Transactions on Ecology and the Environment vol 7, 1995 WIT Press, ISSN

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Joan Atkinson
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1 Non-Fickian tracer dispersion in a karst conduit embedded in a porous medium A.P. Belov*, T.C. Atkinson* "Department of Mathematics and Computing, University of Glamorgan, Pontypridd, Mid Glamorgan, CF3 7 1DL, UK ^School ofenvironmental Sciences, University ofeast Anglia, 77% ABSTRACT Longitudinal dispersion of a tracer slug by the flow along a karst conduit is often assumed to obey Pick's Law, governed by a single dispersion coefficient. Tracer breakthrough curves have often been interpreted on this basis despite empirical evidence of non-fickian asymmetry and tailing of the tracer cloud. A possible cause of this tailing is that tracer diffuses through the wall of the conduit into slow-moving groundwater in the fractured or porous aquifer in which the conduit is embedded. The effects of such behaviour on the time-concentration curve at the end of the conduit are investigated in this study by means of mathematical analysis. An explicit analytical solution to the problem is presented and the future work needed to verify and implement it briefly discussed. 1 INTRODUCTION Karst aquifiers are the most complex and unpredictable of all types of aquifiers in their groundwater flow or contaminant transport. This complexity arises from their structure, which consists of open tunnels, or conduits, embedded in larger volumes of rock matrix. Hydraulic regime in the conduits is normally turbulent. Within the matrix there is generally diffuse laminar flow, through a 3-D network of voids which may be comprised of

2 106 Water Pollution fractures and/or intergranular pores. At a macroscopic scale, the flow in the matrix obeys Darcy's law, but karst aquifiers as a whole show non- Darcian behaviour because of the turbulent flow in the conduits. The distributon of permeability is also extremely inhomogeneous, with very great contrasts between conduits and matrix, but also large contrasts in matrix permeability. These factors make it difficult to model both flow and contaminant transport in karst aquifiers, and most investigations of specific aquifiers have relied heavily on tests using boreholes, wells, measurements of flow in conduits and tracer experiments. An up-to-date summary of the existing state of knowledge is provided by Ford & Williams/ Compared with the study of normal diffuse-flow aquifiers, the theoretical basis for describing and analysing contaminant transport and dispersion in karst aquifiers is in its infancy. Tracer tests are a very frequently used experimental technique for defining and mapping conduit networks. They are sometimes used to deduce conduit volumes, diameters and topology and more rarely to characterise contaminant transport and dispersion, e.g. Atkinson & Smith.^ These studies are usually completely empirical and in very rare cases in which a theoretical foundation is used, the longitudinal dispersion in conduit is assumed to be Fickian, e.g. Smoot et alj In contrast, the literature on fractured aquifiers contain several more complex treatments of longitudinal dispersion in groundwater flow. One important paper with relevance to karst is by Maloszewski & Zuber^ which presents a mathematical analysis of tracer dispersion in plane-parallel fissures embedded in a porous medium. Within the fissures, tracer is envisaged as dispersing longitudinally by a Gaussian process, however, exchange of contaminant may occur between the fissure and pore-water in the matrix, by diffusion across the fissure wall. In this paper, we analyse a similar mechanism of dispersion for a cylindrical karst conduit embedded in a porous medium, and deduce equations which describe theoretically the tracer concentrations within the conduit and in adjacent porous medium, as functions of position and time. 2 THEORETICAL MODEL AND ANALYSIS Fig. 1 shows a coordinate system for the description of a cylindrical conduit of radius R, embedded in infinite porous medium. Such a system is radially

3 Water Pollution 107 symmetrical about the centre line of the conduit, which is chosen as the X- axis. The region 0 < r<r represents the conduit which is surrounded by a porous medium for r>#, where r is the radial coordinate. The nature of the porous medium is in reality likely to be composed of intersecting fractures. We make two simplifying assumptions. First we assume that medium can be treated as continuous. Secondly we assume that tracer may diffuse radially within the porous medium much faster than longitudinally. In reality, this second assumption is likely to be true because tracer concentration in the porous medium will normally show a much smaller gradient parallel to the conduit than radially. Thus, diffusive transport of tracer parallel to the conduit will be much less than radial transport, for a diffusively isotropic medium. Turbulent water flow occurs along the conduit with velocity u. The water in the porous medium is assumed to be static. Tracer is envisaged to be released as an instantaneous slug at time fc=0. The tracer is then mixed by the turbulent motion of water in the conduit by longitudinal and radial turbulent diffusion, with coefficients K± and K^ respectively. Tracer substance reaching the conduit rough walls may enter the porous medium by diffusion, and then spreading radially along the concentration gradient with an empirical dispersion coefficient D, which accounts the influence of porosity, pore tertiosity and geometrical irregularities. Porous medium Conduit. I 7^-xL \J Porous medium Figure 1: Sketch of the coordinate system used to model a cylindrical, water-filled karst conduit in saturated porous medium. This scheme is represented by the pair of coupled equations (l)-(2), in which c represent the tracer concentration in the conduit, and q is the concentration in the pore water of the surrounding medium. Equation (5) stipulates concentration continuity at the boundary, while equation (6) prescribes continuous flux through the conduit wall :

