Analysis of Radionuclide Transport through Fracture Networks by Percolation Theory

Transcription

1 Journal of NUCLEAR SCIENCE and TECHNOLOGY, 28[5], pp. 433~446 (May 1991). 433 Analysis of Radionuclide Transport through Fracture Networks by Percolation Theory Joonhong AHN, Yutaka FURUHAMA, Yadong LI and Atsuyuki SUZUKI Department of Nuclear Engineering, University of Tokyo* Received November 22, 1990 Presented are results of numerical simulations for radionuclide diffusion through fracture networks in geologic layers. Actual fracture networks are expressed as two-dimensional honeycomb percolation lattices. Random-walk simulations of diffusion on percolation lattices are made by the exact-enumeration method, and compared with those from Fickian diffusion with constant and decreasing diffusion coefficients. Mean-square displacement of a random-walker on percolation lattices increases more slowly with time than that for Fickian diffusion with the constant diffusion coefficient. Though the same relation of mean-square displacement vs. time as for the percolation lattices can be obtained for a continuum with decreasing diffusion coefficients, spatial distribution of probability densities of finding the random-walker on the percolation lattice differs from that on a continuum with the decreasing diffusion coefficient. The percolation model results in slow spreading near the origin and fast spreading in the outer region, whereas the decreasing-diffusion coefficient model shows the reverse because of smaller diffusion coefficient in the outer region. We could derive a general formula that can include both Fickian and anomalos diffusion in terms of fractal and fracton dimensionalities and the anomalous diffusion exponent. KEYWORDS: radionuclide migration, translocation percolation theory, anomalous diffusion, diffusion coefficient, high-level radioactive wastes, fracture network, geologic fractures 1. Background I. INTRODUCTION This paper presents the results of numerical simulations for radionuclide transport through fractured geologic media by the percolation theory in the field of statistical physics. Previously, many efforts have been made for predicting radionuclide transport through fractured hard-rock formation to assess the performance of deep geologic repositories for high-level radioactive wastes. For a watersaturated fractured porous medium, two approaches can be considered (see Fig. 1). In one approach it is assumed that regional flow through large volumes of fractured rock can not be analyzed by describing each of the discrete fractures deterministically, and that continuum or equivalent-porous-media analysis can be used(1). If such an equivalent porous medium exists, then we can apply earlier analyses of mass transfer and transport through a porous medium(2)~(5). In the other approach fractures of large aperture are considered to be of special importance to safety assessment of waste disposal. Fractures are divided into principal and minor fractures depending on their apertures. Principal fractures are considered as main paths to the biosphere, whereas minor fractures are considered as pores together with other void spaces in rock matrix. To have an equivalent porous medium, a fractured medium of interest must have the following properties : (1) fracture number density is sufficiently high, (2) fractures have * Hongo, Bunkyo-ku, Tokyo

2 434 J. Nucl. Sci. Technol., Fig. 1 Approaches for transport through fractured geologic media that have been applied in previous studies similar apertures rather than broadly-distributed apertures, (3) fractures have distributed orientations rather than uniform orientation, and (4) the medium has a representative elementary volume (REV)(1), over which we can average various rock properties such as porosity and density. Studies of migration in discrete fractures with matrix diffusion were begun by Neretnieks(6) in early 1980's. He first pointed out the importance of diffusion of radionuclides from the principal fractures to the surrounding rock matrix in the context of the waste disposal. In Neretnieks's model fractures are simplified as planar parallel conduits of infinite extent. Molecular diffusion of the contaminant from fractures to the surrounding rock matrix was pointed out to be an important retention mechanism for transport in fractures. Rasmuson & Neretnieks(7) and Tang et al.(8) studied the effect of longitudinal hydrodynamic dispersion with a mathematical model and parameter sensitivity studies. Sudicky & Frind(9) showed the effect of neighboring parallel fractures by solving the mathematical problem analytically. Effects of radioactive decay chains of multiple members are studied by Chambre et al.(4) and Sudicky & Frincl(10). Effects of arrayed sources intersected by a planar fracture is discussed by Ahn et al.(11) One important feature of transport through fractures which has never been discussed explicitly in the previous analytical studies is the intersection of fractures. Usually a fracture network has been incorporated into analytical models by considering that for a large distance repeated intersections of fractures make the contaminant well-mixed among these fractures, so that one averaged fracture can represent the transport phenomena. The effect of intersection, then, considered to be treated as dispersion in mathematical models. Although these models are rather simple, the approaches are useful for the performance assessment of the high-level radioactive waste repositories if these simplifications are introduced carefully so that the resulting models can give conservative predictions(12). However, proving that the simplified models are actually bounding might not be that easy. Simplifying a complex fracture network to an equivalent porous medium or to a planar - 54-

