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First-passage-time transfer functions for groundwater tracer tests conducted in radially convergent flow
Journal of Contaminant Hydrology 40 Ž2000. 299–310 www.elsevier.comrlocaterjconhyd
First-passage-time transfer functions for groundwater tracer tests conducted in radially convergent flow Matthew W. Becker a
a,)
, Randall J. Charbeneau
b,1
Department of Geology, State UniÕersity of New York at Buffalo, 876 Natural Sciences Complex, Buffalo, NY 14260, USA b Department of CiÕil Engineering, UniÕersity of Texas at Austin, Pickle Research Campus a119, UniÕersity of Texas at Austin, Austin, TX 78712, USA Received 30 March 1998; received in revised form 1 June 1999; accepted 14 June 1999
Abstract Forced-gradient groundwater tracer tests may be conducted using a variety of hydraulic schemes, so it is useful to have simple semi-analytic models available that can examine various injectionrwithdrawal scenarios. Models for radially convergent tracer tests are formulated here as transfer functions, which allow complex tracer test designs to be simulated by a series of simple mathematical expressions. These mathematical expressions are given in Laplace space, so that transfer functions may be placed in series by simple multiplication. Predicted breakthrough is found by numerically inverting the composite transfer function to the time-domain, using traditional computer programs or commercial mathematical software. Transport is assumed to be dictated by a radially convergent or uniform flow field, and is based upon an exact first-passagetime solution of the backward Fokker–Planck equation. These methods are demonstrated by simulating a weak-dipole tracer test conducted in a fractured granite formation, where mixing in the injection borehole is non-ideal. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Groundwater; Solute transport; Tracers; Transfer functions; Laplace transformations; Fractures
) Corresponding author. Tel.: q1-716-645-6800 ext. 3960; fax: q1-716-645-3999; E-mail: [email protected] 1 Tel.: q1-512-471-0070; E-mail: [email protected].
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M.W. Becker, R.J. Charbeneaur Journal of Contaminant Hydrology 40 (2000) 299–310
1. Introduction A groundwater tracer test that employs a forced radially convergent flow field offers advantages for estimating transport properties in porous and fractured media. A radially convergent test promotes recovery of the injected mass, reduces the effect of apparent dispersion due to the flow field, and minimizes the influence of the natural hydraulic gradient. Thus, radially convergent tracer tests are particularly useful where transport, rather than hydraulic properties, are desired. This note offers simple and flexible analytical tools for designing and interpreting groundwater tracer tests conducted in an induced radially convergent flow field. Various test designs are possible using radially convergent flow. For example, tracer may be introduced as a instantaneous pulse, a finite pulse, or a constant concentration. Tracer mass may be followed by a chaser fluid or be flushed from the injection well only by water flowing to the pumping well. The chase fluid may be of finite volume or be added at a constant rate throughout the test Žproducing a ‘‘weak dipole’’ hydraulic field.. In addition to variation in injection design, non-ideal flow behavior may affect the way that tracer is actually introduced to the formation. For example, if tracer is added to the injection well as a pulse, mixing with un-traced water flowing through the well will delay and dilute the tracer as it enters the formation ŽZlotnik and Logan, 1996.. Predictive and interpretive models have been available for radially convergent tests for some time ŽAl-Niami and Rushton, 1978; Hodgkinson and Lever, 1983; Welty and Gelhar, 1994; Moench, 1989, 1991, 1995.. However, these models are generally limited with respect to the variations in test designs that are allowed. The model of Moench Ž1989; 1991; 1995. is widely used to interpret tracer tests. It allows for continuous, instantaneous, or finite concentration injection, and accounts for mixing in both the injection and well bore. However, it does not allow for incomplete mixing Že.g., dead zones in the injection well., reinjection of traced fluid Žrecycle., or other deviations from the aforementioned designs. The model described here employs a transfer function approach to predicting tracer breakthrough. In this approach, each component of the test is represented by a transfer function, so that breakthrough is predicted to be the convolution of these components. Transport in the geologic medium must, therefore, be formulated in terms of flux, rather than concentration. Such a formulation is presented herein. Convolution is accomplished by multiplying the transfer functions in Laplace space, and then inverting the Laplace solution to the time domain. Changing the predictive model is therefore as simple as manipulating algebraic functions. This can be accomplished in a Fortran program, or in mathematical software packages such as Mathematicaw or MathCad w . Two representations of transport are used here. The first is a uniform-flow representation, wherein all streamlines are parallel. The second is a radially convergent flow representation, wherein all streamlines converge upon the extraction well. Some researchers have found that a one-dimensional uniform flow model adequately describes radially convergent tracer tests in fractured media ŽGustafsson and Klockars, 1981; Raven et al., 1988; Maloszewski and Zuber, 1993., possibly due to the channeled nature of transport in these formations ŽRasmuson and Neretnieks, 1986.. Consequently, we offer both the uniform and radially convergent flow transfer functions here. Both
M.W. Becker, R.J. Charbeneaur Journal of Contaminant Hydrology 40 (2000) 299–310
301
transfer functions are obtained using identical boundary conditions, so that the two solutions are directly comparable.
2. The first-passage-time solution Transfer functions are correctly formulated under the framework of the first passage time ŽJury, 1982; Jury et al., 1986.. The basis of the first passage time is the random walk as described by a Markov process in which only local changes of state are permissible Že.g., Cox and Miller, 1972; Gardiner, 1985.. We allow the transition probabilities, characterized by a velocity, Õ, and a dispersion, D, to be functions of position. Cox and Miller Ž1972. ŽSection 5.5. demonstrate how this process results in a particular case of the backward Fokker–Planck equation Žalso known as the backward Kolmogorov equation.: E p Ž x o , x ;t .
sÕ
Et
E p Ž x o , x ;t . E xo
qD
E 2 p Ž x o , x ;t . E x o2
.
Ž 1.
This is an equation of motion of the probability distribution for continuous Markov processes ŽRisken, 1989.. The term, pŽ x o , x;t ., is a transition probability density function where pŽ x o , x;t .d t represents the probability of the particle being at position, x during the time interval t, t q d t, on the condition that it started at position x o at time t s 0. A solution similar to Eq. Ž1. may be obtained for radially convergent flow. We write the backward Fokker–Planck equation in radial coordinates as E p Ž ro ,r ;t . Et
E Ž ro ,r ;t .
sÕ
1E q
Ero
ro Ero
ž
ro D
E p Ž ro ,r ;t . Ero
/
,
Ž 2.
where r is the radial distance and ro is the initial position of the particle. Suppose that for the case of Eq. Ž1., we want to find the first passage time, T, of X Ž t . from x o to a ) x o . A convenient way of finding T is to establish an absorbing boundary at x s a, such that when a particle contacts the absorbing boundary, it is instantaneously and permanently removed from the system. The function pŽ x o , x;t .d t then becomes the probability that X Ž t . s x and that the process does not reach the boundary in the time interval, 0 F t - T. The decrease in cumulative probability over the domain y` F x F a is equivalent to the probability flux out of the domain and may be denoted as: E g Ž t < x o ,a . s y
a
H pŽ x Et y`
o , x ;t
. d x.
