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.     

Incorporating Super‐Diffusion due to Sub‐Grid Heterogeneity to Capture Non‐Fickian Transport

Numerical transport models based on the advection‐dispersion equation (ADE) are built on the assumption that sub‐grid cell transport is Fickian such that dispersive spreading around the average velocity is symmetric and without significant tailing on the front edge of a solute plume. However, anomalous diffusion in the form of super‐diffusion due to preferential pathways in an aquifer has been observed in field data, challenging the assumption of Fickian dispersion at the local scale. This study develops a fully Lagrangian method to simulate sub‐grid super‐diffusion in a multidimensional regional‐scale transport model by using a recent mathematical model allowing super‐diffusion along the flow direction given by the regional model. Here, the time randomizing procedure known as subordination is applied to flow field output from MODFLOW simulations. Numerical tests check the applicability of the novel method in mapping regional‐scale super‐diffusive transport conditioned on local properties of multidimensional heterogeneous media.     

Issues and Options in the Delineation of Well Capture Zones under Uncertainty

The delineation of wellhead protection areas (WHPAs) under uncertainty is still a challenge for heterogeneous porous media. For granular media, one option is to combine particle tracking (PT) with the Monte Carlo approach (PT‐MC) to account for geologic uncertainties. Fractured porous media, however, require certain restrictive assumptions under this approach. An alternative for all types of media is the capture probability (CP) approach, which is based on the solution of the standard advection‐dispersion equation in a backward mode, making use of the analogy between forward and backward transport processes. Within this context, we review the current controversy about the correct form of the conceptual model for transport, finding that the advection‐diffusion model, which represents the diffusive interchange between streamtubes with differing velocities, is more physically realistic than the conventional advection‐dispersion model. For mildly to moderately heterogeneous materials, stochastic theories and simulation experiments show that this process converges at the field scale to an effective advection‐dispersion process that can be simulated with conventional transport models using appropriate macrodispersivity values. For highly heterogeneous materials, stochastic theories do not yet exist but there is no reason why the process should not converge naturally as well. Macrodispersivities appear to be formation‐specific. The advection‐dispersion model can be used for capture zone delineation in heterogeneous granular media. For fractured porous systems, hybrid equivalent porous medium and discrete fracture network or CP‐based approaches may have potential. In general, capture zones delineated by PT without MC will always be too small and should not be used as a basis for land‐use decisions.     

Stochastic analysis of biodegradation fronts in one-dimensional heterogeneous porous media

We consider a one-dimensional model biodegradation system consisting of two reaction–advection equations for nutrient and pollutant concentrations and a rate equation for biomass. The hydrodynamic dispersion is ignored. Under an explicit condition on the decay and growth rates of biomass, the system can be approximated by two component models by setting biomass kinetics to equilibrium. We derive closed form solutions for constant speed traveling fronts for the reduced two component models and compare their profiles in homogeneous media. For a spatially random velocity field, we introduce travel time and study statistics of degradation fronts via representations in terms of the travel time probability density function (pdf) and the traveling front profiles. The travel time pdf does not vary with the nutrient and pollutant concentrations and only depends on the random water velocity. The traveling front profiles are expressed analytically or semi-analytically as functions of the travel time. The problem of nonlinear transport by a random velocity reduces to two subproblems: one being nonlinear transport by a known (unit) velocity, and the other being linear (advective) transport by a random velocity. The approach is illustrated through some examples where the randomness in velocity stems from the spatial variability of porosity.     

Hierarchical simulator of biofilm growth and dynamics in granular porous materials

