WATER RESOURCES RESEARCH, VOL. 27,NO. 1,PAGES 129-131, JANUARY 1991 Comment on "Flow andTracer Transport in a Single Fracture' A Stochastic Model and Its Relation to Some Field Observations"by L. Moreno et al. DANIEL J. GOODE AND ALLEN M. SHAPIRO U.S. Geological Survey, Reston,Virginia Moreno et al. [1988] (hereinafter referred to as MT) used a particle-tracking schemeto investigate the physicsof solute movement in a variable-apertureplanar fracture. The spatially heterogeneous fluid velocity was assumed to be the only mechanism of solute movement; local or pore scale dispersion and molecular diffusion were assumedto be negligible. The particle-tracking scheme used by MT con- sistedof routing particles from node to node in a finite difference grid. In this scheme,the direction of an individual particle is randomly selected and the probability associated with the particle movement in a given direction is propor- tional to the fluid flux in that direction. The same method wasusedby Desbarats [1990] to investigate advective trans- portin aquifers composed of two porousmedia of different hydraulic conductivities. The node-to-noderouting scheme used by MT is a poor model of the physics of advective solute movement in a continuum. In a companion comment [Goode and Shapiro, this issue],we analyze the artificial dispersion introducedby this scheme for the case of uniform flow. Those results are directlyapplicable to the discussionhere. In this comment we show the smearing effect of the node-to-node routing schemeon particle breakthrough, and we show that the spreading indicated by the "transfer matrix" analysis pro- posed by MT is solely an artifact of the node-to-node routing scheme. The differencesin particle breakthroughpresented herebetween the node-to-node routing schemeand a linear velocity interpolation method are indicative of the differ- ences for binary porous media as considered by Desbarats [1990]. Our comments focus only on the errors introduced in employing the node-to-node routing scheme and its impact onsimulating advection-dominated solutemovement. We do not comment onthe con,clusions reached by MT withregard to the physics of their problem. Use of a model that accurately treats advection-dominated solute movement mayor may not influencethe conclusions of these investi- gators; however, we believe that there are more appropriate and available models that can be used to investigateadvec- tion-dominated solute movement. NODE-TO-NODE ROUTING IN A VARIABLE-APERTURE FRACTURE MT applied the node-to-node routing scheme to advective solute transport in a single planarfracture.MT (p. 2037) described the method as follows' "Particles coming to an This paper is not subject to U.S. copyright. Published in 1991 by the American Geophysical Union. Paper number 90WR02310. intersection as distributed in the outlet branches with a probabilityproportional to the flow rates. The residence time for the particle to reside within each square element is determined from the flow rate through this element and the volume involved .... "The only physical processunder study was the advection of solute. MT continued, "In this calculation, we focus on the effects of the different residence times alongthe different pathwaysas the chief sourceof the overall dispersion in the fracture. We therefore do not include the effects of molecular diffusion, matrix diffusion or local dispersion within each channel in our calculations." In MT's work, the nodes are located at the center of the blocks and the aperture is considered to be constant over the block, which leads to the use of harmonic means for interblock conductances. The "outlet branches" are the lines connect- ing nodes; hence each line has a residence time associated with it that is determined by adding the residence times within the two segmentsof the line, one segment in each block. The residence time within each block is defined by equation (11) of MT: biAxAy ti ---- 2 . where bi is the apertureof the block, Ax and Ay are the block dimensions, andIQijlis theabsolute value of the volumetric flux from node i to nodej. This formulation only allows movement along the lines connecting nodes, which may be appropriate when these lines are the discrete fractures of a network [Schwartz et al., 1983]. However, this formulation introduces errors when applied to the continuum of a single planar fracture. As pointed out by Schwartz et al. [1983, p. 1256], "Built into this schemefor partitioning mass at the intersectionsis the assumption that there is perfect mixing at the fracture intersections." In the case of MT, the "fracture intersec- tions" are nodes; hence perfect mixing occurs at each node in the model. Thus, local artificial dispersion is introducedin the model solely as an artifact of the particle-tracking scheme. To illustrate the errors induced by application of the node-to-noderouting scheme of MT to a single fracture, a two-dimensional problem is considered that generally corre- sponds to simulations conducted by MT where block aver- age fracture apertures are generated stochastically. The aperture is a lognormallydistributed random variable, Y, where Y = lOgl0 b and b is the fracture aperture in microme- ters. The mean and variance of Y are E[Y] = !.7 and cr2r = E[YY] - (E[Y]) 2 = (0.43) 2, respectively. In addition, Y is 129