3 Vol. 28, No. 5 (May 1991) 435 parallel infinite fracture is not an exception. We need to investigate radionuclide transport through a fracture network based on more realistic models of fracture networks before we apply the above-mentioned two models to the bounding performance assessment. 2. Nature of Fracture Networks To establish a more realistic model for transport through fracture networks, let us observe nature of fracture networks once more. Fracture networks have been studied for many years by seismologists. Chelidze(13) assumed that crack concentration in rock mass under nearly constant stress increases linearly with time. He found that increase of radon concentration in time in underground water before the earthquake in Tashkent in 1966 could be well explained by the percolation model, which will be explained below. For hydrologists, hydraulic conductivity of a fracture network has been a major problem. de Marsily(14) gave interesting observations. Since 1976, the year of great drought in Northern Europe, a large number of 50 m deep wells have been drilled in two large areas of crystalline rock in France. Even though in both areas many fractures were observed to intersect wells, wells in one area were very successful whereas in the other area the result was quite negative. He claimed that for such a case the concept of REV is useless, and that the connectivity of fractures must be taken into account. Recently, in Stripa Project, Sweden, three discrete-fractureflow models from different institutions are being cross-verified(15). All three models are established starting from the fact that actual fracture networks consist of fractures of finite size. They model fracture segments as ellipses, rectangles, circles and polygons. Fracture segments are generated and dispersed in a finite model space by a computer based on some assumed distribution functions for size, transmissivity and orientation. Preliminary results for water flow through a fracture network show good agreement with measurements at the Stripa mine. These observations indicate that fundamental nature of fracture networks is originated from the fact that a fracture network consists of a large number of finite segments, which implies strong connection between a fracture network and the percolation theory. 3. Purpose of Present Study The purposes of this paper are to show the anomaly of diffusion process in a fracture network, which could affect consequences of the performance assessment of high-level radioactive waste repositories, with the help of the percolation theory and the randomwalk simulations of diffusion process on percolation clusters and to propose a new method for predicting radionuclide transport through a fracture network. II. RANDOM-WALK SIMULA- TIONS FOR DIFFUSION PERCOLATION LATTICES 1. Generation of Percolation Lattices ON The history of the percolation theory can go back to Flory(16) who used it to describe how small molecules form huge macromolecules. His theory corresponds to the current percolation theory for the Bethe lattice. The start of the percolation theory is associated with Broadbent & Hammersley(17) in 1957 which introduced the name and dealt with it more mathematically, using the geometrical and probabilistic concepts. Since then, the percolation theory has been found useful to characterize many disordered systems, such as porous media, fragmentation and fractures, random-resistor insulator systems, forest fire and epidemics(18) Let Zd be a regular lattice in d-dimensional space, and r be a number satisfying 0<r<1. Probability r is called concentration in the percolation theory. Figure 2 shows an example of percolation clusters generated in a two-dimensional honeycomb lattice. For each vertex of Zd, we generate a random number which is normalized so that it uniformly ranges between zero and unity. We declare this vertex to be occupied if the generated random number is less than probability p and empty otherwise, independently of all other vertices. We connect two neighboring occupied vertices by an open edge of

