science. We suggest that future directions of research include the application of direct search solution methods, and developments in stochastic and multi-objective optimization. We present a set of comprehensive test problems for use in the research community as a means for benchmarking and comparing optimization approaches.

The influence of porous medium characteristics and measurement scale on pore-scale distributions of residual nonaqueous-phase liquids

Abstract A series of experiments was performed to characterize the morphologic distribution of nonaqueous-phase liquids (NAPLs) at residual saturation, as a function of porous medium size. Morphologic characterization of NAPL distributions was accomplished using a novel in situ polymerization technique. The porous medium consisted of glass beads. Blob length, volume and shape characteristics were determined for each experiment, and pore size distributions were determined through capillary pressure-saturation experiments. Both the blob lenght and pore size distributions were fitted to a van Genuchten function. Both blob lenght and pressure-saturation data could be scaled with the same averaged porous medium characteristics. The blob length distributions were found to be wider than the pore size distributions. Estimates of representative elementary volumes (REVs) were generated from statistical analysis using a van Genuchten cumulative frequency distribution function for blob lenght and an empirical function for blob volume as a function of blob length. Simulations were also performed using a Monte Carlo method. The size of the REV needed for a given level of prediction of the residual saturation level was found to increase as a function of mean particle volume for the similar used in this study. Extrapolation of the REV analysis suggests that the size of an REV will increase rapidly as uniformity of the medium decreases. If this extrapolation holds true, significant uncertainty would exist in most determination of residual saturation for poorly sorted media that have been reported to date.

Cosolvent-enhanced remediation of residual dense nonaqueous phase liquids: experimental investigation.

The removal of denser than water nonaqueous phase liquids (DNAPLs) trapped at residual saturation is an important problem at many contaminated groundwater sites. Because pump-and-treat technologies have been ineffective in removing DNAPLs, alternative strategies have been suggested, one of which is enhancing the mobilization and dissolution of DNAPLs by flushing with a cosolvent. Tetrachloroethylene (PCE)/methanol/water systems were studied to evaluate the effect of methanol on the remediation of PCE-contaminated porous media. Experimental measurements of interfacial tension, equilibrium phase composition, and phase density at various methanol/water fractions were combined with other published properties to characterize these systems. In methanol flushing experiments, PCE mobilization, non-equilibrium PCE dissolution, and flow bypassing were all observed. The results demonstrate that (a) small-scale heterogeneities may lead to locally high residual DNAPL saturations that are more easily mobilized than DNAPL residuals in homogeneous media ; (b) mass transfer rate coefficients for PCE/methanol/water systems can be predicted to within 30% using an existing correlation developed for systems with similar NAPL emplacement procedures ; and (c) flow bypassing, due to nonuniform distributions of DNAPL residual or dissolution fingering, can occur in even small-scale experiments.

Convergence of iterative split-operator approaches for approximating nonlinear reactive transport problems

Numerical solutions to nonlinear reactive solute transport problems are often computed using split-operator (SO) approaches, which separate the transport and reaction processes. This uncoupling introduces an additional source of numerical error, known as the splitting error. The iterative split-operator (ISO) algorithm removes the splitting error through iteration. Although the ISO algorithm is often used, there has been very little analysis of its convergence behavior. This work uses theoretical analysis and numerical experiments to investigate the convergence rate of the iterative split-operator approach for solving nonlinear reactive transport problems.

Computation of the interfacial area for two-fluid porous medium systems

We develop a method to compute interfacial areas from three-dimensional digital representations of multiphase systems. We approximate the interfaces with the isosurface generated by the standard marching-cube algorithm from the discrete phase distribution. We apply this approach to two-fluid pore-scale simulations by (1) simulating a random packing of spheres that obeys the grain-size distribution and porosity of an experimental porous medium system, and (2) using a previously developed pore-morphology-based model in order to predict the phase distribution for a water-wet porous medium that undergoes primary drainage. The predicted primary drainage curve and interfacial areas are in good agreement with the experimental values reported in the literature, where interfacial areas were measured using interfacial tracers. The energy dissipation during Haines jumps is significant: thus, the mechanical work done on the system is not completely converted into surface energy, and interfacial areas may not be deduced from the primary drainage curve.

Temporal discretisation errors in non-iterative split-operator approaches to solving chemical reaction/groundwater transport models

Split-operator approaches are methods in widespread use for numerical solutions of reaction/transport problems. However, only a couple of analyses of truncation errors resulting from operator splitting have been published. A range of linear, single species models is analysed here, leading to generalisations of earlier findings, in addition to developing new results. Linear retardation, radioactive decay and interphase mass transfer are considered in detail. Operator-splitting errors for both the standard two-step and alternating two-step methods are presented, the latter approach being popular as a way to reduce errors due to operator splitting. An ambiguity in implementing two-step methods is revealed. The truncation error analysis also shows that the application of the usual alternating split-operator method to an equilibrium reaction (linear retardation) actually increases the (first order in time) truncation error relative to the standard method. However, for homogeneous boundary conditions, the same reaction can be solved to second-order accuracy if the model derived from taking the limit of a nonequilibrium, interphase mass transfer process is used. For the linear reactions considered here, the alternating two-step method is always second-order accurate in the temporal discretisation.