Application of Tracers in Arid Zone Hydrology (Proceedings of the Vienna Symposium, August 1994). IAHS Publ. no. 232, 1995. 211

RIETHM: Radioisotope Environmental Tracer Hydrology Model using exponential and dispersion methods

FAHMY M. HUSSEIN Cairo University, Faculty of Agriculture, Soil and Water Department, Cairo, Egypt

Abstract A method for quantitative estimation of aquifer and ground­water hydrological parameters using radioactive environmental isotopes is presented in the form of a menu-driven interactive program, written to solve lumped parameter mathematical problems, using the EXCEL* soft­ware, which can run on Macintosh and PC-compatible computers. The input function and mathematical formulae are stored in a work sheet for updating computed values and the macro selects the calculation pathway. The program solves the direct problem by a selected model using input concentrations, the response function and radioactive decay. The inverse problem could be solved through a trial and error technique. With higher number of tracer data points sampled at longer observation time, a closer estimation of the residence time and the dispersion parameter could be obtained. The program is intended to be used with the assumptions: the system is a homogeneous porous aquifer, water flow is in the steady state, the tracer is "operationally" ideal with respect to the calculated parameters, the tracer natural injection is proportional to water inflow, the tracer output is proportional to water outflow (except for the CFR

dispersive model), the number of the "fitting" parameters is small, the precision of the input function is high enough (either affected or not by thermonuclear tests), the weighting function of the tracer and that of groundwater are similar, the tracer residence time is equivalent to that of groundwater (except for the CFR dispersive model) and sources and sinks of the tracer do not disturb the system. The model has been tested using the input function of an intrusive river in the southern hemisphere. Tritium data collection and interpretation is recommended in these regions in order to make use of the last favourable concentrations of this tracer.

INTRODUCTION

Before and after the large-scale thermonuclear injection of tritium into the atmospheric moisture, this environmental tracer has widely been used in studying many hydrological systems. Beginning from the 1960s, its presence in considerable amounts in groundwater samples is considered as a clear qualitative indication of recent recharge and, in some cases, as a sign of probable recent pollution events. On a quantitative basis, it can be employed in the calculation of the mean residence time of the stored groundwater as well

212 Fahmy M. Hussein

as other aquifer parameters. Works like that of Erikson (1958) show the use of tritium, at its background level, in studying drainage basins with short transit times. Despite the fact that tritium thermonuclear injection belongs to the past, recent studies (e.g. Herrmann, 1986; Maloszewski, 1992) illustrate that the application of tritium output observations in the interpretation of groundwater systems is still an active field of investigation.

This will continue to be so for several years, especially when refined measurements, modelling techniques and time-series tritium data are available. Computer models that deal with groundwater isotope hydrological problems are still few in number. The increasing availability of powerful spreadsheet software on personal computers intro­duces an advantage in user-friendly automation of several types of current hydrological and geochemical isotope investigations such as water-balance (Dexter & Avery, 1991) and some isotopic calculations (Carrol & Rock, 1991). The present work proposes a macro, work sheet and graphic approach (Radioisotope Environmental Tracer Hydro­logy Model: RIETHM) for environmental tritium data use in terms of an explicit solu­tion of the direct problem and an implicit solution of the inverse problem for three conceptual mathematical models which serve extensively in isotope hydrological evalua­tion of aquifer parameters.

ENVIRONMENTAL RADIOISOTOPE HYDROLOGICAL MODELLING

Modelling of groundwater flow and dissolved species transport (Bear & Verruijt, 1987) contributes much to our understanding of the characteristics and behaviour of the aquifer under different conditions and its response to natural and manmade stresses. Water flow and solute transport in the unsaturated zone are active fields of prediction of the spatial and temporal soil-moisture content and potential changes as well as solute and contami­nant transfer in the aeration zone (Hussein, 1990). The prediction of these processes involves the application of a conceptual model written in a mathematical form and mainly solved by convenient numerical methods (Fig. 1) since the analytical solutions are restricted to a few idealized conditions. The wide spectrum of hydrological modelling has been classified by Clark (1973) (cited in Loagne & Green, 1991) in two main categories: (a) type of model; and (b) user point of view. The types includes two sets: (i) high precision models (conceptual and stochastic); and (ii) less precise models (deterministic and empirical or experimental). From the user's point of view, there are the research models (theoretically-based), the engineering models (executive) and the field models (very simplified).