4 436 J. Nuci. Sci. Technol., an arbitrary vertex belonging to the infinite cluster. The Poo is zero for r<rc whereas above rci the strength increases with r asa Pr~ (p-pc)b, P>Pc, ( 1 ) A random-walker can only move through edges that connect with the origin of the system. The rest of the occupied vertices have nothing to do with diffusion. Fig. 2 Honeycomb percolation lattice generated at concentration p=0.70 unit length. Then, nearest neighbors form clusters of edges. The clusters of open edges of Zd represent a hypothetical network of fractures. The resulting process is called a 'site' model since the random blockages in the lattice are associated with vertices. Another type of percolation process is the 'bond' model, where edges rather than vertices are declared to be open or closed at random. It is well known that every bond model may be reformulated as a site model on a different lattice, but that the converse is false(19). Thus, because site models are more general than bond models, we adopt the site model in the following theoretical development. As r increases, the average size of clusters increases. There exists a probability threshold rc at which an infinite cluster appears in an infinite lattice. Because one can only deal with finite systems in computer simulations, one cannot in general obtain a sharply defined threshold unless the threshold is obtained by an exact mathematical proof such as rc=1/2 for site percolation of a two-dimensional triangular lattice. The percolation transition at rc is described by the strength of the infinite network Poo of where the symbol ~ indicates that the relation is loosely defined because the equation is a conjecture based on numerical studies. The symbol can be read as "goes as". It can mean anything from the ratio of the two sides tends to unity to the ratio of the logarithms of the two sides tends to unity. Another important exponent is related to the correlation length x, which can be considered as the mean distance between two vertices belonging to the same cluster. When approaches rci x diverges as r ( 2 ) The exponent b and n are universal and depend only on the space dimensionality d but not on the lattice structure. For example, the exponents are known exactly to be b=5/36 and n=4/3 for d=2, and numerically to be =0.4, and n=0.9 for d=3(20). b For p<pc, there appear clusters of finite size with a typical linear size x(p). Clusters have a statistical internal self-similarity property, which is characteristic of fractal. At P=Pc, x blows up to infinity, which means that the infinite cluster is self-similar everywhere. The fractal dimensionality df of the infinite cluster at p=p, has been shown to be theoretically(28). With the above-mentioned ( 3 ) numerical values for the exponents, the fractal dimension for the infinite cluster at the threshold can be calculated as for a twodimensional space. For p> p there are the infinite cluster and finite clusters. The correlation length in this domain is finite. The finite correlation length for p>pc can be interpreted as a typical length up to which the infinite cluster is statistically self-similar. More specifically, the correlation length can be considered as the limit below which the number Al of occupied vertices in radius R - 56-

5 Vol. 28, No. 5 (May 1991) 437 c an be scaled as M~Rdf for R<x(P) ( 4 ) and above which the medium can be regarded as a homogeneous object ; M~Rd for R>x(p). ( 5 ) Thus, the infinite cluster can be modelled as a collection of fractal unit cells of length (p). The correlation length can be xinter- preted as the size of the representative elementary volume in the hydrological sense, the concept of which is introduced so that actually complex tortuous network can be handled as a continuum. Assuming that a fracture network can be considered as percolation clusters, we may also apply the above theory to a fracture network. Because the above-mentioned theory is known to be applicable for both d=2 and =3, we can take our model space in a twodimensional space for simplicity. We can show anomaly of diffusion process in percolation clusters without loss of generality with a two-dimensional model, and can achieve faster convergence in Monte Carlo simulations than with a three-dimensional model. We adopt in the present study a plane honeycomb lattice as shown in Fig. 2. According to the recent numerical estimates, the threshold pc is known to be for site percolation of the twodimensional honeycomb lattice(20). 2. Ant in Labyrinth Model Diffusion is characterized by the meansquare displacement, <r2(t)> after t steps of random walk and the probability density P(g, t) of finding a random-walker (or, an ant) at g at time step t, having started at the origin initially, which can be numerically obtained by random-walk simulations. For the Euclidean space, we have the well-known result for normal diffusion that <r2(t)>=t, ( 6 ) if we set the diffusion constant to be 1/4 for the d=2 case. For the percolation clusters, because of complexity of the medium, diffusion is expected to be retarded. The anomaly can be expressed by the anomalous diffusion exponent d. as <r2(t)>~t2/dw. (7) The exponent dw becomes 2 for normal diffusion, recovering Eq. ( 6 ), and is greater than 2 for anomalous diffusion. The mean-square displacement can be calculated by ( 8 ) The probability density for normal diffusion on two-dimensional space is the Green's function for the diffusion equation : ( 9 ) where we assume that the lattice constant (the distance travelled in unit time) is equal to unity. Substituting Eq. ( 9 ) into Eq. ( 8 ) recovers Eq. ( 6 ). We will later show numerically that this simple Gaussian behavior does not generalize to anomalous diffusion on the percolation lattice. The number of vertices an ant has visited after t steps of random walks is proportional to <r2(t)>df/2 by Eq. ( 4 ), and so to tdf/dw' by Eq. ( 7 ). Hence, the probability of returning to the origin at t, or the probability of finding the ant at the origin can scale as (10) We define here fracton (or spectral) dimensionality ds as (11) With the fracton dimensionality the probability of finding the ant at the origin can be written as P(0, t)~t-d3/2. (12) The fracton dimensionality characterizes the number of distinct sites S(t) visited by the random walker, as can be seen from Eq. (12): S(t)~td3/2. (13) The fracton dimensionality can be considered - 57-