A new simulator is developed for the prediction of the rate and pattern of growth of biofilms in granular porous media. The biofilm is considered as a heterogeneous porous material that exhibits a hierarchy of length scales. An effective-medium model is used to calculate the local hydraulic permeability and diffusion coefficient in the biofilm, as functions of the local geometric and physicochemical properties. The Navier–Stokes equations and the Brinkman equation are solved numerically to determine the velocity and pressure fields within the pore space and the biofilm, respectively. Biofilm fragments become detached if they are exposed to shear stress higher than a critical value. The detached fragments re-enter into the fluid stream and move within the pore space until they exit from the system or become reattached to downstream grain or biofilm surfaces. A Lagrangian-type simulation is used to determine the trajectories of detached fragments. The spatiotemporal distributions of a carbon source, an electron acceptor and a cell-to-cell signaling molecule are determined from the numerical solution of the governing convection–diffusion–reaction equations. The simulator incorporates growth and apoptosis kinetics for the bacterial cells and production and lysis kinetics for the EPS. The specific growth rate of active bacterial cells depends on the local concentrations of nutrients, mechanical stresses, and a quorum sensing mechanism. Growth-induced deformation of the biofilms is implemented with a cellular automaton approach. In this work, the spatiotemporal evolution of biofilms in the pore space of a 2D granular medium is simulated under high flow rate and nutrient-rich conditions. Transient changes in the pore geometry caused by biofilm growth lead to the formation of preferential flowpaths within the granular porous medium. The decrease of permeability caused by clogging of the porous medium is calculated and is found to be in qualitative agreement with published experimental results.     

Can a Time Fractional‐Derivative Model Capture Scale‐Dependent Dispersion in Saturated Soils?

Time nonlocal transport models such as the time fractional advection‐dispersion equation (t‐fADE) were proposed to capture well‐documented non‐Fickian dynamics for conservative solutes transport in heterogeneous media, with the underlying assumption that the time nonlocality (which means that the current concentration change is affected by previous concentration load) embedded in the physical models can release the effective dispersion coefficient from scale dependency. This assumption, however, has never been systematically examined using real data. This study fills this historical knowledge gap by capturing non‐Fickian transport (likely due to solute retention) documented in the literature (Huang et al. 1995) and observed in our laboratory from small to intermediate spatial scale using the promising, tempered t‐fADE model. Fitting exercises show that the effective dispersion coefficient in the t‐fADE, although differing subtly from the dispersion coefficient in the standard advection‐dispersion equation, increases nonlinearly with the travel distance (varying from 0.5 to 12 m) for both heterogeneous and macroscopically homogeneous sand columns. Further analysis reveals that, while solute retention in relatively immobile zones can be efficiently captured by the time nonlocal parameters in the t‐fADE, the motion‐independent solute movement in the mobile zone is affected by the spatial evolution of local velocities in the host medium, resulting in a scale‐dependent dispersion coefficient. The same result may be found for the other standard time nonlocal transport models that separate solute retention and jumps (i.e., displacement). Therefore, the t‐fADE with a constant dispersion coefficient cannot capture scale‐dependent dispersion in saturated porous media, challenging the application for stochastic hydrogeology methods in quantifying real‐world, preasymptotic transport. Hence improvements on time nonlocal models using, for example, the novel subordination approach are necessary to incorporate the spatial evolution of local velocities without adding cumbersome parameters.     

Biofilm growth in porous media: Experiments,computational modeling at the porescale,and upscaling

Biofilm growth changes many physical properties of porous media such as porosity, permeability and mass transport parameters. The growth depends on various environmental conditions, and in particular, on flow rates. Modeling the evolution of such properties is difficult both at the porescale where the phase morphology can be distinguished, as well as during upscaling to the corescale effective properties. Experimental data on biofilm growth is also limited because its collection can interfere with the growth, while imaging itself presents challenges.In this paper we combine insight from imaging, experiments, and numerical simulations and visualization. The experimental dataset is based on glass beads domain inoculated by biomass which is subjected to various flow conditions promoting the growth of biomass and the appearance of a biofilm phase. The domain is imaged and the imaging data is used directly by a computational model for flow and transport. The results of the computational flow model are upscaled to produce conductivities which compare well with the experimentally obtained hydraulic properties of the medium. The flow model is also coupled to a newly developed biomass–nutrient growth model, and the model reproduces morphologies qualitatively similar to those observed in the experiment.     

