Solute Transport In Fractured Media Research Articles

Fluid flow and solute transport simulations in tight geologic formations: Discrete fracture network and continuous time random walk analyses

Understanding the effect of fracture network heterogeneity on flow and transport in tight formations with low permeability values is of great importance in a wide variety of applications, such as groundwater pollution and storage as well as water production. Although numerous studies have investigated solute transport in fractured media, investigation about scale effects on flow and transport in low-permeability aquifers is limited. More specifically, we are not aware of any study that has addressed how the continuous time random walk (CTRW) model parameters vary with scale and fracture network properties. To this end, we performed a set of numerical simulations in sparse fracture networks of sizes L = 20, 35, and 50 m under two different volumetric fracture counts (p30 = 0.05 and 0.1) using the discrete fracture network approach. The fractures in the networks were elliptical in shape, and their radii followed a truncated power-law distribution with exponents α = 1.5, 2, and 2.5. We simulated fluid flow using the Reynolds equation, and solute transport was simulated using particle tracking. The solute transport behavior was quantified by fitting the solution of the CTRW transport equation to the arrival time distributions. Results showed that as the truncated power law exponent α increased, the permeability of the networks decreased. In all the fracture networks studied, we found non-Fickian solute transport behavior deduced from small values of β(0.69≤β≤1.25), the exponent in the truncated power-law arrival time distribution in the CTRW model. In our fracture networks, the aperture size spanned less than one order of magnitude meaning that the fracture permeability distribution was relatively narrow. Therefore, the observed non-Fickian behavior should be attributed to sparse fracture networks and low connectivity. We found that the effective permeability, keff, at p30 = 0.05 was more scale-dependent compared to that at p30 = 0.1 because the fracture networks with p30 = 0.05 were less connected and, thus, more heterogeneous. Our simulation also disclosed the scale dependence of the CTRW model parameters. We conducted multiple linear regression analysis to link the CTRW model parameters to the network size as well as the geometrical and topological properties of the network. Results of regression analysis showed that all the CTRW model parameters except the average solute velocity were significantly dependent on the system size L (p-value < 0.05). Further investigations using a broader range of fracture network properties, such as broader aperture size distributions, smaller and larger network sizes and volumetric fracture counts are still required.

A probabilistic, flux-conservative particle-based framework for transport in fractured porous media

Advection dominated transport processes in sub-surface formations are characterized by discontinuities in the fields of transported quantities, and realistic predictions are challenging for Eulerian transport schemes because they suffer from numerical diffusion. Henceforth, we have focused on developing a Lagrangian particle-tracking scheme for modeling advective solute transport in fractured media. To this end, we adopt an Embedded Discrete Fracture Model (EDFM) for fractured media with a permeable matrix. The flexibility to use non-conformal fracture–matrix discretizations makes EDFMs a compelling choice in field-scale flow problems. Unaffected by the numerical diffusion, Lagrangian transport schemes complement the potential of EDFMs by allowing the use of sufficiently large grid cell sizes for flow field computations.In an EDFM framework, the inter-continuum fluid mass exchange cannot be quantified by the particle trajectories/pathlines due to the unresolved fracture–matrix interfaces and different dimensionalities of the matrix and fracture discretizations. These constraints motivate the use of a stochastic particle-tracking scheme, and thus, we formulated a pathline-specific probability of inter-continuum particle transfer based on mass conservation of an elementary solute/fluid mass. The particle’s transfer probability is calculated for the maximum residence time period in its associated fracture/matrix control volume, thus making the scheme time-adaptive. In addition, a conditional residence time distribution was derived, which dictates the timestamp of the particle transfer.First, we showcase that the tracking scheme preserves the initially homogeneous solute concentration field, suggesting that the probabilistic rules are consistent with the inter-continuum fluxes. Additionally, we illustrate the estimation of an evolving solute plume and compare the results with those of an Eulerian counterpart. The presented stochastic approach enables straightforward formulations of Lagrangian models for dynamic and sub-grid processes, e.g., solute interactions with the solid phase, the mapping of their effects onto large scale transport and additionally, be included in random walk models for dispersion.