The models frequently encountered in the field of environmental isotope studies are of two conceptual types: (a) distributed parameter models; and (b) black box models (lumped parameter models). The last type ignores the details of what happens inside the probably different components of the natural system and deals only with the comparison between the input and the output content of the tracer and with the interpretation of the shift of the input content towards the output content due to the global effect of some suspected physical and/or chemical change under the constraints of the flow regime. In the lumped parameter models, the number of parameters involved is usually small, so inverse problems could be readily solved and confidence is put in their physical meaning. On the contrary, with distributed parameter models, despite the interest put

RIETHM: Radioisotope Environmental Tracer Hydrology Model 213

Process Concepts

T ! Governing liquations and

Needed Coefficients

Flux and Mass-Balance Equations that do not contain

any information related to v any specific case of flow

The Problem in The Natural System "Water Flow and Solute Transport in Soils"

In a Given Domain at The Macroscopic Level

1A Conceptual Model ( (i.e.a set of assumptions representing our understanding of the real system)

*v «*-

Space & Time Limits

Equations of Initial and Boundary Conditions

G to obtain solution of a particular case of flow

)

* * Mathematical Model -go.

Qa set of partial differential equations)^

Numerical Model Usin; Finite Differerences Finite Elements or Net-Work Method

c Set of Algebric%

Equations

Analytical Solutions J p * -

Comparisons'

pj&- Field or Laboratory Observations h.

Iteration Methods and Programing Language I

Code on a Computer

Comparisons-

-Comparisons-| Approximate [

Solutions

* Equations of boundary conditions express the physical reality of the domain under study, they are imposed by the enviroment.

* * Stochastic treatment may also be needed to account for uncertainties.

Fig. 1 Numeric and analytical solutions of water flow and solute problems in the unsaturated zone.

into solving the direct problem, less confidence is put in them for solving inverse problems due to the large number of fitness parameters which could have no physical meaning (Zuber, 1986).

The quantitative treatment of environmental radioactive tracer data in the interpreta­tion of groundwater systems is also based on conceptual models. Invariably, each conceptual model is written in the from of a mathematical model which is composed of the weighting function, the input function and the radioactive decay term, all under the sign of the convolution integration (equation 2). The weighting function is the major piece of information which distinguishes one model from another (Maloszoweski et al., 1982; Zuber, 1986). It reflects the system response to recharge and demonstrates our concept of how the aquifer (fed by a certain tracer input content) determines the tracer output content (other than the effect of the radioactive decay) as a function of the system of flow. Obviously, the selection of a given weighting function must be consistent with the major groundwater flow conditions and the prevailing hydrogeological characteris­tics of the studied aquifer. A number of idealized situations are more frequently encountered and conceptual models are known for them. Serial and parallel combina­tions of the simple situations are also known (Richter & Szymczak, 1992) with the combined models being more convenient for more realistic situations. However, with the exception of the linear model (which has only a theoretical value but has no real

214 Fahmy M. Hussein

world application) and the piston flow model (which is oversimplified to be used for most natural systems), the exponential and the dispersion models (CFF and CFR) are the most intensively used basic models and they are frequently used as components in most of the combined models. They are currently first tested by most isotope hydrologists before trying the more involved combined models, so there is a need for a user-friendly code which makes the task of testing these models easier.

The use of these models is justified by the adoption of some assumptions about the flow regime such as steady state flow and aquifer homogeneity and the acceptance of a given tracer as if it was an ideal tracer in the sense that its behaviour is identical to the behaviour of the studied fluid.