Ž 3.
The Laplace transform of Eq. Ž3. is defined as `
g Ž xo . s
ys t
H0 e
g Ž t < x o ,a . d t ,
Ž 4.
M.W. Becker, R.J. Charbeneaur Journal of Contaminant Hydrology 40 (2000) 299–310
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where s is the Laplace transform variable. By noting that g Ž0 < x o ,a. s 0, and taking the Laplace transform of Eq. Ž1. we obtain Õ
d g Ž xo . d xo
qD
d2 g Ž x o . d x o2
s sg Ž x o . .
Ž 5.
We impose the condition of an absorbing boundary at x s a in a semi-infinite domain by requiring that g Ž y` . s 0,
Ž 6.
g Ž a . s 1.
Ž 7.
and The first condition ŽEq. Ž6.. specifies that when a particle is introduced at x s y` it will never be absorbed, and the second ŽEq. Ž7.. specifies that when a particle is introduced at x s a, it is absorbed immediately. The Laplace space solution of Eq. Ž5. with boundary conditions Ž6. and Ž7. and x o s 0, is: aÕ
g Ž s . s exp
2D
ž ( 1y
1q
4 Ds Õ2
/
Ž 8.
If it is assumed that coefficient of dispersion is related to the average linear velocity, Õ, and the longitudinal hydrodynamic dispersivity, a , by a < Õ <, the Laplace space backward Fokker–Planck equation in radial form becomes:
a
d 2 g Ž ro . d ro2
y
d g Ž ro . d ro
s
ro s A
g Ž ro . ,
Ž 9.
where the radially convergent flow geometry implies, A Õs
ro
.
Ž 10 .
For transport in porous media, As
Q 2p nB
.
Ž 11 .
and for transport in N fractures of aperture 2b, As
Q 4p Nb
.
Ž 12 .
To obtain the first-passage-time solution for radial flow, the following boundary conditions are applied at an infinite distance from the well, and at the well radius Ž r w ., respectively: g Ž ` . s 0,
Ž 13 .
g Ž r w . s 1.
Ž 14 .
and
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Note that the first condition ŽEq. Ž13.. allows upstream dispersion, which is the appropriate inlet condition for a radially convergent tracer test ŽChen et al., 1996.. The backward Fokker–Planck solution allows the application of a ‘‘semi-infinite’’ transport condition in the case where the actual travel path cannot proceed to infinity, due to the presence of the pumping well. A solution of Eq. Ž9. according to Eqs. Ž13. and Ž14. is suggested by the mathematical solution of the divergent flow case proposed by Moench and Ogata Ž1981. and detailed in Chen Ž1985.. The solution is g Ž ro . s exp
yo y yw
Ai Ž b 1r3 yo .
2
Ai Ž b 1r3 yw .
,
Ž 15 .
where: yo s
ro
a
1 q
4b
,
yw s
rw
a
1 q
4b
bs
,
sa 2 A
and AiŽ. is the Airy function. For purposes of fitting data, it is generally more convenient to work with dimensionless parameters. Transfer functions Ž8. and Ž15. may be re-written in Laplace space as Žrespectively., Pe
g U s exp
2
ž ( 1y
1q
4 sD Pe
/
,
Ž 16 .
and g R s exp
Pe
Ai Ž b D1r3 Yo .
2
Ai Ž b D1r3 Yw .
ž /
,
where Yo s
RPe
1 q
Ry1
4b D
and
Yw s
Pe
1 q
Ry1
4b D
,
Ž 17 .
and the dimensionless variables are given in Table 1.
Table 1 Definition of dimensionless variables Quantity
Uniform flow
Radial flow
Mean first passage time Transport distance Peclet number Dimensionless well separation Dimensionless time Dimensionless Laplace variable Dimensionless flux-averaged concentration
t s ar Õ a Pes1r a None t D s trt s D s st C D s nt QCf Ž t .r Mo
t sp nB Ž ro2 y r w2 .r Q ro y r w Pes Ž ro y r w .r a Rs Ž ro r r w . t D s trt b D s Ž2 s D r Pe 2 .wŽ Ry1.rŽ Rq1.x C D s nt QCf Ž t .r Mo
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M.W. Becker, R.J. Charbeneaur Journal of Contaminant Hydrology 40 (2000) 299–310
To obtain the time-domain solution, Laplace solutions must be inverted. Because no analytical inversion is available for Eqs. Ž16. and Ž17., the inversion must be performed numerically. This is easily accomplished in Fortran using widely available inversion algorithms Že.g., de Hoog et al., 1982., or using algorithms written for commercial mathematical software. For this note, Laplace space functions were inverted to the time
Fig. 1. Comparison of the radially convergent first passage time model ŽRadial FPT. with the previously published models of Moench Ž1989. and Welty and Gelhar Ž1994.. The three models give nearly identical results at higher Peclet numbers Že.g., Pes100., but not at lower Peclet numbers Že.g., Pes10..
M.W. Becker, R.J. Charbeneaur Journal of Contaminant Hydrology 40 (2000) 299–310
305
domain using a MathCad w spreadsheet utilizing an imbedded fast Fourier transform function ŽBecker, 1996.. This method of inversion compared favorably with the algorithm of de Hoog et al. Ž1982. discussed by Moench Ž1991.. The radially convergent solution of Eq. Ž17. is compared to the approximate analytical solution of Welty and Gelhar Ž1994. and the semi-numerical numerical solution of Moench Ž1989; 1991. in Fig. 1. The first-passage-time, Moench Ž1989; 1991. and Welty and Gelhar Ž1994. solutions compare well at Peclet numbers greater than 100, but diverge at lesser Peclet numbers. As noted by the authors, the approximate solution of Welty and Gelhar Ž1994. fails at Peclet numbers much less than 100. The Moench Ž1989; 1991. solution under-predicts breakthrough tailing compared to the first-passagetime solution, because it assumes a closed inlet condition at the injection well, where as the first-passage-time solution imposes an open inlet condition at the injection well.
3. The first-passage-time solution as a transfer function Transfer functions uniquely characterize the response of a linear system to an input function. For example, when the input to the system is mass flux, the output from the system is also mass flux. The output response of a compound system is found by the convolution of the individual transfer functions that represents the contributing sub-systems. Convolution is performed conveniently in Laplace space because multiplication in Laplace space is equivalent to convolution in the time domain. For example, an instantaneous flux of mass, Mo , may be introduced to the uniform flow system Ž8. by multiplying the function in Laplace space by the Dirac pulse, Mo d Ž t ., which in Laplace space is simply, Mo . Inversion of the resulting product results in the time-domain flux-averaged breakthrough from the system: Cf Ž t . s
Mo a Õ'4p Dt 3
exp y
Ž a y Õt . 4 Dt
2
.