Inverse modeling of flow tomography experiments in fractured media

The accurate characterization of the location, hydraulic properties, and connectivity of major fracture zones is essential to model flow and solute transport in fractured media. Cross‐borehole flowmeter tests, which consist of measuring changes in vertical borehole flows when pumping a neighboring borehole, were shown to be an efficient technique to provide information on the properties of the flow zones that connect borehole pairs. The interpretation of such experiments may, however, be quite uncertain when multiple connections exist. In this study, we explore the potential of flow tomography (i.e., sequential cross‐borehole flowmeter tests) for characterizing aquifer heterogeneity. We propose a framework for inverting flow and drawdown data to infer fracture connectivity and transmissivities. We demonstrate that successively exchanging the roles of pumping and observation boreholes improves the quality of available information and reduces the under‐determination of the problem. The inverse method is validated for several synthetic flow scenarios. It is shown to provide a good estimation of connectivity patterns and transmissivities of main flow paths. It also allows the estimation of the transmissivity of fractures that connect the flow paths but do not cross the boreholes, although the associated uncertainty may be high for some geometries.

Particle-tracking simulations of anomalous transport in hierarchically fractured rocks

Complex topology of fracture networks and interactions between transport processes in a fracture and the ambient un-fractured rock (matrix) combine to render modeling solute transport in fractured media a challenge. Classical approaches rely on both strong assumptions of either limited or full diffusion of solutes in the matrix and simplified fracture configurations. We analyze fracture-matrix transport in two-dimensional Sierpinski lattice structures, which display a wide range of matrix block sizes. The analysis is conducted in several transport regimes that are limited by either diffusion or block sizes. Our simulation results can be used to validate the simplifying assumptions that underpin classical analytical solutions and to benchmark other numerical methods. They also demonstrate that both hydraulic and structural properties of fractured rocks control the residence time distribution.

Experimental study of the non-darcy flow and solute transport in a channeled single fracture

The characterization of fracture rocks is always a key issue in understanding the flow and solute transport in fractured media. This article studies the solute transport in a Channeled Single Fracture (CSF), a single fracture with contact in certain areas. The flow in a CSF often has preferential pathways and the transport in a CSF often has Break Through Curves (BTCs) with long tails. The Surface Contact Ratio (SCR), the ratio of the contact area to the total fracture area, is an important indicator for the fracture surface roughness. To study the flow and solute transport in a CSF, a controlled physical model is constructed and a series of flow and tracer test experiments are carried out. Under our experimental conditions, the flow in a CSF is found to follow the Forchheimer equation J = av + bv 2, where J and v are the hydraulic gradient and the average pore velocity, respectively and a and b are two parameters related to the viscous and inertial flow components, respectively. Furthermore, it is found that b decreases with the decrease of SCR. For the solute transport, it is found that the BTCs often deviate from the traditional Fickian behavior, by the early-arrival and the long tailing. More interestingly, the observed BTCs often have a double-peak or a multi-peak, that would be difficult to explain using the existing transport theory such as the Advection-Dispersion Equation (ADE). In addition, the longitudinal dispersion coefficient D L is found to be scale-dependent in a CSF and the D L – l relationship is of exponential type.

Horizontal pre-asymptotic solute transport in a plane fracture with significant density contrasts

We investigate the dispersion of a finite amount of solute after it has been injected into the laminar flow occurring in a horizontal smooth fracture of constant aperture. When solute buoyancy is negligible, the dispersion process eventually leads to the well-known asymptotic Taylor–Aris dispersion regime, in which the solute progresses along the fracture at the average fluid velocity, according to a one-dimensional longitudinal advection–dispersion process. This paper addresses more realistic configurations for which the solute-induced density contrasts within the fluid play an important role on solute transport, in particular at small and moderate times. Flow and transport are coupled, since the solute distribution impacts the variations in time of the advecting velocity field. Transport is simulated using (i) a mathematical description based on the Boussinesq approximation and (ii) a numerical scheme based on a finite element analysis. This enables complete characterization of the process, in particular at moderate times for which existing analytical models are not valid. At very short times as well as very long times, the overall downward advective solute mass flow is observed to scale as the square of the injected concentration. The asymptotic Taylor–Aris effective dispersion coefficient is reached eventually, but vertical density currents, which are significant at short and moderate times, are responsible for a systematic retardation of the asymptotic mean solute position with respect to the frame moving at the mean fluid velocity, as well as for a time shift in the establishment of the asymptotic dispersion regime. These delays are characterized as functions of the Péclet number and another non-dimensional number which we call advective Archimedes number, and which quantifies the ratio of buoyancy to viscous forces. Depending on the Péclet number, the asymptotic dispersion is measured to be either larger or smaller than what it would be in the absence of buoyancy effects. Breakthrough curves measured at distances larger than the typical distance needed to reach the asymptotic dispersion regime are impacted accordingly. These findings suggest that, under certain conditions, density/buoyancy effects may have to be taken into consideration when interpreting field measurement of solute transport in fractured media. They also allow an estimate of the conditions under which density effects related to fracture wall roughness are likely to be significant.

