Theoretical possibilities of the 3H-3He method in investigations of groundwater systems

CATENA

Braunschweig 1983

THEORETICAL POSSIBILITIES OF THE 3 H_3 He METHOD IN INVESTIGATIONS OF GROUNDWATER SYSTEMS

P. Ma~oszewski & A. Zuber, Cracow ABSTRACT Theoretical possibilities of the 3H-3He parent-daughter radiotracei: pair in investigations of groundwater systems are discussed. Output curves of the °He to "H ratio have been calculated for systems which can be approximated by the exponential or dispersion models. Calculations were performed for a .typical tritium input concentration curve. It has been shown thai; contrary to the earlier findings of other authors based on the piston flow model, the "He to °H ratio depends not only on the turnover time but also on the input concentration of tritium and on the assumed hydrodynamic model. Calculations also show that the ratio metho.d does not yield unambiguous results. H0~ever , rrlore informationjs av~tilabi.e i~ a given theoretical model is fitted separately to the He and°H data Qr to, the °He/°H and °H data. Whichever interpretation is chosen, there is no doubt that the °H-°He method is a promising alternative to the tritium method for the nearest several tens of years. This promising alternative results from the fact that in groundwater the peak concentration of the daughter YI~e is delayed in respect to the parent tritium peak concentration.

1. INTRODUCTION Environmental tracers, including environmental radiotracers, have appeared to be particularly useful in preliminary descriptions of systems lacking basic hydrologic data, or for systems where the conventional methods do not yield satisfactory results (e.g. karst formations or fractured rocks). Very often a qualitative interpretation is sufficient, since different conclusions may be drawn for example by the presence of a radioisotope at a given sampling site, or if its concentration differs from that found at other sampling sites, or if its concentration is variable or not in time. Quantitative interpretations are based on mathematical models, which allow various parameters of the investigated systems to be determined. The most common approach is based on the so-called lumped-parameter approach, in which the investigated system is treated as a whole, and the spatial variations of the system are ignored. The main obtainable parameter is the turnover time (the water residence time). This approach is followed here to estimate the potential use of the 3He-3H method. It is a well known experimental fact that in most groundwater systems the tritium concentrations have a tendency to equalize both in time and space. In other words the information obtainable from the commonly applied tritium method is of a limited value because the low variations in tritium concentrations makes the quantitative interpretation a difficult, if not impossible task. This fact initiated an extensive search for new tracers, which could be used instead of, or in combination with the tritium method. Several new possibilities have been recently discovered. Techniques for determining SSKrhave been described by ROZANSKI & FLORKOWSKI (1979) and preliminary theoretical considerations, based on the lumISSN

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(~ Copyright 1983 by Margot Rohdenburg M.A., CATENAVERLAG, Brockenblick8, D-3302 Cremhngen-Destedt, W.Germany

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ped-parameter approach, are given by GRABCZAK et al. (1982) and ZUBER (in press). The freon-11 measurement technique and examples of qualitatively interpreted field studies are described by THOMPSON & HAYES (1979). GRABCZAK et al. (1982) have also described two studies which they have attempted to interpret quantitatively. The 3H-3He method was introduced first to limnology by TORGERSEN et al. (1977 and 1979) who described the criteria for evaluating mass-spectrometrically obtained 3H-3He measurements and the procedures for calculating the 3H-3He ages from the 3He to 3H ratio. TORGERSEN et al. (1979) also suggested that the method can be used for determining the water residence time (age) in closed systems. However their considerations were confined to the piston flow model. This model suggests that the ratio method is highly promising because the results do not depend on the input concentration of tritium, and thus a single measurement should be sufficient for an accurate age determination. Thus the method seemed to be extremely promising. The aim of this work was to evaluate the 3H-3He method on the basis of a more realistic approach, i.e. by applying the exponential and dispersive models. A general description of the lumped-parameter models is given by MALOSZEWSKI & ZUBER (1982), and by ZUBER (in press). Here, the above mentioned models will be used to provide the theoretical background for the 3H-3He method, and to explain when and why the tritium concentrations become equalized. The 3H-3He technique has been described in detail by TORGERSEN et al. (1979). Here it will be reminded that the method involves the collection of about 10 grams of water. The dissolved gases are extracted and purified. 3He, 4He and Ne are then measured massspectrometrically to determine the concentrations in cm 3 STP of gas per gram of water. The remaining water having been totally degassed is sealed in vacuo for ' H analysis by the 3He grow-in method. There may be five components of 3He in groundwater. They are: 3He of saturation - originating at the air-water interface in the infiltration area, °He of oversaturation resulting from air injection which may probably happen in some karstic systems (in open waters such air injections are common), 3He of mantle origin, 3He of crust origin, and the tritiugenic 3He. TORGERSEN et al. (1979) describe how to find the last component in lake waters from the measured total 3He, by making use of Ne and 4He determinations, and by assuming that either mantle or crust components may be present (but not both). It still remains an open question if the determination oftritiugenic 3He is possible with satisfactory accuracy in groundwaters. However the 3H-3He method is worth considering owing to the possibilities it offers.