Ž 18 .
Comparison of Eq. Ž16. with the well-known one-dimensional ‘‘semi-infinite’’ advection–dispersion solution Že.g., Lenda and Zuber, 1970; Kreft and Zuber, 1978. confirms that Cf is the flux-averaged concentration, Õ is the average-linear velocity, and D is the coefficient of longitudinal hydrodynamic dispersion. A transfer-function approach allows a great deal of flexibility in modeling tracer tests because each component of the test system can be represented by a separate transfer function. The expected tracer breakthrough is predicted by convolution of the input function with all of the other transfer functions representing the system. For example, the model of Moench Ž1989; 1991. allows for the prediction of a ‘‘tophat’’ input pulse, mixing in both the injection and pumping well, and transport through the formation. Such a test would be modeled using transfer functions by multiplying in Laplace space, a tophat function, two exponential decay functions, and the radially convergent transport transfer function Ž17.. Laplace space representations for the other functions are readily found in texts concerning mixing tanks Že.g., Levenspiel, 1972. or mathematical references Že.g., Abramowitz and Stegun, 1964.. The practical requirements for placing
306
M.W. Becker, R.J. Charbeneaur Journal of Contaminant Hydrology 40 (2000) 299–310
a subsystems in series is that the transfer function must represent mass flux, and that the integrated area under the transfer functions be equal to 1. 4. Application of the first passage time to a field tracer test Because Laplace space solutions are available for a great variety of functions Že.g., Abramowitz and Stegun, 1964., we will not attempt to provide a list here. Instead, the transfer functions developed above will be used to model a field tracer test conducted in the summer of 1995, in the western foothills of the Sierra Nevada Range near Raymond, California ŽBecker, 1996; Becker et al., 1999.. The design of this test was such that it could not be accurately modeled using previous published analytic or semi-analytic methods. A weak-dipole tracer test was conducted between two wells separated by a distance of 30 m. The weak-dipole design was chosen because it reduces residence time in the injection well bore without creating the transient hydraulic effects that result from the finite chase fluid approach Žsee, e.g., Reimus et al., 1998; Becker et al., 1999.. The pumping flow rate was 7.2 lrmin, and 5.6% of the pumped fluid Ž0.4 lrmin. was reinjected continuously. The borehole radius of both wells was 8.5 cm. Injection and pumping was accomplished through packed intervals of approximately 1 and 3 m, respectively. Tracer mass was introduced as a slug to the injection borehole, which was mixed and monitored by constantly circulating water to the surface using a down-hole pump. A number of tracers were used in the experiments, but for the sake of brevity only fluorescein will be discussed. The test will be represented using the radially convergent model, which assumes that the divergence of streamlines at the injection well was negligible. In reality, streamline divergence at the well does have a small effect on the early breakthrough, and this effect can also be assessed using a transfer function approach. This complication will be considered in a future article, however, so that for now it will be assumed that the flow is approximately radially convergent. Because the concentration in the injection borehole was measured throughout the test, the transfer function representing mixing in the borehole is known independently of the groundwater flow transport transfer function. A concentration record in the injection borehole is extremely useful, because otherwise one must make assumptions about the geometry of the flow field to gauge the influence of borehole dilution resulting from fluid passing through the injection well on its way to the pumping well ŽZlotnik and Logan, 1996.. The concentration history in the borehole ŽFig. 2. demonstrates that through most of the injection, the concentration changes at an exponentially decreasing rate. This is the behavior expected for a perfectly mixed reservoir. At later time, however, the concentration begins to level off. The break in slope precedes arrival of the tracer at the injection well, so it could not be caused by the reinjection of traced fluid. When observed in mixing tanks, such a break in slope is indicative of incomplete mixing or the presents of a ‘‘dead-zone’’ in the tank. Transfer functions that account for a fraction of dead volume in a mixed tank are well known Že.g., Levenspiel, 1972; Nauman and Buffham, 1983.. One model ŽModel L of Levenspiel. assumes that of the total volume, V, and flow rate, Q, of a tank; a fraction
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307
Fig. 2. Fluorescein concentration in the injection well bore fit with a ‘‘dead-zone’’ mixing tank model Ž19.. The early-time slope is the result of ideal mixing and dilution in the active portion of the well bore, and the late-time slope is the result of less active mixing and dilution in a relatively stagnant portion of the well bore.
aV and bQ is associated with a relatively less-mixed subsystem. The transfer function for mass moving through the more active volume, Ž1 y a.V, is given in Laplace space as: gB s
b q as D
Ž 19 .
a Ž 1 y a . Ž m1 y s D . Ž m 2 y s D .
where m1 ,m 2 s
aqb 2 aŽ 1 y a.
)
y1 " 1 y
4 ba Ž 1 y a .
Ž a qb.
2
Fig. 1 displays the concentration of fluorescein in the injection borehole up until the time another dye was introduced, and it is apparent that the dead-zone model Ž19. adequately fits the data when a s 0.05 and b s 0.0025. The expected dimensionless flux-averaged concentration at the pumping well, due to a pulse of mass injected into the imperfectly mixed borehole, is therefore CD s
b q as D a Ž 1 y a . Ž m1 y s D . Ž m 2 y s D .
exp
Pe
Ai Ž b D1r3 Yo .
2
Ai Ž b D1r3 Yw .
ž /
,
Ž 20 .
308
M.W. Becker, R.J. Charbeneaur Journal of Contaminant Hydrology 40 (2000) 299–310
for the radially convergent flow case. Since a portion of the produced fluid was constantly reinjected, it is necessary to account for the recirculation of tracer breakthrough. This is accomplished with ease in Laplace space via the function CUD s
CD 1 q ´ Ž1 y CD .
.
Ž 21 .
Here, CUD is interpreted to be the dimensionless concentration of tracer at the withdrawal well after taking into account the reinjection of a fraction, ´ , of the pumped fluid. It is important to note that this liberal use of convolution operators is valid only because the boundaries on the first-passage-time transport function conserve mass and dispersion from one system to the next. They ensure that flux continuity is preserved between the transport function and the ancillary mixing and source functions, without affecting the boundary characteristics of the transport function. Attempts to find equivalent solutions using continuity arguments or concepts of superposition may or may not lead to identical results Že.g., Novakowski, 1992.. A composite transfer-function model, that accounts for radially convergent flow, stagnancy in the injection well bore, and reinjection of traced water ŽEqs. Ž20. and Ž21.., closely fits the fluorescein breakthrough at the pumping well. The model fit shown in Fig. 3 was achieved by setting Mo to the total recovered mass Žapproximately 48% of the injected mass.. The variable parameters were t and Pe, and the best fit of model to
Fig. 3. Model fit of fluorescein breakthrough between wells separated by 30 m. Imperfect mixing in the well bore ŽFig. 1., radially convergent transport in the formation, and 5.6% reinjection of the traced fluid is model is represented in Laplace space using Eqs. Ž20. and Ž21., with Pes 2.7 and t s95 h.