MATHEMATICAL MODELS

In the case of constant tracer input concentration during time, the output tracer content is calculated using the analytical solution (Erikson, 1958) of the exponential model:

C = '"Put (la) ^output l+\T

and for the residence time T, this formula is written:

r(years) = C'V'~C<"»P" (lb) "^output

For the case of variable input contents (as for tritium and radiocarbon inputs which have been subjected to thermonuclear peaks), the convolution integration (Maloszewski & Zuber, 1982; Zuber, 1986) is used. In its general form it is written as follows:

a

Qf^t, output ~ Q(t-t),output-^(t-t^jnpufi -St'-®* ^ ' | fi(^.

where Q, t, C,t — t' and t' are: discharge, output time (year), tracer content, the year of input, and the time interval (transit time) between the year of input and the that of output, respectively. CM, is the tracer input concentration for the year (t — t'), X is the decay constant of the radioactive tracer, and gt, is the weighting function.

In the steady state, for the construction of the output function using input concentra­tions corresponding to dates separated by unit time interval (one year or one month according to the available input data), this equation reads:

a

Ct,output = C(t_t^inpu[e -gt/-Qt v ;

Introducing a very small value for the X term, this integral could be used for the predic­tion of the output content of a non-radioactive tracer (Richter & Szymczak, 1992) knowing its variable time-series input values.

Integration formulae could be solved by either analytical or numerical methods with numerical methods being in some cases the only possible solutions due to obstacles preventing an analytical solution. In contrast to the elegant and continuous nature of the

R1ETHM: Radioisotope Environmental Tracer Hydrology Model 215

studied domain in analytical solutions (which are applied to limited cases defined by several assumptions), numerical solutions have a discrete nature for both the input and the output and they have more flexible and wider applications (Bear & Verruijt, 1987).

Numerical integration is convenient when we have a discrete set of observations (like the tritium content time series of the input function with each concentration corres­ponding to a certain point in time). Numerical integration begins with the substitution of a simple polynomial for the function to be integrated, (i.e. the polynomial becomes the integrand, Griffith & Smith, 1991).

For one variable function, this substitution leads to the following expression:

b b

JMàx = J g„-iC*)d* (3)

a a

with:

2„-i(*) = a0 + aix + a2x2 + :..+an_lx

n-1 (4)

The polynomial could easily be numerically integrated to yield a reasonable approxi­mate solution, since only summation of the multiplication products involved is needed, as shown by the following expression, which is the basic form of the numerical integra­tion for single variable functions:

b

ôn- lW^ = E W ^ n - A ) (5)

so:

\Ax)àx Ë wfaù (6) =i

with/Çx,) as the observed values at certain point i on the function/(x), and W{ as the weighting coefficient at point i.

Generally, the type of the numerical integration is defined by the spacing of the observation points and the number of these points. A special integration case is noted for the situation where the integrand has an exponential function and the limits of integration lie between zero and infinity. This special case is given by the Gausse-Laguerre integra­tion (Griffith & Smith, 1991):

a n

|e-3y(x)dx = Ç W ^ i ) (7)

with &'x as a weighting function. The evaluation of the/(x) function is made only at the observation points in the

summation term. For the casejfy) = 1, this notation has the following properties:

«•=1 i = i

and the value of Wt decreases rapidly with the increase in number of observation points

216 Fahmy M. Hussein

(whereas in the Newton-Cotes rule, the Wt values are grouped systematically around the integration range midpoint and Wt gives the integration range.

DEFINITION OF WEIGHTING FUNCTIONS

Dispersion models are more realistic in most natural porous aquifers since they account for the physical reality by using the dispersion equation for describing the tracer trans­port. The present program deals with two dispersion models of the four presented by Maloszewski & Zuber (1982) and Zuber (1986), namely the CFF and CFR models (the Da and Db models, respectively, of Richter & Szymczack, 1992). The dispersion para­meter (D/vx) is the dimensionless hydrological parameter characteristic of the flow domain in groundwater. In the lumped parameter models, it has a special meaning: the ratio between the dispersion constant (D/v) (which may be thought of as a measure of the aquifer heterogeneity or the length of the recharge zone (along the flow lines) and the distance (x) between the recharge and the outflow (pumping) zones. For environ­mental tracers (D/v) is the longitudinal dispersivity:

D/v = (DJT V) + ad + b (with D/v = b) (8)

where Dm is the coefficient of molecular diffusion; r is the tortuosity factor; d is the mean grain diameter; a is a constant; b is characteristic of the aquifer, and v is the Darcy velocity. For small scale artificial tracer injection, the b term is dropped out. D/V is a dispersion constant (longitudinal dispersivity); D/vx is a dimensionless dispersion parameter and D is the dispersion coefficient. D and Dm have the same dimensions [L2 T1] .