6 348 J. Nucl. Sci. Technol., as a kind of fractal dimensionality for the object consisting of vertices visited by the random walker. For the Euclidean space, the two dimensionalities degenerate with the space dimensionality, i.e. d=df=ds. 3. Relation between Percolation and Geologic Parameters When applying the above theory to a fracture network, we have to establish a relation between percolation concentration p and measurable geologic parameters. This problem has been solved by Long et al.(21), where actual fracture networks are idealized by collection of random line segments distributed on a model space. Orientation and length of line segments are determined by geostatistical data of actual fractures. The solution is briefly outlined here. Because the permeability of random fracture networks increases ad infinitum by adding fractures to the system, there is no upperbound on permeability, and the fracture system which corresponds to the lattice with p=1 is difficult to identify. Long et al. claimed that by grouping the random fracture networks of the same linear fracture frequency ll (the number of fractures intersecting a line of unit length corrected for orientation bias), the maximum permeability, which corresponds to the lattice with p=1, becomes equivalent to the Snow permeability(22), which is defined for rock containing the system of multiple parallel planar fractures of uniform aperture and infinite extent. To establish the relation between p and the geologic parameters, first, the connectivity of fractures, which is interpreted x as the average number of intersections per fracture, is expressed in terms of the geologic parameters. Second, the connectivity is related to percolation concentration p. The first relation is obtained as x=lilh, (14) where I is the average length of fractures. The ll is obtained by ll =IlA, (15) where la is the number of fractures on a unit area. The H is called the orientation correction factor, which is defined as (16) where g(t) is the probability density function for orientation. Thus, the connectivity is expressed in terms of the geologic parameters such as the probability density functions for orientation and length, and the number density of fractures. For the relation between p and z, starting from the assumption that a piece of a line segment between two adjacent intersections corresponds to one edge of the percolation lattice, they derived the following relation by comparing the average length of linearlyconnected edges in the percolation lattice of concentration p with the average length of random line segments : (17) They also obtained the relation between the coordination number Z averaged over the entire medium and connectivity z as (18) The coordination number z is defined as the number of pieces of line segments branching from one intersection. For the perfect lattice, i.e. p=1, all linear fractures are of infinite length by the definition of Snow permeability, and the coordination number z becomes 4 because four lines branch out from every intersection. For p<1, the network consists of line segments of finite length. A piece of line segment braching out from one intersection to the end of the line segment does not count for the averaged coordination number because they do not contribute for mass transport. So the coordination number is less than four, as given by Eq. (18). Robinson(23) showed numerically that the two-dimensional random network becomes percolate at connectivities between 3.2 and 3.8. If we use 3.6 tentatively for Eqs. (17) and (18), - 58-