Anthropogenic contaminants as tracers in fractured chalk aquifer: transport mechanisms and analytical modeling

One of the major problems encountered in characterizing fluid movement and solute transport in fractured media is how to deal with the high degree of complexity and heterogeneity in these media. To understand the physical transport mechanisms of contaminant in a single fracture, tracer tests were carried out using an artificially fractured chalk block (17 × 15 × 14 cm3). Sorptive and diffusive properties of chalk were evaluated by performing batch and diffusion experiments. The simulation of the breakthrough curves (BTCs) with the single fissure dispersion model could not give reasonable fit. A multi-channel dispersion model with the superposition of BTCs of at least two different flow paths resulted in an acceptable fit. Those simulations indicate the existence of flow channeling effects within the fracture. Comparing the BTCs of nitrate and eosin implied possible adsorption and/or degradation of nitrate within the fracture and the matrix. High dispersivity values are observed which might be explained as a result of spreading of the tracer pulses by diffusion into different flow channels and/or stagnant water zones. The description of the observed BTCs with a multi-channel model is still a theoretical suggestion and needs more investigations to be confirmed.

Solute transport in fractured media – analysis of non-reversibility in tracer tests

Abstract. This paper describes several small-scale (laboratory) experiments designed to simulate solute transport through fractured formations. A block of granite was broken to produce a fracture similar to those found in natural environments. Seven holes were drilled in the block to intersect the fracture. Later these holes functioned as either inlet or outlet points. All the possible combinations of pairs of inlet-outlet points were used to set up the tracer tests that provided the data analysed in this paper. The results indicate that reverse tracer tests do not necessarily provide symmetric results. Under some circumstances, the non-reversibility might be used to detect differences in the morphology of the fracture. The results also indicate that it is possible to estimate reasonably well the volume available for the circulation of the fluid by using transport models that neglect diffusion.

On mapping fracture networks onto continuum

Discrete fracture network (DFN) and stochastic continuum (SC) are two common modeling approaches used for simulating fluid flow and solute transport in fractured media. Fracture continuum approaches combine the merits of each approach; details of the fracture network are preserved, and a computationally efficient grid is utilized for the solution of fluid flow by assigning a conductivity contrast between the grid cells representing the rock matrix and those representing fractures. In this paper, we propose a fracture continuum approach for mapping individual fractures onto a finite‐difference grid as conductivity fields. We focus on several issues that are associated with this approach, such as enhanced connectivity between fractures that would otherwise not be in connection in a DFN simulation and the influence of grid cell size. To addresses these issues, both DFN and the proposed approach are used to solve for fluid flow through two‐dimensional, randomly generated fracture networks in a steady‐state, single‐phase flow system. The DFN flow solution is used as a metric to evaluate the robustness of the method in translating discrete fractures onto grid cell conductivities on four different regularly spaced grids: 1 × 1 m, 2 × 2 m, 5 × 5 m, and 10 × 10 m. Two correction factors are introduced to ensure equivalence between the total flow of the grid and the original fracture network. The first is dependent on the fracture alignment with the grid and is set to account for the difference between the length of the flow path on the grid and that of the fracture. The other correction is applied for areas in the grid with high fracture density and accounts for the artificial degree of connectivity that exists on the grid but not in the DFN. Fifteen different cases are studied to evaluate the effect of fracture statistics on the results of the proposed approach and by taking average results of 100 realizations in each case in a stochastic Monte Carlo framework. The flow equation is solved for the DFN, and total flow is obtained. The flow is also solved separately for the four‐grid resolution levels, and comparisons between the DFN and the grid total flows are made for the different cases and the different grid resolution levels. The approach performed relatively well in all cases for the fine‐grid resolution, but an overestimation of grid flow is observed in the coarse‐grid resolution, especially for cases wherein the network connectivity is controlled by small fractures. This overestimation shows minor variation from one realization to another within the same case. This allowed us to develop an approach that depends on solving limited number of DFN simulations to obtain this overestimation factor. Results indicate that the proposed approach provides improvements over existing approaches and has a potential to provide a link between DFN and SC models.