The exponential model can be used as an approximation of the dispersion model when the dispersion parameter (D/vx) is high whereas the piston flow model could be considered as an approximation of the dispersion model when the D/vx is small (and only under constant tracer input concentrations).

In the present program, the lower and the upper limit of the integration are defined and changed on the work sheet by the control of the instructions in the macro-sheet within a loop. The tritium input CM, corresponding to any given input year is read from the input function zone in the work sheet and used in the loop progress. The program is accompanied by instructions from the macro (given in Appendix A, whereas the functions and formulae used in the work sheet are given in Appendix B). The selection of an appropriate gt, function corresponding to the model required is controlled by the user with two IF statements (in column J of the work sheet). (The Appendices and work­sheet can be obtained by contacting the author.)

Three types of the weighting function are used in the program, namely: that of the exponential model:

Sr-IEXO] = Ce""7) / T (9)

that of the CFF dispersion model:

8t,[CFF] —Alt (10)

RIETHM: Radioisotope Environmental Tracer Hydrology Model 111

with: AirtD vxT .exp

T(l-t/T)2vx ADt

where D is the dispersion coefficient (m2 year"1), v is the transit velocity = xlT; x is the distance from the recharge zone to the outflow (sampling) site of the abstracted groundwater; t is the time interval between the year of input and that of output; and Tis the mean residence time; and

that of the CFR dispersion model:

&',[CFR] = (2^ - B)IT

where A is as given above;

(11)

B = exp 2D ~D

•erfc i+tn

AtD' vxT

r 'A

and erfc is the error function complement. The CFF model is used when the measurement of the observed tritium is carried out

on samples taken from a pumping well, whereas the CFR model is used when the samples are taken from a piezometer. In the present program, a rigorous analytical solution of the erfc is used, after Lerman (1979). The columns F and Win the work sheet are used in the calculation of erfc(x) and column Y selects the appropriate erfc(x) corresponding to the range in which the x value exists.

For the case: 0 < x < 3, the analytical solution is:

erfcQc) 1 „5x8

(12) (1 + axx+a^x2+Oy>c3 + a4x

4 + x3)8

whereaj = 0.14112821; a2 = 0.08864027; a3 = 0.02743349; a4 = -0.00039446; and a5 = 0.00328975. For the case where x > 3, the analytical solution is:

_v2

erfc(x) = IT

, 3 + o 2 v 5

15 105 . + .

2x> 2 V 2 V 2 V 23x i 4 ^ 9

945 ^ 10365 6^13 •)5„11 26X

(13)

In the practical treatment of diffusion-controlled processes, the diffusing substance is initially distributed in given regions. As there is a clear description of diffusion starting from a very thin plane (line source) with a mathematical formula that involves an exponential distribution, the mathematical representation of diffusion starting from a region could be obtained by integration of a successive infinite number of line sources (Crank, 1957) giving rise to summation of the effect of a series of exponential distribu­tions. The resulting form is similar to a well known mathematical function called the error function complement erfc. The above-mentioned summation of an endless number of line sources (each including an exponential distribution) is the appearance of the erfc in the solution of diffusion-controlled problems.

In groundwater hydrology, there is no need for the mean age (ratio between the total age of all the individuals in the system and the total number of these individuals). Rather, a mean transit time (equations (14a), (14b), (14c)) as a ratio between the volu-

218 Fahmy M. Hussein

metric mobile water in the reservoir and the volumetric annual rate of flow (or annual recharge rate under steady-state conditions) is needed. The interest in the mean turnover time (mean residence time) is justified by the importance of the age of the mobile water leaving the system by outflow and the lack of interest in the age of all the existing water. Turnover time is used in almost all conceptual hydrological models, including those of atmospheric moisture, lakes, ocean water and dissolved species (Harte, 1985) with possible different values in functions of the processes involved in the same system.