7 Vol. 28, No. 5 (May 1991) 439 we obtain 2.89 for the coordination number and for threshold concentration. The average coordination number 2.89, obtained for a two-dimensional random line network is less than three, and might be close to that for the honeycomb percolation lattice with p<1, because the coordination number of a honeycomb lattice with p=1 is three. The numerical value of the threshold concentration, 0.643, should be compared with values of the threshold concentration for bond percolation of various kinds of lattices. For bond percolation of a honeycomb lattice, the threshold concentration is exactly obtained as 0.653, which is close to Therefore, it can be said that a honeycomb lattice has properties similar to a two-dimensional random fracture network. (Note that earlier the different value is given for pc of site percolation of a honeycomb lattice. Here we must compare random line networks with bond percolation models, so we have used the value of pc for bond percolation of a honeycomb lattice. However, the simulations shown below have been performed based on the site percolation model. So, we used for the threshold concentration later in this paper.) 4. Exact-enumeration Method This technique was first used by Ben- Avraham & Havlin(24) in the early studies of anomalous diffusion on fractals and on percolation clusters. With the method, we do not have to make hundreds of thousands of Monte Carlo simulations for the process of diffusion on each realization of honeycomb percolation lattices with a given probability. Instead, for one realization of a percolation lattice, average of all possible random walks starting from a given origin on this lattice can be obtained. The exact enumeration of random walks can greatly reduce the error bars and the computation time, compared with Monte Carlo simulations. The main idea of the exact-enumeration method is that the probability of an ant being at any vertex i at some time t is determined solely by the probabilities of being at the nearest neighbors of vertex i at time t -1. We first store the lattice on which diffusion is to take place in a matrix, keeping track of the nearest neighbors of each vertex. To calculate the diffusion, we have two matrices, and M2{j}t., which store the M1{ probability distribution function P(r, t) of the ant j}t at times t and t', where {j} represents the set of all vertices in the lattice. Thus, given the distribution function M1{j}t at time t, the distribution function at time 5+1 is given by (19) Here nn{ j} denotes the nearest neighbors of {j}, and W[nn{j},j] is the probability of the ant to step from nn{j} to j. The W[nn{j},j] depends on the kind of ant in use. There exist different kinds of ants by defining different transition probabilities W[nn{j},j] from vertex nn{j} to vertex j. The blind nt cannot know prior to the next step which pathways are actually open, and allocates the same probability for all the possible directions. So, the blind ant can wait at its present position with a probability equal to the number of blocked pathways divided by the coordination number of the lattice (for the honeycomb lattice 3). The myopic ant, on the other hand, can know before it moves which pathways are blocked and which are open. It allocates the same probability only for open edges, and moves one of the open paths. The two ants are obviously different, although previous computer simulations show that their walks converge to yield the same anomalous-diffusion exponents on fractals(18). The blind ant converges more rapidly to the asymptotic regime, possibly because of a lower probability of entering dead ends. We choose the blind ant for the simulation. Having obtained M2{j},+ one can go back and calculate the distribution function at a time t+2 by (20) In this fashion the probability distribution P(rj, t)ocm{j}t is obtained, where rf is the spatial coordinate of vertex j

8 440 J. Nucl. Sci. Technol., III. NUMERICAL RESULTS AND DISCUSSIONS We generated random numbers based on the Knuth algorithm(25). The random numbers are normalized so that they range between 0 and unity. We set the length of the edge in the lattice to be unity. There are 50 layers of unit hexagons concentrically, with 15,000 vertices and 22,203 edges in the lattice, as shown in Fig. 2. The system extends to the radius of 86. We generate 100 realizations of honeycomb percolation lattices for probability p=0.66, 0.70 and For each realization, we made random-walk simulations by the exact-enumeration method, generating the probability density function P(r, t), as a function of time and vertex numbers. Profiles of probability density functions against time and vertices are averaged over 100 simulations for each concentration p. With the averaged probability density functions, we can calculate the mean-square displacement as a function of time for each p. For p=0.66, smaller than the percolation threshold of site percolation of a honeycomb lattice, the correlation length is calculated by Eq. ( 2 ) as 86, which is equal to the radius of the system. Generated clusters have nearly the same size as the system. For p,-0.70, slightly greater than the threshold, x becomes as large as 1,690. The system is much smaller than the average cluster size. Therefore, the medium has self-similarity property. For p= 0.8, the correlation length is 21, which suggests that the lattice of radius 86 might be treated as a continuum. Figure 3 shows the changes of mean-square displacement in time for percolation lattices and normal Fickian diffusion. Symbols plotted Various kinds of marks represent the results for percolation lattices. The solid line is for Fickian diffusion with a constant diffusion coefficient. The result for p=1 shows fairly good agreement with the solid line. The dashed line is for diffusion on a continuum with variable diffusion coefficient given by Eq. (21). Fig. 3 Mean-square displacement of random-walker after t steps of random walk - 60-

