WATER RESOURCESRESEARCH, VOL. 29, NO. 7, PAGES2463-2464, JULY t993 Reply T. M. KRISHNAM00RTHY, R. N. N.•m, ANDT. P. SARMA Health Physics Division,BhabhaAtomic ResearchCentre, Bombay FIRST RESPONSE TO C.-H. KANG In any model, the selection of a coordinate system de- pends primarily on the assumptions involved rather than the geometry ofthe system. Though a circular fracture isas- sumed, ourmodel employs planar diffusion in the x and z directions. Thediffusional losses of activity from the circular fracture to theporous host rock is assumed to beoccurring through a width element of 2•rb (open up the circular fracture) along the z axis.The coupled equations (la) and (2a) and the material balance equation (8) areinternally consistent with each other for such a system. Theemphasis ofKrishnamoorthy et al. [1992] is on the material balance which facilitates evaluation of the absolute concentration of radionuclides in fracture water; however, the approach will yield conservative results forthe same problem. Tang et al. [1981] hadalready addressed similar views now raised by Kang [thisissue]. Our assumptions are in accord withthose of Tanget al., where thedirection of mass flux in theporous rockis considered perpendicular to the fracture axis neglecting lateral diffusion. Such an assumption of orthogonality isreasonable when the migration inthe host matrix is governed by molecular diffusion and the transport along the fracture ismuch faster than that within the matrix. As Tang etal.pointed out, this results inthe simplification of the basically two-dimensional system to twoorthogonally coupled one-dimensional systems. In this simplified form, the problem is much more amenable tosolution by analytical techniques. In another similar work, Grisak and Pickens [1980] used a finite element technique to calculate concen- trations in both the fracture and the matrix considering two-dimensional mass distribution in theporous matrix. The excellent agreement between these two models observed by Tang et al. [1981] strongly supports the validity of the orthogonality assumption. We had already realized thepoint raised by Kang [this issue] regarding the use of cylindrical coordinates for this problem. Thecoupled equations aswellasthematerial balance equation shown byKang [this issue] are valid only for such a system. It isnot appropriate touse the material balance equation suggested by Kang [this issue] for our model. In fact, wehave derived solutions using Bessel functions for this problem in cylindrical coordinates. They are some- what complicated and need thorough testing prior toappli- cation. As soon as the data generation on the concentration profile of radionuclides in fracture water is obtained, these solutions will be communicated. The near-field processes consider the migration of radio- nuclides through the bentonire clay used as backfill materials Copyright 1993 bythe American Geophysical Union. Paper number93WR00577. 0043.1397/93/93WR-00577502.00 around the canisters within the repository.These processes include advection, diffusion, andsorption by clay.Primarily, the backfill materials act as an additional barrierbecause of their sorption capacity and retard the movement ofradionu- clides from the repository. Thesenear-field processes are not considered in the formulationof Krishnamoorthy et al. [1992] and thestatement in their introduction is intended to conveythis meaning. The formulation shown to compute the flux of radionu- clides asa function of fracture length and timeis intended to indicate the rateof activity thatwould enter a tapping point (users'point). The fractures caused by mechanical andthermal distur- bances can be of manytypes, namely, planar,circular, rectangular orirregular. Some approximations inthis regard are acceptable when they lead to simple solutions without causing serious errors. SECOND RESPONSE TO C.-H. KANG 1. Kang [this issue] has got the fight concept ofphysics. However, his insistence on the use ofcylindrical coordinates in the governing equations and the material balance equation is misplaced. It was stated clearly thata complete analysis of therock repository system consisting of a circular fracture can be performed incylindrical coordinates (as suggested by Kang) if one does notliketo introduce anyformof conservatism. Also it was mentioned thatfor diffusive loss, a portion of the rock matrix (having a width of2•rb and volume element 2 z-b dx dz) was only considered as a conservative approach. The coupled equations (la) and •,2a}, and the material balance equation {8) are internally consistent with each other for such a system. 2. It seems thatKang[this issue] misinterprets Tang et al.'s [1981] assumptions. Tang et al.'s [1981, p. 563] state- ment is as follows: The general analytical solution is based on anumber of assump- tions whose v'alidity depends on the configuration ofthe partic- ular system. Conditions under which these assumptions may be questioned are those where the diffusive loss into the porous matrix islarge, as such conditions tend to produce relatively short penetration distances down the fracture and relatively large penetration distances into the matrix. This would tend to result in (1) agenerally two-dimensional mass distribution in the porous matrix, (2) arelatively greater importance of the varia- tion in concentration across thethickness of the fracture, (3} a relatively greater importance of mass influx at the top surface of the matrix. These factors are neglected in the analytical solu- tion. Thus it isclear that Tang etal.'s equations became indepen- dent of y, asit was neglected. 3. The fluxequation (25) is solved for the required boundary condition CflL, t) = Cp(b, L, t), implying 2463