M.W. Becker, R.J. Charbeneaur Journal of Contaminant Hydrology 40 (2000) 299–310
309
data resulted when Pe s 2.7 and t s 95 h. An adequate fit could also be obtained using the uniform-flow transfer function, using slightly different parameters, indicating that the choice of a radial- or uniform-flow transport model must be made a priori, rather than based upon the simulation of the data. 5. Conclusions We present a new radially convergent transport model, developed from a first-passage-time solution of the backward Fokker–Planck equation. This probabilistic equation of motion allows one to work backward in time to produce a ‘‘semi-infinite’’ solution, even though the path of the particle cannot physically proceed to infinity. The semi-infinite solution is important, because it produces a transfer function that relates the output and input flux of a linear system. Groundwater tracer tests are generally too complex to be represented by a single transfer function. More typically, the test must be represented by a composite function, produced by the convolution of component transfer functions that represent various aspects of the injection, transport, and withdrawal of the traced fluid. When represented mathematically in Laplace space, the composite transfer function model may be found by multiplying component transfer functions. In the example application to an actual weak-dipole tracer test, imperfect mixing in the injection well bore and radially convergent transport in the rock are modeled with transfer functions, while partial reinjection of traced water is accounted for using an algebraic expression in Laplace space. The flexibility of this transfer function approach allows the rapid assessment of proposed tracer test designs and conceptual models of transport. Various scenarios can be evaluated by manipulating and exchanging transfer functions in Laplace space, and then inverting the composite solution to the time domain using numerical methods. Calculations of predicted breakthrough curves may be accomplished using widely available mathematical software such as packages such as Mathematicaw or MathCad w . Acknowledgements The authors would like express their appreciation for the internal and external reviewers that examined earlier versions of this manuscript. The tracer tests at the Raymond field site were performed as part of a cooperative effort between Los Alamos and Lawrence Berkeley National Laboratories. This work was supported by the Yucca Mountain Site Characterization Office as part of the Civilian Radioactive Waste Management Program. This project is managed by the US Department of Energy, Yucca Mountain Site Characterization Project. References Abramowitz, M., Stegun, I.A., 1964. Handbook of Mathematical Functions. Dover Publications, New York, 1046 pp. Al-Niami, A.N.S., Rushton, K.R., 1978. Radial dispersion to an abstraction well. J. Hydrol. 39, 287–300.
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Becker, M.W., 1996. Tracer tests in a fractured rock and their first passage time models. PhD Dissertation, The University of Texas, Austin, TX, 210 pp. Becker, M.W., Reimus, P.W., Vilks, P., 1999. Transport and attenuation of carboxylate-modified-latex microspheres in fractured rock laboratory and field tracer tests. Ground Water 37, 387–395. Chen, C.-S., 1985. Analytical and approximate solutions to radial dispersion from an injection well to a geological unit with simultaneous diffusion into adjacent strata. Water Resour. Res. 21, 1069–1076. Chen, J.-S., Liu, C.-W., Chen, C.-S., Yeh, H.-D., 1996. A Laplace transform solution for tracer tests in a radially convergent flow field with upstream dispersion. J. Hydrol. 183, 263–275. Cox, D.R., Miller, H.D., 1972. The Theory of Stochastic Processes. Chapman and Hall, New York, 398 pp. de Hoog, F.R., Knight, J.H., Stokes, A.N., 1982. An improved method for numerical inversion of Laplace transforms. SIAM J. Sci. Stat. Comput. 3, 357–366. Gardiner, C.W., 1985. The Handbook of Stochastic Methods Žfor Physics, Chemistry and the Natural Sciences., 2nd edn. Springer-Verlag, New York, 442 pp. Gustafsson, E., Klockars, C.E., 1981. Studies on groundwater transport in fractured crystalline rock under controlled conditions using nonradioactive tracers. Geological Survey of Sweden, Report Number SKBFrKBS-TR-81-07. Hodgkinson, D.P., Lever, D.A., 1983. Interpretation of a field experiment on the transport of sorbed and non-sorbed tracers through a fracture in crystalline rock. Radioact. Waste Manage. Nucl. Fuel Cycle 4, 129–158. Jury, W.A., 1982. Simulation of solute transport using a transfer function model. Water Resour. Res. 18, 363–368. Jury, W.A., Sposito, G., White, R.E., 1986. A transfer function model of solute transport through soil: 1. Fundamental concepts. Water Resour. Res. 22, 243–247. Kreft, A., Zuber, A., 1978. On the physical meaning of the dispersion equation and its solutions for different initial and boundary conditions. Chem. Eng. Sci. 33, 1471–1480. Lenda, A., Zuber, A., 1970. Tracer dispersion in groundwater experiments. Isotope Hydrology. International Atomic Energy Agency ŽIAEA., Vienna, pp. 619–641. Levenspiel, O., 1972. Chemical Reaction Engineering, 2nd edn. Wiley, New York, 325 pp. Maloszewski, P., Zuber, A., 1993. Tracer experiments in fractured rocks: matrix diffusion and the validity of models. Water Resour. Res. 29, 2723–2735. Moench, A.F., 1989. Convergent radial dispersion: a Laplace transform solution for aquifer tracer testing. Water Resour. Res. 25, 439–447. Moench, A.F., 1991. Convergent radial dispersion: a note on evaluation of the Laplace transform solution. Water Resour. Res. 27, 3261–3264. Moench, A.F., 1995. Convergent radial dispersion in a double-porosity aquifer with fracture skin: analytical solution and application to a field experiment in fractured chalk. Water Resour. Res. 31, 1823–1835. Moench, A.F., Ogata, A., 1981. A numerical inversion of the Laplace transform solution to radial dispersion in a porous medium. Water Resour. Res. 17, 250–252. Nauman, E.B., Buffham, B.A., 1983. Mixing in Continuous Flow Systems. Wiley, New York, 272 pp. Novakowski, K.S., 1992. An evaluation of boundary conditions for one-dimensional solute transport: 1. Mathematical development. Water Resour. Res. 28, 2399–2410. Rasmuson, A., Neretnieks, I., 1986. Radionuclide transport in fast channels in crystalline rock. Water Resour. Res. 22, 1247–1256. Raven, K.G., Novakowski, K.S., Lapcevic, P.A., 1988. Interpretation of field tracer tests of a single fracture using a transient solute storage model. Water Resour. Res. 24, 2019–2032. Reimus, P.W., Haga, M.J., Callahan, T.J., Anghel, I., Turin, H.J., Counce, D., 1998. C-Holes update report: reinterpretation of the reactive tracer test in the Bullfrog Tuff and results of laboratory experiments. Yucca Mountain Project Milestone SP32E2M4SZ, Los Alamos National Laboratory, Los Alamos, NM. Risken, H., 1989. The Fokker–Planck Equation, 2nd edn. Springer-Verlag, New York. Welty, C., Gelhar, L.W., 1994. Evaluation of longitudinal dispersivity from nonuniform flow tracer tests. J. Hydrol. 153, 71–102. Zlotnik, V.A., Logan, J.D., 1996. Boundary conditions for convergent radial tracer tests and effect of well bore mixing volume. Water Resour. Res. 32, 2323–2328.
