Efficient flow and transport simulations in reconstructed 3D pore geometries

Upscaling pore-scale processes into macroscopic quantities such as hydrodynamic dispersion is still not a straightforward matter for porous media with complex pore space geometries. Recently it has become possible to obtain very realistic 3D geometries for the pore system of real rocks using either numerical reconstruction or micro-CT measurements. In this work, we present a finite element–finite volume simulation method for modeling single-phase fluid flow and solute transport in experimentally obtained 3D pore geometries. Algebraic multigrid techniques and parallelization allow us to solve the Stokes and advection–diffusion equations on large meshes with several millions of elements. We apply this method in a proof-of-concept study of a digitized Fontainebleau sandstone sample. We use the calculated velocity to simulate pore-scale solute transport and diffusion. From this, we are able to calculate the a priori emergent macroscopic hydrodynamic dispersion coefficient of the porous medium for a given molecular diffusion D_m of the solute species. By performing this calculation at a range of flow rates, we can correctly predict all of the observed flow regimes from diffusion dominated to convection dominated.     

Direct numerical simulation of pore-scale flow in a bead pack: Comparison with magnetic resonance imaging observations

A significant body of current research is aimed at developing methods for numerical simulation of flow and transport in porous media that explicitly resolve complex pore and solid geometries, and at utilizing such models to study the relationships between fundamental pore-scale processes and macroscopic manifestations at larger (i.e., Darcy) scales. A number of different numerical methods for pore-scale simulation have been developed, and have been extensively tested and validated for simplified geometries. However, validation of pore-scale simulations of fluid velocity for complex, three-dimensional (3D) pore geometries that are representative of natural porous media is challenging due to our limited ability to measure pore-scale velocity in such systems. Recent advances in magnetic resonance imaging (MRI) offer the opportunity to measure not only the pore geometry, but also local fluid velocities under steady-state flow conditions in 3D and with high spatial resolution. In this paper, we present a 3D velocity field measured at sub-pore resolution (tens of micrometers) over a centimeter-scale 3D domain using MRI methods. We have utilized the measured pore geometry to perform 3D simulations of Navier–Stokes flow over the same domain using direct numerical simulation techniques. We present a comparison of the numerical simulation results with the measured velocity field. It is shown that the numerical results match the observed velocity patterns well overall except for a variance and small systematic scaling which can be attributed to the known experimental uncertainty in the MRI measurements. The comparisons presented here provide strong validation of the pore-scale simulation methods and new insights for interpretation of uncertainty in MRI measurements of pore-scale velocity. This study also provides a potential benchmark for future comparison of other pore-scale simulation methods. © 2012 Elsevier Science. All rights reserved.     

Comparison of continuum and pore-scale models of nutrient biodegradation under transverse mixing conditions

Recent studies indicate that during in situ bioremediation of contaminated groundwater, degradation occurs primarily along transverse mixing zones. Classical reactive-transport models overpredict the amount of degradation because solute spreading and mixing are not distinguished. Efforts to correct this have focused on modifying both dispersion and reaction terms, but no consensus on the best approach has emerged. In this work, a pore-scale model was used to simulate degradation along a transverse mixing zone between two required nutrients, and a continuum model with fitted parameters was used to match degradation rates from the pore-scale model. The pore-scale model solves for the flow field, concentration field, and biomass development within pore spaces of porous medium. For the continuum model, the flow field and biomass distributions are assumed to be homogeneous, and the fitting parameters are the transverse dispersion coefficient (D_T) and maximum substrate utilization rate (k_S,c). Results from the pore-scale model show that degradation rates near the system inlet are limited by the reaction rate, while degradation rates downgradient are limited by transverse mixing. For the continuum model, the value of D_T may be adjusted so that the degradation rate with distance matches that from the pore-scale model in the mixing-limited region. However, adjusting the value of k_S only improves the fit to pore-scale results within the reaction-limited region. Comparison with field and laboratory experiments suggests that the length of the reaction rate-limited region is small compared to the length scale over which degradation occurs. This indicates that along transverse mixing zones in the field, values of k_S are unimportant and only the value of D_T must be accurately fit.     