A Boltzmann-based mesoscopic model for contaminant transport in flow systems

The objective of this paper is to demonstrate the formulation of a numerical model for mass transport based on the Bhatnagar–Gross–Krook (BGK) Boltzmann equation. To this end, the classical chemical transport equation is derived as the zeroth moment of the BGK Boltzmann differential equation. The relationship between the mass transport equation and the BGK Boltzmann equation allows an alternative approach to numerical modeling of mass transport, wherein mass fluxes are formulated indirectly from the zeroth moment of a difference model for the BGK Boltzmann equation rather than directly from the transport equation. In particular, a second-order numerical solution for the transport equation based on the discrete BGK Boltzmann equation is developed. The numerical discretization of the first-order BGK Boltzmann differential equation is straightforward and leads to diffusion effects being accounted for algebraically rather than through a second-order Fickian term. The resultant model satisfies the entropy condition, thus preventing the emergence of non-physically realizable solutions including oscillations in the vicinity of the front. Integration of the BGK Boltzmann difference equation into the particle velocity space provides the mass fluxes from the control volume and thus the difference equation for mass concentration. The difference model is a local approximation and thus may be easily included in a parallel model or in accounting for complex geometry. Numerical tests for a range of advection–diffusion transport problems, including one- and two-dimensional pure advection transport and advection–diffusion transport show the accuracy of the proposed model in comparison to analytical solutions and solutions obtained by other schemes.     

Continuous time random walks for non-local radial solute transport

This study formulates and analyzes continuous time random walk (CTRW) models in radial flow geometries for the quantification of non-local solute transport induced by heterogeneous flow distributions and by mobile–immobile mass transfer processes. To this end we derive a general CTRW framework in radial coordinates starting from the random walk equations for radial particle positions and times. The particle density, or solute concentration is governed by a non-local radial advection–dispersion equation (ADE). Unlike in CTRWs for uniform flow scenarios, particle transition times here depend on the radial particle position, which renders the CTRW non-stationary. As a consequence, the memory kernel characterizing the non-local ADE, is radially dependent. Based on this general formulation, we derive radial CTRW implementations that (i) emulate non-local radial transport due to heterogeneous advection, (ii) model multirate mass transfer (MRMT) between mobile and immobile continua, and (iii) quantify both heterogeneous advection in a mobile region and mass transfer between mobile and immobile regions. The expected solute breakthrough behavior is studied using numerical random walk particle tracking simulations. This behavior is analyzed by explicit analytical expressions for the asymptotic solute breakthrough curves. We observe clear power-law tails of the solute breakthrough for broad (power-law) distributions of particle transit times (heterogeneous advection) and particle trapping times (MRMT model). The combined model displays two distinct time regimes. An intermediate regime, in which the solute breakthrough is dominated by the particle transit times in the mobile zones, and a late time regime that is governed by the distribution of particle trapping times in immobile zones. These radial CTRW formulations allow for the identification of heterogeneous advection and mobile-immobile processes as drivers of anomalous transport, under conditions relevant for field tracer tests.     

Two‐dimensional solute transport for periodic flow in isotropic porous media: an analytical solution

In this article, a mathematical model is presented for the dispersion problem in finite porous media in which the flow is two‐dimensional, the seepage flow velocity is periodic, and dispersion parameter is proportional to the flow velocity. In addition to these, first‐order decay and zero‐order production parameters have also been considered directly proportional to the velocity. Retardation factor is taken into account in the present problem. First‐type boundary condition of periodic nature is considered at the extreme end of the boundary. Mixed‐type boundary condition is assumed at the origin of the domain. A classical mathematical substitution transforms the original advection–dispersion equation into diffusion equation in terms of other dependent and independent variables, with constant coefficients. Laplace transform technique is used to obtain the analytical solution. Copyright © 2011 John Wiley & Sons, Ltd.     

An exact solution of solute transport by one-dimensional random velocity fields

The problem of one-dimensional transport of passive solute by a random steady velocity field is investigated. This problem is representative of solute movement in porous media, for example, in vertical flow through a horizontally stratified formation of variable porosity with a constant flux at the soil surface. Relating moments of particle travel time and displacement, exact expressions for the advection and dispersion coefficients in the Focker-Planck equation are compared with the perturbation results for large distances. The first- and second-order approximations for the dispersion coefficient are robust for a lognormal velocity field. The mean Lagrangian velocity is the harmonic mean of the Eulerian velocity for large distances. This is an artifact of one-dimensional flow where the continuity equation provides for a divergence free fluid flux, rather than a divergence free fluid velocity.     