First-passage-time transfer functions for groundwater tracer tests conducted in radially convergent flow Matthew W. Becker a
a,)
, Randall J. Charbeneau
b,1
Department of Geology, State UniÕersity of New York at Buffalo, 876 Natural Sciences Complex, Buffalo, NY 14260, USA b Department of CiÕil Engineering, UniÕersity of Texas at Austin, Pickle Research Campus a119, UniÕersity of Texas at Austin, Austin, TX 78712, USA Received 30 March 1998; received in revised form 1 June 1999; accepted 14 June 1999
Abstract Forced-gradient groundwater tracer tests may be conducted using a variety of hydraulic schemes, so it is useful to have simple semi-analytic models available that can examine various injectionrwithdrawal scenarios. Models for radially convergent tracer tests are formulated here as transfer functions, which allow complex tracer test designs to be simulated by a series of simple mathematical expressions. These mathematical expressions are given in Laplace space, so that transfer functions may be placed in series by simple multiplication. Predicted breakthrough is found by numerically inverting the composite transfer function to the time-domain, using traditional computer programs or commercial mathematical software. Transport is assumed to be dictated by a radially convergent or uniform flow field, and is based upon an exact first-passagetime solution of the backward Fokker–Planck equation. These methods are demonstrated by simulating a weak-dipole tracer test conducted in a fractured granite formation, where mixing in the injection borehole is non-ideal. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Groundwater; Solute transport; Tracers; Transfer functions; Laplace transformations; Fractures
) Corresponding author. Tel.: q1-716-645-6800 ext. 3960; fax: q1-716-645-3999; E-mail: [email protected] 1 Tel.: q1-512-471-0070; E-mail: [email protected].
0169-7722r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 7 7 2 2 Ž 9 9 . 0 0 0 6 1 - 3
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M.W. Becker, R.J. Charbeneaur Journal of Contaminant Hydrology 40 (2000) 299–310
1. Introduction A groundwater tracer test that employs a forced radially convergent flow field offers advantages for estimating transport properties in porous and fractured media. A radially convergent test promotes recovery of the injected mass, reduces the effect of apparent dispersion due to the flow field, and minimizes the influence of the natural hydraulic gradient. Thus, radially convergent tracer tests are particularly useful where transport, rather than hydraulic properties, are desired. This note offers simple and flexible analytical tools for designing and interpreting groundwater tracer tests conducted in an induced radially convergent flow field. Various test designs are possible using radially convergent flow. For example, tracer may be introduced as a instantaneous pulse, a finite pulse, or a constant concentration. Tracer mass may be followed by a chaser fluid or be flushed from the injection well only by water flowing to the pumping well. The chase fluid may be of finite volume or be added at a constant rate throughout the test Žproducing a ‘‘weak dipole’’ hydraulic field.. In addition to variation in injection design, non-ideal flow behavior may affect the way that tracer is actually introduced to the formation. For example, if tracer is added to the injection well as a pulse, mixing with un-traced water flowing through the well will delay and dilute the tracer as it enters the formation ŽZlotnik and Logan, 1996.. Predictive and interpretive models have been available for radially convergent tests for some time ŽAl-Niami and Rushton, 1978; Hodgkinson and Lever, 1983; Welty and Gelhar, 1994; Moench, 1989, 1991, 1995.. However, these models are generally limited with respect to the variations in test designs that are allowed. The model of Moench Ž1989; 1991; 1995. is widely used to interpret tracer tests. It allows for continuous, instantaneous, or finite concentration injection, and accounts for mixing in both the injection and well bore. However, it does not allow for incomplete mixing Že.g., dead zones in the injection well., reinjection of traced fluid Žrecycle., or other deviations from the aforementioned designs. The model described here employs a transfer function approach to predicting tracer breakthrough. In this approach, each component of the test is represented by a transfer function, so that breakthrough is predicted to be the convolution of these components. Transport in the geologic medium must, therefore, be formulated in terms of flux, rather than concentration. Such a formulation is presented herein. Convolution is accomplished by multiplying the transfer functions in Laplace space, and then inverting the Laplace solution to the time domain. Changing the predictive model is therefore as simple as manipulating algebraic functions. This can be accomplished in a Fortran program, or in mathematical software packages such as Mathematicaw or MathCad w . Two representations of transport are used here. The first is a uniform-flow representation, wherein all streamlines are parallel. The second is a radially convergent flow representation, wherein all streamlines converge upon the extraction well. Some researchers have found that a one-dimensional uniform flow model adequately describes radially convergent tracer tests in fractured media ŽGustafsson and Klockars, 1981; Raven et al., 1988; Maloszewski and Zuber, 1993., possibly due to the channeled nature of transport in these formations ŽRasmuson and Neretnieks, 1986.. Consequently, we offer both the uniform and radially convergent flow transfer functions here. Both
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transfer functions are obtained using identical boundary conditions, so that the two solutions are directly comparable.
2. The first-passage-time solution Transfer functions are correctly formulated under the framework of the first passage time ŽJury, 1982; Jury et al., 1986.. The basis of the first passage time is the random walk as described by a Markov process in which only local changes of state are permissible Že.g., Cox and Miller, 1972; Gardiner, 1985.. We allow the transition probabilities, characterized by a velocity, Õ, and a dispersion, D, to be functions of position. Cox and Miller Ž1972. ŽSection 5.5. demonstrate how this process results in a particular case of the backward Fokker–Planck equation Žalso known as the backward Kolmogorov equation.: E p Ž x o , x ;t .
sÕ
Et
E p Ž x o , x ;t . E xo
qD
E 2 p Ž x o , x ;t . E x o2
.
Ž 1.
This is an equation of motion of the probability distribution for continuous Markov processes ŽRisken, 1989.. The term, pŽ x o , x;t ., is a transition probability density function where pŽ x o , x;t .d t represents the probability of the particle being at position, x during the time interval t, t q d t, on the condition that it started at position x o at time t s 0. A solution similar to Eq. Ž1. may be obtained for radially convergent flow. We write the backward Fokker–Planck equation in radial coordinates as E p Ž ro ,r ;t . Et
E Ž ro ,r ;t .
sÕ
1E q
Ero
ro Ero
ž
ro D
E p Ž ro ,r ;t . Ero
/
,
Ž 2.
where r is the radial distance and ro is the initial position of the particle. Suppose that for the case of Eq. Ž1., we want to find the first passage time, T, of X Ž t . from x o to a ) x o . A convenient way of finding T is to establish an absorbing boundary at x s a, such that when a particle contacts the absorbing boundary, it is instantaneously and permanently removed from the system. The function pŽ x o , x;t .d t then becomes the probability that X Ž t . s x and that the process does not reach the boundary in the time interval, 0 F t - T. The decrease in cumulative probability over the domain y` F x F a is equivalent to the probability flux out of the domain and may be denoted as: E g Ž t < x o ,a . s y
a
H pŽ x Et y`
o , x ;t
. d x.