Chaotic advection at the pore scale: Mechanisms,upscaling and implications for macroscopic transport

The macroscopic spreading and mixing of solute plumes in saturated porous media is ultimately controlled by processes operating at the pore scale. Whilst the conventional picture of pore-scale mechanical dispersion and molecular diffusion leading to persistent hydrodynamic dispersion is well accepted, this paradigm is inherently two-dimensional (2D) in nature and neglects important three-dimensional (3D) phenomena. We discuss how the kinematics of steady 3D flow at the pore scale generate chaotic advection—involving exponential stretching and folding of fluid elements—the mechanisms by which it arises and implications of microscopic chaos for macroscopic dispersion and mixing. Prohibited in steady 2D flow due to topological constraints, these phenomena are ubiquitous due to the topological complexity inherent to all 3D porous media. Consequently 3D porous media flows generate profoundly different fluid deformation and mixing processes to those of 2D flow. The interplay of chaotic advection and broad transit time distributions can be incorporated into a continuous-time random walk (CTRW) framework to predict macroscopic solute mixing and spreading. We show how these results may be generalised to real porous architectures via a CTRW model of fluid deformation, leading to stochastic models of macroscopic dispersion and mixing which both honour the pore-scale kinematics and are directly conditioned on the pore-scale architecture.     

Effects of compound-specific dilution on transient transport and solute breakthrough: A pore-scale analysis

This pore-scale modeling study in saturated porous media shows that compound-specific effects are important not only at steady-state and for the lateral displacement of solutes with different diffusivities but also for transient transport and solute breakthrough. We performed flow and transport simulations in two-dimensional pore-scale domains with different arrangement of the solid grains leading to distinct characteristics of flow variability and connectivity, representing mildly and highly heterogeneous porous media, respectively. The results obtained for a range of average velocities representative of groundwater flow (0.1–10 m/day), show significant effects of aqueous diffusion on solute breakthrough curves. However, the magnitude of such effects can be masked by the flux-averaging approach used to measure solute breakthrough and can hinder the correct interpretation of the true dilution of different solutes. We propose, as a metric of mixing, a transient flux-related dilution index that allows quantifying the evolution of solute dilution at a given position along the main flow direction. For the different solute transport scenarios we obtained dilution breakthrough curves that complement and add important information to traditional solute breakthrough curves. Such dilution breakthrough curves allow capturing the compound-specific mixing of the different solutes and provide useful insights on the interplay between advective and diffusive processes, mass transfer limitations, and incomplete mixing in the heterogeneous pore-scale domains. The quantification of dilution for conservative solutes is in good agreement with the outcomes of mixing-controlled reactive transport simulations, in which the mass and concentration breakthrough curves of the product of an instantaneous transformation of two initially segregated reactants were used as measures of reactive mixing.     

Does pore water velocity affect the reaction rates of adsorptive solute transport in soils? Demonstration with pore-scale modelling

Modelling adsorptive solute transport in soils needs a number of parameters to describe its reaction kinetics and the values of these parameters are usually determined from batch and displacement experiments. Some experimental results reveal that when describing the adsorption as first-order kinetics, its associated reaction rates are not constants but vary with pore water velocity. Explanation of this varies but an independent verification of each explanation is difficult because simultaneously measuring the spatiotemporal distributions of dissolved and adsorbed solutes in soils is formidable. Pore-scale modelling could play an important role to address this gap and has received increased attention over the past few years. This paper investigated the transport of adsorptive solute in a simple porous medium using pore-scale modelling. Fluid flow through the void space of the medium was assumed to be laminar and in saturated condition, and solute transport consisted of advection and molecular diffusion; the sorption and desorption occurring at the fluid–solid interface were modelled as linear first-order kinetics. Based on the simulated spatiotemporal distribution of dissolved and adsorbed solutes at pore scale, volumetric-average reaction kinetics at macroscopic scale and its associated reactive parameters were measured. Both homogeneous adsorption where the reaction rates at microscopic scale are constant, and heterogeneous adsorption where the reaction rates vary from site to site, were investigated. The results indicate that, in contrast to previously thought, the macroscopic reaction rates directly measured from the pore-scale simulations do not change with pore velocity under both homogeneous and heterogeneous adsorptions. In particular, we found that for the homogeneous adsorption, the macroscopic adsorption remains first-order kinetic and can be described by constant reaction rates, regardless of flow rate; whilst for the heterogeneous adsorption, the macroscopic adsorption kinetics continues not to be affected by flow rate but is no longer first-order kinetics that can be described by constant reaction rates. We discuss how these findings could help explain some contrary literature reports over the dependence of reaction rates on pore water velocity.     