.

THE MODELS

If a given volume of water has been separated since the recharge time, the following relation holds for the parent tritium concentration c T (t) = c T (0) exp(-2t),

(1)

and for the daughter helium concentration c He(t) = cT (0) [1-exp(-2t)],

(2)

where CT(0 ) is the initial tritium concentration, 2 is the tritium decay constant equal to (17.7a)-', and t is the time variable defined as the age of water. Dividing Eq.2 by Eq. 1 leads to

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(Eq. 6 in TORGERSEN et al. 1979) 4.01 x 1014 CHe/c T = [1-exp(-2t)] / exp(-2t) = exp(,;tt) - 1, (3) where c He is expresses in cm 3 STP per gram of water, and c T in commonly used tritium units (TU). In the piston flow model (PFM) it is assumed that all the flow lines have the same transit time through the system and that there is no dispersion. The lack of dispersion means that an imaginary parcel of water moves from the recharge point to the measuring point without any mixing or exchange with other parcels. In such a case the age of water appearing in Eqs 1 to 3 is equal to the turnover time, T. The turnover time is defined as T = V/Q, where V is the volume of water contained in the system between the recharge and measuring points, and Q is the volumetric flow rate through the system. However, it is well known that the PFM is an approximation applicable only in the cases of a constant concentration in the recharging water. Thus, for an evaluation of the potential applicability of the 3He method it is necessary to consider other more realistic models. Detailed discussions of the lumped-parameter (or black-box) models applicable to the interpretation of environmental radioisotopes in groundwaters as well as the schematic presentation of systems which can be approximated by particular models have been given recently by MALOSZEWSKI & ZUBER (1982) and ZUBER (in press). Here only two models will be considered briefly. The exponential model (EM) assumes the exponential distribution of transit times of water through the system. The tracer is assumed to move with particular flow lines without any mixing, thus having the same distribution of transit times as the water in the system. The turnover time, equal to the mean transit time, is the only parameter of this model. The dispersion model (DM) assumes that the distribution of transit times of the tracer is described by an adequate solution to the dispersion equation. This distribution may result both from different transit times of particular flow lines and from hydrodynamic dispersion. There are two possible versions of the dispersion model depending on the sampling modes. Usually sampling is performed in outflowing or abstracted water. In such a case the sample is naturally weighted by the volumetric flow rates of particular flow lines contributing to the sampled water. Following MALOSZEWSKI & ZUBER (1982) this case is denoted as the CFF case whereas the corresponding model is denoted as DM-CFF. There is also a possibility that the samples are taken at different depths of a drilled hole, and the average value is calculated by weighting over the penetrated depths. This case is denoted as CFR case and the corresponding model as DM-CFR. Arbitrary sampling (e.g. samples taken from a drilling well at a given depth) does not represent a given system as a whole, and thus the interpretation by the lumped-parameter approach is then doubtful. The c He to c T ratio for any model can be calculated with the aid of the convolution integral. For the parent tritium it has the form oo

c T (t) = I c Tin (t-t') g(t') exp(-,;tt') dr', 0 and for the daughter helium 4.01 x 1014 c He (t) = '~0 c Tin (t-t') [1- exp(-2t')] g(t') dr'

(4)

(5)

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where c Tin (t) is variable tritium concentration at the entrance to the system (tritium input concentration, and g(t) is the weighting function describing a given model (MA~OSZEWSKI & ZUBER 1982). Dividing Eq. 5 by Eq. 4 gives 4.01 x 1014 CHe (t) / cT(t) = = (CONV.INT., 2 = 0) / (CONV.1NT., 2 = 1/17.7a) - 1,

(6)

where CONV.INT. stays for the right hand side of Eq. 4. Putting the weighting function for the PFM into Eq. 6 leads to Eq. 3 for any CTin (t), whereas for the EM and a constant CTin (t) another simple formula is obtained, namely 4.01 x 1014 CHe/c T = ). T.