„ . , . , s volume ofmobilewaterin aquifer(m3) n A „ \ Residence fime(year) = - ——-—- — U4&;

vol. annual rate ofGWflow or recharge (m3 year " ')

Residencetime(y^r) = thickness ofmobilewater in aquifer (m) (Uh)

annual rate ofGWflow or recharge (m year l)

Residencetime(y&i) = (thickness of aquifer (m)) * (effective porosity) ( U c )

annual rate ofGWflow or recharge (myear l)

Annualrecharge(my^) = (thickness of aquifer (m)) * (effective porosity) ( 1 4 d )

residence rz'me(year)

QUANTITATIVE INTERPRETATION IN THE PROGRAM "RIETHM"

The quantitative interpretation of the isotope hydrological models used in this program are based on: (a) The solution of the direct problem where the user introduces the suggested value(s)

of the aquifer parameter(s). The macro displays two option screens for model selection: if the the exponential model is selected, the program asks for the residence time (T) in years. When selecting dispersion models, the user is asked to choose either the CFF dispersion model or the CFR dispersion model. In either of two last cases, in addition to the residence time, the user is asked for the suggested value of the dispersion parameter (Dlvx). Solving the direct problem consists of producing a predicted time-series of tritium output (the output function) which corresponds to the mathematical formulae of a selected conceptual model.

(b) A trial and error solution of the inverse problem. The solution of the inverse problem is more important than solving the direct problem since the investigator has no a priori information on the value of the aquifer parameter (s). Best fit procedures could be used (Richter & Szymczak, 1992) in order to obtain the least deviation between the time-series observations and the output function curve predicted by the selected conceptual model. When the best fit is obtained, it is assumed that the values of the aquifer parameter(s) (used in predicting the output function curve) are good estimates of these unknown physical parameter(s). Instead of the best fit method, the user of the present program is invited to use a trial and error technique by suggesting several sets of estimates of the aquifer parameter(s) and running the program as many times as needed in order to obtain a set of output function curves among which a curve which fairly coincides with the analytical tritium contents (observed for samples taken from the aquifer studied) could be found.

RIETHM: Radioisotope Environmental Tracer Hydrology Model 219

The ultimate purpose of the fitting of the time-series set of environmental tritium data to a certain conceptual hydrological model (the mathematical expression that predicts the output function curve) is the estimation of the residence time (and, in the case of the dispersion models, the estimation of the dispersion parameter). In cases where a paleo-water pole is suggested to outflow in the pumped water or when a combined model is used, a mixing parameter could be estimated (for the paleo-pole fraction or for a basic model contribution, respectively).

When residence time has been estimated, it provides the user with a direct device to estimate the annual recharge rate, which is usually a difficult term to estimate through the non-isotopic hydrological methods. The isotopic data provides a unique chance to tackle this difficulty. However, the fitting of one point (or a few points) of the isotopic observations (as is done in some cases) to the output curve predicted by a certain conceptual model is risky, in particular when fitting problems could result when the observation point(s) are in the decreasing sector of the peak and when slight modifica­tion in the sample collection conditions and/or the model dispersion parameter term gives rise to contradictory residence times. Moreover, erroneous evaluations could be obtained when only the exponential model is used due to ignoring the role of the disper­sion phenomenon.

The exponential model (first introduced by Erikson, 1958) is the conceptual hydro-logical representation of an idealized case of an unconfined aquifer which receives homogeneous infinite long-term recharge over all the area of its surface with the aquifer being small, highly porous and homogeneous in depth and formation. Residence time in the unsaturated zone should be trivial with respect to the total residence time. These conditions generally lead to short residence times with flow lines having different velocities, and the groundwater contains modern and older components. In this case, the residence time distribution is exponential with the modern components being dominant (in contrast to the linear distribution in the linear model where the contributions of the modern and the old components are equal). The exponential model is equivalent to the good mixing model known in industry and lake studies (Maloszewski & Zuber, 1982), but bearing in mind that mixing takes place only at the site of groundwater outflow or at the abstraction well, the exponential model (which assumes the equity of the tracer transit time and water residence time) is accepted.

The above criteria exclude the use of the exponential model in many cases such as: confined aquifers, coastal springs fed by high altitude precipitation, semi-confined aquifers headed by a thick semi-permeable layer, and in cases of effective impact of molecular diffusion and of partial penetration of the aquifer by the abstraction well. However, cases of partial penetration, non-trivial residence time in the unsaturated zone and semi-confined aquifers headed by a considerably thick semi-permeable top layer are very frequent. For these cases, the combined exponential-piston flow model (the EPMr

model of Richter & Szymczak, 1992) is more flexible.