An exact solution of solute transport by one-dimensional random velocity fields

The problem of one-dimensional transport of passive solute by a random steady velocity field is investigated. This problem is representative of solute movement in porous media, for example, in vertical flow through a horizontally stratified formation of variable porosity with a constant flux at the soil surface. Relating moments of particle travel time and displacement, exact expressions for the advection and dispersion coefficients in the Focker-Planck equation are compared with the perturbation results for large distances. The first- and second-order approximations for the dispersion coefficient are robust for a lognormal velocity field. The mean Lagrangian velocity is the harmonic mean of the Eulerian velocity for large distances. This is an artifact of one-dimensional flow where the continuity equation provides for a divergence free fluid flux, rather than a divergence free fluid velocity.     

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.     

Analytical solutions of one-dimensional macrodispersion in stratified porous media by the quadrupole method: convergence to an equivalent homogeneous porous medium

In this article, the quadrupole method is implemented in order to simulate the effects of heterogeneities on one dimensional advective and diffusive transport of a passive solute in porous media. Theoretical studies of dispersion in heterogeneous stratified media can bring insight into transport artefacts linked to scale effects and apparent dispersion coefficients. The quadrupole method is an efficient method for the calculation of transient response of linear systems. It is based here on the Laplace transform technique. The analytical solutions that can be derived by this method assists understanding of upscaled parameters relevant to heterogeneous porous media.First, the method is developed for an infinite homogeneous porous medium. Then, it is adapted to a stratified medium where the fluid flow is perpendicular to the interfaces. The first heterogeneous medium studied is composed of two semi-infinite layers perpendicular to the flow direction each having different transport properties. The concentration response of the medium to a Dirac injection is evaluated. The case studied emphasises the importance in the choice of the boundary conditions.In the case of a periodic heterogeneous porous medium, the concentration response of the medium is evaluated for different numbers of unit-cells. When the number of unit cells is great enough, depending on the transport properties of each layer in the unit cell, an equivalent homogeneous behaviour is reached. An exact determination of the transport properties (equivalent dispersion coefficient) of the equivalent homogeneous porous medium is given.     

A transport phase diagram for pore-level correlated porous media

Transport in porous media is often characterized by the advection–dispersion equation, with the dispersion coefficient as the most important parameter that links the hydrodynamics to the transport processes. Morphological properties of any porous medium, such as pore size distribution, network topology, and correlation length control transport. In this study we explore the impact of correlation length on transport regime using pore-network modelling. Earlier direct simulation studies of dispersion in carbonate and sandstone rocks showed larger dispersion compared to granular homogenous sandpacks. However, in these studies, isolation of the impact of correlation length on transport regime was not possible due to the fundamentally different pore morphologies and pore-size distributions. Against this limitation, we simulate advection–dispersion transport for a wide range of Péclet numbers in unstructured irregular networks with “different” correlation lengths but “identical” pore size distributions and pore morphologies. Our simulation results show an increase in the magnitudes of the estimated dispersion coefficients in correlated networks compared to uncorrelated ones in the advection-controlled regime. The range of the Péclet numbers which dictate mixed advection–diffusion regime considerably reduces in the correlated networks. The findings emphasize the critical role of correlation length which is depicted in a conceptual transport phase diagram and the importance of accounting for the micro-scale correlation lengths into predictive stochastic pore-scale modelling.     

Shear dispersion in a fracture with porous walls

We present an analytical expression for the shear dispersion during solute transport in a coupled fracture–matrix system. The dispersion coefficient is obtained in a fracture with porous walls by taking into account an accurate boundary condition at the interface between the matrix and fracture, and the results were compared with those in a non-coupled system. The analysis presented identifies three regimes: diffusion-dominated, transition, and advection-dominated. The results showed that it is important to consider the exchange of solute between the fracture and matrix in development of the shear dispersion coefficient for the transition and advection-dominated regimes. The new dispersion coefficient is obtained by imposing the continuity of concentrations and mass fluxes along the porous walls. The resulting equivalent transport equation revealed that the effective velocity in a fracture increases while the dispersion coefficient decreases due to mass transfer between the matrix and fracture. A larger effective advection term leads to greater storage of mass in the matrix as compared with the classical double-porosity model with a non-coupled dispersion coefficient. The findings of this study can be used for modeling of tracer tests as well as fate, transport, and remediation of groundwater contaminants in fractured rocks.     