(7)

In general, for a constant tracer input (CTin(t) = const = CTin ), Eq. 6 simplivies to Eq. 8, which also directly results from the mass balance. 4 x 1014 CHe = CTin ( 1 - e l-/CTi n )

(8)

Eq. 8 permits an easy estimation ofc He for any model with the turnover time corresponding to the time from before the nuclear bomb experiments, when the tritium input concentration was constant. Graphs of c/c o given in Fig. 2 of MALOSZEWSKI & ZUBER (1982) can be used for that purpose. However it is well known that the tritium concentration in precipitation decreased after the peak concentration in 1963 with strong seasonal variations caused by the so-called spring injection of tritium from the stratosphere to the atmosphere. Because of these seasonal variations it is necessary to calculate mean yearly concentrations weighted by the monthly ammounts of precipitation and monthly infiltration coefficients. Such a calculation is often strongly simplified by assuming that the summer ira"titration coefficient is a given fraction (a) of the winter infiltration coefficient. For intance a = 0 means no infiltration in the summer months (in the northern hemisphere it is April to September), and equal infiltration coefficients in the winter months, ~ = 1 means equal infiltration coefficients throughout the whole year.

3.

THEORETICAL CONCENTRATION CURVES

In Fig. 1 examples of typical tritium concentration curves are presented for the exponential model. All these curves exhibit a maximum in the sixties and a tendency of the concentrations to equalization in the eighties. In other words, the aquifers which can be approximated by the exponential model can not be interpreted by the tritium method. In Fig. 2 the 3He to 3H ratio is presented for the exponential model. The future tritium concentrations were estimated by extrapolation. With this method the ratio curves exhibit tendency to increasing for many years in the near future. It is worth mentioning that the ratio method seems to be only somewhat dependent on the assumption related to the infiltration coefficient (two extreme approximations related to the infiltration coefficients are shown, i.e. a = 0 a n d a = 1). ComparisonofFig. 2with Fig. 1 leadstoaconclusionthattheexponential

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systems with turnover times up to 50 years wiil not be interpretable in the eighties by the tritium model, whereas they should be interpretable by the 3He/3H ratio method. In the case of systems which can be approximated by the dispersion model the situation is more complex because, as mentioned earlier, this model is characterized by two parameters: the turnover time (T), and the dispersion parameter (D/vx). In the case of a high dispersion parameter D/vx, (see Fig. 3) the tritium concentration curves exhibit a similar character to that for the exponential model. In the case of a low dispersion parameter (see Fig. 4), long-term observations should lead to determining the adequate model because different models are characterized by different concentrations curves. However, the short-term observations may also be ambiguous because of close values of the concentrations for a number of models. The 3He to 3H ratio method (see examples in Fig. 5) should be helpful because this method exhibits different ratios for some of the cases which are undistinguishable by the tritium method. However, Fig. 5 shows clearly that there are other cases in which the 3He/3H method is not unambiguous either. For instance, for T = 50 years and D/vx = 1 the ratio curve is practically the same as for T = 20 years and D/vx = 0.2. In other words the use of the 3He/3H method alone is not sufficient for an unambiguous determination of parameters. It is suggested here to interpret field data by seeking a model whose parameters fit both the experimental tritium concentrations and the experimental 3He/3H ratio. Performed calculations show that for the dispersion model the 3He to 3H ratio is also nearly independent of the tt coefficient.

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Another interpretational possibility exists if direct 3He values are used instead of the 3He to 3H ratio. Performed calculations show different 3He concentrations curves for models which are not distinguishable by the tritium method or by the 3He/3H method. However, 3He concentrations depend strongly on the assumptions related to the infiltration coefficient, whereas in the case of the 3He to 3H ratio this dependence is strongly damped. Thus it seems

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that whenever the infiltration coefficients remain unknown the best alternative is the combined use of the 3He/3H and 3H methods.