DISCUSSION, MODEL RESULTS AND OBSERVATIONS

Non-isotope groundwater techniques are not convenient for providing information on many important characteristics of the aquifer studied and the stored groundwater, such as those concerning water origin and history, turnover time, the spatial heterogeneity of

220 Fahmy M. Hussein

the aquifer parameters, the relationship of the inputs and outputs, mixing between different water bodies outflowing at the observation point, ... etc. The use of environ­mental isotope tracers is helpful in this respect since it presents pieces of information which could not be obtained otherwise, in particular when its employment is considered in a global and conjunctive framework of applications using both the conventional and non-conventional methods. Tritium data is helpful in the quantitative interpretation of groundwater mean transit time and recharge rate when a conceptual model is used to first predict the tracer output function.

The present model has been applied to the case of the Nile water input function which has been constructed using tritium measurements at Cairo (Swailem, 1969) and extrapolations (Salem, 1990). Examples of the application of RIETHM to the Nile basin are given in Figs 2, 3, 4 and 5. Cases of aquifers which have a relatively high dispersion parameter (for example, D/vx = 2, Figs 3(a),(b), 5(a),(b)) such as areas with high recharge rate close to the outflow site were tested. The relatively high dispersion para­meter could be due to the dominance of flow lines with short residence times. In these cases, it is observed that the program predicts relatively wide differences between the CFF and the CFR dispersion models in the high sector of the tritium output peak with the peaks of the CFR model being in a lower position than the position of the CFF peak. The difference in the output content for any two moderate residence times (e.g. between T = 30 and 40 years) for either the CFF or the CFR case is also distinguished in the years of the peak climax. The position of the climax of the CFR peak is slightly shifted to the right compared to the position of the CFF peak on the time-scale axis (Fig. 3(a)). Both the CFR and CFF peaks have little symmetry. On the end-tail of the both of these peaks, a net smoothing is observed, which is consistent with the remarks of Zuber, (1986). This means that in the particular case of the Nile-fed terrain, for aquifers having a relatively high dispersion parameter, the quantitative use of tritium data is generally restricted to situations where older time-series tritium data are available. The use of present-day tritium data alone without previous data of older years will not be highly informative (regardless as to whether the samples are taken from pumping wells, CFF case, or

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995

Fig. 2 Output function predicted by exponential model with residence times T = 30 and 40 years.

RIETHM: Radioisotope Environmental Tracer Hydrology Model 221

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995

Fig. 3 Output function predicted by dispersion models (Cpp and CJR) with residence time T = 30 and 40 years and a relatively high dispersion parameter (D/vx = 2); (b) is an out-set of (a).

piezometers, CFR case) due to the low resolution which makes it difficult to distinguish between the values of the residence times.

On the other hand, for the case of aquifers with a relatively low dispersion parameter (D/vx = 0.2, Fig. 4), there is still a good chance for distinguishing between groundwaters of not very different residence times, since there is a good resolution on the end-tail sector of the tritium output peak. Moreover, regardless of sampling from pumping wells or from piezometers from the same system (same residence time), the samples could be informative enough. However, this will not be so for a long time to come (despite the expected well symmetry of the peaks under the low dispersion para­meter conditions), so there is an urgent need for close monitoring of the tritium levels for aquifers in areas with low recharge close to the outflow site where there is a dominance of flow lines with relatively long residence times. This monitoring must be carried out immediately and in the next few years or the chance of application of tritium

222 Fahmy M. Hussein

data in quantitative interpretation will be definitively lost, particularly for aquifers with residence times longer than 30-40 years.

The computer files include a macro sheet file containing instructions and tables of definitions that control the start up of the menu-driven program RIETHM. A work sheet file is included which provides the field of input and output data storage and optional updating. Three graphics are also provided with the current version.

For more technical information such as program structure and procedures, and possible utilization of the computer code the reader is encouraged to contact the author in the above-mentioned address.