Ž 3.
The Laplace transform of Eq. Ž3. is defined as `
g Ž xo . s
ys t
H0 e
g Ž t < x o ,a . d t ,
Ž 4.
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where s is the Laplace transform variable. By noting that g Ž0 < x o ,a. s 0, and taking the Laplace transform of Eq. Ž1. we obtain Õ
d g Ž xo . d xo
qD
d2 g Ž x o . d x o2
s sg Ž x o . .
Ž 5.
We impose the condition of an absorbing boundary at x s a in a semi-infinite domain by requiring that g Ž y` . s 0,
Ž 6.
g Ž a . s 1.
Ž 7.
and The first condition ŽEq. Ž6.. specifies that when a particle is introduced at x s y` it will never be absorbed, and the second ŽEq. Ž7.. specifies that when a particle is introduced at x s a, it is absorbed immediately. The Laplace space solution of Eq. Ž5. with boundary conditions Ž6. and Ž7. and x o s 0, is: aÕ
g Ž s . s exp
2D
ž ( 1y
1q
4 Ds Õ2
/
Ž 8.
If it is assumed that coefficient of dispersion is related to the average linear velocity, Õ, and the longitudinal hydrodynamic dispersivity, a , by a < Õ <, the Laplace space backward Fokker–Planck equation in radial form becomes:
a
d 2 g Ž ro . d ro2
y
d g Ž ro . d ro
s
ro s A
g Ž ro . ,
Ž 9.
where the radially convergent flow geometry implies, A Õs
ro
.
Ž 10 .
For transport in porous media, As
Q 2p nB
.
Ž 11 .
and for transport in N fractures of aperture 2b, As
Q 4p Nb
.
Ž 12 .
To obtain the first-passage-time solution for radial flow, the following boundary conditions are applied at an infinite distance from the well, and at the well radius Ž r w ., respectively: g Ž ` . s 0,
Ž 13 .
g Ž r w . s 1.
Ž 14 .
and
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Note that the first condition ŽEq. Ž13.. allows upstream dispersion, which is the appropriate inlet condition for a radially convergent tracer test ŽChen et al., 1996.. The backward Fokker–Planck solution allows the application of a ‘‘semi-infinite’’ transport condition in the case where the actual travel path cannot proceed to infinity, due to the presence of the pumping well. A solution of Eq. Ž9. according to Eqs. Ž13. and Ž14. is suggested by the mathematical solution of the divergent flow case proposed by Moench and Ogata Ž1981. and detailed in Chen Ž1985.. The solution is g Ž ro . s exp
yo y yw
Ai Ž b 1r3 yo .
2
Ai Ž b 1r3 yw .
,
Ž 15 .
where: yo s
ro
a
1 q
4b
,
yw s
rw
a
1 q
4b
bs
,
sa 2 A
and AiŽ. is the Airy function. For purposes of fitting data, it is generally more convenient to work with dimensionless parameters. Transfer functions Ž8. and Ž15. may be re-written in Laplace space as Žrespectively., Pe
g U s exp
2
ž ( 1y
1q
4 sD Pe
/
,
Ž 16 .
and g R s exp
Pe
Ai Ž b D1r3 Yo .
2
Ai Ž b D1r3 Yw .
ž /
,
where Yo s
RPe
1 q
Ry1
4b D
and
Yw s
Pe
1 q
Ry1
4b D
,
Ž 17 .
and the dimensionless variables are given in Table 1.
Table 1 Definition of dimensionless variables Quantity
Uniform flow
Radial flow
Mean first passage time Transport distance Peclet number Dimensionless well separation Dimensionless time Dimensionless Laplace variable Dimensionless flux-averaged concentration
t s ar Õ a Pes1r a None t D s trt s D s st C D s nt QCf Ž t .r Mo
t sp nB Ž ro2 y r w2 .r Q ro y r w Pes Ž ro y r w .r a Rs Ž ro r r w . t D s trt b D s Ž2 s D r Pe 2 .wŽ Ry1.rŽ Rq1.x C D s nt QCf Ž t .r Mo
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To obtain the time-domain solution, Laplace solutions must be inverted. Because no analytical inversion is available for Eqs. Ž16. and Ž17., the inversion must be performed numerically. This is easily accomplished in Fortran using widely available inversion algorithms Že.g., de Hoog et al., 1982., or using algorithms written for commercial mathematical software. For this note, Laplace space functions were inverted to the time
Fig. 1. Comparison of the radially convergent first passage time model ŽRadial FPT. with the previously published models of Moench Ž1989. and Welty and Gelhar Ž1994.. The three models give nearly identical results at higher Peclet numbers Že.g., Pes100., but not at lower Peclet numbers Že.g., Pes10..
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domain using a MathCad w spreadsheet utilizing an imbedded fast Fourier transform function ŽBecker, 1996.. This method of inversion compared favorably with the algorithm of de Hoog et al. Ž1982. discussed by Moench Ž1991.. The radially convergent solution of Eq. Ž17. is compared to the approximate analytical solution of Welty and Gelhar Ž1994. and the semi-numerical numerical solution of Moench Ž1989; 1991. in Fig. 1. The first-passage-time, Moench Ž1989; 1991. and Welty and Gelhar Ž1994. solutions compare well at Peclet numbers greater than 100, but diverge at lesser Peclet numbers. As noted by the authors, the approximate solution of Welty and Gelhar Ž1994. fails at Peclet numbers much less than 100. The Moench Ž1989; 1991. solution under-predicts breakthrough tailing compared to the first-passagetime solution, because it assumes a closed inlet condition at the injection well, where as the first-passage-time solution imposes an open inlet condition at the injection well.
3. The first-passage-time solution as a transfer function Transfer functions uniquely characterize the response of a linear system to an input function. For example, when the input to the system is mass flux, the output from the system is also mass flux. The output response of a compound system is found by the convolution of the individual transfer functions that represents the contributing sub-systems. Convolution is performed conveniently in Laplace space because multiplication in Laplace space is equivalent to convolution in the time domain. For example, an instantaneous flux of mass, Mo , may be introduced to the uniform flow system Ž8. by multiplying the function in Laplace space by the Dirac pulse, Mo d Ž t ., which in Laplace space is simply, Mo . Inversion of the resulting product results in the time-domain flux-averaged breakthrough from the system: Cf Ž t . s
Mo a Õ'4p Dt 3
exp y
Ž a y Õt . 4 Dt
2
.