9 Vol. 28, No. 5 (May 1991) 441 discretely in the graph represent anomalous diffusion on percolation lattices. The solid line shows the theoretical relation Eq. ( 6 ) for the normal Fickian diffusion. At p=1, both the random-walk simulation and the theoretical relation agree very well with each other, showing the correctness of the simulation. The slopes of the lines fitted to the discrete symbols give anomalous diffusion exponent 2/dw, as summarized in Table 1. Table 1 Summary of dimensionalities In the cases of p<1, diffusion becomes slower than Fickian. For p slightly greater than the percolation threshold, diffusion pathways become quite complex ; there are deadends as well as bottleneck-type narrow paths, beyond which probability densities are significantly lower, resulting in slow diffusion. In the perfect lattice, i.e. p=1, the shortest distance between two distinct points in the lattice is identical to the Euclidean distance between the two points. For percolation lattices, the shortest distance of the pathway between two points on the same cluster is greater than or equal to the Euclidean distance because the ant must make detour if there are missing passways on the shortest pathway. Therefore, in some cases the ant takes hundreds of time steps until it reaches the neighbor vertex of the current vertex if the edge between these vertices is missing. There are also multiple pathways of different length which connect two distinct points on the same lattice. As p increases, closed pathways become open so that dead-ends, bottlenecks, and multiple linkages of different vertices become less. The dashed line represents the change of 2> in time for a continuum with a diffusion <r coefficient which is prescribed as a function of the distance from the origin of the system as (21) The exponent is determined by the result by Refs. (26) and (27). The diffusion equation for a cylindrical coordinate system with angular and azimuthal symmetries, is solved by the finite element method. The dashed line has a slope of 0.70, which is identical to that for the case of p=0.70. This agreement indicates the possibility of mapping diffusion on a percolation lattice onto diffusion in Euclidean space. In Fig. 4, cumulative numbers of occupied vertices are plotted against the Euclidean distance from the origin of the system for p= 0.66, 0.7 and 0.8. The slope of each fitting line gives the fractal dimensionality df, which is listed in Table 1. The value for p=0.70 is close to the theoretical value obtained by Eq. ( 3 ). With Eq. (11), the fracton dimensionalities can be calculated as listed in Table 1. In Fig. 5, plotted are the probability density functions as a function of squared Euclidean distance from the center of the system, for t=200 and 1,000. By Eq. ( 9 ), straight lines should be obtained for normal diffusion with a fixed diffusion coefficient in the plot of 1n(P(r, t)) vs. r2. With the variable diffusion - coefficient, probability densities are much smaller than those for constant diffusion co

10 442 J. Nucl. Sci. Technol., The slope of the fitted lines give the fractal dimensionality. Fig. 4 Cumulative number of occupied vertices at distance g from center of system Fig. 5 Probability density functions P(r, t) for normal Fickian diffusion with constant diffusion coefficient and with variable diffusion coefficient, defined in Eq. (20), and for anomalous diffusion on percolation lattices of p=0.66, 0.70 and 0.80, as a function of square distance from center of system, at t=-200 and 1,