Lattice Boltzmann Models for Flow and Transport in Saturated Karst

Flow and transport simulation in karst aquifers remains a significant challenge for the ground water modeling community. Darcy's law–based models cannot simulate the inertial flows characteristic of many karst aquifers. Eddies in these flows can strongly affect solute transport. The simple two-region conduit/matrix paradigm is inadequate for many purposes because it considers only a capacitance rather than a physical domain. Relatively new lattice Boltzmann methods (LBMs) are capable of solving inertial flows and associated solute transport in geometrically complex domains involving karst conduits and heterogeneous matrix rock. LBMs for flow and transport in heterogeneous porous media, which are needed to make the models applicable to large-scale problems, are still under development. Here we explore aspects of these future LBMs, present simple examples illustrating some of the processes that can be simulated, and compare the results with available analytical solutions. Simulations are contrived to mimic simple capacitance-based two-region models involving conduit (mobile) and matrix (immobile) regions and are compared against the analytical solution. There is a high correlation between LBM simulations and the analytical solution for two different mobile region fractions. In more realistic conduit/matrix simulation, the breakthrough curve showed classic features and the two-region model fit slightly better than the advection-dispersion equation (ADE). An LBM-based anisotropic dispersion solver is applied to simulate breakthrough curves from a heterogeneous porous medium, which fit the ADE solution. Finally, breakthrough from a karst-like system consisting of a conduit with inertial regime flow in a heterogeneous aquifer is compared with the advection-dispersion and two-region analytical solutions.     

Simulating conservative tracers in fractured till under realistic timescales

Discrete-fracture and dual-porosity models are infrequently used to simulate solute transport through fractured unconsolidated deposits, despite their more common application in fractured rock where distinct flow regimes are hypothesized. In this study, we apply four fracture transport models--the mobile-immobile model (MIM), parallel-plate discrete-fracture model (PDFM), and stochastic and deterministic discrete-fracture models (DFMs)--to demonstrate their utility for simulating solute transport through fractured till. Model results were compared to breakthrough curves (BTCs) for the conservative tracers potassium bromide (KBr), pentafluorobenzoic acid (PFBA), and 1,4-piperazinediethanesulfonic acid (PIPES) in a large-diameter column of fractured till. Input parameters were determined from independent field and laboratory methods. Predictions of Br BTCs were not significantly different among models; however, the stochastic and deterministic DFMs were more accurate than the MIM or PDFM when predicting PFBA and PIPES BTCs. DFMs may be more applicable than the MIM for tracers with small effective diffusion coefficients (De) or for short timescales due to differences in how these models simulate diffusion or incorporate heterogeneities by their fracture networks. At large scales of investigation, the more computationally efficient MIM and PDFM may be more practical to implement than the three-dimensional DFMs, or a combination of model approaches could be employed. Regardless of the modeling approach used, fractures should be incorporated routinely into solute transport models in glaciated terrain.     

Upscaling geochemical reaction rates using pore-scale network modeling

Geochemical reaction rate laws are often measured using crushed minerals in well-mixed laboratory systems that are designed to eliminate mass transport limitations. Such rate laws are often used directly in reactive transport models to predict the reaction and transport of chemical species in consolidated porous media found in subsurface environments. Due to the inherent heterogeneities of porous media, such use of lab-measured rate laws may introduce errors, leading to a need to develop methods for upscaling reaction rates. In this work, we present a methodology for using pore-scale network modeling to investigate scaling effects in geochemical reaction rates. The reactive transport processes are simulated at the pore scale, accounting for heterogeneities of both physical and mineral properties. Mass balance principles are then used to calculate reaction rates at the continuum scale. To examine the scaling behavior of reaction kinetics, these continuum-scale rates from the network model are compared to the rates calculated by directly using laboratory-measured reaction rate laws and ignoring pore-scale heterogeneities. In this work, this methodology is demonstrated by upscaling anorthite and kaolinite reaction rates under simulation conditions relevant to geological CO₂ sequestration. Simulation results show that under conditions with CO₂ present at high concentrations, pore-scale concentrations of reactive species and reaction rates vary spatially by orders of magnitude, and the scaling effect is significant. With a much smaller CO₂ concentration, the scaling effect is relatively small. These results indicate that the increased acidity associated with geological sequestration can generate conditions for which proper scaling tools are yet to be developed. This work demonstrates the use of pore-scale network modeling as a valuable research tool for examining upscaling of geochemical kinetics. The pore-scale model allows the effects of pore-scale heterogeneities to be integrated into system behavior at multiple scales, thereby identifying important factors that contribute to the scaling effect.     

Transport of nonsorbing solutes in a streambed with periodic bedforms

Previous studies of hyporheic zone focused largely on the net mass transfer of solutes between stream and streambed. Solute transport within the bed has attracted less attention. In this study, we combined flume experiments and numerical simulations to examine solute transport processes in a streambed with periodic bedforms. Solute originating from the stream was subjected to advective transport driven by pore water circulation due to current–bedform interactions as well as hydrodynamic dispersion in the porous bed. The experimental and numerical results showed that advection played a dominant role at the early stage of solute transport, which took place in the hyporheic zone. Downward solute transfer to the deep ambient flow zone was controlled by transverse dispersion at the later stage when the elapsed time exceeded the advective transport characteristic time t_c (= L/u_c with L being the bedform length and u_c the characteristic pore water velocity). The advection-based pumping exchange model was found to predict reasonably well solute transfer between the overlying water and streambed at the early stage but its performance deteriorated at the later stage. With dispersion neglected, the pumping exchange model underestimated the long-term rate and total mass of solute transfer from the overlying water to the bed. Therefore both advective and dispersive transport components are essential for quantification of hyporheic exchange processes.     