Reactive Solute Transport in a Nonstationary, Fractured Porous Medium: A Dual-Porosity Approach

A numerical method of moment is developed for solute flux through a nonstationary, fractured porous medium. Solute flux is described as a space-time process where time refers to the solute flux breakthrough and space refers to the transverse displacement distribution at a control plane. A first-order mass diffusion model is applied to describe interregional mass diffusion between fracture (advection) and matrix (nonadvection) regions. The chemical is under linear equilibrium sorption in both fracture and matrix regions. Hydraulic conductivity in the fracture region is assumed to be a spatial random variable. In this study, the general framework of Zhang et al.(2000) is adopted for solute flux in a nonstationary flow field. A time retention function related to physical and chemical sorption in the dual-porosity medium is developed and coupled with solute advection along random trajectories. The mean and variance of total solute flux are expressed in terms of the probability density function of the parcel travel time and transverse displacement. The influences of various factors on solute transport are investigated. These factors include the interregional mass diffusion rate between fracture and matrix regions, chemical sorption coefficients in both regions, water contents in both regions, and location of the solute source. In comparison with solute transport in a one-region medium, breakthrough curves of the mean and variance of the total solute flux in a two-region medium have lower peaks and longer tails. As compared with the classical stochastic studies on solute transport in fractured media, the numerical method of moment provides an approach for applying the stochastic method to study solute transport in more complicated fractured media.

Comment on “Determination of particle transfer in random walk particle methods for fractured porous media” by H. H. Liu et al.

Comment on “Determination of particle transfer in random walk particle methods for fractured porous media” by H. H. Liu et al.

A semi-analytical model for the simulation of solute transport in a network of fractures having random orientations

A semi-analytical model is developed which accounts for solute transport through a multi-dimensional network of fractures having random orientations and sparse distribution. Transport processes including hydrodynamic dispersion, matrix diffusion, retardation, decay and solute transfer at fracture intersections, are considered. The model is derived by application of the Laplace transform to the governing transport equation for each fracture element. The resulting ordinary differential equations are linked using robust descriptions of mass conservation at the fracture intersections. Solute concentrations are determined by numerically inverting the transformed equations in a sequential fashion. The model is verified using a formal mass balance and through comparison to existing solutions for solute transport in fractured media. To illustrate the use of the model, solute transport in a hypothetical network domain was simulated based on fracture conditions measured at a real field setting in a granitic rock.

The Channel Network Model and Field Applications

AbstractThe Channel Network model describes the fluid flow and solute transport in fractured media. The model is based on field observations, which indicate that flow and transport take place in a threedimensional network of connected channels. The channels are generated in the model from observed stochastic distributions and solute transport is modelled taking into account advection and rock interactions, such as matrix diffusion and sorption within the rock. The most important site-specific data for the Channel Network model are the conductance distribution of the channels and the flow-wetted surface. The latter is the surface area of the rock in contact with the flowing water. These parameters may be estimated from hydraulic measurements. For the Äspö site, several borehole data sets are available, where a packer distance of 3 metres was used. Numerical experiments were performed in order to study the uncertainties in the determination of the flowwetted surface and conductance distribution. Synthetic data were generated along a borehole and hydraulic tests with different packer distances were simulated.The model has previously been used to study the Long-term Pumping and Tracer Test (LPT2) carried out in the Äspö Hard Rock Laboratory (HRL) in Sweden, where the distance travelled by the tracers was of the order hundreds of metres. Recently, the model has been used to simulate the tracer tests performed in the TRUE experiment at HRL, with travel distance of the order of tens of metres. Several tracer tests with non-sorbing and sorbing species have been performed.

A hierarchical model for solute transport in fractured media

A hierarchical concept for modeling solute transport in a well‐connected fracture network is presented. The approach exploits the presence of dominating fractures to separate a fracture network into a small group of primary fractures that are represented explicitly and a much larger group of fractures that are included within network blocks. The primary fracture network is identified prior to a flow solution using deterministic rules that link fracture characteristics with the likelihood that a fracture may dominate its local flow regime. Equivalent conductance matrices unique to each network block closely approximate the hydraulic connections within the network blocks. This hierarchical structure leads to the formulation of much smaller systems of equations for calculating the flow regime when compared to conventional discrete network simulation. A particle‐tracking scheme is developed that takes advantage of the hierarchical structure in modeling solute transport. Fracture systems considered include Poisson models; war‐zone models; and a fractal, Levy‐Lee model. As the range in the length scales of the fracture sets becomes broader or the variation in the transmissivity of individual fractures increases, it becomes more likely that individual fractures will dominate their local flow regime, that transport through a fracture network becomes less like that through a granular porous medium, and that a hierarchical model will provide a better approximation of transport. Poisson models with a bimodal length scale, and war‐zone and fractal models, which lead to multiscale networks, can be effectively represented by a hierarchical model. For these systems the loss in accuracy due to approximations underlying the separation of the network into primary fractures and network blocks is small relative to the uncertainty inherent in the prediction of transport through fractured media. The hierarchical approach provides a better simulation strategy than one based on elimination of the least‐transmissive fractures from the model structure.