11 Vol. 28, No. 5 (May 1991) 443 efficients because of the decreasing diffusion coefficient at the outer region. In Fig. 5, three cases for percolation lattices are also shown. With a greater p, the profile extends more toward the outer regions, which means faster diffusion for greater p. The curves for p=0.7 in Fig. 5 show a concave shape; in the region near the center, the probability density decreases much more steeply than in the outer region. The broken curves for the variable diffusion coefficient, on the other hand, show a convex shape. Thus, even though the profile for the meansquare displacement vs. time is the same as that for the continuum model with the variable diffusion coefficient, spatial probability density profiles for percolation lattices are quite different. Due to this difference, we cannot utilize the mapping from the fracture network to the continuum with the diffusion coefficient expressed by Eq. (21), because radionuclides can reach a further point than predicted by the continuum model. This anomalous behavior might have been observed in the previous experiments for measurement of diffusion coefficients. For example, concentration profiles of tracers in rock samples often show very similar concave profiles against the depth of rock samples, and usually two diffusion coefficients are determined ; one for some fast diffusion mechanism and one for slow diffusion mechanism such as done in Ref. (28). However, this could result from disordered pore-space structure of rock samples, which include many dead-end pores and narrow pathways. There is a possibility that the percolation model can give a theoretical background for diffusion in such disordered media. We have showed so far that the simple Gaussian behavior does not generalize to diffusion on the percolation lattice. In the rest of this paper, we consider general formulation including both Fickian diffusion, i.e. Eq. ( 9 ), and anomalous diffusion. To do this, we first begin with the analogy of Eq. ( 9 ). We assume that the probability density is a function of r/t1/dw from Fig. 3. Therefore, we have (23) The density of vertices on the percolation lattice imbedded in the d-dimensional space is proportional to rdf-d by the definition of fractal dimensionality Eqs. ( 4 ) and ( 5 ). We assume that P(r, t) is proportional to the density of vertices. Equation (23), then, is written as (24) At r=0, we consider that the density is unity because there is only one vertex at the origin. Using the probability density at r =0, Eq. (12), as the normalization factor, we can finally write the function as Note that, for the normal diffusion case d= the (25) f =dw=ds=2, we obtain Eq. ( 9) if we assume For the functional form for P, we assume function P(x)=exp(-xu), (26) where u is an exponent obtained so that plot becomes straight line of slope of unity. Note that for the normal Fickian diffusion, u=2. Figures 6~8 show that by setting u=1.672, 1.68 and 1.89 for p=0.66, 0.70 and 0.80, respectively, we can obtain fairly good fit to the straight line of unit slope. The fitting becomes better for p closer to unity, and the fitting exponent u approaches 2. IV. CONCLUSIONS In this study, we have made numerical simulations for radionuclide diffusion through a fracture network in hard-rock formation, which is expected to function as the final barrier for radioactivity confined in a deep geologic repository. We have simulated actual fracture networks as a two-dimensional honey- a - 63-

12 444 J. Nucl. Sci. Technol., Fig. 6 Determination of functional form for probability densities at p=0.66 Fig. 7 Determination of functional form for probability densities at P=

13 Vol. 28, No. 5 (May 1991) 445 Fig. 8 Determination of functional form for probability densities at p=0.08 comb percolation lattice. Random-walk simulation for diffusion on the percolation lattice have been made for percolation concentrations p=0.66, 0.7, 0.8 and unity, by the exactenumeration method. The results for the percolation lattices have been compared with those from normal Fickian diffusion with constant and with variable diffusion coefficients. From these comparisons based on the numerical studies, we make the following concluding remarks : (1) Mean-square displacement of a randomwalker on percolation lattices increases more slowly with time than that for Fickian diffusion with a constant diffusion coefficient. Anomalous diffusion exponents become larger than 2 for percolation lattices. (2) As p increases, or the density of fractures increases, diffusion approaches the normal Fickian diffusion. (3) As far as the profile of mean-square displacement vs. time concerns, diffusion on the percolation lattice can be mapped onto diffusion on a continuum with a variable diffusion coefficient, which is expressed, for example, as Eq. (21). (4) However, spatial distribution of probability densities of finding the randomwalker on the percolation lattice is quite different from that on a continuum with the variable diffusion coefficient. The percolation model yields a concave shape ; there can be observed steep decrease in the vicinity of the origin and gradual decrease in the outer region. With the variable-diffusion coefficient model, the profile looks like convex. Diffusion in the vicinity of the origin is relatively fast because of larger diffusion coefficient. Probability densities decrease quite rapidly with increasing distance from the origin because of the decreasing diffusion coefficient in the outer region. Practically the random-walker can never reach the outer edge of the simulation space. (5) The simple Gaussian behavior does not generalize to diffusion on the percolation - 65-