Lagrangian simulations of unstable gravity-driven flow of fluids with variable density in randomly heterogeneous porous media

A new Lagrangian particle model based on smoothed particle hydrodynamics (SPH) is developed and used to simulate Darcy scale flow and transport in porous media. The method has excellent conservation properties and treats advection exactly. The Lagrangian method is used in stochastic analysis of miscible density-driven fluid flows. Results show that heterogeneity significantly increases dispersion and slows development of Rayleigh–Taylor instability. The presented numerical examples illustrate the advantages of Lagrangian methods for stochastic transport simulations.     

Balancing the source terms in a SPH model for solving the shallow water equations

A shallow flow generally features complex hydrodynamics induced by complicated domain topography and geometry. A numerical scheme with well-balanced flux and source term gradients is therefore essential before a shallow flow model can be applied to simulate real-world problems. The issue of source term balancing has been exhaustively investigated in grid-based numerical approaches, e.g. discontinuous Galerkin finite element methods and finite volume Godunov-type methods. In recent years, a relatively new computational method, smooth particle hydrodynamics (SPH), has started to gain popularity in solving the shallow water equations (SWEs). However, the well-balanced problem has not been fully investigated and resolved in the context of SPH. This work aims to discuss the well-balanced problem caused by a standard SPH discretization to the SWEs with slope source terms and derive a corrected SPH algorithm that is able to preserve the solution of lake at rest. In order to enhance the shock capturing capability of the resulting SPH model, the Monotone Upwind-centered Scheme for Conservation Laws (MUSCL) is also explored and applied to enable Riemann solver based artificial viscosity. The new SPH model is validated against several idealized benchmark tests and a real-world dam-break case and promising results are obtained.     

Characterizing hyporheic transport processes — Interpretation of electrical geophysical data in coupled stream–hyporheic zone systems during solute tracer studies

Quantifying hyporheic solute dynamics has been limited by our ability to assess the magnitude and extent of stream interactions with multiple domains: mobile subsurface storage (MSS, e.g., freely flowing pore water) and immobile subsurface storage (ISS, e.g., poorly connected pore water). Stream-tracer experiments coupled with solute transport modeling are frequently used to characterize lumped MSS and ISS dynamics, but are limited by the ability to sample only “mobile” water and by window of detection issues. Here, we couple simulations of near-surface electrical resistivity (ER) methods with conservative solute transport to directly compare solute transport with ER interpretations, and to determine the ability of ER to predict spatial and temporal trends of solute distribution and transport in stream–hyporheic systems. Results show that temporal moments from both ER and solute transport data are well correlated for locations where advection is not the dominant solute transport process. Mean arrival time and variance are especially well-predicted by ER interpretation, providing the potential to estimate rate-limited mass transport (i.e. diffusive) parameters from these data in a distributed domain, substantially increasing our knowledge of the fate and transport of subsurface solutes.     

Transport in chemically and mechanically heterogeneous porous media IV: large-scale mass equilibrium for solute transport with adsorption

In this article we consider the transport of an adsorbing solute in a two-region model of a chemically and mechanically heterogeneous porous medium when the condition of large-scale mechanical equilibrium is valid. Under these circumstances, a one-equation model can be used to predict the large-scale averaged velocity, but a two-equation model may be required to predict the regional velocities that are needed to accurately describe the solute transport process. If the condition of large-scale mass equilibrium is valid, the solute transport process can be represented in terms of a one-equation model and the analysis is simplified greatly. The constraints associated with the condition of large-scale mass equilibrium are developed, and when these constraints are satisfied the mass transport process can be described in terms of the large-scale average velocity, an average adsorption isotherm, and a single large-scale dispersion tensor. When the condition of large-scale mass equilibrium is not valid, two equations are required to describe the mass transfer process, and these two equations contain two adsorption isotherms, two dispersion tensors, and an exchange coefficient. The extension of the analysis to multi-region models is straight forward but tedious.     

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.     

Pore scale modeling of immiscible and miscible fluid flows using smoothed particle hydrodynamics

A numerical model based on smoothed particle hydrodynamics (SPH) was developed and used to simulate immiscible and miscible fluid flows in porous media and to study effects of pore scale heterogeneity and anisotropy on such flows.