In this work, a new solution is developed for the problem of contaminant transport in a single fracture-matrix system, where the first-order reaction rate constants are different in both fracture and matrix. It takes a form of convolution with three functions as a basis to consider different transport mechanisms separately. The statistical nature of the three functions, as well as the interpretation of the solution as a marginal probability distribution in the case of no first-order reactions, allows us to develop a simple Time Domain Random Walk (TDRW) algorithm to calculate the breakthrough curves at a given point of observation downstream the fracture. Compared with the existing versions of the TDRW algorithm, it is superior not only in the physical reasoning and statistical interpretations but also in its numerical implementations. In addition, the developed algorithm can not only be used to estimate the distribution profile of the contaminant concentration along the fracture but also the concentration within the matrix, since the analytical solution to contaminant concentration in the matrix also takes a convolution form of three functions. Also, the distribution profile of contaminant concentration within the matrix can readily be determined by the use of our TDRW algorithm. To validate the developed algorithm, three benchmark cases are considered for either nuclide or colloid transport through a fractured rock. The results show that TDRW algorithm is superior to the Gaussian quadrature solution, but similar to inverse Laplace transform solution, in computational expense when nearly identical results are obtained. However, the Monte Carlo nature of the TDRW algorithm implies that the accuracy of the computational result is related to the number of particles applied in the simulation, which might make the obtained results fluctuated.

Three cationic tracers, Sr2+, Co2+ and Cs+ were tested with a modified electromigration device by applying 2V, 3V and 4V voltage gradients over an intact Grimsel granodiorite rock sample. An ideal plug-flow model and an advection-dispersion model were applied to analyze the breakthrough curves. Matrix characterization by C-14-PMMA autoradiography and scanning electron microscopy showed that in the centimeter scale of Grimsel granodiorite rock, the interconnected matrix porosity forms a well-connected network for diffusion. Micrometer-scale fissures are transecting biotite and chlorite minerals, indicating sorption of the studied cations. The ideal plug-flow model indicated that the effective diffusion coefficients (D-e values) for Sr2+, Co2+ and Cs+ tracer ions within the Grimsel granodiorite rock were 3.20 x 10(-13) m(2)/s, 1.23 x 10(-13) m(2)/s and 2.25 x 10(-12) m(2)/s, respectively. D-e values were also derived from the advection-dispersion model, from which 2.86 x 10(-13) m(2)/s, 1.35 x 10(-13) m(2)/s and 2.26 x 10(-12) m(2)/s were calculated. The diffusion speed for the tracers was in the sequence of Cs+ > Sr2+ > Co2+ that is in the same sequence as their diffusion in diluted water. The distribution coefficients (K-d values) calculated from the models covered the range of two magnitudes (from 10(-7) m(3)/kg to 10(-5) m(3)/kg). The result indicated that the sorption process of the studied elements did not reach equilibrium during the electromigration process, mainly due to the too much acceleration of the migration speed by the voltage gradients applied over the rock sample.

This study shows a comparison and analysis of results from a modelling exercise concerning a field experiment involving the transport and retention of different radionuclide tracers in crystalline rock. This exercise was performed within the Swedish Nuclear Fuel and Waste Management Company (SKB) Task Force on Modelling of Groundwater Flow and Transport of Solutes (Task Force GWFTS). Task 9B of the Task Force GWFTS was the second subtask within Task 9 and focused on the modelling of experimental results from the Long Term Sorption Diffusion Experiment in situ tracer test. The test had been performed at a depth of about 410m in the Aspo Hard Rock Laboratory. Synthetic groundwater containing a cocktail of radionuclide tracers was circulated for 198 days on the natural surface of a fracture and in a narrow slim hole drilled in unaltered rock matrix. Overcoring of the rock after the end of the test allowed for the measurement of tracer distribution profiles in the rock from the fracture surface (A cores) and also from the slim hole (D cores). The measured tracer activities in the rock samples showed long profiles (several cm) for non-or weakly-sorbing tracers (Cl-36, Na-22), but also for many of the more strongly-sorbing radionuclides. The understanding of this unexpected feature was one of the main motivations for this modelling exercise. However, re-evaluation and revision of the data during the course of Task 9B provided evidence that the anomalous long tails at low activities for strongly sorbing tracers were artefacts due to cross-contamination during rock sample preparation. A few data points remained for Cs-137, Ba-133, Ni-63 and Cd-109, but most measurements at long distances from the tracer source (>10mm) were now below the reported detection limits. Ten different modelling teams provided results for this exercise, using different concepts and codes. The tracers that were finally considered were Na-22, Cl-36, Co-57, Ni-63, Ba-133, Cs-137, Cd-109, Ra-226 and Np-237. Three main types of models were used: i) analytical solutions to the transport-retention equations, ii) continuum -porous-medium numerical models, and iii) microstructure-based models accounting for small-scale heterogeneity (i.e. mineral grains, porosities and/or microfracture distributions) and potential centimetre-scale fractures. The modelling by the different teams led to some important conclusions, concerning for instance the presence of a disturbed zone (a few mm in thickness) next to the fracture surface and to the wall of the slim hole and the role of micro-fractures and cm-scale fractures in the transport of weakly sorbing tracers. These conclusions could be reached after the re-evaluation and revision of the experimental data (tracer profiles in the rock) and the analysis of the different sets of model results provided by the different teams.