14 446 J. Nucl Sci Technol., lattice. We have derived a general formula that can include both Fickian and anomalous diffusion as Eq. (25). (6) The previous experimental results for measurement of diffusion coefficients, where two diffusion coefficients are determined for one rock sample with the assumption that there are some fast and slow diffusion mechanisms, could be explained by disorderedness of pore spaces of rock samples, which include many dead-end pores, and could be expressed by the single formula as Eq. (25). Obvious extensions for the present study are to investigate anomalous diffusion on three-dimensional percolation lattices and to include effects of advection on radionuclide transport in fracture networks, on which we are currently working. These will be reported in future. - REFERENCES- (1) LONG, J. C. S.: Verification and characterization of continuum behaviour of fractured rock at AECL underground research laboratory, LBL , (1981). (2) HARADA, M., et al.: Migration of radionuclides through sorbing media ; Analytical solutions-i, LBL-10500, (1980). (3) PIGFORD, T. H., et al.: Migration of radionuclides through sorbing media ; Analytical solutions-ii, LBL-11616, (1980). (4) CHAMBRE, P. L., et al.: Analytical performance models for geologic repositories, LBL-14842, (1982). (5) BURKHOLDER, H. C., et al.: Nucl. Technol., 31, 202 (1976). (6) NERETNIEKS, I.: J. Geophys. Res., 85, 4379 (1980). (7) RASMUSON, A., NERETNIEKS, I.: ibid., 86, 3746 (1981). (8) TANG, D. H., et al.: Water Resources Res., 17, 555 (1981). (9) SUDICKY, E. A., FRIND, E. O.: ibid., 18, 1634 (1982). (10) idem: ibid., 20, 2021 (1984). (11) AHN, J., et al.: Intermediate-field transport of contaminants ; Multiple areal sources in fractured rock and point sources in porous rock, Waste Management, in press. AHN, J., SUZUKI, A.: (12) Rad, Waste Manage. Nucl. Fuel Cycle, 14( (1990). (13) CHELIDZE, T. L.: Phys. Earth Planet. Int., 28, 93 (1982). (14) de MARSILY, G.: Hydrogeology of rocks of low permeability, Int. Assoc. Hydrogeologists, Memories, Tucson, Arizona, Vol. 17, 267 (1985). DERSHOWITZ, W., et al.: Fracture (15) flow code cross-verification plan, Stripa Project, (1989). FLORY, (16) P. J.: J. Am. Chem. Soc., 63, 3091 (1941). BROADBENT, (17) S. R., HAMMERSLEV, J. M.: Proc. Camb. Phil. Soc., 53, 629 (1957). (18) HAVLIN, S., BEN-AVRAHAM, D.: Adv. Phys., 36[6], 695 (1987). GRIMMETT, (19) G. : "Percolation", (1989), Springer- Verlag, New York. 20) STAUFFER, D.: ( "Introduction to Percolation Theory", (1985), Taylor & Francis, London. ) LONG, J. C. S., et al.: Fluid flow in (21 fractured rock ; Theory and application, "Transport Processes in Porous Media", (BEAR, J., CORAPCIO- GLU, M. Y., eds.), (1989), Kluwer Academic Publ. (22) SNOW, D. T.: A parallel plate model of fractured permeable media, Ph. D. Dissertation, Univ. of Calif., Berkeley, (1965). ROBINSON, P. C.: Connectivity (23) of fracture systems ; A percolation theory approach, HL82/960, AERE Harwell (1982). BEN-AVRAHAM, (24) D., HAVLIN, S.: J. Phys. A: Math. Gen., 15, L691 (1982). (25) PRESS, H. W., et al.: "Numerical Recipes; The Art of Scientific Computing", (1986), Cambridge Univ. Press. GEFEN, (26) Y., et al.: Phys. Rev, Lett., 50[1], 77 (1983). (27) ORBACH, R. : Science, 231, 814 (1986).(28 ) IDEMITSU, K., et al.: Preprint 1990 Fall Mtg. of AESJ, (in Japanese), M