4 108 Water Pollution (2) c(r, M), = o = V>(r, x), q(r, i), = = 0, (3)-(4) where c^(r, x) is the initial tracer concentration in the channel water. After transformation (%, f) -» (z, ), where z=%- ut, the problem (l)-(6) can be rewritten in a slightly different form and solved by the following steps. External problem (2) with boundary condition of (5) is solved Weber's transformation, Budak et ap : using 00 g(a, t) =Jq(r, t) W(r, A) rdr, W(r, A) = M\R)N^(X r) - N,(XR)J^(\ r), and JQ(Z) and NQ(Z) are the Bessel functions of the first and second kind. Assuming the time dependent boundary condition (7) q(r, i;z), = fl = M(*), ft (t) = c(r, z, t), (8) solution of the boundary problem (2) is written using Duhamel's integral: =- The internal problem (1) is be solved with the boundary condition (6), which is converted into the time- dependent boundary condition of the second kind: (10) with the q(r, t) from the equation (9). The initial concentration ( >(r, z) is taken as the instant input of an amount Q of tracer at the point XQ = ZQ at the axis of the channel:

5 Water Pollution 109 y(r,z) = g^6(z_%), (11) where S(x) is Dirac's delta function. The Green function for the second kind of boundary condition is G(r, z, r', %, t) = G(z, ^ t) G^r, r', t), (12) 2 where //& are the roots of equation Jj(//) = 0 and a*. = ^2~dl- The solution of the problem (1) with the initial condition (11) is as follows: cfr, z, t) = j-, oo R(_ h -oo o 0 \ f / / /\ r Z/(0) 1 7 / j / /i o\ G(r, z, r, %^) { (^(r, z ) - -^ ) cfz r (fr, (13) After integration and some further transformations equation (13) yields : ^ ' ^ ^ fc = 0 Jo(C) Applying definitions (8) and (9) and Laplace transform to equation (14) it follows that where c (r, z, s) = ^ #i(r) + i/(4#2(r, 4 + Z,(r,z, 5), (16)

6 110 Water Pollution From equation (9) it follows a Laplace transform of the boundary condition 00 i/(s) - -u(s z\^2l I V(S) - (1(8, Z) ^ J J_,2( where /2(s, z) c(j?, z, s). The concentration of tracer in the water within the porous medium now can be obtained from equation (9) using explicit expression for p,(f) which is given by equations (A.5)-(A.7) in the Appendix. The tracer concentration in the channel water can now be found from equation (14) applying for v(f) definition (10) and using the expression for g(r, f) which is known from equation (9). This scheme yields formulas for c(r, z, f) and <?(r, f) which are explicit, but large and clumsy, so they are not given in full here. They can easily be used for calculations after transforming back to initial coordinate system (z, t) <- (z, ), where x z + ut. 3 CONCLUSIONS The explicit expressions which are deduced for the tracer concentration in the rock medium and consequently concentration in the conduit water will allow relatively simple calculations for downstream values in both matrix and channel. This is an important development of theory for use in interpreting tracer tests. The current task is to render the mathematical solution convenient for use in the form of a simple computer program to perform explicit calculations, which are very repetitive in practice. The next step will be to test the model's ability to describe actual tracer break through curves, especially their commonly observed 'tailing', non-fickian properties. A very important is the model's ability to describe the progressive evolution of a tracer cloud using such data as shown in Fig. 2 which illustrates dye- concentration curves c(t) at three values of x for a flooded karst conduit in Norway. Preliminary analysis indicates that z dependence produce assymetry of the concentration trends. After evaluation of the flow physical parameters (K^ K^ u, D, R) this model can be regarded as suitable for use in prediction or routine modelling calculations. This paper presents the first steps in a long process of future research.

7 Water Pollution REFERENCES 1. Abramovitz, A., Stegun, I. Handbook of Mathematical Functions, Dover, New-York, Atkinson, T.C., Smith, D.I. Rapid groungwater flow in the fissures in the chalk, 1974, 0. J. E^^. GW., 7, Budak, B.M., Samarsky, A.A., Tikhonov, A.N. A Collection of Problems on Mathematical Physics, Pergamon Press, London, Ford, D.C., Williams, P.W. Karst Geomorphology and Hydrology, Chapman & Hall, London, Maloszewski, P., Zuber, A. On the theory of tracer experiments in fissured rocks within the porous matrix, 1986, J. HydroL, 79, Smart, C.C., Ford, D.C. Structure and function of a conduit aquifier, 1986, CVmo^on, J. EarA 56., 23 (7), Smoot, J.L., et al Quantitative dye technique for distribution of solute transport characteristics of groundwater flow in karst terrain, (ed. B.S. Beck), pp. 269 to 275, Proceedings of 2^ Multidisciplinary Conference on Suikholes and the Environmental Impacts of Karst, Orlando, Florida, APPENDIX The Laplace transformant of ^(t) which follows from equations (16) and (19) has the form / \ L(R, z, s) / A 1 \ where 7 y (A3) "^ 8D* and IP D/R*. Applying convolution theorem to the equations (A.l)- (A.4) it possible to find an approximate expression for fi(t):

8 112 Water Pollution = 0 (A.5) This follows to explicit formulas for q(r, t) and c(r, z, ^) from equations (9) and (14). The integrals 4({, z, t) and /^(2, t) are given in Abramovitz & Stegun*: dr (A.6) (A.7) Time since Injection (s) Figure 2: Concentration of tracer at three distances from instantaneous injection in a 560 ra long karst conduit in Norway (unpublished results). The 'tailing' curves to be explained by the advective flow/exchange model.

The distortion observed in the bottom channel of Figure 1 can be predicted from the full transport equation, C t + u C. y D C. z, (1) x D C.

The distortion observed in the bottom channel of Figure 1 can be predicted from the full transport equation, C t + u C. y D C. z, (1) x D C. 1 8. Shear Dispersion. The transport models and concentration field solutions developed in previous sections assume that currents are spatially uniform, i.e. u f(,y,). However, spatial gradients of velocity,