The SKB GroundWater Flow and Transport of Solutes Task Force is an international forum in the area of conceptual and numerical modeling of groundwater flow and solute transport in fractured rocks relevant for the deep geological disposal of radioactive waste. Two in situ matrix diffusion experiments in crystalline rock (gneiss) were performed at POSIVA’s ONKALO underground facility in Finland. Synthetic groundwater containing several conservative and sorbing radiotracers was injected at one end of a borehole interval and flowed along a thin annulus toward the opposite end. Several teams performed predictive modeling of the tracer breakthrough curves using “conventional” modeling approaches (constant diffusion and sorption in the rock, no or minimum rock heterogeneity). Supporting information, derived from small-scale laboratory experiments, was provided. The teams were free to implement different concepts, use different codes, and apply the transport and retention parameters that they considered to be most suited (i.e., not a benchmark exercise). The main goal was the comparison of the different sets of results and the analysis of the possible differences for this relatively simple experimental setup with a well-defined geometry. Even though the experiment was designed to study matrix diffusion, the calculated peaks of the breakthrough curves were very sensitive to the assumed magnitude of dispersion in the borehole annulus. However, given the very different timescales for advection and matrix diffusion, the tails of the curves provided information concerning diffusion and retention in the rock matrix regardless of the magnitude of dispersion. In addition, although the task was designed to be a blind modeling exercise, the model results have also been compared to the measured experimental breakthroughs. Experimental results tend to show relatively small activities, wide breakthroughs, and early first arrivals, which are somewhat similar to model results using large dispersivity values. 

To determine the diffusion and sorption propertiesof radionuclides in intact crystalline rocks, a newelectromigration devicewas built and tested by running with I-and Se(IV) ions. By introducing a potentiostatto impose a constant voltageover the studied rock sample, the electromigration device cangive more stable and accurateexperimental resultsthan those from the traditional electromigration devices.In addition, the variation in the pHofthe background electrolytes wasminimised by adding a small amount of NaHCO3as buffers.To interpret the experimental results with moreconfidence, anadvection-dispersion model was also developed in thisstudy, which accounts for the most important mechanisms governing ionic transport in the electromigration experiments.Data analysis of the breakthrough curves by the advection-dispersion model, instead of the traditional ideal plug-flowmodel,suggest that the effective diffusivitiesof I-and Se(IV)are (1.15±0.06) ×10-13m2/s and (3.50±0.86) ×10-14m2/s, respectively. The results also show thatI-is more mobile than Se(IV) ions when migrating through the sameintact rock sampleand that theirsorption properties are almost identical.

An advection-dispersion model was developed for interpreting the experimental results of electromigration in granitic rock cores. The most important mechanisms governing the movement of the tracer ions, i.e. electromigration, electroosmosis and dispersion were taken into account by the advection-dispersion model, but the influence of aqueous chemistry was ignored. An analytical solution in the Laplace domain was derived and then applied to analyze the measured results of a series of experiments, performed in an updated experimental device using different applied voltages. The modelling results suggested that both studied tracers, i.e. iodide and selenite, are effectively non-sorbing in the intact rock investigated. The effective diffusivities and formation factors evaluated from the model were also found to be in good agreement with data reported in literature and the associated uncertainties are much smaller than those obtained from the classical ideal plug-flow model, which accounts only for the dominant effect of electromigration on ionic transport. To explore further how the quality of parameter identifications would be influenced by neglect of aqueous chemistry, a reactive transport model was also implemented, which may be regarded as a multi-component version of the advection-dispersion model. The analysis showed that the advection-dispersion model works equally well as the reactive transport model but requires much less computational demand. It can, therefore, be used with great confidence to interpret the experimental results of electromigration for studies of diffusion and sorption behavior of radionuclides in intact rock cores.

In order to facilitate the assessment of the safety and function of deep geological repositories for radioactive waste, several models have been developed to describe water flow and transport of solutes in fractured crystalline rock. The rock around the repository is described and modelled as a network of water-bearing fractures.

The first part of the work concerns analytical solutions of the mathematical models, first developed in the 1980s to describe nuclide transport with seeping water in the fractures where the nuclides can also diffuse in and out of the pores into the rock matrix. A new simple analytical solution is described in which the interaction between matrix diffusion and hydrodynamic dispersion could be decoupled, which makes the interaction between the processes visible while making the solution more manageable. In addition, another dispersion mechanism caused by the presence of independent transport paths is easily handled with the new model. This makes it possible to treat both dispersion mechanisms with the same formalism. This makes the new model more useful in interpreting field experiments with tracer as well as for long-term simulation of nuclide migration in rock.