4. CONCLUSIONS Theoretical considerations of this work showed that the equalization of tritium concentrations takes place in many groundwater systems. The two most common mathematical models, the exponential and dispersive models, showed that this equalization is particularly dominant for short turnover-times, say below 30 to 50 years. For longer turnover times the tritium content is also not very variable in time, but the concentrations are lower than the most common values, thus giving some possibility for the age determination. In dispersive systems with a low dispersion parameter, longer turnover times should be still interpretable. Summarizing the above statements, the theoretical calculations confirmed the experimental evidence of the equalization of tritium in groundwaters, but also showed the existence of some exceptions, which may be of practical importance. Calculations performed for the tritiugenic 3He showed, on the other hand, that the 3H3He method is a highly promising tool as an extension of the tritium method for the nearest several tens of years. Namely, the output signal of the 3He to 3H ratio will be increasing for most groundwater systems, showing at the same time a distinct differentiation. However it has been shown that the identification of a given model for a given system will be possible only ifa combined interpretation of the 3He/3H and 3H, or 3He and 3H is applied. It has also been proven that the 3He/3H ratio depends slightly on the assumptions related to the infiltration coefficient. Summarizing the above statements concerning the 3H-3He method, the theoretical calculations showed that the method offers possibilities unobtainable at present from the tritium method, and is worthy of further development. However the interpretation is not so simple as suggested by TORGESEN et al. (1979), because contrary to the statement of these authors a single determination is not sufficient for the age determination. In this work the direct problem of interpretation has been solved, i.e., for known, or assumed parameters the output concentrations have been calculated. Such solutions serve for estimating the potential applicability of the method and may serve in future for planning the sampling schedules. If the tritiugenic 3He proves to be distinguishable from other 3He components, Eqs 5 and 6 together with adequate weighting functions will serve for solving the inverse problems, i.e. for known input and output concentrations the parameters of a given groundwater system will be obtainable. It is the authors' opinion that theoretical works should be initiated in early stages of the development of a given method. For instance, in the case of the well established tritium and 14Cmethods, experimental progress exceeded the development of mathematical models, causing many problems of interpretation and controversions, as for example discussed by MALOSZEWSKI & ZUBER (1982) and by ZUBER (in press). Similarly, many problems of interpretation might result if the conclusions discussed above by TORGERSEN et al. (1979) are accepted.

REFERENCES

GRABCZAK, J., ZUBER, A., MALOSZEWSKI,P., ROZANSKI, K., WEISS, W., & gLIWKA, I. 1982): New mathematical models for the interpretation of environmental tracers in groundwaters and the combined use of tritium, C-14~Kr-85, He-3,and freon-11 methods. Beitr~igezur

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Geologie der Schweiz -Hydrologie, Bd. 28 II, Bern, 395-406. MALOSZEWSKI, P. & ZUBER, A. (1982): Determining the turnover time of groundwater systems with the aid of environmental tracers: I. Models and their applicability. Journal of Hydrology ,57, 207-231. ROZANSKI, K. & FLORKOWSKI, T. (1979): Krypton-85 dating of groundwater. In: Isotope Hydrology 1978, IAEA, Vienna, Vol. II, 949-961. THOMPSON, G.M. & HAYES, J.M. (1979): Trichlorofluoromethane in groundwater - apossible tracer and indicator of groundwater age. Water Resources Research 15, 546-554. TORGERSEN, T., TOP, Z., CLARKE, W.B., JENKINS, W.J. & BROECKER, W.S. (1977): A new method for physical limnology - tritium-helium-3 ages - results for lakes Erie, Huron, and Ontario. Limnology and Oceanography 22, 181-193. TORGERSEN, T., CLARKE, W.B. & JENKINS, W.J. (1979): The tritium/helium-3 method in hydrology. In: Isotope Hydrology 1978. IAEA, Vienna, Vol. II, 917-930. ZUBER, A. (in print): Mathematical models for interpretation of environmental radioisotope data in groundwater systems. In: Handbook of Environmental Isotope Geochemistry (Eds.: P. Fritz and J.Ch. Fontes) Vol. I. The Terrestrial Environment, part B. Elsevier, Amsterdam.

Address of authors: P. Maloszewski and A. Zuber, institute of Nuclear Physics Radzlkowskiego 152, Cracow 23, Poland