WATER RESOURCES RESEARCH, VOL. 31, NO. 2, PAGES 303-312, FEBRUARY 1995 Issues in single-fracture transport modeling: Scales, algorithms, and grid types Robert P. Ewing and Dan B. Jaynes National Soil Tilth Laboratory, Agricultural Research Service, U.S. Departmentof Agriculture Ames, Iowa Abstract. Transportin single fractures has recently been intensely examined. Potential applications of thesestudies includenuclear wastestorage and infiltration of rainwater into soil desiccation cracks. We modeledhydrodynamic dispersion in singlefractures, using a variable-aperture model and particle-tracking techniques. We examined issues of scale of heterogeneity, particle-tracking method,and grid topology. Hydrodynamic dispersion tendsto zero as the scale of the transportpath increases in relation to the scale of heterogeneity. Sincethis is not observed in nature, it implieseither that fractures have fractal structure or that hydrodynamic dispersion alone doesnot account for all the dispersion that occurs in fractures. Dispersion and retardationas simulated usinga node- to-nodeor mixingtype algorithmare greaterthan when they are simulated usingan interpolation algorithm, and the difference cannotbe attributedto molecular diffusion. Differences in conductivity and dispersion betweendifferentgrid types (serial,parallel, square, and randomfield) are related to the coordination number (degreeof connectedness) of the grid, with lower coordination numbergridshaving higher dispersion. Introduction Interestin flow and transport in fractured porous mediahas increased sharplyin the past decade.While safe disposal of nuclearwaste has been the primary impetus,there has also been a growing awareness that fractures are relatively common andaffect transport in many geologic porous media. Bales etal. [1989] examined survival andtransport of virus in fractured tuff from the point of view of public groundwater supplysafety. Gburek and Urban [1990] showed that shallow fracturedbed- rock strongly influences the local groundwater qualityand sup- ply. $chwille [1988] conducted extensive laboratorytests to characterize the behaviorof chlorinated hydrocarbons in frac- tured media. Our immediate interest in fractured media (shrink-swell soils) relates to agricultural chemicalsand water quality. Shrink-swell type soils crackon drying, typically at ped bound- aries. On wetting, the cracks canclose in a few minutes. Blake et al. [1973] and Bouma and Dekker [1978] showthat water moving into shrinkage cracks caninfiltratethrough the fracture faces, but most watertends to pondandinfiltrateat the bottom of the fracture. In regions characterized by hot summers with intermittent, high-intensity rainfall, pesticides applied at the soil surface may be washedto the bottom of the soil fracture network by a single thunderstorm. This canbypass the biolog- icallyactivesoil matrix and rapidlymovethe pesticides to the groundwater table. Dispersion at low Reynolds number flowis oftenbrokeninto three components: molecular diffusion, Taylor dispersion [Tay- lor, 1953], and hydrodynamic dispersion. Molecular dispersion stems from the randomness of the fluid and is time or rate This paper is not subject to U.S. copyright. Published in 1995 by the AmericanGeophysical Union. Paper number94WR02674. dependent. Taylor dispersion interacts strongly with molecular diffusion and sois alsorate dependent. Thesetwo components of dispersion are not considered in this paper. Hydrodynamic dispersion, however,stems from the randomness of the me- dium, which suggests that it is constant over the time spanof the experiment (assuming that the fluid doesnot change the medium). Furthermore, whererandomness is a property of the medium rather than the fluid, the system maybe describable by usingpercolation theory [Berkowitz and Balberg,1993]. This impliesthat the transportcharacteristics may be sensitive to changes in both scaleand topology. The goal of this investigation is to assess hydrodynamic dis- persion in an individual fracture;subsequent studies will ad- dress the other components of dispersion, as well as fracture network effects. Hydrodynamic dispersion in singlefractures hasbeen recently modelednumerically by Tsangand cowork- ers [Tsang and Tsang, 1987; Morenoet al., 1988; Tsang et al., 1988;Tsang and Tsang, 1989; Preuss and Tsang, 1990;Tsang et al., 1991].They modeleda fracture as the space between two parallelplates, where eachplate consisted of a 20 x 20 grid of squares. The aperture betweenthe plates was held constant within eachindividual grid square. Apertures were assigned to the individual grid squares by using random field methods [Mantoglou and Wilson, 1981],which createan artificialcon- tinuous surfacewith predeterminedgeostatistical properties suchas correlation length • (J in the notation of Russo and Bresler [1981]). Conductivities of the individual grid squares werecalculated byusing the so-called cubic law [Witherspoon et al., 1980],andthe flow field wassolved by using sparse matrix methods. Dispersion was then calculated by usingparticle- tracking methods [Warren and Skiba, 1964]. The matrix was considered impermeable, so diffusion in the matrix was ig- nored. The approach of Tsang and coworkers wasthe first to nu- merically model transport in heterogeneous fractures, and it 303