Monte Carlo evaluation of microbial-mediated contaminant reactions in heterogeneous aquifers

Monte Carlo simulations are conducted to evaluate microbial-mediated contaminant reactions in an aquifer comprised of spatially variable microbial biomass concentrations, aquifer hydraulic conductivities, and initial electron donor/acceptor concentrations. A finite element simulation model is used that incorporates advection, dispersion, and Monod kinetic expressions to describe biological processes. Comparisons between Monte Carlo simulations of heterogeneous systems and simulations using homogeneous formulation of the same two-dimensional transport problem are presented. For the assumed set of parameters, physical aquifer heterogeneity is found to have a minor effect on the mass of contaminant biodegraded/transformed when compared to a homogeneous system; however, it noticeably changes the dispersion, skewness, and peakness of contaminant concentration distributions. Similarly, for low microbial growth rate, given favorable microbial growth characteristics, biological heterogeneity has minor effect on the mass of contaminant biodegraded/transformed when compared to a homogeneous system. On the other hand, when higher effective growth rates are assumed, biological heterogeneity and spatial heterogeneities in essential electron donor/acceptors reduce the efficiency of biotic contaminant reactions; consequently, model simulations derived from heterogeneous biomass distributions predict remediation time scales that are longer than those simulated for homogeneous systems. When correlations between physical aquifer and biological heterogeneities are considered, the assumed correlation affects predicted mean and variance of contaminant concentration and biomass distributions. For example, an assumed negative correlation between hydraulic conductivity and the initial biomass distribution produces a plume where less efficient biotic contaminant reactions occur at the leading edge of the plume; this is consistent with less degradation/transformation occurring over regions of higher groundwater velocities. However, the presence and absence of these correlations do not appear to affect the efficiency of microbial-mediated contaminant attenuation.     

Lateral velocity variation related correction in asymptotic true‐amplitude one‐way propagators

It was mathematically proved that the asymptotic true‐amplitude one‐way wave equation could provide the same amplitude as the full‐wave equation in heterogeneous lossless media in the sense of high‐frequency asymptotics. Much work has been done on the vertical velocity variation related amplitude correction term but the lateral velocity variation related term has not received much attention, even being excluded in some asymptotic true‐amplitude one‐way propagator formulations. Here we analyse the effects of different amplitude correction terms in the asymptotic true‐amplitude one‐way propagator, especially the effect related to the lateral velocity variation, by comparing the wavefield amplitude from the one‐way propagator with that from full‐wave modelling. We derive a dual‐domain wide‐angle screen type asymptotic true‐amplitude one‐way propagator and evaluate two implementations of the amplitude correction. Numerical examples show that the lateral velocity variation related correction term can play a significant role in the asymptotic true‐amplitude one‐way propagator. Optimization of the expansion coefficients in the asymptotic true‐amplitude one‐way propagator can improve both the amplitude and phase accuracy for wide‐angle waves.     

Relaxation and reversibility of extended Taylor dispersion from a Markovian–Lagrangian point of view

Taylor dispersion in a two-dimensional (2D) stratified velocity field describes a transition, called relaxation, from convective behaviour for short times, towards Fickian behaviour for large times and is partially reversible upon reversal of the flow direction. In 2D the physics are assumed to be governed by the unidirectional convection diffusion equation (2D uCDE). The approximate height-averaged 1D Generalised Telegraph Equation (GTE) catches an essential part of the longitudinal spreading. Contrary to the 1D Fickian approach, it explicitly accounts for the transient reversible nature [Camacho J. Purely global model for Taylor dispersion. Phys Rev E 1993/2;48(1); Berentsen CWJ, Verlaan ML, van Kruijsdijk CPJW. Upscaling and reversibility of Taylor dispersion in heterogeneous porous media. Phys Rev E 2005;71:046308].