The second part of the work is about molecular diffusion in the rock matrix itself, which is a central mechanism in the model above. One way to measure diffusion and sorption in rock pieces is to force ions through the pores of the rock by means of electromigration. The method previously used has been improved by adding a potentiostat and a pH buffer. The experimental results become more stable.

To better interpret the results, a general model for transport in the rock matrix was developed. The model includes electromigration, electroosmosis and dispersion in the pore system. The effective pore diffusivity and matrix formation factor can be determined from the experiments. The results show that the developed electromigration method can be used to provide high quality experimental data.

För att underlätta bedömning av säkerhet och funktion hos djupa geologiska förvar för radioaktivt avfall har flera modeller utvecklats för att beskriva vattenflöde och transport av lösta ämnen i kristallint berg med sprickor. Berget kring förvaret beskrivs och modelleras som ett nätverk av vattenförande sprickor.

Den fösta delen av arbetet handlar om analytiska lösningar av de matematiska modellerna, utvecklades på 1980-talet för att beskriva nuklidtransport med sipprande vatten i sprickorna där nukliderna även kan diffundera in och ut ur porerna in bergmatrisen. En ny enkel analytisk lösning beskrivs i vilken samverkan mellan hydrodynamisk dispersion och matrisdiffusion kunnat frikopplas, vilket gör att samverkan mellan processerna synliggörs samtidigt som lösningen är mer hanterbar. Dessutom kan en annan dispersionsmekanism orsakad av närvaron av oberoende transportvägar med lätthet hanteras med den nya modellen. Detta gör det möjligt att behandla både dispersionsmekanismer med samma formalism. Detta gör den nya lösningen mer användbar vid tolkningen av fältförsök med spårämnen liksom för långsiktig simulering av nuklidspridning i berg.

Den andra delen av arbetet handlar om molekylär diffusion i bergmatrisen vilket är en central mekanism i modellen ovan. Ett sätt att mäta diffusion och sorption i bergstycken bygger på att driva in joner i bergets porer av med hjälp elektromigration. Den tidigare använda metoden har förbättrats genom att lägga till en potentiostat och pH-buffert. De experimentella resultaten blir därvid mer stabila.

För att bättre tolka resultaten utvecklades en generell modell för transport i bergmatrisen. Modellen inbegriper elektromigration, elektroosmos och dispersion i porsystemet. Den effektiva por-diffusiviteten och matrisens formationsfaktor kan bestämmas ur experimenten. Resultaten visar att den utvecklade elektromigreringsmetoden kan användas för att ge experimentella data av hög kvalitet.

A new simple and robust solution, based on uniform steady-state flow velocity, is developed for the problem of solute transport in a fracture-matrix system with a finite penetration depth of a radioactive contaminant into the rock matrix. The solution is an extension of Liu et al. (2017) to finite penetration depth and an alternative solution strategy to the problem solved by Sudicky et al. (1982). The solution takes the form of a convolution of two functions. The first function describes the probability density function of the residence time distribution of a conservative solute resulting merely from advection and Fickian dispersion. The second function is actually the impulse response of the fracture-matrix system in the case of a plug flow without any hydrodynamic dispersion. As a result, the effects of Fickian dispersion and matrix diffusion on solute transport are decoupled, and thus the resulting breakthrough curve can be analyzed in terms of those two functions. In addition to this, the derived Péclet numbers of those two functions, based on temporal moments, are also found to be associated with the derived Péclet number of the resulting breakthrough curve. By comparing the Péclet numbers of those two functions, the contribution of Fickian dispersion and matrix diffusion to solute spreading is determined in a straightforward way. This can aid to find out the dominating mechanism on solute transport, and therefore the performance of breakthrough curve.

A simple and robust solution is developed for the problem of solute transport along a single fracture in a porous rock. The solution is referred to as the solution to the single-flow-path model and takes the form of a convolution of two functions. The first function is the probability density function of residence-time distribution of a conservative solute in the fracture-only system as if the rock matrix is impermeable. The second function is the response of the fracture-matrix system to the input source when Fickian-type dispersion is completely neglected; thus, the effects of Fickian-type dispersion and matrix diffusion have been decoupled. It is also found that the solution can be understood in a way in line with the concept of velocity dispersion in fractured rocks. The solution is therefore extended into more general cases to also account for velocity variation between the channels. This leads to a development of the multi-channel model followed by detailed statistical descriptions of channel properties and sensitivity analysis of the model upon changes in the model key parameters. The simulation results obtained by the multi-channel model in this study fairly well agree with what is often observed in field experiments—i.e. the unchanged Peclet number with distance, which cannot be predicted by the classical advection-dispersion equation. In light of the findings from the aforementioned analysis, it is suggested that forced-gradient experiments can result in considerably different estimates of dispersivity compared to what can be found in natural-gradient systems for typical channel widths.