Ž 18 .
Comparison of Eq. Ž16. with the well-known one-dimensional ‘‘semi-infinite’’ advection–dispersion solution Že.g., Lenda and Zuber, 1970; Kreft and Zuber, 1978. confirms that Cf is the flux-averaged concentration, Õ is the average-linear velocity, and D is the coefficient of longitudinal hydrodynamic dispersion. A transfer-function approach allows a great deal of flexibility in modeling tracer tests because each component of the test system can be represented by a separate transfer function. The expected tracer breakthrough is predicted by convolution of the input function with all of the other transfer functions representing the system. For example, the model of Moench Ž1989; 1991. allows for the prediction of a ‘‘tophat’’ input pulse, mixing in both the injection and pumping well, and transport through the formation. Such a test would be modeled using transfer functions by multiplying in Laplace space, a tophat function, two exponential decay functions, and the radially convergent transport transfer function Ž17.. Laplace space representations for the other functions are readily found in texts concerning mixing tanks Že.g., Levenspiel, 1972. or mathematical references Že.g., Abramowitz and Stegun, 1964.. The practical requirements for placing
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a subsystems in series is that the transfer function must represent mass flux, and that the integrated area under the transfer functions be equal to 1. 4. Application of the first passage time to a field tracer test Because Laplace space solutions are available for a great variety of functions Že.g., Abramowitz and Stegun, 1964., we will not attempt to provide a list here. Instead, the transfer functions developed above will be used to model a field tracer test conducted in the summer of 1995, in the western foothills of the Sierra Nevada Range near Raymond, California ŽBecker, 1996; Becker et al., 1999.. The design of this test was such that it could not be accurately modeled using previous published analytic or semi-analytic methods. A weak-dipole tracer test was conducted between two wells separated by a distance of 30 m. The weak-dipole design was chosen because it reduces residence time in the injection well bore without creating the transient hydraulic effects that result from the finite chase fluid approach Žsee, e.g., Reimus et al., 1998; Becker et al., 1999.. The pumping flow rate was 7.2 lrmin, and 5.6% of the pumped fluid Ž0.4 lrmin. was reinjected continuously. The borehole radius of both wells was 8.5 cm. Injection and pumping was accomplished through packed intervals of approximately 1 and 3 m, respectively. Tracer mass was introduced as a slug to the injection borehole, which was mixed and monitored by constantly circulating water to the surface using a down-hole pump. A number of tracers were used in the experiments, but for the sake of brevity only fluorescein will be discussed. The test will be represented using the radially convergent model, which assumes that the divergence of streamlines at the injection well was negligible. In reality, streamline divergence at the well does have a small effect on the early breakthrough, and this effect can also be assessed using a transfer function approach. This complication will be considered in a future article, however, so that for now it will be assumed that the flow is approximately radially convergent. Because the concentration in the injection borehole was measured throughout the test, the transfer function representing mixing in the borehole is known independently of the groundwater flow transport transfer function. A concentration record in the injection borehole is extremely useful, because otherwise one must make assumptions about the geometry of the flow field to gauge the influence of borehole dilution resulting from fluid passing through the injection well on its way to the pumping well ŽZlotnik and Logan, 1996.. The concentration history in the borehole ŽFig. 2. demonstrates that through most of the injection, the concentration changes at an exponentially decreasing rate. This is the behavior expected for a perfectly mixed reservoir. At later time, however, the concentration begins to level off. The break in slope precedes arrival of the tracer at the injection well, so it could not be caused by the reinjection of traced fluid. When observed in mixing tanks, such a break in slope is indicative of incomplete mixing or the presents of a ‘‘dead-zone’’ in the tank. Transfer functions that account for a fraction of dead volume in a mixed tank are well known Že.g., Levenspiel, 1972; Nauman and Buffham, 1983.. One model ŽModel L of Levenspiel. assumes that of the total volume, V, and flow rate, Q, of a tank; a fraction
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Fig. 2. Fluorescein concentration in the injection well bore fit with a ‘‘dead-zone’’ mixing tank model Ž19.. The early-time slope is the result of ideal mixing and dilution in the active portion of the well bore, and the late-time slope is the result of less active mixing and dilution in a relatively stagnant portion of the well bore.
aV and bQ is associated with a relatively less-mixed subsystem. The transfer function for mass moving through the more active volume, Ž1 y a.V, is given in Laplace space as: gB s
b q as D
Ž 19 .
a Ž 1 y a . Ž m1 y s D . Ž m 2 y s D .
where m1 ,m 2 s
aqb 2 aŽ 1 y a.
)
y1 " 1 y
4 ba Ž 1 y a .
Ž a qb.
2
Fig. 1 displays the concentration of fluorescein in the injection borehole up until the time another dye was introduced, and it is apparent that the dead-zone model Ž19. adequately fits the data when a s 0.05 and b s 0.0025. The expected dimensionless flux-averaged concentration at the pumping well, due to a pulse of mass injected into the imperfectly mixed borehole, is therefore CD s
b q as D a Ž 1 y a . Ž m1 y s D . Ž m 2 y s D .
exp
Pe
Ai Ž b D1r3 Yo .
2
Ai Ž b D1r3 Yw .
ž /
,
Ž 20 .
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for the radially convergent flow case. Since a portion of the produced fluid was constantly reinjected, it is necessary to account for the recirculation of tracer breakthrough. This is accomplished with ease in Laplace space via the function CUD s
CD 1 q ´ Ž1 y CD .
.
Ž 21 .
Here, CUD is interpreted to be the dimensionless concentration of tracer at the withdrawal well after taking into account the reinjection of a fraction, ´ , of the pumped fluid. It is important to note that this liberal use of convolution operators is valid only because the boundaries on the first-passage-time transport function conserve mass and dispersion from one system to the next. They ensure that flux continuity is preserved between the transport function and the ancillary mixing and source functions, without affecting the boundary characteristics of the transport function. Attempts to find equivalent solutions using continuity arguments or concepts of superposition may or may not lead to identical results Že.g., Novakowski, 1992.. A composite transfer-function model, that accounts for radially convergent flow, stagnancy in the injection well bore, and reinjection of traced water ŽEqs. Ž20. and Ž21.., closely fits the fluorescein breakthrough at the pumping well. The model fit shown in Fig. 3 was achieved by setting Mo to the total recovered mass Žapproximately 48% of the injected mass.. The variable parameters were t and Pe, and the best fit of model to
Fig. 3. Model fit of fluorescein breakthrough between wells separated by 30 m. Imperfect mixing in the well bore ŽFig. 1., radially convergent transport in the formation, and 5.6% reinjection of the traced fluid is model is represented in Laplace space using Eqs. Ž20. and Ž21., with Pes 2.7 and t s95 h.