CONCLUSIONS

The proposed model has research-oriented capabilities. It provides the investigator with a user-friendly computer interface which can be readily used in the preparation of sets of output function data and graphic display. It solves the direct problem under different work hypotheses concerning the values of aquifer parameters. Recognition of the mean transit time of groundwater in a given system could be obtained when close coincidence (fitness) between the tritium time-series observations and a given predicted output function curve is obtained. After solving the inverse problem by this trial and error tech­nique, the annual recharge rate could be readily estimated using the available informa­tion on the aquifer depth ad the effective porosity. Moreover, the model is a user-friendly device through its on-screen computer code display of the formulae involved, extensive help statements accompanying the macro instructions and its open structure available for further modification both in the macro instructions and the work sheet framework, including the insertion of a non-linear curve best-fit technique for the straightforward solution of the inverse problem.

As shown by the examples presented for groundwater on the banks of intrusive river systems in the southern hemisphere (predicted tritium output curves for Nile-fed

70 •*- FF30

60 f-O- FR30

' • - FF40

a- FR40 50

40 "•

TU

30 -•

20

10 "•

? ^ \

- J aa no aananaonnau y . , . . ( -+- - T - t - -+-, 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995

Fig. 4 Output function predicted by dispersion models (Cpp and Cpj ) with residence time T = 30 and 40 years and a relatively low dispersion parameter (D/vx = 0.2).

R1ETHM: Radioisotope Environmental Tracer Hydrology Model 223

2 0 0 J * - E30yr 1 8 0 ::<>- E40yr

160

140

120

TU 100

- • - FF30,D/vx=0.2 •n- FR30,D/vx=0.2 -*r- FF40,D/vx=0.2 •*- FR40,D/vx=0.2

•X- FF30,D/vx=2 -*- FR30,D/vx=2 — FF40,D/vx=2

80 ± « - FR40,D/Vx=2 X X

X X rai3:Da.B:x.,,

:*S0tX

1950 1960 1965 1970 1975 1980 1985 1990 1995

70

60

TU

• - FF30,D/vx= '-•a- FR30,D/vx=

50 --àr- FF40,D/vx= : •*- FR40,D/vx=

4 0 " X - FF30,D/vx=

30

20

10

••- E30gr •O- E40yr

* - FR30,D/vx= .— FF40.D/vx= •0- FR40,D/vx=

<*#&«£

1 J ° / / / n i .."

# f , , , i 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995

Fig. 5 Output function predicted by exponential and dispersion models (Cpp and Cpj,) with residence time T = 30 and 40 and relatively high and low dispersion parameters (D/vx = 2 and 0.2) (composite of Figs 2, 3 and 4; (b) is an out-set of (a)).

aquifers), the difficulty at present (almost 30 years after the large-scale thermonuclear tritium injections) is with the decreasing tritium contents approaching the natural back­ground level. This situation calls for the obligatory use of refined tritium enrichment devices in order to obtain highly accurate measurements of the diminishing tritium contents. Tritium measurements in the same groundwater abstraction points in aquifers of these regions as well as in the coastal deserts of the arid-to semiarid regions in the southern hemisphere are still needed to complete older time-series observations in order to obtain more accurate interpretations using models like that proposed in the present study. For the southern hemisphere, in cases where no tritium data for the years 1970-1980 are available, a valuable chance for the estimation of residence time for aquifers using the advantageous tritium data technique and modelling methods is definitely lost. However, close temporal monitoring during the next few years is recommended in order to catch the information which could be provided by the end-tail data of the tritium peak, in particular in sites where relatively high tritium contents are still observed.

224 Fahmy M. Hussein

REFERENCES

Bear, J. & Verruijt, A. (1987) Modeling Groundwater Flow and Pollution. Reidel, Dordrecht, The Netherlands.

Carrol, G. W. & Rock, N. M. S. (1991) ISOCALC: A Simple Rb-Sr and Sm-Nd isotopic calculator for the Apple Macintosh. Computers Geosci, 17, 465-467.

Crank, J. (1957) Mathematics of Diffusion. Oxford, UK.

Dexter, L. R. & Avery, C. C. (1991) Using spreadsheet software in water-balance modeling. Computers Geosci. 17, 527-536.

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