Solute transport in fractured media — the important mechanisms for performance assessment

The most important mechanisms that control the release of contaminants from a repository for nuclear or chemical waste have been studied. For the time scale of interest for the disposal of nuclear and even chemical waste, diffusion into the rock matrix is an important factor which retards and dilutes the contaminants. It is found that the water flow-rate distribution and the flow-wetted surface are the entities that primarily determine the solute transport. If the diffusion into the rock matrix is negligible, the solute transport is determined by the water flow rate and the flow porosity. This is shown using analytical solutions obtained for simple geometries, such as the flow in a fracture with constant aperture or a channel. Similar results are obtained for more complex systems, such as flow in a fracture with variable aperture and through a network of channels. It is also found that the use of a retardation factor relating the travel times of interacting and noninteracting solutes is inappropriate and may be misleading.

Transport of solutes in fractured media, based on a channel network model : B. Gylling, L. Moreno & I. Neretnieks, in: Groundwater quality: remediation and protection. proc. conference, Prague, 1995, ed K. Kovar & J. Krasny, (IAHS; Publication, 225), 1995, pp 107–113

Abstract Fluid flow and solute transport in fractured media take place mainly through preferential paths in the fractures. Solutes are transported through these paths or channels and they may diffuse into the stagnant water in the porous matrix. This is a very important interaction mechanism for long times. They also may be sorbed within the rock matrix. A new type of model concept for describing flow and transport in fractured rock has been devised which specifically emphasizes the matrix interaction. The flowing water in the rock is envisaged to take place in a three-dimensional network of channels with stochastic properties. The important transport parameters are summarized and how they may be obtained in the field is described. A long term pumping test with tracers at the Swedish Âspo rock laboratory has been predicted using this model. The main tracer test was predicted reasonably well without using any adjustable parameters. INTRODUCTION AND BACKGROUND Fluid flow and solute transport in fractured media occur mainly in the fractures. Due to the variable aperture and heterogeneities on the fracture surfaces the fluid flow will seek out preferential paths. Solutes are transported through these paths. The solutes may migrate from the water in the fracture into the stagnant water in the porous matrix. They also may be sorbed on the micro surfaces within the rock matrix. These two mechanisms strongly influence the residence time distribution, RTD, in fractured media. We call the flow in the preferred paths channelling. The channels need not be physical entities, the water may seek out other paths if the gradient is changed. For a given gradient one can, however, visualize the flow paths as more or less distinct channels. This is often seen in drifts and tunnels in fractured rocks. Intuitively we can conceive that in a narrow channel with large flow there is little surface area (Flow Wetted Surface, FWS) over which the solute can interact with the porous matrix. In a channel with a large FWS and low flow, the solute may have time to diffuse into the porous matrix and thus access more water volume where it can reside. The RTD can be considerably influenced by this process (Neretnieks, 1980). A new type of model concept for describing flow and transport in fractured rock has been devised. The flowing water in the rock is envisaged to take place in a three-dimensional network of channels with stochastic properties. The channels are characterized by lengths, widths, apertures and transmissivities. The rock matrix is assumed to be porous and the solute can diffuse in and out of the matrix. For long term

Solute transport through fractured media: 1. The effect of matrix diffusion

Solute transport through fractured media is described by numerically combining advective‐dispersive transport, which is dominant in the fractures, and diffusive transport, which is usually dominant in the unfractured matrix. Transport is considered in a manner conceptually similar to ‘double‐porosity’ or ‘intra‐aggregate’ transport models. A finite element model is developed for simulating nonreactive and reactive solute transport by advection, mechanical dispersion, and diffusion in a unidirectional flow field. The effect of the value of the solute diffusion coefficient in the matrix (termed the matrix diffusion coefficient) is illustrated by solute breakthrough curves and concentration profiles in the fracture as well as in the matrix. The illustrated conditions are similar to the laboratory tracer study on fractured till described in the accompanying paper (Grisak et al.). The effects on solute transport of fracture aperture size, water velocity in the fracture, matrix porosity, matrix distribution coefficient, and dispersivity in the fracture are illustrated with breakthrough curves and concentration profiles. The net effect of large matrix diffusion coefficients and/or large distribution coefficients in the matrix is to reduce significantly the effective solute velocity in the fracture. Reduction in the effective solute velocity can also be seen to be possible in materials with low matrix porosities such as crystalline rocks. The aperture size, matrix porosity, matrix diffusion coefficient, and distribution coefficient, all are important in determining the relative amounts of solute transported in the fracture and stored in the matrix. Implications with regard to aquifer recharge, groundwater chemistry, contaminant transport, tracer tests, and groundwater age dating in fractured media are discussed. The numerical model and the laboratory tracer test data provide considerable insight into the processes controlling solute transport in fractured media.