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data resulted when Pe s 2.7 and t s 95 h. An adequate fit could also be obtained using the uniform-flow transfer function, using slightly different parameters, indicating that the choice of a radial- or uniform-flow transport model must be made a priori, rather than based upon the simulation of the data. 5. Conclusions We present a new radially convergent transport model, developed from a first-passage-time solution of the backward Fokker–Planck equation. This probabilistic equation of motion allows one to work backward in time to produce a ‘‘semi-infinite’’ solution, even though the path of the particle cannot physically proceed to infinity. The semi-infinite solution is important, because it produces a transfer function that relates the output and input flux of a linear system. Groundwater tracer tests are generally too complex to be represented by a single transfer function. More typically, the test must be represented by a composite function, produced by the convolution of component transfer functions that represent various aspects of the injection, transport, and withdrawal of the traced fluid. When represented mathematically in Laplace space, the composite transfer function model may be found by multiplying component transfer functions. In the example application to an actual weak-dipole tracer test, imperfect mixing in the injection well bore and radially convergent transport in the rock are modeled with transfer functions, while partial reinjection of traced water is accounted for using an algebraic expression in Laplace space. The flexibility of this transfer function approach allows the rapid assessment of proposed tracer test designs and conceptual models of transport. Various scenarios can be evaluated by manipulating and exchanging transfer functions in Laplace space, and then inverting the composite solution to the time domain using numerical methods. Calculations of predicted breakthrough curves may be accomplished using widely available mathematical software such as packages such as Mathematicaw or MathCad w . Acknowledgements The authors would like express their appreciation for the internal and external reviewers that examined earlier versions of this manuscript. The tracer tests at the Raymond field site were performed as part of a cooperative effort between Los Alamos and Lawrence Berkeley National Laboratories. This work was supported by the Yucca Mountain Site Characterization Office as part of the Civilian Radioactive Waste Management Program. This project is managed by the US Department of Energy, Yucca Mountain Site Characterization Project. References Abramowitz, M., Stegun, I.A., 1964. Handbook of Mathematical Functions. Dover Publications, New York, 1046 pp. Al-Niami, A.N.S., Rushton, K.R., 1978. Radial dispersion to an abstraction well. J. Hydrol. 39, 287–300.
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Becker, M.W., 1996. Tracer tests in a fractured rock and their first passage time models. PhD Dissertation, The University of Texas, Austin, TX, 210 pp. Becker, M.W., Reimus, P.W., Vilks, P., 1999. Transport and attenuation of carboxylate-modified-latex microspheres in fractured rock laboratory and field tracer tests. Ground Water 37, 387–395. Chen, C.-S., 1985. Analytical and approximate solutions to radial dispersion from an injection well to a geological unit with simultaneous diffusion into adjacent strata. Water Resour. Res. 21, 1069–1076. Chen, J.-S., Liu, C.-W., Chen, C.-S., Yeh, H.-D., 1996. A Laplace transform solution for tracer tests in a radially convergent flow field with upstream dispersion. J. Hydrol. 183, 263–275. Cox, D.R., Miller, H.D., 1972. The Theory of Stochastic Processes. Chapman and Hall, New York, 398 pp. de Hoog, F.R., Knight, J.H., Stokes, A.N., 1982. An improved method for numerical inversion of Laplace transforms. SIAM J. Sci. Stat. Comput. 3, 357–366. Gardiner, C.W., 1985. The Handbook of Stochastic Methods Žfor Physics, Chemistry and the Natural Sciences., 2nd edn. Springer-Verlag, New York, 442 pp. Gustafsson, E., Klockars, C.E., 1981. Studies on groundwater transport in fractured crystalline rock under controlled conditions using nonradioactive tracers. Geological Survey of Sweden, Report Number SKBFrKBS-TR-81-07. Hodgkinson, D.P., Lever, D.A., 1983. Interpretation of a field experiment on the transport of sorbed and non-sorbed tracers through a fracture in crystalline rock. Radioact. Waste Manage. Nucl. Fuel Cycle 4, 129–158. Jury, W.A., 1982. Simulation of solute transport using a transfer function model. Water Resour. Res. 18, 363–368. Jury, W.A., Sposito, G., White, R.E., 1986. A transfer function model of solute transport through soil: 1. Fundamental concepts. Water Resour. Res. 22, 243–247. Kreft, A., Zuber, A., 1978. On the physical meaning of the dispersion equation and its solutions for different initial and boundary conditions. Chem. Eng. Sci. 33, 1471–1480. Lenda, A., Zuber, A., 1970. Tracer dispersion in groundwater experiments. Isotope Hydrology. International Atomic Energy Agency ŽIAEA., Vienna, pp. 619–641. Levenspiel, O., 1972. Chemical Reaction Engineering, 2nd edn. Wiley, New York, 325 pp. Maloszewski, P., Zuber, A., 1993. Tracer experiments in fractured rocks: matrix diffusion and the validity of models. Water Resour. Res. 29, 2723–2735. Moench, A.F., 1989. Convergent radial dispersion: a Laplace transform solution for aquifer tracer testing. Water Resour. Res. 25, 439–447. Moench, A.F., 1991. Convergent radial dispersion: a note on evaluation of the Laplace transform solution. Water Resour. Res. 27, 3261–3264. Moench, A.F., 1995. Convergent radial dispersion in a double-porosity aquifer with fracture skin: analytical solution and application to a field experiment in fractured chalk. Water Resour. Res. 31, 1823–1835. Moench, A.F., Ogata, A., 1981. A numerical inversion of the Laplace transform solution to radial dispersion in a porous medium. Water Resour. Res. 17, 250–252. Nauman, E.B., Buffham, B.A., 1983. Mixing in Continuous Flow Systems. Wiley, New York, 272 pp. Novakowski, K.S., 1992. An evaluation of boundary conditions for one-dimensional solute transport: 1. Mathematical development. Water Resour. Res. 28, 2399–2410. Rasmuson, A., Neretnieks, I., 1986. Radionuclide transport in fast channels in crystalline rock. Water Resour. Res. 22, 1247–1256. Raven, K.G., Novakowski, K.S., Lapcevic, P.A., 1988. Interpretation of field tracer tests of a single fracture using a transient solute storage model. Water Resour. Res. 24, 2019–2032. Reimus, P.W., Haga, M.J., Callahan, T.J., Anghel, I., Turin, H.J., Counce, D., 1998. C-Holes update report: reinterpretation of the reactive tracer test in the Bullfrog Tuff and results of laboratory experiments. Yucca Mountain Project Milestone SP32E2M4SZ, Los Alamos National Laboratory, Los Alamos, NM. Risken, H., 1989. The Fokker–Planck Equation, 2nd edn. Springer-Verlag, New York. Welty, C., Gelhar, L.W., 1994. Evaluation of longitudinal dispersivity from nonuniform flow tracer tests. J. Hydrol. 153, 71–102. Zlotnik, V.A., Logan, J.D., 1996. Boundary conditions for convergent radial tracer tests and effect of well bore mixing volume. Water Resour. Res. 32, 2323–2328.